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abstract: 'We study aggregation as a mechanism for the creation of complex networks. In this evolution process vertices merge together, which increases a number of highly connected hubs. We study a range of complex network architectures produced by the aggregation. Fat-tailed (in particular, scale-free) distributions of connections are obtained both for networks with a finite number of vertices and growing networks. We observe a strong variation of a network structure with growing density of connections and find the phase transition of the condensation of edges. Finally, we demonstrate the importance of structural correlations in these networks.'
author:
- 'M. J. Alava'
- 'S. N. Dorogovtsev'
title: Complex networks created by aggregation
---
Introduction {#s-introduction}
============
Fat-tailed distributions of connections characterize the complex architectures of many real-world networks [@ab02; @dmbook03; @n03; @pvbook04]. Several mechanisms may accounted for this form of degree distributions of networks. (Degree is the total number of connections of a vertex.) The most popular concepts imply self-organization [@p76; @ba99; @baj99]. The self-organization mechanism is responsible for fat-tailed distributions in a wide circle of evolving systems (see, e.g., Refs. [@y25; @s55]), and not only in networks.
Usually, a very particular preferential attachment version of the self-organization mechanism is discussed, so that highly connected vertices preferentially attract new connections [@p76; @ba99; @baj99], but there are other possibilities. In this paper, we consider agglomeration as a competing possibility. It is known that aggregation processes effectively generate power-law distributions (see, e.g., Ref. [@t89] for an example). In networks, the analogue is the merging of vertices. By this mechanism, vertices accumulate their connections (agglomeration of edges). This increases a number of highly connected hubs and so gives a chance to arrive at a fat-tailed degree distribution.
Evidently, the merging of vertices should take place in cellular networks (merging proteins) as well as in many other real-world networks. For example, in various networks of economic relations, merging and splitting of enterprises are basic elements of the evolution. The same is valid for networks of software components, electronic circuits, networks of relations between social groups, organizations, institutions, and parties, networks of subjects, networks of notions, etc. Simple evolving networks with merging vertices have quite recently been simulated, Ref. [@ktms04], and the generation of fat-tailed degree distribution has been successfully demonstrated. (For a similar process in bipartite graphs, see Ref. [@srtm04]).
In the present paper we provide a comprehensive description of the process of the creation of complex network architectures by the merge of vertices. It is impossible to obtain a uniform picture for all networks of this type. So, we describe a set of typical types of behaviors by considering a line of basic network models, which can be studied analytically. These models may be generalized in a natural way to include clustering and the condensation of clustering. Many other variations are also possible.
All the models that we study in this paper generate fat-tailed degree distributions. We consider both non-equilibrium networks with a fixed number of vertices, and networks, where this number grows. We use the mean degree $\overline{k}$ of a network as a relevant parameter. Then the variation of network architectures with $\overline{k}$ is essentially characterized by the $\gamma(\overline{k})$ dependence. So, our main results are presented in the form: exponent $\gamma$ vs. the mean degree $\overline{k}$.
In most of the networks in this paper, the evolution is due to two parallel processes: (i) the merging of vertices and (ii) random attachment of new vertices. However, we also discuss networks, where the second channel of the evolution is splitting (fragmentation) of vertices.The range of scenarios is wide, but in most of them we find a phase with the condensate of edges (in other words, gelation). Above some critical value $\overline{k}_c$ of the mean degree, a finite fraction of edges is attached to a few vertices or to a single vertex. This condensation, unlike the situation described in Ref. [@bb01], occurs in the homogeneous networks. In the “normal phase”, a degree distribution is of a power-law form, $P(k) \propto k^{-\gamma}$. Moreover, we observe that, rather unexpectedly, even in the condensation phase, normal vertices have a scale-free degree distribution. A resulting picture may be complicated by the presence of correlations, which is typical for non-equilibrium networks. We demonstrate the importance of degree–degree correlations, in the most succinct of these network models. The paper is organized as is follows. In Sec. \[s-results\], we describe the models and present in detail our results for each of them. The complete final information can be obtained from this section. In Sec. \[s-derivations\] we present details of our analytical calculations and simulations.
Models and results {#s-results}
==================
In this section we describe basic models of networks evolving due to aggregation processes and present our results. For sake of brevity we consider only undirected networks, i.e., networks with undirected edges.
Network $O$ {#ss-o}
-----------
This is the simplest model. The evolution starts from a given configuration of vertices and connecting edges. Loops of length one are allowed. At each time step (see Fig. \[f1\]):
- $q \geq 1$ new bare vertices are added to the network.
- Two randomly chosen vertices merge.
Obviously, the final state of the network is a set of bare vertices plus a single vertex with all the edges (actually, loops of length one) attached. This is what we call the condensate of edges.
Network A {#ss-a}
---------
The (large) number of vertices of this network, $N$, does not change during the evolution. Initially, there is an arbitrary configuration of $N$ vertices connected by some number of links. At each time step (see Fig. \[f2\]):
- A new vertex is added to the network. This vertex is attached to a randomly chosen vertex.
- Two randomly chosen vertices merge.
So, the result of the merging of vertices of degrees $k'$ and $k''$ is a vertex of degree $k = k'+k''$. The total degree of the network linearly grows, $K(t) = K(t=0) + 2t$, as well as its mean degree. Network becomes more and more dense with time.
The resulting degree distribution is of a power-law (an asymptotics) form with exponent equal to $3/2$:
$$P(k) \sim k^{-3/2}
\, .
\label{e1}$$
The main part of the distribution is stationary, but its low-degree part and a cut-off in the high-degree range change with time. This ensure the growth of the mean degree with time. Condensation of edges is absent.
Note that we assume that this network is sparse. In principle, we allow multiple connections and 1-loops. However, we believe that in the sparse network regime, they are not important if we are not interested in the position of the cutoff of the degree distribution.
This network was simulated earlier [@ktms04]. Our analytical results confirm the observations in Ref. [@ktms04].
Network B {#ss-b}
---------
This is a growing version of model A. At each time step (see Fig. \[f3\]):
- $q>1$ new vertices are added to the network. Each of these vertices is attached to a randomly chosen vertex by $m$ edges.
- Two randomly chosen vertices merge.
The total number of vertices now grows: $N(t)= N(t=0)+(q-1)t$. The total degree is $K(t) = K(t=0)+2qmt$. So, the average degree approaches the finite value $\overline{k}=2qm/(q-1)$.
The condensation of edges is absent in this model. Unlike model A, the stationary degree distribution of this growing network is a rapidly decreasing function. If, however, the rate of the growth is low, $q-1 \ll 1$, then the power-law dependence (\[e1\]) is realized in the range of degrees below a size-independent cutoff, $k \ll m/(q-1)^2$.
Network C {#ss-c}
---------
In this growing network, at each time step (see Fig. \[f4\]):
- $q>1$ new vertices are added to the network. Each of these vertices is attached to a randomly chosen vertex by $m$ edges.
- Simultaneously, a randomly chosen vertex merges with its randomly chosen neighbor, and the connecting edge disappears.
Rule (2) means the preferential choice: the second vertex in the network is chosen with probability proportional to its degree. The number of vertices and the total degree grow as $N(t)= N(t=0)+(q-1)t$ and $K(t)= K(t=0)+2(qm-1)t$, respectively. So, the mean degree approaches the value
$$\overline{k} \cong 2\frac{qm-1}{q-1}
%%= 2(m + \frac{m-1}{q-1})
>2m
\, .
\label{e2}$$
It turns out that if the network is sufficiently dense, namely if
$$\overline{k} > \overline{k}_c = 2m(1 + \sqrt{1-1/m})
\, ,
\label{e3}$$
than a finite fraction of edges is condensed on a single vertex (or, maybe, on a few vertices). This takes place when the rate of the grows is low:
$$q < q_c = 1 + \sqrt{1-1/m}
\, .
\label{e4}$$
The fraction of edges in the condensate
$$M = \frac{\overline{k}^2 - 4\overline{k}m + 4m}{\overline{k}(\overline{k}-2m)} = \frac{2qm-q^2m-1}{qm-1}
\, .
\label{e5}$$
behaves as $M \propto (\overline{k}-\overline{k}_c) \propto (q_c-q)$ near the condensation point (see Fig. \[f5\]). One can see that all the edges are in the condensate in the limit of $\overline{k}\to\infty$. (Note, however, that we consider a sparse network.)
One can easily understand this condensation phenomenon. For the evolution of the number of edges in the condensate, $K_h(t)=Mt$, that is, the “macroscopic” number of edges attached to the hub, one can immediately write the following equation:
$$\frac{dK_h}{dt} = \frac{K_h}{K}\left[\frac{K-K_h}{(q-1)t} - 2 \right]
\, .
\label{e5a}$$
Here, $K(t)$ is the total degree of the network, and the second factor on the right hand side of the equation is simply the mean degree $\overline{k}_\text{n}$ of “normal” vertices (i.e., the hub is excluded) minus $2$. Indeed, according to rule (2) of the model, the probability that the hub will be chosen for the merging is $K_h/K$. Each act of merging, in average, increases the number of connections of the hub by $\overline{k}_\text{n}-2$, which explains the form of Eq. (\[e5a\]). Consequently,
$$M = \frac{M}{2(qm-1)}\left[\frac{2(qm-1)-M}{q-1} - 2\right]
\, .
\label{e5a}$$
One can see that this equation has a non-zero solution $M$ \[exactly of the form (\[f5\])\] only if the growth rate parameter $q$ is less than the critical value $q_c$ given by expression (\[f4\]). Note that similar equations may be written for for networks with splitting vertices (see below).
“Normal vertices” (i.e., with “microscopic” numbers of connections) have stationary power-law degree distributions (asymptotics) both in the normal and in the condensed phases. The $\gamma$ exponent of the degree distribution $P(k) \sim k^{-\gamma}$ is
$$\gamma =
2 + \frac{4\overline{k}m - \overline{k}^2 - 4m}{(\overline{k}-2)(\overline{k}-2m)} =
2 + \frac{mq^2 - 2qm + 1}{q(m-1)}
> 2
%%\, .
\label{e6}$$
in the phase without condensate ($\overline{k}<\overline{k}_c$, i.e. $q>q_c$) and
$$\gamma =
2 + \frac{\overline{k}^2 + 4m - 4\overline{k}m}{2\overline{k}(m-1)} =
2 + \frac{2qm - mq^2 - 1}{(mq-1)(q-1)}
> 2
%%\, .
\label{e7}$$
in the condensation phase, where $\overline{k}>\overline{k}_c$ ($q<q_c$). Fig. \[f6\] shows how the exponent of the degree distribution varies with the mean degree. Note that in both phases, near the condensation point, $\gamma-2 \propto |\overline{k}-\overline{k}_c| \propto |q-q_c|$.
In the critical point, the degree distribution has the form:
$$P(k) \sim \frac{1}{k^2\ln^2 k}
\, .
\label{e8}$$
Thus, this network is scale-free both is the normal and in the condensation phases. The power-law form of the degree distribution in the condensation phase is rather unexpected.
Let us explain this remark in more detail. A close analogy of the problem under consideration is the emergence of the giant connected component in a growing network, where a phase transition with the Berezinskii-Kosterlitz-Thouless singularity takes place [@chkns01; @dms01; @kkkr02; @bb03]. In that case, the evolution equation for the size distribution of connected components is very similar to the evolution equation for the degree distribution in our case (see the next section). In this analogy, the giant connected component is analogous to the condensate of edges attached to a vertex, and the size distribution of finite connected components is analogous to the degree distribution in our case. The point is that the size distribution of connected components was found to be rapidly decreasing in the phase with the giant component (see Ref. [@dms01]), while in contrast, in the present situation, the degree distribution is scale-free in the phase with the condensate.
The paper [@ktms04] was mainly devoted to the simulation of the “static” version of quite similar network without multiple connections and loops of length $1$, $N=\text{const}$ with a growing number of connections, the sparse network regime. Scale-free degree distributions with exponents exceeding $2$, without any condensation, were reported. We do not consider precisely this situation here, since we focus on stationary degree distributions. These take place in the growing network.
Network D {#ss-d}
---------
At each time step (see Fig. \[f7\]):
- $q>1$ new vertices are added to the network. Each of these vertices is attached to a randomly chosen vertex by $m$ edges.
- Simultaneously, the end vertices of a randomly chosen edge merge together, and this edge disappears.
At first sight, this model is close to model C. The numbers of vertices and connections grow in the same way, and the mean degree is the same, Eq. (\[e2\]). The only difference is the way in which the merging vertices are chosen. One can treat this merging process as transformation of random edges with their end vertices into single vertices.
In fact, there is an essential difference since now the choice of vertices, in principle, depends on correlations between the degrees of the nearest neighbor vertices in the network. Let us compare models C and D once again: (i) The evolution of model C produces degree-degree correlations but is not governed directly by them. (ii) The evolution of model D depends on the degree–degree correlations and, in its turn, produces these correlations. Strict calculations taking into account degree–degree correlations should be rather cumbersome. Furthermore, in related networks, studied in Ref. [@ktms04], correlations have not been found. So, we applied a simplifying ansatz: we assumed that correlations may be neglected.
=69.5mm
It turns out that with this assumption, equations for the degree distribution have a reasonable solution only if the mean degree of the network is below some value: $$\overline{k} < \overline{k}_c = 2.204m - 0.1115 + O(1/m)
\, ,
\label{e9}$$ or, equivalently, $q>q_c \cong 10.815-4.446/m$. In this region, our calculations provide a power-law degree distribution. The $\gamma$ exponent of this distribution approaches infinity at the minimal possible mean degree $2m$ (i.e., $q \to \infty$) and near $\overline{k}_c$ behaves as
$$\gamma - 3 \sim \sqrt{\overline{k}_c - \overline{k}} \sim \sqrt{q-q_c}
\,
\label{e10}$$
(see Fig. \[f8\]). This is in sharp contrast to the behavior $\gamma(\overline{k})$ near the critical point of network C (see Fig. \[f6\]). Note the value $3$ of exponent $\gamma$ at $k=\overline{k}_c$.
Above $\overline{k}_c$, with our assumption that the correlations are absent, the only solution was found to be pathological. This proves the importance of the correlations in this network, which may be especially important in the condensation phase. For studying these degree–degree correlations, we resort to numerical simulations following the rules of model D. The degree distributions of the resulting networks are shown in Fig. \[f10\]. Power-law-like degree distributions were observed both in the condensation phase and in the normal one. Fitting has given values of the $\gamma$ exponent slightly above $3$ at the studied values of the parameter $q$. We obtained the dependence of the mean degree of the nearest neighbors of a vertex on the degree $k$ of this vertex, $\overline{k}_{nn}(k)$. Of particular note here is that $\overline{k}_{nn}(k)$ has to be computed with care: the loops of length 1 are not to be considered for the vertex whose nearest neighbors are under study. However, we take into account the nearest neighbors with one loops. This may be especially important in the condensation phase. The obtained dependences $\overline{k}_{nn}(k)$ are shown in Fig. \[f10b\]. The main conclusions are as follows:
- the network is correlated, and the degree–degree correlations are strong; the correlations are of assortative type; one reason for this is the fact that for nodes with a high $k$ the self-loops contribute strongly to the degree emphasizing such tendencies.
- the correlations are present both in the phase with the condensate and without it, Fig. \[f10b\] demonstrates a non-monotonous dependence of $\overline{k}_{nn}(k)$ on $q$;
- the condensation phase transition takes place at higher values of the mean degree than $\overline{k}_c$, given by Eq. (\[e10\]);
- the observed values of the $\gamma$ exponent of the degree distribution are restricted from above.
Splitting vertices {#ss-split}
------------------
Instead of the process of the random attachment of new vertices in models A–C, one can introduce another channel of evolution—splitting (fragmentation) of vertices. In this paper we only touch upon two possibilities (see Fig. \[f11\]):
- A randomly chosen vertex of degree $k$ splits into a pair of unconnected vertices of degrees $k'+k''=k$ in such a way that all possible resulting configurations are realized with equal probabilities.
- A randomly chosen vertex of degree $k$ splits into a pair of vertices of degrees $k'+k''=k+2$, connected by an edge. Again, we assume that all possible resulting configurations occur with equal probabilities.
The number of these splittings per time step, in principle, may be not equal to $1$.
Derivations {#s-derivations}
===========
In this section we describe details of our analytical calculations. We use an analytical technique similar to that for aggregation processes in more traditional systems [@bk95; @kb00] (for various aspects of the aggregation processes, see, e.g., Refs. [@t89; @rdcb02; @rm01; @ha02]).
Network A {#ss-a_d}
---------
The evolution equation for the average number $\overline{N}(k,t)$ of vertices of degree $k$ in the network A at time $t$ has the following form:
$$\begin{aligned}
&&
\!\!\!\!\!\!\!
\overline{N}(k,t+1) =
\overline{N}(k,t) + \delta_{k,1}
+
\frac{1}{N}\overline{N}(k-1,t) - \frac{1}{N}\overline{N}(k,t)
\nonumber
\\[5pt]
&&
+
\left(\frac{1}{N}\right)^2 \sum_{k'+k''=k}\overline{N}(k',t)\overline{N}(k'',t) - \frac{2}{N}\overline{N}(k,t)
\, .
\label{e11}\end{aligned}$$
Here the average is over the statistical ensemble. (A random network is a statistical ensemble: a set of configurations with their statistical weights.) Three first terms on the right-hand side of Eq. (\[e11\]) describe the process of the addition of a new vertex and the attachment it to a randomly chosen vertex. The two last terms describe the merging of a pair of randomly selected vertices. The factor $2$ in the last term is due to the fact that two vertices merge together.
Note that we assume that our networks are large. So, a merging pair almost surely has no common nearest neighbors, and we can ignore the emergence of multiple connections during merging, if we do not interested in the cutoff region of the degree distribution.
The total number of vertices in the network, $N$, is constant, so the evolution equation for the degree distribution $P(k,t) = \overline{N}(k,t)/N$ has the form:
$$\begin{aligned}
&&
\frac{\partial P(k,t)}{\partial t} = \delta_{k,1}
+
P(k-1,t) - P(k,t)
\nonumber
\\[5pt]
&&
+
\sum_{k'+k''=k}P(k',t)P(k'',t) - 2P(k,t)
\, .
\label{e12}\end{aligned}$$
(as is usual, asymptotically long times are considered). Assuming a stationary form of the degree distribution (except of a time-dependent cutoff and of, maybe, a low-degree part of the distribution), we arrive at a stationary equation. The $Z$-transform of the degree distribution (a generating function) is
$$n(z) \equiv
\sum_{k=0} z^k P(k)
\, .
\label{e13}$$
So, in a $Z$-transformed form, we have
$$0 = n^2(z) + (z-3)n(z) + z
\, .
\label{e14}$$
The solution of this equation is
$$n(z) = \frac{1}{2} [3-z - \sqrt{(9-z)(1-z)}]
\, .
\label{e15}$$
This gives $n(1)=1$, as it should be with $\sum_k P(k)=1$. This root of the equation is chosen, since it must be $0<n'(0)=P(1)<1$. The equation gives $n'(0)=P(1)=1/3$.
Near $z=1$,
$$n(z) \cong \text{analytical terms}+(1-z)^{3/2-1}
\, .
\label{e16}$$
This corresponds to the degree distribution $P(k) \sim k^{-3/2}$, Eq. (\[e1\]), since the form of a $Z$-transform near $z=1$ and the asymptotics of its original are related to each other in the following way
$$\begin{aligned}
n(z\sim 1) \!\!\!\!\!&\cong&\!\!\!\!\! \text{analytical terms}+(1-z)^{\gamma-1}
\nonumber
\\[5pt]
&\longleftrightarrow& \
P(k \gg 1) \sim k^{-\gamma}
\, ,
\label{e17}\end{aligned}$$
if $\gamma$ is non-integer. The result is rather typical not only for aggregation processes but also for general non-equilibrium networks, where the mean degree linearly grows with time (see discussion in Ref. [@dmbook03]).
Initially, we have assumed that the resulting degree distribution is stationary. In principle, one can made a more general assumption, e.g., for brevity, $P(k,t) = t^a f(k)$, where $a$ is some exponent, and $f(k)$ is an arbitrary function of $k$. After the substitution of this form into Eq. (\[e12\]) and $Z$-transformation, we find that the solution, $n(z,t)$, depends only on $z$, that is stationary.
Network B {#ss-b_d}
---------
The evolution equation for the mean number of vertices of degree $k$ in network B is
$$\begin{aligned}
&&
\overline{N}(k,t+1) =
\overline{N}(k,t) + q\delta_{k,m}
\nonumber
\\[5pt]
&&
+
\frac{qm}{N(t)}\overline{N}(k-1,t) - \frac{qm}{N(t)}\overline{N}(k,t)
\nonumber
\\[5pt]
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!+
\frac{1}{N^2(t)} \sum_{k'+k''=k}\overline{N}(k',t)\overline{N}(k'',t) - \frac{2}{N(t)}\overline{N}(k,t)
\, .
\label{e18}\end{aligned}$$
Here, the number of vertices $N(t) \cong (q-1)t$. The evolution equation for the degree distribution $P(k,t) \cong \overline{N}(k,t)/[(q-1)t]$ is
$$\begin{aligned}
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!(q-1)[t\partial_tP(k,t) + P(k,t)] = q\delta_{k,m}
+ qmP(k-1,t)
\nonumber
\\[5pt]
&&
\!\!\!\!\!\!\!\!\!\!\!\!\!- qmP(k,t)+ \sum_{k'+k''=k}P(k',t)P(k'',t) - 2P(k,t)
\, .
\label{e19}\end{aligned}$$
Assuming a stationary degree distribution, in a $Z$-transformed form, this is
$$0 = q z^m + n^2(z) + qmz n(z) - [q(m+1)+1]n(z)
\,
\label{e20}$$
\[compare with Eq. (\[e14\]) for $q=1$\]. The solution of Eq. (\[e20\]) is analytical at $z=1$, so that the degree distribution is a rapidly decreasing function. Suppose, however, that the rate $q$ is close to $1$. Then deviations from non-analytical behavior (\[e16\]) of $n(z)$ are observed only in the range $1-z \lesssim (q-1)^2/m$. This results in a size-independent cutoff $k_{\text{cut}} \sim m/(q-1)^2$ of the power-law dependence $P(k) \sim k^{-3/2}$.
Network C {#ss-c_d}
---------
The evolution equation for the mean number of vertices of degree $k$ in this network is of the form:
$$\begin{aligned}
&&
\overline{N}(k,t+1) =
\overline{N}(k,t) + q\delta_{k,m}
\nonumber
\\[5pt]
&&
+
\frac{qm}{N(t)}\overline{N}(k-1,t) - \frac{qm}{N(t)}\overline{N}(k,t)
\\[5pt]
&&
+ \frac{1}{N^2(t)\overline{k}} \sum_{k'+k''=k+2}\overline{N}(k',t)k''\overline{N}(k'',t)
\nonumber
\\[5pt]
&&
- \frac{1}{N(t)}\overline{N}(k,t) - \frac{k}{N(t)\overline{k}}\overline{N}(k,t)
\nonumber
\, .
\label{e21}\end{aligned}$$
Note that while deriving this equation, we did not assume the absence of correlations between the nearest neighbor vertices. Rule (2) of the model ensures that the second vertex for merging (a random nearest neighbor of a random vertex) is chosen with the probability proportional to the degree of this vertex. This is true irrespective of the presence of correlations in the network.
For the stationary degree distribution, in a $Z$-transformed form, we have
$$\begin{aligned}
&&
0 = q z^m + \frac{1}{z^2\overline{k}}n(z)zn'(z) - \frac{1}{\overline{k}}zn'(z)
\nonumber
\\[5pt]
&&
+ qmz n(z) - q(m+1)n(z)
\, .
\label{e22}\end{aligned}$$
\[Note that the $Z$-transform of the convolution with $\sum_{k'+k''=k+2}P(k')k''P(k'')$ is $$\frac{1}{z^2}\{n(z)zn'(z) - P(0)0P(0) -z(P(0)1P(1)+P(1)0P(0))\}
\, .$$ We have $P(0)=0$, so that this expression simplifies.\] Recall that
$$q = \frac{\overline{k}-2}{\overline{k}-2m} > 1
\, .
\label{e23}$$
Equation (\[e22\]) is the Abel equation of the second kind. In a canonical form it looks as
$$[z^2-n(z)]n'(z) =
\overline{k}q [z^{m+1} + (mz^2 - (m+1)z)n(z)]
\, .
\label{e24}$$
For the detailed analysis of a very similar equation for the size distribution of finite connected components in growing networks, see Ref. [@dms01]. Note, however, that in general terms, Eq. (\[e24\]) has a different solution than that in Ref. [@dms01].
If $P(k=0)=0$, Eq. (\[e24\]) implies
$$n(z\sim0) \cong
\frac{\overline{k}(\overline{k}-2)}
%%{\overline{k}(m+1)-2\overline{k}-2m^2}
{\overline{k}^2(m+1)-\overline{k}(2m+1)-2m^2}
z^m
\, .
\label{e25}$$
For obtaining a large $k$ asymptotics of $P(k)$, one must find the solution $n(z)$ of Eq. (\[e24\]) with the initial condition (\[e25\]) near $z=0$. One can check that this solution arrives at $1$ at $z=1$: $n(1)=1$.
The linearization of Eq. (\[e24\]) near $z=1$ shows that $n(z)$ linearly approaches $z=1$, with a derivative $n'(1)$, which satisfies the equation: $$[-2 + n'(1)]n'(1) = \frac{\overline{k}(\overline{k}-2)}{\overline{k}-2m}[-2m + n'(1)]
\, .
\label{e26}$$ The meaning of $n'(1)$ is the mean degree of a vertex with degree $k=o(N)$, that is the mean degree of a “normal” vertex. The solution of this square equation is $$n'(1) = \frac{\overline{k}^2 - 4m - |-\overline{k}^2 + 4\overline{k}m - 4m|}{2(\overline{k}-2m)}
\, .
\label{e27}$$ That is, there is a point $\overline{k}_c$ (and the corresponding $q_c$), satisfying the equation $$\overline{k}_c^2 - 4\overline{k}_cm + 4m=0
\, ,
\label{e28}$$ where the expression for $n'(1)$ changes its form. $\overline{k}_c$ and $q_c$ are given by Eqs. (\[e3\]) and (\[e4\]), respectively.
At $\overline{k}<\overline{k}_c$, the slope $n'(1)$ equals the mean degree of the network, $\overline{k}$. That is, all the vertices of the network are “normal”. Above $\overline{k}_c$ (i.e., at $q<q_c$) $n'(1) = 2qm = 2m(\overline{k}-2)/(\overline{k}-2m) < \overline{k}$. This means that a fraction $M = [\overline{k}-n'(1)]/\overline{k}$ \[see expression (\[e5\])\] is in the condensed state. The variation of the solution of Eq. (\[e24\]) with $\overline{k}$ is shown schematically in Fig. \[f14\].
The large-degree asymptotic behavior of the degree distribution of “normal vertices” is obtained by analyzing the form of the solution of Eq. (\[e24\]) near $z=1$. We pass to new variables $z = 1-\xi$, $n(z) = 1 - n'(1)\xi + v(\xi)$ in Eq. (\[e24\]). The inspection of the resulting equation shows that at $\overline{k} \neq \overline{k}_c$, the contribution $v(\xi)$ is a power-law function at small $\xi$, $v(\xi) \propto \xi^a$, where exponent $a$ is greater than $1$. The substitution of this form into the equation allows us to obtain $a$. Using the correspondence (\[e17\]) results in the formulas (\[e6\]) and (\[e7\]) for the $\gamma$ exponent of the degree distribution in the normal and condensed phases, respectively (see Fig. \[f6\]).
In the critical point, $\overline{k} = \overline{k}_c$, the solution of the equation for $v(\xi)$ has a more complex form with an additional logarithmic factor (for more detail, see Ref. [@dms01]). The resulting solution $n(z\sim1)$ is
$$n(z) \cong 1 - \overline{k}_c (1-z) +
\text{const}\,\frac{1-z}{\ln[\text{const}(1-z)]}
\, .
\label{e29}$$
The asymptotics of the original of this $Z$-transform is
$$\begin{aligned}
&&
P(k) = \oint_c \frac{dz}{2\pi i}z^{-k-1} P(k)
\nonumber
\\[5pt]
&&
\propto
\oint_c \frac{dz}{2\pi i} \frac{1-z}{\ln[\text{const}(1-z)]}z^{-k-1}
\cong
\oint_{c'} \frac{ds}{2\pi i} \frac{s}{\ln s} e^{sk}
\nonumber
\\[5pt]
&&
=
\int_0^{\infty} \frac{ds}{2\pi i}\,s \left(\frac{1}{\ln s} - \frac{1}{\ln s + 2\pi i}\right)e^{-sk}
\nonumber
\\[5pt]
&&
\cong \int_0^{\infty} \frac{ds}{2\pi i}\,s \frac{2\pi i}{\ln^2 s}e^{-sk}
\sim \frac{1}{k^2\ln^2 k}
\, .
\label{e30}\end{aligned}$$
Here the contour $c$ is around $0$, within the unit circle. The contour $c$ is deformed to the contour $c'$, which comes from $-\infty$ to $+0$ along the cut of $\ln$ and then returns to $-\infty$ by the other shore of the cut. This deformation was made to ensure the decrease of the exponent. Thus, we have arrived at the degree distribution (\[e8\]) at the critical point.
One can show that at $\overline{k} \neq \overline{k}_c$, in the large networks, the degree distribution has two regions. (i) The power-law dependence with exponents (\[e6\]) and (\[e7\]) is realized in the range of degrees $\ln k > 2(\overline{k}^2 - 4\overline{k} + 4m)/|\overline{k}^2 - 4\overline{k}m + 4m|$. (ii) At smaller degrees, the critical dependence (\[e8\]) is present.
Network D {#ss-d_d}
---------
Merging vertices in this model are the ends of randomly selected edges. In this case, for the strict description of the network evolution one has to solve an equation for a joint distribution $P(k',k'')$ of the degrees of the nearest neighbor vertices. This is an essentially more hard problem than the analysis of Eq. (\[e12\]), (\[e19\]), and (\[e21\]) for $P(k)$ or $\overline{N}(k)$ in previous sections. Instead of making these cumbersome calculations, we ignore the possibility of the degree–degree correlations and check whether this ansatz leads to reasonable results or not. This simplification allows us to consider a more simple evolution equation.
Assuming the absence of correlations between degrees of the nearest neighbors of network D results in the following rate equation:
$$\begin{aligned}
&&
\overline{N}(k,t+1) =
\overline{N}(k,t) + q\delta_{k,m}
\nonumber
\\[5pt]
&&
+
\frac{qm}{N(t)}\overline{N}(k-1,t) - \frac{qm}{N(t)}\overline{N}(k,t)
\nonumber
\\[5pt]
&&
+ \frac{1}{N^2(t)\overline{k}^2} \sum_{k'+k''=k+2}k'\overline{N}(k',t)k''\overline{N}(k'',t)
\nonumber
\\[5pt]
&&
- 2\frac{k}{N(t)\overline{k}}\overline{N}(k,t)
\, .
\label{e31}\end{aligned}$$
For brevity, we consider only the case $m>2$ (some of the resulting formulas may be different in the cases of $m=1,2$). So, for a stationary degree distribution, we have in a $Z$-transformed form:
$$\begin{aligned}
&&
0 = q z^m + \frac{1}{z^2\overline{k}^2}[zn'(z)]^2
- \frac{2}{\overline{k}}zn'(z)
\nonumber
\\[5pt]
&&
+ qmz n(z) - [q(m+1)-1]n(z)
\, .
\label{e32}\end{aligned}$$
This is a nonlinear differential equation of the first order. It crucially differs from the corresponding Eq. (\[e22\]) for the model C, and provides a different set of behaviors.
Near $z=0$, the solution of Eq. (\[e32\]) is of the following form:
$$n(z \sim 0) \cong
\frac{\overline{k}(\overline{k}-2)}{2\overline{k}(m-1) - 4m^2 + \overline{k}^2 m} z^m
\, .
\label{e33}$$
This asymptotics is used as a boundary condition for Eq. (\[e32\]).
For a numerical analysis, the following form of Eq. (\[e32\]) is more convenient:
$$\begin{aligned}
&&
n'(z) = kz - k\left\{z^2 - \frac{1}{\overline{k}-2m}[\overline{k}^2(\overline{k}-2)z^m \right.
\nonumber
\\[5pt]
&&
\left.
\phantom{\frac{1}{|}}+ ((\overline{k}-2)mz-\overline{k}+2m)n(z)] \right\}^{1/2}
\, ,
\label{e34}\end{aligned}$$
Note that only the solution with ‘minus’ is reasonable. One can check that at the boundary $z=1$, if $n(1)=1$, then Eq. (\[e32\]) gives $n'(1)=\overline{k}$ and vice versa, if $n'(1)=\overline{k}$, then Eq. (\[e32\]) gives $n(1)=1$. On the other hand, the numerical solution of Eq. (\[e34\]) shows that the solution with $n(1)=1$ exists only at sufficiently low $\overline{k}$. Above some value $\overline{k}_c$, the solutions of Eq. (\[e32\]) or Eq. (\[e34\]) turn out to be greater than $1$ at some $z$: $n(1)>1$ (see Fig. \[f15\], compare with Fig. \[f14\]). But this is impossible, since surely $\sum_k P(k) = 1$. This contradiction indicates that our assumption does not hold at least for $\overline{k}>\overline{k}_c$.
Let us study the analytical structure of the solution $n(z)$, $z$ near $1$, for $\overline{k}<\overline{k}_c$. As in Sec. \[ss-c\_d\], introducing new variables, $z=1-\xi$, $n(z) = 1 - \overline{k}\xi + v(\xi)$, we pass to an equation for $v(\xi)$. It turns out, however, that unlike Sec. \[ss-c\_d\], this equation has no solution $v(\xi) \sim \xi^a$ with exponent $1<a<2$. So, we have to search for the solution in the analytical form $v(\xi)=C\xi^2$. Substituting this form into the equation gives
$$\begin{aligned}
&&
%%C = \frac{1}{4}\frac{\overline{k}}{\overline{k}-2m}
%%\left[
%%\overline{k}+\overline{k}m-4m - \right.
C = \frac{\overline{k}(\overline{k}+\overline{k}m-4m)}{4(\overline{k}-2m)}
\nonumber
\\[5pt]
&&
%%\left.
%%\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
%%\sqrt{[(m\!+\!1)\overline{k}\!-\!4m]^2 - 2(\overline{k}\!-\!2)(\overline{k}\!-\!2m)m(2\overline{k}\!+\!m\!-\!1)}
%%\right]
\!\!\!\!\!\!\!\!\!\!\!\!\!
\times
\left[
1 - \sqrt{1 - 2\frac{(\overline{k}-2)(\overline{k}-2m)m(2\overline{k}+m-1)}{(\overline{k}+\overline{k}m-4m)^2}\,}
\right]
%\,
\!.
\label{e35} \end{aligned}$$
(One can check that only the sign “minus” at the square root leads to reasonable final values of the exponent of the degree distribution.) $C$ is real only if $\overline{k} \leq \overline{k}_c$, where $\overline{k}_c$ is the point where the square root in Eq. (\[e35\]) is zero. $\overline{k}_c$ is exactly the point above which the solution $n(z)$ is not reasonable \[i.e., where $n(1)>1$\]. The resulting expression for $\overline{k}_c$ is cumbersome but for $m \gg 1$ we have
$$\overline{k}_c \cong \frac{7+\sqrt{113}}{8}m - \frac{3-11/\sqrt{113}}{7+\sqrt{113}}
%%= 2.20377m - 0.111469
\, ,
\label{e36}$$
which leads to the formula (\[e9\]). So, if $\overline{k} \leq \overline{k}_c$, we can substitute $v(\xi)=C\xi^2+w(\xi)$ into the equation for $v(\xi)$. An inspection of the resulting equation for $w(\xi)$ shows that $w(\xi) \cong D\xi^b$, where exponent $b>2$. One can easily find $b$:
$$b = \frac{\overline{k}^2(m-1)}{(\overline{k}-2m)(2C-\overline{k})}
\,
\label{e36a}$$
with $C$ given by Eq. (\[e35\]), and by using the correspondence rule (\[e17\]) we readily obtain the exponent of the degree distribution: $$\begin{aligned}
&&
\gamma = 3 +
\nonumber
\\[7pt]
&&
%%\!\!\!\!\!\!\!\!\frac{2\sqrt{(\overline{k}+\overline{k}m-4m)^2-2(\overline{k}-2)(\overline{k}-2m)m(2\overline{k}+m-1)}}{\overline{k}(m\!-\!1)\!-\!\sqrt{(\overline{k}\!+\!\overline{k}m\!-\!4m)^2\!-\!2(\overline{k}\!-\!2)(\overline{k}\!-\!2m)m(2\overline{k}\!+\!m\!-\!1)}}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\frac{2\sqrt{1 - 2\dfrac{(\overline{k}-2)(\overline{k}-2m)m(2\overline{k}+m-1)}{(\overline{k}\!+\!\overline{k}m\!-\!4m)^2}}}
{\dfrac{\overline{k}(m\!-\!1)}{\overline{k}\!+\!\overline{k}m\!-\!4m}\!-\!\sqrt{1\!-\!2\dfrac{(\overline{k}\!-\!2)(\overline{k}\!-\!2m)m(2\overline{k}\!+\!m\!-\!1)}{(\overline{k}\!+\!\overline{k}m\!-\!4m)^2}}}
\, .
\label{e37} \end{aligned}$$ This $\gamma(\overline{k})$ dependence is shown in Fig. \[f8\].
The analytical results for network D were obtained in the framework of the simplifying ansatz: we ignored degree–degree correlations. The simulation of this network has shown that the correlations are significant (see Sec. \[ss-d\]) and that our analytical predictions for the normal must be corrected. Furthermore, the simulations have allowed us to describe the structure of the network in the condensation phase.
Splitting vertices {#ss-splitr_d}
------------------
Here we only show the contributions due to splitting processes to evolution equations for the degree distribution. Splitting process (1) of Sec. \[ss-split\] generates with equal probabilities all possible configurations of two unconnected vertices \[see Fig. \[f11\] (a)\]. For example, the splitting of a vertex of degree $4$ produces: (i) with probability $1/8$, a pair of vertices of degrees $0$ and $4$, (ii) with probability $4/8$, a pair of vertices of degrees $1$ and $3$, and (iii) with probability $3/8$, a pair of vertices of degrees $2$ and $2$. One can easily write down the probability of resulting configurations for any degree of the splitting vertex. This allows us to find the probability that a vertex of degree $k$ will emerge due to the splitting of a vertex of degree $q$. Finally, instead, e.g., of the contribution $\delta_{k,1} + P(k-1,t)-P(k,t)$ due to the process of random attachment of a vertex, we have the following terms: $$\sum_{q=0}\frac{1}{2^{q-1}} {q \choose k}P(q,t) - P(k,t)
\, .
\label{e38}$$ Note that actually only terms $q \geq k$ are nonzero in this sum. In a $Z$-transformed form, this is $$2 n\left(\frac{1+z}{2}\,,t\right) - n(z,t)
\, .
\label{a39}$$
In splitting process (2) of Sec. \[ss-split\], an emerging pair of vertices is interconnected by an extra edge. This leads, instead of the contribution of the form (\[e38\]), to the following terms: $$\sum_{q=0}\frac{1}{2^{q-1}} {q \choose k-1}P(q,t) - P(k,t)
\, ,
\label{e40}$$ where only terms $q \geq k-1$ are nonzero in this sum. In a $Z$-transformed form, Eq. (\[e40\]) looks as $$2 z\, n\left(\frac{1+z}{2}\,,t\right) - n(z,t)
\, .
\label{e41}$$
Formulas (\[e38\])–(\[e41\]) may be used to modify any of the evolution equations in the preceding sections. The resulting equations, in a $Z$-transformed form, will be non-local due to the $n((1+z)/2)$ term. We do not analyze these functional equations in the present paper.
Discussion and summary {#s-summary}
======================
Several remarks are necessary. We have studied a representative set of models allowing an analytical solution if no strong correlations are present. For sake of simplicity and brevity, we considered only models, generating stationary (at least is some range of degrees) degree distributions. We have indicated that we had to ignore correlations in one of networks, and have demonstrated that this neglect created some problems. In addition, we ignored multiple connections and loops of length $1$, which, in principle, emerge during the evolution of these networks.
The effect of multiple connections and loops of length $1$ in related network models was discussed in Ref. [@bk03]. Several features of the network structure depend on the presence (or absence) of these configurations of edges. In particular, the position of a size-dependent cutoff of the degree distribution may change. The number of vertices, attracting the condensate of edges may also change. We did not considered these details (cutoffs, the precise number of vertices attracting the condensate, etc.). So, the strict accounting for multiple connections and one-loops should not essentially change the conclusions of this paper.
One should emphasize the difference of the condensation transitions considered in this work from those which were observed earlier in networks. We have shown that in networks C and D, scale-free degree distributions are realized both in the normal and in the condensation phases. In contrast, in equilibrium networks, a scale-free degree distribution exists only in the condensation phase, and a degree distribution is rapidly decreasing in the normal phase [@dmbook03; @zjj02].
The merging of vertices is only one of processes in the networks under discussion. This process reduces the number of vertices, so other processes, increasing this number, must be present. In this paper, as a parallel process, we used an injection of new vertices (with subsequent attachment to existing vertices) and fragmentation (splitting). This subject has been analyzed in the aggregation literature [@vz88; @stc92; @stc94]. Also, one can implement duplication (copying) or partial copying of vertices [@kkkr02; @dms02; @spsk02] or other processes. One should note, that correlations are important in low-dimensional aggregation processes. In this respect we have an analogy with this case. In the usual aggregation theory the analysis proceeds by considering the scaling properties of the aggregation or fragmentation kernels (see e.g. [@ha02] and references therein). For models such as those considered in our paper these are not known a priori and in the presence of degree correlations arise “self-consistently”.
As is natural, a uniform picture for all networks created by aggregation processes cannot be obtained in principle. However, the models that we have considered in the present paper, reveal a number of basic features:
- The aggregation easily generates network architectures where hubs play a profound role.
- The aggregation often leads to gelation, or, in other words, to condensation of edges in these networks. We have found the condensation point at some mean degree value and have traced the variation of network structure in the normal and the condensation phases.
- These networks are evolving networks, and so their structure is characterized by strong correlations, in particular, by strong degree–degree correlations of assortative type.
These features, which we observed by using demonstrative models, should be present in more complex networks of this kind.
MJA is grateful to the Center of Excellence program of the Academy of Finland for support. SD would like to acknowledge the hospitality of the Helsinki University of Technology. A part of the work was made when one of the authors (SD) attended the Exystence Thematic Institute on Networks and Risks (Collegium Budapest, June 2004). SD was partially supported by project POCTI/FAT/46241/2002 and POCTI/MAT/46176/2002. SD thanks Kim Sneppen for a useful discussion in Budapest, and MJA thanks Supriya Krishnamurty for useful comments.
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|
---
abstract: |
Let $\mathcal{G}=
\left[\smallmatrix A & M\\
N & B \endsmallmatrix \right]$ be a generalized matrix algebra defined by the Morita context $(A, B, _AM_B, _BN_A, \Phi_{MN},
\Psi_{NM})$. In this article we mainly study the question of whether there exist proper Jordan derivations for the generalized matrix algebra $\mathcal{G}$. It is shown that if one of the bilinear pairings $\Phi_{MN}$ and $\Psi_{NM}$ is nondegenerate, then every antiderivation of $\mathcal{G}$ is zero. Furthermore, if the bilinear pairings $\Phi_{MN}$ and $\Psi_{NM}$ are both zero, then every Jordan derivation of $\mathcal{G}$ is the sum of a derivation and an antiderivation. Several constructive examples and counterexamples are presented.
address:
- 'Li: Department of Information and Computing Sciences, Northeastern University at Qinhuangdao, Qinhuangdao, 066004, P.R. China'
- 'van Wyk: Department of Mathematics, Stellenbosch University, Private Bag X1, Matieland 7602, Stellenbosch, South Africa'
- 'Wei: School of Mathematics, Beijing Institute of Technology, Beijing, 100081, P. R. China'
author:
- 'Yanbo Li, Leon van Wyk and Feng Wei'
date: 'September 13, 2011'
title: Jordan Derivations and Antiderivations of Generalized Matrix Algebras
---
[^1]
Introduction {#xxsec1}
============
Let us begin with the definition of generalized matrix algebras given by a Morita context. Let $\mathcal{R}$ be a commutative ring with identity. A Morita context consists of two $\mathcal{R}$-algebras $A$ and $B$, two bimodules $_AM_B$ and $_BN_A$, and two bimodule homomorphisms called the pairings $\Phi_{MN}: M\underset {B}{\otimes} N\longrightarrow A$ and $\Psi_{NM}: N\underset {A}{\otimes} M\longrightarrow B$ satisfying the following commutative diagrams: $$\xymatrix{ M \underset {B}{\otimes} N \underset{A}{\otimes} M
\ar[rr]^{\hspace{8pt}\Phi_{MN} \otimes I_M} \ar[dd]^{I_M \otimes
\Psi_{NM}} && A
\underset{A}{\otimes} M \ar[dd]^{\cong} \\ &&\\
M \underset{B}{\otimes} B \ar[rr]^{\hspace{10pt}\cong} && M }
\hspace{4pt}{\rm and}\hspace{4pt} \xymatrix{ N \underset
{A}{\otimes} M \underset{B}{\otimes} N
\ar[rr]^{\hspace{8pt}\Psi_{NM}\otimes I_N} \ar[dd]^{I_N\otimes
\Phi_{MN}} && B
\underset{B}{\otimes} N \ar[dd]^{\cong}\\ &&\\
N \underset{A}{\otimes} A \ar[rr]^{\hspace{10pt}\cong} && N
\hspace{2pt}.}$$ Let us write this Morita context as $(A, B, _AM_B, _BN_A, \Phi_{MN},
\Psi_{NM})$. If $(A, B, _AM_B,$ $ _BN_A,$ $ \Phi_{MN}, \Psi_{NM})$ is a Morita context, then the set $$\left[
\begin{array}
[c]{cc}A & M\\
N & B\\
\end{array}
\right]=\left\{ \left[
\begin{array}
[c]{cc}a& m\\
n & b\\
\end{array}
\right] \vline a\in A, m\in M, n\in N, b\in B \right\}$$ form an $\mathcal{R}$-algebra under matrix-like addition and matrix-like multiplication, where We assume that at least one of the two bimodules $M$ and $N$ is distinct from zero. Such an $\mathcal{R}$-algebra is called a *generalized matrix algebra* of order 2 and is usually denoted by $\mathcal{G}=
\left[\smallmatrix A & M\\
N & B \endsmallmatrix \right]$. This kind of algebra was first introduced by Morita in [@Morita], where the author investigated Morita duality theory of modules and its applications to Artinian algebras.
Let $\mathcal{R}$ be a commutative ring with identity, $A$ be a unital algebra over $\mathcal{R}$ and $\mathcal{Z}(A)$ be the center of $A$. Recall that an $\mathcal{R}$-linear map $\Theta_{\rm d}$ from $A$ into itself is called a *derivation* if $\Theta_{\rm
d}(ab)=\Theta_{\rm d}(a)b+a\Theta_{\rm d}(b)$ for all $a, b\in A$. Further, an $\mathcal{R}$-linear map $\Theta_{\rm Jord}$ from $A$ into itself is called a *Jordan derivation* if $\Theta_{\rm
Jord}(a^2)=\Theta_{\rm Jord}(a)a+a\Theta_{\rm Jord}(a)$ for all $a\in A$. Every derivation is obviously a Jordan derivation. The inverse statement is not true in general. Those Jordan derivations which are not derivations are said to be *proper*. An $\mathcal{R}$-linear map $\Theta_{\rm antid}$ from $A$ into itself is called an *antiderivation* if $\Theta_{\rm
antid}(ab)=\Theta_{\rm antid}(b)a+b\Theta_{\rm antid}(a)$ for all $a, b\in A$.
In 1957 Herstein [@Herstein] proved that every Jordan derivation from a prime ring of characteristic not $2$ into itself is a derivation. This result has been generalized to different rings and algebras in various directions (see e.g. [@Benkovic; @Bresar1; @Bresar2; @Bresar4; @HanWei; @HouQi; @LiBenkovic; @XiaoWei1; @ZhangYu] and references therein). Zhang and Yu [@ZhangYu] showed that every Jordan derivation on a triangular algebra is a derivation. Xiao and Wei [@XiaoWei1] extended this result to the higher case and obtained that any Jordan higher derivation on a triangular algebra is a higher derivation. Johnson [@Johnson] considered a more challenging question for which Banach algebras $A$ there are no proper Jordan derivations from $A$ into an arbitrary Banach $A$-bimodule $M$. It turned out that this is true for some important classes of algebras (in particular, for the algebra of all $n\times
n$ complex matrices). Motivated by Johnson’s work, Benkovic investigated the structure of Jordan derivations from the upper triangular matrix algebra $\mathcal{T}_n(\mathcal{R})$ into its bimodule and proved that every Jordan derivation from $\mathcal{T}_n(\mathcal{R})$ into its bimodule is the sum of a derivation and an antiderivation. Recently, Li, Xiao and Wei [@LiXiao; @LiWei; @XiaoWei2] jointly studied linear maps of generalized matrix algebras, such as derivations, Lie derivations, commuting maps and semicentralizing maps. Our main purpose is to develop the theory of linear maps of triangular algebras to the case of generalized matrix algebras, which has a much broader background. People pay much less attention to linear maps of generalized matrix algebras, to the best of our knowledge there are fewer articles dealing with linear maps of generalized matrix algebras except for [@BobocDascalescuvanWyk; @LiXiao; @LiWei; @XiaoWei2].
The problem that we address in this article is to study whether there exist proper Jordan derivations for generalized matrix algebras. The outline of this article is as follows. The second section presents two basic examples of generalized matrix algebras which we will revisit later. In the third section we describe the general form of Jordan derivations and antiderivations on generalized matrix algebras. We observe that any antiderivation on a class of generalized matrix algebra is zero (see Proposition \[xxsec3.10\]). Furthermore, it is shown that every Jordan derivation on another class of generalized matrix algebras is the sum of a derivation and an antiderivation (see Theorem \[xxsec3.11\]).
Examples of Generalized Matrix Algebras {#xxsec2}
=======================================
We have presented many examples of generalized matrix algebras in [@LiWei], such as standard generalized matrix algebras and quasi-hereditary algebras, generalized matrix algebras of order $n$, inflated algebras, upper and lower triangular matrix algebras, block upper and lower triangular matrix algebras, nest algebras. For later discussion convenience, we have to give another two new generalized matrix algebras.
Generalized matrix algebras from smash product algebras {#xxsec2.1}
-------------------------------------------------------
Let $H$ be a finite dimensional Hopf algebra over a field $\mathbb{K}$ with comultiplication $\Delta: H\longrightarrow
H\bigotimes H$, counit $\varepsilon: H\longrightarrow \mathbb{K}$ and antipode $S: H\longrightarrow H$. Clearly, $S$ is bijective. Moreover, the space of left integrals $\int_H l=\left\{x\in
H|hx=\varepsilon(h)x, \forall h\in H\right\}$ is one-dimensional. We substitute the “sigma notation" for $\Delta$ in the present article. Now assume that $A$ is an $H$-module algebra, that is, $A$ is a $\mathbb{K}$-algebra which is a left $H$-module, such that
1. $h\cdot(ab)=\underset{(h)}{\sum}(h_1\cdot a) (h_2\cdot b)$ and
2. $h\cdot 1_A=\varepsilon(h)1_A$.
for all $h\in H, a, b\in A$. Then the *smash product algebra* $A\# H$ is defined as follows, for any $a, b\in A, h, k\in H$:
1. as a $\mathbb{K}$-space, $A\#H=A\otimes H$. We write $a\#h$ for the element $a\otimes h$
2. multiplication is given by $(a\#h)(b\#k)=\underset{(h)}{\sum} a(h_1\cdot b)\#h_2k$.
The *invariants subalgebra* of $H$ on $A$ is the set $A^H=\{x\in A|h\cdot x=\varepsilon(h)x, \forall h\in H\}$. $A$ is a left $A\#H$-module in the standard way, that is $$a\#h\rightarrow b=a(h\cdot b)$$ for all $a, b\in A$ and $h\in H$. For a given $t\in \int l$, then $th\in \int l$ for all $h\in H$. Since $\int l$ is one-dimensional, there exists $\alpha\in H^*$ such that $th=\alpha(h)t$ for all $h\in
H$. It is easy to see that $\alpha$ is multiplicative, and it is a group-like element of $H^*$. Hence $$h^\alpha=\alpha\rightarrow h=\underset{(h)}{\sum}\alpha(h_2)h_1,
\quad \forall h\in H$$ defines an automorphism on $H$. Thus $A$ is a right $A\#H$-module via $$a\leftarrow b\#h=S^{-1}h^\alpha\cdot (ab), \quad \forall a\in A,
b\#h\in A\#H.$$
The close relationship between $A\# H$ and $A^H$ enables us to formalize the following generalized matrix algebra. Now $A$ is a left (or right) $A^H$-module simply by left (or right) multiplication. Simultaneously, $A$ is also a left (or right) $A\#H$-module. Thus $M=_{A^H}A_{A\#H}$ and $N=_{A\#H}A_{A^H}$, together with the maps $$\begin{aligned}
\Psi_{NM}: A\otimes_{A^H} A &
\longrightarrow A\#H \hspace{3pt} {\rm defined \hspace{3pt} by}\hspace{3pt} & \Psi_{NM}(a,b)&=(a\#t)(b\#1)\\
\Phi_{MN}: A\otimes_{A\#H} A^H & \longrightarrow A^H \hspace{3pt}
{\rm defined \hspace{3pt} by}\hspace{3pt} & \Phi_{NM}(a,b)&=t\cdot
(ab)
\end{aligned}$$ give rise to a new generalized matrix algebra $$\mathcal{G}_{\rm SPA}=\left[
\begin{array}
[c]{cc}A^H & M\\
N & A\#H\\
\end{array}
\right].$$ We refer the reader to [@Montgomery] about the basic properties of $\mathcal{G}_{\rm SPA}$.
Generalized matrix algebras from group algebras {#xxsec2.2}
-----------------------------------------------
Let $A$ be an associative algebra over a field $\mathbb{K}$ and $G$ be a finite group of automorphisms acting on $A$. The *fixed ring* $A^G$ of the action $G$ on $A$ is the set $\left\{a\in
A|a^g=a, \forall g\in G\right\}$. The *skew group algebra* $A*G$ is the set of all formal sums $\sum _{g\in G}a_g g,
\hspace{2pt} a_g\in A$. The addition operation is componentwise and the multiplication operation is defined distributively by the formula $$ag\cdot bh=ab^{g^{-1}} gh$$ for all $a, b\in A$ and $g, h\in G$. Clearly, $A$ is a left and right $A^G$-module. $A$ can also be viewed as a left or right $A*G$-module as follows: for any $x=\sum_{g\in G}a_gg\in A*G$ and $a\in A$, we define $ x\cdot a=\sum_{g\in G} a_ga^{g^{-1}} $ and $
a\cdot x=\sum_{g\in G} (aa_g)^g. $ Then we obtain a generalized matrix algebra $$\mathcal{G}_{\rm GA}=\left[
\begin{array}
[c]{cc}A^G & M\\
N & A*G\\
\end{array}
\right],$$ where $M=_{A^G}A_{A*G}$ and $N=_{A*G}A_{A^G}$. The bilinear pairings $\Phi_{MN}$ and $\Psi_{NM}$ can be established via $$\begin{aligned}
\Phi_{MN}: A\otimes_{A*G} A & \longrightarrow A^G \\
(x, y) &\longmapsto \sum_{g\in G}(xy)^g
\end{aligned}$$ and $$\begin{aligned}
\Psi_{NM}: A\otimes_{A^G} A & \longrightarrow A*G\\
(x, y)& \longmapsto \sum_{g\in G}xy^{g^{-1}}g.
\end{aligned}$$
Jordan Derivations of Generalized Matrix Algebras {#xxsec3}
=================================================
Let $\mathcal{G}$ be a generalized matrix algebra of order $2$ based on the Morita context $(A, B, _AM_B, _BN_A, \Phi_{MN}, \Psi_{NM})$ and let us denote it by $$\mathcal{G}:=\left[
\begin{array}
[c]{cc}A & M\\
N & B
\end{array}
\right].$$ Here, at least one of the two bimodules $M$ and $N$ is distinct from zero. The main aim of this section is to show that any Jordan derivation on a class of generalized matrix algebras is the sum of a derivation and an antiderivation. Our motivation originates from the following several results. Benkovic [@Benkovic] proved that every Jordan derivation from the algebra of all upper triangular matrices into its bimodule is the sum of a derivation and an antiderivation. Ma and Ji [@MaJi] extended this result to the case of generalized Jordan derivations and obtained that every generalized Jordan derivation from the algebra of all upper triangular matrices into its bimodule is the sum of a generalized derivation and an antiderivation. Zhang and Yu in [@ZhangYu] showed that every Jordan derivation on a triangular algebra is a derivation. Therefore, it is appropriate to describe and characterize Jordan derivations of $\mathcal{G}$. Note that the forms of derivations and Lie derivations of $\mathcal{G}$ were given in [@LiWei].
[@LiWei Proposition 4.2]\[xxsec3.1\] An additive map $\Theta_{\rm d}$ from $\mathcal{G}$ into itself is a derivation if and only if it has the form $$\begin{aligned}
& \Theta_{\rm d}\left(\left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\right) \\
=& \left[
\begin{array}
[c]{cc}\delta_1(a)-mn_0-m_0n & am_0-m_0b+\tau_2(m)\\
n_0a-bn_0+\nu_3(n) & n_0m+nm_0+\mu_4(b)\\
\end{array}
\right] , (\bigstar1)\\
& \forall \left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\in \mathcal{G},
\end{aligned}$$ where $m_0\in M, n_0\in N$ and $$\begin{aligned} \delta_1:& A \longrightarrow A, &
\tau_2: & M\longrightarrow M, & \nu_3: & N\longrightarrow N , &
\mu_4: & B\longrightarrow B
\end{aligned}$$ are all $\mathcal{R}$-linear maps satisfying the following conditions:
1. $\delta_1$ is a derivation of $A$ with $\delta_1(mn)=\tau_2(m)n+m\nu_3(n);$
2. $\mu_4$ is a derivation of $B$ with $\mu_4(nm)=n\tau_2(m)+\nu_3(n)m;$
3. $\tau_2(am)=a\tau_{2}(m)+\delta_1(a)m$ and $\tau_2(mb)=\tau_2(m)b+m\mu_4(b);$
4. $\nu_3(na)=\nu_3(n)a+n\delta_1(a)$ and $\nu_3(bn)=b\nu_3(n)+\mu_4(b)n.$
\[xxsec3.2\] An additive map $\Theta_{\rm Jord}$ from $\mathcal{G}$ into itself is a Jordan derivation if and only if it is of the form $$\begin{aligned}
& \Theta_{\rm Jord}\left(\left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\right)\\ =& \left[
\begin{array}
[c]{cc}\delta_1(a)-mn_0-m_0n+\delta_4(b) & am_0-m_0b+\tau_2(m)+\tau_3(n)\\
n_0a-bn_0+\nu_2(m)+\nu_3(n) & \mu_1(a)+n_0m+nm_0+\mu_4(b)\\
\end{array}
\right] ,(\bigstar2)\\
& \forall \left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\in \mathcal{G},
\end{aligned}$$ where $m_0\in M, n_0\in N$ and $$\begin{aligned} \delta_1:& A \longrightarrow A, & \delta_4:& B \longrightarrow A, & \tau_2:
& M\longrightarrow M, & \tau_3: & N\longrightarrow M,\\
\nu_2: & M\longrightarrow N, & \nu_3: & N\longrightarrow N & \mu_1:
& A\longrightarrow B & \mu_4: & B\longrightarrow B
\end{aligned}$$ are all $\mathcal {R}$-linear maps satisfying the following conditions:
1. $\delta_1$ is a Jordan derivation on $A$ and $\delta_1(mn)=-\delta_4(nm)+\tau_2(m)n+m\nu_3(n);$
2. $\mu_4$ is a Jordan derivation on $B$ and $\mu_4(nm)=-\mu_1(mn)+n\tau_2(m)+\nu_3(n)m;$
3. $\delta_4(b^2)=2\delta_4(b)=0$ for all $ b\in B$ and $\mu_1(a^2)=2\mu_1(a)=0$ for all $a\in A;$
4. $\tau_2(am)=a\tau_2(m)+\delta_1(a)m+m\mu_1(a)$ and $\tau_2(mb)=\tau_2(m)b+m\mu_4(b)+\delta_4(b)m;$
5. $\nu_3(bn)=b\nu_3(n)+\mu_4(b)n+n\delta_4(b)$ and $\nu_3(na)=\nu_3(n)a+n\delta_1(a)+\mu_1(a)n;$
6. $\tau_3(na)=a\tau_3(n)$, $\tau_3(bn)=\tau_3(n)b$, $n\tau_3(n)=0$, $\tau_3(n)n=0;$
7. $\nu_2(am)=\nu_2(m)a$, $\nu_2(mb)=b\nu_2(m)$, $m\nu_2(m)=0$, $\nu_2(m)m=0.$
Suppose that the Jordan derivation $\Theta_{\rm Jd}$ is of the form $$\begin{aligned}
& \Theta_{\rm Jord}\left(\left[
\begin{array}
[c]{cc}a & m \\
n & b \\
\end{array}
\right]\right) \\
= &\left[
\begin{array}
[c]{cc}\delta_1(a)+\delta_2(m)+\delta_3(n)+\delta_4(b) & \tau_1(a)+\tau_2(m)+\tau_3(n)+\tau_4(b) \\
\nu_1(a)+\nu_2(m)+\nu_3(n)+\nu_4(b) & \mu_1(a)+\mu_2(m)+\mu_3(n)+\mu_4(b) \\
\end{array}
\right] ,
\end{aligned}$$ for all $\left[\smallmatrix a & m\\
n & b
\endsmallmatrix \right]\in \mathcal{G}$, where $\delta_1,\delta_2,\delta_3,\delta_4$ are $\mathcal{R}$-linear maps from $A, M, N, B$ to $A$, respectively; $\tau_1,\tau_2$, $\tau_3,\tau_4$ are $\mathcal{R}$-linear maps from $A, M, N, B$ to $M$, respectively; $\nu_1,\nu_2,\nu_3,\nu_4$ are $\mathcal{R}$-linear maps from $A, M, N, B$ to $N$, respectively; $\mu_1,\mu_2,\mu_3,\mu_4$ are $\mathcal{R}$-linear maps from $A, M,
N, B$ to $B$, respectively.
For any $G\in \mathcal{G}$, we will intensively employ the Jordan derivation equation $$\Theta_{\rm Jord}(G^2)=G\Theta_{\rm Jord}(G)+\Theta_{\rm Jord}(G)G.
\eqno(3.1)$$ Taking $G=\left[\smallmatrix a & 0\\
0 & 0 \endsmallmatrix \right]$ into $(3.1)$ we have $$\begin{aligned}
\Theta_{\rm Jord}(G^2)=& \left[
\begin{array}
[c]{cc}\delta_1(a^2) & \tau_1(a^2)\\
\nu_1(a^2) & \mu_1(a^2)\\
\end{array}
\right]
\end{aligned}
\eqno(3.2)$$ and $$\begin{aligned}
&G\Theta_{\rm Jord}(G)+\Theta_{\rm Jord}(G)G= \left[
\begin{array}
[c]{cc}a\delta_1(a)+\delta_1(a)a & a\tau_1(a)\\
\nu_1(a)a & 0\\
\end{array}
\right].
\end{aligned}
\eqno(3.3)$$ By $(3.2)$ and $(3.3)$ we know that $\delta_1$ is a Jordan derivation of $A$, $$\tau_1(a^2)=a\tau_1(a), \quad\quad\nu_1(a^2)=\nu_1(a)a \eqno(3.4)$$ and $$\mu_1(a^2)=0.\eqno(3.5)$$ for all $a\in A$. Similarly, putting $G=\left[\smallmatrix 0 & 0\\
0 & b \endsmallmatrix \right]$ in $(3.1)$ gives $$\begin{aligned}
\Theta_{\rm Jord}(G^2)=& \left[
\begin{array}
[c]{cc}\delta_4(b^2) & \tau_4(b^2)\\
\nu_4(b^2) & \mu_4(b^2)\\
\end{array}
\right]
\end{aligned}
\eqno(3.6)$$ and $$\begin{aligned}
&G\Theta_{\rm Jord}(G)+\Theta_{\rm Jord}(G)G= \left[
\begin{array}
[c]{cc}0 & \tau_4(b)b\\
b\nu_4(b) & b\mu_4(b)+\mu_4(b)b\\
\end{array}
\right].
\end{aligned}
\eqno(3.7)$$ Combining $(3.6)$ with $(3.7)$ yields that $\mu_4$ is a Jordan derivation of $B$, $$\tau_4(b^2)=\tau_4(b)b, \quad\quad\nu_4(b^2)=b\nu_4(b)\eqno(3.8)$$ and $$\delta_4(b^2)=0.\eqno(3.9)$$ for all $b\in B$.
Let us choose $G=\left[\smallmatrix 0 & m\\
0 & 0 \endsmallmatrix \right]$ in $(3.1)$. Then $$\begin{aligned}
\Theta_{\rm Jord}(G^2)=& \left[
\begin{array}
[c]{cc}0 & 0\\
0 & 0\\
\end{array}
\right]
\end{aligned}
\eqno(3.10)$$ and $$\begin{aligned}
&G\Theta_{\rm Jord}(G)+\Theta_{\rm Jord}(G)G= \left[
\begin{array}
[c]{cc}m\nu_2(m) & m\mu_2(m)+\delta_2(m)m\\
0 & \nu_2(m)m\\
\end{array}
\right].
\end{aligned}
\eqno(3.11)$$ The relations $(3.10)$ and $(3.11)$ jointly imply that $$m\nu_2(m)=0,
\quad\quad\nu_2(m)m=0 \eqno(3.12)$$ and $$\delta_2(m)m+m\mu_2(m)=0 \eqno(3.13)$$ for all $m\in M$. Likewise, if we choose $G=\left[\smallmatrix 0 & 0\\
n & 0 \endsmallmatrix \right]$, then $$\begin{aligned}
\Theta_{\rm Jord}(G^2)=& \left[
\begin{array}
[c]{cc}0 & 0\\
0 & 0\\
\end{array}
\right]
\end{aligned}
\eqno(3.14)$$ and $$\begin{aligned}
&G\Theta_{\rm Jord}(G)+\Theta_{\rm Jord}(G)G= \left[
\begin{array}
[c]{cc}\tau_3(n)n & 0\\
n\delta_3(n)+\mu_3(n)n & n\tau_3(n)\\
\end{array}
\right].
\end{aligned}
\eqno(3.15)$$ It follows from $(3.14)$ and $(3.15)$ that $$n\tau_3(n)=0, \quad\quad \tau_3(n)n=0\eqno(3.16)$$ and $$\mu_3(n)n+n\delta_3(n)=0 \eqno(3.17)$$ for all $n\in N$. Let us consider $G=\left[\smallmatrix 1 & m\\
0 & 0 \endsmallmatrix \right]$ in $(3.1)$ and set $\tau_1(1)=m_0$ and $\nu_1(1)=n_0$. Since $\delta_1$ is a Jordan derivation of $A$, $\delta_1(1)=0$. Moreover, $(3.5)$ implies that $\mu_1(1)=0$. Therefore $$\begin{aligned}
\Theta_{\rm Jord}(G^2)=& \left[
\begin{array}
[c]{cc}\delta_2(m) & m_0+\tau_2(m) \\
n_0+\nu_2(m) & \mu_2(m) \\
\end{array}
\right].
\end{aligned}
\eqno(3.18)$$ On the other hand, from $(3.12)$ and $(3.13)$ we have that $$\begin{aligned}
&G\Theta_{\rm Jord}(G)+\Theta_{\rm Jord}(G)G= \left[
\begin{array}
[c]{cc}2\delta_2(m)+mn_0 & m_0+\tau_2(m)\\
n_0+\nu_2(m) & n_0m\\
\end{array}
\right].
\end{aligned}
\eqno(3.19)$$ By $(3.18)$ and $(3.19)$ we arrive at $$\delta_2(m)=-mn_0 \quad {\rm and} \quad\mu_2(m)=n_0m \eqno(3.20)$$ for all $m\in M$. Let us take $G=\left[\smallmatrix 1 & 0\\
n & 0 \endsmallmatrix \right]$ in $(3.1)$. Applying $(3.16)$ and $(3.17)$ leads to $$\mu_3(n)=nm_0\quad{\rm and}\quad\delta_3(n)=-m_0n \eqno(3.21)$$ for all $n\in N$. Furthermore, if we choose $G=\left[\smallmatrix 1 & 0\\
0 & b \endsmallmatrix \right]$ in $(3.1)$, then it follows from $(3.8)$ and $(3.9)$ that $2\delta_4(b)=0$, $$\nu_4(b)=-bn_0\quad {\rm and} \quad\tau_4(b)=-m_0b \eqno(3.22)$$ for all $b\in B$. Taking $G=\left[\smallmatrix a & 0\\
0 & 1 \endsmallmatrix \right]$ into $(3.1)$ and using $(3.4)$ and $(3.5)$ we obtain $2\mu_1(a)=0$, $$\tau_1(a)=am_0\quad{\rm and}\quad\nu_1(a)=n_0a \eqno(3.23)$$ for all $a\in A$. Let us put $G=\left[\smallmatrix a & m\\
0 & 0 \endsmallmatrix \right]$ in $(3.1)$. Then the relations $(3.5)$, $(3.19)$ and $(3.23)$ imply that $$\begin{aligned}
\Theta_{\rm Jord}(G^2)=& \left[
\begin{array}
[c]{cc}\delta_1(a^2)+\delta_2(am) & a^2m_0+\tau_2(am) \\
n_0a^2+\nu_2(am) & n_0am \\
\end{array}
\right].
\end{aligned}
\eqno(3.24)$$ On the other hand, by the relations $(3.4)$, $(3.12)$, $(3.13)$, $(3.20)$ and $(3.23)$ we get $$\begin{aligned}
& G\Theta_{\rm Jord}(G)+\Theta_{\rm Jord}(G)G\\ =& \left[
\begin{array}
[c]{cc}a\delta_1(a)+\delta_1(a)a+amn_0 & a^2m_0+a\tau_2(m)+\delta_1(a)m+m\mu_1(a)\\
n_0a^2+\nu_2(m)a & n_0am\\
\end{array}
\right].
\end{aligned}
\eqno(3.25)$$ Combining $(3.24)$ with $(3.25)$ yields $\nu_2(am)=\nu_2(m)a$ and $$\tau_2(am)=a\tau_2(m)+\delta_1(a)m+m\mu_1(a)$$ for all $a\in A, m\in M$. Similarly, taking $G=\left[\smallmatrix a & 0\\
n & 0 \endsmallmatrix \right]$ into $(3.1)$ gives $\tau_3(na)=a\tau_3(n)$ and $$\nu_3(na)=\nu_3(n)a+n\delta_1(a)+\mu_1(a)n$$ for all $n\in N, a\in A$. Let us choose $G=\left[\smallmatrix 0 & m\\
0 & b \endsmallmatrix \right]$ in $(3.1)$. We will get $\nu_2(mb)=b\nu_2(m)$ and $$\tau_2(mb)=\tau_2(m)b+m\mu_4(b)+\delta_4(b)m$$ for all $m\in M, b\in B$. Putting $G=\left[\smallmatrix 0 & 0\\
n & b \endsmallmatrix \right]$ in $(3.1)$ and employing the same computational approach we conclude that $\tau_3(bn)=\tau_3(n)b$ and $\nu_3(bn)=b\nu_3(n)+\mu_4(b)n+n\delta_4(b)$ for all $b\in B, n\in
N$. Finally, let us set $G=\left[\smallmatrix 0 & m\\
n & 0 \endsmallmatrix \right]$ in $(3.1)$. We have that $\delta_1(mn)=-\delta_4(nm)+\tau_2(m)n+m\nu_3(n)$ and $\mu_4(nm)=-\mu_1(mn)+n\tau_2(m)+\nu_3(n)m$ for all $m\in M, n\in
N$.
If $\Theta_{\rm Jord}$ has the form $(\bigstar2)$ and satisfies conditions $(1)-(7)$, the assertion that $\Theta_{\rm Jord}$ is a Jordan derivation of $\mathcal{G}$ will follow from direct computations. We complete the proof of this proposition.
From now on, we always assume in this section that $M$ is faithful as a left $A$-module and also as a right $B$-module, but no any constraint conditions concerning the bimodule $N$. Then we have the following:
\[xxsec3.3\] Let $\mathcal{G}$ be a $2$-torsion free generalized matrix algebra over the commutative ring $\mathcal{R}$. An additive map $\Theta_{\rm Jord}$ form $\mathcal{G}$ into itself is a Jordan derivation of $\mathcal{G}$ if and only if it has the form $$\begin{aligned}
& \Theta_{\rm Jord}\left(\left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\right)\\ =& \left[
\begin{array}
[c]{cc}\delta_1(a)-mn_0-m_0n & am_0-m_0b+\tau_2(m)+\tau_3(n)\\
n_0a-bn_0+\nu_2(m)+\nu_3(n) & n_0m+nm_0+\mu_4(b)\\
\end{array}
\right] , (\bigstar3)\\
& \forall \left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\in \mathcal{G},
\end{aligned}$$ where $m_0\in M, n_0\in N$ and $$\begin{aligned} \delta_1:& A \longrightarrow A, & \tau_2:
& M\longrightarrow M, & \tau_3: & N\longrightarrow M,\\
\nu_2: & M\longrightarrow N, & \nu_3: & N\longrightarrow N & \mu_4:
& B\longrightarrow B
\end{aligned}$$ are all $\mathcal {R}$-linear maps satisfying conditions
1. $\delta_1$ is a derivation on $A$ and $\delta_1(mn)=\tau_2(m)n+m\nu_3(n);$
2. $\mu_4$ is a derivation on $B$ and $\mu_4(nm)=n\tau_2(m)+\nu_3(n)m;$
3. $\tau_2(am)=a\tau_2(m)+\delta_1(a)m$ and $\tau_2(mb)=\tau_2(m)b+m\mu_4(b);$
4. $\nu_3(na)=\nu_3(n)a+n\delta_1(a)$ and $\nu_3(bn)=b\nu_3(n)+\mu_4(b)n;$
5. $\tau_3(na)=a\tau_3(n)$, $\tau_3(bn)=\tau_3(n)b$, $n\tau_3(n)=0$, $\tau_3(n)n=0;$
6. $\nu_2(am)=\nu_2(m)a$, $\nu_2(mb)=b\nu_2(m)$, $m\nu_2(m)=0$, $\nu_2(m)m=0.$
Let $\Theta_{\rm Jord}$ be a Jordan derivation of $\mathcal{G}$. Then $\Theta_{\rm Jord}$ has the form of $(\bigstar2)$ and satisfies all additional conditions $(1)-(7)$ of Proposition \[xxsec3.2\]. Since $\mathcal{G}$ is a $2$-torsion free generalized matrix algebra, $\delta_4=0$ and $\mu_1=0$ by condition $(3)$ of Proposition \[xxsec3.2\]. Condition $(3)$ of Proposition \[xxsec3.2\] vanishes in the present case. Condition $(4)$ of Proposition \[xxsec3.2\] correspondingly becomes $$\tau_2(am)=a\tau_2(m)+\delta_1(a)m$$ and $$\tau_2(mb)=\tau_2(m)b+m\mu_4(b).$$ Clearly, we only need to prove that $\delta_1$ is a derivation of $A$ and that $\mu_4$ is a derivation of $B$. Then for arbitrary elements $a_1, a_2\in A$, we have $$\tau_2(a_1a_2m)=a_1a_2\tau_2(m)+\delta_1(a_1a_2)m \eqno(3.26)$$ and $$\begin{aligned}
\tau_2(a_1a_2m) & =a_1\tau_2(a_2m)+\delta_1(a_1)a_2m\\
& =a_1a_2\tau_2(m)+a_1\delta_1(a_2)m+\delta_1(a_1)a_2m.
\end{aligned}\eqno(3.27)$$ Combining $(3.26)$ and $(3.27)$ gives $$\delta_1(a_1a_2)m=a_1\delta_1(a_2)m+\delta_1(a_1)a_2m.\eqno(3.28)$$ Note that $M$ is faithful as left $A$-module. Relation $(3.28)$ implies that $$\delta_1(a_1a_2)=a_1\delta_1(a_2)+\delta_1(a_1)a_2$$ for all $a_1, a_2\in A$. So $\delta_1$ is a derivation of $A$. Similarly, we can show that $\mu_4$ is a derivation of $B$.
Conversely, if an additive map $\Theta_{\rm Jord}$ of $\mathcal{G}$ is of the form $(\bigstar3)$ and satisfies all additional conditions $(1)-(6)$, then the fact that is a Jordan derivation of $\mathcal{G}$ will follow from direct computations.
In view of Herstein’s result and recent intensive works [@Bresar1; @Bresar2; @Bresar4; @Johnson; @MaJi; @Zhang; @XiaoWei1; @ZhangYu], the following question is at hand.
\[xxsec3.4\] Is each Jordan derivation on a generalized matrix algebra $\mathcal{G}$ a derivation, or equivalently, do there exist proper Jordan derivations on generalized matrix algebras?
The following counterexample provides an explicit answer to the above question. It is shown that Jordan derivations of generalized matrix algebras need not be derivations. Equivalently, there indeed exist proper Jordan derivations on certain generalized matrix algebras.
\[xxsec3.5\] Let $\mathcal{G}=\left[\smallmatrix A & M\\
N & B \endsmallmatrix \right]$ be a generalized matrix algebra of order $2$ over the commutative ring $\mathcal{R}$. For arbitrary $X=\left[\smallmatrix a_1 & m_1\\
n_1 & b_1 \endsmallmatrix \right]\in \mathcal{G}, Y=\left[\smallmatrix a_2 & m_2\\
n_2 & b_2 \endsmallmatrix \right]\in \mathcal{G}$, we define the sum $X+Y$ as usual. The multiplication $XY$ is given by the rule $$XY=\left[
\begin{array}
[c]{cc}a_1a_2 & a_1m_2+m_1b_2\\
n_1a_2+b_1n_2 & b_1b_2\\
\end{array}
\right].\eqno(\spadesuit)$$ Such kind of generalized matrix algebras are called *trivial generalized matrix algebras*. That is, the bilinear pairings $\Phi_{MN}=\Psi_{NM}=0$ are both zero. Let us establish an $\mathcal{R}$-linear map $$\begin{aligned}
\Gamma_{\rm Jord}: \mathcal{G} & \longrightarrow
\mathcal{G}\\
\left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right] & \longrightarrow \left[
\begin{array}
[c]{cc}0 & m+n\\
m-n & 0\\
\end{array}
\right], \hspace{3pt} \forall \left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\in \mathcal{G}.
\end{aligned}$$ By straightforward computations, we know that $\Gamma_{\rm Jord}$ is a Jordan derivation of $\mathcal{G}$, but not a derivation.
On the other hand, we can also define two $\mathcal{R}$-linear maps $$\begin{aligned}
\Theta_1: \mathcal{G} & \longrightarrow
\mathcal{G}\\
\left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right] & \longrightarrow \left[
\begin{array}
[c]{cc}0 & m\\
-n & 0\\
\end{array}
\right], \hspace{3pt} \forall \left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\in \mathcal{G}
\end{aligned}$$ and $$\begin{aligned}
\Theta_2: \mathcal{G} & \longrightarrow
\mathcal{G}\\
\left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right] & \longrightarrow \left[
\begin{array}
[c]{cc}0 & n\\
m & 0\\
\end{array}
\right], \hspace{3pt} \forall \left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\in \mathcal{G}.
\end{aligned}$$ It is easy to see that $\Theta_1$ is a derivation of $\mathcal{G}$ and $\Theta_2$ is an anti-derivation of $\mathcal{G}$. Therefore $\Gamma_{\rm Jord}$ is the sum of the derivation $\Theta_1$ and the anti-derivation $\Theta_2$.
As a matter of fact, there exist some generalized matrix algebras whose multiplication satisfies the rule $(\spadesuit)$. Let $\mathcal{R}^\prime$ be an associative ring with identity and $\mathcal{Z(R^\prime)}$ be its center. Let us consider the usual $2\times 2$ matrix ring $
\left[\smallmatrix \mathcal{R}^\prime & \mathcal{R}^\prime\\
\mathcal{R}^\prime & \mathcal{R}^\prime \endsmallmatrix \right]$. It will become a generalized matrix algebra under the usual addition and the following multiplication rule $$\left[
\begin{array}
[c]{cc}a & c\\
d & b\\
\end{array}
\right]\left[
\begin{array}
[c]{cc}e & g\\
h & f\\
\end{array}
\right]=\left[
\begin{array}
[c]{cc}ae+sch & ag+cf\\
de+bh & sdg+bf\\
\end{array}
\right],$$ where $s\in \mathcal{Z(R^\prime)}$. A trivial generalized matrix algebra arises in the case of $s=0$. The usual $2\times 2$ matrix ring is produced when $s=1$.
In view of Example \[xxsec3.5\] and our main motivation, we now begin to describe the forms of anti-derivations on the generalized matrix algebra $\mathcal{G}$. We will see below, Example \[xxsec3.5\] can be lifted and extracted to a more general conclusion.
\[xxsec3.6\] An additive map $\Theta_{\rm antid}$ from $\mathcal{G}$ into itself is an antiderivation if and only if it has the form $$\begin{aligned}
& \Theta_{\rm antid}\left(\left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\right) \\=& \left[
\begin{array}
[c]{cc}0 & am_0-m_0b+\tau_3(n)\\
n_0a-bn_0+\nu_2(m) & 0\\
\end{array}
\right] , (\bigstar4) \\ & \forall \left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\in \mathcal{G},
\end{aligned}$$ where $m_0\in M, n_0\in N$ and $$\begin{aligned} \tau_3: & N\longrightarrow M, &
\nu_2: & M\longrightarrow N
\end{aligned}$$ are $\mathcal{R}$-linear maps satisfying the following conditions:
1. $[a, a']m_0=0$, $m_0[b, b']=0$, $n_0[a, a']=0$, $[b,
b']n_0=0$ for all $a'\in A, b'\in B;$
2. $m_0n=0$, $nm_0=0$, $mn_0=0$, $n_0m=0;$
3. $\tau_3(na)=a\tau_3(n)$, $\tau_3(bn)=\tau_3(n)b$, $n\tau_3(n')=0$, $\tau_3(n)n'=0$ for all $n'\in N;$
4. $\nu_2(am)=\nu_2(m)a$, $\nu_2(mb)=b\nu_2(m)$, $m\nu_2(m')=0$, $\nu_2(m)m'=0$ for all $m'\in M$.
Suppose that the Jordan derivation $\Theta_{\rm antid}$ is of the form $$\begin{aligned}
& \Theta_{\rm antid}\left(\left[
\begin{array}
[c]{cc}a & m \\
n & b \\
\end{array}
\right]\right) \\
= &\left[
\begin{array}
[c]{cc}\delta_1(a)+\delta_2(m)+\delta_3(n)+\delta_4(b) & \tau_1(a)+\tau_2(m)+\tau_3(n)+\tau_4(b) \\
\nu_1(a)+\nu_2(m)+\nu_3(n)+\nu_4(b) & \mu_1(a)+\mu_2(m)+\mu_3(n)+\mu_4(b) \\
\end{array}
\right] ,
\end{aligned}$$ for all $\left[\smallmatrix a & m\\
n & b
\endsmallmatrix \right]\in \mathcal{G}$, where $\delta_1,\delta_2,\delta_3,\delta_4$ are $\mathcal{R}$-linear maps from $A, M, N, B$ to $A$, respectively; $\tau_1,\tau_2$, $\tau_3,\tau_4$ are $\mathcal{R}$-linear maps from $A, M, N, B$ to $M$, respectively; $\nu_1,\nu_2,\nu_3,\nu_4$ are $\mathcal{R}$-linear maps from $A, M, N, B$ to $N$, respectively; $\mu_1,\mu_2,\mu_3,\mu_4$ are $\mathcal{R}$-linear maps from $A, M,
N, B$ to $B$, respectively.
For any $G_1, G_2\in \mathcal{G}$, we will intensively employ the antiderivation equation $$\Theta_{\rm antid}(G_1G_2)=\Theta_{\rm antid}(G_2)G_1+G_2\Theta_{\rm
antid}(G_1). \eqno(3.29)$$
Taking $G_1=\left[\smallmatrix a & 0\\
0 & 0 \endsmallmatrix \right]$ and $G_2=\left[\smallmatrix a' & 0\\
0 & 0 \endsmallmatrix \right]$ into $(3.29)$ yields $$\begin{aligned}
\Theta_{\rm antid}(G_1G_2)=& \left[
\begin{array}
[c]{cc}\delta_1(aa') & \tau_1(aa')\\
\nu_1(aa') & \mu_1(aa')\\
\end{array}
\right]
\end{aligned}
\eqno(3.30)$$ and $$\begin{aligned}
&\Theta_{\rm antid}(G_2)G_1+G_2\Theta_{\rm antid}(G_1)= \left[
\begin{array}
[c]{cc}\delta_1(a')a+a'\delta_1(a) & a'\tau_1(a)\\
\nu_1(a')a & 0\\
\end{array}
\right].
\end{aligned}
\eqno(3.31)$$ It follows from $(3.30)$ with $(3.31)$ that $\delta_1$ is an antiderivation of $A$, $\mu_1=0$ and $$\nu_1(aa')=\nu_1(a')a \eqno(3.32)$$ for all $a, a'\in A$. Let us set $a'=1$ in $(3.32)$ and denote $\nu_1(1)$ by $n_0$. Then $\nu_1(a)=n_0a.$ Furthermore, $(3.32)$ implies that $n_0aa'=n_0a'a$ for all $a,a'\in A$, that is, $n_0[a,
a']=0$ for all $a,a'\in A$. If we denote $\tau_1(1)$ by $m_0$, then we obtain $\tau_1(a)=am_0$ and $[a, a']m_0=0$ for all $a,a'\in A$.
Let us choose $G_1=\left[\smallmatrix 0 & 0\\
0 & b \endsmallmatrix \right]$ and $G_2=\left[\smallmatrix 0 & 0\\
0 & b' \endsmallmatrix \right]$ in $(3.29)$. By the same computational approach we conclude that $\mu_4$ is an antiderivation of $B$, $\delta_4=0$ and $$\tau_4(b)=\tau_4(1)b, \quad \nu_4(b)=b\nu_4(1), \quad \tau_4(1)[b,
b']=0, \quad [b, b']\nu_4(1)=0 \eqno(3.33)$$ for all $b,b'\in B$. We claim that $\tau_4(1)=-m_0$. In fact, this can be obtained by taking $G_1=\left[\smallmatrix 0 & 0\\
0 & 1 \endsmallmatrix \right]$ and $G_2=\left[\smallmatrix 1 & 0\\
0 & 0 \endsmallmatrix \right]$ in $(3.29)$. Likewise, we assert that $\nu_4(1)=-n_0$. Thus the relation $(3.33)$ becomes $$\tau_4(b)=-m_0b, \quad \nu_4(b)=-bn_0, \quad m_0[b, b']=0, \quad [b,
b']n_0=0$$ for all $b,b'\in B$.
Putting $G_1=\left[\smallmatrix 1 & 0\\
0 & 0 \endsmallmatrix \right]$ and $G_2=\left[\smallmatrix 0 & m\\
0 & 0 \endsmallmatrix \right]$ in $(3.29)$ and using the fact $\mu_1=0$ gives $$\begin{aligned}
\Theta_{\rm antid}(G_1G_2)=& \left[
\begin{array}
[c]{cc}\delta_2(m) & \tau_2(m)\\
\nu_2(m) & \mu_2(m)\\
\end{array}
\right]
\end{aligned}
\eqno(3.34)$$ and $$\begin{aligned}
&\Theta_{\rm antid}(G_2)G_1+G_2\Theta_{\rm antid}(G_1)= \left[
\begin{array}
[c]{cc}\delta_2(m)+mn_0 & 0\\
\nu_2(m) & 0\\
\end{array}
\right].
\end{aligned}
\eqno(3.35)$$ Combining $(3.34)$ with $(3.35)$ leads to $$mn_0=0,\quad \tau_2=0,\quad \mu_2=0$$ for all $m\in M$. Interchanging $G_1$ and $G_2$ we will get $$\delta_2=0,\quad n_0m=0$$ for all $m\in M$.
If we take $G_1=\left[\smallmatrix 1 & 0\\
0 & 0 \endsmallmatrix \right]$ and $G_2=\left[\smallmatrix 0 & 0\\
n & 0 \endsmallmatrix \right]$ into $(3.29)$, then $$\begin{aligned}
\Theta_{\rm antid}(G_1G_2)=& \left[
\begin{array}
[c]{cc}0 & 0\\
0 & 0\\
\end{array}
\right]
\end{aligned}
\eqno(3.36)$$ and $$\begin{aligned}
&\Theta_{\rm antid}(G_2)G_1+G_2\Theta_{\rm antid}(G_1)= \left[
\begin{array}
[c]{cc}\delta_3(n) & 0\\
\nu_3(n) & 0\\
\end{array}
\right].
\end{aligned}
\eqno(3.37)$$ will follow from the fact $\delta_1(1)=0$. By $(3.36)$ and $(3.37)$ we obtain that $$\delta_3=0,\quad \nu_3=0.\eqno(3.38)$$ Interchanging $G_1$ and $G_2$ again yields $$\mu_3=0, \quad m_0n=0 \eqno(3.39)$$ for all $n\in N$. In order to get $nm_0=0$, we only need to put $G_1=\left[\smallmatrix 0 & 0\\
0 & 1 \endsmallmatrix \right]$ and $G_2=\left[\smallmatrix 0 & 0\\
n & 0 \endsmallmatrix \right]$ in $(3.29)$.
Taking $G_1=\left[\smallmatrix 0 & 0\\
n & 0 \endsmallmatrix \right]$ and $G_2=\left[\smallmatrix a & 0\\
0 & 0 \endsmallmatrix \right]$ into $(3.29)$ and applying $(3.38)$ and $(3.39)$ we arrive at $$\begin{aligned}
\Theta_{\rm antid}(G_1G_2)=& \left[
\begin{array}
[c]{cc}0 & \tau_3(na)\\
0 & 0\\
\end{array}
\right].
\end{aligned}
\eqno(3.40)$$ The fact $\mu_1=0$ and $(3.39)$ imply that $$\begin{aligned}
&\Theta_{\rm antid}(G_2)G_1+G_2\Theta_{\rm antid}(G_1)= \left[
\begin{array}
[c]{cc}0 & a\tau_3(n)\\
0 & 0\\
\end{array}
\right].
\end{aligned}
\eqno(3.41)$$ The relations $(3.40)$ and $(3.41)$ jointly show that $\tau_3(na)=a\tau_3(n)$ for all $a\in A, n\in N$. Likewise, if we choose $G_1=\left[\smallmatrix 0 & 0\\
0 & b \endsmallmatrix \right]$ and $G_2=\left[\smallmatrix 0 & 0\\
n & 0 \endsmallmatrix \right]$ in $(3.29)$, then $\tau_3(bn)=\tau_3(n)b$ for all $b\in B, n\in N$. The equalities $\nu_2(am)=\nu_2(m)a$ and $\nu_2(mb)=b\nu_2(m)$ can be obtained by analogous discussions and the details are omitted here.
Let us consider $G_1=\left[\smallmatrix 0 & 0\\
n & 0 \endsmallmatrix \right]$ and $G_2=\left[\smallmatrix 0 & 0\\
n' & 0 \endsmallmatrix \right]$ in $(3.29)$. Then we get $n\tau_3(n')=0$ and $\tau_3(n)n'=0$ for all $n, n'\in N$. Putting $G_1=\left[\smallmatrix 0 & m\\
0 & 0 \endsmallmatrix \right]$ and $G_2=\left[\smallmatrix 0 & m'\\
0 & 0 \endsmallmatrix \right]$ in $(3.29)$ yields $m\nu_2(m')=0$ and $\nu_2(m)m'=0$ for all $m, m'\in M$.
Taking $G_1=\left[\smallmatrix 0 & m\\
0 & 0 \endsmallmatrix \right]$ and $G_2=\left[\smallmatrix a & m\\
0 & 0 \endsmallmatrix \right]$ into $(3.29)$. Then $\delta_1(a)m=0$ for all $a\in A, m\in
M$. Putting $G_1=\left[\smallmatrix 0 & m\\
0 & b \endsmallmatrix \right]$ and $G_2=\left[\smallmatrix 0 & m\\
0 & 0 \endsmallmatrix \right]$ in $(3.29)$. Then $m\mu_4(b)=0$ for all $b\in B, m\in M$. It follows from the faithfulness of $M$ that $\delta_1=0$ and $\mu_4=0$.
Conversely, suppose that $\Theta_{\rm antid}$ is of the form $(\bigstar4)$ and satisfies conditions $(1)-(4)$. Then the fact that $\Theta_{\rm antid}$ is a antiderivation of $\mathcal{G}$ will follow by direct computations.
Let us next observe the antiderivations of a class of generalized matrix algebras.
\[xxsec3.7\] Let $\mathcal{G}=
\left[\smallmatrix A & M\\
N & B \endsmallmatrix \right]$ be a generalized matrix algebra originating from the Morita context $(A, B, _AM_B,$ $ _BN_A,
\Phi_{MN}, \Psi_{NM})$. The bilinear form $\Phi_{MN}: M\underset
{B}{\otimes} N\longrightarrow A$ (resp. $\Psi_{NM}: N\underset
{A}{\otimes} M\longrightarrow B$) is called *nondegenerate* if for any $0\neq m\in M$ and $0\neq n\in N $, $\Phi_{MN}(m, N)\neq
0$ and $\Phi_{MN}(M, n)\neq 0$ (resp. $\Psi_{NM}(n, M)\neq 0$ and $\Psi_{NM}(N, m)\neq 0$).
\[xxsec3.8\] Let $H$ be a finite dimensional Hopf algebra over filed $\mathbb{K}$ and $A$ be an $H$-module algebra. Let $A^H$ be the invariant subalgebra of $H$ on $A$, and $A\# H$ be the smash product algebra of $A$ and $H$. We now consider the generalized matrix algebra $$\mathcal{G}_{\rm SPA}=\left[
\begin{array}
[c]{cc}A^H & M\\
N & A\#H\\
\end{array}
\right]$$ defined in Example \[xxsec2.1\], where $M=_{A^H}A_{A\#H}$ and $N=_{A\#H}A_{A^H}$. Suppose that $M$ is a faithful right (or left) $A\#H$-module. By [@CohenFischmanMontgomery Proposition 2.13] we know that the bilinear form $\Phi_{MN}$ will be nondegenerate. In this case, we easily check that there is indeed no nonzero antiderivations on $\mathcal{G}_{\rm SPA}$.
\[xxsec3.9\] Let $\mathbb{K}$ be a field and $A$ be an associative algebra over $\mathbb{K}$. Let $G$ be a group and $A*G$ be the skew group algebra over $\mathbb{K}$. Suppose that $A^G$ is the fixed ring of the action $G$ on $A$. We now revisit the generalized matrix algebra $$\mathcal{G}_{\rm GA}=\left[
\begin{array}
[c]{cc}A^G & M\\
N & A*G\\
\end{array}
\right]$$ in Example \[xxsec2.2\], where $M=_{A^G}A_{A*G}$ and $N=_{A*G}A_{A^G}$. For an arbitrary element $n\in N$, we define $$n^\bot=\left\{m\in M|\Psi_{NM}(n, m)=0\right\}.$$ Similarly, for an arbitrary element $m\in M$, we define $$m^\bot=\left\{n\in N|\Psi_{NM}(n, m)=0\right\}.$$ Then $n^\bot$ is a $G$-invariant right ideal of $A$ contained in $r_A(n)$, where $r_A(n)$ is the right annihilator of $n$ in $A$. Indeed, let $m\in n^\bot$ and $g\in G$, then $\Psi_{NM}(n,
m^g)=\Psi_{NM}(n, m\cdot g)=\Psi_{NM}(n, m)g=0$. Hence $n^\bot$ is $G$-invariant, the rest is obvious. Similarly, we can show that $m^\bot$ is a $G$-invariant left ideal of $A$ contained in $l_A(m)$, where $l_A(m)$ is the left annihilator of $m$ in $A$.
In particular, if $A$ is a semiprime $\mathbb{K}$-algebra, then $r_A(n)\neq A$ and $l_A(m)\neq A$. This shows that the bilinear form $\Psi_{NM}$ is nondegenerate. Furthermore, if we assume that the module $N$ is faithful as a left $A*G$-module, then the bilinear form $\Phi_{MN}$ will be also nondegenerate. Indeed, let $\Phi_{MN}(m, N)=0$ for some $m\in M$. Then, $0=N\cdot \Phi_{MN}(m,
N)=\Psi_{NM}(N, m)\cdot N$. By faithfulness and nondegeneracy of $\Psi_{NM}$ we deduce that $m=0$. If one of the bilinear pairings $\Phi_{MN}$ and $\Psi_{NM}$ is nondegenerate, then there is no nonzero antiderivations on $\mathcal{G}_{\rm GA}$, which is similar to Example \[xxsec3.8\].
In order to ensure the semiprimeness of the $\mathbb{K}$-algebra $A$ and the nondegeneracy of the bilinear forms $\Phi_{MN}$ and $\Psi_{NM}$, $A$ may be one of the following algebras:
1. the quantized enveloping algebra $U_q(
\mathfrak{sl}_2(\mathbb{K}))$ over the field $\mathbb{K}$,
2. the quantum $n\times n$ matrix algebra $\mathcal{O}_q(M_n(\mathbb{K}))$ over the field $\mathbb{K}$,
3. the quantum affine $n$-space $\mathcal{O}_q(\mathbb{K}^n)$ over the field $\mathbb{K}$,
4. the double affine Hecke algebra $\widetilde{H}$ over the field $\mathbb{K}$.
5. the Iwasawa algebra $\Omega_G$ over the finite field $\mathbb{F}_p$.
In view of Proposition \[xxsec3.6\], Example \[xxsec3.8\] and Example \[xxsec3.9\] we immediately have
\[xxsec3.10\] Let $\mathcal {G}$ be a generalized matrix algebra over the commutative ring $\mathcal{R}$ and $\Theta_{\rm antid}$ be an $\mathcal {R}$-linear map from $\mathcal{G}$ into itself. If one of the bilinear forms $\Phi_{MN}: M\underset {B}{\otimes}
N\longrightarrow A$ and $\Psi_{NM}: N\underset {A}{\otimes}
M\longrightarrow B$ is nondegenerate, then $\Theta_{\rm antid}$ is an antiderivation of $\mathcal {G}$ if and only if $\Theta_{\rm
antid}=0$.
We will end this section by investigating properties of Jordan derivations of generalized matrix algebras with zero bilinear pairings. Such kind of generalized matrix algebras draw our attention, which is due to Haghany’s work and Example \[xxsec3.5\]. Haghany in [@Haghany] studied hopficity and co-hopficity for generalized matrix algebras with zero bilinear parings. As you see in Example \[xxsec3.5\], those generalized matrix algebras exactly have zero bilinear pairings.
\[xxsec3.11\] Let $\mathcal{G}$ be a $2$-torsion free generalized matrix algebra over the commutative ring $\mathcal{R}$. If the bilinear pairings $\Phi_{MN}$ and $\Psi_{NM}$ are both zero, then every Jordan derivation of $\mathcal{G}$ can be expressed as the sum of a derivation and an antiderivation.
Let $\Theta_{\rm Jord}$ be a Jordan derivation of $\mathcal{G}$. By Corollary \[xxsec3.3\] we know that $\Theta_{\rm Jord}$ is of the form $$\begin{aligned}
& \Theta_{\rm Jord}\left(\left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\right)\\ =& \left[
\begin{array}
[c]{cc}\delta_1(a)-mn_0-m_0n & am_0-m_0b+\tau_2(m)+\tau_3(n)\\
n_0a-bn_0+\nu_2(m)+\nu_3(n) & n_0m+nm_0+\mu_4(b)\\
\end{array}
\right] , (\bigstar3)\\
& \forall \left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\in \mathcal{G}.
\end{aligned}$$ It follows from Proposition \[xxsec3.1\] and Proposition \[xxsec3.6\] that there exist a derivation $\Theta_{\rm d}^\prime$ and an antiderivation $\Theta_{\rm antid}^\prime$ such that $$\begin{aligned}
\Theta_{\rm Jord}\left(\left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\right) & = \left[
\begin{array}
[c]{cc}\delta_1(a)-mn_0-m_0n & am_0-m_0b+\tau_2(m)\\
n_0a-bn_0+\nu_3(n) & n_0m+nm_0+\mu_4(b)\\
\end{array}
\right]\\
& +\left[
\begin{array}
[c]{cc}0 & \tau_3(n)\\
\nu_2(m) & 0\\
\end{array}
\right]\\
&=\Theta_{\rm d}^\prime\left( \left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\right)+\Theta_{\rm antid}^\prime\left(\left[
\begin{array}
[c]{cc}a & m\\
n & b\\
\end{array}
\right]\right)
\end{aligned}$$ for all $
\left[\smallmatrix a & m\\
n & b \endsmallmatrix \right]\in \mathcal{G}$. This shows that $\Theta_{\rm Jord}$ can be expressed the sum of a derivation $\Theta_{\rm d}^\prime$ and an antiderivation $\Theta_{\rm
antid}^\prime$, which is the desired result.
\[xxsec3.12\] The Jordan derivation $\Gamma_{\rm Jord}$ constructed in Example \[xxsec3.5\] can be expressed as the sum of a derivation $\Theta_1$ and an antiderivation $\Theta_2$.
\[xxsec3.13\] Let $\mathcal{R}$ be a $2$-torsion free commutative ring with identity and $T_n(\mathcal{R})(n\geq 2)$ be the upper (or lower) triangular matrix algebra over $\mathcal{R}$. Clearly, $T_n(\mathcal{R})(n\geq 2)$ is a generalized matrix algebra with zero pairings. In view of Theorem \[xxsec3.11\], every Jordan derivation on $T_n(\mathcal{R})(n\geq 2)$ can be written as the sum of a derivation and an antiderivation. By [@Benkovic Corollary 1.2] we assert that the part of antiderivation is zero. This leads to the fact that every Jordan derivation on $T_n(\mathcal{R})(n\geq 2)$ is a derivation [@ZhangYu].
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[^1]: This work is partially supported by the National Natural Science Foundation of China (Grant No. 10871023).
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---
abstract: 'We calculate the rates of formation and detection of ultracold Cs$_2$ molecules obtained from the photoassociation of ultracold atoms through the double-well ${\ensuremath{0_{\mathrm{g}}^-}}(6{\ensuremath{\mathrm{S}_{1/2}}}+ 6{\ensuremath{\mathrm{P}_{3/2}}})$ state. We concentrate on two features previously observed experimentally and attributed to tunneling between the two wells \[Vatasescu et al 2000 Phys. Rev. A **61** 044701\]. We show that the molecules obtained are in strongly bound levels ($v''''=5,6$) of the metastable ${\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}}(6{\ensuremath{\mathrm{S}_{1/2}}}+ 6{\ensuremath{\mathrm{S}_{1/2}}})$ ground electronic state.'
address:
- '$^1$ Institute of Space Sciences, MG-23, RO-77125, Magurele-Bucharest, Romania'
- '$^2$ Department of Physics, Ume[å]{} University, SE-901 87 Ume[å]{}, Sweden'
- '$^3$ Laboratoire Aimé Cotton, CNRS, Bâtiment 505, Univ. Paris-Sud, 91405 Orsay-Cedex, France'
author:
- 'Mihaela Vatasescu$^1$, Claude M Dion$^2$ and Olivier Dulieu$^3$'
title: Efficient formation of strongly bound ultracold cesium molecules by photoassociation with tunneling
---
Introduction
============
The control of elementary interactions between atoms or molecules in the gas phase is a long term concern for researchers, namely in order to find ways towards the full control of a chemical reaction, giving the ability to choose the reaction path unambiguously from a well-defined internal state towards a desired final state. In this context, laser cooling and trapping of atoms has opened an entirely new field of investigation, as they can be brought almost to rest, in a well-defined internal state. A spectacular example is provided by the observation of Bose-Einstein condensation in alkali gases [@anderson1995; @bradley1995; @davis1995], recently followed by the demonstration of quantum degeneracy in a fermionic alkali gas [@demarco1999].
Slowing and cooling of molecules, despite the inherent difficulty caused by the complex internal molecular structure, represents an increasingly active research field with many achievements since the first observation of ultracold Cs$_2$ molecules by photoassociation of laser-cooled cesium atoms [@fioretti1998]. Two approaches have been proven to be very efficient, namely Stark deceleration [@bethlem2000a] and buffer gas cooling [@weinstein1998a]. Other promising methods rely on phase space filtering of a molecular beam [@rangwala2003], on billiard-like collisions [@elioff2003], or on a gas expansion out of a rotating nozzle [@gupta1999]. A major breakthrough has been the observation of molecular Bose-Einstein condensates, using magnetic field tunability of Feshbach resonances [@jochim2003; @zwierlein2003]. Also, two groups recently measured for the first time rates for collisions between ultracold cesium atoms and cesium molecules [@staanum2006; @zahzam2006].
Among the various approaches, the photoassociation (PA) process is very attractive, as it starts from a pair of ultracold atoms which absorbs a photon to form an ultracold molecule in a short-lived excited electronic state [@thorsheim1987]. The main drawback is that the stabilization of the excited molecule into stable electronic states hardly provides an ensemble of molecules in a well-defined internal state, as it relies on spontaneous emission which populates many vibrational levels. Some attempts to use stimulated emission have been reported [@tsai1997; @tolra2001], but they were limited to the transition into high-lying levels. Indeed, due to the poor spatial overlap of the wave function of the photoassociated level — predominant at large interatomic distances — with the wave function of the lowest vibrational levels of the stabilized molecules, it is very difficult to create a significant proportion of ultracold molecules in their absolute ground state. A first indication of the possibility to form ultracold molecules in their absolute ground state has been provided in [@nikolov2000] for K$_2$ molecules using a two-step photoassociation process. Very recently, a multistep excitation/deexcitation scheme has been set up to produce a fair amount of RbCs ultracold molecules in the $v=0$ level of the electronic ground state [@sage2005].
The purpose of this paper is to describe a possible way to fill these objectives, in the particular case of the formation of ultracold Cs$_2$ molecules, using a single-step excitation scheme. It results from the cooperative action of photoassociation at large distances, tunneling and resonant spin-orbit coupling at short distances, and final stabilization by spontaneous decay towards a few low-lying vibrational levels of a stable molecular state. We first invoked such a process for the interpretation of so-called “giant lines” observed in photoassociation spectra in [@vatasescu2000], hereafter referred to as paper I. One of us also performed a time-dependent analysis of the tunneling process in such ultracold conditions, helpful for the determination of the characteristic times of the tunneling motion [@vatasescu2002]. From the present calculations involving the available extensive photoassociative spectroscopy of Cs$_2$, we demonstrate that the ultracold molecules are created with a very narrow distribution of vibrational levels, peaking at $v''=5,6$ of the theoretical potential curve of the Cs$_2$ metastable triplet state. This could be the first example of the formation of ultracold molecules left mainly in a single deep vibrational level via single-photon PA. The possibility to use this mechanism to create the initial state of cesium dimers in ultracold collisions with cesium atoms has been discussed in ref.[@staanum2006].
We present our calculation of the photoassociation rate, of the cold molecule formation rate, and of the molecular ion signal resulting from the photoionization of the stabilized cold molecules. This emphasizes the role of the detection process in the interpretation of the strong intensity of these giant lines observed in the photoassociation spectrum.
We start in section \[sec:facts\] by recalling the main facts concerning the tunneling effects in the photoassociation of Cs$_2$. We then present the model used to calculate the molecule formation rates (section \[sec:model\]), followed by the results of the simulations for the Cs$_2^+$ ionization signals (section \[sec:results\]). Finally, concluding remarks are given in section \[sec:conclu\].
Experimental and theoretical facts on giant lines and tunneling in the PA of cesium {#sec:facts}
===================================================================================
We briefly recall below the main results of paper I. The PA process between two ultracold cesium atoms in a magneto-optical trap (MOT), shown in figure \[fig:potS0g\], is written as $$\begin{aligned}
\mathrm{Cs}(6\mathrm{s}^{2}\mathrm{S}_{1/2}, F=4) +
\mathrm{Cs}(6\mathrm{s}^{2}\mathrm{S}_{1/2}, F=4) +
\hbar(\omega_{\mathrm{D2}} - {\ensuremath{\Delta_{\mathrm{L}}}}) \nonumber \\
\rightarrow
\mathrm{Cs}_{2}({\ensuremath{0_{\mathrm{g}}^-}}(6\mathrm{s}^{2}\mathrm{S}_{1/2} +
6\mathrm{p}^{2}\mathrm{P}_{3/2}; v,J)).\end{aligned}$$ The pair of cold cesium atoms absorbs a photon detuned by ${\ensuremath{\Delta_{\mathrm{L}}}}$ to the red of the D2 atomic transition frequency $\omega_{\mathrm{D2}}$, to populate rovibrational levels of the external well of the double-well-shaped ${\ensuremath{0_{\mathrm{g}}^-}}$ molecular state correlated to the $6{\ensuremath{\mathrm{S}_{1/2}}}+
6{\ensuremath{\mathrm{P}_{3/2}}}$ dissociation limit (hereafter labeled ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$). These levels are detected by two well-established methods: first through the fluorescence variations of the atomic trap induced by their spontaneous decay towards bound levels or towards the dissociation continuum of the lowest ${\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}}(6{\ensuremath{\mathrm{S}_{1/2}}}+6{\ensuremath{\mathrm{S}_{1/2}}})$ state, second through the resonant two-photon ionization (R2PI) of the ultracold molecules formed in this metastable triplet state into Cs$_2^+$ molecular ions. The latter method delivers a very neat spectrum with excellent signal-to-noise ratio. Typical experimental conditions are temperatures between 20 and $200\ \mu\mathrm{K}$, a mean atomic density of about $4 \times 10^{10}\ \mathrm{cm}^{-3}$ with a peak density of $10^{11}\ \mathrm{cm}^{-3}$, the number of atoms being estimated between 2 to $5\times 10^{7}$ atoms, and intensities of 50 up to 500 W/cm$^2$ for the the photoassociation laser [@drag2000]. The full R2PI spectrum for the ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ has been analyzed in [@comparat1999], yielding a very precise determination of the corresponding potential curve through the Rydberg-Klein-Rees (RKR) analysis, and through an approach based on the asymptotic modeling of the atom-atom interactions [@gutterres2002]. Vibrational levels from $v_{\mathrm{ext}}=0$ to $v_{\mathrm{ext}}=132$ have been identified [@fioretti1999] in the ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ outer well, with a rotational structure (up to $J=4$) clearly visible up to $v_{\mathrm{ext}}=72$.
In addition, two intense structures with a large rotational splitting (hereafter referred to as “giant lines”, following paper I, and labeled G$_1$ and G$_2$) are superimposed on lines associated with levels of the ${\ensuremath{0_{\mathrm{g}}^-}}$ external well (figure \[fig:potS0g\]). The most intense line within the G$_1$ (G$_2$) feature was assigned to $J=3$ ($J=0, 1$) at ${\ensuremath{\Delta_{\mathrm{L}}}}=-2.14$ cm$^{-1}$ (${\ensuremath{\Delta_{\mathrm{L}}}}=-6.15$ cm$^{-1}$) superimposed on $v_{\mathrm{ext}}=103$ ($v_{\mathrm{ext}}=80$), while weak lines assigned up to $J=6$ were detected providing a rotational constant $B_v^{\mathrm{G}_1} = 137\pm 4$ MHz ($B_v^{\mathrm{G}_2}=243\pm 8$ MHz).
![Scheme of the photoassociation (PA) process and subsequent spontaneous emission (SE) of the ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ excited molecule to the metastable ${\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}}(6{\ensuremath{\mathrm{S}_{1/2}}}+6{\ensuremath{\mathrm{S}_{1/2}}})$ state. The ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ state is coupled to the ${\ensuremath{0_{\mathrm{g}}^-}}(5{\ensuremath{\mathrm{D}_{3/2}}})$ via spin-orbit interaction. Vibrational levels involved in the tunneling through the ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ barrier, as well as the deep ${\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}}$ vibrational levels which can be reached by spontaneous decay are pictured with lines (not to scale). The recorded experimental spectrum is recalled in the two insets, for both the G$_1$ and G$_2$ features, showing their large rotational structure, and the neighboring levels $v_{\mathrm{ext}}$ of the ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ external well.[]{data-label="fig:potS0g"}](fig1_pot.eps){width="0.7\columnwidth"}
With the help of a coupled-channel model, the G$_1$ and G$_2$ features have been assigned in paper I to levels which tunnel through the potential barrier between the two wells of the ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ state, as schematized in figure \[fig:potS0g\]. Their large rotational structure is induced by the vibrational motion in two coupled vibrational levels of the ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ internal well, which is coupled to several levels in the external well (only two of them are drawn in figure \[fig:potS0g\] for better clarity). The tunneling effect, unusual for a heavy molecule like Cs$_2$, is then an efficient mechanism to transfer the vibrational motion of the photoassociated state from large interatomic distances towards the inner zone: in contrast with the long-range molecular states usually reached by PA, spontaneous emission of the tunneling levels can stabilize the photoassociated molecules into low vibrational levels of the ${\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}}$ metastable state, creating then ultracold molecules with a rather cold vibrational motion. Let us note that the internal state of the molecules created has not been probed experimentally yet.
The ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ outer well is represented by the asymptotic model of [@gutterres2002]. As there is no available spectroscopic determination of the ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ inner well, this external well is matched to the curve computed by Spies and Meyer [@spies1989], through a potential barrier whose height and position have been adjusted in paper I in order to reproduce the experimental observations. The potential barrier then culminates at 2 cm$^{-1}$ above the $6{\ensuremath{\mathrm{S}_{1/2}}}+6{\ensuremath{\mathrm{P}_{3/2}}}$ dissociation limit. Furthermore, according to [@spies1989], the ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ state is coupled in the inner well region to the next ${\ensuremath{0_{\mathrm{g}}^-}}(6{\ensuremath{\mathrm{S}_{1/2}}}+5{\ensuremath{\mathrm{D}_{3/2}}})$ state (hereafter labeled ${\ensuremath{0_{\mathrm{g}}^-}}(5{\ensuremath{\mathrm{D}_{3/2}}})$) through a non-adiabatic coupling generated by spin-orbit interaction. The ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ and ${\ensuremath{0_{\mathrm{g}}^-}}(5{\ensuremath{\mathrm{D}_{3/2}}})$ curves then exhibit a well-localized avoided crossing around $10 a_0$, which we transformed into a real crossing by linearizing the curves around the crossing to determine the corresponding standard Landau-Zener coupling parameters. We imposed a finite range for the coupling using the Gaussian form $A \exp\left[ -(r-r_0)^2/w^2 \right]$ with $r_0=10.02$, $w=2$, and $A=0.000258$ (all in atomic units). We recall that the absence of resonant coupling between levels of the internal wells would result to a single tunneling level, instead of the two levels assigned in the experiment [@vatasescu2000].
For each value of $J$, the radial Schrödinger equation is solved with the Mapped Fourier Grid Hamiltonian (MFGH) method [@kokoouline1999], providing accurate vibrational energies $E_{vJ}$, wave functions $\chi_{vJ}(R)$, and rotational constants $B_{vJ} = \langle \chi_{vJ} | \hbar^2 /(2 \mu R^2) | \chi_{vJ}\rangle$ (where $\mu=121135.83\ \mathrm{a.u.}$ is the Cs$_2$ reduced mass) of the two coupled state. It is well known that tunneling through a potential barrier is very sensitive to its shape and to the position in energy with respect to the top of the barrier. These effects are magnified here as the two tunneling levels of the ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ and ${\ensuremath{0_{\mathrm{g}}^-}}(5{\ensuremath{\mathrm{D}_{3/2}}})$ internal wells, hereafter referred to as $v_{6\mathrm{P}}$ and $v_{5\mathrm{D}}$, respectively, interact resonantly with the dense — but not continuous — energy level spectrum of the external well. Our calculations predict that the tunneling effect will be observable in the detection signal if the tunneling probability is maximum, which is achieved by adjusting the relative position of the two internal wells to create an almost half-and-half mixing of the $v_{6\mathrm{P}}$ and $v_{5\mathrm{D}}$ radial wave functions. Our model actually presents many tunable parameters for such an adjustment: indeed, the internal well of the potentials, as well as their coupling, are unknown experimentally, so that their fine tuning is quite tedious. We also noted that the tunneling effect is very sensitive to the value of $J$. In order to facilitate the convergence, we stopped the adjustment when we found maximal tunneling for an arbitrary $J$ value. In other words, we also considered the rotational number $J$ as a tunable parameter, which we will label as $\bar{J}$ in the following. An example of the resulting wave functions is shown in figure \[fig:wfG2J05\]), where the maximal tunneling effect is found for the value $\bar{J}=5$. This is clearly not in agreement with the rotational quantum number of the experiment, but we only need *radial* wave functions for the rate calculations of the next section, which will be evaluated without any $J$ dependence. The corresponding numerical data is available on request from the authors.
![Two-channel vibrational wave functions $\chi_{\bar{v},\bar{J}}(R)$ (panels b-e) associated with the radial motion in the ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ double well (in black) and in the ${\ensuremath{0_{\mathrm{g}}^-}}(5{\ensuremath{\mathrm{D}_{3/2}}})$ well (in red). The corresponding potentials are drawn with full black and dashed red lines, respectively, to make more visible the extension of the wave functions within the corresponding potentials. The $\bar{J}=5$ value (panels b and c) allows maximal tunneling for the G$_2$ resonance at $-5.87$ cm$^{-1}$, almost degenerate with a level of the external well at $-5.89$ cm$^{-1}$: the wave functions have significant amplitude over the barrier range. The $\bar{J}=0$ case (panels d and e) is also shown for comparison: the energy matching is less favorable and the wave functions hardly penetrate the potential barrier. Panel (a) displays a typical radial wave function of the initial collisional state for a temperature of $150\
\mu\mathrm{K}$, in the ${\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}}(6{\ensuremath{\mathrm{S}_{1/2}}}+6{\ensuremath{\mathrm{S}_{1/2}}})$ state, designed here with a scattering length of about $2370 a_0$.[]{data-label="fig:wfG2J05"}](fig2_wfG2J05.eps){width="0.7\columnwidth"}
Even if in the adjustment procedure above we used $\bar{J}$ as a parameter, figure \[bvJ.eps\] shows that the selectivity of the tunneling process is very strongly dependent on the rotational level. The value of $B_{vJ}$ defined above is very sensitive to the repartition of the wave function inside the two wells, and panel (a) in figure \[bvJ.eps\] reflects the pattern shown in figure \[fig:wfG2J05\], where the value for $\bar{J}=5$ is intermediate between the rotational constant of levels in the internal well ($\approx 0.008$ cm$^{-1}$) and of levels of the external well ($\approx0.0005$ cm$^{-1}$). Similarly, the $\bar{J}=3$ line for G$_1$ also reflects the extension of the wave function over both wells. The $\bar{J}=5$ line displays the same behavior, which can be understood since the rotational structure of G$_1$ has the same magnitude as the energy spacing between vibrational levels of the external well, so that it is resonant with $v_{\mathrm{ext}}=104$ in our calculations. This aspect was not present in the model we set up in paper I and confirms what is observed experimentally. Indeed, due to the ultracold temperature, the s-wave regime is expected to dominate the initial collision between the two Cs atoms (yielding $J=0, 1, 2$) with probably a small contribution of the p wave (yielding $J=1, 2,
3$). The presence of higher values of $J$ at such low temperatures is not yet fully understood [@fioretti1999]. The relative intensities of the $J=0,1,2$ experimental lines approximately reflects their $2J+1$ degeneracy, and the $J=2$ line has indeed the strongest intensity for most of the PA lines assigned to the levels of the ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ external well. The lines with larger $J$ values are weaker, due to the small contribution of higher partial waves involved in the collision. We see that this hierarchy is not preserved for G$_1$ and G$_2$: the intense lines are associated with those rotational levels which are indeed resonant with a level of the internal well, i.e., $J=3$ and $J=0,1$, respectively.
![Rotational constants $B_{vJ}=\langle \chi_{\bar{v},\bar{J}}
|\hbar^2(2 \mu R^2)| \chi_{\bar{v},\bar{J}}\rangle$ for rovibrational levels of the coupled ${\ensuremath{0_{\mathrm{g}}^-}}(6{\ensuremath{\mathrm{S}_{1/2}}}+6{\ensuremath{\mathrm{P}_{3/2}}})$ and ${\ensuremath{0_{\mathrm{g}}^-}}(6{\ensuremath{\mathrm{S}_{1/2}}}+5{\ensuremath{\mathrm{D}_{3/2}}})$ potentials, as function of their binding energy. (a) G$_2$ structure; (b) G$_1$ structure. The levels with an important tunneling effect are marked by arrows.[]{data-label="bvJ.eps"}](fig3_bvJ.eps){width="0.7\columnwidth"}
Photoassociation and cold molecule formation rates for tunneling levels {#sec:model}
=======================================================================
The metastable ${\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}}(6{\ensuremath{\mathrm{S}_{1/2}}}+6{\ensuremath{\mathrm{S}_{1/2}}})$ state is represented by the potential curve of [@foucrault1992], matched beyond $20a_0$ to the standard asymptotic expansion $\sum{C_n/R^n}$ ($n=6,8,10$), with $C_n$ coefficients taken from [@marinescu1994]. The repulsive wall of the potential is slightly changed in order to reproduce a large and positive scattering length [@kerman2001]. We work here with $a_{\mathrm{T}} = 2370 a_0$, which satisfactorily reproduces the intensity envelope of the PA spectrum of [@fioretti1999]. The initial scattering radial wave function is computed through a standard Numerov integration at an energy $E/k_{\mathrm{B}}T = 150\
\mu\mathrm{K}$ above the $6{\ensuremath{\mathrm{S}_{1/2}}}+6{\ensuremath{\mathrm{S}_{1/2}}}$ asymptote, ignoring the hyperfine structure (figure \[fig:wfG2J05\]a). Let us note that the accuracy of the $a_{\mathrm{T}}$ and $C_n$ values is not crucial for the following calculation of the rates, which can be determined experimentally typically within a factor of 2.
The photoassociation and cold molecule formation rates are evaluated according to the perturbative model already presented in previous papers [@drag2000; @pillet1997; @dion2002]. Briefly, the photoassociation rate $R_{\mathrm{PA}}(\bar{v},\bar{J};T)$ per atom (expressed in s$^{-1}$) for an initial continuum state with energy $E=k_{\mathrm{B}}T$ and wave function $\psi_E(R)$ into a vibrational level $\bar{v}$ with a radial wave function $\chi_{\bar{v},\bar{J}}({\ensuremath{0_{\mathrm{g}}^-}}; R)$ of the coupled ${\ensuremath{0_{\mathrm{g}}^-}}$ states (where $\bar{v}$ stands for the quantum number $v$ of the ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ external well levels, or for G$_1$ or G$_2$), at a detuning $\Delta_v$ below $6{\ensuremath{\mathrm{S}_{1/2}}}+6{\ensuremath{\mathrm{P}_{3/2}}}$ limit is expressed as $$R_{\mathrm{PA}}(\bar{v},\bar{J};T) = \left(\frac{3}{2\pi}
\right)^{3/2}\frac{h}{2} n_{\mathrm{at}} \lambda^3_{\mathrm{th}}
\mathrm{e}^{-\frac{E}{k_{\mathrm{B}} T}} \mathcal{A}
K^2 \left| \left\langle \psi_E \left|
\chi_{\bar{v},\bar{J}}({\ensuremath{0_{\mathrm{g}}^-}}) \right\rangle \right. \right|^2,
\label{eq:parate}$$ where $n_{\mathrm{at}}$ is the atomic density and $\lambda_{\mathrm{th}} = h\sqrt{1/(3\mu k_B T)}$ is the thermal de Broglie wavelength. The atomic Rabi frequency 2$K$ is related to the intensity $I$ of the laser through $K^2 = \frac{\Gamma}{8}
\frac{I}{I_0}$, where $\Gamma/2\pi=5.22$ MHz is the natural width of the $6{\ensuremath{\mathrm{P}_{3/2}}}$ atomic level. At the PA wavelength considered, $\lambda_{\mathrm{PA}}$, the saturation intensity $I_0=\pi hc
\Gamma/(3\lambda_{\mathrm{PA}}^3)$ is $1.1\ \mathrm{mW/cm}^{2}$. The angular factor $\mathcal A=125/3888$ includes hyperfine degeneracies, assuming an initial $(F=4)+(F=4)$ hyperfine state. As mentioned above, we actually solve the Schrödinger equation for every chosen value of the rotational quantum number $\bar{J}$ taken as an effective parameter, so we don’t include the degeneracy factor $2J+1$ in the rate formulas.
The two components of the PA level wave function $\left|
\chi_{\bar{v},\bar{J}}({\ensuremath{0_{\mathrm{g}}^-}};R) \right\rangle$ (see figure \[fig:wfG2J05\]) are denoted by $\chi_{\bar{v},\bar{J}}^{6\mathrm{P}}(R)$ and $\chi_{\bar{v},\bar{J}}^{5\mathrm{D}}(R)$, such that $$O(E,\bar{v},\bar{J}) \equiv \left| \left\langle \psi_E \left|
\chi_{\bar{v},\bar{J}}({\ensuremath{0_{\mathrm{g}}^-}}) \right\rangle \right. \right|^2 =
\left| \left\langle \psi_E \left| \chi_{\bar{v},\bar{J}}^{6\mathrm{P}}
\right\rangle \right. + \left\langle \psi_E \left|
\chi_{\bar{v},\bar{J}}^{5\mathrm{D}} \right\rangle \right. \right|^2 .
\label{eq:paov}$$ Following [@dion2001], the rate $R_{\mathrm{mol}}({\ensuremath{\Delta_{\mathrm{L}}}})$ for cold molecule formation after PA in the $\bar{v}$ level is obtained by multiplying the PA rate with the branching ratio $R_{\mathrm{br}}(\bar{v},\bar{J})$ of the $\bar{v}$ level towards the bound levels $v''({\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}})$) of the metastable triplet state, and neglecting here the $R$ dependence of the dipole transition function for the spontaneous decay step, $$R_{\mathrm{mol}}({\ensuremath{\Delta_{\mathrm{L}}}})= R_{\mathrm{PA}}({\ensuremath{\Delta_{\mathrm{L}}}}) R_{\mathrm{br}}(\bar{v},\bar{J}),
\label{eq:molrate}$$ with $$R_{\mathrm{br}}(\bar{v},\bar{J}) = \sum_{v''} \left| \left\langle
\chi_{\bar{v},\bar{J}}({\ensuremath{0_{\mathrm{g}}^-}}) \left| \phi_{v''}({\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}})
\right\rangle \right. \right|^2.
\label{eq:brrate}$$ Figure \[fig:ov-J05\]a displays the overlap between the continuum wave function and the bound states of coupled potentials, equation (\[eq:paov\]), lying in the energy interval between $-9$ and $-1.5$ cm$^{-1}$, for both $J=0$ and $J=5$. As expected, G$_1$ and G$_2$ have smaller overlap with the initial continuum than the external well levels, due to the weaker probability of wave functions to be localized at large distances. In contrast with the PA rate, the branching ratio of the G$_1$ and G$_2$ wave functions is enhanced due to their good localization at short distances (figure \[fig:ov-J05\]b). As it was often emphasized, a good production of cold molecules requires a favorable ratio between free-bound and bound-bound transitions. The product $O(E,\bar{v},\bar{J}) \times R_{\mathrm{br}}(\bar{v},\bar{J})$ is represented in figure \[fig:ov-J05\]c. If for G$_1$ the efficiency seems certain, we see that for G$_2$ this balance is fragile and depends indeed on the $\bar{J}$ value which makes tunneling effective. On the right axis of this figure is reported the rate for cold molecule formation, equation (\[eq:molrate\]), for typical experimental conditions $n_{\mathrm{at}}=10^{11}$ cm$^{-3}$ and a PA laser intensity $I= 100\ \mathrm{W/cm}^2$. It is found two times and eight times larger than the rate for the $v_{\mathrm{ext}}$ levels for G$_2$ and G$_1$, respectively. Even if in the experiment the intensities of the G$_2$ and G$_1$ lines are found to be comparable (see insets in figure \[fig:potS0g\]), our results confirm that the cold molecule formation rate is indeed larger than the rate for the $v_{\mathrm{ext}}$ levels. The relative intensity of the G$_2$ and G$_1$ lines may be influenced by the two-photon ionization used for the detection (see next section). Let us note also that the rate computed for the $v_{\mathrm{ext}}$ levels is in agreement with the one reported in [@drag2000], where $R_{\mathrm{mol}} \approx 0.2\
\mathrm{s}^{-1}$ was measured at a detuning of 7 cm$^{-1}$, for a PA laser intensity $I= 55\ \mathrm{W/cm}^2$.
![(a) Overlap $O(E,\bar{v},\bar{J})$ between the continuum wave function and vibrational wave functions of coupled ${\ensuremath{0_{\mathrm{g}}^-}}$ potentials lying in the detuning interval between $-9$ and $-1.5\
\mathrm{cm}^{-1}$. (b) Branching ratio $R_{\mathrm{br}}(\bar{v},\bar{J})$ of the same levels to the ${\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}}$ vibrational states. (c) Product of the two previous quantities (left axis) and corresponding cold molecule formation rate $R_{\mathrm{mol}}({\ensuremath{\Delta_{\mathrm{L}}}})$ (right axis) assuming an atomic density $n_{\mathrm{at}}=10^{11}$ cm$^{-3}$ and a PA laser intensity $I=100\ \mathrm{W/cm}^2$. Note that in this panel the contribution of the nearby levels at $-5.87$ cm$^{-1}$ and $-5.89$ cm$^{-1}$ shown in figure \[fig:wfG2J05\] are added together into a single rate value around $-5.9$ cm$^{-1}$. Both $\bar{J}=0$ (black crosses) and $\bar{J}=5$ (red open circles) are shown. The levels corresponding to G$_1$ and G$_2$ are indicated. The arrows identify the levels for which we calculated the vibrational distribution of the produced cold molecules displayed in the figure \[fig:G1-2J5dist\].[]{data-label="fig:ov-J05"}](fig4_ov-J05.eps){width="0.7\columnwidth"}
The rates for formation of cold molecules in individual levels $v''$ of the metastable ${\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}}$ state, $$R_{\mathrm{mol}}({\ensuremath{\Delta_{\mathrm{L}}}},v'') = R_{\mathrm{PA}}({\ensuremath{\Delta_{\mathrm{L}}}}) \left| \left\langle
\chi_{\bar{v},\bar{J}}({\ensuremath{0_{\mathrm{g}}^-}}) \left| \phi_{v''}({\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}})
\right\rangle \right. \right|^2,
\label{eq:molrate_v}$$ are represented in figure \[fig:G1-2J5dist\] for the same range of detunings and same experimental conditions as in figure \[fig:ov-J05\]c, for both the G$_1$ and G$_2$ features, considering the levels indicated by arrows in figure \[fig:ov-J05\]b corresponding to $\bar{J}=5$. Both vibrational distributions peak markedly at $v''=5,6$, confirming the efficiency of the PA into the tunneling resonances to produce cold molecules in very low vibrational levels.
![Rate of formation of cold molecules in the vibrational levels $v''({\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}})$, via photoassociation into (a) the G$_1$ and (b) the G$_2$ states. In the latter case, rates for both levels contributing to G$_2$ are given, i.e., the level at $-5.87$ cm$^{-1}$ tunneling from the inner wells (black closed circles) and the external level tunneling from the outer well at $-5.89$ cm$^{-1}$ (red open circles). The calculation is made for $n_{\mathrm{at}}=10^{11}$ cm$^{-3}$ and an intensity $I= 100\
\mathrm{W/cm}^2$ for the PA laser.[]{data-label="fig:G1-2J5dist"}](fig5_G1-2J5dist.eps){width="0.7\columnwidth"}
Modeling the ionization signal {#sec:results}
==============================
In cesium PA experiments [@fioretti1998; @vatasescu2000], the cold molecules are detected via a two-photon ionization process, where the first photon is resonant with rovibrational levels of molecular states correlated to the $6{\ensuremath{\mathrm{S}_{1/2}}}+5{\ensuremath{\mathrm{D}_{3/2}}}$ dissociation limit. Therefore, the ionization signal is very sensitive to the wavelength chosen for the ionization laser. Following our previous studies of this process [@dion2002; @dion2001], we simulate the contributions of G$_1$ and G$_2$ to the Cs$_2^+$ ions signal obtained after PA in a ${\ensuremath{0_{\mathrm{g}}^-}}(6{\ensuremath{\mathrm{S}_{1/2}}}+6{\ensuremath{\mathrm{P}_{3/2}}})$ level detuned by ${\ensuremath{\Delta_{\mathrm{L}}}}$, spontaneous emission towards $v''({\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}})$ bound levels, and ionization with a photon of frequency $\nu_{\mathrm{ion}}$ of these levels through absorption into levels $v'$ of the $(2)^3\Pi_{\mathrm{g}}(6{\ensuremath{\mathrm{S}_{1/2}}}+5\mathrm{D}_{5/2})$ potential. In this model, the second step of the ionization process is described as a uniform ionization probability of the $v'$ levels into Cs$_2^+$ ions, and will not be considered here. The ionization signal $S_{\mathrm{ion}}({\ensuremath{\Delta_{\mathrm{L}}}})$ is expressed as $$S_{\mathrm{ion}}({\ensuremath{\Delta_{\mathrm{L}}}}) = N_{\mathrm{PA}}({\ensuremath{\Delta_{\mathrm{L}}}}) \sum_{v''}
P_{\mathrm{ion}}(v'') \left| \left\langle \chi_{\bar{v},\bar{J}}({\ensuremath{0_{\mathrm{g}}^-}})
\left| \phi_{v''}({\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}}) \right\rangle \right. \right|^2.
\label{eq:sions}$$ As in the cold molecules rate calculation, we neglect the $R$ dependence of the dipole transition moment involved in the spontaneous decay step. In equation (\[eq:sions\]), the number of molecules $N_{\mathrm{PA}}({\ensuremath{\Delta_{\mathrm{L}}}})$ accumulated during the photoassociation step is defined by multiplying the PA rate $R_{\mathrm{PA}}({\ensuremath{\Delta_{\mathrm{L}}}})$ with the number of atoms $n_{\mathrm{PA}}$ in the photoassociation area with the residence time $t_{\mathrm{PA}}$ of the cold molecules within the trapping region: $$N_{\mathrm{PA}}({\ensuremath{\Delta_{\mathrm{L}}}})=R_{\mathrm{PA}}({\ensuremath{\Delta_{\mathrm{L}}}}) n_{\mathrm{PA}} t_{\mathrm{PA}}.$$ In the Cesium PA experiment, typical values for these parameters are $n_{\mathrm{PA}}=5 \times 10^{7}$ and $t_{\mathrm{PA}} \approx 10$ ms. The excitation probability $P_{\mathrm{ion}}(v'')$ of the $v''$ levels into vibrational levels $v'$ of the $(2)^3\Pi_{\mathrm{g}}(6{\ensuremath{\mathrm{S}_{1/2}}}+5\mathrm{D}_{5/2})$ potential is calculated as $$P_{\mathrm{ion}}(v'') = \sum_{v'} \left| \left\langle
\phi_{v'}(^3\Pi_{\mathrm{g}}) \right| D \left| \phi_{v''}({\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}})
\right\rangle \right|^2 f(\nu_{v''v'}),$$ where $D$ is the $R$-dependent dipole moment for the ${\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}}\to
{^3\Pi_{\mathrm{g}}}(6s+5\mathrm{D}_{5/2})$ transition calculated in [@spies1989], and$$f(\nu_{v''v'}) = \exp \left[ - \ln(2)
\frac{(\nu_{v''v'}-\nu_{\mathrm{ion}})^2}{(\delta \nu)^2} \right]$$ is a parametric function accounting for the resolution of the experiment, estimated at $\delta \nu \approx 30$ GHz [@dion2002], and $\nu_{v''v'}$ is the transition frequency between $v''$ and $v'$ levels.
![Cs$_2^+$ ions signal (in arbitrary units) as a function of the frequency of the ionization laser, calculated with equation (\[eq:sions\]) for detunings ${\ensuremath{\Delta_{\mathrm{L}}}}$ corresponding to G$_1$ (right column) and G$_2$ (left column). Signals corresponding to photoassociation in vibrational levels located close to G$_1$ and G$_2$ and belonging to the external well of the ${\ensuremath{0_{\mathrm{g}}^-}}(6{\ensuremath{\mathrm{S}_{1/2}}}+6{\ensuremath{\mathrm{P}_{3/2}}})$ potential are also shown. The energy of the $6{\ensuremath{\mathrm{S}_{1/2}}}+5\mathrm{D}_{5/2}$ dissociation limit is taken as being $14597.08\ \mathrm{cm}^{-1}$ above the $6{\ensuremath{\mathrm{S}_{1/2}}}+6{\ensuremath{\mathrm{S}_{1/2}}}$ origin.[]{data-label="fig:ionJ5"}](fig6_ionJ5.eps){width="0.7\columnwidth"}
The ion signals as a function of the ionization frequency $\nu_{\mathrm{ion}}$ obtained for G$_1$ and G$_2$ are displayed in figure \[fig:ionJ5\], together with those corresponding to PA in neighboring vibrational levels $v_{\mathrm{ext}}$ of the ${\ensuremath{0_{\mathrm{g}}^-}}(6{\ensuremath{\mathrm{S}_{1/2}}}+6{\ensuremath{\mathrm{P}_{3/2}}})$ external well. As expected, large amounts of molecular ions are detected only for specific frequencies corresponding to the resonance condition in the first step. The signals coming from G$_1$ and G$_2$ are clearly more intense than those of the surrounding levels, in agreement with the experimental spectrum [@vatasescu2000], because of the increased efficiency for forming cold molecules in bound levels of the triplet state. The signal corresponding to G$_1$, having a maximum of $0.03$ in figure \[fig:ionJ5\]e, is five times bigger that the simulated G$_2$ signal, whose maximum reaches $0.006$ if we take the sum of the two contributing levels (figures \[fig:ionJ5\]a,b). This ratio between the intensities of the simulated “giant lines” is larger than in the experimental spectrum where the ratio is only two [@fioretti1999]. This could be due to the fact that the ionization spectra for the tunneling resonances do not match perfectly: the differences in vibrational distributions $v''({\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}})$ of G$_1$ and G$_2$ (see figure \[fig:G1-2J5dist\]) make it possible to chose an ionization frequency (e.g., $h \nu_{\mathrm{ion}} \approx 14101$ or 14117 cm$^{-1}$) such that the ratio between the two ion signals is greatly modified, as shown in figure \[fig:ionJ5\_overlay\].
![Same as figure \[fig:ionJ5\]: we superimposed predicted ion signals from panels a, b, and e of this figure, slightly shifted from each other for better visibility. Note that the signal for G$_1$ is shown at 1/10th of its actual value.[]{data-label="fig:ionJ5_overlay"}](fig7_ionJ5_overlay.eps){width="0.7\columnwidth"}
Conclusion {#sec:conclu}
==========
As demonstrated through several studies [@fioretti1998; @vatasescu2000; @dion2001], cesium atoms are well-suited for ultracold molecule formation through photoassociation, mainly due to the peculiarity of some of the excited states of the Cs$_2$ molecule. Of particular interest is the ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{S}_{1/2}}}+{\ensuremath{\mathrm{P}_{3/2}}})$ state, which is composed of two potential wells separated by a barrier whose height culminates at an energy close to the dissociation asymptote. This allows for tunneling between the two wells, enabling a pair of atoms initially far apart to come close together in a photoassociated molecule.
We have shown here that this tunneling process leads to strongly bound ultracold cesium molecules in the metastable ${\ensuremath{\mathrm{a}^3\Sigma_{\mathrm{u}}^{+}}}({\ensuremath{\mathrm{S}_{1/2}}}+{\ensuremath{\mathrm{S}_{1/2}}})$ ground electronic state. Indeed, it is mostly vibrational levels around $v''
=5,6$ that are populated, in contrast to what can be reached for non-tunneling states of the external ${\ensuremath{0_{\mathrm{g}}^-}}({\ensuremath{\mathrm{P}_{3/2}}})$ well ($v'' > 23$ [@drag2000]). The tunneling efficiency is seen to be markedly dependent upon the rotational state $J$, in agreement with what was observed experimentally [@vatasescu2000].
Finally, we have investigated that the experimental detection scheme, resonant two-photon ionization, influences the observation of ultracold Cs$_2$ molecules. Evidence would suggest that the detection process affects the relative intensities seen in the signal, as was the case previously [@dion2001]. In other words, the molecules produced are not all ionized with the same efficiency, and it is therefore important to consider this step in the analysis of photoassociation experiments.
This work has been performed in the framework of the Research and Training Network of the European Union “Cold Molecules” (contract number HPRN-CT-2002-00290). O.D. and M.V. acknowledge partial support from the INTERCAN and QUDEDIS networks of the European Science Foundation.
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|
---
abstract: 'The electrons found in Dirac materials are notorious for being difficult to manipulate due to the Klein phenomenon and absence of backscattering. Here we investigate how spatial modulations of the Fermi velocity in two-dimensional Dirac materials can give rise to localization effects, with either full (zero-dimensional) confinement or partial (one-dimensional) confinement possible depending on the geometry of the velocity modulation. We present several exactly solvable models illustrating the nature of the bound states which arise, revealing how the gradient of the Fermi velocity is crucial for determining fundamental properties of the bound states such as the zero-point energy. We discuss the implications for guiding electronic waves in few-mode waveguides formed by Fermi velocity modulation.'
author:
- 'C. A. Downing'
- 'M. E. Portnoi'
title: Localization of massless Dirac particles via spatial modulations of the Fermi velocity
---
\[intro\]Introduction
=====================
It is a truth universally acknowledged, that a single electron in possession of a gapless, linear spectrum, must be in want of a bound state [@Klein]. There is a considerable resurgence in the importance of such massless fermions in condensed matter physics due to the rise of so-called Dirac materials [@Wehling], whose charge carriers behave according to quasi-relativistic wave equations. Celebrated examples in two dimensions include graphene or the surface states of topological insulators. An important property that is inherent to such Dirac particles is the absence of backscattering [@Katsnelson], which whilst leading to large electron mobilities, presents a considerable difficulty in localizing Dirac electrons [@Trauzettel; @Martino; @Bardarson; @Hartmann2010; @Rozhkov; @Giavaras2012; @Zalipaev2013; @Optimal; @Mag2] and hence building practical digital devices with a well-defined on/off logical state [@Yung2013].
One interesting method proposed to manipulate these somewhat elusive quasi-relativistic charge carriers is to consider systems with a spatially-varying Fermi velocity, $v_F = v_F(\mathbf{r})$ [@deJuan; @Peres; @Amorim], such that so-called velocity barriers may form. The resulting ballistic electron transport in such systems has already been extensively studied [@Concha; @Raoux; @Krstajic; @Pellegrino; @Liu2013; @Cheraghchi], as has the effects of applying external electric [@Liu1] and magnetic fields [@Yuan2] and a superlattice geometry [@Esmailpour; @Lima1; @Lima2; @Bezerra]. There are immediately apparent strong analogies in both acoustics and especially optics, where phenomena such as super-collimation has been envisaged [@Yuan; @Wang1].
Energy-dependent Fermi velocity renormalization has already been seen in experiments with graphene at energies close to the Dirac point [@Elias; @Siegel]. Here we instead consider the problem of Dirac particles that can be described with a spatially modulated Fermi velocity. This situation arises theoretically from both elasticity theory with tight-binding calculations, as well as quantum field theory in curved space [@Sturla]. Experimentally, a spatially dependent Fermi velocity may occur due to ripples in the material [@Martin; @Polini], the use of different substrates [@Luican; @Hwang2012], superlattices [@Park; @Gibertini; @Yan], atomic scale defects induced by ion irradiation [@Tapaszto], straining the material [@Huang2010; @Jang], by placing a grounded plane of metal nearby [@Raoux] or by judiciously applying a uniform electric field [@Diaz2017]. Indeed, a spatial dependence of the Fermi velocity has recently been observed in two different experiments [@Yan; @Jang]. Moreover, our work is relevant for a large range of artificial Dirac materials, which advantageously allow one precise control over the velocity of the hosted Dirac-like particles. Examples of artificial Dirac systems include: cold atoms in an optical lattice [@Zhu2007]; flexural waves in thin plates [@Torrent2013]; microwaves in a lattice of dielectric resonators [@Bellec2013]; and plasmons in metallic nanoparticles [@Downing2017].
Previously, most of the theoretical attention on this topic has been focused on the scattering of two-dimensional (2D) massless Dirac fermions on square velocity barriers, either single [@Yuan], double [@Liu2] or multiple [@Esmailpour; @Lima1; @Lima2]. Here we address the bound state problem for 2D Dirac particles and furthermore we consider non-square velocity distributions, which are arguably more realistic. Indeed, smooth electrostatic and magnetostatic potential barriers are known to lead to effects not found in their sharp barrier counterparts [@Stone2012; @Mag].
In this work, we reveal the dependence of the supported bound state energy levels on the velocity barrier parameters and find how the presence of a finite zero-point (ground state) energy is critically dependent on the gradient of the velocity barrier. For one-dimensional (1D) confinement, whilst exponentially growing velocity barriers have a finite threshold energy for the ground state, algebraically growing velocity barriers (growing at most linearly) have a vanishing ground state energy. This 1D geometry of velocity barrier acts as an electron waveguide for massless quasirelativistic particles [@Beenakker2009; @Zhang2009; @Myoung2011; @Stone2012; @Hartmann2014], with the bound modes propagating along the channel created by the confining barrier. Such systems are of great interest to experimentalists hoping to realize electron optics based on Dirac materials such as graphene [@Chen2016]. For zero-dimensional localization (0D), we show how true confinement in radial velocity barriers acting as nanoscale quantum dots is indeed possible, and demonstrate how both an infinite velocity barrier and a smooth, algebraically growing radial barrier possess a nonvanishing zero-point energy. The proposed bound states may be revealed in experiments via quantum transport measurements, where the signature of a confined mode is a jump in the conductance.
It has been shown by Peres [@Peres] that in order to maintain Hermitian operators, the relevant (and Sturm-Liouville) 2D Dirac Hamiltonian for this problem is $$\label{intro00}
\hat{H} = \sqrt{v_F(\mathbf{r})} \pmb{\sigma} \cdot \mathbf{\hat{p}} \sqrt{v_F(\mathbf{r})},$$ where $\pmb{\sigma} = (\sigma_x, \sigma_y)$ are the spin matrices of Pauli. Eq. acts on a two-component spinor wavefunction $\Phi (\mathbf{r})$, and the eigenvalues $E$ are found via $\hat{H} \Phi = E \Phi$. Continuity of the probability current leads to the following boundary condition at an interface $\mathbf{r} = \mathbf{R}$ $$\label{intro01}
\sqrt{v_F(\mathbf{r})} \Phi (\mathbf{r}) \Bigr|_{\mathbf{r} = \mathbf{R}+\pmb{\delta}} =
\sqrt{v_F(\mathbf{r})} \Phi (\mathbf{r}) \Bigr|_{\mathbf{r} = \mathbf{R}-\pmb{\delta}},$$ which is directly analogous to what occurs in heterostructures defined by a position-dependent mass [@Cole]. In what follows, we make the following assignment of the auxiliary spinor for convenience $\Psi (\mathbf{r}) = \sqrt{v_F(\mathbf{r})} \Phi (\mathbf{r})$. We solve Eq. for various spatial profiles of the Fermi velocity $v_F(\mathbf{r})$ to unveil the general properties of bound states in such systems, with the ultimate aim of proposing on/off logical states in Dirac materials.
The rest of this work is organized as follows. We study in Sec. \[sec2\] a series of different 1D velocity barriers giving rise to bound states. Localization in radially symmetric 2D velocity barriers is discussed in Sec. \[sec3\]. Finally, we draw some conclusions in Sec. \[conc\].
\[sec2\]Velocity barrier channels
=================================
We shall consider several toy models of 1D velocity barriers $v_F = v_F(x)$, each with drastically different spatial profiles, yet all unified by their integrability. Namely, we investigate the following 1D velocity barrier channels: square (Sec. \[subssec1\]), exponential (Sec. \[subssec2\]), linear (Sec. \[subssec3\]) and square root (Sec. \[subssec4\]), as shown graphically in Fig. \[fig1\]. This range of models allows us to identify how the shape of the velocity barrier influences the properties of the bound states.
Working in Cartesian coordinates $(x, y)$, we begin by making the following ansatz for the auxillary spinor $\Psi (x,y) = (L_y)^{-1/2} e^{i q_y y} \left[ \psi_1(x), \psi_2(x) \right]^T$, due to translational invariance in the $y$ direction. Here $q_y$ is the wavenumber along the formed trench and $(L_y)^{-1/2}$ is the length of the material in the $y$-direction.
![(Color online) Profiles of the considered 1D velocity barriers: the square (short-dashed gray line), the exponential (solid red line), the linear (long-dashed blue line) and the square root (dot-dashed green line) channels respectively. Here we take $v_1 / v_0 = 2$ for the square velocity barrier.[]{data-label="fig1"}](fig1){width="40.00000%"}
\[subssec1\]The square velocity barrier
---------------------------------------
![(Color online) Eigenenergies of massless Dirac fermions in a square velocity barrier, as a function of the barrier parameter $v_0 / v_1$, calculated via Eq. . The lowest four states can be seen, from the ground state (red triangular markers), to higher states (blue diamond markers, green square markers and orange circular markers respectively). The region of allowed bound states, arising from Eq. , is denoted by solid black lines. Here we take $q_y d = 1$.[]{data-label="fig2"}](fig2){width="40.00000%"}
Firstly, we revisit the square model velocity barrier [@Peres; @Concha; @Raoux; @Krstajic]. Whilst there has been much focus on scattering on such a barrier, instead we shall focus our investigation on the associated bound states that may be supported. We consider a square barrier of width $d$, given by $$\label{eqbox00}
v_F(x)= \begin{cases} v_0, \quad -\tfrac{d}{2} \le x \le \tfrac{d}{2}, \\
v_1, \quad x > \tfrac{d}{2}, \quad x < -\tfrac{d}{2}, \end{cases}$$ where the velocity parameters satisfy $v_1 > v_0$, as plotted in Fig. \[fig1\] as the short-dashed gray line. Naturally appearing in this problem are the following wavenumbers $$\label{eqbox01}
\kappa = \left(q_y^2 - \left( \tfrac{E}{\hbar v_1} \right)^2 \right)^{1/2}, \quad k = \left(\left( \tfrac{E}{\hbar v_0} \right)^2 - q_y^2 \right)^{1/2},$$ which arise when describing the spinor wavefunction in its evanescent and propagating stages respectively. These wavevectors automatically restrict the region of bound states to the fan $$\label{eqbox013343443}
\hbar v_0 |q_y| < |E| < \hbar v_1 |q_y|,$$ since it is required that both $\kappa$ and $k$ are real quantities. The energies of the bound states in the square barrier follow from the Hamiltonian with the boundary condition Eq. , and are determined by the transcendental equation $$\label{eqbox02}
\tan (kd) = \frac{2 \frac{v_1}{v_0}\frac{\kappa}{k}}{1 + \left( 1 - \frac{v_1}{v_0} \right)^2 \left( \frac{q_y}{k} \right)^2 - \left( \frac{v_1}{v_0} \frac{\kappa}{k} \right)^2 }.$$ We plot in Fig. \[fig2\] the eigensolutions of Eq. , as a function of the barrier strength parameter $v_0/v_1$, showing the four lowest-lying bound states. As is common in 1D barrier problems, there is always at least one bound state, even for arbitrarily weak barriers with $v_0/v_1 \to 1$. Upon taking this limit, one can see from Eq. that the left hand side of the equation will grow monotonically between $0 \le \tan (kd) \le q_y d \left( v_1^2/v_0^2 - 1\right)^{1/2}$. This left hand side will always be intercepted as the right hand side of Eq. shrinks to zero as $|E| \to \hbar v_1 |q_y|$ and hence a bound state solution is guaranteed.
In the ultrastrong barrier limit $v_1 \to \infty$, Eq. must be replaced with the equation $\tan (kd) + k/q_y = 0$. Now there are an increasing number of bound states $N$, which can be estimated from $$\label{eqbox03}
N = \left \lfloor \Upsilon \frac{q_y d}{\pi} \right \rfloor, \quad \Upsilon = \left( v_1^2/v_0^2 - 1\right)^{1/2},$$ where $\left \lfloor ... \right \rfloor$ is the floor function. Inverting Eq. tells us the threshold velocity barrier strengths above which new bound states appear, via the approximate relation $$\label{eqbox033}
v_0/v_1 \simeq \left( 1 + \left( \frac{\pi N}{q_y d} \right)^2 \right)^{-1/2}.$$ This feature of a changing number of bound states with a modulation of $v_0/v_1$ is shown in Fig. \[fig2\]. This phenomenon of losing successive bound states into the continuum as $v_0/v_1$ increases towards unity is superficially reminiscent of the ‘fall-into-the-center’ phenomenon in relativistic quantum mechanics [@Zeldovich]. The analogue of this so-called atomic collapse effect in Dirac material physics (in 1D) sees the lowest lying states diving into the continuum as the band-gap of the Dirac material is reduced [@Hartmann2011; @Collapse]. A notable distinction here is that whilst conventionally the bound states with fewest nodes are successively lost, for the square velocity barrier the nodeless ground state is always present and instead the highest lying (highly nodal) states are successively lost with increasing $v_0/v_1$. For example, in Fig. \[fig2\] and with $q_y d = 1$, the third, second and first excited states (orange circular markers, green square markers and blue diamond markers respectively) disappear one-by-one as $v_0/v_1$ increases at $v_0/v_1 \simeq 0.11, 0.16, \text{and}~0.30$ in turn, whilst the ground state (red triangular markers) persists even as $v_0/v_1 \to 1$.
In what follows, in Secs. \[subssec2\]- \[subssec4\], we move on to investigating non-square velocity barriers, with an emphasis on the bound modes propagating along the formed Dirac electron waveguides with wavenumber $q_y >0$.
\[subssec2\]The exponential velocity barrier
--------------------------------------------
As a first example of a smooth velocity barrier, let us study the exponential velocity barrier, plotted in Fig. \[fig1\] as the solid red line, and defined by $$\label{eqcusp00}
v_F(x)= v_0 e^{|x|/d}.$$ Here $v_0$ is the minimal Fermi velocity, found at the center of the barrier, and $d$ is the length scale of the problem, from which arises the key dimensionless parameter $$\label{eqcusp000}
\lambda = \frac{|E|d}{\hbar v_0},$$ which is useful for describing the eigenvalues. Upon solving the coupled equations formed from the Hamiltonian , one finds the following spinor wavefunction $\psi(x)$ in region I $(x>0)$ with the help of the independent variable change $\xi = \lambda e^{-x/d}$ $$\label{eqcusp01}
\psi_{I}(x)
= \tfrac{c_{I}}{\sqrt{d}} e^{-x/2d}
\left(
\begin{array}{c}
J_{q_y d + 1/2} \left( \xi \right) \\ \text{sgn} (E) J_{q_y d - 1/2} \left( \xi \right)
\end{array}
\right),$$ in terms of the Bessel function of the first kind $J_{\alpha}(\xi)$ and where $c_{I}$ is some constant. The solution in region II $(x<0)$ is found by interchanging the top and bottom wavefunction components, and making the replacements $x \to -x$ and $c_I \to c_{II} = \pm \text{sgn} (E) c_I$. Applying the boundary condition Eq. at the interface $x=0$, one finds the spectrum of bound states is determined via the transcendental equation $$\label{eqcusp02}
J_{q_y d - 1/2} \left( \lambda \right) = \pm J_{q_y d + 1/2} \left( \lambda \right),$$ where the $\pm$ corresponds with the $\pm$ in the definition of $c_{II}$. Eq. can be solved with standard root-finding methods or indeed graphically, the result of which is shown in Fig. \[fig3\]. There we plot the six lowest lying bound states as a function of transversal wavevector $q_y d$, revealing an approximately linear dependence between energy level and transversal momentum, as well as an approximately constant level spacing.
One also notices there is a threshold magnitude of energy at which the first bound state appears. This zero-point energy can be quantified by taking the limit $q_y d \to 0$ in Eq. , such that one arrives at the analytic expression $$\label{eqcusp03}
\lambda_{n, \pm} = \pi \left( n \pm \tfrac{1}{4} \right),$$ where $n$ is a nonnegative (positive) integer when the $+ (-)$ sign is taken. Explicitly, the ground state energy is $\lambda_0 = \pi / 4 \simeq 0.785$. In this small wavevector limit, the energy level separation is a universal constant $\Delta \lambda = \pi /2 \simeq 1.57$.
The characteristics of this velocity barrier suggest it can act as a few-mode, Dirac electronic waveguide in direct analogy to the channeling of photons along optical fibers. The ability to substantially reduce the number of bound modes propagating along the velocity barrier also acts to reduce electronic losses due to scattering, which is enhanced for multimode waveguides supporting many different modes propagating at several different velocities.
![Progression of the six lowest bound states energies with traversal momentum $q_y d$, for massless Dirac particles in a cusp-like velocity barrier, calculated via Eq. .[]{data-label="fig3"}](fig3){width="40.00000%"}
\[subssec3\]The linear velocity barrier
---------------------------------------
Now we look at the linear velocity barrier, governed by the parameters $v_0$ and $d$ and shaped by $$\label{eqlinear00}
v_F(x)= v_0 (1+|x|/d),$$ as sketched as the long-dashed blue line in Fig. \[fig1\]. We define the useful dimensionless quantity to measure the energies $$\label{eqlinear000}
\gamma = \frac{E d}{\hbar v_0}.$$ Proceeding in a similar manner to Sec. \[subssec2\], one finds the following spinor wavefunction in region I $(x>0)$ with the aid of the variable $\xi = 2 q_y d(1+\tfrac{x}{d})$ $$\begin{gathered}
\label{eqlinear01}
\psi_{I}(x) = \tfrac{c_{I}}{\sqrt{d}} \left(1+\tfrac{x}{d}\right)^{i \gamma} e^{- q_y x} \times \\
\left(
\begin{array}{c}
U\left( 1+i \gamma, 1+ 2 i \gamma, \xi \right)
\\ \gamma^{-1} U\left( i \gamma, 1+ 2 i \gamma, \xi \right)
\end{array}
\right),\end{gathered}$$ where $c_{I}$ is a normalization constant and $U(\alpha, \beta, \xi)$ is the Tricomi function or confluent hypergeometric equation of the second kind [@Gradshteyn]. The solution in region II $(x<0)$ is found by interchanging the top and bottom wavefunction components, and making the replacements $x \to -x$ and $c_I \to c_{II} = \pm c_I$. Ensuring a conserved probability current via Eq. yields the eigenvalue equation $$\label{eqlinear02}
U\left( i \gamma, 1+ 2 i \gamma, 2 q_y d \right) = \pm \gamma U\left( 1+i \gamma, 1+ 2 i \gamma, 2 q_y d \right) ,$$ where the $\pm$ is associated with the $\pm$ in the definition of $c_{II}$. Although Eq. must be solved numerically, it is an exact expression and can be solved with any desired accuracy. The result of such computations is shown in Fig. \[fig4\] for the six lowest-lying states as a function of transversal wavevector $q_y d$. Most noticeable is the absence of threshold bound state energies as $q_y d \to 0$, markedly different from the sharper, exponential cusp profile encountered in Sec. \[subssec2\]. Instead, there is a plethora of bound states with vanishing transversal momenta. This model suggests that the fundamental change from an exponentially to a linear algebraically growing velocity barrier manifests itself in the loss of the finite threshold effect, which may be important for the design of velocity barriers as guides of electron waves.
![Progression of the six lowest bound states energies with traversal momentum $q_y d$, for massless Dirac particles in a linear velocity barrier, calculated via Eq. .[]{data-label="fig4"}](fig4){width="40.00000%"}
\[subssec4\]The square root velocity barrier
--------------------------------------------
Finally, we consider a weakly growing square root velocity barrier, graphed in Fig. \[fig1\] as the dot-dashed green line, and with the functional form $$\label{eqsqrt00}
v_F(x)= v_0 \sqrt{1+|x|/d}.$$ One may obtain the spinor wavefunction in region I $(x>0)$ in an analogous fashion to Sec. \[subssec3\]. We obtain, with the assistance of the variable $\xi = 2 q_y d(1+\tfrac{x}{d})$, the solution $$\begin{gathered}
\label{eqsqrt01}
\psi_{I}(x) = \tfrac{c_{I}}{\sqrt{d}} \left(1+\tfrac{x}{d}\right)^{1/2} e^{- q_y x} \times \\
\left(
\begin{array}{c}
U\left( 1- \tfrac{\gamma^2}{2 q_y d}, \tfrac{3}{2}, \xi \right)
\\ \gamma^{-1} \left( 2 q_y d \right)^{1/2} U\left( \tfrac{1}{2} - \tfrac{\gamma^2}{2 q_y d}, \tfrac{3}{2}, \xi \right)
\end{array}
\right),\end{gathered}$$ where $\gamma$ is defined in Eq. and $c_{I}$ is fixed via normalization. The solution in region II $(x<0)$ is found by interchanging the top and bottom wavefunction components, and making the replacements $x \to -x$ and $c_I \to c_{II} = \pm c_I$. The eigenvalues are wholly governed by the expression $$\label{eqsqrt02}
\gamma^{-1} \left( 2qd \right)^{1/2} U\left( \tfrac{1}{2} - \tfrac{\gamma^2}{2 q d}, \tfrac{3}{2}, 2qd \right) = \pm U\left( 1- \tfrac{\gamma^2}{2 q d}, \tfrac{3}{2}, 2qd \right) ,$$ which is tractable with standard root-searching procedures. Solutions of Eq. are shown in Fig. \[fig5\] for the six lowest-lying states. As was the case for the linear barrier in Sec. \[subssec3\], the system observes the vanishing zero-point energy phenomenon as $q_y d \to 0$, which is a hallmark of algebraic velocity barriers growing at most linearly.
Upon comparing the corresponding energy level versus transverse momentum dependences in Fig. \[fig3\], Fig. \[fig4\] and Fig. \[fig5\] one notices as the velocity barrier growth becomes shallower the bound state energies lower and the inter-energy spacing reduces. As soon as the velocity barrier is growing asymptotically linearly, there is no longer a threshold effect as $q_y d \to 0$ for the zero-point energy, a characteristic which persists for sub-linear velocity barriers. These features are important for the design of few electron mode guiding devices, which require robust on and off logical states.
We should mention that we mostly considered velocity barrier models growing asymptotically. If these models were modified such that the spatial profile of $v_F(x)$ instead saturated at some large but finite value, the corresponding expressions presented in this work \[c.f. Eq. , Eq. and Eq. \] will slightly overestimate the magnitudes of the bound state energies. However, the low-lying states, which are the main focus of this work, are unaffected for all practical purposes by this lack of a saturation.
![Progression of the six lowest bound states energies with traversal momentum $q_y d$, for massless Dirac particles in a square root velocity barrier, calculated via Eq. .[]{data-label="fig5"}](fig5){width="40.00000%"}
\[sec3\]Radial velocity barriers
================================
It is worthwhile to also consider axially symmetric velocity barriers $v_F = v_F(r)$ with the Hamiltonian to try to achieve total (0D) confinement. We separate the variables in polar coordinates $(r, \theta)$ with the ansatz $\Psi (r, \theta) = (2 \pi)^{-1/2} \left[ e^{i m \theta} \psi_1(r), i e^{i (m+1) \theta} \psi_2(r) \right]^T$. Here $m=0,\pm1, \pm2...$ is related to angular momentum quantum number. Explicitly, the spinor wavefunction satisfies $J_z \Psi = (m+1/2) \Psi$, where the total angular momentum operator $J_z = -i \hbar \partial_{\theta} + \hbar \sigma_z /2$.
Now, a finite circular velocity barrier cannot trap particles (unlike the equivalent 1D case described in Sec. \[subssec1\]) because the wavefunctions are always described in terms of standard Bessel functions. These functions are non-square integrable, as was already encountered in the case of massless Dirac fermions in an electrostatic barrier [@Chaplik], since they map onto the scattering states of the Schrodinger equation. However, an infinite sharp velocity barrier can lead to bound states, as we shall see in Sec. \[subssec5\], and more significantly so too does an algebraically smooth velocity barrier model of the inverted Lorentzian type [@Downing], which is shown to be integrable in Sec. \[subssec6\]. Both of these aforementioned models exhibit a finite zero-point (ground state) energy. Furthermore, we note there is a general property that states with $m = 0, -1$ are marginally non-square-integrable in the auxiliary spinor $\Psi$ and so may correspond to extended states (rather than bound states) in the full spinor $\Phi$ depending on the asymptotics of the velocity barrier.
\[subssec5\]The infinite velocity barrier
-----------------------------------------
The simplest integrable model is the infinite velocity barrier, defined using the radial distance $R$ as $$\label{squarecircle1}
v_F(r)= \begin{cases} v_0, \quad r \le R, \\
\infty, \quad r > R. \end{cases}$$ The Hamiltonian with this sharp velocity barrier leads to a countably infinite number of bound states, described by $$\label{squarecircle2}
E_{n, m} = \pm \frac{\hbar v_0}{R} \alpha_{\sigma, n}, \quad \sigma =
\begin{cases}
m, & \text{if } m \ge 1, \\
-m-1, & \text{if } m \le -2.
\end{cases}$$ where $\alpha_{\sigma, n}$ is the $n$th positive zero of the Bessel function of the first kind, satisfying $ J_{\sigma}(\alpha_{\sigma, n}) = 0$. It is understood that the eigenvalues respect two symmetries: firstly, for every solution with $E$ there is a solution with $-E$; and secondly the eigenvalues are degenerate with the quantum number replacement $m \to -(m+1)$. The energy levels as a function of angular momentum $m$ are plotted in Fig. \[fig6\] as orange circles. How bound states with increasingly high angular momenta appear at higher energies and the threshold for the first bound state to appear, are both features which can be clearly seen. The explicit threshold energy at which the first bound state appears is $E_{1, 1} = E_{1, -2} \simeq 3.83 \hbar v_0/R$.
Notably the $m = 0, -1$ states are associated with non-square integrable auxiliary spinors $\Psi$, which is a common feature for all radial problems of this type (namely those with velocity barriers strengths tending towards infinity as $r \to \infty$). This is because these states have minimal angular momentum and so are highly susceptible to the Klein tunneling phenomenon of perfect transmission at normal incidence. Nevertheless, these extended states are associated with radial wavefunctions which do decay algebraically ($\sim 1/r$), and so may be important for studies of resonant scattering. These special modes appear at the degenerate level $E_{1, 0} = E_{1, -1} \simeq 2.40 \hbar v_0/R$ in the infinite velocity barrier.
![(Color online) Energy levels as a function of $m$ for the infinite velocity barrier (orange circles) calculated via Eq. ; and the inverted Lorentzian velocity barrier (green triangles) calculated via Eq. .[]{data-label="fig6"}](fig6){width="40.00000%"}
\[subssec6\]The inverted Lorentzian velocity barrier
----------------------------------------------------
Let us now consider a smooth, algebraic model given by the inverted Lorentzian $$\label{eqsquare00}
v_F(r)= v_0 (1+r^2/R^2).$$ Working in the variable $\xi = r^2/R^2$, one can construct the radial part of the spinor wavefunction $\psi(r)$ in terms of Gauss hypergeometric functions $$\begin{gathered}
\label{eqsquare01}
\psi(r) = \tfrac{c}{R} \left(1+\xi\right)^{-p_m/2} \times \\
\left(
\begin{array}{c}
\xi^{\tfrac{|m|}{2}} {}_2 F_1 \left( \left[\tfrac{p_m}{2} + \tfrac{E R}{2\hbar v_0} \right], \left[\tfrac{p_m}{2} - \tfrac{E R}{2\hbar v_0} \right], 1+|m|, \tfrac{\xi}{1+\xi} \right)
\\ \Upsilon \xi^{\tfrac{|1+m|}{2}} {}_2 F_1 \left( \left[\tfrac{p_m}{2} + \tfrac{E R}{2\hbar v_0} \right], \left[\tfrac{p_m}{2} - \tfrac{E R}{2\hbar v_0} \right], p_m-|m|, \tfrac{\xi}{1+\xi} \right)
\end{array}
\right)\end{gathered}$$ where $c$ is a normalization constant and where we have made use of the following number $$\label{eqsquare023434}
p_m = 1 + |m| + |1+m|,$$ and with the prefactor $\Upsilon = \tfrac{1}{2(1+m)} \tfrac{E R}{\hbar v_0}$ when $m\ge0$ and $\Upsilon = 2m \tfrac{\hbar v_0}{E R}$ when $m<0$ respectively. One notices the low angular momentum states $m=0, -1$ are (marginally) non-square integrable in the auxiliary spinors $\Psi$ from the $r \to \infty$ asymptotics of the radial wavefunction, as was foreshadowed in Sec. \[subssec5\]. These extended states, which reside at the degenerate level $E_{0, 0} = E_{0, -1} = \pm 2 \hbar v_0/R$, are fully normalizable in the full spinor $\Phi$ due to the presence of the function Eq. .
Terminating the hypergeometric function in Eq. by setting either of the first two arguments of the hypergeometric function to be a negative integer or zero, one finds the eigenvalues of the system $$\label{eqsquare02333333333}
E_{n, m} = \pm \left( 2n + p_m \right) \frac{\hbar v_0}{R}, \quad n=0,1,2...$$ Fig. \[fig6\] displays as green triangles the energy levels as a function of $m$. Most notable is the threshold bound state energy of $E_{0, 1} = E_{0, -2} = \pm 4 \hbar v_0/R$ at which the first confined mode with $m \ne 0, -1$ is found. This confirms a finite zero-point energy arises even for this smooth, algebraically growing velocity barrier model and is not an artefact of the minimal model of Sec. \[subssec5\].
These proposals for truly bound states in radial velocity barriers joins a small list of setups which can confine massless Dirac fermions in quantum-dot-like systems. Recent experimental work has focused on combined electric and magnetic fields [@Freitag] and electron whispering gallery modes [@ZhaoY], rather than Fermi velocity-induced effects.
\[subssec66\]Zero-energy states in radial velocity barriers
-----------------------------------------------------------
Zero-energy states associated with Dirac Hamiltonians are of great interest due to their importance for topological and Majorana physics. Radial velocity barriers also admit zero-energy state ($E=0$) solutions, which form the degenerate ground state of the system. We consider velocity barriers with the short range behavior $v_F(r \sim 0) \sim r^0$, such that the spinor solutions to the eigenproblem for nonnegative $m$ take the form $$\begin{gathered}
\label{eqzero01}
\Phi \propto
\left(
\begin{array}{c}
e^{i m \theta} r^m / \sqrt{v_F(r)}
\\ 0
\end{array}
\right), \quad m = 0, 1, 2...\end{gathered}$$ In order to be a normalizable solution for a certain state $m$, the velocity barrier must grow asymptotically faster than $v_F(r \to \infty) \sim r^{2 (1+m)}$, limiting the degeneracy of the ground state. Meanwhile, the eigenvector for negative $m$ is given by $$\begin{gathered}
\label{eqzero02}
\Phi \propto
\left(
\begin{array}{c}
0
\\ e^{i (m+1) \theta} r^{-(m+1)} / \sqrt{v_F(r)}
\end{array}
\right), ~ m = -1, -2, -3...\end{gathered}$$ which is square integrable for a state with quantum number $m$ as long as the velocity barrier grows at large distances more rapidly than $v_F(r \to \infty) \sim r^{-2 m}$. These wavefunctions and display the chiral property of total suppression of the electronic probability density on one of the the two sublattices, dependent on the sign of the angular momentum $m$.
\[conc\]Conclusion
==================
We have studied the appearance and nature of bound states of 2D massless Dirac fermions, which is a nontrivial task due to the phenomena of Klein tunneling, that may arise in several different velocity barrier configurations, including trench-like and radial geometries. We have shown how velocity barrier channels growing linearly or sub-linearly support bound modes for arbitrarily small transversal wavevectors, whereas algebraically faster (or indeed exponentially) growing barriers posses a finite zero-point energy. This geometry is a candidate to observe the ballistic guiding of few-mode electronic waves. In contrast, a radial velocity barrier growing algebraically has been shown to have a threshold energy at which the first bound state appears, as has a simple circular radial barrier model. Extended states with quantum number $m = 0, -1$ are not always square-integrable but do decay algebraically and so may be consequential for resonant scattering.
These results open up an intriguing avenue to explore in the ongoing quest to achieve trapping and guiding of massless Dirac particles [@Lee2016; @Gutierrez2016]. With the ongoing improvements in Fermi velocity engineering, particularly via the fabrication of Dirac materials embedded in various substrates [@Hwang2012] and controllably strained devices [@Downs2016], we hope that velocity waveguides and traps, as well as the predicted threshold behavior and confinement-deconfinement transitions of the bound states, can be demonstrated in the laboratory in the near future.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge financial support from the CNRS, as well as the EU H2020 RISE project CoExAN (Grant No. H2020-644076), EU FP7 ITN NOTEDEV (Grant No. FP7-607521), and the FP7 IRSES projects CANTOR (Grant No. FP7-612285), QOCaN (Grant No. FP7-316432), and InterNoM (Grant No. FP7-612624). We would like to thank Jenny Zhao for several illuminating discussions.
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abstract: 'The COSY-11 collaboration has measured the total cross section for the $pp\rightarrow pp\eta^{\prime}$ and $pp\rightarrow pp \eta$ reactions in the excess energy range from Q = 1.5 MeV to Q = 23.6 MeV and from Q = 0.5 MeV to Q = 5.4 MeV, respectively. Measurements have been performed with the total luminosity of 73 nb$^{-1}$ for the $pp \rightarrow pp\eta$ reaction and 1360 nb$^{-1}$ for the $pp \rightarrow pp \eta^{\prime}$ one. Recent results are presented and discussed.'
address: |
$^{(a)}$ Institute of Physics, Jagellonian University, PL–30059 Cracow, Poland\
$^{(b)}$ IKP, Westf. Wilhelms-Universität, Wilhelm-Klemm-Stra[ß]{}e 9, D–48149 M[ü]{}nster, Germany\
$^{(c)}$ Institute of Nuclear Physics, ul. Radzikowskiego 152, PL-31-342 Cracow, Poland\
$^{(d)}$ Institut für Kernphysik, Forschungszentrum Jülich, D–52425 J[ü]{}lich, Germany\
$^{(e)}$ IUCF, Milo B. Samson Lane, Bloomington, IN 47405, USA\
$^{(f)}$ ZEL, Forschungszentrum J[ü]{}lich, D–52425 J[ü]{}lich, Germany\
$^{(g)}$ Institute of Physics, Silesian University, PL-40-007 Katowice, Poland\
$^{(h)}$ Egyptian Atomic Energy Authority, 101 Sharia Kaser El-Aini, 13759 Cairo, Egypt\
author:
- 'P. Moskal$^{(a)}$, H. -H. Adam$^{(b)}$, J. T. Balewski$^{(c,d,e)}$, A. Budzanowski$^{(c)}$, C. Goodman$^{(e)}$, D. Grzonka$^{(d)}$, L. Jarczyk$^{(a)}$, M. Jochmann$^{(f)}$, A. Khoukaz$^{(b)}$, K. Kilian$^{(d)}$, P. Kowina$^{(g)}$, M. Köhler$^{(f)}$, T. Lister$^{(b)}$, W. Oelert$^{(d)}$, C. Quentmeier$^{(b)}$, R. Santo$^{(b)}$, G. Schepers$^{(b,d)}$, U. Seddik$^{(h)}$, T. Sefzick$^{(d)}$, S. Sewerin$^{(d)}$, J. Smyrski$^{(a)}$, A. Strza[ł]{}kowski$^{(a)}$, M. Wolke$^{(d)}$, P. W[ü]{}stner$^{(f)}$'
date:
title: ' Heavy Meson Production at COSY - 11 '
---
=-1.cm
Introduction {#introduction .unnumbered}
============
The word heavy used in the title requires a short explanation. The reason is rather historical, and it seems now that heavy are all mesons but not pions. The talk will concern the production of the $\eta$ and $\eta^{\prime}$ mesons, and since $\eta^{\prime}$ is even heavier than $\eta$ the discussion concerning this meson will constitute the major part of the presentation.
Last year, for the first time total cross sections for the production of the $\eta^{\prime}$ meson in the collision of protons close to the reaction threshold have been published [@hiboupl; @moskalprl]. Two independent experiments performed at the accelerators SATURNE and COSY have delivered consistent results.
The first remarkable inference derived from these experiments was that the total cross sections for the $pp \rightarrow pp\eta^{\prime}$ reaction are by about a factor of fifty smaller than the cross sections for the $pp \rightarrow pp\eta$ reaction at the corresponding values of the excess energy. Trying to explain this large difference Hibou et al. [@hiboupl] showed that the one-pion-exchange model with the parameters adjusted to fit the total cross section for the $pp \rightarrow pp\eta$ reaction underestimate the $\eta^{\prime}$ data by about a factor of two. This discrepancy suggests that short-range production mechanisms as for example heavy meson exchange, mesonic currents [@nakayama], or more exotic processes like the production via a fusion of gluons [@bass] may contribute significantly in the creation of $\eta$ and $\eta^{\prime}$ mesons [@wilk]. Especially that the momentum transfer required to create these mesons is much larger compared to the pion production, and already in case of pions a significant contribution from the short-range heavy meson exchange is necessary in order to obtain agreement with the experiments [@haiden; @horowitz].
The second interesting observation was that the energy dependence of the total cross section for the $pp \rightarrow pp\eta$ and $pp \rightarrow pp\eta^{\prime}$ reactions does not follow the predictions based on the phase space volume and the proton-proton final state interaction, which is the case in the $\pi^{0}$ meson production [@meyer90; @meyer92]. Moreover, for $\eta$ and $\eta^{\prime}$ mesons the deviation from this prediction were qualitatively different. Namely, the close to threshold cross sections for the $\eta$ meson are strongly enhanced compared to the model comprising only the proton-proton interaction [@caleneta] in contrary to the observed suppression in the case of the meson $\eta^{\prime}$. The energy dependence of the total cross section for the $pp \rightarrow pp\eta$ reaction can be, however, explained when the $\eta$-proton attractive interaction is taken into account [@ulf95; @moalem1]. Albeit $\eta$-proton interaction is much weaker than the proton-proton one (compare scattering length a$_{p\eta} = 0.751$ fm + $i$ 0.274 fm [@greenwycech] with a$_{pp} = -7.83$ fm [@naisse]) it becomes important through the interference terms between the various final pair interactions [@moalem1]. By analogy, the steep decrease of the total cross section when approaching a kinematical threshold for the $pp \rightarrow pp\eta^{\prime}$ reaction could have been explained assuming a repulsive $\eta^{\prime}$-proton interaction [@moskalacta; @baruej]. This interpretation, however, should rather be excluded now in view of the new COSY-11 data which will be presented in the next chapters.
Possible production mechanisms {#possible-production-mechanisms .unnumbered}
==============================
The theoretical studies of the mechanisms accounting for the $\pi^{0}$ and $\eta$ mesons creation in the close to threshold $pp\rightarrow pp\pi^{0}(\eta)$ reactions have shown that the short-range component of the N-N force and the off-shell pion rescattering dominate the production process of the $\pi^{0}$ meson [@haiden; @horowitz; @hernandez], whereas the $\eta$ meson is predominantly produced through the excitation of the intermediate baryonic resonance [@faldtwilk1; @germondwilk; @laget; @moalem; @caleneta]. However, the comparison of the experimentally determined $\eta$ and $\eta^{\prime}$ total cross section ratio with the predictions based on the one-pion-exchange model indicates that we are still far from the full understanding of the dynamics of the discussed processes. In particular, at present there is not much known about the relative contribution of the possible reaction mechanisms to the production of the meson $\eta^{\prime}$. It is expected that similarly as in the case of pions the $\eta^{\prime}$ meson can be produced as depicted in Figures \[grafy\]a,b,c,d. However, because of the much larger four-momentum transfer, short-range mechanisms, like heavy meson exchange (Figure \[grafy\]c) or depicted in Figure \[grafy\]e production via a mesonic current, where the $\eta^{\prime}$ is created in a fusion of exchanged virtual $\omega$, $\rho$, or $\sigma$ mesons shell contribute even more significantly. Recently Nakayama et al. [@nakayama], studied contributions from the [*nucleonic* ]{} (Fig. \[grafy\]b), [*nucleon resonance* ]{} (Fig. \[grafy\]d), and [*mesonic* ]{} (Fig. \[grafy\]e) currents and found that each one separately could describe the absolute values and energy dependence of the close to threshold $\eta^{\prime}$ data points [@hiboupl; @moskalprl] after an appropriate adjustment of the ratio of the pseudoscalar to the pseudovector coupling. This rather pessimistic conclusion means that it is not possible to judge about the mechanisms responsible for the $\eta^{\prime}$ meson production from the total cross section alone.
Moreover, the possible gluonium admixture in the meson $\eta^{\prime}$ makes the study even more complicated but certainly also more interesting. Figure \[grafy\]f depicts appropriate short-range mechanism which may lead to the creation of the flavour singlet state via a fusion of gluons emitted from the exchanged quarks of the colliding protons [@kolacosynews]. Albeit the quark content of $\eta$ and $\eta^{\prime}$ mesons is very similar, this manner of the production should contribute primarily in the creation of the meson $\eta^{\prime}$. This is due to the small pseudoscalar mixing angle ($\Theta_{PS} \approx -15^{o}$) [@bram97] which implies that the $\eta^{\prime}$ meson is predominantly a flavour singlet state and is expected to contain a significant admixture of gluons. Further, it is almost two times heavier than $\eta$ and hence its creation requires much larger momentum transfer which is more probable to be realized in the short-range interactions. Unfortunately, at present there are no theoretical calculations concerning this mechanism. Now, since the effective coupling constant describing the $\eta^{\prime}$-proton-proton vertex is not known, it is even not possible to determine the contribution from the simplest possible production mechanism where the $\eta^{\prime}$ is supposed to be emitted as a bremsstrahlung radiation from one of the colliding protons as it is shown in Figure \[grafy\]a. Therefore, investigations of the $\eta^{\prime}$ production have to deal with a few problems at the same time. Namely: unknown reaction mechanism, unknown coupling constant, and unknown proton-$\eta^{\prime}$ interaction. In the next section the present status of the knowledge about the effective NN$\eta^{\prime}$ coupling constant will be given.
$NN\eta^{\prime}$ coupling constant {#couplconstsec .unnumbered}
-----------------------------------
In the effective Lagrangian approach [@zhan95; @mukh96] the strength of the nucleon-$\eta^{\prime}$ coupling is driven by the the $NN\eta^{\prime}$ coupling constant $g_{NN\eta^{\prime}}$, which comprises the information about the structure of the $\eta^{\prime}$ meson and the nucleon. The knowledge of the coupling constant is necessary in the calculation of the production cross section if one considers the Feynman diagrams as illustrated in Figure \[grafy\]. The main difficulty in the determination of this quantity is due to the fact that usually the direct production on the nucleon is either associated with the production through baryonic resonances, as in the case of the $\gamma p \rightarrow \eta^{\prime} p$ reaction [@ploetzke], or with the exchange of other mesons. Therefore, if the direct production mechanism is not dominant it is not possible to extract the $NN\eta^{\prime}$ coupling without the clear understanding of the other mechanisms. However, it would be very interesting to determine the $g_{NN\eta^{\prime}}$ coupling constant and to compare it with the calculations performed on the quark level assuming the $\eta^{\prime}$ meson structure. First theoretical considerations concerning this issue have been published last year [@lehmann].
Assuming that the $\eta$ and $\eta^{\prime}$ mesons are mixtures of the SU(3) singlet and octet states, one can relate the $NN\eta$ and $NN\eta^{\prime}$ coupling constants by the following equation [@zhan95; @moskalphd]: $$g_{NN\eta^{\prime}} = \frac{sin\Theta + \sqrt{2}cos\Theta}
{cos\Theta - \sqrt{2}sin\Theta} \cdot g_{NN\eta}
\stackrel{\Theta=-15.5^{\circ}}
{=\!\!\!=\!\!\!=\!\!\!=\!\!\!=\!\!\!
=\!\!\!=\!\!\!=\!\!\!=\!\!\!=\!\!\!
=\!\!\!=\!\!\!=\!\!\!=\!\!\!=\!\!\!
=\!\!\!=\!\!\!=\!\!\!=} 0.82 \cdot g_{NN\eta}.
\label{getaetap}$$ The measurements of the $\gamma p \rightarrow p \eta$ [@benm95; @benm91] reaction have yielded that: $ 0.2~\leq g_{NN\eta}~\leq~6.2,
$ whereas the comparison of the $\pi^{-}p\rightarrow \eta n$ and $\pi^{-}p\rightarrow \pi^{0} n$ reaction cross sections implies [@benm95]: $ 5.7~\leq~g_{NN\eta}~\leq~9.0
$. The above inequalities and equation \[getaetap\] lead to the following range for the $g_{NN\eta^{\prime}}$ value: $ {\bf 4.7~\leq~g_{NN\eta^{\prime}}~\leq~5.1},
\label{getapvalue}
$ which is to be compared to the $\eta^{\prime}$ coupling determined from the fits to low energy nucleon-nucleon scattering in the one-boson-exchange models amounting to $g_{NN\eta^{\prime}}=7.3$ [@nagels].
On the other hand, the $g_{NN\eta^{\prime}}$ coupling constant determined via dispersion methods [@greinkroll] turns out to be smaller than 1, $ g_{NN\eta^{\prime}} < 1 $, which is in contradiction to the above estimations.
The $g_{NN\eta^{\prime}}$ coupling constant is also related to the issue of the total quark contribution to the proton spin ($\Delta\Sigma$). The approximate equation derived in reference [@efre90] reads: $ \Delta\Sigma = \Delta u + \Delta d + \Delta s
=\frac{\sqrt{3} f_{\eta^{\prime}}}{2M} g_{NN\eta^{\prime}},
\label{sigmacoupling}
$ where, $f_{\eta^{\prime}}\approx 166~MeV$ [@efre90] denotes the $\eta^{\prime}$ decay constant and M the proton mass. $\Delta u$, $\Delta d$ and $\Delta s$ are the contributions from up, down and strange quarks, respectively [^1]. The total contribution of the quarks to the proton spin amounts to $\Delta\Sigma=0.38^{+0.09}_{-0.10}$ [@SMC]. Applying this value in the above equation one obtains $ g_{NN\eta^{\prime}} = 2.48^{+0.59}_{-0.65} $, which is consistent with the upper limit $(g_{NN\eta^{\prime}}~\leq~2.5)$ set from the comparison of the measured total cross section values for the $pp \rightarrow pp\eta^{\prime}$ reaction with the calculations based on the effective Lagrangian approach, where only a direct production has been considered [@moskalprl].
The present estimations for $g_{NN\eta^{\prime}}$ inferred from different experiments are widely spread from 0.2 to 7.3 and are not consistent with each others. Therefore more effort is needed on experimental as well as theoretical side to fix this important parameter.
The COSY - 11 Experiment {#the-cosy-11-experiment .unnumbered}
========================
The experiments were performed at the cooler synchrotron COSY-J[ü]{}lich [@maie97] which accelerates protons up to a momentum of $3500~MeV/c$. The threshold momenta for the $pp \rightarrow pp\eta$ and $pp \rightarrow pp\eta^{\prime}$ reactions are equal to ${\bf 1981.6~MeV/c}$ and ${\bf 3208.3~MeV/c}$, respectively. About $2\cdot 10^{10}$ accelerated protons circulate in the ring passing $~1.6\cdot 10^{6}$ times per second through the $H_{2}$ cluster target [@domb97; @khou96] installed in front of one of the dipole magnets, as depicted schematically in Figure \[tarczawiazka\]. \[aabc\] The target is realized as a beam of $H_{2}$ molecules grouped inside clusters of up to $10^{5}$ atoms.
At the intersection point of the cluster beam with the COSY proton beam the collision of protons may result for example in the production of the $\eta^{\prime}$ meson. The ejected protons of the $pp\rightarrow pp \eta(\eta^{\prime})$ reaction, having smaller momenta than the beam protons, are separated from the circulating beam by the magnetic field. Further they leave the vacuum chamber through a thin exit foil and are registered by the detection system consisting of drift chambers and scintillation counters as depicted in Figure \[tarczawiazka\].
The measurement of the track direction by means of the drift chambers, and the knowledge of both the dipol magnetic field and the target position allow to reconstruct the momentum vector for each registered particle. The time of flight measured between the S1 (S2) - and the S3 scintillators gives the particle velocity. Having momentum and velocity for each particle one can calculate its mass, and hence identify it.
In the first step of the data analysis events with two tracks in drift chambers were preselected, and the mass of each particle was evaluated. Figure \[invmass\] shows the squared mass of two simultaneously detected particles. A clear separations is seen into groups of events with two protons, two pions, proton and pion and also deuteron and pion. Thus, this spectrum allowed for a software selection of events with two registered protons.
The knowledge of the momenta of both protons before and after the reaction allows to calculate the mass of an unobserved particle or system of particles created in the reaction. Figure \[miss\]a depicts the missing mass spectrum obtained for the $pp \rightarrow pp X$ reaction at the excess energy value of Q = 5.8 MeV above the $\eta^{\prime}$ meson production threshold. Most of the entries in this spectrum originate in the multi-pion production, forming a continuous background to the well distinguishable peaks accounting for the $\omega$ and $\eta^{\prime}$ mesons production, which can be seen at mass values of 782 MeV/c$^{2}$ and 958 MeV/c$^{2}$, respectively. The signal of the $pp\rightarrow pp\eta^{\prime}$ reaction is better to be seen in the Figure \[miss\]b, where the missing mass distribution only in the vicinity of the kinematical limit is presented. Figure \[miss\]c shows the missing mass spectrum for the measurement at Q = 7.6 MeV together with the multi-pion background as combined from the measurements at different excess energies [@moskalnewrep]. Subtraction of the background leads to the spectrum with a clear peak at the mass of the meson $\eta^{\prime}$ as shown by the solid line in Figure \[miss\]d. The dashed histogram in this figure corresponds to the Monte-Carlo simulations where the beam and target conditions were deduced from the measurements of the elastically scattered protons [@moskalnewrep].
1.0cm
(12.2,15.0) (-0.5,0.0)[ ]{} (5.0,3.7)[ ]{} (7.0,0.0)[ ]{} (12.5,3.7)[ ]{} (-0.5,5.0)[ ]{} (5.0,8.7)[ ]{} (7.0,5.0)[ ]{} (12.5,8.7)[ ]{} (-0.5,10.0)[ ]{} (1.0,13.8)[ ]{} (7.0,10.0)[ ]{} (12.5,13.8)[ ]{}
The scale of the simulated distribution was adjusted to fit the data, but the consistency of the widths is a measure of understanding of the detection system and the target-beam conditions. Histograms from a measurement at Q = 1.5 MeV shown in Figures \[miss\]e,f demonstrate the achieved accuracy at the COSY-11 detection system. The width of the missing mass distribution (Fig. \[miss\]f), which is now close to the natural width of the $\eta^{\prime}$ meson ($\Gamma_{\eta^{\prime}}=0.203$ MeV [@pdb98]), is again well reproduced by the Monte-Carlo simulations.
Results {#results .unnumbered}
=======
Total cross section {#total-cross-section .unnumbered}
-------------------
Determination of (a) number of the produced $\eta^{\prime}$ events from the presented above missing mass distributions, (b) luminosity from the simultaneous measurements of the elastically scattered protons, and (c) detection system acceptance by means of the Monte-Carlo simulations allows for the calculations of the total cross section for the $pp\rightarrow pp\eta^{\prime}$ reaction. The total cross section for the $pp\rightarrow pp\eta$ reaction was determined by the same method, however, with a much bigger signal to background ratio (40/1) due to the larger total cross section values [@smyrski].
Figure \[etap-eta\] shows the compilation of the total cross sections for the $\eta$ and $\eta^{\prime}$ meson production together with the new COSY-11 data shown as filled squares ($\eta$) and filled circles ($\eta^{\prime}$). The COSY-11 data on the $\eta$ production were taken changing continously during the measurement cycle a momentum of the uncooled proton beam. This technique allowed for the precise determination of the total cross section energy dependence near the kinematical threshold. The obtained result confirmed the enhancement of the close to threshold total cross section values compared to the predictions based on the phase space factors and the proton-proton FSI which was earlier observed by the PINOT [@pinot], WASA [@caleneta], and SPES III [@hiboupl; @bergdolt] collaborations.
New COSY - 11 data concerning $\eta^{\prime}$ meson production are shown in Figure \[etap-eta\] as filled circles. These measurements on the $pp \rightarrow pp\eta^{\prime}$ reaction were performed with the stochastically cooled proton beam and the integrated luminosity of 1360 nb$^{-1}$. Statistical and systematical errors are separated by dashes. The systematical error of the energy equals to 0.44 MeV constitutes of the 0.3 MeV due to the uncertainty in the detection system [@moskalabsolut] and 0.14 MeV due to the uncertainty in the $\eta^{\prime}$ meson mass [@pdb98]. The systematical error of the cross section values, including the overall normalization uncertainty, amounts to 15 $\%$ [@moskalprl; @moskalphd]. It is worth to stress again, that SPES III and COSY - 11 results obtained at different laboratories are in a perfect agreement.
The dashed-dotted line in Figure \[etap-eta\] shows the energy dependence predicted by Fäldt and Wilkin [@faldtwilk] normalized now to the COSY-11 data points, and the solid line corresponds to the predictions based on a one-pion-exchange model adjusted to fit the close to threshold $pp\rightarrow pp\eta$ data (dashed line) [@hiboupl]. The factor two discrepancy suggests that the short-range mechanisms may play a prominent role in the production of these mesons [@wilk; @bass]. However, recent calculations performed by Nakayama et al. [@nakayama] indicate that the determination of the total cross section close to threshold is surely not sufficient to establish the contributions from different mechanisms to the overall production amplitude. Specifically, the primary production amplitude for processes studied by these authors (Fig. \[grafy\]b,d,e) does not change significantly within the present experimental accuracy for the excess energies below Q = 30 MeV. Therefore, the energy dependence of the total cross section for Q $\leq$ 30 MeV should be quite well described by the integral of the phase space volume weighted by the squared amplitude of the final state interaction among the outgoing particles. And indeed, as shown in Figure \[etap-fsi\], the data are in a good agreement with this model even without considering the $\eta^{\prime}$-proton interaction. This leads to the conclusion that the $\eta^{\prime}$-proton interaction is too weak to influence considerably, within the experimental error bars, the total cross section energy-dependence.
Primary production amplitudes {#primary-production-amplitudes .unnumbered}
-----------------------------
The cross section for the reaction $pp \rightarrow ppX$ can be expressed as: $$\sigma_{pp\rightarrow ppX}=
\frac{\displaystyle \int phase~space \cdot |M_{pp\rightarrow ppX}|^{2} }
{flux~factor},
\label{eqcrossform}$$ where, $M_{pp\rightarrow ppX}$ denotes the transition matrix element for the $pp\rightarrow ppX$ reaction, and X stands for $\pi^{0},\eta$ or $\eta^{\prime}$ mesons. In analogy with the [*Watson-Migdal*]{} approximation [@wats52] for two body processes, it can be assumed that the complete transition amplitude of a production process $M_{pp\rightarrow ppX}$ factorizes approximately as [@moalem1]: $$M_{pp \rightarrow ppX } \approx M_{0} \cdot M_{FSI}$$ where, $M_{0}$ accounts for all possible production processes, and $M_{FSI}$ describes the elastic interaction of protons and X meson in the exit channel. Making further assumptions that only the proton-proton interaction is present in the exit channel ($ M_{FSI} = M_{pp\rightarrow pp} $) and that the primary production amplitude does not change with the excess energy, it is possible to calculate $|M_{0}|$. The enhancement from the proton-proton interaction, $|M_{pp\rightarrow pp}|^{2}$, was estimated as an inverse of the squared Jost function, with Coulomb interaction being taken into account [@druzhinin]. The $|M_{pp\rightarrow pp}|^{2}$ is a dimensionless factor which turns to zero with vanishing relative protons momentum k, peaks sharply at k$\approx$25 MeV/c and approaches asymptotically unity for large proton-proton relative momenta.
1.0cm
(12.2,12.0) (-0.5,0.0)[ ]{} (6.8,0.0)[ ]{} (3.0,6.2)[ ]{}
Figure \[m0\] compares the extracted absolute values for the modulus of the primary production amplitude for the near-threshold production of the $\eta$, $\pi^{0}$ and $\eta^{\prime}$ mesons. The quantity $|M_{0}|$ is normalized to unity at the point of highest excess energy, for each meson separately. If the performed assumptions in the derivation of $|M_{0}|$ were fulfilled the obtained values would be equal to one as depicted by the solid line. It can be seen, however, that in the case of the $\eta$ meson, $|M_{0}|$ grows with decreasing excess energy reflecting attractive $\eta$-proton interaction. In the data for the $\pi^{0}$ production, apart from the two closest-to-threshold points[^2], one can notice a tiny grow of $|M_{0}|$ when the excess energy decreases from Q = 20 MeV to Q = 2 MeV. This may be cause by the small $\pi$-proton interaction. The deviation from the constant is much smaller than in the $\eta$ meson case since, the S-wave $\pi$-proton interaction is much weaker than the $\eta$-proton one. Similarly, neglecting the two lowest points for the $\eta^{\prime}$ meson, one observes about 20 $\%$ increase of $|M_{0}|$ when approaching the threshold. This may indicate a small attractive $\eta^{\prime}$-proton interaction. Anyhow, with the new COSY - 11 data points the possible $\eta^{\prime}$-proton repulsive interaction must be excluded.
Instead of conclusion the article of Bernard et al. [@bernard] is recommended where the threshold matrix-elements for the $pp\rightarrow pp\pi^{0}(\eta,\eta^{\prime})$ reactions were evaluated in a fully relativistic Feynman diagrammatic approach as reported by N. Kaiser at this conference.
Acknowledgments {#acknowledgments .unnumbered}
===============
P.M. acknowledges the hospitality and financial support from the Forschungszentrum J[ü]{}lich and the Foundation for Polish Science.
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[^1]: Contribution of quarks heavier than the $strange$ quark are usually not considered, but I. Halperin and A. Zhitnitsky suggested [@halp97] that the intrinsic charm component of proton may also carry a significant amount of the proton spin. The quark and gluon contributions to the proton spin are widely discussed in the literature [@efre90; @SMC; @SMC2; @SMC3; @elli96; @shor90; @hugh88; @ashm88] based on measurements of the spin asymmetries in deep-inealstic scattering of polarised muons on polarised protons.
[^2]: Due to the steep falling of the total cross section near-threshold already a small change of the energy (0.2 MeV) lifts the points significantly up. Moreover, for the very low energies nuclear and Coulomb scattering are expected to compete. The limit is approximately at 0.8 MeV of the proton energy in the rest system of the other proton, where the Coulomb penetration factor C$^{2}$ is equal to 0.5 [@jackson]. Thus, one should be careful, at small excess energies, where the approximately treated Coulomb interaction dominates.
|
---
abstract: |
Motivated by robotic surveillance applications, this paper studies the novel problem of maximizing the return time entropy of a Markov chain, subject to a graph topology with travel times and stationary distribution. The return time entropy is the weighted average, over all graph nodes, of the entropy of the first return times of the Markov chain; this objective function is a function series that does not admit in general a closed form.
The paper features theoretical and computational contributions. First, we obtain a discrete-time delayed linear system for the return time probability distribution and establish its convergence properties. We show that the objective function is continuous over a compact set and therefore admits a global maximum; a unique globally-optimal solution is known only for complete graphs with unitary travel times. We then establish upper and lower bounds between the return time entropy and the well-known entropy rate of the Markov chain. To compute the optimal Markov chain numerically, we establish the asymptotic equality between entropy, conditional entropy and truncated entropy, and propose an iteration to compute the gradient of the truncated entropy. Finally, we apply these results to the robotic surveillance problem. Our numerical results show that, for a model of rational intruder over prototypical graph topologies and test cases, the maximum return time entropy chain performs better than several existing Markov chains.
author:
- 'Xiaoming Duan, , Mishel George, Francesco Bullo, [^1] [^2]'
bibliography:
- 'alias.bib'
- 'Main.bib'
- 'FB.bib'
title: |
**Markov Chains with Maximum Return Time Entropy\
for Robotic Surveillance**
---
Introduction
============
#### Problem description and motivation {#problem-description-and-motivation .unnumbered}
Given a Markov chain, the first return time of a given node is the first time that the random walker returns to the starting node; this is a discrete random variable with infinite support and whose randomness is measured by its entropy. In this paper, given a strongly connected directed graph with integer-valued travel times (weights) and a prescribed stationary distribution, we study Markov chains with maximum return time entropy. Here the return time entropy of a Markov chain is a weighted average of the entropy of different states’ return times with weights equal to the stationary distribution.
This optimization problem is motivated by robotic applications. We design stochastic surveillance strategies with an entropy maximization objective in order to thwart intruders who plan their attacks based on observations of the surveillance agent. The randomness in the first return time is desirable because an intelligent intruder observing the inter-visit times of the surveillance agent is confronted with a maximally unpredictable return pattern by the surveillance agent.
#### Literature review {#literature-review .unnumbered}
Ekroot *et al.* studied the entropy of Markov trajectories in [@LE-TMC:93], i.e., the entropy of paths with specified initial and final states. The authors establish an equivalence relationship between the entropy of return Markov trajectories (paths with the same initial and final states) and the entropy rate of the Markov chains. Compared with [@LE-TMC:93], we study here the return time random variable, by lumping return trajectories with the same length. Importantly, our formulation incorporates travel times, as motivated by robotic applications.
The problem of designing robotic surveillance strategies has been widely studied [@NA-SK-GAK:08; @SA-EF-SLS:14; @HX-BF-FF-BD-AP-MT-MD-FW-AR-MN-JM:17; @JY-SK-DR:15]. Stochastic surveillance strategies, which emphasize the unpredictability of the movement of the patroller, are desirable since they are capable of defending against intelligent intruders who aim to avoid detection/capture. One of the main approaches to the design of robotic stochastic surveillance strategies is to adopt Markov chains; e.g., see the early reference [@JG-JB:05] and the more recent [@BA-SDB:15; @SB-SJC-FYH:17; @GC-AS:11; @NN-AR-JVH-VI:16]. Srivastava *et al.* [@KS-DMS-MWS:09] justified the Markov chain-based stochastic surveillance strategy by showing that for the deterministic strategies, in addition to predictability, it is also hard to specify the visit frequency. However, for the finite state irreducible Markov chains, the visit frequency is embedded naturally in the stationary distribution. Patel *et al.* [@RP-PA-FB:14b] studied the Markov chains with minimum weighted mean hitting time where weights are travel times on edges. For the class of reversible Markov chains, they formulated the problem as a convex optimization problem. An extension of the mean hitting time to the multi-agent case was studied in [@RP-AC-FB:14k]. Asghar *et al.* [@ABA-SLS:16] introduced different intruder models and designed a pattern search-based algorithm to solve for a Markov chain that minimizes the expected reward of the intruders. Recently, George *et al.* [@MG-SJ-FB:17b] studied and quantified the unpredictability of the Markov chains and designed the maxentropic surveillance strategies by maximizing the entropy rate of Markov chains [@EA-NCM:14; @YC-TG-MP-AT:17a]. Compared with [@MG-SJ-FB:17b], our problem formulation features a new notion of entropy, a directed graph topology, and travel times; these three features render the results potentially more widely applicable and more relevant (see also the performance comparison among multiple Markov chains later in the paper).
#### Contributions {#contributions .unnumbered}
In this paper, we propose a new metric that measures the unpredictability of the Markov chains over a directed graph with travel times. This novel formulation is of interest in the general study of Markov chains as well as for its applications to robotic surveillance. The main contributions of this paper are sixfold. First, we introduce and analyze a discrete-time delayed linear system for the return time probabilities of the Markov chains. This system incorporates integer-valued travel times on the directed graph. Second, we propose to characterize the unpredictability of a Markov chain by the return time entropy and formulate an entropy maximization problem. Third, we prove the well-posedness of the return time entropy maximization problem, i.e., the objective function is continuous over a compact set and thus admits a global maximum. For the case of unitary travel times, we derive an upper bound for the return time entropy and solve the problem analytically for the complete graph. Fourth, we compare the return time entropy with the entropy rate of Markov chains; specifically, we prove that the return time entropy is lower bounded by the entropy rate and upper bounded by the number of nodes times of the entropy rate. Fifth, in order to compute Markov chains with maximum return time entropy numerically, we truncate the return time entropy and show that the truncated entropy is asymptotically equivalent to both the original objective and the practically useful conditional return time entropy. We also characterize the gradient of the truncated return time entropy and use it to implement a gradient projection method. Sixth, we apply our solution to different prototypical robotic surveillance scenarios and test cases and show that, for a model of rational intruder, the Markov chain with maximum return time entropy outperforms several existing Markov chains.
#### Paper organization {#paper-organization .unnumbered}
This paper is organized as follows. We formulate the return time entropy maximization problem in Section \[sec:ProblemFormulation\]. We establish the properties of the return time entropy in Section \[sec:properties\]. The approximation analysis and the gradient formulas are provided in Section \[sec:approximation\]. We present the simulation results regarding the robotic surveillance problem in Section \[sec:Simulation\]. Section \[sec:Conclusion\] concludes the paper.
Notation and useful lemmas {#notation-and-useful-lemmas .unnumbered}
--------------------------
Let $\real$, $\mathbb{Z}_{\geq0}$, and $\mathbb{Z}_{>0}$ denote the set of real numbers, nonnegative and positive integers, respectively. Let $\mathbb{1}_n$ and $\mathbb{0}_n$ denote column vectors in $\real^n$ with all entries being $1$ and $0$. $I_n\in \real^{n\times n}$ is the identity matrix. $\mathbb{e}_i$ denotes the $i$-th vector in the standard basis, whose dimension will be made clear when it appears. $[S]$ denotes a diagonal matrix with diagonal elements being $S$ if $S$ is a vector, or being the diagonal of $S$ if $S$ is a square matrix. Let $\otimes$ denote the Kronecker product. $\operatorname{vec}(\cdot)$ is the vectorization operator that converts a matrix into a column vector. The following lemmas are useful.
(A uniform bound for stable matrices [@MHAD-RBV:85 Proposition D.3.1])\[lemma:SolutionBound\] Assume the matrix subset $\mathcal{A}\subset\real^{n\times{n}}$ is compact and satisfies $$\nonumber
\rho_{\mathcal{A}} := \max_{A\in\mathcal{A}}\rho(A)<1.$$ Then for any $\lambda\in(\rho_{\mathcal{A}},1)$ and for any induced matrix norm $\|\cdot\|$, there exists $c>0$ such that $$\nonumber
\|A^k\| \leq c\lambda^k, \quad \text{for all } A\in\mathcal{A}\text{ and } k\in\mathbb{Z}_{\geq0} .$$
(Weierstrass M-test [@WR:76 Theorem 7.10])\[lemma:Weierstrass\] Given a set $\mathcal{X}$, consider the sequence of functions $\{{f_k: \mathcal{X} \rightarrow \real}\}_{k\in\mathbb{Z}_{>0}}$. If there exists a sequence of scalars $\{M_k\in\real\}_{k\in\mathbb{Z}_{>0}}$ satisfying $\sum_{k=1}^\infty M_k<\infty$ and $$\nonumber
|f_k(x)|\leq M_k, \quad\text{for all } x\in \mathcal{X},k\in \mathbb{Z}_{>0},$$ then $\sum_{k=1}^\infty f_k$ converges uniformly on $\mathcal{X}$.
(Geometric distribution generates maximum entropy [@SG-AS:85])\[lemma:MaximumEntropy\] Given a discrete random variable $Y \in \mathbb{Z}_{>0}$ and $\mathbb{E}[Y] = \mu\geq 1$, the probability distribution with maximum entropy is $$\nonumber
\mathbb{P}[Y=k] = (1-\frac{1}{\mu})^{k-1} \frac{1}{\mu},\quad k\in\mathbb{Z}_{>0},$$ with entropy $$\label{eq:maximumentropy}
\mathbb{H}(Y) = \mu \log \mu -(\mu-1)\log(\mu-1).$$
Problem formulation {#sec:ProblemFormulation}
===================
We start by reviewing the basics of discrete-time Markov chains. A finite-state discrete-time Markov chain with state space ${\{1,\dots, n\}}$ is a sequence of random variables taking values in ${\{1,\dots, n\}}$ and satisfying the Markov property. Let $X_k$ be the random variable at time $k\in\mathbb{Z}_{\geq0}$, then a time-homogeneous Markov chain satisfies, for all $i,j\in{\{1,\dots, n\}}$ and $k\in\mathbb{Z}_{\geq0}$, $
\mathbb{P}(X_{k+1}=j\,|\,X_{k}=i,\dots,X_{1}=i_{1},X_{0}=i_0) =\mathbb{P}(X_{k+1}=j\,|\,X_{k}=i)=p_{ij},
$ where $p_{ij}$ is the transition probability from state $i$ to state $j$ and $P=\{p_{ij}\}\in\real^{n\times{n}}$ is the transition matrix satisfying $P\geq 0$ and $P\mathbb{1}_n=\mathbb{1}_n$; see [@JGK-JLS:76], [@JRN:97]. A probability distribution $\bm{\pi}\in\real^n$ is *stationary* for the Markov chain with transition matrix $P$ if it satisfies $\bm{\pi}\geq0$, $\bm{\pi}^\top\mathbb{1}_n=1$ and $\bm{\pi}^\top=\bm{\pi}^\top P$. A Markov chain is *irreducible* if its transition diagram is a strongly connected graph. A Markov chain that satisfies the detailed balance equation $[\bm{\pi}]P=P^\top[\bm{\pi}]$ is *reversible*. A discrete-time Markov chain is also referred to as a random walk on a graph.
Return time of random walks {#sec:returntime}
---------------------------
In this paper, we consider a strongly connected directed weighted graph $\mathcal{G}=\{V,\mathcal{E},W\}$, where $V$ denotes the set of $n$ nodes ${\{1,\dots, n\}}$, $\mathcal{E}\subset V\times V$ denotes the set of edges, and $W\in\mathbb{Z}_{\geq0}^{n\times n}$ is the integer-valued weight (travel time) matrix with $w_{ij}$ being the one-hop travel time from node $i$ to node $j$. If $(i,j)\notin\mathcal{E}$, then $w_{ij}=0$; if $(i,j)\in\mathcal{E}$, then $w_{ij}\geq 1$. Let $w_{\max}=\max_{i,j}\{w_{ij}\}$ be the maximum travel time.
Given the graph $\mathcal{G}=\{V,\mathcal{E},W\}$, let $X_k
\in {\{1,\dots, n\}}$ denote the location of a random walk on $\mathcal{G}$ following a transition matrix $P$ at time $k \in \mathbb{Z}_{\geq0}$. For any pair of nodes $i,j\in V$, the *first hitting time* from $i$ to $j$, denoted by $T_{ij}$, is the first time the random walk reaches node $j$ starting from node $i$, that is $$\label{eq:passagetime}
T_{ij}=\min\Big\{\sum_{k'=0}^{k-1}w_{X_{k'}X_{k'+1}}\,|\,X_0=i,X_k=j,k\geq1\Big\}.$$ In particular, the *return time* $T_{ii}$ of node $i$ is the first time the random walk returns to node $i$ starting from node $i$. Let the $(i,j)$-th element of the *first hitting time probability matrix* $F_k$ denote the probability that the random walk reaches node $j$ for the first time in exactly $k$ time units starting from node $i$, i.e., $F_k(i,j)=\mathbb{P}(T_{ij}=k)$.
Return time entropy of random walks
-----------------------------------
For an irreducible Markov chain, the return time $T_{ii}$ of each state $i$ is a well-defined random variable over $\mathbb{Z}_{>0}$. We define the *return time entropy of state $i$* by $$\begin{aligned}
\mathbb{H}(T_{ii})&=-\sum\limits_{k=1}^\infty \mathbb{P}(T_{ii}=k)\log \mathbb{P}(T_{ii}=k) \nonumber \\
&=-\sum\limits_{k=1}^\infty F_k(i,i)\log F_k(i,i), \label{eq:ret-time-entropy}\end{aligned}$$ where the logarithm is the natural logarithm and $0\log0=0$.
(Coprime travel times) The return time entropy of states does not change when we scale the travel times on all edges simultaneously by the same factor. Therefore, we assume the weights on the graph are coprime.
(The set of Markov chains $\epsilon$-conforming to a graph)\[def:MCconforming\] Given a strongly connected directed weighted graph $\mathcal{G}=\{V,\mathcal{E},W\}$ with $n$ nodes and the stationary distribution ${\bm{\pi}}>0$, pick a minimum edge weight $\epsilon>0$, the set of Markov chains *$\epsilon$-conforming* to $\mathcal{G}$ is defined by $$\begin{aligned}
\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon=\{P\in \real^{n\times n}\,|\,&p_{ij}\geq \epsilon \text{ if } (i,j)\in \mathcal{E},\\
&p_{ij}=0 \text{ if } (i,j)\notin \mathcal{E},\\
&P\mathbb{1}_n=\mathbb{1}_n,{\bm{\pi}}^\top P={\bm{\pi}}^\top\}.
\end{aligned}$$
(Return time entropy) Given a set $\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$, define the *return time entropy* function ${\mathbb{J}}:\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon
\mapsto \real_{\geq 0}$ by $$\label{eq:weightedentropy}
{\mathbb{J}}(P)=\sum_{i=1}^n\pi_i\mathbb{H}(T_{ii}).$$
(The expectation of the first return time)\[rmk:expectedreturntime\] For an irreducible Markov chain defined over a weighted graph with travel times, [@RP-PA-FB:14b Theorem 6] states $$\label{eq:ETii}
\mathbb{E}[T_{ii}]=\frac{\bm{\pi}^\top(P\circ W)\mathbb{1}_n}{\pi_i},$$ where $\circ$ is the Hadamard element-wise product. For unitary travel times, this formula reduces to the usual ${\mathbb{E}}[T_{ii}]=1/\pi_i$. In both cases, the first return times expectations are inversely proportional to the entries of $\bm{\pi}$.
In general, it is difficult to obtain the closed-form expression for the return time entropy function.
(Two special cases with unitary travel times)\[example:closedform\] The elementary proofs of the following results are omitted in the interest of brevity.
1. (Two-node complete graph case) Given a two-node complete graph $\mathcal{G}$ with unit weights, if the transition matrix $P\in\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$ has the following form $$P =\begin{bmatrix}
p_{11}&p_{12}\\
p_{21}&p_{22}
\end{bmatrix},$$ then the return time entropy function is $$\begin{aligned}
{\mathbb{J}}(P)&=-2\pi_1p_{11}\log(p_{11})-2\pi_2p_{22}\log(p_{22})\\
&\quad-2\pi_1p_{12}\log(p_{12})-2\pi_2p_{21}\log(p_{21}).
\end{aligned}$$
2. (Complete graph case with special structure) Given an $n\geq2$-node complete graph $\mathcal{G}$ with unit weights and the stationary distribution $\bm{\pi}=\frac{1}{n}\mathbb{1}_n$, if the transition matrix $P\in\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$ has the form $$P=(a-b)I_n+b\mathbb{1}_n\mathbb{1}_n^\top,$$ for any $a\geq 0$ and $b>0$ satisfying $a+(n-1)b=1$, then the return time entropy function is $$\begin{aligned}
{\mathbb{J}}(P)&=-a\log(a)-(n-1)b\log\big((n-1)b^2\big)\\
&\quad -(n-1)(1-b)\log(1-b).
\end{aligned}$$
In this paper, we are interested in the following problem.
\[prob:weightedentropy\] (Maximization of the return time entropy) Given a strongly connected directed weighted graph $\mathcal{G}=\{V,\mathcal{E},W\}$ and the stationary distribution $\bm{\pi}>0$, pick a minimum edge weight $\epsilon>0$, the maximization of the return time entropy is as follows. $$\begin{aligned}
& \text{maximize}
& &{\mathbb{J}}(P)\\
& \text{subject to}
&& P\in\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon\end{aligned}$$
Properties of the return time entropy {#sec:properties}
=====================================
Dynamical model for hitting time probabilities
----------------------------------------------
In this subsection, we characterize a dynamical model for the first hitting time probabilities and establish several important properties of the model.
(Linear dynamics for the first hitting time probabilities)\[thm:linear-dynamics\] Consider a transition matrix $P\in\mathbb{R}^{n\times n}$ that is nonnegative, row-stochastic and irreducible. Then
1. the hitting time probabilities $F_k$, $k\in\mathbb{Z}_{>0}$, satisfy the discrete-time delayed linear system with a finite number of impulse inputs: $$\begin{gathered}
\operatorname{vec}(F_k) =\operatorname{vec}(P\circ \mathbf{1}_{\{k\mathbb{1}_n\mathbb{1}_n^\top=W\}}) \\
+\sum_{i=1}^n\sum_{j=1}^np_{ij}(E_j\otimes\mathbb{e}_i\mathbb{e}_j^\top)\operatorname{vec}(F_{k-w_{ij}}),\label{eq:dynamicswithtravel}
\end{gathered}$$ where $E_{i}=[\mathbb{1}_n-\mathbb{e}_i]\in\mathbb{R}^{n\times n}$, and the initial conditions are $\operatorname{vec}(F_{k})=\mathbb{0}_{n^2}$ for all $k\leq0$;
2. if the weights are unitary, i.e., $w_{ij}\in\{0,1\}$, then the hitting time probabilities satisfy $$\label{eq:RecursiveInVec}
\operatorname{vec}(F_k) =(I_n\otimes P)(I_{n^2} - [\operatorname{vec}(I_n)])\operatorname{vec}(F_{k-1}),$$ where the initial condition is $F_1 = P$.
By definition in , $F_k(i,j)$ satisfies the following recursive formula for $k\in \mathbb{Z}_{>0}$ $$\label{eq:RecursiveOri}
F_k(i,j) = p_{ij}\mathbf{1}_{\{k=w_{ij}\}}+\sum\limits_{h=1,h\neq j}^np_{ih}F_{k-w_{ih}}(h,j),$$ where $\mathbf{1}_{\{\cdot\}}$ is the indicator function and $F_k(i,j)=0$ for all $k\leq0$ and $i,j\in V$.
Let $D_k(i)\in\mathbb{R}^{n\times n}$ be a matrix associated with node $i$ at time $k$ that has the form $$D_k(i)=\sum_{j\in\mathcal{N}_i}\mathbb{e}_j\mathbb{e}_j^\top F_{k-w_{ij}},$$ where $\mathcal{N}_i$ is the set of out-going neighbors of node $i$. Then, can be written in the following matrix form $$\label{eq:Fkmatrix}
F_k=P\circ \mathbf{1}_{\{k\mathbb{1}_n\mathbb{1}_n^\top=W\}} + \sum_{i=1}^n\mathbb{e}_i\mathbb{e}_i^\top P (D_k(i)-[D_k(i)]).$$ Vectorizing both sides of , we have $$\begin{gathered}
\operatorname{vec}(F_k) =\operatorname{vec}(P\circ \mathbf{1}_{\{k\mathbb{1}_n\mathbb{1}_n^\top=W\}}) \\
+\sum_{i=1}^n(I_n\otimes\mathbb{e}_i\mathbb{e}_i^\top P)(I_{n^2}-[\operatorname{vec}(I_n)])\operatorname{vec}(D_k(i)).\end{gathered}$$ Note that $$\operatorname{vec}(D_k(i)) = \sum_{j\in\mathcal{N}_i}(I_n\otimes\mathbb{e}_j\mathbb{e}_j^\top)\operatorname{vec}(F_{k-w_{ij}}),$$ and $$(I_{n^2}-[\operatorname{vec}(I_n)])(I_n\otimes\mathbb{e}_j\mathbb{e}_j^\top)=E_j\otimes\mathbb{e}_j\mathbb{e}_j^\top.$$ Therefore, we have .
Moreover, if the travel times are unitary, then $F_1=P$ and $$\label{eq:sumindividual}
\sum_{i=1}^n\sum_{j=1}^np_{ij}(E_j\otimes\mathbb{e}_i\mathbb{e}_j^\top)=(I_n\otimes P)(I_{n^2} - [\operatorname{vec}(I_n)]).$$ Thus, equation follows.
The dynamical system can be transformed to an equivalent homogeneous linear system by restarting the system at $k=w_{M}$ with same system matrices and appropriate initial conditions. Moreover, we can augment the system and obtain a discrete-time linear system without delays. This equivalent augmented system is useful for example in studying stability properties. For $k\geq1$, we have $$\label{eq:augmentedsystem}
\begin{bmatrix}
\operatorname{vec}(F_{k+w_{\max}})\\
\operatorname{vec}(F_{k+w_{\max}-1})\\
\vdots\\
\operatorname{vec}(F_{k+1})\\
\end{bmatrix}=\Psi\begin{bmatrix}
\operatorname{vec}(F_{k+w_{\max}-1})\\
\operatorname{vec}(F_{k+w_{\max}-2})\\
\vdots\\
\operatorname{vec}(F_{k})
\end{bmatrix},$$ where $$\label{eq:sysmatrix}
\Psi=\begin{bmatrix}
\Phi_1&\Phi_2&\cdots&\cdots&\Phi_{w_{\max}}\\
I_{n^2}&\mathbb{0}_{n^2\times n^2}&\cdots&\cdots&\mathbb{0}_{n^2\times n^2}\\
\mathbb{0}_{n^2\times n^2}&I_{n^2}&\cdots&\cdots&\mathbb{0}_{n^2\times n^2}\\
\vdots&\vdots&\ddots&\cdots&\mathbb{0}_{n^2\times n^2}\\
\mathbb{0}_{n^2\times n^2}&\cdots&\cdots&I_{n^2}&\mathbb{0}_{n^2\times n^2}\\
\end{bmatrix},$$ and for $h\in[1,w_{\max}]$, $$\label{eq:phimatrix}
\Phi_h=\sum_{i=1}^n\sum_{j=1}^np_{ij}(E_j\otimes\mathbb{e}_i\mathbb{e}_j^\top)\mathbf{1}_{\{w_{ij}=h\}}.$$ The initial conditions for can be computed using . For brevity, we denote $
\begin{bmatrix}
\operatorname{vec}(F_{k+w_{\max}-1})&
\cdots&
\operatorname{vec}(F_{k})
\end{bmatrix}^\top$ by $\operatorname{vec}(\tilde{F}_k)^\top$.
(Properties of the linear dynamics for the first hitting time probabilities) \[lemma:systempropertiestravel\] If $P\in \real^{n\times n}$ is nonnegative, row-stochastic and irreducible, then
1. the matrix $(I_n\otimes P)(I_{n^2} - [\operatorname{vec}(I_n)])$ is row-substochastic with $\rho\big((I_n\otimes P)(I_{n^2} - [\operatorname{vec}(I_n)])\big)<1$.
2. the delayed discrete-time linear system with a finite number of impulse inputs is asymptotically stable;
3. $\operatorname{vec}(F_k)\geq 0$ for $k\in \mathbb{Z}_{>0}$ and $\sum_{k=1}^{\infty}\operatorname{vec}(F_k) = \mathbb{1}_{n^2\times 1}$.
Regarding (i), note that the matrix $(I_n\otimes P)(I_{n^2} - [\operatorname{vec}(I_n)])$ is block diagonal with the $i$-th block being $PE_{i}$. Since $P$ is irreducible, there is at least one positive entry in each column of $P$. Therefore $PE_{i}$’s are row-substochastic and so is $(I_n\otimes P)(I_{n^2} - [\operatorname{vec}(I_n)])$. By [@RP-AC-FB:14k Lemma 2.2], $\rho(PE_{i})<1$ for all $i\in{\{1,\dots, n\}}$ and $\rho((I_n\otimes P)(I_{n^2} - [\operatorname{vec}(I_n)]))=\max_{i}\rho(PE_{i})<1$.
Regarding (ii), since we can rewrite as with appropriate initial conditions and $\Phi_i$’s are nonnegative, by the stability criterion for delayed linear systems [@AH-ET:98 Theorem 1], is asymptotically stable if $$\rho\Big(\sum_{i=1}^{w_{\max}}\Phi_i\Big)=\rho\big((I_n\otimes P)(I_{n^2} - [\operatorname{vec}(I_n)])\big)<1,$$ which is true by (i).
Regarding (iii), first note that all the system matrices are nonnegative, thus $\operatorname{vec}(F_k)\geq 0$ for all $k\in \mathbb{Z}_{>0}$. Moreover, due to (ii), the delayed linear system is asymptotically stable. Summing both sides of over $k$, we have
$$\begin{aligned}
\sum_{k=1}^{\infty}\operatorname{vec}(F_k)&=\operatorname{vec}(P) + \sum_{i=1}^n\sum_{j=1}^np_{ij}(E_j\otimes\mathbb{e}_i\mathbb{e}_j^\top)\sum_{k=1}^{\infty}\operatorname{vec}(F_k)\\
&=\operatorname{vec}(P) + (I_n\otimes P)(I_{n^2} - [\operatorname{vec}(I_n)])\sum_{k=1}^{\infty}\operatorname{vec}(F_k),\end{aligned}$$
which implies that $\sum_{k=1}^{\infty}\operatorname{vec}(F_k) = \mathbb{1}_{n^2\times 1}$.
Well-posedness of the optimization problem
------------------------------------------
We here show that the function ${\mathbb{J}}$ is continuous over the compact set $\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$. Then, by the extreme value theorem, ${\mathbb{J}}$ has a (possibly non-unique) maximum point in the set and thus Problem \[prob:weightedentropy\] is well-posed.
\[lemma:continuity\] (Continuity of the return time entropy function) Given the compact set $\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$, the following statements hold:
1. there exist constants $\lambda_{\max}\in(0,1)$ and $c>0$ such that $$\nonumber
F_k(i,i)\leq c\lambda_{\max}^k, \quad\text{for all }
k\in\mathbb{Z}_{>0}, i\in{\{1,\dots, n\}};$$
2. the return time entropy functions $\mathbb{H}(T_{ii})$, $i\in{\{1,\dots, n\}}$, and ${\mathbb{J}}(P)$ are continuous on the compact set $\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$; and
3. Problem \[prob:weightedentropy\] is well-posed in the sense that a global optimum exists.
Regarding (i), for $k\geq w_{M}+1$, since the spectral radius $\rho(\Psi)$ is a continuous function of $\Psi$ [@CDM:01 Example 7.1.3], where $\Psi$ is given in , and $\Psi$ is a continuous function of $P$, $\rho(\Psi)$ is a continuous function of $P$. Hence, by Lemma \[lemma:systempropertiestravel\](ii) and the extreme value theorem, there exists a $\rho_{\max}<1$ such that $$\rho_{\max} = \max\limits_{P \in\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon} \rho(\Psi)<1.$$ Therefore, for $k\geq w_{M}+1$ and $i\in{\{1,\dots, n\}}$, by Lemma \[lemma:SolutionBound\], there exist $c_1>0$ and $\rho_{\max}<\lambda_{\max}<1$ such that $$\begin{aligned}
F_k(i,i)&\leq\|\operatorname{vec}{(\tilde{F}_{k-w_{\max}+1})}\|_\infty\\
&=\|(\Psi)^{k-w_{\max}}\operatorname{vec}(\tilde{F}_{1})\|_\infty\\
&\leq\|(\Psi)^{k-w_{\max}}\|_\infty\|\operatorname{vec}(\tilde{F}_{1})\|_\infty\\
&\leq c_1\lambda_{\max}^{k-w_{\max}}=\frac{c_1}{\lambda_{\max}^{w_{\max}}}\lambda_{\max}^{k}.\end{aligned}$$ Let $c=\max\{\frac{c_1}{\lambda_{\max}^{w_{\max}}},\frac{1}{\lambda_{\max}^{w_{\max}}}\}$, then we have for $k\geq w_{M}+1$, $$\begin{aligned}
F_k(i,i)\leq\frac{c_1}{\lambda_{\max}^{w_{\max}}}\lambda_{\max}^{k}<c\lambda_{\max}^{k}.\end{aligned}$$ For $k\leq w_{M}$, $$\begin{aligned}
c\lambda_{\max}^k\geq c\lambda_{\max}^{w_{\max}}\geq 1\geq F_{k}(i,i).\end{aligned}$$ Therefore, we have (i).
Regarding (ii), due to (i), there exists a positive integer $K$ that does not depend on the elements of $\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$ such that when $k\geq K$, $c\lambda_{\max}^k\leq e^{-1}$. Since $x\mapsto
-x\log x$ is an increasing function for $x\in[0,e^{-1}]$, when $k\geq K$, $$-F_k(i,i)\log F_k(i,i)\leq-c\lambda_{\max}^k\log (c\lambda_{\max}^k)\coloneqq M_k.$$ For $k< K$, $-F_k(i,i)\log F_k(i,i)\leq e^{-1}\coloneqq M_k$. Then
$$\sum_{k=1}^{K-1} M_k =\frac{K-1}{e},$$
and
$$\begin{aligned}
\label{eq:tailbound}
\sum_{k=K}^\infty M_k&=-\sum_{k=K}^\infty c\lambda_{\max}^k\log (c\lambda_{\max}^k)\nonumber\\
&=-c\log c\sum_{k=K}^\infty \lambda_{\max}^k-c\log (\lambda_{\max})\sum_{k=K}^\infty k\lambda_{\max}^k\nonumber\\
&=-c\Big(\frac{\lambda_{\max}^K}{1-\lambda_{\max}}\log (c\lambda_{\max}^K) \nonumber\\
&\qquad\quad+\frac{\lambda_{\max}^{K+1}}{(1-\lambda_{\max})^2}\log (\lambda_{\max})\Big).\end{aligned}$$
Hence, $$\begin{aligned}
\sum_{k=1}^\infty M_k&=\sum_{k=1}^{K-1} M_k+\sum_{k=K}^\infty M_k<\infty,\end{aligned}$$ which holds for any $i$ and any transition matrix in the compact set $\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$. By Lemma \[lemma:Weierstrass\], the series $-\sum_{k=1}^\infty F_k(i,i)\log F_k(i,i)$ converges uniformly. Since the the limit of a uniformly convergent series of continuous function is continuous [@WR:76 Theorem 7.12], $\mathbb{H}(T_{ii})$ is a continuous function on $\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$. Finally, ${\mathbb{J}}(P)$ is a finite weighted sum of continuous functions $\mathbb{H}(T_{ii})$, thus ${\mathbb{J}}(P)$ is a continuous function.
Regarding (iii), because ${\mathbb{J}}$ is a continuous function over the compact set $\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$, the extreme value theorem ensures that Problem \[prob:weightedentropy\] admits a global optimum solution (possibly non-unique) and is therefore well-posed.
Optimal solution for complete graphs with unitary travel times
--------------------------------------------------------------
We here provide (1) an upper bound for the return time entropy with unitary travel times based on the principle of maximum entropy and (2) the optimal solution to Problem \[prob:weightedentropy\] for the complete graph case with unitary travel times.
(Maximum achieved return time entropy in a complete graph with unitary weights) Given a strongly connected graph $\mathcal{G}$ with unitary weights and the compact set $\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$,
1. the return time entropy function is upper bounded by $$\begin{aligned}
{\mathbb{J}}(P)\leq-\sum_{i=1}^n(\pi_i \log {\pi_i} + (1-{\pi_i})\log(1-{\pi_i}));\end{aligned}$$
2. when the graph $\mathcal{G}$ is complete, the upper bound is achieved and the transition matrix that maximizes the return time entropy ${\mathbb{J}}(P)$ is given by $P={{\mathbbold{1}}_{n}}\bm{\pi}^\top$.
Regarding (i), by Remark \[rmk:expectedreturntime\], in the case of unitary travel times, we have ${\mathbb{E}}[T_{ii}]=1/\pi_i$. Thus, $T_{ii}$ is a discrete random variable with fixed expectation, whose entropy is bounded as shown in Lemma \[lemma:MaximumEntropy\]. For any transition matrix $P\in\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$, the return time entropy function ${\mathbb{J}}(P)$ satisfies
$$\begin{aligned}
{\mathbb{J}}(P)&=\sum_{i=1}^n\pi_i\mathbb{H}(T_{ii})\leq\sum_{i=1}^n\pi_i\max_{T_{ii}}\{\mathbb{H}(T_{ii})\} \\
&=\sum_{i=1}^n\pi_i\big(\frac{1}{\pi_i} \log \frac{1}{\pi_i} -(\frac{1}{\pi_i}-1)\log(\frac{1}{\pi_i}-1)\big)\\
&=-\sum_{i=1}^n\big(\pi_i \log {\pi_i} + (1-{\pi_i})\log(1-{\pi_i})\big),\end{aligned}$$
where the third line uses .
Regarding (ii), when the graph is complete and $P={{\mathbbold{1}}_{n}}\bm{\pi}^\top$, the return time $T_{ii}$ follows the geometric distribution: $$\nonumber
\prob(T_{ii}=k)=\pi_i(1-\pi_i)^{k-1}.$$ Then by Lemma \[lemma:MaximumEntropy\], we obtain the results.
Relations with the entropy rate of Markov chains
------------------------------------------------
Given an irreducible Markov chain $P$ with $n$ nodes and stationary distribution $\bm{\pi}$, the *entropy rate of $P$* is given by $${{\mathbb{H}_{\textup{rate}}}}(P)=-\sum_{i=1}^n\pi_i\sum_{j=1}^np_{ij}\log p_{ij}.$$ We next study the relationship between the return time entropy ${\mathbb{J}}$ with unitary travel times and the entropy rate ${{\mathbb{H}_{\textup{rate}}}}$.
\[thm:relations\] (Relations between the return time entropy with unitary travel times and the entropy rate) For all $P$ in the compact set $\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$ where $\mathcal{G}$ has unitary travel times, the return time entropy ${\mathbb{J}}(P)$ and the entropy rate ${{\mathbb{H}_{\textup{rate}}}}(P)$ satisfy $$\label{eq:relations}
{{\mathbb{H}_{\textup{rate}}}}(P)\leq {\mathbb{J}}(P) \leq n{{\mathbb{H}_{\textup{rate}}}}(P),$$ where $n$ is the number of nodes in the graph $\mathcal{G}$.
We prove this theorem The proof of the following theorem follows from Lemmas \[lemma:upperbounds\_entropy\] and Lemma \[lemma:lowerbound\_entropyrate\] below.
Theorem \[thm:relations\] establishes a large gap, possibly of size $O(n)$, between ${{\mathbb{H}_{\textup{rate}}}}(P)$ and ${\mathbb{J}}(P)$ and, thereby, optimizing ${{\mathbb{H}_{\textup{rate}}}}$ and ${\mathbb{J}}$ are two different matters altogether.
First, we show that the return time entropy is upper bounded by $n$ times of the entropy rate. As in [@LE-TMC:93], we define a *Markov trajectory from state $i$ to state $j$* to be a path with initial state $i$, final state $j$, and no intervening state equal to $j$. Let $\mathcal{T}_{ij}$ be the set of all Markov trajectories from state $i$ to state $j$. Let $\mathbb{P}(\ell)$ denote the probability of a Markov trajectory $\ell\in\mathcal{T}_{ij}$; clearly $\sum_{\ell\in\mathcal{T}_{ij}}\mathbb{P}(\ell)=1$. Let $L_{ij}$ be the Markov trajectory random variable that takes value $\ell$ in $\mathcal{T}_{ij}$ with probability $\mathbb{P}(\ell)$. Finally, we define the entropy of $L_{ij}$ by $$\mathbb{H}(L_{ij})=-\sum_{\ell\in\mathcal{T}_{ij}}\mathbb{P}(L_{ij}=\ell)\log\mathbb{P}(L_{ij}=\ell).$$
(Entropy of Markov trajectories [@LE-TMC:93 Theorem 1])\[lemma:MarkovTrajectories\] For an irreducible Markov chain with transition matrix $P$, the entropy $\mathbb{H}(L_{ii})$ of the random Markov trajectory from state $i$ back to state $i$ is given by $$\mathbb{H}(L_{ii})=\frac{{{\mathbb{H}_{\textup{rate}}}}(P)}{{\pi}_i}.$$
Through the entropy of the Markov trajectories, we are able to establish the upper bound of the return time entropy in .
(Upper bound of the return time entropy by $n$ times of the entropy rate)\[lemma:upperbounds\_entropy\] Given the compact set $\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$,
1. the return time entropy is upper bounded by $$\label{eq:JHconnection}
{\mathbb{J}}(P)\leq n{{\mathbb{H}_{\textup{rate}}}}(P), \quad\text{for all } P\in\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon;$$
2. the equality in holds if and only if any node of the graph $\mathcal{G}$ has the property that all distinct first return paths have different length, i.e., the return paths are distinguishable by their lengths, and in this case, $$\operatorname*{argmax}_{P\in\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon}{\mathbb{J}}(P)=\operatorname*{argmax}_{P\in\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon}{{\mathbb{H}_{\textup{rate}}}}(P).$$
Regarding (i), the return time random variable $T_{ii}$ is defined by lumping the trajectories in $\mathcal{T}_{ii}$ with the same length, $$\label{eq:time_trajectory}
\mathbb{P}(T_{ii}=k)=\sum_{\ell\in\mathcal{T}_{ii},|\ell|=k}\mathbb{P}(L_{ii}=\ell),$$ where $|\ell|$ denotes the length of the path $\ell$. Note that
$$\begin{aligned}
\label{eq:inequality_upperbound}
-\mathbb{P}(&T_{ii}=k)\log\mathbb{P}(T_{ii}=k)\nonumber\\
&=-\big(\sum_{\ell\in\mathcal{T}_{ii},|\ell|=k}\mathbb{P}(L_{ii}=\ell)\big)\log\big(\sum_{\ell\in\mathcal{T}_{ii},|\ell|=k}\mathbb{P}(L_{ii}=\ell)\big)\nonumber\\
&\leq-\sum_{\ell\in\mathcal{T}_{ii},|\ell|=k}\mathbb{P}(L_{ii}=\ell)\log\mathbb{P}(L_{ii}=\ell),\end{aligned}$$
where we used that $(x+y)\log(x+y)\geq x\log x+y\log y$ for $x,y\geq0$. Since both the return time entropy and the entropy of Markov trajectories are absolutely convergent, we have
$$\begin{aligned}
\mathbb{H}(T_{ii})&=-\sum_{k=1}^\infty\mathbb{P}(T_{ii}=k)\log\mathbb{P}(T_{ii}=k)\\
&\leq-\sum_{k=1}^\infty\sum_{\ell\in\mathcal{T}_{ii},|\ell|=k}\big(\mathbb{P}(L_{ii}=\ell)\log\mathbb{P}(L_{ii}=\ell)\big)\\
&=\mathbb{H}(L_{ii}),\end{aligned}$$
which along with Lemma \[lemma:MarkovTrajectories\] imply $${\mathbb{J}}(P)\leq n{{\mathbb{H}_{\textup{rate}}}}(P).$$
Regarding (ii), the inequality in comes from the inequality . If any node of the graph $\mathcal{G}$ has the property that all distinct first return paths have different length, then the summation on the right hand side of only has one term and the inequality in becomes an equality. On the other hand, if for some node of $\mathcal{G}$, there are distinct return paths that have the same length, then one needs to lump the paths with the same length and the inequality in becomes strict. Moreover, if the equality holds, then ${\mathbb{J}}(P)$ is a constant $n$ times of ${{\mathbb{H}_{\textup{rate}}}}(P)$ and thus they have the same maximizer.
For the two-node case in Examples \[example:closedform\](i), the return time entropy is twice the entropy rate. This is not a coincidence since the 2-node complete graph satisfies the property in Lemma \[lemma:upperbounds\_entropy\](ii). Figure \[fig:examplegraph\] illustrates a graph with $4$ nodes that also satisfies the property in Lemma \[lemma:upperbounds\_entropy\](ii).
at (0, 1) (nodeone) ; at (1.5, 1) (nodetwo) ; at (0, 0) (nodethree) ; at (1.5, 0) (nodefour) ; (nodefour) edge\[bend right=15, auto=right\] node (nodethree) (nodetwo) edge\[bend left=15, auto=right\] node (nodeone) (nodetwo) edge\[bend right=0, auto=right\] node (nodefour)
(nodethree) edge\[bend right=15\] node (nodefour) (nodethree) edge\[bend right=0\] node (nodeone)
(nodeone) edge\[bend left=15, auto=right\] node (nodetwo);
In the rest of this subsection, we show that the return time entropy is lower bounded by the entropy rate as shown in .
(Lower bound of the return time entropy by the entropy rate)\[lemma:lowerbound\_entropyrate\] Given the compact set $\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$,
1. the return time entropy is lower bounded by $$\label{eq:JHlowerbound}
{\mathbb{J}}(P)\geq {{\mathbb{H}_{\textup{rate}}}}(P), \quad\text{for all } P\in\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon;$$
2. the equality in holds if and only if $P$ is a permutation matrix.
Regarding (i), note that the first hitting time $T_{ij}$ from state $i$ to state $j$ as defined in is a random variable , whose entropy is $\mathbb{H}(T_{ij})$. Then by definition, we have in the case of unitary travel times,
$$\begin{aligned}
\mathbb{H}&(T_{ij})=-\sum\limits_{k=1}^\infty \mathbb{P}(T_{ij}=k)\log \mathbb{P}(T_{ij}=k)\\
&=-p_{ij}\log p_{ij} - (\sum_{k_1\neq j}p_{ik_1}p_{k_1j})\log(\sum_{k_1\neq j}p_{ik_1}p_{k_1j}) \\
&\quad-(\sum_{k_1,k_2\neq j}p_{ik_1}p_{k_1k_2}p_{k_2j})\log(\sum_{k_1,k_2\neq j}p_{ik_1}p_{k_1k_2}p_{k_2j})\\
&\quad -\cdots\\
&\quad-(\sum_{k_1\cdots k_m\neq j}p_{ik_1}\cdots p_{k_mj})\log(\sum_{k_1\cdots k_m\neq j}p_{ik_1}\cdots p_{k_mj})\\
&\quad-\cdots.\end{aligned}$$
Since $x\mapsto -x\log x$ is a concave function, for $x_i\geq0$ and for coefficients $\alpha_i\geq 0$ satisfying $\sum_{i=1}^n\alpha_i=1$, we have $$\label{eq:concavefunction}
-(\sum_{i=1}^n\alpha_ix_i)\log(\sum_{i=1}^n\alpha_ix_i)\geq -\sum_{i=1}^n \alpha_i (x_i\log x_i).$$ Thus, for $m\geq 1$,
$$\begin{aligned}
\label{eq:entropyconcavity}
-\mathbb{P}(&T_{ij}=m+1)\log \mathbb{P}(T_{ij}=m+1)\nonumber\\
&=-(\sum_{k_1\cdots k_m\neq j}p_{ik_1}\cdots p_{k_mj})\log(\sum_{k_1\cdots k_m\neq j}p_{ik_1}\cdots p_{k_mj})\nonumber\\
&=- (\sum_{k_1\neq j}p_{ik_1}\sum_{k_2\cdots k_m\neq j}p_{k_1k_2}\cdots p_{k_mj}+p_{ij}\cdot0)\nonumber\\
&\qquad \cdot\log (\sum_{k_1\neq j}p_{ik_1}\sum_{k_2\cdots k_m\neq j}p_{k_1k_2}\cdots p_{k_mj} + p_{ij}\cdot0)\nonumber\\
&\geq- \sum_{k_1\neq j}p_{ik_1}(\sum_{k_2\cdots k_m\neq j}p_{k_1k_2}\cdots p_{k_mj}\nonumber\\
&\qquad\qquad\qquad\quad \cdot\log (\sum_{k_2\cdots k_m\neq j}p_{k_1k_2}\cdots p_{k_mj}))\nonumber\\
&=-\sum_{k_1\neq j}p_{ik_1}\mathbb{P}(T_{k_1j}=m)\log \mathbb{P}(T_{k_1j}=m),\end{aligned}$$
where the inequality uses equation . Summing both sides of over $m$ for $m\geq 1$, we have $$\begin{gathered}
\label{eq:passagetimeentropy}
\mathbb{H}(T_{ij}) \geq -p_{ij}\log p_{ij} + \sum\nolimits_{k_1\neq j}p_{ik_1}\mathbb{H}(T_{k_1j}) \\
=-p_{ij}\log p_{ij} + \sum\nolimits_{k_1=1}^np_{ik_1}\mathbb{H}(T_{k_1j})-p_{ij}\mathbb{H}(T_{jj}).\end{gathered}$$ Let $\mathbb{H}(T)$ be a matrix whose $(i,j)$-th element is $\mathbb{H}(T_{ij})$. Then equation can be put in the matrix form $$\begin{aligned}
\label{eq:passagetimeentropy_matrix}
\mathbb{H}(T) \geq -P\circ\log P + P\mathbb{H}(T)-P[\mathbb{H}(T)],\end{aligned}$$ where the inequality and the $\log$ function are entry-wise. Multiplying $\bm{\pi}^\top$ from the left and $\mathbb{1}_n$ from the right on both sides of , we have $$\begin{aligned}
\bm{\pi}^\top [\mathbb{H}(T)]\mathbb{1}_n\geq -\bm{\pi}^\top (P\circ\log P) \mathbb{1}_n,\end{aligned}$$ which is ${\mathbb{J}}(P)\geq {{\mathbb{H}_{\textup{rate}}}}(P)$.
Regarding (ii), if $P$ is a permutation matrix, then ${\mathbb{J}}(P)={{\mathbb{H}_{\textup{rate}}}}(P)=0$. On the other hand, if $P$ is not a permutation matrix, then there exist $2$ or more nonzero elements on at least one row of $P$. In this case, the inequality in is strict for that row for some $m$, which carries over to . Thus, ${\mathbb{J}}(P)>{{\mathbb{H}_{\textup{rate}}}}(P)$.
Truncated return time entropy and its optimization via gradient descent {#sec:approximation}
=======================================================================
We now introduce the truncated and conditional return time entropy and setup a gradient descent algorithm.
The truncated and conditional return time entropies
---------------------------------------------------
In practical applications, we may discard events occurring with extremely low probability. In what follows, we study the return time distribution and its entropy conditioned upon the event that the return time is upper bounded. We first introduce a *truncation accuracy* parameter $0<\eta\ll1$ that upper bounds the cumulative probabilities of very large return times and we define a *duration* $N_\eta\in\mathbb{Z}_{>0}$ by $$\label{eq:duration}
N_\eta = \ceil[\Big]{\frac{w_{\max}}{\eta\pi_{\min}}} -1,$$ where $\pi_{\min}=\min_{i\in{\{1,\dots, n\}}}\{\pi_i\}$ and $\ceil{\cdot}$ is the ceiling function. It is an immediate consequence of the Markov’s inequality that, given the fixed stationary distribution $\bm{\pi}$, for all $i\in{\{1,\dots, n\}}$, $$\mathbb{P}(T_{ii}\geq N_\eta+1)\leq \frac{\mathbb{E}[T_{ii}]}{N_\eta+1}\leq\frac{w_{\max}}{\pi_i(N_\eta+1)}\leq \eta,$$ where we used $$\mathbb{E}[T_{ii}]=\frac{\bm{\pi}^\top(P\circ W)\mathbb{1}_n}{\pi_i}\leq\frac{w_{\max}}{\pi_i}.$$
We now define the conditional return time and its entropy.
(Conditional return time and its entropy) Given $P\in \mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$ and a duration $N_\eta$, the conditional return time $T_{ii}\,|\,T_{ii}\leq N_\eta$ of state $i$ is defined by
$$\begin{aligned}
T_{ii}\,|\,T_{ii}\leq N_\eta&=\min\Big\{\sum_{k'=0}^{k-1}w_{X_{k'}X_{k'+1}}\,|\,\sum_{k'=0}^{k-1}w_{X_{k'}X_{k'+1}}\leq N_\eta,\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad X_0=i,X_k=i, k\geq1\Big\}.
\end{aligned}$$
with probability mass function $$\nonumber
\mathbb{P}(T_{ii}=k\,|\,T_{ii}\leq N_\eta)=\frac{F_k(i,i)}{\sum_{k=1}^{N_\eta} F_k(i,i)}.$$ Moreover, the conditional return time entropy function ${{\mathbb{J}}_{\textup{cond,$\eta$}}}:\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon
\mapsto \real_{\geq 0}$ is
$$\begin{aligned}
{{\mathbb{J}}_{\textup{cond,$\eta$}}}(P)&=\sum_{i=1}^n\pi_i\mathbb{H}(T_{ii}\,|\,T_{ii}\leq N_\eta)\\
&=-\sum_{i=1}^n\pi_i\sum\limits_{k=1}^{N_\eta} \frac{F_k(i,i)}{\sum\limits_{k=1}^{N_\eta} F_k(i,i)}\log \frac{F_k(i,i)}{\sum\limits_{k=1}^{N_\eta} F_k(i,i)}.\end{aligned}$$
Given the duration $N_\eta$, ${{\mathbb{J}}_{\textup{cond,$\eta$}}}(P)$ is a finite sum of continuously differentiable functions and thus more tractable than the original return time entropy function ${\mathbb{J}}(P)$. Next, we introduce a truncated entropy that is even simpler to evaluate.
(Truncated return time entropy function)\[def:Jtrunc\] Given a compact set $\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$ and the duration $N_\eta$, define the *truncated return time entropy function* ${{\mathbb{J}}_{\textup{trunc,$\eta$}}}:\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon
\mapsto \real_{\geq 0}$ by $${{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P)=-\sum_{i=1}^n\pi_i\sum\limits_{k=1}^{N_\eta} F_k(i,i)\log F_k(i,i).$$
The following lemma shows that, for small $\eta$, the truncated return time entropy ${{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P)$ is a good approximation for the conditional return time entropy ${{\mathbb{J}}_{\textup{cond,$\eta$}}}(P)$. Furthermore, when $\eta$ is sufficiently small, the truncated return time entropy ${{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P)$ is also a good approximation for the original return time entropy function ${\mathbb{J}}(P)$.
\[lemma:trunccondrelation\] (Approximation bounds) Given $P\in \mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$ and the truncation accuracy $\eta$, we have
1. the conditional return time entropy is related to the truncated return time entropy by $$\label{eq:trunccondrelation}
{{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P)+ \log (1-\eta)<{{\mathbb{J}}_{\textup{cond,$\eta$}}}(P)<\frac{{{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P)}{1-\eta};$$
2. ${\mathbb{J}}(P)\geq {{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P)$ holds trivially and if $$\label{eq:etacondition}
\eta \leq \frac{w_{\max}\log\lambda_{\max}}{\pi_{\min}(\log\lambda_{\max}-\log c-1)},$$ then $$\label{eq:errorbound}
{\mathbb{J}}(P)-{{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P) \leq \frac{c \log(\lambda_{\max}^{-1}) }{(1-\lambda_{\max})^2} ( 1+ N_\eta ) \lambda_{\max}^{N_\eta} ,$$ where $c$ and $\lambda_{\max}$ are given as in Lemma \[lemma:continuity\](i);
3. $\displaystyle
{\mathbb{J}}(P)
=
\lim_{\eta\to 0^+} {{\mathbb{J}}_{\textup{cond,$\eta$}}}(P)
=
\lim_{\eta\to 0^+} {{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P) $.
Regarding (i), for ${{\mathbb{J}}_{\textup{cond,$\eta$}}}(P)$, we have
$$\begin{aligned}
{{\mathbb{J}}_{\textup{cond,$\eta$}}}(P)&=-\sum_{i=1}^n\pi_i\sum\limits_{k=1}^{N_\eta} \frac{F_k(i,i)}{\sum\limits_{k=1}^{N_\eta} F_k(i,i)}\log \frac{F_k(i,i)}{\sum\limits_{k=1}^{N_\eta} F_k(i,i)}\\
&=-\sum_{i=1}^n\pi_i\big(\frac{\sum\limits_{k=1}^{N_\eta} F_k(i,i)\log F_k(i,i)}{\sum\limits_{k=1}^{N_\eta} F_k(i,i)}- \log {\sum\limits_{k=1}^{N_\eta} F_k(i,i)}\big).\end{aligned}$$
On one hand,
$$\begin{aligned}
\label{eq:Jcondupper}
{{\mathbb{J}}_{\textup{cond,$\eta$}}}(P)&>-\sum_{i=1}^n\pi_i\big(\sum\limits_{k=1}^{N_\eta} F_k(i,i)\log F_k(i,i)- \log {\sum\limits_{k=1}^{N_\eta} F_k(i,i)}\big)\nonumber\\
&\geq-\sum_{i=1}^n\pi_i\sum\limits_{k=1}^{N_\eta} F_k(i,i)\log F_k(i,i)+ \log (1-\eta).\end{aligned}$$
On the other hand,
$$\begin{aligned}
\label{eq:Jcondlower}
\begin{split}
{{\mathbb{J}}_{\textup{cond,$\eta$}}}(P)&<-\sum_{i=1}^n\pi_i\frac{1}{\sum\limits_{k=1}^{N_\eta} F_k(i,i)}\sum\limits_{k=1}^{N_\eta} F_k(i,i)\log F_k(i,i)\\
&\leq-\frac{1}{1-\eta}\sum_{i=1}^n\pi_i\sum\limits_{k=1}^{N_\eta} F_k(i,i)\log F_k(i,i).
\end{split}\end{aligned}$$
Combining and , we have .
Regarding (ii), if $\eta$ satisfies , we have $c\lambda_{\max}^{N_\eta}\leq e^{-1}$. Then, following the same arguments as in the proof of Lemma \[lemma:continuity\](ii) and replacing $K$ in with $N_\eta$, we have
$$\begin{aligned}
&{\mathbb{J}}(P)-{{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P) \nonumber\\
&\leq -c\left(\frac{\lambda_{\max}^{N_\eta}}{1-\lambda_{\max}}\log (c\lambda_{\max}^{N_\eta})+\frac{\lambda_{\max}^{N_\eta+1}}{(1-\lambda_{\max})^2}\log (\lambda_{\max})\right)\\
&\leq - \frac{c \lambda_{\max}^{N_\eta} }{(1-\lambda_{\max})^2} \left( N_\eta \log(\lambda_{\max})+ \lambda_{\max}\log (\lambda_{\max})+ \log (c) \right) \\
&\leq - \frac{c \lambda_{\max}^{N_\eta} }{(1-\lambda_{\max})^2} \left( N_\eta \log(\lambda_{\max})+ \log (\lambda_{\max}) \right) \\
&= \frac{c \log(\lambda_{\max}^{-1}) }{(1-\lambda_{\max})^2} ( 1+ N_\eta ) \lambda_{\max}^{N_\eta} . \label{eq:errorbound}\end{aligned}$$
Regarding (iii), the results follow from and , respectively. Specifically, in , since $0<\lambda_{\max}<1$, the error ${\mathbb{J}}(P) - {{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P)$ goes to $0$ exponentially fast as $\eta$ goes to $0$ ($N_\eta \to \infty$).
The gradient of the truncated return time entropy
-------------------------------------------------
Lemma \[lemma:trunccondrelation\] establishes how ${{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P)$ is a good approximation to both of ${\mathbb{J}}(P)$ and ${{\mathbb{J}}_{\textup{cond,$\eta$}}}(P)$. Since it is also easier to compute ${{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P)$ than the other two quantities, we focus on optimizing ${{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P)$ by computing its gradient.
For $k\in\mathbb{Z}_{>0}$, define $G_k=\frac{\partial \operatorname{vec}(F_k)}{\partial
\operatorname{vec}(P)}\in\mathbb{R}^{n^2\times n^2}$ and note $$\label{eq:Gkanotherform}
{G_k} = \begin{bmatrix}
\frac{\partial \operatorname{vec}(F_k)}{\partial p_{11}}&
\frac{\partial \operatorname{vec}(F_k)}{\partial p_{21}}&
\cdots &
\frac{\partial \operatorname{vec}(F_k)}{\partial p_{(n-1)n}}&
\frac{\partial \operatorname{vec}(F_k)}{\partial p_{nn}}
\end{bmatrix}.$$
(Gradient of the truncated return time entropy function)\[lemma:gradient\] Given $P\in\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$, the matrix sequence $G_k$ in satisfies the iteration for $k\in\mathbb{Z}_{>0}$,
$$\begin{aligned}
\label{eq:GradientRecursive}
G_{k} &= [\operatorname{vec}(\mathbf{1}_{\{k\mathbb{1}_n\mathbb{1}_n^\top=W\}})] +\sum_{i=1}^{w_{\max}}\Phi_i G_{k-i}\nonumber\\
&\quad+ \sum_{i=1}^n\sum_{j=1}^n (E_{j}F_{k-w_{ij}}^\top\otimes I_n)[\operatorname{vec}(\mathbb{e}_{i}\mathbb{e}_{j}^\top)]\mathbf{1}_{\{w_{ij}>0\}},
\end{aligned}$$
where the initial conditions are $G_{k}=\mathbb{0}_{n^2\times n^2}$ for $k\leq0$. Moreover, the vectorization of the gradient of ${{\mathbb{J}}_{\textup{trunc,$\eta$}}}$ satisfies
$$\begin{gathered}
\label{eq:Gradient}
\operatorname{vec}\Big(\frac{\partial {{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P)}{\partial P}\Big) = \\
-\sum_{i=1}^n\pi_i\sum_{k=1}^{N_\eta}
\frac{\partial \big(F_k(i,i)\log F_k(i,i)\big)}{\partial F_k(i,i)}
G_k^\top\mathbb{e}_{(i-1)n+i},\end{gathered}$$
where $\mathbb{e}_{(i-1)n+i}\in \real^{n^2}$ and
$$\frac{\partial F_k(i,i)\log F_k(i,i)}{\partial F_k(i,i)} = \begin{cases}
1+\log(F_k(i,i)), &\quad \text{if}\enspace F_k(i,i)>0, \\
0, &\quad \text{if}\enspace F_k(i,i)= 0.
\end{cases}$$
For $k\in\mathbb{Z}_{>0}$, according to , we have for $p_{uv}>0$,
$$\begin{aligned}
\frac{\partial \operatorname{vec}(F_k)}{\partial p_{uv}}&=\operatorname{vec}(\mathbb{e}_u\mathbb{e}_v^\top)\mathbf{1}_{\{k=w_{uv}\}}\\
&\quad+(E_v\otimes\mathbb{e}_u\mathbb{e}_v^\top)\operatorname{vec}(F_{k-w_{uv}})\\
&\quad+\sum_{i=1}^n\sum_{j=1}^np_{ij}(E_j\otimes\mathbb{e}_i\mathbb{e}_j^\top)\frac{\partial \operatorname{vec}(F_{k-w_{ij}})}{\partial p_{uv}},\end{aligned}$$
where the second term on the right hand side satisfies $$\begin{aligned}
\nonumber
(E_v\otimes\mathbb{e}_u\mathbb{e}_v^\top)\operatorname{vec}(F_{k-w_{uv}})
&=\operatorname{vec}(\mathbb{e}_u\mathbb{e}_v^\top F_{k-w_{uv}}E_v)\\
&=(E_vF_{k-w_{uv}}^\top\otimes I_n)\operatorname{vec}(\mathbb{e}_{u}\mathbb{e}_v^\top).\end{aligned}$$ Stacking $\frac{\partial\operatorname{vec}(F_{k})}{\partial p_{uv}}$’s in a matrix as , we obtain .
Since ${{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P)$ only involves $F_k(i,i)$ for $i={\{1,\dots, n\}}$, we only need the corresponding columns in $G_k^\top$ to compute the gradient, which is realized by multiplying the standard unit vector as in .
Iteration is an exponentially stable discrete-time delayed linear system subject to and a finite number of impulse inputs and an exponentially vanishing input. Hence, the state $G_k\to0$ exponentially fast as $k\to\infty$.
Optimizing the truncated entropy via gradient projection
--------------------------------------------------------
Motivated by the previous analysis, we consider the following problem.
\[probLP\] (Maximization of the truncated return time entropy) Given a strongly connected directed graph $\mathcal{G}$ and the stationary distribution $\bm{\pi}$, pick a minimum edge weight $\epsilon>0$ and a truncation accurate parameter $\eta>0$, the maximization of the truncated return time entropy function is as follows. $$\begin{aligned}
& \text{maximize}
& &{{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P)\\
& \text{subject to}
&& P\in\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon\end{aligned}$$
To solve numerically this nonlinear program, we exploit the results in Lemma \[lemma:gradient\] and adopt the gradient projection method as presented in [@DPB:16 Chapter 2.3]:
select: minimum edge weight $\epsilon\ll1$, truncation accuracy $\eta\ll 1$, and initial condition $P_0$ in $\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon$ $\{G_k\}_{k\in{\{1,\dots, N_\eta\}}} :=$ solution to iteration at $P_s$ $\Delta_s := $ gradient of ${{\mathbb{J}}_{\textup{trunc,$\eta$}}}(P_s)$ via equation $P_{s+1}:= \operatorname{projection}_{\mathcal{P}_{\mathcal{G},\bm{\pi}}^\epsilon}(P_s + \text{(step size)} \cdot \Delta_s)$
We analyze the computational complexity of this algorithm. To compute step [[`3:`]{}]{}, we need to evaluate the right-hand side of equation by computing three terms. For the first term, we need to do $m$ comparisons, where $m$ is the number of edges in the graph (i.e., the number of variables in the transition matrix), and it takes $O(m)$ elementary operations. For the second term, note that the matrices $\Phi_i\in\real^{n^2\times{n^2}}$ introduced in equation can be precomputed and is block diagonal with $n$ blocks of size $n\times{n}$. Also note that $G_k\in\real^{n^2\times{n^2}}$ has only $m$ nonzero columns. Thus, we need $O(w_{\max}mn^3)$ operations. For the third term, $F_k$ is updated by equation , which requires $O(w_{\max}n^3)$ and is the main computational cost. Therefore, it takes $O(w_{\max}mn^3)$ to compute one update of iteration . Thus, it takes $O(N_{\eta}w_{\max}mn^3)$ elementary operations to complete step [[`3:`]{}]{}. In step [[`5:`]{}]{}, we need to solve a least square problem with linear equalities and inequalities constraints; which requires $O(m^3)$ [@SB-LV:04].
Numerical results {#sec:Simulation}
=================
In this section, we provide numerical results on the computation of the maximum return time entropy chain (Subsection \[subsec:comp-MRE\]) and its application to robotic surveillance problems (Subsection \[subsec:app-robotics\]). We compute and compare three chains:
1. the Markov chain that maximizes the return time entropy (solution of Problem \[prob:weightedentropy\]), abbreviated as the *MaxReturnEntropy chain*. This chain may be computed for a directed graph with arbitrary integer-valued travel times. Since we do not have a way to solve Problem \[prob:weightedentropy\] directly, the MaxReturnEntropy chain is approximated by the solution of Problem \[probLP\], which is solved via the gradient projection method. Unless otherwise stated, we choose truncation accuracy $\eta =
0.1$. Note that is quite conservative and the actual probabilities being discarded is much less than $0.1$.
2. the Markov chain that maximizes the entropy rate, abbreviated as the *MaxEntropyRate chain*. This chain can be computed for a directed graph with unitary weights via solving a convex program. Further, if the graph is undirected, the MaxEntropyRate chain can be computed efficiently using the method in [@MG-SJ-FB:17b];
3. the Markov chain that minimizes the (weighted) Kemeny constant, abbreviated as the *MinKemeny chain*. This chain may be computed for a directed graph with arbitrary travel times via solving a nonlinear nonconvex program. We compute this chain using the solver implemented in the KNITRO/TOMLAB package.
Computation, comparison and intuitions {#subsec:comp-MRE}
--------------------------------------
We divide this subsection into two parts. In the first part, we first compare $3$ chains on graphs that have unitary travel times. We then summarize several observations in computing the MaxReturnEntropy chain. Finally, we visualize and plot the chains as well as the return time distributions. In the second part, we compare the MaxReturnEntropy chain with the MinKemeny chain on a realistic map taken from [@SA-EF-SLS:14 Section 6.2] with travel times.
x
[**[Chains on graphs with unitary travel times]{}**]{}
*Comparison*: We consider $2$ simple undirected graphs and solve for the MaxReturnEntropy chain, the MaxEntropyRate chain and the MinKemeny chain for each case. We compare the return time entropy, the entropy rate, and the Kemeny constant of these chains in Table \[comptable\]. The stationary distribution of the ring graph is set to be $\bm{\pi}=[1/12,1/6,\dots,1/12,1/6]^\top$, and the stationary distribution of of grid is proportional to the degree of nodes. To evaluate the value of ${\mathbb{J}}(P)$, we set $\eta=10^{-2}$. From the table, we notice that the MaxReturnEntropy chain has the highest value of the return time entropy in both cases. It also has relatively good performance in terms of the entropy rate and the Kemeny constant, which indicates that the MaxReturnEntropy chain is potentially a good combination of speed (expected traversal time) and unpredictability. Furthermore, it is clear that , which characterizes the relationship between the entropy rate and the return time entropy, holds.
Graph Markov chains ${\mathbb{J}}(P)$ ${{\mathbb{H}_{\textup{rate}}}}(P)$
------- ------------------ ------------------- ------------------------------------- ---------
MaxReturnEntropy 2.4927 0.8698 10.0479
MaxEntropyRate 2.3510 0.9883 19.5339
MinKemeny 1.9641 0.4621 6.1667
MaxReturnEntropy 3.6539 0.9491 16.3547
MaxEntropyRate 3.2844 1.4021 30.8661
MinKemeny 2.0990 0.2188 10.0938
: Comparison between different chains on different graphs[]{data-label="comptable"}
*Observations*: In computing the MaxReturnEntropy chain, we observe some interesting properties of our problem. First, when solving Problem \[probLP\] by the gradient projection method with different initial conditions, we found different optimal solutions, and they have slightly different optimal values. This suggests that Problem \[prob:weightedentropy\] is unlikely to be a convex problem. Secondly, the global optimal solution to Problem \[prob:weightedentropy\] is possibly not unique in general. For instance, for an undirected ring graph with even number of nodes and certain stationary distribution, exchanging the probability of going right and that of going left for all nodes does not change the return time entropy. Thirdly, the optimal solution to Problem \[prob:weightedentropy\] is likely to be nonreversible because none of the approximate optimal solutions we have encountered are reversible. This again indicates that the MaxReturnEntropy chain is a good combination of unpredictability and speed. Fourth, even if we set the edge weight $\epsilon=0$, the MaxReturnEntropy chain is always irreducible.
*Intuitions*: In order to provide intuitions for the maximization of the return time entropy, we compare and plot the chains as well as the return time distribution of a same node on the $8$-node ring graph and the $4\times4$ grid graph in Fig. \[Fig:returndistribution\] and Fig. \[Fig:returndistribution\_grid\], respectively. Since the stationary distribution is fixed and identical for all chains in each case, the expectations of the probability mass functions in each figure are the same. From the figures, we note that for the MaxReturnEntropy chain, the return time distribution is reshaped so that the distribution is more spread out and it is more difficult to predict the return time. In contrast, the return time distribution for the MinKemeny chain has a predictable pattern and the return time probability is constantly $0$ for some time intervals. Moreover, from the visualization of the chains, we notice that the MaxReturnEntropy chain has a net flow on the graph, which again indicates its nonreversibility.
[**[MaxReturnEntropy and MinKemeny on a realistic map]{}**]{}
In this part, we compare the MaxReturnEntropy chain with the MinKemeny chain on a realistic map with travel times. The problem data is taken from [@SA-EF-SLS:14 Section 6.2]: a small area in San Francisco (SF) is modeled by a fully connected directed graph with $12$ nodes and by-car travel times on edges measured in seconds. The map is shown in Fig. \[fig:SF\_location\_crime\_map\]. The importance of the a location (node) is characterized by the the number of crimes recorded at that place during a specific period, and the surveillance agent should visit the places with higher crime rate more often. The visit frequency is set to be $ [\frac{133}{866}, \frac{90}{866},\frac{89}{866},\frac{87}{866},
\frac{83}{866}, \frac{83}{866},\frac{74}{866}, \frac{64}{866},
\frac{48}{866}, \frac{43}{866}, \frac{38}{866}, \frac{34}{866}]^\top $. For simplicity, we quantize the travel times by treating a minute as one unit of time, i.e., dividing the travel times by $60$ and round the result to the smallest integer that is larger than it, and by doing so, we have $w_{\max}=9$. The pairwise travel times are recorded in Table \[tb:pairwiseTime\].
![San Francisco (SF) crime map from [@SA-EF-SLS:14 Section 6.2].[]{data-label="fig:SF_location_crime_map"}](SF_crime_map.pdf){width="0.8\linewidth" height=".15\textheight"}
[ ccccccccccccc ]{}
------------------------------------------------------------------------
Location&A&B&C&D&E&F&G&H&I&J&K&L\
------------------------------------------------------------------------
A&1& 3& 3& 5& 4& 6& 3& 5& 7& 4& 6& 6\
B&3& 1& 5& 4& 2& 4& 4& 5& 5& 3& 5& 5\
C&3& 5& 1& 7& 6& 8& 3& 4& 9& 4& 8& 7\
D&6& 4& 7& 1& 5& 6& 4& 7& 5& 6& 6& 7\
E&4& 3& 6& 5& 1& 3& 5& 5& 6& 3& 4& 4\
F&6& 4& 8& 5& 3& 1& 6& 7& 3& 6& 2& 3\
G&2& 5& 3& 5& 6& 7& 1& 5& 7& 5& 7& 8\
H&3& 5& 2& 7& 6& 7& 3& 1& 9& 3& 7& 5\
I&8& 6& 9& 4& 6& 4& 6& 9& 1& 8& 5& 7\
J&4& 3& 4& 6& 3& 5& 5& 3& 7& 1& 5& 3\
K&6& 4& 8& 6& 4& 2& 6& 6& 4& 5& 1& 3\
L&6& 4& 6& 6& 3& 3& 6& 4& 5& 3& 2& 1\
First, we compare three key metrics of the MaxReturnEntropy chain and MinKemeny chain. The results are reported in Table \[tb:SFcompare\]. It can be observed that the MaxReturnEntropy chain is much better than the MinKemeny chain regarding the return time entropy and the entropy rate. This better performance in terms of the unpredictability is obtained at the cost of being slower as indicated by the larger weighted Kemeny constant.
Markov chains ${\mathbb{J}}(P)$ ${{\mathbb{H}_{\textup{rate}}}}(P)$
------------------ ------------------- ------------------------------------- ---------
MaxReturnEntropy 5.0078 1.7810 63.6007
MinKemeny 2.4678 0.6408 24.2824
: Comparison between different chains on SF crime map[]{data-label="tb:SFcompare"}
We also plot the return time distribution of location A in Fig. \[fig:SF\_returndistr\]. Apparently, the MaxReturnEntropy chain spreads the return time probabilities over the possible return times and it is hard to predict the exact time the surveillance agent comes back to the location. In contrast, the MinKemeny chain tries to achieve fast traversal on the graph and the return times distribute over a few intervals.
Application to the Robotic Surveillance Problem {#subsec:app-robotics}
-----------------------------------------------
In this subsection, we provide simulation results in the application of robotic surveillance.
*Setup*: Consider the scenario where a single agent performs the surveillance task by moving randomly according to a Markov chain on the road map. The intruder is able to observe the local behaviors of the surveillance agent, e.g., presence/absence and duration between visits, and he/she plans and decides the time of attack so as to avoid being captured. It takes a certain amount of time for the intruder to complete an attack, which is called the *attack duration* of the intruder. A successful detection/capture happens when the surveillance agent and the intruder are at the same location and the intruder is attacking.
*Intruder model (success probability maximizer with bounded patience)*: Consider a rational intruder that exploits the return time statistics of the Markov chains and chooses an optimal attack time so as to minimize the probability of being captured. The intruder picks a node $i$ to attack randomly according to the stationary distribution, and it collects and learns the probability distribution of node $i$’s first return time. Suppose the intruder and the surveillance agent are at the same node $i$ at the beginning and the attack duration of the intruder is $\tau$. If the intruder observes that the surveillance agent leaves the node and does not come back for $s$ periods, he/she can attack with the probability of being captured given by $$\label{eq:condcumureturn}
\sum_{k=1}^{\tau}\mathbb{P}(T_{ii}=s+k\,|\,T_{ii}> s).$$ Mathematically speaking, is the conditional cumulative return probability for the surveillance agent. Specifically for $s=0$, is the capture probability when the intruder attacks immediately after the agent leaves the node. Then, the optimal time of attack $s_i$ for the intruder is given by $$\label{eq:attackinstant}
s_i=\operatorname*{argmin}_{0\leq s\leq S_i}\{\sum_{k=1}^{\tau}\mathbb{P}(T_{ii}=s+k\,|\,T_{ii}> s)\}.$$ The reason there is an upper bound $S_i$ on $s$ is that the event $T_{ii}> s$ happens with very low probability when $s$ is large, and the intruder may be unwilling to wait for such an event to happen. Let $\delta\in(0,1)$ be the *degree of impatience* of the intruder, then $S_i$ can be chosen as the minimal positive integer such that the following holds, $$\mathbb{P}(T_{ii}\geq S_i) \leq \delta,$$ where a smaller $\delta$ implies a larger $S_i$ and a more patient intruder. In other words, when $\delta$ is small, the intruder is willing to wait for a rare event to happen. Note that the value of $S_i$ is also dependent on the node $i$ that the intruder chooses to attack, and thus the argmin in is over different ranges when the intruder attacks different nodes. In summary, the intruder is dictated by two parameters: the attack duration $\tau$ and the degree of impatience $\delta$, and the strategy for the intruder is as follows: waits until the event that the surveillance agent leaves and does not come back for the first $s_i$ steps happens, then attacks immediately.
From the surveillance point of view, the probability of capturing the rational intruder when he/she attacks node $i$ is $$\mathbb{P}_i(\text{Capture}) = \sum_{k=1}^{\tau}\mathbb{P}(T_{ii}=s_i+k\,|\,T_{ii}> s_i),$$ and the performance of the Markov chains can be evaluated by the total probability of capture as follows $$\label{eq:prob_capture}
\mathbb{P}(\text{Capture}) = \sum_{i=1}^{n}\pi_i\mathbb{P}_i(\text{Capture}).$$
*Simulation results*: Designing an optimal defense mechanism for the rational intruder is an interesting yet challenging problem in its own. Instead, we use the MaxReturnEntropy chain as a heuristic solution and compare its performance with other chains. In the following, we consider two types of graphs: the grid graph and the SF crime map. The degree of impatience of the intruder is set to be $\eta = 0.1$ in this part.
We first consider a $4\times4$ grid and plot the probability of capture defined by for the chains in comparison in Fig. \[fig:comparison\_grid\]. It can be observed that, when defending against the rational intruder described above, the MaxReturnEntropy chain outperforms all other chains when the attack duration of the intruder is small or moderate. The unpredictability in the return time prevents the rational intruder from taking advantage of the visit statistics learned from the observations. The MinKemeny chain, which emphasizes a faster traversal, has a hard time capturing the intruder when the attack duration of the intruder is small. This is because the agent moves in a relatively more predictable way, and the return time statistics may have a pattern that could be exploited. The MaxEntropyRate chain has the in-between performance.
![Performance of different chains on a $4\times4$ grid.[]{data-label="fig:comparison_grid"}](grid_4x4.pdf)
For the SF crime map, we use the same problem data as described in Subsection \[subsec:comp-MRE\]. Since the MaxEntropyRate chain does not generalize to the case when there are travel times, we compare the performance of the MaxReturnEntropy chain and the MinKemny chain. Again, The MaxReturnEntropy chain outperforms the MinKemeny chain when the attack duration of the intruder is relatively small.
![Performance of different chains on the SF crime map.[]{data-label="fig:SF_crime_map"}](SF_intruder_comp_s_60.pdf)
*Summary*: The simulation results presented in this subsection demonstrate that the MaxReturnEntropy chain is an effective strategy against the intruder with reasonable amount of knowledge and level of intelligence, particularly when the attack duration of the intruder is small or moderate. With the property of both unpredictability and speed, the MaxReturnEntropy chain should also work well in a much more broader range of scenarios.
Conclusion {#sec:Conclusion}
==========
In this paper, we proposed and optimized a new metric that quantifies the unpredictability of Markov chains over a directed strongly connected graph with travel times, i.e., the return time entropy. We characterized the return time probabilities and showed that optimizing the return time entropy is a well-posed problem. For the case of unitary travel times, we established an upper bound for the return time entropy by using the maximum entropy principle and obtained an analytic solution for the complete graph. We connected the return time entropy with the well-known entropy rate of Markov chains and showed that the return time entropy is lower bounded by the entropy rate and upper bounded by $n$ times the entropy rate. In order to solve the optimization problem numerically, we approximated the return time entropy as well as a practically useful conditional return time entropy by the truncated return time entropy. We derived the gradient of the truncated return time entropy and proposed to solve the problem by the gradient projection method. We applied our results to the robotic surveillance problem and found that the chain with maximum return time entropy is a good trade-off between speed and unpredictability, and it performs better than several existing chains against a rational intruder.
A number of problems are still open. First of all, a simple closed-form expression for the return time entropy would enable us to establish more properties of the objective function and thus make the optimization problem more tractable. Second, it is interesting to design a best Markov chain directly that defends against the intruder model proposed in this paper. Third, how to generalize the results to the case of multiple robots remains to be investigated. Fourth, we believe there are more application scenarios for Markov chains where the return time entropy is an appropriate quantity to optimize.
[^1]: This work has been supported in part by Air Force Office of Scientific Research award FA9550-15-1-0138.
[^2]: Xiaoming Duan, Mishel George, and Francesco Bullo are with the Mechanical Engineering Department and the Center of Control, Dynamical Systems and Computation, UC Santa Barbara, CA 93106-5070, USA. [{xmduan,mishel,bullo}@engineering.ucsb.edu]{}
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**Mass inequalities for baryons with heavy quarks**
Marek Karliner$^a$[^1] and Jonathan L. Rosner$^b$[^2]
$^a$ [*School of Physics and Astronomy*]{}
*Faculty of Exact Sciences*
*Tel Aviv University, Tel Aviv 69978, Israel*
$^b$ [*Enrico Fermi Institute and Department of Physics*]{}
*University of Chicago, 5620 S. Ellis Avenue, Chicago, IL 60637, USA*
> ABSTRACT
>
> Baryons with one or more heavy quarks have been shown, in the context of a nonrelativistic description, to exhibit mass inequalities under permutations of their quarks, when spin averages are taken. These inequalities sometimes are invalidated when spin-dependent forces are taken into account. A notable instance is the inequality $2E(Mmm) > E(MMm) + E(mmm)$, where $m =
> m_u = m_d$, satisfied for $M = m_b$ or $M = m_c$ but not for $M = m_s$, unless care is taken to remove effects of spin-spin interactions. Thus in the quark-level analog of nuclear fusion, the reactions $\Lambda_b \Lambda_b
> \to \Xi_{bb}N$ and $\Lambda_c \Lambda_c \to \Xi_{cc}^{++}n$ are exothermic, releasing respectively 138 and 12 MeV, while $\Lambda \Lambda \to \Xi N$ is endothermic, requiring an input of between 23 and 29 MeV. Here we explore such mass inequalities in the context of an approach, previously shown to predict masses successfully, in which contributions consist of additive constituent-quark masses, spin-spin interactions, and additional binding terms for pairs each member of which is at least as heavy as a strange quark.
INTRODUCTION \[sec:intro\]
==========================
Quantum chromodynamics (QCD) predicts the existence of baryons containing not only one, but two or three heavy quarks ($c$ or $b$). The LHCb Collaboration at CERN has discovered the first doubly-heavy baryon, a $\Xi_{cc}^{++} = ccu$ state in the decay modes $\Lambda_c K^- \pi^+ \pi^+$ [@Aaij:2017ueg] and $\Xi_c^+ \pi^+$ [@Aaij:2018gfl], at a mass $M(\Xi_{cc}^{++}) = 3621.24 \pm 0.65 \pm 0.31$ MeV, very close to that predicted in Ref. [@Karliner:2014gca]. (Hints of a possible isospin partner $\Xi_{cc}^+$ decaying to $\Lambda_c K^- \pi^+$ have also been found [@Aaij:2019jfq; @KR2019].) In this approach, one adds up constituent-quark masses and spin-spin interactions [@DeRujula:1975qlm; @Lipkin:1978eh; @Gasiorowicz:1981jz] and corrects for the additional binding in any quark pair both of whose members are at least as heavy as a strange quark [@Karliner:2014gca].
Under some circumstances hadrons satisfy mass inequalities associated with permutations of their quarks [@Bertlmann:1979zs; @Richard:1983mu; @Nussinov:1983hb; @Weingarten:1983uj; @Witten:1983ut; @Lieb:1985aw; @Martin:1986da; @Anwar:2017toa]. For example, under some conditions one expects $2E(Mmm) > E(MMm) + E(mmm)$, where $E$ denotes the mass, $m = m_u = m_d$, and $M$ is the mass of a heavy quark, to apply to spin-averaged states \[cf. Eq. (3) in Ref. [@Martin:1986da]\]. This inequality may be invalidated when spin-dependent forces are taken into account; it holds for $M = m_b$ or $M = m_c$ but not for $M = m_s$. Thus, in the quark-level analog of nuclear fusion [@Karliner:2017elp], the reactions $\Lambda_b
\Lambda_b \to \Xi_{bb}N$ and $\Lambda_c \Lambda_c\to\Xi_{cc}^{++}n$ are exothermic, releasing respectively 138 and 12 MeV, while $\Lambda \Lambda
\to \Xi N$ is endothermic, requiring energy input of between 23 and 29 MeV, depending on which members of the $N$ and $\Xi$ doublets one uses.
Here we give some examples of inequalities involving baryon masses in our constituent-quark approach. We outline in Sec. \[sec:rels\] some of the relations and their origin. In Section \[sec:light\] we treat light-quark systems in which the strange quark plays the role of a heavy quark. Baryons with one and two heavy quarks ($c,b$) are described in Secs. \[sec:one\] and \[sec:two\], respectively, while Sec. \[sec:concl\] concludes.
Inequalities and their origin {#sec:rels}
=============================
A number of mass inequalities involving ground-state mesons and baryons were noted by Nussinov [@Nussinov:1983hb]: respectively, \[eqn:neq\] m\_[x |y]{} > (m\_[x |x]{} + m\_[y |y]{}) , m\_[x y y]{} > (m\_[xxy]{} + m\_[yyy]{}) . We shall motivate these relations in a simple case where quark masses enter through their nonrelativistic kinetic energy, but they are much more general (see many of the references quoted above, in particular [@Martin:1986da]). Consider systems governed by the Hamiltonians H\_[ij]{} = + V() , \_[ij]{} . Then, since = + , we have $H_{12} = \frac{1}{2} (H_{11} + H_{22})$. The ground-state energy in the 12 system is E\_[12]{} = [Min]{}\_\_[12]{}|H\_[12]{}|\_[12]{} = \_[12]{}| (H\_[11]{} + H\_[22]{} )|\_[12]{} . Now \_[12]{}|H\_[11]{}|\_[12]{} > E\_1 ; \_[12]{}|H\_[22]{}|\_[12]{} > E\_2 , since $\psi_{12}$ is not the ground state (assumed non-degenerate) of $H_{11}$ or $H_{22}$. Hence we have the result $E_{12} > (E_1+E_2)/2$, implying Eq. (\[eqn:neq\]).
When spin-spin interactions are taken into account, the energy shift due to the interaction of quarks $i$ and $j$ may be written \[eqn:hfs\] E\_[ij,[HFs]{}]{}=b\_i \_j /(m\_i m\_j) . Inclusion of such terms in the Hamiltonian in general spoils the relation
Baryons with light quarks {#sec:light}
=========================
If we adopt the semi-empirical model of hadron masses [@Karliner:2014gca; @Karliner:2019lau] in which the constituent $u,d,s$ quarks have constituent masses of several hundred MeV, the only corrections to the sum of quark masses, are the sum of hyperfine terms (\[eqn:hfs\]) and terms $B(ss)$ expressing the stronger binding of one strange quark with another. Here the tilde stands for a mass without hyperfine interaction terms. The Gell-Mann–Okubo relation derived under the assumption that SU(3) breaking is linear in hypercharge, then becomes an inequality = 2 m\_q + m\_s - B(ss)/2 < = 2 m\_q + m\_s , and the second inequality in Eq. (\[eqn:neq\]) is satisfied. (It turns out that the experimental values of octet baryon masses [@Tanabashi:2018oca] also satisfy this relation, with $[M(N) + M(\Xi)]/2 = 1128.6$ MeV and $[3M(\Lambda) + M(\Sigma)]/4 = 1135.0$ MeV.)
Baryons with one heavy quark {#sec:one}
============================
The masses of ground state baryons $cqq$, $cqs$, and $css$ containing a single charmed quark satisfy the inequality /2 < M(\_c) , thanks to the binding correction $B(ss) = 9.2$ MeV [@Karliner:2019lau]:[^3] /2 & = & m\_q+m\_s+ m\_c -B(cs)-B(ss)/2\
< M(\_c) & = & m\_q + m\_s + m\_c - B(cs) .
The corresponding relation for states containing a $b$ quark is /2 & = & m\_q+m\_s+ m\_b -B(bs)-B(ss)/2\
< M(\_b) & = & m\_q + m\_s + m\_b - B(bs) , with $B(bs) = 41.8$ MeV [@Karliner:2014gca] common to both sides. Again, the term $B(ss)$ turns the equality into an inequality.
Baryons with two or more heavy quarks {#sec:two}
=====================================
In this section we discuss inequalities between binding energies. In our convention the binding energies are positive, but they contribute to the hadron mass with a minus sign, so the direction of the inequalities between binding energies is flipped vs. inequalities between hadron masses.
We have seen that the second of Eqs. (\[eqn:neq\]) holds for baryons with zero or one heavy ($c,b$) quarks. Inequalities also hold in our semi-empirical approach when two quarks are heavy. For example [@Karliner:2014gca; @Karliner:2019lau], $B(ss) = 9.2$ MeV, $B(cs) = 35.0$ MeV, $B(cc) = 129.0$ MeV, so B(cs) < \[B(ss) + B(cc)\]/2 . Similarly $B(bs) = 41.8$ MeV, $B(bb) = 281.4$ MeV, so B(bs) < \[B(ss) + B(bb)\]2 . It was also found [@Karliner:2014gca] that $B(bc) = 170.8$ MeV, so B(bc) = 170.8 [MeV]{} < \[B(cc) + B(bb)\]/2 = 205.2 [MeV]{} . A related inequality for the spin-weighted averages $\bar M$ of heavy $c \bar
c$, $b \bar b$, and $b \bar c$ ground-state mesons, 2 < |M(b |c) , is comfortably satisfied: The left-hand side is $[3068.7 + 9444.9]/2=6256.8$ MeV, while the right-hand side is likely [@Eichten:2019gig] to be tens of MeV above the mass $M(B_c) = 6274.9 \pm 0.8$ MeV [@Tanabashi:2018oca] of the spin-zero pseudoscalar $b \bar c$ ground state.
Conclusions {#sec:concl}
===========
A semi-empirical method [@Karliner:2014gca] successfully determines hadron masses, including the mass of the first doubly charmed baryon. This method sums constituent-quark masses, quark-quark hyperfine interactions, and terms $B(qq')$ expressing the binding of quarks both of which are at least as heavy as the strange quark. These binding terms are seen to satisfy inequalities $B(qq') < [B(qq) + B(q'q')]/2$, with the consequence that when hyperfine contributions are removed, baryons satisfy the inequality $m_{x y y} >
\frac{1}{2}(m_{xxy} + m_{yyy})$ [@Bertlmann:1979zs; @Richard:1983mu; @Nussinov:1983hb; @Weingarten:1983uj; @Witten:1983ut; @Lieb:1985aw; @Martin:1986da; @Anwar:2017toa]. This constitutes a useful consistency check of the semi-empirical method, and enables rough estimates, independent of potential models, of unseen hadron masses.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
The research of M.K. was supported in part by NSFC-ISF grant No. 0603219411. We thank J.-M. Richard for a helpful discussion.
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[^1]: [[email protected]]{}
[^2]: [[email protected]]{}
[^3]: A term $B(cs) =
35.0$ MeV [@Karliner:2014gca] is common to both sides.
|
---
abstract: 'The regular fractional transformations of the extended quaternionic space have been recently introduced as variants of the classical linear fractional transformations. These variants have the advantage of being included in the class of slice regular functions, introduced by Gentili and Struppa in 2006, so that they can be studied with the useful tools available in this theory. We first consider their general properties, then focus on the regular Möbius transformations of the quaternionic unit ball ${{\mathbb{B}}}$, comparing the latter with their classical analogs. In particular we study the relation between the regular Möbius transformations and the Poincaré metric of ${{\mathbb{B}}}$, which is preserved by the classical Möbius transformations. Furthermore, we announce a result that is a quaternionic analog of the Schwarz-Pick lemma.'
author:
- |
Cinzia Bisi[^1]\
Università degli Studi di Ferrara\
Dipartimento di Matematica e Informatica\
Via Machiavelli 35, 44121 Ferrara, Italy\
[email protected]
- |
Caterina Stoppato$^*$ [^2]\
Università degli Studi di Milano\
Dipartimento di Matematica “F. Enriques”\
Via Saldini 50, 20133 Milano, Italy\
[email protected]\
bibliography:
- 'BisiStoppatoArXiv.bib'
date:
title: |
**Regular vs. classical\
Möbius transformations of the quaternionic unit ball**
---
Classical Möbius transformations and\
Poincaré distance on the quaternionic unit ball {#classicM}
===============================================
A classical topic in quaternionic analysis is the study of Möbius transformations. It is well known that the set of *linear fractional transformations* of the extended quaternionic space ${{\mathbb{H}}}\cup \{\infty\} \cong \mathbb{HP}^1$ $$\label{IndianaM}
\mathbb{G}=\left\{g(q)=(aq+b)\cdot (cq+d)^{-1} \ \left|\ \begin{bmatrix}
a & b \\
c & d
\end{bmatrix} \in GL(2,{{\mathbb{H}}}) \right\}\right.$$ is a group with respect to the composition operation. We recall that $GL(2,{{\mathbb{H}}})$ denotes the group of $2 \times 2$ invertible quaternionic matrices, and that $SL(2,{{\mathbb{H}}})$ denotes the subgroup of those such matrices which have unit Dieudonné determinant (for details, see [@poincare] and references therein). It is known in literature that $\mathbb{G}$ is isomorphic to $PSL(2,{{\mathbb{H}}})= SL(2,{{\mathbb{H}}}) / \{\pm {{\mathrm{Id}}}\}$ and that all of its elements are conformal maps. Among the works that treat this matter, even in the more general context of Clifford Algebras, let us mention [@ahlforsmoebius; @maass; @vahlen]. The alternative representation $$\label{InvM}
\mathbb{G}=\left\{F_A(q)=(qc+d)^{-1} \cdot (qa+b) \ \left|\ A=\begin{bmatrix}
a & c \\
b & d
\end{bmatrix} \in GL(2,{{\mathbb{H}}}) \right\}\right.$$ is also possible, and the anti-homomorphism $GL(2,{{\mathbb{H}}}) \to \mathbb{G}$ mapping $A$ to $F_A$ induces an anti-isomorphism between $PSL(2,{{\mathbb{H}}})$ and $\mathbb{G}$. The group $\mathbb{G}$ is generated by the following four types of transformations:
- $L_1(q)=q+b,\,\,\, b \in \mathbb{H};$
- $L_2(q)=q \cdot a, \,\,\, a \in \mathbb{H}, |a|=1;$
- $L_3(q)= r \cdot q = q \cdot r, \,\,\, r \in \mathbb{R}^+ \setminus \{ 0 \};$
- $L_4(q)=q^{-1}.$
Moreover, if $\mathcal{S}_i$ is the family of all real $i$-dimensional spheres, if $\mathcal{P}_i$ is the family of all real $i$-dimensional affine subspaces of $\mathbb{H}$ and if $\mathcal{F}_i=\mathcal{S}_i \cup \mathcal{P}_i$ then $\mathbb{G}$ maps elements of $\mathcal{F}_i$ onto elements of $\mathcal{F}_i,$ for $i=3,2,1$. At this regard, see [@wilker]; detailed proofs of all these facts can be found in [@poincare].
The subgroup $\mathbb{M}\leq\mathbb{G}$ of *(classical) Möbius transformations* mapping the quaternionic open unit ball $$\mathbb{B}=\{q \in {{\mathbb{H}}}\ |\ |q|<1\}$$ onto itself, has also been studied in detail. Let us denote $H =
\begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}$ and $$Sp(1,1) = \{C \in GL(2, {{\mathbb{H}}}) \ |\ \overline C ^t H C = H\}\subset SL(2, {{\mathbb{H}}}).$$
An element $g \in \mathbb{G}$ is a classical Möbius transformation of ${{\mathbb{B}}}$ if and only if $g(q)=(qc+d)^{-1} \cdot (qa+b)$ with $\begin{bmatrix}
a & c\\
b & d
\end{bmatrix}\in Sp(1,1)$. This is equivalent to $$g(q) = v^{-1} (1-q\bar q_0)^{-1}(q-q_0)u$$ for some $u,v \in \partial {{\mathbb{B}}}, q_0 \in {{\mathbb{B}}}$.
For a proof, see [@poincare]. Hence, $\mathbb{M}$ is anti-isomorphic to $Sp(1,1)/\{\pm {{\mathrm{Id}}}\}$. Since $\mathbb{G}$ leaves invariant the family $\mathcal{F}_1$ of circles and affine lines of $\mathbb{H}$, and since the elements of $\mathbb{G}$ are conformal, the group $\mathbb{M}$ of classical Möbius transformations of ${{\mathbb{B}}}$ preserves the following class of curves.
\[non-euclid\] If $q_1 \neq q_2 \in {{\mathbb{B}}}$ are $\mathbb{R}-$linearly dependent, then the diameter of ${{\mathbb{B}}}$ through $q_1,q_2$ is called the *non-Euclidean line* through $q_1$ and $q_2$. Otherwise, the *non-Euclidean line* through $q_1$ and $q_2$ is defined as the unique circle through $q_1,q_2$ that intersects $\partial \mathbb{B}= {{\mathbb{S}}}^3$ orthogonally.
The formula $$\label{poincaredist}
\delta_{\mathbb{B}}(q_1,q_2)
=\frac{1}{2}\log\left(\frac{1+|1-q_1\overline{q_2}|^{-1}|q_1-q_2|}{1-|1-q_1\overline{q_2}|^{-1}|q_1-q_2|}\right)$$ (for $q_1,q_2 \in{{\mathbb{B}}}$) defines a distance that has the non-Euclidean lines as its geodesics. The elements of $\mathbb{M}$ and the map $q \mapsto \overline{q}$ are all isometries for $\delta_{\mathbb{B}}$.
We refer the reader to [@ahlfors1988]; a detailed presentation can be found in [@poincare].
So far, we mentioned properties of the classical Möbius transformations that are completely analogous to the complex case. However, the analogy fails on one crucial point. The group $\mathbb{M}$ is not included in the best known analog of the class of holomorphic functions: the set of Fueter regular functions, i.e., the kernel of $\frac{\partial}{\partial x_0}+i\frac{\partial}{\partial x_1}+ j\frac{\partial}{\partial x_2}+k\frac{\partial}{\partial x_3}$ (see [@sudbery]). For instance, none of the rotations $q \mapsto aq$ with $a \in {{\mathbb{H}}}, a \neq 0$ is Fueter regular, nor are any of the transformations listed as (i),(ii),(iii),(iv) in our previous discussion. The variant of the Fueter class considered in [@perotti2009], defined as the kernel of $\frac{\partial}{\partial x_0}+i\frac{\partial}{\partial x_1}+ j\frac{\partial}{\partial x_2}-k\frac{\partial}{\partial x_3}$, includes part of the group, for instance the rotations $q \mapsto qb$ for $b \in {{\mathbb{H}}}, b \neq 0$, but not all of it (for instance $q \mapsto kq$ is not in the kernel, nor is $q \mapsto q^{-1}$).
A more recent theory of quaternionic functions, introduced in [@cras; @advances], has proven to be more comprehensive. The theory is based on the next definition.
Let $\Omega$ be a domain in ${{\mathbb{H}}}$ and let $f : \Omega \to {{\mathbb{H}}}$ be a function. For all $I \in {{\mathbb{S}}}= \{q \in {{\mathbb{H}}}\ |\ q^2 = -1\}$, let us denote $L_I = {{\mathbb{R}}}+ I {{\mathbb{R}}}$, $\Omega_I = \Omega \cap L_I$ and $f_I = f_{|_{\Omega_I}}$. The function $f$ is called (Cullen or) *slice regular* if, for all $I \in {{\mathbb{S}}}$, the function $\bar \partial_I f : \Omega_I \to {{\mathbb{H}}}$ defined by $$\bar \partial_I f (x+Iy) = \frac{1}{2} \left( \frac{\partial}{\partial x}+I\frac{\partial}{\partial y} \right) f_I (x+Iy)$$ vanishes identically.
By direct computation, the class of slice regular functions includes all of the generators we listed as (i),(ii),(iii),(iv). It does not contain the whole group $\mathbb{G}$ (nor its subgroup $\mathbb{M}$), because composition does not, in general, preserve slice regularity. However, [@moebius] introduced the new classes of *(slice) regular fractional transformations* and *(slice) regular Möbius transformations* of ${{\mathbb{B}}}$, which are nicely related to the classical linear fractional transformations. They are presented in detail in sections \[regularfractional\] and \[regularmoebius\].
One of the purposes of the present paper is, in fact, to compare the slice regular fractional transformations with the classical ones. Furthermore, we take a first glance at the role played by slice regular Möbius transformations in the geometry of ${{\mathbb{B}}}$. In section \[differential\], we undertake a first study of their differential properties: we prove that they are not, in general, conformal, and that they do not preserve the Poincaré distance $\delta_{{\mathbb{B}}}$. In section \[schwarzpick\], we overview the main results of [@schwarzpick]. Among them is a quaternionic analog of the Schwarz-Pick lemma, which discloses the possibility of using slice regular functions in the study of the intrinsic geometry of ${{\mathbb{B}}}$.
Regular fractional transformations {#regularfractional}
==================================
This section surveys the algebraic structure of slice regular functions, and its application to the construction of regular fractional transformations. From now on, we will omit the term ‘slice’ and refer to these functions as regular, [*tout court*]{}. Since we will be interested only in regular functions on Euclidean balls $$B(0,R) = \{q \in {{\mathbb{H}}}\ |\ |q| <R\},$$ or on the whole space ${{\mathbb{H}}}= B(0,+\infty)$, we will follow the presentation of [@zeros; @poli]. However, we point out that many of the results we are about to mention have been generalized to a larger class of domains in [@advancesrevised].
Fix $R$ with $0<R\leq + \infty$ and let $\mathcal{D}_R$ be the set of regular functions $f:B(0,R)\to {{\mathbb{H}}}$. Then $\mathcal{D}_R$ coincides with the set of quaternionic power series $f(q) =\sum_{n \in {{\mathbb{N}}}} q^n a_n$ (with $a_n \in {{\mathbb{H}}}$) converging in $B(0,R)$. Moreover, defining the *regular multiplication* $*$ by the formula $$\left(\sum_{n \in {{\mathbb{N}}}} q^n a_n\right) * \left(\sum_{n \in {{\mathbb{N}}}} q^n b_n\right)= \sum_{n \in {{\mathbb{N}}}} q^n \sum_{k=0}^n a_k b_{n-k},$$ we conclude that $\mathcal{D}_R$ is an associative real algebra with respect to $+,*$.
The ring $\mathcal{D}_R$ admits a classical ring of quotients $$\mathcal{L}_R = \{f^{-*}*g \ |\ f,g \in \mathcal{D}_R, f \not \equiv 0\}.$$ In order to introduce it, we begin with the following definition.
\[conjugate\] Let $f(q) = \sum_{n \in {{\mathbb{N}}}} q^n a_n$ be a regular function on an open ball $B = B(0,R)$. The [regular conjugate]{.nodecor} of $f$, $f^c : B \to {{\mathbb{H}}}$, is defined as $f^c(q) = \sum_{n \in {{\mathbb{N}}}} q^n \bar a_n$ and the [symmetrization]{.nodecor} of $f$, as $f^s = f * f^c = f^c*f$.
Notice that $f^s(q) = \sum_{n \in {{\mathbb{N}}}} q^n r_n$ with $r_n = \sum_{k = 0}^n a_k \bar a_{n-k} \in {{\mathbb{R}}}$. Moreover, the zero-sets of $f^c$ and $f^s$ have been fully characterized.
\[conjugatezeros\] Let $f$ be a regular function on $B = B(0,R)$. For all $x,y \in {{\mathbb{R}}}$ with $x+y{{\mathbb{S}}}\subseteq B$, the regular conjugate $f^c$ has as many zeros as $f$ in $x+y{{\mathbb{S}}}$. Moreover, the zero set of the symmetrization $f^s$ is the union of the $x+y{{\mathbb{S}}}$ on which $f$ has a zero.
We are now ready for the definition of regular quotient. We denote by $${{\mathcal{Z}}}_h = \{q \in B \ |\ h(q) = 0\}$$ the zero-set of a function $h$.
\[quotient\] Let $f,g : B = B(0,R) \to {{\mathbb{H}}}$ be regular functions. The *left regular quotient* of $f$ and $g$ is the function $f^{-*} * g$ defined in $B \setminus {{\mathcal{Z}}}_{f^s}$ by $$f^{-*} * g (q) = {f^s(q)}^{-1} f^c * g(q).$$ Moreover, the *regular reciprocal* of $f$ is the function $f^{-*} = f^{-*} * 1$.
Left regular quotients proved to be regular in their domains of definition. If we set $(f^{-*}*g)*(h^{-*}*k) = (f^{s}h^{s})^{-1} f^c*g*h^c*k$ then $(\mathcal{L}_R,+,*)$ is a division algebra over $\mathbb{R}$ and it is the classical ring of quotients of $(\mathcal{D}_R,+,*)$ (see [@rowen]). In particular, $\mathcal{L}_R$ coincides with the set of *right regular quotients* $$g*h^{-*} (q) = {h^s(q)}^{-1} g * h^c(q).$$ The definition of regular conjugation and symmetrization is extended to $\mathcal{L}_R$ setting $(f^{-*}*g)^c = g^c*(f^c)^{-*}$ and $(f^{-*}*g)^s(q) = {f^s(q)}^{-1}g^s(q)$. Furthermore, the following relation between the left regular quotient $f^{-*} * g(q)$ and the quotient $f(q)^{-1} g (q)$ holds.
\[quotients\] Let $f,g$ be regular functions on $B=B(0,R)$. Then $$f*g(q) = f(q) g(f(q)^{-1}qf(q)),$$ and setting $T_f(q) = f^c(q)^{-1} q f^c(q)$ for all $q \in B \setminus {{\mathcal{Z}}}_{f^s}$, $$f^{-*}*g(q) = f(T_f(q))^{-1} g(T_f(q)),$$ for all $q \in B \setminus {{\mathcal{Z}}}_{f^s}$. For all $x,y \in {{\mathbb{R}}}$ with $x+y{{\mathbb{S}}}\subset B \setminus {{\mathcal{Z}}}_{f^s}$, the function $T_f$ maps $x+y{{\mathbb{S}}}$ to itself (in particular $T_f(x) = x$ for all $x \in {{\mathbb{R}}}$). Furthermore, $T_f$ is a diffeomorphism from $B \setminus {{\mathcal{Z}}}_{f^s}$ onto itself, with inverse $T_{f^c}$.
We point out that, so far, no simple result relating $g*h^{-*}(q)$ to $g(q) h(q)^{-1}$ is known.
This machinery allowed the introduction in [@moebius] of regular analogs of linear fractional transformations, and of Möbius transformations of ${{\mathbb{B}}}$. To each $A=\begin{bmatrix}
a & c \\
b & d
\end{bmatrix} \in GL(2, {{\mathbb{H}}})$ we can associate the *regular fractional transformation* $${{\mathcal{F}}}_A (q) = (qc+d)^{-*}*(qa+b).$$ By the formula $(qc+d)^{-*}*(qa+b)$ we denote the aforementioned left regular quotient $f^{-*}*g$ of $f(q) = qc+d$ and $g(q) = qa +b$. We denote the $2 \times 2$ identity matrix as ${{\mathrm{Id}}}$.
Choose $R>0$ and consider the ring of quotients of regular quaternionic functions in $B(0,R)$, denoted by $\mathcal{L}_R$. Setting $$f.A = (f c + d)^{-*}*(f a+ b)$$ for all $f \in \mathcal{L}_R$ and for all $A =
\begin{bmatrix}
a & c \\
b & d
\end{bmatrix} \in GL(2,{{\mathbb{H}}})$ defines a right action of $GL(2,{{\mathbb{H}}})$ on $\mathcal{L}_R$. A left action of $GL(2,{{\mathbb{H}}})$ on $\mathcal{L}_R$ is defined setting $$^{t}A.f=(a*f+b)*(c*f+d)^{-*}.$$ The stabilizer of any element of $\mathcal{L}_R$ with respect to either action includes the normal subgroup $N=\left \{t \cdot {{\mathrm{Id}}}\ |\ t \in {{\mathbb{R}}}\setminus \{0\} \right \}\unlhd GL(2,{{\mathbb{H}}})$. Both actions are faithful when reduced to $PSL(2,{{\mathbb{H}}}) = GL(2,{{\mathbb{H}}})/N$.
The statements concerning the right action are proven in [@moebius], the others can be similarly derived. The two actions coincide in one special case.
\[hermitian\] For all Hermitian matrices $A = \begin{bmatrix}
a & \bar b \\
b & d
\end{bmatrix}$ with $a,d \in {{\mathbb{R}}}, b \in {{\mathbb{H}}}$, $$f.A = (f \bar b + d)^{-*}*(f a+ b) =(a*f+b)*(\bar b*f+d)^{-*}= A^{t}.f$$
We observe that $$(f \bar b + d)^{-*}*(f a+ b) =(a*f+b)*(\bar b*f+d)^{-*}$$ if, and only if, $$(f a+ b)*(\bar b*f+d) =(f \bar b + d)*(a*f+b),$$ which is equivalent to $$a f* \bar b*f + |b|^2 f+ ad f + db = a f* \bar b*f +ad f+ |b|^2 f +db.$$
In general, a more subtle relation holds between the two actions.
\[leftright\] For all $A \in GL(2,{{\mathbb{H}}})$ and for all $f\in \mathcal{L}_R$, by direct computation $$\left(f.A\right)^c=\bar{A}^t.f^c$$ As a a consequence, if $A\in GL(2, {{\mathbb{H}}})$ is Hermitian then $(f.A)^c= f^c.\bar{A}$.
Interestingly, neither action is free, not even when reduced to $PSL(2,{{\mathbb{H}}})$. Indeed, the stabilizer of the identity function with respect to either action of $GL(2,{{\mathbb{H}}})$ equals $$\{c\cdot {{\mathrm{Id}}}\ |\ c \in {{\mathbb{H}}}\setminus \{0\} \},$$ a subgroup of $GL(2,{{\mathbb{H}}})$ that strictly includes $N$ and is not normal. As a consequence, the set of regular fractional transformations $${{\mathfrak{G}}}= \{{{\mathcal{F}}}_A \ |\ A \in GL(2,{{\mathbb{R}}})\} = \{{{\mathcal{F}}}_A \ |\ A \in SL(2,{{\mathbb{R}}})\},$$ which is the orbit of the identity function $id = {{\mathcal{F}}}_{{{\mathrm{Id}}}}$ under the right action of $GL(2,{{\mathbb{H}}})$ on $\mathcal{L}_\infty$, does not inherit a group structure from $GL(2,{{\mathbb{H}}})$.
The set ${{\mathfrak{G}}}$ of regular fractional transformations is also the orbit of $id$ with respect to the left action of $GL(2,{{\mathbb{H}}})$ on $\mathcal{L}_\infty$.
Let $A=\begin{bmatrix}
a & c \\
b & d
\end{bmatrix} \in GL(2,{{\mathbb{H}}})$, and let us prove that ${{\mathcal{F}}}_A = id.A$ can also be expressed as $C.id$ for some $C \in GL(2,{{\mathbb{H}}})$. If $c=0$ then $${{\mathcal{F}}}_A(q) = d^{-1}*(qa+b) = (d^{-1}a)*q+ d^{-1}b,$$ else $${{\mathcal{F}}}_A(q) = {{\mathcal{F}}}_{c^{-1}A}(q) = (q-p)^{-*}*(q\alpha +\beta) = [(q-p)^s]^{-1} (q-\bar p)*(q\alpha + \beta)$$ for some $p, \alpha, \beta \in {{\mathbb{H}}}$. If $p= x+Iy$ then there exists $\tilde p \in x+y{{\mathbb{S}}}$ and $\gamma, \delta \in {{\mathbb{H}}}$ such that $ (q-\bar p)*(q\alpha + \beta) = (q \gamma + \delta)*(q-\tilde p)$; additionally, $(q-p)^s = (q-\tilde p)^s$ (see [@zeros] for details). Hence, $${{\mathcal{F}}}_A(q) = [(q-\tilde p)^s]^{-1} (q \gamma + \delta)*(q-\tilde p) = (q \gamma + \delta) * (q-\overline{\tilde p})^{-*} = (\gamma* q + \delta) * (q-\overline{\tilde p})^{-*},$$ which is of the desired form. Similar manipulations prove that for all $C \in GL(2,{{\mathbb{H}}})$, the function $C.id$ equals ${{\mathcal{F}}}_A = id.A$ for some $A \in GL(2,{{\mathbb{H}}})$.
We now state an immediate consequence of the previous lemma and of remark \[leftright\].
The set ${{\mathfrak{G}}}$ of regular fractional transformations is preserved by regular conjugation.
Regular Möbius transformations of ${{\mathbb{B}}}$ {#regularmoebius}
==================================================
The regular fractional transformations that map the open quaternionic unit ball ${{\mathbb{B}}}$ onto itself, called *regular Möbius transformations of ${{\mathbb{B}}}$*, are characterized by two results of [@moebius].
For all $A \in SL(2, {{\mathbb{H}}})$, the regular fractional transformation ${{\mathcal{F}}}_A$ maps ${{\mathbb{B}}}$ onto itself if and only if $A \in Sp(1,1)$, if and only if there exist (unique) $u \in \partial {{\mathbb{B}}}, a \in {{\mathbb{B}}}$ such that $${{\mathcal{F}}}_A(q) = (1-q \bar a)^{-*}*(q-a)u.$$ In particular, the set $\mathfrak{M}= \{f \in {{\mathfrak{G}}}\ |\ f({{\mathbb{B}}}) = {{\mathbb{B}}}\}$ of the regular Möbius transformations of ${{\mathbb{B}}}$ is the orbit of the identity function under the right action of $Sp(1,1)$.
The class of regular bijective functions $f : {{\mathbb{B}}}\to {{\mathbb{B}}}$ coincides with the class $\mathfrak{M}$ of regular Möbius transformations of ${{\mathbb{B}}}$.
As a consequence, the right action of $Sp(1,1)$ preserves the class of regular bijective functions from ${{\mathbb{B}}}$ onto itself. We now study, more in general, the effect of the actions of $Sp(1,1)$ on the class $${\frak{Reg}({{\mathbb{B}}},{{\mathbb{B}}})}= \{f:{{\mathbb{B}}}\to {{\mathbb{B}}}\ |\ f \mathrm{\ is\ regular}\}.$$
If $f \in {\frak{Reg}({{\mathbb{B}}},{{\mathbb{B}}})}$ then for all $a \in {{\mathbb{B}}}$ $$(1-f \bar a)^{-*}*(f-a) =(f-a)*(1-\bar a*f)^{-*}.$$ Moreover, the left and right actions of $Sp(1,1)$ preserve ${\frak{Reg}({{\mathbb{B}}},{{\mathbb{B}}})}$.
The fact that $(1-f\bar a)^{-*}*(f-a)=(f-a)*(1-\bar a*f)^{-*}$ is a consequence of proposition \[hermitian\].
Let us turn to the second statement, proving that for all $a \in {{\mathbb{B}}}, u,v \in \partial {{\mathbb{B}}}$, the function $$\tilde f =v^{-1}*(1-f\bar a)^{-*}*(f-a) u = (v-f\bar av)^{-*}*(f-a) u$$ is in ${\frak{Reg}({{\mathbb{B}}},{{\mathbb{B}}})}$. The fact that for all $a \in {{\mathbb{B}}}, u,v \in \partial {{\mathbb{B}}}$ the function $u*(f-a)*(1-\bar a*f)^{-*} *v^{-*}$ belongs to ${\frak{Reg}({{\mathbb{B}}},{{\mathbb{B}}})}$ will then follow from the equality just proven.
The function $\tilde f$ is regular in ${{\mathbb{B}}}$ since $h=v- f \bar av$ has no zero in ${{\mathbb{B}}}$ (as a consequence of the fact that $|a|<1$, $|f|<1$ and $|v|=1$). Furthermore, $$\tilde f = (v- (f \circ T_h) \bar av)^{-1}(f\circ T_h-a)u=$$ $$v^{-1}(1- (f \circ T_h) \bar a)^{-1}(f\circ T_h-a)u =v^{-1}(M_a \circ f \circ T_h)u$$ where $T_h$ and $M_a(q) = (1-q \bar a)^{-1}(q-a)$ map ${{\mathbb{B}}}$ to itself bijectively and $u,v \in \partial{{\mathbb{B}}}$. Hence, $\tilde f=v^{-*}*(1-f\bar a)^{-*}*(f-a) *u \in {\frak{Reg}({{\mathbb{B}}},{{\mathbb{B}}})}$, as desired.
As a byproduct, we obtain that the orbit of the identity function under the left action of $Sp(1,1)$ equals $\mathfrak{M}$.
If $f\in {\frak{Reg}({{\mathbb{B}}},{{\mathbb{B}}})}$ then its regular conjugate $f^c$ belongs to ${\frak{Reg}({{\mathbb{B}}},{{\mathbb{B}}})}$ as well. Furthermore, $f^c$ is bijective (hence an element of $\mathfrak{M}$) if and only if $f$ is.
Suppose $f^c(p) = a \in {{\mathbb{H}}}\setminus {{\mathbb{B}}}$ for some $p = x+Iy \in {{\mathbb{B}}}$. Then $p$ is a zero of the regular function $f^c-a$. By theorem \[conjugatezeros\], there exists $\tilde p \in x+y{{\mathbb{S}}}\subset {{\mathbb{B}}}$ such that $(f^c-a)^c = f-\bar a$ vanishes at $\tilde p$. Hence, $f({{\mathbb{B}}})$ includes $\bar a \in {{\mathbb{H}}}\setminus{{\mathbb{B}}}$, a contradiction with the hypothesis $f({{\mathbb{B}}}) \subseteq {{\mathbb{B}}}$.
As for the second statement, $f^c$ is bijective if and only if, for all $a \in {{\mathbb{B}}}$, there exists a unique $p \in {{\mathbb{B}}}$ such that $f^c(p) =a$. Reasoning as above, we conclude that this happens if and only if for all $a \in {{\mathbb{B}}}$, there exists a unique $\tilde p \in {{\mathbb{B}}}$ such that $f(\tilde p) = \bar a$. This is equivalent to the bijectivity of $f$.
Differential and metric properties of regular Möbius transformations {#differential}
====================================================================
The present section is concerned with two natural questions:
1. whether the regular Möbius transformations are conformal (as the classical Möbius transformations);
2. whether they preserve the quaternionic Poincaré distance defined on ${{\mathbb{B}}}$ by formula (\[poincaredist\]).
For a complete description of the Poincaré metric, see [@poincare]. In order to answer question (a), we will compute for a regular Möbius transformation the series development introduced by the following result of [@expansion]. Let us set $$U(x_0+y_0{{\mathbb{S}}},r) = \{q \in {{\mathbb{H}}}\ |\ |(q-x_0)^2+y_0^2| < r^2\}$$ for all $x_0,y_0 \in {{\mathbb{R}}}$, $r>0$.
Let $f$ be a regular function on $\Omega=B(0,R)$, and let $U(x_0+y_0{{\mathbb{S}}},r) \subseteq \Omega$. Then for each $q_0 \in x_0+y_0 {{\mathbb{S}}}$ there exists $\{A_n\}_{n \in {{\mathbb{N}}}} \subset {{\mathbb{H}}}$ such that $$\label{expansion}
f(q) = \sum_{n \in {{\mathbb{N}}}} [(q-x_0)^2+y_0^2]^n [A_{2n}+(q-q_0) A_{2n+1}]$$ for all $q \in U(x_0+y_0{{\mathbb{S}}},r)$. As a consequence, $$\lim_{t\to 0} \frac{f(q_0+tv)-f(q_0)}{t} = v A_1 + (q_0v - v\bar q_0)A_2.$$ for all $v \in T_{q_0}\Omega\cong {{\mathbb{H}}}$.
If $q_0 \in L_I$ and if we split the tangent space $T_{q_0}\Omega\cong {{\mathbb{H}}}$ as ${{\mathbb{H}}}= L_I \oplus L_I^\perp$, then the differential of $f$ at $q_0$ acts by right multiplication by $A_1$ on $L_I^\perp$ and by right multiplication by $A_1+2Im(q_0)A_2$ on $L_I$. Furthermore, if for all $q_0 \in \Omega$ the *remainder* $R_{q_0}f$ is defined as $$R_{q_0}f(q) = (q-q_0)^{-*}*(f(q)-f(q_0))$$ then the coefficients of are computed as $A_{2n} = (R_{\bar q_0}R_{q_0})^nf(q_0)$ and $A_{2n+1}= R_{q_0}(R_{\bar q_0}R_{q_0})^nf(\bar q_0)$.
Let us recall the definition of the *Cullen derivative* $\partial_cf$, given in [@advances] as $$\label{cullen}
\partial_cf(x+Iy)=\frac{1}{2}\left(\frac{\partial}{\partial x}-I\frac{\partial}{\partial y}\right)f(x+Iy)$$ for $I \in {{\mathbb{S}}},\ x,y \in {{\mathbb{R}}}$, as well as the definition of the *spherical derivative* $$\label{spherical}
\partial_sf(q) = (2Im(q))^{-1}(f(q)-f(\bar q))$$ given in [@perotti]. We can make the following observation.
If $f$ is a regular function on $B(0,R)$ and if holds then $\partial_cf(q_0) = R_{q_0}f(q_0) = A_1+2Im(q_0)A_2$ and $\partial_sf(q_0)= R_{q_0}f(\bar q_0) = A_1$.
In the case of the regular Möbius transformation $${{\mathcal{M}}}_{q_0}(q) = (1-q\bar q_0)^{-*}*(q-q_0) = (q-q_0)*(1-q\bar q_0)^{-*},$$ clearly $R_{q_0}{{\mathcal{M}}}_{q_0}(q) = (1-q\bar q_0)^{-*}$, so that we easily compute the coefficients $A_n$.
Let $q_0 = x_0+y_0I \in {{\mathbb{B}}}$. Then the expansion of ${{\mathcal{M}}}_{q_0}$ at $q_0$ has coefficients $$\begin{aligned}
A_{2n-1} &=& \frac{\bar q_0^{2n-2}}{(1-|q_0|^2)^{n-1}(1-\bar q_0^2)^{n}}\\
A_{2n} &=& \frac{\bar q_0^{2n-1}}{(1-|q_0|^2)^n(1-\bar q_0^2)^n}\end{aligned}$$ for all $n\geq 1$. As a consequence, for all $v \in {{\mathbb{H}}}$, $$\frac{\partial{{\mathcal{M}}}_{q_0}} {\partial v}(q_0)=v (1-\bar q_0^2)^{-1}+(q_0 v-v \bar q_0) \frac{\bar q_0}{(1-|q_0|^2)(1-\bar q_0^2)}.$$
We have already observed that $R_{q_0}{{\mathcal{M}}}_{q_0}(q) = (1-q\bar q_0)^{-*}$, so that $A_1 = R_{q_0}{{\mathcal{M}}}_{q_0}(\bar q_0) = (1-\bar q_0^2)^{-1}$. Moreover, $$R_{\bar q_0}R_{q_0}{{\mathcal{M}}}_{q_0}(q) = (q-\bar q_0)^{-*}*\left[R_{q_0}{{\mathcal{M}}}_{q_0}(q)- A_{1}\right]=$$ $$=(q-\bar q_0)^{-*}*\left[(1-q\bar q_0)^{-*} - A_{1}\right] =$$ $$= (1-q\bar q_0)^{-*}* (q-\bar q_0)^{-*}*\left[(1-\bar q_0^2)- (1-q\bar q_0)\right]A_1=$$ $$= (1-q\bar q_0)^{-*} \bar q_0 A_1.$$ The thesis follows by induction, proving that for all $n \geq 1$ $$(R_{\bar q_0}R_{q_0})^{n}{{\mathcal{M}}}_{q_0}(q) =(1-q\bar q_0)^{-*} \bar q_0 A_{2n-1},$$ $$R_{q_0}(R_{\bar q_0}R_{q_0})^{n}{{\mathcal{M}}}_{q_0}(q) = (1-q\bar q_0)^{-*} \bar q_0 A_{2n}$$ by means of similar computations.
We are now in a position to answer question (a).
For each $q_0 = x_0 + I y_0\in {{\mathbb{B}}}\setminus {{\mathbb{R}}}$, the differential of ${{\mathcal{M}}}_{q_0}$ at $q_0$ acts by right multiplication by $\partial_c{{\mathcal{M}}}_{q_0}(q_0)=(1-|q_0|^2)^{-1}$ on $L_I$ and by right multiplication by $\partial_s{{\mathcal{M}}}_{q_0}(q_0)=(1-\bar q_0^2)^{-1}$ on $L_I^\perp$. Since these quaternions have different moduli, ${{\mathcal{M}}}_{q_0}$ is not conformal.
We now turn our attention to question (b): whether or not regular Möbius transformations preserve the quaternionic Poincaré metric on ${{\mathbb{B}}}$ described in section \[classicM\] and in [@poincare]. We recall that this metric was constructed to be preserved by the classical (non regular) Möbius transformations of ${{\mathbb{B}}}$. Thanks to theorem \[quotients\], we observe what follows.
If ${{\mathcal{M}}}_{q_0}(q) = (1-q\bar q_0)^{-*}*(q-q_0)$ and $M_{q_0}(q) = (1-q\bar q_0)^{-1}(q-q_0)$ then $${{\mathcal{M}}}_{q_0}(q)= M_{q_0}(T(q)).$$ where $T(q) = (1-q q_0)^{-1} q (1-q q_0)$ is a diffeomorphism of ${{\mathbb{B}}}$ with inverse $T^{-1}(q) = (1-q \bar q_0)^{-1} q (1- q \bar q_0)$.
Thus, a generic regular Möbius transformation of ${{\mathbb{B}}}$ $$q \mapsto {{\mathcal{M}}}_{q_0}(q) u = M_{q_0}(T(q)) u$$ (with $u \in \partial {{\mathbb{B}}}$) is an isometry if and only if $T$ is. An example shows that this is not the case.
Let $q_0=\frac{I_0}{2}$ for some $I_0 \in {{\mathbb{S}}}$. Then $T(q) = (1-q q_0)^{-1} q (1-q q_0)$ is not an isometry for the Poincaré metric defined by formula . Indeed, if $J_0 \in {{\mathbb{S}}}, J_0 \perp I_0$ then setting $q_1=\frac{J_0}{2}$ we have $$\delta_{{\mathbb{B}}}(q_0,q_1) > \delta_{{\mathbb{B}}}(T(q_0),T(q_1))$$ since $$\frac{|q_1-q_0|^2}{|1-q_1\bar q_0|^2} = \frac{|2J_0-2I_0|^2}{|4+J_0I_0|^2} =\frac{8}{17}$$ while, computing $T(q_0) = q_0 = \frac{I_0}{2}$ and $T(q_1) = \frac{8I_0+15J_0}{34}$, we conclude that $$\frac{|T(q_1)-q_0|^2}{|1-T(q_1)\bar q_0|^2} = 4\frac{|-3I_0+5J_0|^2}{|20-5K_0|^2}= \frac{8}{25}.$$ Thus, the regular Möbius transformations do not have a definite behavior with respect to $\delta_{{\mathbb{B}}}$: we have proven that $T$ (hence ${{\mathcal{M}}}_{q_0}$) is not an isometry, nor a dilation; the same computation proves that $T^{-1}(q) = (1-q \bar q_0)^{-1} q (1- q \bar q_0)$ (hence ${{\mathcal{M}}}_{\bar q_0}$) is not a contraction.
The previous discussion proves that the study of regular Möbius transformations cannot be framed into the classical study of ${{\mathbb{B}}}$, and that it requires further research. On the other hand, the theory of regular functions provides working tools that were not available for the classical Möbius transformations. These tools lead us in [@schwarzpick] to an analog of the Schwarz-Pick lemma, which we will present in the next section.
The Schwarz-Pick lemma for regular functions {#schwarzpick}
============================================
In the complex case, holomorphic functions play a crucial role in the study of the intrinsic geometry of the unit disc $\Delta = \{z \in {{\mathbb{C}}}\ |\ |z|<1\}$ thanks to the Schwarz-Pick lemma [@pick2; @pick1].
Let $f:\Delta \to \Delta$ be a holomorphic function and let $z_0 \in \Delta$. Then $$\left|\frac{f(z)-f(z_0)}{1-\overline{f(z_0)}f(z)}\right| \leq \left|\frac{z-z_0}{1-\bar z_0 z}\right|,$$ for all $z \in \Delta$ and $$\frac{|f'(z_0)|}{1-|f(z_0)|^2} \leq \frac{1}{1-|z_0|^2}.$$ All inequalities are strict for $z \in \Delta \setminus \{z_0\}$, unless $f$ is a Möbius transformation.
This is exactly the type of tool that is not available, in the quaternionic case, for the classical Möbius transformations. To the contrary, an analog of the Schwarz-Pick lemma is proven in [@schwarzpick] for quaternionic regular functions. To present it, we begin with a result concerning the special case of a function $f \in {\frak{Reg}({{\mathbb{B}}},{{\mathbb{B}}})}$ having a zero.
\[teo1\] If $f : {{\mathbb{B}}}\to {{\mathbb{B}}}$ is regular and if $f(q_0)=0$ for some $q_0 \in {{\mathbb{B}}}$, then $$|{{\mathcal{M}}}_{q_0}^{-*}*f(q)| \leq 1$$ for all $q \in {{\mathbb{B}}}$. The inequality is strict, unless ${{\mathcal{M}}}_{q_0}^{-*}*f(q) \equiv u$ for some $u \in \partial {{\mathbb{B}}}$.
A useful property of the moduli of regular products is also proven in [@schwarzpick]:
\[modulusproduct\] Let $f,g,h: B(0,R) \to {{\mathbb{H}}}$ be regular functions. If $|f|\leq |g|$ then $|h*f| \leq |h*g|$. Moreover, if $|f|< |g|$ then $|h*f| < |h*g|$ in $B \setminus {{\mathcal{Z}}}_h$.
The property above allows us to derive from theorem \[teo1\] the perfect analog of the Schwarz-Pick lemma in the special case $f(q_0)=0$. We recall that $\partial_c f$ denotes the Cullen derivative of $f$, defined by formula , while $\partial_s f$ denotes the spherical derivative, defined by formula .
\[schwarzp\] If $f : {{\mathbb{B}}}\to {{\mathbb{B}}}$ is regular and if $f(q_0)=0$ for some $q_0 \in {{\mathbb{B}}}$ then $$|f(q)| \leq |{{\mathcal{M}}}_{q_0}(q)|$$ for all $q \in {{\mathbb{B}}}$. The inequality is strict at all $q \in {{\mathbb{B}}}\setminus \{q_0\}$, unless there exists $u \in \partial {{\mathbb{B}}}$ such that $f(q) = {{\mathcal{M}}}_{q_0}(q) \cdot u$ at all $q \in {{\mathbb{B}}}$. Moreover, $|R_{q_0}f(q)| \leq |(1-q\bar q_0)^{-*}|$ in ${{\mathbb{B}}}$ and in particular $$\begin{aligned}
|\partial_c f (q_0)| \le \frac{1}{1-|q_0|^2}\\
|\partial_s f(q_0)|
\leq \frac{1}{|1-\overline{q_0}^2|}.\end{aligned}$$ These inequalities are strict, unless $f(q) = {{\mathcal{M}}}_{q_0}(q) \cdot u$ for some $u \in \partial {{\mathbb{B}}}$.
Finally, the desired result is obtained in full generality.
\[MainSchwarzPick\] Let $f : {{\mathbb{B}}}\to {{\mathbb{B}}}$ be a regular function and let $q_0 \in {{\mathbb{B}}}$. Then $$\label{MainSchwarzPickEq}
|(f(q)-f(q_0))*(1-\overline{f(q_0)}*f(q))^{-*}| \leq |(q-q_0)*(1-\bar q_0*q)^{-*}|,$$ $$|R_{q_0}f(q)*(1-\overline{f(q_0)}*f(q))^{-*}| \leq |(1-\bar q_0*q)^{-*}|$$ in ${{\mathbb{B}}}$. In particular, $$\label{culderiv}
|\partial_c f *(1-\overline{f(q_0)}*f(q))^{-*}|_{|_{q_0}} \leq \frac{1}{1-|q_0|^2}.$$ Apart from at $q_0$ (which reduces to $0\leq0$), all inequalities are strict unless $f$ is a regular Möbius transformation.
This promising result makes it reasonable to expect that regular functions play an important role in the intrinsic geometry of the quaternionic unit ball. Therefore, it encourages to continue the study of regular Möbius transformations and of their differential or metric properties.
[^1]: Partially supported by GNSAGA of the INdAM and by FIRB “Geometria Differenziale Complessa e Dinamica Olomorfa”.
[^2]: Partially supported by FSE and by Regione Lombardia.
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---
abstract: |
In this paper the theory of uniformly convex metric spaces is developed. These spaces exhibit a generalized convexity of the metric from a fixed point. Using a (nearly) uniform convexity property a simple proof of reflexivity is presented and a weak topology of such spaces is analyzed. This topology called co-convex topology agrees with the usualy weak topology in Banach spaces. An example of a $CAT(0)$-spaces with weak topology which is not Hausdorff is given. This answers questions raised by Monod 2006, Kirk and Panyanak 2008 and Espínola and Fernández-León 2009.
In the end existence and uniqueness of generalized barycenters is shown and a Banach-Saks property is proved.
address: 'Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany'
author:
- Martin Kell
bibliography:
- 'bib.bib'
title: Uniformly convex metric spaces
---
[^1]
,
In this paper we summarize and extend some facts about convexities of the metric from a fixed point and give simpler proofs which also work for general metric spaces. In its simplest form this convexity of the metric just requires balls to be convex or that $x\mapsto d(x,y)$ is convex for every fixed $y\in X$. It is easy to see that both conditions are equivalent on normed spaces with strictly convex norm. However, in [@Busemann1979] (see also [@Foertsch2004 Example 1]) Busemann and Phadke constructed spaces whose balls are convex but its metric is not. Nevertheless, a geometric condition called non-positive curvature in the sense of Busemann (see [@Bridson1999; @Bacak2014a]) implies that both concepts are equivalent, see [@Foertsch2004 Proposition 1].
The study of stronger convexities for Banach spaces [@Clarkson1936] has a long tradition. In the non-linear setting so called $CAT(0)$-spaces are by now well-understood, see [@Bridson1999; @Bacak2014a]. Only recently Kuwae [@Kuwae2013] based on [@Noar2011] studied spaces with a uniformly $p$-convexity assumption similar to that of Banach spaces. Related to this are Ohta’s convexities definitions [@Ohta2007] which, however, seem more restrictive than the ones defined in this paper.
In the first section of this article an overview of convexities of the metric and some easy implications are given. Then existence of the projection map onto convex subsets and existence and uniqueness of barycenters of measures is shown. For this we give simple proofs using an old concept introduced by Huff in [@Huff1980].
In the third section we introduce weak topologies. The lack of a naturally defined dual spaces similar to Banach space theory requires a more direct definition either via convex sets, i.e. the co-convex topology (first appeared in [@Monod2006]), or via asymptotic centers (see historical remark at the end of [@Bacak2014a Chapter 3]). Both topologies might not be equivalent and/or comparable. For $CAT(0)$-spaces it is easy to show that the convergence via asymptotic centers is stronger than the co-convex topology. However, the topologies do not agree in general, see Example \[ex:cone-hilbert\]. With this example we answer questions raised in [@Kirk2008] and [@Espinola2009].
In the last sections, we use the results show existence of generalized barycenters and prove the Banach-Saks property. The proof extends a proof recently found by Yokota [@Yokota] in the setting of $CAT(1)$-spaces with small diameter. In the end a discussion about further extending convexities is given.
Convexity of the metric {#convexity-of-the-metric .unnumbered}
=======================
Let $(X,d)$ be a complete metric space. We say that $(X,d)$ admits midpoints if for every $x,y\in X$ there is an $m(x,y)$ such that $d(x,m(x,y))=d(y,m(x,y))=\frac{1}{2}d(x,y)$. One easily sees that each such space is a geodesic space.
Now for $p\in[1,\infty)$ and all non-negative real numbers $a,b$ we define the $p$-mean $$\mathcal{M}^{p}(a,b):=\left(\frac{1}{2}a^{p}+\frac{1}{2}b^{p}\right)^{\frac{1}{p}}.$$ Furthermore, the case $p=\infty$ can be defined as a limit, i.e. $\mathcal{M}_{t}^{\infty}(a,b)=\max\{a,b\}$.
\[$p$-convexity\]Suppose the metric space $(X,d)$ admits midpoints. Then it is called $p$-convex for some $p\in[1,\infty]$ if for each triple $x,y,z\in X$ and each midpoint $m(x,y)$ of $x$ and $y$ $$d(m(x,y),z)\le\mathcal{M}^{p}(d(x,z),d(y,z)).$$
It is called strictly $p$-convex for $p\in(1,\infty]$ if the inequality is strict whenever $x\ne y$ and strictly $1$-convex if the inequality is strict whenever $d(x,y)>|d(x,z)-d(y,z)|$.
\(1) In [@Foertsch2004] Foertsch defined $1$-convexity and $\infty$-convexity under the name distance convexity and ball convexity.
A $p$-convex space is $p'$-convex for all $p'\ge p$ and it is easy to see that balls are convex iff the space is $\infty$-convex. Furthermore, one sees that any strictly $\infty$-convex space is uniquely geodesic.
Instead of just requiring convexity from a fixed point one can assume that assume a convexity of $t\mapsto d(x_{t},y_{t})$ there $x_{t}$ and $y_{t}$ are constant speed geodesics. This gives the following condition.
\[$p$-Busemann curvature\] A metric space $(X,d)$ admitting midpoints is said to satisfy the $p$-Busemann curvature condition for some $p\in[1,\infty]$ if for all triples quadruples $x_{0},x_{1},y_{0},y_{1}\in X$ with midpoints $x_{\frac{1}{2}}=m(x_{0},x_{1})$ and $y_{\frac{1}{2}}=m(y_{0},y_{1})$ it holds $$d(x_{\frac{1}{2}},y_{\frac{1}{2}})\le\mathcal{M}^{p}(d(x_{0},y_{0}),d(x_{1},y_{1})).$$ In such a case we will say that $(X,d)$ is $p$-Busemann.
It is not difficult to show (see e.g. [@Bacak2014a Proposition 1.1.5]) that in case $p\in[1,\infty)$ this is equivalent to the more traditional form: for each triples $x,y,z\in X$ with midpoints $m_{1}=m(x,z)$ and $m_{2}=m(y,z)$ it holds $$d(m_{1},m_{2})^{p}\le\frac{1}{2}d(x,y)^{p}.$$ In particular for $p=1$, this is Busemann’s original non-positive curvature assumption. In this case we will just say that $(X,d)$ is Busemann. Busemann’s condition can be used to show equivalence of all (strict/uniform) $p$-convexity, see Corollary \[cor:Busemann-conv\]. Currently we cannot prove that on $p$-Busemann spaces (strict/uniform) $p$-convexity is equivalent to (strict/uniform) $p'$-convexity for all $p'\ge p$. However, it can be used to get a $p$-Wasserstein contraction of $2$-barycenters if Jensen’s inequality holds on the space, see Proposition \[prop:Jensen-p-Busemann\] below.
In [@Foertsch2004] Foertsch also defines uniform distance/ball convexity. We adapt his definition as follows:
\[uniform $p$-convexity\] Suppose $(X,d)$ admits midpoints and let $p\in[1,\infty]$. Then we say it is uniformly $p$-convex if for all $\epsilon>0$ there is a $\rho_{p}(\epsilon)\in(0,1)$ such that for all triples $x,y,z\in X$ satisfying $d(x,y)>\epsilon\mathcal{M}^{p}(d(x,z),d(y,z))$ for $p>1$ and $d(x,y)>|d(x,z)-d(y,z)|+\epsilon\mathcal{M}^{1}(d(x,z),d(y,z))$ for $p=1$ it holds $$d(m(x,y),z)\le(1-\rho_{p}(\epsilon))\mathcal{M}^{p}(d(x,z),d(y,z)).$$
\(1) W.l.o.g. we assume that $\rho_{p}$ is monotone in $\epsilon$ so that $\rho_{p}(\epsilon)\to0$ requires $\epsilon\to0$.
\(2) Uniform $p$-convexity for $p\in(1,\infty)$ is equivalent to the existence of a $\tilde{\rho}_{p}(\epsilon)>0$ such that $$d(m(x,y),z)^{p}\le(1-\tilde{\rho}_{p}(\epsilon))\mathcal{M}^{p}\left(d(x,z),d(x,z)\right)^{p},$$ just let $\tilde{\rho}_{p}(\epsilon)=1-(1-\rho_{p}(\epsilon))^{p}$.
\(3) The usual definition of uniform convexity for functions is as follows: A function $f$ is uniformly convex if for $x,y\in X$ with midpoint $m$: $$f(m)\le\frac{1}{2}f(x)+\frac{1}{2}f(y)-\omega(d(x,y)),$$ where $\omega$ is the modulus of convexity and $\omega(r)>0$ if $r>0$. For $p\ge2$ and $\omega(r)=Cr^{p}$ and $f=d(\cdot,z)^{p}$ one recovers Kuwae’s $p$-uniform convexity [@Kuwae2013]. However, in this form one does not see whether $p$-convexity implies $p'$-convexity. Furthermore, one gets a restriction that $\omega(r)\ge Cr^{2}$, i.e. the cases $p\in(1,2)$ are essentially excluded. And whereas our definition is multiplicative, matching the fact that $Cd(\cdot,\cdot)$ is also a metric, the usual uniform convexity is only multiplicative by adjusting the modulus of convexity.
\(1) Every $CAT(0)$-space is uniformly $2$-convex with $\tilde{\rho}(\epsilon)=\left(\frac{\epsilon}{2}\right)^{2}$. More generally any $R_{\kappa}$-domain of a $CAT(\kappa)$-space is uniformly $2$-convex with $\rho(\epsilon)=c_{\kappa}\epsilon^{2}$.
\(2) Every $p$-uniformly convex space as defined in [@Noar2011; @Kuwae2013] is uniformly $p$-convex with $\rho(\epsilon)=c_{k}\epsilon{}^{p}$.
A uniformly $p$-convex metric space $(X,d)$ is uniformly $p'$-convex for all $p'\ge p$.
First note that $\mathcal{M}^{p}(a,b)\le\mathcal{M}^{p'}(a,b)$.
Assume first that $1<p<p'\le\infty$. If $x,y,z\in X$ is a triple satisfying the condition for $p'$ then it also satisfies the condition for $p$ and thus for $m=m(x,y)$ $$\begin{aligned}
d(m,z) & \le & (1-\rho_{p}(\epsilon))\mathcal{M}^{p}(d(x,z),d(y,z))\\
& \le & (1-\rho_{p}(\epsilon))\mathcal{M}^{p'}(d(x,z),d(y,z)).\end{aligned}$$ Hence setting $\rho_{p'}(\epsilon):=\rho_{p}(\epsilon)$ gives the result.
For $p=1$ we skip the case $p'=\infty$ as this was proven in [@Foertsch2004 Proposition 1]: Let $x,y,z\in X$ be some triple with $d(x,z)\ge d(y,z)$. If $$d(x,y)>|d(x,z)-d(y,z)|+\frac{\epsilon}{2}\mathcal{M}^{1}(d(x,z),d(y,z))$$ then $$\begin{aligned}
d(m(x,y),z) & \le & \left(1-\rho_{1}\left(\frac{\epsilon}{2}\right)\right)\mathcal{M}^{1}(d(x,z),d(y,z))\\
& \le & \left(1-\rho_{1}\left(\frac{\epsilon}{2}\right)\right)\mathcal{M}^{p'}(d(x,z),d(y,z)).\end{aligned}$$ So assume $$d(x,y)\le d(x,z)-d(y,z)+\frac{\epsilon}{2}\mathcal{M}^{p'}(d(x,z),d(y,z)).$$ If $d(x,z)-d(y,z)\le\frac{\epsilon}{2}\mathcal{M}^{p'}(d(x,z),d(y,z))$ then $$d(x,y)\le\epsilon\mathcal{M}^{p'}(d(x,z),d(y,z)).$$ Hence we can assume $d(x,z)-d(y,z)>\frac{\epsilon}{2}\mathcal{M}^{p'}(d(x,z),d(y,z))$. Now for $p'\ge2$ Clarkson’s inequality $$\left(\frac{1}{2}a+\frac{1}{2}b\right)^{p'}+c_{p'}(a-b)^{p'}\le\frac{1}{2}a^{p'}+\frac{1}{2}b^{p'}$$ holds. Thus using $1$-convexity and our assumption we get $$\begin{aligned}
d(m,z)^{p'} & \le & \left(\frac{1}{2}d(x,z)+\frac{1}{2}d(y,z)\right)^{p'}\\
& \le & \frac{1}{2}d(x,z)^{p'}+\frac{1}{2}d(y,z)^{p'}-c_{p'}(d(x,z)-d(y,z))^{p'}\\
& \le & \left(1-c_{p}\left(\frac{\epsilon}{2}\right)^{p'}\right)\mathcal{M}^{p'}(d(x,z),d(y,z))^{p'}.\end{aligned}$$ Choosing $\rho_{p'}(\epsilon)=\min\{\rho_{1}(\frac{\epsilon}{2}),1-(1-c_{p}(\frac{\epsilon}{2})^{p'})^{\frac{1}{p'}}\}$ gives the result.
For $1<p'<2$ we use the other Clarkson inequality $$\left(\frac{1}{2}a+\frac{1}{2}b\right)^{q}+c_{p'}(a-b)^{q}\le\left(\frac{1}{2}a^{p}+\frac{1}{2}b^{p}\right)^{q}$$ where $\frac{1}{q}+\frac{1}{p'}=1$. By similar arguments we get $$d(m,z)^{q}+c_{p'}\left(\frac{\epsilon}{2}\right)^{\frac{q}{p'}}(\mathcal{M}^{p'}(d(x,z),d(y,z))^{q}\le(\mathcal{M}^{p'}(d(x,z),d(y,z))^{q}.$$ Choosing in this case $$\rho_{p'}(\epsilon)=\min\left\{ \rho_{1}\left(\frac{\epsilon}{2}\right),1-\left(1-c_{p'}\left(\frac{\epsilon}{2}\right)^{\frac{q}{p'}}\right)^{\frac{1}{q}}\right\}$$ finishes the proof.
\[cor:Busemann-conv\]Assume $(X,d)$ is Busemann. Then $(X,d)$ is (strictly/uniformly) $p$-convex for some $p\in[1,\infty]$ iff it is (strictly/uniformly) $p$-convex for all $p\in[1,\infty]$.
This is just a using [@Foertsch2004 Proposition 1] who proved that (strict/uniform) $\infty$-convexity implies (strict/uniform) $1$-convexity.
Any $CAT(0)$-space is both Busemann and uniformly $2$-convex, hence uniformly $p$-convex for every $p\in[1,\infty]$.
Convex subsets and reflexivity {#convex-subsets-and-reflexivity .unnumbered}
==============================
In a geodesic metric space, we say that subset $C\subset X$ is convex if for each $x,y\in C$ and each geodesic $\gamma$ connecting $x$ and $y$ also $\gamma\subset C$. Given any subset $A\subset X$ we define the convex hull of $A$ as follows: $G_{0}=A$ then for $n\ge1$ $$G_{n}=\bigcup_{x,y\in G_{n-1}}\{\gamma_{t}\,|\,\gamma\mbox{ is a geodesic connecting \ensuremath{x\:}and \ensuremath{y\:}and \ensuremath{t\in[0,1]\}}}$$ $$\conv A=\bigcup_{n\in\mathbb{N}}G_{n}.$$ The closed convex hull is just the closure $\overline{\conv A}$ of $\conv A$.
The projection map onto (convex) sets can be defined as follows: Given a non-empty subset $C$ of $X$ define $r_{C}:X\to[0,\infty)$ by $$r_{C}(x)=\inf_{c\in C}d(x,c)$$ and $P_{C}:X\to2^{C}$ by $$P_{C}(x)=\{c\in C\,|\, r_{C}(x)=d(x,c)\}.$$ In case $|P_{C}(x)|=1$ for all $x\in X$ we say that the set $C$ is Chebyshev. In that case, just assume $P_{C}$ is a map from $X$ to $C$.
It is well-known that a Banach space is reflexive iff any decreasing family of closed bounded convex subsets has non-empty intersection. Thus it makes sense for general metric spaces to define reflexivity as follows.
\[Reflexivity\] A metric space $(X,d)$ is said to be reflexive if for every decreasing family $(C_{i})_{i\in I}$ of non-empty bounded closed convex subsets, i.e. $C_{i}\subset C_{j}$ whenever $i>j$ where $I$ is a directed set then it holds $$\bigcap_{i\in I}C_{i}\ne\varnothing.$$
It is obvious that any proper metric space is reflexive. The following was defined in [@Huff1980]. We will simplify Huff’s proof of [@Huff1980 Theorem 1] to show that nearly uniform convexity implies reflexivity using a proof via the projection map, see e.g. [@Bacak2014a Proofs of 2.1.12(i) and 2.1.16]. However, since the weak topology (see below) is not necessarily Hausdorff, we cannot show that nearly uniform convexity also implies the uniform Kadec-Klee property.
We say that a family of points $(x_{i})_{i\in I}$ is $\epsilon$-separated if $d(x_{i},x_{j})\ge\epsilon$ for $i\ne j$, i.e. $$\sep((x_{i})_{i\in I})=\inf d(x_{i},x_{j})\ge\epsilon.$$
\[Nearly uniformly convex\] A $\infty$-convex metric space $(X,d)$ is said to be nearly uniformly convex, if for any $R>0$ for any $\epsilon$-separated infinite family $(x_{i})_{i\in I}$ with $d(x_{i},y)\le r\le R$ there is a $\rho=\rho(\epsilon,R)>0$ such that $$B_{(1-\rho)r}(y)\cap\overline{\conv(x_{i})_{i\in I}}\ne\varnothing.$$
Note that uniform $\infty$-convexity implies nearly uniform convexity, an even stronger statement is formulated in Theorem \[thm:NUC\]. However, not every nearly uniformly convex space is uniformly convex, see [@Huff1980].
For every closed convex subset $C$ of a nearly uniformly convex metric space the projection $P_{C}$ has non-empty compact images, i.e. $P_{c}(x)$ is non-empty and compact for every $x\in X$.
If $(X,d)$ is nearly uniformly convex and strictly $\infty$-convex then every closed convex set is Chebyshev.
\[Proof of the Theorem\]Let $C$ be a closed convex subset, $x\in X$ be arbitrary and set $r=r_{C}(x)$. For each $n\in N$ there is an $x_{n}\in C$ such that $r\le d(x,x_{n})\le r+\frac{1}{n}$. In particular, $d(x,x_{n})\to r$ as $n\to\infty$ . If $r=0$ or every subsequence of $(x_{n})$ admits a convergent subsequence we are done.
So assume $(x_{n})$ w.l.o.g. that $(x_{n})$ is $\epsilon$-separated for some $\epsilon>0$. By nearly uniform convexity there is a $\rho=\rho(\epsilon)>0$ such that $$A_{n}=B_{(1-\rho)(r+\frac{1}{n})}(x)\cap\overline{\conv(x_{m})_{m\ge n}}\ne\varnothing.$$ For sufficiently large $n$ and some $0<\rho'<\rho$ we also have $B_{(1-\rho)(r+\frac{1}{n})}(x)\subset B_{(1-\rho')r}(x)$, i.e. $d(x,y)<r$ for some $y\in A_{n}$. But this contradicts the fact that $\overline{\conv(x_{m})_{m\ge n}}\subset C$, i.e. $d(x,y)\ge r$ for all $y\in A_{n}$.
A nearly uniformly convex metric space is reflexive.
Let $(C_{i})_{i\in I}$ be a non-increasing family of bounded closed convex subsets of $X$ and let $x\in X$ be some arbitrary point. For each $i\in I$ define $r_{i}=\inf_{y\in C_{i}}d(x,y)$. Since $(C_{i})_{i\in I}$ is non-increasing so the net $(r_{i})_{i\in I}$ is non-decreasing and bounded, hence convergent to some $r$. By the previous theorem there are $x_{i}\in C_{i}$ such that $d(x,x_{i})=r_{i}$. If $r=0$ or $(x_{i})_{i\in I}$ admits a convergent subnet we are done.
So assume there is an $\epsilon$-separated subnet $(x_{i'})_{i'\in I'}$ for some $\epsilon>0$. Now nearly uniform convexity implies that for some $\rho=\rho(\epsilon)>0$ $$\varnothing\ne A_{i}=B_{(1-\rho)r}(x)\cap\overline{\conv(x_{j})_{j\ge i}}\subset C_{i}.$$ Since the subnet $(r_{i'})$ is also convergent to $r$ there is some $i$ and $0<\rho'<\rho$ such that $$B_{(1-\rho)r}(x)\subset B_{(1-\tilde{\rho})r_{i}}(x).$$
However, this implies that $d(x,y_{i})<r_{i}$ for all $y_{i}\in A_{i}$ contradicting the definition of $r_{i}$.
In order to use reflexivity to characterize the weak topology defined below better we need the following equivalent description. We say that a collection of sets $(C_{i})_{i\in I}$ has the finite intersection property if any finite subcollection has non-empty intersection, i.e. for every finite $I'\subset I$, $\cap_{i\in I'}C_{i}\ne\varnothing$.
\[cor:reflex-equiv\]The space $(X,d)$ is reflexive iff every collection $(C_{i})_{i\in I}$ of closed bounded convex subsets with finite intersection property satisfies $$\bigcap_{i\in I}C_{i}\ne\varnothing.$$
The if-direction is obvious. So assume $(X,d)$ is reflexive and $(C_{i})_{i\in I}$ be a collection of closed bounded convex subsets with finite intersection property.
Let $\mathcal{I}$ be the set of finite subsets of $I$. This set directed by inclusion and the sets $$\tilde{C}_{\mathbf{i}}=\bigcap_{i\in\mathbf{i}}C_{i}$$ are non-empty closed and convex. Furthermore, the family $(\tilde{C}_{\mathbf{i}})_{\mathbf{i}\in\mathcal{I}}$ is decreasing. By reflexivity $$\bigcap_{i\in I}C_{i}=\bigcap_{\mathbf{i}\in\mathcal{I}}\tilde{C}_{\mathbf{i}}\ne\varnothing.$$
Assume $(X,d)$ is strictly $\infty$-convex and nearly uniformly convex. Then the midpoint map $m$ is continuous.
Since $(X,d)$ is strictly $\infty$-convex we see that geodesics are unique. Thus the midpoint map by $m:(x,y)\mapsto m(x,y)$ is well-defined. Now if $(x_{n},y_{n})\to(x,y)$ then for all $\epsilon>0$ the sequence $m_{n}=m(x_{n},y_{n})$ eventually enters the closed convex and bounded set $$A_{\epsilon}=B_{\frac{1}{2}d(x,y)+\epsilon}(x)\cap B_{\frac{1}{2}d(x,y)+\epsilon}(y).$$ By uniform $\infty$-convexity $\bigcap_{\epsilon>0}A_{\epsilon}$ is non-empty and contains only the point $m(x,y)$.
We only need to show that $\diam A_{\epsilon}\to0$ as $\epsilon\to0$. Now assume there is a sequence $x_{n}\in A_{\frac{1}{n}}$ that is not Cauchy, so assume it is $\delta$-separated for some $\delta>0$. Then by nearly uniform convexity there is a $\rho(\delta)>0$ $$B_{(1-\rho)\left(\frac{1}{2}d(x,y)+\epsilon\right)}(x)\cap\overline{\conv(x_{n})}\ne\varnothing.$$ And thus $$\bigcap_{m}\overline{\conv(x_{n})_{n\ge m}}\cap B_{(1-\rho(\delta))\frac{1}{2}d(x,y)}(x)\ne\varnothing.$$ But this contradicts the fact that $$\bigcap_{m}\overline{\conv(x_{n})_{n\ge m}}\subset\bigcap_{m}A_{\frac{1}{m}}\subset B_{\frac{1}{2}d(x,y)}(y)$$ is disjoint from $B_{(1-\rho)\left(\frac{1}{2}d(x,y)+\epsilon\right)}(x)$.
A strictly $\infty$-convex, nearly uniformly convex metric space is contractible.
Take a fixed point $x_{0}\in X$ and define the map $$\Phi_{t}(x)=\gamma_{xx_{0}}(t)$$ where $\gamma_{xx_{0}}$ is the geodesic connecting $x$ and $x_{0}$. Now proof of previous theorem also shows that $t$-midpoints are continuous, in particular $\Phi_{t}$ is continuous.
Weak topologies {#weak-topologies .unnumbered}
===============
In Hilbert and Banach spaces the concept of weak topologies can be introduced with the help of dual spaces. Since for general metric spaces there is (by now) no concept of dual spaces, a direct definition needs to be introduced. As it turns out the first topology agrees with the usual weak topology, see Corollary \[cor:Banach-weak-co-convex\].
Co-convex topology {#co-convex-topology .unnumbered}
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The first weak topology on metric spaces is the following. It already appeared in [@Monod2006]. As it turns out, this topology is agrees with the weak topology on any Banach space, see Corollary \[cor:Banach-weak-co-convex\] below.
\[Co-convex topology\] Let $(X,d)$ be a metric space. Then the co-convex topology $\tau_{co}$ is the weakest topology containing all complements of closed convex sets.
Obviously this topology is weaker than the topology induced by the metric and since point sets are convex the topology satisfies the $T_{1}$-separation axiom, i.e. for each two points $x,y\in X$ there is an open neighborhood $U_{x}$ containing $x$ but not $y$. Furthermore, the set of weak limit points of a sequence $(x_{n})_{n\in\mathbb{N}}$ is convex if the space is $\infty$-convex. A useful characterization of the limit points is the following:
\[lem:convex-leaving\]A sequence of points $x_{n}$ converges weakly to $x$ iff for all subsequences $(x_{n'})$ it holds $$x\in\overline{\conv(x_{n'})}.$$ The set of limit point $\Lim(x_{n})$ is the non-empty subset $$\bigcap_{(i_{n})\subset I_{\inf}}\overline{\conv(x_{i_{n}})}$$ where $I_{\inf}$ is the set of sequences of increasing natural numbers.
The same statement holds for also for nets. Below we will make most statments only for sequences if in fact they also hold for nets.
This follows immediately from the fact that $$A(x_{n'})=\overline{\conv(x_{n'})}$$ is closed, bounded and convex and thus weakly closed.
First suppose $x\notin A(x_{n'})$ for some subsequence $(x_{n'})$. By definition $x_{n}\overset{\tau_{co}}{\to}y$ implies that $x_{n}$ eventually leaves every closed bounded convex sets not containing $y$. Since $(x_{m'})\subset A(x_{n'})$ for $m'\ge n'$, we conclude $(x_{n'})$ cannot converge weakly to $x$.
Conversely, if $(x_{n})$ does not converge to $x$ then there is a weakly open set $U\in\tau_{co}$ such that $(x_{n})\not\subset U$ and $x\in U$. In particular, for some subsequence $(x_{n'})$ it holds $(x_{n'})\subset X\backslash U$. Since $\tau_{co}$ is generated by complements of closed convex sets we can assume $U=X\backslash C$ for some closed convex subset $C$. Therefore, $(x_{n'})\subset C$ and thus $A(x_{n'})\subset C$, i.e. $x\notin A(x_{n'})$.
For any weakly convergent sequence $(x_{n})$ and countable subset $A$ disjoint from $\Lim(x_{n})$ there is a subsequence $(x_{n'})$ such that $$A\cap\bigcap_{m\in\mathbb{N}}\overline{\conv(x_{n'})_{n'\ge m}}=\varnothing.$$
First note, by the lemma above there is a subsequence $(x_{m_{n}^{(0)}})$ of $(x_{n})$ such that $y_{0}\in A$ is not contained in $\overline{\conv(x_{m_{n}^{(0)}})}$. Now inductively constructing $(x_{m_{n}^{(k)}})$ avoiding $y_{k}$ using the sequence $(x_{m_{n}^{(k-1)}})$ we can choose the diagonal sequence $m_{n}=m_{n}^{(n)}$ such that $$y\notin\bigcap_{m\in\mathbb{N}}\overline{\conv(x_{m_{n}})_{m_{n}\ge m}}$$ for all $y\in A$.
\[cor:Banach-weak-co-convex\]On any Banach space $X$ the co-convex topology $\tau_{co}$ agrees the weak topology $\tau_{w}$. In particular, $\tau_{co}$ is Hausdorff.
By Corollary \[cor:quasi-conv+conc\] below any linear functional $\ell\in X^{*}$ is $\tau_{co}$-continuous. Hence $x_{n}\overset{\tau_{co}}{\to}x$ implies $x_{n}\overset{\tau_{w}}{\to}x$. The converse follows from that fact that for any subsequence$(x_{n'})$ the set $\overline{\conv(x_{n'})}$ is $\tau_{w}$-closed and $x_{n'}\overset{\tau_{w}}{\to}x$. Therefore, $x\in\overline{\conv(x_{n'})}$ which implies $x_{n}\overset{\tau_{co}}{\to}x$ by Lemma \[lem:convex-leaving\] above.
Now similar to Banach spaces, one can easy show that reflexivity implies weak compactness of bounded closed convex subsets.
Bounded closed convex subsets are weakly compact iff the space is reflexive.
By Alexander sub-base theorem it suffices to show that each open cover $(U_{i})_{i\in I}$ of $B$, where $U_{i}$ is a complement of a closed convex set, has a finite subcover. For this, note that $U_{i}=X\backslash C_{i}$ and the cover property of $U_{i}$ is equivalent to $$\bigcap_{i\in I}B\cap C_{i}=\varnothing.$$ If we assume that there is no finite subcover then the collection $(B\cap C_{i})_{i\in I}$ has finite intersection property. But then Corollary \[cor:reflex-equiv\] yields $\bigcap_{i\in I}B\cap C_{i}\ne\varnothing$, which is a contradiction.
Conversely, assume $(X,d)$ is not reflexive but any bounded closed convex subset is weakly compact. Then there $(C_{i})_{i\in I}$ is a decreasing family of non-empty bounded closed convex subsets such that $\cap_{i\in I}C_{i}=\varnothing$. Assume w.l.o.g. that $I$ has a minimal element $i_{0}$. Then $U_{i}=X\backslash C_{i}$ is an open cover of $C_{i_{0}}$, i.e. $$C_{i_{0}}\subset\bigcup_{i\in I}U_{i}.$$ Since $(C_{i})_{i\in I}$ is decreasing, $(U_{i})_{i\in I}$ is increasing. By weak compactness, finitely many of there are sufficient to cover $C_{i_{0}}$. Since $(U_{i})_{i\in I}$ is increasing, there exists exactly one $i_{1}\in I$ such that $C_{i_{0}}\subset U_{i_{1}}=X\backslash C_{i_{1}}$. But then $C_{i_{1}}=\varnothing$ contradicting our assumption.
Note that on general spaces the co-convex topology is not necessary Hausdorff. Even in case of $CAT(0)$-spaces one can construct an easy counterexample.
\[Euclidean Cone of a Hilbert space\]\[ex:cone-hilbert\] For the construction of Euclidean cones see [@Bridson1999 Chapter I.5]. Let $(H,d_{H})$ be an infinite-dimensional Hilbert space and $d_{H}$ be the induced metric. The Euclidean cone over $(H,d_{h})$ is defined as the set $C(H)=H\times[0,\infty)$ with the metric $$d((x,t),(x',t'))^{2}:=t^{2}+t'^{2}-2tt'\cos(d_{\pi}(x,x'))$$ where $d_{\pi}(x,x')=\min\{\pi,d_{H}(x,x')\}$. By [@Bridson1999 Theorem II-3.14] $(C(H),d)$ is a $CAT(0)$-space and thus uniformly $p$-convex for any $p\in[1,\infty]$. In particular, bounded closed convex subsets are compact w.r.t. the co-convex topology. Note that in $(H,d_{H})$ the co-convex topology agrees with the usual weak topology. Now let $((e_{n},1))_{n\in\mathbb{N}}$ be a sequence in $C(H)$. We claim that for any subsequence $((e_{n'},1))$ we have $$\bigcap_{m\in\mathbb{N}}\overline{\conv((e_{n'},1))_{n'\ge m}}=\{(\mathbf{0},r)\,|\, r\in[a,b]\}$$ with $a<b$ where it is easy to see that $a$ and $b$ do not depend on the subsequence. Any point in that intersection is a limit point of $((e_{n},1))$ which implies that $\tau_{co}(C(H))$ is not Hausdorff. To see this, note that the projection $p$ onto the line $\{(\mathbf{0},r)\,|\, r\ge0\}$ has the following form $$p((x,r))=(\mathbf{0},r\cos(d_{H}(x,\mathbf{0})))$$ for $d(x,0)\le\frac{\pi}{2}$. In particular, $d((e_{n},1))=(\mathbf{0},\cos(1))$. Using the weak sequential convergence defined below, this means that $(e_{n},1)\overset{w}{\to}(\mathbf{0},\cos(1))$, in particular.
$$(\mathbf{0},\cos(1))\in\bigcap_{m\in\mathbb{N}}\overline{\conv((e_{n'},1))_{n'\ge m}}.$$ Now we will show that the sequence of midpoints $l_{mn}$ of $(e_{n},1)$ and $(e_{m},1)$ with $m\ne n$ converges weakly sequentially to some point $(\mathbf{0},r)$ with $r>\cos(1)$. This immediately implies that $\bigcap_{m\in\mathbb{N}}\overline{\conv((e_{n'},1))_{n'\ge m}}$ contains more than one point and each is a limit point of $(e_{n},1)$ w.r.t. the co-convex topology.
To show that $l_{mn}$ does not weakly sequentially converge to $(\mathbf{0},\cos(1))$ we just need to show that $p(l_{mn})\ne(\mathbf{0},\cos(1))$. By the calculus of Euclidian cones the points $l_{mn}$ have the following form $$l_{mn}=\left(\frac{e_{m}-e_{n}}{2},r_{\frac{1}{2}}\right)$$ where $r_{\frac{1}{2}}$ is the (positive) solution of the equation $$r^{2}+1-2r\cos\left(\frac{\sqrt{2}}{2}\right)=\frac{1}{4}\left(2-2\cos(\sqrt{2})\right),$$ i.e. $r_{\frac{1}{2}}=\cos\left(\frac{\sqrt{2}}{2}\right)$.
Then the projection has the form $$\begin{aligned}
p(l_{mn}) & = & \left(\mathbf{0},r_{\frac{1}{2}}\cos\left(\left\Vert \frac{e_{n}-e_{m}}{2}\right\Vert \right)\right)\\
& = & \left(\mathbf{0},\cos\left(\frac{\sqrt{2}}{2}\right)^{2}\right).\end{aligned}$$ Since $\cos(1)<\cos(\frac{\sqrt{2}}{2})^{2}$ we see that $l_{mn}\not\to(\mathbf{0},\cos(1))$ w.r.t. weak sequential convergence and $$(\mathbf{0},\cos(\frac{\sqrt{2}}{2})^{2})\in\bigcap_{m\in\mathbb{N}}\overline{\conv((e_{n'},1))}.$$ And this obviously does not depend on the subsequence.
Note that this space also violates the property $(N)$ defined in [@Espinola2009], more generally any cone over a (even proper) $CAT(1)$-space which is not the sphere gives a counterexample. The example also gives a negative answer to Question 3 of [@Kirk2008]. This topology is also a counterexample to topologies similar to Monod’s $\mathcal{T}_{w}$ topology: Let $\tau_{w}^{p}$ be the weakest topology making all maps $x\mapsto d(x,y)^{p}-d(x,z)^{p}$ for $y,z\in X$ continuous. For Hilbert spaces and $p=2$ this is the weak topology, (compare to [@Monod2006 18. Example] which should be $p=2$). For the space $(C(H),d))$ one can show that each $\tau_{w}^{p}$ is strictly stronger that the weak sequential convergence.
\[weak lower semicontinuity\] A function $f:X\to(-\infty,\infty]$ is said to be weakly l.s.c. at a given point $x\in\dom f$ if $$\liminf f(x_{i})\ge f(x)$$ whenever $(x_{i})$ is a net converging to $x$ w.r.t. $\tau_{co}$. We say $f$ is weakly l.s.c. if it is weakly l.s.c. at every $x\in\dom f$.
A priori it is not clear if $\tau_{co}$ is first-countable and thus the continuity needs to be stated in terms of nets. In that case it boils down to $\liminf_{n\to\infty}f(x_{n})\ge f(x)$.
\[prop:co-convex-lscts\]Assume $(X,d)$ is $\infty$-convex. Then every lower semicontinuous quasi-convex function is weakly lower semicontinuous. In particular, the metric is lower semicontinuous.
A function is quasi-convex iff its sublevels are convex, i.e. whenever $z$ is on a geodesic connecting $x$ and $y$ then $f(z)\le\max\{f(x),f(y)\}$.
By definition of the co-convex topology, if $x_{i}\overset{\tau_{co}}{\to}x$ and $x_{i}\in C$ for some closed convex subset $C$ then $x\in C$. Now assume $f$ is not weakly lower semicontinuous at $x$, i.e. $$\liminf f(x_{i})<f(x).$$ Then there is a $\delta>0$ such that $$x_{i}\in A_{\delta}=\{y\in X\,|\, f(y)\le f(x)-\delta\}$$ for all $i\ge i_{0}$. By quasi-convexity and lower semicontinuity the set $A_{\delta}$ is closed convex and thus $x\in A_{\delta}$ which is a contradiction. Hence $f$ is weakly lower semicontinuous.
A function $\ell:X\to\mathbb{R}$ is called quasi-monotone iff it is both quasi-convex and quasi-concave. Similarly $\ell$ is called linear iff it is both convex and concave. A linear function is obviously quasi-monotone. The converse is not true in general: Every $CAT(0)$-spaces with property $(N)$ (see [@Espinola2009]) admits such functionals; for $x,y\in X$ just set $\ell(x')=d(P_{[x,y]}x',x)$ where $P_{[x,y]}$ is the projection onto the geodesic connecting $x$ and $y$.
\[cor:quasi-conv+conc\]Assume $(X,d)$ is $\infty$-convex. Then every continuous quasi-monotone function is weakly continuous.
Just note that the previous theorem implies that a quasi-monotone function is both weakly lower and upper semicontinuous.
In order to get the Kadec-Klee property one needs to find limit points which are easily representable.
\[countable reflexive\] A reflexive metric space $(X,d)$ is called countable reflexive if for each weakly convergent sequence $(x_{n})$ there is a subsequence $(x_{n'})$ such that $$\Lim(x_{n})=\bigcap_{m\in\mathbb{N}}\overline{\conv(x_{n'})_{n'\ge m}}.$$
By diagonal procedure it is easy to see that one only needs to show that for each $\epsilon>0$ there is a subsequence $(x_{n'})$ such that $$B_{\epsilon}(\Lim(x_{n}))\supset\bigcap_{m\in\mathbb{N}}\overline{\conv(x_{n'})_{n'\ge m}}.$$
\[lem:coun-refl-quasi-mono\]Any reflexive Banach space is countable reflexive. More generally any reflexive metric space admitting quasi-monotone functions separating points is countable reflexive. In this case the co-convex topology is Hausdorff.
If $x_{n}\overset{\tau_{co}}{\to}x$ and $x\ne y\in\bigcap_{m}\overline{\conv(x_{n})_{n\ge m}}$ then there is a quasi-monotone functional $\ell$ such that $\ell(y)>\ell(x)$. Since $\ell$ is weakly continuous we have $\ell(x_{n})\to x$ and thus by quasi-convexity of $\ell$ also $\ell(y)>\ell(x')$ for all $x'\in\overline{\conv(x_{n})_{n\ge m}}$ with $m\in\mathbb{N}$ sufficiently large $m\in\mathbb{N}$ However, this contradicts $y\in\bigcap_{m}\overline{\conv(x_{n})_{n\ge m}}$ and also shows that $\tau_{co}$ is Hausdorff.
\[Nearly uniform convexity\]\[thm:NUC\] Let $(X,d)$ be nearly uniformly convex and countable reflexive. Then for any $\epsilon$-separated sequence $(x_{n})$ in $B_{R}(y)$ there is a weak limit point of $(x_{n})$ contained in the ball $B_{(1-\rho)R}(y)$.
If $(x_{n})$ is $\epsilon$-separated with $d(x_{n},y)\le R$ and assume w.l.o.g. that $(x_{n})$ is chosen such that $$\Lim(x_{n})=\bigcap_{m\in\mathbb{N}}C_{m}$$ where $C_{m}=\overline{\conv(x_{n})_{n\ge m}}$. We know by nearly uniform convexity there is a $\rho>0$ such that $$\tilde{C}_{m}=B_{(1-\rho)R}(y)\cap C_{m}\ne\varnothing.$$ Since $\tilde{C}_{m}$ is non-decreasing closed convex and non-empty, we see by reflexivity that $\cap_{m}\tilde{C}_{m}\ne\varnothing$ and hence $B_{(1-\rho)R}(y)\cap\Lim(x_{n})\ne\varnothing$.
\[Kadec-Klee property\]Let $(X,d)$ be strictly $\infty$-convex, nearly uniformly convex and countable reflexive. Suppose some fixed $y\in X$ and for each weak limit point $x$ of $(x_{n})$ one has $d(x_{n},y)\to d(x,y)$ then $(x_{n})$ has exactly one limit point and $(x_{n})$ converges strongly, i.e. norm plus weak convergence implies strong convergence.
If, in addition, $\tau_{co}$ is Hausdorff then $x_{n}\overset{\tau_{co}}{\to}x$ and $d(x_{n},y)\to d(x,y)$ implies $x_{n}\to x$.
\[Proof of the Theorem\]Since strong convergence implies weak and norm convergence, we only need to show the converse. For this let $x_{n}$ be some weakly convergent sequence. Note that $d(x,y)=\lim d(x_{n},y)=const$ for all limit points $x$ of $(x_{n})_{n}$. Since $x\mapsto d(\cdot,y)$ is strictly quasi-convex and the set of limit points is convex, there can be at most one limit point, i.e. $x_{n}\overset{\tau_{co}}{\to}x$ for a unique $x\in X$. If $d(x,y)=0$ then $x=y$ and $x_{n}\to x$ strongly.
Now assume $d(x,y)=R>0$. If $(x_{n})$ is not Cauchy then there is a subsequence $(x_{n'})$ still weakly converging to $x$ which is $\epsilon$-separated for some $\epsilon>0$. By the Theorem \[thm:NUC\] there is a limit point $x^{*}$ of $(x_{n'})$ such that $d(x^{*},y)<R$. But this contradicts the fact that $x^{*}=x$ and $d(x,y)=R$. Therefore, $x=y$.
A “topology” via asymptotic centers {#a-topology-via-asymptotic-centers .unnumbered}
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A more popular notion of convergence is the weak sequential convergence. Note, however, it is an open problem whether this “topology” is actually generated by a topology, see [@Bacak2014a Question 3.1.8.]. Given a sequence $(x_{n})$ in $X$ define the following function $$\omega(x,(x_{n}))=\limsup_{n\to\infty}d(x,x_{n}).$$
Assume $(X,d)$ is uniformly $\infty$-convex. Then function $\omega(\cdot,(x_{n}))$ has a unique minimizer.
It is not difficult to see that the sublevels of $\omega(\cdot,(x_{n}))$ are closed bounded and convex. This reflexivity implies existence of minimizers. Assume $x,x'$ are minimizers and $x_{\frac{1}{2}}$ their midpoint. If $\omega(x,(x_{n}))=0$ then obviously $x_{n}\to x=x'$. So assume $\omega(x,(x_{n})_{n})=c>0$. Then we can choose a subsequence $(x_{n'})_{n'}$ such that $\lim_{n'\to\infty}d(x,x_{n'})$ and $\lim_{n'\to\infty}d(x',x_{n'})$ exists and are equal. If $x\ne x'$ then there is an $\epsilon>0$ such that $d(x,x')\ge2\epsilon c$. This yields $$\begin{aligned}
\limsup_{n'\to\infty}d(x_{\frac{1}{2}},x_{n'}) & \le & (1-\rho(\epsilon))\lim_{n'\to\infty}\max\{d(x,x_{n'}),d(x',x_{n'})\}\\
& < & c.\end{aligned}$$ But this contradicts $x$ and $x'$ be minimizers. Hence $x=x'$.
The minimizer of $\omega(\cdot,(x_{n}))$ is called the asymptotic center. With the help of this we can define the weak sequential convergence as follows.
\[Weak sequential convergence\] We say that a sequence $(x_{n})_{n\in\mathbb{N}}$ converges weakly sequentially to a point $x$ if $x$ is the asymptotic center for each subsequence of $(x_{n})$. We denote this by $x_{n}\overset{w}{\to}x$.
For $CAT(0)$-spaces it is easy to see that $x_{n}\overset{w}{\to}x$ implies $x_{n}\overset{\tau_{co}}{\to}x$, i.e. the weak topology is weaker than the weak sequential convergence (see [@Bacak2014a Lemma 3.2.1]). Later we will show that the weak sequential limits can be strongly approximated by barycenters, which can be seen as a generalization of the Banach-Saks property (see below). If, in addition, the barycenter of finitely many points is in the convex hull of those points, one immediate gets that $x_{n}\overset{w}{\to}x$ implies $x_{n}\overset{\tau_{co}}{\to}x$.
Each bounded sequence $(x_{n})$ has a subsequence $(x_{n'})$ such that $x_{n'}\overset{w}{\to}x$.
The proof can be found in [@Bacak2014a Proposition 3.2.1]. Since it is rather technical we leave it out.
A different characterization of this convergence can be given as follows (see [@Bacak2014a Proposition 3.2.2]).
Assume $(X,d)$ is uniformly $\infty$-convex. Let $(x_{n})$ be a bounded sequence and $x\in X$. The the following are equivalent:
1. The sequence $(x_{n})$ converges weakly sequentially to $x$
2. For every geodesic $\gamma:[0,1]\to X$ with $x\in\gamma_{[0,1]}$, we have $P_{\gamma}x_{n}\to x$ as $n\to\infty$.
3. For every $y\in X$, we have $P_{[x,y]}x_{n}\to x$ as $n\to\infty$.
(i)$\Longrightarrow$(ii): Let $\gamma$ be some geodesic containing $x$. If $$\lim d(P_{\gamma}x_{n},x)\ge0$$ then there is a subsequence $(x_{n'})$ such that $$P_{\gamma}y_{n}\to y\in\gamma_{[0,1]}\backslash\{x\}.$$ But then $d(P_{\gamma}x_{n'},x_{n'})<d(x,x_{n'})$ which implies $$\limsup_{n\to\infty}d(y,x_{n'})=\limsup_{n\to\infty}d(P_{\gamma}x_{n'},x_{n'})\le\limsup d(x,x_{n'})$$ and contradicts uniqueness of the asymptotic center of $(x_{n'})$.
(ii)$\Longrightarrow$(iii): Trivial
(iii)$\Longrightarrow$(i): Assume $(x_{n})$ does not converge weakly sequentially to $x$. Then for some subsequence $x_{n'}\overset{w}{\to}y\in X\backslash\{x\}$. Then by the part above $P_{[x,y]}x_{n'}\to y$. But this contradicts the assumption $P_{[x,y]}x_{n'}\to x$. Hence $x_{n}\overset{w}{\to}x$.
\[Opial property\] \[cor:Opial\]Assume $(X,d)$ is uniformly $\infty$-convex and $(x_{n})$ some bounded sequence with $x_{n}\overset{w}{\to}x$. Then $$\liminf d(x,x_{n})<\liminf d(y,x_{n})$$ for all $y\in X\backslash\{x\}$.
Barycenters in convex metric spaces {#barycenters-in-convex-metric-spaces .unnumbered}
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Wasserstein space {#wasserstein-space .unnumbered}
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For $p\in[1,\infty)$ the $p$-Wasserstein space of a metric space $(X,d)$ is defined as the set $\mathcal{P}_{p}(X)$ of all probability measures $\mu\in\mathcal{P}(X)$ such that $$\int d^{p}(x,x_{0})d\mu(x)$$ for some fixed $x_{0}\in X$. Note that by triangle inequality this definition is independent of $x_{0}$. We equip this set with the following metric $$w_{p}(\mu,\nu)=\left(\inf_{\pi\in\Pi(\mu,\nu)}\int d^{p}(x,y)d\pi(x,y)\right)^{\frac{1}{p}}$$ where $\Pi(\mu,\nu)$ is the set of all coupling measures $\pi\in\mathcal{P}(X\times X)$ such that $\pi(A\times X)=\mu(A)$ and $\pi(X\times B)=\nu(B)$. It is well-known [@Villani2009] that $(\mathcal{P}_{p}(X),w_{p})$ is a complete metric space if $(X,d)$ is complete and that it is a geodesic space if $(X,d)$ is geodesic. Furthermore, by Hölder inequality one easily sees that $w_{p}\le w_{p'}$ whenever $p\le p'$ so that the limit $$w_{\infty}(\mu,\nu)=\lim w_{p}(\mu,\nu)$$ is well-defined and defines a metric on the space $\mathcal{P}_{\infty}(X)$ of probability measures with bounded support. An equivalent description of $w_{\infty}$ can be given as follows (see [@Champion2008]): For a measure $\pi\in\Pi(\mu,\nu)$ let $C(\pi)$ be the $\pi$-essiential support of $d(\cdot,\cdot)$, i.e. $$C(\pi)={\pi-\esup}_{(x,y)\in X\times X}d(x,y).$$ Then $$w_{\infty}(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\nu)}C(\pi).$$
For a fixed point $y\in X$ the distance of $\mu$ to the delta measure $\delta_{y}$ has the following form $$w_{p}^{p}(\mu,\delta_{y})=\int d^{p}(x,y)d\mu(x)$$ and $$w_{\infty}(\mu,\delta_{y})=\sup_{x\in\supp\mu}d(x,y)$$ where $\supp\mu$ is the support of $\mu$.
Existence and uniqueness of barycenters {#existence-and-uniqueness-of-barycenters .unnumbered}
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\[lem:measure-conv\]Assume $(X,d)$ is $p$-convex then $y\mapsto\int d^{p}(x,y)d\mu(x)$ is convex for $p\in[1,\infty)$ whenever $\mu\in\mathcal{P}_{p}(X)$. In case $p>1$ strict $p$-convexity even implies strict convexity. Furthermore, if $\mu$ is not supported on a single geodesic then $y\mapsto\int d(x,y)d\mu(x)$ is strictly convex if $(X,d)$ is strictly $1$-convex.
\(1) It is easy to see that for a measure supported on a geodesic the functional $y\mapsto\int d(x,y)d\mu(x)$ cannot be strictly convex on that geodesic.
\(2) The same holds for the functional $F_{w}(y):=\int d^{p}(x,y)-d^{p}(x,w)d\mu(x)$ as defined in [@Kuwae2013]
Let $y_{0},y_{1}\in X$ be two point in $X$ and $y_{t}$ be any geodesic connecting $y_{0}$ and $y_{1}$. Then by $p$-convexity $$d^{p}(x,y_{t})\le(1-t)d^{p}(x,y_{0})+td^{p}(x,y_{1})$$ which implies convexity of the functional and similarly strict convexity if $p>1$.
If $\mu$ is not supported on a single geodesic then there is a subset of positive $\mu$-measure disjoint from $\{y_{t}|t\in[0,1]\}$ such that $d(x,y_{t})<(1-t)d(x,y_{0})+td(x,y_{1})$. In particular, $y\mapsto\int d(x,y)d\mu(x)$ is strictly convex.
Assume $(X,d)$ is uniformly $\infty$-convex with modulus $\rho$. Let $\mu\in\mathcal{P}_{\infty}(X)$ then the function $F:y\mapsto w_{\infty}(\mu,\delta_{y})$ is uniformly quasi-convex, i.e. whenever $d(y_{0},y_{1})>\epsilon\max\{F(y_{0}),F(y_{1})\}$ for some $\epsilon>0$ then $$F(y_{\frac{1}{2}})\le(1-\rho(\epsilon))\max\{F(y_{0}),F(y_{1})\}.$$
In contrast to the cases $1<p<\infty$ strict $\infty$-convexity is not enough.
Note that $F$ has the following equivalent form $$F(y)=\sup_{x\in\supp\mu}d(x,y).$$ Take any $y_{0},y_{1}\in X$ with $d(y_{0},y_{1})>\epsilon\max\{F(y_{0}),F(y_{1})\}$. Let $x_{n}$ be a sequence such that $F(y_{\frac{1}{2}})=\lim_{n\to\infty}d(x_{n},y_{\frac{1}{2}})$. By uniform $\infty$-convexity we have $$\begin{aligned}
\lim_{n\to\infty}d(x_{n},y_{\frac{1}{2}}) & \le & (1-\rho(\epsilon))\max\{d(x_{n},y_{0}),d(x_{n},y_{1})\}\\
& \le & (1-\rho(\epsilon))\max\{F(y_{0}),F(y_{1})\}.\end{aligned}$$
The following was defined in [@Kuwae2013].
\[$p$-barycenter\] For $p\in[1,\infty]$ the $p$-barycenter of a measure $\mu\in\mathcal{P}_{p}(X)$ is defined as the point $y\in X$ such that $w_{p}(\mu,\delta_{y})$ is minimal. If $p<\infty$ and $\mu$ has only $(p-1)$-moments, i.e. $\int d^{p-1}(x,y)d\mu(x)<\infty$, then the $p$-barycenter can be defined as the minimizer of the functional $F_{w}(y)$ above. If the $p$-barycenter is unique we denote it by $b_{p}(\mu)$.
\(1) This functional $F_{w}(y)$ is well-defined since $$|F_{w}(y)|\le pd(y,w)\int(d(x,y)+d(x,w))^{p-1}d\mu(x).$$ Furthermore, $F_{w}(y)-F_{w'}(y)$ is constant and thus the minimizer(s) are independent of $w\in X$.
\(2) The $\infty$-barycenters are also called circumcenter. In case $\mu$ consists of three points it was recently used in [@Bacak2014] to define a new curvature condition. From the section above, the $\infty$-barycenter only depends on the support of the measure $\mu$. Hence the $\infty$-barycenter of any bounded set $A$ can be defined as $$b_{\infty}(A)=\argmin_{y\in X}\sup_{x\in A}d(x,y).$$ The proofs below work without any change.
On any $p$-convex, reflexive metric space $(X,d)$ every measure $p$-moment has $p$-barycenter.
Define $$A_{r}^{p}=\{y\in X\,|\, w_{p}(\mu,\delta_{y})\le r\},$$ which is a closed convex subset of $X$ which is non-empty for $r>m_{\mu}^{p}=\inf_{y\in X}w_{p}(\mu,\delta_{y})$. If it is bounded then by reflexivity $$A_{m_{\mu}}=\bigcap_{r>m_{\mu}}A_{r}\ne\varnothing.$$ In this case minimality implies $w_{\infty}(\mu,\delta_{y})=m_{\mu}$ for all $y\in A_{m_{\mu}}$.
In case $p=\infty$ note that $y\mapsto w_{\infty}(\mu,\delta_{y})=\sup_{x\in\supp\mu}d(x,y)$ is finite iff $\mu$ has bounded support in which case $A_{r}$ is bounded as well.
The cases $p\in(1,\infty)$ where proven in [@Kuwae2013 Proposition 3.1], the assumption on properness can be dropped using reflexivity. For convenience we include the short proof: If $\mu\in\mathcal{P}_{p}(X)$ then $w_{p}(\mu,\delta_{y_{0}})\le R$. Now take any $y\in X$ and assume $w_{p}(\mu,\delta_{y})\le r$. Since $(X,d)$ is isometrically embedded into $(\mathcal{P}_{p}(X),w_{p})$ by the map $y\mapsto\delta_{y}$ we have $$\begin{aligned}
d(y_{0},y)=w_{p}(\delta_{y_{0}},\delta_{y}) & \le & w_{p}(\mu,\delta_{y_{0}})+w_{p}(\mu,\delta_{y})\\
& \le & R+r,\end{aligned}$$ i.e. $y\in B_{R+r}(y_{0})$ which implies $A_{r}$ is bounded. Using a similar argument one can also show that $A_{r}$ is bounded if $\mu$ is only in $\mathcal{P}_{p-1}(X)$.
Let $p\in[1,\infty]$ and $(X,d)$ be a strictly $p$-convex if $p\in[1,\infty)$ and uniformly $\infty$-convex if $p=\infty$. Then $p$-barycenters are unique for $p>1$. In case $p=1$, all measure admitting $1$-barycenters which are not supported on a single geodesic have a unique $1$-barycenter.
The $p$-product of finitely many metric spaces $\{(X_{i},d_{i})\}_{i=1}^{n}$ is defined the metric space $(X,d)$ with $X=\times_{i=1}^{n}X_{i}$ and $$d(x,y)=\left(\sum_{i=1}^{n}d_{i}^{p}(x_{i},y_{i})\right)^{\frac{1}{p}}.$$ A minor extension of [@Foertsch2004 Theorem 1] shows that for $p\in(1,\infty)$ the space $(X,d)$ is strictly $p$-convex if all $(X_{i},d_{i})$ are if $1<p<\infty$ and projections onto the factors of geodesic in $(X,d)$ are geodesics in $(X_{i},d_{i})$.
Let $\{(X_{i},d_{i})\}_{i=1}^{n}$ be finitely many strictly $p$-convex reflexive metric spaces and $(X,d)$ be the $p$-product of $\{(X_{i},d_{i})\}_{i=1}^{n}$. If $\mu\in\mathcal{P}_{p}(X)$ then $b_{p}(\mu)=(b_{p}(\mu_{i}))$ where $\mu_{i}$ are the marginals of $\mu$.
If is not difficult to see that $$w_{p}^{p}(\mu,\delta_{y})=\sum_{i=1}^{n}w_{p}^{p}(\mu_{i},\delta_{y_{i}}).$$ Thus by existence for the factors we know $$\inf_{y\in X}w_{p}(\mu,\delta_{y})\le\sum_{i=1}^{n}w_{p}^{p}(\mu_{i},\delta_{b_{p}(\mu_{i})})=w_{p}^{p}(\mu,(b_{p}(\mu_{i}))).$$ Conversely, suppose there is a $y$ such that $w_{p}(\mu,\delta_{y})\le w_{p}(\mu,b_{p}(\mu))$. Since it holds $w_{p}(\mu_{i},\delta_{y_{i}})\ge w_{p}(\mu_{i},b_{p}(\mu_{i}))$ we see that $y$ is a minimizer of $y\mapsto w_{p}(\mu,\delta_{y})$. Since $(X,d)$ is strictly $p$-convex, $y=b_{p}(\mu)$.
Jensen’s inequality {#jensens-inequality .unnumbered}
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The classical Jensen’s inequality states that on a Hilbert space $H$ for any measure $\mu\in\mathcal{P}_{1}(H)$ and any convex lower semicontinuous function $\varphi\in L^{1}(H,\mu)$ it holds $$\varphi\left(\int xd\mu(x)\right)\le\int\varphi(x)d\mu(x).$$ With the help of barycenters Jensen’s inequality can be stated as follows.
\[Jensen’s inequality\] A metric space $(X,d)$ is said to admit the $p$-Jensen’s inequality if for all measure $\mu$ admitting a (unique) barycenter $b_{p}(\mu)$ and for every lower semicontinuous function $\varphi\in L^{p-1}(X,\mu)$ it holds $$\varphi(b_{p}(\mu))\le b_{p}(\varphi_{*}\mu)$$ where $\varphi_{*}\mu\in\mathcal{P}(\mathbb{R})$ is the push-forward of $\mu$ via $\varphi$.
For $p=2$ this boils down to $$b_{2}(\varphi_{*}\mu)=\int\varphi d\mu.$$
Using the existence proofs above one can adapt Kuwae’s proof of [@Kuwae2013 Theorem 4.1] to show that Jensen’s inequality holds spaces satisfying the condition $(\mathbf{B})$, i.e. for any two geodesics $\gamma,\eta$ with $\{p_{0}\}=\gamma\cap\eta$ and $\pi_{\gamma}(y)=p_{0}$ for $y\in\eta\backslash\{p_{0}\}$ it holds $\pi_{\eta}(x)=p_{0}$ for all $x\in\eta$. Kuwae states this as $\eta\bot_{p_{0}}\gamma$ implies $\gamma\bot_{p_{0}}\eta$.
Using Busemann’s non-positive curvature condition one can then show that if each measure in $\mathcal{P}_{1}(X)$ admits unique barycenters and Jensen’s inequality holds on the product space then a Wasserstein contraction holds, i.e. $$d(b_{2}(\mu),b_{2}(\nu))\le w_{1}(\mu,\nu).$$
If instead the $p$-Busemann holds one still gets the following:
\[prop:Jensen-p-Busemann\]Let $(X,d)$ be $p$-Busemann for some $p\in[1,\infty)$. If the $2$-Jensen’s inequality for holds on the $2$-product $X\times X$ then $$d(b_{2}(\mu),b_{2}(\nu))\le w_{p}(\mu,\nu).$$
Since $(x,y)\mapsto d(x,y)^{p}$ is convex on $X\times X$ we have by Jensen’s inequality $$d(b_{2}(\mu),b_{2}(\nu))^{p}\le\int d^{p}(x,y)d\pi(x,y)$$ for any $\pi\in\Pi(\mu,\nu)$. Hence $d(b_{2}(\mu),b_{2}(\nu))\le w_{p}(\mu,\nu)$.
Banach-Saks {#banach-saks .unnumbered}
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The classical Banach-Saks property for Banach spaces is stated as follows: Any bounded sequence has a subsequence $(x_{m_{n}})$ such that sequence of Cesàro means $$\frac{1}{N}\sum_{n=1}^{N}x_{m_{n}}$$ converges strongly. In a general metric space there is no addition of two elements defined. Furthermore, convex combinations do not commute (are not associative), i.e. if $(1-\lambda)x\oplus\lambda y$ denotes the point $x_{\lambda}$ on the geodesic connecting $x$ and $y$ then in general $$\frac{2}{3}\left(\frac{1}{2}x\oplus\frac{1}{2}y\right)\oplus\frac{1}{3}z\ne\frac{1}{3}x\oplus\frac{2}{3}\left(\frac{1}{2}y\oplus\frac{1}{2}z\right),$$ so that $\frac{1}{N}\bigoplus_{n=1}^{N}x_{n}$ does not make sense.
For Hilbert spaces the point $\frac{1}{N}\bigoplus_{n=1}^{N}x_{n}$ agrees with the $2$-barycenter of the measure $$\mu_{N}=\frac{1}{N}\sum_{n=1}^{N}\delta_{x_{n}}.$$ Since this is well-defined on general metric spaces the Banach-Saks property can be formulated as follows.
\[$p$-Banach-Saks\] Let $p\in[1,\infty]$ and suppose for any sequence $(x_{n})$ in a metric space $(X,d)$ the measures $\mu_{n}$ admit a unique $p$-barycenter. Then $(X,d)$ is said to satisfy the $p$-Banach-Saks property if every sequence $(x_{n})$ there is a subsequence $(x_{m_{n}})$ such that the sequence of $p$-barycenters of the measures $\tilde{\mu}_{N}=\frac{1}{N}\sum_{n=1}^{N}\delta_{x_{m_{n}}}$ converges strongly.
Since Hilbert spaces satisfy the (traditional) Banach-Saks property they also satisfy the $p$-Banach-Saks property. Yokota managed in [@Yokota Theorem C] (see also [@Bacak2014a Theorem 3.1.5]) to show that any $CAT(1)$-domain with small radius, in particular any $CAT(0)$-space, satisfy the $2$-Banach-Saks property and if $x_{n}\overset{w}{\to}x$ then $b_{p}(\tilde{\mu}_{N})\to x$ where $\tilde{\mu}_{N}$ is defined above. We will adjust his proof to show that for $p\in(1,\infty)$ any uniformly $p$-convex space satisfies the $p$-Banach-Saks property and the limit of the chosen subsequence agrees with the weak sequential limit. Since a $CAT(1)$-domain with small radius is uniformly $p$-convex, our result generalizes Yokota’s when restricted to that convex subset.
We leave the proof of the following statement to the reader.
If $(X,d)$ is uniformly $p$-convex and $x_{n}\to x$ then $b_{p}(\mu_{N})\to x$ where $\mu_{N}=\frac{1}{N}\sum_{n=1}^{N}\delta_{x_{n}}$.
\[lem:unif-conv\]Assume $(X,d)$ be a metric space admitting midpoints and let $f:X\to\mathbb{R}$ be a uniformly convex function with modulus $\omega$. If $f$ attains its minimum at $x_{m}\in X$ then $$f(x)\ge f(x_{\min})+\frac{1}{2}\omega(d(x,x_{\min})).$$
Let $x\in X$ be arbitrary and $m$ be the midpoint of $x$ and $x_{m}$. Then $$f(x_{\min})\le f(m)\le\frac{1}{2}f(x_{\min})+\frac{1}{2}f(x)-\frac{1}{4}\omega(d(x,x_{\min})).$$
\[thm:Banach-Saks\]Let $(X,d)$ be a uniformly $p$-convex metric space. If $x_{n}\overset{w}{\to}x$ then there is a subsequence $(x_{m_{n}})$ such that $b_{p}(\tilde{\mu}_{N})\to x$ where $\tilde{\mu}_{N}=\frac{1}{N}\sum_{n=1}^{N}\delta_{x_{m_{n}}}$. In particular, $(X,d)$ satisfies the $p$-Banach-Saks property.
By the previous lemma we can assume that $(x_{n})$ is $2\epsilon$-separated for some $\epsilon>0$, since otherwise there is a strongly convergent subsequence fulfilling the statement of the theorem. Furthermore, assume w.l.o.g. that $d(x_{n},x)\to r$. Then $$\lim_{N\to\infty}\frac{1}{N}\sum d(x,x_{n})^{p}\to r^{p}.$$
For any measure $\mu\in\mathcal{P}_{p}(X)$ define $$\Var_{\mu,p}(y)=\int d(x,y)^{p}d\mu(x)$$ and $$V(\mu)=\inf_{y\in X}\Var_{\mu,p}(y)$$ Furthermore, for any finite subset $I\subset\mathbb{N}$ define $$\mu_{I}=\frac{1}{|I|}\sum_{i\in I}\delta_{x_{i}}.$$ Let $R>0$ be such that $d(x_{n},x)\le R$. Note that $b_{p}(\mu_{I})\in B_{2R}(x)$ for any finite $I\subset\mathbb{N}$. Furthermore, if $(y_{t})_{t\in[0,1]}$ is a geodesic in $B_{2R}(x)$ with $d(y_{0},y_{1})\ge3\delta R$ then $d(y_{0},y_{1})\ge\delta\mathcal{M}^{p}(d(x_{n},y_{0}),d(x_{n},y_{1}))$ and by uniform $p$-convexity $$d(x_{n},y_{\frac{1}{2}})^{p}\le(1-\tilde{\rho}_{p}(\delta))\left(\frac{1}{2}d(x_{n},y_{0})+\frac{1}{2}d(x_{n},y_{1})\right).$$ Thus there is a monotone function $\tilde{\omega}:(0,\infty)\to(0,\infty)$ such that $$d(x_{n},y_{\frac{1}{2}})^{p}\le(1-\tilde{\omega}(d(y_{0},y_{1}))\left(\frac{1}{2}d(x_{n},y_{0})^{p}+\frac{1}{2}d(x_{n},y_{1})^{p}\right).$$ This implies $$\Var_{\mu_{I},p}(y_{\frac{1}{2}})\le(1-\tilde{\omega}(d(y_{0},y_{1}))\left(\frac{1}{2}\Var_{\mu_{I},p}(y_{0})+\frac{1}{2}\Var_{\mu_{I},p}(y_{1})\right)$$
By $2\epsilon$-separation of $(x_{n})$ for any finite $I\subset\mathbb{N}$ there is at most one $i\in I$ such that $d(x_{i},y)\le\epsilon$. Hence if $|I|\ge2$ $$V(\mu_{I})\ge\frac{1}{2}\epsilon^{p}.$$ Combining this with the above inequality we see that for $\omega(r)=2\epsilon^{p}\tilde{\omega}(r)$ $$\Var_{\mu_{I},p}(y_{\frac{1}{2}})\le\frac{1}{2}\Var_{\mu_{I},p}(y_{0})+\frac{1}{2}\Var_{\mu_{I},p}(y_{1})-\frac{1}{4}\omega(d(y_{0},y_{1})).$$ This implies $$\Var_{\mu_{I},p}(y_{t})\le(1-t)\Var_{\mu_{I},p}(y_{0})+t\Var_{\mu_{I},p}(y_{1})-t(1-t)\omega(d(y_{0},y_{1})).$$ i.e. the functions $\Var_{\mu_{I},p}:B_{2R}(X)\to\mathbb{R}$ are uniformly convex with modulus $\omega$.
The next steps follow directly from the proofs of [@Yokota Theorem C] and [@Bacak2014a Theorem 3.1.5] we include the whole proof for convenience of the reader.
Step 1: set $I_{k}^{N}=\{(k-1)2^{N},\ldots,k2^{N}\}\subset\mathbb{N}$ for any $k,N\in\mathbb{N}$. We claim that if $$\sup_{N\in\mathbb{N}}\liminf_{k\to\infty}V(\mu_{I_{k}^{N}})=r^{p}$$ then $b_{n}=b_{p}(\mu_{I_{n}^{0}})$ converges strongly to $x$. To see this note that for any $\epsilon>0$ there is an $N\in\mathbb{N}$ such that $$\liminf_{k\to\infty}V(I_{k}^{N})\ge r^{p}-\epsilon.$$ Then $$\begin{aligned}
\liminf_{n\to\infty}\Var_{\mu_{I_{n}^{0}},p}(b_{n}) & \ge & \liminf_{k\to\infty}V(\mu_{I_{k}^{N}})\\
& \ge & r^{p}-\epsilon.\end{aligned}$$ Since $\epsilon>0$ is arbitrary we see that $\liminf_{n\to\infty}\Var_{\mu_{I_{n}^{0}},p}(b_{n})\ge r^{p}$.
By uniform convexity of $\Var_{\mu_{I_{n}^{0}},p}$ and Lemma \[lem:unif-conv\] we also have $$\Var_{\mu_{I_{n}^{0}},p}(x)\ge\Var_{\mu_{I_{n}^{0}},p}(b_{n})+\frac{1}{2}\omega(d(x,b_{n})).$$ Since the left hand side converges to $r^{p}$ and $\limsup_{n\to\infty}\Var_{\mu_{I_{n}^{0}},p}(b_{n})\ge r^{p}$. This implies that $\limsup\omega(d(x,b_{n}))\to0$, i.e. $d(x,b_{n})\to0$.
Step 2: We will select a subsequence of $(x_{n})$ such the assumption of the claim in Step 1 are satisfied. Set $J_{k}^{0}=\{k\}$ for $k\in\mathbb{N}$. We construct a sequence of set $J_{k}^{N}$ for $N\in\mathbb{N}$ of cardinality $2^{N}$ such that $J_{k}^{N}=J_{l}^{N-1}\cup J_{m}^{m-1}$ for some $m,l\in N$. Furthermore,$\max J_{k}^{N}<J_{k+1}^{N}$ and $$\lim_{k\to\infty}V(\mu_{J_{k}^{N}})=V_{N}:=\limsup_{l,m\to\infty}V(\mu_{J_{l}^{N-1}\cup J_{m}^{N-1}}).$$ It is not difficult to see that $V_{N}\le V_{N+1}\le r^{p}$ for every $N\in\mathbb{N}$. We will show that $V_{N}\to r^{p}$ as $N\to\infty$. It is not difficult to see that this follows from the claim below.
For every $\epsilon'>0$ there exists a $\delta>0$ such that whenever $V_{N}<(r-\epsilon)^{p}$ then $V_{N+1}>V_{N}+\delta$.
\[Proof of claim\] Fix $N\in\mathbb{N}$ and for $l\in\mathbb{N}$ let $b_{l}^{N}$ be the $p$-barycenter of $\mu_{J_{k}^{N}}$. By assumption there is an $l\in\mathbb{N}$ such that $V(\mu_{J_{k}^{N}})<(r-\epsilon')^{p}$ for all $k\ge l$. By the Opial property, Corollary \[cor:Opial\], there is a large $m>l$ such that $d(b_{l}^{N},x_{i})>r$ for $i\in J_{m}^{N}$.
This implies $$\frac{1}{2^{N}}\sum_{i\in J_{m}^{N}}d(b_{l}^{N},x_{i})^{p}>r^{p}>(r-\epsilon)^{p}>V(\mu_{J_{m}^{N}})=\frac{1}{2^{N}}\sum_{i\in J_{m}^{N}}d(b_{m}^{N},x_{i})$$ and hence $$2\max\left\{ d(b_{m}^{N},b_{m\cup l}^{N}),d(b_{l}^{N},b_{m\cup l}^{N})\right\} \ge d(b_{m}^{N},b_{l}^{N})>\epsilon'$$ where $b_{m\cup l}^{N}$ is the $p$-barycenter of $\mu_{J_{m}^{N}\cup J_{l}^{N}}$. By uniform convexity of $\Var_{\mu_{I},p}$ and Lemma \[lem:unif-conv\] we get $$\begin{aligned}
V(\mu_{J_{l}^{N}\cup J_{m}^{N}}) & = & \frac{1}{2^{N+1}}\left(\sum_{i\in J_{l}^{N}}d(b_{l\cup m}^{N},x_{i})^{p}+\sum_{i\in J_{l}^{N}}d(b_{l\cup m}^{N},x_{i})^{p}\right)\\
& \ge & \frac{1}{2}\left[V(\mu_{J_{l}^{N}})+V(\mu_{J_{m,}^{N}})+\omega(d(b_{m}^{N},b_{m\cup l}^{N}))+\omega(d(b_{l}^{N},b_{m\cup l}^{N}))\right]\\
& \ge & \frac{1}{2}\left[V(\mu_{J_{l}^{N}})+V(\mu_{J_{m,}^{N}})+2\omega(\epsilon')\right].\end{aligned}$$
To finish the proof of the theorem, note that $\bigcap_{N}\bigcup_{k}J_{k}^{N}\subset\mathbb{N}$ is infinite. Denoting its elements in increasing order $(n_{1},n_{2},\ldots)$ we see that the sequence $(x_{n_{k}})$, after naming, satisfies the assumption needed in Step 1.
Assume $(X,d)$ is uniformly $p$-convex and that for any sequence $(x_{n})$ the $p$-barycenter of $\mu_{I}$ for any finite $I\subset\mathbb{N}$ is in the convex hull of the point $\{x_{i}\}_{i\in I}$. Then whenever $x_{n}\overset{w}{\to}x$ implies $x\in\overline{\conv(x_{n})}$. In particular, it holds $x_{n}\overset{\tau_{co}}{\to}x$, i.e. the co-convex topology is weaker that weak sequential topology.
The assumption of the corollary are satisfied on any $CAT(0)$-space, see [@Bacak2014a Lemma 2.3.3] for the $2$-barycenter. More generally Kuwae’s condition $(\mathbf{A})$ is enough as well, see [@Kuwae2013 Remark 3.7 (2)]. In particular, it holds on all spaces that satisfy Jensen’s inequality.
Generalized Convexities {#generalized-convexities .unnumbered}
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Let $L:(0,\infty)\to(0,\infty)$ be a strictly increasing convex function such that $L(1)=1$ and $L(r)\to0$ as $r\to0$. Then $L$ has the following form $$L(r)=\int_{0}^{r}\ell(s)ds$$ where $\ell$ is a positive monotone function. As an abbreviation we also set $L_{\lambda}(r)=L(\frac{r}{\lambda})$ for $\lambda>0$.
Given $L$ we define the $L$-mean of two non-negative numbers $a,b\in[0,\infty)$ as follows $$\mathcal{M}^{L}(a,b)=L^{-1}\left(\frac{1}{2}L\left(a\right)+\frac{1}{2}L\left(b\right)\right)\left\{ t>0\,|\,\frac{1}{2}L\left(\frac{a}{t}\right)+\frac{1}{2}L\left(\frac{b}{t}\right)\le1\right\} .$$
\[$L$-convexity\] A metric space admitting midpoints is said to be $L$-convex if for any triple $x,y,z\in X$ it holds $$d(m(x,y),z)\le\mathcal{M}^{L}(d(x,z),d(y,z)).$$ If the inequality is strict whenever $x\ne y$ then the space is said to be strictly $L$-convex.
In a similar way one can use a more elaborate definition of mean: For $L$ as above define the Orlicz mean $$\tilde{\mathcal{M}}^{L}(a,b)=\inf\left\{ t>0\,|\,\frac{1}{2}L\left(\frac{a}{t}\right)+\frac{1}{2}L\left(\frac{b}{t}\right)\le1\right\} .$$ Now Orlicz $L$-convexity can be defined by using $\tilde{\mathcal{M}}^{L}$ instead of $\mathcal{M}^{L}$. It is not clear if this definition is meaningful. The existence theorem for Orlicz-Wasserstein barycenters below only uses $L$-convexity.
It is easy to see that for $p\in(1,\infty)$ (strict) $p$-convexity is the same as (Orlicz) $L$-convexity for $L(r)=r^{p}$. However, because $L$ needs to be strictly convex, the cases $1$-convexity and $\infty$-convexity are not covered, but can be obtained as limits. Strict convexity of $L$ also implies that the inequality above is strict whenever $d(x,y)=|d(x,z)-d(y,z)|$, i.e. the condition $d(x,y)>|d(x,z)-d(y,z)|$ is not needed for strict $L$-convexity.
Suppose $\Phi$ is a convex function with $\Phi(1)=1$ and $\Phi(r)\to0$ as $r\to0$. Then any (strictly) (Orlicz) $L$-convex metric space is (strictly) (Orlicz) $\Phi\circ L$-convex and (strictly) $\infty$-convex. Also, any (strictly) $1$-convex space is strictly (Orlicz) $L$-convex.
The proof of this lemma follows directly from convexity of $\Phi$. Similarly one can define uniform convexity.
\[uniform $p$-convexity\]A strictly $L$-convex metric space is said to be uniformly $L$-convex if for all $\epsilon>0$ there is a $\rho_{L}(\epsilon)\in(0,1)$ such that for all triples $x,y,z\in X$ satisfying $d(x,y)>\epsilon\mathcal{M}^{L}(d(x,z),d(y,z))$ it holds $$d(m(x,y),z)\le(1-\rho_{L}(\epsilon))\mathcal{M}^{L}(d(x,z),d(y,z)).$$
Using $L$ one can also define an Orlicz-Wasserstein space $(\mathcal{P}_{L}(X),w_{L})$, see [@Sturm2011] and [@Kell2013a Appendix] for precise definition and further properties. Since $L(1)=1$ the natural embedding $x\to\delta_{x}$ is an isomorphism. For $\mu\in\mathcal{P}_{L}(X)$ and $y\in X$ the metric $w_{L}$ has the following form $$w_{L}(\mu,y)=\inf\{t>0\,|\,\int L\left(\frac{d(x,y)}{t}\right)d\mu(x)\le1\}.$$ Note that by [@Sturm2011] the infimum is attained if $w_{L}(\mu,y)>0$.
Now the $L$-barycenter $b_{L}(\mu)$ of a measure $\mu\in\mathcal{P}_{L}(X)$ can be defined as $$b_{L}(\mu)=\argmin_{y\in X}w_{L}(\mu,\delta_{y}).$$
Assume $(X,d)$ is reflexive and strictly $L_{\lambda}$-convex for any $\lambda>0$. Then any measure $\mu\in\mathcal{P}_{L}(X)$ admits a unique barycenter.
Since $L(r)=r^{p}$ is homogeneous, one sees that $(X,d)$ is strictly $ $$L$-convex iff it is strictly $L_{\lambda}$-convex for some $\lambda>0$.
By our assumption we see that for any $\lambda>0$ $$\int L\left(\frac{d(x,y_{\frac{1}{2}})}{\lambda}\right)d\mu(x)\le\frac{1}{2}\int L\left(\frac{d(x,y_{0})}{\lambda}\right)d\mu(x)+\frac{1}{2}\int L\left(\frac{d(x,y_{1})}{\lambda}\right)d\mu(x).$$ with strict inequality whenever $y_{0}\ne y_{1}$. Hence, if $F_{\mu}(y_{0}),F_{\mu}(y_{1})\le\Lambda$ then $$\int L\left(\frac{d(x,y_{\frac{1}{2}})}{\Lambda}\right)d\mu(x)\le1,$$ i.e. $F_{\mu}(y_{\frac{1}{2}})\le\Lambda$. Furthermore, the strict inequality is strict if $y_{0}\ne y_{1}$, i.e. $F_{\mu}$ is strictly quasi-convex and can have at most one minimizer.
This now implies that the sublevels of $F_{\mu}$ are convex. Closedness follows from continuity of $F_{\mu}$. In order to see that they are also bounded, just note that $w_{L}$ is a metric, i.e. implies that $|w_{L}(\mu,\delta_{y_{0}})-w_{L}(\delta_{y_{0}},\delta_{y})|\le w_{L}(\mu,\delta_{y})$. Thus $F_{\mu}(y_{0})\le R$ for all $y\in X\backslash B_{2R}(y)$ it holds $$F_{\mu}(y)=w_{L}(\mu,\delta_{y})>R.$$
reflexivity implies now existence of $L$-barycenters.
In a similar way one can obtain a Banach-Saks theorem for spaces which are uniformly $L_{\lambda}$-convex for each $\lambda>0$ such that the moduli $(\rho_{L_{\lambda}})_{\lambda\in(0,\infty)}$ are equicomparable for compact subsets of $(0,\infty)$. The proof then follows along the line of Theorem \[thm:Banach-Saks\].
[^1]: The author wants to thank Prof. Jürgen Jost and the MPI MiS for providing a stimulating research environment. Also thanks to Miroslav Ba[č]{}ák for helpful explanations of $CAT(0)$-space and remarks on an early version of the paper which simplified some of the statements.
|
---
abstract: 'We explore the possibility of using machine learning to identify interesting mathematical structures by using certain quantities that serve as fingerprints. In particular, we extract features from integer sequences using two empirical laws: Benford’s law and Taylor’s law and experiment with various classifiers to identify whether a sequence is, for example, nice, important, multiplicative, easy to compute or related to primes or palindromes.'
author:
- |
Chai Wah Wu\
IBM Research AI\
IBM T. J. Watson Research Center\
P. O. Box 218\
Yorktown Heights, NY 10598\
`[email protected]`\
date: 'September 9, 2018'
title: 'Can machine learning identify interesting mathematics? An exploration using empirically observed laws '
---
Introduction
============
Machine learning has made significant strides in solving classification problems in several domains that humans excel at, for instance in image processing and speech recognition. There has been some effort to classify scientific knowledge as well [@Akritidis2013] by analyzing the text of scientific articles. So far, there has been much less progress in terms of classification using only the mathematical equations and quantities in scientific knowledge. Part of the difficulty is that there is less leeway in the interpretation of mathematics; the same numbers, symbols and equations can have completely different meaning based on the specific way these objects are composed on the page. On the other hand, classical logic-based AI and symbolic computer algebra systems have been more successful in this regard [@Kushman2014].
The purpose of this paper is to investigate the following perhaps simpler problem: can machine learning identify qualitative attributes of scientific knowledge, i.e. can we tell whether a scientific result is elegant, simple or interesting? We will start our investigation by restricting the domain to mathematical sequences of numbers.
Online Encyclopedia of Integer Sequences
========================================
The object of study in this paper are sequences of integers. One reason for choosing them to study is that many fundamental mathematical ideas are captured in these structures. Another reason is that there exists an extensive database of integer sequences that has been edited and curated for over 50 years: the Online Encyclopedia of Integer Sequences (OEIS) [@oeis]. The OEIS was created by Neil Sloane in 1964 and has grown to over 300,000 sequences as of this writing with thousands of volunteers from the OEIS community editing and adding metadata and references to these sequences. To each sequence are associated keywords assigned by the community members. Some examples of keywords are:‘nice’, ‘core’, ‘base’, ‘hard’, etc. A complete set of keywords and their definitions can be found at <http://oeis.org/wiki/Keywords>. Many classical sequences are in the database, such as the sequence of primes, the binomial coefficients, the Fibonacci numbers, etc. There is a range of complexity, ranging from sequences that are very easy to compute (such as the sequence of odd numbers [A005408](https://oeis.org/A005408)), hard to compute (such as the number of nonsingular $n\times n$ 0-1 matrices [A055165](https://oeis.org/A055165)) to sequences for which it is not known whether it is finite or not (such as the list of Mersenne primes [A000668](https://oeis.org/A000668)).
Empirical laws
==============
Since the number of terms of each sequence that are available for analysis can vary, it is desirable to have a fixed number of features that can be computed on sequences of any finite length. An objective of this paper is to study whether empirical laws can serve this purpose. In particular, we look at 2 empirically observed laws that have appeared in the literature. Empirical laws are not mathematical theorems per se, but are empirical observations of relationships that seem to apply to many natural and man-made data sets (e.g. Moore’s law in electrical engineering [@Moore1965b] or the 80/20 Pareto principle in economics), but why these occur so frequently are typically still not completely understood. As these empirical laws are discovered because many data sets of interest seem to abide by them, they are a good starting point for finding features for classification.
Benford’s law
-------------
Benford’s law [@Benford1938] states that in a set of numerical data, terms with a small leading digit tend to occur more frequency. More precisely, in base $b$, terms with leading digit $d$ occurs with probability equal to $\log_{b}(\frac{d+1}{d})$. In particular we will define the discrete distribution $ \{d_i\}$ where $d_i = \log_{10}(\frac{i+1}{i})$ and $i=1,\cdots 9$ to be the [*Benford*]{} distribution $b(i)$.[^1]
This empirical law was first observed by Simon Newcomb [@Newcomb1881] who noted that in logarithms tables there were more numbers starting with 1 than with any other digit. This was noted later by Frank Benford who analyzed it for other data sets. Recently, Benford’s law has been shown to apply to several integer sequences [@Huerlimann2009].
Taylor’s law
------------
Another empirical law was defined by Lionel Taylor in 1961 [@Taylor1961] who noted that in ecology, the mean $\mu$ and the variance $v$ in species data appear to satisfy a power law: $$v = T_a\mu^{T_b}
\label{eqn:taylor}$$ where $T_a$ and $T_b$ are positive constants.
Taylor’s law has been observed in many naturally observed data sets [@Xiao2015; @Reuman2017] and in integer sequences such as the list of primes [@Cohen2016] and binomial coefficients [@Demers2018].
The data set
============
For the data set, we selected 40,000 sequences randomly from OEIS each with at least 990 terms accessible in the database.[^2] On average, approximately 1 in 4 sequences in the OEIS contains over 990 terms in their entry. Although most sequences in OEIS are defined as infinite sequences, it is not always easy to compute many terms. For each sequence, we collect all the terms that are available in OEIS and compute several quantities for each sequence.
Checking for Benford’s law
--------------------------
To check for Benford’s law, we compute the following features for each sequence: $b_d(i)$ is the proportion of terms with leading digit $i$ for $i = 1,\cdots 9$.
For each sequence $\{a(n)\}$,[^3] we compared the proportion of terms $b_d(i)$ with the Benford distribution $b(i)$ by using 4 different statistical distances: (1) the Kullback-Leibler (KL) divergence $D_{KL}(b_d||b) = \sum_i b_d(i)\log \frac{b_d(i)}{b(i)}$, (2) the Kolmogorov-Smirnov (KS) statistic $KL(b_d,d)$, (3) the Wasserstein distance (or earth mover’s distance) $WD(b_d,b)$ and (4) the total variation distance $TV(b_d,d)$.
Fig. \[fig:KL\] shows the KL divergence of the various sequences. We see that for many of the sequences, the KL-divergence is small and in the range \[0,0.2\]. The KL-divergence between the uniform distribution and $b(i)$ is 0.191, indicating that for most sequences $b_d(i)$ is a decreasing function. There is a cluster of sequences with KL-divergence about 1.2. These are due to sequences whose terms all start with the digit $1$, such as sequences that expresses the terms in binary notation (e.g. OEIS sequence A035526) for which $b_d(i) = 1$ if $i=1$ and $0$ otherwise (a distribution we will denote by $\delta_9$) as the corresponding KL-divergence is $\log\left(\frac{1}{\log_{10}(2)}\right) = \log(\log_2(10)) \approx 1.2005$. Fig. \[fig:KS\] shows the KS statistic which shows a similar behavior to Fig. \[fig:KL\].
We also computed the Wasserstein distance $WD(b_d,b)$ between $b_d$ and $b$ for these sequences. As the Wasserstein distance take into account a permutation of the digits (i.e. the Wasserstein distance does not change if the values of $b_d$ or $b$ are permuted), $WD(b_d,b) \leq WD(\delta_{9},b) = \frac{2(1-\log_{10}(2))}{9} \approx 0.1553$, the plot in Fig. \[fig:WD\] shows relatively smaller values.
Finally, in Fig. \[fig:TV\] we show the total variation distance of $b_d$ and $b$ which is similar to Fig. \[fig:KS\]. There are some sequences (i.e. [A000038](https://oeis.org/A000038)) which is all zero except for a single term which has a total variation distance close to $1$.
[0.5]{}
![$4$ statistical distances between $b_d$ and the benford distribution $b$. []{data-label="fig:benford"}](KL.png){width="2.9in"}
[0.5]{}
![$4$ statistical distances between $b_d$ and the benford distribution $b$. []{data-label="fig:benford"}](ks-statistic.png){width="2.9in"}
[0.5]{}
![$4$ statistical distances between $b_d$ and the benford distribution $b$. []{data-label="fig:benford"}](wasserstein.png){width="2.9in"}
[0.5]{}
![$4$ statistical distances between $b_d$ and the benford distribution $b$. []{data-label="fig:benford"}](totalvariation.png){width="2.9in"}
These figures show that $b_d$ is relatively close to $b$ for many sequences which implies that they adhere to Benford’s law.
Checking for Taylor’s law
-------------------------
To check for Taylor’s law, we compute the following quantities for each sequence $\{a(n)\}$: $\mu(n) = \frac{1}{n}\sum_{i=1}^n a(i)$ and $v(n) = \frac{1}{n-1}\sum_{i=1}^n (a(i) - \mu(i))^2$ with $v(1) = 0$.
Note that we use the sample variance as we interpret the sequence as samples from an experimental process. Using the population variance instead ($\frac{1}{n}\sum_{i=1}^n (a(i)-\mu(i))^2 = (n-1)v(n)/n$) gives very similar results as the total number of terms for each sequence is relatively large.
We fitted $\log v$ against $\log \mu$ with a linear regressor to obtain the following features: slope $s$, intercept $b$ and correlation coefficient $r$. When $r$ is close to $1$, Taylor’s law (Eq. \[eqn:taylor\]) is closely satisfied with slope $s = T_b$ and the intercept $b = \log(T_a)$.
Figure \[fig:cc\] shows the correlation coefficient $r$ against the sequences number. We see that for many (but not all) sequences the correlation coefficient $r$ is close to $1$. We also notice sequences where $r$ is negative, indicating a negative correlation. In this case $s$ is negative, corresponding to a negative exponent $T_b$ in Taylor’s law. This is quite different from the original form of Taylor’s law where $T_b >0$ and observed in general data sets [@Xiao2015; @Reuman2017; @Cohen2016; @Demers2018].
![Correlation coefficient $r$ of $\log(v)$ versus $\log(\mu)$.[]{data-label="fig:cc"}](corrcoef.png){width="5in"}
Figure \[fig:slope-vs-r\] shows $r$ plotted against $s$, along with a regressor derived from the RANSAC algorithm [@Fischler1987] with a slope of approximately $2$. Since $\frac{s}{r} = \frac{Sl_v}{Sl_\mu}$, where $Sl_v$ and $Sl_\mu$ are the standard deviation of $\log(v)$ and $\log(\mu)$ respectively, this implies that for many sequences $Sl_v \approx 2Sl_\mu$. This is further accentuated by the inliers in Fig. \[fig:slope-vs-r\], which represents about $50\%$ of the sequences considered, which matches the regressor line with slope $2$ with a very high correlation coefficient of $0.999$.
![Slope $s$ vs correlation coefficient $r$. The inliers (about half of the sequences) matches the regressor line with slope $2.001$, intercept = $0.003$ with a correlation coefficient equal to $0.999$.[]{data-label="fig:slope-vs-r"}](slope-vs-r.png){width="5in"}
For each sequence, we also compute $p_z$ which is the proportion of terms that are positive resulting in a total of $14$ features: $s$, $b$, $r$ and $p_z$, and $b_d(i)$ for $i=0,\cdots 9$. The feature $b_d(0)$ denotes the proportion of zero terms in the sequence. Note that $p_z$ and the features $b_d(i)$ related to Benford’s law is invariant under permutation of the terms of the sequence, whereas the features related to Taylor’s law ($s$, $b$ and $r$) do not and we think that both types of features are necessary to properly classify a sequence.
Classifiers for identifying OEIS sequences {#sec:classify}
==========================================
The above results show that many, but not all sequences satisfy to some degree Benford’s law (BL) and Taylor’s law (TL), suggesting that BL and TL could be used to identify whether a sequence would be of interest to OEIS editors and users. For instance, we could argue that if $s \approx 2r$, then the sequence is a candidate for inclusion in OEIS. To test this idea we generated approximately 40,000 sequences of 2000 random integers and calculated the $14$ features for these random sequences as well. We add these to the OEIS sequences to obtain a dataset of features from 80,000 sequences and randomly choose 70,000 for training and 10,000 for testing. We implemented a random forest classifier [@Breiman2001] with 665 trees and other parameters obtained via hyperparameter optimization. Preprocessing the data with a Principal Component Analysis, we were able to obtain the following performance metrics in distinguishing OEIS sequences from random sequences: accuracy: 0.999, precision: 0.9984, recall: 0.9996, F$_1$ score: 0.9990.
On the other hand, even though it appears relatively easy to distinguish OEIS sequences from random sequences, the complement of OEIS is hard to define precisely. In fact, almost any integer sequence can be submitted to the OEIS and included if the editors deemed it interesting mathematically. Furthermore, a small perturbation to any sequence in the OEIS will unlikely be in the database, but this will not be detected by this classifier. But given the sequences in the database so far we could draw the conclusion that such interesting sequences tend to satisfy BL and TL (or at least distinguishable by the features derived from BL and TL). The purpose of the next sections is to see if the parameters derived from the sequences to test for adherence to BL and TL can be used to further categorize sequences within OEIS.
Classifiers for identifying keywords in OEIS sequences
======================================================
We first identify the following labels for each sequence.
Sequence labels
---------------
We note for each sequence the absence or presence of the following OEIS keywords: ‘nice’, ‘core’, ‘easy’, ‘mult’. They describe sequences that are “nice”, important, easy to compute and multiplicative (in the number theory sense) respectively. In addition we added the keywords ‘prime’, ‘binomial’, ‘palindrome’[^4] if these words appear in the title or in the comments section of the sequence in the OEIS database. We also added a keyword ’other’ to denote the absence of any of the above keywords. Thus we have total of $8$ labels for each sequence. Each sequence can have more than one label.
We train different types of classifiers to analyze the dataset. A total of 35000 sequences will be use for training and validation. The test set consists of 5000 sequences. Preprocessing based on statistics of the training set are applied to normalize the training and test set. Note that these classifiers are not determining whether the sequences satisfy Benford’s Law and Taylor’s Law (and we have seen that some sequences don’t), but whether the features derived from these laws are useful in classification based on the keywords.
Neural network
--------------
The neural network has 6 dense layers with 933 neurons and about 140,000 trainable parameters, and using ReLu activation functions, except the output layer which uses a sigmoid activation function. A dropout layer with probability 0.25 is inserted after each input and hidden layer. We train this neural network for 40000 epochs with a batch size of 32. We use 31500 sequences for training in each epoch and 3500 sequences for validation.
Random forest ensemble classifier
---------------------------------
We will consider 2 types of ensemble classifiers: The random forest classifier and the extra trees ensemble classifier. For both these classifiers, the hyperopt-sklearn module is used to tune the hyperparameters. A standard scaling normalizes the data based on the variance and mean of the training set. The random forest consists of 744 trees, and all features have similar Gini importance.
Extra trees ensemble classifier
-------------------------------
The extra trees (extremely randomized trees) classifier [@Geurts2006] is a generalization and an improvement of the random forest classifier. The number of trees is 1059 and again all features have similar Gini importance.
Baseline classifier
-------------------
As a baseline, we also construct a random classifier, where each predicted label is chosen with a probability derived from the training set.
Since some labels occur much more frequently than other labels, the (subset) accuracy for such a unbalanced problem is generally not the best metric [@Valverde-Albacete2014], and therefore as in Section \[sec:classify\] we will also compute the precision, recall and F$_1$-score for each classifier.
Experimental results
====================
The performance of these various models in predicting each label is shown in Figure \[fig:performance\], where we plotted the (subset) accuracy, precision, recall and F$_1$-score of each model[^5]. Since each sequence can have multiple labels, these quantities are computed for each label and are averaged among the labels weighted by their support. In Figure \[fig:performanceclass\] we plot these quantities for each of the labels for each of the models. We find that the extra trees ensemble classifier performs the best, followed by a random forest classifier, and then a deep neural network. All of them performed better than the baseline classifier. Furthermore, they all had problems classifying labels that are not well supported: ‘nice’, ‘core’, ‘palindrome’, ‘binomial’ and ‘mult’, a well known problem of multilabel data sets that are not balanced. Note that for the label “palindrome” the extra trees classifier has a nontrivial recall and precision unlike the other classifiers for which these quantities are either $0$ or undefined. The neural network classifier has perfect precision (no false positives and at least one true positive) for ‘nice’ sequences and the extra trees classifier has perfect precision of ‘palindrome’ sequences. Note also that the scores for the ‘mult’ labels are significantly higher for the 3 classifiers versus the baseline classifier, suggesting that multiplicative sequences can be detected using these empirical laws. Can this conclusion be a consequence of the definition of multiplicative?
![Accuracy, precision, recall and F$_1$ score of 3 classifiers compared with the random classifier.[]{data-label="fig:performance"}](performance.pdf){width="5in"}
[0.5]{}
![Performance (precision, recall and F$_{1}$ score) of various classifiers on each label.[]{data-label="fig:performanceclass"}](rg.pdf){width="3in"}
[0.5]{}
![Performance (precision, recall and F$_{1}$ score) of various classifiers on each label.[]{data-label="fig:performanceclass"}](nn.pdf){width="3in"}
[0.5]{}
![Performance (precision, recall and F$_{1}$ score) of various classifiers on each label.[]{data-label="fig:performanceclass"}](rf.pdf){width="3in"}
[0.5]{}
![Performance (precision, recall and F$_{1}$ score) of various classifiers on each label.[]{data-label="fig:performanceclass"}](et.pdf){width="3in"}
The performance of the models are not stellar, but it is better than the baseline classifier. Part of this could be due to the fact that a small number of features (14) are used in the classifiers. As the OEIS database consists of sequences that people have submitted and the editors approved, it is biased towards sequences which people have found to be interesting or useful mathematically. This means that the ability to predict sequences with the “prime” label does not mean it was able to classify all prime-related sequences, but merely that it was able to classify prime sequences that are interesting or relevant. Furthermore, labels such as ‘nice’ and ‘easy’ are subjective and can vary depending on the person who assigned the label, and such issues are common in classification tasks such as sentiment analysis [@Jain2016].
In some cases the models were able to find related sequences. For instance sequence A059260 was in the training set which included the labels ‘nice’ and ‘binomial’. The sequence A059259 which was in the test set is a related sequence which enumerated the triangle of terms in a different order and the extra trees classifier also predicted the labels ‘nice’ and ‘binomial’ even though these labels were not assigned to A059259. The extra trees classifier predicted the label ‘binomial’ for sequence A080575 which is appropriate since it list the triangle of multinomial coefficients. Similarly, sequence A182009 which is an approximation (and is almost identical to) of sequence A033810 (which is in the training set), the extra trees classifier also predicted the labels of A033810 for sequence A182009.
Conclusions
===========
It is difficult to define what constitute interesting mathematical structures. If it is parsimony of representation, then perhaps something like Kolmogorov complexity would be an appropriate metric. We take the alternative approach that since many data sets follow empirical laws, these laws are harbingers of interesting integer sequences. In our experiments, we use OEIS as a proxy of what mathematical sequences are of interest to us. The experimental results point to the possibility of classifying interesting or relevant integer sequences using derived parameters based on empirical laws. They seem to indicate that we can differentiate mathematically interesting OEIS sequences from random sequences and that certain characteristics of these sequences can be identified solely based on the numbers in the sequence. A possible explanation for this is that the empirical laws are capturing inherent salient properties of numerical data that are interesting or important to study. Future work include training a deeper network, and adding other features perhaps from other empirical laws (such as Zipf’s law [@Zipf1935][^6] which is a general form of Benford’s law) to see if the performance improves. Some other ideas to consider include looking at the mean and variance of $n$-th order differences and how well a sequence fits a polynomial equation or a linear recurrence relationship. An interesting open question to investigate is why so many OEIS sequences follows $s = 2r$ so closely.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Neil Sloane for introducing us to Taylor’s law and Benford’s law and for his encouragement and helpful comments to a draft of this manuscript.
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[^1]: This is the base $10$ version of Benford’s law which has been verified for many experimental data sets, and it appears to hold in other bases as well.
[^2]: The reason for this odd number (990) is because the terms in the OEIS database are (for the most part) limited to 1000 digits, and some sequences such as “smallest prime containing at least $n$ consecutive identical digits” (OEIS sequence [A034388](https://oeis.org/A034388)) will have slightly less than 1000 terms in the database.
[^3]: As mentioned before, since the sequences are generally infinite, we mean here all the terms of the sequence that are available in the OEIS database.
[^4]: A number is called a [*palindrome*]{} if it is the same when read left to right or right to left. Examples include the numbers 1348431 and 9889. When the base is not specified, the number is assumed to be written in base 10.
[^5]: Precision, recall and F$_1$ score is set to $0$ when it is undefined (i.e. the denominator is $0$).
[^6]: Since most sequences have distinct terms, in order to test Zipf’s law some methods of grouping the terms into disjoint sets are needed.
|
---
abstract: 'In recent algorithms that use deformation in order to compute the number of points on varieties over a finite field, certain differential equations of matrices over $p$-adic fields emerge. We present a novel strategy to solve this kind of equations in a memory efficient way. The main application is an algorithm requiring quasi-cubic time and only quadratic memory in the parameter $n$, that solves the following problem: for $E$ a hyperelliptic curve of genus $g$ over a finite field of extension degree $n$ and small characteristic, compute its zeta function. This improves substantially upon Kedlaya’s result which has the same quasi-cubic time asymptotic, but requires also cubic memory size.'
author:
- |
Hendrik Hubrechts\
\
\
\
bibliography:
- 'bibliography.bib'
date: 'February 19, 2010'
title: Memory efficient hyperelliptic curve point counting
---
**AMS (MOS) Subject Classification Codes**: 11G20, 11Y99, 12H25, 14F30, 14G50, 14Q05.
Introduction and results {#sec:intro}
========================
Originally motivated by cryptography (see [@CohenFrey] for an overview), in recent years much effort was put in finding algorithms that compute zeta functions of varieties over finite fields. The most efficient algorithms often use deformation, i.e. they ‘deform’ the input variety to another variety that is easier to handle. This use of deformation originates from the work of Lauder [@LauderDeformation] in computing zeta functions of higher dimensional varieties. In that paper, the deformation step allowed him to reduce the dependency on the dimension in the algorithms. Also Tsuzuki [@TsuzukiKloosterman] came up with this idea in the context of the computation of Kloosterman sums. Later Gerkmann [@GerkmannEC] and the present author [@HubrechtsHECOdd; @HubrechtsHECEven] showed that even in dimension one, profit can be drawn from the use of deformation.
Central in all these applications stands a certain $p$-adic differential equation, namely the Picard-Fuchs equation of the associated connection. In the present paper we give an algorithm that allows us to compute a particular solution of this equation in a memory efficient way. Combining this with a well chosen deformation, from a general hyperelliptic curve to one defined over the prime field, yields our main result (Theorem \[thm:OddChar\]), stating that for a hyperelliptic curve of genus $g$ over the finite field ${\ensuremath{\mathbb{F}}}_{p^n}$, $p$ odd, the zeta function can be computed in time and memory (where we assume $p$ fixed and count bit operations) $${\ensuremath{\widetilde{\mathcal{O}}}}(n^3g^{6.376})\quad\text{respectively}\quad {\ensuremath{\mathcal{O}}}(n^2g^4(\log g)^{2}).$$ This result can be compared to Kedlaya’s algorithm [@KedlayaCountingPoints] that has ${\ensuremath{\widetilde{\mathcal{O}}}}(n^3g^4)$ as time and ${\ensuremath{\widetilde{\mathcal{O}}}}(n^3g^3)$ as space requirements. The crucial improvement is hence that our algorithm only requires an amount of memory quadratic in $n$. Note that the ${\ensuremath{\widetilde{\mathcal{O}}}}$ is the soft-Oh notation as defined in [@ModernCompAlg Definition 25.8] (it is essentially a big-Oh notation that ignores logarithmic factors).
Later on Denef and Vercauteren [@DenefVercauteren] extended Kedlaya’s algorithm to characteristic two. Combining this with our new result yields ‘on average’ the same result as in odd characteristic, see Theorem \[thm:EvenChar\] in Section \[sec:evenChar\]. The result for the ‘general case’ is slightly worse.
For any small characteristic it is also possible to compute the zeta functions of $n$ curves within a one dimensional linear family all at once in time ${\ensuremath{\mathcal{O}}}(n^{3+\rho})$ for arbitrary $\rho>0$. This result is presented in Section \[ssec:manyCurves\], and in Section \[ssec:hypersurfaces\] an additional application concerning hypersurfaces is explained. We note that all our complexities are bitwise unless mentioned otherwise.
Before we prove these results, we give in Section \[sec:diffEq\] a very general form of the algorithms involved, including a thorough investigation of the error propagation during the computations, all of which is concluded in Theorem \[thm:solvingDiffEq\].
The author wishes to thank Wouter Castryck, Filip Cools, Jan Denef, Jan Tuitman and the referee for their helpful remarks.
The differential equation {#sec:diffEq}
=========================
In this section we will define the differential equation referred to above. Given some conditions on the coefficients and on certain local solutions of this equation, we can present two algorithms that solve it, together with their complexity analysis.
A general kind of $p$-adic differential equation {#ssec:diffEq}
------------------------------------------------
Let $p$ be a prime number and $\mathbb{K}$ a degree $n$ field extension of the field of $p$-adic numbers ${\ensuremath{\mathbb{Q}}}_p$. We denote with ${\ensuremath{\textnormal{ord}}}$ the valuation on $\mathbb{K}$ normalized to ${\ensuremath{\textnormal{ord}}}(p)=1$, and $\mathcal{O}_{\mathbb{K}}:=\{x\in \mathbb{K}\,|\,{\ensuremath{\textnormal{ord}}}(x)\geq 0\}$ is the ring of integers of $\mathbb{K}$. Let $m>0$ be an integer. If we say that we are working in $\mathbb{K}$ modulo $p^m$, we mean that we use absolute precision, i.e. two numbers are considered equal when their difference has valuation at least $m$. We will also use power series in ${\ensuremath{\Gamma}}$ modulo ${\ensuremath{\Gamma}}^\ell$ for some integer $\ell>0$. The dimension of the square matrices that we will encounter is denoted by $d$. If $A({\ensuremath{\Gamma}})$ is a ($d\times d$) matrix over $\mathbb{K}[[{\ensuremath{\Gamma}}]]$, we will always use $A_i\in\mathbb{K}^{d\times d}$ in the following sense: $A({\ensuremath{\Gamma}})=\sum_iA_i{\ensuremath{\Gamma}}^i$, and in order to ease notation we will often write $A$ instead of $A({\ensuremath{\Gamma}})$. The valuation ${\ensuremath{\textnormal{ord}}}(A({\ensuremath{\Gamma}}))={\ensuremath{\textnormal{ord}}}(A)$ is defined to be $\inf_i({\ensuremath{\textnormal{ord}}}(A_i))$ when this infimum exists, and $-\infty$ otherwise. We say that a series $\sum_iA_i{\ensuremath{\Gamma}}^i$ with $A_i\in \mathbb{K}^{d\times d}$ is $(x,y)$-log convergent for real numbers $x\geq 0$ and $y$ if for every $i\geq 0$ $${\ensuremath{\textnormal{ord}}}(A_i)\geq -x\lceil\log_p(i+1)\rceil-y.$$ This implies in particular that such a series converges on the open unit disk in $\mathbb{K}$. The following easy lemma can be found as Lemma 15 in [@HubrechtsHECEven].
\[lem:convergenceLogProduct\] If $\sum_i A_i{\ensuremath{\Gamma}}^i$ and $\sum_iB_i{\ensuremath{\Gamma}}^i$ converge $(x,y)$-log resp. $(x',y')$-log, then their product has $(x+x',y+y')$-log convergence.
Let $A, B, X, Y$ be matrices over $\mathbb{K}[{\ensuremath{\Gamma}}]$ such that $A_0$ and $B_0$ are invertible. We define $\Delta$ as the $\mathbb{K}$-linear operator acting on $\mathbb{K}[[{\ensuremath{\Gamma}}]]^{d\times d}$ by $$\label{eq:Delta}
K\quad\mapsto\quad\Delta K=A \frac{d K}{d {\ensuremath{\Gamma}}} B+A K X+Y K B.$$
Requirements for the equation {#ssec:assumptions}
-----------------------------
In the proof of Theorem \[thm:solvingDiffEq\] below, we show that for every boundary condition $K_0$, a unique solution $K\in\mathbb{K}[[{\ensuremath{\Gamma}}]]^{d\times d}$ of $\Delta K=0$ exists. Our goal is to compute an approximation modulo $p^m$ of this unique solution $K$, where we assume that we know $K_0,A,B,X$ and $Y$ up to arbitrary precision, and in addition that $K_0$ (and hence $K({\ensuremath{\Gamma}})$) are invertible. Write $\zeta$ for an upper bound for the following three values: $$\deg A+\deg B,\ \ \deg A+\deg X+1,\ \ \deg Y+\deg B+1.$$ We note that $A({\ensuremath{\Gamma}})$ and $B({\ensuremath{\Gamma}})$ are invertible in $\mathbb{K}[[{\ensuremath{\Gamma}}]]^{d\times d}$ and require that there exists some $\alpha\in{\ensuremath{\mathbb{R}}}_{\geq 0}$ such that $${\ensuremath{\textnormal{ord}}}(A),\ {\ensuremath{\textnormal{ord}}}(A^{-1}),\ {\ensuremath{\textnormal{ord}}}(B),\ {\ensuremath{\textnormal{ord}}}(B^{-1}),\ {\ensuremath{\textnormal{ord}}}(K),\ {\ensuremath{\textnormal{ord}}}( K^{-1})\geq -\alpha.$$ We write $C$ for the unique solution of $A\frac{d C}{d{\ensuremath{\Gamma}}}+YC=0$, $C_0=1$, over $\mathbb{K}[[{\ensuremath{\Gamma}}]]^{d\times d}$ and $D$ for the solution of $\frac{dD}{d{\ensuremath{\Gamma}}}B+DX=0$, $D_0=1$. Then we require the existence of constants ${\ensuremath{\gamma}}\geq 0$ and $\delta$ such that $$\label{eq:CDlogConv}
C,\ C^{-1},\ D,\ D^{-1}\text{ have }({\ensuremath{\gamma}},\delta)\text{-log convergence.}$$ Finally we define $\psi:=5(\alpha+\delta)$ and we require that ${\ensuremath{\textnormal{ord}}}(X),\ {\ensuremath{\textnormal{ord}}}(Y)\geq -\psi$. As a last assumption we need that $K({\ensuremath{\Gamma}})$ modulo $p^m$ consists of polynomials of degree less than $\ell$, so that our approximation will be a finite object.
Note that although these conditions combined seem to be quite severe, they are met by a suitably adapted version of the differential equations appearing in the intended point counting algorithms.
Solving the differential equation {#ssec:solution}
---------------------------------
Due to precision loss during the execution of the algorithm, we have to work initially with a higher precision than $p^m$. Therefore we define $\varepsilon := m+(5{\ensuremath{\gamma}}+1)\lceil\log_p\ell\rceil+\psi$ and we work modulo $p^\varepsilon$. Let $\omega$ be an exponent for matrix multiplication, meaning that we can compute the product of two $d\times d$ matrices over $\mathbb{K}$ using ${\ensuremath{\mathcal{O}}}(d^\omega)$ arithmetic operations in $\mathbb{K}$. We may take $\omega = 2.376$, see [@CoppersmithWinograd].
\[thm:solvingDiffEq\] Suppose that we know $K_0,A,A_0^{-1},B,B_0^{-1},X$ and $Y$ modulo $p^\epsilon$ and that all assumptions of Section \[ssec:assumptions\] are met. Then we can compute $K({\ensuremath{\Gamma}})$ modulo $p^m$ using ${\ensuremath{\widetilde{\mathcal{O}}}}(\ell\zeta d^\omega n \varepsilon\log_2p)$ bit operations and with memory requirements ${\ensuremath{\mathcal{O}}}(\ell d^2n\varepsilon\log_2p)$ bits. Moreover, we can compute $K(1)$ modulo $p^m$ in the same amount of time but with memory only ${\ensuremath{\mathcal{O}}}(\zeta d^2n\varepsilon\log_2p)$ bits.
<span style="font-variant:small-caps;">Proof.</span> We first give the algorithm, then we will determine how much precision is lost throughout the computations, and finally we will do a resource analysis.\
Let ${\ensuremath{\mathcal{K}}}(\Gamma)$ and ${\ensuremath{\mathcal{K}}}(1)$ denote the approximations of $K({\ensuremath{\Gamma}})$ resp. ${\ensuremath{\mathcal{K}}}(1)$ that we compute using the following algorithm. Define the operator $\Delta'$ on $\mathbb{K}[[{\ensuremath{\Gamma}}]]^{d\times d}$ by $$\label{eq:DeltaPrime}
K\quad\mapsto\quad\Delta' K=A_0^{-1}A \frac{d K}{d {\ensuremath{\Gamma}}} BB_0^{-1}+A_0^{-1}A K XB_0^{-1}+A_0^{-1}Y K BB_0^{-1},$$ then clearly the equation $\Delta K=0$ is equivalent to $\Delta'K=0$. Writing down the coefficient of ${\ensuremath{\Gamma}}^k$ in $\Delta' K=0$ gives an equality of the form $$\sum_{a+b+c=k}\left( (b+1)A_0^{-1}A_aK_{b+1}B_cB_0^{-1}+A_0^{-1}A_aK_bX_cB_0^{-1}+A_0^{-1}Y_aK_bB_cB_0^{-1} \right)=0.$$ If we isolate $K_{k+1}$, this yields $$\label{eq:inductiveSoln}
(k+1)K_{k+1} = f_k\left(K_k,K_{k-1},\ldots,K_{k-(\zeta-1)}\right)$$ for some easy to construct linear polynomial $f_k$ defined over $\mathbb{K}^{d\times d}$ (where we put $K_i=0$ for $i<0$). This recursion relation allows us to calculate ${\ensuremath{\mathcal{K}}}({\ensuremath{\Gamma}})$ as follows: put ${\ensuremath{\mathcal{K}}}_i:=0$ for $i<0$, ${\ensuremath{\mathcal{K}}}_0:=K_0 \bmod p^\varepsilon$ and for $k=0,1,\ldots,\ell-2$ compute $$\label{eq:inductiveComp}
{\ensuremath{\mathcal{K}}}_{k+1} := \left[\frac 1{k+1}f_k\left({\ensuremath{\mathcal{K}}}_k,{\ensuremath{\mathcal{K}}}_{k-1},\ldots,{\ensuremath{\mathcal{K}}}_{k-(\zeta-1)}\right)\right]\bmod p^\varepsilon.$$ Finally we define ${\ensuremath{\mathcal{K}}}({\ensuremath{\Gamma}}):=\sum_{i=0}^{\ell- 1}{\ensuremath{\mathcal{K}}}_i{\ensuremath{\Gamma}}^i$. Note that equation (\[eq:inductiveSoln\]) implies that the solution $K({\ensuremath{\Gamma}})$ of $\Delta K=0$ exists and is unique.
Computing ${\ensuremath{\mathcal{K}}}(1)$ uses the same idea, but in order to save memory we work as follows. Define ${\ensuremath{\mathcal{L}}}_0:={\ensuremath{\mathcal{K}}}_0$ at the start, and put ${\ensuremath{\mathcal{L}}}_k := {\ensuremath{\mathcal{L}}}_{k-1}+{\ensuremath{\mathcal{K}}}_k\bmod p^{\varepsilon}$ each time that a new ${\ensuremath{\mathcal{K}}}_k$ is computed. This way we only have to store the last $\zeta$ matrices ${\ensuremath{\mathcal{K}}}_k,{\ensuremath{\mathcal{K}}}_{k-1},\ldots,{\ensuremath{\mathcal{K}}}_{k-(\zeta-1)}$ and ${\ensuremath{\mathcal{L}}}_k$ in order to find ${\ensuremath{\mathcal{K}}}_{k+1}$ and ${\ensuremath{\mathcal{L}}}_{k+1}$. After $\ell-1$ steps we will end with ${\ensuremath{\mathcal{L}}}_{\ell-1}=\sum_{i=0}^{\ell-1}{\ensuremath{\mathcal{K}}}_i\mod p^\varepsilon$.\
Every time that we multiply non integral elements of $\mathbb{K}$, we can expect a certain loss in $p$-adic precision. We will now show that working modulo $p^\varepsilon$ suffices to conclude that ${\ensuremath{\mathcal{K}}}({\ensuremath{\Gamma}})\equiv K({\ensuremath{\Gamma}})\bmod p^m$ and ${\ensuremath{\mathcal{K}}}(1)\equiv K(1)\bmod p^m$. We follow the reasoning of Section 3.5 in [@HubrechtsHECEven], but now in a more general setting[^1].
With the appropriate recursion modulo $(p^\varepsilon,{\ensuremath{\Gamma}}^\ell)$ similar to (\[eq:inductiveComp\]), but now for $A_0^{-1}A\frac{dC}{d{\ensuremath{\Gamma}}}+A_0^{-1}YC=0$ and $C_0=1$ we can compute ${\ensuremath{\mathcal{C}}}$, and in an analogous way ${\ensuremath{\mathcal{D}}}$ as approximation to $D$. We remark that $\mathcal{C}$ and $\mathcal{D}$ are only needed for the analysis, not in the actual computation. Rewriting (\[eq:inductiveComp\]) implies that $$(k+1){\ensuremath{\mathcal{K}}}_{k+1} - f_k\left(\mathcal{K}_k,\mathcal{K}_{k-1},\ldots, \mathcal{K}_{k-(\zeta-1)}\right) = p^\varepsilon\cdot(\text{integral error matrix}),$$ and if we sum over all $k$ this gives $\Delta' \mathcal{K}=p^\varepsilon\mathcal{E}_{\ensuremath{\mathcal{K}}}$, where $\mathcal{E}_{\ensuremath{\mathcal{K}}}$ is a matrix over $\mathcal{O}_{\mathbb{K}}[[{\ensuremath{\Gamma}}]]$. Similarly we can find integral matrices $\mathcal{E}_{\ensuremath{\mathcal{C}}}$ and $\mathcal{E}_{\ensuremath{\mathcal{D}}}$ such that $A_0^{-1}A\frac{d\mathcal{C}}{d{\ensuremath{\Gamma}}}+A_0^{-1}Y\mathcal{C}=p^\varepsilon \mathcal{E}_{\ensuremath{\mathcal{C}}}$ and $\frac{d\mathcal{D}}{d{\ensuremath{\Gamma}}}BB_0^{-1}+\mathcal{D}X B_0^{-1}=p^\varepsilon \mathcal{E}_{\ensuremath{\mathcal{D}}}$. Our goal is proving that the polynomial $$\label{eq:errorProp}\text{$p^{-\varepsilon}(\mathcal{K}-K)\bmod {\ensuremath{\Gamma}}^\ell$ has $(5{\ensuremath{\gamma}}+1,\psi)$-log convergence.}$$ This would imply that the valuation of $(\mathcal{K}-K)\bmod{\ensuremath{\Gamma}}^\ell$ is at least $-(5{\ensuremath{\gamma}}+1)\lceil\log_p\ell\rceil-\psi+\varepsilon=m$, and hence the computed $\mathcal{K}$ agrees with the actual solution $K$ modulo $(p^m,{\ensuremath{\Gamma}}^\ell)$. Moreover, as $K\bmod p^m$ has degree less than $\ell$, we even have ${\ensuremath{\mathcal{K}}}\equiv K\bmod p^m$, as required. It is easy to see that this also implies that ${\ensuremath{\mathcal{L}}}_{\ell-1}\equiv K(1)\bmod p^m$.\
We follow the proofs of Lemma 17 and 19 of [@HubrechtsHECEven]. Let $L$ be a matrix such that $p^\varepsilon LD={\ensuremath{\mathcal{D}}}-D$. This gives $$p^\varepsilon\mathcal{E}_{\ensuremath{\mathcal{D}}}=p^\varepsilon\frac{d(LD)}{d{\ensuremath{\Gamma}}}BB_0^{-1}+({\ensuremath{\mathcal{D}}}-D)XB_0^{-1} =p^\varepsilon\frac{dL}{d{\ensuremath{\Gamma}}}DBB_0^{-1},$$ or $dL/d{\ensuremath{\Gamma}}=\mathcal{E}_{\ensuremath{\mathcal{D}}}B_0 B^{-1}D^{-1}$. We note that $L_0=0$ and integrate to find $$p^{-\varepsilon}({\ensuremath{\mathcal{D}}}-D)=LD=\left(\int \mathcal{E}_{\ensuremath{\mathcal{D}}}B_0B^{-1}D^{-1}d{\ensuremath{\Gamma}}\right)D.$$ By Lemma \[lem:convergenceLogProduct\] we see that $\mathcal{E}_{\ensuremath{\mathcal{D}}}B_0B^{-1}D^{-1}$ has $(\gamma,2\alpha+\delta)$-log convergence, and as integrating is not worse than adding 1 to the logarithmic factor, we find that $p^{-\varepsilon}({\ensuremath{\mathcal{D}}}-D)$ has $(2\gamma+1,2(\alpha+\delta)$-log convergence. Working similarly we find the same for $p^{-\varepsilon}(\mathcal{C}-C)$. Note that this implies that ${\ensuremath{\mathcal{C}}}$ and ${\ensuremath{\mathcal{D}}}$, which are polynomials of degree less than $\ell$, are both $(\gamma,\delta)$-log convergent.
From the calculation $$\Delta(CK_0D)=A\frac{d C}{d{\ensuremath{\Gamma}}}K_0DB+ACK_0\frac{dD}{d{\ensuremath{\Gamma}}}B+ACK_0DX+YCK_0DB=0$$ we conclude that $K=C K_0D$. Choose $L'$ such that $p^\varepsilon CL'K_0D={\ensuremath{\mathcal{K}}}-{\ensuremath{\mathcal{C}}}{\ensuremath{\mathcal{K}}}_0{\ensuremath{\mathcal{D}}}$, then $$p^{-\varepsilon}(\Delta'{\ensuremath{\mathcal{K}}}-\Delta'({\ensuremath{\mathcal{C}}}{\ensuremath{\mathcal{K}}}_0{\ensuremath{\mathcal{D}}}))=A_0^{-1}AC\frac{dL'}{d{\ensuremath{\Gamma}}}K_0DBB_0^{-1}.$$ Note that again $L_0'=0$, so that if we isolate $\frac{dL'}{d{\ensuremath{\Gamma}}}$ and integrate, we find $$L'=p^{-\varepsilon}\int C^{-1}A^{-1}A_0(\Delta'{\ensuremath{\mathcal{K}}}-\Delta'({\ensuremath{\mathcal{C}}}{\ensuremath{\mathcal{K}}}_0{\ensuremath{\mathcal{D}}}))B_0B^{-1}D^{-1}K_0^{-1}d{\ensuremath{\Gamma}}.$$ We know that $\Delta'{\ensuremath{\mathcal{K}}}=p^\varepsilon\mathcal{E}_{\ensuremath{\mathcal{K}}}$ and verify that $$\Delta'({\ensuremath{\mathcal{C}}}{\ensuremath{\mathcal{K}}}_0{\ensuremath{\mathcal{D}}})=p^\varepsilon(\mathcal{E}_{\ensuremath{\mathcal{C}}}{\ensuremath{\mathcal{K}}}_0{\ensuremath{\mathcal{D}}}BB_0^{-1}+A_0^{-1}A{\ensuremath{\mathcal{C}}}{\ensuremath{\mathcal{K}}}_0\mathcal{E}_{\ensuremath{\mathcal{D}}}).$$ This gives that $p^{-\varepsilon}({\ensuremath{\mathcal{K}}}-{\ensuremath{\mathcal{C}}}{\ensuremath{\mathcal{K}}}_0{\ensuremath{\mathcal{D}}})$ equals $$C\left[\int C^{-1}A^{-1}\left(A_0\mathcal{E}_KB_0-A_0\mathcal{E}_C {\ensuremath{\mathcal{K}}}_0\mathcal{D}B-A\mathcal{C}{\ensuremath{\mathcal{K}}}_0\mathcal{E}_DB_0\right) B^{-1}D^{-1}d{\ensuremath{\Gamma}}\right]D,$$ and hence has $(5\gamma+1,5(\alpha+\delta))$-log convergence. We conclude from $$p^{-\varepsilon}(\mathcal{K}-K)=p^{-\varepsilon} (\mathcal{K}-\mathcal{C}{\ensuremath{\mathcal{K}}}_0\mathcal{D})+p^{-\varepsilon}(\mathcal{C}-C) {\ensuremath{\mathcal{K}}}_0\mathcal{D}+ p^{-\varepsilon}C{\ensuremath{\mathcal{K}}}_0(\mathcal{D}-D)$$ that $p^{-\varepsilon}(\mathcal{K}-K)$ has $(5{\ensuremath{\gamma}}+1,5(\alpha+\delta))=(5{\ensuremath{\gamma}}+1,\psi)$-log convergence.\
In order to prove the theorem we need to bound the time and memory requirements of the algorithm. We will assume fast arithmetic, see e.g. [@BernsteinFastMultiplication], which means that all basic ring operations in $\mathbb{K}$ can be performed in time essentially linear and memory linear in the object size. All elements of $\mathbb{K}$ that appear in the algorithm have valuation no less than $-\varepsilon$, hence modulo $p^\varepsilon$ all elements have bit size ${\ensuremath{\mathcal{O}}}(n\varepsilon\log_2p)$. Computing with these numbers requires then ${\ensuremath{\widetilde{\mathcal{O}}}}(n\varepsilon\log_2p)$ bit operations.
If we use (\[eq:inductiveComp\]) literally, each computation of some $\mathcal{K}_k$ requires ${\ensuremath{\mathcal{O}}}(\zeta^2)$ matrix multiplications. However, the right hand side of (\[eq:inductiveComp\]) is essentially the coefficient of ${\ensuremath{\Gamma}}^k$ in a sum of three products of (matrix) polynomials of degree $\mathcal{O}(\zeta)$, and — using fast multiplication methods for polynomials over arbitrary algebras, see [@Kaminski] or [@CantorKaltofen] — can thus be computed using only ${\ensuremath{\widetilde{\mathcal{O}}}}(\zeta)$ matrix multiplications. As we need $K({\ensuremath{\Gamma}})\bmod {\ensuremath{\Gamma}}^\ell$ this gives in total a time requirement of ${\ensuremath{\widetilde{\mathcal{O}}}}(\ell\zeta d^\omega n\varepsilon\log_2p)$ for both algorithms. Moreover, the size of $K({\ensuremath{\Gamma}})$ determines the memory requirements for the first algorithm, hence we need ${\ensuremath{\mathcal{O}}}(\ell d^2n\varepsilon\log_2p)$ bits of space. For the second algorithm only ${\ensuremath{\mathcal{O}}}(\zeta)$ matrices over $\mathbb{K}$ have to be kept in memory, and this gives ${\ensuremath{\mathcal{O}}}(\zeta d^2n\varepsilon\log_2p)$ space.$\blacksquare$
\[note:note1\] Let ${\ensuremath{\gamma}}'\in\mathcal{O}_{\mathbb{K}}$, then it is in a similar way possible to compute $K({\ensuremath{\gamma}}')$ modulo $p^m$ with the same time and space requirements as for $K(1)$ in the theorem.
\[cor:lotsOfCurves\] Let ${\ensuremath{\gamma}}_1,\ldots,{\ensuremath{\gamma}}_\ell\in\mathcal{O}_{\mathbb{K}}$ be given with accuracy $m' := m + \alpha$, then with the same assumptions as in Theorem \[thm:solvingDiffEq\] we can compute all matrices $K({\ensuremath{\gamma}}_1),\ldots,K({\ensuremath{\gamma}}_\ell)$ mod $p^m$ in ${\ensuremath{\widetilde{\mathcal{O}}}}(\ell \zeta d^\omega n\varepsilon\log_2p)$ bit operations and ${\ensuremath{\mathcal{O}}}(\ell d^2n\varepsilon\log_2p)$ bits of memory.
<span style="font-variant:small-caps;">Proof.</span> We will use *fast multipoint evaluation*. Let $f(x)$ be a polynomial of degree less than $\ell$ over a ring $R$. In Section 10.1 of [@ModernCompAlg] is explained how to evaluate $f(x)$ in $\ell$ elements of $R$ at once in such a way that it requires only ${\ensuremath{\widetilde{\mathcal{O}}}}(\ell)$ arithmetic operations in $R$. Hence, taking $R=\mathcal{O}_{\mathbb{K}}$ we can compute $f({\ensuremath{\gamma}}_1),\ldots,f({\ensuremath{\gamma}}_\ell)$ modulo $p^{m'}$ in time ${\ensuremath{\widetilde{\mathcal{O}}}}(\ell nm'\log_2p)$ and space ${\ensuremath{\mathcal{O}}}(\ell nm'\log_2p)$.
We use Theorem \[thm:solvingDiffEq\] to compute $K({\ensuremath{\Gamma}})$ modulo $p^{m}$. As $K({\ensuremath{\Gamma}})$ need not be integral, we work with $p^\alpha K({\ensuremath{\Gamma}})\bmod p^{m'}$, a matrix polynomial over $\mathcal{O}_{\mathbb{K}}$ of degree less than $\ell$. Now we can use the above result to find $p^\alpha K({\ensuremath{\gamma}}_1), \ldots, p^\alpha K({\ensuremath{\gamma}}_\ell)$ modulo $p^{m'}$ in time ${\ensuremath{\widetilde{\mathcal{O}}}}(d^2\ell nm'\log_2p)$ and space ${\ensuremath{\mathcal{O}}}(d^2\ell n m'\log_2p)$. Taking the maximum of this result (note that $m'\leq \varepsilon$) and the complexities of Theorem \[thm:solvingDiffEq\] concludes the proof.[$\blacksquare$]{}
Hyperelliptic curves in odd characteristic {#sec:oddChar}
==========================================
In this section we will use some results from our paper [@HubrechtsHECOdd] about the application of deformation in point counting. Let $p$ be an odd prime and suppose we are given a hyperelliptic curve $\bar E_1$ over ${\ensuremath{\mathbb{F}}}_{p^n}$ of genus $g$ in Weierstrass form $$y^2=\bar Q_1(x) = x^{2g+1}+\sum_{i=0}^{2g}\bar a_i x^i\quad\in{\ensuremath{\mathbb{F}}}_{p^n}[x],$$ where $\bar Q_1$ is squarefree. The purpose of this section is to compute the zeta function of this curve in a memory efficient way, using Theorem \[thm:solvingDiffEq\]. The basic idea is to deform this equation to one defined over ${\ensuremath{\mathbb{F}_p}}$, which will give us a differential equation of the kind considered in the previous section. In [@HubrechtsHECOdd] this was done by taking a family $y^2=\bar Q(x,{\ensuremath{\Gamma}})$ over ${\ensuremath{\mathbb{F}_p}}$ or a small extension field and substituting some ${\ensuremath{\bar{\gamma}}}\in{\ensuremath{\mathbb{F}}}_{p^n}$ for ${\ensuremath{\Gamma}}$. This method however does not allow us to compute the zeta function of a general hyperelliptic curve over ${\ensuremath{\mathbb{F}}}_{p^n}$. In this paper we let $\bar Q(x,{\ensuremath{\Gamma}})$ be defined over ${\ensuremath{\mathbb{F}}}_{p^n}$ and we then specialize to ${\ensuremath{\Gamma}}=1$. Combining this with Theorem \[thm:solvingDiffEq\] yields our memory efficient algorithm. We assume $p$ to be fixed in all complexity estimates of this section.
Overview of the deformation theory {#ssec:overviewDeform}
----------------------------------
Let $\bar Q_0(x) = x^{2g+1}+\sum_{i=0}^{2g}\bar b_i x^i\in{\ensuremath{\mathbb{F}}}_p[x]$ define a hyperelliptic curve $\bar E_0$ of genus $g$, for example $\bar Q_0(x):=x^{2g+1}+1$ if $p\nmid 2g+1$ and $\bar Q_0(x):=x^{2g+1}+x$ otherwise. We write ${\ensuremath{\mathbb{Q}}}_{p^n}$ for the unique unramified degree $n$ extension of ${\ensuremath{\mathbb{Q}_p}}$, $\sigma$ denotes the $p$th power Frobenius automorphism on ${\ensuremath{\mathbb{Q}}}_{p^n}$ and ${\ensuremath{\mathbb{Z}}}_{p^n}$ is the ring of integers of ${\ensuremath{\mathbb{Q}}}_{p^n}$. We recall that the *Teichmüller lift* ${\ensuremath{\gamma}}\in{\ensuremath{\mathbb{Z}}}_{p^n}$ of $\bar{\ensuremath{\gamma}}\in{\ensuremath{\mathbb{F}}}_{p^n}$ is the unique root of unity that reduces to $\bar{\ensuremath{\gamma}}$ modulo $p$. Further on we will also need to extend $\sigma$ with $\sigma({\ensuremath{\Gamma}}):={\ensuremath{\Gamma}}^p$; the projection ${\ensuremath{\mathbb{Z}}}_{p^n}\to{\ensuremath{\mathbb{F}}}_{p^n}$ is always denoted with $\bar\ $ and an algebraic closure of a field $k$ is denoted as $k^\text{alg\,cl}$.
Let $a_i\in{\ensuremath{\mathbb{Z}}}_{p^n}$ and $b_i\in{\ensuremath{\mathbb{Z}_p}}$ be (arbitrary) lifts of the coefficients $\bar a_i$ and $\bar b_i$, which gives us also lifts $Q_1$ and $Q_0$ of $\bar Q_1$ resp. $\bar Q_0$ — monic polynomials of degree $2g+1$. We define the polynomial $$Q(x,{\ensuremath{\Gamma}}) := x^{2g+1}+\sum_{i=0}^{2g}\left[(a_i-b_i){\ensuremath{\Gamma}}+b_i\right]x^i,$$ which gives a hyperelliptic curve $\bar E_{{\ensuremath{\bar{\gamma}}}}\leftrightarrow y^2=\bar Q(x,{\ensuremath{\bar{\gamma}}})$ for almost all ${\ensuremath{\bar{\gamma}}}\in{\ensuremath{\mathbb{F}}}_{p^n}^\text{alg\,cl}$, and makes our notation consistent regarding $\bar E_1$ and $\bar E_0$: $$\bar E_1\ \longleftrightarrow\ y^2=\bar Q(x,1)=\bar Q_1(x)\ \ \ \text{ and }\ \ \ \bar E_0 \longleftrightarrow\ y^2=\bar Q(x,0)=\bar Q_0(x).$$ We now give a short overview of the theory in [@HubrechtsHECOdd]. Let $r({\ensuremath{\Gamma}})$ be the resultant $$r({\ensuremath{\Gamma}}) := \text{Res}_x\left(Q(x,{\ensuremath{\Gamma}}); \frac{\partial}{\partial x}Q(x,{\ensuremath{\Gamma}})\right),$$ then it is clear from the construction of $Q(x,{\ensuremath{\Gamma}})$ that $r(0)$ and $r(1)$ are units in ${\ensuremath{\mathbb{Z}}}_{p^n}$ and $\rho := \deg r({\ensuremath{\Gamma}})\leq 4g$. Suppose $r({\ensuremath{\Gamma}})=\sum_{i=0}^\rho r_i{\ensuremath{\Gamma}}^i$; with $\rho'$ the degree of $\bar r({\ensuremath{\Gamma}})$ we define $\tilde r({\ensuremath{\Gamma}}):=\sum_{i=0}^{\rho'}r_i{\ensuremath{\Gamma}}^i$. We defined in Sections 3.2 and 3.3 of [@HubrechtsHECOdd] a ring $S$ and an $S$-module $T$, which can be represented as (where $\dagger$ means overconvergent completion): $$\begin{aligned}
S &:= {\ensuremath{\mathbb{Q}}}_{p^n}\left[{\ensuremath{\Gamma}},\tilde r({\ensuremath{\Gamma}})^{-1}\right]^\dagger,\\
T &:= \cfrac{{\ensuremath{\mathbb{Q}}}_{p^n}\left[x,y,y^{-1},{\ensuremath{\Gamma}},\tilde r({\ensuremath{\Gamma}})^{-1}\right]^\dagger} {(y^2-Q(x,{\ensuremath{\Gamma}}))}.\end{aligned}$$ Let $d:T\to Tdx$ be the differential $\frac{\partial}{\partial x}dx$ and $\nabla:T\to Td{\ensuremath{\Gamma}}$ the connection $\frac{\partial}{\partial {\ensuremath{\Gamma}}}d{\ensuremath{\Gamma}}$ such that $d({\ensuremath{\Gamma}})=\nabla(x)=0$. Then we showed that a certain submodule $H_{MW}^-$ of $Tdx/d(T)$ is a free $S$-module of rank $2g$, which, after substituting for ${\ensuremath{\Gamma}}$ any Teichmüller lift ${\ensuremath{\gamma}}\in{\ensuremath{\mathbb{Q}}}_{p^n}$ which is no zero modulo $p$ of $r({\ensuremath{\Gamma}})$, gives the same $2g$-dimensional ${\ensuremath{\mathbb{Q}}}_{p^n}$-vector space as Kedlaya’s $A^\dagger\otimes{\ensuremath{\mathbb{Q}}}_{p^n}$, defined in Section 3 of [@KedlayaCountingPoints]. We also constructed a Frobenius map $F_p$ on $H_{MW}^-$ which after the specialization ${\ensuremath{\Gamma}}\leftarrow{\ensuremath{\gamma}}$ again equals Kedlaya’s. The following diagram is well defined and commutes: $$\label{eq:diagram}\begin{CD}
H_{MW}^- @>{\nabla}>> H_{MW}^-d{\ensuremath{\Gamma}}\\
@VV{F_p}V @VV{F_p}V\\
H_{MW}^- @>{\nabla}>> H_{MW}^-d{\ensuremath{\Gamma}}.
\end{CD}$$ We have the $S$-basis $\{\frac{x^idx}{\sqrt{Q}}\}_{i=0}^{2g-1}$ for $H_{MW}^-$ and can hence define $(2g\times 2g)$-matrices over $S$ for our operators, namely $G({\ensuremath{\Gamma}})$ for $\nabla$ and $F({\ensuremath{\Gamma}})$ for $F_p$, e.g. $F_p(x^idx/\sqrt{Q})=\sum_j F_{ij}({\ensuremath{\Gamma}})x^jdx/\sqrt{Q}$. As Kedlaya showed in [@KedlayaCountingPoints], the main step in computing the zeta function of $\bar E_1$ is to compute $F(1)$ up to a certain precision.
Let $H({\ensuremath{\Gamma}}):=r({\ensuremath{\Gamma}})G({\ensuremath{\Gamma}})$, then the equation, derived from (\[eq:diagram\]), at the end of Section 3.6 in [@HubrechtsHECOdd] reads $$rr^\sigma\frac{dF}{d{\ensuremath{\Gamma}}}+r^\sigma FH-p{\ensuremath{\Gamma}}^{p-1}rH^\sigma F=0.$$ Here we use $r^\sigma$ and $H^\sigma$ for $r^\sigma({\ensuremath{\Gamma}}^p)$ respectively $H^\sigma({\ensuremath{\Gamma}}^p)$. Substituting $K:=r^MF$ for some integer $M\geq 0$ that is made more precise below, this becomes $$r^\sigma \frac{dK}{d{\ensuremath{\Gamma}}} r + r^\sigma K(H-M\frac{d r}{d{\ensuremath{\Gamma}}})+(-p{\ensuremath{\Gamma}}^{p-1}H^\sigma)Kr=0,$$ which is of the form $\Delta K=0$ explained in Section \[ssec:diffEq\] above, with $d=2g$, $A=r^\sigma$, $B=r$, $X=H-M\frac{d r}{d{\ensuremath{\Gamma}}}$ and $Y=-p{\ensuremath{\Gamma}}^{p-1}H^\sigma$.
Computing the zeta function {#ssec:computeZeta}
---------------------------
We will now determine the constants $\ell,\zeta,m$ and $\varepsilon$ in order to apply Theorem \[thm:solvingDiffEq\]. From Proposition 16 and Lemma 18 in [@HubrechtsHECOdd] it follows that with $\alpha:=(2g-1)(\log_p g +2)+g = {\ensuremath{\mathcal{O}}}(g\log g)$ we have $${\ensuremath{\textnormal{ord}}}(F)={\ensuremath{\textnormal{ord}}}(K)\geq -\alpha\quad\text{ and }\quad {\ensuremath{\textnormal{ord}}}(F^{-1})={\ensuremath{\textnormal{ord}}}(K^{-1})\geq -\alpha.$$ Proposition 17 of [@HubrechtsHECOdd] (with $\kappa:=\deg_{{\ensuremath{\Gamma}}}Q(x,{\ensuremath{\Gamma}})=1$) shows that $\deg H\leq 8g$ and as a consequence we can take $$\zeta := \max\{(p+1)\rho,p\rho+8g+1,p+8pg+\rho\} = {\ensuremath{\mathcal{O}}}(g).$$ We note in passing that this Proposition 17 also implies that ${\ensuremath{\textnormal{ord}}}(H)\geq\frac{-10g}{p-1}$ and hence the conditions ${\ensuremath{\textnormal{ord}}}(X),{\ensuremath{\textnormal{ord}}}(Y)\geq -\psi$ at the end of Section \[ssec:assumptions\] will be met.
We need $F(1)$ modulo $p^m$ with $m$ defined as $N_b$ in Section 4 of [@HubrechtsHECOdd], namely (with $a=1$) $$m := \left\lceil \frac{ng}2+(2g+1)\log_p2\right\rceil+ n\left\lfloor\log_p(g)+2\right\rfloor+\lfloor 2gn(\log_pg+3)\rfloor={\ensuremath{\mathcal{O}}}(ng\log g).$$ The exponent $M:=p(2m+4)+(p-1)/2={\ensuremath{\mathcal{O}}}(ng\log g)$ of $r({\ensuremath{\Gamma}})$ and the precision $\ell$ are given by Proposition 16 in [@HubrechtsHECOdd]: $$\ell := (2m+5)(8g+2)p+1={\ensuremath{\mathcal{O}}}(ng^2\log g).$$ Next we need ${\ensuremath{\gamma}}$ and $\delta$ such that (\[eq:CDlogConv\]) holds. For the solution $C$ of $A\frac{dC}{d{\ensuremath{\Gamma}}}+YC=0$ we can find this in Proposition 20 of [@HubrechtsHECOdd]: the matrix $C$ in that proposition does not correspond to $C$ in this paper, but the result and proof are completely the same. The conclusion is that $$C\text{ and }C^{-1}\text{ have }(2g\log_pg+g,0)\text{-log convergence.}$$ We have that $K=CK_0D$, and as a consequence $D=K_0^{-1}C^{-1}K$ and $D^{-1}=K^{-1}CK_0$ have $(2g\log_pg+g,2\alpha)$-log convergence. Hence we can take ${\ensuremath{\gamma}}:=2g\log_pg+g={\ensuremath{\mathcal{O}}}(g\log g)$ and $\delta:=2\alpha$. Now with $\psi = 2\alpha+5\delta=12\alpha$ we find $$\varepsilon = m+(5{\ensuremath{\gamma}}+1)\lceil\log_p\ell\rceil+\psi = {\ensuremath{\mathcal{O}}}(ng(\log g)^2).$$ The analysis in [@HubrechtsHECOdd], namely Steps 1, 2 and 5 of Section 6.3, shows that the time and space requirements for computing $r$, $H$ and $K_0$ will not have any influence on the result, and as a consequence we can apply Theorem \[thm:solvingDiffEq\] to find $K(1)\bmod p^m$ in time $${\ensuremath{\widetilde{\mathcal{O}}}}(\ell\zeta g^\omega n\varepsilon) = {\ensuremath{\widetilde{\mathcal{O}}}}(g^{4+\omega}n^3)$$ and with memory requirements $$\label{eq:memoryOdd}
{\ensuremath{\mathcal{O}}}(\zeta g^2n\varepsilon)={\ensuremath{\mathcal{O}}}(g^4(\log g)^2n^2).$$
To conclude the algorithm we still need to approximate the matrix $\mathcal{F}$ of $F_p^n$. First we compute $F(1)=r(1)^{-M}K(1)$ and then $$\mathcal{F} = F(1)^{\sigma^{n-1}}\cdot F(1)^{\sigma^{n-2}}\cdots F(1)^\sigma\cdot F(1),$$ which can be certainly done in time ${\ensuremath{\widetilde{\mathcal{O}}}}(g^3n^3+g^{1+\omega}n^2)$ and memory ${\ensuremath{\mathcal{O}}}(n^2g^3)$ as explained in [@KedlayaCountingPoints] (see however Section \[ssec:manyCurves\] for a much faster method). The numerator of the zeta function equals $\det(1-\mathcal{F}t)$ and can be found in time ${\ensuremath{\widetilde{\mathcal{O}}}}(g^{2+\omega}n^2)$. These last complexities can all be found in Step 8 of Section 6.3 of [@HubrechtsHECOdd], where the memory requirements are bounded by (\[eq:memoryOdd\]). This results in the following theorem.
\[thm:OddChar\] There exists an explicit and deterministic algorithm to compute the zeta function of any hyperelliptic curve of genus $g$ over ${\ensuremath{\mathbb{F}}}_{p^n}$, with $p$ odd, that uses ${\ensuremath{\widetilde{\mathcal{O}}}}(n^3g^{4+\omega})$ bit operations and bit space ${\ensuremath{\mathcal{O}}}(n^2g^4(\log_pg)^2)$.
Additional applications {#sec:manyCurves1}
=======================
Hyperelliptic curves in even characteristic {#sec:evenChar}
-------------------------------------------
By an argument similar to the one explained in the previous section, we can prove the following result.
\[thm:EvenChar\] There exists an explicit and deterministic algorithm that computes the zeta function of any hyperelliptic curve of genus $g$ over ${\ensuremath{\mathbb{F}}}_{2^n}$ using ${\ensuremath{\widetilde{\mathcal{O}}}}(n^3g^{4+\omega+3\tau})$ bit operations and ${\ensuremath{\mathcal{O}}}(n^2g^{4+\tau}(\log g)^{2+\tau})$ bits of memory. Here $\tau=0$ for almost all curves and $\tau=1$ in the general case.
In order to show this, one uses the results from [@HubrechtsHECEven], in turn partly inspired by Denef and Vercauteren’s article [@DenefVercauteren]. We want to point out that $\tau$ in the theorem above will be 0 precisely when all $m_i$ are equal to 1 (or are bounded by $\mathcal{O}(1)$) in the notation of the beginning of Section 4 of [@DenefVercauteren]. More details can be found in Section 5.3 of [@HubrechtsThesis].
Many curves at once {#ssec:manyCurves}
-------------------
If we choose a family defined over ${\ensuremath{\mathbb{F}}}_{p^n}$ as in Section \[ssec:overviewDeform\] (or in an analogous way in characteristic 2), e.g. given by $\bar E_{{\ensuremath{\Gamma}}}:y^2=\bar Q(x,{\ensuremath{\Gamma}})$, and ${\ensuremath{\bar{\gamma}}}_1,\ldots,{\ensuremath{\bar{\gamma}}}_{g^2n}\in{\ensuremath{\mathbb{F}}}_{p^n}$, we can compute the zeta functions of the curves $\bar E_{{\ensuremath{\bar{\gamma}}}_i}$ all at once in a very efficient way. There are three main steps needed in order to achieve this. First, computing the Teichmüller lifts of all $\bar {\ensuremath{\gamma}}_i$ modulo $p^\varepsilon$ can be done in time ${\ensuremath{\mathcal{O}}}(g^2n(n\varepsilon)^{1+\rho}) or {\ensuremath{\mathcal{O}}}((n^3g^3)^{1+\rho})$ for any $\rho>0$ as shown in [@HubrechtspAdicArithmetic Proposition 6]. Second, we compute all the matrices of the $p$th power Frobenius using Corollary \[cor:lotsOfCurves\] above in time ${\ensuremath{\widetilde{\mathcal{O}}}}(n^3g^{5+\omega})$. And third, in order to retrieve the matrices of the $q$th power Frobenius and hence the zeta functions, we can use Kedlaya’s trick [@KedlayaCountingPoints Section 5] combined with Proposition 3 of [@HubrechtspAdicArithmetic], resulting in a time complexity bounded by ${\ensuremath{\mathcal{O}}}((g^2n)(n^2g^{1+\omega})^{1+\rho})$. Noting that all these algorithms are deterministic we conclude with the following theorem:
\[thm:ManyCurves\]Suppose we are given a family $\bar E_{\ensuremath{\Gamma}}:y^2=\bar Q(x,{\ensuremath{\Gamma}})$ over ${\ensuremath{\mathbb{F}}}_{p^n}$ and $\bar{\ensuremath{\gamma}}_1,\ldots,\bar{\ensuremath{\gamma}}_{g^2n}\in{\ensuremath{\mathbb{F}}}_{p^n}$ such that $\bar E_0$ is defined over ${\ensuremath{\mathbb{F}}}_p$ and all $\bar E_{\bar {\ensuremath{\gamma}}_i}$ and $\bar E_0$ are hyperelliptic curves of genus $g$. Choose $\rho>0$. There exists an explicit and deterministic algorithm that computes the zeta functions of all curves $\bar E_{\bar {\ensuremath{\gamma}}_i}$ that requires ${\ensuremath{\widetilde{\mathcal{O}}}}(n^{3+\rho}g^{5+\omega})$ bit operations.
In an obvious way a similar algorithm can be shown to exist for characteristic 2.
This result could be interesting if one wants to find a curve with a special property, as is the case in cryptography. For example, suppose that we want to find a curve over ${\ensuremath{\mathbb{F}}}_{p^n}$ with $N$ as order of its jacobian, such that $N$ has a very large prime factor. Then we can expect that we have to try ${\ensuremath{\mathcal{O}}}(n)$ curves in order to find such a curve, and this is exactly something we can do very efficiently as explained above.\
In [@CastryckHubrechtsVercauteren] deformation is used for the computation of the zeta function of $C_{a,b}$ curves, and as explained in Section 5.4 of that paper similar results as above apply.
Hypersurfaces {#ssec:hypersurfaces}
-------------
Finally we can also use our memory efficient algorithm for solving differential equations in the context of hypersurfaces. For example, Lauder gives in [@LauderDeformation] an algorithm that computes the zeta function of certain hypersurfaces satisfying an ‘almost diagonal’ equation over ${\ensuremath{\mathbb{F}}}_{p^n}$. As he uses deformation as the main step in this result, the memory requirements drop from cubic to quadratic in $n$ using the result in this paper. Gerkmann discusses in [@GerkmannHypersurfaces] several deformation strategies for smooth projective surfaces, and an important step in there is again solving such a differential equation. Although the improvements depend on the type of algorithm considered, most algorithms presented in [@GerkmannHypersurfaces] will profit from Theorems \[thm:solvingDiffEq\] and \[thm:ManyCurves\].
[^1]: Gerkmann [@GerkmannHypersurfaces] independently has found a similar control of the error propagation.
|
---
abstract: |
A galaxy group catalog is constructed from the 2MASS Redshift Survey (2MRS) with the use of a halo-based group finder. The halo mass associated with a group is estimated using a ‘GAP’ method based on the luminosity of the central galaxy and its gap with other member galaxies. Tests using mock samples shows that this method is reliable, particularly for poor systems containing only a few members. On average 80% of all the groups have completeness $>0.8$, and about 65% of the groups have zero contamination. Halo masses are estimated with a typical uncertainty $\sim 0.35\,{\rm
dex}$. The application of the group finder to the 2MRS gives 29,904 groups from a total of 43,246 galaxies at $z \leq 0.08$, with 5,286 groups having two or more members. Some basic properties of this group catalog is presented, and comparisons are made with other groups catalogs in overlap regions. With a depth to $z\sim 0.08$ and uniformly covering about 91% of the whole sky, this group catalog provides a useful data base to study galaxies in the local cosmic web, and to reconstruct the mass distribution in the local Universe.
author:
- 'Yi Lu, Xiaohu Yang, Feng Shi, H.J. Mo, Dylan Tweed, Huiyuan Wang, Youcai Zhang, Shijie Li, S.H. Lim'
title: GALAXY GROUPS IN THE 2MASS Redshift Survey
---
Introduction {#sec:intro}
============
One important goal in modern cosmology is to establish the relationship between galaxies and dark matter halos in which galaxies form and reside. Understanding this galaxy-halo connection can provide important information about the underlying processes governing galaxy formation and evolution. Theoretically, there are several ways to study this relationship. The first is to use numerical simulations [@Springel2005; @Wadsley2004; @Bryan1995; @Kravtsov2002; @Teyssier2002; @Springel2010] or semi-analytical models [@vdBosch2002; @Kang2005; @Croton2006]. These approaches incorporate various physical processes that are potentially important for galaxy formation and evolution, such as gas cooling, star formation, feedback mechanisms, and so on. However, many processes in such modeling have to be approximated by sub-grid implementations and simple parameterizations, and so the results obtained are still questionable and sometimes fail to match observations. An alternative method to establish the galaxy-dark matter halo connection is to adopt an empirical approach. Models in this category includes the halo occupation model [e.g. @Jing1998; @Peacock2000; @Berlind2002; @Zheng2005], the conditional luminosity functions [e.g. @Yang2003; @vdBosch2003; @Yan2003; @Tinker2005; @Zheng2005; @Cooray2006; @vdBosch2007; @Yang2012], halo abundance matching [e.g. @Mo1999; @Vale2004; @Conroy2006; @Behroozi2010; @Guo2010; @Trujillo2011], and parametric model fitting [@Lu2014; @Lu2015b]. By construction, the empirical approach can produce much better fits to the observational data than numerical simulations and semi-analytical models, and so the galaxy-halo relationship established in this way is more accurate. Yet another way of to establish the galaxy-dark matter halo connection is to identify galaxy systems (clusters and groups, collectively referred to as groups in the following) to represent dark halos. With a well-defined galaxy group catalog, one can not only study the relationship between halos and galaxies [e.g. @Yang2005a; @Yang2008; @Lan2016; @Erfanianfar2014; @Rodriguez2015; @Jiang2016], but also investigate how dark matter halos trace the large-scale structure of the universe [e.g. @Yang2005b; @Yang2005c; @Tal2014]. In addition to these statistical studies, a well-defined group sample can also be used to reconstruct the current and initial cosmic density fields, so as to study not only the structures but also the formation histories of the cosmic web [e.g. @Wang2012; @Wang2013; @Wang2014].
The quality of a group sample depends on the group finder used to identify individual groups. During the past two decades, numerous group catalogs have been constructed from various observations, including the 2-degree Field Galaxy Redshift Survey (2dFGRS) [@Eke2004; @Yang2005a], the DEEP2 survey [@Crook2007] and the Sloan Digital Sky Survey (SDSS) [e.g. @Berlind2006; @Yang2007; @Tago2010; @Nurmi2013]. The group finders adopted in these investigations range from the traditional friends-of-friends (FOF) algorithm [e.g. @Davis1985], to the hybrid matched filter method [@Kim2002] and the “MaxBCG" method [@Koester2007]. Although the accuracy of a particular group finder depends on the properties of the observational sample, all group finders need to handle the same observational effects, such as redshift distortion that impacts the clustering pattern of galaxies, and the variations of the mean inter-galaxy separation due to apparent magnitude limit.
In this paper, we present our construction of a galaxy group catalog from the 2MASS Redshift Survey (2MRS), which is complete roughly to $K_s = 11.75$ and covers 91% of the sky [@Huchra2012]. Several group catalogs have already been constructed from 2MRS. @Crook2007 constructed a group catalog using galaxies with a magnitude limit at $K_s = 11.25$ and a FOF algorithm similar to that of @HuchraGeller1982. @Tully2015 built a group catalog in the volume between 3,000 and $10,000 {\>{\rm km}\,{\rm s}^{-1}}$ using a methodology similar to that of @Yang2005a. @Tempel2016 constructed a group catalog to larger distances using a FOF algorithm. Our goal here is to obtain a reliable and uniform galaxy group catalog using all galaxies in the 2MRS brighter than $K_s = 11.75$ to a redshift $z
= 0.08$. By involving a new halo mass estimation method, we are trying to obtain a better representive halo distributions in the local Universe.
The group finder to be used is the halo-based group finder developed by @Yang2005a, which groups galaxies within their host dark matter halos. This group finder is suitable to study the relation between galaxies and dark matter haloes over a wide range of halo masses, from rich clusters of galaxies to poor galaxy groups. It has been tested with mock galaxy surveys, and has been applied quite successfully to several galaxy catalogs [@Yang2005a; @Weinmann2006a; @Yang2007]. The essential idea behind this group finder is to use the relationships between halo mass and its size and velocity dispersion when deciding the membership of a group. Thus an accurate estimate of the halo mass for a candidate galaxy group is a key step. As shown in @Yang2007, for relative deep surveys, such as the SDSS, the group total luminosity (or stellar mass) provides a reliable ranking of the halo mass. In this case, halo masses can be estimated reliably by matching the rank of the characteristic luminosity of a group to that of halo mass given by a halo mass function. However, as pointed out in @Lu2015, for a shallow survey, such as the 2MRS, where only a few bright member galaxies in a group can be observed, the characteristic group luminosity is no longer the best choice to estimate the halo mass [see also @Old2014; @Old2015 for the halo mass estimation comparisons on cluster scales]. Instead, they proposed a method that is based on the luminosity of the central galaxy, $L_c$, and a luminosity ‘GAP’, $L_{gap}$, where the central galaxy is defined to be the brightest in a group, and the luminosity gap is defined as $\log
L_{\rm gap} = \log L_c - \log L_s$, with $L_s$ being the luminosity of the satellite galaxy of some rank (e.g. the brightest, or second brightest satellite). The performance of the halo mass estimate is found to be enhanced by using the ‘GAP’ information. Comparisons between the true halo masses and the masses estimated with the ‘GAP’ method in mock catalogs show a typical dispersion of $\sim 0.3 {\rm
dex}$.
In this paper, we modify the halo-based group finder developed by @Yang2005a by using the ‘GAP’ information. The structure of the paper is as follows. §\[sec:data\] describes the samples used in this paper, including the 2MRS galaxy sample and a mock galaxy sample used to evaluate the performance of our group finder. In §\[sec:mass\] we describe our modified halo-based group finder. The performance of our group finder, including completeness, contamination, purity is discussed in §\[sec\_test\], together with the reconstruction of the halo mass function. In §\[sec:catalog\] the properties of the group catalog constructed from 2MRS are detailed and compared to the mock group catalog, and to the SDSS DR7 galaxy group catalog constructed by @Yang2007 [@Yang2012] in the overlapping region. Finally, we summarize our results in §\[sec:summary\]. Unless stated otherwise, we adopt a $\Lambda$CDM cosmology with parameters that are consistent with the nine-year data release of the WMAP mission (hereafter WMAP9 cosmology): $\Omega_{\rm
m} = 0.282$, $\Omega_{\Lambda} = 0.718$, $\Omega_{\rm b} = 0.046$, $n_{\rm s}=0.965$, $h=H_0/(100 {\>{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}}) = 0.697$ and $\sigma_8 =
0.817$ [@Hinshaw2013].
DATA {#sec:data}
====
![Luminosity functions in $K_s$ band. Red points are obtained from the 2MRS galaxy sample, while black solid line represents the one obtained from the MOCK sample. The error bars are estimated using 1000 boot-strap re-sampling. []{data-label="fig:lf"}](f1.eps){height="7.0cm" width="7.5cm"}
[lcccc]{}
\
MOCKt & mock 2MRS catalog & none & simulation & $M_{\rm t}$\
MOCKg & mock 2MRS catalog & halo-based & Gap model & $M_{\rm g}$\
2MRS & 2MRS catalog & halo-based & Gap model & $M_{\rm 2MRS}$ \[tab:samples\]
The 2MRS galaxy catalog
-----------------------
The 2MASS Redshift Survey (2MRS) is based on the Two Micron All Sky Survey [@Jarrett2000; @Jarrett2003] and is complete to a limiting magnitude of $K_s = 11.75$, and $\sim 97.6\%$ of the galaxies brighter than the limiting magnitude have measured redshifts. The survey covers $\sim 91\%$ of the full sky; only $\sim 9\%$ of the sky close to the Milky Way plane is excluded [@Huchra2012]. The catalog contains about 43,533 galaxies extending out to $\sim 30,000 {\>{\rm km}\,{\rm s}^{-1}}$. For our analysis we only use the 43,246 galaxies with $z \leq 0.08$. Among these, 25 entries have negative redshifts ($-0.001 \leqslant z < 0.0$) which are caused by the peculiar velocities of galaxies. In our analysis, all redshifts are corrected to the Local Group rest frame according to @Karachentsev1996. We also use the distance information provided by @Karachentsev2013 for some nearby galaxies, including 22 galaxies with negative redshifts in our 2MRS catalog, to reduce effects caused by peculiar velocities. Corrections of Virgo infall are made to 15 galaxies in the front and back of the Virgo cluster according to @Karachentsev2014. Since the redshifts of our 2MRS galaxies are low, no attempt is made to apply any $K$- or $E$-corrections to galaxy luminosities.
From this catalog, we first measure the galaxy luminosity function (LF) in the $K_s$ band. We adopt the commonly used $1/V_{\rm max}$ algorithm [@Schmidt1968; @Felten1976], in which each galaxy is assigned a weight given by the maximum co-moving volume within which the galaxy could be observed. Fig \[fig:lf\] shows the galaxy luminosity function so obtained from our 2MRS sample, with the error bars estimated from 1,000 bootstrap re-samplings. We have fitted the LF to a Schechter function [@Schechter1976] and the best fit Schechter parameters are $\log\phi^*=1.08\times 10^{-2}$, $\alpha=-1.02$ and $M^*=-23.55$. These values are consistent with those obtained by @Crook2007 and @Tully2015.
The mock 2MRS galaxy catalog {#sec:mock2MRS}
----------------------------
{height="8.0cm" width="15.0cm"}
![The halo mass function of our true sample (MOCKt, dots with error bars). The halo mass function given by @Tinker2008 is also plotted in the same panel for comparison using black solid line. []{data-label="fig:hmf0"}](f3.eps){height="7.5cm" width="7.5cm"}
We construct a mock 2MRS galaxy catalog to test the performance of our group finder and the reliability of the final galaxy group catalog. The mock catalog is constructed as follows.
First, we use a high-resolution simulation carried out at the High Performance Computing Center, Shanghai Jiao Tong University, using L-GADGET, a memory-optimized version of GADGET-2 [@Springel2005]. A total of $3072^3$ dark matter particles were followed in a periodic box of $500{\>h^{-1}{\rm {Mpc}}}$ on a side [@Li2016]. The adopted cosmological parameters are consistent with those from WMAP-9. Each particle in the simulation has a mass of $3.4\times10^{8}
{\>h^{-1}\rm M_\odot}$. Dark matter halos were identified using the standard FOF algorithm [@Davis1985] with a linking length of $b=0.2$ times the mean inter particle separation.
Next, the halos are populated with galaxies of different luminosities. We use the conditional luminosity function [CLF, @Yang2003], which is defined to be the average number of galaxies, as a function of luminosity, that reside in a halo of a given mass, to link galaxies with dark matter haloes. We make use of the set of CLF parameters provided by @Cacciato2009 to generate model galaxies with $r$ band luminosities. Following the observational definition, the central galaxy is defined as the brightest member and is assumed to be located at the center of the corresponding halo. Its velocity follows the velocity of the dark matter halo center. Other galaxies, referred to as satellite galaxies, are distributed spherically following a NFW [@NavarroFrenkWhite1997] profile where the concentration model of @Zhao2009 was adopted. Their velocities are assumed to be the sum of the velocity of the host halo center plus a random velocity drawn from a Gaussian distribution with dispersion given by the virial velocity dispersion of the halo. We refer the reader to @Lu2015 and @Yang2004 for details. In general, one can also populate/generate mock galaxies using more sophisticated methods, e.g., based on sub-halos or halo merger trees, where the galaxies are not spherically distributed. However, as we have tested in @Weinmann2006b, our group finder is not very sensitive to the somewhat non-spherical distribution of galaxies.
In order to convert the $r$ band magnitude to the $K_s$ band, we first measure the [*cumulative*]{} luminosity function separately for both the mock and 2MRS samples. Assuming that galaxies more luminous in the $r$ band are also more luminous in the $K_s$, we assign a $K_s$ band luminosity (absolute magnitude) to each galaxy. In practice, we relate $M_r$ and $M_{K_s}$ through abundance matching: $$\label{eq:ab2}
\int_{-\infty}^{M_r} \phi_r (M_r') dM_r'
=
\int_{-\infty}^{M_{K_s}} \phi_{K_s} (M_{K_s}') dM_{K_s}'\,,$$ where $\phi_r (M_r)$ and $\phi_{K_s} (M_{K_s})$ are the luminosity functions of galaxies in $M_r$ and $M_{K_s}$, respectively.
Finally, we place a virtual observer at the center of our simulation box and define a ($\alpha$, $\delta$)-coordinate frame, and remove all galaxies that are located outside the survey region ($\sim 9\%$ of the total sky). We then assign to each galaxy a redshift and an apparent magnitude according to its distance and luminosity, and select only galaxies that are brighter than the magnitude limit $K_s=11.75$. Here again, no K+E corrections are made to galaxy luminosities. In total, we have 41,876 galaxies in our mock 2MRS catalog. The black solid curve in Fig. \[fig:lf\] shows the $K_s$ band luminosity function estimated from our mock sample.
Apart from the luminosity function of galaxies, we set out to measure the true halo mass function in the mock 2MRS catalog. Because of the survey magnitude limit, faint galaxies formed in low mass halos might not be observed at high redshift, i.e., low mass halos can only be detected below a redshift limit. To properly estimate the halo mass function, one needs to have a complete sample of groups (halos), i.e. to obtain the limiting redshift for a given mass of halos, within which the selection of groups is complete. Unlike the luminosity for which the limiting redshift can be directly calculated from the magnitude limit, we use an empirical way to get the limiting redshift for halos. First, we calculate the number densities of halos in small redshift and halo mass bins and plot them in the $\log M_{\rm t}$-$z$ plane using color bars (see Fig. \[fig:colorbar\]). We can see that the number density of halos of given mass drops sharply above certain redshift. Here we define the limiting redshift for a given mass halos as the redshift at which this rapid drop in density occurs. The smooth line in Fig. \[fig:colorbar\] shows the limiting redshift as a function of $\log M_{\rm t}$ we use, which clearly represents a conservative cut to ensure completeness. Once a limiting redshift is adopted, we can calculate the halo mass function using only halos (and volume) below this redshift. Fig. \[fig:hmf0\] shows the halo mass function obtained in this way with dots and error bars. Here again, the error bars are estimated using 1000 bootstrap re-samplings. For comparison, we also show, using the solid line, the theoretical model predictions given by @Tinker2008. Compared with the model prediction, the data points are slightly lower at intermediate to low mass range, which is mainly due to cosmic variance, since the overall halo mass function in the whole simulation box is quite consistent with theoretical predictions [see @Li2016].
Since we have both the true halo and the galaxy membership informations in our mock 2MRS sample, we can use it to test the performance of our group finder. Together with the 2MRS observational sample, there are three galaxy group catalogs involved in this paper. We refer to the two catalogs related with the mock 2MRS samples as ‘MOCKt’ and ‘MOCKg’ respectively. The former catalog indicates the true group memberships in the FOF dark matter halos obtained directly from the simulation, where we use $M_{\rm t}$ to represent the [*true*]{} halo mass. The latter is constructed using our group finder, where the halo masses are estimated using our ‘GAP’ related mass estimator and are named as $M_{\rm g}$. Finally, the group catalog constructed from the 2MRS data is referred to as ‘2MRS’ and the related halo mass are named as $M_{\rm 2MRS}$ . For clarity, we list the differences of the three definitions in Table. \[tab:samples\], including the group finders that were used to identify galaxy groups, and the methods used to estimate the halo masses. Apart from the above three specific halo mass definitions, we use $M_h$ to represent the general halo masses, including those used in theoretical model predictions [e.g. @Tinker2008].
The modified halo-based group finder {#sec:mass}
====================================
One of the key steps in the halo-based group finder [@Yang2005a; @Yang2007] is to have accurate estimates of the halo masses of candidate galaxy groups. As demonstrated in @Yang2007, halo mass is tightly correlated with the total luminosity of member galaxies. In practice, however, one can only estimate a characteristic luminosity which is the sum of the luminosities of member galaxies brighter than some given limit. For a relatively deep survey such as the SDSS, where the limit can be set sufficiently low, the characteristic luminosity is a good proxy of the total luminosity and so can be used to indicate halo mass. For a shallow survey like the 2MRS, on the other hand, only a few (in most cases one or two) brightest member galaxies in the halos can be observed. The characteristic luminosity is no longer the best halo mass estimator, and an alternative is needed. In this paper, we implement the ‘GAP’ method proposed by @Lu2015.
The GAP halo mass estimator {#sec:gap}
---------------------------
![$L_c-M_h(M_{\rm g})$ relation given by the MOCKg sample using abundance matching between the cumulative luminosity function of galaxies and the halo mass function. ‘Round 1’ relation is obtained using all mock galaxies \[blue dashed line, labelled as MOCKg(1)\] while ‘Round 2’ is obtained using central galaxies only \[red line labelled as MOCKg(2)\]. The true $L_c-M_h(M_{\rm t})$ relation given by the MOCKt sample is plotted with black solid points with error bars which indicate the $16\%-84\%$ percentiles of the distributions around the median values.[]{data-label="fig:MhLcmock"}](f4.eps){height="8.0cm" width="9.0cm"}
{height="10.5cm" width="11.5cm"}
[lccccc]{}
\
MEMBER 2 & $ 10.81^{+ 0.18}_{- 0.19}$ & $ 0.36^{+ 1.60}_{- 0.26}$ & $ -15.44^{
+ 3.35}_{- 7.86}$ & $ 10.39^{+ 0.11}_{- 0.24}$ & $ 1.94^{+ 0.83}_{- 0.4
1}$\
\
MEMBER 3 & $ 10.21^{+ 0.39}_{- 0.10}$ & $ 0.23^{+ 0.54}_{- 0.14}$ & $ -13.40^{
+ 1.21}_{- 3.97}$ & $ 9.90^{+ 0.36}_{- 0.10}$ & $ 2.21^{+ 0.32}_{- 0.2
7}$\
\
MEMBER 4 & $ 9.98^{+ 0.32}_{- 0.18}$ & $ 0.20^{+ 0.25}_{- 0.09}$ & $ -13.39^{
+ 1.10}_{- 2.76}$ & $ 9.81^{+ 0.30}_{- 0.17}$ & $ 2.45^{+ 0.24}_{- 0.1
5}$\
\
MEMBER 5 & $ 9.77^{+ 0.33}_{- 0.07}$ & $ 0.13^{+ 0.15}_{- 0.01}$ & $ -13.67^{
+ 1.18}_{- 0.94}$ & $ 9.67^{+ 0.28}_{- 0.07}$ & $ 2.54^{+ 0.15}_{- 0.0
8}$\
\[tab:DeltaMmock\]
In the ‘GAP’ method, one first needs to estimate the $L_c$-$M_h$ relation. For MOCKt samples, since every groups have the true central galaxies and true halo masses from the simulation, we can obtain this relation directly. Hereafter we refer the $L_c-M_h$ relation obtained directly from the simulation as the intrinsic (true) relation. On the other hand, for observational samples, one can obtain this relation from the conditional luminosity function model [e.g. @Yang2003] or from halo abundance matching [e.g. @Mo1999; @ValeOstriker2006; @Conroy2006; @Behroozi2010; @Guo2010]. Here we adopt the latter and assume that there is a monotonic relation between the luminosity of central galaxy and the mass of dark matter halo, so that a more luminous galaxy resides a more massive halo. We can then get an initial estimate of the dark matter halo mass for each central galaxy from $$\label{eq:ab3}
\int_{L_c}^\infty n_c (L_c') dL_c' =
\int_{M_h}^\infty n_h (M_h') dM_h'\,,$$ where, $n_c (L_c)$ is the number density of central galaxies with luminosity $L_c$ and $n_h (M_h)$ is the number density of halos (or halo mass function) with mass $M_h$. In this paper, we adopt theoretical halo mass function given by @Tinker2008. Note that, in this abundance matching approach, we also need to know whether a galaxy is a central or a satellite. Since we are trying to find galaxy groups within the observation (the 2MRS in our case), we can easily separate galaxies into centrals and satellites with the help of group memberships. As we will show later, although the $L_c-M_h$ relation we obtain may deviates from the true one, especially at the massive end, the deviation can be compensated to some extent by our ‘GAP’-based correction factor.
Our modeling of the $L_c-M_h$ relation using Eq. (\[eq:ab3\]) is carried out via the following two steps. First, before we are able to separate galaxies into centrals and satellites with the help of group memberships, we assume that all of them are centrals [as shown in @Yang2008 more than 60% of the galaxies are centrals]. To show the performance, We have applied this to our mock 2MRS sample, and obtain the ‘Round 1’ $L_c-M_{\rm g}$ relation, which is shown in Fig. \[fig:MhLcmock\] as the blue dashed line.
For comparison, we also plot, as black solid points, the true median $L_c-M_{\rm t}$ relation obtained from the true centrals and true halo masses in the simulation, with error bars indicate the $16\%-84\%$ percentiles of the distributions. Compared to the true relation, we see that, the Round 1 relationship shows a general agreement with the true one, with a slight over-prediction of the halo masses at the bright end and slight under-prediction at the faint end. The deviation at the massive end is caused by the Malmquist bias in the $L_c-M_{\rm g}$ relation which can be corrected by the ‘GAP’ [see @Lu2015]. The deviation at the faint end is caused by the inclusion of all the galaxies (including satellites) in our abundance matching. As we apply our group finder to the galaxy catalog in the next step, the group membership will enable us to separate galaxies into centrals and satellites. We can then limit the application of the abundance matching to centrals only, and improve the $L_c-M_h$ relation. After two to three iterations we converge to a new set of group memberships and a new $L_c-M_{\rm g}$ relationship, which is referred to as ’Round 2’ and shown as the solid red line in Fig. \[fig:MhLcmock\]. After this step, there is no longer any systematic deviation of the $L_c-M_{\rm g}$ relationship relative to the true one at the low mass end.
With the $L_c-M_{\rm g}$ obtained in this step, we can estimate the ‘luminosity gap’, which is defined as the luminosity ratio between the central and a satellite galaxy in the same halo, $\log L_{\rm gap}=
\log(L_c/L_s)$ [see @Lu2015]. The halo mass is then estimated using the relation, $$\label{eq:Mfunc}
\log M_{\rm g}(L_c,L_{\rm gap}) = \log M_{\rm g}(L_c) + \Delta \log
M_{\rm g}(L_c,L_{\rm gap}) \,.$$ This halo mass estimator consists of two parts. The first part is an empirical relation between $M_{\rm g}$ and $L_c$ derived from Eq. (\[eq:ab3\]) which is represented by the first term on the right side. Another part is the amount of correction to that relation, which is represented by the second term $\Delta \log
M_{\rm g}(L_c,L_{\rm gap})$. In order to model this correction term, we use the following functional form, $$\label{eq:DM_func}
\Delta \log M_{\rm g}(L_c,L_{\rm gap}) = \eta_a \exp(\eta_b \log L_{\rm gap}) + \eta_c\,.
\label{eq:delM}$$ The parameters $\eta_a$, $\eta_b$ and $\eta_c$ all depend on $L_c$ as: $$\begin{aligned}
\label{eq:eta_abc}
\eta_a(L_c)~ &=&~ \exp(\log L_c-\beta_1) \nonumber \\
\eta_b(L_c)~ &=&~ \alpha_2(\log L_c +\beta_2) \\
\eta_c(L_c)~ &=&~ -(\log L_c - \beta_3)^{\gamma_3} \nonumber\end{aligned}$$ which is specified by five free parameters.
For a given $L_c - M_{\rm g}$ relation, we fit the model to the true halo masses $M_{\rm t}$ of our galaxy systems (groups) in our mock sample to have the minimum variances between $\log M_{\rm t}$ and $\log M_{\rm g}(L_c,L_{\rm gap})$ [see @Lu2015 for details]. Table \[tab:DeltaMmock\] presents the set of best fit values of these parameters. Since the (mock) 2MRS sample is shallow, we provide the parameters up to 5 group members. As an illustration, Fig. \[fig:M\_correct\] shows the performance of this halo mass estimator. In the top two panels, the original $L_c-M_t$ relations are shown as the open circles; the GAP-corrected relations are shown as the solid points, with the left panel showing results for $L_s=L_2$ and the right for $L_s=L_5$. To see the improvement, we define a ‘pre-corrected’ halo mass, $$\label{eq:M'_h}
\log M'_{\rm t} = \log M_{\rm t} - \Delta \log M_g(L_c,L_{\rm gap})\,.$$ If the correction term $\Delta \log M_{\rm g}(L_c,L_{\rm gap})$ can perfectly describe the scatter in the original relation $L_c-M_{\rm
t}$, then there would be no scatter in the $L_c-M'_{\rm t}$. We can see that, the scatter in the $L_c-M'_{\rm t}$ is significantly reduced compare to that in the $L_c-M_{\rm t}$ relation. For massive halos/groups, this improvement is quite notable where the scatter is reduced almost by a factor of two. The bottom panel of Fig. \[fig:M\_correct\] shows the standard deviation $\sigma$ of the halo mass $\log M_{\rm g}(L_c,L_{\rm gap})$ obtained by Eq.(\[eq:Mfunc\]) from the true halo mass $\log M_{\rm t}$. In both @Lu2015 and this paper, we find that using $L_5$ gives the best correction to the halo mass. As shown @Lu2015, such a correction factor is quite independent of the galaxy formation model used to construct the mock catalog. In this paper we use the set of best fit parameters only up to the fifth ranked member (see below).
The Group Finder {#sec:gf}
----------------
The group finder adopted here is similar to that developed by @Yang2005a. It uses the general properties of dark matter haloes, namely size and velocity dispersion, to iteratively find galaxy groups. Tests show that this group finder is powerful in linking galaxies with dark matter halos, even in the case of single member groups. As we pointed out earlier, the halo mass estimation adopted in @Yang2005a [@Yang2007] is based on the ranking of the characteristic group luminosity and proves to be quite reliable for surveys like the 2dFGRS and SDSS. For the 2MRS considered here, we use the ‘GAP’-corrected estimator described above. The modified group finder with this halo mass estimator consists of the following main steps:
\
**Step 1: Start the halo-based group finder.**
In the earlier version of the halo-based group finder, the first step is to use the FOF algorithm [@Davis1985] with very small linking lengths in redshift space to find potential groups. Here we assume all galaxies in our catalog are candidate groups. The halo mass of each candidate group is calculated using the $L_c-M_h$ relation obtained in Eq. (\[eq:ab3\]) (Round 1).
\
**Step 2: Update group memberships using halo information.**
After assigning halo masses to all the candidates, groups are sorted according to their halo masses. Starting from the most massive one, we estimate the size and velocity dispersion of the dark matter halo, using the halo mass currently assigned to it. A dark matter halo is defined to have an over-density of 180. For the WMAP9 cosmology adopted here, the radius is approximately $$r_{180} = 1.33 {\>h^{-1}{\rm {Mpc}}}\left( \frac{M_h}{10^{14}{\>h^{-1}\rm M_\odot}}\right)^{1/3}
\left(1+z_{\rm group}\right)^{-1},$$ Here, $z_{\rm group}$ is the redshift of the group center. The line-of-sight velocity dispersion of the halo is $$\label{eq:sigma}
\sigma = 418{\>{\rm km}\,{\rm s}^{-1}}\left( \frac{M_h}{10^{14}{\>h^{-1}\rm M_\odot}}\right)^{0.3367}\,.$$ Finally, following @Yang2007 [hereafter Y07], we use the luminosity weighted center of member galaxies as the new group center.
Then, one can assign new member galaxies to the group according to the tentative group center, tentative estimates of halo size and velocity dispersion obtained in the above steps. The phase-space distribution of galaxies is assumed to follow that of dark matter, and the group center is assumed to coincide with the center of halo. We use the following function of the projected distance $R$ and $\Delta z = z - z_{\rm group}$ to describe the number density of galaxies at $z$ in the redshift space around the group center at redshift $z_{\rm group}$: $$P_M(R,\Delta z) = {H_0\over c} {\Sigma(R)\over {\bar \rho}} p(\Delta z) \,,$$ where $c$ is the speed of light and $\bar{\rho}$ is the average density of the Universe. We assume the projected surface density, $\Sigma(R)$, is given by a (spherical) NFW [@NavarroFrenkWhite1997] profile: $$\Sigma(R)= 2~r_s~\bar{\delta}~\bar{\rho}~{f(R/r_s)}\,,$$ where $r_s$ is the scale radius, and the shape function is $$\label{fx}
f(x) = \left\{
\begin{array}{lll}
\frac{1}{x^{2}-1}\left(1-\frac{{\ln
{\frac{1+\sqrt{1-x^2}}{x}}}}{\sqrt{1-x^{2}}}\right) & \mbox{if $x<1$} \\
\frac{1}{3} & \mbox{if $x=1$} \\
\frac{1}{x^{2}-1}\left(1-\frac{{\rm
atan}\sqrt{x^2-1}}{\sqrt{x^{2}-1}}\right) & \mbox{if $x>1$}
\end{array} \right.\,.$$ The normalization of the profile depends on the concentration $c_{180}=r_{180}/r_s$ as: $$\bar{\delta} = {180 \over 3} {c_{180}^3 \over {\rm ln}(1 + c_{180}) -
c_{180}/(1+c_{180})} \,,$$ where the concentration model of @Zhao2009 is adopted. The redshift distribution of galaxies within the halo is assumed to have a normal distribution, and can be described as follows, $$p(\Delta z)= {1 \over \sqrt{2\pi}} {c \over \sigma (1+z_{\rm group})} \exp
\left [ \frac {-(c\Delta z)^2} {2\sigma^2(1+z_{\rm group})^2}\right ] \,,$$ where $\sigma$ is the rest-frame velocity dispersion given by equation (\[eq:sigma\]). So defined, the three-dimensional density in redshift space is $P_M(R,\Delta z)$. Then, we apply the following procedures to assign a galaxy to a particular group. For each galaxy we loop over all groups, and compute the distances $R$ and $\Delta z$ between the galaxy and the group center. An appropriately chosen background level $B=10$ is applied to the density contrast for galaxies to be assigned to a group. If, according to this criterion, a galaxy can be assigned to more than one group it is only assigned to the one with the highest $P_M(R,\Delta z)$. Finally, if all members of two groups can be assigned to one, they are merged into a single group. Note that in our group finder, the background parameter $B=10$ is set to ensure the balance between the interlopers and completeness of group memberships. A lower $B$ value will increase both the completeness of the group memberships and the number of interlopers, especially in massive groups. Thus for those who care most about the completeness of the group memberships only, a lower value of $B$ (e.g. $B=5$) can be used [see @Yang2005a].
\
**Step 3: Update halo mass with ‘GAP’ correction.**
Once the new membership to a group is obtained, we use the new central and satellite galaxy system to estimate the halo mass using the ‘GAP’ method described by Eq. (\[eq:Mfunc\]). For each candidate group, we use the $L_c-M_h$ relation and the luminosity gap $\log L_{\rm
gap}$ between the central galaxy and the faintest satellite (if the group contains less than 5 members) or the fifth brightest galaxy (if the group has membership equal to or larger than 5), to estimate the halo mass. In practice, we only apply the luminosity gap correction for centrals in the luminosity range $10.5 \leq \log L_c \leq
11.7$. As shown in the top panels of Fig. \[fig:M\_correct\], fainter ($\log L_c \leq 10.5$) central galaxies are basically isolated. For $\log L_c \geq 11.7$, we found that using the value of $L_c$ directly in the GAP leads to over-correlation. Thus, for these systems we set $\log L_c = 11.7$ to estimate the GAP correction. In addition, since our galaxy sample is magnitude limited to $K_s = 11.75$, our method also suffers from a ‘missing satellite’ problem, in that some groups do not contain any satellites brighter than the magnitude limit. As an attempt to partly correct for this, we assume that each galaxy group that contains only one member (a central) has a potential member satellite galaxy with an apparent magnitude $K_s = 11.75$, which corresponds to a limiting luminosity $L_{\rm limit}$ at the distance of the group. A ‘GAP’ correction, $\log L_c - \log L_{\rm limit}$, is also applied to all groups of single membership with $\log L_c - \log
L_{\rm limit}\geq 0.5$, and the final halo mass of such a group is set to be the average value between this mass and the original mass based on the central galaxy alone.
\
**Step 4: Update the $L_c - M_h$ relation and Iterate.**
Once all the groups have been updated for new memberships, we can distinguish between centrals and satellites. We use the updated central galaxy sample to update the $L_c-M_h$ relation (Round 2) to be used to assign halo masses to tentative groups. We iterate Steps 2-4 until convergence is reached. Typically three iterations are needed to achieve convergence. Our final catalog is the collection of all the converged groups with information about their positions, galaxy memberships, and halo masses.
\
**Step 5: Update the final halo masses of groups.**
Once all the groups (memberships) have been finalized, we perform a final update of the halo masses of groups using an abundance matching method so that the halo mass function of the groups is consistent with theoretical predictions [e.g. @Yang2007]. In performing the halo abundance matching, we measure the [*cumulative*]{} halo mass functions of groups following the procedures described in section \[sec:mock2MRS\].
Test with Mock catalogs {#sec_test}
=======================
In this section, we test the performances of our group finder, both in halo masses and group memberships it assigns, by comparing the groups selected by our group finder (MOCKg) with the true groups defined by simulation (MOCKt) in our mock 2MRS sample.
Completeness, Contamination and Purity {#sec:compl}
--------------------------------------
![The top, middle and bottom panels show the cumulative distributions of completeness, $f_c$ (the fraction of true members), contamination, $f_i$ (the fraction of interlopers), and purity, $f_p$ (ratio between the number of true members and the total number of group members). These values are number weighted. Different lines represent the results for groups in halos of different masses, as indicated. Results are plotted for groups with at least 2 members, since groups with only 1 member have, by definition, $f_i = 0$.[]{data-label="fig:compl"}](f6.eps){height="16.0cm" width="8.0cm"}
{height="5.5cm" width="15.5cm"}
Starting from a total of 41,876 galaxies in the mock 2MRS sample, our group finder returns 32,368 galaxy groups, among which 4,225 have 2 or more members, and the rest only one member. We follow @Yang2005a [@Yang2007] to assess the performance of the group finder. The procedure is as follows. First, for each group, $k$, in the MOCKg sample, we identify the halo with ID, $h_k$, in MOCKt according to its brightest member. Then, we define the total number of true members belong to halo $h_k$ to be $N_t$. Among $N_t$, the number of true members that belong to the group $k$ is written as $N_s$. The number of interlopers (group members that belong to a different halo) in the group $k$ is defined as $N_i$, while the total number of group members selected by our group finder in MOCKg is assumed to be $N_g$, and $N_g = N_i + N_s$. If our group finder is perfect, it should have $N_i = 0$ and $N_t = N_s =
N_g$. With these numbers, we can define the following three quantities:
- COMPLETENESS: $f_c \equiv N_s/N_t$;
- CONTAMINATION: $f_i \equiv N_i/N_t$;
- PURITY: $f_p \equiv N_t/N_g$.
Here, $f_p = 1/(f_c+f_i)$. If the group is incomplete, then the COMPLETENESS $f_c < 1$. For the CONTAMINATION $f_i$, it can be larger than unity. Finally, for PURITY $f_p$, when the number of interlopers is larger than the number of missed true members, $f_p <
1$, on the other hand, if the number of missed true members is lager than the interlopers, then $f_p > 1$. If our group finder is perfect, then it should have $f_c =f_p =1$ and $f_i=0$ for all the groups. Note also that the value for the background level $B=10$ was tuned to maximize the average value of $f_{\rm c} (1 - f_{\rm
i})$, as described in @Yang2005a.
Fig. \[fig:compl\] shows the reliability of our group catalog constructed from the mock 2MRS galaxy sample. Following Y07, here, only groups in MOCKg with richness $N_g \geq 2$ are included since the single groups with only one member always have zero contamination $f_i=0$ as defined above. The upper panel shows the cumulative distributions of the COMPLETENESS $f_c$. The groups with different true halo mass are represented with different lines as indicated. The fractions of groups with 100 percent completeness ($f_c = 1$) range from $\sim 85\%$ to $\sim65\%$ depends on halo mass, which shows that more massive groups tend to have lower completeness fraction. Since the massive groups with larger velocity dispersions tend to have larger $f_i$ due to contamination of foreground and background galaxies, meanwhile, the purpose of our group finder is to maximizing the average value of $f_c(1-f_i)$, a compromise between $f_c$ and $f_i$, a background level $B=10$ is thus chosen. A smaller $B$ value will increase both $f_c$ and $f_i$ values in more massive groups, which is not preferably adopted in our investigation. Overall, more than 90% of our groups have COMPLETENESS $f_c > 0.6$. For groups with $\log M_{\rm g} \le
14.0$, about 80% of all groups have $f_c >0.8$; only for massive halos with $\log M_{\rm g} > 14.0$ is this fraction a little lower, $\sim 75-80\%$.
The middle panel of Fig. \[fig:compl\] shows the cumulative distribution of the CONTAMINATION $f_i$. The fraction of groups with $f_i = 0$ ranges from 60% to 80%, depending on the halo mass, while $\sim 85\%$ of all the groups have $f_i< 0.5$. The interlopers producing the contamination are either nearby field galaxies or the member galaxies of nearby massive groups, especially for systems that are along the same line of sight. Although the results for different halo masses are similar, groups in the lowest mass bin seems to have the highest fraction of interlopers.
Finally, the cumulative distribution of the PURITY $f_p$ is shown in the lower panel. On average, the number of groups which have $f_p
< 1$ is about the same with that have $f_p > 1$. The break at $f_p=1$ indicates that the number of recovered group members is about the same as the number of the true members. Thus, the sharper the break is, the better. The ideal case, if our group finder is perfect, it should be a step function at $f_p = 1$. As one can see, only for massive haloes there is a small fraction, $\sim 10\%$, with $f_p < 0.5$, and a significant fraction, $\sim15\%$, with $f_p >
1.5$.
We also calculate the COMPLETENESS, CONTAMINATION and PURITY in terms of the total luminosity rather than the number of member galaxies as shown in Fig. \[fig:compl\]. Although not explicitly shown here, the corresponding results are very similar to those in terms of the number of member galaxies.
We now turn to the global properties of groups. First, we examine the *global completeness*, $f_{halo}$ which defined to be the fraction of halos in the MOCKt whose brightest members have actually been identified as the brightest (central) galaxies of the corresponding groups in the MOCKg. The left panel of Fig. \[fig:fhalo\] shows $f_{halo}$ as a function of the true halo mass, obtained from our MOCKt sample for halos with $N_t \geq 1$ (solid blue line) and $N_t \geq 3$ (dashed red line), respectively. As one can see, more than 90% of all the true halos with masses $\geq 10^{13}h^{-1}M_{\bigodot}$ are selected by our group finder, almost independent of their richness. There is a weak trend with halo mass, in the sense that the performance of the group finder, in terms of $f_{halo}$, is better for more massive halos. The other global properties we examined are the richness and redshift distributions of galaxy groups. Shown in the middle and right panel of Fig. \[fig:fhalo\] are the two resulting distributions for the MOCKg and MOCKt catalogs, respectively, and good agreement is clearly seen between MOCKg and MOCKt.
Halo masses of galaxy groups {#sec:mockhmf}
----------------------------
{height="8.5cm" width="13.0cm"}
An important aspect of our group finder is the assignment of halo masses to the groups. An accurate halo mass estimate is not only important in determining group memberships according to common dark halos, but also in the applications of our group catalog to the investigations of galaxy populations in halos and large-scale structure traced by galaxy groups. As described above, our halo mass estimate is based on the ranking of the ‘GAP’ corrected luminosities of central galaxies, and our test in §\[sec:gap\] using true halo masses and group membership information in MOCKt shows that this halo mass estimate is unbiased and has scatter typically of 0.35 to 0.2 dex for halos with masses between $\sim 10^{13}$ to $\sim
10^{15}{\>h^{-1}\rm M_\odot}$. However, in real applications, the halo mass estimate also suffers from survey selection effects, contamination and incompleteness of group memberships, and so on. The accuracy of the mass estimate is expected to be reduced. Here we check the accuracy of our halo mass estimates in MOCKg catalog which is constructed from the mock 2MRS sample using our group finder.
Fig. \[fig:Mhtestmock\] shows the comparison between the true halo mass $M_{\rm t}$ and the estimated halo mass $M_{\rm g}$ from the galaxy group catalog we constructed from the mock 2MRS sample. An estimated group in MOCKg is paired with a true one in MOCKt if they both contains the same central galaxy, and we compare the halo mass assigned by our group finder in MOCKg with the true halo mass in MOCKt. Note that because of the contamination of our group finder (merger or fragmentation), only about $\ga 90\%$ groups are paired and shown here (cf. the left panel of Fig. \[fig:fhalo\]). The left panel of Fig. \[fig:Mhtestmock\] shows the comparison for all groups while the right panels for MOCKg groups with more than one member $N_g\ge 1$. The corresponding standard deviations are plotted in the bottom two panels, with different lines representing groups of different richness. Fig. \[fig:Mhtestmock\] shows that the deviation is typically between 0.2 - 0.45 dex for all groups, with some dependence on halo mass. For $N_g\ge 1$, the scatter appears to be the largest for halos with $M_{\rm t}\sim 10^{13}{\>h^{-1}\rm M_\odot}$. The bottom right panel shows that the mass estimate is improved as the group richness increases. For groups with $N \geq 3$, the scatter is about 0.35 dex, which is comparable to that obtained by Y07 for SDSS groups.
The number distribution of groups as a function of halo mass recovered is another important test of the group finder. In Fig. \[fig:hmf\] we show, as the solid points with error bars (obtained by 1000 bootstrap re-samplings), the number distribution of groups as a function of halo mass obtained from the MOCKg catalog selected by our group finder. For comparison, the distribution of true halos given by the MOCKt catalog is shown as the solid line. We see that the number distribution obtained with our group finder matches fairly well with the true halos. The slight over prediction of the number of groups at the intermediate mass range in the MOCKg is caused by the fact that we have forced the final halo mass function of groups to agree with theoretical model prediction, while the real mass function of MOCKt may deviate from the theoretical prediction due to cosmic variance (cf. Fig. \[fig:hmf0\]).
![The number distribution of groups/halos as a function of halo mass. The results obtained from the group catalog constructed from the mock galaxy sample (MOCKg) are represented by solid points. The error bars are estimated from 1000 bootstrap re-samplings. The black solid curve represents the results obtained using the true halos in the mock 2MRS sample (MOCKt). []{data-label="fig:hmf"}](f9.eps){height="7.5cm" width="7.5cm"}
The 2MRS Galaxy Group Catalog {#sec:catalog}
=============================
{height="7.0cm" width="14cm"} {height="7.0cm" width="14cm"}
{height="6.0cm" width="15.0cm"}
We apply our modified group finder to the 2MRS galaxy catalog in exactly the same way with MOCKg catalog as described in the last section. In the following, we describe our catalog and present some of its basic properties. We also make comparisons with the SDSS groups in the overlapping region, and discuss how some known nearby structures are represented in our catalog.
Basic Properties
----------------
[lcccccc]{}
\
2MRS $0.01\le z \le 0.03$ & $20921$ & $12879$ & $10004$ & $2875$ & $1495$ & $61$\
2MRS $0.0 \le z \le 0.08$ & $43246$ & $29904$ & $24618$ & $5286$ & $8484$ & $1103$\
MOCKg & $41876$ & $32368$ & $28143$ & $4225$ & $8085$ & $1098$\
MOCKt & $41876$ & $34846$ & $31879$ & $2967$ & $6103$ & $867$ \[tab:properties\]
Our modified halo-based group finder identifies 29,904 groups from a total of 43,246 2MRS galaxies in the redshift range $z \leq
0.08$. Among the groups selected, 5,286 have two or more members; 2,208 are triplets; and 1,189 have four or more members. Fig. \[fig:distribution\] shows the distribution of all groups in the 2MRS catalog. In the upper panel, the red points represent groups in the redshift range $0.0 < z \leq 0.02$, while green and blue points represent groups in $0.02 < z \leq 0.03$ and $0.03 < z \leq 0.08$, respectively. In the lower panel, red, green and blue points represent groups in mass ranges $\log M_{\rm 2MRS}\ge 13.5$, $13.5 \ge
\log M_{\rm 2MRS} \ge 12.5$ and $12.5 \ge \log M_{\rm 2MRS}$, respectively. One can see from the lower panel that more massive groups seem to locate preferentially denser regions.
Table \[tab:properties\] lists the number of groups in the 2MRS within two redshift ranges: $z = 0.01 - 0.03$ and $z \leq 0.08$, with single member or with more than one member. We also list the number of massive groups with estimated halo masses in two mass ranges, $14.0
\ge \log M_{\rm 2MRS} \ge 13.0$ and $\log M_{\rm 2MRS} \ge 14.0$. For comparison, the number of the mock groups constructed by our group finder (MOCKg) and given by simulation (MOCKt) in the redshift range $z \leq 0.08$ are also listed in Table \[tab:properties\]. We show in the left panel of Fig. \[fig:distri\] the richness distribution of groups in 2MRS which are shown as the dots with error bards. Compare to the MOCKg which is shown as the solid line, the 2MRS sample tends to contain more rich groups with $N_g>32$. However, the total number of such rich systems is small and the statistic is rather poor. The middle panel of Fig. \[fig:distri\] shows the redshift distribution of groups, where the redshift of each group is the luminosity-weighted average of the redshifts of its member galaxies. Here we see that the 2MRS sample contains slightly less groups at low redshift and slightly more groups at high redshift than the mock 2MRS sample.
One of the purposes of constructing the 2MRS galaxy group catalog is to populate the local Universe with well estimated dark matter halos for our subsequent reconstructions of the local density field [e.g. @Wang2014]. We check the number distribution of galaxy groups as a function of halo mass in the 2MRS volume which are shown in the right panel of Fig. \[fig:distri\] using black points with error bars. For comparison, the ones obtained form the MOCKg groups are also plotted in this figure. The good agreement between MOCKg and 2MRS indicate that the number distribution of groups as a function of halo mass beyond the redshift completeness $z_{\rm limit}$ are also quite similar in the MOCKg and 2MRS samples.
Comparisons with previous results
---------------------------------
{height="12.0cm" width="13.5cm"}
[lccc]{}
\
Abell2199 & 14.84 & 15.26 & 14.81$^{1}$\
Coma & 14.76 & 15.23 & 14.85$^{2}$\
Abell2634 & 14.59 & 14.90 & 14.61$^{3}$\
Perseus & 14.59 & 15.07\
Norma & 14.30 & 15.10 & 15.00$^{4}$\
Virgo & 14.37 & 15.04 & 14.43-14.90$^{5}$\
Abell1367 & 14.34 & 14.80 \[tab:mhcompare\]
In a recent study, @Tully2015 [hereafter T15] identified galaxy groups from the 2MRS using a modified version of the halo-based group finder developed by @Yang2005a, with halo masses estimated from a scaling relation to the characteristic group luminosity. Tully identified 13,606 groups from a total 24,044 galaxies in the velocity range $3000 - 10,000 {\>{\rm km}\,{\rm s}^{-1}}$, among which 3,461 have more than one member. In comparison, our group catalog uses a different halo mass estimator and extends to a larger redshift range. In particular, we have used a realistic mock catalog to quantify the reliability of our group finder and the group masses it gives.
To compare with T15, we list in Table. \[tab:properties\] the properties of groups in the redshift range $0.01 \leq z\leq 0.03$, which is comparable to the redshift range used in T15. For a total of 20,921 galaxies, we identified 12,889 groups, which matches well with the results of T15. The richest group has 184 member galaxies in our catalog, which is consistent with 180 member galaxies given by T15. We list the estimated halo masses of some prominent nearby groups in Table. \[tab:mhcompare\], including groups in the Perseus-Pisces filament, Leo cluster, Norma cluster, Virgo and Coma clusters. For comparison, the results given by T15 for the same groups are also listed in the table. In general, the halo masses given by T15 tend to be larger than the masses we obtain. We suspect that this is caused by different definitions of halo masses. To investigate this further, we looked into the literature for the halo masses of the groups in question, and the results are also shown in Table. \[tab:mhcompare\]. In general our mass estimates match well with the values given in the literature. The only exception is the Norma cluster, for which our mass estimate is significantly lower. However, Norma is located near the Milky Way Zone of Avoidance, and is severely obscured by the interstellar dust at the optical wavelengths. It is unclear if this is also a significant problem in the near infrared data used here.
We further test our 2MRS group catalogs by comparing with an existing group catalog. This group catalog used here was constructed by Y07 from the New York University Value-Added Galaxy Catalog [@Blanton2005b NYU-VAGC; ] based on the SDSS Data Release 7. A total of 639,359 galaxies with redshifts $0.01 \leq z \leq 0.20$ and redshift completenesses $C > 0.7$ were selected for constructing their group catalog. They found a total of 472,416 groups, among which 23,700 have three or more members. For our comparison, we first cross match the 2MRS galaxies with te SDSS DR7 galaxies according to their coordinates in the sky. With the assumption that galaxies located within $5 \arcsec$ of one another in the sky, and with a redshift difference of $\Delta z < 5\times10^{-4}$ (corresponding to a velocity of 150 km/s) are the same one, we got a total of 4,528 galaxy pairs, among which 2,938 galaxy pairs are centrals in both group catalogs.
We investigate the estimated halo masses assigned to the same halo in the two galaxy group catalogs. Here halos from the two group catalogs are matched if they have the same central galaxy according to the matched galaxy pairs. Note that, for the SDSS galaxy groups, the halo masses are estimated by the ‘RANK’ method, which estimates the halo mass of a candidate galaxy group according to its characteristic luminosity, $L_{-19.5}$, defined as the total luminosity of member galaxies brighter than a given luminosity threshold ${\>^{0.1}{\rm M}_r-5\log h}= -19.5$. The top panels in Fig. \[fig:Mcompare\] show the estimated halo masses for all the 2,938 matched central galaxy pairs given by the 2MRS and SDSS DR7 group catalogs, respectively. The top left and top right panels plot the same thing, except that the two mass axes are flipped: SDSS mass versus 2MRS mass in the left and 2MRS mass versus SDSS mass in the right). Ideally, the two estimated halo masses should be the same, so that all the data points would lie on the red solid line ($\log M_{\rm 2MRS} = \log M_{\rm SDSS}$). We can see that the two halo masses estimated are tightly correlated with each another, with no obvious systematic bias (see the middle two panels which show the deviations from the perfect line). The typical scatter is $\sim
0.4{\rm dex}$ in medium to massive halo mass range, and $\sim 0.2$ - $0.3 {\rm dex}$ for low mass groups, as shown in the two lower panels. This scatter is roughly consistent with the one shown in Fig. \[fig:Mhtestmock\] between the group and true halo masses estimated from the mock 2MRS catalogs.
SUMMARY {#sec:summary}
=======
In this paper, we have implemented, tested and applied a modified version of the halo-based group finder developed in @Yang2005a [@Yang2007] to extract galaxy groups from the 2MRS. Covering uniformly about 91% of the sky, the 2MRS provides the best available representation of the structures in local universe, and so a group catalog constructed from it is useful for many purposes. However, 2MRS is quite shallow; in many cases only a few brightest members within a halo can be observed. To deal with this limit, we have updated the halo mass estimate used in the previous group finder with a new method based on ‘GAP’. This ‘GAP’ estimator consists of two parts: (i) a relation between the luminosity of the central galaxy, $L_c$, and the halo mass, $M_h$, inferred iteratively from abundance matching between the luminosity of [*central*]{} galaxies and the masses of dark matter halos; (2) a luminosity gap correction factor obtained from the luminosity difference between the central galaxy and a faint satellite galaxy.
In order to evaluate the performance of our modified group finder, we have constructed mock 2MRS galaxy samples based on the observed $K_s$-band luminosity function. The group catalog obtained from the mock 2MRS galaxy catalog shows a 100% completeness for about $65\%$ of the most massive groups to $\sim85\%$ for groups with halo masses $\log M_h < 10^{14}{\>h^{-1}\rm M_\odot}$. On average, about 80% of the groups have 80% completeness. In terms of interlopers, about $65\%$ of the groups identified have none, and an additional 20% have an interloper fraction lower than 50%. Further tests on the halo mass estimation show that the deviation of the halo mass between the selected groups and the true halos is $\sim 0.35 {\rm dex}$ over the entire mass range. These tests demonstrate that the modified group finder is reliable for the 2MRS sample.
Applying the modified halo-based group finder to the 2MRS, we have obtained a group catalog with a depth to $z \leq 0.08$ and covering $91\%$ of the whole sky. This 2MRS group catalog contains a total of 29,904 groups, among which 24,618 are singles and 5,286 have more than one member. Some of the basic properties of the group catalog are presented, including the distributions in richness, in redshift and in halo mass. This catalog provides a useful data base to study galaxies in different environments. In particular, it can be used to reconstruct the mass distribution in the local Universe, as we will do in a forthcoming paper.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the anonymous referee for helpful comments that greatly improved the presentation of this paper. This work is supported by 973 Program (No. 2015CB857002), national science foundation of China (grants Nos. 11203054, 11128306, 11121062, 11233005, 11073017), NCET-11-0879, the Strategic Priority Research Program “The Emergence of Cosmological Structures" of the Chinese Academy of Sciences, Grant No. XDB09000000 and the Shanghai Committee of Science and Technology, China (grant No. 12ZR1452800). We also thank the support of a key laboratory grant from the Office of Science and Technology, Shanghai Municipal Government (No. 11DZ2260700). HJM would like to acknowledge the support of NSF AST-1517528.
A computing facility award on the PI cluster at Shanghai Jiao Tong University is acknowledged. This work is also supported by the High Performance Computing Resource in the Core Facility for Advanced Research Computing at Shanghai Astronomical Observatory.
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A polynomial training algorithm for
calculating perceptrons of optimal stability
(published in J. Phys. A [**28**]{}(8), p. 2173–2181, 1995)
Jorg Imhoff
Universität Heidelberg, Institut für Theoretische Physik,
Philosophenweg 19, D-69120 Heidelberg
(e-mail: [email protected])
plus 1pt
Spin glass models of neural networks and their application as an associative memory have been of great interest in the last years \[1-10\]. One major issue of the field is the question of training networks, that is the construction of a synaptic matrix in order to store given information. In this paper I am going to present a training algorithm that is able to find solutions of the perceptron problem of optimal stability in finite time. Unlike other algorithms, as Minover presented by Krauth and Mézard \[5\] or AdaTron by Anlauf and Biehl \[6\], this algorithm does not only approximate optimal solutions but actually finds them. Furthermore, there are no divergent timescales in the solution of the problem. Minover and AdaTron both have diverging training times as the critical storage capacity $\alpha_c$ is approached \[6,7\], whereas this algorithm does not. Therefore it can also be used beyond $\alpha_c$ in the region of broken replica symmetry, where it finds local optima of negative stability. A similar algorithm was proposed by Ruján \[8\], which also finds optimal perceptrons in finite time, but cannot advance beyond $\alpha_c$.
Like the pseudo-inverse solution of the perceptron problem \[9,10\] this algorithm uses inversion of pattern correlation matrices for searching (optimal) perceptron couplings. As matrix inversion has to be done repeatedly, the algorithm was called Recomi — peated rrelation atrix nversion. As was shown by Opper \[7\] the problem of finding an optimal perceptron is the problem of finding the subset of embedded training patterns with minimal local fields. Recomi is able to find this subset of patterns iteratively in finite time. The coupling vector is then just the pseudo-inverse of the respective pattern correlation matrix.
I consider a network of $N+1$ neurons $S_i=\pm 1$, $i=1,\ldots,N+1$, coupled through synaptic efficacies $J_{ij}$ (without taking self couplings into account, i.e. $J_{ii}=0 \quad\forall i$). The dynamics of the system is taken to be a simple zero-temperature Monte Carlo process:
$$S_i(t+1) = \sgn \left( \sum_{j(\neq i)} J_{ij} S_j(t) \right)$$
The purpose of perceptron training algorithms is to find couplings $J_{ij}$ such that $p$ patterns $\vect\eta^\mu=(\eta^\mu_1, \ldots,
\eta^\mu_{N+1})^T$, $\eta^\mu_i=\pm 1$, $\mu=1,\ldots,p$, become fixed points of the dynamics. That is
$$\eta^\mu_i \sum_{j(\neq i)} J_{ij} \eta^\mu_j \geq \kappa > 0,
\qquad i=1,\ldots,N+1; \qquad \mu=1,\ldots,p.$$
The problem can be reformulated by looking at the single neurons (or simple perceptrons) of the network, e.g. neuron $N+1$. With
$$\xi^\mu_i \definition \eta^\mu_{N+1} \eta^\mu_i,
\qquad i=1,\ldots,N; \qquad \mu=1,\ldots,p
\label{xi}$$
one now has to find couplings $J_i$, $i=1,\ldots,N$, such that
$$h_\mu = \sum_i J_i \xi^\mu_i \geq \kappa >0, \qquad \mu=1,\ldots,p.$$
If the norm of $\vect J$ is fixed, e.g. $|\vect J|=1$, it is possible to define what is meant by “optimal solutions” of the given problem:
$$\hbox{maximize\ \ } \kappa = \min_\mu \{ h_\mu \}
\hbox{\ \ under the constraint\ \ } \left| \vect J \right| = 1
\label{optimization-problem}$$
With maximal $\kappa$ one expects to have maximum stability against input noise, i.e. maximal basins of attraction in a network of neurons.
From the point of view of mathematical optimization it suitable to reformulate the problem. With $ \vect J \longrightarrow \vect J / |\kappa| $ one gets an equivalent formulation of problem (\[optimization-problem\]): $$\begin{aligned}
\hbox{minimize\ \ } \left| \vect J \right| &
\hbox{under the constraints} \quad h_\mu = \vect J^T \vect\xi^\mu
\geq +1 \quad \forall\mu & \hbox{(for $\kappa>0$)}
\label{optim-prob-pos-kappa} \\
\hbox{maximize\ \ } \left| \vect J
\right| & \hbox{under the constraints} \quad h_\mu = \vect J^T
\vect\xi^\mu \geq -1 \quad \forall\mu & \hbox{(for
$\kappa<0$)} \label{optim-prob-neg-kappa}\end{aligned}$$
I will use this formulation of the problem later in this article. Applying the Kuhn-Tucker theorem of optimization theory \[11\] it can be shown \[7\] (see also \[6\]) that an optimal solution, for $\kappa>0$, can always be written in the form
$$\vect J = \sum_{\mu\in\Gamma} x_\mu \vect\xi^\mu,
\quad \hbox{where\ \ } x_\mu \geq 0 \quad \forall \mu\in\Gamma
\label{positive-embeddings}$$
with $$h_\mu = \vect J^T \vect\xi^\mu
\cases{ = \kappa & $\mu\in\Gamma$ \cr
> \kappa & else \cr
}
\label{and-the-fields}$$
For $\kappa<0$ the same argument holds for all local optima, but with $x_\mu
\leq 0$ $\forall \mu\in\Gamma$. $\Gamma$ is the set of “embedded” patterns, $\Gamma \subseteq \{1,\ldots,p\}$. The $x_\mu$ are called the embedding strengths of solution $\vect J$. Anlauf and Biehl have also shown \[6\] that for $\kappa>0$ this solution is unique (which is in general not the case for $\kappa<0$). I.e. two solutions $\vect J$ and $\vect J^\ast$ of the form (\[positive-embeddings\])(\[and-the-fields\]) are always identical $\vect J
\equiv \vect J^\ast$. Note that if $\{ \xi_\mu | \: \mu\in\Gamma \}$ is a set of linearly independent vectors — e.g. if the patterns are in general position and $\card(\Gamma) \leq N$ — the choice of the $x_\mu$ is unambiguous. On the other hand, if one has a solution of the form (\[positive-embeddings\])(\[and-the-fields\]) it must be the global optimum of the problem.
In the following sections I am going to describe the Recomi algorithm. Recomi can solve the stated problem of finding optimal perceptrons of the form (\[positive-embeddings\])(\[and-the-fields\]) in finite time, if the training patterns are in general position, i.e. if every subset $\{\xi_\mu\}$ with not more than $N$ elements ($\card(\{\xi_\mu\}) \leq N$) is linearly independent. It does so in not more than ${\cal O}(N^4)$ floating point operations. There is no divergence of learning times at the critical storage capacity $\alpha_c=2$ (for unbiased random patterns), where $\alpha=p/N$. I am going to show this numerically. In the last section I will deduce some important constituents of a proof of convergence — unfortunately a full proof cannot yet be given. I will analyze there the properties of locally stable solutions of the optimization problems (\[optim-prob-pos-kappa\]) and (\[optim-prob-neg-kappa\]). It can be shown that Recomi always stops in a local optimum. If an optimal solution with $\kappa>0$ exists, Recomi must stop there. Otherwise it is going to stop in one of the locally stable solutions with $\kappa<0$.
Description of the algorithm {#description-of-the-algorithm .unnumbered}
============================
Recomi is an iterative algorithm. It calculates coupling vectors $\vect J^{(t)}
= \sum_\mu x_\mu^{(t)} \vect\xi^\mu$ and finds after a finite number of iterations a solution of the form (\[positive-embeddings\])(\[and-the-fields\]), if it exists. As we will see later, the algorithm must be initialized with positive embedding strengths $x_\mu^{(0)}\geq 0$, e.g. Hebbian couplings $\vect J^{(0)} = \sum_\mu
\vect\xi^\mu$. For numerical stability $\vect J^{(t)}$ is normalized to 1 after each iteration. Let $C_\Gamma$ be the correlation matrix of the patterns in $\Gamma\subseteq\{1,\ldots,p\}$:
$$C_\Gamma = \left( {\vect\xi^\mu}^T \vect\xi^\nu
\right)_{\mu,\nu\in\Gamma}$$
Iteration loop {#iteration-loop .unnumbered}
--------------
Let $\vect J^{(t)}$ be given (from now on I drop the index $t$): $$\begin{aligned}
\vect J & = & \sum_{\mu=1}^p x_\mu \vect\xi^\mu
\qquad \left( |\vect J| = 1 \right) \\
\kappa & = & \min_\mu \left\{ h_\mu \right\} =
\min_\mu \left\{ \vect J^T \vect\xi^\mu \right\}\end{aligned}$$
Let $\Gamma$ be the subset of patterns with minimal local field $h_\mu$:
$$\Gamma = \left\{ \: \mu \; \big| \; h_\mu = \kappa \: \right\}$$
We now want to alter $\vect J$ $$\vect J \quad \longrightarrow \quad
\vect J' = \sum_{\mu=1}^p \left( x_\mu +
\eps \Delta x_\mu \right) \vect\xi^\mu$$ so that for all patterns in $\Gamma$ the local fields grow equally $$h_\mu' = {\vect J'}^T \vect\xi^\mu = \kappa+\eps
\qquad \forall\mu\in\Gamma.$$
We therefore choose $\Delta\vect x$ to be the pseudo-inverse \[9,10\] of the patterns in $\Gamma$: $$\Delta x_\mu = \cases
{ \sum_{\nu\in\Gamma} \left( C_\Gamma^{-1} \right)_{\mu\nu} &
$\mu\in\Gamma$ \cr \qquad 0 & else \cr }.
\label{def-of-dx}$$
If the training patterns $\vect\xi_\mu$ are in general position, $C_\Gamma$ becomes singular if and only if the number of patterns in $\Gamma$, $\card(\Gamma)$, is greater than $N$. Then Recomi must stop, with $\vect J^{(t)}$ being the best solution found. Nevertheless Recomi is able to find optimal solutions as I will show in the last section of this paper.
Now we want to determine the learning rate $\eps$ in a way that [*all *]{} local fields $h_\mu'$ are greater or equal $\kappa+\eps$:
$$h_\mu' = {\vect J'}^T \vect\xi^\mu \geq \kappa+\eps
\qquad \forall\mu\in\{1,\ldots,p\}.
\label{all_local_fields}$$
$\eps=\eps_\mu$ is the value of the learning rate whith which we get the equality $h_\mu'=\kappa+\eps$ for pattern $\mu$: $$\eps_\mu = { h_\mu - \kappa \over 1 -
\sum_{\nu\in\Gamma} C_{\mu\nu} \Delta x_\nu }.$$
To fulfill eqn. (\[all\_local\_fields\]) $\:\eps$ must be smaller or equal to all relevant, i.e. all positive, $\eps_\mu$. We therefore define the set $\Phi$: $$\Phi \;\;=\;\; \left\{ \;\; \eps_\mu \;\; \bigg| \;\;
\mu\notin\Gamma \quad \hbox{and} \quad
0 < \eps_\mu < \infty \;\; \right\}.$$
If $\Phi$ is not empty we can determine $\eps$ as $$\eps = \min \Phi.$$
If $\Phi$ is empty, we set $\eps=\infty$, i.e. $\vect J' =
\sum_{\mu\in\Gamma} \Delta x_\mu \vect\xi^\mu$, and stop the iteration.
Now $\vect J^{(t+1)} = \vect J' / |\vect J'|$ and we continue at the beginning of the iteration loop. It is easy to show that always $\kappa^{(t+1)} =
(\kappa^{(t)}+\eps)/|\vect J'| > \kappa^{(t)}$ (see Appendix). If no solution with positive $\kappa$ can be found the algorithm typically stops with $\vect
J' = \vect 0$, as will be shown later. (It should be noted that this is the most sensitive part of the algorithm. Rounding errors must be controlled when calculating the norm of $\vect J'$.) Then $\vect J^{(t)}$ is taken as the best solution found by Recomi.
Optimal Recomi {#optimal-recomi .unnumbered}
--------------
The algorithm I have described so far does not yet find optimal solutions of the form (\[positive-embeddings\])(\[and-the-fields\]). As the changes of embedding strenghts $\Delta x_\mu$ might be negative in eqn. (\[def-of-dx\]) the $x_\mu$ might also become negative in the end. But already this version of the algorithm does find nearly optimal solutions $\kappa>0$, as can be seen in fig. \[fig1\], where I compare results for unbiased random patterns ($N=100$) with Gardner’s result \[3\]. Therefore I refer to this version of Recomi as “nearly optimal Recomi”.
To find optimal solutions of the form (\[positive-embeddings\])(\[and-the-fields\]) it is necessary to start with positive embedding strengths $x_\mu\geq 0$, and to make sure that they stay positive throughout the iteration, i.e. $\Delta x_\mu\geq 0$. This is possible by altering eqn. (\[def-of-dx\]). $\Gamma$ must be replaced by a subset $\Gamma'\subseteq\Gamma$ with the following properties:
$$\Gamma'\subseteq\Gamma$$
$$\Delta x_\mu = \sum_{\nu\in\Gamma'} \left( C_{\Gamma'}^{-1}
\right)_{\mu\nu} \geq 0 \qquad \forall\mu\in\Gamma'$$
$$\left( \sum_{\nu\in\Gamma'} \Delta x_\nu \vect\xi^\nu \right)^T
\vect\xi^\mu \geq 1 \qquad \forall\mu\in\Gamma$$
It is always possible to find such a subset $\Gamma'$ (as long as $C_\Gamma$ itself is regular), because $\sum_{\nu\in\Gamma'} \Delta x_\nu \vect\xi^\nu$ then is the (unique) optimal perceptron for the correct mapping of the patterns $\mu\in\Gamma$.
$\Gamma'$ can easily be determined. The following algorithm proved to work in all cases tested (about ${\cal O}(10^5)$ algorithm runs). I cannot yet prove its convergence analytically. This has to be done in later work. To find $\Gamma'$ one can proceed as follows:
1.)
2.)
3.)
By replacing $\Gamma$ by $\Gamma'$ in eqn. (\[def-of-dx\]) Recomi is able to find optimal solutions. I refer to this improved version of the algorithm as “optimal Recomi”. In fig. \[fig2\] I check for unbiased random binary patterns ($N=100$), how often the algorithm stops in optimal solutions with $\kappa>0$, and in locally optimal solutions with $\kappa<0$. For every value of $\alpha=p/N$ 100 different pattern sets are tested. In very rare cases (not in this figure) the algorithm only gets close to but does not reach optimal solutions: trying to invert nearly singular correlation matrices can cause failure of the inversion subroutines.
In fig. \[fig1\] I compare results for unbiased and biased random binary patterns with Gardner’s result \[3\]. The patterns $\eta^\mu_i$ are chosen with a probability distribution $p(\eta^\mu_i) = {(1-m)\over 2}
\,\delta(\eta^\mu_i+1) + {(1+m)\over 2} \,\delta(\eta^\mu_i-1)$, using $m=0$ (unbiased) and $m=0.8$ (biased), and the $\xi^\mu_i$ calculated according to eqn. (\[xi\]). Within the error bounds there is no difference to be seen between optimal and nearly optimal solutions below $\alpha_c$ ($\kappa>0$). In the range of replica symmetry breaking $\alpha>\alpha_c$ ($\kappa<0$) optimal Recomi clearly performs better than the simpler version of the algorithm. Here it cannot be expected that the algorithm finds a global stability optimum, as it gets trapped in one of the many local optima, which will be shown in the last section of this paper. Note that for the biased patterns ($m=0.8$) at $N=100$ one still has to take finite size effects into account: the measured points are all optimal solutions, but yet still lie a little bit below the Gardner curve. Also note that the theoretical lines are all calculated in replica symmetric approximation, i.e. they must be corrected for negative $\kappa$, where replica symmetry is no longer valid.
In fig. \[fig3\] I train perceptrons of different sizes $N$ with unbiased random binary patterns. Convergence time is plotted against system size $N$ for different values of the storage capacity $\alpha$. The most expensive part of the algorithm, in the large $N$ limit, is matrix inversion, which is of ${\cal
O}(N^3)$ for each single inversion. Nearly optimal Recomi therefore is, in the worst case, of ${\cal O}(\sum_{i=1}^N i^3)={\cal O}(N^4)$, as $\card(\Gamma)$ grows at least by one in each iteration step. For optimal Recomi one cannot give such a simple derivation of convergence times, as $\card(\Gamma)$ can also shrink in the learning process. But here convergence time is also bounded from above by ${\cal O}(N^4)$: In fig. \[fig3\] I count the number of floating point operations ($+-*/$) optimal Recomi needs to find solutions. As below $N=100$ convergence time is still dominated by other operations apart from matrix inversion, I only plot the matrix inversion part here. All other operations are of ${\cal O}(N^3)$ or below. Just as predicted for nearly optimal Recomi the optimal version of the algorithm converges in ${\cal O}(N^4)$ or less floating point operations.
In fig. \[fig4\] I plot convergence time (i.e. number of floating point operations) against the storage capacity $\alpha$. Again the perceptron ($N=100$) was trained with unbiased random binary patterns. There is no divergence at $\alpha=\alpha_c=2$. For small $\alpha$ the two versions of the algorithm differ only little, as nearly optimal Recomi also often finds optimal solutions (see also fig. \[fig1\]). For larger values of $\alpha$ the convergence times evolve different.
Analysis of local stability optima: towards a proof of convergence {#analysis-of-local-stability-optima-towards-a-proof-of-convergence .unnumbered}
==================================================================
I cannot yet give a full proof of convergence of Recomi, but some major components can already be deduced. For this reason I want to consider the role of local stability optima. It is useful here to use the problem formulations eqn. (\[optim-prob-pos-kappa\]) (for $\kappa>0$) and eqn. (\[optim-prob-neg-kappa\]) (for $\kappa<0$). If I write a $\pm$-sign in the following text, the $+$ always refers to the case $\kappa>0$ and the $-$ to $\kappa<0$.
The Problem (\[optim-prob-pos-kappa\])(\[optim-prob-neg-kappa\]) can now be formulated as $$\begin{aligned}
\hbox{minimize} & f(\vect x) = \pm \; \vect J^T \vect J =
\pm \; \vect x^T C \vect x & \nonumber \\
\hbox{under the constraints} & h_\mu = \vect J^T \vect\xi^\mu =
(C \vect x)_\mu \geq \pm 1 & \quad \mu = 1,\ldots,p.
\label{optim-prob}\end{aligned}$$
$\Gamma$ is the set of patterns with minimal local field: $$\Gamma = \left\{ \: \mu \; \big| \; h_\mu=(C \vect x)_\mu=\pm 1 \: \right\}.$$
Let $\Omega$ be the set of all possible search directions $\Delta\vect x$, which do not violate the inequality constraints eqn. (\[optim-prob\]):
$$\Omega = \left\{ \: \Delta\vect x \in \real^p \; \bigg| \; (C\Delta\vect x)_\mu
\geq 0 \;\;\;\; \forall\mu\in\Gamma \: \right\}$$
A solution $\vect J$ is locally optimal if and only if $$\Big[ \nabla_x f(\vect x) \Big]^T \Delta\vect x =
\pm\; 2 \: \vect x^T C \Delta\vect x
\geq 0 \qquad \forall \Delta\vect x \in \Omega
\label{local-opt}$$
I now prove the important theorem, that if there is a solution with positive stability $\kappa>0$ there cannot be locally stable solutions $\vect J$ with negative stability $\kappa<0$ and $x_\mu\geq 0$ $\forall\mu$, $\sum_\mu
x_\mu>0$:
If there is a solution with $\kappa>0$ there must be a solution of the form (e.g. the optimal perceptron) $$\vect J^\ast = \sum_\mu x_\mu^\ast \vect\xi^\mu, \quad\hbox{with\ \ }
{\vect J^\ast}^T \vect\xi^\mu = (C\vect x^\ast)_\mu \geq\kappa^\ast >0
\quad\forall\mu$$
Let us assume $\vect J = \sum_\mu x_\mu \vect\xi^\mu$ is locally optimal with $\kappa = \min_\mu \{\vect J^T \vect\xi^\mu\} < 0$ and $x_\mu\geq
0$ $\forall\mu$, $\sum_\mu x_\mu>0$. That means (eqn. (\[local-opt\])): $$\vect x^T C \Delta\vect x \leq 0 \qquad \forall \Delta\vect x \in \Omega
\label{contra}$$
As $(C\vect x^\ast)_\mu\geq\kappa^\ast>0$ $\forall\mu$, we have: $$\Delta\vect x \definition \vect x^\ast \in \Omega$$ $$\vect x^T C \Delta\vect x = \vect x^T C \vect x^\ast =
\sum_\mu x_\mu (C\vect x^\ast)_\mu \geq \kappa^\ast \sum_\mu x_\mu > 0$$
in contradiction to eqn. (\[contra\])! Therefore such a vector $\vect J$ cannot exist. We will see below that optimal Recomi always stops in (local) optima which by definition of the algorithm are of the form $x_\mu\geq
0$ $\forall\mu$ and $\sum_\mu x_\mu>0$. So if there is any solution with $\kappa>0$ Recomi can only stop in the global optimum of the problem, because then there are no other optima of that form.
To show this, I have to make several assumptions, which I cannot prove yet: a) The algorithm described in section “Optimal Recomi” for deriving $\Gamma'$ really always works. b) The size of $\Gamma$, $\card(\Gamma)$, grows not more than by one in each iteration step, especially not from $\card(\Gamma)<N$ to $\card(\Gamma)>N$. c) Recomi really terminates in finite time. About this last point one can only say that $\kappa^{(t)}$ is a strictly monotonical function of $t$ (see Appendix), i.e. there is always an attractor of the training dynamics.
If these three assumptions are correct, Recomi stops in a (local) optimum, which is the global one, if solutions $\kappa>0$ exist. To show this I have to consider the three possible ways the algorithm does stop: 1) $\Phi$ is empty, i.e. $\eps$ becomes infinit. 2) $\vect J'$ is zero. 3) $C_\Gamma$ is singular.
1\) $\Phi$ is empty: This is the most simple case. Then, by definition, $\vect
J' = \sum_{\mu\in\Gamma'} \Delta x_\mu \vect\xi^\mu$, which is an optimal solution of the form (\[positive-embeddings\])(\[and-the-fields\]). This is the usual way Recomi stops if solutions $\kappa>0$ exist.
2\) $\vect J'$ is zero: Then $\vect J^{(t)}=-\eps \sum_{\mu\in\Gamma'} \Delta
x_\mu \vect\xi^\mu$. Applying the Kuhn-Tucker theorem this is a locally stable solution for $\kappa<0$ (just like (\[positive-embeddings\])(\[and-the-fields\]) for $\kappa>0$). As $\vect
J^{(t)}$ is coded in the form $x_\mu\geq 0$ $\forall\mu$ and $\sum_\mu x_\mu>0$ there cannot be solutions with $\kappa>0$ as was shown above. This is the usual way Recomi stops if no solutions $\kappa>0$ exist.
3\) $C_\Gamma$ is singular: Then $\card(\Gamma)>N$ (because the training patterns are in general position). According to our assumption, $\card(\Gamma)$ must have been $N$ in the iteration step before. $\Gamma'$ must have been equal to $\Gamma$ because otherwise $\card(\Gamma)$ would not have grown. As $\{
\xi^\mu | \mu\in\Gamma \}$ does span $\real^N$, $\vect J^{(t-1)}$ is completely determined by the local fields ${\vect J^{(t-1)}}^T \vect\xi^\mu$ $\mu\in\Gamma$, i.e. $\vect J^{(t-1)} \sim \sum_{\mu\in\Gamma} \Delta x_\mu
\vect\xi^\mu$, which is a local optimum. Therefore case 3) does in principal never occur, the algorithm stops before in 1) or 2).
In practice case 3) does occur, as sometimes nearly singular correlation matrices cannot be inverted by the inversion subroutines because of numerical restrictions.
Conclusion {#conclusion .unnumbered}
==========
In this article I presented a perceptron learning algorithm, which is able to find the optimal perceptron in finite time, i.e. in ${\cal O}(N^4)$ floating point operations. The algorithm even works beyond the critical storage capacity $\alpha_c$, where it finds solutions of negative stability that are locally optimal. Calculating the stability curve $\kappa(\alpha)$ for random training patterns exactly reproduces Gardner’s predictions \[3\]. A full prove of convergence could not yet be given, but major constituents were already shown. As the algorithm works very reliably, it can be expected that a full proof of convergence can be found. Furthermore it is planned to generalize the algorithm to two layer perceptrons with fixed output. First results are very promising, yet it cannot be expected that the algorithm finds globally optimal solutions, because replica symmetry breaking effects are very strong in this case.
Appendix {#appendix .unnumbered}
========
In this appendix I will show that $\kappa^{(t)}$ is a strictly monotonical function of $t$: $$\kappa^{(t+1)} = { \kappa^{(t)} + \eps \over |\vect J'| }$$ $$\varrho \definition \left( \sum_\mu \Delta x_\mu \vect\xi^\mu \right)^2 =
\sum_{\mu\nu} \Delta x_\mu (C_\Gamma)_{\mu\nu} \Delta x_\nu =
\sum_{\mu\nu\lambda} \Delta x_\mu (C_\Gamma)_{\mu\nu}
(C^{-1}_\Gamma)_{\nu\lambda} =
\sum_\mu \Delta x_\mu \geq 0$$ $${\vect J'}^T \vect J' = \left( \vect J^{(t)} +
\eps \sum_{\mu\in\Gamma} \Delta x_\mu \vect\xi^\mu \right)^2 =
1 + 2 \eps \kappa^{(t)} \varrho + \eps^2 \varrho
\geq 0 \qquad\forall\eps\in\real$$ $$\hbox{e.g.} \quad \eps = -\kappa^{(t)} \quad\Longrightarrow\quad
1 - {\kappa^{(t)}}^2 \varrho \geq 0$$ $${d\over d\eps} \kappa^{(t+1)} =
\left(1 + 2 \eps \kappa^{(t)} \varrho + \eps^2 \varrho\right)^{-3/2}
\left(1 - {\kappa^{(t)}}^2 \varrho\right) \geq 0$$
${d\over d\eps} \kappa^{(t+1)} = 0$ if and only if Recomi stops in a (local) optimum: $${d\over d\eps} \kappa^{(t+1)} = 0 \iff 1 - {\kappa^{(t)}}^2 \varrho = 0
\iff \vect J^{(t)} = \kappa^{(t)} \sum_{\mu\in\Gamma} \Delta x_\mu \vect\xi^\mu$$
That means $\kappa^{(t+1)}>\kappa^{(t)}$ as long as Recomi has not terminated. qed.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank Prof. H. Horner, Dr. R. Kühn and Prof. H. G.Bock for valuable discussions. This work was sponsored by the IWR (Interdisciplinary Center for Scientific Computing), Universität Heidelberg.
References {#references .unnumbered}
==========
|
---
abstract: 'A quantum mechanical approach and local response theory are applied to study plasmons propagating in nanometer-thin gold slabs sandwiched between different dielectrics. The metal slab supports two different kinds of modes, classified as long-range and short-range plasmons. Quantum spill-out is found to significantly increase the imaginary part of their mode indices, and, surprisingly, even for slabs wide enough to approach bulk the increase is 20%. This is explained in terms of enhanced plasmonic absorption, which mainly takes place at narrow peaks located near the slab surface.'
author:
- 'Enok J. H. Skjølstrup'
- Thomas Søndergaard
- 'Thomas G. Pedersen'
bibliography:
- 'mybib.bib'
title: 'Quantum spill-out in nanometer-thin gold slabs: Effect on plasmon mode index and plasmonic absorption'
---
Introduction {#sec:introduction}
============
Recently, it was found that the effect of quantum spill-out in nanometer-thin gaps in gold has a significant impact on the propagation of surface plasmon polaritons (SPPs) in such structures. In the limit of vanishing gap, the SPP mode index was found to converge to the refractive index of bulk gold, [@spill-out] while classical models neglecting spill-out find a diverging mode index [@narrow_gap1; @gap2]. In addition, it was discovered in Ref. that spill-out significantly increases the plasmonic absorption in these gaps. Furthermore, the predicted reflectance from an ultrasharp groove array is in much better agreement with measurements [@black_gold] than the classical model. [@optics_multiple; @GSP]
In this paper, we study the opposite geometry, i.e., a nanometer-thin gold slab surrounded by different dielectrics. Such a structure supports long-range and short-range SPPs, which are $p$-polarized electromagnetic waves bound to and propagating along the slab [@slab1; @slab2; @slab5; @leaking1; @slab9]. For a nanometer-thin slab, the short-range mode is strongly bound, meaning that a large part of the field profile is located in the slab region, while the long-range mode is weakly bound with most of its field profile located in the dielectric regions. The magnetic fields of the modes are symmetric and antisymmetric, respectively, if the metal slab is sandwiched between identical dielectrics, while the symmetry is broken when sandwiched between different dielectrics[@slab5]. Applications of such SPPs are found in, e.g., plasmonic lenses for biosensors and as mode couplers into dielectric or plasmonic waveguides [@slab6; @slab8]. In addition, plasmonic structures find applications within e.g. solar cells[@application1], and furthermore, they can be applied to squeeze light below the diffraction limit [@control3; @control4], and can be utilized in lasers[@application2].
Refs. applied a classical model neglecting quantum spill-out, such that the dielectric function takes one value in the metal region and another value in the dielectric region, thus changing abruptly at the interfaces, while quantum effects have been included in, e.g., Refs.. Furthermore, gold films with thicknesses down to 1 nm have recently been fabricated [@maniyara].
Here, we examine the effect of quantum spill-out on plasmons propagating in nanometer-thin gold slabs. Local response theory is applied to calculate the mode indices and associated electromagnetic fields. We show in the following that spill-out significantly increases the imaginary part of the mode index, even for slabs wide enough to approach bulk. This is explained in terms of strong plasmonic absorption mainly taking place a few Å from the slab surface, a phenomenon not found in classical models.
Quantum dielectric function {#sec:quantum}
===========================
In the vicinity of the gold slab, the electron density and the effective potential arising from the free electrons (in the $s,p$ band) are significantly modified due to electron tunnelling through the surface barrier. To capture this effect, we calculate the electron density using Density-Functional Theory (DFT) in the jellium model [@jellium1; @jellium2] (see Appendix A for further description). The optical cross sections of metal nanowires [@nanowires1; @nanowires2; @nanowires3; @nanowires4], metal clusters and spheres [@clusters1; @clusters2; @nanowires1; @response1] have previously been calculated by applying such a DFT model in the jellium approximation. Likewise the plasmon resonance of metal dimers and semiconductor nanocrystals have been calculated in Refs. , while Ref. studied the plasmonic properties in ultra-thin metal films. In addition, Ref. examined the role of electron spill-out and non-local effects on the plasmon dispersion relation for gap plasmons propagating between two gold surfaces as well as plasmons propagating in gold slabs surrounded by air. It was found in that paper, that spill-out has a significant impact while the influence from non-local effects was minor. In Refs. , only the real part of the parallel wave number (analogous to mode index) was considered, with no studies of the dependence of the slab (or gap) width. In the present paper, in contrast, we compute both the real and imaginary parts of the mode index, and furthermore, investigate in detail how they depend on the slab width. Non-local effects in metal dimers and cylinders were studied in Refs. , where it was found that these effects slightly blue-shift the plasmon resonances. In this paper, similarly to Refs. we ignore the non-local effects and thereby treat the dielectric function as a local response.
The electron density $n$ across a gold slab of width $d=1$ nm is shown in Fig. \[fig:density\_epsilon\](a) in units of the bulk gold density $n_0$, where the geometry is chosen such that the $x$-axis is perpendicular to the slab, while the plasmons are propagating in the $y$-direction.
![(a) Electron density in units of the bulk gold density $n_0$ across a gold slab of width 1 nm. (b) Real and imaginary part of the dielectric function $\varepsilon$ across the same gold slab at a wavelength of 775 nm, where the shaded areas show the position of the surrounding dielectrics. The solid black (blue) curve is for a slab surrounded by air (glass) on both sides. The imaginary part is unaffected by the kind of dielectric. In both figures, the colored areas represent the ion charge.[]{data-label="fig:density_epsilon"}](density_epsilon){width="8cm"}
The colored area in the figure shows the position of the ion charge in the jellium model, and spill-out is clearly seen to occur as the electron density contains an exponential tail that stretches $\sim$0.3 nm into the dielectric region. In addition, for the charge to be conserved, the density inside the slab is also affected by spill-out. As the slab gets wider, the electron density near the slab boundary contains Friedel oscillations in agreement with Refs. where the electron density at a single interface between gold and air was studied.
The electron density is applied to calculate the local dielectric function $\varepsilon$ across the structure by a method analogous to Ref. . In the bulk, the electron density $n_0$ implies a bulk plasma frequency of $\omega_{p,\textrm{bulk}}=\sqrt{n_0e^2/(m_e \varepsilon_0)}$, which gives rise to a Drude response $\varepsilon_{p,\textrm{bulk}}(\omega)=1-\omega_{p,\textrm{bulk}}^2/(\omega^2+i\omega\Gamma)$ [@nanooptik]. Bound electrons in the lower lying $d$ bands also contribute to the dielectric function[@nanooptik], but in contrast to the free electrons, we assume that they are entirely located in the jellium region, thus not tunnelling through the potential barrier. The response from the bound electrons is calculated from the experimental response of bulk gold, $\varepsilon_{\textrm{gold}}(\omega)$ from Ref. as $\varepsilon_{\textrm{bound}}(\omega)=\varepsilon_{\textrm{gold}}(\omega)-\varepsilon_{p,\textrm{bulk}}(\omega)$. The final dielectric function in the vicinity of a gold slab with a jellium region spanning from $x=-d/2$ to $x=d/2$, is therefore given by $$\begin{aligned}
\varepsilon(\omega,x)&=1-\frac{\omega_p^2(x)}{\omega^2+i\Gamma \omega} + (\varepsilon_{\text{s}}(x)-1)\theta(\vert x \vert-d/2) \nonumber \\
&+\varepsilon_{\textrm{bound}}(\omega)\theta(d/2-\vert x \vert). \label{eq:eps}\end{aligned}$$ Here, the first term describes the local Drude response of free electrons with position dependent plasma frequency $\omega_p(x)=\sqrt{n(x)e^2/(m_e \varepsilon_0)}$ determined by the electron density $n(x)$ calculated using DFT. Also, $\hbar \Gamma=65.8$ meV has been applied for the damping term[@nanooptik]. The dielectric substrate and superstrate, which in general can be different, are described by $\varepsilon_{\text{s}}(x)$, and the Heavyside step function $\theta$ in the first line makes sure that the dielectric function sufficiently far from the slab equals the correct values in the substrate and superstrate. Hence, it has been assumed that the electron density across the slab does not depend on the kind of substrate and superstrate it is surrounded by. Lastly, the abrupt behaviour assumed for the bound electron term is modelled with the step function $\theta$ in the second line of Eq. .
An example of a dielectric function is seen in Fig. \[fig:density\_epsilon\](b) for a slab width of 1 nm at a wavelength of 775 nm, where again the colored area shows the position of the ion charge, while the shaded areas depict the dielectric substrate and superstrate. The black curve shows the dielectric function, when there is air on both sides of the slab, while the blue curve corresponds to applying glass ($\varepsilon_s=2.25$) as both substrate and superstrate. For a gold slab placed instead on a glass substrate with air as superstrate, the blue curve to the left and the black curve to the right of the slab describe the dielectric function in the glass and air, respectively. The real part of the dielectric function is clearly seen to jump at the slab boundary due to the step function in the second line of Eq. . Although it is difficult to see in the figure, the imaginary part of the dielectric function also jumps across the interfaces. Since the substrate and superstrate are assumed lossless, the imaginary part of the dielectric function is unaffected by these materials.
Mode index of propagating plasmons
==================================
The magnetic field of the SPPs only has a $z$-component and is, for a constant slab width, given by [@slab6] $$\begin{aligned}
\vec{H}_m(\vec{r})=\hat{z}H_m(x,y)=\hat{z}\exp(ik_0\beta_my)H_m(x), \label{eq:H}\end{aligned}$$ where the subscripts $m=\{l,s\}$ indicate that the field and associated complex mode index $\beta$ can be either long-range or short-range, respectively, in agreement with Refs. . In Eq. , $k_0=2\pi/\lambda$ is the free space wavenumber, and $H_m(x)$ is the transverse magnetic field distribution. Both the mode index and the transverse magnetic field depend strongly on $d$, especially for the short-range mode, as will be shown below.
The mode index is calculated by the same type of transfer matrix method as presented in detail in Ref. (see Appendix B for classification of modes).
![Real (solid and dashed lines) and imaginary (dotted and dash-dotted lines) parts of the mode index vs. slab width $d$ at a wavelength of 775 nm. Results are shown for spill-out (SO) included (solid and dash-dotted lines) and neglected (dashed and dotted lines). (a) is for the short-range mode, where the blue and red lines are for slabs surrounded by glass (gg) and air (aa), respectively. (b) is for the short-range mode for a slab surrounded by glass and air (ga), and the inset shows a zoom for $d$ below 1 nm. (c) is for the long-range mode for a slab surrounded by glass (gg), where the real and imaginary parts are shown on the left and right $y$-axes, respectively, as indicated by the arrows. []{data-label="fig:modeindex"}](modeindex){width="8cm"}
Figure \[fig:modeindex\] shows the mode indices as a function of $d$ at a wavelength of 775 nm. For the blue and red curves, the geometric structure is symmetric, while it is asymmetric for the green curves, as indicated in the text above each subfigure. This implies that the magnetic fields associated with the blue and red curves are symmetric and antisymmetric for the long-range and short-range mode, respectively, while the symmetry of the associated fields is broken for the corresponding green curves. For the short-range mode, this will be demonstrated in the next section.
Figure \[fig:modeindex\](a) shows the short-range mode indices for the two symmetric structures, which are in agreement with previous studies [@slab1; @slab2; @slab5] when spill-out is neglected. The real parts are almost unaffected by spill-out, while it plays a significant role for the imaginary parts, as will be elaborated upon below. A similar calculation of the mode index with and without spill-out for gap plasmons propagating in narrow gaps in gold showed that the mode index when including spill-out converges to the refractive index of bulk gold in the limit of vanishing gap width[@spill-out], while neglecting spill-out leads to an unphysically diverging mode index [@narrow_gap1; @gap2]. For plasmons bound to the slab, the mode index when neglecting spill-out also diverges unphysically in the limit of vanishing slab thickness[@slab1; @slab2; @slab5]. This is not the case with spill-out included, as plasmonic modes only exist when the real part of the metal dielectric constant is negative in some region along the direction normal to the slab[@nanooptik]. It is found that the slab has to be of sub-atom thickness ($\sim0.3$ Å) in order for the electron density in the model to become so delocalized that the real part of the dielectric constant is everywhere positive. Hence, with spill-out included the mode index does not diverge in the limit of vanishing slab thickness. Instead, plasmonic modes cease to exist for slabs below a cut-off thickness in the sub-atom range. However, since a gold atom has a diameter of roughly 0.3 nm[@black_gold], we only consider slab widths larger than this value.
For an asymmetric structure with glass as substrate and air as superstrate, the short-range mode index is shown in Fig. \[fig:modeindex\](b). Here it is difficult to see the difference in mode index with and without spill-out when the slab width exceeds 1 nm. Therefore the mode index when neglecting spill-out is only included in the inset showing results for $d$ below 1 nm, where $\beta$ has the same behaviour as for the symmetric structure in Fig. \[fig:modeindex\](a). Although it is hard to see in the figure, the imaginary part is small but non-zero for all slab widths. The mode index has converged when $d=100$ nm, and for such a wide slab the plasmon behaves as if bound to a single interface between glass and gold. [@nanooptik]
The asymmetric structure also supports long-range modes, but only for slab thicknesses above a certain threshold[@slab2]. The long-range mode is mainly bound to the air-gold interface, with a mode index that is lower than the refractive index of the glass substrate. This implies that the normal component of the wavevector becomes real (with a very small imaginary part due to loss in the gold) on the glass side of the structure, leading to a wave propagating in the substrate, thus not a truly bound mode [@leaking1]. Hence, the wave will leak out into the substrate, where conservation of momentum determines the leakage angle [@sondergaard2]. However, if the dielectric constants of the substrate and superstrate are not too different, it is possible to obtain a long-range mode that is truly bound to both interfaces (see e.g. Fig. 3 in Ref. ). The phenomenon of leaky modes can be examined using leakage radiation microscopy, see e.g. Refs. .
Figure \[fig:modeindex\](c) shows the mode index of the corresponding long-range mode for a gold slab surrounded by glass. Without spill-out, the mode index for an ultra-thin slab is very close to the refractive index of the substrate, which implies that the mode is weakly bound. The mode is therefore long-range with most of its field profile located in the dielectric regions, which will be illustrated in the next section. As the slab width increases to 200 nm, the mode index without spill-out has converged to $\sqrt{\varepsilon_{\text{gold}}\varepsilon_{\text{glass}}/(\varepsilon_{\text{gold}}+\varepsilon_{\text{glass}})}$, which is the mode index of a plasmon bound to a single interface between gold and glass [@nanooptik]. In addition, the corresponding short-range mode index when neglecting spill-out in Fig. \[fig:modeindex\](a) converges to the same value for $d=200$ nm (not shown) in agreement with Refs. . With spill-out included, the long-range mode index for small slabs in Fig. \[fig:modeindex\](c) is also close to the refractive index of the substrate, and the imaginary part is very low. As for the short-range mode in Fig. \[fig:modeindex\](a,b), especially the real part of the mode index is almost the same with and without spill-out, as seen by comparing the solid and dashed lines. But importantly, spill-out significantly increases the imaginary part of the mode index, even for slab widths up to 200 nm, as seen by comparing the dotted and dash-dotted lines in Fig. \[fig:modeindex\](c).
To further illustrate the effect of spill-out in a symmetric structure, the ratio between mode indices with and without spill-out is shown in Fig. \[fig:ratio\](a,b) for the short-range and long-range modes, respectively.
![Ratio between mode indices with and without spill-out at a wavelength of 775 nm for a slab surrounded by glass (gg). In (a), the mode is short-range, and the inset shows the real part of the transverse magnetic field for $d=200$ nm, as indicated by the arrow, where the colored area represents the ion charge. In (b), the mode is long-range and the inset shows a zoom for $d$ below 1 nm. []{data-label="fig:ratio"}](ratio_both){width="8cm"}
For both types of modes, the real part of the mode index is almost unaffected by spill-out, as the ratios shown by the solid lines in Fig. \[fig:ratio\] have converged to 1.0004 when $d=50$ nm. For the short-range mode the ratio between the imaginary parts is approximately 3.0 for a slab width of 0.3 nm, while it converges to $\sim$1.2 for $d=200$ nm. The real part of the normalized magnetic field profile when spill-out is included is shown in the inset of Fig. \[fig:ratio\](a) for a slab width of 200 nm, as indicated by the arrow. It behaves as two decoupled plasmons bound to the interfaces between glass and gold, as the field profiles bound to the individual interfaces do not interact for such a wide slab.
For the long-range mode, the corresponding ratio between the imaginary parts is extremely high for small $d$ as seen in the inset in Fig. \[fig:ratio\](b). However, as $d$ increases the ratio decreases monotonically and converges to $\sim$1.2 when $d=200$ nm. This is an important result showing that quantum spill-out increases the imaginary part of the mode index by 20%, even for relatively thick slabs that can readily be fabricated [@slab6; @leaking4] and approach bulk gold. It is highly surprising that spill-out plays such a significant role for wide slabs, as it only modifies the electron density in a region very close to the ion charge. In addition, it is noticed that the ratios between the imaginary parts of the two modes converge to the same value when the slab is wide enough, as in this case the field profiles bound to the individual interfaces are decoupled, similarly to classical models [@slab2; @slab5; @slab6]. Furthermore, the short-range mode indices with and without spill-out in the asymmetric structure in Fig. \[fig:modeindex\](b), converge to the same values as for the long-range mode in Fig. \[fig:modeindex\](c), as both modes behave as bound to a single interface between gold and glass. Hence, spill-out also increases the imaginary part of the mode index by 20% in an asymmetric structure.
Ref. applied a dielectric function analogous to Eq. to study the effect of spill-out on plasmons propagating in a magnesium slab ($r_s=2.66$ Bohr) surrounded by silicon and air. With the present method, the real part of the calculated mode index agrees well with values estimated from Fig. 5 in Ref. , showing quantitative agreement between that paper and the method presented here. Likewise, the mode index calculated in the present paper agrees well with values estimated from Figs. 6 and S9 in Ref. regarding plasmons propagating in gold slabs surrounded by air.
Field profile and plasmonic absorption {#sec:fields}
======================================
Once the mode indices have been computed, the magnetic field from Eq. is calculated using the same transfer matrix method as described in Refs. . Applying the same phase convention as in Ref. , the normalized real part of the short-range transverse magnetic field $H_s(x)$ across a gold slab of 0.3 nm is shown in Fig. \[fig:field\_penetration\](a) at a wavelength of 775 nm.
![(a) Normalized real part of $H_s(x)$ across a slab of width 0.3 nm. (b) shows the decay length of the long-range mode for a gold slab surrounded by glass (gg) with and without spill-out. The inset shows the real part of the magnetic field profiles for a slab of width 0.3 nm as indicated by the arrow. In both (a) and (b), the colored areas represent the ion charge and the wavelength is 775 nm.[]{data-label="fig:field_penetration"}](H_field_penetration){width="8cm"}
The associated imaginary parts of the fields are not shown as they are small compared to the real parts, similarly to classical models [@slab1].
When neglecting spill-out, the slope of the magnetic field, corresponding to the normal component of the electric field, becomes discontinuous across the slab surfaces in agreement with Refs. . With spill-out included, the slope is still discontinuous due to the abrupt change in the bound electron term in Eq. , although it is difficilt to see in Fig. \[fig:field\_penetration\](a). But in the vicinity of the slab surface the field profiles behave more smoothly, and their maximum positions are slightly shifted into the dielectric region. We have checked that the appropriate boundary conditions regarding electromagnetic fields across an interface [@nanooptik] are satisfied. Further away from the slab, the field profiles with and without spill-out become almost identical. Consequently, the decay lengths into the dielectrics, calculated as $1/{\text{Im}}{(k_x)}$, where $k_x=k_0\sqrt{\varepsilon_s-\beta^2}$ is the wavenumber in the $x$-direction, are very similay and both are on the order of a few nm. This illustrates that the short-range mode is strongly bound to the slab, as it decays very rapidly into the dielectrics [@slab5], and thereby has a large part of its field profile located in the slab region. Notice, that as the real part of the mode index is much higher than its imaginary part, the decay length mostly depend on the real part of the mode index. Fig. \[fig:field\_penetration\](a) demonstrates that the short-range magnetic field is antisymmetric for the two symmetric structures shown by the blue and red curves, while this is no longer the case for an asymmetric structure, as shown by the green curves. When the slab width increases, the field profile broadens, as shown in the inset of Fig. \[fig:ratio\](a) for a 200 nm wide slab. For such a wide slab, the field profiles with and without spill-out are almost identical, and both behave as two decoupled plasmons bound to the interfaces between glass and gold.
As mentioned above, the long-range mode is weakly bound. Consequently, the electromagnetic fields for a few-nm slab have decay lengths of several micrometers, as shown for a gold slab surrounded by glass in Fig. \[fig:field\_penetration\](b). The long decay length implies that most of the field profiles are located in the dielectric regions. The field profiles are broader when spill-out is neglected, as also seen in the inset showing the real part of the magnetic fields across a slab of width 0.3 nm, i.e., the same slab as in Fig. \[fig:field\_penetration\](a). Including spill-out effectively implies a broader slab (see Fig. \[fig:density\_epsilon\](a)), which means that the fields become slightly more localized with a shorter decay length. For slabs wider than 3 nm, the decay lengths with and without spill-out are very similar, and both converge to the decay length of a plasmon bound to a single interface between gold and glass (not shown).
As argued above, spill-out plays almost no role for the decay length for slabs of a few nm. On the other hand, it significantly increases the imaginary part of the mode index as shown in Fig. \[fig:ratio\]. This leads us to investigate how spill-out affects the electric field and plasmonic absorption across the slab. First, the electric field is calculated from the magnetic field in Eq. as [@nanooptik] $$\begin{aligned}
\vec{E}_{m}(x,y)=\frac{i}{\omega \varepsilon_0\varepsilon(x,y)}{\vec\nabla \times \big}[\hat{z}H_{m}(x,y)\big]. \label{eq:E}\end{aligned}$$ The electric field is subsequently used to calculate the absorption density defined as $$\begin{aligned}
A_{m}(x,y)=\vert \vec{E}_{m}(x,y)\vert^2 {\text{Im}}{(\varepsilon(x,y))}. \label{eq:abs}\end{aligned}$$ By considering the time average of the Poynting vector, $\langle \vec{S}\rangle=1/2{\text{Re}}(\vec{E}\times \vec{H}^*)$[@nanooptik], it can be shown that conservation of energy implies that the plasmonic absorption and the imaginary part of the mode index are related in the following way $$\begin{aligned}
{\text{Im}}(\beta)=\frac{c\varepsilon_0\int A_m(x,y){\,\mathrm{d}}x}{2\int {\text{Re}}(\vec{E}_m(x,y)\times \vec{H}_m^*(x,y))\cdot \hat{y}{\,\mathrm{d}}x}. \label{eq:relation}\end{aligned}$$ We have checked that this relation is satisfied for both types of modes with and without spill-out. The normalized absorption density is shown in Fig. \[fig:abs\] for the short-range mode across the slab of width 0.3 nm at a wavelength of 775 nm.
![Normalized absorption density for the short-range mode across a gold slab of width 0.3 nm at a wavelength of 775 nm. The solid and dashed lines show the absorption density when spill-out is included and neglected, respectively, where the red and blue lines are for slabs surrounded by air (aa) and glass (gg), respectively. []{data-label="fig:abs"}](Abs_antisym){width="8cm"}
If spill-out is neglected, absorption can only take place in the gold as the surrounding dielectrics are assumed lossless. In this case, the absorption density is almost unaffected by the kind of surrounding dielectric, why Fig. \[fig:abs\] only shows it for a slab surrounded by air. But with spill-out included, strong plasmonic absorption occurs, and the absorption density mostly consists of two narrow peaks located in the dielectric regions close to the interfaces. At these positions, similarly to Refs. , the real part of the dielectric function is zero (at the wavelength 775 nm), while its imaginary part is small but nonzero, which ensures that the peaks in the absorption density are finite. The narrow peaks are found a few Å outside the ion charge, and the same is found for the long-range mode (not shown). The contribution from these peaks leads to enhanced plasmonic absorption, as they are a consequence of electron spill-out, and therefore not found in classical models. For a slab surrounded by glass, the peaks occur slightly closer to the slab, as the real part of the dielectric function has its zero shifted slightly compared to the case with a slab surrounded by air (see Fig. \[fig:density\_epsilon\](b)). The same phenomenon was found in Ref. for a magnesium slab surrounded by silicon and air. The peaks in absorption density due to spill-out were recently discussed in Ref. , where they were found to significantly reduce the reflectance from an ultrasharp groove array in much better agreement with measurements[@black_gold] compared to classical models [@GSP; @optics_multiple].
The increased absorption loss due to spill-out will manifest itself as decreased propagation lengths in fabricated plasmonic structures. Losses in such structures have been studied in, e.g., Refs. , where it was found that the measured propagation length of plasmons propagating in a 70 nm silver film deposited on glass is significantly shorter than the one calculated using classical models[@losses3; @losses4]. Hence, these works together with Ref. also support the finding that a classical model is not sufficient to correctly describe losses occurring in plasmonic waveguides.
Conclusion {#sec:conclusion}
==========
In conclusion, we have applied a quantum mechanical approach and local response theory to study the propagation of plasmons in nanometer-thin gold slabs surrounded by different dielectrics. The effect of spill-out is found to be small on the real part of the mode indices but remarkably increases the corresponding imaginary part, and even for slabs wide enough to approach bulk the increase is 20%. This is explained in terms of enhanced plasmonic absorption mainly taking place at narrow peaks located a few Å outside the ion charge. It is highly surprising that spill-out plays such a significant role for wide slabs, as it only modifies the electron density in a region very close to the ion charge. For slab widths above a few nanometer, the decay length of the fields into the dielectrics is almost unaffected by spill-out, as it mostly depends on the real part of the mode index. Furthermore, in contrast to classical models, the short-range mode index does not diverge in the limit of vanishing slab thickness when spill-out is included. Instead, plasmonic modes cease to exist for slab widths below a cut-off thickness in the sub-atom region.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work is supported by Villum Kann Rasmussen (VKR) center of excellence Quscope.
Appendix A: Calculation of electron density {#appendix-a-calculation-of-electron-density .unnumbered}
===========================================
In this appendix, we discuss in more detail how the electron density is calculated. Within the jellium model it is assumed that the charge of the gold ions is smeared out, such that their charge density is constant within the slab[@jellium1; @jellium2]. The characteristic spill-out, as seen in Fig. \[fig:density\_epsilon\](a), stems from the distribution of free electrons in the vicinity of this positive background. The Kohn-Sham equations [@jellium1] are solved self-consistently within the local density approximation (LDA), [@kohanoff] applying the Perdew-Zunger parametrization [@perdew] for the correlation term. The applied Wigner-Seitz radius for gold is $r_s=3.01$ Bohr[@jellium2].
It is found that 2,500 basis functions on the form $\sin(m\pi(x/L+1/2))$ are sufficient to describe the density for slab widths up to 200 nm. The length $L$ is 1 nm larger than the slab width $d$, and the slab is centered at $x=0$. As in Ref. , the density is said to converge when a variation in Fermi energy between two iterations below $10^{-7}$ Ha is achieved. Furthermore, in the Anderson mixing scheme [@anderson], the mixing parameter $\alpha$ must be below a certain threshold which strongly decreases with $d$. It is found that $\alpha\leq 5\cdot 10^{-4}$ is necessary for slab thicknesses up to 20 nm. The potentials for wider slabs can afterwards be constructed from the potential of the 20 nm slab, as the oscillations in potential near its center are negligible, meaning that the effective potential near the center can be seen as constant. This constant potential is added in the central region of wider slabs.
Appendix B: Classification of plasmonic modes {#appendix-b-classification-of-plasmonic-modes .unnumbered}
=============================================
In this short appendix, we discuss how the plasmonic modes are classified. The mode index is calculated by the same type of transfer matrix method as presented in detail in Ref. . However, only one type of plasmonic mode was studied in that paper, why the classification of modes was not presented there. A structure matrix $\mathcal{S}$ is constructed, which relates the magnetic fields to the left and right of the structure, and a mode index is found when the matrix element $\mathcal{S}_{11}$ is zero. The mode is classified by the sign of $\mathcal{S}_{21}$, where positive and negative signs correspond to long-range and short-range modes, respectively, and $\mathcal{S}_{21}$ is exactly $\pm 1$ for symmetric structures. Expressions for the matrix elements can be found in Refs. . In addition, the $x$-axis is divided into tiny segments, each modelled as having a constant dielectric function. Similarly to Ref. , we find that segments of $2.7\cdot 10^{-4}$ nm are sufficient to avoid discretization errors.
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abstract: 'In a recent paper, Castro (J. Math. Phys., 49, 042501, 2008)[@1] has mentioned that in two papers, Mitra (Found. Phys. Lett., 13, 543, 2002)[@2] and ( MNRAS, 369, 492, 2006)[@3], (i) attempted to show that a neutral point particle has zero gravitational mass, but (ii) his intended proof was faulty and can be “bypassed”. In reality, none of these two papers offered any such proof and on the other hand this proof could be found in (Adv. Sp. Res. 38(12), 2917, 2006)[@4; @5] not considered/cited by Castro. This shows that Castro critisized Mitra’s “proof” without really going through it. We briefly revisit this “proof” to show that it is indeed correct and does not suffer from perceived shortcomings. It is reminded that Arnowitt, Deser & Misner’s (ADM) previous work[@6] work suggested that the gravitational mass of a [*neutral point particle*]{} is zero. It is pointed out that that this result has important implications for Castro’s work. Further the mass formula $M=\int 4\pi \rho r^2 dr$ used by Castro may not be valid for the radial gauge he used, and the “spacetime void” inferred by Castro may be an artifact of the discontinuos radial gauge used by him.'
author:
- Abhas Mitra
date:
-
-
title: 'Comments on “The Euclidean gravitational action as black hole entropy, singularities, and spacetime voids”'
---
=10000
Introduction
============
The general general relativistic solution for the spacetime around a point mass may be given by[@7; @8] ds\^2=(1 - \_0/ R) dt\^2 -(1-\_0/ R)\^[-1]{} dr\^2 - R(r)\^2(d\^2 + \^2d\^2) where $r$ is a general radial coordinate. The variable $R(r)$ in the angular part is the circumference or area coordinate which means that the invariant area of a two sphere around the center of symmetry is given by $4 \pi R^2$ independent of the precise form of $R(r)$. This solution involves an integration constant $\alpha_0$ and comparison with Newtonian solutions suggest that $\alpha_0$ is related to the gravitational mass of the “point mass” in the following way: \_0 = [2 GM\_0c\^2]{} If we take $G=c=1$, we will have $\alpha_0 = 2 M_0$. The subscript $0$ here denotes “point mass” vis-a-vis an [*extendended*]{} static spherical body of gravitational mass $M$ and corresponding $\alpha = 2GM/ c^2$.
Choice of various forms of the function $R(r)$ corresponds to various choices of radial gauge and accordingly one can have various solutions. The simplest choice of the gauge here is r = R And this choice was due to Hilbert: ds\^2=(1 - \_0/ R) dT\^2 -(1-\_0/ R)\^[-1]{} dR\^2 - R\^2(d\^2 + \^2d\^2) though, in text books, the above solution is ascribed to Schwarschild. This Hilbert gauge is most natural and physical in the sense that the invariant surface area of a sphere around the point of symmetry is $4 \pi R^2$ rather than $4\pi r^2$. In this gauge, the “point particle”, the source of gravity, is located at $R=0$ and its [*surface area*]{} is $A_p =0$. Note, the time interval, for this particular form of the metric has been designated by $dT$.
The Brillouin radial gauge, on the other hand, is given by[@7] R(r) = r + \_0 while the actual Schwarzschild gauge is given by[@8] R(r)\^3 = r\^3 + \_0\^3; or r\^3 = R\^3 -\_0\^3 In these latter cases, the point mass is believed to be sitting at the origin of the coordinate system, i.e., at $r=0$, and in such pictures, one has R(0) = \_0 so that, [*the point mass has an invariant surface area*]{} of A\_p = 4 R(0)\^2 = 4 \_0\^2 >0 ! if one would really assume that \_0 >0 So, for all such radial gauges, one would arrive at a rather grotesque picture where the [*point mass has a finite surface are*]{}! And only for the Hilbert gauge, one obtains the physical scenario where a point mass has zero surface area. Thus from such a view point, only the Hilbert gauge (which however is ascribed to Schwarzschild in the literature) leads to a physically meaningful solution for a “point mass” (atleast if one would indeed assume $\alpha_0 >0$). In fact Castro[@1] acknowledges this point: “How is it [*possible*]{} for a point mass sitting at $r=0$ to have a non-zero area .. and a [*zero*]{} volume simultaneously?”
Hilbert’s radial gauge however gives rise to a finite horizon area $A_H = 4 \pi \alpha_0^2$ if one would assume the [*integration constant*]{} $\alpha_0 >0$. However, Castro would like to consider a situation where both $A_H \to 0$, $A_p \to 0$. Thus he chose the following radial gauge which seems to be some sort of modification of Brillouin’s gauge:
R(r) = r + \_0 (r); r = R -\_0 (r) where $\Theta(r)$ is the Heaviside step function.
Castro also correctly points out that for a point particle scalar curvature must be a Dirac-$\delta$ function rather than ${\cal R} =0$ as is mentioned in the literature.
It may however be reminded that the purpose of this “comment” is not to review or critisize Castro’s work in general except for some passing comments. On the other hand, we would like to focus attention on Castro’s criticism of Mitra’s work in pp.11-12. Here Castro implied that, Mitra’s purported proof that $\alpha_0 =0$ (for a neutral point mass) is incorrect. First it is pointed out that Mitra’s proof to this effect is not at all presented in the papers cited[@2; @3] by Castro. On the other hand, Castro did not cite the papers[@4; @5] where such a proof was actually presented. First, this shows that Castro may not have at all gone through the said “proof” before dismissing it. For the benefit of the readers, it will be shown that this “proof” is correct and Castro’s related criticism is misplaced. Further, it is emphasized that this proof removes much of the concern of Castro’s work. It will be reminded that previous work of Arnowitt, Deser and Misner[@6] too suggests that a neutral point particle should have $M_0 =0$. Some basic error in Castro’s work would be also pointed.
Invariance of Proper 4-Volume
=============================
For any curvilinear coordinate transformations, one has d= d\^4 x =Invariant where $g = \det g_{ik}$ For instance see. Eq.(2.5.6), pp. 36 of Carmeli[@9] or pp. 99 of Weinberg[@10] or pp. 223 of Landau-Lifshitz[@11]. This invariance has got nothing to do with choice of radial or any other gauge impllied by Castro, and on the other hand, it is a property of tensor transformations.
If the determinant corresponging to the spatial section is $h$, one would have -g = h g\_[00]{} so that d\^4 x = ( dx\^1 dx\^2 dx\^3) ( dx\^0) = d[V]{} dwhere $d{\cal V}$ is the proper 3-volume element and $d\tau$ is the proper time element. Thus the invariant d\^4x = d= Proper Spacetime Volume Element
In fact it is valid in arbitrary dimentions too. However, if one is concerned with a situation which involves only spatial dimensions so that $g$ is positive then $-g$ of Eq.(11) should be replaced by $+g$. For an illustration see pp.170-171 of Hartle[@12]. Now by using this invariance of $d\Omega$, we revisit our proof that a neutral point particle has zero gravitational mass this simple straight forward proof below:
The Proof Overlooked by Castro
------------------------------
First consider the fact that for the diagonal vacuum Hilbert metric(4), one has g = -R\^4 \^2Now let us recall the Eddington-Finkelstein metric form of the vaccum Hilbert metric [@13; @14] $$\begin{aligned}
ds^2 = \left(1 - {\alpha_0 \over R}\right)dT_*^2 \pm {2\alpha_0\over R} dT_* dR \\ \nonumber
- \left(1 +{\alpha_0\over R}\right)dR^2 - R^2(d\theta^2 + \sin^2\theta d\phi^2)\end{aligned}$$ where T\_\*=T\_0 ( [R\_0]{} -1), R\_\*=R, \_\* =; \_\* =The corresponding metric coefficients are $$\begin{aligned}
g_{{T_*} R_*}=-(1-\alpha_0/R), ~g_{R_* R_*}=(1+\alpha_0/R) \\ \nonumber
g_{{T_*} R_*}=g_{R_* T_*}
=\alpha_0/R\end{aligned}$$ In this case too, the determinant is same as $g$: g\_\* = -g\_[\_\* \_\*]{} g\_[\_\* \_\*]{}(g\^2\_[T\_\* R\_\*]{} -g\_[T\_\* T\_\*]{} g\_[R\_\* R\_\*]{}) = -R\^4 \^2 =g Now let us apply the principle of [*invariance of 4-volume*]{} for the coordinate systems ($t$, $R$, $\theta$, $\phi$) and ($t_*$, $R_*$, $\theta_*$, $\phi_*$): d= dT\_\* dR\_\* d\_\* d\_\* = dT dR d dSince $g_*= g, ~ R_* =R, ~ \theta_* =\theta, ~\phi_* =\phi$, we obtain dT\_\* = dT But from Eq.(17), we have, dT\_\* = dT + [\_0R -\_0]{} Eqs.(21) and (22) can be satisfied iff \_0 =0 so that, for neutral “point mass” one has a zero gravitational mass: $M_0 =0$.
It is important to note that if instead of a “point particle” we would be considering a [*finite*]{} static spherical body of gravitational mass $M$, the above derivation would be invalid even though by Birchoff’s theorem, the exterior spacetime would still be described by metric. This is so because, in such a case, exterior spacetime would be described only by metric(4) and [*not*]{} by metric(16). This happens because the coordinate transformation(17) is obtained by integrating the [*vacuum*]{} null geodesic all the way from $R=0$. And this is allowed only when the spacetime is indeed due to a point mass and not by a finite body. For a finite body, the [*interior metric would be different*]{} from what is indicated by Eq(4). and hence metric(16) which presumes the interior to be vacuum cannot be invoked.
Since, for a point mass $\alpha_0 =0$, the horizon area is zero and it contracts to the origin of the coordinate system without the application of any quantum mechanics or quantum gravity. Note that the original motivation of Castro too was see that horizon has zero area and lies at the origin of the coordinate system.
In the first line of pp.12, Castro[@1] writes that “the measure $4\pi R^2 dR~dt$ is not invariant when we change the radial gauges...”
Again this statement is both confusing and incorrect. It must be borne in mind that if there is any change in the spatial coordinates (due to change of gauge or other reasons), the change actually takes place in “spacetime” description and hence the time coordinate too may undergo some transformation. For instance, let us consider metrics (1) and (4) related through the coordinate transformations (5) or (6). It transpires that for all such cases, the metric determinant is same: g\_H =g\_B= g\_S = -R\^4 \^2 Then invariance of proper 4-volume element means dt dr d d = dT dR d dThus the Hilbert time element is related to other time elements in the following way: dt = [dRdr]{} dT For the Brillouin case, from Eq.(5), see that, $dR/dr =1$ and hence $dt =dT$. But, from Eq.(6), note that this is not so in the Schwarzschild case:
dt = [r\^2R\^2]{} dT
Inevitability of this result
============================
We know that the gravity at the horizon is supposed to be “so strong that even lighy cannot escape it”. Let us first measure the strength or weakness of gravity by tKretschmann scalar, and for the given problem, one has K = [12 \_0\^2R\^6]{} Note that the denominator here contains area coordinate $R$ rather than any general $r$ and $K =\infty$ only at $R=0$. This suggests that the genuine physical singularity lies at $R=0$ rather than at any general $r=0$. In turn, this tells that, the source of gravity is at $R=0$ rather than at arbitrary $r=0$. And this is again a strong argument that only the Hilbert gauge gives a physically meaningful picture. And of course, since $K$ is an invariant like $d\Omega$, it is independent of choice of coordinates and gauges.
At the EH $R=\alpha_0$, one finds K\^[EH]{} = [12\_0\^4]{}
But now note that, if the integration constant $\alpha_0$ would be considered to be a free parameter, one can let $K$ to be arbitrarily small. For instance, for sufficiently large value of $\alpha_0$, one may have a situation where K\^[EH]{} K\^[Earth]{} But if the weak gravity of Earth, Sun or Moon or even a Neutron star cannot trap light, how, a still waeker gravity would do so? Further, one can let K\^[EH]{} 0 when \_0And this would mean that [*light would get trapped in zero gravity*]{}! Obviously, this would be unphysical, and consequently, $\alpha_0$ cannot be a free parameter. And as shown above, $\alpha_0 =0$ so that, actually K\^[EH]{} = and this is reason that even light gets trapped at the EH.
Upper Limit on Proper Acceleration
----------------------------------
In his paper, Castro has correctly mentioned that quantum gravity should lead to an upper limit on proper acceletation: a\_[Planck]{} = [c\^2l\_P]{} where $l_P$ is Planck length. But in the spacetime around a “point particle”, the proper acceleration experienced by a test particle is [@12] (pp. 459) a= = [ G M\_0R\^2 ]{} where $a^i$ is the 4-acceleration. Again note that since $a$ is a scalar/invariant, this result is independent of coordinates and gauges. At the EH, $R=\alpha_0$, and hence, one would have a\^[EH]{} = Since $a$ is a physical scalar and is measurable, one expects that, it can blow up only at the genuine physical singularity which lies at $R=0$ and where $K=\infty$. This, in turn, demands that coordinate radius of the horizon $\alpha_0 =0$ so that the horizon and the central singularity are synonymous. Recall that we have already obtained this result $\alpha_0 =0$ from the invariance of $d\Omega$.
Vanishing of Curvature Scalar
-----------------------------
By definition, for a point particle, the density and hence components of the energy momentum tensor $T_i^k$ would be singular at the location of the point particle. Since, $K^{EH}$ diverges at $R=0$ rather than at an arbitrary $r=0$, we expect a Dirac-$\delta$ singularity in the curvature scalar at $R=0$[@15] (R =0)= -(8 G/c\^4) T\_i\^i = - 4 GM\_0(R) /R\^2 c\^2 Depending on the model of the stress energy tensor one adopts, the numerical factors may differ in the foregoing expression[@16]. However, if one would calculate ${\cal R}$ using the Hilbert metric(4), one would obtain =0 One necessary condition for the mutual consistentcy of Eqs. (36) and (37) is $M_0 =0$.
Interior Solution
=================
If the point particle is question would be replaced by an extended distribution of mass energy, then, within this distribution, one must consider suitable [*interior solution*]{}. But the metrics (1) or (4) [*do not represent*]{} such interior solutions. As far as static spherically symmetric interior solutions are concerned, all discussions are based on Hilbert radial gauge $r =R= cirumference ~ coordinate$. In literature, however, this is known as “Schwarzschild Interior Solution”. It is only for this unique radial gauge, the gravitational mass of the body is given by M = 4 (R) R\^2 dR To the best of the knowledge, no corresponding study has ever been made for other choices of radial gauge. And in general, if $r \neq R$, there is no known formula for $M$. Thus, in general, M 4 (r) r\^2 dr if $r$ is an arbitrary radial coordinate. In his paper, Castro has replaced the “point particle” by a mass distribution extending upto $r=\infty$: (r) = M [e\^[-r\^2/4 \^2]{}(4 \^2)\^[3/2]{}]{} and his $r \neq R$. Yet, he has used the mass formula appropriate only for Hilbert gauge $r=R$.
Spacetime Void?
===============
Castro[@1] considers a continuous classical spacetime metric. He does not explicitly use any quantum gravity where one might expect some spacetime irregularity on the scale of $l_P$. Yet Castro infers some “spacetime void” in the spacetime around a point particle. This might be due to the [*discontinuos*]{} radial gauge (10) used by Castro where = 1 + \_0 (r) and from Eq.(26), we see, dt\_[Castro]{} = dT (1 + \_0 (r) ) However, if the result $\alpha_0 =0$ would be taken into consideration, such strange notion of [*spacetime void*]{} would get naturally eliminated.
Conclusions
===========
$\bullet$ The mass formula used by Castro is inappropriate for any radial gauge where $r \neq R$.
$\bullet$ Invariance of proper 4-volume is a propery of tensor calculus and coordinate transformation. And this invariance indeed leads to $M_0 =0$ (for a neutral point particle). And when one recognizes this result, for a point particle [*all radial gauges become the same Hilbert gauge*]{}. Since the invariant surface area of two-spheres symmetric around the centre of symmetry, is $A = 4 \pi R^2$ (rather than $4 \pi r^2$), then it is most matural that the source of gravity is located at $R=0$ rather than at any arbitrary $r=0$. And this demands $M_0 =0$.
Long back, Arnowitt, Deser and Misner[@6] noted that
“ Thus as the interaction energy grows more negative, were a point reached where the total energy vanished, there could be no further interaction energy, in contrast to the negative infinite self-energy of Newtonian theory. General relativity effectively replaces $m_0$ by $m$ in the interaction term: $m=m_0 - (1/2) G m^2/\epsilon$. Solving for $m$ yields $m = G^{-1} [-\epsilon + (\epsilon^2 + 2 G m_0 \epsilon)^{1/2}]$, [ which shows that ${\mathbf m \to 0}$ as $\epsilon \to 0$]{}”.
“The correct result ${\mathbf m=0}$, thus indicates the lack of validity of the perturbation approach”[@1]. Here $\epsilon$ may be viewed as the radius of a collapsing body having a “bare mass” $m_0$ and a gravitational mass $m$ near the singularity.
What these authors mean is the following thing:
The fundamental source of “mass energy” is the fundamental interactions like electromagnetic, strong and weak. Gravitation is structure of spacetime and is not a fundamental interaction in the above sense. However gravity couples to all mass energies and produces “dressing” in the form of negative self-gravitational energy $E_g$. As the object tends to become a “point particles”, $E_g \to unbounded$ and thus tends to nullify the entire bare mass. If the body would be neutral with nil electric, weak or strong “charge”, the final gravitational mass of the resultant “point particle” would be zero.
So, it is the infinite non-linearity of gravity and consequent growth of (negative) self-gravitational energies which may be leading to a net dressed (i.e., gravitational) massed energy which is nil.
Thus the comment by Castro that Mitra’s proof is incorrect and can be “bypassed” is invalid.
Then the gauge used by Castro too becomes none other than Hilbert gauge. Then both the horizon and the point particle have zero surface area: $A =A_p =0$. And this is what Castro wanted to be. In such a case, the mass formula, used by him would become appropriate. And the [*spacetime void*]{} inferred by him in a purely classical theory based on continuous metric would vanish too.
[99]{} C. Castro, J. Math. Phys., 49, 042501 (2008) A. Mitra,Found. Phys. Lett., 13, 543 (2000) A. Mitra, Mon. Not. R. Astron. Soc., 369, 492 (2006) A. Mitra, Advan. Sp. Res., 38, 2917 (2006) A. Mitra, [*Proc. XIth Marcel-Grossmann Conf. On General Relativity*]{}, Vol. 3, pp.1968 (World Sc., Singapore, 2008) R. Arnowitt, S. Deser, & C.W. Misner, in [*Gravitation: an introduction to current research*]{}, Louis Witten ed. (Wiley 1962), chapter 7, pp 227 (gr-qc/0405109), also, PRL, 4(7), 375 (1960)
L. Abrams, Can. J. Phys., 67, 919 (1989) S. Antoci & D.E. Liebscher, gr-qc/012084
M.C. Carmeli, [*Classical Fields: General Relativity & Gauge Theory*]{}, (World Scientific, Singapore, 1993)
S. Weinberg, [*Gravitation and Cosmology: Principles and Applications of General Theory of Relativity*]{}, (John Wiley, New York, 1972) L.D. Landau and E.M. Lifshitz, [*The Classical Theory of Fields*]{}, (Pergamon Press, Oxford, 1975)
J.B. Hartle, [*Gravity: An Introduction to Einstein’s General Relativity*]{}, (Pearson, Delhi, 2005) L.D. Landau and E.M. Lifshitz, [*The Classical Theory of Fields*]{}, (Pergamon Press, Oxford, 1962)
H. Stephani, [*General Relativity: An introduction to the theory of gravitational field*]{} (Cam. Univ. Press 1996) F.R. Tangherlini, PRL, 6(3), 147 (1961) J.M. Heinzle & R. Steinbauer, JMP, 43(3), 1943 (2002)
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abstract: 'We report the successful manufacture and characterization of a microwave resonant cylindrical cavity made of bulk MgB$_2$ superconductor ($T_c \approx 38.5~$K), which has been produced by the Reactive Liquid Mg Infiltration technique. The quality factor of the cavity for the TE$_{011}$ mode, resonating at 9.79 GHz, has been measured as a function of the temperature. At $T=4.2$ K, the unloaded quality factor is $\approx 2.2 \times 10^5$; it remains of the order of $\times 10^5$ up to $T \sim 30$ K. We discuss the potential performance improvements of microwave cavities built from bulk MgB$_2$ materials produced by reactive liquid Mg infiltration.'
address:
- 'EDISON SpA R & D, Foro Buonaparte 31, I-20121 Milano (Italy)'
- 'CNISM and Dipartimento di Scienze Fisiche e Astronomiche, Università di Palermo, Via Archirafi 36, I-90123 Palermo (Italy)'
author:
- 'G. Giunchi'
- 'A. Agliolo Gallitto, G. Bonsignore, M. Bonura, M. Li Vigni'
title: 'Superconducting Microwave Cavity Made of Bulk MgB$_2$'
---
The very low surface resistance of superconducting materials makes them particularly suitable for designing high-performance microwave (mw) devices, with considerably reduced sizes. The advent of high-temperature superconductors (HTS) further improved the expectancy for such applications, offering a potential reduction of the cryogenic-refrigeration limit, with respect to low-temperature superconductors. A comprehensive review on the mw device applications of HTS was given by Lancaster [@lanc]. Among the various devices, the superconducting resonant cavity is one of the most important applications in the systems requiring high selectivity in the signal frequency, such as filters for communication systems [@pand], particle accelerators [@padam; @collings], equipments for material characterization at mw frequencies [@lanc; @zhai].
Nowadays, one of the commercial applications of HTS electronic devices are planar-microstrip filters for transmission line, based on YBa$_2$Cu$_3$O$_7$ thin or thick films [@lanc; @hein], whose manufacturing process allows having an high degree of device miniaturization. Nevertheless, the need of high performance, in many cases, overcomes the drawback of the device sizes, as, e.g., in the satellite-transmission systems, radars, particle accelerators, and demands for cavities with the highest quality factors. Since the discovery of HTS, several attempts have been done to manufacture mw cavities made of these materials in bulk form [@pand; @zaho; @lanc92]; however, limitations in the performance were encountered. Firstly, because of the small coherence length of HTS, grain boundaries in these materials are weakly coupled giving rise to reduction of the critical current and/or nonlinear effects [@golo], which worsen the device performance; furthermore, the process necessary to obtain bulk HTS in a performing textured form is very elaborated. For these reasons, in several applications, such as particle accelerators and equipments for mw characterization of materials, most of the superconducting cavities are still manufactured by Nb, requiring liquid helium as refrigerator.
Since the discovery of superconductivity at 39 K in MgB$_2$ [@naga], several authors have indicated this material as promising for technological applications [@collings; @bugo; @HeinProc]. Indeed, it has been shown that bulk MgB$_2$, contrary to oxide HTS, can be used in the polycrystalline form without a significant degradation of its critical current [@bugo; @HeinProc; @larbalestier]. This property has been ascribed to the large coherence length, which makes the material less susceptible to structural defects like grain boundaries. Actually, it has been shown that in MgB$_2$ only a small amount of grain boundaries act as weak links [@Samanta; @Rowell; @Khare; @agli2]. Furthermore, MgB$_2$ can be processed very easily as high-density bulk material [@giun03], showing very high mechanical strength. Due to these amazing properties, MgB$_2$ has been recommended for manufacturing mw cavities [@collings; @taji], and investigation is carried out to test the potential of different MgB$_2$ materials for this purpose. However, papers discussing the realization and/or characterization of mw cavities made of MgB$_2$ have not yet been reported.
Recently, we have investigated the mw response of MgB$_2$ samples prepared by different methods, in the linear and nonlinear regimes [@agli2; @agli]. Our results have shown that the residual surface resistance strongly depends on the preparation technique and the purity and/or morphology of the components used in the synthesis process. In particular, the investigation of small plate-like samples of MgB$_2$ prepared by Reactive Liquid Mg Infiltration (RLI) process, has highlighted a weak nonlinear response, as well as relatively small values of the residual surface resistance. Furthermore, bulk samples produced by RLI maintain the surface staining unchanged for years, without controlled-atmosphere protection. This worthwhile property is most likely related to the high density, and consequently high grain connectivity, achieved with the RLI process, as well as to the small and controlled amount of impurity phases [@giun-physicaC]. On the contrary, samples prepared by other techniques, though exhibiting lower values of the residual surface resistance, need to be kept in protected atmosphere to avoid their degradation. These interesting results have driven us to build a mw resonant cavity using MgB$_2$ produced by the RLI process. In this work, we discuss the properties of the first mw resonant cavity made of bulk MgB$_2$.
As the first attempt to apply the MgB$_2$ to the cavity-filter technology, we have manufactured a simple cylindrical cavity and have investigated its microwave response in a wide range of temperatures. All the parts of the cavity, cylinder and lids, are made of bulk MgB$_2$ with $T_c \approx 38.5$ K and density $\approx 2.33~\mathrm{g/cm^3}$. The MgB$_2$ material has been produced by the RLI process [@giun03; @giun-cryo06], which consists in the reaction of B powder and pure liquid Mg inside a sealed stainless steel container. In particular, the present cylindrical cavity (inner diameter 40 mm, outer diameter 48 mm, height 42.5 mm) was cut by electroerosion from a thicker bulk MgB$_2$ cylinder prepared as described in Sec. 4.3 of Ref. [@giun-cryo06], internally polished up to a surface roughness of about 300 nm. A photograph of the parts, cylinder and lids, composing the superconducting cavity is shown in Fig. 1.
![Photograph of the cylinder and the lids composing the bulk MgB$_2$ cavity. The holes in one of the lids are used to insert the coupling loops.[]{data-label="cavity"}](cavity_color.eps){width="60.00000%"}
As it is well known, resonant cylindrical cavities support both TE and TM modes. In the TE$_{01n}$ modes, the wall currents are purely circumferential and no currents flow between the lids and the cylinder, requiring no electrical contact between them; for this reason, the TE$_{01n}$ are the most extensively used modes. The TE$_{01n}$ modes are degenerate in frequency with the TM$_{11n}$ modes and this should be avoided to have a well defined field configuration. In order to remove the degeneracy, we have incorporated a “mode trap" in the form of circular grooves (1 mm thick, 2 mm wide) on the inside of the cylinder at the outer edges. This shifts the resonant frequency of the TM$_{11n}$ modes downwards, leaving the TE$_{01n}$ modes nearly unperturbed.
Two small loop antennas, inserted into the cavity through one of the lids, couple the cavity with the excitation and detection lines. The loop antenna was constructed on the end of the lines, soldering the central conductor to the outer shielding of the semirigid cables.
The ratio between the energy stored in the cavity and the energy dissipated determines the quality factor, $Q$, of the resonant cavity. When the cavity is coupled to an external circuit, besides the power losses associated with the conduction currents in the cavity walls, additional losses out of the coupling ports occur. The overall or loaded $Q$ (denoted by $Q^L$) can be defined by $$\frac{1}{Q^L}=\frac{1}{Q^U}+\frac{1}{Q^R}\,,$$ where $Q^U$ is the so-called unloaded $Q$, determined by the cavity-wall losses and $Q^R$ is due to the effective losses through the external coupling network.
The resonant frequency and unloaded quality factor of a cylindrical cavity resonating in the TE$_{01n}$ mode are given by [@lanc; @lanc92] $$f_{01n} = \frac{1}{2 \pi \sqrt{\epsilon \mu}}
\sqrt{\left(\frac{n\pi}{d}\right)^2+\left(\frac{z_{01}}{a}\right)^2}\,,$$
$$Q_{01n}^U = \frac{1}{R_s}\,\sqrt{\frac{\mu}{\epsilon}}
\frac{\left[(z_{01}d)^2 + (n \pi a)^2\right]^{3/2}}{2z_{01}^2 d^3
+ 4 n^2\pi ^2 a^3}\,,$$
where $\mu$ and $\epsilon$ are the permeability and dielectric constant of the medium filling the cavity; $a$ and $d$ are the radius and length of the cavity; $R_s$ is the surface resistance of the material from which the cavity is built; $z_{01} = 3.83170$ is the first zero of the derivative of the zero-order Bessel function.\
$Q^U$ can be determined by taking into account the coupling coefficients, $\beta_1$ and $\beta_2$ for both the coupling lines; these coefficients can be calculated by directly measuring the reflected power at each line, as described in Ref. [@lanc], Chap. IV. Thus, $Q^U$ can be calculated as $$Q^U = (1+\beta_1+\beta_2)Q^L\,.$$
The frequency response of the cavity has been measured in the range of frequencies 8 $\div$ 13 GHz by an *hp*-8719D Network Analyzer. Transmission by two probes has been successfully used for measuring the loaded quality factor in a wide range of temperatures. Among the various modes detected, two of them have shown the highest quality factors; at room temperature and with the cavity filled by helium gas, the resonant frequencies of these modes are 9.79 GHz and 11.54 GHz; according to Eq. (2), they correspond to the TE$_{011}$ and TE$_{012}$ modes. The coupling coefficients $\beta_1$ and $\beta_2$ have been measured as a function of the temperature; they result $\approx 0.2$ at $T =
4.2$ K and reduce to $\approx 0.05$ when the superconductor goes to the normal state. At $T=4.2$ K (without liquid helium inside the cavity) the unloaded quality factors, determined by using Eq. (4), are $Q_{011}^U\approx 220000$ and $Q_{012}^U\approx
190000$; both decrease by a factor $\approx 20$ when the material goes to the normal state.
Fig. \[quality\] shows the temperature dependence of the measured (loaded) and the calculated (unloaded) $Q$ values for the TE$_{011}$ mode, $Q_{011}^L$ and $Q_{011}^U$; in the same plot (right scale) it is shown the mw surface resistance deduced from $Q_{011}^U$ using Eq. (3). As one can note, the quality factor maintains values of the order of $10^5$ up to $T \approx 30$ K and reduces by a factor $\approx 20$ at $T=T_c$.
![Temperature dependence of the loaded and unloaded quality factor, $Q_{011}^L$ and $Q_{011}^U$, (left scale); mw surface resistance $R_s$ deduced from $Q_{011}^U$ (right scale).[]{data-label="quality"}](QvsT_color.eps){width="65.00000%"}
The results of Fig. \[quality\] have been obtained at low input power level ($\approx -15$ dBm). In order to reveal possible nonlinear effects, we have investigated the TE$_{011}$ resonant curve at higher power levels. In this case, to avoid possible heating effects, the measurements have been performed with the cavity immersed into liquid helium. At input power level of $\approx 15$ dBm, we have observed that the quality factor reduces by about 10%, indicating that, at these input power levels, nonlinear effects on the surface resistance are weak.
Our results show that MgB$_2$ produced by RLI is a very promising material for building mw resonant cavities. We have obtained a quality factor higher than those reported in the literature for mw cylindrical cavities manufactured by HTS, both bulk and films [@pand; @hein; @zaho; @lanc92]. Moreover, $Q$ takes on values of the order of $10^5$ from $T=4.2$ K up to $T\approx 30$ K, temperature easily reachable by modern closed-cycle cryocoolers.
We would remark that this is the first attempt to realize a superconducting cavity made of bulk MgB$_2$; we expect that the performance would improve if the cavity were manufactured with material produced by liquid Mg infiltration in micrometric B powder. The present cavity is made of MgB$_2$ material obtained using crystalline B powder with grain mean size $\approx
100~\mu$m. On the other hand, previous studies have shown that the grain size of the B powder, used in the RLI process, affects the morphology [@giun-IEEE-06] and the superconducting characteristics [@agli; @giun-IEEE-06] of the material, including the mw properties.
Investigation of the microwave response of bulk MgB$_2$ obtained by the RLI method has been performed in the linear and nonlinear regimes [@agli; @agli2]. In the linear regime, we have measured the temperature dependence of the mw surface impedance [@agli] at 9.4 GHz; the results have shown that the sample obtained using microcrystalline B powder ($\approx 1~\mu$m in size) exhibits smaller residual surface resistance ($< 0.5~\mathrm{m} \Omega$) than those measured in samples prepared by crystalline B powder with larger grain sizes [@agli]. Since the residual surface resistance obtained from the $Q_{011}^U$ data is $R_s(4.2~\mathrm{K})\approx 3.5~\mathrm{m}\Omega$, we infer that by using microcrystalline B powder in the RLI process the $Q$ factor could increase by one order of magnitude.
In the nonlinear regime (input peak power $\sim 30$ dBm), we have investigated the power radiated at the second-harmonic frequency of the driving field [@agli2]. Since it has been widely shown that the second-harmonic emission by superconductors at low temperatures is due to nonlinear processes in weak links [@golo; @samoilova; @lee], these studies allow to check the presence of weak links in the samples. Our results have shown that MgB$_2$ samples produced by RLI exhibit very weak second-harmonic emission at low temperatures. In particular, the sample obtained using microcrystalline B powder does not show detectable second-harmonic signal in a wide range of temperatures, from $T=4.2$ K up to $T \approx 35$ K [@agli2]. So, we infer that eventual nonlinear effects in the cavity response can be reduced by using microcrystalline B powder.
Because of the shorter percolation length of the liquid Mg into very fine B powder ($1~\mu$m in size), the production of massive MgB$_2$ samples by RLI using microcrystalline B powder turns out to be more elaborated. In this work, we have devoted the attention to explore the potential of bulk MgB$_2$ materials prepared by RLI for manufacturing mw resonant cavities; work is in progress to improve the preparation process in order to manufacture large specimens using microcrystalline B powder.
In summary, we have successfully built and characterized a mw resonant cavity made of bulk MgB$_2$. We have measured the quality factor of the cavity for the TE$_{011}$ mode as a function of the temperature, from $T=4.2$ K up to $T \approx 45$ K. At $T=4.2$ K, the unloaded quality factor is $Q_{011}^U \approx 2.2 \times
10^5$; it maintains values of the order of $10^5$ up to $T \sim
30$ K and reduces by a factor $\approx 20$ when the superconductor goes to the normal state. To our knowledge, these $Q$ values are larger than those obtained in HTS bulk cavities in the same temperature range. The results show that the RLI process provides a useful method for designing high-performance mw cavities, which may have large scale application. We have also indicated a way to further improve the MgB$_2$ mw cavity technology.\
\
The authors acknowledge Yu. A. Nefyodov and A. F. Shevchun for critical reading of the manuscript.
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abstract: |
We construct quasiequilibrium sequences of black hole-neutron star binaries in general relativity. We solve Einstein’s constraint equations in the conformal thin-sandwich formalism, subject to black hole boundary conditions imposed on the surface of an excised sphere, together with the relativistic equations of hydrostatic equilibrium. In contrast to our previous calculations we adopt a flat spatial background geometry and do not assume extreme mass ratios. We adopt a $\Gamma=2$ polytropic equation of state and focus on irrotational neutron star configurations as well as approximately nonspinning black holes. We present numerical results for ratios of the black hole’s irreducible mass to the neutron star’s ADM mass in isolation of $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS} =1$, 2, 3, 5, and 10. We consider neutron stars of baryon rest mass $M_{\rm B}^{\rm NS}/M_{\rm
B}^{\rm max} =$ 83% and 56%, where $M_{\rm B}^{\rm max}$ is the maximum allowed rest mass of a spherical star in isolation for our equation of state. For these sequences, we locate the onset of tidal disruption and, in cases with sufficiently large mass ratios and neutron star compactions, the innermost stable circular orbit. We compare with previous results for black hole-neutron star binaries and find excellent agreement with third-order post-Newtonian results, especially for large binary separations. We also use our results to estimate the energy spectrum of the outgoing gravitational radiation emitted during the inspiral phase for these binaries.
author:
- Keisuke Taniguchi
- 'Thomas W. Baumgarte'
- 'Joshua A. Faber'
- 'Stuart L. Shapiro'
date: 'October 24, 2007'
title: 'Quasiequilibrium black hole-neutron star binaries in general relativity'
---
Introduction
============
Coalescing compact binaries composed of neutron stars and/or black holes are among the most promising sources of gravitational waves for ground-based laser interferometers [@LIGO; @GEO; @TAMA; @VIRGO]. Observations of short gamma-ray bursts (SGRBs) by the [*Swift*]{} and [*HETE-2*]{} satellites (see, e.g. [@Barge06] and references therein) suggest that their central engine may well be the merger remnant of compact binaries that contain a neutron star, namely the remnants of black hole-neutron star (BHNS) binaries or binary neutron star (BNS) systems [@ShibaT06; @FaberBST06; @PriceR06; @OechsJ06]. Motivated by these considerations, significant theoretical effort has gone into the modeling of these binaries.
So far, most studies of BHNS binaries have been performed within the framework of Newtonian gravity in either some or all aspects of the calculation (see, e.g. [@Chand69; @Fishb73; @LaiRS93; @LaiW96; @TanigN96; @Shiba96; @UryuE99; @WiggiL00; @IshiiSM05] for quasiequilibrium calculations and [@Mashh75; @CarteL83; @Marck83; @LeeK99; @Lee00; @RosswSW04; @KobayLPM04] for dynamical simulations). More recently, several groups have initiated studies of BHNS binaries in a fully relativistic framework, both for quasiequilibrium models [@Mille01; @BaumgSS04; @TanigBFS05; @TanigBFS06; @Grand06] and dynamical simulations [@FaberBSTR06; @FaberBST06; @SopueSL06; @LofflRA06; @ShibaU06].
In our own effort, we initially studied BHNS quasiequilibrium models under the assumption of extreme mass ratios, where the mass of the black hole is much greater than that of the neutron star [@BaumgSS04; @TanigBFS05; @FaberBSTR06; @FaberBST06]. This assumption is very appealing computationally and therefore a natural first step in a systematic study of these binaries, but breaks down for the astrophysically more interesting comparable-mass binaries. For the latter, the neutron star is tidally disrupted outside of the black hole’s innermost stable circular orbit (ISCO) (see, e.g. [@IshiiSM05]) and may form an accretion disk, which is considered crucial for launching of a gamma-ray burst. The observation of such a process would provide a wealth of astrophysical information [@Valli00]. In a subsequent study of quasiequilibrium BHNS models [@TanigBFS06], we therefore relaxed the assumption of extreme mass ratios and studied comparable-mass binaries (see also [@Grand06; @ShibaU06]).
We construct quasiequilibrium models by solving Einstein’s constraint equations in the conformal thin-sandwich formalism (see, e.g. [@York99; @Cook00; @BaumgS03]). These equations constrain only some of the gravitational fields; others, namely the conformal background solution, are freely specifiable and have to be chosen before the equations can be solved (see Section \[Sec:IIA\] below). In our previous studies [@BaumgSS04; @TanigBFS05; @TanigBFS06] we chose a background solution describing a single black hole in Kerr-Schild coordinates (in [@TanigBFS05] we also include results for a spatially flat background for extreme mass ratio binaries). This seemed appealing in a number of ways; for example, it allows for black hole spin and the coordinate system extends smoothly into the black hole interior. However, in [@TanigBFS06] we found sizable deviations from post-Newtonian (PN) results which we could attribute, at least in part, to the choice of the background solution [@TanigBFS05; @Grand06]. Moreover, the recent black hole boundary conditions of Cook and Pfeiffer [@CookP04] (see also [@CaudiCGP06]) allow for black hole rotation and penetrating coordinates even for a spatially flat background solution, which simplifies the equations significantly.
In this paper we therefore revisit quasiequilibrium models of BHNS binaries. We now use a spatially flat background solution instead of a Kerr-Schild metric (compare [@Grand06]), and we also adopt a new decomposition of the field variables which significantly reduces the numerical error that was still present in [@TanigBFS06] (see Appendix \[appendix:eq\]). We construct sequences of BHNS binaries in quasicircular orbits for fixed black hole irreducible masses and neutron star baryon rest masses, and focus on irrotational neutron stars orbiting approximately nonspinning black holes (we will implement a more accurate condition for nonspinning black holes in the future, following [@CaudiCGP06]). We find excellent agreement with post-Newtonian results for large binary separations. We track cusp formation to locate the onset of tidal disruption, and, for large mass ratios and neutron star compactions, find turning-points on a binding energy curve to locate the ISCO. For those sequences that encounter an ISCO we find reasonable agreement with the ISCO of test particles around Schwarzschild black holes.
The paper is organized as follows. We briefly review the basic equations in Section II. We present numerical results in Section III, and provide a comparison with the results of [@TanigBFS05; @TanigBFS06; @Grand06] in Section IV. In Section V we briefly summarize our findings.
Throughout this paper we adopt geometric units with $G=c=1$, where $G$ denotes the gravitational constant and $c$ the speed of light. Latin and Greek indices denote purely spatial and spacetime components, respectively.
Formulation
===========
In this Section we briefly review the equations we solve to obtain a BHNS binary in quasiequilibrium. For a more detailed discussion we refer to the review articles [@Cook00; @BaumgS03] as well as Sec. II of [@GourgGTMB01] for the hydrostatics.
Gravitational field equations {#Sec:IIA}
-----------------------------
The line element in $3+1$ form is written as $$\begin{aligned}
ds^2 &=& g_{\mu \nu} dx^{\mu} dx^{\nu} \nonumber \\
&=& -\alpha^2 dt^2 + \gamma_{ij} (dx^i +\beta^i dt)
(dx^j +\beta^j dt),\end{aligned}$$ where $g_{\mu \nu}$ is the spacetime metric, $\alpha$ the lapse function, $\beta^i$ the shift vector, and $\gamma_{ij}$ the spatial metric induced on a spatial slice $\Sigma$. The spatial metric $\gamma_{ij}$ is further decomposed according to $\gamma_{ij} = \psi^4
\tilde{\gamma}_{ij}$, where $\psi$ denotes the conformal factor and $\tilde{\gamma}_{ij}$ the background spatial metric. We also decompose the extrinsic curvature $K^{ij}$ into a trace $K$ and a traceless part $\tilde{A}^{ij}$ according to $$K^{ij} = \psi^{-10} \tilde{A}^{ij} +{1 \over 3} \gamma^{ij} K.$$ The Hamiltonian constraint then becomes $$\label{eq:ham}
\tilde{\nabla}^2 \psi = -2\pi \psi^5 \rho +{1 \over 8} \psi
\tilde{R} +{1 \over 12} \psi^5 K^2 - {1 \over 8} \psi^{-7}
\tilde{A}_{ij} \tilde{A}^{ij}.$$ Here $\tilde{\nabla}^2 = \tilde \gamma^{ij} \tilde \nabla_i \tilde
\nabla_j$ is the covariant Laplace operator, $\tilde \nabla_i$ the covariant derivative, $\tilde{R}_{ij}$ the Ricci tensor, and $\tilde{R}=\tilde{\gamma}^{ij} \tilde{R}_{ij}$ the scalar curvature, all associated with the conformal background metric $\tilde{\gamma}_{ij}$.
We employ the conformal thin-sandwich decomposition of the Einstein equations [@York99]. In this decomposition, we use the evolution equation for the spatial metric to express the traceless part of the extrinsic curvature in terms of the time derivative of the background metric, $\tilde{u}_{ij}\equiv\partial_t
\tilde{\gamma}_{ij}$, and the gradients of the shift vector. Under the assumption of equilibrium, i.e., $\tilde{u}_{ij}=0$ in a corotating coordinate system, the traceless part of the extrinsic curvature reduces to $$\tilde{A}^{ij} ={\psi^6 \over 2\alpha} \Bigl( \tilde{\nabla}^i
\beta^j +\tilde{\nabla}^j \beta^i -{2 \over 3} \tilde{\gamma}^{ij}
\tilde{\nabla}_k \beta^k \Bigr). \label{eq:taij}$$ Inserting Eq. (\[eq:taij\]) into the momentum constraint we obtain $$\begin{aligned}
\label{eq:mom}
&&\tilde{\nabla}^2 \beta^i +{1 \over 3} \tilde{\nabla}^i
(\tilde{\nabla}_j \beta^j) +\tilde{R}^i_j \beta^j \nonumber \\
&&= 16\pi \alpha \psi^4 j^i
+ 2\tilde{A}^{ij} \tilde{\nabla}_j (\alpha \psi^{-6})
+{4 \over 3} \alpha \tilde{\gamma}^{ij} \tilde{\nabla}_j K.\end{aligned}$$ For the construction of quasiequilibrium data it is also reasonable to assume $\partial_t K =0$ in a corotating coordinate system. The trace of the evolution equation for the extrinsic curvature then yields $$\begin{aligned}
\label{eq:Kdot}
\tilde{\nabla}^2 \alpha &=& 4\pi \alpha \psi^4 (\rho +S)
+{1 \over 3} \alpha \psi^4 K^2 + \psi^4 \beta^i \tilde{\nabla}_i K
\nonumber \\
&&+ \alpha \psi^{-8} \tilde{A}_{ij} \tilde{A}^{ij}
- 2 \tilde{\gamma}^{ij} \tilde{\nabla}_i \alpha \tilde{\nabla}_j
\ln \psi.\end{aligned}$$
Equations (\[eq:ham\]), (\[eq:mom\]) and (\[eq:Kdot\]) provide equations for the lapse function $\alpha$, the shift vector $\beta^i$, and the conformal factor $\psi$, while $\tilde A^{ij}$ can be found from Eq. (\[eq:taij\]). The conformally related spatial metric $\tilde{\gamma}_{ij}$ and the trace of the extrinsic curvature $K$ remain freely specifiable, and have to be chosen before we can solve the above equations (note that we have already set to zero the time derivatives of these quantities, which are also freely specifiable). In [@TanigBFS06] we identified these quantities with the corresponding quantities for a Schwarzschild black hole in Kerr-Schild coordinates. Instead, we now assume a flat background $\tilde{\gamma}_{ij} = \eta_{ij}$, where $\eta_{ij}$ denotes a flat spatial metric, and maximal slicing $K=0$. In Cartesian coordinates, Eqs. (\[eq:ham\]), (\[eq:mom\]) and (\[eq:Kdot\]) then reduce to $$\begin{aligned}
&&\underline{\Delta} \psi = -2\pi \psi^5 \rho - {1 \over 8} \psi^{-7}
\tilde{A}_{ij} \tilde{A}^{ij}, \label{eq:ham_constr} \\
%%%
&&\underline{\Delta} \beta^i +{1 \over 3} \partial^i
(\partial_j \beta^j) = 16\pi \alpha \psi^4 j^i
+ 2\tilde{A}^{ij} \partial_j (\alpha \psi^{-6}),
\label{eq:mom_constr} \\
%%%
&&\underline{\Delta} \alpha = 4\pi \alpha \psi^4 (\rho +S)
+ \alpha \psi^{-8} \tilde{A}_{ij} \tilde{A}^{ij} \nonumber \\
&&\hspace{30pt}- 2 \eta^{ij} \partial_i \alpha \partial_j \ln \psi,
\label{eq:trace_evolv}\end{aligned}$$ where $\underline{\Delta}$ and $\partial_i$ denote the flat Laplace operator and the flat partial derivative, while Eq. (\[eq:taij\]) becomes $$\tilde{A}^{ij} ={\psi^6 \over 2\alpha} \Bigl( \partial^i
\beta^j +\partial^j \beta^i -{2 \over 3} \eta^{ij}
\partial_k \beta^k \Bigr).$$ For numerical purposes we further decompose the variables and their equations into parts associated with the black hole and the neutron star. For details of this decomposition we refer to Appendix \[appendix:eq\], but we note that we have found this decomposition crucial for improving the accuracy of the calculations.
The matter terms on the right-hand side of Eqs. (\[eq:ham\_constr\]), (\[eq:mom\_constr\]), and (\[eq:trace\_evolv\]) are derived from the projections of the stress-energy tensor $T_{\mu \nu}$ into the spatial slice $\Sigma$. Assuming an ideal fluid, we have $$T_{\mu \nu} = (\rho_0 +\rho_i +P) u_{\mu} u_{\nu} + P g_{\mu \nu},$$ where $u_{\mu}$ is the fluid 4-velocity, $\rho_0$ the baryon rest-mass density, $\rho_i$ the internal energy density, and $P$ the pressure. Denoting the future-oriented unit normal to $\Sigma$ as $n_{\mu}$, the relevant projections of $T_{\mu \nu}$ are $$\begin{aligned}
&&\rho = n_{\mu} n_{\nu} T^{\mu \nu}, \\
&&j^i = -\gamma^i_{\mu} n_{\nu} T^{\mu \nu}, \\
&&S_{ij} = \gamma_{i \mu} \gamma_{j \nu} T^{\mu \nu}, \\
&&S = \gamma^{ij} S_{ij}.\end{aligned}$$
Hydrostatic equations
---------------------
The matter in the neutron star interior has to satisfy the relativistic equations of hydrodynamics. For stationary configurations, the relativistic Euler equation can be integrated once to yield $$h \alpha {\gamma \over \gamma_0} = {\rm constant},$$ where $h=(\rho_0 +\rho_i +P)/\rho_0$ is the fluid specific enthalpy, and $\gamma$ and $\gamma_0$ are Lorenz factors between the fluid and the rotating frame, and the rotating frame and the inertial frame (see Sec. II.C. of [@TanigBFS05] for the definitions). For irrotational fluids, the fluid 3-velocity with respect to the inertial observer, $U^i$, can be expressed in terms of the gradient of a velocity potential $\Psi$ as $$U^i = {\psi^{-4} \over \alpha u^t h} \tilde{\nabla}^i \Psi,$$ where $u^t$ is the time component of the fluid 4-velocity $u^{\mu}$. Having taken into account the expression of the 3-velocity, the equation of continuity becomes $${\rho_0 \over h} \nabla^{\mu} \nabla_{\mu} \Psi +(\nabla^{\mu} \Psi)
\nabla_{\mu} \Bigl( {\rho_0 \over h} \Bigr) = 0,$$ where $\nabla_{\mu}$ denotes the covariant derivative associated with $g_{\mu \nu}$. We refer to [@GourgGTMB01] for a more detailed derivation of the hydrostatic equations, and to [@TanigG02; @TanigG03; @BejgeGGHTZ05] for BNS applications.
Equation of state
-----------------
We adopt a polytropic equation of state in the form $$P =\kappa \rho_0^{\Gamma},$$ where $\Gamma$ denotes the adiabatic index and $\kappa$ a constant. In this paper, we focus on the case $\Gamma=2$. Using this equation of state, the definition of the specific enthalpy $h=(\rho_0 +\rho_i +P)/\rho_0$, and a thermodynamic relation (Gibbs-Duhem relation) $${d h \over h} = {dP \over \rho_0 +\rho_i +P},$$ we obtain the internal energy density $$\rho_i ={\kappa \over \Gamma -1} \rho_0^{\Gamma}.$$
Since dimensions enter the problem only through the constant $\kappa$, it is convenient to rescale all dimensional quantities with respect to the polytropic length scale $$R_{\rm poly} \equiv \kappa^{1/(2\Gamma-2)}. \label{eq:poly_units}$$
Boundary conditions
-------------------
In order to solve the gravitational field equations (\[eq:ham\_constr\]), (\[eq:mom\_constr\]), and (\[eq:trace\_evolv\]), we have to set appropriate boundary conditions on two different boundaries: outer boundaries at spatial infinity and inner boundaries on the black hole horizons.
The boundary conditions at spatial infinity follow from the assumption of asymptotic flatness. With the help of a radial coordinate transformation $u=1/r$ in the external computational domain our computational grid extends to spatial infinity [@BonazGM98; @GourgGTMB01], and we can impose the exact boundary conditions $$\begin{aligned}
&&\psi |_{r \rightarrow \infty} = 1, \\
&&\beta^i |_{r \rightarrow \infty} = (\mbox{\boldmath $\Omega$}
\times \mbox{\boldmath $R$})^i, \\
&&\alpha |_{r \rightarrow \infty} = 1,\end{aligned}$$ where $\Omega$ is the orbital angular velocity of the binary system measured at infinity, and $\mbox{\boldmath $R$}=(X,~Y,~Z)$ is a Cartesian coordinate which origin is located at the center of mass of the binary system. Here, we express the shift vector $\beta^i$ in a corotating coordinate system that we adopt throughout our calculation. In an inertial coordinate system, the shift vector would tend to zero at spatial infinity, while in the corotating coordinate system of the numerical code the shift vector diverges at spatial infinity. For computational purposes it is therefore convenient to write the shift vector as a sum of the rotational shift term $\beta^i_{\rm rot} \equiv (\mbox{\boldmath $\Omega$} \times
\mbox{\boldmath $R$})^i$ and a residual part (which tends to zero at spatial infinity), and solve the equations only for the latter (see Appendix \[appendix:eq\] for a detailed explanation of the decomposition of the metric quantities and related equations).
The inner boundary conditions arise from the excision of the black hole interior. The assumption that the black hole is in equilibrium leads to a set of boundary conditions for the conformal factor and shift vector [@CookP04] (see also [@Cook02; @CaudiCGP06] as well as the related isolated horizon formalism, e.g. [@AshteK04; @GourgJ06]). The boundary condition for the conformal factor is $$\tilde{s}^k \tilde{\nabla}_k \ln \psi \Bigl|_{\cal S}
=-{1 \over 4} (\tilde{h}^{ij} \tilde{\nabla}_i \tilde{s}_j
-\psi^2 J) \Bigl|_{\cal S},$$ where $s^i \equiv \psi^{-2} \tilde{s}^i$ is the outward pointing unit vector normal to the excision surface and $h_{ij}$ is the induced metric on the excision surface, $h_{ij} \equiv \psi^4 \tilde{h}_{ij} = \gamma_{ij} - s_i s_j$. The quantity $J$ is computed from the projection of the extrinsic curvature $K_{ij}$ as $J \equiv h^{ij} K_{ij}$. The boundary condition on the normal component of the shift vector is $$\beta_{\perp} |_{\cal S} = \alpha |_{\cal S}, \label{eq:beta_perp}$$ while the tangential components have to satisfy $$\beta_{\parallel}^i |_{\cal S} = \epsilon^{i}_{jk} \Omega_r^j x^k.
\label{eq:beta_para}$$ Here $\Omega_r^j$ is the black hole spin angular velocity vector, which we take to be aligned with the $Z$-axis, and $x^k$ is a Cartesian coordinate centered on the black hole. To construct approximately nonspinning black holes, we set the shift vector according to Eqs. (\[eq:beta\_perp\]) and (\[eq:beta\_para\]) with $\Omega_r = \Omega$. This assignment corresponds to a “leading-order approximation” in the language of [@CaudiCGP06], and we plan to implement their more accurate condition for nonspinning black holes in the future. According to [@CookP04], the boundary condition on the lapse function can be chosen freely. In this paper, we choose a Neumann boundary condition $$\label{eq:lapse_bound}
{d \alpha \over dr} \Bigl|_{\cal S} =0$$ on the excised surface, where $r$ is a radial isotropic coordinate.
Orbital angular velocity
------------------------
In this paper we take the rotational axis of the binary system to be the $Z$-axis, and the line connecting the black hole and neutron star centers to be the $X$-axis. To impose a quasicircular orbit we require a force balance along the $X$-axis at the center of the neutron star, which results in the condition [@GourgGTMB01] $${\partial \ln h \over \partial X} \Bigl|_{(X_{\rm NS},Y_{\rm NS},0)}
= 0.$$ Here $X_{\rm NS}$ and $Y_{\rm NS}$ denote the $X$ and $Y$-coordinates of the center of the neutron star relative to the rotational axis of the binary system. Imposing this condition determines the orbital angular velocity $\Omega$. We confirm that the orbital angular velocity obtained by this method agrees with that obtained by requiring the enthalpy at two points on the neutron star’s surface to be equal [@BaumgSS04] within one part in $10^{-6}$.
Global quantities
-----------------
It is reasonable to require that the irreducible mass of the black hole and the baryon rest mass of the neutron star are conserved during the inspiral of the BHNS binaries. For such a constant-mass sequence we then monitor the Arnowitt-Deser-Misner (ADM) mass, the Komar mass, and the total angular momentum, all of which are defined globally as follows.
The irreducible mass of the black hole is defined as $$M_{\rm irr}^{\rm BH} \equiv \sqrt{A_{\rm EH} \over 16\pi},$$ where $A_{\rm EH}$ is the proper area of the event horizon. In practice we approximate this area with that of the apparent horizon, $A_{\rm AH}$, which is computed from an integral on the excision surface ${\cal S}$, $$A_{\rm AH} = \int_{\cal S} \psi^4 d^2 x.$$ The baryon rest mass of the neutron star is $$M_{\rm B}^{\rm NS} = \int \rho_0 u^t \sqrt{-g} d^3 x,$$ where $g$ is the determinant of $g_{\mu \nu}$. In our case we have $\sqrt{-g} = \alpha \psi^6$, so that $$M_{\rm B}^{\rm NS} = \int \rho_0 \alpha u^t \psi^6 d^3 x.$$ In the polytropic units of Eq. (\[eq:poly\_units\]) we normalize the baryon rest mass according to $$\bar{M}_{\rm B}^{\rm NS} \equiv {M_{\rm B}^{\rm NS} \over
R_{\rm poly}},$$ whereby $\bar{M}_{\rm B}^{\rm NS}$ is the rest mass of the polytrope for polytropic constant $\kappa = 1$.
We express the ADM mass in isotropic Cartesian coordinates as $$M_{\rm ADM} =-{1 \over 2\pi} \oint_{\infty} \partial^i \psi dS_i.$$ The Komar mass can be written as $$M_{\rm Komar} ={1 \over 4\pi} \oint_{\infty} \partial^i \alpha dS_i,$$ where we use the fact that the shift vector falls off sufficiently rapidly. The total angular momentum is $$J_i ={1 \over 16\pi} \epsilon_{ijk} \oint_{\infty} (X^j K^{kl} -X^k
K^{jl}) dS_l,$$ where $X^i$ is a spatial Cartesian coordinate relative to the center of mass of the binary system. Finally, the linear momentum is $$P^i ={1 \over 8\pi} \oint_{\infty} K^{ij} dS_j,$$ where we have assumed maximal slicing $K=0$.
During the iteration, we require that the linear momentum vanishes. We enforce this by changing the position of the black hole and neutron star relative to the location of the axis of rotation. Stated differently, this condition fixes the location of the axis of rotation.
We define the binding energy of the binary system as $$E_{\rm b} = M_{\rm ADM} - M_0, \label{eq:bindene}$$ where $M_0$ is the ADM mass of the binary system at infinite orbital separation, as defined by the sum of the irreducible mass of the isolated black hole and the ADM mass of an isolated neutron star with the same baryon rest mass, $$M_0 \equiv M_{\rm irr}^{\rm BH} + M_{\rm ADM,0}^{\rm NS}.$$
In order to measure a global error in the numerical results, we define the virial error as the fractional difference between the ADM mass and Komar mass, $$\delta M \equiv \Bigl| {M_{\rm ADM} - M_{\rm Komar} \over M_{\rm
ADM}} \Bigr|. \label{eq:virial}$$
Numerical results
=================
Our numerical code is based on the spectral method [Lorene]{} library routines developed by the Meudon relativity group [@Lorene]. Our computational grid depends on the orbital separation, the mass of the neutron star, and the mass ratio. For example, the grid is divided into 10 (8) domains for the black hole (neutron star) at large separations for a neutron star mass of $\bar{M}_{\rm B}^{\rm
NS}=0.15$ and mass ratio $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm
NS}=5$. Each domain for the black hole is covered by $N_r \times
N_{\theta} \times N_{\phi}=41 \times 33 \times 32$ collocation points and those for the neutron star by $25 \times 17 \times 16$, where $N_r$, $N_{\theta}$, and $N_{\phi}$ denote the collocation points in the radial, polar, and azimuthal directions, respectively. We use a larger number of collocation points for the black hole domains than the neutron star domains because the source terms of the gravitational field equations for the black hole vary slightly around the neutron star position. The amplitude of this excess in the source terms is small but significantly affects the accuracy of the results. In order to resolve this source in the black hole domains, which are centered on the black hole, we need a higher angular resolution. On the other hand, the source terms of the gravitational field equations for the neutron star have large contributions only near the neutron star, and drop off monotonically away from the neutron star. Thus a smaller angular resolution is sufficient in the neutron star domains, which are centered on the neutron star. (See Appendix \[appendix:eq\] for the decomposition of the gravitational field equations into the black hole and neutron star parts.)
As stated above, we choose $\Gamma=2$ for the adiabatic index of the polytropic equation of state in this paper, and compute several different constant-mass inspiral sequences. We consider neutron stars of rest mass $\bar{M}_{\rm B}^{\rm NS}=0.15$ and 0.10, with corresponding compactions of $M_{\rm ADM,0}^{\rm NS}/R_0=0.145$ and 0.0879. Here $R_0$ is the areal radius of the neutron star in isolation. The maximum baryon rest mass for spherical $\Gamma=2$ polytropes in isolation is $\bar M_{\rm
B}^{\rm max} = 0.180$. Our more compact polytrope with $\bar M_{\rm
B}^{\rm NS}=0.15$ has a compaction that is reasonably realistic (e.g. it would apply to binaries containing neutron stars with baryon rest mass $1.5M_\odot$ and isolated spherical neutron stars with a maximum baryon rest mass of $1.8M_\odot$); we consider the less compact model for purposes of comparison.
We also consider mass ratios $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm
NS}=1$, 2, 3, 5, and 10. Note again that we fix the irreducible mass of the black hole and the baryon rest mass of the neutron star for the construction of constant-mass sequences. For the definition of the mass ratio, however, we use the ADM mass of a spherical isolated neutron star $M_{\rm ADM,0}^{\rm NS}$, since this turns out to be more convenient for comparisons with third-order post-Newtonian results.
We summarize our results in Tables \[table:seqM015\] and \[table:seqM01\] in Appendix \[appendix:table\].
![Contours of the lapse function $\alpha$ in the equatorial plane for our innermost configuration of a binary of mass ratio $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=3$ and neutron star mass $\bar{M}_{\rm B}^{\rm NS}=0.15$. The minimum value of the lapse, $\alpha \simeq 0.453$, is located on the excised surface and the maximum value, $\alpha \simeq 0.909$, at the right edge of the figure. The value of the lapse at the point of maximum baryon rest-mass density is about 0.558. The contour curves are spaced by $\delta \alpha \simeq
0.03$. The cross “$\times$” indicates the position of the rotation axis.[]{data-label="fig:lapse"}](fig1.eps){width="8cm"}
In Fig. \[fig:lapse\] we present contours of the lapse function $\alpha$ for a binary of mass ratio $M_{\rm irr}^{\rm BH}/M_{\rm
ADM,0}^{\rm NS}=3$ and neutron star mass $\bar{M}_{\rm B}^{\rm
NS}=0.15$ at the smallest binary separation for which our code converged. The thick solid circle on the left-hand side marks the position of the excised surface (the apparent horizon), while that on the right-hand side marks the position of the neutron star surface. The value of the lapse function $\alpha$ on the excised surface is not uniform because we use the Neumann boundary condition (\[eq:lapse\_bound\]) rather than a Dirichlet boundary condition. The approximate value of the lapse on the excision surface is $\alpha|_{\cal S} \sim 0.45-0.48$.
![Fractional binding energy $E_{\rm b}/M_0$ as a function of $\Omega M_0$ for binaries of mass ratio $M_{\rm irr}^{\rm
BH}/M_{\rm ADM,0}^{\rm NS}=1$. The solid line with filled circles shows results for neutron stars of mass $\bar{M}_{\rm B}^{\rm
NS}=0.15$, and the dashed line with squares for neutron stars of mass of 0.10. The dash-dotted line denotes the results of the third post-Newtonian approximation [@Blanc02]. These sequences end due to cusp formation – and hence the onset of tidal disruption – before the binary reaches the ISCO at the turning point of the binding energy.[]{data-label="fig:bindene1"}](fig2.eps){width="8cm"}
![Same as Fig. \[fig:bindene1\] but for sequences of mass ratio $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=2$.[]{data-label="fig:bindene2"}](fig3.eps){width="8cm"}
![Same as Fig. \[fig:bindene1\] but for sequences of mass ratio $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=3$.[]{data-label="fig:bindene3"}](fig4.eps){width="8cm"}
![Same as Fig. \[fig:bindene1\] but for sequences of mass ratio $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=5$. For the more compact neutron star the binary now encounters an ISCO before the neutron star is tidally disrupted.[]{data-label="fig:bindene5"}](fig5.eps){width="8cm"}
![Same as Fig. \[fig:bindene1\] but for sequences of mass ratio $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=10$.[]{data-label="fig:bindene10"}](fig6.eps){width="8cm"}
In Figs. \[fig:bindene1\] – \[fig:bindene10\] we plot the binding energy $E_{\rm b}/M_0$ versus the orbital angular velocity $\Omega
M_0$ for sequences with mass ratios 1, 2, 3, 5 and 10. We find excellent agreement with third-order post-Newtonian approximations [@Blanc02], especially for large binary separations. For more compact neutron stars with $\bar{M}_{\rm B}^{\rm NS}=0.15$, we also find a turning point in the sequences with mass ratios $M_{\rm irr}^{\rm BH}/M_{\rm
ADM,0}^{\rm NS}=5$ and 10. Such turning points mark the ISCO, inside of which no stable circular orbits can exist. For irrotational binaries this instability is dynamical (see, e.g. [@LaiRS93]). Binaries with larger neutron star compaction and larger mass ratios therefore encounter an ISCO before being tidally disrupted. For all other binaries, the sequences end due to cusp formation – and hence at the onset of tidal disruption – before the binary encounters an ISCO.
Qualitatively, this behavior can be understood very easily from a simple Newtonian analysis. The binary separation at the onset of tidal disruption, $d_{\rm tid}$, can be estimated by equating the gravitational and tidal forces on a test particle on the surface of the neutron star $$\label{isco_scaling}
\frac{d_{\rm tid}}{M_{\rm BH}} \sim
\left( \frac{M_{\rm NS}}{M_{\rm BH}} \right)^{2/3}
\frac{R_{\rm NS}}{M_{\rm NS}}.$$ If this is smaller than the binary separation at the ISCO, at $d_{\rm
ISCO}/M_{\rm BH} = 6$ for a test particle in Schwarzschild coordinates, the binary encounters the ISCO before being tidally disrupted. This happens for large mass ratios $M_{\rm BH}/M_{\rm NS}$ and large compactions $M_{\rm NS}/R_{\rm NS}$ – in accordance with our findings.
In the test-particle limit the location of the turning point should coincide with that of test particle orbiting a Schwarzschild black hole of mass $M_0$, $\Omega M_0 = 6^{-3/2} \simeq 0.068$. We find similar but slightly larger values, $\Omega M_0 \sim 0.082$ and $\sim 0.084$ for $M_{\rm irr}^{\rm
BH}/M_{\rm ADM,0}^{\rm NS}=5$ and 10. Locating the minimum of a numerically generated curve always introduces some additional error, and it is possible that this deviation is entirely a numerical artifact. However, it is also possible that the deviation is a consequence of our “leading-order approximation” of the black hole spin. This approximation leads to a small but non-zero black hole spin, so that effectively we locate the ISCO around a Kerr instead of a Schwarzschild black hole. Further evidence for this hypothesis is the fact that we do not find turning points of the angular momentum that coincide with those of the binding energy. In [@CookP04], the authors similarly found that the binding energy and angular momentum for “leading-order” irrotational black hole binaries did not have simultaneous turning points. In [@CaudiCGP06], however, the authors show that an improved condition for nonspinning black holes leads to a very good agreement between the binding energy and angular momentum turning points. We plan to implement this improved condition in the near future.
![Virial error $\delta M$ versus $\Omega M_0$ for our binary sequences.[]{data-label="fig:virial"}](fig7a.eps "fig:"){width="8cm"}\
![Virial error $\delta M$ versus $\Omega M_0$ for our binary sequences.[]{data-label="fig:virial"}](fig7b.eps "fig:"){width="8cm"}
As one measure of the error in our numerical calculations we compute the virial expression (\[eq:virial\]). As shown in Fig. \[fig:virial\], all our configurations have relative virial errors of less than $10^{-3}$, generally on the order of several $\times 10^{-4}$. The errors are smaller at intermediate binary separations, and somewhat larger both at larger binary separations (where the angular resolution becomes worse) and smaller binary separations (where the larger tidal deformation of the neutron star causes larger numerical error).
From the virial error we can estimate the error in the binding energy as follows. We assume that the relative error in the ADM mass is of the same order as the relative virial error, $$\Bigl| {M_{\rm ADM} - M_{\rm ADM,true} \over M_{\rm ADM,true}}
\Bigr| \sim \delta M, \label{eq:errorADM}$$ where $M_{\rm ADM,true}$ is the true value of the ADM mass. From the definition of the binding energy Eq. (\[eq:bindene\]), we can write the relative difference of the binding energy from its true value $E_{\rm b,true}$ as $$\Bigl| {E_{\rm b} - E_{\rm b,true} \over E_{\rm b,true}} \Bigr| \sim
\delta M \Bigl| {M_{\rm ADM,true} \over M_{\rm ADM,true} -M_0}
\Bigr|. \label{eq:error}$$ Since the term $|M_{\rm ADM,true}/(M_{\rm ADM,true} -M_0)|$ on the right-hand side is of order of $10^2$ for the binary separations that we calculate, we conclude that the relative error in the binding energy is approximately $10^2 \delta M$. For our least accurate sequences this results in a relative error of several $\times
10^{-2}$ and for our most accurate sequences it is a few $\times
10^{-3}$.
![Minimum of the mass-shedding indicator $\chi_{\rm min}$ as a function of $\Omega M_0$.[]{data-label="fig:chi"}](fig8a.eps "fig:"){width="8cm"}\
![Minimum of the mass-shedding indicator $\chi_{\rm min}$ as a function of $\Omega M_0$.[]{data-label="fig:chi"}](fig8b.eps "fig:"){width="8cm"}
----------------------------------------------- --------------------------------- ---------------------------------
$M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}$ $\bar{M}_{\rm B}^{\rm NS}=0.10$ $\bar{M}_{\rm B}^{\rm NS}=0.15$
1 0.0179 0.0382
2 0.0255 0.0550
3 0.0334 0.0728
5 0.0488 0.103
10 0.0825 ——
----------------------------------------------- --------------------------------- ---------------------------------
: Estimated orbital angular velocity at the tidal disruption point.
\[table:omega\]
Using a spectral code we cannot follow our sequences all the way to cusp formation (see our discussion in [@TanigBFS05]). To monitor the formation of a cusp we therefore define a mass-shedding indicator [@GourgGTMB01] $$\chi \equiv {(\partial (\ln h)/\partial r)_{\rm eq}
\over (\partial (\ln h)/\partial r)_{\rm pole}},$$ the ratio of the radial derivative of the enthalpy in the equatorial plane at the surface to that to the polar direction at the surface. Given our choice of boundary conditions the minimum point of $\chi$ is not on the $X$ axis (which connects the neutron star with the black hole), but slightly away from the axis. In Fig. \[fig:chi\] we therefore graph the minimum value of $\chi$, searched on the surface of the neutron star, as a function of $\Omega M_0$. For spherical stars at infinite separation we have $\chi_{\rm min}=1$, while $\chi_{\rm min}=0$ indicates the formation of a cusp and hence tidal breakup. Our spectral code stops converging before reaching $\chi_{\rm min}=0$, but the sharp drop in $\chi_{\rm min}$ is an indication of the formation of a cups. Extrapolating to $\chi_{\rm
min}=0$ from the last three points (indicated by the thin dotted lines in Fig. \[fig:chi\]) we estimate the orbital angular velocity at the onset of tidal disruption (see Table \[table:omega\]). We did not include the sequence with mass ratio $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=10$ and neutron star mass $\bar{M}_{\rm B}^{\rm NS}=0.15$ in this analysis, since the binary encounters an ISCO long before the neutron star is tidally disrupted (see also Table \[table:seqM015\]). For the sequence with $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=10$ and $\bar{M}_{\rm B}^{\rm NS}=0.10$ the extrapolation of $\chi_{\rm min}$ suggests that tidal disruption occurs at $\Omega M_0 \simeq 0.0825$ (the last entry in Table \[table:omega\]), a value very similar to the values of the ISCO in those two sequences for which we found a turning point. This suggests that the neutron star in this binary may be tidally disrupted just as the orbit becomes unstable.
![Decrease in the maximum density parameter $\delta q_{\rm max}$ as a function of $\Omega M_0$.[]{data-label="fig:dqmax"}](fig9a.eps "fig:"){width="8cm"}\
![Decrease in the maximum density parameter $\delta q_{\rm max}$ as a function of $\Omega M_0$.[]{data-label="fig:dqmax"}](fig9b.eps "fig:"){width="8cm"}
In [@TanigBFS06], where we computed similar BHNS binary sequences but with a Kerr-Schild background, we also considered a binary of mass ratio $M_{\rm irr}^{\rm BH}/M_{\rm B}^{\rm NS}=5$ and neutron star mass $\bar{M}_{\rm B}^{\rm NS}=0.10$. There we estimated that tidal disruption would set in at $\Omega M_0 \simeq 0.046$. We compute a sequence with the same definition of the mass ratio $M_{\rm irr}^{\rm BH}/M_{\rm B}^{\rm NS}=5$ for comparison, and find tidal disruption at approximately $\Omega M_0 \sim 0.053$, a value within 15% of our earlier value.
As the orbital separation decreases, the maximum density of the neutron star decreases. We define the dimensionless density parameter $$q \equiv \frac{P}{\rho_0}$$ and monitor the decrease in the maximum density $$\delta q_{\rm max} \equiv {q_{\rm max} - q_{\rm max,0} \over
q_{\rm max,0}},$$ where $q_{\rm max,0}$ denotes that of an isolated neutron star with the same baryon rest mass. For $\bar{M}_{\rm B}^{\rm NS}=0.15$ (0.10) we have $q_{\rm max,0}=0.12665$ (0.058827). Note that $\delta q_{\rm max}$ is negative for all configurations considered in this study. In Fig. \[fig:dqmax\] we therefore graph $-\delta q_{\rm max}$ as a function of the orbital angular velocity $\Omega M_0$ on a logarithmic scale.
Within Newtonian gravity, the decrease in the central density scales, to leading order, with $$\delta q \propto d^{-\sigma} \propto \Omega^{2\sigma/3},$$ where $d$ is the orbital separation, and for irrotational binaries the index $\sigma$ takes the value $\sigma = 6$ (see, e.g. [@TanigN00] for an analytic derivation and [@GourgGTMB01] for comparison with numerical results). Here the leading order term is caused by a quadrupole deformation in response to the companion’s tidal field. We see from Fig. \[fig:dqmax\] that the power-law index $\sigma$ is indeed close to 6 for large binary separations. For intermediate binary separations, however, the index is generally smaller, $\sigma\sim 2-3$. For the equal-mass binary $\sigma$ remains close to 6 for the entire sequence, but for increasing mass ratios $\sigma$ decreases at intermediate binary separations. We plan to investigate this behavior with the help of semi-analytic models in a future publication. Finally, for small binary separations, just before tidal disruption sets in, $\sigma$ increases again. This has also been seen for BNS systems [@TanigG02; @TanigG03] and may be caused by higher order multipole deformations.
![The energy spectrum $dE_{\rm b}/df$ for the sequences with $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=1$ as a function of the dimensionless gravitational wave frequency $f M_0$. The solid, dashed, dash-dotted, and dotted curves show the fits to the $\bar{M}_{\rm B}^{\rm NS}=0.15$ and $\bar{M}_{\rm B}^{\rm NS}=0.1$ sequences and the 3PN and Newtonian expressions, respectively. Asterisks denote the approximate point where a cusp forms and tidal disruption begins, terminating the sequence, at frequencies given by Table \[table:omega\].[]{data-label="fig:dedf1"}](fig10.eps){width="8cm"}
![Same as Fig. \[fig:dedf1\], but for the sequences with $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=3$.[]{data-label="fig:dedf3"}](fig11.eps){width="8cm"}
![Same as Fig. \[fig:dedf1\], but for the sequences with $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=5$. Note that the sequence with $\bar{M}_{\rm B}^{\rm NS}=0.15$ reaches a minimum binding energy, where $dE_{\rm b}/df=0$, prior to tidal disruption.[]{data-label="fig:dedf5"}](fig12.eps){width="8cm"}
Finally, we turn our attention to the energy spectrum of gravitational waves emitted from a BHNS binary, which is calculated from the Fourier transform of the gravitational wave strains averaged over all angles (see, e.g. [@Zhuge94]), $$\frac{dE}{df}=\frac{\pi}{2}(4\pi
r^2)f^2\left<|\tilde{h}_+(f)|^2+
|\tilde{h}_\times(f)|^2\right>,$$ where the brackets denote angle averaging and $$\tilde{h}(f)=\int_{-\infty}^\infty h(t)e^{2\pi ift}dt,$$ where $f$ is the gravitational wave frequency, rather than the orbital frequency. From the results of our quasiequilibrium sequences, we can compute approximate gravitational wave energy spectra using the methods described in [@FaberGRT02]. To do so, we assume that gravitational waves from the binary are emitted coherently with $f=\Omega/\pi$ at any given moment in time, where $\Omega$ is the standard orbital angular velocity, and further assume that the system is adiabatic and infall velocities negligible, so that the gravitational wave energy losses drive the system gradually through a series of quasiequilibrium configurations. Under these assumptions, we recover a simple expression for the gravitational wave energy spectrum, $$\frac{dE}{df} \approx
-\pi \left( \frac{dE_{\rm b}}{d\Omega}\right)_{\rm QE},$$ i.e., we can calculate the energy spectrum by numerically differentiating the binding energy along the quasiequilibrium sequence with respect to the angular frequency. This method has been shown to reproduce closely the actual energy spectrum calculated from a fully dynamical binary neutron star merger calculation in the frequency regime of interest [@FaberGR04]. While we can numerically extrapolate up to the frequency at which we find a minimum in the binding energy, representing the ISCO, our results do not apply inside the separation at which a cusp forms and tidal disruption begins (see Table \[table:omega\]). Dynamically, this would lead to the beginning of mass transfer and a rapid evolution away from quasiequilibrium, which in turn would produce a qualitatively different gravitational wave signal in both the time and frequency domains.
Instead of taking derivatives of our numerical data for the binding energy directly (see Tables \[table:seqM015\] and \[table:seqM01\]), we first fit these data to a fourth-order polynomial of the form $$\frac{E_{\rm b}}{M_0}=a_{\rm N} x
\left(1+a_1 x+a_2x^2+a_3x^3+a_4x^4\right)
\label{eq:fit}$$ where $x\equiv (\Omega M_0)^{2/3}=(\pi
f M_0)^{2/3}$. Given this parametrization of the polynomial, the coefficients $a_i$ can be interpreted as post-Newtonian terms of order $i$. The leading order, Newtonian term, is given by $$a_{\rm N}=-\frac{\nu}{2}\equiv
-\frac{M_{\rm irr}^{\rm BH} M_{\rm ADM,0}^{\rm NS}}{2M_0^2}.$$ Here $\nu$ is the symmetric mass ratio, i.e., the ratio of the reduced mass of the system to the total mass. We tabulate the coefficients $a_i$, together with the exact 3PN coefficients for point-mass systems as listed in [@Blanc02], for the different binary sequences in Table \[table:dedf\]. We also include the “cutoff frequencies” $f_n$, denoting the frequency at which the spectral energy is $n$% of its Newtonian counterpart, $$\begin{aligned}
\frac{1}{M_0^2}\frac{dE_{\rm b}}{df}(f_n) &=&
\frac{n}{100}\times \left(\frac{1}{M_0^2}
\frac{dE_{\rm b}}{df}\right)_{\rm N} \nonumber \\
&=&\frac{n}{100} \frac{2\pi a_{\rm N}}{3}
\left(\pi f_n M_0\right)^{-1/3},
\label{eq:omegacut}\end{aligned}$$ for $n= 75, 50$ and 25.
In Figs. \[fig:dedf1\] – \[fig:dedf5\], we show the results of the energy spectrum of gravitational waves. As is expected, binaries of comparable mass terminate with the formation of a cusp, rather than at an ISCO denoting a minimum in the binding energy of the binary. For the smallest black hole mass we consider, $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=1$, shown in Fig. \[fig:dedf1\], tidal disruption occurs before we see significant deviations from either the lowest-order Newtonian point-mass results or the exact 3PN point-mass expression. As the black hole mass and tidal disruption frequency both increase, so do the deviations in the gravitational wave energy spectrum away from the Newtonian point-mass result. This is especially true for the more compact neutron star models with $\bar{M}_{\rm B}^{\rm NS}=0.15$. For a mass ratio $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=5$, shown in Fig. \[fig:dedf5\], the less compact neutron star model with $\bar{M}_{\rm
B}^{\rm NS}=0.1$ (dashed curve) shows significant deviations from the 3PN result at the tidal disruption point, whereas the more compact model with $\bar{M}_{\rm B}^{\rm NS}=0.15$ (solid curve) does reach the ISCO before the onset of tidal disruption. This is also the case for the most extreme mass ratio we consider, $M_{\rm
irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=10$, for which any physically reasonably compact neutron star model reaches an ISCO prior to the tidal disruption point, as had been confirmed by our previous dynamical calculations [@FaberBST06].
We note that the gravitational wave spectrum should produce an important constraint on the neutron star compaction, regardless of how the sequence terminates. As the compaction decreases, tidal effects grow in importance, and tend to increases the binding energy of the system (lowering the magnitude of the negative binding energy), especially at smaller separations, implying that the gravitational wave spectrum will be cutoff at lower frequencies.
[l|cccc|ccc]{} & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $f_{75} M_0$ & $f_{50} M_0$ & $f_{25} M_0$\
\
3PN & -0.771 & -2.78 & -0.967 & & 1.05(-2) & 2.14(-2) & 3.15(-2)\
0.15 & 0.155 & -23.6 & 156. & -401. & [**9.80(-3)**]{} & [*1.96(-2)*]{} & [*2.49(-2)*]{}\
0.10 & 1.42 & -128. & 2.45(3) & -1.66(4) & 5.48(-3) & [*6.81(-3)*]{} & [*7.64(-3)*]{}\
\
3PN & -0.769 & -2.85 & -2.02 & & 1.02(-2) & 2.05(-2) & 2.98(-2)\
0.15 & 1.66 & -54.6 & 386. & -1.02(3) & [**9.22(-3)**]{} & 1.76(-2) & [*2.15(-2)* ]{}\
0.10 & 1.77 & -105. & 1.43(3) & -6.94(3) & 7.48(-3) & [*9.72(-3)*]{} & [*1.10(-2)*]{}\
\
3PN & -0.766 & -2.93 & -3.34 & & 9.89(-3) & 1.95(-2) & 2.80(-2)\
0.15 & 1.28 & -37.0 & 210. & -494. & [**9.57(-3)**]{} & [**1.75(-2)**]{} & 2.28(-2)\
0.10 & 1.47 & -74.2 & 815. & -3.33(3) & [**8.57(-3)**]{} & [*1.20(-2)*]{} & [*1.38(-2)*]{}\
\
3PN & -0.762 & -3.05 & -5.20 & & 9.51(-3) & 1.84(-2) & 2.60(-2)\
0.15 & 0.802 & -26.1 & 138. & -357. & [**1.01(-2)**]{} & [**1.72(-2)**]{} & [**2.21(-2)**]{}\
0.10 & 1.03 & -47.7 & 423. & -1.47(3) & [**9.54(-3)**]{} & 1.45(-2) & [*1.71(-2)*]{}\
\
3PN & -0.757 & -3.18 & -7.36 & & 9.14(-3) & 1.73(-2) & 2.42(-2)\
0.15 & -0.934 & -14.7 & 122. & -365. & [**6.94(-3)**]{} & [**1.71(-2)**]{} & [**2.30(-2)**]{}\
0.10 & 0.305 & -28.5 & 212. & -643. & [**9.35(-3)**]{} & [**1.67(-2)**]{} & [**2.07(-2)**]{}\
\[table:dedf\]
Comparison with previous results
================================
We can compare our numerical results with those from previous attempts to model BHNS binaries. In particular, we compare with our previous work that assumed extreme mass ratios [@TanigBFS05], our comparable-mass results for a Kerr-Schild background [@TanigBFS06], and the results of [@Grand06] for comparable masses and a spatially flat background, as in this study.
Comparison with extreme mass ratio results
------------------------------------------
![Comparison of our new results to those calculated assuming an extreme mass ratio for a spatially flat background geometry [@TanigBFS05] for the minimum of the mass-shedding indicator $\chi_{\rm min}$. Here $M_{\rm BH}$ denotes either the black hole irreducible mass (for our new results) or the background mass (for the results of [@TanigBFS05]); see text.[]{data-label="fig:vsextr"}](fig13.eps){width="8cm"}
In Fig. \[fig:vsextr\], we compare our results for a mass ratio $M_{\rm irr}^{\rm BH}/M_{\rm B}^{\rm NS}=10$ with those for $M_{\rm BH}/M_{\rm B}^{\rm NS}=10$ in [@TanigBFS05] for a spatially flat background, but under the assumption of an extreme mass ratio. Since the tidal effects of the neutron star on the black hole are neglected in the extreme mass-ratio approximation, it is impossible to evaluate the black hole’s irreducible mass. In [@TanigBFS05], therefore, we instead fixed the background mass of the black hole $M_{\rm BH}$. Since the difference between $M_{\rm irr}^{\rm BH}$ and $M_{\rm BH}$ scales with the binary’s binding energy (see [@DenniBP06] for an analytic, leading-order treatment), the relative difference is at most a few percent for this mass ratio.
Our present results agree well with those calculated assuming an extreme mass ratio. The difference in the orbital angular velocity for the same $\chi_{\rm min}$ is order 10%. We can explain this 10% discrepancy as follows: for a given value of $\chi_{\rm min}$, the tidal deformation of the neutron star by the black hole is approximately the same in the two cases. This implies that the orbital separation is very similar as well. However, the orbital angular velocity in our calculation here is given in terms of the distance from the center of mass of the binary system to the center of the neutron star, while for an extreme mass ratio it is given by the binary separation. This is because with the extreme mass-ratio assumption, we assumed that the binary’s center of mass is located at the center of the black hole. The difference between these two definitions of distance, along with a change in the black hole mass we use in the definition of the mass ratio, accounts for the discrepancy in the orbital angular velocities. To Newtonian order, there is a difference between $\Omega =\sqrt{M_0/d^3}$ in our calculation here and $\Omega =\sqrt{M_{\rm BH}/d^3}$ for the extreme mass ratio. The relative difference between these angular velocities is about 10% for the mass ratio in question.
Comparison with Kerr-Schild coordinates results
-----------------------------------------------
![Comparison of our new results to those calculated using a Kerr-Schild background [@TanigBFS06] for the total angular momentum. The dash-dotted line denotes the results of the third post-Newtonian approximation [@Blanc02].[]{data-label="fig:isovsks"}](fig14.eps){width="8cm"}
In Fig. \[fig:isovsks\], we compare our new results for the total angular momentum with those in [@TanigBFS06], for a mass ratio $M_{\rm irr}^{\rm BH}/M_{\rm B}^{\rm NS}=5$ and neutron star mass $\bar{M}_{\rm B}^{\rm NS}=0.10$. The difference between our treatment here and that in [@TanigBFS06] is the choice of the spatial background metric $\tilde{\gamma}_{ij}$. In this paper, we assume a flat background, $\tilde{\gamma}_{ij} = \eta_{ij}$, while in [@TanigBFS06] we chose a Kerr-Schild metric. We see from Fig. \[fig:isovsks\] that the behavior of the sequence agrees to within 5% [@angmom] throughout most of the sequence.
Comparison with previous flat background results
------------------------------------------------
In [@Grand06] (hereafter G06), Grandclément computed similar sequences of BHNS binaries, adopting an approach that in some ways is similar to ours. Both codes are based on the [Lorene]{} spectral method library, but are otherwise completely independent. In particular, we impose different inner boundary conditions on the black hole’s excision surface, use different conditions to make the black hole nonspinning, and adopt a different decomposition of the equations and variables. We summarize these differences between our implementations in Table \[table:code\].
Here G06
---------------------------------------------------- ------------------------------ ---------------------------------------
$\tilde{\gamma}_{ij}$ $\eta_{ij}$ $\eta_{ij}$
Ex. BC: $\psi$, $\beta^i$ Cook & Pfeiffer [@CookP04] Gourgoulhon [*et.al.*]{} [@GourgGB02]
Ex. BC: $\alpha$ $d \alpha/dr|_{\cal S} =0$ $\alpha|_{\cal S}=0$
$\Omega_r$ of BH Cook & Pfeiffer [@CookP04] Caudill [*et.al.*]{} [@CaudiCGP06]
Dec. of (\[eq:ham\_constr\])-(\[eq:trace\_evolv\]) See Appendix \[appendix:eq\] Gourgoulhon [*et.al.*]{} [@GourgGB02]
: Comparison between our implementation and that of G06. Here “Ex. BC” stands for the excision boundary condition, and “Dec. of (\[eq:ham\_constr\]) – (\[eq:trace\_evolv\])” stands for the decomposition of the relevant equations and variables.
\[table:code\]
![Comparison between our results and those of G06 for the binding energy as a function of $\Omega M_0$, for a binary of mass ratio $M_{\rm irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=5$. The solid line with filled circles marks our results for a neutron star compaction of $M_{\rm ADM,0}^{\rm NS}/R_0=0.150$, which corresponds to the baryon rest mass of $\bar{M}_{\rm B}^{\rm NS}=0.153$. The dashed line with triangles represents those of G06 for a neutron star compaction of 0.150. The dash-dotted line denotes the results of the third post-Newtonian approximation [@Blanc02].[]{data-label="fig:bind_compare"}](fig15.eps){width="8cm"}
To compare our results with those of G06, we show in Fig. \[fig:bind\_compare\] the binding energy as a function of $\Omega M_0$ for two sequences with the same physical parameters. Both sequences are for a binary with mass ratio $M_{\rm irr}^{\rm
BH}/M_{\rm ADM,0}^{\rm NS}=5$. Our results are for a neutron star rest mass of $\bar{M}_{\rm B}^{\rm NS}=0.153$, which corresponds to a compaction of $M_{\rm ADM,0}^{\rm NS}/R_0=0.150$. We find agreement with the third-order post-Newtonian result to within 3% except for configurations inside the turning point, and hence inside the ISCO. We also include the binding energy for a neutron star compaction of 0.150 as computed by G06 (and as found in Fig. 3 of G06). Clearly this curve shows a much larger deviation from both our results and the post-Newtonian ones, by about a factor of two. We speculate that this deviation could be caused by the larger numerical error in the calculations of G06. According to G06, the relative virial error is as large as 2%. If the masses themselves carry an error of 2%, the binding energy may have an error as large as 100% or more, as we have outlined in the paragraph including Eqs. (\[eq:errorADM\]) and (\[eq:error\]). This is consistent with the deviations found in Fig. \[fig:bind\_compare\].
Summary
=======
We construct quasiequilibrium sequences of BHNS binaries in general relativity. We solve the constraint equations of general relativity, decomposed in the conformal thin-sandwich formalism and subject to the black hole boundary conditions of [@CookP04], together with the relativistic equations of hydrostatic equilibrium. In contrast to our earlier approach in [@TanigBFS06], where we solved these equations for a Kerr-Schild background, we now adopt a spatially flat background. We also employ a new decomposition of the equations and variables that leads to a significant improvement in the numerical accuracy.
We construct constant-mass sequences of mass ratios $M_{\rm
irr}^{\rm BH}/M_{\rm ADM,0}^{\rm NS}=1$, 2, 3, 5, and 10 for neutron star baryon rest masses of $\bar{M}_{\rm B}^{\rm NS}=0.15$ and 0.10. We find excellent agreement between our calculated binding energies and the third-order post-Newtonian approximation, especially for large binary separations, but also agreeing to within 5% at small separations. We locate the onset of tidal disruption and, for large mass ratios and neutron star compactions, the ISCO.
We find some evidence that our “leading-order approximation” for constructing nonspinning black holes leads to small errors, and plan to implement an improved condition in the near future.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
J.A.F. gratefully acknowledges support through NSF Grant AST-0401533. This paper was supported in part by NSF Grants PHY-0205155 and PHY-0345151 as well as NASA Grant NNG04GK54G to University of Illinois at Urbana-Champaign, and NSF Grant PHY-0456917 to Bowdoin College.
Decomposition of the gravitational field variables {#appendix:eq}
==================================================
For the construction of binary configurations with codes that are based on spherical polar coordinates it is natural to use two computational domains, each one centered on one of the binary companions. An equation of the form $$\underline{\Delta} \phi = \mbox{RHS}$$ can then be split into two equations $$\begin{aligned}
\label{split}
\underline{\Delta} \phi_{\rm BH} &=& \mbox{RHS}_{\rm BH}
\label{split1} \\
\underline{\Delta} \phi_{\rm NS} &=& \mbox{RHS}_{\rm NS},
\label{split2}\end{aligned}$$ where $\phi = \phi_{\rm BH} + \phi_{\rm NS}$ and $\mbox{RHS} = \mbox{RHS}_{\rm BH} + \mbox{RHS}_{\rm NS}$. The two equations (\[split1\]) and (\[split2\]) can then be solved on two computational domains, one centered on the black hole and the other centered on the neutron star.
Clearly, the separation of the source terms $\mbox{RHS}$ into its two parts $\mbox{RHS}_{\rm BH}$ and $\mbox{RHS}_{\rm NS}$ is not unique. One guiding principle is to move those parts of $\mbox{RHS}$ that are large in the neighborhood of the black hole into $\mbox{RHS}_{\rm BH}$, and likewise for the neutron star. Another principle is that each of the source terms should asymptotically coincide with that for the isolated black hole and neutron star when the orbital separation is large. In addition, we use this decomposition freedom to deal with the fact that metric quantities are not defined inside the excised parts of the black hole. It is not clear how to treat such functions if they appear in $\mbox{RHS}_{\rm NS}$. One approach is to artificially fill the resulting “hole” in the function with a smooth function (see, e.g. [@GourgGB02]). Instead, we carefully separate the source terms in such a way that no metric quantities that are excised in the black hole interior affect the right-hand side of the neutron star equation.
Specifically, we first decompose the metric quantities as $$\begin{aligned}
&&\alpha = \alpha_{\rm BH} +\alpha_{\rm NS}, \\
&&\beta^i = \beta^i_{\rm BH} +\beta^i_{\rm NS}
+\beta^i_{\rm rot}, \\
&&\psi = \psi_{\rm BH} +\psi_{\rm NS},\end{aligned}$$ where $\beta^i_{\rm rot}$ is the rotational shift vector. We then decompose the gravitational field equations as follows. Equation (\[eq:ham\_constr\]) becomes $$\begin{aligned}
&&\underline{\Delta} \psi_{\rm BH} = -{1 \over 8} \psi^{-7}
\tilde{A}_{ij} \tilde{A}^{ij} \nonumber \\
&&\hspace{20pt}+{1 \over 8} (\psi_{\rm NS}+c_{\psi})^{-7}
\tilde{A}_{ij}^{\rm NS} \tilde{A}^{ij}_{\rm NS}, \\
%%%
&&\underline{\Delta} \psi_{\rm NS} = -2 \pi \psi^5 \rho
-{1 \over 8} (\psi_{\rm NS}+c_{\psi})^{-7}
\tilde{A}_{ij}^{\rm NS} \tilde{A}^{ij}_{\rm NS}, \label{app:ham_ns}\end{aligned}$$ Eq. (\[eq:mom\_constr\]) is decomposed as $$\begin{aligned}
&&\underline{\Delta} \beta^i_{\rm BH} +{1 \over 3} \partial^i
(\partial_j \beta^j_{\rm BH}) =2 \tilde{A}^{ij} \partial_j
(\alpha \psi^{-6}) \nonumber \\
&&\hspace{20pt}-2 \tilde{A}^{ij}_{\rm NS} \partial_j
[(\alpha_{\rm NS}+c_{\alpha})
(\psi_{\rm NS}+c_{\psi})^{-6}], \\
%%%
&&\underline{\Delta} \beta^i_{\rm NS} +{1 \over 3} \partial^i
(\partial_j \beta^j_{\rm NS}) =16 \pi \alpha \psi^4 j^i
\nonumber \\
&&\hspace{20pt}+2 \tilde{A}^{ij}_{\rm NS} \partial_j
[(\alpha_{\rm NS}+c_{\alpha})
(\psi_{\rm NS}+c_{\psi})^{-6}], \label{app:mom_ns}\end{aligned}$$ and Eq. (\[eq:trace\_evolv\]) is written as $$\begin{aligned}
&&\underline{\Delta} \alpha_{\rm BH} =\alpha \psi^{-8}
\tilde{A}_{ij} \tilde{A}^{ij} -2 \eta^{ij} \partial_i \alpha
\partial_j \ln \psi \nonumber \\
&&\hspace{20pt}-(\alpha_{\rm NS}+c_{\alpha})
(\psi_{\rm NS}+c_{\psi})^{-8}
\tilde{A}_{ij}^{\rm NS} \tilde{A}^{ij}_{\rm NS} \nonumber \\
&&\hspace{20pt}+2 \eta^{ij} \partial_i (\alpha_{\rm NS}+c_{\alpha})
\partial_j \ln (\psi_{\rm NS}+c_{\psi}), \\
%%%
&&\underline{\Delta} \alpha_{\rm NS} =4 \pi \alpha \psi^4
(\rho +S) \nonumber \\
&&\hspace{20pt}+(\alpha_{\rm NS}+c_{\alpha})
(\psi_{\rm NS}+c_{\psi})^{-8}
\tilde{A}_{ij}^{\rm NS} \tilde{A}^{ij}_{\rm NS} \nonumber \\
&&\hspace{20pt}-2 \eta^{ij} \partial_i (\alpha_{\rm NS}+c_{\alpha})
\partial_j \ln (\psi_{\rm NS}+c_{\psi}), \label{app:trace_ns}\end{aligned}$$ where we define $\bar{A}^{ij}_{\rm NS}$ as $$\tilde{A}^{ij}_{\rm NS} \equiv {(\psi_{\rm NS}+c_{\psi})^6
\over 2 (\alpha_{\rm NS}+c_{\alpha})} \Bigl( \partial^i
\beta^j_{\rm NS} +\partial^j \beta^i_{\rm NS}
-{2 \over 3} \eta^{ij} \partial_k \beta^k_{\rm NS} \Bigr).$$
Note that the neutron star equations (\[app:ham\_ns\]), (\[app:mom\_ns\]), and (\[app:trace\_ns\]) contain metric quantities like $\alpha$ and $\psi$ that are not defined inside the excised black hole, but only in products with matter quantities that vanish outside the neutron star. As a consequence, the neutron star equations are regular everywhere. All singular, or un-defined terms have been moved to the black hole equations, where they are harmless since they are excised from the computational grid.
We finally point out that the outer boundary conditions of the total lapse function and total conformal factor are asymptotically flat, i.e., $\alpha|_{r \rightarrow \infty}=1$ and $\psi|_{r \rightarrow \infty}=1$. This implies that each part of the metric quantities, i.e., $\alpha_{\rm BH}$, $\alpha_{\rm NS}$, $\psi_{\rm BH}$, and $\psi_{\rm NS}$, cannot go to unity individually at infinity. We set the outer boundary conditions for these quantities as $\alpha_{\rm BH}|_{r \rightarrow \infty}=0.5$, $\alpha_{\rm NS}|_{r \rightarrow \infty}=0.5$, $\psi_{\rm BH}|_{r \rightarrow \infty}=0.5$, and $\psi_{\rm NS}|_{r \rightarrow \infty}=0.5$ for the convenience of the computation. We therefore insert constants $c_{\alpha}=0.5$ and $c_{\psi}=0.5$ to ensure that the total quantities take on their proper asymptotic values.
Tables of sequences {#appendix:table}
===================
We summarize our results in Tables \[table:seqM015\] and \[table:seqM01\]. In these tables, we tabulate the orbital angular velocity $\Omega$, binding energy $E_{\rm b}$, total angular momentum $J$, decrease in the maximum density parameter $\delta q_{\rm max}$, minimum of the mass-shedding indicator $\chi_{\rm min}$, and fractional difference $\delta M$ between the ADM mass and the Komar mass along a sequence.
[rcccccc]{}\
$d/M_0$&$\Omega M_0$&$E_{\rm b}/M_0$&$J/M_0^2$& $\delta q_{\rm max}$&$\chi_{\rm min}$&$\delta M$\
15.28 & 0.0153 & -7.34(-3) & 1.107 & -9.56(-5) & 0.961 & 6.66(-6)\
13.10 & 0.0191 & -8.36(-3) & 1.048 & -4.16(-4) & 0.934 & 2.83(-5)\
10.91 & 0.0246 & -9.71(-3) & 0.985 & -1.45(-3) & 0.871 & 1.95(-5)\
9.82 & 0.0285 & -1.06(-2) & 0.952 & -2.87(-3) & 0.803 & 5.10(-6)\
9.17 & 0.0313 & -1.11(-2) & 0.933 & -4.56(-3) & 0.731 & 9.44(-6)\
8.73 & 0.0335 & -1.15(-2) & 0.920 & -6.43(-3) & 0.648 & 2.73(-5)\
8.51 & 0.0347 & -1.18(-2) & 0.914 & -7.80(-3) & 0.575 & 4.76(-5)\
8.30 & 0.0359 & -1.20(-2) & 0.907 & -1.03(-2) & 0.435 & 1.42(-4)\
\
$d/M_0$&$\Omega M_0$&$E_{\rm b}/M_0$&$J/M_0^2$& $\delta q_{\rm max}$&$\chi_{\rm min}$&$\delta M$\
20.37 & 0.0101 & -5.15(-3) & 1.089 & -3.65(-5) & 0.992 & 4.45(-4)\
11.64 & 0.0224 & -8.30(-3) & 0.891 & -1.19(-3) & 0.952 & 1.33(-4)\
8.73 & 0.0334 & -1.04(-2) & 0.815 & -3.49(-3) & 0.870 & 3.13(-5)\
8.00 & 0.0376 & -1.10(-2) & 0.796 & -5.06(-3) & 0.822 & 2.39(-5)\
7.28 & 0.0427 & -1.17(-2) & 0.777 & -7.99(-3) & 0.742 & 3.70(-5)\
6.84 & 0.0464 & -1.22(-2) & 0.765 & -1.12(-2) & 0.661 & 6.78(-5)\
6.55 & 0.0491 & -1.26(-2) & 0.758 & -1.46(-2) & 0.566 & 1.14(-4)\
6.40 & 0.0506 & -1.27(-2) & 0.754 & -1.70(-2) & 0.472 & 1.80(-4)\
\
$d/M_0$&$\Omega M_0$&$E_{\rm b}/M_0$&$J/M_0^2$& $\delta q_{\rm max}$&$\chi_{\rm min}$&$\delta M$\
19.65 & 0.0105 & -4.50(-3) & 0.900 & -1.01(-4) & 0.995 & 6.33(-4)\
13.10 & 0.0189 & -6.41(-3) & 0.778 & -1.15(-3) & 0.981 & 4.37(-4)\
9.82 & 0.0282 & -8.09(-3) & 0.709 & -2.79(-3) & 0.952 & 2.87(-4)\
7.64 & 0.0398 & -9.70(-3) & 0.661 & -5.86(-3) & 0.888 & 1.71(-4)\
6.55 & 0.0489 & -1.07(-2) & 0.638 & -9.61(-3) & 0.808 & 1.31(-4)\
6.01 & 0.0548 & -1.12(-2) & 0.626 & -1.34(-2) & 0.735 & 1.36(-4)\
5.46 & 0.0620 & -1.17(-2) & 0.616 & -2.08(-2) & 0.597 & 1.98(-4)\
5.35 & 0.0636 & -1.18(-2) & 0.614 & -2.31(-2) & 0.543 & 2.32(-4)\
\
$d/M_0$&$\Omega M_0$&$E_{\rm b}/M_0$&$J/M_0^2$& $\delta q_{\rm max}$&$\chi_{\rm min}$&$\delta M$\
18.19 & 0.0117 & -3.56(-3) & 0.642 & -3.87(-4) & 0.997 & 7.20(-4)\
11.64 & 0.0221 & -5.19(-3) & 0.550 & -2.12(-3) & 0.988 & 6.02(-4)\
7.28 & 0.0422 & -7.44(-3) & 0.481 & -7.28(-3) & 0.945 & 4.12(-4)\
5.46 & 0.0617 & -8.71(-3) & 0.454 & -1.43(-2) & 0.856 & 2.86(-4)\
4.59 & 0.0769 & -9.12(-3) & 0.443 & -2.15(-2) & 0.736 & 2.26(-4)\
$\dagger$ 4.37 & 0.0817 & -9.14(-3) & 0.442 & -2.49(-2) & 0.682 & 2.22(-4)\
4.23 & 0.0852 & -9.13(-3) & 0.441 & -2.81(-2) & 0.635 & 2.27(-4)\
4.08 & 0.0889 & -9.08(-3) & 0.440 & -3.23(-2) & 0.566 & 2.36(-4)\
\
$d/M_0$&$\Omega M_0$&$E_{\rm b}/M_0$&$J/M_0^2$& $\delta q_{\rm max}$&$\chi_{\rm min}$&$\delta M$\
11.91 & 0.0212 & -2.82(-3) & 0.326 & -2.39(-3) & 0.997 & 6.21(-4)\
7.94 & 0.0372 & -3.80(-3) & 0.291 & -5.21(-3) & 0.991 & 3.81(-4)\
6.35 & 0.0501 & -4.39(-3) & 0.276 & -7.14(-3) & 0.978 & 2.92(-4)\
5.56 & 0.0596 & -4.73(-3) & 0.269 & -8.57(-3) & 0.963 & 2.69(-4)\
4.57 & 0.0764 & -5.10(-3) & 0.261 & -1.19(-2) & 0.925 & 2.80(-4)\
$\dagger$ 4.25 & 0.0835 & -5.14(-3) & 0.259 & -1.40(-2) & 0.904 & 2.96(-4)\
3.97 & 0.0906 & -5.10(-3) & 0.258 & -1.67(-2) & 0.879 & 3.18(-4)\
3.58 & 0.1028 & -4.82(-3) & 0.257 & -2.37(-2) & 0.823 & 3.72(-4)\
\[table:seqM015\]
[rcccccc]{}\
$d/M_0$&$\Omega M_0$&$E_{\rm b}/M_0$&$J/M_0^2$& $\delta q_{\rm max}$&$\chi_{\rm min}$&$\delta M$\
31.84 & 0.00533 & -3.74(-3) & 1.498 & -3.84(-5) & 0.976 & 4.38(-6)\
25.47 & 0.00738 & -4.59(-3) & 1.362 & -2.00(-4) & 0.949 & 1.44(-5)\
20.70 & 0.00997 & -5.55(-3) & 1.250 & -7.50(-4) & 0.894 & 8.77(-6)\
17.51 & 0.01267 & -6.44(-3) & 1.171 & -2.21(-3) & 0.797 & 1.38(-6)\
15.92 & 0.01453 & -7.00(-3) & 1.130 & -4.31(-3) & 0.683 & 1.03(-5)\
15.28 & 0.01541 & -7.24(-3) & 1.114 & -5.87(-3) & 0.596 & 1.77(-5)\
14.97 & 0.01588 & -7.37(-3) & 1.106 & -6.98(-3) & 0.530 & 2.41(-5)\
14.81 & 0.01613 & -7.43(-3) & 1.102 & -7.68(-3) & 0.486 & 2.89(-5)\
\
$d/M_0$&$\Omega M_0$&$E_{\rm b}/M_0$&$J/M_0^2$& $\delta q_{\rm max}$&$\chi_{\rm min}$&$\delta M$\
33.96 & 0.00483 & -3.15(-3) & 1.364 & -5.89(-5) & 0.990 & 1.19(-4)\
21.23 & 0.00958 & -4.85(-3) & 1.121 & -4.27(-4) & 0.952 & 2.48(-5)\
16.98 & 0.01321 & -5.91(-3) & 1.027 & -1.18(-3) & 0.899 & 1.50(-6)\
13.80 & 0.01774 & -7.07(-3) & 0.951 & -3.57(-3) & 0.785 & 2.83(-6)\
12.74 & 0.01986 & -7.55(-3) & 0.925 & -5.82(-3) & 0.699 & 1.62(-5)\
12.10 & 0.02134 & -7.88(-3) & 0.910 & -8.20(-3) & 0.614 & 3.27(-5)\
11.68 & 0.02244 & -8.10(-3) & 0.899 & -1.06(-2) & 0.523 & 5.11(-5)\
11.46 & 0.02303 & -8.22(-3) & 0.894 & -1.23(-2) & 0.454 & 6.41(-5)\
\
$d/M_0$&$\Omega M_0$&$E_{\rm b}/M_0$&$J/M_0^2$& $\delta q_{\rm max}$&$\chi_{\rm min}$&$\delta M$\
19.10 & 0.0111 & -4.53(-3) & 0.905 & -6.79(-4) & 0.961 & 8.29(-5)\
15.92 & 0.0144 & -5.31(-3) & 0.844 & -1.27(-3) & 0.929 & 5.01(-5)\
12.74 & 0.0198 & -6.41(-3) & 0.779 & -3.13(-3) & 0.849 & 2.51(-5)\
11.94 & 0.0217 & -6.76(-3) & 0.762 & -4.22(-3) & 0.810 & 2.41(-5)\
11.14 & 0.0239 & -7.14(-3) & 0.745 & -5.97(-3) & 0.753 & 2.86(-5)\
10.35 & 0.0264 & -7.57(-3) & 0.728 & -9.05(-3) & 0.665 & 4.46(-5)\
10.03 & 0.0276 & -7.75(-3) & 0.721 & -1.10(-2) & 0.611 & 5.70(-5)\
9.55 & 0.0295 & -8.04(-3) & 0.711 & -1.52(-2) & 0.484 & 8.75(-5)\
\
$d/M_0$&$\Omega M_0$&$E_{\rm b}/M_0$&$J/M_0^2$& $\delta q_{\rm max}$&$\chi_{\rm min}$&$\delta M$\
16.98 & 0.0131 & -3.73(-3) & 0.638 & -1.15(-3) & 0.975 & 1.64(-4)\
12.74 & 0.0197 & -4.79(-3) & 0.575 & -2.57(-3) & 0.936 & 1.20(-4)\
10.61 & 0.0254 & -5.55(-3) & 0.542 & -4.54(-3) & 0.884 & 9.89(-5)\
9.55 & 0.0294 & -6.02(-3) & 0.524 & -6.55(-3) & 0.834 & 9.23(-5)\
8.49 & 0.0345 & -6.56(-3) & 0.507 & -1.04(-2) & 0.748 & 9.67(-5)\
7.96 & 0.0376 & -6.85(-3) & 0.499 & -1.40(-2) & 0.676 & 1.11(-4)\
7.43 & 0.0413 & -7.17(-3) & 0.491 & -2.00(-2) & 0.556 & 1.41(-4)\
7.33 & 0.0421 & -7.23(-3) & 0.489 & -2.17(-2) & 0.518 & 1.49(-4)\
\
$d/M_0$&$\Omega M_0$&$E_{\rm b}/M_0$&$J/M_0^2$& $\delta q_{\rm max}$&$\chi_{\rm min}$&$\delta M$\
11.58 & 0.0224 & -3.02(-3) & 0.330 & -3.52(-3) & 0.976 & 1.58(-4)\
8.11 & 0.0365 & -4.00(-3) & 0.297 & -8.37(-3) & 0.925 & 1.25(-4)\
6.95 & 0.0448 & -4.43(-3) & 0.286 & -1.27(-2) & 0.876 & 1.16(-4)\
5.79 & 0.0568 & -4.88(-3) & 0.276 & -2.22(-2) & 0.770 & 1.24(-4)\
5.50 & 0.0607 & -4.98(-3) & 0.273 & -2.67(-2) & 0.723 & 1.34(-4)\
5.21 & 0.0650 & -5.07(-3) & 0.271 & -3.30(-2) & 0.657 & 1.45(-4)\
4.98 & 0.0688 & -5.13(-3) & 0.270 & -3.99(-2) & 0.579 & 1.54(-4)\
4.93 & 0.0698 & -5.14(-3) & 0.269 & -4.16(-2) & 0.553 & 1.55(-4)\
\[table:seqM01\]
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|
---
abstract: 'While most steps in the modern object detection methods are learnable, the region feature extraction step remains largely hand-crafted, featured by RoI pooling methods. This work proposes a general viewpoint that unifies existing region feature extraction methods and a novel method that is end-to-end learnable. The proposed method removes most heuristic choices and outperforms its RoI pooling counterparts. It moves further towards *fully learnable object detection*.'
author:
- 'Jiayuan Gu$^{1}$[^1], Han Hu$^2$, Liwei Wang$^1$, Yichen Wei$^2$ and Jifeng Dai$^2$'
bibliography:
- 'egbib.bib'
title: Learning Region Features for Object Detection
---
Introduction
============
A noteworthy trait in the deep learning era is that many hand-crafted features, algorithm components, and design choices, are replaced by their data-driven and learnable counterparts. The evolution of object detection is a good example. Currently, the leading region-based object detection paradigm [@girshick2014rich; @he2014spatial; @girshick2015fast; @ren2015faster; @dai2016rfcn; @lin2016feature; @dai2017deformable; @he2017mask] consists of five steps, namely, image feature generation, region proposal generation, region feature extraction, region recognition, and duplicate removal. Most steps become learnable in recent years, including image feature generation [@girshick2015fast], region proposal [@szegedy2014scalable; @erhan2014scalable; @ren2015faster], and duplicate removal [@hosang2017learning; @hu2018relation]. Note that region recognition step is learning based in nature.
The region feature extraction step remains largely hand-crafted. The current practice, RoI (regions of interest) pooling [@girshick2015fast], as well as its variants [@he2014spatial; @he2017mask], divides a region into regular grid bins, computes features of the bin from the image features located nearby to the bin via heuristic rules (avg, max, bilinear interpolation [@he2017mask; @dai2017deformable], etc), and concatenates such features from all the bins as the region features. The process is intuitive and works well, but is more like rules of thumb. There is no clear evidence that it is optimal in some sensible way.
The recent work of deformable RoI pooling [@dai2017deformable] introduces a bin-wise offset that is adaptively learnt from the image content. The approach is shown better than its RoI pooling counterpart. It reveals the potential of making the region feature extraction step *learnable*. However, its form still resembles the regular grid based pooling. The learnable part is limited to bin offsets only.
This work studies *fully learnable* region feature extraction. It aims to improve the performance and enhance the understanding of this step. It makes the following two contributions.
First, a general viewpoint on region feature extraction is proposed. The feature of each bin (or in a general sense, part) of the region is formulated as a weighted summation of image features on different positions over the whole image. Most (if not all) previous region feature extraction methods are shown to be specialization of this formulation by specifying the weights in different ways, mostly hand-crafted.
Based on the viewpoint, the second contribution is a learnable module that represents the weights in terms of the RoI and image features. The weights are affected by two factors: the geometric relation between the RoI and image positions, as well as the image features themselves. The first is modeled using an attention model as motivated by [@vaswani2017attention; @hu2018relation]. The second is exploited by simply adding one convolution layer over the input image features, as motivated by [@dai2017deformable].
The proposed method removes most heuristic choices in the previous RoI pooling methods and moves further towards *fully learnable object detection*. Extensive experiments show that it outperforms its RoI pooling counterparts. While a naive implementation is computationally expensive, an efficient sparse sampling implementation is proposed with little degradation in accuracy. Moreover, qualitative and quantitative analysis on the learnt weights shows that it is feasible and effective to learn the spatial distribution of such weights from data, instead of designing them manually.
A General Viewpoint on Region Feature Extraction {#sec.general_viewpoint}
================================================
Image feature generation step outputs feature maps $\mathbf{x}$ of spatial size $H\times W$ (usually $16\times$ smaller than that of the original image due to down sampling of the network [@ren2015faster]) and $C_f$ channels. Region proposal generation step finds a number of regions of interest (RoI), each a four dimensional bounding box $b$.
In general, the region feature extraction step generates features $\mathbf{y}(b)$ from $\mathbf{x}$ and an RoI $b$ as
$$\mathbf{y}(b) = \text{RegionFeat}(\mathbf{x}, b).$$
Typically, $\mathbf{y}(b)$ is of dimension $K\times C_f$. The channel number is kept the same as $C_f$ in $\mathbf{x}$ and $K$ represents the number of *spatial parts* of the region. Each part feature $\mathbf{y}_k(b)$ is a partial observation of the region. For example, $K$ is the number of bins ([[*e.g.*]{}]{}, $7\times 7$) in the current RoI pooling practice. Each part is a bin in the regular grid of the RoI. Each $\mathbf{y}_k(b)$ is generated from image features in $\mathbf{x}$ within the bin.
The concepts above can be generalized. A part does not need to have a regular shape. The part feature $\mathbf{y}_k(b)$ does not need to come from certain spatial positions in $\mathbf{x}$. Even, the union of all the parts does not need to be the RoI itself. A general formulation is to treat the part feature as the weighted summation of image features $\mathbf{x}$ over all positions within a support region $\Omega_b$, as $$\label{eq.weighted_average_part_feature}
\mathbf{y}_k(b) = \sum_{p \in \Omega_b}w_k(b,p,\mathbf{x})\odot \mathbf{x}(p).$$
Here, $\Omega_b$ is the supporting region. It could simply be the RoI itself or include more context, even the entire image. $p$ enumerates the spatial positions within $\Omega_b$. $w_k(b,p,x)$ is the weight to sum the image feature $\mathbf{x}(p)$ at the position $p$. $\odot$ denotes element-wise multiplication. Note that the weights are assumed normalized, [[*i.e.*]{}]{}, $\sum_{p \in \Omega_b}w_k(b,p,x) = 1$.
We show that various RoI pooling methods [@girshick2015fast; @he2014spatial; @he2017mask; @dai2017deformable] are specializations of Eq. (\[eq.weighted\_average\_part\_feature\]). The supporting region $\Omega_b$ and the weight $w_k(\cdot)$ are realized differently in these methods, mostly in hand-crafted ways.
#### Regular RoI Pooling [@girshick2015fast]
The supporting region $\Omega_b$ is the RoI itself. It is divided into regular grid bins ([[*e.g.*]{}]{}, $7 \times 7$). Each part feature $\mathbf{y}_k(b)$ is computed as max or average of all image features $\mathbf{x}(p)$ where $p$ is within the $k^{th}$ bin.
Taking averaging pooling as an example, the weight in Eq. (\[eq.weighted\_average\_part\_feature\]) is
$$w_k(b, p) =
\begin{cases}
1/|R_{bk}| & \text{ if $p \in R_{bk}$} \\
0 & \text{else}
\end{cases}
\label{eq.weight_in_roi_pooling}$$
Here, $R_{bk}$ is the set of all positions within the $k^{th}$ bin of the grid.
The regular pooling is flawed in that it cannot distinguish between very close RoIs due to spatial down sampling in the networks, [[*i.e.*]{}]{}, the spatial resolution of the image feature $\mathbf{x}$ is usually smaller ([[*e.g.*]{}]{}, $16\times$) than that of the original image. If two RoIs’ distance is smaller than $16$ pixels, their $R_{bk}$s are the same, and so are their features.
#### Spatial Pyramid Pooling [@he2014spatial]
Because it simply applies the regular RoI pooling on different levels of grid divisions, it can be expressed via simple modification of Eq. (\[eq.weighted\_average\_part\_feature\]) and (\[eq.weight\_in\_roi\_pooling\]). Details are irrelevant and omitted here.
#### Aligned RoI Pooling [@he2017mask]
It remedies the quantization issue in the regular RoI pooling above by bilinear interpolation at fractionally sampled positions within each $R_{bk}$. For simplicity, we assume that each bin only samples one point, [[*i.e.*]{}]{}, its center $(u_{bk}, v_{bk})$[^2]. Let the position $p=(u_p,v_p)$. The weight in Eq. (\[eq.weighted\_average\_part\_feature\]) is
$$w_k(b, p) = g(u_p, u_{bk}) \cdot g(v_p, v_{bk}),
\label{eq.weight_in_roi_aligned_pooling}$$
where $g(a,b) = \max(0, 1-|a-b|)$ denotes the 1-D bilinear interpolation weight. Note that the weight in Eq. (\[eq.weight\_in\_roi\_aligned\_pooling\]) is only non-zero for the four positions immediately surrounding the sampling point $(u_{bk}, v_{bk})$.
Because the weight in Eq. (\[eq.weight\_in\_roi\_aligned\_pooling\]) depends on the bin center $(u_{bk}, v_{bk})$, the region features are sensitive to even subtle changes in the position of the RoI. Thus, aligned pooling outperforms its regular pooling counterpart [@he2017mask].
Note that everything till now is hand-crafted. Also, image feature $\mathbf{x}$ is not used in $w_k(\cdot)$ in Eq. (\[eq.weight\_in\_roi\_pooling\]) and (\[eq.weight\_in\_roi\_aligned\_pooling\]).
#### Deformable RoI pooling [@dai2017deformable]
It generalizes aligned RoI pooling by learning an offset $(\delta u_{bk}, \delta v_{bk})$ for each bin and adding it to the bin center. The weight in Eq. (\[eq.weight\_in\_roi\_aligned\_pooling\]) is extended to
$$w_k(b,p,\mathbf{x}) = g(u_p, u_{bk} + \delta u_{bk}) \cdot g(v_p, v_{bk} + \delta v_{bk}).$$
The image feature $\mathbf{x}$ appears here because the offsets are produced by a learnable submodule applied on the image feature $\mathbf{x}$. Specifically, the submodule starts with a regular RoI pooling to extract an initial region feature from image feature, which is then used to regress offsets through an additional learnable fully connected (fc) layer.
As the weight and the offsets depend on the image features now and they are learnt end-to-end, object shape deformation is better modeled, adaptively according to the image content. It is shown that deformable RoI pooling outperforms its aligned version [@dai2017deformable]. Note that when the offset learning rate is zero, deformable RoI pooling strictly degenerates to aligned RoI pooling.
Also note that the supporting region $\Omega_b$ is no longer the RoI as in regular and aligned pooling, but potentially spans the whole image, because the learnt offsets could be arbitrarily large, in principle.
More Related Works
------------------
Besides the RoI pooling methods reviewed above, there are more region feature extraction methods that can be thought of specializations of Eq. (\[eq.weighted\_average\_part\_feature\]) or its more general extension.
#### Region Feature Extraction in One-stage Object Detection [@liu2016ssd; @redmon2016you; @lin2017focal]
As opposed to the two-stage or region based object detection paradigm, another paradigm is one-stage or dense sliding window based. Because the number of windows (regions) is huge, each region feature is simply set as the image feature on the region’s center point, which can be specialized from Eq. (\[eq.weighted\_average\_part\_feature\]) as $K=1$, $\Omega_b = \{\text{center}(b)\}$. This is much faster but less accurate than RoI pooling methods.
#### Pooling using Non-grid Bins [@chen2017masklab; @wu2017interpretable]
These methods are similar to regular pooling but change the definition of $R_{bk}$ in Eq. (\[eq.weight\_in\_roi\_pooling\]) to be non-grid. For example, MaskLab [@chen2017masklab] uses triangle-shaped bins other than rectangle ones. It shows better balance in encoding center-close and center-distant subregions. In Interpretable R-CNN [@wu2017interpretable], the non-grid bins are generated from the grammar defined by an AND-OR graph model.
#### MNC [@dai2016mnc]
It is similar as regular RoI pooling. The difference is that only the bins inside the mask use Eq. (\[eq.weight\_in\_roi\_pooling\]) to compute weights. The weights of the bins outside are zeros. This equals to relax the normalization assumption on $w_k$.
#### Position Sensitive RoI Pooling [@dai2016rfcn; @li2017light]
It is similar as regular RoI pooling. The difference is that each bin only corresponds to a subset of channels in the image feature $\mathbf{x}$, instead of all channels. This can be expressed by extending Eq. (\[eq.weighted\_average\_part\_feature\]) as $$\label{eq.weighted_average_part_feature_extension}
\mathbf{y}_k(b) = \sum_{p \in \Omega_b}w_k(b,p,\mathbf{x}_k)\odot \mathbf{x}_k(p),$$ where $\mathbf{x}_k$ only contains a subset of channels in $\mathbf{x}$, according to the $k^{\text{th}}$ bin.
![Illustration of the proposed region feature extraction module in Eq. (\[eq.weighted\_average\_part\_feature\]) and (\[eq.att\_weight\]).[]{data-label="fig.algorithm_illustration"}](figure/fig1_module_v4.pdf){width="98.00000%"}
Learning Region Features {#sec.learning_region_features}
========================
Regular and aligned RoI pooling are fully hand-crafted. Deformable RoI pooling introduces a learnable component, but its form is still largely limited by the regular grid. In this work, we seek to learn the weight $w_k(b,p,\mathbf{x})$ in Eq. (\[eq.weighted\_average\_part\_feature\]) with minimum hand crafting.
Intuitively, we consider two factors that should affect the weight. First, the geometric relation between the position $p$ and RoI box $b$ is certainly critical. For example, positions within $b$ should contribute more than those far away from it. Second, the image feature $\mathbf{x}$ should be adaptively used. This is motivated by the effectiveness of deformable RoI pooling [@dai2017deformable].
Therefore, the weight is modeled as the exponential of the sum of two terms
$$\label{eq.att_weight}
w_k(b,p,\mathbf{x}) \propto \exp(G_k(b,p)+A_k(\mathbf{x},p)).$$
The first term $G_k(b,p)$ in Eq. (\[eq.att\_weight\]) captures *geometric relation* as
$$\label{eq.geo_att}
G_k(b,p) = \langle W^{\text{box}}_{k}\cdot \mathcal{E}^{\text{box}}(b), W^{\text{im}}\cdot \mathcal{E}^{\text{im}}(p) \rangle.$$
There are three steps. First, the box and image positions are embedded into high dimensional spaces similarly as in [@vaswani2017attention; @hu2018relation]. The embedding is performed by applying sine and cosine functions of varying wavelengths to a scalar $z$, as
$$\mathcal{E}_{2i}(z) = \sin \Large ( \frac{z}{1000^{2i/C_{\mathcal{E}}}} \Large ), \quad \mathcal{E}_{2i+1}(z) = \cos \Large (\frac{z}{1000^{2i/C_{\mathcal{E}}}} \Large ).$$
The embedding vector $\mathcal{E}(z)$ is of dimension $C_{\mathcal{E}}$. The subscript $i$ above ranges from $0$ to $C_{\mathcal{E}}/2-1$. The image position $p$ is embedded into a vector $\mathcal{E}^{\text{im}} (p)$ of dimension $2\cdot C_{\mathcal{E}}$, as $p$ has two coordinates. Similarly, each RoI box $b$ is embedded into a vector $\mathcal{E}^{\text{box}}(b)$ of dimension $4\cdot C_{\mathcal{E}}$.
Second, the embedding vectors $\mathcal{E}^{\text{im}}(p)$ and $\mathcal{E}^{\text{box}}(b)$ are linearly transformed by weight matrices $W^{\text{im}}$ and $W^{\text{box}}_k$, respectively, which are learnable. The transformed vectors are of the same dimension $C_g$. Note that the term $W^{\text{box}}_{k}\cdot \mathcal{E}^{\text{box}}(b)$ has high complexity because the $\mathcal{E}^{\text{box}}(b)$’s dimension $4\cdot C_{\mathcal{E}}$ is large. In our implementation, we decompose $W^{\text{box}}_k$ as $W^{\text{box}}_k = \hat{W}^{\text{box}}_k V^{\text{box}}$. Note that $V^{\text{box}}$ is shared for all the parts. It does not have subscript $k$. Its output dimension is set to $C_{\mathcal{E}}$. In this way, both computation and the amount of parameters are reduced for the term $W^{\text{box}}_{k}\cdot \mathcal{E}^{\text{box}}(b)$.
Last, the inner product of the two transformed vectors is treated as the geometric relation weight.
Eq. (\[eq.geo\_att\]) is basically an attention model [@vaswani2017attention; @hu2018relation], which is a good tool to capture dependency between distant or heterogeneous elements, e.g., words from different languages [@vaswani2017attention], RoIs with variable locations/sizes/aspect ratios [@hu2018relation], and etc, and hence naturally bridges the target of building connections between 4D bounding box coordinates and 2D image positions in our problem. Extensive experiments show that the geometric relations between RoIs and image positions are well captured by the attention model.
The second term $A_k(\mathbf{x},p)$ in Eq. (\[eq.att\_weight\]) uses the *image features* adaptively. It applies an $1\times 1$ convolution on the image feature,
$$\label{eq.app_att}
A_k(\mathbf{x},p) = W^{\text{app}}_k \cdot \mathbf{x}(p),$$
where $ W^{\text{app}}_k$ denotes the convolution kernel weights, which are learnable.
The proposed region feature extraction module is illustrated in Figure \[fig.algorithm\_illustration\]. During training, the image features $\mathbf{x}$ and the parameters in the module ($W^{\text{box}}_k$, $W^{\text{im}}$, and $W^{\text{app}}_k$) are updated simultaneously.
Complexity Analysis and an Efficient Implementation {#sec.compute_complexity}
---------------------------------------------------
[c|l|c || c|l|c]{} & &
---------
typical
values
---------
: **Top**: description and typical values of main variables. **Bottom**: computational complexity of the proposed method. $\dag$Using default maximum sample numbers as in Eq. (\[eq.stride\_b\_in\]) and (\[eq.stride\_b\_out\]), the average actual sample number is about 200. See also Table \[table.aggregation\_point\_sampling\]. \*Note that we decompose $W^{\text{box}}_k$ as $W^{\text{box}}_k = \hat{W}^{\text{box}}_k V^{\text{box}}$, and the total computational cost is the sum of two matrix multiplications $V^{\text{box}} \cdot \mathcal{E}^{\text{box}}$ (the multiplication result is denoted as $\hat{\mathcal{E}}^{\text{box}}$) and $\hat{W}^{\text{box}}_k \cdot \hat{\mathcal{E}}^{\text{box}}$. See also Section \[sec.learning\_region\_features\] for details.[]{data-label="table.complexity"}
& & &
---------
typical
values
---------
: **Top**: description and typical values of main variables. **Bottom**: computational complexity of the proposed method. $\dag$Using default maximum sample numbers as in Eq. (\[eq.stride\_b\_in\]) and (\[eq.stride\_b\_out\]), the average actual sample number is about 200. See also Table \[table.aggregation\_point\_sampling\]. \*Note that we decompose $W^{\text{box}}_k$ as $W^{\text{box}}_k = \hat{W}^{\text{box}}_k V^{\text{box}}$, and the total computational cost is the sum of two matrix multiplications $V^{\text{box}} \cdot \mathcal{E}^{\text{box}}$ (the multiplication result is denoted as $\hat{\mathcal{E}}^{\text{box}}$) and $\hat{W}^{\text{box}}_k \cdot \hat{\mathcal{E}}^{\text{box}}$. See also Section \[sec.learning\_region\_features\] for details.[]{data-label="table.complexity"}
\
$|\Omega_b|$ & size of support region & hundreds & $N$ & $\#$RoIs & 300\
$H$ & height of image feature $\mathbf{x}$ & dozens & $K$ & $\#$parts/bins & 49\
$W$ & width of image feature $\mathbf{x}$ & dozens & $C_{\mathcal{E}}$ & embed dim. in Eq. (\[eq.geo\_att\]) & 512\
$C_f$ & $\#$channels of image feature $\mathbf{x}$ & 256 & $C_g$ & transform dim. in Eq. (\[eq.geo\_att\]) & 256\
[l|c|c|c]{} &
---------------
computational
complexity
---------------
: **Top**: description and typical values of main variables. **Bottom**: computational complexity of the proposed method. $\dag$Using default maximum sample numbers as in Eq. (\[eq.stride\_b\_in\]) and (\[eq.stride\_b\_out\]), the average actual sample number is about 200. See also Table \[table.aggregation\_point\_sampling\]. \*Note that we decompose $W^{\text{box}}_k$ as $W^{\text{box}}_k = \hat{W}^{\text{box}}_k V^{\text{box}}$, and the total computational cost is the sum of two matrix multiplications $V^{\text{box}} \cdot \mathcal{E}^{\text{box}}$ (the multiplication result is denoted as $\hat{\mathcal{E}}^{\text{box}}$) and $\hat{W}^{\text{box}}_k \cdot \hat{\mathcal{E}}^{\text{box}}$. See also Section \[sec.learning\_region\_features\] for details.[]{data-label="table.complexity"}
&
-------------------
*naive*
($|\Omega_b|$=HW)
-------------------
: **Top**: description and typical values of main variables. **Bottom**: computational complexity of the proposed method. $\dag$Using default maximum sample numbers as in Eq. (\[eq.stride\_b\_in\]) and (\[eq.stride\_b\_out\]), the average actual sample number is about 200. See also Table \[table.aggregation\_point\_sampling\]. \*Note that we decompose $W^{\text{box}}_k$ as $W^{\text{box}}_k = \hat{W}^{\text{box}}_k V^{\text{box}}$, and the total computational cost is the sum of two matrix multiplications $V^{\text{box}} \cdot \mathcal{E}^{\text{box}}$ (the multiplication result is denoted as $\hat{\mathcal{E}}^{\text{box}}$) and $\hat{W}^{\text{box}}_k \cdot \hat{\mathcal{E}}^{\text{box}}$. See also Section \[sec.learning\_region\_features\] for details.[]{data-label="table.complexity"}
&
---------------------------
*efficient*
($|\Omega_b|$=200$^\dag$)
---------------------------
: **Top**: description and typical values of main variables. **Bottom**: computational complexity of the proposed method. $\dag$Using default maximum sample numbers as in Eq. (\[eq.stride\_b\_in\]) and (\[eq.stride\_b\_out\]), the average actual sample number is about 200. See also Table \[table.aggregation\_point\_sampling\]. \*Note that we decompose $W^{\text{box}}_k$ as $W^{\text{box}}_k = \hat{W}^{\text{box}}_k V^{\text{box}}$, and the total computational cost is the sum of two matrix multiplications $V^{\text{box}} \cdot \mathcal{E}^{\text{box}}$ (the multiplication result is denoted as $\hat{\mathcal{E}}^{\text{box}}$) and $\hat{W}^{\text{box}}_k \cdot \hat{\mathcal{E}}^{\text{box}}$. See also Section \[sec.learning\_region\_features\] for details.[]{data-label="table.complexity"}
\
(P1) [ transform position embedding in Eq. (\[eq.geo\_att\])]{} & $2 H W C_{\mathcal{E}} C_g$ & 0.59G & 0.59G\
(P2) [transform RoI box embedding in Eq. (\[eq.geo\_att\])]{} & $N C_{\mathcal{E}} (KC_g+4C_{\mathcal{E}})$\* & 2.1G & 2.1G\
(P3) inner product in Eq. (\[eq.geo\_att\]) & $N K |\Omega_b| C_g$ & 7.2G & **0.72G**\
(P4) appearance usage in Eq. (\[eq.app\_att\]) & $H W K C_f$ &0.03G & 0.03G\
(P5) weighted aggregation in Eq. (\[eq.weighted\_average\_part\_feature\]) & $N K |\Omega_b| C_f$ & 7.2G & **0.72G**\
sum & & 17.1G & 4.16G\
The computational complexity of the proposed region feature extraction module is summarized in Table \[table.complexity\]. Note that $A_k(x,p)$ and $W^{\text{im}}\cdot \mathcal{E}^{\text{im}}(p)$ are computed over all the positions in the image feature $\mathbf{x}$ and shared for all RoIs.
A *naive* implementation needs to enumerate all the positions in $\Omega_b$. When $\Omega_b$ spans the whole image feature $\mathbf{x}$ densely, its size is $H\times W$ and typically a few thousands. This incurs heavy computational overhead for step 3 and 5 in Table \[table.complexity\]. An *efficient* implementation is to sparsely sample the positions in $\Omega_b$, during the looping of $p$ in Eq. (\[eq.weighted\_average\_part\_feature\]). Intuitively, the sampling points within the RoI should be denser and those outside could be sparser. Thus, $\Omega_b$ is split into two sets as $\Omega_b=\Omega_b^{\text{In}}\cup \Omega_b^{\text{Out}}$, which contain the positions within and outside of the RoI, respectively. Note that $\Omega_b^{\text{Out}}$ represents the context of the RoI. It could be either empty when $\Omega_b$ is the RoI or span the entire image when $\Omega_b$ does, too.
Complexity is controlled by specifying a maximum number of sampling positions for $\Omega_b^{\text{In}}$ and $\Omega_b^{\text{Out}}$, respectively (by default, 196 for both). Given an RoI $b$, the positions in $\Omega_b^{\text{In}}$ are sampled at stride values $stride_{\text{x}}^{b}$ and $stride_{\text{y}}^{b}$, in x and y directions, respectively. The stride values are determined as $$\label{eq.stride_b_in}
stride_{\text{x}}^{b} = \lceil W_b/\sqrt{196} \rceil \text{ AND } stride_{\text{y}}^{b} = \lceil H_b/\sqrt{196} \rceil,$$ where $W_b$ and $H_b$ are the width and height of the RoI. The sampling of $\Omega_b^{\text{Out}}$ is similar. Let $stride^{\text{out}}$ be the stride value, it is derived by, $$\label{eq.stride_b_out}
stride^{\text{out}} = \lceil \sqrt{HW/196} \rceil.$$
The sparse sampling of $\Omega_b$ effectively reduces the computational overhead. Especially, notice that many RoIs have smaller area than the maximum sampling number specified above. So the actual number of sampled positions of $\Omega_b^{\text{In}}$ in those RoIs is equal to their area, thus even smaller.
Experiments show that the accuracy of sparse sampling is very close to the naive dense sampling (see Table \[table.aggregation\_point\_sampling\]).
Experiments
===========
All experiments are performed on COCO detection datasets [@lin2014coco]. We follow the COCO 2017 dataset split: 115k images in the *train* split for training; 5k images in the *minival* split for validation; and 20k images in the *test-dev* split for testing. In most experiments, we report the accuracy on the *minival* split.
State-of-the-art Faster R-CNN [@ren2015faster] and FPN [@lin2016feature] object detectors are used. ResNet-50 and ResNet-101 [@he2016deep] are used as the backbone image feature extractor. By default, Faster R-CNN with ResNet-50 is utilized in ablation study.
For Faster R-CNN, following the practice in [@dai2016rfcn; @dai2017deformable], the *conv4* and *conv5* image features are utilized for region proposal generation and object detection, respectively. The RPN branch is the same as in [@ren2015faster; @dai2016rfcn; @dai2017deformable]. For object detection, the effective feature stride of *conv5* is reduced from 32 pixels to 16 pixels. Specifically, at the beginning of the *conv5* block, stride is changed from 2 to 1. The dilation of the convolutional filters in the *conv5* block is changed from 1 to 2. On top of the *conv5* feature maps, a randomly initialized $1 \times 1$ convolutional layer is added to reduce the dimension to 256-D. The proposed module is applied on top to extract regional features, where 49 bins are utilized by default. Two fully-connected (fc) layers of 1024-D, followed by the classification and the bounding box regression branches, are utilized as the detection head. The images are resized to 600 pixels at the shorter side if the longer side after resizing is less than or equal to 1000; otherwise resized to 1000 pixels at the longer side, in both training and inference [@girshick2015fast].
For FPN, a feature pyramid is built upon an input image of single resolution, by exploiting multi-scale feature maps generated by top-down and lateral connections. The RPN and Fast R-CNN heads are attached to the multi-scale feature maps, for proposing and detecting objects of varying sizes. Here we follow the network design in [@lin2016feature], and just replace RoI pooling by the proposed learnable region feature extraction module.
The images are resized to 800 pixels at the shorter side if the longer side after resizing is less than or equal to 1333; otherwise resized to 1333 pixels at the longer side, in both training and inference.
SGD training is performed on 4 GPUs with 1 image per GPU. Weight decay is $1 \times 10^{-4}$ and momentum is $0.9$. The added parameters in the learnable region feature extraction module, $W^{\text{box}}_k$, $W^{\text{im}}$, and $W_k^{\text{app}}$, are initialized by random Gaussian weights ($\sigma = 0.01$), and their learning rates are kept the same as the existing layers. In both Faster R-CNN and FPN, to facilitate experiments, separate networks are trained for region proposal generation and object detection, without sharing their features. In Faster R-CNN, 6 and 16 epochs are utilized to train the RPN and the object detection networks, respectively. The learning rates are set as $2\times 10^{-3}$ for the first $\frac{2}{3}$ iterations and $2\times 10^{-4}$ for the last $\frac{1}{3}$ iterations, for both region proposal and object detection networks. In FPN, 12 epochs are utilized to train both the RPN and the object detection networks, respectively. For both networks training, the learning rates start with $5\times 10^{-3}$ and decay twice at 8 and 10.667 epochs, respectively. Standard NMS with IoU threshold of 0.5 is utilized for duplication removal.
[l|c|cc|ccc]{}
--------
method
--------
: Comparison of three region feature extraction methods using different support regions. Accuracies are reported on COCO detection *minival* set. \*It is not clear how to exploit the whole image for regular and aligned RoI pooling methods. Hence the corresponding accuracy numbers are omitted.[]{data-label="table.aggregation_range"}
&
-----
mAP
-----
: Comparison of three region feature extraction methods using different support regions. Accuracies are reported on COCO detection *minival* set. \*It is not clear how to exploit the whole image for regular and aligned RoI pooling methods. Hence the corresponding accuracy numbers are omitted.[]{data-label="table.aggregation_range"}
& mAP$_{50}$ & mAP$_{75}$ &
---------
mAP$_S$
---------
: Comparison of three region feature extraction methods using different support regions. Accuracies are reported on COCO detection *minival* set. \*It is not clear how to exploit the whole image for regular and aligned RoI pooling methods. Hence the corresponding accuracy numbers are omitted.[]{data-label="table.aggregation_range"}
&
---------
mAP$_M$
---------
: Comparison of three region feature extraction methods using different support regions. Accuracies are reported on COCO detection *minival* set. \*It is not clear how to exploit the whole image for regular and aligned RoI pooling methods. Hence the corresponding accuracy numbers are omitted.[]{data-label="table.aggregation_range"}
&
---------
mAP$_L$
---------
: Comparison of three region feature extraction methods using different support regions. Accuracies are reported on COCO detection *minival* set. \*It is not clear how to exploit the whole image for regular and aligned RoI pooling methods. Hence the corresponding accuracy numbers are omitted.[]{data-label="table.aggregation_range"}
\
\
regular RoI pooling & 29.8 & 52.2 & 29.9 & 10.4 & 32.6 & 47.8\
aligned RoI pooling & 32.9 & 54.0 & 34.9 & 13.9 & 36.9 & 48.8\
ours & **33.4** & 54.5 & 35.2 & 13.9 & 37.3 & 50.4\
\
regular RoI pooling & 30.1 & 53.2 & 30.6 & 10.6 & 33.3 & 47.4\
aligned RoI pooling & 32.8 & 54.6 & 35.1 & 14.2 & 37.0 & 48.5\
ours & **33.8** & 55.1 & 35.8 & 14.2 & 37.8 & 51.1\
\
regular RoI pooling\* & - & - & - & - & - & -\
aligned RoI pooling\* & - & - & - & - & - & -\
ours & **34.3** & 56.0 & 36.4 & 15.4 & 38.1 & 51.9\
Ablation Study
--------------
### Effect of supporting region $\Omega$.
It is investigated in Table \[table.aggregation\_range\]. Three sizes of the supporting region $\Omega$ are compared: the RoI itself, the RoI expanded with twice the area (with the same center), and the whole image range. Regular and aligned RoI pooling are also compared[^3].
There are two observations. First, our method outperforms the other two pooling methods. Second, our method steadily improves from using larger support regions, indicating that exploiting contextual information is helpful. Yet, using larger support regions, e.g., $2\times$ RoI region, has minor and no improvements for regular and aligned RoI pooling, respectively, when compared to using $1\times$ RoI region. Moreover, it is unclear how to exploit the whole image for regular and aligned pooling in a reasonable way.
### Effect of sparse sampling.
Table \[table.aggregation\_point\_sampling\] presents the results of using different numbers of sampling positions for efficient implementation. By utilizing proper number of sampling positions, the accuracy can be very close to that of naive dense enumeration. And the computational overhead can be significantly reduced thanks to the sparse sampling implementation. By default, 196 maximum sampling positions are specified for both $\Omega_b^{\text{In}}$ and $\Omega_b^{\text{Out}}$. The mAP score is 0.2 lower than that of dense enumeration. In runtime, large RoIs will have fewer sampling positions for $\Omega_b^{\text{Out}}$ and small RoIs will have fewer sampling positions than the maximum threshold for $\Omega_b^{\text{In}}$. The average counted sampling positions in runtime are are around 114 and 86 for $\Omega_b^{\text{In}}$ and $\Omega_b^{\text{Out}}$, respectively, as shown in Table \[table.aggregation\_point\_sampling\]. The corresponding computational cost is 4.16G FLOPS, which coarsely equals that of the 2-fc head (about 3.9G FLOPs).
For all the following experiments, our method will utilize the sparse sampling implementation with 196 maximum sampling positions for both $\Omega_b^{\text{In}}$ and $\Omega_b^{\text{Out}}$.
[cc|c|cc|ccc|ccc]{}
----------------------------------------
$|\Omega_b^{\text{Out}}|_{\text{max}}$
----------------------------------------
: Detection accuracy and computational times of *efficient* method using different number of sample points. The average samples $|\Omega_b^{\text{Out}}|_\text{avg}$ and $|\Omega_b^{\text{In}}|_\text{avg}$ are counted on COCO *minival* set using 300 ResNet-50 RPN proposals. The bold row ($|\Omega_b^{\text{Out}}|_{\text{max}}=196$, $|\Omega_b^{\text{In}}|_{\text{max}}=14^2$) are used as our default maximum sample point number. \**full* indicates that all image positions are used without any sampling.[]{data-label="table.aggregation_point_sampling"}
& $|\Omega_b^{\text{In}}|_{\text{max}}$ &
-----
mAP
-----
: Detection accuracy and computational times of *efficient* method using different number of sample points. The average samples $|\Omega_b^{\text{Out}}|_\text{avg}$ and $|\Omega_b^{\text{In}}|_\text{avg}$ are counted on COCO *minival* set using 300 ResNet-50 RPN proposals. The bold row ($|\Omega_b^{\text{Out}}|_{\text{max}}=196$, $|\Omega_b^{\text{In}}|_{\text{max}}=14^2$) are used as our default maximum sample point number. \**full* indicates that all image positions are used without any sampling.[]{data-label="table.aggregation_point_sampling"}
&
------------
mAP$_{50}$
------------
: Detection accuracy and computational times of *efficient* method using different number of sample points. The average samples $|\Omega_b^{\text{Out}}|_\text{avg}$ and $|\Omega_b^{\text{In}}|_\text{avg}$ are counted on COCO *minival* set using 300 ResNet-50 RPN proposals. The bold row ($|\Omega_b^{\text{Out}}|_{\text{max}}=196$, $|\Omega_b^{\text{In}}|_{\text{max}}=14^2$) are used as our default maximum sample point number. \**full* indicates that all image positions are used without any sampling.[]{data-label="table.aggregation_point_sampling"}
&
------------
mAP$_{75}$
------------
: Detection accuracy and computational times of *efficient* method using different number of sample points. The average samples $|\Omega_b^{\text{Out}}|_\text{avg}$ and $|\Omega_b^{\text{In}}|_\text{avg}$ are counted on COCO *minival* set using 300 ResNet-50 RPN proposals. The bold row ($|\Omega_b^{\text{Out}}|_{\text{max}}=196$, $|\Omega_b^{\text{In}}|_{\text{max}}=14^2$) are used as our default maximum sample point number. \**full* indicates that all image positions are used without any sampling.[]{data-label="table.aggregation_point_sampling"}
&
---------
mAP$_S$
---------
: Detection accuracy and computational times of *efficient* method using different number of sample points. The average samples $|\Omega_b^{\text{Out}}|_\text{avg}$ and $|\Omega_b^{\text{In}}|_\text{avg}$ are counted on COCO *minival* set using 300 ResNet-50 RPN proposals. The bold row ($|\Omega_b^{\text{Out}}|_{\text{max}}=196$, $|\Omega_b^{\text{In}}|_{\text{max}}=14^2$) are used as our default maximum sample point number. \**full* indicates that all image positions are used without any sampling.[]{data-label="table.aggregation_point_sampling"}
&
---------
mAP$_M$
---------
: Detection accuracy and computational times of *efficient* method using different number of sample points. The average samples $|\Omega_b^{\text{Out}}|_\text{avg}$ and $|\Omega_b^{\text{In}}|_\text{avg}$ are counted on COCO *minival* set using 300 ResNet-50 RPN proposals. The bold row ($|\Omega_b^{\text{Out}}|_{\text{max}}=196$, $|\Omega_b^{\text{In}}|_{\text{max}}=14^2$) are used as our default maximum sample point number. \**full* indicates that all image positions are used without any sampling.[]{data-label="table.aggregation_point_sampling"}
&
---------
mAP$_L$
---------
: Detection accuracy and computational times of *efficient* method using different number of sample points. The average samples $|\Omega_b^{\text{Out}}|_\text{avg}$ and $|\Omega_b^{\text{In}}|_\text{avg}$ are counted on COCO *minival* set using 300 ResNet-50 RPN proposals. The bold row ($|\Omega_b^{\text{Out}}|_{\text{max}}=196$, $|\Omega_b^{\text{In}}|_{\text{max}}=14^2$) are used as our default maximum sample point number. \**full* indicates that all image positions are used without any sampling.[]{data-label="table.aggregation_point_sampling"}
& $|\Omega_b^{\text{Out}}|_\text{avg}$ & $|\Omega_b^{\text{In}}|_\text{avg}$ & FLOPS\
*full*\* & $7^2$ & 33.4 & 55.6 & 35.3 & 14.1 & 37.2 & 50.7 & 1737 & 32 & 15.3G\
*full* & $14^2$ & 34.2 & 56.2 & 36.3 & 15.0 & 38.5 & 51.3 & 1737 & 86 & 15.7G\
*full* & $21^2$ & 34.1 & 56.0 & 35.9 & 14.5 & 38.3 & 51.1 & 1737 & 158 & 16.2G\
*full* & *full* & 34.3 & 56.0 & 36.4 & 15.4 & 38.1 & 51.9 & 1737 & 282 & 17.1G\
100 & $14^2$ & 33.8 & 55.5 & 35.9 & 14.3 & 38.0 & 50.8 & 71 & 86& 3.84G\
**196** & $\mathbf{14^2}$ & 34.1 & 55.6 & 36.3 & 14.5 & 38.3 & 51.1 & **114** & **86** & **4.16G**\
400 & $14^2$ & 34.0 & 55.7 & 36.0 & 14.4 & 38.4 & 51.0 & 194 & 86 & 4.72G\
625 & $14^2$ & 34.1 & 55.7 & 36.1 & 14.5 & 38.0 & 51.3 & 432 & 86& 6.42G\
*full* & $14^2$ & 34.2 & 56.2 & 36.3 & 15.0 & 38.5 & 51.3 & 1737 & 86 & 15.7G\
[l|c|cc|ccc]{}
--------
method
--------
: Effect of geometric and appearance terms in Eq. (\[eq.att\_weight\]) for the proposed region feature extraction module. Detection accuracies are reported on COCO *minival* set.[]{data-label="table.attentional_weight"}
&
-----
mAP
-----
: Effect of geometric and appearance terms in Eq. (\[eq.att\_weight\]) for the proposed region feature extraction module. Detection accuracies are reported on COCO *minival* set.[]{data-label="table.attentional_weight"}
& mAP$_{50}$ & mAP$_{75}$ &
---------
mAP$_S$
---------
: Effect of geometric and appearance terms in Eq. (\[eq.att\_weight\]) for the proposed region feature extraction module. Detection accuracies are reported on COCO *minival* set.[]{data-label="table.attentional_weight"}
&
---------
mAP$_M$
---------
: Effect of geometric and appearance terms in Eq. (\[eq.att\_weight\]) for the proposed region feature extraction module. Detection accuracies are reported on COCO *minival* set.[]{data-label="table.attentional_weight"}
&
---------
mAP$_L$
---------
: Effect of geometric and appearance terms in Eq. (\[eq.att\_weight\]) for the proposed region feature extraction module. Detection accuracies are reported on COCO *minival* set.[]{data-label="table.attentional_weight"}
\
regular RoI pooling & 29.8 & 52.2 & 29.9 & 10.4 & 32.6 & 47.8\
aligned RoI pooling & 32.9 & 54.0 & 34.9 & 13.9 & 36.9 & 48.8\
deformable pooling & 34.0 & 55.3 & 36.0 & 14.7 & 38.3 & 50.4\
our (geometry) & **33.2** & 55.2 & 35.4 & 14.2 & 37.0 & 50.0\
our (geometry+appearance) & **34.1** & 55.6 & 36.3 & 14.5 & 38.3 & 51.1\
### Effect of geometric relation and appearance feature terms.
Table \[table.attentional\_weight\] studies the effect of geometric relation and appearance feature terms in Eq. (\[eq.att\_weight\]) of the proposed module. Using geometric relation alone, the proposed module is slightly better than aligned RoI pooling, and is noticeably better than regular RoI pooling. By further incorporating the appearance feature term, the mAP score rises by 0.9 to 34.1. The accuracy is on par with deformable RoI pooling, which also exploits appearance features to guide the region feature extraction process.
### Comparison on stronger detection backbones.
We further compare the proposed module with regular, aligned and deformable versions of RoI pooling on stronger detection backbones, where FPN and ResNet-101 are also utilized.
Table \[table.backbones\] presents the results on COCO *test-dev* set. Using the stronger detection backbones, the proposed module also achieves on par accuracy with deformable RoI pooling, which is noticeably better than aligned and regular versions of RoI pooling. We achieve a final mAP score of 39.9 using FPN+ResNet-101 by the proposed fully learnable region feature extraction module.
[l|l|c|cc|ccc]{}
----------
backbone
----------
: Comparison of different algorithms using different backbones. Accuracies on COCO *test-dev* are reported.[]{data-label="table.backbones"}
& method &
-----
mAP
-----
: Comparison of different algorithms using different backbones. Accuracies on COCO *test-dev* are reported.[]{data-label="table.backbones"}
& mAP$_{50}$ & mAP$_{75}$ &
---------
mAP$_S$
---------
: Comparison of different algorithms using different backbones. Accuracies on COCO *test-dev* are reported.[]{data-label="table.backbones"}
&
---------
mAP$_M$
---------
: Comparison of different algorithms using different backbones. Accuracies on COCO *test-dev* are reported.[]{data-label="table.backbones"}
&
---------
mAP$_L$
---------
: Comparison of different algorithms using different backbones. Accuracies on COCO *test-dev* are reported.[]{data-label="table.backbones"}
\
& regular RoI pooling & 29.9 & 52.6 & 30.1 & 9.7 & 31.9 & 46.3\
& aligned RoI pooling & 33.1 & 54.5 & 35.1 & 13.9 & 36.0 & 47.4\
& deformable RoI pooling & 34.2 & 55.7 & 36.7 & 14.5 & 37.4 & 48.8\
& our & **34.5** & 56.4 & 36.4 & 14.6 & 37.4 & 50.3\
& regular RoI pooling & 32.7 & 53.6 & 23.7 & 11.4 & 35.2 & 50.0\
& aligned RoI pooling & 35.6 & 57.1 & 38.0 & 15.3 & 39.3 & 51.0\
& deformable RoI pooling & **36.4** & 58.1 & 39.3 & 15.7 & 40.2 & 52.1\
& our & **36.4** & 58.6 & 38.6 & 15.3 & 40.2 & 52.2\
& regular RoI pooling & 35.9 & 59.0 & 38.4 & 19.6 & 38.8 & 45.4\
& aligned RoI pooling & 36.7 & 59.1 & 39.4 & 20.9 & 39.5 & 46.3\
& deformable RoI pooling & 37.7 & 60.6 & 40.9 & 21.3 & 40.7 & 47.4\
& our & **37.8** & 60.9 & 40.7 & 21.3 & 40.4 & 48.0\
& regular RoI pooling & 38.5 & 61.5 & 41.8 & 21.4 & 42.0 & 49.2\
& aligned RoI pooling & 39.1 & 61.4 & 42.3 & 21.5 & 42.5 & 50.2\
& deformable RoI pooling & **40.0** & 62.7 & 43.5 & 22.4 & 43.4 & 51.3\
& our & 39.9 & 63.1 & 43.1 & 22.2 & 43.4 & 51.6\
[p[0.99]{}]{} ![Qualitative analysis of learnt weights. For visualization, all weights are normalized by the maximum value over all image positions and half-half matted with the original image.[]{data-label="figure.att_weights"}](figure/fig2_initial_final_att_weights_v3.pdf "fig:"){width="99.00000%"}\
[(a) The initial (**left**) and final (**right**) weights $w_k(*)$ in Eq. (\[eq.att\_weight\]) of two given RoIs (the red boxes). The center images show the maximum value of all $K=49$ weight maps. The smaller images around show 4 individual weight maps. ]{}\
\
![Qualitative analysis of learnt weights. For visualization, all weights are normalized by the maximum value over all image positions and half-half matted with the original image.[]{data-label="figure.att_weights"}](figure/fig3_geo_app_att_v2.pdf "fig:"){width="99.00000%"}\
[(b) Example results of geometric weights (**top**), appearance weights (**median**) and final weights (**bottom**).]{}\
What is learnt? {#sec.learnt_weights}
===============
#### Qualitative Analysis
The learnt weights $w_k(*)$ in Eq. (\[eq.att\_weight\]) are visualized in Figure \[figure.att\_weights\] (a). The supporting region $\Omega$ is the whole image.
Initially, the weights $w_k(*)$ are largely random on the whole image. After training, weights in different parts are learnt to focus on different areas on the RoI, and they mostly focus on the instance foreground.
To understand the role of the geometric and appearance terms in Eq. (\[eq.att\_weight\]), Figure \[figure.att\_weights\] (b) visualizes the weights when either of them is ignored. It seems that the geometric weights mainly attend to the RoI, while the appearance weight focuses on all instance foreground.
#### Quantitative Analysis
For each part $k$, the weights $w_k(*)$ are treated as a probability distribution over all the positions in the supporting region $\Omega$, as $\sum_{p \in \Omega}w_k(b,p,\mathbf{x}) = 1$. KL divergence is used to measure the discrepancy between such distributions.
We firstly compare the weights in different parts. For each ground truth object RoI, KL divergence value is computed between all pairs of $w_{k_1}(*)$ and $w_{k_2}(*)$, $k_1,k_2=1,...,49$. Such values are then averaged, called *mean KL between parts* for the RoI. Figure \[figure.att\_weights\_quantitative\] (left) shows its value averaged over objects of three sizes (as defined by COCO dataset) during training. Initially, the weights of different parts are largely indistinguishable. Their KL divergence measure is small. The measure grows dramatically after the first test. This indicates that *the different parts are learnt to focus on different spatial positions*. Note that the divergence is larger for *large* objects, which is reasonable.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Quantitative analysis of learnt weights. The two plots are *mean KL between parts* (**left**) and *KL of mask* (**right**) during training, respectively. Note that we test KL divergence every two epochs since our training framework saves model weights using such frequency.[]{data-label="figure.att_weights_quantitative"}](figure/kl_leftbin_rightmask_v4.pdf "fig:"){width="99.00000%"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
We then investigate how the weights resemble the instance foreground, by comparing them to the ground-truth instance foreground mask in COCO. Towards this, for each ground truth object RoI, the weights from all the parts are aggregated together by taking the maximum value at each position, resulting in a “max pooled weight map". The map is then normalized as a distribution (sum is 1). The ground truth object mask is filled with 1 and 0. It is also normalized as a distribution. KL divergence between these two distributions is called *KL of mask*. Figure \[figure.att\_weights\_quantitative\] (right) shows this measure averaged over objects of three sizes during training. It quickly becomes small, indicating that *the aggregation of all part weights is learnt to be similar as the object mask*.
The second observation is especially interesting, as it suggests that learning the weights as in Eq. (\[eq.att\_weight\]) is related to instance segmentation, in some implicit manner. This is worth more investigation in the future work.
[^1]: This work is done when Jiayuan Gu is an intern at Microsoft Research Asia.
[^2]: In practical implementation [@he2017mask], multiple ([[*e.g.*]{}]{}, 4) points are sampled within each bin and their features are averaged as the bin feature. This is beneficial as more image position features get back-propagated gradients.
[^3]: Deformable RoI pooling [@dai2017deformable] is omitted as it does not have a fixed support region.
|
---
abstract: |
Hadwiger’s conjecture asserts that any graph contains a clique minor with order no less than the chromatic number of the graph. We prove that this well-known conjecture is true for all graphs if and only if it is true for squares of split graphs. Since all split graphs are chordal, this implies that Hadwiger’s conjecture is true for all graphs if and only if it is true for squares of chordal graphs. It is known that $2$-trees are a class of chordal graphs. We further prove that Hadwiger’s conjecture is true for squares of $2$-trees. In fact, we prove the following stronger result: for any $2$-tree $T$, its square $T^2$ has a clique minor of order $\chi(T^2)$ for which each branch set is a path, where $\chi(T^2)$ is the chromatic number of $T^2$. As a corollary, we obtain that the same statement holds for squares of generalized $2$-trees, and so Hadwiger’s conjecture is true for such squares, where a generalized $2$-tree is a graph constructed recursively by introducing a new vertex and making it adjacent to a single existing vertex or two adjacent existing vertices in each step, beginning with the complete graph of two vertices.
*Key words*: Hadwiger’s conjecture; minors; split graph; chordal graph; 2-tree; generalized 2-tree; square of a graph
*AMS subject classification*: 05C15, 05C83
author:
- |
L. Sunil Chandran$^{1}$[^1], Davis Issac$^{2}$ and Sanming Zhou$^{3}$[^2]\
\
[$^{1}$ Indian Institute of Science, Bangalore -560012, India]{}\
[`[email protected]`]{}\
[$^{2}$ Max Planck Institute for Informatics, Saarland Informatics Campus, Germany]{}\
[`[email protected]`]{}\
[$^{3}$ School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia]{}\
[`[email protected]`]{}
bibliography:
- 'hadwiger.bib'
title: 'Hadwiger’s conjecture and squares of chordal graphs'
---
0.4
Introduction {#sec:introduction}
============
A graph $H$ is called a *minor* of a graph $G$ if a graph isomorphic to $H$ can be obtained from a subgraph of $G$ by contracting edges. An *$H$-minor* is a minor isomorphic to $H$, and a *clique minor* is a $K_t$-minor for some positive integer $t$, where $K_t$ is the complete graph of order $t$. The *Hadwiger number* of $G$, denoted by $\eta(G)$, is the largest integer $t$ such that $G$ contains a $K_t$-minor. A graph is called *$H$-minor free* if it does not contain an $H$-minor.
In 1937, Wagner [@wagner] proved that the Four Color Conjecture is equivalent to the following statement: If a graph is $K_5$-minor free, then it is $4$-colorable. In 1943, Hadwiger [@hadwiger1943klassifikation] proposed the following conjecture which is a far reaching generalization of the Four Color Theorem.
For any integer $t \ge 1$, every $K_{t+1}$-minor free graph is $t$-colorable.
Hadwiger’s conjecture is well known to be a challenging problem. Bollobás, Catlin and Erdős [@Bollobs1980195] describe it as “one of the deepest unsolved problems in graph theory”. Hadwiger himself [@hadwiger1943klassifikation] proved the conjecture for $t = 3$. (The conjecture is trivially true for $t=1,2$.) In view of Wagner’s result [@wagner], Hadwiger’s conjecture for $t = 4$ is equivalent to the Four Color Conjecture, the latter being proved by Appel and Haken [@appel1977everyi; @appel1977everyii] in 1977. In 1993, Robertson, Seymour and Thomas [@robertson1993hadwiger] proved that Hadwiger’s conjecture is true for $t = 5$. The conjecture remains unsolved for $t \ge 6$, though for $t=6$ Kawarabayashi and Toft [@kawarabayashi2005any] proved that any graph that is $K_7$-minor free and $K_{4,4}$-minor free is $6$-colorable.
Hadwiger’s conjecture has been proved for several classes of graphs, including line graphs [@reed2004hadwiger], proper circular arc graphs [@belkale2009hadwiger], quasi-line graphs [@Chudnovsky:2008:HCQ:1400140.1400143], 3-arc graphs [@WXZ], complements of Kneser graphs [@XZ], and powers of cycles and their complements [@li2007hadwiger]. Since Hadwiger’s conjecture is equivalent to the statement that $\eta(G) \ge \chi(G)$ for any graph $G$, where $\chi(G)$ is the chromatic number of $G$, there is also an extensive body of work on the Hadwiger number. See for example [@chandran2008hadwiger] for a study of the Hadwiger number of the Cartesian product of two graphs and [@GHPP] for a recent work on the Hadwiger number of graphs with small chordality.
It would be helpful if one could reduce Hadwiger’s conjecture for all graphs to some special classes of graphs. In this paper we establish a result of this type. A graph is called a *split graph* if its vertex set can be partitioned into an independent set and a clique. The *square* of a graph $G$, denoted by $G^2$, is the graph with the same vertex set as $G$ such that two vertices are adjacent if and only if the distance between them in $G$ is equal to $1$ or $2$. The first main result in this paper is as follows.
\[thm:hadsquare\] Hadwiger’s conjecture is true for all graphs if and only if it is true for squares of split graphs.
A graph is called a *chordal graph* if it contains no induced cycles of length at least $4$. Since split graphs form a subclass of the class of chordal graphs, Theorem \[thm:hadsquare\] implies:
\[coro:hadsquare\] Hadwiger’s conjecture is true for all graphs if and only if it is true for squares of chordal graphs.
These results show that squares of chordal or split graphs capture the complexity of Hadwiger’s conjecture, though they may not make the conjecture easier to prove.
In light of Corollary \[coro:hadsquare\], it would be interesting to study Hadwiger’s conjecture for squares of some subclasses of chordal graphs in the hope of getting new insights into the conjecture. As a step towards this, we prove that Hadwiger’s conjecture is true for squares of $2$-trees. It is well known that chordal graphs are precisely those graphs that can be constructed by recursively applying the following operation a finite number of times beginning with a clique: Choose a clique in the current graph, introduce a new vertex, and make this new vertex adjacent to all vertices in the chosen clique. If we begin with a $k$-clique and choose a $k$-clique at each step, then the graph constructed this way is called a *$k$-tree*, where $k$ is a fixed positive integer. The second main result in this paper is as follows.
\[thm:2tree\] Hadwiger’s conjecture is true for squares of $2$-trees. Moreover, for any $2$-tree $T$, $T^2$ has a clique minor of order $\chi(T^2)$ for which all branch sets are paths.
A graph is called a *generalized $2$-tree* if it can be obtained by allowing one to join a fresh vertex to a clique of order $1$ or $2$ instead of exactly $2$ in the above-mentioned construction of $2$-trees. (This notion is different from the concept of a partial $2$-tree which is defined as a subgraph of a $2$-tree.) The class of generalized $2$-trees contains all $2$-trees as a proper subclass. The following corollary is implied by (and equivalent to) Theorem \[thm:2tree\].
\[coro:2tree\] Hadwiger’s conjecture is true for squares of generalized $2$-trees. Moreover, for any generalized $2$-tree $G$, $G^2$ has a clique minor of order $\chi(G^2)$ for which all branch sets are paths.
In general, in proving Hadwiger’s conjecture it is interesting to study the structure of the branch sets forming a clique minor of order no less than the chromatic number. Theorem \[thm:2tree\] and Corollary \[coro:2tree\] provide this kind of information for squares of $2$-trees and generalized $2$-trees respectively.
A *quasi-line graph* is a graph such that the neighborhood of every vertex can be partitioned into two (not necessarily non-empty) cliques. We call a graph $G$ a *generalized quasi-line graph* if for any $\emptyset \ne S \subseteq V(G)$ there exists a vertex $u \in S$ such that the set of neighbors of $u$ in $S$ can be partitioned into two vertex-disjoint (not necessarily non-empty) cliques. It is evident that quasi-line graphs are trivially generalized quasi-line graphs, but the converse is not true. We observe that the square of any 2-tree is a generalized quasi-line graph but not necessarily a quasi-line graph. Chudnovsky and Fradkin [@Chudnovsky:2008:HCQ:1400140.1400143] proved that Hadwiger’s conjecture is true for all quasi-line graphs. It would be interesting to investigate whether Hadwiger’s conjecture is true for all generalized quasi-line graphs. Theorem \[thm:2tree\] can be thought as a step towards this direction: It shows that Hadwiger’s conjecture is true for a special class of generalized quasi-line graphs that is not contained in the class of quasi-line graphs.
All graphs considered in the paper are finite, undirected and simple. As usual the vertex and edge sets of a graph $G$ are denoted by $V(G)$ and $E(G)$, respectively. If $u$ and $v$ are adjacent in $G$, then $uv$ denotes the edge joining them. As usual we use $\chi(G)$ and $\omega(G)$ to denote the chromatic and clique numbers of $G$, respectively. A proper coloring of $G$ using exactly $\chi(G)$ colors is called an *optimal coloring* of $G$. A graph $G$ is *$t$-colorable* if $t \ge \chi(G)$.
An $H$-minor of a graph $G$ can be thought as a family of $t = |V(H)|$ vertex-disjoint connected subgraphs $G_1, \ldots, G_t$ of $G$ such that the graph constructed in the following way is isomorphic to $H$: Identify all vertices of each $G_i$ to obtain a single vertex $v_i$, and draw an edge between $v_i$ and $v_j$ if and only if there exists at least one edge of $G$ between $V(G_i)$ and $V(G_j)$. Each subgraph $G_i$ in the family is called a [*branch set*]{} of the minor $H$. This equivalent definition of a minor will be used throughout the paper.
Proof of Theorem \[thm:hadsquare\] {#sec:hadsquare}
==================================
It suffices to prove that if Hadwiger’s conjecture is true for squares of all split graphs then it is also true for all graphs.
So we assume that Hadwiger’s conjecture is true for squares of split graphs. Let $G$ be an arbitrary graph with at least two vertices. Since deleting isolated vertices does not affect the chromatic or Hadwiger number, without loss of generality we may assume that $G$ has no isolated vertices. Construct a split graph $H$ from $G$ as follows: For each vertex $x$ of $G$, introduce a vertex $v_x$ of $H$, and for each edge $e$ of $G$, introduce a vertex $v_e$ of $H$, with the understanding that all these vertices are pairwise distinct. Denote $$S = \{v_x: x \in V(G)\},\;\, C = \{v_e: e \in E(G)\}.$$ Construct $H$ with vertex set $V(H) = S \cup C$ in such a way that no two vertices in $S$ are adjacent, any two vertices in $C$ are adjacent, and $v_x \in S$ is adjacent to $v_e \in C$ if and only if $x$ and $e$ are incident in $G$. Obviously, $H$ is a split graph as its vertex set can be partitioned into the independence set $S$ and the clique $C$.
*Claim 1:* The subgraph of $H^2$ induced by $S$ is isomorphic to $G$.
In fact, for distinct $x, y \in V(G)$, $v_x$ and $v_y$ are adjacent in $H^2$ if and only if they have a common neighbor in $H$. Clearly, this common neighbor has to be from $C$, say $v_e$ for some $e \in E(G)$, but this happens if and only if $x$ and $y$ are adjacent in $G$ and $e=xy$. Therefore, $v_x$ and $v_y$ are adjacent in $H^2$ if and only if $x$ and $y$ are adjacent in $G$. This proves Claim 1.
*Claim 2:* In $H^2$ every vertex of $S$ is adjacent to every vertex of $C$.
This follows from the fact that $C$ is a clique of $H$ and $x$ is incident to at least one edge in $G$.
*Claim 3:* $\chi(H^2) = \chi(G) + |C|$.
In fact, by Claim 1 we may color the vertices of $S$ with $\chi(G)$ colors by using an optimal coloring of $G$ (that is, choose an optimal coloring $\phi$ of $G$ and assign the color $\phi(x)$ to $v_x$ for each $x \in V(G)$). We then color the vertices of $C$ with $|C|$ other colors, one for each vertex of $C$. It is evident that this is a proper coloring of $H^2$ and hence $\chi(H^2) \le \chi(G) + |C|$. On the other hand, since $C$ is a clique, it requires $|C|$ distinct colors in any proper coloring of $H^2$. Also, by Claim 2 none of these $|C|$ colors can be assigned to any vertex of $S$ in any proper coloring of $H^2$, and by Claim 1 the vertices of $S$ needs at least $\chi(G)$ colors in any proper coloring of $H^2$. Therefore, $\chi(H^2) \ge \chi(G) + |C|$ and Claim 3 is proved.
*Claim 4:* $\eta(H^2) = \eta(G) + |C|$.
To prove this claim, consider the branch sets of $G$ that form a clique minor of $G$ with order $\eta(G)$, and take the corresponding branch sets in the subgraph of $H^2$ induced by $S$. Take each vertex of $C$ as a separate branch set. Clearly, these branch sets produce a clique minor of $H^2$ with order $\eta(G) + |C|$. Hence $\eta(H^2) \ge \eta(G) + |C|$.
To complete the proof of Claim 4, consider an arbitrary clique minor of $H^2$, say, with branch sets $B_1,B_2,\ldots,B_k$. Define $B_i' = B_i$ if $B_i \cap C = \emptyset$ (that is, $B_i \subseteq S$) and $B_i' = B_i \cap C$ if $B_i \cap C \ne \emptyset$. It can be verified that $B_1',B_2',\ldots,B_k'$ also produce a clique minor of $H^2$ with order $k$. Thus, if $k > \eta(G) + |C|$, then there are more than $\eta(G)$ branch sets among $B_1',B_2',\ldots,B_k'$ that are contained in $S$. In view of Claim 1, this means that $G$ has a clique minor of order strictly bigger than $\eta(G)$, contradicting the definition of $\eta(G)$. Therefore, any clique minor of $H^2$ must have order at most $\eta(G) + |C|$ and the proof of Claim 4 is complete.
Since we assume that Hadwiger’s conjecture is true for squares of split graphs, we have $\eta(H^2) \ge \chi(H^2)$. This together with Claims 3-4 implies $\eta(G) \ge \chi(G)$; that is, Hadwiger’s conjecture is true for $G$. This completes the proof of Theorem \[thm:hadsquare\].
Proof of Theorem \[thm:2tree\] {#sec:2tree}
==============================
Prelude {#subsec:pre}
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By the definition of a $k$-tree given in the previous section, a 2-tree is a graph that can be recursively constructed by applying the following operation a finite number of times beginning with $K_2$: Pick an edge $e=uv$ in the current graph, introduce a new vertex $w$, and add edges $uw$ and $vw$ to the graph. We say that $e$ is *processed* in this *step* of the construction. We also say that $w$ is a *vertex-child* of $e$, each of $uw$ and $vw$ is an *edge-child* of $e$, $e$ is the *parent* of each of $w, uw$ and $vw$, and $uw$ and $vw$ are *siblings* of each other. An edge $e_2$ is said to be an *edge-descendant* of an edge $e_1$, if either $e_2 = e_1$, or recursively, the parent of $e_2$ is an *edge-descendant* of $e_1$. A vertex $v$ is said to be a *vertex-descendant* of an edge $e$ if $v$ is a *vertex-child* of an *edge-descendant* of $e$.
An edge $e$ may be processed in more than one step. If necessary, we can change the order of edge-processing so that $e$ is processed in consecutive steps but the same $2$-tree is obtained. *So without loss of generality we may assume that for each edge $e$ all the steps in which $e$ is processed occur consecutively.*
We now define a *level* for each edge and each vertex as follows. Initially, the level of the first edge and its end-vertices is defined to be $0$. Inductively, any vertex-child or edge-child of an edge with level $k$ is said to have level $k+1$. Observe that two edges that are siblings of each other have the same level. If there exists a pair of edges $e, f$ with levels $i, j$ respectively such that $i < j$ and the batch of consecutive steps where $e$ is processed is immediately after the batch of consecutive steps where $f$ is processed, then we can move the batch of steps where $e$ is processed to the position immediately before the processing of $f$ without changing the structure of the $2$-tree. We repeat this procedure until no such a pair of edges exists. So without loss of generality we may assume that a *breadth-first ordering* is used when processing edges, that is, edges of level $i$ are processed before edges of level $j$ whenever $i<j$.
To prove Theorem \[thm:2tree\], we will prove $\eta(T^2) \ge \chi(T^2)$ for any $2$-tree $T$. In the simplest case where $\chi(T^2)=2$, this inequality is true as $T^2$ has at least one edge and so contains a $K_2$-minor. Moreover, in this case both branch sets of this $K_2$-minor are singletons (and so are paths of length $0$).
In what follows $T$ is an arbitrary 2-tree with $\chi(T^2) \ge 3$. Denote by $T_i$ the $2$-tree obtained after the $i^{\textrm{th}}$ step in the construction of $T$ as described above. Then there is a unique positive integer $\ell$ such that $\chi(T^2)=\chi(T^2_{\ell})=\chi(T^2_{{\ell}-1})+1$. Define $$G=T_{\ell}.$$ We will prove that ${\eta({G^2})}\ge {\chi({G^2})}$ and $G^2$ has a clique minor of order ${\chi({G^2})}$ for which each branch set is a path. Once this is achieved, we then have $\eta(T^2) \ge \eta(G^2) \ge \chi(G^2) = \chi(T^2)$ and $T^2$ contains a clique minor of order $\chi(T^2)$ whose branch sets are paths, as required to complete the proof of Theorem \[thm:2tree\].
Given $X \subseteq V(G)$, define $$N(X) = \{v \in V(G) \setminus X: \text{$v$ is adjacent in $G$ to at least one vertex in $X$}\}.$$ Define $$N[X] = N(X) \cup X,\;\, N_2[X]=N[N[X]],\;\, N_2(X)=N_2[X]\setminus X.$$ In particular, for $x \in V(G)$, we write $N(x)$, $N[x]$, $N_2(x)$, $N_2[x]$ in place of $N(\{x\})$, $N[\{x\}]$, $N_2(\{x\})$, $N_2[\{x\}]$, respectively.
Denote by ${\ell_{max}}$ the maximum level of any edge of $G$. Then the maximum level of any vertex in $G$ is also ${\ell_{max}}$. Observe that the level of the last edge processed is ${\ell_{max}}-1$, and none of the edges with level ${\ell_{max}}$ has been processed at the completion of the $\ell$-th step, due to the breadth-first ordering of processing edges. Obviously, ${\ell_{max}}\le \ell$.
If $l_{max} = 0$ or $1$, then $G^2$ is a complete graph and so $\chi(G^2) = \omega(G^2) = \eta(G^2)$. Moreover, $G^2$ contains a clique minor of order $\chi(G^2)$ for which each branch set is a path of length $0$. Hence the result is true when $l_{max} = 0$ or $1$.
We assume $l_{max} \ge 2$ in the rest of the proof. We will prove a series of lemmas that will be used in the proof of Theorem \[thm:2tree\]. See Figure \[fig:relations\] for relations among some of these lemmas.
![Lemmas to be proved and their relations.[]{data-label="fig:relations"}](relations.pdf)
Pivot coloring, pivot vertex and its proximity
----------------------------------------------
\[pivot\] There exist an optimal coloring $\mu$ of $G^2$ and a vertex $p$ of $G$ at level $l_{max}$ such that $p$ is the only vertex with color $\mu(p)$.
Let $v$ be the vertex introduced in step $\ell$. Then $v$ has level $l_{max}$. By the definition of $G = T_{\ell}$, there exists a proper coloring of $T^2_{\ell-1}$ using $\chi(G^2)-1$ colors. Extend this coloring to $G^2$ by assigning a new color to $v$. This extended coloring $\phi$ is an optimal coloring of $G^2$ under which $v$ is the only vertex with color $\phi(v)$.
Note that, apart from the pair $(\phi, v)$ in the proof above, there may be other pairs $(\mu, p)$ with the property in Lemma \[pivot\].
In the remaining part of the paper, we will use the following notation and terminology (see Figure \[fig:mainfig\] for an illustration):
- $\mu, p$: an optimal coloring of $G^2$ and a vertex of $G$, respectively, as given in Lemma \[pivot\]; we fix a pair $(\mu, p)$ such that the minimum level of the vertices in $N(p)$ is as large as possible; we call this particular $\mu$ the *pivot coloring* and this particular $p$ the *pivot vertex*;
- $uw$: the parent of $p$;
- $t$: the vertex such that $w$ is a child of $ut$, so that the level of $ut$ is ${\ell_{max}}-2$, and $uw$ and $wt$ are siblings with level ${\ell_{max}}-1$ (the existence of $t$ is ensured by the fact ${\ell_{max}}\ge 2$);
- $B$: the set of vertex-children of $wt$;
- $C$: the set of vertex-children of $uw$;
- $\mu(X) = \{\mu(x): x \in X\}$, for any subset $X \subseteq V(G)$;
- when we say the color of a vertex, we mean the color of the vertex under the coloring $\mu$, unless stated otherwise.
\[lemma:allcolors\] All colors used by $\mu$ are present in $N_2[p]$.
If there is a color $c$ used by $\mu$ that is not present in $N_2\left[ p \right]$, then we can re-color $p$ with $c$. Since $p$ is the only vertex with color $\mu(p)$ under $\mu$, we then obtain a proper coloring of $G^2$ with $\chi(G^2)-1$ colors, which is a contradiction.
![Vertex subsets of the 2-tree $G$ used in the proof of Theorem \[thm:2tree\].[]{data-label="fig:mainfig"}](mainsets)
\[unprocessed\] $N(b)=\{w,t\}$ for any $b\in B$, and $N(c)=\{u,w\}$ for any $c\in C$.
Since both $bw$ and $bt$ have level ${\ell_{max}}$, they have not been processed at the completion of the $\ell^{\textrm{th}}$ step. Hence the first statement is true. The second statement can be proved similarly.
Define $$F = (N(u)\cap N(t)) \setminus \{w\}$$ $$C' = \{x\in N(t): \mu(x) \in \mu(C)\}$$ $$A = N(t)\setminus (B \cup F \cup C' \cup \{u, w\}).$$ Note that $C' \subseteq N(t)\setminus (B \cup F \cup \{u, w\})$ and $\{A, C'\}$ is a partition of $N(t)\setminus (B \cup F \cup \{u, w\})$.
\[lemma:AA\] $\mu(A)\subseteq \mu(N(u)\setminus (C \cup F \cup \{w,t\}))$.
Let $a\in A$. Clearly, $\mu(a) \notin \mu(N_2(a))$. On the other hand, $\mu(a) \in \mu(N_2\left[ p \right])$ by Lemma \[lemma:allcolors\]. So $\mu(a) \in \mu (N_2[p]\setminus N_2(a))$. Since $N_2[p]\setminus N_2(a)\subseteq N(u)\setminus (F \cup \{w, t\})$, it follows that $\mu(a) \in \mu(N(u) \setminus (F \cup \{w,t\}))$. By the definition of $A$, we also have $\mu(a) \notin \mu(C)$. Therefore, $\mu(a) \in \mu(N(u)\setminus (C \cup F \cup \{w,t\}))$.
By Lemma \[lemma:AA\], for each color $c\in \mu(A)$, there is a $c$-colored vertex in $N(u)\setminus (C \cup F \cup \{w,t\})$. On the other hand, no two vertices in $N(u)$ can have the same color. So each color in $\mu(A)$ is used by exactly one vertex in $N(u)$. Let $$A' = \{x \in N(u): \mu(x) \in \mu(A)\}.$$ Then $A' \subseteq N(u)\setminus (C \cup F \cup \{w,t\})$ and $$\mu(A')=\mu(A).$$ Since no two vertices in $A$ ($A'$, respectively) are colored the same, the relation $\mu(a) = \mu(a')$ defines a bijection $a \mapsto a'$ from $A$ to $A'$. We call $a$ and $a'$ the mates of each other and denote the relation by $$a = {\mathtt{mate}}(a'),\;\, a' = {\mathtt{mate}}(a).$$ Note that $a \ne a'$ as $A$ and $A'$ are disjoint. Define $$Q=N(u)\setminus (A' \cup C \cup F \cup \{ w,t\}).$$ Then $\{A', Q\}$ is a partition of $N(u)\setminus (C \cup F \cup \{w,t\})$.
Define $$D = \{x\in B: \mu(x)\notin \mu(N(u))\}$$ $$Q' = B \setminus D$$ Then $A',A,C,C',D,F,Q,Q',\{u,w,t\}$ are pairwise disjoint. See Figure \[fig:mainfig\] for an illustration of these sets.
A few lemmas
------------
\[lemma:empty\] Suppose $D=\emptyset$. Then ${\eta({G^2})}\ge {\chi({G^2})}$. Moreover, ${\chi({G^2})}= \omega(G^2)$ and so $G^2$ contains a clique minor of order ${\chi({G^2})}$ for which each branch set is a singleton.
Since $D=\emptyset$, we have $N_2[p]=N[u]\cup Q'$. So by Lemma \[lemma:allcolors\] all colors of $\mu$ are present in $N[u]\cup Q'$. However, $\mu(Q')\subseteq \mu(N[u])$ by the definition of $Q'$. So all colors of $\mu$ are present in $N[u]$. Since $N[u]$ is a clique of $G^2$, it follows that ${\chi({G^2})}= |N[u]| \le {\omega(G^2)}$. Therefore, ${\chi({G^2})}={\omega(G^2)}\le {\eta({G^2})}$.
\[lemma:duniquecolor\] For any $d\in D$, no vertex in $N_2[p]$ other than $d$ is colored $\mu(d)$.
Suppose that there is a vertex in $N_2[p] \setminus \{d\}$ with color $\mu(d)$. Such a vertex must be in $N_2[p]\setminus N_2[d]$. However, $N_2[p]\setminus N_2[d]=Q\cup A'$, but $\mu(d)\notin \mu(Q)$ by the definition of $D$ and $\mu(d)\notin \mu(A')=\mu(A)$ as $A\subseteq N_2[d]$. This contradiction proves the result.
\[lemma:QQ\] Suppose $D \ne \emptyset$. Then $\mu(Q)=\mu(Q')$.
We prove $\mu(Q')\subseteq \mu(Q)$ first. By the definition of $Q'$, $\mu(Q')\subseteq \mu(N(u))$. Clearly, $\mu(Q')\cap \mu(N_2(Q'))=\emptyset$, and $\mu(Q')\cap \mu(A')=\emptyset$ as $\mu(A')=\mu(A)$. Hence $\mu(Q') \subseteq \mu(N(u) \setminus (N_2(Q') \cup A'))$. However, $N(u)\setminus (N_2(Q') \cup A') = Q$. Therefore, $\mu(Q')\subseteq \mu(Q)$.
Now we prove $\mu(Q)\subseteq \mu(Q')$. Suppose otherwise. Say, $q \in Q$ satisfies $\mu(q) \notin \mu(Q')$. Since $D \ne \emptyset$ by our assumption, we may take a vertex $d \in D$. We claim that $\mu(q)\notin N_2(d)$. This is because $N_2(d)\setminus N_2[q]\subseteq A \cup C' \cup Q' \cup D $, but $\mu(q)\notin \mu(A)=\mu(A')$, $\mu(q)\notin \mu(C')\subseteq \mu(C)$, $\mu(q)\notin \mu(Q')$, and $\mu(q)\notin \mu(D)$ by the definition of $D$. So we can recolor $d$ with $\mu(q)$. By Lemma \[lemma:duniquecolor\], we can then recolor $p$ with $\mu(d)$. In this way we obtain a proper coloring of $G^2$ with $\chi(G^2)-1$ colors, which is a contradiction. Hence $\mu(Q)\subseteq \mu(Q')$.
\[lemma:Aempty\] Suppose $D \ne \emptyset$ but $A=\emptyset$. Then ${\eta({G^2})}\ge {\chi({G^2})}$. Moreover, ${\chi({G^2})}= \omega(G^2)$ and so $G^2$ contains a clique minor of order ${\chi({G^2})}$ for which each branch set is a singleton.
Since $A=\emptyset$, we have $A'=\emptyset$ and $\mu(N_2[p])=\mu(N[w]\cup F)$ by Lemma \[lemma:QQ\]. By Lemma \[lemma:allcolors\], $|\mu(N_2[p])|=\chi(G^2)$. On the other hand, $N[w]\cup F$ is a clique of $G^2$ and so $|\mu(N[w] \cup F)| \le \omega(G^2)$. So $\chi(G^2)=|\mu(N_2[p])|=|\mu(N[w]\cup F)| \le \omega(G^2)$, and therefore $\chi(G^2) = \omega(G^2) \le\eta(G^2)$.
*Due to Lemmas \[lemma:empty\] and \[lemma:Aempty\], in the rest of the proof we assume without mentioning explicitly that $D \ne \emptyset$ and $A \ne \emptyset$ (so that $A' \ne \emptyset$).*
\[lmax2\] The following hold:
- ${\ell_{max}}\ge 3$;
- the level of $u$ is ${\ell_{max}}-2$.
\(a) We have assumed ${\ell_{max}}\ge 2$. Suppose ${\ell_{max}}=2$ for the sake of contradiction. Since $A'\neq\emptyset$ and $D\neq \emptyset$ by our assumption, we can take $a'\in A'$ and $d\in D$. Since ${\ell_{max}}=2$, we have that $ut$ is the only edge with level $0$, and moreover $V(G)=N[\{u,t\}]$.
We claim that no vertex in $N_2[a']$ is colored $\mu(d)$ under the coloring $\mu$. Suppose otherwise. Say, $d_1$ is such a vertex. Then $d_1\neq d$ as $a'\notin N(\{w,t\})$. We have $d_1\notin N(u)$ by the definition of $D$. We also have $d_1 \notin N(t)$ for otherwise two distinct vertices in $N(t)$ have the same color. Thus, $d_1 \notin N(u) \cup N(t) = N[\{u,t\}] = V(G)$, a contradiction. Therefore, no vertex in $N_2[a']$ is colored $\mu(d)$.
So we can recolor $a'$ with color $\mu(d)$ but retain the colors of all other vertices. In this way we obtain another proper coloring of $G^2$. Observe that $a'$ was the only vertex in $N_2[p]$ with color $\mu(a')$ under $\mu$ as $N_2[p] \subseteq {N_2[a']} \cup N_2(a)$, where $a = {\mathtt{mate}}(a') \notin N_2[p]$. Since $a'$ has been recolored $\mu(d)$, we can recolor $p$ with $\mu(a')$ to obtain a proper coloring of $G^2$ using fewer colors than $\mu$, but this contradicts the optimality of $\mu$.
\(b) Suppose otherwise. Since the level of $ut$ is ${\ell_{max}}-2$, the level of $t$ must be ${\ell_{max}}-2$ and the level of $u$ must be smaller than ${\ell_{max}}-2$. Since $D\neq \emptyset$ by our assumption, we may take a vertex $d \in D$. Denote by $\mu'$ the coloring obtained by exchanging the colors of $d$ and $p$ (while keeping the colors of all other vertices). By Lemma \[lemma:duniquecolor\], $\mu'$ is a proper coloring of $G^2$. Note that $d$ is the only vertex with color $\mu'(d) = \mu(p)$ under the coloring $\mu'$. The minimum level of a vertex in $N(d)$ is ${\ell_{max}}-2$, and the minimum level of a vertex in $N(p)$ is smaller than ${\ell_{max}}-2$ since the level of $u$ is smaller than ${\ell_{max}}-2$. However, this means that we would have selected respectively $\mu'$ and $d$ as the pivot coloring and pivot vertex instead of $\mu$ and $p$, which is a contradiction.
In the sequel we fix a vertex $s \in F$ such that $ut$ is a child of $st$. The existence of $s$ is ensured by Lemma \[lmax2\]. Note that the level of $st$ is $l_{max}-3$, and $us$ is the sibling of $ut$ and has level ${\ell_{max}}-2$.
Bichromatic paths
-----------------
*Given a proper coloring $\phi$ of $G^2$ and two distinct colors $r$ and $g$, a path in $G^2$ is called a *$(\phi,r,g)$-bichromatic path* if its vertices are colored $r$ or $g$ under the coloring $\phi$.*
\[lemma:pathD\] For any $a'\in A'$ and $d\in D$, there exists a $(\mu,\mu(a'),\mu(d))$-bichromatic path from $a'$ to ${\mathtt{mate}}(a')$ in $G^2$.
Let $a={\mathtt{mate}}(a')$. Denote $r = \mu(a')$ ($=\mu(a)$) and $g = \mu(d)$. Then $r \ne g$ as $d \in N_2(a)$. Consider the subgraph $H$ of $G^2$ induced by the set of vertices with colors $r$ and $g$ under $\mu$. Let $H'$ be the connected component of $H$ containing $a'$. It suffices to show that $a$ is contained in $H'$.
Suppose to the contrary that $a \not \in V(H')$. Define $$\mu'(v) = \left\{
\begin{array}{ll}
\mu(v), & \mbox{if } v\in V(G)\setminus (V(H')\cup \{p\}) \\
r, & \mbox{if } v = p \\
r, & \mbox{if $v\in V(H')$ and $\mu(v)=g$} \\
g, & \mbox{if $v\in V(H')$ and $\mu(v)=r$.}
\end{array}
\right.$$ In particular, $\mu'(a')=g$. We will prove that $\mu'$ is a proper coloring of $G^2$, which will be a contradiction as $\mu'$ uses less colors than $\mu$. Since exchanging colors $r$ and $g$ within $H'$ does not destroy the properness of the coloring, in order to prove the properness of $\mu'$, it suffices to prove that $N_2(p)$ does not contain any vertex with color $\mu'(p)$ under $\mu'$. Suppose otherwise. Say, $v\in N_2(p)$ satisfies $\mu'(v)=\mu'(p)=r$. Consider first the case when $v\in V(H')$. In this case, we have $\mu(v)=g$, and so $v=d$ since by Lemma \[lemma:duniquecolor\], $d$ is the only vertex in $N_2[p]$ with color $g$ under $\mu$. On the other hand, $d\notin V(H')$ as $a\notin V(H')$ is the only vertex in $N_2[d]$ with color $r$ under $\mu$. Hence $v\notin V(H')$, which is a contradiction. Now consider the case when $v\notin V(H')$. In this case, we have $\mu(v)=r$. Since $N_2[p]\subseteq {N_2[a']}\cup N_2(a)$, $a'$ is the only vertex in $N_2[p]$ with color $r$ under $\mu$. So $v=a'\in V(H')$, which is again a contradiction.
\[lemma:l-1\] For any edge $e=xy$ with level ${\ell_{max}}-2$ and any vertex-descendant $z$ of $e$, we have $N_2(z)\subseteq N[\{x, y\}]$.
Consider an arbitrary vertex $v$ in $N_2(z)$. Since the level of $e$ is ${\ell_{max}}-2$, there are only two possibilities for $z$. The first possibility is that $z$ is a vertex-child of $e$. In this possibility, either $v$ is a vertex-child of $x z$ or $yz$, or $v \in \{x, y\}$, or $v \in N(x) \cup N(y)$; in each case we have $v \in N[\{x, y\}]$. The second possibility is that $z$ is the vertex-child of an edge-child of $e$. Without loss of generality we may assume that $z$ is the vertex-child of $x q$, where $q$ is a vertex-child of $e$. Then either $v$ is a vertex-child of $yq$, or $v \in N(x)$; in each case we have $v \in N[\{x, y\}]$.
\[lemma:n2adjacentuts\] The following hold:
- $N_2(A'\cup Q)\subseteq N[\{u,t,s\}]$;
- if $v\in N_2(A'\cup Q)$ and $\mu(v)\in \mu(B)$, then $v\in N(\{u,s\})$;
- if $v\in N_2(A'\cup Q)$ and $\mu(v)\in \mu(D)$, then $v\in N(s)$.
\(a) Any vertex $x\in A'\cup Q$ is a vertex-descendant of $ut$ or $us$. Since the levels of $ut$ and $us$ are both ${\ell_{max}}-2$, by Lemma \[lemma:l-1\], if $x$ is a vertex-descendant of $ut$ then $N_2(x)\subseteq N[\{u,t\}]$, and if $x$ is a vertex-descendant of $us$ then $N_2(x)\subseteq N[\{u,s\}]$. Therefore, $N_2(x)\subseteq N[\{ u,t,s\}]$.
\(b) Consider $v\in N_2(x)$ for some $x\in A' \cup Q$ such that $\mu(v)\in \mu(B)$. Since $v\in N[\{u,t,s\}]$ by (a), it suffices to prove $v\notin N[t]$. Suppose otherwise. Since $\mu(v)\in \mu(B)$, if $v \not \in B$, then both $v \in N[t]$ and another neighbor of $t$ in $B$ have color $\mu(v)$, a contradiction. Hence $v\in B$. Since $N_2(x)\cap B=\emptyset$, we then have $v\notin N_2(x)$, but this is a contradiction.
\(c) By (b), every vertex $v\in N_2(A'\cup Q)$ with $\mu(v)\in \mu(D)$ must be in $N[\{u,s\}]$. If $v \in N[u]$, then $\mu(v) \in \mu(N[u])$ and so $\mu(v) \notin \mu(D)$ by the definition of $D$, a contradiction. Hence $v \notin N[u]$ and therefore $v\in N(s)$.
Define $$D' = \{x\in N(s): \mu(x)\in\mu(D)\}.$$
\[lemma:pathd-adj\] The following hold:
- $\mu(D')=\mu\left( D \right)$;
- for any $a'\in A'$ and $d'\in D'$, there exists a $(\mu,\mu(a'),\mu(d'))$-bichromatic path in $G^2$ from $a'$ to ${\mathtt{mate}}(a')$ such that $d'$ is adjacent to $a'$ in this path.
Let $d$ be an arbitrary vertex in $D$. Let $a_1'$ and $a_2'$ be arbitrary vertices in $A'$. By Lemma \[lemma:pathD\], there exists a $(\mu,\mu(a'_1),\mu(d))$-bichromatic path $P_1$ from $a_1'$ to ${\mathtt{mate}}(a_1')$, and there exists a $(\mu,\mu(a'_2),\mu(d))$-bichromatic path $P_2$ from $a_2'$ to ${\mathtt{mate}}(a_2')$. Note that $P_1$ and $P_2$ each has at least three vertices. Let $d_1$ be the vertex adjacent to $a_1'$ in $P_1$ and $d_2$ the vertex adjacent to $a_2'$ in $P_2$. Clearly, $\mu(d_1)=\mu(d_2)=\mu(d)$. By Lemma \[lemma:n2adjacentuts\](c), both $d_1$ and $d_2$ are in $N(s)$, and hence $d_1\in N_2[d_2]$. This together with $\mu(d_1)=\mu(d_2)$ implies $d_1=d_2$. Thus, for any $d\in D$, there exists $d'\in N(s)$ with $\mu(d')=\mu(d)$ such that for each $a'\in A'$, there exists a $(\mu,\mu(a'),\mu(d))$-bichromatic path from $a'$ to ${\mathtt{mate}}(a')$ that passes through the edge $a' d'$. Both statements in the lemma easily follow from the statement in the previous sentence.
Since no two vertices in $D$ ($D'$, respectively) are colored the same, by Lemma \[lemma:pathd-adj\] we have $|D| = |D'|$ and every $d' \in D'$ corresponds to a unique $d \in D$ such that $\mu(d) = \mu(d')$, and vice versa. We call $d$ and $d'$ the mates of each other, written $d = {\mathtt{mate}}(d')$ and $d' = {\mathtt{mate}}(d)$. Lemma \[lemma:pathd-adj\] implies the following results (note that ${\mathtt{mate}}(a')$ is adjacent to ${\mathtt{mate}}(d')$ in $G^2$).
\[corollary:dadja’\] The following hold:
- each $a'\in A'$ is adjacent to each $d'\in D'$ in $G^2$;
- for any $a'\in A'$ and $d'\in D'$, there exists a $(\mu,\mu(a'),\mu(d'))$-bichromatic path from $d'$ to ${\mathtt{mate}}(d')$ in $G^2$.
Bridging sets, bridging sequences, and re-coloring
--------------------------------------------------
\[bridging-set\] [*An ordered set $\{x_1, x_2, \ldots, x_k\}$ of vertices of $G^2$ is called a *bridging set* if for each $i$, $1\le i\le k$, $x_i \in N(s) \setminus D'$ and there exists a vertex $q_i \in Q$ such that $\mu(q_i)=\mu(x_i)$ and $q_i$ is not adjacent in $G^2$ to at least one vertex in $D'\cup \{x_1,x_2, \ldots, x_{i-1}\}$. Denote $q_i = bp(x_i)$ and call it the *bridging partner* of $x_i$. We also fix one vertex in $D'\cup \{x_1, x_2, \ldots, x_{i-1}\}$ not adjacent to $q_i$ in $G^2$, denote it by $bn(q_i)$, and call it the *bridging non-neighbor* of $q_i$. (If there are more than one candidates, we fix one of them arbitrarily as the bridging non-neighbor.)*]{}
In the definition above we have $bp(x_i) \neq x_i$ for each $i$, for otherwise $bp(x_i)$ would be adjacent in $G^2$ to all vertices in $N(s)$ and so there is no candidate for the bridging non-neighbor of $bp(x_i)$, contradicting the definition of a bridging set.
In the following we take $L$ to be a fixed bridging set with maximum cardinality.
[*Given $z \in D' \cup L$, the *bridging sequence* of $z$ is defined as the sequence of distinct vertices $s_1, s_2, \ldots, s_j$ such that $s_1=z$, $s_j\in D'$, and for $2\le i\le j $, $s_i$ is the bridging non-neighbor of the bridging partner of $s_{i-1}$.*]{}
By Definition \[bridging-set\], it is evident that the bridging sequence of every $z \in D' \cup L$ exists. In particular, for $d \in D'$, the bridging sequence of $d$ consists of only one vertex, namely $d$ itself.
\[lemma:bpbn\] Let $q\in L$, $x=bp(q)$ and $y=bn(x)$. If there exists $v\in N_2(x)$ such that $\mu(v)=\mu(y)$, then $y\in L$ and $v=bp(y)$.
Since $\mu(v)=\mu(y)\in \mu\left( B \right)$, by Lemma \[lemma:n2adjacentuts\](b), $v$ must be in $N(\{s, u\})$. If $v\in N(s)$, then $v=y$, but this cannot happen as $y=bn(x)\notin N_2(x)$. Hence $v\in N(u)$ and so $\mu(v)\notin \mu(D')$. Therefore, $\mu(y)\notin \mu(D')$, which implies $y\in L$. Since the only vertex in $N(u)$ with color $\mu(y)$ is $bp(y)$, we obtain $v=bp(y)$.
\[bridging-recoloring\]
*Given a vertex $z\in D' \cup L$ with bridging sequence $s_1,s_2,\ldots, s_j$, define the *bridging re-coloring* $\psi_z$ of $\mu$ with respect to $z$ by the following rules:*
- $\psi_z(x)=\mu(x)$ for each $x\in V(G) \setminus \left\{ bp(s_i):1\le i< j \right\}$;
- \[constr-step2-bridging-recoloring\] $\psi_z(bp(s_i))=\mu(s_{i+1})$ for $1\le i< j$.
Observe that for $i \neq j$ we have $\mu(s_i)\neq \mu(s_j)$ as $s_i, s_j \in N(s)$. So each color is used at most once for recoloring in (b) above.
\[psi-optimal\] For any $z\in D' \cup L$, $\psi_z$ is an optimal coloring of $G^2$.
Since $\psi_z$ only uses colors of $\mu$, it suffices to prove that it is a proper coloring of $G^2$. Let $s_1, s_2, \ldots, s_j$ be the bridging sequence of $z$. Suppose to the contrary that $\psi_z$ is not a proper coloring of $G^2$. Then by the definition of $\psi_z$ there exists $1 \le i \le j-1$ such that $\psi_z(bp(s_i)) \in \psi_z(N_2(bp(s_i)))$. Denote $x=bp(s_i)$. Then there exists $v\in N_2(x)$ such that $\psi_z(v)=\psi_z(x)=\mu(s_{i+1})$. Since $x$ is the only vertex that has the color $\mu(s_{i+1})$ under $\psi_z$ and a different color under $\mu$, we have $\mu(v)=\mu(s_{i+1})$. Since $s_{i+1}=bn(x)$, by Lemma \[lemma:bpbn\] we have $s_{i+1}\in L$ and $v=bp(s_{i+1})$. However, $\psi_z(bp(s_{i+1}))=\mu(s_{i+2})\neq \mu(s_{i+1})$ by the definition of $\psi_z$. Therefore, $\psi_z(v)\neq \mu(s_{i+1})$, which is a contradiction.
\[rgnotrecolored\] Let $a'\in A'$, $z \in L$, $r=\mu(a')$ and $g=\mu(z)$. Then for any $x \in V(G) \setminus \{bp(z)\}$, $\psi_z(x)\in \{r,g\}$ if and only if $\mu(x)\in \{r,g\}$, whilst $\mu(bp(z)) \in \{r, g\}$ but $\psi_q(bp(z) \notin \{r, g\}$.
This follows from the definition of $\psi_z$ and the fact that $r, g \notin \mu(\{s_2, s_3, \ldots, s_j\})$.
\[lemma:qpath\] For any $a'\in A'$ and $q\in L$, there exists a $(\mu, \mu(a'),\mu(q))$-bichromatic path from $a'$ to ${\mathtt{mate}}(a')$ in $G^2$ which contains the edge $a' q$.
Denote $\mu(a')=r$, $\mu(q)=g$ and $a={\mathtt{mate}}(a')$. In view of Lemma \[rgnotrecolored\], it suffices to prove that there exists a $(\psi_q, r,g)$-bichromatic path from $a'$ to $a$ in $G^2$ which uses the edge $a' q$. Consider the subgraph $H$ of $G^2$ induced by the set of vertices with colors $r$ and $g$ under $\psi_q$. Denote by $H'$ the connected component of $H$ containing $a'$.
We first prove that $a \in V(H')$. Suppose otherwise. Define a coloring $\phi$ of $G^2$ as follows: for each $v \in V(H')$, if $\psi_q(v)=r$ then set $\phi(v) = g$, and if $\psi_q(v)=g$ then set $\phi(v) = r$; set $\phi(p) = r$ and $\phi(x) = \psi_q(x)$ for each $x \in V(G) \setminus (V(H') \cup \{p\})$. We claim that $\phi$ is a proper coloring of $G^2$. To prove this it suffices to show $r\notin \phi(N_2(p))$ because exchanging the two colors within $V(H')$ does not destroy properness of the coloring. Suppose to the contrary that there exists a vertex $v\in N_2(p)$ such that $\phi(v)=r$. If $v\in V(H')$, then $\psi_q(v)=g$ and so $v\neq bp(q)$ by the definition of $\psi_q$. Also $\mu(v)=g$ by Lemma \[rgnotrecolored\]. The only vertices in $N_2[p]$ with color $g$ under $\mu$ are $bp(q)$ and one vertex in $Q'$, say, $q'$. Since $v\neq bp(q)$, we have $v=q'$. Since $a\in N_2(q')$, we get $a\in V(H')$, which is a contradiction. If $v\notin V(H')$, then $\psi_q(v)=r$, and by Lemma \[rgnotrecolored\], $\mu(v)=r$. However, the only vertex in $N_2[p]$ with color $r$ is $a'$ but it has color $g$ under $\psi_q$, which is a contradiction. Thus $\phi$ is a proper coloring of $G^2$. Recall that $p$ is the only vertex in $G$ with color $\mu(p)$ under $\mu$. By the definition of $\psi_q$, $p$ remains to be the only vertex with color $\mu(p)$ under $\psi_q$. Hence $\phi$ uses one less color than $\psi_q$ as it does not use the color $\psi_q(p)=\mu(p)$. This is a contradiction as by Lemma \[psi-optimal\] $\psi_q$ is an optimal coloring of $G^2$. Therefore, $a \in V(H')$.
Now that $a \in V(H')$, there is a $(\psi_q, r, g)$-bichromatic path from $a'$ to $a$ in $G^2$. We show that in this path $a'$ has to be adjacent to $q$. Suppose otherwise. Say, $v\neq q$ is adjacent to $a'$ in this path. Then $\psi_q(v)=g$, and by Lemma \[rgnotrecolored\], $\mu(v)=g$. By Lemma \[lemma:n2adjacentuts\](b), $v\in N(\{u,s\})$. Since $v\neq q$, we have $v\not \in N(s)$. Hence, $v\in N(u)$, which implies $v=bp(q)$. Since $\psi_q(bp(q))\neq g$ by the definition of $\psi_q$, it follows that $\psi_q(v)\neq g$, but this is a contradiction. This completes the proof.
\[qadja’\] Each $a'\in A'$ is adjacent to each $q\in L$ in $G^2$.
We now extend the definition of mate to the set $L$. For each $q\in L$, define ${\mathtt{mate}}(q)$ to be the vertex in $Q'$ with the same color as $q$ under the coloring $\mu$. We now have the following corollary of Lemma \[lemma:qpath\].
\[cor:pathq\] For any $a'\in A'$ and $q\in L$, there is a $(\mu,\mu(a'),\mu(q))$-bichromatic path from $q$ to ${\mathtt{mate}}(q)$.
This follows because ${\mathtt{mate}}(a')$ is adjacent to ${\mathtt{mate}}(q)$ in $G^2$.
Define $$bp(L) = \{bp(q): q\in L\}.$$ Then $bp(L)\subseteq Q$, $\mu(bp(L))=\mu(L)$, and $\mu(L \cup (Q \setminus bp(L)))=\mu(Q)=\mu(Q')$.
\[lemma:q3conn-q1d’\] For any $q \in Q \setminus bp(L)$, $D' \cup L \subseteq N_2[q]$.
Suppose otherwise. Say, $q \in Q \setminus bp(L)$ and $z\in (D' \cup L) \setminus N_2[q]$.
Consider first the case when $\mu(q)\in \mu(N(s))$, say, $\mu(q)=\mu(x)$ for some $x \in N(s)$. Then $x \neq q$ for otherwise $z\in N_2[q]$. Also, $x\notin L$ for otherwise, $bp(x)$ and $\mu(q)$ are adjacent in $G^2$ but has the same color under $\mu$. We also know $x\notin D'$ as $\mu(D')\cap \mu(N[u])=\emptyset$. Hence $L\cup \{x\}$ is a larger bridging set than $L$ by taking $bp(x)=q$ and $bn(q)=z$. This contradicts the assumption that $L$ is a bridging set with maximum cardinality.
Henceforth we assume that $\mu(q)\notin \mu(N(s))$. Since $A' \ne \emptyset$ by our assumption, we can take a vertex $a' \in A'$. Define a coloring $\phi$ of $G^2$ as follows: set $\phi(q) = \psi_z(z)=\mu(z)$, $\phi(a') = \psi_z(q)=\mu(q)$ and $\phi(p) = \psi_z(a')=\mu(a')$, and color all vertices in $V(G) \setminus \{q, a', p\}$ in the same way as in $\psi_z$. Clearly, $\phi$ uses less colors than $\psi_z$ as it does not use the color $\psi_z(p)$. Since by Lemma \[psi-optimal\], $\psi_z$ is an optimal coloring of $G^2$, $\phi$ cannot be a proper coloring of $G^2$. Hence one of the following three cases must happen. In each case, we will obtain a contradiction and thus complete the proof. Note that, by the definition of $Q \setminus bp(L), A', L$ and $D'$, the colors $\mu(z)$, $\mu(q)$ and $\mu(a')$ used by $\phi$ are pairwise distinct.
*Case 1:* There exists $v\in N_2(q)$ such that $\phi(v)=\phi(q)=\mu(z)$.
In this case $q$ is the only vertex with color $\mu(z)$ under $\phi$ that has a different color under $\psi_z$. Since $v\neq q$, $\psi_z(v)=\phi(v)=\mu(z)$. Since $\mu(z)$ is not a color that was recolored to some vertex during the construction of $\psi_z$, we have $\mu(v)=\psi_z(v)=\mu(z)$. By Lemma \[lemma:n2adjacentuts\](b), $v\in N(u\cup s)$. If $v\in N(s)$, then $v=z$, which is a contradiction as $z\notin N_2[q]$. Thus, $v\in N(u)$, which implies $v=bp(z)$ as $bp(z)$ is the only vertex in $N(u)$ with color $\mu(z)$ under $\mu$. However, $\phi(bp(z))=\psi_z(bp(z))=\mu(bn(bp (z)))\neq\mu(z) = \phi(v)$, which is a contradiction.
*Case 2:* There exists $v\in N_2(a')$ such that $\phi(v)=\phi(a')=\mu(q)$.
In this case $a'$ is the only vertex with color $\mu(q)$ under $\phi$ that has a different color under $\psi_z$. Since $v\neq a'$, $\psi_z(v)=\phi(v)=\mu(q)$. Since $\mu(q)$ is not a color that was recolored to some vertex during the construction of $\psi_z$, we have $\mu(v)=\psi_z(v)=\mu(q)$. By Lemma \[lemma:n2adjacentuts\](b), $v\in N(u\cup s)$. As $\mu(q)\notin \mu(N(s))$ by our assumption, we have $v\notin N(s)$. So $v\in N(u)$ which implies $v=q$. On the other hand, by the construction of $\phi$, we have $\phi(q)=\mu(z)\neq \mu(q)$, which means $\phi(q)\neq \phi(v)$, which is a contradiction to $v=q$.
*Case 3:* There exists $v\in N_2(p)$ such that $\phi(v)=\phi(p)=\mu(a')$.
In this case $p$ is the only vertex with color $\mu(a')$ under $\phi$ that has a different color under $\psi_z$. Since $v\neq p$, $\psi_z(v)=\phi(v)=\mu(a')$. Since $\mu(a')$ is not a color that was recolored to some vertex during the construction of $\psi_z$, we have $\mu(v)=\psi_z(v)=\mu(a')$. Note that $a'$ is the only vertex in $N_2(p)$ with color $\mu(a')$ under $\mu$. However, $\phi(a')=\mu(q')\neq \mu(a')$, which is a contradiction.
Finale
------
Denote by $a_1', a_2', \ldots, a_{k}'$ the vertices in $A'$ and $z_1, z_2, \ldots, z_{\ell}$ the vertices in $D'\cup L$, where $k = |A'|$ and $\ell = |D'\cup L|$.
*Case A:* $k \le \ell$.
In this case, by Lemmas \[lemma:pathd-adj\] and \[lemma:qpath\], for each $1 \le i \le k$, we can take a $(\mu,\mu(a_i'),\mu(z_i))$-bichromatic path $P_i$ from $a_i$ to ${\mathtt{mate}}(a_i)$. Define ${\mathcal{B}}$ to be the family of the following branch sets: each vertex in $N[w]$ is a singleton branch set, each vertex in $F$ is a singleton branch set, and each $V(P_i)$ for $1\le i\le k$ is a branch set.
*Case B:* $\ell < k$.
In this case, by Corollaries \[corollary:dadja’\](b) and \[cor:pathq\], for each $1 \le i \le l$, we can take a $(\mu,\mu(a_i),\mu(z_i))$-bichromatic path $P_i$ from $z_i$ to ${\mathtt{mate}}(z_i)$. Define ${\mathcal{B}}$ to be the family of the following branch sets: each vertex in $N[u]\setminus bp(L)$ is a singleton branch set, and each $V(P_i)$ for $1\le i\le l$ is a branch set.
In either case above, the paths $P_1, P_2, \ldots, P_n$ (where $n = \min\{k, l\}$) are pairwise vertex-disjoint because the colors of the vertices in $P_i$ and $P_j$ are distinct for $i \neq j$. Therefore, the branch sets in ${\mathcal{B}}$ are pairwise vertex-disjoint in either case.
\[lemma:brset-connected\] Each pair of branch sets in ${\mathcal{B}}$ are joined by at least one edge in $G^2$.
Consider Case A first. It is readily seen that $N(w) \cup F$ is a clique of $G^2$. Hence the singleton branch sets in ${\mathcal{B}}$ are pairwise adjacent. For $1\le i\le k$, each vertex in $N(w)\cup F$ is adjacent to $a'_i$ or ${\mathtt{mate}}(a'_i)$ in $G^2$. Hence each singleton branch set is adjacent to each path branch set. For $1\le i, j\le k$ with $i \ne j$, we have $a'_j\in N_2[a'_i]$ and thus the branch sets $V(P_i)$ and $V(P_j)$ are joined by at least one edge.
Now consider Case B. Since $N(u)\setminus bp(L)$ is a clique of $G^2$, the singleton branch sets in ${\mathcal{B}}$ are pairwise adjacent. All vertices in $N(u)\setminus (bp(L)\cup A' \cup (Q \setminus bp(L)))$ are adjacent to ${\mathtt{mate}}(z_i)$ in $G^2$ for $1\le i\le l$. By Corollaries \[qadja’\] and \[corollary:dadja’\](a), all vertices in $A'$ are adjacent to $z_i$ in $G^2$ for $1\le i\le l$. By Lemma \[lemma:q3conn-q1d’\], all vertices in $Q \setminus bp(L)$ are adjacent to $z_i$ in $G^2$ for $1\le i\le l$. Hence each singleton branch set is joined to each path branch set by at least one edge. Since $z_i\in N(s)$ for $1\le i\le l$, the path branch sets are pairwise joined by at least one edge.
\[lemma:brsetcolors\] $|{\mathcal{B}}|\ge \chi(G^2)$.
By Lemma \[lemma:allcolors\], all colors used by $\mu$ are present in $\mu(N_2[p])$. In Case A, all colors in $\mu(N_2[p]) \setminus \mu(A)$ are present in $N(w)\cup F$, the set of singleton branch sets in ${\mathcal{B}}$. Hence $|{\mathcal{B}}| \ge |N_2[p]|-|\mu(A)| + k = (\chi(G^2) - k) + k = \chi(G^2)$. In Case B, all colors in $\mu(N_2[p]) \setminus \mu(D' \cup L)$ are present in $N(u)\setminus bp(L)$, the set of singleton branch sets in ${\mathcal{B}}$. Hence $|{\mathcal{B}}| \ge |N_2[p]|-|\mu(D' \cup L)|+l=(\chi(G^2)-l)+l = \chi(G^2)$.
Theorem \[thm:2tree\] follows from Lemmas \[lemma:brset-connected\] and \[lemma:brsetcolors\] immediately.
Proof of Corollary \[coro:2tree\]
=================================
We now prove Corollary \[coro:2tree\] using Theorem \[thm:2tree\]. It can be easily verified that if $G$ is a generalized 2-tree with small order, say at most $4$, then $G^2$ has a clique minor of order $\chi(G^2)$ for which each branch set is a path. Suppose by way of induction that for some integer $n \ge 5$, for any generalized $2$-tree $H$ of order less than $n$, $H^2$ has a clique minor of order $\chi(H^2)$ for which each branch set is a path. Let $G$ be a generalized $2$-tree with order $n$. If $G$ is a $2$-tree, then by Theorem \[thm:2tree\], the result in Corollary \[coro:2tree\] is true for $G^2$. Assume that $G$ is not a $2$-tree. Then at some step in the construction of $G$, a newly added vertex $v$ is made adjacent to a single vertex $u$ in the existing graph. (Note that $v$ may be adjacent to other vertices added after this particular step.) This means that $u$ is a cut vertex of $G$. Thus $G$ is the union of two edge-disjoint subgraphs $G_1, G_2$ with $V(G_1)\cap V(G_2) = \{u\}$. Since both $G_1$ and $G_2$ are generalized $2$-trees, by the induction hypothesis, for $i=1,2$, $G_i^2$ has a clique minor of order $\chi(G_i^2)$ for which each branch set is a path. It is evident that $G^2$ is the union of $G_1^2$, $G_2^2$ and the clique induced by the neighbourhood $N_G(u)$ of $u$ in $G$. Denote $N_i = N_{G_i}(u)$ for $i=1,2$. Then in any proper coloring of $G_i^2$, the vertices in $N_i$ need pairwise distinct colors. Without loss of generality we may assume $\chi(G_1^2) \le \chi(G_2^2)$. If $|N_G(u)| = |N_1|+|N_2| \le \chi(G_2^2) - 1$, then we can color the vertices in $N_1$ using the colors that are not present at the vertices in $N_2$ in an optimal coloring of $G_2^2$. Extend this coloring of $N_1$ to an optimal coloring of $G_1^2$. One can see that we can further extend this optimal coloring of $G_1^2$ to obtain an optimal coloring of $G^2$ using $\chi(G_2^2)$ colors. Thus, if $|N_G(u)| \le \chi(G_2^2) - 1$, then $\chi(G^2) = \chi(G_2^2)$. Moreover, the above-mentioned clique minor of $G_2^2$ is a clique minor of $G^2$ with order $\chi(G^2)$ for which each branch set is a path. On the other hand, if $|N_G(u)| \ge \chi(G_2^2)$, then one can show that $\chi(G^2) = |N_G(u)|$ and $N_G(u)$ induces a clique minor of $G^2$, with each branch set a singleton. In either case we have proved that $G^2$ has a clique minor of order $\chi(G^2) = \max\{\chi(G_1^2), \chi(G_2^2), |N_G(u)|\}$ for which each branch set is a path. This completes the proof of Corollary \[coro:2tree\].
Concluding remarks
==================
![A 2-tree $G$ with $\omega(G^2)=2\lambda+5$ and $\chi(G^2)=3\lambda+3$.[]{data-label="fig:counter-eg"}](counter-eg)
We have proved that for any $2$-tree $G$, $G^2$ has a clique minor of order $\chi(G^2)$. Since large cliques played an important role in our proof of this result, it is natural to ask whether $G^2$ has a clique of order close to $\chi(G^2)$, say, $\omega(G^2) \ge c \chi(G^2)$ for a constant $c$ close to $1$ or even $\omega(G^2) = \chi(G^2)$. Since the class of $2$-trees contains all maximal outerplanar graphs, this question seems to be relevant to Wegner’s conjecture, which asserts that for any planar graph $G$ with maximum degree $\Delta$, $\chi(G^2)$ is bounded from above by $7$ if $\Delta=3$, by $\Delta + 5$ if $4\le\Delta\le 7$, and by $(3\Delta/2) + 1$ if $\Delta\ge 8$. This conjecture has been studied extensively, but still it is wide open. In the case of outerplanar graphs with $\Delta=3$, the conjecture was proved by Li and Zhou in [@LZ] (as a corollary of a stronger result). In [@Lih2003303], Lih, Wang and Zhu proved that for any $K_4$-minor free graph $G$ with $\Delta\geq 4$, $\chi(G^2)\le (3\Delta/2) + 1$. Since $2$-trees are $K_4$-minor free, this bound holds for them. Combining this with $\omega(G^2)\ge \Delta(G)$, we then have $\omega(G^2) \ge 2(\chi(G^2)-1)/3$ for any $2$-tree $G$. It turns out that the factor $2/3$ here is the best one can hope for: In Figure \[fig:counter-eg\], we give a $2$-tree whose square has clique number $2\lambda+5$ and chromatic number $3\lambda+3$.
In view of Theorem \[thm:2tree\], the obvious next step would be to prove Hadwiger’s Conjecture for squares of $k$-trees for a fixed $k\ge 3$. Since squares of $2$-trees are generalized quasi-line graphs, another related problem would be to prove Hadwiger’s Conjecture for the class of generalized quasi-line graphs or some interesting subclasses of it. It is also interesting to work on Hadwiger’s conjecture for squares of some other special classes of graphs such as planar graphs.
[^1]: Part of the work was done when this author was visiting Max Planck Institute for Informatics, Saarbruecken, Germany supported by Alexander von Humboldt Fellowship.
[^2]: Research supported by ARC Discovery Project DP120101081.
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abstract: 'The aim of this contribution is to provide a short introduction to recently investigated models in which our accessible universe is a four-dimensional submanifold, or brane, embedded in a higher dimensional spacetime and ordinary matter is trapped in the brane. I focus here on the gravitational and cosmological aspects of such models with a single extra-dimension.'
address: |
Institut d’Astrophysique de Paris,\
98bis Boulevard Arago, 75014 Paris, France
author:
- 'DAVID LANGLOIS[^1]'
title: 'GRAVITATIONAL AND COSMOLOGICAL PROPERTIES OF A BRANE-UNIVERSE'
---
The traditional view on extra-dimensions is the Kaluza-Klein picture: the matter fields live in compact (usually flat) extra-dimensions, and their Fourier expansion along the extra-coordinates lead to an infinite collection of so-called Kaluza-Klein modes, which can be interpreted as four-dimensional fields. Their mass spectrum is very specific since it is discrete with a mass gap of the order of $R^{-1}$, where $R$ is the size of the extra dimensions (common for all extra-directions in the simplest cases). The non-observation of Kaluza-Klein modes in present collider experiments therefore provides an upper bound on the size of the extra-dimensions: R1 ([TeV]{})\^[-1]{}.
Superstring theory requires extra-dimensions to be consistent at the quantum level and the Kaluza-Klein compactification was invoked to get rid of the superfluous six extra-dimensions, until a new picture on extra-dimensions emerged recently, such as in the Horava-Witten eleven-dimensional supergravity[@hw]. In this context, the ordinary matter fields are not supposed to be defined everywhere but, in contrast, are assumed to be [*confined*]{} in a submanifold, called [*brane*]{}, embedded in a higher dimensional space.
The Horava-Witten model was followed by the radical proposal of Arkani-Hamed, Dimopoulos and Dvali[@add], who, in order to solve the hierarchy problem, suggested that we live confined in a three-brane surrounded by $n\geq 2$ (flat and compact) extra-dimensions with a size $R$ as large as the millimeter scale. One could then explain the huge value of the Planck mass, with respect to the TeV scale, simply as [*a projection effect*]{}, the relation between the fundamental (higher-dimensional) Planck mass $M_{(4+n)}$ and the usual four-dimensional Planck mass $M_{(4)}$ being given by M\_[(4)]{}\^2\~M\_[(4+n)]{}\^[2+n]{} R\^n. \[vol\_extra\] The absence of any observed deviation from ordinary Newton’s law gives an upper bound on the compactification radius [@grav_exp], presently of the order of a fraction of millimiter ($R \lesssim 0.2 \, {\rm mm}$).
Another proposal, even more interesting from the point of view of general relativity and cosmology, is due to Randall and Sundrum [@rs99b]. They consider only one extra-dimension but take into account the self-gravity of the brane endowed with a tension $\sigma$. They moreover assume the presence of a negative cosmological constant $\Lambda$ in the bulk (thus Anti-de Sitter). Provided the tension of the brane is adjusted so that =[1]{}, \[rs\] they find a static solution (and mirror-symmetric with respect to the brane) of the five-dimensional Einstein equations, described by the metric ds\^2=e\^[-2|y|/]{}\_ dx\^dx\^+dy\^2, where $\eta_{\mu\nu}$ is the usual Minkowski metric. Linearized gravity in the brane can be worked out explicitly in this model [@rs99b; @gt99] and one finds that the effective gravitational potential reads V(r)=[G\_[(4)]{}r]{}(1+ [2\^23r\^2]{}), where the four-dimensional gravitational coupling is given by 8G\_[(4)]{}=\^2/. \[G4\] The (approximate) recovery of the usual Newton’s law is due to the presence of a zero mode which is (exponentionally) localized near the brane. The corrections to the usual potential, significant only at the AdS scale $\ell$ and below, arise from a continuum of massive graviton modes.
Let us now turn to cosmology. The main motivation for exploring cosmology in models with extra-dimensions is that the new effects might be significant only at very high energies, i.e. in the very early universe. Let us thus consider a five-dimensional spacetime with three-dimensional isotropy and homogeneity, which contains a three-brane corresponding to our universe. The metric can be written in the form ds\^2=- n(t,y)\^2 dt\^2+a(t,y)\^2 \_[ij]{}dx\^idx\^j+dy\^2, \[metric\] with the brane (here spatially flat) located at $y=0$.
The energy-momentum tensor can be decomposed into a bulk energy-momentum tensor, which is assumed to vanish here, and a brane energy-momentum tensor, the latter being of the form T\^A\_[ B]{}= S\^A\_[ B]{}(y)= {-\_b, p\_b, p\_b, p\_b, 0}(y), where the delta function expresses the confinement of matter in the brane and where $\rho_b$ and $P_b$ are respectively the total energy density and pressure in the brane and depend only on time. The presence of the brane induces a jump of the extrinsic curvature tensor (defined by $K_{AB}=h_{A}^C\nabla_C n_B$, where $n^A$ is the unit vector normal to the brane) related to the brane matter content according to the Israel junction conditions =\^2 S\^A\_[ B]{}. \[israel\] Still assuming, for simplicity, mirror (i.e. $Z_2$) symmetry, these junction conditions applied to the cosmological metric (\[metric\]) yield the two conditions ([n’n]{})\_[0\^+]{}=[\^26]{}(3p\_b+2\_b), ([a’a]{})\_[0\^+]{}=-[\^26]{}\_b. \[junction\] The bulk Einstein equations can be integrated [@bdel99], and the substitution of (\[junction\]) at the location of the brane gives the [*generalized Friedmann equation*]{}
H\_0\^2=[\^436]{}\_b\^2+[6]{} +[a\^4]{}, \[fried\] where $\C$ is an integration constant, the subscript ‘$0$’ denoting evaluation at $y=0$. The most remarkable feature of (\[fried\]) is that the energy density of the brane enters quadratically on the right hand side in contrast with the standard four-dimensional Friedmann equation where the energy density enters linearly. As for the energy conservation equation it is unchanged in this five-dimensional setup and still reads \_b+3H(\_b+p\_b)=0.
In the simplest case where $\Lambda=0$ and $\C=0$, the evolution of the scale factor is given[@bdl99] by $a\sim t^{1/4}$ (instead of $a\sim t^{1/2}$) for radiation and $a\sim t^{1/3}$ (instead of $a\sim t^{2/3}$) for pressureless matter. Such behaviour is problematic because it cannot be reconciled with the primordial nucleosynthesis scenario, wich relies on the competition between microphysical reaction rates and the expansion rate of the universe.
Brane cosmology can however be made consistent with nucleosynthesis [@cosmors] if, following Randall and Sundrum, one introduces a negative cosmological constant in the bulk as well as a tension in the brane so that the total energy density in the brane, $\rho_b$, splits into \_b=+, where $\rho$ is the usual cosmological energy density. Substituting this decomposition into (\[fried\]), one gets H\^2= ([\^436]{}\^2-[1\^2]{}) +[\^418]{}+[\^436]{}\^2+[a\^4]{}. If one fine-tunes the brane tension and the bulk cosmological cosmological constant as in (\[rs\]), the first term on the right hand side vanishes. The second term then becomes the dominant term if $\rho$ is small enough and [*one thus recovers the usual Friedmann equation at low energy*]{}, with the identification $
8\pi G= {\kappa^4}\sigma/6$, which is consistent with (\[rs\]) and (\[G4\]).
The third term on the right hand side, quadratic in the energy density, provides a [*high-energy correction*]{} to the Friedmann equation which becomes significant when the value of the energy density approaches the value of the tension $\sigma$ and dominates at higher energy densities. Finally, the radiation-like term, proportional to $\C$, is related to the bulk Weyl tensor. It must be small enough during nucleosynthesis in order to satisfy the constraints on the number of extra light degrees of freedom.
Our model is valid if nucleosynthesis takes place in the low energy regime, i.e. $\sigma^{1/4} \gtrsim 1 \ {\rm MeV}$, which implies $M \equiv \kappa^{-2/3}\gtrsim 10^4 \ {\rm GeV}$ for the fundamental mass. However, the requirement to recover ordinary gravity down to scales of the submillimeter order gives the tighter constraint M10\^8 [GeV]{}.
With this viable [*homogeneous*]{} cosmological model, the next step is to investigate the perturbations from homogeneity, to check whether brane cosmology can be made compatible with the current observations and more interestingly, whether it can provide deviations from the standard predictions which might be tested in the future. Both questions are still unanswered today. One can however get a first flavour of the possible modifications by analyzing the equations obtained from the linearized five-dimensional equations [@l00b]. One finds equations similar to the standard evolution equations for the perturbations supplemented with corrective terms, which are of two types: new terms due to the modified Friedmann equation, which become negligible in the low energy regime $\rho\ll\sigma$; source terms, which come from the bulk perturbations and which cannot be determined solely from the evolution inside the brane.
[0]{}
P. Horava and E. Witten, Nucl. Phys. B [**460**]{}, 506 (1996); Nucl. Phys. B [**475**]{}, 94 (1996)
N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys.Lett. B429, (1998) 263; Phys. Rev. [**D59**]{} (1999) 086004.
C. D. Hoyle et al., Phys. Rev. Lett. [**86**]{}, 1418 (2001). L. Randall and R. Sundrum, Phys. Rev. Lett. [**83**]{} (1999) 4690. J. Garriga and T. Tanaka, Phys. Rev. Lett. [**84**]{} (2000) 2778.
P. Binétruy, C. Deffayet, U. Ellwanger and D. Langlois, Phys. Lett. B [**477**]{} (2000) 285. P. Binétruy, C. Deffayet and D. Langlois, Nucl. Phys. B [**565**]{} (2000) 269. C. Csaki, M. Graesser, C. Kolda and J. Terning, Phys. Lett. B [**462**]{}, 34 (1999); J. M. Cline, C. Grojean and G. Servant, Phys. Rev. Lett. [**83**]{}, 4245 (1999).
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[^1]: [email protected]
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---
abstract: 'We propose a generalisation of concurrent Kleene algebra [@Hoa09] that can take account of probabilistic effects in the presence of concurrency. The algebra is proved sound with respect to a model of automata modulo a variant of rooted $\eta$-simulation equivalence. Applicability is demonstrated by algebraic treatments of two examples: algebraic may testing and Rabin’s solution to the choice coordination problem.'
author:
- Annabelle McIver$^1$
- 'Tahiry Rabehaja$^{1}$'
- 'Georg Struth$^{3}$'
bibliography:
- 'references-comp.bib'
title: Weak Concurrent Kleene Algebra with Application to Algebraic Verification
---
Introduction
============
Kleene algebra generalises the language of regular expressions and, as a basis for reasoning about programs and computing systems, it has been used in applications ranging from compiler optimisation, program refinement, combinatorial optimisation and algorithm design [@Con71; @Koz94; @Koz00a; @Koz00b; @Mci06]. A number of variants of the original axiom system and language of Kleene algebra have extended its range of applicability to include probability [@Mci05] with the most recent being the introduction of a concurrency operator [@Hoa09]. Main benefits of the algebraic approach are that it captures some essential aspects of computing systems in a simple and concise way and that the calculational style of reasoning it supports is very suitable for automated theorem proving.
In this paper we continue this line of work and propose *weak concurrent Kleene algebra*, which extends the abstract probabilistic Kleene algebra [@Mci05] with the concurrency operator of concurrent Kleene algebra [@Hoa09] and thus supports reasoning about concurrency in a context of probabilistic effects. This extension calls for a careful evaluation of the axiom system so that it accurately accounts for the interactions of probabilistic choice, nondeterministic choice and the treatment of concurrency. For example probabilistic Kleene algebra accounts for the presence of probability in the *failure* of the original distributive law $x(y + z) = xy + xz$ which is also absent in most process algebras. That is because when the terms $x, y, z$ are interpreted as probabilistic programs, with $xy$ meaning “first execute $x$ and then $y$" and $+$ interpreted as a nondeterministic choice, the expression on the left hand side exhibits a greater range of nondeterminism than the right in the case that $x$ includes probabilistic behaviours. For example if $x$ is interpreted as a program which flips a bit with probability $1/2$ then the following nondeterministic choice in $y+z$ can always be resolved so that $y$ is executed *if and only if* the bit was indeed flipped. This is not a behaviour amongst those described by $xy + xz$, where the nondeterminism is resolved before the bit is flipped and therefore its resolution is unavoidably independent of the flipping. Instead, in contexts such as these, distributivity be replaced by a weaker law: $$\label{eq:subdistributivity}
\textrm{Sub-distributivity:} \qquad xy + xz ~\leq ~ x(y + z)~.$$ Elsewhere [@Rab11] we show that this weakening of the original axioms of Kleene algebra results in a complete system relative to a model of nondeterministic automata modulo simulation equivalence.
The behaviour of the concurrency operator of concurrent Kleene algebra [@Hoa09] is captured in particular by the *Interchange law*: $$(x \| y) (u \| v) ~ \leq ~ (x u) \| (y v)$$ which expresses that there is a lesser range of nondeterministic executions on the left where, for example, the execution of $u$ is constrained to follow a complete execution of $x$ run concurrently with $y$ but on the right it is not.
Our [**first contribution**]{} is the construction of a concrete model of abstract probabilistic automata (where the probability is at the action level) over which to interpret terms composed of traditional Kleene algebra together with concurrent composition. In this interpretation, each term represents an automaton. For example in Equation (\[eq:subdistributivity\]), $x,y$ and $z$ are automata and so is $xy + xz$. We show that the axiom system of concurrent Kleene algebra weakened to allow for the presence of probability is sound with respect to those probabilistic automata. Our use of probabilistic automata is similar to models where the resolution of probability and nondeterminism can be interleaved; concurrent composition of automata models CSP synchronisation [@Hoa78] in that context. Finally we use a notion of rooted $\eta$-simulation to interpret the inequality $\leq$ used in algebraic inequations.
Our [**second contribution**]{} is to explore some applications of our axiomatisation of weak concurrent Kleene algebra, to explain our definition of rooted $\eta$-simulation in terms of may testing [@Nic83], and to demonstrate the proof system on Rabin’s distributed consensus protocol [@Rab82].
One of the outcomes of this study is to expose the tensions between the various aspects of system execution. Some of the original concurrent Kleene algebra axioms [@Hoa09] required for the concurrency operator now fail to be satisfiable in the presence of probabilistic effects and synchronisation supported by the interchange law. For example, the term $1$ from Kleene algebra (interpreted as “do nothing") can no longer be a neutral element for the concurrency operator $\|$ — we only have the specific equality $1 \| 1 = 1$ and not the more general $1 \| x = x$. In fact we chose to preserve the full interchange law in our choice of axioms because it captures so many notions of concurrency already including exact parallel and synchronisation, suggesting that it is a property about general concurrent interactions.
A feature of our approach is to concentrate on broad algebraic structures in order to understand how various behaviours interact rather than to study precise quantitative behaviours. Thus we do not include an explicit probabilistic choice operator in the signature of the algebra — probability occurs explicitly only in the concrete model as a special kind of asynchronous probabilistic action combined with internal events (events that the environment cannot access). This allows the specification of complex concurrent behaviour to be simplified using applications of weak distributivity embodied by Equation (\[eq:subdistributivity\]) and/or the interchange law as illustrated by our case study. Finally we note that the axiomatisation we give is entirely in terms of first-order expressions and therefore is supported by first-order reasoning. Thus all of our algebraic proofs has been implemented within the Isabelle/HOL theorem proving environment. These proof can be found in a repository of formalised algebraic theorems. [^1]
In Section \[sec:axiomatisation\] we explore the axiomatisation of the new algebra. It is essentially a mixture of probabilistic and concurrent Kleene algebras. Sections \[sec:concrete-model\] and \[sec:soundness\] are devoted to showing the consistency of our approach. A concrete model based on automata and $\eta$-simulation is constructed. In section \[sec:probabilistic-aut\], we compare our approach with probabilistic automata (automata that exhibit explicit probability) and probabilistic simulation. We conclude that, up to some constraint, the concrete model is a very special case of that more general model. In sections \[sec:algebraic-testing\] and \[sec:rabin-protocol\], we present some applications, in particular an algebraic version of may testing is studied and variations of the specification of Rabin’s protocol are explored.
In this paper $x,y,$ etc represent algebraic expressions or variables. Terms are denoted $s,t,$ etc. Letters $a,b,$ etc stand for actions and $\tau$ represents an internal action. An automaton associated with a term or an expression is usually denoted by the same letter. Other notation is introduced as we need it.
In this extended abstract we can only explain the main properties of weak concurrent Kleene algebra and sketch the construction of the automaton model. Detailed constructions and proofs of all statements in this paper can be found in an appendix.
Axiomatisation {#sec:axiomatisation}
==============
A Kleene algebra is a structure that encodes algebraically the sequential behaviour of a system. It is generally presented in the form of an idempotent [^2] semiring structure $(K,+,\cdot,0,1)$ where $x\cdot y$ (sequential composition) is sometimes written using juxtaposition $xy$ in expressions. The term $0$ is the neutral element of $+$ and $1$ is the neutral element of $\cdot$. The semiring is then endowed with a unary Kleene star $*$ representing finite iteration to form a Kleene algebra. This operator is restricted by the following axioms: $$\begin{aligned}
\textrm{Left unfold:}\qquad 1 + xx^* & = & x^*, \label{eq:unfold}\\
\textrm{Left induction:}\hspace{1.5mm}\qquad xy\leq y&\Rightarrow &x^*y\leq y,\label{hf:linduction}\end{aligned}$$ where $x\leq y$ if and only if $x + y = y$. In the sequel our interpretations will be over a version of probabilistic automata. In particular we will interpret $\leq$ and $=$ as $\eta$-simulations.
Often, the dual of (\[eq:unfold\]-\[hf:linduction\]) i.e. $1 + x^*x = x^*$ and $yx\leq y\Rightarrow yx^*\leq y$ are also required. However, (\[eq:unfold\]) and (\[hf:linduction\]) are sufficient here and the dual laws follow from continuity of sequential composition for finite automata.
In a Kleene algebra, the semiring structure supports two distributivity laws: $$\begin{aligned}
\textrm{Left distributivity:}\qquad\hspace{0.8mm} xy + xz & = & x(y + z), \label{eq:ldist}\\
\textrm{Right distributivity:}\qquad (x + y)z & = & xz + yz. \label{eq:rdist}\end{aligned}$$ Equation (\[eq:ldist\]) however is not valid in the presence of probability. For example, compare the behaviour of probabilistic choice in the diagrams below. Here, $\flip_p$ denotes the process that flips a $p$-biased coin, which we can represent by a probabilistic automaton (details are given in Section \[sec:concrete-model\]).
\[fig:dist\] $$\xymatrix{
&\ar[dl]_{\flip_p}\ar[dr]^{\flip_p}& &\hspace{0.5cm}&&\ar[d]_{\flip_p}&\\
\ar[d]_x &&\ar[d]^y &&&\ar[dl]_x\ar[dr]^y &\\
&& && &&
}$$
In the right diagram, the choice between $a$ and $b$ can be based on the outcome of the coin flip but such resolution is not possible in the left-hand diagram. We express the greater range of possible outcomes by the general inequation (\[eq:subdistributivity\]), specifically here it becomes $$\label{eq:lsubdist}
(\flip_p)y + (\flip_p)z \leq (\flip_p)(y + z).~\footnote{We have abused notation in this example by using $\flip_p$ to represent both an action and an automaton which performs that action.}$$
As mentioned above, the zero of a Kleene algebra satisfies: $$\begin{aligned}
\textrm{Left annihilation:}\qquad 0x & = & 0, \label{eq:lzero}\\
\textrm{Right annihilation:}\qquad x0 & = & 0. \label{eq:rzero}\end{aligned}$$
In our interpretation that includes concurrency, we assume that $0$ captures *deadlock*. However, axiom (\[eq:rzero\]) is no longer appropriate because we should be able to differentiate between the process doing an action and deadlocking from a process that is just deadlocked.
A weak probabilistic Kleene algebra is a structure $(K,+,\cdot,*,0,1)$ that satisfies the axioms of Kleene algebra except there is no left distributivity (it is replaced by (\[eq:subdistributivity\])) and Equation (\[eq:rzero\]) does not hold generally.
A concurrency operator was added to Kleene algebra by Hoare et al [@Hoa09]. Our concurrency operator $\|$ satisfies the following standard axioms: $$\begin{aligned}
\textrm{Associativity:}\qquad x \| (y \| z) & = & (x \| y) \| z, \label{eq:par-assoc}\\
\textrm{Commutativity:}\hspace{1.3cm} x \| y & = & y \| x, \label{eq:par-comm}\\
\textrm{One-idempotence:}\hspace{1.35cm} 1 \| 1 & = & 1.\label{eq:par-1}\end{aligned}$$
In [@Hoa09], $\|$ satisfies the identity $1\|x = x$ which we do not have here because in the concrete model, we will interpret $\|$ as the synchronisation operator found in CSP [@Hoa78]. However, we still maintain the instance of that law in the special case $x = 1$ (see axiom (\[eq:par-1\])) where $1$ is interpreted as “do nothing".
Next we have the axioms dealing the interaction of $\|, +$ and $\cdot$. $$\begin{aligned}
\textrm{Monotonicity :}\qquad x \| y + x \| z & \leq & x \| (y + z) \label{eq:par-dist}\\
\textrm{Interchange-law:}\hspace{0.3cm} (x \| y) (u \| v) & \leq & (x u) \| (y v)\label{eq:exchange-law}\end{aligned}$$
The interchange law is the most interesting axiom of concurrent Kleene algebra. In fact it allows the derivation of many properties involving $\|$. To illustrate this in the probabilistic context, consider a probabilistic vending machine $\mathtt{VM}$ which we describe as the expression $$\mathtt{VM}\ =\ \coin\cdot\flip_p\cdot(\tau_h\cdot(\tea+1) + \tau_t\cdot(\coffee+1))$$ where $\coin,\tea,\coffee,\tau_h,\tau_t$ and $\flip_p$ are all represented by automata. That is the vending machine accepts a coin and then decides internally whether it will enable the button coffee or tea. The decision is determined by the action $\flip_p$ [^3] which (as explained later) enables either $\tau_h$ or $\tau_t$. The actions $\tau_t$ and $\tau_h$ are internal and the user cannot access them. Now, a user who wants to drink tea is specified as $$\mathtt{U}\ =\ \coin\cdot(\tea+1).$$ The system becomes $\mathtt{U}\|\mathtt{VM}$ where the concurrent operation is CSP like and synchronises on $\coin,\tea$ and $\coffee$. The interchange law together with the other axioms and some system assumptions imply the following inequation: $$\label{eq:vm}
\mathtt{U}\|\mathtt{VM}\ \geq\ \coin\cdot\flip_p\cdot(\tau_h\cdot(\tea+1) + \tau_t)$$ which is proved automatically in our repository. In other words, the user will only be satisfied with *probability at least* $p$ since the right-hand side equation says that the tea action can only be enabled provided that $\tau_h$ is enabled, and in turn that is determined by the result of the $\flip_p$ action.
Now we are ready to define our algebra.
A weak concurrent Kleene algebra is a weak probabilistic Kleene algebra $(K,+,\cdot,*,0,1)$ with a concurrency operator $\|$ satisfying (\[eq:par-assoc\]-\[eq:exchange-law\])
We assume the operators precedence $*<\cdot<\|<+$.
\[pro:elementary-consequences\] Let $s,t$ be terms, the following equations holds in weak concurrent Kleene algebra.
1. All the operators are monotonic.
2. $(s^*\|t^*)^* = s^*\|t^*$.\[eq:star-idem\]
3. $(s\|t)^*\leq s^*\| t^*$.\[eq:subdist\]
4. $(s + t)^* = (s^*t^*)^*$.
Concrete Model {#sec:concrete-model}
==============
Semantic Space {#subsec:semantic-space}
--------------
We use nondeterministic automata to construct a concrete model. An automaton is denoted by a tuple $$(P,\lra,i,F)$$ where $P$ is a set of states. The set $\lra\subseteq P\times\Sigma\times P$ is a transition relation and we write $x\trans{a } y$ when there is a transition, labelled by $a$, from state $x$ to state $y$. The alphabet $\Sigma$ is left implicit and considered to be fixed for every automaton. The state $i\in P$ is the initial state and $F\subseteq P$ is the set of final states of the automaton. In the sequel, we will denote an automaton $(P,\lra,i,F)$ by its set of states $P$ when no confusion is possible.
The actions in the alphabet $\Sigma$ are categorised into three kinds:
- *internal*: actions that will be “ignored" by the simulation relation (as in $\tau_h$ and $\tau_t$). Internal actions are never synchronised by $\|$.
- *external*: actions that *can* be synchronised. Probabilistic actions are external (as in $\flip_p$) but they are *never* synchronised.
- *synchronised*: external actions that will be synchronised when applying $\|$ (as in $\coin,\tea$ and $\coffee$). These actions are determined by a set of external actions $A$. More specifically, $\|$ refers to $\pr{A} $ which we assume is fixed and given beforehand.
The special case of probabilistic choice is modelled by combining probabilistic and internal actions. That is a process that does $a$ with probability $p$ and does $b$ with probability $1-p$ is interpreted as the following automaton
$$\xymatrix{
& \ar[d]^{\flip_p} & \\
& \ar[dl]_{\tau_h}\ar[dr]^{\tau_l} &\\
\ar[d]_{a}&&\ar[d]^{b}\\
& &
}$$
where $\flip_p\in\Sigma$ represents the action of flipping a $p$-biased coin which produces head with probability $p$ and tail with probability $1-p$. The internal actions $\tau_t$ and $\tau_h$ are enabled according to the result of $\flip_p$. Hence only one of $\tau_h$ and $\tau_t$ will be enabled just after the coin flip. Since $\tau_t$ and $\tau_h$ are internal actions, the choice is internal and based upon the outcome of $\flip_p$. The important facts here are that the choice after $\flip_p$ is internal so could be based on the probabilistic outcome of $\flip_p$ and that the environment cannot interfere with that choice. These two behavioural characteristics are what we consider to be the most general features of probability in a concurrent setting and they are those which we axiomatise and record in our concrete model.
Next, we impose some conditions on the automata to ensure soundness.
- \[hc:reachable\] reachability: every state of the automaton is reachable by following a finite path from the initial state.
- \[hc:initial\] initiality: there is no transition that leads to the initial state. This means that $a^*$ corresponds to the automata associated to $1 + aa^*$ rather than a self loop labeled by $a\in\Sigma$.
We denote by $\aut$ the set of automata satisfying these two conditions. The next step is to define the operators that act on $\aut$. We use the standard inductive construction found in [@Coh09; @Gla90; @Rab11] and the diagrams illustrating the constructions are given in the appendix.
- **Deadlock: $0$**\
This is the automaton that has only one state, namely the initial state, and no transition at all. It is the tuple $(\{i\}, \emptyset,i,\emptyset)$.
- **Skip: $1$**\
This is the automaton that has only one state $i$ which is both initial and final. This automaton has no transition i.e. is denoted by $(\{i\},\emptyset,i,\{i\})$.
- **Single action:**\
The automata associated with $a$ is $i\trans{a} \circ$ where $i$ is the initial state and $\circ$ is a final state. It is the tuple $(\{i,\circ\},\{i\trans{a} \circ\}, i, \{\circ\})$.
- **Addition: $P+Q$**\
This automaton is obtained using the standard construction of identifying the initial states of $P$ and $Q$. (This is possible due to the initiality property.)
- **Multiplication: $PQ$ (or $P\cdot Q$)**\
This automaton is constructed in the standard way of identifying copies of the initial state of $Q$ with final states of $P$.
- **Concurrency: $P\pr{A} Q$**\
This automaton is constructed as in CSP [@Hoa78]. It is a sub-automaton of the Cartesian product of $P$ and $Q$. The initial state is $(i_P,i_Q)$ and final states are reachable elements of $F_P\times F_Q$. Notice that the set $A$ never contains probabilistic actions. Further explanation about $\pr{A} $ is given below.
- **Kleene star: $P^*$**\
This automaton is the result of repeating $P$ allowing a successful termination after each (possibly empty) full execution of $P$. The initial state of $P^*$ is final and copies of the initial state of $P$ are identified with the final states of $P$.
All automata begin with an initial state and end in some final or deadlock state. Our main use of final states is in the construction of sequential composition and Kleene star.
The concurrency operator $\pr{A} $ synchronises transitions labeled by an action in $A$ and interleaves the others (including internal transitions). As in CSP, a synchronised transition waits for a corresponding synchronisation action from the other argument of $\pr{A} $. This is another reason we do not have $1\pr{\{a\}} P = P$ because if $P = i_P\trans{a} \circ$ and $i_P$ is not a final state, then $$1\pr{\{a\}} P = (\{(i,i_P)\}, \emptyset, (i,i_P),\emptyset) = 0.$$
\[pro:hc-welldef\] These operations of weak concurrent Kleene algebra are well defined on $\aut$ that is if $P,Q\in\aut$ then $P+Q, PQ, P\pr{A} Q$ and $P^*$ are elements of $\aut$.
The proof consists of checking that $P+Q, PQ, P\|Q$ and $P^*$ satisfy the reachability and initiality conditions whenever $P$ and $Q$ satisfy the same conditions. (See Proposition \[apro:stability\] in the appendix).
In the sequel, whenever we use an unframed concurrency operator $\|$, we mean that the frame $A$ has been given and remains fixed.
Equivalence {#subsec:notion-of-equality}
-----------
The previous subsection has given us the objects and operators needed to construct our concrete model. Next we turn to the interpretation of equality for our concrete interpretation.
Following the works found in [@Coh09; @Rab11; @Mil71], we again use a simulation-like relation to define valid equations in the concrete model. More precisely, due to the presence of internal actions, we will use an *$\eta$-simulation* as the basis for our equivalence.
Before we give the definition of simulation, we need the following notation. Given the state $x$ and $y$, we write $x\Rightarrow y$ if there exists a path, possibly empty, from $x$ to $y$ such that it is labelled by internal actions only. This notation is also used in [@Gla90] with the same meaning.
\[df:sim\] Let $P,Q$ be automata, a relation $S\subseteq P\times Q$ (or $S:P\rightarrow Q)$ is called **$\eta$-simulation** if
- $(i_P,i_Q)\in S$,
- if $(x,y)\in S$ and $x\trans{a} x'$ then
- if $a$ is internal then there exits $y'$ such that $y\Rightarrow y'$ and $(x',y')\in S$,
- if $a$ is external then there exists $y_1$ and $y'$ in $Q$ such that $y\Rightarrow y_1\trans{a} y'$ and $(x,y_1)\in S$ and $(x',y')\in S$.
- if $(x,y)\in S$ and $x\in F_P$ then $y\in F_Q$.
A simulation $S$ is **rooted** if $(i_P,y)\in S$ implies $y = i_Q$. If there is a rooted simulation from $P$ to $Q$ then we say that $P$ is simulated by $Q$ and we write $P\leq Q$. Two processes $P$ and $Q$ are **simulation equivalent** if $P\leq Q$ and $Q\leq P$, and we write $P\equiv Q$. In the sequel, rooted any $\eta$-simulation will be referred simply as a simulation.
Relations satisfying Definition \[df:sim\] are also $\eta$-simulation in the sense of [@Gla90] where property (a) is replaced by: $$\label{pr:property-a}
\textrm{if } a \textrm{ is internal then } (x',y)\in S.$$ The identity relation (drawn as dotted arrow) in the following diagram is a simulation relation satisfying Definition \[df:sim\], but it is not a simulation in the sense of [@Gla90].
$$\xymatrix{
\ar[d]^\tau\ars[rr]&&\ar[d]^\tau\\
\circ\ars[rr]&&\circ
}$$
We need the identity relation to be a simulation here because in our proof of soundness, more complex simulations are constructed from identity relations.
\[pro:eta-sim-welldef\] The following statements hold.
1. The relational composition of two rooted $\eta$-simulations is again a rooted $\eta$-simulation. That is, if $S,T$ are rooted $\eta$-simulations then $S\circ T$ is also a rooted $\eta$-simulation, where $\circ$ denotes relational composition.
2. The simulation relation $\leq$ is a preorder on $\aut$.
Proposition \[pro:eta-sim-welldef\] is proven in Proposition \[apro:sim-equivalence\] of the appendix.
Therefore, $\equiv$ as determined by Definition \[df:sim\] is an equivalence. In fact, we prove in the following proposition that it is a congruence with respect to $+$.
\[pro:sim-cong\] The equivalence relation $\equiv$ is a congruence with respect to $+$ and $P\leq Q$ iff $P + Q\equiv Q$.
The proof adapts and extends the one found in [@Gla90] and the specialised version for our case is Proposition \[apro:sim-congruence\] in the appendix.
It is well documented that $\eta$-simulation is not a congruence without the rootedness condition [@Gla90]. A typical example is given by the expressions $\tau a + \tau b$ and $\tau(a +b)$. The automata associated to these expressions are equivalent under non-rooted $\eta$-simulation.
The manipulation of probabilistic actions is also an important facet of our model. We assume that probabilistic actions are not synchronised and in that respect they are similar to internal actions. However probabilistic actions cannot be treated as internal as the following examples illustrates. Consider the action $\flip_{1/2}$ which flips a fair coin. If $\flip$ is an internal action then the inequality $$(\flip_{1/2})(\tau a + \tau b)\leq (\flip_{1/2})\tau a + (\flip_{1/2})\tau b$$ would be valid when interpreted in the concrete model. In other words, we would have the following simulation:
$$\xymatrix{
&\ar[d]_{\flip_{1/2}}\ars[rrrr]&&& &\ar[dl]_{\flip_{1/2}}\ar[dr]^{\flip_{1/2}}&\\
&\ar[ld]_{\tau}\ars[rrurr]\ar[dr]^{\tau}&&& \ar[d]_{\tau}&&\ar[d]^{\tau}\\
\ar[d]_a\ars@/_/[rrrr]\ars@/_/[uurrrrr]&&\ar[d]^b\ars[uurrr]\ars@/_/[rrrr]&& \ar[d]^a&&\ar[d]_b\\
\ars@/_/[rrrr]&&\ars@/_/[rrrr]&& &&
}$$
But this relationship (which implies distributivity of $\flip_p$ through $+$) does not respect the desired behaviour of probability which, as we explained earlier, satisfies only a weaker form of distributivity. Whence, we assume that probabilistic actions such as $\flip_{1/2}$ are among the external actions which will never be synchronised.
Soundness {#sec:soundness}
=========
In this section, we prove that the set $\aut$ endowed with the operators defined in Subsection \[subsec:semantic-space\] modulo rooted $\eta$-simulation equivalence (Subsection \[subsec:notion-of-equality\]) forms a weak concurrent Kleene algebra.
The first part is to prove that $\aut$ is a weak probabilistic Kleene algebra.
\[pro:pka-soundness\] $(\aut,+,\cdot,*,0,1)$ is a weak probabilistic Kleene algebra.
The proof consists of detailed verifications of the axioms for weak probabilistic Kleene algebra (see Proposition \[apro:weak-pka\] in the appendix).
The second part consists of proving that $\|$ satisfies the equations (\[eq:par-assoc\]-\[eq:exchange-law\]). Associativity depends heavily on the fact that both concurrent compositions involved in $x\|y\|z$ have the same frame set. For instance, let $\Sigma = \{a,b,c\}$. The identities $$(a\pr{\{a\}} b ) \pr{\{c\}} a = ab0 + ba0$$ and $$a\pr{\{a\}} (b \pr{\{c\}} a) = ab + ba$$ are valid in the concrete model. Hence, the first process will always go into a deadlock state though the second one will always terminate successfully. Therefore, to have associativity, the concurrency operator must have a fixed frame.
\[pro:soundness-par\] $(\aut, +,\cdot,\pr{A} ,1 )$ satisfies equations (\[eq:par-assoc\]- \[eq:exchange-law\]) modulo rooted $\eta$-simulation equivalence for any set of synchronisable actions $A\subseteq\Sigma$ (i.e. no probabilistic actions).
Associativity is mainly a consequence of the fact that there is only one frame for $\|$. The other axioms need to be checked thoroughly (see Proposition \[apro:parallel-algebra\]).
Our soundness result directly follows from these two propositions.
$(\aut,+,\cdot,\pr{A} ,*,0,1)$ is a weak concurrent Kleene algebra for any set of synchronisable actions $A\subseteq\Sigma$.
In this theorem, the frame $A$ is fixed beforehand. In other words, a model of weak concurrent Kleene algebra is constructed for each possible choice of $A$. In particular, if $A$ is empty then the concurrency operator is interleaving all actions i.e. no actions are synchronised. This particular model satisfies the identity $1\pr{\emptyset} x = x$ of the original concurrent Kleene algebra found in [@Hoa09].
The sequential and concurrent composition actually have stronger properties in the concrete model. If we consider finite automata only — automata with finitely many states and transitions— then we show that these two operators are *conditionally Scott continuous* in the sense of [@Rab11] (see Proposition \[pro:mult-cont\] and \[pro:par-continuous\] in the appendix).
Relationship to Probabilistic Processes {#sec:probabilistic-aut}
=======================================
Firstly, it is shown in [@Mci04] that a probabilistic choice $a\pc{p} b$ simulates the nondeterministic choice $a+b$. A similar result also holds in our setting. In the absence of internal transitions, simulation has been also defined elsewhere [@Coh09; @Gla90; @Rab11] which we will refer to as strong simulation. Recall that $(\flip_p)a+ (\flip_p)b\leq(\flip_p)(a+b)$ is a general property of probabilistic Kleene algebra [@Mci05] so it is valid under strong simulation equivalence [@Coh09; @Rab11]. Due to the absence of internal actions, the middle part of the diagram of Figure \[fig:figure1\] does not exist with respect to strong simulation equivalence.
In the context of Definition \[df:sim\], the right-hand simulation of Figure \[fig:figure1\] is the refinement of probabilistic choice by nondeterminism. This example gives an explicit distinction between $(\flip_p)(a+b)$ and $(\flip_p) a+ (\flip_p) b$ by considering the fact that the choice in $(\flip_p) (a + b)$ can depend on the probabilistic outcome of $(\flip_p)$, but this is not the case for $(\flip_p) a + (\flip_p)b$.
$$\xymatrix{
& & &&& \ar[d]^{\flip_p}\ars[rrrr] & && &\ar[d]^{\flip_p}&\\
&\ar[dl]_{\flip_p}\ar[dr]^{\flip_p}\ars[urrrr]& &&& \ar[dl]_{\tau}\ar[dr]^{\tau}\ars[rrrr] & && &\ar[ld]_{a}\ar[rd]^b &\\
\ar[d]_a\ars[urrrrr]\ars@/_/[rrrr]&&\ar[d]^b\ars@/_/[rrrr]\ars[urrr] &&\ar[d]_{a}\ars[urrrrr]&&\ar[d]^{b}\ars[urrr] && & &\\
\ars@/_/[rrrr] &&\ars@/_/[rrrr] && \ars@/_/[urrrr]& &\ars@/_/[urrrr] && & &
}$$
Secondly, we discuss about the relationship between our concrete model and probabilisitic automata. Remind that our interpretation of probability lies in the use of actions that implicitly contain probabilistic information. In its most general form, a probabilistic choice between $n$ possibilities can be written as
$$\flip_{p_1,\dots,p_n}\cdot(\tau_1\cdot a_1+\dots+\tau_n\cdot a_n)$$ where $\sum_ip_i = 1$. In this algebraic expression, we implicitly ensure that each guard $\tau_i$ is enabled with a corresponding probability $p_i$. Therefore, if these $\tau_i$’s are not found directly after the execution of the probabilistic action then matching them with the corresponding $p_i$ becomes a difficult task. We call $p$-automaton [^4] a transition system as per the definition of Subsection \[subsec:semantic-space\] such that if a probabilistic action has associated $\tau$ transitions then all of them follow that action directly.
Another complication also arises from the use of these $\tau_i$’s. Consider the following two processes $$\flip_{p_1,p_2}\cdot(\tau_1\cdot a+\tau_2\cdot b)$$ and $$\flip_{p_1,p_2}\cdot(\tau_1\cdot b + \tau_2\cdot a)$$ where $p_1+p_2 = 1$. We can construct a (bi)simulation relation between the corresponding automata though the probabilities of doing an $a$ are different. Hence we need to modify the definition of $\eta$-simulation (Definition \[df:sim\]) to account for these particular structure.
\[df:p-sim\] A $p$-simulation $S$ between two $p$-automata $P,Q$ is a $\eta$-simulation such that if
- $x\trans{\flip_{p_1,\dots,p_n}} x'\trans{\tau_i} x_i''$ is a transition in $P$,
- $y\trans{\flip_{p_1,\dots,p_n}} y'\trans{\tau_i} y_i''$ is a transition in $Q$,
- and $(x,y)\in S$
then $(x_i'',y_i'')\in S$, for each $i=1,\dots,n$.
This definition ensures that the probability of doing a certain action from $y$ is greater than doing that action from $x$. With similar proofs as in the previous Sections, we can show that the set of $p$-automata modulo $p$-simulation forms again a weak concurrent Kleene algebra. We denote $p$-$\aut$ the set of $p$-automata modulo $p$-simulation.
We will now show that this definition is a very special case of probabilistic simulation on probabilistic automata. To simplify the comparison, we assume that $\tau$ transitions occur only as part of these probabilistic choices in $p$-automata.
A probabilistic automaton is defined as a tuple $(P,\lra,\Delta,F)$ where $P$ is a set of states, $\lra$ is a set of labelled transitions from state to distributions [^5] of states i.e. $\lra\subseteq P\times\Sigma\times\D P$, $\Delta$ is the initial distribution and $F\subseteq P$ is a set of final states.
The notion of simulation also exists for probabilistic automata [@Seg94] and, in particular, simulation and failure simulation is discussed in [@Den07] where they are proven to be equivalent to may and must testing respectively.
To give a proper definition of probabilistic simulation, we need the following notations which are borrowed from [@Den07] and [@Gla90]. Given a relation $R\subseteq P\times\D Q$, the lifting of $R$ is a relation $\hat{R}\subseteq \D P\times \D Q$ such that $\phi \hat{R} \psi$ iff:
- $\phi = \sum_xp_x\delta_x$, [^6]
- for each $x\in\supp(\phi)$ (the support of $\phi$) there exists $\psi_x\in\D Q$ such that $x R\psi_x$,
- $\psi = \sum_xp_x\psi_x$.
Similarly, the lifting of a transition relation $\trans{\tau} $ is denote $\trans{\hat{\tau}} $ whose reflexive transitive closure is denote $\ttrans{\hat{\tau}} $. For each external action $a$, we write $\ttrans{\hat{a}} $ for the sequence $\ttrans{\hat{\tau}} \trans{a} $.
\[df:probsim\] A probabilistic simulation $S$ between two probabilistic automata $P$ and $Q$ is a relation $S\subseteq R\times\D Q$ such that:
- $(\Delta_P,\Delta_Q)\in \hat{S}$,
- if $(x,\psi)\in S$ and $x\trans{a} \phi$ then there exists $\psi'\in\D Q$ such that $\psi\ttrans{\hat{a}} \psi'$ and $(\phi,\psi')\in\hat{S}$ (for every $a\in\Sigma\cup\{\tau\}$).
- if $x\in F_P$ and $(x,\psi)\in S$ then $\supp(\psi)\subseteq F_Q$.
we denote by $\paut$ the set of probabilistic automata modulo simulation equivalence.
We can now construct a mapping $\epsilon:\praut\rightarrow \paut$ such that each instance of structure similar to $\flip_{p_1,\dots, p_n}\cdot(\tau_1\cdot a_1 + \dots + \tau_n\cdot a_n)$ is collapsed into probabilistic transitions. More precisely, let $P\in\praut$ and $\lra$ be its transition relation. The automaton $\epsilon(P)$ has the same state space as $P$ (up to accessibility with respect to the transitions of $\epsilon(P)$). The initial distribution of $\epsilon(P)$ is $\delta_{i_P}$ and the set of final states of $\epsilon(P)$ is $F_P$ again [^7] .
The set of transitions $\lra_{\epsilon(P)}$ is constructed as follow. Let $x\trans{a} x'$ be a transition of $P$, there are two possible cases:
- if $a$ is probabilistic i.e. of the form $\flip_{p_1,\dots,p_n}$ and is followed by the $\tau_i$’s, then the transition $$x\trans{\tau} p_1\delta_{x_1'}+\dots+p_n\delta_{x_n'}$$ is in $\lra_{\epsilon(P)}$ where $x'\trans{\tau_i} x_i'$ is a transition in $P$.
- else the transition $x\trans{a} x'$ is in $\lra_{\epsilon(P)}$.
We now prove that $\epsilon$ is a monotonic function from $\praut$ to $\paut$.
\[pro:cor\] If $P\leq Q$ then $\epsilon(P)\leq\epsilon(Q)$.
Assume that $S$ is a $p$-simulation from $P$ to $Q$. Consider the exact same relation but restricted to the state space of $\epsilon(P)$ and $\epsilon(Q)$. We show that this restriction is a probabilistic simulation.
- Obviously, $(\delta_{i_P},\delta_{i_Q})\in \hat{S}$.
- Let $x\trans{a} \phi$ and $(x,\psi)\in\hat{S}$. Since $\tau$ transitions only occur as part of probabilistic choices, we have two possibilities:
- $x\trans{\tau} p_1\delta_{x_1'}+\dots +p_n\delta_{x_n'}$ is a transition of $\epsilon(P)$ and $(x,\psi)\in S$ where $\psi = \delta_y$. Since $(x,y)$ belongs to the original $S$. In this case, $y\trans{\tau} p_1\delta_{y_1'}+\dots +p_n\delta_{y_n'}$ is a transition of $\epsilon(Q)$ and each $(x_i',y_i')$ belongs to the original $S$ (Definition of $p$-simulation).
- $x\trans{a} x'$ and $a$ is an external action. Therefore there are two possibilities again, $y\trans{\tau_i} y_i\trans{a} y'$ or $y\trans{a} y'$. In both cases, we have $(x',y')\in S$.
- Conservation of final states follows easily from the fact that $S$ is a $p$-simulation.
Since our Definition (\[df:probsim\]) implies the definition of probabilistic simulation in [@Den07], we conclude that maximal probability of doing a particular action in $p$-automata is increased by $p$-simulation. This remark provides a formal justification of our earlier example. That is, Equation (\[eq:vm\]) ensures that the maximal probability that a buyer will be satisfied when using the probabilistic vending machine is at least $1/2$ because the maximal probability of a trace containing $\tea$ in the automata described by $$\coin\cdot\flip\cdot(\tau_h\cdot(\tea+1) + \tau_t$$ is $1/2$.
In the proof of proposition \[pro:cor\], the simulation constructed is a very particular case of probabilistic simulation so it is too weak to establish certain relationships between $p$-automata. For instance, the automaton represented by $a\pc{p} (a\pc{q} b)$ should be equivalent to $a\pc{p+q-pq} b$ but Definition \[df:p-sim\] will not provide such equality. This line of research is part of our future work where we will study proper probabilistic automata and simulations against weak concurrent Kleene algebra.
Algebraic Testing {#sec:algebraic-testing}
=================
In this section, we describe an algebraic treatment of *testing*. Testing is a natural ordering for processes that was studied first in [@Nic83]. The idea is to “measure" the behaviour of the process with respect to the environment. In other words, given two processes $x$ and $y$ and a set of test processes $T$, the goal is to compare the processes $x\|t$ and $y\|t$ for every $t\in T$. In our case, the set $T$ will contain all processes.
We consider a function $o$ from the set of terms to the set of internal expressions $I = \{x\ |\ x\leq 1\}$. The function $o:T_\Sigma\rightarrow I$ is defined by $$\begin{array}{lll}
o(x) = x\textrm{ if }x\in I & &o(st) = o(s)o(t)\\
o(a) = \tau\textrm{ for any a }\in\Sigma-I && o(s^*) = 1\\
o(s+t) = o(s)+ o(t) &&o(s\|t)\leq o(s)o(t)
\end{array}$$
In the model, the function $o$ is interpreted by substituting each external action with the internal action $\tau$ ($o(a) = \tau$ for any $a\in\Sigma-I$). Then any final state is labelled by $1$ and deadlock states are labelled by $0$. Inductively, we label a state that leads to some final state by $1$, else it is labelled by $0$. This is motivated by the fact that $x0=0$ for any $x\in I$ so each transition leading to *deadlock states only* will be removed. Therefore, only states labelled by $1$ will remain and the transitions between them. Hence, $o(s)\neq 0$ iff the resulting automaton contains at least one state labelled by $1$. In other words, $o(s) = 0$ iff $x$ *must not terminate successfully*.
Without loss of generality (by considering automata modulo simulation), we assume that $\tau$ is the only internal action in $\Sigma$ and it satisfies $\tau\tau = \tau$. This equation is valid in the concrete model.
The existence of a well-defined function $o$ satisfying these conditions depends on our definition of simulation. That is, we can show that if $P\leq Q$ then $o(P)\leq o(Q)$ where we have abused notation by writing $o(P)$ as the application of $o$ on the term associated to $P$. A detailed discussion about this can be found in the appendix under Remark \[rem:remark-o\].
The *may testing order* is given by $$x\may{} y\quad \textrm{ iff }\quad \forall t\in T_\Sigma.\left[o(y\| t) = 0\Rightarrow o(x\| t) = 0\right].~\footnote{Notice $\|$ should be framed because some external actions are not synchronised. But in the setting of testing, we can also assume that all external actions are synchronised which permits to follow up all external actions present in the process.}$$
We now provide some results about algebraic may testing. It follows from monotonicity of $\|$ with respect to $\leq$ (Proposition \[pro:elementary-consequences\]) that may ordering $\may{} $ is weaker than the rooted $\eta$-simulation order.
$x\leq y$ implies $x\may{} y$.
In fact, $\may{} $ is too weak compared to $\leq$: may testing is equivalent to language equivalence. Given a term $s$, the language $Tr(s)$ of $s$ is the set of finite words formed by external actions and are accepted by the automata represented by $s$. In other word, it is the set of finite traces in the sense of CSP which lead to final states. The precise definition of this language equivalence can be found in the appendix and so is the proof of the following proposition (Proposition \[apro:may-language\] of the appendix).
\[pro:may-equals-language\] In $\aut$, $\may{} $ reduces to language equivalence.
We have shown that $\may{} $ is equivalent to language equivalence and hence it is weaker than our simulation order. This is also a consequence of the fact that our study of may testing is done in a qualitative way because the probabilities are found implicitly within actions. A quantitative study of probabilistic testing orders can be found in [@Den07].
Case Study: Rabin’s Choice Coordination {#sec:rabin-protocol}
=======================================
The problem of choice coordination is well known in the area of distributed systems. It usually appears in the form of processes voting for a common goal among some possibilities. Rabin has proposed a probabilistic protocol which solves the problem [@Rab82] and a sequential specification can be found in [@Mci04].
We specify the protocol in our algebra and prove that a fully concurrent specification is equivalent to a sequential one. Once this has been done, the full verification can proceed by reusing the techniques for sequential reasoning [@Mci04].
The protocol consists of a set of tourists and two places: a church $C$ and a museum $M$. Each tourist has a notepad where he keeps track of an integer $k$. Each place has a board where tourists can read and write. We denote by $L$ (resp. $R$) the value on the church board (resp. museum board).
In this section, we use $\cdot$ again for the sequential composition to make the specifications clearer.
- The church is specified as $C = (c!L)^*\cdot(c?L)$ where the channel $c$ represents the church’s door. $c!L$ means that the value of $L$ is available to be read in the channel $c$ and $c?L$ waits for an input which is used as value for $L$ in the subsequent process.
In other words, each tourist can read as many times as they want from the church board but write on it only once. Repeated writing will be considered in the specification of the protocol.
Similarly, the museum is specified as $M = (m!R)^*\cdot(m?R)$.
- Each tourist is specified as $P(\alpha,k)$ where $\alpha\in\{c,m\}$ is the door before which the tourist currently stands and $k$ is the actual value written on his notepad. A detailed description of $P$ can be found in the appendix but roughly, we have $$P(\alpha,k) = (\alpha?K)\cdot\mathtt{rabin}\cdot[\alpha:=\underline{\alpha}]~\footnote{Any action written within square brackets will denote internal action (see appendix for the detailed specification).}$$ where $\underline c = m$ and $\underline m = c$. In other words, the tourist reads the value on the place specified by $\alpha$, executes Rabin’s protocol `rabin` and then goes to the other place. Notice that the process $\texttt{rabin}$ contains the probabilistic component of Rabin’s protocol. Essentially, it describes the rules that are used by each tourist to update their actual value for $k$ with respect to the value on the board and vice versa.
The whole specification of the protocol executed by each tourist is described by the automata of Figure \[fig:rabin\]
$$\xymatrix{
&P(\alpha,k)\ar[d]_{\alpha?K}&&\\
&\ar[dl]_{[K=here]}\ar[dr]^{[K\neq here]}&&\\
\bullet & \ar[l]^{\alpha!here}& \ar[l]_{[k>K]}\ar[dl]^{[k<K]}\ar[d]^{[k=K]}&\\
& \ar[dl]^{[k:=K]} &\ar[d]^{\flip_{1/2}}& \\
\ar[d]_{\alpha!k}& &\ar[dl]_{\tau_h}\ar[dr]^{\tau_t} &\\
\ar[d]_{[\alpha := \underline\alpha]}& \ar[dr]_{[k := K+2]}& &\ar[dl]^{[k:=\overline{K+2}]}\\
\circ&&\ar[d]^{\alpha!k}&\\
& &\ar[d]^{[\alpha := \underline{\alpha}]} &\\
& &\circ &
}$$
We are ready to specify the whole system. Assume we have two tourists $P$ and $Q$ (our result generalises easily to $n$ tourists). The tourists’ joint action is specified as $(P + Q)^*$. This ensures that when a tourist has started his turn by reading the board, he will not be interrupted by any other tourist until he is done and goes inside the current place or to the other place. This condition is crucial for the protocol to work properly.
The actions of the locations process are specified by $(M+C)^*$ which ensures that each tourist can be at one place at a time only — this is a physical constraint. Now, the whole system is specified by $$\label{eq:spec}
\mathtt{init}\cdot\left([P(\alpha,u) + Q(\beta,v)]^*\pr{\{c,m\} } (M + C)^*\right)$$
where $\mathtt{init}$ is the initialisation of the values on the boards, notepads and initial locations. Specification \[eq:spec\] describes the most arbitrary behaviour of the tourists compatible with visiting and interacting with the locations in the manner described above. Rabin’s design of the protocol means that this behaviour is equivalent to a serialised execution where first one location is visited, followed by the other. We can write that behaviour behaviour as $[((P+Q)\|M)^*((P+Q)\|C)^*]^*$, where (for this section only) we denote the concurrency operator by $\|$ instead of $\pr{\{c,m\}} $ to make the notation lighter. The next theorem says that this more uniform execution is included in $S= [P(\alpha,u) + Q(\beta,v)]^*\| (M + C)^*$, described by Specification \[eq:spec\].
\[pro:dupl\] We have $$S \geq [((P+Q)\|M)^*((P+Q)\|C)^*]^*$$
The proof is a simple application of Proposition \[pro:elementary-consequences\]. Theorem \[pro:dupl\] means $S$ could execute all possible actions related to door $M$, and then those at door $C$, and then back to door $M$ and so one. In fact, we can also prove the converse i.e. Proposition \[pro:dupl\] could be strengthen to equality. But for that, we need the continuity of the operators $\cdot$ and $\|$.
\[thm:rabin\] In the concrete model, the specification of Rabin’s protocol satisfies $$S = [((P+Q)\|M)^*((P+Q)\|C)^*]^*$$
The proof of this theorem depends heavily on the fact that the concurrent and sequential compositions are continuous in the the concrete model. The complete proof can be found in the appendix.
In the proof, if we stopped at the distribution over $\|$, we obtain the equivalent specification $$S = [(P+Q)\|M + (P+Q)\|C]^*$$ which describes a simpler situation where P or Q interacts at the Museum or at the Church. This is similar to the sequential version found in [@Mci04], which can be treated by standard probabilistic invariants to complete a full probabilistic analysis of the protocol.
Conclusion
==========
An algebraic account of probabilistic and concurrent system has been presented in this paper. The idea was to combine probabilistic and concurrent Kleene algebra. A soundness result with respect to automata and rooted $\eta$-simulation has been provided. The concrete model ensures not only the consistency of the axioms but provides also a semantic space for systems exhibiting probabilistic, nondeterministic and concurrent behaviour. We also showed that the model has stronger properties than just the algebraic axiomatisation. For instance, sequential and concurrent compositions are both continuous in the case of finite automata.
We provided some applications of the framework. An algebraic account of may testing has been discussed in Section \[sec:algebraic-testing\]. It was shown that may ordering reduces to language equivalence.
We also provided a case study of Rabin’s solution to the choice coordination problem. A concurrent specification was provided and it was shown to be structurally equivalent to the sequential one given in [@Mci04].
Though the algebra was proven to be powerful enough to derive non-trivial properties for concrete protocols, the concrete model still needs to be refined. For instance, the inclusion of tests is important especially for the construction of probabilistic choices. Tests need to be introduced carefully because their algebraic characterisation are subtle due to presence of probability. We also need to improve and refine the manipulation of quantitative properties in the model as part of our future work.
Finally, it is customary to motivate automated support for algebraic approaches. The axioms system for weak concurrent Kleene algebra is entirely first-order, therefore proof automation is supported and automatised version of our algebraic proofs can be found in our repository.
$$$$
Appendix {#appendix .unnumbered}
========
The following proofs, diagrams, remarks and other results are only included to add further clarification of the contents of the present paper. It is left to the discression of the reviewers to choose whether they will read these proofs or not.
Diagrams, Theorems and Proofs
=============================
**Diagram of the Operators:** The construction are done inductively from $0,1$ and elements of the alphabet $\Sigma$.
- **Deadlock:** $0$.\
This is the automaton that has only one state, no transition and no final state.
- **Skip:** $1$\
This is the automaton $\circ$ which has only one state which is both initial and final and has no transition.
- **Single action:**\
The automaton associated to $a\in\Sigma$ is $i\trans{a} \circ$ where $i$ is the initial state and $\circ$ is a final state.
- **Addition:** $P+Q$.\
This is constructed by identifying the initial states of $P$ and $Q$. This construction is allowed because of the initiality condition (Figure \[fig:p+q\]).
$$\xymatrix{
& i_P\ar[dl]_a\ar[d]^b & + & i_Q\ar[d]_c\ar[dr]^d & & = & P'_1& \ar[l]_a\ar[dl]^bi_{P+Q}\ar[r]^d\ar[dr]_c&Q'_2\\
P'_1&P'_2 & & Q'_1 & Q'_2& & P'_2& & Q'_1
}$$
- **Multiplication:** $PQ$.\
This is constructed by identifying each final state of $P$ with the initial state of $Q$ (Figure \[fig:pq\]).
$$\xymatrix{
& P'\ar[d]^a & \cdot & i_Q\ar[d]_c\ar[dr]^d & & = & & P'\ar[d]^a&\\
\dots&\ar[l]_{\dots}\circ& & Q'_1 & Q'_2& & \dots& \ar[l]_{\dots}\bullet \ar[d]_c\ar[dr]^d&\\
& & & & & & & Q'_1& Q'_2
}$$
- **Concurrency:** $P\pr{A} Q$\
This is constructed as a sub-automaton of the Cartesian product of $P$ and $Q$ following CSP [@Hoa78]. Assuming $a\in A$ and $b,d\notin A$, the concurrent composition $P\pr{A } Q$ is inductively constructed as in Figure \[fig:p|q\]
$$\xymatrix{
& i_P\ar[dl]_a\ar[d]^b & \pr{A} & i_Q\ar[d]_a\ar[dr]^d & & = & & \ar[dl]_a(i_{P},i_Q)\ar[d]^b\ar[dr]^d&\\
P'_1&P'_2 & & Q'_1 & Q'_2& &P'_{1}\ \!\! \pr{A} Q_1& P'_{2}\ \!\!\pr{A} Q & P\pr{A } Q'_2
}$$
Notice that $A\subseteq\Sigma$ is a set of synchronised action and does not contain any (strictly) probabilistic actions such as $\flip(p)$, for $p\in]0,1[$.
- **Kleene star:** $P^*$\
This is the result of repeating $P$ allowing a successful termination after each (possibly empty) full execution of $P$. In the diagram of Figure \[fig:p\*\], we just picture one transition from the initial state and one final state. The construction needs to be performedfor each initial transition and final state.
$$\xymatrix{
(i_P\ar[r]^a& P'\ar[r]^b& \circ )^* & = & \circ\ar[r]^a & P'\ar@/^/[r]^b& \circ\ar@/^/[l]^a}$$
Notice the initial state of $P^*$ is a final state too.
\[apro:stability\] These operations are well defined on $\aut$ that is if $P,Q\in\aut$ then $P+Q, PQ, P\pr{A} Q$ and $P^*$ are elements of $\aut$.
The proof is by induction on the structure of the automata $P$ and $Q$. For the base case, it is obvious that $0,1$ and $i\trans{a} \circ$ satisfy the reachability and initiality conditions.
Let $P,Q\in\aut$. It is easy to see from the diagrams that $P+Q,P\|Q$ and $P^*$ belongs to $\aut$ too. $PQ$ satisfies the initiality condition because the initial state is $i_P$. For reachability, let $x\in Q$. Then $x$ is reachable from $i_Q$ which in turn is reachable from $i_P$ by the definition of sequential composition.
\[apro:sim-equivalence\] The following statements hold.
1. The relational composition of two rooted $\eta$-simulations is again a rooted $\eta$-simulation. That is, if $S,T$ are rooted $\eta$-simulations then $S\circ T$ is also a rooted $\eta$-simulation, where $\circ$ denotes relational composition.
2. The simulation relation $\leq$ is a preorder on $\aut$.
<!-- -->
1. Let $S:P\rightarrow Q$ and $T:Q\rightarrow R$ be simulations and let us show that $ST:P\rightarrow R$ is a simulation.
- Evidently, $(i,i)\in ST$.
- Let $(x,z)\in S T$ and $x\trans{a} x'$. By definition of the relational composition there exists $y\in Q$ such that $(x,y)\in S$ and $(y,z)\in T$.
- if $a$ is internal, there exists $y'\in Q$ such that $y\Rightarrow y'$ and $(x',y')\in S$. Since $y\Rightarrow y'$ consists of a sequence of finite internal transition, there exists $z'\in T$ such that $(y',z')\in T$ and $z\Rightarrow z'$. Hence $(x',z')\in ST$ and $z\Rightarrow z'$.
- If $a$ is external, there exists $y_1,y'\in Q$ such that $y\Rightarrow y_1\trans{a} y'$ and $(x,y_1)\in S$ and $(x',y')\in S$. Since $(y,z)\in T$ and $y\Rightarrow y_1$, there exists $z_1\in R$ such that $(y_1,z_1)\in T$ and $z\Rightarrow z_1$. Again, since $T$ is a simulation and $y_1\trans{a} y'$, there exists $z_2,z'\in R$ such that $z_1\Rightarrow z_2\trans{a} z'$ and $(y_1,z_2)\in T$ and $(y',z')\in T$. Hence, by transitivity of $\Rightarrow$, we have $z\Rightarrow z_2\trans{a} z'$ and $(x,z_2)\in ST$ and $(x',z')\in ST$.
- Let $(x,z)\in ST$ and $x\in F_P$, there exists $y\in Q$ such that $(x,y)\in S$ and $(y,z)\in T$. So $y\in F_Q$ and hence $z\in F_R$.
- Let $(i,z)\in ST$, there exists $y\in Q$ such that $(i,y)\in S$ and $(y,z)\in T$. So $y=i$ and hence $z=i$.
2. For reflexivity, the identity relation is a rooted $\eta$-simulation and transitivity follows from 1.
\[apro:sim-congruence\] The equivalence relation $\equiv$ is a congruence with respect to $+$ and $P\leq Q$ iff $P + Q\equiv Q$.
Let $S:P\rightarrow Q$ and $S':P'\rightarrow Q'$ be routed $\eta$-simulations. We show that $S\cup S':P+P'\rightarrow Q+Q'$ is again a routed $\eta$-simulation.
- Since initial states are identified in the construction of $+$, we have $(i_{P+P'},i_{Q+Q'}) = (i_P,i_Q)\in S\cup S'$.
- Let $(x,y)\in S\cup S'$ and $x\trans{a} x'$ be a transition of $P$ (the case where this transition belongs to $P'$ is dealt with the exact same way). We have two cases:
1. if $(x,y)\in S$, then either $a$ is internal and hence $(x',y)\in S$ (so in $S\cup S'$ too) or there exists $y_1,y'\in Q$ such that $y\Rightarrow y_1\trans{a} y'$ is a path in $Q$ and $(x,y_1)\in S$ and $(x',y')\in S$. By definition of $+$, $y\Rightarrow y_1\rightarrow y'$ is again a path in $Q+Q'$ such that $(x,y_1)\in S\cup S'$ and $(x',y')\in S\cup S'$.
2. if $(x,y)\in S'$, then $x = i_P$ because $x\trans{a} x'$ is assumed to be a transition in $P$. Since the initial states are merged, $(i_{P'},y)\in S'$ and therefore $y = i_{Q'} = i_{Q+Q'} = i_Q$. Therefore $(x,y)\in S$ and we are back to Case 1.
- Let $(x,y)\in S\cup S'$ and $x\in F_P$ (the case $x\in F_{P'}$ is similar). We have two cases again, $(x,y)\in S$ and $y\in F_{P'}$. Or $(x,y)\in S'$ and then $x\in P\cap P'$. Hence $x=i$ and we are back to the first case again.
- $S\cup S'$ is rooted because $S$ and $S'$ are both rooted.
Now assume $P\leq Q$. Then $P + Q\leq Q + Q\equiv Q$ follows from the fact that $\leq$ is a congruence and the idempotence of $+$ in Proposition \[pro:pka-soundness\]. Moreover, since $id_Q:Q\rightarrow Q$ is a simulation, we have $P+Q\equiv Q$. Conversely, assume $P + Q\equiv Q$, since $id_P:P\rightarrow P + Q$ is a simulation we have $P\leq Q$ by transitivity of $\leq$.
\[apro:weak-pka\] $(\aut,+,\cdot,*,0,1)$ is a weak probabilistic Kleene algebra.
Associativity and commutativity of $+$ and $0+x = x$ follows easily from the fact that $+$ is base on $\cup$.
- Idempotence of $+$: since the union is made disjoint, we assume $P_c$ is a copy of $P$ where every states is indexed by $c$. Then $id_P:P\rightarrow P + P_c$ is a simulation and $\{(y,x)\ |\ y = x \textrm{ or } y = x_c\}$ is a simulation from $P+P_c$ to $P$.
- Associativity of $\cdot$: associativity follows from the same proof found in [@Rab11] because identity relations are simulation and our multiplication here is exactly the $\eps$-free version of the multiplication there.
- $1$ is neutral for $\cdot$: it follows easily from the construction that $1P = P$ and $P1 = P$.
- Subdistributivity \[eq:subdist\]: to show that $PQ + PR\leq P(Q+R)$, it suffices to show that $PR\leq P(Q+R)$ and derive the result from idempotence of $+$. Remind that $id_P$ and $id_Q$ are simulation so it suffices to show that $id_P\cup id_Q:PQ\rightarrow P(Q+R)$ is again a simulation. Obviously, $(i,i)\in S$ and it is rooted and conserves final states. Moreover $\lra_{P(Q+R)}\supseteq \lra_{PQ}$. Hence $id_P\cup id_Q$ is a simulation.
- Right distributivity \[eq:rdist\]: let $P_c$ be a disjoint copy of $P$, then the relation $$S = \{(x,y)\ |\ y = x\textrm{ or } y = x_c\}\nonumber$$ from $(Q+R)P$ to $QP_c + RP$ is rooted and preserves final states. Let $(x,y)\in S$ and $x\trans{a} x'\in\lra_{(Q+R)P}$. Remind that $$\begin{aligned}
\lra_{(Q+R)P} = \lra_Q\cup\lra_R\cup\lra_P-\{i\trans{a} z\in\lra_P\} \nonumber\\
\cup\{z\trans{a} z' \ |\ z\in F_Q\cup F_R \textrm{ and } i\trans{a} z'\in\lra_P\nonumber\}\end{aligned}$$
If the transition belongs to the first three sets then we are done, else we can assume $x\in F_Q$ and $i\trans{a} x'\in\lra_P$ i.e. $y = x$ and $x'\in P$. We have $$\lra_{QP_c+RP} \supseteq \{z\trans{a} z'_c \ |\ z\in F_Q \textrm{ and } i\trans{a} z'_c\in\lra_{P_c}\}$$ so $x\trans{a} x'_c\in \lra_{QP + RP}$ and $(x',x'_c)\in S$. Similarly, we can prove that if $(x,y)\in S$ and $y\trans{a} y'$ then there exists $x'$ such that $(x'y')\in S$ and $x\trans{a} x'$. Hence $S$ is a bisimulation.
- Left unfold \[eq:unfold\]: Let $x_*$ be a state in $P^*$ and $x$ the corresponding state in the unfolded version $(1 + PP^*)$ i.e. $x$ is considered as a state of $P$. The rooted version of relation $S = id_{P^*}\cup\{(x_*,x)\}$ is a rooted $\eta$-bisimulation from $P^*$ to $1 + PP^*$.
- Left induction \[hf:linduction\]: as in (10), the proof is again similar to [@Rab11] because rooted $\eta$-simulation are stable by union.
\[apro:parallel-algebra\] $(\aut, +,\cdot,\pr{A} ,1 )$ satisfies equations (\[eq:par-assoc\]- \[eq:exchange-law\]) modulo rooted $\eta$-simulation equivalence for any set of synchronisable actions $A\subseteq\Sigma$ (i.e. no probabilistic actions).
$1\pr{A} 1 = 1 $ follows directly from the definition of $\pr{A} $ and the simulation used for the commutativity is $\{((x,y),(y,x))\ |\ x\in P\textrm{ and } y\in Q\}$.
- For associativity, we show that if $(x,(y,z))\trans{a} (x',(y',z'))\in\lra_{P\pr{A} (Q\pr{A} R)}$ then $((x,y),z)\trans{a} ((x',y'),z')\in\lra_{(P\pr{A} Q)\pr{A} R}$.
- If $a\notin A$, then
- $x\trans{a} x'$ and $y = y', z = z'$. So $(x,y)\trans{a} (x',y)\in \lra_{P\pr{A} Q}$ and hence $((x,y),z)\trans{a} ((x',y),z)\in \lra_{(P\pr{A} Q)\pr{A} R}$ because $a\notin A$.
- or $x = x'$ and $(y,z)\trans{a} (y',z')$. Since $a\notin A$:
- $y\trans{a} y'$ and $z = z'$, hence $((x,y),z)\trans{a} ((x,y'),z)\in \lra_{(P\pr{A} Q)\pr{A} R}$,
- or $y = y'$ and $z\trans{a} z'$ and hence $((x,y),z)\trans{a} ((x,y),z')\in \lra_{(P\pr{A} Q)\pr{A} R}$.
- If $a\in A$, then $x\trans{a} x'$ and $(y,z)\trans{a} (y',z')$. Since $a$ is again synchronised in $Q\pr{A} R$, $y\trans{a} y'$ and $z\trans{a} z'$. So $(x,y)\trans{a} (x',y')\in \lra_{P\pr{A} Q}$ and hence $((x,y),z)\trans{a} ((x',y'),z')\in \lra_{(P\pr{A} Q)\pr{A} R}$.
Since $\pr{A} $ is commutative, we deduce that $\lra_{(P\pr{A} Q)\pr{A} R} = \lra_{P\pr{A} (Q\pr{A} R)}$ so the identity relations could again be used for the simulation.
- To prove monotonicity, we consider the relation $S:P \pr{A} Q + {P_c} \pr{A} R\to P\pr{A} (Q + R)$ as in the case of multiplication i.e. $S = \{((x,y),(x,y)),((x_c,y),(x,y))\ |\ x\in P\wedge y\in Q\cup R\}$ and $x_c$ is the copy of the state $x\in P$ in $P_C$. Let $((x_c,y),(x,y))\in S$ (the case $((x,y),(x,y))\in S$ is easier and can be handled in the same way) and $(x_c,y)\trans{a} (x'_c,y')\in\lra_{P\pr{A} Q+P\pr{A} R}$. By definition of $+$, that transition belongs to $\lra_{P\pr{A} Q}$ or $\lra_{P\pr{A} R}$. Since the first component is a copy of $x$, we have $(x_c,y)\trans{a} (x'_c,y')\in\lra_{P\pr{A} R}$ that is $y,y'\in R$.
- if $a\notin A$, then
- $x_c\trans{a} x'_c$ and $y = y'$, so $x\trans{a} x'\in\lra_{P}$ and hence $(x,y)\trans{a} (x',y)\in\lra_{P\pr{A} (Q+R)}$ and $((x'_c,y),(x',y))\in S$ by definition of $S$.
- or $x_c= x'_c$ and $y\trans{a} y'$, so $(x,y)\trans{a} (x,y')\in\lra_{P\pr{A} (Q+R)}$ and $((x,y'),(x_c,y'))\in S$.
- if $a\in A$, then $x_c\trans{a} x'_c$ and $y\trans{a} y\in\lra_{R}'$. So $x\trans{a} x'\in\lra_{P}$ and hence $(x,y)\trans{a} (x',y')\in\lra_{P\pr{A} (Q+R)}$ and $((x'_c,y'),(x',y'))\in S$.
- Firstly notice that the set of states of $(P\|Q)(P'\|Q')$ (where the frame $A$ of the concurrency operator is left implicit) is a subset of $(P\times Q)\cup(P'\times Q')$ which is in turn a subset of $(P\cup P')\times (Q\cup Q')$. Hence we consider the injection $id$ of the former set to the later one and the relation defined in Figure \[fig:sim-exchangelaw\]
\[fig:sim-exchangelaw\] $$\begin{aligned}
S = id & \cup & \{((x',i),(x',y))\ |\ y\in F_Q\textrm{ and } i\trans{a} x'\in \lra_{P'} \textrm{ and } i\in Q'\}\nonumber\\
& \cup &\{((i,y'),(x,y'))\ |\ x\in F_P\textrm{ and } i\trans{a} y'\in \lra_{Q'} \textrm{ and } i\in P'\}\nonumber\end{aligned}$$
We show that $S$ is a simulation in our sense.
- Since $(i,i) = i$ is related to itself. In particular, $S$ is rooted because $x'\neq i$ in the second set in the definition of $S$ (resp. for the third set) and HCI.
- Let $(x,y)\in (P\|Q)(P'\|Q')$ such that $(x,y)\trans{a} (x',y')$. We have the following cases.
- The transition is in $\lra_{P\|Q}$, in which case $(x,y),(x',y')\in P\times Q$.
- if $a\notin A$ then $x\trans{a} x'\in\lra_P$ and $y = y'$ or $x = x'$ and $y\trans{a} y'\in\lra_Q$. By definition of the sequential composition again, these transitions belong to $\lra_{PP'}$ or $\lra_{QQ'}$ respectively. Hence $(x,y)\trans{a} (x',y')\in\lra_{PP'\|QQ'}$.
- if $a\in A$ then $x\trans{a} x'\in\lra_P$ and $y\trans{a} y'\in\lra_Q$. As in the previous case, the considered transition exists in $PP'\|QQ'$
- The transition is in $\lra_{P'\|Q'-\{(i,i)\}}$. This case is similar to the previous one because because $x\neq i$ and $y\neq i$ as states of $P'$ and $Q'$ respectively.
- It is a linking transition i.e. $(x,y)\in F_{P\|Q}$ and $(i,i)\trans{a} (x',y')\in\lra_{P'\|Q'}$. Then $x\in F_P$ and $y\in F_Q$ and we have two cases:
- if $a\notin A$, then $i\trans{a} x'\in\lra_{P'}$ and $y' = i$ or $x' = i$ and $i\trans{a} y'\in\lra_{Q'}$. In the first case, the definition of $S$ implies that $((x',y'),(x',y))\in S$ and since $a\notin A$, we have $(x,y)\trans{a} (x',y)\in\lra_{PP'\|QQ'}$. Similarly for the other case.
- if $a\in A$, then $i\trans{a} x'\in\lra_{P'}$ and $i\trans{a} y'\in\lra_{Q'}$. Then $x\trans{a} x'\in\lra_{PP'}$ and $y\trans{a} y'\in\lra_{QQ'}$. Hence $(x,y)\trans{a} (x',y') \in\lra_{PP'\|QQ'}$.
- Let $((x',i),(x',y))\in S$ as in the above definition of $S$ and $(x',i)\trans{a} (x'',y'')\in\lra_{P'\|Q'}$.
- If $a\notin A$, then $x'\trans{a} x''\in\lra_{P'}$ and $y''=i$ or $x' = x''$ and $i\trans{a} y''\in\lra_{Q'}$. In the first case, $(x',y)\trans{a} (x'',y)\in\lra_{PP'\|QQ'}$ because $a\notin A$ and $(x'',y)\in S$ because $y'' = i$. In the second case, $y\trans{a} y''\in\lra_{QQ'}$ and hence $(x',y)\trans{a} (x'',y'')\in\lra_{PP'\|QQ'}$ because $x' = x''$.
- If $a\in A$, then $x'\trans{a} x''\in\lra_{P'}$ and $i\trans{a} y''\in\lra_{Q'}$. By definition of sequential composition, $y\trans{a} y''\in\lra{QQ'}$ and since $a\in A$, $(x',y)\trans{a} (x'',y'')\in\lra_{PP'\|QQ'}$.
- The case $((i,y'),(x,y'))\in S$ is similar.
- Let $((x,y),(u,v))\in S$ such that $(x,y)\in F_{(P\|Q)(P'\|Q')}$. That is, $$x\in (F_{P'}-\{i\})\cup o_{P'\|Q'}((i,i))F_{P}\subseteq (F_{P'}-\{i\})\cup o_{P'}(i)F_{P}$$ and similarly for $y$. Hence, if tuple belongs to $id$ then we are done. Assume $i\trans{a} x\in\lra_{P'}$ and $y = i$ (the other case is proved in exactly the same way), then $u = x$ and $v\in F_Q$. Since $(x,i)$ is a final state, we have $F_{QQ'} = (F_{Q'}-\{i\})\cup F_{Q}$ and $x'\in F_{P'}-\{i\}$. Hence $(u,v)\in F_{PP'}\times F_{QQ'}$.
Finally, since simulation preserves reachability, the reachable part of $(P\|Q)(P'\|Q')$ is simulated by the reachable part of $PP'\|QQ'$.
\[pro:mult-cont\] The sequential composition is (conditionally) continuous from the left and the right in $\aut_f$. That is, if $(P_i)_i$ is a $\leq$-directed set of finite automata with limit $P$ then $\sup_i P_iK = PK$ and $\sup_i KP_i = KP$.
We denote $\aut_f$ the set of finite automata satisfying the reachability and initiality conditions. $\aut_f$ is a subalgebra of $\aut$. The proof is similar to our proof in [@Rab11]. The only difference is from the manipulation of $0$ (because we do not have $x0 = 0$ in this setting) and hence Proposition \[pro:mult-cont\] is a generalised version of the continuity in [@Rab11].
We first define a notion of residuation on $\aut_f$. For automata $P$ and $Q$ we define the automaton $P/Q$ with initial state $i_{P/Q} = i_P$, final states $F_{P/Q} = \{x\in P\ |\ Q\leq P_x\}$, where $P_x$ is constructed from $P$ by making its initial state into $x$. We make the resulting automaton reachable by discarding all states not reachable from $x$. Notice that $P_x$ does not necessarily satisfy HCI. In this case, we unfold each transition from $x$ once and isolate $x$ but keeping a disjoint copy of it to make sure that the resulting automata is bisimulation equivalent to the non-rooted version.
We now show that $RQ\leq P\textrm{ iff } R\leq P/Q$. Assume $S$ is a simulation from $RQ$ to $P$. That means $S$ generates a simulation from $Q$ to $P_x$ for some $x$. It follows from the definition of $P/Q$ that $S$ generates a simulation from $R$ to $P/Q$, since the state $x$ become final state of $G/H$ and they are images of the final states of $K$ under the simulation generated by $S$.
For the converse direction, suppose that $S$ is a simulation from $R$ to $P/Q$. By Theorem \[pro:pka-soundness\], multiplication is isotone, hence $RQ\leq (P/Q)Q$, and it remains to show that $(P/Q)Q\leq P$.
First, if $F_{P/Q}$ is empty and then $R$ has no final state either and $RQ = R$ by definition of sequential composition. Hence $RQ = R\leq P/Q\leq P$.
Assume $F_{P/Q}$ is not empty and let $S'$ be a simulation from $RQ$ to $(P/Q)Q$. By construction of $P/Q$, we know that there exists a simulation $S_x$ from $Q$ to $P_x$ for all final states $x\in F_{P/Q}$. Moreover, there is a relation $T:P/Q\rightarrow P$ satisfying all properties of simulation except the final state property, namely a restriction of the identity relation $id_P$. We can show that $T' = (\cup_x S_x)\cup T$ is indeed a simulation from $(P/Q)Q$ to $P$ and $S'\circ T'$ is a simulation from $RQ$ to $P$.
It then follows from general properties of Galois connections that $(\cdot H)$ is (conditionally) completely additive, hence right continuous.
It remains to show left continuity. Let $(Q_i)_i$ be a directed set of automata such that $\sup_iQ_i = Q$ and let $P$ be any automaton. Then $\sup_i(PQ_i) \leq PQ$ because multiplication is monotone and it remains to show $PQ \leq \sup_i(PQ_i)$. Let us assume that $\sup_i(PQ_i)\leq R$. We will show that $PQ\leq R$.
By definition of supremum, $PQ_i\leq R$ for all $i$, hence there is a set of states $X_i = \{x\in R\ |\ Q_i\leq R_x\}$, that is, the set of all those states in $R$ from which $Q_i$ is simulated. Obviously, $X_i\subseteq X_j$ if $Q_j\leq Q_i$ in the directed collection. But since $R\in\aut_f$ has only finitely many states, there must be a minimal set $X$ in that directed set such that all $Q_i$ are simulated by $R_x$ for some $x\in X$. Therefore $Q = \sup_iQ_i\leq R_x$ for all $x\in X$. There exists a simulation $S_X: PQ_i\rightarrow R$ for some $i$ such that the residual automaton $R/Q_i$ has precisely $X$ as its set of final states. We can thus take the union of $S_X$ restricted to $P$ with all simulations yielding $Q\leq R_x$ for all $x\in X$ and verify that this is indeed a simulation of $PQ$ to $R$.
We denote $L(P) = \{t \ |\ t \textrm{ is a tree and } t\leq P\}$ the tree language associated to $P$. We have
\[lem:tree-lang\] $P\leq Q$ iff $L(P)\subseteq L(Q)$.
A specialized version of this theorem could be found in [@Rab11]. In this paper, we prove it for our rooted $\eta$-simulation.
By transitivity of simulation, we have $P\leq Q$ implies $L(P)\subseteq L(Q)$ so it suffices to show the converse.
Let $L(P)\subseteq L(Q)$ and consider the relation $S:P\rightarrow Q$ such that $(x,y)\in S$ iff $L(P_x)\subseteq L(Q_y)$, where $P_x$ is the automata constructed from $P$ with initial state $x$ as in the previous proof. We now show that the rooted version of $S$ is a simulation.
- Since $L(P)\subseteq L(Q)$, we have $(i_P,i_Q)\in S$.
- Let $(x,y)\in S$ and $x\in F_P$, then $1\in L(P_x)\subseteq L(Q_y)$. Hence $y\in F_Q$.
- Let $(x,y)\in S$, $L(P_x)\subseteq L(Q_y)$ and $x\trans{a} x'$ be a transition of $P$. There are two cases:
- $a$ is internal: for any tree $t$, $at\leq t$. Hence $L(P_{x'})\subseteq L(Q_y)$ i.e. $(x',y)\in S$.
- $a$ is external: assume for a contradiction that for each $y'_i\in Q$ such that $y\Rightarrow y_1\trans{a} y'_i$, there exists $t_i\in L(P_{x'})$ such that $t_i\notin L(Q_{y'_i})$. Since $Q\in \aut_f$, there are only finitely many such $y'_i$. By definition of $\eta$-simulation, $a(\sum_i t_i)\in L(P_x)$ and from it follows from the hypothesis that $a(\sum_i t_i)\in L(Q_y)$ i.e. $a(\sum_i t_i)\leq Q_y$. It follows from the definition of $\eta$-simulation that there exists $y'_j$ such that $y\Rightarrow y_1\trans{a} y'_j$ and $\sum_i t_i\leq y'_j$ which implies $t_j\leq\sum_it_i\leq y_j$, a contradiction.
- Making the relation $S$ rooted does not affect the well-definedness of $S$ as a simulation because the automata $P,Q$ are rooted.
\[pro:par-continuous\] The concurrency operator $\|$ is (conditionally) continuous in $\aut_f$.
We need to show that for any $\leq$-directed sequence $(Q_i)_i\subseteq \aut_f$ such that $\sup_i Q_i = Q$, we have $\sup_i (P\| Q_i) = P\| Q$, where the frame is left implicit.
Firstly, we show that $$L(P\| Q) = \downarrow(L(P)\| L(Q)) = \downarrow\{t\| t' \ |\ t\in L(P)\wedge t'\in L(Q)\}$$ where $\downarrow X$ is the down closure of $X$. Since $\| $ is monotone, $t\|t'\in L(P\| Q)$. Conversely, let $t\in L(P\pr{A} Q)$. By unfolding $P$ and $Q$ up to the depth of $t$, we can find two tree $t_P,t_Q$ such that $t\leq {t_P} \| t_Q $ and hence $t\in \downarrow (L(P)\| L(Q))$.
Secondly, we have $$\begin{aligned}
L(P\| Q) & = & \downarrow\{ t\| t'\ |\ t\in P\wedge t'\in \cup_iL(Q_i)\}\nonumber\\
& = & \downarrow \cup_i\{t\| t' \ |\ t\in P\wedge t'\in L(Q_i)\}\nonumber\\
& = & \cup_i\downarrow\{ t \|t'\ |\ t\in P\wedge t'\in L(Q_i)\}\nonumber\\
& = & \cup_iL(P\|Q_i)\nonumber\end{aligned}$$ and directedness ensures that $L(Q) = \cup_iL(Q_i)$ [^8]. Therefore, Lemma \[lem:tree-lang\] ensures that $P\|Q = \sup_i(P\|Q_i)$.
\[rem:remark-o\] The following remarks ensures the existence of an $o$ function satisfying the properties listed in Section \[sec:algebraic-testing\]
1. The axiom $\tau\tau=\tau$ ensures that $o(x)\in\{0,\tau,1\}$ for any $x\in T_\Sigma$. If $o(x) = 0$ then $x$ will *never terminate successfully*. If $o(x) = 1$, then $x$ *may terminate successfully* without the execution of any action [^9], and if $o(x) = \tau$ then $x$ *may terminate successfully* after the execution of some action.
2. The interpretation of $o$ in the concrete model respects simulation. In fact, let $P,Q$ be the automata representing some terms in $T_\Sigma$ and $S:P\rightarrow Q$ be a simulation. After replacing each action in $P,Q$ by $\tau$, $S$ remains a simulation by Propriety (a) of Definition \[df:sim\]. Therefore
- if $o(P) = 1$ then the initial state of $P$ is final and so is the initial state of $Q$,
- if $o(P) = \tau$ then the initial state of $P$ leads to some final state and so is the initial state of $Q$ i.e. $1\leq o(Q)$,
- if $o(P) = 0$ then we are done,
and in all three cases $o(P)\leq o(Q)$. Hence, it is safe to assume that $o$ is well defined on $T_\Sigma$ modulo the axioms of weak concurrent Kleene algebra. In particular, $o$ is monotonic with respect to the restriction of the natural order of the algebra on $I$.
3. The last property $o(x\|y)\leq o(x)o(y)$ is in general a strict inequality. For instance, if $a,b$ are synchronised actions then $o(a\|b) = o(0) = 0$ but $o(a)o(b) = \tau\tau = \tau$.
\[apro:may-language\] In $\aut$, $\may{} $ reduces to language equivalence.
Remind that we assume there is only one non-trivial internal action, namely $\tau$, and it satisfies $\tau\tau = \tau$.
Firstly, the language of the automata associated to $x$ is given by $$\begin{aligned}
Tr(x) & = & \{t\ |\ t \textrm{ is linear, loop-free, has only $\tau$ as} \nonumber \\
& &\quad \textrm{non-synchronised action and } t\leq_\tau x\}\nonumber\end{aligned}$$ where $x\leq_\tau y$ if there is a simulation between the automata represented by $x$ and $y$ such that all non-synchronised actions are replaced by $\tau$. This ensures for instance that $Tr(\flip) = \{\tau\}$.
Remind that $Tr(x\| y) = Tr(x) \cap Tr(y)$ [^10] because elements of $Tr(x)$ are of the form $w\tau$ or $w$ (modulo the equivalence from $\leq_\tau$) where $w$ is a word formed of synchronised actions only.
For the direct implication, assume $x\may{} y$ and let $t\in Tr(x)$. Then $o(x\|t)\neq 0$ and since $x\|t\leq y\|t$, we have $o(y\| t)\neq 0$. Since $t$ has synchronised actions only (or possibly ends with $\tau$) and $o(y\| t)\neq 0$, then $t\in Tr(y)$ that is $Tr(x)\subseteq Tr(y)$.
Conversely, let $Tr(x)\subseteq Tr(y)$ and $z\in T_\Sigma$. $Tr(x\|z) = Tr(x)\cap Tr(z)\subseteq Tr(y)\cap Tr(z) = Tr(y\|z)$. So if $o(y\|z) = 0$ then $y\| z$ has no final state and hence $Tr(y\| z) = \{0\}$. Hence $Tr(x\|z) = \{0\}$ i.e. $x\|z$ has no final state that is $o(x\| z) = 0$.
Specification of Rabin’s Protocol.
==================================
Remind that $P(\alpha,k)$ is the specification of a tourist in from of the door $\alpha\in\{m,c\}$ and has $k$ written on his notepad (Figure \[fig:app-rabin\]).
$$\xymatrix{
&P(\alpha,k)\ar[d]_{\alpha?K}&&\\
&\ar[dl]_{[K=here]}\ar[dr]^{[K\neq here]}&&\\
\bullet & \ar[l]^{\alpha!here}& \ar[l]_{[k>K]}\ar[dl]^{[k<K]}\ar[d]^{[k=K]}&\\
& \ar[dl]^{[k:=K]} &\ar[d]^{\flip_{1/2}}& \\
\ar[d]_{\alpha!k}& &\ar[dl]_{\tau_h}\ar[dr]^{\tau_t} &\\
\ar[d]_{[\alpha := \underline\alpha]}& \ar[dr]_{[k := K+2]}& &\ar[dl]^{[k:=\overline{K+2}]}\\
\circ&&\ar[d]^{\alpha!k}&\\
& &\ar[d]^{[\alpha := \underline{\alpha}]} &\\
& &\circ &
}$$
Any action of the form $[a]$ are considered internal. The symbol $\circ$ denotes final states and $\bullet$ is a deadlock state. In this protocol, deadlock state is used to specify that the tourist has come to a decision and the common place would be the value of $\alpha$ when the deadlock state is reached.
In the concrete model, the specification of Rabin’s protocol satisfies $$S = [((P+Q)\|M)^*((P+Q)\|C)^*]^*$$
Firstly, notice that if $\cdot$ is (conditionally) continuous then $x^* = \sup_{n\in\mathbb{N}}(1+x)^n$. The proof relies on the fact that $f_x^n(0) = (1 + x)^n$ where $f_x(y) = 1 + x\cdot y$ and the result follows by taking the limit.
The above property allows us to express $x^*$ as the limit of finite iterations of $x$ interleaved with successful termination. We have $$\begin{aligned}
(P + Q)^*\|(M+C)^* & = & \sup_m(1 + P + Q)^m\|\sup_n(1 + M + C)^n\nonumber\\
& = & \sup_m\sup_n\left[(1 + P+Q)^m\|(1 + M + C)^n\right]\nonumber
$$
The processes $P$ and $Q$ are essentially delimited by $\alpha?K$ and $\alpha!K$ which ensures the following properties of the system $$\begin{aligned}
X \cdot A\|Y\cdot B & = & [X\|Y]\cdot [A\|B]\label{eq:p1}\\
X\cdot A\|1 & = & 0 \label{eq:p2}\\
Y\cdot B\|1 & = & 0\label{eq:p2'}\end{aligned}$$ for every processes $A,B$ and where $X = P+Q$ is the collection of tourists and $Y = M+C$ is the collection of places.
In particular, $$1\|(1 + X)^n = 1\|(1 + X)^{n-1} + 1\|X\cdot(1 + X)^n = 1\|(1 + X)^{n-1}$$ and by induction, since $1\|1 = 1$, $$\label{eq:p3}
1\|(1 + X)^n = 1$$ for every $n\in\mathbb{N}$. Similarly, $1\|(1 + Y)^n = 1$.
On the other hand, let us denote $T_{m,n} = (1 + X)^m\|(1 + Y)^n$, then $$\begin{aligned}
T_{m,n}& = & \left[(1 +X)^{m-1} + X\cdot(1+X)^{m-1}\right]\|\nonumber\\
& & \qquad \left[(1 + Y)^{n-1} + Y\cdot (1+Y)^{n-1}\right]\nonumber\\
& = & T_{m-1,n-1} + X\cdot(1 + X)^{m-1}\|(1 + Y)^{n-1} + \nonumber \\
& & \qquad(1+X)^{m-1}\|Y\cdot (1+Y)^{m-1} + \nonumber \\
& & \qquad [X\|Y]\cdot T_{m-1,n-1}\nonumber\\
& = & (1 + X\|Y)\cdot T_{m-1,n-1} + U_{m-1,n-1} + \nonumber\\
& &\qquad V_{m-1,n-1}\nonumber\end{aligned}$$ where $$\begin{aligned}
U_{m-1,n-1} & = & X\cdot(1 + X)^{m-1}\|[(1 + Y)^{n-1}\nonumber\\
& = & U_{m-1,n-2} + [X\|Y]\cdot T_{m-1,n-2}\nonumber\\
& = & U_{m-1,n-3} + [X\|Y]\cdot T_{m-1,n-3} + \nonumber\\
& &\qquad [X\|Y]\cdot T_{m-1,n-3}\nonumber\end{aligned}$$
Since the sequence $(1+Y)^n$ is monotone, $T_{m,n}\leq T_{m,n'}$ for every $n\leq n'$ and therefore $U_{m-1,n-1}\leq U_{m-1,0} + [X\|Y]\cdot T_{m-1,n-1}$. But Property \[eq:p2\] implies that $U_{m-1,0} = 0$.
Similarly, $V_{m-1,n-1} \leq [X\|Y]\cdot T_{m-1,n-1}$. Hence $$T_{m,n} = (1 + [X\|Y])\cdot T_{m-1,n-1}.$$ By induction, we show that $$T_{m,n} = (1 + [X\|Y])^{\inf(m,n)}$$ because $T_{0,n} = T_{m,0} = 1$ by Equation \[eq:p3\].
Finally, we have $$\begin{aligned}
X^*\|Y^* & = & \sup_m\sup_n (1+X)^m\|(1+Y)^n\nonumber\\
& = & \sup_n (1 + [X\|Y])^n\nonumber\\
& = & (X\|Y)^*\nonumber\end{aligned}$$
[^1]: <http://staffwww.dcs.shef.ac.uk/people/G.Struth/isa/>
[^2]: Idempotence refers to the operation $+$ i.e. $x + x = x$.
[^3]: i.e. the automaton that performs a $\flip_p$ action.
[^4]: The name $p$-automata describes probabilistic automata and as we will see later on, there is a relationship between the two of them.
[^5]: We assume that all distributions are finitely supported.
[^6]: We denote by $\delta_x$ the point distribution concentrated on $x$.
[^7]: Notice that by assuming the structure $\flip_{p_1,\dots,p_n}\cdot(\tau_1\cdot a_1 + \dots + \tau_n\cdot a_n$, the state between the flip action the corresponding $\tau$ transitions is never a final state. Hence we are safe to use $F_P$ as the final state of $\epsilon(P)$
[^8]: $\cup_iL_i\subseteq Q$ is obvious and the converse could be proven by showing that if $t\in L(Q)$ and $t\notin L(Q_i)$ for every $i$, then there exists $Q'$ constructed from $Q$ “minus" some part of $t$ such that $Q_i\leq Q'< Q$ for any $i$.
[^9]: A rigorous proof of this fact could be done by induction on the structure of $x$.
[^10]: A small difference from CSP is that we consider only words terminating to final states but since $F_{x\|y} = F_x\times F_y$, we are safe to use most of the general properties found in CSP such as $$Tr(x\| y) = \{t \ |\ t_{|x}\in Tr(x) \wedge t_{|y} \in Tr(y) \}$$.
|
---
abstract: 'Let $X$ and $Y$ be Banach spaces, let $\mathcal{A}(X)$ stands for the algebra of approximable operators on $X$, and let $P\colon\mathcal{A}(X)\to Y$ be an orthogonally additive, continuous $n$-homogeneous polynomial. If $X^*$ has the bounded approximation property, then we show that there exists a unique continuous linear map $\Phi\colon\mathcal{A}(X)\to Y$ such that $P(T)=\Phi(T^n)$ for each $T\in\mathcal{A}(X)$.'
address:
- |
Departamento de An' alisis Matem' atico\
Facultad de Ciencias\
Universidad de Granada\
18071 Granada, Spain
- |
Departamento de An' alisis Matem' atico\
Facultad de Ciencias\
Universidad de Granada\
18071 Granada, Spain
- |
Departamento de An' alisis Matem' atico\
Facultad de Ciencias\
Universidad de Granada\
18071 Granada, Spain
author:
- 'J. Alaminos'
- 'M. L. C. Godoy'
- 'A.R. Villena'
title: Orthogonally additive polynomials on the algebras of approximable operators
---
[^1]
Introduction
============
Throughout all algebras and linear spaces are complex. Of course, linearity is understood to mean complex linearity.
Let $A$ be an algebra and let $Y$ be a linear space. A map $P\colon A\to Y$ is said to be *orthogonally additive* if $$a,b\in A, \ ab=ba=0 \ \Rightarrow \ P(a+b)=P(a)+P(b) .$$ Let $X$ and $Y$ be linear spaces, and let $n\in\mathbb{N}$. A map $P\colon X\to Y$ is said to be an *$n$-homogeneous polynomial* if there exists an $n$-linear map $\varphi\colon X^n\to Y$ such that $P(x)=\varphi(x,\dotsc,x)$ $(x\in X)$. Here and subsequently, $X^n$ stands for the $n$-fold Cartesian product of $X$. Such a map is unique if it is required to be symmetric. This is a consequence of the so-called polarization formula which defines $\varphi$ through $$\varphi(x_1,\ldots,x_n)=
\frac{1}{n!\,2^n}\sum_{\epsilon_1,\ldots,\epsilon_n=\pm 1}
\epsilon_{1}\cdots\epsilon_{n} P(\epsilon_1x_1+\cdots+\epsilon_{n}x_{n}).$$ Further, in the case where $X$ and $Y$ are normed spaces, the polynomial $P$ is continuous if and only if the symmetric $n$-linear map $\varphi$ associated with $P$ is continuous.
Let $A$ be a Banach algebra. Given $n\in\mathbb{N}$, a Banach space $Y$, and a continuous linear map $\Phi\colon A\to Y$, the map $a\mapsto \Phi(a^n)$ is a typical example of continuous orthogonally additive $n$-homogeneous polynomial, and a standard problem consists in determining whether these are precisely the canonical examples of continuous orthogonally additive $n$-homogeneous polynomials on $A$. In the case where $A$ is a $C^*$-algebra, it is shown in [@P] that every continuous $n$-homogeneous polynomial $P\colon A\to Y$ can be represented in the form $$P(a)=\Phi(a^n) \quad (a\in A)$$ for some continuous linear map $\Phi\colon A\to Y$ (see [@P2; @P3] for the case where $A$ is a $C^*$-algebra and $P$ is a holomorphic map). The references [@A1; @A2; @V] discuss the case where $A$ is a commutative Banach algebra. This paper is concerned with the problem of representing the continuous orthogonally additive homogeneous polynomials in the case where $A$ is the algebra $\mathcal{A}(X)$ of *approximable operators* on a Banach space $X$. Here, $\mathcal{B}(X)$ is the Banach algebra of continuous linear operators on $X$, $\mathcal{F}(X)$ is the two-sided ideal of $\mathcal{B}(X)$ consisting of finite-rank operators, and $\mathcal{A}(X)$ is the closure of $\mathcal{F}(X)$ in $\mathcal{B}(X)$ with respect to the operator norm.
Let $X$ be a Banach space. Then $X^*$ denotes the dual of $X$. For $x\in X$ and $f\in X^*$, we write $x\otimes f$ for the operator defined by $(x\otimes f)(y)=f(y)x$ for each $y\in X$. Let $n\in\mathbb{N}$. Then we write $\mathbb{M}_n$ for the full matrix algebra of order $n$ over $\mathbb{C}$, and $\mathfrak{S}_n$ for the symmetric group of order $n$.
Orthogonally additive polynomials on the algebra of finite-rank operators
=========================================================================
\[l1526\] Let $\mathcal{M}$ be a Banach algebra isomorphic to $\mathbb{M}_k$ for some $k\in\mathbb{N}$, let $Y$ be a Banach space, and let $P\colon\mathcal{M}\to Y$ be an orthogonally additive $n$-homogeneous polynomial. Then there exists a unique linear map $\Phi\colon\mathcal{M}\to
Y$ such that $$\label{m00}
P(a)=\Phi(a^n)$$ for each $a\in\mathcal{M}$. Further, if $\varphi\colon\mathcal{M}^n\to Y$ is the symmetric $n$-linear map associated with $P$ and $e$ is the identity of $\mathcal{M}$, then $$\label{m0}
\Phi(a)=\varphi(a,e,\dotsc,e)$$ for each $a\in \mathcal{M}$.
Let $\Psi\colon\mathcal{M}\to\mathbb{M}_k$ be an isomorphism. Since $\mathbb{M}_k$ is a $C^*$-algebra and the map $P\circ\Psi^{-1}\colon\mathbb{M}_k\to Y$ is easily seen to be an orthogonally additive $n$-homogeneous polynomial, [@P Corollary 3.1] then shows that there exists a unique linear map $\Theta\colon\mathbb{M}_k\to Y$ such that $P(\Psi^{-1}(M))=\Theta(M^n)$ for each $M\in\mathbb{M}_k$. It is a simple matter to check that the map $\Phi=\Theta\circ\Psi$ satisfies . Now the polarization of yields $$\varphi(a_1,\ldots,a_n)=
\frac{1}{n!}
\Phi\left(\sum_{\sigma\in\mathfrak{S}_n}a_{\sigma(1)}\cdots a_{\sigma(n)}\right)$$ for each $(a_1,\ldots,a_n)\in \mathcal{M}$, whence $\varphi(a,e,\dotsc,e)=\Phi(a)$ for each $a\in\mathcal{M}$.
\[l1525\] Let $X$ be a Banach space and let $T_1,\dotsc,T_m\in\mathcal{F}(X)$. Then there exists a subalgebra $\mathcal{M}$ of $\mathcal{F}(X)$ such that $T_1,\dotsc,T_m\in\mathcal{M}$ and $\mathcal{M}$ is isomorphic to $\mathbb{M}_k$ for some $k\in\mathbb{N}$.
We can certainly assume that $T_1,\dotsc,T_m$ are rank-one operators. Write $T_j=x_j\otimes f_j$ with $x_j\in X$ and $f_j\in X^*$ for each $j\in \left\{1,\dotsc,m\right\}$.
We claim that there exist $y_1,\dotsc,y_k\in X$ and $g_1,\dotsc g_k\in X^*$ such that $$\label{e1559}
\begin{split}
x_1,\dotsc,x_m\in\text{span}\left(\{y_1,\dotsc,y_k\}\right)&,\\
f_1,\dotsc,f_m\in\text{span}\left(\{g_1,\dotsc,g_k\}\right)&,
\end{split}$$ and $$\label{e1558}
g_i(y_j)=\delta_{ij} \quad \left(i,j\in\{1,\dotsc,k\}\right).$$ Let $\{g_1,\dotsc,g_l\}$ be a basis of the linear span of $\{f_1,\dotsc,f_m\}$, and let $y_1\dotsc,y_l\in X$ be such that $g_i(y_j)=\delta_{ij}$ ($i,j\in\{1,\dotsc,l\}$). Let $U$ be the linear span of $\{y_1,\dotsc,y_l\}$ in $X$, let $Q\colon X\to X/U$ the quotient map, and let $$U^\perp=\{f\in X^*\colon f(y_1)=\dots=f(y_l)=0\}.$$ If $\{x_1,\dotsc,x_m\}\subset\{y_1,\dotsc,y_l\}$, then our claim follows. We now assume that $\{x_1,\dotsc,x_m\}\not\subset\{y_1,\dotsc,y_l\}$. Let $y_{l+1},\dotsc,y_k\in X$ be such that $\{Q(y_{l+1}),\dotsc,Q(y_k)\}$ is a basis of the linear span of the set $\{Q(x_1),\dotsc,Q(x_m)\}$ in $X/U$. Since the map $f\mapsto f\circ Q$ defines an isometric isomorphism from $\bigl(X/U\bigr)^*$ onto $U^\perp$, it follows that there exist $g_{l+1},\dotsc,g_k\in U^\perp$ such that $g_i(y_j)=\delta_{ij}$ for all $i,j\in\{l+1,\dotsc,k\}$. It is a simple matter to check that the sets $\{y_1,\dotsc,y_k\}$ and $\{g_1,\dotsc,g_k\}$ satisfy the requirements and .
Let $\mathcal{M}$ be the subalgebra of $\mathcal{F}(X)$ generated by the set $\bigl\{y_i\otimes g_j\colon i,j\in\{1,\dotsc,k\}\bigr\}$. By , $T_1,\dotsc,T_m\in\mathcal{M}$. From we conclude that the algebra $\mathcal{M}$ is isomorphic to the full matrix algebra $\mathbb{M}_n$. Actually, the map $T\mapsto\bigl[g_i(T(y_j))\bigr]_{i,j}$ defines an isomorphism from $\mathcal{M}$ onto $\mathbb{M}_k$ that takes the operator $y_i\otimes g_j$ into the standard matrix unit $E_{i,j}$ ($i,j\in\{1,\dotsc,k\}$).
From Lemmas \[l1526\] and \[l1525\] we see immediately that each orthogonally additive $n$-homogeneous polynomial $P$ on $\mathcal{F}(X)$ can be represented in the standard way on any finitely generated subalgebra of $\mathcal{F}(X)$. The issue is whether the pieces $\Phi_\mathcal{M}$ (where $\mathcal{M}$ ranges over the finitely generated subalgebras of $\mathcal{F}(X)$) fit together to give a linear map $\Phi$ representing the polynomial $P$ on the whole $\mathcal{F}(X)$.
\[crf\] Let $X$ and $Y$ be Banach spaces, and let $P\colon \mathcal{F}(X)\to Y$ be an orthogonally additive $n$-homogeneous polynomial. Then there exists a unique linear map $\Phi\colon\mathcal{F}(X)\to Y$ such that $P(T)=\Phi(T^n)$ for each $T\in\mathcal{F}(X)$.
Let $T\in\mathcal{F}(X)$. Lemma \[l1525\] shows that there exists a subalgebra $\mathcal{M}$ of $\mathcal{F}(X)$ such that $T\in\mathcal{M}$ and $\mathcal{M}$ is isomorphic to $\mathbb{M}_k$ for some $k\in\mathbb{N}$. Then Lemma \[l1526\] yields a unique linear map $\Phi_\mathcal{M}\colon\mathcal{M}\to Y$ such that $P(S)=\Phi_{\mathcal{M}}(S^n)$ for each $S\in\mathcal{M}$. Then we set $\Phi(T)=\Phi_\mathcal{M}(T)$. We now show that $\Phi$ is well-defined. Assume that $\mathcal{M}_1$ and $\mathcal{M}_2$ are subalgebras of $\mathcal{F}(X)$ with the properties that $T\in\mathcal{M}_j$ and $\mathcal{M}_j$ is isomorphic to a full matrix algebra ($j=1,2$). Then, according to Lemma \[l1525\], there exists a subalgebra $\mathcal{N}$ of $\mathcal{F}(X)$ such that $\mathcal{M}_1,\mathcal{M}_2\subset\mathcal{N}$ and $\mathcal{N}$ is isomorphic to a full matrix algebra. The uniqueness of the represention asserted in Lemma \[l1526\] gives that $\Phi_\mathcal{N}$ equals $\Phi_{\mathcal{M}_j}$ when restricted to $\mathcal{M}_j$ for $j=1,2$. Accordingly, we have $\Phi_{\mathcal{M}_1}(T)=\Phi_\mathcal{N}(T)=\Phi_{\mathcal{M}_2}(T)$.
Let $S$, $T\in\mathcal{F}(X)$ and let $\alpha,\beta\in\mathbb{C}$. By Lemma \[l1525\] there exists a subalgebra $\mathcal{M}$ of $\mathcal{F}(X)$ such that $S,T\in\mathcal{M}$ and $\mathcal{M}$ is isomorphic to a full matrix algebra. Then $$\Phi(\alpha S+\beta T)=
\Phi_\mathcal{M}(\alpha S+\beta T)=
\alpha\Phi_\mathcal{M}(S)+\beta\Phi_\mathcal{M}(T)=
\alpha\Phi(S)+\beta\Phi(T),$$ and, since $T^n\in\mathcal{M}$, $$P(T)=\Phi_\mathcal{M}(T^n)=\Phi(T^n).$$ This shows that $\Phi$ is linear and gives a representation of $P$. It should be pointed out that the polarization of this representations yields $$\label{m1}
\varphi(T_1,\ldots,T_n)=
\frac{1}{n!}
\Phi\left(\sum_{\sigma\in\mathfrak{S}_n}T_{\sigma(1)}\cdots T_{\sigma(n)}\right)$$ for each $(T_1,\ldots,T_n)\in \mathcal{F}(X)$, where $\varphi$ is the symmetric $n$-linear map associated with $P$.
Our final task is to prove the uniqueness of the map $\Phi$. Suppose that $\Psi\colon\mathcal{F}(X)\to Y$ is a linear map such that $P(T)=\Psi(T^n)$ for each $T\in\mathcal{F}(X)$. The polarization of this identity gives $$\label{m2}
\varphi(T_1,\ldots,T_n)=
\frac{1}{n!}
\Psi\left(\sum_{\sigma\in\mathfrak{S}_n}T_{\sigma(1)}\cdots T_{\sigma(n)}\right)$$ for each $(T_1,\ldots,T_n)\in \mathcal{F}(X)$. Let $T\in\mathcal{F}(X)$. On account of Lemma \[l1525\], there exists $S\in\mathcal{F}(X)$ such that $TS=ST=T$. From and we obtain $\Phi(T)=\varphi(T,S,\dotsc,S)=\Psi(T)$.
It is not clear at all whether or not the linear map $\Phi$ given in the preceding result is continuous in the case where the polynomial $P$ is continuous.
Orthogonally additive polynomials on the algebra of approximable operators
==========================================================================
Let $A$ be a Banach algebra, let $(e_\lambda)_{\lambda\in\Lambda}$ be a bounded approximate identity for $A$ of bound $C$, and let $\mathcal{U}$ be an ultrafilter on $\Lambda$ containing the order filter on $\Lambda$ (which will be associated with $(e_\lambda)_{\lambda\in\Lambda}$ and fixed throughout). Let $Y$ be a dual Banach space and let $Y_*$ be a predual of $Y$. It follows from the Banach-Alaoglu theorem that each bounded subset of $Y$ is relatively compact with respect to the $\sigma(Y,Y_*)$-topology on $Y$. Consequently, each bounded net $(y_\lambda)_{\lambda\in\Lambda}$ in $Y$ has a unique limit with respect to the $\sigma(Y,Y_*)$-topology along the ultrafilter $\mathcal{U}$, and we write $\lim_{\mathcal{U}} y_\lambda$ for this limit.
Let $\varphi\colon A^n\to Y$ be a continuous $n$-linear map. For each $a_1,\ldots,a_{n-1}\in A$ and $\lambda\in\Lambda$, we have $$\label{e1908}
\begin{split}
\Vert\varphi(a_1,\ldots,a_{n-1},e_\lambda)\Vert
& \le
\Vert\varphi\Vert\Vert a_1\Vert\cdots\Vert a_{n-1}\Vert\Vert e_\lambda\Vert\\
&\le
C\Vert\varphi\Vert\Vert a_1\Vert\cdots\Vert a_{n-1}\Vert .
\end{split}$$ Hence the net $(\varphi(a_1,\ldots,a_{n-1},e_\lambda))_{\lambda\in\Lambda}$ is bounded and therefore we can define the map $\varphi'\colon A^{n-1}\to Y$ by $$\varphi'(a_1,\ldots,a_{n-1})=\lim_{\mathcal{U}}\varphi(a_1,\ldots,a_{n-1},e_\lambda)$$ for each $(a_1,\ldots,a_{n-1})\in A^{n-1}$. The linearity of the limit along an ultrafilter on a topological linear space gives the $(n-1)$-linearity of $\varphi'$. Moreover, from we deduce that $$\Vert\varphi'(a_1,\ldots,a_{n-1})\Vert\le C\Vert\varphi\Vert\Vert a_1\Vert\cdots\Vert a_{n-1}\Vert$$ for each $(a_1,\ldots,a_{n-1})\in A^{n-1}$, which gives the continuity of $\varphi'$ and $\Vert\varphi'\Vert\le C\Vert\varphi\Vert$. Further, it is clear that if the map $\varphi$ is symmetric, then the map $\varphi'$ is symmetric.
\[l177\] Let $A$ be a Banach algebra with a bounded approximate identity $(e_\lambda)_{\lambda\in\Lambda}$, let $Y$ be a dual Banach space, and let $\varphi\colon A^n\to Y$ be a continuous symmetric $n$-linear map with $n\ge 2$. Suppose that $$\varphi(a_1,\ldots,a_n)=
\frac{1}{n!}
\sum_{\sigma\in\mathfrak{S}_n}\varphi'(a_{\sigma(1)},\ldots,a_{\sigma(n-1)}a_{\sigma(n)})$$ for each $(a_1,\dotsc,a_n)\in A^n$. Then there exists a continuous linear map $\Phi\colon A\to Y$ such that $$\varphi(a_1,\ldots,a_n)=
\frac{1}{n!}
\Phi\left(\sum_{\sigma\in\mathfrak{S}_n}a_{\sigma(1)}\cdots a_{\sigma(n)}\right)$$ for each $(a_1,\ldots,a_n)\in A^n$.
The proof is by induction on $n$. The result is certainly true if $n=2$. Assume that the result is true for $n$, and let $\varphi\colon A^{n+1}\to Y$ be a continuous symmetric $(n+1)$-linear map such that $$\varphi(a_1,\ldots,a_n,a_{n+1})=
\frac{1}{(n+1)!}
\sum_{\sigma\in\mathfrak{S}_{n+1}}\varphi'(a_{\sigma(1)},\ldots,a_{\sigma(n-1)},a_{\sigma(n)}a_{\sigma(n+1)})$$ for each $(a_1,\dotsc,a_{n+1})\in A^{n+1}$. We claim that $$\label{1112}
\varphi'(a_1,\ldots,a_n)=
\frac{1}{n!}
\sum_{\sigma\in\mathfrak{S}_n}\varphi''(a_{\sigma(1)},\ldots,a_{\sigma(n-1)}a_{\sigma(n)})$$ for each $(a_1,\dotsc,a_{n})\in A^n$. Here $\varphi''$ stands for the $(n-1)$-linear map $(\varphi')'$. Indeed, for all $(a_1,\dotsc,a_n)\in A^n$ and $\lambda\in\Lambda$, we have $$\begin{aligned}
\varphi(a_1,\ldots,a_n,e_\lambda)
& =
\frac{1}{(n+1)!}\sum_{\tau\in\mathfrak{S}_{n}}\varphi'(a_{\tau(1)},\ldots,a_{\tau(n-1)},a_{\tau(n)}e_\lambda)\\
& \quad {}+ \frac{1}{(n+1)!}\sum_{\tau\in\mathfrak{S}_{n}}\varphi'(a_{\tau(1)},\ldots,a_{\tau(n-1)},e_\lambda a_{\tau(n)})\\
& \quad {}+
\frac{1}{(n+1)!}\sum_{\tau\in\mathfrak{S}_{n}}\varphi'(a_{\tau(1)},\ldots,e_\lambda ,a_{\tau(n-1)}a_{\tau(n)})\\
&\quad {}+\dotsb+
\frac{1}{(n+1)!}\sum_{\tau\in\mathfrak{S}_{n}}\varphi'(e_\lambda,a_{\tau(1)},\ldots ,a_{\tau(n-1)}a_{\tau(n)}).\\\end{aligned}$$ Since $(e_\lambda)_{\lambda\in\Lambda}$ is a bounded approximate identity for $A$, the nets $(a_ke_\lambda)_{\lambda\in\Lambda}$ and $(e_\lambda a_k)_{\lambda\in\Lambda}$ converge to $a_k$ in norm for each $k\in\{1,\dotsc,n\}$, and so, taking limits along $\mathcal{U}$ on both sides of the above equation (and using the continuity of $\varphi'$), we see that $$\begin{aligned}
\varphi'(a_1,\ldots,a_n)
& =
\lim_\mathcal{U}
\varphi(a_1,\ldots,a_n,e_\lambda)\\
& =
\frac{1}{(n+1)!}\sum_{\tau\in\mathfrak{S}_{n}}\lim_\mathcal{U}\varphi'(a_{\tau(1)},\ldots,a_{\tau(n-1)},a_{\tau(n)}e_\lambda)\\
& \quad {}+
\frac{1}{(n+1)!}\sum_{\tau\in\mathfrak{S}_{n}}\lim_\mathcal{U}\varphi'(a_{\tau(1)},\ldots,a_{\tau(n-1)},e_\lambda a_{\tau(n)})\\
& \quad {}+
\frac{1}{(n+1)!}\sum_{\tau\in\mathfrak{S}_{n}}\lim_\mathcal{U}\varphi'(a_{\tau(1)},\ldots,e_\lambda ,a_{\tau(n-1)}a_{\tau(n)})\\
& \quad {}+\dotsb +
\frac{1}{(n+1)!}\sum_{\tau\in\mathfrak{S}_{n}}\lim_\mathcal{U}\varphi'(e_\lambda,a_{\tau(1)},\ldots ,a_{\tau(n-1)}a_{\tau(n)})\\
& =
\frac{1}{(n+1)!}\sum_{\tau\in\mathfrak{S}_{n}}\varphi'(a_{\tau(1)},\ldots,a_{\tau(n-1)},a_{\tau(n)})\\
& \quad {}+
\frac{1}{(n+1)!}\sum_{\tau\in\mathfrak{S}_{n}}\varphi'(a_{\tau(1)},\ldots,a_{\tau(n-1)},a_{\tau(n)})\\
& \quad {}+
\frac{1}{(n+1)!}\sum_{\tau\in\mathfrak{S}_{n}}\varphi''(a_{\tau(1)},\ldots ,a_{\tau(n-1)}a_{\tau(n)})\\
& \quad {}+\dotsb+
\frac{1}{(n+1)!}\sum_{\tau\in\mathfrak{S}_{n}}\varphi''(a_{\tau(1)},\ldots ,a_{\tau(n-1)}a_{\tau(n)})\\
& =
\frac{1}{(n+1)!} 2n!\,\varphi'(a_1,\ldots,a_{n-1},a_n)\\
& \quad {}+
\frac{1}{(n+1)!}(n-1)\sum_{\tau\in\mathfrak{S}_{n}}\varphi''(a_{\tau(1)},\ldots ,a_{\tau(n-1)}a_{\tau(n)}).\\\end{aligned}$$ We thus get $$\Bigl(1-\frac{2}{n+1}\Bigr)\varphi'(a_1,\ldots,a_n)=
\frac{n-1}{(n+1)!}\sum_{\tau\in\mathfrak{S}_{n}}\varphi''(a_{\tau(1)},\ldots ,a_{\tau(n-1)}a_{\tau(n)}),$$ which proves our claim.
By and the inductive hypothesis, there exists a continuous linear map $\Phi\colon A\to Y$ such that $$\varphi'(a_1,\ldots,a_n)=
\frac{1}{n!}\Phi\left(
\sum_{\tau\in\mathfrak{S}_n}a_{\tau(1)}\cdots a_{\tau(n)}\right)$$ and thus $$\begin{aligned}
\varphi'(a_1,\ldots,a_n) & =
\frac{1}{n!} \sum_{\tau\in\mathfrak{S}_{n-1}}\Phi(a_{\tau(1)}\cdots a_{\tau(n-1)}a_n)+
\frac{1}{n!}\sum_{\tau\in\mathfrak{S}_{n-1}}\Phi(a_{\tau(1)}\cdots a_na_{\tau(n-1)})\\
& \quad {}+\cdots +
\frac{1}{n!}\sum_{\tau\in\mathfrak{S}_{n-1}}\Phi\left(a_na_{\tau(1)}\cdots a_{\tau(n-1)}\right)\end{aligned}$$ for each $(a_1,\ldots,a_n)\in A^n$. Therefore, for each $(a_1,\ldots,a_n,a_{n+1})\in A^{n+1}$, we have $$\begin{gathered}
\varphi(a_1,\ldots,a_n,a_{n+1})
=
\frac{1}{(n+1)!}\sum_{\sigma\in\mathfrak{S}_{n+1}}
\varphi'(a_{\sigma(1)},\ldots,a_{\sigma(n-1)},a_{\sigma(n)}a_{\sigma(n+1)})\\
=
\frac{1}{(n+1)!}\sum_{\sigma\in\mathfrak{S}_{n+1}}
\frac{1}{n!}\sum_{\tau\in\mathfrak{S}_{n-1}}
\Phi(a_{\sigma(\tau(1))}\cdots a_{\sigma(\tau(n-1))}a_{\sigma(n)}a_{\sigma(n+1)})\\
+
\frac{1}{(n+1)!}\sum_{\sigma\in\mathfrak{S}_{n+1}}
\frac{1}{n!}\sum_{\tau\in\mathfrak{S}_{n-1}}
\Phi(a_{\sigma(\tau(1))}\cdots a_{\sigma(n)}a_{\sigma(n+1)}a_{\sigma(\tau(n-1))})\\
\qquad {}+\cdots+
\frac{1}{(n+1)!}\sum_{\sigma\in\mathfrak{S}_{n+1}}
\frac{1}{n!}\sum_{\tau\in\mathfrak{S}_{n-1}}
\Phi(a_{\sigma(n)}a_{\sigma(n+1)}a_{\sigma(\tau(1))}\ldots a_{\sigma(\tau(n-1))}) \\
=
\frac{1}{(n+1)!}\frac{1}{n!}
\sum_{\tau\in\mathfrak{S}_{n-1}}\sum_{\sigma\in\mathfrak{S}_{n+1}}
\Phi(a_{\sigma(\tau(1))}\cdots a_{\sigma(\tau(n-1))}a_{\sigma(n)}a_{\sigma(n+1)})\\
{} +
\frac{1}{(n+1)!}\frac{1}{n!}
\sum_{\tau\in\mathfrak{S}_{n-1}}\sum_{\sigma\in\mathfrak{S}_{n+1}}
\Phi(a_{\sigma(\tau(1))}\cdots a_{\sigma(n)}a_{\sigma(n+1)}a_{\sigma(\tau(n-1))})\\
\qquad {}+\cdots+
\frac{1}{(n+1)!}\frac{1}{n!}
\sum_{\tau\in\mathfrak{S}_{n-1}}\sum_{\sigma\in\mathfrak{S}_{n+1}}
\Phi(a_{\sigma(n)}a_{\sigma(n+1)}a_{\sigma(\tau(1))}\cdots a_{\sigma(\tau(n-1))})\\
=
\frac{1}{(n+1)!}\frac{1}{n!}
\sum_{\tau\in\mathfrak{S}_{n-1}}\sum_{\sigma\in\mathfrak{S}_{n+1}}
\Phi(a_{\sigma(1)}\cdots a_{\sigma(n-1)}a_{\sigma(n)}a_{\sigma(n+1)}) \\
{}+
\frac{1}{(n+1)!}\frac{1}{n!}
\sum_{\tau\in\mathfrak{S}_{n-1}}\sum_{\sigma\in\mathfrak{S}_{n+1}}
\Phi(a_{\sigma(1)}\cdots a_{\sigma(n-1)}a_{\sigma(n)}a_{\sigma(n+1)}) \\
\qquad {}+\cdots+
\frac{1}{(n+1)!}\frac{1}{n!}
\sum_{\tau\in\mathfrak{S}_{n-1}}\sum_{\sigma\in\mathfrak{S}_{n+1}}
\Phi(a_{\sigma(1)}\cdots a_{\sigma(n-1)}a_{\sigma(n)}a_{\sigma(n+1)})\\
=
\frac{1}{(n+1)!}\sum_{\sigma\in\mathfrak{S}_{n+1}}
\Phi(a_{\sigma(1)}\cdots a_{\sigma(n-1)}a_{\sigma(n)}a_{\sigma(n+1)}),\end{gathered}$$ and the induction continues.
\[rfc3\] Let $\xi_1,\dotsc,\xi_n,\zeta$ be noncommuting indeterminates and let $\pi_n$ be the polynomial defined by $$%\label{l1}
\pi_n(\xi_1,\dotsc,\xi_n)=
\sum_{\sigma\in \mathfrak{S}_n}\xi_{\sigma(1)}\dotsm\xi_{\sigma(n)}.$$ Then the following identities hold: $$\begin{gathered}
\label{l2}
\sum_{\sigma\in \mathfrak{S}_n}\pi_n(\xi_{\sigma(1)},\dotsc,\xi_{\sigma(n)}\zeta)
= (n-1)!\sum_{\sigma\in \mathfrak{S}_n}
\Bigl[
\xi_{\sigma(1)}\dotsm \xi_{\sigma(n-1)}\xi_{\sigma(n)}\zeta
\\ {}+
\xi_{\sigma(1)}\dotsm \xi_{\sigma(n-1)}\zeta\xi_{\sigma(n)}
+ \dotsb +
\xi_{\sigma(1)}\zeta \dotsm \xi_{\sigma(n-1)}\xi_{\sigma(n)}
\Bigr],\end{gathered}$$ $$\begin{gathered}
\label{l23}
\sum_{\sigma\in \mathfrak{S}_n}\pi_n(\xi_{\sigma(1)},\dotsc,\zeta\xi_{\sigma(n)})
= (n-1)!\sum_{\sigma\in \mathfrak{S}_n}
\Bigl[
\xi_{\sigma(1)}\dotsm \xi_{\sigma(n-1)}\zeta \xi_{\sigma(n)}
\\ {} +
\xi_{\sigma(1)}\dotsm \zeta\xi_{\sigma(n-1)} \xi_{\sigma(n)}
+
\cdots {}+ \zeta \xi_{\sigma(1)} \dotsm \xi_{\sigma(n-1)}\xi_{\sigma(n)}
\Bigr],\end{gathered}$$ and $$\begin{gathered}
\label{l4}
\sum_{\sigma\in \mathfrak{S}_n}\pi_n(\xi_{\sigma(1)},\dotsc,\xi_{\sigma(n-1)}\xi_{\sigma(n)},\zeta) =
(n-1)!\sum_{\sigma\in \mathfrak{S}_n}
\Bigl[
\xi_{\sigma(1)}\dotsm \xi_{\sigma(n-1)}\xi_{\sigma(n)}\zeta
\\ +
\zeta \xi_{\sigma(1)} \dotsm \xi_{\sigma(n-1)}\xi_{\sigma(n)}
\Bigr]\\
+(n-2)(n-2)!\sum_{\sigma\in \mathfrak{S}_n}
\Bigl[
\xi_{\sigma(1)}\dotsm \xi_{\sigma(n-1)}\zeta \xi_{\sigma(n)}
+\dotsb+
\xi_{\sigma(1)}\zeta \dotsm \xi_{\sigma(n-1)}\xi_{\sigma(n)}
\Bigr].\end{gathered}$$
It is clear that $$\label{r14}
\pi_n(\xi_1,\dotsc,\xi_n)
=
\sum_{\tau\in \mathfrak{S}_{n-1}}
\Bigl[
\xi_{\tau(1)}\dotsm \xi_{\tau(n-1)}\xi_n
+\dotsb+
\xi_n\xi_{\tau(1)}\dotsm \xi_{\tau(n-1)}
\Bigr].$$ Therefore, for each $\sigma\in \mathfrak{S}_n$, we have $$\begin{aligned}
\pi_n(\xi_{\sigma(1)},\dotsc,\xi_{\sigma(n)}\zeta)
& =
\sum_{\tau\in \mathfrak{S}_{n-1}}
\Bigl[
\xi_{\sigma(\tau(1))}\dotsm \xi_{\sigma(\tau(n-1))}\xi_{\sigma(n)}\zeta
\\
& \qquad {}+\cdots+
\xi_{\sigma(n)}\zeta\xi_{\sigma(\tau(1))}\dotsm \xi_{\sigma(\tau(n-1))}
\Bigr].\end{aligned}$$ We thus get $$\begin{gathered}
\sum_{\sigma\in \mathfrak{S}_n}
\pi_n(\xi_{\sigma(1)},\dotsc,\xi_{\sigma(n)}\zeta) \\*
=
\sum_{\tau\in \mathfrak{S}_{n-1}}\sum_{\sigma\in \mathfrak{S}_n}
\Bigl[
\xi_{\sigma(\tau(1))}\dotsm \xi_{\sigma(\tau(n-1))}\xi_{\sigma(n)}\zeta
+\cdots+
\xi_{\sigma(n)}\zeta\xi_{\sigma(\tau(1))}\dotsm \xi_{\sigma(\tau(n-1))}
\Bigr] \\
=
\sum_{\tau\in \mathfrak{S}_{n-1}}
\Bigl[
\sum_{\sigma\in \mathfrak{S}_n}\xi_{\sigma(1)}\dotsm \xi_{\sigma(n-1)}\xi_{\sigma(n)}\zeta
+\dotsb+
\sum_{\sigma\in \mathfrak{S}_n}\xi_{\sigma(1)}\zeta \dotsm \xi_{\sigma(n-1)}\xi_{\sigma(n)}
\Bigr]\\
=
(n-1)!
\sum_{\sigma\in \mathfrak{S}_n}
\Bigl[
\xi_{\sigma(1)}\dotsm \xi_{\sigma(n-1)}\xi_{\sigma(n)}\zeta
+\dotsb+
\xi_{\sigma(1)}\zeta \dotsm \xi_{\sigma(n-1)}\xi_{\sigma(n)}
\Bigr],\end{gathered}$$ which gives . In the same way we can check .
From we now see that $$\begin{aligned}
\pi_n(\xi_1,\dotsc,\xi_n)
& =
\sum_{\tau\in \mathfrak{S}_{n-2}}
\Bigl[
\xi_{\tau(1)}\dotsm\xi_{\tau(n-2)}\xi_{n-1}\xi_n+
\xi_{\tau(1)}\dotsm\xi_{n-1}\xi_{\tau(n-2)}\xi_n \\
& \qquad {}+ \dotsb +
\xi_{n-1}\xi_{\tau(1)}\dotsm\xi_{\tau(n-2)}\xi_n\Bigr]\\
&\quad {}+\sum_{\tau\in\mathfrak{S}_{n-2}}
\Bigl[
\xi_{\tau(1)}\dotsm\xi_{\tau(n-2)}\xi_n\xi_{n-1}+
\xi_{\tau(1)}\dotsm\xi_{n-1}\xi_n\xi_{\tau(n-2)}\\
& \qquad {}+ \dotsb +
\xi_{n-1}\xi_{\tau(1)}\dotsm\xi_n\xi_{\tau(n-2)}\Bigr]\\
&\quad {}+\sum_{\tau\in\mathfrak{S}_{n-2}}
\Bigl[
\xi_{\tau(1)}\dotsm\xi_n\xi_{\tau(n-2)}\xi_{n-1}+
\xi_{\tau(1)}\dotsm\xi_n\xi_{n-1}\xi_{\tau(n-2)} \\
& \qquad+
%\quad\quad\quad\quad
\dotsb+
\xi_{n-1}\xi_{\tau(1)}\dotsm\xi_n\xi_{\tau(n-3)}\xi_{\tau(n-2)}\Bigr]\\
& \quad {} + \dotsb + {}\\
& \quad {}+\sum_{\tau\in \mathfrak{S}_{n-2}}
\Bigl[
\xi_n\xi_{\tau(1)}\dotsm\xi_{\tau(n-2)}\xi_{n-1}+
\xi_n\xi_{\tau(1)}\dotsm\xi_{n-1}\xi_{\tau(n-2)}\\
&\qquad {}+ \dotsb+
\xi_n\xi_{n-1}\xi_{\tau(1)}\dotsm\xi_{\tau(n-3)}\xi_{\tau(n-2)}\Bigr].\\\end{aligned}$$ Therefore, for each $\sigma\in \mathfrak{S}_{n}$, we have $$\begin{aligned}
&\pi_n(\xi_{\sigma(1)},\dotsc,\xi_{\sigma(n-1)}\xi_{\sigma(n)},\zeta)\\
= &
\sum_{\tau\in \mathfrak{S}_{n-2}}
\Bigl[
\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(\tau(n-2))}\xi_{\sigma(n-1)}\xi_{\sigma(n)}\zeta+
\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(n-1)}\xi_{\sigma(n)}\xi_{\sigma(\tau(n-2))}\zeta\\
&
\qquad {}+\dotsb+
\xi_{\sigma(n-1)}\xi_{\sigma(n)}\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(\tau(n-2))}\zeta\Bigr]\\
& {}+\sum_{\tau\in\mathfrak{S}_{n-2}}
\Bigl[
\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(\tau(n-2))}\zeta\xi_{\sigma(n-1)}\xi_{\sigma(n)}+
\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(n-1)}\xi_{\sigma(n)}\zeta\xi_{\sigma(\tau(n-2))}\\
&
\qquad {}+\dotsb+
\xi_{\sigma(n-1)}\xi_{\sigma(n)}\xi_{\sigma(\tau(1))}\dotsm\zeta\xi_{\sigma(\tau(n-2))}\Bigr]\\
&
{}+ \sum_{\tau\in\mathfrak{S}_{n-2}}
\Bigl[
\xi_{\sigma(\tau(1))}\dotsm\zeta\xi_{\sigma(\tau(n-2))}\xi_{\sigma(n-1)}\xi_{\sigma(n)}+
\xi_{\sigma(\tau(1))}\dotsm\zeta\xi_{\sigma(n-1)}\xi_{\sigma(n)}\xi_{\sigma(\tau(n-2))}\\
&
\qquad {}+\dotsb+
\xi_{\sigma(n-1)}\xi_{\sigma(n)}\xi_{\sigma(\tau(1))}\dotsm\zeta\xi_{\sigma(\tau(n-3))}\xi_{\sigma(\tau(n-2))}\Bigr]\\
& {}+\dots +{}\\
& + \sum_{\tau\in\mathfrak{S}_{n-2}}
\Bigl[
\zeta\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(\tau(n-2))}\xi_{\sigma(n-1)}\xi_{\sigma(n)}+
\zeta\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(n-1)}\xi_{\sigma(n)}\xi_{\sigma(\tau(n-2))}\\
&
\qquad {}+\dotsb+
\zeta\xi_{\sigma(n-1)}\xi_{\sigma(n)}\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(\tau(n-3))}\xi_{\sigma(\tau(n-2))}\Bigr].\\\end{aligned}$$ We thus get $$\begin{aligned}
&\sum_{\sigma\in \mathfrak{S}_n}\pi_n (\xi_{\sigma(1)},\dotsc,\xi_{\sigma(n-1)}\xi_{\sigma(n)},\zeta)
\\
=\sum_{\tau\in\mathfrak{S}_{n-2}}
\Bigl[ &
\sum_{\sigma\in \mathfrak{S}_n}\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(\tau(n-2))}\xi_{\sigma(n-1)}\xi_{\sigma(n)}\zeta \\
& {}+ \sum_{\sigma\in \mathfrak{S}_n}\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(n-1)}\xi_{\sigma(n)}\xi_{\sigma(\tau(n-2))}\zeta \\
& {}+\dotsb+ \sum_{\sigma\in \mathfrak{S}_n}\xi_{\sigma(n-1)}\xi_{\sigma(n)}\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(\tau(n-2))}\zeta\Bigr] \\
{}+ \sum_{\tau\in\mathfrak{S}_{n-2}}
\Bigl[ &
\sum_{\sigma\in\mathfrak{S}_n}\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(\tau(n-2))}\zeta\xi_{\sigma(n-1)}\xi_{\sigma(n)}\\
& {} + \sum_{\sigma\in\mathfrak{S}_n}\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(n-1)}\xi_{\sigma(n)}\zeta\xi_{\sigma(\tau(n-2))} \\
& {}+\dotsb+ \sum_{\sigma\in\mathfrak{S}_n}\xi_{\sigma(n-1)}\xi_{\sigma(n)}\xi_{\sigma(\tau(1))}\dotsm\zeta\xi_{\sigma(\tau(n-2))}\Bigr] \\
{}+{} \sum_{\tau\in\mathfrak{S}_{n-2}}
\Bigl[ &
\sum_{\sigma\in\mathfrak{S}_n}\xi_{\sigma(\tau(1))}\dotsm\zeta\xi_{\sigma(\tau(n-2))}\xi_{\sigma(n-1)}\xi_{\sigma(n)} \\
& {}+ \sum_{\sigma\in\mathfrak{S}_n}\xi_{\sigma(\tau(1))}\dotsm\zeta\xi_{\sigma(n-1)}\xi_{\sigma(n)}\xi_{\sigma(\tau(n-2))} \\
& {}+\dotsb+ \sum_{\sigma\in\mathfrak{S}_n}\xi_{\sigma(n-1)}\xi_{\sigma(n)}\xi_{\sigma(\tau(1))}\dotsm\zeta\xi_{\sigma(\tau(n-3))}
\xi_{\sigma(\tau(n-2))}\Bigr] \\
& {}+\dotsb +{} \\
%\vdots\quad\quad& \\
{}+\sum_{\tau\in\mathfrak{S}_{n-2}}
\Bigl[&
\sum_{\sigma\in\mathfrak{S}_n}\zeta\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(\tau(n-2))}\xi_{\sigma(n-1)}\xi_{\sigma(n)} \\
& {}+ \sum_{\sigma\in\mathfrak{S}_n}\zeta\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(n-1)}\xi_{\sigma(n)}\xi_{\sigma(\tau(n-2))} \\
& {}+\dotsb+ \sum_{\sigma\in\mathfrak{S}_n}\zeta\xi_{\sigma(n-1)}\xi_{\sigma(n)}\xi_{\sigma(\tau(1))}\dotsm\xi_{\sigma(\tau(n-3))}\xi_{\sigma(\tau(n-2))}\Bigr]\\
=
\sum_{\tau\in \mathfrak{S}_{n-2}}
\Bigl[
&\sum_{\sigma\in\mathfrak{S}_n}\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\xi_{\sigma(n)}\zeta+
\sum_{\sigma\in\mathfrak{S}_n}\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\xi_{\sigma(n)}\zeta\\
& {}+\dotsb+
\sum_{\sigma\in\mathfrak{S}_n}\xi_{\sigma(1)}\xi_{\sigma(2)}\xi_{\sigma(3)}\dotsm\xi_{\sigma(n)}\zeta\Bigr] \\
{}+ \sum_{\tau\in\mathfrak{S}_{n-2}}
\Bigl[
&\sum_{\sigma\in\mathfrak{S}_n}\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\zeta\xi_{\sigma(n-1)}\xi_{\sigma(n)}+
\sum_{\sigma\in\mathfrak{S}_n}\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\zeta\xi_{\sigma(n)}\\
& {}+\dotsb +
\sum_{\sigma\in\mathfrak{S}_n}\xi_{\sigma(1)}\xi_{\sigma(2)}\xi_{\sigma(3)}\dotsm\zeta\xi_{\sigma(n)}\Bigr] \\
{}+ \sum_{\tau\in\mathfrak{S}_{n-2}}
\Bigl[
&\sum_{\sigma\in\mathfrak{S}_n}\xi_{\sigma(1)}\dotsm\zeta\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\xi_{\sigma(n)}+
\sum_{\sigma\in \mathfrak{S}_n}\xi_{\sigma(1)}\dotsm\zeta\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\xi_{\sigma(n)}\\
& {}+\dotsb +
\sum_{\sigma\in\mathfrak{S}_n}\xi_{\sigma(1)}\xi_{\sigma(2)}\xi_{\sigma(3)}\dotsm\zeta\xi_{\sigma(n-1)}\xi_{\sigma(n)}\Bigr] \\
{} +\dotsb +{} \\
%\vdots\quad\quad&\\
{}+\sum_{\tau\in\mathfrak{S}_{n-2}}
\Bigl[
&\sum_{\sigma\in\mathfrak{S}_n}\zeta\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\xi_{\sigma(n)}+
\sum_{\sigma\in\mathfrak{S}_n}\zeta\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\xi_{\sigma(n)}\\
& {}+\dotsb+
\sum_{\sigma\in\mathfrak{S}_n}\zeta\xi_{\sigma(1)}\xi_{\sigma(2)}\xi_{\sigma(3)}\dotsm\xi_{\sigma(n-1)}\xi_{\sigma(n)}\Bigr]\\
= \sum_{\tau\in\mathfrak{S}_{n-2}}\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
&(n-1)\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\xi_{\sigma(n)}\zeta\Bigr]\\
{}+ \sum_{\tau\in\mathfrak{S}_{n-2}}\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
&\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\zeta\xi_{\sigma(n-1)}\xi_{\sigma(n)}+
(n-2)
\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\zeta\xi_{\sigma(n)}\Bigr] \\
{}+ \sum_{\tau\in\mathfrak{S}_{n-2}}\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
& 2\xi_{\sigma(1)}\dotsm\zeta\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\xi_{\sigma(n)}+
(n-3)
\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\zeta\xi_{\sigma(n-1)}\xi_{\sigma(n)}\Bigr]\\ {}+ \dotsb +{} \\
%\vdots\quad\quad&\\
{} +\sum_{\tau\in\mathfrak{S}_{n-2}}\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
& k\xi_{\sigma(1)}\dotsm\zeta\xi_{\sigma(n-k)}\dotsm\xi_{\sigma(n)}+
(n-k-1)
\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-k)}\zeta\dotsm\xi_{\sigma(n)}\Bigr] \\
{}+\dotsb +{}\\
%\vdots\quad\quad&\\
{}+\sum_{\tau\in\mathfrak{S}_{n-2}}\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
& (n-1)\zeta\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\xi_{\sigma(n)}\Bigr]\\
= (n-2)!\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
& (n-1)\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\xi_{\sigma(n)}\zeta\Bigr] \\
{}+ (n-2)!\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
&\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\zeta\xi_{\sigma(n-1)}\xi_{\sigma(n)}+
(n-2)
\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\zeta\xi_{\sigma(n)}\Bigr]\\
{}+ (n-2)!\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
& 2\xi_{\sigma(1)}\dotsm\zeta\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\xi_{\sigma(n)}+
(n-3)
\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\zeta\xi_{\sigma(n-1)}\xi_{\sigma(n)}\Bigr]\\
{}+ \dotsb + {}\\
%\vdots\quad\quad&\\
+ (n-2)!\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
& k\xi_{\sigma(1)}\dotsm\zeta\xi_{\sigma(n-k)}\dotsm\xi_{\sigma(n)}+
(n-k-1)
\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-k)}\zeta\dotsm\xi_{\sigma(n)}\Bigr] \\
{}+\dotsb+{}\\
%\vdots\quad\quad&\\
+(n-2)!\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
&(n-1)\zeta\xi_{\sigma(1)}\dotsm\xi_{\sigma(n-2)}\xi_{\sigma(n-1)}\xi_{\sigma(n)}\Bigr],\end{aligned}$$ which gives .
\[main\] Let $X$ and $Y$ be Banach spaces, and let $P\colon\mathcal{A}(X)\to Y$ be a continuous $n$-homogeneous polynomial. Suppose that $X^*$ has the bounded approximation property. Then the following conditions are equivalent:
1. the polynomial $P$ is orthogonally additive, i.e., $P(S+T)=P(S)+P(T)$ whenever $S$, $T\in\mathcal{A}(X)$ are such that $ST=TS=0$;
2. the polynomial $P$ is orthogonally additive on $\mathcal{F}(X)$, i.e., $P(S+T)=P(S)+P(T)$ whenever $S$, $T\in\mathcal{F}(X)$ are such that $ST=TS=0$;
3. there exists a unique continuous linear map $\Phi\colon\mathcal{A}(X)\to Y$ such that $P(T)=\Phi(T^n)$ for each $T\in\mathcal{A}(X)$.
It is clear that $(1)\Rightarrow(2)$ and that $(3)\Rightarrow(1)$. We will henceforth prove that $(2)\Rightarrow(3)$.
Since $P$ is orthogonally additive on $\mathcal{F}(X)$, Corollary \[crf\] yields a linear map $\Phi_0\colon\mathcal{F}(X)\to Y$ such that $$\label{crf1}
P(T)=\Phi_0(T^n) \quad (T\in\mathcal{F}(X)).$$ It seems to be appropriate to emphasize that we don’t know whether or not $\Phi_0$ is continuous, despite the continuity of $P$.
Let $\varphi\colon\mathcal{A}(X)^n\to Y$ be the continuous symmetric $n$-linear map associated with $P$. By considering the natural embedding of $Y$ into the second dual $Y^{**}$ we regard $Y$ as a closed linear subspace of $Y^{**}$, and we regard $\varphi$ as a continuous $n$-linear map from $\mathcal{A}(X)^n$ into $Y^{**}$ henceforth. Since $X^*$ has the bounded approximation property, it follows that $\mathcal{A}(X)$ has a bounded approximate identity $(E_\lambda)_{\lambda\in\Lambda}$ ([@D Theorem 2.9.37(iii) and (v)]) and we may assume that $(E_\lambda)_{\lambda\in\Lambda}$ lies in $\mathcal{F}(X)$. Let $\varphi'\colon\mathcal{A}(X)^{n-1}\to Y^{**}$ be as defined in the beginning of this section. The polarization of the identity $\eqref{crf1}$ yields $$\label{e1547}
\varphi(T_1,\dotsc,T_n)=
\frac{1}{n!}\Phi_0\bigl(\pi_n(T_1,\dotsc,T_n)\bigr) \quad ((T_1,\dotsc,T_n)\in\mathcal{F}(X)^n)$$ where $\pi_n$ is the noncommutative polynomial introduced in Lemma \[rfc3\].
We claim that $$\label{f1720}
\varphi(T_1,\dotsc,T_n)=
\frac{1}{n!}
\sum_{\sigma\in\mathfrak{S}_n}
\varphi'(T_{\sigma(1)},\dotsc,T_{\sigma(n-1)}T_{\sigma(n)})$$ for each $(T_1,\dotsc,T_n)\in\mathcal{A}(X)^n$. Since $\mathcal{A}(X)$ is the closed linear span of the rank-one operators and both $\varphi$ and $\varphi'$ are continuous, we are reduced to proving in the case where $T_1,\dotsc,T_n$ are rank-one operators. Let $T_1,\dotsc,T_n\in\mathcal{F}(X)$ be rank-one operators, and let $\lambda\in\Lambda$. On account of , , , and , we have $$\begin{gathered}
\label{l15}
\sum_{\sigma\in\mathfrak{S}_n}\varphi(T_{\sigma(1)},\dotsc,T_{\sigma(n)}E_\lambda)\\
= \frac{1}{n}\Phi_0\left(\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
T_{\sigma(1)}\dotsm T_{\sigma(n-1)}T_{\sigma(n)}E_\lambda
+\dotsb+
T_{\sigma(1)}E_\lambda \dotsm T_{\sigma(n-1)}T_{\sigma(n)}
\Bigr]\right) ,\end{gathered}$$ $$\begin{gathered}
\label{l16}
\sum_{\sigma\in\mathfrak{S}_n}\varphi(T_{\sigma(1)},\dotsc,E_\lambda T_{\sigma(n)}) \\
= \frac{1}{n}\Phi_0\left(\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
T_{\sigma(1)}\dotsm T_{\sigma(n-1)}E_\lambda T_{\sigma(n)}
+\cdots+
E_\lambda T_{\sigma(1)} \dotsm T_{\sigma(n-1)}T_{\sigma(n)}
\Bigr]\right),\end{gathered}$$ $$\begin{gathered}
\label{l17}
\sum_{\sigma\in\mathfrak{S}_n}\varphi(T_{\sigma(1)},\dotsc,T_{\sigma(n-1)}T_{\sigma(n)},E_\lambda) \\
= \frac{1}{n}\Phi_0\left(\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
T_{\sigma(1)}\dotsm T_{\sigma(n-1)}T_{\sigma(n)}E_\lambda
+
E_\lambda T_{\sigma(1)} \dotsm T_{\sigma(n-1)}T_{\sigma(n)}
\Bigr]\right)\\
{}+\frac{n-2}{n(n-1)}\Phi_0\left(\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
T_{\sigma(1)}\dotsm T_{\sigma(n-1)}E_\lambda T_{\sigma(n)}
+\dotsb+
T_{\sigma(1)}E_\lambda \dotsm T_{\sigma(n-1)}T_{\sigma(n)}
\Bigr]\right).\end{gathered}$$ Adding and , and then subtracting we can assert that $$\begin{gathered}
\label{l18}
\sum_{\sigma\in\mathfrak{S}_n}\varphi(T_{\sigma(1)},\dotsc,T_{\sigma(n)}E_\lambda)+
\sum_{\sigma\in\mathfrak{S}_n}\varphi(T_{\sigma(1)},\dotsc,E_\lambda T_{\sigma(n)})\\
{}- \sum_{\sigma\in\mathfrak{S}_n}
\varphi(T_{\sigma(1)},\dotsc,T_{\sigma(n-1)}T_{\sigma(n)},E_\lambda)\\
= \frac{1}{n-1}\Phi_0\left(\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
T_{\sigma(1)}\dotsm T_{\sigma(n-1)}E_\lambda T_{\sigma(n)}
+\dotsb+
T_{\sigma(1)}E_\lambda \dotsm T_{\sigma(n-1)}T_{\sigma(n)}
\Bigr]\right).\end{gathered}$$ Let $\mathcal{M}$ be the subalgebra of $\mathcal{F}(X)$ generated by $T_1,\dotsc,T_n$. Then $\mathcal{M}$ is finite-dimensional, a property which implies that the restriction of $\Phi_0$ to $\mathcal{M}$ is continuous, of course. Further, it is straightforward to check that $T_iST_j\in\mathcal{M}$ for all $S\in\mathcal{B}(X)$ and $i,j\in\{1,\dotsc,n\}$. Accordingly, for each $\sigma\in\mathfrak{S}_n$, all the nets $$\bigl(T_{\sigma(1)}\dotsb T_{\sigma(n-1)}E_\lambda T_{\sigma(n)}\bigr)_{\lambda\in\Lambda}, \, \dotsc \,
,\bigl(T_{\sigma(1)}E_\lambda\dotsb T_{\sigma(n-1)} T_{\sigma(n)}\bigr)_{\lambda\in\Lambda}$$ lie in $\mathcal{M}$, and, since $(E_\lambda)_{\lambda\in\Lambda}$ is a bounded approximate identity for $\mathcal{A}(X)$, each of them converges to $T_{\sigma(1)}\dotsb T_{\sigma(n-1)}T_{\sigma(n)}$ in the operator norm. Therefore, taking limits along $\mathcal{U}$ on both sides of the equation (and using the continuity of $\Phi_0$ on $\mathcal{M}$), we see that $$\begin{gathered}
\label{l19}
\sum_{\sigma\in\mathfrak{S}_n}\varphi(T_{\sigma(1)},\dotsc,T_{\sigma(n)})+
\sum_{\sigma\in\mathfrak{S}_n}\varphi(T_{\sigma(1)},\dotsc,T_{\sigma(n)})-
\sum_{\sigma\in\mathfrak{S}_n}
\varphi'(T_{\sigma(1)},\dotsc,T_{\sigma(n-1)}T_{\sigma(n)})\\ =
\frac{1}{n-1}\Phi_0\left(\sum_{\sigma\in\mathfrak{S}_n}
\Bigl[
T_{\sigma(1)}\dotsm T_{\sigma(n-1)} T_{\sigma(n)}
+\dotsb+
T_{\sigma(1)} \dotsm T_{\sigma(n-1)}T_{\sigma(n)}
\Bigr]\right)\\ =
\Phi_0\left(\sum_{\sigma\in\mathfrak{S}_n}
T_{\sigma(1)}\dotsm T_{\sigma(n-1)} T_{\sigma(n)}\right).\end{gathered}$$ By using the symmetry of $\varphi$ and in we obtain $$2n!\varphi(T_1,\dotsc,T_n) - \sum_{\sigma\in\mathfrak{S}_n}
\varphi'(T_{\sigma(1)},\dotsc,T_{\sigma(n-1)}T_{\sigma(n)})=
n!\varphi(T_1,\dotsc,T_n),$$ which yields .
Having disposed of identity , we can now apply Lemma \[l177\] to obtain a continuous linear map $\Phi\colon\mathcal{A}(X)\to Y^{**}$ such that $$\label{aa}
\varphi(T_1,\ldots,T_n)=
\frac{1}{n!}
\Phi\left(\sum_{\sigma\in\mathfrak{S}_n}T_{\sigma(1)}\cdots T_{\sigma(n)}\right)$$ for each $(T_1,\ldots,T_n)\in \mathcal{A}(X)$. Our next objective is to show that the range of $\Phi$ lies actually in $Y$. Let $T\in\mathcal{F}(X)$. On account of Lemma \[l1525\], there exists $S\in\mathcal{F}(X)$ such that $TS=ST=T$. From we see that $\Phi(T)=\varphi(T,S,\dotsc,S)\in Y$. Since $\mathcal{F}(X)$ is dense in $\mathcal{A}(X)$, $\Phi$ is continuous, and $Y$ is closed in $Y^{**}$, it may be concluded that $\Phi(\mathcal{A}(X))\subset Y$.
Our final task is to prove the uniqueness of the map $\Phi$. Suppose that $\Psi\colon\mathcal{A}(X)\to Y$ is a continuous linear map such that $P(T)=\Psi(T^n)$ for each $T\in\mathcal{A}(X)$. By Corollary \[crf\], $\Psi(T)=\Phi(T)\bigl(=\Phi_0(T)\bigr)$ for each $T\in\mathcal{F}(X)$. Since $\mathcal{F}(X)$ is dense in $\mathcal{A}(X)$, and both $\Phi$ and $\Psi$ are continuous, it follows that $\Psi(T)=\Phi(T)$ for each $T\in\mathcal{A}(X)$.
Let $X$ and $Y$ be as in Theorem \[main\], let $P\colon\mathcal{A}(X)\to Y$ be a continuous orthogonally additive $n$-homogeneous polynomial, and let $\varphi\colon\mathcal{A}(X)^n\to Y$ be the continuous symmetric $n$-linear map associated with $P$. Suppose that $(E_\lambda)_{\lambda\in\Lambda}$ is any bounded approximate identity for $\mathcal{A}(X)$. We already know that there exists a continuous linear map $\Phi\colon\mathcal{A}(X)\to Y$ such that $P(T)=\Phi(T^n)$ for each $T\in\mathcal{A}(X)$. The polarization of this representation gives $$\label{oo}
\varphi(T_1,\dotsc,T_n)=\frac{1}{n!}\Phi\left(\sum_{\sigma\in\mathfrak{S}_n}T_{\sigma(1)}\cdots T_{\sigma(n)}\right)$$ for each $(T_1,\ldots,T_n)\in \mathcal{A}(X)$. If $T\in\mathcal{A}(X)$, then each of the nets $$\bigl(TE_\lambda\dotsb E_\lambda\bigr)_{\lambda\in\Lambda},
\bigl(E_\lambda T\dotsb E_\lambda\bigr)_{\lambda\in\Lambda},\dotsc,
\bigl(E_\lambda\dotsb TE_\lambda\bigr)_{\lambda\in\Lambda},
\bigl(E_\lambda\dotsb E_\lambda T\bigr)_{\lambda\in\Lambda},$$ converges to $T$ in the operator norm, and using and the continuity of $\Phi$ we see that the net $\bigl(\varphi(T,E_\lambda,\dotsc,E_\lambda)\bigr)_{\lambda\in\Lambda}$ converges to $\Phi(T)$ in norm. Accordingly, the map $\Phi$ is necessarily given by $$\label{ooo}
\Phi(T)=\lim_{\lambda\in\Lambda}\varphi(T,E_\lambda,\dotsc,E_\lambda) \quad (T\in\mathcal{A}(X)),$$ where there is no need for taking the limit with respect to any weak topology along any ultrafilter. Note the similarity of with .
[99]{}
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[^1]: The first and the third named authors were supported by MINECO grant MTM2015–65020–P and Junta de Andalucía grant FQM–185. The second named author was supported by Beca de iniciación a la investigación of Universidad de Granada.
|
---
abstract: 'In this paper searches for flaring astrophysical neutrino sources and sources with periodic emission with the IceCube neutrino telescope are presented. In contrast to time integrated searches, where steady emission is assumed, the analyses presented here look for a time dependent signal of neutrinos using the information from the neutrino arrival times to enhance the discovery potential. A search was performed for correlations between neutrino arrival times and directions as well as neutrino emission following time dependent lightcurves, sporadic emission or periodicities of candidate sources. These include active galactic nuclei, soft $\gamma$-ray repeaters, supernova remnants hosting pulsars, micro-quasars and X-ray binaries. The work presented here updates and extends previously published results to a longer period that covers four years of data from 2008 April 5 to 2012 May 16 including the first year of operation of the completed 86-string detector. The analyses did not find any significant time dependent point sources of neutrinos and the results were used to set upper limits on the neutrino flux from source candidates.'
address:
- 'III. Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany'
- 'School of Chemistry & Physics, University of Adelaide, Adelaide SA, 5005 Australia'
- 'Dept. of Physics and Astronomy, University of Alaska Anchorage, 3211 Providence Dr., Anchorage, AK 99508, USA'
- 'CTSPS, Clark-Atlanta University, Atlanta, GA 30314, USA'
- 'School of Physics and Center for Relativistic Astrophysics, Georgia Institute of Technology, Atlanta, GA 30332, USA'
- 'Dept. of Physics, Southern University, Baton Rouge, LA 70813, USA'
- 'Dept. of Physics, University of California, Berkeley, CA 94720, USA'
- 'Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA'
- 'Institut für Physik, Humboldt-Universität zu Berlin, D-12489 Berlin, Germany'
- 'Fakultät für Physik & Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany'
- 'Physikalisches Institut, Universität Bonn, Nussallee 12, D-53115 Bonn, Germany'
- 'Université Libre de Bruxelles, Science Faculty CP230, B-1050 Brussels, Belgium'
- 'Vrije Universiteit Brussel, Dienst ELEM, B-1050 Brussels, Belgium'
- 'Dept. of Physics, Chiba University, Chiba 263-8522, Japan'
- 'Dept. of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand'
- 'Dept. of Physics, University of Maryland, College Park, MD 20742, USA'
- 'Dept. of Physics and Center for Cosmology and Astro-Particle Physics, Ohio State University, Columbus, OH 43210, USA'
- 'Dept. of Astronomy, Ohio State University, Columbus, OH 43210, USA'
- 'Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark'
- 'Dept. of Physics, TU Dortmund University, D-44221 Dortmund, Germany'
- 'Dept. of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA'
- 'Dept. of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2E1'
- 'Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany'
- 'Département de physique nucléaire et corpusculaire, Université de Genève, CH-1211 Genève, Switzerland'
- 'Dept. of Physics and Astronomy, University of Gent, B-9000 Gent, Belgium'
- 'Dept. of Physics and Astronomy, University of California, Irvine, CA 92697, USA'
- 'Dept. of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA'
- 'Dept. of Astronomy, University of Wisconsin, Madison, WI 53706, USA'
- 'Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center, University of Wisconsin, Madison, WI 53706, USA'
- 'Institute of Physics, University of Mainz, Staudinger Weg 7, D-55099 Mainz, Germany'
- 'Université de Mons, 7000 Mons, Belgium'
- 'Technische Universität München, D-85748 Garching, Germany'
- 'Bartol Research Institute and Dept. of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA'
- 'Department of Physics, Yale University, New Haven, CT 06520, USA'
- 'Dept. of Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK'
- 'Dept. of Physics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA'
- 'Physics Department, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA'
- 'Dept. of Physics, University of Wisconsin, River Falls, WI 54022, USA'
- 'Oskar Klein Centre and Dept. of Physics, Stockholm University, SE-10691 Stockholm, Sweden'
- 'Dept. of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA'
- 'Dept. of Physics, Sungkyunkwan University, Suwon 440-746, Korea'
- 'Dept. of Physics, University of Toronto, Toronto, Ontario, Canada, M5S 1A7'
- 'Dept. of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA'
- 'Dept. of Astronomy and Astrophysics, Pennsylvania State University, University Park, PA 16802, USA'
- 'Dept. of Physics, Pennsylvania State University, University Park, PA 16802, USA'
- 'Dept. of Physics and Astronomy, Uppsala University, Box 516, S-75120 Uppsala, Sweden'
- 'Dept. of Physics, University of Wuppertal, D-42119 Wuppertal, Germany'
- 'DESY, D-15735 Zeuthen, Germany'
author:
- 'M. G. Aartsen'
- 'M. Ackermann'
- 'J. Adams'
- 'J. A. Aguilar'
- 'M. Ahlers'
- 'M. Ahrens'
- 'D. Altmann'
- 'T. Anderson'
- 'C. Arguelles'
- 'T. C. Arlen'
- 'J. Auffenberg'
- 'X. Bai'
- 'M. Baker'
- 'S. W. Barwick'
- 'V. Baum'
- 'R. Bay'
- 'J. J. Beatty'
- 'J. Becker Tjus'
- 'K.-H. Becker'
- 'S. BenZvi'
- 'P. Berghaus'
- 'D. Berley'
- 'E. Bernardini'
- 'A. Bernhard'
- 'D. Z. Besson'
- 'G. Binder'
- 'D. Bindig'
- 'M. Bissok'
- 'E. Blaufuss'
- 'J. Blumenthal'
- 'D. J. Boersma'
- 'C. Bohm'
- 'F. Bos'
- 'D. Bose'
- 'S. Böser'
- 'O. Botner'
- 'L. Brayeur'
- 'H.-P. Bretz'
- 'A. M. Brown'
- 'N. Buzinsky'
- 'J. Casey'
- 'M. Casier'
- 'E. Cheung'
- 'D. Chirkin'
- 'A. Christov'
- 'B. Christy'
- 'K. Clark'
- 'L. Classen'
- 'F. Clevermann'
- 'S. Coenders'
- 'D. F. Cowen'
- 'A. H. Cruz Silva'
- 'J. Daughhetee'
- 'J. C. Davis'
- 'M. Day'
- 'J. P. A. M. de André'
- 'C. De Clercq'
- 'H. Dembinski'
- 'S. De Ridder'
- 'P. Desiati'
- 'K. D. de Vries'
- 'G. de Wasseige'
- 'M. de With'
- 'T. DeYoung'
- 'J. C. D[í]{}az-Vélez'
- 'J. P. Dumm'
- 'M. Dunkman'
- 'R. Eagan'
- 'B. Eberhardt'
- 'T. Ehrhardt'
- 'B. Eichmann'
- 'J. Eisch'
- 'S. Euler'
- 'P. A. Evenson'
- 'O. Fadiran'
- 'A. R. Fazely'
- 'A. Fedynitch'
- 'J. Feintzeig'
- 'J. Felde'
- 'K. Filimonov'
- 'C. Finley'
- 'T. Fischer-Wasels'
- 'S. Flis'
- 'K. Frantzen'
- 'T. Fuchs'
- 'T. K. Gaisser'
- 'R. Gaior'
- 'J. Gallagher'
- 'L. Gerhardt'
- 'D. Gier'
- 'L. Gladstone'
- 'T. Glüsenkamp'
- 'A. Goldschmidt'
- 'G. Golup'
- 'J. G. Gonzalez'
- 'J. A. Goodman'
- 'D. Góra'
- 'D. Grant'
- 'P. Gretskov'
- 'J. C. Groh'
- 'A. Gro[ß]{}'
- 'C. Ha'
- 'C. Haack'
- 'A. Haj Ismail'
- 'P. Hallen'
- 'A. Hallgren'
- 'F. Halzen'
- 'K. Hanson'
- 'D. Hebecker'
- 'D. Heereman'
- 'D. Heinen'
- 'K. Helbing'
- 'R. Hellauer'
- 'D. Hellwig'
- 'S. Hickford'
- 'J. Hignight'
- 'G. C. Hill'
- 'K. D. Hoffman'
- 'R. Hoffmann'
- 'A. Homeier'
- 'K. Hoshina'
- 'F. Huang'
- 'W. Huelsnitz'
- 'P. O. Hulth'
- 'K. Hultqvist'
- 'S. In'
- 'A. Ishihara'
- 'E. Jacobi'
- 'J. Jacobsen'
- 'G. S. Japaridze'
- 'K. Jero'
- 'M. Jurkovic'
- 'B. Kaminsky'
- 'A. Kappes'
- 'T. Karg'
- 'A. Karle'
- 'M. Kauer'
- 'A. Keivani'
- 'J. L. Kelley'
- 'A. Kheirandish'
- 'J. Kiryluk'
- 'J. Kläs'
- 'S. R. Klein'
- 'J.-H. Köhne'
- 'G. Kohnen'
- 'H. Kolanoski'
- 'A. Koob'
- 'L. Köpke'
- 'C. Kopper'
- 'S. Kopper'
- 'D. J. Koskinen'
- 'M. Kowalski'
- 'A. Kriesten'
- 'K. Krings'
- 'G. Kroll'
- 'M. Kroll'
- 'J. Kunnen'
- 'N. Kurahashi'
- 'T. Kuwabara'
- 'M. Labare'
- 'J. L. Lanfranchi'
- 'D. T. Larsen'
- 'M. J. Larson'
- 'M. Lesiak-Bzdak'
- 'M. Leuermann'
- 'J. Lünemann'
- 'J. Madsen'
- 'G. Maggi'
- 'K. B. M. Mahn'
- 'R. Maruyama'
- 'K. Mase'
- 'H. S. Matis'
- 'R. Maunu'
- 'F. McNally'
- 'K. Meagher'
- 'M. Medici'
- 'A. Meli'
- 'T. Meures'
- 'S. Miarecki'
- 'E. Middell'
- 'E. Middlemas'
- 'N. Milke'
- 'J. Miller'
- 'L. Mohrmann'
- 'T. Montaruli'
- 'R. Morse'
- 'R. Nahnhauer'
- 'U. Naumann'
- 'H. Niederhausen'
- 'S. C. Nowicki'
- 'D. R. Nygren'
- 'A. Obertacke'
- 'A. Olivas'
- 'A. Omairat'
- 'A. O’Murchadha'
- 'T. Palczewski'
- 'L. Paul'
- 'Ö. Penek'
- 'J. A. Pepper'
- 'C. Pérez de los Heros'
- 'C. Pfendner'
- 'D. Pieloth'
- 'E. Pinat'
- 'J. Posselt'
- 'P. B. Price'
- 'G. T. Przybylski'
- 'J. Pütz'
- 'M. Quinnan'
- 'L. Rädel'
- 'M. Rameez'
- 'K. Rawlins'
- 'P. Redl'
- 'I. Rees'
- 'R. Reimann'
- 'M. Relich'
- 'E. Resconi'
- 'W. Rhode'
- 'M. Richman'
- 'B. Riedel'
- 'S. Robertson'
- 'J. P. Rodrigues'
- 'M. Rongen'
- 'C. Rott'
- 'T. Ruhe'
- 'B. Ruzybayev'
- 'D. Ryckbosch'
- 'S. M. Saba'
- 'H.-G. Sander'
- 'J. Sandroos'
- 'M. Santander'
- 'S. Sarkar'
- 'K. Schatto'
- 'F. Scheriau'
- 'T. Schmidt'
- 'M. Schmitz'
- 'S. Schoenen'
- 'S. Schöneberg'
- 'A. Schönwald'
- 'A. Schukraft'
- 'L. Schulte'
- 'O. Schulz'
- 'D. Seckel'
- 'Y. Sestayo'
- 'S. Seunarine'
- 'R. Shanidze'
- 'M. W. E. Smith'
- 'D. Soldin'
- 'G. M. Spiczak'
- 'C. Spiering'
- 'M. Stamatikos'
- 'T. Stanev'
- 'N. A. Stanisha'
- 'A. Stasik'
- 'T. Stezelberger'
- 'R. G. Stokstad'
- 'A. Stö[ß]{}l'
- 'E. A. Strahler'
- 'R. Ström'
- 'N. L. Strotjohann'
- 'G. W. Sullivan'
- 'M. Sutherland'
- 'H. Taavola'
- 'I. Taboada'
- 'A. Tamburro'
- 'S. Ter-Antonyan'
- 'A. Terliuk'
- 'G. Te[š]{}ić'
- 'S. Tilav'
- 'P. A. Toale'
- 'M. N. Tobin'
- 'D. Tosi'
- 'M. Tselengidou'
- 'E. Unger'
- 'M. Usner'
- 'S. Vallecorsa'
- 'N. van Eijndhoven'
- 'J. Vandenbroucke'
- 'J. van Santen'
- 'S. Vanheule'
- 'M. Vehring'
- 'M. Voge'
- 'M. Vraeghe'
- 'C. Walck'
- 'M. Wallraff'
- 'Ch. Weaver'
- 'M. Wellons'
- 'C. Wendt'
- 'S. Westerhoff'
- 'B. J. Whelan'
- 'N. Whitehorn'
- 'C. Wichary'
- 'K. Wiebe'
- 'C. H. Wiebusch'
- 'D. R. Williams'
- 'H. Wissing'
- 'M. Wolf'
- 'T. R. Wood'
- 'K. Woschnagg'
- 'D. L. Xu'
- 'X. W. Xu'
- 'Y. Xu'
- 'J. P. Yanez'
- 'G. Yodh'
- 'S. Yoshida'
- 'P. Zarzhitsky'
- 'J. Ziemann'
- 'M. Zoll'
title: Searches for Time Dependent Neutrino Sources with IceCube Data from 2008 to 2012
---
triggered searches, multi-messenger searches, multi-wavelength campaigns, blazars, active galaxies, $\gamma$-ray bursts, soft $\gamma$-ray repeaters, X-ray binaries
Introduction {#intro}
============
The cosmic ray spectrum spans ten decades in energy up to $10^{11}$ GeV per particle and despite being extensively studied for many years the origin and the acceleration mechanism remains uncertain. Cosmic rays consist of hadrons, mainly protons and in part also ionized nuclei with an energy dependent composition. As a result of interactions of cosmic rays with matter and ambient photons close to the acceleration sites pions are produced and decays of the charged pions and their daughter muons then produce neutrinos. Such astrophysical neutrinos are a unique and valuable messenger in astro-particle physics because of their properties and the specifics of their production mechanisms. In particular, they carry information about the origin and spectrum of cosmic rays. In contrast to the cosmic rays, neutrinos are not deflected in magnetic fields, nor do they interact on the way to Earth. As a consequence their trajectories point back to their origin. A detection of neutrinos from a given site would therefore stand as proof of accelerated hadrons, identifying it as a source of cosmic rays.
This paper updates and expands the IceCube time dependent searches and results for flaring sources [@mike; @ic59-time; @crab] and periodic sources [@periodical]. Time dependent astrophysical neutrino signals can be better observed by using event times in addition to the direction and energy used in standard IceCube point source searches, because the additional information improves the rejection of the atmospheric muon and neutrino events which form the dominant background.
If several events are coming from the same astrophysical point source they should be spatially concentrated around the emitting source and they should have a harder spectrum than muons and neutrinos produced in atmospheric showers. Assuming a Fermi acceleration model [@FermiAcc; @FermiAcc2] a differential neutrino spectrum close to $E^{-2}$ will be produced. Additional effects like the acceleration of muons in the cosmic ray sources [@mu_acc1; @mu_acc2] can modify the spectrum. In case of sufficiently high acceleration gradients (above 160 keV/m) often required for extremly short flares, the daughter muons from charged pion decays can be accelerated before they decay. Thus the energy of the neutrinos from the muon decays will be enhanced. The impact on the overall neutrino spectra strongly depends on various properties of the source, e.g. the magnetic field strength and the amount of matter in the acceleration region. Conventional atmospheric neutrinos at energies above 100 GeV have a much softer spectrum which asymptotically approaches a spectrum one power steeper than the primary spectrum, first in the vertical direction and at higher energies also for larger zenith angles [@ic59_diffuse; @nufluxes].
Time dependent searches can be performed in an ‘untriggered’ way, meaning that the correlation of event times is investigated and no additional information from independent observations is used. The full parameter space of time, energy and direction of measured events is scanned looking for clusters in time and space of high energy events among the background of atmospheric events, hereby called the “All-Sky Time Scan”. This search is the most generic one but it is subject to a large trial factor that penalizes the significance of the signal with respect to the background. Hence, in addition, more specific searches are carried out using a multi-messenger approach. The basic assumption is that neutrinos and $\gamma$-rays are correlated because they have a common origin in the astrophysical sources that accelerate the cosmic rays. The “Search for Triggered Multi-Messenger Flares” makes use of lightcurves measured by $\gamma$-ray experiments. This analysis is triggered by multi-wavelength measurements (alerts for flares from TeV and X-ray experiments) as well as the Fermi-LAT lightcurves which provide continuous monitoring of selected sources or of flares above a certain photon flux level [@Fmonitored].
The paper outline is as follows. In Section \[sec:sources\] the possible sources of astrophysical neutrinos are briefly discussed and in Section \[sec:detsamp\] the IceCube detector and the details of the data samples used are introduced. Section \[sec:llh\] gives a detailed description of the general likelihood method applied while the specifics, which are different for each search, are given at the beginning of the corresponding sections.
In the following sections, the results for the various searches are presented. First in Section \[sec:All-Sky Time Scan\] the untriggered “All-Sky Time Scan” looking for any cluster of high energy neutrino events from any direction in the sky is presented. For this search the data taken by the 59-string configuration of IceCube (IC-59), 79-string configuration (IC-79) and the first year of the full IceCube data taking with 86 strings (IC-86I) were used. For each of the three years time dependent skymap scans were performed.
Then in Section \[sec:fflares\] the results of the “Search for Triggered Multi-Messenger Flares” are discussed. The data taken with the IC-59, IC-79 and IC-86I configurations of IceCube were combined. Hence, this represents a long term study for flaring objects. In Section \[sec:sflares\] the results of the “Search for Triggered Flares with Sporadic Coverage” are presented. These were triggered by higher energy $\gamma$-ray observations in the TeV range.
In Section \[sec:per\] the “Search for Periodic Neutrino Emissions from Binary Systems” is presented. The search was performed in the phase domain. For this search data from four IceCube data taking seasons, from the IC-40 to the IC-86I configuration were used.
Finally, in Section \[sec:Conclusions\] we conclude with a brief outlook.
Potential Flaring Sources of Neutrinos {#sec:sources}
======================================
The focus of this paper is on active galactic nuclei (AGNs), and particularly blazars, i.e. AGNs with jets pointing toward us, which are interesting due to their variability and the high power emitted during bursts [@AGNs1; @AGNs2; @AGNs3]. The analyses described in the following sections are sensitive also to short flares (of the duration of a few seconds) and hence, to some extent, to $\gamma$-ray bursts. More sensitive searches for neutrinos from $\gamma$-ray bursts are presented elsewhere [@naturepaper; @grb_ic40]. Blazars exhibit sudden sequences of multiple flares that may last from minutes to months and are observed in various wavelengths from radio to $\gamma$-rays. Correlations between various bands have been observed in numerous multi-wavelength campaigns, particularly between X-rays and $\gamma$ TeV emissions. A well-studied case is that of the close-by TeV blazar Mrk 421 (see e.g. [@mrk2010; @magic_mrk421]). In other instances optical flares have triggered TeV flare observations [@optical] .
The Spectral Energy Distribution (SED) of AGNs is characterized by two broad peaks. The lower energy one is believed to originate from synchrotron emission of the charged particles in the jet. In leptonic models the higher energy one is generally explained by inverse Compton scattering of either the synchrotron seed photons (Synchrotron Self Compton - SSC, see e.g. [@blazar1]) or external seed photons (External Compton - EC [@blazar2; @blazar3]) by the electrons and positrons in the jet.
In the simplest case, both SSC and EC mechanisms predict that flaring at TeV energy should be accompanied by a simultaneous flaring in the synchrotron peak and so a connection between bands from optical to $\gamma$-ray is expected. If the synchrotron peak is located far from the optical, as in the case of High-frequency Peaked Blazars (HBLs), then the synchrotron flares should be visible at other wavelengths, usually X-rays [@mrk2010; @magic_mrk421].
Alternatively, hadronic models suggest production of neutrinos and $\gamma$-rays from pion decays [@blazar4; @blazar5; @blazar6]. In these models, the high energy peak is due to proton synchrotron emission or decay of neutral pions formed in cascades by the interaction of high-energy proton beam with the radiation or gas clouds surrounding the source [@blazar6]. In this scenario, a strong correlation between the $\gamma$-ray and the neutrino fluxes is expected. Observations of neutrinos would clearly distinguish between leptonic and hadronic models.
For some observations it has been claimed that hadronic processes could explain flares better than leptonic processes [@reimer]. Orphan flares, i.e. a TeV flare without a lower energy counterpart, challenge leptonic models [@orphan1; @orphan2; @orphan3]. Non-observation of significant X-ray activity could naturally be interpreted as due to the suppression of electron acceleration and inverse Compton scattering and dominance of very high energy (VHE) $\gamma$-ray production from meson decays in hadronic models.
In addition to flares, another possible time dependent signature is a periodicity in the neutrino emission. In the case of binary systems, a modulation of the neutrino emission could occur due to the relative geometry of the emitted beam with respect to the large mass star and the observer. A particularly interesting case is that of micro-quasars, which are radio-jet X-ray binaries that can include either a neutron star or a black hole. A similar modulation is observed for three binary systems in TeV $\gamma$-rays: LS 5039 [@mqso1], LS I +61 303 [@mqso2], and HESS J0632+057 [@mqso3]. These three binary systems are found to be periodic at GeV and TeV energies, although the emission in the two bands seems anti-correlated [@mqso_anti]. This anti-correlation is in fact a generic feature of models in which the GeV (TeV) emission is enhanced (reduced) when the highly relativistic electrons are moving in the direction of the observer and encounter the seed photons head-on. Neutrino emission from micro-quasars has been described in various papers [@mqso_nu1; @mqso_nu2; @mqso_nu3; @mqso_nu4; @mqso_nu5]. It can be assumed a neutrino signal would be correlated to the periodic $\gamma$-ray emission from the binary system due to the system’s rotation, but it is not clear in which phase with respect to the $\gamma$-rays the neutrinos are to be expected. Hence the values for the period are taken from multi-wavelength observations and the phase is fitted as a free parameter.
The IceCube Detector and the Data Samples {#sec:detsamp}
=========================================
The IceCube observatory is a Cherenkov detector searching for high energy neutrinos. It is an array of 5,160 Digital Optical Modules (DOMs) deployed deep in the Antarctic ice near the South Pole. The purpose of the detector is to allow observations of neutrinos of astrophysical origin and atmospheric muons and neutrinos induced by cosmic rays at energies around and above the *knee* ($\mathrm{\sim 3 \times 10^{15}\:eV}$). Cherenkov light produced by secondary leptons from neutrino interactions in the vicinity of the detector is used to indirectly detect these neutrinos. Only charged-current muon neutrino events were considered for the studies presented here, because of the large range of the secondary muons, which provides good angular resolution and a high rate by including events that start outside the detector. The pointing information relies on the secondary muon direction, which at energies above TeV differs from the original neutrino direction by less than the angular resolution of the detector [@jakerameez].
The DOMs are spherical, pressure resistant glass housings, each containing a Hamamatsu photomultiplier tube (PMT) of 25 cm diameter and the associated electronics necessary for waveform digitization [@PMT; @PMTel]. These DOMs are connected together in 86 vertical strings of 60 DOMs each and the strings are lowered into 60 cm wide holes drilled into the ice using hot water to instrument a kilometer-cubed volume in a depth range from 1.5 km down to 2.5 km. At the center of the detector the strings are placed closer to each other, forming a sub-array called DeepCore. This sub-array enhances the sensitivity of the detector to neutrinos of energies below the standard IceCube threshold of 100 GeV [@DeepCore]. The IceCube detector has been in operation even before full completion in December 2010. The partial detector layouts (resp. data samples) used for this paper are labeled IC-40, IC-59, IC-79 and IC-86I, where the number corresponds to the number of operational stings. The data samples used in the studies presented here are described in detail in previous papers reporting time integrated searches [@ic40PS; @juanan; @jakerameez].
The analyses in the present paper used identical detector simulation and corresponding estimates of neutrino effective area and point spread function based on Monte Carlo studies as for the time integrated analyses [@jakerameez]. Table \[tab:livetimes\] summarizes the four data samples consisting of both up- and down-going muons which have different origin; while the up-going muons are mostly from interactions of atmospheric neutrinos that have passed through the Earth, the down-going muons are mainly from meson decays in atmospheric showers caused by cosmic rays.
Unbinned Time Dependent Likelihood Method {#sec:llh}
=========================================
The unbinned time dependent likelihood method which was used for the analyses in this paper has been applied in previous analyses [@method1] and [@method2].
The likelihood function is defined as:
$$\label{eq:likelihood}
\mathcal{L}(\gamma, n_{\mathrm{s}},...) = \prod_{j} \mathcal{L}^{j} (\gamma, n^{j}_{\mathrm{s}},...) = \prod_{j}\;\prod_{i \in j} \left[\frac{n^{j}_{\mathrm{s}}}{N^{j}}\mathcal{S}^{j}_{i} + \left(1- \frac{n^{j}_{\mathrm{s}}}{N^{j}}\right)\mathcal{B}^{j}_{i} \right],$$
where $i \in j$ indicates that the $i^{\mathrm{th}}$ event belongs to the $j^{\mathrm{th}}$ sample (IC-40, IC-59, IC-79 or IC-86I), $N^j$ is the total number of events in the $j^{\mathrm{th}}$ sample, $\mathcal{S}^j_i$ and $\mathcal{B}^{j}_i$ is the signal and background probability density function (PDF) respectively, $\gamma$ is the spectral index of the differential spectrum $dN/dE \propto E^{-\gamma}$ and is assumed to be the same for all the data samples. The total number of signal events $n_{\mathrm{s}}$ is the sum of the signal events coming from each sample $n^{j}_{\mathrm{s}}$. The fractional contribution of each $n_{\mathrm{s}}^j$ to the total $n_{\mathrm{s}}$ is fixed by the relative neutrino effective areas of each of the configurations (determined by detector simulation), and varies depending on the spectral index and the declination. The likelihood function in equation (\[eq:likelihood\]) is a function of the global common parameters $\gamma$ and $n_{\mathrm{s}}$ and the parameters specific for each time dependent search.
The signal PDF $\mathcal{S}^j_i$ is given by:
$$\label{eq:signalpdf}
\mathcal{S}^j_i = P^{\mathrm{signal},j}_{i} (|\vec{x}_i-\vec{x}_{\mathrm{s}}|,\sigma_{i})\cdot \mathcal{E}^{\mathrm{signal},j}_{i} (E_i, \delta_i, \gamma) \cdot \mathcal{T}^{\mathrm{signal},j}_{i},$$
where $\vec{x}_{\mathrm{s}}$ is the source direction, $\vec{x}_i$ is the reconstructed direction of the event, $\sigma_{i}$ is the angular error estimate for the reconstruction, and $E_i$ and $\delta_i$ are the reconstructed energy and declination respectively. The term $P^{\mathrm{signal},j}_{i}$ is the spatial PDF (the point spread function) and $\mathcal{E}^{\mathrm{signal},j}_{i}$ is the energy PDF. These two terms are the same as for the IceCube time integrated searches and can be found in [@juanan]. The time dependent signal PDF $\mathcal{T}^{\mathrm{signal},j}_{i}$ is specific for each of the different signal hypotheses and will be described in the corresponding sections.
Since the data are background dominated, the background PDF $\mathcal{B}^{j}_i$ can be estimated from the data itself and it has the form:
$$\label{eq:backgroundpdf}
\mathcal{B}^j_i = P^{\mathrm{bkg},j}_{i} (\delta_{i}) \cdot \mathcal{E}^{\mathrm{bkg},j}_{i} (E_i, \delta_i) \cdot \mathcal{T}^{\mathrm{bkg},j}_{i}.$$
For sufficiently long time scales, the spatial PDF for the background $P_i^{\mathrm{bkg},j}(\delta_i)$ is uniform in right ascension as the Earth rotation averages over detector effects and follows the distribution of the data with declination. For short time scales (less than a day) the spatial PDF is no longer uniform in right ascension and shows a pattern when the neutrino directions align with the directions of geometrical symmetries of the non-uniform detector array. The effect disappears as the time scale grows large enough to allow the Earth rotation to average out the differences. The energy density function $\mathcal{E}^{\mathrm{bkg},j}_{i}$ can be found in [@juanan]. The time probability $\mathcal{T}^{\mathrm{bkg},j}_{i}$ is taken to be flat since it was demonstrated in [@mike] that the seasonal modulations of background atmospheric muons and neutrinos are negligible compared to possible measurable signals.
For all the searches presented here the test statistic $TS$ is defined as the maximum likelihood ratio:
$$TS = -2 \log\Big[\frac{\mathcal{L}(n_{\mathrm{s}}=0)}{\mathcal{L}(\hat{n}_{\mathrm{s}},\hat{\gamma}_{\mathrm{s}},...)}\Big],
\label{eq:ts}$$
where $\mathcal{L}(n_{\mathrm{s}}=0)$ is the likelihood of the background-only (null) hypothesis and $\mathcal{L}(\hat{n}_{\mathrm{s}},\hat{\gamma}_{\mathrm{s}},...)$ is the likelihood evaluated with the best-fit values for the signal-plus-background hypothesis. The parameters denoted with carets are the best fit values of $n_{\mathrm{s}}$ and $\gamma_{\mathrm{s}}$ which are present in all searches[^1] while the parameters specific to each time dependent search (represented by the ellipsis above) will be discussed in each search section. Estimates for the values of these parameters can be derived from the maximization of the likelihood function defined in equation (\[eq:likelihood\]).
The test statistic $TS$ as defined in equation (\[eq:ts\]) for the background-only data is expected to follow a $\chi^2$-distribution with number of degrees of freedom reflecting the number of fitted parameters. This behavior of the $TS$ can be used to estimate the pre-trial p-value as the probability of the $TS$ value from the $\chi^2$-distribution.
Because our samples are background dominated, in order to translate the pre-trial p-values into post-trial p-values the time scrambled IceCube data from the same period can be used to generate background samples. The time scrambling procedure assigns to the events a random time within the period while keeping all other event properties (energy, local reconstructed coordinates, etc.) unchanged. In this way many different samples describing the background are obtained. About one thousand of these sets were generated and for each of them a p-value was obtained. Then the distribution of these p-values for the time scrambled data were compared to the p-value obtained for real data. From this the probability is derived that a random fluctuation will result in deviation at least as significant as our result for real data and this probability is called the post-trial p-value. The obtained post-trial p-value is stable with respect to deviations of background-only $TS$ from the assumed $\chi^2$ distribution and accounts for trial factors related to looking at many positions at the sky.
In analyzing the results it is convenient to characterize events by their time integrated weight $w_i$, defined as the ratio of the signal and background PDFs from equation (\[eq:signalpdf\]) and (\[eq:backgroundpdf\]) without the time PDF term:
$$\label{eq:signalweight}
w_i=\frac{P^{\mathrm{signal},j}_{i} (|\vec{x}_i-\vec{x}_{\mathrm{s}}|,\sigma_{i})\cdot \mathcal{E}^{\mathrm{signal},j}_{i} (E_i, \delta_i, \gamma)}{P^{\mathrm{bkg},j}_{i} (\delta_{i}) \cdot \mathcal{E}^{\mathrm{bkg},j}_{i} (E_i, \delta_i)},$$
which will be later used to visualize the result in a way helpful for understanding whether the significance comes from the spatial or the time clustering.
For the time dependent searches[^2] it is natural to express the discovery potentials, sensitivities, and upper limits in terms of fluence, defined for an $E^{-2}$ spectrum as:
$$\label{eq:fluence}
f =\int^{t_{\mathrm{max}}}_{t_{\mathrm{min}}} dt \int^{E_{\mathrm{max}}}_{E_{\mathrm{min}}}dE \times E\frac{dN}{dE} = \Delta t \int^{E_{\mathrm{max}}}_{E_{\mathrm{min}}}dE \times E\frac{\Phi_0}{E^2}=\Delta t \Phi_0 \int^{E_{\mathrm{max}}}_{E_{\mathrm{min}}}\frac{1}{E}dE$$
where $\Phi_0$ is the normalization of an average flux during the emission. The emission duration $\Delta t$ is defined by the time PDF. The integration limits $E_{\mathrm{min}}$ and $E_{\mathrm{max}}$ are set to the 5% and 95% resp. energy percentile of the event sample for the given declination band.
All-Sky Time Scan {#sec:All-Sky Time Scan}
=================
This is the most generic time dependent search among the ones presented in this paper. It is optimized to look for neutrino emission from a point-like source with limited duration. Because it aims for one-time flares it does not benefit from adding multiple datasets together. Here the results for the IC-59, IC-79 and IC-86I IceCube data are presented separately. The results of a similar search for the IC-40 configuration can be found in [@mike].
Method {#untrig_method}
------
The “All-Sky Time Scan” is an untriggered search since it is performed only using the neutrino data itself. For this search the whole sky[^3] was divided into a grid of 0.1$^\circ \times 0.1^\circ$, much finer than the angular resolution of the detector, and the likelihood was maximized at each grid point. The expression for the likelihood is given by equation (\[eq:likelihood\]) (with each detector configuration analyzed separately). In equation (\[eq:signalpdf\]) the signal PDF $\mathcal{T}^{\mathrm{signal}}_{i}$ was chosen of the following form:
$$\mathcal{T}^{\mathrm{signal}}_{i}=\frac{1}{\sqrt{2\pi}\sigma_{\mathrm{T}}}exp\left(-\frac{(t_i-T_0)^2}{2\sigma^2_{\mathrm{T}}}\right),
\label{eq:time_pdf_allSky}$$
where $t_i$ is the arrival time of the $i^{\mathrm{th}}$ event, $T_0$ and $\sigma_{\mathrm{T}}$ are the mean and the width of a Gaussian time structure, respectively. The Gaussian function was used as a smooth, general parametrization of a limited duration increase in the emission of a source. The best fit values of these parameters were obtained maximizing the likelihood function described in Section \[sec:llh\].
Since within the limited livetime of a given sample it is possible to accommodate a larger number of flares as their duration decreases, an undesired bias towards finding a short flare is introduced. This could also be interpreted as a hidden extra trial factor affecting short flares. To account for it the test statistic $TS$ from equation (\[eq:ts\]) was modified. As explained in [@method2], an additional marginalization term $T/\sqrt{2\pi}\sigma$ that penalizes short flares compared to long ones was introduced. The test statistic including this modification is thus:
$$TS = -2 \log\Big[\frac{T}{\sqrt{2\pi}\hat{\sigma}_{\mathrm{T}}}\times\frac{\mathcal{L}(n_{\mathrm{s}}=0)}{\mathcal{L}(\hat{n}_{\mathrm{s}},\hat{\gamma}_{\mathrm{s}},\hat{\sigma}_{\mathrm{T}},\hat{T}_0)}\Big],
\label{eq:ts_allSky}$$
where $\hat{n}_{\mathrm{s}},\hat{\gamma}_{\mathrm{s}},\hat{\sigma}_{\mathrm{T}},\hat{T}_0$ are the best-fit values and $T$ is the total livetime of the data taking period (either IC-59, IC-79 or IC-86I). For details on the numerical maximization procedure see [@mike].
The expected performance of this approach in terms of discovery potential and sensitivity is shown in Figure \[fig:untriggered\_discPot\]. The figure compares this time dependent search with the standard time integrated all sky search for given declination. The “All-Sky Time Scan” search is better for flares with $\sigma_{\mathrm{T}}$ shorter than hundred days in terms of discovery potential and for flares with $\sigma_{\mathrm{T}}$ shorter than 6 hours in terms of sensitivity. As the flare duration $\sigma_{\mathrm{T}}$ gets shorter, the sensitivity levels out at around three events. For the calculation of the discovery potential at least two events are required in order to identify a flare. To generate a sample according to a Poissonian distribution with 50% of the cases having two or more events, the Poissonian mean has to be equal to 1.68. Therefore for the shortest timescales the discovery potential will asymptotically approach this value causing it to drop below the sensitivity.
Results {#untrig_rez}
-------
The maximization of the likelihood at each grid point in the sky results in a map of $TS$ values or, equivalently, a map of pre-trial p-values that serve as an estimate of the local significance of the best-fit parameters. To address the question of whether any excess in the sky is significant, a correction for the trial factor involved in searching the entire sky had to be used. This was done as described at the end of Section \[sec:llh\] by repeating the analysis on time-scrambled data.
### IC-59 Results
Figure \[fig59\_p\] shows the IC-59 skymap of pre-trial p-values for the all-sky search. The most significant point in the IC-59 data was found at (RA, Dec.) = (21.35$^{\circ}$, -0.25$^{\circ}$). The peak occurred on MJD 55259 (2010 March 4), and had a width parameter $\hat{\sigma}_{\mathrm{T}}$ of 5.5 days, a soft spectral index of $\hat{\gamma}=3.9$, and 14.5 fitted signal events. The pre-trial p-value was $2.04\times10^{-7}$; a value at least as significant as this was found somewhere in the sky in 14 out of 1000 scrambled maps. Thus the post-trial p-value was 1.4%, which was low but not significant evidence of an actual flare. When this analysis was repeated on the data for the following years (these results are presented in the following sections) the IC-59 hot spot significance decreased and it was not seen with high significance any more. Figure \[fig59\_flareevts\] shows the time integrated event weights $w_i$ at the position of maximum significance plotted throughout the year, a clustering near the time of the best-fit $\hat{T}_0$ is clearly visible. When a bin of radius $2^{\circ}$ and 13 days in time (the FWHM of the Gaussian) centered on the peak is considered, 13 events are found compared to an expected background of 1.7.
![The time integrated event weights $w_i$, defined in equation (\[eq:signalweight\]), evaluated for the IC-59 data at the location of the most significant flare. The best-fit Gaussian time PDF is shown in black (arbitrary scaling). See Table \[tab:all\_sky\_rez\] for details.[]{data-label="fig59_flareevts"}](hammer_ic59_AllSkyTDep_Unblinded.eps){width="\textwidth"}
![The time integrated event weights $w_i$, defined in equation (\[eq:signalweight\]), evaluated for the IC-59 data at the location of the most significant flare. The best-fit Gaussian time PDF is shown in black (arbitrary scaling). See Table \[tab:all\_sky\_rez\] for details.[]{data-label="fig59_flareevts"}](AllSky_UntriggeredTimeStuffIc59_improved.eps){width="5.0in"}
### IC-79 Results
Figure \[fig:ic79\_untriggered\_skyMap\] shows the IC-79 skymap of pre-trial p-values. The most significant deviation from the background-only hypothesis was found at (RA, Dec.)=(343.45$^{\circ}$, -31.65$^{\circ}$). The mean of the fitted Gaussian flare was at MJD 55466 (2010 September 27) with a width parameter $\hat{\sigma}_{\mathrm{T}}$ of 1.8 days, a soft spectral index of $\hat{\gamma}=3.95$, and 7.2 fitted signal events. The large blue spot in the upper right quadrant in Figure \[fig:ic79\_untriggered\_skyMap\] was caused by two events arriving very close in time, the consequence of which was an increased significance over a wider area to which those two events contribute.
The pre-trial p-value obtained for the IC-79 hotspot was $1.07\times10^{-5}$. The post-trial p-value was 66%, consistent with a background-only hypothesis.
Figure \[fig:ic79\_untriggered\_selectedEv\] shows the time integrated event weights at the position of maximum significance plotted throughout the year with a hint of clustering recognizable at the fitted time.
![IC-79 skymap in equatorial coordinates showing the pre-trial p-value for the best-fit flare hypothesis tested in each direction of the sky. The strongest Gaussian-like signal was found at (RA, Dec.) = (343.45$^{\circ}$, -31.65$^{\circ}$), with post-trial significance of p=66%, see Table \[tab:all\_sky\_rez\] for details. The black curve is the galactic plane.[]{data-label="fig:ic79_untriggered_skyMap"}](hammer_ic79_AllSkyTDep_Unblinded.eps){width="\textwidth"}
![The time integrated event weights $w$, defined in equation (\[eq:signalweight\]), evaluated for the IC-79 data at the location of the most significant flare. The best-fit Gaussian time PDF is shown in black (arbitrary scaling). See Table \[tab:all\_sky\_rez\] for details.[]{data-label="fig:ic79_untriggered_selectedEv"}](AllSky_UntriggeredTimeStuffIc79_improved.eps){width="5.in"}
### IC-86I Results
Figure \[fig:ic86I\_untriggered\_skyMap\] shows the skymap of the pre-trail p-values obtained for the IC-86I data, the most significant point was at (RA, Dec.)=(235.95$^{\circ}$, 42.95$^{\circ}$), and the fitted Gaussian parameters are MJD 55882 for the mean and 7.57 days for the width parameter $\hat{\sigma}_{\mathrm{T}}$. A spectral index of $\hat{\gamma}=2.0$, and 13.1 signal events were fitted. A pre-trial p-value of $1.06\times10^{-5}$ translates into a post-trial p-value of 63%. The time integrated event weights for the region around the most significant point through the year are shown in Figure \[fig:ic86I\_untriggered\_selectedEv\].
![The time integrated event weights $w$, defined in equation (\[eq:signalweight\]), evaluated for the IC-86I data at the location of the most significant flare. The best-fit Gaussian time PDF is shown in black (arbitrary scaling). See Table \[tab:all\_sky\_rez\] for details.[]{data-label="fig:ic86I_untriggered_selectedEv"}](hammer_ic86I_AllSkyTDep_Unblinded.eps){width="\textwidth"}
![The time integrated event weights $w$, defined in equation (\[eq:signalweight\]), evaluated for the IC-86I data at the location of the most significant flare. The best-fit Gaussian time PDF is shown in black (arbitrary scaling). See Table \[tab:all\_sky\_rez\] for details.[]{data-label="fig:ic86I_untriggered_selectedEv"}](AllSky_UntriggeredTimeStuffIc86I_improved.eps){width="5.0in"}
Search for Triggered Multi-Messenger Flares {#sec:fflares}
===========================================
This search targets a set of astronomical objects which were observed to be in a flaring state by Fermi LAT [@bib:Fermiweb] during the time period analyzed in this study. The tested hypothesis is that the neutrino emission follows the intensity of the photon emission. If the photon emission has a hadronic origin, then the accelerated hadrons (protons or nuclei) interact with matter and produce neutral and charged pions, which produce $\gamma$-rays and neutrinos, respectively, when they decay.
Since Fermi LAT provides continuous monitoring and the data is publicly available it is possible to make use of the measured lightcurves. A continuation of the search made during the previous period of IceCube data-taking [@mike; @ic59-time] is presented here and it was extended including the whole IC-59, IC-79 and IC-86I combined data samples. Also a more advanced de-noising method was implemented, which is described below, to better reconstruct the Fermi-LAT lightcurves.
Method {#fflares_method}
------
This search was performed only for selected objects in the sky. The criteria for selecting these objects (FSRQs, BL Lacs, etc.) were based on the Fermi-LAT photometric measurements. The Fermi LAT monitored source list [@Fmonitored] was taken as a starting point. The first step was to retrieve the raw lightcurves from the Fermi Public Release data, using the analysis tools made available by the Fermi-LAT Collaboration. For each source the *Fermi Science Tools v9r31p1* package [@bib:FermiTools] was used to select photons within $2^\circ$ of the source and to calculate the exposure. Photon events with zenith angles greater than $105^\circ$ were excluded to avoid contamination due to the Earth’s albedo. For each source lightcurves with one day binning were obtained.
The denoised lightcurve was used as time dependent signal PDF, $\mathcal{T}^{\mathrm{signal},j}_{i}$. The “Bayesian Blocks” method [@bib:BBlocks; @bib:BBlocks1] was applied to de-noise the lightcurves, implemented for the purpose of this analysis in the version described in [@bib:BBlocks]. A simplified explanation of the method is that it splits the time axis of the lightcurve into blocks for which the flux variations are assumed to be compatible with Poisson distributions with a constant mean within each block. The criterion for deciding whether to split a section into two blocks or not is based on comparing how well the variations follow a single Poisson distribution or two Poisson distributions with different means.
In order to optimize the performance of the Bayesian Blocks method on IceCube data the optimal value of the method parameter $F_{\mathrm{B}}$ had to be found. This parameter affects the method behavior in the following way. If the log-likelihood of two Poisson distributions is larger than the log-likelihood value for a single Poisson distribution by at least $F_{\mathrm{B}}$ the split is made. Too small values of $F_{\mathrm{B}}$ will cause denoised lightcurves to follow almost every point in the lightcurve while for too large values of $F_{\mathrm{B}}$ the denoised result will ignore important structures of the lightcurve. In Figure \[fig:BBlocksWrong\] two examples of the denoised lightcurves are shown for $F_{\mathrm{B}}$ being outside of the optimal range. To determine the value of the parameter $F_{\mathrm{B}}$ to be used in this analysis, a series of tests were performed to evaluate the performance of the Bayesian Blocks method as function of $F_{\mathrm{B}}$.
Realistic Fermi-LAT lightcurves were simulated in order to optimize the $F_{\mathrm{B}}$ parameter, using Fermi-LAT exposure data and assuming constant flux levels with Poissonian fluctuations. Folding in the Poisson distributed flux with the exposure gave a model of the background fluctuations including the correct statistical errors. On top of this background Gaussian shaped flares with random mean in time were injected. The background level was set to $0.5 \times 10^{-6}\:\mathrm{photons\:cm^{-2} s^{-1}}$ and was not varied. Instead the properties of the Gaussian flares were varied, i.e. strength of the flare, the time and the duration. The amplitude values were taken to be either equal to the background level or twice the background level. The tested widths were two days, five days and ten days. In addition, combinations of two injected flares were also tested.
For each of the tested combinations of a flare with some amplitude and duration, hundreds of random instances of lightcurves with injected flares and background were produced. For each of these the lightcurve was denoised, scanning over different values of $F_{\mathrm{B}}$, and two quantities were calculated as a function of $F_{\mathrm{B}}$: the rate of finding a fake flare, i.e. a flare at a position where none was injected; and the rate of finding the injected flare. For this purpose “successful flare finding” was defined as the denoised lightcurve reaching above a three standard deviations upward fluctuation of the background. In Figure \[fig:sim2flares\] an example is shown of two flares of different duration being injected and successfully recognized by the Bayesian Blocks method.
![Example of the simulated background with two injected flares. In this example the noise level is $0.5\times10^{-6}\:\mathrm{photons\:cm^{-2} s^{-1}}$ and the red horizontal line indicates a three standard deviations upward fluctuation of the background. Two flares were injected on top of the background with widths of two and ten days with amplitudes of $10^{-6}\:\mathrm{photons\:cm^{-2} s^{-1}}$. The solid magenta line is the result of the Bayesian Blocks denoising procedure with the value of $F_{\mathrm{B}}=5.0$. Both flares are clearly visible and both were successfully recognized by the method.[]{data-label="fig:sim2flares"}](sim_example_OK.eps){width="4.in"}
After evaluating the $F_{\mathrm{B}}$ scans for the different simulated combinations of background and injected flares the value of $F_{\mathrm{B}}$ to be used in this analysis was set to 5. Using this value the rate of fake flares drops significantly while the rate of finding the injected flares is still high. Figure \[fig:FBscan\] shows an example of the $F_{\mathrm{B}}$ scan and in Figure \[fig:BBlocksRight\] an example of the denoising method applied to a real lightcurve for one of the candidate sources with the chosen value of $F_{\mathrm{B}}=5$ is shown.
![Example of the performance of the Bayesian Blocks method as a function of the $F_{\mathrm{B}}$ parameter for a single flare injected with width of two days and amplitude of $10^{-6}\:\mathrm{photons\:cm^{-2} s^{-1}}$. The blue squares indicate the rate (number flares per trial) at which fake flares are found. In red circles the rate for finding the injected flare is shown. The value of $F_{\mathrm{B}}$ chosen to be used for the analysis is 5.0, shown as vertical dashed line in the plot.[]{data-label="fig:FBscan"}](perftest_biggermarker.eps){width="4.in"}
![Example of a denoised lightcurve (solid line) together with the original data (black data points) for blazar Ton 599.[]{data-label="fig:BBlocksRight"}](Ton_OK.eps){width="4.in"}
At this point some selection criteria were applied on the Fermi LAT monitored list of sources, with the aim of selecting lightcurves with flaring behavior and significant enhancement of the flux over the average level. The first criterion was that the denoised lightcurve flux must reach above $10^{-6}\:\mathrm{photons\:cm^{-2} s^{-1}}$ during IC-86I[^4]. The second selection criterion aims at identifying denoised lightcurves exhibiting significant variations in time. In order to quantify this an 11-day running mean[^5] was calculated for each of the candidate sources. The maximum spread of the running mean was then divided by the mean over the entire 3-year data taking period, resulting in a measure of the maximum relative time variation. Only sources for which this maximum relative time variation was greater than 0.5 were selected.
After selecting a set of potentially interesting neutrino sources the statistical approach described in Section \[sec:llh\] was applied. The general form of the likelihood function given by equation (\[eq:likelihood\]) was used and the signal PDF as defined in equation (\[eq:signalpdf\]) was obtained using the denoised Fermi LAT lightcurves. For each candidate source the likelihood function was maximized with respect to the number of signal events $n_{\mathrm{s}}$, the power law index $\gamma$, the time lag $D_{\mathrm{t}}$ and flux threshold $f_{\mathrm{th}}$. The time lag parameter allows for a time offset between the photon lightcurve and the neutrino PDF of up to $\pm$0.5 days. Since neutrinos are expected to be produced in the high activity states of the sources the flux threshold was varied during the maximization procedure. Once the threshold changes, the time PDF is redefined, setting it equal to zero below the threshold and normalizing to unity what was left above the threshold. The test statistic for this search is given by the maximum likelihood ratio:
$$TS = -2 \log\Big[\frac{\mathcal{L}(n_{\mathrm{s}}=0)}{\mathcal{L}(\hat{n}_{\mathrm{s}},\hat{\gamma}_{\mathrm{s}},\hat{D}_{\mathrm{t}},\hat{f}_{\mathrm{th}})}\Big],
\label{eq:ts_bflares}$$
where $\hat{n}_{\mathrm{s}},\hat{\gamma}_{\mathrm{s}},\hat{D}_{\mathrm{t}},\hat{f}_{\mathrm{th}}$ are the best-fit values for the number of signal events, the power law index, the time lag and time PDF threshold.
Results {#fflares_rez}
-------
Using the selection criteria above, the list of sources in Table \[tab:bflrares\] was selected. The most significant deviation from the background-only hypothesis was observed for the Quasar PKS 2142-75 at (RA, Dec.)=(326.8$^{\circ}$, -75.6$^{\circ}$). To evaluate the post-trial p-value time-scrambled samples of the IceCube events were generated and the analysis was repeated on them for all the selected sources. Then the most significant result for each of the scrambled datasets was identified and its significance compared to the p-value for our most significant IceCube result. In 77% of the scrambled sets the p-value was equal to or smaller than the most significant p-value observed in the non-scrambled data and therefore it is well compatible with background fluctuations.
Figure \[fig:SelectedSignalRegPKS2142\] shows, for PKS 2142-75, the best-fit flux threshold together with the denoised lightcurve and the IceCube event weights $w_i$ defined in equation (\[eq:signalweight\]). In this figure one can see that the fit prefers a high flux threshold value, therefore reducing the time PDF to be non-zero only in a narrow time interval, leading to a low best-fit signal strength of $\hat{n}_{\mathrm{s}}=1.9$
![The denoised Fermi LAT lightcurve for PKS 2142-75 is shown with the red line and the red dashed horizontal line indicates the fit result for the flux threshold. For the lightcurve and the flux threshold the red scale on the right is used on the y-axis. The blue vertical lines are drawn at the times of measured IceCube events and the height indicates the event weights $w_i$ defined in Eg. \[eq:signalweight\] on the left scale. Only events in the periods when the lightcurve is above the best-fit flux threshold contribute to the significance.[]{data-label="fig:SelectedSignalRegPKS2142"}](Lc_PKS_2142_75_326_803__75_604_1day_ndof2_unbl.eps){width="5.in"}
[|l|r|r|r|r|r|r|r|r|r|r|r|]{}
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[[** – continued from previous page**]{}]{}\
& & & & & & & & & & &\
\
& & & & & & & & & & &\
PKS 2142-75 & -75.6 & 326.8 & 0.02 & 1.9 & 3.95 & -0.40 & 1.32e-06 & 5 & 5.0 & 7.8 & 8.10\
PKS 0235-618 & -61.6 & 39.2 & - & 0 & -&-&-&-&-&-&-\
PMN J1038-5311 & -53.2 & 159.7 & 0.47 & 0.97 & 2.25 & 0.50 & 0 & 1075 & 5.2 & 7.8 & 9.13\
Fermi J1717-5156 & -51.9 & 259.4 & 0.50 & 0.32 & 3.95 & 0.15 & 0 & 1075 & 5.2 & 7.8 & 8.38\
PKS 2326-502 & -49.9 & 352.3 & - & 0 & -&-&-&-&-&-&-\
PKS 1424-41 & -42.1 & 217.0 & 0.34 & 3.02 & 2.95 & -0.37 & 8.15e-07 & 910 & 5.2 & 7.9 & 9.73\
PKS 0920-39 & -40.0 & 140.7 & 0.10 & 6.38 & 3.25 & -0.44 & 7.93e-07 & 684 & 5.1 & 7.8 & 11.60\
PKS 0426-380 & -37.9 & 67.2 & - & 0 & -&-&-&-&-&-&-\
PKS 0402-362 & -36.1 & 61.0 & - & 0 & -&-&-&-&-&-&-\
PMN J2250-2806 & -28.1 & 342.7 & - & 0 & -&-&-&-&-&-&-\
PKS 2255-282 & -28.0 & 344.5 & - & 0 & -&-&-&-&-&-&-\
PKS 1244-255 & -25.8 & 191.7 & 0.08 & 0.98 & 3.22 & 0.03 & 1.73e-06 & 2 & 5.1 & 7.9 & 3.51\
PKS 1622-253 & -25.5 & 246.4 & 0.49 & 0.54 & 2.75 & -0.20 & 2.28e-06 & 82 & 5.1 & 7.9 & 2.37\
PMN J1626-2426 & -24.4 & 246.8 & 0.30 & 3.28 & 2.15 & -0.08 & 1.98e-06 & 269 & 5.2 & 7.8 & 3.58\
PKS 0454-234 & -23.4 & 74.3 & 0.16 & 1.40 & 2.05 & -0.30 & 2.38e-06 & 1 & 5.1 & 7.8 & 2.92\
PKS 1830-211 & -21.1 & 278.4 & - & 0 & -&-&-&-&-&-&-\
PMN J2345-1555 & -15.9 & 356.3 & - & 0 & -&-&-&-&-&-&-\
Fermi J1532-1321 & -13.4 & 233.2 & 0.45 & 2.56 & 2.85 & 0.40 & 0 & 1075 & 4.9 & 7.8 & 2.57\
PKS 1730-130 & -13.1 & 263.3 & - & 0 & -&-&-&-&-&-&-\
PKS 0727-11 & -11.7 & 112.6 & 0.09 & 7.74 & 3.55 & -0.06 & 8.64e-07 & 1069 & 4.8 & 7.8 & 3.43\
PKS 1346-112 & -11.5 & 207.4 & 0.18 & 1.37 & 2.21 & -0.50 & 1.50e-06 & 2 & 4.8 & 7.8 & 1.49\
PKS 1510-089 & -8.8 & 228.2 & 0.45 & 1.68 & 3.56 & -0.17 & 0 & 1075 & 4.5 & 7.7 & 1.63\
3C 279 & -5.8 & 194.0 & - & 0 & -&-&-&-&-&-&-\
PKS 2320-035 & -3.3 & 350.9 & 0.12 & 2.67 & 2.50 & 0.28 & 1.34e-06 & 4 & 3.5 & 7.3 & 0.40\
PMN J0948+0022 & 0.4 & 147.2 & 0.27 & 7.12 & 3.95 & 0.50 & 0 & 1075 & 3.5 & 7.2 & 0.72\
3C 273 & 2.1 & 187.3 & 0.26 & 1.65 & 1.85 & -0.36 & 1.12e-06 & 201 & 3.4 & 7.0 & 0.38\
PMN J0505+0416 & 4.3 & 76.4 & - & 0 & -&-&-&-&-&-&-\
J123939+044409 & 4.7 & 189.9 & - & 0 & -&-&-&-&-&-&-\
OG 050 & 7.5 & 83.2 & - & 0 & -&-&-&-&-&-&-\
CTA 102 & 11.7 & 338.2 & 0.26 & 3.58 & 3.95 & 0.10 & 7.14e-07 & 128 & 3.2 & 6.4 & 0.41\
3C 454.3 & 16.1 & 343.5 & 0.28 & 1.94 & 3.65 & -0.45 & 1.99e-06 & 10 & 3.2 & 6.3 & 0.34\
OX 169 & 17.7 & 325.9 & 0.20 & 2.70 & 3.95 & -0.50 & 9.45e-07 & 29 & 3.2 & 6.2 & 0.44\
OJ 287 & 20.1 & 133.7 & - & 0 & -&-&-&-&-&-&-\
PKS B1222+216 & 21.4 & 186.2 & - & 0 & -&-&-&-&-&-&-\
Crab Pulsar & 22.0 & 83.6 & - & 0 & -&-&-&-&-&-&-\
4C 28.07 & 28.8 & 39.5 & - & 0 & -&-&-&-&-&-&-\
Ton 599 & 29.2 & 179.9 & - & 0 & -&-&-&-&-&-&-\
B2 1520+31 & 31.7 & 230.5 & 0.42 & 3.75 & 2.15 & 0.44 & 0 & 1075 & 3.0 & 5.8 & 0.78\
B2 1846+32 & 32.3 & 282.1 & 0.37 & 1.57 & 3.95 & 0.37 & 8.58e-07 & 81 & 3.0 & 5.8 & 0.43\
B2 0619+33 & 33.4 & 95.7 & - & 0 & -&-&-&-&-&-&-\
1H 0323+342 & 34.2 & 51.2 & 0.47 & 1.52 & 2.15 & -0.25 & 5.40e-07 & 1051 & 3.0 & 5.8 & 0.71\
4C 38.41 & 38.1 & 248.8 & 0.23 & 11.55& 2.55 & 0.50 & 0 & 1075 & 3.0 & 5.7 & 1.19\
3C 345 & 39.7 & 250.4 & 0.09 & 4.0 & 2.18 & 0.16 & 1.07e-06 & 61 & 3.0 & 5.7 & 0.83\
NGC 1275 & 41.5 & 50.0 & 0.15 & 2.67 & 3.95 & -0.40 & 1.25e-06 & 24 & 3.0 & 5.7 & 0.60\
BL Lac & 42.3 & 330.7 & - & 0 & -&-&-&-&-&-&-\
B3 1343+451 & 44.9 & 206.4 & 0.30 & 0.82 & 1.55 & 0.19 & 1.29e-06 & 14 & 3.0 & 5.6 & 0.50\
4C 49.22 & 49.5 & 178.4 & 0.11 & 4.72 & 3.95 & 0.25 & 4.29e-07 & 99 & 3.0 & 5.6 & 0.66\
NRAO 676 & 50.8 & 330.4 & - & 0 & -&-&-&-&-&-&-\
BZU J0742+5444 & 54.7 & 115.7 & - & 0 & -&-&-&-&-&-&-\
S4 1849+67 & 67.1 & 282.3 & 0.25 & 11.70& 3.75 & 0.18 & 2.90e-07 & 1046 & 2.9 & 5.2 & 1.27\
S5 0836+71 & 70.9 & 130.4 & 0.23 & 3.13 & 3.95 & -0.32 & 5.96e-07 & 109 & 2.9 & 5.2 & 0.75\
PKS 0716+714 & 71.4 & 110.4 & - & 0 & -&-&-&-&-&-&-\
S5 1803+78 & 78.5 & 270.2 & 0.19 & 3.98 & 3.95 & 0.12 & 7.92e-07 & 26 & 2.9 & 5.2 & 0.95\
Search for Triggered Flares with Sporadic Coverage {#sec:sflares}
==================================================
In the searches presented in the previous section neutrino emission was assumed to follow the $\gamma$-ray photon flux of the source, using lightcurves extracted from Fermi-LAT data in the energy range from 100 MeV to 300 GeV as templates. A complementary search was performed to cover a potentially interesting group of sources that did not pass the selection criteria in Section \[fflares\_method\]. These are sources which exhibited flares in the TeV range, but which did not show significant activity in the lower energy range covered by the Fermi-Lat lightcurves. For these flares Astronomer’s Telegrams (ATel) were issued by imaging air-Cherenkov telescopes such as H.E.S.S., MAGIC or VERITAS. As explained in Section \[sec:sources\], such orphan flares, exhibiting TeV emission while lacking emissions in the lower energy region, may be interesting for hadronic models, if the lack of emission is real and not due to limited exposure of experimental observations [@orphan1; @orphan2; @orphan3]. This search was performed for the IC-79 and IC-86I periods only since the data from previous periods have already been analyzed [@mike].
Method {#oflares_method}
------
Once more the general form of equation (\[eq:likelihood\]) was used for the likelihood function but here the time dependent part of the signal PDF was a normalized box function. The box shaped time PDF was defined as being equal to zero in the whole period except for the flare time reported by the ATel plus a one day margin before and after. The use of detailed lightcurves was impossible in these cases because there is no continuous monitoring of TeV observations, like Fermi-LAT provides at lower energies.
Since the duration of the flare was fixed, the likelihood function was maximized only with respect to the spectral index $\gamma$ and the number of signal events $n_{\mathrm{s}}$.
Results {#results}
-------
In Table \[tab:table\_box\] the candidate sources are listed. These were selected from the reports found in the ATel indicated in the table. Two sources, 1ES 0806+524 during IC-79 and PG1553+113 during IC-86I, showed a positive fluctuation over the background but the p-values are compatible with the background.
[-1in]{}[-1in]{}
-- ---------------- ------------ ------------------------ ------ ------ ------
1ES 0806+524 3192 55615 < T <55618 0.78 3.95 0.24
HESS J0632+057 3153, 3161 55598 < T <55602 0 - -
1ES 1215+303 3100 55562 < T <55565 0 - -
PG1553+113 4069 56036 < T <56039 0.8 3.95 0.23
BL Lacertae 3459 55739 < T <55742 0 - -
-- ---------------- ------------ ------------------------ ------ ------ ------
: Source candidates selected for the “Search for Triggered Flares with Sporadic Coverage”. The best-fit values for the number of signal events $\hat{n}_{\mathrm{s}}$ and the spectral index $\hat{\gamma}_{\mathrm{s}}$ are listed. The p-values are pre-trial, the dashes indicate that the source best-fit was for zero signal events.[]{data-label="tab:table_box"}
Conclusions {#sec:Conclusions}
===========
Searches described within this paper have found no evidence for the existence of flaring or periodic neutrino sources. Both the untriggered search looking for neutrino flares anywhere in the sky and the triggered search looking for neutrino flares coinciding with $\gamma$-ray flares reported by the Fermi LAT returned results consistent with the background-only hypothesis. No evidence has been found for periodic neutrino emission by binary systems either. These analyses include data from the first year of operation of the completed IceCube detector, taken between May 2011 and May 2012. IceCube will continue to run in this configuration for the foreseeable future and the resultant signal build-up will increase the sensitivity to steady point sources of neutrinos [@jakerameez]. A similar improvement is expected in the sensitivity of the detector towards periodic neutrino signals. For flare searches, on the other hand, additional years of data with the full detector improve the chances to see rarer (rather than weaker) neutrino emission from outbursts which we have not been fortunate enough to witness yet.
In this paper we have demonstrated the viability of long term monitoring of sources of interest triggered by multiwavelength information from other experiments. As the detector matures, detector operations, data acquisition and processing will become more automated and we will soon be able to carry out this monitoring near-realtime - reducing the time delay between the trigger and the results.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge the support from the following agencies: U.S. National Science Foundation-Office of Polar Programs, U.S. National Science Foundation-Physics Division, University of Wisconsin Alumni Research Foundation, the Grid Laboratory Of Wisconsin (GLOW) grid infrastructure at the University of Wisconsin - Madison, the Open Science Grid (OSG) grid infrastructure; U.S. Department of Energy, and National Energy Research Scientific Computing Center, the Louisiana Optical Network Initiative (LONI) grid computing resources; Natural Sciences and Engineering Research Council of Canada, WestGrid and Compute/Calcul Canada; Swedish Research Council, Swedish Polar Research Secretariat, Swedish National Infrastructure for Computing (SNIC), and Knut and Alice Wallenberg Foundation, Sweden; German Ministry for Education and Research (BMBF), Deutsche Forschungsgemeinschaft (DFG), Helmholtz Alliance for Astroparticle Physics (HAP), Research Department of Plasmas with Complex Interactions (Bochum), Germany; Fund for Scientific Research (FNRS-FWO), FWO Odysseus programme, Flanders Institute to encourage scientific and technological research in industry (IWT), Belgian Federal Science Policy Office (Belspo); University of Oxford, United Kingdom; Marsden Fund, New Zealand; Australian Research Council; Japan Society for Promotion of Science (JSPS); the Swiss National Science Foundation (SNSF), Switzerland; National Research Foundation of Korea (NRF); Danish National Research Foundation, Denmark (DNRF).
[^1]: The two parameters, $n_{\mathrm{s}}$ and $\gamma_{\mathrm{s}}$, are present in all searches but their best fit values $\hat{n}_{\mathrm{s}}$ and $\hat{\gamma}_{\mathrm{s}}$ can be different for the different searches.
[^2]: Except for the “Search for Periodic Neutrino Emission from Binary Systems” in Section \[sec:per\] which benefits from the signal being accumulated over many periods of the binary system.
[^3]: Since for declinations above +85$^\circ$ and below -85$^\circ$ the off-source region is very small and the statistics for data scrambling is too limited, this region is excluded from the analysis.
[^4]: The requirement was for the flux to reach above $10^{-6}\:\mathrm{photons\:cm^{-2} s^{-1}}$ specifically during IC-86I and not during the whole analyzed time window (IC-59, IC-79 and IC-86I together) because the IC-59 and IC-79 data were analyzed before, finding no significant results [@ic59-time].
[^5]: The value of 11-day for the running mean comes from the fact that the denoised lightcurve has one day binning. Thus for calculating the mean we take the value for the current bin and five bins before and after for 11-day in total.
|
---
abstract: 'We introduce a class of variational states to describe quantum many-body systems. This class generalizes matrix product states which underly the density-matrix renormalization group approach by combining them with weighted graph states. States within this class may (i) possess arbitrarily long-ranged two-point correlations, (ii) exhibit an arbitrary degree of block entanglement entropy up to a volume law, (iii) may be taken translationally invariant, while at the same time (iv) local properties and two-point correlations can be computed efficiently. This new variational class of states can be thought of as being prepared from matrix product states, followed by commuting unitaries on arbitrary constituents, hence truly generalizing both matrix product and weighted graph states. We use this class of states to formulate a renormalization algorithm with graph enhancement (RAGE) and present numerical examples demonstrating that improvements over density-matrix renormalization group simulations can be achieved in the simulation of ground states and quantum algorithms. Further generalizations, e.g., to higher spatial dimensions, are outlined.'
author:
- 'R. Hübener, C. Kruszynska, L. Hartmann, and W. Dür'
- 'F. Verstraete'
- 'J. Eisert and M.B. Plenio'
title: Renormalization algorithm with graph enhancement
---
Strongly correlated quantum systems give rise to a number of intriguing phenomena in condensed matter systems such as the existence of rare-earth magnetic insulators or high-temperature superconductors. The classical description of such quantum many-body systems is difficult, as entanglement and interactions cannot be neglected. A full solution of the underlying microscopic model is unfeasible due to the exponential growth of the dimension of the Hilbert space with the number of constituent particles. In turn, numerical variational approaches, like the [*density-matrix renormalization group*]{} (DMRG) technique [@Wh91; @Sc04], make use of an important observation. Typically, ground or thermal states do not occupy the exponentially large Hilbert space, but a much smaller subspace. DMRG can indeed be seen as a variation over the polynomially sized set of matrix product states (MPS) [@FNW92; @Rommer95; @Vi03; @Ve04], approximating the true ground state iteratively. This approach is expected to work particularly well in one-dimensional gapped systems, in which correlation functions decay exponentially and the entanglement entropy saturates at larger block sizes, satisfying an “area law” [@Area].
Any such variational method, however, has its limitations. For example, in a critical one-dimensional system, the MPS description is no longer economical, with other variational sets potentially being more appropriate. When it comes to time evolution, area laws may be replaced by volume laws [@Calabrese], and a DMRG picture can become very expensive. [*Projected entangled pair states*]{} (PEPS) form higher-dimensional analogues of [*matrix product states*]{} MPS [@Ve04b]. This approach is very promising but still in development. For critical systems, [*multi-scale entanglement renormalization*]{} or [*contractor renormalization*]{} [@MERA; @Flow] are promising candidates also in two dimensions, but are not easily reconcilable with translational invariance. [*Weighted graph states*]{} (WGS) [@Du05; @Ca05; @Weighted] are a family of states that can embody long-range correlations in any spatial dimensions, but do not seem to grasp short-range properties as well as MPS do [@Weighted1].
With these observations in mind, one of the key questions seems to be the following. How far can one go with the efficient classical description of quantum many-body systems? Can MPS for example be generalized to a larger class of states encompassing some of the above approaches while retaining all of their convenient features? Can one have additional long-range correlations while still being able to efficiently compute local properties and correlation functions? In particular, given the complementary strengths of the MPS and the WGS approach it is natural to attempt a unification of the two approaches. This work shows that indeed the two pictures can be combined to form a new enlarged variational set, while retaining all of the desirable structural elements of its ancestors. We first define the set, discuss variations, sketch generalizations and finally demonstrate applicability and performance as well as limitations in ground state approximations and simulations of quantum algorithms.
[*Renormalization algorithm with graph enhancement. –*]{} We start from MPS of a quantum chain of length $N$, consisting of $d$-level systems, as used in DMRG [@FNW92; @Rommer95; @Vi03; @Ve04] $$|\psi(A)\rangle := \sum_{s_1,\ldots,s_N=0}^{d-1}
\text{tr} [A_{s_1}^{(1)}\ldots A_{s_N}^{(N)} ]
|s_1,\ldots,s_N\rangle$$ where the $A_{s_n}^{(n)}$ are complex $D\times D$ matrices. For open boundary conditions, the left- and rightmost matrices can be taken to be vectors. For simplicity of notation, but in a way that can be trivially generalized, we now fix $d=2$. MPS have correlation functions $\langle Z^{(j)} Z^{(j+k)} \rangle
- \langle Z^{(j)} \rangle\langle Z^{(j+k)}\rangle$ exponentially decaying in $k$ and satisfy an area law [@Area] by construction [^1]. An area law in 1D implies that any Renyi entropy $S_\alpha$ of the reduced state of a block of $L$ contiguous spins will eventually saturate ($S_\alpha(\rho_L)= O(1)$); many ground states possess this property and hence a good and economical MPS approximation of them is possible [@Schuch].
Now we go beyond this picture and apply to the MPS any set of commuting unitaries between any two constituents, irrespective of the distance. More specifically, we consider the adjacency matrix $\Phi$ of a weighted simple graph with $\Phi_{k,l}\in [0,2\pi)$ and apply without loss of generality the corresponding [*phase gates*]{} $U(\Phi_{k,l}):= |0,0\rangle\langle 0,0|+ |0,1\rangle\langle 0,1|+
|1,0\rangle\langle 1,0|+ |1,1\rangle\langle 1,1|e^{i \Phi_{k,l}}$ between the particles $k,l$ in the chain. Finally, we apply local rotations $V_j\in U(2)$, to arrive at the variational class of states defined by $$\begin{gathered}
|\psi(A,\Phi,V)\rangle :=\prod_{j=1}^N V_j^{(j)}
\prod_{k,l} U^{(k,l)}(\Phi_{k,l})\\
\times
\sum_{s_1, \ldots , s_N}
\text{tr}
[A_{s_1}^{(1)}\ldots A_{s_N}^{(N)}]
|s_1, \ldots , s_N\rangle,\label{RAGE}\end{gathered}$$ which then forms the basis of the [*renormalization group algorithm with graph enhancement*]{} (RAGE). The above set clearly embodies a large variational class. By definition, for $\Phi=0$ and $V_j={\mathbbm{1}}$, it includes the MPS. It also includes superpositions of WGS as first considered in Ref. [@Weighted], $$\begin{aligned}
|\varphi\rangle\!\!
&=& \!\! \sum_m \alpha_m \prod_{j=1}^N V_j^{(j)}\!\!\!\!\!\!\!
\sum_{s_1,\ldots,s_N=0}^1
e^{-i \mathbf{s}^T\Phi \mathbf{s}+ \mathbf{d}_m^T \mathbf{s}}
|s_1,\ldots ,s_N\rangle
\label{defwgraph}\\
&=& \prod_{j=1}^N V_j^{(j)} \prod_{k,l} U^{(k,l)}(\Phi_{k,l})
\sum_m \alpha_m |\eta_{m,1}\rangle\otimes
\ldots\otimes|\eta_{m,N}\rangle\nonumber\end{aligned}$$ where $\mathbf{d}_m=(d_{m,1},\dots, d_{m,N})$, $\mathbf{s}=(s_1,\dots, s_N)$, $|\eta_{m,n}\rangle := |0\rangle + e^{d_{m,n}} |1\rangle$ and $U(\Phi_{m,n})$ are defined as above, and which can be shown to be of the form of Eq. (\[RAGE\]). For simplicity, and w.l.o.g, we will often set $V_j={\mathbbm{1}}$ subsequently.
[*Main properties of RAGE states. –*]{} To start with, RAGE states have a polynomially sized description, where the MPS and the WGS part are fully determined by $O(ND^2)$ and $O(N^2)$ real parameters respectively. Furthermore
[*(i) Volume law for the entanglement entropy:*]{} By having a collection of maximally entangled qubit pairs across a boundary, the von-Neumann entropy of a block of length $L$ can be taken to scale as $S(\rho_L) = O(L)$. Encompassing graph states, our class can hence maximize the entanglement entropy.
[*(ii) Translational invariance:*]{} Whenever the MPS part is translationally invariant, $\Phi$ is a cyclic matrix, and $V_j$ is the same for all $j$, the whole state $|\varphi\rangle$ is manifestly translationally invariant. There exist other translational invariant states that do not have this simple form. The key feature, though, is that unlike for multiscale entanglement renormalization [@MERA], there exists this natural subset of states for which translational invariance is guaranteed to be exactly fulfilled, while at the same time a volume law for block-wise entanglement is possible [@Du05; @Ca05].
[*(iii) Completeness:*]{} As MPS already form a complete set in Hilbert space (if one allows $D$ to scale as $O(2^N)$, one can represent any pure state in $({{{\mathbbm{C}}}}^2)^{\otimes N}$) and this remains true for the RAGE set.
[*Efficient computation of local properties and correlation functions. –*]{} The previous properties are all very natural and desirable, and especially (i) cannot be achieved efficiently with MPS alone. However, as will be shown, this does not prevent us from computing local properties and correlation functions efficiently – which is the key feature of this set.
To compute expectation values of observables with small support we use the relevant reduced density matrix $\rho_{\cal S}$, which may be computed efficiently with an effort of $O(ND^5)$ in the total size $N$ of the system ${\cal S}\subset \{1,\dots, N\}$. Controlled phase gates acting exclusively on qubits that are traced out make no contribution, while those that act on the spins in ${\cal S}=\{m_1,\dots, m_{|{\cal S}|}\}$ amount to a redefinition of the observables that are local to ${\cal S}$. Therefore we redefine $\Phi$ such that the above simplifications are enforced, $\omega_{k,l}=\Phi_{k,l} \text{ if } k\in{\cal S},l\in \bar{\cal S}$ or $k \in \bar {\cal S}, l \in {\cal S}$, where $\bar {\cal S}=\{1,\ldots,N\}\setminus{\cal S}$, and $\omega_{k,l}=0$ otherwise. We also define $E^{(j)}_{k,l}:= A_{k}^{(j)}\otimes (A_{l}^{(j)})^\ast$, where $\ast$ denotes complex conjugation. The reduced density matrix $\rho_{\cal S}$ (up to phase gates in ${\cal S}$) is then found to be $$\begin{aligned}
\rho_{\cal S}&=&
\sum_{{s_1,\ldots,s_N=0}\atop{r_1,\ldots,r_N=0}}^1
\text{tr} [ E_{s_1,r_1}^{(1)}
\ldots E_{s_N,r_N}^{(N)}
] \text{tr}_{\bar {\cal{S}}}
[ (\prod_{k,l}
U^{(k,l)}(\omega_{k,l}))\nonumber\\
&&\times
|s_1,\ldots ,s_N\rangle\langle r_1,\ldots , r_N|
(\prod_{k,l} U^{(k,l)\dagger}(\omega_{k,l}))
]\nonumber\\
&=&
\sum_{{s_1,\ldots,s_N=0}\atop{r_1,\ldots,r_N=0}}^1
\text{tr} [ E_{s_1,r_1}^{(1)}
\ldots E_{s_N,r_N}^{(N)}
] |s_{m_1},\dots, s_{m_{|\cal{S}|}}\rangle
\nonumber\\
&&\times \langle r_{m_1},\dots, r_{m_{|\cal{S}|}}|
\prod_{k\in {\cal{S}}, l \in{\bar {\cal{S}}}}
e^{i \omega_{k,l}(\delta_{s_k,1} - \delta_{r_k,1})\delta_{s_l,1}}.
\nonumber\end{aligned}$$ The key of the above argument is that the effect of the phases is a mere modification of the transfer operators of the MPS by a phase factor, the phase depending on the matrix element in question. Thus, the evaluation of expectation values is performed using (products of) transfer operators associated with the single sites. The reduced state can then be written as $$\begin{aligned}
\rho_{\cal S}&=&
\sum_{
{s_{m_1},\dots, s_{m_{|\cal S|}}=0}
\atop
{r_{m_1},\ldots,r_{m_{|\cal S|}}=0}}^1
\text{tr} \bigl[ \prod_{k=1}^N T_{s_k,r_k}^{(k)}(\{ s_{m_p},r_{m_p}:m_p\in{\cal S} \} )
\bigr]
\\
&\times&| s_{m_1},\dots, s_{m_{|\cal S|}}\rangle\langle r_{m_1},\dots, r_{m_{|\cal S|}}|,\end{aligned}$$ where now $T_{s_k,r_k}^{(k)}(\{ s_{m_p},r_{m_p}\}):=
E^{(k)}_{s_k,r_k}$ if $k\in {\cal S}$, which are the unmodified transfer operators of MPS, and $$\begin{aligned}
T_{s_k,r_k}^{(k)}(\{ s_{m_p},r_{m_p}\}) &:=& \sum_{l=0}^1
B_{l}^{(k)}(\{s_{m_p}\},s_k)\nonumber\\
&\otimes & (B_{l}^{(k)}(\{r_{m_p}\},r_k))^\ast \end{aligned}$$ if $k\in {\bar {\cal S}}$, which are the transfer operators modified by phases, $B_{l}^{(k)}(\{ s_{m_p}\},s_k):=
A_{l}^{(k)}\prod_{m_p\in {\cal S}}
e^{i\omega_{m_p,k}\delta_{s_{m_p}, 1}\delta_{s_k,1}}$. Grouped in this way, the reduced density operator can indeed be evaluated efficiently. In fact, for each $\{ r_{m_p},s_{m_p}\}$ the effort to compute the entry of the reduced state is merely $O(N D^5)$, as one has to multiply $N$ transfer matrices of dimension $D^2\times D^2$, just as in the case of MPS. This procedure is inefficient in $|{\cal S}|$, with an exponential scaling effort. However, any Hamiltonian with two-body (possibly long-ranged) interactions can be treated efficiently term by term. [*Efficient updates. –*]{} Besides procedures for the efficient computation of reduced density matrices, and therefore expectation values, we need a variational principle to improve the trial states. We will focus on local variational approaches to approximate ground states by minimizing the energy $$\label{Energy}
E:=\frac{\langle \psi(A,\Phi,V) |H |
\psi(A,\Phi,V)\rangle}{\langle \psi(A,\Phi,V) |
\psi(A,\Phi,V)\rangle},$$ on the approximation of time evolution and on the simulation of quantum circuits. The search for ground states is well known to be related to imaginary-time evolution.
[*Static updates. –*]{} The MPS part can be updated as in variants of DMRG [@Ve04]. The expression $\langle \psi(A,\Phi,V) |H |
\psi(A,\Phi,V)\rangle$ is (as in MPS) a quadratic form in each of the entries of the matrices $A^{(k)}_0$, $A^{(k)}_1$ for each site $k=1,\dots, N$. An optimal local update can therefore be found by means of solving generalized eigenvalue problems with an effort of $O(D^3)$. Local rotations can be incorporated by parametrizing single qubit rotations on spin $k$ with real parameters $x_k\in {{\mathbbm{R}}}^4$ as $V_k = \sum_{j=1}^4 x_{k,j} M_j$ with $M=({\mathbbm{1}}, \sigma_z, \sigma_y,\sigma_x)$ being the vector of the Pauli matrices including the identity. Again, the local variation of $x_k$ in (\[Energy\]) is a generalized eigenvalue problem in $x_k$ for each site $k=1,\dots, N$. To optimize the phases of the WGS, one may first define the new Hamiltonian $H_V := (\prod_{j=1}^N V_j^{(j)\dagger}) H (\prod_{k=1}^N V_k^{(k)})$. The optimal phase gates between any pair of spins $j,k\in\{1,\dots,N\}$ can be computed efficiently as the procedure amounts to a quadratic function of a single variable $z=e^{i\Phi_{j,k}}$. To summarize, an update of $|\psi(A,\Phi,V)\rangle$ to minimize (\[Energy\]) corresponds to a sweeping over such local variations, each of which is efficiently possible, with an effort of $O(MND^3)$ for $M$ sweeps. An element that is not present for MPS alone is that one can make a choice whether one adapts an MPS part or the adjacency matrix for an identical change in the physical state. In practice, we have supplemented this procedure with a gradient-based global optimization, making use of the fact that the gradient can be explicitly computed.
We have applied the RAGE-method to proof-of-principle 1D and 2D models, where the adjacency matrix is allowed to connect any constituents in the lattice. Fig. \[SimulationFigure\](a) shows results for the 2D Ising model with transversal magnetic field, $H=\sum_{\langle a,b\rangle }
\sigma_z^{(a)}\sigma_z^{(b)} + B\sum_a\sigma_x^{(a)}$, comparing the achievable accuracy of MPS (using a one-dimensional path in the 2D lattice) and the RAGE-method for a fixed total number of free parameters. The RAGE-method gives a significantly better accuracy regarding ground state energy and two-point correlations, already for a very small number of parameters. For other models, we see a similar improvement of RAGE over MPS, although in some cases (e.g., for a 2D Heisenberg model) the overall accuracy is still not very satisfactory, possibly related to local minima encountered in the procedure. This new class of states does allow for new features like long-range correlations and a violation of an area law, but in turn, breaks the local $SU(2)$ gauge invariance. It is also clear from the simulations that the limitation of the underlying 1D structure of the MPS cannot always be fully overcome by the graph enhancement. The full potential in numerical performance in identifying ground states is yet to be explored. There exist, however, a number of interesting parent Hamiltonians where the RAGE method should be particularly well suited, e.g., perturbations of models which have a WGS as an exact or approximate ground state. We mention Kitaev’s model (and perturbations thereof) on a hexagonal lattice which has the toric code state – a WGS – as ground state.
![(a) 2D Ising model on a $4\times4$ periodic lattice . We compare the achievable accuracy with RAGE (red, solid) and MPS (green, dotted) with $D=4$ with exact results (blue, dashed). Two-point correlations as function of $B$ are shown. The inset depicts the energy for different [*total*]{} numbers of parameters and $B=2$ in comparison with the exact ground state (blue, dashed) as well as the first excited state (light blue, dashed). (b) Comparison of MPS (blue, dashed) and RAGE (red, solid) with $D = 2$ for the simulation of a random quantum circuit [@Biham] on $N = 14$ qubits. Application of a random local phase gate followed by a random controlled-phase gate with random uniform phase in $[0, 2\pi)$ constitutes one block. For given $k$ we apply this block $k$ times to a randomly chosen initial MPS state. $500$ such runs are determined, and in each the fidelity with the exact state is computed. The average over $500$ realizations is then plotted.[]{data-label="SimulationFigure"}](jointfigure2.eps){width=".49\textwidth"}
\[Ising\]
[*Time evolution and simulation of quantum circuits. –*]{} We have also considered time evolution, more specifically the evolution of a quantum state in a quantum circuit. Here, sequences of elementary gates are applied, e.g., two-qubit phase gates and arbitrary single-qubit rotations. This method can be easily adapted to Hamiltonian (real or imaginary) time evolution. We now show how to efficiently obtain an optimal approximation of the resulting state after the application of an elementary gate. It turns out to be useful to restrict the variational family by setting $V_j^{(j)}={\mathbbm{1}}$, although an extension to arbitrary $V$ is possible. For phase gates, this update is particularly simple, as only a change in the adjacency matrix $\Phi$ is required. It is part of the strength of the scheme that phase gates between arbitrary constituents are already included in the variational set. The update of a local unitary will require some more attention:
Consider an initial state vector $|\psi(A,\Phi,{\mathbbm{1}})\rangle$, to which a single qubit unitary operation $U$ is applied – acting, e.g., on the first qubit. The goal is now to find the best approximation $|\psi(A',\Phi',{\mathbbm{1}})\rangle$ which maximizes $$\begin{aligned}
O:= \frac{|\langle \psi(A',\Phi',{\mathbbm{1}})|U_1 |\psi(A,\Phi,{\mathbbm{1}})\rangle|^2}
{\langle \psi(A',\Phi',{\mathbbm{1}})|\psi(A',\Phi',{\mathbbm{1}})\rangle
}.
\label{inner}\end{aligned}$$ It appears natural to vary only phases that directly affect qubit 1, i.e., $\Phi'_{j,k}= \Phi_{j,k}$ if $j\not=1$. In this case, one can rewrite (\[inner\]) in such a way that the optimal MPS part $A'$ can be obtained analytically by solving a set of linear equations, while the optimization of a single phase $\Phi'_{1,k}$ leads to a simple quadratic form. In practice, an alternating sweeping of both kinds of local variational methods is required. We have tested this method for a random quantum circuit (see Fig. \[SimulationFigure\](b)) and compared the achievable accuracy with MPS. Again, we obtain an improvement due to the WGS. [*Extensions. –*]{} A similar construction as illustrated for MPS also works for unifying WGS with other underlying tensor network descriptions. Similarly, one can use arbitrary clifford circuits instead of the WGS and can still efficiently contract. More precisely, whenever an exact or approximate evaluation of expectation values of arbitrary product observables (i.e., tensor products of local operators) for a state described by a tensor network is possible, then *local* observables (i.e., observables with a small support) can be efficiently computed for the unified family of such tensor network states and WGS (or clifford circuits), following an approach similar as in Eq.(\[defwgraph\]). While this certainly restricts the set of computable quantities (e.g., string-order parameters can no longer be evaluated), it still suffices to compute expectation values of all *local* Hamiltonians and hence one obtains a variational method for a ground-state approximation or simulation of quantum circuits.
[*Conclusions. –*]{} To summarize, we have introduced a new variational class of states to describe quantum many-body systems. These states have a number of desirable properties. Correlation functions can be computed efficiently, systematic improvements of the approximation within the class are possible and the states carry long-range correlations and violate entanglement area laws, as being encountered in critical systems or in quenched quantum systems undergoing time evolution. We have applied the RAGE ansatz to condensed matter and quantum computation problems, where we find an improvement over MPS. From a fundamental perspective the key question is where exactly the boundaries for the efficient classical description of quantum systems might lie. In fact, intriguingly, the entanglement content of the state cannot be taken as an indicator for the “complexity of a state” [@Kr07]. Delineating this boundary will reveal more about the structure of quantum mechanics from a complexity point of view and holds the potential for new improved algorithms and methods for the description of quantum systems.
[*Acknowledgements. –*]{} We thank S. Anders, T.J. Osborne and C.M. Dawson for illuminating discussions and the FWF, the EU (QAP, OLAQUI, SCALA), the EPSRC QIP-IRC, the Royal Society, Microsoft Research, and a EURYI for support.
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[^1]: In our notation the the brackets in the upper position denote the support of an operator.
|
---
abstract: 'The weak coupling expansion is applied to the single flavour Schwinger model with Wilson fermions on a symmetric toroidal lattice of finite extent. We develop a new analytic method which permits the expression of the partition function as a product of pure gauge expectation values whose zeroes are the Lee–Yang zeroes of the model. Application of standard finite–size scaling techniques to these zeroes recovers previous numerical results for the small and moderate lattice sizes to which those studies were restricted. Our techniques, employable for arbitrarily large lattices, reveal the absence of accumulation of these zeroes on the real hopping parameter axis at constant weak gauge coupling. The consequence of this previously unobserved behaviour is the absence of a zero fermion mass phase transition in the Schwinger model with single flavour Wilson fermions at constant weak gauge coupling.'
---
=msbm10 scaled 0
Dublin Preprint: TCDMATH 98-14\
[The Phase Structure of the Weakly\
Coupled Lattice Schwinger Model]{}\
[**R. Kenna[^1],\
School of Mathematics, Trinity College Dublin, Ireland\
\
C. Pinto and J.C. Sexton\
School of Mathematics, Trinity College Dublin, Ireland\
and\
Hitachi Dublin Laboratory, Dublin, Ireland** ]{}\
\
December 1998
#### Introduction
In lattice gauge theory, there has been considerable discussion on the phase structure of gauge theories with Wilson fermions. A system of free Wilson fermions exhibits a second order phase transition at $(\beta,\kappa)=(\infty,1/(2d))$, $\beta$ and $\kappa$ being the usual inverse gauge coupling squared and hopping parameters respectively and $d$ being the lattice dimensionality. Recent discussions concern the extent to which this phase transition extends into the $(\beta,\kappa)$ plane, the expectation being that there is a line of phase transitions (the chiral limit) extending to the strong coupling limit $\beta = 0$. In the simplest model to which this applies, the lattice Schwinger model [@Sc62], there is also a second order phase transition in the strongly coupled limit [@GaLa95Karsch]. This critical line, which is expected to connect these two phase transitions, is also expected to recover massless physics and therein lies the importance in determining its nature and position.
While earlier numerical results casted doubt on the above scenario [@GaLa92], more recent ones support the hypothesis that the critical line extends over the whole range of couplings [@GaLa94; @HiLa98; @AzDi96]. To our knowledge, only two groups have attempted to determine the phase diagram in the lattice Schwinger model. While in rough agreement regarding the location of the critical line, they differ in their conclusions regarding its quantitive critical properties. Using Lee–Yang zeroes [@LY], finite–size scaling techniques [@IPZ] as well as PCAC relations, the results of [@GaLa94; @HiLa98] support the free boson scenario, where the model lies in the same universality class of the Ising model, with correlation length critical exponent $\nu = 1$ and the specific heat (chiral susceptibility) exponent $\alpha = 0$. This is not in agreement with [@AzDi96] where finite–size scaling of Lee–Yang zeroes on larger lattices provides evidence for $\nu = 2/3$, $\alpha = 2/3$.
The precise analytical determination of the phase structure in the weakly coupled regime is the primary motivation for this work. We contend that the Lee–Yang zeroes in the Schwinger model with fixed weak gauge coupling display very unusual size–dependent behaviour, do not accumulate on the real hopping parameter axis and may not, in fact, lead to a phase transition. The behaviour of these zeroes is, however, such as to mimic the appearance of a phase transition when a finite–size analysis is restricted to small or moderate lattice sizes.
#### Lattice Schwinger Model
We consider a finite $d=2$ dimensional lattice with spacing $a$ and $N$ sites in each direction. For the fermion fields, we impose antiperiodic boundary conditions in the temporal ($1$-) direction and periodic boundary conditions in the spatial ($2$-) direction. Lattice sites are labelled $x_\mu = n_\mu a$ where $ n_\mu = -N/2, \dots, N/2 - 1$. With these mixed boundary conditions the momenta for the Fourier transformed fermion fields are $ k_\mu = 2\pi {\hat{k}}_\mu /Na $, where $ {\hat{k}}_1 = -N/2+1/2, \dots, N/2 - 1/2$ and $ {\hat{k}}_2 = -N/2, \dots, N/2 - 1$.
The action for gauge invariant lattice QED with Wilson fermions is $ S_{\rm{QED}} = S_G[\phi] + S_F^{(W)}[\phi,\psi,\bar{\psi}]$, where $
S_F^{(W)} [\phi,\psi,\bar{\psi}] = \sum_{m,n}
\bar{\psi} (m) M^{(W)}(m,n) \psi (n)
$ and $$M^{(W)}(m,n) =
\frac{1}{2\kappa} \delta_{m,n}
-
\frac{1}{2}\sum_\mu{
\left\{
( r - \gamma_\mu ) U_\mu(m)\delta_{m+\hat{\mu},n}
+ ( r + \gamma_\mu ) U^\dagger_\mu(n)\delta_{m-\hat{\mu},n}
\right\}
}
\quad .
\label{3.5}$$ This Wilson fermion matrix can be written in terms of free and interacting parts as $M^{(W)} = M^{(0)} + M^{({\rm{int}})} $ where $M^{(0)}$ is given by $U_\mu = 1$ in (\[3.5\]). Here $U_\mu (n) = \exp{i \phi_\mu(n)} = \exp{(i e_0 a A_\mu (n))}$ is the link variable. The Wilson parameter, $r$, is henceforth set to unity. The partition function is $
Z_{\rm{QED}} =
\int{
{\cal{D}} U {\cal{D}} \bar{\psi} {\cal{D}} \psi
\exp{(
\{
-S_{\rm{G}} -S^{(W)}_{\rm{F}}
\}
)}
}
\propto
\int{
{\cal{D}} U
\exp{
\{-S_{\rm{G}}
\}
}
\det{M^{(W)}}
}
\propto
\langle
\det{M^{(W)}}
\rangle
$, the integration over the Grassmann Fermionic fields having been performed. Here $\langle {\cal{O}} \rangle $ is the pure gauge expectation value of the quantity ${\cal{O}}$ and the pure gauge action is $
S_G[\phi]=
\beta\sum_P{\left[1-\frac{1}{2}(U_P+U_P^\dagger)\right]}
$, where $U_P$ is the usual product of link variables around a fundamental plaquette and where $\beta = 1/e_0^2 a^2$. In the weak coupling expansion we employ the Feynman gauge for the calculation of pure gauge expectation values. (There, for large enough $N$, the gauge propagator in momentum space is given approximately by $\langle \tilde{\phi}_\mu (\hat{p}) \tilde{\phi}_\nu (-\hat{p})\rangle
= \delta_{\mu \nu} (N^d a^{2d}/2 \beta)
\sum_{\rho,p\ne 0}{(1 -\cos{p_\rho a})}$ with the infra–red zero–momentum mode excluded as standard [@irzm].)
For $\beta = \infty$, the only gauge configurations which contribute have $S_G = 0$. We assume for the following analysis that all gauge configurations satisfying this condition belong to a single equivalence class under the group of gauge transformations. Since the fermion determinant is gauge invariant under the same group of gauge transformations it is sufficient to evaluate the full partition function on this single configuration to determine the behaviour of the system at $\beta = \infty$ [@GaLa94]. Therefore the partition function is simply proportional to $ \det{M^{(0)}} $. Employing lattice Fourier transforms, the free fermion partition function can be written in terms of its $dN^d$ eigenvalues by solving the $d\times d$ dimensional eigenvalue problem of the form $M^{(0)}(p) | \lambda^{(0)}_\alpha (p) \rangle = \lambda^{(0)}_\alpha (p)
| \lambda^{(0)}_\alpha (p) \rangle $ where $\alpha=1,\dots,d$. In $d=2$ dimensions, the resulting eigenvalues are $
\lambda^{(0)}_\alpha (p)
=
1/2\kappa
- \sum_{\rho=1}^2 \cos{p_\mu a}
+ i (-)^\alpha \sqrt{ \sum_{\rho=1}^2{\sin^2{p_\rho a}}}
$.
With mixed boundary conditions, the eigenvalues in the free fermion case are either two–fold ($\hat{p}_2 = 0$ or $-N/2$) or four–fold degenerate. In this case, and with $\eta = 1/2\kappa$, the Lee–Yang zeroes are $\eta^{(0)}_\alpha(p) =
\sum_{\rho=1}^2 r \cos{p_\mu a}
- i (-)^\alpha \sqrt{\sum_{\rho=1}^2{\sin{p_\rho a}}}
$. The lowest zero with finite real part in $\kappa$ corresponds to $\hat{p} = (\pm 1/2,0)$ and for a symmetric lattice is $ \eta^{(0)}_\alpha (\pm 1/2,0)
=
1 + \cos( \pi/N)
- i (-)^\alpha {\sin{( \pi/N)}}$. Pinching of the positive real finite hopping parameter axis occurs at the massless point $\kappa_c = 1/2d $ and application of finite–size scaling [@IPZ] to the imaginary parts of the zero gives the critical exponent $ \nu = 1$ ($\alpha = 0$ then follows from hyperscaling). Therefore the free fermion model is in the same universality class as the Ising model in two dimensions and describes free bosons. In the presence of a gauge field the position and nature of this phase transition will change and a quantitive investigation into the nature of that change is the central aim of this paper.
#### The Weak Coupling Expansion
The weak coupling expansion is formally obtained by expanding the link variables $U_\mu (n)$ as a power series in $e_0$. The weak coupling expansion of the matrix $M^{({\rm{int}})}$ in (\[3.5\]) is $ M^{({\rm{int}})} = M^{(1)} + M^{(2)} +
{\cal{O}}(e_0^3)$ where $M^{(n)}$ is of order $e_0^n$. The fermion determinant $\det{M^{(W)}} = \det{M^{(0)}} \times \det{{M^{(0)}}^{-1} M^{(W)}}
= \det{M^{(0)}} \exp{ {\rm{tr}}
\ln{(1 + {M^{(0)}}^{-1} M^{({\rm{int}})})}}$ can then be expanded giving the following ‘traditional’ additive form for the weak coupling expansion [@Rothe]: $$\frac{
\langle \det{M^{(W)}} \rangle
}{
\det{M^{(0)}}
}
=
1
+
\sum_{i=1}^{dN^d}{
\frac{t_i}{\eta-\eta_i^{(0)}}
}
-
\frac{1}{2}
\sum_{i\ne j}^{dN^d}{
\frac{
t_{ij}
}{
(\eta-\eta_i^{(0)})
(\eta-\eta_j^{(0)})
}
}
+ \dots
\quad .
\label{additive}$$ in which $ t_i = \langle M_{ii}^{({\rm{int}})} \rangle $ and $ t_{ij} = t_{ji}
= \langle M_{ij}^{({\rm{int}})} M_{ji}^{({\rm{int}})}\rangle
-
\langle M_{ii}^{({\rm{int}})} M_{jj}^{({\rm{int}})}\rangle
$. Here the index $i$ generically stands for the combination of Dirac index and momenta $(\alpha,p)$ which label fermionic matrix elements so that $M_{ij}^{({\rm{int}})}$ represents $\langle \lambda_\alpha^{(0)} (q) | M^{({\rm{int}})}(q,p)
| \lambda_\beta^{(0)} (p) \rangle $. We note at this point that the expansion is analytic in $\eta$ with poles at $\eta = \eta_i^{(0)}$. We note further that $t_i=\langle M_{ii}^{({\rm{int}})} \rangle$ at order $e$ is proportional to the pure gauge expectation value of the gauge field $\phi_\mu$ and is zero.
An alternative formulation of the partition function may be obtained by writing the Wilson fermion matrix as $M^{(W)} = \eta
+ H $ where $H$ is the hopping matrix. The fermion determinant $\det M^{(W)} = \det(\eta + H)$, is a polynomial in $\eta$ since for finite lattice size these matrices are of finite dimension. Therefore the pure gauge expectation value of the fermion determinant is also a polynomial in $\eta$ and as such may be written $
\langle \det M^{(W)} \rangle = \prod_{i=1}^{dN^d}
(\eta - \eta_i)
$. Here $\eta_i$ represent $\eta_\alpha (p)$ and are the Lee–Yang zeroes in the complex $1/2 \kappa$ plane. We may write a ‘multiplicative’ weak coupling expansion as $$\frac{
\langle \det{M^{(W)}} \rangle
}{
\det{M^{(0)}}
}
= \prod_{i=1}^{dN^d}
\frac{(\eta - \eta_i)}{(\eta - \eta_i^{(0)})}
= \prod_{i=1}^{dN^d}\left(
1 - \frac{\Delta_i}{\eta - \eta_i^{(0)}}
\right)
\quad ,
\label{multiplicative}$$ where $\Delta_i = \eta_i - \eta_i^{(0)}$ are the shifts that occur in the zeroes when the gauge field is turned on and are the quantities to be determined. Note again that (\[multiplicative\]) is analytic in $\eta$ with poles at $\eta_i^{(0)}$. Since the two expressions (\[additive\]) and (\[multiplicative\]) must be equal, the residues of the poles must be identical.
Let $t_i = t_i^{(2)} + \dots$, $t_{ij} = t_{ij}^{(2)} + \dots$ and $\Delta_i = \eta_i^{(1)} + \eta_i^{(2)}
+ \dots $ where $\eta_i^{(1)}$ is ${\cal{O}}(e)$ and $t_i^{(2)}$, $t_{ij}^{(2)} $ and $\eta_i^{(2)}$ are ${\cal{O}}(e^2)$. Assume the $n_j^{\rm{th}}$ free eigenvalues are degenerate for $j=1,\dots,D_n$ where $D_n$ is $2$ or $4$ and $\lambda^{(0)}_{n_j} \equiv
\lambda^{(0)}_{n}$. At this stage we make no assumptions regarding the other eigenvalues. The traditional weak coupling expansion (\[additive\]) and its ‘multiplicative’ counterpart (\[multiplicative\]) have single poles at $\eta = \eta^{(0)}_n$, the residues of which are easily calculated to $ {\cal{O}}( e^2 )$. Equating the two residues order by order in $e$ gives $$\sum_{j=1}^{D_n}{ \eta_{n_j}^{(1)} } = 0
\quad , \quad \quad \quad
\sum_{j=1}^{D_n}{ \eta_{n_j}^{(2)} }
=
D_n \bar{\Delta}_n^{(2)}
\equiv
\sum_{j=1}^{D_n}{ \left[
- t_{n_j}^{(2)}
+
\sum_{i, \lambda_i^{(0)}\neq \lambda_n^{(0)} }{
\frac{t_{i n_j}^{(2)} }{
\eta_n^{(0)}-\eta_i^{(0)}
}
}
\right]
}
\quad .
\label{spe2}$$ In this latter equation, $ \bar{\Delta}_n^{(2)}$ is the average of two zeroes which in the free fermion case were two–fold degenerate and is ${\cal{O}}(e^2)$. Equating the residues of the corresponding double poles gives no extra information at this order.
In the non–degenerate case $D_n=1$, (\[spe2\]) is sufficient and necessary to fully determine the shifts in the positions of the zeroes $\eta_{n_j}^{(1)}$ and $\eta_{n_j}^{(2)}$. This then amounts to a new method to uniquely express the partition function and all derivable thermodynamic functions in multiplicative (as opposed to the traditional ‘additive’) form. Thus, we have expressed the pure gauge expectation value of a product of fermion eigenvalues as a product of pure gauge expectation values.
In the $D_n$–fold degenerate case, (\[spe2\]) gives for the expansion of the ‘multiplicative’ weak coupling expression (\[multiplicative\]) to ${\cal{O}}(e^2)$, $$\prod_i{\left(
1 - \frac{\Delta_i}{\eta - \eta_{n_i}^{(0)}}
\right)}
=
1 + \sum_i{\frac{t_i^{(2)}}{\lambda_i^{(0)}}}
+
\sum_{i,j,\lambda_i^{(0)}\neq \lambda_j^{(0)}}{
\frac{t_{ij}^{(2)}}{\lambda_i^{(0)}(\lambda_i^{(0)}-\lambda_j^{(0)})}
+\sum_{i \neq j}{
\frac{\Delta_i \Delta_j}{\lambda_i^{(0)} \lambda_j^{(0)}}
}
+ \dots
}
\quad .$$ This is identical to the traditional weak coupling expansion (\[additive\]) to ${\cal{O}}(e^2)$ if $$\sum_i{\left( \frac{\eta_i^{(1)}}{\lambda_i^{(0)}} \right)^2}
=
\sum_{
i\neq j,\lambda_i^{(0)}=\lambda_j^{(0)}
}{
\frac{t_{ij}^{(2)}}{\left(\lambda_i^{(0)}\right)^2 }
}
\quad .
\label{MVAe2'}$$ Setting $\eta=\eta_n^{(0)} + \epsilon$ where $\epsilon \neq 0$ and separating out the contributions coming from the $D_n$–fold degenerate cases $n_j$ now gives that $$\sum_{i=1}^{D_n} (\eta_{n_i}^{(1)})^2
=
\sum_{i=1}^{D_n} \sum_{j=1,j\neq i}^{D_n}
t^{(2)}_{n_i n_j} + {\cal{O}}(\epsilon)
\quad .
\label{eta1-4}$$ Now the ‘multiplicative’ and ‘additive’ expressions for the partition function coincide up to ${\cal{O}}(e^2)$ everywhere in the complex $1/2 \kappa $ plane arbitrarily close to any pole and have the same poles and the same residues (up to ${\cal{O}}(e^2)$) at those poles. The partition function zeroes are $
\eta_{n_i} = \eta_n^{(0)} + \eta_{n_i}^{(1)} + \bar{\Delta}_n^{(2)}
$ where in the free field case the ${n_i}^{\rm{th}}$ zero is in the $n^{\rm{th}}$ degeneracy class. In terms of the Dirac and momenta labels, we can write the erstwhile 2–fold degenerate lowest zeroes as $$\eta_{\alpha}(\pm p_1,0) = \eta_\alpha^{(0)}(|p_1|,0)
\pm \eta_{\alpha}^{(1)}(|p_1|,0) + \bar{\Delta}_\alpha^{(2)}(|p_1|,0)
\quad .
\label{zeroes}$$ In this way, the shifts in the positions of the erstwhile two–fold degenerate zeroes are also determined to ${\cal{O}}(e^2)$ and the ${\cal{O}}(e^2)$–shift is the shift in the average position of the two zeroes while their relative separation is ${\cal{O}}(e)$. The calculation of Lee–Yang zeroes is now reduced to the calculation of the pure gauge expectation values $\Delta_\alpha (p)$ which we do in Feynman gauge.
#### Results and Conclusions
The main plot in Figure 1 is a finite–size scaling plot for the lowest zero for $\beta=10$ and $N=8,10,...,62$ coming from our weak coupling analysis (circles). The corresponding data from the numerical analysis of [@AzDi96] are also included (crosses). The two lines are the linear fit to the lowest zeroes for lattice sizes $16,20,24$ of [@AzDi96] and a fit to the second zero. These yield slopes $-1.5$ and $-1.4$ respectively, corresponding to $\nu \approx 2/3$. The insert contains a similar plot at $\beta = 5$ where the circles are our results for $N=2,4,...,42$ and the squares are the results of [@GaLa94] for the lowest zero for $N=2,4,8$. The line is the best linear fit to this latter data and yields $\nu$ compatable with $1$. Fitting to our small lattice data also gives $\nu \approx 1$.
Therefore, confining the finite–size scaling analysis to the range $N=2$ – $8$ yields a slope compatable with $\nu=1$ in agreement with [@GaLa94]. A corresponding analysis with $N=8$–$24$ gives a steeper slope, compatable with $\nu=2/3$ in agreement with [@AzDi96]. It is clear from the figure, however, that the curve does not in fact settle to a finite–size scaling line. Instead as $N$ increases, the lowest zero crosses the real $1/2\kappa$ axis. The second zero exhibits the same behaviour as demonstrated by the upper curve in Figure 1. The first two zeroes therefore fail to accumulate and do not contribute to critical behaviour [@LY; @Abe; @Sa94]. One interprets these zeroes as isolated singularities of thermodynamic functions, having measure zero in the thermodynamic limit.
It would be extremely difficult to detect the isolated nature of these Lee–Yang zeroes from numerical data alone. A standard numerical technique for determination of the phase diagram – assuming accumulation of zeroes – is to approximate the infinite volume critical point by the real part of the lowest zero (then a pseudocritical point) for some reasonably large lattice. Plotted against $\beta$ this appproximates the phase diagram. In Figure 2 we present such a plot for $N=24$ (circles) to compare with the results of [@AzDi96] (crosses) also at $N=24$. The phase diagram of [@HiLa98] for the Schwinger model with [*[two]{}*]{} fermion flavours coming from a separate PCAC based analysis is also included (squares) for comparison.
Now, the ${\cal{O}}(e)$ shift in the free field lowest zero is, in fact, $
\eta_{\alpha}^{(1)} (\pm 1/2,0)
=
\pm
( i / (Na)^d)
\sqrt{
\langle
|\phi_\mu(1,0)|
\rangle
}
=
\pm i /( 2 \sqrt{\beta} N|\sin{p_1a}|)
$, the leading finite–size scaling behaviour of which is $N$–independent. The crossing of the real axis by the first two zeroes is therefore due to the $N$ dependency of their ${\cal{O}}(e^2)$ average. In the four fold degenerate case, the leading $N$ behaviour of the zeroes is again determined by that of their average. The four fold degenerate ${\cal{O}}(e^2)$ averages of small momentum zeroes also cross the real axis and do not accumulate.
Although we cannot logically exclude the possibility that some zero which was far from the real axis in the free fermion case (or some combination of such zeroes) conspires to accumulate on the real $1/2\kappa$ axis in the usual way, such a situation seems unlikely. Nor can we exclude the possiblilty that just as the subdominant terms in the weak coupling expansion (the ${\cal{O}}(e^2)$ averages) correspond to the dominant $N$ behaviour, so too could higher order terms in the weak coupling expansion correspond to superdominant $N$ effects. This seems again unlikely as the average zeroes coming from our weak coupling expansion agree very well with those of [@AzDi96] (see [@us] for more details). Moreover, superdominant $N$ effects from subdominant weak coupling terms would spell disaster for lattice (and indeed continuum) perturbation theory and is an unlikely scenario in this – a superrenormalisable – model.
Although the possibility of existence of isolated singularities and non–accumulation of partition function zeroes has been known for a long time [@LY; @Abe; @Sa94], this is to our knowledge the first instance where such behaviour has been observed. In conclusion, in the free case, there is a phase transition precipitated by the accumulation of Lee–Yang zeroes on the real hopping parameter axis. In the weakly coupled regime at fixed $\beta$, this accumulation no longer occurs. Instead, the movement of zeroes for small and moderately sized lattices mimics phase transition like behaviour. As the lattice size becomes large, however, these zeroes move across the real axis, and do not give rise to a phase transition.
[**[Acknowledgements]{}**]{}: We would like to thank the following for discussions. N. Christ, R. Mawhinney, M.P. Fry, H. Gausterer, I. Hip, A.C. Irving, C.B. Lang, R. Teppner.
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[^1]: Supported by EU TMR Project No. ERBFMBI-CT96-1757 and a Forbairt European Presidency Post Doctoral Fellowship.
|
---
abstract: 'We present a theoretical framework well suited to analyze Ballistic Electron Emission Microscopy (BEEM) experiments. At low temperatures and low voltages, near the threshold value of the Schottky barrier, the BEEM current is dominated by the elastic component. Using a Keldysh Green’s functions method, we analyze the injected distribution of electrons and the subsequent propagation through the metal. Elastic scattering by the lattice results in the formation of focused beams and narrow lines in real space. To obtain the current injected in the semiconductor, we compute the current distribution in reciprocal space and, assuming energy and $k_{\parallel}$ conservation, we match states to the projected conduction band minima of the semiconductor. Our results show an important focalization of the injected electron beam and explain the similarity between BEEM currents for Au/Si(111) and Au/Si(100).'
address:
- ' Instituto de Ciencia de Materiales (CSIC), Cantoblanco, E-28049 Madrid (SPAIN) '
- ' Lehrstuhl fur Festkörperphysik, University of Erlangen-Nürnberg (Germany) '
- ' Dept. de Fisica Teorica de la Materia Condensada (UAM), Universidad Autonoma de Madrid, E-28049 Madrid (SPAIN) '
author:
- 'P.L. de Andres'
- 'K. Reuter'
- 'F.J. Garcia-Vidal, D. Sestovic and F. Flores'
title: ' A theoretical analysis of Ballistic Electron Emission Microscopy: k-space distributions and spectroscopy. '
---
0.0cm
PACS numbers: 61.16.Ch, 72.10.Bg, 73.20.At (Appl. Surf. Sci., in press)\
INTRODUCTION
============
Among the many exciting applications of the Scanning Tunneling Microscope (STM) one has recently become established as a technique in its own: Ballistic Electron Emission Microscopy (BEEM) [@kaiser; @mario]. In the standard version of this technique, the STM acts as a microscopic gun injecting a very narrow and coherent beam of electrons on a metalic layer deposited on a semiconductor. The electrons are subsequently propagated through the metalic layer and finally are detected in the semiconductor, passed the metal-semiconductor interfacial Schottky barrier. Both, spectral resolution and a lateral nanometric geometrical resolution, allow a detailed study of the buried interface that cannot be easily obtained by other techniques. Besides the interest of obtaining accurate information on the metal-semiconductor interface, other processes of technological importance have a considerable influence on the final BEEM current, and can be studied by applying this technique. A relevant example is the current attenuation length, determined by various inelastic processes: clearly these quantities are of paramount importance to design and operate very small devices.
The main obstacle on this road is just the complexity of the technique itself. The general case is that many different physical processes contribute in quite different ways to the BEEM current, resulting in a complex mixing of interfering effects that preclude an easy determination of each of them separately. The only way out is to construct a realistic theory that can reliably handle each factor, trying to avoid spurious correlations of fitted values. Unfortunately, this is often the case when a parametrized approach, like the popular E-space Monte-Carlo technique, is applied to obtain different values related between them, transfering uncertainties and errors from one place to the other without control. In particular, the elastic interaction of electrons with the lattice has been described as equivalent to a random walk, that cannot mimic the real interaction that is known to produce gaps in certain directions. Free propagation of carriers on distances of about $20$ Å in forbidden directions are not desirable, but possible in those Monte-Carlo simulations. Also, the simulations are very sensitive to the initial tunneling distribution, that is taken into account only through planar tunneling theory and WKB approximation.
As mentioned before, an important capability of the technique is its ability to measure different attenuation lengths related to inelastic interactions. The main inelastic processes are the electron-electron and the electron-phonon interaction. Lowering the temperature to $77$ K the electron-phonon interaction is already very low and one can ignore it to concentrate on the electron-electron interaction only. A very interesting result obtained using this approach is that the standard lifetime derived considering the dynamically screened Coulomb potential in an electron gas with a density related to gold cannot adequately explain the BEEM experiments[@bell111]. This result seems independent of the particular set of parameters used in the E-space Monte-Carlo simulation, and has been also confirmed by fully [*ab initio*]{} k-space Monte-Carlo[@ulrich]. Because of the importance of this problem, we shall discuss it further in the context of our [*ab initio*]{} Green’s function calculation.
In this paper we describe an [*ab initio*]{} Green’s function calculation that can be used to compared with experimental results. In the pure elastic case (ballistic current) this formalism does not use any free parameter, while to take into account the inelastic electron-electron interaction we make use of a mean free path parametrization proposed by Bell[@bell111], that let us to choose a single value to get good agreement with experiments. We notice that this is a rather similar situation to other fields, like the Low Energy Electron Diffraction, where first principles Green’s functions methods are supplemented with parametrized values of the imaginary part of the self-energy (typically a constant value) to include inelastic effects in the theory[@pendry]. This approach has been proven succesful in those fields, and we hope it will introduce for BEEM a new way to compare theory and experiment. The paper is organized as follows: in section II we describe a theoretical framework covering three important steps on a BEEM experiment: the initial tunneling injection, the propagation of electrons through the metalic layer, and the transmission of electrons at the two-dimensional plane defining the interface between the metal and the semiconductor. Finally, in section III, our theoretical results are compared with spectroscopic I(V) experimental data taken under conditions compatible with our main assumptions by other authors.
THEORY
======
Our theoretical approach is based on a hamiltonian describing separately the STM tip (T), the metal (M), and their interaction (I):
$$\hat H = \hat H_{T} + \hat H_{M} + \hat H_{I}$$
where the tip is defined by $\hat H_{T} = \sum \epsilon_{\alpha}
\hat n_{\alpha} + \sum \hat T_{\alpha \beta}
c_{\alpha}^{\dagger} c_{\beta}$, the metal is given by $\hat H_{M} = \sum \epsilon_{i}
\hat n_{i} + \sum \hat T_{i j}
c_{i}^{\dagger} c_{i}$, and the coupling term between them is $\hat H_{I} =
\sum \hat T_{\alpha j}
c_{\alpha}^{\dagger} c_{i}$ (where greek indexes, $\alpha, \beta$, describe orbitals on the tip and latin indexes, $i, j$, refer to the metal). The interaction hamiltonian has been written in terms of a hopping matrix that couples the atomic orbitals in the tip with the atomic orbitals in the metal. Because for BEEM work the tip-sample distance is usually large, we make the approximation of considering only tunneling between s-orbitals. The hamiltonian describing the metal has been written on a tight-binding approximation with parameters given by Papaconstantopoulos[@constan] for gold.
Currents between the tip and the sample and within the metal base are computed applying a Keldysh formalism[@keldysh]:
$$J_{i j} =
\int d \omega Tr \{ \hat T_{i j}
(\hat G_{ij}^{+-} - \hat G_{ji}^{+-}) \}$$
that is well suited for a non-equilibrium problem like the tunneling one, but also allow to write an elegant expresion for more standard scenarios like the propagation through the metal layer. The central objects in this formalism, $\hat G_{ij}^{+-}$, are obtained from a Dyson-like equation through the retarded and advanced Green functions of the uncoupled parts of the system (the tip and the sample), $\hat g^{R}$ and $\hat g^{A}$, and the hopping matrix $\hat T$[@prl; @phscr]. In that context, it has been shown previously how the current between two atoms $i, j$ in the metal can be obtained by:
$$J_{ij} = {2 e \over \pi \hbar} \Re \int_{-\infty}^{\infty}
d \omega \lbrack f_{T}(\omega) - f_{S}(\omega) \rbrack \times
Tr \sum_{m \alpha \beta n} \lbrack
\hat T_{ij} \hat g_{jm}^{R} \hat T_{m \alpha}
( \hat g_{\alpha \beta}^{A} -
\hat g_{\alpha \beta}^{R} )
\hat T_{\beta n} \hat g_{ni}^{A}
\rbrack$$
where $T_{ij}$ is the hopping matrix between atoms $i,j$, $g_{jm}^{r}$ and $g_{mj}^{a}$ are the retarded and advanced Green functions linking sites $j$ and $m$, $( \hat g_{\alpha \beta}^{A} -
\hat g_{\alpha \beta}^{R} )$ is related to the density of states matrix on the active atom at the tip (from now on assumed for simplicity to be $0$), and the trace implies a summation over the orbitals forming the chosen basis. Eq. 3 describes the coherent propagation of electrons between the surface sites ($m$ or $n$), and the atoms inside the crystal ($i$ and $j$), and ingredient that we find of paramount importance to describe appropriately the BEEM current in the metal.
In this formalism the initial tunneling current is obtained as the current between the last atom in the tip (0) and all the atoms active for tunneling in the metal (m). In the same Keldysh formalism, we can write:
$$J_{T}= \int d \omega Tr
\sum_{m}[ \hat{T}_{0m} \hat{G}^{+,-}_{m0}(\omega) -
\hat{T}_{m0} \hat{G}^{+,-}_{0m}(\omega) ]$$
that for long tip-sample distances and the simpler case where only one atom is active for tunneling in the metal yields[@alvaro]:
$$J_{T} = \frac{4 \pi e}{\hbar}
\int_{-\infty}^{\infty} Tr [ \hat{T}_{01}
\hat{\rho}_{11}(\omega) \hat{T}_{10} \hat{\rho}_{00}(\omega)]
[f_{T}(\omega)-f_{S}(\omega)] d \omega$$
where the further simplification of allowing tunneling to only one atom in the metal (m=1) has been made. This formula shows the dependence of the tunneling current with the hopping matrix linking the tip to the surface, $T_{01}$, the density of states at the two active sites, $\rho_{00}$ and $\rho_{11}$, and the different Fermi distribution functions. This approach can be used to compute topographic images of a given surface (either at constant intensity or constant height) and already has been succesfully used to compare with experiments[@fernando].
The current distribution in reciprocal space is necessary to match wavefunctions across the metal-semiconductor interface in order to calculate the BEEM current in the semiconductor. It is possible to show that in k-space the current per energy unit between in the metal, is given by an expression formally identical to Eq. 3.:
$$J_{ab}(k_{\parallel}) =
{2 e \over \pi \hbar} \Re
Tr \lbrack
\hat T_{ab} (k_{\parallel})
\hat g_{b1}^{R} (k_{\parallel}) \hat T_{10}
\hat \rho_{00}
\hat T_{01} \hat g_{1a}^{A} (k_{\parallel})
\rbrack$$
where the various quantities are two-dimensional Fourier transforms of the respective objects appearing in the real space current between of the metalic lattice $i, j$:
$$\hat T_{i,j} (\vec r_{\parallel}) =
\sum_{k_{\parallel}}
e^{- i \vec k_{\parallel} \vec r_{\parallel}}
\hat T_{a,b} (\vec k_{\parallel})$$
The summation is perfomed over a set of special points covering the two-dimensional Brillouin zone[@rafa].
A proper calculation of the quantum mechanical transmission coefficient, $T$, across the interface can be done by projecting and matching metal and semiconductor bulk states into the interface[@stiles]. This procedure yields, as expected, a square root variation of that coefficient with the energy (measured w.r.t. the semiconductor conduction band minimum). Usually, a much easier model is applied in the literature where the transmission coefficient over a step-like potential barrier (taken as the Schottky barrier) is considered: this brings the correct behaviour with energy, but because this is a one-dimensional problem the variation of $T$ with $k_{\parallel}$ is not well represented. An intermediate level of sophistication, improving significantly the model, would be to describe the semiconductor within the Jones zone approximation and to match the wavefunctions logarithmic derivatives given a particular orientation for the semiconductor. In particular, this approach is quite reasonable around the semiconductor energy gap and can describe satisfactorily the region close to the conduction band minima. We have followed this approach using a Surface Green’s function matching formalism[@elices; @moliner] (equivalent to the mentioned wave function matching), considering the neighbourghood of the projected $\Gamma-X$ direction ($\overline{M}$) in the Si(111) and Si(100) surfaces: these are the regions where the conduction band minima of Si are projected. In the parallel direction, where the kinematic restriction of conserving $k_{\parallel}$ and having enough energy to be injected in the semiconductor is applied, a simple parabolic approximation with effective masses taken from the literature is used. This defines an ellipsoidal region around the minimum of the semiconductor conduction band where the two-dimensional transmission coefficient described above changes continously from its value at the center of the ellipse (the origin where the $\vec k_{\parallel}$ is measured in the semiconductor) to zero at the boundaries of that region. This approach increases transmission at the interace by values between $\frac{15}{100}$ and $\frac{40}{100}$, depending on the energy.
For the particular case of a (111) orientation and in the neighbourhood of the $\overline{M}$ point we have[@elices]:
$$\gamma = {(1-\frac{2}{3} h L)(1-\frac{5}{6} h L) \over
(1+\frac{2}{3} h L)(1+\frac{6}{6} h L) }$$
where $h={ 2 \pi \sqrt{3}\over a}$, and ${1 \over L} = \sqrt{2 (E- {\kappa^{2} \over 2}}$.
defining, $$a = {e-{\kappa^{2} \over 2} - v +
\sqrt{(e-{\kappa^{2} \over 2})^{2} - v^{2}} \over v}$$ with $v \approx 2 eV$ to represent the silicon gap. Finally, the reflection amplitude is given by:
$$r = {
{1+\gamma \over 1-\gamma} -
{a-1 \over a+1}
\over
{1+\gamma \over 1-\gamma} +
{a-1 \over a+1}
}$$
RESULTS
=======
k-space and r-space distribution.
---------------------------------
In this section, our formalism is applied to the case of Au grown on Si(111) or Si(100) orientations. From previous structural work performed with LEED, Auger and STM[@structure], we assume that Au grows on (111) crystalline directions, except for the first few layers near the interface that may present some disorder.
First of all, we observe the formation of narrow beams and lines in real space (e.g. see Figs. 1 and 2). This effect was previously predicted on the basis of a semi-classical calculation of the Green’s functions necessary to compute Eq. 3[@prl], and it is now confirmed working with a more accurate, full quantum mechanical, approximation based on a decimation technique[@guinea]. The same effect has also been observed when the propagation of electrons on (100) directions is considered[@phscr]. In fact, within our formalism, it is possible to follow the propagation through the material layer-by-layer. In this way, we can observe the gradual construction of Bloch states inside the solid, seen in particular by the formation of gaps in characteristic directions like the (111) or (100) depending on the energy. This is the case when Figs. 1 and 2 are compared: close to the surface (1) electrons spread in all directions (including the (111)), but far away (2) the propagation in that direction is inhibited.
Secondly, we study the current distribution in k-space (e.g. see Fig. 3). In the purely elastic case, the energy is a real quantity and the wavefunction does not decay. However, it is usual in any Green’s function formalism to add a very small imaginary part to the energy to create a region in the complex plane free of poles to ensure the proper analytic behaviour. In this work, we discuss the case of a very small imaginary part added to the energy ($\eta=0.001$ eV), while we leave a detailed discussion of a finite $\eta$, representing electron-electron inelastic interaction, for a forthcoming paper[@karsten]. A typical current distribution under these conditions can be seen in Fig. 3. This distribution does not present noticeable changes when observed at different layers. The symmetry found is six-fold, also obtained on other similar quantum-mechanical calculations[@stiles]. This result is expected because these currents reflect the projected density of states on a given layer. It is interesting to notice, however, the different symmetry found in real space (three-fold), reflecting the symmetry of the whole lattice.
Spectroscopy: Au on Si(111) and Si(100).
----------------------------------------
An interesting feature of BEEM is its ability to bring spectroscopic information of the interface. A very simple model shows how the spectroscopy is an integral on energies of different factors[@mario]. Therefore, the different dependences with energy of the these factors can be checked against the experiment through comparison of I(V) curves.
We apply our formalism to compute the I(V) curves of the Au/Si(111) ($75 \AA$) and Au/Si(100) ($100 \AA$) taken at low temperature (77 K)[@bell111; @bell100]. These are favourable cases to be compared with a pure elastic theory, and in some way mark the correctness of the name ballistic given to the technique: from a direct comparison between experiment and theory (e.g. see Fig. 4) we conclude that the ballistic assumption breaks down at voltages higher than $1.3$ eV. The theory has been calculated using k-space distributions like the ones shown in Fig. 3, with a very small $\eta$ (0.001 eV), and assuming that the electrons are injected only in the first pass. The introduction of an electron-electron inelastic interaction has the main effect of taking out intensity due to the important loss involved (typically half of the energy is lost). In Fig. 4 we also show the effect of introducing such a finite $\eta$ to match the ballistic intensity to the experiment. Our best fit has been obtained using the energy dependence proposed by Bell[@bell111] and for a mean free path of $155$ Åat $1$ eV. While not a particular effort to optimize these values has been made, both $D=75$ Å and $D=300$ Å can be explained at the same time using this approach. It is worth commenting that the standard mean free paths values derived from the gas electron theory[@quinn] cannot describe correctly the data because apparently underestimate the electron-electron interaction resulting in longer mean free paths. Finally we should comment on the role of the multiple reflections inside the metalic layer. As mentioned above, the ballistic result is obtained from direct injection, but in a more appropriated description multiple reflections must be taken into account, as shown by Bell[@bell111]. Because mean free paths are longer at lower energies their effect is more important near the threshold, but on average for a width like $D=75$ Åthree to four reflections are enough. This brings about the model for reflections at the surface and at the interface. We have considered perfectly specular reflection and completely diffuse reflection: differences are not dramatic because of the focusing effects introduced by the lattice, but the diffuse reflection model produces the best agreement with the experiments.
In the past, it has been difficult to understand why Au/Si(111) and Au/Si(100) produced so similar BEEM currents in spite of their different projected band structure. This is specially difficult without realizing that the elastic interaction of electrons with the lattice results in k-space current distributions that accumulates around the two-dimensional Brillouin zone edge. To test the similarity between the two interfaces we compare in Fig 5 our results for the two orientations with the experiments. The ratio between the intensity for the 111 orientation to the intensity for the 100 orientation is plotted against the tip voltage. The same trend is clearly observed in theory and experiment, and the main discrepancy (about $\frac{30}{100}$) is seen for low voltages, where the different Schottky barrier for the two orientations produce a bigger uncertainty on the comparison. This clearly shows how the different projected band structure of the two orientations for silicon yields similar BEEM-currents without having to resort to non-conserving parallel momentum due to scattering processes of the electrons at the interface[@ludeke].
CONCLUSIONS
===========
We present an [*ab initio*]{} theory based on Green’s functions techniques and a tight-binding hamiltonian to compute the current distributions for a BEEM experiment in real and reciprocal space. Focusing of the propagating electrons is determined by the elastic interaction with the periodic lattice. The resulting narrow lines and beams are of the order of 2-3 atomic interspacing, explaining the observed nanometric resolution of the technique. In reciprocal space we observe a six-fold distribution that determines the pure ballistic current. Comparison of experimental results taken on thin layers and at low temperature with the elastic theory ($\eta=0.001$ eV) shows the influence of inelastic processes in the high energy range ($\approx 1.5$ eV). Finally, an inelastic electron-electron interaction is taken into account resulting in a good agreement between the theory and the experimental data.
ACKNOWLEDGMENTS
===============
We acknowledge financial support from the Spanish CICYT under contracts number PB94-53 and PB92-0168C. K.R. is grateful for financial support from SFB292 (Germany). We are grateful with Prof. P. Kocevar and Dr. U. Hohenester for many interesting discussions and their effort to build a [*first-principles*]{} k-space Monte-Carlo approach to the BEEM problem, and with Prof. K. Heinz for his continued interest.
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|
---
abstract: |
We study weak amenability for locally compact quantum groups in the sense of Kustermans and Vaes. In particular, we focus on non-discrete examples. We prove that a coamenable quantum group is weakly amenable if there exists a net of positive, scaling invariant elements in the Fourier algebra $A({\mathbb{G}})$ whose representing multipliers form an approximate identity in $C_0({\mathbb{G}})$ that is bounded in the ${M_0A}({\mathbb{G}})$ norm; the bound being an upper estimate for the associated Cowling-Haagerup constant.
As an application, we find the appropriate approximation properties of the extended quantum $SU(1,1)$ group and its dual. That is, we prove that it is weakly amenable and coamenable. Furthermore, it has the Haagerup property in the quantum group sense, introduced by Daws, Fima, Skalski and White.
address: 'M. Caspers, Laboratoire de Mathématiques, Université de Franche-Comté, 16 Route de Gray, 25030 Besançon, France'
author:
- Martijn Caspers
date: |
.\
The author was supported by the ANR project: ANR-2011-BS01-008-01.
title: 'Weak amenability of locally compact quantum groups and approximation properties of extended quantum $SU(1,1)$'
---
Introduction
============
Locally compact quantum groups have been introduced by Kustermans and Vaes in their papers [@KusVae], [@KusVaeII]. They form a category larger than locally compact groups, admitting a full Pontrjagin duality theorem.
A class of examples occurs as deformations of the algebra of continuous or measurable functions on a locally compact group, for example the deformations of simple compact Lie groups. Furthermore, suitable deformations of $E(2)$ and $SU(1,1)$ have been constructed.
These examples give rise to the question which approximation properties these quantum groups and their duals have. For the deformation of $E(2)$ (see [@Wor], [@Jacobs]) it is known that it is both [*amenable*]{} and [*coamenable*]{}. Here, amenability is the (or at least one of the) quantum generalization(s) of group amenability, whereas coamenability is the dual notion. Deformations of compact simple Lie groups are coamenable (c.f. [@Banica Corollary 6.2]) and trivially amenable.
Recall that [*coamenability*]{} asserts the existence of a bounded approximate identity in the convolution algebra $L^1({\mathbb{G}})$ of a locally compact quantum group ${\mathbb{G}}$. [*Amenability*]{} can for example be characterized by the fact that the universal and reduced dual quantum groups $C^\ast_u({\mathbb{G}})$ and $C^\ast_r({\mathbb{G}})$ are equal.
Amenability and coamenability of a quantum group are dual to each other, in the sense that for a compact quantum group it is known that it is coamenable if and only if the Pontrjagin dual quantum group is amenable, see the work of Ruan [@RuanAm] and Tomatsu [@Tomatsu]. The question whether or not this generalizes to arbitrary quantum groups remains open.
For groups subsequently weaker approximation properties than amenability have been introduced, in particular [*weak amenability*]{} [@CanHaa]. These notions have been generalized to quantum groups of Kac type by Kraus and Ruan [@Ruan]. In particular, equivalent intrinsic approximation properties of the C$^\ast$-algebra and von Neumann algebra of a discrete Kac algebra have been found. For example weak amenability of $\mathbb{G}$ corresponds to the completely bounded approximation property (CBAP) of $C^\ast_r(\mathbb{G})$.
Let us mention that in [@Freslon] Freslon proved that the free orthogonal and free unitary quantum groups are weakly amenable. The result was extended to the non-Kac deformations in [@ComFreYam] in terms of the CBAP. The current paper deals with a weak amenability result for non-discrete quantum groups, which is of different flavor, basically since it does not have an interpretation in terms of the CBAP.
Various equivalent notions of the Haagerup property for locally compact quantum groups were recently investigated by Daws, Fima, Skalski and White [@DawFimSkaWhi]. Examples so far come from Brannan’s result [@Bran] on free orthogonal and free unitary quantum groups and Lemeux [@Lem] for quantum reflexion groups. Beyond the discrete case, the only known (non-classical) examples are amenable or follow from coamenability of the dual.
The ${SU_q(1,1)_{{\rm ext}}}$ group was first established as a proper von Neumann algebraic quantum group by Koelink and Kustermans [@KoeKus]. One of the main difficulties was to prove the coassociativity of the comultiplication. More recently a novel way of obtaining ${SU_q(1,1)_{{\rm ext}}}$ was found by De Commer [@ComII] (see also [@ComI]), avoiding the proof of the coassociativity. In [@GroKoeKus] Groenevelt, Koelink and Kustermans obtain the Plancherel decomposition of the multiplicative unitary of ${SU_q(1,1)_{{\rm ext}}}$. Part of this paper is based on this achievement. We expand below in more detail.
Let us mention that the construction of non-compact operator algebraic quantum groups is one of the major open questions. However, there is reasonable hope that the techniques of [@ComI] allow a passage to a larger class of examples. In fact, in [@ComIII] it was shown that also a version of $E(2)$ can be recovered.
We summarize the main results of this paper. Section \[Sect=CompactaApproximation\] provides a sufficient condition for a quantum group to be weakly amenable. Explaining the notation and definitions in the subsequent sections, we state:
[**Theorem 1**]{}. Let ${\mathbb{G}}$ be a coamenable locally compact quantum group. Suppose that there exists a net $\{ a_i \}$ of positive elements in the Fourier algebra $A({\mathbb{G}})$, whose representing elements in $C_0({\mathbb{G}})$ are invariant under the scaling group and such that they form an approximate identity for $C_0({\mathbb{G}})$ with $\Vert a_i \Vert_{{M_0A}({\mathbb{G}})}$ bounded. Then, ${\mathbb{G}}$ is weakly amenable.
The theorem is based on the arguments of De Canniere and Haagerup [@CanHaa] and is typically applicable if the trivial corepresentation lies nicely in the representation spectrum of a quantum group. Compared to [@CanHaa], besides technical difficulties, we encounter two new phenomena in the quantum case. It appears that for quantum groups [*coamenability*]{} plays an important role. Also, it turns out that we need to work with [*modular* ]{} multipliers (we state the definition later). Recall that classical groups are always coamenable and the modular assumption is trivially satisfied.
As an example, we find the appropriate approximation properties of ${SU_q(1,1)_{{\rm ext}}}$ and its dual. Our proofs rely on $q$-analysis of little $q$-Jacobi functions that are special cases of $_2 \varphi _1$-hypergeometric series. As a first result, we find:
[**Theorem A.**]{} ${SU_q(1,1)_{{\rm ext}}}$ is coamenable.
Let us mention that coamenability was recently used in relation with idempotent states on locally compact quantum groups [@PekSal]. It turns out that ${SU_q(1,1)_{{\rm ext}}}$ falls within the category of examples, answering a question in [@PekSal Section 3].
As a consequence of [**Theorem A**]{}, we are able to prove the following.
[**Theorem B.**]{} ${SU_q(1,1)_{{\rm ext}}}$ is weakly amenable with Cowling-Haagerup constant 1.
The proof of [**Theorem B**]{} relies on computations based on the Plancherel decomposition [@GroKoeKus] and then follows the proof of Haagerup and De Canniere [@CanHaa] in order to apply [**Theorem 1**]{}. As in the classical case (see also [@CanHaa Remark 3.8]), we believe that our proofs are adaptable to deformations of Lie groups of which the identity operator lies nicely in the spectrum of corepresentations. Finally, we find that:
[**Theorem C.**]{} ${SU_q(1,1)_{{\rm ext}}}$ has the Haagerup property.
These are the first (genuine) examples of non-discrete, non-amenable quantum groups that: are weakly amenable/have the Haagerup property. Also the dual of ${SU_q(1,1)_{{\rm ext}}}$ has the Haagerup property, as will follow from [**Theorem A**]{}.
The structure of this paper is as follows. Section \[Sect=Lcqg\] recalls the definition of a locally compact quantum group. In Section \[Sect=CompactaApproximation\] we recall the definition of amenability and prove [**Theorem 1**]{}. In Section \[Sect=SUone\] we recall the necessary preliminaries on ${SU_q(1,1)_{{\rm ext}}}$. Section \[Sect=Coamenability\] proves [**Theorem A**]{}. Sections \[Sect=Spherical\] and \[Sect=WeakAmenability\] prove [**Theorem B**]{}. Section \[Sect=Haagerup\] proves [**Theorem C**]{}. In the Appendix, we prove certain density properties and we prove convergence properties of basic hypergeometric series.
General notation {#general-notation .unnumbered}
----------------
$\mathbb{N}$ denotes the natural numbers excluding 0. For weight theory we refer to [@TakII]. If $\varphi$ is a normal, semi-finite, faithful weight on a von Neumann algebra $L^\infty({\mathbb{G}})$, then we denote $J, \nabla, \sigma$ for the modular conjugation, modular operator and modular automorphism group. We use the formal notation $L^2({\mathbb{G}})$ for the GNS-space and $L^2({\mathbb{G}}) \cap L^\infty({\mathbb{G}})$ for the elements $x \in L^\infty({\mathbb{G}})$ for which $\varphi(x^\ast x) < \infty$. We use $L^1({\mathbb{G}})$ for the predual of $L^\infty({\mathbb{G}})$ and $L^1({\mathbb{G}})^+$ for its positive part. We use the Tomita algebra, $$\begin{split}
{\mathcal{T}}_\varphi = \{ x \in L^\infty({\mathbb{G}}) \mid &\: x \textrm{ is analytic for } \sigma \textrm{ and } \\
&\: \sigma_z(x) \textrm{ and } \sigma_z(x)^\ast \textrm{ are in } L^2({\mathbb{G}}) \cap L^\infty({\mathbb{G}}) \}.
\end{split}$$ We freely use Tomita-Takesaki theory, but let us at least recall the following standard facts [@TakII]. Let $\varphi$ be a normal, semi-finite, faithful weight with GNS-representation $(L^2({\mathbb{G}}), \pi, \Lambda)$. The domain of $\Lambda$ equals $L^2({\mathbb{G}}) \cap L^\infty({\mathbb{G}})$ and $\Lambda$ is $\sigma$-weakly/weak (or equivalently $\sigma$-strong-$\ast$/norm) closed. ${\mathcal{T}}_\varphi$ is a $\sigma$-weak/norm core for $\Lambda$.
For $c \in L^\infty({\mathbb{G}})$ and $\omega \in L^1({\mathbb{G}})$ we set $(c \cdot \omega)(x) = \omega(xc), x\in L^\infty({\mathbb{G}})$. Also, for $a,b \in L^2({\mathbb{G}}) \cap L^\infty({\mathbb{G}})$, we set $(a\varphi b^\ast)(x) = \varphi(b^\ast xa), x \in L^\infty({\mathbb{G}})$. For $\xi, \eta \in L^2({\mathbb{G}})$ we set $\omega_{\xi, \eta}(x) = \langle x \xi, \eta \rangle, x \in L^\infty({\mathbb{G}})$ and $\omega_\xi = \omega_{\xi, \xi}$.
Tensor products are always von Neumann algebraic, unless clearly stated otherwise.
Locally compact quantum groups {#Sect=Lcqg}
==============================
For the Kustermans-Vaes definition of a locally compact quantum group, see [@KusVae] and [@KusVaeII]. We would recommend [@DaeleLcqg] as an introduction, giving an almost self-contained approach to the theory. For a broader introduction we refer to [@Tim].
Throughout the paper $\mathbb{G}$ is a locally compact quantum group. It consists of a von Neumann algebra $L^\infty({\mathbb{G}})$ and comultiplication $\Delta: L^\infty({\mathbb{G}}) \rightarrow L^\infty({\mathbb{G}}) \otimes L^\infty({\mathbb{G}})$ which is implemented by the left multiplicative unitary $W \in B(L^2({\mathbb{G}}) \otimes L^2({\mathbb{G}}))$, $$\Delta(x) = W^\ast (1 \otimes x) W, \qquad x \in L^\infty({\mathbb{G}}).$$ Moreover, there exist normal, semi-finite, faithful Haar weights $\varphi$ and $\psi$ on $L^\infty({\mathbb{G}})$ that satisfy the left and right invariance axioms, $$({\textrm{id}}\otimes \varphi )\Delta(x) = \varphi(x) \: 1_{L^\infty({\mathbb{G}})}, \qquad
(\psi \otimes {\textrm{id}})\Delta(x) = \psi(x) \: 1_{L^\infty({\mathbb{G}})}, \qquad x \in L^\infty({\mathbb{G}})^+.$$ We have a GNS-construction $(L^2({\mathbb{G}}), \pi, \Lambda)$ with respect to the left Haar weight. We omit $\pi$ in the notation. The left and right Haar weight are related by the modular element $\delta$, affiliated with $L^\infty({\mathbb{G}})$, by the formal identification $\psi(\: \cdot \:) = \varphi(\delta^{\frac{1}{2}} \: \cdot \: \delta^{\frac{1}{2}} )$, see [@VaeRad]. The triple $(L^2({\mathbb{G}}), {\textrm{id}}, \Gamma)$ denotes the GNS-construction for $\psi$.
The quantum group ${\mathbb{G}}$ comes with an unbounded antipode $S: ( {\rm Dom}(S) \subseteq L^\infty({\mathbb{G}}) ) \rightarrow L^\infty({\mathbb{G}})$ that has a unique polar decomposition $S = R \circ \tau_{-i/2}$. Here, $R: L^\infty({\mathbb{G}}) \rightarrow L^\infty({\mathbb{G}})$ is the unitary antipode and $\tau: \mathbb{R} \rightarrow {\rm Aut}(L^\infty({\mathbb{G}}))$ is the scaling group. We define [@KusUniv Section 4], $$L^1({\mathbb{G}})^\sharp = \left\{ \omega \in L^1({\mathbb{G}}) \mid \exists \theta \in L^1({\mathbb{G}}) \textrm{ s.t. } (\theta \otimes {\textrm{id}})(W) = (\omega \otimes {\textrm{id}})(W)^\ast \right\}.$$ In case $\omega \in L^1({\mathbb{G}})^\sharp$, the corresponding $\theta \in L^1({\mathbb{G}})$ satisfies $\theta(x) = \overline{\omega( S(x)^\ast )}$ for every $x \in {\rm Dom}(S)$. Conversely, if ${\rm Dom}(S) \rightarrow \mathbb{C}: x \mapsto \overline{\omega( S(x)^\ast )}$ extends boundedly to $L^\infty({\mathbb{G}})$, then $\omega \in L^1({\mathbb{G}})^\sharp$ and we denote this extension by $\omega^\ast$. The scaling constant $\nu\in \mathbb{R}^+$ is then defined by $\varphi \circ \tau_t = \nu^{-t} \varphi$.
There exists a Pontrjagin dual quantum group $\hat{{\mathbb{G}}}$ and all its associated objects will be equipped with a hat. The left multiplicative unitary $W \in L^\infty({\mathbb{G}}) \otimes L^\infty(\hat{{\mathbb{G}}})$ fully determines ${\mathbb{G}}$ as well as $\hat{{\mathbb{G}}}$. We have $\hat{W} = \Sigma W^\ast \Sigma$, where $\Sigma: L^2({\mathbb{G}}) \otimes L^2({\mathbb{G}}) \rightarrow L^2({\mathbb{G}}) \otimes L^2({\mathbb{G}})$ is the flip on the Hilbert space level. $L^\infty({\mathbb{G}})$ is the $\sigma$-strong-$\ast$ closure of, $$\left\{ ({\textrm{id}}\otimes \omega)(W) \mid \omega \in
B(L^2({\mathbb{G}}))_\ast \right\},$$ and $L^\infty(\hat{{\mathbb{G}}})$ is the $\sigma$-strong-$\ast$ closure of, $$\left\{ (\omega \otimes {\textrm{id}}) (W) \mid \omega \in
B(L^2({\mathbb{G}}))_\ast \right\}.$$ For $\omega \in
L^1({\mathbb{G}}), \theta \in L^1(\hat{{\mathbb{G}}})$, we use the standard notation, $$\lambda(\omega) = (\omega
\otimes {\textrm{id}}) (W), \qquad \hat{\lambda}(\theta) = ({\textrm{id}}\otimes \theta)(W^\ast).$$
The dual left Haar weight $\hat{\varphi}$ on $L^\infty(\hat{{\mathbb{G}}})$ is constructed as follows. We let $\mathcal{I}$ be the set of $\omega \in L^1({\mathbb{G}})$, such that $\Lambda(x) \mapsto \omega(x^\ast),
x \in L^2({\mathbb{G}}) \cap L^\infty({\mathbb{G}})$ extends to a bounded functional on $L^2({\mathbb{G}})$. By the Riesz theorem, for every $\omega \in \mathcal{I}$, there is a unique vector denoted by $\xi(\omega) \in L^2({\mathbb{G}})$ such that, $$\omega(x^\ast) = \langle \xi(\omega), \Lambda(x) \rangle, \quad x \in
L^2({\mathbb{G}}) \cap L^\infty({\mathbb{G}}).$$ The dual left Haar weight $\hat{\varphi}$ is defined to be the unique normal, semi-finite, faithful weight on $L^\infty(\hat{{\mathbb{G}}})$, with GNS-construction $(L^2({\mathbb{G}}), \iota, \hat{\Lambda})$ such that $\lambda(\mathcal{I})$ is a $\sigma$-strong-$\ast$/norm core for $\hat{\Lambda}$ and $\hat{\Lambda}(\lambda(\omega)) = \xi(\omega),
\omega \in \mathcal{I}$.
We mention that one can also construct a dual right Haar weight, but we do not use it in this paper.
There is a collection of relations between the objects we introduced so far and they can all be found in [@KusVaeII]. We record them here. $P$ is defined by $P^{it} = \nu^{\frac{t}{2}} \Lambda(\tau_t(x)) $. $$\begin{array}{lll}
\varphi \circ R = \psi & \nabla^{is } \delta^{it } = \nu^{ist} \delta^{it } \nabla^{is } & \tau_t(x) = \hat{\nabla}^{it} x \hat{\nabla}^{-it} \\
\varphi \circ \tau_t = \nu^{-t} \varphi & \hat{\nabla}^{is } \delta^{it } = \delta^{it } \hat{\nabla}^{is } &
R(x) = \hat{J} x^\ast \hat{J} \\
\psi \circ \tau_t = \nu^{-t} \psi & \hat{\nabla}^{it} \nabla^{is} = \nu^{ist} \nabla^{is} \hat{\nabla}^{it} & \nabla^{it} = \hat{P}^{it} J \hat{\delta}^{it} J \\
& \hat{J} \delta \hat{J} = \delta^{-1} &
\end{array}$$ Furthermore, $$(\tau_t \otimes \hat{\tau}_t)(W) = W, \qquad (R \otimes \hat{R})(W) = W^\ast.$$
Underlying ${\mathbb{G}}$ there exist reduced C$^\ast$-algebraic quantum groups of which the C$^\ast$-algebras are defined by, $$C_0({\mathbb{G}}) = \textrm{clo} \left\{ ({\textrm{id}}\otimes \omega)(W) \mid \omega \in B(L^2({\mathbb{G}}))_\ast
\right\},$$ and $$C^\ast_r({\mathbb{G}}) = \textrm{ clo} \left\{ (\omega
\otimes {\textrm{id}})(W) \mid \omega \in B(L^2({\mathbb{G}}))_\ast
\right\},$$ where the closures are norm closures in $B(L^2({\mathbb{G}}))$. It is worth mentioning that in this paper, when working with $C_0({\mathbb{G}})$, we always add the assumption that it is coamenable (see Lemma \[Lem=CoamenabilityEquivalence\] for the definition of coamenability). Hence, there is no distinction between reduced and universal in our notation for the C$^\ast$-algebra of ${\mathbb{G}}$.
There is a universal C$^\ast$-algebraic quantum group $C^\ast_u({\mathbb{G}})$ whose C$^\ast$-algebra is the univeral completion of $L^1({\mathbb{G}})^\sharp$ equipped with its Banach $\ast$-algebra structure, [@KusUniv]. We use $\mathcal{V}$ for Kusterman’s universal multiplicative unitary. For coamenable quantum groups it is contained in $\mathcal{M}(C_0({\mathbb{G}}) \otimes C_u^\ast({\mathbb{G}}))$ (with $\mathcal{M}$ the multiplier algebra and $\otimes$ the minimal tensor product). We use the universal objects briefly in the proof of Theorem \[Thm=Haagerup\], explaining its universal properties further.
Corepresentations {#corepresentations .unnumbered}
-----------------
An operator $U \in L^\infty({\mathbb{G}}) \otimes B({\mathcal{H}}_U)$ is called a [*corepresentation*]{} if $(\Delta \otimes {\textrm{id}})(U) = U_{13} U_{23}$. We call a corepresentation $U$ unitary (resp. invertible) if $U$ is unitary (resp. invertible) as an operator. In case ${\mathbb{G}}= (L^\infty(G), \Delta_G)$ is a commutative quantum group, then every (bounded) corepresentation is automatically invertible. For arbitrary quantum groups this is more subtle, c.f. [@BraDawSam]. Recall that $W$ is a corepresentation that is moreover unitary.
The Fourier algebra and multipliers {#the-fourier-algebra-and-multipliers .unnumbered}
-----------------------------------
For operator spaces, we refer to [@EffRua]. For a locally compact quantum group ${\mathbb{G}}$, the predual $L^1({\mathbb{G}})$ carries a natural operator space structure. Pulling back the comultiplication to $L^1({\mathbb{G}})$ yields a [*convolution product*]{} $\Delta_\ast: L^1({\mathbb{G}}) \hat{\otimes} L^1({\mathbb{G}})\rightarrow L^1({\mathbb{G}})$ which will usually be denoted by $\ast$. In this way $L^1(\hat{{\mathbb{G}}})$ becomes a completely contractive Banach algebra. We will use $A({\mathbb{G}}) = \hat{\lambda}(L^1(\hat{{\mathbb{G}}}))$. And $\Vert \hat{\lambda}(\omega) \Vert_{A({\mathbb{G}})} := \Vert \omega \Vert_{L^1(\hat{{\mathbb{G}}})}$ ($\hat{\lambda}$ is injective). $A({\mathbb{G}})^+$ is the subset that corresponds to $L^1(\hat{{\mathbb{G}}})^+$.
An operator $b \in L^\infty({\mathbb{G}})$ is called a [*left Fourier multiplier*]{} if for every $\omega \in L^1(\hat{{\mathbb{G}}})$ there exists a $\theta \in L^1(\hat{{\mathbb{G}}})$ such that $\hat{\lambda}(\theta) = b \hat{\lambda}(\omega)$. In this case left multiplication with $b$ defines a bounded map $A({\mathbb{G}}) \rightarrow A({\mathbb{G}})$ by the closed graph theorem. In case this map is completely bounded, we write $b \in {M_0A}({\mathbb{G}})$, i.e. $b$ is a completely bounded left Fourier multiplier. In this paper all multipliers will be left multipliers and therefore we will drop the indication ‘[*left*]{}’ in our terminology. For $b \in {M_0A}({\mathbb{G}})$, we have, $$\label{Eqn=MultiplierLinftyNorm}
\Vert b \Vert_{L^\infty({\mathbb{G}})} \leq \Vert b \Vert_{{M_0A}({\mathbb{G}})}.$$ It is also useful to remark that $A({\mathbb{G}}) \subseteq {M_0A}({\mathbb{G}})$ and for $\omega \in A({\mathbb{G}})$ we have, $$\Vert \omega \Vert_{A({\mathbb{G}})} \geq \Vert \omega \Vert_{{M_0A}({\mathbb{G}})}.$$ A completely bounded multiplier $b \in {M_0A}({\mathbb{G}})$ is called [*positive*]{} if it maps $A({\mathbb{G}})^+$ to $A({\mathbb{G}})^+$. [*Complete positivity*]{} is then defined by being positive on every matrix level of the operator space structure.
A sufficient condition for weak amenability of locally compact quantum groups {#Sect=CompactaApproximation}
=============================================================================
We call a locally compact quantum group ${\mathbb{G}}$ [*weakly amenable*]{} if there exists a $C \in \mathbb{R}$ and a net $\{b_k\}$ in $A({\mathbb{G}})$ such that $\Vert b_k \Vert_{{M_0A}({\mathbb{G}})} \leq C$ and for every $c \in A({\mathbb{G}})$, we have, $$\label{Eqn=ApproximationThing}
\Vert b_k c - c \Vert_{A({\mathbb{G}})} \rightarrow 0.$$ The infimum over all $C \in \mathbb{R}$ such that such a net $b_k$ exists is called the [*Cowling-Haagerup*]{} constant; notation $\Lambda({\mathbb{G}})$.
The definition of weak amenability first appears in [@CanHaa], where it is proved that $SO_0(n,1)$ is weakly amenable. In Section \[Sect=WeakAmenability\] we give a more elaborate discussion of examples of weakly amenable (quantum) groups.
In concrete examples can often be hard to check. In this section, we give a sufficient criterium for weak amenability based on [@CanHaa]. This is Theorem \[Thm=CompactaApproximation\].
Recall that a locally compact quantum group ${\mathbb{G}}$ is called [*coamenable*]{} if there exists a state $\epsilon$ on $C_0({\mathbb{G}})$ such that $(\epsilon \otimes {\textrm{id}})(W) = 1$, see also Lemma \[Lem=CoamenabilityEquivalence\] for equivalent definitions. We also need [*modular*]{} multipliers.
We call a multiplier $b \in {M_0A}({\mathbb{G}})$ [*modular*]{} if for every $t \in \mathbb{R}$ we have $\tau_t(b) = b$.
Let $b \in {M_0A}({\mathbb{G}})$ be a modular multiplier. Consider the mapping $L^1(\hat{{\mathbb{G}}}) \rightarrow L^1(\hat{{\mathbb{G}}}): \omega \rightarrow \hat{\lambda}^{-1}(b\hat{\lambda}(\omega))$ and let $\Phi_b: L^\infty(\hat{{\mathbb{G}}}) \rightarrow L^\infty(\hat{{\mathbb{G}}})$ be its dual. Then, $\Phi_b$ commutes with the modular automorphsim group $\hat{\sigma}$. Indeed, using Lemma \[Lem=ModularAppendix\] in the second and fourth equation, $$\begin{split}
& \langle \hat{\sigma}_t(\Phi_b(x)), \omega \rangle_{L^\infty(\hat{{\mathbb{G}}}), L^1(\hat{{\mathbb{G}}})} \\
=& \langle x, \hat{\lambda}^{-1}(b\hat{\lambda}(\omega \circ \hat{\sigma}_t)) \rangle_{L^\infty(\hat{{\mathbb{G}}}), L^1(\hat{{\mathbb{G}}})} \\
= & \langle x, \hat{\lambda}^{-1}(b \tau_{-t}(\hat{\lambda}(\omega )) \delta^{it} ) \rangle_{L^\infty(\hat{{\mathbb{G}}}), L^1(\hat{{\mathbb{G}}})} \\
= & \langle x, \hat{\lambda}^{-1}(\tau_{-t}( b \hat{\lambda}(\omega )) \delta^{it} ) \rangle_{L^\infty(\hat{{\mathbb{G}}}), L^1(\hat{{\mathbb{G}}})} \\
= & \langle \hat{\sigma}_t( x) ,\hat{\lambda}^{-1}( b \hat{\lambda}(\omega )) \rangle_{L^\infty(\hat{{\mathbb{G}}}), L^1(\hat{{\mathbb{G}}})} \\
= & \langle \Phi_b(\hat{\sigma}_t(x)) , \omega \rangle_{L^\infty(\hat{{\mathbb{G}}}), L^1(\hat{{\mathbb{G}}})}.
\end{split}$$ This justifies the terminology [*modular*]{}.
For any $c \in A({\mathbb{G}})$ we denote $\omega_c \in L^1(\hat{{\mathbb{G}}})$ for the functional such that $\hat{\lambda}(\omega_c) = c$.
\[Lem=L1Approx\] Let ${\mathbb{G}}$ be a coamenable quantum group. Let $b_k \in {M_0A}({\mathbb{G}})$ be a net of positive multipliers, such that for every $c \in C_0({\mathbb{G}})$ we have $b_k c \rightarrow c$ in the norm of $C_0({\mathbb{G}})$. Then, for every $c \in A({\mathbb{G}})^+$ we have $\Vert b_k c \Vert_{A({\mathbb{G}})} \rightarrow \Vert c \Vert_{A({\mathbb{G}})}$.
Take $c \in A({\mathbb{G}})^+$ so that $\omega_c$ is a positive functional. In that case also $\omega_{b_k c}$ is positive. Since ${\mathbb{G}}$ is coamenable, by definition there is a bounded positive functional $\epsilon \in C_0({\mathbb{G}})^\ast$ such that $(\epsilon \otimes {\textrm{id}})(W) = 1$. Then, also $(\epsilon \otimes {\textrm{id}})(W^\ast) = 1$. Then, using the positivity of $\omega_{b_k c}$ in the first equality and in addition the positivity of $\epsilon$ in the third equality, $$\begin{split}
\Vert b_k c \Vert_{A({\mathbb{G}})} = & \omega_{b_kc}(1) = \langle (\epsilon \otimes {\textrm{id}})(W^\ast), \omega_{b_k c} \rangle_{L^\infty(\hat{{\mathbb{G}}}), L^1(\hat{{\mathbb{G}}})} \\=& \langle \epsilon, ({\textrm{id}}\otimes \omega_{b_k c} )(W^\ast)\rangle_{C_0({\mathbb{G}})^\ast, C_0({\mathbb{G}})}= \epsilon(b_k c).
\end{split}$$ Similarly, $\Vert c \Vert_{A({\mathbb{G}})} = \epsilon(c)$. Taking the limit $k \rightarrow \infty$, $$\Vert b_k c \Vert_{A({\mathbb{G}})}
= \epsilon(b_k c) \rightarrow \epsilon(c) =
\Vert c \Vert_{A({\mathbb{G}})}.$$ This proves the lemma.
The following result is the main theorem of this section. It relies on Lemma \[Lem=Technical\], which we shall prove in the remainder of this section. Throughout the proof we use convergences in the norm of $L^\infty({\mathbb{G}})$, which in fact take place in the smalller C$^\ast$-algebra $C_0({\mathbb{G}})$. We will not incorporate this in the notation. Recall also that the dense subspace, $$L^1({\mathbb{G}})^\flat \subseteq L^1({\mathbb{G}}),$$ is defined in Lemma \[Lem=TomitaTransform\]. Recall also that for $\omega \in L^1({\mathbb{G}})^\flat$ and $z \in \mathbb{C}$ we denote $\omega_{[z]}\in L^1({\mathbb{G}})$ for the unique functional such that $\lambda(\omega_{[z]}) = \hat{\sigma}_z( \lambda(\omega))$. In Lemma \[Lem=TomitaTransform\] we proved that $\mathbb{C} \rightarrow L^1({\mathbb{G}}): z \mapsto \omega_{[z]}$ is analytic for every $\omega \in L^1({\mathbb{G}})^\flat$.
\[Thm=CompactaApproximation\] Let $\mathbb{G}$ be a coamenable quantum group. Let $b_k \in A({\mathbb{G}})$ be a sequence of positive modular multipliers such that $\Vert b_k \Vert_{{M_0A}({\mathbb{G}})}$ is bounded and such that for every $c \in C_0({\mathbb{G}})$ we have $\Vert b_k c - c \Vert_{L^\infty({\mathbb{G}})} \rightarrow 0$. Then, for every $c \in A({\mathbb{G}})$ as $k \rightarrow \infty$, $$\label{Eqn=Approx}
\Vert b_k c - c \Vert_{A({\mathbb{G}})} \rightarrow 0.$$ That is, ${\mathbb{G}}$ is weakly amenable with Cowling-Haagerup constant smaller than or equal to $\limsup_{k \in K} \Vert b_k \Vert_{{M_0A}({\mathbb{G}})}$.
For clarity, let us make the following remark on the convergence . Note that each $b_k$ is in $A({\mathbb{G}})$ and not in ${M_0A}({\mathbb{G}})$. So each $b_k$ corresponds to a functional $\hat{\omega}_k \in L^1(\hat{{\mathbb{G}}})^+$ and similarly $c \in A({\mathbb{G}})$ corresponds to a $\omega_c \in L^1(\hat{{\mathbb{G}}})$. The convergence can be restated at the $L^1(\hat{{\mathbb{G}}})$-level, namely $\Vert \omega_k \ast \omega_c - \omega_c \Vert_{L^1(\hat{{\mathbb{G}}})} \rightarrow 0$.
Since $\Vert b_k \Vert_{{M_0A}({\mathbb{G}})}$ is bounded, it follows from a $3\epsilon$-argument that we may prove for a dense set of $c \in A({\mathbb{G}})^+$. By Lemma \[Lem=GapLemma\], in order to prove our theorem, we may assume that $\omega_c(x) = \langle x \hat{\Lambda}(d), \hat{\Lambda}(d) \rangle =: (d \hat{\varphi} d^\ast)(x)$, with $d = \lambda((\omega^\ast \ast \omega)_{[-i/2]}), \omega \in L^1({\mathbb{G}})^\flat$. In particular $\hat{\sigma}_{i/2}(d) = \lambda(\omega^\ast \ast \omega) \geq 0$.
Moreover, using Lemma \[Lem=Technical\], we may assume that there exist $d_k \in {\mathcal{T}}_{\hat{\varphi}}$ such that $\Vert \omega_{b_kc} - d_k \hat{\varphi} d_k^\ast \Vert_{L^1(\hat{{\mathbb{G}}})} < \frac{1}{k}$ and such that $\hat{\sigma}_{i/2}(d_k)\hat{\sigma}_{i/2}(d_k)^\ast$ is a bounded sequence in $L^\infty(\hat{{\mathbb{G}}})$ and that in fact $\hat{\sigma}_{i/2}(d_k) \in L^\infty(\hat{{\mathbb{G}}})^+$. We sometimes write $\hat{\sigma}_{i/2}(d_k)^\ast$ to clarify our equalities, even though this is a positive operator.
It suffices then to prove that: $$\label{Eqn=AlternativeConvergence}
\Vert d_k \hat{\varphi} d_k^\ast - d \hat{\varphi} d^\ast \Vert_{L^1(\hat{{\mathbb{G}}})} \rightarrow 0.$$ Since for $x \in L^\infty(\hat{{\mathbb{G}}})$ we have that $(d_k \hat{\varphi} d_k^\ast)(x) = \langle x \hat{\Lambda}(d_k), \hat{\Lambda}(d_k) \rangle$ and similarly with $d_k$ replaced by $d$, it suffices to prove that $\hat{\Lambda}(d_k) \rightarrow \hat{\Lambda}(d)$ in the norm of $L^2(\hat{{\mathbb{G}}})$. Now, $$\begin{split}
& \Vert \hat{\Lambda}(d_k) - \hat{\Lambda}(d) \Vert_{L^2(\hat{{\mathbb{G}}})}^2
= \langle \hat{\Lambda}(d_k) - \hat{\Lambda}(d), \hat{\Lambda}(d_k) - \hat{\Lambda}(d) \rangle
\\
= & \langle \hat{\Lambda}(d_k) , \hat{\Lambda}(d_k) \rangle + \langle \hat{\Lambda}(d) , \hat{\Lambda}(d) \rangle - 2 \Re \left( \langle \hat{\Lambda}(d_k) ,\hat{\Lambda}(d) \rangle \right).
\end{split}$$ Hence, it suffices to prove that, $$\label{Eqn=SufficientCondition}
\Vert \hat{\Lambda}(d_k) \Vert_{L^2(\hat{{\mathbb{G}}})} \rightarrow \Vert \hat{\Lambda}(d) \Vert_{L^2(\hat{{\mathbb{G}}})} \quad \textrm{ and } \quad \langle \hat{\Lambda}(d_k) - \hat{\Lambda}(d),\hat{\Lambda}(d) \rangle \rightarrow 0.$$
For the left condition of , we find $$\begin{split}
& \left| \Vert \hat{\Lambda}(d_k) \Vert_{L^2(\hat{{\mathbb{G}}})}^2 - \Vert \hat{\Lambda}(d) \Vert_{L^2(\hat{{\mathbb{G}}})}^2 \right|
= \left| (d_k \hat{\varphi} d_k^\ast)(1) - (d \hat{\varphi} d^\ast)(1) \right| \\
\leq & \left| \omega_{b_k c}(1) - \omega_{c}(1) \right| + \frac{1}{k}
= \left| \Vert b_k c \Vert_{A({\mathbb{G}})} - \Vert c \Vert_{A({\mathbb{G}})} \right| + \frac{1}{k},
\end{split}$$ and from from Lemma \[Lem=L1Approx\] (this is where we use that ${\mathbb{G}}$ is coamenable) it follows that this expression converges to 0. Hence, it remains to check the right condition of .
Let $\theta \in L^1({\mathbb{G}})^+$. Set $x = \lambda(\theta)$, then as $k \rightarrow \infty$, $$\begin{split}
& \vert \hat{\varphi}(d^\ast x d) - \hat{\varphi}(d_k^\ast x d_k)\vert
\leq \vert \omega_c(x) - \omega_{b_k c}(x) \vert + \frac{1}{k} \Vert x \Vert_{L^\infty(\hat{{\mathbb{G}}})} \\
= & \vert \theta(c) - \theta(b_k c) \vert + \frac{1}{k} \Vert x \Vert_{L^\infty(\hat{{\mathbb{G}}})} \rightarrow 0.
\end{split}$$ Since we proved that $d_k \hat{\varphi} d_k^\ast$ is a bounded sequence in $L^1(\hat{{\mathbb{G}}})$ (i.e. the left part of ) and the span of $\lambda(\theta)$ with $\theta \in L^1({\mathbb{G}})^+$ is norm dense in $C_r^\ast({\mathbb{G}})$, we find that in fact, $$\label{Eqn=PartialGoToZero}
\vert \hat{\varphi}(d^\ast x d) - \hat{\varphi}(d_k^\ast x d_k)\vert \rightarrow 0 \qquad \textrm{ for every } x \in C^\ast_r({\mathbb{G}}).$$
Let $e,f \in C^\ast_r({\mathbb{G}})$ be such that $e,f \in {\mathcal{T}}_{\hat{\varphi}}$. The existence of such elements is guaranteed by Lemma \[Lem=TomitaTransform\]. Moreover, such elements are norm dense in $C^\ast_r({\mathbb{G}})$. Then, $$\begin{split}
& \hat{\varphi}(d_k^\ast e^\ast f d_k)
= \hat{\varphi}( \hat{\sigma}_{i/2}(fd_k) \hat{\sigma}_{-i/2}(d_k^\ast e^\ast) ) \\
= & \langle \hat{\sigma}_{i/2}(d_k)\hat{\sigma}_{i/2}(d_k)^\ast \hat{\Lambda}(\hat{\sigma}_{-i/2}(e^\ast) ), \hat{\Lambda}(\hat{\sigma}_{i/2}(f)^\ast ) \rangle,
\end{split}$$ and similarly, $$\begin{split}
\hat{\varphi}(d^\ast e^\ast f d) = \langle \hat{\sigma}_{i/2}(d)\hat{\sigma}_{i/2}(d)^\ast \hat{\Lambda}(\hat{\sigma}_{-i/2}(e^\ast) ), \hat{\Lambda}(\hat{\sigma}_{i/2}(f)^\ast ) \rangle.
\end{split}$$ By , we see that $\vert \hat{\varphi}(d^\ast e^\ast f d) - \hat{\varphi}(d_k^\ast e^\ast f d_k) \vert \rightarrow 0$ and since we assumed that $\hat{\sigma}_{i/2}(d_k)\hat{\sigma}_{i/2}(d_k)^\ast$ is a bounded sequence, we find that for all vectors $\xi, \eta \in L^2({\mathbb{G}})$, $$\langle \hat{\sigma}_{i/2}(d_k)\hat{\sigma}_{i/2}(d_k)^\ast \xi, \eta \rangle \rightarrow \langle \hat{\sigma}_{i/2}(d)\hat{\sigma}_{i/2}(d)^\ast \xi, \eta \rangle.$$ Approximating $t \mapsto \sqrt{t}$ with polynomials on the compact interval from $0$ to the number $\sup_k \Vert\hat{\sigma}_{i/2}(d_k)\hat{\sigma}_{i/2}(d_k)^\ast\Vert_{L^\infty({\mathbb{G}})}$, this then implies that, $$\langle \vert \hat{\sigma}_{i/2}(d_k)^\ast \vert \xi, \eta \rangle \rightarrow \langle \vert \hat{\sigma}_{i/2}(d)^\ast\vert \xi, \eta \rangle.$$ Henceforth, using positivity of $\hat{\sigma}_{i/2}(d_k)$ and $\hat{\sigma}_{i/2}(d)$ (see the assumptions in the first paragraph of our proof), $$\langle \hat{\sigma}_{i/2}(d_k)^\ast \xi, \eta \rangle \rightarrow \langle \hat{\sigma}_{i/2}(d)^\ast \xi, \eta \rangle.$$ Now, let again $e, f \in \mathcal{T}_{\hat{\varphi}}$. Then, $$\begin{split}
&\langle \hat{\Lambda}(ef), \hat{\Lambda}(d_k) \rangle \\
= & \hat{\varphi}(d_k^\ast ef ) \\
= & \hat{\varphi}(\hat{\sigma}_{i/2}(f) \hat{\sigma}_{-i/2}(d_k^\ast) \hat{\sigma}_{-i/2}(e) )\\
= & \langle \hat{\sigma}_{i/2}(d_k)^\ast \hat{\Lambda}(\hat{\sigma}_{-i/2}(e) ), \hat{\Lambda}(\hat{\sigma}_{i/2}(f)^\ast ) \rangle \\
\rightarrow & \langle \hat{\sigma}_{i/2}(d)^\ast \hat{\Lambda}(\hat{\sigma}_{-i/2}(e) ), \hat{\Lambda}(\hat{\sigma}_{i/2}(f)^\ast ) \rangle \\
= & \langle \hat{\Lambda}(ef), \hat{\Lambda}(d) \rangle,
\end{split}$$ where the last equation follows by following the first equations backwards but with $d$ instead of $d_k$. Using the proved fact that $\Vert \hat{\Lambda}(d_k)\Vert_{L^2({\mathbb{G}})}$ is bounded, and the standard fact that $\hat{\Lambda}({\mathcal{T}}_{\hat{\varphi}}^2)$ is dense in $L^2(\hat{{\mathbb{G}}})$ we find that, $$\langle \hat{\Lambda}(d_k) - \hat{\Lambda}(d), \xi\rangle \rightarrow 0,$$ for all vectors $\xi \in L^2({\mathbb{G}})$. In all, we have proved and hence conclude the theorem.
We now prove the technical lemmas that are needed to make the assumptions at the beginning of the proof of Theorem \[Thm=CompactaApproximation\].
\[Lem=GapLemma\] The functionals on $L^\infty(\hat{{\mathbb{G}}})$ given by $$a \mapsto \langle a \hat{\Lambda}( \lambda( (\omega^\ast \ast \omega)_{[-i/2]} )), \hat{\Lambda}( \lambda( (\omega^\ast \ast \omega)_{[-i/2]} )) \rangle,$$ with $\omega \in L^1({\mathbb{G}})^\flat$ are dense in $L^1(\hat{{\mathbb{G}}})^+$.
For $x, y \in L^2(\hat{{\mathbb{G}}}) \cap L^\infty(\hat{{\mathbb{G}}} )$, there exists a unique functional $\hat{\varphi}_{y^\ast x} \in L^1(\hat{{\mathbb{G}}})$ such that for $e, f \in {\mathcal{T}}_{\hat{\varphi}}$ we have [@Terp Proposition 4], $$\label{Eqn=NewFunctEqn}
\begin{split}
\hat{\varphi}_{y^\ast x} (e^\ast f) = & \langle \hat{J} x^\ast y \hat{J} \hat{\Lambda}(f), \hat{\Lambda}(e) \rangle \\
= & \langle \hat{J} f^\ast e \hat{J} \hat{\Lambda}(x), \hat{\Lambda}(y) \rangle.
\end{split}$$ From the second line of we infer that $\{ \hat{\varphi}_{x^\ast x} \mid x \in L^2(\hat{{\mathbb{G}}}) \cap L^\infty(\hat{{\mathbb{G}}}) \}$ is dense in $L^1(\hat{{\mathbb{G}}})^+$. (Indeed, every positive normal functional on $L^\infty(\hat{{\mathbb{G}}})$ is of the form $\omega_{\xi, \xi} = \omega_{\hat{J} \hat{J} \xi, \hat{J} \hat{J} \xi}$ and $\hat{J} \xi$ can be approximated with $\hat{\Lambda}(x)$ with $x \in L^2(\hat{{\mathbb{G}}}) \cap L^\infty(\hat{{\mathbb{G}}})$). Since we have $\hat{\varphi}_{x^\ast x} = \hat{\varphi}_{\vert x \vert^2 }$, we see that the functionals given by, $$a \mapsto \langle \hat{J} a^\ast \hat{J} \hat{\Lambda}(x), \hat{\Lambda}(x) \rangle,$$ with $x \in L^2(\hat{{\mathbb{G}}}) \cap L^\infty(\hat{{\mathbb{G}}})^+$ are dense in $L^1(\hat{{\mathbb{G}}})^+$.
Now, let $x \in L^2(\hat{{\mathbb{G}}}) \cap L^\infty(\hat{{\mathbb{G}}})^+$ with spectral decomposition $x = \int_{0}^\infty \lambda dE_x(\lambda)$. Put $x_n = \int_{1/n}^\infty \lambda dE_x(\lambda)$. Then $\Vert x - x_n \Vert_{L^\infty(\hat{{\mathbb{G}}})} \rightarrow 0$ and clearly $x_n \leq x$. Moreover, this implies that $x_n \in L^2(\hat{{\mathbb{G}}}) \cap L^\infty(\hat{{\mathbb{G}}})^+$, and $$\Vert \hat{\Lambda}(x_n) - \hat{\Lambda}(x) \Vert^2_{L^2(\hat{{\mathbb{G}}})} = \hat{\varphi}( x^2) + \hat{\varphi}(x_n^2) - \hat{\varphi}(x x_n) - \hat{\varphi}(x_n x) \rightarrow 0,$$ since $\hat{\varphi}$ is normal and $\sup_n (x_n^2) = x^2 = \sup_n (x_n x)$. The conclusion is that the functionals $$a \mapsto \langle \hat{J} a^\ast \hat{J} \hat{\Lambda}(x), \hat{\Lambda}(x) \rangle,$$ with $$\label{Eqn=XCondition}
x \in L^2(\hat{{\mathbb{G}}}) \cap L^\infty(\hat{{\mathbb{G}}})^+ \quad \textrm{ such that } x = \int_{1/n}^\infty \lambda dE_x(\lambda) \quad \textrm{ for some } \quad n \in \mathbb{N},$$ are dense in $L^1(\hat{{\mathbb{G}}})^+$.
Now, let $x$ indeed satisfy the condition . Put $y = \sqrt{x}$. Then, $y \leq \sqrt{n} x$, so that $y \in L^2(\hat{{\mathbb{G}}}) \cap L^\infty(\hat{{\mathbb{G}}})^+$. Since $\lambda( L^1({\mathbb{G}})^\flat ) $ forms a $\sigma$-strong-$\ast$/norm core for $\hat{\Lambda}$, c.f. Lemma \[Lem=TomitaTransform\], we can find a net $\{ \omega_i \}$ in $L^1({\mathbb{G}})^\flat$, such that $\lambda(\omega_i)$ is a (bounded) net converging to $y$ in the $\sigma$-strong-$\ast$ topology and $\xi(\omega_i) = \hat{\Lambda}(\lambda(\omega_i)) \rightarrow \hat{\Lambda}(y)$ in norm. Then, a $2\epsilon$-estimate shows that $\hat{\Lambda}( \lambda( \omega_i^\ast \ast \omega_i) ) = \lambda(\omega_i)^\ast \hat{\Lambda}(\lambda(\omega_i)) \rightarrow y^\ast \hat{\Lambda}(y) = \hat{\Lambda}(x)$ in norm. From this it follows that the functionals, $$a \mapsto \langle \hat{J} a^\ast \hat{J} \hat{\Lambda}( \lambda( \omega^\ast \ast \omega ) ), \hat{\Lambda}(\lambda(\omega^\ast \ast \omega)) \rangle,$$ with $\omega \in L^1({\mathbb{G}})^\flat$ are dense in $L^1(\hat{{\mathbb{G}}})^+$. But, using that $\hat{J} = \hat{\nabla}^{1/2} \hat{J} \hat{\nabla}^{1/2}$ and $\lambda(\omega^\ast \ast \omega) \geq 0$, $$\begin{split}
& \langle \hat{J} a^\ast \hat{J} \hat{\Lambda}( \lambda( \omega^\ast \ast \omega ) ), \hat{\Lambda}(\lambda(\omega^\ast \ast \omega)) \rangle \\
= & \langle a \hat{J} \hat{\Lambda}( \lambda( \omega^\ast \ast \omega ) ), \hat{J} \hat{\Lambda}(\lambda(\omega^\ast \ast \omega)) \rangle \\
= & \langle a \hat{\nabla}^{1/2} \hat{\Lambda}( \lambda( \omega^\ast \ast \omega ) ), \hat{\nabla}^{1/2} \hat{\Lambda}(\lambda(\omega^\ast \ast \omega)) \rangle \\
= & \langle a \hat{\Lambda}( \lambda( (\omega^\ast \ast \omega)_{[-i/2]} ) ), \hat{\Lambda}(\lambda( (\omega^\ast \ast \omega)_{[-i/2]} )) \rangle,
\end{split}$$ which concludes our lemma.
\[Lem=Modular\] Let $\omega \in L^1({\mathbb{G}})^\flat$ and let $b \in {M_0A}({\mathbb{G}})$ be a modular multiplier. We have $b \cdot \omega \in {\mathcal{I}}$ and $\lambda(b \cdot \omega )$ is analytic for $\hat{\sigma}$. For every $z \in \mathbb{C}$, we have $\hat{\sigma}_z(\lambda(b\cdot \omega)) = \lambda(b \cdot \omega_{[z]})$.
By Lemma \[Lem=ModularAppendix\] we find that we must have $\theta_{[t]}(x) = \theta(\delta^{it} \tau_{-t}( x ))$ for every $t \in \mathbb{R}$ and $\theta \in L^1({\mathbb{G}})^\flat$. Applying this to $\theta = b \cdot \omega$, this means that $$\begin{split}
& \langle x, \theta_{[t]} \rangle = \langle \delta^{it} \tau_{-t}( x ), \theta \rangle =
\langle \delta^{it} \tau_{-t}( x ), b \cdot \omega \rangle =
\langle \delta^{it} \tau_{-t}( x ) b, \omega \rangle \\ = &
\langle \delta^{it} \tau_{-t}( x b), \omega \rangle =
\langle xb, \omega_{[t]} \rangle =
\langle x, b \cdot \omega_{[t]} \rangle.
\end{split}$$ So that $$\label{Eqn=AnalEqn}
\hat{\sigma}_t(\lambda(b\cdot \omega)) = \hat{\sigma}_t(\lambda(\theta)) = \lambda(\theta_{[t]}) = \lambda(b \cdot \omega_{[t]}).$$ Since, $t \mapsto \omega_{[t]}$ extends analytically to $\mathbb{C} \rightarrow L^1({\mathbb{G}})$ it follows that is analytic on $\mathbb{C}$. This proves the lemma, since $\mathcal{I}$ is a left $L^\infty({\mathbb{G}})$-module [@KusVae Result 8.6].
\[Lem=L1CohenTypeLemma\] Let $b_k \in {M_0A}({\mathbb{G}})$ be a bounded sequence of multipliers such that for every $c \in C_0({\mathbb{G}})$ we have $\Vert b_k c - c \Vert_{L^\infty({\mathbb{G}})} \rightarrow 0$. For every $\omega \in L^1({\mathbb{G}})$, we have $\Vert b_k \cdot \omega - \omega \Vert_{L^1({\mathbb{G}})} \rightarrow 0$.
Let $\xi, \eta \in L^2({\mathbb{G}})$ and let $c \in C_0({\mathbb{G}})$. Then, $$\begin{split}
\Vert b_k \cdot \omega_{ c \xi, \eta} - \omega_{ c \xi, \eta} \Vert_{L^1({\mathbb{G}})} = & \Vert \omega_{b_k c \xi, \eta} - \omega_{c \xi, \eta} \Vert_{L^1({\mathbb{G}})}\\
\leq & \Vert \xi \Vert_{L^2({\mathbb{G}})} \Vert \eta \Vert_{L^2({\mathbb{G}})} \Vert b_k c - c \Vert_{L^\infty({\mathbb{G}})} \rightarrow 0.
\end{split}$$ Let $\xi, \eta \in L^2({\mathbb{G}})$ again be arbitrary. By Cohen factorization, there exists a $c \in C_0({\mathbb{G}})$ and $\xi' \in L^2({\mathbb{G}})$ such that $\xi = c \xi'$ with $\Vert c \Vert \leq 1$. The lemma then follows from the previous computation.
Let $x,y \in L^2({\mathbb{G}}) \cap L^\infty({\mathbb{G}})$. It is proved in [@Terp Proposition 4] that there exists a unique functional $\varphi_{y^\ast x } \in L^1({\mathbb{G}})$ such that for every $e,f \in {\mathcal{T}}_{\hat{\varphi}}$ we have, $$\label{Eqn=MagicFunctional}
\varphi_{y^\ast x }(e^\ast f) = \langle J x^\ast y J \Lambda(f), \Lambda(e) \rangle = \langle J f^\ast e J \Lambda(x), \Lambda(y) \rangle.$$ We now prove the following technical lemma. It plays a crucial role in Theorem \[Thm=CompactaApproximation\]. Remark \[Rmk=TechnicalNature\] explains its nature.
\[Lem=Technical\] Let $b_k \in {M_0A}({\mathbb{G}})$ be a bounded sequence of positive modular multipliers such that for every $c \in C_0({\mathbb{G}})$ we have $\Vert b_k c - c \Vert_{L^\infty({\mathbb{G}})} \rightarrow 0$. Let $\omega \in L^1({\mathbb{G}})^{\flat}$ and set $d = \lambda(\omega)$. In particular, $d \in {\mathcal{T}}_{\hat{\varphi}}$ and $d\hat{\varphi}d^\ast \in L^1(\hat{{\mathbb{G}}})$ is well defined. Let $\hat{\theta}_k \in L^1(\hat{{\mathbb{G}}})$ be such that $\hat{\lambda}(\hat{\theta}_k) = b_k \hat{\lambda}(d \hat{\varphi} d^\ast)$. Then, for every $k$, there exists an operator $d_k \in {\mathcal{T}}_{\hat{\varphi}}$ such that $$\label{Eqn=TechnicalConvergence}
\Vert d_k \hat{\varphi} d_k^\ast - \hat{\theta}_k \Vert_{L^1(\hat{{\mathbb{G}}})} < \frac{1}{k},$$ and moreover, $\hat{\sigma}_{i/2}(d_k) \hat{\sigma}_{i/2}(d_k)^\ast$ is bounded in $L^\infty({\mathbb{G}})$ and $\hat{\sigma}_{i/2}(d_k) \in L^\infty({\mathbb{G}})^+$.
Note that the key properties we need are boundedness of $\hat{\sigma}_{i/2}(d_k) \hat{\sigma}_{i/2}(d_k)^\ast$ and positivity of $\hat{\sigma}_{i/2}(d_k) \in L^\infty({\mathbb{G}})^+$. The following is the main point of our set-up.
[**Claim 1:**]{} For every $k \in \mathbb{N}$, there exists a unique element $x_k \in L^\infty(\hat{{\mathbb{G}}})$ such that for every $e,f \in {\mathcal{T}}_{\hat{\varphi}}$ we have, $$\label{Eqn=XkFormula}
\hat{\theta}_k(e^\ast f) = \langle \hat{J} x_k^\ast \hat{J} \hat{\Lambda}(f), \hat{\Lambda}(e) \rangle.$$ Moreover, $x_k$ is positive and the sequence is bounded in $L^\infty(\hat{{\mathbb{G}}})$.
[**Proof of claim 1:**]{} If such $x_k$ exists, then it is unique by . Since $b_k$ is a positive multiplier, $\hat{\theta}_k$ is a positive functional. Then, $x_k$ is positive, since for $e \in {\mathcal{T}}_{\hat{\varphi}}$, $$\langle x_k \hat{J} \hat{\Lambda}(e), \hat{J} \hat{\Lambda}(e) \rangle =\langle \hat{J} x_k^\ast \hat{J} \hat{\Lambda}(e),\hat{\Lambda}(e) \rangle= \hat{\theta}_k(e^\ast e) \geq 0.$$ We use the notation $\omega' = (\omega^\ast)_{[-i]}$. We claim that we may take: $$x_k :=
\lambda\left( b_k \cdot\left( \left( \omega \ast \omega' \right)_{[i/2]}\right)\right).$$ It follows from Lemma \[Lem=L1CohenTypeLemma\] that $b_k \cdot\left( \left( \omega \ast \omega' \right)_{[i/2]}\right)$ is a bounded sequence in $L^1({\mathbb{G}})$ and hence $x_k$ is a bounded sequence in $L^\infty(\hat{{\mathbb{G}}})$. In fact, $x_k \in L^2(\hat{{\mathbb{G}}}) \cap L^\infty(\hat{{\mathbb{G}}}) $ by Lemma \[Lem=Modular\]. We claim that, $$\label{Eqn=Dualizing}
\xi( \omega \ast \omega') = \hat{\Lambda}( \lambda(\omega) \lambda(\omega') ) = \hat{\xi}( \lambda(\omega) \hat{\varphi} \lambda(\omega) ^\ast) = \hat{\xi}( d \hat{\varphi} d^\ast).$$ Indeed, the first and last equation follow by definition. The second equation follows from the fact that for $e\in {\mathcal{T}}_{\hat{\varphi}}$ we have, $$\label{Eqn=L2Check}
\begin{split}
&(\lambda(\omega) \hat{\varphi} \lambda(\omega) ^\ast)(e^\ast) = \hat{\varphi}(\lambda(\omega) ^\ast e^\ast \lambda(\omega)) = \hat{\varphi}(e^\ast \lambda(\omega)\hat{\sigma}_{-i}(\lambda(\omega) ^\ast ))\\
=& \hat{\varphi}(e^\ast \lambda(\omega)\lambda(\omega')) = \langle \hat{\Lambda}(\lambda(\omega)\lambda(\omega')), \hat{\Lambda}(e) \rangle.
\end{split}$$ Using the fact that ${\mathcal{T}}_{\hat{\varphi}}$ is a $\sigma$-weak/norm core for $\hat{\Lambda}$, this yields that holds for all $e \in L^2(\hat{{\mathbb{G}}}) \cap L^\infty(\hat{{\mathbb{G}}})$, and hence follows.
We now prove . Let $e, f \in {\mathcal{T}}_{\hat{\varphi}}$. Then, explaining the equations below, we have $$\begin{split}
& \langle \hat{J} x_k^\ast \hat{J} \hat{\Lambda}(f), \hat{\Lambda}(e) \rangle \\
= & \langle x_k \hat{\Lambda}(\hat{\sigma}_{i/2}(e)^\ast ), \hat{\Lambda}( \hat{\sigma}_{i/2} (f)^\ast )\\
= & \hat{\varphi}(\hat{\sigma}_{i/2}(f) x_k \hat{\sigma}_{i/2}(e)^\ast ) \\
= & \hat{\varphi}(\hat{\sigma}_{i/2}(e^\ast f) x_k ) \\
= & \langle \hat{\Lambda}(x_k), \hat{\Lambda}( \hat{\sigma}_{-i/2}(f^\ast e) ) \rangle\\
= & \langle b_k \xi(\left( \omega \ast \omega' \right)_{[i/2]} ), \hat{\Lambda}(\hat{\sigma}_{-i/2}(f^\ast e))\rangle\\
= & \langle b_k \hat{\nabla}^{-\frac{1}{2}} \xi( \omega \ast \omega' ),\hat{\nabla}^{\frac{1}{2}} \hat{\Lambda}(f^\ast e)\rangle\\
= & \langle b_k \xi( \omega \ast \omega' ), \hat{\Lambda}(f^\ast e)\rangle\\
= & \langle b_k \hat{\xi}(d \hat{\varphi} d^\ast), \hat{\Lambda}(f^\ast e) \rangle \\
= & \langle \Lambda(b_k \hat{\lambda}(d\hat{\varphi} d^\ast) ), \hat{\Lambda}(f^\ast e) \rangle \\
= & \langle \Lambda(\hat{\lambda}(\hat{\theta}_k) ), \hat{\Lambda}(f^\ast e) \rangle \\
= & \langle \hat{\xi}(\theta_k), \hat{\Lambda}(f^\ast e) \rangle \\
= & \hat{\theta}_k(e^\ast f).
\end{split}$$ The first four equations follow from Tomita-Takesaki theory; the fifth equation is the definition of $x_k$ and the fact that $\xi(b_k\cdot \theta) = b_k \xi(\theta)$ for every $\theta \in {\mathcal{I}}$ ([@KusVae Result 8.6]); the sixth equation is Tomita-Takesaki theory; the seventh equation is Tomita-Takesaki theory and the fact that since $b_k$ is modular, we have $\tau_t(b_k) = \hat{\nabla}^{it} b_k \hat{\nabla}^{-it} = b_k$ for $t \in \mathbb{R}$; the eight equation is ; the remaing equations follow by the definitions of their objects. Note that in the eleventh equation we have used Lemma \[Lem=L2Thingy\]. This proves the claim.
[**Claim 2:**]{} There exists a sequence $y_k \in L^2(\hat{{\mathbb{G}}}) \cap L^\infty(\hat{{\mathbb{G}}})$ that is bounded with respect to the norm of $L^\infty(\hat{{\mathbb{G}}})$ and such that $\Vert \hat{\varphi}_{y_{k}^\ast y_{k}} - \hat{\theta}_k \Vert_{L^1(\hat{{\mathbb{G}}})} \rightarrow 0$. See for the definition of $\hat{\varphi}_{y_{k}^\ast y_{k}}$. Moreover, we may take $y_k \in L^\infty(\hat{{\mathbb{G}}})^+$.
[**Proof of claim 2:**]{} Let $\{e_j\}$ be a net in ${\mathcal{T}}_{\hat{\varphi}}$ such that $e_j \rightarrow 1$ in the $\sigma$-weakly topology and such that $\Vert \hat{\sigma}_z(e_j) \Vert_{L^\infty(\hat{{\mathbb{G}}})} \leq e^{\Im(z)^2}$, see [@Terp Lemma 9]. For every $k$, we let $y_{k,j} \in L^2(\hat{{\mathbb{G}}}) \cap L^\infty(\hat{{\mathbb{G}}})$ be such that, $$\label{Eqn=PositiveEqn}
y_{k,j}^\ast y_{k,j} = \hat{\sigma}_{i/2}(e_j) x_k \hat{\sigma}_{i/2}(e_j)^\ast.$$ The proof of [@Terp Theorem 8, p. 331-332] yields that, $$\begin{split}
&\Vert y_{k,j}^\ast y_{k,j} \Vert_{L^\infty(\hat{{\mathbb{G}}})} < e^{\frac{1}{2}} \Vert x_k \Vert_{L^\infty(\hat{{\mathbb{G}}})}, \\
&\lim_{j \in J} \Vert \hat{\varphi}_{y_{k,j}^\ast y_{k,j}} - \hat{\theta}_k \Vert_{L^1(\hat{{\mathbb{G}}})}= 0.
\end{split}$$ For every $k$, we choose $j_k \in J$ such that $\Vert \hat{\varphi}_{y_{k,j_k}^\ast y_{k,j_k}} - \hat{\theta}_k \Vert_{L^1(\hat{{\mathbb{G}}})} < \frac{1}{k}$. We then set $y_k := y_{k, j_k}$. Note that we could have choosen $y_k$ to be positive in since $x_k$ was positive.
[**Claim 3:**]{} There exists a sequence $z_k \in {\mathcal{T}}_{\hat{\varphi}}$ that is bounded with respect to the norm of $L^\infty(\hat{{\mathbb{G}}})$ and such that $\Vert \hat{\varphi}_{z_{k}^\ast z_{k}} - \hat{\theta}_k \Vert_{L^1(\hat{{\mathbb{G}}})} \rightarrow 0$. Morever, we may take $z_k \in L^\infty(\hat{{\mathbb{G}}})^+$.
[**Proof of claim 3:**]{} Let $y_k$ be as in [**Claim 2**]{}. We assume that $y_k \in L^\infty(\hat{{\mathbb{G}}})^+$.
We then define, using a $\sigma$-weak integral, $$z_{k,n} = \sqrt{\frac{n}{\pi}} \int_{-\infty} ^\infty e^{-nt^2} \hat{\sigma}_t(y_k) dt\quad \in L^\infty({\mathbb{G}})^+,$$ and by standard arguments (see for instance the proof of [@Terp Lemma 9]) we see that $z_{k,n} \in {\mathcal{T}}_{\hat{\varphi}}$. Moreover, $$\hat{\Lambda}(z_{k,n}) = \sqrt{\frac{n}{\pi}} \int_{-\infty} ^\infty e^{-nt^2} \hat{\nabla}^{it} \hat{\Lambda}(y_k) dt,$$ where the integral is a Bochner integral in $L^2(\hat{{\mathbb{G}}})$ and as $n \rightarrow \infty$ we find that $\hat{\Lambda}(z_{k,n}) \rightarrow \hat{\Lambda}(y_k)$ in norm. Recalling that, $$\begin{split}
\hat{\varphi}_{z_{k,n}^\ast z_{k,n}}(e^\ast f) = & \langle \hat{J} f^\ast e \hat{J} \hat{\Lambda}(z_{k,n}), \hat{\Lambda}(z_k) \rangle, \\
\hat{\varphi}_{y_{k}^\ast y_{k}}(e^\ast f) = & \langle \hat{J} f^\ast e \hat{J} \hat{\Lambda}(y_{k}), \hat{\Lambda}(y_k) \rangle, \\
\end{split}$$ this implies that taking the limit $n \rightarrow \infty$, we get $\Vert \hat{\varphi}_{z_{k,n}^\ast z_{k,n}} - \hat{\varphi}_{y_{k}^\ast y_{k}} \Vert_{L^1(\hat{{\mathbb{G}}})} \rightarrow 0$. [**Claim 3**]{} then follows from [**Claim 2**]{}.
[**Proof of Lemma \[Lem=Technical\].**]{} Let $z_k$ be as in [**Claim 3**]{}. We assume that $z_k \in L^\infty(\hat{{\mathbb{G}}})^+$. We set $d_k = \hat{\sigma}_{-i/2}(z_k)$. It follows from [**Claim 3**]{} that $d_k$ satisfies the required boundedness and positivity properties in the statement of our lemma. That is, we require that $\hat{\sigma}_{i/2}(d_k) = z_k$ is bounded and positive, which follows from [**Claim 3**]{}. Finally, for $e, f \in {\mathcal{T}}_{\hat{\varphi}}$ we find, $$\begin{split}
(d_k \hat{\varphi} d_k^\ast)(e^\ast f) = & \hat{\varphi}( d_k^\ast e^\ast f d_k ) \\
= & \langle \hat{\Lambda}(fd_k), \hat{\Lambda}(ed_k) \rangle \\
= & \langle \hat{J} \hat{\nabla}^{\frac{1}{2}} d_k^\ast \hat{J} \hat{\nabla}^{\frac{1}{2}} \hat{\Lambda}(f), \hat{J} \hat{\nabla}^{\frac{1}{2}} d_k^\ast \hat{J} \hat{\nabla}^{\frac{1}{2}} \hat{\Lambda}(e) \rangle \\
= & \langle \hat{J} \hat{\sigma}_{i/2}(d_k) \hat{\sigma}_{i/2}(d_k)^\ast \hat{J} \hat{\Lambda}(f), \hat{\Lambda}(e) \rangle \\
= & \langle \hat{J} z_k z_k^\ast \hat{J} \hat{\Lambda}(f), \hat{\Lambda}(e) \rangle
\end{split}$$ Hence, we find that $d_k \hat{\varphi} d_k^\ast= \hat{\varphi}_{z_{k} z_{k}^\ast} = \hat{\varphi}_{z_{k}^\ast z_{k}}$. Then, follows from [**Claim 3**]{}.
Theorem \[Thm=CompactaApproximation\] and Lemma \[Lem=Technical\] are true if sequences are replaced by nets as it directly follows from the proofs.
\[Rmk=TechnicalNature\] Since the statement and the proof of Lemma \[Lem=Technical\] are technical in nature, it is useful to comment on its origin.
Suppose that ${\mathbb{G}}$ comes from an abelian group $G$. For simplicity, suppose that $\omega \in L^1(G)$ is a compactly supported function. Let $d = \mathcal{F}(\omega)$ be its Fourier transform. In this case $d \hat{\varphi} d^\ast$ corresponds to a function $f \in L^1(\hat{G})$.
The proof of Lemma \[Lem=Technical\], Claim 1 proceeds as follows. Take the Fourier transform $\hat{\mathcal{F}}(f) \in L^\infty(G)$ of $f$. One shows that in fact, $\hat{\mathcal{F}}(f) \in L^1(G)$. Then we multiply $\hat{\mathcal{F}}(f)$ with the multiplier $b_k \in L^\infty(G)$. Next, we take the dual Fourier transform, resulting in the function $(\mathcal{F} \circ b_k \circ \hat{\mathcal{F}})(f) \in L^\infty(\hat{G})$. Because of our choices, $b_k \hat{\mathcal{F}}(f)$ turns out to be a bounded sequence of functions in $L^1(G)$ and henceforth $(\mathcal{F} \circ b_k \circ \hat{\mathcal{F}})(f)\: (= x_k \textrm{ of Claim 1})$ is a bounded sequence in $L^\infty(\hat{G})$. Claims 2 and 3 are then standard approximation methods.
Claim 1 of the proof of Lemma \[Lem=Technical\] is proved in exactly the way described above. Note that in the process we used that $\hat{\mathcal{F}}(f) \in L^\infty(G)$ is in fact in $L^1(G)$. For quantum groups the intersection of $L^\infty({\mathbb{G}})$ and $L^1({\mathbb{G}})$ has a proper interpretation in terms of compatible couples of non-commutative $L^p$-spaces, see [@CasLpf Section 3]. We use these ideas implicitly while passing from $L^\infty({\mathbb{G}})$ to $L^1({\mathbb{G}})$
Let us make the following heuristic comment on why we need the modular condition on the multipliers $b_k$. We use the language of compatible couples of non-commutative $L^p$-space for which we refer to [@Kos].
Along the proof of Lemma \[Lem=Technical\], we use a transition between the left injection and the symmetric injection of non-commutative $L^p$-spaces. It was shown in [@CasLpf Theorem 7.1] that Fourier transforms only exist for the left injection. However, in the proof of Theorem \[Thm=CompactaApproximation\] we need to work with an injection that has the property that an $x \in L^1({\mathbb{G}}) \cap L^\infty({\mathbb{G}})$ is positive in $L^1({\mathbb{G}})$ if and only if it is positive in $L^\infty({\mathbb{G}})$. The left injection does not have this property, but the symmetric injection does. The transition between the left and symmetric injection causes that we need the modular condition on our multipliers $b_k$.
We do not know if the modular assumption on $b_k$ is strictly necessary. However, in our example this condition is easy to check. We expect that in similar examples for which Theorem \[Thm=CompactaApproximation\] is applicable this will not be different.
Basic hypergeometric series and ${SU_q(1,1)_{{\rm ext}}}$ {#Sect=SUone}
=========================================================
This section recalls the preliminaries on basic hypergeometric series [@GasRah] and the definition of the extended quantum $SU(1,1)$ group [@KoeKus], [@GroKoeKus]. Though that some of the definitions below can be extended for other $q$, we always assume that $0 < q < 1$. For $k \in \mathbb{N} \cup \{0,\infty\}$, we set the $q$-factorial, $$(a; q)_k = \prod_{l=0}^{k-1} (1-aq^l), \qquad (a_0, \ldots, a_n; q)_k = (a_0;q)_k \cdot \ldots \cdot (a_n;q)_k.$$ We need the following [*$\theta$-product identity*]{}. For $a \in \mathbb{C} \backslash \{ 0 \}, k \in \mathbb{Z}$, $$\label{EqnIntThePro}
(aq^k, q^{1-k}/a ; q)_\infty = (-a)^{-k} q^{-\frac{1}{2}k(k-1)} (a, q/a;q)_\infty,$$ Recall that the basic hypergeometric $_2 \! \varphi_1$-function is expressed by: $$_2 \! \varphi_1\left( \begin{array}{c} a, b \\ c \end{array};q, z \right) = \prod_{k=0}^\infty \frac{(a,b;q)_k}{(c, q;q)_k} z^{k}.$$ We use the following notation, following [@GroKoeKus]:
------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------
$\mu: \mathbb{C} \backslash \{ 0 \} \rightarrow \mathbb{C} \backslash \{ 0 \}: y \mapsto \frac{1}{2} (y + y^{-1})$ , $\chi: -q^{\mathbb{Z}} \cup q^{\mathbb{Z}}: p \mapsto ^q\!\!\log(\vert p\vert)$,
$\nu: -q^{\mathbb{Z}} \cup q^{\mathbb{Z}} \rightarrow \mathbb{R}: t \mapsto q^{\frac{1}{2}(\chi(t) - 1)(\chi(t)-2) }$, $\kappa: \mathbb{R} \rightarrow \mathbb{R}: x \mapsto {{\rm sgn}}(x) x^2$,
$c_q = (\sqrt{2} q (q^2, -q^2; q^2)_\infty)^{-1}$.
------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------
Furthermore, we set $I_q = q^{\mathbb{Z}} \cup -q^{\mathbb{N}}$ and anticipating to the definition of ${SU_q(1,1)_{{\rm ext}}}$, we set $$L^2({\mathbb{G}}) = L^2(\mathbb{Z}) \otimes L^2(I_q) \otimes L^2(I_q),$$ where each tensor component is understood with respect to the counting measure. It has a canonical basis $f_{m,p,t}$, with $m \in \mathbb{Z}, p,t \in I_q$.
The quantum group ${SU_q(1,1)_{{\rm ext}}}$ was established in the Kustermans-Vaes setting by Koelink and Kustermans [@KoeKus]. Its Plancherel decomposition was obtained by Koelink, Kustermans and Groenevelt in [@GroKoeKus]. It is worth mentioning that ${SU_q(1,1)_{{\rm ext}}}$ is constructed by first defining its multiplicative unitary and then reconstructing a von Neumann algebraic quantum group. Theorem-Definition \[Dfn=DfnSUone\] defines ${SU_q(1,1)_{{\rm ext}}}$. Its complete definition is rather involved, while in our proofs, we only use the Plancherel decomposition of its multiplicative unitary. Therefore, we [*define*]{} ${SU_q(1,1)_{{\rm ext}}}$ by means of this decomposition. defines the principal series corepresentations, while defines the discrete series corepresentations. The only fact we use about the discrete series is that their matrix coefficients are analytic extensions of the matrix coefficients of the principal series. This fact can be derived from a comparison of the actions of the generators of the universal enveloping Lie algebra [@GroKoeKus Lemma 10.1 and Eqn. (92)] on the representation spaces.
Theorem-Definition \[Dfn=DfnSUone\] uses the $C$-function defined in [@GroKoeKus Lemma 9.4]. In our proofs we compute special values of this functions and give further references.
For direct integrals [@DixVna] is the standard reference.
\[Dfn=DfnSUone\] Let $0 < q < 1$. Then, we define ${\mathbb{G}}= {SU_q(1,1)_{{\rm ext}}}$ as follows.
1. \[Item=Principal\] For every $x \in [-1, 1], p \in q^{\mathbb{Z}}$, let $$\label{Eqn=HSpace}
{\mathcal{H}}_{p,x} = {\textrm{span}}\{ e^{{\epsilon}, \eta}_m(p,x) \mid {\epsilon}, \eta \in \{+, -\}, m \in \textrm{$\frac{1}{2}$} \mathbb{Z}\}.$$ where the vectors $e^{{\epsilon}, \eta}_m(p,x)$ are by definition orthonormal. There exists a unitary operator $W_{p,x} \in B(L^2({\mathbb{G}}) \otimes {\mathcal{H}}_{p,x })$ determined by, $$\label{Eqn-CCoefficient}
\begin{split}
&({\textrm{id}}\otimes \omega_{e^{{\epsilon},\eta}_m(p,x), e^{{\epsilon}',\eta'}_{m'}(p,x)})\left(W_{p,x} \right) f_{m_0, p_0, t_0} \\
= & C(\eta{\epsilon}x;m',{\epsilon}',\eta';{\epsilon}{\epsilon}'\vert p_0\vert
p^{-1}q^{-m-m'},p_0,m-m') \\
& \times \quad \delta_{sgn(p_0),\eta\eta'} f_{m_0 - m +
m',{\epsilon}{\epsilon}' \vert p_0\vert p^{-1} q^{-m-m'} , t_0},
\end{split}$$ where the $C$-function is given in terms of basic hypergeometric series in [@GroKoeKus Lemma 9.4].
2. \[Item=Discrete\] For every $p \in q^{\mathbb{Z}}$, let the countable discrete set $\sigma_d(\Omega_p)$ be the discrete spectrum of the Casimir operator [@GroKoeKus Definition 4.5, Theorem 4.6] restricted to the space defined in [@GroKoeKus Theorem 5.7]. Let ${\mathcal{H}}_{p,x}$ be the Hilbert space spanned by the orthonormal vectors $e^{{\epsilon}, \eta}_m(p,x)$ defined in [@GroKoeKus Proposition 5.2] (the notation is $\mathcal{L}_{p,x}$ instead of ${\mathcal{H}}_{p,x}$ here and the span of these vectors forms a subspace of ). There exists a unitary operator $W_{p,x} \in B(L^2({\mathbb{G}}) \otimes {\mathcal{H}}_{p,x })$ determined by .
3. There exists a unique locally compact quantum group ${SU_q(1,1)_{{\rm ext}}}$ with multiplicative unitary $W \in B(L^2({\mathbb{G}}) \otimes L^2({\mathbb{G}}) )$ that is explicitly given in terms of the direct integral decomposition, $$\label{Eqn=Plancherel}
W = \bigoplus_{p \in q^\mathbb{Z}} \left( \int^\oplus_{[-1,1]} W_{p,x} \oplus \bigoplus_{x \in \sigma_d(\Omega_p)} W_{p,x} \right).$$
For every $p \in q^\mathbb{Z}$ and $x \in [-1,1] \cup \sigma_d(\Omega_p)$ the operator $W_{p,x}$ defined in Theorem \[Dfn=DfnSUone\] is a unitary [*corepresentation*]{} of ${SU_q(1,1)_{{\rm ext}}}$. See, [@GroKoeKus Proposition 5.2, Lemma 10.8]. The corepresentations are not mutually inequivalent.
In , the measure on $[-1,1]$ is understood as the Askey-Wilson measure. For our purposes we need only that this is a measure equivalent to the Lebesgue measure.
\[Rmk=TypeI\] It follows that the von Neumann algebra of the dual of ${SU_q(1,1)_{{\rm ext}}}$ as a (proper) subalgebra of: $$\bigoplus_{p \in q^\mathbb{Z}} \left( \int^\oplus_{[-1,1]} B({\mathcal{H}}_{p,x}) \oplus \bigoplus_{x \in \sigma_d(\Omega_p)} B({\mathcal{H}}_{p,x}) \right).$$ (In fact in [@CasKoe Proposition B.2] it is proved that it is of type I.) It follows that finite linear combinations of inner product functionals with respect to vectors of the form, $$\int^\oplus_{[-1, 1]\cup \sigma_d(\Omega_p)} g(x) e^{{\epsilon}, \eta}_m(p,x) dx \in L^2({\mathbb{G}}), \quad {\epsilon}, \eta \in \{ -,+\}, p \in q^{\mathbb{Z}}, m \in \mathbb{Z},$$ and $g$ a square integrable function on $[-1, 1]\cup \sigma_d(\Omega_p)$, form a separating set of functionals for the dual von Neumann algebra of ${SU_q(1,1)_{{\rm ext}}}$.
Coamenability {#Sect=Coamenability}
=============
The main result of this section is that $SU_q(1,1)_{{\rm ext}}$ is coamenable. There are various equivalent notions of coamenability, see [@BedTus]. We recall the following four. Whereas is commonly used in the literature, we prove of Lemma \[Lem=CoamenabilityEquivalence\] in the main theorem of this section. was used in the proof of Lemma \[Lem=L1Approx\]. For $\xi, \eta \in L^2({\mathbb{G}})$, we set $\omega_{\xi, \eta} \in L^1({\mathbb{G}})$ by $\omega_{\xi, \eta}(x) = \langle x \xi, \eta\rangle$. Furthermore, $\omega_\xi = \omega_{\xi, \xi}$.
\[Lem=CoamenabilityEquivalence\] Let $\mathbb{G}$ be a locally compact quantum group. The following are equivalent:
1. \[Item=CoamI\] $L^1(\mathbb{G})$ has a bounded approximate identity. That is, there exists a bounded net $\{ \omega_i \}_i$ in $L^1(\mathbb{G})$ such that for every $\omega \in L^1(\mathbb{G})$ we have $\Vert \omega_i \ast \omega - \omega \Vert_{L^1(\mathbb{G})} \rightarrow 1$.
2. \[Item=CoamII\] There exists a net of unit vectors $\{ \xi_i \}_i$ in $L^2(\mathbb{G})$ such that $(\omega_{\xi_i} \otimes id)(W) \rightarrow 1$ in the $\sigma$-weak topology of $L^\infty(\hat{\mathbb{G}})$.
3. \[Item=CoamIII\] There exists a net of unit vectors $\{ \xi_i \}_i$ in $L^2(\mathbb{G})$ such that $(\omega_{\xi_i} \otimes id)(W^\ast) \rightarrow 1$ in the topology induced by a separating set of vectors in $L^1({\mathbb{G}})$.
4. \[Item=CoamIV\] There exists a state $\epsilon: C_0({\mathbb{G}}) \rightarrow \mathbb{C}$ such that $(\epsilon \otimes id)(W) = 1$.
If $\mathbb{G}$ satisfies these criteria, then it is called [*coamenable*]{}. The notation is consistent in the sense that the nets $\{ \xi_i \}_i$ in and can be taken the same.
if and only if if and only if is proved in [@BedTus]. if and only if follows from a standard convexity argument and the fact that $(\omega_{\xi_i} \otimes id)(W^\ast) = (\omega_{\xi_i} \otimes id)(W)^\ast$.
\[Thm=Coamenability\] Let ${\mathbb{G}}= {SU_q(1,1)_{{\rm ext}}}$. For $n \in \mathbb{N}, p_1 \in q^{\mathbb{Z}}$ we define the unit vector, $$\xi_{n,p_1} = \frac{1}{\sqrt{2n+1}} \sum_{k=-n}^n f_{0, p_1 q^{k}, 1} \in L^2(\mathbb{G}).$$ Let $I = \mathbb{N} \times q^\mathbb{Z}$ and define a net structure on $I$ by saying that for $(n, p_1), (n', p_1') \in I$ we have $(n, p_1) \leq (n', p_1')$ if and only if $n \leq n'$ and $p_1q^n \leq p_1' q^{n'}$. Then, $
(\omega_{\xi_{n,p_1}} \otimes id)(W)\rightarrow 1
$ $\sigma$-weakly. That is, $\mathbb{G}$ is coamenable.
We start with computing explicit matrix coefficients of $W^\ast$. We take $p \in q^{\mathbb{Z}}$ and let $x \in [-1, 1] \cup \sigma_d(\Omega_p)$ so that $W_{p,x}$ is a corepresentation weakly contained in $W$ (i.e. it occurs on the Plancherel decomposition of $W$). For $p_1 \in q^\mathbb{Z}, k \in \mathbb{N}, m \in \mathbb{Z}, {\epsilon}, \eta \in \{-1, 1\}$, we find using [@GroKoeKus Lemma 10.7] and its short proof, that $$\begin{split}
& (\omega_{f_{0, p_1, 1}, f_{0, p_1 q^k, 1} } \otimes {\textrm{id}})(W^\ast_{p,x}) e_m^{{\epsilon}, \eta}(p,x) \\
= \: & \delta_{2m, k-\chi(p)} C({\epsilon}\eta x; m, {\epsilon}, \eta; p_1, p_1 q^k, 0) e_m^{{\epsilon}, \eta}(p,x).
\end{split}$$ To prevent tedious notation, we concentrate on $p = 1$ only. The reader may verify that for different $p\in q^{\mathbb{Z}}$ one gets similar (shifted) expressions. In case $ -n \leq m \leq 0$, $$(\omega_{\xi_{n,p_1}} \otimes id)(W^\ast_{1,x}) e_{m }^{{\epsilon}, \eta}(1,x)
= \frac{1}{2n+1} \sum_{p_0 = p_1 q^{n+2m} }^{ p_1q^{-n}}\!\!\! \!\! C({\epsilon}\eta x; m, {\epsilon}, \eta; p_0, p_0 q^{2m}, 0) e_{m }^{{\epsilon}, \eta}(1,x),$$ and in case $n \geq m \geq 0$, $$(\omega_{\xi_{n,p_1}} \otimes id)(W^\ast_{1,x}) e_m^{{\epsilon}, \eta}(p,x) =
\frac{1}{2n+1} \!\! \sum_{p_0 = p_1 q^{n-2m}}^{p_1q^{-n}}\!\!\! \!\! C({\epsilon}\eta x; m, {\epsilon}, \eta; p_0, p_0 q^{2m}, 0) e_m^{{\epsilon}, \eta}(p,x),$$ where the sums over $p_0$ take values in $q^{\mathbb{Z}}$ and hence, there are exactly $2(n-\vert m\vert )+1$ summands. Since $(2(n-\vert m\vert )+1)/(2n+1) \rightarrow 1$ as $n \rightarrow \infty$, it suffices to prove that $$C({\epsilon}\eta x; m, {\epsilon}, \eta; p_1, p_1 q^{2m}, 0) \rightarrow 1, \qquad \textrm{ as } p_1 \rightarrow \infty,$$ uniformly in $x$ on compact sets of $[-1, 1] \cup \sigma_d(\Omega_1)$, since by the Plancherel decomposition, this entails of Lemma \[Lem=CoamenabilityEquivalence\]. Indeed, as the separating set of functionals, one can for example take direct integrals over a compact index of finite linear combinations of inner product functionals with respect to the vectors $e^{{\epsilon},\eta}_m(p,x)$ (see also Remark \[Rmk=TypeI\]).
Let $\lambda \in \mathbb{C}$ be such that $\mu(\lambda) = x$. Then, see [@GroKoeKus Lemma 9.1] for the $S$-function, $$\label{Eqn=CoamenabilityCompI}
\begin{split}
& C({\epsilon}\eta x; m, {\epsilon}, \eta, p_1, p_1q^{2m}, 0)\\
= & S(-\lambda; p_1, p_1 q^{2m}, 0) \\
= & p_1^2 q^k \nu(p_1)\nu(p_1 q^{2m}) c_q^2 \sqrt{(-\kappa(p_1), -\kappa(p_1 q^{2m}); q^2 )_\infty } \\
& \times \:\frac{(q^2, -q^2/\kappa(p_1 q^{2m}), \lambda q^3/p_1^2q^{2m}, p_1^2 q^{{2m}-1}/\lambda, -q^{1-{2m}}/\lambda; q^2)_\infty )}{( p_1^2 q^{{2m}-1}/\lambda, \lambda q^3/p_1^2q^{2m}, -q^{1-{2m}}/\lambda; q^2)_\infty } \\
& \times \:(q^2; q^2)_\infty \: _2 \! \varphi_1\left( \begin{array}{c} -q^{1+{2m}}/\lambda, -\lambda q^{1+{2m}} \\ q^2 \end{array};q^2, -q^2/\kappa(p_1q^{2m}) \right)\\
= & p_1^2 q^{2m} \nu(p_1)\nu(p_1 q^{2m}) c_q^2 \sqrt{(-\kappa(p_1), -\kappa(p_1 q^{2m}); q^2 )_\infty } \\
& \times \: (q^2, -q^2/\kappa(p_1 q^{2m}); q^2)_\infty )
(q^2; q^2)_\infty \\
& \times \: _2 \! \varphi_1\left( \begin{array}{c} -q^{1+{2m}}/\lambda, -\lambda q^{1+{2m}} \\ q^2 \end{array};q^2, -q^2/\kappa(p_1q^{2m}) \right)
\end{split}$$ As $p_1 \rightarrow \infty$ we have that, $$_2 \! \varphi_1\left( \begin{array}{c} -q^{1+2m}/\lambda, -\lambda q^{1+2m} \\ q^2 \end{array};q^2, -q^2/\kappa(p_1q^{2m}) \right)\rightarrow 1,$$ uniformly on compact sets in $\lambda$ (such that $\mu(\lambda) \in [-1,1] \cup \sigma_d(\Omega_p)$). (That the convergence is uniform is well known. Alternatively, it can be derived from the Arzela-Ascoli theorem, which implies that it is enough to have a bounded sequence of locally analytic functions that converges pointwise). Hence, it remains to check that the coefficient of this function in converges to 1 as $p_1 \rightarrow \infty$. We find, putting $p_0 = q^l$ in the third equality, $$\label{Eqn=CoamenabilityCompII}
\begin{split}
& p_1^2 q^{2m} \nu(p_1)\nu(p_1 q^{2m}) c_q^2 \sqrt{(-\kappa(p_1), -\kappa(p_1 q^{2m}); q^2 )_\infty } \\
& \times \: (q^2, -q^2/\kappa(p_1 q^{2m}); q^2)_\infty )
(q^2; q^2)_\infty \\
= & c_q^2 (q^2; q^2)_\infty p_1^2 q^{2m} \nu(p_1)\nu(p_1 q^{2m}) \sqrt{( -\kappa(p_1 q^{2m}), -q^2/ \kappa(p_1 q^{2m}) ; q^2 )_\infty } \\
&\times \:
\sqrt{( -\kappa(p_1 ), -q^2/ \kappa(p_1 q^{2m}) ; q^2 )_\infty } \\
= & c_q^2 (q^2; q^2)_\infty p_1^2 q^{2m} \nu(p_1)\nu(p_1 q^{2m}) \sqrt{( -\kappa(p_1 q^{2m}), -q^2/ \kappa(p_1 q^{2m}) ; q^2 )_\infty } \\
& \times \:
\sqrt{( -\kappa(p_1 ), -q^2/ \kappa(p_1 ) ; q^2 )_\infty ( -q^2/p_1^2q^{4m};q^2 )_{2m} } \\
= & c_q^2 (q^2; q^2)_\infty p_1^2 q^{2m} \nu(p_1)\nu(p_1 q^{2m}) \sqrt{ q^{-({2m}+l)({2m}+l-1)} ( -1, -q^2 ; q^2 )_\infty } \\
& \times \:
\sqrt{ q^{-l(l-1)} (- 1, -q^2 ; q^2 )_\infty ( -q^2/p_1^2q^{4m};q^2 )_{2m} } \\
= & c_q^2 q^2 (q^2; q^2)_\infty^2 (-1,-q^2; q^2)_\infty \sqrt{ ( -q^2/p_1^2q^{4m};q^2 )_{2m} }\\
= & \sqrt{ ( -q^2/p_1^2q^{4m};q^2 )_{2m} }
\end{split}$$ Here, the first and second equality are elementary rearrangements of the terms, the third equality follows from the $\theta$-product identity, the fourth equality follows from an elementary computation using the replacement $p_1 = q^l$, the last equality follows from the definition of $c_q$ and the $\theta$-product identity. If $p_1 \rightarrow \infty$, we find that $ \sqrt{ ( -q^2/p_1^2q^{4m};q^2 )_{2m} } \rightarrow 1$ and the theorem follows.
\[Rmk=Reflexion\] It was pointed out to the author that reflexion of a quantum group (see [@ComI]) preserves coamenability. This can be proved from an unpublished result due to De Commer [@ComPhD Section 7.6]. Here, a Morita equivalence of both the universal and reduced C$^\ast$-algebraic quantum group and its reflection is established. Coamenability of ${SU_q(1,1)_{{\rm ext}}}$ follows then from [@ComII] and the well known fact that $SU_q(2)$ is coamenable. Since these results from [@ComPhD] are unpublished we give a direct proof here. It also gives the counit explicitly.
Spherical functions {#Sect=Spherical}
===================
Throughout the section we put ${\mathbb{G}}= {SU_q(1,1)_{{\rm ext}}}$. We construct matrix coefficients of ${\mathbb{G}}$ that form an approximate identity of $C_0({\mathbb{G}})$. For convenience, we set the corepresentations, where $z = ib$ with $0 \leq b \leq -\frac{\pi}{ \log(q)}$ (natural logarithm), $$V_z = W_{1, \mu(q^z)}, \quad {\mathcal{H}}_{z} = {\mathcal{H}}_{1,\mu(q^z) },$$ So $V_z \in L^\infty({\mathbb{G}}) \otimes B({\mathcal{H}}_z)$. We define the unit vector, $$\label{Eqn=State}
f_z = \frac{1}{2} \sqrt{2} \left( e^{+,-}_{0}(1,\mu(q^z)) + e^{-,+}_{0}(1,\mu(q^z)) \right) \in {\mathcal{H}}_{z},$$ and set, $$\label{Eqn=AzFunction}
\begin{split}
a_z & = ({\textrm{id}}\otimes \omega_{f_z, f_z})\left( V_z \right).
\end{split}$$ These are spherical matrix coefficients associated with the irreducible components of $V_z$, see [@Cas].
It follows from that $f_{m_0,p_0,t_0}$ is an eigenvector for $a_z$, where the eigenvalue is independent of $m_0$ and $t_0$. We let $a_z(p_0)$ be the eigenvalue of the vector $f_{m_0, p_0, t_0}$ for $a_z$. We will regard $a_z$ as a function on $I_q$ as well as an operator in $L^\infty({\mathbb{G}})$.
\[Prop=Analytic\]\[Thm=Approx\] We collect the following properties for $a_z$.
1. \[Item=AnalyticI\] There is a simply connected neighbourhood ${\mathcal{G}}$ of $i \mathbb{R} \cup [0, 1)$ such that for every $p_0 \in I_q$, the function $z\mapsto a_z(p_0)$ extends analytically to ${\mathcal{G}}$. Moreover, for every $\alpha > 1$ we can choose ${\mathcal{G}}= {\mathcal{G}}_\alpha$ such that $a_z$ is the matrix coefficient of a (possibly non-unitary) invertible corepresentation $V_z$ of ${\mathbb{G}}$ with, $$\Vert V_z \Vert \leq \alpha \quad \textrm{ and } \quad \Vert V_z^{-1} \Vert \leq \alpha.$$
2. \[Item=AnalyticII\] For every $z \in i \mathbb{R}$ we have $a_{z+ \frac{2i\pi}{\log(q)}} = a_{z}$.
3. \[Item=AnalyticIII\] For every $p_0 \in I_q $, $$\lim_{z \rightarrow 1} a_z(p_0) = 1.$$ Moreover, this convergence is uniform on $I_q \cap [1, \infty)$.
The first claim was observed in [@GroKoeKus Section 10.3]. That is, that $z\mapsto a_z(p_0)$ extends analytically to a neighbourhood of $i [0, -\frac{\pi }{\log(q)}] \cup [0, 1)$. The fact that the analytic domain ${\mathcal{G}}$ can be extended to a neighbourhood of the whole imaginary axis, is a consequence of the fact that $$\mu(q^{i t \log(q)}) = \mu(q^{i (2\pi - t) \log(q)}) = \mu(q^{i (t+2\pi) \log(q)}),$$ and using the Schwartz reflection principle. To apply this principle, we need to check that $a_z(p_0)$ takes real values for $z \in \frac{i \pi\mathbb{Z}}{ \log(q)} + (-\varepsilon, \varepsilon)$ for certain $\varepsilon >0$. But this follows from the explicit expressions of $a_z(p_0)$ which where computed along the proof of Proposition \[Prop=TwoPhiOneApproximation\].
Recall that for a Hilbert space ${\mathcal{H}}$, the invertible operators form an open subset of $B({\mathcal{H}})$. Since $z \mapsto V_z$ extends analytically to a neighbourhood of $i \mathbb{R} \cup [0, 1)$, and for $z \in i \mathbb{R} \cup [0, 1)$ the corepresentation $V_z$ is unitary [@GroKoeKus Section 10.3], we may choose the neighbourhood ${\mathcal{G}}_\alpha$ small; i.e. such that $\Vert V_z \Vert \leq \alpha$ and $\Vert V_z^{-1} \Vert \leq \alpha$.
The property follows directly from the symmetry argument in .
We postpone this proof to the appendix, see Proposition \[Prop=TwoPhiOneApproximation\].
It is unknown what the exact domain of ${\mathcal{G}}$ in Proposition \[Prop=Analytic\] is, we merely know its existence. Let us also indicate that as $q \rightarrow 1$, the periodicity tends to infinity, resulting in the classical limit. This was already observed in [@MasudaEtAl].
\[Thm=ApproxSU\] Let ${\mathbb{G}}= {SU_q(1,1)_{{\rm ext}}}$, let $a_z$ be the matrix coefficient defined by and its analytic extension to ${\mathcal{G}}_\alpha$ as in Proposition \[Prop=Analytic\]. For every $c \in C_0({\mathbb{G}})$ we have $\Vert a_z c - c \Vert_{L^\infty({\mathbb{G}})} \rightarrow 0$ as $z \rightarrow 1$ in ${\mathcal{G}}_\alpha$.
Recall that $L^\infty({\mathbb{G}})$ is given by $L^\infty(\mathbb{T}) \otimes B(L^2(I_q))$ which acts on the first two legs of $L^2({\mathbb{G}}) = L^2(\mathbb{Z}) \otimes L^2(I_q) \otimes L^2(I_q)$ by identifying $L^2(\mathbb{Z})$ with $L^2(\mathbb{T})$ under the Fourier transform, c.f. [@KoeKus Lemma 2.4]. In this representation $a_z$ corresponds to an operator in $1 \otimes B(L^2(I_q))$ and in fact the canonical basis $1 \otimes \delta_{p_0} \in L^2(\mathbb{T}) \otimes L^2(I_q)$ with $p_0 \in I_q$ forms a complete set of eigenvectors of $a_z$ with respective eigenvalues $a_z(p_0)$.
For $i \in \{ 1, 2\}$ we define, $$v_i = \int^\oplus_{[-1, 1]\cup \sigma_d(\Omega_p)} g_i(x) e^{{\epsilon}_i, \eta_i}_{m_i}(p_i,x) dx \in L^2({\mathbb{G}}),$$ for some ${\epsilon}_i, \eta_i \in \{ -,+\}, p_i \in q^{\mathbb{Z}}, m_i \in \mathbb{Z}$ and $g_i$ a square integrable function on $[-1, 1]\cup \sigma_d(\Omega_p)$. Set $x = ({\textrm{id}}\otimes \omega_{v_1, w_2})(W)$. Then, $x \in C_0({\mathbb{G}})$ and in fact the linear span of such elements $x$ forms a norm dense subset of $C_0({\mathbb{G}})$ see Remark \[Rmk=TypeI\].
Fix $m_0 \in \mathbb{Z}$, $t_0 \in I_q$. From one sees that there is at most one $m_2 \in \mathbb{Z}$ and $p_2 \in q^{\mathbb{Z}}$ and $\pm$ either $+$ or $-$ such that the following function is non-zero: $$\label{Eqn=HelpfulCoef}
\Phi_x: I_q \rightarrow \mathbb{C}: p_0 \mapsto \langle x f_{m_0, p_0, t_0}, f_{m_0+m_2, \pm p_0p_2, t_0} \rangle.$$ Using that the Haar weight, $${\rm Tr}(\: \cdot \:) \otimes \sum_{p_0 \in I_q} p_0^{-2} \langle \: \cdot \: \delta_{p_0}, \delta_{p_0}\rangle,$$ restricts to a semi-finite weight on $C_0({\mathbb{G}})$ we must have $\Phi_x(p_0) \rightarrow 0$ as $p_0 \rightarrow 0$, since else $x$ cannot be approximated in norm by square integrable elements of $C_0({\mathbb{G}})$.
Consider the projections $P_0: L^2(I_q) \rightarrow L^2(I_q \cup (-1, 1))$ and $P_1: L^2(I_q) \rightarrow L^2(I_q \cup [1, \infty))$. We decompose $x = P_0 x + P_1 x$ where $P_0 x, P_1 x \in L^\infty({\mathbb{G}})$. We need to prove that, $$\Vert a_z P_0 x - P_0 x \Vert_{L^\infty({\mathbb{G}})} \rightarrow 0 \textrm{ and } \Vert a_z P_1x - P_1x \Vert_{L^\infty({\mathbb{G}})} \rightarrow 0.$$ The right convergence follows since $\Vert a_z P_1 - P_1\Vert_{L^\infty({\mathbb{G}})} \rightarrow 0$ by Proposition \[Prop=Analytic\]. For the left convergence, it follows from that it suffices to show that $\Phi_{a_z x - x}(p_0) \rightarrow 0$ as $z \rightarrow 1$ uniformly for $p_0 \in (-1, 1) \cap I_q$. But this follows from Proposition \[Prop=Analytic\], the fact that $\Phi_x(p_0)$ tends to zero as $p_0 \rightarrow 0$ and the fact that $a_z(p_0)$ is bounded in $p_0$.
Weak amenability {#Sect=WeakAmenability}
================
Recall that we defined weak amenability in Section \[Sect=CompactaApproximation\]. Here, we prove that ${SU_q(1,1)_{{\rm ext}}}$ is weakly amenable. From this point, the proof is essentially the same as [@CanHaa Theorem 3.7]. The necessary modifications to lift the arguments from groups to quantum groups are presented in this section. In particular, we use coamenability as established in Section \[Sect=Coamenability\] to apply Theorem \[Thm=CompactaApproximation\].
Let us mention that in [@CowHaa] it is proved that every real rank one simple Lie group $G$ with finite center is weakly amenable with Cowling-Haagerup constant depending on the local isomorphism class of $G$. If the rank is greater than 1, $G$ is not weakly amenable [@Haa].
The most important examples of weakly amenable quantum groups come from Freslon’s result [@Freslon], showing that the free orthogonal and free unitary quantum groups of Kac type are weakly amenable.
The following lemma is the quantum group analogue of [@CanHaa Theorem 2.2]. It follows directly from [@BraDawSam Corollary 4.8]. See also [@Daws Proposition 4.1].
\[Lem=FellAbsorption\] Let ${\mathbb{G}}$ be a locally compact quantum group. Let $U \in L^\infty(\mathbb{G}) \otimes B({\mathcal{H}}_U)$ be an invertible corepresentation of ${\mathbb{G}}$. Let $\omega \in B({\mathcal{H}}_U)_\ast$ and set $a = (id \otimes \omega)(U)^\ast$. Then, $a \in {M_0A}({\mathbb{G}})$. Moreover, $\Vert a \Vert_{{M_0A}({\mathbb{G}})} \leq \Vert U \Vert \Vert U^{-1} \Vert \Vert \omega \Vert$.
The slice maps $\pi: L^1({\mathbb{G}}) \rightarrow L^\infty({\mathbb{G}}): \omega \mapsto (\omega \otimes id)(U)$ and $\check{\pi}: L^1({\mathbb{G}}) \rightarrow L^\infty({\mathbb{G}}): \omega \mapsto (\omega \otimes id)(U^{-1})$ are completely bounded and $\Vert \pi \Vert_{{\mathcal{CB}}} = \Vert U \Vert, \Vert \check{\pi} \Vert_{{\mathcal{CB}}} = \Vert U^{-1} \Vert$. Indeed, recall from [@EffRua Chapter 7] that, $${\mathcal{CB}}(L^1({\mathbb{G}}), B({\mathcal{H}})) \simeq (L^1({\mathbb{G}}) \hat{\otimes} {\mathcal{T}}({\mathcal{H}}))^\ast \simeq L^\infty({\mathbb{G}}) \otimes B({\mathcal{H}}),$$ where $\hat{\otimes}$ is the operator space projective tensor product. Moreover, the correspondence is given such that $\pi$ corresponds to $U$ and $\check{\pi}$ to $U^{-1}$. Hence, [@BraDawSam Corollary 4.8] yields that $a \in {M_0A}({\mathbb{G}})$ with bound $\Vert a \Vert_{{M_0A}({\mathbb{G}})} \leq \Vert U \Vert \Vert U^{-1} \Vert \Vert \omega \Vert$.
\[Cor=SUqOneMultipliers\] Let ${\mathbb{G}}= {SU_q(1,1)_{{\rm ext}}}$, $\alpha > 1$ and let $a_z = ({\textrm{id}}\otimes \omega_{f_z})(V_z) \in L^\infty({\mathbb{G}})$ with $z \in {\mathcal{G}}_\alpha$ be as in Proposition \[Prop=Analytic\]. Then, $a_z \in {M_0A}({\mathbb{G}})$ and $\Vert a_z \Vert_{{M_0A}({\mathbb{G}})} \leq \alpha^2$.
By Lemma \[Lem=FellAbsorption\] applied to the contragredient corepresentation of $V_z$, we see that $a_z \in {M_0A}({\mathbb{G}})$ and $\Vert a_z \Vert_{{M_0A}({\mathbb{G}})} \leq \alpha^2$.
\[Lem=ComP\] Let ${\mathbb{G}}= {SU_q(1,1)_{{\rm ext}}}$, $\alpha > 1$ and let $a_z \in L^\infty({\mathbb{G}})$ with $z \in {\mathcal{G}}_\alpha$ be as in Proposition \[Prop=Analytic\]. For $z \in [0,1]$ the multiplier $a_z \in {M_0A}({\mathbb{G}})$ is completely positive.
Recall that $a_z = ({\textrm{id}}\otimes \omega_{f_z})(V_z)$ and that for $z \in [0,1]$ the corepresentation $V_{z}$ is unitary (see [@GroKoeKus Section 10.3]). For every $\omega \in L^1({\mathbb{G}})^\sharp$ it follows that $$(\omega^\ast \otimes \omega)(V_z) = \left((\omega \otimes {\textrm{id}})(V_z)\right)^\ast (\omega \otimes {\textrm{id}})(V_z ) \geq 0.$$ Using [@DawSal Proposition 20] for coamenable quantum groups, we see that $a_z$ is in fact completely positive.
Along the following proof we use the well-known fact that a function $f: \mathbb{C} \rightarrow \mathcal{X}$ with $\mathcal{X}$ a Banach space is norm analytic if and only if for a separating set of functionals $\beta \in \mathcal{X}^\ast$ the function $z \mapsto \langle f, \beta \rangle_{\mathcal{X}, \mathcal{X}^\ast}$ is analytic.
Let ${\mathbb{G}}= SU_q(1,1)_{{\rm ext}}$. There exists a net $\{ a_i \}$ in $A({\mathbb{G}})$ such that for every $b \in A({\mathbb{G}})$ we have $\Vert a_i b - b \Vert_{A({\mathbb{G}})} \rightarrow 0$. Moreover, the net can be choosen such that, $$\limsup_{i \in I} \Vert a_i \Vert_{{M_0A}(\mathbb{G})} = 1.$$ That is, ${\mathbb{G}}$ is weakly amenable with Cowling-Haagerup constant $\Lambda({\mathbb{G}}) = 1$.
Let $\alpha > 1$ and let ${\mathcal{G}}_\alpha$ and $a_z$ be as in Proposition \[Prop=Analytic\]. By Corollary \[Cor=SUqOneMultipliers\] we find that $a_z \in {M_0A}({\mathbb{G}})$ with $\Vert a_z \Vert_{{M_0A}({\mathbb{G}})} \leq \alpha^2$. Define a path $\gamma_0: \mathbb{R} \rightarrow {\mathcal{G}}_\alpha: s \mapsto is$. For $k \in \mathbb{N}$, we fix a (continuous) path $\gamma_k: \mathbb{R} \rightarrow {\mathcal{G}}_\alpha$ that satisfies the following two criteria: (1) $\gamma_k(0) = 1- \frac{1}{k}$, (2) $\vert \gamma_1(s) - \gamma_0(s) \vert < 37$ for all $s \in \mathbb{R}$. Define, $$b_{k,n} = \sqrt{\frac{n}{\pi}} \int_{\gamma_k} e^{-n (z-1+\frac{1}{k})^2} a_{z} dz.$$ We claim that the integral exists as a Bochner integral in ${M_0A}({\mathbb{G}})$. Indeed, for $\omega \in L^1({\mathbb{G}})$ we see that $z \mapsto e^{-n (z-1+\frac{1}{k})^2}\omega( a_{z} )$ is analytic. Using [@HuNeuRua Theorem 3.4] (but then for left multipliers), this shows that $z \mapsto e^{-n (z-1+\frac{1}{k})^2} a_{z} $ is analytic in ${M_0A}({\mathbb{G}})$. Moreover, $b_{k,0} \in A({\mathbb{G}})$. We also set, $$b_k = a_{1-1/k}, \qquad k \in \mathbb{N}.$$ Recall that $\Vert a_z \Vert_{{M_0A}({\mathbb{G}})} \leq \alpha^2$ for every $z \in {\mathcal{G}}$. It follows from Cauchy’s theorem and assertion (2) on our path $\gamma_k$ that, $$\label{Eqn=CauchyTheorem}
b_{k,n} = \sqrt{\frac{n}{\pi}} \int_{\gamma_k} e^{-n (z-1+\frac{1}{k})^2} a_{z} dz = \sqrt{\frac{n}{\pi}} \int_{\gamma_0} e^{-n (z-1+\frac{1}{k})^2} a_{z} dz = b_{k,0} \in A({\mathbb{G}}),$$ where the integrals are understood within ${M_0A}({\mathbb{G}})$. Here, the second equality of is explained in full detail in [@CanHaa p. 482] and we leave the mutatis mutandis copy to the reader. Also, for every $k \in \mathbb{N}$, we find that as $n \rightarrow \infty$, $$\begin{split}
&\Vert b_{k,n} - b_k \Vert_{{M_0A}({\mathbb{G}})} \\
= &
\Vert \sqrt{\frac{n}{\pi}} \int_{\gamma_k} e^{-n(z-1+\frac{1}{k})^2} a_z dz - a_{1-1/k} \Vert_{{M_0A}({\mathbb{G}})} \\
\leq & \Vert \sqrt{\frac{n}{\pi}} \int_{\gamma_k} e^{-n(z-1+\frac{1}{k})^2} a_z - a_{1-1/k} dz \Vert_{{M_0A}({\mathbb{G}})} \\ & \: +\:
\Vert \left( \sqrt{\frac{n}{\pi}} \int_{\gamma_k} e^{-n(z-1+\frac{1}{k})^2} dz -1 \right)a_{1-1/k} \Vert_{{M_0A}({\mathbb{G}})} \\
\leq & \sqrt{\frac{n}{\pi}} \int_{\gamma_k} \vert e^{-n (z-1+\frac{1}{k})^2}\vert\: \Vert a_{z} -a_{1-1/k} \Vert_{{M_0A}({\mathbb{G}})} dz \\ & \: +\:
\vert \sqrt{\frac{n}{\pi}} \int_{\gamma_k} e^{-n(z-1+\frac{1}{k})^2} dz -1 \vert \Vert a_{1-1/k} \Vert_{{M_0A}({\mathbb{G}})} \\
\rightarrow &0.
\end{split}$$ This implies that for every $c \in A({\mathbb{G}})$ as $n \rightarrow \infty$, $$\Vert b_{k,n} c - b_k c \Vert_{A({\mathbb{G}})} \rightarrow 0.$$ Using Theorem \[Thm=ApproxSU\] we see furthermore that for every $c \in C_0({\mathbb{G}})$ we have, $$\Vert b_k c - c\Vert_{A({\mathbb{G}})} \rightarrow 0.$$ Note that $b_k$ is a positive multiplier by Lemma \[Lem=ComP\] and Theorem \[Thm=Coamenability\] shows that ${\mathbb{G}}$ is coamenable. Theorem \[Thm=CompactaApproximation\] then concludes our proof. Since we could choose $\alpha>1$ arbitrary, and $\Vert a_z \Vert_{{M_0A}({\mathbb{G}})} \leq \alpha^2$, we find that $\Lambda({\mathbb{G}}) \leq 1$.
Haagerup property {#Sect=Haagerup}
=================
Recently, the Haagerup property was introduced conceptually for locally compact quantum groups by Daws, Fima, Skalski and White, see [@DawFimSkaWhi]. Let us state the definition that is most convenient for us. See [@DawFimSkaWhi Theorem 5.5.(iii)].
Let ${\mathbb{G}}$ be a locally compact quantum group. ${\mathbb{G}}$ has the [*Haagerup property*]{} if there exists a net of states $\{\mu_i \}$ on $C^\ast_u({\mathbb{G}})$ such that $a_i := ({\textrm{id}}\otimes \mu_i)(\mathcal{V}) \in L^\infty({\mathbb{G}})$ satisfies the property that for every $c \in C_0({\mathbb{G}})$ we have $\Vert a_i c - c \Vert_{L^\infty({\mathbb{G}})} \rightarrow 0$.
Let us comment on the existing examples. Firstly, we recall the following proposition from [@DawFimSkaWhi Proposition 5.2].
\[Prop=ImpliesHaagerup\] Let ${\mathbb{G}}$ be a locally compact quantum group. If ${\mathbb{G}}$ is coamenble, then $\hat{{\mathbb{G}}}$ has the Haagerup property.
The question whether amenability of ${\mathbb{G}}$ implies Haagerup property remains open [@DawFimSkaWhi Remark 5.3].
Examples of quantum groups with the Haagerup property were found amongst the amenable and coamenable quantum groups. These include quantum $E(2)$, quantum $ax+b$ and quantum $az+b$ and their duals. See [@DawFimSkaWhi Example 5.4] and references given there.
Non-amenable examples so far come from discrete quantum groups. For the free orthogonal and free unitary quantum groups of Kac type, the Haagerup property was proved by Brannan [@Bran]. For quantum reflexion groups, the Haagerup property was found by Lemeux [@Lem].
As a consequence of what we have proved so far, we see that ${SU_q(1,1)_{{\rm ext}}}$ is a non-compact, non-amenable quantum group that has the Haagerup property.
\[Thm=Haagerup\] Let ${\mathbb{G}}= {SU_q(1,1)_{{\rm ext}}}$. ${\mathbb{G}}$ and $\hat{{\mathbb{G}}}$ have both the Haagerup property.
$\hat{{\mathbb{G}}}$ has the Haagerup property since ${\mathbb{G}}$ is coamenable, c.f. Theorem \[Thm=Coamenability\] and Proposition \[Prop=ImpliesHaagerup\].
Recall from that for $z \in [0,1)$ there exists a unitary corepresentation of ${\mathbb{G}}$, $$V_z \in L^\infty({\mathbb{G}}) \otimes B({\mathcal{H}}_{z}).$$ Let $C_z$ be the the C$^\ast$-algebra generated by $V_z$. It is the norm closure of the space spanned by slices $(\omega \otimes {\textrm{id}})(V_z)$ with $\omega \in L^1({\mathbb{G}})$. By [@KusUniv Proposition 5.3], there exists a non-degenerate $\ast$-homomorphism, $$\pi_z: C^\ast_u({\mathbb{G}}) \rightarrow \mathcal{M}(C_z) \quad \textrm{ such that } \quad V_z = ({\textrm{id}}\otimes \pi_z)(\mathcal{V}).$$ Here, $\mathcal{M}(C_z)$ is the multiplier algebra of $C_z$. Let the unit vector $f_z \in {\mathcal{H}}_{z}$ be as in and set $a_z = ({\textrm{id}}\otimes \omega_{f_z})(V_z)$ as in . Using the fact that $C_z$ acts non-degenerately on ${\mathcal{H}}_z$, it follows that $\omega_{f_z}$ is a state on $C_z$. Since $\pi_z$ is non-degenerate, it preserves bounded approximate identities. For a state $\omega$ on a C$^\ast$-algebra $C$, it follows from Cohen’s factorization theorem that $\Vert \omega \Vert = \omega(1) = \lim_j \omega(e_j)$, $\{ e_j\}$ being a bounded approximate identity of $C$ (here $\omega(1)$ is interpreted in the multiplier algebra of $C$). Hence, $\mu_z = \omega_{f_z} \circ \pi_z$ is a state on $C^\ast_u({\mathbb{G}})$ and $a_z = ({\textrm{id}}\otimes \mu_z)(\mathcal{V})$. Theorem \[Thm=ApproxSU\] proves that $\Vert a_z c - c\Vert_{L^\infty({\mathbb{G}})} \rightarrow 1$ for every $c \in C_0({\mathbb{G}})$ as $z \rightarrow 1$ (limit over the domain $[0,1)$).
We prove the necessary technical results, which we have not found explicitly in the literature. Firstly, we have the following reformulation of a result of [@KusVae].
\[Lem=ModularAppendix\] For every $t \in \mathbb{R}$ we have, $$(\sigma_t \otimes {\textrm{id}})(W) = ({\textrm{id}}\otimes \hat{\tau}_{-t})(W) (1 \otimes \hat{\delta}^{-it} ) \quad {\rm and } \quad
({\textrm{id}}\otimes \hat{\sigma}_t)(W) = (\delta^{it} \otimes 1) (\tau_{-t} \otimes {\textrm{id}})(W).$$
\[Lem=TomitaTransform\] Let $L^1({\mathbb{G}})^\natural$ be the set of $\omega \in L^1({\mathbb{G}})$ such that the following inclusions hold, $$(1) \:\: \omega \in L^1({\mathbb{G}})^\sharp, \quad (2) \:\:\omega \in {\mathcal{I}}, \quad (3)\:\: \omega^\ast \in {\mathcal{I}}.$$ Let $L^1({\mathbb{G}})^\flat$ be the set of all $\omega \in L^1({\mathbb{G}})^\natural$ such that $\lambda(\omega) \in {\mathcal{T}}_{\hat{\varphi}}$ and for every $z \in \mathbb{C}$ there exists a functional $\omega_{[z]} \in L^1({\mathbb{G}})^\natural$ with $\lambda(\omega_{[z]}) = \hat{\sigma}_z(\lambda(\omega))$. Then, $L^1({\mathbb{G}})^\flat$ is a $\ast$-algebra, dense in $L^1({\mathbb{G}})$. Moreover, $\lambda(L^1({\mathbb{G}})^\flat)$ is a $\sigma$-strong-$\ast$/norm core for $\hat{\Lambda}$. Furthermore, $\mathbb{C} \rightarrow L^1({\mathbb{G}}): z \mapsto \omega_{[z]}$ is analytic.
By [@KusVaeII Proposition 2.6] we see that $L^1({\mathbb{G}})^\natural$ is a $\ast$-subalgebra of $L^1({\mathbb{G}})^\sharp$, norm dense in $L^1({\mathbb{G}})$. And $\lambda(L^1({\mathbb{G}})^\natural)$ forms a $\sigma$-strong-$\ast$/norm core for $\hat{\Lambda}$.
Define the norm continuous one-parameter group $\rho$ acting on $L^1({\mathbb{G}})$ by $\rho_t(\omega)(x) = \omega(\delta^{-it}\tau_{-t}(x))$, where $t \in \mathbb{R}$, see [@KusVae Notation 8.7]. For $\omega \in {\mathcal{I}}$ we have $\rho_t(\omega) \in {\mathcal{I}}$ and $\hat{\Lambda}(\lambda( \rho_t(\omega) )) = P^{-it} J \delta^{it} J \hat{\Lambda}(\lambda(\omega))$ for every $t \in \mathbb{R}$, c.f. [@KusVae Lemma 8.8]. By [@KusVae Proposition 7.12] we have $S(\delta^{it})^\ast = \delta^{it}$. Since $\tau_t$ and $R$ commute for every $t \in \mathbb{R}$ we have that $S = R \circ \tau_{-i/2}$ commutes with $\tau_t$. Hence, for $\omega \in L^1({\mathbb{G}})^\sharp$ and $x \in D(S)$ we have that $$\begin{split}
& \langle S(x)^\ast, \rho_t(\omega) \rangle = \langle \delta^{-it} \tau_{-t} (S(x)^\ast), \omega \rangle = \langle \delta^{-it} S( \tau_{-t}(x) )^\ast, \omega \rangle \\
= &
\langle S(\delta^{-it} \tau_{-t}(x))^\ast , \omega \rangle
=
\overline{ \langle \delta^{-it} \tau_{-t}(x), \omega^\ast \rangle } = \overline{ \langle x, \rho_t(\omega^\ast) \rangle }
\end{split}$$ It follows that $\rho_t(\omega) \in L^1({\mathbb{G}})^\sharp$ with $\rho_t(\omega)^\ast = \rho_t(\omega^\ast)$.
Now, let $\omega \in L^1({\mathbb{G}})^\natural$. For each $t \in \mathbb{R}$, we have that $\rho_t(\omega) \in {\mathcal{I}}$ and $\rho_t(\omega) \in L^1({\mathbb{G}})^\sharp$. And also, $\rho_t(\omega)^\ast = \rho_t(\omega^\ast) \in {\mathcal{I}}$ as $\omega^\ast \in {\mathcal{I}}$. Thus $\rho_t(\omega) \in L^1({\mathbb{G}})^\natural$.
We now use a standard smearing argument, which we take from [@KusVaeII Lemma 2.5]. Set, $$\omega(n, z) = \frac{n}{\sqrt{\pi}} \int_{-\infty}^\infty e^{-n^2 (t+z)^2} \rho_t(\omega) dt.$$ From the closedness of the mapping $\omega \mapsto \hat{\Lambda}(\lambda(\omega))$ it follows that $\omega(n, z) \in {\mathcal{I}}$ with, $$\hat{\Lambda}( \lambda( \omega(n, z) )) = \frac{n}{\sqrt{\pi}} \int_{-\infty}^\infty e^{-n^2 (t+z)^2} P^{-it} J \delta^{it} J \hat{\Lambda}( \lambda(\omega) ) dt.$$ From the previous paragraph, we see that $\omega(n, z) \in L^1({\mathbb{G}})^\sharp$ with $\omega(n,z)^\ast = (\omega^\ast)(n, \overline{z})$. Hence, $\omega(n, z) \in L^1({\mathbb{G}})^\natural$. Moreover, $\omega(n, 0)$ is analytic for $\rho$ and $\rho_z( \omega(n, 0) ) = \omega(n, -z)$.
Now, [@KusVae Proposition 8.9] shows that $\lambda(\rho_t(\omega)) = \hat{\sigma}_t( \lambda(\omega))$. Hence, from the smearing techniques, we find that $\lambda(\omega(n, 0))$ is analytic with respect to $\hat{\sigma}$. Also, $\omega(n, z) \in {\mathcal{I}}$ and $\omega(n,z)^\ast \in {\mathcal{I}}$ for all $z$, so that $\lambda(\omega(n, z))$ is in ${\mathcal{T}}_{\hat{\varphi}}$. Since $\omega(n, 0) \rightarrow \omega$ in norm, we see that $L^1({\mathbb{G}})^\flat$ is dense in $L^1({\mathbb{G}})$. Moreover, we find that $\lambda(\omega(n, 0)) \rightarrow \lambda(\omega)$ in norm (and hence in the $\sigma$-strong-$\ast$ topology), and $\hat{\Lambda}(\lambda(\omega(n, 0) )) \rightarrow \hat{\Lambda}(\lambda(\omega))$ in norm. So indeed, $\lambda(L^1({\mathbb{G}})^\flat)$ is a $\sigma$-strong-$\ast$/norm core for $\hat{\Lambda}$. That $L^1({\mathbb{G}})^\flat$ is a $\ast$-algebra follows from [@KusVae Result 8.6] and the relation $(\omega \ast \theta)^\ast = \theta^\ast \ast \omega^\ast$.
It remains to prove that [*for every*]{} $\omega \in L^1({\mathbb{G}})^\flat$, the map $z \mapsto \omega_{[z]}$ is analytic. But for any $\theta \in L^1(\hat{{\mathbb{G}}})$, the map $z \mapsto \theta(\lambda(\omega_{[z]} )) = \theta(\hat{\sigma}_z(\lambda(\omega )))$ is analytic, which proves the claim since $\theta \circ \lambda$ with $\theta \in L^1(\hat{{\mathbb{G}}})$ forms a set of separating functionals on $L^1({\mathbb{G}})$.
\[Lem=L2Thingy\] Let $\theta \in L^1({\mathbb{G}})$ be such that $\lambda(\theta) \in L^2(\hat{{\mathbb{G}}}) \cap L^\infty(\hat{{\mathbb{G}}})$. Then, $\theta \in {\mathcal{I}}$.
{#Sect=AppB}
Here, we prove the necessary results on convergences of basic hypergeometric series.
\[Lem=LambdaFraction\] For every $k \in \mathbb{N} \cup \{ \infty\}$, we have the limit, $$\frac{(q/\lambda, q/\lambda; q^2)_k }{(1/\lambda^2; q^2)_k} \rightarrow 0,$$ as $\lambda \rightarrow q$.
For $k=1$ and $k=2$ this is trivial. For $k \geq 3$ we need to be more careful, because the $q$-factorial in the denominator becomes 0 as $\lambda \rightarrow q$. However, we have, $$\begin{split}
& \frac{(q/\lambda, q/\lambda; q^2)_k }{(1/\lambda^2; q^2)_k} = \prod_{i=0}^{k-1} \frac{(1- \frac{q}{\lambda} q^i)^2}{ 1 - \frac{1}{\lambda^2} q^i}, \\
= & \frac{(1-q/\lambda)^2}{(1- q^2/\lambda^2)} \frac{(1-q^2/\lambda)^2}{(1- 1/\lambda^2)} \frac{(1-q^3/\lambda)^2}{(1- q/\lambda^2)} \prod_{i=3}^{k-1} \frac{(1- \frac{q}{\lambda} q^i)^2}{ 1 - \frac{1}{\lambda^2} q^i}\\
= & \frac{ 1-q/\lambda }{1 + q^2/\lambda^2} \frac{(1-q^2/\lambda)^2}{(1- 1/\lambda^2)} \frac{(1-q^3/\lambda)^2}{(1- q/\lambda^2)} \prod_{i=3}^{k-1} \frac{(1- \frac{q}{\lambda} q^i)^2}{ 1 - \frac{1}{\lambda^2} q^i}.
\end{split}$$ As $\lambda \rightarrow q$, this term goes to zero.
\[Prop=TwoPhiOneApproximation\] Let $a_z: I_q \rightarrow \mathbb{C}$ be the function defined in . Let ${\mathcal{G}}_\alpha$ be the domain of Proposition \[Prop=Analytic\]. Then $a_z(p_0) \rightarrow 1$ pointwise as $z \rightarrow 1$ (limit in ${\mathcal{G}}_\alpha$). Moreover, this convergence is uniform on $I_q \cap [1, \infty)$.
Firstly, we record the following expression using [@GroKoeKus Lemma 9.4]. For every $p_0 \in I_q$, and $x = \mu(q^z) = \mu(\lambda)$ with $z \in {\mathcal{G}}_\alpha$, we have, $$\begin{split}
a_z(p_0) = & \frac{1}{2} C(-x; 0, +, -; p_0,p_0, 0) + \frac{1}{2} C(-x; 0, -, +; p_0,p_0, 0) \\
= & (-1)^{\frac{1}{2}(1-{{\rm sgn}}(p_0) ) } S(\lambda; p_0, p_0, 0) \\
& \: \times \: \left(\frac{A(-\lambda; 1, 0, {{\rm sgn}}(p_0), -{{\rm sgn}}(p_0)) }{A(-\lambda; 1, 0, +, -) } + \frac{A(-\lambda; 1, 0,-{{\rm sgn}}(p_0), {{\rm sgn}}(p_0)) }{A(-\lambda; 1, 0, -, +) } \right).
\end{split}$$ The fractions of the $A$-functions can be computed from [@GroKoeKus Appendix B.6] and one finds directly that, $$a_z(p_0) = (-1)^{\frac{1}{2}(1-{{\rm sgn}}(p_0) ) } S(\lambda; p_0, p_0, 0).$$ Next, we separate three cases. We prove that $a_z \rightarrow 1$ on each of the domains $q^{-\mathbb{N} \cup \{0 \}}$, $q^{\mathbb{N}}$, and $ - q^{\mathbb{N}}$ and that the convergence is uniform on the first one. Reason for the separation of cases is that we need to consider analytic extensions of basic hypergeometric $_2 \varphi _1$-series using the transformation formula [@GasRah Eqn. (III.32)] on different domains.
[**Case 1:**]{} The domain $p_0 \in q^{-\mathbb{N} \cup \{0 \}}$. In this case, using exactly the same computations as in and for $m= 0$, we see that, $$\begin{split}
a_z(p_0) = & \: _2 \! \varphi_1\left( \begin{array}{c} q/\lambda, \lambda q \\ q^2 \end{array};q^2, -q^2/\kappa(p_0) \right).
\end{split}$$ In case $z \rightarrow 1$ we see that $\lambda \rightarrow q$ and $a_z(p_0) \rightarrow 1$ uniformly for $p_0 \in q^{-\mathbb{N} \cup \{0 \}}$ (that the convergence is uniform follows directly from the defining power series of the basic hypergeometric series, which terminates for $\lambda = q$ as a constant function).
[**Case 2:**]{} The domain $p_0 \in q^{\mathbb{N}}$. In this case, we find using the expression [@GroKoeKus Lemma 9.1], the $\theta$-product formula and the transformation formula [@GasRah Eqn. (III.32)], $$\begin{split}
& a_z(p_0) \\
=
& S(\lambda, p_0, p_0, 0) \\
= & p_0^2 \nu(p_0)^2 c_q^2 (-p_0^2; q^2)_\infty (q^2, -q^2/p_0^2; q^2)_\infty (q^2; q^2)_\infty
\: _2 \! \varphi_1\left( \begin{array}{c} q/\lambda, \lambda q \\ q^2 \end{array};q^2, -q^2/\kappa(p_0) \right) \\
= & \: _2 \! \varphi_1\left( \begin{array}{c} q/\lambda, \lambda q \\ q^2 \end{array};q^2, -q^2/\kappa(p_0) \right)\\
= &
\frac{(q \lambda, q\lambda, -q^3/\lambda\kappa(p_0), - \lambda \kappa(p_0)/q; q^2 )_\infty }{(q^2, \lambda^2, -q^2/\kappa(p_0), -\kappa(p_0); q^2 )_\infty } \: _2 \! \varphi_1\left( \begin{array}{c} q/\lambda, q/\lambda \\ q^2/\lambda^2 \end{array};q^2, -\kappa(p_0) \right) \\
& \: + \:
\frac{(q /\lambda, q/\lambda, -q^3 \lambda / \kappa(p_0), - \kappa(p_0)/q \lambda; q^2 )_\infty }{(q^2, 1/ \lambda^2, -q^2/\kappa(p_0), -\kappa(p_0); q^2 )_\infty } \: _2 \! \varphi_1\left( \begin{array}{c} q \lambda, q \lambda \\ q^2 \lambda^2 \end{array};q^2, -\kappa(p_0) \right)\\
\end{split}$$ By Lemma \[Lem=LambdaFraction\] we see that the second summand goes to 0 if $\lambda \rightarrow q$. Hence, we find formally that, $$\label{Eqn=BasicHypComp}
\begin{split}
&\lim_{z \rightarrow 1} a_z(p_0) \\
= & \lim_{\lambda \rightarrow q}
\frac{(q \lambda, q\lambda, -q^3/\lambda\kappa(p_0), - \lambda \kappa(p_0)/q; q^2 )_\infty }{(q^2, \lambda^2, -q^2/\kappa(p_0), -\kappa(p_0); q^2 )_\infty } \\
& \: \times \: \: _2 \! \varphi_1\left( \begin{array}{c} q/\lambda, q/\lambda \\ q^2/\lambda^2 \end{array};q^2, -\kappa(p_0) \right), \\
= & \lim_{\lambda \rightarrow q} \: _2 \! \varphi_1\left( \begin{array}{c} q/\lambda, q/\lambda \\ q^2/\lambda^2 \end{array};q^2, -\kappa(p_0) \right), \\
\end{split}$$ however we have to justify the existence of the (latter) limit. In order to do this, we have for $k \in \mathbb{N}$, $$\frac{(q/\lambda, q/\lambda ; q^2)_k }{ (q^2/\lambda^2; q^2)_k } = \frac{ (1-\frac{q}{\lambda} )^2 }{1- \frac{q^2}{\lambda^2} } \prod_{i=1}^{k-1} \frac{ (1-\frac{q}{\lambda} q^i)^2 }{1- \frac{q^2}{\lambda^2} q^i } =
\frac{ 1-\frac{q}{\lambda} }{1 + \frac{q}{\lambda} } \prod_{i=1}^{k-1} \frac{ (1-\frac{q}{\lambda} q^i)^2 }{1- \frac{q^2}{\lambda^2} q^i }.$$ As $\lambda \rightarrow q$ this expression goes to zero. This means that the coefficients of the basic hypergeometric series in go to 0 as $\lambda \rightarrow q$, resulting in the constant function one. Thus $\lim_{z \rightarrow 1} a_z(p_0) = 1$ pointwise for $p_0 \in q^{\mathbb{N}}$.
[**Case 3:**]{} The domain $p_0 \in -q^{\mathbb{N}}$. We find that using [@GroKoeKus Lemma 9.1] and the transformation formula [@GasRah Eqn. (III.32)] that, $$\label{Eqn=BasicSeriesCompII}
\begin{split}
& - a_z(p_0) \\
= & p_0^2 \nu(p_0)^2 c_q^2 (- \kappa(p_0) ; q^2)_\infty
\frac{( q^2, -q^2/\kappa(p_0), -\lambda q^3/p_0^2, - p_0^2/q\lambda; q^2 )_\infty}{( p_0^2/q\lambda, \lambda q^3/p_0^2; q^2 )_\infty } (q^2; q^2)_\infty\\
& \: \times \: \: _2 \! \varphi_1\left( \begin{array}{c} q/\lambda, q \lambda \\ q^2 \end{array};q^2, -q^2/\kappa(p_0) \right) \\
= & p_0^2 \nu(p_0)^2 c_q^2 (- \kappa(p_0) ; q^2)_\infty (q^2; q^2)_\infty^2 \frac{(q^2/p_0^2, -\lambda q^3/p_0^2, -p_0^2/q\lambda; q^2 )_\infty }{ (p_0^2/q\lambda, \lambda q^3/p_0^2; q^2)_\infty } \\
& \: \times \: \left(
\frac{( \lambda q, \lambda q, -q^3/\lambda \kappa(p_0), -\lambda \kappa(p_0) /q; q^2 )_\infty}{( q^2, \lambda^2, -q^2/ \kappa(p_0), - \kappa(p_0) ; q^2 )_\infty } \: _2 \! \varphi_1\left( \begin{array}{c} q/\lambda, q /\lambda \\ q^2/\lambda^2 \end{array};q^2, -\kappa(p_0) \right)\right. \\
& \: + \: \left.
\frac{( q/\lambda , q/\lambda , -q^3\lambda/ \kappa(p_0), -\kappa(p_0) / \lambda q; q^2 )_\infty}{( q^2, 1/\lambda^2, -q^2/ \kappa(p_0), - \kappa(p_0) ; q^2 )_\infty } \: _2 \! \varphi_1\left( \begin{array}{c} q \lambda, q \lambda \\ q^2 \lambda^2 \end{array};q^2, -\kappa(p_0) \right)
\right)
\end{split}$$ As in [**Case 2**]{} it follows from Lemma \[Lem=LambdaFraction\] that the second summand within the large brackets of tends to zero as $\lambda \rightarrow q$. Therefore, using that $p_0 := -q^k$ is negative, and the $\theta$-product identity various times, $$\begin{split}
& \lim_{z \rightarrow 1} - a_z(p_0) \\
= & \lim_{\lambda \rightarrow q} p_0^2 \nu(p_0)^2 c_q^2 (- \kappa(p_0) ; q^2)_\infty (q^2; q^2)_\infty^2 \frac{(q^2/p_0^2, -\lambda q^3/p_0^2, -p_0^2/q\lambda; q^2 )_\infty }{ (p_0^2/q\lambda, \lambda q^3/p_0^2; q^2)_\infty } \\
& \: \times \:
\frac{( \lambda q, \lambda q, -q^3/\lambda \kappa(p_0), -\lambda \kappa(p_0) /q; q^2 )_\infty}{( q^2, \lambda^2, -q^2/ \kappa(p_0), - \kappa(p_0) ; q^2 )_\infty } \: _2 \! \varphi_1\left( \begin{array}{c} q/\lambda, q /\lambda \\ q^2/\lambda^2 \end{array};q^2, -\kappa(p_0) \right) \\
= & \lim_{\lambda \rightarrow q} p_0^2 \nu(p_0)^2 c_q^2 (p_0^2 ; q^2)_\infty (q^2; q^2)_\infty^2 (-q^4/p_0^2, -p_0^2/q^2; q^2)_\infty \frac{(\lambda q, \lambda q;q^2)_\infty }{ (q^2, \lambda^2;q^2)_\infty } \\
& \: \times \: \frac{(q^3/\lambda p_0^2, \lambda p_0^2/q; q^2)_\infty }{(q^3\lambda/ p_0^2, p_0^2/q \lambda ; q^2)_\infty } \frac{1}{(p_0^2; q^2)_\infty} \: _2 \! \varphi_1\left( \begin{array}{c} q/\lambda, q /\lambda \\ q^2/\lambda^2 \end{array};q^2, -\kappa(p_0) \right) \\
= & \lim_{\lambda \rightarrow q} p_0^2 \nu(p_0)^2 c_q^2 (p_0^2 ; q^2)_\infty (q^2; q^2)_\infty (-q^4/p_0^2, -p_0^2/q^2; q^2)_\infty \frac{(q^2, q^2; q^2)_\infty}{ (q^2;q^2)_\infty} \\
& \: \times \: \frac{1-\frac{q^2}{p_0^2}}{1-\frac{p_0^2}{q^2}} \frac{1}{(p_0^2; q^2)_\infty} \: _2 \! \varphi_1\left( \begin{array}{c} q/\lambda, q /\lambda \\ q^2/\lambda^2 \end{array};q^2, -\kappa(p_0) \right)\\
= & p_0^2 \nu(p_0)^2 c_q^2 (q^2; q^2)_\infty (-q^4/p_0^2, -p_0^2/q^2; q^2)_\infty \frac{(q^2, q^2; q^2)_\infty}{ (q^2;q^2)_\infty} \frac{q^2}{p_0^2} \frac{p_0^2 - q^2}{q^2 - p_0^2} \\
= & - q^2 q^{(k-1)(k-2)} c_q^2 (q^2; q^2)_\infty^2 (-1, -q^2; q^2)_\infty q^{-(k-1)(k-2)} \\
= & -c_q^2 (q^2; q^2)_\infty^2 (-1, -q^2; q^2)_\infty q^2\\
= & -1.
\end{split}$$
[**Acknowledgements.**]{} The author thanks Wolter Groenevelt for useful correspondence on the corepresentation spectrum of extended quantum $SU(1,1)$. The author thanks Erik Koelink for several indispensable discussions. The author also thanks the IMAPP institute at the Radboud Universiteit Nijmegen for their hospitality; the final part of this paper was completed there. We thank Yuki Arano for pointing out Remark \[Rmk=Reflexion\]. We thank the anonymous referee for several improvements of the manuscript.
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abstract: 'Dujmović et al. (FOCS 2019) recently proved that every planar graph is a subgraph of the strong product of a graph of bounded treewidth and a path. Analogous results were obtained for graphs of bounded Euler genus or apex-minor-free graphs. These tools have been used to solve longstanding problems on queue layouts, non-repetitive colouring, $p$-centered colouring, and adjacency labelling. This paper proves analogous product structure theorems for various non-minor-closed classes. One noteable example is $k$-planar graphs (those with a drawing in the plane in which each edge is involved in at most $k$ crossings). We prove that every $k$-planar graph is a subgraph of the strong product of a graph of treewidth $O(k^5)$ and a path. This is the first result of this type for a non-minor-closed class of graphs. It implies, amongst other results, that $k$-planar graphs have non-repetitive chromatic number upper-bounded by a function of $k$. All these results generalise for drawings of graphs on arbitrary surfaces. In fact, we work in a much more general setting based on so-called shortcut systems that are of independent interest. This leads to analogous results for map graphs, string graphs, graph powers, and nearest neighbour graphs.'
author:
- 'Vida Dujmović [^1], Pat Morin [^2], and David R. Wood[^3]'
---
Introduction {#Introduction}
============
The starting point for this work is the following ‘product structure theorem’ for planar graphs[^4] by @dujmovic.joret.ea:planar.
\[PlanarProduct\] Every planar graph is a subgraph of:
$H\boxtimes P$ for some graph $H$ of treewidth at most $8$ and for some path $P$,
$H\boxtimes P \boxtimes K_3$ for some graph $H$ of treewidth at most $3$ and for some path $P$.
Here $\boxtimes$ is the strong product[^5], and treewidth[^6] is an invariant that measures how ‘tree-like’ a given graph is; see \[ProductExample\] for an example. Loosely speaking, \[PlanarProduct\] says that every planar graph is contained in the product of a tree-like graph and a path. This enables combinatorial results for graphs of bounded treewidth to be generalised for planar graphs (with different constants).
![Example of a strong product. \[ProductExample\]](ProductExample)
\[PlanarProduct\] has been the key tool in solving the following well-known open problems:
@dujmovic.joret.ea:planar use it to prove that planar graphs have bounded queue-number (resolving a conjecture of @HLR92 from 1992).
@dujmovic.esperet.ea:planar use it to prove that planar graphs have bounded non-repetitive chromatic number (resolving a conjecture of @AGHR-RSA02 from 2002).
@debski.felsner.ea:improved use it to make dramatic improvements to the best known bounds for $p$-centered colourings of planar graphs.
@bonamy.gavoille.ea:shorter use it to find shorter adjacency labellings of planar graphs (improving on a sequence of results going back to 1988 [@kannan.naor.ea:implicit-stoc; @kannan.naor.ea:implicit]), and @DEJGMM have since used it to find asymptotically optimal adjacency labellings of planar graphs.
The result of @DEJGMM implies that, for every integer $n>0$, there is a *universal* graph $U_n$ with $n^{1+o(1)}$ vertices such that every $n$-vertex planar graph is an induced subgraph of $U_n$.
All of these results hold for any graph class that has a product structure theorem analogous to \[PlanarProduct\]; that is, for any graph class $\mathcal{G}$ where every graph in $\mathcal{G}$ is a subgraph of $H\boxtimes P\boxtimes K_\ell$ where $H$ has bounded treewidth, $P$ is a path, and $\ell$ is bounded[^7]. These applications motivate finding product structure theorems for other graph classes. @dujmovic.joret.ea:planar prove product structure theorems for graphs of bounded Euler genus[^8] and for apex-minor-free graphs[^9], and @DEMWW do so for graphs in any minor-closed class and with bounded maximum degree. See \[Generalisations\] for more precise statements and see [@DHGLW] for a survey on this topic.
The purpose of this paper is to prove product structure theorems for several non-minor-closed classes of interest. Our results are the first of this type for non-minor-closed classes.
$k$-Planar Graphs
-----------------
We start with the example of $k$-planar graphs. A graph is *$k$-planar* if it has a drawing in the plane in which each edge is involved in at most $k$ crossings. Such graphs provide a natural generalisation of planar graphs, and are important in graph drawing research; see the recent bibliography on 1-planar graphs and the 140 references therein [@kobourov.liotta.ea:annotated]. It is well-known that the family of $k$-planar graphs is not minor-closed. Indeed, 1-planar graphs may contain arbitrarily large complete graph minors [@dujmovic.eppstein.ea:structure]. Hence the above results are not applicable for $k$-planar graphs. We extend \[PlanarProduct\] as follows.
\[kPlanarProduct\] Every $k$-planar graph is a subgraph of $H\boxtimes P\boxtimes K_{18k^2+48k+30}$, for some graph $H$ of treewidth $\binom{k+4}{3}-1$ and for some path $P$.
This theorem has applications in diverse areas, including queue layouts [@dujmovic.joret.ea:planar], non-repetitive colouring [@dujmovic.esperet.ea:planar], $p$-centered colouring [@debski.felsner.ea:improved], and adjacency labelling [@DEJGMM], which we explore in \[Applications\]. For example, we prove that $k$-planar graphs have bounded non-repetitive chromatic number (for fixed $k$). Prior to the recent work of @dujmovic.esperet.ea:planar, it was even open whether planar graphs have bounded non-repetitive chromatic number.
Shortcut Systems
----------------
Although $k$-planar graphs are the most high-profile target for a generalization of \[PlanarProduct\], we actually prove a substantially stronger result than \[kPlanarProduct\] using the following definition. A non-empty set $\mathcal{P}$ of non-trivial paths in a graph $G$ is a *$(k,d)$-shortcut system* (for $G$) if:
every path in $\mathcal{P}$ has length at most $k$,[^10] and
for every $v\in V(G)$, the number of paths in $\mathcal{P}$ that use $v$ as an internal vertex is at most $d$.
Each path $P\in\mathcal{P}$ is called a *shortcut*; if $P$ has endpoints $v$ and $w$ then it is a *$vw$-shortcut*. Given a graph $G$ and a $(k,d)$-shortcut system $\mathcal{P}$ for $G$, let $G^{\mathcal{P}}$ denote the supergraph of $G$ obtained by adding the edge $vw$ for each $vw$-shortcut in $\mathcal{P}$.
This definition is related to $k$-planarity because of the following observation:
\[AddDummy\] Every $k$-planar graph is a subgraph of $G^{\mathcal{P}}$ for some planar graph $G$ and some $(k+1,2)$-shortcut system ${\mathcal{P}}$ for $G$.
The proof of \[AddDummy\] is trivial: Given a $k$-plane embedding of a graph $G'$, create a planar graph $G$ by adding a dummy vertex at each crossing point. For each edge $vw\in E(G')$ there is a path $P$ in $G$ between $v$ and $w$ of length at most $k+1$. Let ${\mathcal{P}}$ be the set of such paths $P$. For each vertex $v$ of $G$, at most two paths in ${\mathcal{P}}$ use $v$ as an internal vertex (since no original vertex of $G'$ is an internal vertex of a path in ${\mathcal{P}}$). Thus ${\mathcal{P}}$ is a $(k+1,2)$-shortcut system for $G$, such that $G'\subseteq G^{\mathcal{P}}$. This idea can be pushed further to obtain a rough characterisation of $k$-planar graphs, which is interesting in its own right, and is useful for showing that various classes of graphs are $k$-planar (see \[Characterisation\]).
The following theorem is the main contribution of the paper. It says that if a graph class $\mathcal{G}$ has a product structure theorem, then the class of graphs obtained by applying a shortcut system to graphs in $\mathcal{G}$ also has a product structure theorem.
\[ShortcutProduct\] Let $G$ be a subgraph of $H\boxtimes P \boxtimes K_\ell$, for some graph $H$ of treewidth at most $t$ and for some path $P$. Let ${\mathcal{P}}$ be a $(k,d)$-shortcut system for $G$. Then $G^{\mathcal{P}}$ is a subgraph of $J\boxtimes P\boxtimes K_{d\ell(k^3+3k)}$ for some graph $J$ of treewidth at most $\binom{k+t}{t}-1$ and some path $P$.
Theorems \[PlanarProduct\](b) and \[ShortcutProduct\] and \[AddDummy\] imply \[kPlanarProduct\] with $K_{6(k^3+3k)}$ instead of $K_{18k^2+48k+30}$. Some further observations presented in \[sec-k-planar\] lead to the improved result.
\[ShortcutProduct\] is applicable for many graph classes in addition to $k$-planar graphs. Here is one example. The *$k$-th power* of a graph $G$ is the graph $G^k$ with vertex set $V(G^k):=V(G)$, where $vw\in E(G^k)$ if and only if $\operatorname{dist}_G(v,w){\leqslant}k$.[^11] If $G$ has maximum degree $\Delta$, then $G^k = G^{\mathcal{P}}$ for some $(k,2k\Delta^{k})$-shortcut system ${\mathcal{P}}$; see \[PowerShortcut\]. Theorems \[PlanarProduct\](b) and \[ShortcutProduct\] then imply:
\[kPowerBasic\] For every planar graph $G$ with maximum degree $\Delta$ and for every integer $k{\geqslant}1$, $G^k$ is a subgraph of $H\boxtimes P\boxtimes K_{6k^2(k^2+3)\Delta^{k}}$ for some graph $H$ of treewidth at most $\binom{k+3}{3}-1$ and some path $P$.
\[Examples\] presents further examples of graph classes that can be constructed using shortcut systems, including map graphs, string graphs, and $k$-nearest neighbour graphs. \[ShortcutProduct\] implies product structure theorems for each of these classes. All of the above-mentioned applications also hold for these examples.
Generalisations {#Generalisations}
---------------
As mentioned above, product structure theorems have been established for several minor-closed classes in addition to planar graphs. The first generalises \[PlanarProduct\] for graphs of bounded Euler genus.
\[GenusProduct\] Every graph of Euler genus $g$ is a subgraph of:
$H \boxtimes P$ for some graph $H$ of treewidth at most $2g+8$ and some path $P$.
$H \boxtimes P \boxtimes K_{\max\{2g,3\}}$ for some graph $H$ of treewidth at most $4$ and for some path $P$.\[temp\]
@dujmovic.joret.ea:planar generalised \[GenusProduct\] for apex-minor-free graphs as follows.
\[ApexMinorFree\] For every apex graph $X$, there exists $c\in\mathbb{N}$ such that every $X$-minor-free graph is a subgraph of $H\boxtimes P$ for some graph $H$ with $\operatorname{tw}(H){\leqslant}c$ and some path $P$.
The assumption that $X$ is apex is needed in \[ApexMinorFree\], since if the class of $X$-minor-free graphs has a product structure theorem analogous to \[PlanarProduct\], then $X$ is apex [@dujmovic.joret.ea:planar]. On the other hand, @DEMWW proved a product structure theorem for bounded degree graphs in any minor-closed class.
\[MinorFreeDegree\] For every graph $X$ there exists $c\in\mathbb{N}$ such that for every $\Delta\in\mathbb{N}$, every $X$-minor-free graph $G$ with maximum degree at most $\Delta$ is a subgraph of $H\boxtimes P$ for some graph $H$ with $\operatorname{tw}(H) {\leqslant}c\Delta$ and for some path $P$.
Layered Partitions {#LayeredPartitions}
------------------
While strong products enable concise statements of the theorems in \[Introduction\], to prove such results it is helpful to work with layerings and partitions, which we now introduce.
A *layering* of a graph $G$ is a sequence $\mathcal{L}=\langle L_0,L_1,\ldots\rangle$ such that $\{L_0,L_1,\ldots\}$ is a partition of $V(G)$ and for every edge $vw\in E(G)$, if $v\in L_i$ and $w\in L_j$ then $|j-i|{\leqslant}1$. For any partition $\mathcal{P}=\{S_1,\ldots,S_p\}$ of $V(G)$, a *quotient graph* $H=G/\mathcal{P}$ has a $p$-element vertex set $V(H)=\{x_1,\ldots,x_p\}$ and $x_ix_j\in E(H)$ if and only if there exists an edge $vw\in E(G)$ such that $v\in S_i$ and $w\in S_j$. To highlight the importance of the quotient graph $H$, we call $\mathcal{P}$ an *$H$-partition* and write this concisely as $\mathcal{P}=\{S_x : x\in V(H)\}$ so that each element of $\mathcal{P}$ is indexed by the vertex it creates in $H$.
For any partition $\mathcal{P}$ of $V(G)$ and any layering $\mathcal{L}$ of $G$ we define the *layered width* of $\mathcal{P}$ with respect to $\mathcal{L}$ as $\max\{|L\cap P|: L\in\mathcal{L},\, P\in\mathcal{P}\}$. For any partition $\mathcal{P}$ of $V(G)$, we define the *layered width* of $\mathcal{P}$ as the minimum, over all layerings $\mathcal{L}$ of $G$, of the layered width of $\mathcal{P}$ with respect to $\mathcal{L}$.
@dujmovic.joret.ea:planar introduced the study of partitions with bounded layered width such that the quotient has some additional desirable property, like small treewidth. Dujmović et al. define a class $\mathcal{G}$ of graphs to *admit bounded layered partitions* if there exist $t,\ell\in\mathbb{N}$ such that every graph $G\in \mathcal{G}$ has an $H$-partition of layered width at most $\ell$ for some graph $H=H(G)$ of treewidth at most $t$.
These definitions relate to strong products as follows.
\[PartitionProduct\] For every graph $H$, a graph $G$ has an $H$-partition of layered width at most $\ell$ if and only if $G$ is a subgraph of $H \boxtimes P \boxtimes K_\ell$ for some path $P$.
As an example of the use of layered partitions, to prove \[PlanarProduct\](a), @dujmovic.joret.ea:planar showed that every planar graph has an $H$-partition of layered width $1$ for some planar graph $H$ of treewidth at most $8$. The proof is constructive and gives a simple quadratic-time algorithm for finding the corresponding partition and layering. At the core of their work is the elegant proof by @pilipczuk.siebertz:polynomial-soda of the following result:
\[ps\] Every planar triangulation $G$ has an $H$-partition $\mathcal{P}$ such that $\operatorname{tw}(H){\leqslant}8$ and $G[P]$ is a shortest path in $G$ for each $P\in\mathcal{P}$.
Indeed, the above-mentioned result of @dujmovic.joret.ea:planar is a slight strengthening of \[ps\], where for each $P\in\mathcal{P}$ no two vertices of $P$ have the same distance to some fixed root vertex $r$.
The following result is the main technical contribution of the paper. Loosely speaking, it shows that if a graph $G$ admits bounded layered partitions, then so too does $G^{\mathcal{P}}$ for every shortcut system ${\mathcal{P}}$ of $G$.
[thm]{}[mmg]{} \[ShortcutPartition\] Let $G$ be a graph having an $H$-partition of layered width $\ell$ in which $H$ has treewidth at most $t$ and let ${\mathcal{P}}$ be a $(k,d)$-shortcut system for $G$. Then $G^{\mathcal{P}}$ has a $J$-partition of layered width at most $d\ell(k^3+3k)$ for some graph $J$ of treewidth at most $\binom{k+t}{t}-1$.
Note that \[ShortcutPartition\] is equivalent to \[ShortcutProduct\] by \[PartitionProduct\].
Shortcut Systems {#Structure}
================
The purpose of this section is to prove our main technical result, \[ShortcutPartition\]. This theorem shows how, given a $(k,d)$-shortcut system $\mathcal{P}$ of a graph $G$, a $H$-partition of $G$ can be used to obtain a $J$-partition of $G^{\mathcal{P}}$ where the layered width does not increase dramatically and the treewidth of $J$ is not much more than the treewidth of $H$.
For convenience, it will be helpful to assume that $\mathcal{P}$ contains a length-1 $vw$-shortcut for every edge $vw\in E(G)$. Since $G^{\mathcal{P}}$ is defined to be a supergraph of $G$, this assumption has no effect on $G^{\mathcal{P}}$ but eliminates special cases in some of our proofs.
For a tree $T$ rooted at some node $x_0\in V(T)$, we say that a node $a\in V(T)$ is a *$T$-ancestor* of $x\in V(T)$ (and $x$ is a *$T$-descendant* of $a$) if $a$ is a vertex of the path, in $T$, from $x_0$ to $x$. Note that each node $x\in V(T)$ is a $T$-ancestor and $T$-descendant of itself. We say that a $T$-ancestor $a\in V(T)$ of $x\in V(T)$ is a *strict* $T$-ancestor of $x$ if $a\neq x$. The *$T$-depth* of a node $x\in V(T)$ is the length of the path, in $T$, from $x_0$ to $x$. For each node $x\in V(T)$, define $$T_x := T[\{y\in V(T):\mbox{$x$ is a $T$-ancestor of $y$}\}]$$ to be the maximal subtree of $T$ rooted at $x$.
We begin with a standard technique that allows us to work with a normalised tree decomposition:
For every graph $H$ of treewidth $t$, there is a rooted tree $T$ with $V(T)=V(H)$ and a width-$t$ $T$-decomposition $(B_x:x\in V(T))$ of $H$ that has following additional properties:
[\[t:subtree-root\]]{} for each node $x\in V(H)$, the subtree $T[x]:=T[\{y\in V(T):x\in B_y\}]$ is rooted at $x$; and consequently
[\[t:ancestor-edge\]]{}[\[t:last\]]{} for each edge $xy\in E(H)$, one of $x$ or $y$ is a $T$-ancestor of the other.
That [(T\[t:subtree-root\])]{} implies [(T\[t:ancestor-edge\])]{} is a standard observation: If two subtrees intersect, then one contains the root of the other. Thus, it suffices to construct width-$t$ tree decomposition that satisfies [(T\[t:ancestor-edge\])]{}.
Begin with any width-$t$ tree decomposition $(B_x:x\in V(T_0))$ of $H$ that uses some tree $T_0$. Select any node $x\in V(T_0)$, add a leaf $x_0$, with $B_{x_0}=\emptyset$, adjacent to $x$ and root $T_0$ at $x_0$. (The purpose of $x_0$ is to ensure that every node $x$ for which $B_x$ is non-empty has a parent.) Let $f:V(H)\to V(T)$ be the function that maps each $x\in V(H)$ onto the root of the subtree $T_0[x]:=T_0[\{y\in V(T_0): x\in B_y\}]$. If $f$ is not one-to-one, then select some distinct pair $x,y\in V(H)$ with $a:=f(x)=f(y)$. Subdivide the edge between $a$ and its parent in $T$ by introducing a new node $a'$ with $B_{a'}=B_{a}\setminus\{x\}$. This modification reduces the number of distinct pairs $x,y\in V(H)$ with $f(x)=f(y)$, so repeatedly performing this modification will eventually produce a tree decomposition $(B_x:x\in V(T_0))$ of $H$ in which $f$ is one-to-one.
Next, consider any node $a\in V(T_0)$ such that there is no vertex $x\in V(H)$ with $f(x)=a$. In this case, $B_{a}\subseteq B_{a'}$ where $a'$ is the parent of $a$ since any $x\in B_a\setminus B_{a'}$ would have $f(x)=a$. In this case, contract the edge $aa'$ in $T_0$, eliminating the node $a$. Repeating this operation will eventually produce a width-$t$ tree decomposition of $(B_x:x\in V(T_0))$ where $f$ is a bijection between $V(H)$ and $V(T_0)$. Renaming each node $a\in V(T_0)$ as $f^{-1}(a)$ gives a tree decomposition $(B_x:x\in V(T))$ with $V(T)=V(H)$. By the definition of $f$, the tree decomposition $(B_x:x\in V(T))$ satisfies [(T\[t:subtree-root\])]{}.
![The sets $Y_x$, $F_x$, and $V_x$ associated with $x\in V(T)$ and the ancestors $a_1,\ldots,a_{t'}$ of $X$ such that $F_x \subseteq \bigcup_{i=1}^{t'} Y_{a_i}$.[]{data-label="fig:generalized-tripod"}](figs/tripoddo)
The following lemma shows how to interpret an $H$-partition of $G$ and a tree decomposition of $H$ as a hierarchical decomposition of $G$; refer to \[fig:generalized-tripod\].
\[generalized-tripod\] Let $G$ be a graph; let $\mathcal{L}:=\langle L_1,\ldots,L_h\rangle$ be a layering of $G$; let $\mathcal{Y}:=(Y_x: x\in V(H))$ be an $H$-partition of $G$ of layered width at most $\ell$ with respect to $\mathcal{L}$ where $H$ has treewidth at most $t$; and let $\mathcal{T}:=(B_x:x\in V(T))$ be a tree decomposition of $H$ satisfying the conditions of . For each $x\in V(T)$, let $V_x := \bigcup_{y\in V(T_x)} Y_y$, $F_x:=\{w\in V(G): vw\in E(G), v\in V_x,\, w\not\in V_x\}$, and $N_x:=V_x\cup F_x$. Then,
[\[y:separator\]]{} For each $x\in V(T)$, there is no edge $vw\in E(G)$ with $v\in V_x$ and $w\in V(G)\setminus N_x$.
[\[y:ancestor-edge\]]{} For each $x\in V(T)$, there is a set $\{a_1,\ldots,a_{t'}\}$ of $t'{\leqslant}t$ strict $T$-ancestors of $x$ such that $F_x \subseteq \bigcup_{i=1}^{t'} Y_{a_i}$.
Before proving \[generalized-tripod\] we point out more properties that are immediately implied by it:
[\[y:y-subsets\]]{} $Y_x\subseteq V_x$ for every $x\in V(T)$.
[\[y:containment-i\]]{} $V_x\subseteq V_a$ for every $T$-ancestor $a$ of $x$.
[\[y:containment-ii\]]{}$N_x\subseteq N_a$ for every $T$-ancestor $a$ of $x$.
Property [(Y\[y:y-subsets\])]{} follows from the fact that $V_x$ is the union of several sets, one of which is $Y_x$. Property [(Y\[y:containment-i\])]{} follows from the definition of $V_x$ and the fact that $V(T_x)\subseteq V(T_a)$. To show Property [(Y\[y:containment-ii\])]{} first note that, by [(Y\[y:containment-i\])]{} it suffices to consider vertices $w\in F_x=N_x\setminus V_x$. By definition, every vertex $w\in F_x$ is adjacent, in $G$, to a vertex $v\in V_x$. By [(Y\[y:containment-i\])]{}, $v\in V_a$, so $w$ is either in $V_a$ or $w$ satisfies the condition $vw\in E(G)$, $v\in V_a$, and $w\not\in V_a$, so $w\in F_a$. In either case $w\in N_a=V_a\cup F_a$. Note that none of [(Y\[y:y-subsets\])]{}–[(Y\[y:containment-ii\])]{} depends on [(Y\[y:ancestor-edge\])]{} (which is important, since [(Y\[y:containment-i\])]{} is used to establish [(Y\[y:ancestor-edge\])]{} in the following proof).
Property [(Y\[y:separator\])]{} is immediate from the definitions of $F_x$ and $N_x$. In particular, $(N_x,V(G)\setminus V_x)$ is a separation of $G$ with $F_x=N_x\cap(V(G)\setminus V_x)$.
To establish Property [(Y\[y:ancestor-edge\])]{}, consider some vertex $w\in F_x$. Since $w\in F_x$, there exists an edge $vw\in E(G)$ with $v\in V_x$ and $w\not\in V_x$. Since $v\in V_x$, $v\in Y_{x'}$ for some $T$-descendant $x'$ of $x$ (possibly $x=x'$). Since $\mathcal{Y}$ is a partition, $w\in Y_{a}$ for some $a\not\in V(T_x)$. Since $vw\in E(G)$, we have $x'a\in E(H)$. By [(T\[t:ancestor-edge\])]{}, one of $a$ or $x'$ is a $T$-ancestor of the other. Since $w\in Y_a\subseteq V_a$ and $w\not\in V_x\supseteq V_{x'}$, [(Y\[y:containment-i\])]{} rules out the possibility that $x'$ is a $T$-ancestor of $a$. Therefore, $a$ is a $T$-ancestor of $x$ which is a $T$-ancestor of $x'$. Let $z_0,\ldots,z_r$ be the path in $T$ from $z_0:=x'$ to $z_r:=a$. For each $i\in\{0,\ldots,r\}$, at least one of $a$ or $x'$ is in $B_{z_i}$, since $x'a\in E(H)$. However, by [(T\[t:subtree-root\])]{} $x'$ is not contained in $B_{x_i}$ for any $i\in\{1,\ldots,r\}$. Therefore $a\in B_{x_i}$ for each $i\in\{0,\ldots,r\}$. In particular, $a$ is contained in $B_x$. Property [(Y\[y:ancestor-edge\])]{} now follows from the fact that $|B_x|{\leqslant}t+1$ and $B_x$ contains $x$.
We are now ready to prove our main result, which we restate here for convenience:
Apply \[generalized-tripod\] to $G$ and let $\mathcal{L}$, $\mathcal{Y}$, $\mathcal{T}$, $T$, $Y_x$, $V_x$, $F_x$, and $N_x$ be defined as in \[generalized-tripod\], where the partition $\mathcal{Y}$ has width $\ell$ with respect to the layering $\mathcal{L}$.
For a node $x\in V(T)$, we say that a shortcut $P\in\mathcal{P}$ *crosses* $x$ if $Y_x$ contains an internal vertex of $P$, that is, $P=(v_0,\ldots,v_r)$ and $\{v_1,\ldots,v_{r-1}\}\cap Y_x\neq\emptyset$. We say that a vertex $v\in V(G)$ *participates* in $x$ if $v\in Y_x$, or $\mathcal{P}$ contains a shortcut $P$ with $v\in V(P)$ and $P$ crosses $x$. We let $X_v$ denote the set of nodes $x\in V(T)$ such that $v$ participates in $x$.
\[x-v-ancestor\] For any $v\in V(G)$ there exists a (unique) node $a(v)\in X_v$ such that $a(v)$ is a $T$-ancestor of every node in $X_v$.
Let $Z := \{v\} \cup \{\{v_1,\ldots,v_{r-1}\}:(v_0,\ldots,v_r)\in\mathcal{P}, v\in \{v_0,\ldots,v_r\}\}$. Then $G[Z]$ is connected because $Z$ is the union of (vertex sets of) paths in $G$, each of which contains $v$.
We claim that $v$ participates in a node $x\in V(T)$ if and only if $Z\cap Y_x\neq\emptyset$. If $v$ participates in $x$ then either $v\in Y_x$, so $Z\cap Y_x\supseteq\{v\}$; or $v\in \{v_0,\ldots,v_r\}$ for some shortcut $(v_0,\ldots,v_r)\in\mathcal{P}$ that crosses $x$, so $Z\cap Y_x\supseteq \{v_i\}$ for some $i\in\{1,\ldots,r-1\}$. In the other direction, if $Z\cap Y_x\neq\emptyset$, then either $Z\cap Y_x\supseteq \{v\}$, so $v\in Y_x$; or $Z\cap Y_x\supseteq \{v_i\}$ where $i\in\{1,\ldots,r\}$, $(v_0,\ldots,v_r)\in{\mathcal{P}}$ and $v\in\{v_0,\ldots,v_r\}$, so $v\in V(P)$ for a path $P=(v_0,\ldots,v_r)\in\mathcal{P}$ that crosses $x$.
Let $X_H:=\{x\in V(H): Z\cap Y_x\neq\emptyset\}$. The connectivity of $G[Z]$ implies that $H[X_H]$ is connected. Choose $a(v)\in X_H$ to be the member of $X_H$ that does not have a strict $T$-ancestor in $X_H$. Transitivity of the $T$-ancestor relationship, [(T\[t:ancestor-edge\])]{}, and connectivity of $H[X_H]$ implies that such an $a(v)$ exists and is a $T$-ancestor of every node $x\in X_H$, as required.
For each $x\in V(T)$, define $S_x := \{v\in V(G): a(v)= x\}$. Observe that $\mathcal{S}:=(S_x : x\in V(T))$ is a partition of $V(G)$. We let $J:=G^\mathcal{P}/\mathcal{S}$ denote the resulting quotient graph and we let $V(J)\subseteq V(T)$ in the obvious way, so that each $x\in V(J)$ is the vertex obtained by contracting $S_x$ in $G^{\mathcal{P}}$. (Nodes $x\in V(T)$ with $S_x=\emptyset$ do not contribute a vertex to $J$.)
From this point onward, the plan is to show that:
$\mathcal{S}$ has small layered width with respect to the layering $\mathcal{L}$ and that
$J$ has small treewidth.
First we need to understand the relationship between sets $S_x\in\mathcal{S}$ and the related sets $Y_x$, $V_x$, $F_x$, and $N_x$.
\[s-subset\] For every $x\in V(T)$, $S_x\subseteq V_x$.
For the sake of contradiction, assume otherwise, so there exists some $v\in S_x\setminus V_x$. By [(Y\[y:y-subsets\])]{}, $Y_x\subseteq V_x$, so $v\not\in Y_x$. Therefore, $\mathcal{P}$ contains a path $P$, with $v\in V(P)$, that crosses $x$. The path $P$ contains a proper subpath $v_0,v_1,\ldots,v_{r}$ such that $v=v_0$ and $v_r\in Y_x$. Since $v\not\in V_x$ and $v_r\in Y_x\subseteq V_x$, [(Y\[y:separator\])]{}, implies that $v_i\in F_x$ for some $i\in\{0,\ldots,r-1\}$. Now [(Y\[y:ancestor-edge\])]{} implies $v_i\in Y_a$ for some strict $T$-ancestor $a$ of $x$. Therefore, either $v\in Y_a$ or $P$ crosses $a$. But this implies that $a(v)$ is a $T$-ancestor of $a$, which is a strict $T$-ancestor of $x$, contradicting the assumption that $v\in S_x$.
Next we complete Step (i) and show that $\mathcal{S}$ has small layered width with respect to the layering $\mathcal{L}=\langle L_1,\ldots,L_h\rangle$:
\[general-width\] For each $i\in\{1,\ldots,h\}$ and each $x\in V(J)$, $|S_x\cap L_i|{\leqslant}d\ell(k^2+3)$.
Recall that $S_x$ is defined by vertices that participate in $x$, and these are vertices that are either in $Y_x$ or in shortcuts that cross $Y_x$. We say that a vertex $w\in Y_x$ *contributes* a vertex $v\in S_x$ if $v$ participates in $x$. We upper bound the number of vertices in $S_x\cap L_i$ by upper-bounding the number of vertices contributed to $S_x\cap L_i$ by each $w\in Y_x$. Refer to \[contribute\]. If $w\in Y_x\cap L_i$ and no path in $\mathcal{P}$ includes $w$ as an internal vertex then $w$ contributes at most one vertex, itself, to $S_x\cap L_i$.
![A path $P$ containing an internal vertex $w\in Y_x\cap L_{i-j}$.[]{data-label="contribute"}](figs/contribute)
Otherwise, consider some path $P\in\mathcal{P}$ that contains $w$ as an internal vertex. If $w\in L_{i}$, then $P$ contributes at most $k+1$ vertices to $S_x\cap L_i$. If $w\in L_{i-1}\cup L_{i+1}$, then $P$ contributes at most $k$ vertices to $S_x\cap L_i$. If $w\in L_{i-j}\cup L_{i+j}$ for $j{\geqslant}2$, then $P$ contributes at most $k-j$ vertices to $S_x\cap L_i$.
For any $j$, the number of vertices $w\in L_{i+j}\cap Y_x$ is at most $\ell$. Each such vertex $w$ is an internal vertex of at most $d$ paths in $\mathcal{P}$. Therefore, $$|S_x\cap L_i|{\leqslant}d\ell \, \Big(k+1 + 2k + \sum_{j=2}^k 2(k-j)\Big) = d\ell(k^2 +3) \enspace . \qedhere$$
We now proceed with Step (ii), showing that $J$ has small treewidth. To accomplish this, we construct a small width tree-decomposition $\mathcal{C}:=(C_x:x\in V(T))$ of $J$ using the same tree $T$ used in the tree decomposition $\mathcal{T}:=(B_x:x\in V(T))$ of $H$. The following claim will be useful in showing that the resulting decomposition has small width.
\[i-ancestor\] For each edge $xy\in E(J)$, one of $x$ or $y$ is a $T$-ancestor of the other.
Suppose, for the sake of contradiction, that neither $x$ nor $y$ is a $T$-ancestor of the other. Since $xy\in E(J)$, $G^\mathcal{P}$ contains an edge $vw$ with $v\in S_x$ and $w\in S_y$. Since $vw\in E(G^{\mathcal{P}})$, $\mathcal{P}$ contains a $vw$-shortcut $P$.[^12] By \[s-subset\], $v\in V_x$ and $w\in V_y$. By [(Y\[y:containment-i\])]{}, if neither $x$ nor $y$ is a $T$-ancestor of the other, then $V_x$ and $V_y$ are disjoint. By [(Y\[y:ancestor-edge\])]{}, $N_x$ and $V_y$ are also disjoint. By [(Y\[y:separator\])]{} $P$ contains an internal vertex $v'\in F_x$. By [(Y\[y:ancestor-edge\])]{}, $v'\in Y_a$ for some strict $T$-ancestor $a$ of $x$. But this implies that $a(v)=a'$ so $v\in S_{a'}$ for some $T$-ancestor $a'$ of $a$, contradicting the assumption that $v\in S_x$.
\[general-bag-size\] The graph $J$ has a tree decomposition in which every bag has size at most $\binom{k+t}{t}$.
For the tree decomposition $(C_x:x\in V(T))$ of $J$ we use the same tree $T$ used in the tree decomposition $(B_x:x\in V(T))$ of $H$. For each node $x$ of $T$, we define $C_x$ as follows: $C_x$ contains $x$ as well as every $T$-ancestor $a$ of $x$ such that $J$ contains an edge $ax'$ where $x$ is a $T$-ancestor of $x'$ (including the possibility that $x=x'$). \[i-ancestor\] ensures that, for every edge $ax'\in E(J)$, $a,x'\in C_{x'}$. The connectivity of $T[a]:=T[\{x\in V(T):a\in C_x\}]$ follows from the fact that, for every node $x'\in T[a]$, every node $x$ on the path in $T$ from $x'$ to $a$ is also a node of $T[a]$. Therefore $(C_x:x\in V(T))$ is indeed a tree decomposition of $J$. It remains to bound the size of each bag $C_x$.
Consider an arbitrary node $x\in V(T)$ where $x_0,\ldots,x_r$ is the path from the root $x_0$ of $T$ to $x_r:=x$. To avoid triple-subscripts in what follows, we abuse notation slightly by using $V_i$, $F_i$, and $N_i$, as shorthands for $V_{x_i}$, $F_{x_i}$ and $N_{x_i}$, respectively.
If $x_\delta\in C_x$, it is because $x_\delta x'\in E(J)$ for some $T$-descendant $x'$ of $x$. This implies $G^{\mathcal{P}}$ contains an edge $vw$ with $v\in S_{x'}$ and $w\in S_{x_\delta}=S_\delta$. This implies that $\mathcal{P}$ contains a $vw$-shortcut $P_{vw}$. Let $v'$ be the second-last vertex of $P_{vw}$ (so $v'w\in E(G)$).
Since $w\in S_{\delta}$, $a(w)=x_\delta$, so $w$ participates in $x_\delta$, so at least one of the following is true:
There exists $w'\in V(G)$ such that $\mathcal{P}$ contains a $ww'$-shortcut $P_{ww'}$ that has an internal vertex in $Y_{\delta}$; or
$w\in Y_\delta$. In this case, we define $P_{ww'}$ to be the path of length 0 that contains only $w=w'$.
Let $w''$ denote the first vertex of $P_{ww'}$ contained in $Y_{\delta}$.
Let $w_0,w_1,\ldots,w_p$ be the path that begins $w_0:=v'$ and then follows the subpath of $P_{ww'}$ that begins at $w_1:=w$ and ends at $w_p:=w''$. For each $i\in\{0,\ldots,p\}$, let $s_i=\max\{j\in\{0,\ldots,r\}: \{w_0,\ldots,w_i\}\subseteq V_{j}\}\}$, and let $a_i=x_{s_i}$. Note that $s_0,\ldots,s_p$ is a non-increasing sequence and $a_0,\ldots,a_p$ is a sequence of nodes of $T$ whose distance from the root, $x_0$, of $T$ is non-increasing.
We claim that $a_0=x_r$. Since $v\in S_{x'}$, $a(v)=x'$, so $V(P_{vw})\subseteq V_{x'}$, otherwise $v$ participates in $a$ for some node $a$ not in the subtree $T_{x'}$ rooted at $x'$, but this contradicts \[i-ancestor\] since $a(v)=x'$ is a $T$-ancestor of every node in which $v$ participates. Since $v'=w_0\in V(P_{vw})$, $\{w_0\}\subseteq V_{x'}\subseteq V_{r}$, so $s_0=r$ and $a_0=x_r$.
We claim that $a_p=x_\delta$—that is, $s_p=\delta$. To see this, first observe that, for each $i\in\{1,\ldots,p\}$, $w_i\in V_{\delta}$ since, otherwise, an internal vertex of $P_{ww'}$ belong to $F_\delta$, which would imply (by [(Y\[y:ancestor-edge\])]{}) that $w\in S_{\delta'}$ for some $\delta' < \delta$, contradicting the assumption that $w\in S_\delta$. Therefore $s_p{\geqslant}\delta$. To see that $s_p<\delta+1$, observe that either $w=w''\in Y_\delta$ or $P_{ww'}$ contains an internal vertex $w''$ in $Y_\delta$. By the definition of $V_x$, $V_{\delta-1}$ does not contain $w''$, so $s_p<\delta+1$.
Let $H^+$ denote the supergraph of $H$ with vertex set $V(T)$ and in which $xy\in E(H^+)$ if and only there exists some $z\in V(T)$ such that $x,y\in B_z$. We claim that $a_0,\ldots,a_p$ is a lazy walk[^13] in $H^+$. Indeed, if $a_i\neq a_{i+1}$ for some $i\in\{0,\ldots,p-1\}$ then this is precisely because $w_i\in V_{a_i}$ but $w_{i+1}\not\in V_{a_i}$. By definition, $w_i\in Y_{a_i'}$ for some $T$-descendant $a_i'$ of $a_i$. By [(Y\[y:separator\])]{}, $w_{i+1}\in F_{a_i}$ so by [(Y\[y:ancestor-edge\])]{} $w_{i+1}\in Y_{a_i''}$ for some strict $T$-ancestor $a_i''$ of $a_i$. Since $w_iw_{i+1}\in E(G)$, $a_i'a_i''\in E(H)$. By [(T\[t:subtree-root\])]{}, $a_i''\in B_{a_i''}$ and $a_i''\in B_{a_i'}$. Since $a_i$ is on the path from $a_i'$ to $a_i''$ in $T$ this implies that $a_i''\in B_{a_i}$. Therefore $a_ia_i''\in E(H^+)$ as claimed.
Thus, $a_0,\ldots,a_p$ is a lazy walk in $H^+$ of length $p{\leqslant}k$ where the distance $s_i$ between $a_i$ and the root $x_0$ of $T$ is non-decreasing. By removing repeated vertices this gives a path in the directed graph $\overrightarrow{H}^+$ obtained by directing each edge $xy\in E(H^+)$ from its $T$-descendant $x$ towards its $T$-ancestor $y$. Finally, we are in a position to appeal to [@pilipczuk.siebertz:polynomial-soda Lemma 24] which states that the number of nodes in $\overrightarrow{H}^+$ that can be reached from any node $x$ by a directed path of length at most $k$ is at most $\binom{k+t}{t}$.
At this point, the proof is complete, but let us summarize. For each node $x\in V(T)$, $C_x$ contains only $T$-ancestors of $x$ (including $x$ itself). For each ancestor $x_\delta$ of $x$ contained in $C_x$, there is path from $x$ to $x_\delta$ in $\overrightarrow{H}^+$ of length at most $k$. The number of ancestors of $x$ that can be reached by paths of length at most $k$ in $\overrightarrow{H}^+$ is at most $\binom{k+t}{t}$. Therefore $|C_x|{\leqslant}\binom{k+t}{t}$, as required.
At this point, the proof of \[ShortcutPartition\] is almost immediate from \[general-width,general-bag-size\], except that the layering $\mathcal{L}$ of $G$ may not be a valid layering of $G^{\mathcal{P}}$. In particular, $G^{\mathcal{P}}$ may contain edges $vw$ with $v\in L_i$ and $w\in L_{i+j}$ for any $j\in\{0,\ldots,k\}$. To resolve this, we use a new layering $\mathcal{L}':=\langle L_0',\ldots,L_h'\rangle$ in which $L_i'=\bigcup_{j=ki}^{ki+k-1} L_i$. This increases the layered width given by \[general-width\] from $d\ell(k^2+3)$ to $d\ell(k^3+3k)$. Therefore $G$ has an $H$-partition of layered width at most $d\ell(k^3+3k)$ in which $H$ has treewidth at most $\binom{k+t}{t}$, completing the proof of \[ShortcutPartition\].
Allowing Crossings {#sec-k-planar}
==================
This section applies our main results for shortcut systems to prove graph product structure theorem for graphs drawn with a bounded number of crossings per edge. Then we show how to improve the bounds in this case.
$k$-Planar Graphs
-----------------
We first formally define $k$-planar graphs. An *embedded graph* $G$ is a graph with $V(G)\subset\R^2$ in which each edge $vw\in E(G)$ is a closed curve[^14] in $\R^2$ with endpoints $v$ and $w$ and not containing any vertex of $G$ in its interior. A *crossing* in an embedded graph $G$ is a triple $(p,vw,xy)$ with $p\in\R^2$, $vw,xy\in E(G)$ and such that $p\in (vw\cap xy)\setminus\{v,w,x,y\}$. An embedded graph $G$ is *$k$-plane* if each edge of $G$ takes part in at most $k$ crossings. A (not necessarily embedded) graph $G'$ is *$k$-planar* if there exists a $k$-plane graph $G$ isomorphic to $G'$. Under these definitions, $0$-planar graphs are exactly planar graphs and $0$-plane graphs are exactly plane graphs.
As mentioned in \[Introduction\], \[PlanarProduct,ShortcutProduct\] imply a product structure theorem for $k$-planar graphs. We get improved bounds as follows.
Let $G$ be a $k$-plane graph. We will assume, for ease of exposition, that any point $p\in\R^2$ is involved in at most one crossing $(p,vw,xy)$ of $G$. This assumption is justified since it can be enforced by a slight deformation of the edges of $G$ and the resulting (deformed) graph is also $k$-plane.
As in the proof of \[AddDummy\], let $G_0$ be the plane graph obtained by adding a dummy vertex at each crossing in $G$. In this way, each edge $vw\in E(G)$ corresponds naturally to a path $P_{vw}$ of length at most $k+1$ in $G_0$. Let ${\mathcal{P}}:= \{P_{vw}: vw\in E(G)\}$. Observe that ${\mathcal{P}}$ is a $(k+1,2)$-shortcut system for $G_0$ and that $G_0^{{\mathcal{P}}}\supseteq G$. Specifically, $G_0^{{\mathcal{P}}}$ contains every edge and vertex of $G$ as well as the dummy vertices in $V(G_0)\setminus V(G)$ and their incident edges.
Since $G_0$ is planar, \[PlanarProduct\](b) and \[PartitionProduct\] implies that $G_0$ has an $H$-partition of layered width 3 for some planar graph $H$ of treewidth at most 3. Applying \[ShortcutPartition\] to $G_0$ and $\mathcal{P}$ immediately implies that $G$ (an arbitrary $k$-planar graph) has an $H$-partition of layered width $6((k+1)^3+3(k+1))$ for some graph $H$ of treewidth at most $\binom{k+4}{3}-1$.
We can reduce the layered width of the $H$-partition of $G$ from $O(k^3)$ to $O(k^2)$ by observing that the dummy vertices in $V(G_0)\setminus V(G)$ do not contribute to the layered width of this partition. In this setting, the proof of \[general-width\] is simpler since each vertex $w\in Y_x$ contributes at most two vertices to $L_i\cap Y_x$. More precisely, each path $P\in\mathcal{P}$ containing an internal (dummy) vertex $w\in Y_x\cap (L_{i-j}\cup L_{i+j})$ contributes: (i) at most two vertices to $S_x\cap L_i$ for $j\in\{0,\ldots,\floor{(k+1)/2}\}$; (ii) at most one vertex to $S_x\cap L_j$ for $j\in\{\floor{(k+1)/2}+1,\ldots,k+1\}$; or (iii) no vertices to $S_x\cap L_j$ for $j > k+1$. Redoing the calculation at the end of the proof of \[general-width\] then yields $$\begin{aligned}
|S_x\cap Y_i| {\leqslant}d\ell\left(
2
+ 4\left\lfloor\tfrac{k+1}{2}\right\rfloor
+ 2\left\lceil\tfrac{k+1}{2}\right\rceil
\right)
=
d\ell\left(
2 + 2(k+1) + 2\left\lfloor\tfrac{k+1}{2}\right\rfloor
\right)
{\leqslant}d\ell(3k+5)
= 18k+30 \enspace .
\end{aligned}$$ With this change, the layered width of the partition given by \[ShortcutPartition\] becomes $(18k+30)(k+1)=18k^2+48k+30$. The result follows from \[PartitionProduct\].
$1$-Planar Graphs {#sec-1-planar}
-----------------
In the important special case of 1-planar graphs we obtain better constants and an additional property (planarity) of $H$.
\[1-planar\] Every 1-planar graph is a subgraph of $H\boxtimes P\boxtimes K_{30}$ for some planar graph $H$ with treewidth at most 3 and for some path $P$.
Let $G$ be an edge-maximal 1-plane multigraph with no two parallel edges on the boundary of a single face. Here, edge-maximal should be taken to mean that, if any two vertices $v$ and $w$ appear on a common face[^15] $F$, then there is an edge $vw\in E(G)$ that is contained in the boundary of $F$. We assume that no two edges incident to a common vertex cross each other since, in a 1-plane graph, such a crossing can always be removed by a local modification to obtain an isomorphic 1-plane graph in which the two edges do not cross.[^16]
A *kite* in $G$ is the subgraph $K=G[\{v,w,x,y\}]$ induced by the endpoints of a pair of crossing edges $vw,xy\in E(G)$. It follows from edge-maximality that every kite is isomorphic to the complete graph $K_4$. The edges $vw$ and $xy$ are called *spars* of $K$. The cycle $vxwy$ is called the *sail* of $K$. It follows from edge-maximality that none of the edges $vx$, $xw$, $wy$, or $yv$ are crossed by any other edges of $G$. Thus any edge that is a spar of a kite $K$ is not part of a sail of any kite $K'$. Observe that any spar of $K$ is incident on exactly four *kite faces* of $G$, each of which has three edges and two vertices of $G$ on its boundary.
The 1-plane graph $G$ has a plane triangulation $G'$ as a subgraph that can be obtained by removing one spar from each kite in $G$. Observe that, for any spar $xy\in E(G)\setminus E(G')$ that crosses $vw\in E(G')$, $G'$ contains the path $vxw$ (and $vyw$). It follows that $\operatorname{dist}_{G'}(v,w){\leqslant}2$.
Our proof of \[1-planar\] follows quickly from the following technical lemma, which is an extension of the analagous result for plane graphs [@dujmovic.joret.ea:planar].
\[induction\] The setup:
Let $G$ and $G'$ be defined as above.
Let $T$ be a BFS spanning tree of $G'$ rooted at some vertex $r$.
For each integer $j{\geqslant}0$, let $L_j=\{v\in V(G):\operatorname{dist}_T(r,v)=j\}$.
Let $F$ be a cycle in $G'$ with $r$ in the exterior of $F$ and such that
No edge of $F$ is crossed by any edge of $G$; and
$V(F)$ can be partitioned into $P_1,\ldots,P_k$, for some $k\in\{1,2,3\}$ such that for each $i\in\{1,\ldots,k\}$,
$F[P_i]$ is a path; and
$|V(P_i)\cap L_j| {\leqslant}15$ for all integers $j{\geqslant}0$.
Let $N$ and $N'$ be the subgraphs of $G$ and $G'$ consisting only of those edges and vertices contained in $F$ or the interior of $F$.
Then $N$ has an $H$-partition $\mathcal{P}=\{S_x : x\in V(H)\}$ such that:
$H$ is planar;
for all integers $j{\geqslant}0$, and all $x\in V(H)$, $|S_x\cap L_j|{\leqslant}15$;
for each $i\in\{1,\ldots,k\}$, there exists some $x_i\in V(H)$ such that $P_i=S_{x_i}$; and
$H$ has a tree decomposition in which every bag has size at most 4 and such that some bag contains $x_1,\ldots,x_k$.
This proof is very similar to the proof of Lemma 14 by @dujmovic.joret.ea:planar. Rather than duplicate every detail of that proof here, we focus on the differences and refer the reader to the original proof for the remaining details.
The proof is by induction on the number of vertices of $N$. First note that $N'$ is a near-triangulation. If $k=3$, set $R_i := P_i$ for each $i\in\{1,2,3\}$. Otherwise, as in [@dujmovic.joret.ea:planar], split $P_1,\ldots,P_k$ to partition $V(F)$ into three sets $R_1$, $R_2$, and $R_3$ such that each $F[R_i]$ is a non-empty path and each $R_i$ contains vertices from exactly one of $P_1,\ldots,P_k$.
Next, as in [@dujmovic.joret.ea:planar], use Sperner’s Lemma to find an inner face $\tau=v_1v_2v_3$ of $N'$ such that, $T$ contains disjoint vertical paths $Q_1,Q_2,Q_3$ such that each $Q_i$ begins at $v_i$, ends at some vertex in $R_i$, and whose internal vertices (if any) are contained in $N'-V(F)$.
Let $\overline{Y}$ denote the subgraph of $N'$ consisting of vertices and edges of $Q_1$, $Q_2$, $Q_3$, and $\tau$. Let $\overline{Y}^+$ denote the subgraph of $N$ consisting of the vertices and edges of $\overline{Y}$ plus the vertices and edges of every kite formed by a crossing between an edge of $G$ and an edge of $\overline{Y}$.
We claim that, for each integer $i{\geqslant}0$, $|V(\overline{Y}^+)\cap L_i|{\leqslant}15$. First observe that, since $Q_1,Q_2,Q_3$ are each vertical paths in $T$, $\overline{Y}$ contains at most three vertices of $L_i$, each incident on at most two edges of $\overline{Y}$. Since $\operatorname{dist}_{G'}(v,w){\leqslant}2$ for each $vw\in E(G)$, any vertex $x\in V(\overline{Y}^+)\setminus V(\overline{Y})\cap L_i$, is incident to an edge $xy\in E(G)$ that crosses one of the at most six edges in $\overline{Y}$ having an endpoint in $L_i$. These at most six edges have at most 12 endpoints. Therefore $|V(\overline{Y}^+)\setminus V(\overline{Y})\cap L_i|{\leqslant}6\times 2=12$, so $|V(\overline{Y}^+)\cap L_i|{\leqslant}12+3=15$.
Let $M$ and $M^+$ denote the subgraph of $G$ containing the edges and vertices of $\overline{Y}$, respectively $\overline{Y}^+$, and the edges and vertices of $F$. The graph $M^+$ has some number of bounded faces, all contained in the interior of $F$. Some of the bounded faces of $M^+$ are kite faces. Let $F_1,\ldots,F_m$ be the non-kite bounded faces of $M^+$.
We claim that, for each $i\in\{1,\ldots,m\}$, the boundary of $F_i$ is a cycle in $G'$ that contains no spars. Otherwise, some edge $vw$ contributes to the boundary of $F_i$ but is crossed by an edge $xy\in E(G)$. Then, $vw\not\in E(F)$ since no edge of $F$ is crossed by any edge of $G$. Therefore $vw\in E(\overline{Y}^+)$ so $xy\in E(Y^+)$. But then the only faces of $M^+$ incident to $vw$ are kite faces. In particular $vw$ cannot be incident to the non-kite face $F_i$.
Observe that each of the faces $F_1,\ldots,F_m$ is contained in a single internal face of $M$. Let $Y^+ := \overline{Y}^+-F$. Therefore, $V(F_i)$ can be partitioned into at most three sets $P_1'$, $P_2'$, and $P_3'$ where $P_1'\subset V(Y^+)$, $P_2'\subseteq P_a$, $P_3'\subseteq P_b$ for some $a,b\in\{1,2,3\}$, and $F_i[P_j']$ is a path, for each $j\in\{1,2,3\}$.
Finally, the subgraph $N_i$ of $G$ consisting of the edges and vertices of $G$ contained in $F_i$ or its interior does not contain one of the three vertices of $\tau$. Therefore, we can apply induction using the cycle $F_i$ and the partition $P_1',P_2',P_3'$ of $V(C_i)$ to obtain the desired $H$-partition and tree decomposition of $N_i$.
The proof finishes in the same way as the proof in [@dujmovic.joret.ea:planar]. The paths $P_1,\ldots,P_k$, and $S=V(Y^+)$ become elements of the $H$-partition. Elements in each of the $H$-partitions of $N_1,\ldots,N_3$ that intersect $P_1,\ldots,P_k$, or $V(\overline{Y}^+-F)$ are discarded and all the resulting sets are combined to obtain an $H$-partition of $G$. The desired tree decomposition of $G$ is obtained in exactly the same way as in the proof of Lemma 14 in [@dujmovic.joret.ea:planar], except that now each node $x$ has a child for each face $F_i$ of $M^+_x$ that contains a vertex of $G$ in its interior.
The planarity of $H$ comes from two properties:
$G/\mathcal{P}$ and $G^+/\mathcal{P^+}$ are isomorphic, where $G^+$ is the triangulation obtained by adding dummy vertices at each crossing in $G$ and $\mathcal{P}^+$ is the partition we obtain by adding a dummy vertex $z$ to $\overline{Y}^+$ if $\overline{Y}^+$ contains an edge $vw$ that contains $z$ in its interior.
$G^+[\overline{Y}^+-F]$ is connected. To see this, first observe that $\overline{Y}-F$ is connected, and then observe that every vertex of $\overline{Y}^+$ is either a vertex of $\overline{Y}$ or adjacent to a vertex of $\overline{Y}$.
Since $G^+$ is planar, the second point implies that $H=G^+/\mathcal{P}$ is planar.
Using \[induction\], the proof of \[1-planar\] is now straightforward.
Given a 1-plane graph $G$, add edges to make it edge-maximal so that it has an outer face $F=v_1v_2v_3$. Next, add a vertex $r$ adjacent to $v_1$, $v_2$, and $v_3$ to obtain an edge-maximal 1-plane graph $\overline{G}$ with one vertex $r$ of degree 3 on its outer face.
Let $G'$ be the plane graph obtained by removing one spar from each kite of $\overline{G}$ and let $T$ be a BFS tree of $G'$ rooted at $r$. Now apply \[induction\] with $G=\overline{G}$, $G'$, $F$, and $P_i=\{v_i\}$ for each $i\in\{1,2,3\}$. This gives an $H$-partition $\{S_x:x\in V(H)\}$ of $\overline{G}-\{r\}\supseteq G$ in which $H$ is planar and has treewidth at most 3.
Use the layering $\mathcal{L}=\langle L_0',L_1'\ldots\rangle$ where $L_i'=L_{2i}\cup L_{2i+1}$ for each integer $i{\geqslant}0$. That this is a layering of $G$ follows from the fact that $\operatorname{dist}_{G'}(v,w){\leqslant}2$ for every edge $vw\in E(G)$. Since $|L_j\cap S_x|{\leqslant}15$ for every integer $j{\geqslant}0$, $|L_i'\cap S_x|{\leqslant}30$ for every integer $i{\geqslant}0$ and every $x\in V(H)$. The result follows from \[PartitionProduct\].
$(g,k)$-Planar Graphs
---------------------
The definition of $k$-planar graphs naturally generalises for other surfaces. A graph $G$ drawn on a surface $\Sigma$ is $(\Sigma,k)$-plane if every edge of $G$ is involved in at most $k$ crossings. A graph $G$ is $(g,k)$-planar if it is isomorphic to some $(\Sigma,k)$-plane graph, for some surface $\Sigma$ with Euler genus at most $g$. \[AddDummy\] immediately generalises as follows:
\[gAddDummy\] Every $(g,k)$-planar graph $G$ is a subgraph of $G_0^{\mathcal{P}}$ for some graph $G_0$ of Euler genus at most $g$ and some $(k+1,2)$-shortcut system ${\mathcal{P}}$ for $G_0$. Moreover, $V(G) \subseteq V(G_0)$ and for every edge $vw \in E(G)$ there is a $vw$-path $P$ in $G_0$ of length at most $k+1$, such that every internal vertex in $P$ has degree at most $4$ in $G_0$.
Theorems \[ShortcutProduct\] and \[GenusProduct\](b) imply a product structure theorem for $(g,k)$-planar graphs. The obtained bounds are improved by the following result, which is proved using exactly the same approach used in the proof of \[kPlanarProduct\] (applying \[GenusProduct\](b) instead of \[PlanarProduct\](b)). We omit repeating the details.
\[gkPlanarProduct\] Every $(g,k)$-planar graph is a subgraph of $H\boxtimes P \boxtimes K_\ell$ for some graph $H$ with $\operatorname{tw}(H) {\leqslant}\binom{k+5}{4}-1$, where $\ell:=\max\{2g,3\}\cdot(6k^2+16k+10)$.
Prior to this work, the strongest structural description of $k$-planar or $(g,k)$-planar graphs (or any of the other classes presented in \[Examples\]) was in terms of layered treewidth, which we now define. A *layered tree decomposition* $(\mathcal{L},\mathcal{T})$ consists of a layering $\mathcal{L}$ and a tree decomposition $\mathcal{T}$ of $G$. The layered width of $(\mathcal{L},\mathcal{T})$ is $\max\{|L\cap B|: L\in \mathcal{L},\, B\in \mathcal{T}\}$. The *layered treewidth* of $G$ is the minimum layered width of any layered tree decomposition of $G$. @dujmovic.morin.ea:layered proved that planar graphs have layered treewidth at most 3, that graphs of Euler genus $g$ have layered treewidth at most $2g+3$, and more generally that a minor-closed class has bounded layered treewidth if and only if it excludes some apex graph. @dujmovic.eppstein.ea:structure show that every $k$-planar graph has layered treewidth at most $6(k+1)$, and more generally that every $(g,k)$-planar graph has layered treewidth at most $(4g+6)(k+1)$. It follows from this result that $(g,k)$-planar graphs have treewidth $O(\sqrt{(g+1)(k+1)n})$ and thus have balanced separators of the same order, which can also be concluded from the work of @FP08. In related work, @grigoriev.bodlaender:algorithms used structural results to obtain approximation algorithms for $(g,k)$-planar graphs, and @PachToth97 determined the maximum number of edges in a $k$-planar graph (up to a constant factor).
If a graph class admits bounded layered partitions, then it also has bounded layered treewidth. In particular, if $\mathcal{P}=(P_x:x\in V(H))$ is an $H$-partition of $G$ of layered width $\ell$ with respect to some layering $\mathcal{L}$ of $G$ and $(B_x:x\in V(T))$ is a width-$t$ tree decomposition of $H$, then setting $C_x = \bigcup_{y\in B_x} P_y$ for each $x\in V(T)$ gives a tree decomposition $(C_x:x\in V(T))$ of $G$ that has layered treewidth $(t+1)\ell$ [@dujmovic.joret.ea:planar]. Therefore, any property that holds for graphs of bounded layered treewidth also holds for $G$. What sets layered partitions apart from layered treewidth is that they lead to constant upper bounds on the queue-number and non-repetitive chromatic number, whereas for both these parameters, the best known upper bound obtainable via layered treewidth is $O(\log n)$; see \[Applications\].
Rough Characterisation {#Characterisation}
----------------------
\[gAddDummy\] shows that $(g,k)$-planar graphs can be obtained by a shortcut system applied to a graph of Euler genus $g$, where internal vertices on the paths have bounded degree. This observation and the following converse result together provide a rough characterisation of $(g,k)$-planar graphs, which is interesting in its own right, and is useful for showing that various classes of graphs are $(g,k)$-planar.
\[DrawG\] Fix integers $g{\geqslant}0$ and $k,\Delta{\geqslant}2$. Let $G_0$ be a graph of Euler genus at most $g$. Let $G$ be a graph with $V(G) \subseteq V(G_0)$ such that for every edge $vw \in
E(G)$ there is a $vw$-path $P_{vw}$ in $G_0$ of length at most $k$, such that every internal vertex on $P_{vw}$ has degree at most $\Delta$ in $G_0$. Then $G$ is $(g, 2k(k+1)\Delta^{k} )$-planar.
For a vertex $x$ of $G_0$ with degree at most $\Delta$, and for $i\in\{1,\dots,k-1\}$, say a vertex $v$ is *$i$-close* to $x$ if there is a $vx$-path $P$ in $G_0$ of length at most $i$ such that every internal vertex in $P$ has degree at most $\Delta$ in $G_0$. For each edge $vw$ of $G$, say that $vw$ *passes through* each internal vertex on $P_{vw}$. Say $vw$ passes through $x$. Then $v$ is $i$-close to $x$ and $w$ is $j$-close to $x$ for some $i,j\in\{1,\dots,k-1\}$ with $i+j{\leqslant}k$. At most $\Delta^{i}$ vertices are $i$-close to $x$. Thus, the number of edges of $G$ that pass through $x$ is at most $$\sum_{i=1}^{k-1} \sum_{j=1}^{k-i} \Delta^i \Delta^j
= \sum_{i=1}^{k-1} \Delta^i \sum_{j=1}^{k-i} \Delta^j
< \sum_{i=1}^{k-1} \Delta^i 2 \Delta^{k-i}
= \sum_{i=1}^{k-1} 2\Delta^k
< 2k \Delta^k \enspace.$$ Draw each edge $vw$ of $G$ alongside $P_{vw}$ in $G_0$, so that every pair of edges cross at most once. Every edge of $G$ that crosses $vw$ passes through a vertex on $P_{vw}$ (including $v$ and/or $w$ if they too have degree at most $\Delta$). Since $P_{vw}$ has at most $k+1$ vertices, and less than $2k\Delta^{k}$ edges of $G$ pass through each vertex on $P_{vw}$, the edge $vw$ is crossed by less than $2k(k+1)\Delta^{k}$ edges in $G$. Hence $G$ is $(g, 2k(k+1)\Delta^{k} )$-planar.
Applications {#Applications}
============
Here we discuss some of the consequences of the above theorems for $k$-planar and $(g,k)$-planar graphs.
Queue Layouts
-------------
For an integer $k{\geqslant}0$, a *$k$-queue layout* of a graph $G$ consists of a linear ordering $\preceq$ of $V(G)$ and a partition $\{E_1,E_2,\dots,E_k\}$ of $E(G)$, such that for $i\in\{1,2,\dots,k\}$, no two edges in $E_i$ are nested with respect to $\preceq$. That is, it is not the case that $v\prec x \prec y \prec w$ for edges $vw,xy\in E_i$. The *queue-number* of a graph $G$, denoted by $\operatorname{qn}(G)$, is the minimum integer $k$ such that $G$ has a $k$-queue layout. Queue-number was introduced by @HLR92, who famously conjectured that planar graphs have bounded queue-number. @dujmovic.joret.ea:planar recently proved this conjecture using \[PlanarProduct\] and the following lemma. Indeed, resolving this question was the motivation for the development of \[PlanarProduct\].
\[qn\] If $G\subseteq H \boxtimes P \boxtimes K_\ell$ then $\operatorname{qn}(G) {\leqslant}3 \ell \, \operatorname{qn}(H) + \floor{\tfrac{3}{2}\ell}
{\leqslant}3 \ell \, 2^{\operatorname{tw}(H)} - \ceil{\tfrac{3}{2}\ell}$.
\[qn,gkPlanarProduct\] imply that $(g,k)$-planar graphs have queue-number at most $g 2^{O(k^4)}$. Note that @dujmovic.joret.ea:planar previously proved the bound of $O(g^{k+2})$ using \[GenusProduct\] and an ad-hoc method. Our result provides a better bound when $g>2^{k^3}$. In the case of 1-planar graphs we can improve further. @ABGKP18 proved that every planar graph with treewidth at most $3$ has queue-number at most $5$. Thus the graph $H$ in \[1-planar\] has queue-number at most $5$. \[qn,1-planar\] then imply:
Every 1-planar graph has queue-number at most $3 \times 30 \times 5 + \floor{\tfrac{3}{2} \times 30} = 495$.
Non-Repetitive Colouring
------------------------
The next two applications are in the field of graph colouring. For our purposes, a *$c$-colouring* of a graph $G$ is any function $\phi\colon V(G)\to C$, where $C$ is a set of size at most $c$. A $c$-colouring $\phi$ of $G$ is *non-repetitive* if, for every path $v_1,\ldots,v_{2h}$ in $G$, there exists $i\in\{1,\ldots,h\}$ such that $\phi(v_i)\neq\phi(v_{i+h})$. The *non-repetitive chromatic number* $\pi(G)$ of $G$ is the minimum integer $c$ such that $G$ has a non-repetitive $c$-colouring. This concept, introduced by @AGHR-RSA02, has since been widely studied; see [@dujmovic.esperet.ea:planar] for more than 40 references. Up until recently the main open problem in the field has been whether planar graphs have bounded non-repetitive chromatic number, first asked by @AGHR-RSA02. @dujmovic.esperet.ea:planar recently solved this question using \[PlanarProduct\] and the following lemma.
\[non-repetitive\] If $G\subseteq H\boxtimes P \boxtimes K_\ell$ then $\pi(G){\leqslant}\ell\, 4^{\operatorname{tw}(H)+1}$.
\[non-repetitive,kPlanarProduct,1-planar,gkPlanarProduct\] imply the following result:
For every $1$-planar graph $G$, $\pi(G){\leqslant}30\times 4^4=7680$.
For every $k$-planar graph $G$, $\pi(G){\leqslant}(18k^2+48k+30) 4^{\binom{k+4}{3}}$.
For every $(g,k)$-planar graph $G$, $ \pi(G){\leqslant}\max\{2g,3\}\cdot(6k^2+16k+10) 4^{\binom{k+5}{4}}.$
Prior to the current work, the strongest upper bound on the non-repetitive chromatic number of $n$-vertex $k$-planar graphs was $O(k\log n)$ [@dujmovic.morin.ea:layered].
Centered Colourings
-------------------
A $c$-colouring $\phi$ of $G$ is *$p$-centered* if, for every connected subgraph $X\subseteq G$, $|\{\phi(v):v\in V(X)\}| > p$ or there exists some $v\in V(X)$ such that $\phi(v)\neq \phi(w)$ for every $w\in V(X)\setminus\{v\}$. In words, either $X$ receives more than $p$ colours or some vertex in $X$ receives a unique colour. Let $\chi_p(G)$ be the minimum integer $c$ such that $G$ has a $p$-centered $c$-colouring. Centered colourings are important since they characterise classes of bounded expansion, which is a key concept in the sparsity theory of @Sparsity.
@pilipczuk.siebertz:polynomial-soda and @debski.felsner.ea:improved use \[ps\] and \[PlanarProduct\](b), respectively, to show that $\chi_p(G)$ is polynomial in $p$ when $G$ is planar or of bounded Euler genus. We use the following lemma due to @pilipczuk.siebertz:polynomial-arxiv [Lemma 15].
\[p-centered-treewidth\] Every graph $H$ of treewidth at most $t$ has $\chi_p(H){\leqslant}\binom{p+t}{t}$.
The following lemma is implicitly in the work of @debski.felsner.ea:improved [Proof of Theorem 2.1]. We include the proof for completeness.
\[p-centered\] If $G\subseteq H\boxtimes P \boxtimes K_\ell$ then $\chi_p(G){\leqslant}\ell (p+1)\, \chi_p(H)$.
By \[PartitionProduct\], $G$ has an $H$-partition $(\mathcal{L}=\langle L_0,L_1,\ldots\rangle, \mathcal{P}=(B_x:x\in V(H))$ with layered width at most $\ell$. Use a product colouring $\phi:V(G)\to \{1,\ldots,\ell\}\times\{0,\ldots,p\}\times\{1,\ldots,\chi_p(H)\}$. For each integer $i{\geqslant}0$ and each $x\in V(H)$, assign the colour $\phi(v):=(\alpha(v),\beta(v),\gamma(v))$ to each vertex $v\in L_i\cap B_x$ such that:
$\alpha(v)$ is unique among $\{\phi(w): w\in L_i\cap B_x\}$, which is possible since $|L_i\cap B_x|{\leqslant}\ell$,
$\beta(v)= i\bmod (p+1)$, and
$\gamma(v)=\gamma(x)$ where $\gamma:V(H)\to\{1,\ldots,\chi_p(H)\}$ is a $p$-centered colouring of $H$.
To show this is a $p$-centered colouring, consider some connected subgraph $X\subseteq G$.
First suppose that there exists $v,w\in V(X)$ with $v\in L_i$ and $w\in L_j$ with $j-i{\geqslant}p$. Since $X$ is connected, $X$ contains a path from $v$ to $w$. By the definition of layering, this path contains at least one vertex from $L_{i'}$ for each $i'\in\{i,i+1,\ldots,j\}$. Therefore, $|\{\beta(v'):v'\in V(X)\}|{\geqslant}j-i+1 > p$, so $X$ receives more than $p$ distinct colours.
Otherwise, $V(X)\subseteq L_{i}\cup\cdots\cup L_{i+s}$ for some $s<p$. Let $H':=H[\{x\in V(H):B_x\cap V(X)\neq\emptyset]$. If $|\{\gamma(x):x\in V(H')\}| > p$ then $|\{\gamma(v):v\in V(X)\}|> p$ so $|\{\phi(v):v\in V(X)\}|> p$ and we are done. Otherwise, since $\gamma$ is a $p$-centered colouring of $H$, there must exist some $x\in V(H')$ such that $\gamma(x)\neq\gamma(y)$ for every $y\in V(H')\setminus\{x\}$. For any $v,w\in B_x$ with $v\neq w$, either $v,w\in L_{i'}$ for some $i'\in\{i,i+1,\ldots,i+s\}$ in which case $\alpha(v)\neq\alpha(w)$; or $v\in L_{i'}$ and $w\in L_{i''}$ with $|i'-i''|< p$, in which case $\beta(v)\neq\beta(w)$. Therefore every vertex $v\in B_x$ receives a colour $\phi(v)$ distinct from every colour in $\{\phi(z):z\in X\setminus\{x\}\}$. Therefore, every vertex in $B_x$ receives a colour distinct from every other vertex in $X$.
\[p-centered,p-centered-treewidth,kPlanarProduct,1-planar,gkPlanarProduct\] immediately imply the following results, for every $p{\geqslant}2$:
For every $1$-planar graph $G$, $ \chi_p(G){\leqslant}5 (p+3)(p+2)(p+1)^2$.
For every $k$-planar graph $G$, $\displaystyle \chi_p(G){\leqslant}(18k^2+48k+30)(p+1) \binom{p+ \binom{k+4}{3}-1}{ \binom{k+4}{3}-1}$.
For every $(g,k)$-planar graph $G$, $\displaystyle \chi_p(G) {\leqslant}\max\{2g,3\}\cdot(6k^2+16k+10) (p+1) \binom{p+\binom{k+5}{4}-1}{\binom{k+5}{4}-1}$.
Prior to the current work, the strongest known upper bounds on the $p$-centered chromatic number of $(g,k)$-planar graphs $G$ were doubly-exponential in $p$, as we now explain. @dujmovic.eppstein.ea:structure proved that $G$ has layered treewidth $(4g+6)(k+1)$. Van den Heuvel and Wood [@vdHW17] showed that this implies that $G$ has $r$-strong colouring number at most $(4g + 6)(k + 1)(2r + 1)$. By a result of @zhu:colouring, $G$ has $r$-weak colouring number at most $( (4g + 6)(k + 1)(2r + 1) )^r$, which by another result of @zhu:colouring implies that $G$ has $p$-centered chromatic number at most $( (4g+6)(k+1)(2^{p-1} + 1) )^{2^{p-2}}$. The above results are substantial improvements, providing bounds on $\chi_p(G)$ that are polynomial in $p$ for fixed $g$ and $k$.
Examples {#Examples}
========
This section describes several examples of graph classes that can be obtained from a shortcut system typically applied to graphs of bounded Euler genus.
Map Graphs
----------
Map graphs are defined as follows. Start with a graph $G_0$ embedded in a surface of Euler genus $g$, with each face labelled a ‘nation’ or a ‘lake’, where each vertex of $G_0$ is incident with at most $d$ nations. Let $G$ be the graph whose vertices are the nations of $G_0$, where two vertices are adjacent in $G$ if the corresponding faces in $G_0$ share a vertex. Then $G$ is called a *$(g,d)$-map graph*. A $(0,d)$-map graph is called a (plane) *$d$-map graph*; see [@FLS-SODA12; @CGP02] for example. The $(g,3)$-map graphs are precisely the graphs of Euler genus at most $g$; see [@dujmovic.eppstein.ea:structure]. So $(g,d)$-map graphs generalise graphs embedded in a surface, and we now assume that $d{\geqslant}4$ for the remainder of this section.
There is a natural drawing of a map graph obtained by positioning each vertex of $G$ inside the corresponding nation and each edge of $G$ as a curve passing through the corresponding vertex of $G_0$. It is easily seen that each edge is in at most $\floor{\frac{d-2}{2}}\ceil{\frac{d-2}{2}}$ crossings; see [@dujmovic.eppstein.ea:structure]. Thus $G$ is $(g,\floor{\frac{d-2}{2}}\ceil{\frac{d-2}{2}})$-planar. Also note that \[DrawG\] with $k=2$ implies that $G$ is $(g, O(d^{2}) )$-planar. \[gkPlanarProduct\] then establishes a product structure theorem for map graphs, but we get much better bounds by constructing a shortcut system directly. The following lemma is reminiscent of the characterisation of $(g,d)$-map graphs in terms of the half-square of bipartite graphs [@CGP02; @dujmovic.eppstein.ea:structure].
\[MapShortcut\] Every $(g,d)$-map graph $G$ is a subgraph of $G_1^{\mathcal{P}}$ for some graph $G_1$ with Euler genus at most $g$ and some $(2,\tfrac12 d(d-3) )$-shortcut system ${\mathcal{P}}$ for $G_1$.
Let $G$ be a $(g,d)$-map graph. So there is a graph $G_0$ embedded in a surface of Euler genus $g$, with each face labelled a ‘nation’ or a ‘lake’, where each vertex of $G_0$ is incident with at most $d$ nations. Let $N$ be the set of nations. Then $V(G)=N$ where two vertices are adjacent in $G$ if the corresponding nation faces of $G_0$ share a vertex. Let $G_1$ be the graph with $V(G_1):=V(G_0) \cup N$, where distinct vertices $v,w\in N$ are adjacent in $G_1$ if the boundaries of the corresponding nations have an edge of $G_0$ in common, and $v\in V(G_0)$ and $w\in N$ are adjacent in $G_1$ if $v$ is on the boundary of the nation corresponding to $w$. Observe that $G_1$ embeds in the same surface as $G_0$ with no crossings, and that each vertex in $V(G_0)$ has degree at most $d$ in $G_1$. Consider an edge $vw\in E(G)$. If the nations corresponding to $v$ and $w$ share an edge of $G_0$, then $vw$ is an edge of $G_1$. Otherwise, $v$ and $w$ have a common neighbour $x$ in $V(G_0)$. In the latter case, let $P_{vw}$ be the path $(v,x,w)$. Let ${\mathcal{P}}$ be the set of all such paths $P_{vw}$. Each vertex $x\in V(G_0)$ is the middle vertex on at most $\tfrac12 d(d-3)$ paths in ${\mathcal{P}}$. Thus ${\mathcal{P}}$ is a $(2,\tfrac12 d(d-3))$-shortcut system for $G_1$, and by construction, $G \subseteq G_1^{\mathcal{P}}$.
Theorems \[PlanarProduct\](b), \[ShortcutProduct\] and \[GenusProduct\](b) and \[MapShortcut,qn,p-centered,p-centered-treewidth,non-repetitive\] imply the following results.
\[PlaneMapPartition\] Every $d$-map graph $G$:
is a subgraph of $H \boxtimes P \boxtimes K_{21d(d-3)}$ for some path $P$ and for some graph $H$ with $\operatorname{tw}(H){\leqslant}9$,
has queue-number $\operatorname{qn}(G) < 32225\, d(d-3)$.
has $p$-centered chromatic number $\chi_p(G) {\leqslant}21d(d-3) (p+1) \binom{p+9}{9}$,
has non-repetitive chromatic number $ \pi(G) {\leqslant}21 \cdot 4^{10} d(d-3)$.
\[MapPartition\] For integers $g{\geqslant}0$ and $d{\geqslant}4$, if $\ell:= 7d(d-3)\, \max\{2g,3\}$ then every $(g,d)$-map graph $G$:
is a subgraph of $H \boxtimes P \boxtimes K_{\ell}$ for some path $P$ and for some graph $H$ with $\operatorname{tw}(H){\leqslant}14$,
has queue-number $\operatorname{qn}(G) < 49151\, \ell $.
has $p$-centered chromatic number $\chi_p(G) {\leqslant}\ell (p+1)\, \binom{p+14}{14}$,
has non-repetitive chromatic number $ \pi(G) {\leqslant}4^{15}\,\ell $.
These results give the first constant upper bound on the non-repetitive chromatic number of map graphs, the first polynomial bounds on the $p$-centered chromatic number of map graphs, and the best known bounds on the queue-number of map graphs.
String Graphs
-------------
A *string graph* is the intersection graph of a set of curves in the plane with no three curves meeting at a single point; see [@PachToth-DCG02; @FP10; @FP14] for example. For an integer $\delta{\geqslant}2$, if each curve is in at most $\delta$ intersections with other curves, then the corresponding string graph is called a *$\delta$-string graph*. A *$(g,\delta)$-string* graph is defined analogously for curves on a surface of Euler genus at most $g$.
\[StringShortcut\] Every $(g,\delta)$-string graph $G$ is a subgraph of $G_0^{\mathcal{P}}$ for some graph $G_0$ with Euler genus at most $g$ and some $(\delta+1,\delta+1 )$-shortcut system ${\mathcal{P}}$ for $G_0$.
Let $\mathcal{C}=\{C_v:v\in V(G)\}$ be a set of curves in a surface of Euler genus at most $g$ whose intersection graph is $G$. Let $G_0$ be the graph obtained by adding a vertex at the intersection point of every pair of curves in $\mathcal{C}$ that intersect, where two such consecutive vertices on a curve $C_v$ are adjacent in $G_0$. For each vertex $v\in V(G)$, if $C_v$ intersects $k{\leqslant}\delta$ other curves, then introduce a new vertex called $v$ on $C_v$ between the $\floor{\frac{k}{2}}$-th vertex already on $C_v$ and the $\floor{\frac{k}{2}+1}$-th such vertex. For each edge $vw$ of $G$, there is a path $P_{vw}$ of length at most $2\ceil{\frac{\delta}{2}}{\leqslant}\delta+1$ in $G_0$ between $v$ and $w$. Let ${\mathcal{P}}$ be the set of all such paths $P_{vw}$. Consider a vertex $x$ in $G_0$ that is an internal vertex on some path in ${\mathcal{P}}$. Then $x$ is at the intersection of $C_v$ and $C_w$ for some edge $vw\in E(G)$. If some path $P\in {\mathcal{P}}$ passes through $x$, then $P=P_{vu}$ for some edge $vu$ incident to $v$, or $P=P_{wu}$ for some edge $wu$ incident to $w$. At most $\ceil{\frac{\delta}{2}}$ paths in ${\mathcal{P}}$ corresponding to edges incident to $v$ pass through $x$, and similarly for edges incident to $w$. Thus at most $2\ceil{\frac{\delta}{2}}{\leqslant}\delta+1$ paths in ${\mathcal{P}}$ use $x$ as an internal vertex. Thus ${\mathcal{P}}$ is a $(\delta+1,\delta+1)$-shortcut system for $G_0$, and by construction, $G \subseteq G_0^{\mathcal{P}}$.
A similar proof to that of \[StringShortcut\] shows that every $(g,\delta)$-string graph is $(g,2\delta^2)$-planar. Theorems \[PlanarProduct\](b), \[ShortcutProduct\] and \[GenusProduct\](b) and \[StringShortcut,qn\] imply:
\[StringPartition\] For integers $g{\geqslant}0$ and $\delta{\geqslant}2$, let $\ell:= \max\{2g,3\} \,(\delta^4 + 4 \delta^3 + 9 \delta^2 + 10 \delta + 4)$, and $t:= \binom{ \delta+5}{4}-1$ if $g{\geqslant}1$ and $t:= \binom{ \delta+4}{3}-1$ if $g=0$. Then every $(g,\delta)$-string graph:
is a subgraph of $H\boxtimes P \boxtimes K_{\ell}$ for some path $P$ and for some graph $H$ with treewidth $t$,
has queue-number $\operatorname{qn}(G^k) {\leqslant}3 \ell \, 2^t - \ceil{\tfrac{3}{2}\ell}$,
Our results also give bounds on the non-repetitive chromatic number and the $p$-centered chromatic number of $(g,\delta)$-string graphs, but the bounds are weak, since such graphs $G$ have maximum degree at most $2\delta$, implying that $\pi(G) {\leqslant}(4+o(1))\delta^2$ and $\chi_p(G){\leqslant}p(64\delta)^2$ by results of @DJKW16 and @debski.felsner.ea:improved, respectively.
Powers of Bounded Degree Graphs {#Powers}
-------------------------------
Recall that the *$k$-th power* of a graph $G$ is the graph $G^k$ with vertex set $V(G^k):=V(G)$, where $vw\in E(G^k)$ if and only if $\operatorname{dist}_G(v,w){\leqslant}k$. If $G$ is planar with maximum degree $\Delta$, then $G^k$ is $2k(k+1)\Delta^{k}$-planar by \[DrawG\]. Thus we can immediately conclude that bounded powers of planar graphs of bounded degree admit bounded layered partitions. However, the bounds we obtain are improved by the following lemma that constructs a shortcut system directly.
\[PowerShortcut\] If a graph $G$ has maximum degree $\Delta$, then $G^k = G^{\mathcal{P}}$ for some $(k,2k \Delta^{k})$-shortcut system ${\mathcal{P}}$.
For each pair of vertices $x$ and $y$ in $G$ with $\operatorname{dist}_G(x,y)\in\{1,\dots,k\}$, fix an $xy$-path $P_{xy}$ of length $\operatorname{dist}_G(x,y)$ in $G$. Let ${\mathcal{P}}:=\{P_{xy}: \operatorname{dist}_G(x,y)\in\{1,\dots,k\} \}$. Say $P_{xy}$ uses some vertex $v$ as an internal vertex. If $\operatorname{dist}_G(v,x)=i$ and $\operatorname{dist}_G(v,y)=j$, then $i,j\in\{1,\dots,k-1\}$ and $i+j{\leqslant}k$. The number of vertices at distance $i$ from $v$ is at most $\Delta^i$. Thus the number of paths in ${\mathcal{P}}$ that use $v$ as an internal vertex is at most $$\sum_{i=1}^{k-1} \sum_{j=1}^{k-i} \Delta^i\Delta^j
= \sum_{i=1}^{k-1} \Delta^i \sum_{j=1}^{k-i} \Delta^j
< \sum_{i=1}^{k-1} \Delta^i ( 2 \Delta^{k-i} )
< 2k \Delta^k\enspace.$$ Hence ${\mathcal{P}}$ is a $(k, 2k \Delta^k)$-shortcut system.
\[ShortcutProduct,PowerShortcut\] imply:
\[PowerProduct\] Let $G$ be a subgraph of $H\boxtimes P\boxtimes K_\ell$ with maximum degree $\Delta$, for some graph $H$ of treewidth at most $t$ and for some path $P$. Then for every integer $k{\geqslant}1$, the $k$-th power $G^k$ is a subgraph of $J\boxtimes P\boxtimes K_{2k \ell \Delta^{k}(k^3+3k)}$ for some graph $J$ of treewidth at most $\binom{k+t}{t}-1$ and some path $P$.
Theorems \[PlanarProduct\](b), \[GenusProduct\](b) and \[ShortcutProduct\] and \[DrawG,qn,p-centered,p-centered-treewidth,non-repetitive,PowerShortcut\] imply the following result, which with $g=0$ implies \[kPowerBasic\] in the introduction.
\[PowerGenus\] For integers $g{\geqslant}0$ and $k,\Delta{\geqslant}1$, let $\ell:= \max\{2g,3\} (2k^4+6k^2) \Delta^{k}$, and $t:= \binom{k+4}{4}-1$ if $g{\geqslant}1$ and $t:= \binom{k+3}{3}-1$ if $g=0$. Then for every graph $G$ of Euler genus $g$ and maximum degree $\Delta$,
$G^k$ is a subgraph of $H\boxtimes P \boxtimes K_{\ell}$ for some path $P$ and for some graph $H$ with treewidth $t$
$G^k$ is $(g, 2k(k+1)\Delta^{k} )$-planar,
$G^k$ has queue-number $\operatorname{qn}(G^k) {\leqslant}3 \ell \cdot 2^t - \ceil{\tfrac{3}{2}\ell}$.
$G^k$ has $p$-centered chromatic number $\chi_p(G^k) {\leqslant}\ell (p+1)\, \binom{p+t}{t}$,
$G^k$ has non-repetitive chromatic number $ \pi(G^k) {\leqslant}\ell \, 4^{t+1}$.
This result is the first constant upper bound on the queue-number of bounded powers of graphs with bounded degree and bounded Euler genus. For every graph $G$, since $G^k$ has maximum degree at most $\Delta^k$, a result of @DJKW16 implies that $\pi(G^k) {\leqslant}(1+o(1))\Delta^{2k}$. \[PowerGenus\] improves upon this bound when $k,g\ll\Delta$. Similarly, a result of @debski.felsner.ea:improved implies that $\chi_p(G^k){\leqslant}1024p\Delta^{2k}$ and \[PowerGenus\] improves upon this bound when $p,k,g\ll\Delta$.
\[qn,p-centered,p-centered-treewidth,non-repetitive,MinorFreeDegree,PowerProduct\] imply the following analogous result for powers of graphs in any minor-closed class with bounded maximum degree.
\[PowerMinor\] For every graph $X$ there exists an integer $c$ such that for all integers $k,\Delta{\geqslant}1$, if $t:= 2k\Delta^{k}(k^3+3k)\binom{k+c\Delta}{c\Delta}-1$ and $G$ is an $X$-minor-free graph with maximum degree $\Delta$, then:
$G^k$ is a subgraph of $H\boxtimes P$ for some graph $H$ with treewidth $t$ and for some path $P$.
$G^k$ has queue-number at most $3\cdot 2^t-2$,
$G^k$ has $p$-centered chromatic number at most $(p+1)\binom{p+t}{t}$.
$k$-Nearest-Neighbour Graphs
----------------------------
In this section, we show that $k$-nearest neighbour graphs of point sets in the plane are $O(k^2)$-planar. For two points $x,y\in\R^2$, let $d_2(x,y)$ denote the Euclidean distance between $x$ and $y$. The $k$-nearest-neighbour graph of a point set $P\subset\R^2$ is the geometric graph $G$ with vertex set $V(G)=P$, where the edge set is defined as follows. For each point $v\in P$, let $N_k(v)$ be the set of $k$ points in $P$ closest to $v$. Then $vw\in E(G)$ if and only if $w\in N_k(v)$ or $v\in N_k(w)$. (The edges of $G$ are straight-line segments joining their endpoints.) See [@ProximityGraphs] for a survey of results on $k$-nearest neighbour graphs and other related proximity graphs.
The following result, which is immediate from @abrego.munroy.ea:on [Corollary 4.2.6] states that $k$-nearest-neighbour graphs have bounded maximum degree:
\[k-nn-max-degree\] The degree of every vertex in a $k$-nearest-neighbour graph is at most $6k$.
We make use of the following well-known observation (see for example, @bose.morin.ea:routing [Lemma 2]):
\[convex\] If $v_0,\ldots,v_3$ are the vertices of a convex quadrilateral in counterclockwise order then there exists at least one $i\in\{0,\ldots,3\}$ such that $\max\{d_2(v_i,v_{i-1}), d_2(v_i,v_{i+1})\} < d_2(v_{i-1},v_{i+1})$, where subscripts are taken modulo 4.
\[nearest-neighbour\] Every $k$-nearest-neighbour graph is $O(k^2)$-planar.
Let $G$ be a $k$-nearest-neighbour graph and consider any edge $vw\in E(G)$. Let $xy\in E(G)$ be an edge that crosses $vw$. Note that $vxwy$ are the vertices of a convex quadrilateral in (without loss of generality) counterclockwise order. Then we say that
$xy$ is of Type $v$ if $\max\{d_2(v,x), d_2(v,y)\}< d_2(x,y)$;
$xy$ is of Type $w$ if $\max\{d_2(w,x), d_2(w,y)\}< d_2(x,y)$; or
$xy$ is of Type C otherwise.
If $xy$ is of Type C, then \[convex\] implies that $\max\{d_2(x,v),d_2(x,w)\} < d_2(v,w)$ without loss of generality. In this case, we call $x$ a Type C vertex. We claim that $V(G)$ contains at most $k-1$ Type C vertices. Indeed, more than $k-1$ Type C vertices would contradict the fact that $vw\in E(G)$ since every Type C vertex is closer to both $v$ and $w$ than $d_2(v,w)$.
Next oberve that, if $xy$ is of Type $v$, then at least one of $xv$ or $yv$ is in $E(G)$ in which case we call $x$ (respectively $y$) a Type $v$ vertex. By \[k-nn-max-degree\], there are at most $6k$ Type $v$ vertices. Similarly, there are at most $6k$ Type $w$ vertices.
Thus, in total, there are at most $13k-1$ Type $v$, Type $w$, and Type C vertices. By \[k-nn-max-degree\], each of these vertices is incident with at most $6k$ edges that cross $vw$. Therefore, there are at most $78k^2-6k$ edges of $G$ that cross $vw$. Since this is true for every edge $vw\in E(G)$, $G$ is $(78k^2-6k)$-planar.
Note that \[nearest-neighbour\] is tight up to the leading constant: Every $k$-nearest neighbour graph on $n{\geqslant}k+1$ vertices has at least $kn/2$ edges and at most $kn$ edges. For $k{\geqslant}7$, the Crossing Lemma [@ajtai.chvatal.ea:crossing-free; @leighton:complexity] implies that the total number of crossings is therefore $\Omega(k^3n)$ so that the average number of crossings per edge is $\Omega(k^2)$.
\[nearest-neighbour,kPlanarProduct,qn,p-centered,p-centered-treewidth\] imply:
\[k-nn\] For every integer $k{\geqslant}1$ there exists integers $t{\leqslant}O(k^6)$ and $\ell{\leqslant}O(k^4)$ such that every $k$-nearest-neighbour graph:
is a subgraph of $H\boxtimes P \boxtimes K_\ell$ for some graph $H$ with treewidth $t$ and some path $P$,
has queue-number at most $2^{O(k^6)}$, and
has $p$-centered chromatic number at most $\ell (p+1)\binom{p+t}{t}$.
\[non-repetitive,k-nn\] also give bounds on the non-repetitive chromatic number of a $k$-nearest neighbour graph $G$. However, the bound is weak, since $G$ has maximum degree at most $6k$, implying that $\pi(G) {\leqslant}(36+o(1))k^2$ by a result of @DJKW16.
Wrapping Up
===========
As mentioned in \[Introduction\], @DEJGMM prove results about adjacency labelling schemes for planar graphs. Stated in terms of universal graphs, their main theorem is interpreted as follows:
\[Universal\] For every fixed integer $t$ and every integer $n>0$ there exists a graph $U_n$ with $n^{1+o(1)}$ vertices such that for every graph $H$ of treewidth at most $t$ and path $P$, every $n$-vertex subgraph of $H\boxtimes P$ is isomorphic to an induced subgraph of $U_n$.
Combining this with \[gkPlanarProduct,k-nn,PowerMinor,StringPartition,MapPartition\] yields the following:
\[UniversalUniversal\] For every fixed graph $X$ and all fixed integers $d,\delta,\Delta,g,k>0$, and every integer $n>0$, there exists a graph $U_n$ with $n^{1+o(1)}$ vertices such that $U_n$ contains all of the following graphs as induced subgraphs:
every $n$-vertex $(g,k)$-planar graph;
every $n$-vertex $(g,d)$-map graph;
every $n$-vertex $(g,\delta)$-string graph;
every $n$-vertex graph $G^k$ where $G$ is $X$-minor-free and has maximum degree at most $\Delta$;
every $k$-nearest neighbour graph of $n$ points in $\R^2$.
We finish with an open problem. \[kPlanarProduct\] shows that every $k$-planar graph is a subgraph of $H\boxtimes P \boxtimes K_\ell$ for some graph $H$ with treewidth $O(k^3)$ where $\ell{\leqslant}O(k^2)$. Does there exist a function $\ell:\N\to\N$ and a universal constant $C$ such that every $k$-planar graph is a subgraph of $H\boxtimes P \boxtimes K_{\ell(k)}$ for some graph $H$ with treewidth at most $C$? Perhaps $C=3$. Note that $C{\geqslant}3$ even for planar graphs [@dujmovic.joret.ea:planar].
= =
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[^1]: School of Computer Science and Electrical Engineering, University of Ottawa, Ottawa, Canada (`[email protected]`). Research supported by NSERC and the Ontario Ministry of Research and Innovation.
[^2]: School of Computer Science, Carleton University, Ottawa, Canada (`[email protected]`). Research supported by NSERC and the Ontario Ministry of Research and Innovation.
[^3]: School of Mathematics, Monash University, Melbourne, Australia (`[email protected]`). Research supported by the Australian Research Council.
[^4]: In this paper, all graphs are finite and undirected. Unless specifically mentioned otherwise, all graphs are also simple. For any graph $G$ and any set $S$ (typically $S\subseteq V(G)$), let $G[S]$ denote the graph with vertex set $V(G)\cap S$ and edge set $\{uv\in E(G) : u,v\in S\}$. We use $G-S$ as a shorthand for $G[V(G)\setminus S]$. We use $G'\subseteq G$ to denote subgraph containment; that is, $V(G')\subseteq V(G)$ and $E(G')\subseteq E(G)$.
[^5]: The *strong product* of graphs $A$ and $B$, denoted by $A\boxtimes B$, is the graph with vertex set $V(A)\times V(B)$, where distinct vertices $(v,x),(w,y)\in V(A)\times V(B)$ are adjacent if $v=w$ and $xy\in E(B)$, or $x=y$ and $vw\in E(A)$, or $vw\in E(A)$ and $xy\in E(B)$.
[^6]: A *tree decomposition* $\mathcal{T}$ of a graph $G$ consists of a tree $T$ and a collection $\mathcal{T}=(B_x:x\in V(T))$ of subsets of $V(G)$ indexed by the nodes of $T$ such that (i) for every $vw\in E(G)$, there exists some node $x\in V(T)$ with $v,w\in B_x$; and (ii) for every $v\in V(G)$, the induced subgraph $T[v] := T[\{x: v\in B_x\}]$ is connected. The *width* of the tree decomposition $\mathcal{T}$ is $\max\{|B_x|:x\in V(T)\}-1$. The *treewidth* $\operatorname{tw}(G)$ of a graph $G$ is the minimum width of a tree decomposition of $G$. Treewidth is the standard measure of how similar a graph is to a tree. Indeed, a connected graph has treewidth 1 if and only if it is a tree. Treewidth is of fundamental importance in structural and algorithmic graph theory; see [@Reed03; @HW17; @Bodlaender-TCS98] for surveys.
[^7]: It is easily seen that $\operatorname{tw}(H\boxtimes K_\ell) {\leqslant}(\operatorname{tw}(H)+1)\ell-1$, so we may assume that $\ell=1$ in this definition.
[^8]: The *Euler genus* of the orientable surface with $h$ handles is $2h$. The *Euler genus* of the non-orientable surface with $c$ cross-caps is $c$. The *Euler genus* of a graph $G$ is the minimum integer $g$ such that $G$ embeds in a surface of Euler genus $g$. Of course, a graph is planar if and only if it has Euler genus 0; see [@mohar.thomassen:graphs] for more about graph embeddings in surfaces.
[^9]: A graph $M$ is a *minor* of a graph $G$ if a graph isomorphic to $M$ can be obtained from a subgraph of $G$ by contracting edges. A class $\mathcal{G}$ of graphs is *minor-closed* if for every graph $G\in\mathcal{G}$, every minor of $G$ is in $\mathcal{G}$. A minor-closed class is *proper* if it is not the class of all graphs. For example, for fixed $g{\geqslant}0$, the class of graphs with Euler genus at most $g$ is a proper minor-closed class. A graph $G$ is $t$-apex if it contains a set $A$ of at most $t$ vertices such that $G-A$ is planar. A 1-apex graph is *apex*. A minor-closed class $\mathcal{G}$ is apex-minor-free if some apex graph is not in $\mathcal{G}$.
[^10]: A path of length $k$ consists of $k$ edges and $k+1$ vertices. A path is *trivial* if it has length 0 and *non-trivial* otherwise.
[^11]: For a graph $G$ and two vertices $v,w\in V(G)$, $\operatorname{dist}_G(v,w)$ denotes the length of a shortest path, in $G$, with endpoints $v$ and $w$. We define $\operatorname{dist}_G(v,w):=\infty$ if $v$ and $w$ are in different connected components of $G$.
[^12]: Recall that we have made the assumption that $\mathcal{P}$ contains a length-$1$ $vw$-shortcut for each edge $vw\in E(G)$.
[^13]: A *lazy walk* in a graph $H$ is a walk in the pseudograph $H'$ obtained by adding a loop at each vertex of $H$.
[^14]: A closed curve in a surface $\Sigma$ is a continuous function $f:[0,1]\to \Sigma$. The points $f(0)$ and $f(1)$ are called the *endpoints* of the curve. When there is no danger of misunderstanding we treat a curve $f$ as the point set $\{f(t):0{\leqslant}t{\leqslant}1\}$.
[^15]: The *faces* of an embedded graph $G$ are the connected components of $\R^2\setminus \bigcup_{vw\in E(G)} vw$. We say that a vertex $v\in V(G)$ appears on a face $F$ if $v$ is contained in the closure of $F$.
[^16]: While this is true for 1-plane graphs it is not true for $k$-plane graphs with $k{\geqslant}3$; the uncrossing operation can increase the number of crossings on a particular edge from $k$ to $2(k-1)$.
|
---
abstract: 'This letter presents a new approach using the cosmic peculiar velocity field to characterize the morphology and size of large scale structures in the local Universe. The algorithm developed uses the three-dimensional peculiar velocity field to compute flow lines, or streamlines. The local Universe is then partitioned into volumes corresponding to gravitational basins, also called watersheds, among the different end-points of the velocity flow lines. This new methodology is first tested on numerical cosmological simulations, used as benchmark for the method, and then applied to the [*Cosmic-Flows*]{} project observational data in order to to pay particular attention to the nearby superclusters including ours. More extensive tests on both simulated and observational data will be discussed in an accompanying paper.'
author:
- |
Alexandra Dupuy$^{1}$, Helene M. Courtois$^{1}$, Florent Dupont$^{2}$, Florence Denis$^{2}$, Romain Graziani$^{1}$, Yannick Copin$^{1}$, Daniel Pomarède$^{3}$, Noam Libeskind$^{1,4}$, Edoardo Carlesi$^{4}$, Brent Tully$^{5}$, Daniel Guinet$^{1}$\
$^{1}$University of Lyon, UCB Lyon 1, CNRS/IN2P3, IPN Lyon, 69622 Villeurbanne, France\
$^{2}$Univ Lyon, LIRIS, UMR 5205 CNRS, Université Claude Bernard Lyon 1, 43 bd du 11 Novembre 1918, 69622 Villeurbanne CEDEX, France\
$^{3}$Institut de Recherche sur les Lois Fondamentales de l’Univers, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, France\
$^{4}$Leibniz–Institut für Astrophysik, Potsdam, An der Sternwarte 16, D–14482 Potsdam, Germany\
$^{5}$Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA\
bibliography:
- 'bibliothese.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: Partitioning the universe into gravitational basins using the cosmic velocity field
---
\[firstpage\]
large-scale structure of Universe
Introduction
============
The current cosmological paradigm, known as the $\Lambda$CDM model, lends itself to a hierarchical classification scheme, wherein small dwarf galaxies merge to form galaxies like the Milky Way, which are bound together into galaxy groups, which are in turn part of galaxy clusters. Such structures exist within the [*cosmic web*]{} [@Bond:1996aa], a multi-scale network of clusters, filaments, sheets and voids which constitutes the large scale distribution of matter in the universe. The term supercluster is used to define an ensemble of such knots, filaments, walls, and voids.
Hence, the measured large-scale distribution of matter in the universe can be used to measure constrain our cosmological model. Upcoming observational (such as Euclid, DESI or LSST) and computational (such as MultiDark) projects are based upon this idea that the study of the distribution (and motion) of galaxies can be a probe of cosmology. However, to infer this and cosmological parameters directly from the distribution of galaxies, a robust definition and quantification of the large-scale structures are required.
Many methods for classifying the local large-scale structures using galaxy redshift surveys have been developped in recent years. For example [@Sousbie:2008aa; @Sousbie:2011aa] introduced the [*DisPerSE*]{} algorithm which identifies the skeleton of clusters, filaments, walls and voids directly from the distribution of galaxies in observational catalogs such as the Sloan Digitized Sky survey (SDSS). [*SpineWeb*]{} [@Aragon-Calvo:2007aa; @Aragon-Calvo:2010aa] is a method based on the watershed segmentation of the density field derived from the observed galaxy redshift distribution, allowing for a multiscale morphology filter to identify clusters, filaments and walls. Similarly, [*Bisous*]{} [@Tempel:2014aa] studies the redshift galaxy distribution to find patterns and regularities in galaxy filaments. A new exciting approach has recently been suggested by [@Leclercq:2017aa], who advocates using the gravitational velocity field derived from redshift surveys such as SDSS to identify structures.
On the other hand, anticipating the exponential growth of kinematical (peculiar velocity) galaxy data, like $Cosmic-Flows$, 6DF, TAIPAN, WALLABY, a number of methods were developed based on computing the tidal tensor field (Hessian of the gravitational potential) to classify large scale structures (LSS) [@Hahn:2007aa; @Forero-Romero:2009aa]. Similarly [@Hoffman:2012aa] used the velocity shear tensor to classify volumes as knots, filaments, sheets, voids, according to the tensor’s eigenvalues.
A third category is combining redshift and peculiar velocity data. For example, $NEXUS$ [@Cautun:2013aa] is a method which uses the density field, the velocity divergence and the velocity shear fields to identify automatically volumes of space as cosmic web structures. See [@Libeskind:2018aa] for a comprehensive review and comparison between all these, and other, methods.
Recently, [@Tully:2014aa] suggested that one could additionally employ kinematics to define superclusters, as non-virialized objects that will dissipate with the expansion. Superclusters are not gravitationally bound – their sheer size and existence within an expanding Universe prohibits that, and so these previous methods do not fully capture the scale of these structures. However if the expansion of the Universe is instantaneously frozen, regions of space show gravitationally induced coherent inward motions, known as “Basins of attraction” (BoA). This simple definition allows also one to identify regions that are dual to basins of attraction - basins of repulsion (BoR), namely volumes of space with gravitationally coherent outward motions such as the dipole repeller [@Hoffman:2017aa] and the CMB Cold Spot repeller [@Courtois:2017aa]. In order to identify such structures we must observe the gravitational velocity (also called peculiar velocity) of a representative sample of galaxies in a large enough region of space.
This article proposes a different approach to all current methods of cosmic web classifications by focusing on this kinematical new definition of superclusters. This new tool can prove superclusters as robust probes of the cosmology when applied to larger upcoming redshift surveys spanning a good fraction of the universe age, like Euclid. The article is organized as follows: in Section \[sec:method\] the methodology is explained, in Section \[sec:tests\] it is it is tested on $\Lambda$CDM cosmological simulations and in Section \[sec:observations\] it is used on observed peculiar velocity data.
Methodology {#sec:method}
===========
Streamlines
-----------
A stream- or flow line, in a time-independent velocity field, represents the curve that is always tangent to the local value of the velocity field. It can be parametrized in terms of a coordinate $\tau$, which represents the position along the streamline $s$, as follows:
$$\label{eq:streamline1}
\frac{d\vec{s}}{d\tau} = \vec{v}(\vec{s}),$$
where $\vec{s}(\tau=0) = \vec{s}_0$ corresponds to the initial condition and is the [*seed point*]{} of the streamline. The streamline $\vec{s}(\tau)$ is obtained by integrating equation \[eq:streamline1\] over $\tau$. Thus, a set of streamline points $\{\vec{s}_i\}_{i \in [0,n]}$ is generated, where $n$ is the number of integration steps $\Delta\tau$.
A streamline is therefore obtained step by step by starting from a seed point (each voxel) and integrating spatially the components of the velocity field. For clarity purposes, the end of a streamline, i.e the location $\vec{s}_n$ at which the computation of $\vec{s}$ is stopped, is noted $\vec{s}_\mathrm{stop}$ and called the “stop point” throughout this article.
Streamlines converge towards the critical points of the velocity field, which correspond to the positions at which the velocity is zero, i.e, the stop point. In a peculiar velocity field, we define such locations as attractors (centers of attracting regions) or repellers (centers of emptying regions). In order to visualize these structures, one can specify the direction in which streamlines are generated. On the one hand, a streamline generated in the [*forward*]{} direction is obtained by integrating the velocity field $\vec{v}$ from a given seed point, and the stop point corresponds to a sink, or attractor. On the other hand, a streamline generated in the [*backward*]{} direction is achieved by integrating $-1 \times \vec{v}$ from a given seed point, allowing us to visualize emptying volumes.This is equivalent to setting the step $\Delta\tau$ negative.
In the study presented here, the velocity field is discretised in a regular cubic grid. Each voxel (cell) is used to generate a streamline. Furthermore, streamlines are generated by the use of the `Streamtracer` function of the VTK library [@VTK:2006]. The integration step $\Delta\tau$ is set to the resolution of the computation grid (size of a single voxel), as this value is the typical step size which allows to obtain converged results and well-defined basins’ borders (see tests in accompanying paper). Streamlines are generated by the use of the fourth order Runge-Kutta method (RK4). The computation of a streamline is stopped if the maximal number of integrations, set beforehand, is reached. This parameters controls the length of streamlines. Tests described in the accompanying paper show that results converge with a typical length of streamlines equivalent approximately to the size of the box in which the velocity field is computed. The velocity field is interpolated by the VTK `Streamtracer` function in order to be evaluated at the positions of the streamlines’ points at each integration step. This interpolation and parameters related to the computation of streamlines will be studied in the accompanying paper along other studies like grid resolution effects.
Identification of basins
------------------------
Once streamlines are computed as described in the previous section, we apply the following algorithm in order to partition regions of space into basins of attraction and repulsion.
Consider a peculiar velocity field $\vec{v}$ discretised into a regular, cubic three-dimensional grid of size $N^3$. $N^3$ thus represents the dimensionality of the discretised velocity field and sets the scale on which the velocity field is computed. A three-dimensional grid $M_\mathrm{stop}$ (also of dimensionality $N^3$) is constructed by counting the number of streamlines whose final destination $\vec{s}_\mathrm{stop}$ is in a given cell. Local maxima of $M_\mathrm{stop}$ correspond thus to the locations of the center’s of attractors and repellers (depending on the direction of integration). A local maximum of $M_\mathrm{stop}$ is detected if the number of streamlines which end in a given cell is (a) greater than the neighbouring cells and (b) greater than a threshold value. This procedure results in $N_{\rm max}$ local maxima $\{\vec{x}_{1}, \vec{x}_{2}, ... , \vec{x}_{N_{\rm max}}\}$ labeled (arbitrarily) as $\{l_{1}, l_{2}, ... , l_{N_{\rm max}}\} \neq 0$. Therefore, voxels in $M_\mathrm{stop}$ are either given a value of 0, if they are not local maxima, or $\{l_{i}\}$ if they are.
Let us now consider another three-dimensional grid $B$ (also of dimensionality $N^3$). The purpose of $B$ is to assign each voxel to a given basin center, $\{\vec{x}_{i}\}$. Each voxel that is not a local maxima is allocated to the basin in which its streamline stops. Note that in our algorithm each voxel represents the starting point of a streamline which ends at one of the local maxima $\{\vec{x}_{i}\}$. In other words voxels are all (1) starting sites for streamlines, but may also (2) host other streamlines or (3) be the end points of a streamline(s). It is the linking of the starting points voxels with the stopping point voxels (local maxima) which define the basins.
\
Figure \[test\_BOA\] displays the result of the segmentation algorithm on a simple distribution of 6 artificial attractors in a volume.
Tests on constrained simulations {#sec:tests}
================================
The method described above is first tested in an ideal setting, namely by using numerical cosmological simulations. The simulation considered in this section is an $N$-body $\Lambda$CDM simulation with $256^3$ dark matter particles of mass $6.57 \times 10^{11}$ M$_\odot/h$, which has been run with the simulation code GADGET-2 [@Springel:2005aa] as part of the Constrained Local UniversE Simulations (CLUES) and Cosmicflows collaborations [@Yepes:2009aa; @Gottloeber:2010aa; @Courtois:2012aa; @Sorce:2016aa]. This simulation provides the density field $\rho_\mathrm{simu}$ and the three cartesian components of the peculiar velocity field $\vec{v}_\mathrm{simu}$. These two fields have been computed in a box of dimensions $256^3$ and of width 500[Mpc/$h$]{}. A $\Lambda$CDM Planck 2013 [@Ade:2013zuv] cosmology has been assumed and the [*Cosmicflows-2*]{} catalog [CF2, @Tully:2013aa] has been used to generate the initial conditions using the constrained realization technique [@Hoffman:1992aa] and the reverse Zeldovich approximation [@Doumler:2013aa].
We can compute an image of the streamlines, i.e.an histogram of the number density of streamlines that cross a voxel in space.
We basically count the number of streamlines that intersect the voxels. The matrix $S$ thus includes values ranging from 1 (only one streamline passes through a voxel, i.e the streamline starting from this voxel) to a huge number (voxels located at attractors or repellers positions, i.e locations to which lots of streamlines converge).
The figure \[simu\_BOA\] presents four panels in supergalactic coordinates SGX-SGY at SGZ = 0 Mpc/$h$. The “image”, i.e.density, of streamlines is displayed in the two top panels. On the left for the forward velocity field, and on the right for the backward velocity field. Two examples of basins of attraction (A and B) and repulsion (C and D), were computed with the segmentation methodology. The bottom panels display those superclusters and emptying regions boundary outlines as solid perimeters on top of the matter density $\rho$ of the simulation. The usual filamentary structure of the matter density can be observed in red, with the highest peaks, i.e.clusters and filaments, in yellow. It is quite striking that from the density alone the boundaries of A and B superclusters, and C and D emptying regions, cannot be identified; however they appear very clearly as well-defined large scale structures in the images of streamlines.
The algorithm finds 87 basins of attractions and 76 basins of repulsion in the $(500$ Mpc/$h)^3$ simulated local universe. This number is related to the voxel size; the grid resolution used here is the “linear” scale for the growth of structures. Peculiar velocities in Cosmic-flows catalogs do not probe non-linear scales typically below 4 Mpc/$h$. As some example, the simulation predicts local superclusters volumes as: Laniakea $5\times10^5 ($Mpc/$h)^3 $, Coma supercluster $1\times10^6\ ($Mpc/$h)^3 $ (in red on Figure \[rho\]), Perseus-Pisces supercluster $7\times10^5 ($Mpc/$h)^3$.
Once a supercluster size and location are kinematically defined, one can compute its total mass. Similarly it is possible to compute the mass of emptying regions. The size of a supercluster varies from $250-500 \times10^3 ($Mpc/$h)^3$, while emptying regions cover volumes of $250-700 \times10^3 ($Mpc/$h)^3$. The typical mass of a supercluster is $5\times10^{16} $M$\odot/h$.\
On Figure \[rho\] (see animation here : <https://vimeo.com/305959931/b85e36f10a>) can be appreciated the density field of the $\Lambda$CDM simulation as red isosurfaces, and the image (or density) of streamlines derived from the peculiar velocity field of the same simulation as yellow isosurfaces. Two gravitational basins are displayed as colored filled contours: one basin of attraction in red (which corresponds to the Coma superclusters, see explanations above), and one basin of repulsion in blue. One can easily see that the isosurfaces of the image of streamlines connect the densities $\rho$ of the simulation belonging to the same gravitational basin.
Testing nearby superclusters {#sec:observations}
============================
In this section, the segmentation methodology is applied to velocity fields reconstructed from observational data, i.e the [*Cosmicflows-2*]{} [CF2, @Tully:2013aa] and [*Cosmicflows-3*]{} [CF3, @Tully:2016aa] catalogs. The velocity fields computed from the CF2 and CF3 datasets allow to study two nearby superclusters: Laniakea and Perseus-Pisces respectively.
The CF2 reconstructed velocity field is computed using the Wiener Filter algorithm [@Zaroubi:1999aa], in a box of dimensions $256^3$ and of width 320 Mpc/$h$. The left panel of Figure \[BOAcosmicflows\] displays as a green solid line the boundary of Laniakea, resulting from the segmentation of the [*local*]{} CF2 three-dimensional velocity field, after filtering out the tidal components caused by distant structures. The methodology applied here to separate the local and tidal flows is the same as the one used in [@Tully:2014aa] in order to define our home supercluster. Indeed, adjacent large scale structures such as Coma, Perseus-Pisces, Shapley and Hercules cannot be fully captured due to the lack of data coverage in the CF2 catalog. The local flow is then extracted by computing the velocity field from the overdensity field contained in a sphere of 100 Mpc/$h$ radius centered on the Norma supercluster and ignoring all the structures outside. For the sake of comparison, the manually found boundary of the Laniakea supercluster as defined in [@Tully:2014aa] is shown in the left panel of Figure \[BOAcosmicflows\] as a yellow solid line. These two frontiers of the supercluster look similar with respectively enclosed volumes of $2.3 \times 10^6$ (Mpc/$h$)$^3$ (automatic – green solid line) and $1.7 \times 10^6$ (Mpc/$h$)$^3$ (manual – yellow). The data, definition and methodology used to identify Laniakea are the same as in [@Tully:2014aa], however the algorithm finds now automatically gravitational basins, instead of manually.
The velocity field reconstructed from the CF3 data is derived in a box of dimensions $256^3$ and of width 500 Mpc/$h$, by forward-modeling the data and computing a set of about a thousand constrained realizations converging with a Monte-Carlo-Markov chain, as proposed by [@Graziani:2018aa], instead of deriving just one single Wiener Filter reconstruction as done for CF2. In this case, the segmentation methodology is applied to the [*full*]{} velocity field, i.e the local and tidal components are not separated. On the right panel of Figure \[BOAcosmicflows\] can be appreciated the frontier of our neighbouring supercluster Perseus-Pisces displayed as a green solid line. This automatically segmented region encloses a volume of $8.8 \times 10^6($Mpc/$h$)$^3$. This volume is however overestimated, as the basin of Perseus-Pisces is not well-defined in the outer region due to the lack of data in this direction. In order to better define the limits of this gravitational basin, one would need to observe more distant structures behind Perseus-Pisces. This letter will be soon followed by a more detailed analysis of the automated segmentation derived from the CF3 reconstructed velocity field. The partitioning analysis will provide identifications of other nearby superclusters like Hercules, Shapley, etc., as well as errors estimates.
It is to be noted that although the Cosmicflows datasets, especially CF2, are very coarse, these volumes, as seen with tests in the previous section, are realistic for a $\Lambda$CDM universe. For example, Laniakea’s and Perseus-Pisces’ volumes are found to be, in a $\Lambda$CDM constrained simulation, $5 \times 10^5$ (Mpc/$h$)$^3$ and $7 \times 10^5$ (Mpc/$h$)$^3$ respectively (see Section \[sec:tests\]).
This automated segmentation of volumes also brings the new capacity of estimating errors on supercluster sizes by the use of constrained realizations (CRs) of the reconstructed velocity fields. In addition to the CF2 and CF3 velocity fields, twenty realizations of each velocity field are considered throughout this section. Here, the segmentation algorithm is applied to the full velocity fields of the CRs, for both CF2 and CF3 datasets. The black crosses and black solid lines displayed in Figure \[BOAcosmicflows\] represent the location of the central attractor and boundaries of the Laniakea (right) and Perseus-Pisces (left) superclusters obtained from the automated segmentation applied on 20 constrained realizations of the CF2 and CF3 velocity fields respectively. The Wiener Filter is by definition the mean over Constrained Realizations (CRs) of the local universe. We define the mean positions, and their associated error, of the basins as the mean positions over CRs (resp. the standard deviation). We obtain: Laniakea (-58$\pm$6, +6$\pm$10, +21$\pm$15) Mpc/$h$, Perseus-Pisces supercluster (33$\pm$10, -7$\pm$10, 31$\pm$9) Mpc/$h$.
In the case of CF2, the tidal and local flows have to be separated to identify Laniakea on observational data (see above). This methodology can lead to different velocity fields and thus different gravitational basins depending on the parameters considered. However, here, this methodology has not been applied to the CF2 CRs. Applying the same approach to both data and CRs may lead to a better agreement between the two.
Conclusions
===========
This letter proposes an automated way of partitioning the universe into gravitational basins, by constructing and following streamlines directly from the gravitational (peculiar) velocity field at a resolution above non-linearities (galaxy collisions, clusters of galaxies) and at a given instant in cosmic time.
As the Figures \[simu\_BOA\] and \[rho\] explicitly show, it is difficult to identify basins of attraction or repulsion from the density field only. Moreover the full density field is not available from observations. This methodology based on the peculiar velocity field brings a new capacity to the field of identifying physically connected large scale structures.
The typical size of basins of attraction has been found to be a volume of the order $10^5-10^6 ($Mpc/$h)^3$ both in the observed and simulated $\Lambda$CDM universe. Figure \[BOAcosmicflows\] shows the current results for the segmentation of the observed Local Universe. The Laniakea and Perseus-Pisces superclusters are easily identified by their basins of attractions, however, at the most distant scales, one can see that both CF2 and CF3 reconstructions are still clearly showing artefacts.
The present-day, current, velocity field is considered throughout this letter. Such a velocity field does not necessarily point along the direction that the actual mass transport takes place. Thus, even if all the flow lines converge to one point, it does not mean that all the mass associated to those flow lines will actually (when taking into account the time evolution of the velocity field) end up in that point. An analysis at other redshifts is included in the accompanying paper, allowing us to study the evolution of gravitational basins over time (basins’ number and total mass).
As stated above, this automated segmentation method will be tested further in a more extensive article which will follow. A future goal would be to quantify the capacity of the segmentation method at deriving physical quantities usually unaccessible, such as the total mass enclosed in a gravitational basin, as seen in Section \[sec:tests\].
Acknowledgements {#acknowledgements .unnumbered}
================
Authors acknowledge support from the Institut Universitaire de France and the CNES. AD and NIL acknowledges financial support of the Project IDEXLYON at the University of Lyon under the Investments for the Future Program (ANR-16-IDEX-0005).
\[lastpage\]
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---
abstract: 'We present the Alternating Anderson-Richardson (AAR) method: an efficient and scalable alternative to preconditioned Krylov solvers for the solution of large, sparse linear systems on high performance computing platforms. Specifically, we generalize the recently proposed Alternating Anderson-Jacobi (AAJ) method (Pratapa et al., *J. Comput. Phys.* (2016), 306, 43–54) to include preconditioning, discuss efficient parallel implementation, and provide serial MATLAB and parallel C/C++ implementations. In serial applications to nonsymmetric systems, we find that AAR is comparably robust to GMRES, using the same preconditioning, while often outperforming it in time to solution; and find AAR to be more robust than Bi-CGSTAB for the problems considered. In parallel applications to the Helmholtz and Poisson equations, we find that AAR shows superior strong and weak scaling to GMRES, Bi-CGSTAB, and Conjugate Gradient (CG) methods, using the same preconditioning, with consistently shorter times to solution at larger processor counts. Finally, in massively parallel applications to the Poisson equation, on up to 110,592 processors, we find that AAR shows superior strong and weak scaling to CG, with shorter minimum time to solution. We thus find that AAR offers a robust and efficient alternative to current state-of-the-art solvers, with increasing advantages as the number of processors grows.'
address:
- 'College of Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA'
- 'Physics Division, Lawrence Livermore National Laboratory, Livermore, CA 94550, USA'
author:
- Phanish Suryanarayana
- 'Phanisri P. Pratapa'
- 'John E. Pask'
bibliography:
- 'Manuscript.bib'
title: 'Alternating Anderson-Richardson method: An efficient alternative to preconditioned Krylov methods for large, sparse linear systems'
---
Linear systems of equations, Parallel computing, Anderson extrapolation, Richardson iteration, Electronic structure calculations
Introduction
============
Linear systems of equations are encountered in the gamut of applications areas within computational physics, from quantum to continuum to celestial mechanics. The strategies adopted for solving such systems can be broadly classified into two categories: direct methods [@davis2006direct] and iterative methods [@Saad_text]. For relatively small system sizes, direct methods such as QR decomposition and LU factorization are generally the preferred approaches. However, as the size of the system increases, direct methods become inefficient—in terms of both computational cost and storage requirements—due to poor scaling with system size relative to iterative approaches, particularly Krylov subspace based techniques such as the Generalized Minimal Residual (GMRES) [@saad1986gmres] and Conjugate Gradient (CG) [@shewchuk1994introduction] methods. Therefore, such iterative approaches are often the methods of choice for the solution of large-scale linear systems of equations.
A number of physical applications require the repeated solution of large, sparse linear systems. For example, in real-space quantum molecular dynamics calculations [@jing1994ab; @shimojo2005embedded; @osei2014accurate; @suryanarayana2017SQDFT] or electronic structure calculations with exact exchange [@perdew1996rationale; @lin2016adaptively], the Poisson equation may be solved hundreds of thousands of times within a single simulation. Therefore, it is critical to reduce the time to solution as far as possible in such situations, a goal typically achieved through parallel computing, wherein the number of floating point operations per second increases linearly with the number of processors. However, the cost associated with inter-processor communication, especially global communication, limits the parallel efficiency of linear solvers, which in turn limits the reduction in wall time that can be achieved in practice [@de1994communication; @duff1999developments; @yang2003improved]. Therefore, there is wide interest in developing algorithms that scale well on modern large-scale parallel computers which can contain tens to hundreds of thousands of computational cores or more [@zuo2010improved; @ghysels2013hiding; @mcinnes2014hierarchical].
Krylov subspace methods such as GMRES and CG have limited parallel scalability due to the large number of global operations inherent to them [@de1995reducing; @ghysels2014hiding]. In this context, the classical Richardson and Jacobi fixed-point iterations [@hackbusch1994iterative; @Saad_text] are ideally suited for massive parallelization by virtue of the strict locality of operations required, i.e., they do not require the calculation of any dot products, other than those required for the calculation of the residual [@golub2001closer; @barrett1994templates]. However, they suffer from extremely large prefactors and poor scaling with system size compared to Krylov subspace methods, which has made them unattractive on even the largest modern platforms. This has motivated the development of strategies that significantly accelerate the convergence of the basic Richardson/Jacobi iterations while maintaining their underlying parallel scalability and simplicity to the maximum extent possible [@Mittal2014; @pratapa2016anderson]. Such approaches include the Chebyshev acceleration technique [@Saad_text] and the recently developed Scheduled Relaxation Jacobi (SRJ) method [@Mittal2014]. However, Chebyshev acceleration requires the computation of the extremal eigenvalues of the coefficient matrix, which may be computationally expensive. Moreover, the current formulation of the SRJ method is applicable only to linear systems arising from the discretization of elliptic equations using second-order finite-differences. For such reasons, Krylov subspace methods have remained the methods of choice in general for the solution of large, sparse linear systems.
Recently, we proposed to employ Anderson extrapolation [@Anderson1965][^1] at periodic intervals within the classical Jacobi iteration, resulting in the so called Alternating Anderson-Jacobi (AAJ) method [@pratapa2016anderson]. This strategy was found to accelerate the Jacobi iteration by orders of magnitude, to the point in fact of outperforming GMRES significantly in serial computations without preconditioning[^2]. In the present work, we generalize the AAJ method to include preconditioning, discuss efficient parallel implementation, and provide serial MATLAB and parallel C/C++ implementations. In serial applications to nonsymmetric systems, we find that AAR is comparably robust to GMRES, using the same preconditioning, while often outperforming it in time to solution. In parallel applications to the Helmholtz and Poisson equations, on up to 110,592 processors, we find that AAR shows superior strong and weak scaling to GMRES, Bi-CGSTAB, and Conjugate Gradient (CG) methods, using the same preconditioning, with consistently shorter times to solution at larger processor counts. We thus find that AAR offers a robust and efficient alternative to current state-of-the-art solvers, with increasing advantages as the number of processors grows.
The remainder of this paper is organized as follows. In Section \[Sec:AAR\], we describe the preconditioned AAR method. We demonstrate the efficiency and parallel scaling of the method in Section \[Sec:Examples\]. Finally, we conclude in Section \[Sec:Conclusions\].
Alternating Anderson-Richardson method {#Sec:AAR}
======================================
In this section, we present the preconditioned Alternating Anderson-Richardson (AAR) method for the solution of large, sparse linear systems: $$\begin{aligned}
& & \bA \bx = \bb \,, \label{Eqn:LinearSystem} \\
& & \bA \in \mathbb{C}^{N\times N}\,, \,\,\, \bx \in \mathbb{C}^{N\times 1}\,, \,\,\, \bb \in \mathbb{C}^{N\times 1}\,, \nonumber\end{aligned}$$ where $\mathbb{C}$ is the set of complex numbers. This approach generalizes the Alternating Anderson-Jacobi (AAJ) method presented previously [@pratapa2016anderson] to include preconditioning and therefore accelerate convergence. In this work, we summarize and focus on the incorporation of preconditioning and the development of an efficient parallel formulation and implementation; a more complete discussion of the underlying alternating Anderson approach found in our previous work [@pratapa2016anderson].
In the AAR method, Anderson extrapolation [@Anderson1965] is performed periodically within a preconditioned Richardson fixed-point iteration [@Saad_text] to accelerate its convergence, while maintaining its parallel scalability to the maximum extent possible. Since the method makes no assumptions about the symmetry of $\bA$, it is applicable to symmetric and nonsymmetric systems alike. Moreover, it is amenable to the three types of preconditioning: left, right, and split [@Saad_text; @benzi2002preconditioning]. For the sake of simplicity, we choose left preconditioning in the present work. Mathematically, the linear system in Eq. \[Eqn:LinearSystem\] is solved using a fixed-point iteration of the form $$\label{Eqn:UpdateAAR}
\bx_{k+1} = \bx_k + \mathbf{B}_k \bff_k \,, \quad k=0, 1, \ldots$$ where the matrix $\mathbf{B}_k \in \mathbb{C}^{N \times N}$ can be written as $$\label{Eqn:Bk}
\mathbf{B}_k = \begin{cases}
\omega \bI & \mbox{if } (k+1)/p \not\in \mathbb{N} \,,\quad\text{(Richardson)}\\
\beta \mathbf{I} - (\mathbf{X}_k + \beta \mathbf{F}_k)(\mathbf{F}_k^{T}\mathbf{F}_k)^{-1} \mathbf{F}_k^{T}& \mbox{if } (k+1)/p \in \mathbb{N} \,.\quad\text{(Anderson)}
\end{cases}$$ Above, $\omega \in \mathbb{C}$ is the relaxation parameter used in the Richardson update, $\beta \in \mathbb{C}$ is the relaxation parameter used in the Anderson update, $\bI \in \mathbb{R}^{N\times N}$ is the identity matrix, the superscript $T$ denotes the conjugate transpose, and $p$ is the frequency of Anderson extrapolation. Additionally, $\bX_k \in \mathbb{C}^{N\times m}$ and $\bF_k \in \mathbb{C}^{N\times m}$ contain the iteration and residual histories at the $k^{th}$ iteration: $$\begin{aligned}
\bX_k & = & \begin{bmatrix} (\bx_{k-m+1} - \bx_{k-m}) & (\bx_{k-m+2} - \bx_{k-m+1}) & \ldots & (\bx_k-\bx_{k-1}) \end{bmatrix} \,, \label{Eq:Xk}\\
\bF_k & = & \begin{bmatrix} (\bff_{k-m+1} - \bff_{k-m}) & (\bff_{k-m+2} - \bff_{k-m+1}) & \ldots & (\bff_k-\bff_{k-1}) \end{bmatrix} \,, \label{Eq:Fk}\end{aligned}$$ where $m+1$ is the number of iterates used for Anderson extrapolation,[^3] and the residual vector $$\label{Eqn:Residual}
\bff_k = \mathbf{M}^{-1} (\bb - \bA \bx_k) \,,$$ with preconditioner $\mathbf{M}^{-1}\in \mathbb{C}^{N\times N}$. As discussed in the context of AAJ [@pratapa2016anderson], the key to the robustness and efficiency of the method is the Anderson extrapolation step which minimizes the $l_2$ norm of the residual in the column space of $\bF_k$, yielding consistent and substantial reductions with increasing history length $m$;[^4] while the key to the parallel scalability of the method is that the extrapolation is performed only periodically, thus reducing nonlocal communications significantly.
We summarize the AAR method in Fig. \[Fig:flowchart\], wherein $\bx_0$ denotes the initial guess, $r_k$ denotes the relative $l_2$ norm of the residual vector (i.e., $r_{k} = {\| \bA \bx_k - \bb \|}/{\| \bb \|}$), and $\epsilon$ is the tolerance specified for convergence. In order to enhance parallel scalability, we reduce global communications by checking for convergence of the fixed-point iteration (i.e., calculating $r_k$) only during Anderson extrapolation steps.
![The preconditioned Alternating Anderson-Richardson (AAR) method.[]{data-label="Fig:flowchart"}](./AAR_flowchart.eps){width="60.00000%"}
The key difference between the AAR and AAJ [@pratapa2016anderson] methods lies in the choice of residual vector Eq. \[Eqn:Residual\]. In the AAR method, any available preconditioner $\mathbf{M}^{-1}$ can be employed, whereas in AAJ, $\mathbf{M} = \mathbf{D}$ is the diagonal part of $\mathbf{A}$. The AAR method thus generalizes the AAJ method in the sense that the AAJ method is recovered for the particular choice of preconditioner $\mathbf{M}^{-1} = \mathbf{D}^{-1}$, i.e., the classical Jacobi preconditioner. Furthermore, just as the AAJ method can be understood as a generalization of the Jacobi [@Saad_text] and Anderson-Jacobi (AJ) [@Anderson1965; @pratapa2016anderson] methods, the AAR method can be understood as a generalization of the Richardson [@Saad_text] and Anderson-Richardson (AR) [@rohwedder2011analysis; @walker2011anderson; @potra2013characterization] methods. Specifically, the AR method is recovered for $p=1$, while the Richardson iteration is obtained in the limit $p \rightarrow \infty$.
As discussed in the context of AAJ [@pratapa2016anderson], the convergence of the AAR method can be understood though its connection to GMRES. First, we note that with complete history (i.e., $m=\infty$), AAR is equivalent to AR for $\omega \neq 0$ and $p \geq 1$ since, upon extrapolation, the residual norm is minimized over the same Krylov subspace regardless of previous extrapolations (in exact arithmetic).[^5] Second, it has been shown [@rohwedder2011analysis; @walker2011anderson; @potra2013characterization] that AR with complete history is equivalent to GMRES without restart, in the sense that the iterates of one can be readily obtained from those of the other (in exact arithmetic).[^6] Hence, in the above sense, AAR with complete history is equivalent to GMRES without restart and so must show corresponding convergence.
With finite history and restarts, however, as typical in practice to reduce storage and/or orthogonalization costs, the convergence rates of both AAR and GMRES are generally reduced. And in this context, as demonstrated in Section \[Sec:Examples\], we typically find shorter times to solution for AAR than for GMRES, with increasing advantages for AAR in parallel calculations as the number of processors grows. As discussed in the context of AAJ [@pratapa2016anderson], this may be due in part to the fact that AAR retains and minimizes over the most recent $m$-vector history at each extrapolation, while GMRES begins anew at each restart. The key advantage of AAR, however, in parallel calculations in particular, is that the majority of iterations are simple, computationally local Richardson iterations, with Anderson extrapolations only every $p$ iterations. A study of the mathematical properties of AAR in relation to GMRES and AR can be found in the recent work of Lupo Pasini [@pasini2018].
Finally, in finite precision, other considerations come into play. For example, while for complete history and exact arithmetic, the iterates produced by AAR upon extrapolation are independent of $\omega$ and $p$, this no longer holds with finite history and floating point arithmetic. Nevertheless, as shown in the context of AAJ [@pratapa2016anderson], the dependence is generally weak over a broad range of values so that the method is generally insensitive to the particular choice of values within the range. Similar insensitivity is found for the Anderson extrapolation parameter $\beta$, though larger values can accelerate convergence in better-conditioned (or well preconditioned) problems, consistent with findings in the nonlinear context [@banerjee2016periodic]. Finally, while in exact arithmetic, a larger history length $m$ must generally improve convergence, by providing a larger subspace over which to minimize, in finite precision, the increasing condition number of the matrix $\mathbf{F}_k^{T}\mathbf{F}_k$ in Eq. \[Eqn:Bk\] with increasing $m$ limits the effective history length in practice. Given the general insensitivity of the method to the particular choice of parameter values, we use the same default set $\{\omega,\beta,m,p \} = \{0.6, 0.6, 9, 8\}$ for all systems in the present work. While possible to optimize for a particular application area, we have found these to be sufficient in a broad range of applications, with available preconditioning in particular, as demonstrated in Section \[Sec:Examples\].
Results and discussion {#Sec:Examples}
======================
In this section, we demonstrate the efficiency and scaling of the preconditioned Alternating Anderson-Richardson (AAR) method in the solution of large, sparse linear systems of equations. Specifically, we consider an assortment of nonsymmetric systems from Matrix Market[^7] as well as Poisson and complex-valued Helmholtz equations arising in real-space electronic structure calculations, and use the default parameters $\{\omega,\beta,m,p\} = \{0.6, 0.6, 9, 8\}$ in AAR for all systems.
Matrix Market: assortment of nonsymmetric systems
-------------------------------------------------
In order to demonstrate the robustness and efficiency of AAR, we first study the relative performance of AAR, GMRES, and Bi-CGSTAB in MATLAB[^8] for nonsymmetric linear systems available in the Matrix Market repository.[^9] Specifically, we consider ten matrices that arise in various areas of computational physics, including oil reservoir modeling, fluid dynamics, and the study of plasmas. We use the default MATLAB parameters for GMRES and Bi-CGSTAB, with a vector of all ones as the starting guess $\bx_0$ in all cases. The simulations are performed on a workstation with the following configuration: Intel Xeon Processor E3-1220 v3 (Quad Core, 3.10GHz Turbo, 8MB), 16GB (2x8GB) 1600MHz DDR3 ECC UDIMM.
In Table \[tab:matmark\], we present the timings (in seconds) obtained for achieving a convergence tolerance of $\epsilon = 10^{-6}$ for two cases: (i) Jacobi preconditioner and (ii) ILU(0) preconditioner. We first note that the simple Jacobi preconditioner is insufficient to obtain convergence for a number of these systems, though AAR is able to converge more systems than GMRES and Bi-CGSTAB. On the other hand, we see that ILU(0) preconditioning is sufficient to obtain convergence for all ten systems for AAR, all but one for GMRES, but only half the systems for Bi-CGSTAB. Moreover, we see that when ILU(0) preconditioning is employed, AAR outperforms GMRES significantly for all but one system; while Bi-CGSTAB can outperform both AAR and GMRES significantly when it converges, but even then outperforms AAR in only two of those five cases. We thus find that AAR shows comparable robustness to GMRES while often outperforming it, while Bi-CGSTAB, though highly competitive when it converges, shows considerably less robustness in the applications considered.
Importantly, we note that while the default parameters for AAR work well for a wide range of applications, as demonstrated above, the flexibility to tune the parameters can be leveraged to tailor AAR for particular applications areas. For example, by simply tuning $\beta$, the solution times for the utm3060, utm1700b, and sherman5 applications above can be brought down to 0.087 s, 0.026 s, and 0.010 s for $\beta$ = 1.7, 1.3, and 0.8, respectively. More importantly, however, by virtue of the superior parallel scaling of AAR, times to solution can be brought down substantially further still relative to standard solvers such as GMRES and Bi-CGSTAB, as we now show.
------------ ------- -------- ------- ----------- ------- ------- -----------
Matrix $N$
AAR GMRES Bi-CGSTAB AAR GMRES Bi-CGSTAB
utm3060 3060 15.104 - 1.805 0.283 0.191 0.043
utm1700a 1700 - - - 0.004 0.013 0.006
utm1700b 1700 4.733 - - 0.045 0.159 0.023
fidap029 2870 0.004 0.019 0.007 0.005 0.016 0.008
sherman5 3312 0.028 0.095 0.016 0.014 0.025 0.014
mcfe 765 - 0.392 - 0.007 0.018 -
memplus 17758 0.521 1.344 - 0.735 2.185 -
add32 4960 0.021 0.070 - 0.021 0.061 -
mcca 180 0.019 0.023 0.008 0.013 0.021 -
fs\_680\_3 680 0.211 - - 0.004 - -
------------ ------- -------- ------- ----------- ------- ------- -----------
: Time taken in seconds by AAR, GMRES, and Bi-CGSTAB in MATLAB for nonsymmetric linear systems from Matrix Market. The symbol ‘-’ is used to indicate that convergence was not achieved within $1000$ sec.[]{data-label="tab:matmark"}
Orbital-free Density Functional Theory: Helmholtz equation {#sec:helmholtz}
----------------------------------------------------------
Next, we study the relative performance of AAR, GMRES with standard restarts [@saad1986gmres], GMRES with augmented restarts [@baker2005technique] (LGMRES), and Bi-CGSTAB [@van1992bi] in PETSc [@Petsc1; @Petsc2][^10]. We consider the periodic Helmholtz problem arising in real-space orbital-free Density Functional Theory (OF-DFT) calculations [@Choly2002; @ghosh2016higher; @Suryanarayana2014524]: $$-\frac{1}{4\pi} \nabla^2 V(\br) + Q \, V(\br) = P \, \rho^{\alpha}(\br) \,\,\, \text{in} \,\,\, \Omega,\quad \begin{cases} V(\br)=V(\br+L_i\hat{\be}_i) \,\,\, \text{on} \,\,\, \partial\Omega \,, \\ \hat{\be}_i\cdot \nabla V(\br)=\hat{\be}_i\cdot \nabla V(\br+L_i\hat{\be}_i) \,\,\, \text{on} \,\,\, \partial\Omega \,, \end{cases} \label{Eqn:Helmholtz_p}$$ where $V(\br)$ is the kernel potential [@wang1998orbital; @wang1999orbital], $\rho(\br)$ is the electron density, $\alpha=\frac{5}{6}+\frac{\sqrt{5}}{6}$, $P=0.003277-i0.009081$, $Q=-0.134992-i0.070225$, $i=\sqrt{-1}$, and $\Omega$ is a cuboidal domain with side lengths $L_i$, unit vectors $\hat{\be}_i$ along each edge, and boundary $\partial \Omega$. The equation is discretized using sixth-order accurate finite-differences on a uniform grid with mesh-size $h$. The computations are parallelized by decomposing the domain into cubical subdomains of equal size, with communication between processors handled via Message Passing Interface (MPI) in the PETSc framework.
In the Anderson extrapolation step of AAR, we perform only one global communication call (i.e., `MPI_Allreduce`) to simultaneously determine $r$ and the complete matrix $\mathbf{F}_k^{T}\mathbf{F}_k$. Since the matrix $\mathbf{F}_k^{T}\mathbf{F}_k$ is generally ill-conditioned, we compute its inverse using the Moore-Penrose pseudoinverse [@laub2005matrix]. We employ the default PETSc parameters for GMRES, LGMRES, and Bi-CGSTAB. In all the simulations, we use a vector of all zeros as the starting guess $\bx_0$ and a convergence tolerance of $\epsilon = 10^{-6}$ on the relative residual. We perform the calculations on a computer cluster consisting of $16$ nodes with the following configuration: Altus 1804i Server - 4P Interlagos Node, Quad AMD Opteron 6276, 16C, 2.3 GHz, 128GB, DDR3-1333 ECC, 80GB SSD, MLC, 2.5" HCA, Mellanox ConnectX 2, 1-port QSFP, QDR, memfree, CentOS, Version 5, and connected through InfiniBand cable.
We first consider a $3 \times 3 \times 3$ aluminum supercell based on a face-centered cubic (FCC) unit cell having lattice constant $7.78$ Bohr, with atoms randomly displaced from ideal positions. We discretize the domain using a finite-difference grid with a mesh-size of $h=0.486$ Bohr, which is sufficient to achieve chemical accuracy in the energy and atomic forces. For the resulting linear system, we employ block Jacobi preconditioning with ILU(0) factorization on each block. Fig. \[Fig:strong:h\] shows the wall time taken by AAR, GMRES, LGMRES, and Bi-CGSTAB on 1, 8, 27, 64, 216, 512, and 1000 cores. We observe that even though the performances of all approaches are similar at low core counts (a consequence of requiring similar number of iterations), AAR starts demonstrating superior performance as the core count is increased. In particular, the minimum wall time achieved by AAR is a factor of $1.38$, $1.43$, and $1.90$ smaller than GMRES, LGMRES, and Bi-CGSTAB, respectively. This is a consequence of the significantly less global communication in AAR compared to the other methods.
We next periodically replicate the above $3 \times 3 \times 3$ aluminum system along one direction by factors of $1$, $2$, $4$, $8$, and $16$, i.e., we generate $3 \times 3 \times 3$, $6 \times 3 \times 3$, $12 \times 3 \times 3$, $24 \times 3 \times 3$, and $48 \times 3 \times 3$ supercells. Correspondingly, we choose 64, 128, 256, 512, and 1024 computational cores. Again, we select $h = 0.486$ Bohr and employ block Jacobi preconditioning with ILU(0) factorization on each block. We plot the results of this weak scaling study in Fig. \[Fig:weak:h\], from which we obtain $\mathcal{O}(N^{1.28})$, $\mathcal{O}(N^{1.57})$, $\mathcal{O}(N^{1.51})$, and $\mathcal{O}(N^{1.59})$ scaling with system size for AAR, GMRES, LGMRES, and Bi-CGSTAB, respectively. Notably, all approaches demonstrate slightly superlinear scaling even though the number of iterations remain constant. This is due to the increased cost of global communications at larger core counts. Therefore, the performance of AAR relative to the other methods is expected to further improve as the number of cores increases.
It is worth noting that the performance gap between AAR and other approaches increases as the linear system becomes less well conditioned, as demonstrated in previous work for AAJ vs. GMRES in serial computations [@pratapa2016anderson]. In order to verify this result for AAR in parallel computations, we consider the Helmholtz equation for a $1 \times 1 \times 1$ Al supercell with lattice constant of $7.78$ Bohr and randomly displaced atoms. To demonstrate the effect of conditioning, we choose a simple Jacobi preconditioner $\mathbf{M}^{-1} = \mathbf{D}^{-1}$, where $\mathbf{D}$ is the diagonal part of $\mathbf{A}$. In Fig. \[Fig:app:Helmholtz\], we present the strong scaling of AAR, GMRES, LGMRES, and Bi-CGSTAB for mesh-sizes of $h=0.216$ and $h=0.108$ Bohr. We observe that as the mesh gets finer and condition number of $\mathbf{A}$ becomes larger, the speedup of AAR over the other methods increases at both small and large core counts. Specifically, for $h=0.216$ Bohr, the minimum wall time taken by AAR is $3.70$, $3.26$, and $3.28$ times smaller than GMRES, LGMRES, and Bi-CGSTAB, respectively. The corresponding numbers for $h=0.108$ Bohr are $6.13$, $4.08$, and $3.04$, respectively. Therefore, we conclude that as the solution of the linear system becomes more challenging (e.g., in the absence of an effective preconditioner), the speedup of AAR over Krylov subspace approaches like GMRES and Bi-CGSTAB is expected to become more substantial in both the serial and parallel settings.[^11]
Density Functional Theory: Poisson equation {#sec:poisson}
-------------------------------------------
Next, we study the relative performance of AAR and Conjugate Gradient (CG) methods for solving the periodic Poisson problem arising in real-space Density Functional Theory (DFT) calculations [@Pask2005; @suryanarayana2013coarse; @Pask2012; @ghosh2016sparc2]: $$-\frac{1}{4\pi} \nabla^2 \phi(\br) = \rho(\br) + b(\br) \,\,\, \text{in} \,\,\, \Omega,\quad \begin{cases} V(\br)=V(\br+L_i\hat{\be}_i) \,\,\, \text{on} \,\,\, \partial\Omega \,, \\ \hat{\be}_i\cdot \nabla V(\br)=\hat{\be}_i\cdot \nabla V(\br+L_i\hat{\be}_i) \,\,\, \text{on} \,\,\, \partial\Omega \,, \end{cases} \label{Eqn:poisson_p}$$ where $\phi(\br)$ is the electrostatic potential, $\rho(\br)$ is the electron density, $b(\br)$ is the pseudocharge density [@Pask2005; @Phanish2010; @Phanish2011; @ghosh2016sparc], and $\Omega$ is a cuboidal domain with side lengths $L_i$, unit vectors $\hat{\be}_i$ along each edge, and boundary $\partial \Omega$. The Poisson equation is discretized using sixth-order accurate finite-differences on a uniform grid with mesh-size $h=0.486$ Bohr, which is sufficient to achieve chemical accuracy in the energy and atomic forces. The computations are parallelized by decomposing the domain into cubical subdomains of equal size, with communication between processors handled via Message Passing Interface (MPI).
In the Anderson extrapolation step of AAR, we again perform only one global communication call (i.e., `MPI_Allreduce`) to simultaneously determine $r$ and the complete matrix $\mathbf{F}_k^{T}\mathbf{F}_k$, whose inverse in computed using the Moore-Penrose pseudoinverse. In all calculations, we again choose a vector of all zeros as the starting guess $\bx_0$, and a convergence tolerance of $\epsilon = 10^{-6}$ on the relative residual. Calculations on up to 1,024 cores were carried out on the same computer cluster as for the Helmholtz problem in Section \[sec:helmholtz\]. Larger calculations, up to 110,592 cores, were carried out on the Vulcan IBM BG/Q machine at the Lawrence Livermore National Laboratory, consisting of 24,576 compute nodes, with 16 computational cores and 16 GB memory per node, for a total of 393,216 cores and 1.6 PB memory.
We first consider a $3 \times 3 \times 3$ Al supercell based on a FCC unit cell with lattice constant 7.78 Bohr, with atoms randomly displaced from ideal positions, and again employ block Jacobi preconditioning with ILU(0) factorization on each block (default in PETSc) in the solution of the resulting linear systems. In Fig. \[Fig:strong:p\], we plot the wall time taken by AAR and CG as implemented in PETSC[^12] on 1, 8, 27, 64, 216, 512, and 1000 computational cores. At small core counts, CG demonstrates better performance than AAR by virtue of requiring fewer iterations to achieve convergence. However, as the number of cores is increased, the performance of AAR relative to CG improves, with the minimum wall time taken by AAR being a factor of $1.31$ smaller than CG by virtue of the lesser global communication required by AAR.
We next perform a weak scaling study by periodically replicating the above $3 \times 3 \times 3$ system along one direction by factors of $1$, $2$, $4$, $8$, and $16$, i.e., we generate $3 \times 3 \times 3$, $6 \times 3 \times 3$, $12 \times 3 \times 3$, $24 \times 3 \times 3$, and $48 \times 3 \times 3$ supercells. Correspondingly, we choose 64, 128, 256, 512, and 1024 computational cores. Again, we employ block Jacobi preconditioning with ILU(0) factorization on each block. In Fig. \[Fig:weak:p\], we plot the wall time taken by AAR and CG for the resulting systems, from which we obtain the weak scaling with system size to be $\mathcal{O}(N^{1.33})$ and $\mathcal{O}(N^{1.38})$, respectively. As before, even though the number of iterations does not vary with system size, the increasing cost associated with global communications results in superlinear scaling for both approaches[^13]. Therefore, the performance of AAR relative to CG is expected to further improve as core counts are increased, a result which we verify next.
To assess the efficiency and scaling of AAR in larger-scale parallel calculations, up to 110,592 cores, we consider strong and weak scaling on the Vulcan IBM BG/Q machine at the Lawrence Livermore National Laboratory. We implement AAR and CG using C and MPI directly[^14], and choose a simple Jacobi preconditioner for parallel scalability. For the strong scaling study, we choose a $12\times 12 \times 12$ Al supercell with atoms randomly displaced, and a maximum of 110,592 computational cores. For the weak scaling study, we go from $6\times 6 \times 6$ supercell on 1728 processors to $24\times 24 \times 24$ supercell on 110,592 processors, with atomic displacements, electron density, and pseudocharge density periodically repeated from the $6\times 6 \times 6$ system. We present the results obtained in Fig. \[Fig:Poisson:llnl\]. We find that the minimum wall time achieved by AAR within the number of cores available for this study is $1.91$ times smaller than that achieved by CG. In addition, the weak scaling with system size for AAR and CG are $\mathcal{O}(N^{1.01})$ and $\mathcal{O}(N^{1.07})$, respectively. This demonstrates again the increased advantage of AAR over current state-of-the-art Krylov solvers in parallel computations as the number of processors is increased.
Concluding remarks {#Sec:Conclusions}
==================
We generalized the recently proposed Alternating Anderson-Jacobi (AAJ) method to include preconditioning and make it particularly well suited for scalable high-performance computing, and demonstrated its efficiency and scaling in the solution of large, sparse linear systems on parallel computers. Specifically, the AAR method employs Anderson extrapolation at periodic intervals within a preconditioned Richardson iteration to accelerate convergence while maintaining its underlying parallel scalability and simplicity to the maximum extent possible. In serial applications to nonsymmetric systems, we find that AAR is comparably robust to GMRES, using the same preconditioning, while often substantially outperforming it in time to solution; and find AAR to be more robust than Bi-CGSTAB for the problems considered. In parallel applications to the Helmholtz and Poisson equations, we find that AAR shows superior strong and weak scaling to GMRES, Bi-CGSTAB, and Conjugate Gradient (CG) methods, using the same preconditioning, with consistently shorter times to solution at larger processor counts. Finally, in massively parallel applications to the Poisson equation, on up to 110,592 processors, we find that AAR shows superior strong and weak scaling to CG, with shorter minimum time to solution.
Our findings suggest that the AAR method provides an efficient and scalable alternative to current state-of-the-art preconditioned Krylov solvers for the solution of large, sparse linear systems on high performance computing platforms, with increasing advantage as the number of processors is increased. Moreover, the method is simple and general, applying to symmetric and nonsymmetric systems, real and complex alike. Additional mathematical analysis which provides further insights into the performance of the AAR method and therefore enables the development of effective preconditioners tailored to it will enable still larger-scale applications, and so constitutes a potentially fruitful direction for future research.
Acknowledgements
================
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. We gratefully acknowledge support from the Laboratory Directed Research and Development Program. P.S. and P.P. also gratefully acknowledge the support of National Science Foundation under Grant Number 1333500.
[^1]: Anderson’s extrapolation has been successfully utilized for accelerating the convergence of non-linear fixed-point iterations arising in electronic structure calculations [@pulay1980convergence], coupled fluid-structure transient thermal problems [@ganine2013nonlinear], as well as neutronics and plasma physics [@willert2014leveraging]. In the context of linear systems of equations, Anderson’s technique bears a close connection to the GMRES method [@rohwedder2011analysis; @walker2011anderson; @potra2013characterization].
[^2]: In the context of electronic structure calculations, the analogue of the AAJ method for nonlinear fixed-point iterations—referred to as Periodic Pulay [@banerjee2016periodic]—is found to significantly accelerate the convergence of the self-consistent field (SCF) method.
[^3]: In the initial iterations, $\bx_j$ in Eq. \[Eq:Xk\] and $\bff_j$ in Eq. \[Eq:Fk\] with $j<0$ are omitted or can be set to null vectors if a pseudoinverse is used to evaluate $(\mathbf{F}_k^{T}\mathbf{F}_k)^{-1}$ in Eq. \[Eqn:Bk\].
[^4]: In practice, $m$ is limited by available memory and/or finite precision effects as the matrix $\mathbf{F}_k^{T}\mathbf{F}_k$ in Eq. \[Eqn:Bk\] becomes ill-conditioned.
[^5]: Excluding potential differences in stagnation [@pasini2018].
[^6]: Excluding potential differences in stagnation [@potra2013characterization].
[^7]: <http://math.nist.gov/MatrixMarket/>
[^8]: <https://www.mathworks.com/>
[^9]: The MATLAB implementation of AAR is available as part of the code accompanying this paper. For GMRES and Bi-CGSTAB, the inbuilt functions in MATLAB are utilized.
[^10]: The PETSc implementation of AAR for complex-valued systems is available as part of the code accompanying this paper. For GMRES, LGMRES, and Bi-CGSTAB, the inbuilt functions in PETSc are utilized.
[^11]: The upturns in strong scaling plots at largest core counts arise due to insufficient computational work per core relative to local inter-processor communications when the chosen computation is spread beyond a certain number of cores. This indicates the strong scaling limit of the current implementation for the chosen problem size.
[^12]: The PETSc implementation of AAR for real-valued systems is available as part of the code accompanying this paper. For CG, the inbuilt function in PETSc is utilized.
[^13]: Another factor contributing to the superlinear scaling for both AAR and CG is the inefficiency of communications between processors in the computer cluster used for the study.
[^14]: The standalone implementation of AAR for real-valued systems using C and MPI directly is available as part of the code accompanying this paper.
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---
abstract: 'Light emission from carbon nanotubes is expected to be dominated by excitonic recombination. Here we calculate the properties of excitons in nanotubes embedded in a dielectric, for a wide range of tube radii and dielectric environments. We find that simple scaling relationships give a good description of the binding energy, exciton size, and oscillator strength.'
author:
- 'Vasili Perebeinos, J. Tersoff, and Phaedon Avouris$^*$'
title: Scaling of excitons in carbon nanotubes
---
The optical properties of carbon nanotubes have received increasing experimental and theoretical attention. Optical absorption and emission spectra of carbon nanotubes have been studied by a number of groups [@Li; @Connell; @Bachilo; @Hagen; @Lebedkin; @Lefebvre]; and electro-optical devices have already appeared [@Misewich; @Marcus]. Initial attempts to explain the experimental observations naturally took independent-electron theory as their starting point. However, theoretically it is now clear that emission is dominated by excitonic recombination [@Ando; @Pedersen; @Kane; @Louie].
A number of theoretical approaches have been used to describe these excitons. One approach involves variational calculations [@Pedersen; @Kane]. While valuable, these have been limited to an effective-mass approximation, and do not address issues of spectral weight. The most accurate description is provided by an [*ab initio*]{} solution of the Bethe-Salpeter equation using GW-corrected quasi-particle energies [@Louie]. However, it is not currently feasible to apply this computationally intensive approach to a wide range of nanotube sizes or environments.
Here we use an intermediate level of theory to provide a broad overview of the exciton properties. We calculate the excitonic properties of nanotubes embedded in dielectric media, for the range of tube radii and dielectric constants most relevant to potential applications. We find that the exciton size, binding energy, and oscillator strength all exhibit robust (though approximate) scaling relationships. The relationships obtained for the excitonic properties can be used to better understand and optimize the operation of nanotube opto-electronic devices.
The proper procedure for the calculation of excitons has been described in detail in Ref. [@Rohlfing]. It involves solving the Bethe-Salpeter equation, $$\begin{aligned}
\Delta_{k}A_{k}^S+\sum_{k'}{\cal K}_{k,k'}A_{k'}^S=\Omega_S
A_{k}^S \label{eq34}\end{aligned}$$ where the kernel ${\cal K}_{k,k'}$ describes the interaction between all possible electron-hole pairs of total momentum $q_{exc}$, and $\Delta_k$ is the quasiparticle energy for a non-interacting electron and hole with wavevector $k$ and $q_{exc}-k$. The exciton momentum $q_{exc}$ is equal to that of the exciting photon, and is hereafter approximated by $q_{exc}=0$. We approximate the quasiparticle energies by eigenvalues of the tight-binding Hamiltonian [@Saito; @Louie2] ($t=3.0$ eV), with any additional self-energy corrections restricted to the so-called “scissors operator”, in which the self-energy is approximated by a rigid shift of the conduction band relative to the valence band. Since the quasi-particle bandstructures are not well known for nanotubes of varying diameters and embedding media, we report only properties that are not affected by the magnitude of this shift.
For the optically active singlet excitons, the interaction has two contributions, direct (${\cal K}^d$) and exchange (${\cal K}^x$): $$\begin{aligned}
{\cal K}_{k,k'}&=&{\cal K}_{k,k'}^d+2{\cal K}^x_{k,k'}
\label{eq35}\end{aligned}$$ where the direct (exchange) term is evaluated with the screened (bare) Coulomb interactions [@Rohlfing]. The unscreened Coulomb interaction between carbon p$_z$ orbitals is modelled by the Ohno potential, which realistically describes organic polymer systems: $$\begin{aligned}
V(r_{ij})=\frac{U}{\sqrt{
\left(\frac{4\pi\varepsilon_0}{e^2}Ur_{ij}\right)^2+1}}\end{aligned}$$ where $r_{ij}$ is the distance between sites $i$ and $j$, and $U=11.3$ eV is the energy cost to place two electrons on a single site ($r_{ij}=0$). Our results are not sensitive to the value of $U$ when the size of the exciton is large.
An ideal calculation would include the nonlocal dielectric response of both the nanotube itself and the medium in which it is embedded. Here, for computational simplicity, we replace this complicated response function with a single dielectric constant $\varepsilon$ [@dielectric]. This is most accurate for narrow tubes, and for embedding media with large dielectric constants. In this regime, the exciton length along the tube is large relative to the tube radius, and most of the dielectric screening occurs in the surrounding medium. The screening is then well described by the dielectric constant $\varepsilon$ of the nanotube environment. For isolated nanotubes, or tubes in low-$\varepsilon$ media, this treatment is not very accurate. Fortunately, it is most accurate in precisely the regime of greatest technological interest. Modulated electro-optical devices [@Misewich; @Marcus] are most practical for relatively narrow tubes embedded in SiO$_2$ ($\varepsilon \sim 4$) or higher-$\varepsilon$ materials.
To calculate the optical properties, we evaluate the imaginary part of the dielectric function for light polarized along the nanotube axis [@DelSole]: $$\begin{aligned}
\epsilon_2(\omega)&=&\frac{8\pi^2
e^2}{V_0m_e^2}\sum_S\left|\sum_{k}A_{k}^S\frac{P_{cv}(k)}
{\Delta_k}\right|^2 \delta(\hbar\omega-\Omega_S) \label{eps2}\end{aligned}$$ where $P$ is the dipole matrix element [@dipole]. The optical response Eq. (\[eps2\]) is the same as derived in the presence of the $GW$ non-local potential [@DelSole]. $\varepsilon_2(\omega)$ obeys a sum rule, where $\int\varepsilon_2d\omega\propto\sum_k
P_{cv}^2(k)/\Delta^2_k$ is a constant independent of the strength of the screened interaction $e^2/\varepsilon$.
We solve the BSE equation (\[eq34\]) by direct diagonalization, choosing a $k$ sample sufficient to converge the low-energy optical spectra (and [*a fortiori*]{} the binding energies). We calculate the binding energy, size, and spectral function for singlet excitons. The binding energy of the first optically active exciton vs. $\varepsilon$ is shown on Fig. \[fig1\]a for four zig-zag tubes with diameters $d=1.0-2.5$ nm. (There is another singlet state 3-5 meV lower in energy, but it is optically silent by symmetry.)
![\[fig1\] (a) Binding energy of first optically active exciton, vs. $\varepsilon$, in four semiconducting zig-zag tubes: (13,0), (19,0), (25,0), and (31,0). (b) Scaling of binding energy of first and second exciton (red dots) in semiconducting tubes with all possible chirality (156 tubes with $d=1.0-2.5$ nm, $\varepsilon=2-15$). Here $R$ and $m$ are in a.u. The black solid line is the best fit to Eq. () for $\varepsilon=4-15$ (3% RMS error over this range), corresponding to $\alpha=1.40$ and ${\rm A_b}=24.1$ eV.](Fig1new_2.eps){height="2.34in" width="2.85in"}
The dependence of binding energy $E_{b1}$ on $\varepsilon$ in Fig. \[fig1\] can be fitted well with a power law, with the exponent being almost independent of the tube diameter. This suggests a more general power law scaling, which we can motivate in an effective mass approximation as follows. Given a variation wavefunction described by a single parameter $L$ that scales the size along the tube axis, the exciton binding energy is: $$\begin{aligned}
E_{L}&\propto&\frac{\hbar^2}{2mL^2}-\frac{e^2}{\epsilon
R}f\left(\frac{L}{R}\right)=\frac{\hbar^2}{mR^2}
g\left(\frac{L}{R},\frac{mR}{\varepsilon}\right) \label{eq46}\end{aligned}$$ Here the first term is the kinetic energy, and $m$ is the effective mass. The second term is the potential energy, which depends on the exciton size [*via*]{} the dimensionless function $f(L/R)$. Then the exciton binding energy is $$\begin{aligned}
E_{b}&=&\min_{L}\left( E_{L} \right)
=\frac{\hbar^2}{mR^2} h\left(\frac{mR}{\varepsilon}\right)
\label{eq47a}\\
E_{b}&\approx& {\rm A_b} R^{\alpha-2}
m^{\alpha-1}\epsilon^{-\alpha}
\label{eq47}\end{aligned}$$ where we approximate the function $h$ by a power law over the range of interest, with empirical parameters $\alpha$ and $A_b$. The effective mass $m$ depends on the tube indices [@Saito] (i.e. on radius and chirality).
In 3D semiconductors, the potential energy is $\propto 1/L$, and the energy is minimized when the exciton size $L_S\propto\varepsilon / m$, so the binding energy scales as $E_b\propto m/\epsilon^2$. This corresponds to true scaling, with a power law $\alpha=2$ in Eq. (\[eq47\]). In the case of nanotubes, the power-law scaling is only an approximation. Nevertheless, for the most important range of tube sizes and dielectric constants, the behavior is rather well described by a power-law scaling in $R$ and $\varepsilon$ with a single value of $\alpha$. Indeed, all the binding energies for the first and second excitons in semiconducting tubes, with all possible chiralities, ($d=1.0-1.5$ nm) collapse onto a single curve shown in Fig. \[fig1\]b. Similar energy scaling was reported by Pedersen [@Pedersen] in a variational effective-mass model. In the range $\varepsilon\gtrsim 4$, where our approach is most reliable, we obtain the best fit with $\alpha=1.40$.
The second exciton that is optically active derives primarily from the second band of the nanotube. It falls within the continuum of the first band, and so becomes a resonance with a finite lifetime [@Louie]. By artificially turning off the interband coupling, we determine that this coupling has very little effect on the exciton energy.
Considerable attention has been focused on the ratio $E_{b2}/E_{b1}$ between the binding energies of the first and second excitons [@Kane; @Louie]. ($E_{b2}$ is defined relative to the second-band quasiparticle gap. Note that the exciton [*formation*]{} energies involve also the [*quasiparticle*]{} bandgaps.) The scaling relation of Eq. (\[eq47\]) predicts $E_{b2}/E_{b1}=(m_2/m_1)^{\alpha-1}$, where $m_2$ and $m_1$ are the effective masses of the first and second bands. In the case of zig-zag tubes: $$\begin{aligned}
m_1&=&\frac{\hbar^2\Delta_1}{3a^2t^2}
\left(1+\sigma\frac{\Delta_1}{2t}\right)^{-1}
\nonumber \\
m_2&=&\frac{\hbar^2\Delta_2}{3a^2t^2}
\left(1-\sigma\frac{\Delta_2}{2t}\right)^{-1}
\label{mass}\end{aligned}$$ Here $\Delta_1$ and $\Delta_2$ are the tight-binding bandgaps; $a$ is the graphene lattice constant; and for tube indeces (n,0), $\sigma$=1 if mod(n,3)=1 and $\sigma$=-1 if mod(n,3)=2.
It is common to treat the gap values in the infinite-radius limit, $\Delta_2^{\infty}=2\Delta_1^{\infty}=2ta/\sqrt{3}R$. This is rather accurate (within 5%) for tubes with $d=1.0-2.5$. On the other hand, for the same range of diameters, the effective mass ratio $m_2/m_1$ varies from 3.4 to 1.3, approaching the infinite-radius limit $m_2/m_1\rightarrow 2$ much more slowly than the gap ratio. Thus caution must be used in discussing available experimental data in terms of the $R\rightarrow\infty$ limit [@Kane]. In particular, for the (8,0) tube $m_2/m_1=0.96$, and according to Eq. (\[eq47\]) the binding energies of the first two excitons should be very similar. Indeed, the accurate first-principles calculations by Spataru et. al. [@Louie] find the binding energies of the first and second excitons (A’$_1$ and C’$_1$ in [@Louie]) to be 0.99 and 1.00 eV. Using $\varepsilon=1.93$ to best reproduce this, our calculations give $E_{b1}=0.99$ eV and $E_{b2}=1.05$ eV. In contrast, the (10,0) tube has $m_2/m_1=4.14$, and for the same $\varepsilon=1.93$ we find $E_{b2}/E_{b1}=1.41$. \[The simple $m^{\alpha-1}$ scaling is not accurate for such small $\varepsilon$.\] It is important to note that effective mass dependence similar to Eq. (\[mass\]) holds also for chiralities other from zig-zag tubes. Thus we expect exciton properties in tubes of index (m,n) to depend primarily on whether mod(n-m,3)=1 or 2, independent of the chiral angle.
To quantify the exciton size, we use the root-mean-square (RMS) distance between electron and hole, $L_S$. The size $L_1$ of the first exciton is shown in Fig. \[fig2\]a as a function of $\varepsilon$, for four different tube diameters. The size is approximately linear in $\varepsilon$. From Eq. (\[eq46\]), $L_S
/ R$ is expected to be a function of $mR/\varepsilon$. Combining this with the observed linear dependence on $\varepsilon$, we anticipate that the exciton size will obey the scaling relationship: $$\begin{aligned}
\frac{L_1}{R}={\rm A_L}+{\rm B_L}\frac{\varepsilon}{Rm_{1}}
\label{LL1}\end{aligned}$$ This is confirmed in Fig. \[fig2\]b, which shows a linear dependence of $L_1/R$ on $\varepsilon/m_{1}R$ in all semiconducting tubes with $d=1.0-2.5$ nm, for all chiralities.
![\[fig2\] (a) First exciton RMS e-h separation $L_1$ in four zig-zag tubes (13,0), (19,0), (25,0), and (31,0). The solid lines are the best linear fits. The slope $k$ scales approximately as $m^{-1}$: the product $km_1$ (in a.u.) equals 0.167, 0.160, 0.157, and 0.155 for tubes with diameter 1.0, 1.5, 2.0, and 2.5 nm, respectively. (b) The linear scaling of $L_1/R$ with $\varepsilon/mR$ in semiconducting tubes of all possible chirality with $d=1.0-2.5$ nm, and $\varepsilon=2-15$. The solid line is the best fit to Eq. () to the results in the range $\varepsilon\ge 4$, giving $A_L=2.13$ and $B_L=0.174$ (for $m$ and $R$ in a.u.). The RMS discrepancy between the fit and the full calculations is 2% for the subset of data having $\varepsilon\geq 4$ or $\varepsilon/mR^2\geq 9.5$.](Fig2new_2.eps){height="2.34in" width="2.85in"}
The exciton size directly affects observable quantities such as the exciton oscillator strength and the radiative lifetime. The exciton oscillator strength is proportional to the probability to find an electron and a hole at the same position [@Elliot]. In 3D semiconductors this is inversely proportional to the exciton volume $1/L^3$. In the case of nanotubes the electron and hole wavefunctions are confined in two dimensions and therefore the oscillator strength should be inversely proportional to the exciton size $L_1$.
Typical optical absorption spectra are shown on Fig. \[fig3\]a-c, calculated for a $(19,
0)$ tube in different dielectric media. As $\varepsilon$ increases, the spectral function converges to the non-interacting limit. For $\varepsilon = 10$, the spectral weight transfer to the first and second excitons, as a fraction of the total spectral weights for first and second bands in the non-interacting limit, are 71% and 55 % respectively. The second exciton resonance is more bound than the first, by 63 meV vs. 43 meV. The higher spectral weight transfer to the first exciton is due not to stronger binding, but rather to the smaller band gap $\Delta_1$.
The probability argument [@Elliot] along with Eq. (\[eps2\]) suggest the following scaling relation for the exciton oscillator strength: $$\begin{aligned}
\frac{I_1}{I_0}=\frac{\rm A_I}{\Delta^2L_1R}\left(1-\frac{{\rm
B_I}R}{L_1}\right) \label{ii0}\end{aligned}$$ where $I_0$ is the spectral weight of the first band for non-interacting particles. The second term is a higher order correction due to the band mixing and the non-constant band-to-band optical density. Figure \[fig3\]d confirms this scaling. While $L_1$ is not directly observable, the figure is virtually unchanged if we use the actual calculated $L_1$ values instead of the scaling expression Eq. (\[LL1\]) for $L_1$. Thus $I_1 / I_0$ obeys rather well an explicit scaling relationship with $\varepsilon/Rm$.
![\[fig3\] Absorption spectra $\varepsilon_2$ in (19,0) tube in dielectric environment (a) $\varepsilon=\infty$ (equivalent to no e-h interaction), (b) $\varepsilon=10$, (c) $\varepsilon=4$. ($E_s$ is the unknown self-energy shift.) Note expanded scales for dotted lines in continuum region in (b) and (c). The fractional spectral weight transfer to the first exciton is $I_1/I_0=$0, 0.71, and 0.95 respectively. Spectra are broadened with a Gaussian width of 0.0125 eV. (d) The scaling of spectral weight transfer to the first exciton, $I_1$, according to Eq. (), in all semiconducting tubes with $d=1.0-2.5$ nm and $\varepsilon=2-15$ and all possible chiralities. The best fit to Eq. () (RMS difference 3.5%) is obtained with ${\rm A_I}=1.22$ eV$^2$nm$^2$ and ${\rm B_I}=1.61$.](Fig3new2_2.eps){height="2.34in" width="2.85in"}
The radiative lifetime of the excitons in carbon nanotubes is a key factor for possible applications in photonics and optoelectronics. The radiative lifetime of an exciton is inversely proportional to its oscillator strength [@Stern]. In the regime of large binding ($\varepsilon\lesssim 3$), $I_1\approx I_0$. Then the oscillator strength per atom is almost independent of tube diameter and chirality, and is equal to $f_0\approx \left( 0.014 {\rm eV}^{-1}\right) E_{\rm exc}$.
We emphasize however that the radiative lifetime and luminescence efficiency of nanotubes involve other factors as well. In principle, a single exciton coherently captures spectral weight from a macroscopic region [@Hanamura]; so the lifetime of an exciton actually depends on the coherence length in the nanotube, which in turn depends on environment and temperature. (If the coherence length is sufficiently large, other lengthscales such as tube length or photon wavelength can become important.) Another important factor is that electrons and holes are relatively unlikely to form optically active excitons, because there are far more excitons that are optically inactive. These include triplet and other dipole-forbidden excitons at [*lower energy*]{} than the optically active exciton. Most importantly, only a tiny fraction of excitons have a total momentum compatible with photon emission, so phonon scattering plays an important role [@finiteQ].
In conclusion, we have calculated optical spectra of carbon nanotubes including the electron-hole Coulomb interaction by solving the Bethe-Salpeter equation (\[eq34\]) in a tight-binding wavefunction basis set. We find scaling relations with respect to the tube radius and dielectric constant $\varepsilon$, for the binding energy Eq. (\[eq47\]), exciton size Eq. (\[LL1\]), and oscillator strength Eq. (\[ii0\]). Thus the absorption and emission properties depend on the dielectric media in which the nanotube is placed. We find a strong dependence on tube index (chirality), but only via the effective mass. This depends strongly on whether mod(n-m,3)=1 or 2, but is otherwise insensitive to the chiral angle.
The authors thank M. Freitag, T. Heinz, M. Hybertsen, S. G. Louie, G. Mahan, and F. Wang for helpful discussions.
Electronic address: [email protected] Z. M. Li, Z. K. Tang, H. J. Liu, N. Wang, C. T. Chan, R. Saito, S. Okada, G. D. Li, J. S. Chen, N. Nagasawa, and S. Tsuda, Phys. Rev. Lett. [**87**]{}, 127401 (2003). M. J. O’Connell, S. M. Bachilo, C. B. Huffman, V. C. Moore, M. S. Strano, E. H. Haroz, K. L. Rialon, P. J. Boul, W. H. Noon, C. Kittrell, J. Ma, R. H. Hauge, R. B. Weisman, and R. E. Smalley, Science, [**297**]{}, 593 (2002). S. M. Bachilo, M. S. Strano, C. Kittrell, R. H. Hauge, R. E. Smalley, R. B. Weisman, Science, [**298**]{}, 2361 (2002). A. Hagen and T. Hertel, Nano Lett. [**3**]{}, 383 (2003) S. Lebedkin, F. Hennrich, T. Skipa, and M. M. Kappes, J. Phys. Chem. B [**107**]{}, 1949 (2003). J. Lefebvre, Y. Homma, and P. Finnie, Phys. Rev. Lett.[**90**]{}, 217401 (2003) J. A. Misewich, R. Martel, Ph. Avouris, J. C. Tsang, S. Heinze, J. Tersoff, Science [**300**]{}, 783 (2003). M. Freitag, Y. Martin, J. A. Misewich, R. Martel, and Ph. Avouris, Nano Lett. [**3**]{}, 1067 (2003). T. Ando, J. Phys. Soc. Japan [**66**]{}, 1066 (1996). T. G. Pedersen, Phys. Rev. B [**67**]{}, 073401 (2003). C.L. Kane and E. J. Mele, Phys. Rev. Lett. [**90**]{}, 207401 (2003). C. D. Spataru, S. Ismail-Beigi, L. X. Benedict, and S. G. Louie, cond-mat/0310220, Applied Physics A M. Rohlfing and S. G. Louie, Phys. Rev. B [**62**]{}, 4927 (2000). R. Saito and H. Kataura, in [*Carbon Nanotubes: Synthesis, Structure, Properties and Application*]{}, edited by M. S. Dresselhaus, G. Dresselhaus, P. Avouris (Springer-Verlag, Heidelberg 2001), Vol. 80. This approach neglects s-p hybridization, which becomes increasingly important for narrower tubes. Fortunately, the associated error is small for the entire range of tubes and excitation energies considered here \[except perhaps for the (8,0) and (10,0) tubes which are mentioned in the text but not including in the scaling analysis\]; see X. Blase, L. X. Benedict, E. L. Shirley, and S. G. Louie, Phys. Rev. Lett. [**72**]{}, 1878 (1994). If the exciton binding energy is much smaller the optical phonon energy in the dielectric medium, then the static dielectric constant $\varepsilon_0$ should be used in the scaling relations obtained here; but if much larger, then the screening $\varepsilon_{\infty}$ (i.e. for frequencies well above the phonon frequency) is used. A general treatment would require explicit inclusion of exciton-phonon coupling. R. Del Sole and R. Girlanda, Phys. Rev. B [**48**]{}, 11789 (1993). L. G. Johnson and G. Dresselhaus, Phys. Rev. B [**7**]{}, 2275 (1973). R. J. Elliott, Phys. Rev. [**108**]{}, 1384 (1957). F. Stern, in [*Solid State Physics*]{}, edited by F. Seitz and D. Turnbull (Academic Press Inc., New York, 1963), Vol. 15, Sec. 36. E. Hanamura, Phys. Rev. B [**38**]{}, 1228 (1988). D. S. Citrin, Phys. Rev. Lett. [**69**]{}, 3393 (1992).
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author:
- |
Felix Finster\
Harvard University, Department of Mathematics
date: |
“uberarbeitete und leicht gek”urzte Fassung\
Dezember 1996
title: Ableitung von Feldgleichungen aus dem Prinzip des fermionischen Projektors
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=0.3cm =0.3cm =-1cm
makefntext\#1[1em to 1.8em[$^{\@thefnmark}$]{}\#1]{}
\[section\] \[Def\][Theorem]{} \[Def\][Beispiel]{} \[Def\][Proposition]{} \[Def\][Korollar]{} \[Def\][Lemma]{} \[Def\][Zusammenfassung]{} \[Def\][Bemerkung]{} \[Def\][Satz]{}
|
---
abstract: 'Molecular spintroinic device based on a single-molecule magnet is one of the ultimate goals of semiconductor nanofabrication technologies. It is thus necessary to understand the electron transport properties of a single-molecule magnet junction. Here we study the negative differential conductance and super-Poissonian shot noise properties of electron transport through a single-molecule magnet weakly coupled to two electrodes with either one or both of them being ferromagnetic. We predict that the negative differential conductance and super-Poissonian shot noise, which can be tuned by a gate voltage, depend sensitively on the spin polarization of the source and drain electrodes. In particular, the shot noise in the negative differential conductance region can be enhanced or decreased originating from the different formation mechanisms of negative differential conductance. The effective competition between fast and slow transport channels is responsible for the observed negative differential conductance and super-Poissonian shot noise. In addition, we further discuss the skewness and kurtosis properties of transport current in the super-Poissonian shot noise regions. Our findings suggest a tunable negative differential conductance molecular device, and the predicted properties of high-order current cumulants are very interesting for a better understanding of electron transport through single-molecule magnet junctions.'
author:
- 'Hai-Bin Xue'
- 'Jiu-Qing Liang'
- 'Wu-Ming Liu'
title: 'Negative differential conductance and super-Poissonian shot noise in single-molecule magnet junctions'
---
Introduction {#introduction .unnumbered}
============
Electronic transport through a single-molecule magnet (SMM) has been intensively studied both experimentally[Heersche,Jo,Grose,Loth,Zyazin,Roch,Komeda,Kahle,Vincent,Thiele]{} and theoretically[Romeike01,Romeike02,Timm01,Elste01,Elste02,Timm02,Wegewijs,Gonzalez01,Gonzalez02,Misiorny02,Misiorny03,Xiejap12,Misiorny04,Xiejap13,WangRN]{} due to its applications in molecular spintronics[@Lapo], but these investigations were focused mainly on the differential conductance or average current. Although the shot noise of electron transport through a SMM has not yet been observed experimentally, new techniques based on carbon nanotubes have been proposed for its possible realization[@Gruneis]. Recently, the current noise properties of electron transport through a SMM have been attracting much theoretical research interests[Romeike03,Imura,Misiorny01,Xuejap10,Xuepla,Xuejap11,WangRQ,Aguado]{} due to they can provide a deeper insight into the nature of transport mechanisms that cannot be obtained by measuring the differential conductance or average current[Blanter,Nazarov]{}. For example, the super-Poissonian shot noise can be used to reveal the information about the internal level structure of the SMM, the left-right asymmetry of the SMM-electrode coupling[@Xuejap10; @Xuepla], and the angle between the applied magnetic field and the SMM’s easy axis[Xuejap11]{}; and distinguish the two types of different nonequilibrium dynamics mechanisms, namely, the quantum tunneling of magnetization process and the thermally excited spin relaxation[@WangRQ]. In particular, the frequency-resolved shot noise spectrum of artificial SMM, e.g., a CdTe quantum dot doped with a single $S=5/2$ Mn spin, can allow one to separately extract the hole and Mn spin relaxation times via the Dicke effect[Aguado]{}.
Among these observed or predicted characteristics, the negative differential conductance (NDC) is especially concerned due to the SMM’s potential applications in a new generation of molecule-based memory devices and logic circuits. On the other hand, the shot noise is usually the sub-Poissonian statistics in non-interacting fermion systems originating from the Pauli exclusion principle. Thus, the super-Poissonian shot noise is another important characteristic of transport current. For the SMM weakly coupled to two normal metal electrodes, the NDC formation mechanism originates essentially from the non-equilibrium electron occupation of the system eigenstates entering bias voltage window[@Timm01; @Xuejap10], namely, the increased current magnitudes of the new opened transport channels do not compensate the decreased current magnitude(s) of the already opened transport channel(s), and the shot noise in this NDC region is obviously enhanced even up to a super-Poissonian shot noise value. In particular, the occurrence of super-Poissonian shot noise depends on the effective competition between different transport channels, thus, the SMM’s internal level structure and the left-right asymmetry of the SMM-electrode coupling, which can tune the SMM transport channels, have an important influence on the super-Poissonian shot noise properties[@Xuejap10; @Xuepla; @Xuejap11]. Whereas for the SMM weakly coupled to two electrodes with either one or both of them being ferromagnetic, the spin polarization of the source and drain electrodes play an important role in the forming speed of the correlated SMM eigenstates involved in the electron tunneling processes, and thus have a remarkable influence on the transport channels entering bias voltage window[@Misiorny01; @Misiorny02; @LuoW]. Consequently, the spin polarization of the source and drain electrodes will have an significant impact on the NDC and super-Poissonian shot noise properties of this SMM system. However, the influences of the spin polarization of the source and drain electrodes on the NDC and super-Poissonian shot noise in the SMM system have not yet been revealed.
The goal of this report is thus to study the influences of the spin polarization of the source and drain electrodes and the applied gate voltage on the NDC and super-Poissonian shot noise in a SMM weakly coupled to two electrodes with either one or both of them being ferromagnetic, and discuss the underlying mechanisms of the observed NDC and super-Poissonian shot noise. It is demonstrated that the gate-voltage-controlled NDC and super-Poissonian shot noise depend sensitively on the spin polarization of the source and drain electrodes. In particular, whether the shot noise in the NDC region being enhanced or not is associated with the formation mechanism of the NDC. Moreover, the skewness and kurtosis in the super-Poissonian shot noise regions show the crossovers from a large positive (negative) to a large negative (positive) values, which also depend on the spin polarization of the source and drain electrodes. These observed characteristics are very interesting for a better understanding of electron transport through single-molecule magnet junctions and will allow for experimental tests in the near future.
Results {#results .unnumbered}
=======
Single-molecule magnet junction {#single-molecule-magnet-junction .unnumbered}
-------------------------------
The SMM junction consists of a SMM weakly coupled to two electrodes, see Fig. 1. The SMM is characterized by the lowest unoccupied non-degenerate molecular orbital (LUMO), the phenomenological giant spin $\overrightarrow{S}$, and the uniaxial anisotropy. The SMM Hamiltonian is thus described by $$H_{\text{SMM}}=(\varepsilon _{d}-eV_{g})\hat{n}+\frac{U}{2}\hat{n}(\hat{n}%
-1)-J\,\vec{s}\cdot \vec{S}-K_{2}(S_{z})^{2}-B_{z}(s^{z}+S^{z}), \label{model}$$Here, the first two terms depict the LUMO, $\hat{n}\equiv d_{\uparrow
}^{\dag }d_{\uparrow }+d_{\downarrow }^{\dag }d_{\downarrow }$ and $U$ are respectively the electron number operator and the Coulomb repulsion between two electrons in the LUMO, with $d_{\sigma }^{\dag }$ ($d_{\sigma }$) being the electron creation (annihilation) operators with spin $\sigma $ and energy $\varepsilon _{d}$ (which can be tuned by a gate voltage $V_{g}$) in the LUMO. The third term describes the exchange coupling between the conduction electron spin $\vec{s}\equiv \sum_{\sigma \sigma ^{\prime
}}d_{\sigma }^{\dag }\left( \vec{\sigma}_{\sigma \sigma ^{\prime }}\right)
d_{\sigma ^{\prime }}$ in the LUMO and the SMM spin $\vec{S}$, with $\vec{%
\sigma}\equiv $ $(\sigma _{x},\sigma _{y},\sigma _{z})$ being the vector of Pauli matrices. The forth term stands for the anisotropy energy of the SMM whose easy-axis is $Z$-axis ($K_{2}>0$). The last term denotes Zeeman splitting. For simplicity, we assume an external magnetic field $%
\overrightarrow{B}$ is applied along the easy axis of the SMM.
The relaxation in the two electrodes is assumed to be sufficiently fast so that their electron distributions can be described by equilibrium Fermi functions. The two electrodes are thus modeled as noninteracting Fermi gases and the corresponding Hamiltonians read $$H_{\text{Leads}}=\sum_{\alpha \mathbf{k}s}\varepsilon _{\alpha \mathbf{k}%
s}a_{\alpha \mathbf{k}s}^{\dag }a_{\alpha \mathbf{k}s}, \label{Leads}$$where $a_{\alpha \mathbf{k}\sigma }^{\dag }$ ($a_{\alpha \mathbf{k}\sigma }$) is the electron creation (annihilation) operators with energy $%
\varepsilon _{\alpha \mathbf{k}\sigma }$, momentum $\mathbf{k}$ and spin $s$ in $\alpha $ ($\alpha =L,R$) electrode, and the index $s=+\left( -\right) $ denotes the majority (minority) spin states with the density of states $%
g_{\alpha }^{s}$. The electrode polarization is characterized by the orientation of the polarization vector $\mathbf{p}_{\alpha }$ and its magnitude is defined as $p_{\alpha }=(g_{\alpha }^{\uparrow }-g_{\alpha }^{\downarrow
})/(g_{\alpha }^{\uparrow }+g_{\alpha }^{\downarrow })$. Here, the polarization vectors $\mathbf{p}_{L}$ (left electrode) and $\mathbf{p}_{R}$ (right electrode) are parallel to the spin quantization $Z$ axis, and spin-up $\uparrow $ and spin-down $\downarrow $ are respectively defined to be the majority spin and minority spin of the ferromagnet. The tunneling between the SMM and the two electrodes are thus described by $$H_{\text{tun}}=\sum_{\alpha \mathbf{k}\sigma }\left( t_{\alpha \mathbf{k}%
\sigma }a_{\alpha \mathbf{k}\sigma }^{\dag }d_{\sigma }+\text{H.c.}\right) ,
\label{tunneling}$$Here, for the ferromagnetic electrode case, the electronic tunneling rates depend on the conduction-electron spin, namely, $\Gamma _{\alpha }^{\uparrow }=2\pi
|t_{\alpha }|^{2}g_{\alpha }^{\uparrow }=(1+p_{\alpha })\Gamma _{\alpha }/2$ and $\Gamma _{\alpha }^{\downarrow }=2\pi |t_{\alpha }|^{2}g_{\alpha
}^{\downarrow }=(1-p_{\alpha })\Gamma _{\alpha }/2$, where the tunneling amplitudes $t_{\alpha }$ and the density of the state $g_{\alpha }^{\sigma }$ are assumed to be independent of wave vector and energy, and $\Gamma
_{\alpha }=\Gamma _{\alpha }^{\uparrow }+\Gamma _{\alpha }^{\downarrow }$; while for the normal-metal electrode case, $p_{\alpha }=0$, thus, $\Gamma
_{\alpha }^{\uparrow }=\Gamma _{\alpha }^{\downarrow }=$ $\Gamma _{\alpha }/2
$.
In addition, we assume that the bias voltage is symmetrically entirely dropped at the SMM-electrode tunnel junctions, i.e., $\mu _{L}=-\mu
_{R}=V_{b}/2$, which implies that the levels of the SMM are independent of the applied bias voltage, and choose meV as the unit of energy. In the Coulomb blockade regime, the occurrence or absence of super-Poissonian shot noise is related to the sequential tunneling gap $\epsilon _{se}$ that being the energy difference between the ground state of charge $N$ and the first excited state of charge $N-1$, and the vertical energy gap $\epsilon
_{co} $ between the ground state of charge $N$ and the first excited state of the same charge[@Aghassi01]. In the present work, we only study the electron transport above the sequential tunneling threshold, namely, $%
V_{b}>2\epsilon _{se}$. In this bias voltage region, the conduction electrons have sufficient energy to overcome the Coulomb blockade and tunnel sequentially through the SMM. It should be noted that the transport current in the Coulomb blockade regime is exponentially suppressed and the co-tunneling tunneling process is dominant in the electron transport, thus, the normalized shot noise will deviate from the present results when taking co-tunneling into account. The parameters of the SMM are chosen as: $S=2$, $\varepsilon
_{d}=0.2$, $U=0.1$, $J=0.1$, $K_{2}=0.04$, $B=0.08$, $\Gamma _{L}=\Gamma
_{R}=\Gamma=0.002$ and $k_{B}T=0.02$.
We first study numerically the effects of the spin polarization of the two electrodes and the applied gate voltage on the NDC and super-Poissonian shot noise in the three different electrode configurations (see Fig. 1), namely, (i) the ferromagnetic lead (Source) - SMM - normal-metal lead (Drain) (i.e., the F-SMM-N system), (ii) the normal-metal lead (Source) - SMM - ferromagnetic lead (Drain) (i.e., the N-SMM-F system), (iii) the ferromagnetic lead (Source) - SMM - ferromagnetic lead (Drain) (i.e., the F-SMM-F system).
The ferromagnetic lead (Source) - SMM - normal-metal lead (Drain) {#the-ferromagnetic-lead-source---smm---normal-metal-lead-drain .unnumbered}
-----------------------------------------------------------------
For the F-SMM-N system considered here, the conduction electron will tunnel into the SMM from the ferromagnetic lead and then tunnel out of the SMM onto the normal-metal lead. The strengths of tunneling coupling of the SMM with two electrodes can be expressed as $%
\Gamma _{L}^{\uparrow }=\Gamma (1+p_{L})/2$, $\Gamma _{L}^{\downarrow
}=\Gamma (1-p_{L})/2$ and $\Gamma _{R}^{\uparrow }=\Gamma _{R}^{\downarrow
}=\Gamma /2$. Since only the energy eigenvalues of singly-occupied and doubly-occupied eigenstates $\epsilon ^{\pm }\left( 1,m\right) $ and $%
\epsilon \left( 2,m\right) $ depend on the gate voltage $V_{g}$, the transition between the singly- and doubly-occupied eigenstates, or between the empty- and singly-occupied eigenstates first entering bias voltage window can be tuned by the gate voltage[@Timm01]. For example, for a relatively small or negative gate voltage, the transition from the singly- to empty-occupied eigenstates first takes place; while for a large enough gate voltage that from the double- to singly-occupied eigenstates first occurs.
Figures 2(a) and 2(b), 2(e) and 2(f) show the average current and shot noise as a function of the bias voltage for different gate voltages $V_{g}$ with $p_{L}=0.3$ and $p_{L}=0.9$. For a large enough spin polarization of source electrode $p_{L}$, the super-Poissonian shot noise is observed when the transition from the doubly- and singly-occupied eigenstates first participates in the electron transport with the bias voltage increasing, see the short dashed, short dash-dotted and thick dashed lines in Fig. 2(f), whereas for the QD system the super-Poissonian noise dose not appear[@Lindebaum]. This characteristic can be understood in terms of the effective competition between fast and slow transport channels[Xuejap10,Xuepla,Xuejap11,Aghassi01,Safonov,Djuric,Aghassi02,Xueepjb,Xueaip,Xueaop]{} and the forming speed of the new correlated eigenstates[@Xueaip]. The current magnitudes of the SMM transport channels can be expressed as[Timm01,Xuejap10,Xuejap11]{}$$I_{\left\vert n,m\right\rangle \longrightarrow \left\vert
n-1,m-1/2\right\rangle }=C_{\left\vert n-1,m-1/2\right\rangle ,\left\vert
n,m\right\rangle }\Gamma _{R}^{\uparrow }n_{R}^{\left( -\right) }\left(
\epsilon _{\left\vert n,m\right\rangle }-\epsilon _{\left\vert
n-1,m-1/2\right\rangle }-\mu _{R}\right) P_{\left\vert n,m\right\rangle },
\label{channelup}$$$$I_{\left\vert n,m\right\rangle \longrightarrow \left\vert
n-1,m+1/2\right\rangle }=C_{\left\vert n-1,m+1/2\right\rangle ,\left\vert
n,m\right\rangle }\Gamma _{R}^{\downarrow }n_{R}^{\left( -\right) }\left(
\epsilon _{\left\vert n,m\right\rangle }-\epsilon _{\left\vert
n-1,m+1/2\right\rangle }-\mu _{R}\right) P_{\left\vert n,m\right\rangle },
\label{channeldown}$$where $C_{\left\vert n-1,m\pm 1/2\right\rangle ,\left\vert n,m\right\rangle
}=\left\vert \left\langle n-1,m\pm 1/2\right\vert d_{\sigma }\left\vert
n,m\right\rangle \right\vert ^{2}$ is a constant which related to the two SMM eigenstates but independent of the applied bias voltage, and $%
P_{\left\vert n,m\right\rangle }$ is the occupation probability of the SMM eigenstate $\left\vert n,m\right\rangle $. Here, the Fermi function $%
n_{R}^{\left( +\right) }\left( \epsilon _{\left\vert n,m\right\rangle
}-\epsilon _{\left\vert n-1,m-1/2\right\rangle }-\mu _{R}\right) $ changes very slowly with increasing bias voltage above the sequential tunneling threshold, namely, $n_{R}^{\left( +\right) }\left( \epsilon _{\left\vert
n,m\right\rangle }-\epsilon _{\left\vert n-1,m-1/2\right\rangle }-\mu
_{R}\right) \simeq 0$, thus, $n_{R}^{\left( -\right) }\left( \epsilon
_{\left\vert n,m\right\rangle }-\epsilon _{\left\vert n-1,m-1/2\right\rangle
}-\mu _{R}\right) \simeq 1$. The current magnitude of the SMM transport channel is thus mainly determined by the occupation probability $P_{\left\vert n,m\right\rangle }$ and $\Gamma _{R}^{\sigma }$.
In order to give a qualitative explanation for the underlying mechanism of the observed super-Poissonian shot noise, we plot the occupation probabilities of the SMM eigenstates as a function of bias voltage for $p_{L}=0.9$ and $%
V_{g}=0.6$ in Fig. 3. With increasing bias voltage, the transport channel $%
\left\vert 2,2\right\rangle \overset{\downarrow }{\longrightarrow }%
\left\vert 1,5/2\right\rangle $ begins to participate in the electron transport. When the bias voltage increases up to about 0.6 meV, the new transport channel $\left\vert 2,2\right\rangle \overset{\uparrow }{\longrightarrow }%
\left\vert 1,3/2\right\rangle ^{-}$ enters the bias voltage window. In this situation, the conduction electron can tunnel out SMM via the two transport channels $\left\vert 2,2\right\rangle \overset{\downarrow }{\longrightarrow }%
\left\vert 1,5/2\right\rangle $ and $\left\vert 2,2\right\rangle \overset{%
\uparrow }{\longrightarrow }\left\vert 1,3/2\right\rangle ^{-}$. For the F-SMM-N system, the electron tunneling between the SMM and the drain electrode (normal-metal lead) is independent of the conduction electron spin, thus the tunneling process mainly relies on the forming speed of the new doubly-occupied eigenstate $\left\vert 2,2\right\rangle $. In the case of $\Gamma _{L}^{\uparrow }\gg \Gamma _{L}^{\downarrow }$, a new doubly-occupied eigenstate $\left\vert 2,2\right\rangle $ can be quickly formed when the spin-up electron tunnels out of the SMM; whereas for the case of the spin-down electron tunneling out of the SMM, the forming of the corresponding new doubly-occupied eigenstate $\left\vert 2,2\right\rangle $ takes a relatively longer time. Thus, for a large enough $p_{L}$, the fast transport channel $\left\vert 2,2\right\rangle \overset{\uparrow }{%
\longrightarrow }\left\vert 1,3/2\right\rangle ^{-}$ can be modulated by the correlated slow channel $\left\vert 2,2\right\rangle \overset{\downarrow }{%
\longrightarrow }\left\vert 1,5/2\right\rangle $, which leads to the bunching effect of the conduction electrons being formed, and is responsible for the formation of the super-Poissonian noise. When $V_{b}>0.9$, the transport channels $\left\vert 1,5/2\right\rangle \overset{\uparrow }{%
\longrightarrow }\left\vert 0,2\right\rangle $ and $\left\vert
1,3/2\right\rangle ^{-}\overset{\downarrow }{\longrightarrow }\left\vert
0,2\right\rangle $ enter the bias volatge window, so that the two successive electron tunneling processes $\left\vert 2,2\right\rangle \overset{%
\downarrow }{\longrightarrow }\left\vert 1,5/2\right\rangle \overset{%
\uparrow }{\longrightarrow }\left\vert 0,2\right\rangle $ and $\left\vert
2,2\right\rangle \overset{\uparrow }{\longrightarrow }\left\vert
1,3/2\right\rangle ^{-}\overset{\downarrow }{\longrightarrow }\left\vert
0,2\right\rangle $ can be formed. Consequently, the formed active competition between the fast-and-slow transport channels is suppressed even destroyed with the current magnitudes of the two new transport channels increasing, which leads to the super-Poissonian shot noise being decreased and even to the sub-Poissonian.
The normal-metal lead (Source) - SMM - ferromagnetic lead (Drain) {#the-normal-metal-lead-source---smm---ferromagnetic-lead-drain .unnumbered}
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In the N-SMM-F system, the strengths of tunnel coupling between the SMM and the two electrodes are described by $\Gamma _{L}^{\uparrow }=\Gamma
_{L}^{\downarrow }=\Gamma /2$, $\Gamma _{R}^{\uparrow }=\Gamma (1+p_{R})/2$, $\Gamma _{R}^{\downarrow }=\Gamma (1-p_{R})/2$. It is demonstrated that the NDC is observed for a small enough or negative gate voltage, and a relatively large spin polarization of drain electrode $p_{R}$, see the solid and dashed lines in Figs. 4(a) and 4(e), especially for a large enough spin polarization $p_{R}$ a strong NDC takes place, see the solid and dashed lines in Fig. 4(e). Moreover, the shot noise can be significantly enhanced and reaches up to a super-Poissonian value when the magnitude of the total current begins to decrease, but the super-Poissonian value in the NDC region is then decreased with further increasing the bias voltage, see the solid and dashed lines in Figs. 4(b) and 4(f). While for a large enough gate voltage, the peaks of super-Poissonian shot noise are observed for a relatively large spin polarization $p_{R}$, see the short dash-dotted and thick dashed lines in Figs. 4(b) and 4(f). The observed NDC and super-Poissonian shot noise characteristics can also be attributed to the mechanism of the fast-and-slow transport channels. Here, we take the $%
V_{g}=-0.1$ and $V_{g}=0.6$ cases with $p_{R}=0.9$ as examples to illustrate these characteristics.
For the $V_{g}=-0.1$ case, the transition from singly-occupied to empty eigenstates $\left\vert 1,5/2\right\rangle \overset{\uparrow }{%
\longrightarrow }\left\vert 0,2\right\rangle $ first participates in the electron transport with increasing the bias voltage, see Figs. 5(a) and 5(b). When the bias voltage is larger than 0.33 meV, the SMM has a small probability of forming the empty-occupied eigenstate $\left\vert 0,-2\right\rangle $, see the thick solid line in Fig. 5(a). If the spin-down electron tunnels into the SMM, the singly-occupied eigenstate $\left\vert 1,-5/2\right\rangle $ can be formed. In this case, for a large enough spin polarization $p_{R}$, namely, $\Gamma _{R}^{\uparrow }\gg \Gamma _{R}^{\downarrow }$, the spin-down electron will remain for a relatively long time in the SMM, so that the electron tunneling processes via the fast transport channels $\left\vert
1,5/2\right\rangle \overset{\uparrow }{\longrightarrow }\left\vert
0,2\right\rangle $ and $\left\vert 1,-3/2\right\rangle ^{-}\overset{\uparrow
}{\longrightarrow }\left\vert 0,-2\right\rangle $ can be blocked and the conduction electrons appear the bunching effect. On the other hand, the current magnitude of the formed fast transport channel $\left\vert
1,5/2\right\rangle \overset{\uparrow }{\longrightarrow }\left\vert
0,2\right\rangle $ begins to decrease with increasing the bias voltage up to about 4.25 meV, while that of the two new opened transport channels $%
\left\vert 1,-3/2\right\rangle ^{-}\overset{\uparrow }{\longrightarrow }%
\left\vert 0,-2\right\rangle $ and $\left\vert 1,-5/2\right\rangle \overset{%
\downarrow }{\longrightarrow }\left\vert 0,-2\right\rangle $ increase. Since the occupation probabilities of the eigenstates $\left\vert
1,-3/2\right\rangle ^{-}$ and $\left\vert 1,-5/2\right\rangle $, $%
P_{\left\vert 1,-3/2\right\rangle ^{-}}$ and $P_{\left\vert
1,-5/2\right\rangle }$ are much smaller that $P_{\left\vert
1,5/2\right\rangle }$, see Fig. 5(b), thus, for the $\Gamma _{R}^{\uparrow
}\gg \Gamma _{R}^{\downarrow }$ case the decreased current magnitude of transport channel $\left\vert 1,5/2\right\rangle \overset{\uparrow }{%
\longrightarrow }\left\vert 0,2\right\rangle $ is much larger than the increased current magnitudes of transport channels $\left\vert
1,-3/2\right\rangle ^{-}\overset{\uparrow }{\longrightarrow }\left\vert
0,-2\right\rangle $ and $\left\vert 1,-5/2\right\rangle \overset{\downarrow }%
{\longrightarrow }\left\vert 0,-2\right\rangle $. Thus, a strong NDC is observed, see the solid line in Fig. 4(e). Moreover, the active competition between the fast channel of current decreasing and the slow channels of current increasing can also obviously enhance the shot noise. Consequently, the shot noise is significantly enhanced by the above two mechanisms and reaches up to a very large value of super-Poissonian shot noise before the occupation probabilities $P_{\left\vert 1,-3/2\right\rangle ^{-}}$ and $P_{\left\vert
1,-5/2\right\rangle }$ are larger than $P_{\left\vert 1,5/2\right\rangle }$, and $P_{\left\vert 0,-2\right\rangle }$ is larger than $P_{\left\vert
0,2\right\rangle }$. With the bias voltage further increasing, the value of super-Poissonian shot noise is decreased quickly but still remains the super-Poissonian distribution. This originates from the fact that the transport channels $%
\left\vert 1,-3/2\right\rangle ^{-}\overset{\uparrow }{\longrightarrow }%
\left\vert 0,-2\right\rangle $ and $\left\vert 1,-5/2\right\rangle \overset{%
\downarrow }{\longrightarrow }\left\vert 0,-2\right\rangle $ can form a new effective competition between the fast and slow transport channels. When the occupation probability $P_{\left\vert 2,2\right\rangle }$ is larger than $P_{\left\vert
0,2\right\rangle }$ ($V_{b}\simeq 0.9$ meV), the active competition between the transport channels $\left\vert 1,-3/2\right\rangle ^{-}\overset{\uparrow
}{\longrightarrow }\left\vert 0,-2\right\rangle $ and $\left\vert
1,-5/2\right\rangle \overset{\downarrow }{\longrightarrow }\left\vert
0,-2\right\rangle $ is destroyed by the new transport channel $\left\vert
2,-2\right\rangle \overset{\uparrow }{\longrightarrow }\left\vert
1,-5/2\right\rangle $ due to the electron tunneling process via the transport channel $\left\vert 1,-5/2\right\rangle $ $\overset{\uparrow }{%
\longrightarrow }\left\vert 2,-2\right\rangle \overset{\uparrow }{%
\longrightarrow }\left\vert 1,-5/2\right\rangle $ can occur, which is responsible for the super-Poissonian shot noise being decreased to the sub-Poissonian distribution.
As for the $V_{g}=0.6$ case, the transport channel $\left\vert
2,2\right\rangle \overset{\downarrow }{\longrightarrow }\left\vert
1,5/2\right\rangle $, which is a slow transport channel for the $\Gamma
_{R}^{\uparrow }\gg \Gamma _{R}^{\downarrow }$ case, first participates in the electron transport, see Figs. 5(c) and 5(d). With the bias voltage increasing up to about $0.4$ meV, the fast electron tunneling process via the transport channel $%
\left\vert 2,-2\right\rangle \overset{\uparrow }{\longrightarrow }\left\vert
1,-5/2\right\rangle $ takes place, thus, the effective competition between fast and slow transport channels can form, and the shot noise is rapidly enhanced and reaches up to a relatively large super-Poissonian value. However, the new transport channels $\left\vert 2,-2\right\rangle \overset{\downarrow }{%
\longrightarrow }\left\vert 1,-3/2\right\rangle ^{-}$ and $\left\vert
2,-1\right\rangle \overset{\uparrow }{\longrightarrow }\left\vert
1,-3/2\right\rangle ^{-}$ can be quickly opened with the bias voltage further increasing, then the fast transport channel $\left\vert
2,-2\right\rangle \overset{\uparrow }{\longrightarrow }\left\vert
1,-5/2\right\rangle $ will be weakened when a spin-down electron tunnels out the SMM through the transition from the eigenstates $\left\vert
2,-2\right\rangle $ to $\left\vert 1,-3/2\right\rangle ^{-}$, so that the formed effective competition between fast and slow transport channels is suppressed and even destroyed. Moreover, when the transport channel $\left\vert 2,2\right\rangle \overset%
{\downarrow }{\longrightarrow }\left\vert 1,5/2\right\rangle $ does not participate in the quantum transport originating from the occupation probabilities $%
P_{\left\vert 2,2\right\rangle }$ and $P_{\left\vert 1,5/2\right\rangle }$ being approaching zero, the two transport channels $\left\vert
2,-2\right\rangle \overset{\uparrow }{\longrightarrow }\left\vert
1,-5/2\right\rangle $ and $\left\vert 2,-2\right\rangle \overset{\downarrow }%
{\longrightarrow }\left\vert 1,-3/2\right\rangle ^{-}$ can not form a new effective competition between fast and slow transport channels due to a relatively fast electron tunneling process via $\left\vert 2,-2\right\rangle \overset{\downarrow }{%
\longrightarrow }\left\vert 1,-3/2\right\rangle ^{-}\overset{\uparrow }{%
\longrightarrow }$ $\left\vert 2,-1\right\rangle \overset{\uparrow }{%
\longrightarrow }\left\vert 1,-3/2\right\rangle ^{-}$ can take place. Consequently, the super-Poissonian shot noise is decreased quickly to a sub-Poissonian value and displays a sharp peak.
The ferromagnetic lead (Source) - SMM - ferromagnetic lead (Drain) {#the-ferromagnetic-lead-source---smm---ferromagnetic-lead-drain .unnumbered}
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We now consider the F-SMM-F system, the strengths of the spin-dependent SMM-electrode coupling are characterized by $\Gamma _{\alpha }^{\uparrow
}=\Gamma (1+p_{\alpha })/2$ and $\Gamma _{\alpha }^{\downarrow }=\Gamma
(1-p_{\alpha })/2$, here we set $p_{L}=p_{R}=p$. For a small enough or negative gate voltage and relatively large spin polarization of the source and drain electrodes $p$, an obvious NDC is observed but weaker than that in the N-SMM-F system, especially for a large enough spin polarization $p$, see the solid and dashed lines in Figs. 4(a) and 6(a), and 4(e) and 6(e). While for a relatively large gate voltage, such as $V_{g}\geq 0.4$ meV, a weak NDC can be observed for a large enough spin polarization $p$, but that in the N-SMM-F system does not occur. Interestingly, for a small enough or negative gate voltage, the shot noise in the NDC region is dramatically enhanced and reaches up to a super-Poissonian value, see the solid and dashed lines in Figs. 6(b) and 6(f); whereas for a large enough gate voltage the formed super-Poissonian shot noise in the NDC region is decreased, see the short dashed, short dash-dotted and thick dashed lines in Fig. 6(f). This characteristic depends on the formation mechanism of the NDC, which is illustrated by the examples of $V_{g}=-0.1$ and $V_{g}=0.6$ with $%
p=0.9$.
For a negative gate voltage $V_{g}=-0.1$, the fast transport channel $%
\left\vert 1,5/2\right\rangle \overset{\uparrow }{\longrightarrow }%
\left\vert 0,2\right\rangle $ first enters the bias voltage window. When the bias voltage increases up to about $0.48$ meV, the new spin-up electron tunneling processes, namely, $\left\vert 1,-3/2\right\rangle ^{-}\overset{\uparrow }{%
\longrightarrow }\left\vert 0,-2\right\rangle $, $\left\vert
1,-1/2\right\rangle ^{-}\overset{\uparrow }{\longrightarrow }\left\vert
0,-1\right\rangle $, $\left\vert 1,1/2\right\rangle ^{-}\overset{\uparrow }{%
\longrightarrow }\left\vert 0,0\right\rangle $, $\left\vert
1,3/2\right\rangle ^{-}\overset{\uparrow }{\longrightarrow }\left\vert
0,1\right\rangle $, and the spin-dowm electron tunneling processes, namely, $%
\left\vert 1,-5/2\right\rangle \overset{\downarrow }{\longrightarrow }%
\left\vert 0,-2\right\rangle $, $\left\vert 1,-3/2\right\rangle ^{-}\overset{%
\downarrow }{\longrightarrow }\left\vert 0,-1\right\rangle $, $\left\vert
1,-1/2\right\rangle ^{-}\overset{\downarrow }{\longrightarrow }\left\vert
0,0\right\rangle $, $\left\vert 1,1/2\right\rangle ^{-}\overset{\downarrow }{%
\longrightarrow }\left\vert 0,1\right\rangle $, $\left\vert
1,3/2\right\rangle ^{-}\overset{\downarrow }{\longrightarrow }\left\vert
0,2\right\rangle $ begin to participate in the quantum transport, see Figs. 7(a) and 7(b). This leads to the current magnitude of the fast transport channel $%
\left\vert 1,5/2\right\rangle \overset{\uparrow }{\longrightarrow }%
\left\vert 0,2\right\rangle $ decrease, but the increased current magnitudes of the new opened transport channels are too small to compensate the decreased current magnitude of $\left\vert 1,5/2\right\rangle \overset{%
\uparrow }{\longrightarrow }\left\vert 0,2\right\rangle $. Thus, a NDC region can form, in which the corresponding shot noise is rapidly enhanced by the active competition between the fast channel of current decreasing and the slow channels of current increasing, and reaches up to a large super-Poissonian value, see the solid line in Fig. 6(f). With further increasing the bias voltage, the formed active competition between the fast channel of current decreasing and the slow channels of current increasing is weakened and even disappears, but the effective competition between the spin-up and spin-down electron tunneling processes is still valid due to $%
\Gamma _{L}^{\uparrow }\gg \Gamma _{L}^{\downarrow }$ and $\Gamma
_{R}^{\uparrow }\gg \Gamma _{R}^{\downarrow }$, thus, the value of the formed super-Poissonian begins to continually decrease but still remains super-Poissonian distribution. When the bias voltage increases up to $0.8$ meV, the current magnitudes of the transport channels originating from the transitions between the double- and singly-occupied eigenstates are already larger than that of the some transport channels originating from the transitions between the singly- and empty-occupied eigenstates, for example, $\left\vert 2,2\right\rangle \overset{\uparrow }{\longrightarrow }\left\vert
1,3/2\right\rangle ^{+}$. In this case, the formed effective competition between the fast and slow transport channels is suppressed and finally destroyed due to these transport channels via the transitions from the double- to singly-occupied eigenstates entering the bias voltage. Consequently, the super-Poissonian shot noise is decreased quickly up to a sub-Poissonian value, see the solid line in Fig. 6(f).
Compared with the $V_{g}=-0.1$ case, for $V_{g}=0.6$ the transport channel $%
\left\vert 2,2\right\rangle \overset{\downarrow }{\longrightarrow }%
\left\vert 1,5/2\right\rangle $ first participates in the quantum transport. When the bias voltage increases up to about $4.8$ meV, the spin-up transport channels $\left\vert 2,-2\right\rangle \overset{\uparrow }{\longrightarrow }%
\left\vert 1,-5/2\right\rangle $, $\left\vert 2,-1\right\rangle \overset{%
\uparrow }{\longrightarrow }\left\vert 1,-3/2\right\rangle ^{-}$, $%
\left\vert 2,0\right\rangle \overset{\uparrow }{\longrightarrow }\left\vert
1,-1/2\right\rangle ^{-}$, $\left\vert 2,1\right\rangle \overset{\uparrow }{%
\longrightarrow }\left\vert 1,1/2\right\rangle ^{-}$, $\left\vert
2,2\right\rangle \overset{\uparrow }{\longrightarrow }\left\vert
1,3/2\right\rangle ^{-}$, and the spin-down transport channels $\left\vert
2,-2\right\rangle \overset{\downarrow }{\longrightarrow }\left\vert
1,-3/2\right\rangle ^{-}$, $\left\vert 2,-1\right\rangle \overset{\downarrow
}{\longrightarrow }\left\vert 1,-1/2\right\rangle ^{-}$, $\left\vert
2,0\right\rangle \overset{\downarrow }{\longrightarrow }\left\vert
1,1/2\right\rangle ^{-}$, $\left\vert 2,1\right\rangle \overset{\downarrow }{%
\longrightarrow }\left\vert 1,3/2\right\rangle ^{-}$ can be opened, while the current magnitude of the transport channel $\left\vert 2,2\right\rangle
\overset{\downarrow }{\longrightarrow }\left\vert 1,5/2\right\rangle $ begin to decrease, see Figs. 7(c) and 7(d). For the $\Gamma _{R}^{\uparrow }\gg \Gamma
_{R}^{\downarrow }$ case, the decreased current magnitude of the spin-down transport channel $\left\vert 2,2\right\rangle \overset{\downarrow }{%
\longrightarrow }\left\vert 1,5/2\right\rangle $ is smaller than the increased current magnitudes of the new opened transport channels, thus, the NDC does not appear. Whereas the active competition between the fast channel of current decreasing and the slow channels of current increasing in a relatively small bias voltage range can form but soon be destroyed, so that the shot noise is significantly enhanced up to a very large super-Poissonian value, then this value begins to decrease but still remains super-Poissonian distribution due to the effective competition between the spin-up and spin-down electron tunneling processes being still valid, see the thick dashed line in Fig. 6(f). In particular, it is interesting note thatthe current magnitudes of the transport channels $\left\vert 2,2\right\rangle
\overset{\downarrow }{\longrightarrow }\left\vert 1,5/2\right\rangle $ and $%
\left\vert 2,2\right\rangle \overset{\uparrow }{\longrightarrow }\left\vert
1,3/2\right\rangle ^{-}$ increase with further increasing the bias voltage, while the current magnitudes of the other transport channels $\left\vert
2,-2\right\rangle \overset{\uparrow }{\longrightarrow }\left\vert
1,-5/2\right\rangle $, $\left\vert 2,-1\right\rangle \overset{\uparrow }{%
\longrightarrow }\left\vert 1,-3/2\right\rangle ^{-}$, $\left\vert
2,0\right\rangle \overset{\uparrow }{\longrightarrow }\left\vert
1,-1/2\right\rangle ^{-}$, $\left\vert 2,1\right\rangle \overset{\uparrow }{%
\longrightarrow }\left\vert 1,1/2\right\rangle ^{-}$, $\left\vert
2,-2\right\rangle \overset{\downarrow }{\longrightarrow }\left\vert
1,-3/2\right\rangle ^{-}$, $\left\vert 2,-1\right\rangle \overset{\downarrow
}{\longrightarrow }\left\vert 1,-1/2\right\rangle ^{-}$, $\left\vert
2,0\right\rangle \overset{\downarrow }{\longrightarrow }\left\vert
1,1/2\right\rangle ^{-}$ and $\left\vert 2,1\right\rangle \overset{%
\downarrow }{\longrightarrow }\left\vert 1,3/2\right\rangle ^{-}$ decrease. This feature leads to the occurrence of a weak NDC. In this NDC bias voltage range, however, the super-Poissonian shot noise value continually decreases, see the thick dashed line in Fig. 6(f). When the transport channels originating from the transitions from the singly- to empty-occupied eigenstates enter the bias voltage, the physical mechanism of decreasing super-Poissonian shot noise is the same as the $V_{g}=-0.1$ case, namely, the formed effective competition between the spin-up and spin-down electron tunneling processes is weakened even destroyed by these current increased transport channels. This is responsible for the super-Poissonian shot noise being decreased to a sub-Poissonian value.
We now study the skewness and kurtosis properties of the transport current in the super-Poissonian shot noise bias voltage regions. It is well known that the skewness and kurtosis (both its magnitude and sign) characterize, respectively, the asymmetry of and the peakedness of the probability distribution around the average transferred-electron number $\bar{n}$ during a time interval t, thus that provide further information for the counting statistics beyond the shot noise. In the N-SMM-F system with a given small enough or negative gate voltage, for a relatively large $p_{R}$, the skewness shows a crossover from a large negative to a relatively small positive values, while the kurtosis shows a crossover from a large positive to a relatively small negative values, see the solid and dashed lines in Figs. 4(c) and 4(d); whereas for a large enough $p_{R}$, the transition of the skewness from a large negative to a large positive values takes place and forms a Fano-like resonance, see the solid, dashed and dotted lines in Fig. 4(g), while the transitions of the kurtosis from a large positive to a large negative values and then from the large negative to a large positive values take place, and form the double Fano-like resonances, see the solid, dashed and dotted lines in Fig. 4(h). In contrast with a small enough or negative gate voltage, for a large enough gate voltage, the skewness and kurtosis for a relatively large $p_{R}$ show, respectively, the crossovers from a large positive to a relatively small negative values and from a small positive to a relatively large negative values, see the short dash-dotted and thick dashed lines in Figs. 4(c) and 4(d); while for a large enough $p_{R}$, the skewness and kurtosis show, respectively, the crossovers from a small positive to a relatively large negative values and from a small negative large to a relatively large positive values, see the short dash-dotted and thick dashed lines in Figs. 4(g) and 4(h), but the variations in the magnitudes of the skewness and kurtosis are much smaller than that for a small enough or negative gate voltage, see Figs. 4(g) and 4(h). As for the F-SMM-F system with a given relatively large $p$, the skewness for a small enough or negative gate voltage shows a large negative value, see the solid, dashed and dotted lines in Figs. 6(c) and 6(g), whereas for a large enough gate voltage that shows a large positive value, see the short dashed, short dash-dotted and thick dashed lines in Figs. 6(c) and 6(g); while the kurtosis shows the double-crossover from a large positive to a relatively small negative values and then from the negative to a large positive values but the latter has a remarkable variation in the magnitude of the kurtosis, see Figs. 6(d) and 6(h). Moreover, we found that the magnitudes of the skewness and kurtosis are more sensitive to the active competition between the fast channels of current decreasing and the corresponding slow channels of current increasing than the shot noise, see the short dashed, short dash-dotted and thick dashed lines in Figs. 2, 4 and 6.
Discussion {#discussion .unnumbered}
==========
In summary, we have studied the NDC and super-Poissonian shot noise properties of electron transport through a SMM weakly coupled to two electrodes with either one or both of them being ferromagnetic, and analyzed the skewness and kurtosis properties of the transport current in the super-Poissonian shot noise regions. It is demonstrated that the occurrences of the NDC and super-Poissonian shot noise depend sensitively on the spin polarization of the soure and drain electrodes and the applied gate voltage. For the F-SMM-N system, when the transition from the double- to singly-occupied eigenstates first enters the bias voltage window, which corresponds to a large enough gate voltage, the super-Poissonian shot noise is observed for a large enough spin polarization of left electrode $p_{L}$. As for the N-SMM-F system, the NDC and super-Poissonian shot noise can be observed for a relatively large spin polarization of right electrode $p_{R}$ and a small enough or negative gate volatge, especially for a large enough $p_{R}$ a strong NDC and a very large value of the super-Poissonian shot noise appear, and the shot noise in the NDC region is first enhanced up to a super-Poissonian value and then is decreased but still remains super-Poissonian distribution for a large enough $p_{R}$; while for a large enough gate voltage and a relatively large $p_{R}$ the super-Poissonian shot noise is only observed. Compared with the N-SMM-F system, for the F-SMM-F system a relatively weak NDC and a large super-Poissonian shot noise bias voltage range are observed; whereas the formed super-Poissonian shot noise in the NDC region is continually decreased for a large enough gate voltage and spin polarization of left and right electrodes $p$. Furthermore, the transitions of the skewness and kurtosis from a large positive (negative) to a large negative (positive) values are also observed, which can provide a deeper and better understanding of electron transport through single-molecule magnet junctions. The observed NDC and super-Poissonian shot noise in the SMM system can be qualitatively attributed to the effective competition between the fast and slow transport channels, and the NDC properties suggest a gate-voltage-controlled NDC molecular device.
Methods {#methods .unnumbered}
=======
The SMM-electrode coupling is assumed to be sufficiently weak, so that the sequential tunneling is dominant. The transitions are well described by the quantum master equation of a reduced density matrix spanned by the eigenstates of the SMM. Under the second order Born approximation and Markov approximation, the particle-number-resolved quantum master equation for the reduced density matrix is given by [@Li1; @Li2; @WangSK] $$\dot{\rho}^{\left( n\right) }\left( t\right) =-i\mathcal{L}\rho ^{\left(
n\right) }\left( t\right) -\frac{1}{2}\mathcal{R}\rho ^{\left( n\right)
}\left( t\right) , \label{Master1}$$with$$\begin{aligned}
\mathcal{R}\rho ^{\left( n\right) }\left( t\right) & =%
%TCIMACRO{\dsum \limits_{\mu=\uparrow,\downarrow}}%
%BeginExpansion
{\displaystyle\sum\limits_{\mu=\uparrow,\downarrow}}
%EndExpansion
\left[ d_{\mu }^{\dagger }A_{\mu }^{\left( -\right)
}\rho ^{\left( n\right) }\left( t\right) +\rho ^{\left( n\right) }\left(
t\right) A_{\mu }^{\left( +\right) }d_{\mu }^{\dagger }-A_{L\mu }^{\left(
-\right) }\rho ^{\left( n\right) }\left( t\right) d_{\mu }^{\dagger }\right.
\notag \\
& \left. -d_{\mu }^{\dagger }\rho ^{\left( n\right) }\left( t\right) A_{L\mu
}^{\left( +\right) }-A_{R\mu }^{\left( -\right) }\rho ^{\left( n-1\right)
}\left( t\right) d_{\mu }^{\dagger }-d_{\mu }^{\dagger }\rho ^{\left(
n+1\right) }\left( t\right) A_{R\mu }^{\left( +\right) }\right] +H.c.,
\label{Master2}\end{aligned}$$where $A_{\mu }^{\left( \pm \right) }=\sum_{\alpha =L,R}A_{\alpha \mu
}^{\left( \pm \right) }$, $A_{\alpha \mu }^{\left( \pm \right) }=\Gamma
_{\alpha }^{\mu }n_{\alpha }^{\left( \pm \right) }\left( -\mathcal{L}\right)
d_{\mu }$, $n_{\alpha }^{\left( +\right) }=f_{\alpha }$ and $n_{\alpha }^{\left(
-\right) }=1-f_{\alpha }$ ($f_{\alpha }$ is the Fermi function of the electrode $\alpha $). Liouvillian superoperator $\mathcal{L}$ is defined as $%
\mathcal{L}\left( \cdots \right) =\left[ H_{\text{SMM}},\left( \cdots
\right) \right] $. $\rho ^{\left( n\right) }\left( t\right) $ is the reduced density matrix of the SMM conditioned by the electron numbers arriving at the right electrode up to time $t$. In order to calculate the first four cumulants, one can define $S\left( \chi ,t\right) =\sum_{n}\rho ^{\left(
n\right) }\left( t\right) e^{in\chi }$. According to the definition of the cumulant generating function $e^{-F\left( \chi \right) }=\sum_{n}$Tr$\left[
\rho ^{\left( n\right) }\left( t\right) \right] e^{in\chi }=\sum_{n}P\left(
n,t\right) e^{in\chi }$, we evidently have $e^{-F\left( \chi \right) }=$Tr$%
\left[ S\left( \chi ,t\right) \right] $, where the trace is over the eigenstates of the SMM. Since Eq. (\[Master1\]) has the following form $$\dot{\rho}^{\left( n\right) }=A\rho ^{\left( n\right) }+C\rho ^{\left(
n+1\right) }+D\rho ^{\left( n-1\right) }, \label{formalmaster}$$$S\left( \chi ,t\right) $ satisfies $$\dot{S}=AS+e^{-i\chi }CS+e^{i\chi }DS\equiv \mathcal{L}_{\chi }S.
\label{formalmaster1}$$In the low frequency limit, the counting time is much longer than the time of electron tunneling through the SMM. In this case, $F\left( \chi \right) $ can be expressed as[Bagrets,Groth,Flindt01,Kieblich]{} $$F\left( \chi \right) =-\lambda _{1}\left( \chi \right) t, \label{CGFformal}$$where $\lambda _{1}\left( \chi \right) $ is the eigenvalue of $\mathcal{L}%
_{\chi }$ which goes to zero for $\chi \rightarrow 0$. According to the definition of the cumulants, one can express $\lambda _{1}\left( \chi \right)
$ as $$\lambda _{1}\left( \chi \right) =\frac{1}{t}\sum_{k=1}^{\infty }C_{k}\frac{%
\left( i\chi \right) ^{k}}{k!}. \label{Lambda}$$Here, the first four cumulants $C_{k}$ are directly related to the transport characteristics. For example, the first-order cumulant (the peak position of the distribution of transferred-electron number) $C_{1}=%
\bar{n}$ gives the average current $\left\langle I\right\rangle =eC_{1}/t$. The zero-frequency shot noise is related to the second-order cumulant (the peak-width of the distribution) $S=2e^{2}C_{2}/t=2e^{2}\left( \overline{n^{2}%
}-\bar{n}^{2}\right) /t$. The third-order $C_{3}=\overline{\left( n-\bar{n}%
\right) ^{3}}$ and four-order $C_{4}=\overline{\left( n-\bar{n}\right) ^{4}}$characterize, respectively, the skewness and kurtosis of the distribution. Here, $\overline{\left( \cdots \right) }=\sum_{n}\left( \cdots \right)
P\left( n,t\right) $. In general, the shot noise, skewness and kurtosis are represented by the Fano factor $F_{2}=C_{2}/C_{1}$, $F_{3}=C_{3}/C_{1}$ and $%
F_{4}=C_{4}/C_{1}$, respectively.
The low order cumulants can be calculated by the Rayleigh–Schrödinger perturbation theory in the counting parameter $\chi $. In order to calculate the first four current cumulants we expand $L_{\chi }$ to four order in $\chi $$$L_{\chi }=L_{0}+L_{1}\chi +\frac{1}{2!}L_{2}\chi ^{2}+\frac{1}{3!}L_{3}\chi
^{3}+\frac{1}{4!}L_{4}\chi ^{4}\cdots . \label{matirxL}$$and define the two projectors[@Flindt01; @Flindt02; @Flindt03] $P=P^{2}=\left\vert \left. 0\right\rangle
\right\rangle \left\langle \left\langle \tilde{0}\right. \right\vert $ and $%
Q=Q^{2}=1-P$, obeying the relations $PL_{0}=L_{0}P=0$ and $%
QL_{0}=L_{0}Q=L_{0}$. Here, $\left\vert \left. 0\right\rangle \right\rangle $ being the steady state $\rho ^{stat}$ is the right eigenvector of $L_{0}$, namely, $L_{0}\left\vert \left. 0\right\rangle \right\rangle =0$, and $%
\left\langle \left\langle \tilde{0}\right. \right\vert \equiv \hat{1}$ is the corresponding left eigenvector. In view of $L_{0}$ being regular, we also introduce the pseudoinverse according to $R=QL_{0}^{-1}Q$, which is well-defined due to the inversion being performed only in the subspace spanned by $Q$. After a careful calculation, $\lambda _{1}\left( \chi \right) $ is given by $$\begin{aligned}
\lambda _{1}\left( \chi \right) & =\left\langle \left\langle \tilde{0}%
\right. \right\vert L_{1}\left\vert \left. 0\right\rangle \right\rangle \chi
\notag \\
& +\frac{1}{2!}\left[ \left\langle \left\langle \tilde{0}\right. \right\vert
L_{2}\left\vert \left. 0\right\rangle \right\rangle -2\left\langle
\left\langle \tilde{0}\right. \right\vert L_{1}RL_{1}\left\vert \left.
0\right\rangle \right\rangle \right] \chi ^{2} \notag \\
& +\frac{1}{3!}\left[ \left\langle \left\langle \tilde{0}\right. \right\vert
L_{3}\left\vert \left. 0\right\rangle \right\rangle -3\left\langle
\left\langle \tilde{0}\right. \right\vert \left(
L_{2}RL_{1}+L_{1}RL_{2}\right) \left\vert \left. 0\right\rangle
\right\rangle \right. \notag \\
& \left. -6\left\langle \left\langle \tilde{0}\right. \right\vert
L_{1}R\left( RL_{1}P-L_{1}R\right) L_{1}\left\vert \left. 0\right\rangle
\right\rangle \right] \chi ^{3}+ \notag \\
& +\frac{1}{4!}\left[ \left\langle \left\langle \tilde{0}\right. \right\vert
L_{4}\left\vert \left. 0\right\rangle \right\rangle -6\left\langle
\left\langle \tilde{0}\right. \right\vert L_{2}RL_{2}\left\vert \left.
0\right\rangle \right\rangle \right. \notag \\
& -4\left\langle \left\langle \tilde{0}\right. \right\vert \left(
L_{3}RL_{1}+L_{1}RL_{3}\right) \left\vert \left. 0\right\rangle \right\rangle
\notag \\
& -12\left\langle \left\langle \tilde{0}\right. \right\vert L_{2}R\left(
RL_{1}P-L_{1}R\right) L_{1}\left\vert \left. 0\right\rangle \right\rangle
\notag \\
& -12\left\langle \left\langle \tilde{0}\right. \right\vert L_{1}R\left(
RL_{2}P-L_{2}R\right) L_{1}\left\vert \left. 0\right\rangle \right\rangle
\notag \\
& -12\left\langle \left\langle \tilde{0}\right. \right\vert L_{1}R\left(
RL_{1}P-L_{1}R\right) L_{2}\left\vert \left. 0\right\rangle \right\rangle
\notag \\
& -24\left\langle \left\langle \tilde{0}\right. \right\vert L_{1}R\left(
R^{2}L_{1}PL_{1}P-RL_{1}PL_{1}R-L_{1}R^{2}L_{1}P\right. \notag \\
& \left. \left. -RL_{1}RL_{1}P+L_{1}RL_{1}R\right) L_{1}\left\vert \left.
0\right\rangle \right\rangle \right] \chi ^{4}+\cdots . \label{matrixLambda}\end{aligned}$$From Eqs. (\[Lambda\]) and (\[matrixLambda\]) we can identify the first four current cumulants:$$C_{1}/t=\left\langle \left\langle \tilde{0}\right. \right\vert
L_{1}\left\vert \left. 0\right\rangle \right\rangle /i, \label{current}$$$$C_{2}/t=\left[ \left\langle \left\langle \tilde{0}\right. \right\vert
L_{2}\left\vert \left. 0\right\rangle \right\rangle -2\left\langle
\left\langle \tilde{0}\right. \right\vert L_{1}RL_{1}\left\vert \left.
0\right\rangle \right\rangle \right] /i^{2}, \label{shot noise}$$$$\begin{aligned}
& C_{3}/t=\left[ \left\langle \left\langle \tilde{0}\right. \right\vert
L_{3}\left\vert \left. 0\right\rangle \right\rangle -3\left\langle
\left\langle \tilde{0}\right. \right\vert \left(
L_{2}RL_{1}+L_{1}RL_{2}\right) \left\vert \left. 0\right\rangle
\right\rangle \right. \notag \\
& \left. -6\left\langle \left\langle \tilde{0}\right. \right\vert
L_{1}R\left( RL_{1}P-L_{1}R\right) L_{1}\left\vert \left. 0\right\rangle
\right\rangle \right] /i^{3}. \label{skewness}\end{aligned}$$$$\begin{aligned}
& C_{4}/t=\left[ \left\langle \left\langle \tilde{0}\right. \right\vert
L_{4}\left\vert \left. 0\right\rangle \right\rangle -6\left\langle
\left\langle \tilde{0}\right. \right\vert L_{2}RL_{2}\left\vert \left.
0\right\rangle \right\rangle \right. \notag \\
& -4\left\langle \left\langle \tilde{0}\right. \right\vert \left(
L_{3}RL_{1}+L_{1}RL_{3}\right) \left\vert \left. 0\right\rangle \right\rangle
\notag \\
& -12\left\langle \left\langle \tilde{0}\right. \right\vert L_{2}R\left(
RL_{1}P-L_{1}R\right) L_{1}\left\vert \left. 0\right\rangle \right\rangle
\notag \\
& -12\left\langle \left\langle \tilde{0}\right. \right\vert L_{1}R\left(
RL_{2}P-L_{2}R\right) L_{1}\left\vert \left. 0\right\rangle \right\rangle
\notag \\
& -12\left\langle \left\langle \tilde{0}\right. \right\vert L_{1}R\left(
RL_{1}P-L_{1}R\right) L_{2}\left\vert \left. 0\right\rangle \right\rangle
\notag \\
& -24\left\langle \left\langle \tilde{0}\right. \right\vert L_{1}R\left(
R^{2}L_{1}PL_{1}P-RL_{1}PL_{1}R-L_{1}R^{2}L_{1}P\right. \notag \\
& \left. \left. -RL_{1}RL_{1}P+L_{1}RL_{1}R\right) L_{1}\left\vert \left.
0\right\rangle \right\rangle \right] /i^{4}. \label{kurtosis}\end{aligned}$$The above four equations are the starting point of the numerical calculation.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the NKBRSFC under grants Nos. 2011CB921502, 2012CB821305, NSFC under grants Nos. 11204203, 61405138, 11275118, 61227902, 61378017, 11434015, SKLQOQOD under grants No. KF201403, SPRPCAS under grants No. XDB01020300.
Author Contributions {#author-contributions .unnumbered}
====================
H. B. X. conceived the idea and designed the research and performed calculations. J. Q. L. and W. M. L. contributed to the analysis and interpretation of the results and prepared the manuscript.
Competing Interests {#competing-interests .unnumbered}
===================
The authors declare no competing financial interests.
Correspondence {#correspondence .unnumbered}
==============
Correspondence and requests for materials should be addressed to Hai-Bin Xue or Wu-Ming Liu.
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---
abstract: 'We determine all continuous solutions $g\colon I\to I$ of the polynomial-like iterative equation $g^3(x)=3g(x)-2x$, where $I\subset\mathbb R$ is an interval. In particular, we obtain an answer to a problem posed by Zoltán Boros (during the Fiftieth International Symposium on Functional Equations, 2012) of determining all continuous functions $f\colon (0,+\infty)\to (0,+\infty)$ satisfying $f^3(x)=\frac{[f(x)]^3}{x^2}$.'
address: 'Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland'
author:
- 'Szymon Draga, Janusz Morawiec'
title: 'On a Zoltán Boros’ problem connected with polynomial-like iterative equations'
---
Introduction
============
Given an interval $I\subset\mathbb R$ we are interested in determining all continuous functions ${g\colon I\to I}$ satisfying $$\label{E}
g^3(x)=3g(x)-2x.$$ Here and throughout the paper $g^n$ denotes the $n$-th iterate of a given self-mapping $g\colon I\to I$; i.e., $g^0={\rm id}_I$ and $g^k=g\circ g^{k-1}$ for all integers $k\geq 1$.
There are two reasons to find all continuous solutions $g\colon I\to I$ of equation . The first one is to answer a problem posed by Zoltán Boros (see [@B]) of determining all continuous functions $f\colon (0,+\infty)\to (0,+\infty)$ satisfying $$\label{be}
f^3(x)=\frac{[f(x)]^3}{x^2}.$$
The second reason is that equation belongs to the class of important and intensively investigated iterative functional equations; i.e., the class of polynomial-like iterative equations of the form $$\label{ple}
\sum_{n=0}^Na_ng^n(x)=F(x),$$ where $a_n$’s are given real numbers, $F\colon I\to I$ is a given function and $g\colon I\to I$ is the unknown function. For the theory of equation and its generalizations we refer the readers to books [@KCG1990; @T1981], surveys [@BJ2001; @ZYZ1995], and some recent papers [@CZ2009; @G2014; @GZ2013; @LZ2013; @MZ2000; @T2010; @TZ2004; @X2004; @XZ2007a; @XZ2007b; @ZNX2006; @ZXZ2013].
Equation represents a linear dependence of iterates of the unknown function and looks like a linear ordinary differential equation with constant coefficients, expressing the linear dependence of derivatives of the unknown function. The difference between these two equations is that linear ordinary differential equations with constant coefficients have a complete theory for finding their solutions, in contrast, even to a very interesting subclass of homogeneous polynomial-like iterative equations of the form $$\label{sple}
\sum_{n=0}^Na_ng^n(x)=0.$$ The difficulties in solving equation , and hence also equation , comes from the fact that the iteration operator $g\to g^n$ is non-linear.
The problem of finding all continuous solutions of equation for a given positive integer $N$ seems to be very difficult. It is completely solved in [@N1974] (see also [@MZ1998]) for $N=2$, but it is still open even in the case where $N=3$ (see [@M1989]). It turns out that the nature of continuous solutions of equation depends deeply on the behavior of complex roots of its characteristic equation $$\label{che}
\sum_{n=0}^N\alpha_nr^n=0.$$ This characteristic equation is motivated by the Euler’s idea for differential equations; it is obtained by putting $g(x)=rx$ into to determine all its linear solutions. There are some results describing all continuous solutions of equation with $N\geq 3$ in very particular cases where the complex roots of equation fulfill special conditions (see [@YZ2004; @ZG; @ZZ2010]). Note that the characteristic equation of equation is of the form $$r^3-3r+2=0,$$ and it has two roots: $r_1=1$ of multiplicity $2$ and $r_2=-2$ of multiplicity $1$. Therefore, none of known results can be used to determine all continuous solutions $g\colon I\to I$ of equation .
Preliminary
===========
It is easy to check that the identity function, defined on an arbitrary set $A\subset\mathbb R$, is a continuous solution of equation . Thus equation has the unique solution $g\colon I\to I$ in the case where $I\subset\mathbb R$ is an interval degenerated to a single point. Therefore, from now on, fix a non-degenerated interval $I\subset\mathbb R$; open or closed or closed on one side, possible infinite.
\[lem11\] Assume that $g\colon I\to I$ is a continuous solution of equation . Then $g$ is strictly monotone. Moreover, if $I\neq\mathbb R$, then $g$ is strictly increasing.
Fix $x,y\in I$ and assume that $g(x)=g(y)$. Then by we obtain $$x=\frac{3g(x)-g^3(x)}{2}=\frac{3g(y)-g^3(y)}{2}=y.$$ Since $g$ is continuous, it follows that it is strictly monotone.
Assume now that $I\neq\mathbb R$ and suppose that, contrary to our claim, $g$ is strictly decreasing. Put $a=\inf I$ and $b=\sup I$. If $b=+\infty$, then $$-\infty<a\leq\lim_{x\to b}g^3(x)=\lim_{x\to b}\big(3g(x)-2x\big)=-\infty,$$ a contradiction. Similarly, if $a=-\infty$, then $$+\infty>b\geq\lim_{x\to a}g^3(x)=\lim_{x\to a}\big(3g(x)-2x\big)=+\infty,$$ a contradiction. Therefore, we have proved that $a,b\in\mathbb R$. Put $c=\inf g(I)$. Since $I$ is non-degenerated and $g$ is strictly decreasing, we have $a\leq c<b$. Moreover, by we have $$c\leq\lim_{x\to b}g^3(x)=\lim_{x\to b}\big(3g(x)-2x\big)=3c-2b,$$ a contradiction.
\[lem12\] Every continuous solution $g\colon\mathbb R\to\mathbb R$ of equation maps bijectively $\mathbb R$ onto $\mathbb R$.
According to Lemma \[lem11\] it is enough to show that $\lim_{x\to +\infty}g(x)\in\{-\infty,+\infty\}$ and $\lim_{x\to -\infty}g(x)\in\{-\infty,+\infty\}$.
Suppose, towards a contradiction, that $\lim_{x\to +\infty}g(x)\in\mathbb R$. By the continuity of $g$ we have $$g^2\big(\lim_{x\to +\infty}g(x)\big)=\lim_{x\to +\infty}g^3(x)=\lim_{x\to +\infty}\big(3g(x)-2x\big)=-\infty,$$ a contradiction. In the same manner we obtain $\lim_{x\to -\infty}g(x)\in\{-\infty,+\infty\}$.
\[lem21\] Define sequences $(a_n)_{n\in\mathbb N_0}$, $(b_n)_{n\in\mathbb N_0}$ and $(c_n)_{n\in\mathbb N_0}$ by putting $$a_0=0,\hspace{3ex}b_0=3,\hspace{3ex}c_0=-2$$ and $$a_{n+1}=b_n,\hspace{3ex}b_{n+1}=3a_n+c_n,\hspace{3ex}c_{n+1}=-2a_n
\hspace{3ex}\hbox{for every }n\in\mathbb N_0.$$ Furthermore, assume that $g\colon I\to I$ solves . Then $$\label{en}
g^{n+3}(x)=a_ng^2(x)+b_ng(x)+c_nx$$ for all $n\in\mathbb N_0$ and $x\in I$.
Moreover, for every $n\in\mathbb N_0$, the following assertions hold:
- $a_n+b_n+c_n=1$;
- $b_{n+1}-b_n=\sum_{k=0}^{n+3}(-2)^k$;
- $b_n=\frac{1}{9}[(-2)^{n+4}+3n+11]$.
The proof is by induction on $n\in\mathbb N_0$.
To prove the main part of the lemma if is enough to observe that putting $g(x)$ instead of $x$ in and making use of we obtain $$g^{n+4}(x)=a_ng^3(x)+b_ng^2(x)+c_ng(x)=b_ng^2(x)+(3a_n+c_n)g(x)-2a_nx$$ for every $x\in I$.
\(i) Since $a_0+b_0+c_0=1$ and $a_{n+1}+b_{n+1}+c_{n+1}=a_n+b_n+c_n$, the assertion follows.
\(ii) From assertion (i) we have $$b_{n+1}-b_n=1-a_{n+1}-c_{n+1}-b_n=-2[b_n-b_{n-1}]+1.$$ Now we need only to observe that $b_1-b_0=-5=\sum_{k=0}^{3}(-2)^k$.
\(iii) Clearly, $b_0=3=\frac{1}{9}[(-2)^4+11]$. Moreover, by assertion (ii) we have $$b_{n+1}=(b_{n+1}-b_n)+b_n=\frac{1-(-2)^{n+4}}{3}+b_n=\frac{1}{9}[(-2)^{n+5}+3(n+1)+11],$$ which completes the proof.
From now on $(a_n)_{n\in\mathbb N_0}$, $(b_n)_{n\in\mathbb N_0}$ and $(c_n)_{n\in\mathbb N_0}$ will stand for the sequences defined in the foregoing lemma.
\[lem22\] Assume that $g\colon I\to I$ is a continuous solution of equation . Then $$\label{22}
\lim_{n\to\infty}\frac{g^{n+3}(x)}{b_n}=-\frac{1}{2}g^2(x)+g(x)-\frac{1}{2}x \hspace{3ex}\hbox{ for every }x\in I.$$
From Lemma \[lem21\] we conclude that $\lim_{n\to\infty}\frac{1}{b_n}=0$ and $\lim_{n\to\infty}\frac{a_n}{b_n}=\lim_{n\to\infty}\frac{c_n}{b_n}=-\frac{1}{2}$. Dividing both sides of by $b_n$ and next tending with $n$ to infinity we obtain .
\[lem23\] Assume that $g\colon I\to I$ is a continuous solution of equation . If for every $x\in I$ the sequence $(g^n(x))_{n\in\mathbb N}$ converges to a real number, then $$\label{23}
g(x)=x \hspace{3ex}\hbox{ for every }x\in I.$$
As in the previous proof we obtain $\lim_{n\to\infty}\frac{1}{b_n}=0$. Then Lemma \[lem22\] implies $$\label{24}
g^2(x)-g(x)=g(x)-x \hspace{3ex}\hbox{ for every }x\in I.$$ By a simple induction we obtain $$\label{25}
g^{n+1}(x)-g^n(x)=g(x)-x \hspace{3ex}\hbox{ for all }n\in\mathbb N\hbox{ and }x\in I.$$ Finally, tending with $n$ to infinity in we come to .
\[lem24\] Assume that $g\colon I\to I$ is a continuous solution of equation .
- If for some $x\in I$ we have $g^2(x)-2g(x)+x\neq 0$, then $$\label{26}
\lim_{n\to\infty}\frac{g^{n+4}(x)}{g^{n+3}(x)}=-2.$$
- If $g$ is increasing, then holds.
\(i) Lemmas \[lem22\] and \[lem21\] yield $$\lim_{n\to\infty}\frac{g^{n+4}(x)}{g^{n+3}(x)}=\lim_{n\to\infty}\frac{g^{n+4}(x)}{b_{n+1}} \frac{b_n}{g^{n+3}(x)}\frac{b_{n+1}}{b_n}=\lim_{n\to\infty}\frac{b_{n+1}}{b_n}=-2.$$
\(ii) If $g$ is increasing, then the sequence $(g^n(x))_{n\in\mathbb N}$ is monotone for every $x\in I$. Hence $\lim_{n\to\infty}g^n(x)$ exists and it equals either a real number or $\pm\infty$. In both the cases cannot be satisfied. Then by assertion (i) we see that holds. In consequence holds.
Main results
============
We are now in a position to find all continuous solutions $g\colon I\to I$ of equation . We will do it in three steps.
\[thm31\] Assume that $I$ is bounded. If $g\colon I\to I$ is a continuous solution of equation , then holds.
By Lemma \[lem11\] we see that $g$ is strictly increasing. This jointly with boundedness of $I$ yields that for every $x\in I$ the sequence $(g^n(x))_{n\in\mathbb N}$ converges to a real number. Lemma \[lem23\] completes the proof.
\[thm32\] Assume that $I$ is a half-line. If $g\colon I\to I$ is a continuous solution of equation , then there exist $c\in\mathbb R$ such that $$\label{31}
g(x)=x+c \hspace{3ex}\hbox{ for every }x\in I.$$ Moreover, $c\leq 0$ in the case where $\inf I=-\infty$ and $c\geq 0$ in the case where $\sup I=+\infty$.
From Lemma \[lem11\] we see that $g$ is strictly increasing. By replacing the function $g$ by the function $\overline{g}\colon -I\to -I$ given by $\overline{g}(x)=-g(-x)$ if necessary, we can assume that $\sup I=+\infty$.
By Theorem \[thm31\] we can also assume that $g$ has no fixed point in int$I$. Indeed, if $g(x_0)=x_0$ for some $x_0\in{\rm int}I$, then $g(x)=x$ for every $x\in (-\infty,x_0]\cap I$, by Theorem \[thm31\], and $g$ solves on $(x_0,+\infty)$.
First, we prove that $g(x)>x$ for every $x\in{\rm int}I$. Suppose the contrary; i.e., $g(x)<x$ for every $x\in{\rm int}I$. Then for every $x\in I$ the sequence $(g^n(x))_{n\in\mathbb N}$ converges to $\inf I$. Since $\inf I$ is a real number, it follows, by Lemma \[lem23\], that holds; a contradiction.
By Lemma \[lem24\] we see that holds. From , Lemma \[lem21\] and we obtain $$g^{n+3}(x)=b_{n-1}(2g(x)-x)+b_ng(x)-2b_{n-2}x=(n+3)g(x)-(n+2)x$$ for all $n\in\mathbb N_0$ and $x\in I$. Hence $$\label{32}
\frac{g^{n+3}(y)-g^{n+3}(x)}{y-x}=(n+3)\frac{g(y)-g(x)}{y-x}-(n+2)$$ for all $n\in\mathbb N_0$ and $x,y\in I$ with $x\neq y$.
Fix $x\in {\rm int}I$ and choose $y\in I$ and $k\in\mathbb N$ such that $x<y$ and $y<g^k(x)$; it is possible because $\lim_{k\to\infty}g^k(x)=+\infty$. Then by the monotonicity of $g$ and we obtain $$0<\frac{g^{n+3}(y)-g^{n+3}(x)}{y-x}\leq\frac{g^{n+k+3}(x)-g^{n+3}(x)}{y-x}=\frac{k(g(x)-x)}{y-x},$$ and hence $\lim_{n\to\infty}\frac{g^{n+3}(y)-g^{n+3}(x)}{(n+3)(y-x)}=0$. Thus, dividing both sides of by $(n+3)$ and next tending with $n$ to infinity, we obtain $$\frac{g(y)-g(x)}{y-x}=1.$$ This jointly with continuity of $g$ gives with $c=g(y)-y>0$.
In conclusion, we have proved that holds with some $c>0$ in the case where $g$ has no fixed point in int$I$; otherwise holds with $c=0$.
\[thm33\] Assume that $I=\mathbb R$. If $g\colon I\to I$ is a continuous solution of equation , then there exists $c\in\mathbb R$ such that either holds or $$\label{33}
g(x)=-2x+c \hspace{3ex}\hbox{ for every }x\in I.$$
From Lemma \[lem12\] we see that either $g$ is an increasing bijection from $\mathbb R$ onto $\mathbb R$ or it is a decreasing bijection from $\mathbb R$ onto $\mathbb R$.
First, we consider the case where $g$ is an increasing bijection.
If there exists $x_0\in\mathbb R$ such that $g(x_0)=x_0$, then both the functions $g|_{(-\infty,x_0]}$ and $g|_{[x_0,+\infty)}$ satisfy equation . Then applying Theorem \[thm32\] we conclude that holds with $c=0$.
If $g(x)\neq x$ for every $x\in\mathbb R$, then either $g(x)>x$ for every $x\in\mathbb R$ or $g(x)<x$ for every $x\in\mathbb R$. Assume that $g(x)>x$ for every $x\in\mathbb R$. Fix $y\in\mathbb R$ and observe that the function $g|_{[y,+\infty)}$ satisfies equation . By Theorem \[thm32\] there exists $c>0$ such that $g(x)=x+c$ for every $x\in[y,+\infty)$. Letting with $y$ to $-\infty$ we conclude that holds. In the same way we can prove that holds with some $c<0$ in the case where $g(x)<x$ for every $x\in\mathbb R$.
Secondly, we consider the case where $g$ is a decreasing bijection.
By Lemma \[lem12\] we see that the formula $G=g^{-1}$ defines a strictly decreasing bijection $G\colon\mathbb R\to\mathbb R$. Putting $g^{-3}(x)$ in place of $x$ in we conclude that $$\label{34}
G^3(x)=\frac{3}{2}G^2(x)-\frac{1}{2}x$$ for every $x\in\mathbb R$.
Fix $x\in\mathbb R$ and define a sequence $(x_n)_{n\in\mathbb N_0}$ putting $$x_0=x\hspace{3ex}\hbox{and}\hspace{3ex}x_n=G(x_{n-1})\hspace{2ex}\hbox{for every }n\in\mathbb N.$$ By we have $$\label{35}
x_{n+3}=\frac{3}{2}x_{n+2}-\frac{1}{2}x_n$$ for every $n\in\mathbb N_0$. It is clear that we can find unique real constants $A$, $B$ and $C$ (depending on $x$) such that $$\label{36}
x_n=A\cdot n+B+C\cdot\left(-\frac{1}{2}\right)^n$$ for $n\in\{0,1,2\}$. According to we conclude, by a simple induction, that holds for every $n\in\mathbb N_0$. Since $G$ is strictly decreasing, it follows that the sequence $(x_n)_{n\in\mathbb N_0}$ is anti-monotone; i.e., the expression $(-1)^n(x_{n+1}-x_n)$ does not change its sign. This forces $A=0$, and hence $2G^2(x)-G(x)-x=2x_2-x_1-x_0=2B+\frac{1}{2}C-B+\frac{1}{2}C-B-C=0$.
In conclusion, we have proved that $$\label{37}
2G^2(x)-G(x)-x=0$$ for every $x\in\mathbb R$. Putting $g^2(x)$ in place of $x$ in we obtain $$2x-g(x)-g^2(x)=0$$ for every $x\in\mathbb R$. Finally, applying Theorem 9 from [@N1974] we conclude that holds.
A Zoltán Boros problem
======================
In this section we answer the question posed by Zoltán Boros of determining all continuous solutions $f\colon (0,+\infty)\to (0,+\infty)$ of equation . In fact, we determine all continuous solutions $f\colon J\to J$ of equation , where $J$ is a subinterval of the half-line $(0,+\infty)$; open or closed or closed on one side, possible infinite or degenerated to a single point.
The proof of the next lemma is very easy, so we omit it.
\[lem41\] If $J\subset\mathbb (0,+\infty)$ is an interval and $f\colon J\to J$ is a solution of equation , then the formula $g=\log\circ f\circ \exp$ defines a function acting from $\log J$ into itself such that holds for every $x\in \log J$.
Conversely, if $I\subset\mathbb R$ is an interval and $g\colon I\to I$ is a solution of equation , then the formula $f=\exp\circ g\circ\log$ defines a function acting from $\exp I$ into itself such that holds for every $x\in \exp I$.
Lemma \[lem41\] and Theorems \[thm31\]–\[thm33\] give the following answer to the question of Zoltán Boros.
\[thm42\] Assume $J\subset(0,+\infty)$ is an interval and let $f\colon J\to J$ be a continuous solution of equation .
- If $J$ is bounded and $0\not\in{\rm cl }J$, then $f(x)=x$ for every $x\in J$.
- If $J$ is bounded and $0\in{\rm cl }J$, then there exists $c\in(0,1]$ such that $$\label{41}
f(x)=cx\hspace{3ex}\hbox{for every }x\in J.$$
- If $J$ is unbounded and $0\not\in{\rm cl }J$, then there exists $c\in[1,+\infty)$ such that holds.
- If $J=(0,+\infty)$, then there exists $c\in(0,+\infty)$ such that either holds or $$f(x)=\frac{c}{x^2}\hspace{3ex}\hbox{ for every }x\in (0,+\infty).$$
Acknowledgment {#acknowledgment .unnumbered}
--------------
This research was supported by University of Silesia Mathematics Department (Iterative Functional Equations and Real Analysis program).
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|
---
author:
- 'Ayan Banerjee,'
- Takol Tangphati
- and Phongpichit Channuie
title: 'Strange Quark Stars in 4D Einstein-Gauss-Bonnet Gravity'
---
Introduction {#intro:Sec}
============
In modern gravity theories, higher derivative gravity (HDG) theories have attracted considerable attention, as an alternative theories beyond GR. Among many impressive outcomes, HDG shows quite different aspects from that in four dimensions, and Einstein–Gauss–Bonnet (EGB) theory [@Lanczos:1938sf] is one of them. The EGB theory is a natural extension of GR to higher dimensions, which emerges as a low energy effective action of heterotic string theory [@Wiltshire:1985us; @Boulware:1985wk; @Wheeler:1986]. As the string theory yields additional higher order curvature correction terms to the Einstein action [@Callan:1985ia]. Interestingly, the EGB Lagrangian is a linear combination of Euler densities continued from lower dimensions, has been widely studied from astrophysics to cosmology. The EGB theory which contains quadratic powers of the curvature is a special case of Lovelocks’ theory of gravitation (LG) [@Lovelock; @Lovelock:1972vz] and is free of ghost. In $4 D$ spacetime EGB and GR are equivalent, as the Gauss-Bonnet (GB) term does not give any contribution to the dynamical equations.
According to the recent theoretical developments, Glavan and Lin [@Glavan:2019inb] proposed a 4-dimensional EGB gravity theory by rescaling the coupling constant $\alpha \to \alpha/(D -4)$, and then taking the limit $D \to 4$, a non-trivial black hole solution was found. It was suggested that one can bypass the Lovelock’s theorem and the GB term gives rise to a non-trivial contribution to the gravitational dynamics. However, it seems that regularization procedure was originally be traced back to Tomozawa [@Tomozawa:2011gp] with finite one-loop quantum corrections to Einstein gravity. One can say that this interesting proposal has opened up a new window for several novel predictions, though the validity of this theory is at present under debate and doubts. The spherically symmetric black hole solutions and their physical properties have been discussed [@Glavan:2019inb] that claims to differ from the standard vacuum-GR Schwarzschild BH. Furthermore, rotating and non-rotating black hole solutions and their physical properties have been discussed, see [@Ghosh:2020syx; @Konoplya:2020juj; @Kumar:2020uyz; @Kumar:2020xvu; @Zhang:2020sjh; @Liu:2020vkh; @li04; @li05; @we03]. Beside that geodesics motion and shadow [@Zeng:2020dco], the strong/ weak gravitational lensing by black hole [@Islam:2020xmy; @Kumar:2020sag; @Heydari-Fard:2020sib; @Jin:2020emq], spinning test particle [@zh03], thermodynamics AdS black hole [@sa03], Hawking radiation [@Zhang:2020qam; @Konoplya:2020cbv], quasinormal modes [@Churilova:2020aca; @Mishra:2020gce; @ar04], and wormhole solutions [@Jusufi:2020yus; @liu20], were extensively analyzed. It has attracted a great deal of recent attention, see [@Jusufi:2020qyw; @Yang:2020jno; @Ma:2020ufk; @si20] for more. More recently, the study of the possible existence of thermal phase transition between AdS to dS asymptotic geometries in vacuum in the context of novel 4D Einstein-Gauss-Bonnet (EGB) gravity has been proposed in Ref.[@Samart:2020sxj].
Hence, the 4$D$ EGB gravity witnessed significant attention that includes finding astrophysical solutions and investigating their properties. In particular, the mass-radius relations are obtained for realistic hadronic and for strange quark star EoS [@Doneva:2020ped]. Precisely speaking, we are interested to investigate the behaviour of compact star namely strange quark stars in regularized 4$D$ EGB gravity. Matter at densities exceeding that of nuclear matter will have to be discussed in terms of quarks. As mentioned in Ref. [@Haensel1986] that for quark matter models massive neutron stars may exist in the form of strange quark stars. Usually the quark matter phase is modeled in the context of the MIT bag model as a Fermi gas of $u$, $d$, and $s$ quarks. At finite densities and zero or small temperature, quark matter can exhibit substantial rich phase structures resulting from different pairing mechanisms due to the coupling of color, flavor and spin degrees of freedom, see e.g. Refs. [@RAJAGOPAL2001; @alford2007; @Alford2002]. In addition, a variety of different condensates underlying fundamental descriptions may be plausible.
An expectation is that quark matter might play an important role in cosmology and in astrophysics, see e.g. [@Alford2004; @far84]. On the one hand, in cosmology, it may provide an explanation of a source of density fluctuation and as a consequence of how galaxies form generated by the quark-hadron transition. On the other hand, in astrophysics, quark matter is an interplay between general relativistic effects and the equation of state of nuclear particle physics. These objects are present in the form of the stellar equilibrium including neutron stars with a quark core, super massive stars, white dwarfs and even strange quark stars. Nevertheless, in all possible applications of quark matter from cosmology and astrophysics, our lack of knowledge of the exact equation posses the main source of uncertainties in describing stars. In order to study the stable/unstable configurations and even other physical properties of stars, the realistic equations of state (EoS) have to be proposed. The color-flavor locked phase appearing in three flavor (up, down, strange) matter posses the importance of condensates [@alf01; @raj01; @lugo02; @Steiner:2002gx] and is shown to be the asymptotic ground state of quark matter at low temperature [@Schafer:1999jg]. For instant, the authors of Ref.[@Banerjee:2020stc] studied a class of static and spherically symmetric compact objects made of strange matter in the color flavor locked (CFL) phase in 4$D$ EGB gravity.
The structure of the present work is as follows: after the introduction in Sec.\[intro:Sec\], we quickly review how to derive the field equations in the context of 4$D$ EGB gravity and show that it makes a nontrivial contribution to gravitational dynamics in 4$D$ in Sec.\[sec2\]. In Sec.\[sec3\] we discuss a class of static and spherically symmetric compact objects invoking the equation of state parameters in quark matter phases invoking massless quark and cold star approximations. In Sec.\[sec4\], we discuss the numerical procedure used to solve the field equations. In the same section, we report the general properties of the spheres in terms of the massless quark and cold star approximations. We analyzed the energy conditions as well as other properties of the spheres, such as sound velocity and adiabatic stability. Finally, we conclude our findings in the last section.
Basic equations of EGB gravity {#sec2}
==============================
Let us start from the general action of EGB gravity in $D$-dimensions and also derive the equations of motion. The action takes the form $$\label{action}
\mathcal{I}_{G}=\frac{c^4}{16 \pi G}\int d^{D}x\sqrt{-g}\left[ R +\frac{\alpha}{D-4} \mathcal{L}_{\text{GB}} \right]
+\mathcal{S}_{\text{matter}},$$ where $g$ denotes the determinant of the metric $g_{\mu\nu}$ and $\alpha$ is the Gauss-Bonnet coupling constant. The Ricci scalar $R$ provides the general relativistic part of the action. The Einstein-Gauss-Bonnet Lagrangian $\mathcal{L}_{\text{GB}}$ is given by $$\mathcal{L}_{\text{GB}}=R^{\mu\nu\rho\sigma} R_{\mu\nu\rho\sigma}- 4 R^{\mu\nu}R_{\mu\nu}+ R^2\label{GB}.$$
We add also the matter action $\mathcal{S}_{\text{matter}}$ which induces the energy momentum tensor $T_{\mu\nu}$. If the above action, Eq. (\[action\]), is varied with respect to $g_{\mu \nu}$, one obtains the field equations $$\label{GBeq}
G_{\mu\nu}+\frac{\alpha}{D-4} H_{\mu\nu}= \frac{8 \pi G}{c^{4}} T_{\mu\nu}~,~~~\mbox{where}~~~~~~T_{\mu\nu}= -\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\mathcal{S}_m\right)}{\delta g^{\mu\nu}},$$ with $G_{\mu\nu}$ is the Einstein tensor and $H_{\mu\nu}$ is the contribution of the Gauss-Bonnet term with the following expression $$\begin{aligned}
&& G_{\mu\nu} = R_{\mu\nu}-\frac{1}{2}R~ g_{\mu\nu},\nonumber\\
&& H_{\mu\nu} = 2\Bigr( R R_{\mu\nu}-2R_{\mu\sigma} {R}{^\sigma}_{\nu} -2 R_{\mu\sigma\nu\rho}{R}^{\sigma\rho} - R_{\mu\sigma\rho\delta}{R}^{\sigma\rho\delta}{_\nu}\Bigl)- \frac{1}{2}~g_{\mu\nu}~\mathcal{L}_{\text{GB}},\label{FieldEq}\end{aligned}$$ where $R_{\mu\nu}$ is the Ricci tensor, $R$ and $R_{\mu\sigma\nu\rho}$ are the Ricci scalar and the Riemann tensor, respectively. Additionally, GB terms is total derivative in 4$D$ space-time, and hence do not contribute to the field equations. However, it was proposed that re-scaling the coupling constant as $ \alpha/(D-4)$, maximally symmetric spacetimes with curvature scale ${\cal K}$, the variation of the Gauss-Bonnet [@Ghosh:2020vpc], yield $$\label{gbc}
\frac{g_{\mu\sigma}}{\sqrt{-g}} \frac{\delta \mathcal{L}_{\text{GB}}}{\delta g_{\nu\sigma}} = \frac{\alpha (D-2) (D-3)}{2(D-1)} {\cal K}^2 \delta_{\mu}^{\nu},$$ with this re-scaled coupling constant [@Glavan:2019inb], the Eq. (\[gbc\]) does not vanish in $D=4$ [@Glavan:2019inb]. For solution describing stellar objects, we use the regularization process (see Ref. [@Glavan:2019inb; @Cognola:2013fva], where obtained spherically symmetric solutions are also exactly same as of other regularised theories [@Lu:2020iav; @Hennigar:2020lsl; @Casalino:2020kbt; @Ma:2020ufk].
Here, we consider static spherically symmetric $D$-dimensional metric anstaz with two independent functions of radial coordinate, which is: $$\begin{aligned}
\label{metric}
ds^2_{D}= - e^{2\Phi(r)}c^{2}dt^2 + e^{2\Lambda(r)}dr^2 + r^{2}d\Omega_{D-2}^2, \end{aligned}$$ where $d\Omega_{D-2}^2$ is the metric on the unit $(D-2)$-dimensional sphere and $\Phi(r)$ and $\Lambda(r)$ are functions of $r$, only. The energy momentum tensor $T_{\mu\nu}$ is a perfect fluid matter source and describe the interior of a star, which in this study is written as $$\begin{aligned}
T_{\mu\nu} = (\epsilon+P)u_{\nu} u_{\nu} + P g_{\nu \nu}, \label{em}\end{aligned}$$ where $P=P(r)$ is the pressure, $\epsilon=\epsilon(r)$ is the energy density of matter, and $u_{\nu}$ is the contravariant $D$-velocity. On using the metric (\[metric\]) with stress tensor (\[em\]), in the limit $D \to 4$, the $tt$, $rr$ and hydrostatic continuity equations (\[GBeq\]) read: $$\begin{aligned}
\label{DRE1}
&& \frac{2}{r} \frac{d\Lambda}{dr} = e^{2\Lambda} ~ \left[\frac{8\pi G}{c^4} \epsilon - \frac{1-e^{-2\Lambda}}{r^2}\left(1- \frac{\alpha(1-e^{-2\Lambda})}{r^2}\right)\right]\left[1 + \frac{2\alpha(1-e^{-2\Lambda})}{r^2}\right]^{-1}, \\
&& \frac{2}{r} \frac{d\Phi}{dr} = e^{2\Lambda} ~\left[\frac{8\pi G}{c^4} P + \frac{1-e^{-2\Lambda}}{r^2} \left(1- \frac{\alpha(1-e^{-2\Lambda})}{r^2} \right) \right] \left[1 + \frac{2\alpha(1-e^{-2\Lambda})}{r^2}\right]^{-1},\label{DRE2} \\
&& \frac{dP}{dr} = - (\epsilon + P) \frac{d\Phi}{dr}. \label{DRE3}\end{aligned}$$ As usual, the asymptotic flatness imposes $\Phi(\infty)=\Lambda(\infty)=0$ while the regularity at the center requires $\Lambda(0)=~0$.
It is advantageous to define the gravitational mass within the sphere of radius $r$, such that $e^{-2\Lambda} =1-\frac{2G m(r)}{c^2 r}$. Now, we are ready to write the Tolman-Oppenheimer-Volkoff (TOV) equations in a form we want to use. So, using (\[DRE2\]-\[DRE3\]), we obtain the modified TOV as $${dP \over dr} = -{G\epsilon(r) m(r) \over c^{2}r^2}\frac{\left[1+{P(r) \over \epsilon(r)}\right]\left[1+{4\pi r^3 P(r) \over c^{2}m(r)}-{2G\alpha m(r) \over c^{2}r^3}\right]}{\left[1+{4G\alpha m(r) \over c^{2}r^3}\right]\left[1-{2Gm(r) \over c^{2}r}\right]}. \label{e2.11}$$ If we take the $\alpha \to 0$ limit, the above equation reduces to the standard TOV equation of GR. Replacing the last equality in Eq. (\[DRE1\]), we obtain the gravitational mass: $$m'(r)=\frac{6 \alpha G m(r)^2+4 \pi r^6 \epsilon (r)}{4 \alpha G r m(r)+c^2 r^4}, \label{e2.12}$$ using the initial condition $m(0)=0$. Then we use the dimensionless variables $P(r)=\epsilon_{0}{\bar P}(r)$ and $\epsilon(r)=\epsilon_{0}{\bar \epsilon}(r)$ and $m(r)=M_{\odot}{\bar M}(r)$, with $\epsilon_{0}=1\,{\rm MeV}/{\rm fm}^{3}$. As a result, the above two equations become $$\begin{aligned}
{d{\bar P}(r) \over dr} &=& -{G{\bar \epsilon}(r) M_{\odot}{\bar M}(r) \over c^{2}r^2}\frac{\left[1+{{\bar P}(r) \over {\bar \epsilon}(r)}\right]\left[1 + {4\pi r^3 \epsilon_{0}{\bar P}(r) \over c^{2}M_{\odot}{\bar M}(r)}-{2G\alpha M_{\odot}{\bar M}(r) \over c^{2}r^3}\right]}{\left[1+{4G\alpha M_{\odot}{\bar M}(r) \over c^{2}r^3}\right]\left[1-{2GM_{\odot}{\bar M}(r) \over c^{2}r}\right]} \nonumber \\
&=& - \frac{c_1 {\bar \epsilon}(r) {\bar M}(r)}{r^2} \frac{\left[1+{{\bar P}(r) \over {\bar \epsilon}(r)}\right] \left[1+{c_2 r^3 {\bar P}(r) \over {\bar M}(r)}-{2 c_1 \alpha {\bar M}(r) \over r^3}\right] }{\left[1+{4 c_1 \alpha {\bar M}(r) \over r^3}\right]\left[1-{2 c_1 {\bar M}(r) \over r}\right]}, \label{e2.11d}\end{aligned}$$ and $$\begin{aligned}
M_{\odot}\frac{d{\bar M}(r)}{dr} &=& \frac{6 \alpha G M^{2}_{\odot}{\bar M}(r)^2 + 4 \pi r^6 \epsilon_{0}{\bar \epsilon}(r)}{4 \alpha G r M_{\odot}{\bar M}(r) + c^2 r^4} \label{e2.12d} \nonumber \\
\frac{d{\bar M}(r)}{dr} &=& \frac{6 c_1 \alpha {\bar M}(r)^2 + c_2 r^6 {\bar \epsilon}(r)}{4 c_1 \alpha r {\bar M}(r) + r^4}, \label{mr}\end{aligned}$$ where $c_1 \equiv \frac{G M_{\odot}}{c^2} = 1.474 \text{ km}$ and $c_2 \equiv \frac{4 \pi \epsilon_0}{M_{\odot} c^2} = 1.125 \times 10^{-5} \; \text{km}^{-3}$. The relationship between mass $M$ and radius $R$ can be straightforwardly illuminated using Eq. (\[mr\]) with a given EoS. Therefore, the final two Eqs. (\[e2.11d\]) and (\[mr\]) can be numerically solved for a given EoS $P=P(\epsilon)$. In the next section, we will discuss the strange matter hypothesis.
Equation of state and numerical techniques {#sec3}
==========================================
To understand, what kind of matter compact stars may be built up from, assuming an EoS is the most important step, which encompasses all the information regarding the stellar inner structure. Here, we solve the hydrostatic equilibrium Eq. (\[e2.11d\])- (\[e2.12d\]) numerically for a specific EoS, $\epsilon$ = $f(P)$, where $\epsilon$ is the energy density and $p$ is the pressure. Since each possible EoS, there is a unique family of stars, parametrized by, say, the central density and the central pressure. The standard procedure is to derive the expressions $P=f(\rho)$ and $\epsilon =g(\rho)$, with $\rho$ being the baryon density, and then obtain an $\epsilon-P$ pair for every value of $\rho$. Fitting a curve to this data results in the EoS.
The bag model is a simple tool when we work with quark stars and is invoked in this present work. It is worth noting that in a simple model of free quarks in a bag, analytical expressions of pressure, energy density, and other relevant parameters can be derived. For a Fermi gas of quarks, we can separate into two cases: (1) $T\neq 0,\,m=0$ and (2) $T=0,\,m\neq0$. The pressure, energy density, and baryon number density are given by [@Glendenning2000] $$\begin{aligned}
P &=&\sum_{f} \frac{1}{3} \frac{\gamma_f}{2\pi^2}\int^{\infty}_{0} k\frac{\partial E_f(k)}{\partial k}\left[n(k,\mu_f) +n(k,-\mu_f)\right]k^2 dk-B, \label{setp}\\
\epsilon &=& \sum_{f} \frac{\gamma_f}{2\pi^2}\int^{\infty}_{0} E_f(k)\left[n(k,\mu_f) +n(k,-\mu_f)\right]k^2 dk+B, \label{setep}\\
\rho &=&\sum_{f} \frac{1}{3} \frac{\gamma_f}{2\pi^2}\int^{\infty}_{0} \left[n(k,\mu_f) -n(k,-\mu_f)\right]k^2 dk,\label{setrho}\end{aligned}$$ where $E_f(k)=\left(m_f^2+k^2\right)^{1/2}$ is the is the quark kinetic energy, $$\begin{aligned}
n(k, \mu_f)= \left(\exp\left[E_f(k)-\mu_f\right]/T+1\right)^{-1} \notag\,,\end{aligned}$$ is the Fermi distribution function for temperature $T$ , and the quark degeneracy for each flavor is $\gamma_f=2_{\mbox{spin}} \times 3_{\mbox{spin}}$, with $B$ is the MIT Bag constant that represents the positive energy shift per unit volume in the deconfined vacuum relative to the confined vacuum. The factor $1/3$ is due to there are three quarks per baryon. Note that the electric charge density can also be computed.
Massless quark approximation
----------------------------
Here, we consider the first of the limiting cases outlined above. As mentioned in Ref. [@Glendenning2000], one can obtain the analytical expressions if quarks are massless while the temperature is finite. Invoking the standard integrals and applying some algebraic computation, we end up with $$\begin{aligned}
P &=& \sum_{f} \left( \frac{7}{60} \pi^2 T^4 +\frac{1}{2} T^2 \mu_f ^2 +\frac{1}{4 \pi^2 } \mu_f ^4\right)-B,\label{pres}\\
\epsilon &=& 3p+4B,\label{EoS}\\
\rho &=& \sum_{f} \frac{1}{3}\left( T^2 \mu_f + \frac{\mu_f ^3}{\pi^2}\right).\end{aligned}$$ Interestingly we obtain a simple form of an EoS given in Eq. (\[EoS\]). What we have to do next is to solve three equations with four unknown functions, which are $m(r),\,\Phi(r) ,\, P(r)\,$ and $\epsilon(r)$. Notice that the EoS for the massless quark approximations explicitly depends on the bag constant $B$ and the pressure $P(r)$. Due to the long range effects of confinement of quarks, the stability of strange quark stars is represented by the bag constant, $B$. We then consider the re-scaled TOV equations Eq. (\[e2.11d\]) and mass function Eq. (\[mr\]). Therefore, the mass is measured in the solar mass unit ($M_{\odot}$), radius in ${\rm km}$, while energy density and pressure are in ${\rm MeV}/{\rm fm}^{3}$. The bag constant $B$ is also in ${\rm MeV}/{\rm fm}^{3}$. In the present analysis, we treat the values of $B$ and $\alpha$ as free constant parameters. Since, the parameter $B$ can vary from $57$ to $94 \,{\rm MeV}/{\rm fm}^{3}$ [@Witten]. For the study of quark matter with massless strange quark, we consider $B= 70 \,{\rm MeV}/{\rm fm}^{3}$.
Given the set of differential equations (\[e2.11d\]) and (\[mr\]) together with the EoS (\[EoS\]), we apply numerical approach for integrating and calculate the maximum mass and other properties of the strange quark matter star. To do so, one can consider the boundary conditions $P(r_{0}) = P_{c}$ and $M(R) = M$, and integrates Eq. (\[e2.11d\]) outwards to a radius $r = R$ in which fluid pressure $P$ vanishes for $P(R)=0$. This leads to the strange star radius $R$ and mass $M = m(R)$. The initial radius $r_{0}= 10^{-5}$ and mass $m(r_{0})= 10^{-30}$ are set to very small numbers rather than zero to avoid discontinuities, as they appear in denominators within the equations.
We start from the center of the star for a certain value of central pressure, $P({r_{0}})=800 \,{\rm MeV}/{\rm fm}^{3}$ and the radius of the star is identified when the pressure vanishes or drops to a very small value. For such a choice, we plot pressure and density versus distance from the center of strange star (see Fig. \[f1\]). At that point we recorded the mass-radius relation of the star in Fig. \[f2\]. As one can see, the mass-radius ($M-R$) relation depends on the choice of the value of coupling constant $\alpha$. For $\alpha >0$ the mass of star for given radius increases with fixed value of $B$. In all the presented cases, we can note that there is significantly different for positive and negative values of $\alpha$, but $\alpha =0$ case is equivalent to pure general relativity. Moreover, it can be seen from Table \[ta11\] and comparing the results to GR, one may obtain maximum mass for strange stars with positive $\alpha $. Therefore, we argue that a confirmed determination of a compact star with 2$M_{\odot}$, which are actually very close to the ones of realistic neutron star models [@Haensel:1986qb].
Cold star approximation
-----------------------
This section contains a discussion of the zero temperature ($T=0$) and $m\neq0$. Then the expressions given in Eqs. (\[setp\]-\[setrho\]) can be further simplified. To be more accuracy, we add the electrons to the system with their statistical weights ($=2$) due to the spin. Performing the standard calculations, we obtain [@Glendenning2000] $$\begin{aligned}
P &=& -B+\sum_{f} \Bigg[\frac{1}{4\pi^2}\Bigg(\mu_{f} k_{f}\Big(\mu_{f}^2 -\frac{5}{2}m_{f}^2 \Big) +\frac{3}{2} m_f^4 \ln\Big(\frac{\mu_{f}+ k_{f}}{m_f}\Big)\Bigg)\Bigg]\nonumber\\&&\quad\quad+\frac{1}{12\pi^2}\Bigg[\mu_{e} k_{e}\Big(\mu_{e}^2 -\frac{5}{2}m_{e}^2 \Big) +\frac{3}{2} m_e^4 \ln\Big(\frac{\mu_{e}+ k_{e}}{m_e}\Big)\Bigg],\label{eq37}\\
\epsilon &=& B+\sum_{f} \Bigg[\frac{3}{4\pi^2}\Bigg(\mu_{f} k_{f}\Big(\mu_{f}^2 -\frac{1}{2}m_{f}^2 \Big) +\frac{1}{2} m_f^4 \ln\Big(\frac{\mu_{f}+ k_{f}}{m_f}\Big) \Bigg)\Bigg]\nonumber\\&&\quad\quad+\frac{1}{4\pi^2}\Bigg[\mu_{e} k_{e}\Big(\mu_{e}^2 -\frac{1}{2}m_{e}^2 \Big) +\frac{1}{2} m_e^4 \ln\Big(\frac{\mu_{e}+ k_{e}}{m_e}\Big) \Bigg], \label{eq38}\\
\rho &=&\sum_{f}\frac{k_{f}^3}{3\pi^2},\label{eq39}\end{aligned}$$ where $k_{f}$ is the Fermi momentum for flavor $f$ with $k_f=\left(\mu_{f}^2-m_{f}^2 \right)^{1/2}$ and $k_e=\left(\mu_{e}^2-m_{e}^2 \right)^{1/2}$. Notice that there are four independent variables appeared in the above equations, i.e. $\mu_{u},\,\mu_{d},\,\mu_{s}$ and $\mu_{e}$. Strange stars are composed of $uds$ quarks. Hence, we constrain the chemical potentials of the quarks to a single independent variable $\mu$ such that $\mu_{d}=\mu_{s}=\mu$ and $\mu_{u}+\mu_{e}=\mu$. Thus, the two independent variables $\mu$ and $\mu_{e}$, two equations are necessary to produce a set of chemical potentials and solve the system for a pair of values for $\epsilon$ and $P$.
It seems like we end up with the higher order/nonlinear EoS and the TOV equations are very hard to be solved in this particular situation. This is because the quark chemical potentials increase when we increase the baryon number density, while the electron chemical potential is neglectable. Therefore, from Eqs. (\[eq37\]-\[eq38\]) we only focus a simple linear relationships of $\epsilon$ and $P$, which can be simplified to $\epsilon = 3.05P(r) + 368.00$, see Appendix \[apa1\] for more details .
The input data for the numerical calculation are similar to aforementioned. The pressure and density versus radial distance from the center of cold star i.e. quark matter at zero temperature are represented in Fig. \[f3\]. All curves in Fig. \[f3\], note that the pressure and density are maximum at the center and decrease monotonically towards the boundary. In turn, to study the mass-radius relation and the mass vs. central density for cold quark matter EoS are given for 5 representative values of $\alpha$ in Figs. \[f4\] and \[f5\], respectively. For a given central density, the star mass grows with increasing $\alpha$. The maximum mass increases with increasing value of $\alpha$ and we find that, for $\alpha=5$, the maximum mass becomes $M_{\mbox{max}}$ = 1.80 $M_{\odot}$. At that point we recorded the mass of the star, 1.52$M_{\odot}$ when $\alpha=0$ in GR. For more clarity, the properties of stars with maximal mass are tabulated in Tables \[ta11\] and are compared to GR ($\alpha=0$). Finally, we compare between the two mass-radius relationships (see Fig. \[f6\]). Interestingly, all values of $M_{\mbox{max}}$ for massless quarks are higher than Chandrasekhar limit, which is about 1.4$M_{\odot}$.
---------- --------------- ------ ------------------------ --------------- ------ ------------------------
$\alpha$ $M_{max}$ $R$ $\epsilon_c$ $M_{max}$ $R$ $\epsilon_c$
$(M_{\odot})$ (km) (10$^{15}$ g/cm$^{3})$ $(M_{\odot})$ (km) (10$^{15}$ g/cm$^{3})$
-5.0 1.50 8.98 4.78 1.24 7.78 5.02
-2.5 1.61 9.04 4.78 1.38 7.92 5.02
0 1.74 9.14 4.78 1.52 8.09 5.02
2.5 1.87 9.25 4.78 1.66 8.27 5.02
5.0 1.99 9.37 4.78 1.80 8.44 5.02
---------- --------------- ------ ------------------------ --------------- ------ ------------------------
: We summarize the parameters of the strange quark stars using various values of the 4D EGB coupling constant, $\alpha$. We show the maximum mass of the stars $M$ in a unit of the solar mass $M_{\odot}$ with their radius $R$ in km and the central energy density $\epsilon_c$ for both the massless quark model and the cold star approximation.[]{data-label="ta11"}
Structural properties of strange stars {#sec4}
=======================================
For completeness, we would also like to explore the physical properties in the interior of the fluid sphere.
Energy conditions
-----------------
In this section, we will discuss the energy conditions (ECs). The ECs are local inequalities, depending on $T_{\mu\nu}$, that capture the idea of *energy should be positive* for strange star. Most notable weak energy condition (WEC), i.e. $T_{\mu\nu} U^{\mu}U^{\nu}$, requires that $$\label{EC1}
\epsilon(r)\geq 0 ~~\text{and}~~ \epsilon(r)+ P(r)\geq 0,$$ where $U^{\mu}$ is a timelike vector and follows that if WEC is satisfied then NEC also satisfied. NEC is the assertion that for any null vector $k^{\mu}$, we should have $T_{\mu\nu} k^{\mu} k^{\nu}\geq 0$. In addition to its role in constraining the set of physical spacetimes, the NEC demands $T_{\mu\nu} k^{\mu} k^{\nu}\geq 0$ for null vector $k^{\mu}$. Since, the NEC is the simplest energy condition to deal with algebraically.
Whereas, the strong energy condition (SEC) asserts that $\left(T_{\mu\nu}-\frac{1}{2} T g_{\mu\nu}\right) U^{\mu} U^{\nu}\geq 0$ for any timelike vector $U^{\mu}$. The SEC asserts that gravity is attractive, $$\label{EC3}
\epsilon(r)+ \sum P_{i}(r)\geq 0, ~~\implies~~ \epsilon(r)+3 P(r)\geq 0.$$ Note that the SEC does not imply the WEC, but it follows that any violation of the NEC also violates the SEC and WEC. Finally, using the inequalities (\[EC1\]-\[EC3\]), we plot Fig. \[f7\] for different values of $\alpha$. We see from Fig.\[f7\] that all energy conditions are satisfied, and our assumed EoS is suitable for modelling viable strange stars.
Speed of sound and Le Chatelier’s principle
-------------------------------------------
In a perfect fluid described by an EoS of the form $P = P(\epsilon)$, we are going to consider the speed of sound propagation, $c^2 =dP/ d\rho $. In order to preserve the causality, one expects that the velocity of sound ($v$) should be less than the light’s velocity ($c$). Thus one can put a constraint using the following expression $ 0\leq v^2= dP/ d\rho \leq c^2$, i.e $dP/ d\epsilon \leq 1$.
Now, we are going to use Eqs. and for obtaining the speed of sound inside the fluid sphere. It is seen that for massless case $dP/ d\epsilon \approx 0.333$, whereas for cold star we obtain $dP/ d\epsilon \approx 0.328$. We can, then, say that the speed of sound is constant everywhere within the strange star and less than unity.
In [@Glendenning:2000dh], it was suggested that matter of star satisfies $dP/ d\epsilon \geq 0$ which is a necessary condition that a body is stable both as a whole, and also with respect to the non-equilibrium elementary regions with spontaneous contraction or expansion (Le Chatelier’s principle). The results of such calculations for the given parameters $P$ and $\epsilon$ are shown in Fig. \[ff1\]. Thus, Le Chatelier’s principle is established.
The stability criterion and the adiabatic indices
--------------------------------------------------
Now we will discuss the adiabatic index ($\gamma$) based on our theoretical quark matter models. The adiabatic index is related to the thermodynamical quantity. In order to solve the instability problem, Chandrasekhar [@Chandrasekhar] introduced a criterion for dynamical stability based on the variational method. To be more specific, the final expression for the adiabatic index reads $$\label{adi}
\gamma \equiv \left(1+\frac{\epsilon}{P}\right)\left(\frac{dP}{d\epsilon}\right)_S,$$ where $dP/ d\epsilon$ is the speed of sound in units of speed of light and the subscript $S$ indicates the derivation at constant entropy. The Eq. is a dimensionless quantity measuring the stiffness of the EoS.
Therefore, we need to study the effects of $\gamma$ corresponding to the respective EoSs. In particular, the EoS related to neutron star matter, this value lies between 2 to 4 [@Haensel]. In [@Glass], authors have shown that $\gamma$ should exceed $4/3$ in a stable polytropic star by an amount that depends on that ratio $\epsilon/P$ at the centre of the star. In resent studies it was shown that for dynamical stability $\gamma $ should be more than 4/3 $(i.e~ \gamma > 1.33)$. In Table \[ta11\], we present a set of parameters used in the present work, and Fig. \[f8\] shows that stellar model is stable against the radial adiabatic infinitesimal perturbations. Additionally, increasing values of $\gamma$ mean the growth of pressure for a given increase in energy density, i.e. a stiffer EoS. In view of the above, we conclude that massless quark matter gives more stiffer EoS compare to cold and dense quark matter.
Conclusions and astrophysical implications {#sec6}
===========================================
In this paper, we investigate the features of $4D$ Einstein-Gauss-Bonnet gravity in an extreme circumstances such as those arising within highly compact static spherically symmetric bodies. We considered the self-bound strange matter hypothesis. The interesting part of this theory is that the resulting regularized $4D$ EGB gravity has nontrivial dynamics and free from the Ostrogradsky instability.
There exist considerable evidences that the possible existence of compact stars are partially or totally made up of quark matter. But the existence of quark stars is still controversial and its EoS is also uncertain. Here, we first considered the static spherically symmetric $D$-dimensional metric and derived corresponding field equations taking a limit of $D\rightarrow 4$ at the level of field equations. We then numerically solved field equations for strange matter hypothesis. To clarify the astrophysical implications of our work, we discuss two important scenarios. Firstly, we considered quark matter phases consisting of massless quarks, and secondly quark matter at zero temperature.
To gain better understanding of the physical properties, we quantified the maximal mass from the central density and mass-radius relation of the stellar structure. The mass-radius results are graphically shown which strictly depends on the values of the coupling constant and the chosen EoS. Then, we showed that for $\alpha \to 0$ limit, the obtained TOV in $4D$ EGB gravity reduces to the standard Einstein theory, and solutions are compared in Table \[ta11\]. The main difference between the two classes is that the maximum stable mass of the massless quark star is almost the same as that for a cold quark star for increasing value of $\alpha$. Since negative $\alpha$ reduces the maximum mass of a compact star for a given EoS. Furthermore, we obtained the interesting results of their physical properties such as velocity of sound, energy conditions and adiabatic stability. It is worth noting that the obtained solutions are in favour of the physical requirements (see from Figs. \[f7\]-\[f8\]).
Finally, it is notable that the investigation for other compact objects such as neutron star and white dwarf using the same context and its modified TOV equation are interesting subjects. However, we will leave these interesting topics for our future work.
Acknowledgments {#acknowledgments .unnumbered}
===============
PC is thankful for the financial support from Thailand Research Fund (TRF) under a contract No. MRG63xxxxx.
Linear approximation of EoS for the cold star approximation {#apa1}
===========================================================
In order to study the effect of the non-vanishing strange quark mass on the EoS for uncharged quark matter with two massless flavors (up & down quarks) and one massive flavor (strange quark), we have to include electrons with the chemical potential $\mu_{e}$. We impose the chemical equilibrium of three quarks which maintain the following interactions [@Fraga:2004gz]: $$\begin{aligned}
d &\rightarrow& u+e+{\bar \nu}_{e}\,,\nonumber\\u+e &\rightarrow& d+\nu_{e}\,,\nonumber\\
s &\rightarrow& u+e+{\bar \nu}_{e},\nonumber\\u+e &\rightarrow& s+\nu_{e}\,,\nonumber\\
s+u &\rightarrow& d+u\,.\label{inters}\end{aligned}$$ As mentioned in Ref. [@Fraga:2004gz], neutrinos are rapidly lost, so one may set their chemical potential to zero. As a result, we assume the chemical potentials of the above system as a single independent variable, $\mu$: $$\begin{aligned}
\mu_{d}=\mu_{s}=\mu,\quad\mu_{u}+\mu_{e}=\mu\,,\label{mu}\end{aligned}$$ where $\mu_{d},\,\mu_{s}$ and $\mu_{u}$ are the up, down and strange quark chemical potentials, respectively. The EoS describes the relation between the energy density ($\epsilon$) and pressure (P) for the above system which have been already given in Eqs. (\[eq37\]-\[eq38\]). Basically, the pressure and the energy density are solved in terms of the baryon density, $\rho$. For the cold star approximation, these three variables, ($\epsilon,\,P,\,\rho$), are written in terms of the quark mass $m_f$ and chemical potential $\mu_f$. The charge neutrality of quarks taking account of electrons must satisfy the following condition: $$\begin{aligned}
\tilde{q} = \sum_f \left( \tilde{q}_f \frac{k_f^3}{\pi^2} \right) - \frac{k_e^3}{3 \pi^2} = 0. \label{qcharge}\end{aligned}$$ In order to obtain the EoS, we have to solve the system of equations; Eqs. (\[eq37\]-\[eq39\]) and Eq. (\[qcharge\]). What we have to do first is to quantify the relations between the chemical potentials ($\mu_u$, $\mu_d$, $\mu_s$, $\mu_c$) and the baryon density $\rho$. In this step, we use the numerical calculation considering the baryon density and the conservation of charge of quarks; Eq. (\[eq37\]) and Eq. (\[qcharge\]), respectively.
It is worth noting that the effects of the finite strange quarks mass on the energy density ($\epsilon$) and the pressure ($P$) for neutral quark matter including electrons have been already examined in Ref. [@Fraga:2004gz]. The results showed that there was a sizable difference in the energy density and the pressure between zero strange quark mass and non-vanishing strange quark mass. However, the EoS in this case basically shows a non-linear behavior between $\epsilon$ and $P$, and as a result this non-linearity is very hard to be solved.
Fortunately, it has been also noticed from Ref. [@Fraga:2004gz] that the resulting EoS $\epsilon=\epsilon(P)$ can be approximated by a non-ideal bag model which is written in the following form: $$\begin{aligned}
\epsilon = a\,B+b\,P, \label{app}\end{aligned}$$ with $a$ and $b$ being arbitrary constants. In our case, we find that $\epsilon = 3.05\,P+368$ which we have used $B=70\,{\rm MeV}/{\rm fm}^{3}$. Using the numerical calculations, the chemical potentials and the number density $\rho$ are simultaneously obtained. After substituting the results into Eq. (\[eq37\]) and Eq. (\[eq38\]), we finally end up with the EoS displaying a relationship between the energy density and pressure. However, the linear behavior is maintained by the relation given in Eq.\[app\].
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|
---
abstract: |
The electronic energy of [$\rm H_2^+$]{}in magnetic fields of up to $B=0.2
B_0$ (or 4.7 $\times 10^4$ Tesla) is investigated. Numerical values of the magnetic susceptibility for both the diamagnetic and paramagnetic contributions are reported for arbitrary orientations of the molecule in the magnetic field. It is shown that both diamagnetic and paramagnetic susceptibilities grow with inclination, while paramagnetic susceptibility is systematically much smaller than the diamagnetic one. Accurate two-dimensional Born-Oppenheimer surfaces are obtained with special trial functions. Using these surfaces, vibrational and rotational states are computed and analysed for the isotopologues [$\rm H_2^+$]{}and [$\rm D_2^+$]{}.
address:
- 'Université de Reims Champagne-Ardenne, Groupe de Spectrométrie Moléculaire et Atmosphérique (UMR CNRS 7331), U.F.R. Sciences Exactes et Naturelles, Moulin de la Housse B.P. 1039,F-51687 Reims Cedex 2, France'
- 'Université de Reims Champagne-Ardenne, Groupe de Spectrométrie Moléculaire et Atmosphérique (UMR CNRS 7331), U.F.R. Sciences Exactes et Naturelles, Moulin de la Housse B.P. 1039,F-51687 Reims Cedex 2, France'
- 'Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 México, D.F., Mexico'
- 'Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 México, D.F., Mexico'
author:
- Héctor Medel Cobaxin
- Alexander Alijah
- Juan Carlos López Vieyra
- 'Alexander V. Turbiner'
title: '[$\rm H_2^+$]{}in a weak magnetic field'
---
[*Keywords*]{}: Variational method, weak magnetic fields, magnetic susceptibility, rovibrational states
Introduction
============
Since the pioneering work of de Melo [*et al.*]{} [@Melo:76676] on the molecular ion [$\rm H_2^+$]{}in strong magnetic fields, $B\geq B_0$ ($B_0=2.35\times10^9$ Gauss $=2.35\times10^5$ T), many studies have been conducted for this system under such conditions (see for example [@Kappes:954542]-[@Song:13064305] and references therein) where the electronic energy of the ground and first excited states as well as some rotation-vibrational states have been studied. [$\rm H_2^+$]{}has been used as a test system for the investigation of the validity of approximations commonly made in field-free molecular physics, such as the Born-Oppenheimer approximation [@Schmelcher:88672; @Schmelcher:886066]. Though [$\rm H_2^+$]{}can be considered a benchmark molecule for the development of appropriate theoretical methods for the accurate computation of molecular structure and properties in magnetic fields that may be extended to more complex systems [@Turbiner:06309; @Turbiner:07267; @Turbiner:07053408; @Turbiner:10042503], only few studies have been reported in the range of [*small*]{} fields ($0\leq B \leq
B_0$) [@Wille:87L417; @Larsen:821295], where electronic energies of the ground state and physical features such as a qualitative evolution of the rotational levels as function of the field are presented.
The goal of the present study is to investigate the electronic ground state of the molecular system [$\rm H_2^+$]{}placed in a weak magnetic field, $B\leq 0.2 B_0$. Overall in this domain of field strength, the effects of the magnetic field cannot be treated accurately via perturbation theory. In the first part of the present work we use physically motivated, specially tailored trial functions [@Turbiner:03012504; @Turbiner:04053413; @Turbiner:06309] to obtain sufficiently accurate estimates of the electronic energy over a range of field strengths up to $B=B_0$ and different inclinations of the molecular axis with respect to the field direction. In the second part we investigate the vibrational and rotational structure of [$\rm H_2^+$]{}and [$\rm D_2^+$]{}in the external magnetic field.
Hamiltonian
===========
We consider a homonuclear molecular ion formed by two nuclei of charge $q$ separated by a distance $R$, and one electron $e$ placed in a uniform magnetic field $\vec{B}$ oriented along the $z$-axis. The reference point for coordinates is chosen to be at the midpoint of the line connecting the nuclei which in turn forms an angle $\theta$ with respect to the magnetic field direction (see Figure \[htpfig\]).
![The homonuclear molecular ion consisting of two centers of charge $q$ separated by a relative distance $R$, and one electron $e$. The system is placed in a uniform magnetic field $\vec{B}$ along $z$-axis.[]{data-label="htpfig"}](figure1.ps){width="40.00000%"}
In the framework of non-relativistic quantum mechanics, [*i.e.*]{} neglecting spin interactions, following pseudoseparation of the center of mass motion [^1] and resorting to the Born-Oppenheimer approximation of order zero, [*i.e.*]{} neglecting terms of order smaller than $1/\cal{M}$ ($\cal{M}$ is the total mass of the system), the Hamiltonian that describes the system is given by $$\label{AHamil}
\hat{\cal H}= \frac{2}{M_s}\left[\hat{P}_R-\frac{q}{2}\vec{\cal A}_R\right]^2 + \frac{1}{2m_e}\left[\hat{p}-e\vec{A}_e\right]^2+\frac{q^2}{R}+q e\left(\frac{1}{r_1}+\frac{1}{r_2}\right),$$ where $M_s$ is the total mass of the nuclei, $q$ is the nuclear charge, $\hat{P}_R = -i \hbar \nabla_{R}$ is the momentum operator and $\vec{A}_R$ ($\vec{A}_R=\frac{1}{2}\,\vec{B}\times \vec{R}$) is the vector potential for the relative motion ${\vec R}$ of the nuclei; $e$ and $m_e$ are the electronic charge and mass, respectively; $\hat{p}=
-i \hbar \nabla_{r}$ and $\vec{A}_e$ are the momentum operator and vector potential for the electron which is at the position $\vec{r} =(x,y,z)$; $r_1$ and $r_2$ are the distances between the electron and each of the nuclei. In the Hamiltonian the first term is the kinetic energy of the nuclear relative motion in a magnetic field; the remaining terms correspond to the electronic Hamiltonian, written in the Born-Oppenheimer approximation of zero order. In the remainder of this article, atomic units shall be used, [*i.e.*]{} distances are measured in Bohr, $a_0=1\,$ a.u., energies in Hartrees, $E_h=1\,$ a.u. and $\hbar=|e|=m_e=1$.
Solving the electronic Schrödinger equation
===========================================
To solve the electronic Schrödinger equation an appropriate gauge for $\vec{A}_e$ must be chosen. Though the problem is in principle gauge invariant this is not the case if the equation is solved approximately [@Larsen:821295; @Kobe:833710; @Larsen:07042502]. We have therefore adopted the strategy of introducing a variational parameter, $\xi$, in the definition of the gauge which is then varied together with the variational parameters of the wave function. For a magnetic field directed along the $z$-axis $\vec{B}=B\hat{z}$, a suitable vector potential is $$\label{gengauge}
\vec{A}_e=B\left[(\xi-1)y,\xi x,0\right],$$ where $\xi$ is the parameter of the family of Coulomb gauges. With $\xi=0$ the Landau gauge is obtained, while $\xi=1/2$ corresponds to the symmetric gauge. Substituting into we obtain the electronic Hamiltonian (the last three terms in ) in the form $$\begin{aligned}
\label{ElectHamil}
\hat{\cal H}_{elec}=&-&\frac{1}{2}\nabla^2-iB\left[ (\xi-1)y \partial_x+ \xi x \partial_y\right] \\\nonumber
&+&\frac{1}{2}B^2\left[\xi^2x^2+(1-\xi)^2y^2 \right]+\frac{q^2}{R}-\left(\frac{1}{r_1}+\frac{1}{r_2}\right).\end{aligned}$$ As usual, in (\[ElectHamil\]) the contribution to the energy due to the Coulomb interaction between the nuclei, [*i.e.*]{} $q^2/R$, is treated classically. Hence, $R$ is considered an external parameter.
Trial functions {#trialf}
---------------
A set of physically adequate real trial functions introduced in [@Turbiner:03012504; @Turbiner:04053413; @Turbiner:06309] are used to calculate the total energy of the electronic Hamiltonian (\[ElectHamil\]). Thus, the trial function employed in the present study is a linear superposition of three particular functions, $$\label{Trialf}
\Psi=A_1\Psi_1+A_2\Psi_2+A_3\Psi_3\,,$$ where $$\label{psitotal}
\Psi_1= \ e^{ - \alpha_1 (r_1+r_2)}e^{-B[\beta_{1x}\xi x^2 +\beta_{1y}(1-\xi)y^2]}\,,$$ is a Heitler-London type function, $$\Psi_2= \left(e^{ - \alpha_2 r_1}+e^{ - \alpha_2 r_2}\right)e^{-B[\beta_{2x}\xi x^2 +\beta_{2y}(1-\xi)y^2]}\,,$$ is a Hund-Mulliken type function, and $$\Psi_3= \left(e^{ - \alpha_3 r_1-\alpha_4 r_2}+e^{ - \alpha_3 r_2-\alpha_4 r_1}\right)e^{-B[\beta_{3x}\xi x^2 +\beta_{3y}(1-\xi)y^2]}\,,$$ is a Guillemin-Zener type function, all multiplied with exponential terms that correspond to the lowest Landau orbital.
Without loss of generality one of the linear parameters $A_{1,2,3}$ may be set equal to one, hence the trial function consists of 13 variational parameters. For the parallel configuration the parameters are not independent and must obey the symmetry relations $\beta_{1x}=\beta_{1y}$, $\beta_{2x}=\beta_{2y}$ and $\beta_{3x}=\beta_{3y}$, reducing the number of variational parameters to ten. The trial function defined in this way is expected to provide an accurate approximation to the exact electronic wave function of the ground state of molecular ion [$\rm H_2^+$]{}for a large variety of strengths and inclinations of the magnetic field.
Calculations are performed using the minimization package MINUIT from CERN-LIB. Numerical integrations were done with a relative accuracy of $\sim 10^{-6}$ using the adaptive NAG-LIB (D01FCF) routine.
Results
-------
Using the trial function (\[psitotal\]) presented in Section \[trialf\], two-dimensional potential energy surfaces of the electronic energy have been obtained variationally as function of the internuclear distance, $R$, and the inclination $\theta$ (see Figure \[htpfig\]).
As examples we show in Figures \[B02Plots\] and \[B05Plots\] sections of the potential surface at different inclinations, $\theta=0^{\circ}, 45^{\circ}$ and $90^{\circ}$, for $B=0.2 B_0$ and $B=0.5 B_0$.
The most stable configuration is achieved for parallel orientation of the molecule, which is a well known result for $B\geq
10^5\,$T [@Turbiner:06309]. At perpendicular orientation an energy ridge shows up which can be interpreted as barrier of a hindered rotation. This fact is shown in more detail in Figure \[h2pET\] where the electronic energy is plotted as function of the inclination for the fields $B =0.1\, B_0$ and $0.2 \, B_0$. It is worth noting that at large distance $R\gg R_{eq}$, when the system separates to a proton and a hydrogen atom, the potential surface exhibits a relative maximum at $\sim 45^{\circ}$ inclination which is due to interaction of the proton charge with the quadrupole moment of the atom [@Potekhin:01065402]. As the sign of the interaction term is angular dependent, a barrier is built up as the molecule is oriented from parallel towards perpendicular configuration [@Turbiner:06309]. With increasing field strength and inclination, the internuclear distance at equilibrium becomes smaller while the rotational barrier is increased. Data are presented in Table \[TableBTRE\]. In Figure \[h2pRT\] the equilibrium distance $R_{eq}(B,\theta)$ is plotted as a function of $\theta$ for the field strengths $B = 0.1 \, B_0$ and $0.2\, B_0$.
--------- -------------- -------------- -------------- --------- -------------- -------------- --------------
$B/B_0$ $\theta$ $R_{eq}/a_0$ Energy/$E_h$ $B/B_0$ $\theta$ $R_{eq}/a_0$ Energy/$E_h$
$0$ $-$ 1.9971 -0.602625
$0^{\circ}$ 1.9920 -0.601029 $0^{\circ}$ 1.8705 -0.550864
$45^{\circ}$ 1.9897 -0.600785 $45^{\circ}$ 1.8201 -0.543923
$90^{\circ}$ 1.9882 -0.600613 $90^{\circ}$ 1.7968 -0.539131
$0^{\circ}$ 1.9786 -0.596311 $0^{\circ}$ 1.8399 -0.534186
$45^{\circ}$ 1.9687 -0.595361 $45^{\circ}$ 1.7800 -0.525296
$90^{\circ}$ 1.9637 -0.594678 $90^{\circ}$ 1.7535 -0.519216
$0^{\circ}$ 1.9566 -0.588667 $0^{\circ}$ 1.8096 -0.515853
$45^{\circ}$ 1.9379 -0.586615 $45^{\circ}$ 1.7411 -0.504917
$90^{\circ}$ 1.9283 -0.585161 $90^{\circ}$ 1.7112 -0.497503
$0^{\circ}$ 1.9301 -0.578360 $0^{\circ}$ 1.7799 -0.496041
$45^{\circ}$ 1.9013 -0.574889 $45^{\circ}$ 1.7027 -0.482994
$90^{\circ}$ 1.8862 -0.572447 $90^{\circ}$ 1.6721 -0.474219
$0^{\circ}$ 1.9019 -0.565667 $0^{\circ}$ 1.7563 -0.474937
$45^{\circ}$ 1.8610 -0.560550 $45^{\circ}$ 1.6687 -0.459670
$90^{\circ}$ 1.8413 -0.556976 $90^{\circ}$ 1.6348 -0.449532
--------- -------------- -------------- -------------- --------- -------------- -------------- --------------
: Total energy at equilibrium distance as function of the field strength and inclination.
\[TableBTRE\]
Magnetic Susceptibility
-----------------------
An important quantity that describes the response of the molecular system with respect to the external field is the magnetic susceptibility. It is defined via a Taylor expansion of the electronic energy in powers of the magnetic field $B$ $$\label{etaylor}
E(\vec{B})= E(0)-\sum_{\alpha}c_{\alpha}B_{\alpha}-\frac{1}{2}\sum_{\alpha\beta}\chi_{\alpha\beta}B_{\alpha}B_{\beta}+\ldots$$ For the electronic ground state, when the spin contributions are neglected, the first coefficient, $c_\alpha$, vanishes. The coefficient tensor $\chi_{\alpha\beta}$ is the magnetic susceptibility. The response of a molecule to an external magnetic field leads to a classification into two types (see for example [@Landau3]): diamagnetic and paramagnetic.
In the electronic Hamiltonian there are two terms containing the magnetic field, $B$, a linear and a quadratic one. Correspondingly, there are two contributions to the susceptibility: a paramagnetic contribution originating from the linear term of the Hamiltonian when treated by second order perturbation theory in $B$, and a diamagnetic contribution coming from the quadratic term in the first order in perturbation theory in $B$.
At first, let us proceed to the diamagnetic susceptibility. The diamagnetic susceptibility term $\chi^d$ can be expressed as the expectation value with respect to the field-free wavefunction at equilibrium distance. Thus, in the symmetric gauge, $\vec{A}_e=\frac{1}{2}\vec{B}\times\vec{\rm r}$, the expression of the diamagnetic susceptibility tensor is $$\chi_{\alpha\beta}^d= -\frac{1}{4}\left[\langle\vec{r}^{2}\rangle\delta_{\alpha\beta}-\langle r_{\alpha}r_{\beta}\rangle\right],$$ where $\vec{r}=(x,y,z)$ is the position vector of the electron and $r_{\alpha}$, $\alpha=1,2,3$, its components. If the magnetic field direction is chosen along the $z$-axis, $\vec{B}=B\hat{z}$, the tensor $\chi^d$ contains a single non-zero component, $\chi_{zz}^d \equiv
\chi^d$, $$\label{chi1}
\chi^d=-\frac{1}{4}\langle\rho^2\rangle=-\frac{1}{4}\left[\langle x^2\rangle+\langle y^2\rangle\right]\,.$$ Let us now consider the molecule in the $x$-$z$ plane (it can be regarded as the definition of the $x$-direction). For different orientations of the molecule with respect to the , the expectation values change according to a rotation by the angle $\theta$ around the $y$-axis ($x\to x\cos\theta+z\sin\theta$, $y\to
y$, $z\to z\cos\theta-x\sin\theta$) $$\label{chi2}
\chi^d=-\frac{1}{4}\left[\langle x^2\rangle_0\left(1+\cos^2\theta\right)+\langle z^2\rangle_0\sin^2\theta\right],$$ where $\langle x^2\rangle_0$ and $\langle z^2\rangle_0$ are the expectation values at zero inclination, and where we have used the fact that $\langle x^2\rangle_0=\langle y^2\rangle_0$ and $\langle
xz\rangle_0=0$. In Table \[Tablechi\], the numerical values of the expectation values of the squares of the components of the position vector of the electron, and the diamagnetic susceptibility, $\chi^d$, are presented, at equilibrium distance, as function of $\theta$ and compared with results obtained by Hegstrom [@Hegstrom:7917] for parallel and perpendicular orientations.
-------------- ---------------------- ---------------------- ---------------------- ------------------------- ------------------------ -------------------------
$\theta$ $\langle x^2\rangle$ $\langle y^2\rangle$ $\langle z^2\rangle$ $\chi^d$ $\chi^p$ $\chi$
$0^{\circ}$ 0.64036 0.64036 1.11131 -0.32018 0.00000 -0.32018
-0.3209[@Hegstrom:7917] — -0.3209
$15^{\circ}$ 0.67192 0.64037 1.07976 -0.32807 0.00022 -0.32785
$30^{\circ}$ 0.75810 0.64035 0.99359 -0.34961 0.00216 -0.34745
$45^{\circ}$ 0.87583 0.64040 0.87583 -0.37906 0.00992 -0.36914
$60^{\circ}$ 0.99357 0.64038 0.75811 -0.40849 0.02062 -0.38787
$75^{\circ}$ 1.07971 0.64042 0.67196 -0.43003 0.03090 -0.39913
$90^{\circ}$ 1.11125 0.64041 0.64041 -0.43792 0.03447 -0.40345
-0.4382[@Hegstrom:7917] 0.0378[@Hegstrom:7917] -0.4004[@Hegstrom:7917]
-------------- ---------------------- ---------------------- ---------------------- ------------------------- ------------------------ -------------------------
: \[Tablechi\]Expectation values of $x^2$, $y^2$ and $z^2$ for $B=0$, evaluated at the equilibrium distance $R=R_{eq}=1.9971 \, a_0$, and diamagnetic, paramagnetic and total susceptibility ($\chi^d$, $\chi^p$, $\chi$) as function of $\theta$. $\chi^p$, obtained as $\chi^p=\chi-\chi^d$, is included for convenience.
For strong fields, higher powers of $B$ might need to be considered in the expansion . With our variational method, evaluation of higher order terms is straightforward as the trial functions depend parametrically on the field strength. To this end we define the function $X(\vec{B})=-\langle \rho^2\rangle_B/4$, where the expectation value is taken with the optimized, $B$-dependent trial function. In the limiting case when $B\to0$, the diamagnetic susceptibility is recovered, $X(\vec{B})\to\chi^d$. Numerical results of $X(\vec{B})$ were obtained at the equilibrium distances for $B=0.0,
0.01, \dots 0.2\, B_0$ and $\theta=0^{\circ}, 15^{\circ}, \dots
90^{\circ} $ and fitted ($rms = 1.6 \times 10^{-4}$) to a simple functional form of the field strength $B$ and inclination $\theta$, $$\begin{aligned}
\label{chia}
X_a(\vec{B})=&&-0.43795+0.013498B+0.37103B^2\\\nonumber
&&+\left(0.11774-0.01577B-0.141B^2\right)\cos^2\theta.\end{aligned}$$ This surface is plotted in Figure \[Chisurface\],
while cuts at constant orientation angles are presented in Figure \[Chicuts\].
It can be seen that the $X(\vec{B})$ is a smooth function of the orientation angle, $\theta$, and the field strength, $B$, and tends to the magnetic susceptibility $X(\vec{B})\to \chi^d$ as the field tends to zero. The points on the ordinate represent the value of the magnetic susceptibility $\chi^d$ for various orientations and agree with the corresponding data obtained from the model , within the accuracy. At weak fields $\lesssim $ $0.04\,B_0$, $X$ is close to $\chi^d$ as given by . It indicates that perturbation theory in $B$ can be applied and can provide sufficiently accurate results. Eventually the diamagnetic susceptibility can be given ($rms = 1.6 \times 10^{-4}$) as $$\label{chia2}
\chi^d=-0.43795 + 0.11774 \cos^2\theta\,,$$ (c.f. (\[chia\])). Turning now to the total susceptibility $\chi$. In principle, it can be obtained directly using the Taylor expansion of the energy potential curve $E=E(B,R,\theta)$ for fixed $\theta$ in powers of $B$, but taken into account that the equilibrium distance $R_{eq}(B,\theta)$ evolves in $B$. It is a quite complicated procedure. It is much easier to calculate numerically the energy evolution with $B$ at minimum of the energy potential curve at fixed inclination. Then interpolate this curve $E(B)$ near the origin, $B=0$ using a polynomial of finite degree in $B$. The total susceptibility $\chi$ will be related to the coefficient $E^{(2)}$ in front of the $B^2$ term. Numerical values of the total susceptibility, defined as $\chi=-2E^{(2)}$ for different inclination $\theta$, are presented in Table \[Tablechi\]. They can be fitted accurately ($rms=3.45\times
10^{-4}$) to the following expression, $$\label{chi3}
\chi=-0.41067+0.08260\cos^2{\theta}+0.007620\cos^2{2\theta}\,.$$ Hence, for arbitrary inclination, the diamagnetic and total susceptibility can be obtained using the expressions (\[chia2\]) and . Finally, the paramagnetic contribution to the susceptibility can be evaluated as the difference $\chi^p=\chi
- \chi^d$, see data in Table \[Tablechi\]. In general, the paramagnetic susceptibility is much smaller that the diamagnetic part. It grows with inclination.
Concerning our statement that standard first order perturbation theory based on the field-free $H^{(0)}$ problem should be applicable up to $B \approx 0.04 \, B_0$, we may now add that at least 92% of the total susceptibility is recovered in this way. The energy correction quadratic in $B$ is accurate to $\sim 6\times 10^{-5} \, E_h$.
Solving the nuclear Schrödinger equation
========================================
Substituting in the Hamiltonian the electronic part (last three terms) by the potential energy surface, $\tilde{V}(R,\theta)$, we obtain the nuclear Hamiltonian. In the symmetric gauge, it can be written as $$\label{NucHamil}
\hat{\cal H}_{nuc}=\frac{2}{M_s}\hat{P}_R^2-\frac{1}{M_s}\vec{B}\cdot\hat{L}_R+\frac{1}{8M_s}\left[ B^2\vec{R}^2-(\vec{B}\cdot\vec{R})^2\right]\ +\tilde{V}(R,\theta) ,$$ where $\hat{L}_R=\vec{R}\times\hat{P}_R$ is the angular momentum operator of the molecular frame. Transforming the Hamiltonian in spherical coordinates yields $$\label{NucHamil2}
\hat{\cal H}_{nuc}=-\frac{2}{M_s}\frac{1}{R}\frac{\partial^2}{\partial R^2}R +\frac{2}{M_sR^2}\hat{L}_R^2-\frac{1}{M_s}B\hat{L}_z
+\frac{1}{8M_s}B^2R^2\sin^2{\theta}+\tilde{V}(R,\theta),$$ where $\hat{L}_z$ is the projection of angular momentum along $z$-axis and $\theta$ the angle between the molecular and the $z$-axis.
We have solved the nuclear Schrödinger equation with Hamiltonian numerically. To this end the Hamiltonian is divided in the two separate terms $$\label{H1}
\hat{\cal H}_{1}=-\frac{2}{M_s}\frac{1}{R}\frac{\partial^2}{\partial R^2}R
+\tilde{V}(R,\theta^{\prime})
+\frac{1}{8M_s}B^2R^2\sin^2{\theta^{\prime}}$$ and $$\label{H2}
\hat{\cal H}_{2}= \frac{2}{M_sR^2}\hat{L}_R^2-\frac{1}{M_s}B\hat{L}_z
+\frac{1}{8M_s}B^2R^2\left(\sin^2\theta-\sin^2\theta^{\prime}\right)
+\tilde{V}(R,\theta)-\tilde{V}(R,\theta^{\prime})$$ which roughly correspond to a vibrational part of the molecule with reference orientation angle $\theta^{\prime}$, and a rotational part. The rovibrational wave function is then expanded in terms of vibrational and rotational basis functions as $$\label{rv_basis}
\Psi(R,\theta,\phi) = \sum_{v,L} c_{v,L} \frac{ \xi_{v}(R;\underline{\theta}^{\prime})}{R}
Y_L^M(\theta,\phi)$$ where $\xi_{v}(R;\underline{\theta}^{\prime})$ are solutions of , obtained by numerical integration using the renormalized Numerov algorithm, and $Y_L^M(\theta,\phi)$ are spherical harmonics. The matrix elements of Hamiltonian in this basis are $$\begin{aligned}
\label{Hnuc_mat1}
\left\langle v^{\prime} L^{\prime} M | \hat{\cal H}_{nuc} | v L M \right\rangle & = &
E_v(\theta^{\prime}) \delta_{L^{\prime} L}\delta_{v^{\prime} v}
+ \frac{2}{M_s} \left\langle v^{\prime} | \frac{1}{R^2} | v \right\rangle L(L+1) \delta_{L^{\prime}L} \nonumber \\
& - & \frac{B M }{M_s} \delta_{L^{\prime} L}\delta_{v^{\prime} v} \\ \nonumber
& + & \frac{B^2}{8M_s} \left\langle L^{\prime} M | \sin^2\theta-\sin^2\theta^{\prime} | LM \right\rangle
\left\langle v^{\prime} | R^2 | v \right\rangle \nonumber \\
& + & \left\langle v^{\prime} L^{\prime} M | \tilde{V}(R,\theta)-\tilde{V}(R,\theta^{\prime})
| v L M \right\rangle \nonumber\end{aligned}$$ with $E_v(\theta^{\prime})$ the eigenvalue of the vibrational operator . To evaluate these matrix elements, the potential $\tilde{V}(R,\theta)$ is presented as that of a hindered rotator, $$\begin{aligned}
\label{pot}
\tilde{V}(R,\theta) & = & \tilde{V}(R,0)
+ \sum_n \frac{V_{90,n}(R)}{2} \left[ 1 - \cos (2 n \theta) \right] \nonumber \\
& \approx & \tilde{V}(R,0) + V_{90}(R) \sin^2 \theta\end{aligned}$$ $V_{90}(R)=\tilde{V}(R,90)-\tilde{V}(R,0)$ is the barrier height for a given value of $R$.
Limitation of the above expansion to just one term is a good approximation of the potential at the field strengths considered in the present work, as we have verified numerically. The $rms$ values for fits of $V(R_{eq},\theta)$, using increments $\Delta \theta =
5^{\circ}$, are $rms=7.98\times 10^{-5}$ and $rms=3.66\times 10^{-6}$ for the one and two-term approximations and $B=0.2 B_0$. For $B=0.1
B_0$ the fitting error is reduced by a factor of four, approximately. Figure \[rotap\] shows the performance of the two approximations.
The one-term approximation thus represents the potential energy surface to within the accuracy of the raw data of the electronic energy. An appealing feature is that just two slices, at $\theta=0^{\circ}$ and $\theta=90^{\circ}$ of the surface are needed explicitly. Within the one-term approximation and choosing the reference orientation $\theta^{\prime}=0$, the matrix elements can be evaluated readily as [^2] $$\begin{aligned}
\label{Hnuc_mat2}
\left\langle v^{\prime} L^{\prime} M | \hat{\cal H}_{nuc} | v L M \right\rangle & = &
E_v \delta_{L^{\prime} L}\delta_{v^{\prime} v}
+ \frac{2}{M_s} \left\langle v^{\prime} | \frac{1}{R^2} | v \right\rangle L(L+1) \delta_{L^{\prime}L} \nonumber \\
& - & \frac{B M }{M_s} \delta_{L^{\prime} L}\delta_{v^{\prime} v} \nonumber \\
& + & \left[\frac{B^2}{12M_s} \langle v^{\prime} | R^2 | v \rangle
+\frac{2}{3}\langle v^{\prime} | V_{90}(R) | v \rangle \right] \delta_{L^{\prime}L} \nonumber \\
& - &\left[\frac{B^2}{12M_s} \langle v^{\prime} | R^2 | v \rangle
+\frac{2}{3}\langle v^{\prime} | V_{90}(R) | v \rangle \right] \nonumber \\
& \times & (-1)^{M}\sqrt{(2L'+1)(2L+1)} \nonumber \\
& \times & \left(
\begin{array}{ccc}
L&2&L'\\
0&0&0
\end{array}
\right)
\left(
\begin{array}{ccc}
L&2&L'\\
M&0&-M
\end{array}
\right)\end{aligned}$$ The terms in parentheses are Wigner $3j$-symbols. The matrix is diagonal in $M$ as expected, since $M$ is an exact quantum number. $L$-functions are coupled in steps of 2, conserving parity.
Results
-------
For the isotopologues [$\rm H_2^+$]{}and [$\rm D_2^+$]{}we have computed the rovibrational eigenvalues of the nuclear Hamiltonian for the four lowest vibrational states and rotational excitation up to $L=5$ with respect to the field-free case. Two levels of approximation are considered: a simplified model in which only the diagonal terms with respect to the vibrational basis are retained, and a second model which consists in diagonalizing the Hamiltonian in the full basis. These data are presented in Tables \[Table:rotv0\] to \[Table:Drotv3\]. The results obtained at the two levels of approximation agree to within $10^{-5} \, E_h$.
If spin effects are neglected, rovibrational states of [$\rm H_2^+$]{}in a magnetic field can be classified in terms of three quantum numbers: the vibrational quantum number, $\nu$, the projection of the angular momentum of the molecular frame on the field axis, $M$, and the $z$-parity, $\pi$. The latter quantum number is due to the fact that positive and negative $z$-directions of the field are equivalent. If the wave function is reflected at the plane $z=0$, $\theta$ is mapped to $\pi - \theta$ and $Y_L^M(\pi-\theta,\phi) = (-1)^{L+M}
Y_L^M(\theta,\phi)$. The $z$-parity of the state is thus $\pi=(-1)^{L+M}$. The nuclear wave function of a system of two fermions must be antisymmetric with respect to an exchange of the nuclei. The vibrational part of the wave function is symmetric for even vibrational quanta, $v=0,2, \dots$ and antisymmetric for odd, $v=1,3,\dots$. The symmetry of the rotational part can be derived from the properties of the spherical harmonics with respect to inversion, ($\theta, \phi \rightarrow \theta+\pi, \phi+\pi$), $Y_L^M(\theta+\pi,\phi+\pi) = (-1)^{L} Y_L^M(\theta,\phi)$. Hence for even $v$ the rotational functions must have odd parity, while for odd $v$ they must have even parity, just as in the field-free case. The expression for the $z$-parity is $$\label{zpar}
\pi = (-1)^{M+v+1} = \left\{
\begin{array}{rl}
-(-1)^M & {\rm for} ~v~ {\rm even} \\
(-1)^M & {\rm for} ~v~ {\rm odd}
\end{array}
\right.$$ For [$\rm D_2^+$]{}, a system with two bosonic nuclei, vibrational and rotational parts of the wavefunction must have the same parity. In this case, the $z$-parity is given by $$\label{Dzpar}
\pi = (-1)^{M+v} = \left\{
\begin{array}{rl}
(-1)^M & {\rm for} ~v~ {\rm even} \\
-(-1)^M & {\rm for} ~v~ {\rm odd}
\end{array}
\right.$$ The calculated rovibrational states in Tables \[Table:rotv0\] to \[Table:Drotv3\] are labeled with the exact quantum numbers. Graphical analysis of the [$\rm H_2^+$]{}rovibrational states, Figure \[rovibplot\], shows that they remain grouped according to the field-free quantum number $L$ which is explained by the fact that all rovibrational states are located above the rotational barrier, for the two isotopologues. $B=0.2 B_0$, or 47000 Tesla, is a strong field but of modest size in atomic units, hence $L$ may be considered a good quantum number. The main effect of the magnetic field in this region of field strengths is on the electronic energy.
Conclusions
===========
We have investigated the problem of [$\rm H_2^+$]{}and [$\rm D_2^+$]{}in an external magnetic field of up to $B = 0.2 B_0$ or 4.7$\times 10^4$ T by exact and approximate methods. This includes a thorough analysis of the electronic energy as function of field strength and orientation of the molecule with respect to the external field as well as of the rovibrational structure of [$\rm H_2^+$]{}and [$\rm D_2^+$]{}. The electronic problem has been solved by the variational method with physically adequate trial functions. It is shown that both diamagnetic and paramagnetic susceptibilities grow with inclination, while paramagnetic susceptibility is systematically much smaller than the diamagnetic one. Evaluation of the magnetic susceptibility shows that first-order perturbation theory based on zero-field trial functions may no longer be accurate at a field strength of above $B
\approx 0.04 B_0$. To solve the ro-vibrational problem, the hindered rotor approximation, in which the potential energy surface is approximated by a zero-inclination potential curve as a function of the internuclear distance $R$ and a simple parametrization of the rotational barrier in the angular coordinate $\theta$ gives results accurate to about $10^{-5} \; E_h$, which is comparable in accuracy with the rigid-rotor approximation to separate vibrational and rotational motion. Some of the approximations have been used before, by other authors, at much higher field strengths, were they are less accurate. The findings of the present paper provide a basis for future investigations dealing with higher field strengths and different molecules such as $\rm H_2$.
Acknowledgements {#acknowledgements .unnumbered}
================
H.M.C. is grateful to Consejo Nacional de Ciencia y Tecnología, Mexico, for a postdoctoral grant (CONACyT grant no 202139). This work was also supported by a Mexican-French binational research grant (CONACyT-CNRS grant no 26218) and by the Computer Center ROMEO of the University of Reims Champagne-Ardenne. Research by J.C.L.V and A.V. T. was supported in part by CONACyT grant 116189 and DGAPA IN109512.
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[^1]: For further information see [@Avron:78431; @Kappes:954542; @Schmelcher:88672; @Schmelcher:886066]
[^2]: We use $\sin^2{\theta}=\frac{2}{3}-\frac{2}{3}\sqrt{\frac{4\pi}{5}}Y_2^0(\theta,\phi)$ and the expression for the scalar product of three spherical harmonics, the Gaunt coefficients, $\int d\Omega
Y_L^{M*}(\Omega)Y_{l_1}^{m_1}(\Omega)Y_{l_2}^{m_2}(\Omega)=(-1)^M\sqrt{\frac{(2l_1+1)(2l_2+1)(2L+1)}{4\pi}}
\left(
\begin{array}{ccc}
l_1&l_2&L\\
0&0&0
\end{array}
\right)
\left(
\begin{array}{ccc}
l_1&l_2&L\\
m_1&m_2&-M
\end{array}
\right)$. The Condon-Shortly phase [@MagnascoBook] convention has been adopted.
|
---
abstract: 'Satisfiability solvers are increasingly playing a key role in software verification, with particularly effective use in the analysis of security vulnerabilities. String processing is a key part of many software applications, such as browsers and web servers. These applications are susceptible to attacks through malicious data received over network. Automated tools for analyzing the security of such applications, thus need to reason about strings. For efficiency reasons, it is desirable to have a solver that treats strings as first-class types. In this paper, we present some theories of strings that are useful in a software security context and analyze the computational complexity of the presented theories. We use this complexity analysis to motivate a byte-blast approach which employs a Boolean encoding of the string constraints to a corresponding Boolean satisfiability problem.'
author:
- |
Susmit Jha Sanjit A. Seshia Rhishikesh Limaye\
EECS Department, UC Berkeley\
{jha,sseshia,rhishi}@eecs.berkeley.edu
bibliography:
- 'strings.bib'
title: |
On the Computational Complexity\
of Satisfiability Solving for String Theories
---
Introduction
============
Many security-critical applications such as Web servers routinely process strings as an essential part of their functionality. They take strings as inputs, screen them using filters, manipulate them and use them for operations such as database queries. It is pertinent to verify that these programs do not have vulnerabilities which can be used to compromise system security. Verification and structured testing techniques to validate security of such applications often rely on using constraint solvers. The frequent use of string operations in these applications has motivated several groups to explore the possibility of designing a constraint solver which treats strings as first-class types. Such a specialized solver for strings would further facilitate the use of constraint solving for analysis of security applications with string operations.
Software applications use various string predicates and functions which are often made available to the developers as libraries. A satisfiability solver for string constraints must be able to handle these predicates and functions. From the string constraints and predicates available in high level programming languages such as C, JAVA and C++, we identify a set of core predicates and functions. Many other more complicated string-manipulating functions can be expressed as some simple composition of these functions. We use these predicates and functions to define a theory of strings.
The main contribution of this paper is an analysis of the complexity of several fragments of the theory of strings. We show that fairly small and simple-looking fragments are NP-complete. In light of the progress in SAT solving and SMT solving for bit-vector arithmetic, these results indicate that a SAT-based approach is reasonable to satisfiability solving of string constraints.
Conclusion and Future Work
==========================
The analysis of different fragments of the theory of strings presented in this paper shows that the satisfiability problem for even small non-trivial fragments is NP-complete. Thus, it is unlikely that an efficient (polynomial-time) algorithm for checking the satisfiability of the strings would be found. Hence, a simple approach based on Boolean encoding of string constraints to propositional logic is, in principle, as effective as any other technique for solving string constraints. This justifies a “byte-blast” approach to solving string constraints which relies on encoding strings as bit-vectors and using an off-the-shelf bit-vector SMT solver. Further, these hardness results underline the importance of using domain knowledge about string constraints arising out of security applications. We believe, in practice, word-level reasoning over strings that exploits such domain knowledge through pragmatic approaches such as abstraction-refinement might prove to be very effective in making an efficient and scalable for theory of strings. The key challenge in developing an SMT solver for theory of strings is identification of such properties of string constraints arising from real code.
Inspired by the success of abstraction-refinement based approaches for SMT solving (e.g., [@kroening-cav04; @bryant-tacas07; @ganesh-cav07]), we believe such an approach would be useful for the theory of strings also. We identify the abstraction techniques that we believe would be especially useful in the context of a theory of strings:
[*Length abstraction:*]{} To our knowledge, this approach has been first published by Bjorner et al [@bjorner-tr08]. It operates by creating an over-approximation of the actual formula by abstracting each string constraint with a corresponding length constraint. The resulting integer linear arithmetic formula is solved to obtain candidate lengths for the strings in the original formula, with a possible refinement needed if these candidate lengths turn out to be too small. We believe that this general idea can be used but with some guidance to the solver to not simply generate the smallest lengths.
[*Position abstraction:*]{} We have observed that, in the security applications of interest, *string-containment* is a widely used predicate and the encoding the choice of position of containment adds significant complexity to the constraint satisfaction problem. For large string-lengths, a standard byte-blast approach which reduces the string constraints to bit-vector formula would require the SAT solver to branch over a large set of choices of positions. We hypothesize based on our observations of string constraints generated by colleagues in security applications [@juan-tr09], that the position and order of containment of substrings is often not critical to finding a satisfying assigment. Hence, an effective approach to construct under-approximation of the string formula would be fixing some heuristic ordering of containment constraints. If the formula with this fixed ordering is unsatisfiable, the unsat core generated by the SAT solver can be used to selectively refine the ordering.
The overall approach we envisage will be similar to the iterative construction of over- and under-approximate formulas as performed in prior work on model checking [@mcmillan-tacas03] and SMT solving for bit-vector arithmetic [@bryant-tacas07]. It would be interesting to evaluate how such an approach based on abstraction-refinement performs for string formulas generated in practice from security applications.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Juan Caballero and Dawn Song for many helpful discussions on formalizing fragments of the theory of strings that are relevant for software security.
|
---
abstract: 'The strong decays of the $5^1S_0$ $q\bar{q}$ states are evaluated in the $^3P_0$ model with two types of space wave functions. Comparing the model expectations with the experimental data for the $\pi(2360)$, $\eta(2320)$, $X(2370)$, and $X(2500)$, we suggest that the $\pi(2360)$, $\eta(2320)$, and $X(2500)$ can be assigned as the members of the $5^1S_0$ meson nonet, while the $5^1S_0$ assignment for the $X(2370)$ is not favored by its width. The $5^1S_0$ kaon is predicted to have a mass of about 2418 MeV and a width of about 163 MeV or 225 MeV.'
author:
- 'Shi-Chen Xue'
- 'Guan-Ying Wang'
- 'Guan-Nan Li'
- En Wang
- 'De-Min Li'
title: ' The possible members of the $5^1S_0$ meson nonet'
---
Introduction
============
[\[Introduction\]]{}
In the framework of quantum chromodynamics (QCD), apart from the ordinary $q\bar{q}$ states, other exotic states such as glueballs, hybrids, and tetraquarks are permitted to exist in meson spectra. To identify these exotic states, one needs to distinguish them from the background of ordinary $q\bar{q}$ states, which requires one to understand well the conventional $q\bar{q}$ meson spectroscopy both theoretically and experimentally. To be able to understand the nature of a newly observed state, it is natural and necessary to exhaust the possible $q\bar{q}$ description before restoring to more exotic assignments.
[c|l]{} Isospin & States\
$I=1$ & $\pi$, $\pi(1300)$, $\pi(1800)$, $\pi(2070)$, $\pi(2360)$\
\*
-------
$I=0$
-------
: \[tab:pseu\]The pseudoscalar states reported experimentally.
& $\eta$, $\eta(1295)$, $\eta(1760)$, $\eta(2010)$, $\eta(2100)$, $\eta(2190)$, $\eta(2320)$\
& $\eta'$, $\eta(1475)$, $X(1835)$[^1], $\eta(2225)$, $X(2500)$\
$I=1/2$& $K$, $K(1460)$, $K(1830)$\
As shown in Table \[tab:pseu\], many pseudoscalar states have been accumulated experimentally [@PDG2016; @Ablikim:2016hlu]. Among these states, the assignments of the $\pi$, $\eta$, $\eta'$, and $K$ as the members of the $1^1S_0$ meson nonet and the $\pi(1300)$, $\eta(1295)$, $\eta(1475)$, and $K(1460)$ as the members of the $2^1S_0$ meson nonet have been widely accepted [@PDG2016]. In our previous works, we suggested that the $\pi(1800)$, $\eta(1760)$, $X(1835)$, and $K(1830)$ can be identified as members of the $3^1S_0$ meson nonet [@Li:2008mza], the $\pi(2070)$, $\eta(2100)$, and $\eta(2225)$ can be identified as the members of $4^1S_0$ meson [@Li:2008et], and the $X(2500)$ is the mainly $s\bar{s}$ member of the $5^1S_0$ meson nonet [@Pan:2016bac], where the mixing of the $X(2500)$ and its isoscalar partner is not considered and other members of the $5^1S_0$ meson nonet are not discussed. In this work, we shall address the possible SU(3) multiplet partners of the $X(2500)$. With the assignment of the $X(2500)$ as the $s\bar{s}$ member of the $5^1S_0$ nonet, one can expect that other members of the $5^1S_0$ nonet should be lighter than the $X(2500)$. Along this line, considering that other pseudoscalar states have discussed in our previous works [@Li:2008mza; @Li:2008et], we shall focus on the $\pi(2360)$ and $\eta(2320)$ shown in Table \[tab:pseu\], and check whether they can be explained as the $5^1S_0$ $q\bar{q}$ states or not. Study on the pseudoscalar radial $q\bar{q}$ excitations in the mass region of $2.3\sim 2.6 $ GeV is especially interesting because the pseudoscalar glueball is predicted to exist in this mass region [@Morningstar:1999rf; @Hart:2001fp; @Chen:2005mg].
The $\pi(2360)$ was observed in $\bar{p}p \to 3\pi^0,\pi^0\eta, \pi^0\eta^\prime, \eta \eta \pi^0$, and its mass and width are $2360\pm 25$ MeV and $300^{+100}_{-50}$ MeV, respectively [@Anisovich:2001pn; @Anisovich:2001pp]. The $\pi(5^1S_0)$ mass is expected to be $2316 \pm 40$ MeV in a relativistic independent quark model [@Khruschov:2005zc] or $2385$ MeV in a relativistic quark model [@Ebert:2009ub], both consistent with the $\pi(2360)$ mass. Thus, the $\pi(2360)$ appears a good candidate for the $\pi(5^1S_0)$ based on its measured mass.
![The $\pi$, $\eta$, and $\eta^\prime$-trajectories with $M^2_n = M^2_0+(n-1)\mu^2$ by fitting to the experimental masses of the mesons. $\pi$-trajectory: $M^2_0=0.019480\pm0.000001~\mbox{GeV}^2$, $\mu^2=1.5387\pm 0.0165~\mbox{GeV}^2$, $\chi^2/\mbox{d.o.f}=49.7/(5-2)$; $\eta$-trajectory: $M^2_0=0.30015\pm0.00002~\mbox{GeV}^2$, $\mu^2=1.3511\pm 0.0084 ~\mbox{GeV}^2$, $\chi^2/\mbox{d.o.f}=27.9/(5-2)$; $\eta^\prime$-trajectory: $M^2_0=0.91734\pm0.00115~\mbox{ GeV}^2$, $\mu^2=1.2723\pm 0.0092~\mbox{GeV}^2$, $\chi^2/\mbox{d.o.f}=19.9/(5-2)$. The meson masses used to fit are taken from Refs. [@PDG2016; @Ablikim:2016hlu].[]{data-label="fig:piRegge"}](reggepi.eps "fig:")![The $\pi$, $\eta$, and $\eta^\prime$-trajectories with $M^2_n = M^2_0+(n-1)\mu^2$ by fitting to the experimental masses of the mesons. $\pi$-trajectory: $M^2_0=0.019480\pm0.000001~\mbox{GeV}^2$, $\mu^2=1.5387\pm 0.0165~\mbox{GeV}^2$, $\chi^2/\mbox{d.o.f}=49.7/(5-2)$; $\eta$-trajectory: $M^2_0=0.30015\pm0.00002~\mbox{GeV}^2$, $\mu^2=1.3511\pm 0.0084 ~\mbox{GeV}^2$, $\chi^2/\mbox{d.o.f}=27.9/(5-2)$; $\eta^\prime$-trajectory: $M^2_0=0.91734\pm0.00115~\mbox{ GeV}^2$, $\mu^2=1.2723\pm 0.0092~\mbox{GeV}^2$, $\chi^2/\mbox{d.o.f}=19.9/(5-2)$. The meson masses used to fit are taken from Refs. [@PDG2016; @Ablikim:2016hlu].[]{data-label="fig:piRegge"}](reggeetaetap.eps "fig:")
A series of the papers of Anisovich [@Anisovich:2000kxa; @Anisovich:2001ig; @Anisovich:2002us; @Anisovich:2003tm; @Anisovich:2004vj] indicate that with a good accuracy, the light $q\bar{q}$ meson states with different radial excitations fit to the following quasi-linear $(n,M^2_n)$-trajectories $$M^2_n=M^2_0+(n-1)\mu^2, \label{eq:regge}$$ where $M_n$ denotes the mass of the meson with radial quantum number $n$, $M^2_0$ and $\mu^2$ are the parameters of the corresponding trajectory. The relation of Eq. (\[eq:regge\]) can be derived from the Regge phenomenology [@collins; @Li:2007px]. One can use this relation to roughly estimate the masses for higher radial excitations. As displayed in Fig. \[fig:piRegge\], we find that in the $(n,M^2_n)$ plane, the three pseudoscalar meson groups, \[$\pi$, $\pi(1300)$, $\pi(1800)$, $\pi(2070)$, $\pi(2360)$\], \[$\eta$, $\eta(1295)$, $\eta(1760)$, $\eta(2100)$, $\eta(2320)$\], and \[$\eta^\prime$, $\eta(1475)$, $X(1835)$, $\eta(2225)$, $X(2500)$\], approximately populate the $\pi$, $\eta$, and $\eta^\prime$-linear trajectories, respectively. With the assignment that the \[$\pi(1800)$, $\eta(1760)$, $X(1835)$\] and \[$\pi(2070)$, $\eta(2100)$, $\eta(2225)$ \] belong to the $3^1S_0$ and $4^1S_0$ meson nonets, respectively, one can naturally expect that the $\pi(2360)$, $\eta(2320)$, and $X(2500)$ could belong to the $5^1S_0$ nonet based on their masses.
Both the mass and width of a resonance are related to its inner structure. Although the masses of the $\pi(2360)$, $\eta(2320)$, and $X(2500)$ are consistent with them belonging to the $5^1S_0$ meson nonet, their decay properties also need to be compared with model expectations in order to identify the possible candidates for the $5^1S_0$ meson nonet. Below, we shall evaluate their strong decays in the framework of the $^3P_0$ model.
This paper is organized as follows. In Sec. \[sec:formalisms\], we present the $^3P_0$ model parameters used in our calculations. The results and discussions are given in Sec. \[sec:result\]. Finally, a short summary is given in Sec. \[sec:summary\].
Model and Parameters {#sec:formalisms}
====================
The $^3P_0$ model has been widely used to study the strong decays of hadrons in literature [@Roberts:1992js; @Blundell:1996as; @Barnes:1996ff; @Barnes:2002mu; @Close:2005se; @Barnes:2005pb; @Zhang:2006yj; @Ding:2007pc; @Li:2008mza; @Li:2008we; @Li:2008et; @Li:2008xy; @Li:2009rka; @Li:2009qu; @Li:2010vx; @Lu:2014zua; @Pan:2016bac; @Lu:2016bbk; @Wang:2017pxm]. In the $^3P_0$ model, the meson strong decay takes place by producing a quark-antiquark pair with vacuum quantum number $J^{PC}=0^{++}$. The newly produced quark-antiquark pair, together with the $q\bar{q}$ within the initial meson, regroups into two outgoing mesons in all possible quark rearrangement ways. Some detailed reviews on the $^3P_0$ model can be found in Refs. [@Li:2008mza; @Li:2008et; @Li:2008we; @Roberts:1992js; @Blundell:1996as]. Here we give the main ingredients of the $^3P_0$ model briefly.
Following the conventions in Ref. [@Li:2008mza], the transition operator $T$ of the decay $A\rightarrow BC$ in the $^3P_0$ model is given by $$\begin{aligned}
T=-3\gamma\sum_m\langle 1m1-m|00\rangle\int
d^3\boldsymbol{p}_3d^3\boldsymbol{p}_4\delta^3(\boldsymbol{p}_3+\boldsymbol{p}_4)\nonumber\\
{\cal{Y}}^m_1\left(\frac{\boldsymbol{p}_3-\boldsymbol{p}_4}{2}\right
)\chi^{34}_{1-m}\phi^{34}_0\omega^{34}_0b^\dagger_3(\boldsymbol{p}_3)d^\dagger_4(\boldsymbol{p}_4),\end{aligned}$$ where the $\gamma$ is a dimensionless parameter denoting the production strength of the quark-antiquark pair $q_3\bar{q}_4$ with quantum number $J^{PC}=0^{++}$. $\boldsymbol{p}_3$ and $\boldsymbol{p}_4$ are the momenta of the created quark $q_3$ and antiquark $\bar{q}_4$, respectively. $\chi^{34}_{1,-m}$, $\phi^{34}_0$, and $\omega^{34}_0$ are the spin, flavor, and color wave functions of $q_3\bar{q}_4$, respectively. The solid harmonic polynomial ${\cal{Y}}^m_1(\boldsymbol{p})\equiv|\boldsymbol{p}|^1Y^m_1(\theta_p, \phi_p)$ reflects the momentum-space distribution of the $q_3\bar{q_4}$.
The $S$ matrix of the process $A\rightarrow BC$ is defined by $$\begin{aligned}
\langle BC|S|A\rangle=I-2\pi i\delta(E_A-E_B-E_C)\langle BC|T|A\rangle,\end{aligned}$$ where $|A\rangle$ ($|B\rangle$,$|C\rangle$) is the mock meson defined by [@Hayne:1981zy] $$\begin{aligned}
&&|A(n^{2S_A+1}_AL_{A}\,\mbox{}_{J_A M_{J_A}})(\boldsymbol{P}_A)\rangle
\equiv \nonumber\\
&& \sqrt{2E_A}\sum_{M_{L_A},M_{S_A}}\langle L_A M_{L_A} S_A
M_{S_A}|J_A
M_{J_A}\rangle\nonumber\\
&&\times \int d^3\boldsymbol{p}_A\psi_{n_AL_AM_{L_A}}(\boldsymbol{p}_A)\chi^{12}_{S_AM_{S_A}}
\phi^{12}_A\omega^{12}_A\nonumber\\
&&\times \left|q_1\left({\scriptstyle
\frac{m_1}{m_1+m_2}}\boldsymbol{P}_A+\boldsymbol{p}_A\right)\bar{q}_2
\left({\scriptstyle\frac{m_2}{m_1+m_2}}\boldsymbol{P}_A-\boldsymbol{p}_A\right)\right\rangle.\end{aligned}$$ Here $m_1$ and $m_2$ ($\boldsymbol{p}_1$ and $\boldsymbol{p}_2$) are the masses (momenta) of the quark $q_1$ and the antiquark $\bar{q}_2$, respectively; $\boldsymbol{P}_A=\boldsymbol{p}_1+\boldsymbol{p}_2$, $\boldsymbol{p}_A=\frac{m_2\boldsymbol{p}_1-m_1\boldsymbol{p}_2}{m_1+m_2}$; $\chi^{12}_{S_AM_{S_A}}$, $\phi^{12}_A$, $\omega^{12}_A$, and $\psi_{n_AL_AM_{L_A}}(\boldsymbol{p}_A)$ are the spin, flavor, color, and space wave functions of the meson $A$ composed of $q_1\bar{q}_2$ with total energy $E_A$, respectively. $n_A$ is the radial quantum number of the meson $A$. $\boldsymbol{S}_A=\boldsymbol{s}_{q_1}+\boldsymbol{s}_{\bar{q}_2}$, $\boldsymbol{J}_A=\boldsymbol{L}_A+\boldsymbol{S}_A$, $\boldsymbol{s}_{q_1}(\boldsymbol{s}_{\bar{q}_2})$ is the spin of $q_1(\bar{q}_2)$, and $\boldsymbol{L}_A$ is the relative orbital angular momentum between $q_1$ and $\bar{q}_2$. $\langle L_A M_{L_A} S_AM_{S_A}|J_AM_{J_A}\rangle$ denotes a Clebsch-Gordan coefficient.
The transition matrix element $\langle BC|T|A\rangle$ can be written as $$\begin{aligned}
\langle BC|T|A\rangle=\delta^3(\boldsymbol{P}_A-\boldsymbol{P}_B-\boldsymbol{P}_C){\cal{M}}^{M_{J_A}M_{J_B}M_{J_C}}(\boldsymbol{P}),\end{aligned}$$ where ${\cal{M}}^{M_{J_A}M_{J_B}M_{J_C}}
(\boldsymbol{P})$ is the helicity amplitude. In the center of mass frame of meson A, the helicity amplitude is $$\begin{aligned}
&&{\cal{M}}^{M_{J_A}M_{J_B}M_{J_C}}(\boldsymbol{P})=\gamma\sqrt{8E_AE_BE_C} \sum_{M_{L_A},M_{S_A}}
\nonumber\\&&\times \sum_{M_{L_B},M_{S_B}} \sum_{M_{L_C},M_{S_C}}\sum_m
\langle L_A M_{L_A} S_AM_{S_A}|J_AM_{J_A}\rangle\nonumber\\
&&\times\langle L_B M_{L_B} S_BM_{S_B}|J_BM_{J_B}\rangle\langle L_C M_{L_C} S_CM_{S_C}|J_CM_{J_C}\rangle\nonumber\\
&&\times\langle 1m1-m|00\rangle\langle \chi^{14}_{S_BM_{S_B}}\chi^{32}_{S_CM_{S_C}}|\chi^{12}_{S_AM_{S_A}}\chi^{34}_{1-m}\rangle\nonumber\\
&&\times[f_1I(\boldsymbol{P},m_1,m_2,m_3)\nonumber\\
&&+(-1)^{1+S_A+S_B+S_C}f_2I(-\boldsymbol{P},m_2,m_1,m_3)],
\label{helicity}\end{aligned}$$ with $f_1=\langle \phi^{14}_B\phi^{32}_C|\phi^{12}_A\phi^{34}_0\rangle$ and $f_2=\langle \phi^{32}_B\phi^{14}_C|\phi^{12}_A\phi^{34}_0\rangle$, and $$\begin{aligned}
I(\boldsymbol{P},m_1,m_2,m_3)=&&\int d^3\boldsymbol{p}\psi^*_{n_BL_BM_{L_B}}\left({\scriptstyle
\frac{m_3}{m_1+m_3}}\boldsymbol{P}_B+\boldsymbol{p}\right)\nonumber\\&&\times\psi^*_{n_CL_CM_{L_C}}\left({\scriptstyle
\frac{m_3}{m_2+m_3}}\boldsymbol{P}_B+\boldsymbol{p}\right)\nonumber\\&&\times\psi_{n_AL_AM_{L_A}}\left(\boldsymbol{P}_B+\boldsymbol{p}\right){\cal{Y}}^m_1(\boldsymbol{p}),
\label{overlap space}\end{aligned}$$ where $\boldsymbol{P}={\boldsymbol{P}}_B=-{\boldsymbol{P}}_C$, $\boldsymbol{p}=\boldsymbol{p}_3$, $m_3$ is the mass of the created quark $q_3$.
The partial wave amplitude ${\cal{M}}^{LS}(\boldsymbol{P})$ can be given by [@Jacob:1959at], $$\begin{aligned}
{\cal{M}}^{LS}(\boldsymbol{P})&=&
\sum_{\renewcommand{\arraystretch}{.5}\begin{array}[t]{l}
\scriptstyle M_{J_B},M_{J_C},\\\scriptstyle M_S,M_L
\end{array}}\renewcommand{\arraystretch}{1}\!\!
\langle LM_LSM_S|J_AM_{J_A}\rangle \nonumber\\
&&\langle
J_BM_{J_B}J_CM_{J_C}|SM_S\rangle\nonumber\\
&&\times\int
d\Omega\,\mbox{}Y^\ast_{LM_L}{\cal{M}}^{M_{J_A}M_{J_B}M_{J_C}}
(\boldsymbol{P}). \label{pwave}\end{aligned}$$
Various $^3P_0$ models exist in literature and typically differ in the choices of the pair-production vertex, the phase space conventions, and the meson wave functions employed. In this work, we restrict to the simplest vertex as introduced originally by Micu [@Micu:1968mk] which assumes a spatially constant pair-production strength $\gamma$, adopt the relativistic phase space, and employ two types of meson space wave functions: the simple harmonic oscillator (SHO) wave functions and the relativized quark model (RQM) wave functions [@Godfrey:1985xj].
With the relativistic phase space, the decay width $\Gamma(A\rightarrow BC)$ can be expressed in terms of the partial wave amplitude $$\begin{aligned}
\Gamma(A\rightarrow BC)= \frac{\pi
|\boldsymbol{P}|}{4M^2_A}\sum_{LS}|{\cal{M}}^{LS}(\boldsymbol{P})|^2, \label{width1}\end{aligned}$$ where $|\boldsymbol{P}|=\sqrt{[M^2_A-(M_B+M_C)^2][M^2_A-(M_B-M_C)^2]}{2M_A}$, and $M_A$, $M_B$, and $M_C$ are the masses of the mesons $A$, $B$, and $C$, respectively.
The parameters used in the $^3P_0$ model calculations involve the $q\bar{q}$ pair production strength $\gamma$, the parameters associated with the meson wave functions, and the constituent quark masses. In the SHO wave functions case (case A), we follow the parameters used in Ref. [@Close:2005se], where the SHO wave function scale is $\beta=\beta_A=\beta_B=\beta_C=0.4$ GeV, the constituent quark masses are $m_u=m_d=330$ MeV, $m_s=550$ MeV, and $\gamma=8.77$ obtained by fitting to 32 well-established decay modes. In the RQM wave functions case (case B), we take $m_u=m_d=220$ MeV, and $m_s=419$ MeV as used in the relativized quark model of Godfrey and Isgur [@Godfrey:1985xj], and $\gamma=15.28$ by fitting to the same decay modes used in Ref. [@Close:2005se] except for three decay modes without the specific branching ratios $K^{\ast\prime}\rightarrow \rho K$, $K^{\ast\prime}\rightarrow K^\ast\pi$, and $a_2\rightarrow \rho\pi$ [@PDG2016]. The meson flavor wave functions follow the conventions of Refs. [@Godfrey:1985xj; @Barnes:2002mu]. We assume that the $a_0(1450)$, $K^\ast_0(1430)$, $f_0(1370)$, and $f_0(1710)$ are the ground scalar mesons as in Refs. [@Barnes:2002mu; @Ackleh:1996yt; @Barnes:1996ff]. Masses of the final state mesons are taken from Ref. [@PDG2016].
Results and discussions {#sec:result}
=======================
$\pi(2360)$
-----------
The decay widths of the $\pi(2360)$ as the $\pi(5^1S_0)$ are listed in Table \[tab:2360\]. The $\pi(5^1S_0)$ total width is predicted to be about 281 MeV in case A or 285 MeV in case B, both in agreement with the $\pi(2360)$ width of $\Gamma=300^{+100}_{-50}$ MeV [@Anisovich:2001pn; @Anisovich:2001pp]. The dependence of the $\pi(5^1S_0)$ width on the initial mass is shown in Fig. \[Fig:2360\]. Within the $\pi(2360)$ mass errors ($2360\pm 25$ MeV), in both cases, the predicted width of the $\pi(5^1S_0)$ always overlaps with the $\pi(2360)$ width. Therefore, the measured width for the $\pi(2360)$ supports that the $\pi(2360)$ can be identified as the $\pi(5^1S_0)$. The flux-tube model calculations in Ref. [@Wang:2017iai] also favor this assignment.
It is noted that for some decay modes such as $\pi\rho$, $\pi\rho(1700)$, $\pi(1300)\rho$, $\rho h_1(1170)$ $\omega b_1(1235)$, $\pi\rho_3(1690)$, and $K K^*_3(1780)$, the predictions in case A are similar with those in case B, while for other modes such as the $\pi f_0(1370)$, $\eta a_0(1450)$, $K K^\ast_0(1430)$, $\pi \rho(1450)$, $K K^*(1410)$, $K K^\ast(1680)$, $K(1460)K^\ast$, $\pi f_2(1270)$, $K K^\ast_2(1430)$, $\rho a_2(1320)$, and $\pi \rho_3(1990)$, there are some big variations between cases A and B. The similar behavior also exists in the flux-tube model (a variant of the $^3P_0$ model) calculations with different space wave functions [@Blundell:1995ev; @Kokoski:1985is].
As shown in Eqs. (\[helicity\]) and (\[overlap space\]), the partial width from the $^3P_0$ model depends on the overlap integrals of flavor, spin, and space wave functions of initial and final states. For a given decay mode, the overlap integrals of the flavor and spin wave functions of initial and final mesons are identical in both RQM and SHO cases, therefore, the partial width difference between the RQM and SHO cases results from the different choices of meson space wave functions. Generally speaking, the different space wave functions would lead to different decay widths. Especially, if the overlap is near to the nodes of space wave functions, the decay width would strongly depend on the details of wave functions, and the small wave function difference could generate a large discrepancy of the decay width. However, for some modes, the possibility that the different wave functions can give the similar decay widths also exists. To our knowledge, there is no some rules to judge whether the RQM and SHO wave functions can give the similar or different results before the numerical calculations.
The difference between the predictions in case A and those in case B provides a chance to distinguish among different meson space wave functions. At present, we are unable to conclude which type of wave function is more reasonable due to the lack of the branching ratios for the $\pi(2360)$. However, as suggested by Ref. [@Lu:2016mbb], we should keep in mind that it is essential to treat the wave functions accurately in the $^3P_0$ model calculations.
[c|ccc]{} \*
---------
Channel
---------
: \[tab:2360\] Decay widths of the $\pi(2360)$ as the $\pi(5^1S_0)$ with two types of wave functions (in MeV). The initial state mass is set to 2360 MeV.
& \*
------
Mode
------
: \[tab:2360\] Decay widths of the $\pi(2360)$ as the $\pi(5^1S_0)$ with two types of wave functions (in MeV). The initial state mass is set to 2360 MeV.
&\
& &SHO &RQM\
$0^-\rightarrow 0^-0^+$ & $ \pi f_0(1370) $ & 1.31 & 44.22\
& $\eta a_0(1450) $ & 0.17 & 11.07\
& $K K^*_0(1430) $ & 0.08 & 7.03\
$0^-\rightarrow 0^-1^- $ & $\pi\rho$ & 1.67 & 1.67\
& $\pi\rho(1450)$ & 0.002 & 25.98\
& $\pi\rho(1700)$ & 2.35 & 2.83\
& $\pi(1300)\rho$ & 23.54 & 29.81\
& $K K^*$ & 0.16 & 5.11\
& $K K^*(1410)$ & 25.01 & 0.80\
& $K K^*(1680)$ & 2.10 & 0.0006\
& $K(1460) K^*$ & 0.77 & 0.01\
$0^-\rightarrow 1^-1^+ $ & $\rho a_1(1260)$ & 13.58 & 34.96\
& $\rho h_1(1170)$ & 10.64 & 17.45\
& $\omega b_1(1235) $ & 10.95 & 8.73\
& $K^* K_1(1270)$ & 8.49 & 0.08\
& $K^* K_1(1400)$ & 11.70 & 3.50\
$0^-\rightarrow 1^-1^- $ & $\rho \omega$ & 2.05 & 5.31\
& $\rho \omega(1420)$ & 37.15 & 5.29\
& $\omega\rho(1450)$ & 36.95 & 5.61\
& $K^* K^*$ & 0.40 & 4.75\
& $K^* K^*(1410)$ & 20.64 & 0.69\
$0^-\rightarrow 0^-2^+ $ & $\pi f_2(1270) $ & 1.61 & 24.08\
& $\eta a_2(1320) $ & 4.07 & 2.99\
& $\eta^\prime a_2(1320) $ & 0.93 & 0.47\
& $K K^*_2(1430) $ & 12.67 & 0.0004\
$0^-\rightarrow 1^-2^+ $ & $\rho a_2(1320)$ & 27.57 & 3.17\
& $K^* K^*_2(1430)$ & 0.32 & 0.06\
$0^-\rightarrow 0^-3^- $ &$\pi\rho_3(1690)$ & 21.09 & 21.49\
&$\pi\rho_3(1990)$ & 3.42 & 18.31\
&$K K^*_3(1780)$ & 0.05 & 0.07\
$0^-\rightarrow 0^-4^+ $ &$\pi f_4(2050) $ & 0.01 & 0.12\
& 281.46 & 285.65\
&\
![The dependence of the $\pi(5^1S_0)$ total width on the initial state mass in the $^3P_0$ model with two types of wave functions. The yellow band denotes the measured width for the $\pi(2360)$ [@Anisovich:2001pn; @Anisovich:2001pp].[]{data-label="Fig:2360"}](pi2360.eps)
$\eta(2320)$ and $X(2500)$ {#sec:eta2320}
---------------------------
The $\eta(2320)$ was observed in $\bar{p}p\rightarrow \eta\eta\eta$ process, and its mass and width are $2320\pm 15$ MeV and $230\pm 35$ MeV [@Anisovich:2000ix]. The predicted $\eta(5^1S_0)$ mass in the relativistic quark model is about $2385$ MeV [@Ebert:2009ub], close to the $\eta(2320)$ mass. In the presence of the $X(2500)$ as the isoscalar member of the $5^1S_0$ meson nonet [@Pan:2016bac], we shall discuss the possibility of the $\eta(2320)$ as the isoscalar partner of the $X(2500)$.
In a meson nonet, the two physical isoscalar states can mix. The mixing of the two isoscalar states can be parametrized as $$\begin{aligned}
&&\eta(5^1S_0)=\cos\phi~ n\bar{n}-\sin\phi~ s\bar{s},\\
&&X(2500)=\sin\phi~ n\bar{n}+\cos\phi~ s\bar{s},\end{aligned}$$ where $n\bar{n}=(u\bar{u}+d\bar{d})/\sqrt{2}$ and $s\bar{s}$ are the pure $5\,^1S_0$ nonstrange and strange states, respectively, and the $\phi$ is the mixing angle. Accordingly, the partial widths for the $\eta(5^1S_0)$ and $X(2500)$ can be expressed as $$\begin{aligned}
\Gamma(\eta(5^1S_0)\rightarrow
BC)&=&\frac{\pi~P}{4M^2_{\eta(5^1S_0)}}\sum_{LS}|\cos\phi
{\cal{M}}^{LS}_{n\bar{n}\rightarrow BC}\nonumber \\ && -\sin\phi
{\cal{M}}^{LS}_{s\bar{s}\rightarrow BC}|^2,
\label{w1}\\
\Gamma(X(2500)\rightarrow
BC)&=&\frac{\pi~P}{4M^2_{X(2500)}}\sum_{LS}|\sin\phi
{\cal{M}}^{LS}_{n\bar{n}\rightarrow BC} \nonumber \\ &&
+\cos\phi
{\cal{M}}^{LS}_{s\bar{s}\rightarrow BC}|^2. \label{w2}\end{aligned}$$
[c|ccc]{} \*
---------
Channel
---------
& \*
------
Mode
------
&$\eta(2320)$ & $X(2500)$\
& &$\Gamma_i$ &$\Gamma_i$\
$0^-\rightarrow 0^-0^+$ & $\pi a_0(1450) $ & $3.69 c^2 $ & $3.84 s^2 $\
& $\eta f_0(1370) $ & $0.40 c^2 $ & $0.79 s^2 $\
& $\eta^\prime f_0(1370)$ & $-$ & $0.22 s^2 $\
& $\eta f_0(1710) $ & $2.48 s^2 $ & $0.20 c^2 $\
&$K K^*_0(1430) $ & $0.0006 c^2 -0.07cs+2.25s^2 $ & $1.17 c^2 +1.98cs+0.84s^2 $\
$0^-\rightarrow 0^-1^- $ & $K K^*$ & $0.06 c^2 +0.35cs+0.51s^2 $ & $0.06 c^2 +0.39cs+0.65s^2 $\
& $K K^*(1680)$ & $1.51 c^2 +5.40cs+4.82s^2 $ & $2.29 c^2 -5.11cs+2.85s^2 $\
& $K K^*(1410)$ & $27.47 c^2 -3.84cs+0.13s^2 $ & $13.83 c^2 +29.02cs+15.23s^2 $\
& $K(1460) K^*$ & $-$ & $24.94 c^2 -54.68cs+25.40s^2 $\
& $K K^*(1830)$ & $-$ & $72.21 c^2 -63.56cs+135.49s^2 $\
$0^-\rightarrow 0^-2^+ $ & $\pi a_2(1320) $ & $10.99 c^2 $ & $0.85 s^2 $\
& $\eta f_2(1270) $ & $4.05 c^2 $ & $1.60s^2 $\
& $\eta^\prime f_2(1270) $ & $0.96 c^2 $ & $4.86 s^2 $\
& $\eta f^\prime_2(1525) $ & $6.53 s^2 $ & $9.61c^2 $\
& $K K^*_2(1430) $ & $10.90 c^2 -18.89cs+8.19s^2 $ & $0.43 c^2 +5.34cs+16.51s^2$\
$0^-\rightarrow 0^-3^-$ &$K K^*_3(1780)$ & $0.007 c^2 +0.05cs+0.10s^2 $ & $9.25 c^2 -5.95cs+0.96s^2 $\
$0^-\rightarrow 0^-4^+ $ & $\pi a_4(2040) $ & $0.02 c^2$ & $0.83 s^2 $\
$0^-\rightarrow 1^-1^- $ & $\rho \rho$ & $2.63 c^2 $ & $3.78 s^2 $\
& $\rho \rho(1450)$ & $84.95 c^2 $ & $100.69 s^2 $\
& $\omega \omega$ & $0.83 c^2 $ & $1.26 s^2 $\
& $\omega \omega(1420)$ & $24.43 c^2 $ & $35.69s^2 $\
& $\phi\phi$ & $1.16 s^2 $ & $0.01c^2 $\
& $K^* K^*$ & $0.63 c^2 +1.97cs+0.54s^2 $ & $1.37 c^2 -0.35cs+0.02s^2 $\
& $K^* K^*(1410)$ & $3.30c^2 -11.29cs+9.65s^2 $ & $42.44c^2 +104.60cs+64.44s^2 $\
$0^-\rightarrow 1^-1^+ $ & $\rho b_1(1235)$ & $31.05 c^2 $ & $28.29 s^2 $\
&$\omega h_1(1170)$ & $10.87 c^2 $ & $7.57 s^2 $\
& $K^* K_1(1270)$ & $5.34 c^2 -6.82cs+2.22s^2$ & $8.76 c^2 +11.89cs+18.55s^2 $\
& $K^* K_1(1400)$ & $10.88 c^2 +9.29cs+2.51s^2 $ & $18.13 c^2 +0.85cs+4.14s^2 $\
$0^-\rightarrow 1^-2^+ $ &$K^* K^*_2(1430)$ & $0.0004c^2 +0.002cs+0.003s^2$ & $18.59 c^2 -20.81cs+5.83s^2 $\
& $234.93 c^2 -10.07cs+42.09s^2 $ & $227.34 c^2 +3.70cs+481.18s^2 $\
& $230\pm 35$ [@Anisovich:2000ix] & $230^{+64+56}_{-35-33}$ [@Ablikim:2016hlu]\
[c|ccc]{} \*
---------
Channel
---------
& \*
------
Mode
------
&$\eta(2320)$ & $X(2500)$\
& &$\Gamma_i$ &$\Gamma_i$\
$0^-\rightarrow 0^-0^+$ & $\pi a_0(1450) $ & $93.36 c^2 $ & $141.51 s^2 $\
& $\eta f_0(1370) $ & $13.30 c^2 $ & $21.64 s^2 $\
& $\eta^\prime f_0(1370) $ & $-$ & $8.02 s^2 $\
& $\eta f_0(1710) $ & $1.46s^2 $ & $3.42 c^2 $\
& $K K^*_0(1430) $ & $6.33c^2 -34.96cs+48.24s^2 $ & $84.74 c^2 +55.93cs+9.23s^2 $\
$0^-\rightarrow 0^-1^- $ & $K K^*$ & $4.81 c^2 -4.59cs+1.10s^2$ & $0.97 c^2 +4.78cs+5.85s^2 $\
& $K K^*(1680)$ & $0.003 c^2 +0.01cs+0.02s^2 $ & $4.81 c^2 -0.58cs+0.02s^2 $\
& $K K^*(1410)$ & $0.49 c^2 +6.77cs+23.54s^2 $ & $10.53 c^2 -9.13cs+1.98s^2 $\
& $K(1460) K^*$ & $-$ & $0.08 c^2 +0.02cs+0.001s^2 $\
& $K K^*(1830)$ & $-$ & $38.40 c^2 -32.26cs+6.78s^2$\
$0^-\rightarrow 0^-2^+ $ & $\pi a_2(1320) $ & $50.99 c^2 $ & $88.50 s^2 $\
& $\eta f_2(1270)$ & $2.88 c^2 $ & $10.95s^2 $\
& $\eta^\prime f_2(1270) $ & $0.47c^2 $ & $0.04 s^2 $\
& $\eta f^\prime_2(1525) $ & $0.01 s^2 $ & $0.12c^2 $\
& $K K^*_2(1430) $ & $0.01 c^2 +0.56cs+6.22s^2 $ & $24.99 c^2 +3.60cs+0.13s^2 $\
$0^-\rightarrow 0^-3^- $ & $K K^*_3(1780)$ & $0.004 c^2 +0.10cs+0.55s^2 $ & $15.06 c^2 -4.27cs+0.30s^2 $\
$0^-\rightarrow 0^-4^+ $ & $\pi a_4(2040) $ & $0.22 c^2 $ & $6.69 s^2 $\
$0^-\rightarrow 1^-1^- $ & $\rho \rho$ & $11.20 c^2 $ & $0.67 s^2 $\
& $\rho \rho(1450)$ & $29.40 c^2 $ & $32.06 s^2 $\
& $\omega \omega$ & $4.07 c^2 $ & $0.33 s^2 $\
& $\omega \omega(1420)$ & $10.48 c^2 $ & $7.14s^2 $\
& $\phi\phi$ & $1.10 s^2 $ & $1.71c^2 $\
& $K^* K^*$ & $4.25 c^2 -1.47cs+0.13s^2 $ & $3.23 c^2 -8.78cs+5.96s^2 $\
& $K^* K^*(1410)$ & $0.16 c^2 -2.28cs+7.86s^2 $ & $8.59 c^2 -2.35cs+0.16s^2 $\
$0^-\rightarrow 1^-1^+ $ & $\rho b_1(1235)$ & $17.12 c^2 $ & $96.26 s^2 $\
& $\omega h_1(1170)$ & $10.12 c^2 $ & $45.91 s^2 $\
& $K^* K_1(1270)$ & $0.09 c^2 -2.07cs+12.55s^2 $ & $53.22 c^2 +0.48cs+0.001s^2 $\
& $K^* K_1(1400)$ & $2.54 c^2 -2.24cs+0.54s^2 $ & $0.56 c^2 +3.12cs+4.41s^2 $\
$0^-\rightarrow 1^-2^+ $ &$K^* K^*_2(1430)$ & $0.00007 c^2 +0.001cs+0.007s^2 $ & $7.26 c^2 -4.08cs+0.47s^2 $\
&$262.31 c^2 -40.16cs+103.32s^2 $ & $257.71 c^2 +6.48cs+495.09s^2 $\
& $230\pm 35$ [@Anisovich:2000ix] & $230^{+64+56}_{-35-33}$ [@Ablikim:2016hlu]\
![The dependence of the $\eta(2320)$ and $X(2500)$ total widths on the $\phi$ in the $^3P_0$ model with two types of wave functions: (a) with the SHO wave functions (b) with the RQM wave functions. The blue and green bands denote the measured widths for the $X(2500)$ and $\eta(2320)$, respectively [@Ablikim:2016hlu; @Anisovich:2000ix]. []{data-label="Fig:eta2320"}](eta23201.eps "fig:") ![The dependence of the $\eta(2320)$ and $X(2500)$ total widths on the $\phi$ in the $^3P_0$ model with two types of wave functions: (a) with the SHO wave functions (b) with the RQM wave functions. The blue and green bands denote the measured widths for the $X(2500)$ and $\eta(2320)$, respectively [@Ablikim:2016hlu; @Anisovich:2000ix]. []{data-label="Fig:eta2320"}](eta23202.eps "fig:")
Under the mixing of $\eta(2320)$ and $X(2500)$, their decays in the case A are listed in Table \[tab:mixing1\] and those in the case B are listed in Table \[tab:mixing2\]. The dependence of the $\eta(2320)$ and $X(2500)$ total widths on the mixing angle $\phi$ is displayed in Fig. \[Fig:eta2320\]. In order to simultaneously reproduce the measured widths for the $\eta(2320)$ and $X(2500)$, the mixing angle $\phi$ is required to satisfy $-0.5\leq \phi \leq 0.45$ radians in case A or $-0.69\leq \phi \leq 0.59$ radians in case B. Below, we shall estimate the value of $\phi$ to check whether it satisfies these constraints based on the mass-squared describing the mixing of two isoscalar mesons.
In the $n\bar{n}$ and $s\bar{s}$ bases, the mass-squared matrix describing the $\eta(2320)$ and $X(2500)$ mixing can be expressed as [@Li:2008mza; @Li:2008et; @Li:2001qg]
$$\begin{aligned}
M^2=\left(\begin{array}{cc}
M^2_{n\bar{n}}+2A_m&\sqrt{2}A_mX\\
\sqrt{2}A_mX&M_{s\bar{s}}^2+A_mX^2
\end{array}\right),
\label{matrix}\end{aligned}$$
where $M_{n\bar{n}}$ and $M_{s\bar{s}}$ are the masses of the pure $5^1S_0$ $n\bar{n}$ and $s\bar{s}$, respectively, $A_m$ denotes the total annihilation strength of the $q\bar{q}$ pair for the light flavors $u$ and $d$, $X$ describes the SU(3)-breaking ratio of the nonstrange and strange quark masses via the constituent quark mass ratio $m_u/m_s$. Since the $n\bar{n}$ is the orthogonal partner of the $\pi(5^1S_0)$, one can expect that $n\bar{n}$ degenerates with $\pi(5\,^1S_0)$ in effective quark masses. Here we take $M_{n\bar{n}}=M_{\pi(5^1S_0)}=M_{\pi(2360)}$. The $M_{s\bar{s}}$ can be obtained from the Gell-Mann-Okubo mass formula $M^2_{s\bar{s}}=2M^2_{ K(5^1S_0)}-M^2_{n\bar{n}}$.
The masses of the two physical states $\eta(2320)$ and $X(2500)$ can be related to the matrix $M^2$ by the unitary matrix $$\begin{aligned}
U=\left(
\begin{array}{cc}
\cos\phi&-\sin\phi\\
\sin\phi&\cos\phi
\end{array}\right),\end{aligned}$$ which satisfies $$\begin{aligned}
U M^2 U^\dagger=\left(\begin{array}{cc}
M^2_{\eta(2320)}&0\\
0&M^2_{X(2500)}\end{array}\right). \label{diag}\end{aligned}$$
From Eqs. (\[matrix\]) and (\[diag\]), one can have $$\begin{aligned}
&& 8X^2(M^2_{K(5^1S_0)}-M^2_{\pi(2360)})^2 \nonumber \\
&=&\left[4M^2_{K(5^1S_0)}-(2-X^2)M^2_{\pi(2360)}-(2+X^2)M^2_{\eta(2320)}\right]\nonumber\\
&\times&\left[(2-X^2)M^2_{\pi(2360)}+(2+X^2)M^2_{X(2500)}-4M^2_{K(5^1S_0)}\right], \nonumber \\
\label{msch}\end{aligned}$$ and $$\begin{aligned}
A_m&=&(M^2_{X(2500)}-2M^2_{K(5^1S_0)}+M^2_{\pi(2360)})(M^2_{\eta(2320)}-2M^2_{K(5^1S_0)} \nonumber \\ && +M^2_{\pi(2360)})/\left[2(M^2_{\pi(2360)}-M^2_{K(5^1S_0)})X^2\right].
\label{am}\end{aligned}$$ Equation (\[msch\]) is the generalized Schwinger’s nonet mass formula [@Li:2001qg]. If the SU(3)-breaking effect is not considered, i.e., $X=1$, Eq. (\[msch\]) can be reduced to original Schwinger’s nonet mass formula [@Schwinger:1964zza]. With the masses of the $\pi(2360)$, $\eta(2320)$, and $X(2500)$, from Eqs. (\[msch\]) and (\[am\]), we have $$\begin{aligned}
M_{K(5\,^1S_0)}=2.418~\mbox{GeV}, A_m=-0.085~ {\mbox{GeV}}^2
\label{eq: Kmass1}\end{aligned}$$ for $X=m_u/m_s=330/550$ as used in case A, and $$\begin{aligned}
M_{K(5\,^1S_0)}=2.418~ \mbox{GeV}, A_m=-0.111~ {\mbox{GeV}}^2
\label{eq:Kmass2}\end{aligned}$$ for $X=m_u/m_s=220/419$ as used in case B.
Then the unitary matrix $U$ can be given by $$\begin{aligned}
U=\left(\begin{array}{cc}
\cos\phi&-\sin\phi\\
\sin\phi&\cos\phi\end{array}\right)=\left(\begin{array}{cc}
+0.995&+0.102\\
-0.102&+0.995
\end{array}\right)
\label{mixangle1}\end{aligned}$$ for $X=330/550$, and $$\begin{aligned}
U=\left(\begin{array}{cc}
\cos\phi&-\sin\phi\\
\sin\phi&\cos\phi\end{array}\right)=\left(\begin{array}{cc}
+0.994&+0.109\\
-0.109&+0.994
\end{array}\right)
\label{mixangle2}\end{aligned}$$ for $X=220/419$.
Equations (\[mixangle1\]) and (\[mixangle2\]) consistently give $\phi=-0.1$ radians, which makes both the $\eta(2320)$ and $X(2500)$ widths in agreement with experimental data. Also, both Eqs. (\[mixangle1\]) and (\[mixangle2\]) indicate that the $\eta(2320)$ is mainly the $n\bar{n}$, consistent with the $\pi(2360)$ nearly degenerating with the $\eta(2320)$, while the $X(2500)$ is mainly the $s\bar{s}$, consistent with our previous analysis [@Pan:2016bac]. Therefore, the $\eta(2320)$ and $X(2500)$, together with the $\pi(2360)$, appear to be the convincing $5\,^1S_0$ states.
In above discussions, we focus on the possibility of the pseudoscalar states $\pi(2360)$, $\eta(2320)$, and $X(2500)$ as the $5^1S_0$ mesons. Apart from the states listed in Table \[tab:pseu\], the $X(2120)$ and $X(2370)$ also probably are the $J^{PC}=0^{-+}$ resonances. The $X(2120)$ and $X(2370)$ were observed by the BESIII collaboration in the $\pi^+\pi^-\eta^\prime$ invariant mass spectrum and their spin parities are not determined [@Ablikim:2010au]. Based on the observed decay mode $\pi^+\pi^-\eta^\prime$, the possible $J^{PC}$ for the $X(2120)$ and $X(2370)$ are $0^{-+}$, $1^{++}$, $\cdots$. The natures of the $X(2120)$ and $X(2370)$ are not clear [@Wang:2017iai; @Yu:2011ta; @Liu:2010tr; @Chen:2011kp; @Wang:2010vz]. Since the $X(2370)$ mass is also close to the quark model expectation for the $\eta(5^1S_0)$ mass [@Ebert:2009ub], we shall discuss the possibility of the $X(2370)$ as the isoscalar partner of the $X(2500)$.
With the $X(2370)$-$X(2500)$ mixing, the decay widths for the $X(2370)$ are listed in Table \[tab:2370\]. The dependence of the $X(2370)$ and $X(2500)$ total widths on the mixing angle is plotted in Fig. \[Fig:eta2370\]. Obviously, the $X(2370)$ width can not be reproduced in the whole region of the mixing angle. Therefore, our calculations do not support the $5^1S_0$ assignment for the $X(2370)$. Other calculations from the $^3P_0$ model suggest that the $X(2370)$ is unlikely to be the $4^1S_0$ $q\bar{q}$ state [@Liu:2010tr; @Chen:2011kp]. If the $X(2370)$ turns out to have $J^{PC}=0^{-+}$ in future, in order to explain its properties, more complicate scheme such as the $q\bar{q}$-glueball mixing may be necessary, since the $X(2370)$ mass is close to the pseudoscalar glueball mass of about $2.3-2.6$ GeV predicted by the lattice QCD [@Morningstar:1999rf; @Hart:2001fp; @Chen:2005mg].
[c|ccc]{} \*
---------
Channel
---------
& \*
------
Mode
------
&\
& &SHO &RQM\
$0^-\rightarrow 0^-0^+$ & $\pi a_0(1450) $ & $4.16 c^2 $ & $110.41 c^2 $\
& $\eta f_0(1370) $ & $0.62 c^2 $ & $16.15 c^2 $\
& $\eta^\prime f_0(1370) $ & $1.52 c^2 $ &$4.73 c^2 $\
& $\eta f_0(1710) $ & $1.63s^2 $ & $2.22 s^2 $\
& $K K^*_0(1430) $ & $0.15 c^2 -1.14cs+2.19s^2 $ & $7.33 c^2 -42.31cs+61.07s^2 $\
$0^-\rightarrow 0^-1^- $ & $K K^*$ & $0.21 c^2 +0.36cs+0.16s^2 $ & $5.22 c^2 -5.22cs+1.30s^2 $\
& $K K^*(1680)$ & $2.29 c^2 +6.74cs+4.94s^2 $ & $0.00006 c^2 -0.009cs+0.36s^2 $\
& $K K^*(1410)$ & $23.78 c^2 -17.56cs+3.24s^2 $ & $0.95 c^2 +9.21cs+22.38s^2 $\
& $K(1460) K^*$ & $3.67 c^2 +12.09cs+9.95s^2 $ & $0.002 c^2 +0.65cs+3.11s^2 $\
& $K K^*(1830)$ & $16.65 c^2 +48.04cs+34.66s^2 $ & $2.59 c^2 +21.80cs+45.82s^2 $\
$0^-\rightarrow 0^-2^+ $ & $\pi a_2(1320) $ & $6.35 c^2 $ & $67.47 c^2 $\
& $\eta f_2(1270) $ & $3.26 c^2 $ & $5.52c^2 $\
& $\eta^\prime f_2(1270) $ & $2.47 c^2 $ & $0.59 c^2 $\
& $\eta f^\prime_2(1525) $ & $8.40 s^2 $ & $0.0002s^2 $\
& $K K^*_2(1430) $ & $13.36 c^2 -15.98cs+4.78s^2 $ & $0.0007 c^2 -0.19cs+13.13s^2 $\
$0^-\rightarrow 0^-3^- $ & $K K^*_3(1780)$ & $0.10 c^2 +0.71cs+1.30s^2 $ & $0.04 c^2 +0.81cs+3.97s^2 $\
$0^-\rightarrow 0^-4^+ $ & $\pi a_4(2040) $ & $0.11 c^2 $ & $1.02 c^2 $\
$0^-\rightarrow 1^-1^- $ & $\rho \rho$ & $3.29 c^2 $ & $6.14 c^2 $\
& $\rho \rho(1450)$ & $119.77 c^2 $ & $6.80 c^2 $\
& $\omega \omega$ & $1.07 c^2 $ & $2.33 c^2 $\
& $\omega \omega(1420)$ & $38.85 c^2 $ & $3.87c^2 $\
& $\phi\phi$ & $0.43 s^2 $ & $1.41s^2 $\
& $K^* K^*$ & $0.32 c^2 +1.49cs+1.74s^2 $ & $4.94 c^2 +1.86cs+0.18s^2 $\
& $K^* K^*(1410)$ & $28.51 c^2 -78.14cs+53.54s^2 $ & $0.79 c^2 -6.84cs+14.88s^2 $\
$0^-\rightarrow 1^-1^+ $ & $\rho b_1(1235)$ & $33.12 c^2 $ & $34.57 c^2 $\
& $\omega h_1(1170)$ & $10.51 c^2 $ & $19.70 c^2 $\
& $K^* K_1(1270)$ & $9.91 c^2 -10.15cs+5.03s^2 $ & $0.07 c^2 -2.30cs+18.79s^2 $\
& $K^* K_1(1400)$ & $10.91 c^2 +5.54cs+7.06s^2 $ & $3.71 c^2 -6.82cs+0.01s^2 $\
$0^-\rightarrow 1^-2^+ $ &$K^* K^*_2(1430)$ & $0.69 c^2 +3.30cs+3.98s^2 $ & $0.11 c^2 +1.69cs+6.39s^2 $\
& $335.65 c^2 -44.70cs+143.03s^2 $ & $305.01 c^2 -27.66cs+195.00s^2 $\
&\
![The total widths of the $\eta(2320)$ and $X(2500)$ dependence on the $\phi$ in the $^3P_0$ model with two types of wave functions: (a) with the SHO wave functions (b) with the RQM wave functions. The blue and green band denote the measured widths for the $X(2500)$ and $X(2370)$, respectively [@Ablikim:2016hlu; @Ablikim:2010au].[]{data-label="Fig:eta2370"}](x23701.eps "fig:") ![The total widths of the $\eta(2320)$ and $X(2500)$ dependence on the $\phi$ in the $^3P_0$ model with two types of wave functions: (a) with the SHO wave functions (b) with the RQM wave functions. The blue and green band denote the measured widths for the $X(2500)$ and $X(2370)$, respectively [@Ablikim:2016hlu; @Ablikim:2010au].[]{data-label="Fig:eta2370"}](x23702.eps "fig:")
$K(5^1S_0)$ {#subsec:dvector}
-----------
[c|ccc]{} \*
---------
Channel
---------
: \[tab:2420\]Decay widths of the $K(2418)$ as the $5^1S_0$ state in the $^3P_0$ model with two types of wave functions (in MeV). The initial state mass is set to 2418 MeV.
& \*
------
Mode
------
: \[tab:2420\]Decay widths of the $K(2418)$ as the $5^1S_0$ state in the $^3P_0$ model with two types of wave functions (in MeV). The initial state mass is set to 2418 MeV.
&\
& &SHO &RQM\
$0^-\rightarrow 0^-0^+$ & $\pi K^*_0(1430) $ & 0.36 & 32.62\
& $K a_0(1450) $ & 0.96 & 19.08\
& $\eta K^*_0(1430) $ & 0.03 & 0.36\
& $\eta^\prime K^*_0(1430) $ & 2.12 & 4.71\
& $K f_0(1370)$ & 0.31 & 8.48\
& $K f_0(1710)$ & 0.33 & 1.64\
$0^-\rightarrow 0^-1^- $ & $\pi K^* $ & 0.07 & 0.05\
& $K \rho $ & 0.14 & 0.10\
& $\pi K^*(1680) $ & 0.05 & 2.67\
& $K \rho(1700)$ & 1.84 & 0.35\
& $\pi K^*(1410) $ & 5.75 & 0.10\
& $K \rho(1450)$ & 0.06 & 8.50\
& $\pi(1300)K^*$ & 8.11 & 7.11\
& $K(1460) \rho$ & 9.69 & 2.14\
& $\eta K^* $ & 0.42 & 0.001\
& $\eta^\prime K^* $ & 0.02 & 0.10\
& $\eta K^*(1410) $ & 0.03 & 9.51\
& $\eta^\prime K^*(1410) $ & 0.21 & 0.04\
& $\eta K^*(1680) $ & 2.00 & 0.03\
& $K \phi$ & 0.36 & 0.93\
& $K \phi(1680)$ & 10.41 & 0.04\
& $\eta(1475)K^*$ & 5.17 & 0.03\
& $K \omega$ & 0.05 & 0.03\
& $K \omega(1420)$ & 0.03 & 2.81\
& $\eta(1295)K^*$ & 2.54 & 2.61\
& $K \omega(1650)$ & 0.49 & 0.32\
& $K(1460) \omega$ & 3.39 & 0.58\
$0^-\rightarrow 0^-2^+ $ & $\pi K^*_2(1430) $ & 0.19 & 13.75\
& $K a_2(1320)$ & 1.04 & 7.24\
& $\eta K^*_2(1430) $ & 0.13 & 0.04\
& $\eta^\prime K^*_2(1430) $ & 0.27 & 0.23\
& $K f_2^\prime(1525)$ & 5.74 & 0.05\
& $K f_2(1270)$ & 0.15 & 2.86\
$0^-\rightarrow 0^-3^- $ & $\pi K^*_3(1780) $ & 9.77 & 6.14\
& $K \rho_3(1690)$ & 3.18 & 3.66\
& $\eta K^*_3(1780) $ & 0.28 & 0.91\
& $K \phi_3(1850)$ & 0.03 & 0.007\
& $K \omega_3(1670)$ & 1.36 & 1.42\
$0^-\rightarrow 0^-4^+ $ & $\pi K^*_4(2045) $ & 0.07 & 0.79\
$0^-\rightarrow 1^-1^- $ & $K^* \rho$ & 1.47 &0.16\
& $K^*(1410) \rho$ & 12.60 & 13.47\
& $K^* \rho(1450)$ & 19.21 & 6.47\
& $K^* \phi$ & 0.03 & 1.93\
& $K^* \omega$ & 0.49 & 0.04\
& $K^*(1410) \omega$ & 4.55 & 3.91\
& $K^* \omega(1420)$ & 5.94 & 2.21\
$0^-\rightarrow 1^-1^+ $ &$K^* b_1(1235)$ & 6.66 & 4.15\
&$K^* a_1(1260)$ & 4.35 & 8.46\
&$\rho K_1(1270)$ & 3.47 & 24.99\
&$\rho K_1(1400)$ & 8.29 & 0.19\
&$\phi K_1(1270)$ & 2.45 & 0.06\
&$K^* h_1(1380)$ & 2.40 & 0.63\
&$K^* f_1(1420)$ & 2.69 & 1.20\
&$\omega K_1(1270)$ & 1.16 & 7.95\
&$\omega K_1(1400)$ & 2.74 & 0.03\
&$K^* h_1(1170)$ & 2.01 & 2.92\
&$K^* f_1(1285)$ & 1.21 & 1.73\
$0^-\rightarrow 1^-2^+ $ &$K^* a_2(1320)$ & 8.40 & 1.71\
&$\rho K^*_2(1430) $ & 8.92 & 0.44\
&$\omega K^*_2(1430) $ & 2.92 & 0.23\
&$K^* f^\prime_2(1525)$ & 0.00006 & 0.000008\
&$K^* f_2(1270)$ & 3.05 & 0.07\
& 163.38 & 224.98\
As mentioned in Sec. \[sec:eta2320\], with the $\pi(2360)$, $\eta(2320)$, and $X(2500)$ as the members of $5^1S_0$ meson nonet, from Eq. (\[msch\]), the $K(5^1S_0)$ mass is predicted to be about 2418 MeV as shown in Eqs. (\[eq: Kmass1\]) and (\[eq:Kmass2\]). At present, no candidate for the $I(J^P)=1/2(0^{-})$ state around 2418 MeV is reported experimentally. It is noted that with our estimated masses for the $K(4^1S_0)$ and $K(5^1S_0)$, $M_{K(4^1S_0)}=2153\pm 20$ MeV [@Li:2008et] and $M_{K(5^1S_0)}=2418\pm 49$ MeV, the $K$, $K(1460)$, $K(1830)$, $K(2153)$, and $K(2418)$ approximately populate a trajectory as shown in Fig. \[fig:regge\_kaon\], which indicates that the $K(2153)$ and $K(2418)$ could be the good candidates for the $4^1S_0$ and $5^1S_0$ kaons, respectively.
![The $K$-trajectory with $M^2_n=M^2_0+(n-1)\mu^2$. $M^2_0=0.24669\pm 0.00002 \mbox{GeV}^2$, $\mu^2=1.456\pm 0.026\mbox{GeV}^2$, $\chi^2/\mbox{d.o.f}=1.897/(4-2)$. In our fit, the we don’t use the data of $K(1460)$ since the $K(1460)$ mass error is not given experimentally. The masses of the $K$ and $K(1830)$ are taken from Ref. [@PDG2016]. The masses of the $K(2153)$ and $K(2418)$ are taken to be $2153\pm 20$ MeV and $2418\pm 49$ MeV, respectively. The $K(1460)$ mass is taken to be 1460 MeV [@Daum:1981hb].[]{data-label="fig:regge_kaon"}](reggekaon.eps)
The decay widths of the $K(2418)$ as the $5^1S_0$ kaon are listed in Table \[tab:2420\]. The total width of the $K(5^1S_0)$ is predicated to be about 163 MeV in case A or 225 MeV in case B. This could be of use in looking for the candidate for the $5^1S_0$ kaon experimentally.
SUMMARY AND CONCLUSION {#sec:summary}
======================
In this work, we have discussed the possible members of the $5^1S_0$ meson nonet by analysing the masses and calculating the strong decay widths in the $^3P_0$ model with the SHO and RQM meson space wave functions. Both the mass and width for the $\pi(2360)$ are consistent with the quark model expectations for the $\pi(5^1S_0)$. In the presence of the $X(2500)$ as the $5^1S_0$ isoscalar state, the possibility of the $\eta(2320)$ and $X(2370)$ as the isoscalar partner of the $X(2500)$ is discussed. The $X(2370)$ width can not be reproduced for any value of the mixing angle $\phi$, thus, the assignment of the $X(2370)$ as the $5^1S_0$ isoscalar state is not favored by its width. Both the $\eta(2320)$ and $X(2500)$ widths can be reproduced with $-0.5\leq \phi \leq 0.45$ radians for the SHO wave functions or $-0.69\leq \phi \leq 0.59$ for the RQM wave functions. The assignment of the $\pi(2360)$, $\eta(2320)$, and $X(2500)$ as the members of the $5^1S_0$ nonet not only gives $\phi=-0.1$ radians, which naturally accounts for the $\eta(2320)$ and $X(2500)$ widths, but also shows that the $5^1S_0$ kaon has a mass of about 2418 MeV. The $K$, $K(1460)$, $K(1830)$, $K(2153)$, and $K(2418)$ approximately populate a common trajectory. The $K(2418)$ is predicted to have a width of about 163 MeV for SHO wave functions or 225 MeV for the RQM wave functions. We tend to conclude that the $\pi(2360)$, $\eta(2320)$, $X(2500)$, together with the unobserved $K(2418)$, construct the $5^1S_0$ meson nonet.
Our numerical results show that the $^3P_0$ model predictions depend on the choice of meson space wave functions. It is essential to treat the wave functions accurately in the $^3P_0$ model calculations. The difference between the predictions in SHO case and those in RQM case provides a chance to distinguish among different meson space wave functions. To conclude which type of wave function is preferable, the further experimental study on the decays of $\pi(2360)$, $\eta(2320)$, and $X(2500)$ is needed. Also, in our calculations, all the states are assumed to be $q\bar{q}$. It is noted that some resonances such as $h_1(1170)$, $h_1(1380)$, $f_1(1285)$, $b_1(1235)$, $a_1(1260)$, and $K_1(1270)$, can also be explained as the dynamically generated resonances [@Roca:2005nm; @Geng:2006yb; @GarciaRecio:2010ki], which means they might have large hadron-molecular components in their wave functions. If so, both the SHO and RQM wave functions derived from the simple $q\bar{q}$ picture, would be not appropriate and could lead to the big discrepancies between the experiments and the $^3P_0$ model predictions. To test this point, the further experimental information about the partial widths is also needed.
[**ACKNOWLEDGEMENTS**]{}\
We would like to thank Dr. Qi-Fang Lü for valuable discussions. This work is partly supported by the National Natural Science Foundation of China under Grant Nos. 11505158, 11605158, the China Postdoctoral Science Foundation under Grant No.2015M582197, the Postdoctoral Research Sponsorship in Henan Province under Grant No.2015023, and the Academic Improvement Project of Zhengzhou University.
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---
abstract: 'I. Caprini’s, G. Colangelo’s, and H. Leutwyler’s (CCL) article “Mass and Width of the Lowest Resonance in QCD”, Phys. Rev. Lett. 96, 132001 (2006) \[arXiv:hep-ph/0512364\], is critically reviewed. The present comment is devoted to complement a recent experimental discussion (D.V. Bugg, J. Phys. G 34, 151 (2007) \[arXiv:hep-ph/0608081\]) of short-comings in the CCL analysis, by presenting theoretical arguments pointing at a serious flaw in the theoretical formalism used by CCL, and also at the unlikeliness of their tiny error bars in the $\sigma$-meson mass and width. The criticism made in the comment applies analogously to the analysis on the $\kappa$-meson mass performed in the article “The K0\*(800) scalar resonance from Roy-Steiner representations of pi K scattering” published as S. Descotes-Genon and B. Moussallam, Eur. Phys. J. C 48, 553 (2006) \[arXiv:hep-ph/0607133\].'
author:
- Frieder Kleefeld
title: Comment on Mass and Width of the Lowest Resonance in QCD
---
[**Comment on Mass and Width of the Lowest Resonance in QCD**]{}\
In a recent Letter [@Caprini:2005zr], in which *no* use is made of QCD, I. Caprini, G. Colangelo, and H. Leutwyler (CCL) repeated an unmentioned analysis of $\pi\pi$ scattering from 1973 [@Pennington:1973xv], based on the Roy equations (REs), to make out a case for the existence of a scalar $I=0$ resonance $f_0(441)$, listed in the PDG tables [@Yao:2006px] as $f_0(600)$ and known as $\sigma$-meson. The primary aspect resulting of the CCL analysis is the claimed [*model-*]{} and [*parametrization-independent*]{} determination of a $\sigma$-pole mass of $(441^{+16}_{-8}-\frac{1}{2}\,i\,544^{+18}_{-25})$ MeV implying unprecedented small error bars. Moreover, the latter result is incompatible with very recent experimental findings, i.e., $(500\pm 30-\,i\,(264\pm 30))$ MeV [@Bugg:2006gc] and $(541\pm 39-\,i\,(252\pm 42))$ MeV [@Ablikim:2004qn; @Bugg:2005xz], as well as with a combined theoretical analysis yielding $((476$–$628)-\,i\,(226$–$346))$ MeV [@vanBeveren:2006ua]. The present comment will be devoted to complement a recent experimental discussion [@Bugg:2006gc] of short-comings in the CCL analysis, by presenting theoretical arguments pointing at a serious flaw in the theoretical formalism used by CCL, and also at the unlikeliness of their tiny error bars in the $\sigma$ mass and width. The simplest way to identify this flaw in Ref. [@Caprini:2005zr] also present in the corresponding results [@Descotes-Genon:2006uk] of S. Descotes-Genon and B. Moussallam (DM) on the scalar meson $K^\ast_0(800)$ in the context of Roy-Steiner equations (RSEs), is to recall a warning statement by G.F. Chew and S. Mandelstam (CM) from 1960 (see footnote 6 of Ref. [@Chew:1960iv]). CM state that if a strongly interacting particle with the same quantum numbers as a pair of pions should be found, then corresponding poles must be added to the double-dispersion representation, whether or not the new particle is interpreted as a two-pion bound state. It should be emphasized that this statement does not only apply to possible bound-state (BS) poles of the S- or T-matrix in the physical sheet (PS) of the complex $s$-plane, but also to any kind of virtual BS poles and resonance poles in the unphysical sheet (US). This is justified from first principles by reviewing briefly how dispersion relations (DRs) are to be derived on the basis of Cauchy’s integral formula $t(s)=(2\pi i)^{-1}\oint dz \; t(z)/(z-s)$ which holds for a function $t(s)$ *analytic* in the domain encirculated by the closed integration contour. As the so-called “matching point” of CCL (and DM) is located *in the US*, the closed integration contour yielding the REs/RSEs must extend *also to the US* where the S- and T-matrix poles for scalar isoscalar $\pi\pi$-scattering are found. Excluding these poles situated at $s_j$ ($j=1,\ldots, n$) from the integration contour and assuming $t(s\rightarrow \infty)\rightarrow 0$ sufficiently fast one obtains the well known (here) unsubtracted DRs $t(s)= \sum_{j=1}^n r_j/(s-s_j) - \frac{1}{\pi} \int_{L,R} dz \,\mbox{Im}[t(z)]/(s-z+i\varepsilon)$, where L/R denotes the left-/right-hand cut, and $r_j$ is the residue of $t(s)$ at the corresponding pole $s_j$. According to CCL, REs/RSEs are twice-subtracted DRs yielding $$\begin{aligned}
t(s) & = & t(s_0) + (s-s_0) \; t^\prime(s_0) + \sum\limits_{j=1}^n \frac{(s_0-s)^2 \,r_j}{(s_0-s_j)^2(s-s_j)} \nonumber \\ & - & \frac{1}{\pi} \int_{L,R} dz\;\frac{(s_0-s)^2 \; \mbox{Im}[t(z)]}{(s_0-z+i\varepsilon)^2(s-z+i\varepsilon)} \; , \label{eq1}\end{aligned}$$ where the subtraction point $s_0$ used by CCL appears to be the $\pi\pi$ threshold, as CCL perform the identification $t(s_0)=a_0^0$ and $t^\prime(s_0)=(2a_0^0-5a^2_0)/(12 m^2_\pi)$ with $a^I_0$ being S-wave scattering lengths for isospin $I=0,2$. It is now easy to see that the REs/RSEs considered by CCL and DM [*disregard the pole terms*]{} (PTs) in the DRs (yielding $r_j = 0$), despite the presence of poles in the US that are claimed to exist by observing respective S-matrix zeros in the PS. As the $s$-dependence of the disregarded $\sigma$- and $f_0(980)$-PTs in the vicinity of the $\pi\pi$- and $KK$-theshold is clearly [*non-linear*]{}, it is to be expected on grounds of dispersion theory that the S-matrix poles predicted by CCL will [*not*]{} coincide with the actual ones to be determined yet by CCL for self-consistency reasons. An analogous statement applies to the results of DM. Moreover will the inclusion of PTs in REs/RSEs not only reinstate dispersion theoretic self-consistency, yet also yield a [*significant*]{} change in the resulting $\sigma$- and $K^\ast_0(800)$-pole positions, which unfortunately will enter now via the PTs as unknown parameters the REs/RSEs to be solved. Hence the inclusion of PTs in REs/RSEs will yield an uncertainty of pole positions which is likely to be of the order of the one estimated in Ref. [@Bugg:2006gc] and therefore much larger than the error bars presently claimed by CCL and DM being even without taking into account PTs for at least two reasons clearly [*parametrization-dependent*]{}: (1) the extrapolation of the two particle phase space to the complex $s$-plane and below threshold invoked by CCL and DM is known to be speculative and even unphysical as it yields e.g. in the approach of DM scattering below the pseudo-threshold; (2) standard chiral perturbation theory (ChPT) disregarding (yet) non-perturbative PTs relates claimed values for scalar scattering lengths and their (too) tiny error bars entering REs/RSEs lacking (yet) PTs to scalar square radii $\left<r_S^2\right>$ the presently used (too) high values of which yield chiral symmetry breaking (ChSB) of the order of 6-8% being much larger than $3$% as observed in Nature. A revision of the analysis of CCL and DM by taking into account PTs in REs/RSEs and ChPT would be highly desirable to reconcile their results with Refs. [@Bugg:2006gc]-[@vanBeveren:2006ua] and to improve the poor description of the resonance $K^\ast (892)$ in the approach of DM.\
Frieder Kleefeld ${}^{1,2,3}$\
${}^{1}$ Present address: Pfisterstr. 31, 90762 Fürth, Germany\
${}^{2}$ Doppler & Nucl. Physics Institute (Dep. Theor. Phys.), Academy of Sciences of Czech Republic; collaborator of the CFIF, Instituto Superior Técnico, LISBOA, Portugal\
${}^{3}$ Electronic address: [[email protected]]{}
[21]{} I. Caprini [*et al.*]{}, Phys. Rev. Lett. 96, 132001 (2006). M. R. Pennington [*et al.*]{}, Phys. Rev. D [**7**]{}, 1429,2591 (1973). W. M. Yao [*et al.*]{} \[PDG\], J. Phys. G [**33**]{}, 1 (2006). D. V. Bugg, J. Phys. G [**34**]{}, 151 (2007). M. Ablikim [*et al.*]{} \[BES\], Phys. Lett. B [**598**]{}, 149 (2004). D. V. Bugg, AIP Conf. Proc. [**814**]{}, 78 (2006). E. van Beveren [*et al.*]{}, Phys. Lett. B [**641**]{}, 265 (2006). S. Descotes-Genon [*et al.*]{}, Eur. Phys. J. C [**48**]{}, 553 (2006). G. F. Chew, S. Mandelstam, Phys. Rev. [**119**]{}, 467 (1960).
This work has been supported by the FCT of the *Ministério da Ciência, Tecnologia e Ensino Superior* of Portugal, under contracts POCI/FP/63437/2005, PDCT/FP/63907/2005 and the Czech project LC06002.\
Conversations with E. van Beveren, D. V. Bugg, , A. Moussallam, G. Rupp, M. D. Scadron and M. Znojil are gratefully acknowledged.
|
---
abstract: 'Memory and forgetting constitute two sides of the same coin, and although the first has been rigorously investigated, the latter is often overlooked. A number of experiments under the realm of psychology and experimental neuroscience have described the properties of forgetting in humans and animals, showing that forgetting exhibits a power-law relationship with time. These results indicate a counter-intuitive property of forgetting, namely that old memories are more stable than younger ones. We have devised a phenomenological model that is based on the principle of retroactive interference, driven by a multi-dimensional valence measure for acquired memories. The model has only one free integer parameter and can be solved analytically. We performed recognition experiments with long streams of words were performed, resulting in a good match to a five-dimensional version of the model.'
address: 'Weizmann Institute of Science, Department of Neurobiology, Rehovot, 7610001, Israel'
author:
- Antonios Georgiou
- Mikhail Katkov
- Misha Tsodyks
bibliography:
- 'sample.bib'
title: 'Retroactive Interference Model of Power-Law Forgetting'
---
Introduction {#introduction .unnumbered}
============
Memory has been often associated solely with the property of persistence, that is the ability to retain and retrieve information with the passage of time. Another equally important characteristic of memory however, is transience, or in other words, the ability to forget and discard information that could be no longer relevant. This process is considered crucial for memory and it is hypothesized to be essential for adaptive behavior [@Richards2017]. Traditionally, since Ebbinghaus‘s seminal study [@Ebbinghaus1885], forgetting has been described with the use of the retention curve. This curve is a continuous function of time $R(\tau)$, which denotes the probability that a memory of age $\tau$ still exists (i.e not yet forgotten). The shape of the retaining function has been investigated through the examination of experimental data over the last century, with some of the authors proposing universal power-law decay [@Wixted1990; @Wixted1991; @Kahana2002] (but see [@Fisher2018]): $$\label{eq:retention_prop}
R(\tau)\propto \tau^{-\alpha}$$ Conversely, we can define a *forgetting* rate function $F(\tau)$, expressing the probability that an available memory of age $\tau$, will be forgotten within the next time interval $dt$. The two functions are related under the equation: $$\label{eq:forgetting}
R(\tau)\approx\prod_{t=0}^{t=\tau/dt}(1-F(t)dt) \approx e^{-\int_{0}^{\tau}F(t)dt}$$ which simply expresses the fact that in order to remain for time $\tau$, the memory has to remain for each of the intervening time intervals. Equation \[eq:forgetting\] can be inverted to $$\label{eq:2}
F(\tau)=-\frac{R'(\tau)}{R(\tau)}$$ This equation means that while the retention function is monotonically decreasing with time (if we assume that extinguished memories cannot be reinstated), the forgetting function could in principle be both decreasing and increasing, depending on the decay speed of the retention function. In particular, an exponential retention function is a borderline case which results in a forgetting function that is independent of time, i.e. all memories have the same probability to be forgotten, irrespective of their age.
Substituting into , we get that for power-law decay of retention, the forgetting rate will decay in time at an inversely proportional manner, regardless of the value of the exponent $\alpha$: $$\label{eq:4}
F(\tau)\propto\frac{\alpha}{\tau}$$ In other words, memories that are older are more resilient (have lower probability to be forgotten at any given moment). Most neural network models of memory however, treat forgetting as a process that is linear in time, i.e the older the memory, the more probable it is to be forgotten at any given moment, which is the core idea of forgetful learning and memory palimpsests [@Nadal1986].
The interest in mathematical forms of memory curves was encouraged by the hope that they may shed light int the mechanisms of remembering and forgetting. While this is not necessarily true, as different mechanisms could result in similar mathematical forms of forgetting (see below), memory models should still be at least broadly compatible with observations, and ideally provide some insights into their computational advantages to potentially alternative schemes. Mechanisms that are usually considered in relation to forgetting are passive decay of memories, interference and consolidation (see e.g. [@Wixted2004-R]). Decay theories state that memories are degraded with time, and are completely forgotten when a threshold is reached. On the other hand, the more popular interference theories suggest that prior (proactive) or subsequent (retroactive) learning disrupts memory consolidation and therefore memories are forgotten (For a review of both cases see [@Wixted2004-R]). A simple and elegant mathematical model of the first type is presented in [@Kahana2002]. While it would appear that passive decay of memory strength should result in new memories gradually replacing the older ones, Kahana and Adler showed that when new memories are characterized by variable initial strengths and decay rates, and are forgotten when the strength dips below threshold, retention function converges to $1/\tau$ scaling in the limit of large $\tau$. Importantly, the necessary condition for this property is that the distribution of decay rates extends all the way towards zero, i.e. some memories don’t decay with time. For example, in the simplest case when memory decay is characterized by a linear function: $S(t) = a - bt$ with random positive $a$ and $b$, one can show that asymptotic scaling for the probability that a memory is still available at time $\tau$ after acquisition is given by $$R(\tau) \approx \frac{P_b(0)}{\tau}<a>,$$ where $P_b(b)$ is the probability density of the decay rate $b$ and $<a>$ is the average value of the initial memory strength (see Appendix for a derivation). The condition that $P_b(0)>0$ also means that the average life-span of a memory is infinite. We conclude that while this study to some extent demystifies the power-law scaling of retention curves, the assumption about the passive decay of memories seems to contradict the well-documented effect of memory interference [@Wixted2004] and the model does not provide any mechanistic reasons for why some memories decay with time and some not, which is a crucial requirement for power-law forgetting. An alternative model that combines passive decay and interference was proposed in [@Lewandowsky], where memories are characterized by a ratio of time since their acquisition to that of other memories. Recall probability is assumed to depend on its ’distinctiveness’, defined as an inverse of acquisition time ratios averaged over all other memories . On one hand, interference is involved since different memories interact to determine their distinctiveness; on the other hand, when time passes without any new memories being acquired, distinctiveness of all memories, and hence their recollection, declines, indicating that passive decay is also effectively at play. The authors show that this mathematical model accounts for experimental retention curves as well as other well-known phenomena in recall literature, such as recency-to-primacy gradient. However, the model rests on several strong assumptions, e.g. the time since acquisition of each memory has to somehow be encoded, and a particular mathematical form of distinctiveness measure has no mechanistic underpinning.
In the current contribution, we aim at a forgetting mechanism that would be compatible with realistic retention curves, contain as few assumptions as possible, and could have a clear functional interpretation. To this end, we propose a phenomenological model that parallels the concept of retroactive interference and captures the statistical properties of forgetting that were previously discussed. Similar to [@Kahana2002] it simplifies the memory retention as a binary process (available/forgotten) and introduces the notion of memory strength (that is generalized to be a multi-dimensional valence). However it does not assume an independent and passive decay of memory valence, rather it proposes a specific process of forgetting based on the idea of retroactive interference. The process proposed has a clear functional interpretation of trying to keep important memories while discarding less important ones. Besides valence dimensionality, the model has no free parameters to tune and results in an asymptotic scaling of retention function similar to that of [@Kahana2002] without any additional assumptions. Furthermore, we performed a series of recognition experiments designed to quantitatively test the predictions of the model and found that it provides a good match to the data.
The Model {#the-model .unnumbered}
=========
![ [**Interference model of forgetting.**]{} [**1-D.**]{} Each item is represented as a thin vertical bar. The height of the bar corresponds to the valence of an item. The top row bars above the black line represent items that are stored in memory just before the acquisition of a new item, shown on the right (Sample). All the items that have smaller valence (bar height) than the new item are discarded from memory (crossed by red bar), and the new item is added. Bottom row represents the memory content, after the new memory is acquired. [**2-D.**]{} Same as 1-D, but each memory item has 2 valences, represented by the width and height of a rectangular. In this case, all the items that have both valences smaller than the corresponding valences of the new item are discarded. []{data-label="fig:cartoon"}](cartoon_smaller.pdf){width="0.7\linewidth"}
To illustrate the main idea of our model, let us first consider a system that continuously acquires new memory items, each characterized by the scalar value $v$ (valence), considered to be a measure of its importance and independently sampled from a distribution $P(v)$. The form of this distribution can be arbitrary, but we assume that it is not changing with time. For simplicity, we assume that memories are acquired at a constant rate (one new memory per time unit). Each time a new item is sampled, all the previously stored items that have a smaller valence are discarded (’forgotten’; see Fig. \[fig:cartoon\], upper panel). Therefore, the total number of stored items will increase if items with relatively small valences are sampled but can suddenly decrease if the sampled element is very potent. This process can be regarded as a crude approximation to retroactive interference. By the construction of the model, at any given moment the valence of the stored units will be a monotonically increasing function of their age, since units are retained only if following units have a smaller valence and are discarded otherwise. Therefore, the probability that a unit will be forgotten at the next time step is strictly a decaying function of its age, i.e. one of the most counter-intuitive features of memory retention is inherently captured by the model. Mathematically, the retention function $R(t)$ is defined as the probability that a memory item is still retained in memory $t$ time steps after its acquisition, and can be computed as: $$\label{eq:5}
R(t)=\int_{-\infty}^{\infty}dv P(v)\bigg[1-\int_{v}^{\infty}dv'P(v')\bigg]^{t}$$ where the term in square brackets expresses the chance that an item with valence $v$ survives the acquisition of one extra item. With the change of variables $v\rightarrow y(v) = \int_{v}^{\infty}dv' P(v')$, equation (\[eq:5\]) converts to: $$\label{eq:6}
R(t)=\int_{0}^{1}(1-y)^{t} dy=\frac{1}{t+1}$$ We see that this simple model exhibits the uniform power-law scaling of memory retention for all times. Importantly, there are no free parameters that affect the retention properties of the model, in particular the form of the probability distribution of valences, $P(v)$ has no effect on the model behavior. The $1/t$ scaling of the retention curve implies that the average number of memories does not saturate with time but continues to grow, which is an attractive feature of the model. However, if we compute the average number of items in memory after a long time $T$ from the beginning of the acquisition process, we obtain $$\label{eq:7}
N(T)=\sum_{t=1}^{T}\frac{1}{t+1}\approx \log (T)$$ i.e. the number of stored items is very small in relation to the total number of sampled ones. In particular, even after $T=10^8$ time units (several years of learning if one assumes a new memory acquisition per second), no more than twenty memories are retained, which is clearly not a reasonable estimate. Furthermore, the assumption that there is a single metric of importance is also unrealistic. Each piece of acquired information might be very important in one context but trivial in another (see also Discussion below). This idea can be easily translated into the model by introducing a multidimensional valence distribution, where each component of the sample $\textbf{v}$ represents its valence on a different domain. The forgetting rule in this case is expanded to all dimensions, and for an item to be forgotten, it is required that the newly acquired sample has a larger valence value on all axes (see \[fig:cartoon\], lower panel, for 2-dimensional case). The retention function in this extended model cannot be expressed in a closed form but can be iteratively computed with the following scheme: $$\label{eq:recursive}
R_n(t) = \frac{1}{t+1}\sum_{k=0}^t R_{n-1}(k)$$ with n being the number of dimensions and $R_1(t)$ being the retention curve of the 1-dimensional model (equation \[eq:6\] above, for detailed derivation see \[ap:1\]). Repeated application of equation \[eq:recursive\] allows the exact calculation of the retention curve for arbitrary $n$. Assuming large t, this expression approximates to $$R_n(t) \approx \frac{1}{(n-1)!}\frac{\log ^{n-1}(t+1)}{(t+1)},
\label{eq:ret}$$ which has the same scaling as in the one-dimensional case with a logarithmic correction. This correction aggregated over a long time $T$, leads to the total number of retained memories given by $$N_n(T) \ =\sum\limits_{t=1}^T R_n(t) \approx \frac{1}{n!}\log ^n(T).$$
Figure \[fig:theory\] shows the plots for $R(t)$ and $N(T)$ for several values of dimensionality. For example, we see that for $n=5$ the number of retained memories after $T=10^8$ steps of acquisition is around few tens of thousands, which appears to be a reasonable estimate [@Landauer1986].
The above analysis shows that in the multidimensional case, the retention curves deviate from simple power-law functions due to logarithmic corrections. One can still approximate the retention curve with a power-law function with a slowly changing exponent: $$R_n(t) \approx c(t) t^{\alpha (t)}$$ where the exponent $\alpha(t)$ can be estimated as $$\alpha(t) = \frac{d(log(R_n(t))}{d(log(t))}$$ (see Fig. \[fig:theory\]). One can see that the scaling exponent is slowly reduced to $-1$ for very large times, remaining significantly above that asymptotic value even for times as large as $10^8$. The asymptotic expression of the exponent can be derived by the asymptotic expression for the retention curve, Eq. \[eq:ret\], resulting in $$\alpha(t) \approx -1 + \frac{n-1}{log(t)}.
\label{eq:alpha_approx}$$
![[**Power fit of theoretical retention curves.**]{} [**A-E**]{} Theoretical retention curve computed with equation for $n=5$, plotted for different time windows. In the inset, the estimated value of the power $\alpha$ for the corresponding window is shown. [**F**]{} The dependence of $\alpha$ on time (orange curve). The value of $\alpha$ very slowly approaches $-1$, such that even for $T=10^8$ it is still about $-0.8$. For comparison the asymptotic estimate of $\alpha$ given by equation is shown in blue. []{data-label="fig:alpha"}](Kahanaplot_multi_v5_.pdf){width="\linewidth"}
Experiment {#experiment .unnumbered}
==========
To test whether the model conforms with human memory performance and to estimate the number of dimensions for the valence distribution of memory, we designed an experimental protocol based on the two-alternative-forced-choice delayed recognition task [@Standing1973]. The experiment was performed on Amazon’s Mechanical Turk$ $ platform. Participants were presented with a sequence of $500$ words, intermittent with recognition attempts. During recognition inquiries participants were prompted to select between two words on the basis of which word they remembered as having previously appeared: one choice constituted a word presented earlier in the sequence (either 2 or 10 words before the recognition attempt or one of the first 25 presented) and another one a lure (see Figure \[fig:exp\_design\]). Following [@Standing1973] we make a simplifying assumption that if a previously presented word is still in memory, the participant will provide a correct answer, otherwise the response is going to stem from guessing. The experimental results are shown in Figure \[fig:exp\]. Figure \[fig:avg\_ret\_all\] shows the results for all 471 participants in the experiment. One can observe that the probability to recognize a word decays towards chance level (50% correct recognitions) as a function of lag between presentation and inquiry (green line). The probability of recognizing the word presented 10 (10-back task) or 2 (2-back task) positions before the recognition prompt also declines as the experiment furthers in time (blue and orange lines). This could result from either proactive interference (when previously memorized items interfere with an acquisition of new words), or general fatigue accompanied with diminished attention, leading to disruption of new word acquisition. Since the short-term memory capacity is estimated to be 3–5 items [@Cowan2007], we conjectured that the last 2 words, if acquired, should stay in short-term memory. We therefore selected 197 participants who exhibited perfect performance on the 2-back recognition task (see \[fig:avg\_ret\_filt\]). Indeed, these participants show no decline of performance for the 10-back test either, indicating the absence of forward interference. Their retention performance (green line) is in agreement with the theoretical prediction for $n=5$ (dashed green line).
![ [**Experimental protocol.**]{} A series of vertical bars represents word presentations. Pairs of horizontal bars represent a delayed recognition task, where participants were presented with one word shown to them previously and one lure word. Participants were requested to click on the word they felt that they saw before. In total 500 words were presented and all first 25 words were queried at different moments. Additionally, participants undertook recognition tasks for the second (25) and tenth (25) back word from the time of inquiry resulting in a total of 75 tasks per participant. The 2/10 back conditions were conducted in blocks, i.e no question from the first 25 words was asked between presentation of the item to be queried and that query itself. []{data-label="fig:exp_design"}](expcartoon.pdf)
[.0]{}
[.0]{}
![ [**Experimental results.**]{} [**A**]{} Recognition performance for all participants who passed the qualification task (see Methods). [**B**]{} Recognition performance for participants that were perfect in the 2-back task. The experimental retention curve (recognition performance vs the lag between word presentation and delayed recognition task positions, solid green curve) declines for both groups. One can observe that performance in the 2-back task (solid blue curves, time here indicates the position of the recognition task) is declining with time, It is generally assumed that working memory capacity is 3-5 items, therefore the drop of performance in the 2-back task indicates that a progressively smaller fraction of words was acquired towards the end of the trial, due to fatigue, loss of attention, or other reasons. Selecting the group of participants (in panel [**B**]{}) that are perfect in the 2-back task we ensured that all the words were acquired during presentation. For this group of participants, the performance in the 10-back task (orange solid curves, time indicates query position) remains constant throughout the experiment (compare with faint orange line representing mean performance in the 10-back task). The dashed lines represent theoretical retention curves computed with equation \[eq:retention\] corrected for guessing (see Methods), for different number of dimensions $n$. []{data-label="fig:exp"}](everythingplot2.pdf){width="\linewidth"}
Discussion {#discussion .unnumbered}
==========
We proposed a simple phenomenological model of forgetting that is broadly compatible with power-law retention curves reported in the earlier literature and with focused recognition experiments performed specifically for this study. In particular, the model results in power-law retention curves with exponents that very slowly decline towards $-1$, remaining significantly above this asymptotic value for all realistic time lags that can be measured experimentally. As opposed to previous quantitative models [@Kahana2002; @Lewandowsky], it is founded on a single computational principle that has a clear functional meaning; namely, we assume that the system tries to maintain important memories at the expense of less important ones, and to this end, each newly acquired element erases already stored ones that are less significant. Importance is evaluated by a multi-dimensional valence measure such that memories that remain are characterized by relatively higher valence measures in one or more dimensions. The nature of these differential valence dimensions is not specified in the model. For example, the memory of an event could have five domains (who, when, where, what and why), each of them defining a different axis of importance. If that event involves a very relevant person (therefore a high value on the ’who’ axis), it would be likely to be retained in memory, even if what happened was relatively insignificant. Another way to view a piece of information as a multidimensional element, comes from the work on semantic representations of words. In particular, it has been shown that the same word pertaining to different conceptual groups, activates different parts of the brain according to the contextual associations made upon acquisition [@Huth2016].
Similar to [@Kahana2002], the average life-span of memories in the model diverges due to accumulation of very strong memories, and hence the process never reaches a steady state, with the number of memories increasing, albeit with monotonically decreasing speed. Besides the number of dimensions, the model does not have a single free parameter, hence the observation that it fits the experimental results so well is quite surprising. It shows that retroactive interference, which is well documented in psychological studies [@Wixted2004], is by itself sufficient to account for power-law retention curves. Consolidation is thus not critical for this property of memory, which does not preclude its role in other aspects of memory that were not addressed in this study.
Some of the assumptions of the model clearly oversimplify the memory system (some of them are mentioned in Introduction). In particular, the valence of each memory is supposed to be stable for the duration of memory, and the distribution of valences, while not constrained in the model, is supposed to be stationary. In real life, one could imagine that some memories’ importance could be altered in time while the distribution of new memory valences could also potentially change, for example due to aging or other life changes. It would be interesting to consider how the system would adapt to these changes by slowly replacing memories that become less relevant by the more relevant ones.
Finally, we believe that we proposed a model with a minimal set of assumptions that is compatible with power-law forgetting curves. However, since other models, based on different principles, can also account for the same observations, we want to conclude the paper by encouraging further experimental and theoretical studies, in particular designing ’critical experiments’, that would try to disambiguate different models and uncover the true mechanisms of forgetting.
Methods {#methods .unnumbered}
=======
Participants, Stimuli and Procedure {#participants-stimuli-and-procedure .unnumbered}
-----------------------------------
A total of 900 participants were recruited to undertake a series of recognition tasks, designed to be performed utilizing Amazon’s Mechanical Turk$^\circledR$ (mTurk) platform (https://www.mturk.com). Ethics approval was obtained by the IRB(Institutional Review Board) of the Weizmann Institute of Science and each participant accepted an informed consent form before participation. Participants were first required to complete the qualification task and if they met criteria described below they were allowed to participate in the main experiment (471 people). Participation was compensated at 10 cents for the qualification task and 30 cents for the regular task.
#### Delayed Recognition Tasks
All tasks performed in this study were two alternative forced choice (2AFC) delayed recognition tasks. Experiments were initiated with participants clicking on a ’Start Experiment’ button. A stream of words was presented sequentially utilizing the standard interface on mTurk’s website for Human Intelligence Tasks, using a custom HTML file with embedded Javascript. Each word was briefly flashed for a duration of $1s$ followed by a blank screen of $0.5s$. The words were displayed centrally on a white screen in black font. At random points during the trial and once in the end, after all words were presented, the presentation of words paused and participants were given a choice of two words in the form of vertically aligned buttons. Each button was randomly assigned with a word, one that was previously presented during the trial and one new word (lure). Participants were instructed to select the button containing the word they remembered seeing. After the selection, presentation resumed automatically. The list of presented words for each participant was randomly generated by sampling without replacement from a pool of 751 words which was produced by selecting English words[@Healey2014] that exhibited a frequency larger than ten per million[@Medler2005]. Each participant performed only one qualification and one main task trials.
#### Qualification
In previous experience we encountered that many workers on mTurk platform are not following instructions. Therefore, each participant was presented with simpler and shorter task first. A recognition delayed task with one hundred words in a stream was presented to participants. In 25 recognition tasks participants were questioned about word that was presented just before the last one (2-back task). We reasoned that two last presented words should stay in short-term memory if participants are attending to stimuli and following instruction. Therefore, we informed people who performed the qualification task with a success rate of more than 95% that they may perform the main experiment. The rest were compensated for participation in qualification experiment.
#### Main Task
Similarly to the qualification task participants had to attend to a stream of words, in this case five hundred in total. During the trial, at seventy four random points (excluding the first 25 words) plus at the end of the list, they were prompted for a delayed recognition of a previously shown word versus a lure word. Twenty five of them requested a recognition of the second-back word as in the qualification, twenty five for the tenth-back and twenty five for the first twenty five words presented. Recognition tasks were randomly intermixed.
Analysis {#analysis .unnumbered}
--------
In Figure \[fig:exp\] the lag was computed as the difference between a query position and a presentation position in the stream of words. For example, if before the $100^{\mbox{th}}$ word there was a recognition task related to $15^{\mbox{th}}$ word the lag is $85$. In the figure the mean fraction of correct recognition is shown for lag bins with equal population of measurements (197) per bin, averaged across all participants having questions with query lags inside the bin. Not all participants had queries for all bins.
### Correction for guessing {#correction-for-guessing .unnumbered}
In computing the theoretical performance for the recognition task we assumed that if a person is remembering the presented word then she/he would correctly point out to the presented word. In the case where participants do not remember the word, we assume they are guessing, and therefore choosing with equal probability. Therefore, one may express recognition performance as $$p(t) = R(t) + \frac{1}{2}(1-R(t)) = \frac{1+R(t)}{2}.$$ where $p(t)$ is fraction of correct responses in recognition task plotted in Figure \[fig:avg\_ret\_filt\] (dashed curves) and $R(t)$ is retention probability of a memory acquired $t$ time steps before testing.
Solution of Kahana model
========================
We analyze the version of Kahana model [@Kahana2002] with linear decay of memory strength: $S(t)=a-bt$ with positive random coefficients $a$ and $b$. Other types of passive decay produce similar results. For simplicity we assume that memory is forgotten when its strength dips below zero. The probability that a memory is still available time $t$ after its inception is given by $$R(t)=Prob(a-bt>0)=\int_0^\infty db P_b(b) \int _{bt}^\infty da P_a(a)$$ where $P_a$ and $P_b$ is the probability density of $a$ and $b$, respectively. Introducing the new variable $b \rightarrow bt $, and taking the limit $t \gg 1$, one obtains $$\begin{aligned}
R(t)&=\frac{1}{t}\int_0^\infty db P_b(b/t) \int _{b}^\infty da P_a(a) \\
&\approx \frac{P_b(0)}{t} \int_0^\infty db \int _{b}^\infty da P_a(a) \\
&=\frac{p_b(0)}{t}<a>\end{aligned}$$ where the third line is obtained by integration by part of the previous line, $<a>$ stands for the average value of $a$. Finally, we note that the $1/t$ scaling of retention implies that the probability density of the life-span of a memory, $P_{life}(t)$ scales as $1/t^2$ asymptotically for large $t$: $$P_{life}(t)=-\frac{d}{dt}R(t)\sim \frac{1}{t^2}$$ which it turn implies that the average memory life-span is infinite.
Multidimensional Retention Function Derivation {#ap:1}
==============================================
In the multidimensional case we can define the retention function as: $$R_n(t) = \idotsint_{-\infty}^\infty P(v_1)P(v_2)\dotsi P(v_n)d(v_1)d(v_2)\dotsi d(v_n) \Big[1-\int_{v_1}^\infty \dotsi \int_{v_n}^\infty dv'_1 dv'_2 \dotsi dv'_n P(v'_1)P(v'_2)\dotsi P(v'_n)\Big]^t$$ With a change of variable $v_i \rightarrow y(v_i)$ we get: $$R_n(t) = \idotsint_0^1 dy_1 \dotsi dy_n (1-y_1 \dotsi y_n)^t$$ And computing the first integral: $$\begin{aligned}
R_n(t) &= \idotsint_0^1 dy_1\dotsi dy_{n-1} \left[\left. \frac{(1-y_1\dotsi y_n)^{t+1}}{(t+1) (y_1 \dotsi y_{n-1})}\right|_0^1 \right] \\
&= \frac{1}{t+1}\int_0^1 dy_1\dotsi dy_{n-1} \left [\frac{(1-y_1\dotsi y_{n-1})^{t+1}-1}{(y_1 \dotsi y_{n-1})} \right] \\
&= \frac{1}{t+1}\int_0^1 dy_1\dotsi dy_{n-1} \left[ \frac{1-(1-y_1\dotsi y_{n-1})^{t+1}}{1-(1-y_1 \dotsi y_{n-1})} \right]\end{aligned}$$ The term on the bracket is a progression and by expanding it we get: $$R_n(t) = \frac{1}{t+1}\idotsint_0^1 dy_1\dotsi dy_{n-1} \big[1+(1-y_1\dotsi y_{n-1})+(1-y_1\dotsi y_{n-1})^2 + \dotso + (1-y_1\dotsi y_{n-1})^t)\big]$$ Rearranging terms: $$R_n(t) = \frac{1}{t+1} \Big[ \idotsint_0^1dy_1\dotsi dy_{n-1}(1-y_1\dotsi y_{n-1})^t + \idotsint_0^1dy_1\dotsi dy_{n-1}(1-y_1\dotsi y_{n-1})^{t-1} + \dotso + \idotsint_0^1dy_1\dotsi dy_{n-1}\Big]$$ And by definition: $$\label{eq:retention}
R_n(t) = \frac{1}{t+1} \big[ R_{n-1}(t) + R_{n-1}(t-1) + \dotso + R_{n-1}(0) \big] = \frac{1}{t+1}\sum_{k=0}^t R_{n-1}(k)$$ with $R_1(t)=\frac{1}{t+1}$ as derived in the text. In the limit of large $t$, one can replace the sums in equation by corresponding integrals, which results in the approximate expression .
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by the EU-H2020-FET 1564 and Foundation Adelis and EU - M-GATE 765549. We thank Michelangelo Naim for the help in designing and conducting Amazon Mechanical Turk$ $ experiments.
Additional information {#additional-information .unnumbered}
======================
**Competing interests** The authors have no competing interests or other interests that might be perceived to influence the results and/or discussion reported in this paper.
|
---
abstract: |
Building upon recent results obtained in [@Causley2013a; @Causley2013; @Causley_Christlieb_Cho], we describe an efficient second order, A-stable scheme for solving the wave equation, based on the method of lines transpose (MOL$^T$), and the resulting semi-discrete (i.e. continuous in space) boundary value problem. In [@Causley2013a], A-stable schemes of high order were derived, and in [@Causley_Christlieb_Cho] a high order, fast $\mathcal{O}(N)$ spatial solver was derived, which is matrix-free and is based on dimensional-splitting.
In this work, are interested in building a wave solver, and our main concern is the development of boundary conditions. We demonstrate all desired boundary conditions for a wave solver, including outflow boundary conditions, in 1D and 2D. The scheme works in a logically Cartesian fashion, and the boundary points are embedded into the regular mesh, without incurring stability restrictions, so that boundary conditions are imposed without any reduction in the order of accuracy. We demonstrate how the embedded boundary approach works in the cases of Dirichlet and Neumann boundary conditions. Further, we develop outflow and periodic boundary conditions for the MOL$^T$ formulation. Our solver is designed to couple with particle codes, and so special attention is also paid to the implementation of point sources, and soft sources which can be used to launch waves into waveguides.
[*Keywords*]{}: Method of Lines Transpose, Tranverse Method of Lines, Implicit Methods, Boundary Integral Methods, Alternating Direction Implicit Methods, ADI schemes
address:
- 'Mathematics Department, Kettering University, Flint, MI 48504'
- 'Department of Mathematics, Michigan State University, East Lansing, MI 48824'
- 'Department of Mathematics, Michigan State University, East Lansing, MI 48824'
author:
- Matthew Causley
- Andrew Christlieb
- Eric Wolf
bibliography:
- 'ref.bib'
title: 'Method of lines transpose: an efficient A-stable solver for wave propagation'
---
[^1]
Introduction
============
Numerical solutions of the wave equation have been an area of investigation for many decades. The wave equation is ubiquitous in the physical world, arising in acoustics, electromagnetics, and fluid dynamics. Our main interest is in electromagnetic wave propagation, where traditional finite difference methods such as the finite-difference time-domain (FDTD) method are often used to solve Maxwell’s equations. When the classical Yee scheme is used, the Courant-Friedrichs-Lewy (CFL) stability criterion restricts the time step to scale with the smallest cells in the domain. This becomes computationally prohibitive in a variety of interesting problems, such as electromagnetic scattering or waveguide design, where complex geometries need to be embedded in a Cartesian mesh, leading to small spatial cells near the boundaries. Alternatively, problems with multiple and disparate temporal or spatial scales, such as those presented by plasma simulations, require time steps and mesh spacings which are on the order of the shortest temporal and spatial scales. Due to the large number of charged particles in a typical simulation [@Birdsall1976], and the high dimensionality of their distribution space, time steps are computationally prohibitive, and the amount of them must be minimized.
As a result, A-stable Maxwell solvers have been developed, which remove the CFL restriction and restore the ability of the user to define a time step which is based on the physical problem, rather than on its spatial discretization. Notable advances include the introduction of the FDTD-ADI [@Fornberg; @Fornberga; @Namiki2000; @Smithe2009] algorithm, as well as several semi-implicit time split schemes. However, it remains difficult to preserve both accuracy, and A-stability when non-rectangular domains are required.
Alternatively, Maxwell’s equations can be reformulated (e.g., using the scalar and vector potential formulation), so that each field component independently satisfies the second order wave equation. Method of lines (MOL) formulations of the wave equation are well-studied, and often lead to conditionally stable schemes. But when the discretization is first performed in time, following the method of lines transpose (MOL$^T$), the wave equation is found at discrete time levels by solving a semi-discrete boundary value problem. In [@Alpert2000; @Alpert2002], the MOL$^T$ is used to produce an exact integral formulation of the wave equation, where the solution is determined by a convolution over the domain of dependence against a space-time Green’s function $G_d(x,t)$, in dimensions $d = 1,2, 3$. However, more commonly the semi-discrete solution is obtained, in which the modified Helmholtz equation must be solved, and the corresponding semi-discrete Green’s function $G(x,\Delta t)$ is the Yukawa, or modified Helmholtz kernel. The resulting boundary integrals methods can then be solved at each time step using fast summation algorithms, such as tree-codes [@Barnes1986; @Christlieb2004; @Li2009; @Lindsay2001], or the fast multipole method (FMM) [@greengard1987fast; @Cheng1999; @Coifman1993; @Li2006; @Gimbutas2002; @Li2009; @Cheng2006]. These methods often scale as $O(N)$ or $O(N\log N)$, but require substantial storage or precomputing stages.
It is worth noting however that in one spatial dimension, the modified Helmholtz kernel is a simple decaying exponential. This fact has been combined with alternate dimension implicit (ADI) splitting to achieve an A-stable wave solver with computational complexity of $O(N \log N)$ [@Bruno2010; @Lyon2010], and $O(N)$ [@Causley2013; @Causley2013a], respectively. The scaling presented in [@Bruno2010; @Lyon2010] is due to the Fourier continuation method, which makes use of the fast Fourier transform to compute a periodic extension of the the convolution integral, where the period is taken sufficiently longer than the domain, so that boundary conditions can be satisfied.
In our approach [@Causley2013a; @Causley2013; @Causley_Christlieb_Cho], the one-dimensional convolution is instead computed strictly over the computational domain, using polynomial interpolation. In [@Causley2013], the solution is proven to be A-stable, second order accurate, and that the matrix formed by the discrete convolution can be applied in $\mathcal{O}(N)$ operations. However, the discrete convolution matrix was formed using piecewise linear integration, and it was found that a Lax-type correction was required to ensure convergence of the scheme in the semi-discrete limit $\Delta t\to 0$, where $\Delta x$ is fixed. This issue was addressed in [@Causley_Christlieb_Cho], where spatial discretization was extended to orders $M\geq 2$, on non-uniform grids while retaining $\mathcal{O}(N)$ complexity. In [@Causley2013a], the scheme was extended to higher orders in time using a novel approach, successive convolution. These higher order schemes were proven to be A-stable.
The purpose of this present work is to develop a robust embedded boundary approach for complex geometry for the MOL$^T$ formulation. Because this paper is focused on developing a range of boundary conditions for the wave equation, we limit our attention to the 2nd order accurate solver. In addition to addressing standard closed (i.e., Dirichlet, Neumann and periodic) boundary conditions, we also develop an open, or outflow boundary condition which is suitable for our implicit solver. The method is extended to higher spatial dimensions using the factorization developed in [@Causley2013a], which is similar to but different from the traditional ADI splitting. This means that solutions are computed line-by-line along dimensional sweeps leveraging the $O(N)$ 1D solver to construct a high dimensional $O(N$) implicit method.
The rest of this paper is laid out as follows. In section \[sec:molt\], we use the method of lines transpose (MOL$^T$) to reformulate the 2D wave equation as an implicit, semi-discrete modified Helmholtz equation. The Helmholtz operator is inverted analytically using dimensional splitting, and we recast the solution as a series of one-dimensional boundary integral equations. We specifically show how to obtain a fully discretized, second order accurate solution by performing spatial quadrature, using a fast $\mathcal{O}(N)$ convolution algorithm.
In section \[sec:artificial\_dissipation\], we modify the time-centered scheme of [@Causley2013a] to introduce artificial dissipation. This will be used to stabilize the embedded boundary method in our 2D algorithm, presented in section \[sec:BC\]. We demonstrate our solution to be fast, second order accurate, and A-stable, even on non-rectangular domains with numerical results presented in section \[sec:results\]. We conclude the body of the paper with several remarks in section \[sec:conclusion\].
We also include in Appendix \[sec:wave\_summary\], a table summarizing the main algorithmic aspects of our wave solver. In Appendix \[sec:sources\], we show how point sources, which may represent charged plasma particles, or soft point sources launched into a waveguide, can be implemented.
Integral solution using MOL$^T$ {#sec:molt}
===============================
We now develop a dimensionally split algorithm for solving the initial value problem $$\begin{aligned}
\nabla^2 u - \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} &= -S(\mathbf{x},t), \quad \mathbf{x} \in \Omega, \quad t>0 \label{eqn:prob2} \\
u(\mathbf{x},0) &= f(\mathbf{x}), \quad \mathbf{x} \in \Omega \nonumber \\
u_t(\mathbf{x},0) &= g(\mathbf{x}), \quad \mathbf{x} \in \Omega \nonumber,\end{aligned}$$ with consistent boundary conditions. We utilize the method of lines transpose (MOL$^T$) to perform a second order accurate temporal discretization, as was shown in [@Causley2013a]. In particular, the discretization is time-centered, and implicit in the spatial derivatives $$\left(1 - \frac{\nabla^2}{\alpha^2}\right) \left[u^{n+1}-2u^n+u^{n-1} +\beta^2 u^n\right] (x,y)= \beta^2 \left(u^n + \frac{1}{\alpha^2} S^n \right),$$ where $$\alpha = \frac{\beta}{c\Delta t}, \qquad \beta>0,$$ and $\beta\leq 2$ is chosen to enforce A-stability [@Causley2013a]. Inversion of this operator leads to a boundary integral formulation, where the Green’s function is the Yukawa potential, defined for 2D in terms of the modified Bessel function $K_0(r)$. However we will employ our previously developed [@Causley2013a; @Causley2013] splitting algorithm, which has $\mathcal{O}(N)$ complexity $$\label{eqn:Helmholtz_Equation_2D}
\mathcal{L}_x \mathcal{L}_y \left[u^{n+1}-2u^n+u^{n-1} +\beta^2 u^n\right] (x,y)= \beta^2 \left(u^n + \frac{1}{\alpha^2} S^n \right).$$ Here the subscripts denote the spatial component of univariate modified Helmholtz operators, $$\label{eqn:HelmholtzL}
\mathcal{L}_x[u]: = \left(1 - \frac{\partial_{xx}}{\alpha^2}\right)u(x,y), \qquad \mathcal{L}_y[u]: = \left(1 - \frac{\partial_{yy}}{\alpha^2}\right)u(x,y).$$ The modified Helmholtz operators are formally inverted using the Green’s function. We define convolution with this Green’s function by the integral operator $$\label{eqn:Iu}
I_x[u](x,y) := \frac{\alpha}{2}\int_a^b u(x',y)e^{-\alpha|x-x'|}dx', \quad a\leq x \leq b,$$ so that $$\label{eqn:L_Inverse}
\mathcal{L}_x^{-1}[u](x,y):= \underbrace{I_x[u](x)}_{\text{Particular Solution}}+ \underbrace{\vphantom{I_x[u](x)} A e^{-\alpha(x-a)} + B e^{-\alpha(b-x)}}_{\text{Homogeneous Solution}},$$ where the coefficients $A$ and $B$ of the homogeneous solution are determined by applying boundary conditions. A similar definition holds for $\mathcal{L}_y^{-1}$. Formally inverting both operators to the right hand side we find the explicit equation $$\label{eqn:2D_explicit}
u^{n+1}-2u^n+u^{n-1} = -\beta^2 \mathcal{D}_{xy}[u^n] + \beta^2\mathcal{L}_{x}^{-1}\mathcal{L}_{y}^{-1}\left[\frac{1}{\alpha^2} S^n \right](x,y),$$ where the multidimensional operator is now $$\mathcal{D}_{xy}[u]: = u - \mathcal{L}_x^{-1}\mathcal{L}_y^{-1}[u].$$ Since the application of $\mathcal{L}_x^{-1}$ is done for fixed $y$, and vice versa for $\mathcal{L}_y^{-1}$, the operator $\mathcal{D}_{xy}$ can be constructed in a line-by-line fashion, similar to ADI algorithms. It was also proven in [@Causley2013a] that the scheme is A-stable, for $0< \beta \leq 2$.
In the continuous case, $\mathcal{L}_x$ and $\mathcal{L}_y$ (and their inverses) commute, so $\mathcal{D}_{xy} = \mathcal{D}_{yx}$. However in practice the discretize operators will not commute, and some small spatial anisotropy is introduced. This can be controlled by applying both operators $\mathcal{D}_{xy}$ and $\mathcal{D}_{yx}$, and then averaging the result.
Similar to a traditional ADI formulation of the wave equation, this factorization produces a fourth order splitting error term, $$\mathcal{L}_x\mathcal{L}_y = \left(1-\frac{\partial_{xx}}{\alpha^2}\right)\left(1-\frac{\partial_{yy}}{\alpha^2}\right) = \left(1-\frac{\partial_{xx}+\partial_{yy}}{\alpha^2}+\frac{\partial_{xx}\partial_{yy}}{\alpha^4}\right),$$ which can be compensated for by adding a term to the right hand side $$\label{eqn:Helmholtz_Equation_2D_corrected}
\mathcal{L}_x \mathcal{L}_y \left[u^{n+1}-2u^n+u^{n-1} +\beta^2 u^n\right] = \beta^2 \left(u^n + \frac{1}{\alpha^2} S^n \right) +\beta^2\left(\mathcal{L}_x -1 \right)\left(\mathcal{L}_y -1 \right)[u^n].$$ Note our use of the identity $(\mathcal{L}_x - 1)(\mathcal{L}_y - 1) = \partial_{xx}\partial_{yy}/\alpha^4$. Formally inverting both operators to the right hand side, the solution can be rearranged and found as $$\label{eqn:2D_explicit_corrected}
u^{n+1}-2u^n+u^{n-1} = - \beta^2 \mathcal{C}[u^n ](x,y) +\beta^2\mathcal{L}_{x}^{-1}\mathcal{L}_{y}^{-1}\left[ \frac{1}{\alpha^2} S^n \right](x,y),$$ where the convolution operator $\mathcal{C}$ is the operator defined in [@Causley2013a] as $$\label{eqn:C_def}
\mathcal{C}:= \mathcal{L}_x^{-1} \mathcal{D}_y + \mathcal{L}_y^{-1}\mathcal{D}_x = \mathcal{D}_{xy} -\mathcal{D}_x\mathcal{D}_y.$$ This form was used in [@Causley2013a] to achieve schemes of higher order through successive convolution, where removing splitting errors is of paramount concern.
Fast convolution algorithm for the integral solution {#sec:Fast}
----------------------------------------------------
In [@Causley2013], the particular solution was discretized in space using the weighted midpoint and trapezoidal rules, which amounted to replacing $u$ with a piecewise constant and linear approximation, respectively. More recently [@Causley_Christlieb_Cho], we have detailed the spatial discretization of $I_x$ to arbitrary order, while providing a means for its rapid evaluation. In this work we focus on the second order accurate implementation of this algorithm, and reiterate the relevant details.
This particular solution is first decomposed into a left and right oriented integral, split at $y=x$ so that $$\label{eqn:ILR}
I[u](x) = I^L[u](x)+I^R[u](x),$$ where $$I^L[u](x) = \frac{\alpha}{2} \int_a^x e^{-\alpha(x-y)}u(y)dy, \quad I^R[u](x)= \frac{\alpha}{2}\int_{x}^b e^{-\alpha(y-x)}u(y)dy.$$ These quantities can be updated locally, using exponential recursion $$\begin{aligned}
\label{eqn:IL_def}
I^L[u](x) &= I^L[u](x-\delta_L) e^{-\alpha \delta_L}+ J^L[u](x), \quad J^L[u](x):= \frac{\alpha}{2} \int_{0}^{\delta_L} u(x-y) e^{-\alpha y}dy, \\
\label{eqn:IR_def}
I^R[u](x) &= I^R[u](x+\delta_R) e^{-\alpha \delta_R}+ J^R[u](x), \quad J^R[u](x):= \frac{\alpha}{2} \int_{0}^{\delta_R} u(x+y) e^{-\alpha y}dy.\end{aligned}$$ The recursive updates and are exact (in space), and making $\delta_L$ and $\delta_R$ small (typically, $\delta_L =\delta_R = \Delta x$) effectively localizes the contribution of the integrals. Based on these observations, we now outline the fast convolution algorithm. In [@Causley_Christlieb_Cho], we derived a spatial quadrature of general order $M \geq 2$ for irregular grids. Here, we will utilize a second-order accurate method on a uniform grid, which we explicitly develop.
Consider the domain $(a,b)$ discretized by uniform grid points $x_{1} = a < x_{2} < \cdots < x_{N+1} = b$ of width $\Delta x = \frac{b-a}{N} = x_{j+1}-x_{j}$, $j=1,...,N$. Suppose, given a function $f$ compactly supported in $(a,b)$, we are to evaluate the convolution operator at each grid points, that is compute $I_{j} = I[f](x_{j}) = I^{L}[f](x_{j})+I^{R}[f](x_{j}) = I^{L}_{j}+I^{R}_{j}$, in $O(N)$ operations. We proceed by evaluating the local integrals $J^{L}_{j} = J^{L}[f](x_{j})$ and $J^{L}_{j} = J^{L}[f](x_{j})$, as in and . The local integrals may be evaluated with quadrature, or, if possible, analytically. A second-order accurate quadrature is given by
$$\begin{aligned}
J^{L}_{j} &\approx P f(x_{j})+Qf(x_{j-1})+R(f(x_{j+1})-2f(x_{j})+f(x_{j-1})) \\
J^{R}_{j} &\approx P f(x_{j})+Qf(x_{j+1})+R(f(x_{j+1})-2f(x_{j})+f(x_{j-1}))\end{aligned}$$
where defining $\nu = \alpha {\Delta x}$ and $d = e^{-\nu}$, the quadrature weights are given by $$\begin{aligned}
P&= 1-\frac{1-d}{\nu} \\
Q&= -d+\frac{1-d}{\nu} \\
R&= \frac{1-d}{\nu^{2}}-\frac{1+d}{2\nu}.\end{aligned}$$
We summarize our fast method in Algorithm \[alg:fast\_conv\].
1. Compute $J^{L}_{j+1}$ and $J^{R}_{j}$ for $j=1,...,N$ via quadrature or analytical integration.
2. Initialize $I^{L}_{1} = 0$ and $I^{R}_{N+1}=0$, and perform the exponential recursion, $I^{L}_{j+1} = J^{L}_{j+1} + e^{-\alpha \Delta x} I^{L}_{j}$ for $j=1,...,N$ and $I^{R}_{N-j+1} = J^{R}_{N-j+1} + e^{-\alpha \Delta x} I^{R}_{N-j+2}$ for $j=1,...,N$.
Artificial dissipation {#sec:artificial_dissipation}
======================
As will be discussed in section \[sec:embedded\_neumann\_bc\], it is necessary to include some artificial dissipation in the numerical scheme to maintain stability with embedded boundary methods for Neumann boundary conditions. We first present a version of the wave solver based on a backwards difference formula (BDF) time discretization, leading to what we call a diffusive scheme, which is dissipative. This method has a larger truncation error than a centered scheme, does not possess a means to tune the level of dissipation, and also has an implicit source term (at time level $n+1$), which is problematic for application in the context of particle-in-cell (PIC) methods for the simulation of plasmas. The second-order centered (dispersive) scheme given above is therefore preferable, but are non-dissipative in their original forms. We give a method for adding tunable artificial dissipation terms into the centered scheme, while maintaining A-stability.
Diffusive wave solver
---------------------
We substitute the following backward difference formula (BDF) discretization: $$\begin{aligned}
u_{tt}^{n+1} &= \frac{2u^{n+1}-5u^n+4u^{n-1}-u^{n-2}}{\Delta t^2} - \frac{11\Delta t^2}{12} u_{tttt}(x,\eta)\end{aligned}$$ into the wave equation $\frac{1}{c^{2}}u_{tt}-\nabla^{2} u = S(x,t)$.
Rearranging, defining $\alpha = \sqrt{2}/(c \Delta t)$ and dividing by $\alpha^{2}$ gives the semi-discrete scheme $$\begin{aligned}
\left(-{\frac{1}{\alpha^{2}}}\Delta+1\right) u^{n+1} = \frac{1}{2}\left(5u^{n}-4u^{n-1}+u^{n-2}\right) + {\frac{1}{\alpha^{2}}}S(x,t^{n+1}) + O({\Delta t}^{4}).\end{aligned}$$
This method is A-stable and dissipative, but does not possess a mechanism for tuning the dissipation, has an inconvenient implicit source term (at time level $n+1$), and typically has a larger truncation error compared to the centered scheme.
Artificial dissipation in centered schemes
------------------------------------------
### Artificial Dissipation in 1D
We give a modified form of the centered version of the implicit wave solver with tunable artificial dissipation that retains the property of unconditional stability. We let $\epsilon$ denote a small artificial dissipation parameter, and $\mathcal{D}_x[u] = u-\mathcal{L}^{-1}[u] = u(x)-\frac{\alpha}{2}\int_{-\infty}^{\infty} e^{-\alpha|x-x'|}u(x') \, dx'$ be defined as usual.
Ignoring sources, we have the second order scheme with dissipation,
$$\begin{aligned}
u^{n+1}-2u^{n}+u^{n-1} = -\beta^{2}\mathcal{D}_x[u^{n}]+ \epsilon \mathcal{D}_x^{2}[u^{n-1}].\end{aligned}$$
We now prove the unconditional stability of this scheme for prescribed values of $\beta$. As in [@Causley2013a], we pass to the high-frequency limit.
We obtain the Von Neumann polynomial $\rho^{2}-(2-\beta^{2})\rho+(1-\epsilon)$. We can check that the roots of this polynomial will be complex if $0 < \beta \leq \sqrt{2+2\sqrt{1-\epsilon}}$, and that in this case the roots satisfy
$$\begin{aligned}
|\rho|^{2} &= \frac{1}{4}\left((2-\beta^{2})^{2}+4(1-\epsilon)-(2-\beta^{2})\right) \\
&= 1-\epsilon < 1\end{aligned}$$
which shows both the stability and dissipative nature of the scheme. $\qed$
We note that the maximum allowed value of $\beta$ is slightly smaller than what is allowed by the corresponding scheme without dissipation. A more detailed analysis shows that the effective damping rate is $\left(\frac{k^{2}}{k^{2}+\alpha^{2}}\right)^{2} \epsilon$, meaning that high frequencies are more rapidly damped than low frequencies.
### Artificial Dissipation in 2D
Using the notation defined in [@Causley2013a], and again ignoring sources, we have the second order scheme with dissipation,
$$\begin{aligned}
u^{n+1}-2u^{n}+u^{n-1} = -\beta^{2}\mathcal{C}[u^{n}]+ \epsilon \mathcal{C}^{2}[u^{n-1}],\end{aligned}$$
where now $\mathcal{D} = 1-\mathcal{L}_{x}^{-1}\mathcal{L}_{y}^{-1}$ and $\mathcal{C} = \mathcal{L}_{y}^{-1}\mathcal{D}_{x}+\mathcal{L}_{x}^{-1}\mathcal{D}_{y}$. For further details on these operators, see [@Causley2013a]. Numerical experiments indicate that the 2D scheme with artificial dissipation is indeed unconditionally stable with the same maximum value of $\beta$ as with the 1D schemes.
Boundary conditions {#sec:BC}
===================
We will now discuss the implementation of boundary conditions. Since our algorithm is dimensionally split, we first develop the boundary conditions in one spatial dimension where the solution reduces to $$\label{eqn:Integral_Solution_Full}
u^{n+1}-2u^n+u^{n-1} = -\beta^2 \mathcal{D}_{x}[u^n] + \beta^2\mathcal{L}_{x}^{-1}\left[\frac{1}{\alpha^2} S^n \right](x), \qquad a\leq x \leq b,$$ and where we consider the following boundary conditions $$\begin{aligned}
\label{eqn:Dirichlet}
\text{Dirichlet:}\qquad & u(a,t) = U_L(t), \quad u(b,t) = U_R(t), \\
\label{eqn:Neumann}
\text{Neumann:}\qquad & u_x(a,t) = V_L(t), \quad u_x(b,t) = V_R(t), \\
\label{eqn:Periodic}
\text{Periodic}\qquad & u(a,t) = u(b,t), \quad u_x(a,t) = u_x(b,t), \\
\label{eqn:outflow}
\text{Outflow:}\qquad & u_t(a,t) =c u_x(a,t), \quad u_t(b,t)=-c u_x(b,t).\end{aligned}$$ Once these boundary conditions have been derived, we use them to build a boundary solver in 2D.
Boundary conditions in one dimension {#sec:bc_1d}
------------------------------------
In 1D, this homogeneous solution requires the determination of two coefficients from the imposed boundary conditions and the endpoint values of the particular (integral) solution by solving a $2\times2$ linear system. We now show how to impose several common boundary conditions in 1D. These methods are extended to the 2D case in Section \[sec:bc\_2d\].
### 1D Dirichlet boundary conditions
Let us begin with Dirichlet boundary conditions . Evaluating the semi-discrete solution at $x = a$ and $b$, we find $$\begin{aligned}
U_L(t_{n+1})= 2U_L(t_n)-U_L(t_{n-1}) - \beta^2 \left(U_L(t_n) - I\left[u^n+\frac{1}{\alpha^2}S^n \right](a) - A - B e^{-\alpha(b-a)}\right), \\
U_R(t_{n+1})= 2U_R(t_n)-U_R(t_{n-1}) - \beta^2 \left(U_R(t_n) - I\left[u^n+\frac{1}{\alpha^2}S^n \right](b)-A e^{-\alpha(b-a)} - B\right),\end{aligned}$$ which, after solving for the unknown coefficients can be written as $$\begin{aligned}
A^n +\mu B^n &= -w_a^D, \\
\mu A^n + B^n &= -w_b^D,\end{aligned}$$ with $$\begin{aligned}
w_a^D &= I\left[u^n+\frac{1}{\alpha^2}S^n \right](a) -U_L(t^{n}) - \frac{U_L(t^{n+1}) - 2 U_L(t^{n}) +U_L(t^{n-1}) }{\beta^2}, \\
w_b^D &= I\left[u^n+\frac{1}{\alpha^2}S^n \right](b) -U_R(t^{n}) -\frac{U_R(t^{n+1}) - 2 U_R(t^{n}) +U_R(t^{n-1}) }{\beta^2},\end{aligned}$$ and $\mu = e^{-\alpha(b-a)}$. Homogeneous boundary conditions are recovered upon setting $U_L(t) = U_R(t) = 0$. Solving the resulting linear system for the unknowns $A^n$ and $B^n$ gives $$\begin{aligned}
\label{eqn:wh_Dirichlet}
A = -\left(\frac{w_a^D - \mu w_b^D }{1-\mu^2}\right), \quad B = - \left(\frac{w_b^D - \mu w_a^D}{1-\mu^2}\right).\end{aligned}$$
### 1D Neumann boundary conditions
For Neumann conditions, first observe that all dependence on $x$ in the integral solution is on the Green’s function, which is a simple exponential function. Using this, we obtain the following identities $$\begin{aligned}
\label{eqn:DtN}
I'(a) = \alpha I(a), \quad I'(b) = -\alpha I(b).\end{aligned}$$ Now, differentiating the semi-discrete solution , and applying the Neumann boundary conditions at $x=a$ and $b$ yields $$\begin{aligned}
V_L(t_{n+1})=2 V_L(t_n)-V_L(t_{n-1}) - \alpha \beta^2 \left(\frac{1}{\alpha} V_L(t_n) - I\left[u^n+\frac{1}{\alpha^2}S^n \right](a) + A - B e^{-\alpha(b-a)}\right), \\
V_R(t_{n+1})=2 V_R(t_n)-V_R(t_{n-1}) - \alpha \beta^2 \left(\frac{1}{\alpha} V_R(t_n) + I\left[u^n+\frac{1}{\alpha^2}S^n \right](b) + A e^{-\alpha(b-a)} - B\right),\end{aligned}$$ which, after solving for the unknown coefficients can be written as $$\begin{aligned}
A^n - \mu B^n &= w_a^N, \\
-\mu A^n + B^n &= w_b^N,\end{aligned}$$ with $$\begin{aligned}
w_a^N &= I\left[u^n+\frac{1}{\alpha^2}S^n \right](a) -\frac{1}{\alpha}V_L(t^n) - \frac{V_L(t^{n+1}) -2 V_L(t^{n}) +V_L(t^{n-1})}{\alpha\beta^2}, \\
w_b^N &= I\left[u^n+\frac{1}{\alpha^2}S^n \right](b) +\frac{1}{\alpha}V_R(t^n) + \frac{V_R(t^{n+1}) -2 V_R(t^{n}) +V_R(t^{n-1})}{\alpha\beta^2}.\end{aligned}$$ Upon solving the linear system we obtain $$\begin{aligned}
\label{eqn:wh_Neumann}
A = \left(\frac{w_a^N +\mu w_b^N}{1-\mu^2}\right), \quad B = \left(\frac{w_b^N+\mu w_a^N}{1-\mu^2}\right).\end{aligned}$$
The cases of applying mixed boundary conditions at $x=a$ and $b$ are not considered here, but the details follow from an analogous procedure to that demonstrated above.
### 1D Periodic boundary conditions
We impose periodic boundary conditions, by assuming that $$u^n(b) = u^n(a), \quad u^n_x(a) = u^n_x(b), \quad n\geq 0.$$ Enforcing this in the semi-discrete solution then yields $$\begin{aligned}
I[u^n](a) + A + B\mu &= I[u^n](b) + A\mu + B, \\
\alpha \left(I[u^n](a) - A + B\mu\right) &= \alpha\left(- I[u^n](b) - A\mu + B\right),\end{aligned}$$ where we have used the identity applied to derivatives of $I$. Solving this linear system is accomplished quickly by dividing the second equation by $\alpha$, and either adding or subtracting it from the first equation, to produce $$\begin{aligned}
\label{eqn:wh_Periodic}
A = \frac{I[u^n](b)}{1-\mu}, \quad B = \frac{I[u^n](a)}{1-\mu}.\end{aligned}$$
### 1D Outflow boundary conditions {#sec:Outflow}
When computing wave phenomena, whether we are interested in finite or infinite domains, it is often the case that we must restrict our attention to some smaller subdomain $\Omega$ of the problem, which does not include the physical boundaries. We say that $\Omega$ is the *computational domain*, and that the boundary $\partial \Omega$ is the non-physical, or *artificial boundary*. Under these circumstances, it is necessary to enforce an outflow, or non-reflecting boundary condition, which allows the wave to leave the computational domain, but not incur (non-physical) reflections at the artificial boundary.
For this reason, let us consider the free space solution $$\mathcal{L}^{-1}[u](x) = \frac{\alpha}{2} \int_{-\infty}^\infty u(y)e^{-\alpha|x-y|}dy,$$ but where we are only interested in evaluating this expression for $x \in \Omega = [a,b]$. Then the contributions can be decomposed as $$\begin{aligned}
\mathcal{L}^{-1}[u](x) =& I[u](x) + \frac{\alpha}{2}\int_{-\infty}^a u(y)e^{-\alpha(x-y)}dy + \frac{\alpha}{2} \int_b^\infty u(y) e^{-\alpha(y-x)}dy \nonumber \\
\label{eqn:Transmission}
=& I[u](x) + A e^{-\alpha(x-a)} + B e^{-\alpha(b-x)},\end{aligned}$$ where the homogeneous coefficients are $$\begin{aligned}
A =& \frac{\alpha}{2}\int_{-\infty}^a u(y)e^{-\alpha(a-y)}dy, \\
B =& \frac{\alpha}{2}\int_b^\infty u(y) e^{-\alpha(y-b)}dy,\end{aligned}$$ which we observe do not depend on $x$. Since the coefficients $A$ an $B$ are the contributions of the integral to the left and right of $[a,b]$ respectively, they can be thought of as transmission conditions (rather than boundary conditions). We make use of this fact to develop outflow boundary conditions, and it will serve as a key idea in our planed follow-up work on a domain decomposition algorithm and multi-core computing with our implicit wave solver.
For the one-dimensional wave equation the exact outflow boundary conditions turn out to be local in space and time. We emphasize that this is only the case in one spatial dimension, but we shall utilize this fact to obtain an outflow boundary integral solution from the integral equation . We extend the support of our function to $(-\infty,\infty)$, and extend the definition of the outflow boundary conditions to the domains exterior to $[a,b]$ $$\begin{aligned}
u_t+cu_x = 0, \quad x\geq b, \\
u_t-cu_x = 0, \quad x\leq a.\end{aligned}$$
Next, assume the initial conditions have some compact support; for simplicity we will take this support to be $\Omega_0 = [a,b]$. Then after a time $t=t_n$, the domain of dependence of $u^n(x)$ is $\Omega_t = [a-ct_n,b+c t_n]$, since the propagation speed is $c$. Now the free space solution becomes $$\begin{aligned}
\mathcal{L}^{-1}[u^n](x) =& \frac{\alpha}{2} \int_{a-c t_n}^{b+c t_n} e^{-\alpha|x-y|} u^n(y) dy \nonumber \\
=& I[u^n](x)+ A^ne^{-\alpha(x-a)} + B^ne^{-\alpha(b-x)}\end{aligned}$$ with coefficients $$\begin{aligned}
\label{eqn:A_Out_Def}
A^n &= \frac{\alpha}{2}\int_{a-ct_n}^{a} e^{-\alpha(a-y)} u^n(y) dy, \\
\label{eqn:B_Out_Def}
B^n &= \frac{\alpha}{2}\int_{b}^{b+ct_n} e^{-\alpha(y-b)} u^n(y) dy.\end{aligned}$$
At first glance, these coefficients are not at all helpful, as they require computing integrals along spatial domains which not only are outside of the computational domain, but also grow linearly in time. However, we will now make use of the extended boundary conditions to turn these spatial integrals into time integrals, which exist at precisely the endpoints $x=a$ and $b$ respectively. Consider first $x>b$. By assumption, this region contains only right traveling waves, $u(x,t) = u(x-ct)$, and by tracing backward along a characteristic ray we find $$u(b+y,t) = u\left(b,t-\frac{y}{c}\right), \quad y>0.$$ Thus, $$\begin{aligned}
B^n &= \frac{\alpha}{2} \int_{0}^{ct_n} e^{-\alpha y} u(b+y,t_n) dy \\
&= \frac{\alpha c}{2} \int_{0}^{t_n} e^{-\alpha c s} u\left(b,t_n-s\right) ds\end{aligned}$$ and so $B^n$ is equivalently represented by a convolution in time, rather than space. Now, knowing the history of $u$ at $x=b$ is sufficient to impose outflow boundary conditions. Furthermore, we find in analog to equation , a temporal recurrence relation due to the exponential $$\begin{aligned}
B^n &= \frac{\alpha c}{2} \int_{0}^{\Delta t} e^{-\alpha c s} u\left(b,t_n-s\right) ds + e^{-\alpha c \Delta t}\left(\frac{\alpha c}{2}\int_0^{t_{n-1}} e^{-\alpha c s} u\left(b,t_{n-1}-s\right) ds \right) \\
&= \frac{\beta}{2} \int_{0}^{1} e^{-\beta z} u\left(b,t_n-z\Delta t\right) dz +e^{-\beta} B^{n-1},\end{aligned}$$ where $\beta = \alpha c \Delta t$, by definition . Thus, the coefficient $B^n$, which imposes an outflow boundary condition at $x=b$, can be computed locally in both time and space. To maintain second order accuracy, we fit $u$ with a quadratic interpolant $$u(b,t_n-z\Delta t) \approx p(z) = u^n(b) -\frac{z}{2}\left(u^{n+1}(b)-u^{n-1}(b)\right) + \frac{z^2}{2}\left(u^{n+1}(b)-2u^n(b)+u^{n-1}(b)\right)$$ and integrate the expression analytically to arrive at $$\label{eqn:Outflow_Update_Intermediate}
B^n = e^{-\beta}B^{n-1} +\gamma_0 u^{n+1}(b)+ \gamma_1 u^n(b)+ \gamma_2 u^{n-1}(b)$$ where $$\begin{aligned}
\gamma_0 =& \frac{E_2(\beta)-E_1(\beta) }{4} = \frac{(1-e^{-\beta})}{2\beta^2}-\frac{(1+e^{-\beta})}{4\beta} \\
\gamma_1 =& \frac{E_0(\beta)-E_2(\beta) }{2} = -\frac{(1-e^{-\beta})}{\beta^2}+ \frac{1}{\beta}e^{-\beta} +\frac{1}{2} \\
\gamma_2 =& \frac{E_2(\beta)+E_1(\beta)}{4} = \frac{(1-e^{-\beta})}{2\beta^2}+\frac{(1-3e^{-\beta})}{4\beta} - \frac{e^{-\beta}}{2}.\end{aligned}$$ In this outflow update equation , the quantities $u^{n+1}(b)$ and $B^n$ are both unknown. In order to determine these values, we must also evaluate the update equation for $u^{n+1}$ at $x=b$ $$u^{n+1}(b) = 2u^{n}(b)-u^{n-1}(b) + \beta^2\left(-u^n(b) + I[u^n](b)+A^n \mu + B^n\right), \quad \mu = e^{-\alpha(b-a)}.$$ We now use these two equations to solve for $u^{n+1}(b)$, and eliminate it from the outflow update equation , so that $$\label{eqn:Outflow_Update_B}
-\Gamma_0 \mu A^n + (1-\Gamma_0)B^n = e^{-\beta}B^{n-1} +\Gamma_0 I[u^n](b) +\Gamma_1u^n(b)+ \Gamma_2 u^{n-1}(b)$$ where $$\Gamma_0 = \beta^2 \gamma_0, \quad
\Gamma_1 = \gamma_1-\gamma_0(\beta^2-2), \quad
\Gamma_2 = \gamma_2-\gamma_0$$
While this procedure could be avoided by omitting $u^{n+1}(b)$ in the interpolation stencil, it turns out to be necessary to obtain convergent outflow boundary conditions.
Likewise, upon considering $x<a$, we find $$\label{eqn:Outflow_Update_A}
(1-\Gamma_0) A^n -\Gamma_0 \mu B^n = e^{-\beta}A^{n-1} +\Gamma_0 I(a) +\Gamma_1u^n(a)+ \Gamma_2 u^{n-1}(a).$$ Solving the resulting linear system produces $$\label{eqn:Outflow_Update}
A^n = \frac{(1-\Gamma_0)w_a^{\text{Out}} +\mu \Gamma_0w_b^{\text{Out}}} {(1-\Gamma_0)^2-(\mu \Gamma_0)^2}, \quad
B^n = \frac{(1-\Gamma_0)w_b^{\text{Out}} +\mu \Gamma_0w_a^{\text{Out}}} {(1-\Gamma_0)^2-(\mu \Gamma_0)^2},$$ where $$\begin{aligned}
w_a^{\text{Out}} &= e^{-\beta}A^{n-1} +\Gamma_0 I[u^n](a) +\Gamma_1u^n(a)+ \Gamma_2 u^{n-1}(a), \\
w_b^{\text{Out}} &= e^{-\beta}B^{n-1} +\Gamma_0 I[u^n](b) +\Gamma_1u^n(b)+ \Gamma_2 u^{n-1}(b)\end{aligned}$$
Boundary conditions in two dimensions {#sec:bc_2d}
-------------------------------------
We now describe our approach for imposing boundary conditions up to second-order accuracy in two dimensions for our MOL$^T$ formulation of the wave equation. Boundary conditions must be supplied for the intermediate sweep variable $w$. Since $w = u + O(c^{2}\Delta t^{2})$, for second order accuracy it suffices for $w$ to inherit the boundary condition imposed on the main solution variable $u$. In the case of rectangular, grid-aligned boundaries, the 1D boundary correction terms can be imposed in a line-by-line fashion. Typical applications of periodic and outflow boundary conditions can be imposed in this manner. In the case of complex boundary geometries that are not grid-aligned, Dirichlet boundary conditions can be imposed in a similar line-by-line fashion (by including irregular boundary points as the end points of the sweep lines), but Neumann boundary conditions require more careful treatment due to the resulting coupling of grid lines. Practically speaking, we only anticipate needing to impose Dirichlet and Neumann boundary conditions on complex boundary geometries, with outflow and periodic boundary conditions being imposed in a line-by-line approach using the 1D results from section \[sec:bc\_1d\] on rectangular domains. Hence, we will limit our discussion here to the implementation of Dirichlet and Neumann boundary conditions in two dimensions.
### Dirichlet boundary conditions in two dimensions
Discretization of a general smooth domain $\Omega$ is accomplished by embedding it in a regular Cartesian mesh of say $N_y$ horizontal ($x$) lines and $N_x$ vertical ($y$) lines, and additionally incorporating the termination points of each line, which will lie on the boundary. For example, the lines and boundary points for a circle are shown in Figure \[fig:sweep\].
Thus, a line $y = y_k$, which is discretized in $x$, and has endpoints $a_k$ and $b_k$, determined as the intersection of the line with the boundary curve. Analogously we define endpoints of the line $x = x_j$ as $c_j$ and $d_j$, and now have $$\begin{aligned}
\label{eqn:x_sweep}
&\mathcal{L}_x^{-1}[u]_k(x)= \frac{\alpha}{2}\int_{a_k}^{b_k} e^{-\alpha|x-x'|} u(x',y)dx' + A_k e^{-\alpha(x-a_k)} + B_k e^{-\alpha(b_k-x)}, \\
\label{eqn:y_sweep}
&\mathcal{L}_y^{-1}[u]_j(y) = \frac{\alpha}{2}\int_{c_j}^{d_j} e^{-\alpha|y-y'|} u(x,y')dy' + C_j e^{-\alpha(y-c_j)} + D_j e^{-\alpha(d_j-y)},\end{aligned}$$ for $1\leq k \leq N_y$ and $1\leq j \leq N_x$ respectively. If the boundary is defined using a level set function $C(x,y) = 0$, then the points $x = a_k, b_k$ can be found by solving $C(x,y_k) = 0$, and following from the analogous approach, the endpoints $y = c_j, d_j$ corresponding to $x = x_j$ are found.
Once the domain has been discretized and the lines have been defined, it remains to compute the discrete form of the scheme . Since the one-dimensional convolution algorithm presented in Section \[sec:Fast\] is formulated for non-uniform grid points, the embedded boundary points do not affect the implementation of the horizontal and vertical sweeps. Thus, the only point of consideration that remains is that of boundary conditions.
First we consider an intermediate variable $w^{(1)}(x,y)$, defined by $$w^{(1)}: = \mathcal{L}_y \left[u^{n+1}-2u^n+u^{n-1}+\beta^2 u^n\right],$$ and which, according to the implicit scheme , can be seen to satisfy $$\mathcal{L}_x[w^{(1)}] = \beta^2\left(u^n + \frac{1}{\alpha^2} S^n\right).$$ Boundary conditions must be applied to $w^{(1)}$ at the boundary points which terminate each line $y = y_k$, defined as $(a_k,y_k)$ and $(b_k,y_k)$. Since $u^{n+1},$ $u^{n},$ and $u^{n-1},$ will be prescribed at these boundary points, we see that the boundary condition will be of the form $$w^{(1)}(a_k,y_k) = \lim_{(x,y)\to (a_k,y_k)} \left(1 - \frac{\partial_{yy}}{\alpha^2}\right)\left(u^{n+1}-2u^n+u^{n-1}+\beta^2 u^n\right)(x,y),$$ and the order of the limit and the partial derivatives commute only when partial derivatives of the boundary data can be constructed. But this is the case only when the tangent line at the boundary is in the $y$-direction, which is precisely why we must restrict our attention to the aforementioned cases for boundary conditions.
Upon introducing a second intermediate variable $w^{(2)}(x,y)$, defined by $$w^{(2)}: = u^{n+1}-2u^n+u^{n-1}+\beta^2 u^n,$$ we make the observation that $$\mathcal{L}_y[w^{(2)}] = w^{(1)}.$$ Boundary conditions are now applied to $w^{(2)}$ along the lines $x = x_j$, at the boundary points $(x_j,c_j)$ and $(x_j,d_j)$, where now no difficulties remain in considering $$w^{(2)}(x_j,c_j) = \lim_{(x,y) \to (x_j,c_j)} \left(u^{n+1}-2u^n+u^{n-1}+\beta^2 u^n\right)(x,y),$$ since the right hand side will be fully prescribed. We summarize this procedure in Algorithm \[alg3\].
1. Initialize temporary variables $w^{(1)}$ and $w^{(2)}$, which are the same size $u^{n}$.
2. For each horizontal line $y = y_k$, for $1\leq k \leq N_y$, create the temporary variable $w^{(1)}_{k}(x)$, defined by $$\mathcal{L}_x[w^{(1)}_{k}](x) = \beta^2 \left(u^n +\frac{1}{\alpha^2}S^n\right)(x,y_k).$$ The homogeneous coefficients are determined by the fact that $$w^{(1)} = \mathcal{L}_y[u^{n+1} - 2u^n + u^{n-1} + \beta^2 u^n ],$$ which is applied at $x = a_k, b_k$.
3. For each vertical line $x = x_j$, for $1\leq j \leq N_x$, create the temporary variable $w^{(2)}_{j}(y)$, defined by $$\mathcal{L}_x[w^{(2)}_{j}](y) = w^{(1)}_{k}(x).$$ The homogeneous coefficients are determined by the fact that $$w^{(2)} = [u^{n+1} - 2u^n + u^{n-1} + \beta^2 u^n ],$$ which is applied at $y = c_j, d_j$.
4. Solve for the update $$u^{n+1} = 2u^n - u^{n-1} -\beta^2 u^n +w^{(2)}.$$
5. The dimensional splitting error is corrected by adding $$\beta^2\mathcal{D}_x\mathcal{D}_y[u^n] = \beta^2\mathcal{D}_y[u^n] + w^{(1)}-w^{(2)}.$$
### Neumann boundary conditions in two dimensions {#sec:embedded_neumann_bc}
In implementing Neumann boundary conditions for boundary geometries conforming to grid lines, such as a rectangular domain, we can directly impose a two-point boundary correction. One way to extend this method to a general polygonal domain would be to use multiple overset grids, each aligned with a boundary segment, which communicate with the interior grid through interpolation on a ghost cell region, though we do not pursue that approach in this work.
For curved boundaries, an alternative approach is that of an embedded boundary method, which involves determining the Dirichlet values at the endpoints of each $x$- and $y$-sweep lines that result in the approximate satisfaction of the Neumann boundary condition (in effect, constructing an approximate Neumann-to-Dirichlet map). We present the implementation of an embedded boundary method for Neumann boundary conditions for the implicit wave solver on a curved boundary geometry. The approach taken here follows the work in [@kreiss2004difference], which proposes an embedded boundary method for Neumann boundary conditions with a finite difference method for the wave equation. The analysis in that work suggests that, on the continuous level, the modified equations and boundary conditions resulting from typical truncation error terms possess unstable boundary layer solutions, so that the addition of a dissipative term is necessary to achieve a stable method. This is consistent with our experience in the implementation described here, with the embedded boundary method becoming unstable when applied to the non-dissipative dispersive solver, but remaining stable for the diffusive solver, which is dissipative.
In the following, we briefly describe the two-point boundary correction method for a grid-aligned rectangular boundary. We then describe the embedded boundary method for a 1D problem. This method requires an iterative procedure, which we show, in the setting of the 1D problem, to be a convergent contraction mapping with a rate of convergence that depends on the CFL number. We describe the implementation of the embedded boundary method in 2D, and finally give numerical results.
### Description of the Two-Point Boundary Correction Method
In a rectangular domain where the boundaries conform to grid lines, it is straightforward to impose the two-point boundary correction terms in a line-by-line fashion, since in this case, the grid lines are not coupled through the normal derivative. As this is a simple extension of the 1D boundary correction algorithm, we do not elaborate further.
### Description of the Embedded Boundary Method and Proof of Convergence of the Iterative Solution in 1D
We consider the situation of a one-dimensional domain $\left\lbrace x_{B}<x \right\rbrace$ with a single boundary point not aligned with the grid points, as displayed in Figure \[fig:boundary\_geom\_1d\]. We have grid points $x_{0},x_{1},...$ with uniform grid spacing $x_{i+1}-x_{i} = \Delta x$, boundary location $x_{B}$, and ghost point location $x_{G}=x_{0}$. We define interior points to be any grid points lying within the domain (including the boundary), and exterior points to be any grid points lying outside of the domain. We define a ghost point to be any exterior point for which at least one of the neighboring points $x_{i \pm 1}$ is an interior point. We neglect the right boundary in the present analysis for simplicity, though it can be extended to the case with both boundaries. We consider applying the diffusive version of the wave solver, having calculated the convolution integral $I(x)$, and now needing to find the value of the coefficient $A$ such that the solution $u(x)$ at the next time step is given by $$\begin{aligned}
u(x) = I(x) + A e^{-\alpha (x-x_{G})}, \quad x \geq x_{0}\end{aligned}$$
Given the value of the convolution integral and the solution at the ghost point, $I_{G} = I(x_{G})$ and $u_{G}=u(x_{G})$, respectively, the coefficient may be computed as $A = u_{G}-I_{G}$. We now describe the procedure for determining the value of the solution at the ghost point, $u_{G}$, that leads to a solution consistent with homogeneous Neumann boundary conditions to second-order accuracy. We construct a quadratic interpolant using the boundary condition and interior interpolation points $x_{I} = x_{B}+ \Delta s_{I}$ and $x_{II} = x_{B}+2\Delta s_{I}$, lying in between grid points $x_{m}$ and $x_{m+1}$, and $x_{n}$ and $x_{n+1}$, respectively. The interpolation distance will be chosen such that $\Delta x < \Delta s_{I} < (3/2) \Delta x$. We define the distances $\xi_{G} = x_{B}-x_{G}$, $\xi_{I} = x_{I}-x_{B} = \Delta s_{I}$, and $\xi_{II} = x_{II}-x_{B} = 2\Delta s_{I}$, and construct a quadratic Hermite-Birkhoff [@birkhoff1906general] interpolant $P(\xi)$ by imposing the conditions $P'(0) = 0$, $P(\xi_{I}) = u_{I}$ and $P(\xi_{II}) = u_{II}$. We then obtain the following second-order approximation to the ghost point value, given by
$$\begin{aligned}
u_{G} &= P(\xi_{G}) + O(\Delta x^{2}) = u_{I} \frac{\xi_{II}^{2}-\xi_{G}^{2}}{\xi_{II}^{2}-\xi_{I}^{2}}+ u_{II} \frac{\xi_{G}^{2}-\xi_{I}^{2}}{\xi_{II}^{2}-\xi_{I}^{2}} + O(\Delta x^{2}) \\
&= \gamma_{I} u_{I}+\gamma_{II} u_{II}.\end{aligned}$$
As the coefficients $\gamma_{I} = \frac{\xi_{II}^{2}-\xi_{G}^{2}}{\xi_{II}^{2}-\xi_{I}^{2}} > 0$ and $\gamma_{II} = \frac{\xi_{G}^{2}-\xi_{I}^{2}}{\xi_{II}^{2}-\xi_{I}^{2}} < 0$ are $O(1)$, we only need supply second-order accurate approximations to $u_{I}$ and $u_{II}$ to maintain overall second-order accuracy. (The coefficients would be $O(1/\Delta x)$ in the case of nonhomogeneous Neumann boundary conditions, which would require third-order accurate approximations to $u_{I}$ and $u_{II}$. For simplicity, we consider only the case of homogeneous Neumann boundary conditions in the present work.) Such approximations may be obtained through linear interpolation, giving
$$\begin{aligned}
u_{I} &= \sigma_{I} u_{m} + (1-\sigma_{I}) u_{m+1} \\
u_{II} &= \sigma_{II} u_{n} + (1-\sigma_{II}) u_{n+1}\end{aligned}$$
where $\sigma_{I} = \frac{x_{m+1}-x_{I}}{\Delta x}$, $\sigma_{II} = \frac{x_{n+1}-x_{II}}{\Delta x}$, and $u_{j} = u(x_{j})$ are the values of the function at the uniform gridpoints for $j=m,m+1,n,n+1$.
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Hence, to determine the ghost point value $u_{G}$ that leads to a solution consistent with homogeneous Neumann boundary conditions, we should solve the following system of equations.
$$\begin{aligned}
u_{G} &= \gamma_{I} (\sigma_{I} u_{m} + (1-\sigma_{I}) u_{m+1}) + \gamma_{II} (\sigma_{II} u_{n} + (1-\sigma_{II}) u_{n+1}) \\
u_{m} &= I_{m} + (u_{G}-I_{G})e^{-\alpha(x_{m}-x_{0})} \\
u_{m+1} &= I_{m+1} + (u_{G}-I_{G})e^{-\alpha(x_{m+1}-x_{0})} \\
u_{n} &= I_{n} + (u_{G}-I_{G})e^{-\alpha(x_{n}-x_{0})} \\
u_{n+1} &= I_{n+1} + (u_{G}-I_{G})e^{-\alpha(x_{n+1}-x_{0})} \end{aligned}$$
with $\gamma_{I}$, $\gamma_{II}$, $\sigma_{I}$ and $\sigma_{II}$ defined as above, and where $I_{j} = I(x_{j})$ is the convolution integral evaluated at uniform grid points for $j=m,m+1,n,n+1$. This system can be solved formally for $u_{G}$, giving
$$\begin{aligned}
u_{G} &= \left[ \gamma_{I} \left(\sigma_{I}(I_{m}-I_{G} e^{-\alpha(x_{m}-x_{G})})+(1-\sigma_{I})(I_{m+1}-I_{G} e^{-\alpha(x_{m+1}-x_{G})})\right) + \right. \\
& \quad \left. \gamma_{II} \left(\sigma_{II}(I_{n}-I_{G} e^{-\alpha(x_{n}-x_{G})})+(1-\sigma_{II})(I_{n+1}-I_{G} e^{-\alpha(x_{n+1}-x_{G})})\right) \right] \div \\
& \quad\left[ 1- \gamma_{I} \left(\sigma_{I}e^{-\alpha(x_{m}-x_{G})}+(1-\sigma_{I})e^{-\alpha(x_{m+1}-x_{G})}\right) - \right. \\
& \quad \left. \gamma_{II} \left(\sigma_{II}e^{-\alpha(x_{n}-x_{G})}+(1-\sigma_{II})e^{-\alpha(x_{n+1}-x_{G})}\right) \right]\end{aligned}$$
To show that this solution formula is well-defined, we argue that $$\begin{aligned}
0< K &:= \gamma_{I} \left(\sigma_{I}e^{-\alpha(x_{m}-x_{G})}+(1-\sigma_{I})e^{-\alpha(x_{m+1}-x_{G})}\right) + \\
& \quad + \gamma_{II} \left(\sigma_{II}e^{-\alpha(x_{n}-x_{G})}+(1-\sigma_{II})e^{-\alpha(x_{n+1}-x_{G})}\right) < 1\end{aligned}$$
for the relevant values of $m$, $n$ and $\xi_{G}$, $\xi_{I}$, and $\xi_{II}$. We define $d = e^{-\alpha \Delta x}$, and noting that $0 < d < 1$, $m<n$, $\xi_{G} < \xi_{I} = \Delta s_{I} < \xi_{II} = 2\Delta s_{I}$, $\gamma_{I}>0$, $\gamma_{II}<0$, $0 \leq \sigma_{I} \leq 1$, and $0 \leq \sigma_{II} \leq 1 $, we obtain
$$\begin{aligned}
K &= \gamma_{I} d^{m} \left[\sigma_{I}+(1-\sigma_{I})d\right] + \gamma_{II} d^{n} \left[\sigma_{II}+(1-\sigma_{II})d\right] \\
&\leq \gamma_{I} d^{m} + \gamma_{II} d^{n+1} \\
&= \frac{d^{m} \xi_{II}^{2}-d^{n+1}\xi_{I}^{2}+\xi_{G}^{2}(d^{n+1}-d^{m})}{\xi_{II}^{2}-\xi_{I}^{2}} \\
&\leq \frac{d^{m} \xi_{II}^{2}-d^{n+1}\xi_{I}^{2}}{\xi_{II}^{2}-\xi_{I}^{2}} \\
&= \frac{4\Delta s_{I}^{2} d^{m}-d^{n+1} \Delta s_{I}^{2}}{4\Delta s_{I}^{2}-\Delta s_{I}^{2}} = \frac{4d^{m}-d^{n+1}}{3}\end{aligned}$$
Now, since $\Delta x < \Delta s_{I} < (3/2) \Delta x$, we can see that it is the case that either $m=1$ and $n=2$ or $3$, or that $m=2$ and $n=3$. It is then a matter of some simple calculus to check that that the functions $f_{m,n} (x) = (4x^{m}-x^{n+1})/3$ satisfy $f_{m,n}(x)<1$ for $0<x<1$ and the given combinations of $m$ and $n$. This proves that $K<1$ for the relevant values of the parameters, so that the solution for $u_{G}$ is well-defined. We note, however, that $K$ approaches 1 as $d$ approaches 1, that is, as the CFL number becomes large. Thus, we may expect an ill-conditioned system when the CFL number is very large.
To obtain the lower bound on $K$, we observe $$\begin{aligned}
K &= \gamma_{I} d^{m} \left[\sigma_{I}+(1-\sigma_{I})d\right] + \gamma_{II} d^{n} \left[\sigma_{II}+(1-\sigma_{II})d\right] \\
&\geq \gamma_{I} d^{m+1} + \gamma_{II} d^{n} \\
&= \frac{d^{m+1}(\xi_{II}^{2}-\xi_{G}^{2})+d^{n}(\xi_{G}^{2}-\xi_{I}^{2})}{\xi_{II}^{2}-\xi_{I}^{2}} \\
&= \frac{d^{m+1}(4\xi_{I}^{2}-\xi_{G}^{2})+d^{n}(\xi_{G}^{2}-\xi_{I}^{2})}{\xi_{II}^{2}-\xi_{I}^{2}} \\
&= \frac{(d^{m+1}-d^{n})(\xi_{I}^{2}-\xi_{G}^{2})+3d^{m+1}\xi_{I}^{2}}{\xi_{II}^{2}-\xi_{I}^{2}} > 0 \end{aligned}$$
In the two-dimensional case, the line-by-line solution method couples the ghost point values, and a general explicit solution formula is impossible to write down. In principle, one may write out and directly solve a linear system to obtain the ghost point values. Instead, we propose an iterative solution method that avoids the formation of a matrix. We now describe this iterative solution method and prove its convergence in the context of the one-dimensional problem described above.
Suppose we have the convolution integral evaluated at the gridpoints, $I_{j}$, and a $k$-th iterate for the ghost point value, $u_{G}^{k}$. Then we may obtain the next iterate by the formulas $$\begin{aligned}
u_{m}^{k+1} &= I_{m} + (u_{G}^{k}-I_{G})e^{-\alpha(x_{m}-x_{0})} \\
u_{m+1}^{k+1} &= I_{m+1} + (u_{G}^{k}-I_{G})e^{-\alpha(x_{m+1}-x_{0})} \\
u_{n}^{k+1} &= I_{n} + (u_{G}^{k}-I_{G})e^{-\alpha(x_{n}-x_{0})} \\
u_{n+1}^{k+1} &= I_{n+1} + (u_{G}^{k}-I_{G})e^{-\alpha(x_{n+1}-x_{0})} \\
u_{G}^{k+1} &= \gamma_{I} (\sigma_{I} u_{m}^{k+1} + (1-\sigma_{I}) u_{m+1}^{k+1}) + \gamma_{II} (\sigma_{II} u_{n}^{k+1} + (1-\sigma_{II}) u_{n+1}^{k+1})\end{aligned}$$ where quantities are defined as above. Now, to prove the convergence of the interation, we show it is contractive. Taking the difference of two iterates, we have $$\begin{aligned}
|u_{G}^{k+1}-u_{G}^{k}| &= |\gamma_{I} (\sigma_{I} (u_{m}^{k+1}-u_{m}^{k}) + (1-\sigma_{I}) (u_{m+1}^{k+1}-u_{m+1}^{k})) + \\
& \quad \gamma_{II} (\sigma_{II} (u_{n}^{k+1}-u_{n}^{k}) + (1-\sigma_{II}) (u_{n+1}^{k+1}-u_{n+1}^{k}))| \\
& = |\gamma_{I} \left(\sigma_{I}(u_{G}^{k}-u_{G}^{k-1})e^{-\alpha(x_{m}-x_{G})}+(1-\sigma_{I})(u_{G}^{k}-u_{G}^{k-1})e^{-\alpha(x_{m+1}-x_{G})}\right) + \\
& \quad + \gamma_{II} \left(\sigma_{II}(u_{G}^{k}-u_{G}^{k-1})e^{-\alpha(x_{n}-x_{G})}+(1-\sigma_{II})(u_{G}^{k}-u_{G}^{k-1})e^{-\alpha(x_{n+1}-x_{G})}\right)| \\
&\leq K |u_{G}^{k}-u_{G}^{k-1}|\end{aligned}$$ where $0 < K < 1$ as defined above. Hence, the Contraction Mapping Theorem implies that the iteration converges to a unique fixed point (which is the solution given in the formula above). We note again that $K$ approaches 1 as the CFL number becomes large, so that the rate of convergence will become slower for larger CFL numbers.
### Description of the Method in 2D
We now describe the implementation of the embedded Neumann boundary condition in the 2D case. We consider the situation displayed in Figure \[fig:boundary\_geom\_2d\], in which we need to determine the value of our unknown $u_{G}$ at the ghost point location $(x_{G},y_{G})$. In the 2D case, we define a ghost point to be any exterior point $(x_{i},y_{j})$ for which at least one of the neighboring points $(x_{i \pm 1},y_{j})$ or $(x_{i},y_{j\pm 1})$ is an interior point. Similarly to the 1D case, we will construct a quadratic Hermite-Birkhoff boundary interpolant $P(\xi)$ along the direction normal to the boundary, which intersects the boundary curve $\Gamma$ at location $(x_{B},y_{B})$, and supply the interior interpolation point values $u_{I}$ and $u_{II}$, at points $(x_{I},y_{I})$ and $(x_{II},y_{II})$, respectively, by further interpolation from interior grid points. These points are selected along the normal, in analogy to the 1D case, such that $\xi_{I} = |(x_{I},y_{I})-(x_{B},y_{B})| = \Delta s_{I}$ and $\xi_{II} = |(x_{II},y_{II})-(x_{B},y_{B})| = 2 \Delta s_{I}$, where we will typically take $\Delta s_{I} = \sqrt{2} \Delta x$. We construct a quadratic Hermite-Birkhoff interpolant $P(\xi)$ by imposing the conditions $P'(0) = 0$, $P(\xi_{I}) = u_{I}$ and $P(\xi_{II}) = u_{II}$. Defining further the distance from the boundary to the ghost point $\xi_{G} = |(x_{G},y_{G})-(x_{B},y_{B})|$, we obtain, as in the 1D case, the following second-order approximation to the ghost point value, given by
$$\begin{aligned}
u_{G} &= P(\xi_{G}) + O(\Delta x^{2}) = u_{I} \frac{\xi_{II}^{2}-\xi_{G}^{2}}{\xi_{II}^{2}-\xi_{I}^{2}}+ u_{II} \frac{\xi_{G}^{2}-\xi_{I}^{2}}{\xi_{II}^{2}-\xi_{I}^{2}} + O(\Delta x^{2}) \\
&= \gamma_{I} u_{I}+\gamma_{II} u_{II}.\end{aligned}$$
where the coefficients are defined as $\gamma_{I} = \frac{\xi_{II}^{2}-\xi_{G}^{2}}{\xi_{II}^{2}-\xi_{I}^{2}} > 0$ and $\gamma_{II} = \frac{\xi_{G}^{2}-\xi_{I}^{2}}{\xi_{II}^{2}-\xi_{I}^{2}} < 0$. In the 2D case, we find approximations to $u_{I}$ and $u_{II}$ through bilinear interpolation. This is in contrast to [@kreiss2004difference], who find the intersection of the normal with grid lines, then interpolate along the grid lines. We have also implemented a second-order accurate version of this approach and compared to the bilinear interpolation scheme proposed here. We have found that the two schemes behave similarly, however the bilinear interpolation scheme is slightly more accurate and simpler to code, not requiring to handle separate cases of intersection with horizontal and vertical grid lines. The bilinear interpolation scheme is standard, but we give it here for completeness. If the interpolation point $u_{I}$ lies in a cell with corners $(x_{i},y_{j})$, $(x_{i+1},y_{j})$, $(x_{i+1},y_{j+1})$ and $(x_{i},y_{j+1})$, then we have the following approximation for $u_{I}$:
$$\begin{aligned}
u_{I} &= w_{1}u_{i,j}+w_{2}u_{i+1,j}+w_{3}u_{i+1,j+1}+w_{4}u_{i,j+1}\end{aligned}$$
where $w_{1} = \frac{(x_{i+1}-x_{I})(y_{j+1}-y_{I})}{\Delta x \Delta y}$, $w_{2} = \frac{(x_{I}-x_{i})(y_{j+1}-y_{I})}{\Delta x \Delta y}$, $w_{3} = \frac{(x_{I}-x_{i})(y_{I}-y_{j})}{\Delta x \Delta y}$ and $w_{4} = \frac{(x_{i+1}-x_{I})(y_{I}-y_{j})}{\Delta x \Delta y}$. With this interpolation scheme established, we now outline the algorithm for the 2D dimensionally-split wave solver.
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The above interpolation procedure applies regardless of the variety of the wave solver that it is used with, provided that the wave solver has sufficient dissipation to maintain stability. We now describe the rest of the embedded boundary algorithm in the context of the diffusive wave solver, though it it may be similarly applied to the the dispersive scheme with artificial dissipation described above. In analogy to the iteration presented in the 1D case, the 2D iterative algorithm proceeds by using the interpolation scheme to provide values at the ghost points, which in turn provide new values for the boundary correction coefficients, which are then used to update the values at the interior grid points, comprising one full iteration. It should be noted that not all interior grid points need be updated in the iteration, only those near the boundary that lie within the boundary interpolation stencils.
Using values from previous time steps, the initial guess for the interior grid points in the boundary interpolation stencils may be given by extrapolation in time, as $u^{n+1,0} = 2u^{n}-u^{n-1}$ (linear extrapolation) or $u^{n+1,0} = 3u^{n}-3u^{n-1}+u^{n-2}$ (quadratic extrapolation). Either extrapolated initial guess provides a modest reduction in the number of iterations required versus a zero initial guess, with only a slight further reduction in the number of iterations going from linear to quadratic extrapolation. An effective stopping criterion for iteration is $|u^{n+1,l+1}-u^{n+1,l}|_{\infty} < \epsilon$, where $\epsilon$ is some chosen tolerance, which may be chosen to be quite small, as the iteration is a fixed point interation. In the numerical example, we choose a tolerance of $10^{-15}$, and we achieve convergence in less than 40 iterations at a CFL number of 2.
In applying the diffusive version of the wave solver, we assume we have previous time steps $u^{n}$, $u^{n-1}$ and $u^{n-2}$. We have to solve the modified Helmholtz equation with homogeneous Neumann boundary conditions,
$$\begin{aligned}
\left(1-\frac{1}{\alpha^{2}}\nabla^{2}\right)u^{n+1} = \frac{1}{2}\left(5u^{n}-4u^{n-1}+u^{n-2}\right) & \mbox{ in } \Omega \\
\frac{\partial u}{\partial n} = 0 & \mbox{ on } \Gamma = \partial \Omega\end{aligned}$$
where $\alpha = \frac{\sqrt{2}}{c \Delta t}$. We apply dimensional splitting to find
$$\begin{aligned}
\left(1-\frac{1}{\alpha^{2}}\nabla^{2}\right)u^{n+1} = \left(1-\frac{1}{\alpha^{2}}\partial_{xx}\right)\left(1-\frac{1}{\alpha^{2}}\partial_{yy}\right)u^{n+1} + O\left(\frac{1}{\alpha^{4}}\right)\end{aligned}$$
so we define $w = \left(1-\frac{1}{\alpha^{2}}\partial_{yy}\right)u$, and noting that $w = u+O\left(\left(c\Delta t \right)^{2}\right)$ so that $\frac{\partial w}{\partial n} = \frac{\partial u}{\partial n}+O\left(\left(c\Delta t \right)^{2}\right)$, we obtain the following approximate system
$$\begin{aligned}
\left(1-\frac{1}{\alpha^{2}}\partial_{xx}\right)w^{n+1} = \frac{1}{2}\left(5u^{n}-4u^{n-1}+u^{n-2}\right) & \mbox{ in } \Omega \\
\frac{\partial w}{\partial n}^{n+1} = 0 & \mbox{ on } \Gamma = \partial \Omega \\
\left(1-\frac{1}{\alpha^{2}}\partial_{yy}\right)u^{n+1} = w^{n+1} & \mbox{ in } \Omega \\
\frac{\partial u}{\partial n}^{n+1} = 0 & \mbox{ on } \Gamma = \partial \Omega\end{aligned}$$
We now suppose our domain is embedded in a uniform Cartesian grid, with horizontal grid lines corresponding to $y=y_{k}$, $1\leq k \leq N_{y}$ and vertical grid lines corresponding to $x=x_{j}$, $1\leq j \leq N_{x}$. The embedded boundary algorithm will be applied when calculating the intermediate variable $w^{n+1}$ in horizontal line sweeps as well as the solution variable $u^{n+1}$ in vertical line sweeps. The iterations for these two variables are separate; first, the iterative procedure is applied to $w$ to convergence, and then this value of $w$ is used to compute $u$, and the iterative procedure is applied to $u$ to convergence. However, in each iteration, the grid lines are coupled through the interpolation scheme, so that all grid lines must be iterated together. The overall iterative algorithm is described in \[alg:eb\_algorithm\], with details specified for the iteration on $w$. The iteration on $u$ is very similar, and so we omit the details.
1. **(Initialization of ghost points)** Perform the interpolation scheme described above to obtain the values of $u^{n}$, $u^{n-1}$, and $u^{n-2}$ at the ghost points, which are the endpoints of the horizontal and vertical grid lines.
2. **(Evaluation of particular solution)** For each horizontal line $y = y_{k}$, for $1\leq k \leq N_{y}$, with ghost (end) points $x=a_{k}$ and $b_{k}$ find the particular solution $w^{n+1}_{p,k}$ for the intermediate variable $w^{n+1}_{k}(x)$ by evaluating the discrete convolution operator $$w^{n+1}_{p,k}(x_{j}) = \frac{\alpha}{4}\int_{a_{k}}^{b_{k}}[5u^{n}-4u^{n-1}+u^{n-2}](x',y_{k})e^{-\alpha|x_{j}-x'|} \, dx'$$
for each grid point $x_{j}$ in the horizontal line, including the ghost points.
3. **(Boundary correction initialization)** For each horizontal line $y = y_{k}$, set the initial guess for the intermediate variable via extrapolation, $w^{n+1,0}_{k} = 3w^{n}_{k}-3w^{n-1}_{k}+w^{n-2}_{k}$, on the interior points within the boundary interpolation stencil.
4. **(Boundary correction iteration)** For each horizontal line $y = y_{k}$, perform the interpolation scheme using the interior values of $w^{n+1,l}_{p,k}$ to find the ghost point values. Using these ghost point values, apply the boundary correction on each line to obtain the updated intermediate variable,
$$w^{n+1,l+1}_{k}(x_{j}) = w^{n+1}_{p,k}(x_{j})+A_{k}e^{-\alpha(x_{j}-a_{k})} + B_{k}e^{-\alpha(b_{k}-x_{j})}$$
for the values of $x_{j}$ lying within the boundary interpolation stencil, where $A_{k} = \frac{w^{n+1,l}_{k}(a_{k})-w^{n+1}_{p,k}(a_{k})-\mu_{k} \left(w^{n+1,l}_{k}(b_{k})-w^{n+1}_{p,k}(b_{k})\right)}{1-\mu_{k}^{2}}$, $B_{k} = \frac{w^{n+1,l}_{k}(b_{k})-w^{n+1}_{p,k}(b_{k})-\mu_{k} \left(w^{n+1,l}_{k}(a_{k})-w^{n+1}_{p,k}(a_{k})\right)}{1-\mu_{k}^{2}}$, $\mu_{k} = e^{-\alpha(b_{k}-a_{k})}$. Check for convergence, and if converged, store the intermediate variable $w^{n+1}$.
5. Repeat this process for the vertical line sweeps, using the intermediate variable $w^{n+1}$ to calculate the particular solution for $u^{n+1}$, then apply the bounday correction interation.
Numerical Results {#sec:results}
=================
Double Circle Cavity
--------------------
In this example, we solve the wave equation with homogeneous Dirichlet boundary conditions on a 2D domain $\Omega$ which is, as in Figure \[fig:dblcirc\_geom\], the union of two overlapping disks, with centers $P_{1}=\left(- \gamma,0\right)$ and $P_{2}=\left(\gamma,0\right)$, respectively, and each with radius $R$: $$\nonumber
\Omega = \left\{\left(x,y\right) : |\left(x,y\right)-P_{1}| < R \right\}\cup \left\{\left(x,y\right) : |\left(x,y\right)-P_{2}|<R \right\}$$ where $|\left(x,y\right)| = \sqrt{x^{2}+y^{2}}$ is the usual Euclidean vector norm, and $\gamma < R$.
(-1.8,0) – (1.8,0) node \[above\] [$x$]{}; (0,-1.1) – (0,1.1) node \[right\] [$y$]{}; (0,-.8) arc (-126.9:126.9:1); (0,.8) arc (53.1:306.9:1); (.6,0) – (.6,1); (.6,.5) node\[anchor=west\] [$R$]{};
in [-.6,.6]{} [ (,-1pt)–(,1pt); ]{}
(.6,0) node\[anchor=north\] [$\gamma$]{}; (-.6,0) node\[anchor=north\] [$-\gamma$]{};
(.6,0) node\[anchor=south west\] [$P_{2}$]{}; (-.6,0) node\[anchor=south east\] [$P_{1}$]{};
This geometry is of interest due to, for example, its similarity to that of the radio frequency (RF) cavities used in the design of linear particle accelerators, and presents numerical difficulties due to the curvature of, and presence of corners in, the boundary. Our method avoids the staircase approximation used in typical finite difference methods to handle curved boundaries, which reduces accuracy to first order and may introduce spurious numerical diffraction.
As initial conditions, we choose $$u\left(x,y,0\right) =
\begin{cases}
-\cos^{6}\left(\frac{\pi}{2}\left(\frac{|(x,y)-P_{1}|}{0.8 \gamma}\right)^{2}\right) & |\left(x,y\right)-P_{1}| < 0.8 \gamma \\
\cos^{6}\left(\frac{\pi}{2}\left(\frac{|(x,y)-P_{2}|}{0.8 \gamma}\right)^{2}\right) & |\left(x,y\right)-P_{2}| < 0.8 \gamma \\
0 & \text{otherwise}
\end{cases}$$ and $$u_{t}\left(x,y,0\right)=0$$ for $(x,y) \in \Omega$. Selected snapshots of the evolution are given in Figure \[fig:dblcirc\_plots\], and the results of a refinement study are given in Table \[tab:refinement\_dblcirc\]. The discrete $L^{2}$ error was computed against a well-refined numerical reference solution ($\Delta x = \num{0.00021875}$); the error displayed in the table is the maximum over time steps with $t \in [0.28,0.29]$. For this example, $R=0.3$, $\gamma=0.2$, $c=1$, and the CFL is 2.
$\Delta x$ $\Delta y$ $\Delta t$ $L^{2}$ error $L^{2}$ order
------------------ ---------------------- --------------------- --------------------- ---------------
$\num{0.0070}$ $\num{0.0043333333}$ $\num{0.008666667}$ $\num{0.006143688}$ $-$
$\num{0.0035}$ $\num{0.0021666667}$ $\num{0.004333333}$ $\num{0.001682923}$ $1.8681$
$\num{0.00175}$ $\num{0.0010833333}$ $\num{0.00216666}$ $\num{0.000435945}$ $1.9488$
$\num{0.000875}$ $\num{0.0005416667}$ $\num{0.00108333}$ $\num{0.000105150}$ $2.0517$
: Refinement study for the double circle cavity with Dirichlet BC. For the numerical reference solution, $\Delta x = \num{.00021875}$, $\Delta y = \num{.00013542}$, and $\Delta t = \num{.00027083}$.[]{data-label="tab:refinement_dblcirc"}
Symmetry on a Quarter Circle
----------------------------
With the goal of testing the capabilities of our boundary conditions as well as circular geometry, we construct standing modes on a circular wave guide of radius $R$, in two different ways. First, we solve the Dirichlet problem, with initial conditions $$u(x,y,0) = J_0\left(z_{20}\frac{r}{R} \right), \quad u_t(x,y,0) = 0,$$ and exact solution $u = J_0\left(z_{20}\frac{r}{R} \right)\cos\left(z_{20}\frac{ct}{R}\right)$, where $J_0$ is the Bessel function of order 0, and $z_{20} = 5.5218$ is the $2$-nd zero. Secondly, we use the symmetry of the mode to construct the solution restricted to the second quadrant, with homogeneous Neumann boundary conditions taken along the $x$ and $y$ axes.
In both cases, the solution converges to second order. An overlay of the two are shown in Figure \[fig:Bessel\_Quarter\], demonstrating the close agreement.
Bessel Mode with Neumann Boundary Conditions
--------------------------------------------
Here we present a numerical example of the embedded boundary method for homogeneous Neumann boundary conditions given in Section \[sec:embedded\_neumann\_bc\]. We apply the method to a circular domain, for which analytical solutions exist. We consider a radially-symmetric Bessel mode with homogeneous Neumann boundary conditions, with an analytic solution given by
$$\begin{aligned}
u(r,t) &= J_{0}\left(Z_{0}\frac{r}{R}\right)\cos\left(Z_{0}\frac{ct}{R}\right),\end{aligned}$$
where $J_{0}$ is the Bessel function of the first kind of order 0, $r = \sqrt{x^{2}+y^{2}}$, $R$ is the radius of the domain, and $Z_{0}\approx3.8317$ is the smallest nonzero root of $J_{0}'$ (so that $\frac{\partial u}{\partial \mathbf{n}}(R,t) = \frac{\partial u}{\partial r}(R,t) = \frac{Z_{0}}{R}J_{0}' \left(Z_{0}\right)\cos\left(Z_{0}\frac{ct}{R}\right) =0$). In this example, we take radius $R = \pi/2$ and wave speed $c=1$. An example of the embedded boundary grid used is given in Figure \[fig:eb\_grid\_plot\]. We perform a refinement study with a fixed CFL number of 2, with the results in Figure \[fig:eb\_refinement\_study\] indicating the expected second-order convergence. We set the iteration tolerance to $10^{-15}$, and we see convergence of the boundary correction iteration in fewer than 40 iterations. We note some oscillation of the $L^{\infty}$ error about the line giving second-order accuracy, which we believe to be due to the grid points moving with respect to the boundary through the refinement, causing some variation in the maximum error.
Periodic Slit Diffraction Grating
---------------------------------
In this example, we apply our method to model an infinite, periodic diffraction grating under an incident plane wave. Diffraction gratings are periodic structures used in optics to separate different wavelengths of light, much like a prism. The high resolution that can be achieved with diffraction gratings makes them useful in spectroscopy, for example, in the determination of atomic and molecular spectra. Our numerical experiment, depicted in Figure \[fig:slit\_geom\], demonstrates the use of our method with multiple boundary conditions and nontrivial geometry in a single simulation to capture complex wave phenomena.
An idealized slit diffraction grating consists of a reflecting screen of vanishing thickness, with open slits of aperture width $a$, spaced distance $d$ apart, measured from the end of one slit to the beginning of the next (that is, the periodicity of the grating is $d$). We impose an incident plane wave of the form $u_{inc}(x,y,t)=\cos{\left(\omega t + ky\right)}$, where $k = 2 \pi / a$ and $\omega = k/c$, where $c$ is the wave speed. Periodic BCs at $x=\pm d/2$ (determining the periodicity of the grating), and homogeneous Dirichlet BCs are imposed at the screen. We also test outflow boundary conditions in multiple dimensions, which are imposed at $y=\pm L_{y}/2$.
(-1,0)–(-0.2,0); (0.2,0)–(1,0);
(-0.8,-0.8) – (-0.6,-0.8) node\[right\] [$x$]{}; (-0.8,-0.8) – (-0.8,-0.6) node\[above\][$y$]{};
(-0.2,0) – (0.2,0) node \[black,midway,xshift=0,yshift=10pt\] [$a$]{}; (-1,0) – (1,0) node \[black,midway,xshift=0,yshift=15pt\] [$d$]{};
(0,-.7) – (0,-.2) node \[right,align=center,midway\] [ [$u_{\mbox{\scriptsize inc}}$]{} ]{}; (-1,-1)–(-1,1)–(1,1)–(1,-1)–cycle;
(0,-1) node [Outflow BC]{}; (0,1) node [Outflow BC]{}; (1,0) node [Periodic BC]{}; (-1,0) node [Periodic BC]{};
In Figure \[fig:slit\_plots\], we observe the time evolution of the incident plane wave passing through the aperture, and the resulting interference patterns as the diffracted wave propagates across the periodic boundaries. The outflow boundary conditions allow the waves to propagate outside the domain. While a rigorous analysis of the efficacy of our outflow BCs is the subject of future work, the results look quite reasonable, as no spurious reflections are seen at the artificial boundaries.
Point Sources and Outflow Boundary Conditions
---------------------------------------------
As a final example, we illustrate some of the more interesting features of our solver. We launch two point sources (from points not on the mesh), and employ Dirichlet, periodic and outflow boundary conditions along the edges of the domain. As can be seen in Figure \[fig:Source\], the point sources propagate perfectly, despite not being placed at grid points.
Conclusion {#sec:conclusion}
==========
In this paper we have presented a fast, A-stable, second order scheme for solving the wave equation. Using the method of lines transpose (MOL$^T$), the solution can be interpreted in the semi-discrete sense as a boundary integral solution, posed as a convolution against an exponential kernel. We have exploited this fact to develop a matrix-free, $O(N)$ fast spatial convolution algorithm, capable of embedding boundary points in a regular Cartesian mesh, without affecting the accuracy or stability.
In addition to demonstrating second order accuracy for wave propagation in a variety of non-Cartesian geometries, with time steps in excess of the usual CFL restriction, we have also developed a novel method for implementing outflow boundary conditions, as well as methods for launching waves, using soft sources, from points located at arbitrary locations (e.g., not located at a mesh point) inside the domain, which is of interest in particle-wave simulations such as those required in studying plasma. Several topics warrant further investigation. We are developing a domain decomposition approach to multi-core computing with our implicit wave solver based on the properties of the exponential kernel, where the subdomains require only pointwise data communication from adjoining edges. Future work will investigate the implementation of higher-order accurate boundary conditions for complex boundary geometries and outflow boundary conditions, and further, apply these methods to Maxwell’s equations both in electromagnetic scattering and plasma physics problems.
Summary Table For Second-Order Wave Solver {#sec:wave_summary}
==========================================
------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Wave Equation Dispersive Scheme, $\alpha = \frac{2}{c \Delta t}$:
$\frac{1}{c^{2}}\frac{\partial u}{\partial t} - \nabla^{2} u = S(x,t)$ $\left(-\frac{1}{\alpha^{2}}\nabla^{2}+1 \right)\left(u^{n+1}+2u^{n}+u^{n-1}\right) = 4u^{n} + \frac{4}{\alpha^{2}}S(x,t^{n})$
To Diffusive Scheme, $\alpha = \frac{\sqrt{2}}{c \Delta t}$:
Modified Helmholtz Equation $\left(-\frac{1}{\alpha^{2}}\nabla^{2}+1 \right)u^{n+1} = \frac{1}{2}\left(5u^{n}-4u^{n-1}+u^{n-2}\right) + \frac{1}{\alpha^{2}} S(x,t^{n+1})$
Dimensionally Split $\left(-\frac{1}{\alpha^{2}}\nabla^{2}+1 \right)u=f$ $\Rightarrow$
Modified Helmholtz Equation $\left(-\frac{1}{\alpha^{2}}\frac{\partial^{2}}{\partial x^{2}}+1 \right)\left(-\frac{1}{\alpha^{2}}\frac{\partial^{2}}{\partial y^{2}}+1 \right)u = f$ $\Rightarrow$
(2D) $\left(-\frac{1}{\alpha^{2}}\frac{\partial^{2}}{\partial x^{2}}+1 \right)w = f$, $ \left(-\frac{1}{\alpha^{2}}\frac{\partial^{2}}{\partial y^{2}}+1 \right)u = w$
1D Integral Solution $\left(-\frac{1}{\alpha^{2}}\frac{d^{2}}{dx^{2}}+1 \right)u = f$ on $(a,b)$ $\Rightarrow$
$u(x) = \frac{\alpha}{2}\int_{a}^{b} f(x')e^{-\alpha|x-x'|} \, dx' + Ae^{-\alpha(x-a)} + Be^{-\alpha(b-x)}$
$= I[f](x) + Ae^{-\alpha(x-a)} + Be^{-\alpha(b-x)}$
1D BC Correction Coefficients
Dirichlet: $A = \frac{(u_{a}-I_{a})-\mu (u_{b}-I_{b})}{1-\mu^{2}}$, $B = \frac{(u_{b}-I_{b})-\mu (u_{a}-I_{a})}{1-\mu^{2}}$
$u(a) = u_{a}$, $u(b) = u_{b}$
Neumann: $A = \frac{\mu(v_{b}+\alpha I_{b})-(v_{a}-\alpha I_{a})}{\alpha \left(1-\mu^2\right)}$, $B = \frac{(v_{b}+\alpha I_{b})-\mu(v_{a}-\alpha I_{a})}{\alpha \left(1-\mu^2\right)}$
$u'(a) = v_{a}$, $u'(b) = v_{b}$
Periodic: $A = \frac{I_{b}}{1-\mu}$, $B = \frac{I_{a}}{1-\mu}$
$u(a) = u(b)$, $u'(a) = u'(b)$
$I_{a} = I[f](a)$, $I_{b} = I[f](b)$, $\mu = e^{-\alpha(b-a)}$
Fast Convolution Algorithm $a = x_{0} < x_{1} < \cdots < x_{N} = b$,
$x_{j+1} - x_{j}+ \Delta x$, $j=0,...,N-1$
$I_{j} = I[f](x_{j}) = \frac{\alpha}{2}\int_{a}^{b} f(x')e^{-\alpha|x_{j}-x'|} \, dx' = I^{L}_{j} + I^{R}_{j}$
$I^{L}_{j} = \frac{\alpha}{2}\int_{a}^{x_{j}} f(x')e^{-\alpha|x_{j}-x'|} \, dx'$, $I^{R}_{j} = \frac{\alpha}{2}\int_{x_{j}}^{b} f(x')e^{-\alpha|x_{j}-x'|} \, dx'$
$I^{L}_{0}=0$, $I^{L}_{j} = e^{-\alpha \Delta x} I^{L}_{j-1} + J^{L}_{j}$,
$J^{L}_{j} = \frac{\alpha}{2}\int_{x_{j-1}}^{x_{j}} f(x')e^{-\alpha|x_{j}-x'|} \, dx'$, $j=1,...,N$
$I^{R}_{N}=0$, $I^{R}_{j} = e^{-\alpha \Delta x} I^{R}_{j+1} + J^{R}_{j}$,
$J^{R}_{j} = \frac{\alpha}{2}\int_{x_{j}}^{x_{j+1}} f(x')e^{-\alpha|x_{j}-x'|} \, dx'$, $j=N-1,...,0$
Second Order Quadrature $J^{R}_{j} = Pf(x_{j})+Qf(x_{j+1}) + R(f(x_{j+1})-2f(x_{j})+f(x_{j-1}))$
$J^{L}_{j} = Pf(x_{j})+Qf(x_{j-1}) + R(f(x_{j+1})-2f(x_{j})+f(x_{j-1}))$
$P = 1-\frac{1-d}{\nu}$, $Q = -d+\frac{1-d}{\nu}$, $R = \frac{1-d}{\nu^{2}}-\frac{1+d}{2\nu}$
$\nu = \alpha {\Delta x}$, $d = e^{-\nu}$
------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Treatment of point sources, and soft sources {#sec:sources}
============================================
We now consider the inclusion of source terms. We present the algorithm in 1D for simplicity, and observe that the extensions to multiple dimensions are analogous to those shown for dimensional splitting presented in section \[sec:molt\].
We are predominantly interested in the case where $S(x,t)$ consists of a large number of time dependent point sources. However, it is often the case that in electromagnetics problems, a soft source is prescribed to excite waves of a prescribed frequency, or range of frequencies, within the domain. A soft source is so named because, although incident fields are generated at a prescribed fixed spatial location, no scattered fields are generated.
The implementation of a soft source $\sigma(t)$ at $x = x_s$ is accomplished by prescribing the source condition $$\label{eqn:condition}
u(x_s,t) = \sigma(t).$$ However, it can be shown that if we set $$\begin{aligned}
S(x,t)= \frac{2}{c}\sigma'(t) \delta(x-x_s)\end{aligned}$$ and insert it into the wave equation, then the soft source condition is satisfied, and the solutions are equivalent. Thus, a soft source is nothing more than a point source, whose time-varying field is integrated by the wave equation.
Upon convolving this source term with the Green’s function according to , we find $$\begin{aligned}
I\left[\frac{1}{\alpha^2}S\right](x)=& \frac{1}{2\alpha}\int_a^b \left(\frac{2}{c}\sigma'(t_n)\delta(x-x_s) \right)e^{-\alpha|x-y|}dy \\
=& \frac{\Delta t}{\beta} \sigma'(t_n) e^{-\alpha|x-x_s|},\end{aligned}$$ where the definition of $\alpha = \beta/(c\Delta t)$ has been utilized.
It is often the case that taking the analytical derivative $\sigma'(t_n)$ is to be avoided, for various reasons. In this case, any finite difference approximation which is of the desired order of accuracy can be substituted.
Likewise for general point sources, $$S(x,t) = \sum_{i} \tilde{\sigma}_i(t) \delta(x-x_i)$$ the corresponding form of the source term is $$\label{eqn:Delta_Source}
I\left[\frac{1}{\alpha^2}S\right](x) = \frac{c \Delta t}{2\beta}\sum_i \tilde{\sigma}_i(t_n) e^{-\alpha|x-x_i|}$$ Therefore, it suffices to consider delta functions both for the implementation of soft sources, as well as including time dependent point sources.
[^1]: This work has been supported in part by AFOSR grants FA9550-11-1-0281, FA9550-12-1-0343 and FA9550-12-1-0455, NSF grant DMS-1115709, and MSU Foundation grant SPG-RG100059.
|
---
abstract: 'In fiber-optic communications, evaluation of mutual information (MI) is still an open issue due to the unavailability of an exact and mathematically tractable channel model. Traditionally, lower bounds on MI are computed by approximating the (original) channel with an auxiliary forward channel. In this paper, lower bounds are computed using an auxiliary backward channel, which has not been previously considered in the context of fiber-optic communications. Distributions obtained through two variations of the stochastic digital backpropagation (SDBP) algorithm are used as auxiliary backward channels and these bounds are compared with bounds obtained through the conventional digital backpropagation (DBP). Through simulations, higher information rates were achieved with SDBP compared with DBP, which implies that tighter lower bound on MI can be achieved through SDBP.'
author:
- 'Naga V. Irukulapati, , Marco Secondini, , Erik Agrell, , Pontus Johannisson, and Henk Wymeersch, [^1] [^2]'
title: 'Tighter Lower Bounds on Mutual Information for Fiber-Optic Channels'
---
Introduction {#secIntro}
============
Shannon proved that reliable communication through a noisy channel is possible with channel coding, as long as the information rate is less than [the]{} channel capacity [@Shannon1948]. Reliable communication means that coding schemes exist that can make the probability of error arbitrarily small. Furthermore, if the information rate is greater than the channel capacity, then regardless of the coding scheme, the probability of error cannot be made zero. The channel capacity for an additive white Gaussian noise (AWGN) channel is known exactly and has been derived in [@Shannon1948 Sec. 24]. However, the evaluation of the channel capacity for the fiber-optic channel (FOC) accounting for the dispersion, nonlinearity, and noise is still an open problem due to the unavailability of an exact and mathematically tractable channel model. Therefore, accurately predicting the capacity of the FOC has been the focus of much recent research [@Mitra2001; @Essiambre2010Capacity; @Ellis2010; @Agrell2014; @Kramer2015; @Agrell2015b].
For a fixed input distribution, [the]{} mutual information (MI) gives a lower bound on the channel capacity. MI is also used for predicting the [post-forward-error-correction]{} (FEC) [bit-error rate]{} (BER) between the input and output of the discrete-time channel. MI is shown to be a better metric than the pre-FEC BER for estimating the post-FEC BER in soft-decision FEC systems [@Wan2006; @Franceschini2006; @Brueninghaus2005; @Leven2011]. [For the FOC, ]{}the most commonly used approach to lower-bound the MI is to approximate the original forward channel with an auxiliary forward channel (AFC) [[@Djordjevic2005; @Colavolpe2011; @Leven2011; @Secondini2013; @Fehenberger2015a; @Eriksson2016; @Liga2016]]{}. A receiver that is optimal for an AFC is used to process the data generated from the original forward channel [and to compute an information rate. This rate is achievable by that receiver and, for this reason, is often referred to as an achievable information rate (AIR)]{} [@Ganti2000; @Arnold2006; @Secondini2013]. [However, in some scenarios, only an auxiliary backward channel (ABC)—i.e., an approximation for the posterior distribution of the channel input given its output—is available, while a corresponding AFC does not exist or is computationally too complex to find. In such cases, an alternative approach to lower-bound the MI is through a direct use of the ABC. This approach will be applied in the context of the FOC for the first time in this paper.]{} [The concept of ABC was used for the first time (to the best of our knowledge) in the context of universal decoding for memoryless channels with deterministic interference [@Merhav1993].]{} An ABC is used instead of an AFC to maximize the lower bound using an iterative procedure [@Sadeghi2009]. [A receiver that is optimal for an ABC is used to process the data generated from the original forward channel and to compute an AIR that is achievable by that receiver. ]{} In this paper, distributions obtained from two variations of the stochastic digital backpropagation (SDBP) algorithm are used as ABCs, namely from symbol-by-symbol SDBP (SBS-SDBP) [@Irukulapati2014TCOM] and Gaussian message passing SDBP (GMP-SDBP) [@Wymeersch2015SPAWC]. Through simulations, the AIR computed using these two distributions of SDBP was observed to be higher than the AIR obtained using the conventional digital backpropagation (DBP) algorithm, implying that a tighter lower bound on the MI can be obtained using SDBP.
### Organization of the paper {#organization-of-the-paper .unnumbered}
MI is mathematically introduced in Sec. \[secMI\]. [Lower bounds on the MI using an AFC and an ABC and their Monte Carlo (MC) estimation are described in Sec. \[secMILowerBounds\]]{}. Computation of AIRs for the FOC is considered in Sec. \[secAuxSDBP\], where SBS-SDBP and GMP-SDBP are described briefly, leading to a discussion on how ABCs are obtained using these two approaches. AIRs computed using these two versions of SDBP are then compared with DBP. Numerical results are presented and discussed in Sec. \[secNumSim\], followed by conclusions in Sec. \[secConcl\].
Mutual Information {#secMI}
==================
For a discrete-time channel with memory, the MI between random vectors ${\mathbf{X}}\triangleq (X_1, X_2,\ldots,X_{K})$ and ${\mathbf{Y}}\triangleq (Y_1, Y_2,\ldots,Y_{J})$ with ${J}\ge {K}$ is defined as[^3]$$\begin{aligned}
\nonumber I({\mathbf{X}}; {\mathbf{Y}})&= \int_{\mathcal X}\int_{\mathcal Y}p({\mathbf{x}}, {\mathbf{y}})\log \frac{p({\mathbf{y}}|{\mathbf{x}})}{p({\mathbf{y}})} \mathrm{d}{\mathbf{x}}\mathrm{d}{\mathbf{y}}\\
\label{eqnMIXnYnpyx}&\triangleq \mathbb E_{{\mathbf{X}},{\mathbf{Y}}}\bigg[\log\frac{p({\mathbf{Y}}|{\mathbf{X}})}{p({\mathbf{Y}})}\bigg], $$ where ${\mathbf{x}}= (x_1,x_2,\ldots,x_{K}) \in {\mathcal X}$ is a realization of ${\mathbf{X}}$ drawn from the input distribution $p({\mathbf{x}})$, and ${\mathbf{y}}= (y_1,y_2,\ldots,y_{J}) \in {\mathcal Y}$ is a realization of the corresponding output random vector ${\mathbf{Y}}$. $p({\mathbf{y}}|{\mathbf{x}})$ is the channel conditional distribution, $p({\mathbf{y}})=\int_{{\mathcal X}} p({\mathbf{x}})p({\mathbf{y}}|{\mathbf{x}})\mathrm{d}{\mathbf{x}}$ is the output distribution, and $\mathbb E_{{\mathbf{X}},{\mathbf{Y}}}\{.\}$ is expectation over the joint distribution $p({\mathbf{x}},{\mathbf{y}})=p({\mathbf{y}}|{\mathbf{x}})p({\mathbf{x}})$.
Alternatively, the MI can also be written as $$\begin{aligned}
\label{eqnMIXnYnpxy} I({\mathbf{X}}; {\mathbf{Y}})&= \mathbb E_{{\mathbf{X}},{\mathbf{Y}}}\bigg[\log\frac{p({\mathbf{X}}|{\mathbf{Y}})}{p({\mathbf{X}})}\bigg],\end{aligned}$$ where $p({\mathbf{x}}|{\mathbf{y}})=p({\mathbf{y}}|{\mathbf{x}})p({\mathbf{x}})/p({\mathbf{y}})$. The information rate between the ergodic processes for the channel with memory is [@Arnold2006] $$\begin{aligned}
\label{eqnMIrate}
I^{\textrm{mem}} = \lim_{{K},{J}\rightarrow \infty} \frac{1}{{K}}I({\mathbf{X}}; {\mathbf{Y}}).\end{aligned}$$
It is often the case, especially for the FOC, that the channel distributions, $p({\mathbf{y}}|{\mathbf{x}})$ and $p({\mathbf{x}}|{\mathbf{y}})$, are not known in closed form. Hence, the MI of (\[eqnMIXnYnpyx\]) and (\[eqnMIXnYnpxy\]) and subsequently the information rate of (\[eqnMIrate\]) cannot be computed in closed form. The information rate of (\[eqnMIrate\]) can, in principle, be estimated through simulations using the BCJR algorithm [@Arnold2006]. However, the complexity of this simulation-based technique increases exponentially with memory and, hence, is infeasible for systems with a large memory such as the FOC [@Radosevic2011]. In such instances, upper and lower bounds on (\[eqnMIrate\]) are calculated. A useful and practical approach to lower-bound the MI is by using a concept known as mismatched decoding [@Fischer1978; @Merhav1994]. In mismatched decoding, the original distributions $p({\mathbf{y}}|{\mathbf{x}})$ or $p({\mathbf{x}}|{\mathbf{y}})$ are approximated with auxiliary distributions and the rates computed using these auxiliary distributions are a lower bound on the MI, as will be detailed in Sec. \[subSecLBAFC\] and \[secRevAuxCh\]. The [better]{} the auxiliary distribution approximates the original distribution, the tighter are these bounds. Currently, [all lower bounds on the MI for the FOC are based on AFCs]{} [@Secondini2013; @Fehenberger2015a; @Eriksson2016]. In this paper, [we use an ABC to compute lower bounds on the MI for the FOC and also]{} investigate if the existing bounds can be further improved, thereby obtaining tighter bounds on the MI.
Before proceeding, we define four entities, similar to [@Sadeghi2009 Fig. 1], that will be used throughout the paper:
- $p({\mathbf{y}}|{\mathbf{x}})$ is the original forward channel;
- $p({\mathbf{x}}|{\mathbf{y}})$ is the original backward channel;
- $q({\mathbf{y}}|{\mathbf{x}})$ is an AFC;
- $r({\mathbf{x}}|{\mathbf{y}})$ is an ABC.
We define a backward channel by reversing the usual meaning of ${\mathbf{X}}$ and ${\mathbf{Y}}$, i.e., looking at ${\mathbf{X}}$ as being the output of some channel which is fed by ${\mathbf{Y}}$, which in turn is produced by some source [@Merhav1993; @Sadeghi2009]. The input and output alphabets of the AFC match the input and output alphabets of the original forward channel. Similarly, the input and output alphabets of the ABC match the input and output alphabets of the original backward channel. Note that original backward channel is associated with the original forward channel as $p({\mathbf{x}}|{\mathbf{y}})=p({\mathbf{y}}|{\mathbf{x}})p({\mathbf{x}})/p({\mathbf{y}})$. However, it is not necessary that such a relation exists between auxiliary channels, i.e., $r({\mathbf{x}}|{\mathbf{y}})$ can be any conditional distribution which does not correspond[^4] to any AFC $q({\mathbf{y}}|{\mathbf{x}})$.
Lower bounds on mutual information {#secMILowerBounds}
==================================
Lower bounds using an AFC $q({\mathbf{y}}|{\mathbf{x}})$ and an ABC $r({\mathbf{x}}|{\mathbf{y}})$ have been derived in [@Ganti2000; @Arnold2006; @Sadeghi2009] and are briefly described here for completeness.
Lower Bounds using Auxiliary Forward Channel $q({\mathbf{y}}|{\mathbf{x}})$ {#subSecLBAFC}
---------------------------------------------------------------------------
For any input distribution $p({\mathbf{x}})$, original forward channel $p({\mathbf{y}}|{\mathbf{x}})$, and AFC $q({\mathbf{y}}|{\mathbf{x}})$, a lower bound on the MI is [@Arnold2006 Eq. (41)] $$\begin{aligned}
\label{eqnIqyxleqIxy}
I({\mathbf{X}};{\mathbf{Y}}) \geq I_q({\mathbf{X}};{\mathbf{Y}}) & \triangleq \mathbb E_{{\mathbf{X}},{\mathbf{Y}}}\bigg[\log\frac{q({\mathbf{Y}}|{\mathbf{X}})}{q({\mathbf{Y}})}\bigg] \\
\nonumber &= \int_{{\mathcal X}}\int_{{\mathcal Y}} p({\mathbf{x}}, {\mathbf{y}})\log{\frac{q({\mathbf{y}}|{\mathbf{x}})}{q({\mathbf{y}})}} \mathrm{d}{\mathbf{x}}\mathrm{d}{\mathbf{y}},\end{aligned}$$ where $q({\mathbf{y}})\triangleq \int_{{\mathcal X}} p({\mathbf{x}})q({\mathbf{y}}|{\mathbf{x}})\mathrm{d}{\mathbf{x}}$ is the output distribution obtained by connecting the original source $p({\mathbf{x}})$ to the AFC. It has been shown that the lower bound, $I_q({\mathbf{X}};{\mathbf{Y}})$ of (\[eqnIqyxleqIxy\]), can be achieved by using an optimal detector, i.e., [a maximum a posteriori detector]{}, designed for the AFC and used as a receiver for the original forward channel [@Fischer1978; @Ganti2000], i.e., the decisions $\hat{{\mathbf{x}}}$ are taken as $$\begin{aligned}
\label{eqnMAPqyx}
\hat{{\mathbf{x}}}({\mathbf{y}})=\arg \max_{{\mathbf{x}}} q({\mathbf{y}}|{\mathbf{x}})p({\mathbf{x}}).\end{aligned}$$ [By achievability, we mean that if the data rate is lower than the rate calculated using [(\[eqnIqyxleqIxy\])]{}, there exist coding schemes that that can make the error probability arbitrarily low. ]{} [Since the data generated by the original forward channel is processed by a receiver that is optimized for a different AFC, this approach is known as mismatched decoding.]{}
Using the [Kullback–Leibler]{} divergence (KLD) [@Cover2006 Sec. 8.1], it can be easily verified [[@Arnold2006 Eq. (34)–(41)]]{} that (\[eqnIqyxleqIxy\]) is indeed a lower bound, $$\begin{aligned}
\label{eqnKLDqyx}
I({\mathbf{X}};{\mathbf{Y}}) - I_q({\mathbf{X}};{\mathbf{Y}})&=\mathbb E_{{\mathbf{X}},{\mathbf{Y}}}\bigg[\log\frac{p({\mathbf{Y}}|{\mathbf{X}})}{p({\mathbf{Y}})}\frac{q({\mathbf{Y}})}{q({\mathbf{Y}}|{\mathbf{X}})}\bigg] \\ &=\mathbb E_{{\mathbf{X}},{\mathbf{Y}}}\bigg[\log\frac{p({\mathbf{X}},{\mathbf{Y}})}{p({\mathbf{Y}})r_q({\mathbf{X}}|{\mathbf{Y}})}\bigg] \\ \label{eqnKLDgeq0} &=D(p({\mathbf{x}},{\mathbf{y}})||p({\mathbf{y}})r_q({\mathbf{x}}|{\mathbf{y}})) \geq 0,\end{aligned}$$ where $$\begin{aligned}
\label{eqnrqbyAFC}
r_q({\mathbf{x}}|{\mathbf{y}}) \triangleq \frac{p({\mathbf{x}})q({\mathbf{y}}|{\mathbf{x}})}{q({\mathbf{y}})}\end{aligned}$$ is the ABC induced by the AFC $q({\mathbf{y}}|{\mathbf{x}})$. A sufficient condition for the inequality (\[eqnKLDgeq0\]) to hold is that $p({\mathbf{y}}) r_q({\mathbf{x}}|{\mathbf{y}})$ is a joint distribution, i.e., $\int_{\mathcal X}\int_{\mathcal Y}p({\mathbf{y}}) r_q({\mathbf{x}}|{\mathbf{y}}) \mathrm d{\mathbf{x}}\mathrm d{\mathbf{y}}=1$ [@Cover2006 Th. 8.6.1]. [This condition is fulfilled for any combination of $p({\mathbf{y}}|{\mathbf{x}})$ and $q({\mathbf{y}}|{\mathbf{x}})$, and can be verified by using $q({\mathbf{y}})= \int_{{\mathcal X}} p({\mathbf{x}})q({\mathbf{y}}|{\mathbf{x}})\mathrm{d}{\mathbf{x}}$ in $r_q({\mathbf{x}}|{\mathbf{y}})$ of (\[eqnrqbyAFC\]).]{}
Lower Bounds using Auxiliary Backward Channel $r({\mathbf{x}}|{\mathbf{y}})$ {#secRevAuxCh}
----------------------------------------------------------------------------
There are instances such as SDBP, where an ABC $r({\mathbf{x}}|{\mathbf{y}})$ is known while the corresponding AFC $q({\mathbf{y}}|{\mathbf{x}})$ is unknown. In such cases, if (\[eqnIqyxleqIxy\]) is to be used to compute a lower bound on the MI, an AFC $q({\mathbf{y}}|{\mathbf{x}})$ has to be computed corresponding to a given ABC $r({\mathbf{x}}|{\mathbf{y}})$ and a given $p({\mathbf{x}})$. There are two challenges with this approach. Firstly, given $r({\mathbf{x}}|{\mathbf{y}})$ and $p({\mathbf{x}})$, no general method exists to compute $q({\mathbf{y}}|{\mathbf{x}})$, and depending on the input and output alphabets, there may not always exist a corresponding $q({\mathbf{y}}|{\mathbf{x}})$ or there may exist multiple solutions. Secondly, even if a solution exists to get an AFC corresponding to an ABC and $p({\mathbf{x}})$, it may be computationally very difficult to compute it. In this section, we provide an alternate approach to lower-bound the MI using $r({\mathbf{x}}|{\mathbf{y}})$, without the explicit knowledge of a corresponding AFC.
For any input distribution $p({\mathbf{x}})$ and any conditional distribution $r({\mathbf{x}}|{\mathbf{y}})$, lower bounds on the MI can be derived as [@Sadeghi2009 Eq. (39)] $$\begin{aligned}
\label{eqnIqxyleqIxy}
I({\mathbf{X}};{\mathbf{Y}}) \geq I_r({\mathbf{X}};{\mathbf{Y}}) \triangleq \mathbb E_{{\mathbf{X}},{\mathbf{Y}}}\bigg[\log\frac{r({\mathbf{X}}|{\mathbf{Y}})}{p({\mathbf{X}})}\bigg] ,\end{aligned}$$ where, similarly to (\[eqnIqyxleqIxy\]), the averaging is computed with respect to the joint distribution $p({\mathbf{x}},{\mathbf{y}})$. The lower bound (\[eqnIqxyleqIxy\]) can be also proved using KLD as $$\begin{aligned}
\label{eqnKLDRxy}
I({\mathbf{X}};{\mathbf{Y}})- I_r({\mathbf{X}};{\mathbf{Y}})&=\mathbb E_{{\mathbf{X}},{\mathbf{Y}}}\bigg[\log{\frac{p({\mathbf{X}},{\mathbf{Y}})}{p({\mathbf{Y}})r({\mathbf{X}}|{\mathbf{Y}})}}\bigg] \\
\label{eqnKLDRrgeq0} &=D(p({\mathbf{x}},{\mathbf{y}})||{p({\mathbf{y}})r({\mathbf{x}}|{\mathbf{y}})}) \geq 0 $$ For the inequality in (\[eqnKLDRrgeq0\]) to hold, $p({\mathbf{y}})r({\mathbf{x}}|{\mathbf{y}})$ should be a joint distribution, which is always true as long as $r({\mathbf{x}}|{\mathbf{y}})$ is chosen as a conditional distribution. Note that if the ABC is given by $r_q({\mathbf{x}}|{\mathbf{y}})$, i.e., induced by the AFC according to (\[eqnrqbyAFC\]), then (\[eqnIqyxleqIxy\]) and (\[eqnIqxyleqIxy\]) provide the same lower bound, i.e., $I_q({\mathbf{X}};{\mathbf{Y}})=I_r({\mathbf{X}};{\mathbf{Y}})$. However, it should be noted that (\[eqnIqxyleqIxy\]) can be used as a lower bound on the MI for any arbitrary conditional distribution $r({\mathbf{x}}|{\mathbf{y}})$, which is not necessarily related to any $q({\mathbf{y}}|{\mathbf{x}})$. It has been shown that $I_r({\mathbf{X}};{\mathbf{Y}})$ is achievable using an optimal detector, i.e., a maximum a posteriori detector, designed for the ABC $r({\mathbf{x}}|{\mathbf{y}})$ [@Ganti2000],[@Sadeghi2009 Eq. [(42)–(43)]{}]. [The decisions $\hat{{\mathbf{x}}}$ are taken as]{} $$\begin{aligned}
\label{eqnMAPrxy}
\hat{{\mathbf{x}}}=\arg \max_{{\mathbf{x}}} r({\mathbf{x}}|{\mathbf{y}}).\end{aligned}$$
### Remark 1 {#remark-1 .unnumbered}
[Either maximizing $I_q$ of (\[eqnIqyxleqIxy\]) over all possible AFCs $q({\mathbf{y}}|{\mathbf{x}})$ or maximizing $I_r$ of (\[eqnIqxyleqIxy\]) over all possible ABCs $r({\mathbf{x}}|{\mathbf{y}})$ leads to the true MI [(\[eqnMIXnYnpyx\])–(\[eqnMIXnYnpxy\])]{}.]{}
Monte Carlo Estimation of AIR
-----------------------------
By combining (\[eqnMIrate\]) and (\[eqnIqyxleqIxy\]), and (\[eqnMIrate\]) and (\[eqnIqxyleqIxy\]), we have $$\begin{aligned}
\label{eqnIrateyx} I^{\textrm{mem}}_q &\triangleq \lim_{{K},{J}\rightarrow \infty}\frac{1}{{K}}I_q({\mathbf{X}}; {\mathbf{Y}}) = \lim_{{K},{J}\rightarrow \infty} \frac{1}{{K}} \mathbb E_{{\mathbf{X}},{\mathbf{Y}}}\bigg[\log\frac{q({\mathbf{Y}}|{\mathbf{X}})}{q({\mathbf{Y}})}\bigg], \\
\label{eqnIratexy} I^{\textrm{mem}}_r &\triangleq \lim_{{K},{J}\rightarrow \infty}\frac{1}{{K}}I_r({\mathbf{X}}; {\mathbf{Y}}) = \lim_{{K},{J}\rightarrow \infty}\frac{1}{{K}} \mathbb E_{{\mathbf{X}},{\mathbf{Y}}}\bigg[\log\frac{r({\mathbf{X}}|{\mathbf{Y}})}{p({\mathbf{X}})}\bigg].\end{aligned}$$
The state-of-the-art method for the estimation of (\[eqnIrateyx\]) and (\[eqnIratexy\]) is a simulation based on MC averages [@Arnold2001; @Arnold2006]. This method utilizes the asymptotic equipartition propery for ergodic processes, which states that, for long enough sequence length ${K}$, $\log(p({\mathbf{a}}))/{K}$ for a single realization ${\mathbf{a}}$ converges to the expectation $E_{\mathbf A}[\log (p({\mathbf{a}}))/{K}]$ with probability one as ${K}\rightarrow \infty$ [@Cover2006 Ch. 3]. The channel is simulated ${N_\textrm{mc}}$ times, each time by generating input ${\mathbf{x}}$ with different random seed, say ${\mathbf{x}}^{(n)}$ for the $n$th MC run, to get a corresponding output ${\mathbf{y}}^{(n)}$ from the FOC for each MC run. The lower bound on the MIs (\[eqnIrateyx\]) and (\[eqnIratexy\]) can then be estimated as $$\begin{aligned}
\label{eqnMIMC_qyx} \hat{I}_q^{\textrm{mem}} &= \frac{1}{{N_\textrm{mc}}} \sum_{n=1}^{{N_\textrm{mc}}} \left\{\frac{1}{{K}} \log \frac{q({\mathbf{y}}^{(n)}|{\mathbf{x}}^{(n)})}{q({\mathbf{y}}^{(n)})}\right\}, \\
\label{eqnMIMC_Rxy} \hat{I}_r^{\textrm{mem}} &= \frac{1}{{N_\textrm{mc}}} \sum_{n=1}^{{N_\textrm{mc}}} \left\{\frac{1}{{K}} \log \frac{r({\mathbf{x}}^{(n)}|{\mathbf{y}}^{(n)})}{p({\mathbf{x}}^{(n)})}\right\}. $$
(-3.6,-1.3) rectangle (6.1,0.15);
(-3.6,-1.2) – (-3.6,1.6) node at (-2.5,1.8) [GMP-SDBP]{}; (-3.6,1.6) – (6.1,1.6) ; (6.1,0.4) – (6.1,1.6) ; (-0.8,0.4) – (6.1,0.4) ; (-0.8,-1.2) – (-0.8,0.4) ; (-3.6,-1.2) – (-0.8,-1.2) ;
(-0.8,0.4) rectangle (6.1,1.6); (-3.6,-1.2) rectangle (-0.8,1.6);
(-3.6,-1.3) rectangle (6.1,0.15) node at (-2.5,0.4) [SBS-SDBP]{};
(-0.25,0.5) rectangle (2,1.5) node at (0.9,1) [GMP]{}; (-1,1) – (-0.25,1); (2,1) – (3.75,1) node at (3,1.2) [$\tilde{r}_k(\cdot|{\mathbf{y}})$]{}; (3.75,0.5) rectangle (6,1.5) node at (4.9,1.2) [Evaluate in $\Omega$]{} node at (4.9,0.8) [+ normalize]{}; (6,1) – (7,1) node at (6.8,1.3) [$r_k(\cdot|{\mathbf{y}})$]{}; (-1,-0.5) – (-1,1); (-0.25,0) rectangle (2,-1) node at (0.9,-0.25) [MF+samp.+]{} node at (0.9,-0.75) [Gauss. approx.]{}; (-1.5,-0.5) – (-0.25,-0.5); (2,-0.5) – (3.75,-0.5) node at (3,-0.25) [$\tilde{r}_k(\cdot|{\mathbf{y}})$]{}; (3.75,0) rectangle (6,-1) node at (4.9,-0.3) [Evaluate in $\Omega$]{} node at (4.9,-0.7) [+ normalize]{}; (6,-0.5) – (7,-0.5) node at (6.8,-0.2) [$r_k(\cdot|{\mathbf{y}})$]{}; (-1.5,0) rectangle (-3.5,-1) node at (-2.5,-0.5) [SDBP]{}; (-5,-0.5) – (-3.5,-0.5) node at (-4.25,-0.25) [${\mathbf{y}}$]{} node at (-4.25,-0.75) [${\mathbf{y}}\in \mathbb C^{{J}}$]{} ; (-5,0) rectangle (-7,-1) node at (-6,-0.5) [Fiber link]{}; (-7.5,-0.5) – (-7,-0.5) node at (-7.25,-0.25) [${\mathbf{s}}$]{} node at (-7.25,-1.25) [${\mathbf{s}}\in \mathbb C^{J}$]{} ; (-7.5,0) rectangle (-9.5,-1) node at (-8.5,-0.5) [Pulse shaper]{}; (-11,-0.5) – (-9.5,-0.5) node at (-10.25,-0.25) [${\mathbf{x}}$]{} node at (-10.25,-0.75) [${\mathbf{x}}\in \Omega^{K}$]{} node at (-10.25,-1.25) [$\Omega \subseteq \mathbb C$]{} ;
(-9.5,0.25) – coordinate\[below=10pt\](C) (-5,0.25) node at (-7.2,0.8) [FOC]{};
(-1.5,-1.5) rectangle (-3.5,-2.5) node at (-2.5,-2) [DBP]{}; (-4.3,-2) – (-3.5,-2); (-4.3,-2) – (-4.3,-0.5); (-1.5,-2) – (-0.25,-2) ; (2,-1.5) rectangle (-0.25,-2.5) node at (1,-2) [MF+samp.]{}; (2,-2) – (3.75,-2) node at (2.8,-1.8) [${\mathbf{z}}$]{} node at (3,-2.3) [${\mathbf{z}}\in \mathbb C^{{K}}$]{}; (6,-1.5) rectangle (3.75,-2.5) node at (4.9,-1.8) [Estimate pars.]{} node at (5,-2.2) [for Gauss.]{}; (6,-2) – (7,-2) node at (6.8,-1.7) [$q(z_k|x_k)$]{};
Computation of AIR for the FOC {#secAuxSDBP}
==============================
The computation of AIRs for the FOC using different auxiliary channels is abstracted in Fig. \[figDiffAuxChanlSDBP\]. The input data ${\mathbf{x}}$ is sent through a pulse shaper followed by the fiber link to get the output of the FOC, ${\mathbf{y}}$. This output is fed either to SDBP or DBP, which aim to undo the impairments induced by the FOC. The effect of pulse shaping is reversed using either the output from DBP or through one of the two techniques for SDBP to get the auxiliary channels. In the section, these processes will be detailed.
Computation of AIR using DBP
----------------------------
The traditional approach of computing the AIR is by assuming the AFC to be memoryless[^5]. This assumption is justified by using a post-processing block, such as a DBP block for nonlinear compensation, after the FOC as part of the AFC. The output statistics of the FOC and the DBP has been considered memoryless with additive Gaussian noise with the same variance in all dimensions [@Secondini2013; @Fehenberger2015a]. Recently, the correlation between the in-phase and quadrature components was accounted for in the Gaussian assumption [@Eriksson2016]. In that study, it was shown that significant gains in the AIR are possible depending on the scenarios such as inline dispersion compensation at high powers. We will use both these approaches in benchmarking the results. As shown in Fig. \[figDiffAuxChanlSDBP\], let ${\mathbf{z}}=[z_1, z_2,\ldots,z_K]$, with $z_i \in \mathbb C$ for $i=1,2,\ldots,K$, be the signal after DBP, matched filtering (MF) and sampling. According to the [data-processing inequality]{} [@Cover2006 Ch. 2], the information content of a signal cannot be increased after post-processing and hence we have $I({\mathbf{X}}; {\mathbf{Z}})\leq I({\mathbf{X}}; {\mathbf{Y}})$, where $$I({\mathbf{X}}; {\mathbf{Z}})=\mathbb E_{{\mathbf{X}},{\mathbf{Z}}}\bigg[\log\frac{p({\mathbf{Z}}|{\mathbf{X}})}{p({\mathbf{Z}})}\bigg]$$ with $p({\mathbf{z}})=\int_{{\mathbf{x}}} p({\mathbf{x}})p({\mathbf{z}}|{\mathbf{x}})\mathrm{d}{\mathbf{x}}$. Similar to the lower bound (\[eqnIqyxleqIxy\]) using AFC, we can define $$\begin{aligned}
\label{eqnIqzxleqIxy}
I_q({\mathbf{X}};{\mathbf{Z}}) \triangleq \mathbb E_{{\mathbf{X}},{\mathbf{Z}}}\bigg[\log\frac{q({\mathbf{Z}}|{\mathbf{X}})}{q({\mathbf{Z}})}\bigg] \leq I({\mathbf{X}};{\mathbf{Z}}),\end{aligned}$$ with $q({\mathbf{z}})=\int_{{\mathcal X}} p({\mathbf{x}})q({\mathbf{z}}|{\mathbf{x}})\mathrm{d}{\mathbf{x}}$. Using DBP as detector, $q({\mathbf{z}}|{\mathbf{x}})$ is commonly assumed to factorize into marginal distributions, i.e., $q({\mathbf{z}}|{\mathbf{x}})=\prod_{k=1}^K q(z_k|x_k)$ and $q({\mathbf{z}})=\prod_{k=1}^K q(z_k)$. By using these factorizations in (\[eqnIqzxleqIxy\]) and using an equivalent of (\[eqnMIMC\_qyx\]) for ${\mathbf{z}}$ as output, we have $$\begin{aligned}
\label{eqnMCmemlessXZ}
\hat{I}_q(X;Z) = \frac{1}{{N_\textrm{mc}}}\sum_{n=1}^{{N_\textrm{mc}}} \left\{ \frac{1}{{K}}\sum_{k=1}^{{K}} \log\frac{q(z_k^{(n)}|x_k^{(n)})}{q(z_k^{(n)})} \right \},\end{aligned}$$ where, similarly to the approach in [@Eriksson2016], $q(z_k|x_k)$ is [assumed]{}[^6] to be [a Gaussian distribution with a different mean and covariance matrix for each possible value of $x_k$. In particular, the real and imaginary components of $z_k$ are taken to be either]{} independent and identically distributed Gaussian (iidG) or correlated Gaussian (CG). In both these variations, a training phase is employed to obtain mean and variance corresponding to each of the constellation points. This training phase is visualized as the ‘Estimate pars. for Gauss.’ block in Fig. \[figDiffAuxChanlSDBP\], referring to the estimation of parameters for the Gaussian distribution. The means are obtained using [@Eriksson2016 Eq. (8)], and variances for iidG and CG are obtained using [@Eriksson2016 Eq. (9)] and [@Eriksson2016 Eq. (10)], respectively.
### Remark 2 {#remark-2 .unnumbered}
[Note that even though AIRs for DBP are computed using an AFC, the same AIRs are also obtained using an ABC induced by this AFC, i.e., $q({\mathbf{z}}|{\mathbf{x}})p({\mathbf{x}})/q({\mathbf{z}})$.]{}
Computation of AIR using SDBP
-----------------------------
DBP is not an optimal processing strategy for nonlinear compensation. Indeed, some residual memory due to signal–noise interaction is present even after DBP is performed. As SDBP accounts for this memory, it may lead to tighter bounds on the MI.
The theory behind SDBP is based on factor graphs and message passing, and is derived and explained in detail in [@Irukulapati2014TCOM], while improved versions of SDBP are found in [@Irukulapati2014ECOC; @Wymeersch2015SPAWC; @Irukulapati2015JLT]. A short summary of SDBP is provided here for completeness. SDBP compensates not only for linear and nonlinear effects existing in the fiber but also accounts for the noise from the amplifiers. The main idea of SDBP is to represent the uncertainty present in the unobserved signals at each stage of the FOC. These unobserved signals are signals after each of the linear and nonlinear blocks of the split-step Fourier method (SSFM) and also the signals after the amplifiers. Uncertainty is captured through a collection of ${N_p}$ waveforms, which represents a distribution. These ${N_p}$ waveforms are passed through the inverse of each of the blocks of the FOC, starting from the received signal ${\mathbf{y}}$. SDBP can thus be viewed as an algorithm that takes an input ${\mathbf{y}}\in \mathbb C^{{J}}$ and returns a vector of ${N_p}$ outputs, where each of these ${N_p}$ outputs is in $\mathbb C^{{J}}$, and describes the knowledge the receiver has regarding the variable ${\mathbf{s}}$ in Fig. \[figDiffAuxChanlSDBP\]. To account for the effect of pulse shaping, two different approaches, SBS-SDBP and GMP-SDBP, are proposed in [@Irukulapati2014TCOM] and [@Wymeersch2015SPAWC], respectively, and are explained briefly below.
In the first approach, SBS-SDBP, the output after SDBP is passed through MF followed by sampling[^7]. Corresponding to each symbol $x_k$, ${N_p}$ outputs are approximated with a multivariate Gaussian distribution, $\tilde{r}_k(\cdot|{\mathbf{y}})$. In the second approach, referred to as GMP-SDBP in this paper, all ${N_p}$ outputs from SDBP are first approximated with a multivariate Gaussian distribution and then Gaussian message passing (GMP) is applied according to [@Loeliger2007FG Table III] instead of an MF, and $\tilde{r}_k(\cdot|{\mathbf{y}})$ is obtained. In SBS-SDBP and GMP-SDBP, the distribution $\tilde{r}_k(\cdot|{\mathbf{y}})$ is evaluated at $x_k \in \Omega$ and normalized to get an ABC as shown in Fig. \[figDiffAuxChanlSDBP\]. By assuming that the input distribution $p({\mathbf{x}})$ [is the]{} product of its marginals, i.e., $p_{{\mathbf{X}}}({\mathbf{x}})=\prod_{k=1}^{{K}}p_{X}(x_k)$ and by assuming that the ABC is factorized as $$\begin{aligned}
\label{eqnRxyRxky}
r({\mathbf{x}}|{\mathbf{y}}) \triangleq \prod_{k=1}^{{K}} r_k(x_k|{\mathbf{y}}),\end{aligned}$$ (\[eqnMIMC\_Rxy\]) becomes $$\begin{aligned}
\label{eqnMIMC_Irxvecy}
\hat{I}_r^{\textrm{mem}} = \frac{1}{{N_\textrm{mc}}}\sum_{n=1}^{{N_\textrm{mc}}} \left\{ \frac{1}{{K}}\sum_{k=1}^{{K}} \log\frac{r_k(x_k^{(n)}|{\mathbf{y}}^{(n)})}{p_X(x_k^{(n)})} \right \},\end{aligned}$$ where $r_k(x_k^{(n)}|{\mathbf{y}}^{(n)})$ is obtained by either SBS-SDBP or GMP-SDBP.
### Effect of Number of Particles in SDBP {#effect-of-number-of-particles-in-sdbp .unnumbered}
Given an output sequence ${\mathbf{y}}$, SDBP provides an ABC $r_k(\cdot|{\mathbf{y}})$. However, in practice, this distribution is dependent on the number of particles, ${N_p}$. Specifically, the statistics of $r_k(\cdot|{\mathbf{y}})$ are better captured with a high number of particles, and the statistics do not change significantly after a certain ${N_p}>{N_p}'$. In other words, when ${N_p}<{N_p}'$, distributions obtained from SDBP can be dependent on the particular noise realization used in SDBP. However, the bounds obtained are still valid although they are not [as]{} tight as with ${N_p}>{N_p}'$. [In order to prove this, we consider the noise realizations added in SDBP ${\mathbf{w}}\in \mathbb C^{{J}\times {N_p}\times N}$ as an additional variable independent of ${\mathbf{x}}$ provided by the detector itself, where $N$ is the number of spans in the fiber link. The ABC provided by SDBP can then be written as $r_k(\cdot|{\mathbf{y}},{\mathbf{w}})$. By plugging this in the ABC lower bound (\[eqnIqxyleqIxy\]), an achievable lower bound $I_r({\mathbf{X}};{\mathbf{Y}},{\mathbf{W}})$ can be computed between the input ${\mathbf{X}}$ and output pairs ${\mathbf{Y}}$, ${\mathbf{W}}$ as $$\begin{aligned}
\label{eqnIrxywleqIxy}
I_r({\mathbf{X}};{\mathbf{Y}},{\mathbf{W}}) &= \iiint p({\mathbf{x}}, {\mathbf{y}},{\mathbf{w}})\log{\frac{r({\mathbf{x}}|{\mathbf{y}},{\mathbf{w}})}{p({\mathbf{x}})}} \mathrm{d}{\mathbf{x}}\mathrm{d}{\mathbf{y}}\mathrm{d}{\mathbf{w}}, \\
\label{eqnIrxywleqIxy_2}&=\int p({\mathbf{w}}) \mathrm{d}{\mathbf{w}}\iint p({\mathbf{x}}, {\mathbf{y}})\log{\frac{r({\mathbf{x}}|{\mathbf{y}},{\mathbf{w}})}{p({\mathbf{x}})}} \mathrm{d}{\mathbf{x}}\mathrm{d}{\mathbf{y}},\\
\label{eqnIrxywleqIxy_3}&= \int p({\mathbf{w}}) \mathrm{d}{\mathbf{w}}I_{r_{\mathbf{w}}}({\mathbf{X}};{\mathbf{Y}}) \\
\label{eqnIrxywleqIxy_4}&\leq \int p({\mathbf{w}}) \mathrm{d}{\mathbf{w}}I({\mathbf{X}};{\mathbf{Y}}) = I({\mathbf{X}};{\mathbf{Y}}),\end{aligned}$$ where in (\[eqnIrxywleqIxy\_2\]), we factorized the joint distribution due to the fact that ${\mathbf{w}}$ does not provide any additional information about ${\mathbf{x}}$ (as it is independent of [${\mathbf{x}}$]{} given ${\mathbf{y}}$). $r_{\mathbf{w}}$ in (\[eqnIrxywleqIxy\_3\]) is to explicitly show that the ABC is dependent on ${\mathbf{w}}$ and the inequality in (\[eqnIrxywleqIxy\_4\]) follows from (\[eqnIqxyleqIxy\]) for any ABC $r$. It can be noticed that the inequality in (\[eqnIrxywleqIxy\_4\]) is valid irrespective of ${N_p}$. ]{}
Numerical Results and Discussion {#secNumSim}
================================
System Parameters
-----------------
The FOC used in this paper is identical to [@Irukulapati2015JLT Fig. 1] with a single-channel system comprising a single-polarization transmitter[^8] and a fiber link [consisting of $N$ spans. Each span of the fiber link consists of a transmission fiber of length $L$]{}, which is a standard single-mode fiber (SMF) simulated using SSFM, and a fiber Bragg grating (FBG) for optical [dispersion-managed]{} (DM) links. In between fiber spans, there are EDFAs that compensate for the losses in the fiber. The transmitter uses a root raised cosine pulse shaper with a roll-off factor of 0.25 and truncation length of 16 symbol periods. The modulation format is 64-QAM and the symbol rate, $R_\text{s}$, is either $14$ GBd or $28$ GBd. The parameters used for the SMF are [a dispersion coefficient of $D=16$ ps/(nm km), a Kerr nonlinearity parameter of $\gamma=1.3~\text{(W km)}^{-1}$, and an attenuation of $0.2$ dB/km,]{} which are according to the ITU-T G.652 standard. Propagation in the fiber is simulated using the SSFM with a segment length [@Zhang2008stepSize] of $\Delta = (\epsilon L_\text{N} L_\text{D}^{2})^{1/3},$ where $\epsilon = 10^{-4}$, $L_\text{N}=1/(\gamma P)$ is the nonlinear length, $L_\text{D} = {2 \pi c}/(R_\text{s}^{2}|D|\lambda^2)$ is the dispersion length, $\lambda$ is the wavelength, $c$ is the speed of light, and $P$ is the average input power to each fiber span. We used the same segment lengths for simulating the channel and for both DBP and SDBP. An FBG with an insertion loss of $3$ dB and perfect dispersion compensation for the preceding SMF is used. The noise figure is $5.5$ dB for each of the amplifiers. [Ideal band-pass filters with an equivalent low-pass bandwidth]{} equal to the symbol rate are used in the EDFAs and at the input of the receiver. The [number]{} of particles used in the SDBP approach is ${N_p}=500$ for both SBS-SDBP and GMP-SDBP. The input distribution is assumed to be uniform, i.e., $p_X(x_k)=1/|\Omega|$ for $x_k \in \Omega$.
Results
-------
[[![Gains in AIR for 64-QAM, $N=30$ over DBP-CG for 14 GBd (solid line) and 28 GBd (dashed line).[]{data-label="fig_AIRSDBP_2"}](GainMI_diffL_64QAM_FBG_14G_SBS_GMP_SDBP_DBP_iidG_CG_N30_Np500_Nsymb1024_index5_47.eps "fig:"){width="50.00000%"} ]{}]{}
Fig. \[fig\_AIRSDBP\] shows the lower bounds on the MI, AIR, as a function of input power, obtained through different auxiliary channels for 14 GBd over a 64-QAM FBG link. For reference, the capacity of the AWGN channel and the constrained capacity of an equivalent AWGN channel with 64-QAM are also shown. We notice a trend for the DBP approaches (diamond and square markers) to have lower AIRs than the SDBP approaches (plus and circle markers), and also that the AIRs behave differently for DBP and SDBP. Specifically, we can observe from the DBP curves that up to about $0$ dBm power, the behavior of this link is approximately linear. At higher powers [(beyond $1$ dBm for DBP and $4$ dBm for SDBP)]{}, nonlinearity comes into play and the AIR decreases. However, the AIR for DBP decreases faster than that of SDBP, which means that SDBP performs better in the nonlinear regime. This is an expected behavior because SDBP accounts for the nonlinear signal–noise interactions, which DBP does not account for. If we compare AIRs between DBP techniques, DBP-CG (square markers) has better AIR than DBP-iidG (diamond markers), as the former accounts for the correlation between the in-phase and quadrature components. Also, we can observe that GMP-SDBP has better AIR than SBS-SDBP, as GMP-SDBP is a more principled way of computing a specific message, whereas SBS-SDBP is a heuristic approach.
Lower bounds on the MI similar to Fig. \[fig\_AIRSDBP\] were computed for SBS-SDBP, GMP-SDBP, DBP-CG, and DBP-iidG by varying the number of spans for 14 GBd, 64-QAM, and $L=100$ km. For each $N$, the maximum AIR is computed for these four approaches, and the gains are plotted in Fig. \[fig\_AIRSDBP\_3\]. The DBP-CG and DBP-iidG approaches are used to benchmark the performance of GMP-SDBP and SBS-SDBP. The difference between the maximum AIR for SBS-SDBP or GMP-SDBP and the maximum AIR for DBP-CG (resp. DBP-iidG) is shown in dashed (resp. solid) lines. The gains of GMP-SDBP (resp. SBS-SDBP) are shown using red circle (resp. black diamond) markers. All the curves have a maximum at some intermediate point as the gain between SDBP and DBP will be zero at small $N$ (both AIRs saturate to the maximum value), and at large $N$, both AIRs vanish. The gains are lower for both GMP-SDBP and SBS-SDBP when DBP-CG is used as a benchmark technique. This is expected, and is in line with the conclusions of [@Eriksson2016], as DBP-CG has better AIR than DBP-iidG. We can observe that using GMP-SDBP, 0.7 bits/symbol higher AIR is obtained compared to DBP-CG.
The span length is varied by keeping $N=30$ for the 64-QAM FBG link. Similarly to Fig. \[fig\_AIRSDBP\_3\], the highest AIR is computed, and the gains over DBP-CG are plotted in Fig. \[fig\_AIRSDBP\_2\]. We plot the gains for 14 GBd (diamond markers) and 28 GBd (circle markers) and observe that the gains for 14 GBd are higher than for 28 GBd. This behavior was observed also in our previous research [@Irukulapati2014TCOM; @Irukulapati2015JLT]. The SDBP accounts for [signal–noise]{} interactions, which decreases as we increase the symbol rate. Therefore, the performance of SDBP approaches DBP with increasing symbol rate.
Discussion
----------
The computation of lower bounds on the MI using either an AFC or an ABC are two different ways with their own advantages and disadvantages. If an AFC is available, then $q({\mathbf{y}})$ has to be calculated first, which may involve some integrals, and (\[eqnIqyxleqIxy\]) is used to lower-bound the MI. Lower bounds on the MI can be obtained using an ABC by using any conditional distribution $r({\mathbf{x}}|{\mathbf{y}})$, i.e., by removing the constraint that $r_q({\mathbf{x}}|{\mathbf{y}})$ in (\[eqnKLDgeq0\]) is induced by an AFC. That is, if an ABC is available, then (\[eqnIqxyleqIxy\]) can be used to lower-bound the MI without explicitly finding an AFC. The true MI can in theory be obtained either by maximizing over all possible AFCs in (\[eqnIqyxleqIxy\]), or by maximizing over all possible ABCs in (\[eqnIqxyleqIxy\]). In this paper, we obtained ABCs from the SDBP algorithm and results indicate that GMP-SDBP gives the tightest lower bound on the MI compared to DBP-iidG, DBP-CG, and SBS-SDBP.
We will discuss the extension of the results for dual polarization and comment on the complexity. Firstly, computation of AIRs through the ABC using (\[eqnMIMC\_Irxvecy\]) is applicable for dual polarization also.[SBS-SDBP has been developed for dual polarization [@Irukulapati2014TCOM] and GMP-SDBP can be extended to dual polarization. A single polarization was used for computing AIRs with GMP-SDBP in this paper for computational simplicity.]{} We note that the polarization mode dispersion for a dual-polarization transmitter degrades the performance of both DBP and SDBP, as observed in [@Irukulapati2014TCOM], and hence the AIRs of Fig. \[fig\_AIRSDBP\] will be lowered. However, we conjecture that the relative gains of SDBP compared to DBP would be similar to what we have shown in Fig. \[fig\_AIRSDBP\] for a single polarization. Secondly, the number of segments per span used in the simulation of the fiber using SSFM is the same as the ones used for DBP and SDBP. There exist many low-complexity variations of DBP, where the number of segments is optimized for real-time implementation. Low-complexity variations of SDBP can be derived, e.g., by optimizing the number of particles or segments per span [@Irukulapati2014TCOM]. Tighter lower bounds may possibly be obtained than those reported in the paper. Here we present two different methods. Firstly, in the GMP-SDBP, we used a linear Gaussian message passing algorithm to account for the effect of the pulse shaper, which may not be an optimal strategy for the FOC. We conjecture that when the distribution is represented in a particle form, techniques other than linear Gaussian message passing might yield even tighter bounds than those presented in the paper. Secondly, tighter bounds on the MI can be obtained by extending the principle used from AFC to ABC to a more general technique. The required [property]{} to lower-bound the MI using an AFC and also an ABC is $D\geq0$ in (\[eqnKLDgeq0\]) and (\[eqnKLDRrgeq0\]). This principle can be extended by allowing $p({\mathbf{y}})$ in (\[eqnKLDgeq0\]) to be an arbitrary probability density function over ${\mathbf{y}}$ [that is not necessarily induced by $p({\mathbf{y}}|{\mathbf{x}})$ or $q({\mathbf{y}}|{\mathbf{x}})$]{} [@Sadeghi2009].
Conclusion {#secConcl}
==========
Traditionally, lower bounds on the MI were computed using an AFC. In this paper, we computed lower bounds using an ABC for the first time for the FOC. These bounds are achievable by a maximum a posteriori detector based on the ABC. Two different distributions obtained through the SDBP algorithm are used as ABCs for estimation of AIRs. Both these distributions have better AIR in comparison to the state-of-the-art method of using DBP. Through simulations, it was also found that up to 0.7 bit/symbol higher AIR is obtained using GMP-SDBP compared to DBP. This means that in comparison to the DBP approach, tighter lower bounds can be obtained using the SDBP approach.
Acknowledgments
===============
The authors would like to thank Rahul Devassy, Kamran Keykhosravi, Assoc. Prof. Giulio Colavolpe, Tobias Fehenberger, and Dr. Amina Piemontese for fruitful discussions.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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[^1]: N. V. Irukulapati, H. Wymeersch, and E. Agrell are with the Dept. of Signals and Systems at Chalmers University of Technology, 41296 Gothenburg Sweden. M. Secondini is with the Institute of Communication, Information, and Perception Technologies, Scuola Superiore Sant’Anna, Pisa, Italy. P. Johannisson is with Acreo Swedish ICT AB, 40014 Gothenburg, Sweden. Corresponding author’s email: [email protected]
[^2]: This research was supported by the Swedish Research Council (VR) under grant 2013-5642 and the European Research Council under grant no. 258418 (COOPNET). The simulations were performed in part on resources provided by the Swedish National Infrastructure for Computing (SNIC) at C3SE.
[^3]: [All logarithms in this paper are in base $2$; therefore, MI will be measured in bits. To simplify the notation, we used $p({\mathbf{y}}|{\mathbf{x}})$, $p({\mathbf{x}})$, and $p({\mathbf{y}})$ instead of explicitly writing $p_{{\mathbf{Y}}|{\mathbf{X}}}({\mathbf{y}}|{\mathbf{x}})$, $p_{{\mathbf{X}}}({\mathbf{x}})$, and $p_{{\mathbf{Y}}}({\mathbf{y}})$. So, $p({\mathbf{y}}|{\mathbf{x}})$, $p({\mathbf{x}})$, $p({\mathbf{y}})$, and their auxiliary counterparts refer to different distributions. For the limit in (\[eqnMIrate\]) to exist, we assume the existence of sequences of distributions $p({\mathbf{y}}|{\mathbf{x}})$, $p({\mathbf{x}})$, and $p({\mathbf{y}})$ for ${K}=1,2,\ldots$]{}
[^4]: To highlight this difference, we chose to use $r({\mathbf{x}}|{\mathbf{y}})$ for an ABC instead of $q({\mathbf{x}}|{\mathbf{y}})$.
[^5]: When the output of the channel $y_i$ at discrete time $i$, given the channel input $x_i$ at time $i$, is independent of channel inputs and outputs at all other times, we call the channel as memoryless.
[^6]: When superscripts for indicating MC run $n$ are omitted as in $q(z_k|x_k)$, it should be interpreted as applying to any general MC run.
[^7]: There is residual memory left after SBS-SDBP [@Irukulapati2014ECOC] as MF followed by sampling is a linear technique and may not be the optimal processing for the nonlinear FOC [@Liga2015; @Agrell2015RSTA]. This residual memory was accounted for by using the Viterbi algorithm on the samples obtained after MF, and was shown to have improved performance compared to SBS-SDBP [@Irukulapati2015JLT].
[^8]: The theory remains the same for dual-polarization FOC also. As GMP-SDBP involves inverting a very high-dimensional matrix, we simulated only for single-polarization in this paper.
|
---
abstract: 'We prove an identity conjectured by Adin and Roichman involving the descent set of $\lambda$-unimodal cyclic permutations. These permutations appear in formulas for characters of certain representations of the symmetric group. Such formulas have previously been proven algebraically. In this paper, we present a combinatorial proof for one such formula and discuss the consequences for the distribution of the descent set on cyclic permutations.'
author:
- Kassie Archer
bibliography:
- 'lambdabib.bib'
title: 'Descents of $\lambda$-unimodal cycles in a character formula'
---
Introduction
============
Given a composition $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k)$ of $n$, we say a permutation is *$\lambda$-unimodal* if the permutation, written in its one-line notation, is the concatenation of unimodal segments of length $\lambda_i$. (See Figure \[Fig:1\] for an example.) These permutations and their descent sets appear in the formulas for certain characters of representations of the symmetric group [@char; @arc; @note]. These formulas are of the same form found in Theorem \[thm:main\], where the sum occurs over $\lambda$-unimodal permutations with some extra property which varies based on the character. Known formulas for characters include sums over $\lambda$-unimodal permutations which are involutions, are in some given Knuth class, or have a given Coxeter length [@char].
We prove a formula of this type originally conjectured by Ron Adin and Yuval Roichman [@yuv]. Suppose $\chi$ is the character of the representation on ${\mathcal{S}}_n$ induced from the primitive linear representation on a cyclic subgroup generated by an $n$-cycle. Let $\chi_\lambda$ be its value on the conjugacy class of type $\lambda$. Denote by $S(\lambda)$ the set of partial sums $\{\lambda_1, \lambda_1+\lambda_2, \ldots, \lambda_1 + \cdots + \lambda_k\}$ and by ${\mathcal{C}}(\lambda)$ the set of $\lambda$-unimodal permutations which are also *cyclic*, meaning they can be written in cycle-notation as a single $n$-cycle. Theorem \[thm:main\], which we will prove in Section \[sec:pf\], is the main result of this paper.
\[thm:main\] For every composition $\lambda$, $$\label{eq:main}
\chi_\lambda = \sum_{\pi \in {\mathcal{C}}(\lambda)} (-1)^{|\operatorname{Des}(\pi) \setminus S(\lambda)|}.$$
It is an simple exercise to show (see Proposition \[prop:ind\]) that the character described above takes the following values. $$\label{eq:chi}
\chi_\lambda = \begin{cases} (k-1)! d^{k-1} \mu(d) & \text{if } \lambda = (d^k) \\ 0 & \text{otherwise} \end{cases}$$ where $\mu$ denotes the number-theoretic Möbius function, which takes values $\mu(d) = (-1)^t$ when $d$ is the square-free product of $t$ distinct primes, and $\mu(d) = 0$ otherwise.
Previously, these types of character formulas have been proven algebraically [@Roi1; @Roi2; @Roi3]. Here, we prove Theorem \[thm:main\] using combinatorial methods by showing that the sum on the right hand side of Equation takes the same values as $\chi_\lambda$ in Equation . To do this, we use a relationship between $\lambda$-unimodal permutations and primitive words developed in [@archer].
In Section \[sec:cons\], we will see that Theorem \[thm:main\] implies interesting results about the distribution of the descent set on ${\mathcal{C}}_n$, the set of cyclic permutations. For example, the descent sets of elements of ${\mathcal{C}}_n$ are equi-distributed with the descent sets of the standard Young tableaux that form a basis to the representation described above. Additionally, the number of permutations of ${\mathcal{S}}_{n-1}$ with descent set $D$ is equal to the number of permutations of ${\mathcal{C}}_{n}$ whose descent set is either $D$ or $D \cup [n-1]$. Different proofs of these consequences can alternatively be found in [@note] and [@sergi], respectively.
Finally, there is a special case of Theorem \[thm:main\] when $\lambda = (n)$ in terms of more well-known properties and statistics of permutations. This gives the result $$\sum_{\substack{\pi \in {\mathcal{C}}_n \\ \text{unimodal}}} (-1)^{\operatorname{des}(\pi)} = \mu(n),$$ of which the author knows of no other proof.
Background
==========
Definitions and notation {#subset:defn}
------------------------
Let ${\mathcal{S}}_n$ denote the set of permutations on $[n] = \{1,2,\ldots, n\}$. We write permutations in their one-line notation as $\pi = \pi_1\pi_2\cdots \pi_n$. A *cyclic* permutation is a permutation $\pi \in {\mathcal{S}}_n$ which can be written in its cycle notation as a single $n$-cycle. For example, the permutation $\pi= 36578124$ is cyclic since it can be written in cycle notation as $\pi = (13584726)$. We denote the set of cyclic permutations of size $n$ as ${\mathcal{C}}_n$. The *descent set* of a permutation is the set $\operatorname{Des}(\pi) = \{i : 1\leq i\leq n-1, \pi_i>\pi_{i+1}\}$. The size of the descent set (or number of descents) is denoted by $\operatorname{des}(\pi)$.
We say that a word $x_1 x_2 \cdots x_n$ is *unimodal* if there is some $m$, with $1\leq m \leq n$, for which $$x_1<x_2 < \cdots< x_m>x_{m+1}> \cdots >x_n.$$ That is, the word is increasing, then decreasing. For example, the word $367841$ is unimodal.
A *composition of $n$* is an ordered list $\lambda = (\lambda_1, \lambda_2, \ldots,\lambda_k)$ so that $\lambda_1+ \lambda_2+\cdots + \lambda_k = n$. For the remainder of the paper, we will assume that $\lambda$ is a composition of $n$ of length $k$. We say a permutation $\pi \in {\mathcal{S}}_n$ is *$\lambda$-unimodal*[^1] if when one breaks $\pi$ into contiguous segments of lengths $\lambda_i$, each segment is unimodal. That is, defining the partial sums of $\lambda$ by $s_i(\lambda) = \lambda_1 + \lambda_2 + \cdots + \lambda_i$, the segment $\pi_{s_{i-1}(\lambda) + 1} \ldots \pi_{s_{i}(\lambda)}$ of $\pi$ is unimodal for all $1\leq i \leq k$. See Figure \[Fig:1\] for an example of a $\lambda$-unimodal permutation.
(0,0) rectangle (9,9); (0,0) grid (9,9); (0,0) – (1,1); (0,1) – (1,0); (1,3) – (2,4); (1,4) – (2,3); (2,8) – (3,9); (2,9) – (3,8); (3,6) – (4,7); (3,7) – (4,6); (4,4) – (5,5); (4,5) – (5,4); (5,2) – (6,3); (5,3) – (6,2); (6,1) – (7,2); (6,2) – (7,1); (7,7) – (8,8); (7,8) – (8,7); (8,5) – (9,6); (8,6) – (9,5);
A *word* of length $n$ on $m$ letters is a string $s = s_1s_2\ldots s_n$ where $s_i \in \{0,1,\ldots, m-1\}$. A *necklace* of length $n$ on $m$ letters is an equivalence class of words $[s]$ so that $t = t_1t_2\ldots t_n \sim s = s_1s_2\ldots s_n$ if and only if $t_1t_2\ldots t_n = s_is_{i+1} \ldots s_ns_1\ldots s_{i-1}$ for some $1\leq i \leq n$, that is, $t$ is some cyclic rotation of $s$. For example, $1101 \sim 1011\sim 0111\sim 1110$. Denote by $W_m(n)$ the set of words of length $n$ on $m$ letters and by $N_m(n)$ the set of necklaces of length $n$ on $m$ letters.
We call a word $s$ (or a necklace $[s]$) *primitive* if there is no strictly smaller word $q$ so that $s = q^r$ for some $r>1$, where $q^r$ denotes the concatenation of $q$ with itself $r$ times. We denote the number of primitive necklaces of length $n$ on $m$ letters by $L_m(n)$. We let $a_t(s) = |\{j \in [n] : s_j = t\}|$, that is the number of copies of $t$ in word $s$ and we let $o(s) = \sum_{\text{odd } t} a_t(s)$, that is the number of odd letters in word $s$. We denote by $L(a_1, a_2, \ldots, a_m)$ the size of the set of primitive necklaces $[s]$ so that $a_t(s) = a_t$. The enumeration of this set where $\sum_i a_i = n$ is well-known.
Suppose that $a_1+ a_2+ \cdots a_m = n$ and that $a_i\geq 1$ for all $1\leq i \leq m$. Then, $$\label{eq:L}
L(a_1, a_2\ldots, a_m) =
\frac{1}{n}\sum_{\ell \mid\gcd(a_1, \ldots, a_m)}
\mu(\ell) \frac{(n/\ell)!}{(\frac{a_1}{\ell})!(\frac{a_2}{\ell})! \cdots(\frac{a_m}{\ell})!}.$$
Define the set $N_\lambda \subseteq N_{2k}(n)$ to be the set of necklaces $[s]$ so that $a_{2t}(s) + a_{2t+1}(s) = \lambda_{t+1}$ for all $0\leq t \leq k-1$ and so that $[s]$ is either primitive or $s = q^2$ for some primitive word $q$ so that $o(q)$ is odd. For example, the word $s = 00121\in N_{(4,1)}$ since $a_0(s) + a_1(s) = 2+2 = 4$ and $a_2(s) + a_3(s) = 1+0 = 1$ and $s$ is primitive. For another example, $t = 0213302133$ is in $N_{(4,6)}$ since $a_0(t) + a_1(t) = 2+2 = 4$ and $a_2(t) + a_3(t) = 2+4 = 6$ and also $t = (02133)^2$ where $o(02133) = 3$ and $02133$ is primitive.
Let $N_\lambda^{(m)}$ be the set of elements $[s] \in N_\lambda$ where $o(s) = m$. Denote by ${\mathcal{L}(\lambda,m)}$ the number of primitive necklaces in $N_\lambda^{(m)}$. We will often use the notation $d = \gcd(\lambda_1, \ldots, \lambda_k)$. If $2\mid d$, let $\lambda/2$ denote the composition of $n/2$ obtained by dividing each part of $\lambda$ by 2.
\[lem:N=bigL\] For $m, n \geq 1$ and $d = \gcd(\lambda_1, \ldots, \lambda_k)$. $$\bigl| N_\lambda^{(m)} \bigr| = \begin{cases} {\mathcal{L}(\lambda,m)} + {\mathcal{L}(\lambda/2,m/2)} &
\text{if } d \text{ is even and } m = 2\bmod 4 \\ {\mathcal{L}(\lambda,m)} & \text{otherwise.}\end{cases}$$
The elements of $N_\lambda^{(m)}$ which are not primitive are exactly those of the form $[s]$ where $s = q^2$ and $q$ is primitive with $o(q) = m/2$ is odd. Therefore, we must have that $n$ is even and $m = 2 \bmod 4.$ Additionally, if $n$ is even and $m = 2 \bmod 4$, given any primitive word $q$ of length $n/2$ with $a_{2t}(q) + a_{2t+1}(q) = \lambda_{t+1}/2$ and $o(q) = m/2$, we will have $q^2 \in N_\lambda^{(m)}.$
Notice that by definition, we can write ${\mathcal{L}(\lambda,m)}$ in the following way: $$\label{eq:bigL}
{\mathcal{L}(\lambda,m)} = \sum_{\substack{\sum_t i_t = m \\ 0\le i_t\leq \lambda_t}}
L(\lambda_1-i_1,i_1, \ldots, \lambda_k-i_k, i_k).$$ The next lemma demonstrates a useful symmetry of ${\mathcal{L}(\lambda,m)}$.
\[lem:bigL\] For all $0\leq m \leq n$ and any composition $\lambda$ of $n$, $${\mathcal{L}(\lambda,m)} = {\mathcal{L}(\lambda,n-m)}.$$
Given a primitive word $s \in N_\lambda$ with $o(s) = m$, we can construct a primitive word $s' \in N_\lambda$ with $o(s) = n-m$ by letting $s'_i = 2t$ if $s_i = 2t+1$ and letting $s'_i = 2t+1$ if $s_i = 2t$ for all $0\leq t \leq k-1$. Doing this switches odd letters with even letters and also ensures $\lambda_{t+1} = a_{2t}(s) + a_{2t+1}(s) = a_{2t}(s') + a_{2t+1}(s')$.
Periodic patterns
-----------------
In order to prove Theorem \[thm:main\], we must define a mapping $\operatorname{\Pi}_\lambda$ from $N_\lambda$ to $\lambda$-unimodal cyclic permutations. The mapping we describe here is a special case of the mapping $\operatorname{\Pi}_\sigma$ defined in [@archer].
Define a map $\Sigma: W_{2k}(n) \to W_{2k}(n)$ which takes a word $s_1s_2\ldots s_n$ to the word $s_2s_3\ldots s_ns_1$. For example, $\Sigma(00100011) = 01000110$, $\Sigma^2(00100011) = 10001100$, etc. We will define an ordering on words in $W_{2k}(n)$ denoted by $\prec$. Suppose that $s= s_1s_2\ldots s_n$ and $s' = s'_1s'_2\ldots s'_n$ and that for some $1\leq i \leq n$, we have $s_1\ldots s_{i-1} = s'_1\dots s'_{i-1}$ and $s_i \neq s'_i$. Then we say that $s\prec s'$ if either $o(s_1\ldots s_{i-1})$ is even and $s_i<s'_i$ or if $o(s_1\ldots s_{i-1})$ is odd and $s_i>s'_i$, where $<$ is the ordering on $\{0,1,2,\ldots,2k\}$ inherited from the integers.
For example, suppose we want to compare $s = 0010111$ and $s' = 0010001$. Notice that $s_1s_2s_3s_4 = s'_1s'_2s'_3s'_4=0010$, $s_5=1$ and $s_5'=0$. Since $o(0010)$ is odd and $s_5>s'_5$, it must be true that $s\prec s'$.
We are now prepared to define a mapping $\operatorname{\Pi}_\lambda: N_\lambda \to {\mathcal{C}}(\lambda)$ where ${\mathcal{C}}(\lambda)$ are the $\lambda$-unimodal cyclic permutations. Suppose first that $[s]\in N_\lambda$ is primitive. Choose a representative $s\in[s]$. Take $\pi = \pi_1\pi_2\ldots \pi_n\in {\mathcal{S}}_n$ to be the permutation (which we call the *pattern*) so that $\pi$ is in the same relative order as the sequence $$s, \Sigma(s), \Sigma^2(s), \ldots, \Sigma^{n-1}(s)$$ with respect to the ordering $\prec$. Then take ${\hat{\pi}}= (\pi_1\pi_2\ldots \pi_n)$, the cyclic permutation obtained by sending $\pi_1$ to $\pi_2$, $\pi_2$ to $\pi_3$, etc. Then we say that $\operatorname{\Pi}_\lambda([s]) = {\hat{\pi}}$. Notice that the choice of representative $s \in [s]$ does not matter.
If $[s]\in N_\lambda$ is not primitive, then $s$ is of the form $q^2$ for primitive $q$ where $o(q)$ is odd. Defining the pattern $\pi$ in this case is a little different since for any representative $s\in[s]$, $\Sigma^{i}(s) = \Sigma^{\frac{n}{2}+i}(s)$ and so extend our definition of $\prec$ to associate a permutation of length $n$ to it. For a given $s\in[s]$ we can choose $s\prec\Sigma^{\frac{n}{2}}(s)$ to be true, which in turn forces the following relations for $1\leq i \leq \frac{n}{2}$: that $\Sigma^i(s) \prec \Sigma^{\frac{n}{2}+i}(s)$ whenever $o(s_1\ldots s_i)$ is even and $\Sigma^i(s) \succ \Sigma^{\frac{n}{2}+i}(s)$ whenever $o(s_1\ldots s_i)$ is odd. Since $o(s_1\ldots s_{\frac{n}{2}})$ is odd, it will follow that $\Sigma^{\frac{n}{2}}(s)\succ \Sigma^{n}(s) = s$, which is consistent with our original choice.
We can thus define a permutation $\pi = \pi_1\pi_2\ldots \pi_n$ which is in the same relative order as $$s, \Sigma(s), \Sigma^2(s), \ldots, \Sigma^{n-1}(s)$$ with respect to $\prec$. As before, we take ${\hat{\pi}}= (\pi_1\pi_2\ldots \pi_n)$ and $\operatorname{\Pi}_\lambda([s]) = {\hat{\pi}}$. Again, ${\hat{\pi}}$ does not depend on the choice of representative $s\in[s]$.
Let us see an example. Suppose $\lambda = (3,6)$ and $s = 321132202$ is our choice of representative. Certainly, $\pi_8=1$ since $\Sigma^7(s) = 023211322$ must be less than any other cyclic shift of $s$ since it is the only one that starts with 0 (the smallest number in our set $\{0,1,2,3\}$. Additionally, $\pi_3$ and $\pi_4$ must be 2 and 3 in some order since $\Sigma^2(s)$ and $\Sigma^3(s)$ both start with 1. To determine which is smaller, we compare them using the ordering $\prec$. $\Sigma^2(s) =113220232$ and $\Sigma^3(s) =132202321$. Comparing the two, we see that the first place they disagree is the second position and $o(1)$ is odd. Comparing the values of the first place they disagree we determine that $\pi_3>\pi_4$. By repeating this process, we obtain $\pi = 953286417$ and ${\hat{\pi}}= (953286417) = 782134965$. Notice that ${\hat{\pi}}$ is $(3,6)$-unimodal since $782$ and $134965$ are both unimodal segments. If we had chosen a different representative, for example, $s = 322023211$, then we would have obtained $\pi = 864179532$ and ${\hat{\pi}}= (864179532) = 782134965$. Notice that the choice of representative did not matter.
For an example when $[s]$ is not primitive, consider $\lambda = (4,4)$ and $s = 02210221$. Notice that $s = (0221)^2$ where $0221$ is primitive and $o(0221) = 1$ is odd. Therefore $[s] \in N_\lambda$. In this case, we choose $s\prec \Sigma^3(s)$. From that choice, it follows that $\pi = 17532864$ and thus ${\hat{\pi}}= (17532864) = 78213456$. Notice that ${\hat{\pi}}$ is $(4,4)$-unimodal since $7821$ and $3456$ are both unimodal segments.
\[remark: prec\] Notice that the way $\prec$ is defined, we must have that if $s\prec s'$ and $s_1 = s'_1$, then $\Sigma(s)\prec\Sigma(s')$ if $s_1$ is even and $\Sigma(s)\succ\Sigma(s')$ if $s_1$ is odd.
Counting Lemmas
---------------
Here we include a few combinatorial lemmas we will need in the proof of Theorem \[thm:main\].
\[lem:bincoef\] Suppose $p$ is a $r$-degree polynomial. For $r<n$, $$\label{eq:p_r}
\sum_{i = 0}^{n} (-1)^{i} p(i) {{n}\choose{i}} = 0.$$
Given the Binomial Theorem, $$(x+1)^n = \sum_{i = 0}^n{{{n}\choose{i}}} x^i,$$ if $r<n$, then we can take $r$ derivatives of both sides to obtain: $$\frac{n!}{(n-r)!}(x+1)^{n-r} = \sum_{i = r}^n{{{n}\choose{i}}} x^{i-r} p_r(i),$$ where $p_r(i) = i(i-1)\cdots (i-r+1)$. Plugging in $x = -1$, we obtain Equation for each $p_r$. (Since $p_r(i) = 0$ for all $i<r$, we can start indexing at $i = 0$.) Any polynomial can be written as a linear combination of these polynomials and so Equation must hold for all polynomials of degree at most $n-1$.
\[lem:incex\] For any $d,k\geq 1$, $$\sum_{i = 1}^{k} (-1)^{i+k} {{di}\choose{k}} {{k}\choose{i}} = d^k.$$
We claim that both sides of this formula count the number of ways to pick a single element from each of $k$ different boxes containing $d$ objects each. The right hand side clearly counts the ways to do this. On the left hand side, we choose $i$ boxes to consider, ${{k}\choose{i}}$, then choose $k$ elements from this collection of boxes (possibly pulling many from the same box) in ${{di}\choose{k}}$ ways. Using inclusion exclusion, we find the number of ways to choose $k$ objects from $k$ different boxes.
\[lem:long\] Suppose $\gamma_1 + \gamma_2 + \cdots + \gamma_k = r$. Then, $$\sum_{\substack{0\le a_t\leq \gamma_t \\ a_1+\cdots +a_k = i}}
\frac{(r)!}{ (\gamma_1-a_1)!(a_1)!\cdots (\gamma_k-a_k)!(a_k)!} = \frac{(r)!}{\gamma_1!\cdots \gamma_k!}
{{r}\choose{i}}.$$
Both sides count the number of ways to separate $r$ objects into $k$ different sets of size $\gamma_t$ for $1\leq t \leq k$ and then coloring $i$ of the total objects.
Relationship between necklaces and $\lambda$-unimodal permutations {#sec:necklaces}
==================================================================
The following theorem is a special case of Theorem 2.1 in [@archer]. For that reason, we provide a short proof without all of the details. A complete proof of a more general statement can be found in [@archer].
\[lem:map PPat\] For any $[s] \in N_\lambda$, we have $\operatorname{\Pi}_\lambda([s]) \in \mathcal{C}(\lambda)$. Additionally, the map $\operatorname{\Pi}_\lambda: N_\lambda \to \mathcal{C}(\lambda)$ is surjective.
Suppose $[s] \in N_\lambda$, $\pi$ is the pattern of $s'\in [s]$, and ${\hat{\pi}}= \operatorname{\Pi}_\lambda([s])$. For all $0\leq t \leq 2k$, define $e_t = |\{j \in [n] : s'_j <t \}|$. We claim that for any $0\leq t <2k$, if $e_{t} < \pi_i<\pi_j\leq e_{t+1}$, then $\pi_{i+1}<\pi_{j+1}$ if $t$ is even and $\pi_{i+1}>\pi_{j+1}$ if $t$ is odd, where here we let $\pi_{n+1} :=\pi_1$. Indeed, since $\pi_i<\pi_j$, it must be true that $\Sigma^{i-1}(s)\prec\Sigma^{j-1}(s)$. Additionally, since $e_{t} < \pi_i,\pi_j\leq e_{t+1}$, we must have $s_i = s_j = t$. By Remark \[remark: prec\], it follows that $\Sigma^i(s)\prec\Sigma^j(s)$ (and thus $\pi_{i+1}<\pi_{j+1}$) if $t$ is even and $\Sigma^i(s)\succ \Sigma^j(s)$ (and thus $\pi_{i+1}>\pi_{j+1}$) if $t$ is odd.
From this, it will follow that the segment ${\hat{\pi}}_{e_t+1}\ldots {\hat{\pi}}_{e_{t+1}}$ is increasing if $t$ is even and decreasing if $t$ is odd. For if $e_{t} < a<b\leq e_{t+1}$, we only need to show that ${\hat{\pi}}_a<{\hat{\pi}}_b$ if $t$ is even and ${\hat{\pi}}_a>{\hat{\pi}}_b$ if $t$ is odd. First, notice that ${\hat{\pi}}_{\pi_i} = \pi_{i+1}$. Therefore, take $i$ and $j$ so that $\pi_i = a$ and $\pi_j= b$. Then $e_{t} < \pi_i<\pi_j\leq e_{t+1}$, and thus ${\hat{\pi}}_a = \pi_{i+1}<\pi_{j+1} = {\hat{\pi}}_b$ if $t$ is even and ${\hat{\pi}}_a = \pi_{i+1}>\pi_{j+1}= {\hat{\pi}}_b$ if $t$ is odd. Finally, since $e_{t} = \sum_{r<t} a_r(s)$, it follows that $e_{2t+2}-e_{2t} = a_{2t}(s) + a_{2t+1}(s) = \lambda_{t+1}$. Therefore the segment ${\hat{\pi}}_{e_{2t}+1} \ldots{\hat{\pi}}_{e_{2t+1}} {\hat{\pi}}_{e_{2t+1}+1}\ldots {\hat{\pi}}_{e_{2t+2}}$ has length $\lambda_{t+1}$ and is unimodal for all $0\leq t<k$. Therefore ${\hat{\pi}}\in {\mathcal{C}}(\lambda)$.
To see that the map is surjective let ${\hat{\pi}}\in {\mathcal{C}}(\lambda)$ be arbitrary and let $\pi =\pi_1\ldots \pi_n$ be such that ${\hat{\pi}}= (\pi_1\pi_2\ldots \pi_n)$. Since ${\hat{\pi}}\in {\mathcal{C}}(\lambda)$, there is some sequence $0= e_0\leq e_1\leq \cdots \leq e_{2k} = n$ so that (1) the segment ${\hat{\pi}}_{e_t+1}\ldots {\hat{\pi}}_{e_{t+1}}$ is increasing if $t$ is even and decreasing if $t$ is odd, and (2) $e_{2t+2}-e_{2t} = \lambda_{t+1}$ for all $0\leq t \leq k-1$. The word $s = s_1s_2\ldots s_n$ you obtain by setting $s_i = t$ if $e_t<\pi_i\leq e_{t+1}$ is a word in $N_\lambda$ so that $\operatorname{\Pi}_\lambda([s]) = {\hat{\pi}}$. Therefore, the map is surjective.
It is a nontrivial fact that any choice of $0= e_0\leq e_1\leq \cdots \leq e_{2k} = n$ will result in a primitive word (or a 2-periodic word) and thus lies in $N_\lambda$. The proof of this remark is omitted from the proof above, but can follow from contradiction (using several cases).
The next theorem describes the relationship between the number of odd letters in an element of $N_\lambda$ and the number of descents of its image under $\operatorname{\Pi}_\lambda$. This will prove useful when we rewrite the sum in Theorem \[thm:main\] as a sum over necklaces.
Recall that $N_\lambda^{(m)}$ denotes the set of elements $[s] \in N_\lambda$ where $o(s) = m$. Let $\operatorname{\Pi}_\lambda^{(m)}$ be the map $\operatorname{\Pi}_\lambda$ restricted to the set $N_\lambda^{(m)}$. Also, let ${\mathcal{C}}_\lambda(m)$ be the set of $\lambda$-unimodal cycles $\tau$ with $|\operatorname{Des}(\tau)\setminus S(\lambda)| = m$, where $S(\lambda)$ is the set of partial sums of $\lambda$ and let ${c_{\lambda}(m)} = |{\mathcal{C}}_\lambda(m)|$.
The map $$\operatorname{\Pi}_\lambda^{(m)}: N_\lambda^{(m)} \to \bigcup_{j = 0}^k{\mathcal{C}}_\lambda(m-j)$$ is surjective. Moreover, for $\tau \in {\mathcal{C}}_\lambda(m-j)$, the size of the preimage $\big(\operatorname{\Pi}_\lambda^{(m)}\big)^{-1}(\tau)$ is ${{{k}\choose{j}}}$.
For some $\pi = \pi_1\pi_2\ldots \pi_n$, we have $\tau = {\hat{\pi}}$. We know from Lemma \[lem:map PPat\] that for given ${\hat{\pi}}\in {\mathcal{C}}(\lambda)$, there is a word $s = s_1s_2\ldots s_n \in N_\lambda$ so that $\operatorname{\Pi}([s]) = {\hat{\pi}}$. We obtain this word by finding a sequence $0= e_0\leq e_1\leq \cdots \leq e_{2k} = n$ so that the segment ${\hat{\pi}}_{e_t+1}\ldots {\hat{\pi}}_{e_{t+1}}$ is increasing if $t$ is even and decreasing if $t$ is odd. However, this sequence is not unique. Certainly, we must have that $e_{2t+2}-e_{2t}= \lambda_{t+1}$ for all $0\leq t \leq k-1$ and so $e_{2t}$ is fixed for every $ 0 \leq t \leq k$. However, for $e_{2t+1}$, we have exactly two choices for all $0\leq t \leq k-1$. This is because the unimodal segment of length $\lambda_t \neq 0$ can be broken up in exactly two ways, where the corner could be included in either the increasing segment or the decreasing segment.
Suppose $\tau = {\hat{\pi}}$ has exactly $m-j$ descents, not including those from $S(\lambda)$. Then we must have at least $m-j$ odd letters. For example, in a unimodal permutation $24587631$, there are 4 descents and the decreasing segment is either length 4 or 5. Choose $j$ of the $k$ corners to include in the decreasing part of each unimodal segment. This will add exactly $j$ odd letters to the minimum number resulting in exactly $m$ odd letters in a given representative $s\in[s]$. Clearly, there are ${{{k}\choose{j}}}$ ways to do this.
\[cor:Nlambda\] Suppose $\lambda$ is a composition of $n$ with $k$ parts and $0\leq m\leq n$. Then, $$\bigl| N_\lambda^{(m)} \bigr| = \sum_{j = 0}^k {{{k}\choose{j}}} {c_{\lambda}(m-j)}.$$
Using the above relationship, we can use generating functions to find an equation for ${c_{\lambda}(m)}$ in terms of $\bigl| N_\lambda^{(m)} \bigr|$.
\[lem:2\] For a composition $\lambda$ of $n$ with $k$ parts and for $0\leq m\leq n$, $$\label{eq:odd al}
{c_{\lambda}(m)} = \sum_{j=0}^{m} (-1)^{m-j} {{m-j+k-1}\choose{m-j}} \bigl| N_\lambda^{(j)} \bigr|.$$
We have a formula for $\bigl| N_\lambda^{(m)} \bigr|$ in terms of ${c_{\lambda}(m)}$ by Corollary \[cor:Nlambda\]. Since ${{k}\choose{j}} =0$ when $j>k$, we can rewrite it as $$\bigl| N_\lambda^{(m)} \bigr| = \sum_{j = 0}^m {{k}\choose{j}} {c_{\lambda}(m-j)} =
\sum_{j=0}^m {{k}\choose{m-j}} {c_{\lambda}(j)}.$$ We can write this relationship in terms of the generating functions for $\bigl| N_\lambda^{(m)} \bigr|$ and ${c_{\lambda}(m)}$. $$\sum_{m\geq 0} \bigl| N_\lambda^{(m)} \bigr| x^m= \Bigg(\sum_{j=0}^k {{k}\choose{j}}x^j\Bigg)
\Bigg(\sum_{m\geq 0} {c_{\lambda}(m)}x^m\Bigg) =(1+x)^k \Bigg(\sum_{m\geq 0} {c_{\lambda}(m)}x^m\Bigg).$$ By multiplying both sides of this equation by $(1+x)^{-k}$, it follows that $$\sum_{m\geq 0} {c_{\lambda}(m)} x^m= \Bigg(\sum_{j\geq 0} (-1)^j {{j+k-1}\choose{k-1}} x^j\Bigg)
\Bigg(\sum_{m\geq 0} \bigl| N_\lambda^{(m)} \bigr| x^m\Bigg).$$ Equation follows.
Proof of the main result {#sec:pf}
========================
Notice that using Equation and the definition of ${c_{\lambda}(m)}$, we can rewrite the statement of Theorem \[thm:main\] in the following way: $$\label{eq:re}
\sum_{m= 0}^{n-k} (-1)^m {c_{\lambda}(m)} = \begin{cases} (k-1)! d^{k-1} \mu(d)& \text{if }\lambda = (d^k), \\
0 & \text{otherwise.} \end{cases}$$ We will prove two cases in the next two theorems, when $d = \gcd(\lambda_1,\lambda_2,\cdots, \lambda_c)$ is odd and when $d$ is even. Combining Theorems \[thm:dodd\] and \[thm:deven\] will give us a proof of Equation and thus a proof of Theorem \[thm:main\].
\[thm:dodd\] When $d = \gcd(\lambda_1,\cdots,\lambda_k)$ is odd, $$\label{eq:dodd}
\sum_{m = 0}^{n-k} (-1)^m {c_{\lambda}(m)} = \begin{cases} (k-1)! d^{k-1} \mu(d)& \text{if }\lambda = (d^k), \\
0 & \text{otherwise.} \end{cases}$$
Using Lemma \[lem:2\], we can expand the left hand side of Equation by plugging in the right hand side of Equation for ${c_{\lambda}(m)}$. By Lemma \[lem:N=bigL\], we know that when $d$ is odd, $\bigl| N_\lambda^{(j)} \bigr| = {\mathcal{L}(\lambda,j)}$. We switch the order of summation which allows us to simplify the equation to a single sum. The binomial identity used here to simplify the equation is easily checked. $$\sum_{j = 0}^{n-k} \sum_{m=j}^{n-k} (-1)^{j} {{m-j+k-1}\choose{m-j}} {\mathcal{L}(\lambda,j)} =
\sum_{j = 0}^{n-k} (-1)^{j} {{n-j}\choose{k}} {\mathcal{L}(\lambda,j)}.$$ Notice that since Lemma \[lem:bigL\] implies ${\mathcal{L}(\lambda,n-j)} = {\mathcal{L}(\lambda,j)}$, then we can perform a change of variables by setting $j:=n-j$ to obtain the following formula. We then expand this formula using Equations and . $$\begin{aligned}
\sum_{j = k}^{n} (-1)^{n-j} &{{j}\choose{k}} {\mathcal{L}(\lambda,j)} \\
&= \sum_{j = 1}^{n} (-1)^{n-j} {{j}\choose{k}} \sum_{\substack{0\le i_t\leq \lambda_t \\ i_1+\cdots +i_k = j}}
\frac{1}{n}\sum_{\ell \mid \gcd(d,i_1, \cdots, i_k)} \mu(\ell) \frac{(n/\ell)!}{(\frac{\lambda_1-i_1}{\ell})!
(\frac{i_1}{\ell})!\cdots (\frac{\lambda_k-i_k}{\ell})!(\frac{i_k}{\ell})!}.
\end{aligned}$$ Notice that for convenience, we write the sum on the right starting at 1. This does not change the formula since ${{{j}\choose{k}}} = 0$ whenever $j<k$.
Since $\ell \mid\gcd(d,i_1,i_2, \cdots, i_k)$, certainly $\ell\mid d$ and thus $\ell$ must always be odd. Also, for any fixed $\ell$ and for any choice of $i_1, \ldots, i_k$, we have that $\ell\mid i_t$ for $1\leq t \leq k$ and thus $\ell \mid j$. We can rewrite the above formula, now moving the sum over $\ell\mid d$ to the front. We make the following substitutions for the indices in our equation, letting $i$, $r$, $a_t$, and $\gamma_t$ be such that $j = \ell i$, $n = \ell r$, $i_t = \ell a_t$, $\lambda_t = \ell\gamma_t$ for all $1\leq t\leq k$. After these substitutions, our resulting formula is the following: $$\frac{1}{n} \sum_{\ell\mid d} \mu(\ell) \sum_{i = 1}^{r} (-1)^{\ell r-\ell i} {{\ell i}\choose{k}}
\sum_{\substack{0\le a_t\leq \gamma_t \\ a_1+\cdots + a_k = i}}
\frac{(r)!}{(\gamma_1-a_1)!(a_1)!\cdots (\gamma_\ell-a_k)!(a_k)!}.$$ We simplify this formula by noticing first that the right-most sum is equal to $\frac{(r)!}{\gamma_1!\cdots \gamma_k!} {{r}\choose{i}}$ by Lemma \[lem:long\]. Also, since $\ell$ is odd, we have $(-1)^{\ell r - \ell i} = (-1)^{i+r}$ We therefore obtain: $$\frac{1}{n} \sum_{\ell\mid d} \mu(\ell) \frac{(r)!}{\gamma_1!\cdots \gamma_k!} \sum_{i = 1}^{r} (-1)^{i+r}
{{\ell i}\choose{k}} {{r}\choose{i}}.$$ Since ${{\ell i}\choose{k}}$ is a degree $k$ polynomial in $i$, the right-most sum is zero when $r > k$ by Lemma \[lem:bincoef\]. When $\frac{n}{\ell} = r \leq k$, we have $n \leq \ell k$, but $\ell \mid d$ and in turn $d \mid \lambda_t$ for all $1\leq t \leq k$ where $\lambda_1+\cdots +\lambda_k = n$. Therefore, we must have that $\ell = d = \lambda_t$ for all $1\leq t \leq k$. It follows that if $\lambda \neq (d^k)$, then $\sum_{m=0}^{n-k} (-1)^m {c_{\lambda}(m)} = 0$.
If $\lambda = (d^k)$, then by the above argument, we must have that $r = k$ and $\ell = d$. Substituting these values into the formula gives us: $$\mu(d) \frac{(k)!}{k d} \sum_{i = 1}^{k} (-1)^{i+k} {{di}\choose{k}} {{k}\choose{i}}.$$ By Lemma \[lem:incex\], we know that $\sum_{i = 1}^{k} (-1)^{i+k} {{di}\choose{k}} {{k}\choose{i}} = d^k.$ Equation follows.
\[thm:deven\] When $d = \gcd(\lambda_1,\cdots, \lambda_k)$ is even, $$\label{eq:deven}
\sum_{m = 0}^{n-k} (-1)^m {c_{\lambda}(m)} = \begin{cases} (k-1)! \cdot d^{k-1} \mu(d) & \text{if } \lambda = (d^k), \\
0 & \text{otherwise.} \end{cases}$$
Let $\lambda' = \lambda/2$, the composition of $n/2$ of length $k$ where $\lambda'_i = \lambda_i/2$. We use Lemmas \[lem:N=bigL\] and \[lem:2\] to expand Equation . $$\label{eq}
\sum_{m = 0}^{n-k} \sum_{j=0}^{m} (-1)^{j} {{m-j+k-1}\choose{m-j}} {\mathcal{L}(\lambda,j)} +
\sum_{m = 0}^{n-k} \sum_{\substack{0\le i \leq m \\ i = 2 \bmod 4}} {{m-i+k-1}\choose{m-i}}
{\mathcal{L}(\lambda',\frac{i}{2})}.$$ Notice that the left most sum in Equation looks similar to the formula in the proof of Theorem \[thm:dodd\]. Through the same steps, we find that the left most sum in Equation can be written as: $$\frac{1}{n} \sum_{\ell\mid d} \mu(\ell) \frac{(r)!}{\gamma_1!\cdots \gamma_k!}
\sum_{i = 1}^{r} (-1)^{\ell i} {{\ell i}\choose{k}} {{r}\choose{i}}.$$ From this point, the proof is different than the proof of Theorem \[thm:dodd\], because $\ell$ could be even. We split the sum into the two cases when $\ell$ is either even or odd. $$\frac{1}{n} \sum_{\substack{\ell\mid d\\ \ell \text{ even}}} \mu(\ell) \frac{(r)!}{\gamma_1!\cdots \gamma_k!}
\sum_{i = 1}^{r} {{\ell i}\choose{k}} {{r}\choose{i}} + \frac{1}{n}
\sum_{\substack{\ell\mid d\\ \ell \text{ odd}}} \mu(\ell) \frac{(r)!}{\gamma_1!\cdots \gamma_k!}
\sum_{i = 1}^{r} (-1)^{i} {{\ell i}\choose{k}} {{r}\choose{i}}.$$ As before, by Lemma \[lem:bincoef\] the right hand sum is zero except possibly when $r = k$ and $\ell=d$. However $d$ is even and $\ell$ is odd, so this can never be the case. Therefore, the right hand sum must always be zero. For reasons we’ll see later, let $n' = n/2$, $d' = d/2$ and $\ell' = \ell/2$. We only need to consider $\ell'$ odd, since if it were even, $\mu(\ell) = 0$. If $\ell'$ odd, then we have that $\mu(\ell) = \mu(2)\mu(\ell') = -\mu(\ell')$. Therefore, the first sum in Equation can now be written as: $$\label{eq:lhs}
-\frac{1}{n} \sum_{\substack{\ell'\mid d'\\ \ell' \text{ odd}}} \mu(\ell') \frac{r!}{\gamma_1!\cdots \gamma_k!}
\sum_{i = 1}^{r} {{2\ell'i}\choose{k}} {{r}\choose{i}}.$$ Now, consider the second sum in equation . As usual, we change the order of summation to simplify the equation to one summation. For simplicity, we change our variable to $j := (n-i)/2$. Let $n' = n/2$. Using Lemma \[lem:bigL\], which says that ${\mathcal{L}(\lambda',j)} = {\mathcal{L}(\lambda',n'-j)}$, we obtain the following formula: $$\sum_{\substack{0\le j \leq n' \\ n'-j \text{ odd}}} {{2j}\choose{k}} {\mathcal{L}(\lambda',j)}.$$ There are two cases: when $n'$ is even and when $n'$ is odd. If $n'$ is even, $j$ is odd and if $n'$ is odd, $j$ is even. The two cases are very similar with only slight changes in a few details. Here, we will do the case when $n'$ is even. The sum above can be rewritten as a sum over odd $j$. Using Equations and , we can expand this formula. $$\sum_{\substack{0\le j \leq n' \vspace{.05cm} \\ j \text{ odd}}} {{2j}\choose{k}}
\sum_{\substack{0\le i_t\leq \lambda'_t \\ i_1+\cdots +i_k = j}}\frac{1}{n'}
\sum_{\ell \mid \gcd(d',i_1, \cdots, i_k)} \mu(\ell)
\frac{(n'/\ell)!}{(\frac{\lambda'_1-i_1}{\ell})!(\frac{i_1}{\ell})! \cdots (\frac{\lambda'_k-i_k}{\ell})!(\frac{i_k}{\ell})!}.$$ Since $\ell \mid\gcd(d',i_1,i_2, \cdots, i_k)$, certainly $\ell\mid d'$. Also, for any fixed $\ell$ and for any choice of $i_1, \ldots, i_k$, we have that $\ell\mid i_t$ for $1\leq t \leq k$ and thus $\ell \mid j$. Since $j$ is always odd, we must also always have that $\ell$ is odd. We can rewrite the above formula, now moving the sum over odd $\ell\mid d$ to the front. We make the following substitutions for the indices in our equation, letting $i$, $r$, $a_t$, and $\gamma_t$ be such that $j = \ell i$, $n' = \ell r$, $i_t = \ell a_t$, $\lambda'_t = \ell\gamma_t$ for all $1\leq t\leq k$. After these substitutions, our resulting formula is the following: $$\sum_{\substack{\ell\mid d' \\ \ell \text{ odd}}}
\sum_{\substack{ 0\le i \leq r \vspace{.05cm} \\ i \text{ odd}}} {{2\ell i}\choose{k}} \mu(\ell) \frac{1}{n'}
\sum_{\substack{0\le a_t\leq \gamma_t \\ a_1+\cdots + a_k = i}}
\frac{(r)!}{ (\gamma_1-a_1)!(a_1)!\cdots (\gamma_k-a_k)!(a_k)!}$$ We simplify this formula by noticing first that the right-most sum is equal to $\frac{(r)!}{\gamma_1!\cdots \gamma_k!} {{r}\choose{i}}$ by Lemma \[lem:long\]. We obtain the following formula for the right hand sum of Equation : $$\label{eq:rhs}
\frac{1}{n'} \sum_{\substack{\ell\mid d' \\ \ell \text{ odd}}} \mu(\ell) \frac{(r)!}{\gamma_1!\cdots \gamma_k!}
\sum_{\substack{0\le i \leq r \\ i \text{ odd}}} {{2\ell i}\choose{k}} {{r}\choose{i}}.$$ Finally, we combine the two summations from Equation which we have found to be equal to Equations and . After combining like terms, we obtain the formula: $$\begin{aligned}
\frac{1}{n'} \sum_{\substack{\ell\mid d'\\ \ell \text{ odd}}} \mu(\ell) \frac{r!}{\gamma_1!\cdots \gamma_k!} \cdot
& \frac{1}{2}\left[ 2\sum_{\substack{i \text{ odd}\\ i \leq r}} {{2\ell i}\choose{k}} {{r}\choose{i}}-
\sum_{i = 1}^{r} {{2\ell i}\choose{k}} {{r}\choose{i}}\right] \\
& = \sum_{\substack{\ell\mid d'\\ \ell \text{ odd}}} \mu(\ell) \frac{r!}{\gamma_1!\cdots \gamma_k!}
\cdot \frac{1}{2\ell r} \left[ - \sum_{i = 1}^{r} (-1)^i {{2\ell i}\choose{k}} {{r}\choose{i}}\right] .
\end{aligned}$$ As before, by Lemma \[lem:bincoef\], the right most term vanishes except when $r \leq k$. Since $n'/\ell =r\leq k$, we have $n' \leq \ell k$. But $\ell \mid d'$ and $d'\mid \lambda'_t$ for all $1\leq t \leq k$ where $\sum_t\lambda'_t = n'$. Therefore, we must have that $\ell = d' = \lambda'_t$ for all $1\leq t \leq k$. It follows that $\lambda' = (d'^k)$ and therefore we must have that $\lambda = (d^k)$. It follows that if $\lambda \neq (d^k)$, then the sum $\sum_{m=0}^{n-k}(-1)^m {c_{\lambda}(m)} = 0$.
Suppose now that $\lambda = (d^k)$. Notice that $d' = \ell$ is odd. For $d'$ odd, we thus obtain: $$- \frac{1}{2kd'} \mu(d') k! \cdot \sum_{i = 1}^{k} (-1)^i {{2d'i}\choose{k}} {{k}\choose{i}} =
\frac{1}{kd} \mu(d) k! \cdot \sum_{i = 1}^{k} (-1)^i {{di}\choose{k}} {{k}\choose{i}}.$$ Recall we are dealing with the case when $n'$ is even and thus $d'$ is odd. Therefore, we must have $k$ even. Therefore, by Lemma \[lem:incex\], the rightmost sum is $d^k$. Equation follows.
Distribution of the descent set of ${\mathcal{C}}_n$ {#sec:cons}
====================================================
In this section, we prove some consequences of Theorem \[thm:main\] using representation theory. We will let $\mathcal{U}(\lambda)$ denote the set of $\lambda$-unimodal permutations.
Consider the one-dimensional representation $\psi$ on $H = \langle (123\cdots n) \rangle$ obtained by letting $\psi(g^i) = \zeta_n^i$ where $g$ is a generator of $H$ and $\zeta_n$ is an $n$th root of unity. We also use $\psi$ to denote the character for this representation since it coincides with the representation itself. Denote by $\rho$ the representation on ${\mathcal{S}}_n$ induced from the representation on $H\leq {\mathcal{S}}_n$ described above. In the next proposition, we prove that the character of $\rho$ denoted by $\chi$ satisfies Equation on each conjugacy class of cycle type $\lambda$ of ${\mathcal{S}}_n$.
\[prop:ind\] Recall that $\chi_\lambda$ denotes the value of $\chi$ on conjugacy class $\lambda.$ For any conjugacy class $\lambda$, $$\chi_\lambda = \begin{cases} (k-1)! d^{k-1} \mu(d) & \text{if } \lambda = (d^k) \\ 0 & \text{otherwise.} \end{cases}$$
The induced character on ${\mathcal{S}}_n$ is defined by $$\chi(\tau) = \frac{1}{|H|} \sum_{\sigma \in {\mathcal{S}}_n} \psi(\sigma^{-1}\tau\sigma)$$ where we define $\psi(\sigma^{-1}\tau\sigma) = 0$ when $\sigma^{-1}\tau\sigma \notin H$.
Since $\sigma^{-1}\tau\sigma$ is in the same conjugacy class as $\tau$, it will have the same cycle type. Notice that $H$ must only contain permutations with cycle type $(d^k)$ for some $k$. Therefore, if $\tau$ has cycle type which is not of the form $(d^k)$, then every $\psi(\sigma^{-1}\tau\sigma) $ contributes 0 and so $\chi(\tau) = 0$. Thus, we see that $\chi_\lambda = 0$ whenever $\lambda \neq (d^k)$ for some $k$.
It is well known that $$\sum_{\gcd(i,n) = 1} \zeta_n^i = \mu(n).$$ Suppose $\tau$ has cycle type $\lambda = (d^k)$. Then $\psi(\sigma^{-1}\tau\sigma)$ will always be either $0$ or a primitive $d^{\text{th}}$ root of unity since permutations in $H$ with cycle type $d^k$ are the generators of the cyclic subgroup of $H$ of size $d$. Since $|H|= n = dk$ and $\sum_{\gcd(d,i) = 1} \zeta_d^i = \mu(d)$, it is enough to show that $|\{ \sigma \in {\mathcal{S}}_n : \sigma^{-1}\tau\sigma = \tau'\}| = k! d^k$ for each $\tau' \in H$ with cycle type $(d^k)$.
Suppose we have two elements $g$ and $h$ so that $g^{-1} \tau g = \tau'$ and $h^{-1} \tau h = \tau'$. Then we must have that $g^{-1} \tau g = h^{-1} \tau h$, which means that $gh^{-1}\tau hg^{-1} = \tau$ and so $hg^{-1}$ is in the centralizer of $\tau$, $C(\tau)$. Therefore, if $g$ is such that $g^{-1} \tau g = \tau'$, then every possible $h$ such that $h^{-1} \tau h = \tau'$ must be of the form $ag$ for some $a$ in $C(\tau)$. Therefore $|\{ \sigma \in {\mathcal{S}}_n : \sigma^{-1}\tau\sigma = \tau'\}| = |C(\tau)|$. By the orbit-stabilizer theorem, this is equal to $|{\mathcal{S}}_n|/|\{\pi \in {\mathcal{S}}_n : \pi \text{ has cycle type } (d^k)\}$. Using the well-known formula for the size of conjugacy classes in ${\mathcal{S}}_n$, we find that indeed $|C(\tau)| = k! d^k$.
We will show that Theorem \[thm:main\] implies that the descent sets of elements of ${\mathcal{C}}_n$ are equi-distributed with the descent sets of the standard Young tableaux that form a basis to the representation $\rho$. That is to say, for any given $D\subseteq[n-1]$, $$|\{\pi \in {\mathcal{C}}_n : \operatorname{Des}(\pi) = D\}| = |\{T \in \mathcal{B}_\rho : \operatorname{Des}(T) = D\}|$$ where $\mathcal{B}_\rho$ is the basis of representation $\rho$ and the descent set of a given standard Young tableau $T$ is defined to be $\operatorname{Des}(T) = \{1\leq i \leq n-1 : i+1 \text{ lies strictly south of } i\}$.
To prove this, we must first introduce a few definitions from [@char]. We say that a subset $D\subset[n-1]$ is *$\lambda$-unimodal* if $D\setminus S(\lambda)$ is the disjoint union of intervals of the form $[s_{t-1}(\lambda) + \ell_t, s_{t}(\lambda)]$ where $1\leq \ell_t \leq \lambda_t$ for all $1\leq t\leq k$. Note that a permutation $\pi\in{\mathcal{S}}_n$ is $\lambda$-unimodal if and only if its descent set $D$ is $\lambda$-unimodal.
Consider a given set of combinatorial objects, $\mathcal{B}$, and descent map $\operatorname{Des}: \mathcal{B} \to \mathcal{P}([n-1])$ which sends each element $b\in \mathcal{B}$ to a subset $\operatorname{Des}(B) \subseteq [n-1]$. If $\rho'$ is some complex representation of ${\mathcal{S}}_n$, the we call $\mathcal{B}$ a *fine set* for $\rho'$ if the character of $\rho'$ satisfies: $$\chi^{\rho'}_\lambda = \sum_{b\in \mathcal{B}^\lambda} (-1)^{|\operatorname{Des}(b)\setminus S(\lambda)|}$$ where $\mathcal{B}^\lambda$ are the elements of $\mathcal{B}$ whose descent set is $\lambda$-unimodal. For example, Theorem \[thm:main\] proves that ${\mathcal{C}}_n$ is a fine set for the representation $\rho$ defined above.
The following two propositions from [@char] will be useful.
\[prop:cor6.7\] If sets $\mathcal{B}_1$ and $\mathcal{B}_2$ are both fine sets for the same representation, then their descent sets are equi-distributed.
\[prop:thm2.1\] Any Knuth class ${\mathcal{C}}$ of shape $\nu$ is a fine set for the irreducible representation ${\mathcal{S}}^\nu$ of ${\mathcal{S}}_n$.
Here, a Knuth class is the set of permutations which result in the same insertion tableau when performing the Robinson–Schensted–Knuth (RSK) algorithm. For reference, see [@Stanley2].
Denote by $B_\rho$ the set (or possibly, multiset) of standard Young tableaux which form a basis to the representation $\rho$. Recall that any representation of ${\mathcal{S}}_n$ can be written as the direct sum of some irreducible representations ${\mathcal{S}}^\nu$ (possibly with multiplicity); that is, $$\rho = \bigoplus_{\nu \in V} {\mathcal{S}}^\nu.$$ Then the basis of $\rho$ is then defined to be all tableaux $B_\rho = \{T \in \operatorname{SYT}(\nu) : \nu \in V\}$ with multiplicities.
\[prop:basis is fine\] $B_\rho$ is a fine set.
Suppose $\chi^\nu$ is the ${\mathcal{S}}_n$-character of the irreducible representation ${\mathcal{S}}^\nu$. By Proposition \[prop:thm2.1\], we have that $$\chi^\nu_\lambda = \sum_{\pi \in {\mathcal{C}}\cap \mathcal{U}(\lambda)} (-1)^{|\operatorname{Des}(\pi)\setminus S(\lambda)|}$$ where ${\mathcal{C}}$ is any Knuth class of shape $\nu$. In performing RSK, the descent set of the permutations is the same as the descent set of the recording tableau $Q$ (see [@schu]). Therefore, we can rewrite this sum over $\lambda$-unimodal permutations in a given Knuth class as a sum over all tableaux $Q$ of shape $\nu$ whose descent set is $\lambda$-unimodal. $$\chi^\nu_\lambda = \sum_{Q \in \operatorname{SYT}(\nu)\cap \mathcal{T}(\lambda)} (-1)^{|\operatorname{Des}(Q)\setminus S(\lambda)|}$$ where $\mathcal{T}(\lambda)$ is the set of tableaux whose descent set is $\lambda$-unimodal and $\operatorname{SYT}(\nu)$ is the set of standard Young tableaux of shape $\nu$.
Finally, $\rho$ can be written as the direct sum of some irreducible representations ${\mathcal{S}}^\nu$ and the character of the representation is the sum of the irreducible characters for the representations that appear in this direct sum. Therefore, the theorem follows.
\[thm:equi\] The descent sets of elements of ${\mathcal{C}}_n$ are equi-distributed with the descent sets of the standard Young tableaux in $B_\rho$.
By Theorem \[thm:main\], ${\mathcal{C}}_n$ is a fine set for the representation $\rho$. By Proposition \[prop:basis is fine\], the set of standard Young tableaux that form a basis of the representation $\rho$ is also a fine set for $\rho$. Therefore, the theorem follows from Proposition \[prop:cor6.7\].
It should be noted that this statement of equi-distribution of descent sets in Theorem \[thm:equi\] is a special case of Theorem 2.2 in [@note], which itself is a reformulation of Theorem 2.1 in [@gessel].
As another consequence of Theorem \[thm:main\], we recover Theorem \[thm:sn-1cn\], a result of Elizalde [@sergi] stating that the number of permutations of ${\mathcal{S}}_{n-1}$ with descent set $D$ is equal to the number of permutations of ${\mathcal{C}}_{n}$ whose descent set is either $D$ or $D\cup[n-1]$. In [@sergi], this is proved directly with a complicated bijection. In the proof of Theorem \[thm:sn-1cn\], we show that the result also follows from Theorem \[thm:main\] and Propositions \[prop:regular\] and \[prop:sn is fine\].
\[prop:regular\] The restriction of $\rho$ to ${\mathcal{S}}_{n-1}$ is isomorphic to the regular representation.
The primitive linear representation $\rho$ of $H$ acts on ${\mathbb{C}}v$ by $(12\ldots n) \cdot v \mapsto e^{2\pi i/n} v$. Since $H$ is cyclic, this determines the action. The induced representation $\rho'$ on ${\mathcal{S}}_n$ is then an action on ${\mathbb{C}}\{\sigma_i v\}$ where the $\sigma_i$ are all representatives from the distinct cosets of $H \leq {\mathcal{S}}_n$. For a given element $\pi \in {\mathcal{S}}_n$ and coset representative $\sigma_i$, there must be some $\tau \in H$ and $j$ so that $\pi\sigma_i = \sigma_j \tau$. The action of the representation is that $\pi \in {\mathcal{S}}_n$ acts on basis element $\sigma_i v$ by $$\pi \cdot \sigma_i v = \sigma_j \rho(\tau) v.$$
We can take the coset representatives $\sigma_i$ to be the elements of ${\mathcal{S}}_n$ which fix $n$. There are exactly $(n-1)!$ such permutations and $\{\sigma_i H\}$ are all distinct cosets, which follows from the fact that for $i \neq j$, $\sigma_i^{-1}\sigma_j$ fixes $n$ and therefore can only be in $H$ if $\sigma_i^{-1}\sigma_j$ is the identity. Notice that these coset representatives form a subgroup isomorphic to ${\mathcal{S}}_{n-1}$.
If we take the restriction of $\rho'$ to ${\mathcal{S}}_{n-1}$, we act on ${\mathbb{C}}\{\sigma_i v\}$ by the elements of ${\mathcal{S}}_{n-1}$, which are exactly the coset representatives. We have that $\sigma_j \cdot \sigma_i v = \sigma_k v$ for some $k$, since $\tau$ is the identity and thus $\rho(\tau) = 1$. If we set $v = 1$, this action is exactly the action by left multiplication on ${\mathbb{C}}{\mathcal{S}}_{n-1}$, which is the regular representation.
\[prop:sn is fine\] ${\mathcal{S}}_n$ is a fine set for the regular representation on ${\mathcal{S}}_n$.
It is well-known (for example, see [@Sagan]) that the character of the regular representation $\chi^R$ of ${\mathcal{S}}_n$ takes values $$\chi^R(\pi) = \begin{cases} n! & \pi =1 \\ 0 & \text{otherwise.}\end{cases}$$ Therefore, from the definition of a fine set, it suffices to show that $$\label{eqn:reg}
\sum_{\pi \in \mathcal{U}(\lambda)} (-1)^{|\operatorname{Des}(\pi) \setminus S(\lambda)|} =
\begin{cases} n! & \lambda = (1^n) \\ 0 & \text{otherwise.}\end{cases}$$ If $\lambda = 1^n$, then every permutation is $\lambda$-unimodal and each one contributes 1 to the sum since $|\operatorname{Des}(\pi) \setminus S(\lambda)| = |\operatorname{Des}(\pi) \setminus [n-1]| = 0$. Therefore, when $\lambda = (1^n)$, the sum is indeed $n!$.
Now, suppose $\lambda \neq (1^n)$. Consider the following map $\varphi: \mathcal{U}(\lambda) \to \mathcal{U}(\lambda)$. Take $i\geq1$ to be the smallest positive integer such that $\lambda_i>1$. Then $\pi_i\pi_{i+1}\ldots\pi_{i-1+\lambda_i}$ is a unimodal segment of length $\lambda_i>1$. By switching the positions of the largest and second largest elements of this segment, we obtain a unimodal segment with either one more or one less descent. Let $\varphi(\pi)$ be the permutation you obtain by making this change. Then $\varphi$ is an involution on $\mathcal{U}(\lambda)$ which changes $|\operatorname{Des}(\pi) \setminus S(\lambda)|$ by one. Therefore, there must be a bijection between $\lambda$-unimodal permutations with $|\operatorname{Des}(\pi) \setminus S(\lambda)|$ odd and those with $|\operatorname{Des}(\pi) \setminus S(\lambda)|$ even. It follows that the sum must be 0 in this case.
\[thm:sn-1cn\] The number of permutations in ${\mathcal{S}}_{n-1}$ with descent set $D$ is equal to the number of permutations in ${\mathcal{C}}_{n}$ whose descent set is either $D$ or $D\cup[n-1]$.
By Proposition \[prop:sn is fine\], we know that ${\mathcal{S}}_{n-1}$ a fine set for the regular representation of ${\mathcal{S}}_{n-1}$. Denote the restriction of the character $\chi$ to ${\mathcal{S}}_{n-1}$ by $\chi^r$. Consider the injection $\iota: {\mathcal{S}}_{n-1} \to {\mathcal{S}}_n$ sending $\tau \in {\mathcal{S}}_{n-1}$ to $\pi\in{\mathcal{S}}_n$ defined by $\pi(i) = \tau(i)$ for $1\leq i\leq n-1$ and $\pi(n) = n$. This allows us to think of ${\mathcal{S}}_{n-1}$ as a subgroup of ${\mathcal{S}}_n$. We have $\chi^r(\tau) = \chi(\tau)$ when $\tau \in {\mathcal{S}}_{n-1} \leq {\mathcal{S}}_n$, and $\chi^r(\tau) = 0$ when $\tau \in{\mathcal{S}}_n\setminus {\mathcal{S}}_{n-1}$. Recall that by Theorem \[thm:main\], $$\chi_\lambda = \sum_{\pi \in {\mathcal{C}}(\lambda)} (-1)^{|\operatorname{Des}(\pi) \setminus S(\lambda)|}.$$ Suppose $\lambda^r$ is a composition of $n-1$ and $\lambda = (\lambda_1^r, \lambda_2^r, \ldots, \lambda_{k-1}^r, 1)$. Additionally, for $\pi \in {\mathcal{S}}_n$, let $\operatorname{Des}^r(\pi)$ denote the *$r$-descent set* defined to be $\operatorname{Des}(\pi)\setminus\{n-1\}$. Then any element of ${\mathcal{C}}(\lambda)$ is $\lambda^r$-unimodal with respect to $\operatorname{Des}^r$, that is to say, the $r$-descent set of an element of ${\mathcal{C}}(\lambda)$ is $\lambda^r$-unimodal as a set. It follows that $$\chi_{\lambda^r}^r = \sum_{\pi \in {\mathcal{C}}(\lambda)} (-1)^{|\operatorname{Des}^r(\pi) \setminus S(\lambda^r)|}.$$ Therefore, it follows that ${\mathcal{C}}_n$ (equipped with the $r$-descent set) is a fine set for the restricted representation. By Proposition \[prop:cor6.7\], it follows that for a given $D\subseteq[n-2]$, the number of permutations in ${\mathcal{S}}_{n-1}$ with descent set $D$ is equal to the number of permutations in ${\mathcal{C}}_n$ with $r$-descents set $D$, from which the theorem follows.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author would like to thank Yuval Roichman, Sergi Elizalde, and Zajj Daugherty for helpful discussions.
[^1]: In [@char], these permutations are called *$\mu$-unimodal permutations*. In this paper, we use $\lambda$ for the composition and reserve $\mu$ for the Möbius function.
|
---
abstract: 'Convolutional Neural Networks (CNNs) have recently been shown to excel at performing visual place recognition under changing appearance and viewpoint. Previously, place recognition has been improved by intelligently selecting relevant spatial keypoints within a convolutional layer and also by selecting the optimal layer to use. Rather than extracting features out of a particular layer, or a particular set of spatial keypoints within a layer, we propose the extraction of features using a subset of the channel dimensionality within a layer. Each *feature map* learns to encode a different set of weights that activate for different visual features within the set of training images. We propose a method of calibrating a CNN-based visual place recognition system, which selects the subset of feature maps that best encodes the visual features that are consistent between two different appearances of the same location. Using just 50 calibration images, all collected at the beginning of the current environment, we demonstrate a significant and consistent recognition improvement across multiple layers for two different neural networks. We evaluate our proposal on three datasets with different types of appearance changes - afternoon to morning, winter to summer and night to day. Additionally, the dimensionality reduction approach improves the computational processing speed of the recognition system.'
author:
- |
Stephen Hausler, Adam Jacobson and Michael Milford\
Queensland University of Technology, Australia\
[email protected] [^1]
bibliography:
- 'FeatFilt.bib'
title: 'Feature Map Filtering: Improving Visual Place Recognition with Convolutional Calibration'
---
Pre-print of article that will appear in Proceedings of the Australasian Conference on Robotics and Automation 2018.
Please cite this paper as:
Stephen Hausler, Adam Jacobson, and Michael Milford. Feature Map Filtering: Improving Visual Place Recognition with Convolutional Calibration. Proceedings of Australasian Conference on Robotics and Automation, 2018.
bibtex:
@inproceedings{hausler2018FeatFilt,\
author = {Hausler, Stephen and Jacobson, Adam and Milford, Michael},\
title = {Feature Map Filtering: Improving Visual Place Recognition with Convolutional Calibration},\
booktitle = {Proceedings of Australasian Conference on Robotics and Automation (ACRA)},\
year = {2018},\
}
Introduction
============
Visual Place Recognition, the ability to localize using just a visual sensor, is challenging due to the significant appearance change that visual scenes experience on a regular basis, including day to night, summer to winter and morning to afternoon. Both hand-crafted features, such as SURF [@BH2008] and HOG [@DN2005], and deep learnt networks have been used to attempt to solve the VPR challenge [@NT2018; @CM2008; @SN2015; @AR2018]. Both viewpoint and appearance robustness has been demonstrated when convolutional neural networks (CNNs) are used for visual place recognition [@SN2015]. This is especially the case when a CNN is trained for recognizing a specific environment [@AR2018; @FeatReweight2017; @ConvLearn2017]. However, this performance has the disadvantage of requiring training for all the environmental conditions that the robot is expected to experience, where-as for practical autonomy, the robot should be able to automatically, and swiftly, adjust its neural parameters to suit the current conditions.
We propose a novel solution to achieve this, by calibrating a convolutional neural network for the current environment. In state-of-the-art approaches, a neural network is re-trained for the specific environment by selecting a set of images from the new environment and re-training the model using these images [@AR2018; @FeatReweight2017; @ConvLearn2017]. However, this requires a significant time and processing cost, so much so that typical robot platforms do not have the capability to re-train the neural model online and in real-time. We propose a method that enables a fast, computationally cheap process of *filtering* the collection of feature maps within a layer of a deep convolutional neural network (see Fig. \[TitlePage\]). When a network is trained on a diverse set of images, each feature map encodes a different type of abstraction from this collection of images. For example, one map in a late convolutional layer might learn to ‘fire’ upon regions of an image containing a building. We propose a calibration procedure which removes the feature maps that do not suit the recognition between the current environment and the learnt environment. This is achieved by minimizing the L2-distance between two identical locations that appear significantly different due to a change in the environment, while maximizing the distance between two different locations that look visually similar due to having the same environmental conditions.
We demonstrate the versatility of our approach by experimenting with two different CNN architectures, HybridNet [@CZ2017] and AlexNet trained on ImageNet [@AlexNet], across three different datasets which demonstrate different types of appearance variations.
The paper proceeds as follows. In Section 2, we review prior uses of convolutional neural networks for the visual place recognition task and previous methods of neural network simplification. Section 3 presents our approach, explaining our calibration procedure and computational methods in detail. Section 4 details the setup of our experimental datasets and Section 5 evaluates the performance of feature map filtering, compared to not filtering. Section 6 provides intuition as to why feature map filtering works and Section 7 summarizes our contributions and provides suggestions for future work.
Related Work
============
In early experiments using convolutional neural networks for place recognition, a feature vector is produced from a particular layer of the network, using all the information that is encoded in the activations of that layer [@SN2015]. However, such a whole-image approach is sensitive to viewpoint variations. This was addressed by developing a landmark extraction algorithm and computing the neural responses to each landmark region in a scene [@Sunderhauf2015]. Intelligently selecting the useful information within an image is a valuable method of improving the localization performance. Rather than finding regions, LoST [@Lost2018] creates a feature vector by extracting semantically meaningful keypoints within the feature map spatial region. [@LookOnce2017] finds keypoints by observing the activations out of a late convolutional layer, while [@Zetao2018] trains a soft attention mask to select salient regions within an image to improve the selection of features used to formulate the feature vector. These keypoint feature vectors consist of the activations across all the feature maps within that layer at the spatial location of the keypoint, even if some of the feature maps are encoding visual information that is counter-productive to localizing in the current environment.
Several experiments compared the performance across different layers [@SN2015; @CZ2017], while a number of experiments use multiple layers simultaneously [@VehicleMultiLayer; @Zetao2018], to improve the visual recognition performance beyond the performance of a single layer. Different layers have been found to encode different types of visual features, such as color and texture in early layers, and objects and scenes in later layers [@NetDissect].
The literature discussed in the previous paragraphs optimizes either the choice of layer to use, or the choice of spatial locations across the feature map stack. The third dimension to optimize is the choice of feature maps themselves within the stack of feature maps that comprise a layer. [@FeatPruneFirst2017] proposes that a CNN can be simplified by pruning the selection of feature maps, which attains comparable performance while improving the computational speed of a forward-pass through the network. [@FeatPrune2018] suggests an improvement by using linear discriminant analysis to calculate the discriminability score for each feature map. They are able to remove a greater number of feature maps without causing a major reduction in accuracy. [@ReThink] re-weights feature maps using a feedback process to improve the classification performance. However, they only re-weight feature maps and don’t completely remove any feature maps. The concept of improving visual place recognition by discriminatively selecting a subset of the feature maps within a convolutional layer is a gap in the literature.
Recent literature on network dissection has provided evidence that individual feature maps encode specific visual features that are relatable to the classifier outputs [@NetDissect]. In their work, the hidden convolutional layers are probed by testing an individual feature map on a pixel-wise semantic segmentation task. They discover that individual feature maps activate for different objects, scenes, textures and colors. This research underpins the motivation for this work - for example, if a particular feature map activates to man-made lighting, this feature map will confuse the localization between night and day and is better removed from the feature vector.
Proposed Approach
=================
We propose a novel method of calibrating a convolutional neural network for the current environment. Our calibration procedure removes the feature maps that do not suit the recognition between the current environment, and the learnt environment. This is achieved by minimizing the L2-distance between two identical locations that appear significantly different due to a change in the environment, while maximizing the distance between two different locations that look visually similar due to having the same environmental conditions (see Fig. \[MethodDiagram\]). This is termed triplet loss in literature and like previous work, we also use the L2-distance as our calibration optimization metric [@FaceNetFirstTriplet; @LightCNN; @AR2018].
Calibration Procedure
---------------------
For each calibration scene, we extract deep-learnt features for the currently viewed scene, the corresponding reference image and a randomly selected image elsewhere in the database of reference images. We use a total of 50 calibration images, all extracted from the beginning of the query dataset - this is to mimic the real world situation where the calibration is performed prior to the robot beginning the navigation of its environment. This calibration can be achieved using pre-defined maneuvers, such as the methods described in [@Jacobson2015]. These calibration triplets are used to perform feature map filtering, as explained in the following sections.
Extracting Deep Learnt Features
-------------------------------
In a convolutional neural network, a convolutional layer tensor consists of dimension $W \times H \times C$, where $W$ and $H$ are the width and height of the data matrix and $C$ is the number of channels, otherwise termed the number of feature maps. To reduce the dimensionality of this $W \times H \times C$ feature vector, we use maximum pyramid spatial pooling [@CZ2017], which was chosen as it has both viewpoint robustness and provides a significant dimensionality reduction while keeping the key features in each feature map. In our version of pyramid spatial pooling, we convert each $W \times H$ map into a vector of length 5, consisting of the maximum activation in each map and the maximum activation in each of the four quadrants of each $W \times H$ map.
Out of a stack of feature maps within a convolutional layer, certain feature maps will activate to certain visual features in an image. For example, a feature map in a network might fire on regions of an image containing vehicles. However, in the context of visual place recognition, activations on vehicles has a negative effect, because vehicles are dynamic objects and not temporally static. This applies to other time-varying features, like snow in winter. Our goal is to search through the stack of feature maps to find the worst feature maps. We define the worst feature maps as feature maps that contain activations that vary across a change in appearance when the location does not change. We perform this search on the spatially pooled features in each feature map, for improved viewpoint robustness.
Filtering Feature Maps
----------------------
We use a Greedy algorithm [@Fegaras1998] to determine which subset of the feature map stack suits the current environmental conditions. Combinatorial optimization problems are typically NP-hard, with a variety of techniques employed to produce approximate solutions in related problems such as sensor selection [@SensorSelect2009]. In our method, using Greedy causes the worst feature map to be filtered at each iteration of the algorithm, until a local maximum is reached. We chose Greedy as it runs in polynomial time and was found to converge to a satisfactory position.
To measure the feature map performance, we select each feature map individually and remove it from the feature vector before calculating the L2 (Euclidean) distance between both the images from the same location and the two images from the reference traverse. This results in two distance scores, one for the same location at different times of day and one for different locations at the same time of day. The result is a vector of difference scores across a different feature map being removed.
$$D(q_i^j,r_i^j) = \sqrt{ \sum^M_{k=1} (q_i^j(k) - r_i^j(k))^2}$$
where $M$ is the dimension of the filtered query feature vector $q_i^j$. $$D(j) = D(r_i^j,n_i^j) - D(q_i^j,r_i^j)\quad \forall j$$ where $r_i^j$ represents the current location filtered reference feature vector and $n_i^j$ represents the filtered feature vector from a random image somewhere else within the reference image database. $j$ denotes the index of the currently filtered feature map.
We then find the maximum distance: $$maxval = \underset{1 \leq j \leq N}{\max} D(j)$$ $$worstFmap = \underset{1 \leq j \leq N}{\operatorname*{argmax}}\:\: D(j)$$ where N is the number of remaining feature maps.
The index of the maximum distance represents the feature map to be removed to achieve the greatest L2 difference between the images from the same location and the images from different locations. With this chosen feature map, we modify the original feature vector and remove this worst performing feature map before repeating the above algorithm for this new, filtered, feature vector.
We iterate in this fashion until a local maximum is reached, that is, the largest L2 difference between the images at same location and the images at different locations (with the images at the same location being closer in L2 space than the images at different locations). In our initial experiments we observed that the gradient towards the local maximum becomes very small prior to reaching the maximum and a significant number of feature maps are filtered out. As an alternate, less aggressive filtering algorithm, we added a gradient minimum cut-off threshold, which we set to 0.1. When removing the worst-performing feature map, if the difference between the previous iteration difference score and the current difference is less than 0.1, we stop the iteration and use the current set of remaining feature maps.
For improved robustness and to prevent outliers, we use multiple calibration images. The choice of filtered feature maps is stored for all images and after the calibration procedure is finished, the number of times a particular feature map is removed is summed across all 50 calibration images. We then find the set of feature maps that were least chosen to be filtered out, and the number of final feature maps is equal to the maximum number of remaining feature maps in the set of calibration images. This heuristic was chosen based on the principle that the choice of remaining feature maps needs to be able to encode all the features within all the calibration images, else minor variations in the current environment will cause key visual features to be missed. The filtering procedure is designed to only remove the feature maps that are irrelevant or damaging to the ability to match between the two appearances of the same location.
Place Recognition Validation Algorithm
--------------------------------------
We developed a single-frame place recognition algorithm to evaluate the improvement gained by using feature map filtering. The features extracted from both the query images and the reference database only include the particular feature maps that were chosen by the feature map filter calibration algorithm. Each query image is compared to the reference database using the cosine distance metric to create a difference vector with length equal to the number of reference templates. We then normalise the difference vector to the range 0.001 to 0.999, where 0.001 denotes a poor match and 0.999 denotes the best matching template. We calculate the quality of the best matching template using a method originally proposed in SeqSLAM [@MM2012], where the quality score is the ratio between the score at the best matching template and the next highest score outside a window around the best matching template. Precision and Recall scores are then calculated across a swept set of quality threshold values.
Experimental Method
===================
We demonstrate our approach on three benchmark datasets, which have been extensively tested in recent literature [@SRAL; @NT2018; @GS2018]. Each dataset is briefly described in the sections below and visually shown in Figure \[DatasetExamples\].
**St Lucia** – consists of multiple vehicular traverses through the suburb of St Lucia, Brisbane across five different times of day [@LuciaDatasetRef]. We use the early morning traverse (190809\_0845) as the reference dataset and the late afternoon video (180809\_1545) as the query, with significant appearance change occurring between morning and afternoon. For the query traverse, we use 1000 images out of the original 15 FPS video. The dataset provides GPS ground truth and we use a ground-truth tolerance of 30 meters. For the calibration procedure, we extract 50 frames from the first 690 frames of the 15 FPS video. The query traverse is started after the last frame of the calibration procedure.
**Nordland** – The Nordland dataset [@NordlandDatasetRef] is recorded from a train travelling for 728 km through Norway across four different seasons. We use the Summer route as the reference dataset and the Winter traverse as the recognition route, using a 2000 image subset of the original videos. For the ground truth we compare the query traverse frame number to the matching database frame number, with a ground-truth tolerance of 10 frames, since the two traverses are aligned frame-by-frame. The 50 calibration images are collected from the videos immediately prior to the section we use for the 2000 image subset.
**Oxford RobotCar** - RobotCar was recorded over a year across different times of day, seasons and routes [@RobotCar]. We use an approximately 2 km route through Oxford, matching from an overcast day (2014-12-09-13-21-02) to night on the next day (2014-12-10-18-10-50). We down sample the original frame rate by a factor of three and start both traverses at the same location, corresponding to 1534 query images. We use a ground truth tolerance of 40 meters, consistent with a recent publication [@GS2018]. Calibration images are collected from the dataset prior to commencing the place recognition experiment.
Results
=======
To produce our results, we run our filtering algorithm on layers Conv3 through to Conv6 of HybridNet and layers Conv2 through to Conv5 of AlexNet. By experimenting on multiple layers, the layer where filtering provides the greatest value can be found. The place recognition performance is evaluated using a single-frame matching algorithm and the F1 score metric is used to quantitatively measure the performance. In Tables \[HNetLuciaMapCounts\] to \[ANetOxforMapCounts\], we compare the number of feature maps pre and post filtering and display the percentage filtered across different layers, networks and datasets.
St Lucia
--------
---------------------------------------------------- -- -- --
**Layer & **Map Count &**Filtered Map Count & **%\
Conv-3 & 384 & 199 & 52%\
Conv-4 & 384 & 223 & 58%\
Conv-5 & 256 & 153 & 60%\
Conv-6 & 256 & 162 & 63%\
********
---------------------------------------------------- -- -- --
: Number of feature maps pre-filtering and post-filtering for HybridNet on St Lucia[]{data-label="HNetLuciaMapCounts"}
For HybridNet on St Lucia, filtering the stack of feature maps improves the localization performance across all layers (Fig. \[F1BarLuciaHNet\]). This is to be expected when framed with respect to the original training data. HybridNet was trained on a collection of security cameras over time in disparate locations [@CZ2017], thus certain feature maps would have learnt to encode visual features that enable matching between summer and winter while others learn to match from morning to afternoon. Since the class output of HybridNet classifies images to a particular location, this encoding is consistent even at higher network layers. When filtering is applied to AlexNet, unlike HybridNet, not all layers find a major improvement after filtering. Only Conv2 and Conv3 find a significant improvement using filtering (Fig. \[F1BarLuciaANet\]). Also, a larger number of feature maps are filtered for the same gradient cut-off threshold. Since AlexNet is trained on a wider variety of images that are not applicable to visual place recognition (such as images of clothing), a larger proportion of feature maps need to be removed in the higher network layers (refer to Tables \[HNetLuciaMapCounts\] and \[ANetLuciaMapCounts\]).
---------------------------------------------------- -- -- --
**Layer & **Map Count &**Filtered Map Count & **%\
Conv-2 & 256 & 129 & 50%\
Conv-3 & 384 & 174 & 45%\
Conv-4 & 384 & 169 & 44%\
Conv-5 & 256 & 113 & 44%\
********
---------------------------------------------------- -- -- --
: Number of feature maps pre-filtering and post-filtering for AlexNet on St Lucia[]{data-label="ANetLuciaMapCounts"}
Nordland
--------
Like our experiment on the St Lucia dataset, when filtering is applied to HybridNet, our recognition performance improves consistently across all four layers (Fig. \[F1BarNordHNet\]). In this dataset, which has a greater appearance change, a larger number of feature maps are filtered for all four layers (see Table \[HNetNordlandMapCounts\]). From the results we can also infer that the higher network layers are more appearance invariant, since proportionally less feature maps require filtering.
---------------------------------------------------- -- -- --
**Layer & **Map Count &**Filtered Map Count & **%\
Conv-3 & 384 & 150 & 39%\
Conv-4 & 384 & 167 & 43%\
Conv-5 & 256 & 125 & 49%\
Conv-6 & 256 & 150 & 59%\
********
---------------------------------------------------- -- -- --
: Number of feature maps pre-filtering and post-filtering for HybridNet on Nordland[]{data-label="HNetNordlandMapCounts"}
--------------------------------------------------- -- -- --
**Layer & **Map Count &**Filtered Map Count &**%\
Conv-2 & 256 & 90 & 35%\
Conv-3 & 384 & 132 & 34%\
Conv-4 & 384 & 160 & 42%\
Conv-5 & 256 & 111 & 43%\
********
--------------------------------------------------- -- -- --
: Number of feature maps pre-filtering and post-filtering for AlexNet on Nordland[]{data-label="ANetNordlandMapCounts"}
When AlexNet is applied to the Nordland dataset, a larger proportion of feature maps require filtering (see Table \[ANetNordlandMapCounts\]). As can be seen in Figure \[ANetF1BarNord\], for Conv2, Conv3 and Conv4, feature map filtering improves the baseline place recognition performance. The improvement is particularly apparent for Conv2. In related works [@SN2015; @Sunderhauf2015; @CZ2017], Conv2 is not considered for place recognition and our baseline results reflect the typically poor performance using Conv2. However, when filtering is used, the place recognition performance exceeds that of Conv5 baseline.
Oxford RobotCar
---------------
It is worth noting that for the same gradient cut-off threshold, more feature maps are filtered on the Oxford RobotCar dataset (see Table \[HNetOxforMapCounts\]). We hypothesize that this is because this dataset has the greatest appearance variation of night to day. Filtering only improves Conv3 by a noticeable margin on the Oxford dataset (refer to Fig. \[F1BarOxfordHNet\]). A possible explanation for this is a mismatch between the scene categories observed in the calibration images and the scenes observed in other sections of the dataset. For example, the calibration route occurs through an urban street with no vegetation, while later in the dataset, the road travels past a park. Conv3 encodes more generic visual features which are captured during the calibration route.
---------------------------------------------------- -- -- --
**Layer & **Map Count &**Filtered Map Count & **%\
Conv-3 & 384 & 117 & 30%\
Conv-4 & 384 & 137 & 36%\
Conv-5 & 256 & 112 & 44%\
Conv-6 & 256 & 134 & 52%\
********
---------------------------------------------------- -- -- --
: Number of feature maps pre-filtering and post-filtering for HybridNet on Oxford RobotCar[]{data-label="HNetOxforMapCounts"}
---------------------------------------------------- -- -- --
**Layer & **Map Count &**Filtered Map Count & **%\
Conv-2 & 256 & 62 & 24%\
Conv-3 & 384 & 142 & 37%\
Conv-4 & 384 & 138 & 36%\
Conv-5 & 256 & 88 & 34%\
********
---------------------------------------------------- -- -- --
: Number of feature maps pre-filtering and post-filtering for AlexNet on Oxford RobotCar[]{data-label="ANetOxforMapCounts"}
When our calibration procedure is applied to AlexNet, the same trend continues - the larger appearance variation causes a greater proportion of feature maps to be filtered out (refer to Table \[ANetOxforMapCounts\]). For the Conv2 layer, three quarters of the original stack of feature maps are removed and in doing so, the maximum F1 score increases from 0.41 to 0.69. This is further evidence that our proposed approach is successfully finding the feature maps that are consistent across the appearance change. Again, the higher level layers gain no localization benefit from feature map filtering, however an improvement is still made to the compute time.
Discussion
==========
Localization Improvement
------------------------
A key result from our experimentation is that filtering provides a considerable improvement to earlier convolutional layers. Early layers have been shown to encode simple visual features while later layers encode objects and regions that are associated with the final class outputs [@NetDissect]. Our results show that filtering object types has less of an advantage, since objects within a scene are typically less affected by environmental changes than lower level visual features, such as the color of the leaves of a tree. When an early layer is filtered, filters that encode a visual feature that is impacted by the change in environment is removed, leaving only the visual features that remain consistent over time. The feature maps selected by our approach can be visually seen in Figure \[DeepDream\]. We also show examples where our filtering approach enables localization when the baseline of not filtering causes an incorrect place hypothesis (see Fig. \[Recognition\_Panel\]).
Computational Improvement
-------------------------
Our improved F1 scores across most layers on both HybridNet and AlexNet is particularly significant when compared to the quantity of feature maps that are removed. As can be seen in the six tables, our filter algorithm filters, on average, 51% of all feature maps when HybridNet is used and 61% when AlexNet is used. This is a significant reduction of information and yet we achieve improved localization performance and significantly improve the place recognition computation time. For example, using Conv3 of HybridNet requires an average of 68 ms to match a query image to a reference database of 1442 images (on a standard desktop PC). When filtering is used, this drops to 43.9ms, 64% of the original time per frame. This is even more apparent with Conv2 of AlexNet on the Oxford RobotCar dataset, where the processing time halves from 81ms to 41ms.
Conclusion and Future Work
==========================
This paper proposes a novel method of performing convolutional network calibration for visual place recognition, without requiring any computationally intensive re-training of the neural network parameters. We achieve this by filtering the set of feature maps produced by a layer within a CNN, by minimizing the L2 distance between the current scene and the corresponding reference image while maximizing the distance between the reference image and another reference image elsewhere in the database. Our feature map filtering approach has two key advantages: improved localization ability in changing environments, and improved computation speed. Our results demonstrate a considerable localization improvement for earlier network layers, with the greatest improvement on the Oxford RobotCar dataset, matching from night to day, using the Conv3 layer on HybridNet and the Conv2 layer on AlexNet. Our calibration procedure resulted in an improvement in HybridNet’s Conv3 F1 score from 0.56 to 0.81 and AlexNet’s Conv2 F1 score from 0.41 to 0.69.
Future work will devise a method of performing feature map filtering in real-time, without requiring any prior calibration. This could be achieved by devising a method of classifying the type of visual feature a particular feature map is activating to and specifically filtering the set of classes that are only occurring in the query traverse and not present anywhere in the reference traverse (such as street lighting at night-time). Also, our feature map calibration strategy using Greedy could be replaced with an alternative heuristic, to further improve the optimization quality. Finally, feature map filtering may also have applications in other computer vision tasks, as this approach could be used to quickly prepare a deep, generically trained CNN for a very specific task without re-training the network weights.
[^1]: SH is supported by a Research Training Program Stipend and ARC Future Fellowship FT140101229. AJ is supported by an Advance Queensland Innovation Partnership, Caterpillar and Mining3. MM is with the Australian Centre for Robotic Vision and was partially supported by an ARC Future Fellowship FT140101229.
|
---
abstract: 'A super-modular category is a unitary pre-modular category with Müger center equivalent to the symmetric unitary category of super-vector spaces. Super-modular categories are important alternatives to modular categories as any unitary pre-modular category is the equivariantization of a either a modular or super-modular category. Physically, super-modular categories describe universal properties of quasiparticles in fermionic topological phases of matter. In general one does not have a representation of the modular group $\mathrm{SL}(2,\mathbb{Z})$ associated to a super-modular category, but it is possible to obtain a representation of the (index 3) $\theta$-subgroup: $\Gamma_\theta<\mathrm{SL}(2,\mathbb{Z})$. We study the image of this representation and conjecture a super-modular analogue of the Ng-Schauenburg Congruence Subgroup Theorem for modular categories, namely that the kernel of the $\Gamma_\theta$ representation is a congruence subgroup. We prove this conjecture for any super-modular category that is a subcategory of modular category of twice its dimension, i.e. admitting a minimal modular extension. Conjecturally, every super-modular category admits (precisely 16) minimal modular extensions and, therefore, our conjecture would be a consequence.'
address:
- 'Microsoft Research Station Q, University of California, Santa Barbara, CA U.S.A.'
- 'Department of Mathematics, Texas A&M University, College Station, TX U.S.A.'
- 'Microsoft Research Station Q and Department of Mathematics, University of California, Santa Barbara, CA U.S.A.'
- 'Department of Mathematics, Texas A&M University, College Station, TX U.S.A.'
author:
- Parsa Bonderson
- 'Eric C. Rowell'
- Zhenghan Wang
- Qing Zhang
bibliography:
- 'bibliography.bib'
title: 'Congruence Subgroups and Super-Modular Categories'
---
[^1]
Introduction
============
A key part of the data for a modular category ${\mathcal{C}}$ is the $S$ and $T$ matrices encoding the non-degeneracy of the braiding and the twist coefficients, respectively. We will denote by ${\tilde{S}}$ the unnormalized matrix obtained as the invariants of the Hopf link so that ${\tilde{S}}_{0,0}=1$, while $S=\frac{{\tilde{S}}}{D}$ will denote the (unitary) normalized $S$-matrix where $D^2=\dim({\mathcal{C}})$ is the categorical dimension and $D>0$. Later, we will use the same conventions for any pre-modular category (for which $S$ may not be invertible). The diagonal matrix $T:=\theta_i\delta_{i,j}$ has finite order (Vafa’s theorem, see [@bakalov2001lectures]) for any pre-modular category. For a modular category the $S$ and $T$ matrices satisfy (see e.g. [@bakalov2001lectures Theorem 3.1.7]):
1. $S^2=C$ where $C_{i,j}=\delta_{i,j^*}$ (so $S^4=C^2=I$)
2. $(ST)^3=\frac{D_+}{D}S^2$ where $D_+=\sum_i{\tilde{S}}_{0,i}^2\theta_i$
3. $TC=CT$.
These imply that from any modular category ${\mathcal{C}}$ of rank $r$ (i.e. with $r$ isomorphism classes of simple objects) one obtains a projective unitary representation of the modular group $\rho:SL(2,{\mathbb{Z}})\rightarrow \operatorname{PSU}(r)$ defined on generators by: ${\mathfrak{s}}=\begin{pmatrix} 0 & -1\\1&0\end{pmatrix}\rightarrow S$ and ${\mathfrak{t}}=\begin{pmatrix}1 & 1\\0&1\end{pmatrix}\rightarrow T$ composed with the canonical projection $\pi_r:\operatorname{U}(r)\rightarrow \operatorname{PSU}(r)$. By rescaling the $S$ and $T$ matrices, $\rho$ may be lifted to a linear representation of $SL(2,{\mathbb{Z}})$, but these lifts are not unique. This representation has topological significance: one identifies the modular group with the mapping class group $\operatorname{Mod}(\Sigma_{1,0})$ of the torus (${\mathfrak{t}}$ and ${\mathfrak{s}}{\mathfrak{t}}^{-1}{\mathfrak{s}}^{-1}$ correspond to Dehn twists about the meridian and parallel) and this projective representation is the action of the mapping class group on the Hilbert space associated to the torus by the modular functor obtained from ${\mathcal{C}}$.
A subgroup $H<\operatorname{SL}(2,{\mathbb{Z}})$ is called a **congruence subgroup** if $H$ contains a principal congruence subgroup $\Gamma(n):=\{A\in \operatorname{SL}(2,{\mathbb{Z}}): A\equiv I\pmod{n}\}$ for some $n\geq 1$. Since $\Gamma(n)$ is the kernel of the reduction modulo $n$ map $\operatorname{SL}(2,{\mathbb{Z}})\rightarrow \operatorname{SL}(2,{\mathbb{Z}}/n{\mathbb{Z}})$, any congruence subgroup has finite index. The **level** of a congruence subgroup $H$ is the minimal $n$ so that $\Gamma(n)<H$. More generally, for $G<\operatorname{SL}(2,{\mathbb{Z}})$ we say $H<G$ is a congruence subgroup if $G\cap \Gamma(n)<H$ with the level of $H$ defined similarly.
The connection between topology and number theory found through the representation above is deepened by the following Congruence Subgroup Theorem:
\[nstheorem\] Let ${\mathcal{C}}$ be a modular category of rank $r$ with $T$ matrix of order $N$. Then the projective representation $\rho:\operatorname{SL}(2,{\mathbb{Z}})\rightarrow \operatorname{PSU}(r)$ has $\ker(\rho)$ a congruence subgroup of level $N$.
In particular the image of $\rho$ factors over $\operatorname{SL}(2,{\mathbb{Z}}/N{\mathbb{Z}})$ and hence is a finite group. This fact has many important consequences: for example, it is related to rank-finiteness [@BNRW1] and can be used in classification problems [@BNRW2].
A **super-modular** category is a unitary ribbon fusion category whose Müger center is equivalent, as a unitary symmetric ribbon fusion category, to the category $\operatorname{sVec}$ of super-vector spaces (equipped with its unique structure as a unitary spherical symmetric fusion category). Super-modular categories (or slight variations) have been studied from several perspectives, see [@Bon; @DMNO; @16fold; @BCT; @KLW] for a few examples. An algebraic motivation for studying these categories is the following: any unitary braided fusion category is the equivariantization [@DGNO] of either a modular or super-modular category (see [@Sawin Theorem 2]). Physically, super-modular categories provide a framework for studying fermionic topological phases of matter [@16fold]. Topological motivations include the study of spin 3-manifold invariants ([@Sawin; @Bl; @BM]) and $(3+1)$-TQFTs ([@WW]).
We restrict to unitary categories both for mathematical convenience and for their physical significance. On the other hand, there is a non-unitary version $\operatorname{sVec}^{-}$ of $\operatorname{sVec}$: the underlying (non-Tannakian) symmetric fusion category is the same, but with the other possible spherical structure, which leads to negative dimensions. We could define super-modular categories more generally as pre-modular categories ${\mathcal{B}}$ with Müger center equivalent to either of $\operatorname{sVec}$ or $\operatorname{sVec}^{-}$. However, we do not know of any examples ${\mathcal{B}}$ with ${\mathcal{B}}^{\prime}\cong\operatorname{sVec}^{-}$ that are not simply of the form ${\mathcal{C}}\boxtimes\operatorname{sVec}^{-}$ for some modular category ${\mathcal{C}}$ (A. Bruguières asked the second and third authors for such an example in 2016).
One interesting feature of super-modular categories ${\mathcal{B}}$ is that their $S$ and $T$ matrices have tensor decompositions ([@BNQ Appendix],[@16fold Theorem III.5]): $$\label{stfactored}
S=
\frac{1}{\sqrt{2}}\begin{pmatrix}
1&1\\1&1\\
\end{pmatrix}{\otimes}{\hat{S}}, \quad T= \begin{pmatrix}
1&0\\0&-1\\
\end{pmatrix}{\otimes}{\hat{T}}$$ where ${\hat{S}}$ is unitary and ${\hat{T}}$ is a diagonal (unitary) matrix, depending on $r/2-1$ sign choices. Two naive questions motivated by the above are: 1) Do ${\hat{S}}$ and a choice of ${\hat{T}}$ provide a (projective) representation of $\operatorname{SL}(2,{\mathbb{Z}})$? and 2) Is the group generated by ${\hat{S}}$ and a choice of ${\hat{T}}$ finite? Of course if ${\mathcal{B}}=\operatorname{sVec}\boxtimes {\mathcal{D}}$ for some modular category ${\mathcal{D}}$ (*split super-modular*) then the answer to both is yes. More generally, as Example \[exnofinite\] below illustrates, the answer to both questions is no.
The physical and topological applications of super-modular categories motivate a more refined question as follows. The consideration of fermions on a torus [@MooreVafa] leads to the study of spin structures on the torus $\Sigma_{1,0}$: there are three even spin structures $(A,A),(A,P),(P,A)$ and one odd spin structure $(P,P)$, where $A,P$ denote antiperiodic and periodic boundary conditions. The full mapping class group $\operatorname{Mod}(\Sigma_{1,0})=\operatorname{SL}(2,{\mathbb{Z}})$ permutes the even spin structures: ${\mathfrak{s}}$ interchanges $(P,A)$ and $(A,P)$, and preserves $(A,A)$, whereas ${\mathfrak{t}}$ interchanges $(A,A)$ and $(P,A)$ and preserves $(A,P)$. Note that both ${\mathfrak{s}}$ and ${\mathfrak{t}}^2$ preserve $(A,A)$, so that the index $3$ subgroup $\Gamma_\theta:=\langle {\mathfrak{s}},{\mathfrak{t}}^2\rangle<\operatorname{SL}(2,{\mathbb{Z}})$ is the spin mapping class group of the torus equipped with spin structure $(A,A)$. The spin mapping class group of the torus with spin structure $(A,P)$ or $(P,A)$ is similarly generated by ${\mathfrak{s}}^2$ and ${\mathfrak{t}}$, which is projectively isomorphic to ${\mathbb{Z}}$. On the other hand, $\Gamma_\theta$ is projectively the free product of ${\mathbb{Z}}/2{\mathbb{Z}}$ with ${\mathbb{Z}}$ ([@Rademacher]). Now the matrix ${\hat{T}}^2$ is unambiguously defined for any super-modular category ${\mathcal{B}}$, and in [@16fold Theorem II.7] it is shown that ${\mathfrak{s}}\rightarrow {\hat{S}}$ and ${\mathfrak{t}}^2\rightarrow{\hat{T}}^2$ defines a projective representation $\hat{\rho}$ of $\Gamma_\theta$. We propose the following:
\[mainconj\] Let ${\mathcal{B}}$ be a super-modular category of rank $2k$ and ${\hat{S}}$ and ${\hat{T}}^2$ the corresponding matrices as in equation (\[stfactored\]). Then the projective representation $\hat{\rho}:\Gamma_\theta\rightarrow \operatorname{PSU}(k)$ given by $\hat{\rho}({\mathfrak{s}})=\pi_k({\hat{S}})$ and $\hat{\rho}({\mathfrak{t}}^2)=\pi_k({\hat{T}}^2)$ has kernel a congruence subgroup.
In particular if this conjecture holds then $\hat{\rho}(\Gamma_\theta)$ is finite. We do not know what to expect the level of $\ker{\hat{\rho}}$ to be (in terms of, say, the order of ${\hat{T}}^2$), but we provide some examples below.
An important outstanding conjecture ([@DNO Question 5.15], [@16fold Conjecture III.9], see also [@M2 Conjecture 5.2]) is that every super-modular category ${\mathcal{B}}$ has a *minimal modular extension*: that is, ${\mathcal{B}}$ can be embedded in a modular category ${\mathcal{C}}$ of dimension $\dim({\mathcal{C}})=2\dim({\mathcal{B}})$. One may characterize such ${\mathcal{C}}$: they are called *spin modular categories* ([@BBC]), see Section \[spinsection\] below. Our main result proves Conjecture \[mainconj\] for super-modular categories admitting minimal modular extensions.
Preliminaries
=============
Super-Modular Categories
------------------------
Whereas one may always define an $S$-matrix for any ribbon fusion category ${\mathcal{B}}$, it may be degenerate. This failure of modularity is encoded it the subcategory of transparent objects called the **Müger center** ${\mathcal{B}}^\prime$. Here an object $X$ is called **transparent** if all the double braidings with $X$ are trivial: $c_{Y,X}c_{X,Y}=Id_{X{\otimes}Y}$. By a theorem of Bruguières [@Brug] the simple objects in ${\mathcal{B}}^\prime$ are those $X$ with ${\tilde{S}}_{X,Y}=d_Xd_Y$ for all simple $Y$, where $d_Y=\dim(Y)={\tilde{S}}_{{\mathbf{1}},Y}$ is the categorical dimension of the object $Y$. The Müger center is obviously **symmetric**, that is, $c_{Y,X}c_{X,Y}=Id_{X{\otimes}Y}$ for all $X,Y\in{\mathcal{B}}^\prime$. Symmetric fusion categories have been classified by Deligne [@Del], in terms of representations of supergroups. In the case that ${\mathcal{B}}^\prime\cong\operatorname{Rep}(G)$ (i.e. is Tannakian), the modularization (de-equivariantization) procedure of Bruguières [@Brug] and Müger [@M3] yields a modular category ${\mathcal{B}}_G$ of dimension $\dim({\mathcal{B}})/|G|$. Otherwise, by taking a maximal Tannakian subcategory $\operatorname{Rep}(G)\subset {\mathcal{B}}^\prime$ the de-equivariantization ${\mathcal{B}}_G$ has Müger center $({\mathcal{B}}_G)^\prime\cong \operatorname{sVec}$, the symmetric fusion category of super-vector spaces. Generally, a braided fusion category ${\mathcal{B}}$ with ${\mathcal{B}}^\prime\cong\operatorname{sVec}$ as symmetric fusion categories is called **slightly degenerate** [@DGNO].
The symmetric fusion category $\operatorname{sVec}$ has a unique spherical structure compatible with unitarity and has $S-$ and $T-$matrices: $S_{\operatorname{sVec}}=
\frac{1}{\sqrt{2}}\begin{pmatrix}
1&1\\1&1\\
\end{pmatrix}$ and $T_{\operatorname{sVec}}=
\begin{pmatrix}
1&0\\0&-1\\
\end{pmatrix}$.
From this point on we will assume that all our categories are unitary, so that $\operatorname{sVec}$ is a unitary symmetric fusion category. A unitary slightly degenerate ribbon category will be called **super-modular**. In other terminology, we say ${\mathcal{B}}$ is super-modular if its Müger center is generated by a **fermion**, that is, an object $\psi$ with $\psi^{\otimes 2}\cong{\mathbf{1}}$ and $\theta_\psi=-1$.
Equation (\[stfactored\]) shows that the $S$ and $T$ matrices of any super-modular category can be expressed as (Kronecker) tensor products: $S=S_{\operatorname{sVec}}\otimes{\hat{S}}$ and $T=T_{\operatorname{sVec}}\otimes{\hat{T}}$ with ${\hat{S}}$ uniquely determined and ${\hat{T}}$ determined by some sign choices. The projective group generated by ${\hat{S}}$ and ${\hat{T}}$ may be infinite for all choices of ${\hat{T}}$ as the following example illustrates:
\[exnofinite\] Consider the modular category $ \operatorname{SU}(2)_{6} $. The label set is $ I = \{0, 1, 2, 3, 4, 5, 6\} $. The subcategory $\operatorname{PSU}(2)_{6} $ is generated by 4 simple objects with even labels: $ X_{0}= {\mathbf{1}}, X_{2}, X_{4}, X_{6}$. We have $ \hat{S}=\dfrac{1}{\sqrt{4+2\sqrt{2}}}\begin{pmatrix}1 & 1+\sqrt{2}\\ 1+\sqrt{2} & -1\end{pmatrix} $ and $\hat{T}=\begin{pmatrix}1 & 0\\ 0 & \pm i\end{pmatrix} $. For either choice of ${\hat{T}}$ the eigenvalues of ${\hat{S}}{\hat{T}}$ are not roots of unity: one checks that they satisfy the irreducible polynomial $x^{16}-x^{12}+\frac{1}{4}x^8-x^4+1$, which has non-abelian Galois group and is not monic over ${\mathbb{Z}}$.
The $\theta$-subgroup of $\operatorname{SL}(2,{\mathbb{Z}})$
------------------------------------------------------------
The index $3$ subgroup $\Gamma_\theta<\operatorname{SL}(2,{\mathbb{Z}})$ generated by ${\mathfrak{s}}$ and ${\mathfrak{t}}^2$ has a uniform description (see e.g. [@Koh]): $$\Gamma_\theta=\{\begin{pmatrix} a& b\\ c&d\end{pmatrix}\in \operatorname{SL}(2,{\mathbb{Z}}): ac\equiv bd\equiv 0\pmod{2}\}.$$ The notation $\Gamma_\theta$ comes from the fact that Jacobi’s $\theta$ series $\theta(z):=\sum_{n=-\infty}^{\infty} e^{n^2\pi \mathrm{i}z}$ is a modular form of weight $1/2$ on $\Gamma_\theta$. Moreover,$\Gamma_\theta$ is isomorphic to $\Gamma_0(2)$, the Hecke congruence subgroup of level $2$ defined as those matrices in $\operatorname{SL}(2,{\mathbb{Z}})$ that are upper triangular modulo $2$, and $\Gamma(2)$ is a subgroup of both $\Gamma_0(2)$ and $\Gamma_\theta$. In particular $\Gamma_0(2)$ and $\Gamma_\theta$ are distinct, yet isomorphic, congruence subgroups of level $2$. An explicit isomorphism $\vartheta: \Gamma_\theta\rightarrow \Gamma_0(2)$ is given by $\vartheta(\mathfrak{g})=M\mathfrak{g}M^{-1}$ where $M=\begin{pmatrix} 1 & 1\\0 &2\end{pmatrix}$. This can be verified directly, via: $$M\begin{pmatrix} a& b\\ c&d\end{pmatrix}M^{-1}=\begin{pmatrix} a+c&\frac{d+b-a-c}{2}
\\2c&d-c\end {pmatrix}.$$ Observe that $\vartheta(\Gamma(n))=\Gamma(n)$ for any $n$, and for $n$ even $\Gamma(n)\lhd\Gamma_\theta$. In particular, we see that $\Gamma_\theta/\Gamma(n)<\operatorname{SL}(2,{\mathbb{Z}})/\Gamma(n)$ is isomorphic to an index 3 subgroup of $\operatorname{SL}(2,{\mathbb{Z}}/n{\mathbb{Z}})$ that is not normal. Suppose $\varphi:\Gamma_\theta\rightarrow H$ has kernel a congruence subgroup, i.e. $\Gamma(n)<\ker(\varphi)$. The congruence level of $\ker(\varphi)$, i.e. the minimal $n$ with $\Gamma(n)<\ker(\varphi)$, is the minimal $n$ so that $\Gamma_\theta/\Gamma(n)\twoheadrightarrow \varphi(\Gamma_\theta)$. The following provides a characterization of such quotients:
\[index 3\] Suppose that $n=2^kq$ with $k\geq 1$ and $q$ odd. Denote by $P_k$ a $2$-Sylow subgroup of $\operatorname{SL}(2,{\mathbb{Z}}/2^k{\mathbb{Z}})$. Then, $$\Gamma_\theta/\Gamma(n)\cong P_k\times \operatorname{SL}(2,{\mathbb{Z}}/q{\mathbb{Z}}).$$
By the Chinese Remainder Theorem, non-normal index $3$ subgroups of $$\operatorname{SL}(2,{\mathbb{Z}}/n{\mathbb{Z}})\cong\prod_{p\mid n}\operatorname{SL}(2,{\mathbb{Z}}/p^{\ell_p}{\mathbb{Z}})$$ correspond to non-normal index $3$ subgroups of $\operatorname{SL}(2,{\mathbb{Z}}/p^{\ell_p}{\mathbb{Z}})$ where $n=\prod_{p\mid n}p^{\ell_p}$ is the prime factorization of $n$. Any $2$-Sylow subgroup of $\operatorname{SL}(2,{\mathbb{Z}}/2^k{\mathbb{Z}})$ has index $3$ and is not normal (since reduction modulo $2$ gives a surjection to $\operatorname{SL}(2,{\mathbb{Z}}/2{\mathbb{Z}})\cong \mathfrak{S}_3$) so it is enough to show that this fails for $\operatorname{SL}(2,{\mathbb{Z}}/p^{k}{\mathbb{Z}})$ with $p>2$.
In general, if $H<G$ is a non-normal subgroup of index $3$ then the (transitive) left action of $G$ on the coset space $G/H$ provides a homomorphism to the symmetric group on 3 letters: $\phi:G\rightarrow \mathfrak{S}_3$. If $\phi(G)=\mathfrak{A}_3$ (the alternating group on $3$ letters) then we would have $\ker(\phi)=H\lhd G$. Thus $\phi(G)=\mathfrak{S}_3$, so that any such group $G$ must have an irreducible $2$ dimensional representation with character values $2,-1,0$.
By [@Nobs; @Eholzer] we see that for $p>2$, the groups $\operatorname{SL}(2,{\mathbb{Z}}/p^k{\mathbb{Z}})$ only have $2$-dimensional irreducible representations for $p=3,5$, and each of these representations factor over the reduction modulo $p$ map $\operatorname{SL}(2,{\mathbb{Z}}/p^k{\mathbb{Z}})\twoheadrightarrow\operatorname{SL}(2,{\mathbb{Z}}/p{\mathbb{Z}})$. By inspection neither $\operatorname{SL}(2,{\mathbb{Z}}/3{\mathbb{Z}})$ nor $\operatorname{SL}(2,{\mathbb{Z}}/5{\mathbb{Z}})$ have $\mathfrak{S}_3$ as quotients.
Main Results
============
In this section we prove Conjecture \[mainconj\] for any super-modular category that admits a minimal (spin) modular extension.
Spin Modular Categories {#spinsection}
-----------------------
A **spin modular category** ${\mathcal{C}}$ is a modular category with a (chosen) fermion. Let ${\mathcal{C}}$ be a spin modular category, with fermion $\psi$, (unnormalized) $S$-matrix ${\tilde{S}}$ and $T$-matrix $T$. Proposition II.3 of [@16fold] provides a number of useful symmetries of ${\tilde{S}}$ and $T$:
1. ${\tilde{S}}_{\psi,\alpha}=\epsilon_\alpha d_\alpha$, where $\epsilon_\alpha=\pm 1$ and $\epsilon_\psi=1$.
2. $\theta_{\psi \alpha}=-\epsilon_\alpha \theta_\alpha$.\[thetasign\]
3. ${\tilde{S}}_{\psi\alpha,\beta}=\epsilon_\beta {\tilde{S}}_{\alpha,\beta}$.\[smatrixsign\]
We have a canonical ${\mathbb{Z}}/2{\mathbb{Z}}$-grading ${\mathcal{C}}_0\oplus {\mathcal{C}}_1$ with simple objects $X\in{\mathcal{C}}_0$ if $\epsilon_X=1$ and $X\in{\mathcal{C}}_1$ when $\epsilon_X=-1$. The trivial component ${\mathcal{C}}_0$ is a super-modular category, since ${\mathcal{C}}_0^\prime=\langle \psi \rangle\cong \operatorname{sVec}$.
Since $\theta_X=-\epsilon_X\theta_{\psi X}$ it is clear that $\psi X\not\cong X$ for $X\in{\mathcal{C}}_0$. However, objects in ${\mathcal{C}}_1$ may be fixed by ${-}\otimes \psi$ or not. This provides another canonical decomposition ${\mathcal{C}}_1={\mathcal{C}}_v\oplus {\mathcal{C}}_\sigma$ as abelian categories, where a simple object $X\in{\mathcal{C}}_v\subset{\mathcal{C}}_1$ if $X\psi\not\cong X$ and $X\in{\mathcal{C}}_\sigma\subset{\mathcal{C}}_1$ if $X\psi\cong X$. Finally, using the action of ${-}{\otimes}\psi$ we make a (non-canonical) decomposition of ${\mathcal{C}}_0={\breve{{\mathcal{C}}}}_0\oplus\psi{\breve{{\mathcal{C}}}}_0$ and ${\mathcal{C}}_v={\breve{{\mathcal{C}}}}_v\oplus\psi{\breve{{\mathcal{C}}}}_v$ so that when $X\in{\breve{{\mathcal{C}}}}_0$ we have $X\psi\in\psi{\breve{{\mathcal{C}}}}_0$ and similarly for ${\mathcal{C}}_v$. Notice that for $X\in{\mathcal{C}}_0$ we have $X^*\not\cong\psi\otimes X$ since $\theta_X=\theta_{X^*}$, so that we may ensure $X$ and $X^*$ are both in ${\breve{{\mathcal{C}}}}_0$ or both in $\psi{\breve{{\mathcal{C}}}}_0$. On the other hand, for $Y\in{\mathcal{C}}_v$ it is possible that $X^*\cong \psi\otimes X$–for example, this occurs for $SO(2)_1$.
As in [@BCT] we choose an ordered basis $\Pi=\Pi_0\bigsqcup \psi\Pi_0\bigsqcup \Pi_v\bigsqcup\psi\Pi_v\bigsqcup\Pi_\sigma$ for the Grothendieck ring of ${\mathcal{C}}$ that is compatible with the above partition ${\mathcal{C}}={\breve{{\mathcal{C}}}}_0\oplus\psi{\breve{{\mathcal{C}}}}_0\oplus{\breve{{\mathcal{C}}}}_v\oplus\psi{\breve{{\mathcal{C}}}}_v\oplus{\mathcal{C}}_\sigma$. Using [@16fold Proposition II.3] we have the block matrix decomposition for the $S$ and $T$ matrices:
$$S=\begin{pmatrix}\frac{1}{2}{\hat{S}}& \frac{1}{2}{\hat{S}}& A&A &X\\
\frac{1}{2}{\hat{S}}& \frac{1}{2}{\hat{S}}& -A &-A &-X\\
A^T&-A^T & B& -B&0\\
A^T&-A^T &-B & B &0\\
X^T &-X^T &0 &0 &0
\end{pmatrix}\quad T=\begin{pmatrix} \hat{T}&0& 0&0&0\\
0&-\hat{T} & 0 &0 &0\\
0&0 &\hat{T_{v}}& 0&0\\
0&0 &0 & {\hat{T}}_{v} &0\\
0 &0 &0 &0 &{T_{\sigma}}
\end{pmatrix}.$$ Here $B$ and ${\hat{S}}$ are symmetric matrices, and each of ${\hat{T}},{\hat{T}}_v$ and $T_\sigma$ are diagonal matrices.
Now consider the following ordered partitioned basis:
1. $\Pi_0^+:=\{X_i+\psi X_i: X_i\in\Pi_0\}$,
2. $\Pi_0^-:=\{X_i-\psi X_i: X_i\in\Pi_0\}$,
3. $\Pi_v^+:=\{Y_i+\psi Y_i: Y_i\in\Pi_v\}$,
4. $ \Pi_\sigma:=\{Z_i\in\Pi_\sigma\}$ and
5. $\Pi_v^-:=\{Y_i-\psi Y_i: Y_i\in\Pi_v\}$.
With respect to this partitioned basis, the $S$ and $T$ matrices have the block form: $$S^\prime=\begin{pmatrix}{\hat{S}}& 0& 0&0&0\\
0 & 0 & 2A &X &0\\
0&2A^T &0& 0&0\\
0&2X^T &0 & 0 &0\\
0 &0 &0 &0 &2B
\end{pmatrix}\quad T^\prime=\begin{pmatrix}0 &\hat{T}& 0&0&0\\
\hat{T} & 0 & 0 &0 &0\\
0&0 &\hat{T_{v}}& 0&0\\
0&0 &0 & T_{\sigma} &0\\
0 &0 &0 &0 &\hat{T_{v}}
\end{pmatrix}.$$ From this choice of basis one sees that the representation $\rho$ restricted to $\Gamma_\theta=\langle {\mathfrak{s}},{\mathfrak{t}}^2\rangle$ has 3 invariant (projective) subspaces, spanned by $\Pi_0^+,\Pi_0^-\cup \Pi_v^+\cup \Pi_\sigma$ and $\Pi_v^-$ respectively. In particular we have a surjection $\rho(\Gamma_\theta)\twoheadrightarrow\hat{\rho}(\Gamma_\theta)$, mapping the image of $S$ in $\operatorname{PSU}(|\Pi|)$ to the image of ${\hat{S}}$ in $\operatorname{PSU}(|\Pi_0^+|)$. We can now prove:
Suppose that ${\mathcal{B}}$ is a super-modular category with minimal modular extension ${\mathcal{C}}$ so that ${\mathcal{B}}={\mathcal{C}}_0$. Assume further that the $T$-matrix of ${\mathcal{C}}$ has order $N$. Then $\hat{\rho}:\Gamma_\theta\rightarrow \operatorname{PSU}(k)$ has $\ker(\hat{\rho})$ a congruence subgroup of level at most $N$.
Let $S$ and $T$ be the $S$-matrix and $T$-matrix of ${\mathcal{C}}$. Consider the projective representation $\rho$ of $\operatorname{SL}(2,{\mathbb{Z}})$ defined by $\rho({\mathfrak{s}})=S$ and $\rho({\mathfrak{t}})=T$. By Theorem \[nstheorem\], $\ker(\rho)$ is a congruence subgroup of level $N$, i.e. $\Gamma(N)<\ker(\rho)$. Now the restriction of $\rho_{\mid \Gamma_\theta}$ to $\Gamma_\theta$ has $\ker(\rho_{\mid \Gamma_\theta})=\ker(\rho)\cap \Gamma_\theta\supset \Gamma(N)\cap\Gamma_\theta$. However, since ${\mathcal{C}}$ contains a fermion $N$ is even, so $\Gamma(N)<\Gamma(2)<\Gamma_\theta$ hence $\Gamma(N)\cap\Gamma_\theta=\Gamma(N)$. It follows that $\Gamma(N)<\ker(\rho_{\mid \Gamma_\theta})$. The discussion above now implies $\Gamma(N)<\ker(\rho_{\mid \Gamma_\theta})<\ker(\hat{\rho})$ as we have a surjection $\rho(\Gamma_\theta)\twoheadrightarrow\hat{\rho}(\Gamma_\theta)$. Thus, we have shown that $\ker(\hat{\rho})$ is a congruence subgroup of level at most $N$, and in particular $\hat{\rho}$ has finite image.
Further Questions
-----------------
The charge conjugation matrix $C$ in the basis above has the form $C^\prime_{i,j}=\pm\delta_{i,j^*}$. Since we have arranged that $X_i\in\Pi_0$ implies $X_i^*\in\Pi_0$, $C^\prime_{i,j}=-1$ can only occur for $i=j\in\Pi_v^-$: if $(W-\psi W)^*=-(W-\psi W)$ for some simple object $W$, then $W^*=\psi W$. We see that this can only happen if $W\in{\mathcal{C}}_v$ by comparing twists. Under this change of basis, we have $(S^\prime)^2=\dim({\mathcal{C}})C^\prime$ and $(S^\prime T^\prime)^3=\frac{D_+}{D}(S^\prime)^2$. It would be interesting to explore the extra relations among the various submatrices of $S^\prime$ and $T^\prime$.
The 16 spin modular categories of dimension $4$ are of the form $\operatorname{SO}(n)_1$ (where $\operatorname{SO}(n)_1\cong \operatorname{SO}(m)_1$ if and only if $n\cong m\pmod{16}$). For $n$ odd $\operatorname{SO}(n)_1$ has rank $3$ whereas for $n$ even $\operatorname{SO}(n)_1$ has rank $4$. For example, the Ising modular category corresponds to $n=1$ and $\operatorname{SO}(2)_1$ has fusion rules like the group ${\mathbb{Z}}_4$. For any modular category ${\mathcal{D}}$ and $1\leq n \leq 16$ the spin modular category $\operatorname{SO}(n)_1\boxtimes {\mathcal{D}}$ with fermion $(\psi,{\mathbf{1}})$ has either ${\mathcal{C}}_\sigma=\emptyset$ or ${\mathcal{C}}_v=\emptyset$. An interesting problem is to classify spin modular categories with either ${\mathcal{C}}_\sigma=\emptyset$ or ${\mathcal{C}}_v=\emptyset$, particularly those with no $\boxtimes$-factorization.
A Case Study
============
Our result gives an upper bound on the level of $\ker(\hat{\rho})$ for super-modular categories ${\mathcal{B}}$ with minimal modular extensions ${\mathcal{C}}$: the level of $\ker(\hat{\rho})$ is at most the order of the $T$-matrix of ${\mathcal{C}}$. The actual level can be lower: for a trivial example we consider the super-modular category $\operatorname{sVec}$. In this case ${\hat{S}}={\hat{T}}^2=I$ so the level $\ker(\hat{\rho})$ is $1$, yet the order of the $T$ matrix for its (16) minimal modular extensions can be $2,4,8$ or $16$. More generally for any split super-modular category ${\mathcal{B}}={\mathcal{D}}\boxtimes\operatorname{sVec}\subset {\mathcal{D}}\boxtimes \operatorname{SO}(n)_1={\mathcal{C}}$ (with fermion $({\mathbf{1}},\psi)$) the ratio of the levels of the kernels of the $\operatorname{SL}(2,{\mathbb{Z}})$ (for ${\mathcal{C}}$) and $\Gamma_\theta$ (for ${\mathcal{B}}$, i.e. ${\mathcal{D}}$) representations can be $2^k$ for $0\leq k\leq 4$.
To gain further insight we consider a family of non-split super-modular categories obtained from the spin modular category (see [@16fold Lemma III.7]) $ \operatorname{SU}(2)_{4m+2} $. This has modular data:\
$$\tilde{S}_{i,j}:= \dfrac{\sin \left({\frac{(i+1)(j+1)\pi}{4m+4}}\right)}{\sin ({\frac{\pi}{4m+4}})}, \quad
T_{j,j}:= e^{\frac{\pi \mathrm{i}(j^{2}+2j)}{8m+8}}$$
where $ 0\leq i,j \leq 4m+2 $. Since $T$ has order $16(m+1)$, Theorem \[nstheorem\] implies that the image of the projective representation $\rho:\operatorname{SL}(2,{\mathbb{Z}})\rightarrow \operatorname{PSU}(4m+3)$ defined via the normalized $S$-matrix $S$ and $T$ factors over $\operatorname{SL}(2,{\mathbb{Z}}/N{\mathbb{Z}})$ where $N=16(m+1)$.
The super-modular subcategory $\operatorname{PSU}(2)_{4m+2} $ has simple objects labeled by even $i,j$. The factorization (\[stfactored\]) yields the following: $$\label{PSU2 S and T}{\hat{S}}_{i,j}= \dfrac{\sin \left({\frac{(2i+1)(2j+1)\pi}{4m+4}}\right)}{\Xi\sin ({\frac{\pi}{4m+4}})}, \quad {\hat{T}}_{j,j}=e^{\frac{\pi \mathrm{i}(j^2+j)}{2m+2}}$$ for $0\leq i,j \leq m$, where $\Xi=\frac{\sqrt{\frac{m+1}{2}}}{\sin \left(\frac{\pi }{4 m+4}\right)} $. In [@16fold] all $16$ minimal modular extensions of $\operatorname{PSU}(2)_{4m+2}$ are explicitly constructed and each has $T$-matrix of order $16(m+1)$ so that the kernel of the corresponding projective $\operatorname{SL}(2,{\mathbb{Z}})$ representation is a congruence subgroup of level $16(m+1)$. The image $\hat{\rho}(\Gamma_\theta)$ is somewhat unwieldy in general; computational experiments suggest that $\hat{\rho}(\Gamma_\theta)\cong A_m^\prime\rtimes H$ where $H$ is an abelian group and $A_m^\prime:=[A_m,A_m]$ is the commutator subgroup. For this reason we study the subgroup $A_m^\prime\lhd \hat{\rho}(\Gamma_\theta)$. Another well-behaved related group is the quotient $A_m/Z(A_m)$. Since the representation $\hat{\rho}$ is not irreducible in general, the center $Z(A_m)$ may be larger than the subgroup of scalar matrices in $A_m$. However we have a surjection $\hat{\rho}(\Gamma_\theta)\twoheadrightarrow A_m/Z(A_m)$. Moreover, $\hat{\rho}(\Gamma_\theta)$ is a subquotient of $\operatorname{SL}(2,{\mathbb{Z}}/N{\mathbb{Z}})$ where $N=16(m+1)$. Indeed, if we write $m+1=2^nq$ for some odd number $q$ then the Chinese remainder theorem implies that $\operatorname{SL}(2,{\mathbb{Z}}/N{\mathbb{Z}})\cong \operatorname{SL}(2,{\mathbb{Z}}/2^{(n+4)}{\mathbb{Z}})\times \operatorname{SL}(2,{\mathbb{Z}}/q{\mathbb{Z}})$. Our computations suggests the following conjecture, with cases verified using Magma software [@magma] indicated in parentheses. A sample of the results of these computations are found in Table \[table1\]. The notation $\langle n,k\rangle$ indicates the $k$th group of order $n$ in the GAP [@GAP] library of small groups. In the last column, we sometimes give a slightly different description than is indicated in part (f) below. We include the groups $\hat{\rho}(\Gamma_\theta)$, $A_m^\prime:=[A_m,A_m]$ and $\overline{A}_m:=A_m/Z(A_m)$. As $\hat{\rho}$ is not necessarily irreducible, we have $\hat{\rho}(\Gamma_\theta)\twoheadrightarrow\overline{A}_m$. The congruence level of $\ker{\hat{\rho}}$ is computed using Lemma \[index 3\].
\[bigsuconj\] Let $A_m$ be the subgroup of $\operatorname{SU}(k)$ generated by ${\hat{S}}$ and ${\hat{T}}^2$ associated with $\operatorname{PSU}(2)_{4m+2}$, the quotient ${\overline{A}}_m:=A_m/Z(A_m)$ and the commutator subgroup $A_m^\prime:=[A_m,A_m]$. Then
1. When $m+1=q$ is odd, ${\overline{A}}_m={\overline{A}}_{q-1}\cong \operatorname{PSL}(2,{\mathbb{Z}}/q{\mathbb{Z}})$ (verified for $2\leq m\leq 18$).
2. When $m+1=2^n$ we have $|{\overline{A}}_m|=|{\overline{A}}_{2^n-1}|=2^{3n+1}$ (verified for $1\leq n\leq 5$).
3. If we write $m+1=2^nq$ where $q$ is odd, then ${\overline{A}}_m\cong {\overline{A}}_{2^n-1}\times {\overline{A}}_{q-1}$ (verified for $1\leq m\leq 14$).
4. If we write $m+1=2^nq$ where $q$ is odd $|{\overline{A}}_m|=2^{3n+1}q^3\prod_{p|q}\frac{p^2-1}{2p^2}$ (primes $p$) (verified for $1\leq m\leq 21$).
5. For $5\leq m+1=p$ prime $A_{p-1}^\prime\cong \operatorname{SL}(2,{\mathbb{Z}}/p{\mathbb{Z}})$ (verified for $4\leq m\leq 12$).
6. If we write $m+1=2^nq$ where $q$ is odd, then $A_m^\prime\cong A_{2^n-1}^\prime\times A_{q-1}^\prime$ (verified for $1\leq m\leq 14$).
7. For $m+1\not\equiv 0\pmod{4}$, we have $A_m^\prime\lhd\hat{\rho}(\Gamma_\theta)$ and $\hat{\rho}(\Gamma_\theta)$ is an iterated semidirect product of $A_m^\prime$ with cyclic group actions (verified for $1\leq m\leq 14$). In general, $\ker(\hat{\rho})$ is a congruence subgroup of level $4(m+1)$ (verified for $1\leq m\leq 12$).
[ | c | c | c| c|c|]{} $m$ & $|{\overline{A}}_m|$ & ${\overline{A}}_m$&$A_m^\prime$&$\hat{\rho}(\Gamma_\theta)$\
$1$ & $ 2^4 $ & $ D_{16}$& ${\mathbb{Z}}/8{\mathbb{Z}}$&$D_{16}=A_1^\prime\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$\
$2$ & $ 12 $ & $ \operatorname{PSL}(2,{\mathbb{Z}}/3{\mathbb{Z}})$&${\mathbf{Q}}_8$&$(A_2^\prime\rtimes{\mathbb{Z}}/3{\mathbb{Z}})\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$$\operatorname{SL}(2,{\mathbb{Z}}/3{\mathbb{Z}})\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$\
$3$ & $ 2^7$ & $\langle 128,71\rangle$ &$\langle 64,184\rangle$& $\langle 128,71\rangle$\
$4$ & $ 60 $ & $ \operatorname{PSL}(2,{\mathbb{Z}}/5{\mathbb{Z}}) $& $\operatorname{SL}(2,{\mathbb{Z}}/5{\mathbb{Z}})$&$A_4^\prime\rtimes {\mathbb{Z}}/2{\mathbb{Z}}$\
$5$ & $ 2^4\cdot 12$ & $ D_{16}\times \operatorname{PSL}(2,{\mathbb{Z}}/3{\mathbb{Z}}) $ &${\mathbb{Z}}/8{\mathbb{Z}}\times {\mathbf{Q}}_8$&$A_5^\prime\rtimes{\mathbb{Z}}/6{\mathbb{Z}}$$({\mathbb{Z}}/8{\mathbb{Z}}\times \operatorname{SL}(2,{\mathbb{Z}}/3{\mathbb{Z}}))\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$\
$6$ & $ 168$ & $ \operatorname{PSL}(2,{\mathbb{Z}}/7{\mathbb{Z}}) $&$ \operatorname{SL}(2,{\mathbb{Z}}/7{\mathbb{Z}}) $ &$A_6^\prime\rtimes {\mathbb{Z}}/2{\mathbb{Z}}$\
$7$ & $ 2^{10} $ & $\overline{A}_{7} $ &$|\cdot|=2^9$&$\overline{A}_7$\
$8$ & $ 324$ & $\operatorname{PSL}(2,{\mathbb{Z}}/9{\mathbb{Z}}) $&$({\mathbb{Z}}/3{\mathbb{Z}})^3\rtimes {\mathbf{Q}}_8$&$(A_8^\prime\rtimes{\mathbb{Z}}/3{\mathbb{Z}})\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$\
$9$ & $ 2^4\cdot 60$ & $ D_{16}\times \operatorname{PSL}(2,{\mathbb{Z}}/5{\mathbb{Z}}) $& $ {\mathbb{Z}}/8{\mathbb{Z}}\times \operatorname{SL}(2,{\mathbb{Z}}/5{\mathbb{Z}})$&$A_9^\prime\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$\
$10$ & $660$ & $\operatorname{PSL}(2,{\mathbb{Z}}/11{\mathbb{Z}}) $& $\operatorname{SL}(2,{\mathbb{Z}}/11{\mathbb{Z}})$& $A_{10}^\prime\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$\
$11$ & $2^7\cdot 12$ &$\langle 128,71\rangle\times \operatorname{PSL}(2,{\mathbb{Z}}/3{\mathbb{Z}})$ &$\langle64,184\rangle\times{\mathbf{Q}}_8$&$\operatorname{SL}(2,{\mathbb{Z}}/3{\mathbb{Z}})\rtimes\langle 128,71\rangle$\
$12$ & $1092 $ & $ \operatorname{PSL}(2,{\mathbb{Z}}/13{\mathbb{Z}}) $ &$\operatorname{SL}(2,{\mathbb{Z}}/13{\mathbb{Z}})$&$\operatorname{SL}(2,{\mathbb{Z}}/13{\mathbb{Z}})\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$\
$13$ & $2^4\cdot 168 $ & $D_{16}\times \operatorname{PSL}(2,{\mathbb{Z}}/7{\mathbb{Z}})$&${\mathbb{Z}}/8{\mathbb{Z}}\times \operatorname{SL}(2,{\mathbb{Z}}/7{\mathbb{Z}})$& $A_{13}^\prime\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$\
$14$ & $720$ & $ \operatorname{PSL}(2,{\mathbb{Z}}/15{\mathbb{Z}})$&${\mathbf{Q}}_8\times \operatorname{SL}(2,{\mathbb{Z}}/5{\mathbb{Z}})$&$(A_{14}^\prime\rtimes{\mathbb{Z}}/3{\mathbb{Z}})\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$$\operatorname{SL}(2,{\mathbb{Z}}/15{\mathbb{Z}})\rtimes{\mathbb{Z}}/2{\mathbb{Z}}$\
$15$ & $ 2^{13} $ &$ $&&\
$16$ & $ 2448 $ & $\operatorname{PSL}(2,{\mathbb{Z}}/17{\mathbb{Z}}) $&&\
$17$ & $ 2^4\cdot 324 $ & $ $&&\
$18$ & $ 3420$ &$ \operatorname{PSL}(2,{\mathbb{Z}}/19{\mathbb{Z}}) $&&\
$19$ & $ 2^7\cdot 60 $& $ $&&\
$20$ & $12\cdot 168 $ & $ $&&\
$21$ & $ 2^4\cdot 660 $ & $ $ &&\
Appendix: Magma Code {#appendix-magma-code .unnumbered}
====================
For our computational experiments we used the symbolic algebra software Magma [@magma]. In this appendix we give some basic pseudo-code and some sample Magma code to illustrate how we found the image of $\hat{\rho}(\Gamma_\theta)$ in our case study, so that the interested reader can do similar explorations. Given an integer $m$, the $(m+1)\times (m+1)$ ${\hat{S}}$ and ${\hat{T}}^2$ matrices obtained from $\operatorname{PSU}(2)_{4m+2}$ are given in equation (\[PSU2 S and T\]). In order to use the Magma software we express the entries of ${\hat{S}}$ and ${\hat{T}}^2$ in the cyclotomic field ${\mathbf{Q}}(\omega)$, where $\omega$ is an $(8m+8)$-th root of unity. For this we must write $\sin \left({\frac{(2i+1)(2j+1)\pi}{4m+4}}\right)$ and $\sqrt{2(m+1)}$ in terms of $\omega$ for which we use the result of generalized form of quadratic Gauss sums [@berndt1981determination].
Here is the pseudocode to find $\hat{\rho}(\Gamma_\theta)$ for $\operatorname{PSU}(2)_{4m+2}$:\
**algorithm** projective image：\
**input:** integer $m$\
**output:** $\hat{\rho}(\Gamma_\theta) $\
**set** $ K$ to be the cyclotomic field ${\mathbf{Q}}(\omega)$, where $\omega$ is an $(8m+8)$-th root of unity.\
**set** $M=2(m+1)$\
**initialize** S and T2 to be $(m+1)\times (m+1)$ zero matrices over K.\
**initialize** $\alpha=0$.\
**step 1: calculate $\alpha$**\
**if** $M\equiv 0\pmod{4}$ **return** $\alpha=\sum_{n=0}^{M-1} \omega^{4n^2}/(1+\omega^M)$\
**else** Consider $M/2=m+1 \pmod{4}$. Notice there are only two cases: $m+1\equiv 1$ (mod 4) and $m+1\equiv 3\pmod{4}$.\
**if** $m+1\equiv 1 \pmod{4}$ **return** $\alpha=\frac{\omega^{m + 1} - \omega^{-(m + 1)}}{\omega^{2m + 2}}\sum_{n=0}^{m} \omega^{8n^2}$\
**else** **return** $\alpha=\frac{\omega^{m + 1} - \omega^{-(m + 1)}}{\omega^{2m + 2}}\sum_{n=0}^{m} \omega^{8n^2}/(\omega^M)$\
**return** $\alpha=2/\alpha$.\
**step 2: define the entries**\
**for** $1\leq i,j\leq m+1$, $S_{i,j}=\alpha \dfrac{\omega^{(2i-1)(2j-1)}-\omega^{-(2i-1)(2j-1)}}{2(\omega^M)}$ and $T2_{j,j}=\omega^{(2(j-1))^2+4(j-1)}$\
**step 3: find the projective image**\
**set** $A$ to be the matrix group generated by $S$ and $T2$ defined above, and $ZK$ the group of scalar matrices over $K$. The projective image of $A$ is then $A/(ZK\cap A)$.\
The following code can be used in Magma [@magma] to find the $\hat{\rho}(\Gamma_\theta)$ in this case, and slight modifications will give the other headings of Table \[table1\]:
m:=1;
K<w>:= CyclotomicField(8*m+8);
GL:=GeneralLinearGroup(m+1,K);
M:=2*(m+1);
alpha:=0;
if M mod 4 eq 0 then
for n:=0 to M-1 do
alpha:=alpha + w^(4*(n^2));
end for;
alpha:=alpha/(w^M+1);
else
if (m+1) mod 4 eq 1 then
for n:=0 to m do
alpha:= alpha + w^(8*n^2);
end for;
else
for n:=0 to m do
alpha:=alpha + w^(8*(n^2));
end for;
alpha:=alpha/(w^M);
end if;
alpha:=((w^(m + 1) - w^(-(m + 1)))/(w^(2*m + 2)))*alpha;
end if;
alpha:=2/alpha;
S:=ZeroMatrix(K,m+1,m+1);
for i:=1 to m+1 do
for j:=1 to m+1 do
S[i,j]:=(w^((2*i-1)*(2*j-1))-w^(-(2*i-1)*(2*j-1)))/(2*(w^M));
S[i,j]:=S[i,j]*alpha;
end for;
end for;
T2:=ZeroMatrix(K,m+1,m+1);
for j:=1 to m+1 do
T2[j,j]:=w^((2*(j-1))^2+4*(j-1));
end for;
A:=MatrixGroup<m+1,K|S,T2>;
ZK:=MatrixGroup<m+1,K|w*IdentityMatrix(K,m+1)>;
F:=(A/(A meet ZK));
[^1]: E. Rowell and Q. Zhang were partially supported by NSF grant DMS-1410144, and Z. Wang by NSF grant DMS-1411212. The authors thank M. Cheng, M. Papanikolas and Z. Sunic for valuable discussions.
|
---
abstract: 'We study the Loschmidt echo for a system of electrons interacting through mean-field Coulomb forces. The electron gas is modeled by a self-consistent set of hydrodynamic equations. It is observed that the quantum fidelity drops abruptly after a time that is proportional to the logarithm of the perturbation amplitude. The fidelity drop is related to the breakdown of the symmetry properties of the wave function.'
author:
- 'G. Manfredi'
- 'P.-A. Hervieux'
title: Loschmidt echo in a system of interacting electrons
---
[*Introduction*]{}.—In a famous controversy with L. Boltzmann, J. Loschmidt pointed out that, if one reverses the velocities of all particles in a physical system, the latter would evolve back to its initial state, thus violating the second law of thermodynamics. The main objection to this argument is that velocity reversal is a very unstable operation and tiny errors in the reversal quickly restore normal entropy increase.
More recently, the original idea of Loschmidt was revived in the context of quantum information theory. Indeed, any attempt at coding information using quantum bits is prone to failure if a small coupling to an uncontrollable environment destroys the unitary evolution of the wave function (decoherence) [@zurek]. In order to estimate the robustness of the system against perturbations from the environment, the following procedure has been suggested. The system is allowed to evolve under the action of an unperturbed Hamiltonian until time $T$; then it is evolved backwards in time until $2T$ with the original Hamiltonian plus a small perturbation (the ‘environment’). The square of the scalar product of the initial and final states defines the quantum fidelity of the system (Loschmidt echo) and has been the object of intense study in recent years. Jalabert and Pastawski [@jalabert] have proven that, for perturbations that are classically weak but quantum-mechanically strong, the fidelity decay rate only depends on the classical Lyapunov exponent of the unperturbed system. This universal behavior was later corroborated by numerical simulations [@jacquod; @cucchietti] and experiments [@echo-exp]. For weaker perturbations, the decay rate is still exponential, but perturbation-dependent (Fermi golden rule regime). For still weaker perturbations, the decay is Gaussian [@jacquod].
An equivalent approach to the Loschmidt echo was proposed earlier by Peres [@peres]. In order to study the separation of classical trajectories, it is customary to compare the evolution of two slightly different initial conditions. Peres noted that one could just as well compare the [*same*]{} initial condition evolving in two slightly different Hamiltonians, an unperturbed one $H_0$ and a perturbed one $H=H_0+\delta H$. The fidelity at time $t$ is then defined as the square of the scalar product of the wave functions evolving with $H_0$ and $H$ respectively: $F(t)
= ~\vline \langle \psi_{H_0}(t) \vline ~\psi_H(t) \rangle
\vline^{~2}$. The latter approach is the one adopted throughout the present paper.
Virtually all theoretical investigations of the Loschmidt echo consider one-particle systems evolving in a given (usually chaotic) Hamiltonian. The aim of the present work is to explore the more realistic case of a system of many interacting particles, particularly electrons. In order to obtain a tractable model, we shall assume that the electrons interact via the electrostatic mean field, their dynamics being described by a set of one-dimensional (1D) hydrodynamic equations. As many experimental studies on quantum information involve the manipulation of charged particles, our approach may shed some light on the robustness of such systems against perturbations from the environment.
[*Model*]{}.—The physical properties of our model are best illustrated by considering its classical counterpart, the so-called ‘cold plasma’ model [@cold-plasma; @bertrand]. In the latter, the electron population is described by a phase-space distribution function of the type: $f(x,v,t) =
n(x,t)~\delta[v-u(x,t)]$, where $n$ and $u$ are respectively the electron density and average velocity, and $\delta$ denotes the Dirac delta function. The support of such a distribution function in the 2D phase space $(x,v)$ is a 1D curve defined by the relation $v=u(x,t)$. The electron distribution evolves according to the (collisionless) Vlasov equation. The ions are motionless with uniform equilibrium density $n_0$ and periodic boundary conditions (with $-L/2 \le x \le L/2$) are assumed for all variables.
As long as there is no particle overtaking in the phase space, each position $x$ corresponds to a well-defined velocity $u(x,t)$. In this case, the Vlasov equation can be reduced to a closed set of pressureless hydrodynamic equations: $$\begin{aligned}
\label{eq:contin} \frac{\partial\,n}{\partial\,t} &+&
\frac{\partial\,(nu)}{\partial\,x} = 0, \\
\label{eq:force} \frac{\partial\,u}{\partial\,t} &+&
u\frac{\partial\,u}{\partial\,x} =
\frac{e}{m}\frac{\partial\,\phi}{\partial\,x},\end{aligned}$$ where $-e$ an $m$ are respectively the electron charge and mass, and $\phi(x,t)$ is the electric potential obeying Poisson’s equation $$\label{eq:poisson} \frac{\partial^{2}\phi}{\partial\,x^2} =
\frac{e}{\varepsilon_0}\left(n - n_{0}\right).$$ When particles overtake each other, the function $x \to u(x)$ becomes multivalued. The above hydrodynamic description then breaks down, although the microscopic Vlasov model remains valid.
A trivial stationary solution of Eqs. (\[eq:contin\])-(\[eq:force\]) is given by $n=n_0$, $u=0$. The electron dynamics can be excited by modulating the initial velocity: $u(x,t=0^+) = V_0 \cos(k_0 x)$, with $k_0=2\pi/L$, which is equivalent to applying an instantaneous electric field at time $t=0$. The classical dynamics is determined by a single dimensionless parameter, namely the normalized wave number of the initial perturbation $K_0 = k_0 V_0/\omega_p$, where $\omega_p =
(e^2 n_0/m \varepsilon_0)^{1/2}$ is the electron plasma frequency. Note that $K_0$ can be viewed as the ratio of kinetic to potential (electric) energy. It can be shown that two different regimes exist. When $K_0<1$, electric repulsion dominates, so that the electrons never overtake each other in the phase space. In this case, the hydrodynamic equations (\[eq:contin\])-(\[eq:force\]) can be solved analytically and the solution displays nonlinear oscillations at the plasma frequency. When $K_0>1$, the analytical solution breaks down and the dynamics must be described by the microscopic Vlasov equation. In the extreme case $K_0 \gg 1$, the dynamics becomes again integrable, because it reduces to that of free-streaming electrons. For moderate values of $K_0$ (but still larger than unity), the electrons can be alternately free streaming and trapped by the self-consistent (SC) potential. This regime corresponds to the formation of complex vortices in the phase space and leads to a chaotic dynamics, as was pointed out for the similar scenario of nonlinear Landau damping [@valen]. In this work, we will be mainly interested in the chaotic regime and use the value $K_0=2$.
Quantum corrections to the hydrodynamic equations (\[eq:contin\])-(\[eq:force\]) were previously derived [@qfluid]. For fermions at zero temperature (a case that is relevant to electrons in metals), the momentum conservation equation (\[eq:force\]) should be modified as follows: $$\label{eq:force_q} \frac{\partial\,u}{\partial\,t} +
u\frac{\partial\,u}{\partial\,x} =
\frac{e}{m}\frac{\partial\,\phi}{\partial\,x}
+\frac{\hbar^2}{2m^2} \frac{\partial}{\partial
x}\left(\frac{\partial_x^{2}\sqrt{n}} {{\sqrt{n}}}\right) -
\frac{1}{mn} \frac{\partial\,P_F}{\partial\,x}.$$ The second term on the right-hand side is the Bohm potential: this is a dispersive term that prevents the breakdown of the quantum hydrodynamics even when $K_0>1$ [@bertrand]. The third term is the Fermi pressure, which in 1D can be written as: $P_F/P_0 =
(n/n_0)^3$, where the equilibrium pressure is given by the usual formula, $P_0=\frac{2}{5}n_0 E_F$. $E_F$ is the Fermi energy computed with the equilibrium density.
The continuity equation (\[eq:contin\]) and the quantum momentum equation (\[eq:force\_q\]) can be written in the form of a single nonlinear Schr[ö]{}dinger equation by introducing the effective wave function $\Psi(x,t) = \sqrt{n(x,t)}\exp{(iS(x,t)/\hbar)}$, where $S(x,t)$ is defined according to the relation $m u =
\partial_x S$, and $n=|\Psi|^2$. The wave function $\Psi$ obeys the equation $$\label{eq:nlse} i\hbar\frac{\partial\Psi}{\partial\,t} = -
\,\frac{\hbar^2}{2m}~\frac{\partial^2 \Psi}{\partial x^2} -
e\phi\Psi + \frac{3}{5} E_F \frac{|\Psi|^4}{n_0^2}~ \Psi.$$
Equation (\[eq:nlse\]) with Poisson’s equation (\[eq:poisson\]) constitute the mathematical model used throughout this Letter. The equilibrium Hamiltonian $H_0$ is time dependent, as it depends self-consistently on the wave function, but conserves both the total mass and the total energy. The initial condition is analog to the classical one described in the preceding paragraphs and can be easily derived from the velocity perturbation $u(x,t=0^+)$ by using the relation between $S$ and $u$. Two more dimensionless parameters (in addition to $K_0$) intervene in the quantum model: (i) the normalized Planck constant $h=\hbar \omega_p/m V_0^2$, which measures the importance of quantum effects; and (ii) the normalized Fermi velocity $v_F/V_0$. The latter affects very little the results (provided it is not too large), and will be fixed at $v_F/V_0 = 0.1$ in the forthcoming simulations.
![\[fig:fig1\] Frequency spectrum of the potential energy, for an unperturbed evolution with $K_0=2$ and $h=0.05$.](fig_1.eps){height="4.5cm"}
The numerical solution of Eq. (\[eq:nlse\]) is obtained through a splitting scheme that separates the kinetic and potential parts of the Hamiltonian. Derivatives are computed with centered differences. The resulting algorithm is second-order accurate both in space and time.
[*Results*]{}.—First, we characterize the spectral properties of the unperturbed Hamiltonian. We consider a case with $K_0=2$ and $h=0.05$ and plot in Fig. 1 the frequency spectrum of the time history of the electrostatic energy. The spectrum is broad and virtually flat in the range $0 < \omega \lesssim 3 \omega_p$. The dynamics is therefore sufficiently irregular to enable us to compare our results to those obtained for a single-particle chaotic Hamiltonian.
In order to study the behavior of the quantum fidelity, we need to compare the evolution of $\Psi$ obtained with the unperturbed and perturbed Hamiltonians. We use a static perturbation consisting of a sum of a large number of uncorrelated waves: $\delta H(x)/mV_0^2
= \epsilon \sum_{j=N_{\rm min}}^{N_{\rm max}} \cos(k_j x
+\alpha_j)$, where $\epsilon$ is the amplitude of the perturbation, $k_j = j~ (2\pi/L)$, and the $\alpha_j$ are random phases. The wave number spectrum of the perturbation (i.e. the values of $N_{\rm min}$ and $N_{\rm max}$) affect only very weakly the behavior of the fidelity: therefore, we will focus our analysis on the dependence of the fidelity on the amplitude $\epsilon$.
![\[fig:fig2\] Fidelity decay for $K_0=2$, $h=0.05$, and perturbation $\epsilon =10^{-9}$.](fig_2_rev.eps){height="4.5cm"}
A typical result for the quantum fidelity is presented in Fig. 2. Contrarily to most single-particle cases, the fidelity does not decay exponentially. Instead, it remains equal to unity until a critical time $\tau_c$, after which it decays abruptly within a few units of $\omega_p^{-1}$. This behavior is generic and was observed for all set of parameters that were studied, provided the dynamics is sufficiently irregular. Numerical tests showed that the value of the critical time is independent on the time step. However, as the underlying dynamics is chaotic, evolutions computed using different time steps will inevitably diverge for long times. Therefore, the details of the evolution for $t \gg
\tau_c$ are not quantitatively meaningful, and simply indicate that the fidelity has dropped to very small values.
The observed drop in the quantum fidelity is related to a sudden symmetry breaking of the wave function. Indeed, the evolution equations (\[eq:poisson\])-(\[eq:nlse\]) for the unperturbed Hamiltonian are invariant under the transformation $x \to -x$. If the initial condition is an even function of $x$, this symmetry is thus preserved in time, i.e. $\Psi(x,t) = \Psi(-x,t),~\forall t$. But the perturbation $\delta H$ possesses no particular symmetry, and one would expect that the symmetry of the initial state quickly deteriorates. The symmetry properties can be conveniently measured by the following quantity: $$\Sigma(t) = ~\frac{2}{n_0 L} ~{\vline \int_0^{L/2} \Psi(x,t)
~\Psi^\star(-x,t) ~dx \vline}^{~2}, \label{eq:symm}$$ which is equal to unity when $\Psi(x,t) = \Psi(-x,t)$. The evolution of $\Sigma(t)$ for the perturbed Hamiltonian is plotted in Fig. 3. The drop of the quantum fidelity happens virtually at the same time as the breaking of the wave function symmetry. This behavior is also generic across a wide range of parameters.
![\[fig:fig3\] Evolution of the symmetry $\Sigma(t)$ for the same case as in Fig. 2.](fig_3_rev.eps){height="4.5cm"}
We further investigated the dependence of the critical time $\tau_c$ on the perturbation amplitude $\epsilon$, for various values of the normalized Planck constant $h$. The critical time is defined as the time at which the fidelity has dropped to 10% of its maximum value, i.e. $F(\tau_c)=0.1$. Figure 4 shows that, for small values of $h$, $\tau_c$ depends logarithmically on the perturbation amplitude, i.e. $\tau_c \sim -t_0 \ln \epsilon$, with $\omega_p t_0 \simeq 4.3$ (this is the straight line depicted in Fig. 4). This logarithmic dependence appears to be universal (both in slope and absolute value), at least for the values of $K_0$ and $v_F/V_0$ adopted in these runs. For larger values of Planck’s constant ($h \gtrsim 0.2$), this behavior is less neat, particularly for small perturbations.
![\[fig:fig4\] Critical time $\tau_c$ (in units of $\omega_p^{-1}$) versus perturbation amplitude $\epsilon$, for $h=0.05$ (stars), $h=0.025$ (diamonds), and $h=0.0125$ (triangles). The solid line represents the curve $\tau_c \sim -t_0
\ln \epsilon$, with $\omega_p t_0 = 4.3$.](fig_4.eps){height="4.25cm"}
A similar pattern was observed for a chaotic quantum map [@casati]: in that case, the fidelity stays equal to unity until a critical time, after which it starts to decay exponentially at the classical Lyapunov rate. No sudden drop was observed, as is the case for our simulations.
In order to better evaluate the impact of the SC field, we performed some simulations where all nonlinear terms have been suppressed in Eqs. (\[eq:poisson\])-(\[eq:nlse\]). The first nonlinearity comes from the Fermi pressure and can be removed simply by setting $E_F=0$. The SC nonlinearity comes from the fact that the electric potential depends on the wave function through the electron density $n=|\Psi|^2$. To remove this nonlinearity, we define the electron density independently of $\Psi$, as the sum of traveling plane waves: $n=n_{\rm ext} \equiv n_0[1+\delta
\sum_{j=1}^{25} k_j^2 \cos(k_j x -\omega_p t+\alpha'_j)]$, where $\delta$ is the amplitude of the density fluctuations, and the $\alpha'_j$ are random phases. This definition is plugged into Poisson’s equation to yield the electric potential. As the resonances of the plane waves overlap in phase space, the resulting (time-dependent) Hamiltonian $H_0$ is likely to display chaotic regions [@reson]. The fidelity decay is studied by perturbing the Hamiltonian in the same way as in the SC case.
In Fig. 5 we plot the quantum fidelity for $\delta=0.5$, $h=0.025$, and several values of the perturbation $\epsilon$. The fidelity decay is exponential and begins at $t=0$. The decay rate is approximately proportional to the square of the perturbation, which shows that we are in the so-called Fermi golden rule regime [@jacquod; @casati]. However, contrarily to Ref. [@casati], no plateau was observed for short times.
![\[fig:fig5\] Fidelity decay for a purely external Hamiltonian: $\epsilon=10^{-3}$ (stars); $\epsilon=2 \times
10^{-3}$ (triangles); $\epsilon=5 \times10^{-4}$ (diamonds). Time is rescaled to the square of the perturbation $\epsilon$.](fig_5.eps){height="4cm" width="6cm"}
![\[fig:fig6\] Fidelity decay for a mixed external and SC Hamiltonian, for $\beta=0$ (solid line); $\beta=0.01$ (dotted); $\beta=0.03$ (dashed); $\beta=0.1$ (dot-dashed); $\beta=0.3$ (dot-dot-dashed).](fig_6.eps){height="4cm" width="6cm"}
Finally, we studied a case where both the SC and the external fields are present. This is accomplished by defining the electron density as: $n(x,t)= n_{\rm ext} + \beta (|\Psi|^2-n_0)$. By varying $\beta$ and $\delta$, we can move continuously from a purely ‘external’ regime ($\beta=0$) to a purely SC one ($\delta=0$, $\beta=1$). We concentrate on a case with $\delta=0.5$, $h=0.025$, and $\epsilon=10^{-3}$, and vary the value of $\beta$ (Fig. 6). For short times, the fidelity decays exponentially with the same rate as in the purely ‘external’ case. Subsequently, the decay becomes faster, and for large values of $\beta$ we almost recover the abrupt drop of Fig. 2.
[*Discussion*]{}.— The present work is a first attempt at studying quantum fidelity decay in a system of electrons interacting through their SC electric field. Our numerical results show that the quantum fidelity can display a rapid decrease. Such effect is probably related to the fact that the unperturbed Hamiltonian $H_0$ depends on the wave function. When the perturbation $\delta H$ induces a small change in $\Psi$, $H_0$ is itself modified, which in turns affects $\Psi$, and so on. Because of such nonlinear loop, the perturbed and unperturbed solutions can diverge very fast (typically, within a few $\omega_p^{-1}$). In contrast, for the single-particle dynamics, $H_0$ is fixed and the solutions only diverge because of the small perturbation $\delta H$.
In summary, it appears that the ‘natural’ response of a SC system to a perturbation is a sudden drop of the fidelity after a quiescent period rather than an exponential decay. Further studies will be needed to understand whether these results extend to more complex models, going beyond the mean-field approach adopted here.
We acknowledge fruitful discussions with R. A. Jalabert.
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|
---
author:
- |
E.L. Bratkovskaya\
[*Frankfurt Institute for Advanced Studies,*]{}\
[*Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany*]{}
title: '**Signals of the QGP phase transition - a view from microscopic transport models** '
---
[**Abstract**]{}\
In this contribution the results from various transport models on different observables - considered as possible signals of the phase transition from hadronic matter to the quark-gluon plasma (QGP) - are briefly reviewed.
Introduction {#s1}
============
The phase transition from partonic degrees of freedom (quarks and gluons) to interacting hadrons is a central topic of modern high-energy physics. In order to understand the dynamics and relevant scales of this transition laboratory experiments under controlled conditions are presently performed with ultra-relativistic nucleus-nucleus collisions. Hadronic spectra and relative hadron abundancies from these experiments reflect important aspects of the dynamics in the hot and dense zone formed in the early phase of the reaction.
Estimates based on the Bjorken formula [@bjorken] for the energy density achieved in central Au+Au collisions suggest that the critical energy density for the formation of a quark-gluon plasma (QGP) is by far exceeded during a few fm/c in the initial phase of Au+Au collisions at Relativistic Heavy-Ion Collider (RHIC) energies, but sufficient energy densities ($\sim$ 0.7-1 GeV/fm$^3$ [@Karsch]) might already be achieved at Alternating Gradient Synchrotron (AGS) energies of $\sim$ 10 $A\cdot$GeV [@exita]. More recently, lattice QCD calculations at finite temperature and quark chemical potential $\mu_q$ [@Fodor] show a rapid increase of the thermodynamic pressure $P$ with temperature above the critical temperature $T_c$ for a phase transition to the QGP. The crucial question is, however, at what bombarding energies the conditions for the phase transition are fulfilled. Thus, it is very important to perform an ’energy scan’ of different observables in order to find an ’anomalous’ behavior that might be attributed to a phase transition.
In addition to the strong interactions in the initial stage of the reaction - attributed to the QGP - there are also strong (pre-)hadronic interactions after/during the hadronization phase. Thus it becomes very important to know the impact of such (pre-) hadronic interactions on the final observables. The relevant information on this issue can be provided by microscopic transport models based on a nonequilibrium description of the nuclear dynamics [@HORST].
In this contribution I present the compilation of HSD results on two of the possible signals of the phase transition: strangeness and charm. The HSD (Hadron-String-Dynamics) transport approach [@Geiss; @Cass99] employs hadronic and string degrees of freedom and takes into account the formation and multiple rescattering of hadrons; it thus dynamically describes the generation of pressure in the early phase - dominated by strings - and the hadronic expansion phase. The HSD transport approach is matched to reproduce the nucleon-nucleon, meson-nucleon and meson-meson cross section data in a wide kinematic range. It also provides a good description of particle production in p+A reactions [@Sibbi] as well electroproduction of hadrons off nuclei [@Falter]. In order to obtain a model independent conclusion, we also address the results from the UrQMD model [@URQMD1; @URQMD2] which has similar underlying concepts as HSD but differs in the actual realizations.
Strangeness signals of the QGP {#s2}
==============================
As has been proposed in 1982 by Rafelski and Müller [@Rafelski1] the strangeness degree of freedom might play an important role in distinguishing hadronic and partonic dynamics. In 1999 Gaździcki and Gorenstein [@SMES] - within the statistical model - have predicted experimental observables which should show an anomalous behaviour at the phase transition: the ’kink’ – an enhancement of pion production in central Au+Au (Pb+Pb) collisions relative to scaled $pp$ collisions; the ’horn’ – a sharp maximum in the $K^+/\pi^+$ ratio at 20 to 30 A$\cdot$GeV; the ’step’ – an approximately constant slope of $K^\pm$ spectra starting from 20 to 30 A$\cdot$GeV. Indeed, such “anomalies” have been observed experimentally by the NA49 Collaboration [@NA49_new; @NA49_T].
In Refs. [@Weber02; @Brat03PRL; @Bratnew] we have investigated the hadron production as well as transverse hadron spectra in nucleus-nucleus collisions from 2 $A\cdot$GeV to 21.3 $A\cdot$TeV within the independent transport approaches UrQMD and HSD. The comparison to experimental data demonstrates that both approaches agree quite well with each other and with the experimental data on hadron production. The enhancement of pion production in central Au+Au (Pb+Pb) collisions relative to scaled $pp$ collisions (the ’kink’) is well described by both approaches without involving any phase transition. However, the maximum in the $K^+/\pi^+$ ratio at 20 to 30 A$\cdot$GeV (the ’horn’) is missed by $\sim$ 40% [@Weber02; @Bratnew] – cf. Fig. \[Fig1ab\] (l.h.s.). A comparison to the transverse mass spectra from $pp$ and C+C (or Si+Si) reactions shows the reliability of the transport models for light systems [@Brat03PRL]. For central Au+Au (Pb+Pb) collisions at bombarding energies above $\sim$ 5 A$\cdot$GeV, however, the measured $K^{\pm}$ $m_{T}$-spectra have a larger inverse slope parameter than expected from the calculations. The approximately constant slope of $K^\pm$ spectra at SPS (the ’step’) is not reproduced either [@Brat03PRL; @Bratnew] – cf. Fig. \[Fig1ab\] (r.h.s.). The HSD calculations also demonstrate that the ’partonic’ Cronin effect plays a minor role at AGS and SPS energies for the parameter $T$. The slope parameters from $pp$ collisions (r.h.s. in Fig. \[Fig1ab\]) are seen to increase smoothly with energy both in the experiment (full squares) and in the transport calculations (full lines with open circles) and are significantly lower than those from central Au+Au reactions for $\sqrt{s} > 3.5$ GeV.
Thus the pressure generated by hadronic interactions in the transport models above $\sim$ 5 A$\cdot$GeV is lower than observed in the experimental data. This finding suggests that the additional pressure - as expected from lattice QCD calculations at finite quark chemical potential and temperature - might be generated by strong interactions in the early pre-hadronic/partonic phase of central Au+Au (Pb+Pb) collisions.
Charm signals of the QGP {#s3}
========================
The microscopic HSD transport calculations (employed here) provide a suitable space-time geometry of the nucleus-nucleus reaction and a rather reliable estimate for the local energy densities achieved. The energy density $\varepsilon({\bf r};t)$ – which is identified with the matrix element $T^{00}({\bf r};t)$ of the energy momentum tensor in the local rest frame at space-time $({\bf r},t)$ – reaches up to 30 GeV/fm$^3$ in a central Au+Au collision at $\sqrt{s}$ = 200 GeV [@Olena2].
According to present knowledge the charmonium production in heavy-ion collisions, [*i.e.*]{} $c\bar{c}$ pairs, occurs exclusively at the initial stage of the reaction in primary nucleon-nucleon collisions. The parametrizations of the total charmonium cross sections ($i = \chi_c, J/\Psi, \Psi^\prime$) from $NN$ collisions as a function of the invariant energy $\sqrt{s}$ used in this work are taken from [@Cass99; @Cass00; @brat03; @Cass01]. We recall that (as in Refs. [@brat03; @Cass01; @Geiss99; @Cass97; @CassKo]) the charm degrees of freedom in the HSD approach are treated perturbatively and that initial hard processes (such as $c\bar{c}$ or Drell-Yan production from $NN$ collisions) are ‘precalculated’ to achieve a scaling of the inclusive cross section with the number of projectile and target nucleons as $A_P \times A_T$ when integrating over impact parameter. For fixed impact parameter $b$ the $c\bar{c}$ yield then scales with the number of binary hard collisions $N_{bin}$ ([*cf.*]{} Fig. 8 in Ref. [@Cass01]).
In the QGP ‘threshold scenario’, e.g the geometrical Glauber model of Blaizot et al. [@Blaizot] as well as the percolation model of Satz [@Satzrev], the QGP suppression ‘(i)’ sets in rather abruptly as soon as the energy density exceeds a threshold value $\varepsilon_c$, which is a free parameter. This version of the standard approach is motivated by the idea that the charmonium dissociation rate is drastically larger in a quark-gluon-plasma (QGP) than in a hadronic medium [@Satzrev].
On the other hand, the extra suppression of charmonia in the high density phase of nucleus-nucleus collisions at SPS energies [@NA50aa; @NA60] has been attributed to inelastic comover scattering ([*cf.*]{} [@Cass99; @Cass00; @Cass97; @Olena; @Capella; @Vogt99; @Gersch; @Kahana; @Spieles; @Gerland] and Refs. therein) assuming that the corresponding $J/\Psi$-hadron cross sections are in the order of a few mb [@Haglin; @Konew; @Sascha]. In these models ‘comovers’ are viewed not as asymptotic hadronic states in vacuum but rather as hadronic correlators (essentially of vector meson type) that might well survive at energy densities above 1 GeV/fm$^3$. Additionally, alternative absorption mechanisms might play a role such as gluon scattering on color dipole states as suggested in Refs. [@Kojpsi; @Rappnew; @Blaschke1; @Blaschke2] or charmonium dissociation in the strong color fields of overlapping strings [@Geiss99].
The explicit treatment of initial $c\bar{c}$ production by primary nucleon-nucleon collisions and the implementation of the comover model - involving a single matrix element $M_0$ fixed by the data at SPS energies - as well as the QGP threshold scenario in HSD are described in Refs. [@Olena2; @Olena] (see Fig. 1 of Ref. [@Olena] for the relevant cross sections). We recall that the ‘threshold scenario’ for charmonium dissociation is implemented as follows: whenever the local energy density $\varepsilon(x)$ is above a threshold value $\varepsilon_j$ (where the index $j$ stands for $J/\Psi, \chi_c, \Psi^\prime$), the charmonium is fully dissociated to $c + \bar{c}$. The default threshold energy densities adopted are $\varepsilon_1 = 16$ GeV/fm$^3$ for $J/\Psi$, $\varepsilon_2 = 2$ GeV/fm$^3$ for $\chi_c$, and $\varepsilon_3 =2 $ GeV/fm$^3$ for $ \Psi^\prime$. Two more scenarios were implemented similarly to the ‘comover suppression’ and the ‘threshold melting’ by adding the only additional assumption – that the comoving mesons (including the $D$-mesons) exist only at energy densities below some energy density $\epsilon _{cut}$, which is a free parameter. We use $\epsilon _{cut}=1$ GeV/fm$^3$, [*i.e.*]{} of the order of the critical energy density.
In the following, we compare our calculations to the experimental data at the top RHIC energy of $\sqrt{s}=200$ GeV. We recall that the experimentally measured nuclear modification factor $R_{AA}$ is given by, $$R_{AA}=\frac{ d N (J/\Psi) _{AA} / d y }{ N_{coll} \cdot d N
(J/\Psi) _{pp} / d y },$$ where $d N (J/\Psi) _{AA} / d y $ denotes the final yield of $J/\Psi$ in $A A$ collisions, $d N (J/\Psi) _{pp} / d y$ is the yield in elementary $p p$ reactions and $N_{coll}$ is the number of binary collisions.
Due to the very high initial energy densities reached (corresponding to $T \geq 2 T_c$), in the threshold melting scenario all initially created $J/\Psi$, $\Psi'$ and $\chi _c$ mesons melt. However, the PHENIX collaboration has found that at least 20% of $J/\Psi$ do survive at RHIC [@PHENIX]. Thus, the importance of charmonium recreation is shown again. In HSD, we account for $J/\Psi$ recreation via the $D \bar D$ annihilation processes as explained in detail in [@Olena2; @Olena]. Note that in our approach, the cross sections of charmonium recreation in $D
+ \bar D \to J/\Psi + meson$ processes is fixed by detailed balance from the comover absorption cross section $J/\Psi + meson
\to D + \bar D$. But even after both these processes are added to the threshold melting mechanism, the centrality dependence of the $R_{AA} (J/\Psi)$ cannot be reproduced in the ‘threshold melting’ scenario, especially for peripheral collisions (cf. Fig. \[RHICthreshold\]). This holds for both possibilities: with (r.h.s. of Fig. \[RHICthreshold\]) and without (center of Fig. \[RHICthreshold\]) energy-density cut $\epsilon_{cut}$, below which $D$-mesons and comovers do exist and can participate in $D + \bar D \leftrightarrow J/\Psi + meson$ reactions.
Comover absorption scenarios give generally a correct dependence of the yield on the centrality. If an existence of D-mesons at energy densities above 1 GeV/fm$^3$ is assumed, the amplitude of suppression of $J/\Psi$ at mid-rapidity is also well reproduced (see the line for ‘comover without $\epsilon_{cut}$’ scenario in Fig.\[RHICcomover\], l.h.s.). Note that this line corresponds to the prediction made in the HSD approach in [@brat04]. On the other hand, the rapidity dependence of the comover result is wrong, both with and without $\epsilon _{cut}$. If hadronic correlators exist only at $\epsilon < \epsilon _{cut}$, comover absorption is insufficient to reproduce the $J/\Psi$ suppression even at mid-rapidity (Fig. \[RHICcomover\], r.h.s.). But its contribution to the charmonium suppression is, nevertheless, substantial. The difference between the theoretical curves marked ‘comover + $\epsilon _{cut}$’ and the data shows the maximum possible suppression that can be attributed to a deconfined medium.
We mention that there are also alternative explanations of the experimental data for the anomalous $J/\Psi$ suppression in A+A collisions: e.g. formation of charmonia only at the phase boundary as advocated by Andronic [*et al.*]{} [@Andronic] in the statistical hadronization model.
Summary
=======
Summarizing this contribution, I want to point out that strange hadron production in central Au+Au (or Pb+Pb) collisions is quite well described in the independent transport approaches HSD and UrQMD [@Weber02]. The exception are the pion rapidity spectra at the highest AGS energy and lower SPS energies, which are overestimated by both models. As a consequence the HSD and UrQMD transport approaches underestimate the experimental maximum (’horn’) of the $K^+/\pi^+$ ratio at $\sim$ 20-30 A$\cdot$GeV [@Weber02]. The inverse slope parameters $T$ for $K^\pm$ mesons from the HSD and UrQMD transport models are practically independent of system size from $pp$ up to central Pb+Pb collisions and show only a slight increase with collision energy, but no ’step’ in the $K^\pm$ transverse momentum slopes as suggested by Gaździcki and Gorenstein [@SMES] in 1999 and found experimentally by the NA49 Collaboration. The rapid increase of the inverse slope parameters of kaons for collisions of heavy nuclei (Au+Au) found experimentally in the AGS energy range, however, is not reproduced by both models (see Fig. \[Fig1ab\]). Since the pion transverse mass spectra – which are hardly effected by collective flow – are described sufficiently well at all bombarding energies [@Bratnew], the failure has to be attributed to a lack of pressure. I have argued - based on lattice QCD calculations at finite temperature and baryon chemical potential $\mu_B$ [@Karsch; @Fodor] as well as the experimental systematics in the chemical freeze-out parameters [@Cleymans] - that this additional pressure should be generated in the early phase of the collision, where the ’transverse’ energy densities in the transport approaches are higher than the critical energy densities for a phase transition (or cross-over) to the QGP. The interesting finding of the analysis is, that pre-hadronic degrees of freedom might already play a substantial role in central Au+Au collisions at AGS energies above $\sim$ 5 $A\cdot$GeV.
The formation and suppression dynamics of $J/\Psi$, $\chi_c$ and $\Psi^\prime$ mesons has, furthermore, been studied within the HSD transport approach for $Au+Au$ reactions from FAIR to top RHIC energies of $\sqrt{s}$ = 200 GeV. It is found that both the ‘comover absorption’ and ‘threshold melting’ concepts fail severely at RHIC energies [@Olena2] whereas both models perform quite well at SPS energies. The failure of the ’hadronic comover absorption’ model goes in line with its underestimation of the collective flow $v_2$ of leptons from open charm decay [@brat05]. This suggests that the dynamics of $c, \bar{c}$ quarks at RHIC energies are dominated by strong pre-hadronic/partonic interactions of charmonia with the medium in a strong QGP (sQGP), which cannot be modeled by ‘hadronic’ scattering or described appropriately by color screening alone.
The evidence for creating a ’new state of matter’, the sQGP, in Au+Au collisions at RHIC is overwhelming and is additionally supported by the strong suppression of high $p_T$ hadrons and ’far-side’ jets - both observations being insufficiently described by string/hadronic models [@Ca1; @Ca2] - as well as the quark-number scaling of elliptic flow $v_2(p_T)$. The question now reads: what are the properties of the sQGP?
[**Acknowledgements:**]{} I finally like to thank Mark Gorenstein for the numerous discussions and fruitful collaboration over the last years. Also a lot of thanks to my coauthors M. Bleicher, W. Cassing, O. Linnyk, H. Stöcker and H. Weber who contributed to the results presented here.
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address: 'Polyfold Lab, UC Berkeley, Evans Hall, Berkeley CA 94720-3840'
author:
- Benjamin Filippenko
- Zhengyi Zhou
- Katrin Wehrheim
title: Counterexamples in Scale Calculus
---
[H]{}
i c ø ł
Ł ¶
=
\[section\] \[theorem\][Conjecture]{} \[theorem\][Example]{} \[theorem\][Notation]{} \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Corollary]{} \[theorem\][Definition]{} \[theorem\][Remark]{} \[theorem\][Technical Remark]{} \[theorem\][Exercise]{} \[theorem\][Question]{} \[theorem\][Fact]{} \[theorem\][Claim]{}
=-0.5in =-0.5in
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abstract: 'Double-layers (DLs) were observed in the expanding region of an inductively coupled plasma with $\text{Ar}/\text{SF}_6$ gas mixtures. No DL was observed in pure argon or $\text{SF}_6$ fractions below few percent. They exist over a wide range of power and pressure although they are only stable for a small window of electronegativity (typically between 8% and 13% of $\text{SF}_6$ at 1mTorr), becoming unstable at higher electronegativity. They seem to be formed at the boundary between the source tube and the diffusion chamber and act as an internal boundary (the amplitude being roughly 1.5$\frac{kT_e}{e}$)between a high electron density, high electron temperature, low electronegativity plasma upstream (in the source), and a low electron density, low electron temperature, high electronegativity plasma downstream.'
author:
- 'N. Plihon'
- 'C.S. Corr'
- 'P. Chabert'
bibliography:
- '095508APL.bib'
title: Double layer formation in the expanding region of an inductively coupled electronegative plasma
---
Double layers (DLs) have been studied over the past decades theoretically, numerically and experimentally (see [@raadu] and references therein). The biggest part of the literature treats the case of electropositive plasmas, however, DLs were also found in electronegative plasmas [@kouznetsovjap99; @sheri99; @sheridlpop99; @merlinopofb90]. More recently, Charles and co-workers [@charlesapl03; @charlespop04] have observed a current-free DL in the expanding region of a helicon wave excited plasma at very low pressures (typically less than a millitorr). A strongly diverging static magnetic field seemed to be required in order to reach the conditions for DL formation. Their system also had an abrupt change in radius at the boundary between the source and the diffusion chambers, which could possibly be a source of DL formation [@andrews71].
In this letter we show that a DL can be formed in a system that has a similar geometry to that of Charles and co-workers, but without the use of a static diverging magnetic field. However, this was possible only with a minimum percentage of an electronegative gas (namely $\text{SF}_6$) added to argon. Moreover, this electronegative DL was mostly unstable. It was found to be stable for a very narrow range of $\text{SF}_6$ mixtures.
The reactor is shown schematically in figure \[plihonfig1\]. It was originally designed to operate in the helicon regime. For the work presented in this letter, the system was operated without a static magnetic field, i.e. in the inductive mode. The reactor consists of a source chamber sitting on top of a 32 cm diameter, 26 cm long aluminum diffusion chamber. Hence, there is an expanding plasma underneath the source region. The source is a 14 cm diameter, 30 cm long and 0.7 cm thick pyrex cylinder surrounded by a double saddle field type helicon antenna [@bos70]. The fan-cooled antenna is powered through a close-coupled L-type matching network by an rf power supply operating at 13.56 MHz and capable of delivering up to 2 kW forward power. The input power was recorded as the difference between the forward and reflected powers. The pyrex cylinder is housed in an aluminum cylinder of 20 cm diameter and 30 cm long. A metal grid attached to the other end of the source tube confines the plasma from a turbomolecular pump that routinely achieves base pressures of $10^{-6}$ mbar. The partial gas pressures of Ar and $\text{SF}_6$ are determined by controlling the flows.
All measurements reported here were made along the revolution axis ($z$ axis) of the discharge. Two types of electrostatic probes were used for measurements. The first is a nickel planar probe with guard ring biased at the same potential as the probe, to measure the real saturated positive ion current. The diameter of the collecting area was 4 mm and the diameter of the outer ring was 8 mm. The second is a passively compensated Langmuir probe (LP) [@can77], of 0.25 mm diameter and 6 mm long platinum wire tip. The LP was used to find the plasma potential, electron densities and electron temperature from measurements of the probe I(V) characteristics using a Smartsoft data acquisition system [@hop86]. The electronegativity, $\alpha = n_-/n_e$, and consequently the ion densities (electro-neutrality $n_+=n_-+n_e$ was assumed), were measured according to the double-probe technique described in [@chabPSST99]. This technique, which relies on the theory developed in [@sheri99], allows to deduce $\alpha$ from the ratio of the cylindrical probe current at the plasma potential to the positive ion saturation current measured by the planar probe, $R=I(V_p)/I_{\rm sat+}$. The technique requires an estimation of the ratio of the electron temperature to the negative ion temperature $\gamma = T_e/T_-$ and the positive ion mass $m_+$, both difficult to measure in the gas mixture studied here. We chose $\gamma=15$, as is commonly thought to be a reasonable value in low pressure electronegative discharges, and $m_+=40$ since (i) $\text{Ar}^+$ may be dominant since we used small percentages of $\text{SF}_6$ in argon (ii) we expect a fairly high dissociation degree of $\text{SF}_6$ and therefore $\text{SF}_x^+$ ions with $x\ll6$ (low mass ions). As a consequence of these estimations, the absolute values of $\alpha$ should be regarded as indicative. However, we believe that spatial gradients of $\alpha$, or relative variations with operating conditions (pressure, power, mixture) are correctly captured by the technique.
Stable DLs are accessible for pressures from 0.3 to 10 mTorr when carefully adjusting the $\text{SF}_6$ concentration. The minimum power required to obtain a stationary DL increases with increasing pressure, with no upper limit observed (at 1 mTorr, DLs are observed above 200W; at 10 mTorr, above 1400W)
All results presented here are for a gas pressure of 1 mTorr and an input power of 600W. Figure \[plihonfig2\]a shows the axial evolution of the plasma potential and the electronegativity for a $\text{SF}_6$ concentration of 6%. The dashed line represents the position of the interface between the source and the diffusion chamber. The plasma potential decreases continuously from the source to the diffusion chamber, as expected for an expanding plasma which exhibits a gradient in the electron density, while the electronegativity remains roughly constant along the axis. There is evidently no DL. For these conditions of pressure and power, the transition towards the formation of the DL is observed to occur at about 8% $\text{SF}_6$ concentration; with no DLs observed in pure argon or for $\text{SF}_6$ concentrations below 8%. Above this concentration, the plasma potential and particles gradients are drastically changed as shown in figure \[plihonfig2\]b. The plasma potential presents a sharp drop at around z = 22cm on the axis, that is about 4 cm below the interface between the two chambers. The potential difference between the source chamber and the diffusion chamber seems to be at least 10V, although the sharp drop seems to be around 5V (which is $1.5\frac{kT_{e}}{e}$ with the downstream electron temperature). This sharp drop is preceded by a strong but smoother gradient that resembles a pre-sheath. From visual observation, it seems that the DL has a spherical shape that is attached to the boundary between the source and the diffusion chamber and that expands into the diffusion chamber (refer to the dashed gray line on Figure \[plihonfig1\]).
The electronegativity is also profoundly affected. It presents a sharp maximum at the DL position, with a slow decay downstream (below the DL in the diffusion chamber) and a much faster decay upstream. The variations of $\alpha$ are directly related to the change in the electron density, as shown on figure \[plihonfig3\]a. The electron density is strongly affected by the sudden drop in potential, whereas both the positive and negative ion densities seem to decrease continuously from the source to the diffusion chamber. The electron temperature changes significantly when crossing the DL. The DL acts as an internal boundary (or sheath), which separates two plasmas; a high electron density, high electron temperature, low electronegativity plasma upstream, and a low electron density, low electron temperature, high electronegativity plasma downstream.
As the $\text{SF}_6$ concentration is increased, the upstream plasma moves further into the diffusion chamber (for a 11% $\text{SF}_6$ mixture, the spherical shape of the DL being more elongated, the position of the DL on the axis is z = 18cm) and the plasma potential drop becomes less abrupt, gradually replaced by a larger region of strong gradient of potential upstream, before entering the DL itself. The downstream plasma potential remains mostly constant at about 15 V.
Downstream and upstream of the DL, the electrons remain in Boltzmann equilibrium, with temperatures given by the slope of $\ln(n_e)$ as a function of $V_p$ being 3.2 eV downstream and 4.5 eV upstream, which is very close to the temperatures (from LP processing) given in Figure \[plihonfig3\]b. On the contrary, negative ions are far from Boltzmann equilibrium, and are present both sides of the DL. Since they cannot cross it from upstream to downstream (their temperature is much too small), they must be created downstream, i.e. the attachment rate must be strong in this region. This may be due to the relatively low electron temperature, and also to higher neutral gas density because of colder neutral gas (inductive discharges are known to produce significant gas heating near the coil). The negative ions created in the big buffer region downstream from the DL would then be accelerated toward the source through the DL. Unlike attachment, ionization is probably mainly located in the source region where the the electron temperature is high. Hence, positive ions are mainly produced in the source region and are accelerated downstream through the DL. The DL is therefore crossed by two ion streams in opposite directions.
The origin of the DL formation remains unclear. From our data and from visual observation, we can postulate that the DL is formed at the boundary between the two chambers and diffuses in the diffusion chamber, as proposed in earlier work for electropositive gases [@andrews71]. However, this geometric feature is not sufficient to explain our observations since we did not observe the DL in pure argon. We can postulate that the $\text{SF}_6$ addition has two main effect that contributes to the DL formation. First, the positive ions will more easily reach the ion sound limitation (a necessary condition to form a DL) since it is well known that the ion sound speed is lowered in electronegative plasmas[@braith88]. Second, the attachment process is a very efficient loss term for electrons during the plasma expansion, which makes steeper $n_e$ gradients and therefore higher potential gradients. This effect may be compared to the strongly divergent magnetic field used by Charles and co-workers [@charlesapl03], which also acts as a loss process for electrons during the expansion.
For the typical conditions considered so far (1 mTorr and 600W), the DLs were stable from 8% to 13% $\text{SF}_6$. Above 13%, the DL becomes unstable, and periodic oscillations of the charged particle densities, plasma potential and electron temperature are observed. It seems that the unstable regime is characterized by a periodic formation and propagation of a double layer. This instability has strong similarities with the downstream instabilities observed and modeled by Tuszewski and co-workers [@Tuszewskipop03]. However, the association between the downstream instabilities and a DL was not clearly established by these authors. This issue will be treated in a separate publication.
We have observed double-layer formation in the expanding region of an inductively coupled electronegative plasma. The DL’s were not observed in pure argon. They are stable for a small window of electronegativity and become unstable at higher electronegativity. They seem to have a spherical shape and be formed at the boundary between the source and the diffusion chambers. They act as an internal boundary between a high electron density, high electron temperature, low electronegativity plasma upstream, and a low electron density, low electron temperature, high electronegativity plasma downstream. They exist in a wide range of pressure and power and without a strongly divergent magnetic field, which can be seen as an advantage for space plasma propulsion. However, the voltage drop in the DL is about three times smaller than the DL’s described by Charles and co-workers [@charlesapl03].
![\[plihonfig1\]Schematic of the experimental setup.](095508APL1)
![\[plihonfig2\] Spatial evolution of the plasma potential, $V_p$, and the electronegativity, $\alpha = \frac{n_-}{n_e}$, in (a) the no DL case (6% $\text{SF}_6$ mixture), and (b) the DL case (9% $\text{SF}_6$ mixture), at 1 mTorr, 600 W.](095508APL2)
![\[plihonfig3\] Spatial evolution of the particles densities and electron temperature for a 9% $\text{SF}_6$ mixture at 1 mTorr, 600W.](095508APL3)
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---
abstract: 'In this paper we consider square functions (also called Littlewood-Paley $g$-functions) associated to Hankel convolutions acting on functions in the Bochner Lebesgue space $L^p((0,\infty ),\mathbb{B})$, where $\mathbb{B}$ is a UMD Banach space. As special cases we study square functions defined by fractional derivatives of the Poisson semigroup for the Bessel operator $\Delta _\lambda =-x^{-\lambda }\frac{d}{dx}x^{2\lambda }\frac{d}{dx}x^{-\lambda }$, $\lambda >0$. We characterize the UMD property for a Banach space $\mathbb{B}$ by using $L^p((0,\infty ),\mathbb{B})$-boundedness properties of $g$-functions defined by Bessel-Poisson semigroups. As a by product we prove that the fact that the imaginary power $\Delta_\lambda^{i\omega}$, $\omega \in \mathbb{R}\setminus\{0\}$, of the Bessel operator $\Delta _\lambda $ is bounded in $L^p((0,\infty ),\mathbb{B})$, $1<p<\infty$, characterizes the UMD property for the Banach space $\mathbb{B}$. As applications of our results for square functions we establish the boundedness in $L^p((0,\infty ),\mathbb{B})$ of spectral multipliers $m(\Delta _\lambda )$ of Bessel operators defined by functions $m$ which are holomorphic in sectors $\Sigma_\vartheta$.'
address: ' Jorge J. Betancor, Alejandro J. Castro and Lourdes Rodríguez-Mesa Departamento de Análisis Matemático, Universidad de La Laguna, Campus de Anchieta, Avda. Astrofísico Francisco Sánchez, s/n, 38271, La Laguna (Sta. Cruz de Tenerife), Spain'
author:
- 'Jorge J. Betancor'
- 'Alejandro J. Castro'
- 'L. Rodríguez-Mesa'
title: Square functions and spectral multipliers for Bessel operators in UMD spaces
---
Introduction {#sec:intro}
============
Square functions (also called Littlewood-Paley $g$-functions) were considered in the works of Littlewood, Paley, Zygmund and Marcinkiewicz during the decade of the 30’s in the last century. These functions were introduced to get new equivalent norms, for instance, in $L^p$-spaces. By using these new equivalent norms the boundedness of some operators, for instance multipliers, can be established.
Suppose that $(\Omega, \Sigma, \mu)$ is a $\sigma$-finite measure space and $\{T_t\}_{t>0}$ is a symmetric diffusion semigroup of operators in the sense of Stein ([@Ste1]). For every $k\in \mathbb{N}$, the square function $g_k$ associated to $\{T_t\}_{t>0}$ is defined by $$g_k(\{T_t\}_{t>0})(f)(x)
=\left(\int_0^\infty \left| t^{k}\partial_t^k T_t(f)(x)\right|^2 \frac{dt}{t}\right)^{1/2}.$$
In [@Ste1 p. 120] it is shown that, for every $k\in \mathbb{N}$ and $1<p<\infty$, there exists $C>0$ such that $$\label{equiv}
\frac{1}{C}\|f-E_0(f)\|_{L^p(\Omega ,\mu )}
\leq \| g_k(\{T_t\}_{t>0})(f) \|_{L^p(\Omega ,\mu )}
\leq C\|f\|_{L^p(\Omega ,\mu )},\quad f\in L^p(\Omega ,\mu ),$$ where $E_0(f)=\lim_{t\rightarrow \infty}T_t(f)$ is the projection onto the fixed point space of $\{T_t\}_{t>0}$. As application of it can be proved that Laplace transform type multipliers associated with $\{T_t\}_{t>0}$ are bounded from $L^p(\Omega ,\mu )$ into itself, $1<p<\infty$ ([@Ste1 p. 121]). Note that when $E_0=0,$ says that by defining for every $1<p<\infty$ and $k\in \mathbb{N}$, $$||| f |||_k
=\| g_k(\{T_t\}_{t>0})(f) \|_{L^p(\Omega ,\mu )},\quad f\in L^p(\Omega ,\mu ),$$ $|||\cdot |||_k$ is an equivalent norm to the usual norm in $L^p(\Omega,\mu )$. Meda in [@Me1] extends the property to symmetric contraction semigroups $\{T_t\}_{t>0}$ with $E_0=0$ and he applies it to get the boundedness in $L^p$ of spectral multipliers $m(\mathcal{L})$ where the operator $\mathcal{L}$ is the infinitesimal generator of $\{T_t\}_{t>0}$ and $m$ is an holomorphic and bounded function in a sector $\Sigma_\vartheta =\{z\in \mathbb{C}: |\mbox{Arg }z|<\vartheta \}$. Here $1<p<\infty$ and $\vartheta \in [0,\pi /2]$ are connected.
We consider the functions $$\label{2.1}
{\mathbb{W}}_t(z)
=\frac{e^{-|z|^2/4t}}{(4\pi t)^{n/2}},\quad z\in \mathbb{R}^n\mbox{ and }t>0,$$ and $${\mathbb{P}}_t(z)
=b_n\frac{t}{(t^2+|z|^2)^{(n+1)/2}},\quad z\in \mathbb{R}^n\mbox{ and }t>0,$$ where $b_n=\pi ^{-(n+1)/2}\Gamma ((n+1)/2)$.
As it is well-known the classical heat semigroup $\{\mathbb{W}_t\}_{t>0}$ is defined by $${\mathbb{W}}_t(f)(x)
=\int_{\mathbb{R}^n} {\mathbb{W}}_t(x-y)f(y)dy,\quad x\in \mathbb{R}^n \mbox{ and } t>0,$$ and the classical Poisson semigroup $\{{\mathbb{P}}_t\}_{t>0}$ is given by $${\mathbb{P}}_t(f)(x)
=\int_{\mathbb{R}^n} {\mathbb{P}}_t(x-y)f(y)dy,\quad x\in \mathbb{R}^n \mbox{ and } t>0 ,$$ for every $f\in L^p(\mathbb{R}^n)$. $\{{\mathbb{W}}_t\}_{t>0}$ and $\{{\mathbb{P}}_t\}_{t>0}$ are generated by $-\Delta$ and $-\sqrt{\Delta}$, respectively, where $\Delta =-\sum_{j=1}^n \frac{\partial
^2}{\partial x_i^2}$ denotes the Laplace operator. The classical heat and Poisson semigroups are the first examples of diffusion semigroups having trivial fixed point space. For every measurable function $g:\mathbb{R}^n\longrightarrow \mathbb{C}$, we define $g_t(x)=t^{-n}g(x/t)$, $x\in \mathbb{R}^n$ and $t>0$.
Let $k\in \mathbb{N}$. We can write $$g_k(\{{\mathbb{W}}_t\}_{t>0})(f)(x)
= \left(\int_0^\infty |(\varphi _{\sqrt{t}}*f)(x)|^2\frac{dt}{t}\right)^{1/2},\quad x\in \mathbb{R}^n,$$ being $\varphi (x)=\partial _t^kG_{\sqrt{t}}(x)_{|t=1}$ and $G(x)=(4\pi )^{-n/2}e^{-|x|^2/4}$, $x\in \mathbb{R}^n$. Also, we have that $$g_k(\{{\mathbb{P}}_t\}_{t>0})(f)(x)
=\left(\int_0^\infty |(\phi _t*f)(x)|^2\frac{dt}{t}\right)^{1/2},\quad x\in \mathbb{R}^n,$$ where $\phi (x)=\partial _t^k {\mathbb{P}}_{t}(x)_{|t=1}$, $x\in \mathbb{R}^n$.
If $\psi \in L^2(\mathbb{R}^n)$ we consider the square function defined by $$g_\psi (f)(x)
=\left(\int_0^\infty |(\psi _t*f)(x)|^2\frac{dt}{t}\right)^{1/2},\quad x\in \mathbb{R}^n.$$ Thus, $g_\psi$ includes as special cases $g_k(\{{\mathbb{W}}_t\}_{t>0})$ and $g_k(\{{\mathbb{P}}_t\}_{t>0})$.
In the sequel, if $f\in \mathcal{S}(\mathbb{R}^n)$, the Schwartz class, we denote by $\widehat{f}$ the Fourier transform of $f$ given by $$\widehat{f}(y)
=\frac{1}{(2\pi )^{n/2}}\int_{\mathbb{R}^n}f(x)e^{-ix\cdot y}dx,\quad y\in \mathbb{R}^n.$$ As it is well-known the Fourier transform can be extended to $L^2(\mathbb{R}^n)$ as an isometry in $L^2(\mathbb{R}^n)$.
\[ThA\] Suppose that $\psi \in L^2(\mathbb{R}^n)$ satisfies the following properties:
\(i) if $\alpha =(\alpha _1,...,\alpha_n)\in \{0,1\}^n$ and $|\alpha |=\sum_{j=1}^n\alpha _j\leq 1+[n/2]$, the distributional derivative $\frac{\partial ^{|\alpha |}}{\partial x_1^{\alpha _1}...\partial x_n^{\alpha _n}}\widehat{\psi}$ is represented by a measurable function and $$\sup_{|z|=1}\int_0^\infty t^{2|\alpha |}\left|\left(\frac{\partial ^{|\alpha |}}{\partial x_1^{\alpha _1}...\partial x_n^{\alpha _n}}\widehat{\psi}\right)(tz)\right|^2\frac{dt}{t}<\infty,$$
\(ii) $\displaystyle \inf_{|z|=1}\int_0^\infty |\widehat{\psi }(tz)|^2\frac{dt}{t}>0$.
Then, for every $1<p<\infty$, there exists $C>0$ such that $$\frac{1}{C}\|f\|_{L^p(\mathbb{R}^n)}
\leq \|g_\psi (f)\|_{L^p(\mathbb{R}^n)}\leq C\|f\|_{L^p(\mathbb{R}^n)},\quad f\in L^p(\mathbb{R}^n).$$
Also, square functions can be defined by using functional calculus for operators (see, for instance, [@LeMer2] and [@Me2]). Note that if $A$ is the infinitesimal generator of the semigroup $\{T_t\}_{t>0}$ we can write, for every $k\in \mathbb{N}$, $$t^k\partial _t^kT_t
=F_k(tA),\quad t>0,$$ where $F_k(z)=(-z)^ke^{-z}$, $z\in \mathbb{C}$.
Suppose that $\mathbb{B}$ is a Banach space and $T$ is a linear bounded operator from $L^p(\Omega ,\mu )$ into itself where $1<p<\infty$. We define $T\otimes I_\mathbb{B}$ on $L^p(\Omega ,\mu )\otimes \mathbb{B}$ in the usual way. If $T$ is positive, $T\otimes I_\mathbb{B}$ can be extended to $L^p(\Omega ,\mu ,\mathbb{B})$ as a bounded operator from $L^p(\Omega ,\mu ,\mathbb{B})$ into itself, and to simplify we continue denoting this extension by $T$.
The question is to give a definition for the square functions, when we consider functions taking values in a Banach space $\mathbb{B}$, defining equivalent norms in the Bochner-Lebesgue space $L^p(\Omega, \mu ,\mathbb{B})$, $1<p<\infty$.
Let $\{T_t\}_{t>0}$ be a symmetric diffusion semigroup on a $\sigma$-finite measure space $(\Omega, \Sigma ,\mu )$. We denote by $\{P_t\}_{t>0}$ the subordinated semigroup to $\{T_t\}_{t>0}$, that is, $$P_t(f)
=\frac{t}{2\sqrt{\pi}}\int_0^\infty \frac{e^{-t^2/(4u)}}{u^{3/2}}T_u(f)du,\quad t>0.$$ Thus $\{P_t\}_{t>0}$ is also a symmetric diffusion semigroup. The classical Poisson semigroup is the subordinated semigroup of the classical heat semigroup.
In order to define $g$-functions in a Banach valued setting, the more natural way is to replace the absolute value in the scalar definition by the norm in $\mathbb{B}$. This is the way followed, for instance, in [@MTX] and [@Xu] where they work with square functions defined by subordinated semigroups $\{P_t\}_{t>0}$ as follows $$g_{1,\mathbb{B}}(\{P_t\}_{t>0})(f)(x)
=\left(\int_0^\infty \|t\partial_tP_t(f)(x)\|_{\mathbb{B}}^2\frac{dt}{t}\right)^{1/2},\quad x\in \Omega .$$ Actually in [@MTX] and [@Xu] generalized square functions are considered where the $L^2$-norm is replaced by the $L^q$-norm, $1<q<\infty$. As a consequence of [@MTX Theorems 5.2 and 5.3] (see also [@Kw]) we deduce that for a certain $1<p<\infty$ there exists $C>0$ such that, for every $f\in L^p({{\mathbb{R}^n}},\mathbb{B})$, $$\frac{1}{C} \|f\|_{L^p({{\mathbb{R}^n}},\mathbb{B})}
\leq \|g_{1,\mathbb{B}}(\{{\mathbb{P}}_t\}_{t>0})(f) \|_{L^p({{\mathbb{R}^n}})}
\leq C \|f\|_{L^p({{\mathbb{R}^n}},\mathbb{B})},$$ if, and only if, $\mathbb{B}$ is isomorphic to a Hilbert space.
In order to get new equivalent norms for $L^p(\Omega, \mu ,\mathbb{B})$ by using square functions, for Banach spaces $\mathbb{B}$ which are not isomorphic to Hilbert spaces, stochastic integrals and $\gamma$-radonifying operators have been considered. We point out the papers of Bourgain [@Bou], Hytönen [@Hy], Hytönen, Van Neerven and Portal [@HNP], Hytönen and Weis [@HW], Kaiser [@Ka], and Kaiser and Weis [@KaWe], amongst others. In this work we use $\gamma$-radonifying operators. We recall some definitions and properties about this kind of operators that will be useful in the sequel.
Assume that $\mathbb{B}$ is a Banach space and $H$ is a Hilbert space. We choose a sequence $\{\gamma _j\}_{j\in \mathbb{N}}$ of independent standard Gaussian variables defined on some probability space $(\Omega, \mathcal{F},\rho)$. By $\mathbb{E}$ we denote the expectation with respect to $\rho$. A linear operator $T:H\longrightarrow \mathbb{B}$ is said to be $\gamma$-summing ($T\in \gamma ^\infty (H,\mathbb{B})$) when $$\|T\|_{\gamma ^\infty (H,\mathbb{B})}
=\sup\left(\mathbb{E} \Big\|\sum_{j=1}^k\gamma _jT(h_j)\Big\|_\mathbb{B}^2\right)^{1/2}<\infty ,$$ where the supremum is taken over all the finite family $\{h_j\}_{j=1}^k$ of orthonormal vectors in $H$. $\gamma ^\infty (H,\mathbb{B})$ endowed with the norm $\|\cdot\|_{\gamma ^\infty (H,\mathbb{B})}$ is a Banach space. We say that a linear operator $T:H\longrightarrow \mathbb{B}$ is $\gamma$-radonifying (shortly, $T \in \gamma(H,{\mathbb{B}})$) when $T\in \overline{\mathcal{F}(H,\mathbb{B})}^{\gamma ^\infty (H,\mathbb{B})}$, where $\mathcal{F}(H,\mathbb{B})$ denotes the span of finite range operators from $H$ to ${\mathbb{B}}$. If $\mathbb{B}$ does not contain isomorphic copies of $c_0$ then, $\gamma (H,\mathbb{B})=\gamma ^\infty (H,\mathbb{B})$ ([@HJ], [@Kw1] and [@Nee Theorem 5.9]). Note that if $\mathbb{B}$ is UMD, $\mathbb{B}$ does not contain isomorphic copies of $c_0$. If $H$ is separable and $\{h_n\}_{n\in \mathbb{N}}$ is an orthonormal basis of $H$, a linear operator $T:H\longrightarrow \mathbb{B}$ is $\gamma$-radonifying, if and only if, the series $\sum_{j=1}^\infty \gamma _jT(h_j)$ converges in $L^2(\Omega ,\mathbb{B})$ and, in this case, $$\|T\|_{\gamma ^\infty (H,\mathbb{B})}
=\left(\mathbb{E}\Big\|\sum_{j=1}^\infty \gamma _jT(h_j)\Big\|_\mathbb{B}^2\right)^{1/2}.$$ In the sequel we write $\|\cdot\|_{\gamma (H,\mathbb{B})}$ to refer to $\|\cdot\|_{\gamma ^\infty (H,\mathbb{B})}$ when acts on $\gamma (H,\mathbb{B})$.
Throughout this paper we always consider $H=L^2((0,\infty ),dt/t)$. Suppose that $f:(0,\infty )\longrightarrow \mathbb{B}$ is a measurable function such that $S\circ f\in H$, for every $S\in \mathbb{B}^*$, the dual space of $\mathbb{B}$. Then, there exists a bounded linear operator $T_f:H\longrightarrow \mathbb{B}$ (shortly, $T_f\in L(H,\mathbb{B})$) such that $$\langle S,T_f(h)\rangle_{{\mathbb{B}}^*,{\mathbb{B}}}
=\int_0^\infty \langle S,f(t)\rangle_{{\mathbb{B}}^*,{\mathbb{B}}} h(t)\frac{dt}{t},\quad h\in H\mbox{ and }S\in \mathbb{B}^*.$$ We say that $f\in \gamma ((0,\infty ),dt/t,\mathbb{B})$ provided that $T_f\in \gamma (H,\mathbb{B})$. If $\mathbb{B}$ does not contain isomorphic copies of $c_0$, then the space $\{T_f: f\in \gamma ((0,\infty ),dt/t,\mathbb{B})\}$ is dense in $\gamma (H,\mathbb{B})$. It is usual to identify $f$ and $T_f$.
Banach spaces with the UMD property play an important role in our results. The Hilbert transform $\mathcal{H}(f)$ of $f\in L^p (\mathbb{R})$, $1\leq p<\infty$, is defined by $$\mathcal{H}(f)(x)
=\lim_{\varepsilon \rightarrow 0^+} \frac{1}{\pi} \int_{|x-y|>\varepsilon }\frac{f(y)}{x-y}dy,\quad \mbox{ a.e. }x\in\mathbb{R}.$$ As it is well-known the Hilbert transform defines a bounded linear operator from $L^p(\mathbb{R})$ into itself, $1<p<\infty$, and from $L^1(\mathbb{R})$ into $L^{1,\infty }(\mathbb{R})$. $\mathcal{H}$ is defined on $L^p(\mathbb{R})\otimes \mathbb{B}$, $1\leq p<\infty$, in a natural way. We say that $\mathbb{B}$ is a UMD space when the Hilbert transform can be extended to $L^p(\mathbb{R},\mathbb{B})$ as a bounded operator from $L^p(\mathbb{R},\mathbb{B})$ into itself for some (equivalently, for every) $1<p<\infty$. There exist a lot of characterizations of UMD Banach spaces. The papers of Bourgain [@Bou] and Burkholder [@Bu3] have been fundamental in the development of the theory of Banach spaces with the UMD property. UMD Banach spaces are the suitable setting to analyze vector valued Littlewood-Paley functions.
Kaiser and Weis [@KaWe] (see also [@Ka]) considered, for every $\psi \in L^2(\mathbb{R}^n)$, the operator (usually called wavelet transform associated to $\psi$) $\mathcal{W}_\psi$ defined by $$\mathcal{W}_\psi (f)(t,x)
=(\psi_t *f)(x),\quad x\in \mathbb{R}^n \text{ and } t>0,$$ for every $f\in \mathcal{S}(\mathbb{R}^n,\mathbb{B})$, the ${\mathbb{B}}$-valued Schwartz space.
The following result was established in [@KaWe Theorem 4.2].
\[ThB\] Suppose that $\mathbb{B}$ is a UMD Banach space with Fourier type $r\in (1,2]$ and $\psi \in L^2(\mathbb{R}^n)$ satisfies the following two conditions:
\(i) If $\alpha =(\alpha_1,...,\alpha _n)\in \{0,1\}^n$ and $|\alpha |\leq 1+[n/r]$, the distributional derivative $\frac{\partial ^{|\alpha |}}{\partial x_1^{\alpha _1}...\partial x_n^{\alpha _n}}\widehat{\psi}$ is represented by a measurable function and $$\sup_{|z|=1}\int_0^\infty t^{2|\alpha |}\left|\left(\frac{\partial ^{|\alpha |}}{\partial x_1^{\alpha _1}...\partial x_n^{\alpha _n}}\widehat{\psi}\right)(tz)\right|^2\frac{dt}{t}<\infty.$$
\(ii) $\displaystyle \inf_{|z|=1}\int_0^\infty |\widehat{\psi }(tz)|^2\frac{dt}{t}>0$.
Then, for every $1<p<\infty$ there exists $C>0$ such that $$\frac{1}{C}\|f\|_{L^p(\mathbb{R}^n,\mathbb{B})}
\leq \|\mathcal{W}_\psi (f)\|_{L^p(\mathbb{R}^n,\gamma (H,\mathbb{B}))}\leq C\|f\|_{L^p(\mathbb{R}^n,\mathbb{B})},\quad f\in \mathcal{S}(\mathbb{R}^n,\mathbb{B}).$$
Note that since $\gamma (H,\mathbb{C})=H$, Theorem \[ThB\] can be seen as a vector-valued generalization of Theorem \[ThA\]. We recall that every UMD Banach space has Fourier type greater than 1 ([@Bou1]) and the complex plane $\mathbb{C}$ has Fourier type 2.
Our objective in this paper is to get new equivalent norms for $L^p((0,\infty ),\mathbb{B})$, when $\mathbb{B}$ is a UMD Banach space, by using square functions involving Hankel convolutions and Poisson semigroups associated with Bessel operators. These square functions allow us to obtain new characterizations of UMD Banach spaces. We also describe the UMD property by the boundedness in $L^p((0,\infty ),\mathbb{B})$, $1<p<\infty$, of the imaginary power $\Delta _\lambda^{i\omega}$, $\omega \in \mathbb{R}\setminus\{0\}$, of the Bessel operator $\Delta _\lambda =-x^{-\lambda }\frac{d}{dx}x^{2\lambda }\frac{d}{dx}x^{-\lambda}$, on $(0,\infty )$. As a consequence of our results about square functions in the Bessel setting we obtain $L^p((0,\infty ),\mathbb{B})$-boundedness properties for spectral multipliers associated with Bessel operators.
If $J_\nu$ denotes the Bessel function of the first kind and order $\nu >-1$, we have that $$\label{BesselJ}
\Delta_{\lambda ,x}(\sqrt{xy}J_{\lambda -1/2}(xy))=y^2\sqrt{xy}J_{\lambda -1/2}(xy),\quad x,y\in (0,\infty ).$$ Here and in the sequel, unless otherwise stated, we assume that $\lambda >0$. The Hankel transform $h_\lambda (f)$ of $f\in L^1(0,\infty )$ is defined by $$h_\lambda (f)(x)
=\int_0^\infty \sqrt{xy}J_{\lambda -1/2}(xy)f(y)dy,\quad x\in (0,\infty ).$$ This transform plays in the Bessel setting the same role as the Fourier transformation in the classical (Laplacian) setting (see ).
We consider the space $\mathcal{S}_\lambda (0,\infty )$ of all those smooth functions $\phi$ on $(0,\infty )$ such that, for every $m,k \in \mathbb{N}$, $$\eta _{m,k}^\lambda (\phi )
=\sup_{x\in (0,\infty )} x^m\left|\left(\frac{1}{x}\frac{d}{dx}\right)^k(x^{-\lambda }\phi (x))\right|<\infty .$$ If $\mathcal{S}_\lambda (0,\infty )$ is endowed with the topology generated by the family $\{\eta _{m,k}^\lambda \}_{m,k\in \mathbb{N}}$ of seminorms, $\mathcal{S}_\lambda (0,\infty )$ is a Fréchet space and $h_\lambda $ is an isomorphism on $\mathcal{S}_\lambda (0,\infty )$ ([@Ze Lemma 8]). Moreover, $h_\lambda ^{-1}=h_\lambda$ on $\mathcal{S}_\lambda (0,\infty)$. The Hankel transformation $h_\lambda$ can be also extended to $L^2(0,\infty )$ as an isometry ([@Ti p. 214 and Theorem 129]).
By adapting the results in [@Hi], we define the Hankel convolution $f\# _\lambda g$ of $f,g\in L^1((0,\infty ), x^\lambda dx)$ by $$(f\# _\lambda g)(x)
=\int_0^\infty f(y) \;_\lambda \tau _x(g)(y)dy,\quad x\in (0,\infty ),$$ where the Hankel translation $\;_\lambda \tau _x(g)$ of $g$ is given by $$\;_\lambda \tau _x(g)(y)
=\frac{(xy)^\lambda }{\sqrt{\pi}2^{\lambda -1/2}\Gamma (\lambda )}\int_0^\pi (\sin \theta )^{2\lambda -1}\frac{g(\sqrt{(x-y)^2+2xy(1-\cos \theta )})}{((x-y)^2+2xy(1-\cos \theta ))^{\lambda /2}}d\theta ,\quad x,y\in (0,\infty ).$$ Note that there is not a group operation $\circ$ on $(0,\infty )$ for which $$\;_\lambda \tau _x(g)(y)
=g(x\circ y),\quad x,y\in (0,\infty ).$$ The following interchange formula holds $$\label{6.1}
h_\lambda (f\# _\lambda g)
=x^{-\lambda }h_\lambda (f)h_\lambda (g),\quad f,g\in L^1((0,\infty ), x^\lambda dx).$$ If $\psi$ is a measurable function on $(0,\infty )$ we define $$\psi _{(t)}(x)
=\psi _{(t)}^\lambda (x)=\frac{1}{t^{\lambda +1}}\psi \Big(\frac{x}{t}\Big),\quad t,x\in (0,\infty ).$$ If $\psi \in \mathcal{S}_\lambda (0,\infty )$ and $\mathbb{B}$ is a Banach space, we define the operator (Hankel wavelet transform) $\mathcal{W}_{\psi ,\mathbb{B}}^\lambda $ as follows: $$\mathcal{W}_{\psi ,\mathbb{B}}^\lambda (f)(t,x)
=(\psi _{(t)} \# _\lambda f)(x),\quad t,x\in (0,\infty ),$$ for every $f\in L^p((0,\infty ),\mathbb{B})$, $1<p<\infty$.
We establish in our first result a Hankel version of Theorem \[ThB\].
\[ThBHankel\] Let $\mathbb{B}$ be a UMD Banach space, $\lambda >0$ and $1<p<\infty$. Suppose that $\psi \in \mathcal{S}_\lambda (0,\infty )$ is not identically zero and $\int_0^\infty x^\lambda \psi(x) dx=0$. Then, there exists $C>0$ such that $$\frac{1}{C}\|f\|_{L^p((0,\infty ),\mathbb{B})}
\leq \|\mathcal{W}_{\psi ,\mathbb{B}}^\lambda (f)\|_{L^p((0,\infty ),\gamma (H,\mathbb{B}))}
\leq C \|f\|_{L^p((0,\infty ),\mathbb{B})},$$ for every $f\in L^p((0,\infty ),\mathbb{B})$.
Harmonic analysis associated with Bessel operators was firstly analyzed by Muckenhoupt and Stein [@MS]. Recently, that study has been completed (see [@BBFMT], [@BCR1] and [@BFMT]).
The Poisson semigroup $\{P_t^\lambda \}_{t>0}$ associated to the operator $\Delta _\lambda $ is defined as follows: $$P_t^\lambda (f)(x)
=\int_0^\infty P_t^\lambda (x,y)f(y)dy,\quad t,x\in (0,\infty ),$$ for every $f\in L^p(0,\infty )$, $1\leq p<\infty$. The Poisson kernel $P_t^\lambda (x,y)$, $t,x,y\in (0,\infty )$, is defined by (see [@Wei]) $$P_t^\lambda (x,y)
=\frac{2\lambda (xy)^ \lambda t}{\pi }\int_0^\pi \frac{(\sin \theta )^{2\lambda -1}}{((x-y)^2+t^2+2xy(1-\cos \theta ))^{\lambda +1}}d\theta ,\quad t,x,y\in (0,\infty ).$$ For every $t>0$, we can write $$P_t^\lambda (f)
=K_{(t)}^\lambda \# _\lambda f,\quad f\in L^p(0,\infty ), \ 1\leq p<\infty ,$$ where $$K^\lambda (x)
=\frac{2^{\lambda +1/2}\Gamma (\lambda +1)}{\sqrt{\pi }}\frac{x^\lambda }{(1+x^2)^{\lambda +1}},\quad x\in (0,\infty ).$$ Then, for every $k\in \mathbb{N}$ and $f\in L^p(0,\infty )$, $1<p<\infty $, $$g_k(\{P_t^\lambda \}_ {t>0})(f)(x)
=\left(\int_0^\infty |(t^k\partial ^k_tK_{(t)}^\lambda \# _\lambda f)(x)|^2\frac{dt}{t}\right)^{1/2},\quad x\in (0,\infty ).$$ $g$-functions in the Bessel setting were studied in [@BFS].
In [@BFMT] it was considered the square function defined by $$g_{1,\mathbb{B}}(\{P_t^\lambda \}_ {t>0})(f)(x)
=\left(\int_0^\infty \|t\partial _tP_t^\lambda (f)(x)\|_\mathbb{B}^2\frac{dt}{t}\right)^{1/2},\quad x\in (0,\infty ),$$ for every $f\in L^p((0,\infty ),\mathbb{B})$, $1<p<\infty$. According to [@BFMT Theorems 2.4 and 2.5] and [@Kw] we have that, $\mathbb{B}$ is isomorphic to a Hilbert space if, and only if, for some (equivalently, for every ) $1<p<\infty$, there exists $C>0$ for which $$\frac{1}{C}\|f\|_{L^p((0,\infty ),\mathbb{B})}
\leq \|g_{1, \mathbb{B}}(\{P_t^\lambda \}_{t>0})(f)\|_{L^p(0,\infty )}
\leq C \|f\|_{L^p((0,\infty ),\mathbb{B})},\quad f\in L^p((0,\infty ),\mathbb{B}).$$ Note that the semigroup $\{P_t^\lambda \}_{t>0}$ is not Markovian. Hence, the results in [@MTX] do not imply those in [@BFMT]. Also the theory developed in [@Hy] do not apply for the Bessel Poisson semigroup.
In [@SW] Segovia and Wheeden defined a fractional derivative as follows. Suppose that $F:\Omega \times (0,\infty )\longrightarrow \mathbb{C}$ is a good enough function, where $\Omega \subset \mathbb{R}^n$, and $\beta >0$. The $\beta$-derivative $\partial_t^\beta F$ is defined by $$\partial _t^\beta F(x,t)
=\frac{e^{-i\pi (m-\beta)}}{\Gamma (m-\beta )}\int_0^\infty \partial_t^mF(x,t+s)s^{m-\beta -1}ds,\quad x\in \Omega, \ t\in (0,\infty ),$$ where $m\in \mathbb{N} $ and $m-1\leq \beta <m$. By using this fractional derivative, Segovia and Wheeden obtained characterizations of Sobolev spaces.
If $\mathbb{B}$ is a Banach space and $\beta >0$ we define the operator $G_{P,\mathbb{B}}^{\lambda ,\beta }$ by $$G_{P,\mathbb{B}}^{\lambda ,\beta }(f)(x)
=t^\beta \partial _t^\beta P_t^\lambda (f)(x),\quad t,x\in (0,\infty ),$$ for every $f\in L^p((0,\infty ),\mathbb{B})$, $1<p<\infty $.
We now prove that the operators $G_{P,\mathbb{B}}^{\lambda ,\beta }$ allow us to get new equivalent norms in $L^p((0,\infty ),\mathbb{B})$ provided that $\mathbb{B}$ is a UMD space.
\[boundedness\] Let $\mathbb{B}$ be a UMD Banach space, $\lambda ,\beta >0$ and $1<p<\infty$. Then, there exists $C>0$ such that $$\label{7.1}
\frac{1}{C}\|f\|_{L^p((0,\infty ),\mathbb{B})}
\leq \|G_{P, \mathbb{B}}^{\lambda ,\beta }(f)\|_{L^p((0,\infty ),\gamma (H,\mathbb{B}))}
\leq C \|f\|_{L^p((0,\infty ),\mathbb{B})},$$ for every $f\in L^p((0,\infty ),\mathbb{B})$.
For every $\lambda>0$ the Poisson semigroup $\{P_t^\lambda\}_{t>0}$ is generated by $-\sqrt{\Delta_\lambda}$ in $L^p(0,\infty)$, $1<p<\infty$. According to [@NoSt3 Proposition 6.1], $\{P_t^\lambda\}_{t>0}$ is contractive in $L^p(0,\infty)$, $1<p<\infty$, provided that $\lambda \geq 1$. Then, by [@Tag Theorem 6.1], equivalence follows from [@NVW2 Proposition 2.16] (see also [@Maa Lemma 2.3]) when $\lambda \geq 1$, $\beta>0$, $1<p<\infty$ and ${\mathbb{B}}$ is a UMD Banach space. In Theorem \[boundedness\], is established for every $\lambda>0$. Our proof (see Section \[sec:Proof2\]) does not use functional calculus arguments. We exploit the fact that the Bessel operator $\Delta_\lambda$ is, in some sense, a nice perturbation of the Laplacian operator $-d^2/dx^2$. We connect the $g$-function operator $G_{P, \mathbb{B}}^{\lambda ,\beta }$ with the corresponding operator associated with the classical Poisson semigroup and then we apply Theorem \[ThB\].
We also consider square functions associated with Bessel Poisson semigroups involving derivative with respect to $x$. If $\mathbb{B}$ is a Banach space we define, for every $f\in L^p((0,\infty ),\mathbb{B})$, $1<p<\infty$, $$\mathcal{G}_{P,\mathbb{B}}^\lambda (f)(t,x)
=tD_{\lambda ,x}^* P_t^{\lambda +1}(f)(x),\quad x,t\in (0,\infty ),$$ where $D_\lambda ^*=-x^{-\lambda }\frac{d}{dx}x^\lambda $.
\[boundedness2\] Let $\mathbb{B}$ be a UMD Banach space, $\lambda >0$ and $1<p<\infty$. Then, the operator $\mathcal{G}_{P,\mathbb{B}}^\lambda $ is bounded from $L^p((0,\infty ),\mathbb{B})$ into $L^p((0,\infty ),\gamma(H,\mathbb{B}))$.
The operators $G_{P,\mathbb{B}}^{\lambda ,1}$ and $\mathcal{G}_{P,\mathbb{B}}^\lambda$ are connected by certain Cauchy-Riemann type equations and Riesz transforms associated with Bessel operators. These relations allow us to get new characterizations of UMD Banach spaces. Also, the equivalence of $L^p$-norms in Theorem \[boundedness\] characterizes UMD Banach spaces. In order to see this last property we need first to describe UMD Banach spaces by using $L^p$-boundedness of the imaginary power $\Delta_\lambda^{i\omega}$, $\omega \in \mathbb{R}\setminus\{0\}$, of Bessel operators (see Proposition \[ImagBess\]).
\[Th1.4\] Let $\mathbb{B}$ be a Banach space and $\lambda >0$. The following assertions are equivalent.
\(i) $\mathbb{B}$ is UMD.
\(ii) For some (equivalently, for every) $1<p<\infty$, there exists $C>0$ such that, $$\label{A2}
\frac{1}{C} \|f\|_{L^p((0,\infty ),\mathbb{B})}
\leq \|G_{P, \mathbb{B}}^{\lambda ,1 }(f)\|_{L^p((0,\infty ),\gamma (H,\mathbb{B}))}, \quad f\in L^p(0,\infty )\otimes \mathbb{B},$$ and $$\label{A3}
\|\mathcal{G}_{P, \mathbb{B}}^{\lambda }(f)\|_{L^p((0,\infty ),\gamma (H,\mathbb{B}))}
\leq C \|f\|_{L^p((0,\infty ),\mathbb{B})}, \quad f\in L^p(0,\infty )\otimes \mathbb{B}.$$
\(iii) For some (equivalently, for every) $1<p<\infty$ and $\beta >0$, there exists $C>0$ such that, for $\delta =\beta$ and $\delta =\beta +1$, $$\label{A1}
\frac{1}{C}\|f\|_{L^p((0,\infty ),\mathbb{B})}
\leq \|G_{P, \mathbb{B}}^{\lambda ,\delta }(f)\|_{L^p((0,\infty ),\gamma (H,\mathbb{B}))}
\leq C \|f\|_{L^p((0,\infty ),\mathbb{B})}, \quad f\in L^p(0,\infty )\otimes \mathbb{B}.$$
Inspired in [@Me1 Theorem 1] as application of the result in Theorem \[boundedness\] we give sufficient conditions in order that spectral multipliers associated with Bessel operators are bounded in $L^p((0,\infty ),\mathbb{B})$, $1<p<\infty$.
If $f\in \mathcal{S}_\lambda (0,\infty)$ from we deduce that $$h_\lambda (\Delta _\lambda f)(x)
=x^2h_\lambda (f)(x),\quad x\in (0,\infty ).$$ We define $$\Delta _\lambda f
=h_\lambda (x^2h_\lambda (f)),\quad f\in D(\Delta _\lambda ),$$ where the domain $D(\Delta _\lambda )$ of $\Delta _\lambda $ is $$D(\Delta _\lambda )
=\{f\in L^2(0,\infty ): x^2h_\lambda (f)\in L^2(0,\infty )\}.$$ Suppose that $m\in L^\infty (0,\infty )$. The spectral multiplier $m(\Delta _\lambda )$ is defined by $$\label{M1}
m(\Delta _\lambda )(f)
=h_\lambda (m(x^2)h_\lambda ),\quad f\in L^2(0,\infty ).$$ Since $h_\lambda $ is bounded in $L^2(0,\infty )$, it is clear that $m(\Delta _\lambda)$ is bounded from $L^2(0,\infty )$ into itself. At this point the question is to give conditions on the function $m$ which imply that the operator $m(\Delta _\lambda )$ can be extended from $L^2(0,\infty )\cap L^p(0,\infty )$ to $L^p(0,\infty )$ as a bounded operator from $L^p(0,\infty )$ into itself for some $p\in (1,\infty )\setminus\{2\}$.
In [@BCC1] and [@BMR] Laplace transform type Hankel multipliers were investigated. A function $m$ is called of Laplace transform type when $$m(y)
=y\int_0^\infty e^{-yt}\psi (t)dt,\quad y\in (0,\infty ),$$ for some $\psi\in L^\infty (0,\infty )$. If $m$ is of Laplace transform type then the operator $m(\Delta _\lambda )$ defined in can be extended to $L^p(0,\infty )$ as a bounded operator from $L^p(0,\infty )$ into itself, $1<p<\infty$, and from $L^1(0,\infty )$ into $L^{1,\infty }(0,\infty )$ ([@BCC1], [@BMR] and [@Ste1 p. 121]).
Let $\omega \in \mathbb{R}\setminus\{0\}$. The imaginary power $\Delta_\lambda ^{i\omega}$ of $\Delta _\lambda$ is defined by $$\Delta_\lambda^{i\omega}(f)
=h_\lambda (y^{2i \omega}h_\lambda (f)),\quad f\in L^2(0,\infty ).$$ Since $$y^{ i \omega}
=y\int_0^\infty e^{-yt}\frac{t^{-i\omega}}{\Gamma (1-i \omega )}dt, \quad y\in (0,\infty ),$$ the operator $\Delta_\lambda^{i\omega}$ is a Laplace transform type Hankel multiplier.
In Proposition \[ImagBess\] (Section \[sec:Proof4\]) we show that a Banach space $\mathbb{B}$ is UMD if, and only if, the operator $\Delta_\lambda^{i\omega}$, $\omega \in \mathbb{R}$, is a bounded operator from $L^p((0,\infty ),\mathbb{B})$ into itself, for some (equivalently, for every) $1<p<\infty$. This is a Bessel version of [@Gue Theorem, p. 402].
In the following theorem we establish a Banach valued version of [@Me1 Theorem 1] for the Bessel operator.
If $m\in L^\infty (0,\infty )$ we define, for every $n\in \mathbb{N}$, $$m_n(t,y)
=(ty)^ne^{-ty/2}m(y^2),\quad t,y \in (0,\infty ),$$ and $\mathcal{M}_n(t,u)$, $t\in (0,\infty )$, $u\in\mathbb{R}$, represents the Mellin transform of $m_n$ with respect to the variable $y$, that is, $$\mathcal{M}_n(t,u)
=\int_0^\infty m_n(t,y)y^{-iu-1}dy,\quad u\in \mathbb{R} \text{ and } t>0.$$
\[Th1.5\] Let $\mathbb{B}$ be a UMD Banach space, $\lambda >0$ and $m \in L^\infty(0,\infty)$. Suppose that for some $1<p<\infty$ and $n\in \mathbb{N}$ the following property holds $$\label{M2}
\int_{\mathbb{R}}\sup _{t>0}|\mathcal{M}_n(t,u)|\|\Delta_\lambda ^{iu/2}\|_{L^p((0,\infty ),\mathbb{B})\rightarrow L^p((0,\infty ),\mathbb{B})}du<\infty .$$ Then, $m(\Delta _\lambda )$ can be extended from $\mathcal{S}_\lambda (0,\infty )\otimes \mathbb{B}$ to $L^p((0,\infty ),\mathbb{B})$ as a bounded operator from $L^p((0,\infty ),\mathbb{B})$ into itself.
We now specify some conditions over the function $m$ and the UMD Banach space $\mathbb{B}$ for which is satisfied. As in [@Me1 Theorem 3] we consider $m\in L^\infty (0,\infty )$ that extends to a bounded analytic function in a sector $\Sigma_\vartheta =\{z\in \mathbb{C}:|\mbox{Arg }\;z|<\vartheta \}$. In this case, we have that $$\sup _{t>0}|\mathcal{M}_n(t,u)|
\leq Ce^{\pi |u|/2}(1+|u|),\quad u\in \mathbb{R}.$$ By [@Cow Corollary 1] (see also [@BCC1 Corollary 1.2]) we can obtain that, for every $1<p<\infty$, $$\label{M3}
\|\Delta _\lambda ^{iu}\|_{L^p(0,\infty )\rightarrow L^p(0,\infty )}
\leq C(1+|u|^3\log |u|)^{|1/p-1/2|}\exp \left(\pi |1/p-1/2||u|\right),\quad u\in \mathbb{R},$$ where $C>0$ depends on $p$ but it does not depend on $u$.
Even when we consider the usual Laplacian operator instead of the Bessel operator $\Delta _\lambda$, it is not known if holds when the functions take values in a UMD Banach space (see, for instance, [@TagPhD Corollary 2.5.3]). In order to get an estimate as , replacing $L^p(0,\infty )$ by $L^p((0,\infty ),\mathbb{B})$ we need to strengthen the property of the Banach spaces as follows. $\mathbb{B}$ must be isomorphic to a closed subspace of a complex interpolation space $[{\mathbb{H}},X]_\theta$, where $0<\theta <1$, ${\mathbb{H}}$ is a Hilbert space and $X$ is a UMD Banach space. When $\mathbb{B}$ satisfies this property for some $\theta \in (0,1)$ we write ${\mathbb{B}}\in I_\theta (\mathpzc{H}, UMD)$. The class $\cup _{\theta \in (0,1)}I_\theta (\mathpzc{H}, UMD)$ includes all UMD lattices ([@Rub Corollary on p. 216]) and it also includes the Schatten ideals $C_p$, $p\in (1,\infty )$ (see [@DDP]). It is clear that $\mathbb{B}$ is UMD provided that ${\mathbb{B}}\in \cup _{\theta \in (0,1)}I_\theta (\mathpzc{H}, UMD)$.
As far as it is known, it is an open problem whether every UMD Banach space is in $\cup _{\theta \in (0,1)}$ $I_\theta (\mathpzc{H}, UMD)$ ([@Rub Problem 4 on p. 220]). This class of Banach spaces has been used, for instance, in [@Hy], [@MTX] and [@TagPhD], and also it plays a central role in the vector-valued version of Carleson’s theorem recently established in [@HyLa].
\[Th1.6\] Let $\lambda \geq 1$. Suppose that $m$ is a bounded holomorphic function in $\Sigma_\vartheta$ for certain $\vartheta \in (0,\pi )$, and that the Banach space $\mathbb{B}$ is in $I_\theta (\mathpzc{H}, UMD)$, for some $\theta \in (0,\vartheta/\pi)$. Then, the spectral multiplier $m(\Delta _\lambda )$ can be extended to $L^q((0,\infty ),\mathbb{B})$ as a bounded operator from $L^q((0,\infty ),\mathbb{B})$ into itself, for every $q\in [2/(1+\theta), 2/(1-\theta)]$.
In the following sections of this paper we present proofs for our theorems. Throughout this paper $C$ and $c$ always denote positive constants, not necessarily the same in each occurrence.
Proof of Theorem \[ThBHankel\] {#sec:Proof1}
==============================
{#subsec:2.1}
Firstly we prove that there exists $C>0$ such that $$\label{Lp1}
\|\mathcal{W}_{\psi ,\mathbb{B}}^\lambda (f)\|_{L^p((0,\infty ),\gamma (H, \mathbb{B}))}
\leq C\|f\|_{L^p((0,\infty ),\mathbb{B})},$$ for every $f\in L^p((0,\infty ),\mathbb{B})$.
We choose $\phi \in\mathcal{S}(\mathbb{R})$ such that $\phi (x^2)= x^{-\lambda }\psi (x)$, $x\in (0,\infty )$ (see [@EG p. 85]). Then, we can write $$\begin{aligned}
\;_\lambda \tau _x(\psi _{(t)} )(y)
= &\frac{(xy)^\lambda t^{-\lambda -1}}{\sqrt{\pi }2^{\lambda -1/2}\Gamma (\lambda )}\int_0^\pi
\psi \left(\frac{\sqrt{(x-y)^2+2xy(1-\cos \theta )}}{t}\right)\\
& \times ((x-y)^2+2xy (1-\cos \theta ))^{-\lambda /2}(\sin \theta )^{2\lambda -1}d\theta \\
= &\frac{(xy)^\lambda t^{-2\lambda -1}}{\sqrt{\pi }2^{\lambda -1/2}\Gamma (\lambda)}\int_0^\pi (\sin \theta )^{2\lambda -1}
\phi \left(\frac{(x-y)^2+2xy(1-\cos \theta )}{t^2}\right)d\theta ,\quad t,x,y \in (0,\infty ).\end{aligned}$$
We define the function $\Phi $ as follows: $$\Phi (x)
=\frac{1}{\sqrt{\pi }2^{\lambda +1/2}\Gamma (\lambda )}\int_0^\infty u^{\lambda -1}\phi (x^2+u)du,\qquad x\in \mathbb{R}.$$
It is not hard to see that $\Phi \in \mathcal{S}(\mathbb{R})$. Hence, since $\widehat{\Phi}(0)=0$ (see [@BCR1 (17)]), $\Phi $ satisfies conditions $(C1)$ and $(C2)$ in [@KaWe p. 111] ($(i)$ and $(ii)$ in Theorem \[ThA\]).
We consider the operator $$\mathcal{W}_{\Phi ,\mathbb{B}}(f)(t,x)
=(\Phi _t*f)(x),\quad f\in L^p(\mathbb{R},\mathbb{B}), \ t\in (0,\infty )\mbox{ and }x\in \mathbb{R}.$$
According to [@KaWe Theorem 4.2] (Theorem \[ThB\]) we have that, for every $f\in \mathcal{S}(\mathbb{R})\otimes \mathbb{B}$, $$\label{Lp2}
\|\mathcal{W}_{\Phi ,\mathbb{B}}(f)\|_{L^p(\mathbb{R},\gamma (H,\mathbb{B}))}
\leq C\|f\|_{L^p(\mathbb{R},\mathbb{B})}.$$ We are going too see that the inequality holds for every $f\in L^p(\mathbb{R},\mathbb{B})$. Let $f\in L^p(\mathbb{R},\mathbb{B})$. We choose a sequence $(f_n)_{n\in \mathbb{N}}\subset \mathcal{S}(\mathbb{R})\otimes \mathbb{B}$ such that $f_n\longrightarrow f$, as $n\rightarrow \infty$, in $L^p(\mathbb{R},\mathbb{B})$. According to , by defining $$\widetilde{\mathcal{W}}_{\Phi ,\mathbb{B}}(f)
=\lim_{n\rightarrow \infty }\mathcal{W}_{\Phi ,\mathbb{B}}(f_n),$$ where the limit is understood in $L^p(\mathbb{R},\gamma (H,\mathbb{B}))$, we have that $$\|\widetilde{\mathcal{W}}_{\Phi ,\mathbb{B}}(f)\|_{L^p(\mathbb{R},\gamma (H,\mathbb{B}))}
\leq C\|f\|_{L^p(\mathbb{R},\mathbb{B})}.$$ Also, there exists an increasing sequence $(n_k)_{k\in \mathbb{N}}\subset \mathbb{N}$ and a subset $\Omega$ of $\mathbb{R}$ such that $$\widetilde{\mathcal{W}}_{\Phi ,\mathbb{B}}(f)(x)
=\lim_{k \rightarrow \infty }\mathcal{W}_{\Phi ,\mathbb{B}}(f_{n_k})(\cdot,x),\quad x\in \Omega ,$$ where the limit is understood in $\gamma (H,\mathbb{B})$, and $|\mathbb{R}\setminus \Omega |=0$.
For every $\varepsilon >0$, $$\mathcal{W}_{\Phi ,\mathbb{B}}(f_n)(\cdot,x)\longrightarrow \mathcal{W}_{\Phi ,\mathbb{B}}(f)(\cdot,x), \quad \text{as } n\rightarrow \infty,
\text{ in } L^2((\varepsilon ,\infty ), dt/t, \mathbb{B}),$$ uniformly in $x\in \mathbb{R}$. Indeed, let $\varepsilon >0$. By using Minkowski’s inequality we get $$\begin{aligned}
& \left(\int_\varepsilon ^\infty \|\mathcal{W}_{\Phi ,\mathbb{B}}(f_n)(t,x)-\mathcal{W}_{\Phi , \mathbb{B}}(f)(t,x)\|_\mathbb{B}^2\frac{dt}{t} \right)^{1/2}\\
& \qquad \leq \int_\mathbb{R}\|f_n(y)-f(y)\|_\mathbb{B}\left(\int_\varepsilon ^\infty \frac{1}{t^3}\Big|\Phi \Big(\frac{|x-y|}{t}\Big)\Big|^2dt\right)^{1/2}dy\\
& \qquad \leq C\int_\mathbb{R}\|f_n(y)-f(y)\|_\mathbb{B}\left(\int_\varepsilon ^\infty \frac{1}{(t+|x-y|)^3}dt\right)^{1/2}dy\\
& \qquad \leq C\int_\mathbb{R}\frac{\|f_n(y)-f(y)\|_\mathbb{B}}{\varepsilon +|x-y|}dy
\leq C\|f_n-f\|_{L^p(\mathbb{R},\mathbb{B})}\left(\int_\mathbb{R}\frac{dy}{(\varepsilon +|y|)^{p'}}\right)^{1/p'} \\
& \qquad \leq C\varepsilon ^{-1/p}\|f_n-f\|_{L^p(\mathbb{R},\mathbb{B})},\quad n\in \mathbb{N}\mbox{ and }x\in \mathbb{R},\end{aligned}$$ where $p'$ is the conjugated exponent of $p$, that is, $p'=p/(p-1)$.
Let $S\in \mathbb{B}^*$. Since $\gamma (H,\mathbb{B})$ is continuously contained in the space $L(H,\mathbb{B})$ of linear bounded operators from $H$ into $\mathbb{B}$, for every $x\in \Omega $ and $h\in L^2((0,\infty ),dt/t)$ with $\mbox{supp}\; (h)\subset (0,\infty )$, we have that $$\begin{aligned}
\langle S, [\widetilde{\mathcal{W}}_{\Phi ,\mathbb{B}}(f)(x)](h)\rangle _{\mathbb{B}^ *,\mathbb{B}}
=&\lim_{k\rightarrow \infty }\langle S,[\mathcal{W}_{\Phi ,\mathbb{B}}(f_{n_k})(\cdot ,x)](h)\rangle _{\mathbb{B}^ *,\mathbb{B}}\\
=&\lim_{k\rightarrow \infty}\int_0^\infty \langle S,\mathcal{W}_{\Phi ,\mathbb{B}}(f_{n_k})(t,x)\rangle _{\mathbb{B}^ *,\mathbb{B}}h(t)\frac{dt}{t}\\
=&\int_0^\infty \langle S, \mathcal{W}_{\Phi ,\mathbb{B}}(f)(t,x)\rangle _{\mathbb{B}^ *,\mathbb{B}}h(t)\frac{dt}{t}.\end{aligned}$$ Hence, for every $x\in \Omega $, $\langle S, \mathcal{W}_{\Phi ,\mathbb{B}}(f)(\cdot , x)\rangle _{\mathbb{B}^ *,\mathbb{B}}\in L^2((0,\infty ),dt/t)$ and $$\langle S, [\widetilde{\mathcal{W}}_{\Phi ,\mathbb{B}}(f)(x)](h)\rangle _{\mathbb{B}^ *,\mathbb{B}}
=\int_0^\infty \langle S, \mathcal{W}_{\Phi ,\mathbb{B}}(f)(t,x) \rangle_{{\mathbb{B}}^*,{\mathbb{B}}} h(t)\frac{dt}{t},\quad h\in H.$$ We conclude that $\widetilde{\mathcal{W}}_{\Phi ,\mathbb{B}}(f)(x)=\mathcal{W}_{\Phi ,\mathbb{B}}(f)(\cdot,x)$, $x\in \Omega$, as elements of $\gamma (H,\mathbb{B})$. Hence, holds for every $f \in L^p({\mathbb{R}},{\mathbb{B}})$.
Suppose now that $f\in L^p((0,\infty ),\mathbb{B})$. By defining the function $f_o$ as the odd extension of $f$ to $\mathbb{R}$, we have that $$\begin{aligned}
\mathcal{W}_{\Phi ,\mathbb{B}}(f_o)(t,x)
= &\frac{1}{t}\int_{-\infty }^{+\infty }\Phi \Big( \frac{x-y}{t}\Big)f_o(y)dy \\
= -& \frac{1}{t}\int_0^\infty \Phi \Big( \frac{x+y}{t}\Big)f(y)dy+\frac{1}{t}\int_0^\infty \Phi \Big( \frac{x-y}{t}\Big)f(y)dy \\
= &L_{\Phi ,\mathbb{B}}^1(f)(t,x)+L_{\Phi ,\mathbb{B}}^2(f)(t,x),\quad x\in \mathbb{R} \mbox{ and } t\in (0,\infty ).\end{aligned}$$ We get by using Minkowski’s inequality $$\begin{aligned}
& \|L_{\Phi ,\mathbb{B}}^1(f)(\cdot ,x)\|_{L^2((0,\infty ),dt/t,\mathbb{B})}
\leq \int_0^\infty \|f(y)\|_\mathbb{B}\left(\int_0^\infty \frac{1}{t^3}\Big|\Phi \Big(\frac{x+y}{t}\Big)\Big|^2dt\right)^{1/2}dy\\
& \qquad \leq C\int_0^\infty \frac{\|f(y)\|_\mathbb{B}}{x+y}dy
\leq C \left[H_0(\|f\|_\mathbb{B})(x)+H_\infty (\|f\|_\mathbb{B})(x) \right],\quad x\in (0,\infty ),\end{aligned}$$ where $H_0$ and $H_\infty $ denote the Hardy operators defined by $$H_0(g)(x)
=\frac{1}{x}\int_0^xg(y)dy,\quad x\in (0,\infty ),$$ and $$H_\infty (g)(x)
=\int_x^\infty \frac{g(y)}{y}dy, \quad x\in (0,\infty ).$$ Since $H_0$ and $H_\infty$ are bounded operators from $L^p(0,\infty )$ into itself ([@HLP p. 244, (9.9.1) and (9.9.2)]), $L^1_{\Phi ,\mathbb{B}}$ is a bounded operator from $L^p((0,\infty ),\mathbb{B})$ into $L^p((0,\infty ),L^2((0,\infty ),dt/t,\mathbb{B}))$. By taking into account that $\gamma (H,\mathbb{C})=H$, we deduce that, for every $x\in (0,\infty )$, $L_{\Phi ,\mathbb{B}}^1(f)(\cdot ,x)\in \gamma (H,\mathbb{B})$ and $L_{\Phi ,\mathbb{B}}^1$ defines a bounded operator from $L^p((0,\infty ),\mathbb{B})$ into $L^p((0,\infty ),\gamma (H,\mathbb{B}))$.
By using now we conclude that the operator $L^2_{\Phi ,\mathbb{B}}$ is bounded from $L^p((0,\infty ),\mathbb{B})$ into $L^p((0,\infty ),\gamma (H,\mathbb{B}))$.
Inequality will be proved once we establish that $$\label{11.1}
\left\| \left[ \mathcal{W}_{\psi,{\mathbb{B}}}^\lambda - L^2_{\Phi ,\mathbb{B}} \right](f)\right\|_{L^p((0,\infty),\gamma(H,{\mathbb{B}}))}
\leq C \|f\|_{L^p((0,\infty),{\mathbb{B}})}, \quad f \in L^p((0,\infty),{\mathbb{B}}).$$ In order to do this, we study the function $$K_\lambda (t,x,y)
=\;_\lambda \tau _x(\psi _{(t)} )(y)-\Phi _t(x-y),\quad t,x,y\in (0,\infty ).$$ Firstly we write $$\;_\lambda \tau _x(\psi _{(t)} )(y)
=H_{\lambda ,1}(t,x,y)+H_{\lambda ,2}(t,x,y),\quad t,x,y\in (0,\infty ),$$ where $$H_{\lambda ,1}(t,x,y)
=\frac{(xy)^\lambda t^{-2\lambda -1}}{\sqrt{\pi }2^{\lambda -1/2}\Gamma (\lambda )}\int_0^{\pi /2}(\sin \theta )^{2\lambda -1}
\phi \left(\frac{(x-y)^2+2xy(1-\cos \theta )}{t^2}\right)d\theta, \quad t,x,y\in (0,\infty ).$$ We have that, for every $x \in (0,\infty)$, $$\begin{aligned}
& \|H_{\lambda ,2}(\cdot,x,y) \|_H \\
& \qquad \leq C(xy)^\lambda \left(\int_0^\infty t^{-4\lambda -3}\left(\int_{\pi /2}^\pi
(\sin \theta )^{2\lambda -1}\left|\phi \left(\frac{(x-y)^2+2xy(1-\cos \theta )}{t^2}\right)\right|d\theta\right)^2 dt\right)^{1/2}\\
& \qquad \leq C(xy)^\lambda
\left\{\begin{array}{l}
\displaystyle \frac{1}{|x-y|^{2\lambda +1}} \Big(\int_0^\infty u^{-4\lambda -3} \Big(\int_{\pi /2}^\pi (\sin \theta )^{2\lambda -1}\\
\displaystyle \qquad \times \left|\phi \left(\frac{(x-y)^2+2xy(1-\cos \theta )}{(x-y)^2u^2}\right)\right|d\theta\Big)^2 du\Big)^{1/2}, \quad y\in (0,x/2)\cup (2x,\infty ),\\
\\
\displaystyle \frac{1}{(xy)^{\lambda +1/2}} \Big(\int_0^\infty u^{-4\lambda -3}\Big(\int_{\pi /2}^\pi (\sin \theta )^{2\lambda -1}\\
\displaystyle \qquad \times \left|\phi \left(\frac{(x-y)^2+2xy(1-\cos \theta )}{xyu^2}\right)\right|d\theta\Big)^2 du\Big)^{1/2}, \quad y\in (x/2,2x).
\end{array} \right.\end{aligned}$$ Then, since $\phi \in \mathcal{S}(\mathbb{R})$, it follows that $$\begin{aligned}
\label{Lp3}
& \|H_{\lambda ,2}(\cdot,x,y)\|_H
\leq C \frac{(xy)^\lambda }{|x-y|^{2\lambda +1}}\left(\int_1^\infty u^{-4\lambda -3} du+\int_0^1du\right)^{1/2}
\leq C\left\{\begin{array}{ll}
\displaystyle \dfrac{1}{x},&0<y<x/2,\\
&\\
\displaystyle \dfrac{1}{y},&y>2x>0,
\end{array}\right.\end{aligned}$$ and $$\label{Lp4}
\|H_{\lambda ,2}(\cdot,x,y)\|_H
\leq C\frac{1}{(xy)^{1/2}}\left(\int_1^\infty u^{-4\lambda -3} du+\int_0^1du\right)^{1/2}\leq \frac{C}{x}, \quad y\in (x/2,2x).$$ By proceeding in a similar way we can see that $$\label{Lp5}
\|H_{\lambda ,1}(\cdot,x,y)\|_H
\leq C\left\{\begin{array}{ll}
\displaystyle \frac{1}{x},&0<y<x/2,\\
&\\
\displaystyle \frac{1}{y},&y>2x>0,
\end{array}\right.$$ and also $$\label{Lp6}
\left\|\Phi_t(x-y)\right\|_H\leq \frac{C}{|x-y|}\leq C\left\{\begin{array}{ll}
\displaystyle \frac{1}{x},&0<y<x/2,\\
&\\
\displaystyle \frac{1}{y},&y>2x
\end{array}\right., \quad x\in (0,\infty ).$$ Suppose now that $x\in (0,\infty )$ and $x/2<y<2x$. We split the difference $H_{\lambda ,1}(t,x,y)-\Phi _t(x-y)$, $t\in (0,\infty )$, as follows $$\begin{aligned}
& H_{\lambda ,1}(t,x,y)-\Phi _t(x-y)
=\frac{(xy)^\lambda t^{-2\lambda -1}}{\sqrt{\pi }2^{\lambda -1/2}\Gamma (\lambda )}\int_0^{\pi /2}
[(\sin \theta )^{2\lambda -1}-\theta ^{2\lambda -1}]\phi \left(\frac{(x-y)^2+2xy(1-\cos \theta )}{t^2}\right)d\theta\\
& \qquad \qquad + \frac{(xy)^\lambda t^{-2\lambda -1}}{\sqrt{\pi }2^{\lambda -1/2}\Gamma (\lambda )}\int_0^{\pi /2}
\theta ^{2\lambda -1}\left[\phi \left(\frac{(x-y)^2+2xy(1-\cos \theta )}{t^2}\right)-
\phi \left(\frac{(x-y)^2+xy\theta ^2}{t^2}\right)\right]d\theta\\
& \qquad \qquad +\frac{(xy)^\lambda t^{-2\lambda -1}}{\sqrt{\pi }2^{\lambda -1/2}\Gamma (\lambda )}\int_0^{\pi /2}\theta ^{2\lambda -1}
\phi \left(\frac{(x-y)^2+xy\theta ^2}{t^2}\right)d\theta-\Phi _t(x-y)\\
& \qquad = H_{\lambda ,1,1}(t,x,y)+H_{\lambda ,1,2}(t,x,y) +H_{\lambda ,1,3}(t,x,y), \quad t,x,y\in (0,\infty ).\end{aligned}$$ By using the mean value theorem we get $$\begin{aligned}
& \|H_{\lambda ,1,1}(\cdot,x,y)\|_H\\
& \qquad \leq C(xy)^\lambda \left\{\int_0^\infty t^{-4\lambda -3}\left(\int_0^{\pi /2}\theta ^{2\lambda +1}\left|\phi \left(\frac{(x-y)^2+2xy(1-\cos \theta )}{t^2}\right)\right| d\theta \right)^2dt\right\}^{1/2}\\
& \qquad \leq \frac{C}{(xy)^{1/2}} \left\{\int_0^\infty u^{-4\lambda -3}\left(\int_0^{\pi /2}\theta ^{2\lambda +1}\left|\phi \left(\frac{(x-y)^2+2xy(1-\cos \theta )}{u^2xy}\right)\right| d\theta \right)^2du\right\}^{1/2}\\
& \qquad \leq \frac{C}{(xy)^{1/2}} \left\{\int_1^\infty u^{-4\lambda -3}du+\int_0^1 u^{-4\lambda -3}\left(\int_0^{\pi /2}\theta ^{2\lambda +1}\left(\frac{u^2xy}{(x-y)^2+xy \theta^2}\right)^{\lambda +3/4}d\theta \right)^2du\right\}^{1/2}\\
& \qquad \leq \frac{C}{x},\end{aligned}$$ and $$\begin{aligned}
& \|H_{\lambda ,1,2}(\cdot,x,y)|\|_H\\
& \quad \leq \frac{C}{(xy)^{1/2}} \left\{\int_0^\infty u^{-4\lambda -7}\left(\int_0^{\pi /2}\theta ^{2\lambda -1}\left|\int_{1-\cos \theta}^{\theta ^2/2}\phi '\left(\frac{(x-y)^2+2xyz}{u^2xy}\right)dz\right| d\theta \right)^2du\right\}^{1/2}\\
&\quad \leq \frac{C}{(xy)^{1/2}} \left\{\int_1^\infty u^{-4\lambda -7}du
+\int_0^1 u^{-4\lambda -7}\left(\int_0^{\pi /2}\theta ^{2\lambda -1}\int_{1-\cos \theta}^{\theta ^2/2}\left(\frac{u^2xy}{(x-y)^2+2xyz}\right)^{\lambda +7/4}dzd\theta \right)^2du\right\}^{1/2}\\
&\quad \leq \frac{C}{(xy)^{1/2}} \left\{1+\int_0^1 \left(\int_0^{\pi /2}\theta ^{2\lambda -1}\left|\int_{1-\cos \theta}^{\theta ^2/2}\frac{dz}{z^{\lambda +7/4}}\right|d\theta \right)^2du\right\}^{1/2}\leq\frac{C}{x},\end{aligned}$$ On the other hand, a suitable change of variables allows us to write $$\begin{aligned}
H_{\lambda ,1,3}(t,x,y)
= & \frac{(xy)^\lambda t^{-2\lambda -1}}{\sqrt{\pi }2^{\lambda -1/2}\Gamma (\lambda )}\int_0^{\pi /2}\theta ^{2\lambda -1}
\phi \left(\frac{(x-y)^2+xy\theta ^2}{t^2}\right)d\theta\\
& - \frac{1}{t\sqrt{\pi }2^{\lambda +1/2}\Gamma (\lambda )}\int_0^\infty u^{\lambda -1}\phi \left(\Big(\frac{x-y}{t}\Big)^2+u\right)du\\
= & -\frac{(xy)^\lambda t^{-2\lambda -1}}{\sqrt{\pi }2^{\lambda -1/2}\Gamma (\lambda )}\int_{\pi /2}^\infty \theta ^{2\lambda -1}\phi \left(\frac{(x-y)^2+xy\theta ^2}{t^2}\right)d\theta,\quad t\in (0,\infty ).\end{aligned}$$ Hence, we deduce that $$\begin{aligned}
& \|H_{\lambda ,1,3}(\cdot,x,y)\|_H
\leq \frac{C}{(xy)^{1/2}} \left\{\int_0^\infty u^{-4\lambda -3}\left(\int_{\pi /2}^\infty \theta ^{2\lambda -1}
\left|\phi \left(\frac{(x-y)^2+xy\theta ^2}{xyu^2}\right)\right|d\theta\right)^2du\right\}^{1/2}\\
& \qquad \leq \frac{C}{(xy)^{1/2}} \left\{\int_1^\infty u^{-4\lambda -3}\left(\int_{\pi /2}^\infty \theta ^{2\lambda -1}
\left(\frac{xyu^2}{(x-y)^2+xy\theta ^2}\right)^{\lambda +1/4}d\theta\right)^2du\right.\\
& \qquad \qquad + \left.\int_0^1u^{-4\lambda -3}\left(\int_{\pi /2}^\infty \theta ^{2\lambda -1}
\left(\frac{xyu^2}{(x-y)^2+xy\theta ^2}\right)^{\lambda +3/4}d\theta\right)^2du\right\}^{1/2}
\leq \frac{C}{x}.\end{aligned}$$ By putting together the above estimates we obtain $$\label{Lp7}
\|H_{\lambda ,1}(\cdot ,x,y)-\Phi_t (x-y)\|_H \leq \frac{C}{x},\quad 0<\frac{x}{2}<y<2x.$$ From - we deduce that $$\label{Lp8}
\|K_\lambda (\cdot,x,y)\|_H
\leq \frac{C}{\max \{x,y\}},\quad x,y\in (0,\infty ).$$ By proceeding as in the case of $L^1_{\Phi,{\mathbb{B}}}$, since $\gamma (H,\mathbb{C})=H$, we infer from that the operator $\mathcal{W}_{\psi ,\mathbb{B}}^\lambda -L_{\Phi ,\mathbb{B}}^2$ is bounded from $L^p((0,\infty ),\mathbb{B})$ into $L^p((0,\infty ),\gamma (H,\mathbb{B}))$. Thus is established.
{#subsec:2.2}
Our next objective is to show that there exists $C>0$ such that $$\label{Lp9}
\|f\|_{L^p((0,\infty ),\mathbb{B})}
\leq C\|\mathcal{W}_{\psi ,\mathbb{B}}^\lambda (f)\|_{L^p((0,\infty ),\gamma (H,\mathbb{B}))},$$ for every $f\in L^p((0,\infty ),\mathbb{B})$. It is enough to see for every $f\in\mathcal{S}_\lambda (0,\infty )\otimes \mathbb{B}$. Indeed, suppose that is true for every $f\in \mathcal{S}_\lambda(0,\infty )\otimes \mathbb{B}$. Let $f\in L^p((0,\infty ),\mathbb{B})$. We choose a sequence $(f_n)_{n\in \mathbb{N}}\subset \mathcal{S}_\lambda(0,\infty )\otimes \mathbb{B}$ such that $f_n\longrightarrow f$, as $n\rightarrow \infty$, in $L^p((0,\infty ),\mathbb{B})$. Then, by $$\label{Lp10}
\|f_n\|_{L^p((0,\infty ),\mathbb{B})}
\leq C\|\mathcal{W}_{\psi ,\mathbb{B}}^\lambda (f_n)\|_{L^p((0,\infty ),\gamma (H,\mathbb{B}))}, \quad n \in {\mathbb{N}}.$$ Since, as it was proved in Section \[subsec:2.1\], $\mathcal{W}_{\psi ,\mathbb{B}}^\lambda$ is a bounded operator from $L^p((0,\infty ),\mathbb{B})$ into $L^p((0,\infty ),\gamma (H,\mathbb{B}))$, by letting $n\rightarrow \infty$ in we conclude that $$\|f\|_{L^p((0,\infty ),\mathbb{B})}
\leq C\|\mathcal{W}_{\psi ,\mathbb{B}}^\lambda (f)\|_{L^p((0,\infty ),\gamma (H,\mathbb{B}))}.$$
The following result was established in [@BCR1 after Lemma 2.4].
\[Lema1\] Let $\lambda >0$. If $\psi \in \mathcal{S}_\lambda (0,\infty )$ is not identically zero, then there exists $\phi\in \mathcal{S}_\lambda (0,\infty )$ such that $$\label{Lp10b}
\int_0^\infty h_\lambda (\psi)(y)h_\lambda (\phi )(y)y^{-2\lambda -1}dy=1,$$ where the last integral is absolutely convergent.
In order to see we need to show the next result.
\[Lema2\] Let $\lambda >0$. Suppose that $\psi ,\phi \in \mathcal{S}_\lambda (0,\infty )$ satisfy , being the integral absolutely convergent. If $f,g\in \mathcal{S}_\lambda (0,\infty )$, then, $$\label{Lp11}
\int_0^\infty f(x)g(x)dx
=\int_0^\infty \int_0^\infty (f\#_\lambda \psi _{(t)} )(y)(g\#_\lambda \phi_{(t)} )(y)\frac{dydt}{t}.$$
Let $f,g\in \mathcal{S}_\lambda (0,\infty )$. Note firstly that the integral in the right hand side of is absolutely convergent. Indeed, according to we get $$\begin{aligned}
\int_0^\infty \int_0^\infty |(f\#_\lambda \psi _{(t)} )(y)||(g\#_\lambda \phi _{(t)} )(y)|\frac{dydt}{t}
&\leq \|\mathcal{W}_{\psi ,\mathbb{C}}^\lambda (f)\|_{L^p((0,\infty),H )} \|\mathcal{W}_{\phi ,\mathbb{C}}^\lambda (g)\|_{L^{p'}((0,\infty), H )}\\
& \leq C\|f\|_{L^p(0,\infty )}\|g\|_{L^{p'}(0,\infty )}.
\end{aligned}$$
Plancherel equality and the interchange formula for Hankel transforms lead to $$\begin{aligned}
\int_0^\infty (f\#_\lambda \psi _{(t)} )(y)(g\#_\lambda \phi _{(t)} )(y)dy
& =\int_0^\infty h_\lambda (f\#_\lambda \psi _{(t)} )(y)h_\lambda (g\#_\lambda \phi _{(t)} )(y)dy\\
& =\int_0^\infty h_\lambda (f)(y)h_\lambda (g)(y)(ty)^{-2\lambda }h_ \lambda (\psi)(ty)h_\lambda (\phi)(ty)dy,\quad t\in (0,\infty ).
\end{aligned}$$ Hence, it follows that $$\begin{aligned}
& \int_0^\infty \int_0^\infty (f\#_\lambda \psi _{(t)} )(y)(g\#_\lambda \phi _{(t)} )(y)\frac{dydt}{t} \\
& \qquad = \int_0^\infty \int_0^\infty h_\lambda (f)(y)h_\lambda (g)(y)(ty)^{-2\lambda }h_\lambda (\psi)(ty)h_\lambda (\phi)(ty)\frac{dydt}{t}\\
& \qquad = \int_0^\infty h_\lambda (f)(y)h_\lambda (g)(y)\int_0^\infty h_\lambda (\psi)(ty)h_\lambda (\phi)(ty)(ty)^{-2\lambda } \frac{dtdy}{t}
= \int_0^\infty h_\lambda (f)(y)h_\lambda (g)(y)dy\\
& \qquad =\int_0^\infty f(x)g(x)dx.
\end{aligned}$$
An immediate consequence of Lemma \[Lema2\] is the following.
\[Lema3\] Let ${\mathbb{B}}$ be a Banach space and $\lambda >0$. Suppose that $\psi ,\phi \in \mathcal{S}_\lambda (0,\infty )$ satisfy , being the integral absolutely convergent. If $f\in \mathcal{S}_\lambda (0,\infty )\otimes \mathbb{B}$ and $g\in \mathcal{S}_\lambda (0,\infty )\otimes \mathbb{B}^*$, then, $$\int_0^\infty \langle g(x),f(x)\rangle _{\mathbb{B}^*,\mathbb{B}}dx
=\int_0^\infty \int_0^\infty \langle (g\#_\lambda \phi _{(t)} )(x),(f\#_\lambda \psi _{(t)} )(x)\rangle _{\mathbb{B}^*,\mathbb{B}}\frac{dxdt}{t}.$$
Let $f\in \mathcal{S}_\lambda (0,\infty )\otimes \mathbb{B}$. Since $\mathcal{S}_\lambda (0,\infty )\otimes \mathbb{B}^*$ is dense in $L^{p'}((0,\infty ),\mathbb{B}^*)$, according to [@GLY Lemma 2.3] we have that $$\|f\|_{L^p((0,\infty ),\mathbb{B})}
=\sup_{\substack{g\in \mathcal{S}_\lambda (0,\infty )\otimes \mathbb{B}^* \\ \|g\|_{L^{p'}((0,\infty ),\mathbb{B}^*)}\leq 1}}
\left|\int_0^\infty \langle g(x),f(x)\rangle _{\mathbb{B}^*,\mathbb{B}}dx\right|.$$ By Lemma \[Lema1\] we choose $\psi, \phi\in \mathcal{S}_\lambda (0,\infty )$ such that holds, being the integral absolutely convergent. Since ${\mathbb{B}}^*$ is UMD, it was proved in Section \[subsec:2.1\] that the operator $\mathcal{W}_{\phi ,\mathbb{B}^*}^\lambda $ is bounded from $L^{p'}((0,\infty ),\mathbb{B}^*)$ into $L^{p'}((0,\infty ),\gamma (H, \mathbb{B}^*))$. According to Lemma \[Lema3\] and [@HW Proposition 2.2] we get, for every $g\in \mathcal{S}_\lambda (0,\infty )\otimes \mathbb{B}^*$, $$\begin{aligned}
\left|\int_0^\infty \langle g(x),f(x)\rangle _{\mathbb{B}^*,\mathbb{B}}dx\right|
= &\left|\int_0^\infty \int_0^\infty \langle (g\#_\lambda \phi _{(t)} )(x),(f\#_\lambda \psi _{(t)} )(x)\rangle _{\mathbb{B}^*,\mathbb{B}}\frac{dxdt}{t}\right|\\
\leq & \int_0^\infty \int_0^\infty |\langle (g\#_\lambda \phi _{(t)} )(x),(f\#_\lambda \psi _{(t)} )(x)\rangle _{\mathbb{B}^*,\mathbb{B}}|\frac{dxdt}{t}\\
\leq &\int_0^\infty \|\mathcal{W}_{\phi ,\mathbb{B}^*}^\lambda (g)(\cdot, x)\|_{\gamma (H,\mathbb{B}^*)}\|\mathcal{W}_{\psi ,\mathbb{B}}^\lambda (f)(\cdot, x)\|_{\gamma (H,\mathbb{B})}dx\\
\leq &\|\mathcal{W}_{\phi ,\mathbb{B}^*}^\lambda (g)\|_{L^{p'}((0,\infty ), \gamma (H,\mathbb{B}^*))}\|\mathcal{W}_{\psi ,\mathbb{B}}^\lambda (f)\|_{L^p((0,\infty ), \gamma (H,\mathbb{B}))}\\
\leq &C\|g\|_{L^{p'}((0,\infty ),\mathbb{B}^*)}\|\mathcal{W}_{\psi ,\mathbb{B}}^\lambda (f)\|_{L^p((0,\infty ), \gamma (H,\mathbb{B}))}.\end{aligned}$$ Hence, $$\|f\|_{L^p((0,\infty ),\mathbb{B})}
\leq C\|\mathcal{W}_{\psi ,\mathbb{B}}^\lambda (f)\|_{L^p((0,\infty ), \gamma (H,\mathbb{B}))}.$$ Thus the proof of Theorem \[ThBHankel\] is finished.
Proof of Theorem \[boundedness\] {#sec:Proof2}
================================
{#subsec:3.1}
In this section we prove that $$\label{16.1}
\| G_{P,{\mathbb{B}}}^{\lambda,\beta}(f) \|_{L^p((0,\infty),\gamma(H,{\mathbb{B}}))}
\leq C \|f\|_{L^p((0,\infty),{\mathbb{B}})}, \quad f \in L^p((0,\infty),{\mathbb{B}}),$$ for some $C>0$ independent of $f$.
We define the $g$-operator associated with the classical Poisson semigroup on $\mathbb{R}$ as follows $$G_{P,\mathbb{B}}^\beta(f)(t,x)
=t^\beta \partial _t^\beta {\mathbb{P}}_t(f)(x),\quad x\in \mathbb{R} \mbox{ and } t\in (0,\infty ),$$ for every $f\in L^p(\mathbb{R},\mathbb{B})$.
By [@BCCFR1 Proposition 1] there exists $C>0$ such that $$\|G_{P,\mathbb{B}}^\beta (f)\|_{L^p(\mathbb{R}, \gamma (H,\mathbb{B}))}
\leq C \|f\|_{L^p(\mathbb{R},\mathbb{B})},\quad f\in \mathcal{S} (\mathbb{R})\otimes \mathbb{B}.$$ The arguments developed in the proof of Theorem \[ThBHankel\] allow us to show that $$\label{Lp12}
\|G_{P,\mathbb{B}}^\beta (f)\|_{L^p(\mathbb{R}, \gamma (H,\mathbb{B}))}\leq \|f\|_{L^p(\mathbb{R},\mathbb{B})},\quad f\in L^p (\mathbb{R}, \mathbb{B}).$$
In [@BCCFR1 Lemma 1] it was established that $$t^\beta \partial _t^\beta {\mathbb{P}}_t(z)
=\sum_{k=0}^{(m+1)/2}\frac{c_k}{t}\varphi ^k \left(\frac{z}{t}\right),\quad z\in \mathbb{R} \mbox{ and } t\in (0,\infty ),$$ where $m\in \mathbb{N}$ is such that $m-1\leq \beta <m$, and, for every $k\in \mathbb{N}$, $0\leq k\leq (m+1)/2$, $c_k\in \mathbb{C}$ and $$\varphi ^k(z)
=\int_0^\infty \frac{(1+v)^{m+1-2k}v^{m-\beta -1}}{((1+v)^2+z^2)^{m-k+1}}dv,\quad z\in \mathbb{R}.$$ By proceeding as in the proof of [@BCCFR1 Lemma 1] we can obtain the analogous identity in the Bessel setting $$\label{16.2}
t^\beta \partial _t^\beta P_t^\lambda (x,y)
=\sum_{k=0}^{(m+1)/2}\frac{b_k^\lambda }{t^{2\lambda +1}}(xy)^\lambda \int_0^\pi (\sin \theta )^{2\lambda -1}\varphi^{\lambda ,k}\left(\frac{\sqrt{(x-y)^2+2xy(1-\cos \theta )}}{t}\right)d\theta ,$$ where $m\in \mathbb{N}$ is such that $m-1\leq \beta <m$, and, for every $k\in \mathbb{N}$, $0\leq k\leq (m+1)/2$, $$\varphi ^{\lambda ,k}(z)
=\int_0^\infty \frac{(1+v)^{m+1-2k}v^{m-\beta -1}}{((1+v)^2+z^2)^{\lambda +m-k+1}}dv,\quad z\in (0,\infty ),$$ and $$b_k^\lambda = \frac{2 \lambda(\lambda+1) \cdots (\lambda+m-k)}{(m-k)!} c_k.$$
Let $k\in \mathbb{N}$, $0\leq k\leq (m+1)/2$. We define, for every $f\in L^p(\mathbb{R},\mathbb{B})$, $${\mathbb{P}}_{k}(f)(t,x)
=\int_\mathbb{R}\varphi _t^k(x-y)f(y)dy,\quad t\in (0,\infty )\mbox{ and }x\in \mathbb{R}.$$ Let $f\in L^p((0,\infty),\mathbb{B})$. If $f_o$ denotes the odd extension of $f$ to $\mathbb{R}$, we write $$\begin{aligned}
{\mathbb{P}}_{k}(f_o)(t,x)
= &\int_0^\infty \varphi _t^k(x-y)f(y)dy-\int_0^\infty \varphi _t^k(x+y)f(y)dy\\
= & {\mathbb{P}}_{k,1}(f)(t,x)- {\mathbb{P}}_{k,2}(f)(t,x),\quad t,x\in (0,\infty ).\end{aligned}$$ We have that $$\begin{aligned}
\label{Lp13}
& \|\varphi_t ^k(x+y)\|_H
\leq \int_0^\infty (1+v)^{m+1-2k}v^{m-\beta -1} \nonumber \\
& \qquad \times \left(\int_0^\infty \frac{1}{t^3}\frac{t^{4(m-k+1)}}{((1+v^2)t^2+(x+y)^2)^{2(m-k+1)}}dt\right)^{1/2}dv
\leq \frac{C}{x+y},\quad x,y\in (0,\infty ).\end{aligned}$$ Since $\gamma (H,\mathbb{C})=H$, we deduce that $$\begin{aligned}
\|{\mathbb{P}}_{k,2}(f)(\cdot,x)\|_{\gamma (H,\mathbb{B})}
\leq &\int_0^\infty \|f(y)\|_\mathbb{B}\left\|\frac{1}{t}\varphi ^k\Big(\frac{x+y}{t}\Big)\right\|_Hdt
\leq C\int_0^\infty \frac{\|f(y)\|_\mathbb{B}}{x+y}dy\\
\leq & C \left[H_0(\|f\|_\mathbb{B})(x)+H_\infty (\|f\|_\mathbb{B})(x)\right],\quad x\in (0,\infty ).\end{aligned}$$ Thus, according to [@HLP p. 244, (9.9.1) and (9.9.2)], ${\mathbb{P}}_{k,2}$ is a bounded operator from $L^p((0,\infty ),\mathbb{B})$ into $L^p((0,\infty ),\gamma (H,\mathbb{B}))$.
We define, for every $f\in L^p((0,\infty ),\mathbb{B})$, $$G_{P,\mathbb{B}}^{\beta,-}(f)(t,x)
=\int_0^\infty t^\beta \partial _t^\beta {\mathbb{P}}_t(x+y)f(y)dy,\quad t,x\in (0,\infty ).$$ Since $G_{P,\mathbb{B}}^{\beta, -}=\sum_{k=0}^{(m+1)/2}c_k {\mathbb{P}}_{k,2}$, we conclude that $G_{P,\mathbb{B}}^{\beta, -}$ is a bounded operator from $L^p((0,\infty ),\mathbb{B})$ into $L^p((0,\infty ),\gamma (H,\mathbb{B}))$. Then, according to , if for every $f\in L^p((0,\infty ),\mathbb{B})$, we define $$G_{P,\mathbb{B}}^{\beta,+}(f)(t,x)
=\int_0^\infty t^\beta \partial _t^\beta {\mathbb{P}}_t(x-y)f(y)dy,\quad t,x\in (0,\infty ),$$ the operator $G_{P,\mathbb{B}}^{\beta ,+}$ is also bounded from $L^p((0,\infty ),\mathbb{B})$ into $L^p((0,\infty ),\gamma (H,\mathbb{B}))$.
In order to prove it is enough to show that the difference $G_{P,\mathbb{B}}^{\lambda ,\beta}-G_{P,\mathbb{B}}^{\beta ,+}$ is bounded from $L^p((0,\infty ),\mathbb{B})$ into $L^p((0,\infty ),\gamma (H,\mathbb{B}))$.
By proceeding as in we get, for every $x \in (0,\infty)$, $$\label{Lp14}
\|t^\beta \partial _t^\beta {\mathbb{P}}_t(x-y)\|_H\leq \frac{C}{|x-y|}\leq C\left\{\begin{array}{ll}
\displaystyle \frac{1}{x},&\displaystyle 0<y<\frac{x}{2},\\
&\\
\displaystyle \frac{1}{y},&\displaystyle y>2x.
\end{array}
\right.$$ We split $P_t^\lambda (x,y)$, $t,x,y\in (0,\infty )$, as follows $$\begin{aligned}
P_t^\lambda (x,y)
=&\frac{2\lambda (xy)^\lambda t}{\pi }\int_0^{\pi /2}\frac{(\sin \theta )^{2\lambda -1}}{((x-y)^2+t^2+2xy(1-\cos \theta ))^{\lambda +1}}d\theta \\
& +\frac{2\lambda (xy)^\lambda t}{\pi }\int_{\pi /2}^\pi \frac{(\sin \theta )^{2\lambda -1}}{((x-y)^2+t^2+2xy(1-\cos \theta ))^{\lambda +1}}d\theta\\
=&P_t^{\lambda ,1}(x,y)+P_t^{\lambda ,2}(x,y).\end{aligned}$$ From we have that $$\|t^\beta \partial _t^\beta P_t^{\lambda ,2}(x,y)\|_H
\leq C(xy)^\lambda \int_{\pi /2}^\pi (\sin \theta )^{2\lambda -1}\sum_{k=0}^{(m+1)/2}\left\|\frac{1}{t^{2\lambda +1}}\varphi ^{\lambda ,k}\left(\frac{\sqrt{(x-y)^2+2xy(1-\cos \theta )}}{t}\right)\right\|_Hd\theta ,$$ and, for every $k\in \mathbb{N}$, $0\leq k\leq (m+1)/2$, $$\begin{aligned}
& \left\|\frac{1}{t^{2\lambda +1}}\varphi ^{\lambda ,k}\left(\frac{\sqrt{(x-y)^2+2xy(1-\cos \theta )}}{t}\right)\right\|_H \\
& \qquad \leq C\int_0^\infty (1+v)^{m+1-2k}v^{m-\beta -1}\left(\int_0^\infty \frac{t^{4(\lambda +1+m-k)-4\lambda -3}}{((1+v^2)t^2+(x+y)^2)^{2(\lambda +m-k+1)}}dt\right)^{1/2}dv\\
& \qquad \leq \frac{C}{(x+y)^{2\lambda +1}}\int_0^\infty \frac{v^{m-\beta -1}}{(1+v)^m}dv\left(\int_0^\infty \frac{u^{4(m-k)+1}}{(1+u)^{4(\lambda +m-k+1)}}du\right)^{1/2} \\
& \qquad \leq \frac{C}{(x+y)^{2\lambda +1}},\quad x,y\in (0,\infty )\mbox{ and }\theta \in (\pi/2,\pi ).\end{aligned}$$ Hence, $$\label{Lp15}
\|t^\beta \partial _t^\beta P_t^{\lambda ,2}(x,y)\|_H\leq C\frac{(xy)^\lambda}{(x+y)^{2\lambda +1}}\leq \frac{C}{x+y},\quad x,y\in (0,\infty ).$$ Similar manipulations lead to $$\label{Lp16}
\|t^\beta \partial _t^\beta P_t^{\lambda ,1}(x,y)\|_H\leq \frac{C}{|x-y|}\leq C\left\{\begin{array}{ll}
\displaystyle \frac{1}{x},&\displaystyle 0<y<x/2,\\
&\\
\displaystyle \frac{1}{y},&\displaystyle y>2x>0.\\
\end{array}
\right.$$ We decompose $t^\beta \partial _t^\beta P_t^{\lambda ,1}(x,y)$ as follows: $$\begin{aligned}
& t^\beta \partial _t^\beta P_t^{\lambda ,1}(x,y)
=\sum_{k=0}^{(m+1)/2}\frac{b_k^\lambda }{t^{2\lambda +1}}(xy)^\lambda \left\{\int_0^{\pi /2}[(\sin \theta)^{2\lambda -1}-\theta ^{2\lambda -1}]\varphi ^{\lambda ,k}\left(\frac{\sqrt{(x-y)^2+2xy(1-\cos \theta )}}{t}\right)d\theta \right.\\
& \quad \quad + \int_0^{\pi /2}\theta ^{2\lambda -1}\left[\varphi ^{\lambda ,k}\left(\frac{\sqrt{(x-y)^2+2xy(1-\cos \theta )}}{t}\right)- \varphi ^{\lambda ,k}\left(\frac{\sqrt{(x-y)^2+xy\theta ^2}}{t}\right)\right]d\theta \\
& \quad \quad -\left. \int_{\pi /2}^\infty \theta ^{2\lambda -1}\varphi ^{\lambda ,k}\left(\frac{\sqrt{(x-y)^2+xy\theta ^2}}{t}\right)d\theta +\int_0^\infty \theta ^{2\lambda -1}\varphi ^{\lambda ,k}\left(\frac{\sqrt{(x-y)^2+xy\theta ^2}}{t}\right)d\theta \right\}\\
& \quad =\sum_{k=0}^{(m+1)/2}b_k^\lambda [R_1^{\lambda ,k}(t,x,y)+R_2^{\lambda ,k}(t,x,y)+R_3^{\lambda ,k}(t,x,y)+R_4^{\lambda ,k}(t,x,y)],\quad t,x,y\in (0,\infty ).\end{aligned}$$ Let $k \in {\mathbb{N}}$, $0 \leq k \leq (m+1)/2$. By using the mean value theorem we obtain $$\begin{aligned}
\label{Lp17}
\|R_1^{\lambda ,k}(\cdot,x,y)\|_H
\leq & C(xy)^\lambda \int_0^{\pi /2}\theta ^{2\lambda +1}\int_0^\infty (1+v)^{m+1-2k}v^{m-\beta -1}\nonumber\\
&\times \left(\int_0^\infty \frac{t^{4(\lambda +1+m-k)-4\lambda-3}}{((1+v)^2t^2+(x-y)^2+xy\theta ^2)^{2(\lambda +m-k+1)}}dt\right)^{1/2}dvd\theta \nonumber\\
\leq & C(xy)^\lambda \int_0^{\pi /2}\frac{\theta ^{2\lambda +1}}{((x-y)^2+xy\theta ^2)^{\lambda +1/2}}d\theta \leq \frac{C}{x},\quad 0<\frac{x}{2}<y<2x,\end{aligned}$$ and $$\begin{aligned}
\label{Lp18}
& \|R_2^{\lambda ,k}(\cdot,x,y)\|_H
\leq C(xy)^\lambda \int_0^{\pi /2}\theta ^{2\lambda -1}\int_0^\infty (1+v)^{m+1-2k}v^{m-\beta -1} \Big\{\int_0^\infty t^{4(m-k)+1}\nonumber\\
& \qquad \qquad \times \Big|\frac{1}{((1+v)^2t^2+(x-y)^2+2xy(1-\cos \theta ))^{\lambda +m-k+1}} \nonumber\\
& \qquad \qquad - \frac{1}{((1+v)^2t^2+(x-y)^2+xy\theta ^2)^{\lambda +m-k+1}}\Big|^2dt \Big\}^{1/2}dvd\theta \nonumber\\
& \qquad \leq C(xy)^\lambda \int_0^{\pi /2}\theta ^{2\lambda -1}\int_0^\infty (1+v)^{m+1-2k}v^{m-\beta -1} \nonumber\\
& \qquad \qquad \times \left\{\int_0^\infty t^{4(m-k)+1}\left(\frac{xy\theta ^4}{((1+v)^2t^2+(x-y)^2+xy\theta ^2)^{\lambda +m-k+2}}\right)^2dt\right\}^{1/2}dvd\theta \nonumber\\
& \qquad \leq C(xy)^{\lambda+1} \int_0^{\pi /2}\frac{\theta ^{2\lambda +3}}{((x-y)^2+xy\theta ^2)^{\lambda +3/2}}d\theta
\leq \frac{C}{x},\quad 0<\frac{x}{2}<y<2x.\end{aligned}$$ We have also that $$\begin{aligned}
\label{Lp19}
\|R_3^{\lambda ,k}(\cdot,x,y)\|_H
\leq & C(xy)^\lambda \int_{\pi /2}^\infty \theta ^{2\lambda -1}\int_0^\infty (1+v)^{m+1-2k}v^{m-\beta -1}\nonumber\\
&\times \left(\int_0^\infty \frac{t^{4(m-k)+1}}{((1+v)^2t^2+(x-y)^2+xy\theta ^2)^{2(\lambda +m-k+1)}}dt\right)^{1/2}dvd\theta \nonumber\\
\leq&C(xy)^\lambda \int_{\pi /2}^\infty \frac{\theta ^{2\lambda -1}}{((x-y)^2+xy\theta ^2)^{\lambda +1/2}}d\theta
\leq \frac{C}{x},\quad 0<\frac{x}{2}<y<2x.\end{aligned}$$ Finally, we get that $$\begin{aligned}
& \int_0^\infty \theta^{2\lambda -1}\varphi ^{\lambda ,k}\left(\frac{\sqrt{(x-y)^2+xy\theta ^2}}{t}\right)d\theta \\
& \qquad = \int_0^\infty \theta ^{2\lambda -1}\int_0^\infty \frac{(1+v)^{m+1-2k}v^{m-\beta -1}}{((1+v)^2+[(x-y)^2+xy \theta^2]/t^2)^{\lambda +m-k+1}}dvd\theta \\
& \qquad = t^{2(\lambda +m-k+1)}\int_0^\infty (1+v)^{m+1-2k}v^{m-\beta -1}\int_0^\infty \frac{\theta ^{2\lambda -1}}{((1+v)^2t^2+(x-y)^2+xy\theta ^2)^{\lambda +m-k+1}}d\theta dv\\
& \qquad = \frac{t^{2(\lambda +m-k+1)}}{(xy)^{\lambda }}\int_0^\infty \frac{(1+v)^{m+1-2k}v^{m-\beta -1}}{((1+v)^2t^2+(x-y)^2)^{m-k+1}}dv \int_0^\infty \frac{u^{2\lambda -1}}{(1+u^2)^{\lambda +m-k+1}}du\\
& \qquad = \frac{(m-k)!}{2 \lambda (\lambda+1) \cdots (\lambda+m-k)}\frac{t^{2\lambda}}{(xy)^\lambda}\varphi ^k\Big(\frac{x-y}{t}\Big).\end{aligned}$$ Then, $$\label{Lp20}
\sum_{k=0}^{(m+1)/2}b_k^\lambda R_4^{\lambda ,k}(t,x,y)
=t^\beta \partial _t^ \beta {\mathbb{P}}_t(x-y),\quad t,x,y\in (0,\infty ).$$ By putting together - we conclude that $G_{P,\mathbb{B}}^{\lambda ,\beta}-G_{P,\mathbb{B}}^{\beta ,+}$ is bounded from $L^p((0,\infty ),\mathbb{B})$ into $L^p((0,\infty ),\gamma (H,\mathbb{B}))$, and hence $G_{P,\mathbb{B}}^{\lambda ,\beta}$ is bounded from $L^p((0,\infty ),\mathbb{B})$ into $L^p((0,\infty ),\gamma (H,\mathbb{B}))$.
{#subsec:3.2}
We are going to show that there exists $C>0$ such that, for every $f\in L^p((0,\infty ),\mathbb{B})$, $$\label{Lp21}
\|f\|_{L^p((0,\infty ),\mathbb{B})}\leq C\|G_{P,\mathbb{B}}^{\lambda ,\beta }(f)\|_{L^p((0,\infty ),\gamma (H,\mathbb{B}))}.$$ Since $G_{P,\mathbb{B}}^{\lambda ,\beta}$ is bounded from $L^p((0,\infty ),\mathbb{B})$ into $L^p((0,\infty ),\gamma (H,\mathbb{B}))$ and $\mathcal{S}_\lambda (0,\infty )\otimes \mathbb{B}$ is a dense subspace of $L^p((0,\infty ),\mathbb{B})$, holds for every $f\in L^p((0,\infty ),\mathbb{B})$ whenever it is true for every $f\in \mathcal{S}_\lambda (0,\infty )\otimes \mathbb{B}$.
By proceeding as in Section \[subsec:2.2\], the inequality in can be proved as a consequence of a polarization identity involving the operator $G_{P,\mathbb{B}}^{\lambda ,\beta}$. To show this equality we need previously to establish the following.
\[Lem4\] Let $\lambda$, $\beta>0$. Then, for every $f \in \mathcal{S}_\lambda(0,\infty)$, $$h_\lambda\left( t^\beta \partial_t^\beta P_t^\lambda f \right)(x)
= e^{i\pi \beta }(tx)^\beta e^{-xt} h_\lambda(f)(x), \quad t,x \in (0,\infty).$$
Let $f \in \mathcal{S}_\lambda(0,\infty)$. We have that (see [@EMOT2 §8.5 (19)]) $$h_\lambda(P_t^\lambda f)(x)
= e^{-xt} h_\lambda(f)(x), \quad t,x \in (0,\infty).$$ We choose $m \in {\mathbb{N}}$ such that $m-1 \leq \beta < m$. It is not hard to see that $\partial_t^\beta e^{-xt}=e^{i\pi \beta}x^\beta e^{-xt}$, $t,x \in (0,\infty)$. Then, $$\partial_t^\beta h_\lambda \left( P_t^\lambda f \right)(x)
= e^{i\pi \beta }x^\beta e^{-xt} h_\lambda(f)(x), \quad t,x \in (0,\infty).$$ According to [@GLLNU (4.6)] we can write, for every $t,x,y \in (0,\infty)$ and $\theta \in (0,\pi)$, $$\begin{aligned}
& \partial_t^m \left[ \frac{t}{\left[ (x-y)^2 + 2xy(1-\cos \theta) + t^2\right]^{\lambda+1}} \right]
= - \frac{1}{2\lambda} \partial_t^{m+1} \left[ \frac{1}{\left[ (x-y)^2 + 2xy(1-\cos \theta) + t^2\right]^{\lambda}} \right] \\
& \qquad = \frac{1}{2}\sum_{k =0}^{(m+1)/2} (-1)^{m-k} E_{m+1,k} t^{m+1-2k}
\frac{(\lambda+1)(\lambda+2) \cdot \dots \cdot (\lambda+m-k)}{\left[ (x-y)^2 + 2xy(1-\cos \theta) + t^2\right]^{\lambda+m-k+1}},
\end{aligned}$$ where $$E_{m+1,k}
= \frac{2^{m+1-2k}(m+1)!}{k! (m+1-2k)!}, \quad 0 \leq k \leq \frac{m+1}{2}.$$ Hence, $\partial_t^m \left[ t/\left[ (x-y)^2 + 2xy(1-\cos \theta) + t^2\right]^{\lambda+1}\right]$ is continuous in $(t,x,y,\theta) \in (0,\infty)^3 \times (0,\pi)$. Moreover, for each $t,x,y \in (0,\infty)$ and $\theta \in (0,\pi)$, $$\begin{aligned}
& \left|\partial_t^m \left[ \frac{t}{\left[ (x-y)^2 + 2xy(1-\cos \theta) + t^2\right]^{\lambda+1}} \right] \right|
\leq \frac{C}{\left[(x-y)^2 +t^2\right]^{\lambda+(m+1)/2}}.
\end{aligned}$$ Then, $$\left| \partial_t^m P_{t+s}^\lambda(f)(x) \right|
\leq C \int_0^\infty |f(y) |\frac{(xy)^\lambda}{\left[(x-y)^2 +(t+s)^2\right]^{\lambda+(m+1)/2}} dy, \quad t,x \in (0,\infty),$$ and $\partial_t^\beta P_t^\lambda (f) \in L^1(0,\infty)$, $t>0$. Since the function $\sqrt{z} J_\nu(z)$ is bounded on $(0,\infty)$ when $\nu>-1/2$, the derivation under the integral sing is justified and we get $$h_\lambda\left( \partial_t^\beta P_t^\lambda f \right)(x)
= \partial_t^\beta h_\lambda\left(P_t^\lambda(f)\right)(x)
= e^{i\pi \beta }x^\beta e^{-xt} h_\lambda(f)(x), \quad t,x \in (0,\infty).$$
\[Lem5\] Let ${\mathbb{B}}$ be a UMD Banach space and $\lambda$, $\beta>0$. If $f \in \mathcal{S}_\lambda(0,\infty) \otimes {\mathbb{B}}$ and $g \in \mathcal{S}_\lambda(0,\infty) \otimes {\mathbb{B}}^*$, then $$\label{Lp22}
\int_0^\infty \langle g(x), f(x)\rangle_{{\mathbb{B}}^*,{\mathbb{B}}} dx
= \frac{e^{i2\pi \beta}2^{2\beta}}{{\Gamma}(2\beta)} \int_0^\infty \int_0^\infty \langle t^\beta \partial_t^\beta P_t^\lambda(g)(x) , t^\beta \partial_t^\beta P_t^\lambda(f)(x) \rangle_{{\mathbb{B}}^*,{\mathbb{B}}} \frac{dtdx}{t}.$$
It is enough to show when $f,g \in \mathcal{S}_\lambda(0,\infty)$ and ${\mathbb{B}}={\mathbb{C}}$. Let $f,g \in \mathcal{S}_\lambda(0,\infty)$.
Since $\mathbb{C}$ is a UMD Banach space, as it was proved in Section \[subsec:3.1\], the operator $G_{P,\mathbb{C}}^{\lambda,\beta}$ is bounded from $L^p(0,\infty)$ into $L^p((0,\infty),H)$, $1<p<\infty$. Hence, the integral in the right hand side of is absolutely convergent.
As $h_\lambda$ is an isometry in $L^2(0,\infty)$ ([@Ti p. 214 and Theorem 129]), Lemma \[Lem4\] implies that $t^\beta \partial_t^\beta P_t^\lambda(f) \in L^2(0,\infty)$ and $t^\beta \partial_t^\beta P_t^\lambda(g) \in L^2(0,\infty)$, for every $t>0$. Plancherel equality for Hankel transforms and Lemma \[Lem4\] lead to $$\begin{aligned}
& \int_0^\infty \int_0^\infty t^\beta \partial_t^\beta P_t^\lambda(f)(x) t^\beta \partial_t^\beta P_t^\lambda(g)(x)\frac{dtdx}{t}
= \int_0^\infty \int_0^\infty t^\beta \partial_t^\beta P_t^\lambda(f)(x) t^\beta \partial_t^\beta P_t^\lambda(g)(x)\frac{dxdt}{t} \\
& \qquad = e^{i2\pi\beta} \int_0^\infty \int_0^\infty (tx)^{2\beta} e^{-2xt} h_\lambda(f)(x) h_\lambda(g)(x) \frac{dxdt}{t} \\
& \qquad = e^{i2\pi\beta} \int_0^\infty h_\lambda(f)(x) h_\lambda(g)(x) \int_0^\infty (tx)^{2\beta} e^{-2xt} \frac{dtdx}{t} \\
& \qquad = e^{i2\pi\beta}\frac{{\Gamma}(2\beta)}{2^{-2\beta}} \int_0^\infty h_\lambda(f)(x) h_\lambda(g)(x)dx
= e^{i2\pi\beta} \frac{{\Gamma}(2\beta)}{2^{-2\beta}} \int_0^\infty f(x) g(x)dx.
\end{aligned}$$
By using now Lemma \[Lem5\], the arguments developed in Section \[subsec:2.2\] allow us to show that holds, for every $f \in L^p((0,\infty),{\mathbb{B}})$.
Thus the proof of Theorem \[boundedness\] is completed.
Proof of Theorem \[boundedness2\] {#sec:Proof3}
=================================
The Riesz transform $R_\lambda$ associated with the Bessel operator $\Delta_\lambda$ is the principal value integral operator defined, for every $f \in L^p(0,\infty)$, by $$R_\lambda(f)(x)
= \lim_{\varepsilon \to 0^+} \int_{0, \ |x-y|>\varepsilon}^\infty R_\lambda(x,y) f(y)dy, \quad \text{a.e. } x \in (0,\infty),$$ where $$R_\lambda(x,y)
= \int_0^\infty D_\lambda P_t^\lambda(x,y)dt, \quad x,y \in (0,\infty), \ x \neq y,$$ and $D_\lambda=x^\lambda \frac{d}{dx} x^{-\lambda}$. Main properties of Riesz transform $R_\lambda$ can be encountered in [@BBFMT]. We denote by $R_\lambda^*$ the “adjoint” operator of $R_\lambda$ defined, for every $f \in L^p(0,\infty)$, by $$R_\lambda^*(f)(x)
= \lim_{\varepsilon \to 0^+} \int_{0, \ |x-y|>\varepsilon}^\infty R_\lambda(y,x) f(y)dy, \quad \text{a.e. } x \in (0,\infty).$$ Riesz transforms $R_\lambda$ and $R_\lambda^*$ are bounded from $L^p(0,\infty)$ into itself. Moreover, since ${\mathbb{B}}$ is a UMD Banach space, by defining $R_\lambda$ and $R_\lambda^*$ on $L^p(0,\infty)\otimes {\mathbb{B}}$ in the natural way, $R_\lambda$ and $R_\lambda^*$ can be extended to $L^p((0,\infty),{\mathbb{B}})$ as bounded operators on $L^p((0,\infty),{\mathbb{B}})$ into itself ([@BFMT Theorem 2.1]).
We define, for every $f \in L^p(0,\infty)$, the function ${\mathcal{Q}}_t^\lambda(f)$ by $${\mathcal{Q}}_t^\lambda(f)(x)
= \int_0^\infty {\mathcal{Q}}_t^\lambda(x,y)f(y)dy, \quad t,x \in (0,\infty),$$ where $${\mathcal{Q}}_t^\lambda(x,y)
= \frac{2\lambda(xy)^\lambda}{\pi} \int_0^\pi \frac{(x-y\cos \theta)(\sin \theta)^{2\lambda-1}}{(x^2+y^2+t^2-2xy \cos \theta)^{\lambda+1}} d\theta, \quad t,x,y \in (0,\infty).$$ The following Cauchy-Riemann equations hold $$D_\lambda P_t^\lambda(f)= \partial_t {\mathcal{Q}}_t^\lambda(f), \qquad
D_\lambda^* {\mathcal{Q}}_t^\lambda(f) = \partial_t P_t^\lambda(f), \quad t>0.$$ These relations motivate that ${\mathcal{Q}}_t^\lambda(f)$ is called $\Delta_\lambda$-conjugated to the Poisson integral $P_t^\lambda(f)$.
The adjoint $\Delta_\lambda$-conjugated ${\mathbb{Q}}_t^\lambda(f)$ of $f \in L^p(0,\infty)$ is defined by $${\mathbb{Q}}_t^\lambda(f)(x)
= \int_0^\infty {\mathcal{Q}}_t^\lambda(y,x)f(y)dy, \quad t,x \in (0,\infty).$$ We have that $$D_\lambda^* P_t^{\lambda+1}(f)= \partial_t {\mathbb{Q}}_t^\lambda(f), \qquad
D_\lambda {\mathbb{Q}}_t^\lambda(f) = \partial_t P_t^{\lambda+1}(f), \quad t>0.$$ By using Hankel transform (see [@MS (16.5)]) we can see that, for every $f \in \mathcal{S}_\lambda(0,\infty)$, $$P_t^\lambda(R_\lambda^* f)
= {\mathbb{Q}}_t^\lambda(f), \quad t>0.$$ Then, for every $f \in \mathcal{S}_\lambda(0,\infty)$, $$\label{Lp23}
\partial_t P_t^\lambda(R_\lambda^* f)
= D_\lambda^* P_t^{\lambda+1}(f), \quad t>0.$$ Equality also holds for every $f \in \mathcal{S}_\lambda(0,\infty) \otimes {\mathbb{B}}$. Then, $$\label{Lp24}
\mathcal{G}_{P,{\mathbb{B}}}^{\lambda}(f)
= G_{P,{\mathbb{B}}}^{\lambda,1}(R_\lambda^* f), \quad f \in \mathcal{S}_\lambda(0,\infty) \otimes {\mathbb{B}}.$$
Since $R_\lambda^*$ can be extended to $L^p((0,\infty),{\mathbb{B}})$ as a bounded operator from $L^p((0,\infty),{\mathbb{B}})$ into itself, Theorem \[boundedness\] implies that the operator $\mathcal{G}_{P,{\mathbb{B}}}^{\lambda}$ can be extended from $\mathcal{S}_\lambda(0,\infty) \otimes {\mathbb{B}}$ as a bounded operator from $L^p((0,\infty),{\mathbb{B}})$ into $L^p((0,\infty),\gamma(H,{\mathbb{B}}))$. We denote this extension by $\widetilde{\mathcal{G}}_{P,{\mathbb{B}}}^{\lambda}$.
We define $$\mathcal{G}_{P,{\mathbb{B}}}^{\lambda}(t,x,y)
= t D_\lambda^* P_t^{\lambda+1}(x,y), \quad t,x,y \in (0,\infty).$$ We have that $$\begin{aligned}
\mathcal{G}_{P,{\mathbb{B}}}^{\lambda}(t,x,y)
= & - \frac{2(\lambda+1)}{\pi} t^2 y^{\lambda+1} x^{-\lambda}
\partial_x \left( x^{2\lambda+1} \int_0^\pi \frac{(\sin \theta)^{2\lambda+1}}{\left[ (x-y)^2 + t^2 + 2xy(1-\cos \theta) \right]^{\lambda+2}} d\theta \right) \\
= & - \frac{2(\lambda+1)(2\lambda+1)}{\pi} t^2 x^{\lambda} y^{\lambda+1}
\int_0^\pi \frac{(\sin \theta)^{2\lambda+1}}{\left[ (x-y)^2 + t^2 + 2xy(1-\cos \theta) \right]^{\lambda+2}} d\theta \\
& + \frac{4(\lambda+1)(\lambda+2)}{\pi} t^2 (xy)^{\lambda+1}
\int_0^\pi \frac{[(x-y)+y(1-\cos \theta)](\sin \theta)^{2\lambda+1}}{\left[ (x-y)^2 + t^2 + 2xy(1-\cos \theta) \right]^{\lambda+3}} d\theta,
\quad t,x,y \in (0,\infty).\end{aligned}$$ Then, $$\begin{aligned}
\left| \mathcal{G}_{P,{\mathbb{B}}}^{\lambda}(t,x,y) \right|
\leq & C \sqrt{t} \Big\{ x^\lambda y^{\lambda+1} \left(\int_0^{\pi/2} + \int_{\pi/2}^\pi \right) \frac{(\sin \theta)^{2\lambda+1}}{\left[ (x-y)^2 + t^2 + 2xy(1-\cos \theta) \right]^{\lambda+5/4}} d\theta \\
& + (xy)^{\lambda+1} \left(\int_0^{\pi/2} + \int_{\pi/2}^\pi \right) \frac{(\sin \theta)^{2\lambda+1}}{\left[ (x-y)^2 + t^2 + 2xy(1-\cos \theta) \right]^{\lambda+7/4}} d\theta \Big\} \\
= & \mathcal{G}_{P,{\mathbb{B}}}^{\lambda,1,1}(t,x,y) + \mathcal{G}_{P,{\mathbb{B}}}^{\lambda,1,2}(t,x,y)
+ \mathcal{G}_{P,{\mathbb{B}}}^{\lambda,2,1}(t,x,y) + \mathcal{G}_{P,{\mathbb{B}}}^{\lambda,2,2}(t,x,y) ,\quad t,x,y \in (0,\infty).\end{aligned}$$
Let $\varepsilon >0$. Since, for every $x,y \in (0,\infty)$ and $\theta \in (0,\pi/2)$, $$\left( \int_\varepsilon^\infty \frac{dt}{\left[ (x-y)^2 + t^2 + 2xy(1-\cos \theta) \right]^{2\lambda+5/2}} \right)^{1/2}
\leq \frac{C}{(|x-y|+\varepsilon+\sqrt{xy}\theta)^{2\lambda+2}},$$ and $$\left( \int_\varepsilon^\infty \frac{dt}{\left[ (x-y)^2 + t^2 + 2xy(1-\cos \theta) \right]^{2\lambda+7/2}} \right)^{1/2}
\leq \frac{C}{(|x-y|+\varepsilon+\sqrt{xy}\theta)^{2\lambda+3}},$$ we obtain $$\begin{aligned}
&\Big\| \mathcal{G}_{P,{\mathbb{B}}}^{\lambda,1,1}(\cdot,x,y) \Big\|_{L^2\left((\varepsilon,\infty),dt/t\right)}
+ \Big\| \mathcal{G}_{P,{\mathbb{B}}}^{\lambda,2,1}(\cdot,x,y) \Big\|_{L^2\left((\varepsilon,\infty),dt/t\right)} \\
& \qquad \leq C \left( x^\lambda y^{\lambda+1} \int_0^{\pi/2} \frac{\theta^{2\lambda+1}}{(|x-y|+\varepsilon+\sqrt{xy}\theta)^{2\lambda+2}} d\theta
+ (xy)^{\lambda+1} \int_0^{\pi/2} \frac{\theta^{2\lambda+1}}{(|x-y|+\varepsilon+\sqrt{xy}\theta)^{2\lambda+3}} d\theta \right) \\
& \qquad \leq C \left( \frac{y}{(|x-y|+\varepsilon)^2} + \frac{xy}{(|x-y|+\varepsilon)^3} \right)
\leq C \left\{\begin{array}{ll}
1/(x+\varepsilon), & 0<y<x/2, \\
y/\varepsilon^2+y^2/\varepsilon^3, & x/2<y<2x, \\
1/(y+\varepsilon), & y>2x>0.
\end{array} \right.\end{aligned}$$ Analogously, $$\begin{aligned}
&\Big\| \mathcal{G}_{P,{\mathbb{B}}}^{\lambda,1,2}(\cdot,x,y) \Big\|_{L^2\left((\varepsilon,\infty),dt/t\right)}
+ \Big\| \mathcal{G}_{P,{\mathbb{B}}}^{\lambda,2,2}(\cdot,x,y) \Big\|_{L^2\left((\varepsilon,\infty),dt/t\right)} \\
& \qquad \leq C \left( x^\lambda y^{\lambda+1} \int_{\pi/2}^\pi \frac{(\sin \theta)^{2\lambda+1}}{(x+y+\varepsilon)^{2\lambda+2}} d\theta
+ (xy)^{\lambda+1} \int_{\pi/2}^\pi \frac{(\sin \theta)^{2\lambda+1}}{(x+y+\varepsilon)^{2\lambda+3}} d\theta \right) \\
& \qquad \leq \frac{C}{x+y+\varepsilon},\quad x,y \in (0,\infty).\end{aligned}$$ Hence, for every $x \in (0,\infty)$, $\left\| \mathcal{G}_{P,{\mathbb{B}}}^{\lambda}(\cdot,x,y) \right\|_{L^2\left((\varepsilon,\infty),dt/t\right)} \in L^{p'}(0,\infty)$.
By proceeding now as in Section \[subsec:2.1\] we conclude that $$\mathcal{G}_{P,{\mathbb{B}}}^{\lambda}(f)
= \widetilde{\mathcal{G}}_{P,{\mathbb{B}}}^{\lambda}(f), \quad f \in L^p((0,\infty), {\mathbb{B}}).$$ and the proof of Theorem \[boundedness2\] is completed.
Proof of Theorem \[Th1.4\] {#sec:Proof4}
==========================
Proof of $(i) \Rightarrow (ii)$ and $(i) \Rightarrow (iii)$ {#subsec:4.1}
------------------------------------------------------------
In Theorems \[boundedness\] and \[boundedness2\] it was proved that if ${\mathbb{B}}$ is a UMD Banach space, then , and are satisfied, for every $1<p<\infty$.
Proof of $(ii) \Rightarrow (i)$ {#subsec:4.2}
-------------------------------
Let $1<p<\infty$. Suppose that and hold. Let $f \in \mathcal{S}_\lambda(0,\infty)\otimes {\mathbb{B}}$. Since $R_\lambda^*$ is bounded from $L^p(0,\infty)$ into itself ([@BBFMT Theorem 4.2]), $R_\lambda^*f \in L^p(0,\infty)\otimes {\mathbb{B}}$. According to we obtain $$\begin{aligned}
\|R_\lambda^* f\|_{L^p((0,\infty),{\mathbb{B}})}
\leq & C \|G_{P,{\mathbb{B}}}^{\lambda,1}(R_\lambda^* f)\|_{L^p((0,\infty),\gamma(H,{\mathbb{B}}))}
= C \|\mathcal{G}_{P,{\mathbb{B}}}^{\lambda}(f)\|_{L^p((0,\infty),\gamma(H,{\mathbb{B}}))} \\
\leq & C \| f \|_{L^p((0,\infty),{\mathbb{B}}).}\end{aligned}$$ Since $\mathcal{S}_\lambda(0,\infty) \otimes {\mathbb{B}}$ is dense in $L^p((0,\infty),{\mathbb{B}})$, $R_\lambda^*$ can be extended to $L^p((0,\infty),{\mathbb{B}})$ as a bounded operator from $L^p((0,\infty),{\mathbb{B}})$ into itself. By using [@BFMT Theorem 2.1] we deduce that ${\mathbb{B}}$ is UMD.
Proof of $(iii) \Rightarrow (i)$ {#subsec:4.3}
--------------------------------
Assume now that holds. In order to show that ${\mathbb{B}}$ is UMD we prove previously a characterization of UMD Banach spaces involving $L^p$-boundedness properties of the imaginary powers $\Delta_\lambda^{i\omega}$, $\omega \in \mathbb{R} \setminus \{0\}$, of the Bessel operator $\Delta_\lambda$.
Let $\omega \in \mathbb{R} \setminus \{0\}$. The $i \omega$-power $\Delta_\lambda^{i\omega}$ of $\Delta_\lambda$ is the Hankel multiplier defined by $$\label{23.1}
\Delta_\lambda^{i\omega} f
= h_\lambda \left( y^{2i \omega} h_\lambda(f)\right), \quad f \in L^2(0,\infty).$$ Since $h_\lambda$ is an isometry in $L^2(0,\infty)$, the operator $\Delta_\lambda^{i\omega}$ is bounded from $L^2(0,\infty)$ into itself. Moreover $$y^{2i \omega}
= y^2 \int_0^\infty e^{-y^2 u} \frac{u^{-i \omega}}{{\Gamma}(1-i \omega)} du, \quad y \in (0,\infty),$$ and hence $\Delta_\lambda^{i\omega}$ is a Hankel multiplier of Laplace transform type. This type of Hankel multipliers were studied in [@BCC1] and [@BMR]. Proceeding as in [@BCC1 Theorem 1.2], for every $f \in C_c^\infty(0,\infty)$, we have that $$\label{Lp25}
\Delta_\lambda^{i\omega} f (x)
= \lim_{\varepsilon \to 0^+} \left( \alpha(\varepsilon)f(x) - \int_{0, \ |x-y|>\varepsilon}^\infty K^\lambda_\omega(x,y)f(y)dy\right),
\quad \text{a.e. } x \in (0,\infty),$$ where $$K^\lambda_\omega(x,y)
= \int_0^\infty \frac{t^{-i\omega}}{{\Gamma}(1-i \omega)} \partial_t W_t^\lambda(x,y)dt, \quad x,y \in (0,\infty), \ x \neq y,$$ and $W_t^\lambda(x,y)$ is the Bessel heat kernel $$W_t^\lambda(x,y)
= \frac{1}{\sqrt{2t}} \left( \frac{xy}{2t}\right)^{1/2} I_{\lambda-1/2}\left( \frac{xy}{2t} \right) e^{-(x^2+y^2)/4t}, \quad t,x,y \in (0,\infty).$$ Here $\alpha$ denotes a bounded function on $(0,\infty)$ and $I_\nu$ is the modified Bessel function of the first kind and order $\nu$. By [@BMR Theorem 1.2], $\Delta_\lambda^{i\omega}f$ can be extended to $L^p(0,\infty)$ as a bounded operator from $L^p(0,\infty)$ into itself. Moreover, as in [@BCC1 Theorem 1.4], we can see that this extension, that we will continue denoting by $\Delta_\lambda^{i\omega}$, is given by the limit in for every $f \in L^p(0,\infty)$. The operator $\Delta_\lambda^{i\omega}$, is defined on $L^p(0,\infty) \otimes {\mathbb{B}}$ in the usual way.
The following result is a Bessel version of [@Gue Theorem p. 402].
\[ImagBess\] Let $X$ be a Banach space and $\lambda>0$. Then, $X$ is UMD if and only if, for some (equivalently, for every) $1<q<\infty$, the operator $\Delta^{i\omega}_\lambda$, $\omega \in \mathbb{R}\setminus \{0\}$, can be extended from $L^q(0,\infty) \otimes X$ to $L^q((0,\infty),X)$ as a bounded operator from $L^q((0,\infty),X)$ into itself.
According to [@Gue Theorem p. 402], $X$ is UMD if, and only if, for every $\omega \in \mathbb{R}\setminus\{0\}$ and for some (equivalently, for every) $1<q<\infty$, the $i\omega$-power $\left( - \frac{d^2}{dx^2}\right)^{i \omega}$ of the operator $- \frac{d^2}{dx^2}$ can be extended from $L^q(\mathbb{R}) \otimes X$ to $L^q(\mathbb{R},X)$ as a bounded operator from $L^q(\mathbb{R},X)$ into itself.
We recall that (see [@BCFR Appendix] for a proof) for every $f \in L^q(\mathbb{R})$, $1<q<\infty$, and $\omega \in \mathbb{R}\setminus \{0\}$, $$\left( - \frac{d^2}{dx^2}\right)^{i \omega} f(x)
= \lim_{\varepsilon \to 0^+} \left( \alpha(\varepsilon)f(x) - \int_{\ |x-y|>\varepsilon} K_\omega(x,y)f(y)dy\right),
\quad \text{a.e. } x \in \mathbb{R},$$ where $$K_\omega (x,y)
= -\int_0^\infty \frac{t^{-i\omega}}{{\Gamma}(1-i \omega)} \partial_t {\mathbb{W}}_t(x-y)dt, \quad x,y \in \mathbb{R}, \ x \neq y,$$ and ${\mathbb{W}}_t(z)$ denotes the classical heat kernel . Here $\alpha$ represents the same function that appears in . The operator $\left( - \frac{d^2}{dx^2}\right)^{i \omega}$, $\omega \in \mathbb{R}\setminus\{0\}$, is defined on $L^q(\mathbb{R}) \otimes X$, $1<q<\infty$, in the natural way.
Let $\omega \in {\mathbb{R}}\setminus \{0\}$. We are going to obtain some estimates for the kernels $K^\lambda_\omega(x,y)$ and $K_\omega(x,y)$, $x,y \in (0,\infty )$, that will allow us to get our characterization of the UMD spaces by using imaginary powers of Bessel operators.
Note firstly that $$\label{Lp26}
|K_\omega(x,-y)|
\leq C \int_0^\infty \left| \partial_t {\mathbb{W}}_t(x+y) \right| dt
\leq C \int_0^\infty \frac{e^{-c(x+y)^2/t}}{t^{3/2}} dt
\leq \frac{C}{x+y}, \quad x,y \in (0,\infty).$$ In a similar way we obtain, for every $x \in (0,\infty)$, $$\label{Lp27}
|K_\omega(x,y)|
\leq C \left\{ \begin{array}{ll}
1/x, & 0<y<x/2,\\
1/y, & y>2x.
\end{array} \right.$$ According to [@Leb pp. 108 and 123] if $\nu>-1$, we have that $$\label{Lp28}
I_\nu(z) \sim \frac{z^\nu}{2^\nu {\Gamma}(\nu+1)}, \quad \text{as } z \to 0^+,$$ and $$\label{Lp29}
\sqrt{z} I_\nu(z)
= \frac{e^z}{\sqrt{2\pi}} \left( \sum_{r=0}^n \frac{(-1)^r [\nu,r]}{(2z)^r} + \mathcal{O}\left(\frac{1}{z^{n+1}}\right)\right),$$ where $[\nu,0]=1$ and $$[\nu,r]=\frac{(4\nu^2-1)(4\nu^2-3^2) \cdot \dots \cdot (4\nu^2-(2r-1)^2)}{2^{2r}{\Gamma}(r+1)}, \quad r=1,2, \dots$$ Since (see [@Leb p. 110]), $$\label{24.1}
\frac{d}{dz}\left( z^{-\nu} I_\nu(z)\right)
= z^{-\nu} I_{\nu+1}(z), \quad z \in (0,\infty), \ \nu>-1,$$ it follows that, for every $t,x,y \in (0,\infty)$, $$\begin{aligned}
& \partial_t \left[ W_t^\lambda(x,y) - {\mathbb{W}}_t(x-y) \right]
= \partial_t \left[ {\mathbb{W}}_t(x-y) \left\{ \sqrt{2\pi} \left( \frac{xy}{2t} \right)^{\nu+1/2} \left( \frac{xy}{2t} \right)^{-\nu} I_\nu\left( \frac{xy}{2t}\right)e^{-xy/2t}-1\right\} \right] \\
& \qquad = \partial_t {\mathbb{W}}_t(x-y) \left\{ \sqrt{2\pi} \left( \frac{xy}{2t} \right)^{1/2} I_\nu\left( \frac{xy}{2t}\right)e^{-xy/2t}-1\right\} \\
& \qquad \qquad - \sqrt{2\pi} {\mathbb{W}}_t(x-y) \Big\{ (\nu+1/2) \left( \frac{xy}{2t} \right)^{\nu-1/2} \frac{xy}{2t^2} \left( \frac{xy}{2t} \right)^{-\nu} I_\nu\left( \frac{xy}{2t}\right) \\
& \qquad \qquad + \left( \frac{xy}{2t} \right)^{\nu+1/2} \frac{xy}{2t^2} \left( \frac{xy}{2t} \right)^{-\nu} I_{\nu+1}\left( \frac{xy}{2t}\right)
- \frac{xy}{2t^2} \left( \frac{xy}{2t} \right)^{1/2} I_{\nu}\left( \frac{xy}{2t}\right) \Big\} e^{-xy/2t} \\
& \qquad = \partial_t {\mathbb{W}}_t(x-y) \left\{ \sqrt{2\pi} \left( \frac{xy}{2t} \right)^{1/2} I_\nu\left( \frac{xy}{2t}\right)e^{-xy/2t}-1\right\} \\
& \qquad \qquad - \sqrt{2\pi} {\mathbb{W}}_t(x-y) \frac{xy}{2t^2} e^{-xy/2t} \\
& \qquad \qquad \times \left\{ (\nu+1/2) \frac{2t}{xy} \left( \frac{xy}{2t} \right)^{1/2} I_{\nu}\left( \frac{xy}{2t}\right)
+ \left( \frac{xy}{2t} \right)^{1/2} I_{\nu+1}\left( \frac{xy}{2t}\right)
- \left( \frac{xy}{2t} \right)^{1/2} I_{\nu}\left( \frac{xy}{2t}\right) \right\},
\end{aligned}$$ being $\nu=\lambda-1/2$.
From , we deduce $$\label{Lp30}
\left| \partial_t \left[ W_t^\lambda(x,y)-{\mathbb{W}}_t(x-y) \right] \right|
\leq C \frac{e^{-c(x-y)^2/t}}{t^{3/2}}, \quad t,x,y \in (0,\infty) \text{ and } xy \leq 2t,$$ and by using , $$\label{Lp31}
\left| \partial_t \left[ W_t^\lambda(x,y)-{\mathbb{W}}_t(x-y) \right] \right|
\leq C \frac{e^{-c(x-y)^2/t}}{t^{1/2}xy}, \quad t,x,y \in (0,\infty) \text{ and } xy \geq 2t.$$ Combining and we obtain $$\begin{aligned}
\label{Lp32}
\left| K_\omega(x,y)-K_\omega^\lambda(x,y) \right|
\leq &C \int_0^\infty \left| \partial_t \left[ W_t^\lambda(x,y) - {\mathbb{W}}_t(x-y) \right] \right| dt \nonumber \\
\leq& C \left( \int_0^{xy/2} \frac{e^{-c(x-y)^2/t}}{t^{1/2}xy} dt + \int_{xy/2}^\infty \frac{e^{-c(x^2+y^2)/t}}{t^{3/2}} dt \right) \nonumber\\
\leq& \frac{C}{(xy)^{1/2}} \leq \frac{C}{x}, \quad x/2 < y < 2x, \quad x \in (0,\infty).
\end{aligned}$$ Moreover, and imply that, for each $x \in (0,\infty)$, $$\begin{aligned}
\label{Lp33}
\left| K_\omega^\lambda(x,y) \right|
\leq C \int_0^\infty \frac{e^{-c(x-y)^2/t}}{t^{3/2}}dt
\leq \frac{C}{|x-y|}
\leq C \left\{
\begin{array}{ll}
1/x, & 0<y<x/2, \\
1/y, & y>2x.
\end{array}\right.
\end{aligned}$$ Suppose that $X$ is UMD and $1<q<\infty$. Let $f \in L^q(0,\infty) \otimes X$. We define the function $\tilde{f}$ by $$\tilde{f}(x)
= \left\{
\begin{array}{ll}
0, & x \leq 0, \\
f(x), & x>0.
\end{array}\right.$$ Thus, $\tilde{f} \in L^q({\mathbb{R}})\otimes X$. We have that $$\left( - \frac{d^2}{dx^2} \right)^{i \omega} \tilde{f}(x)
= \lim_{\varepsilon \to 0^+} \left( f(x)\alpha(\varepsilon) - \int_{0, |x-y|>\varepsilon}^\infty K_\omega(x,y)f(y)dy \right), \quad \text{a.e. } x \in (0,\infty),$$ and $$\Delta_\lambda^{i \omega} f(x)
= \lim_{\varepsilon \to 0^+} \left( f(x)\alpha(\varepsilon) - \int_{0, |x-y|>\varepsilon}^\infty K_\omega^\lambda(x,y)f(y)dy \right), \quad \text{a.e. } x \in (0,\infty).$$ Then, , , lead to $$\begin{aligned}
& \Big\| \left( - \frac{d^2}{dx^2} \right)^{i \omega} \tilde{f}(x) - \Delta_\lambda^{i \omega} f(x) \Big\|_X
\leq {\underset{\varepsilon \to 0^+}{\overline{\lim}}} \int_{0, |x-y|>\varepsilon}^\infty \left| K_\omega(x,y) - K_\omega^\lambda(x,y) \right| \|f(y)\|_X dy \\
& \qquad \leq C \left[ H_0(\|f\|_X)(x) + H_\infty(\|f\|_X)(x) \right], \quad \text{a.e. } x \in (0,\infty).
\end{aligned}$$ Hence, according to [@HLP p. 244, (9.9.1) and (9.9.2)], there exists $C>0$ such that $$\Big\| \left( - \frac{d^2}{dx^2} \right)^{i \omega} \tilde{f} - \Delta_\lambda^{i \omega} f \Big\|_{L^q((0,\infty),X)}
\leq C \|f\|_{L^q((0,\infty),X)}, \quad f \in L^q(0,\infty)\otimes X.$$ Moreover, by [@Gue Theorem p. 402], we also have $$\Big\| \left( - \frac{d^2}{dx^2} \right)^{i \omega} \tilde{f} \Big\|_{L^q((0,\infty),X)}
\leq C \|f\|_{L^q((0,\infty),X)}, \quad f \in L^q(0,\infty)\otimes X.$$ We conclude that $$\Big\| \Delta_\lambda^{i \omega} f \Big\|_{L^q((0,\infty),X)}
\leq C \|f\|_{L^q((0,\infty),X)}, \quad f \in L^q(0,\infty)\otimes X.$$
Suppose now that $\Delta_\lambda^{i\omega}$ can be extended from $L^q(0,\infty)\otimes X$ to $L^q((0,\infty),X)$ as a bounded operator from $L^q((0,\infty),X)$ into itself. According to [@Gue Theorem p. 402] in order to see that $X$ is UMD it is sufficient to see that, for a certain $C>0$, $$\Big\| \left( - \frac{d^2}{dx^2} \right)^{i \omega} f \Big\|_{L^q({\mathbb{R}},X)}
\leq C \|f\|_{L^q({\mathbb{R}},X)}, \quad f \in L^q({\mathbb{R}})\otimes X.$$ Let $f \in L^p(\mathbb{R})\otimes X$. By defining $$f_+(x)=f(x), \quad \text{and} \quad f_-(x)=f(-x), \quad x \in (0,\infty),$$ we have that $$\begin{aligned}
& \left( - \frac{d^2}{dx^2} \right)^{i \omega} f(x)
= \lim_{\varepsilon \to 0^+} \left( f_+(x)\alpha(\varepsilon) - \int_{0, |x-y|>\varepsilon}^\infty K_\omega(x,y)f_+(y)dy
- \int_{-\infty}^0 K_\omega(x,y)f(y)dy \right) \\
\qquad & = \lim_{\varepsilon \to 0^+} \left( f_+(x)\alpha(\varepsilon) - \int_{0, |x-y|>\varepsilon}^\infty K_\omega(x,y)f_+(y)dy\right)
- \int_0^\infty K_\omega(x,-y)f_-(y)dy \, \quad \text{a.e. } x \in (0,\infty),
\end{aligned}$$ and $$\begin{aligned}
& \left( - \frac{d^2}{dx^2} \right)^{i \omega} f(x)
= \lim_{\varepsilon \to 0^+} \left( f(x)\alpha(\varepsilon) - \int_{-\infty, |x-y|>\varepsilon}^0 K_\omega(x,y)f(y)dy
- \int_0^\infty K_\omega(x,y)f(y)dy \right) \\
\qquad & = \lim_{\varepsilon \to 0^+} \left( f_-(-x)\alpha(\varepsilon) - \int_{0, |x+y|>\varepsilon}^\infty K_\omega(x,-y)f(-y)dy
- \int_0^\infty K_\omega(x,y)f(y)dy \right)\\
\qquad & = \lim_{\varepsilon \to 0^+} \left( f_-(-x)\alpha(\varepsilon) - \int_{0, |x+y|>\varepsilon}^\infty K_\omega(x,-y)f_-(y)dy\right)
- \int_0^\infty K_\omega(x,y)f_+(y)dy , \quad \text{a.e. } x \in (-\infty,0).
\end{aligned}$$ We consider the operators $$T_{\omega,1}(g)(x)
= \lim_{\varepsilon \to 0^+} \left( g(x)\alpha(\varepsilon) - \int_{0, |x-y|>\varepsilon}^\infty K_\omega(x,y)g(y)dy\right), \quad x\in (0,\infty),$$ and $$T_{\omega,2}(g)(x)
= \int_{0}^\infty K_\omega(x,-y)g(y)dy, \quad x\in (0,\infty),$$ for every $g \in L^q(0,\infty) \otimes X$.
We can write $$\begin{aligned}
\label{Lp34}
\Big\| \left( - \frac{d^2}{dx^2} \right)^{i \omega} f\Big\|_{L^q({\mathbb{R}},X)}^q
= & \Big\| T_{\omega,1}(f_+) \Big\|_{L^q((0,\infty),X)}^q
+ \Big\| T_{\omega,2}(f_-) \Big\|_{L^q((0,\infty),X)}^q \nonumber \\
& + \Big\| T_{\omega,1}(f_-) \Big\|_{L^q((0,\infty),X)}^q
+ \Big\| T_{\omega,2}(f_+) \Big\|_{L^q((0,\infty),X)}^q
\end{aligned}$$ According to we get, for every $g \in L^q(0,\infty) \otimes X$, $$\| T_{\omega,2}(g)(x) \|_X
\leq C \int_0^\infty \frac{\|g(y)\|_X}{x+y}dy
\leq C \left[ H_0(\|g\|_X)(x) + H_\infty(\|g\|_X)(x)\right], \quad x\in (0,\infty).$$ Also, by combining , and we obtain, for each $g \in L^q(0,\infty) \otimes X$, $$\| T_{\omega,1}(g)(x) - \Delta_\lambda^{i\omega}(g)(x) \|_X
\leq C \left[ H_0(\|g\|_X)(x) + H_\infty(\|g\|_X)(x)\right], \quad x\in (0,\infty).$$ Then, by [@HLP p. 244, (9.9.1) and (9.9.2)] it follows that , for every $g \in L^q(0,\infty) \otimes X$, $$\label{Lp35}
\| T_{\omega,2}(g)\|_{L^q((0,\infty),X)} + \| T_{\omega,1}(g) - \Delta_\lambda^{i\omega}(g)\|_{L^q((0,\infty),X)}
\leq C \|g\|_{L^q((0,\infty),X)}.$$ Since $\Delta_\lambda^{i\omega}$ can be extended from $L^q(0,\infty) \otimes X$ to $L^q((0,\infty),X)$ as a bounded operator from $L^q((0,\infty),X)$ into itself, and implies that $$\Big\| \left( - \frac{d^2}{dx^2} \right)^{i \omega} f\Big\|_{L^q({\mathbb{R}},X)}
\leq C \left( \left\| f_+\right\|_{L^q((0,\infty),X)} + \left\| f_-\right\|_{L^q((0,\infty),X)} \right)
\leq C \left\| f \right\|_{L^q({\mathbb{R}},X)},$$ for every $f \in L^q({\mathbb{R}}) \otimes X$.
Let $\beta>0$ and $f \in \mathcal{S}_\lambda(0,\infty)$. According to Theorem \[boundedness\], there exists a set $\Omega \subset (0,\infty)$, such that $|(0,\infty) \setminus \Omega|=0$ and for every $x \in \Omega$, the functions $G_{P,{\mathbb{C}}}^{\lambda,\beta}(\Delta_\lambda^{i\omega}f)(\cdot,x)$ and $G_{P,{\mathbb{C}}}^{\lambda,\beta+1}(f)(\cdot,x)$ are in $H$. Let $x \in \Omega$. We denote by $A_1$ and $A_2$ the linear bounded operators from $H$ into ${\mathbb{C}}$ defined by $$A_1(h)
= \int_0^\infty G_{P,{\mathbb{C}}}^{\lambda,\beta}(\Delta_\lambda^{i\omega}f)(t,x) h(t) \frac{dt}{t}, \quad h \in H,$$ and $$A_2(h)
= \int_0^\infty G_{P,{\mathbb{C}}}^{\lambda,\beta+1}(f)(t,x) h(t) \frac{dt}{t}, \quad h \in H.$$ We also define, for every $h \in H$, $$T_{\omega,\beta}(h)(t)
= \frac{1}{t^\beta} \int_0^t (t-s)^{\beta-1} h(t-s) \phi_\omega(s)ds, \quad t \in (0,\infty),$$ where $\phi_\omega(s)=s^{-2 i \omega}/{\Gamma}(1-2 i \omega)$, $s \in (0,\infty)$. Thus, $T_{\omega,\beta}$ is a linear bounded operator from $H$ into itself. Indeed, Jensen’s inequality leads to $$\begin{aligned}
\|T_{\omega,\beta}(h)\|_H
\leq & \left( \int_0^\infty \frac{1}{t^{2\beta+1}} \left( \int_0^t \left| h(t-s)(t-s)^{\beta-1} \phi_\omega(s) \right| ds \right)^2 dt \right)^{1/2} \\
\leq & C \left( \int_0^\infty \frac{1}{t} \left( \int_0^t \left| h(u) \right| \frac{u^{\beta-1}du}{t^\beta}\right)^2 dt \right)^{1/2}
\leq C \left( \int_0^\infty \frac{1}{t^{\beta+1}} \int_0^t \left| h(u) \right|^2 u^{\beta-1} du dt \right)^{1/2} \\
\leq & C \|h\|_H, \quad h \in H.\end{aligned}$$ We now show that $$\label{Lp36}
A_1(h)
= - A_2(T_{\omega,\beta} h), \quad h \in H.$$ Indeed, let $h \in H$. Since $T_{\omega,\beta} h \in H$, we can write $$\begin{aligned}
A_2(T_{\omega,\beta}h)
& = \int_0^\infty G_{P,{\mathbb{C}}}^{\lambda,\beta+1}(f)(t,x) (T_{\omega,\beta}h)(t) \frac{dt}{t}
= \int_0^\infty t^{\beta+1} \partial_t^{\beta+1} P_t^\lambda (f)(x) (T_{\omega,\beta}h)(t) \frac{dt}{t}.\end{aligned}$$ By Lemma \[Lem4\] we have that $$\label{28.2}
\partial_t^{\beta+1} P_t^\lambda(f)(x)
= e^{i \pi (\beta+1)} h_\lambda \left( y^{\beta+1} e^{-yt} h_\lambda(f)(y) \right)(x), \quad t,x \in (0,\infty).$$ Interchanging the order of integration twice we get $$\begin{aligned}
A_2(T_{\omega,\beta}h)
= & e^{i \pi (\beta+1)} \int_0^\infty t^\beta h_\lambda \left( y^{\beta+1} e^{-yt} h_\lambda(f)(y) \right)(x) (T_{\omega,\beta}h)(t)dt \\
= & e^{i \pi (\beta+1)} h_\lambda \left( h_\lambda(f)(y) y^{\beta+1}\int_0^\infty e^{-yt} t^\beta (T_{\omega,\beta}h)(t) dt \right)(x)\\
= & e^{i \pi (\beta+1)} h_\lambda \left( h_\lambda(f)(y)y^{\beta+1} \int_0^\infty e^{-yt} \int_0^t (t-s)^{\beta-1} h(t-s) \phi_\omega(s) ds dt \right)(x)\\
= & e^{i \pi (\beta+1)} h_\lambda \left( y^{\beta+2 i \omega} h_\lambda(f)(y) \int_0^\infty e^{-yu} u^{\beta-1} h(u) du \right)(x)\\
= & e^{i \pi (\beta+1)} \int_0^\infty h_\lambda \left[ y^{\beta+2 i \omega} h_\lambda(f)(y) e^{-yu} u^{\beta}\right](x) h(u) \frac{du}{u}\\
= & - \int_0^\infty h_\lambda \left[ u^{\beta} e^{i\pi \beta}y^{\beta+2 i \omega} e^{-yu} h_\lambda(f)(y) \right](x) h(u) \frac{du}{u}\\
= & - \int_0^\infty u^\beta \partial_u^\beta P_u^\lambda \left[ h_\lambda( y^{2i\omega}h_\lambda(f)(y)) \right](x) h(u) \frac{du}{u}\\
= & -A_1(h), \quad h \in H,\end{aligned}$$ and is established. Note that the interchanges in the order of integration are justified because the function $\sqrt{z} J_{\lambda-1/2}(z)$ is bounded on $(0,\infty)$ and $h_\lambda(f) \in \mathcal{S}_\lambda(0,\infty)$.
From we deduce that, for every $f \in \mathcal{S}_\lambda(0,\infty) \otimes {\mathbb{B}}$, $$\label{Lp36.5}
G_{P,{\mathbb{B}}}^{\lambda,\beta}(\Delta_\lambda^{i\omega}f)(\cdot,x)
= - G_{P,{\mathbb{B}}}^{\lambda,\beta+1}(f)(\cdot,x) \circ T_{\omega, \beta}, \quad \text{a.e. } x \in (0,\infty),$$ as elements of $L(H,{\mathbb{B}})$, the space of linear bounded operators from $H$ into ${\mathbb{B}}$.
Let $f \in \mathcal{S}_\lambda(0,\infty) \otimes {\mathbb{B}}$. Since $\Delta_\lambda^{i\omega}f \in L^p(0,\infty) \otimes \mathbb{B}$, implies that $$\label{28.1}
G_{P,{\mathbb{B}}}^{\lambda,\beta}(\Delta_\lambda^{i\omega}f)(\cdot,x)
= - G_{P,{\mathbb{B}}}^{\lambda,\beta+1}(f)(\cdot,x) \circ T_{\omega, \beta}, \quad \text{a.e. } x \in (0,\infty),$$ as elements of $\gamma(H,{\mathbb{B}})$. Moreover, according to the ideal property for $\gamma$-radonifying operators [@Nee Theorem 6.2], we get $$\Big\| G_{P,{\mathbb{B}}}^{\lambda,\beta+1}(f)(\cdot,x) \circ T_{\omega, \beta} \Big\|_{\gamma(H,{\mathbb{B}})}
\leq \| T_{\omega, \beta} \|_{L(H,H)} \Big\| G_{P,{\mathbb{B}}}^{\lambda,\beta+1}(f)(\cdot,x) \Big\|_{\gamma(H,{\mathbb{B}})}, \quad \text{a.e. } x \in (0,\infty).$$ Then, and lead to $$\begin{aligned}
\Big\| \Delta_\lambda^{i \omega}(f) \Big\|_{L^p((0,\infty),{\mathbb{B}})}
& \leq C \Big\| G_{P,{\mathbb{B}}}^{\lambda,\beta}(\Delta_\lambda^{i\omega}f) \Big\|_{L^p((0,\infty),\gamma(H,{\mathbb{B}}))}
= C \Big\| G_{P,{\mathbb{B}}}^{\lambda,\beta+1}(f)\circ T_{\omega, \beta} \Big\|_{L^p((0,\infty),\gamma(H,{\mathbb{B}}))} \\
& \leq C \Big\| G_{P,{\mathbb{B}}}^{\lambda,\beta+1}(f)\Big\|_{L^p((0,\infty),\gamma(H,{\mathbb{B}}))}
\leq C \|f\|_{L^p((0,\infty),{\mathbb{B}})}.\end{aligned}$$ Hence, $\Delta_\lambda^{i\omega}$ can be extended from $L^p(0,\infty) \otimes {\mathbb{B}}$ to $L^p((0,\infty),{\mathbb{B}})$ as a bounded operator from $L^p((0,\infty),{\mathbb{B}})$ into itself. By Proposition \[ImagBess\] we conclude that ${\mathbb{B}}$ is UMD, and the proof of Theorem \[Th1.4\] is complete.
Proof of Theorem \[Th1.5\] {#sec:Proof5}
==========================
The Bessel operator $\Delta_\lambda$ is positive in $L^2(0,\infty)$. Then, the square root $\sqrt{\Delta}_\lambda$ of $\Delta_\lambda$ is defined by $$\sqrt{\Delta}_\lambda f
= h_\lambda \left( y h_\lambda(f)\right), \quad f \in D(\sqrt{\Delta}_\lambda),$$ where, since $h_\lambda$ is an isometry in $L^2(0,\infty)$, the domain $D(\sqrt{\Delta}_\lambda)$ of $\sqrt{\Delta}_\lambda$ is the following set $$D(\sqrt{\Delta}_\lambda)
= \{ f \in L^2(0,\infty) : y h_\lambda(f) \in L^2(0,\infty)\}.$$
The Poisson semigroup $\{P_t^\lambda\}_{t>0}$ is the one generated by the operator $-\sqrt{\Delta}_\lambda$. We define $M(y)=m(y^2)$, $y \in (0,\infty)$. It is clear that the $\sqrt{\Delta}_\lambda$-multiplier associated with $M$ coincides with the $\Delta_\lambda$-multiplier defined by $m$. Since the function $M$ satisfies the conditions specified in [@Me1 Theorem 1], from the proof of [@Me1 Theorem 1] we deduce that, for every $n \in {\mathbb{N}}$, and $f \in \mathcal{S}_\lambda(0,\infty)$, $$\label{29.1}
t^{n+1} \partial_t^{n+1} P_t^\lambda \left( M(\sqrt{\Delta}_\lambda)f\right)(x)
= \frac{1}{2\pi} \int_{\mathbb{R}}\mathcal{M}_n(t,u) t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x)du, \quad t,x \in (0,\infty),$$ where $$\mathcal{M}_n(t,u)
= \int_0^\infty y^{-iu-1}M_n(t,y) dy, \quad u \in {\mathbb{R}}\text{ and } t \in (0,\infty),$$ and $$M_n(t,y)
= (ty)^n e^{-ty/2} M(y), \quad t,y \in (0,\infty).$$
We also have that, for every $n \in {\mathbb{N}}$ and $f \in \mathcal{S}_\lambda(0,\infty) \otimes {\mathbb{B}}$, $$t^{n+1} \partial_t^{n+1} P_t^\lambda \left( M(\sqrt{\Delta}_\lambda)f\right)(x)
= \frac{1}{2\pi} \int_{\mathbb{R}}\mathcal{M}_n(t,u) t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x)du, \quad t,x \in (0,\infty).$$ Moreover, according to [@Me1 Theorem 1], $M(\sqrt{\Delta}_\lambda)f \in L^p(0,\infty)\otimes {\mathbb{B}}$, $f \in \mathcal{S}_\lambda(0,\infty) \otimes {\mathbb{B}}$.
Let $n \in {\mathbb{N}}$. We define, for every $u \in {\mathbb{R}}$, the operator $$L_{n,u}(h)(t)
= \mathcal{M}_n(t,u)h(t),\quad t \in (0,\infty).$$ Since $$\sup_{\substack{ u \in {\mathbb{R}}\\ t \in (0,\infty) }} \left| \mathcal{M}_n(t,u) \right|
\leq C \|m\|_{L^\infty(0,\infty)},$$ the family of operators $\{L_{n,u}\}_{u \in {\mathbb{R}}}$ is bounded in $L(H,H)$.
Let $f \in \mathcal{S}_\lambda(0,\infty) \otimes {\mathbb{B}}$. Since $h_\lambda$ is an isometry in $L^2(0,\infty)$, and allow us to write $$t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f\right)(x)
= -\frac{1}{2} h_\lambda \left( ty e^{-ty/2} y^{iu} h_\lambda(f)(y) \right)(x), \quad t,x \in (0,\infty) \text{ and } u \in {\mathbb{R}}.$$ Then, Minkowski’s inequality leads to $$\begin{aligned}
& \left( \int_0^\infty \left\| t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2}f \right)(x) \right\|_{\mathbb{B}}^2 \frac{dt}{t} \right)^{1/2}
\leq C \int_0^\infty \|h_\lambda(f)(y)\|_{\mathbb{B}}\left( \int_0^\infty \left| t y e^{-ty/2} \right|^2 \frac{dt}{t} \right)^{1/2} dy \\
& \qquad \leq C \int_0^\infty \|h_\lambda(f)(y)\|_{\mathbb{B}}dy < \infty, \quad x \in (0,\infty) \text{ and } u \in {\mathbb{R}},\end{aligned}$$ because $h_\lambda(f) \in \mathcal{S}_\lambda(0,\infty) \otimes {\mathbb{B}}$ and the function $\sqrt{z} J_\nu(z)$ is bounded on $(0,\infty)$ when $\nu>-1/2$. We conclude that $$t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) \in \gamma(H,{\mathbb{B}}), \quad u \in {\mathbb{R}}\text{ and } x \in (0,\infty).$$ According to [@Me1 p. 642], we get $$\int_{\mathbb{R}}|\mathcal{M}_n(t,u)| \left\|t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) \right\|_{\mathbb{B}}du
\in L^p\left( (0,\infty), L^2\left( (0,\infty), dt/t \right)\right),$$ and we infer that $$\int_0^\infty \left( \int_{\mathbb{R}}|\mathcal{M}_n(t,u)| \left\|t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) \right\|_{\mathbb{B}}du\right)^2 \frac{dt}{t}
<\infty, \quad \text{a.e. } x \in (0,\infty).$$ If $h \in H$ we have that $$\begin{aligned}
& \int_0^\infty \int_{\mathbb{R}}\mathcal{M}_n(t,u) t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) h(t) \frac{dudt}{t} \\
& \qquad = \int_{\mathbb{R}}\int_0^\infty \mathcal{M}_n(t,u) t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) h(t) \frac{dtdu}{t} ,
\quad \text{a.e. } x \in (0,\infty).\end{aligned}$$ Hence, if $\{h_j\}_{j=1}^k$ is an orthonormal system in $H$ we can write $$\begin{aligned}
& \left( {\mathbb{E}}\Big\| \sum_{j=1}^k \gamma_j
\int_0^\infty \int_{\mathbb{R}}\mathcal{M}_n(t,u) t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) h_j(t) \frac{dudt}{t} \Big\|_{\mathbb{B}}^2\right)^{1/2} \\
& \qquad = \left( {\mathbb{E}}\Big\| \int_{\mathbb{R}}\sum_{j=1}^k \gamma_j
\int_0^\infty \mathcal{M}_n(t,u) t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) h_j(t) \frac{dtdu}{t} \Big\|_{\mathbb{B}}^2\right)^{1/2} \\
& \qquad \leq \int_{\mathbb{R}}\left( {\mathbb{E}}\Big\| \sum_{j=1}^k \gamma_j
\int_0^\infty \mathcal{M}_n(t,u) t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) h_j(t) \frac{dt}{t} \Big\|_{\mathbb{B}}^2\right)^{1/2} du \\
& \qquad \leq \int_{\mathbb{R}}\Big\| \mathcal{M}_n(t,u) t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x)\Big\|_{\gamma(H,{\mathbb{B}})} du,
\quad \text{a.e. } x \in (0,\infty).\end{aligned}$$ Here $\{\gamma_j\}_{j=1}^\infty$ is a sequence of independent Gaussian variables.
We conclude that, for a.e. $x \in (0,\infty)$, $$\label{30.1}
\left\| \int_{\mathbb{R}}\mathcal{M}_n(t,u) t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) du \right\|_{\gamma(H,{\mathbb{B}})}
\leq C \int_{\mathbb{R}}\left\| \mathcal{M}_n(t,u) t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) \right\|_{\gamma(H,{\mathbb{B}})} du.$$
For every $u \in {\mathbb{R}}$ we have that $$\mathcal{M}_n(t,u) t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x)
= t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) \circ L_{n,u}, \quad \text{a.e. } x \in (0,\infty),$$ in the sense of equality in $L(H,{\mathbb{B}})$. According to [@Nee Theorem 6.2] we deduce $$\begin{aligned}
\label{30.2}
& \left\| \mathcal{M}_n(t,u) t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) \right\|_{\gamma(H,{\mathbb{B}})}
\leq \|L_{n,u}\|_{L(H,H)} \left\| t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) \right\|_{\gamma(H,{\mathbb{B}})} \nonumber \\
& \qquad \leq C\sup_{t>0} |\mathcal{M}_n(t,u)| \left\| t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) \right\|_{\gamma(H,{\mathbb{B}})},
\quad \text{a.e. } x \in (0,\infty).\end{aligned}$$
Putting together , , and by taking into account Theorem \[boundedness\] and Proposition \[ImagBess\] we obtain $$\begin{aligned}
\|m(\Delta_\lambda)f\|_{L^p((0,\infty),{\mathbb{B}})}
= & \|M(\sqrt{\Delta_\lambda})f\|_{L^p((0,\infty),{\mathbb{B}})}
\leq C \left\| G_{P,{\mathbb{B}}}^{\lambda, n+1} (M(\sqrt{\Delta_\lambda})f)\right\|_{L^p((0,\infty),\gamma(H,{\mathbb{B}}))} \\
\leq & C \left\| \int_{\mathbb{R}}\mathcal{M}_n(t,u) t \partial_t P_{t/2}^\lambda \left( \Delta_\lambda^{iu/2} f \right)(x) du \right\|_{L^p((0,\infty),\gamma(H,{\mathbb{B}}))} \\
\leq & C \int_{\mathbb{R}}\sup_{t>0} |\mathcal{M}_n(t,u)| \left\| G_{P,{\mathbb{B}}}^{\lambda,1} \left( \Delta_\lambda^{iu/2} f \right) \right\|_{L^p((0,\infty),\gamma(H,{\mathbb{B}}))} du \\
\leq & C \int_{\mathbb{R}}\sup_{t>0} |\mathcal{M}_n(t,u)| \left\| \Delta_\lambda^{iu/2} f \right\|_{L^p((0,\infty),{\mathbb{B}})} du \\
\leq & C \left(\int_{\mathbb{R}}\sup_{t>0} |\mathcal{M}_n(t,u)| \left\| \Delta_\lambda^{iu/2} \right\|_{L^p((0,\infty),{\mathbb{B}}) \to L^p((0,\infty),{\mathbb{B}}) } du \right)
\|f\|_{L^p((0,\infty),{\mathbb{B}})}.\end{aligned}$$ Hence, $m(\Delta_\lambda)$ can be extended from $\mathcal{S}_\lambda(0,\infty) \otimes {\mathbb{B}}$ to $L^p((0,\infty),{\mathbb{B}})$ as a bounded operator from $L^p((0,\infty),{\mathbb{B}})$ into itself.
Proof of Theorem \[Th1.6\] {#sec:Proof6}
==========================
In order to apply Theorem \[Th1.5\] it is necessary to know nice estimations for the norm $$\| \Delta_\lambda^{i\omega} \|_{L^p((0,\infty),{\mathbb{B}}) \to L^p((0,\infty),{\mathbb{B}}) }, \quad \omega \in {\mathbb{R}}\setminus \{0\}.$$
\[Propnorm\] Let $X$ be a UMD Banach space, $\lambda >0$ and $1<p<\infty$. Then, there exists $C>0$ such that $$\| \Delta_\lambda^{i\omega} \|_{L^p((0,\infty),X) \to L^p((0,\infty),X) }
\leq C e^{\pi |\omega|}, \quad \omega \in {\mathbb{R}}.$$ Moreover, if $\lambda \geq 1$ for every $\omega \in {\mathbb{R}}\setminus \{0\}$, $\Delta_\lambda^{i\omega}$ can be extended to $L^1((0,\infty),X)$ as a bounded operator from $L^1((0,\infty),X)$ into $L^{1,\infty}((0,\infty),X)$, and $$\| \Delta_\lambda^{i\omega} \|_{L^1((0,\infty),X) \to L^{1,\infty}((0,\infty),X)}
\leq C e^{\pi |\omega|},$$ where $C>0$ does not depend on $\omega$.
Let $\omega \in {\mathbb{R}}\setminus \{0\}$. According to Proposition \[ImagBess\] the operator $\Delta_\lambda^{i\omega}$ can be extended to $L^p((0,\infty),X)$ as a bounded operator from $L^p((0,\infty),X)$ into itself. Moreover, by , for every $f \in \mathcal{S}_\lambda(0,\infty) \otimes X$, we have that $$G_{P,X}^{\lambda,1}(\Delta_\lambda^{i\omega}f)(\cdot,x)
= - G_{P,X}^{\lambda,2}(f)(\cdot,x) \circ T_{\omega}, \quad \text{a.e. } x \in (0,\infty),$$ as elements of $\gamma(H,X)$, where $$T_\omega(h)(t)
= \frac{1}{t} \int_0^t h(t-s) \frac{s^{-2 i \omega}}{{\Gamma}(1-2 i \omega)}ds, \quad h \in H.$$ As in the proof of Theorem \[Th1.4\] we can see that $$\|T_\omega\|_{L(H,H)}
\leq \frac{1}{|{\Gamma}(1-2 i \omega)|}
\leq e^{\pi |\omega|},$$ and $$\begin{aligned}
\left\| \Delta_\lambda^{i \omega} f \right\|_{L^p((0,\infty),X)}
\leq & C \left\| G_{P,X}^{\lambda,1}(\Delta_\lambda^{i \omega} f) \right\|_{L^p((0,\infty),\gamma(H,X))} \\
\leq & C e^{\pi |\omega|}\left\| G_{P,X}^{\lambda,2}( f) \right\|_{L^p((0,\infty),\gamma(H,X))}
\leq C e^{\pi |\omega|} \left\| f \right\|_{L^p((0,\infty),X)}, \quad f \in \mathcal{S}_\lambda(0,\infty) \otimes X,
\end{aligned}$$ that is, $$\label{Lp36.9}
\| \Delta_\lambda^{i\omega} \|_{L^p((0,\infty),X) \to L^p((0,\infty),X) }
\leq C e^{\pi |\omega|},$$ where $C>0$ does not depend on $\omega$.
We are going to show that $\Delta_\lambda^{i\omega}$ is a $X$-valued Calderón-Zygmund operator. According to and we have that $$\left| \partial_t W_t^\lambda(x,y) \right|
\leq C \frac{e^{-c(x-y)^2/t}}{t^{3/2}}, \quad t,x,y \in (0,\infty).$$ Then, $$\label{Lp37}
\left| K^\lambda_\omega(x,y) \right|
\leq C \int_0^\infty \frac{|t^{-i \omega}|}{|{\Gamma}(1-i \omega)|} \frac{e^{-c(x-y)^2/t}}{t^{3/2}} dt
\leq C \frac{e^{\pi |\omega|/2}}{|x-y|}, \quad x,y \in (0,\infty), \ x \neq y.$$ From we deduce that, for every $f \in \mathcal{S}_\lambda(0,\infty) \otimes X$, $$\int_0^\infty |K^\lambda_\omega(x,y)| \ |f(y)|dy < \infty, \quad x \notin \operatorname{supp}(f).$$ Hence, for each $f \in \mathcal{S}_\lambda(0,\infty) \otimes X$, implies that $$\Delta_\lambda^{i\omega} f(x)
= \int_0^\infty K^\lambda_\omega(x,y)f(y)dy, \quad \text{a.e. } x \notin \operatorname{supp}(f).$$
We can write $$\begin{aligned}
\partial_x \partial_t W_t^\lambda(x,y)
= & \partial_x \partial_t \left[ {\mathbb{W}}_t(x-y) \sqrt{2\pi} \left( \frac{xy}{2t} \right)^{1/2} I_{\lambda-1/2}\left( \frac{xy}{2t}\right)e^{-xy/2t}\right] \\
= & \partial_x \partial_t [{\mathbb{W}}_t(x-y) ]\sqrt{2\pi} \left( \frac{xy}{2t} \right)^{1/2} I_{\lambda-1/2}\left( \frac{xy}{2t}\right)e^{-xy/2t} \\
& + \partial_x [{\mathbb{W}}_t(x-y) ]\sqrt{2\pi} \partial_t \left[\left( \frac{xy}{2t} \right)^{1/2} I_{\lambda-1/2}\left( \frac{xy}{2t}\right)e^{-xy/2t} \right] \\
& + \partial_t [{\mathbb{W}}_t(x-y)] \sqrt{2\pi} \partial_x \left[\left( \frac{xy}{2t} \right)^{1/2} I_{\lambda-1/2}\left( \frac{xy}{2t}\right)e^{-xy/2t} \right] \\
& + {\mathbb{W}}_t(x-y) \sqrt{2\pi} \partial_x \partial_t \left[ \left( \frac{xy}{2t} \right)^{1/2} I_{\lambda-1/2}\left( \frac{xy}{2t}\right)e^{-xy/2t}\right] \\
= & \sum_{j=1}^4 {\mathcal{E}}_j(t,x,y), \quad t,x,y \in (0,\infty).
\end{aligned}$$ Applying and we obtain $$\begin{aligned}
& A) \qquad \partial_t \left[\left( \frac{xy}{2t} \right)^{1/2} I_\nu\left( \frac{xy}{2t}\right)e^{-xy/2t} \right]
= - \frac{xy}{2t^2} \frac{d}{dz} \left[z^{\nu + 1/2} z^{-\nu} I_\nu\left( z \right)e^{-z} \right]_{|_{z=xy/2t}} \\
& \qquad \qquad = - \frac{xy}{2t^2} \left[ (\nu+1/2)z^{\nu-1/2}z^{-\nu} I_{\nu}(z)e^{-z}
+ z^{\nu+1/2}z^{-\nu} I_{\nu+1}(z)e^{-z}
- z^{\nu+1/2}z^{-\nu} I_{\nu}(z)e^{-z} \right]_{|_{z=xy/2t}} \\
& \qquad \qquad = - \frac{1}{\sqrt{2 \pi}} \frac{xy}{2t^2} \left[ \frac{\nu+1/2}{z} \left\{ 1 + \mathcal{O}\left( \frac{1}{z}\right) \right\}
+ 1 - \frac{[\nu+1,1]}{2z} + \mathcal{O}\left( \frac{1}{z^2}\right)
- 1 + \frac{[\nu,1]}{2z} + \mathcal{O}\left( \frac{1}{z^2}\right) \right]_{|_{z=xy/2t}}\\
& \qquad \qquad = \frac{xy}{t^2} \mathcal{O}\left(\left( \frac{t}{xy}\right)^2\right), \quad t,x,y \in (0,\infty);
\end{aligned}$$ $$\begin{aligned}
& B) \qquad \partial_x \left[\left( \frac{xy}{2t} \right)^{1/2} I_\nu\left( \frac{xy}{2t}\right)e^{-xy/2t} \right]
= \frac{y}{t} \mathcal{O}\left(\left( \frac{t}{xy}\right)^2\right), \quad t,x,y \in (0,\infty); \hspace{3cm}
\end{aligned}$$ $$\begin{aligned}
& C) \qquad \partial_x \partial_t \left[\left( \frac{xy}{2t} \right)^{1/2} I_\nu\left( \frac{xy}{2t}\right)e^{-xy/2t} \right]
= \partial_x \left[ - \frac{xy}{2t^2}\frac{d}{dz}\left[ z^{\nu+1/2}z^{-\nu} I_{\nu}(z)e^{-z} \right]_{|_{z=xy/2t}} \right]\\
& \qquad \qquad = - \frac{y}{2t^2} \frac{d}{dz}\left[ z^{\nu+1/2}z^{-\nu} I_{\nu}(z)e^{-z} \right]_{|_{z=xy/2t}}
- \frac{xy^2}{4t^3} \frac{d^2}{dz^2}\left[ z^{\nu+1/2}z^{-\nu} I_{\nu}(z)e^{-z} \right]_{|_{z=xy/2t}} \\
& \qquad \qquad = \frac{y}{t^2} \mathcal{O}\left(\left( \frac{t}{xy}\right)^2\right)
- \frac{xy^2}{4t^3} \frac{d}{dz} \Big[ (\nu+1/2)z^{\nu-1/2}z^{-\nu} I_{\nu}(z)e^{-z}
+ z^{\nu+1/2}z^{-\nu} I_{\nu+1}(z)e^{-z}\\
& \qquad \qquad \quad - z^{\nu+1/2}z^{-\nu} I_{\nu}(z)e^{-z} \Big]_{|_{z=xy/2t}} \\
& \qquad \qquad = \frac{y}{t^2} \mathcal{O}\left(\left( \frac{t}{xy}\right)^2\right)
- \frac{xy^2}{4t^3} \Big[\frac{\nu^2-1/4}{z^2}\sqrt{z}I_{\nu}(z)e^{-z}
+ \frac{2\nu+2}{z}\sqrt{z}I_{\nu+1}(z)e^{-z} \\
& \qquad \qquad \quad + \sqrt{z} I_{\nu+2}(z)e^{-z}
- 2 \sqrt{z}I_{\nu+1}(z)e^{-z}
+ \sqrt{z}I_{\nu}(z)e^{-z}
-\frac{2\nu+1}{z} \sqrt{z} I_{\nu}(z)e^{-z}\Big]_{|_{z=xy/2t}}\\
& \qquad \qquad = \frac{y}{t^2} \mathcal{O}\left(\left( \frac{t}{xy}\right)^2\right)
- \frac{xy^2}{4\sqrt{2\pi}t^3} \Big[ \frac{\nu^2-1/4}{z^2} \left\{ 1 + \mathcal{O}\left( \frac{1}{z}\right)\right\}
+ \frac{2\nu+2}{z} \left\{ 1 - \frac{[\nu+1,1]}{2z} + \mathcal{O}\left( \frac{1}{z^2}\right)\right\}\\
& \qquad \qquad \quad -\frac{2\nu+1}{z}\left\{ 1 - \frac{[\nu,1]}{2z} + \mathcal{O}\left( \frac{1}{z^2}\right)\right\}
+ \left\{ 1 - \frac{[\nu+2,1]}{2z} + \frac{[\nu+2,2]}{4z^2} + \mathcal{O}\left( \frac{1}{z^3}\right)\right\} \\
& \qquad \qquad \quad - 2 \left\{ 1 - \frac{[\nu+1,1]}{2z} + \frac{[\nu+1,2]}{4z^2} + \mathcal{O}\left( \frac{1}{z^3}\right)\right\}
+ \left\{ 1 - \frac{[\nu,1]}{2z} + \frac{[\nu,2]}{4z^2} + \mathcal{O}\left( \frac{1}{z^3}\right)\right\}\Big]_{|_{z=xy/2t}} \\
& \qquad \qquad = \frac{xy^2}{t^3} \mathcal{O}\left( \left( \frac{t}{xy} \right)^3 \right), \quad t,x,y \in (0,\infty).
\end{aligned}$$ Here $\nu=\lambda-1/2$. Then, we deduce\
$\displaystyle \bullet |{\mathcal{E}}_1(t,x,y)|
\leq C \frac{e^{-c(x-y)^2/t}}{t^2}, \quad t,x,y \in (0,\infty)$,\
$\displaystyle \bullet |{\mathcal{E}}_2(t,x,y)|
\leq C \frac{e^{-c(x-y)^2/t}}{t} \frac{xy}{t^2} \frac{t^2}{(xy)^2}, \quad t,x,y \in (0,\infty)$,\
$\displaystyle \bullet |{\mathcal{E}}_3(t,x,y)|
\leq C \frac{e^{-c(x-y)^2/t}}{t^{3/2}} \frac{y}{t} \frac{t^2}{(xy)^2}, \quad t,x,y \in (0,\infty)$,\
and
$\displaystyle \bullet |{\mathcal{E}}_4(t,x,y)|
\leq C \frac{e^{-c(x-y)^2/t}}{t^{1/2}} \frac{xy^2}{t^3} \frac{t^3}{(xy)^3}, \quad t,x,y \in (0,\infty)$.\
We now estimate $$\int_0^{xy/2} |{\mathcal{E}}_j(t,x,y)| dt, \quad j=1,2,3,4.$$ Firstly, we have that $$\begin{aligned}
\int_0^{xy/2} |{\mathcal{E}}_1(t,x,y)| dt
\leq & C \int_0^\infty \frac{e^{-c(x-y)^2/t}}{t^2} dt
\leq \frac{C}{|x-y|^2}, \quad x,y \in (0,\infty), \ x \neq y,
\end{aligned}$$ and also $$\begin{aligned}
\int_0^{xy/2} |{\mathcal{E}}_2(t,x,y)| dt
\leq & C \int_0^{xy/2} \frac{e^{-c(x-y)^2/t}}{txy} dt
\leq C \int_0^\infty \frac{e^{-c(x-y)^2/t}}{t^2} dt
\leq \frac{C}{|x-y|^2}, \quad x,y \in (0,\infty), \ x \neq y.
\end{aligned}$$ To study ${\mathcal{E}}_3$ and ${\mathcal{E}}_4$ we distingue two cases $$\begin{aligned}
& \int_0^{xy/2} \left( |{\mathcal{E}}_3(t,x,y)| + |{\mathcal{E}}_4(t,x,y)| \right) dt
\leq C \int_0^{xy/2} \frac{e^{-c(x-y)^2/t}}{\sqrt{t}} \frac{dt}{x^2y} \\
& \qquad \qquad \leq \left\{ \begin{array}{l}
C \displaystyle \int_0^{xy/2} \frac{e^{-c(x-y)^2/t}}{\sqrt{t}} \frac{1}{x^2y} \left( \frac{xy}{t}\right)^{3/2} dt
\leq C \sqrt{\frac{y}{x}}\int_0^{xy/2} \frac{e^{-c(x-y)^2/t}}{t^2} dt \\
C \displaystyle \int_0^{xy/2} \frac{e^{-c(x-y)^2/t}}{\sqrt{t}} \frac{1}{x^2y} \left( \frac{xy}{t}\right)^{2} dt
\leq C y\int_0^{xy/2} \frac{e^{-c(x-y)^2/t}}{t^{5/2}} dt
\end{array}\right. \\
& \qquad \qquad \leq \left\{ \begin{array}{l}
C \displaystyle \int_0^\infty \frac{e^{-c(x-y)^2/t}}{t^2} dt
\leq \frac{C}{|x-y|^2}, \quad y<2x, \ x \in (0,\infty), \\
C \displaystyle y \int_0^\infty \frac{e^{-c(x-y)^2/t}}{t^{5/2}} dt
\leq C \frac{y}{|x-y|^3} \leq \frac{C}{|x-y|^2}, \quad y \geq 2x, \ x \in (0,\infty).
\end{array}\right.
\end{aligned}$$ Hence, we conclude that $$\label{Lp38}
\int_0^{xy/2} \left| \partial_x \partial_t W_t^\lambda(x,y) \right| dt
\leq \frac{C}{|x-y|^2}, \quad x,y \in (0,\infty), \ x \neq y.$$
According to and by taking in mind the above calculations we get $$\begin{aligned}
\left| \partial_x \partial_t W_t^\lambda(x,y) \right|
\leq & C \frac{e^{-c(x^2+y^2)/t}}{t^2} \left[ \left( \frac{xy}{t}\right)^\lambda + \frac{y}{\sqrt{t}} \left( \frac{xy}{t} \right)^{\lambda-1} \right] \\
\leq & C \frac{e^{-c(x^2+y^2)/t}}{t^2} \left( 1 + \frac{y}{\sqrt{t}} \right), \quad t,x,y \in (0,\infty), \ xy \leq 2t,
\end{aligned}$$ provided that $\lambda \geq 1$. Then, $$\begin{aligned}
\label{Lp39}
\int_{xy/2}^\infty \left| \partial_x \partial_t W_t^\lambda(x,y) \right| dt
\leq & C \int_{0}^\infty \frac{e^{-c(x^2+y^2)/t}}{t^2} \left( 1 + \frac{y}{\sqrt{t}} \right) dt \nonumber \\
\leq & C \left( \frac{1}{x^2+y^2} + \frac{y}{(x^2+y^2)^{3/2}} \right)
\leq \frac{C}{|x-y|^2} , \quad x,y \in (0,\infty), \ x \neq y.
\end{aligned}$$ From and we deduce that $$\label{Lp40}
\left| \partial_x K^\lambda_\omega(x,y)\right|
\leq C \frac{e^{\pi |\omega|/2}}{|x-y|^2}, \quad x,y \in (0,\infty), \ x \neq y.$$ Since $K^\lambda_\omega(x,y)=K^\lambda_\omega(y,x)$, $x,y \in (0,\infty)$, we also have that $$\label{Lp41}
\left| \partial_y K^\lambda_\omega(x,y)\right|
\leq C \frac{e^{\pi |\omega|/2}}{|x-y|^2}, \quad x,y \in (0,\infty), \ x \neq y.$$ By , and , $K^\lambda_\omega$ is a standard Calderón-Zygmund kernel.
By applying now the Calderón-Zygmund theory for Banach valued singular integral we obtain that the operator $\Delta_\lambda^{i\omega}$ can be extended to $L^1((0,\infty),X)$ as a bounded operator from $L^1((0,\infty),X)$ into $L^{1,\infty}((0,\infty),X)$. Moreover, , , and lead to $$\left\| \Delta_\lambda^{i \omega} \right\|_{L^{1}((0,\infty),X) \to L^{1,\infty}((0,\infty),X)}
\leq C e^{\pi |\omega|},$$ where $C>0$ does not depend on $\omega$.
\[Prop norm1\] Let ${\mathbb{H}}$ be a Hilbert space and $\lambda>0$. Then, $\|\Delta_\lambda^{i\omega}\|_{L^2((0,\infty),{\mathbb{H}})}=1$, for every $\omega \in {\mathbb{R}}\setminus \{0\}$.
We consider $f \in L^2(0,\infty)\otimes {\mathbb{H}}$, that is, $f=\sum_{j=1}^n a_j f_j$ where $a_j \in {\mathbb{H}}$ and $f_j \in L^2(0,\infty)$. By using Plancherel equality for Hankel transforms on $L^2(0,\infty)$ we can write $$\begin{aligned}
\int_0^\infty \left\| h_\lambda(f)(x)\right\|_{\mathbb{H}}^2 dx
= & \int_0^\infty \langle h_\lambda(f)(x) , h_\lambda(f)(x) \rangle_{\mathbb{H}}dx \\
= & \sum_{i,j=1}^n \langle a_i , a_j \rangle_{\mathbb{H}}\int_0^\infty h_\lambda(f_i)(x) h_\lambda(f_j)(x) dx
= \sum_{i,j=1}^n \langle a_i , a_j \rangle_{\mathbb{H}}\int_0^\infty f_i(x) f_j(x) dx \\
= & \int_0^\infty \langle f(x) , f(x) \rangle_{\mathbb{H}}dx
= \int_0^\infty \|f(x)\|_{\mathbb{H}}^2 dx.
\end{aligned}$$ Hence, $h_\lambda$ can be extended to $L^2((0,\infty),{\mathbb{H}})$ as a bounded operator from $L^2((0,\infty),{\mathbb{H}})$ into itself. Since $|y^{2i \omega}|=1$, $y \in (0,\infty)$ and $\omega \in {\mathbb{R}}\setminus \{0\}$, by we conclude that, for every $\omega \in {\mathbb{R}}\setminus \{0\}$, $\Delta_\lambda^{i\omega}$ is bounded from $L^2((0,\infty),{\mathbb{H}})$ into itself and $$\|\Delta_\lambda^{i\omega}\|_{L^2((0,\infty),{\mathbb{H}}) \to L^2((0,\infty),{\mathbb{H}})}=1.$$
Let $\omega \in {\mathbb{R}}\setminus \{0\}$ and assume that ${\mathbb{B}}=[{\mathbb{H}},X]_\theta$, where ${\mathbb{H}}$ is a Hilbert space and $X$ is a UMD space, $0<\theta<\vartheta/\pi$. Then, by using the interpolation theorem for vector-valued Lebesgue spaces [@BL Theorem 5.1.2], Propositions \[Propnorm\] and \[Prop norm1\] we deduce that, $\Delta_\lambda^{i\omega}$ is a bounded operator from $L^p((0,\infty),{\mathbb{B}})$ into itself, being $p=2/(1+\theta)$ and $$\begin{aligned}
\|\Delta_\lambda^{i\omega}\|_{L^p((0,\infty),{\mathbb{B}}) \to L^p((0,\infty),{\mathbb{B}})}
\leq & C \|\Delta_\lambda^{i\omega}\|_{L^2((0,\infty),{\mathbb{H}}) \to L^2((0,\infty),{\mathbb{H}})}^{1-\theta} \|\Delta_\lambda^{i\omega}\|_{L^1((0,\infty),X) \to L^{1,\infty}((0,\infty),X)}^{\theta}\\
\leq & C e^{2\pi(1/p-1/2)|\omega|}.\end{aligned}$$ Here $C>0$ does not depend on $\omega$.
Since $\Delta_\lambda^{i\omega}$ is selfadjoint, by using duality and that $[{\mathbb{H}},X]_\theta^*=[{\mathbb{H}}^*,X^*]_\theta$ (see [@Hy p. 1007]) we get $$\|\Delta_\lambda^{i\omega}\|_{L^{p'}((0,\infty),{\mathbb{B}}) \to L^{p'}((0,\infty),{\mathbb{B}})}
\leq C e^{2\pi(1/p-1/2)|\omega|}.$$ Hence, another interpolation leads to $$\label{Lp42}
\|\Delta_\lambda^{i\omega}\|_{L^{q}((0,\infty),{\mathbb{B}}) \to L^{q}((0,\infty),{\mathbb{B}})}
\leq C e^{2\pi(1/p-1/2)|\omega|}, \quad p \leq q \leq p'.$$
Since $m$ is a bounded holomorphic function in $\sum_\vartheta$, the function $M(y)=m(y^2)$, $y \in (0,\infty)$, is bounded and holomorphic in $\sum_{\vartheta/2}$. The proof now can be finished by proceeding as in the proof of [@Me1 Theorem 3] and by using .
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|
---
address:
- 'School of Mathematics and Statistics, University of Melbourne, 813 Swanston Street, Parkville VIC 3010, Australia'
- 'Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain. '
- 'School of Mathematics and Statistics, University of Melbourne, 813 Swanston Street, Parkville VIC 3010, Australia, and Università degli studi di Milano, Via Saldini 50, 20133 Milan, Italy, and Weierstra[ß]{} Institut für Angewandte Analysis und Stochastik, Hausvogteiplatz 11A, 10117 Berlin, Germany.'
author:
- Serena Dipierro
- María Medina
- Enrico Valdinoci
bibliography:
- 'refs.bib'
title: |
Fractional elliptic problems\
with critical growth\
in the whole of ${{\mathbb R}}^n$
---
[ ]{}
Abstract {#abstract .unnumbered}
========
This is a research monograph devoted to the analysis of a nonlocal equation in the whole of the Euclidean space. In studying this equation, we will introduce all the necessary material in the most self-contained way as possible, giving precise reference to the literature when necessary.
In further detail, we study here the following nonlinear and nonlocal elliptic equation in ${{\mathbb R}}^n$ $$(-\Delta)^s u = {\varepsilon }\,h\,u^q + u^p \ {\mbox{ in }}{{\mathbb R}}^n,$$ where $s\in(0,1)$, $n>2s$, ${\varepsilon }>0$ is a small parameter, $p=\frac{n+2s}{n-2s}$, $q\in(0,1)$, and $h\in L^1({{\mathbb R}}^n)\cap L^\infty({{\mathbb R}}^n)$. The problem has a variational structure, and this allows us to find a positive solution by looking at critical points of a suitable energy functional. In particular, in this monograph, we find a local minimum and a different solution of this functional (this second solution is found by a contradiction argument which uses a mountain pass technique, so the solution is not necessarily proven to be of mountain pass type).
One of the crucial ingredient in the proof is the use of a suitable Concentration-Compactness principle.
Some difficulties arise from the nonlocal structure of the problem and from the fact that we deal with an equation in the whole of ${{\mathbb R}}^n$ (and this causes lack of compactness of some embeddings). We overcome these difficulties by looking at an equivalent extended problem.
This monograph is organized as follows.
Chapter \[90f56rfxFFFj\] gives an elementary introduction to the techniques involved, providing also some motivations for nonlocal equations and auxiliary remarks on critical point theory.
Chapter \[intro-mono\] gives a detailed description of the class of problems under consideration (including the main equation studied in this monograph) and provides further motivations.
Chapter \[FAS\] introduces the analytic setting necessary for the study of nonlocal and nonlinear equations (this part is of rather general interest, since the functional analytic setting is common in different problems in this area).
The research oriented part of the monograph is mainly concentrated in Chapters \[ECXMII\], \[7xucjhgfgh345678\] and \[EMP:CHAP\] (as a matter of fact, Chapter \[7xucjhgfgh345678\] may also be of general interest, since it deals with a regularity theory for a general class of equations).
Introduction {#90f56rfxFFFj}
============
This research monograph deals with a nonlocal problem with critical nonlinearities. The techniques used are variational and they rely on classical variational methods, such as the Mountain Pass Theorem and the Concentration-Compactness Principle (suitably adapted, in order to fit with the nonlocal structure of the problem under consideration). The subsequent sections will give a brief introduction to the fractional Laplacian and to the variational methods exploited.
Of course, a comprehensive introduction goes far beyond the scopes of a research monograph, but we will try to let the interested reader get acquainted with the problem under consideration and with the methods used in a rather self-contained form, by keeping the discussion at the simplest possible level (but trying to avoid oversimplifications). The expert reader may well skip this initial overview and go directly to Chapter \[intro-mono\].
The fractional Laplacian {#pruzzo}
------------------------
The operator dealt with in this paper is the so-called fractional Laplacian.
For a “nice” function $u$ (for instance, if $u$ lies in the Schwartz Class of smooth and rapidly decreasing functions), the $s$ power of the Laplacian, for $s\in(0,1)$, can be easily defined in the Fourier frequency space. Namely, by taking the Fourier transform $$\hat u(\xi) = {{{\mathscrF}}} u(\xi)=
\int_{{{\mathbb R}}^n} u(x)\,e^{-2\pi i x\cdot\xi} \,dx,$$ and by looking at the Fourier Inversion Formula $$u(x) = {{{\mathscrF}}}^{-1} \hat u(x)=
\int_{{{\mathbb R}}^n} \hat u(\xi)\,e^{2\pi i x\cdot\xi} \,d\xi,$$ one notices that the derivative (say, in the $k$th coordinate direction) in the original variables corresponds to the multiplication by $2\pi i \xi_k$ in the frequency variables, that is $$\partial_k u(x) =
\int_{{{\mathbb R}}^n} 2\pi i \xi_k \,\hat u(\xi)\,e^{2\pi i x\cdot\xi} \,d\xi
= {{{\mathscrF}}}^{-1} \big( 2\pi i \xi_k \,\hat u\big).$$ Accordingly, the operator $(-\Delta)=-\sum_{k=1}^n \partial^2_{k}$ corresponds to the multiplication by $(2\pi |\xi|)^2$ in the frequency variables, that is $$-\Delta u(x) =
\int_{{{\mathbb R}}^n} (2\pi |\xi|)^2 \,\hat u(\xi)\,e^{2\pi i x\cdot\xi} \,d\xi
= {{{\mathscrF}}}^{-1} \big( (2\pi | \xi|)^2\,\hat u\big).$$ With this respect, it is not too surprising to define the power $s$ of the operator $(-\Delta)$ as the multiplication by $(2\pi |\xi|)^{2s}$ in the frequency variables, that is $$\label{DEF:1}
(-\Delta)^s u(x)
:= {{{\mathscrF}}}^{-1} \big( (2\pi | \xi|)^{2s}\,\hat u\big).$$ Another possible approach to the fractional Laplacian comes from the theory of semigroups and fractional calculus. Namely, for any $\lambda>0$, using the substitution $\tau=\lambda t$ and an integration by parts, one sees that $$\int_0^{+\infty} t^{-s-1}(e^{-\lambda t}-1)\,dt
= \Gamma(-s)\,\lambda^{s},$$ where $\Gamma$ is the Euler’s Gamma-function. Once again, not too surprising, one can define the fractional power of the Laplacian by formally replacing the positive real number $\lambda$ with the positive operator $-\Delta$ in the above formula, that is $$(-\Delta)^{s} :=
\frac{1}{\Gamma(-s)} \int_0^{+\infty} t^{-s-1}(e^{\Delta t}-1)\,dt,$$ which reads as $$\label{DEF:2}
(-\Delta)^{s} u(x) =
\frac{1}{\Gamma(-s)} \int_0^{+\infty} t^{-s-1}(e^{\Delta t}u(x)-u(x))\,dt.$$ Here above, the function $U(x,t)=e^{\Delta t}u(x)$ is the solution of the heat equation $\partial_t U=\Delta U$ with initial datum $U|_{t=0}=u$.
The equivalence between the two definitions in and can be proved by suitable elementary calculations, see e.g. [@Bucur].
The two definitions in and are both useful for many properties and they give different useful pieces of information. Nevertheless, in this monograph, we will take another definition, which is equivalent to the ones in and (at least for nice functions), but which is more flexible for our purposes. Namely, we set $$\label{DEF:3}\begin{split}
(-\Delta)^s u(x)\,&:= c_{n,s}\,PV \int_{{{\mathbb R}}^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy
\\&:=c_{n,s}\,\lim_{r\to0} \int_{{{\mathbb R}}^n\setminus B_r(x)}
\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,\end{split}$$ where $$c_{n,s}:=\frac{2^{2s}\,s\,\Gamma\left( \frac{n}{2}+s\right)}{\pi^{n/2}\,
\Gamma(1-s)}.$$ See for instance [@Bucur] for the equivalence of with and .
Roughly speaking, our preference (at least for what concerns this monograph) for the definition in lies in the following features. First of all, the definition in is more unpleasant, but geometrically more intuitive (and often somehow more treatable) than the ones in and , since it describes an incremental quotient (of differential order $2s$) weighted in the whole of ${{\mathbb R}}^n$. As a consequence, one may obtain a “rough” idea on how $(-\Delta)^s$ looks like by considering the oscillations of the original function $u$, suitably weighted.
Conversely, the definitions in and are perhaps shorter and more evocative, but they require some “hidden calculations” since they involve either the Fourier transform or the heat flow of the function $u$, rather than the function $u$ itself.
Moreover, the definition in has straightforward probabilistic interpretations (see e.g. [@Bucur] and references therein) and can be directly generalized to other singular integrodifferential kernels (of course, in many cases, even when dealing in principle with the definition in , the other equivalent definitions do provide additional results).
In addition, by taking the definition in , we do not need $u$ to be necessarily in the Schwartz Class, but we can look at weak, distributional solutions, in a similar way to the theory of classical Sobolev spaces. We refer for instance to [@DPV] for a basic discussion on the fractional Sobolev spaces and to [@SV-2] for the main functional analytic setting needed in the study of variational problems.
To complete this short introduction to the fractional Laplacian, we briefly describe a simple probabilistic motivation arising from game theory on a traced space (here, we keep the discussion at a simple, and even heuristic level, see for instance [@Bertoin], [@MOL] and the references therein for further details). The following discussion describes the fractional Laplacian occurring as a consequence of a classical random process in one additional dimension (see Figure \[BROWN\]).
![A Brownian motion on ${{\mathbb R}}^{n+1}$ and the payoff on ${{\mathbb R}}^n\times\{0\}$.[]{data-label="BROWN"}](BROWN.pdf){width="12.4cm"}
We consider a bounded and smooth domain $\Omega\subset{{\mathbb R}}^n$ and a nice (and, for simplicity, rapidly decaying) payoff function $f:{{\mathbb R}}^n\setminus\Omega\to [0,1]$. We immerse this problem into ${{\mathbb R}}^{n+1}$, by defining $\Omega_*:=\Omega
\times\{0\}$.
The game goes as follows: we start at some point of $\Omega_*$ and we move randomly in ${{\mathbb R}}^{n+1}$ by following a Brownian motion, till we hit $({{\mathbb R}}^n\setminus\Omega)\times\{0\}$ at some point $p$: in this case we receive a payoff of $f(p)$ livres.
For any $x\in{{\mathbb R}}^n$, we denote by $u(x)$ the expected value of the payoff when we start at the point $(x,0)\in{{\mathbb R}}^{n+1}$ (that is, roughly speaking, how much we expect to win if we start the game from the point $(x,0)\in\Omega\times\{0\}$). We will show that $u$ is solution of the fractional equation $$\label{EQ-LW-iu-1}
\left\{ \begin{matrix}
(-\Delta)^{1/2} u =0 & {\mbox{ in }}\Omega,\\
u =f & {\mbox{ in }}{{\mathbb R}}^n\setminus\Omega.
\end{matrix}
\right.$$ Notice that the condition $u=f$ in ${{\mathbb R}}^n\setminus\Omega$ is obvious from the construction (if we start directly at a place where a payoff is given, we get that). So the real issue is to understand the equation satisfied by $u$.
For this scope, for any $(x,y)\in{{\mathbb R}}^n\times{{\mathbb R}}={{\mathbb R}}^{n+1}$, we denote by $U(x,y)$ the expected value of the payoff when we start at the point $(x,y)$. We observe that $U(x,0)=u(x)$. Also, we define $T:=({{\mathbb R}}^n\setminus\Omega)\times\{0\}$ ($T$ is our “target” domain) and we claim that $U$ is harmonic in ${{\mathbb R}}^{n+1}\setminus T$, i.e. $$\label{EQ-LW-iu-2}
{\mbox{$\Delta U=0$ in ${{\mathbb R}}^{n+1}\setminus T$.}}$$ To prove this, we argue as follows. Fix $P\in {{\mathbb R}}^{n+1}\setminus T$ and a small ball of radius $r>0$ around it, such that $B_r(P)\subset
{{\mathbb R}}^{n+1}\setminus T$. Then, the expected value that we receive starting the game from $P$ should be the average of the expected value that we receive starting the game from another point $Q\in \partial B_r(P)$ times the probability of drifting from $Q$ to $P$. Since the Brownian motion is rotationally invariant, all the points on the sphere have the same probability of drifting towards $P$, and this gives that $$U(P) = \dashint_{\partial B_r(P)} U(Q)\,d{{{\mathscrH}}}^n(Q).$$ That is, $U$ satisfies the mean value property of harmonic functions, and this establishes .
Furthermore, since the problem is symmetric with respect to the $(n+1)$th variable we also have that $U(x,y)=U(x,-y)$ for any $x\in\Omega$ and so $$\label{EQ-LW-iu-3}
{\mbox{$\partial_y U(x,0)=0$ for any~$x\in\Omega$.}}$$ Now we take Fourier transforms in the variable $x\in{{\mathbb R}}^n$, for a fixed $y>0$. [F]{}rom , we know that $\Delta U(x,y)=0$ for any $x\in{{\mathbb R}}^n$ and $y>0$, therefore $$-(2\pi|\xi|)^2 \hat U(\xi,y)+\partial_{yy} \hat U(\xi,y)=0,$$ for any $\xi\in{{\mathbb R}}^n$ and $y>0$. This is an ordinary differential equation in $y>0$, which can be explicitly solved: we find that $$\hat U(\xi,y) = \alpha(\xi)\, e^{2\pi|\xi| y} + \beta(\xi)\, e^{-2\pi |\xi| y},$$ for suitable functions $\alpha$ and $\beta$. As a matter of fact, since $$\lim_{y\to+\infty} e^{2\pi|\xi| y}=+\infty,$$ to keep $\hat U$ bounded we have that $\alpha(\xi)=0$ for any $\xi\in{{\mathbb R}}^n$. This gives that $$\hat U(\xi,y) = \beta(\xi)\, e^{-2\pi|\xi| y}.$$ We now observe that $$\hat u(\xi) =\hat U(\xi, 0)=\beta(\xi),$$ therefore $$\hat U(\xi,y) = \hat u(\xi)\, e^{-2\pi|\xi| y}$$ and so $${{{\mathscrF}}}(\partial_y U)(\xi,y)=
\partial_y \hat U(\xi,y) = -2\pi|\xi|\,\hat u(\xi)\, e^{-2\pi|\xi| y}.$$ In particular, ${{{\mathscrF}}}(\partial_y U)(\xi,0)=-2\pi|\xi|\,\hat u(\xi)$. Hence we exploit (and we also recall ): in this way, we obtain that, for any $x\in\Omega$, $$0= \partial_y U(x,0)= - {{{\mathscrF}}}^{-1} \Big(
2\pi\,|\xi|\,\hat u(\xi)\Big) (x)= -(-\Delta)^{1/2} u(x),$$ which proves .
The Mountain Pass Theorem
-------------------------
Many of the problems in mathematical analysis deal with the construction of suitable solutions. The word “construction” is often intended in a “weak” sense, not only because the solutions found are taken in a “distributional” sense, but also because the proof of the existence of the solution is often somehow not constructive (though some qualitative or quantitative properties of the solutions may be often additionally found).
In some cases, the problem taken into account presents a variational structure, namely the desired solutions may be found as critical points of a functional (this functional is often called “energy” in the literature, though it is in many cases related more to a “Lagrangian action” from the physical point of view).
When the problem has a variational structure, it can be attacked by all the methods which aim to prove that a functional indeed possesses a critical point. Some of these methods arise as the “natural” generalizations from basic Calculus to advanced Functional Analysis: for instance, by a variation of the classical Weierstra[ß]{} Theorem, one can find solutions corresponding to local (or sometimes global) minima of the functional.
In many circumstances, these minimal solutions do not exhaust the complexity of the problem itself. For instance, the minimal solutions happen in many cases to be “trivial” (for example, corresponding to the zero solution). Or, in any case, solutions different from the minimal ones may exist, and they may indeed have interesting properties. For example, the fact that they come from a “higher energy level” may allow them to show additional oscillations, or having “directions” along which the energy is not minimized may produce some intriguing forms of instabilities.
Detecting non-minimal solutions is of course, in principle, harder than finding minimal ones, since the direct methods leading to the Weierstra[ß]{} Theorem (basically reducing to compactness and some sort of continuity) are in general not enough.
As a matter of fact, these methods need to be implemented with the aid of additional “topological” methods, mostly inspired by Morse Theory (see [@Milnor]). Roughly speaking, these methods rely on the idea that critical points prevent the energy graph to be continuously deformed by following lines of steepest descent (i.e. gradient flows).
One of the most important devices to detect critical points of non-minimal type is the so called Mountain Pass Theorem. This result can be pictorially depicted by thinking that the energy functional is simply the elevation ${{{\mathscrE}}}$ of a given point on the Earth. The basic assumption of the Mountain Pass Theorem is that there are (at least) two low spots in the landscape, for instance, the origin, which (up to translations) is supposed to lie at the sea level (say, ${{{\mathscrE}}}(0)=0$) and a far-away place $p$ which also lies at the sea level, or even below (say, ${{{\mathscrE}}}(p){\leqslant}0$).
The origin is also supposed to be surrounded by points of higher elevation (namely, there exist $r$, $a>0$ such that ${{{\mathscrE}}}(u){\geqslant}a$ if $|u| =r$). Under this assumption, any path joining the origin with $p$ is supposed to “climb up” some mountains (i.e., it has to go up, at least at level $a>0$, and then reach again the sea level in order to reach $p$).
Thus, each of the path joining $0$ to $p$ will have a highest point. If one needs to travel in “real life” from $0$ to $p$, then (s)he would like to minimize the value of this highest point, to make the effort as small as possible. This corresponds, in mathematical jargon, to the search of the value $$\label{78ghKKK}
c:=\inf_{\Gamma}\sup_{t\in[0,1]} {{{\mathscrE}}}(g(t)),$$ where $\Gamma$ is the collection of all possible path $g$ such that $g(0)=0$ and $g(1)=p$.
Roughly speaking, one should expect $c$ to be a critical value of saddle type, since the “minimal path” has a maximum in the direction “transversal to the range of mountains”, but has a minimum with respect to the tangential directions, since the competing paths reach a higher altitude.
![The function $z=x^2+y^2-4x^4-2y^4$ ($3$D plot and level sets).[]{data-label="FIG-1"}](FIG-1.pdf "fig:"){height="7.5cm"}\
![The function $z=x^2+y^2-4x^4-2y^4$ ($3$D plot and level sets).[]{data-label="FIG-1"}](FIG-1b.pdf "fig:"){height="7.5cm"}
A possible picture of the structure of this mountain pass is depicted in Figure \[FIG-1\]. On the other hand, to make the argument really work, one needs a compactness condition, in order to avoid that the critical point “drifts to infinity”.
We stress that this loss of compactness for critical points is not necessarily due to the fact that one works in complicate functional spaces, and indeed simple examples can be given even in Calculus curses, see for instance the following example taken from Exercise 5.42 in [@Chierchia]. One can consider the function of two real variables $$f(x,y)= (e^x+e^{-x^2})y^2(2-y^2)-e^{-x^2}+1.$$ By construction $f(0,0)=0$, $$\begin{aligned}
\partial_x f &=& (e^x-2x e^{-x^2})y^2(2-y^2)+2xe^{-x^2}, \\
\partial_y f &=& 2(e^x+e^{-x^2})y(2-y^2)-2(e^x+e^{-x^2})y^3\\
{\mbox{and }}\qquad
D^2f(0,0) &=&\left(
\begin{matrix}
2 & 0 \\
0 & 8
\end{matrix}
\right).\end{aligned}$$ As a consequence, the origin is a nondegenerate local minimum for $f$. In addition, $f(0,\sqrt{2})=0$, so the geometry of the mountain pass is satisfied. Nevertheless, the function $f$ does not have any other critical points except the origin. Indeed, a critical point should satisfy $$\begin{aligned}
\label{R5:LK1}
&& (e^x-2x e^{-x^2})y^2(2-y^2)+2xe^{-x^2}=0 \\
\label{R5:LK2}
{\mbox{and }}&&
2(e^x+e^{-x^2})y(2-y^2)-2(e^x+e^{-x^2})y^3=0.\end{aligned}$$ If $y=0$, then we deduce from that also $x=0$, which gives the origin. So we can suppose that $y\ne0$ and write as $$2(e^x+e^{-x^2})(2-y^2)-2(e^x+e^{-x^2})y^2=0,$$ which, after a further simplification gives $(2-y^2)-y^2=0$, and therefore $y=\pm 1$.
By inserting this into , we obtain that $$0 =
(e^x-2x e^{-x^2})+2xe^{-x^2}= e^x,$$ which produces no solutions. This shows that this example provides no additional critical points than the origin, in spite of its mountain pass structure.
The reason for this is that the critical point has somehow drifted to infinity: indeed $$\lim_{n\to+\infty} \nabla f( -n,1)=0.$$
To avoid this type of pathologies of critical points[^1] drifting to infinity, one requires an assumption that provides the compactness (up to subsequences) of “almost critical” points.
This additional compactness assumption is called in the literature “Palais-Smale condition” and requires that if a sequence $u_k$ is such that ${{{\mathscrE}}}(u_k)$ is bounded and ${{{\mathscrE}}}'(u_k)$ is infinitesimal, then $u_k$ has a convergent subsequence.
We remark that in ${{\mathbb R}}^n$ a condition of this sort is satisfied automatically for proper maps (i.e., for functions which do not take unbounded sets into bounded sets), but in functional spaces the situation is definitely more delicate. We refer to [@Mawhin-Willem] and the references therein for a throughout discussion about the Palais-Smale condition.
![The function $z=(e^x+e^{-x^2})y^2(2-y^2)-e^{-x^2}+1$ ($3$D plot and level sets).[]{data-label="FIG-2"}](FIG-2.pdf "fig:"){height="7.5cm"}\
![The function $z=(e^x+e^{-x^2})y^2(2-y^2)-e^{-x^2}+1$ ($3$D plot and level sets).[]{data-label="FIG-2"}](FIG-2b.pdf "fig:"){height="7.5cm"}
The standard version of the Mountain Pass Theorem is due to [@AmRab], and goes as follows:
\[mpt\] Let ${{{\mathscrH}}}$ be a Hilbert space and let ${{{\mathscrE}}}$ be in $C^1({{{\mathscrH}}},{{\mathbb R}})$. Suppose that there exist $u_0$, $u_1\in {{{\mathscrH}}}$ and $r>0$ such that $$\begin{aligned}
\label{1.5} && \inf_{\|u-u_0\|=r}{{{\mathscrE}}}(u)> {{{\mathscrE}}}(u_0),\\
\label{1.6}&& \|u_1-u_0\|> r \;{\mbox{ and }}\; {{{\mathscrE}}}(u_1){\leqslant}{{{\mathscrE}}}(u_0).\end{aligned}$$ Suppose also that the Palais-Smale condition holds at level $c$, with $$c:=\inf_{\Gamma}\sup_{t\in[0,1]} {{{\mathscrE}}}(g(t)),$$ where $\Gamma$ is the collection of all possible path $g$ such that $g(0)=u_0$ and $g(1)=u_1$.
Then ${{{\mathscrE}}}$ has a critical point at level $c$.
Notice that hypothesis has a strict sign, therefore it requires the existence of a real mountain (i.e., with a strictly positive elevation with respect to $u_0$) surrounding $u_0$.
Mathematically, this condition easily follows if, for instance, we previously know that $u_0$ is a strict local minimum. Unfortunately, in the applications it is not so common to have as much information, being more likely to only know that $u_0$ is just a local, possibly degenerate, minimum. Thus, a natural question arises: what happens if we can cross the mountain through a flat path? That is, what if the separating mountain range has zero altitude? Does the Mountain Pass Theorem hold in this limiting case?
The answer is yes. In [@GG] N. Ghoussub and N. Preiss refined the result of A. Ambrosetti and P. Rabinowitz to overcome this difficulty. Indeed, they proved that the conclusion in Theorem \[mpt\] holds if we replace the hypotheses by $$\begin{aligned}
\label{1.5bis}&& \inf_{\|u-u_0\|=r}{{{\mathscrE}}}(u){\geqslant}{{{\mathscrE}}}(u_0),\\
\label{1.6bis}&& \|u_1-u_0\|> r \;{\mbox{ and }}\; {{{\mathscrE}}}(u_1){\leqslant}{{{\mathscrE}}}(u_0).\end{aligned}$$ Now, the first condition is satisfied if we prove that $u_0$ is just a (not necessarily strict) local minimum. In fact, this will be the version of the Mountain Pass Theorem that we will apply in this monograph since, as we will see in Chapter \[EMP:CHAP\], our paths across the mountain will start from a point that, as far as we know, is only a local minimum.
As a matter of fact, we point out that the results in [@GG] are more general than assumptions and , and they are based on the notion of “separating set”. Namely, one says that a closed set $S$ separates $u_0$ and $u_1$ if $u_0$ and $u_1$ belong to disjoint connected components of the complement of $S$.
With this notion, it is proved that the Mountain Pass Theorem holds if there exists a closed set $F$ such that $F\cap \{{{{\mathscrE}}}{\geqslant}c\}$ separates $u_0$ and $u_1$ (see in particular Theorem (1. bis) in [@GG]).
Let us briefly observe that conditions and indeed imply the existence of a separating set. For this, we first observe that, for any path $g$ which joins $u_0$ and $u_1$, we have that $$\sup_{t\in[0,1]} {{{\mathscrE}}}(g(t)) {\geqslant}{{{\mathscrE}}}(g(0))
={{{\mathscrE}}}(u_0),$$ and so, taking the infimum, we obtain that $c{\geqslant}{{{\mathscrE}}}(u_0)$.
So we can distinguish two cases: $$\begin{aligned}
\label{primo caso} && {\mbox{either $c>{{{\mathscrE}}}(u_0)$}}\\
\label{secondo caso} && {\mbox{or $c={{{\mathscrE}}}(u_0)$}}.\end{aligned}$$ If holds true, than we choose $F$ to be the whole of the space, and we check that the set $F\cap \{{{{\mathscrE}}}{\geqslant}c\}=
\{{{{\mathscrE}}}{\geqslant}c\}$ separates $u_0$ and $u_1$.
To this goal, we first notice that $c>{{{\mathscrE}}}(u_0){\geqslant}{{{\mathscrE}}}(u_1)$, due to and , thus $u_0$ and $u_1$ belong to the complement of $\{{{{\mathscrE}}}{\geqslant}c\}$, which is $\{{{{\mathscrE}}}<c\}$. They cannot lie in the same connected component of such set, otherwise there would be a joining path $g$ all contained in $\{{{{\mathscrE}}}<c\}$, i.e. $$\sup_{t\in [0,1]}{{{\mathscrE}}} (g(t))=
\max_{t\in [0,1]}{{{\mathscrE}}} (g(t)) < c,$$ which is in contradiction with .
This proves that conditions and imply the existence of a separating set in case . Let us now deal with case . In this case, we choose $F=\{ u {\mbox{ s.t. }} \|u-u_0\|=r\}$, where $r$ is given by and . Notice that, by and , we know that ${{{\mathscrE}}}(u){\geqslant}{{{\mathscrE}}}(u_0)=c$ for any $u\in F$, and therefore $F\subseteq \{{{{\mathscrE}}}{\geqslant}c\}$. As a consequence $F\cap \{{{{\mathscrE}}}{\geqslant}c\} = F$, which is a sphere of radius $r$ which contains $u_0$ in its center and has $u_1$ in its exterior, and so it separates the two points, as desired.
The Concentration-Compactness Principle
---------------------------------------
The methods based on concentration and compactness, or compensated compactness are based on a careful analysis, which aims to recover compactness (whenever possible) from rather general conditions. These methods are indeed very powerful and find applications in many different contexts. Of course, a throughout exposition of these techniques would require a monograph in itself, so we limit ourselves to a simple description on the application of these methods in our concrete case.
As we have already noticed in the discussion about the Palais-Smale condition, one of the most important difficulties to take care when dealing with critical point theory is the possible loss of compactness.
Of course, boundedness is the first necessary requirement towards compactness, but in infinitely dimensional spaces this is of course not enough. The easiest example of loss of compactness in spaces of functions defined in the whole of ${{\mathbb R}}^n$ is provided by the translations: namely, given a (smooth, compactly supported, not identically zero) function $f$, the sequence of functions $U_k(x):=U(x-k e_1)$ is not precompact.
One possibility that avoids this possible “drifting towards infinity” of the mass of the sequence is the request that the sequence is “tight”, i.e. the amount of mass at infinity goes to zero uniformly. Of course, the appropriate choice of the norm used to measure such tightness depends on the problem considered. In our case, we are interested in controlling the weighted tail of the gradient at infinity (a formal statement about this will be given in Definition \[defTight\]).
Other tools to use in order to prove compactness often rely on boundedness of sequences in possibly stronger norms. For instance, in variational problems, one often obtains uniform energy bounds which control the sequence in a suitable Sobolev norm: for instance, one may control uniformly the $L^2$-norm of the first derivatives. This, together with the compactness of the measures (see e.g. [@evans]) implies that the squared norm of the gradient converges in the sense of measures (formal details about this will be given in Definition \[convMeasures\]).
The version of the Concentration-Compactness Principle that will be used in this monograph is due to P. L. Lions in [@L1] and [@L2], and will be explicitly recalled in Proposition \[CCP\], after having introduced the necessary functional setting.
The problem studied in this monograph {#intro-mono}
=====================================
Fractional critical problems
----------------------------
A classical topic in nonlinear analysis is the study of the existence and multiplicity of solutions for nonlinear equations. Typically, the equations under consideration possess some kind of ellipticity, which translates into additional regularity and compactness properties at a functional level.
In this framework, an important distinction arises between “subcritical” problems and “critical” ones. Namely, in subcritical problems the exponent of the nonlinearity is smaller than the Sobolev exponent, and this gives that any reasonable bound on the Sobolev seminorm implies convergence in some $L^p$-spaces: for instance, minimizing sequences, or Palais-Smale sequences, usually possess naturally a uniform bound in the Sobolev seminorm, and this endows the subcritical problems with additional compactness properties that lead to existence results via purely functional analytic methods.
The situation of critical problems is different, since in this case the exponent of the nonlinearity coincides with the Sobolev exponent and therefore no additional $L^p$-convergence may be obtained only from bounds in Sobolev spaces. As a matter of fact, many critical problems do not possess any solution. Nevertheless, as discovered in [@BN], critical problems do possess solutions once suitable lower order perturbations are taken into account. Roughly speaking, these perturbations are capable to modify the geometry of the energy functional associated to the problem, avoiding the critical points to “drift towards infinity”, at least at some appropriate energy level. Of course, to make such argument work, a careful analysis of the variational structure of the problem is in order, joint with an appropriate use of topological methods that detect the existence of the critical points of the functional via its geometric features.
Recently, a great attention has also been devoted to problems driven by nonlocal operators. In this case, the “classical” ellipticity (usually modeled by the Laplace operator) is replaced by a “long range, ferromagnetic interaction”, which penalizes the oscillation of the function (roughly speaking, the function is seen as a state parameter, whose value at a given point of the space influences the values at all the other points, in order to avoid sharp fluctuations). The ellipticity condition in this cases reduces to the validity of some sort of maximum principle, and the prototype nonlocal operators studied in the literature are the fractional powers of the Laplacian.
In this research monograph we deal with the problem $$\label{problem}
(-\Delta)^s u = {\varepsilon }\,h\,u^q + u^p \ {\mbox{ in }}{{\mathbb R}}^n,$$ where $s\in(0,1)$ and $(-\Delta)^s$ is the so-called fractional Laplacian (as introduced in Section \[pruzzo\]), that is $$\label{laplacian}
(-\Delta)^s u(x)= c_{n,s}\,PV \int_{{{\mathbb R}}^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy \
{\mbox{ for }}x\in{{\mathbb R}}^n,$$ where $c_{n,s}$ is a suitable positive constant (see [@DPV; @Silv] for the definition and the basic properties). Moreover, $n>2s$, ${\varepsilon }>0$ is a small parameter, $0<q<1$, and $p=\frac{n+2s}{n-2s}$ is the fractional critical Sobolev exponent.
Problems of this type have widely appeared in the literature. In the classical case, when $s=1$, the equation considered here arises in differential geometry, in the context of the so-called Yamabe problem, i.e. the search of Riemannian metrics with constant scalar curvature.
The fractional analogue of the Yamabe problem has been introduced in [@Chang] and was also studied in detail, see e.g. [@JIE13], [@WANG15] and the references therein.
Here we suppose that $h$ satisfies $$\begin{aligned}
&& h\in L^1(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n),\label{h1}\\
{\mbox{and }} && {\mbox{there exists a ball~$B\subset{{\mathbb R}}^n$ such that }}
\inf_B h>0. \label{h2}\end{aligned}$$ Notice that condition implies that $$\label{h0}
h\in L^r({{\mathbb R}}^n) \quad {\mbox{ for any }}r\in(1,+\infty).$$
In the classical case, that is when $s=1$ and the fractional Laplacian boils down to the classical Laplacian, there is an intense literature regarding this type of problems, see [@ABC; @AGP1; @AGP; @ALM; @AM; @AM1; @BertiM; @BN; @CW; @Cing; @Da; @Mal; @Mal2], and references therein. See also [@GMP], where the concave term appears for the first time.
In a nonlocal setting, in [@bcss] the authors deal with problem in a bounded domain with Dirichlet boundary condition. Problems related to ours have been also studied in [@SV-2; @SV-1; @SV].
We would also like to mention that the very recent literature is rich of new contributions on fractional problems related to Riemannian geometry: see for instance the articles [@NEW2a] and [@NEW2b], where a fractional Nirenberg problem is taken into account. Differently from the case treated here, the main operator under consideration in these papers is the square root of the identity plus the Laplacian (here, any fractional power is considered and no additional invertibility comes from the identity part); also, the nonlinearity treated here is different from the one considered in [@NEW2a; @NEW2b], since the lower order terms produce new phenomena and additional difficulties (moreover, the techniques in these papers have a different flavor than the ones in this monograph and are related also to the Leray-Schauder degree).
Furthermore, in [@vecchio], we find solutions to by considering the equation as a perturbation of the problem with the fractional critical Sobolev exponent, that is $$(-\Delta)^s u=u^{\frac{n+2s}{n-2s}} \quad {\mbox{ in }}{{\mathbb R}}^n.$$ Indeed, it is known that the minimizers of the Sobolev embedding in ${{\mathbb R}}^n$ are unique, up to translations and positive dilations, and non-degenerate (see [@vecchio] and references therein, and also [@NEW1] for related results; see also [@WEI] and the references therein for classical counterparts in Riemannian geometry). In particular, in [@vecchio] we used perturbation methods and a Lyapunov-Schmidt reduction to find solutions to that bifurcate[^2] from these minimizers. The explicit form of the fractional Sobolev minimizers was found in [@COZ] and it is given by $$\label{tale}
z(x):= \frac{c_\star}{\big( 1+|x|^2\big)^{\frac{n-2s}{2}}},$$ for a suitable $c_\star>0$, depending on $n$ and $s$.
In order to state our main results, we introduce some notation. We set $$[u]_{\dot H^s({{\mathbb R}}^n)}^2:= \frac{c_{n,s}}{2}\iint_{{{\mathbb R}}^{2n}}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy,$$ and we define the space $\dot{H}^s({{\mathbb R}}^n)$ as the completion of the space of smooth and rapidly decreasing functions (the so-called Schwartz space) with respect to the norm $[u]_{\dot H^s({{\mathbb R}}^n)}+\|u\|_{L^{2^*_s}({{\mathbb R}}^n)}$, where $$2^*_s=\frac{2n}{n-2s}$$ is the fractional critical exponent. Notice that we can also define $\dot{H}^s({{\mathbb R}}^n)$ as the space of measurable functions $u:{{\mathbb R}}^n\to{{\mathbb R}}$ such that the norm $[u]_{\dot H^s({{\mathbb R}}^n)}+\|u\|_{L^{2^*_s}({{\mathbb R}}^n)}$ is finite, thanks to a density result, see e.g. [@DipVAl].
Given $f\in L^\beta({{\mathbb R}}^n)$, where $\beta:=\frac{2n}{n+2s}$, we say that $u\in \dot{H}^s(\mathbb{R}^{n})$ is a (weak) solution to $(-\Delta)^s u=f$ in ${{\mathbb R}}^n$ if $$\frac{c_{n,s}}{2}\iint_{{{\mathbb R}}^{2n}}
\frac{\big( u(x)-u(y)\big)\,\big( \varphi(x)-\varphi(y)\big)}{|x-y|^{n+2s}}
\,dx\,dy=\int_{{{\mathbb R}}^n} f\,\varphi\,dx,$$ for any $\varphi\in \dot{H}^s(\mathbb{R}^{n})$.
Thus, we can state the following
\[TH1\] Let $0<q<1$. Suppose that $h$ satisfies and . Then there exists $\varepsilon_0>0$ such that for all $\varepsilon\in (0,\varepsilon_0)$ problem has at least two nonnegative solutions. Furthermore, if $h{\geqslant}0$ then the solutions are strictly positive.
This result can be seen as the nonlocal counterpart of Theorem 1.3 in [@AGP]. To prove it we will take advantage of the variational structure of the problem. The idea is first to “localize” the problem, via the extension introduced in [@CS] and consider a functional in the extended variables. More precisely, this extended functional will be introduced in the forthcoming formula . It turns out that the existence of critical points of the “extended” functional implies the existence of critical points for the functional on the trace, that is related to problem . The functional in the original variables will be introduced in , see Section \[sec:ext\] for the precise framework.
The proof of Theorem \[TH1\] is divided in two parts. More precisely, in the first part we obtain the existence of the first solution, that turns out to be a minimum for the extended functional introduced in the forthcoming Section \[sec:ext\]. Then in the second part we will find a second solution, by applying the Mountain Pass Theorem introduced in [@AmRab]. We stress, however, that this additional solution is not necessarily of mountain pass type, since, in order to obtain the necessary compactness, one adopts the contradiction assumption that no other solution exists.
Notice that in [@vecchio] we have proved that if $h$ changes sign then there exist two distinct solutions of that bifurcate from a non trivial critical manifold. Here we also show that there exists a third solution that bifurcates from $u=0$. This means that when $h$ changes sign, problem admits at least three different solutions.
Let us point out that, when $h$ changes sign, the solution $u_{1,{\varepsilon }}$ found in [@vecchio] can possibly coincide with the second solution that we construct in this monograph. It would be an interesting open problem to investigate on the Morse index of the second solution found.
So the main point is to show that the extended functional satisfies a compactness property. In particular, for the existence of the minimum, we will prove that a Palais-Smale condition holds true below a certain energy level, see Proposition \[PScond\]. Then the existence of the minimum will be ensured by the fact that the critical level lies below this threshold.
In order to show the Palais-Smale condition we will use a version of the Concentration-Compactness Principle, see Section \[sec:CC\], and for this we will borrow some ideas from [@L1; @L2]. Differently from [@bcss], here we are dealing with a problem in the whole of ${{\mathbb R}}^n$, therefore, in order to apply the Concentration-Compactness Principle, we also need to show a tightness property (see Definition \[defTight\]). Of course, fractional problems may, in principle, complicate the tightness issues, since the nonlocal interaction could produce (or send) additional mass from (or to) infinity.
As customary in many fractional problems, see [@CS], we will work in an extended space, which reduces the fractional operator to a local (but possibly singular and degenerate) one, confining the nonlocal feature to a boundary reaction problem. This functional simplification (in terms of nonlocality) creates additional difficulties coming from the fact that the extended functional is not homogeneous. Hence, we will have to deal with weighted Sobolev spaces, and so we have to prove some weighted embedding to obtain some convergences needed throughout the monograph, see Section \[sec:weighted\].
A further source of difficulty is that the exponent $q$ in is below $1$, hence the associated energy is not convex and not smooth.
In the forthcoming Section \[sec:ext\] we present the variational setting of the problem, both in the original and in the extended variables, and we state the main results of this monograph. In particular, we first introduce the material that we are going to use in order to construct the first solution, that is the minimum solution. Then, starting from this minimum, we introduce a translated functional, that we will exploit to obtain the existence of a mountain pass solution (under the contradictory assumption that no other solutions exist).
An extended problem and statement of the main results {#sec:ext}
-----------------------------------------------------
In this section we introduce the variational setting, we present a related extended problem, and we state the main results of this monograph.
Since we are looking for positive solutions, we will consider the following problem: $$\label{problem-1}
(-\Delta)^s u = {\varepsilon }h u_+^q + u_+^p \quad {\mbox{ in }}{{\mathbb R}}^n.$$ Hence, we say that $u\in\dot{H}^s({{\mathbb R}}^n)$ is a (weak) solution to if for every $v\in\dot{H}^s({{\mathbb R}}^n)$ we have $$\begin{aligned}
&& \frac{c_{n,s}}{2}\iint_{{{\mathbb R}}^{2n}}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy \\&&=
\int_{{{\mathbb R}}^n}h(x) u_+^{q}(x)v(x)\,dx + \int_{{{\mathbb R}}^n}u_+^p(x)v(x).\end{aligned}$$ It turns out that if $u$ is a solution to , then it is nonnegative in ${{\mathbb R}}^n$ (see the forthcoming Proposition \[prop:pos\], and also Section \[sec:positivity\] for the discussion about the positivity of the solutions). Therefore, $u$ is also a solution of .
Notice that problem has a variational structure. Namely, solutions to can be found as critical points of the functional $f_{\varepsilon }:\dot{H}^s({{\mathbb R}}^n)\rightarrow{{\mathbb R}}$ defined by $$\begin{split}\label{f giu}
f_{\varepsilon }(u) :=\,&\frac{c_{n,s}}{4}\iint_{{{\mathbb R}}^{2n}}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy \\
& \quad -\frac{{\varepsilon }}{q+1}\,\int_{{{\mathbb R}}^n}h(x)\,u_+^{q+1}(x)\,dx
-\frac{1}{p+1}\,\int_{{{\mathbb R}}^n}u_+^{p+1}(x)\,dx.
\end{split}$$ However, instead of working with this framework derived from Definition \[laplacian\] of the Laplacian, we will consider the extended operator given by [@CS], that allows us to transform a nonlocal problem into a local one by adding one variable.
For this, we will denote by $\mathbb{R}^{n+1}_+:=\mathbb{R}^n\times(0,+\infty)$. Also, for a point $X\in\mathbb{R}^{n+1}_+$, we will use the notation $X=(x,y)$, with $x\in\mathbb{R}^n$ and $y>0$.
Moreover, for $x\in{{\mathbb R}}^{n}$ and $r>0$, we will denote by $B_r(x)$ the ball in ${{\mathbb R}}^n$ centered at $x$ with radius $r$, i.e. $$B_r(x):=\{x'\in{{\mathbb R}}^n : |x-x'|<r\},$$ and, for $X\in{{\mathbb R}}^{n+1}_+$ and $r>0$, $B^+_r(X)$ will be the ball in ${{\mathbb R}}^{n+1}_+$ centered at $X$ with radius $r$, that is $$B_r^+(X):=\{X'\in{{\mathbb R}}^{n+1}_+ : |X-X'|<r\}.$$
Now, given a function $u:{{\mathbb R}}^n\to{{\mathbb R}}$, we associate a function $U$ defined in ${{\mathbb R}}^{n+1}_+$ as $$\label{poisson}
U(\cdot,z)=u*P_s(\cdot,z), \quad {\mbox{ where }}P_s(x,z):=c_{n,s}\frac{z^{2s}}{(|x|^2+z^2)^{(n+2s)/2}}.$$ Here $c_{n,s}$ is a normalizing constant depending on $n$ and $s$.
Set also $a:=1-2s$, and $$\label{extNorm}
[U]_a^*:=\left(\kappa_s\int_{\mathbb{R}^{n+1}}{y^a|\nabla U|^2\,dX}\right)^{1/2},$$ where $\kappa_s$ is a normalization constant. We define the spaces $$\dot{H}^s_a(\mathbb{R}^{n+1}):=\overline{C_0^\infty(\mathbb{R}^{n+1})}^{[\cdot]_a^*},$$ and $$\begin{split}\label{D123}
\dot{H}^s_a(\mathbb{R}^{n+1}_+):=\{&U:=\tilde{U}|_{\mathbb{R}^{n+1}_+}\mbox{ s.t. }\tilde{U}\in \dot{H}^s_a(\mathbb{R}^{n+1}),\\
&\tilde{U}(x,y)=\tilde{U}(x,-y)\hbox{ a.e. in }\mathbb{R}^n\times\mathbb{R}\},
\end{split}$$ endowed with the norm $$[U]_a:=\left(\kappa_s\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U|^2\,dX}\right)^{1/2}.$$ From now on, for simplicity, we will neglect the dimensional constants $c_{n,s}$ and $\kappa_s$. It is known that finding a solution $u\in \dot{H}^s(\mathbb{R}^n)$ to a problem $$(-\Delta)^su=f(u)\quad\hbox{ in }\mathbb{R}^n$$ is equivalent to find $U\in\dot{H}^s_a(\mathbb{R}^{n+1}_+)$ solving the local problem $$\begin{cases}
\hbox{div}(y^a\nabla U)=0\quad\hbox{ in }\mathbb{R}^{n+1}_+,\\
-\displaystyle \lim_{y\rightarrow 0^+}y^a\,\frac{\partial U}{\partial\nu}=f(u),
\end{cases}$$ and that this extension is an isometry between $\dot{H}^s(\mathbb{R}^n)$ and $\dot{H}^s_a(\mathbb{R}^{n+1}_+)$ (again up to constants), that is, $$\label{equivNorms}
[U]_a=[u]_{\dot{H}^s(\mathbb{R}^n)},$$ where we make the identification $u(x)=U(x,0)$, with $U(x,0)$ understood in the sense of traces (see e.g. [@CS] and [@Bucur]).
Also, we recall that the Sobolev embedding in $\dot{H}^s(\mathbb{R}^n)$ gives that $$S\,\|u\|^2_{L^{2^*_s}(\mathbb{R}^n)}{\leqslant}[u]^2_{\dot{H}^s(\mathbb{R}^n)},$$ where $S$ is the usual constant of the Sobolev embedding of $\dot{H}^s(\mathbb{R}^n)$, see for instance Theorem 6.5 in [@DPV]. As a consequence of this and we have the following result.
\[traceIneq\] Let $U\in \dot{H}^s_a(\mathbb{R}^{n+1}_+)$. Then, $$\label{TraceIneq}
S\,\|U(\cdot,0)\|^2_{L^{2^*_s}(\mathbb{R}^n)}{\leqslant}[U]^2_a.$$
Therefore, we can reformulate problem as $$\label{ExtendedProblem}
\begin{cases}
\hbox{div}(y^a\nabla U)=0\quad\hbox{ in }\mathbb{R}^{n+1}_+,\\
-\displaystyle \lim_{y\rightarrow 0^+}y^a\,\frac{\partial U}{\partial\nu}=\varepsilon h u_+^q+u_+^p.
\end{cases}$$ In particular, we will say that $U\in\dot{H}^s_a(\mathbb{R}^{n+1}_+)$ is a (weak) solution of problem if $$\label{weak sol}
\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla U,\nabla \varphi\rangle\,dX}
=\int_{\mathbb{R}^n}{(\varepsilon h(x)U_+^q(x,0)+U_+^p(x,0))\varphi(x,0)\,dx},$$ for every $\varphi\in\dot{H}^s_a(\mathbb{R}^{n+1}_+)$. Likewise, the energy functional associated to the problem is $$\begin{split}\label{f ext}
{{\mathscrF}}_\varepsilon(U):=&
\frac{1}{2}\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U|^2\,dX}
-\frac{\varepsilon}{q+1}\int_{\mathbb{R}^n}{h(x)U_+^{q+1}(x,0)\,dx}\\
&\quad -\frac{1}{p+1}\int_{\mathbb{R}^n}{U_+^{p+1}(x,0)\,dx}.
\end{split}$$ Notice that for any $U,V\in\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$ we have $$\begin{split}\label{pqoeopwoegi}
& \langle {{\mathscrF}}_\varepsilon'(U),V\rangle =
\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U, \nabla V\rangle\,dX \\
&\quad -
{\varepsilon }\int_{{{\mathbb R}}^n}h(x)U_+^{q}(x,0)\,V(x,0)\,dx -
\int_{{{\mathbb R}}^n}U_+^p(x,0)\,V(x,0)\,dx.
\end{split}$$ Hence, if $U$ is a critical point of ${{\mathscrF}}_{\varepsilon }$, then it is a weak solution of , according to . Therefore $u:=U(\cdot,0)$ is a solution to .
Moreover, if $U$ is a minimum of ${{\mathscrF}}_\varepsilon$, then $u(x):=U(x,0)$ is a minimum of $f_\varepsilon$, thanks to , and so $u$ is a solution to problem .
In this setting, we can prove the existence of a first solution of problem , and consequently of problem .
\[MINIMUM\] Let $0<q<1$ and suppose that $h$ satisfies and . Then, there exists $\varepsilon_0>0$ such that ${{\mathscrF}}_\varepsilon$ has a local minimum $U_\varepsilon\neq 0$, for any ${\varepsilon }<{\varepsilon }_0$. Moreover, $U_\varepsilon\rightarrow 0$ in $\dot{H}^s_a(\mathbb{R}^{n+1}_+)$ when $\varepsilon\rightarrow 0$.
We now set $u_{\varepsilon }:=U_{\varepsilon }(\cdot,0)$, where $U_{\varepsilon }$ is the local minimum of ${{\mathscrF}}_{\varepsilon }$ found in Theorem \[MINIMUM\]. Then, according to , $u_{\varepsilon }$ is a local minimum of $f_{\varepsilon }$, and so a solution to .
Notice that, again by , $$[u_{\varepsilon }]_{\dot{H}^s({{\mathbb R}}^n)}= [U_{\varepsilon }]_a\to 0 \quad {\mbox{ as }}{\varepsilon }\to 0.$$ In this sense, the solution $u_{\varepsilon }$ obtained by minimizing the functional bifurcates from the solution $u=0$.
Furthermore, $u_{\varepsilon }$ is nonnegative, and thus $u_{\varepsilon }$ is a true solution of . Indeed, we can prove the following:
\[prop:pos\] Let $u\in\dot{H}^s({{\mathbb R}}^n)$ be a nontrivial solution of and let $U$ be its extension, according to . Then, $u{\geqslant}0$ and $U>0$.
We set $u_-(x):=-\min\{u(x),0\}$, namely $u_-$ is the negative part of $u$, and we claim that $$\label{u meno}
u_-=0.$$ For this, we multiply by $u_-$ and we integrate over ${{\mathbb R}}^n$: we obtain $$\int_{{{\mathbb R}}^n}(-\Delta)^su\,u_-\,dx=\int_{{{\mathbb R}}^n}\left({\varepsilon }h(x)u_+^q+u_+^p\right)u_-\,dx=0.$$ Hence, by an integration by parts we get $$\label{negggg-1}
\iint_{{{\mathbb R}}^{2n}}\frac{(u(x)-u(y))(u_-(x)-u_-(y))}{|x-y|^{n+2s}}\,dx\,dy=0.$$ Now, we observe that $$\label{neggg}
(u(x)-u(y))(u_-(x)-u_-(y)){\geqslant}|u_-(x)-u_-(y)|^2.$$ Indeed, if both $u(x){\geqslant}0$ and $u(y){\geqslant}0$ and if both $u(x){\leqslant}0$ and $u(y){\leqslant}0$ then the claim trivially follows. Therefore, we suppose that $u(x){\geqslant}0$ and $u(y){\leqslant}0$ (the symmetric situation is analogous). In this case $$\begin{aligned}
&& (u(x)-u(y))(u_-(x)-u_-(y))= -(u(x)-u(y))u(y)\\
&&\qquad\quad =-u(x)u(y)+|u(y)|^2{\geqslant}|u(y)|^2
=|u_-(x)-u_-(y)|^2, \end{aligned}$$ which implies .
From and , we obtain that $$\iint_{{{\mathbb R}}^{2n}}\frac{|u_-(x)-u_-(y)|^2}{|x-y|^{n+2s}}\,dx\,dy{\leqslant}0,$$ and this implies , since $u\in \dot{H}^s({{\mathbb R}}^n)$. Hence $u{\geqslant}0$. This implies that $U>0$, being a convolution of $u$ with a positive kernel.
We can also prove the existence of a second solution of problem , and consequently of problem .
\[TH:MP\] Let $0<q<1$ and suppose that $h$ satisfies and . Then, there exists $\varepsilon_0>0$ such that ${{\mathscrF}}_\varepsilon$ has a second solution $\overline{U}_{\varepsilon}\neq 0$, for any ${\varepsilon }<{\varepsilon }_0$.
To prove the existence of a second solution of problem we consider a translated functional. Namely, we let $U_\varepsilon$ be the local minimum of the functional (already found in Theorem \[MINIMUM\]), and we consider the functional ${{\mathscrI}}_\varepsilon:\dot{H}^s_a(\mathbb{R}^{n+1}_+)\rightarrow \mathbb{R}$ defined as $$\label{def I}
{{\mathscrI}}_\varepsilon(U)=
\frac{1}{2}\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U|^2\,dX}
-\int_{\mathbb{R}^n}{G(x,U(x,0))\,dx},$$ where $$G(x,U):=\int_0^{U}{g(x,t)\,dt},$$ and $$\label{def g small}
g(x,t):=
\begin{cases}
\varepsilon h(x)((U_\varepsilon +t)^q-U_\varepsilon^q )+(U_\varepsilon +t)^p-U_\varepsilon^p,\;\hbox{ if }t{\geqslant}0,\\
0\;\hbox{ if }t< 0.
\end{cases}$$ Explicitly, $$\begin{split}\label{def G}
G(x,U)=&\,\frac{{\varepsilon }\,h(x)}{q+1}
\left((U_{\varepsilon }+U_+)^{q+1}-U_{\varepsilon }^{q+1}\right)
-{\varepsilon }\,h(x)\,U_{\varepsilon }^q U_+ \\
&\qquad +\frac{1}{p+1}\left((U_{\varepsilon }+U_+)^{p+1}-U_{\varepsilon }^{p+1}\right)
-U_{\varepsilon }^pU_+.
\end{split}$$ Moreover, for any $U,V\in\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$, we have that $$\label{DER}
\langle {{\mathscrI}}_\varepsilon'(U),V\rangle =
\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U, \nabla V\rangle\,dX -
\int_{{{\mathbb R}}^n}g(x,U(x,0))\,V(x,0)\,dx.$$ Notice that a critical point of is a solution to the following problem $$\label{alksjaksfhhf}
\begin{cases}
\hbox{div}(y^a\nabla U)=0\quad\hbox{ in }\mathbb{R}^{n+1}_+,\\
-\displaystyle \lim_{y\rightarrow 0^+}y^a\,\frac{\partial U}{\partial\nu}=g(x,U(x,0)).
\end{cases}$$ One can prove that a solution $U$ to this problem is positive, as stated in the forthcoming Lemma \[lemma:max\]. Therefore, $\overline{U}:=U_{\varepsilon }+U>0$, thanks to Proposition \[prop:pos\]. Also, $\overline{U}$ will be the second solution of , and so $\overline{u}:=\overline{U}(\cdot,0)$ will be the second solution to .
\[lemma:max\] Let $U\in\dot{H}_a^s(\mathbb{R}^{n+1}_+)$, $U\neq0$, be a solution to . Then $U$ is positive.
We first observe that, if $U$ is a solution to , then $u:=U(\cdot,0)$ is a solution of $$\label{alksjaksfhhf-1}
(-\Delta)^su=g(x,u) \quad {\mbox{ in }}{{\mathbb R}}^n.$$ Now, we set $u_-(x):=-\min\{u(x),0\}$, namely $u_-$ is the negative part of $u$, and we claim that $$\label{u meno-1}
u_-=0.$$ For this, we multiply by $u_-$ and we integrate over ${{\mathbb R}}^n$: we obtain $$\int_{{{\mathbb R}}^n}(-\Delta)^su\,u_-\,dx=\int_{{{\mathbb R}}^n}g(x,u)\,u_-\,dx.$$ Recalling the definition of $g$ in , we have that $$\int_{{{\mathbb R}}^n}g(x,u)\,u_-\,dx=0.$$ Hence, by an integration by parts we get $$\label{negggg-12}
\iint_{{{\mathbb R}}^{2n}}\frac{(u(x)-u(y))(u_-(x)-u_-(y))}{|x-y|^{n+2s}}\,dx\,dy=0.$$ Now, we observe that $$\label{neggg00}
(u(x)-u(y))(u_-(x)-u_-(y)){\geqslant}|u_-(x)-u_-(y)|^2.$$ Indeed, if both $u(x){\geqslant}0$ and $u(y){\geqslant}0$ and if both $u(x){\leqslant}0$ and $u(y){\leqslant}0$ then the claim trivially follows. Therefore, we suppose that $u(x){\geqslant}0$ and $u(y){\leqslant}0$ (the symmetric situation is analogous). In this case $$\begin{aligned}
&& (u(x)-u(y))(u_-(x)-u_-(y))= -(u(x)-u(y))u(y)\\
&&\qquad\quad =-u(x)u(y)+|u(y)|^2{\geqslant}|u(y)|^2
=|u_-(x)-u_-(y)|^2, \end{aligned}$$ which implies .
From and , we obtain that $$\iint_{{{\mathbb R}}^{2n}}\frac{|u_-(x)-u_-(y)|^2}{|x-y|^{n+2s}}\,dx\,dy{\leqslant}0,$$ and this implies , since $u\in H^s({{\mathbb R}}^n)$. Hence $u{\geqslant}0$. This implies that $U>0$, being a convolution of $u$ with a positive kernel.
The next sections will be devoted to the proof of Theorems \[MINIMUM\] and \[TH:MP\].
More precisely, for this goal some preliminary material from functional analysis is needed. The main analytic tools are contained in Chapter \[FAS\]. Namely, since we will work with an extended functional (that also contains terms with weighted Sobolev norms), we devote Section \[sec:weighted\] to show some weighted Sobolev embeddings and Section \[sec:CC\] to prove a suitable Concentration-Compactness Principle.
The existence of a minimal solution is discussed in Chapter \[ECXMII\]. In particular, in Section \[sec:conv\] we deal with some convergence results, that we need in the subsequent Section \[sec:PS\], where we show that under a given level the Palais-Smale condition holds true for the extended functional. Then, in Section \[concaveFirstSol\] we complete the proof of Theorem \[MINIMUM\].
In Chapter \[7xucjhgfgh345678\], we discuss some regularity and positivity issues about the solution that we constructed. More precisely, in Section \[sec:reg\] we show some regularity results, and in Section \[sec:positivity\] we prove the positivity of the solutions to , making use of a strong maximum principle for weak solutions.
Then, in Chapter \[EMP:CHAP\] we deal with the existence of the mountain pass solution (under the contradictory assumption that the solution is unique). We first show, in Section \[sec:exi\], that the translated functional introduced in Section \[sec:ext\] has $U=0$ as a local minimum (notice that this is a consequence of the fact that we are translating the original functional with respect to its local minimum).
Sections \[sec:prelim\] and \[sec:prelim2\] are devoted to some preliminary results. We will exploit these basic lemmata in the subsequent Section \[sec:PS MP\], where we prove that the above-mentioned translated functional satisfies a Palais-Smale condition.
In Section \[sec:BMMV\] we estimate the minimax value along a suitable path (roughly speaking, the linear path constructed along a suitably cut-off minimizer of the fractional Sobolev inequality). This estimate is needed to exploit the Mountain Pass Theorem via the convergence of the Palais-Smale sequences at appropriate energy levels. With this, in Section \[sec:proof\] we finish the proof of Theorem \[TH:MP\].
Functional analytical setting {#FAS}
=============================
Weighted Sobolev embeddings {#sec:weighted}
---------------------------
For any $r\in(1,+\infty)$, we denote by $L^r({{\mathbb R}}^{n+1}_+,y^a)$ the weighted[^3] Lebesgue space, endowed with the norm $$\|U\|_{L^r({{\mathbb R}}^{n+1}_+,y^a)}:=\left(\int_{{{\mathbb R}}^{n+1}_+} y^a |U|^r\,dX\right)^{1/r}.$$ The following result shows that $\dot{H}^s_a(\mathbb{R}^{n+1}_+)$ is continuously embedded in $L^{2\gamma}({{\mathbb R}}^{n+1}_+,y^a)$.
\[WeightedSob\] There exists a constant $\hat{S}>0$ such that for all $U\in \dot{H}_a^s(\mathbb{R}^{n+1}_+)$ it holds $$\label{SobIneq}
\left(\int_{\mathbb{R}^{n+1}_+}{y^a|U|^{2\gamma}\,dX}\right)^{1/2\gamma}{\leqslant}\hat{S}\left(\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U|^2\,dX}\right)^{1/2},$$ where $\gamma=1+\dfrac{2}{n-2s}$.
Let us first prove the result for $U\in C_0^\infty(\mathbb{R}^{n+1})$. If $s\in(0,1/2)$, inequality is easily deduced from Theorem 1.3 of [@CR]. By a density argument, we obtain that inequality holds for any function $U\in\dot{H}^s_a(\mathbb{R}^{n+1}_+)$. Indeed, if $U\in\dot{H}^s_a(\mathbb{R}^{n+1}_+)$, then there exists a sequence of functions $\{U_k\}_{k\in\mathbb{N}}\in C^\infty_0({{\mathbb R}}^{n+1})$ such that $U_k$ converges to some $\tilde{U}$ in $\dot{H}^s_a(\mathbb{R}^{n+1})$ as $k\to\infty$, where $U=\tilde{U}$ in $\mathbb{R}^{n+1}_+$ and $\tilde{U}$ is even with respect to the $(n+1)$-th variable. Hence, for any $k$, we have $$\begin{split}\label{sobUk}
\left(\int_{\mathbb{R}^{n+1}_+}{y^a|U_k|^{2\gamma}\,dX}\right)^{1/2\gamma}&{\leqslant}\hat{S}\left(\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U_k|^2\,dX}\right)^{1/2}\\
&{\leqslant}\hat{S}\left(\int_{\mathbb{R}^{n+1}}{y^a|\nabla U_k|^2\,dX}\right)^{1/2}.
\end{split}$$ Moreover, given two functions of the approximating sequence, there holds $$\begin{split}
\left(\int_{\mathbb{R}^{n+1}_+}{y^a|U_k-U_m|^{2\gamma}\,dX}\right)^{1/2\gamma}
&{\leqslant}\hat{S}\left(\int_{\mathbb{R}^{n+1}}{y^a|\nabla (U_k-U_m)|^2\,dX}\right)^{1/2}\rightarrow 0,
\end{split}$$ and thus, up to a subsequence, $$\begin{split}
U_k\rightarrow \tilde{U}&\hbox{ in }L^{2\gamma}(\mathbb{R}^{n+1}_+,y^a),\\
U_k\rightarrow \tilde{U}&\hbox{ a.e. in }\mathbb{R}^{n+1}_+.
\end{split}$$ Hence, by Fatou’s Lemma and we get $$\begin{split}\label{density}
\left(\int_{\mathbb{R}^{n+1}_+}{y^a|U|^{2\gamma}\,dX}\right)^{1/2\gamma}&=\left(\int_{\mathbb{R}^{n+1}_+}{y^a|\tilde{U}|^{2\gamma}\,dX}\right)^{1/2\gamma}\\
&{\leqslant}\lim_{k\to+\infty}\left(\int_{\mathbb{R}^{n+1}_+}{y^a|U_k|^{2\gamma}\,dX}\right)^{1/2\gamma}\\
&{\leqslant}\lim_{k\to+\infty}
\hat{S}\left(\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U_k|^2\,dX}\right)^{1/2}\\
&{\leqslant}\lim_{k\to+\infty}
\hat{S}\left(\int_{\mathbb{R}^{n+1}}{|y|^a|\nabla U_k|^2\,dX}\right)^{1/2}\\
&=\hat{S}\left(\int_{\mathbb{R}^{n+1}}{|y|^a|\nabla \tilde{U}|^2\,dX}\right)^{1/2}\\
&=\hat{S}\left(2\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U|^2\,dX}\right)^{1/2},
\end{split}$$ which shows that Proposition \[WeightedSob\] holds true for any function $U\in\dot{H}^s_a(\mathbb{R}^{n+1}_+)$, up to renaming $\hat{S}$.
On the other hand, the case $s=\frac{1}{2}$ corresponds to the classical Sobolev inequality, so we can now concentrate on the range $s\in (1/2,1)$, that can be derived from Theorem 1.2 of [@FKS] by arguing as follows.
Let us denote $$w(X):=|y|^a.$$ Thus, it can be checked that $$\label{Muck}
w\in A_q \hbox{ for every }q\in (2-2s,2],$$ where $A_q$ denotes the class of Muckenhoupt weights of order $q$. Since in particular $w\in A_2$, by Theorem 1.2 of [@FKS], we know that there exist positive constants $C$ and $\delta$ such that for all balls $B_R\subset \mathbb{R}^{n+1}$, all $u\in C_0^\infty(B_R)$ and all $\gamma$ satisfying $1{\leqslant}\gamma{\leqslant}\frac{n+1}{n}+\delta$, one has $$\label{ineqFKS}
\left(\frac{1}{w(B_R)}\int_{B_R}{|U|^{2\gamma}w\,dX}\right)^{1/2\gamma}{\leqslant}CR\left(\frac{1}{w(B_R)}\int_{B_R}{|\nabla U|^2w\,dX}\right)^{1/2}.$$ In particular, it yields $$\begin{aligned}
w(B_R)&=&\int_{B_R}{|y|^a\,dX}=\int_{|x|^2+y^2{\leqslant}R^2}{|y|^a\,dx\,dy}\\
&=&\int_{|\eta|^2+\xi^2{\leqslant}1}{|\xi|^a R^{n+1+a}\,d\xi\,d\eta}
= CR^{a+n+1}=CR^{2-2s+n},\end{aligned}$$ with $C$ independent of $R$. Thus, $$Rw(B_R)^{\frac{1}{2\gamma}-\frac{1}{2}}= CR^{1+\frac{(1-\gamma)(2-2s+n)}{2\gamma}},$$ and plugging this into we get $$\left(\int_{B_R}{|U|^{2\gamma}w\,dX}\right)^{1/2\gamma}{\leqslant}CR^{1+\frac{(1-\gamma)(2-2s+n)}{2\gamma}}\left(\int_{B_R}{|\nabla U|^2w\,dX}\right)^{1/2},$$ where $C$ is a constant independent of $R$. In particular, if we set $\gamma=1+\frac{2}{n-2s}$, then $$1+\frac{(1-\gamma)(2-2s+n)}{2\gamma}=0,$$ and the inequality holds for every ball with the same constant. It remains to check that this value of $\gamma$ is under the hypotheses of Theorem 1.2 of [@FKS], that is, $1{\leqslant}\gamma{\leqslant}\frac{n+1}{n}+\delta$. Keeping track of $\delta$ in [@FKS], this condition actually becomes $$1{\leqslant}\gamma{\leqslant}\frac{n+1}{n+1-\frac{2}{q}},$$ for every $q<2$ such that $w\in A_q$. Thus, by , we can choose any $q\in (2-2s,2)$. Since $\gamma$ is clearly greater than $1$, we have to prove the upper bound, that is, $$1+\frac{2}{n-2s}{\leqslant}\frac{n+1}{n+1-\frac{2}{q}},$$ but this is equivalent to ask $$q{\leqslant}\frac{n-2s+2}{n+1}.$$ Since we can choose $q$ as close as we want to $2-2s$, this inequality will be true whenever $$2-2s<\frac{n-2s+2}{n+1},$$ which holds if and only if $s>\frac{1}{2}$. Summarizing, we have that $$\left(\int_{B_R}{|y|^a|U|^{2\gamma}\,dX}\right)^{1/2\gamma}{\leqslant}C\left(\int_{B_R}{|y|^a|\nabla U|^2\,dX}\right)^{1/2},$$ where $\gamma=1+\frac{2}{n-2s}$ and $C$ is a constant independent of the domain. Choosing $R$ large enough, it yields $$\left(\int_{\mathbb{R}^{n+1}}{|y|^a|U|^{2\gamma}\,dX}\right)^{1/2\gamma}{\leqslant}C\left(\int_{\mathbb{R}^{n+1}}{|y|^a|\nabla U|^2\,dX}\right)^{1/2},$$ Consider now $U\in\dot{H}^s_a(\mathbb{R}^{n+1}_+)$. We perform the same density argument as in the case $s\in (0,1/2)$, with the only difference that instead of we have $$\begin{split}
\left(\int_{\mathbb{R}^{n+1}_+}{y^a|U|^{2\gamma}\,dX}\right)^{1/2\gamma}&=\left(\int_{\mathbb{R}^{n+1}_+}{y^a|\tilde{U}|^{2\gamma}\,dX}\right)^{1/2\gamma}\\
&{\leqslant}\lim_{k\to+\infty}\left(\int_{\mathbb{R}^{n+1}_+}{y^a|U_k|^{2\gamma}\,dX}\right)^{1/2\gamma}\\
&{\leqslant}\lim_{k\to+\infty}\left(\int_{\mathbb{R}^{n+1}}{|y|^a|U_k|^{2\gamma}\,dX}\right)^{1/2\gamma}\\
&{\leqslant}\lim_{k\to+\infty}
\hat{S}\left(\int_{\mathbb{R}^{n+1}}{|y|^a|\nabla U_k|^2\,dX}\right)^{1/2}\\
&=\hat{S}\left(\int_{\mathbb{R}^{n+1}}{|y|^a|\nabla \tilde{U}|^2\,dX}\right)^{1/2}\\
&=\hat{S}\left(2\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U|^2\,dX}\right)^{1/2}.\qedhere
\end{split}$$
We also show a compactness result that we will need in the sequel. More precisely, we prove that $\dot{H}^s_a(\mathbb{R}^{n+1}_+)$ is locally compactly embedded in $L^2({{\mathbb R}}^{n+1}_+,y^a)$. The precise statement goes as follows:
\[lemma:compact\] Let $R>0$ and let ${{\mathscrJ}}$ be a subset of $\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$ such that $$\sup_{U\in{{\mathscrJ}}}\int_{{{\mathbb R}}^{n+1}_+}y^a|\nabla U|^2\,dX<+\infty.$$ Then ${{\mathscrJ}}$ is precompact in $L^2(B^+_R,y^a)$.
We will prove that ${{\mathscrJ}}$ is totally bounded in $L^2(B^+_R,y^a)$, i.e. for any ${\varepsilon }>0$ there exist $M$ and $U_1,\ldots,U_M\in L^2(B^+_R,y^a)$ such that for any $U\in{{\mathscrJ}}$ there exists $i\in\{1,\ldots,M\}$ such that $$\label{alewqewgew}
\|U_i-U\|_{L^2(B^+_R,y^a)}{\leqslant}{\varepsilon }.$$ For this, we fix ${\varepsilon }>0$, we set $$\label{pwqrtt000}
A:=\sup_{U\in{{\mathscrJ}}}\int_{{{\mathbb R}}^{n+1}_+}y^a|\nabla U|^2\,dX<+\infty$$ and we let $$\label{eta def}
\eta:=\left[\frac{{\varepsilon }^2}{2\hat{S}^2A}\left(\frac{a+1}{|B_R|}\right)^{\frac{\gamma-1}{\gamma}}
\right]^{\frac{\gamma}{(\gamma-1)(a+1)}},$$ where $\gamma$ and $\hat{S}$ are the constants introduced in the statement of Proposition \[WeightedSob\], and $|B_R|$ is the Lebesgue measure of the ball $B_R$ in ${{\mathbb R}}^n$.
Now, notice that $$\label{lskfjgrekge}
{\mbox{if $X\in B_R^+\cap\{y{\geqslant}\eta\}$ then
$y^a{\geqslant}\min\{\eta^a,R^a\}$}}.$$ Indeed, if $a{\geqslant}0$ (that is $s\in(0,1/2]$) then $y^a{\geqslant}\eta^a$, while if $a<0$ (that is $s\in(1/2,1)$) then we use that $y{\leqslant}R$, and so $y^a{\geqslant}R^a$, thus proving . Analogously, one can prove that $$\label{lskfjgrekge-1}
{\mbox{if $X\in B_R^+\cap\{y{\geqslant}\eta\}$ then
$y^a{\leqslant}\max\{\eta^a,R^a\}$}}.$$ Therefore, using , we have that, for any $U\in{{\mathscrJ}}$, $$A{\geqslant}\int_{B_R^+\cap\{y{\geqslant}\eta\}}y^a|\nabla U|^2\,dX{\geqslant}\min\{\eta^a,R^a\}\int_{B_R^+\cap\{y{\geqslant}\eta\}}|\nabla U|^2\,dX.$$ Hence, $$\int_{B_R^+\cap\{y{\geqslant}\eta\}}|\nabla U|^2\,dX<+\infty$$ for any $U\in{{\mathscrJ}}$. So by the Rellich-Kondrachov theorem we have that ${{\mathscrJ}}$ is totally bounded in $L^2(B_R^+\cap\{y{\geqslant}\eta\})$. Namely, there exist $\tilde{U}_1,\ldots,\tilde{U}_M\in L^2(B_R^+\cap\{y{\geqslant}\eta\})$ such that for any $U\in{{\mathscrJ}}$ there exists $i\in\{1,\ldots,M\}$ such that $$\label{qeqwptoepyrekh}
\|U_i-U\|_{L^2(B_R^+\cap\{y{\geqslant}\eta\})}{\leqslant}\frac{{\varepsilon }^2}{2\max\{\eta^a,R^a\}}.$$ Now for any $i\in\{1,\ldots,M\}$ we set $$U_i:=
\begin{cases}
\tilde{U}_i & {\mbox{ if }} y{\geqslant}\eta,\\
0 & {\mbox{ if }} y<\eta.
\end{cases}$$ Notice that $U_i\in L^2(B^+_R,y^a)$ for any $i\in\{1,\ldots,M\}$. Indeed, fixed $i\in\{1,\ldots,M\}$, we have that $$\begin{aligned}
\int_{B_R^+}y^a|U_i|^2\,dX &=& \int_{B_R^+\cap \{y<\eta\}}y^a|U_i|^2\,dX +
\int_{B_R^+\cap \{y{\geqslant}\eta\}}y^a|U_i|^2\,dX \\
&=& 0 + \int_{B_R^+\cap \{y{\geqslant}\eta\}}y^a|\tilde{U}_i|^2\,dX \\
&{\leqslant}& \max\{\eta^a,R^a\}\int_{B_R^+\cap \{y{\geqslant}\eta\}}|U_i|^2\,dX <+\infty,\end{aligned}$$ thanks to and the fact that $\tilde{U}_i\in L^2(B_R^+\cap\{y{\geqslant}\eta\})$ for any $i\in\{1,\ldots,M\}$.
It remains to show . For this, we first observe that $$\label{alewqewgew-3}
\|U_i-U\|_{L^2(B^+_R,y^a)}^2 = \int_{B_R^+\cap\{y<\eta\}}y^a |U|^2\,dX +
\int_{B_R^+\cap\{y{\geqslant}\eta\}}y^a |\tilde{U}_i-U|^2\,dX.$$ Using the Hölder inequality with exponents $\gamma$ and $\frac{\gamma}{\gamma-1}$ and Proposition \[WeightedSob\] and recalling and , we obtain that $$\begin{aligned}
\int_{B_R^+\cap\{y<\eta\}}y^a |U|^2\,dX &=&
\int_{B_R^+\cap\{y<\eta\}}y^{\frac{a}{\gamma}} |U|^2 y^{\frac{a(\gamma-1)}{\gamma}}\,dX \\
& {\leqslant}&
\left(\int_{B_R^+\cap\{y<\eta\}}y^a |U|^{2\gamma}\,dX\right)^{\frac{1}{\gamma}}
\left(\int_{B_R^+\cap\{y<\eta\}}y^a \,dX\right)^{\frac{\gamma-1}{\gamma}} \\
&{\leqslant}& \hat{S}^2\int_{{{\mathbb R}}^{n+1}_+}y^a |\nabla U|^2\,dX \,
\left(\frac{|B_R|}{a+1}\right)^{\frac{\gamma-1}{\gamma}}\eta^{\frac{(a+1)(\gamma-1)}{\gamma}}\\
&{\leqslant}& \hat{S}^2 A \left(\frac{|B_R|}{a+1}\right)^{\frac{\gamma-1}{\gamma}}
\eta^{\frac{(a+1)(\gamma-1)}{\gamma}}\\
&=& \frac{{\varepsilon }^2}{2}.\end{aligned}$$ Moreover, making use of and , we have that $$\int_{B_R^+\cap\{y{\geqslant}\eta\}}y^a |\tilde{U}_i-U|^2\,dX {\leqslant}\max\{\eta^a,R^a\}
\int_{B_R^+\cap\{y{\geqslant}\eta\}}|\tilde{U}_i-U|^2\,dX {\leqslant}\frac{{\varepsilon }^2}{2}.$$ Plugging the last two formulas into , we get $$\|U_i-U\|_{L^2(B^+_R,y^a)}^2{\leqslant}\frac{{\varepsilon }^2}{2} +\frac{{\varepsilon }^2}{2}={\varepsilon }^2,$$ which implies and thus concludes the proof of Lemma \[lemma:compact\].
A Concentration-Compactness Principle {#sec:CC}
-------------------------------------
In this section we show a Concentration-Compactness Principle, in the spirit of the original result proved by P. L. Lions in [@L1] and [@L2]. In particular, we want to adapt Lemma 2.3 of [@L2]. See also, [@AGP; @giampiero], where this principle was proved for different problems.
For this, we recall the following definitions:
\[defTight\] We say that a sequence $\{U_k\}_{k\in\mathbb{N}}$ is tight if for every $\eta>0$ there exists $\rho>0$ such that $$\int_{\mathbb{R}^{n+1}_+\setminus B_\rho^+}{y^a|\nabla U_k|^2\,dX}{\leqslant}\eta \quad {\mbox{ for any }}k.$$
\[convMeasures\] Let $\{\mu_k\}_{k\in\mathbb{N}}$ be a sequence of measures on a topological space $X$. We say that $\mu_k$ converges to $\mu$ in $X$ if and only if $$\lim_{k\to+\infty}\int_X{\varphi\,d\mu_k}= \int_X{\varphi\,d\mu},$$ for every $\varphi\in C_0(X)$.
This definition is standard, see for instance Definition 1.1.2 in [@evans]. In particular, we will consider measures on $\mathbb{R}^n$ and $\mathbb{R}^{n+1}_+$.
\[CCP\] Let $\{U_k\}_{k\in\mathbb{N}}$ be a bounded tight sequence in $\dot{H}^s_a(\mathbb{R}^{n+1}_+)$, such that $U_k$ converges weakly to $U$ in $\dot{H}^s_a(\mathbb{R}^{n+1}_+)$. Let $\mu,\nu$ be two nonnegative measures on $\mathbb{R}^{n+1}_+$ and $\mathbb{R}^n$ respectively and such that $$\label{first conv}
\lim_{k\to+\infty}y^a|\nabla U_k|^2=\mu$$ and $$\label{second conv}
\lim_{k\to+\infty}|U_k(x,0)|^{2^*_s}=\nu$$ in the sense of Definition \[convMeasures\].
Then, there exist an at most countable set $J$ and three families $\{x_j\}_{j\in J}\in \mathbb{R}^n$, $\{\nu_j\}_{j\in J}$, $\{\mu_j\}_{j\in J}$, $\nu_j,\mu_j{\geqslant}0$ such that
- $\displaystyle \nu = |U(x,0)|^{2^*_s}+\sum_{j\in J}{\nu_j\delta_{x_j}}$,
- $\displaystyle \mu {\geqslant}y^a|\nabla U|^2+\sum_{j\in J}{\mu_j\delta_{(x_j,0)}}$,
- $\mu_j{\geqslant}S\nu_j^{2/2^*_s}$ for all $j\in J$.
We first suppose that $U\equiv 0$. We claim that $$\label{lemma lions}
\left(\int_{{{\mathbb R}}^n} |\varphi(x,0)|^{2^*_s}\, d\nu\right)^{2/2^*_s}{\leqslant}C\int_{{{\mathbb R}}^{n+1}_+} \varphi^2 \, d\mu, \quad {\mbox{ for any }} \varphi\in C^\infty_0({{\mathbb R}}^{n+1}_+),$$ for some $C>0$. For this, let $\varphi\in C_0^\infty (\mathbb{R}^{n+1}_+)$ and $K:=\,$supp$(\varphi)$. By Proposition \[traceIneq\], we have that $$\label{traceVarphi}
\left(\int_{\mathbb{R}^n}{|(\varphi U_k)(x,0)|^{2^*_s}\,dx}\right)^{2/2^*_s}{\leqslant}C
\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla (\varphi U_k)|^2\,dX},$$ for a suitable positive constant $C$. By , we deduce $$\label{owqurwqhgf}
\lim_{k\to+\infty} \int_{\mathbb{R}^n}{|(\varphi U_k)(x,0)|^{2^*_s}\,dx}=
\int_{\mathbb{R}^n}{|\varphi(x,0)|^{2^*_s}\,d\nu}.$$ On the other hand, the right hand side in can be written as $$\begin{aligned}
\label{convRHS}
\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla (\varphi U_k)|^2\,dX}&=& \int_{\mathbb{R}^{n+1}_+}{y^a\varphi^2|\nabla U_k|^2\,dX}+\int_{\mathbb{R}^{n+1}_+}{y^aU_k^2|\nabla \varphi|^2\,dX}\nonumber\\
&&\qquad +2\int_{\mathbb{R}^{n+1}_+}{y^a\varphi\,U_k\,\langle\nabla \varphi,\nabla U_k\rangle\,dX}.\end{aligned}$$
Now we observe that $$\label{pqotugeohgvg}
[U_k]_a{\leqslant}C$$ for some $C>0$ independent of $k$, and so, by Lemma \[lemma:compact\], we have that, up to a subsequence, $$\label{wjdehqwjpfognjbvn}
{\mbox{$U_k$ converges to~$U=0$ in~$L^2_{\rm loc}({{\mathbb R}}^{n+1}_+,y^a)$ as~$k\to+\infty$.}}$$ Therefore, $$\label{laksjkashgj}
\lim_{k\to+\infty}\int_{\mathbb{R}^{n+1}_+}{y^aU_k^2|\nabla \varphi|^2\,dX}{\leqslant}C \lim_{k\to+\infty}\int_{K}{y^a U_k^2\,dX}=0.$$ Also, by the Hölder inequality and , $$\begin{aligned}
&&\left|\int_{\mathbb{R}^{n+1}_+}{y^a\varphi\,U_k\,\langle\nabla \varphi,\nabla U_k\rangle\,dX}\right|\\ &{\leqslant}&
\left(\int_{\mathbb{R}^{n+1}_+}{y^a|\varphi|^2\,|\nabla U_k|^2\,dX}\right)^{1/2}\,
\left(\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla\varphi|^2\,|U_k|^2\,dX}\right)^{1/2}\\
&{\leqslant}& C\, \left(\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U_k|^2\,dX}\right)^{1/2}\,
\left(\int_{K}{y^a |U_k|^2\,dX}\right)^{1/2}\\
&{\leqslant}& C\,\left(\int_{K}{y^a |U_k|^2\,dX}\right)^{1/2},\end{aligned}$$ where $C$ may change from line to line. Hence, from we have that $$\lim_{k\to+\infty}\int_{\mathbb{R}^{n+1}_+}{y^a\varphi\,U_k\,\langle\nabla \varphi,\nabla U_k\rangle\,dX}=0.$$ Thus, plugging this and into , and using , we obtain $$\lim_{k\to+\infty}\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla (\varphi U_k)|^2\,dX}=
\int_{\mathbb{R}^{n+1}_+}{\varphi^2\,d\mu}.$$ Therefore, taking the limit in as $k\to+\infty$, and using , we get $$\left(\int_{\mathbb{R}^n}{|\varphi(x,0)|^{2^*_s}\,d\nu}\right)^{2/2^*_s}{\leqslant}C \int_{\mathbb{R}^{n+1}_+}{\varphi^2\,d\mu},\qquad\hbox{for all }\varphi\in C_0^\infty(\mathbb{R}^{n+1}_+),$$ which shows in the case $U\equiv 0$.
Let us consider now the case $U\not\equiv 0$. First, we define a function $V_k:=U_k-U$, and we observe that $V_k\in\dot{H}^s_a(\mathbb{R}^{n+1}_+)$, and $$\label{2.3base}
{\mbox{$V_k$ converges weakly to~0 in~$\dot{H}^s_a(\mathbb{R}^{n+1}_+)$
as~$k\to+\infty$.}}$$ Also, we denote by $$\label{2.3bis}
\tilde{\nu}:=\lim_{k\rightarrow\infty}{|V_k(x,0)|^{2^*_s}}\qquad\hbox{ and }\qquad \tilde{\mu}:=\lim_{k\rightarrow\infty}{y^a|\nabla V_k|^2},$$ where both limits are understood in the sense of Definition \[convMeasures\]. Then, we are in the previous case, and so we can apply , that is $$\label{traceVarphiVk}
\left(\int_{\mathbb{R}^n}{|\varphi(x,0)|^{2^*_s}\,d\tilde{\nu}}\right)^{2/2^*_s}{\leqslant}C \int_{\mathbb{R}^{n+1}_+}{\varphi^2\,d\tilde{\mu}},\qquad\hbox{for all }\varphi\in C_0^\infty(\mathbb{R}^{n+1}_+).$$ Furthermore, by [@BL], we know that $$\lim_{k\rightarrow\infty}{\int_{\mathbb{R}^n}{|(\varphi V_k)(x,0)|^{2^*_s}\,dx}}
=\lim_{k\rightarrow\infty}{\int_{\mathbb{R}^n}{|(\varphi U_k)(x,0)|^{2^*_s}\,dx}}-
\int_{\mathbb{R}^n}{|(\varphi U)(x,0)|^{2^*_s}\,dx},$$ that is, recalling , $$\int_{\mathbb{R}^n}{|\varphi(x,0)|^{2^*_s}\,d\tilde{\nu}}=\int_{\mathbb{R}^n}{|\varphi (x,0)|^{2^*_s}\,d\nu}-
\int_{\mathbb{R}^n}{|(\varphi U)(x,0)|^{2^*_s}\,dx}.$$ Therefore $$\label{2.3ter}
\nu=\tilde{\nu}+|U(x,0)|^{2^*_s}.$$
On the other hand, $$\begin{aligned}
\int_{\mathbb{R}^{n+1}_+}{y^a\varphi^2|\nabla U_k|^2\,dX}&=& \int_{\mathbb{R}^{n+1}_+}{y^a\varphi^2|\nabla U|^2\,dX}+\int_{\mathbb{R}^{n+1}_+}{y^a\varphi^2|\nabla V_k|^2\,dX}\\
&&\qquad +2\int_{\mathbb{R}^{n+1}_+}{y^a\varphi^2\langle\nabla V_k,\nabla U\rangle\,dX}.\end{aligned}$$ Now we take the limit as $k\to+\infty$, we use , and , and we obtain $$\int_{\mathbb{R}^{n+1}_+}{\varphi^2\,d\mu} =\int_{\mathbb{R}^{n+1}_+}{y^a\varphi^2|\nabla U|^2\,dX}+ \int_{\mathbb{R}^{n+1}_+}{\varphi^2\,d\tilde{\mu}},$$ i.e., $$\label{2.45}
\mu=\tilde{\mu}+y^a|\nabla U|^2.$$
Now, since inequality is satisfied, we can apply Lemma 1.2 in [@L1] to $\tilde{\nu}$ and $\tilde{\mu}$ (see also Lemma 2.3 in [@L2]). Therefore, there exist an at most countable set $J$ and families $\{x_j\}_{j\in J}\in \mathbb{R}^n$, $\{\nu_j\}_{j\in J}$, $\{\mu_j\}_{j\in J}$, with $\nu_j{\geqslant}0$ and $\mu_j>0$, such that $$\tilde{\nu}=\sum_{j\in J}{\nu_j\delta_{x_j}} \quad{\mbox{ and }}\quad \tilde{\mu}{\geqslant}\sum_{j\in J}{\mu_j\delta_{(x_j,0)}}.$$ So the proof is finished, thanks to and .
Existence of a minimal solution and proof of Theorem \[MINIMUM\] {#ECXMII}
================================================================
Some convergence results in view of Theorem \[MINIMUM\] {#sec:conv}
-------------------------------------------------------
In this section we collect some results about the convergence of sequences of functions in suitable $L^r({{\mathbb R}}^n)$ spaces. We will exploit the following lemmata in the forthcoming Section \[sec:PS\], see in particular the proof of Proposition \[PScond\].
The first result that we prove is the following:
\[PSL-1\] Let $v_k\in L^{2^*_s}({{\mathbb R}}^n,[0,+\infty))$ be a sequence converging to some $v$ in $L^{2^*_s}({{\mathbb R}}^n)$. Then $$\begin{aligned}
\label{R45-2}
&&
\lim_{k\to+\infty}\int_{{{\mathbb R}}^n} |v_k^q(x)-v^q(x)|^{\frac{2^*_s}{q}}\,dx=0
\\
\label{R45-1}
{\mbox{and }}&&
\lim_{k\to+\infty}\int_{{{\mathbb R}}^n} |v_k^p(x)-v^p(x)|^{\frac{2n}{n+2s}}\,dx=0.\end{aligned}$$
For any $t{\geqslant}-1$, let $$f(t):=\frac{|(1+t)^q-1|}{|t|^q}.$$ We have that $(1+t)^q =1+qt+o(t)$ for $t$ close to $0$, and therefore $$f(t)= \frac{|qt+o(t)|}{|t|^q}\to 0$$ as $t\to0$. In addition, $f(-1)=1$ and $$\lim_{t\to+\infty} f(t)=1.$$ As a consequence, we can define $$L:=\sup_{t{\geqslant}-1} f(t)$$ and we have that $L\in[1,+\infty)$. Now we show that $$\label{fdcvbpfp11}
|a^q-b^q|{\leqslant}L|a-b|^q$$ for any $a$, $b{\geqslant}0$. To prove this, we can suppose that $b\ne0$, otherwise we are done, and we write $t:=\frac{a}{b}-1$. Then we have that $$|a^q-b^q| = b^q |(1+t)^q -1|{\leqslant}L b^q |t|^q=L|a-b|^q,$$ which proves .
As a consequence of this and of the convergence of $v_k$, we have that $$\int_{{{\mathbb R}}^n} |v_k^q(x)-v^q(x)|^{\frac{2^*_s}{q}}\,dx
{\leqslant}L\int_{{{\mathbb R}}^n} |v_k(x)-v(x)|^{2^*_s}\,dx \to0,$$ as $k\to+\infty$, which establishes . Now we prove . For this, given $a{\geqslant}b{\geqslant}0$, we notice that $$a^p-b^p=
p\int_b^a t^{p-1}\,dt{\leqslant}pa^{p-1}(a-b){\leqslant}p(a+b)^{p-1}(a-b).$$ By possibly exchanging the roles of $a$ and $b$, we conclude that, for any $a$, $b{\geqslant}0$, $$|a^p-b^p|{\leqslant}p(a+b)^{p-1}|a-b|.$$ Accordingly, for any $a$, $b{\geqslant}0$, $$|a^p-b^p|^{\frac{2n}{n+2s}}
{\leqslant}p^{\frac{2n}{n+2s}} (a+b)^{\frac{2n(p-1)}{n+2s}}
|a-b|^{\frac{2n}{n+2s}} =
p^{\frac{2n}{n+2s}} (a+b)^{\frac{8sn}{(n-2s)(n+2s)}}
|a-b|^{\frac{2n}{n+2s}}.$$ We use this and the Hölder inequality with exponents $\frac{n+2s}{4s}$ and $\frac{n+2s}{n-2s}$ to deduce that $$\begin{aligned}
&& \int_{{{\mathbb R}}^n} \big|v_k^p(x)-v^p(x)\big|^{\frac{2n}{n+2s}}\,dx\\
&{\leqslant}& p^{\frac{2n}{n+2s}}
\int_{{{\mathbb R}}^n} \big(
v_k(x)+v(x) \big)^{\frac{8sn}{(n-2s)(n+2s)}}
\big|
v_k(x)-v(x) \big|^{\frac{2n}{n+2s}}\,dx\\
&{\leqslant}&
p^{\frac{2n}{n+2s}} \left(
\int_{{{\mathbb R}}^n} \big(
v_k(x)+v(x) \big)^{\frac{2n}{n-2s}}
\,dx \right)^{ \frac{4s}{n+2s} }
\left( \int_{{{\mathbb R}}^n}
\big|v_k(x)-v(x) \big|^{\frac{2n}{n-2s}}\,dx\right)^{\frac{n-2s}{n+2s}}\\
&= & p^{\frac{2n}{n+2s}}
\| v_k+v\|_{L^{2^*_s}({{\mathbb R}}^n)}^{ \frac{8sn}{(n-2s)(n+2s)} }
\| v_k-v\|_{L^{2^*_s}({{\mathbb R}}^n)}^{\frac{2n}{n+2s}}.\end{aligned}$$ [F]{}rom the convergence of $v_k$, we have that $\| v_k+v\|_{L^{2^*_s}({{\mathbb R}}^n)}
{\leqslant}\| v_k\|_{L^{2^*_s}({{\mathbb R}}^n)}
+\| v\|_{L^{2^*_s}({{\mathbb R}}^n)}$ is bounded uniformly in $k$, while $\| v_k-v\|_{L^{2^*_s}({{\mathbb R}}^n)}$ in infinitesimal as $k\to+\infty$, therefore now plainly follows.
Next result shows that we can deduce strong convergence in $L^{2^*_s}({{\mathbb R}}^n)$ from the convergence in the sense of Definition \[convMeasures\].
\[PSL-2\] Let $v_k\in L^{2^*_s}({{\mathbb R}}^n,[0,+\infty))$ be a sequence converging to some $v$ a.e. in ${{\mathbb R}}^n$. Assume also that $v_k^{2^*_s}$ converges to $v^{2^*_s}$ in the measure sense given in Definition \[convMeasures\], i.e. $$\label{09ngjhgfnmxxu}
\lim_{k\to+\infty} \int_{{{\mathbb R}}^n}v_k^{2^*_s} \varphi\,dx=
\int_{{{\mathbb R}}^n}v^{2^*_s} \varphi\,dx$$ for any $\varphi\in C_0({{\mathbb R}}^n)$.
In addition, assume that for any $\eta>0$ there exists $\rho>0$ such that $$\label{TI67}
\int_{{{\mathbb R}}^n\setminus B_\rho }v_k^{2^*_s} (x)\,dx <\eta.$$ Then, $v_k\to v$ in $L^{2^*_s}({{\mathbb R}}^n,[0,+\infty))$ as $k\to+\infty$.
First of all, by Fatou’s lemma, $$\label{FA6ichofhav}
\lim_{k\to+\infty} \int_{{{\mathbb R}}^n}v_k^{2^*_s} \,dx{\geqslant}\int_{{{\mathbb R}}^n}v^{2^*_s}\,dx.$$ Now we fix $\eta>0$ and we take $\rho=\rho(\eta)$ such that holds true. Let $
\varphi_\rho\in C^\infty_0(B_{\rho+1},[0,1])$ such that $\varphi_\rho=1$ in $B_\rho$. Then, by $$\int_{{{\mathbb R}}^n}v_k^{2^*_s} \,dx <
\int_{B_\rho }v_k^{2^*_s} \,dx +\eta
{\leqslant}\int_{{{\mathbb R}}^n}v_k^{2^*_s}\varphi_\rho \,dx +\eta .$$ Hence, exploiting , $$\lim_{k\to+\infty} \int_{{{\mathbb R}}^n}v_k^{2^*_s} \,dx
{\leqslant}\lim_{k\to+\infty}
\int_{{{\mathbb R}}^n}v_k^{2^*_s}\varphi_\rho \,dx +\eta =
\int_{{{\mathbb R}}^n}v^{2^*_s}\varphi_\rho \,dx +\eta.$$ Since $\varphi_\rho{\leqslant}1$, this gives that $$\lim_{k\to+\infty} \int_{{{\mathbb R}}^n}v_k^{2^*_s} \,dx
{\leqslant}\int_{{{\mathbb R}}^n}v^{2^*_s}\,dx +\eta.$$ Since $\eta$ can be taken arbitrarily small, we obtain that $$\lim_{k\to+\infty} \int_{{{\mathbb R}}^n}v_k^{2^*_s} \,dx
{\leqslant}\int_{{{\mathbb R}}^n}v^{2^*_s}\,dx .$$ This, together with , proves that $$\lim_{k\to+\infty}
\|v_k\|_{L^{2^*_s}({{\mathbb R}}^n)}^{2^*_s}=
\lim_{k\to+\infty} \int_{{{\mathbb R}}^n}v_k^{2^*_s} \,dx=
\int_{{{\mathbb R}}^n}v^{2^*_s}\,dx=\|v\|_{L^{2^*_s}({{\mathbb R}}^n)}^{2^*_s}.$$ This and the Brezis-Lieb lemma (see e.g. formula (1) in [@BL]) implies the desired result.
Palais-Smale condition for ${{{\mathscrF}}}_{\varepsilon }$ {#sec:PS}
-----------------------------------------------------------
In this section we show that the functional ${{\mathscrF}}_{\varepsilon }$ introduced in satisfies a Palais-Smale condition. The precise statement is contained in the following proposition.
\[PScond\] There exists $\bar C, c_1>0$, depending on $h$, $q$, $n$ and $s$, such that the following statement holds true.
Let $\{U_k\}_{k\in\mathbb{N}}\subset \dot{H}^s_a(\mathbb{R}^{n+1}_+)$ be a sequence satisfying
1. $\displaystyle\lim_{k\to+\infty}{{\mathscrF}}_{\varepsilon }(U_k)=
c_{\varepsilon }$, with $$\label{ceps}
c_{\varepsilon }+c_1\varepsilon^{1/\gamma}+\bar C {\varepsilon }^{\frac{p+1}{p-q}}<
\dfrac{s}{n}S^{\frac{n}{2s}},$$ where $\gamma=1+\frac{2}{n-2s}$ and $S$ is the Sobolev constant appearing in Proposition \[traceIneq\],
2. $\displaystyle\lim_{k\to+\infty}{{\mathscrF}}'_{\varepsilon }(U_k)= 0.$
Then there exists a subsequence, still denoted by $\{U_k\}_{k\in\mathbb{N}}$, which is strongly convergent in $\dot{H}^s_a(\mathbb{R}^{n+1}_+)$ as $k\to+\infty$.
\[rem:3.3-1\] The limit in ii) is intended in the following way $$\begin{aligned}
&& \lim_{k\to+\infty}\|{{\mathscrF}}'_{\varepsilon }(U_k)\|_
{{{\mathscrL}}(\dot{H}^s_a(\mathbb{R}^{n+1}_+),\dot{H}^s_a(\mathbb{R}^{n+1}_+))}
\\ &&\qquad = \lim_{k\to+\infty}\sup_{{V\in \dot{H}^s_a({{\mathbb R}}^{n+1}_+)}\atop{ [V]_a =1 }}
\left|\langle {{\mathscrF}}'_{\varepsilon }(U_k), V\rangle\right|
=0,\end{aligned}$$ where ${{\mathscrL}}(\dot{H}^s_a(\mathbb{R}^{n+1}_+),\dot{H}^s_a(\mathbb{R}^{n+1}_+))$ consists of all the linear functional from $\dot{H}^s_a(\mathbb{R}^{n+1}_+)$ in $\dot{H}^s_a(\mathbb{R}^{n+1}_+)$.
First we show that a sequence that satisfies the assumptions in Proposition \[PScond\] is bounded.
\[lemma bound\] Let ${\varepsilon }$, $\kappa>0$. Let $\{U_k\}_{k\in\mathbb{N}}\subset \dot{H}^s_a(\mathbb{R}^{n+1}_+)$ be a sequence satisfying $$\label{9sd45678trdfghbvcrtyujbv}
|{{{\mathscrF}}}_{\varepsilon }(U_k)| +
\sup_{{V\in \dot{H}^s_a({{\mathbb R}}^{n+1}_+)}\atop{ [V]_a =1 }}
\big|\langle {{{\mathscrF}}}_{\varepsilon }'(U_k),V\rangle\big|{\leqslant}\kappa,$$ for any $k\in{{\mathbb N}}$.
Then there exists $M>0$ such that $$\label{bound}
[U_k]_a{\leqslant}M.$$
If $[U_k]_a=0$ we are done. So we can suppose that $[U_k]_a\ne0$ and use to obtain $$\left|{{\mathscrF}}_{\varepsilon }(U_k)\right|{\leqslant}\kappa ,\hbox{ and }
\left|\langle {{\mathscrF}}'_{\varepsilon }(U_k), U_k/[U_k]_{a}\rangle\right| {\leqslant}\kappa.$$ Therefore, we have that $$\label{lakjdkfeowpt}
{{\mathscrF}}_{\varepsilon }(U_k)-\frac{1}{p+1}\langle {{\mathscrF}}'_{\varepsilon }(U_k), U_k\rangle
{\leqslant}\kappa\,\left(1+ [U_k]_a\right).$$ On the other hand, by the Hölder inequality and Proposition \[traceIneq\], we obtain $$\begin{aligned}
&& {{\mathscrF}}_{\varepsilon }(U_k)-\frac{1}{p+1}\langle {{\mathscrF}}'_{\varepsilon }(U_k), U_k\rangle\\
& =&\left(\frac{1}{2}-\frac{1}{p+1}\right)\int_{\mathbb{R}^{n+1}}{y^a|\nabla U_k|^2\,dX}
-\varepsilon\left(\frac{1}{q+1}-\frac{1}{p+1}\right)\int_{\mathbb{R}^n}{h(x)(U_k)_+^{q+1}(x,0)\,dx}\\
&{\geqslant}& \left(\frac{1}{2}-\frac{1}{p+1}\right)[U_k]_a^2
-\varepsilon C\left(\frac{1}{q+1}-\frac{1}{p+1}\right)
\|h\|_{L^{\frac{p+1}{p-q}}(\mathbb{R}^n)}[U_k]_a^{q+1}.\end{aligned}$$ From this and we conclude that $[U_k]_a$ must be bounded (recall also and that $q+1<2$). So we obtain the desired result.
In order to prove that ${{\mathscrF}}_{\varepsilon }$ satisfies the Palais-Smale condition, we need to show that the sequence of functions satisfying the hypotheses of Proposition \[PScond\] is tight, according to Definition \[defTight\].
First we make the following preliminary observation:
\[lemma basic\] Let $m:=\frac{p+1}{p-q}$. Then there exists a constant $\bar{C}=\bar{C}(n,s,p,q,\|h\|_{L^m(\mathbb{R}^n)})>0$ such that, for any $\alpha>0$, $$\frac{s}{n}\alpha^{p+1}-{\varepsilon }\left(\frac{1}{q+1}-
\frac{1}{2}\right)\|h\|_{L^m(\mathbb{R}^n)}\alpha^{q+1}
{\geqslant}-\bar{C}{\varepsilon }^{\frac{p+1}{p-q}}.$$
Let us define the function $f:(0,+\infty)\rightarrow\mathbb{R}$ as $$f(\alpha):=c_1\alpha^{p+1}-\varepsilon c_2 \alpha^{q+1}, \;\quad c_1:=\frac{s}{n},\;\quad c_2:=\left(\frac{1}{q+1}-
\frac{1}{2}\right)\|h\|_{L^m(\mathbb{R}^n)}.$$ Differentiating, we obtain that $$f'(\alpha)=\alpha^q((p+1)c_1\alpha^{p-q}-\varepsilon (q+1)c_2),$$ and thus, $f$ has a local minimum at the point $$\overline{\alpha}:= c_3\varepsilon^{\frac{1}{p-q}},\quad c_3=c_3(n,s,p,q,\|h\|_{L^m(\mathbb{R}^n)}):=\left(\frac{c_2(q+1)}{c_1(p+1)}\right)^{\frac{1}{p-q}}.$$ Evaluating $f$ at $\overline{\alpha}$, we obtain that the minimum value that $f$ will reach is $$f(\overline{\alpha})=c_4\varepsilon^{\frac{p+1}{p-q}},$$ with $c_4$ a constant depending on $n$, $s$, $p$, $q$ and $\|h\|_{L^m(\mathbb{R}^n)}$. Therefore, there exists $\bar{C}=\bar{C}(n,s,p,q,\|h\|_{L^m(\mathbb{R}^n)})>0$ such that $$f(\alpha){\geqslant}f(\overline{\alpha}){\geqslant}-\bar{C}\varepsilon^{\frac{p+1}{p-q}},$$ for any $\alpha>0$, and this concludes the proof.
The tightness of the sequence in Proposition \[PScond\] is contained in the following lemma:
\[tightness\] Let $\{U_k\}_{k\in\mathbb{N}}\subset \dot{H}^s_a(\mathbb{R}^{n+1}_+)$ be a sequence satisfying the hypotheses of Proposition \[PScond\].
Then for all $\eta>0$ there exists $\rho>0$ such that for every $k\in\mathbb{N}$ it holds $$\int_{\mathbb{R}^{n+1}_+\setminus B_\rho^+}{y^a|\nabla U_k|^2\,dX}
+\int_{\mathbb{R}^n\setminus\{B_\rho\cap\{y=0\}\}}{(U_k)_+^{2^*_s}(x,0)\,dx}
<\eta.$$ In particular, the sequence $\{U_k\}_{k\in\mathbb{N}}$ is tight.
First we notice that holds in this case, due to conditions (i) and (ii) in Proposition \[PScond\]. Hence, Lemma \[lemma bound\] gives that the sequence $\{U_k\}_{k\in\mathbb{N}}$ is bounded in $\dot{H}^s_a(\mathbb{R}^{n+1}_+)$, that is $[U_k]_a{\leqslant}M$. Thus, $$\begin{split}\label{weak convergence-1}
& U_k\rightharpoonup U \quad \hbox{ in }\dot{H}^s_a(\mathbb{R}^{n+1}_+) \quad {\mbox{ as }}k\to+\infty \\
{\mbox{and }}& U_k\rightarrow U\quad \hbox{ a.e. in }\mathbb{R}^{n+1}_+\quad {\mbox{ as }}k\to+\infty.
\end{split}$$
Now, we proceed by contradiction. Suppose that there exists $\eta_0>0$ such that for all $\rho>0$ there exists $k=k(\rho)\in{{\mathbb N}}$ such that $$\label{contrad}
\int_{\mathbb{R}^{n+1}_+\setminus B_\rho^+}{y^a|\nabla U_k|^2\,dX}
+\int_{\mathbb{R}^n\setminus\{B_\rho\cap\{y=0\}\}}{(U_k)_+^{2^*_s}(x,0)\,dx}
{\geqslant}\eta_0.$$ We observe that $$\label{forse0}
k\to+\infty \quad {\mbox{ as }}\rho\to+\infty.$$ Indeed, let us take a sequence $\{\rho_i\}_{i\in\mathbb{N}}$ such that $\rho_i\rightarrow +\infty$ as $i\rightarrow +\infty$, and suppose that $k_i:=k(\rho_i)$ given by is a bounded sequence. That is, the set $F:=\{k_i:\;i\in{{\mathbb N}}\}$ is a finite set of integers.
Hence, there exists an integer $k^\star$ so that we can extract a subsequence $\{k_{i_j}\}_{j\in{{\mathbb N}}}$ satisfying $k_{i_j}=k^\star$ for any $j\in{{\mathbb N}}$. Therefore, $$\label{contrad2}
\int_{\mathbb{R}^{n+1}_+\setminus B_{\rho_{i_j}}^+}{y^a|\nabla U_{k^\star}|^2\,dX}
+\int_{\mathbb{R}^n\setminus\{B_{\rho_{i_j}}\cap\{y=0\}\}}{(U_{k^\star})_+^{2^*_s}(x,0)\,dx}
{\geqslant}\eta_0,$$ for any $j\in {{\mathbb N}}$.
But on the other hand, since $U_{k^\star}$ belongs to $\dot{H}^s_a(\mathbb{R}^{n+1}_+)$ (and so $U_{k^\star}(\cdot,0)\in L^{2^*_s}({{\mathbb R}}^n)$ thanks to Proposition \[traceIneq\]), for $j$ large enough there holds $$\int_{\mathbb{R}^{n+1}_+\setminus B_{\rho_{i_j}}^+}{y^a|\nabla U_{k^\star}|^2\,dX}
+\int_{\mathbb{R}^n\setminus\{B_{\rho_{i_j}}\cap\{y=0\}\}}{(U_{k^\star})_+^{2^*_s}(x,0)\,dx}
{\leqslant}\frac{\eta_0}{2},$$ which is a contradiction with . This shows .
Now, since $U$ given in belongs to $\in\dot{H}^s_a(\mathbb{R}^{n+1}_+)$, by Propositions \[WeightedSob\] and \[traceIneq\], we have that, for a fixed $\varepsilon>0$, there exists $r_{\varepsilon }>0$ such that $$\int_{\mathbb{R}^{n+1}_+\setminus B_{r_{\varepsilon }}^+}{y^a|\nabla U|^2\,dX}
+\int_{\mathbb{R}^{n+1}_+\setminus B_{r_{\varepsilon }}^+}{y^a|U|^{2\gamma}\,dX}
+\int_{\mathbb{R}^n\setminus\{B_{r_{\varepsilon }}\cap\{y=0\}\}}{|U(x,0)|^{2^*_s}\,dx}<\varepsilon.$$ Notice that $$\label{eps to zero}
{\mbox{$r_{\varepsilon }\to +\infty$ as ${\varepsilon }\to 0$.}}$$
Moreover, by and again by Propositions \[WeightedSob\] and \[traceIneq\], we obtain that there exists $\tilde{M}>0$ such that $$\label{boundk}
\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U_k|^2\,dX}+\int_{\mathbb{R}^{n+1}_+}{y^a|U_k|^{2\gamma}\,dX}
+\int_{\mathbb{R}^n}{|U_k(x,0)|^{2^*_s}\,dx}{\leqslant}\tilde{M}.$$
Now let $j_{\varepsilon }\in\mathbb{N}$ be the integer part of $\frac{\tilde{M}}{\varepsilon}$. Notice that $j_{\varepsilon }$ tends to $+\infty$ as ${\varepsilon }$ tends to 0. We also set $$I_l:=\{(x,y)\in\mathbb{R}^{n+1}_+:r+l{\leqslant}|(x,y)|{\leqslant}r+(l+1)\},\;l=0,1,\cdots,j_{\varepsilon }.$$ Thus, from we get $$\begin{aligned}
(j_{\varepsilon }+1)\varepsilon &{\geqslant}&
\frac{\tilde{M}}{{\varepsilon }}{\varepsilon }\\&{\geqslant}&
\sum_{l=0}^{j_{\varepsilon }}\left({\int_{I_l}{y^a|\nabla U_k|^2\,dX}
+\int_{I_l}{y^a|U_k|^{2\gamma}\,dX}
+\int_{I_l\cap\{y=0\}}{|U_k(x,0)|^{2^*_s}\,dx}}\right).\end{aligned}$$ This implies that there exists $\bar{l}\in\{0,1,\cdots, j_{\varepsilon }\}$ such that, up to a subsequence, $$\label{epsBound}
\int_{I_{\bar{l}}}{y^a|\nabla U_k|^2\,dX}+\int_{I_{\bar{l}}}{y^a|U_k|^{2\gamma}\,dX}
+\int_{I_{\bar{l}}\cap\{y=0\}}{|U_k(x,0)|^{2^*_s}\,dx}{\leqslant}\varepsilon.$$
Now we take a cut-off function $\chi\in C^\infty_0({{\mathbb R}}^{n+1}_+,[0,1])$, such that $$\label{3.4bis}
\chi(x,y)=\begin{cases}
1,\quad |(x,y)|{\leqslant}r+\bar{l}\\
0,\quad |(x,y)|{\geqslant}r+(\bar{l}+1),
\end{cases}$$ and $$\label{3.4bisbis}
|\nabla \chi|{\leqslant}2.$$ We also define $$\label{3.4ter}
V_k:=\chi U_k \quad {\mbox{ and }}\quad W_k:=(1-\chi)U_k.$$ We estimate $$\begin{split}\label{math F}
&|\langle {{\mathscrF}}'_{\varepsilon }(U_k)-{{\mathscrF}}'_{\varepsilon }(V_k),V_k\rangle |\\
&\quad = \bigg|\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla U_k,\nabla V_k\rangle\,dX}
-{\varepsilon }\int_{\mathbb{R}^{n}}{h(x) (U_k)_+^q(x,0)\,V_k(x,0)\,dx}\\
&\qquad\quad -\int_{\mathbb{R}^{n}}{(U_k)_+^p(x,0)\,V_k(x,0)\,dx}
-\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla V_k,\nabla V_k\rangle\,dX}\\
&\qquad\quad +{\varepsilon }\int_{\mathbb{R}^{n}}{h(x) (V_k)_+^{q+1}(x,0)\,dx}
+\int_{\mathbb{R}^{n}}{(V_k)_+^{p+1}(x,0)\,dx}\bigg|.
\end{split}$$ First, we observe that $$\begin{split}\label{AA}
&\bigg|\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla U_k,\nabla V_k\rangle\,dX}-\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla V_k,\nabla V_k\rangle\,dX}\bigg|\\
&\qquad{\leqslant}\int_{I_{\overline{l}}}{y^a|\nabla U_k|^2|\chi||1-\chi|\,dX}+\int_{I_{\overline{l}}}{y^a|\nabla U_k||U_k||\nabla\chi|\,dX}\\
&\qquad\qquad +2\int_{I_{\overline{l}}}{y^a|U_k||\nabla U_k||\nabla\chi||\chi|\,dX}+\int_{I_{\overline{l}}}{y^a|U_k|^2|\nabla \chi|^2\,dX}\\
&\qquad =:A_1+A_2+A_3+A_4.
\end{split}$$ By , we have that $A_1{\leqslant}C\varepsilon$, for some $C>0$. Furthermore, by the Hölder inequality, and , we obtain $$\begin{aligned}
A_2 &{\leqslant}& 2\int_{I_{\overline{l}}}{y^a|\nabla U_k||U_k|\,dX}{\leqslant}2\left(\int_{I_{\overline{l}}}{y^a|\nabla U_k|^2\,dX}\right)^{1/2}\left(\int_{I_{\overline{l}}}{y^a| U_k|^2\,dX}\right)^{1/2}\\
&{\leqslant}& 2\varepsilon^{1/2} \left(\int_{I_{\overline{l}}}{y^a|U_k|^{2\gamma}\,dX}\right)^{1/{2\gamma}}
\left(\int_{I_{\overline{l}}}{y^{(a-\frac{a}{\gamma})m}\,dX}\right)^{1/2m},\end{aligned}$$ where $m=\dfrac{n+2-2s}{2}$. Since $\left(a-\dfrac{a}{\gamma}\right)m=a=(1-2s)>-1$, we have that the second integral is finite, and therefore, for $\varepsilon<1$, $$A_2{\leqslant}\tilde{C}\varepsilon^{1/2} \left(\int_{ I_{\overline{l} }}{y^a|U_k|^{2\gamma}\,
dX}\right)^{1/{2\gamma}}{\leqslant}C\varepsilon^{1/2}{\varepsilon }^{1/2\gamma}{\leqslant}C{\varepsilon }^{1/\gamma},$$ where was used once again. In the same way, we get that $A_3{\leqslant}C\varepsilon^{1/\gamma}$. Finally, by , $$A_4{\leqslant}C\left(\int_{I_{ \overline{l} }}{y^a|U_k|^{2\gamma}\,dX}\right)^{1/{\gamma}}
\left(\int_{I_{\overline{l}}}{y^{(a-\frac{a}{\gamma})m}\,dX}\right)^{1/m}{\leqslant}C\varepsilon^{1/\gamma}.$$ Using these informations in , we obtain that $$\bigg|\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla U_k,\nabla V_k\rangle\,dX}
-\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla V_k,\nabla V_k\rangle\,dX}\bigg|
{\leqslant}C{\varepsilon }^{1/\gamma},$$ up to renaming the constant $C$.
On the other hand, since $p+1=2^*_s$, by and , $$\begin{aligned}
\bigg|\int_{\mathbb{R}^n}{( (U_k)_+^p(x,0)\,V_k(x,0)-(V_k)_+^{p+1}(x,0))\,dx}\bigg|
&{\leqslant}&\int_{\mathbb{R}^n}{|1-\chi^p||\chi| |U_k(x,0)|^{p+1}\,dx}\\
&{\leqslant}& C\int_{I_{\overline{l}}\cap\{y=0\}}{|U_k(x,0)|^{2^*_s}\,dx}{\leqslant}C\varepsilon.\end{aligned}$$ In the same way, applying the Hölder inequality, one obtains $$\begin{aligned}
&&\bigg|\int_{\mathbb{R}^n}{h(x)\,((U_k)_+^q(x,0)\,V_k(x,0)-(V_k)_+^{q+1}(x,0))\,dx}\bigg|
\\&&\qquad {\leqslant}\int_{\mathbb{R}^n}{|h(x)|\,|1-\chi^q||\chi| |U_k(x,0)|^{q+1}\,dx}\\
&&\qquad {\leqslant}C\, \|h\|_{L^\infty({{\mathbb R}}^n)}\,
\int_{I_{\overline{l}}\cap\{y=0\}}{|U_k(x,0)|^{2^*_s}\,dx}{\leqslant}C\varepsilon.\end{aligned}$$
All in all, plugging these observations in , we obtain that $$\label{boundV}
|\langle {{\mathscrF}}'_{\varepsilon }(U_k)-{{\mathscrF}}'_{\varepsilon }(V_k),V_k\rangle|
{\leqslant}C\varepsilon^{1/\gamma}.$$ Likewise, one can see that $$\label{boundW}
|\langle {{\mathscrF}}'_{\varepsilon }(U_k)-{{\mathscrF}}'_{\varepsilon }(W_k),W_k\rangle|
{\leqslant}C\varepsilon^{1/\gamma}.$$
Now we claim that $$\label{fprimeV}
|\langle {{\mathscrF}}'_{\varepsilon }(V_k),V_k\rangle|{\leqslant}C\varepsilon^{1/\gamma}+o_k(1),$$ where $o_k(1)$ denotes (here and in the rest of this monograph) a quantity that tends to 0 as $k$ tends to $+\infty$. For this, we first observe that $$\label{bbbb}
[V_k]_a{\leqslant}C,$$ for some $C>0$. Indeed, recalling and using and , we have $$\begin{aligned}
[V_k]_a^2 &=& \int_{{{\mathbb R}}^{n+1}_+}y^a|\nabla V_k|^2\,dX \\
&=& \int_{{{\mathbb R}}^{n+1}_+}y^a|\nabla\chi|^2|U_k|^2\,dX +
\int_{{{\mathbb R}}^{n+1}_+}y^a\,\chi^2|\nabla U_k|^2\,dX + 2\int_{{{\mathbb R}}^{n+1}_+}y^a\,\chi\,U_k
\ \langle \nabla U_k, \nabla\chi\rangle\,dX\\
&{\leqslant}& 4 \int_{I_{\overline{l}} }y^a| U_k|^2\,dX + [U_k]_a^2 +
C\left(\int_{ I_{\overline{l}}}y^a|\nabla U_k|^2\,dX\right)^{1/2}\,
\left(\int_{I_{\overline{l}}}y^a |U_k|^2\,dX\right)^{1/2}\\
&{\leqslant}& C \left(\int_{I_{\overline{l}} }y^a| U_k|^{2\gamma}\,dX\right)^{1/\gamma}
+ [U_k]_a^2 +
C\, [U_k]_a\, \left(\int_{I_{\overline{l}}}y^a |U_k|^{2\gamma}\,dX\right)^{1/2\gamma}, \end{aligned}$$ where the Hölder inequality was used in the last two lines. Hence, from Proposition \[WeightedSob\] and , we obtain .
Now, we notice that $$\begin{aligned}
|\langle {{\mathscrF}}'_{\varepsilon }(V_k),V_k\rangle| {\leqslant}|\langle {{\mathscrF}}'_{\varepsilon }(V_k)-{{\mathscrF}}'_{\varepsilon }(U_k),V_k\rangle| +
|\langle {{\mathscrF}}'_{\varepsilon }(U_k),V_k\rangle| {\leqslant}C\,{\varepsilon }^{1/\gamma} +|\langle {{\mathscrF}}'_{\varepsilon }(U_k),V_k\rangle|,\end{aligned}$$ thanks to . Thus, from and assumption (ii) in Proposition \[PScond\] we get the desired claim in .
Analogously (but making use of ), one can see that $$\label{fprimeW}
|\langle {{\mathscrF}}'_{\varepsilon }(W_k),W_k\rangle|{\leqslant}C\varepsilon^{1/\gamma}+o_k(1),$$
From now on, we divide the proof in three main steps: we first show lower bounds for ${{\mathscrF}}_{\varepsilon }(V_k)$ and ${{\mathscrF}}_{\varepsilon }(W_k)$ (see Step 1 and Step 2, respectively), then in Step 3 we obtain a lower bound for ${{\mathscrF}}_{\varepsilon }(U_k)$, which will give a contradiction with the hypotheses on ${{\mathscrF}}_{\varepsilon }$, and so the conclusion of Lemma \[tightness\].
[*Step 1: Lower bound for ${{\mathscrF}}_{\varepsilon }(V_k)$.*]{} By we obtain that $$\label{skajgfoew}
{{\mathscrF}}_{\varepsilon }(V_k){\geqslant}{{\mathscrF}}_{\varepsilon }(V_k)
-\frac{1}{2}\langle {{\mathscrF}}'_{\varepsilon }(V_k), V_k\rangle-\frac{1}{2}C\varepsilon^{1/\gamma}
+o_k(1).$$ Using the Hölder inequality, it yields $$\begin{split}
{{\mathscrF}}_{\varepsilon }(V_k)&-\frac{1}{2}\langle {{\mathscrF}}'_{\varepsilon }(V_k), V_k\rangle
= \left(\frac{1}{2}-\frac{1}{p+1}\right)\|(V_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}\\
&-{\varepsilon }\left(\frac{1}{q+1}-\frac{1}{2}\right)\int_{\mathbb{R}^n}{h(x)(V_k)_+^{q+1}(x,0)\,dx}\\
&{\geqslant}\frac{s}{n}\|(V_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}
-{\varepsilon }\left(\frac{1}{q+1}-\frac{1}{2}\right)\|h\|_{L^m(\mathbb{R}^n)}\|(V_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{q+1},
\end{split}$$ with $m=\dfrac{p+1}{p-q}$ (recall ). Therefore, from Lemma \[lemma basic\] (applied here with $\alpha:=\|(V_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}$) we deduce that $${{\mathscrF}}_{\varepsilon }(V_k)-\frac{1}{2}\langle {{\mathscrF}}'_{\varepsilon }(V_k), V_k\rangle
{\geqslant}-\bar{C}\,{\varepsilon }^{\frac{p+1}{p-q}}.$$ Going back to , this implies that $$\label{LowerBoundV}
{{\mathscrF}}_\varepsilon(V_k){\geqslant}-c_0{\varepsilon }^{1/\gamma}
-\bar{C}\,{\varepsilon }^{\frac{p+1}{p-q}}+o_k(1).$$\
[*Step 2: Lower bound for ${{\mathscrF}}_{\varepsilon }(W_k)$.*]{} First of all, by the definition of $W_k$ in , Proposition \[traceIneq\] and , we have that $$\begin{split}\label{upBoundWq}
\bigg|{\varepsilon }\int_{\mathbb{R}^n}{ h(x)(W_k)_+^{q+1}(x,0)\,dx }\bigg|
{\leqslant}\,&{\varepsilon }\, \|h\|_{L^m(\mathbb{R}^n)}
\|(W_k)_+(\cdot,0)\|_{L^{2^*_s}(\mathbb{R}^n)}^{q+1}\\
{\leqslant}\,& {\varepsilon }\, C\,\|h\|_{L^m(\mathbb{R}^n)}\|(U_k)_+(\cdot,0)\|_{L^{2^*_s}(\mathbb{R}^n)}^{q+1}\\
{\leqslant}\,& {\varepsilon }\, C\, \|h\|_{L^m(\mathbb{R}^n)}[U_k]_a^{q+1}{\leqslant}C{\varepsilon }.
\end{split}$$ Thus, from we get $$\begin{split}\label{Wbound}
&\bigg|\int_{\mathbb{R}^{n+1}_+} {y^a|\nabla W_k|^2\,dX}
-\int_{\mathbb{R}^n} {(W_k)_+^{p+1} (x,0)\,dx} \bigg|
\\ &\qquad{\leqslant}\left|\langle {{\mathscrF}}'_{\varepsilon }(W_k),W_k\rangle\right| +
\left|{\varepsilon }\int_{\mathbb{R}^n}{ h(x) (W_k)_+^{q+1}(x,0)\,dx}\right|\\
&\qquad {\leqslant}C\varepsilon^{1/\gamma} + o_k(1),
\end{split}$$ where was also used in the last passage. Moreover, notice that $W_k=U_k$ in ${{\mathbb R}}^{n+1}_+\setminus B_{r+\overline{l}+1}$ (recall and ). Hence, using with $\rho:=r+\overline{l}+1$, we get $$\begin{split}\label{espero}
&\int_{\mathbb{R}^{n+1}_+\setminus B^+_{r+\bar{l}+1}}{y^a|\nabla W_k|^2\,dX}
+\int_{\mathbb{R}^n\setminus\{B_{r+\bar{l}+1}\cap\{y=0\}\}}{(W_k)_+^{2^*_s}(x,0)\,dx}\\
&\qquad =\int_{\mathbb{R}^{n+1}_+\setminus B^+_{r+\bar{l}+1}}{y^a|\nabla U_k|^2\,dX}
+\int_{\mathbb{R}^n\setminus\{B_{r+\bar{l}+1}\cap\{y=0\}\}}
{(U_k)_+^{2^*_s}(x,0)\,dx}
{\geqslant}\eta_0,
\end{split}$$ for $k=k(\rho)$. We observe that $k$ tends to $+\infty$ as ${\varepsilon }\to 0$, thanks to and .
From we obtain that either $$\int_{\mathbb{R}^n\setminus\{ B_{r+\bar{l}+1} \cap\{y=0\}\} }
{(W_k)_+^{2^*_s}(x,0)\,dx}
{\geqslant}\frac{\eta_0}{2}$$ or $$\int_{\mathbb{R}^{n+1}_+\setminus B^+_{r+\bar{l}+1}}{y^a|\nabla W_k|^2\,dX}
{\geqslant}\frac{\eta_0}{2}.$$ In the first case, we get that $$\int_{\mathbb{R}^n} { (W_k)_+^{2^*_s}(x,0)\,dx}(x,0){\geqslant}\int_{\mathbb{R}^n\setminus\{B_{r+\bar{l}+1}\cap\{y=0\}\}}
{(W_k)_+^{2^*_s}(x,0)\,dx}(x,0){\geqslant}\frac{\eta_0}{2}.$$ In the second case, taking $\varepsilon$ small (and so $k$ large enough), by we obtain that $$\begin{split}
\int_{\mathbb{R}^n}{(W_k)_+^{p+1}(x,0)\,dx} & {\geqslant}\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla W_k|^2\,dX}-C\varepsilon^{1/\gamma}-o_k(1)
\\ &{\geqslant}\int_{\mathbb{R}^{n+1}_+\setminus B^+_{r+\bar{l}+1}}
{y^a|\nabla W_k|^2\,dX}-C\varepsilon^{1/\gamma}-o_k(1)
\\ & >\frac{\eta_0}{4}.
\end{split}$$ Hence, in both the cases we have that $$\label{lowBoundWp}
\int_{\mathbb{R}^n}{(W_k)_+^{p+1}(x,0)\,dx} >\frac{\eta_0}{4}.$$
Now we define $\psi_k:=\alpha_kW_k$, with $$\alpha_k^{p-1}:=\frac{[W_k]_a^2}{\|(W_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}}.$$ Notice that from we have that $$\begin{aligned}
[W_k]_a^2 &{\leqslant}& \|(W_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}
+\left|{\varepsilon }\int_{{{\mathbb R}}^n}h(x)(W_k)_+^{q+1}(x,0)\,dx\right|
+C\,{\varepsilon }^{1/\gamma} +o_k(1)\\
&{\leqslant}& \|(W_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}
+C\,{\varepsilon }^{1/\gamma} +o_k(1)\end{aligned}$$ where was used in the last line. Hence, thanks to , we get that $$\label{star-1}
\alpha_k^{p-1}{\leqslant}1+C\varepsilon^{1/2\gamma} +o_k(1).$$ Also, we notice that for this value of $\alpha_k$, we have the following: $$[\psi_k]^2_a=\alpha_k^2[W_k]_a^2=\alpha^{p+1}_k
\|(W_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}
=\|(\psi_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}.$$ Thus, by and Proposition \[traceIneq\], we obtain $$\begin{aligned}
&& S{\leqslant}\frac{[\psi_k(\cdot,0)]^2_{\dot{H}^s(\mathbb{R}^n)}}
{\|(\psi_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{2}}
=\frac{[\psi_k]_a^2}{\|(\psi_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{2}}
\\&&\qquad\qquad =\frac{\|(\psi_k)_+(\cdot,0)\|^{p+1}_{L^{p+1}(\mathbb{R}^n)}}
{\|(\psi_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{2}}
=\|(\psi_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{\frac{4s}{n-2s}}.\end{aligned}$$ In the last equality we have used the fact that $p+1=2^*_s$. Consequently, $$\|(W_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}
=\frac{\|(\psi_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}}{\alpha_k^{p+1}}
{\geqslant}S^{n/2s}\frac{1}{\alpha_k^{p+1}}.$$ This together with give that $$\begin{aligned}
S^{n/2s}&{\leqslant}&(1+C\varepsilon^{1/\gamma}+o_k(1))^{\frac{p+1}{p-1}}
\|(W_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}\\
&{\leqslant}& \|(W_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}+C\varepsilon^{1/\gamma}+o_k(1).\end{aligned}$$ Also, we observe that $$\frac12-\frac{1}{p+1}=\frac{s}{n}.$$ Hence, $$\begin{aligned}
{{\mathscrF}}_{\varepsilon }(W_k)-\frac{1}{2}\langle {{\mathscrF}}'_{\varepsilon }(W_k),W_k\rangle
&=&\frac{s}{n}\|(W_k)_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}\\
&&\qquad -{\varepsilon }\left(\frac{1}{q+1}-\frac{1}{2}\right)
\int_{\mathbb{R}^n}{h(x)(W_k)_+^{q+1}(x,0)\,dx}\\
&{\geqslant}&\frac{s}{n}S^{n/2s}-C\varepsilon^{1/\gamma}+o_k(1).\end{aligned}$$ Finally, using also , we get $$\label{LowBoundFW}
{{\mathscrF}}_{\varepsilon }(W_k){\geqslant}\frac{s}{n}S^{n/2s}-C\varepsilon^{1/\gamma}+o_k(1).$$\
[*Step 3: Lower bound for ${{\mathscrF}}_{\varepsilon }(U_k)$.*]{} We first observe that, thanks to , we can write $$\label{adwetperigyrejh}
U_k=(1-\chi)U_k+\chi U_k=W_k+V_k.$$ Therefore $$\begin{split}\label{sumF}
{{\mathscrF}}_{\varepsilon }(U_k) = &\, {{\mathscrF}}_{\varepsilon }(V_k)+{{\mathscrF}}_{\varepsilon }(W_k)
+\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla V_k,\nabla W_k\rangle\,dX} \\
&\quad +\frac{1}{p+1}\int_{\mathbb{R}^n}{(V_k)_+^{p+1}(x,0)\,dx}
\\&\quad
+\frac{{\varepsilon }}{q+1}\int_{\mathbb{R}^n}{h(x)(V_k)_+^{q+1}(x,0)\,dx}\\
&\quad +\frac{1}{p+1}\int_{\mathbb{R}^n}{(W_k)_+^{p+1}(x,0)\,dx}
\\&\quad
+\frac{{\varepsilon }}{q+1}\int_{\mathbb{R}^n}{h(x)(W_k)_+^{q+1}(x,0)\,dx}\\
&\quad -\frac{1}{p+1}\int_{\mathbb{R}^n}{(U_k)_+^{p+1}(x,0)\,dx}
\\&\quad
-\frac{{\varepsilon }}{q+1}\int_{\mathbb{R}^n}{h(x)(U_k)_+^{q+1}(x,0)\,dx}.
\end{split}$$ On the other hand, $$\begin{aligned}
&& \int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla V_k,\nabla W_k\rangle\,dX} \\
&&\qquad = \frac{1}{2}\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla U_k-\nabla V_k,\nabla V_k\rangle\,dX} +\frac{1}{2}\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla U_k-\nabla W_k,\nabla W_k\rangle\,dX}.\end{aligned}$$ Recall also that, according to , $$\begin{aligned}
&&\langle {{\mathscrF}}_\varepsilon'(U_k)-{{\mathscrF}}_\varepsilon'(V_k),V_k
\rangle \\&=&
\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_k-\nabla V_k, \nabla V_k\rangle\,dX \\
&&\qquad - {\varepsilon }\int_{{{\mathbb R}}^n}h(x)(U_k)_+^q(x,0)\,V_k(x,0)\,dx -
\int_{{{\mathbb R}}^n}(U_k(x,0))_+^p\,V_k(x,0)\,dx\\
&&\qquad +{\varepsilon }\int_{{{\mathbb R}}^n}h(x)(V_k)_+^{q+1}(x,0)\,dx +
\int_{{{\mathbb R}}^n}(V_k)_+^{p+1}(x,0)\,dx,\end{aligned}$$ and $$\begin{aligned}
&&\langle {{\mathscrF}}_\varepsilon'(U_k)-{{\mathscrF}}_\varepsilon'(W_k),W_k
\rangle \\&=&
\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_k-\nabla W_k, \nabla W_k\rangle\,dX \\
&&\qquad - {\varepsilon }\int_{{{\mathbb R}}^n}h(x)(U_k)_+^q(x,0)\,W_k(x,0)\,dx -
\int_{{{\mathbb R}}^n}(U_k)_+^p(x,0)\,W_k(x,0)\,dx\\
&&\qquad +{\varepsilon }\int_{{{\mathbb R}}^n}h(x)(W_k)_+^{q+1}(x,0)\,dx +
\int_{{{\mathbb R}}^n}(W_k)_+^{p+1}(x,0)\,dx.\end{aligned}$$ Hence, plugging the three formulas above into we get $$\begin{split}
{{\mathscrF}}_{\varepsilon }(U_k) = &\, {{\mathscrF}}_{\varepsilon }(V_k)+{{\mathscrF}}_{\varepsilon }(W_k)+\frac12
\langle {{\mathscrF}}_\varepsilon'(U_k)-{{\mathscrF}}_\varepsilon'(V_k),V_k
\rangle +\frac12\langle {{\mathscrF}}_\varepsilon'(U_k)-{{\mathscrF}}_\varepsilon'(W_k),W_k
\rangle \\
&\quad +\frac{1}{p+1}\int_{\mathbb{R}^n}{(V_k)_+^{p+1}(x,0)\,dx}
+\frac{{\varepsilon }}{q+1}\int_{\mathbb{R}^n}{h(x)(V_k)_+^{q+1}(x,0)\,dx}\\
&\quad +\frac{1}{p+1}\int_{\mathbb{R}^n}{(W_k)_+^{p+1}(x,0)\,dx}
+\frac{{\varepsilon }}{q+1}\int_{\mathbb{R}^n}{h(x)(W_k)_+^{q+1}(x,0)\,dx}\\
&\quad -\frac{1}{p+1}\int_{\mathbb{R}^n}{(U_k)_+^{p+1}(x,0)\,dx}
-\frac{{\varepsilon }}{q+1}\int_{\mathbb{R}^n}{h(x)(U_k)_+^{q+1}(x,0)\,dx}\\
&\quad +\frac{{\varepsilon }}{2}
\int_{{{\mathbb R}}^n}h(x)(U_k)_+^q(x,0)\,V_k(x,0)\,dx +
\frac12 \int_{{{\mathbb R}}^n}(U_k)_+^p(x,0)\,V_k(x,0)\,dx\\
&\quad -\frac{{\varepsilon }}{2}\int_{{{\mathbb R}}^n}h(x)(V_k)_+^{q+1}(x,0)\,dx -
\frac12\int_{{{\mathbb R}}^n}(V_k)_+^{p+1}(x,0)\,dx\\
&\quad + \frac{{\varepsilon }}{2} \int_{{{\mathbb R}}^n}h(x)(U_k)_+^q(x,0)
\,W_k(x,0)\,dx
+\frac12 \int_{{{\mathbb R}}^n}(U_k)_+^{p}(x,0)\,W_k(x,0)\,dx\\
&\quad -\frac{{\varepsilon }}{2}\int_{{{\mathbb R}}^n}h(x)(W_k)_+^{q+1}(x,0)\,dx - \frac12
\int_{{{\mathbb R}}^n}(W_k)_+^{p+1}(x,0)\,dx.
\end{split}$$ Notice that all the integrals with ${\varepsilon }$ in front are bounded. Therefore, using this and and we obtain that $$\begin{split}\label{qwqweteyryhg}
{{\mathscrF}}_{\varepsilon }(U_k) {\geqslant}&\, {{\mathscrF}}_{\varepsilon }(V_k)+{{\mathscrF}}_{\varepsilon }(W_k)\\
&\quad +\frac{1}{p+1}\int_{\mathbb{R}^n}{(V_k)_+^{p+1}(x,0)\,dx}
+\frac{1}{p+1}\int_{\mathbb{R}^n}{(W_k)_+^{p+1}(x,0)\,dx} \\
&\quad -\frac{1}{p+1}\int_{\mathbb{R}^n}{(U_k)_+^{p+1}(x,0)\,dx}
+\frac12 \int_{{{\mathbb R}}^n}(U_k)_+^{p}(x,0)\,V_k(x,0)\,dx\\
&\quad -
\frac12\int_{{{\mathbb R}}^n}(V_k)_+^{p+1}(x,0)\,dx
+\frac12 \int_{{{\mathbb R}}^n}(U_k)_+^p(x,0)\,W_k(x,0)\,dx\\
&\quad - \frac12
\int_{{{\mathbb R}}^n}(W_k)_+^{p+1}(x,0)\,dx -C{\varepsilon }^{1/\gamma},
\end{split}$$ for some positive $C$. We observe that, thanks to , $$\begin{aligned}
&&\int_{{{\mathbb R}}^n}(U_k)_+^p(x,0)\,V_k(x,0)\,dx + \int_{{{\mathbb R}}^n}(U_k)_+^p(x,0)\,W_k(x,0)\,dx\\
&=& \int_{{{\mathbb R}}^n}(U_k)_+^p(x,0)\,\big(V_k(x,0)+W_k(x,0)\big)\,dx\\
&=&\int_{{{\mathbb R}}^n}(U_k)_+^{p+1}(x,0)\,dx.\end{aligned}$$ Therefore, becomes $$\begin{split}
{{\mathscrF}}_{\varepsilon }(U_k) {\geqslant}&\, {{\mathscrF}}_{\varepsilon }(V_k)+{{\mathscrF}}_{\varepsilon }(W_k)\\
&\quad +\frac{1}{p+1}\int_{\mathbb{R}^n}{(V_k)_+^{p+1}(x,0)\,dx}
+\frac{1}{p+1}\int_{\mathbb{R}^n}{(W_k)_+^{p+1}(x,0)\,dx} \\
&\quad -\frac{1}{p+1}\int_{\mathbb{R}^n}{(U_k)_+^{p+1}(x,0)\,dx}
- \frac12 \int_{{{\mathbb R}}^n}(V_k)_+^{p+1}(x,0)\,dx\\
&\quad +\frac12 \int_{{{\mathbb R}}^n}(U_k)_+^{p+1}(x,0)\,dx- \frac12
\int_{{{\mathbb R}}^n}(W_k)_+^{p+1}(x,0)^{p+1}\,dx -C{\varepsilon }^{1/\gamma}\\
= &\, {{\mathscrF}}_{\varepsilon }(V_k)+{{\mathscrF}}_{\varepsilon }(W_k)\\
&\quad +\left(\frac12-\frac{1}{p+1}\right)\int_{\mathbb{R}^n}\left(
(U_k)_+^{p+1}(x,0)-(V_k)_+^{p+1}(x,0)-(W_k)_+^{p+1}(x,0)\right)\,dx-C{\varepsilon }^{1/\gamma}\\
= &\, {{\mathscrF}}_{\varepsilon }(V_k)+{{\mathscrF}}_{\varepsilon }(W_k)\\
&\quad +\left(\frac12-\frac{1}{p+1}\right)\int_{\mathbb{R}^n}(U_k)_+^{p+1}(x,0)\left(1-\chi^{p+1}(x,0)-(1-\chi(x,0))^{p+1}\right)\,dx-C{\varepsilon }^{1/\gamma},
\end{split}$$ where was used in the last line. Since $p+1>2$ and $$1-\chi^{p+1}(x,0)-(1-\chi(x,0))^{p+1}{\geqslant}0 \quad {\mbox{ for any }}x\in{{\mathbb R}}^n,$$ this implies that $${{\mathscrF}}_{\varepsilon }(U_k) {\geqslant}{{\mathscrF}}_{\varepsilon }(V_k)+{{\mathscrF}}_{\varepsilon }(W_k)-C{\varepsilon }^{1/\gamma}.$$ This, and imply that $${{\mathscrF}}_{\varepsilon }(U_k){\geqslant}\frac{s}{n}S^{n/2s}-c_1\varepsilon^{1/\gamma}-\bar{C}\varepsilon^{\frac{p+1}{p-q}}+o_k(1).$$ Hence, taking the limit as $k\to+\infty$ we obtain that $$c_{\varepsilon }=\lim_{k\to+\infty}{{\mathscrF}}_{\varepsilon }(U_k){\geqslant}\frac{s}{n}S^{n/2s}-c_1\varepsilon^{1/\gamma}-\bar{C}\varepsilon^{\frac{p+1}{p-q}},$$ which is a contradiction with assumption (i) of Proposition \[PScond\]. This concludes the proof of Lemma \[tightness\].
By Lemma \[tightness\], we know that $\{U_k\}_{k\in\mathbb{N}}$ is a tight sequence. Moreover, from Lemma \[lemma bound\] we have that $[U_k]_a{\leqslant}M$, for $M>0$. Hence, also $\{(U_k)_+\}_{k\in{{\mathbb N}}}$ is a bounded tight sequence in $\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$. Therefore, there exists $\overline{U}\in \dot{H}^s_a(\mathbb{R}^{n+1}_+)$ such that $$(U_k)_+\rightharpoonup \overline U \qquad\hbox{ in }\dot{H}^s_a(\mathbb{R}^{n+1}_+).$$
Also, we observe that Theorem 1.1.4 in [@evans] implies that there exist two measures on ${{\mathbb R}}^n$ and ${{\mathbb R}}^{n+1}_+$, $\nu$ and $\mu$ respectively, such that $(U_k)_+^{2^*_s}(x,0)$ converges to $\nu$ and $y^a|\nabla (U_k)_+|^2$ converges to $\mu$ as $k\to+\infty$, according to Definition \[convMeasures\].
Hence, we can apply Proposition \[CCP\] and we obtain that $$\label{point a}
{\mbox{$\displaystyle (U_k)_+^{2^*_s}(\cdot,0)$ converges to
$\nu = \overline U^{2^*_s}(\cdot,0) +\sum_{j\in J}{\nu_j\delta_{x_j}}$ as $k\to+\infty$, with $\nu_j{\geqslant}0$,}}$$ $$\label{point b}
{\mbox{$\displaystyle y^a|\nabla (U_k)_+|^2$ converges to
$\mu{\geqslant}y^a|\nabla\overline U|^2+\sum_{j\in J}{\mu_j\delta_{(x_j,0)}}$ as $k\to+\infty$,
with $\mu_j{\geqslant}0$,}}$$ and $$\label{point c}
\mu_j{\geqslant}S \nu_j^{2/2^*_s} \ {\mbox{ for all }} j\in J,$$ where $J$ is an at most countable set.
We want to prove that $\mu_j=\nu_j=0$ for any $j\in J$. For this, we suppose by contradiction that there exists $j\in J$ such that $\mu_j\neq 0$. We denote $X_j:=(x_j,0)$. Moreover, we fix $\delta>0$ and we consider a cut-off function $\phi_\delta\in C^\infty(\mathbb{R}^{n+1}_+,[0,1])$, defined as $$\phi_\delta(X)=
\begin{cases}
1,\qquad\hbox{ if }X\in B_{\delta/2}^+(X_j),\\
0,\qquad\hbox{ if }X\in (B_{\delta}^+(X_j))^c,
\end{cases}$$ with $|\nabla \phi_\delta|{\leqslant}\frac{C}{\delta}$.
We claim that there exists a constant $C>0$ such that $$\label{fi delta bounded}
[\phi_\delta\,(U_k)_+]_a{\leqslant}C.$$ Indeed, we compute $$\begin{aligned}
[\phi_\delta\,(U_k)_+]_a^2
&=&\int_{{{\mathbb R}}^{n+1}_+}y^a|\nabla(\phi_\delta (U_k)_+)|^2\,dX \\
&=& \int_{B^+_\delta(X_j)}y^a|\nabla\phi_\delta|^2(U_k)_+^2\,dX
+\int_{B^+_\delta(X_j)}y^a\,\phi_\delta^2|\nabla (U_k)_+|^2\,dX\\&&\qquad
+2\int_{B^+_\delta(X_j)}y^a\,\phi_\delta\,(U_k)_+
\langle\nabla\phi_\delta,\nabla (U_k)_+\rangle\,dX\\
&{\leqslant}& \frac{C^2}{\delta^2}\int_{B^+_\delta(X_j)}y^a|U_k|^2\,dX
+\int_{B^+_\delta(X_j)}y^a|\nabla U_k|^2\,dX \\&&\qquad
+\frac{2C}{\delta}\int_{B^+_\delta(X_j)}y^a|U_k|\,|\nabla U_k|\,dX\\
& {\leqslant}& C\,\left(\int_{B^+_\delta(X_j)}y^a|U_k|^{2\gamma}\,dX\right)^{1/\gamma}
+\int_{B^+_\delta(X_j)}y^a|\nabla U_k|^2\,dX \\
&& \qquad +C\,\left(\int_{B^+_\delta(X_j)}y^a|U_k|^{2\gamma}\,dX\right)^{1/2\gamma}
\left(\int_{B^+_\delta(X_j)}y^a|\nabla U_k|^2\,dX\right)^{1/2}\\
&{\leqslant}& C\,M^2,\end{aligned}$$ up to renaming $C$, where we have used Proposition \[WeightedSob\] and Lemma \[lemma bound\] in the last step. This shows .
Hence, from , and (ii) in Proposition \[PScond\] we deduce that $$\begin{split}\label{prima esp}
0 =&\,\lim_{k\rightarrow\infty}{\langle {{\mathscrF}}'_\varepsilon(U_k),\phi_\delta (U_k)_+\rangle}\\
=&\,\lim_{k\rightarrow\infty}\left(\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla U_k,\nabla (\phi_\delta (U_k)_+)\rangle\,dX}\right.\\
&\qquad -\left.\varepsilon\int_{\mathbb{R}^n}{h(x)\phi_\delta(x,0)
(U_k)_+^{q+1}(x,0)\,dx}
-\int_{\mathbb{R}^n}{\phi_\delta(x,0)(U_k)_+^{p+1}(x,0)\,dx}\right)\\
=&\,\lim_{k\rightarrow\infty}\left(\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla (U_k)_+|^2\phi_\delta\,dX}
+\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla (U_k)_+,\nabla \phi_\delta \rangle (U_k)_+\,dX}\right.\\
&\qquad -\left.\varepsilon\int_{\mathbb{R}^n}{h(x)\phi_\delta(x,0)(
U_k)_+^{q+1}(x,0)\,dx}
-\int_{\mathbb{R}^n}{\phi_\delta(x,0)(U_k)_+^{p+1}(x,0)\,dx}\right).
\end{split}$$ Now we recall that $p+1=2^*_s$, and so, using and , we have that $$\begin{aligned}
\label{conv12}
\lim_{k\to+\infty}
\int_{\mathbb{R}^n}{\phi_\delta(x,0)(U_k)_+^{p+1}(x,0)\,dx}&=&
\int_{\mathbb{R}^n}{\phi_\delta(x,0)\,d\nu}\\
\label{conv11}
{\mbox{and }}\ \lim_{k\to+\infty}
\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla (U_k)_+|^2\phi_\delta\,dX}&=&
\int_{\mathbb{R}^{n+1}_+}{\phi_\delta\,d\mu}.\end{aligned}$$ Also, we observe that supp$(\phi_\delta)\subseteq B_{\delta}^+(X_j)$. Moreover $[(U_k)_+(\cdot,0)]_{\dot{H}^s({{\mathbb R}}^n)}=[(U_k)_+]_a{\leqslant}M$, thanks to and Lemma \[lemma bound\]. Finally, the Hölder inequality and Proposition \[traceIneq\] imply that $\|(U_k)_+(\cdot,0)\|_{L^2(B_{\delta}^+(x_j))}{\leqslant}C$, for a suitable positive constant $C$. Hence, we can apply Theorem 7.1 in [@DPV] and we obtain that $$\label{q conv}
{\mbox{$(U_k)_+(\cdot,0)$ converges to $\overline U(\cdot,0)$ strongly
in $L^{r}(B_{\delta}^+(x_j))$ as $k\to+\infty$, for any $r\in[1,2]$.}}$$ Therefore, $$\begin{aligned}
&&\left| \int_{\mathbb{R}^n}{h(x)\phi_\delta(x,0)(U_k)_+^{q+1}(x,0)\,dx} -
\int_{\mathbb{R}^n} {h(x)\phi_\delta(x,0)\overline U^{q+1}(x,0)\,dx} \right|
\\
&&\qquad = \left|\int_{ B_{\delta}^+(X_j) \cap \{y=0\} } h(x)\,\phi_\delta(x,0)
((U_k)_+^{q+1}(x,0)-\overline U^{q+1}(x,0))\,dx\right|
\\
&&\qquad {\leqslant}\|h\|_{L^\infty({{\mathbb R}}^n)}
\left|\int_{ B_{\delta}^+(X_j)\cap \{y=0\} } ((U_k)_+^{q+1}(x,0)-\overline U^{q+1}(x,0))\,dx\right|,\end{aligned}$$ which together with implies that $$\label{conv2}
\lim_{k\to+\infty}
\int_{\mathbb{R}^n}{h(x)\phi_\delta(x,0)(U_k)_+^{q+1}(x,0)\,dx}=
\int_{B_\delta^+(X_j)\cap\{y=0\}}{h(x)\phi_\delta(x,0)\,\overline{U}^{q+1}(x,0)\,dx}.$$ Finally, taking the limit as $\delta\to 0$ we get $$\begin{split}\label{conv222}
&\lim_{\delta\to 0}\lim_{k\to+\infty}
\int_{\mathbb{R}^n}{h(x)\phi_\delta(x,0)(U_k)_+^{q+1}(x,0)\,dx}\\
&\qquad =\,
\lim_{\delta \to 0}\int_{B_\delta^+(X_j)\cap\{y=0\}}{h(x)\phi_\delta(x,0)\,\overline U^{q+1}(x,0)\,dx}=0.
\end{split}$$
Also, by the Hölder inequality and Lemma \[lemma bound\] we obtain that $$\begin{split}\label{conv3}
&\bigg|\int_{\mathbb{R}^{n+1}_+}y^a\langle\nabla (U_k)_+,
\nabla \phi_\delta \rangle (U_k)_+\,dX\bigg|
= \bigg|\int_{B_{\delta}^+(X_j) }y^a\langle\nabla (U_k)_+,\nabla \phi_\delta \rangle (U_k)_+\,dX\bigg|\\
&\qquad{\leqslant}\left(\int_{B_{\delta}^+(X_j)}{y^a|\nabla U_k|^2\,dX}\right)^{1/2}
\left(\int_{B_{\delta}^+(X_j)}{y^a(U_k)_+^2|\nabla \phi_\delta|^2\,dX}\right)^{1/2}\\
&\qquad{\leqslant}M\,\left(\int_{B_{\delta}^+(X_j)}{y^a(U_k)_+^2|\nabla \phi_\delta|^2\,dX}\right)^{1/2}.
\end{split}$$ Notice that, since $\{(U_k)_+\}$ is a bounded sequence in $\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$, using Lemma \[lemma:compact\], we have $$\label{wfrewphyrtyrjh}
\lim_{k\to+\infty}\int_{B_{\delta}^+(X_j)}{y^a(U_k)_+^2|\nabla \phi_\delta|^2\,dX}=
\int_{B_{\delta}^+(X_j)}{y^a\overline U^2|\nabla \phi_\delta|^2\,dX}.$$ Moreover, by the Hölder inequality, $$\label{conv4}
\int_{B_{\delta}^+(X_j)}y^a\overline U^2|\nabla \phi_\delta|^2\,dX{\leqslant}\left(\int_{B_{\delta}^+(X_j)}{y^a\overline U^{2\gamma}\,dX}\right)^{1/\gamma}\left(\int_{B_{\delta}^+(X_j)}{y^a|\nabla \phi_\delta|^{2\gamma'}\,dX}\right)^{1/\gamma'},$$ where $$\label{gamma primo}
\gamma'=\frac{n-2s+2}{2}.$$ Thus, taking into account that $|\nabla\phi_\delta|{\leqslant}\frac{C}{\delta}$, we have $$\begin{aligned}
\left(\int_{B_{\delta}^+(X_j)}{y^a|\nabla\phi_\delta|^{2\gamma'}\,dX}\right)^{1/\gamma'}
&{\leqslant}& \frac{C^2}{\delta^2}
\left(\int_{B_{\delta}^+(X_j)}{y^a\,dX}\right)^{1/\gamma'} \\
&{\leqslant}& \frac{C^2}{\delta^2}\,\delta^{\frac{n+1+a}{\gamma'}}.\end{aligned}$$ We recall and that $a=1-2s$, and we obtain that $$\frac{n+1+a}{\gamma'}-2= 0,$$ and so $$\left(\int_{B_{\delta}^+(X_j)}{y^a|\nabla\phi_\delta|^{2\gamma'}\,dX}\right)^{1/\gamma'}
{\leqslant}C^2.$$ This and give that $$\int_{B_{\delta}^+(X_j)}y^a\overline U^2|\nabla \phi_\delta|^2\,dX{\leqslant}C^2
\left(\int_{B_{\delta}^+(X_j)}{y^a\overline U^{2\gamma}\,dX}\right)^{1/\gamma},$$ Hence $$\lim_{\delta\to 0}\int_{B_{\delta}^+(X_j)}y^a\overline U^2|\nabla \phi_\delta|^2\,dX
{\leqslant}C^2\lim_{\delta\to 0}
\left(\int_{B_{\delta}^+(X_j)}{y^a\overline U^{2\gamma}\,dX}\right)^{1/\gamma}= 0.$$ From this and we obtain $$\label{conv5}
\lim_{\delta\to 0}\lim_{k\to+\infty}\int_{B_{\delta}^+(X_j)}{y^a(U_k)_+^2
|\nabla \phi_\delta|^2\,dX} =
\lim_{\delta\to 0}\int_{B_{\delta}^+(X_j)}{y^a\overline U^2|\nabla \phi_\delta|^2\,dX}=0.$$
Using , , and in , we obtain that $$\begin{split}\label{limk}
0 =&\,\lim_{\delta\to 0}\lim_{k\rightarrow\infty}{\langle {{\mathscrF}}'_\varepsilon(U_k),
\phi_\delta (U_k)_+\rangle}
\\=&\, \lim_{\delta\to 0}\left(\int_{\mathbb{R}^{n+1}_+}{\phi_\delta\,d\mu}
- \int_{\mathbb{R}^n}{\phi_\delta(x,0)\,d\nu}\right)
\\=&\, \lim_{\delta\to 0}\left(\int_{B_\delta^+(X_j)}{\phi_\delta\,d\mu}
- \int_{B_\delta^+(X_j)\cap \{y=0\}}{\phi_\delta(x,0)\,d\nu}\right)\\
{\geqslant}&\, \mu_j-\nu_j,
\end{split}$$ thanks to and . Therefore, recalling , we obtain that $$\nu_j{\geqslant}\mu_j{\geqslant}S\, \nu_j^{2/2^*_s}.$$ Hence, either $\nu_j=\mu_j=0$ or $\nu_j^{1-2/2^*}{\geqslant}S$. Since we are assuming that $\mu_j\neq 0$, the first possibility cannot occur, and so, from the second one, we have that $$\label{maggiore-1} \nu_j{\geqslant}S^{n/2s}.$$
Now, from Lemma \[lemma bound\] we know that $[(U_k)_+(\cdot,0)]_a{\leqslant}M$. Moreover we observe that $q+1<2<2^*_s$. Hence Proposition \[traceIneq\] and the compact embedding in Theorem 7.1 in [@DPV] imply that $$\begin{split}&\|(U_k)_+(\cdot,0)-\overline U(\cdot,0)\|_{L^{2^*_s}(\mathbb{R}^n)}{\leqslant}2M\\
\hbox{ and }\quad & (U_k)_+(\cdot,0)\rightarrow \overline U(\cdot,0)\hbox{ in }L^{q+1}_{loc}(\mathbb{R}^n)
{\mbox{ as }}k\to+\infty.
\end{split}$$ Therefore, recalling and , and using the Hölder inequality, we obtain $$\begin{aligned}
&&\left|\int_{\mathbb{R}^{n}}h(x)\left((U_k)_+(x,0)-\overline U(x,0)\right)^{q+1}\,dx\right|\\
&&\qquad {\leqslant}\int_{B_R}|h(x)|\,|(U_k)_+(x,0)-\overline U(x,0)|^{q+1}\,dx
+\int_{\mathbb{R}^{n}\setminus B_R}|h(x)|\,
|(U_k)_+(x,0)- \overline U(x,0)|^{q+1}\,dx\\
&&\qquad {\leqslant}\|h\|_{L^\infty(\mathbb{R}^n)}\|((U_k)_+-\overline U)(\cdot,0)\|_{L^{q+1}(B_R)}
+\|h\|_{L^\alpha(\mathbb{R}^n\setminus B_R)}\|((U_k)_+-\overline U)(\cdot,0)\|_{L^{2^*_s}(\mathbb{R}^n)}^{q+1}\\
&&\qquad {\leqslant}C\, \|((U_k)_+-\overline U)(\cdot,0)\|_{L^{q+1}(B_R)}
+(2M)^{q+1}\|h\|_{L^\alpha(\mathbb{R}^n\setminus B_R)}, \end{aligned}$$ where $\alpha$ satisfies $\displaystyle \frac{1}{\alpha}=1-\frac{q+1}{2^*_s}$. Hence, letting first $k\to+\infty$ and then $R\to+\infty$, we conclude that $$\label{doppiabis}
\lim_{k\to+\infty}\int_{\mathbb{R}^{n}}h(x){(U_k)_+^{q+1}(x,0)\,dx}=
\int_{\mathbb{R}^n}{h(x)\overline U^{q+1}(x,0)\,dx}.$$
On the other hand, let $\{\varphi_m\}_{m\in\mathbb{N}}\in C_0^\infty(\mathbb{R}^n)$ be a sequence such that $0{\leqslant}\varphi_m{\leqslant}1$ and $\displaystyle \lim_{m\rightarrow\infty}\varphi_m(x)=1$ for all $x\in\mathbb{R}^n$. Thus, by , we have that $$\lim_{k\to+\infty}\int_{\mathbb{R}^n}{(U_k)_+^{p+1}(x,0)\,dx}{\geqslant}\lim_{k\to +\infty}\int_{\mathbb{R}^n}{(U_k)^{p+1}_+(x,0)\varphi_m\,dx}
=\int_{\mathbb{R}^n}{\varphi_m\,d\nu}.$$ Furthermore, by Fatou’s lemma and , $$\lim_{m\rightarrow\infty}{\int_{\mathbb{R}^n}{\varphi_m\,d\nu}}{\geqslant}\int_{\mathbb{R}^n}{\,d\nu}{\geqslant}S^{n/2s}+\int_{\mathbb{R}^n}{\overline U^{p+1}(x,0)\,dx}.$$ So, using the last two formulas we get $$\begin{split}\label{pwqotpytiyoi}
\lim_{k\rightarrow\infty}{\int_{\mathbb{R}^n}{(U_k)_+^{p+1}(x,0)\,dx}} =&\,
\lim_{m\to+\infty}\lim_{k\rightarrow\infty}{\int_{\mathbb{R}^n}{(U_k)_+^{p+1}(x,0)\,dx}} \\
{\geqslant}&\, \lim_{m\to+\infty} {\int_{\mathbb{R}^n}{\varphi_m\,d\nu}}\\
{\geqslant}&\, S^{n/2s}+\int_{\mathbb{R}^n}{\overline U^{p+1}(x,0)\,dx}.
\end{split}$$
Now, since $[U_k]_a{\leqslant}M$ (thanks to Lemma \[lemma bound\]), from (ii) in Proposition \[PScond\] we have that $$\lim_{k\to+\infty}\langle {{\mathscrF}}_\varepsilon'(U_k),U_k\rangle=0,$$ and so, by hypothesis (i) we get $$\label{alklasfhhsgdkjg}
\lim_{k\rightarrow\infty}{\left({{\mathscrF}}_\varepsilon(U_k)
-\frac{1}{2}\langle {{\mathscrF}}_\varepsilon'(U_k),U_k\rangle\right)}=c_{\varepsilon }.$$ On the other hand, $$\begin{aligned}
&&\lim_{k\rightarrow\infty}{\left({{\mathscrF}}_\varepsilon(U_k)
-\frac{1}{2}\langle {{\mathscrF}}_\varepsilon'(U_k),U_k\rangle\right)}\\
&=& \lim_{k\to+\infty}\left(\left(\frac12-\frac{1}{p+1}\right)\int_{{{\mathbb R}}^n}(U_k)_+^{p+1}(x,0)\,dx
-{\varepsilon }\left(\frac{1}{q+1}-\frac12\right)\int_{{{\mathbb R}}^n}h(x)(U_k)_+^{q+1}(x,0)\,dx\right).\end{aligned}$$ We notice that $$\frac12-\frac{1}{p+1}=\frac{s}{n},$$ and so from and we obtain that $$\begin{aligned}
&&\lim_{k\rightarrow\infty}{\left({{\mathscrF}}_\varepsilon(U_k)
-\frac{1}{2}\langle {{\mathscrF}}_\varepsilon'(U_k),U_k\rangle\right)}\\
&{\geqslant}&\frac{s}{n}S^{n/2s}+\frac{s}{n}\int_{\mathbb{R}^n}{\overline U^{p+1}(x,0)\,dx}
-\varepsilon\left(\frac{1}{q+1}-\frac{1}{2}\right)\int_{\mathbb{R}^n}{h(x)\overline U^{q+1}(x,0)\,dx}\\
&{\geqslant}&\frac{s}{n}S^{n/2s}+\frac{s}{n}\|\overline U(\cdot,0)\|^{p+1}_{L^{p+1}({{\mathbb R}}^n)}
-\varepsilon\left(\frac{1}{q+1}-\frac{1}{2}\right)\|h\|_{L^m({{\mathbb R}}^n)}
\|\overline U(\cdot,0)\|^{q+1}_{L^{p+1}({{\mathbb R}}^n)}\\
&{\geqslant}& \frac{s}{n}S^{n/2s}-\bar{C}{\varepsilon }^{\frac{p+1}{p-q}},\end{aligned}$$ where we have applied Lemma \[lemma basic\] with $\alpha:=\|\overline U(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}$ in the last line. This and imply that $$c_{\varepsilon }{\geqslant}\frac{s}{n}S^{n/2s}-\bar{C}{\varepsilon }^{\frac{p+1}{p-q}},$$ which gives a contradiction with .
Therefore, necessarily $\mu_j=\nu_j=0$. Repeating this argument for every $j\in J$, we obtain that $\mu_j=\nu_j=0$ for any $j\in J$. Hence, by , $$\label{chiama-1}
\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}(U_k)_+^{2^*_s}(x,0)\varphi\,dx =
\int_{{{\mathbb R}}^n}\overline U^{2^*_s}(x,0)\varphi\,dx,$$ for any $\varphi\in C_0({{\mathbb R}}^n)$.
Then the desired result will follow. Indeed, we use Lemmata \[PSL-1\] and \[PSL-2\], with $v_k(x):=(U_k)_+(x,0)$ and $v(x):=\overline U(x,0)$. More precisely, condition is guaranteed by , while condition follows from Lemma \[tightness\]. This says that we can use Lemma \[PSL-2\] and obtain that $(U_k)_+(\cdot,0)\to \overline U(\cdot,0)$ in $L^{2^*_s}({{\mathbb R}}^n,[0,+\infty))$. With this, the assumptions of Lemma \[PSL-1\] are satisfied, which in turn gives that $$\begin{aligned}
&&
\lim_{k\to+\infty}\int_{{{\mathbb R}}^n} |(U_k)_+^q(x,0)
-\overline U^q(x,0)|^{\frac{2^*_s}{q}}\,dx=0
\\
{\mbox{and }}&&
\lim_{k\to+\infty}\int_{{{\mathbb R}}^n} |(U_k)_+^p(x,0)
-\overline U^p(x,0)|^{\frac{2n}{n+2s}}\,dx=0.\end{aligned}$$ Therefore, we can fix $\delta\in(0,1)$, to be taken arbitrarily small in the sequel, and say that $$\label{0vjewrsjn029}
\begin{split}
& \int_{{{\mathbb R}}^n} |(U_k)_+^q(x,0)
-(U_m)_+^q(x,0)|^{\frac{2^*_s}{q}}\,dx
\\ &\qquad+ \int_{{{\mathbb R}}^n} |(U_k)_+^p(x,0)
-(U_m)_+^p(x,0)|^{\frac{2n}{n+2s}}\,dx{\leqslant}\delta
\end{split}$$ for any $k$, $m$ large enough, say larger than some $k_\star(\delta)$.
Let us now take $\Phi\in \dot H^s_a({{\mathbb R}}^{n+1}_+)$ with $$\label{11-0dvf67dd}
[\Phi]_a=1.$$ By assumption (ii) in Proposition \[PScond\] we know that for large $k$ (again, say, up to renaming quantities, that $k{\geqslant}k_\star(\delta)$), $$|\langle {{{\mathscrF}}}'_{\varepsilon }(U_k),\Phi\rangle|{\leqslant}\delta.$$ This and say that $$\begin{aligned}
&&\Big| \int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla U_k(X),\nabla \Phi(X)\rangle\,dX\\
&&\qquad -{\varepsilon }\int_{{{\mathbb R}}^n} h(x)\,(U_k)_+^{q}(x,0) \phi(x)\,dx
-\int_{{{\mathbb R}}^n} (U_k)_+^p(x,0)\phi(x)\,dx
\Big|{\leqslant}\delta,\end{aligned}$$ where $\phi(x):=\Phi(x,0)$. In particular, for $k$, $m{\geqslant}k_\star(\delta)$, $$\begin{aligned}
&&\Big| \int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla (U_k(X)-U_m(X)),\nabla \Phi(X)\rangle\,dX\\
&&\qquad\qquad
-{\varepsilon }\int_{{{\mathbb R}}^n} h(x)\,\big( (U_k)_+^q(x,0)-(U_m)_+^q(x,0)\big) \phi(x)\,dx
\\&&\qquad\qquad -\int_{{{\mathbb R}}^n} \big((U_k)_+^p(x,0)-(U_m)_+^p(x,0)\big)\phi(x)\,dx
\Big|{\leqslant}2\delta.\end{aligned}$$ So, using the Hölder inequality with exponents $\frac{2n}{n+2s-q(n-2s)}$ $\frac{2^*_s}{q}=\frac{2n}{q(n-2s)}$ and $2^*_s=\frac{2n}{n-2s}$, and with exponents $\frac{2^*_s}{p}=\frac{2n}{n+2s}$ and $2^*_s$, we obtain $$\begin{aligned}
&& \left| \int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla (U_k(X)-U_m(X)),\nabla \Phi(X)\rangle\,dX\right|
\\&{\leqslant}&
\left|\int_{{{\mathbb R}}^n} h(x)\,\big( (U_k)_+^q(x,0)-(U_m)_+^q(x,0)\big)
\phi(x)\,dx\right| \\&&\qquad +
\left|\int_{{{\mathbb R}}^n} \big((U_k)_+^p(x,0)-(U_m)_+^p(x,0)\big)\phi(x)\,dx
\right|+2\delta\\
&{\leqslant}&
\left[ \int_{{{\mathbb R}}^n} |h(x)|^{\frac{2n}{n+2s-q(n-2s)}} \,dx \right]^{
\frac{n+2s-q(n-2s)}{2n} }\\ &&\qquad\qquad \cdot
\left[ \int_{{{\mathbb R}}^n} \big| (U_k)_+^q(x,0)-(U_m)_+^q(x,0)\big|^{\frac{2^*_s}{q}}
\,dx \right]^{
\frac{q(n-2s)}{2n} }
\left[ \int_{{{\mathbb R}}^n} |\phi(x)|^{2^*_s}\,dx \right]^{\frac{1}{2^*_s}}
\\ &&\qquad+
\left[
\int_{{{\mathbb R}}^n} \big|(U_k)_+^p(x,0)-(U_m)_+^p(x,0)\big|^{
\frac{2n}{n+2s} }\,dx
\right]^{ \frac{n+2s}{2n} }
\left[ \int_{{{\mathbb R}}^n} |\phi(x)|^{2^*_s}\,dx \right]^{\frac{1}{2^*_s}}
+2\delta.\end{aligned}$$ Thus, by and , $$\left| \int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla (U_k(X)-U_m(X)),\nabla \Phi(X)\rangle\,dX\right|{\leqslant}C\, \delta^{ \frac{q(n-2s)}{2n} }\,\|\phi\|_{L^{2^*_s}({{\mathbb R}}^n)}
+ C\,\delta^{ \frac{n+2s}{2n} } \,\|\phi\|_{L^{2^*_s}({{\mathbb R}}^n)}
+2\delta,$$ for some $C>0$. Now, by and , we have that $\|\phi\|_{L^{2^*_s}({{\mathbb R}}^n)}{\leqslant}S^{-1/2}
[\Phi]_a=S^{-1/2}$, therefore, up to renaming constants, $$\left| \int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla (U_k(X)-U_m(X)),\nabla \Phi(X)\rangle\,dX\right|{\leqslant}C\delta^\gamma,$$ for some $C$, $\gamma>0$, as long as $k$, $m{\geqslant}k_\star(\delta)$. Since this inequality is valid for any $\Phi$ satisfying , we have proved that $$[ U_k-U_m]_a{\leqslant}C\delta^\gamma,$$ that says that $U_k$ is a Cauchy sequence in $\dot H^s_a({{\mathbb R}}^{n+1}_+)$, and then the desired result follows.
Proof of Theorem \[MINIMUM\] {#concaveFirstSol}
----------------------------
With all this, we are in the position to prove Theorem \[MINIMUM\].
We recall that holds true. Thus, applying the Hölder inequality and Proposition \[traceIneq\], for $U\in\dot{H}^s_a(\mathbb{R}^{n+1}_+)$ we have $$\begin{aligned}
{{\mathscrF}}_\varepsilon(U)&{\geqslant}&
\frac{1}{2}[U]_a^2-c_1\|U_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}
- c_2{\varepsilon }\,\|h\|_{L^m({{\mathbb R}}^n)}\|U_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{q+1}\\
&{\geqslant}&\frac{1}{2}[U]_a^2-\tilde{c}_1[U]_a^{p+1}-\varepsilon \tilde{c}_2[U]_a^{q+1}.\end{aligned}$$ We consider the function $$\phi(t)=\frac{1}{2}t^2-\tilde{c}_1 t^{p+1}-\varepsilon\tilde{c}_2 t^{q+1}, \qquad t{\geqslant}0.$$ Since $q+1<2<p+1$, we have that for every $\varepsilon>0$ we can find $\rho=\rho(\varepsilon)>0$ satisfying $\phi(\rho)=0$ and $\phi(t)<0$ for any $t\in(0,\rho)$. As a matter of fact, $\rho$ is the first zero of the function $\phi$.
Furthermore, it is not difficult to see that $$\label{rho0}
\rho(\varepsilon)\rightarrow 0\quad \hbox{ as }\varepsilon\rightarrow 0.$$ Thus, there exist $c_0>0$ and ${\varepsilon }_0>0$ such that for all $\varepsilon<\varepsilon_0$, $$\label{behavF}\begin{cases}
{{\mathscrF}}_\varepsilon(U){\geqslant}-c_0\qquad \hbox{ if }[U]_a<\rho({\varepsilon }_0),\\
{{\mathscrF}}_\varepsilon(U)>0\qquad \hbox{ if }[U]_a=\rho({\varepsilon }_0).
\end{cases}$$ Now we take $\varphi\in C_0^\infty(\mathbb{R}^{n+1}_+)$, $\varphi{\geqslant}0$, $[\varphi]_a=1$, and such that supp$(\varphi(\cdot,0))\subset B$, where $B$ is given in condition . Hence, for any $t>0$, $$\begin{aligned}
{{\mathscrF}}_\varepsilon(t\varphi)&=&
\frac{1}{2}t^2-\frac{\varepsilon}{q+1}t^{q+1}
\int_{\mathbb{R}^n}{h(x)\varphi^{q+1}(x,0)\,dx}-
\frac{t^{p+1}}{p+1}\int_{\mathbb{R}^n}{\varphi^{p+1}(x,0)\,dx}
\\&{\leqslant}&
\frac{1}{2}t^2-\frac{\varepsilon}{q+1}t^{q+1}\inf_B h
\int_{B}{\varphi^{q+1}(x,0)\,dx}-
\frac{t^{p+1}}{p+1}\int_{B}{\varphi^{p+1}(x,0)\,dx}
.\end{aligned}$$ This inequality and condition give that, for any ${\varepsilon }<{\varepsilon }_0$ (possibly taking ${\varepsilon }_0$ smaller) there exists $t_0<\rho({\varepsilon }_0)$ such that, for any $t<t_0$, we have $${{\mathscrF}}_\varepsilon{(t\varphi)}<0.$$ This implies that $$i_\varepsilon:=\inf_{U\in\dot{H}^s_a(\mathbb{R}^{n+1}_+), [U]_a< \rho({\varepsilon }_0)}
{{{\mathscrF}}_\varepsilon(U)}<0.$$ This and give that, for $0<\varepsilon<\varepsilon_0$, $$-\infty<-c_0{\leqslant}i_\varepsilon<0.$$
Now we take a minimizing sequence $\{U_k\}$ and we observe that $$\lim_{{\varepsilon }\to 0}\lim_{k\to+\infty} {{{\mathscrF}}_\varepsilon(U_k)}
=\lim_{{\varepsilon }\to 0}i_{\varepsilon }{\leqslant}0<\dfrac{s}{n}S^{\frac{n}{2s}}.$$ Hence, condition is satisfied with $c_{\varepsilon }:=i_{\varepsilon }$, and so we can apply Proposition \[PScond\] and we conclude that $i_\varepsilon$ is attained at some minimum $U_\varepsilon$.
Finally, since $[U_\varepsilon]_a{\leqslant}\rho(\varepsilon_0)$, implies that $U_{\varepsilon }$ converges to 0 in $\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$ as ${\varepsilon }$ tends to 0. This concludes the proof of Theorem \[MINIMUM\].
Regularity and positivity of the solution {#7xucjhgfgh345678}
=========================================
A regularity result {#sec:reg}
-------------------
In this section we show a regularity result[^4] that allows us to say that a nonnegative solution to is bounded.
\[prop:bound\] Let $u\in \dot{H}^s({{\mathbb R}}^n)$ be a nonnegative solution to the problem $$(-\Delta)^su=f(x,u)\qquad\hbox{ in }{{\mathbb R}}^n,$$ and assume that $|f(x,t)|{\leqslant}C(1+|t|^p)$, for some $1{\leqslant}p{\leqslant}2^*_s-1$ and $C>0$. Then $u\in L^\infty({{\mathbb R}}^n)$.
Let $\beta{\geqslant}1$ and $T>0$, and let us define $$\varphi(t)=\begin{cases} 0, \hbox{ if }t{\leqslant}0,\\
t^\beta, \hbox{ if }0<t<T,\\
\beta T^{\beta-1}(t-T)+T^\beta,\hbox{ if }t{\geqslant}T.
\end{cases}$$ Since $\varphi$ is convex and Lipschitz, $$\label{unoqqq}
\varphi(u)\in \dot{H}^s({{\mathbb R}}^n)$$ and $$\label{dueqqq}
(-\Delta)^s\varphi(u){\leqslant}\varphi'(u)(-\Delta)^su$$ in the weak sense.
We recall that and Proposition \[traceIneq\] imply that, for any $u\in\dot{H}^s({{\mathbb R}}^n)$, $$\|u\|_{L^{2^*_s}({{\mathbb R}}^n)}{\leqslant}S^{-2}[u]_{\dot{H}^s({{\mathbb R}}^n)}.$$ Moreover, by Proposition 3.6 in [@DPV] we have that $$[u]_{\dot{H}^s({{\mathbb R}}^n)}= \|(-\Delta)^{s/2}u\|_{L^2({{\mathbb R}}^n)}.$$ Hence, from , an integration by parts and we deduce that $$\begin{aligned}
&&\|\varphi(u)\|_{L^{2^*_s}({{\mathbb R}}^n)}^2{\leqslant}S^{-1}\int_{{{\mathbb R}}^n}|(-\Delta)^{s/2}
\varphi(u)|^2\,dx \\&&\qquad\quad =
S^{-1}\int_{{{\mathbb R}}^n}\varphi(u)(-\Delta)^{s}
\varphi(u)\,dx {\leqslant}S^{-1}\int_{{{\mathbb R}}^n}\varphi(u)\,\varphi'(u)(-\Delta)^{s}u\,dx. \end{aligned}$$ Therefore, from the assumption on $u$ and $f$ we obtain $$\begin{aligned}
\|\varphi(u)\|_{L^{2^*_s}({{\mathbb R}}^n)}^2&{\leqslant}& S^{-1}\int_{{{\mathbb R}}^n}\varphi(u)\,\varphi'(u)(1+u^{2^*_s-1})\,dx \\&=& S^{-1}\left(\int_{{{\mathbb R}}^n}\varphi(u)\,\varphi'(u)\,dx
+\int_{{{\mathbb R}}^n}\varphi(u)\,\varphi'(u)\,u^{2^*_s-1}\,dx\right).\end{aligned}$$ Using that $\varphi(u)\varphi'(u){\leqslant}\beta u^{2\beta-1}$ and $u\varphi'(u){\leqslant}\beta \varphi(u)$, we have $$\label{boundBeta}
\left(\int_{{{\mathbb R}}^n}(\varphi(u))^{2^*_s}\right)^{2/2^*_s}{\leqslant}C\beta\left(\int_{{{\mathbb R}}^n}u^{2\beta-1}\,dx+\int_{{{\mathbb R}}^n}(\varphi(u))^2u^{2^*_s-2}\,dx\right),$$ where $C$ is a positive constant that does not depend on $\beta$. Notice that the last integral is well defined for every $T$ in the definition of $\varphi$. Indeed, $$\begin{split}
\int_{{{\mathbb R}}^n}(\varphi(u))^2u^{2^*_s-2}\,dx&\,=\int_{\{u{\leqslant}T\}}(\varphi(u))^2u^{2^*_s-2}\,dx+\int_{\{u>T\}}(\varphi(u))^2u^{2^*_s-2}\,dx\\
&{\leqslant}T^{2\beta-2}\int_{{{\mathbb R}}^n}u^{2^*_s}\,dx+C\int_{{{\mathbb R}}^n}u^{2^*_s}\,dx<+\infty,
\end{split}$$ where we have used that $\beta>1$ and that $\varphi(u)$ is linear when $u{\geqslant}T$. We choose now $\beta$ in such that $2\beta-1=2^*_s$, and we name it $\beta_1$, that is, $$\label{scelta beta}
\beta_1:=\frac{2^*_s+1}{2}.$$
Let $R>0$ to be fixed later. Attending to the last integral in and applying the Hölder’s inequality with exponents $r:=2^*_s/2$ and $r':=2^*_s/(2^*_s-2)$, $$\begin{split}\label{boundBeta2}
\int_{{{\mathbb R}}^n}(\varphi(u))^2&u^{2^*_s-2}\,dx=\int_{\{u{\leqslant}R\}}(\varphi(u))^2u^{2^*_s-2}\,dx+\int_{\{u>R\}}(\varphi(u))^2u^{2^*_s-2}\,dx\\
&{\leqslant}\int_{\{u{\leqslant}R\}}\frac{(\varphi(u))^2}{u}R^{2^*_s-1}\,dx+\left(\int_{{{\mathbb R}}^n}(\varphi(u))^{2^*_s}\,dx\right)^{2/2^*_s}\left(\int_{\{u>R\}}u^{2^*_s}\,dx\right)^{\frac{2^*_s-2}{2^*_s}}.
\end{split}$$ By the Monotone Convergence Theorem, we can choose $R$ large enough so that $$\left(\int_{\{u>R\}}u^{2^*_s}\,dx\right)^{\frac{2^*_s-2}{2^*_s}}{\leqslant}\frac{1}{2C\beta_1},$$ where $C$ is the constant appearing in . Therefore, we can absorb the last term in by the left hand side of to get $$\left(\int_{{{\mathbb R}}^n}(\varphi(u))^{2^*_s}\,dx\right)^{2/2^*_s}{\leqslant}2C\beta_1\left(\int_{{{\mathbb R}}^n}u^{2^*_s}\,dx+R^{2^*_s-1}\int_{{{\mathbb R}}^n}\frac{(\varphi(u))^2}{u}\,dx\right),$$ where is also used. Now we use that $\varphi(u){\leqslant}u^{\beta_1}$ and once again in the right hand side and we take $T\rightarrow\infty$: we obtain $$\left(\int_{{{\mathbb R}}^n}u^{2^*_s\beta_1}\,dx\right)^{2/2^*_s}{\leqslant}2C\beta_1\left(\int_{{{\mathbb R}}^n}u^{2^*_s}\,dx+R^{2^*_s-1}\int_{{{\mathbb R}}^n}u^{2^*_s}\,dx\right)<+\infty,$$ and therefore $$\label{alkdhghuhu}
u\in L^{2^*_s\beta_1}({{\mathbb R}}^n).$$
Let us suppose now $\beta>\beta_1$. Thus, using that $\varphi(u){\leqslant}u^\beta$ in the right hand side of and letting $T\rightarrow\infty$ we get $$\label{boundBeta3}
\left(\int_{{{\mathbb R}}^n}u^{2^*_s\beta}\,dx\right)^{2/2^*_s}{\leqslant}C\beta \left(\int_{{{\mathbb R}}^n}u^{2\beta-1}\,dx+\int_{{{\mathbb R}}^n}u^{2\beta+2^*_s-2}\,dx\right).$$ Furthermore, we can write $$u^{2\beta-1}=u^au^b,$$ with $a:=\frac{2^*_s(2^*_s-1)}{2(\beta-1)}$ and $b:=2\beta-1-a.$ Notice that, since $\beta>\beta_1$, then $0<a,b<2^*_s$. Hence, applying Young’s inequality with exponents $$r:=2^*_s/a \quad \hbox{ and }\quad r':=2^*_s/(2^*_s-a),$$ there holds $$\begin{split}
\int_{{{\mathbb R}}^n}u^{2\beta-1}\,dx &{\leqslant}\frac{a}{2^*_s}\int_{{{\mathbb R}}^n}u^{2^*_s}\,dx+\frac{2^*_s-a}{2^*_s}\int_{{{\mathbb R}}^n}u^{\frac{2^*_sb}{2^*_s-a}}\,dx\\
&{\leqslant}\int_{{{\mathbb R}}^n}u^{2^*_s}\,dx+\int_{{{\mathbb R}}^n}u^{2\beta+2^*_s-2}\,dx\\
&{\leqslant}C\left(1+\int_{{{\mathbb R}}^n}u^{2\beta+2^*_s-2}\,dx\right),
\end{split}$$ with $C>0$ independent of $\beta$. Plugging this into , $$\left(\int_{{{\mathbb R}}^n}u^{2^*_s\beta}\,dx\right)^{2/2^*_s}{\leqslant}C\beta \left(1+\int_{{{\mathbb R}}^n}u^{2\beta+2^*_s-2}\,dx\right),$$ with $C$ changing from line to line, but remaining independent of $\beta$. Therefore, $$\label{boundBeta4}
\left(1+\int_{{{\mathbb R}}^n}u^{2^*_s\beta}\,dx\right)^{\frac{1}{2^*_s(\beta-1)}}{\leqslant}(C\beta)^{\frac{1}{2(\beta-1)}} \left(1+\int_{{{\mathbb R}}^n}u^{2\beta+2^*_s-2}\,dx\right)^{\frac{1}{2(\beta-1)}},$$ that is (2.6) in [@bcss Proposition 2.2]. From now on, we follow exactly their iterative argument. That is, we define $\beta_{m+1}$, $m{\geqslant}1$, so that $$2\beta_{m+1}+2^*_s-2=2^*_s\beta_m.$$ Thus, $$\beta_{m+1}-1=\left(\frac{2^*_s}{2}\right)^m(\beta_1-1),$$ and replacing in it yields $$\left(1+\int_{{{\mathbb R}}^n}u^{2^*_s\beta_{m+1}}\,dx\right)^{\frac{1}{2^*_s(\beta_{m+1}-1)}}{\leqslant}(C\beta_{m+1})^{\frac{1}{2(\beta_{m+1}-1)}} \left(1+\int_{{{\mathbb R}}^n}u^{2^*_s\beta_m}\,dx\right)^{\frac{1}{2^*_s(\beta_m-1)}}.$$ Defining $C_{m+1}:=C\beta_{m+1}$ and $$A_m:=\left(1+\int_{{{\mathbb R}}^n}u^{2^*_s\beta_m}\,dx\right)^{\frac{1}{2^*_s(\beta_m-1)}},$$ we conclude that there exists a constant $C_0>0$ independent of $m$, such that $$A_{m+1}{\leqslant}\prod_{k=2}^{m+1}C_k^{\frac{1}{2(\beta_k-1)}}A_1{\leqslant}C_0A_1.$$ Thus, $$\|u\|_{L^\infty({{\mathbb R}}^n)}{\leqslant}C_0A_1<+\infty,$$ thanks to . This finishes the proof of Proposition \[prop:bound\].
\[coro:bound\] Let $u\in\dot{H}^s({{\mathbb R}}^n)$ be a solution of and let $U$ be its extension, according to .
Then $u\in L^\infty({{\mathbb R}}^n)$, and $U\in L^\infty({{\mathbb R}}^{n+1}_+)$.
First we observe that $u{\geqslant}0$, thanks to Proposition \[prop:pos\]. Moreover, since $u$ is a solution to , it solves $$(-\Delta)^su=f(x,u) \quad {\mbox{ in }}{{\mathbb R}}^n,$$ where $f(x,t):={\varepsilon }h(x)t^q_++t^p_+$. It is easy to check that $f$ satisfies the hypotheses of Proposition \[prop:bound\]. Hence the boundedness of $u$ simply follows from Proposition \[prop:bound\].
Let us now show the $L^\infty$ estimate for $U$. According to , for any $(x,z)\in{{\mathbb R}}^{n+1}_+$, $$U(x,z)=\int_{{{\mathbb R}}^n}u(x-y)\,P_s(y,z)\,dy.$$ Therefore, $$|U(x,z)|{\leqslant}\|u\|_{L^\infty({{\mathbb R}}^n)}\int_{{{\mathbb R}}^n}P_s(y,z)\,dy=\|u\|_{L^\infty({{\mathbb R}}^n)},$$ for any $(x,z)\in{{\mathbb R}}^{n+1}_+$, which implies the $L^\infty$-bound for $U$, and concludes the proof of the corollary.
Finally, we can prove that a solution to is continuous, as stated in the following:
\[coro:conti\] Let $u\in\dot{H}^s({{\mathbb R}}^n)$ be a solution of and let $U$ be its extension, according to .
Then $u\in C^\alpha({{\mathbb R}}^n)$, for any $\alpha\in(0,\min\{2s,1\})$, and $U\in C(\overline{{{\mathbb R}}^{n+1}_+})$.
The regularity of $u$ follows from Corollary \[coro:bound\] and Proposition 5 in [@SV:weak], being $u$ a solution to . The continuity of $U$ follows from Lemma 4.4 in [@CabS].
A strong maximum principle and positivity of the solutions {#sec:positivity}
----------------------------------------------------------
In this section we deal with the problem of the positivity of the solutions to . We have shown in Proposition \[prop:pos\] that a solution of is nonnegative. Here we prove that if $h{\geqslant}0$ then the solution is strictly positive.
Following is the strong maximum principle for weak solutions needed for our purposes:
\[prop:maxpr\] Let $u$ be a bounded, continuous function, with $u{\geqslant}0$ in ${{\mathbb R}}^n$ and $(-\Delta)^s u{\geqslant}0$ in the weak sense in $\Omega$. If there exists $x_\star\in\Omega$ such that $u(x_\star)=0$, then $u$ vanishes identically in ${{\mathbb R}}^n$.
Let $R>0$ such that $B_R(x_\star)\subset\Omega$. For any $r\in(0,R)$, we consider the solution of $$\label{0dvmnb7whhh12} \left\{
\begin{matrix}
(-\Delta)^s v_r = 0 & {\mbox{ in }} B_r(x_\star),\\
v_r =u & {\mbox{ in }} {{\mathbb R}}^n\setminus B_r(x_\star)
\end{matrix}
\right.$$ Notice that $v_r$ may be obtained by direct minimization and $v_r$ is continuous in the whole of ${{\mathbb R}}^n$ (see e.g. Theorem 2 in [@SV:weak]). Moreover, if $w_r:= v_r-u$, we have that $(-\Delta)^s w_r{\leqslant}0$ in the weak sense in $B_r(x_\star)$, and $w_r$ vanishes outside $B_r(x_\star)$. Accordingly, by the weak maximum principle for weak solutions (see e.g. Lemma 6 in [@SV:weak]), we have that $w_r{\leqslant}0$ in the whole of ${{\mathbb R}}^n$, which gives that $v_r{\leqslant}u$. In particular, $$\label{89990}
v_r(x_\star){\leqslant}u(x_\star)=0.$$ The weak maximum principle for weak solutions and the fact that $v_r=u{\geqslant}0$ outside $B_r(x_\star)$ also imply that $v_r{\geqslant}0$ in ${{\mathbb R}}^n$. This and say that $$\label{89991}
\min_{{{\mathbb R}}^n} v_r=v_r(x_\star)=0.$$ In addition, $v_r$ is also a solution of in the viscosity sense (see e.g. Theorem 1 in [@SV:weak]), hence it is smooth in the interior of $B_r(x_\star)$, and we can compute $(-\Delta)^s v_r(x_\star)$ and obtain from that $$0=\int_{{{\mathbb R}}^n} \frac{ v_r(x_\star+y)+v_r(x_\star-y)-2v_r(x_\star)}{
|y|^{n+2s}}\,dy {\geqslant}0.$$ This implies that $v_r$ is constant in ${{\mathbb R}}^n$, that is $v_r(x)=v_r(x_\star)=0$ for any $x\in{{\mathbb R}}^n$. In particular $0=v_r(x)=u(x)$ for any $x\in {{\mathbb R}}^n\setminus B_r(x_\star)$. By taking $r$ arbitrarily small, we obtain that $u(x)=0$ for any $x\in {{\mathbb R}}^n\setminus\{x_\star\}$, and the desired result plainly follows.
Thanks to Proposition \[prop:maxpr\] we now show the positivity of solutions of .
Let $u\in\dot{H}^s({{\mathbb R}}^n)$, $u\neq0$, be a solution of . Suppose also that $h{\geqslant}0$. Then, $u>0$.
First we observe that $u\in C^{\alpha}({{\mathbb R}}^n)\cap L^\infty({{\mathbb R}}^n)$, for some $\alpha\in(0,1)$, thanks to Corollaries \[coro:bound\] and \[coro:conti\]. Also, by Proposition \[prop:pos\] we have that $u{\geqslant}0$. Moreover, since $u$ is a solution to with $h{\geqslant}0$, then $u$ satisfies $$(-\Delta)^s u {\geqslant}0 \quad {\mbox{ in }}{{\mathbb R}}^n.$$ This means that the hypotheses of Proposition \[prop:maxpr\] are satisfied, and so if $u$ is equal to zero at some point then $u$ must be identically zero in ${{\mathbb R}}^n$. This contradicts the fact that $u\neq0$, and thus implies the desired result.
Existence of a second solution and proof of Theorem \[TH:MP\] {#EMP:CHAP}
=============================================================
In this chapter, we complete the proof of Theorem \[TH:MP\]. The computations needed for this are delicate and somehow technical. Many calculations are based on general ideas of Taylor expansions and can be adapted to other types of nonlinearities (though other estimates do take into account the precise growth conditions of the main term of the nonlinearity and of its perturbation). Rather than trying to list abstract conditions on the nonlinearity which would allow similar techniques to work (possibly at the price of more careful Taylor expansions), we remark that the case treated here is somehow classical and motivated from geometry.
Namely, the power-like nonlinearity $u^p$ is inherited by Riemannian geometry and the critical exponent $p=\frac{n+2s}{n-2s}$ comes from conformal invariance in the classical case (see also the recent fractional contributions in [@NEW2a; @NEW2b] in which the same term is taken into account). The perturbative term $u^q$ is somehow more “arbitrary” and it could be generalized: we stick to this choice both to compare with the classical cases in [@AGP] and references therein and in order to emphasize the role of such perturbation, which is to produce a small, but not negligible, subcritical growth. One of the features of this perturbation is indeed to modify the geometry of the energy functional near the origin, without spoiling the energy properties at infinity. Indeed, roughly speaking, since $p>1$, the unperturbed energy term $u^{p+1}$ has a higher order of homogeneity with respect to the diffusive part of the energy, which is quadratic. Conversely, terms which behave like $u^q$ with $q\in(0,1)$ near the origin induce a negative energy term which may produce (and indeed produces) nonzero local minima (the advantage of having a pure power in the perturbation is that the inclusion in classical Lebesgue spaces becomes explicit, but of course more general terms can be taken into account, at a price of more involved computations).
Once a critical point $U_{\varepsilon }$ of minimal type is produced near the origin (as given by Theorem \[MINIMUM\], whose proof has been completed in Chapter \[ECXMII\]), an additional critical point is created by the behavior of the functional at a large scale. Indeed, while the $(q+1)$-power becomes dominant near zero and the $(p+1)$-power leads the profile of the energy at infinity towards negative values (recall that $q+1<2<p+1$), in an intermediate regime the quadratic part of the energy that comes from fractional diffusion endows the functional with a new critical point along the path joining $U_{\varepsilon }$ to infinity. In Figures \[mafig\] and \[mafig2\] we try to depict this phenomenon with a one dimensional picture, by plotting the graphs of $y=x^2 - |x|^{p+1}-{\varepsilon }\,|x|^{q+1}$, with $p=2$, $q=1/2$ and ${\varepsilon }\in
\left\{0, \,\frac{1}{4},\,\frac{1}{3},\,\frac{5}{14}\right\}$. Of course, the infinite dimensional analysis that follows is much harder than the elementary twodimensional picture, which does not even take into account the possible saddle properties of the critical points “in other directions” and only serves to favor a basic intuition.
![The function $y=x^2 - |x|^{3}-{\varepsilon }\,|x|^{3/2}$, with ${\varepsilon }=0$.[]{data-label="mafig2"}](Plot_maria2.pdf){width="12.4cm"}
![The function $y=x^2 - |x|^{3}-{\varepsilon }\,|x|^{3/2}$, with ${\varepsilon }\in
\left\{\frac{1}{4},\,\frac{1}{3},\,\frac{5}{14}\right\}$. []{data-label="mafig"}](Plot_maria1.pdf){width="12.4cm"}
Existence of a local minimum for ${{\mathscrI}}_{\varepsilon }$ {#sec:exi}
---------------------------------------------------------------
In this section we show that $U=0$ is a local minimum for ${{\mathscrI}}_\varepsilon$.
\[prop:zero\] Let $U_\varepsilon$ be a local positive minimum of ${{\mathscrF}}_\varepsilon$ in $\dot{H}_a^s(\mathbb{R}^{n+1}_+)$. Then $U=0$ is a local minimum of ${{\mathscrI}}_\varepsilon$ in $\dot{H}_a^s(\mathbb{R}^{n+1}_+)$.
Let $U_\varepsilon$ be a local minimum of ${{\mathscrF}}_\varepsilon$ in $\dot{H}_a^s(\mathbb{R}^{n+1}_+)$. Then, there exists $\eta>0$ such that $$\label{UepsMin}
{{\mathscrF}}_\varepsilon(U_\varepsilon+U){\geqslant}{{\mathscrF}}_\varepsilon(U_\varepsilon),
\ \hbox{ if }\;u\in\dot{H}^s_a({{\mathbb R}}^{n+1}_+)\ {\mbox{ s.t. }} [U]_a{\leqslant}\eta.$$ Moreover, since $U_{\varepsilon }$ is a positive critical point of ${{\mathscrF}}_\varepsilon$, we have that, for every $V\in \dot{H}^s_a(\mathbb{R}^{n+1}_+)$, $$\begin{split}\label{UepsSol}
\int_{\mathbb{R}^{n+1}_+}y^a\langle\nabla U_\varepsilon&,\nabla V\rangle\,dX-\int_{\mathbb{R}^n}{U_\varepsilon(x,0)^{p}V(x,0)\,dx}\\
&-\varepsilon\int_{\mathbb{R}^n}{h(x)U_\varepsilon(x,0)^{q}V(x,0)\,dx}=0.
\end{split}$$ Now, we take $U\in\dot{H}_a^s(\mathbb{R}^{n+1}_+)$ such that $$\label{wqsdhjykp}
[U]_a{\leqslant}\eta.$$ From and , we have that $$\begin{aligned}
{{\mathscrI}}_\varepsilon(U) &=& \frac{1}{2}\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U|^2\,dX} \\
&&\qquad -\frac{{\varepsilon }}{q+1}\int_{{{\mathbb R}}^n}h(x)\,\left((U_{\varepsilon }+U_+)^{q+1}-U_{\varepsilon }^{q+1}\right)\,dx +{\varepsilon }\int_{{{\mathbb R}}^n}h(x)\,U_{\varepsilon }^q U_+\,dx \\
&&\qquad -\frac{1}{p+1}\int_{{{\mathbb R}}^n}\left((U_{\varepsilon }+U_+)^{p+1}-U_{\varepsilon }^{p+1}\right)\,dx +\int_{{{\mathbb R}}^n}U_{\varepsilon }^pU_+\,dx.\end{aligned}$$ On the other hand, recalling the definition of ${{\mathscrF}}_{\varepsilon }$ in , we have that $$\begin{aligned}
&&{{\mathscrF}}_\varepsilon(U_{\varepsilon }+U_+) -{{\mathscrF}}_{\varepsilon }(U_{\varepsilon })\\
&=& \frac{1}{2}\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U_+|^2\,dX} +
\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla U_{\varepsilon },\nabla U_+\rangle\,dX} \\
&&\qquad -\frac{{\varepsilon }}{q+1}\int_{{{\mathbb R}}^n}h(x)\,\left((U_{\varepsilon }+U_+)^{q+1}-U_{\varepsilon }^{q+1}\right)\,dx \\
&&\qquad -\frac{1}{p+1}\int_{{{\mathbb R}}^n}\left((U_{\varepsilon }+U_+)^{p+1}-U_{\varepsilon }^{p+1}\right)\,dx \\
&=& \frac{1}{2}\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U_+|^2\,dX} +
\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla U_{\varepsilon },\nabla U_+\rangle\,dX} \\
&&\qquad -\frac{{\varepsilon }}{q+1}\int_{{{\mathbb R}}^n}h(x)\,\left((U_{\varepsilon }+U_+)^{q+1}-U_{\varepsilon }^{q+1}\right)\,dx \\
&&\qquad -\frac{1}{p+1}\int_{{{\mathbb R}}^n}\left((U_{\varepsilon }+U_+)^{p+1}-U_{\varepsilon }^{p+1}\right)\,dx,\end{aligned}$$ where in the last equality we have used the fact that both $U_{\varepsilon }$ and $U_{\varepsilon }+U_+$ are positive. Hence, the last two formulas give that $$\begin{aligned}
{{\mathscrI}}_\varepsilon(U)&=&\frac12\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U_-|^2\,dX}+ {{\mathscrF}}_\varepsilon(U_\varepsilon +U_+)-{{\mathscrF}}_\varepsilon(U_\varepsilon)\\
&&\quad -\int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla U_\varepsilon,\nabla U_+\rangle\,dX}\\
&&\quad+\varepsilon\int_{\mathbb{R}^n}{h(x)U_\varepsilon^{q}U_+\,dx}
+\int_{\mathbb{R}^n}{U_\varepsilon^{p}U_+\,dx}.\end{aligned}$$ Using with $V:=U_+$, we obtain $${{\mathscrI}}_\varepsilon(U)=\frac12\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U_-|^2\,dX}+ {{\mathscrF}}_\varepsilon(U_\varepsilon +U_+)-{{\mathscrF}}_\varepsilon(U_\varepsilon).$$ Moreover, we observe that $[U_+]_a{\leqslant}\eta$, thanks to . Hence, from we deduce that $${{\mathscrI}}_\varepsilon(U){\geqslant}\frac12\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U_-|^2\,dX}{\geqslant}0={{\mathscrI}}_\varepsilon(0).$$ This shows the desired result.
Some preliminary lemmata towards the proof of Theorem \[TH:MP\] {#sec:prelim}
---------------------------------------------------------------
In this section we show some preliminary lemmata, that we will use in the sequel to prove that a Palais-Smale sequence is bounded.
We start with a basic inequality.
For every $\delta>0$ there exists $M_\delta>0$ such that the following inequality holds true for every $\alpha$, $\beta{\geqslant}0$ and $m>0$: $$\label{9sdf5568gg}
(\alpha+\beta)^{m+1} - \alpha^{m+1}-(m+1)\alpha^m \beta-
\beta\big( (\alpha+\beta)^m-\alpha^m\big){\leqslant}\delta \beta^{m+1} +M_\delta \alpha^{m+1}.$$
First of all, we observe that the left hand side of vanishes when $\alpha=0$, therefore we can suppose that $$\label{9ref1wfgfds}
\alpha\ne0.$$ For any $\tau{\geqslant}0$, let $$f(\tau):=
(1+\tau)^{m+1} - 1-(m+1)\tau-
\tau\big( (1+\tau)^m-1\big).$$ We observe that $$\lim_{\tau\to+\infty} \frac{f(\tau)}{\tau^{m+1}}=0$$ therefore there exists $\tau_\delta>0$ such that $\frac{f(\tau)}{\tau^{m+1}}
{\leqslant}\delta$ for any $\tau{\geqslant}\tau_\delta$. Let also $$M_\delta:= \max_{\tau\in[0,\tau_\delta]} f(\tau).$$ Then, by looking separately at the cases $\tau\in[0,\tau_\delta]$ and $\tau\in [\tau_\delta,+\infty)$, we see that $$f(\tau){\leqslant}\delta \tau^{m+1}+M_\delta.$$ As a consequence, recalling and taking $\tau:=\beta/\alpha$, $$\begin{split}
& (\alpha+\beta)^{m+1} - \alpha^{m+1}-(m+1)\alpha^m \beta-
\beta\big( (\alpha+\beta)^m-\alpha^m\big) \\
&\qquad= \alpha^{m+1} \Big[
(1+\tau)^{m+1} - 1-(m+1)\tau-
\tau\big( (1+\tau)^m-1\big)\Big] \\
&\qquad= \alpha^{m+1} f(\tau)\\
&\qquad{\leqslant}\alpha^{m+1} \big( \delta \tau^{m+1}+M_\delta\big)
\\&\qquad= \delta \beta^{m+1}+M_\delta\alpha^{m+1}
.\qedhere\end{split}$$
We recall and , and we have the following estimates.
For any $U\in H^s_a({{\mathbb R}}^{n+1}_+)$ and any $\delta\in(0,1)$, we have that $$\label{LA1-00}
\int_{{{\mathbb R}}^n} G(x,U(x,0))-\frac{1}{p+1} g(x,U(x,0))\,U(x,0)\,dx\\
{\leqslant}C ({\varepsilon }+\delta) \| U_+(\cdot,0)\|_{L^{2^*_s}({{\mathbb R}}^n)}^{2^*_s}
+ C_{\delta,{\varepsilon }},$$ for suitable $C$, $C_{\delta,{\varepsilon }}>0$. Moreover, $$\label{LA2-00}
\int_{{{\mathbb R}}^n} g(x,U(x,0))\,U(x,0)\,dx{\geqslant}\frac{\|U_+(\cdot,0)\|^{2^*_s}_{L^{2^*_s}({{\mathbb R}}^n)}}{8}-C_{\varepsilon },$$ for a suitable $C_{\varepsilon }>0$.
By and , we can write $g=g_1+g_2$ and $G=G_1+G_2$, where $$\begin{aligned}
&& g_1(x,t) :={{\varepsilon }h(x)}
\big((U_{\varepsilon }(x,0)+t_+)^q-U_{\varepsilon }^q(x,0)\big),\\
&& g_2(x,t) := (U_{\varepsilon }(x,0)+t_+)^p-U_{\varepsilon }^p(x,0),\\
&& G_1(x,t) :=\frac{{\varepsilon }h(x)}{q+1}
\big((U_{\varepsilon }(x,0)+t_+)^{q+1}-U_{\varepsilon }^{q+1}(x,0)\big)-{\varepsilon }h(x)
U_{\varepsilon }^q(x,0) \,t_+\\
{\mbox{and }}&& G_2(x,t):=
\frac{1}{p+1} \big((U_{\varepsilon }(x,0)+t_+)^{p+1}-U_{\varepsilon }^{p+1}(x,0)\big)
-U_{\varepsilon }^p(x,0) \,t_+
.\end{aligned}$$ We observe that for any $\tau{\geqslant}0$, $$\begin{aligned}
&& (1+\tau)^q-1 =q\int_0^\tau (1+\theta)^{q-1} \,d\theta
{\leqslant}q\int_0^\tau \theta^{q-1} \,d\theta=
\tau^q\\
{\mbox{and }}&&(1+\tau)^{q+1}-1 =(q+1)
\int_0^\tau (1+\theta)^{q} \,d\theta{\leqslant}(q+1)(1+\tau)^q\tau,\end{aligned}$$ since $q\in(0,1)$. Therefore, taking $\tau:=t_+/U_{\varepsilon }$, $$(U_{\varepsilon }+t_+)^q-U_{\varepsilon }^q=
U_{\varepsilon }^q \big( (1+\tau)^q-1\big){\leqslant}U_{\varepsilon }^q \tau^q=
t_+^q$$ and $$\begin{aligned}
&& (U_{\varepsilon }+t_+)^{q+1}-U_{\varepsilon }^{q+1}=U_{\varepsilon }^{q+1}\big(
(1+\tau)^{q+1}-1\big)\\&&\qquad \qquad {\leqslant}(q+1)U_{\varepsilon }^{q+1}(1+\tau)^q\tau
=(q+1) (U_{\varepsilon }+t_+)^q t_+.\end{aligned}$$ As a consequence, $$\begin{aligned}
&&|g_1|{\leqslant}{\varepsilon }|h| t_+^q \\{\mbox{and }}&&
|G_1|{\leqslant}{\varepsilon }|h|(U_{\varepsilon }+t_+)^q t_+
+{\varepsilon }|h|U_{\varepsilon }^q t_+ {\leqslant}2{\varepsilon }|h|(U_{\varepsilon }+t_+)^q t_+.\end{aligned}$$ Thus we obtain $$|G_1(x,t)|+|g_1(x,t) t|{\leqslant}2{\varepsilon }|h|(U_{\varepsilon }+t_+)^q t_++{\varepsilon }|h| t_+^{q+1}
{\leqslant}3{\varepsilon }|h|(U_{\varepsilon }+t_+)^q t_+.$$ Since $U_{\varepsilon }$ is bounded (recall Corollary \[coro:bound\]), we obtain that $$\label{9dkjcc9gg-1}
|G_1(x,t)|+|g_1(x,t) t|{\leqslant}C{\varepsilon }|h| \,(1+t_+^{q+1}),$$ for some $C>0$. By considering the cases $t_+\in [0,1]$ and $t_+\in [1,+\infty)$, we see that $$t_+^{q+1} {\leqslant}t_+^{p+1} +1,$$ since $q<p$. This and give that $$\label{9dkjcc9gg-2}
|G_1(x,t)|+|g_1(x,t) t|{\leqslant}C{\varepsilon }|h| \,(1+t_+^{p+1}),$$ up to changing the constants. Now we fix $\delta\in(0,1)$. Using with $\alpha:=U_{\varepsilon }$, $\beta:=t_+$ and $m:=p$, we have that $$\begin{aligned}
&& G_2(x,t)-\frac{1}{p+1}g_2(x,t)t\\
&=&
\frac{1}{p+1} \big((U_{\varepsilon }+t_+)^{p+1}-U_{\varepsilon }^{p+1}\big)
-U_{\varepsilon }^p \,t_+
-(U_{\varepsilon }+t_+)^p+U_{\varepsilon }^p
\\ &=& \frac{1}{p+1}\left(
(\alpha+\beta)^{m+1} - \alpha^{m+1}-(m+1)\alpha^m \beta-
\beta\big( (\alpha+\beta)^m-\alpha^m\big)\right)\\
&{\leqslant}& \frac{1}{p+1}\left(
\delta \beta^{m+1} +M_\delta \alpha^{m+1} \right)\\
&{\leqslant}& \delta\,t_+^{p+1} +M_\delta U_{\varepsilon }^{p+1}.\end{aligned}$$ This, together with , implies that $$\begin{aligned}
&& G(x,t)-\frac{1}{p+1} g(x,t)t \\
&=& G_1(x,t)-\frac{1}{p+1}g_1(x,t)t + G_2(x,t)-\frac{1}{p+1}g_2(x,t)t\\
&{\leqslant}& C{\varepsilon }|h| \,(1+t_+^{p+1}) +
\delta\,t_+^{p+1} +M_\delta U_{\varepsilon }^{p+1}\\
&{\leqslant}& C ({\varepsilon }+\delta)t_+^{p+1} + C|h| + M_\delta U_{\varepsilon }^{p+1}.\end{aligned}$$ As a consequence, and recalling that $p+1=2^*_s$, we obtain $$\int_{{{\mathbb R}}^n} G(x,U(x,0))-\frac{1}{p+1} g(x,U(x,0))\,U(x,0)\,dx\\
{\leqslant}C ({\varepsilon }+\delta) \| U_+(\cdot,0)\|_{L^{2^*_s}({{\mathbb R}}^n)}^{2^*_s}
+ C_{\delta,{\varepsilon }}$$ for some $C$, $C_{\delta,{\varepsilon }}>0$. This proves and we now focus on the proof of . To this goal, for any $\tau{\geqslant}0$, we set $\ell(\tau):=\frac{\tau^p}{2}
-(1+\tau)^p+1$. We observe that $\ell(0)=0$ and $$\lim_{\tau\to+\infty} \ell(\tau)=-\infty,$$ therefore $$L:=\sup_{\tau{\geqslant}0} \ell(\tau) \in [0,+\infty).$$ As a consequence, $$(1+\tau)^p-1 = \frac{\tau^p}{2}-\ell(\tau){\geqslant}\frac{\tau^p}{2}-L.$$ By taking $\tau:= \frac{U_+}{U_{\varepsilon }}$, this implies that $$\begin{aligned}
&& g_2(x,U)=(U_{\varepsilon }+U_+)^p-U_{\varepsilon }^p
= U_{\varepsilon }^p \big( (1+\tau)^p-1\big)\\
&&\qquad{\geqslant}U_{\varepsilon }^p \left( \frac{\tau^p}{2}-L \right)=
\frac{U_+^p}{2}-L U_{\varepsilon }^p.\end{aligned}$$ Integrating this formula and using the Young inequality, we obtain $$\label{9cbnmrasuifg}
\int_{{{\mathbb R}}^n} g_2(x,U(x,0))\,U(x,0)\,dx{\geqslant}\frac{\|U_+(\cdot,0)\|^{p+1}_{L^{p+1}({{\mathbb R}}^n)}}{4}-C_{\varepsilon },$$ for some $C_{\varepsilon }>0$. On the other hand, by , we have that $$\begin{aligned}
\left|\int_{{{\mathbb R}}^n} g_1(x,U(x,0))\,U(x,0)\,dx\right|
&{\leqslant}& C{\varepsilon }\int_{{{\mathbb R}}^n} |h(x)| \,(1+U_+^{p+1}(x,0))\,dx\\&{\leqslant}&
C+C{\varepsilon }\|U_+(\cdot,0)\|^{p+1}_{L^{p+1}({{\mathbb R}}^n)}.\end{aligned}$$ By combining this and , we get $$\begin{aligned}
&&\int_{{{\mathbb R}}^n} g(x,U(x,0))\,U(x,0)\,dx \\&=&
\int_{{{\mathbb R}}^n} g_1(x,U(x,0))\,U(x,0)\,dx+
\int_{{{\mathbb R}}^n} g_2(x,U(x,0))\,U(x,0)\,dx\\&{\geqslant}&
\frac{\|U_+(\cdot,0)\|^{p+1}_{L^{p+1}({{\mathbb R}}^n)}}{8}-C_{\varepsilon },\end{aligned}$$ if ${\varepsilon }$ is small enough, up to renaming constants. Recalling that $p+1=2^*_s$, the formula above gives the proof of .
Finally, we recall and and we show the following:
\[cor:bound\] Let ${\varepsilon }$, $\kappa>0$. There exists $M>0$, possibly depending on $\kappa$, ${\varepsilon }$, $n$ and $s$, such that the following statement holds true.
For any $U\in H^s_a({{\mathbb R}}^{n+1}_+)$ such that $$|{{{\mathscrI}}}_{\varepsilon }(U)| +
\sup_{{V\in H^s_a({{\mathbb R}}^{n+1}_+)}\atop{ [V]_a =1 }}
\big|\langle {{{\mathscrI}}}_{\varepsilon }'(U),V\rangle\big|{\leqslant}\kappa$$ one has that $$[U]_a {\leqslant}M.$$
If $[U]_a=0$ we are done, so we suppose that $[U]_a\ne0$ and we obtain that $$\left|
\langle {{{\mathscrI}}}_{\varepsilon }'(U),\frac{U}{[U]_a}\rangle\right|
{\leqslant}\kappa.$$ This and give that $$\label{Rgj67}
\left| [U]_a^2 -\int_{{{\mathbb R}}^n} g(x,U(x,0)) U(x,0)\,dx \right|
{\leqslant}\kappa\,[U]_a.$$ Therefore $$\begin{aligned}
\kappa +\frac{\kappa\,[U]_a}{2} &{\geqslant}&
{{{\mathscrI}}}_{\varepsilon }(U)-\frac{1}{2}\left(
[U]_a^2 -\int_{{{\mathbb R}}^n} g(x,U(x,0)) U(x,0)\,dx\right)\\
&=& -\int_{{{\mathbb R}}^n} G(x,U(x,0)) \,dx+\frac{1}{2}
\int_{{{\mathbb R}}^n} g(x,U(x,0)) U(x,0)\,dx \\
&=& \frac{1}{p+1}
\int_{{{\mathbb R}}^n} g(x,U(x,0)) U(x,0)\,dx
-\int_{{{\mathbb R}}^n} G(x,U(x,0)) \,dx\\
&&\qquad +
\left(\frac{1}{2}-\frac{1}{p+1}\right)
\int_{{{\mathbb R}}^n} g(x,U(x,0)) U(x,0)\,dx.\end{aligned}$$ Consequently, by fixing $\delta\in(0,1)$, to be taken conveniently small in the sequel, and using and , $$\begin{aligned}
\kappa +\frac{\kappa\,[U]_a}{2} &{\geqslant}&
-C ({\varepsilon }+\delta) \| U_+(\cdot,0)\|_{L^{2^*_s}({{\mathbb R}}^n)}^{2^*_s}
- C_{\delta,{\varepsilon }}\\&&\qquad
+ \left(\frac{1}{2}-\frac{1}{p+1}\right)
\left(\frac{\|U_+(\cdot,0)\|^{2^*_s}_{L^{2^*_s}({{\mathbb R}}^n)}}{8}-C_{\varepsilon }\right),\end{aligned}$$ for suitable $C$, $C_{\delta,{\varepsilon }}$ and $C_{\varepsilon }>0$. By taking $\delta$ and ${\varepsilon }$ appropriately small, we thus obtain that $$\kappa +\frac{\kappa\,[U]_a}{2}{\geqslant}\left(\frac{1}{2}-\frac{1}{p+1}\right)
\frac{\|U_+(\cdot,0)\|^{2^*_s}_{L^{2^*_s}({{\mathbb R}}^n)}}{16}-C_{{\varepsilon }},$$ up to renaming the latter constant (this fixes $\delta$ once and for all). That is $$\label{stima su U}
\|U_+(\cdot,0)\|^{2^*_s}_{L^{2^*_s}({{\mathbb R}}^n)} {\leqslant}M_1 ( [U]_a +1),$$ for a suitable $M_1$, possibly depending on $\kappa$, ${\varepsilon }$, $n$ and $s$.
Now we recall and (used here with $\delta:=1$), and we see that $$\begin{aligned}
&&\kappa+\frac{\kappa\,[U]_a}{p+1} \\
&{\geqslant}&
{{{\mathscrI}}}_{\varepsilon }(U)-\frac{1}{p+1}\left(
[U]_a^2 -\int_{{{\mathbb R}}^n} g(x,U(x,0)) U(x,0)\,dx\right)\\
&=& \left( \frac{1}{2}-\frac{1}{p+1}\right)\,[U]_a^2
+ \frac{1}{p+1} \int_{{{\mathbb R}}^n} g(x,U(x,0)) U(x,0)\,dx\\
&&\qquad -\int_{{{\mathbb R}}^n} G(x,U(x,0)) \,dx\\
&{\geqslant}& \left( \frac{1}{2}-\frac{1}{p+1}\right)\,[U]_a^2
-
C \| U_+(\cdot,0)\|_{L^{2^*_s}({{\mathbb R}}^n)}^{2^*_s}
- C_{{\varepsilon }},\end{aligned}$$ for suitable $C$, $C_{\varepsilon }>0$. As a consequence, $$[U]_a^2
{\leqslant}M_2\big( [U]_a + \| U_+(\cdot,0)\|_{L^{2^*_s}({{\mathbb R}}^n)}^{2^*_s}
+1\big),$$ for a suitable $M_2$, possibly depending on $\kappa$, ${\varepsilon }$, $n$ and $s$. Hence, from , $$[U]_a^2
{\leqslant}M_3\big( [U]_a +
+1\big),$$ for some $M_3$, possibly depending on $\kappa$, ${\varepsilon }$, $n$ and $s$. This implies the desired result.
Some convergence results in view of Theorem \[TH:MP\] {#sec:prelim2}
-----------------------------------------------------
In this section we collect two convergence results that we will need in the sequel.
The first one shows that weak convergence to 0 in $\dot H^s_a({{\mathbb R}}^{n+1}_+)$ implies a suitable integral convergence.
\[lemma:conv\] Let $\alpha$, $\beta>0$ with $\alpha+\beta{\leqslant}2^*_s$. Let $V_k\in\dot H^s_a({{\mathbb R}}^{n+1}_+)$ be a sequence such that $V_k$ converges to $0$ weakly in $\dot H^s_a({{\mathbb R}}^{n+1}_+)$. Let $U_o\in \dot H^s_a({{\mathbb R}}^{n+1}_+)$, with $U_o(\cdot,0)\in
L^\infty({{\mathbb R}}^{n})$, and $\psi\in L^{\rm{a}}({{\mathbb R}}^n)\cap L^{\rm{b}}({{\mathbb R}}^n)$, where $$\rm{a}:=\begin{cases}
\frac{2^*_s}{2^*_s-\alpha-\beta} & {\mbox{ if }}\alpha +\beta <2^*_s,\\
+\infty &{\mbox{ if }}\alpha +\beta=2^*_s,
\end{cases}$$ and $$\rm{b}:=\begin{cases}
\frac{2^*_s+\alpha}{2^*_s-\alpha-\beta} & {\mbox{ if }}\alpha +\beta <2^*_s,\\
+\infty &{\mbox{ if }}\alpha +\beta=2^*_s,
\end{cases}$$ Then, up to a subsequence, $$\lim_{k\to+\infty} \int_{{{\mathbb R}}^n} \big|\psi(x)\big|\,\big|(V_k)_+ (x,0)\big|^\alpha
\,\big|U_o (x,0)\big|^\beta\,dx =0.$$
Since weakly convergent sequences are bounded, we have that $[V_k]_a{\leqslant}C_o$, for every $k\in{{\mathbb N}}$ and a suitable $C_o>0$. Accordingly, by , we obtain that $[v_k]_{\dot{H}^s({{\mathbb R}}^n)}{\leqslant}C_o$, where $v_k(x):=V_k(x,0)$. As a consequence, by Theorem 7.1 of [@DPV], we know that, up to a subsequence, $v_k$ converges to some $v$ in $L^\gamma_{\rm loc}({{\mathbb R}}^n)$ for any $\gamma\in
[1,2^*_s)$, and a.e.: we claim that $$\label{zer}
v=0.$$ To prove this, let $\eta\in C^\infty_0({{\mathbb R}}^n)$ and $\psi$ be the solution of $$\label{90567}
{\mbox{$(-\Delta)^s \psi=\eta$ in~${{\mathbb R}}^n$.}}$$ Also, let $\Psi$ be the extension of $\psi$ according to . In particular, ${\rm div}(y^a \nabla\Psi)=0$ in ${{\mathbb R}}^{n+1}_+$, therefore $$\int_{{{\mathbb R}}^{n+1}_+} {\rm div}(y^a V_k\nabla\Psi)\,dX
=\int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla V_k,\nabla\Psi\rangle\,dX.$$ The latter term is infinitesimal as $k\to+\infty$, thanks to the weak convergence of $V_k$ in $\dot H^s_a({{\mathbb R}}^{n+1}_+)$. Thus, using the Divergence Theorem in the left hand side of the identity above, we obtain $$\lim_{k\to+\infty} \int_{{{\mathbb R}}^n} v_k(x) \partial_y \Phi(x)\,dx=0.$$ That is, recalling and the convergence of $v_k$, $$\int_{{{\mathbb R}}^n} v(x) \eta(x)\,dx=
\lim_{k\to+\infty} \int_{{{\mathbb R}}^n} v_k(x) \eta(x)\,dx=0.$$ Since $\eta$ is arbitrary, we have established .
Now we set $u_o(x):=U_o(x,0)$ and we observe that $u_o\in L^{2^*_s}({{\mathbb R}}^n)$, thanks to Proposition \[traceIneq\]. Therefore, we can fix ${\varepsilon }>0$ and find $R_{\varepsilon }>0$ such that $$\int_{{{\mathbb R}}^n\setminus B_{R_{\varepsilon }}} |u_o(x)|^{2^*_s}\,dx{\leqslant}{\varepsilon }.$$ In virtue of , $\| v_k\|_{L^{2^*_s({{\mathbb R}}^n)}}{\leqslant}S^{-1/2} C_o$. Consequently, using the Hölder inequality with exponents $\rm{a}$, $2^*_s/\alpha$ and $2^*_s/\beta$, we deduce that $$\label{679ujj9}
\begin{split}
& \int_{{{\mathbb R}}^n\setminus B_{R_{\varepsilon }}} \big|\psi(x)\big|
\,\big|(V_k)_+ (x,0)\big|^\alpha
\,\big|U_o (x,0)\big|^\beta\,dx
\\ &\qquad{\leqslant}\|\psi\|_{L^{\rm{a}}({{\mathbb R}}^n)}\,
\left[
\int_{{{\mathbb R}}^n\setminus B_{R_{\varepsilon }}}
\big|V_k (x,0)\big|^{2^*_s} \,dx
\right]^{\frac{\alpha}{2^*_s}}\,
\left[
\int_{{{\mathbb R}}^n\setminus B_{R_{\varepsilon }}}
\big|U_o (x,0)\big|^{2^*_s}
\,dx
\right]^{\frac{\beta}{2^*_s}}\\
&\qquad{\leqslant}\|\psi\|_{L^{\rm{a}}({{\mathbb R}}^n)}\,
\big(S^{-1/2} C_o\big)^{\frac{\alpha}{2^*_s}}\,
{\varepsilon }^{\frac{\beta}{2^*_s}}.
\end{split}$$ Now we fix $\gamma:=\frac{\alpha+2^*_s}{2}$. Notice that $\gamma\in(1,2^*_s)$, thus, using the convergence of $v_k$ and , we see that $$\lim_{k\to+\infty} \|v_k\|_{L^\gamma(B_{R_{\varepsilon }})}=0.$$ In addition, $$\int_{{{\mathbb R}}^n} |u_o(x)|^{2^*_s+\alpha}\,dx
{\leqslant}\|u_o\|_{L^\infty({{\mathbb R}}^n)}^{\alpha}
\int_{{{\mathbb R}}^n} |u_o(x)|^{2^*_s}
\,dx{\leqslant}C_*,$$ for some $C_*>0$. Therefore we use the Hölder inequality with exponents $\rm{b}$, $\frac{\gamma}{\alpha}
=\frac{\alpha+2^*_s}{2\alpha}$ and $\frac{2^*_s+\alpha}{\beta}$, and we obtain $$\begin{aligned}
&& \lim_{k\to+\infty}
\int_{B_{R_{\varepsilon }}} \big|\psi(x)\big|
\,\big|(V_k)_+ (x,0)\big|^\alpha
\,\big|U_o (x,0)\big|^\beta\,dx \\
&{\leqslant}& \|\psi\|_{L^{\rm{b}}({{\mathbb R}}^n)}\,
\lim_{k\to+\infty}
\left[ \int_{B_{R_{\varepsilon }}}
\,\big|v_k(x)\big|^\gamma\right]^{\frac{\alpha}{\gamma}}\,
\left[ \int_{{{\mathbb R}}^n}
\big|u_o (x)\big|^{2^*_s+\alpha}\,dx
\right]^{\frac{\beta}{2^*_s+\alpha}}\\
&=&0.\end{aligned}$$ [F]{}rom this and , we see that $$\lim_{k\to+\infty}
\int_{{{\mathbb R}}^n} \big|\psi(x)\big|
\,\big|(V_k)_+ (x,0)\big|^\alpha
\,\big|U_o (x,0)\big|^\beta\,dx{\leqslant}\|\psi\|_{L^{\rm{a}}({{\mathbb R}}^n)}\,
\big(S^{-1/2} C_o\big)^{\frac{\alpha}{2^*_s}}\,
{\varepsilon }^{\frac{\beta}{2^*_s}}.$$ The desired result then follows by taking ${\varepsilon }$ as small as we wish.
As a corollary we have
Let $V_k$, $U_o$ and $\psi$ as in Lemma \[lemma:conv\]. Then $$\label{second}
\| (U_o + (V_k)_+)(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^{p+1}
= \| U_o \|_{L^{p+1}({{\mathbb R}}^n)}^{p+1}
+ \|(V_k)_+(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^{p+1} +o_k(1),$$ and $$\label{usare-1}
\left| \int_{\mathbb{R}^n}\psi(x)\left( (U_o+(V_{k})_+)^{q+1}(x,0)
-U_o^{q+1}(x,0)\right)\,dx\right| {\leqslant}C+o_k(1),$$ for some $C>0$.
Formula \[second\] plainly follows from Lemma \[lemma:conv\], by taking $\psi:=1$ (notice that $p+1=2^*_s$).
To prove , we use Lemma \[lemma:conv\] to see that $$\label{first}\begin{split}
& \int_{\mathbb{R}^n}\psi(x)
\left( (U_o+(V_{k})_+)^{q+1}(x,0)-U_o^{q+1}(x,0)\right)\,dx\\
=\,& \int_{\mathbb{R}^n}\psi(x)(U_o+(V_{k})_+)^{q+1}(x,0)\,dx
-\int_{\mathbb{R}^n}\psi(x)U_o^{q+1}(x,0)\,dx\\
=\,&\int_{\mathbb{R}^n}\psi(x)U_o^{q+1}(x,0)\,dx+
\int_{\mathbb{R}^n}\psi(x)(V_{k})_+^{q+1}(x,0)\,dx +o_k(1)
-\int_{\mathbb{R}^n}\psi(x)U_o^{q+1}(x,0)\,dx\\
=\,&\int_{\mathbb{R}^n}\psi(x)(V_{k})_+^{q+1}(x,0)\,dx +o_k(1).
\end{split}$$ By Hölder inequality with exponents $\frac{2^*_s}{2^*_s-q-1}$ and $\frac{2^*_s}{q+1}$ and by Proposition \[traceIneq\] we have that $$\begin{aligned}
&& \int_{\mathbb{R}^n}\psi(x)(V_{k})_+^{q+1}(x,0)\,dx \\
&&\qquad {\leqslant}\left(\int_{\mathbb{R}^n}|\psi(x)|^{\frac{2^*_s}{2^*_s-q-1}}(x,0)\,dx\right)^{\frac{2^*_s-q-1}{2^*_s}}
\left(\int_{\mathbb{R}^n}(V_{k})_+^{2^*_s}(x,0)\,dx\right)^{\frac{q+1}{2^*_s}}\\
&&\qquad {\leqslant}\|\psi\|_{ L^{ \frac{2^*_s}{2^*_s-q-1}}({{\mathbb R}}^n)}
\left( \int_{\mathbb{R}^n}(V_k)_+^{2^*_s}(x,0)\,dx\right)^{ \frac{q+1}{2^*_s} }\\
&&\qquad {\leqslant}S^{-\frac{q+1}{2}} \|\psi\|_{ L^{\frac{2^*_s}{2^*_s-q-1} }({{\mathbb R}}^n) }[V_k]_a.\end{aligned}$$ Now notice that in this case $\alpha+\beta=q+1<2^*_s$, and so $\psi\in L^{\rm{a}}({{\mathbb R}}^n)$, with $\rm{a}=\frac{2^*_s}{2^*_s-q-1}$, by hypothesis. Moreover, since $V_k$ is a weakly convergent sequence in $\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$, then $V_k$ is uniformly bounded in $\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$. Hence $$\int_{\mathbb{R}^n}\psi(x)(V_{k})_+^{q+1}(x,0)\,dx{\leqslant}C,$$ for a suitable $C>0$. Plugging this information into , we obtain that $$\left| \int_{\mathbb{R}^n}\psi(x)
\left( (U_o+(V_{k})_+)^{q+1}(x,0)-U_o^{q+1}(x,0)\right)\,dx\right| {\leqslant}C+o_k(1),$$ as desired.
Now we show that, under an assumption on the positivity of the limit function, weak convergence in $\dot{H}^s_a({{\mathbb R}}^{n+1})$ implies weak convergence of the positive part.
\[POS\] Assume that $W_m$ is a sequence of functions in $\dot H^s_a({{\mathbb R}}^{n+1}_+)$ that converges weakly in $\dot H^s({{\mathbb R}}^{n+1}_+)$ to $
W\in \dot H^s_a({{\mathbb R}}^{n+1}_+)$.
Suppose also that for any bounded set $K\subset{{\mathbb R}}^{n+1}_+$, we have that $$\inf_K W>0.$$ Then, up to a subsequence, $(W_m)_+\in H^s_a({{\mathbb R}}^{n+1}_+)$ and it also converges weakly in $\dot H^s_a({{\mathbb R}}^{n+1}_+)$ to $W$.
Notice that $|\nabla (W_m)_+|=
|\nabla W_m|\chi_{\{W_m>0\}}{\leqslant}|\nabla W_m|$ a.e., which shows that $(W_m)_+\in H^s_a({{\mathbb R}}^{n+1}_+)$.
We also recall that, since weakly convergent sequences are bounded, $$\label{C0}
\sup_{m\in{{\mathbb N}}} \int_{{{\mathbb R}}^{n+1}_+} y^a |\nabla W_m|^2\,dX{\leqslant}C_o,$$ for some $C_o>0$.
Now we claim that $$\label{DF780}\begin{split}
&{\mbox{for any~$\Phi\in C^\infty_0(\overline{{{\mathbb R}}^{n+1}_+})$, }}\\&
\lim_{m\to+\infty} \int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla (W_m)_+,\nabla\Phi\rangle\,dX
= \int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla W, \nabla\Phi\rangle\,dX.
\end{split}$$ For this, we let $K$ be the support of $\Phi$. Up to a subsequence, we know that $W_m$ converges a.e. to $W$. Therefore, by Egorov Theorem, fixed ${\varepsilon }>0$, there exists $K_{\varepsilon }\subseteq K$ such that $W_m$ converges to $W$ uniformly in $K_{\varepsilon }$ and $|K\setminus K_{\varepsilon }|{\leqslant}{\varepsilon }$. Then, for any $X\in K_{\varepsilon }$, $$W_m(X) {\geqslant}W(X) -\| W_m-W\|_{L^\infty(K_{\varepsilon })}
{\geqslant}\inf_K W -\| W_m-W\|_{L^\infty(K_{\varepsilon })}
{\geqslant}\frac{1}{2}\inf_K W >0,$$ as long as $m$ is large enough, say $m{\geqslant}m_\star(K,{\varepsilon })$.
Accordingly, $\nabla (W_m)_+=\nabla W_m$ a.e. in $K_{\varepsilon }$ if $m{\geqslant}m_\star(K,{\varepsilon })$ and therefore $$\label{9sdvfg898}
\lim_{m\to+\infty} \int_{K_{\varepsilon }} y^a \langle\nabla (W_m)_+, \nabla\Phi\rangle\,dX
= \int_{K_{\varepsilon }} y^a \langle\nabla W, \nabla\Phi\rangle\,dX.$$ Moreover, for any $\eta>0$, the absolute continuity of the integral gives that $$\int_{K\setminus K_{\varepsilon }} y^a |\nabla W|^2\,dX
+\int_{K\setminus K_{\varepsilon }} y^a |\nabla \Phi|^2\,dX{\leqslant}\eta,$$ provided that ${\varepsilon }$ is small enough, say ${\varepsilon }\in(0,{\varepsilon }_\star(\eta))$, for a suitable ${\varepsilon }_\star(\eta)$. As a consequence, recalling , $$\begin{aligned}
&& \left|\int_{K\setminus K_{\varepsilon }} y^a \langle\nabla (W_m)_+,\nabla\Phi\rangle\,dX\right|+
\left|\int_{K\setminus K_{\varepsilon }} y^a \langle\nabla W,\nabla\Phi\rangle\,dX\right|
\\ &{\leqslant}&
\sqrt{\int_{K\setminus K_{\varepsilon }} y^a |\nabla (W_m)_+|^2\,dX}
\cdot\sqrt{\int_{K\setminus K_{\varepsilon }} y^a |\nabla \Phi|^2\,dX}
\\&&\qquad +
\sqrt{\int_{K\setminus K_{\varepsilon }} y^a |\nabla W|^2\,dX}
\cdot\sqrt{\int_{K\setminus K_{\varepsilon }} y^a |\nabla \Phi|^2\,dX}
\\ &{\leqslant}& \sqrt{C_o \eta} +\eta.\end{aligned}$$ Using this and , we obtain that $$\begin{aligned}
&& \left|
\lim_{m\to+\infty} \int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla (W_m)_+, \nabla\Phi\rangle\,dX
-\int_{{{\mathbb R}}^{n+1}_+} y^a\langle \nabla W, \nabla\Phi\rangle\,dX
\right| \\
&=& \left|
\lim_{m\to+\infty} \int_{K} y^a \langle\nabla (W_m)_+,\nabla\Phi\rangle\,dX
-\int_{K} y^a \langle\nabla W, \nabla\Phi\rangle\,dX
\right|\\
&{\leqslant}& \left| \lim_{m\to+\infty} \int_{K\setminus K_{\varepsilon }} y^a \langle\nabla (W_m)_+, \nabla\Phi\rangle\,dX
-\int_{K\setminus K_{\varepsilon }} y^a \langle\nabla W, \nabla\Phi\rangle\,dX
\right|
\\ &{\leqslant}& \sqrt{C_o \eta} +\eta.\end{aligned}$$ By taking $\eta$ as small as we like, we complete the proof of .
Now we finish the proof of Lemma \[POS\] by a density argument. Let $\Phi\in \dot H^s_a({{\mathbb R}}^{n+1}_+)$ and ${\varepsilon }>0$. We take $\Phi_{\varepsilon }\in C^\infty_0({{\mathbb R}}^{n+1}_+)$ such that $$\int_{{{\mathbb R}}^{n+1}_+} y^a |\nabla (\Phi-\Phi_{\varepsilon })|^2\,dX{\leqslant}{\varepsilon }.$$ The existence of such $\Phi_{\varepsilon }$ is guaranteed by . Then, recalling and for the function $\Phi_{\varepsilon }$, we obtain that $$\begin{aligned}
&& \left|
\lim_{m\to+\infty} \int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla (W_m)_+, \nabla\Phi\rangle\,dX
-\int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla W, \nabla\Phi\rangle\,dX
\right| \\
&{\leqslant}&
\left|
\lim_{m\to+\infty} \int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla (W_m)_+, \nabla\Phi_{\varepsilon }\rangle\,dX
-\int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla W, \nabla\Phi_{\varepsilon }\rangle\,dX
\right|
\\&&\qquad + \left(\sqrt{ C_o }+ \sqrt{\int_{{{\mathbb R}}^{n+1}_+} y^a|\nabla W|^2\,dX}\right)\,
\sqrt{
\int_{{{\mathbb R}}^{n+1}_+} y^a |\nabla (\Phi-\Phi_{\varepsilon })|^2\,dX }\\
&{\leqslant}& 0+\left(\sqrt{ C_o }+ \sqrt{\int_{{{\mathbb R}}^{n+1}_+} y^a|\nabla W|^2\,dX}\right)\,\sqrt{{\varepsilon }}.\end{aligned}$$ Accordingly, by taking ${\varepsilon }$ as small as we please, we obtain that $$\lim_{m\to+\infty} \int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla (W_m)_+, \nabla\Phi\rangle\,dX
=\int_{{{\mathbb R}}^{n+1}_+} y^a \langle\nabla W, \nabla\Phi\rangle\,dX,$$ for any $\Phi\in \dot H^s_a({{\mathbb R}}^{n+1}_+)$, thus completing the proof of Lemma \[POS\].
Palais-Smale condition for ${{\mathscrI}}_{\varepsilon }$ {#sec:PS MP}
---------------------------------------------------------
Once we have found a minimum of ${{\mathscrI}}_\varepsilon$, we apply a contradiction procedure to prove the existence of a second critical point.
Roughly speaking, the idea is the following: let us suppose that $U=0$ is the only critical point; thus, we prove some compactness and geometric properties of the functional (based on the fact that the critical point is unique), and these facts allow us to apply the Mountain Pass Theorem, that provides a second critical point. Hence, we reach a contradiction, so $U=0$ cannot be the only critical point of ${{\mathscrI}}_\varepsilon$.
As we did in Proposition \[PScond\] for the minimal solution, also to find the second solution we need to prove that a Palais-Smale condition holds true below a certain threshold, as stated in the following result:
\[PScond2\] There exists $C>0$, depending on $h$, $q$, $n$ and $s$, such that the following statement holds true. Let $\{U_k\}_{k\in\mathbb{N}}\subset \dot{H}^s_a(\mathbb{R}^{n+1}_+)$ be a sequence satisfying
1. $\displaystyle\lim_{k\to+\infty}{{\mathscrI}}_{\varepsilon }(U_k)= c_{\varepsilon }$, with $$\label{ceps2}
c_{\varepsilon }+C {\varepsilon }^{\frac{1}{2\gamma}}<
\dfrac{s}{n}S^{\frac{n}{2s}},$$ where $\gamma=1+\frac{2}{n-2s}$ and $S$ is the Sobolev constant appearing in Proposition \[traceIneq\],
2. $\displaystyle\lim_{k\to+\infty}{{\mathscrI}}'_{\varepsilon }(U_k)= 0.$
Assume also that $U=0$ is the only critical point of ${{\mathscrI}}_\varepsilon$.
Then $\{U_k\}_{k\in\mathbb{N}}$ contains a subsequence strongly convergent in $\dot{H}^s_a(\mathbb{R}^{n+1}_+)$.
\[rem:3.3\] The limit in (ii) is intended in the following way $$\begin{aligned}
&& \lim_{k\to+\infty}\|{{\mathscrI}}'_{\varepsilon }(U_k)\|_
{{{\mathscrL}}(\dot{H}^s_a(\mathbb{R}^{n+1}_+),\dot{H}^s_a(\mathbb{R}^{n+1}_+))}
\\ &&\qquad = \lim_{k\to+\infty}\sup_{V\in\dot{H}^s_a(\mathbb{R}^{n+1}_+),
[V]_{a}=1}
\left|\langle {{\mathscrI}}'_{\varepsilon }(U_k), V\rangle\right|
=0,\end{aligned}$$ where ${{\mathscrL}}(\dot{H}^s_a(\mathbb{R}^{n+1}_+),\dot{H}^s_a(\mathbb{R}^{n+1}_+))$ consists of all the linear functionals from $\dot{H}^s_a(\mathbb{R}^{n+1}_+)$ in $\dot{H}^s_a(\mathbb{R}^{n+1}_+)$.
We observe that a sequence that satisfies the assumptions of Proposition \[PScond2\] is weakly convergent. The precise statement goes as follows:
\[lemma:weak\] Let $\{U_k\}_{k\in\mathbb{N}}\subset \dot{H}^s_a(\mathbb{R}^{n+1}_+)$ be a sequence satisfying the hypotheses of Proposition \[PScond2\]. Assume also that $U=0$ is the only critical point of ${{\mathscrI}}_\varepsilon$.
Then, up to a subsequence, $U_k$ weakly converges to 0 in $\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$ as $k\to+\infty$.
Notice that assumptions (i) and (ii) imply that there exists $\kappa>0$ such that $$|{{\mathscrI}}_{\varepsilon }(U_k)|+\sup_{{V\in H^s_a({{\mathbb R}}^{n+1}_+)}\atop{ [V]_a =1 }}
\big|\langle {{{\mathscrI}}}_{\varepsilon }'(U_k),V\rangle\big|{\leqslant}\kappa.$$ Hence, by Corollary \[cor:bound\] we have that there exists a positive constant $M$ (independent of $k$) such that $$\label{uniformBoundUk}
[U_k]_a{\leqslant}M.$$ Therefore, there exists a subsequence (that we still denote by $U_k$) converging weakly to some function $U_\infty\in \dot{H}^s_a(\mathbb{R}^{n+1}_+)$, that is, $$\label{kqwrwet}
U_k\rightharpoonup U_\infty\quad\hbox{ in }\dot{H}^s_a(\mathbb{R}^{n+1}_+),$$ as $k\to+\infty$. We now claim that $$\label{U fa zero}
U_\infty=0.$$ For this, we first observe that, thanks to and Theorem 7.1 in [@DPV], we have that $$\label{asfewgjgfjr}
U_k(\cdot,0)\rightarrow U_\infty(\cdot,0)\qquad\hbox{ in }L^\alpha_{\rm{loc}}({{\mathbb R}}^n),\qquad 1{\leqslant}\alpha<2^*_s,$$ and so $$\label{punto}
U_k(\cdot,0)\rightarrow U_\infty(\cdot,0) \quad {\mbox{ a. e. }}{{\mathbb R}}^n.$$ Let now $\Psi\in C_0^\infty(\mathbb{R}^{n+1})$, $\psi:=\Psi(\cdot,0)$ and $K:=supp(\psi)$. According to , $$\begin{split}\label{adweterjutrj}
\langle{{\mathscrI}}_{\varepsilon }'(U_k),\Psi\rangle =\,& \int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_k,\nabla\Psi\rangle\,dX -
\int_{{{\mathbb R}}^n}g(x,U_k(x,0))\psi(x)\,dx \\
=\,& \int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_k,\nabla\Psi\rangle\,dX -
\int_{K}g(x,U_k(x,0))\psi(x)\,dx.
\end{split}$$ Thanks to , we have that $$\label{kqwrwet-1}
\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_k,\nabla\Psi\rangle\,dX \to
\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_\infty,\nabla\Psi\rangle\,dX$$ as $k\to+\infty$. Moreover, implies that $$g(x,U_k(\cdot,0))\to g(x,U_\infty(\cdot,0)) \quad {\mbox{ a. e. }}{{\mathbb R}}^n,$$ as $k\to+\infty$. Also, notice that $$(1+r)^p-1{\leqslant}C(1+r^p),$$ for any $r{\geqslant}0$ and for some positive constant $C>0$. Hence, recalling that $U_{\varepsilon }>0$, thanks to Proposition \[prop:pos\], we can use this with $r:=t/U_{\varepsilon }$ and we obtain that $$(U_{\varepsilon }+t)^p-U_{\varepsilon }^p=U_{\varepsilon }^p\left[\left(1+\frac{t}{U_{\varepsilon }}\right)^p-1\right] {\leqslant}C\,U_{\varepsilon }^p\left(1+\frac{t^p}{U_{\varepsilon }^p}\right) =C\left(U_{\varepsilon }^p+t^p\right).$$ This, formulas and give that $$\label{alkdfsjgerphitrji}
|g(x,t)|{\leqslant}C\left(U_{\varepsilon }^p+t^p\right)+{\varepsilon }|h|.$$ Hence, for any $k\in{{\mathbb N}}$, $$\label{wqrfewhghbenb}
|g(x,U_k(x,0))|\,|\psi| {\leqslant}C|\psi| \left(U_{\varepsilon }^p(x,0)+U_k^p(x,0)\right)+{\varepsilon }|h||\psi|.$$ This means that the sequence $g(\cdot,U_k(\cdot,0))$ is bounded by a sequence that is strongly convergent in $L^1_{\rm{loc}}({{\mathbb R}}^n)$. Moreover, by Theorem 4.9 in [@brezis] we have that there exists a function $f\in L^1_{\rm{loc}}({{\mathbb R}}^n)$ such that, up to a subsequence, $$\label{wqrfewhghbenb-1}
C|\psi| \left(U_{\varepsilon }^p(\cdot,0)+U_k^p(\cdot,0)\right)+{\varepsilon }|\psi||h|{\leqslant}|f|.$$ Formulas and together with imply that we can use the Dominated Convergence Theorem (see e.g. Theorem 4.2 in [@brezis]) and we obtain that $$\int_{K}g(x,U_k(x,0))\psi(x)\,dx\to \int_{K}g(x,U_\infty(x,0))\psi(x)\,dx,$$ as $k\to+\infty$. Using this and into we have $$\langle{{\mathscrI}}_{\varepsilon }'(U_k),\Psi\rangle\to
\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_\infty,\nabla\Psi\rangle\,dX -
\int_{{{\mathbb R}}^n}g(x,U_\infty(x,0))\psi(x)\,dx =
\langle{{\mathscrI}}_{\varepsilon }'(U_\infty),\Psi\rangle,$$ as $k\to+\infty$. On the other hand, assumption (ii) implies that $$\langle{{\mathscrI}}_{\varepsilon }'(U_k),\Psi\rangle\to 0$$ as $k\to+\infty$. The last two formulas imply that $$\label{qwrewputhjtynmf}
\langle{{\mathscrI}}_{\varepsilon }'(U_\infty),\Psi\rangle =0, \quad {\mbox{ for any~$\Psi\in C^\infty_0({{\mathbb R}}^{n+1}_+)$}}.$$ Let now $\Psi\in\dot{H}^s_a({{\mathbb R}}^{n+1})$, with $\psi:=\Psi(\cdot,0)$. Then by there exists a sequence of functions $\Psi_m\in C^\infty_0({{\mathbb R}}^{n+1}_+)$, with $\psi_m:=\Psi_m(\cdot,0)$, such that $$\label{wefreypqqpqwpqwpqw}
{\mbox{$\Psi_m\to\Psi$ in~$\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$ as~$m\to+\infty$.}}$$ By Proposition \[traceIneq\] this implies also that $$\label{wefreypqqpqwpqwpqw-1}
{\mbox{$\psi_m\to\psi$ in~$L^{2^*_s}({{\mathbb R}}^{n})$ as~$m\to+\infty$.}}$$ Therefore, from we deduce that for any $m\in{{\mathbb N}}$ $$\label{distri}
0=\langle{{\mathscrI}}_{\varepsilon }'(U_\infty),\Psi_m\rangle =
\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_\infty,\nabla\Psi_m\rangle\,dX -
\int_{{{\mathbb R}}^n}g(x,U_\infty(x,0))\psi_m(x)\,dx.$$ Now, implies that $$\begin{split}
&\left| \int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_\infty,\nabla\Psi_m\rangle\,dX -\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_\infty,\nabla\Psi\rangle\,dX\right|
\\{\leqslant}\, & \int_{{{\mathbb R}}^{n+1}_+}y^a|\nabla U_\infty|\,|\nabla(\Psi_m-\Psi)|\,dX\\
{\leqslant}\, & \sqrt{\int_{{{\mathbb R}}^{n+1}_+}y^a|\nabla U_\infty|^2\,dX} \cdot
\sqrt{\int_{{{\mathbb R}}^{n+1}_+}y^a|\nabla(\Psi_m-\Psi)|^2\,dX}\to 0,
\end{split}$$ as $m\to+\infty$. Moreover, by Hölder inequality with exponents $2^*_s$ and $\frac{2^*_s}{2^*_s-1}$ we get $$\begin{split}
&\left| \int_{{{\mathbb R}}^n}g(x,U_\infty(x,0))\psi_m(x)\,dx- \int_{{{\mathbb R}}^n}g(x,U_\infty(x,0))\psi(x)\,dx\right| \\
{\leqslant}\,& \int_{{{\mathbb R}}^n}|g(x,U_\infty(x,0))|\,|\psi_m(x)-\psi(x)|\,dx\\
{\leqslant}\,& C\int_{{{\mathbb R}}^n}(U_{\varepsilon }^p+U_\infty^p)\,|\psi_m(x)-\psi(x)|\,dx+{\varepsilon }\int_{{{\mathbb R}}^n}|h|\,|\psi_m(x)-\psi(x)|\,dx\\
{\leqslant}\,& C\left(\int_{{{\mathbb R}}^n}(U_{\varepsilon }^p+U_\infty^p)^{\frac{2^*_s}{2^*_s-1}}\,dx\right)^{\frac{2^*_s-1}{2^*_s}}\,
\left(\int_{{{\mathbb R}}^n}|\psi_m(x)-\psi(x)|^{2^*_s}\,dx\right)^{\frac{1}{2^*_s}}\\
& + {\varepsilon }\left(\int_{{{\mathbb R}}^n}|h|^{\frac{2^*_s}{2^*_s-1}}\,dx\right)^{\frac{2^*_s-1}{2^*_s}}\,
\left(\int_{{{\mathbb R}}^n}|\psi_m(x)-\psi(x)|^{2^*_s}\,dx\right)^{\frac{1}{2^*_s}},
\end{split}$$ where we have used . Furthermore, noticing that $\frac{2^*_sp}{2^*_s-1}=2^*_s$, we have that $$\left(U_{\varepsilon }^p+U_\infty^p\right)^{\frac{2^*_s}{2^*_s-1}}{\leqslant}\left(U_{\varepsilon }+U_\infty\right)^{\frac{2^*_sp}{2^*_s-1}}{\leqslant}\left(U_{\varepsilon }+U_\infty\right)^{2^*_s},$$ up to renaming constants. Thus, since $U_{\varepsilon }$, $U_\infty\in\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$, and $h\in L^r({{\mathbb R}}^n)$ for every $1{\leqslant}r{\leqslant}+\infty$, by Proposition \[traceIneq\] we deduce that $$\begin{split}
&\left| \int_{{{\mathbb R}}^n}g(x,U_\infty(x,0))\psi_m(x)\,dx- \int_{{{\mathbb R}}^n}g(x,U_\infty(x,0))\psi(x)\,dx\right| \\
{\leqslant}\,& C\,
\left(\int_{{{\mathbb R}}^n}|\psi_m(x)-\psi(x)|^{2^*_s}\,dx\right)^{\frac{1}{2^*_s}}\to 0
\end{split}$$ as $m\to+\infty$, thanks to . All in all and going back to we obtain that $$0=\lim_{m\to+\infty}\langle{{\mathscrI}}_{\varepsilon }'(U_\infty),\Psi_m\rangle=
\langle{{\mathscrI}}_{\varepsilon }'(U_\infty),\Psi\rangle,$$ and this shows that holds true for any $\Psi\in\dot{H}^s_a({{\mathbb R}}^{n+1})$. Namely, $U_\infty$ is a critical point for ${{\mathscrI}}_{\varepsilon }$. Since $U=0$ is the only critical point of ${{\mathscrI}}_{\varepsilon }$, we obtain the claim in . This concludes the proof of Lemma \[lemma:weak\].
As we did in the first part to obtain the existence of the minimum, (see in particular Lemma \[tightness\]), to prove Proposition \[PScond2\] we first need to show that the sequence is tight, according to Definition \[defTight\]. Then we can prove the following:
\[lemma:tight\] Let $\{U_k\}_{k\in\mathbb{N}}\subset \dot{H}^s_a(\mathbb{R}^{n+1}_+)$ be a sequence satisfying the hypotheses of Proposition \[PScond2\]. Assume also that $U=0$ is the only critical point of ${{\mathscrI}}_\varepsilon$.
Then, for all $\eta>0$ there exists $\rho>0$ such that for every $k\in\mathbb{N}$ there holds $$\int_{\mathbb{R}^{n+1}_+\setminus B_\rho^+}{y^a|\nabla U_k|^2\,dX}
+\int_{\mathbb{R}^n\setminus\{B_\rho\cap\{y=0\}\}}{|U_k(x,0)|^{2^*_s}\,dx}
<\eta.$$ In particular, the sequence $\{U_k\}_{k\in\mathbb{N}}$ is tight.
From Lemma \[lemma:weak\] we have that $$\begin{split}\label{weak convergence}
& U_k\rightharpoonup 0 \quad \hbox{ in }\dot{H}^s_a(\mathbb{R}^{n+1}_+) \quad {\mbox{ as }}k\to+\infty \\
{\mbox{and }}& U_k(\cdot,0)\rightarrow 0\quad \hbox{ a.e. in }\mathbb{R}^{n}\quad {\mbox{ as }}k\to+\infty.
\end{split}$$ Now we proceed by contradiction. That is, we suppose that there exists $\eta_0>0$ such that for every $\rho>0$ there exists $k\in\mathbb{N}$ such that $$\label{contradV2}
\int_{{{\mathbb R}}^{n+1}_+\setminus B_\rho^+} y^a |\nabla U_k|^2\,dX +
\int_{ {{\mathbb R}}^n \setminus\left(B_\rho\cap\{y=0\}\right) } (U_k)_+^{2^*_s}(x,0)\,dx {\geqslant}\eta_0.$$ Proceeding as in , one can prove that actually $k\rightarrow +\infty$ as $\rho\rightarrow +\infty$.
Let $U_\varepsilon$ be the local minimum of the functional ${{\mathscrF}}_\varepsilon$ found in Theorem \[MINIMUM\]. Since $U_{\varepsilon }\in\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$, from Propositions \[WeightedSob\] and \[traceIneq\] we have that for any ${\varepsilon }>0$ there exists $r:=r_{\varepsilon }>0$ such that $$\label{lasfrepungfmh}\begin{split}
&\int_{{{\mathbb R}}^{n+1}_+\setminus B_r^+}y^a|\nabla U_{\varepsilon }|^2\,dX +
\int_{{{\mathbb R}}^{n+1}_+\setminus B_r^+}y^a|U_{\varepsilon }|^{2\gamma}\,dX \\&\qquad+
\int_{{{\mathbb R}}^{n}\setminus \left(B_r^+\cap \{y=0\}\right)}|U_{\varepsilon }(x,0)|^{2^*_s}\,dx <{\varepsilon },
\end{split}$$ where $\gamma:=1+\frac{2}{n-2s}$. Moreover, by and again by Propositions \[WeightedSob\] and \[traceIneq\] we deduce that $$\label{epsBoundV2-3}
\begin{split}
\int_{{{\mathbb R}}^{n+1}_+}y^a&|\nabla U_k|^2\,dX+\int_{{{\mathbb R}}^{n+1}_+}{y^a|U_k|^{2\gamma}\,dX}+\int_{{{\mathbb R}}^n}{|U_k(x,0)|^{2^*_s}\,dx}\\
&+\int_{{{\mathbb R}}^{n+1}_+}{y^a|\nabla (U_k+U_\varepsilon)|^2\,dX}
+\int_{{{\mathbb R}}^{n+1}_+}{y^a(|U_k|+U_\varepsilon)^{2\gamma}\,dX}\\
&+\int_{{{\mathbb R}}^n}{(|U_k(x,0)|+U_\varepsilon(x,0))^{2^*_s}\,dx}{\leqslant}\tilde{M},
\end{split}$$ for some $\tilde{M}>0$.
Now let $j_{\varepsilon }\in{{\mathbb N}}$ be integer part of $\frac{\tilde{M}}{{\varepsilon }}$, and set, for any $l\in\{0,1,\ldots,j_{\varepsilon }\}$ $$I_l:=\{(x,y)\in{{\mathbb R}}^{n+1}_+ : r+l{\leqslant}|(x,y)|{\leqslant}r+l+1\}.$$ Notice that $j_{\varepsilon }\to +\infty$ as ${\varepsilon }\to 0$. Therefore, by we have that $$\begin{aligned}
(j_{\varepsilon }+1){\varepsilon }&{\geqslant}& \frac{\tilde{M}}{{\varepsilon }}{\varepsilon }\\
&{\geqslant}& \sum_{l=0}^{j_{\varepsilon }} \Big(
\int_{I_l}y^a|\nabla U_k|^2\,dX+\int_{I_l}{y^a|U_k|^{2\gamma}\,dX}
+\int_{I_l\cap\{y=0\}}{|U_k(x,0)|^{2^*_s}\,dx}\\
&&\qquad\quad +\int_{I_l}{y^a|\nabla (U_k+U_\varepsilon)|^2\,dX}
+\int_{I_l}{y^a(|U_k|+U_\varepsilon)^{2\gamma}\,dX}\\
&&\qquad\quad +\int_{I_l\cap\{y=0\}}{(|U_k(x,0)|+U_\varepsilon(x,0))^{2^*_s}\,dx} \Big).\end{aligned}$$ This implies that there exists $\bar{l}\in\{0,1,\ldots,j_{\varepsilon }\}$ such that, up to a subsequence, $$\label{epsBoundV2}
\begin{split}
\int_{I_{\bar{l}}}y^a&|\nabla U_k|^2\,dX+\int_{I_{\bar{l}}}{y^a|U_k|^{2\gamma}\,dX}+\int_{I_{\bar{l}}\cap\{y=0\}}{|U_k(x,0)|^{2^*_s}\,dx}\\
&+\int_{I_{\bar{l}}}{y^a|\nabla (U_k+U_\varepsilon)|^2\,dX}
+\int_{I_{\bar{l}}}{y^a(|U_k|+U_\varepsilon)^{2\gamma}\,dX}\\
&+\int_{I_{\bar{l}}\cap\{y=0\}}{(|U_k(x,0)|+U_\varepsilon(x,0))^{2^*_s}\,dx}
{\leqslant}\varepsilon.
\end{split}$$
Let now $\chi\in C^\infty_0({{\mathbb R}}^{n+1}_+,[0,1])$ be a cut-off function such that $$\label{chi}
\chi(x,y)=\begin{cases}
1,\qquad |(x,y)|{\leqslant}r+\bar{l},\\
0,\qquad |(x,y)|{\geqslant}r+\bar{l}+1,
\end{cases}
\quad {\mbox{ and }} \quad |\nabla \chi|{\leqslant}2.$$ Define, for any $k\in{{\mathbb N}}$, $$\label{def Wk}
W_{1,k}:=\chi U_k \quad {\mbox{ and }} \quad W_{2,k}:=(1-\chi)U_k.$$ Hence $W_{1,k}+W_{2,k}=U_k$ for any $k\in{{\mathbb N}}$. Moreover, $$\label{conv Wk}
W_{1,k},\,W_{2,k}\rightharpoonup 0 \quad {\mbox{ in }} \dot{H}^s_a({{\mathbb R}}^{n+1}_+),$$ as $k\to+\infty$. Indeed, for any $\Psi\in\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$ with $[\Psi]_a=1$ and $\delta>0$, we have that $$\left|\int_{\mathbb{R}^{n+1}_+}y^a\langle\nabla U_{k},\nabla\Psi\rangle \,dX\right|{\leqslant}\frac{\delta}{2},$$ for any $k$ sufficiently large, say $k{\geqslant}\bar{k}(\delta)$, thanks to . Moreover, the compactness result in Lemma \[lemma:compact\] implies that $$\int_{I_{\bar{l}} }y^a|U_k|^2\,dX{\leqslant}\frac{\delta^2}{16},$$ for $k$ large enough (say $k{\geqslant}\bar{k}(\delta)$, up to renaming $\bar{k}(\delta)$). Therefore, recalling and and using Hölder inequality, we obtain that $$\begin{aligned}
&&\left|\int_{\mathbb{R}^{n+1}_+}y^a\langle\nabla W_{1,k},\nabla\Psi\rangle \,dX \right|\\&=&
\left|\int_{\mathbb{R}^{n+1}_+}y^a\langle\nabla (\chi\,U_{k}),\nabla\Psi\rangle \,dX\right|\\
&{\leqslant}& \left|\int_{\mathbb{R}^{n+1}_+}y^a\chi \langle\nabla U_{k},\nabla\Psi\rangle \,dX\right| +
\left|\int_{\mathbb{R}^{n+1}_+}y^a U_k \langle\nabla\chi,\nabla\Psi\rangle \,dX\right|\\
&{\leqslant}& \left|\int_{\mathbb{R}^{n+1}_+}y^a\langle\nabla U_{k},\nabla\Psi\rangle \,dX\right|+ \int_{\mathbb{R}^{n+1}_+}y^a |U_k|\, |\nabla\chi|\,|\nabla\Psi|\,dX\\
&{\leqslant}& \frac{\delta}{2} + 2\sqrt{\int_{\mathbb{R}^{n+1}_+}y^a |U_k|^2\,dX}
\cdot\sqrt{\int_{\mathbb{R}^{n+1}_+}y^a |\nabla\Psi|^2\,dX}\\
&{\leqslant}&\frac{\delta}{2}+2\frac{\delta}{4}\\
&=&\delta,\end{aligned}$$ which proves for $W_{1,k}$. The proof for $W_{2,k}$ is similar, and so we omit it.
Furthermore, from and Theorem 7.1 in [@DPV] we have that $$\label{convWi}\begin{split}
& W_{i,k}(\cdot,0)\rightarrow 0\quad {\mbox{ a.e. }}{{\mathbb R}}^n,\\
&{\mbox{and }}(U_\varepsilon+W_{i,k})(\cdot,0)\rightarrow U_\varepsilon(\cdot,0)\ \hbox{ in }L^\alpha_{\rm{loc}}({{\mathbb R}}^n),\;\; \forall\;1{\leqslant}\alpha<2^*_s,\; i=1,2,
\end{split}$$ as $k\to+\infty$. Notice also that there exists a positive constant $C$ (independent of $k$) such that $$\label{lim124}
[U_{\varepsilon }+W_{i,k}]_a{\leqslant}C,$$ for $i\in\{1,2\}$. Let us show only for $W_{1,k}$, being the proof for $W_{2,k}$ similar. From we obtain that $$\begin{aligned}
&& [W_{i,k}]_a^2=\int_{{{\mathbb R}}^{n+1}_+}y^a|\nabla W_{1,k}|^2\,dX
= \int_{{{\mathbb R}}^{n+1}_+}y^a|\nabla(\chi U_k)|^2\,dX\\
&&\qquad {\leqslant}2\int_{{{\mathbb R}}^{n+1}_+}y^a\chi^2 |\nabla U_k|^2\,dX +
2\int_{{{\mathbb R}}^{n+1}_+}y^a |U_k|^2|\nabla\chi|^2\,dX\\
&&\qquad {\leqslant}2M +8\int_{I_{\bar{l}}}y^a |U_k|^2\,dX{\leqslant}2M+8C,\end{aligned}$$ for some $C>0$ independent of $k$, thanks to , and Lemma \[lemma:compact\]. This, together with the fact that $U_{\varepsilon }\in\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$, gives .
Therefore, using hypothesis (ii), $$\label{conv1}
\lim_{k\to+\infty}\langle{{\mathscrI}}_\varepsilon'(U_k),
U_\varepsilon+W_{i,k}\rangle =0,\qquad i=1,2.$$ On the other hand, by , $$\begin{split}\label{laskgprekjn b}
&\left|
\langle {{\mathscrI}}_\varepsilon'(U_k)-{{\mathscrI}}_\varepsilon'(W_{1,k}),
U_\varepsilon+W_{1,k}\rangle \right| \\
{\leqslant}\,& \left| \int_{\mathbb{R}^{n+1}_+} {y^a\langle \nabla (1-\chi)U_{k},\nabla(U_\varepsilon+W_{1,k})\rangle\,dX} \right|\\
& +\left| \varepsilon \int_{\mathbb{R}^n}{h(x)\left( (U_\varepsilon + (W_{1,k})_+ )^q-(U_\varepsilon +(U_k)_+)^q \right)(x,0)(U_\varepsilon+W_{1,k})(x,0)\,dx} \right| \\
&+\left| \int_{\mathbb{R}^n} \left( (U_\varepsilon+(W_{1,k})_+)^p
-(U_\varepsilon+(U_k)_+)^p \right)(x,0)(U_\varepsilon+W_{1,k})(x,0)\,dx \right|\\
=:\,& I_1+I_2+I_3.
\end{split}$$ To estimate $I_1$, notice that $I_1{\leqslant}I_{1,1}+I_{1,2}$, where $$\begin{aligned}
I_{1,1}&:=&\left|\int_{ \mathbb{R}^{n+1}_+} {y^a\langle \nabla (1-\chi)U_{k},\nabla U_\varepsilon\rangle\,dX} \right|\\
{\mbox{and }}\quad I_{1,2}&:=&\left|\int_{\mathbb{R}^{n+1}_+} {y^a\langle \nabla (1-\chi)U_{k},\nabla(\chi U_{k})\rangle\,dX} \right|.\end{aligned}$$ We split further $I_{1,1}$ as $$\begin{aligned}
I_{1,1}{\leqslant}\left|\int_{{{\mathbb R}}^{n+1}_+\setminus B^+_{r+\bar{l}} } {y^a(1-\chi)\langle \nabla U_{k},\nabla U_\varepsilon\rangle\,dX} \right| +
\left|\int_{ I_{\bar{l}}} {y^a U_k\langle \nabla (1-\chi),\nabla U_\varepsilon\rangle\,dX} \right|\\\end{aligned}$$ Since $B^+_r\subset B^+_{r+\bar{l}}$, by Hölder inequality, and we have that $$\begin{aligned}
&&\left|\int_{{{\mathbb R}}^{n+1}_+\setminus B^+_{r+\bar{l}}} {y^a(1-\chi)\langle \nabla U_{k},\nabla U_\varepsilon\rangle\,dX} \right|\\
&&\qquad {\leqslant}\sqrt{\int_{{{\mathbb R}}^{n+1}_+} {y^a|\nabla U_{k}|^2\,dX}}
\cdot\sqrt{\int_{{{\mathbb R}}^{n+1}_+\setminus B^+_r} {y^a|\nabla U_{\varepsilon }|^2\,dX}}
{\leqslant}M{\varepsilon }^{1/2}.\end{aligned}$$ Moreover, by and applying twice the Hölder inequality (first with exponent $1/2$ and then with exponents $\gamma$ and $\frac{\gamma}{\gamma-1}$) we obtain that $$\begin{aligned}
&& \left|\int_{ I_{\bar{l}}} {y^a U_k\langle \nabla (1-\chi),\nabla U_\varepsilon\rangle\,dX} \right|\\
&&\qquad {\leqslant}2\left(\int_{I_{\bar{l}}}y^a|U_k|^2\,dX\right)^{1/2}
\left(\int_{I_{\bar{l}}}y^a|\nabla U_{\varepsilon }|^2\,dX\right)^{1/2}\\
&&\qquad {\leqslant}2\left(\int_{I_{\bar{l}}}y^a|U_k|^2\,dX\right)^{1/2}
\left(\int_{{{\mathbb R}}^{n+1}_+}y^a|\nabla U_{\varepsilon }|^2\,dX\right)^{1/2}\\
&&\qquad {\leqslant}C\left(\int_{I_{\bar{l}}} y^a\,dX\right)^{\frac{\gamma-1}{2\gamma}}
\left(\int_{I_{\bar{l}}}y^a|U_k|^{2\gamma}\,dX\right)^{\frac{1}{2\gamma}}
{\leqslant}C{\varepsilon }^{1/2\gamma},\end{aligned}$$ up to renaming constants, where was used in the last line. Hence, $$I_{1,1}{\leqslant}C{\varepsilon }^{1/2\gamma},$$ for a suitable constant $C>0$. Let us estimate $I_{1,2}$: $$\begin{aligned}
I_{1,2}&{\leqslant}&\left|\int_{I_{\bar{l}}} {y^a|U_k|^2\langle \nabla (1-\chi),\nabla\chi\rangle\,dX} \right|
+ \left|\int_{I_{\bar{l}}} {y^a U_k\chi\langle\nabla (1-\chi),\nabla U_{k}\rangle\,dX} \right|\\
&&\qquad +\left|\int_{I_{\bar{l}}} {y^a\chi(1-\chi)|\nabla U_{k}|^2\,dX} \right|
+\left|\int_{I_{\bar{l}}} {y^a(1-\chi)U_k \langle\nabla U_{k},\nabla\chi\rangle\,dX} \right|.\end{aligned}$$ Thus, in the same way as before, and using once more, we obtain that $I_{1,2}{\leqslant}C{\varepsilon }^{1/2\gamma}$ for some $C>0$. Therefore $$\label{est1}
I_1 {\leqslant}C{\varepsilon }^{1/2\gamma},$$ for some positive constant $C$.
We estimate now $I_2$. For this, we first observe that formulas and give that $$\left|(U_{\varepsilon }+(W_{1,k})_+)^q-(U_{\varepsilon }+(U_k)_+)^q\right| {\leqslant}L|(W_{1,k})_+-(U_k)_+|^q = L (U_k)_+^q|1-\chi|^q,$$ for a suitable constant $L>0$. Consequently, applying Hölder inequality with exponents $\frac{2^*_s}{2^*_s-1-q}$, $\frac{2^*_s}{q}$ and $2^*_s$ we obtain that $$\begin{aligned}
I_2 &{\leqslant}& {\varepsilon }\int_{{{\mathbb R}}^n}|h|\,\left|(U_{\varepsilon }+(W_{1,k})_+)^q-(U_{\varepsilon }+(U_k)_+)^q\right|\,|U_{\varepsilon }+W_{1,k}|\,dx\\
&{\leqslant}& {\varepsilon }L\int_{{{\mathbb R}}^n}|h| (U_k)_+^q |U_{\varepsilon }+W_{1,k}|\,dx\\
&{\leqslant}& {\varepsilon }L \|h\|_{L^{\frac{2^*_s}{2^*_s-1-q}}({{\mathbb R}}^n)}
\left(\int_{{{\mathbb R}}^n}(U_k)_+^{2^*_s}\,dx\right)^{\frac{q}{2^*_s}}
\left( \int_{{{\mathbb R}}^n}|U_{\varepsilon }+W_{1,k}|^{2^*_s}\,dx\right)^{\frac{1}{2^*_s}}
\\
&{\leqslant}& C{\varepsilon }, \end{aligned}$$ for some $C>0$, thanks to , and Proposition \[traceIneq\].
To estimate $I_3$, let us define the auxiliary function $$f(t):=(U_\varepsilon+t\chi (U_k)_+ +(1-t)(U_k)_+)^p,\qquad t\in[0,1].$$ Thus, recalling , we have that $$\begin{split}
& \left|(U_\varepsilon+(W_{1,k})_+)^p-(U_\varepsilon+(U_k)_+)^p\right|
=\left|(U_\varepsilon+\chi(U_k)_+)^p-(U_\varepsilon+(U_k)_+)^p\right|
\\&\qquad =|f(1)-f(0)|=\left|\int_0^1f'(t)\,dt\right|\\
&\qquad {\leqslant}p(1-\chi)(U_k)_+\int_0^1|U_\varepsilon+t\chi (U_k)_+ +(1-t)(U_k)_+|^{p-1}\,dt\\
&\qquad {\leqslant}p(1-\chi)(U_k)_+(U_\varepsilon+(U_k)_+)^{p-1}
{\leqslant}C(1-\chi)(U_k)_+ U_\varepsilon^{p-1}+C(1-\chi)(U_k)_+^p,
\end{split}$$ for a suitable positive constant $C$. Therefore, $$\begin{split}
I_3&{\leqslant}C\Big(\int_{\mathbb{R}^n}(1-\chi(x,0))(U_k)_+(x,0)U_\varepsilon^p(x,0)\,dx \\
&\qquad\quad +\int_{\mathbb{R}^n}(1-\chi(x,0))(U_k)_+^p(x,0) U_\varepsilon(x,0)\,dx \\
&\qquad \quad +
\int_{I_{\bar{l}}} \chi(x,0)(1-\chi(x,0))U_\varepsilon^{p-1}(x,0)(U_k)_+^2(x,0)\,dx
\\&\qquad\quad +\int_{I_{\bar{l}}}\chi(x,0)(1-\chi(x,0))(U_k)_+^{p+1}(x,0)\,dx\Big)\\
&=: I_{3,1}+I_{3,2}+I_{3,3}+I_{3,4}.
\end{split}$$ Concerning $I_{3,1}$ and $I_{3,2}$ we are in the position to apply Lemma \[lemma:conv\] with $V_k:=U_k$, $U_o:=U_{\varepsilon }$ and $\psi:=1-\chi(\cdot,0)$ (notice that $\alpha:=1$ and $\beta:=p$ and $\alpha:=p$ and $\beta:=1$, respectively). So we obtain that both $I_{3,1}=o_k(1)$ and $I_{3,2}=o_k(1)$.
Moreover, using Hölder inequality with exponent $\frac{p+1}{p-1}$ and $\frac{p+1}{2}$, Proposition \[traceIneq\] and , we have $$I_{3,3}{\leqslant}\left(\int_{I_{\bar{l}}}U_{\varepsilon }^{p+1}(x,0)\,dx\right)^{\frac{p-1}{p+1}}
\left(\int_{I_{\bar{l}}}(U_k)_+^{p+1}(x,0)\,dx\right)^{\frac{2}{p+1}} {\leqslant}C\varepsilon^{\frac{2}{p+1}},$$ for a suitable $C>0$. Finally, making use of once again we obtain that $I_{3,4}{\leqslant}C{\varepsilon }$, for some $C>0$. Consequently, putting all these informations together we get $$I_3{\leqslant}C\varepsilon^{\frac{2}{p+1}}+o_k(1).$$ All in all, from we obtain that $$\label{boundW1}
|\langle {{\mathscrI}}_\varepsilon'(U_k)-
{{\mathscrI}}_\varepsilon'(W_{1,k}),U_\varepsilon+W_{1,k}\rangle|{\leqslant}C\varepsilon^{\frac{1}{2\gamma}}
+o_k(1).$$ Likewise, it can be checked that $$\label{boundW2}
|\langle {{\mathscrI}}_\varepsilon'(U_k)-{{\mathscrI}}_\varepsilon'(W_{2,k}),
U_\varepsilon+W_{2,k}\rangle|{\leqslant}C\varepsilon^{\frac{1}{2\gamma}}+o_k(1).$$ Therefore, using this and , $$\label{boundWi}
|\langle {{\mathscrI}}_\varepsilon'(W_{i,k}),U_\varepsilon+W_{i,k}\rangle|
{\leqslant}C\varepsilon^{\frac{1}{2\gamma}} +o_k(1), \qquad i=1,2.$$
From now on we organize the proof in three steps as follows: in the forthcoming Step 1 and 2 we show lower bounds for ${{\mathscrI}}_{\varepsilon }(W_{1,k})$ and ${{\mathscrI}}_{\varepsilon }(W_{2,k})$, respectively. Then, in Step 3 we use these estimates to obtain a lower bound for ${{\mathscrI}}_{\varepsilon }(U_k)$ that will give a contradiction with the assumptions on ${{\mathscrI}}_{\varepsilon }$, and so the desired claim in Lemma \[lemma:tight\] follows.
[*Step 1: Lower bound for ${{\mathscrI}}_{\varepsilon }(W_{1,k})$.*]{} From and we have $$\begin{split}\label{limW1}
&{{\mathscrI}}_\varepsilon(W_{1,k})-
\frac{1}{2}\langle {{\mathscrI}}_\varepsilon'(W_{1,k}),U_\varepsilon+W_{1,k}\rangle\\
=\,& -\frac12\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_{\varepsilon },\nabla W_{1,k}\rangle\,dX \\
&-\frac{\varepsilon}{q+1}\int_{{{\mathbb R}}^n}h(x)\left(
(U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0) -U_\varepsilon^{q+1}(x,0)\right)\,dx \\& +
{\varepsilon }\int_{{{\mathbb R}}^n}h(x)U^q_{\varepsilon }(x,0)(W_{1,k})_+(x,0)\,dx \\
& -\frac{1}{p+1}\int_{{{\mathbb R}}^n}\left(
(U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0) -U_\varepsilon^{p+1}(x,0)\right)\,dx \\& +
\int_{{{\mathbb R}}^n}U^p_{\varepsilon }(x,0)(W_{1,k})_+(x,0)\,dx \\
& +\frac{{\varepsilon }}{2}\int_{{{\mathbb R}}^n}h(x)
\left( (U_{\varepsilon }+(W_{1,k})_+)^{q}(x,0)-U_{\varepsilon }^q(x,0)\right)
\left(U_{\varepsilon }+W_{1,k}\right)(x,0) \,dx \\
& +\frac{1}{2}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(W_{1,k})_+)^{p}(x,0)-U_{\varepsilon }(x,0)\right)
\left(U_{\varepsilon }+W_{1,k}\right)(x,0) \,dx.
\end{split}$$ Thanks to , we have that $$\lim_{k\to+\infty}\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_{\varepsilon },\nabla W_{1,k}\rangle\,dX=0.$$ Moreover, from Lemma \[lemma:conv\] applied here with $V_k:=W_{1,k}$, $U_o:=U_{\varepsilon }$, $\psi:=h$, $\alpha:=1$ and $\beta:=q$ we have that $$\label{ccc96}
\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}h(x)U^q_{\varepsilon }(x,0)(W_{1,k})_+(x,0)\,dx =0.$$ Analogously, by taking $V_k:=W_{1,k}$, $U_o:=U_{\varepsilon }$, $\psi:=1$, $\alpha:=1$ and $\beta:=p$ in Lemma \[lemma:conv\] (notice that in this case $\alpha+\beta=p+1=2^*_s$ and $\psi\in L^{\infty}({{\mathbb R}}^n)$) we obtain that $$\label{ccc95}
\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}U^p_{\varepsilon }(x,0)(W_{1,k})_+(x,0)\,dx =0.$$ Taking the limit as $k\to+\infty$ in and using the last three formulas, we obtain that $$\begin{split}\label{limW1-1}
&\lim_{k\to+\infty}{{\mathscrI}}_\varepsilon(W_{1,k})-
\frac{1}{2}\langle {{\mathscrI}}_\varepsilon'(W_{1,k}),U_\varepsilon+W_{1,k}\rangle\\
=\,& \lim_{k\to+\infty}\Big(-\frac{\varepsilon}{q+1}\int_{{{\mathbb R}}^n}h(x)\left(
(U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0) -U_\varepsilon^{q+1}(x,0)\right)\,dx \\
& \qquad \quad -\frac{1}{p+1}\int_{{{\mathbb R}}^n}\left(
(U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0) -U_\varepsilon^{p+1}(x,0)\right)\,dx \\
& \qquad \quad +\frac{{\varepsilon }}{2}\int_{{{\mathbb R}}^n}h(x)\left(
(U_{\varepsilon }+(W_{1,k})_+)^{q}(x,0)-U_{\varepsilon }^q(x,0)\right)
\left(U_{\varepsilon }+W_{1,k}\right)(x,0) \,dx \\
& \qquad \quad +\frac{1}{2}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(W_{1,k})_+)^{p}(x,0)-U_{\varepsilon }^p(x,0)\right)
\left(U_{\varepsilon }+W_{1,k}\right)(x,0) \,dx\Big).
\end{split}$$ Now we observe that if $x\in{{\mathbb R}}^n$ is such that $W_{1,k}(x,0){\leqslant}0$, then $$(U_{\varepsilon }+(W_{1,k})_+)^{q}(x,0)-U_{\varepsilon }^q(x,0)=U_{\varepsilon }^{q}(x,0)-U_{\varepsilon }^q(x,0)=0,$$ and so $$\begin{aligned}
&&\int_{{{\mathbb R}}^n}h(x)\left(
(U_{\varepsilon }+(W_{1,k})_+)^{q}(x,0)-U_{\varepsilon }^q(x,0)\right)
\left(U_{\varepsilon }+W_{1,k}\right)(x,0) \,dx \\
&=& \int_{{{\mathbb R}}^n}h(x)\left(
(U_{\varepsilon }+(W_{1,k})_+)^{q}(x,0)-U_{\varepsilon }^q(x,0)\right)
\left(U_{\varepsilon }+(W_{1,k})_+\right)(x,0) \,dx\\
&=& \int_{{{\mathbb R}}^n}h(x)
(U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0)\,dx - \int_{{{\mathbb R}}^n} h(x)U_{\varepsilon }^{q+1}(x,0)\,dx \\&&\qquad -
\int_{{{\mathbb R}}^n} h(x)U_{\varepsilon }^q(x,0) (W_{1,k})_+(x,0) \,dx\\
&=& \int_{{{\mathbb R}}^n}h(x)\left((U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0) - U_{\varepsilon }^{q+1}(x,0)\right)\,dx \\&&\qquad-
\int_{{{\mathbb R}}^n} h(x)U_{\varepsilon }^q(x,0) (W_{1,k})_+(x,0) \,dx.\end{aligned}$$ Analogously $$\begin{aligned}
&& \int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(W_{1,k})_+)^{p}(x,0)-U_{\varepsilon }^p(x,0)\right)
\left(U_{\varepsilon }+W_{1,k}\right)(x,0) \,dx\\
&=& \int_{{{\mathbb R}}^n}\left((U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0) - U_{\varepsilon }^{p+1}(x,0)\right)\,dx -
\int_{{{\mathbb R}}^n} U_{\varepsilon }^p(x,0) (W_{1,k})_+(x,0) \,dx.\end{aligned}$$ Therefore, using once more and , from we obtain that $$\begin{split}\label{limW1-2}
&\lim_{k\to+\infty}{{\mathscrI}}_\varepsilon(W_{1,k})-
\frac{1}{2}\langle {{\mathscrI}}_\varepsilon'(W_{1,k}),U_\varepsilon+W_{1,k}\rangle\\
=\,& \lim_{k\to+\infty}\Big(-\frac{\varepsilon}{q+1}\int_{{{\mathbb R}}^n}h(x)\left(
(U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0) -U_\varepsilon^{q+1}(x,0)\right)\,dx \\
& \qquad \quad -\frac{1}{p+1}\int_{{{\mathbb R}}^n}\left(
(U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0) -U_\varepsilon^{p+1}(x,0)\right)\,dx \\
& \qquad \quad +\frac{{\varepsilon }}{2} \int_{{{\mathbb R}}^n}h(x)\left((U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0)
- U_{\varepsilon }^{q+1}(x,0)\right)\,dx \\& \qquad \quad-
\frac{{\varepsilon }}{2}\int_{{{\mathbb R}}^n} h(x)U_{\varepsilon }^q(x,0) (W_{1,k})_+(x,0) \,dx \\
& \qquad \quad +\frac{1}{2}\int_{{{\mathbb R}}^n}\left((U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0) - U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\& \qquad \quad-\frac{1}{2}
\int_{{{\mathbb R}}^n} U_{\varepsilon }^p(x,0) (W_{1,k})_+(x,0) \,dx\Big)\\
=\,& \lim_{k\to+\infty}\Big(-\frac{\varepsilon}{q+1}\int_{{{\mathbb R}}^n}h(x)\left(
(U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0) -U_\varepsilon^{q+1}(x,0)\right)\,dx \\
& \qquad \quad -\frac{1}{p+1}\int_{{{\mathbb R}}^n}\left(
(U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0) -U_\varepsilon^{p+1}(x,0)\right)\,dx \\
& \qquad \quad +\frac{{\varepsilon }}{2} \int_{{{\mathbb R}}^n}h(x)\left((U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0)
- U_{\varepsilon }^{q+1}(x,0)\right)\,dx \\
& \qquad \quad +\frac{1}{2}\int_{{{\mathbb R}}^n}\left((U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0) -
U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
=\,& \lim_{k\to+\infty}\Big(-{\varepsilon }\left(\frac{1}{q+1}-\frac12\right)\int_{{{\mathbb R}}^n}h(x)\left(
(U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0) -U_\varepsilon^{q+1}(x,0)\right)\,dx \\
& \qquad \quad +\left(\frac12-\frac{1}{p+1}\right)\int_{{{\mathbb R}}^n}\left(
(U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0) -U_\varepsilon^{p+1}(x,0)\right)\,dx\Big).
\end{split}$$ Now we claim that $$\label{ewhupytlkvdxsmnvew}
\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}h(x)\left(
(U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0) -U_\varepsilon^{q+1}(x,0)\right)\,dx=0.$$ For this, notice that if $x\in{{\mathbb R}}^n\setminus B_{r+\bar{l}+1}$, then $W_{1,k}(x,0)=0$, thanks to and . Therefore, for any $x\in{{\mathbb R}}^n\setminus B_{r+\bar{l}+1}$ we have that $$(U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0) -U_\varepsilon^{q+1}(x,0) =
U_{\varepsilon }^{q+1}(x,0) -U_\varepsilon^{q+1}(x,0)=0.$$ Thus $$\begin{split}\label{qwtrewgrhgvr}
&\int_{{{\mathbb R}}^n}h(x)\left(
(U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0) -U_\varepsilon^{q+1}(x,0)\right)\,dx \\=\,&
\int_{B_{r+\bar{l}+1}}h(x)\left(
(U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0) -U_\varepsilon^{q+1}(x,0)\right)\,dX.
\end{split}$$ Thanks to , we have that $W_{1,k}(\cdot, 0)$ converges to zero a.e. in ${{\mathbb R}}^n$, and so $(W_{1,k})_+(\cdot, 0)$ converges to zero a.e. in ${{\mathbb R}}^n$, as $k\to+\infty$. Therefore $$U_{\varepsilon }+(W_{1,k})_+^{q+1}(x,0) \to U_\varepsilon^{q+1}(x,0)\quad
{\mbox{ for a.e. }}x\in{{\mathbb R}}^n,$$ as $k\to+\infty$. Moreover the strong convergence of $W_{1,k}(\cdot,0)$ in $L^{q+1}_{\rm{loc}}({{\mathbb R}}^n)$ (due again to ) and Theorem 4.9 in [@brezis] imply that there exists a function $F\in L^{q+1}_{\rm{loc}}({{\mathbb R}}^n)$ such that $|W_{1,k}(x,0)|{\leqslant}|F(x)|$ for a.e. $x\in{{\mathbb R}}^n$. This and the boundedness of $U_{\varepsilon }$ (see Corollary \[coro:bound\]) give that $$h(U_{\varepsilon }+(W_{1,k})_+)^{q+1}{\leqslant}|h|(|U_{\varepsilon }|+|W_{1,k}|)^{q+1}{\leqslant}C|h|(1+|F|^{q+1})
\in L^1(B_{r+\bar{l}+1}),$$ for a suitable $C>0$. Thus, the Dominated Convergence Theorem applies, and together with give the convergence in .
Consequently, from and we obtain that $$\begin{split}\label{limPositive}
&\lim_{k\rightarrow\infty}
\left({{\mathscrI}}_\varepsilon(W_{1,k})-\frac{1}{2}\langle
{{\mathscrI}}_\varepsilon'(W_{1,k}),U_\varepsilon+W_{1,k}\rangle\right)\\
=\,& \left(\frac12-\frac{1}{p+1}\right) \lim_{k\rightarrow\infty}
\int_{\mathbb{R}^n} \left( (U_\varepsilon+(W_{1,k})_+)^{p+1}(x,0)-U_\varepsilon^{p+1}(x,0)\right)
\,dx {\geqslant}0
\end{split}$$ (recall that $p+1=2^*_s>2$). In particular, by and , there holds $$\begin{split}\label{infBoundW1}
{{\mathscrI}}_\varepsilon(W_{1,k})&= {{\mathscrI}}_\varepsilon(W_{1,k})
-\frac{1}{2}\langle {{\mathscrI}}_\varepsilon'(W_{1,k}),U_\varepsilon+W_{1,k}\rangle
+\frac{1}{2}\langle {{\mathscrI}}_\varepsilon'(W_{1,k}),U_\varepsilon+W_{1,k}\rangle \\
&{\geqslant}{{\mathscrI}}_\varepsilon(W_{1,k})
-\frac{1}{2}\langle {{\mathscrI}}_\varepsilon'(W_{1,k}),U_\varepsilon+W_{1,k}\rangle
-C\varepsilon^{\frac{1}{2\gamma}} +o_k(1)\\
&{\geqslant}-C\varepsilon^{\frac{1}{2\gamma}} +o_k(1),
\end{split}$$ where $C$ is a positive constant that may change from line to line.
Formula provides the desired estimate from below for ${{\mathscrI}}_{\varepsilon }(W_{1,k})$. Next step is to obtain an estimate from below for ${{\mathscrI}}_{\varepsilon }(W_{2,k})$ as well.
[*Step 2: Lower bound for ${{\mathscrI}}_{\varepsilon }(W_{2,k})$.*]{} We first observe that formula implies that there exists a constant $L>0$ such that $$|(U_{\varepsilon }+(W_{2,k})_+)^q(x,0) -U_{\varepsilon }^q(x,0)|{\leqslant}L(W_{2,k})_+^q(x,0).$$ Hence $$\begin{split}
& \left|\varepsilon
\int_{\mathbb{R}^n}
h(x)\left((U_\varepsilon+(W_{2,k})_+)^{q}(x,0)-U_\varepsilon^q(x,0)\right)
(U_\varepsilon+W_{2,k})(x,0)\,dx\right|\\
&\qquad {\leqslant}\varepsilon
\int_{\mathbb{R}^n}
|h(x)|\left|(U_\varepsilon+(W_{2,k})_+)^{q}(x,0)-U_\varepsilon^q(x,0)\right|
|(U_\varepsilon+W_{2,k})(x,0)|\,dx\\
&\qquad {\leqslant}\varepsilon L
\int_{\mathbb{R}^n}
|h(x)| (W_{2,k})_+^{q}(x,0)|(U_\varepsilon+W_{2,k})(x,0)|\,dx\\
&\qquad {\leqslant}\varepsilon L\left(
\int_{\mathbb{R}^n} |h(x)| (W_{2,k})_+^{q}(x,0)U_\varepsilon(x,0)\,dx +
\int_{\mathbb{R}^n} |h(x)| (W_{2,k})_+^{q+1}(x,0)\,dx\right).
\end{split}$$ Thanks to Lemma \[lemma:conv\] (applied here with $V_k:=W_{2,k}$, $U_o:=U_{\varepsilon }$, $\psi:=h$, $\alpha:=q$ and $\beta:=1$) we have that $$\lim_{k\to+\infty}\int_{\mathbb{R}^n} |h(x)| (W_{2,k})_+^{q}(x,0)U_\varepsilon(x,0)\,dx=0.$$ Moreover, by Hölder inequality with exponents $\frac{2^*_s}{2^*_s-1-q}$ and $\frac{2^*_s}{q+1}$ we obtain that $$\begin{aligned}
&& \int_{\mathbb{R}^n} |h(x)| (W_{2,k})_+^{q+1}(x,0)\,dx\\&&\qquad {\leqslant}\left(\int_{\mathbb{R}^n} |h(x)|^{\frac{2^*_s}{2^*_s-1-q}}\,dx\right)^{\frac{2^*_s-1-q}{2^*_s}}
\left(\int_{\mathbb{R}^n}(W_{2,k})_+^{2^*_s}(x,0)\,dx\right)^{\frac{q+1}{2^*_s}}
\\&&\qquad {\leqslant}C \left(\int_{\mathbb{R}^n}(U_{k})_+^{2^*_s}(x,0)\,dx\right)^{\frac{q+1}{2^*_s}}
{\leqslant}C [U_k]_a^{q+1}{\leqslant}C,\end{aligned}$$ for some constant $C>0$, where we have also used , Proposition \[traceIneq\] and Corollary \[coro:bound\]. The last three formulas imply that $$\left|\varepsilon
\int_{\mathbb{R}^n}
h(x)\left((U_\varepsilon+(W_{2,k})_+)^{q}(x,0)-U_\varepsilon^q(x,0)\right)
(U_\varepsilon+W_{2,k})(x,0)\,dx\right|{\leqslant}C{\varepsilon }+o_k(1),$$ for a suitable $C>0$. This, together with , and (with $i=2$) gives $$\begin{split}\label{upBoundW2}
& \int_{\mathbb{R}^{n+1}_+}{y^a|\nabla W_{2,k}|^2\,dX}\\
=\,& \langle {{\mathscrI}}_{\varepsilon }'(W_{2,k}, U_{\varepsilon }+W_{2,k}\rangle
- \int_{\mathbb{R}^{n+1}_+}{y^a\langle\nabla W_{2,k}, \nabla U_{\varepsilon }\rangle,dX}\\
&\qquad + {\varepsilon }\int_{{{\mathbb R}}^n}h(x)\left((U_\varepsilon+(W_{2,k})_+)^{q}(x,0)-U_\varepsilon^q(x,0)\right)
(U_\varepsilon+W_{2,k})(x,0)\,dx \\
&\qquad
+ \int_{{{\mathbb R}}^n}\left((U_\varepsilon+(W_{2,k})_+)^{p}(x,0)-U_\varepsilon^p(x,0)\right)
(U_\varepsilon+W_{2,k})(x,0)\,dx \\
{\leqslant}\,& C\varepsilon^{\frac{1}{2\gamma}} +o_k(1)
+\int_{{{\mathbb R}}^n}\left((U_\varepsilon+(W_{2,k})_+)^{p}(x,0)-U_\varepsilon^p(x,0)\right)
(U_\varepsilon+W_{2,k})(x,0)\,dx.
\end{split}$$ Now notice that if $x\in{{\mathbb R}}^n$ is such that $W_{2,k}{\leqslant}0$ then $$(U_\varepsilon+(W_{2,k})_+)^{p}(x,0)-U_\varepsilon^p(x,0) =0,$$ and so $$\begin{aligned}
&& \int_{{{\mathbb R}}^n}\left((U_\varepsilon+(W_{2,k})_+)^{p}(x,0)-U_\varepsilon^p(x,0)\right)
(U_\varepsilon+W_{2,k})(x,0)\,dx \\
&=& \int_{{{\mathbb R}}^n}\left((U_\varepsilon+(W_{2,k})_+)^{p}(x,0)-U_\varepsilon^p(x,0)\right)
(U_\varepsilon+(W_{2,k})_+)(x,0)\,dx\\
&=& \int_{{{\mathbb R}}^n}\left((U_\varepsilon+(W_{2,k})_+)^{p+1}(x,0)
-U_\varepsilon^{p+1}(x,0)\right)\,dx -\int_{{{\mathbb R}}^n}
U_\varepsilon^{p}(x,0)(W_{2,k})_+(x,0)\,dx.\end{aligned}$$ According to Lemma \[lemma:conv\] (applied here with $V_k:=W_{2,k}$, $U_o:=U_{\varepsilon }$, $\psi:=1$, $\alpha:=1$ and $\beta:=p$) we have that $$\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}
U_\varepsilon^{p}(x,0)(W_{2,k})_+(x,0)\,dx=0.$$ Therefore, becomes $$\begin{split}\label{upBoundW2-1}
& \int_{\mathbb{R}^{n+1}_+}{y^a|\nabla W_{2,k}|^2\,dX}\\
{\leqslant}\, & \int_{{{\mathbb R}}^n}\left((U_\varepsilon+(W_{2,k})_+)^{p+1}(x,0)
-U_\varepsilon^{p+1}(x,0)\right)\,dx + C\varepsilon^{\frac{1}{2\gamma}} +o_k(1).
\end{split}$$ Furthermore, it is not difficult to see that that there exist two constants $0<c_1<c_2$ such that $$c_1{\leqslant}\frac{(1+t)^{p+1}-1-t^{p+1}}{t^p+t}{\leqslant}c_2,\qquad t>0$$ Thus, setting $t:=\frac{(W_{2,k})_+}{U_{\varepsilon }}$, one has $$\begin{aligned}
&& (U_\varepsilon+(W_{2,k})_+)^{p+1}-U_\varepsilon^{p+1}=
U_{\varepsilon }^{p+1} \left[ \left( 1+\frac{(W_{2,k})_+}{U_{\varepsilon }} \right)^{p+1}-1\right] \\
&&\qquad {\leqslant}U_{\varepsilon }^{p+1} \left[ c_2 \left( \frac{(W_{2,k})_+^p}{U_{\varepsilon }^p}
+\frac{(W_{2,k})_+}{U_{\varepsilon }} \right) + \frac{(W_{2,k})_+^{p+1}}{U_{\varepsilon }^{p+1}} \right] \\
&&\qquad =c_2 U_{\varepsilon }(W_{2,k})_+^p +c_2 U_{\varepsilon }^p(W_{2,k})_+ +(W_{2,k})_+^{p+1}.\end{aligned}$$ Therefore $$\begin{split}
& \int_{\mathbb{R}^n}((U_\varepsilon+(W_{2,k})_+)^{p+1}(x,0)-
U_\varepsilon^{p+1}(x,0))\,dx \\
{\leqslant}\,& c_2\int_{\mathbb{R}^n}{U_\varepsilon^p(x,0) (W_{2,k})_+(x,0)\,dx}
+c_2 \int_{\mathbb{R}^n}{(W_{2,k})_+^p(x,0)U_\varepsilon(x,0) \,dx}
\\&\qquad+\int_{\mathbb{R}^n}{(W_{2,k})_+^{p+1}(x,0) \,dx}.
\end{split}$$ Applying Lemma \[lemma:conv\] once more, we obtain that $$\begin{aligned}
&&\lim_{k\to+\infty}\int_{\mathbb{R}^n}{U_\varepsilon^p(x,0) (W_{2,k})_+(x,0)\,dx}=0\\
{\mbox{and }} && \lim_{k\to+\infty}\int_{\mathbb{R}^n}{(W_{2,k})_+^p(x,0)U_\varepsilon(x,0) \,dx}=0.\end{aligned}$$ Hence, going back to , we get $$\label{upW2}
\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla W_{2,k}|^2\,dX}{\leqslant}\int_{\mathbb{R}^n}{(W_{2,k})_+^{p+1}(x,0)\,dx} +C\varepsilon^{\frac{1}{2\gamma}}+o_k(1).$$
Now we observe that, thanks to , $W_{2,k}=U_k$ outside $B_{r+\bar{l}+1}$. So, using with $\rho:=r+\bar{l}+1$, we have $$\begin{aligned}
&& \int_{{{\mathbb R}}^{n+1}_+\setminus B_{r+\bar{l}+1}^+} y^a |\nabla W_{2,k}|^2\,dX +
\int_{ {{\mathbb R}}^n \setminus\left(B_{r+\bar{l}+1}\cap\{y=0\}\right) } (W_{2,k})_+^{2^*_s}(x,0)\,dx \\
&&\qquad =
\int_{{{\mathbb R}}^{n+1}_+\setminus B_{r+\bar{l}+1}^+} y^a |\nabla U_k|^2\,dX +
\int_{ {{\mathbb R}}^n \setminus\left(B_{r+\bar{l}+1}\cap\{y=0\}\right) } (U_k)_+^{2^*_s}(x,0)\,dx {\geqslant}\eta_0,\end{aligned}$$ for some $k$ that depends on $\rho$. This implies that either $$\int_{ {{\mathbb R}}^{n+1}_+\setminus B_{r+\bar{l}+1}^+ }{y^a|\nabla W_{2,k}|^2\,dX}{\geqslant}\frac{\eta_0}{2}$$ or $$\int_{\mathbb{R}^n\setminus\{B_{r+\bar{l}+1}\cap \{y=0\}\}}{(W_{2,k})_+^{p+1}(x,0)\,dx}{\geqslant}\frac{\eta_0}{2}.$$ In the first case we have $$\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla W_{2,k}|^2\,dX}{\geqslant}\int_{{{\mathbb R}}^{n+1}_+\setminus B_{r+\bar{l}+1}^+}{y^a|\nabla W_{2,k}|^2\,dX}
{\geqslant}\frac{\eta_0}{2}.$$ From this and it follows $$\label{usare}
\int_{\mathbb{R}^n}{(W_{2,k})_+^{p+1}(x,0) \,dx}> \frac{\eta_0}{4}.$$ In the second case, this inequality holds trivially.
Accordingly, we can define $\psi_k:=\alpha_k W_{2,k}$, where $$\label{def alfa}
\alpha^{p-1}_k:=\frac{[W_{2,k}]_a^2}
{\|(W_{2,k})_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}}.$$ We claim that $$\label{wqftrejhlipppuy}
[W_{2,k}]_a^2{\leqslant}\|(W_{2,k})_+(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^{p+1}+C{\varepsilon }^{\frac{2}{p+1}}+o_k(1),$$ for a suitable positive constant $C$. For this, notice that , and give $$\begin{split}\label{sqwlgergbedsdc}
[W_{2,k}]_a^2 =\,& \langle {{\mathscrI}}_{\varepsilon }'(W_{2,k}),U_{\varepsilon }+W_{2,k}\rangle -
\int_{{{\mathbb R}}^{n+1}_+}y^a\langle \nabla U_{\varepsilon },\nabla W_{2,k}\rangle\,dX \\
&\quad +\varepsilon\int_{\mathbb{R}^n}h(x)\left( (U_\varepsilon+(W_{2,k})_+)^{q}(x,0)-U_{\varepsilon }^q(x,0)
\right) (U_{\varepsilon }+W_{2,k})(x,0)\,dx
\\&\quad +\int_{\mathbb{R}^n}\left( (U_\varepsilon+(W_{2,k})_+)^{p}(x,0)-U_{\varepsilon }^p(x,0)\right)
(U_{\varepsilon }+W_{2,k})(x,0)\,dx\\
{\leqslant}\, & C{\varepsilon }^{\frac{1}{2\gamma}} +o_k(1) \\
&\quad +\varepsilon\int_{\mathbb{R}^n}h(x)\left( (U_\varepsilon+(W_{2,k})_+)^{q}(x,0)-U_{\varepsilon }^q(x,0)
\right) (U_{\varepsilon }+W_{2,k})(x,0)\,dx
\\&\quad +\int_{\mathbb{R}^n}\left( (U_\varepsilon+(W_{2,k})_+)^{p}(x,0)-U_{\varepsilon }^p(x,0)\right)
(U_{\varepsilon }+W_{2,k})(x,0)\,dx.
\end{split}$$ We can rewrite $$\begin{aligned}
&&\int_{\mathbb{R}^n}h(x)\left( (U_\varepsilon+(W_{2,k})_+)^{q}(x,0)-U_{\varepsilon }^q(x,0)
\right) (U_{\varepsilon }+W_{2,k})(x,0)\,dx\\
&=& \int_{\mathbb{R}^n}h(x)\left( (U_\varepsilon+(W_{2,k})_+)^{q+1}(x,0)-U_{\varepsilon }^{q+1}(x,0)\right)\,dx
\\&&\qquad-\int_{{{\mathbb R}}^n}h(x) U_{\varepsilon }^{q}(x,0)(W_{2,k})_+(x,0)\,dx\\
&=& \int_{\mathbb{R}^n}h(x)\left( (U_\varepsilon+(W_{2,k})_+)^{q+1}(x,0)-U_{\varepsilon }^{q+1}(x,0)\right)\,dx
+o_k(1),\end{aligned}$$ where we have applied once again Lemma \[lemma:conv\]. Analogously, $$\begin{aligned}
&&\int_{\mathbb{R}^n}\left( (U_\varepsilon+(W_{2,k})_+)^{p}(x,0)-U_{\varepsilon }^p(x,0)\right)
(U_{\varepsilon }+W_{2,k})(x,0)\,dx\\
&=& \int_{\mathbb{R}^n}\left((U_\varepsilon+(W_{2,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx
+o_k(1).\end{aligned}$$ Plugging these informations into , we obtain that $$\begin{split}\label{sqwlgergbedsdc-1}
[W_{2,k}]_a^2 {\leqslant}&
{\varepsilon }\int_{\mathbb{R}^n}h(x)\left( (U_\varepsilon+(W_{2,k})_+)^{q+1}(x,0)-U_{\varepsilon }^{q+1}(x,0)\right)\,dx\\
&\quad +\int_{\mathbb{R}^n}\left((U_\varepsilon+(W_{2,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx\\
&\quad +C{\varepsilon }^{\frac{1}{2\gamma}} +o_k(1).
\end{split}$$
So using into we obtain $$\begin{split}\label{sqwlgergbedsdc-2}
[W_{2,k}]_a^2 {\leqslant}&
\int_{\mathbb{R}^n}\left((U_\varepsilon+(W_{2,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx\\
&\quad +C{\varepsilon }^{\frac{1}{2\gamma}} +o_k(1),
\end{split}$$ up to renaming constants. Now we use with $V_k:=W_{2,k}$ and $U_o:=U_{\varepsilon }$ and we get $$\begin{aligned}
[W_{2,k}]_a^2 &{\leqslant}& \int_{\mathbb{R}^n}(U_\varepsilon+(W_{2,k})_+)^{p+1}(x,0)\,dx
-\int_{{{\mathbb R}}^n}U_{\varepsilon }^{p+1}(x,0)\,dx \\
&&\qquad +C{\varepsilon }^{\frac{1}{2\gamma}} +o_k(1)\\
&=& \| (U_{\varepsilon }+ (W_{2,k})_+)(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^{p+1}
- \| U_{\varepsilon }\|_{L^{p+1}({{\mathbb R}}^n)}^{p+1}+C{\varepsilon }^{\frac{1}{2\gamma}} +o_k(1)\\
&=&\|(W_{2,k})_+(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^{p+1} +C{\varepsilon }^{\frac{1}{2\gamma}}+o_k(1),\end{aligned}$$ and this shows .
From , and we have that $$\label{star}
\alpha_k^{p-1}= \frac{[W_{2,k}]_a^2}
{\|(W_{2,k})_+(\cdot,0)\|_{L^{p+1}(\mathbb{R}^n)}^{p+1}}
{\leqslant}1+C{\varepsilon }^{\frac{1}{2\gamma}} +o_k(1).$$ Notice also that, with the choice of $\alpha_k$ in , it holds $$[\psi_k]_a^2=\alpha_k^2 [W_{2,k}]_a^2 = \alpha_k^{p+1}
\|(W_{2,k})_+(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^{p+1}= \|(\psi_{k})_+(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^{p+1}.$$ Hence, by and Proposition \[traceIneq\], we have that $$\begin{aligned}
&& S{\leqslant}\frac{[\psi_k(\cdot,0)]_{\dot{H}^s({{\mathbb R}}^n)}^2}{\|(\psi_k)_+(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^2} = \frac{[\psi_k]_{a}^2}{\|(\psi_k)_+(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^2}\\
&&\qquad = \frac{\|(\psi_k)_+(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^{p+1}}{\|(\psi_k)_+(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^2} = \|(\psi_k)_+(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^{p-1}.\end{aligned}$$ Accordingly, $$\|(W_{2,k})_+(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^{p+1}=\frac{\|(\psi_k)_+(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^{p+1}}{\alpha_k^{p+1}}{\geqslant}S^{n/2s}\frac{1}{\alpha_k^{p+1}},$$ where we have used the fact that $p-1=2^*_s-2=\frac{4s}{n-2s}$. This, together with , gives that $$\begin{split}\label{SobW2}
S^{n/2s}{\leqslant}\, & (1+C{\varepsilon }^{\frac{2}{p+1}} +o_k(1))^{\frac{p+1}{p-1}}
\|(W_{2,k})_+(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^{p+1}\\
{\leqslant}\, & \|(W_{2,k})_+(\cdot,0)\|_{L^{p+1}({{\mathbb R}}^n)}^{p+1}
+C{\varepsilon }^{\frac{1}{2\gamma}} +o_k(1).
\end{split}$$ Moreover, by and Lemma \[lemma:conv\] we have that $$\begin{aligned}
&& {{\mathscrI}}_\varepsilon(W_{2,k})-
\frac{1}{2}\langle {{\mathscrI}}'_\varepsilon(W_{2,k}),U_\varepsilon+W_{2,k}\rangle \\
&=& -\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla W_{2,k},\nabla U_{\varepsilon }\rangle\,dX \\
&&\quad -\frac{{\varepsilon }}{q+1}\int_{{{\mathbb R}}^n}h(x)\left( (U_{\varepsilon }+(W_{2,k})_+)^{q+1}(x,0)
-U_{\varepsilon }^{q+1}(x,0)\right)\,dx \\
&&\quad +{\varepsilon }\int_{{{\mathbb R}}^n}h(x)U_{\varepsilon }^q(x,0)(W_{2,k})_+(x,0)\,dx\\
&&\quad -\frac{1}{p+1}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(W_{2,k})_+)^{p+1}(x,0)
-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&&\quad +\int_{{{\mathbb R}}^n}U_{\varepsilon }^p(x,0)(W_{2,k})_+(x,0)\,dx\\
&&\quad +\frac{{\varepsilon }}{2}\int_{{{\mathbb R}}^n}h(x)\left( (U_{\varepsilon }+(W_{2,k})_+)^{q}(x,0)
-U_{\varepsilon }^{q}(x,0)\right)(U_{\varepsilon }+W_{2,k})(x,0))\,dx \\
&&\quad +\frac{1}{2}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(W_{2,k})_+)^{p}(x,0)
-U_{\varepsilon }^{p}(x,0)\right)(U_{\varepsilon }+W_{2,k})(x,0))\,dx \\
&=& -{\varepsilon }\left(\frac{1}{q+1}-\frac12\right)
\int_{{{\mathbb R}}^n}h(x)\left( (U_{\varepsilon }+(W_{2,k})_+)^{q+1}(x,0)
-U_{\varepsilon }^{q+1}(x,0)\right)\,dx \\
&&\quad +\left(\frac12-\frac{1}{p+1}\right)
\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(W_{2,k})_+)^{p+1}(x,0)
-U_{\varepsilon }^{p+1}(x,0)\right)\,dx + o_k(1).\end{aligned}$$ We observe that $\frac12-\frac{1}{p+1}=\frac{s}{n}$. Thus, using we have that $$\begin{aligned}
&& {{\mathscrI}}_\varepsilon(W_{2,k})-
\frac{1}{2}\langle {{\mathscrI}}'_\varepsilon(W_{2,k}),U_\varepsilon+W_{2,k}\rangle \\
&{\geqslant}& \frac{s}{n}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(W_{2,k})_+)^{p+1}(x,0)
-U_{\varepsilon }^{p+1}(x,0)\right)\,dx -C{\varepsilon }+o_k(1),\end{aligned}$$ for some $C>0$. Therefore, using with $V_k:=W_{2,k}$ and $U_o:=U_{\varepsilon }$, we have that $${{\mathscrI}}_\varepsilon(W_{2,k})-\frac{1}{2}\langle {{\mathscrI}}'_\varepsilon(W_{2,k}),U_\varepsilon+W_{2,k}\rangle{\geqslant}\frac{s}{n}\|(W_{2,k})_+\|_{L^{p+1}({{\mathbb R}}^n)}^{p+1}-C{\varepsilon }+o_k(1).$$ Furthermore, by , $${{\mathscrI}}_\varepsilon(W_{2,k})-\frac{1}{2}\langle {{\mathscrI}}'_\varepsilon(W_{2,k}),
U_\varepsilon+W_{2,k}\rangle{\geqslant}\frac{s}{n}S^{n/2s}-C\varepsilon^{\frac{1}{2\gamma}}+o_k(1).$$ This and give the desired estimate for ${{\mathscrI}}_{\varepsilon }(W_{2,k})$, namely $$\label{infBoundW2}
{{\mathscrI}}_\varepsilon(W_{2,k}){\geqslant}\frac{s}{n}S^{n/2s}-C\varepsilon^{\frac{1}{2\gamma}}+o_k(1).$$
[*Step 3: Lower bound for ${{\mathscrI}}_{\varepsilon }(U_k)$.*]{} Now, keeping in mind the estimates obtained in and for ${{\mathscrI}}_{\varepsilon }(W_{1,k})$ and ${{\mathscrI}}_{\varepsilon }(W_{2,k})$ respectively, we will produce an estimate for ${{\mathscrI}}_{\varepsilon }(U_k)$. Indeed, notice first that $U_k=\chi U_k +(1-\chi)U_k=W_{1,k}+W_{2,k}$, thanks to . Hence, recalling , we have $$\begin{aligned}
&&{{\mathscrI}}_\varepsilon(U_k)\\ &=&{{\mathscrI}}_\varepsilon(W_{1,k})+{{\mathscrI}}_\varepsilon(W_{2,k})+
\int_{\mathbb{R}^{n+1}_+}y^a\langle\nabla W_{1,k},\nabla W_{2,k}\rangle\,dX\\
&&\quad -\frac{{\varepsilon }}{q+1}\int_{{{\mathbb R}}^n} h(x)\left((U_{\varepsilon }+(U_k)_+)^{q+1}(x,0)-U_{\varepsilon }^{q+1}(x,0)\right)\,dx \\&&\quad+{\varepsilon }\int_{{{\mathbb R}}^n}h(x)U_{\varepsilon }^q(x,0)(U_k)_+(x,0)\,dx\\
&&\quad -\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(U_k)_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\&&\quad+\int_{{{\mathbb R}}^n}U_{\varepsilon }^p(x,0)(U_k)_+(x,0)\,dx\\
&&\quad +\frac{{\varepsilon }}{q+1}\int_{{{\mathbb R}}^n} h(x)\left((U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0)-U_{\varepsilon }^{q+1}(x,0)\right)\,dx \\&&\quad-{\varepsilon }\int_{{{\mathbb R}}^n}h(x)U_{\varepsilon }^q(x,0)(W_{1,k})_+(x,0)\,dx\\
&&\quad +\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\&&\quad-\int_{{{\mathbb R}}^n}U_{\varepsilon }^p(x,0)(W_{1,k})_+(x,0)\,dx\\
&&\quad +\frac{{\varepsilon }}{q+1}\int_{{{\mathbb R}}^n} h(x)\left((U_{\varepsilon }+(W_{2,k})_+)^{q+1}(x,0)-U_{\varepsilon }^{q+1}(x,0)\right)\,dx\\&&\quad -{\varepsilon }\int_{{{\mathbb R}}^n}h(x)U_{\varepsilon }^q(x,0)(W_{2,k})_+(x,0)\,dx\\
&&\quad +\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(W_{2,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\&&\quad-\int_{{{\mathbb R}}^n}U_{\varepsilon }^p(x,0)(W_{2,k})_+(x,0)\,dx.\end{aligned}$$ Thanks to Lemma \[lemma:conv\] we have that $$\begin{aligned}
&&\lim_{k\to+\infty} \int_{{{\mathbb R}}^n}h(x)U_{\varepsilon }^q(x,0)(U_k)_+(x,0)\,dx=0,\\
&&
\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}U_{\varepsilon }^p(x,0)(U_k)_+(x,0)\,dx,\\
&&\lim_{k\to+\infty} \int_{{{\mathbb R}}^n}h(x)U_{\varepsilon }^q(x,0)(W_{1,k})_+(x,0)\,dx=0,
\\&& \lim_{k\to+\infty}\int_{{{\mathbb R}}^n}U_{\varepsilon }^p(x,0)(W_{1,k})_+(x,0)\,dx=0,\\
&& \lim_{k\to+\infty}\int_{{{\mathbb R}}^n}h(x)U_{\varepsilon }^q(x,0)(W_{2,k})_+(x,0)\,dx
\\ {\mbox{ and }} && \lim_{k\to+\infty}\int_{{{\mathbb R}}^n}U_{\varepsilon }^p(x,0)(W_{2,k})_+(x,0)\,dx=0.\end{aligned}$$ Therefore, $$\begin{aligned}
{{\mathscrI}}_\varepsilon(U_k)&=&{{\mathscrI}}_\varepsilon(W_{1,k})+{{\mathscrI}}_\varepsilon(W_{2,k})+
\int_{\mathbb{R}^{n+1}_+}y^a\langle\nabla W_{1,k},\nabla W_{2,k}\rangle\,dX\\
&&\quad -\frac{{\varepsilon }}{q+1}\int_{{{\mathbb R}}^n} h(x)\left((U_{\varepsilon }+(U_k)_+)^{q+1}(x,0)-U_{\varepsilon }^{q+1}(x,0)\right)\,dx \\
&&\quad -\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(U_k)_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&&\quad +\frac{{\varepsilon }}{q+1}\int_{{{\mathbb R}}^n} h(x)\left((U_{\varepsilon }+(W_{1,k})_+)^{q+1}(x,0)-U_{\varepsilon }^{q+1}(x,0)\right)\,dx \\
&&\quad +\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&&\quad +\frac{{\varepsilon }}{q+1}\int_{{{\mathbb R}}^n} h(x)\left((U_{\varepsilon }+(W_{2,k})_+)^{q+1}(x,0)-U_{\varepsilon }^{q+1}(x,0)\right)\,dx \\
&&\quad +\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(W_{2,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx +o_k(1).\end{aligned}$$ Since the terms with ${\varepsilon }$ in front are bounded (see and notice that it holds also for $U_k$ and $W_{1,k}$), we have that $$\begin{split}\label{ok-1}
{{\mathscrI}}_\varepsilon(U_k){\geqslant}\,&{{\mathscrI}}_\varepsilon(W_{1,k})+{{\mathscrI}}_\varepsilon(W_{2,k})+
\int_{\mathbb{R}^{n+1}_+}y^a\langle\nabla W_{1,k},\nabla W_{2,k}\rangle\,dX\\
&\quad -\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(U_k)_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&\quad +\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&\quad +\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(W_{2,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\&\quad-C{\varepsilon }+ o_k(1).
\end{split}$$ Now notice that $$\begin{split}\label{ok1}
& \int_{\mathbb{R}^{n+1}_+}y^a\langle\nabla W_{1,k},\nabla W_{2,k}\rangle\,dX\\
&\qquad = \frac12\int_{\mathbb{R}^{n+1}_+}y^a\langle\nabla(U_k- W_{1,k}),\nabla W_{1,k}\rangle\,dX
\\&\qquad\qquad+\frac12\int_{\mathbb{R}^{n+1}_+}y^a\langle\nabla(U_k- W_{2,k}),\nabla W_{2,k}\rangle\,dX.
\end{split}$$ Moreover, from we have that for any $i\in\{1,2\}$ $$\begin{split}\label{ok2}
&\langle {{\mathscrI}}'_\varepsilon(U_k)-{{\mathscrI}}_{\varepsilon }'(W_{i,k}),
U_\varepsilon+W_{i,k}\rangle\\
=\,& \int_{\mathbb{R}^{n+1}_+}y^a\langle\nabla(U_k- W_{i,k}),\nabla (W_{i,k}+U_{\varepsilon })\rangle\,dX \\
&\qquad -{\varepsilon }\int_{{{\mathbb R}}^n} h(x)\left((U_{\varepsilon }+(U_k)_+)^q-
U_{\varepsilon }^q\right)(U_{\varepsilon }+W_{i,k})\,dx
\\&\qquad-\int_{{{\mathbb R}}^n}\left((U_{\varepsilon }+(U_k)_+)^p-
U_{\varepsilon }^p\right)(U_{\varepsilon }+W_{i,k})\,dx \\
&\qquad +{\varepsilon }\int_{{{\mathbb R}}^n} h(x)\left((U_{\varepsilon }+(W_{i,k})_+)^q-
U_{\varepsilon }^q\right)(U_{\varepsilon }+W_{i,k})\,dx \\&\qquad
+\int_{{{\mathbb R}}^n}\left((U_{\varepsilon }+(W_{i,k})_+)^p-
U_{\varepsilon }^p\right)(U_{\varepsilon }+W_{i,k})\,dx.
\end{split}$$ We claim that $$\label{speriamo}\begin{split}
&\int_{{{\mathbb R}}^n} h(x)\left((U_{\varepsilon }+(U_k)_+)^q(x,0)-
U_{\varepsilon }^q(x,0)\right)(U_{\varepsilon }+W_{i,k})(x,0)\,dx{\leqslant}C+o_k(1)\\
{\mbox{and }} &\int_{{{\mathbb R}}^n} h(x)\left((U_{\varepsilon }+(W_{i,k})_+)^q(x,0)-
U_{\varepsilon }^q(x,0)\right)(U_{\varepsilon }+W_{i,k})(x,0)\,dx {\leqslant}C+o_k(1),
\end{split}$$ for some $C>0$. Let us prove the first estimate in . For this, we notice that if $x\in{{\mathbb R}}^n$ is such that $U_k(x,0){\geqslant}0$ then also $W_{i,k}(x,0){\geqslant}0$, thanks to the definition of $W_{i,k}$ given in . Hence $$\begin{aligned}
&&\int_{{{\mathbb R}}^n} h(x)\left((U_{\varepsilon }+(U_k)_+)^q(x,0)-
U_{\varepsilon }^q(x,0)\right)(U_{\varepsilon }+W_{i,k})(x,0)\,dx\\
&&\qquad =\int_{{{\mathbb R}}^n} h(x)\left((U_{\varepsilon }+(U_k)_+)^q(x,0)-
U_{\varepsilon }^q(x,0)\right)(U_{\varepsilon }+(W_{i,k})_+)(x,0)\,dx\\
&&\qquad {\leqslant}\int_{{{\mathbb R}}^n} h(x)\left((U_{\varepsilon }+(U_k)_+)^q(x,0)-
U_{\varepsilon }^q(x,0)\right)(U_{\varepsilon }+(U_{k})_+)(x,0)\,dx\\
&&\qquad =\int_{{{\mathbb R}}^n} h(x)\left((U_{\varepsilon }+(U_k)_+)^{q+1}(x,0)-
U_{\varepsilon }^{q+1}(x,0)\right)\,dx\\&&\qquad\qquad-\int_{{{\mathbb R}}^n}h(x)U_{\varepsilon }^{q}(x,0)
(U_{k})_+)(x,0)\,dx\\
&&\qquad {\leqslant}C+o_k(1),\end{aligned}$$ for a suitable $C>0$, thanks to (that holds true also for $U_k$) and Lemma \[lemma:conv\]. Analogously one can prove also the second estimate in , and this finishes the proof of .
Hence, from , , and we get $$\begin{aligned}
&&{{\mathscrI}}_\varepsilon(U_k)\\ &{\geqslant}&{{\mathscrI}}_\varepsilon(W_{1,k})+{{\mathscrI}}_\varepsilon(W_{2,k})+
\frac12 \langle {{\mathscrI}}_{\varepsilon }'(U_k)-{{\mathscrI}}_{\varepsilon }'(W_{1,k}),
U_{\varepsilon }+W_{1,k}\rangle \\&&\quad+
\frac12 \langle {{\mathscrI}}_{\varepsilon }'(U_k)-{{\mathscrI}}_{\varepsilon }'(W_{2,k}),
U_{\varepsilon }+W_{2,k}\rangle\\
&&\quad -\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(U_k)_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&&\quad +\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&&\quad +\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(W_{2,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&&\quad +\frac{1}{2}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_k)_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)(U_{\varepsilon }+W_{1,k})(x,0)\,dx \\
&&\quad -\frac{1}{2}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(W_{1,k})_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)(U_{\varepsilon }+W_{1,k})(x,0)\,dx \\
&&\quad +\frac{1}{2}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_k)_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)(U_{\varepsilon }+W_{2,k})(x,0)\,dx \\
&&\quad -\frac{1}{2}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(W_{2,k})_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)(U_{\varepsilon }+W_{1,k})(x,0)\,dx \\&&\quad-C{\varepsilon }+o_k(1) .\end{aligned}$$ Moreover, the estimates in and give $$\begin{aligned}
{{\mathscrI}}_\varepsilon(U_k)&{\geqslant}&
{{\mathscrI}}_\varepsilon(W_{1,k})+{{\mathscrI}}_\varepsilon(W_{2,k})\\
&&\quad -\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(U_k)_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&&\quad +\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&&\quad +\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(W_{2,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&&\quad +\frac{1}{2}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_k)_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)(U_{\varepsilon }+W_{1,k})(x,0)\,dx \\
&&\quad -\frac{1}{2}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(W_{1,k})_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)(U_{\varepsilon }+W_{1,k})(x,0)\,dx \\
&&\quad +\frac{1}{2}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_k)_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)(U_{\varepsilon }+W_{2,k})(x,0)\,dx \\
&&\quad -\frac{1}{2}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(W_{2,k})_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)(U_{\varepsilon }+W_{1,k})(x,0)\,dx \\&&\quad-C{\varepsilon }^{\frac{1}{2\gamma}} +o_k(1).\end{aligned}$$ Now we use Lemma \[lemma:conv\] once more to see that, for $i\in\{1,2\}$, $$\begin{aligned}
&& \int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(W_{i,k})_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right) (U_{\varepsilon }+W_{i,k})(x,0)\,dx\\
&&\qquad
= \int_{{{\mathbb R}}^n} ( U_{\varepsilon }+(W_{i,k})_+ )^{p+1}(x,0)\,dx
-\int_{{{\mathbb R}}^n} U_{\varepsilon }^{p+1}(x,0)\,dx +o_k(1).\end{aligned}$$ Hence, using this and collecting some terms, we have $$\begin{split}\label{ok forse}
&{{\mathscrI}}_\varepsilon(U_k)\\
{\geqslant}\,&
{{\mathscrI}}_\varepsilon(W_{1,k})+{{\mathscrI}}_\varepsilon(W_{2,k})\\
&\quad -\frac{1}{p+1}\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(U_k)_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&\quad +\frac{1}{2}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_{k})_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)(U_{\varepsilon }+W_{1,k})(x,0)\,dx \\
&\quad +\frac{1}{2}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_{k})_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)(U_{\varepsilon }+W_{2,k})(x,0)\,dx \\
&\quad -\left(\frac12-\frac{1}{p+1}\right)\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&\quad -\left(\frac12-\frac{1}{p+1}\right)
\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(W_{2,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx
\\&\quad-C{\varepsilon }^{\frac{1}{2\gamma}} +o_k(1).
\end{split}$$ Now we claim that $$\label{speriamo-1}
\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_{k})_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)U_{\varepsilon }(x,0)\,dx=0.$$ Indeed, we first observe that for any $a{\geqslant}b{\geqslant}0$ $$a^p-b^p =p\int_b^at^{p-1}\,dt{\leqslant}pa^{p-1}(a-b).$$ Hence, taking $a:=U_{\varepsilon }+(U_{k})_+$ and $b:=U_{\varepsilon }$, we have that $$|(U_{\varepsilon }+(U_k)_+)^p-U_{\varepsilon }^p|{\leqslant}p(U_{\varepsilon }+(U_k)_+)^{p-1}(U_k)_+.$$ Accordingly, $$\begin{aligned}
&&\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_{k})_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)U_{\varepsilon }(x,0)\,dx\\
&&\qquad {\leqslant}p \int_{{{\mathbb R}}^n}(U_{\varepsilon }+(U_{k})_+)^{p-1}(x,0)
(U_k)_+(x,0)U_{\varepsilon }(x,0)\,dx.\\\end{aligned}$$ We now use Hölder inequality with exponents $\frac{2^*_s}{p-1}=\frac{n}{2s}$ and $\frac{n}{n-2s}$ and obtain $$\begin{aligned}
&&\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_{k})_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)U_{\varepsilon }(x,0)\,dx\\
&&\qquad {\leqslant}p \left(\int_{{{\mathbb R}}^n}(U_{\varepsilon }+(U_{k})_+)^{2^*_s}(x,0)\,dx
\right)^{\frac{2s}{n}} \left(\int_{{{\mathbb R}}^n}
(U_k)_+^{\frac{n}{n-2s}}(x,0)U_{\varepsilon }^{\frac{n}{n-2s}}(x,0)\,dx
\right)^{\frac{n-2s}{n}}\\
&&\qquad {\leqslant}C [U_{\varepsilon }+(U_{k})_+]_a^{p-1} \left(\int_{{{\mathbb R}}^n}
(U_k)_+^{\frac{n}{n-2s}}(x,0)U_{\varepsilon }^{\frac{n}{n-2s}}(x,0)\,dx
\right)^{\frac{n-2s}{n}}\\
&&\qquad {\leqslant}C\left(\int_{{{\mathbb R}}^n}
(U_k)_+^{\frac{n}{n-2s}}(x,0)U_{\varepsilon }^{\frac{n}{n-2s}}(x,0)\,dx
\right)^{\frac{n-2s}{n}},\end{aligned}$$ for some positive $C$ that may change from line to line, thanks to Proposition \[traceIneq\] and . Now the desired claim in simply follows by using Lemma \[lemma:conv\] with $V_k:=U_k$, $U_o:=U_{\varepsilon }$, $\psi:=1$, $\alpha:=\frac{n}{n-2s}$ and $\beta:=\frac{n}{n-2s}$ (notice that $\alpha+\beta=2^*_s$).
From we deduce that $$\begin{aligned}
&&\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_{k})_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)(U_{\varepsilon }+W_{1,k})(x,0)\,dx
\\&&\qquad +\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_{k})_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)(U_{\varepsilon }+W_{2,k})(x,0)\,dx\\
&&\qquad = \int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_{k})_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)(U_{\varepsilon }+W_{1,k}+W_{2,k})(x,0)\,dx+o_k(1)\\
&&\qquad = \int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_{k})_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)(U_{\varepsilon }+ U_k)(x,0)\,dx+o_k(1)\\
&&\qquad = \int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_{k})_+)^{p+1}(x,0)
-U_{\varepsilon }^{p+1}(x,0)\right)(x,0)\,dx+o_k(1),\end{aligned}$$ where Lemma \[lemma:conv\] was used once again in the last line. Plugging this information into we obtain $$\begin{aligned}
&&{{\mathscrI}}_\varepsilon(U_k)\\&{\geqslant}&
{{\mathscrI}}_\varepsilon(W_{1,k})+{{\mathscrI}}_\varepsilon(W_{2,k})\\
&&\quad +\left(\frac12-\frac{1}{p+1}\right)
\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(U_k)_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&&\quad -\left(\frac12-\frac{1}{p+1}\right)\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(W_{1,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx \\
&&\quad -\left(\frac12-\frac{1}{p+1}\right)\int_{{{\mathbb R}}^n} \left((U_{\varepsilon }+(W_{2,k})_+)^{p+1}(x,0)-U_{\varepsilon }^{p+1}(x,0)\right)\,dx-C{\varepsilon }^{\frac{1}{2\gamma}} +o_k(1).\end{aligned}$$ Now we use with $U_o:=U_{\varepsilon }$ and $V_k:=U_k$, $V_k:=W_{1,k}$ and $V_k:=W_{2,k}$ respectively, and so $$\begin{aligned}
{{\mathscrI}}_\varepsilon(U_k)&{\geqslant}&
{{\mathscrI}}_\varepsilon(W_{1,k})+{{\mathscrI}}_\varepsilon(W_{2,k})
+\left(\frac12-\frac{1}{p+1}\right)
\int_{{{\mathbb R}}^n}(U_k)_+^{p+1}(x,0)\,dx\\
&&\qquad -\left(\frac12-\frac{1}{p+1}\right)\int_{{{\mathbb R}}^n} (W_{1,k})_+^{p+1}(x,0)\,dx \\
&&\qquad -\left(\frac12-\frac{1}{p+1}\right)\int_{{{\mathbb R}}^n} (W_{2,k})_+^{p+1}(x,0)\,dx-C{\varepsilon }^{\frac{1}{2\gamma}} +o_k(1).\end{aligned}$$ Notice now that for any $x\in{{\mathbb R}}^n$ $$\begin{aligned}
&&(U_k)_+^{p+1}(x,0) - (W_{1,k})_+^{p+1}(x,0)-(W_{2,k})_+^{p+1}(x,0)\\
&=& (U_k)_+^{p+1}(x,0) - \chi^{p+1}(x,0)(U_{k})_+^{p+1}(x,0)-(1-\chi(x,0))^{p+1}(U_{k})_+^{p+1}(x,0)\\
&=&(U_k)_+^{p+1}(x,0)\left(1-\chi^{p+1}(x,0)-(1-\chi)^{p+1}(x,0)\right){\geqslant}0. \end{aligned}$$ This and the fact that $p+1>2$ give $${{\mathscrI}}_\varepsilon(U_k){\geqslant}{{\mathscrI}}_\varepsilon(W_{1,k})+{{\mathscrI}}_\varepsilon(W_{2,k})
-C{\varepsilon }^{\frac{1}{2\gamma}} +o_k(1).$$ Finally, this, together with and , implies that $${{\mathscrI}}_\varepsilon(U_k){\geqslant}\frac{s}{n}S^{n/2s}-C\varepsilon^{\frac{1}{2\gamma}}+o_k(1),$$ up to renaming constants. Therefore, taking the limit as $k\to+\infty$ we have $$c_{\varepsilon }=\lim_{k\to+\infty}{{\mathscrI}}_{\varepsilon }(U_k){\geqslant}\frac{s}{n}S^{n/2s}-C\varepsilon^{\frac{1}{2\gamma}}.$$ This gives a contradiction with and finishes the proof of Lemma \[lemma:tight\].
We are now in the position to show that the functional ${{\mathscrI}}_{\varepsilon }$ introduced in satisfies a Palais-Smale condition.
Thanks to Lemma \[lemma:weak\] we know that the sequence $U_k$ weakly converges to 0 in $\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$ as $k\to+\infty$.
For any $k\in{{\mathbb N}}$, we set $V_k:=U_{\varepsilon }+U_k$, where $U_{\varepsilon }$ is the local minimum of ${{\mathscrF}}_{\varepsilon }$ found Theorem \[MINIMUM\]. Since $U_{\varepsilon }$ is a critical point of ${{\mathscrF}}_{\varepsilon }$, from we deduce that $$\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_{\varepsilon },\nabla U_k\rangle\,dX =
{\varepsilon }\int_{{{\mathbb R}}^n}h(x)U_{\varepsilon }^q(x,0)U_k(x,0)\,dx +
\int_{{{\mathbb R}}^n}U_{\varepsilon }^p(x,0)U_k(x,0)\,dx.$$ Therefore, recalling and we have $$\begin{split}\label{aslpooooooo}
{{\mathscrF}}_{\varepsilon }(V_k) =\,& \frac12\int_{{{\mathbb R}}^{n+1}_+}
y^a|\nabla(U_{\varepsilon }+U_k)|^2\,dX \\
&\quad -\frac{{\varepsilon }}{q+1}\int_{{{\mathbb R}}^n}h(x)(U_{\varepsilon }+U_k)_+^{q+1}(x,0)\,dx
-\frac{1}{p+1}\int_{{{\mathbb R}}^n}(U_{\varepsilon }+U_k)_+^{p+1}(x,0)\,dx\\
=\,& {{\mathscrI}}_{\varepsilon }(U_k)+{{\mathscrF}}_{\varepsilon }(U_{\varepsilon })
+\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_{\varepsilon },\nabla U_k\rangle\,dX\\
&\quad +\frac{{\varepsilon }}{q+1}\int_{{{\mathbb R}}^n}h(x)\left( (U_{\varepsilon }+(U_k)_+)^{q+1}(x,0) -(U_{\varepsilon }+U_k)_+^{q+1}(x,0)\right)\,dx\\
&\quad +\frac{1}{p+1}\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_k)_+)^{p+1}(x,0) -(U_{\varepsilon }+U_k)_+^{p+1}(x,0)\right)\,dx\\
&\quad -{\varepsilon }\int_{{{\mathbb R}}^n}h(x)U_{\varepsilon }^q(x,0)(U_k)_+(x,0)\,dx
-\int_{{{\mathbb R}}^n}U_{\varepsilon }^p(x,0)(U_k)_+(x,0)\\
=\,& {{\mathscrI}}_{\varepsilon }(U_k)+{{\mathscrF}}_{\varepsilon }(U_{\varepsilon })\\
&\quad +\frac{{\varepsilon }}{q+1}\int_{{{\mathbb R}}^n}h(x)\Big( (U_{\varepsilon }+(U_k)_+)^{q+1}(x,0) -(U_{\varepsilon }+U_k)_+^{q+1}(x,0)\\
&\qquad \qquad\quad +(q+1)U_{\varepsilon }^q(x,0)(U_k-(U_k)_+)(x,0)\Big)\,dx\\
&\quad +\frac{1}{p+1}\int_{{{\mathbb R}}^n}\Big( (U_{\varepsilon }+(U_k)_+)^{p+1}(x,0) -(U_{\varepsilon }+U_k)_+^{p+1}(x,0)\\
&\qquad \qquad\quad +(p+1)U_{\varepsilon }^p(x,0)(U_k-(U_k)_+)(x,0)\Big)\,dx.
\end{split}$$ We now claim that $$\label{aslgttrjuytk}
(U_{\varepsilon }+(U_k)_+)^{r+1}(x,0) -(U_{\varepsilon }+U_k)_+^{r+1}(x,0) +(r+1)U_{\varepsilon }^r(x,0)(U_k-(U_k)_+)(x,0){\leqslant}0,$$ for any $x\in{{\mathbb R}}^n$ and $r\in\{p,q\}$. Indeed, the claim is trivially true if $U_k(x,0){\geqslant}0$. Hence we suppose that $U_k(x,0)<0$, and so becomes $$\label{aslgttrjuytk-1}
U_{\varepsilon }^{r+1}(x,0) -(U_{\varepsilon }+U_k)_+^{r+1}(x,0) +(r+1)U_{\varepsilon }^r(x,0)(U_k-(U_k)_+)(x,0){\leqslant}0.$$ Given $a>0$, the function $f(t):=(a+t)^{r+1}_+$, for $t\in{{\mathbb R}}$, is convex, and therefore it satisfies for any $b<0$ $$f(b){\geqslant}f(0) +f'(0)b,$$ that is $$(a+b)^{r+1}_+{\geqslant}a^{r+1}+(r+1)a^r b.$$ Thus, taking $a:=U_{\varepsilon }(x,0)$ and $b:=U_k(x,0)$ we have $$(U_{\varepsilon }+U_k)_+^{r+1}(x,0){\geqslant}U_{\varepsilon }^{r+1}(x,0)+(r+1)U_{\varepsilon }^r(x,0) U_k(x,0),$$ which shows , and in turn .
Accordingly, using into we get $$\label{minore}
{{\mathscrF}}_{\varepsilon }(V_k){\leqslant}{{\mathscrI}}_{\varepsilon }(U_k)+{{\mathscrF}}_{\varepsilon }(U_{\varepsilon }).$$ This and assumption (i) in Proposition \[PScond2\] imply that $$\label{questa}
|{{\mathscrF}}_{\varepsilon }(V_k)|{\leqslant}C,$$ for a suitable $C>0$ independent of $k$.
Now we recall that $U_{\varepsilon }$ is a critical point of ${{\mathscrF}}_{\varepsilon }$. Hence, from we deduce that for any $\Psi\in\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$ with $\psi:=\Psi(\cdot,0)$ $$\label{dopo-1}
\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_{\varepsilon },\nabla\Psi\rangle\,dX
= {\varepsilon }\int_{{{\mathbb R}}^n}h(x)U_{\varepsilon }^q(x,0)\psi(x)\,dx
+\int_{{{\mathbb R}}^n}U_{\varepsilon }^p(x,0)\psi(x)\,dx.$$ Moreover, from and we have that $$\begin{aligned}
&&\langle{{\mathscrF}}_{\varepsilon }'(V_k),\Psi\rangle \\
&=& \int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla V_k,\nabla\Psi\rangle\,dX
-{\varepsilon }\int_{{{\mathbb R}}^n}h(x)(V_k)_+^q(x,0)\psi(x)\,dx
-\int_{{{\mathbb R}}^n} (V_k)_+^p(x,0)\psi(x)\,dx\\
&=&\langle{{\mathscrI}}_{\varepsilon }'(U_k),\Psi\rangle +\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla U_{\varepsilon },\nabla\Psi\rangle\,dX \\
&&\quad +{\varepsilon }\int_{{{\mathbb R}}^n}h(x)\left( (U_{\varepsilon }+(U_k)_+)^q(x,0)
-U_{\varepsilon }^q(x,0)\right)\psi(x)\,dx\\
&&\quad +\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_k)_+)^p(x,0)
-U_{\varepsilon }^p(x,0)\right)\psi(x)\,dx\\
&&\quad -{\varepsilon }\int_{{{\mathbb R}}^n}h(x)(V_k)_+^q(x,0)\psi(x)\,dx
-\int_{{{\mathbb R}}^n} (V_k)_+^p(x,0)\psi(x)\,dx.\end{aligned}$$ Using in the formula above and recalling that $V_k=U_{\varepsilon }+U_k$, we obtain $$\begin{split}\label{lasgtprjutr}
&\langle{{\mathscrF}}_{\varepsilon }'(V_k),\Psi\rangle\,=\,
\langle{{\mathscrI}}_{\varepsilon }'(U_k),\Psi\rangle \\
&\qquad +{\varepsilon }\int_{{{\mathbb R}}^n}h(x)\left( (U_{\varepsilon }+(U_k)_+)^q(x,0)
-(U_{\varepsilon }+U_k)_+^q(x,0)\right)\psi(x)\,dx\\
&\qquad +\int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_k)_+)^p(x,0)
-(U_{\varepsilon }+U_k)_+^p(x,0)\right)\psi(x)\,dx.
\end{split}$$ We claim that $$\begin{split}\label{viene?}
& \lim_{k\to+\infty}\int_{{{\mathbb R}}^n}h(x)\left( (U_{\varepsilon }+(U_k)_+)^q(x,0)
-(U_{\varepsilon }+U_k)_+^q(x,0)\right)\psi(x)\,dx=0\\
{\mbox{and }} & \lim_{k\to+\infty} \int_{{{\mathbb R}}^n}\left( (U_{\varepsilon }+(U_k)_+)^p(x,0)
-(U_{\varepsilon }+U_k)_+^p(x,0)\right)\psi(x)\,dx=0.
\end{split}$$ Notice that if $U_k(x,0){\geqslant}0$ then $$\begin{aligned}
&& (U_{\varepsilon }+(U_k)_+)^q(x,0) -(U_{\varepsilon }+U_k)_+^q(x,0)\\&&\qquad= (U_{\varepsilon }+U_k)^q(x,0)
-(U_{\varepsilon }+U_k)^q(x,0)=0\\
{\mbox{and }} && (U_{\varepsilon }+(U_k)_+)^p(x,0) -(U_{\varepsilon }+U_k)_+^p(x,0)
\\&&\qquad= (U_{\varepsilon }+U_k)^p(x,0)
-(U_{\varepsilon }+U_k)^p(x,0)=0.\end{aligned}$$ Therefore the claim becomes $$\begin{split}\label{viene?-1}
& \lim_{k\to+\infty}\int_{{{\mathbb R}}^n\cap\{U_k(\cdot,0)<0\}}h(x)\left( U_{\varepsilon }^q(x,0)
-(U_{\varepsilon }+U_k)_+^q(x,0)\right)\psi(x)\,dx=0\\
{\mbox{and }} & \lim_{k\to+\infty} \int_{{{\mathbb R}}^n\cap\{U_k(\cdot,0)<0\}}\left( U_{\varepsilon }^p(x,0)
-(U_{\varepsilon }+U_k)_+^p(x,0)\right)\psi(x)\,dx=0.
\end{split}$$ Now, we recall that Lemma \[lemma:weak\] here and the compact embedding in Theorem 7.1 in [@DPV] imply that $U_k(\cdot,0)\to0$ a.e. in ${{\mathbb R}}^n$ as $k\to+\infty$. Moreover, we notice that, by the Hölder inequality with exponents $\frac{2^*_s}{2^*_s-1-q}$, $\frac{2^*_s}{q}$ and $2^*_s$, $$\begin{aligned}
&&\left| \int_{{{\mathbb R}}^n\cap\{U_k(\cdot,0)<0\}}h(x) U_{\varepsilon }^q(x,0)\psi(x)\,dx\right|
\\&&\qquad {\leqslant}\left( \int_{{{\mathbb R}}^n\cap\{U_k(\cdot,0)<0\} } |h(x)|^{\frac{2^*_s}{2^*_s-1-q}}
\,dx\right)^{\frac{2^*_s-1-q}{2^*_s}}
\left( \int_{{{\mathbb R}}^n\cap\{U_k(\cdot,0)<0\} } U_{\varepsilon }^{2^*_s}(x,0)\,dx\right)^{\frac{q}{2^*_s}}
\\&&\qquad\cdot
\left( \int_{{{\mathbb R}}^n\cap\{U_k(\cdot,0)<0\} }|\psi(x)|^{2^*_s} \,dx\right)^{\frac{1}{2^*_s}}\\
&&\qquad {\leqslant}\|h\|_{L^{\frac{2^*_s}{2^*_s-1-q}}({{\mathbb R}}^n)}S^{-q/2}[U_{\varepsilon }]_a^q S^{-1/2}[\Psi]_a{\leqslant}C,\end{aligned}$$ for some $C>0$, thanks to and Proposition \[traceIneq\]. Consequently $$h\left( U_{\varepsilon }^q(\cdot,0)
-(U_{\varepsilon }+U_k)_+^q(\cdot,0)\right)\psi{\leqslant}|h| U_{\varepsilon }^q(\cdot,0)
|\psi|\in L^1({{\mathbb R}}^n\cap\{U_k(\cdot,0)<0\}).$$ Hence, by the Dominated Convergence Theorem we get the first limit in .
To prove the second limit in , we use the Hölder inequality with exponents $\frac{2^*_s}{p}=\frac{2n}{n+2s}$ and $2^*_s=\frac{2n}{n-2s}$ to see that $$\begin{aligned}
&&\left| \int_{{{\mathbb R}}^n\cap\{U_k(\cdot,0)<0\}}U_{\varepsilon }^p(x,0)\psi(x)\,dx\right| \\
&{\leqslant}&
\left( \int_{{{\mathbb R}}^n\cap\{U_k(\cdot,0)<0\} } U_{\varepsilon }^{2^*_s}(x,0)\,dx\right)^{\frac{n+2s}{2n}}
\left( \int_{{{\mathbb R}}^n\cap\{U_k(\cdot,0)<0\} }|\psi(x)|^{2^*_s} \,dx\right)^{\frac{1}{2^*_s}}\\
&{\leqslant}& S^{-p/2}[U_{\varepsilon }]_a^p S^{-1/2}[\Psi]_a{\leqslant}C,\end{aligned}$$ for a suitable $C>0$, where was also used. Therefore, $$\left( U_{\varepsilon }^p(\cdot,0)
-(U_{\varepsilon }+U_k)_+^p(\cdot,0)\right)\psi{\leqslant}U_{\varepsilon }^p(\cdot,0)|\psi|
\in L^1({{\mathbb R}}^n\cap\{U_k(\cdot,0)<0\}).$$ So the second limit in follows from the Dominated Convergence Theorem. This shows and so the proof of is finished.
As a consequence of , and assumption (ii) in Proposition \[PScond2\] we have that $$\label{dlghjtrjh}
{\mbox{${{\mathscrF}}_{\varepsilon }'(V_k)\to0$ as~$k\to+\infty$}}$$ in the sense of Remark \[rem:3.3\]. This, together with and Lemma \[lemma bound\], implies that the sequence $V_k$ is uniformly bounded in $\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$, namely there exists a constant $M>0$ such that $$\label{boundM}
{\mbox{$[V_k]_a{\leqslant}M$ for all~$k\in{{\mathbb N}}$.}}$$ Hence, $V_k$ is weakly convergent in $\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$ to some function $V_0$. Since $V_k=U_{\varepsilon }+U_k$ and $U_k$ weakly converges to 0 in $\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$ as $k\to+\infty$ (see Lemma \[lemma:weak\]), it turns out that $V_0=U_{\varepsilon }$. Also, we recall that $U_{\varepsilon }$ is positive, thanks to Proposition \[prop:pos\]. Therefore, we are in the position to apply Lemma \[POS\] with $W_m:=V_k$ and $W:=U_{\varepsilon }$, and we obtain that $$\label{qwwrtruoiungf}
{\mbox{$(V_k)_+$ weakly converges to $U_{\varepsilon }$ in~$\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$ as~$k\to+\infty$.}}$$ We also show that $$\label{qwwrtruoiungf-1}
{\mbox{the sequence $\{V_k\}_k$ is tight, according to Definition~\ref{defTight}.}}$$ For this, we fix $\eta>0$. Thanks to Lemma \[lemma:tight\], we have that there exists $\rho_1>0$ such that $$\int_{{{\mathbb R}}^{n+1}_+\setminus B_{\rho_1}^+}y^a|\nabla U_k|^2\,dX<\frac{\eta}{4},$$ for any $k\in{{\mathbb N}}$. Moreover, since $U_{\varepsilon }\in\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$, there exists $\rho_2>0$ such that $$\int_{{{\mathbb R}}^{n+1}_+\setminus B_{\rho_2}^+}y^a|\nabla U_{\varepsilon }|^2\,dX<\frac{\eta}{4}.$$ We take $\rho:=\max\{\rho_1,\rho_2\}$, and so the two formulas above give that $$\begin{aligned}
&&\int_{{{\mathbb R}}^{n+1}_+\setminus B_\rho^+}y^a|\nabla V_k|^2\,dX \\
& = & \int_{{{\mathbb R}}^{n+1}_+\setminus B_\rho^+}y^a|\nabla U_k|^2\,dX
+\int_{{{\mathbb R}}^{n+1}_+\setminus B_\rho^+}y^a|\nabla U_{\varepsilon }|^2\,dX
\\&&\qquad+2\int_{{{\mathbb R}}^{n+1}_+\setminus B_\rho^+}y^a\langle\nabla U_k,\nabla U_{\varepsilon }\rangle\,dX\\
&{\leqslant}&\int_{{{\mathbb R}}^{n+1}_+\setminus B_\rho^+}y^a|\nabla U_k|^2\,dX
+\int_{{{\mathbb R}}^{n+1}_+\setminus B_\rho^+}y^a|\nabla U_{\varepsilon }|^2\,dX\\
&&\qquad
+2\sqrt{\int_{{{\mathbb R}}^{n+1}_+\setminus B_\rho^+}y^a|\nabla U_k|^2\,dX}\cdot
\sqrt{\int_{{{\mathbb R}}^{n+1}_+\setminus B_\rho^+}y^a|\nabla U_{\varepsilon }|^2\,dX}\\
&{\leqslant}& \frac{\eta}{4}+\frac{\eta}{4}+2\frac{\sqrt{\eta}}{2}\frac{\sqrt{\eta}}{2} =\eta.\end{aligned}$$ This shows .
Also, Theorem 1.1.4 in [@evans] gives the existence of two measures on ${{\mathbb R}}^n$ and ${{\mathbb R}}^{n+1}_+$, $\nu$ and $\mu$ respectively, such that $(V_k)_+^{2^*_s}(\cdot,0)$ converges to $\nu$ and $y^a|\nabla (V_k)_+|^2$ converges to $\mu$ as $k\to+\infty$, according to Definition 1.1.2 in [@evans] (see also Definition \[convMeasures\]). This, and imply that the hypotheses of Proposition \[CCP\] are satisfied, and so there exist an at most countable set $J$ and three families $\{x_j\}_{j\in J}\in{{\mathbb R}}^n$, $\{\nu_j\}_{j\in J}$ and $\{\mu_j\}_{j\in J}$, with $\nu_j,\mu_j{\geqslant}0$ such that $$\label{lklhjgghf}
{\mbox{$(V_k)_+^{2^*_s}$ converges to~$\nu=U_{\varepsilon }^{2^*_s}+\sum_{j\in J}\nu_j\delta_{x_j}$ as~$k\to+\infty$,}}$$ $$\label{lklhjgghf-2}
{\mbox{$y^a|\nabla(V_k)_+|^2$ converges to~$\mu{\geqslant}y^a|\nabla U_{\varepsilon }|^2+\sum_{j\in J}\mu_j\delta_{(x_j,0)}$ as~$k\to+\infty$}}$$ and $$\label{lklhjgghf-3}
\mu_j{\geqslant}S \nu_j^{2/2^*_s}\quad {\mbox{ for all }} j\in J.$$
We claim now that $\nu_j=\mu_j=0$ for every $j\in J$. To prove this, we argue by contradiction and we suppose that there exists $j\in J$ such that $\mu_j=0$. We denote $X_j:=(x_j,0)$, we fix $\delta>0$ and we take a cut-off function $\phi_\delta\in C^\infty({{\mathbb R}}^{n+1}_+,[0,1])$ such that $$\phi_\delta(X)=\begin{cases}
1,\qquad {\mbox{ if }} X\in B^+_{\delta/2}(X_j),\\
0,\qquad {\mbox{ if }} X\in (B^+_{\delta}(X_j))^c,
\end{cases}
\quad {\mbox{ and }} \quad |\nabla \phi_\delta|{\leqslant}\frac{C}{\delta},$$ for some $C>0$.
Now, it is not difficult to show that the sequence $\phi_\delta(V_k)_+$ is uniformly bounded in $\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$. Therefore, from and we have that $$\begin{split}\label{sixth}
0=\,& \lim_{k\to+\infty} \langle{{\mathscrF}}_{\varepsilon }'(V_k),\phi_\delta(V_k)_+\rangle\\
=\,& \lim_{k\to+\infty}\Big( \int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla V_k,\nabla(\phi_\delta (V_k)_+)\rangle\,dX\\
&\qquad \qquad -{\varepsilon }\int_{{{\mathbb R}}^n}h(x)(V_k)_+^{q+1}(x,0)\phi_\delta(x,0)\,dx
-\int_{{{\mathbb R}}^n}(V_k)_+^{p+1}(x,0)\phi_\delta(x,0)\,dx\Big)\\
=\,& \lim_{k\to+\infty}\Big( \int_{{{\mathbb R}}^{n+1}_+}y^a|\nabla (V_k)_+|^2\phi_\delta \,dX
+\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla (V_k)_+,\nabla\phi_\delta\rangle (V_k)_+\,dX\\
&\qquad \qquad -{\varepsilon }\int_{{{\mathbb R}}^n}h(x)(V_k)_+^{q+1}(x,0)\phi_\delta(x,0)\,dx
-\int_{{{\mathbb R}}^n}(V_k)_+^{p+1}(x,0)\phi_\delta(x,0)\,dx\Big).
\end{split}$$ We recall that $p+1=2^*_s$ and we use and to see that $$\begin{aligned}
\label{qqqqqqqqqqqpppp}
\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}(V_k)_+^{p+1}(x,0)\phi_\delta(x,0)\,dx&=& \int_{{{\mathbb R}}^n}\phi_\delta(x,0)\,d\nu\\
{\mbox{and }} \lim_{k\to+\infty}\int_{{{\mathbb R}}^{n+1}_+}y^a|\nabla (V_k)_+|^2\phi_\delta \,dX &=&
\int_{{{\mathbb R}}^{n+1}_+}\phi_\delta\,d\mu\label{qqqqqqqqqqqpppp-1}.\end{aligned}$$ Also, the weak convergence in , and Theorem 7.1 in [@DPV] imply that $(V_k)_+(\cdot,0)$ strongly converges to $U_{\varepsilon }(\cdot,0)$ in $L^r_{\rm{loc}}({{\mathbb R}}^n)$ as $k\to+\infty$, for any $r\in[1,2^*_s)$. Accordingly, $$\begin{aligned}
&& \left|\int_{{{\mathbb R}}^n}h(x)(V_k)_+^{q+1}(x,0)\phi_\delta(x,0)\,dx-
\int_{{{\mathbb R}}^n}h(x)U_{\varepsilon }^{q+1}(x,0)\phi_\delta(x,0)\,dx\right| \\
&&\qquad {\leqslant}\|h\|_{L^\infty({{\mathbb R}}^n)}
\left| \int_{B_\delta^+(X_j)\cap\{y=0\} }
\left((V_k)_+^{q+1}(x,0)-U_{\varepsilon }^{q+1}(x,0)\right)\,dx\right|\to 0,\end{aligned}$$ as $k\to+\infty$, since $1<q+1<2^*_s$. This implies that $$\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}h(x)(V_k)_+^{q+1}(x,0)\phi_\delta(x,0)\,dx
= \int_{{{\mathbb R}}^n}h(x)U_{\varepsilon }^{q+1}(x,0)\phi_\delta(x,0)\,dx.$$ Taking the limit as $\delta\to0$ we have $$\label{forth}\begin{split}
&\lim_{\delta\to0} \lim_{k\to+\infty}\int_{{{\mathbb R}}^n}h(x)(V_k)_+^{q+1}(x,0)\phi_\delta(x,0)\,dx
\\&\qquad= \lim_{\delta\to0}\int_{B_\delta^+(X_j)\cap\{y=0\} }h(x)U_{\varepsilon }^{q+1}(x,0)\phi_\delta(x,0)\,dx=0.
\end{split}$$ Finally, we claim that $$\label{fifth}
\lim_{\delta\to0}\lim_{k\to+\infty}
\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla (V_k)_+,\nabla\phi_\delta\rangle (V_k)_+\,dX=0.$$ For this, we apply the Hölder inequality and we use to obtain that $$\begin{split}\label{slasgkrehrtohurt}
&\left|\int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla (V_k)_+,\nabla\phi_\delta\rangle (V_k)_+\,dX \right|
\\&\qquad
=\left| \int_{B_\delta^+(X_j)}y^a\langle\nabla (V_k)_+,\nabla\phi_\delta\rangle (V_k)_+\,dX\right|\\
&\qquad
{\leqslant}\left( \int_{B_\delta^+(X_j)}y^a|\nabla (V_k)_+|^2\,dX\right)^{1/2}
\left( \int_{B_\delta^+(X_j)}y^a(V_k)_+^2|\nabla\phi_\delta|^2\,dX\right)^{1/2}\\&\qquad{\leqslant}M\left( \int_{B_\delta^+(X_j)}y^a(V_k)_+^2|\nabla\phi_\delta|^2\,dX\right)^{1/2}.
\end{split}$$ Again by and Lemma \[lemma:compact\], we deduce that $$\begin{aligned}
&& \left| \int_{B_\delta^+(X_j)}y^a(V_k)_+^2|\nabla\phi_\delta|^2\,dX
- \int_{B_\delta^+(X_j)}y^aU_{\varepsilon }^2|\nabla\phi_\delta|^2\,dX\right| \\
&&\quad{\leqslant}\frac{C^2}{\delta^2} \left| \int_{B_\delta^+(X_j)}y^a(V_k)_+^2\,dX -
\int_{B_\delta^+(X_j)}y^aU_{\varepsilon }^2\,dX\right| \to 0,\end{aligned}$$ as $k\to+\infty$. Hence $$\label{sldujhktpriujr}
\lim_{k\to+\infty} \int_{B_\delta^+(X_j)}y^a(V_k)_+^2|\nabla\phi_\delta|^2\,dX=
\int_{B_\delta^+(X_j)}y^aU_{\varepsilon }^2|\nabla\phi_\delta|^2\,dX.$$ Now by the Hölder inequality with exponents $\gamma$ and $\frac{\gamma}{\gamma-1}$ we have that $$\begin{aligned}
&& \int_{B_\delta^+(X_j)}y^aU_{\varepsilon }^2|\nabla\phi_\delta|^2\,dX \\&{\leqslant}&
\left( \int_{B_\delta^+(X_j)}y^aU_{\varepsilon }^{2\gamma}\,dX\right)^{\frac1{\gamma}}
\left(\int_{B_\delta^+(X_j)}y^a|\nabla\phi_\delta|^{\frac{2\gamma}{\gamma-1}}\,dX
\right)^{\frac{\gamma-1}{\gamma}}\\
&{\leqslant}& \frac{C^2}{\delta^2}
\left( \int_{B_\delta^+(X_j)}y^aU_{\varepsilon }^{2\gamma}\,dX\right)^{\frac1{\gamma}}
\left(\int_{B_\delta^+(X_j)}y^a\,dX\right)^{\frac{\gamma-1}{\gamma}}\\
&{\leqslant}& C\delta^{\frac{(n+a+1)(\gamma-1)}{\gamma}-2}
\left( \int_{B_\delta^+(X_j)}y^aU_{\varepsilon }^{2\gamma}\,dX\right)^{\frac1{\gamma}},\end{aligned}$$ up to renaming constants. Since $\frac{(n+a+1)(\gamma-1)}{\gamma}-2=0$, this implies that $$\int_{B_\delta^+(X_j)}y^aU_{\varepsilon }^2|\nabla\phi_\delta|^2\,dX{\leqslant}C
\left( \int_{B_\delta^+(X_j)}y^aU_{\varepsilon }^{2\gamma}\,dX\right)^{\frac1{\gamma}},$$ for a suitable positive constant $C$. Hence, $$\lim_{\delta\to0} \int_{B_\delta^+(X_j)}y^aU_{\varepsilon }^2|\nabla\phi_\delta|^2\,dX =0.$$ This, together with and , proves .
From , , , and we obtain that $$\begin{aligned}
&& 0= \lim_{\delta\to0}\lim_{k\to+\infty} \langle{{\mathscrF}}_{\varepsilon }'(V_k),\phi_\delta(V_k)_+\rangle\\
\\&&\qquad= \lim_{\delta\to0}
\Big( \int_{{{\mathbb R}}^{n+1}_+}\phi_\delta \,d\mu-\int_{{{\mathbb R}}^n}\phi_\delta(x,0)\,d\nu\Big)
{\geqslant}\mu_j-\nu_j.\end{aligned}$$ Therefore, this and give that $\nu_j{\geqslant}\mu_j{\geqslant}S\nu_j^{2/2^*_s}$. Hence, either $\nu_j=\mu_j=0$ or $\nu_j^{1-2/2^*_s}{\geqslant}S$. Since we are in the case $\mu_j\neq0$, the first possibility cannot occur. As a consequence, $$\label{maggiore}
\nu_j{\geqslant}S^{n/2s}.$$
Now, taking the limit as $k\to+\infty$ in and recalling assumption (i) of Proposition \[PScond2\], and , we have that $$\begin{split}\label{vabbe0}
&c_{\varepsilon }+{{\mathscrF}}_{\varepsilon }(U_{\varepsilon })\\
{\geqslant}\, & \lim_{k\to+\infty}\left({{\mathscrF}}_{\varepsilon }(V_k)
-\frac12\langle {{\mathscrF}}_{\varepsilon }'(V_k),V_k\rangle \right)\\
=\,& \lim_{k\to+\infty} \left[\left(\frac12-\frac{1}{p+1}\right)\int_{{{\mathbb R}}^n}(V_k)_+^{p+1}(x,0)\, dx
-{\varepsilon }\left(\frac{1}{q+1}-\frac12\right)\int_{{{\mathbb R}}^n}h(x)(V_k)_+^{q+1}(x,0)\,dx\right].
\end{split}$$ We claim that $$\label{vabbe}
\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}(V_k)_+^{p+1}(x,0)\, dx {\geqslant}S^{n/2s}+
\int_{{{\mathbb R}}^n}U_{\varepsilon }^{p+1}(x,0)\, dx.$$ For this, we take a sequence $\{\varphi_m\}_{m\in{{\mathbb N}}}\in C^\infty_0({{\mathbb R}}^n,[0,1])$ such that $\displaystyle\lim_{m\to+\infty}\varphi_m(x)=1$ for any $x\in{{\mathbb R}}^n$. By we have that $$\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}(V_k)_+^{p+1}(x,0)\, dx {\geqslant}\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}(V_k)_+^{p+1}(x,0)\varphi_m(x)\, dx
= \int_{{{\mathbb R}}^n}\varphi_m(x)\, d\nu.$$ Moreover, thanks to Fatou’s lemma and , $$\lim_{m\to+\infty}\int_{{{\mathbb R}}^n}\varphi_m(x)\,d\nu {\geqslant}\int_{{{\mathbb R}}^n}\,d\mu {\geqslant}S^{n/2s} +
\int_{{{\mathbb R}}^n}U_{\varepsilon }^{p+1}(x,0)\,dx.$$ The last two formulas imply that $$\begin{aligned}
&& \lim_{k\to+\infty}\int_{{{\mathbb R}}^n}(V_k)_+^{p+1}(x,0)\, dx=
\lim_{m\to+\infty}\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}(V_k)_+^{p+1}(x,0)\, dx
\\ &&\qquad{\geqslant}\lim_{m\to+\infty} \int_{{{\mathbb R}}^n}\varphi_m(x)\,d\nu
{\geqslant}S^{n/2s} +
\int_{{{\mathbb R}}^n}U_{\varepsilon }^{p+1}(x,0)\,dx,\end{aligned}$$ which gives the desired result in . We now show that $$\label{vabbe-1}
\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}h(x)(V_k)_+^{q+1}(x,0)\,dx =
\int_{{{\mathbb R}}^n}h(x)U_{\varepsilon }^{q+1}(x,0)\,dx.$$ Indeed, thanks to we know that $[(V_k)_+]_a{\leqslant}M$. Therefore, Proposition \[traceIneq\] and Theorem 7.1 in [@DPV] imply that $$\begin{aligned}
&& \|(V_k)_+(\cdot,0)-U_{\varepsilon }(\cdot,0)\|_{L^{2^*_s}({{\mathbb R}}^n)}{\leqslant}2M\\
{\mbox{and }} && (V_k)_+(\cdot,0)\to U_{\varepsilon }(\cdot,0) \ {\mbox{ in }} L^{q+1}_{\rm{loc}}( {{\mathbb R}}^n )
\ {\mbox{ as }}k\to+\infty. \end{aligned}$$ Thus, we fix $R>0$ and we use , and the Hölder inequality to obtain that $$\begin{aligned}
&&\left| \int_{{{\mathbb R}}^n}h(x)\left((V_k)_+(x,0)-U_{\varepsilon }(x,0)\right)^{q+1}\,dx\right| \\
&&\qquad {\leqslant}\int_{B_R}|h(x)|\,\left|(V_k)_+(x,0)-U_{\varepsilon }(x,0)\right|^{q+1}\,dx \\&&\qquad\qquad+
\int_{{{\mathbb R}}^n\setminus B_R}|h(x)|\,\left|(V_k)_+(x,0)-U_{\varepsilon }(x,0)\right|^{q+1}\,dx\\
&&\qquad {\leqslant}\|h\|_{L^\infty({{\mathbb R}}^n)} \|(V_k)_+(\cdot,0)-U_{\varepsilon }(\cdot,0)\|_{L^{q+1}(B_R)} \\&&\qquad\qquad+
\|h\|_{L^{\frac{2^*_s}{2^*_s-q-1}}({{\mathbb R}}^n\setminus B_R)} \|(V_k)_+(\cdot,0)-U_{\varepsilon }(\cdot,0)\|_{L^{2^*_s}({{\mathbb R}}^n)}\\
&&\qquad {\leqslant}C\|(V_k)_+(\cdot,0)-U_{\varepsilon }(\cdot,0)\|_{L^{q+1}(B_R)}\,dx +
(2M)^{q+1}\|h\|_{L^{\frac{2^*_s}{2^*_s-q-1}}({{\mathbb R}}^n\setminus B_R)}. \end{aligned}$$ Hence, letting first $k\to+\infty$ and then $R\to+\infty$, we obtain .
Also, we observe that $\frac12-\frac{1}{p+1}=\frac{s}{n}$. Using this and plugging and into we obtain that $$\begin{aligned}
&&c_{\varepsilon }+ {{\mathscrF}}_{\varepsilon }(U_{\varepsilon }) \\&{\geqslant}& \frac{s}{n}S^{n/2s} +
\left(\frac12-\frac{1}{p+1}\right)\int_{{{\mathbb R}}^n}U_{\varepsilon }^{p+1}(x,0)\, dx
\\&&\qquad-{\varepsilon }\left(\frac{1}{q+1}-\frac12\right)\int_{{{\mathbb R}}^n} h(x)
U_{\varepsilon }^{q+1}(x,0)\,dx\\
&=& \frac{s}{n}S^{n/2s}+{{\mathscrF}}_{\varepsilon }(U_{\varepsilon }).\end{aligned}$$ Hence $$c_{\varepsilon }{\geqslant}\frac{s}{n}S^{n/2s},$$ and this is a contradiction with .
As a consequence, necessarily $\mu_j=\nu_j=0$ for any $j\in J$. Hence, by $$\label{chiama}
\lim_{k\to+\infty}\int_{{{\mathbb R}}^n}
(V_k)_+^{2^*_s}(x,0)\varphi(x)\,dx =
\int_{{{\mathbb R}}^n} U_{\varepsilon }^{2^*_s}(x,0)\varphi(x)\,dx,$$ for any $\varphi\in C_0({{\mathbb R}}^n)$. Furthermore, by Lemma \[lemma:tight\] and the fact that $U_{\varepsilon }(\cdot,0)\in L^{2^*_s}({{\mathbb R}}^n)$ (thanks to Proposition \[traceIneq\]), we have that for any $\eta>0$ there exists $\rho>0$ such that $$\int_{{{\mathbb R}}^n\setminus B_\rho} (V_k)_+^{2^*_s}(x,0)\,dx <\eta.$$ Thus we are in the position to apply Lemma \[PSL-2\] with $v_k:=(V_k)_+(\cdot,0)$ and $v:=U_{\varepsilon }(\cdot,0)$, and we obtain that $(V_k)_+(\cdot,0)\to U_{\varepsilon }(\cdot,0)$ in $L^{2^*_s}({{\mathbb R}}^n)$ as $k\to+\infty$. Then, by Lemma \[PSL-1\] (again applied with $v_k:=(V_k)_+(\cdot,0)$ and $v:=U_{\varepsilon }(\cdot,0)$) we have that $$\begin{aligned}
&& \lim_{k\to+\infty}\int_{{{\mathbb R}}^n}|(V_k)_+^q(x,0)-U_{\varepsilon }^q(x,0)|^{\frac{2^*_s}{q}}\,dx =0\\
{\mbox{and }} && \lim_{k\to+\infty}\int_{{{\mathbb R}}^n}|(V_k)_+^p(x,0)-U_{\varepsilon }^p(x,0)|^{\frac{2n}{n+2s}}\,dx =0.\end{aligned}$$ Therefore, we can fix $\delta\in(0,1)$ (that we will take arbitrarily small in the sequel), and say that $$\label{pegyhrehehrht}
\begin{split}
&\int_{{{\mathbb R}}^n}|(V_k)_+^q(x,0)-(V_m)_+^q(x,0)|^{\frac{2^*_s}{q}}\,dx \\&\qquad+
\int_{{{\mathbb R}}^n}|(V_k)_+^p(x,0)-(V_m)_+^p(x,0)|^{\frac{2n}{n+2s}}\,dx{\leqslant}\delta
\end{split}$$ for $k$ and $m$ sufficiently large (say bigger that some $k_\star(\delta)$).
We now take $\Psi\in\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$ with $\psi:=\Psi(\cdot,0)$ and such that $$\label{hlkhfjfdsafgh}
[\Psi]_a=1.$$ From we have that, for large $k$ (say $k{\geqslant}k_\star(\delta)$, up to renaming $k_\star(\delta)$), we deduce that $$\left|\langle {{\mathscrF}}_{\varepsilon }'(V_k),\Psi\rangle\right|{\leqslant}\delta.$$ As a consequence of this and , $$\begin{aligned}
&& \left| \int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla V_k,\nabla \Psi\rangle\,dX \right.\\
&&\qquad\left.
-{\varepsilon }\int_{{{\mathbb R}}^n}h(x)(V_k)_+^q(x,0)\psi(x)\,dx -\int_{{{\mathbb R}}^n}(V_k)_+^p(x,0)\psi(x)\,dx\right|{\leqslant}\delta.\end{aligned}$$ In particular, for $k,m{\geqslant}k_\star(\delta)$, $$\begin{aligned}
&&\Big| \int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla(V_k-V_m),\nabla \Psi\rangle\,dX
\\&&\qquad -{\varepsilon }\int_{{{\mathbb R}}^n}h(x)\left((V_k)_+^q(x,0)-(V_m)_+^q(x,0)\right)\psi(x)\,dx
\\&&\qquad -\int_{{{\mathbb R}}^n}\left((V_k)_+^p(x,0)-(V_m)_+^p(x,0)\right)\psi(x)\,dx\Big|{\leqslant}2\delta.\end{aligned}$$ Now we use the Hölder inequality with exponents $\frac{2n}{n+2s-q(n-2s)}$, $\frac{2^*_s}{q}=\frac{2n}{q(n-2s)}$ and $2^*_s=\frac{2n}{n-2s}$, and with exponents $\frac{2^*_s}{p}=\frac{2n}{n+2s}$ and $2^*_s$, and we obtain that $$\begin{aligned}
&&\left| \int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla(V_k-V_m),\nabla \Psi\rangle\,dX \right| \\
&{\leqslant}& {\varepsilon }\left|\int_{{{\mathbb R}}^n}h(x)\left((V_k)_+^q(x,0)-(V_m)_+^q(x,0)\right)\psi(x)\,dx\right|
\\&&\qquad
+ \left|\int_{{{\mathbb R}}^n}\left((V_k)_+^p(x,0)-(V_m)_+^p(x,0)\right)\psi(x)\,dx\right| +2\delta \\
&{\leqslant}& {\varepsilon }\left[\int_{{{\mathbb R}}^n}|h(x)|^{\frac{2n}{n+2s-q(n-2s)}}\,dx
\right]^{\frac{n+2s-q(n-2s)}{2n}}\\ &&\qquad\cdot
\left[ \int_{{{\mathbb R}}^n}\left|(V_k)_+^q(x,0)-(V_m)_+^q(x,0)\right|^{\frac{2^*_s}{q}}\,dx\right]^{\frac{q(n-2s)}{2n}}
\left[\int_{{{\mathbb R}}^n}|\psi(x)|^{2^*_s}\,dx\right]^{\frac{1}{2^*_s}} \\
&&\qquad + \left[ \int_{{{\mathbb R}}^n}\left|(V_k)_+^p(x,0)-(V_m)_+^p(x,0)\right|^{\frac{2n}{n+2s}}\,dx\right]^{\frac{n+2s}{2n}}
\left[ \int_{{{\mathbb R}}^n}|\psi(x)|^{2^*_s}\,dx\right]^{\frac{1}{2^*_s}} +2\delta.\end{aligned}$$ Hence, from and we have that $$\left| \int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla(V_k-V_m),\nabla \Psi\rangle\,dX \right|{\leqslant}C\delta^{\frac{q(n-2s)}{2n}}\|\psi\|_{L^{2^*_s}({{\mathbb R}}^n)} +C\delta^{\frac{n+2s}{2n}}
\|\psi\|_{L^{2^*_s}({{\mathbb R}}^n)} +2\delta,$$ for a suitable positive constant $C$. Now notice that and imply that $\|\psi\|_{L^{2^*_s}({{\mathbb R}}^n)}{\leqslant}S^{-1/2}[\Psi]_a=S^{-1/2}$, and so $$\label{menomale}
\left| \int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla(V_k-V_m),\nabla \Psi\rangle\,dX \right|{\leqslant}C\delta^a,$$ for some $C,a>0$, as long as $k,m{\geqslant}k_\star(\delta)$. Also, $$\nabla(V_k-V_m) = \nabla V_k -\nabla V_m = \nabla(U_k+U_{\varepsilon })-\nabla(U_m+U_{\varepsilon })
=\nabla U_k-\nabla U_m.$$ Hence, plugging this into , we have $$\left| \int_{{{\mathbb R}}^{n+1}_+}y^a\langle\nabla(U_k-U_m),\nabla \Psi\rangle\,dX \right|{\leqslant}C\delta^a.$$ Since this inequality is valid for any $\Psi$ satisfying , we deduce that $$[U_k-U_m]_a{\leqslant}C\delta^a,$$ namely $U_k$ is a Cauchy sequence in $\dot{H}^s_a({{\mathbb R}}^{n+1}_+)$. Then, the desired result plainly follows.
Bound on the minimax value {#sec:BMMV}
--------------------------
The goal of this section is to show that the minimax value (computed along a suitable path) lies below the critical threshold given by Proposition \[PScond2\]. The chosen path will be a suitably cut-off rescaling of the fractional Sobolev minimizers introduced in .
To start with, we set $$\label{67ydv8888kk}
\tilde G(x,U):=\int_0^{U} \tilde g(x,t)\,dt$$ and $$\tilde g(x,t):= \begin{cases}
(U_\varepsilon+t)^q-U_\varepsilon^q,\;\hbox{ if }t{\geqslant}0,
\\ 0\;\hbox{ if }t< 0.
\end{cases}$$ We observe that $$\label{G0poo}
\tilde G(x,0)=0.$$ Also, we see that $\tilde g(x,t){\geqslant}0$ for any $t\in{{\mathbb R}}$, and so $$\label{G0po}
{\mbox{$\tilde G(x,U){\geqslant}0$ for any~$U{\geqslant}0$.}}$$ Moreover, recalling , we write the ball $B$ in as $B_{\mu_0}(\xi)$ for some $\xi\in{{\mathbb R}}^n$ and $\mu_0>0$. We fix a cut-off function $\bar\phi\in C^\infty_0(B_{\mu_0}(\xi),\,[0,1])$ with $$\label{PP0dsfhuw5r6t78y009}
{\mbox{$\bar\phi(x)=1$ for any~$x\in B_{\mu_0/2}(\xi)$.}}$$ The quantities $\xi\in{{\mathbb R}}^n$ and $\mu_0>0$, as well as the cut-off $\bar\phi$, are fixed from now on. Also, if $z$ is as in , given $\mu>0$, we let $$\label{tale2} z_{\mu,\xi}(x):=\mu^{-\frac{n-2s}{2}}
z\left( \frac{x-\xi}{\mu}\right).$$ Let also $\bar Z_{\mu,\xi}$ be the extension of $\bar\phi z_{\mu,\xi}$, according to .
[F]{}rom , we know that $$\label{SO89sd}
S= \frac{[z]_{\dot H^s({{\mathbb R}}^n)}^2}{\|z\|^2_{L^{2^*_s}({{\mathbb R}}^n)}}$$ and $(-\Delta)^s z=z^p$. Thus, by testing this equation against $z$ itself, we obtain that $$[z]_{\dot H^s({{\mathbb R}}^n)}^2 = \|z\|^{2^*_s}_{L^{2^*_s}({{\mathbb R}}^n)},$$ which, together with , gives that $$\|z\|_{L^{2^*_s}({{\mathbb R}}^n)}=S^{\frac{n-2s}{4s}}$$ and so $$[z]_{\dot H^s({{\mathbb R}}^n)}^2 = S^{\frac{n}{2s}}.$$ Moreover, by scaling, we have that $$\label{8dvsuyuscal}
\|z_{\mu,\xi}\|_{L^{2^*_s}({{\mathbb R}}^n)}=\|z\|_{L^{2^*_s}({{\mathbb R}}^n)}=
S^{\frac{n-2s}{4s}}$$ and $$[z_{\mu,\xi}]_{\dot H^s({{\mathbb R}}^n)}^2 =
[z]_{\dot H^s({{\mathbb R}}^n)}^2= S^{\frac{n}{2s}}.$$ [F]{}rom the equivalence of norms in and Proposition 21 in [@SV], we have that $$\label{8dvsuyuscal-2}
[\bar Z_{\mu,\xi}]_a^2
=[\bar\phi z_{\mu,\xi}]_{\dot H^s({{\mathbb R}}^n)}^2{\leqslant}S^{\frac{n}{2s}} + C\mu^{n-2s},$$ for some $C>0$.
This setting is fixed from now on, together with the minimum $u_{\varepsilon }(x)=U_{\varepsilon }(x,0)$ given in Theorem \[MINIMUM\]. Now we show that the effect of the cut-off on the Lebesgue norm of the rescaled Sobolev minimizers is negligible when $\mu$ is small. The quantitative statement is the following:
\[LE:NU90\] We have that $$\int_{{{\mathbb R}}^n} |\bar\phi^{2^*_s}-1|\, z_{\mu,\xi}^{2^*_s} \,dx
{\leqslant}C\mu^{n},$$ for some $C>0$.
We observe that $$\begin{split}
& \int_{{{\mathbb R}}^n\setminus B_{\frac{\mu_0}2}(\xi)} z_{\mu,\xi}^{2^*_s} (x)\,dx
= \mu^{-n}
\int_{{{\mathbb R}}^n\setminus B_{\frac{\mu_0}2}(\xi)}
z^{2^*_s}\left( \frac{x-\xi}{\mu}\right)\,dx \\
&\qquad =
\int_{{{\mathbb R}}^n\setminus B_{\frac{\mu_0}{2\mu}} }
z^{2^*_s}(y)\,dy
{\leqslant}C\,\int_{{{\mathbb R}}^n\setminus B_{\frac{\mu_0}{2\mu}} }
|y|^{-2n}\,dy {\leqslant}C\mu^{n},
\end{split}$$ for some $C>0$ (that may vary from line to line and may also depend on $\mu_0$). As a consequence, recalling , we have that $$\int_{{{\mathbb R}}^n} |\bar\phi^{2^*_s}-1|\, z_{\mu,\xi}^{2^*_s} \,dx
=\int_{{{\mathbb R}}^n\setminus B_{\frac{\mu_0}2}(\xi)}
|\bar\phi^{2^*_s}-1|\, z_{\mu,\xi}^{2^*_s} \,dx
{\leqslant}C\mu^{n}. \qedhere$$
The next result states that we can always “concentrate the mass near the positivity set of $h$”, in order to detect a positive integral out of it.
\[SP\] We have that $$\label{TO:P:SP}
\int_{{{\mathbb R}}^n} h(x)\,
\tilde G\big( x,t\bar\phi(x)\,z_{\mu,\xi}(x)\big)\,dx
{\geqslant}0,$$ for any $\mu>0$ and any $t{\geqslant}0$.
We have that $\bar\phi(x)=0$ if $x\in {{\mathbb R}}^n\setminus
B_{\mu_0}(\xi)$. Thus, using , we have that $$\tilde G\big(x,t\bar\phi(x)\,z_{\mu,\xi}(x) \big)=0$$ for any $x\in {{\mathbb R}}^n\setminus
B_{\mu_0}(\xi)$. Therefore $$\int_{{{\mathbb R}}^n} h(x)\,
\tilde G\big( x,t\bar\phi(x)\,z_{\mu,\xi}(x)\big)\,dx
=\int_{B_{\mu_0}(\xi)} h(x)\,\tilde G
\big( x,t\bar\phi(x)\,z_{\mu,\xi}(x)\big)\,dx.$$ Then, the desired result follows from and .
Now we check that the geometry of the mountain pass is satisfied by the functional ${{\mathscrI}}_\varepsilon$. Indeed, we first observe that Proposition \[prop:zero\] gives that $0$ is a local minimum for the functional $
{{\mathscrI}}_\varepsilon$. The next result shows that the path induced by the function $\bar Z_{\mu,\xi}$ attains negative values, in a somehow uniform way (the uniform estimates in $\mu$ in Lemma \[yu799\] will be needed in the subsequent Corollary \[CC-below-PS\] and, from these facts, we will be able to deduce the mountain pass geometry, check that the minimax values stays below the critical threshold and complete the proof of Theorem \[TH:MP\] in the forthcoming Section \[sec:proof\]). To this goal, it is useful to introduce the auxiliary functional $$\label{def I2}
{{\mathscrI}}^\star_\varepsilon(U):=
\frac{1}{2}\int_{\mathbb{R}^{n+1}_+}{y^a|\nabla U|^2\,dX}-
\int_{\mathbb{R}^n}{G^\star(x,U(x,0))\,dx},$$ where $$G^\star(x,U):=\int_0^{U} g^\star(x,t)\,dt$$ and $$g^\star(x,t):= \begin{cases}
(U_\varepsilon+t)^p-U_\varepsilon^p,\;\hbox{ if }t{\geqslant}0,
\\ 0\;\hbox{ if }t< 0.
\end{cases}$$ By and , we see that $G=G^\star+{\varepsilon }h\tilde G$. Thus, as a consequence of Lemma \[SP\], we have that $$\label{789hgbnjjjjjKK}
{{\mathscrI}}_\varepsilon(t\bar Z_{\mu,\xi})
={{\mathscrI}}_\varepsilon^\star(t\bar Z_{\mu,\xi})
-{\varepsilon }\int_{{{\mathbb R}}^n} h(x)\,\tilde G(x,t\bar Z_{\mu,\xi}(x))\,dx
{\leqslant}{{\mathscrI}}_\varepsilon^\star(t\bar Z_{\mu,\xi}).$$ Then we have:
\[yu799\] There exists $\mu_1\in(0,\mu_0)$ such that $$\lim_{t\to+\infty}
\sup_{\mu\in(0,\mu_1)}
{{\mathscrI}}_\varepsilon^\star(t\bar Z_{\mu,\xi}) =-\infty.$$ In particular, there exists $T_1>0$ such that $$\label{789hgshh67g000io}
\sup_{\mu\in(0,\mu_1)}
{{\mathscrI}}_\varepsilon^\star(t\bar Z_{\mu,\xi}){\leqslant}0$$ for any $t{\geqslant}T_1$.
We observe that, if $U{\geqslant}0$, $$\label{9sg78ujohFgj}
G^\star(x,U)=\int_0^{U} (U_\varepsilon+t)^p-U_\varepsilon^p\,dt
=\frac{(U_\varepsilon+U)^{p+1} -U_\varepsilon^{p+1}}{p+1}
-U_\varepsilon^p\,U.$$ Moreover, $$U_{\varepsilon }^{p}(x,0)\,
\bar Z_{\mu,\xi}(x,0){\leqslant}u_{\varepsilon }^{p+1}(x)+
z_{\mu,\xi}^{p+1}(x).$$ Using this and , we obtain that $$\begin{aligned}
&&G^\star(x,t\bar Z_{\mu,\xi}(x,0))\\
&=& \frac{1}{p+1}\left((U_{\varepsilon }(x,0) +t\bar Z_{\mu,\xi}(x,0))^{p+1}-
U_{\varepsilon }^{p+1}(x,0)\right)
-tU_{\varepsilon }^p (x,0)\,\bar Z_{\mu,\xi}(x,0)
\\ &{\geqslant}&
\frac{1}{p+1}\Big(
(t\bar \phi(x) z_{\mu,\xi}(x))^{p+1}-
u_{\varepsilon }^{p+1}(x)\Big)
-t u_{\varepsilon }^{p+1}(x) -tz^{p+1}_{\mu,\xi}(x)
.\end{aligned}$$ Thus, integrating over ${{\mathbb R}}^n$ and recalling , and the fact that $2^*_s=p+1$, we get $$\label{78990iuii-0}
\begin{split}
& \int_{{{\mathbb R}}^n}
G^\star(x,t\bar Z_{\mu,\xi})\,dx\\ &\qquad{\geqslant}\frac{t^{p+1}}{p+1}\int_{{{\mathbb R}}^n}
\bar\phi^{p+1}(x) z_{\mu,\xi}^{p+1}(x)\,dx
-C-Ct,\end{split}$$ for some $C>0$ (up to renaming constants).
Now we deduce from Lemma \[LE:NU90\] that there exists $\mu_1\in(0,1)$ such that if $\mu\in(0,\mu_1)$ then $$\int_{{{\mathbb R}}^n}
\bar\phi^{p+1}(x) z_{\mu,\xi}^{p+1}(x)\,dx {\geqslant}\frac{ S^{\frac{n}{2s}} }{2}.$$ Now, by inserting this into , we obtain that, if $\mu\in(0,\mu_1)$, then $$\int_{{{\mathbb R}}^n}
G^\star(x,t\bar Z_{\mu,\xi})\,dx{\geqslant}\frac{ S^{\frac{n}{2s}}\,t^{p+1} }{2(p+1)}
-C-Ct.$$ This and give that $${{\mathscrI}}_\varepsilon^\star(t\bar Z_{\mu,\xi}) {\leqslant}\frac{t^2 [\bar Z_{\mu,\xi}]^2_a}{2}
+C(1+t)-\frac{ S^{\frac{n}{2s}}\,t^{p+1} }{2(p+1)}.$$ Hence, recalling , $${{\mathscrI}}_\varepsilon^\star(t\bar Z_{\mu,\xi}) {\leqslant}C(1+t+t^2)-\frac{ S^{\frac{n}{2s}}\,t^{p+1} }{2(p+1)},$$ up to renaming constants, for any $\mu\in(0,\mu_1)$. Since $p+1>2$, the desired claim easily follows.
Now we introduce a series of purely elementary, but useful, estimates.
\[ABC-1\] For any $a$, $b{\geqslant}0$ and any $p>1$, we have that $$\label{9sghjtrf}
(a+b)^p{\geqslant}a^p +b^p.$$ Also, if $a$, $b>0$, we have that $$\label{9sghjtrf-2}
(a+b)^p> a^p +b^p.$$
If either $a=0$ or $b=0$, then is obvious. So we can suppose that $a\ne0$ and $b\ne0$. We let $f(b):=(a+b)^p - a^p-b^p$. Notice that $f'(b)=p\big( (a+b)^{p-1} - b^{p-1}\big)>0$, since $a>0$. Hence $$(a+b)^p-a^p-b^p = f(b)> f(0)=0,$$ since $b>0$, as desired.
The result in Lemma \[ABC-1\] can be made more precise when $p{\geqslant}2$, as follows:
\[ABC-2\] Let $p{\geqslant}2$. Then, there exists $c_p>0$ such that, for any $a$ and $b{\geqslant}0$, $$(a+b)^p{\geqslant}a^p +b^p + c_p\,a^{p-1} b.$$
If $a=0$, then we are done, so we suppose $a\ne0$ and we set $t_o:=b/a$. For any $t>0$, we let $$f(t):=\frac{ (1+t)^p-1-t^p }{ t }.$$ [F]{}rom (used here with $a:=1$ and $b:=t$), we know that $f(t)>0$ for any $t>0$. Moreover $$\lim_{t\to 0} f(t)=
\lim_{t\to 0} \frac{ 1 + pt +o(t)-1-t^p }{ t }=p,$$ hence $f$ can be continuously extended over $[0,+\infty)$ by setting $f(0):=p$. Furthermore, $$\begin{aligned}
&& \lim_{t\to +\infty} f(t)=
\lim_{t\to +\infty} t^{p-1}
\left( \left(\frac{1}{t}+1\right)^p-\frac{1}{t^p}-1\right)
\\&&\qquad=\lim_{t\to +\infty}
t^{p-1}
\left( 1+\frac{p}{t}+o\left(\frac{1}{t}\right)-\frac{1}{t^p}-1\right)
\\ &&\qquad=\lim_{t\to +\infty} t^{p-2}\left( p
+\frac{o\left(\frac{1}{t}\right)}{ \frac{1}{t} }
\right)-\frac{1}{t}
= \left\{ \begin{matrix}
p & {\mbox{ if }} p=2,\\
+\infty & {\mbox{ if }} p>2.
\end{matrix}\right.\end{aligned}$$ In any case, $$\lim_{t\to +\infty} f(t) {\geqslant}f(0)=p,$$ hence $$c_p:=\inf_{[0,+\infty)} f =\min_{[0,+\infty)} f >0.$$ As a consequence, $$\begin{aligned}
&& (a+b)^p- a^p -b^p - c_p\,a^{p-1} b\\
&=& a^p \big( (1+t_o)^p -1- t_o^p -c_p\, t_o\big)\\
&=& a^p t_o \big( f(t_o)-c_p\big)\\
&{\geqslant}& 0,\end{aligned}$$ as desired.
It is worth to stress that the result in Lemma \[ABC-2\] does not hold when $p\in(1,2)$, differently than what is often stated in the literature: as a counterexample, one can take $b=1$ and observe that $$\begin{aligned}
&& \lim_{a\to0} \frac{(a+b)^p- a^p -b^p}{a^{p-1} b}
=\lim_{a\to0} \frac{(a+1)^p- a^p -1}{a^{p-1}}\\
&& \qquad
=\lim_{a\to0} \frac{1+pa+o(a)- a^p -1}{a^{p-1}}
=\lim_{a\to0} pa^{2-p} + a^{1-p} o(a)- a=0\end{aligned}$$ when $p\in(1,2)$. In spite of this additional difficulty, when $p\in(1,2)$ one can obtain a variant of Lemma \[ABC-2\] under an additional assumption on the size of $b$. The precise statement goes as follows:
\[ABC-3\] Let $p\in(1,2)$ and $\kappa>0$. Then, there exists $c_{p,\kappa}>0$ such that, for any $a>0$, $b{\geqslant}0$, with $\frac{b}{a}\in[0,\kappa]$, we have $$(a+b)^p{\geqslant}a^p +b^p + c_{p,\kappa}\,a^{p-1} b.$$
The proof is a variation of the one of Lemma \[ABC-2\]. Full details are provided for the facility of the reader. We set $t_o:=\frac{b}{a}\in [0,\kappa]$. For any $t>0$, we let $$f(t):=\frac{ (1+t)^p-1-t^p }{ t }.$$ [F]{}rom (used here with $a:=1$ and $b:=t$), we know that $f(t)>0$ for any $t>0$. Moreover, $f$ can be continuously extended over $[0,+\infty)$ by setting $f(0):=p$. Therefore $$c_{p,\kappa}:=\min_{[0,\kappa]} f>0.$$ As a consequence, $$\begin{aligned}
&& (a+b)^p- a^p -b^p - c_{p,\kappa}\,a^{p-1} b\\
&=& a^p \big( (1+t_o)^p -1- t_o^p -c_{p,\kappa}\, t_o\big)\\
&=& a^p t_o \big( f(t_o)-c_{p,\kappa}\big)\\
&{\geqslant}& 0,\end{aligned}$$ as desired.
Now we consider the functional introduced in , deal with the path induced by the function $z$ in (suitably scaled and cut-off) and show that the associated mountain pass level for ${{\mathscrI}}^\star_\varepsilon$ lies below the critical threshold $\frac{s}{n} S^{\frac{n}{2s}}$ (see Proposition \[PScond2\]). The precise result goes as follows:
\[7cu889k9\] There exists $\mu_\star\in(0,\mu_0)$ such that if $\mu\in(0,\mu_\star)$ then we have $$\label{8tyunbvc666678}
\sup_{t{\geqslant}0} {{\mathscrI}}^\star_\varepsilon(t\bar Z_{\mu,\xi})<
\frac{s}{n} S^{\frac{n}{2s}}.$$
We will take $\mu_\star{\leqslant}\mu_1$, where $\mu_1>0$ was introduced in Lemma \[yu799\]. We also take $T_1$ as in Lemma \[yu799\]. Then, by , $$\label{8tyunbvc666678-pre}
\sup_{t{\geqslant}T_1}
\sup_{\mu\in(0,\mu_\star)}
{{\mathscrI}}^\star_\varepsilon(t\bar Z_{\mu,\xi})
{\leqslant}\sup_{t{\geqslant}T_1}
\sup_{\mu\in(0,\mu_1)}
{{\mathscrI}}^\star_\varepsilon(t\bar Z_{\mu,\xi})
{\leqslant}0
<\frac{s}{n} S^{\frac{n}{2s}}.$$ Consequently, we have that the claim in holds true if we prove that, for any $\mu\in(0,\mu_\star)$, $$\label{9ikUUYTGp}
\sup_{t\in[0,T_1]} {{\mathscrI}}^\star_\varepsilon(t\bar Z_{\mu,\xi})<
\frac{s}{n} S^{\frac{n}{2s}}.$$ To this goal, we set $$\label{CJ}
m:=\left\{
\begin{matrix}
2 & {\mbox{ if }} n>4s,\\
2^*_s-1 & {\mbox{ if }} n\in(2s,4s],
\end{matrix}
\right.$$ and $$\label{CJ-2}
\Omega:=\left\{
\begin{matrix}
B_{{2}{\sqrt{\mu}}}(\xi)\setminus B_{{\sqrt{\mu}}}(\xi)
& {\mbox{ if }} n>4s,\\
{{\mathbb R}}^n & {\mbox{ if }} n\in(2s,4s].
\end{matrix}
\right.$$ For further reference, we point out that, if $n\in(2s,4s]$, then $m-2=\frac{6s-n}{n-2s}>0$, and so $$\label{CJ-200}
{\mbox{$m-2{\geqslant}0$ for every $n>2s$.}}$$ We claim that, for any $t\in [0,T_1]$, any $\mu\in(0,\mu_\star)$ and any $x\in \Omega$, we have $$\label{FUND-ABC}
G^\star\big(x,t\bar\phi(x) z_{\mu,\xi}(x)\big){\geqslant}\frac{t^{2^*_s}\bar\phi^{2^*_s} (x)\,
z_{\mu,\xi}^{2^*_s}(x)}{2^*_s}+\frac{c\,
u_\varepsilon^{2^*_s-m}(x)
\,t^m\bar\phi^m (x)\,z_{\mu,\xi}^m(x)}{m},$$ for some $c>0$.
To prove it, we distinguish two cases, according to whether $n>4s$ or $n\in(2s,4s]$. If $n>4s$, we take $a:=u_\varepsilon(x)$ and $b{\geqslant}0$, with $b{\leqslant}t\bar\phi(x) z_{\mu,\xi}(x)$, and $x\in\Omega=B_{{2}{\sqrt{\mu}}}
(\xi)\setminus B_{{\sqrt{\mu}}}(\xi)$. Notice that, in this case, $$\label{FJghy67890hgfddfg}
a{\geqslant}\inf_{ B_{2\sqrt{\mu}}(\xi)\setminus B_{{\sqrt{\mu}}} (\xi) }
u_\varepsilon{\geqslant}\inf_{ B_{2}(\xi)}
u_\varepsilon {\geqslant}a_0,$$ for some $a_0>0$. Moreover, from , $$b{\leqslant}t z_{\mu,\xi}(x) =
t\mu^{-\frac{n-2s}{2}} z\left( \frac{x-\xi}{\mu}\right)
= \frac{c_\star t\mu^{-\frac{n-2s}{2}} }{ \left( 1+
\left|\frac{x-\xi}{\mu}\right|^2
\right)^{\frac{n-2s}{2}}}
= \frac{c_\star t\mu^{\frac{n-2s}{2}} }{ \left( \mu^2+
|x-\xi|^2\right)^{\frac{n-2s}{2}}}
.$$ Since $x\in B_{2\sqrt{\mu}}(\xi)\setminus B_{{\sqrt{\mu}}} (\xi)$, we obtain that $|x-\xi|{\geqslant}{\sqrt{\mu}}$ and so $$b{\leqslant}\frac{c_\star t\mu^{\frac{n-2s}{2}} }{ \left( \mu^2+
\mu\right)^{\frac{n-2s}{2}}} {\leqslant}\frac{c_\star t\mu^{\frac{n-2s}{2}} }{
\mu^{\frac{n-2s}{2}}} {\leqslant}c_\star T_1.$$ [F]{}rom this and we obtain that $b/a{\leqslant}\kappa$, for some $\kappa>0$, hence we can apply Lemma \[ABC-3\] and obtain that $$\begin{aligned}
G^\star\big(x,t\bar\phi(x) z_{\mu,\xi}(x)\big)
&=&\int_0^{t\bar\phi(x) z_{\mu,\xi}(x)} \big[(u_{\varepsilon }(x)+b)^p-u_{\varepsilon }^p(x)\big]\,db
\\&=&
\int_0^{t\bar\phi(x) z_{\mu,\xi}(x)} \big[ (a+b)^p-a^p\big]\,db
\\ &{\geqslant}& \int_0^{t\bar\phi(x) z_{\mu,\xi}(x)} \big[
b^p +c_{p,\kappa} a^{p-1}b\big]\,db
\\ &=& \frac{
\big( t\bar\phi(x) z_{\mu,\xi}(x)\big)^{p+1} }{p+1}
+ c_{p,\kappa} u_{\varepsilon }^{p-1}(x)\, \frac{\big(
t\bar\phi(x) z_{\mu,\xi}(x) \big)^2}{2}.\end{aligned}$$ This and complete the proof of when $n>4s$ (recall that $p+1=2^*_s$).
Now we prove when $n\in(2s,4s]$. In this case, we observe that $$p=\frac{n+2s}{n-2s}{\geqslant}2.$$ So we choose $a{\geqslant}0$, with $a{\leqslant}t\bar\phi(x) z_{\mu,\xi}(x)$, and $b:=u_\varepsilon(x)$, and we can use Lemma \[ABC-2\] to obtain that $$\begin{aligned}
G^\star\big(x,t\bar\phi(x) z_{\mu,\xi}(x)\big)
&=&\int_0^{t\bar\phi(x) z_{\mu,\xi}(x)} \big[
(u_{\varepsilon }(x)+a)^p-u_{\varepsilon }^p(x)\big]\,da
\\&=&
\int_0^{t\bar\phi(x) z_{\mu,\xi}(x)} \big[
(a+b)^p-b^p\big]\,da
\\ &{\geqslant}& \int_0^{t\bar\phi(x) z_{\mu,\xi}(x)} \big[
a^p +c_{p} a^{p-1}b\big]\,da
\\ &=&
\frac{ \big( t\bar\phi(x) z_{\mu,\xi}(x)\big)^{p+1} }{p+1}
+c_p \frac{ \big( t\bar\phi(x) z_{\mu,\xi}(x)\big)^{p} }{p}\,
u_{\varepsilon }(x).\end{aligned}$$ This and imply when $n\in(2s,4s]$. With this, we have completed the proof of .
Now we claim that, for any $t\in [0,T_1]$, any $\mu\in(0,\mu_\star)$ and any $x\in {{\mathbb R}}^n$, $$\label{FUND-ABC-2}
G^\star\big(x,t\bar\phi(x) z_{\mu,\xi}(x)\big){\geqslant}\frac{t^{2^*_s}\bar\phi^{2^*_s} (x)\,
z_{\mu,\xi}^{2^*_s}(x)}{2^*_s}
.$$ We remark that is a stronger inequality than , but only holds in $\Omega$, while holds in the whole of ${{\mathbb R}}^n$ (this is an advantadge in the case $n>4s$, according to ). To prove , we use Lemma \[ABC-1\], with $a:=u_{\varepsilon }(x)$ and $b{\geqslant}0$, to see that $$\begin{aligned}
G^\star\big(x,t\bar\phi(x) z_{\mu,\xi}(x)\big)
&=&\int_0^{t\bar\phi(x) z_{\mu,\xi}(x)} \big[(u_{\varepsilon }(x)+b)^p-u_{\varepsilon }^p(x)\big]\,db
\\&=&
\int_0^{t\bar\phi(x) z_{\mu,\xi}(x)}\big[ (a+b)^p-a^p\big]\,db
\\ &{\geqslant}& \int_0^{t\bar\phi(x)z_{\mu,\xi}(x)} b^p\,db
\\ &=& \frac{ \big( t\bar\phi(x)\,z_{\mu,\xi}(x)\big)^{p+1} }{p+1},\end{aligned}$$ and this establishes .
By combining and , we obtain that $$\label{FUNF-78h}
\begin{split}
& \int_{{{\mathbb R}}^n} G^\star\big(x,t\bar\phi(x) z_{\mu,\xi}(x)\big)\,dx\\
&\qquad=
\int_{{{\mathbb R}}^n\setminus\Omega} G^\star\big(x,t\bar\phi(x) z_{\mu,\xi}(x)\big)\,dx
+
\int_{\Omega} G^\star\big(x,t\bar\phi(x) z_{\mu,\xi}(x)\big)\,dx\\
&\qquad{\geqslant}\int_{{{\mathbb R}}^n\setminus\Omega}
\frac{t^{2^*_s}\bar\phi^{2^*_s} (x)\,
z_{\mu,\xi}^{2^*_s}(x)}{2^*_s}\,dx
\\ &\qquad\qquad+\int_{\Omega}
\frac{t^{2^*_s}\bar\phi^{2^*_s} (x)\,
z_{\mu,\xi}^{2^*_s}(x)}{2^*_s}+\frac{c\,
u_\varepsilon^{2^*_s-m}(x)
\,t^m\bar\phi^m (x)\,z_{\mu,\xi}^m(x)}{m}\,dx
\\ &\qquad{\geqslant}\frac{t^{2^*_s}}{2^*_s}
\int_{{{\mathbb R}}^n} \bar\phi^{2^*_s} (x)\,z_{\mu,\xi}^{2^*_s}(x)\,dx
+ \frac{c\,t^m}{m}
\int_{\Omega} u_\varepsilon^{2^*_s-m}(x)
\,\bar\phi^m (x)\,z_{\mu,\xi}^m(x) \,dx.
\end{split}$$ Now we claim that $$\label{89-0fgUIjjk906}
\int_{\Omega} u_\varepsilon^{2^*_s-m}(x)
\,\bar\phi^m (x)\,z_{\mu,\xi}^m(x) \,dx {\geqslant}c' \mu^\beta,$$ for some $c'>0$, where $$\label{CJ-beta}
\beta:=\left\{
\begin{matrix}
\displaystyle\frac{n}{2} & {\mbox{ if }} n>4s,\\
\, \\
\displaystyle\frac{n-2s}{2} & {\mbox{ if }} n\in(2s,4s],
\end{matrix}
\right.$$ To prove this, when $n>4s$ we remark that, for small $\mu$, we have $B_{{2}{\sqrt{\mu}}}(\xi)\subseteq B_{\mu_0/2}(\xi)$, and $\bar\phi=1$ in this set, due to . So, we use and and we have that $$\begin{aligned}
&& \int_{\Omega} u_\varepsilon^{2^*_s-m}(x)
\,\bar\phi^m (x)\,z_{\mu,\xi}^m(x) \,dx
=
\int_{B_{{2}{\sqrt{\mu}}}(\xi)\setminus B_{{\sqrt{\mu}}}(\xi)}
u_\varepsilon^{2^*_s-2}(x)\,z_{\mu,\xi}^2(x) \,dx
\\ &&\qquad{\geqslant}\inf_{B_2(\xi)} u_\varepsilon^{2^*_s-2}
\,\int_{B_{{2}{\sqrt{\mu}}}(\xi)\setminus B_{{\sqrt{\mu}}}(\xi)}
z_{\mu,\xi}^2(x) \,dx
\\ &&\qquad=
\inf_{B_2(\xi)} u_\varepsilon^{2^*_s-2}
\,\mu^{-(n-2s)}\,\int_{B_{{2}{\sqrt{\mu}}}(\xi)\setminus B_{{\sqrt{\mu}}}(\xi)}
z^2\left(\frac{x-\xi}{\mu}\right) \,dx
\\&&\qquad=
\inf_{B_2(\xi)} u_\varepsilon^{2^*_s-2}
\,\mu^{2s}\,\int_{B_{\frac{2}{\sqrt{\mu}}}
\setminus B_{\frac{1}{\sqrt{\mu}}}}
z^2(y)\,dy.\end{aligned}$$ Thus, recalling and taking $\mu$ suitably small, we have that $$\begin{aligned}
&& \int_{\Omega} u_\varepsilon^{2^*_s-m}(x)
\,\bar\phi^m (x)\,z_{\mu,\xi}^m(x) \,dx
{\geqslant}c_1 \mu^{2s}\int_{1/\sqrt{\mu}}^{2/\sqrt{\mu}}
\frac{\rho^{n-1}d\rho}{(1+\rho^2)^{n-2s}}
\\ &&\qquad
{\geqslant}c_1 \mu^{2s}\int_{1/\sqrt{\mu}}^{2/\sqrt{\mu}}
\frac{\rho^{n-1}d\rho}{(2\rho^2)^{n-2s}}
= c_2 \mu^{\frac{n}{2}},\end{aligned}$$ for some $c_1$, $c_2>0$. This proves when $n>4s$.
Now we prove when $n\in(2s,4s]$. For this, we exploit and and we observe that $$\begin{aligned}
&& \int_{\Omega} u_\varepsilon^{2^*_s-m}(x)\,\bar{\phi}^m(x)\, z_{\mu,\xi}^m(x)\,dx
=
\int_{{{\mathbb R}}^n} u_\varepsilon (x)\,\bar{\phi}^{2^*_s-1}(x)\, z_{\mu,\xi}^{2^*_s-1}(x)\,dx
\\ &&\qquad
{\geqslant}\mu^{-\frac{n+2s}{2}}
\int_{B_{2\sqrt{\mu}}(\xi)} u_\varepsilon(x)\,
z^p \left(\frac{x-\xi}{\mu}\right)\,dx
\\ &&\qquad{\geqslant}\mu^{-\frac{n+2s}{2}}
\,\inf_{B_1(\xi)}u_\varepsilon
\,\int_{B_{2\sqrt{\mu}}(\xi)}
z^p\left(\frac{x-\xi}{\mu}\right)\,dx
\\ &&\qquad=
\mu^{\frac{n-2s}{2}}
\,\inf_{B_1(\xi)}u_\varepsilon\,
\int_{B_1}
z^p(y)\,dy
\\ &&\qquad{\geqslant}c' \mu^{\frac{n-2s}{2}},\end{aligned}$$ for some $c'>0$, which establishes when $n\in(2s,4s]$. The proof of is thus complete.
Now, by inserting into , we obtain that $$\label{FUNF-78hIII}
\int_{{{\mathbb R}}^n} G^\star\big(x,t\bar\phi(x) z_{\mu,\xi}(x)\big)\,dx{\geqslant}\frac{t^{2^*_s}}{2^*_s}
\int_{{{\mathbb R}}^n} \bar\phi^{2^*_s} (x)\,z_{\mu,\xi}^{2^*_s}(x)\,dx
+ \frac{c\,t^m\,\mu^\beta}{m},$$ for some $c>0$, up to renaming constants.
By Lemma \[LE:NU90\] and , we conclude that $$\int_{{{\mathbb R}}^n} G^\star\big(x,t\bar\phi(x) z_{\mu,\xi}(x)\big)\,dx{\geqslant}\frac{t^{2^*_s}}{2^*_s}
\int_{{{\mathbb R}}^n} \,z_{\mu,\xi}^{2^*_s}(x)\,dx
+ \frac{c\,\mu^\beta\,t^m}{m}-\frac{C\mu^{n}\,t^{2^*_s}}{2^*_s}.$$ This and give that $$\int_{{{\mathbb R}}^n} G^\star(x,t\bar\phi z_{\mu,\xi})\,dx {\geqslant}\frac{t^{2^*_s} \,S^{\frac{n}{2s}}}{2^*_s}+
\frac{c\,\mu^\beta\,t^m}{m}
-\frac{C \mu^n \,t^{2^*_s}}{2^*_s},$$ for some $c$, $C>0$. As a consequence, recalling , we obtain that $$\begin{aligned}
{{\mathscrI}}^\star_\varepsilon(t\bar Z_{\mu,\xi}) &{\leqslant}&
\frac{t^2\,[\bar Z_{\mu,\xi}]_a^2}{2}
-
\frac{t^{2^*_s} \,S^{\frac{n}{2s}}}{2^*_s}-
\frac{c\,\mu^\beta\,t^m}{m}
+\frac{C \mu^n \,t^{2^*_s}}{2^*_s}
\\ &{\leqslant}&
\frac{t^{2} \,S^{\frac{n}{2s}}}{2}
-
\frac{t^{2^*_s} \,S^{\frac{n}{2s}}}{2^*_s}-
\frac{c\,\mu^\beta\,t^m}{m}
+\frac{C \mu^n \,t^{2^*_s}}{2^*_s}
+\frac{C \mu^{n-2s} \,t^{2}}{2},\end{aligned}$$ and so, up to renaming constants, $$\label{8dfuihgs019}
{{\mathscrI}}^\star_\varepsilon(t\bar Z_{\mu,\xi})
{\leqslant}S^{\frac{n}{2s}} \Psi(t),$$ with $$\Psi(t):=
\frac{t^{2}}{2}
-\frac{t^{2^*_s} }{2^*_s}-
\frac{c\,\mu^\beta\,t^m}{m}
+\frac{C \mu^n \,t^{2^*_s}}{2^*_s}
+\frac{C \mu^{n-2s} \,t^{2}}{2},$$ for some $C$, $c>0$.
Now we claim that $$\label{PSIMA}
\sup_{t{\geqslant}0}\Psi(t)<\frac{s}{n}
,$$ provided that $\mu>0$ is suitably small. To check this, we notice that $\Psi(0)=0$ and $$\lim_{t\to+\infty}\Psi(t)=-\infty,$$ since $2^*_s>\max\{2,m\}$ (recall ). As a consequence, $\Psi$ attains its maximum at some point $T\in[0,+\infty)$. If $T=0$, then $\Psi(T)=0$ and is obvious, so we can assume that $T\in(0,+\infty)$. Accordingly, we have that $\Psi'(T)=0$. Therefore $$0=\frac{\Psi'(T)}{T}=
1-T^{2^*_s-2}-
c\,\mu^\beta\,T^{m-2}
+C \mu^n \,T^{2^*_s-2}
+C \mu^{n-2s}.$$ So we set $$\Phi_\mu(t):=
1-t^{2^*_s-2}-
c\,\mu^\beta\,t^{m-2}
+C \mu^n \,t^{2^*_s-2}
+C \mu^{n-2s}$$ and we have that $T=T(\mu)$ is a solution of $\Phi_\mu(T)=0$. We remark that $$\Phi_\mu'(t)=
-(2^*_s-2)(1-C \mu^n)t^{2^*_s-3}-
c\,\mu^\beta\,(m-2) t^{m-3} <0,$$ since $m-2{\geqslant}0$ and $(2^*_s-2)(1-C \mu^n){\geqslant}0$ for small $\mu$ (recall ). This says that $\Phi_\mu$ is strictly decreasing, hence $T=T(\mu)$ is the unique solution of $\Phi_\mu(T(\mu))=0$. It is now convenient to write $\tau(\mu):=T(\mu^{\frac{1}{\beta}})$ and $\eta:=\mu^\beta$, so that our equation becomes $$\begin{aligned}
&& 0 =\Phi_\mu(T(\mu)) = \Phi_\mu (\tau(\mu^\beta))=\Phi_\mu(\tau(\eta))
\\ &&\qquad=
1-(1-C \eta^{\frac{n}{\beta}})(\tau(\eta))^{2^*_s-2}-
c\,\eta\,(\tau(\eta))^{m-2}
+C \eta^{\frac{n-2s}{\beta}}.\end{aligned}$$ Accordingly, if we differentiate in $\eta$, we have that $$\label{0syufertyuiuytrgb}\begin{split}
& 0=\frac{\partial}{\partial \eta}\left(
1-(1-C \eta^{\frac{n}{\beta}})(\tau(\eta))^{2^*_s-2}-
c\,\eta\,(\tau(\eta))^{m-2}
+C \eta^{\frac{n-2s}{\beta}}
\right) \\
&\qquad=
-(2^*_s-2)(1-C \eta^{\frac{n}{\beta}})(\tau(\eta))^{2^*_s-3}\tau'(\eta)
+C\,\frac{n}{\beta} \eta^{\frac{n}{\beta}-1}
(\tau(\eta))^{2^*_s-2}
\\ &\qquad\qquad-
c\,(\tau(\eta))^{m-2}
-
c(m-2)\,\eta\,(\tau(\eta))^{m-3}\tau'(\eta)
+{\frac{C(n-2s)}{\beta}} \eta^{\frac{n-2s}{\beta}-1}.
\end{split}$$ Now we claim that $$\label{BETA78HGV}
\frac{n-2s}{\beta}-1 >0.$$ Indeed, using , we see that $$\begin{aligned}
\frac{n-2s}{\beta}-1 &=&
\left\{
\begin{matrix}
\displaystyle\frac{2(n-2s)}{n}-1 & {\mbox{ if }} n>4s,\\
\, \\
2-1 & {\mbox{ if }} n\in(2s,4s],
\end{matrix}\right.\\
&=&
\left\{
\begin{matrix}
\displaystyle\frac{n-4s}{n} & {\mbox{ if }} n>4s,\\
\, \\
1 & {\mbox{ if }} n\in(2s,4s],\end{matrix}\right.\end{aligned}$$ which proves .
Now we observe that when $\mu=0$, we have that $T=1$ is a solution of $\Phi_0(t)=0$, i.e. $T(0)=1$ and so $\tau(0)=1$. Hence, we evaluate at $\eta=0$ and we conclude that $$0=
-(2^*_s-2)\tau'(0) - c.$$ We remark that was used here. Then, we obtain $$\tau'(0)=-\frac{c}{2^*_s-2},$$ which gives that $$\tau(\eta) =1-\frac{c\eta}{2^*_s-2} + o(\eta)$$ and so $$T(\mu)=\tau(\mu^\beta)=1-\frac{c\mu^\beta}{2^*_s-2} + o(\mu^\beta)=
1-c_o\mu^\beta+ o(\mu^\beta),$$ for some $c_o>0$. Therefore $$\begin{aligned}
&& \sup_{t{\geqslant}0}\Psi(t) =\Psi(T(\mu))\\
&&\qquad =
(1+C \mu^{n-2s})\frac{(T(\mu))^{2}}{2}
-(1-C \mu^n)\frac{(T(\mu))^{2^*_s} }{2^*_s}-
\frac{c\,\mu^\beta\,(T(\mu))^m}{m}\\
&&\qquad=
(1+C \mu^{n-2s})\frac{(1-c_o\mu^\beta+ o(\mu^\beta))^{2}}{2}
-(1-C \mu^n)\frac{(1-c_o\mu^\beta+ o(\mu^\beta))^{2^*_s} }{2^*_s}
\\ &&\qquad\qquad-
\frac{c\,\mu^\beta\,(1-c_o\mu^\beta+ o(\mu^\beta))^m}{m}\\
&&\qquad=
(1+C \mu^{n-2s})\frac{1-2c_o\mu^\beta}{2}
-(1-C \mu^n)\frac{1-2^*_sc_o\mu^\beta}{2^*_s}-
\frac{c\,\mu^\beta}{m}
+ o(\mu^\beta)\\
&&\qquad=
\frac{1-2c_o\mu^\beta}{2}
-\frac{1-2^*_sc_o\mu^\beta}{2^*_s}-
\frac{c\,\mu^\beta}{m}
+ o(\mu^\beta)
\\ &&\qquad=
\frac{1}{2}-\frac{1}{2^*_s}
-\frac{c\,\mu^\beta}{m}
+ o(\mu^\beta)
\\ &&\qquad<
\frac{1}{2}-\frac{1}{2^*_s}
\\ &&\qquad=\frac{s}{n}
,\end{aligned}$$ and this proves .
Using and , we obtain that $$\sup_{t\in[0,T_1]}
{{\mathscrI}}^\star_\varepsilon(t\bar Z_{\mu,\xi})
{\leqslant}S^{\frac{n}{2s}} \sup_{t{\geqslant}0} \Psi(t)<
S^{\frac{n}{2s}}\cdot\frac{s}{n},$$ which proves and so it completes the proof of Lemma \[7cu889k9\].
By combining with Lemmata \[yu799\] and \[7cu889k9\], we obtain:
\[CC-below-PS\] There exists $\mu_\star>0$ such that if $\mu\in(0,\mu_\star)$ we have that $$\begin{aligned}
&& \lim_{t\to+\infty}
{{\mathscrI}}_\varepsilon(t\bar Z_{\mu,\xi}) =-\infty
\\{\mbox{and }}&&
\sup_{t{\geqslant}0}
{{\mathscrI}}_\varepsilon(t\bar Z_{\mu,\xi})<
\frac{s}{n} S^{\frac{n}{2s}}.\end{aligned}$$
The result in Corollary \[CC-below-PS\] says that the path induced by the function $\bar Z_{\mu,\xi}$ is a mountain pass path which lies below the critical threshold given in Proposition \[PScond2\] (so, from now on, the value of $\mu\in(0,\mu_\star)$ will be fixed so that Corollary \[CC-below-PS\] holds true).
Proof of Theorem \[TH:MP\] {#sec:proof}
--------------------------
In this section we establish Theorem \[TH:MP\]. For this, we argue by contradiction and we suppose that $U=0$ is the only critical point of ${{\mathscrI}}_{\varepsilon }$. As a consequence, the functional ${{\mathscrI}}_{\varepsilon }$ verifies the Palais-Smale condition below the critical level given in , according to Proposition \[PScond2\].
In addition, ${{\mathscrI}}_{\varepsilon }$ fulfills the mountain pass geometry, and the minimax level $c_{\varepsilon }$ stays strictly below the level $\frac{s}{n}S^{\frac{n}{2s}}$, as shown in Proposition \[prop:zero\] and Corollary \[CC-below-PS\].
Hence, for small ${\varepsilon }$, we have that $
c_{\varepsilon }+C {\varepsilon }^{\frac{1}{2\gamma}}$ remains strictly below $
\frac{s}{n}S^{\frac{n}{2s}}$, thus satisfying .
Then, exploiting Proposition \[PScond2\] and the Mountain Pass Theorem in [@AmRab; @GG], we obtain the existence of another critical point, in contradiction with the assumption. This ends the proof of Theorem \[TH:MP\].
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[^1]: We observe that this pathology does not occur for functions in ${{\mathbb R}}$, since, in one variable, the conditions ${{{\mathscrE}}}(0)=0$, ${{{\mathscrE}}}(r)=a>0$ and ${{{\mathscrE}}}(p){\leqslant}0$ imply the existence of another critical point, by Rolle’s Theorem.
[^2]: As a technical observation, we stress that papers like [@NEW1] and [@vecchio] deal with a bifurcation method from a ground state solution that is already known to be non-degenerate, while in the present monograph we find: (i) solutions that bifurcate from zero, (ii) different solutions obtained by a contradiction argument (e.g., if no other solutions exist, then one gains enough compactness to find a mountain pass solution).
Notice that this second class of solutions is not necessarily of mountain pass type, due to the initial contradictory assumption.
In particular, the methods of [@NEW1] and [@vecchio] cannot be applied in the framework of this monograph. An additional difficulty in our setting with respect to [@NEW1] is that we deal also with a subcritical power in the nonlinearity which makes the functional not twice differentiable and requires a different functional setting.
[^3]: Some of the results presented here are valid for more general families of weights, in the setting of Muckenhoupt classes. Nevertheless, we focused on the monomial weights both for the sake of concreteness and simplicity, and because some more general results follow in a straightforward way from the ones presented here. With this respect, for further comments that compare monomial and Muckenhoupt weights, see the end of Section 1 in [@CR].
[^4]: We remark that many of the techniques presented in this part have a very general flavor and do not rely on variational principles, so they can be possibly applied to integrodifferential operators with general kernels and to nonlinear operators with suitable growth conditions, and the method can interplay with tools for viscosity solutions. Nevertheless, rather than trying to exhaust the many possible applications of this theory in different frameworks, which would require a detailed list of cases and different assumptions, we presented here the basic theory just for the fractional Laplacian, both for the sake of simplicity and because in the problem considered in the main results of this monograph (such as Theorems \[TH1\], \[MINIMUM\] and \[TH:MP\]) we treat an equation arising from the variational energy introduced in (hence the general fully nonlinear case cannot be comprised at that level) and extension methods will be exploited in Chapter \[EMP:CHAP\] (recall Section \[sec:ext\]: the case of general integrodifferentiable kernels is not comprised the variational principle discussed there).
|
---
abstract: '[ We demonstrate the utility of the equation-free methodology developed by one of the authors (I.G.K) for the study of scalar conservation laws with disordered initial conditions. The numerical scheme is benchmarked on exact solutions in Burgers turbulence corresponding to process initial data. For these initial data, the kinetics of shock clustering is described by Smoluchowski’s coagulation equation with additive kernel. The equation-free methodology is used to develop a particle scheme that computes self-similar solutions to the coagulation equation, including those with fat tails. ]{}'
author:
- 'Xingjie Li [^1],'
- 'Matthew O. Williams [^2],'
- 'Ioannis G. Kevrekidis [^3],'
- 'Govind Menon [^4]'
bibliography:
- 'lkmw.bib'
title: 'Coarse graining, dynamic renormalization and the kinetic theory of shock clustering'
---
[*Keywords*]{}: Dynamic scaling, equation free method, Smoluchowski’s coagulation equation, sticky particles, Burgers turbulence.
Acknowledgements
================
One of the authors (I.G.K.) would like to remember here Stephen A. Orszag, who suggested this very problem as a challenge for equation free methods a decade ago.
The authors also acknowledge support from the following funding agencies. M.O.W. acknowledges support from NSF DMS 1204783. I.G.K. acknowledges partial support from US-AFOSR through grant number FA9550-12-1-0332 and NSF grant CDSE 1310173. G.M. acknowledges partial support from NSF DMS 1411278.
[^1]: Division of Applied Mathematics, Brown University, 182 George St., Providence, RI 02912, USA. Email: [xingjie\[email protected]]{}.
[^2]: Program in Applied and Computational Mathematics, Princeton University, Fine Hall, Washington Rd., Princeton, NJ 08544, USA. Email: [[email protected]]{}.
[^3]: Department of Chemical and Biological Engineering, Princeton University, 6 Olden St., Princeton, NJ 08544, USA. Email: [[email protected]]{}.
[^4]: Division of Applied Mathematics, Brown University, 182 George St., Providence, RI 02912, USA, Email: [govind\[email protected]]{}.
|
---
abstract: 'Increasing the collection efficiency from solid-state emitters is an important step towards achieving robust single photon sources, as well as optically connecting different nodes of quantum hardware. A metallic substrate may be the most basic method of improving the collection of photons from quantum dots, with predicted collection efficiency increases of up to 50%. The established ‘method-of-images’ approach models the effects of a reflective surface for atomic and molecular emitters by replacing the metal surface with a second fictitious emitter which ensures appropriate electromagnetic boundary conditions. Here, we extend the approach to the case of driven solid-state emitters, where exciton-phonon interactions play a key role in determining the optical properties of the system. We derive an intuitive polaron master equation and demonstrate its agreement with the complementary half-sided cavity formulation of the same problem. Our extended image approach offers a straightforward route towards studying the dynamics of multiple solid-state emitters near a metallic surface.'
author:
- Dale Scerri
- 'Ted S. Santana'
- 'Brian D. Gerardot'
- 'Erik M. Gauger'
bibliography:
- 'references.bib'
title: 'Method of images applied to driven solid-state emitters'
---
=1
\[intro\]Introduction
=====================
The problem of a dipole emitter placed close to a reflective surface has received much interest over the last few decades: seminal work [@DREXHAGE] by Drexhage in 1970 first demonstrated that a reflective interface modifies the intrinsic properties of the emitter, influencing both the emission frequency [@frequencyshift; @Morawitz1969] and the emitter’s excited lifetime [@Morawitz1969; @babiker; @babiker2; @babiker:superradiance; @ficek2005quantum]. Recently, a sound analogue of Drexhage’s experiment has been performed to study the acoustic frequency shifts of a gong struck near a hard wall [@acoustic].
Mirrors have widespread use for directing light from sources that emit across a extended solid angle, for example in the form parabolic reflectors in everyday light sources. On the nanoscale, precise guiding of photons into particular optical modes is of paramount importance for quantum information processing and communication, where on demand single photons are required [@KLM; @SinglePhotonTransport1; @SinglePhotonTransport2; @SinglePhotonTransport3]. Although micron-sized spherical mirrors for open access microcavities [@JasonSmith] have recently enabled the investigation of quantum dot–cavity systems in the strong coupling regime [@Warburton; @Imamoglu], the use of sophisticated mirrors remains a challenge for solid-state quantum emitters that are often embedded in heterogenous layers of substrates with varying refractive indices. This motivates the more straightforward alternative of increasing the photon collection efficiency by placing the emitter above a planar mirroring interface [@SurfaceEnhancedRef1; @SurfaceEnhancedRef2; @SurfaceEnhancedRef3]. Interestingly, the presence of even such a simple mirror also affects the physical properties of the emitter, as discussed above.
In recent years, progress in the synthesis and control of solid-state emitters has enabled experimental investigation of these modified properties of condensed-state emitters including quantum dots (QDs) [@nanowire; @gerardot] as well as perovskite [@imageexcitons] and transition metal dichalcogenide monolayers [@MonolayerRef] deposited on reflective surfaces. Circuit QED analogues of an atom and a variable mirror have also been successfully implemented [@circuit_qed; @circuit_qed2]; these offer the advantage of increased control over the artificial atom’s interaction with the mirror. With improved atom-mirror coupling, Hoi *et al.* managed to collect over $99\%$ of the radiation by coupling a transmon microwave emitter to a 1D superconducting waveguide [@circuit_qed].
![Artistic rendition of a driven quantum dot (QD), depicted as a cyan spheroid, in the proximity of a golden metallic surface. The corresponding ‘image dot’ is shown blurred on the other side ‘below’ of the semiconductor-gold interface. The optical dipoles are depicted as ‘dumbbells’ within the QDs. The vertical red beam represents the laser driving, and the magenta spiralling arrows indicate scattered photons.[]{data-label="blender"}](artistic.pdf){width="\linewidth"}
Several theoretical investigations [@frequencyshift; @Morawitz1969; @babiker; @ficek2005quantum] have shown that an atomic two-level system (TLS) near a reflective surface can be modelled as a pair of emitters: the real one as well as an identical emitter that is placed equidistant from, but on the opposite side of, the interface (see Figs. \[blender\] and \[schematic\]). The basic idea follows that of the electrostatics concept of an image charge to capture the surface charge distribution that ensures meeting the electric field boundary conditions [@jackson]. In the optical case, the ‘method of images’ relies on considering the emission from the combined dipole-image system. This yields the same expression for the modified spontaneous emission (SE) rate which one obtains from a full QED treatment (employing surface-dependent response functions to arrive at the modifications to the emitter’s lifetime and transition frequency)[@agarwal3]. The image dipole treatment has also been applied to model the surface-induced modifications of more complex structures such as molecules [@sers; @barnesreview], multiple dipole emitters [@George1985; @imagedipoleold2; @sanders] and solid state-emitters [@nanowire; @imageexcitons]. To date, however, the latter have largely ignored the vibrational solid state environment and the continuous wave (cw) laser driving typical of a resonance fluorescence (RF) setting.
Motivated by these successes, we here present a full image dipole polaron master equation (ME) treatment of a driven TLS (such as, e.g., a quantum dot) in the proximity of a metal surface (see Fig. \[blender\]). Our calculations extend previous image dipole studies as follows: (i) we consider driven systems, showing how to incorporate a laser driving term into the dipole and image Hamiltonian; (ii) we discuss the need for introducing an additional ‘selection rule’ to prevent unphysical double excitation; (iii) we demonstrate how a solid-state phonon environment can be accounted for – via a single bosonic bath that is perfectly correlated across the real emitter and its image.
We will show that the resulting master equation model remains highly intuitive and possesses appealing simplicity. We establish the correctness of this model by comparing its results to those obtained from an alternative calculation which does not involve fictitious entities or rely on ad-hoc assumptions: the half-sided cavity model. This agreement gives us confidence that the model could also be extended to the case of multiple solid-state emitters near a reflective surface, laying the groundwork for the investigation of collective effects in this setting, where we believe that an image approach will be easier to deploy than both the Green’s function and the half-sided cavity approach.
This Article is organised as follows: We will start by briefly summarising the results from the established Green’s function method for calculating the SE rate of a ‘bare’ dipole emitter. Next, we shall derive a ME for the emitter by treating the metal surface as a half-sided Fabry–P[é]{}rot cavity, providing the benchmark model for a single TLS near the metal surface (see Fig. \[schematic\]a). Finally, we formulate the ME using the method of images (see Fig. \[schematic\]b). We show that, with suitable alterations, the two-body ME reduces to an effective two level system with rates and energy shifts agreeing with the cavity model. Finally, we put our model to use to obtain the RF spectrum of the modified system, featuring a phonon sideband, the Mollow triplet, and the ratio of coherently to incoherently scattered light.
\[Green\]Green’s function approach: Brief summary
=================================================
We begin by summarising the main results of the Green’s function approach for modelling the optical environment of a dipole emitter. This can be applied to obtain the SE rate of an emitter in free space [@principles] as well as in the presence of a metallic surface [@principles; @green; @babiker2]. Whilst this approach gives a closed analytical solution for the case of a single dipole, a numerical route has to be taken to model a system comprised of a larger number of emitters [@sanders; @principles], even in the absence of a driving field and phonon-environments. Therefore, we here limit the discussion to a single ‘bare’ emitter as an independent reference point for the SE rate (and energy shift) in that idealised configuration.
Let the dipole be situated at position $\mathbf{r}_d$, where $\mathbf{r}_d$ is perpendicular to a metal surface containing the origin of the coordinate system. In the Green’s function approach, the emitter is usually modelled as a classical dipole oscillating harmonically with amplitude $\mathbf{x}$ at frequency $\omega_0$ about $\mathbf{r}_d$ [@sanders]. In vacuum, the SE rate can be calculated as $$\label{SEGreen}
\gamma^{pt}_0(\omega_0) = \frac{4 \omega^2_0}{ \pi \epsilon_0 \hbar c^2}\left[ \hat{\mathbf{d}} \cdot \mathrm{Im}\{ \mathbf{G}(\mathbf{r}_d, \mathbf{r}_d; \omega_0) \} \cdot \hat{\mathbf{d}} \right]~,$$ where $\epsilon_0$ is the electric permittivity of vacuum, $c$ is the speed of light, $\hat{\mathbf{d}}$ is a unit vector indicating the direction of the emitter’s dipole moment, and $\mathbf{G}(\mathbf{r}_d, \mathbf{r}_d; \omega_0)$ is the Fourier transform of the dyadic Green’s function at the emitter’s position [@principles]. In Ref. [@sanders], Choquette *et al.* studied the the collective decay rate of $N$ such classical emitters near a planar interface, arriving at a diagonal Green’s function matrix, so that Eq. allows one to find the SE rate for arbitrary dipole orientations.
To obtain the SE rate in a dielectric environment, we consider the following expression for the normalised dissipated power: $$\label{power}
\frac{P}{P_0} = 1 + \frac{6 \pi \epsilon_0 \epsilon_r}{|\mathbf{d}|^2 k^3} \mathrm{Im} \{ \mathbf{d}^* \cdot \mathbf{E}_s(\mathbf{r}_d) \} ~,$$ where $P_0$ is rate of energy dissipation in free space, $\epsilon_r$ and $k$ are the relative permittivity and wave vector magnitude in the dielectric surrounding the emitter, respectively, and $\mathbf{E}_s(\mathbf{r}_d)$ is the scattered electric field at the dipole’s position (which, for a single dipole near the surface, corresponds to the reflected field) [@principles]. The connection between the Green’s function and the decay rate of the dipole emitter is established via the relationship $$\frac{P}{P_0} = \frac{\gamma^{pt}(\omega_0)}{\gamma^{pt}_0(\omega_0)} ~.$$ Rearranging the above then yields an integral expression for the desired SE rate $\gamma^{pt}(\omega_0)$.
We note that the Green’s function method is not limited to ideal metallic interfaces but can also be applied straightforwardly to reflective dielectric interfaces, simply by substituting appropriate dielectric constants into the above relevant expressions [@principles]. In this case, one obtains qualitatively very similar results for a dielectric mirror, especially at larger separations [@principles]. Whilst the method of images fundamentally relies on the assumption of a perfectly conducting interface, it is fair to assume its qualitative predictions will by analogy also carry across to the case of dielectric mirrors.
\[halfcavity\]Half-sided Cavity Model
=====================================
In the previous section, we discussed how to determine the SE rate for an undriven emitter interacting only with a photonic environment. However, in order to fully model a solid-state emitter such as a QD, we need to include interactions between the emitter and its phonon environment [@Machnikowski:Phonons; @Machnikowski:Rabi]. Now we shall derive the polaron ME for a TLS near a metal surface, by modelling the latter as a half-sided Fabry–P[é]{}rot cavity positioned at $z=0$ lying in the $xy$ plane, and the QD positioned at $z = r_d \geq 0$, where $r_d = |\mathbf{r}_d|$. Our calculation follows the general cavity model from Refs. [@demartini; @ficek2005quantum], taking the appropriate limits for the reflectivity and transmittivity of the two mirrors to obtain, effectively, only a single perfectly reflecting surface (see Fig. \[cavityvecs\]).
\[halfcavity:hamiltonian\]Hamiltonian
-------------------------------------
We consider a driven TLS with ground state $\Ket{0}$ and excited state $\Ket{X}$, which is governed by the following Hamiltonian in a rotating frame and after the usual rotating wave approximation ($\hbar = 1$)
$$\label{CavityHamiltonian}
H_S = \delta \Ket{X} \Bra{X} + \frac{\Omega^*_{cav}}{2}\Ket{0} \Bra{X} + \mathrm{H.c.} ~,$$
where H.c. denotes the Hermitian conjugate and $\delta = \omega_0 - \omega_l$ is the detuning between the TLS transition frequency $\omega_0$ and the laser frequency $\omega_l$. $\Omega_{cav}$ is the effective Rabi frequency in the presence of the metal surface, given by $$\label{cavityRabi}
\Omega_{cav} = 2\sqrt{\frac{\omega_l}{2 \epsilon V}}~ \mathbf{d} \cdot \left( \mathbf{e}_{l_-} \mathrm{e}^{-i \mathbf{q}_{l} r} - \mathbf{e}_{l_+} \mathrm{e}^{i \mathbf{q}_{l} r} \right) ~,$$ where $\mathbf{q}_l$ is the laser field wavevector, with polarisation $\mathbf{e}_{l_-}$ ($\mathbf{e}_{l_+}$ after reflection), as shown in Fig. \[cavityvecs\] for the case of the laser beam being perpendicular to the surface. Photon and phonon environments are modelled by the Hamiltonians $$\begin{aligned}
H^{pt}_E &= \sum_{\mathbf{q}, \, \lambda} \nu_\mathbf{q} a^\dagger_{\mathbf{q}\lambda} a_{\mathbf{q}\lambda}~, \\
H^{pn}_E &= \sum_{\mathbf{k}} \omega_\mathbf{k} b^\dagger_\mathbf{k} b_\mathbf{k} ~,\end{aligned}$$ where $b^\dagger_\mathbf{k}$ and $a^\dagger_{\mathbf{q}\lambda}$ ($b_\mathbf{k}$ and $a_{\mathbf{q}\lambda}$) are the $\mathbf{k}$-phonon and $\mathbf{q}\lambda$-photon creation (annihilation) operators, respectively. In the dipole approximation, the photon interaction Hamiltonian is of the form $$H^{pt}_I = -\mathbf{d} \cdot \mathbf{E}(\mathbf{r}_d) (\Ket{0} \Bra{X} + \Ket{X} \Bra{0}) ~
\label{eq:hpt0}$$ with $\mathbf{E}(\mathbf{r})$ being the Schr[ö]{}dinger picture electric field for the half-sided cavity [@ficek2005quantum; @demartini], $$\label{electricfield}
\mathbf{E}(\mathbf{r}) = i \sum_{\mathbf{q}, \lambda} \left[ \mathbf{u}_{\mathbf{q} \lambda}(\mathbf{r}) a_{\mathbf{q} \lambda} - \mathrm{H.c.} \right] ~.$$ The spatial mode functions $\mathbf{u}_{\mathbf{q} \lambda}(\mathbf{r})$ for an ideal half-sided cavity (of perfect reflectivity) are given by
$$\label{cavity_spatial_fns}
\mathbf{u}_{\mathbf{q} \lambda}(\mathbf{r}) = \sqrt{\frac{\omega_{\mathbf{q} \lambda}}{2 \epsilon V}}\left( \mathbf{e}_{\mathbf{q}_- \lambda} \mathrm{e}^{i \mathbf{q}_- r} - \mathbf{e}_{\mathbf{q}_+ \lambda} \mathrm{e}^{i \mathbf{q}_+ r} \right)~.$$
Here, $\mathbf{q}_-$ ($\mathbf{q}_+$) is the incident (reflected) wavevector, with corresponding polarisation $\mathbf{e}_{\mathbf{q}_- \lambda}$ ($\mathbf{e}_{\mathbf{q}_+ \lambda}$). For simplicity, we have assumed that the dipole moment $\mathbf{d}$ of the TLS is real.
The interaction with the phonon bath can be generically represented by the Hamiltonian [@Mahan] $$H^{pn}_I = \Ket{X} \Bra{X}\sum_{\mathbf{k}} g_\mathbf{k} ( b^\dagger_\mathbf{k} + b_\mathbf{k} ) ~,$$ where $g_\mathbf{k}$ is the coupling strength of the TLS’s excited electronic configuration with phonon mode $\mathbf{k}$. We move to the polaron frame by employing the standard Lang–Firsov-type transformation $U = e^S$, $S = \Ket{X}\Bra{X} \sum_{\mathbf{k}} (g_\mathbf{k} / \omega_\mathbf{k}) ( b^\dagger_\mathbf{k} - b_\mathbf{k} )$, obtaining the following transformed system Hamiltonian: $$\begin{aligned}
\label{polaronsystem}
\begin{split}
H_{SP} = \delta' \Ket{X} \Bra{X} &+ \frac{\Omega^*_{cav}}{2}\Ket{0} \Bra{X} B_- \\
&+ \frac{\Omega_{cav}}{2}\Ket{X} \Bra{0} B_+~,
\end{split}\end{aligned}$$ where $\delta' = \delta - \sum_\mathbf{k} g^2_\mathbf{k} / \omega_\mathbf{k}$ (becoming $\delta - \int_0^\infty J_{pn}(\omega) / \omega$ in the continuum limit), and the phonon bath operators $B_\pm$ are defined as $B_\pm = \Pi_\mathbf{k} D_\mathbf{k} (g_\mathbf{k} / \omega_\mathbf{k})$, with $D_\mathbf{k}(\pm \alpha) = \exp[\pm(\alpha b^\dagger_\mathbf{k} -\alpha^* b_\mathbf{k})]$ being the $\mathbf{k}$th mode displacement operator. For numerical results we shall later use a superohmic exciton-phonon spectral density $J_{pn}(\omega)$ with exponential cut-off at frequency $\omega_c$ that is appropriate for self-assembled III-V quantum dots [@Ramsay; @phononrabi2]: $$\label{phonon_spectral_density}
J_{pn}(\omega) = \alpha \omega^3 \mathrm{e}^{-\frac{\omega^2}{\omega_c^2}}~.$$
In the polaron frame the light-mattter interaction Hamiltonian Eq. becomes $$\begin{aligned}
\begin{split}
H^{pt}_{IP} = &i \Ket{0} \Bra{X} B_- \sum_{\mathbf{q}, \lambda} \mathbf{d}\cdot\mathbf{u}^*_{\mathbf{q}\lambda}(\mathbf{r}_d) a^\dagger_{\mathbf{q}\lambda} \\
-&i \Ket{X} \Bra{0} B_+ \sum_{\mathbf{q}, \lambda} \mathbf{d}\cdot\mathbf{u}_{\mathbf{q}\lambda}(\mathbf{r}_d) a_{\mathbf{q}\lambda} ~.
\end{split}\end{aligned}$$ With the definitions $A_1^{pt} = \Ket{0} \Bra{X}$, $A_2^{pt} = A_1^{pt \dagger}$, $B^{pt}_{1/2} \equiv B_\mp$, $C_1 = i \sum_{\mathbf{q}, \lambda} \mathbf{d}\cdot\mathbf{u}^*_{\mathbf{q}\lambda}(\mathbf{r}_d) a^\dagger_{\mathbf{q}\lambda}$, and $C_2 = C^\dagger_1$, we can compactly write the above Hamiltonian as $$\label{eq:compactHpti}
H^{pt}_{IP} = \sum_{i=1}^2 A^{pt}_i \otimes B^{pt}_i \otimes C_i ~,$$ Since the second term in Eq. contains system and environment operators, we identify this as our new exciton-phonon interaction term [@phononreview]. This new interaction term possesses a non-zero expectation value with respect to the thermal equilibrium bath state $\rho^{pn}_E$; tracing out the phonon bath degrees of freedom, we thus obtain $$\begin{aligned}
\mathrm{Tr}_E^{pn}\left[\left(\frac{\Omega^*_{cav}}{2}\Ket{0} \Bra{X} B_- + \frac{\Omega_{cav}}{2}\Ket{X} \Bra{0} B_+\right)\rho^{pn}_E\right] \nonumber \\
= \frac{\Omega^*_{cav}}{2}\langle B \rangle\Ket{0} \Bra{X} + \frac{\Omega_{cav}}{2}\langle B \rangle\Ket{X} \Bra{0} ~,
\end{aligned}$$ where $$\langle B \rangle = \exp\left[ -\frac{1}{2}\int_0^\infty \mathrm{d} \omega \frac{J_{pn}(\omega)}{\omega^2} \coth(\beta \omega / 2) \right] ~.$$ In order to expand perturbatively, we therefore define the system-bath interaction with respect to this value. To this end, we add the expectation value by defining $\mathcal{B}_\pm = B_\pm - \langle B \rangle$ and $\Omega^{pn}_{cav} = \langle B \rangle \Omega_{cav}$ and regrouping our system and interaction Hamiltonian terms, obtaining: $$\begin{aligned}
\label{newinteraction}
H_{SP} &= \delta' \Ket{X}\Bra{X} + \frac{\Omega^{pn *}_{cav}}{2}\Ket{0} \Bra{X} + \frac{\Omega^{pn}_{cav}}{2}\Ket{X} \Bra{0}~, \\
H_{IP}^{pn} &= \frac{\Omega^*_{cav}}{2}\Ket{0} \Bra{X} \mathcal{B}_- + \frac{\Omega_{cav}}{2}\Ket{X} \Bra{0} \mathcal{B}_+ ~,\end{aligned}$$ As for Eq. , we introduce operator labels $B^{pn}_{1/2} = \mathcal{B}_\mp$, $A^{pn}_1 = \Omega^*_{cav} /2 \, \Ket{0} \Bra{X}$ and $A^{pn}_2 = A_1^{pn \dagger}$ to recast the above interaction Hamiltonian into the compact form $$\label{eq:compactHphi}
H_{IP}^{pn} = \sum_{i=1}^2 A^{pn}_i \otimes B^{pn}_i$$ which will prove useful for the derivation of the master equation.
Master Equation {#cavityME}
---------------
Having obtained our Hamiltonian in the polaron frame and partitioned it into system, interaction and environment parts, we can make use of the generically derived microscopic second-order Born-Markov master equation of Ref. [@breuer] (Eqn. 3.118). The interaction terms Eqs. and are of the required form underlying this derivation, and the resultant ME (in the interaction picture) reads: $$\begin{aligned}
\label{generalME}
\diff{}{t} &\rho_{SP}(t) = \\
&-\int_0^\infty \mathrm{d}\tau \; \mathrm{Tr}_E [ H_{IP}(t), [ H_{IP}(t-\tau), \rho_{SP}(t)\otimes\rho_E(0) ] ]~, \nonumber\end{aligned}$$ where $H_{IP}(t) = H^{pn}_{IP}(t) + H^{pt}_{IP}(t)$, and $\mathrm{Tr}_E$ denotes the trace over both environments [@breuer]. It can be easily shown [@phononreview] that the right-handside (RHS) of the above equation can be split into two parts:
$$\begin{aligned}
\label{splitgeneralME}
\diff{}{t} &\rho_{SP}(t) = \\
&-\int_0^\infty \mathrm{d}\tau \mathrm{Tr}^{pn}_E [ H^{pn}_{IP}(t), [ H^{pn}_{IP}(t-\tau), \rho_{SP}(t)\otimes\rho^{pn}_E(0) ] ]~ \nonumber \\
&-\int_0^\infty \mathrm{d}\tau \mathrm{Tr}_E [ H^{pt}_{IP}(t), [ H^{pt}_{IP}(t-\tau), \rho_{SP}(t)\otimes\rho_E(0) ] ]~. \nonumber\end{aligned}$$
Since we assume that the (initial) environmental state is thermal, $\rho_E(0) $ factorises: $\rho_E(0) = \rho^{pn}_E(0) \otimes \rho^{pt}_E(0)$.
### Phonon bath correlations
We proceed by analysing the first term on the RHS of Eq. which captures the influence of phonons on the TLS dynamcis with scattering rates determined by phonon correlation functions [@Ulhaq2013; @PhononRates; @spectrum]. In the ME formalism, the rate $\gamma(\omega)$ of a dissipative process is given by $\gamma(\omega) = 2 \mathrm{Re}\left[ \int_0^\infty \mathrm{d}s K(s) \right]$, where $K(s)$ is the relevant correlation function \[[*c.f.*]{} Eq. (3.137) in Ref. [@breuer]\]. For our phonon dissipator, these functions are given by $$\begin{aligned}
C^{pn}_{ii}(\tau) &= \mathrm{Tr}^{pn}_{E} \left[ \mathcal{B}^\dagger_\pm(\tau) \mathcal{B}_\pm(0) \rho^{pn}_E(0)\right] \nonumber\\
& = \langle B \rangle^2 (\mathrm{e}^{\phi(\tau)} -1)~, \label{phonon_cor_fns_cav:reg}\\
C^{pn}_{ij}(\tau) &= \mathrm{Tr}^{pn}_{E} \left[ \mathcal{B}^\dagger_\pm(\tau) \mathcal{B}_\mp(0) \rho^{pn}_E(0)\right] \nonumber \\
&= \langle B \rangle^2 (\mathrm{e}^{-\phi(\tau)} -1)~, \label{phonon_cor_fns_cav:cross}\end{aligned}$$ where $i, j \in \{ 1,2 \}$, $i \neq j$. After some algebra, we obtain a phonon dissipator of the form $$\begin{aligned}
\begin{split}
&\gamma^{pn}(\omega') \mathcal{L}[\sigma_-] + \gamma^{pn}(-\omega') \mathcal{L}[\sigma_+] \\[10pt]
&- \gamma^{pn}_{cd}(\omega') \mathcal{L}_{cd}[\sigma_-] - \gamma^{pn}_{cd}(-\omega') \mathcal{L}_{cd}[\sigma_+] ~,
\end{split}\end{aligned}$$ where $\mathcal{L}[C] = C \rho_{SP} C^\dagger - \frac{1}{2}\{C^\dagger C, \rho_{SP} \}$ and $\mathcal{L}_{cd}[C] = C \rho_{SP} C - \frac{1}{2}\{C^2, \rho_{SP} \}$. The rates $\gamma^{pn}(\pm\omega')$ and $\gamma_{cd}^{pn}$ are $$\begin{aligned}
\gamma^{pn}(\pm \omega') &= \frac{\left| \Omega_{cav}^{pn} \right|^2}{4} \int_{-\infty}^\infty \mathrm{d}\tau \; \mathrm{e}^{\pm i \omega' \tau} \left( \mathrm{e}^{\phi(\tau)} - 1 \right)~, \\
\gamma^{pn}_{cd}(\omega') &= \frac{\left( \Omega^{pn*}_{cav} \right)^2}{4} \int_{-\infty}^\infty \mathrm{d}\tau \; \cos(\omega' \tau) \left( 1- \mathrm{e}^{-\phi(\tau)} \right)~, \\
\gamma^{pn}_{cd}(-\omega') &= \frac{\left( \Omega_{cav}^{pn} \right)^2}{4} \int_{-\infty}^\infty \mathrm{d}\tau \; \cos(\omega' \tau) \left( 1- \mathrm{e}^{-\phi(\tau)} \right)~,\end{aligned}$$ where $\phi(\tau) = \int_0^\infty \mathrm{d} \omega \frac{J_{pn}(\omega)}{\omega^2} [\coth(\beta \omega / 2)\cos(\omega \tau) - i \sin(\omega \tau)]$. Our rates match the ones obtained by Roy-Choudhury *et al.* [@PhononRates] in previous work[^1]. The rates $\gamma^{pn}(\omega')$ and $\gamma^{pn}(-\omega')$ correspond to enhanced radiative decay and incoherent excitation of the TLS, respectively, whilst $\gamma^{pn}_{cd}(\pm\omega')$ is associated with cross-dephasing [@Ulhaq2013].
### Electromagnetic bath correlations {#EM_bath_cor_cav}
Having arrived at a ‘Lindblad-like’ phonon dissipator[^2], we now turn our attention to the second term of the RHS of Eq. . This term will yield the modified SE rate of the TLS near the cavity, as well as account for the frequency shift via a unitary renormalisation term. As in the previous section, we begin by explicitly printing the correlation functions obtained from Eq. : $$\begin{aligned}
\label{photon_cor_fns_cav}
&C^{pt}_{ij}(\tau) \\
&= \mathrm{Tr}_{E} \left[ \left(B^{pt \dagger}_i(\tau) \otimes C^\dagger_i(\tau) \right) \left( B^{pt}_j(0) \otimes C_j(0) \right) \rho_E(0)\right]~, \nonumber \\
&= \mathrm{Tr}^{pn}_{E} \left[ B^{pt \dagger}_i(\tau) B^{pt}_j(0) \rho^{pn}_E(0)\right] \mathrm{Tr}^{pt}_{E} \left[ C^\dagger_i(\tau) C_j(0) \rho^{pt}_E(0)\right]~, \nonumber\end{aligned}$$ where $i, j \in \{ 1,2 \}$. After substituting for the bath operators, we make use of the following relations [@breuer] $$\begin{aligned}
\mathrm{Tr}^{pt}_{E} \left[ a_{\mathbf{q} \lambda} a_{\mathbf{q}' \lambda'} \rho^{pt}_E(0) \right] &= \mathrm{Tr}^{pt}_{E} \left[ a^\dagger_{\mathbf{q} \lambda} a^\dagger_{\mathbf{q}' \lambda'} \rho^{pt}_E(0) \right] &=&~0 ~,\\
\mathrm{Tr}^{pt}_{E} \left[ a_{\mathbf{q} \lambda} a^\dagger_{\mathbf{q}' \lambda'} \rho^{pt}_E(0) \right] &= \delta_{\mathbf{q}\mathbf{q}'}\delta_{\lambda \lambda'} (1 + N(\nu_\mathbf{q})) &\approx&~\delta_{\mathbf{q}\mathbf{q}'}\delta_{\lambda \lambda'}~, \\[10pt]
\mathrm{Tr}^{pt}_{E} \left[ a^\dagger_{\mathbf{q} \lambda} a_{\mathbf{q}' \lambda'} \rho^{pt}_E(0) \right] &= \delta_{\mathbf{q}\mathbf{q}'}\delta_{\lambda \lambda'} N(\nu_\mathbf{q}) &\approx& ~0~, \end{aligned}$$ where we have assumed that $\forall \omega > 0$, the Planck distribution $N(\omega) \approx 0$[^3]. This means that we only have a single non-vanishing correlation function $C^{pt}_{11}(\tau)$. Following Ref. [@FermiGR2], we consider well-separated photon and phonon correlation times (appropriate for an unstructured photonic environment), so that $C^{pt}_{11}(\tau)$ reduces to the photon bath correlation function in the absence of a phonon bath. The latter is given by $$C^{pt}_{11}(\tau) = \frac{|\mathbf{d}|^2 }{6 \pi^2 \epsilon c^3}\int_0^\infty \mathrm{d}\nu_\mathbf{q} \; \nu^3_\mathbf{q} [1 + \mathcal{F}_{cav}(q r_d)]~,$$ where the term
[0.5]{}
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{width="90.00000%"}
$$\label{Fcav}
\mathcal{F}_{cav}(x) = \frac{3}{2}\left( -\frac{\sin(2 x)}{2 x} - \frac{\cos(2 x)}{(2 x)^2} + \frac{\sin(2 x)}{(2 x)^3} \right)~,$$
describes the influence of the metal surface. The SE rate then evaluates to $$\label{SE_rate_cav}
\gamma_{cav}^{pt}(\omega') = (1+ \mathcal{F}_{cav}(q_0 r_d))\gamma_0^{pt}(\omega') ~,$$ where $\gamma^{pt}_0(\omega')$ is the bare SE rate for an isolated TLS, and is given by $\gamma^{pt}_0(\omega') = |\mathbf{d}|^2 \omega'^3 / 3 \pi \epsilon c^3$. The imaginary part of the correlation tensor has two components: the first term is the usual Lamb shift (whose expression is divergent unless one adopts a full QED approach based on a relativistic Hamiltonian and appropriate renormalisation [@Gardiner]). The second term is the additional energy shift term and takes the form [@agarwal; @agarwal2; @ficek2005quantum] $$\label{energy_shift_cav}
V_{cav} = \frac{1}{2} \mathcal{G}_{cav}(q_0 r_d)\gamma^{pt}_0(\omega')~,$$ where the function $\mathcal{G}_{cav}$ is given by $$\mathcal{G}_{cav}(x) = \frac{3}{2} \left(- \frac{\sin(2 x)}{(2 x)^2} - \frac{\cos(2 x)}{(2 x)^3} + \frac{\cos(2 x)}{2 x}\right)~.$$ Overall, the transition frequency for the TLS in the polaron frame is now given by $$\label{shifted_cavity_frequency}
\tilde{\omega}' = \omega' + V_{cav}~$$ and the final polaron frame ME takes the following form in the Schrödinger picture: $$\begin{aligned}
\label{ME_polaron_cav}
\begin{split}
\diff{}{t} &\rho_{SP} = \\
&-\frac{i}{\hbar} [H'_{SP}, \rho_{SP}(t)] + D_{pn}(\rho_{SP}) +D_{pt}(\rho_{SP}) ~,
\end{split}\end{aligned}$$ where $D_{pn}(\rho_{SP}) = \gamma^{pn}(\omega') \mathcal{L}[\sigma_-] + \gamma^{pn}(-\omega') \mathcal{L}[\sigma_+] - \gamma^{pn}_{cd}(\omega') \mathcal{L}_{cd}[\sigma_-] - \gamma^{pn}_{cd}(-\omega') \mathcal{L}_{cd}[\sigma_+] $ and $D_{pt}(\rho_{SP}) = \gamma^{pt}_{cav}(\omega') \mathcal{L}[\sigma_-]$. $H'_{SP}$ is the system Hamiltonian in the polaron frame including the energy shift from Eq. .
In summary, Eqs. and capture how the presence of a metal surface (here treated as a perfect reflector) alters the SE rate and the transition frequency of the TLS, respectively. Considering our results in the absence of phonons, we find full analytical agreement with the prior literature on the image dipole approach [@ficek2005quantum; @agarwal3], and except for very small separations, we also have excellent numerical agreement with the full QED approach [@agarwal3]. We show this agreement in Fig. \[energyrate\] as a function of the distance of the emitter to the surface. The dashed vertical lines at multiples of $1/8 n$ (where $n$ is the refractive index of the host material, taken to be GaAs in our case), taken from Eqns. and , serve as a guide to the eye for the approximate frequency of oscillation, and demonstrate that multiple periods occur within a wavelength’s separation of emitter to surface. In the limiting case $r_d \rightarrow \infty$, we have $V_{cav} \rightarrow 0$ and $\gamma_{cav}^{pt}(\omega') \rightarrow \gamma_0^{pt}(\omega')$, i.e. we recover the case of an isolated QD as required.
Image Emitter Approach {#image:dot:approach}
======================
Models involving emission from a combination of two identical TLS have been used extensively to study the modifications to the SE rate of an emitter in the proximity of a dielectric or metal surface. After setting up the appropriate Hamiltonian, we shall once more derive a polaron frame ME. We then show that this ME is identical to the one derived using the half-sided cavity approach, provided we disregard certain terms in order to constrain the dynamics of our two emitter model to the ‘right’ subspace.
Setup
-----
We focus on the case where the dipole is oriented parallel to the surface[^4] (as is appropriate for a typical self-assembled QD emitter), implying that the image dipole will be antiparallel [@babiker; @agarwal; @agarwal2; @agarwal3]. In what follows, we shall once again take the [*real*]{} emitter to be situated at a distance $r_d > 0$ along the positive $z$-axis, with the dipole vector oriented in the positive $x$-direction. Hence, the corresponding [*image*]{} dipole is positioned at $z = -r_d$, with its dipole vector being parallel to the negative $x$-axis.
Hamiltonian
-----------
The Hamiltonian of the two driven TLS in a frame rotating with frequency $\omega_l$ is given by $$H_S = \sum_{j=1}^2 \delta \Ket{X_j}\Bra{X_j} + \frac{\Omega^*_j}{2}\Ket{0_j} \Bra{X_j} + \frac{\Omega_j}{2}\Ket{X_j} \Bra{0_j} ~,$$ where the subscript $j=1, 2$ denotes the real and image TLS, respectively. In order to match the boundary conditions required for reflection, we model the classical driving field as two counter-propagating beams, with the secondary ‘reflected’ beam having a $\pi$ phase shift with respect to the original beam. For simplicity, we model these as plane waves propagating along the $z$-axis and polarised in the $x$-direction. In phasor notation, these two waves can be written as $$\begin{aligned}
\begin{split}
\mathbf{E}_1(\mathbf{r}) &= \mathbf{E}_{incident}(\mathbf{r}) = E_0 \mathrm{e}^{i \mathbf{q}_l \cdot \mathbf{r}} \hat{\mathbf{x}} ~, \\
\mathbf{E}_2(\mathbf{r}) &= \mathbf{E}_{reflected}(\mathbf{r}) = -E_0 \mathrm{e}^{-i \mathbf{q}_l \cdot \mathbf{r}} \hat{\mathbf{x}}~,
\end{split}\end{aligned}$$ giving rise to the following Rabi frequencies at the positions $\mathbf{r}_{1,2}$ of the two emitters: $$\begin{aligned}
\begin{split}
\Omega_1 &= 2 \mathbf{d}_1 \cdot (\mathbf{E}_1(\mathbf{r}_1) + \mathbf{E}_2(\mathbf{r}_1))~, \\
\Omega_2 &= 2 \mathbf{d}_2 \cdot (\mathbf{E}_1(\mathbf{r}_2) + \mathbf{E}_2(\mathbf{r}_2)) ~.
\end{split}\end{aligned}$$ Since $\mathbf{r}_2 = -\mathbf{r}_1$ and $\mathbf{d}_2 = -\mathbf{d}_1$, we have $\Omega \coloneqq \Omega_1 = \Omega_2$.
We now turn to the wider electromagnetic environment (excluding the coherent driving field discussed above). The electric field operator can be written as in Eq. but with the spatial mode functions now being replaced by the free-space functions $$\mathbf{u}_{\mathbf{q} \lambda}(\mathbf{r}) = \sqrt{\frac{\omega_{\mathbf{q} \lambda}}{2 \epsilon V}} \mathbf{e}_{\mathbf{q} \lambda} \mathrm{e}^{i \mathbf{q} r} ~.$$ The interaction Hamiltonian of the TLS with the photonic environment is then given by $$\begin{aligned}
\begin{split}
H^{pt}_I =&H^{pt,1}_I + H^{pt,2}_I \\
=&-\sum_{j=1}^2 \mathbf{d}_j \cdot \mathbf{E}(\mathbf{r_j}) (\Ket{0_j} \Bra{X_j} + \Ket{X_j} \Bra{0_j})~.
\end{split}\end{aligned}$$
For the interaction with vibrational modes, we assume that both real and image TLS see the same phonon bath and possess perfectly correlated coupling constants $g_\mathbf{k}$. This ensures the image system exactly follows the dynamics of real dipole, as is required for matching the boundary condition of a perfectly reflecting interface. Thus, our relevant Hamiltonian reads $$\begin{aligned}
\begin{split}
H^{pn}_I =&H^{pn,1}_I + H^{pn,2}_I \\
=&\sum_{j=1}^2 \sum_{\mathbf{k}} \Ket{X_j} \Bra{X_j} g_\mathbf{k} ( b^\dagger_\mathbf{k} + b_\mathbf{k} )~.
\end{split}\end{aligned}$$ Next, we move into the polaron frame with the transformation $\mathrm{e}^{S_1+S_2} = \mathrm{e}^{S_1}\mathrm{e}^{S_2}$, obtaining the transformed Hamiltonians $$\begin{aligned}
H_{SP} = & \sum_{j=1}^2 \delta' \Ket{X_j}\Bra{X_j} + \frac{\Omega^{pn *}}{2}\Ket{0_j} \Bra{X_j} + \mathrm{H.c.}~, \\
H^{pt,j}_{IP} = &i \Ket{0_j} \Bra{X_j} B_- \sum_{\mathbf{q}, \lambda} \mathbf{d}_j\cdot\mathbf{u}^*_{\mathbf{q}\lambda}(\mathbf{r}_j) a^\dagger_{\mathbf{q}\lambda} \nonumber \\
-&i \Ket{X_j} \Bra{0_j} B_+ \sum_{\mathbf{q}, \lambda} \mathbf{d}_j\cdot\mathbf{u}_{\mathbf{q}\lambda}(\mathbf{r}_j) a_{\mathbf{q}\lambda} ~,\nonumber \\
H^{pn,j}_{IP} = &\frac{\Omega^*}{2}\Ket{0_j} \Bra{X_j} \mathcal{B}_- + \frac{\Omega}{2}\Ket{X_j} \Bra{0_j} \mathcal{B}_+ ~.\end{aligned}$$ As in Sec. \[halfcavity\], the latter two can easily be seen to be of the following generic form (with appropriate identifications for the $A, B, C$ operators) which will enable straightforward use of the ME (3.118) from Ref. [@breuer]: $$\begin{aligned}
H^{pn,j}_{IP} = &\sum_{i=1}^2 A^{pn,j}_i \otimes B^{pn,j}_i~, \\
H^{pt,j}_{IP} = &\sum_{i=1}^2 A^{pt,j}_i \otimes B^{pt,j}_i \otimes C^j_i ~.\end{aligned}$$
Master equation {#master-equation}
---------------
The ME for our system can, once again, be written as $$\begin{aligned}
\label{splitgeneralME2}
\diff{}{t} &\rho_{SP}(t) = \\
&-\int_0^\infty \mathrm{d}\tau \mathrm{Tr}^{pn}_E [ H^{pn}_{IP}(t), [ H^{pn}_{IP}(t-\tau), \rho_{SP}(t)\otimes\rho^{pn}_E(0) ] ]~ \nonumber \\
&-\int_0^\infty \mathrm{d}\tau \mathrm{Tr}_E [ H^{pt}_{IP}(t), [ H^{pt}_{IP}(t-\tau), \rho_{SP}(t)\otimes\rho_E(0) ] ]~, \nonumber\end{aligned}$$ however, it now features a larger number of correlation functions due to the presence of the image emitter. Following the general procedure in Sec. \[cavityME\], we shall analyse different contributions in turn to arrive at our final ME of the image emitter model.
### Phonon dissipator
The correlation functions (including cross correlation terms between bath operators of the real and image system) result in the following phonon dissipator $$\begin{aligned}
D_{pn}&(\rho_{SP})= \\
&\sum_{i,j=1}^2 \gamma^{pn}_{ji}(\omega') \left( \sigma^j_- \rho_{SP}(t) \sigma^i_+ - \frac{1}{2}\{ \sigma^i_+ \sigma^j_-, \rho_{SP}(t) \} \right) \nonumber \\
+&\sum_{i,j=1}^2 \gamma^{pn}_{ji}(-\omega') \left( \sigma^j_+ \rho_{SP}(t) \sigma^i_- - \frac{1}{2}\{ \sigma^i_- \sigma^j_+, \rho_{SP}(t) \} \right) \nonumber \\
-&\sum_{i,j=1}^2 \gamma^{pn}_{cd, ji}(\omega') \left( \sigma^j_- \rho_{SP}(t) \sigma^i_- - \frac{1}{2}\{ \sigma^i_- \sigma^j_-, \rho_{SP}(t) \} \right) \nonumber \\
-&\sum_{i,j=1}^2 \gamma^{pn}_{cd, ji}(-\omega') \left( \sigma^j_+ \rho_{SP}(t) \sigma^i_+ - \frac{1}{2}\{ \sigma^i_+ \sigma^j_+, \rho_{SP}(t) \} \right)~,\nonumber \end{aligned}$$ where the rates $\gamma^{pn}_{ji}(\pm\omega')$ and $\gamma^{pn}_{cd, j}$ are given by $$\begin{aligned}
\label{phononratesRealandImage}
\gamma^{pn}_{ji}(\pm \omega') &= \frac{|\Omega^{pn}|^2}{4} \int_{-\infty}^\infty \mathrm{d}\tau \; \mathrm{e}^{\pm i \omega' \tau} \left( \mathrm{e}^{\phi(\tau)} - 1 \right)~, \\
\gamma^{pn}_{cd, ji}(\omega') &= \frac{ (\Omega^{pn*})^2}{4} \int_{-\infty}^\infty \mathrm{d}\tau \; \cos(\omega' t) \left( 1- \mathrm{e}^{-\phi(\tau)} \right)~, \\
\gamma^{pn}_{cd, ji}(-\omega') &= \frac{ (\Omega^{pn})^2}{4} \int_{-\infty}^\infty \mathrm{d}\tau \; \cos(\omega' t) \left( 1- \mathrm{e}^{-\phi(\tau)} \right) ~.\end{aligned}$$ We shall return back to the phonon dissipator when discussing the ME equation in the symmetric-antisymmetric basis, which allows us to derive a model agreeing with the half-sided cavity approach.
### Photon dissipator
We now turn our attention to the photon dissipator term from Eq. . After evaluating the correlation and cross-correlation functions, we obtain the usual expression for two emitters [@ficek2005quantum] in a shared electromagnetic environment, $$\begin{aligned}
\begin{split}
D_{pt}&(\rho_{SP})= \\
&\sum_{i,j=1}^2 \gamma^{pt}_{ji} \left( \sigma^j_- \rho_{SP}(t) \sigma^i_+ - \frac{1}{2}\{ \sigma^i_+ \sigma^j_-, \rho_{SP}(t) \} \right) ~,
\end{split}\end{aligned}$$ where the diagonal terms $\gamma^{pt}_{22}(\omega') = \gamma^{pt}_{11}(\omega') = \gamma_0^{pt}(\omega')$, whilst the off diagonal terms are given by $\gamma^{pt}_{12}(\omega') = \gamma^{pt}_{21}(\omega') = \mathcal{F}_{12}(q_0 \Delta r)\gamma_0^{pt}(\omega')$ with $\Delta r = r_1 - r_2 = 2r_d$, and where $$\begin{aligned}
\label{Fimage}
\begin{split}
\mathcal{F}_{12}(x) = \frac{3}{2}\left( -\frac{\sin(x)}{x} - \frac{\cos(x)}{x^2} + \frac{\sin(x)}{x^3} \right)~.
\end{split}\end{aligned}$$ This is the same function obtained for the half-sided cavity approach \[[*c.f.*]{} Eq. \]. The imaginary part of the correlation function yields the ‘correction’ term to the unitary part of the ME [@breuer; @ficek2005quantum; @agarwal]: its diagonal contribution represents diagonal Lamb shift terms. Their small energetic shifts can be absorbed into the bare TLS transition frequency. We thus focus on the off-diagonal element which is of the form: $$\label{energy_shift_image}
V_{12} = \frac{1}{2} \mathcal{G}_{12}(q \Delta r)\gamma^{pt}_0(\omega')~,$$ where the function $\mathcal{G}_{12}$ is $$\begin{aligned}
\begin{split}
\mathcal{G}_{12}(x) = \frac{3}{2} \left(- \frac{\sin(x)}{x^2} - \frac{\cos(x)}{x^3} + \frac{\cos(x)}{x}\right)~.
\end{split}\end{aligned}$$ Again, this corresponds to the same energy shift term we have previously encountered in Sec. \[EM\_bath\_cor\_cav\]. After diagonalising the Hamiltonian, the frequency of the symmetric excited to ground state transition (in the polaron frame) is then given by $$\tilde{\omega}' = \omega' + V_{12}~,$$ exactly matching the transition frequency Eq. of the half-sided cavity model.
Effective TLS in the energy eigenbasis {#eigbasis}
--------------------------------------
As stated in the introduction, previous literature treating spontaneous emission from initially excited emitters considered the transition from the symmetrically excited to the ground state, as this choice yields matching results with other methods [@ficek2005quantum; @babiker]. We follow this approach and adopt the basis $\{\Ket{e}, \Ket{s}, \Ket{a},\Ket{g}\}$ with $\Ket{s} = (\Ket{0_1} \Ket{X_2} + \Ket{X_1}\Ket{0_2}) / \sqrt{2}$ and $\Ket{a} = (\Ket{0_1} \Ket{X_2} - \Ket{X_1}\Ket{0_2}) / \sqrt{2}$, see Fig. \[lvl\_diag\]. In this basis, our full polaron ME reads: $$\begin{aligned}
\label{fullME}
\begin{split}
\diff{}{t} \rho_{SP}(t) = &-\frac{i}{\hbar} [H'_{SP}, \rho_{SP}(t)] \\
&+ D^{s}_{pn}(\rho_{SP}) +D^{a}_{pt}(\rho_{SP})+D^{s}_{pt}(\rho_{SP}) ~,
\end{split}\end{aligned}$$ where the dissipator terms are explicitly given in Appendix \[app:dissipators\]. Here, $H'_{SP}$ denotes the system diagonalised Hamiltonian \[including the energy shift term Eq. \]. The ME photonic dissipator separates into a symmetric channel ($\Ket{g} \leftrightarrow \Ket{s} \leftrightarrow \Ket{e}$) and an antisymmetric one ($\Ket{g} \leftrightarrow \Ket{a} \leftrightarrow \Ket{e}$). Courtesy of the fully correlated phonon bath, phonons also only act in the symmetric channel.
Since $\Omega_1 = \Omega_2$, the symmetric channel Rabi frequency becomes $\Omega_{sg} \coloneqq (\Omega_1 + \Omega_2)/\sqrt{2} = \sqrt{2}\Omega = \Omega_{cav}$ and hence we obtain the same phonon rates as in the half-sided cavity approach[^5]. Furthermore, the antisymmetric channel Rabi frequency $\Omega_{a} \coloneqq(\Omega_1 - \Omega_2)/\sqrt{2} = 0$, meaning that the laser field is completely decoupled from the antisymmetric state.
Consistency with the Green’s function and half-sided cavity approach demands that we restrict the dynamics of our four-dimensional Hilbert space to the subspace spanned by the states $\{ \Ket{g}, \Ket{s} \}$, i.e. the larger Hilbert space only served to let us calculate the correct properties of this single transition. Fully decoupling the antisymmetric singly and the doubly excited states from the dynamics is achieved by disabling the laser driving on the $\Ket{s} \leftrightarrow \Ket{e}$ transition. For finite temperature photon environments with $N(\omega) \neq 0$, we also need to remove dissipative photon absorption channels, by dropping the antisymmetric dissipator term $D^{a}_{pt}(\rho_{SP})$ from the ME and explicitly removing the dissipative $\Ket{s} \leftrightarrow \Ket{e}$ operator.
The image approach can thus be reduced to an effective TLS model featuring the same Rabi frequency, SE rate, and transition frequency as the half-sided cavity approach – i.e. displaying full equivalence between the two representations.
In Fig. \[schematic2\], we summarise the key results from the previous sections: We show the transition frequency and SE rate for the all four cases considered in this Article alongside their schematic depictions. The driving term is not included as it has no direct influence on the properties of the optical dipole transition.
Resonance Fluorescence Spectrum
===============================
[0.5]{} {width="105.00000%"}
[0.5]{} {width="85.00000%"}
Having included the possibility of laser driving in our model, a natural application is to study the resonance fluorescence (RF) spectrum of a condensed matter TLS near a mirroring surface. We use the ME (after discarding the antisymmetric channel, as argued above) to calculate the spectral function, which is given by the Fourier transform of the (steady-state) first order correlation function $\mathrm{lim}_{t \rightarrow \infty}\langle \mathbf{E}^{(-)}(\mathbf{R}, t) \mathbf{E}^{(+)}(\mathbf{R}, t + \tau) \rangle$, where $\mathbf{E}^{(-)}(\mathbf{R}, t)$ and $\mathbf{E}^{(+)}(\mathbf{R}, t)$ are, respectively, the negative and positive components of the electric field operator evaluated at the position $\mathbf{R}$ of the detector [@ficek2005quantum]. These operators are related to the system operators $\sigma_- = \Ket{0}\Bra{X}$ and $\sigma_+ = \Ket{X}\Bra{0}$, and hence, after applying the polaron transformation, the RF spectral function can be written as $$\begin{aligned}
\label{spectrum}
\begin{split}
S(\omega) \propto \int_{-\infty}^\infty \mathrm{d}\tau &\mathrm{e}^{-i (\omega - \omega') \tau} \times \\
&\langle \sigma_+(\tau) B_+(\tau) \sigma_-(0) B_-(0) \rangle_{s}~, \\
\end{split}\end{aligned}$$ where we have exploited the temporal homogeneity of the stationary correlation function, and where the subscript ‘s’ denotes the trace taken with respect the steady-state density matrix [@breuer]. The correlation function appearing in Eq. involves two timescales, the nanosecond timescale associated with the exciton lifetime, and the shorter picosecond phonon bath relaxation timescale, allowing us to separate the correlation function into the product $\langle \sigma_+(\tau) \sigma_-(0) \rangle_s \langle B_+(\tau) B_-(0) \rangle_{s}$ [@jake]. Substituting the expression for the phonon bath correlation function, we obtain the spectral function $$\begin{aligned}
\label{spectrum_simplified}
\begin{split}
S(\omega) \propto \langle B \rangle^2\int_{-\infty}^\infty \mathrm{d}\tau &\mathrm{e}^{-i (\omega - \omega') \tau} \times \\
&\mathrm{e}^{\phi(\tau)} \langle \sigma_+(\tau) \sigma_-(0) \rangle_{s}~. \\
\end{split}\end{aligned}$$ In the left panel of Fig. \[spectrumplot\], we show the incoherent part of the emission spectrum of our surface-modified system as well as that of a reference TLS (also subject to the same phonon environment). Following Ref. [@gerardot], we take the TLS’s position relative to the surface as $r_d \sim 177$ nm. The reference TLS is driven with ‘free space’ Rabi frequency given by $\Omega^{pn} = 2 \langle B \rangle \mathbf{d}\cdot\mathbf{E}_0$. As expected, the curves differ in the position of the Mollow sidebands and the width of the three peaks, since the former is determined by the effective Rabi frequency and the later depends on the emission rate, which both undergo a change in the presence of a reflective surface. The two insets in the left panel of Fig. \[spectrumplot\] show the much broader phonon sideband, which receives $\sim 16\%$ of the scattered photons for the chosen spectral density at a phonon temperature of T=10 K.
In the right panel of Fig. \[spectrumplot\], we plot the fraction of coherently scattered photons as a function of the renormalised effective Rabi frequency. This ratio is obtained numerically as the (integrated) coherent spectrum divided by the total integrated spectrum. There are two pairs of curves: one with and one without phonons. For the former, the finite area under the phonon sideband means that the coherent fraction does not go to unity even when driving far below saturation. The level at which this fraction plateaus is phonon coupling strength and temperature dependent [@jake]. By contrast, in the absence of phonons, almost all light is coherently scattered at weak enough driving. The close agreement between the two curves in each pair bears testament to the fact that the surface-modified emitter largely behaves like a bare emitter once the effective Rabi frequency has been corrected for (with the slight remaining discrepancy due to modifications of the natural lifetime). Indeed, plotting this ratio directly as a function of the laser driving field amplitude reveals sizeable horizontal shifts between these two curves in each pair (not shown).
Summary and Discussion
======================
We have extended the method of images – traditionally developed for capturing spontaneous emission in atomic ensembles near reflective interfaces – to the case of a driven solid-state emitter near a metal surface. We have developed two approaches: a half-sided cavity and image dipole, and shown that the latter agrees with the former, but only when additional ‘selection rules’ are introduced to constrain the dynamics to the relevant subspace. Both our approaches agree with a Green’s function treatment in the absence of a vibrational environment. Through a rigorous derivation, we find that the emitter can indeed still be described as an effective (phonon-dressed) two-level system with appropriately modified properties, even in the presence of a phonon bath and for a driven system. Our calculated RF spectrum corroborates this observation.
We note that image dipole approach not only necessitates a larger Hilbert space but also involved a more cumbersome ME derivation than the half-sided cavity approach. This begs the questions whether such an image approach remains useful. We submit that the method of images can more easily accommodate larger numbers of emitters near a surface (of varying separation to the surface), as the problem then straightforwardly maps onto the case of several optical dipoles in a shared (free space) electromagnetic environment – a problem which has been studied extensively, see, e.g., Ref. [@ficek2005quantum]. Future work might investigate the role of geometry in configurations with $N>1$ emitters, possibly resulting in the enhancement of Dicke superradiance of an ensemble of solid state emitters [@sanders; @Machnikowski:Superradiance], or the use of mirrors to bring about other collective effects in the light matter interaction, for example inspired by a recent proposal for engineering the quantum-enhanced absorption of light [@superabsorption] or by harnessing sub-radiant collective states [@Scully2015; @Higgins2015].
Another interesting avenue for future work might be the study of charged quantum dots featuring excited trion states. In addition to the optical dipole, the image approach would then feature a separate permanent dipole. To a first approximation, we would expect this second dipole to be static, meaning it would not radiate and only modify the spectrum via energetic shifts. However, one might speculate whether the Coulomb interaction of the three charges involved in the trion state could slightly ‘wiggle’ this dipole, making some radiative contribution to the overall spectrum conceivable.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Peter Kirton, Fabio Biancalana, and David Gershoni for valuable suggestions. D.S. thanks SUPA for financial support, T.S. acknowledges studentship funding from EPSRC under grant no EP/G03673X/1, B. D. G. thanks the Royal Society, and E. M. G. acknowledges support from the Royal Society of Edinburgh and the Scottish Government.
Appendix {#appendix .unnumbered}
========
Eigenbasis Dissipators {#app:dissipators}
======================
The dissipators of Sec. \[eigbasis\] are given by $$\begin{aligned}
\begin{split}
&D^{s}_{pn}(\rho_{SP})\\&= 2 \gamma^{pn}(\omega') \Big[ (S_{se}+S_{gs}) \rho_{SP}(t) (S_{es}+S_{sg})\\
&- \frac{1}{2}\{ (S_{ee}+S_{ss}), \rho_{SP}(t) \} \Big] \\[15pt]
&+ 2 \gamma^{pn}(-\omega') \Big[ (S_{es}+S_{sg}) \rho_{SP}(t) (S_{se}+S_{gs})\\
&- \frac{1}{2}\{ (S_{gg}+S_{ss}), \rho_{SP}(t) \} \Big] \\[15pt]
&- 2 \gamma^{pn}_{cd}(\omega') (S_{se}+S_{gs}) \rho_{SP}(t)(S_{se}+S_{gs}) \\[10pt]
&- 2 \gamma^{pn}_{cd}(-\omega') (S_{es}+S_{sg}) \rho_{SP}(t) (S_{es}+S_{sg})~,
\end{split}\end{aligned}$$ $$\begin{aligned}
\begin{split}
&D^{a}_{pt}(\rho_{SP})\\&= 2 \gamma^{pt}(\omega') \Big[ (S_{se}+S_{gs}) \rho_{SP} (t)(S_{es}+S_{sg})\\
&- \frac{1}{2}\{ (S_{ee}+S_{ss}), \rho_{SP}(t) \} \Big] \\[15pt]
&+ 2 \gamma^{pt}(-\omega') \Big[ (S_{es}+S_{sg}) \rho_{SP}(t) (S_{se}+S_{gs})\\
&- \frac{1}{2}\{ (S_{gg}+S_{ss}), \rho_{SP}(t) \} \Big]
\end{split}\end{aligned}$$ $$\begin{aligned}
\begin{split}
&D^{s}_{pt}(\rho_{SP})\\&= 2 \gamma^{pt}(\omega') \Big[ (S_{se}+S_{gs}) \rho_{SP}(t) (S_{es}+S_{sg})\\
&- \frac{1}{2}\{ (S_{ee}+S_{ss}), \rho_{SP}(t) \} \Big] \\[15pt]
&+ 2 \gamma^{pt}(-\omega') \Big[ (S_{es}+S_{sg}) \rho_{SP}(t) (S_{se}+S_{gs})\\
&- \frac{1}{2}\{ (S_{gg}+S_{ss}), \rho_{SP} (t)\} \Big]~,
\end{split}\end{aligned}$$ with $S_{ij} = \Ket{i}\Bra{j}~; i,j \in \{ g,a,s,e \}$; $\Ket{g}$, $\Ket{a}$, $\Ket{s}$ and $\Ket{e}$ being the ground, antisymmetric, symmetric and doubly excited state of our joint system, respectively.
SE rate and cross Lamb shift terms for dipole perpendicular to the surface
==========================================================================
In the case of a dipole perpendicular to the surface, expressions for the cross Lamb shift term and SE rate similar to the ones used in section \[image:dot:approach\] can be derived from first principles as well, arriving at the expressions $$\begin{aligned}
\label{Fimage}
\begin{split}
\mathcal{F}_{12}(q \Delta r) = 3\left( - \frac{\cos(q \Delta r)}{(q \Delta r)^2} + \frac{\sin(q \Delta r)}{(q \Delta r)^3} \right)~,
\end{split}\end{aligned}$$ and $$\begin{aligned}
\begin{split}
\mathcal{G}_{12}(q \Delta r) = -3\left(\frac{\sin(q \Delta r)}{(q \Delta r)^2} + \frac{\cos(q \Delta r)}{(q \Delta r)^3}\right)~,
\end{split}\end{aligned}$$ instead of the ones used in section \[image:dot:approach\].
[^1]: Ref. [@PhononRates] introduces an additional, phenomenological, pure dephasing term, which we have not included in this paper.
[^2]: Note that we have not performed a secularisation and our ME is therefore not strictly of Lindblad form
[^3]: Only (optical) photon modes with energies close to $\omega_0$ are relevant, for which this approximation is typically justified under ambient conditions. However, the generalisation to a finite temperature photon bath is also straightforward.
[^4]: We discuss modifications for the perpendicular case in the Appendix
[^5]: The last equality holds due to the difference in density of modes appearing in the derivation of the Rabi frequency in both models.
|
---
abstract: 'We propose a novel approach to enable the coexistence between Multi-Input-Multi-Output (MIMO) radar and downlink multi-user Multi-Input-Single-Output (MU-MISO) communication system. By exploiting the constructive multi-user interference (MUI), the proposed approach trades-off useful MUI power for reducing the transmit power, to obtain a power efficient transmission. This paper focuses on two optimization problems: a) Transmit power minimization at the base station (BS) while guaranteeing the receive signal-to-interference-plus-noise ratio (SINR) level of downlink users and the interference-to-noise ratio (INR) level to radar; b) Minimization of the interference from BS to radar for a given requirement of downlink SINR and transmit power budget. To reduce the computational overhead of the proposed scheme in practice, an algorithm based on gradient projection is designed to solve the power minimization problem. In addition, we investigate the trade-off between the performance of radar and communication, and analytically derive the key metrics for MIMO radar in the presence of the interference from the BS. Finally, a robust power minimization problem is formulated to ensure the effectiveness of the proposed method in the case of imperfect Channel State Information (CSI). Numerical results show that the proposed method achieves a significant power saving compared to conventional approaches, while obtaining a favorable performance-complexity trade-off.'
author:
- 'Fan Liu, Christos Masouros, Ang Li, Tharmalingam Ratnarajah, and Jianming Zhou[^1] [^2]'
bibliography:
- 'IEEEabrv.bib'
- 'TSP\_CI.bib'
title: 'Interference Exploitation for Radar and Cellular Coexistence: The Power-Efficient Approach'
---
MU-MISO downlink, radar-communication coexistence, spectrum sharing, constructive interference.
Introduction
============
response to the increasing demand for wireless communication devices and services, the Federal Communications Commission (FCC) has adopted a broadband plan to release an additional 500MHz spectrum that is currently occupied by military and governmental operations, such as air surveillance and weather radar systems[@federal2010connecting]. Since then, spectrum sharing between radar and communication has been regarded as an enabling solution. In [@7131098], a radar information rate has been defined, such that the performance of radar and communication can be discussed using the same metric. Similar work has been done in [@6875553; @7279172], in which radar and communication are unified under the framework of information theory, and the channel capacity between radar and target has been defined by applying the rate distortion theory. Nevertheless, these works focus on the single-antenna systems rather than MIMO systems. At present, several methods considering the spectrum sharing between MIMO radar and communication have been proposed [@6503914; @4058251; @7347464; @6735841; @7485316; @7089157; @6831613; @6636787; @7127473; @7289385; @6956861; @7485158; @6933960; @7485066; @7470514; @7418294; @7178410], since traditional radar will soon be replaced by MIMO radar in the near future due to the advantages of waveform diversity and higher detection capability[@4350230; @li2009mimo]. In [@4058251], the feasibility of combining MIMO radar and Orthogonal Frequency Division Multiplexing (OFDM) communication has been studied. More recently, a novel dual-functional waveform has been reported by [@7347464], where communication bits are embedded in the radar waveform by controlling the sidelobe of the transmit beam patterns for radar. Other related schemes, including modulating information by shuffling the waveform across the radar transmit antennas and Phase Shift Keying (PSK) modulation via different weight vectors, have been proposed in [@7485316; @7485066].\
More relevant to this work, transmit beamforming has been viewed as a promising solution to eliminating the mutual interference between radar and communication. First pioneered by[@6503914], the idea of null space projection (NSP) beamforming has been widely discussed [@7089157; @6831613; @6636787; @7127473; @7289385; @6956861], where the radar waveforms are projected onto the null space of the interference channel matrix from radar transmitter to communication receiver. Optimization-based beamforming has been exploited to solve the problem in [@7485158], where the SINR of radar has been optimized subject to power and capacity constraints of communication. Related work discusses the coexistence between MIMO-Matrix Completion (MIMO-MC) radar and MIMO communication system, where the radar beamforming matrix and communication covariance matrix are jointly optimized [@7470514]. In general, existing works on interference mitigation for radar-communication coexistence mainly consider the scenario between MIMO radar and point-to-point MIMO (P2P MIMO) communication, while few efforts have been taken for the case of radar and multi-user communication. Moreover, none of above works discusses the case of imperfect CSI.\
Motivated by the robust beamforming in the broader area of cognitive radio networks [@4787135; @6373750], the work [@liu2016robust] investigated the robust MIMO beamforming for the coexistence of radar and downlink MU communication, where the radar detection probability was maximized while guaranteeing the transmit power of BS and the receive SINR for each downlink user using Semidefinite Relaxation (SDR) techniques[@5447068; @5447076]. In such optimizations, all the interference from other downlink users is regarded as harmful to the user of interest. Nevertheless, previous works proved that for a downlink MU-MIMO system using PSK modulations, the known interference can act constructively to benefit the symbol decision at downlink users. In [@4801492], partial channel inversion was applied to the BS such that the constructive part of MUI was preserved while the destructive part was eliminated. Further research [@5159472; @5605266] reported that by rotating the interference into the direction of the signal of interest, the MUI was always kept constructive. Moreover, recent works[@6619580; @7103338] showed that by rotating the destructive interference into constructive region using optimization techniques, the receive SINR target for each user was actually relaxed compared to the conventional SDR-based beamformer, thus a significant power saving was obtained. This work has also been applied to cognitive radio transmission to design closed-form precoding solutions[@6193461; @6094141].\
In this paper, we develop a novel precoding optimization approach for the spectrum sharing between MIMO radar and downlink MU-MISO communication based on the concept of constructive interference (CI). By allowing the BS to utilize the known interference as a green signal power, the receive SINR at the users is increased. In fact, for a given SINR constraint using constructive interference, the feasible domain of the optimization problem is extended compared to the conventional SDR-based beamforming. We consider two optimization-based transmit beamforming designs, one is to minimize the transmit power at the BS while guaranteeing the receive SINR at the users and the interference level from BS to radar, the other is to minimize the total interference from BS to radar subject to the SINR constraint per user and transmit power budget. It is worth noting that both problems are convex and can be optimally solved by numerical tools. To efficiently apply the proposed schemes in practice, we design an efficient gradient projection algorithm for power minimization by analyzing the structure of the optimization. To investigate the effect of interference minimization beamforming on the performance of radar, we further derive the analytic form of detection probability and Cramér-Rao bound (CRB) for MIMO radar with the presence of the interference from the BS. By doing so, important trade-offs between the performance of radar and communication are given. Finally, we consider the uncertainty in the estimated channel information, and design a worst-case robust beamformer based on the principle of interference exploitation. For clarity, we list the contributions of this paper as follows:
- We design a power efficient optimization-based beamforming technique for the coexistence of MIMO radar and downlink MU-MISO communication based on exploiting the constructive interference power, where two optimization problems are formulated: a) Power minimization subject to SINR and INR constraints; b) Interference minimization subject to SINR and power constraints. The proposed approach outperforms the conventional SDR-based method.
- We investigate the structure of the power minimization problem, and derive a computationally efficient algorithm to solve it.
- We analytically derive the detection probability and the CRB for MIMO radar when the proposed beamforming scheme is used.
- We derive the robust beamforming design of power minimization for the case of imperfect CSI.
The remainder of this paper is organized as follows. Section II introduces the system model and briefly recalls the conventional SDR-based beamforming problems. Section III describes the concept of CI and formulates the proposed optimization problems using the CI technique. In Section IV, a thorough analysis for the power minimization optimization is present and an efficient algorithm is derived. Section V derives the detection probability and the Cramér-Rao bound of MIMO radar for the proposed scenario. A worst-case approach for imperfect CSI is given for robust power minimization in Section VI, with norm-bounded CSI errors. Numerical results are provided and discussed in Section VII. Finally, Section VIII concludes the paper.\
: Matrices are denoted by bold uppercase letters (i.e., ${\bf{H}}$), bold lowercase letters are used for vectors (i.e., $\pmb\beta$), subscripts indicate the rows of a matrix unless otherwise specified (i.e., ${\bf{h}}_i$ is the *i*-th row of ${\bf{H}}$), scalars are denoted by normal font (i.e., $R_m$), $\text{tr}\left(\cdot\right)$ stands for the trace of the argument, $(\cdot)^T$, $(\cdot)^*$ and $(\cdot)^H$ stand for transpose, complex conjugate and Hermitian transpose respectively, $\operatorname{Re}(\cdot)$ and $\operatorname{Im}(\cdot)$ denote the real and imaginary part of the argument.
System Model and SDR-based Beamforming
======================================
Consider a spectrum sharing scenario where a *K*-user MU-MISO downlink system operates at the same frequency band with a MIMO radar. As can be seen in Fig. 1, the *N*-antenna BS is transmitting signals to *K* single-antenna users while the MIMO radar with $M_t$ transmit antennas and $M_r$ receive antennas is detecting a point-like target in the far-field. Inevitably, these two systems will cause interference to each other. The received signal at the *i*-th downlink user is given as
![Spectrum sharing scenario.[]{data-label="fig:1"}](TSPFig1.eps){width="3.0in"}
$${{y}^C_i[l]} = {\bf{h}}_i^T\sum\limits_{k = 1}^K {{{\bf{t}}_k}{d}_k[l]} + \sqrt{P_R}{\bf{f}}_i^T{{\bf s}_l} + {n}_i[l], i = 1,2,...,K,$$
where $ {\bf{h}}_{i} \in {\mathbb{C}^{\emph{N} \times 1}} $ denotes the communication channel vector, $ {\bf{f}}_{i} \in {\mathbb{C}^{{{\emph{M}}_{t}} \times 1}} $ denotes the interference channel vector from radar to the user, $ {\bf{t}}_{i} \in {\mathbb{C}^{\emph{N} \times 1}} $ denotes the precoding vector, $ {d}_i[l] $ and $ {n}_i[l] \sim {\mathcal{C}}{\mathcal{N}}\left( {0,{\sigma _C^2}} \right)$ stands for the communication symbol and the received noise for the *i*-th user. $ l = 1,2,...,L $ is the symbol index, $ L $ is the length of the communication frame, and $P_R$ is the power of radar signal. Without loss of generality, we assume that the communication symbol is drawn from a normalized PSK constellation, while we note that the proposed concept of interference exploitation has been shown to offer benefits for other modulation formats [@7417066; @7831497]. Hence, the PSK symbol can be denoted as ${d_k}[l] = e ^ {j{\phi _k}[l]} $. It is assumed that $ {\bf{H}}=\left[{\bf{h}}_1, {\bf{h}}_2,...,{\bf{h}}_K \right] $ and $ {\bf{F}}=\left[{\bf{f}}_1, {\bf{f}}_2,...,{\bf{f}}_K \right] $ are flat Rayleigh fading and statistically independent with each other, and can be estimated by the BS through the pilot symbols.\
The second term at the right hand of (1) denotes the interference from radar to the user, where $ {\bf{S}} = \left[ {\bf{s}}_1,{\bf{s}}_2,...,{\bf{s}}_{L_R} \right] \in {\mathbb{C}^{M_t \times L_R}} $ is the radar transmit waveforms. According to the standard assumption in MIMO radar literature [@4350230; @1703855], ${\bf{S}}$ is set to be orthogonal, i.e., $\mathbb{E}\left[ {\bf s}_l{{\bf s}^H_l} \right] = \frac{1}{L_R}\sum\limits_{l = 1}^{L_R} {{\bf s}_l{{\bf s}^H_l}} = {\bf{I}}$, where $\mathbb{E}$ denotes the ensemble average. For notational convenience, it is assumed that the symbol duration of the radar waveform is the same as the communication signal. It should be highlighted that in order to preserve the orthogonality of $\bf S$, radar may utilize codeword that is longer than a typical communication frame. Without loss of generality, we assume $L_R=L$ for the ease of our derivation.\
Based on the above, the receive SINR is given by $${\gamma _i} = \frac{{{{\left| {{\bf{h}}_i^T{{\bf{t}}_i}} \right|}^2}}}{{\sum\limits_{k = 1,k \ne i}^K {{{\left| {{\bf{h}}_i^T{{\bf{t}}_k}} \right|}^2} + {P_R}{{\left\| {{{\bf{f}}_i}} \right\|}^2} + \sigma _C^2} }}, \forall i.$$ And the average transmit power of the BS is $${P_C} = \sum\limits_{k = 1}^K {{{\left\| {{{\bf{t}}_k}} \right\|}^2}}.$$\
With the presence of a point-like target located at direction $\theta$, the echo wave that received by radar at the *l*-th time slot is $${{\bf{y}}^R_l}=\alpha \sqrt{P_R} {\bf{A}}\left( \theta \right){{\bf s}_l} + {{\bf{G}}^T}\sum\limits_{k = 1}^K {{{\bf{t}}_k}{d_k}\left[ l \right]} + {\mathbf{z}}_l,$$ where $ {\bf G }=\left[{\bf{g}}_1,{\bf{g}}_2,...,{\bf{g}}_{M_r} \right] \in {\mathbb{C}}^{N\times M_r}$ is the interference channel matrix between BS and radar RX, and is also assumed to be flat Rayleigh fading and statistically independent with other two channels, and is estimated at the BS, $\alpha \in \mathbb{C}$ is the complex path loss of the path between radar and target, $ {\mathbf{z}}_l=\left[z_1\left[l\right], z_2\left[l\right],..., z_{M_r}\left[l\right]\right]^T \in {\mathbb{C}}^{M_r\times 1} $ is the received noise at the *l*-th time slot with $z_m[l]\sim {\mathcal{C}}{\mathcal{N}}\left( {0,{\sigma _R^2}} \right), \forall m$, $ {\bf{A}}\left( \theta \right) = {{\bf{a}}_R}\left( \theta \right){\bf{a}}_T^T\left( \theta \right) $, in which $ {{\bf{a}}_T}\left( \theta \right) \in {\mathbb{C}^{{M_t} \times 1}} $ and $ {{\bf{a}}_R}\left( \theta \right) \in {\mathbb{C}^{{M_r} \times 1}} $ are transmit and receive steering vectors of the radar antenna array. The model in (4) is assumed to be obtained in a single range-Doppler bin of the radar detector and thus ignores the range and Doppler parameters. In this paper, we apply the basic assumptions in [@1703855] on the radar model, which is $$\begin{gathered}
{M_r} = {M_t} = M, %\hfill \\
\;\;{{\mathbf{a}}_R}\left( \theta \right) = {{\mathbf{a}}_T}\left( \theta \right) = {\mathbf{a}}\left( \theta \right), \hfill \\
{{\mathbf{A}}_{im}}\left( \theta \right) = {{\mathbf{a}}_i}\left( \theta \right){{\mathbf{a}}_m}\left( \theta \right) = e^{ { - j\omega {\tau _{im}}\left( \theta \right)} } \hfill \\
\quad = e^ {\left( { - j\frac{{2\pi }}{\lambda }{{\left[ {\sin \left( \theta \right);\cos \left( \theta \right)} \right]}^T}\left( {{{\mathbf{x}}_i} + {{\mathbf{x}}_m}} \right)} \right)}, \hfill \\
\end{gathered}$$ where $ \omega $ and $ \lambda $ denote the frequency and the wavelength of the carrier, $ {{\bf A}_{im}}\left( \theta \right) $ is the $ i $-th element at the $ m $-th column of the matrix $\bf A$, which is the total phase delay of the signal that transmitted by the *i*-th element and received by the *m*-th element of the antenna array, and $ {{\bf{x}}_i} = \left[ {x_i^1;x_i^2} \right] $ is the location of the *i*-th element of the antenna array. In the above radar signal model, it is assumed that the communication interference is the only interference received by radar. Following the closely related literature, the interference caused by clutter and false targets is not considered[@7089157]. For convenience, we ignore the time index $ l $ in the rest of the paper unless otherwise specified. The interference from the BS on the *m*-th antenna of radar is given by $${u_m} = {\mathbf{g}}_m^T\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}d_k}.$$ We define the INR at the *m*-th receive antenna of radar as $${r_m} = \frac{{{{\left| {{u_m}} \right|}^2}}}{{\sigma _R^2}}. %= \frac{{{{\left| {{\bf{g}}_m^T\sum\limits_{k = 1}^K {{{\bf{t}}_k}} } \right|}^2}}}{{\sigma _R^2}}.$$\
From a conventional perspective, all interference should be treated as harmful when optimizing the performance of the two systems. The power minimization problem of the BS subject to INR and SINR thresholds is formulated as $$\begin{array}{*{20}{l}}
{{{\cal{P}}_0}:\mathop {\min }\limits_{{{\mathbf{t}}_k}} \;\;\sum\limits_{k = 1}^K {{{\left\| {{{\mathbf{t}}_k}} \right\|}^2}} {\text{ }}} \\
\displaystyle {s.t.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{{\left| {{\mathbf{h}}_i^T{{\mathbf{t}}_i}} \right|}^2}}}{{\sum\limits_{k = 1,k \ne i}^K {{{\left| {{\mathbf{h}}_i^T{{\mathbf{t}}_k}} \right|}^2} + {P_R}{{\left\| {{{\mathbf{f}}_i}} \right\|}^2} + \sigma _C^2} }} \ge {\Gamma _i},\forall i,} \\
\displaystyle {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{{{\left| {{\mathbf{g}}_m^T\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}} {d_k} } \right|}^2}}}{{\sigma _R^2}} \le {R_m},\forall m,} \\
\end{array}$$ where $ \Gamma_i $ is the required SINR of the *i*-th communication user, $ R_m $ is the maximum tolerable INR level of the *m*-th receive element of radar. Similarly, we can formulate the optimization problem that minimizes the interference to radar while guaranteeing the BS power budget and the required SINR level at each user, which is given as $$\begin{gathered}
{{\cal{P}}_1}: \mathop {\min }\limits_{{{\mathbf{t}}_k}} {\kern 1pt} {\kern 1pt} \sum\limits_{m = 1}^M {{\kern 1pt} {\kern 1pt} {{\left| {{\mathbf{g}}_m^T\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}} {d_k}} \right|}^2}} \hfill \\
s.t.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{k = 1}^K {{{\left\| {{{\mathbf{t}}_k}} \right\|}^2}} \le P, \hfill \\
\;\;\;\;\;\frac{{{{\left| {{\mathbf{h}}_i^T{{\mathbf{t}}_i}} \right|}^2}}}{{\sum\limits_{k = 1,k \ne i}^K {{{\left| {{\mathbf{h}}_i^T{{\mathbf{t}}_k}} \right|}^2} + {P_R}{{\left\| {{{\mathbf{f}}_i}} \right\|}^2} + \sigma _C^2} }} \ge {\Gamma _i},\forall i, \hfill \\
\end{gathered}$$ where $P$ is the budget of the BS transmit power. Problem ${{\cal{P}}_0}$ and ${{\cal{P}}_1}$ can be transformed into Semidefinite Program (SDP)[@boyd2004convex] with Semidefinite Relaxation techniques, and thus can be solved by numerical tools. We refer readers to [@liu2016robust; @5447068; @5447076] for more details on this topic. As shown in Fig. 1 by red arrows, it is worth noting the above problems ignore the fact that for each user, interference from other users can contribute to the received signal power constructively. In this paper, we aim to show that the solution of these problems is suboptimal from an instantaneous point of view and design a symbol-based beamforming method in accordance to the concept of constructive interference.
Beamforming with Constructive Interference
==========================================
![The principle of constructive interference.[]{data-label="fig:2"}](TSPFig2.eps){width="2.8in"}
As per the model of[@7103338], the instantaneous interference can be divided into two categories, constructive interference and destructive interference. Generally, the constructive interference is defined as the interference that moves the received symbol away from the decision thresholds. The purpose of the CI-based beamforming is to rotate the known interference from other users such that the resultant received symbol falls into the constructive region. This is shown in Fig. 2, where we denote the constructive area of the QPSK symbol by the gray shade. It has been proven in[@7103338] that the optimization will become more relaxed than conventional interference cancellation optimizations due to the expansion of the optimization region. Hence, the performance of the beamformer is improved. Here we consider the instantaneous transmit power, which is given as $${P_T}[l] = {\left\| {\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}e^{ {j\left({\phi _k}[l]-{\phi_1}[l]\right)} }} } \right\|^2},$$ where $d_1[l]=e^{j\phi_1[l]}$ is used as the phase reference. Based on [@7103338], we rewrite the SINR constraints of ${{\cal{P}}_0}$ and ${{\cal{P}}_1}$ in a CI sense, and reformulate the power minimization problem ${{\cal{P}}_0}$ as (11) on the top of the next page as ${{\cal{P}}_2}$, where $\psi=\frac{\pi}{M_p}$, and $M_p$ is the PSK modulation order. Readers are referred to [@7103338] for a detailed derivation of the CI constraints and classification. It should be highlighted that, while here we focus on PSK constellations, the optimizations ${\cal{P}}_2$ onwards can be readily adapted to other constellation formats such as Quadrature Amplitude Modulation (QAM)[@7417066; @7831497]. Note that ${{\cal{P}}_2}$ is convex in contrast to the non-convex ${{\cal{P}}_0}$ and ${{\cal{P}}_1}$. To be more specific, problem ${{\cal{P}}_2}$ is a second-order cone program (SOCP) and can be solved optimally by numerical tools.\
In both ${{\cal{P}}_0}$ and ${{\cal{P}}_2}$, by letting $R_m = 0$, it follows $ {{\mathbf{g}}_m^T\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}} {d_k} } =0 $, which requires the transmitting signal to fall into the null space of the interference matrix $\bf G$ and causes zero interference to radar. This yields the solution with which the radar can achieve the best performance. However, the strict equality will result in a large transmit power at BS. On the other hand, if we let $R_m \to \infty$, the INR constraints will be ineffective, which is equivalent to the typical downlink power minimization in the absence of radar. This trade-off between radar and communication performance will be further evaluated by numerical simulations below.
$$\begin{array}{*{20}{l}}
{{\cal{P}}_2}:\mathop {\min }\limits_{{{\mathbf{t}}_k}} \;\;{\left\| {\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}e^{ {j\left({\phi _k}-{\phi_1}\right)} }} } \right\|^2} \hfill \\
\displaystyle s.t.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left| {\operatorname{Im} \left( {{\mathbf{h}}_i^T\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}e^{j\left( {{\phi _k} - {\phi _i}} \right)}} } \right)} \right| \le \left( {\operatorname{Re} \left( {{\mathbf{h}}_i^T\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}e^{j\left( {{\phi _k} - {\phi _i}} \right)} } } \right) - \sqrt {{\Gamma _i}\left( {\sigma _C^2 + {P_R}{{\left\| {{{\mathbf{f}}_i}} \right\|}^2}} \right)} } \right)\tan {\psi } ,\forall i, \hfill \\
{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt}{\kern 1pt}{\left| {{\mathbf{g}}_m^T\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}} {e^{j\phi_k}}} \right|^2} \le {R_m}\sigma _R^2,\forall m.
\end{array}$$
Following the virtual multicast model in [@7103338], the power minimization problem ${{\cal{P}}_2}$ can be equivalently written as $$\begin{array}{*{20}{l}}
{{\cal{P}}_3}:\mathop {\min }\limits_{\mathbf{w}} {\left\| {\mathbf{w}} \right\|^2} \hfill \\
\displaystyle s.t.{\kern 1pt}\left| {\operatorname{Im} \left( {{\mathbf{{\tilde h}}}_i^T{\mathbf{w}}} \right)} \right| \le \left( {\operatorname{Re} \left( {{\mathbf{{\tilde h}}}_i^T{\mathbf{w}}} \right) - \sqrt {{{\tilde \Gamma }_i}} } \right)\tan {\psi }, \forall i, \hfill \\
\;\;\;\;\;{\kern 1pt}\left| {{\mathbf{\tilde g}}_m^T{\mathbf{w}}} \right| \le \sqrt {{R_m}\sigma _R^2} ,{\kern 1pt} {\kern 1pt} \forall m, \hfill \\
\end{array}$$ where $ {\mathbf{w}} \triangleq \sum\limits_{k = 1}^K {{{\mathbf{t}}_k}{e^{j\left( {{\phi _k} - {\phi _1}} \right)}}}$, ${{\mathbf{{\tilde h}}}_i} \triangleq {{\mathbf{h}}_i}{e^{j\left( {{\phi _1} - {\phi _i}} \right)}}$, ${{\mathbf{\tilde g}}_m} \triangleq {{\mathbf{g}}_m}{e^{j{\phi _1}}}$, ${{\tilde \Gamma }_i}={\Gamma _i}\left( {\sigma _C^2 + {P_R}{{\left\| {{{\mathbf{f}}_i}} \right\|}^2}} \right)$. Similarly, the CI-based interference minimization problem is given by $$\begin{array}{*{20}{l}}
{{\cal{P}}_4}:\mathop {\min }\limits_{\mathbf{w}} \sum\limits_{m = 1}^M {{{\left| {{\mathbf{\tilde g}}_m^T{\mathbf{w}}} \right|}^2}} \hfill \\
\displaystyle s.t.{\kern 1pt}\left| {\operatorname{Im} \left( {{\mathbf{{\tilde h}}}_i^T{\mathbf{w}}} \right)} \right| \le \left( {\operatorname{Re} \left( {{\mathbf{{\tilde h}}}_i^T{\mathbf{w}}} \right) - \sqrt {{{\tilde \Gamma }_i}} } \right)\tan{\psi }, \forall i, \hfill \\
\;\;\;\;\;{\kern 1pt}\;{\left\| {\mathbf{w}} \right\|} \le \sqrt{P}.
\end{array}$$ After obtaining the optimal solution ${\bf{w}}$, the beamforming vectors can be obtained as $${{\bf t}_1} = \frac{{\mathbf{w}}}{K},$$ $${{\bf t}_k} = \frac{{{\mathbf{w}}{e^{j\left( {{\phi _1} - {\phi _k}} \right)}}}}{K},\forall k.$$ Note that both ${{\cal{P}}_3}$ and ${{\cal{P}}_4}$ are convex and can be easily solved by numerical tools. To make the proposed method more realizable in practical scenarios, we will take ${{\cal{P}}_3}$ as an example to derive an efficient algorithm to solve it, and a similar algorithm can be also applied to ${{\cal{P}}_4}$.
Efficient Algorithm for Power Minimization Beamforming
======================================================
Real representation of the problem
----------------------------------
For the ease of our further analysis, we first derive the real representation of the problem. Let us rewrite the related channel vectors and the beamforming vector as follows $$\begin{array}{*{20}{l}}
{{{\mathbf{{\tilde h}}}}_i} = {{{\mathbf{{\tilde h}}}}_{Ri}} + j{{{\mathbf{{\tilde h}}}}_{Ii}}, \hfill \\
{{{\mathbf{\tilde g}}}_m} = {{{\mathbf{\tilde g}}}_{Rm}} + j{{{\mathbf{\tilde g}}}_{Im}}, \hfill \\
{\mathbf{w}} = {{\mathbf{w}}_R} + j{{\mathbf{w}}_I}, \hfill \\
\end{array}$$ where $$\begin{array}{*{20}{l}}
{{{\mathbf{{\tilde h}}}}_{Ri}} = \operatorname{Re} \left( {{{{\mathbf{{\tilde h}}}}_i}} \right),{{{\mathbf{{\tilde h}}}}_{Ii}} = \operatorname{Im} \left( {{{{\mathbf{{\tilde h}}}}_i}} \right), \hfill \\
{{{\mathbf{\tilde g}}}_{Rm}} = \operatorname{Re} \left( {{{{\mathbf{\tilde g}}}_m}} \right),{{{\mathbf{\tilde g}}}_{Im}} = \operatorname{Im} \left( {{{{\mathbf{\tilde g}}}_m}} \right), \hfill \\
{{\mathbf{w}}_R} = \operatorname{Re} \left( {\mathbf{w}} \right),{{\mathbf{w}}_I} = \operatorname{Im} \left( {\mathbf{w}} \right). \hfill \\
\end{array}$$ Then we define the following real-valued vectors and matrices $$\begin{array}{*{20}{l}}
{{{\mathbf{{\bar h}}}}_i} = \left[ {{{{\mathbf{{\tilde h}}}}_{Ri}};{{{\mathbf{{\tilde h}}}}_{Ii}}} \right], \hfill \\
{{\mathbf{w}}_1} = \left[ {{{\mathbf{w}}_I};{{\mathbf{w}}_R}} \right],{{\mathbf{w}}_2} = \left[ {{{\mathbf{w}}_R}; - {{\mathbf{w}}_I}} \right], \hfill \\
{{\pmb{\beta}}_m}{\text{ = }}\left[ {\begin{array}{*{20}{c}}
{{{{\mathbf{\tilde g}}}_{Rm}}}&{{{{\mathbf{\tilde g}}}_{Im}}} \\
{{{{\mathbf{\tilde g}}}_{Im}}}&{ - {{{\mathbf{\tilde g}}}_{Rm}}}
\end{array}} \right],{\mathbf{\Pi }} = \left[ {\begin{array}{*{20}{c}}
{\mathbf{0}}_K&{{\text{ - }}{\mathbf{I}}_K} \\
{\mathbf{I}}_K&{\mathbf{0}}_K
\end{array}} \right],
\end{array}$$ where ${\bf{I}}_K$ and ${\bf{0}}_K$ denote the $K\times K$ identity matrix and all-zero matrix respectively. Thus we obtain $$\begin{array}{*{20}{l}}
\operatorname{Re} \left( {{\mathbf{{\tilde h}}}_i^T{\mathbf{w}}} \right) = {{{\mathbf{{\bar h}}}}_i}^T{{\mathbf{w}}_2}, \hfill \\
\operatorname{Im} \left( {{\mathbf{{\tilde h}}}_i^T{\mathbf{w}}} \right) = {{{\mathbf{{\bar h}}}}_i}^T{{\mathbf{w}}_1} = {{{\mathbf{{\bar h}}}}_i}^T{\mathbf{\Pi }}{{\mathbf{w}}_2} \triangleq {\mathbf{b}}_i^T{{\mathbf{w}}_2}, \hfill \\
{\left| {{\mathbf{\tilde g}}_m^T{\mathbf{w}}} \right|^2} = {\left\| {\left[ {\begin{array}{*{20}{c}}
{{\mathbf{\tilde g}}_{Rm}^T}&{{\mathbf{\tilde g}}_{Im}^T} \\
{{\mathbf{\tilde g}}_{Im}^T}&{ - {\mathbf{\tilde g}}_{Rm}^T}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{{\mathbf{w}}_R}} \\
{ - {{\mathbf{w}}_I}}
\end{array}} \right]} \right\|^2} = {\left\| {{\pmb{\beta }}_m^T{{\mathbf{w}}_2}} \right\|^2}. \hfill \\
\end{array}$$ Finally, the real version of the problem is given as $$\begin{array}{*{20}{l}}
{{\cal{P}}_5}: \mathop {\min }\limits_{{\mathbf{w}}_2} \;\;{\kern 1pt} {\kern 1pt} {\left\| {{{\mathbf{w}}_2}} \right\|^2} \hfill \\
\displaystyle s.t.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathbf{b}}_i^T{{\mathbf{w}}_2} - {{{\mathbf{{\bar h}}}}_i}^T{{\mathbf{w}}_2}\tan {\psi } + \sqrt {{{\tilde \Gamma }_i}} \tan {\psi } \le 0,\forall i, \hfill \\
\displaystyle \;\;\;\;\;\; - {\mathbf{b}}_i^T{{\mathbf{w}}_2} - {{{\mathbf{{\bar h}}}}_i}^T{{\mathbf{w}}_2}\tan{\psi } + \sqrt {{{\tilde \Gamma }_i}} \tan {\psi }\le 0,\forall i, \hfill \\
\;\;\;\;\;\;{\left\| {{\pmb{\beta }}_m^T{{\mathbf{w}}_2}} \right\|^2} \le {R_m}\sigma _R^2,\forall m.
\end{array}$$
The Dual Problem
----------------
In order to reveal the structure of the solution, we formulate the dual problem of ${{\cal{P}}_5}$. Let us define the dual variable that associate with the three constraints in (20) as $ {\mathbf{u}},{\mathbf{v}},{\mathbf{c}}$ respectively, where $u_i \ge 0, v_i \ge 0, c_m \ge 0, \forall i, \forall m$ are the elements of the three dual vectors. The corresponding Lagrangian is given as (21) at the top of the next page.
$$\begin{array}{*{20}{l}}
\displaystyle \mathcal{L}\left( {{{\mathbf{w}}_2},{\mathbf{u}},{\mathbf{v}},{\mathbf{c}}} \right) \hfill \\
\displaystyle = {\left\| {{{\mathbf{w}}_2}} \right\|^2} + \sum\limits_{i = 1}^K {{u_i}\left( {{\mathbf{b}}_i^T{{\mathbf{w}}_2} - {\mathbf{{\bar h}}}_i^T{{\mathbf{w}}_2}\tan \psi + \sqrt {{{\tilde \Gamma }_i}} \tan \psi } \right)} \hfill \\
\displaystyle + \sum\limits_{i = 1}^K {{v_i}\left( { - {\mathbf{b}}_i^T{{\mathbf{w}}_2} - {\mathbf{{\bar h}}}_i^T{{\mathbf{w}}_2}\tan \psi + \sqrt {{{\tilde \Gamma }_i}} \tan \psi } \right)} + \sum\limits_{m = 1}^M {{c_m}\left( {{{\left\| {\pmb \beta _m^T{{\mathbf{w}}_2}} \right\|}^2} - {R_m}\sigma _R^2} \right)} \hfill \\
\displaystyle = {\mathbf{w}}_2^T\left( {{\bf I} + \sum\limits_{m = 1}^M {{c_m}}\pmb \beta _m \pmb \beta _m^T} \right){{\mathbf{w}}_2} + \sum\limits_{i = 1}^K {{\kern 1pt} \left[ {\left( {{u_i} - {v_i}} \right){\mathbf{b}}_i^T - \left( {{u_i} + {v_i}} \right){\mathbf{{\bar h}}}_i^T\tan \psi } \right]} {{\mathbf{w}}_2} %\hfill \\
\displaystyle + \tan \psi \sum\limits_{i = 1}^K {\sqrt {{{\tilde \Gamma }_i}} \left( {{u_i} + {v_i}} \right)} - {R_m}\sigma _R^2\sum\limits_{m = 1}^M {{c_m}}.
\end{array}$$
By the following definitions $$\begin{array}{*{20}{l}}
{\mathbf{{\bar h}}} = \left[ {{{{\mathbf{{\bar h}}}}_1},{{{\mathbf{{\bar h}}}}_2},...,{{{\mathbf{{\bar h}}}}_K}} \right],
{\mathbf{b}} = \left[ {{{\mathbf{b}}_1},{{\mathbf{b}}_2},...,{{\mathbf{b}}_K}} \right], {\mathbf{1}} = \left[ {{{\mathbf{I}}_K};{{\mathbf{I}}_K}} \right], \hfill \\
{\pmb{\lambda }} = \left[ {{\mathbf{u}};{\mathbf{v}}} \right], {\pmb{\beta }} = \left[ {{{\pmb{\beta }}_1},{{\pmb{\beta }}_2},...,{{\pmb{\beta }}_M}} \right], {\mathbf{R}} = \left[ {{R_1},{R_2},...,{R_M}} \right],\hfill \\
{\mathbf{c}} = \left[ {{c_1};{c_2};...;{c_{{M}}}} \right], {\mathbf{\tilde c}} = \left[ {{c_1};{c_1};{c_2};{c_2};...;{c_{{M}}};{c_{{M}}}} \right], \hfill \\
\displaystyle {\mathbf{\tilde \Gamma }} = \left[ {{\tilde\Gamma _1};{\tilde\Gamma _2};...;{\tilde\Gamma _K}} \right], {\mathbf{A}} = \left[ {{\mathbf{{\bar h}}}\tan \psi - {\mathbf{b}},{\mathbf{{\bar h}}}\tan \psi + {\mathbf{b}}} \right], \hfill \\
\end{array}$$ the Lagrangian can be further simplified as $$\begin{gathered}
\mathcal{L}\left( {{{\mathbf{w}}_2},{\mathbf{u}},{\mathbf{v}},{\mathbf{c}}} \right) \hfill \\
= {\mathbf{w}}_2^T\left( {{\mathbf{I}} + {\pmb{\beta }}\operatorname{diag} \left( {{\mathbf{\tilde c}}} \right){{\pmb{\beta }}^T}} \right){{\mathbf{w}}_2} + {{\pmb{\lambda }}^T}{{\mathbf{A}}^T}{{\mathbf{w}}_2} \hfill \\
+ \tan \psi \sqrt {{{{\mathbf{\tilde \Gamma }}}^T}} {{\mathbf{1}}^T}{\pmb{\lambda }} - \sigma _R^2{{\mathbf{R}}^T}{\mathbf{c}}, \hfill \\
\end{gathered}$$ where $\operatorname{diag}(\bf{x})$ denotes the diagonal matrix whose diagonal elements are given by $\bf{x}$. Let $ \frac{{\partial \mathcal{L}}}{{\partial {{\mathbf{w}}_2}}} = 0$, the optimal solution of $\bf w_2$ is given by $${\mathbf{w}}_2^* = - \frac{{{{\left( {{\mathbf{I}} + {\pmb{\beta }}\operatorname{diag} \left( {{\mathbf{\tilde c}}} \right){{\pmb{\beta }}^T}} \right)}^{ - 1}}{\mathbf{A \pmb \lambda }}}}{2},$$ which implies ${\pmb{\lambda}} \ne \bf{0} $, for the reason that ${\pmb{\lambda}} = \bf{0} $ yields the trivial solution of ${\mathbf{w}}_2^*=\bf{0}$. Substituting the optimal ${\mathbf{w}}_2^*$ into the Lagrangian leads to $$\begin{gathered}
\mathcal{L}\left( {{\mathbf{u}},{\mathbf{v}},{\mathbf{c}}} \right) = - \frac{1}{4}{{\pmb{\lambda }}^T}{{\mathbf{A}}^T}{\left( {{\mathbf{I}}{\text{ + }}{\pmb{\beta}}\operatorname{diag} \left( {{\mathbf{\tilde c}}} \right){{\pmb{\beta}}^T}} \right)^{ - 1}}{\mathbf{A\pmb\lambda }} \hfill \\
\quad + \tan \psi \sqrt {{{{\mathbf{\tilde \Gamma }}}^T}} {{\mathbf{1}}^T}{\pmb{\lambda }} - \sigma _R^2{{\mathbf{R}}^T}{\mathbf{c}}. \hfill \\
\end{gathered}$$ Therefore, the dual problem is given as $$\begin{gathered}
{{\cal{P}}_6}:\;\;\mathop{\max} \limits_{\pmb{\lambda}, \mathbf{c}} \;\; - \frac{1}{4}{{\pmb{\lambda }}^T}{{\mathbf{A}}^T}{\left( {{\mathbf{I}}{\text{ + }}{\pmb{\beta}}\operatorname{diag} \left( {{\mathbf{\tilde c}}} \right){{\pmb{\beta}}^T}} \right)^{ - 1}}{\mathbf{A\pmb\lambda }} \hfill \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\; \quad \quad + \tan \psi \sqrt {{{{\mathbf{\tilde \Gamma }}}^T}} {{\mathbf{1}}^T}{\pmb{\lambda }} - \sigma _R^2{{\mathbf{R}}^T}{\mathbf{c}} \hfill \\
s.t.\;\;\;{\pmb{\lambda }} \ge {\mathbf{0}},{\mathbf{c}} \ge {\mathbf{0}}. \hfill \\
\end{gathered}$$ Note that when removing the INR constraints, the dual problem is the same as the original CI-based power minimization problem in [@7103338].
Karush-Kuhn-Tucker Conditions
-----------------------------
Let us first rewrite the dual problem as the following standard convex form $$\begin{gathered}
{{\cal{P}}_7}:\;\;\mathop{\min} \limits_{\pmb{\lambda}, \mathbf{c}} \;\;f\left( {{\pmb{\lambda }},{\mathbf{c}}} \right)= \frac{1}{4}{{\pmb{\lambda }}^T}{{\mathbf{A}}^T}{\left( {{\mathbf{I}}{\text{ + }}{\pmb{\beta}}\operatorname{diag} \left( {{\mathbf{\tilde c}}} \right){{\pmb{\beta}}^T}} \right)^{ - 1}}{\mathbf{A\pmb\lambda }} \hfill \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\; \quad \quad - \tan \psi \sqrt {{{{\mathbf{\tilde \Gamma }}}^T}} {{\mathbf{1}}^T}{\pmb{\lambda }} + \sigma _R^2{{\mathbf{R}}^T}{\mathbf{c}} \hfill \\
s.t.\;\;\;{\pmb{\lambda }} \ge {\mathbf{0}},{\mathbf{c}} \ge {\mathbf{0}}. \hfill \\
\end{gathered}$$ It is easy to observe that the primal problem ${{\cal{P}}_5}$ is a standard Quadratically Constrained Quadratic Program (QCQP), and is convex and the strong duality holds, thus the Karush-Kuhn-Tucker (KKT) Conditions are sufficient for primal and dual optimal variables [@boyd2004convex], which are denoted by ${\mathbf{w}}_2^*$, $\pmb \lambda^*$ and $\bf c^*$ respectively, and they have zero duality gap. Based on the complementary slackness conditions we have $$\begin{gathered}
c_m^*\left( {{\left\| {\pmb \beta _m^T{{\mathbf{w}}_2}} \right\|}^2 - {R_m}\sigma _R^2} \right) = 0,\forall m.\hfill \\
% u_i^*\left( {{\mathbf{b}}_i^T{\mathbf{w}}_2^* - {\mathbf{{\bar h}}}_i^T{\mathbf{w}}_2^*\tan \psi + \sqrt {{{\tilde \Gamma }_i}} \tan \psi } \right) = 0,\forall i; \hfill \\
% v_i^*\left( { - {\mathbf{b}}_i^T{\mathbf{w}}_2^* - {\mathbf{{\bar h}}}_i^T{\mathbf{w}}_2^*\tan \psi + \sqrt {{{\tilde \Gamma }_i}} \tan \psi } \right) = 0,\forall i. \hfill \\
\end{gathered}$$ When removing the INR constraints, the optimization ${{\cal{P}}_5}$ has the same structure with the original CI-based power minimization problem[@7103338], which is given as $$\begin{array}{*{20}{l}}
{{\cal{P}}_8}:{\kern 1pt} \;\;\mathop {\min }\limits_{{\mathbf{w}}_2} \;\;{\kern 1pt} {\kern 1pt} {\left\| {{{\mathbf{w}}_2}} \right\|^2} \hfill \\
\displaystyle s.t.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathbf{b}}_i^T{{\mathbf{w}}_2} - {{{\mathbf{{\bar h}}}}_i}^T{{\mathbf{w}}_2}\tan {\psi } + \sqrt {{{\tilde \Gamma }_i}} \tan {\psi } \le 0,\forall i, \hfill \\
\displaystyle \;\;\;\;\;\; - {\mathbf{b}}_i^T{{\mathbf{w}}_2} - {{{\mathbf{{\bar h}}}}_i}^T{{\mathbf{w}}_2}\tan{\psi } + \sqrt {{{\tilde \Gamma }_i}} \tan {\psi }\le 0,\forall i. \hfill \\
\end{array}$$ The dual problem of ${{\cal{P}}_8}$ is given by $$\begin{gathered}
{{\cal{P}}_9}:\mathop{\min} \limits_{\pmb{\lambda}} \;\;\frac{{{{\left\| {{\mathbf{A\pmb\lambda }}} \right\|}^2}}}{4} - \tan \psi \sqrt {{{{\mathbf{\tilde \Gamma }}}^T}} {{\mathbf{1}}^T}{\pmb{\lambda }} \hfill \\
s.t.\;\;{\pmb{\lambda }} \ge 0, \hfill \\
\end{gathered}$$ and the optimal solution to ${\cal P}_8$ has the structure of $${\mathbf{w}}_2^* = - \frac{{{\mathbf{A\pmb\lambda }}}}{2},$$ where $\bf A$ and $\pmb \lambda$ are defined in (22). By substituting (31) in the INR constraint to obtain $${\left\| {{\pmb{\beta }}_m^T{{\mathbf{w}}_2}} \right\|^2}= \frac{1}{4}{\left\| {{\pmb{\beta }}_m^T{\mathbf{A \pmb\lambda }}} \right\|^2}.$$ Therefore, if ${R_m}\sigma _R^2\ge\frac{1}{4}{\left\| {{\pmb{\beta }}_m^T{\mathbf{A \pmb\lambda }}} \right\|^2}$, (31) is a feasible point for ${{\cal{P}}_5}$. Since (31) is the optimal point of ${{\cal{P}}_8}$, this implies that it is also the optimal point for ${{\cal{P}}_5}$ for the reason that the minimum value of ${{\cal{P}}_5}$ will always be greater than or equal to the minimum value of ${{\cal{P}}_8}$ due to the extra INR constraint. Thus the related INR constraint will always be satisfied, and $c_m=0$. By denoting the optimal solution of ${{\cal{P}}_9}$ by $\pmb \lambda_0$, the following corollary holds immediately.\
*Corollary 1:* If ${R_m}\sigma _R^2>\frac{1}{4}{\left\| {{\pmb{\beta }}_m^T{\mathbf{A \pmb\lambda_0 }}} \right\|^2}$, ${{\cal{P}}_5}$ is equivalent to the original CI problem ${{\cal{P}}_8}$, where $\pmb\lambda_0$ is the optimal solution of ${{\cal{P}}_9}$.
Efficient Gradient Projection Method
------------------------------------
The closed form of the optimal solution to ${{\cal{P}}_7}$ is difficult to derive. Nevertheless, thanks to the simple constraints with only bounds on the variables, it is convenient to apply a gradient projection algorithm to solve the problem[@wright1999numerical]. Let us first derive the gradient of the dual function as follows. By letting ${\mathbf{M}} = {\left( {{\mathbf{I}}{\text{ + }}{\pmb{\beta }}\operatorname{diag} \left( {{\mathbf{\tilde c}}} \right){{\pmb{\beta }}^T}} \right)^{ - 1}}$, the derivative is given as $$\begin{gathered}
\frac{{\partial f}}{{\partial {\pmb{\lambda }}}} = \frac{1}{2}{{\pmb{\lambda }}^T}{{\mathbf{A}}^T}{\mathbf{MA}} - \tan \psi \sqrt {{{{\mathbf{\tilde \Gamma }}}^T}} {{\mathbf{1}}^T}, \hfill \\
\frac{{\partial f}}{{\partial {c_m}}} = - \frac{1}{4}{\left| {{{\pmb{\lambda }}^T}{{\mathbf{A}}^T}{\mathbf{M}}{{\pmb{\beta }}_m}} \right|^2} + \sigma _R^2{R_m},\forall m. \hfill \\
\end{gathered}$$ Thus the gradient is give by $$\begin{gathered}
\triangledown f\left( {{\pmb{\lambda }},{\mathbf{c}}} \right) = {\left[ {\frac{{\partial f}}{{\partial {\pmb{\lambda }}}},\frac{{\partial f}}{{\partial {\mathbf{c}}}}} \right]^T} \hfill \\
\quad = \left[ \begin{gathered}
\frac{1}{2}{{\mathbf{A}}^T}{\mathbf{MA \pmb\lambda }} - \tan \psi {{\mathbf{1}}} \sqrt {{{{\mathbf{\tilde \Gamma }}}}} ; \hfill \\
- \frac{1}{4}{\left| {{{\pmb{\lambda }}^T}{{\mathbf{A}}^T}{\mathbf{M}}{{\pmb{\beta }}_1}} \right|^2} + \sigma _R^2{R_1}; \hfill \\
- \frac{1}{4}{\left| {{{\pmb{\lambda }}^T}{{\mathbf{A}}^T}{\mathbf{M}}{{\pmb{\beta }}_2}} \right|^2} + \sigma _R^2{R_2}; \hfill \\
... \hfill \\
- \frac{1}{4}{\left| {{{\pmb{\lambda }}^T}{{\mathbf{A}}^T}{\mathbf{M}}{{\pmb{\beta }}_M}} \right|^2} + \sigma _R^2{R_M} \hfill \\
\end{gathered} \right]. \hfill \\
\end{gathered}$$ Based on above derivations, the following Algorithm 1 is proposed to solve problem ${{\cal{P}}_7}$, where we use an iterative gradient projection method, and the step size can be decided by the Armijo rule or other backtracking linesearch methods [@wright1999numerical]. After obtaining the optimal ${\bf w}_2$, the beamforming vectors can be calculated by (14) and (15).
------------------------------------------------------------------------
[**[Algorithm 1]{}**]{}
${\mathbf{H}},{\mathbf{G}},{\mathbf{F}},{\mathbf{\Gamma }},{\mathbf{R}},{\sigma _c},{\sigma _R}$. Optimal solution ${{\bf w}^*_2}$ for problem ${{\cal{P}}_5}$.\
Initialize randomly $\pmb \lambda^{(0)} \ge 0, \mathbf c^{(0)} \ge 0 $. In the *i*th iteration, update $\pmb \lambda$ and $\bf c$ by:
$$\left[ {{{\pmb{\lambda }}^{\left(i \right)}},{{\mathbf{c}}^{\left( i \right)}}} \right] = \max \left( {\left[ {{{\pmb{\lambda }}^{\left( i \right)}},{{\mathbf{c}}^{\left( i \right)}}} \right] - {a_i}\triangledown f\left( {{{\pmb{\lambda }}^{\left( {i - 1} \right)}},{{\mathbf{c}}^{\left( {i - 1} \right)}}} \right),{\mathbf{0}}} \right),$$
where the step size $a_i$ is calculated by the backtracking linesearch method. Go back to 2 until convergence. Calculate ${{\bf w}^*_2}$ by
$${\mathbf{w}}_2^* = - \frac{{{{\left( {{\mathbf{I}} + {\pmb{\beta }}\operatorname{diag} \left( {{\mathbf{\tilde c}^{(i)}}} \right){{\pmb{\beta }}^T}} \right)}^{ - 1}}{\mathbf{A \pmb \lambda }^{(i)}}}}{2}.$$
**end**
------------------------------------------------------------------------
Impact on Radar Performance
===========================
SDR-based beamforming
---------------------
The interference from BS to radar will have an impact on radar’s performance, which will lower the detection probability and the accuracy for Direction of Arrival (DoA) estimation. First we consider the detection problem. Note that the target detection process can be described as a Binary Hypothesis Testing problem, which is given by $${{\bf{y}}^R_l}=\left\{ \begin{gathered}
{{\cal H}_1}:\alpha \sqrt{P_R} {\bf{A}}\left( \theta \right){{\bf s}_l} + {{\bf{G}}^T}\sum\limits_{k = 1}^K {{{\bf{t}}_k}{d_k}\left[ l \right]} + {\mathbf{z}}_l,\hfill \\\;\;\;\;\;\;\;\; l = 1,2,...,L, \hfill \\
{{\cal H}_0}:{{\bf{G}}^T}\sum\limits_{k = 1}^K {{{\bf{t}}_k}{d_k}\left[ l \right]} + {\mathbf{z}}_l,\;\;l = 1,2,...,L. \hfill \\
\end{gathered} \right.$$\
Due to the unknown parameters $\alpha$ and $\theta$, we use the Generalized Likelihood Ratio Test (GLRT) method to solve the above problem. Consider the sufficient statistic of the received signal, which is obtained by matched filtering [@1703855], and is given by $$\begin{gathered}
{\mathbf{\tilde Y}} = \frac{1}{{\sqrt L }}\sum\limits_{l = 1}^L {{\bf{y}}^R_l}{{\mathbf{s}}^H_l} \hfill \\
\quad= \alpha {\sqrt {LP_R} }{\mathbf{A}}\left( \theta \right) + \frac{1}{{\sqrt L }}\sum\limits_{l = 1}^L {\left( {{{\mathbf{G}}^T}\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}{d_k}\left[ l \right] + {\mathbf{z}}_l} } \right)} {{\mathbf{s}}^H_l} \hfill \\
\end{gathered}$$ Let ${\mathbf{\tilde y}}$ be the vectorization of ${\mathbf{\tilde Y}}$, we have $$\begin{gathered}
{\mathbf{\tilde y}} = \operatorname{vec} \left( {{\mathbf{\tilde Y}}} \right) \hfill \\
\quad= \alpha {\sqrt {LP_R} }\operatorname{vec} \left( {{\mathbf{A}}\left( \theta \right)} \right) \hfill \\
\quad + \operatorname{vec} \left( \frac{1}{{\sqrt L }}\sum\limits_{l = 1}^L {\left( {{{\mathbf{G}}^T}\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}{d_k}\left[ l \right] + {\mathbf{z}}_l} } \right)} {{\mathbf{s}}^H_l} \right) \hfill \\
\quad\triangleq \alpha {\sqrt {LP} _R}\operatorname{vec} \left( {{\mathbf{A}}\left( \theta \right)} \right) + {\pmb{\varepsilon }}, \hfill \\
\end{gathered}$$ where $\pmb \varepsilon$ is zero-mean, complex Gaussian distributed, and has the block covariance matrix as $${\mathbf{C}} = \left[ {\begin{array}{*{20}{c}}
{{\mathbf{J}} + \sigma _R^2{{\mathbf{I}}_M}}&{}&{\mathbf{0}} \\
{}&{...}&{} \\
{\mathbf{0}}&{}&{{\mathbf{J}} + \sigma _R^2{{\mathbf{I}}_M}}
\end{array}} \right],$$ where ${\mathbf{C}} \in {\mathbb{C}}^{{M^2}\times {M^2}}$, and ${\mathbf{J}}={{\mathbf{G}}^T}\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}{\mathbf{t}}_k^H} {{\mathbf{G}}^*} $. Hence, (35) is equivalent to the following hypothesis: $${\mathbf{\tilde y}} = \left\{ \begin{gathered}
{{\cal H}_1}:\alpha {\sqrt {LP_R} }{\bf d}(\theta) + {\mathbf{\pmb\varepsilon }}, \hfill \\
{{\cal H}_0}:{\mathbf{\pmb\varepsilon }}, \hfill \\
\end{gathered} \right.$$ where ${\bf d}(\theta) = \operatorname{vec} \left( {{\mathbf{A}}\left( \theta \right)} \right)$. As per the standard GLRT decision rule, if $${L_{\tilde{\bf y}}\left({\hat \alpha}, {\hat \theta}\right)}=\frac{{p\left( {{\mathbf{\tilde y}};{\hat \alpha}, {\hat \theta}, {{\cal H}_1}} \right)}}{{p\left( {{\mathbf{\tilde y}}; {{\cal H}_0}} \right)}} > \eta,$$ then ${\cal H}_1$ is chosen, where ${p\left( {{\mathbf{\tilde y}};{\hat \alpha}, {\hat \theta}, {{\cal H}_1}} \right)}$ and ${p\left( {{\mathbf{\tilde y}};{{\cal H}_0}} \right)}$ are the Probability Density Function (PDF) under ${\cal{H}}_1$ and ${\cal{H}}_0$ respectively, $\hat \alpha$ and $\hat \theta$ is the maximum likelihood estimation (MLE) of $\alpha$ and $\theta$ under ${\cal{H}}_1$, and is given by $\left[ {\hat \alpha ,\hat \theta } \right] = \mathop {\max }\limits_{\alpha ,\theta } p\left( {{\mathbf{\tilde y}}\left| {\alpha ,\theta ,{\cal H}_1} \right.} \right)$, $\eta$ is the decision threshold. (39) can be viewed as a hypothesis testing problem for MIMO radar detection in the homogeneous Gaussian clutter with covariance matrix $\bf C$, which has been discussed in[@5545189]. In this case, the GLRT detection statistic is given by $$\begin{array}{*{20}{l}}
\displaystyle \ln {L_{{\mathbf{\tilde y}}}}\left( {\hat \theta } \right) = \frac{{{{\left| {{{\mathbf{d}}^H}\left( {\hat \theta } \right){{\mathbf{C}}^{ - 1}}{\mathbf{\tilde y}}} \right|}^2}}}{{{{\mathbf{d}}^H}\left( {\hat \theta } \right){{\mathbf{C}}^{ - 1}}{\mathbf{d}}\left( {\hat \theta } \right)}}\hfill \\
\displaystyle \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= \frac{{{{\left| {{\text{tr}}\left( {{\mathbf{\tilde Y}}{{\mathbf{A}}^H}\left( {\hat \theta } \right){{{\mathbf{\tilde J}}}^{ - 1}}} \right)} \right|}^2}}}{{{\text{tr}}\left( {{\mathbf{A}}\left( {\hat \theta } \right){{\mathbf{A}}^H}\left( {\hat \theta } \right){{{\mathbf{\tilde J}}}^{ - 1}}} \right)}}\mathop {\gtrless}\limits_{{\cal H}_0}^{{\cal H}_1} \eta,
\end{array}$$ where ${{\mathbf{\tilde J}}}={{\mathbf{J}} + \sigma _R^2{{\mathbf{I}}_M}}$. According to [@kay1998fundamentals2], the asymptotic distribution of (41) is given by $$\ln {L_{{\mathbf{\tilde y}}}}\left( {\hat \theta } \right) \sim \left\{ \begin{gathered}
{{\cal H}_1}:{\mathcal X}_2^2\left( \rho \right), \hfill \\
{{\cal H}_0}:{\mathcal X}_2^2, \hfill \\
\end{gathered} \right.$$ where $ {\mathcal X}_2^2 $ and $ {\mathcal X}_2^2\left( \rho \right) $ are central and non-central chi-squared distributions with two Degrees of Freedom (DoFs), and $\rho$ is the non-central parameter, which is given by $$\begin{gathered}
\rho = {|\alpha| ^2}L{P_R}{\operatorname{vec} ^H}\left( {{\mathbf{A}}\left( \theta \right)} \right){{\mathbf{C}}^{ - 1}}\operatorname{vec} \left( {{\mathbf{A}}\left( \theta \right)} \right) \hfill \\
\;\;\; = {\operatorname{SNR}}_R\sigma _R^2\operatorname{tr} \left( {{\mathbf{A}}\left( \theta \right){{\mathbf{A}}^H}\left( \theta \right){{\left( {{\mathbf{J}} + \sigma _R^2{{\mathbf{I}}_M}} \right)}^{ - 1}}} \right), \hfill \\
\end{gathered}$$ where we define radar SNR as ${\operatorname{SNR}}_R=\frac{{{{\left| \alpha \right|}^2}L{P_R}}}{\sigma_R^2}$ [@1703855]. To maintain a constant false alarm rate $P_{FA}$, $\eta$ is decided by the given $P_{FA}$ under Neyman-Pearson criterion [@kay1998fundamentals2], i.e., $${P_{FA}} = 1 - {{\mathfrak{F}}_{{\cal X}_2^2}}\left( \eta \right),$$ $$\eta={{\mathfrak{F}}^{-1}_{{\cal X}_2^2}}(1-{P_{FA}}),$$ where ${{\mathfrak{F}}^{-1}_{{\cal X}_2^2}}$ is the inverse function of chi-squared Cumulative Distribution Function (CDF) with 2 DoFs. The detection probability is thus given as $${P_D} = 1-{\mathfrak{F}_{\mathcal{X}_2^2\left( \rho \right)}}(\eta)= 1 - {\mathfrak{F}_{\mathcal{X}_2^2\left( \rho \right)}}\left( {\mathfrak{F}_{\mathcal{X}_2^2}^{ - 1}\left( {1 - {P_{FA}}} \right)} \right),$$ where ${\mathfrak{F}_{\mathcal{X}_2^2\left( \rho \right)}}$ is the non-central chi-squared CDF with 2 DoFs.\
It is well-known that the accuracy of parameter estimation can be measured by the Cramér-Rao bound [@kay1998fundamentals1], which is the lower bound for all the unbiased estimators. In our case, the parameters to be estimated are $\theta$ and $\alpha$. The Fisher Information Matrix is partitioned as $${\mathbf{{\pmb \xi} }}\left( {{\mathbf{\tilde y}}} \right) = \left[ {\begin{array}{*{20}{c}}
{{{{{ \xi} }}_{\theta \theta }}}&{{{\mathbf{{\pmb \xi} }}_{\theta \alpha }^T}} \\
{{\mathbf{{\pmb \xi} }}_{\theta \alpha }}&{{{\mathbf{{\pmb \xi} }}_{\alpha \alpha }}}
\end{array}} \right],$$ where $\xi_{\theta \theta }$ is a scalar, ${\pmb \xi} _{\theta \alpha }$ is a vector and ${\pmb \xi} _{\alpha \alpha }$ is a matrix for the reason that $\theta$ is a real parameter while $\alpha$ is complex. The CRB for DoA estimation is given by $$\operatorname{CRB} \left( \theta \right) = {\left( {{{{{\xi} }}_{\theta \theta }} - {\mathbf{{\pmb \xi} }}_{\theta \alpha }^T{\mathbf{{\pmb \xi} }}_{\alpha \alpha }^{ - 1}{{\mathbf{{\pmb \xi} }}_{\theta \alpha }}} \right)^{ - 1}}.$$ By the similar derivation as [@1703855], ${\xi}_{\theta\theta}$, ${\mathbf{{\pmb \xi} }}_{\alpha \alpha }$ and ${\mathbf{{\pmb \xi} }}_{\theta \alpha }$ are given as $$\begin{gathered}
{{\xi} _{\theta \theta }} = 2{\left| \alpha \right|^2}L{P_R}\operatorname{tr} \left( {{\mathbf{\dot A}}\left( \theta \right){{{\mathbf{\dot A}}}^H}\left( \theta \right){{{\mathbf{\tilde J}}}^{ - 1}}} \right),\hfill \\
{{\pmb \xi} _{{\alpha \alpha }}} = 2L{P_R}\operatorname{tr} \left( {{\mathbf{A}}\left( \theta \right){{\mathbf{A}}^H}\left( \theta \right){{{\mathbf{\tilde J}}}^{ - 1}}} \right){{\mathbf{I}}_2},\hfill \\
{{\pmb \xi} _{\theta {\alpha }}} = 2L{P_R}\operatorname{Re} \left( {{\alpha ^*}\operatorname{tr} \left( {{\mathbf{A}}\left( \theta \right){{{\mathbf{\dot A}}}^H}\left( \theta \right){{{\mathbf{\tilde J}}}^{ - 1}}} \right)\left( {1;j} \right)} \right),
\end{gathered}$$ where ${\mathbf{\dot A}}\left( \theta \right) = \frac{{\partial {\mathbf{A}}\left( \theta \right)}}{{\partial \theta }}$. By substituting (49) into (48), we have $$\begin{gathered}
\operatorname{CRB} \left( \theta \right) \hfill \\
= \frac{1}{{2{{\operatorname{SNR} }_R}\sigma _R^2}}\times \hfill \\
\;\;\;\;\frac{{\operatorname{tr} \left( {{\mathbf{A}}{{\mathbf{A}}^H}{{{\mathbf{\tilde J}}}^{ - 1}}} \right)}}{{\operatorname{tr} \left( {{\mathbf{\dot A}}{{{\mathbf{\dot A}}}^H}{{{\mathbf{\tilde J}}}^{ - 1}}} \right)\operatorname{tr} \left( {{\mathbf{A}}{{\mathbf{A}}^H}{{{\mathbf{\tilde J}}}^{ - 1}}} \right) - {{\left| {\operatorname{tr} \left( {{\mathbf{A}}{{{\mathbf{\dot A}}}^H}{{{\mathbf{\tilde J}}}^{ - 1}}} \right)} \right|}^2}}},
\end{gathered}$$
Constructive Interference based Beamforming
-------------------------------------------
The proposed CI-based beamforming should be computed symbol by symbol, which means that the precoding vectors are functions of the time index, thus the corresponding hypothesis testing problem (35) is modified as $${{\bf{y}}^R_l}=\left\{ \begin{gathered}
{{\cal H}_1}:\alpha \sqrt {{P_R}} {\mathbf{A}}\left( \theta \right){\mathbf{s}}_l + {{\mathbf{G}}^T} {\tilde {\bf w}}[l]+ {\mathbf{z}}_l,\hfill \\
\;\;\;\;\;\;\;\;\; l = 1,2,...,L, \hfill \\
{{\cal H}_0}:{{\mathbf{G}}^T} {\tilde {\bf w}}[l] + {\mathbf{z}}_l,\;\;l = 1,2,...,L, \hfill \\
\end{gathered} \right.$$ where ${\tilde {\bf w}}[l]={\bf w}[l]{e^{j{\phi _1}[l]}}$. While the exact analytic form of the distribution for ${\bf w}[l]$ is hard to derive, here we employ the Gaussian detector for SDR beamformer in (41). We note that for CI precoding, ${\bf w}[l]$ is not in general Gaussian, but our results show that this is indeed an affordable approximation, and, even with a Gaussian detector, CI-based beamformer achieves better performance at radar. Following the same procedure of the previous subsection, we have $${\mathbf{J}}=\frac{1}{L}\sum\limits_{l = 1}^L{{{\mathbf{G}}^T}{\tilde {\bf w}}[l]{\tilde {\bf w}^H}[l]{{\mathbf{G}}^*}}=\frac{1}{L}\sum\limits_{l = 1}^L{{{\mathbf{G}}^T}{{\bf w}}[l]{{\bf w}^H}[l]{{\mathbf{G}}^*}} .$$ By substituting (52) into (46) and (50) we obtain the detection probability and the $\operatorname{CRB}(\theta)$ of CI-based beamforming method.
Robust Beamforming for Power Minimization with Bounded CSI Errors
=================================================================
Channel Error Model
-------------------
It is generally difficult to obtain perfect CSI in the practical scenarios. In this section, we study the beamforming design for imperfect CSI. Following the standard assumptions in the related literatures, let us first model the channel vectors as $$\begin{gathered}
{{\mathbf{h}}_i} = {{{\mathbf{\hat h}}}_i} + {{\mathbf{e}}_{hi}}, {{\mathbf{f}}_i} = {{{\mathbf{\hat f}}}_i} + {{\mathbf{e}}_{fi}},\forall i, \hfill \\
{{\mathbf{g}}_m} = {{{\mathbf{\hat g}}}_m} + {{\mathbf{e}}_{gm}},\forall m, \hfill \\
\end{gathered}$$ where $ {\bf{\hat h}}_i $, $ {\bf{\hat g}}_m $ and $ {\bf{\hat f}}_i $ denote the estimated channel vectors known to the BS, $ {\bf{e}}_{hi} $, $ {\bf{e}}_{gm} $ and $ {\bf{e}}_{fi} $ denote the CSI uncertainty within the spherical sets $ {{\cal{U}}_{hi}}=\left\{ {{{\bf{e}}_{hi}}|{{\left\| {{{\bf{e}}_{hi}}} \right\|}^2} \le \delta _{hi}^2} \right\} $, $ {{\cal{U}}_{gm}}=\left\{ {{{\bf{e}}_{gm}}|{{\left\| {{{\bf{e}}_{gm}}} \right\|}^2} \le \delta _{gm}^2} \right\} $ and $ {{\cal{U}}_{fi}}=\left\{ {{{\bf{e}}_{fi}}|{{\left\| {{{\bf{e}}_{fi}}} \right\|}^2} \le \delta _{fi}^2} \right\} $. This model is reasonable for scenarios that CSI is quantized at the receiver and fed back to the BS. Particularly, if the quantizer is uniform, the quantization error region can be covered by spheres of given sizes [@4586299].\
It is assumed that BS has no knowledge about the error vectors except for the bounds of their norms. We therefore consider a worst-case approach to guarantee the solution is robust to all the uncertainties in above spherical sets. It should be highlighted that this is only valid when all the uncertainties lie in the constraints. For the interference minimization problem, we can not formulate a robust problem in the real sense because the uncertainty of the channel $\bf G$ lies in the objective function. However, a weighting minimization method can be applied for the case to obtain a suboptimal result. Readers are referred to [@liu2016robust] for details. Due to the limited space, we designate this as the objective of the future work, and focus on the robust version for power minimization in this paper.
SDR-based Robust Beamforming
----------------------------
The robust version of the SDR-based problem ${{\cal{P}}_0}$ is given by $$\begin{array}{*{20}{l}}
{{{\cal{P}}_{10}}:\mathop {\min }\limits_{ {{{\mathbf{t}}_k}} } \;\;\sum\limits_{k = 1}^K {{{\left\| {{{\mathbf{t}}_k}} \right\|}^2}} } \\
\displaystyle {s.t.\;\;\frac{{{{\left| {{\mathbf{h}}_i^T{{\mathbf{t}}_i}} \right|}^2}}}{{\sum\limits_{k = 1,k \ne i}^K {{{\left| {{\mathbf{h}}_i^T{{\mathbf{t}}_k}} \right|}^2} + {P_R}{{\left\| {{{\mathbf{f}}_i}} \right\|}^2} + \sigma _C^2} }} \ge {\Gamma _i},}
{\text{ }} \\
{\;\;\;\;\;\;\forall {{\mathbf{e}}_{hi}} \in {{\cal{U}}_{hi}},\forall {{\mathbf{e}}_{fi}} \in {{\cal{U}}_{fi}},\forall i,} \\
\displaystyle {\;\;\;\;\;\;\frac{{{{\left| {{\mathbf{g}}_m^T\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}}{d_k} } \right|}^2}}}{{\sigma _R^2}} \le {R_m},\forall {{\mathbf{e}}_{gm}} \in {{\cal{U}}_{gm}},\forall m.} \\
\end{array}$$ The above problem is then reformulated as a worst-case approach, and can be solved by employing the well-known S-procedure [@boyd2004convex]. According to basic linear algebra, we have $${{{\left\| {{{\mathbf{f}}_i}} \right\|}^2} = {{\left\| {{{{\mathbf{\hat f}}}_i} + {{\mathbf{e}}_{fi}}} \right\|}^2} \le {{\left( {\left\| {{{{\mathbf{\hat f}}}_i}} \right\| + \left\| {{{\mathbf{e}}_{fi}}} \right\|} \right)}^2} \le {{\left( {\left\| {{{{\mathbf{\hat f}}}_i}} \right\| + {\delta _{fi}}} \right)}^2}.}$$ Similarly, for the interference power we have $$\begin{gathered}
{\left| {{\mathbf{g}}_m^T\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}} {d_k}} \right|^2} = \sum\limits_{k = 1}^K {\operatorname{tr} \left( {\left( {{\mathbf{\hat g}}_m^* + {\mathbf{e}}_{gm}^*} \right)\left( {{\mathbf{\hat g}}_m^T + {\mathbf{e}}_{gm}^T} \right){{\mathbf{t}}_k}{\mathbf{t}}_k^H} \right)} \hfill \\
= \sum\limits_{k = 1}^K {\operatorname{tr} \left( {\left( {{\mathbf{\hat g}}_m^*{\mathbf{\hat g}}_m^T + {\mathbf{\hat g}}_m^*{\mathbf{e}}_{gm}^T + {\mathbf{e}}_m^*{\mathbf{\hat g}}_m^T + {\mathbf{e}}_{gm}^*{\mathbf{e}}_{gm}^T} \right){{\mathbf{t}}_k}{\mathbf{t}}_k^H} \right)}. \hfill \\
\end{gathered}$$ By using the Cauchy-Schwarz inequality and rearranging the formula, it follows that $$\begin{gathered}
{\left| {{\mathbf{g}}_m^T\sum\limits_{k = 1}^K {{{\mathbf{t}}_k}}{d_k} } \right|^2} \le
\sum\limits_{k = 1}^K {\operatorname{tr} \left( {{\mathbf{\hat g}}_m^*{\mathbf{\hat g}}_m^T{{\mathbf{t}}_k}{\mathbf{t}}_k^H} \right)} {\kern 1pt} \hfill \\
+ \left( {2\left\| {{{{\mathbf{\hat g}}}_m}} \right\|\left\| {{{\mathbf{e}}_{gm}}} \right\| + {{\left\| {{{\mathbf{e}}_{gm}}} \right\|}^2}} \right)\sum\limits_{k = 1}^K {\operatorname{tr} \left( {{{\mathbf{t}}_k}{\mathbf{t}}_k^H} \right)} \hfill \\
\le \sum\limits_{k = 1}^K {\operatorname{tr} \left( {{\mathbf{\hat g}}_m^*{\mathbf{\hat g}}_m^T{{\mathbf{t}}_k}{\mathbf{t}}_k^H} \right)} {\kern 1pt} + \left( {2{\delta _{gm}}\left\| {{{{\mathbf{\hat g}}}_m}} \right\| + \delta _{gm}^2} \right)\sum\limits_{k = 1}^K {\operatorname{tr} \left( {{{\mathbf{t}}_k}{\mathbf{t}}_k^H} \right)}. \hfill \\
\end{gathered}$$ Based on the work [@liu2016robust], we directly give the worst-case formulation of ${\cal{P}}_{10}$ by $$\begin{gathered}
{{\cal P}_{11}}:\mathop {\min }\limits_{{{\mathbf{T}}_i},{s_i}} \sum\limits_{i = 1}^K {\operatorname{tr} \left( {{{\mathbf{T}}_i}} \right)} \hfill \\
s.t.\;\left[ {\begin{array}{*{20}{c}}
{{\mathbf{\hat h}}_i^T{{\mathbf{Q}}_i}{{{\mathbf{\hat h}}}_i^*} - {\Gamma _i}{\beta _i} - {s_i}\delta _{hi}^2}&{{\mathbf{\hat h}}_i^T{{\mathbf{Q}}_i}} \\
{{{\mathbf{Q}}_i}{{{\mathbf{\hat h}}}_i^*}}&{{{\mathbf{Q}}_i} + s _i{\mathbf{I}}}
\end{array}} \right] \succeq 0, \hfill \\
\;\;\;\;\;\;\;{{\mathbf{T}}_i} \succeq 0,{{\mathbf{T}}_i} = {\mathbf{T}}_i^*,{\text{rank}}\left( {{{\mathbf{T}}_i}} \right) = 1,{s_i} \ge 0,\forall i, \hfill \\
\;\;\;\;\;\;\;\sum\limits_{i = 1}^K {\left( \begin{gathered}
\operatorname{tr} \left( {{{{\mathbf{\hat g}}}_m}^*{\mathbf{\hat g}}_m^T{{\mathbf{T}}_i}} \right) + {\zeta_{gm}}\operatorname{tr} \left( {{{\mathbf{T}}_i}} \right) \hfill \\
\end{gathered} \right)} \le {R_m}\sigma _R^2,\forall m, \hfill \\
\end{gathered}$$ where $ {{\bf{T}}_k}={{\bf{t}}_k}{\bf{t}}_k^H $, $ {{\mathbf{Q}}_i} = {{\mathbf{T}}_i} - {\Gamma _i}\sum\limits_{n = 1,n \ne i}^K {{{\mathbf{T}}_n}} $, $ {\zeta_{gm}}={2{\delta _2}\left\| {{{{\mathbf{\hat g}}}_m}} \right\| + \delta _{gm}^2} $ and $ {{\beta _i}={P_R}{{\left( {\left\| {{{{\mathbf{\hat f}}}_i}} \right\| + {\delta _{fi}}} \right)}^2} + \sigma _C^2} $. By dropping the rank constraints on ${\bf T}_i$, the above problem becomes a standard SDP and can be solved by SDR method, after which the beamforming vectors can be obtained by rank-1 approximation or Gaussian randomization [@5447068].
Constructive Interference based Robust Beamforming
--------------------------------------------------
Let us first formulate the robust version of the virtual multicast problem ${{\cal P}_{3}}$ as $$\begin{array}{*{20}{l}}
{{\cal{P}}_{12}}:\mathop {\min }\limits_{\mathbf{w}} \;\;{\kern 1pt} {\kern 1pt} {\left\| {\mathbf{w}} \right\|^2} \hfill \\
\displaystyle s.t.{\kern 1pt}\left| {\operatorname{Im} \left( {{\mathbf{{\tilde h}}}_i^T{\mathbf{w}}} \right)} \right| \le \left( {\operatorname{Re} \left( {{\mathbf{{\tilde h}}}_i^T{\mathbf{w}}} \right) - \sqrt {{{\tilde \Gamma }_i}} } \right)\tan {\psi }, \hfill \\
{\;\;\;\;\;\;\forall {{\mathbf{e}}_{hi}} \in {{\cal{U}}_{hi}},\forall {{\mathbf{e}}_{fi}} \in {{\cal{U}}_{fi}},\forall i,} \\
\;\;\;\;\;{\kern 1pt}\left| {{\mathbf{\tilde g}}_m^T{\mathbf{w}}} \right| \le \sqrt {{R_m}\sigma _R^2}, \forall {{\mathbf{e}}_{gm}} \in {{\cal{U}}_{gm}},\forall m.\\
\end{array}$$ Apparently the robust case for the channel vector ${\bf f}_i$ is the same as (55). Consider the worst case of the INR constraints, which is $$\max\;\;\left| {{\mathbf{\tilde g}}_m^T{\mathbf{w}}} \right| \le \sqrt {{R_m}\sigma _R^2}, \forall {{\mathbf{e}}_{gm}} \in {{\cal{U}}_{gm}},\forall m.$$ Since ${{\mathbf{\tilde g}}_m} \triangleq {{\mathbf{g}}_m}{e^{j{\phi _1}}}$, it is easy to see ${\left\| {{{{\mathbf{\tilde g}}}_m}{\mathbf{w}}} \right\|^2} = {\left\| {{{\mathbf{g}}_m}{\mathbf{w}}} \right\|^2}$. For the convenience of further analysis, we drop the subscript, and denote the interference channel vector by its real and imaginary parts, which is given by $${\mathbf{g}} = {{{{\mathbf{\hat g}}}_R} + j{{{\mathbf{\hat g}}}_I} + {{\mathbf{e}}_{gR}} + j{{\mathbf{e}}_{gI}}}.$$ Let ${\mathbf{\bar g}} = \left[ {{{{\mathbf{\hat g}}}_{R}};{{{\mathbf{\hat g}}}_{I}}} \right], {{{\mathbf{\bar e}}}_g} = \left[ {{{\mathbf{e}}_{gR}};{{\mathbf{e}}_{gI}}} \right]$, the interference from radar can be written as $$\begin{gathered}
{\left| {{\mathbf{\tilde g}}^T{\mathbf{w}}} \right|^2} = {\left\| {\left[ {\begin{array}{*{20}{c}}
{{\mathbf{\hat g}}_R^T + {\mathbf{e}}_{gR}^T}&{{\mathbf{\hat g}}_I^T + {\mathbf{e}}_{gI}^T} \\
{{\mathbf{\hat g}}_I^T + {\mathbf{e}}_{gI}^T}&{ - {\mathbf{\hat g}}_R^T - {\mathbf{e}}_{gR}^T}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
{{{\mathbf{w}}_R}} \\
{ - {{\mathbf{w}}_I}}
\end{array}} \right]} \right\|^2} \hfill \\
= {\left\| \begin{gathered}
{{{\mathbf{\bar g}}}^T}{{\mathbf{w}}_2} + {\mathbf{\bar e}}_g^T{{\mathbf{w}}_2} \hfill \\
{{{\mathbf{\bar g}}}^T}{{\mathbf{w}}_1} + {\mathbf{\bar e}}_g^T{{\mathbf{w}}_1} \hfill \\
\end{gathered} \right\|^2}, \hfill \\
\end{gathered}$$ According to the Cauchy-Schwarz inequality, (62) can be further expanded as $$\begin{gathered}
{\left\| \begin{gathered}
{{{\mathbf{\bar g}}}^T}{{\mathbf{w}}_2} + {\mathbf{\bar e}}_g^T{{\mathbf{w}}_2} \hfill \\
{{{\mathbf{\bar g}}}^T}{{\mathbf{w}}_1} + {\mathbf{\bar e}}_g^T{{\mathbf{w}}_1} \hfill \\
\end{gathered} \right\|^2} \hfill \\
\le {\left| {{{{\mathbf{\bar g}}}^T}{{\mathbf{w}}_2}} \right|^2} + {\left| {{{{\mathbf{\bar g}}}^T}{{\mathbf{w}}_1}} \right|^2} + 2\delta _g^2{\left\| {{{\mathbf{w}}_2}} \right\|^2} \hfill \\
\quad + 2{\delta _g}\left( {\left\| {{{{\mathbf{\bar g}}}^T}{{\mathbf{w}}_2}{\mathbf{w}}_2^T} \right\| + \left\| {{{{\mathbf{\bar g}}}^T}{{\mathbf{w}}_1}{\mathbf{w}}_1^T} \right\|} \right) \hfill \\
\le {\left| {{{{\mathbf{\bar g}}}^T}{{\mathbf{w}}_2}} \right|^2} + {\left| {{{{\mathbf{\bar g}}}^T}{{\mathbf{w}}_1}} \right|^2} + \left( {2\delta _g^2 + 4{\delta _g}\left\| {{\mathbf{\bar g}}} \right\|} \right){\left\| {{{\mathbf{w}}_2}} \right\|^2}, \hfill \\
\end{gathered}$$ and the robust constraint for INR is given by $${\left| {{{{\mathbf{\bar g}}}^T}{{\mathbf{w}}_2}} \right|^2} + {\left| {{{{\mathbf{\bar g}}}^T}{{\mathbf{w}}_1}} \right|^2} + \left( {2\delta _g^2 + 4{\delta _g}\left\| {{\mathbf{\bar g}}} \right\|} \right){\left\| {{{\mathbf{w}}_2}} \right\|^2}\le R\sigma _R^2.$$ For the SINR constraint, note that the corresponding worst case is equivalent to $$\begin{gathered}
\max\;\;\left| {\operatorname{Im} \left( {{\mathbf{{\tilde h}}}_i^T{\mathbf{w}}} \right)} \right| - \operatorname{Re} \left( {{\mathbf{{\tilde h}}}_i^T{\mathbf{w}}} \right)\tan \psi + \sqrt {{{\tilde \Gamma }_i}} \tan \psi \le 0, \hfill \\
\;\;\;\;\;\;\;\;\;\;\forall {{\mathbf{e}}_{hi}} \in {{\cal{U}}_{hi}},\forall {{\mathbf{e}}_{fi}} \in {{\cal{U}}_{fi}},\forall i. \hfill \\
\end{gathered}$$ Let ${{\mathbf{\hat {{\tilde h}}}}_i} = {{\mathbf{\hat h}}_i}{e^{j\left( {{\phi _1} - {\phi _i}} \right)}},{{\mathbf{\tilde e}}_{hi}} = {{\mathbf{e}}_{hi}}{e^{j\left( {{\phi _1} - {\phi _i}} \right)}}$, we have ${{{\mathbf{{\tilde h}}}}_i} = {{{\mathbf{\hat {{\tilde h}}}}}_i} + {{{\mathbf{\tilde e}}}_{hi}}$. Similarly, we drop the subscript and denote the channel vector by its real and imaginary parts, which is $${\mathbf{{\tilde h}}} = {{{\mathbf{\hat {{\tilde h}}}}}_R} + j{{{\mathbf{\hat {{\tilde h}}}}}_I} + {{{\mathbf{\tilde e}}}_{hR}} + j{{{\mathbf{\tilde e}}}_{hI}}.$$ It follows that $$\begin{gathered}
\operatorname{Im} \left( {{\mathbf{{\tilde h}w}}} \right) = \operatorname{Im} \left( {\left( {{{{\mathbf{\hat {\tilde h}}}}_R} + j{{{\mathbf{\hat {\tilde h}}}}_I} + {{{\mathbf{\tilde e}}}_{hR}} + j{{{\mathbf{\tilde e}}}_{hI}}} \right)\left( {{{\mathbf{w}}_R} + j{{\mathbf{w}}_I}} \right)} \right) \hfill \\
= \left[ {{{{\mathbf{\hat {\tilde h}}}}_R},{{{\mathbf{\hat {\tilde h}}}}_I}} \right]\left[ \begin{gathered}
{{\mathbf{w}}_I} \hfill \\
{{\mathbf{w}}_R} \hfill \\
\end{gathered} \right] + \left[ {{{{\mathbf{\tilde e}}}_{hR}},{{{\mathbf{\tilde e}}}_{hI}}} \right]\left[ \begin{gathered}
{{\mathbf{w}}_I} \hfill \\
{{\mathbf{w}}_R} \hfill \\
\end{gathered} \right] \hfill \\
\triangleq {{{\mathbf{\hat {{\bar h}}}}}^T}{{\mathbf{w}}_1} + {\mathbf{\bar e}}_h^T{{\mathbf{w}}_1}, \hfill \\
\end{gathered}$$
$$\begin{gathered}
\operatorname{Re} \left( {{\mathbf{{\tilde h}w}}} \right) = \operatorname{Re} \left( {\left( {{{{\mathbf{\hat {\tilde h}}}}_R} + j{{{\mathbf{\hat {\tilde h}}}}_I} + {{{\mathbf{\tilde e}}}_{hR}} + j{{{\mathbf{\tilde e}}}_{hI}}} \right)\left( {{{\mathbf{w}}_R} + j{{\mathbf{w}}_I}} \right)} \right) \hfill \\
= \left[ {{{{\mathbf{\hat {\tilde h}}}}_R},{{{\mathbf{\hat {\tilde h}}}}_I}} \right]\left[ \begin{gathered}
{{\mathbf{w}}_R} \hfill \\
{-{\mathbf{w}}_I} \hfill \\
\end{gathered} \right] + \left[ {{{{\mathbf{\tilde e}}}_{hR}},{{{\mathbf{\tilde e}}}_{hI}}} \right]\left[ \begin{gathered}
{{\mathbf{w}}_R} \hfill \\
{-{\mathbf{w}}_I} \hfill \\
\end{gathered} \right] \hfill \\
\triangleq {{{\mathbf{\hat {{\bar h}}}}}^T}{{\mathbf{w}}_2} + {\mathbf{\bar e}}_h^T{{\mathbf{w}}_2}. \hfill \\
\end{gathered}$$
By noting that ${\left\| {{{{\mathbf{\bar e}}}_h}} \right\|^2} \le \delta _h^2$, (65) is equivalent to $$\begin{gathered}
\max \left| {{{{\mathbf{\hat {{\bar h}}}}}^T}{{\mathbf{w}}_1} + {\mathbf{\bar e}}_h^T{{\mathbf{w}}_1}} \right| - \left( {{{{\mathbf{\hat {{\bar h}}}}}^T}{{\mathbf{w}}_2} + {\mathbf{\bar e}}_h^T{{\mathbf{w}}_2}} \right)\tan \psi \hfill \\
\;\;\;\;\;\;\; + \sqrt {\tilde \Gamma } \tan \psi \le 0, \forall {\left\| {{{{\mathbf{\bar e}}}_h}} \right\|^2} \le \delta _h^2,\forall {\left\| {{{\mathbf{e}}_f}} \right\|^2} \le \delta _f^2,
\end{gathered}$$ and can be decomposed into the following two constraints: $$\begin{gathered}
\max \;{{{{\mathbf{\hat {{\bar h}}}}}^T}{{\mathbf{w}}_1} + {\mathbf{\bar e}}_h^T{{\mathbf{w}}_1}} - \left( {{{{\mathbf{\hat {{\bar h}}}}}^T}{{\mathbf{w}}_2} + {\mathbf{\bar e}}_h^T{{\mathbf{w}}_2}} \right)\tan \psi \hfill \\
\;\;\;\;\;\;\; + \sqrt {\tilde \Gamma } \tan \psi \le 0, \forall {\left\| {{{{\mathbf{\bar e}}}_h}} \right\|^2} \le \delta _h^2,\forall {\left\| {{{\mathbf{e}}_f}} \right\|^2} \le \delta _f^2,
\end{gathered}$$ $$\begin{gathered}
\max -{{{{\mathbf{\hat {{\bar h}}}}}^T}{{\mathbf{w}}_1} -{\mathbf{\bar e}}_h^T{{\mathbf{w}}_1}} - \left( {{{{\mathbf{\hat {{\bar h}}}}}^T}{{\mathbf{w}}_2} + {\mathbf{\bar e}}_h^T{{\mathbf{w}}_2}} \right)\tan \psi \hfill \\
\;\;\;\;\;\;\; + \sqrt {\tilde \Gamma } \tan \psi \le 0, \forall {\left\| {{{{\mathbf{\bar e}}}_h}} \right\|^2} \le \delta _h^2,\forall {\left\| {{{\mathbf{e}}_f}} \right\|^2} \le \delta _f^2.
\end{gathered}$$ Based on above, the worst-case constraints for (70) and (71) are given by $$\begin{gathered}
{{{\mathbf{\hat {\bar h}}}}^T}{{\mathbf{w}}_1} - {{{\mathbf{\hat {\bar h}}}}^T}{{\mathbf{w}}_2}\tan \psi + {\delta _h}\left( {{{\mathbf{w}}_1} - {{\mathbf{w}}_2}\tan \psi } \right) \hfill \\
\;\; + \sqrt {\Gamma \left( {\sigma _C^2 + {P_R}\left( {{{\left\| {\mathbf{\hat f}} \right\|}} + \delta _f} \right)^2} \right)} \tan \psi \le 0,
\end{gathered}$$ $$\begin{gathered}
- {{{\mathbf{\hat {\bar h}}}}^T}{{\mathbf{w}}_1} - {{{\mathbf{\hat {\bar h}}}}^T}{{\mathbf{w}}_2}\tan \psi + {\delta _h}\left( {{{\mathbf{w}}_1} + {{\mathbf{w}}_2}\tan \psi } \right) \hfill \\
\;\;\;\;\;\;+ \sqrt {\Gamma \left( {\sigma _C^2 + {P_R}\left( {{{\left\| {\mathbf{\hat f}} \right\|}} + \delta _f} \right)^2} \right)} \tan \psi \le 0. \hfill \\
\end{gathered}$$\
The final robust optimization problem is given by $$\begin{array}{*{20}{l}}
{{\cal{P}}_{13}}:\mathop {\min }\limits_{{\mathbf{w}}_1,{\mathbf{w}}_2} \;\;{\kern 1pt} {\kern 1pt} {\left\| {{\mathbf{w}}_1} \right\|^2} \hfill \\
s.t.\;\;\;\; \text{Constraints (64), (72) and (73)}, \forall i, \forall m, \hfill \\
\;\;\;\;\;\;\;\;\;{{\mathbf{w}}_1} = {\mathbf{\Pi }}{{\mathbf{w}}_2}.
\end{array}$$
Numerical Results
=================
In this section, numerical results based on Monte Carlo simulations are shown to validate the effectiveness of the proposed beamforming method. Without loss of generality, all the channel matrices follow the standard complex Gaussian distribution, and are independent and identically distributed (i.i.d). For simplicity, the INR thresholds for different radar antennas, the SINR level for different downlink users and the error bounds for different channels are set to be equal, respectively, i.e., $ R_m = R, \Gamma_i = \Gamma, \delta_{hi} = \delta_{fi} = \delta_{gm} = \delta, \forall i, \forall m $. While it is plausible that the benefits of the proposed scheme extend to various scenarios, here we assume $\alpha=P_R=1$, $ N=8 $, $ K=M=4 $ unless otherwise specified, and explore the results for QPSK and 8PSK modulations. The power of all the noise vectors are set to be 1mW, i.e., ${\sigma_R^2} = {\sigma_C^2} = 0\text{dBm}$. We denote the conventional SDR beamformer as ‘SDR’ in the figures, and the proposed beamformer based on constructive interference as ‘CI’.
Average Transmit Power
----------------------
In Fig. 3, we compare the minimized power for the two beamforming methods under a given INR level of 0dB with the increasing $\Gamma$. Unsurprisingly, the power needed for transmission increases with growing $\Gamma$ for both methods. However, it can be easily seen that the proposed method obtains a lower transmit power for given INR and SINR requirements than the conventional SDR-based method thanks to the exploitation of the constructive interference. Particularly if QPSK modulation is used, the required power for CI-based scheme is less than half of the power needed for SDR-based beamforming. Similar results have been provided in Fig. 4, where the transmit power of different methods with increased $R$ has been given with required SINR fixed at 20dB and 24dB respectively. It is worth noting that there exists a trade-off between the power needed for BS and the INR level received by radar as has been discussed in the previous section. For both SINR levels, the proposed method performs far better than the conventional one especially in all practical INR levels.
![Average transmit power vs. required SINR, with $R = 0\text{dB}$.[]{data-label="fig:3"}](TSPFig3.eps){width="3.0in"}
![Trade-off between BS transmit power and INR level, with $\Gamma = 20\text{dB}$ and 24dB respectively.[]{data-label="fig:4"}](TSPFig4.eps){width="3.0in"}
![Result comparison of Algorithm 1 and CVX-Solver for ${\cal P}_3$, $N = 12, M = 4$, $\Gamma = 20\text{dB}, R=5\text{dB}$, QPSK.[]{data-label="fig:5"}](TSPFig5.eps){width="3.0in"}
Complexity
----------
In order to verify the effectiveness of the proposed efficient algorithm for ${\cal P}_3$, we compare the results obtained by the built-in *SeDuMi* solver in CVX [@grant2008cvx] and Algorithm 1 with increasing downlink users $K$ in Fig. 5, where $N=12, M=4, \Gamma = 20\text{dB}, R=5\text{dB}$. As we can see that the two curves match very well and the difference is less than 0.05mW when $M = 6$. Since it is difficult to analytically derive the complexity of the optimization based beamforming as well as the proposed iteration algorithm, the complexity for ${\cal P}_0$, ${\cal P}_3$ and Algorithm 1 has been compared in terms of average execution time for a growing number of downlink users in Fig. 6, with the same configuration of Fig. 5. Note that it takes less time to solve ${\cal P}_3$ than ${\cal P}_0$ by the CVX solver. This is because the latter needs a rank-1 approximation or Gaussian randomization to obtain the optimal beamforming vectors, which involves extra amount of computations [@5447068]. Nevertheless, the proposed CI-based approach is a symbol-level beamformer, which means that the beamforming vectors should be calculated symbol by symbol while the SDR-based beamforming needs only one-time calculation during a communication frame in slow fading channels. Fortunately, the proposed gradient projection algorithm is far more efficient than the CVX solver and saves nearly 90% of time with respect to the SDR beamformer. In a typical LTE system with 14 symbols in one frame, the total execution time for the gradient projection algorithm will be 140% ($(1-90\%)\times 14=140\%$) of the SDR-based beamforming, but the gain of the saved transmit power is more than 200% as has been shown in Fig. 3 and Fig. 4, which is cost-effective in energy-limited systems.
![Average execution time for optimization ${\cal P}_0$, ${\cal P}_3$ and Algorithm 1, $N = 12, M = 4$, $\Gamma = 20\text{dB}, R=5\text{dB}$, QPSK.[]{data-label="fig:6"}](TSPFig6.eps){width="3.0in"}
![Detection probability vs. radar SNR for different cases, $P = 24\text{dBm}$, $\Gamma = 18\text{dB}$, $\eta=13.5\text{dBm}$, QPSK.[]{data-label="fig:7"}](TSPFig7.eps){width="3.0in"}
![Detection Probability vs. SINR threshold for different cases, $P = 24\text{dBm}$, QPSK.[]{data-label="fig:8"}](TSPFig8.eps){width="3.0in"}
![RMSE vs. SINR threshold for different power budget, ${\text{SNR}}_R=10\text{dB}$, QPSK.[]{data-label="fig:9"}](TSPFig9.eps){width="3.0in"}
Radar Performance
-----------------
Fig. 7-9 demonstrate a series of results for the impact of the proposed scheme on different radar metrics by solving the interference minimization problem ${\cal P}_1$ and ${\cal P}_4$. Here we assume that radar is equipped with an Uniform Linear Array (ULA) with half-wavelength spacing, and *m*-sequences are used as the radar waveform with a length of 40 digits, i.e., $L=40$. The target is set to be located at the direction of $\theta=\pi/5$.\
In Fig. 7, the average detection probability with increased radar SNR for the two methods are given, where the solid line with triangle markers denotes the case without interference from the BS. Among the rest lines, the solid curves and dashed ones denote the simulated and asymptotic detection performance respectively. The parameters are given as $\eta=13.5\text{dBm}$. $\Gamma = 18\text{dB}$, and $P=24\text{dBm}$. As shown in the figure, the simulated results match well with the asymptotic ones for both SDR and CI methods. Once again, we see that the proposed method outperforms the SDR-based method significantly. For instance, the extra gain needed for the SDR method is 4dB compared with the proposed method for a desired $P_D = 0.9$.\
Fig. 8 shows another important trade-off between radar and communication, where the detection probability at the radar with increased SINR threshold of the downlink users are provided for the two methods with the same parameter configuration as Fig. 7. It can be seen that a higher SINR requirement at users leads to a lower $P_D$ for radar, and the proposed method obtains better trade-off curves for both simulated and asymptotic results thanks to the utilization of MUI. The results in Figs. 7 and 8 justify the use of the Gaussian radar detector of (41) for the CI beamformer, which still gives significant performance gains w.r.t the SDR beamformer.\
In Fig. 9, the lower bound of radar DoA estimation is given in terms of the root-mean-square-error (RMSE) with increased SINR threshold and different BS power budget, where $\text{RMSE}(\theta)=\sqrt{\text{CRB}(\theta)}$. As expected, the loose of the communication constraints brings benefits to radar target estimation. It can be also observed that the proposed approach is not only robust to the increasing SINR requirement, but also performs far better than the SDR method.
Robust Designs
--------------
In Fig. 10, the BS transmit power with increasing CSI error bound $\delta$ is shown with $\Gamma = 25\text{dB}, R=30\text{dB}$, where different cases with perfect and imperfect CSI are simulated for both SDR and CI-based beamforming. The legend denotes the channel which suffers from CSI errors for each case, while the rest are assumed perfectly known. Thanks to its relaxed nature, the CI-based beamforming has a higher degree of tolerance for the CSI errors than SDR-based ones. The same trend is also shown in Fig. 11, where we apply a fixed channel error bound $\delta^2=2\times 10^{-4}$ and $R=25\text{dB}$ for all the robust cases to see the variation of the transmit power with an increased SINR level. Since the interference channel between radar and users should first be estimated by the users and then fed back to the BS, the knowledge about $\bf F$ is more likely to be known inaccurately by the BS compared with other two channels. Fortunately, we observe that in both Fig. 10 and Fig. 11, the imperfect channel $\bf F$ requires less transmit power to meet the same SINR level than $\bf H$ and $\bf G$ with CSI errors of the same bound. Hence, the accuracy for the estimation of $\bf F$ can be relatively lower than the other channels.
![Average transmit power vs. error bound for different robust cases, $\Gamma = 25\text{dB}, R = 30\text{dB}$, QPSK.[]{data-label="fig:10"}](TSPFig10.eps){width="2.8in"}
![Average transmit power vs. SINR for different robust cases, $\delta^2=2 \times 10^{-4}, R = 25\text{dB}$, QPSK.[]{data-label="fig:11"}](TSPFig11.eps){width="2.8in"}
Conclusion
==========
This paper proposes a novel optimization-based beamforming approach for MIMO radar and downlink MU-MISO communication coexistence, where multi-user interference is utilized to enhance the performance of communication system and relax the constraints in the optimization problems. Numerical results show that the proposed scheme outperforms the conventional SDR-based beamformers in terms of both power and interference minimization. An efficient gradient projection method is further given to solve the proposed power minimization problem, and is compared with SDR-based solver in the sense of average execution time. While the proposed technique is applied at symbol level, the computation complexity is still comparable with the SDR approach in typical LTE systems. Moreover, the detection probability and the Cramér-Rao bound for MIMO radar in the presence of the interference from BS are analytically derived, and the trade-off between the performance of radar and communication is revealed. Finally, a robust beamformer for power minimization is designed for imperfect CSI cases based on interference exploitation, and obtains significant performance gains compared with conventional schemes.
Acknowledgment {#acknowledgment .unnumbered}
==============
This research was supported by the Engineering and Physical Sciences Research Council (EPSRC) project EP/M014150/1 and the China Scholarship Council (CSC) under Grant No. 201606030054.
[^1]: Manuscript received \*\*\*
[^2]: Fan Liu and Jianming Zhou are with the School of Information and Electronics, Beijing Institute of Technology, Beijing, 100081, China (e-mail: [email protected], [email protected]). Fan Liu, Christos Masouros and Ang Li are with the Department of Electronic and Electrical Engineering, University College London, London, WC1E 7JE, UK (e-mail: [email protected], [email protected]). Tharmalingam Ratnarajah is with the Institute for Digital Communications, School of Engineering, The University of Edinburgh, Edinburgh, EH9 3JL, UK (e-mail: [email protected]).
|
---
abstract: 'We construct noncommutative multidimensional versions of overconvergent power series rings and Robba rings. We show that the category of étale $(\varphi,\Gamma)$-modules over certain completions of these rings are equivalent to the category of étale $(\varphi,\Gamma)$-modules over the corresponding classical overconvergent, resp. Robba rings (hence also to the category of $p$-adic Galois representations of $\mathbb{Q}_p$). Moreover, in the case of Robba rings, the assumption of étaleness is not necessary, so there exists a notion of trianguline objects in this sense.'
author:
- Gergely Zábrádi
title: '$(\varphi,\Gamma)$-modules over noncommutative overconvergent and Robba rings'
---
Introduction
============
In recent years it has become increasingly clear that some kind of $p$-adic version of the local Langlands correspondence should exist. In fact, Colmez [@Co1; @Co2] constructed such a correspondence for ${\mathrm{GL}}_2(\mathbb{Q}_p)$. His construction is done in several steps using $(\varphi,\Gamma)$-modules (the category of which is well-known [@F] to be equivalent to the category of $p$-adic Galois representations of $\mathbb{Q}_p$). We briefly recall Colmez’s correspondence here. Let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $o_K$ and uniformizer $p_K$.
The so-called “Montreal-functor” associates to a smooth $o_K$-torsion representation of the standard Borel subgroup $B_2(\mathbb{Q}_p)$ of ${\mathrm{GL}}_2(\mathbb{Q}_p)$ an $o_K$-torsion $(\varphi,\Gamma)$-module over Fontaine’s ring $\mathcal{O_E}$. If we are given a unitary Banach space representation $\Pi$ over the field $K$ of the group ${\mathrm{GL}}_2(\mathbb{Q}_p)$ then it admits an $o_K$-lattice $L(\Pi)$ which is invariant under ${\mathrm{GL}}_2(\mathbb{Q}_p)$. Hence $L(\Pi)/p_K^r$ is a smooth $o_K$-torsion representation that we restrict now to $B_2(\mathbb{Q}_p)$. The $(\varphi,\Gamma)$-module associated to $\Pi$ is the projective limit (as $r\to\infty$) of the $(\varphi,\Gamma)$-modules associated to $L(\Pi)/p_K^r$ via the Montreal functor. This is generalized in [@SVi] to general reductive groups over $\mathbb{Q}_p$.
The reverse direction, how one adjoins a unitary continuous $p$-adic representation to a $2$-dimensional $(\varphi,\Gamma)$-module $D$ over Fontaine’s ring, is even more subtle. One first constructs a unitary $p$-adic Banach space representation $\Pi(D)$ to each $2$-dimensional *trianguline* $(\varphi,\Gamma)$-module $D$ over $\mathcal{E}=\mathcal{O_E}[p^{-1}]$ using some kind of parabolic induction. This Banach space is well described as a quotient of the space of $p$-adic functions satisfying certain properties by a certain ${\mathrm{GL}}_2(\mathbb{Q}_p)$-invariant subspace (see [@Co3] and [@Br] for details), however, a priori it is not clear whether or not it is nontrivial. On the other hand, there is a general construction of a (somewhat bigger) ${\mathrm{GL}}_2(\mathbb{Q}_p)$-representation $D\boxtimes_\delta\mathbb{P}^1$ which is in fact the space of global sections of a ${\mathrm{GL}}_2(\mathbb{Q}_p)$-equivariant sheaf $U\mapsto D\boxtimes_\delta U$ ($U\subseteq \mathbb{P}^1$ open) on the projective space $\mathbb{P}^1(\mathbb{Q}_p)\cong{\mathrm{GL}}_2(\mathbb{Q}_p)/B_2(\mathbb{Q}_p)$ for any (not necessarily $2$-dimensional) $(\varphi,\Gamma)$-module $D$ and any unitary character $\delta\colon\mathbb{Q}_p^{\times}\to o_K^{\times}$. This sheaf has the following properties: $(i)$ the centre of ${\mathrm{GL}}_2(\mathbb{Q}_p)$ acts via $\delta$ on $D\boxtimes_\delta \mathbb{P}^1$; $(ii)$ we have $D\boxtimes_\delta \mathbb{Z}_p\cong D$ as a module over the monoid $\begin{pmatrix}\mathbb{Z}_p\setminus\{0\}&\mathbb{Z}_p\\0&1\end{pmatrix}$ (where we regard $\mathbb{Z}_p$ as an open subspace in $\mathbb{P}^1=\mathbb{Q}_p\cup\{\infty\}$). (See [@SVZ] for a generalization of this construction to general reductive groups.) Then Colmez shows that in case $D$ is $2$-dimensional and trianguline, then there exists a unitary character $\delta$ (namely $\delta=\chi^{-1}\det D$ where $\chi$ is the cyclotomic character and $\det D$ is the character associated to the $1$-dimensional $(\varphi,\Gamma)$-module $\bigwedge^2 D$ via Fontaine’s equivalence composed with class field theory) such that a certain subspace $D^\natural\boxtimes_\delta \mathbb{P}^1$ (for the definition see [@Co2]) of $D\boxtimes_\delta\mathbb{P}^1$ is isomorphic to the dual of the Banach space representation $\Pi(\check{D})$ associated earlier to the dual $(\varphi,\Gamma)$-module $\check{D}$—therefore showing in particular that the previous construction is nonzero. This subspace makes sense also in case $D$ is not trianguline (nor of rank $2$), but a priori only known to be $B_2(\mathbb{Q}_p)$-invariant. Moreover, whenever $D$ is indecomposable and $2$-dimensional, then the above $\delta$ is unique [@P], and whenenever $D$ is absolutely irreducible and $\geq 3$-dimensional, then there does not exist [@P] such a character $\delta$ (so that the subspace $D^\natural\boxtimes_\delta \mathbb{P}^1$ is ${\mathrm{GL}}_2(\mathbb{Q}_p)$-invariant). Since the construction of $D\mapsto D^{\natural}\boxtimes_\delta \mathbb{P}^1$ behaves well in families (see chapter II in [@Co1]) and the trianguline Galois-representations are Zariski-dense in the deformation space of $2$-dimensional $(\varphi,\Gamma)$-modules with given reduction mod $p$ [@Ki], Colmez [@Co1] shows that this subspace is not only $B_2(\mathbb{Q}_p)$, but also ${\mathrm{GL}}_2(\mathbb{Q}_p)$-invariant for general $2$-dimensional $(\varphi,\Gamma)$-modules. Moreover, for $\delta=\chi^{-1}\det D$ (in this case we omit the subscript $\delta$ from the notation) we have a short exact sequence $$0\to \Pi(\check{D})\to D\boxtimes\mathbb{P}^1\to \Pi(D)\to 0$$ where $\Pi(D)$ is the unitary Banach-space representation associated to $D$ via the $p$-adic Langlands correspondence.
Colmez ([@Co1], chapter V and VI) also identifies the space $\Pi(D)^{an}$ of locally analytic and the space $\Pi(D)^{alg}$ of locally algebraic vectors in the Banach-space representation $\Pi(D)$. These play a crucial role in the proof of the compatibility of the $p$-adic and the classical local Langlands correspondence. In fact, we have $\Pi(D)^{an}=(D^{\dagger}\boxtimes\mathbb{P}^1)/K\cdot (D^{\natural}\boxtimes\mathbb{P}^1)$ where $D^\dagger\boxtimes\mathbb{P}^1$ is the subspace of elements $x\in D\boxtimes\mathbb{P}^1$ such that both ${\mathrm{Res}}_{\mathbb{Z}_p}^{\mathbb{P}^1}(x)$ and ${\mathrm{Res}}_{\mathbb{Z}_p}^{\mathbb{P}^1}\left(\begin{pmatrix}0&1\\1&0\end{pmatrix}x\right)$ lie in the subspace of overconvergent elements $D^\dagger\subset D\cong D\boxtimes\mathbb{Z}_p$. $D^\dagger$ is an étale $(\varphi,\Gamma)$-module over the ring $\mathcal{E}^\dagger$ of overconvergent power series with coefficients in $K$ such that $D\cong \mathcal{E}\otimes_{\mathcal{E}^\dagger}D^\dagger$ [@CC].
Let now $G$ be the group of $\mathbb{Q}_p$-points of a connected $\mathbb{Q}_p$-split reductive group and $P=TN$ a Borel subgroup of $G$. Further denote by $\Phi^+$ the set of positive roots with respect to $P$ and $\Delta\subset \Phi^+$ the set of simple roots. The above noted generalizations of Colmez’s work ([@SVi] and [@SVZ]) both use a certain microlocalisation $\Lambda_\ell(N_0)$ (constructed originally in [@SVe]) of the Iwasawa algebra $\Lambda(N_0)$ of a compact open subgroup $N_0$ of $N$. This can be thought of as the noncommutative analogue of Fontaine’s ring $\mathcal{O_E}$. On the other hand, Colmez’s $p$-adic Langlands correspondence heavily relies on the theory of trianguline $(\varphi,\Gamma)$-modules. A $(\varphi,\Gamma)$-module over the Robba ring is a free module $D_{rig}^\dagger$ over $\mathcal{R}$ together with commuting semilinear actions of the operator $\varphi$ and the group $\Gamma$ such that $\varphi$ takes a basis of the free module to another basis. Such a $(\varphi,\Gamma)$-module $D_{rig}^\dagger$ is said to be étale (or of slope $0$) if there is a basis of $D_{rig}^\dagger$ such that the matrix of $\varphi$ in this basis is an invertible matrix over the subring $\mathcal{O}_{\mathcal{E}}^\dagger\subset\mathcal{R}$ of overconvergent Laurent series. An étale $(\varphi,\Gamma)$-module over $\mathcal{R}$ is *trianguline* if it admits a filtration of (not necessarily étale) $(\varphi,\Gamma)$-modules over $\mathcal{R}$ with subquotients of rank $1$ possibly after a finite base change $E\otimes_K\cdot$. The fact that the Robba ring and the ring of overconvergent Laurent series play such a role in the construction of the $p$-adic Langlands correspondence for ${\mathrm{GL}}_2(\mathbb{Q}_p)$ and also in the identification of the locally analytic vectors is the motivation for the construction of noncommutative analogues of these rings—as they will most probably be needed for a future correspondence for reductive groups other than ${\mathrm{GL}}_2(\mathbb{Q}_p)$.
The motivation of this paper is twofold. On one hand, we reinterpret the ring $\Lambda_\ell(N_0)$ as follows. Instead of localising and completing the Iwasawa algebra $\Lambda(N_0)$, one may construct $\Lambda_\ell(N_0)$ as the projective limit of certain skew group rings over $\mathcal{O_E}$. The only assumptions on the ring $R=\mathcal{O_E}$ such that this new construction of $\Lambda_\ell(N_0)$ can be carried out are that $R$ admits an inclusion $\chi\colon\mathbb{Z}_p\to R^{\times}$ of the additive group $\mathbb{Z}_p$ into its group of invertible elements and an étale action of an operator $\varphi$ that is compatible with $\chi$. The noncommutative ring that is constructed is a completed skew group ring $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$ of a closed normal subgroup $H_1$ of a pro-$p$ group $H_0$ such that $\ell\colon H_0\twoheadrightarrow \mathbb{Z}_p$ is a homomorphism with kernel $H_1$ (hence $H_0/H_1\cong\mathbb{Z}_p$). The main result in this direction is Prop. \[equivcat\] showing that the category of $\varphi$-modules over $R$ is equivalent to the category of $\varphi$-modules over the completed skew group ring $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$. This can be applied also to the ring $R=\mathcal{O}_\mathcal{E}^\dagger$ of overconvergent Laurent series with coefficients in $o_K$ and the Robba ring $\mathcal{R}$. The other motivation (probably also the more important one) is the construction of the right noncommutative analogues of $\mathcal{O}_\mathcal{E}^\dagger$ and $\mathcal{R}$. The elements of the rings $\mathcal{R}{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$ and $\mathcal{O}_{\mathcal{E}}^\dagger{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$, however, are not necessarily convergent in any open annulus since they are obtained by taking an inverse limit. Therefore in section \[partialrobba\] we construct the rings $\mathcal{R}(H_1,\ell)$ and $\mathcal{R}^{int}(H_1,\ell)$ as direct limits of certain microlocalisations of the distribution algebra. The elements of these are convergent in a region of the form $$\left\{\rho_2<|b_\alpha|<1,\ |b_\beta|<|b_\alpha|^{r}\text{ for }\beta\in\Phi^+\setminus\{\alpha\}\right\}$$ for some $p^{-1}<\rho_2<1$ and $1\leq r\in\mathbb{Z}$. In section \[relate\] we show that $\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ (resp. $\mathcal{O}_\mathcal{E}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$) is a certain completion of $\mathcal{R}(N_1,\ell)$ (resp. $\mathcal{R}^{int}(N_1,\ell)$). Note that although the natural map $j_{int}\colon\mathcal{R}^{int}(N_1,\ell)\to\mathcal{O}_\mathcal{E}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ is injective, the map $j\colon \mathcal{R}(N_1,\ell)\to\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ is not. Both rings $\mathcal{R}(N_1,\ell)$ and its integral version admit an étale action of the monoid $T_+=\{t\in T\mid tN_0t^{-1}\subseteq N_0\}$. However, it is an open question whether the categories of étale $T_+$-modules over these rings are equivalent to the étale $T_+$-modules over their completions.
In my opinion, the right noncommutative analogues of the ring $\mathcal{R}$ (resp. $\mathcal{O}_\mathcal{E}^\dagger$) is $\mathcal{R}(N_1,\ell)$ (resp. $\mathcal{R}^{int}(N_1,\ell)$) in the context of $\mathbb{Q}_p$-split reductive groups $G$ over $\mathbb{Q}_p$ as both rings admit an étale action of the monoid $T_+$ and their elements converge in certain polyannuli. However, it might still be useful to also consider the rings $\mathcal{R}{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$ and $\mathcal{O}_{\mathcal{E}}^\dagger{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$, as they can help us compare the category of usual $(\varphi,\Gamma)$-modules with the category of $T_+$-modules over $\mathcal{R}(N_1,\ell)$ (resp. over $\mathcal{R}^{int}(N_1,\ell)$) using the equivalence of categories in Proposition \[equivcat\]. Note that only one variable is inverted in these rings in contrast to the rings constructed in [@SZ]. The reasons for this are the following: $(i)$ this way $\mathcal{R}^{int}(N_1,\ell)$ is a subring of $\Lambda_\ell(N_0)$; $(ii)$ the equivalence of categories in Proposition \[equivcat\] holds for rings in which only one variable is inverted; $(iii)$ all the usual $(\varphi,\Gamma)$-modules are overconvergent, ie. descend to $\mathcal{O}_{\mathcal{E}}^\dagger$ already in one variable. However, if $\mathbb{Q}_p$ is replaced by a finite unramified extension $F$ then one might have to consider Lubin-Tate $(\varphi,\Gamma_F)$-modules (with $\Gamma_F\cong o_F^{\times}$) instead so that the monoid $\varphi^{\mathbb{N}}\Gamma_F$ is isomorphic to $o_F\setminus\{0\}$. These $(\varphi,\Gamma_F)$-modules are not overconvergent in general but they might still correspond to objects over certain multivariable Robba rings (in which all the variables are inverted). For a first result in this direction see [@Be2]. It is plausible to expect that for general reductive groups $G$ over $F$ one has to invert exactly $|F:\mathbb{Q}_p|$ ($\mathbb{Q}_p$-)variables that correspond to the root subgroup $N_\alpha\cong F\cong\mathbb{Q}_p^{|F:\mathbb{Q}_p|}$ for a given simple root $\alpha$.
Acknowledgements
----------------
This research was supported by the Hungarian OTKA grant K-100291. The proof of Prop. \[equivcat\] is a direct generalization of Thm. 8.20 in [@SVZ] and grew out of related discussions with Marie-France Vigneras and Peter Schneider. I express my gratitude to both of them for allowing me to include this proof here and for other valueable discussions. My debt is especially due to Peter Schneider from whom I learnt the most that I know about $p$-adic representation theory. I would like to thank Torsten Schoeneberg for reading the manuscript and for his comments. I also benefited from enlightening conversations with Kiran Kedlaya. Finally, I thank the referee for a careful reading of the manuscript.
Completed skew group rings {#phiring}
==========================
Let $R$ be a commutative ring (with identity) with the following properties:
1. There exists a group homomorphism $\chi\colon\mathbb{Z}_p\hookrightarrow
R^{\times}$.
2. The ring $R$ admits an étale action of the $p$-Frobenius $\varphi$ that is compatible with $\chi$. More precisely there is an injective ring homomorphism $\varphi\colon R\hookrightarrow R$ such that $\varphi(\chi(x))=\chi(px)$ and $$R=\bigoplus_{i=0}^{p-1}\chi(i)\varphi(R).$$ In particular, $R$ is free of rank $p$ over $\varphi(R)$.
We remark first of all that one may iterate $(ii)$ $c$ times for any positive integer $c$ to obtain $$R=\bigoplus_{i=0}^{p^c-1}\chi(i)\varphi^c(R)\ .
\label{iterate}$$ Indeed, by induction we may assume that holds for $c-1$ and obtain $$R=\bigoplus_{k=0}^{p^{c-1}-1}\chi(k)\varphi^{c-1}(R)=
\bigoplus_{k=0}^{p^{c-1}-1}\chi(k)\varphi^{c-1}\left(\bigoplus_{j=0}^{p-1}\chi(j)\varphi(R)\right)=
\bigoplus_{k=0}^{p^{c-1}-1}\bigoplus_{j=0}^{p-1}\chi(k+p^{c-1}j)\varphi^c(R)$$ since $\varphi^{c-1}$ takes direct sums to direct sums as it is injective. Now the claim follows from noting that any integer $0\leq i\leq p^c-1$ can be uniquely written in the form $i=k+p^{c-1}j$ with $0\leq k\leq p^{c-1}-1$ and $0\leq j\leq p-1$.
Moreover, for any $x\in\mathbb{Z}_p$ we have $\chi(p^cx)=\varphi^c(\chi(x))\in\varphi^c(R)^\times$. Hence $\chi(i)\varphi^c(R)=\chi(i+p^cx)\varphi^c(R)$ and we may replace each value of $i$ in the formula by any element in the coset $i+p^c\mathbb{Z}_p$.
\[phiringdef\] We call a ring $R$ with the above properties $(i)$ and $(ii)$ a $\varphi$-ring over $\mathbb{Z}_p$ or often just a $\varphi$-ring.
For example, if $K/\mathbb{Q}_p$ is a finite extension with ring of integers $o$ and uniformizer $p_K$ then the Iwasawa algebra $o{[\hspace{-0.04cm}[}T{]\hspace{-0.04cm}]}$ is a $\varphi$-ring with the homomorphism $$\begin{aligned}
\chi\colon\mathbb{Z}_p&\rightarrow&o{[\hspace{-0.04cm}[}T{]\hspace{-0.04cm}]}\\
1&\mapsto&1+T\end{aligned}$$ and Frobenius $\varphi(T)=(T+1)^p-1$. Similarly with the same $\chi$ and $\varphi$, Fontaine’s ring $\mathcal{O_E}$, its field of fractions $\mathcal{E}$, the Robba ring $\mathcal{R}$ and the rings $\mathcal{E}^\dagger$, $\mathcal{O}_{\mathcal{E}}^{\dagger}$ of overconvergent power series are also $\varphi$-rings (for the definitions see the paragraph before Lemma \[isom\] (for $\mathcal{O_E}$ and $\mathcal{E}$), section \[partialrobba\] (for $\mathcal{R}$, $\mathcal{O}_{\mathcal{E}}^{\dagger}$, and $\mathcal{E}^\dagger$)).
\[polynomial\] For any positive integer $c$ we have a ring isomorphism $$\begin{aligned}
\varphi^c(R)[X]/(X^{p^c}-\chi(p^c))&\overset{\sim}{\longrightarrow}&R\\
X&\longmapsto&\chi(1)\ .\end{aligned}$$
Since the polynomial ring $\varphi^c(R)[X]$ is a free object in the category of commutative $\varphi^c(R)$-algebras, we may extend the natural inclusion homomorphism $f\colon\varphi^c(R)\hookrightarrow R$ given by $(ii)$ to a ring homomorphism $\tilde{f}\colon\varphi^c(R)[X]\rightarrow R$ by any free choice for the value $\tilde{f}(X)$, in particular such that $\tilde{f}(X):=\chi(1)\in R$ and, of course, $\tilde{f}_{|\varphi^c(R)}:=f$. We need to show that $\tilde{f}$ is surjective with kernel equal to the ideal generated by $X^{p^c}-\chi(p^c)$. Note that $\chi(p^c)=\varphi^c(\chi(1))$ lies in $\varphi^c(R)$ so the claim makes sense.
By $(i)$ the map $\chi$ is a group homomorphism, so $\chi(r)=\chi(1)^r=\tilde{f}(X)^r=\tilde{f}(X^r)$ lies in the image of $\tilde{f}$ for any positive integer $r$. Hence we obtain the surjectivity from by noting that $\varphi^c(R)$ also lies in the image of $\tilde{f}$.
Using again $\chi(r)=\tilde{f}(X^r)$ with the choice of $r=p^c$ we see immediately that $X^{p^c}-\chi(p^c)$ lies in the kernel of $\tilde{f}$. Moreover, note that $\varphi^c(R)[X]/(X^{p^c}-\chi(p^c))$ is a free module of rank $p^c$ over $\varphi^c(R)$ with generators the classes of $\{X^r\}_{r=0}^{p^c-1}$ in the quotient. On the other hand, $R$ is also a free module of rank $p^c$ with generators $\{\chi(r)\}_{r=0}^{p^c-1}$ by and these two sets of generators correspond to each other under the map $\tilde{f}$ hence the isomorphism.
Let $H_0$ be a pro-$p$ group of finite rank (therefore a compact $p$-adic Lie group by Corollary 4.3 and Theorem 8.18 in [@DDMS]) without elements of order $p$ admitting a continuous surjective group homomorphism $\ell\colon
H_0\twoheadrightarrow\mathbb{Z}_p$ with kernel $H_1:={\mathrm{Ker}}(\ell)$. We assume further the following
1. $H_0$ also admits an injective group endomorphism $\varphi\colon
H_0\hookrightarrow H_0$ with finite cokernel and compatible with $\ell$ in the sense that $\ell(\varphi(h))=\varphi(\ell(h))=p\ell(h)$. In particular, we have $\varphi(H_1)\subseteq H_1$.
2. $\bigcap_{n\geq 1}\varphi^n(H_0)=\{1\}$ and the subgroups $\varphi^n(H_0)$ form a system of neighbourhoods of $1$ in $H_0$.
We remark first of all that by a Theorem of Serre (Thm. 1.17 in [@DDMS]) any finite index subgroup in $H_0$ is open. Hence the homomorphism $\varphi$ is automatically continuous and the subgroups $\varphi^n(H_0)$ are open.
Note that $H_1$ is a closed subgroup of $H_0$ hence it is also a pro-$p$ group of finite rank. By assumption $(B)$ we also have in particular that the subgroups $\varphi^n(H_1)$ form a system of open neighbourhoods of $1$ in $H_1$. Note that the subgroups $\varphi^n(H_1)$ may not be normal in either $H_1$ or $H_0$. Hence we define the normal subgroups $H_k\lhd
H_0$ as the normal subgroup of $H_0$ generated by $\varphi^{k-1}(H_1)$. Since $H_1$ is normal in $H_0$ we automatically have $H_k\subseteq H_1$ for any $k\geq 1$. Moreover, since the $p$-adic Lie group $H_1$ has a system of neighbourhoods of $1$ containing only characteristic subgroups, the $H_k$ also form a system of neighbourhoods of $1$ in $H_1$. On the other hand, we have by definition that $\varphi(H_k)\subseteq H_{k+1}\subseteq H_k$ for each $k\geq 1$. In particular, we have an induced $\varphi$ action on the quotient group $H_0/H_k$. This is, of course, no longer injective.
Since the group $\mathbb{Z}_p$ is topologically generated by one element, we may find a splitting $\iota\colon\mathbb{Z}_p\hookrightarrow H_0$ for the group homomorphism $\ell$. We fix this splitting $\iota$, too. Assume further that
1. the group homomorphism $\iota$ is $\varphi$-equivariant, ie. we have $\iota(\varphi(x))=\varphi(\iota(x))$ for all $x\in \mathbb{Z}_p$.
We define the skew group ring $R[H_1/H_k,\ell,\iota]$ as follows. We put $$R[H_1/H_k,\ell,\iota]:=\bigoplus_{h\in H_1/H_k}Rh\ . \label{defR[H_1/H_k]}$$ as left $R$-modules. Note that since $H_1$ is a normal subgroup in $H_0$, we also have $H_1/H_k\lhd H_0/H_k$. Therefore we obtain a conjugation action of $\mathbb{Z}_p$ on $H_1/H_k$ given by $$\begin{aligned}
\rho\colon\mathbb{Z}_p&\to&{\mathrm{Aut}}(H_1/H_k)\\
z&\mapsto&(h\mapsto\iota(z)h\iota(z)^{-1}, h\in H_1/H_k)\ .\end{aligned}$$ Since $H_1/H_k$ is a finite $p$-group, $|{\mathrm{Aut}}(H_1/H_k)|<\infty$ and we have an integer $c_k\geq1$ such that $p^{c_k}\mathbb{Z}_p\subseteq{\mathrm{Ker}}(\rho)$. The multiplication is defined so that $\varphi^{c_k}(R)$ commutes with elements $h$ in $H_1/H_k$ and $\chi(i)$ acts on $H_1/H_k$ via $\iota\circ\chi^{-1}$ and conjugation. More precisely for $r_1,r_2\in R$ and $h_1,h_2\in H_1/H_k$ we may write $$r_2=\sum_{i=0}^{p^{c_k}-1}\chi(i)\varphi^{c_k}(r_{i,2})\label{expand}$$ and put $$\label{mult}
(r_1h_1)(r_2h_2):=\sum_{i=0}^{p^{c_k}-1}r_1\chi(i)\varphi^{c_k}(r_{i,2})\left((\iota(i)^{-1}h_1\iota(i))h_2\right)\in\bigoplus_{h\in H_1/H_k}Rh.$$ Note that in case $r_2=1$ we have $(r_1h_1)h_2=r_1(h_1h_2)$ and in case $h_1=1$ we have $r_1(r_2h_2)=(r_1r_2)h_2$. Moreover, by the choice of $c_k$, $\iota(p^{c_k}\mathbb{Z}_p)$ lies in the centre of $H_0/H_k$. So we may use any set of representatives of $\mathbb{Z}_p/p^{c_k}\mathbb{Z}_p$ instead of $\{0,1,\dots,p^{c_k}-1\}$ in in order to compute . Indeed, if $i\equiv i'\pmod{p^{c_k}}$ then we have $\chi(i)\varphi^{c_k}(r_{i,2})=\chi(i')\varphi^{c_k}(\chi(\frac{i-i'}{p^{c_k}})r_{i,2})$ and $\iota(i)^{-1}h_1\iota(i)=\iota(i')^{-1}h_1\iota(i')$.
\[lemmamult\] The multipliction equips $R[H_1/H_k,\ell,\iota]$ with a ring structure.
There exists an easy, but rather long computation showing this. However, there is another, more conceptual description of the ring $R[H_1/H_k,\ell,\iota]$ pointed out by Torsten Schoeneberg that proves this lemma without any computations. Let $S$ be the group ring $S:=\varphi^{c_k}(R)[H_1/H_k]$ and $\sigma$ be the automorphism of $S$ trivial on $\varphi^{c_k}(R)$ and acting by conjugation with $\iota(1)$ on $H_1/H_k$, ie. for $h\in H_1/H_k$ put $\sigma(h):=\iota(1)^{-1}h\iota(1)$. Now define the skew polynomial ring $S[X,\sigma]$ by the relation $aX=X\sigma(a)$ for $a\in S$. Note that by the definition of $\sigma$, the subring $\varphi^{c_k}(R)$ lies in the centre of $S[X,\sigma]$ therefore so does $\chi(p^{c_k})= \varphi^{c_k}(\chi(1))\in \varphi^{c_k}(R)$. On the other hand, we have $aX^{p^{c_k}}=X^{p^{c_k}}\sigma^{p^{c_k}}(a)=X^{p^{c_k}}a$ for all $a\in S$ since $\sigma^{p^{c_k}} $ is the conjugation by the central element $\iota(1)^{p^{c_k}}=\iota(p^{c_k})\in H_0/H_k$ on $H_1/H_k$ and is trivial by definition on $\varphi^{c_k}(R)$ hence $\sigma^{p^{c_k}}={\mathrm{id}}_S$. This shows that $X^{p^{c_k}}-\chi(p^{c_k})$ is central and that $S[X,\sigma](X^{p^{c_k}}-\chi(p^{c_k}))=(X^{p^{c_k}}-\chi(p^{c_k}))S[X,\sigma]$ is a two-sided ideal in $S[X,\sigma]$. So we may form the quotient ring and compute (as left $\varphi^{c_k}(R)$-modules) $$\begin{aligned}
S[X,\sigma]/(X^{p^{c_k}}-\chi(p^{c_k}))\cong\left(\bigoplus_{r=0}^{\infty}\bigoplus_{h\in
H_1/H_k}X^r\varphi^{c_k}(R)h\right)/ (X^{p^{c_k}}-\chi(p^{c_k}))\cong\\
\cong\bigoplus_{h\in
H_1/H_k}\left(\varphi^{c_k}(R)[X]/(X^{p^{c_k}}-\chi(p^{c_k}))\right)h\cong \bigoplus_{h\in
H_1/H_k}Rh\end{aligned}$$ using Lemma \[polynomial\] in the middle. Note that on the component $h=1$ in the above direct sum the identification is even multiplicative as Lemma \[polynomial\] gives an isomorphism of rings, not just $\varphi^{c_k}(R)$-modules. Hence $S[X,\sigma]/(X^{p^{c_k}}-\chi(p^{c_k}))$ contains $R$ as a subring and the isomorphism above is an isomorphism of left $R$-modules. The transport of ring structure gives back the definition of multiplication on the right hand side. Indeed, we have $$\begin{aligned}
(r_1h_1)(r_2h_2)=\sum_{i=0}^{p^{c_k}-1}r_1h_1\chi(i)\varphi^{c_k}(r_{i,2})h_2=\sum_{i=0}^{p^{c_k}-1}r_1h_1X^i\varphi^{c_k}(r_{i,2})h_2=\\
=\sum_{i=0}^{p^{c_k}-1}r_1X^i\sigma^i(h_1)\varphi^{c_k}(r_{i,2})h_2=\sum_{i=0}^{p^{c_k}-1}r_1\chi(i)\varphi^{c_k}(r_{i,2})\left((\iota(i)^{-1}h_1\iota(i))h_2\right)\end{aligned}$$ since $\chi(i)$ corresponds to $X^i$ under the isomorphism in Lemma \[polynomial\].
We further have a natural action of $\varphi$ on $R[H_1/H_k,\ell,\iota]$ coming from the $\varphi$-action on both $R$ and $H_1/H_k$ by putting $\varphi(rh):=\varphi(r)\varphi(h)$ for $r\in R$ and $h\in H_1/H_k$.
The map $\varphi\colon R[H_1/H_k,\ell,\iota]\to R[H_1/H_k,\ell,\iota]$ defined above is a ring homomorphism.
The additivity is clear, so it suffices to check the multiplicativity. Using we compute $$\begin{aligned}
\varphi((r_1h_1)(r_2h_2))=\sum_{i=0}^{p^{c_k}-1}\varphi\left(r_1\chi(i)\varphi^{c_k}(r_{i,2})\right)\varphi\left((\iota(i)^{-1}h_1\iota(i))h_2\right)=\\
=\sum_{i=0}^{p^{c_k}-1}\varphi(r_1)\chi(pi)\varphi^{c_k+1}(r_{i,2})\left((\iota(pi)^{-1}\varphi(h_1)\iota(pi))\varphi(h_2)\right)=\\
=\sum_{i=0}^{p^{c_k}-1}\varphi(r_1)\varphi^{c_k+1}(r_{i,2})\varphi(h_1)\chi(pi)\varphi(h_2)=\varphi(r_1)\varphi(h_1)\sum_{i=0}^{p^{c_k}-1}\chi(pi)\varphi^{c_k+1}(r_{i,2})\varphi(h_2)=\\
=\varphi(r_1h_1)\varphi\left(\sum_{i=0}^{p^{c_k}-1}\chi(i)\varphi^{c_k}(r_{i,2})h_2\right)=\varphi(r_1h_1)\varphi(r_2h_2)\ .\end{aligned}$$
Moreover, the map $\chi$ and the inclusion of the group $H_1/H_k$ in the multiplicative group of $R[H_1/H_k,\ell,\iota]$ are compatible in the sense that they glue together to a $\varphi$-equivariant group homomorphism $\chi_k\colon H_0\to R[H_1/H_k,\ell,\iota]^{\times}$ (with kernel ${\mathrm{Ker}}\chi_k=H_k$) making the diagram $$\label{chicomm}
\xymatrix{
\mathbb{Z}_p\ar[r]_{\iota} \ar[d]_{\chi} &H_0 \ar[d]_{\chi_{k}}\\
R\ar[r]_{\iota_{R,k}} & R[H_1/H_k,\ell]
}$$ commutative where $\iota_{R,k}$ is the natural inclusion of $R$ in $R[H_1/H_k,\ell]$. Indeed, $H_0\cong\iota(\mathbb{Z}_p)\ltimes H_1$, so we put $\chi_k(\iota(i)h):=\chi(i)(hH_k)$ for $i\in\mathbb{Z}_p$, $h\in H_1$ and compute $$\begin{aligned}
\chi_k(\iota(i_1)h_1\iota(i_2)h_2)=\chi_k(\iota(i_1+i_2)\iota(i_2)^{-1}h_1\iota(i_2)h_2)=\chi(i_1+i_2)(\iota(i_2)^{-1}h_1\iota(i_2)h_2)H_k=\\
=\chi(i_1)(h_1H_k)\chi(i_2)(h_2H_k)=\chi_k(\iota(i_1)h_1)\chi_k(\iota(i_2)h_2)\end{aligned}$$ showing that $\chi_k$ is indeed a group homomorphism. The commutativity of the diagram is clear by definition. Moreover, $\chi_k$ is $\varphi$-equivariant, since we have $$\chi_k\circ\varphi(\iota(i)h)=\chi_k(\iota(pi)\varphi(h))=\chi(pi)\varphi(h)H_k=\varphi(\chi(i)hH_k)=\varphi\circ\chi_k(\iota(i)h)\ .$$
The above definition of $R[H_1/H_k,\ell,\iota]$ does not depend on the choice of the section $\iota$ up to natural isomorphism.
Let $\iota'\colon\mathbb{Z}_p\hookrightarrow H_0$ be another section of $\ell$. Note that the integer $c_k$ depends on $\iota$ but we also have another integer $c_k'$ such that $\iota'(p^{c_k'})$ acts trivially by conjugation on $H_1/H_k$, ie. $\iota'(p^{c_k'})$ lies in the centre of $H_0/H_k$. On the other hand, we may choose $m_k\geq 0$ so that $H_1^{p^{m_k}}\subseteq H_k$ since $H_1/H_k$ is a finite $p$-group. From $\ell\circ\iota={\mathrm{id}}_{\mathbb{Z}_p}=\ell\circ\iota'$ we see that $\iota^{-1}\iota'(\mathbb{Z}_p)\subseteq{\mathrm{Ker}}(\ell)=H_1$ hence for any $x\in\mathbb{Z}_p$ we have $$\begin{aligned}
\iota^{-1}(p^{m_k+\max(c_k,c_k')}x)\iota'(p^{m_k+\max(c_k,c_k')}x)= \iota^{-1}(p^{\max(c_k,c_k')}x)^{p^{m_k}}\iota'(p^{\max(c_k,c_k')}x)^{p^{m_k}}=\\
=\left(\iota^{-1}(p^{\max(c_k,c_k')}x)\iota'(p^{\max(c_k,c_k')}x)\right)^{p^{m_k}}\in H_1^{p^{m_k}}\subseteq H_k\ . \end{aligned}$$ Therefore for $m\geq m_k+\max(c_k,c_k')$ the map $$\begin{aligned}
\iota_k'\colon R&\hookrightarrow& R[H_1/H_k,\ell,\iota]\notag\\
\iota'_k\left(\sum_{i=0}^{p^m-1}\chi(i)\varphi^m(r_i)\right)&:=&\sum_{i=0}^{p^m-1}\chi(i)\varphi^m(r_i)(\iota(i)^{-1}\iota'(i))\label{iota'}\end{aligned}$$ extends to an isomorphism $$\begin{aligned}
\iota_k'\colon R[H_1/H_k,\ell,\iota']&\rightarrow& R[H_1/H_k,\ell,\iota]\\
rh&\mapsto&\iota'_k(r)h\end{aligned}$$ of $\varphi$-rings. Indeed, the map $\iota'_k$ is clearly additive and bijective. We claim that it is multiplicative and $\varphi$-equivariant. We first show the latter statement and compute $$\begin{aligned}
\varphi\circ\iota'_k\left(\sum_{i=0}^{p^m-1}\chi(i)\varphi^m(r_i)\right)=\sum_{i=0}^{p^m-1}\varphi(\chi(i)\varphi^m(r_i)(\iota(i)^{-1}\iota'(i)))=\\
\sum_{i=0}^{p^m-1}\chi(pi)\varphi^{m+1}(r_i)(\iota(pi)^{-1}\iota'(pi))
=\iota'_k\left(\sum_{i=0}^{p^m-1}\chi(pi)\varphi^{m+1}(r_i)\right)=\iota'_k\circ\varphi\left(\sum_{i=0}^{p^m-1}\chi(i)\varphi^m(r_i)\right)\ .\end{aligned}$$
Note that since $m\geq \max(c_k,c_k')$ the subring $\varphi^m(R)$ lies in the centre of both $R[H_1/H_k,\ell,\iota]$ and $R[H_1/H_k,\ell,\iota']$. Therefore—in view of the associativity (Lemma \[lemmamult\])—we may compute the multiplication by expanding elements of $R$ to degree $m$. So we write $$r_1=\sum_{j=0}^{p^m-1}\chi(j)\varphi^m(r_{j,1})\ ,\quad
r_2=\sum_{i=0}^{p^m-1}\chi(i)\varphi^m(r_{i,2})\ .$$ Moreover, we may compute using any set of representatives of $\mathbb{Z}_p/p^m\mathbb{Z}_p$ (e.g. $\{j,j+1,\dots,j+p^m-1\}$ instead of $\{0,1,\dots,p^m-1\}$) since $\iota^{-1}\iota'(p^m\mathbb{Z}_p)\subseteq H_k$. Hence we obtain $$\begin{aligned}
\iota'_k\left((r_1h_1)(r_2h_2)\right)=\iota_k'\left(\sum_{i=0}^{p^m-1}r_1\chi(i)\varphi^m(r_{i,2})(\iota'(i)^{-1}h_1\iota'(i)h_2)\right)=\\
=\iota_k'\left(\sum_{i,j=0}^{p^m-1}\chi(j)\varphi^m(r_{j,1})\chi(i)\varphi^m(r_{i,2})(\iota'(i)^{-1}h_1\iota'(i)h_2)\right)=\\
=\sum_{i=0}^{p^m-1}\iota_k'\left(\sum_{j=0}^{p^m-1}\chi(i+j)\varphi^m(r_{j,1}r_{i,2})\right)\iota'(i)^{-1}h_1\iota'(i)h_2=\\
=\sum_{i,j=0}^{p^m-1}\chi(i+j)\varphi^m(r_{j,1}r_{i,2})\iota(i+j)^{-1}\iota'(i+j)\iota'(i)^{-1}h_1\iota'(i)h_2=\\
=\sum_{i,j=0}^{p^m-1}\chi(i+j)\varphi^m(r_{j,1}r_{i,2})\iota(i+j)^{-1}\iota'(j)h_1\iota'(i)h_2)=\\
=\sum_{i,j=0}^{p^m-1}\chi(j)\varphi^m(r_{j,1})\chi(i)\varphi^m(r_{i,r})\iota(i)^{-1}(\iota(j)^{-1}\iota'(j)h_1)\iota(i)(\iota(i)^{-1}\iota'(i)h_2)=\\
=\left(\sum_{j=0}^{p^m-1}\chi(j)\varphi^m(r_{j,1})(\iota(j)^{-1}\iota'(j)h_1)\right)
\left(\sum_{i=0}^{p^m-1}\chi(i)\varphi^m(r_{i,2})(\iota(i)^{-1}\iota'(i)h_2)\right)=\\
\left(\sum_{j=0}^{p^m-1}\varphi^m(r_{j,1})\iota'(j)h_1\right)
\left(\sum_{i=0}^{p^m-1}\varphi^m(r_{i,2})\iota'(i)h_2\right)=\iota'_k(r_1h_1)\iota_k'(r_2h_2)\ .\end{aligned}$$
In view of the above Lemma we omit $\iota$ from the notation from now on. This construction is compatible with the natural surjective homomorphisms $H_1/H_{k+1}\twoheadrightarrow H_1/H_k$ therefore the rings $R[H_1/H_k,\ell]$ form an inverse system for the induced maps. So we may define the completed skew group ring $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$ as the projective limit $$R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}:=\varprojlim_{k}R[H_1/H_k,\ell].$$ We denote by $I_k$ the kernel of the canonical surjective homomorphism from $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$ to $R[H_1/H_k,\ell]$.
Whenever $R$ is a topological ring we equip $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$ with the projective limit topology of the product topologies on each $\bigoplus_{h\in H_1/H_k}Rh$.
The augmentation map $H_1\twoheadrightarrow1$ induces a ring homomorphism $\ell:=\ell_R\colon
R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}\twoheadrightarrow R$. This also has a section $\iota:=\iota_R=\varprojlim\iota_{R,k}\colon R\hookrightarrow R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$ (whenever clear we omit the subscript $_R$), ie. $\ell_R\circ\iota_R={\mathrm{id}}_R$. Moreover, by the group homomorphism $\chi\colon
\mathbb{Z}_p\to R^{\times}$ extends to a group homomorphism $\chi_{H_0}\colon H_0\to
R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}^{\times}$ making the diagram $$\xymatrix{
\mathbb{Z}_p\ar[r]_{\iota} \ar[d]_{\chi} &H_0 \ar[d]_{\chi_{H_0}}\\
R\ar[r]_{\iota_R} & R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}}$$ commutative.
The operator $\varphi$ acts naturally on this projective limit. If $R$ is a topological ring and $\varphi$ acts continuously on $R$ then $\varphi$ also acts continuously on each $R[H_1/H_k,\ell]$ and by taking the limit also on $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$. For an open subgroup $H'$ of a profinite group $H$ we use the notation $J(H/H')$ for a set of representatives of the left cosets of $H'$ in $H$. Similarly, we use $J(H'\setminus H)$ for a set of representatives of the right cosets $H'\setminus H$.
\[cosets\]
1. Let $L\leq K\leq H$ be groups. Then the set $J(H/K)J(K/L)$ (resp. $J(L\setminus K)J(K\setminus H)$) is a set of representatives for the cosets $H/L$ (resp. for $L\setminus H$).
2. Let $K\leq H$ be groups and $N\lhd H$ a *normal* subgroup. Then $J((K\cap N)\setminus N)$ is also a set of representatives for $K\setminus KN$.
These are well-known facts in group theory, however, for the convenience of the reader we recall their proofs here. Note that in $b)$ we need $N$ to be a normal subgroup so that $KN$ is a subgroup of $H$. Also note that $J(K\setminus KN)$ might not lie in $N$ in general.
$a)$ Let $h_1,h_2\in J(H/K)$ and $k_1,k_2\in J(K/L)$. Suppose we have $h_1k_1L=h_2k_2L$. Then we also have $h_1^{-1}h_2\in K$ so $h_1=h_2$, whence $k_1^{-1}k_2\in L$ so $k_1=k_2$. So the elements of the set $J(H/K)J(K/L)$ are in distinct left cosets of $L$. On the other hand, if $hL\in H/L$ is a left coset, then we may first choose $h_1\in
J(H/K)$ so that $h_1^{-1}h\in K$ and then $k_1\in J(K/L)$ so that $k_1^{-1}h_1^{-1}h\in L$, ie. $hL=h_1k_1L$.
$b)$ If $n_1\neq n_2\in J((K\cap N)\setminus N)$ are distinct then $Kn_1\neq Kn_2$ as $n_1n_2^{-1}$ does not lie in $K\cap N$, but it lies in $N$. On the other hand, if $kn\in KN$ then we may find $n_1\in
J((K\cap N)\setminus N)$ such that $nn_1^{-1}\in K\cap N$ hence $knn_1^{-1}\in K$.
The map $\varphi\colon R{[\hspace{-0.04cm}[}H_1,\ell {]\hspace{-0.04cm}]}\rightarrow R{[\hspace{-0.04cm}[}H_1,\ell {]\hspace{-0.04cm}]}$ is injective. Moreover, we have $$R{[\hspace{-0.04cm}[}H_1,\ell {]\hspace{-0.04cm}]}=\bigoplus_{h\in J(\varphi(H_0)\setminus H_0)}\varphi(R{[\hspace{-0.04cm}[}H_1,\ell {]\hspace{-0.04cm}]})h.$$ In particular, $R{[\hspace{-0.04cm}[}H_1,\ell {]\hspace{-0.04cm}]}$ is a free (left) module of rank $[H_0:\varphi(H_0)]$ over itself via $\varphi$.
*Step 1.* Let $k$ be an integer and denote by $A_k$ the kernel of the map $\varphi\colon
R[H_1/H_k,\ell]\to R[H_1/H_k,\ell]$ so that we have a short exact sequence of abelian groups $$0\longrightarrow A_k \longrightarrow R[H_1/H_k,\ell]
\overset{\varphi}{\longrightarrow} \varphi(R[H_1/H_k,\ell])\longrightarrow 0.$$ We are going to show that the sequence $A_k$ satisfies the trivial Mittag-Leffler condition. From this the injectivity of $\varphi$ follows, and we obtain $$\varprojlim_{k}\varphi(R[H_1/H_k,\ell])\cong \varphi(\varprojlim_k
R[H_1/H_k,\ell])=\varphi(R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}).\label{lim}$$ Take a fixed positive integer $k$. Since $\varphi\colon
H_1\rightarrow H_1$ is an open map (bijective and continuous between the compact sets $H_1$ and $\varphi(H_1)$ hence a homeomorphism) and the subgroups $H_l$ form a system of neighbourhoods we find an integer $l> k$ such that $H_k\supseteq\varphi^{-1}(H_{l})$. In view of Lemma \[cosets\] we put $$J(H_1/H_l):=J(H_1/\varphi^{-1}(H_l))J(\varphi^{-1}(H_l)/H_l)$$ for $J(H_1/\varphi^{-1}(H_l))$ and $J(\varphi^{-1}(H_l)/H_l)$ arbitrarily fixed sets of representatives for the cosets of $H_1/\varphi^{-1}(H_l)$ and of $\varphi^{-1}(H_l)/H_l$, respectively.
Let now $\sum_{h\in J(H_1/H_{l})}r_h\chi_l(h)$ be an element in $A_l$ and denote by $f_{k,l}$ the natural surjection from $R[H_1/H_l,\ell]\twoheadrightarrow R[H_1/H_k,\ell]$. We have $$\begin{aligned}
0=\varphi\left(\sum_{h\in J(H_1/H_l)}r_h\chi_l(h)\right)=
\sum_{h_1\in
J(H_1/\varphi^{-1}(H_l))}\sum_{h_2\in
J(\varphi^{-1}(H_l)/H_l)}\varphi(r_{h_1h_2})\chi_l(\varphi(h_1h_2))=\\
=\sum_{h_1}\sum_{h_2}\varphi(r_{h_1h_2})\chi_l(\varphi(h_1))=\sum_{h_1\in J(H_1/\varphi^{-1}(H_l))}\varphi\left(\sum_{h\in
J(H_1/H_l)\cap h_1\varphi^{-1}(H_l)}r_h\right)\chi_l(\varphi(h_1)).\end{aligned}$$ Note that for $h_1\neq h_1'\in J(H_1/\varphi^{-1}(H_l))$ we have $\varphi(h_1)H_l\neq \varphi(h_1')H_l$. Since $R[H_1/H_l,\ell]$ is defined as a direct sum, we obtain $$\varphi\left(\sum_{h\in J(H_1/H_l)\cap h_1\varphi^{-1}(H_l)}r_h\right)=0\
\text{, whence}\
\sum_{h\in J(H_1/H_l)\cap h_1\varphi^{-1}(H_l)}r_h=0$$ for any fixed $h_1\in J(H_1/\varphi^{-1}(H_l))$ as $\varphi$ is injective on $R$. On the other hand, we have $$\begin{aligned}
f_{k,l}\left(\sum_{h\in J(H_1/H_l)}r_h\chi_l(h)\right)
=\sum_{h_1\in J(H_1/H_k)}(\sum_{h\in
J(H_1/H_l)\cap h_1H_k}r_h)\chi_k(h_1)=\sum_{h_1\in J(H_1/H_k)}0\chi_k(h_1)=0\end{aligned}$$ as $h_1H_k$ is a disjoint union of cosets of $\varphi^{-1}(H_l)$ by the choice of $l$. This shows that $f_{k,l}(A_{l})=0$ as claimed. Therefore follows as discussed above.
*Step 2.* Since $\varphi(H_0)\cap H_1$ is open in $H_1$, there exists an integer $k_0\geq 2$ such that for $k\geq k_0$ we have $H_k\subseteq\varphi(H_0)$. (We remark here that we may not be able to take $k_0=2$ because $H_k$ is the *normal* subgroup generated by $\varphi(H_1)$ which does have elements outside $\varphi(H_0)$ in general.) We claim now the decomposition $$\label{flat}
R[H_1/H_k,\ell ]=\bigoplus_{h\in J(\varphi(H_0)\setminus H_0)}\varphi(R[H_1/H_k,\ell ])\chi_k(h)$$ for $k\geq k_0$. Note that since $H_k$ is a normal subgroup of $H_0$ contained in $\varphi(H_0)$ the elements $\chi_k(h)$ above are distinct.
For the proof of we apply Lemma \[cosets\] $b)$ in the situation $K:=\varphi(H_0)$, $N:=H_1$, and $H:=H_0$ to be able to choose $J(\varphi(H_0)\setminus\varphi(H_0)H_1):=J((\varphi(H_0)\cap
H_1)\setminus H_1)$. Moreover, by the injectivity of $\varphi$ on $H_0/H_1$ we see that $\varphi(H_0)\cap H_1=\varphi(H_1)$. On the other hand, $\iota(\{0,1,\dots,p-1\})$ is a set of representatives for the cosets $H_1\varphi(H_0)\setminus H_0$. Therefore (using Lemma \[cosets\] $a)$ with $L:=\varphi(H_0)$, $K:=\varphi(H_0)H_1$, and $H:=H_0$) we may choose $$J(\varphi(H_0)\setminus H_0):=J(\varphi(H_0)\setminus
\varphi(H_0)H_1)J(\varphi(H_0)H_1\setminus H_0)=J(\varphi(H_1)\setminus
H_1)\iota(\{0,1,\dots,p-1\})\ .$$
We are going to use this specific set $J(\varphi(H_0)\setminus H_0)$ in order to compute the right hand side of . Let $\sum_{h\in
J(H_1/H_k)}r_h\chi_k(h)$ be an arbitrary element in $R[H_1/H_k,\ell]$. By the étaleness of the action of $\varphi$ on $R$ (noting that $R$ is commutative) we may uniquely decompose $$r_h=\sum_{i=0}^{p-1}\chi(i)\varphi(r_{i,h})=\sum_{i=0}^{p-1}\varphi(r_{i,h})\chi(i)\ .$$ On the other hand, we write $\iota(i)h\iota(i)^{-1}=\varphi(u_{i,h})v_{i,h}$ with unique $u_{i,h}\in H_1$ and $v_{i,h}\in J(\varphi(H_1)\setminus H_1)$. Therefore we have $$\begin{aligned}
\sum_{h\in J(H_1/H_k)}r_h\chi_k(h)=
\sum_{h\in
J(H_1/H_k)}\sum_{i=0}^{p-1}\varphi(r_{i,h})\chi(i)\chi_k(\iota(i)^{-1}\varphi(u_{i,h})v_{i,h}\iota(i))=\\
=\sum_{h\in
J(H_1/H_k)}\sum_{i=0}^{p-1}\varphi(r_{i,h}\chi_k(u_{i,h}))\chi_k(v_{i,h}\iota(i))\in
\sum_{h\in J(\varphi(H_0)\setminus
H_0)}\varphi(R[H_1/H_k,\ell])\chi_k(h)\end{aligned}$$ as $\chi(i)=\chi_k(\iota(i))$ and $\chi_k\circ\varphi=\varphi\circ\chi_k$ by .
It remains to show that the sum in is indeed direct. For this we may expand any element $x_{i,h}\in R[H_1/H_k,\ell]$ as $$x_{i,h}=\sum_{m\in J(H_1/H_k)}r_{i,h,m}\chi_k(m)$$ and compute $$\begin{aligned}
\sum_{i=0}^{p-1}\sum_{h\in J(\varphi(H_1)\setminus
H_1)}\varphi(x_{i,h})\chi_k(h\iota(i))=\sum_{i,h,m}\varphi(r_{i,h,m})\chi_k(\varphi(m)h\iota(i))=\label{unique}\\
=\sum_{i,h,m}\chi(i)\varphi(r_{i,h,m})\chi_k(\iota(i)^{-1}\varphi(m)h\iota(i))=\notag\\
=\sum_{i,h}\sum_{m_0\in J(\varphi(H_1)/H_k)}\chi(i)\left(\sum_{m\in J(H_1/H_k)\cap\varphi^{-1}(m_0H_k)}\varphi(r_{i,h,m})\right)\chi_k(\iota(i)^{-1}m_0h\iota(i)).\notag\end{aligned}$$ Assume now that the left hand side of is 0. The set $J(\varphi(H_1)/H_k)J(\varphi(H_1)\setminus H_1)$ is a set of representatives of $H_k\setminus H_1$ because $H_k$ is normal in $H_1$ whence $\varphi(H_1)/H_k=H_k\setminus \varphi(H_1)$. This shows that the elements $m_0h$ are distinct in $H_1/H_k$ on the right hand side of . Moreover, the conjugation by $\iota(i)$ is an automorphism of $H_1/H_k$ therefore the elements $\iota(i)^{-1}m_0h\iota(i)$ are also distinct for any fixed $i\in\{0,1,\dots,p-1\}$. On the other hand, by the étaleness of $\varphi$ on $R$ and by we obtain $$R[H_1/H_k,\ell]=\bigoplus_{i=0}^{p-1}\bigoplus_{h_1\in
H_1/H_k}\chi(i)\varphi(R)h\ .$$ Hence we have $$\sum_{m\in J(H_1/H_k)\cap\varphi^{-1}(m_0H_k)}\varphi(r_{i,h,m})=0$$ for any fixed $m_0$, $i$, and $h$. In particular, we also have $$\begin{aligned}
\varphi(x_{i,h})=\sum_{m\in
J(H_1/H_k)}\varphi(r_{i,h,m})\chi_k(\varphi(m))=\\
=\sum_{m_0\in J(\varphi(H_1)/H_k)}\left(\sum_{m\in J(H_1/H_k)\cap\varphi^{-1}(m_0H_k)}\varphi(r_{i,h,m})\right)\chi_k(m_0)=0\end{aligned}$$ showing that the sum in is direct.
*Step 3.* The result follows by taking the projective limit of using .
The above lemma holds for replacing left and right, as well, ie. we also have $$R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}=\bigoplus_{h\in J(H_0/\varphi(H_0))}h\varphi(R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}).$$
Let $S$ be a (not necessarily commutative) ring (with identity) with the following properties:
1. There exists a group homomorphism $\chi\colon H_0\hookrightarrow
S^{\times}$.
2. The ring $S$ admits an étale action of the $p$-Frobenius $\varphi$ that is compatible with $\chi$. More precisely there is an injective ring homomorphism $\varphi\colon S\hookrightarrow S$ such that $\varphi(\chi(x))=\chi(\varphi(x))$ and $$S=\bigoplus_{h\in \varphi(H_0)\setminus H_0}\varphi(S)\chi(h)=\bigoplus_{h\in H_0/\varphi(H_0)}\chi(h)\varphi(S)\ .$$ In particular, $S$ is free of rank $|H_0:\varphi(H_0)|$ as a left, as well as a right module over $\varphi(S)$.
\[phiringH\_0\] We call a ring $S$ with the above properties $(i)$ and $(ii)$ a $\varphi$-ring over $H_0$.
The map $R\mapsto R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ is a functor from the category of $\varphi$-rings over $\mathbb{Z}_p$ to the category of $\varphi$-rings over $H_0$.
\[pronilp\] We have $\varphi(I_k)\subseteq I_{k+1}$ for all $k\geq 1$.
Take $x\in I_k$ and write $x+I_{k+1}\in R[H_1/H_{k+1},\ell]$ as $x+I_{k+1}=\sum_{h\in J(H_1/H_{k+1})}r_h\chi_{k+1}(h)$. Since $x\in
I_k$ we have $$0=\sum_{h\in J(H_1/H_{k+1})}r_h\chi_{k}(h)=\sum_{h_1\in
J(H_1/H_k)}\sum_{h\in J(H_1/H_{k+1})\cap h_1H_k}r_h\chi_k(h_1)$$ hence $\sum_{h\in J(H_1/H_{k+1})\cap h_1H_k}r_h=0$ for any fixed $h_1\in
J(H_1/H_k)$. So we compute $$\begin{aligned}
\varphi(x)+I_{k+1}=\sum_{h_1\in
J(H_1/H_k)}\sum_{h\in J(H_1/H_{k+1})\cap
h_1H_k}\varphi(r_h)\varphi(\chi_k(h))=\\
=\sum_{h_1\in
J(H_1/H_k)}\sum_{h\in J(H_1/H_{k+1})\cap
h_1H_k}\varphi(r_h)\chi_k(\varphi(h_1))=\sum_{h_1\in
J(H_1/H_k)}0\chi_k(\varphi(h_1))=0\end{aligned}$$ since $\varphi(H_k)\subseteq H_{k+1}$ whence $\varphi(h_1)=\varphi(h)$ above.
Recall that Fontaine’s ring $\mathcal{O_E}:=\varprojlim_{n}(o{[\hspace{-0.04cm}[}T{]\hspace{-0.04cm}]}[T^{-1}])/p_K^n$ is defined as the $p$-adic completion of the ring of formal Laurent-series over $o$. It is a complete discrete valuation ring with maximal ideal $p_K\mathcal{O_E}$, residue field $k{(\hspace{-0.06cm}(}T{)\hspace{-0.06cm})}$, and field of fractions $\mathcal{E}=\mathcal{O_E}[p_K^{-1}]$. We are going to show that the completed skew group ring $\mathcal{O_E}{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$ is isomorphic to the previously constructed ([@SVe], see also section 8 of [@SVi], [@SVZ], [@Z]) microlocalized ring $\Lambda_\ell(H_0)$ of the Iwasawa algebra $\Lambda(H_0)$. ($H_0=N_0$ in the notations of [@SVi], [@SVZ], and [@Z].) For the convenience of the reader we recall the definition here. Let $\Lambda(H_0):=o{[\hspace{-0.04cm}[}H_0{]\hspace{-0.04cm}]}$ be the Iwasawa algebra of the pro-$p$ group $H_0$. It is shown in [@CFKSV] that $S:=\Lambda(H_0)\setminus(p_K,H_1-1)$ is a left and right Ore set in $\Lambda(H_0)$ so that the localization $\Lambda(H_0)_S$ exists. The ring $\Lambda_\ell(H_0)$ is defined as the $(p_K,H_1-1)$-adic completion of $\Lambda(H_0)_S$ (the so called “microlocalization”). Note that since $\varphi\colon H_0\to H_0$ is a continuous group homomorphism, it induces a continuous ring homomorphism $\varphi\colon \Lambda(H_0)\to\Lambda(H_0)$ of the Iwasawa algebra. Moreover, since $\varphi(S)\subset S$, $\varphi$ extends to a ring homomorphism $\varphi\colon\Lambda(H_0)_S\to\Lambda(H_0)_S$ and by continuity to its completion $\Lambda_\ell(H_0)$ (see section 8 of [@SVi] for more details).
\[remark\] Let $R$ be a $\varphi$-ring containing (as a $\varphi$-subring) the Iwasawa algebra $o{[\hspace{-0.04cm}[}T{]\hspace{-0.04cm}]}\cong\Lambda(\mathbb{Z}_p)$. Then using we compute $$\begin{aligned}
R[H_1/H_k,\ell ] \cong (R\otimes_{\Lambda(\mathbb{Z}_p)}\Lambda(\mathbb{Z}_p))[H_1/H_k,\ell]\cong
R\otimes_{\Lambda(\mathbb{Z}_p)}(\Lambda(\mathbb{Z}_p)[H_1/H_k,\ell])\\
\cong R\otimes_{\Lambda(\mathbb{Z}_p),\iota}\Lambda(H_0/H_k)\cong
(\varphi^{c_k}(R)\otimes_{\varphi^{c_k}(\Lambda(\mathbb{Z}_p))}\Lambda(\mathbb{Z}_p))\otimes_{\Lambda(\mathbb{Z}_p),\iota}\Lambda(H_0/H_k)\\
\cong\varphi^{c_k}(R)\otimes_{\varphi^{c_k}(\Lambda(\mathbb{Z}_p)),\iota}\Lambda(H_0/H_k)\end{aligned}$$ for any $k\geq 1$.
\[isom\] We have a $\varphi$-equivariant ring-isomorphism $\mathcal{O_E}{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}\cong\Lambda_\ell(H_0)$.
The ring $\Lambda_\ell(H_0)$ is complete and Hausdorff with respect to the filtration by the ideals generated by $(H_k-1)$ since these ideals are closed with intersection zero in the pseudocompact ring $\Lambda_\ell(H_0)$ (cf. Thm. 4.7 in [@SVe]). So it remains to show that $\Lambda_\ell(H_0/H_k)$ is naturally isomorphic to the skew group ring $\mathcal{O_E}[H_1/H_k,\ell]$. At first we show that $\Lambda(H_0/H_k)\cong\Lambda(\mathbb{Z}_p)[H_1/H_k,\ell]$. Both sides are free modules of rank $|H_1/H_k|$ over $\Lambda(\mathbb{Z}_p)$ with generators $h\in H_1/H_k$ so there is an obvious isomorphism between them as $\Lambda(\mathbb{Z}_p)$-modules. Moreover, $\varphi^{c_k}(\Lambda(\mathbb{Z}_p))$ lies in the centre of both rings. However, the obvious map above is also multiplicative since the multiplication on $\Lambda(\mathbb{Z}_p)[H_1/H_k,\ell]$ is uniquely determined by so that is satisfied and $\varphi^{c_k}(\Lambda(\mathbb{Z}_p))$ lies in the centre.
Now by Remark \[remark\] we have $$\begin{aligned}
\mathcal{O_E}[H_1/H_k,\ell ]\cong\varphi^{c_k}(\mathcal{O_E})\otimes_{\varphi^{c_k}(\Lambda(\mathbb{Z}_p)),\iota}\Lambda(H_0/H_k)\end{aligned}$$ for any $k\geq 1$.
Since $\iota(\varphi^{p^{c_k}}(\Lambda(\mathbb{Z}_p)))$ lies in the centre of $\Lambda(H_0/H_k)$, the right hand side above is the localisation of $\Lambda(H_0/H_k)$ inverting the central element $\varphi^{p^{c_k}}(T)$ and taking the $p$-adic completion afterwards (ie. “microlocalisation” at $\varphi^{p^{c_k}}(T)$). However, in a $p$-adically complete ring $T$ is invertible if and only if so is $\varphi^{p^{c_k}}(T)$. Indeed, we have $$T\mid\varphi^{p^{c_k}}(T)=(T+1)^{p^{c_k}}-1=\sum_{i=1}^{p^{c_k}}\binom{p^{c_k}}{i}T^i\in
T^{p^{c_k}}(1+po{[\hspace{-0.04cm}[}T{]\hspace{-0.04cm}]}[T^{-1}])\ .$$ Hence we obtain $$\varphi^{c_k}(\mathcal{O_E})\otimes_{\varphi^{c_k}(\Lambda(\mathbb{Z}_p)),\iota}\Lambda(H_0/H_k)\cong\Lambda_\ell(H_0/H_k)$$ as both sides are the microlocalisation of $\Lambda(H_0/H_k)$ at $T$.
Equivalence of categories {#equiv}
=========================
Let $S$ be a $\varphi$-ring over any pro-$p$ group $H_0$ satisfying $(A)$, $(B)$, and $(C)$ (for now it would suffice to assume that $S$ has an injective ring-endomorphism $\varphi\colon S\to S$). We define a $\varphi$-module over $S$ to be a free $S$-module $D$ of finite rank together with a semilinear action of $\varphi$ such that the map $$\begin{aligned}
1\otimes\varphi\colon S\otimes_{S,\varphi}D&\rightarrow& D\notag\\
r\otimes d&\mapsto& r\varphi(d)\label{etaledef}\end{aligned}$$ is an isomorphism. Note that for rings $S$ in which $p$ is not invertible (such as $S=\mathcal{O_E}$ and $\mathcal{O}^\dagger_{\mathcal{E}}$) this is the definition of an *étale* $\varphi$-module. However, for rings in which $p$ is invertible (such as the Robba ring $\mathcal{R}$) this is the usual definition of a $\varphi$-module. We use this definition for both $S=R$ and $S=R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$—the former being a $\varphi$-ring over $\mathbb{Z}_p$ and the latter being a $\varphi$-ring over $H_0$. We denote the category of $\varphi$-modules over $R$ (resp. over $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$) by $\mathfrak{M}(R,\varphi)$ (resp. by $\mathfrak{M}(R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]},\varphi)$). These are clearly additive categories. However, they are not abelian in general, as the kernel and cokernel might not be a free module over $R$, resp. over $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$.
Note that for modules $M$ over $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$ (with $D=M$) is equivalent to saying that each element $m\in
M$ is uniquely decomposed as $$m=\sum_{u\in J(H_0/\varphi^k(H_0))}u \varphi^{k}(m_{u,k})$$ for $k=1$, or equivalently, for all $k\geq 1$.
There is an obvious functor in both directions induced by $\ell_R$ and $\iota_R$ that we denote by $$\begin{aligned}
\mathbb{D}:=R\otimes_{R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]},\ell}\cdot\colon&\mathfrak{M}(R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]},\varphi)&\rightarrow\mathfrak{M}(R,\varphi)\\
\mathbb{M}:=R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}\otimes_{R,\iota}\cdot\colon&\mathfrak{M}(R,\varphi)&\rightarrow\mathfrak{M}(R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]},\varphi)\ .\end{aligned}$$
The following is a generalization of Thm. 8.20 in [@SVZ]. The proof is also similar, but we include it here for the convenience of the reader.
\[equivcat\] The functors $\mathbb{D}$ and $\mathbb{M}$ are quasi-inverse equivalences of categories.
We first note that since $\ell\circ\iota={\mathrm{id}}_{R}$ we also have $\mathbb{D}\circ\mathbb{M}\cong{\mathrm{id}}_{\mathfrak{M}(R,\varphi)}$. So it remains to show that $\mathbb{D}$ is full and faithful.
For the faithfulness of $\mathbb{D}$ let $f\colon M_{1}\to M_{2}$ be a morphism in ${\mathfrak M}(R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]},\varphi)$ such that $\mathbb{D}(f) =0 $ which means that $f(M_1)\subseteq I_1M_2$. Let $m\in M_{1}$. For any $k\in \mathbb N$ we write $m=\sum_{u\in J(H_0/\varphi^k(H_0))}u \varphi^{k}(m_{u,k})$ and $$f(m)=\sum_{u\in J(H_0/\varphi^k(H_0))}u \varphi^{k}f(m_{u,k})\in
\varphi^{k}(I_1M_{2})\subseteq I_{k+1}M_2$$ by Remark \[pronilp\]. Therefore $f(M_1)\subseteq I_{k+1}M_2$ for any $k\geq0$ and therefore $f=0$ since $M_2$ is a finitely generated free module over $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$ and $\bigcap_{k\geq 0}I_{k+1}=0$ since $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}\cong\varprojlim R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}/I_k$.
Now we prove that for any object $M$ in $\mathfrak{M}(R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]},\varphi)$ we have an isomorphism $\mathbb{M}\circ\mathbb{D}(M)\rightarrow M$. We start with an arbitrary finite $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$-basis $(\epsilon_{i})_{1\leq i\leq d}$ of $M$ (where $d$ is the rank of $M$). As $R$-modules we have $$M=(\bigoplus_{1\leq i\leq d} \iota (R)\epsilon_{i})\oplus
(\bigoplus_{1\leq i\leq d} I_1\epsilon_{i}) \ .$$ It is clear that the $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$-linear map from $M$ to $\mathbb{M}(\mathbb{D}(M))$ sending $\epsilon_{i}$ to $1\otimes (1\otimes
\epsilon_{i})$ is bijective. It is $\varphi$-equivariant if and only if $\bigoplus_{1\leq
i\leq d} \iota (R)\epsilon_{i}$ is $\varphi$-stable which is, of course, not true in general. We always have $$\varphi(\epsilon_{i})=\sum_{1\leq j\leq d}(a_{i,j} +b_{i,j}) \epsilon_{j} \ \ {\rm where } \ a_{i,j}
\in \iota (R) \ , \ b_{i,j}\in I_1 \ .$$ If the $b_{i,j}$ are not all $0$, we will find elements $ x_{i,j}\in I_1$ such that $$\eta_{i} := \epsilon_{i} +\sum_{1\leq j\leq d} x_{i,j} \epsilon_{j}$$ satisfies $$\varphi (\eta_{i}) = \sum_{1\leq j\leq d}a_{i,j}\eta_{j} \ \ {\rm for } \ i\in I \ .$$ The conditions on the matrix $X:=(x_{i,j})_{1\leq i,j\leq d}$ are : $$\varphi ({\mathrm{id}}+X) (A+B) = A ({\mathrm{id}}+X) $$ for the matrices $A:=(a_{i,j})_{1\leq i,j\leq d} \ , B:=(b_{i,j})_{1\leq
i,j\leq d}$. The coefficients of $A$ belong to the commutative ring $\iota (R) $. The matrix $A+B$ is invertible because the $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$-endomorphism $f$ of $M$ defined by $$f(\epsilon_{i})=\varphi (\epsilon_{i}) \ \ {\rm for } \ 1\leq i\leq d \ ,$$ is an automorphism of $M$ as $M$ lies in $\mathfrak{M}(R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]},\varphi)$. Therefore the matrix $A=\ell(A+B)$ is also invertible. We are reduced to solve the equation $$A^{-1}B+A^{-1} \varphi (X) (A+B ) = X$$ in the indeterminate $X$. We are looking for the solution $X$ in the form of an infinite sum $$X= A^{-1}B + \ldots + (A^{-1}\varphi (A^{-1})\ldots \varphi ^{k-1}(A^{-1})
\varphi ^{k}(A^{-1}B) \varphi ^{k-1}(A+B) \ldots \varphi (A+B)(A+ B) )+
\ldots \ .$$ The coefficients of $A^{-1}B $ belong to the two-sided ideal $I_1$ of $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$ and the coefficients of the $k$-th term of the series $$(A^{-1}\varphi (A^{-1})\ldots \varphi ^{k-1}(A^{-1}) \varphi ^{k}(A^{-1}B)
\varphi ^{k-1}(A+B) \ldots \varphi (A+B)(A+ B) )$$ belong to $\varphi^{k}(I_1)\subseteq I_{k+1}$. Hence the series converges since $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}\cong\varprojlim_k R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}/I_k$. Its limit $X$ is the unique solution of the equation. The coefficients of every term in the series belong to $I_1$ and $I_1$ is closed in $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$, hence $x_{i,j}\in I_1$ for $1\leq i,j\leq d$.
We still need to show that the set $(\eta_{i})_{1\leq i\leq d}$ is an $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$-basis of $M$. Similarly to the above equation we may find a matrix $Y$ with coefficients in $I_1$ such that we have $$(A+B)({\mathrm{id}}+Y)=\varphi({\mathrm{id}}+Y)A.$$ Therefore we obtain $$(A+B)({\mathrm{id}}+Y)({\mathrm{id}}+X)=\varphi\left(({\mathrm{id}}+Y)({\mathrm{id}}+X)\right)(A+B)$$ which means that the map $$\begin{aligned}
({\mathrm{id}}+Y)({\mathrm{id}}+X)\colon M&\rightarrow&M\\
\epsilon_i&\mapsto& ({\mathrm{id}}+Y)({\mathrm{id}}+X)\epsilon_i\end{aligned}$$ is a $\varphi$-equivariant map such that $\mathbb{D}(({\mathrm{id}}+Y)({\mathrm{id}}+X))={\mathrm{id}}$, hence $({\mathrm{id}}+Y)({\mathrm{id}}+X)={\mathrm{id}}$ by the faithfulness of $\mathbb{D}$. By a similar computation we also obtain $A({\mathrm{id}}+X)({\mathrm{id}}+Y)=\varphi\left(({\mathrm{id}}+X)({\mathrm{id}}+Y)\right)A$ showing that $({\mathrm{id}}+X)({\mathrm{id}}+Y)$ is a $\varphi$-equivariant endomorphism of $\mathbb{M}\circ\mathbb{D}(M)$ reducing to the identity modulo $I_1$. Hence $({\mathrm{id}}+Y)$ is a twosided inverse to the map $({\mathrm{id}}+X)$, in particular $(\eta_i)_{1\leq i\leq d}$ is an $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$-basis of $M$. So we obtain an isomorphism in $\mathfrak{M}(R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]},\varphi)$, $$\Theta \ : \ M\to\mathbb \mathbb{M}(\mathbb{D}(M)) \ \ , \ \ \Theta
(\eta_{i})=1\otimes (1 \otimes \eta_{i}) \ \ \hbox{for} \ 1\leq i\leq d \ ,$$ such that $\mathbb{D}(\Theta)$ is the identity morphism of $\mathbb{D}(M)$.
Now if $f\colon \mathbb{D}(M_1)\rightarrow\mathbb{D}(M_2)$ then for $$\mathbb{M}(f)\colon M_1\cong\mathbb{M}\circ\mathbb{D}(M_1)\rightarrow
\mathbb{M}\circ\mathbb{D}(M_2)\cong M_2$$ we have $\mathbb{D}\circ\mathbb{M}(f)=f$ therefore $\mathbb{D}$ is full.
There is a small mistake in Lemma 1 of [@Z]. The map $\omega$ is in fact not a $p$-valuation, since assertion $(iii)$ stating that $\omega(g^p)=\omega(g)+1$ is false. It is only true in the weaker form $\omega(g^p)\geq \omega(g)+1$. However, this does not influence the validity of the rest of the paper as $N_{0,n}:=\{g\in N_0\mid \omega(g)\geq n\}$ is still a subgroup satisfying Lemma 2. Alternatively, it is possible to modify $\omega$ so that one truely obtains a $p$-valuation. I would like to take this opportunity to thank Torsten Schoeneberg for pointing this out to me.
Note that in the case of $R=\mathcal{O_E}$ we may end the proof of Proposition \[equivcat\] by saying that ${\mathrm{id}}+X$ is invertible since $X$ lies in $I_1^{d\times d}$ and $\mathcal{O_E}{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}\cong\Lambda_\ell(H_0)$ is $I_1$-adically complete. However, in the general situation $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}$ may not be complete $I_1$-adically. The reason for this is the fact that the ideals $(I_k)_{k\geq 1}$ are only cofinal with the ideals $I_1^k$ whenever $R$ is killed by a power of $p$. Therefore if $R$ is not $p$-adically complete, we do not have $R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}\cong \varprojlim R{[\hspace{-0.04cm}[}H_1,\ell{]\hspace{-0.04cm}]}/I_1^k$ in general. Moreover, in case of $R=\mathcal{O_E}$ Proposition \[equivcat\] holds for not necessarily free modules, as well. See [@SVZ] for the proof of this.
The matrix $Y$ in the proof of Prop. \[equivcat\] is given by a convergent sum of the terms $$-(A+B)^{-1}\varphi((A+B)^{-1})\ldots\varphi^{k-1}((A+B)^{-1}) \varphi ^{k}((A+B)^{-1}B)
\varphi ^{k-1}(A) \ldots \varphi (A)A$$ for $k\geq 0$ and a direct computation also shows that $({\mathrm{id}}+Y)({\mathrm{id}}+X)={\mathrm{id}}=({\mathrm{id}}+X)({\mathrm{id}}+Y)$.
Reductive groups over $\mathbb{Q}_p$ and Whittaker functionals
--------------------------------------------------------------
Let $p$ be a prime number let $\mathbb{Q}_p\subseteq K$ be a finite extension with ring of integer $o_K$, uniformizer $p_K$, and residue field $k=o_K/p_K$. This field will only play the role of coefficients, the reductive groups will all be defined over $\mathbb{Q}_p$. Following [@SVi], let $G$ be the $\mathbb{Q}_p$-rational points of a $\mathbb{Q}_p$-split connected reductive group over $\mathbb{Q}_p$. In particular, $G$ is a locally $\mathbb{Q}_p$-analytic group. Moreover, we assume that the centre of $G$ is connected. We fix a Borel subgroup $P=TN$ in $G$ with maximal split torus $T$ and unipotent radical $N$. Let $\Phi^+$ denote, as usual, the set of positive roots of $T$ with respect to $P$ and let $\Delta\subseteq\Phi^+$ be the subset of simple roots. For any $\alpha\in\Phi^+$ we have the root subgroup $N_{\alpha}\subseteq N$. We recall that $N=\prod_{\alpha\in\Phi^+}N_{\alpha}$ (set-theoretically) for any total ordering of $\Phi^+$. Let $T_0\subseteq T$ be the maximal compact subgroup. We fix a compact open subgroup $N_0\subseteq N$ which is totally decomposed, in other words $N_0=\prod_{\alpha}(N_0\cap N_{\alpha})$ for any total ordering of $\Phi^+$. Hence $P_0:=T_0N_0$ is a group. We introduce the submonoid $T_+\subseteq T$ of all $t\in T$ such that $tN_0t^{-1}\subseteq
N_0$, or equivalently, such that $|\alpha(t)|\leq1$ for any $\alpha\in\Delta$. Obviously, $P_+:=N_0T_+=P_0T_+P_0$ is then a submonoid of $P$.
We fix once and for all isomorphisms of algebraic groups $$\iota_{\alpha}\colon N_{\alpha}\overset{\cong}{\rightarrow}\mathbb{Q}_p$$ for $\alpha\in\Delta$, such that $$\iota_{\alpha}(tnt^{-1})=\alpha(t)\iota_{\alpha}(n)$$ for any $n\in N_{\alpha}$ and $t\in T$. We normalize these isomorphisms so that $\iota_\alpha(N_0\cap N_\alpha)=\mathbb{Z}_p\subset \mathbb{Q}_p$. Since $\prod_{\alpha\in\Delta}N_{\alpha}$ is naturally a quotient of $N/[N,N]$ we may view any homomorphism $$\ell\colon \prod_{\alpha\in\Delta}N_{\alpha}\rightarrow\mathbb{Q}_p$$ as a functional on $N$. We fix once and for all a homomorphism $\ell$ such that we have $\ell(N_0)=\mathbb{Z}_p$. Let $X^*(T):={\mathrm{Hom}}_{alg}(T,\mathbb{G}_m)$ (resp. $X_*(T):={\mathrm{Hom}}_{alg}(\mathbb{G}_m,T)$) be the group of algebraic characters (resp. cocharacters) of $T$. Since we assume that the centre of $G$ is connected, the quotient $X^*(T)/\bigoplus_{\alpha\in\Delta}\mathbb{Z}\alpha$ is free. Hence we find a cocharacter $\xi$ in $X_*(T)$ such that $\alpha\circ\xi={\mathrm{id}}_{\mathbb{G}_m}$ for any $\alpha$ in $\Delta$. It is injective and uniquely determined up to a central cocharacter. We fix once and for all such a $\xi$. It satisfies $$\xi(\mathbb{Z}_p\setminus\{0\})\subseteq T_+$$ and $$\label{compat}
\ell(\xi(a)n\xi(a^{-1}))=a\ell(n)$$ for any $a$ in $\mathbb{Q}_p^{\times}$ and $n$ in $N$ since $\ell$ is a linear functional on the space $\prod_{\alpha\in\Delta}N_\alpha$ and therefore can be written as a linear combination of the isomorphisms $\iota_\alpha\colon
N_\alpha\rightarrow \mathbb{Q}_p$.
For example, if $G={\mathrm{GL}}_n(\mathbb{Q}_p)$, $T$ is the group of diagonal matrices, and $N$ is the group of unipotent upper triangular matrices, then we could choose $\xi\colon \mathbb{G}_m(\mathbb{Q}_p)=\mathbb{Q}_p^\times\to
T=(\mathbb{Q}_p^{\times})^n$, $\xi(x):=\begin{pmatrix}x^{n-1}&&&\\&x^{n-2}&&\\&&\ddots&\\&&&1\end{pmatrix}$.
Put $\Gamma:=\xi(\mathbb{Z}_p^{\times})$ and $s:=\xi(p)$. The element $s$ acts by conjugation on the group $N_0$ such that $\bigcap_ks^kN_0s^{-k}=\{1\}$. We denote this action by $\varphi:=\varphi_s$. This is compatible with the functional $\ell$ in the sense $\ell\circ\varphi=p\ell$ (see section \[phiring\]) by . Therefore we may apply the theory of the preceding sections to any $\varphi$-ring $R$ with the homomorphism $\ell\colon
N_0\rightarrow\mathbb{Z}_p$ and $N_1:={\mathrm{Ker}}(\ell_{\mid N_0})$. We are going to apply the theory of section \[phiring\] in the setting $H_0:=N_0$ and $H_1:=N_1$.
Note that in [@SVi] and [@Z] $\ell$ is assumed to be generic—we do not assume this here, though. We remark that for any $\alpha\in\Delta$ the restriction of $\ell$ to a fixed $N_{\alpha}$ is either zero or an isomorphism of $N_\alpha$ with $\mathbb{Q}_p$ and put $a_\alpha:=\ell(\iota_\alpha^{-1}(1))$. By the assumption $\ell(N_0)=\mathbb{Z}_p$ we obtain $a_\alpha\in\mathbb{Z}_p$ for all $\alpha\in\Delta$ and $a_\alpha\in\mathbb{Z}_p^{\times}$ for at least one $\alpha$ in $\Delta$. We put $T_{+,\ell}:=\{t\in T_+\mid
tN_1t^{-1}\subseteq N_1\}$. The monoid $T_{+,\ell}$ acts on the group $\mathbb{Z}_p$ via $\ell\colon N_0\rightarrow\mathbb{Z}_p$, too.
A $(\varphi,\Gamma)$-ring $R$ is by definition a $\varphi$-ring (in the sense of section \[phiring\]) together with an action of $\Gamma\cong\mathbb{Z}_p^{\times}$ commuting with $\varphi$ and satisfying $\gamma(\chi(x))=\chi(\xi^{-1}(\gamma)x)$. For example $\mathcal{O_E},\mathcal{O}^\dagger_{\mathcal{E}},\mathcal{E}^\dagger,\mathcal{R}$ are $(\varphi,\Gamma)$-rings. Note that the endomorphism ring ${\mathrm{End}}(\mathbb{Z}_p)$ of the $p$-adic integers (as a topological abelian group) is isomorphic to $\mathbb{Z}_p$. On the other hand, the multiplicative monoid $\mathbb{Z}_p\setminus\{0\}$ is isomorphic to $\varphi^{\mathbb{N}}\Gamma$. Now having an action of $\varphi$ and $\Gamma$ on $R$ we obtain an action of $T_{+,\ell}$ on $R$ since the map $\ell\colon N_0\rightarrow\mathbb{Z}_p$ induces a monoid homorphism $T_{+,\ell}\rightarrow\mathbb{Z}_p\setminus\{0\}\cong\varphi^{\mathbb{N}}\Gamma$. We denote the kernel of this monoid homomorphism by $T_{0,\ell}$. Similarly, we have a natural action of $T_{+,\ell}$ on the ring $R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ by conjugation. Indeed, if $t\in T_{+,\ell}$ then since $T$ is commutative we have $$t\varphi^k(N_1)t^{-1}=ts^kN_1s^{-k}t^{-1}=s^ktN_1t^{-1}s^{-k}=\varphi^k(tN_1t^{-1})\subseteq\varphi^k(N_1),$$ whence $tN_kt^{-1}\subseteq N_k$. Hence $t$ acts naturally on the skew group ring $R[N_1/N_k,\ell]$ and by taking the limit we also obtain an action on $R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$. We denote the map on both $R$ and $R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ induced by the action of $t\in T_{+,\ell}$ by $\varphi_t$.
Now a $T_{+,\ell}$-module over $R$ (resp. over $R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$) is a finitely generated free $R$-module $D$ (resp. $R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$-module $M$) with a semilinear action of $T_{+,\ell}$ (denoted by $\varphi_t\colon D\rightarrow D$, resp. $\varphi_t\colon M\rightarrow M$ for any $t\in T_{+,\ell}$) such that the restriction of the $T_{+,\ell}$-action to $s\in T_{+,\ell}$ defines a $\varphi$-module over $R$ (resp. over $R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$). We denote the category of $T_{+,\ell}$-modules over $R$ (resp. over $R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$) by $\mathfrak{M}(R,T_{+,\ell})$ (resp. by $\mathfrak{M}(R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},T_{+,\ell})$).
Let $M$ be in $\mathfrak{M}(R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},T_{+,\ell})$ and $D$ be in $\mathfrak{M}(R,T_{+,\ell})$. Then the maps $$\begin{aligned}
1\otimes\varphi_t\colon R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\otimes_{R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},\varphi_t}M&\rightarrow& M\\
r\otimes m&\mapsto& r\varphi_t(m)\end{aligned}$$ and $$\begin{aligned}
1\otimes\varphi_t\colon R\otimes_{R,\varphi_t}D&\rightarrow& D\\
r\otimes d&\mapsto& r\varphi_t(d)\end{aligned}$$ are isomorphisms for any $t\in T_{+,\ell}$.
We only prove the statement for $M$ (the statement for $D$ is entirely analogous). First note that the subgroups $s^kN_0s^{-k}$ (resp. $s^kN_1s^{-k}$) form a system of neighbourhoods of $1$ in $N$ (resp. in ${\mathrm{Ker}}(\ell)$). On the other hand, if $t$ is in $T_{+,\ell}$ then $$t{\mathrm{Ker}}(\ell_{\mid N})t^{-1}=t\left(\bigcup_{k\in\mathbb{Z}} s^k
N_1s^{-k}\right)t^{-1}=\bigcup_{k\in\mathbb{Z}} s^ktN_1t^{-1}s^{-k}= {\mathrm{Ker}}(\ell_{\mid N}).$$ since $tN_1t^{-1}$ has finite index in $N_1$. Now since $t^{-1}N_0t$ and on $t^{-1}N_1t$ are compact, we find $k_0>0$ so that $t^{-1}N_0t\subseteq s^{-k_0}N_0s^{k_0}$ and $t^{-1}N_1t\subseteq
s^{-k_0}N_1s^{k_0}$ whence $s^{k_0}t^{-1}$ lies in $T_{+,\ell}$. Since $M$ is a $\varphi$-module over $R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$, the map $$\begin{aligned}
1\otimes\varphi_{s^{k_0}}\colon R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\otimes_{R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},\varphi_{s^{k_0}}}M&\rightarrow& M\\
r\otimes m&\mapsto&r\varphi_{s^{k_0}}(m)\end{aligned}$$ is an isomorphism. Moreover, under the identifications $$\begin{aligned}
R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\otimes_{R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},\varphi_t}(R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\otimes_{R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},\varphi_{s^{k_0}t^{-1}}}M)\cong R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\otimes_{R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},\varphi_{s^{k_0}}}M\cong\\ \cong R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\otimes_{R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},\varphi_{s^{k_0}t^{-1}}}(R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\otimes_{R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},\varphi_t}M) \end{aligned}$$ we have $$(1\otimes\varphi_t)\circ(1\otimes(1\otimes\varphi_{s^{k_0}t^{-1}}))=1\otimes\varphi_{s^{k_0}}=(1\otimes\varphi_{s^{k_0}t^{-1}})\circ(1\otimes(1\otimes\varphi_t)),$$ so $1\otimes\varphi_t$ is surjective by the equality on the left and injective by the equality on the right.
Note that the action of $T_{0,\ell}$ on a $T_{+,\ell}$-module $D$ over $R$ is linear since $T_{0,\ell}$ acts trivially on $R$. Therefore this action extends (uniquely) to the subgroup $T_\ell\leq T$ generated by the monoid $T_{0,\ell}$.
By the étaleness of the action of $\varphi_t$ for $t\in T_{0,\ell}$ we see immediately that $\varphi_t$ is an automorphism of $D$ since $\varphi_t\colon
R\rightarrow R$ is the identity map. Therefore $\varphi_t$ has a (left and right) inverse (as a linear transformation of the $R$-module $D$) which we denote by $\varphi_{t^{-1}}$. The remark follows noting that $T_\ell$ consists of the quotients of elements of $T_{0,\ell}$.
In the case when $\ell=\ell_\alpha$ given by the projection of $\prod_{\beta\in\Delta}N_\beta$ to $N_\alpha$ for some fixed simple root $\alpha\in\Delta$ it is clear that $T_{+,\ell}=T_+$ as $N_\beta$ is $T_+$-invariant for each $\beta\in\Phi^+$ and ${\mathrm{Ker}}(\ell)=\prod_{\alpha\neq\beta\in\Phi^+}N_\beta$. Therefore $T_\ell\cong(\mathbb{Q}_p^{\times})^{n-1}$ where $n=\dim T$. This is the case in which a $G$-equivariant sheaf on $G/P$ is constructed in [@SVZ] associated to any object $D$ in $\mathfrak{M}(\mathcal{O_E},T_{+,\ell})$. So an object in $\mathfrak{M}(\mathcal{O_E},T_{+,\ell})$ is nothing else but a $(\varphi,\Gamma)$-module over $\mathcal{O_E}$ with an additional linear action of the group $T_\ell$ (once we fixed the cocharacter $\xi$). In case of $G={\mathrm{GL}}_2(\mathbb{Q}_p)$ this additional action is just an action of the centre $Z=T_\ell$ of $G$. In the work of Colmez [@Co1; @Co2] on the $p$-adic Langlands correspondence for ${\mathrm{GL}}_2(\mathbb{Q}_p)$ the action of $Z$ on an irreducible $2$-dimensional étale $(\varphi,\Gamma)$-module $D$ is given by the determinant (ie. the action of $\mathbb{Q}_p^{\times}\cong Z$ on $\bigwedge^2D$). It is unclear at this point whether the action of $T_\ell$ can be chosen canonically (in a similar fashion) for a given $n$-dimensional irreducible étale $(\varphi,\Gamma)$-module $D$.
As a corollary of Prop. \[equivcat\] we obtain
\[T\_+equiv\] The functors $\mathbb{D}=R\otimes_{R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},\ell}\cdot$ and $\mathbb{M}=R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\otimes_{R,\iota}\cdot$ are quasi-inverse equivalences of categories between $\mathfrak{M}(R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},T_{+,\ell})$ and $\mathfrak{M}(R,T_{+,\ell})$.
Since we clearly have $\mathbb{D}\circ\mathbb{M}\cong{\mathrm{id}}_{\mathfrak{M}(R,T_{+,\ell})}$ and the faithfulness of $\mathbb{D}$ is a formal consequence of Prop. \[equivcat\], it suffices to show that the isomorphism $\Theta\colon
M\rightarrow\mathbb{M}\circ\mathbb{D}(M)$ is $T_{+,\ell}$-equivariant whenever $M$ lies in $\mathfrak{M}(R{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},T_{+,\ell})$. Let $t\in T_{+,\ell}$ be arbitrary and for an $m\in M$ write $m=\sum_{u\in
J(N_0/\varphi_s^k(N_0))}u\varphi_s^k(m_{u,k})$. Since $\mathbb{D}(\Theta)={\mathrm{id}}_{\mathbb{D}(M)}$, we have $(\Theta\circ\varphi_t-\varphi_t\circ\Theta)(M)\subseteq I_1\mathbb{M}\circ\mathbb{D}(M)$. We compute $$\begin{aligned}
(\Theta\circ\varphi_t-\varphi_t\circ\Theta)(m)=\sum_{u\in
J(N_0/\varphi_s^k(N_0))}\varphi_t(u)\varphi_s^k\circ(\Theta\circ\varphi_t-\varphi_t\circ\Theta)(m_{u,k})\subseteq\\ \subseteq\varphi_s^k(I_1\mathbb{M}\circ\mathbb{D}(M))\subseteq
I_{k+1}\mathbb{M}\circ\mathbb{D}(M)\end{aligned}$$ for all $k\geq 0$ showing that $\Theta$ is $\varphi_t$-equivariant.
The case of overconvergent and Robba rings {#microlocal}
==========================================
The locally analytic distribution algebra
-----------------------------------------
Let $p$ be a prime and put $\epsilon_p=1$ if $p$ is odd and $\epsilon_p=2$ if $p=2$. If $H$ is a compact locally $\mathbb{Q}_p$-analytic group then we denote by $D(H,K)$ the algebra of $K$-valued locally analytic distributions on $H$. Recall that $D(H,K)$ is equal to the strong dual of the locally convex vector space $C^{an}(H,K)$ of $K$-valued locally $\mathbb{Q}_p$-analytic functions on $H$ with the convolution product.
Recall that a topologically finitely generated pro-$p$ group $H$ is uniform, if it is powerful (ie. $H/\overline{H^{p^{\epsilon_p}}}$ is abelian) and for all $i\geq 1$ we have $|P_i(H):P_{i+1}(H)|=|H:P_2(H)|$ where $P_1(H)=H$ and $P_{i+1}(H)=\overline{P_i(H)^p[P_i(H),H]}$ (see [@DDMS] for more details). Now if $H$ is uniform, then it has a bijective global chart $$\begin{aligned}
\mathbb{Z}_p^{d}&\rightarrow&H\\
(x_1,\dots,x_{d})&\mapsto&h_1^{x_1}\cdots h_{d}^{x_{d}}\end{aligned}$$ where $h_{1},\dots,h_{d}$ is a fixed (ordered) minimal set of topological generators of $H$. Putting $b_i:=h_{i}-1\in\mathbb{Z}[G]$, ${\bf
b}^{\bf k}:=b_{1}^{k_1}\cdots b_{d}^{k_d}$ for ${\bf k}=(k_i)\in\mathbb{N}^{d}$ we can identify $D(H,K)$ with the ring of all formal series $$\lambda=\sum_{{\bf k}\in\mathbb{N}^{d}}d_{\bf k}{\bf b}^{\bf k}$$ with $d_{\bf k}$ in $K$ such that the set $\{|d_{\bf k}|\rho^{\epsilon_p|{\bf k}|}\}_{\bf k}$ is bounded for all $0<\rho<1$. Here the first $|\cdot|$ is the normalized absolute value on $K$ and the second one denotes the degree of ${\bf k}$, that is $\sum_{i}k_{i}$. For any $\rho$ in $p^{\mathbb{Q}}$ with $p^{-1}<\rho<1$, we have a multiplicative norm $\|\cdot\|_{\rho}$ on $D(H,K)$ [@ST1] given by $$\|\lambda\|_{\rho}:=\sup_{\bf k}|d_{\bf k}|\rho^{\epsilon_p|{\bf k}|}\ .$$ The family of norms $\|\cdot\|_{\rho}$ defines the Fréchet topology on $D(H,K)$. The completion with respect to the norm $\|\cdot\|_{\rho}$ is denoted by $D_{[0,\rho]}(H,K)$.
Microlocalization {#partialrobba}
-----------------
Let $G$ be the group of $\mathbb{Q}_p$-points of a $\mathbb{Q}_p$-split connected reductive group with a fixed Borel subgroup $P=TN$. We also choose a simple root $\alpha$ for the Borel subgroup $P$ and let $\ell=\ell_\alpha$ be the functional given by the projection $$\ell_\alpha\colon N\rightarrow
N/[N,N]\rightarrow\prod_{\beta\in\Delta}N_\beta\rightarrow N_\alpha\overset{\iota_\alpha}{\rightarrow}\mathbb{Q}_p\ .$$ Therefore we have $T_{+,\ell}=T_+$ as $N_\beta$ is $T_+$-invariant for each $\beta\in\Phi^+$. We assume further that $N_0$ is *uniform*.
Let us begin by recalling the definition of the classical Robba ring for the group $\mathbb{Z}_p$. The distribution algebra $D(\mathbb{Z}_p,K)$ of $\mathbb{Z}_p$ can clearly be identified with the ring of power series (in variable $T$) with coefficients in $K$ that are convergent in the $p$-adic open unit disc. Now put $$\mathcal{A}_{[\rho,1)}:=\hbox{the ring of all Laurent series
}f(T)=\sum_{n\in\mathbb{Z}}a_nT^n\hbox{ that converge for
}\rho\leq|T|<1.$$ For $\rho\leq\rho'$ we have a natural inclusion $\mathcal{A}_{[\rho,1)}\hookrightarrow\mathcal{A}_{[\rho',1)}$ so we can form the inductive limit $$\mathcal{R}:=\varinjlim_{\rho\rightarrow1}\mathcal{A}_{[\rho,1)}$$ defining the Robba ring. $\mathcal{R}$ is a $(\varphi,\Gamma)$-ring over $\mathbb{Z}_p$ with the maps $\chi\colon\mathbb{Z}_p\rightarrow\mathcal{R}^{\times}$ and $\varphi\colon\mathcal{R}\rightarrow\mathcal{R}$ such that $\chi(1)=1+T$, $\varphi(T)=(T+1)^p-1$, and $\gamma(T)=(1+T)^{\xi^{-1}(\gamma)}-1$ for $\gamma\in\Gamma$.
Recall that the ring $$\mathcal{O}_{\mathcal{E}}^\dagger:=\{\sum_{n\in\mathbb{Z}}a_nT^n\mid a_n\in o_K\text{
and there exists a }\rho<1\text{ s.t.\ }|a_n|\rho^n\to 0\text{ as }n\to-\infty\}$$ is called the ring of overconvergent power series. It is a subring of both $\mathcal{O_E}$ and $\mathcal{R}$. We put $\mathcal{E}^\dagger:=K\otimes_{o_K}\mathcal{O}_{\mathcal{E}}^\dagger$ which is also a subring of the Robba ring. These rings are also $(\varphi,\Gamma)$-rings.
The rings $\mathcal{O}_{\mathcal{E}}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ and $\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ constructed in the previous sections are only overconvergent (resp. Robba) in the variable $b_\alpha$ for the fixed simple root $\alpha$. In all the other variables $b_\beta$ they behave like the Iwasawa algebra $\Lambda(N_1)$ since we took the completion with respect to the ideals generated by $(N_k-1)$. Moreover, in the projective limit $\mathcal{O}_{\mathcal{E}}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\cong
\varprojlim_k\mathcal{O}_{\mathcal{E}}^\dagger[N_1/N_k,\ell]$ the terms are not forced to share a common region of convergence. In this section we construct the rings $\mathcal{R}^{int}(N_1,\ell)$ and $\mathcal{R}(N_1,\ell)$ with better analytic properties.
We start with constructing a ring $\mathfrak{R}_0=\mathfrak{R}_0(N_0,K,\alpha)$ as a certain microlocalization of the distribution algebra $D(N_0,K)$. We fix the topological generator $n_{\alpha}$ of $N_0\cap N_{\alpha}$ such that $\ell_{\alpha}(n_{\alpha})=1$. This is possible since we normalized $\iota_\alpha\colon N_\alpha\overset{\sim}{\to}\mathbb{Q}_p$ so that $\iota_\alpha(N_0\cap N_\alpha)=\mathbb{Z}_p$. Further, we fix topological generators $n_{\beta}$ of $N_0\cap
N_{\beta}$ for each $\alpha\neq\beta\in\Phi^+$. Since $N_0$ is uniform of dimension $|\Phi^+|$, the set $A:=\{n_{\beta}\mid\beta\in\Phi^+\}$ is a minimal set of topological generators of the group $N_0$. Moreover, $A\setminus\{n_{\alpha}\}$ is a minimal set of generators of the group $N_1={\mathrm{Ker}}(\ell)\cap N_0$. Further, we put $b_{\beta}:=n_\beta-1$. For any real number $p^{-1}<\rho<1$ in $p^{\mathbb{Q}}$ the formula $\|b_\beta\|_\rho:=\rho$ (for all $\beta\in\Phi^+$) defines a multiplicative norm on $D(N_0,K)$. The completion of $D(N_0,K)$ with respect to this norm is a Banach algebra which we denote by $D_{[0,\rho]}(N_0,K)$. Let now $p^{-1}<\rho_1<\rho_2<1$ be real numbers in $p^{\mathbb{Q}}$. We take the generalized microlocalization (cf. the Appendix of [@SZ]) of the Banach algebra $D_{[0,\rho_2]}(N_0,K)$ at the multiplicatively closed set $\{(n_{\alpha}-1)^i\}_{i\geq1}$ with respect to the pair of norms $(\rho_1,\rho_2)$. This provides us with the Banach algebra $D_{[\rho_1,\rho_2]}(N_0,K,\alpha)$. Recall that the elements of this Banach algebra are equivalence classes of Cauchy sequences $((n_\alpha-1)^{-k_n}x_n)_n$ (with $x_n\in D_{[0,\rho_2]}(N_0,K)$) with respect to the norm $\|\cdot\|_{\rho_1,\rho_2}:=\max(\|\cdot\|_{\rho_1},\|\cdot\|_{\rho_2})$.
Letting $\rho_2$ tend to $1$ we define $D_{[\rho_1,1)}(N_0,K,\alpha):=\varprojlim_{\rho_2\rightarrow
1}D_{[\rho_1,\rho_2]}(N_0,K,\alpha)$. This is a Fréchet-Stein algebra (the proof is completely analogous to that of Theorem 5.5 in [@SZ], but it is not a formal consequence of that). However, we will not need this fact in the sequel so we omit the proof. Now the partial Robba ring $\mathfrak{R}_0:=\mathfrak{R}_0(N_0,K,\alpha):=\varinjlim_{\rho_1\rightarrow1}D_{[\rho_1,1)}(N_0,K,\alpha)$ is defined as the injective limit of these Fréchet-Stein algebras. We equip $\mathfrak{R}_0$ with the inductive limit topology of the Fréchet topologies of $D_{[\rho_1,1)}(N_0,K,\alpha)$. By the following parametrization the partial Robba ring can be thought of as a skew Laurent series ring on the variables $b_{\beta}$ ($\beta\in\Phi^+$) with certain convergence conditions such that only the variable $b_\alpha$ is invertible. Note that in [@SZ] a “full” Robba ring is constructed such that all the variables $b_\beta$ are invertible. We denote the corresponding “fully” microlocalized Banach algebras by $D_{[\rho_1,\rho_2]}(N_0,K)$. In all these rings we will often omit $K$ from the notation if it is clear from the context.
The microlocalization of quasi-abelian normed algebras (Appendix of [@SZ]) is somewhat different from the microlocalisation constructing $\Lambda_\ell(N_0)$ where first a localization (with respect to an Ore set) is constructed and then the completion is taken. The set we are inverting here does not satisfy the Ore property, so the localization in the usual sense does not exist. However, we may complete and localize at the same time in order to obtain a microlocalized ring directly.
In order to be able to work with these rings we will show that their elements can be viewed as Laurent series. The discussion below is completely analogous to the discussion before Prop. A.24 in [@SZ]. However, for the convenience of the reader, we explain the method specialized to our case here. We introduce the affinoid domain $$A_{\alpha}[\rho_1,\rho_2] := \{(z_\beta)_{\beta\in\Phi^+} \in
\mathbb{C}_p^{\Phi^+} : \rho_1
\leq |z_{\alpha}| \leq \rho_2, 0\leq |z_\beta/z_\alpha|\leq 1\
\hbox{for}\ \alpha\neq\beta\in\Phi^+\} \ .$$ This has the affinoid subdomain $$X^{\Phi^+}_{[\rho_1,\rho_2]} := \{(z_\beta)_{\beta\in\Phi^+} \in \mathbb{C}_p^{\Phi^+} : \rho_1 \leq |z_{\beta_1}| = \ldots = |z_{\beta_{|\Phi^+|}}| \leq \rho_2\}$$ (where $\{\beta_1,\dots,\beta_{|\Phi^+|}\}=\Phi^+$) as defined in [@SZ] (Prop. A.24).
The ring $\mathcal{O}_K(A_\alpha[\rho_1,\rho_2])$ of $K$-analytic functions on $A_\alpha[\rho_1,\rho_2]$ is the ring of all Laurent series $$f(\mathbf{Z}) = \sum_{{{\bf k}}\in \mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}} d_{{\bf k}}\mathbf {Z}^{{\bf k}}$$ with $d_{{\bf k}}\in K$ and such that $\lim_{{{\bf k}}\rightarrow
\infty} |d_{{\bf k}}| \rho^{{{\bf k}}} = 0$ for any $\rho_1 \leq \rho \leq
\rho_2$. Here $$\mathbf{Z}^{{\bf k}}:= \prod_{\beta\in\Phi^+}Z_\beta^{k_\beta}
\qquad\text{and}\qquad \rho^{{\bf k}}:=\rho^{\sum_{\beta\in\Phi^+}k_\beta}$$ and ${\bf k}\to \infty$ means that $\sum_{\beta\in\Phi^+}|k_\beta|\to\infty$. This is the subring of $\mathcal{O}_K(X^{\Phi^+}_{[\rho_1,\rho_2]})$ consisting of elements in which the variables $Z_\beta$ appear only with nonnegative exponent for all $\alpha\neq\beta\in\Phi^+$.
Since $X^{\Phi^+}_{[\rho_1,\rho_2]}\subseteq A_\alpha[\rho_1,\rho_2]$, we clearly have $\mathcal{O}_K(A_\alpha[\rho_1,\rho_2])\subseteq
\mathcal{O}_K(X^{\Phi^+}_{[\rho_1,\rho_2]})$. Moreover, the power series in $\mathcal{O}_K(A_\alpha[\rho_1,\rho_2])$ converge for $z_\beta=0$ ($\beta\neq \alpha$), hence these variables appear with nonnegative exponent. On the other hand, if we have a power series $f(\mathbf{Z})\in \mathcal{O}_K(X^{\Phi^+}_{[\rho_1,\rho_2]})$ such that the variables $Z_\beta$ have nonnegative exponent for all $\alpha\neq\beta\in\Phi^+$ then it also converges in the region $
A_\alpha[\rho_1,\rho_2]$ as we have the trivial estimate $|\prod_{\beta\in\Phi^+}z_\beta^{k_\beta}|\leq
|z_\alpha|^{\sum_{\beta\in\Phi^+}k_\beta}$ in this case.
Since $\rho^{{\bf k}}\leq
\max(\rho_1^{{\bf k}},\rho_2^{{\bf k}})$ for any $\rho_1 \leq \rho \leq \rho_2$ and any ${{\bf k}}\in \mathbb
{Z}^{(\alpha)}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}$ the convergence condition on $f$ is equivalent to $$\lim_{{{\bf k}}\rightarrow
\infty} |d_{{\bf k}}| \rho_1^{{{\bf k}}} = \lim_{{{\bf k}}\rightarrow \infty}
|d_{{\bf k}}| \rho_2^{{{\bf k}}} = 0 \ .$$ The spectral norm on the affinoid algebra $\mathcal{O}_K(A_\alpha[\rho_1,\rho_2])$ (for the definition of these notions see [@FvdP]) is given by $$\begin{aligned}
\|f\|_{A_\alpha[\rho_1,\rho_2]} & = \sup_{\rho_1 \leq \rho \leq
\rho_2} \max_{{{\bf k}}\in \mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}} |d_{{\bf k}}|\rho^{{\bf k}}\\
& = \max( \max_{{{\bf k}}\in \mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}} |d_{{\bf k}}|\rho_1^{{\bf k}},
\max_{{{\bf k}}\in \mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}} |d_{{\bf k}}| \rho_2^{{\bf k}}) \ .\end{aligned}$$ Setting ${{\bf b}}^{{\bf k}}:= \prod_{\beta\in\Phi^+}b_\beta^{ k_\beta} $ for some fixed ordering of $\Phi^+$ and for any ${{\bf k}}= ( k_\beta)_{\beta\in\Phi^+} \in \mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}$ we claim that $f({{\bf b}}) := \sum_{{{\bf k}}\in
\mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}} d_{{\bf k}}{{\bf b}}^{{\bf k}}$ converges in $D_{[\rho_1,\rho_2]}(N_0,K,\alpha)$ for $f\in\mathcal{O}_K(A_\alpha[\rho_1,\rho_2])$. As a consequence of Prop. A.21 and Lemma A.7.iii in [@SZ] we have $$\|{{\bf b}}^{{\bf k}}\|_{\rho_1,\rho_2} = \max(\rho_1^{{\bf k}},\rho_2^{{\bf k}})$$ for any ${{\bf k}}\in \mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}$. Hence $$\lim_{{{\bf k}}\rightarrow
\infty} \|d_{{\bf k}}{{\bf b}}^{{\bf k}}\|_{\rho_1,\rho_2} = \lim_{{{\bf k}}\rightarrow
\infty} \max(|d_{{\bf k}}| \rho_1^{{\bf k}}, |d_{{\bf k}}| \rho_2^{{\bf k}}) = \max(
\lim_{{{\bf k}}\rightarrow \infty} |d_{{\bf k}}| \rho_1^{{\bf k}},
\lim_{{{\bf k}}\rightarrow \infty} |d_{{\bf k}}| \rho_2^{{\bf k}}) = 0 \ .$$ Therefore $$\begin{aligned}
\mathcal{O}_K(A_\alpha[\rho_1,\rho_2]) & \longrightarrow
D_{[\rho_1,\rho_2]}(N_0,K,\alpha) \\
f & \longmapsto f({{\bf b}})\end{aligned}$$ is a well defined $K$-linear map. In order to investigate this map we introduce the filtration $$F^iD_{[\rho_1,\rho_2]}(N_0,K,\alpha) := \{ e \in D_{[\rho_1,\rho_2]}(N_0,K,\alpha) : \|e\|_{\rho_1,\rho_2}
\leq |p|^i \} \qquad\text{for $i \in \mathbb{R}$}$$ on $D_{[\rho_1,\rho_2]}(N_0,K,\alpha)$. Since $K$ is discretely valued and $\rho_1,\rho_2 \in
p^{\mathbb{Q}}$ this filtration is quasi-integral in the sense of [@ST1] §1. The corresponding graded ring $gr^\cdot
D_{[\rho_1,\rho_2]}(N_0,K,\alpha)$, by Prop. A.21 in [@SZ], is commutative. We let $\sigma(e) \in gr^\cdot D_{[\rho_1,\rho_2]}(N_0,K,\alpha)$ denote the principal symbol of any element $e \in D_{[\rho_1,\rho_2]}(N_0,K,\alpha)$.
\[4\]
- $gr^\cdot D_{[\rho_1,\rho_2]}(N_0,K,\alpha)$ is a free $gr^\cdot
K$-module with basis $\{ \sigma({{\bf b}}^{{\bf k}}) : {{\bf k}}\in
\mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}} \}$.
- The map $$\begin{aligned}
\mathcal{O}_K(A_\alpha[\rho_1,\rho_2]) & \xrightarrow{\;\cong\;}
D_{[\rho_1,\rho_2]}(N_0,K,\alpha) \\
f & \longmapsto f({{\bf b}})\end{aligned}$$ is a $K$-linear isometric bijection.
Since $\{ b_\alpha^{-l}\mu
: l \geq 0 , \mu \in D_{[0,\rho_1]}(N_0,K) \}$ is dense in $D_{[\rho_1,\rho_2]}(N_0,K,\alpha)$ every element in the graded ring $gr^\cdot D_{[\rho_1,\rho_2]}(N_0,K,\alpha)$ is of the form $\sigma(b_\alpha^{-l}\mu)$. Suppose that $\mu =
\sum_{{{\bf k}}\in\mathbb{N}_0^{d}} d_{{{\bf k}}}{{\bf b}}^{{{\bf k}}}$. Then $b_\alpha^{-l}\mu =
\sum_{{{\bf k}}\in\mathbb{N}_0^{d}}
d_{{{\bf k}}}b_\alpha^{-l}{{\bf b}}^{{{\bf k}}}$ and, using Lemma A.7.iii [@SZ] we compute $$\begin{aligned}
\|b_\alpha^{-l}\mu\|_{\rho_1,\rho_2} = \max(
\|b_\alpha\|_{\rho_1}^{-l} \|\mu\|_{\rho_1},
\|b_\alpha\|_{\rho_2}^{-l}\|\mu\|_{\rho_2})
= \max( \max_{{{\bf k}}\in \mathbb{N}_0^d} |d_{{\bf k}}| \rho_1^{{{\bf k}}- l}, \max_{{{\bf k}}\in \mathbb{N}_0^d} |d_{{\bf k}}| \rho_2^{{{\bf k}}- l}) \\
= \max_{{{\bf k}}\in \mathbb{N}_0^d} |d_{{\bf k}}| \max(\rho_1^{{{\bf k}}- l}, \rho_2^{{{\bf k}}- l})
= \max_{{{\bf k}}\in \mathbb{N}_0^d} |d_{{\bf k}}|
|b_\alpha^{-l}{{\bf b}}^{{\bf k}}|_{\rho_1,\rho_2} \ .\end{aligned}$$ It follows that $gr^\cdot D_{[\rho_1,\rho_2]}(N_0,K,\alpha)$ as a $gr^\cdot
K$-module is generated by the principal symbols $\sigma(b_\alpha^{-l}{{\bf b}}^{{\bf k}})$ with ${{\bf k}}\in
\mathbb{N}_0^d$, $l\geq 0$. But it also follows that, for a fixed $l\geq
0$, the principal symbols $\sigma(b_\alpha^{-l}{{\bf b}}^{{\bf k}})$ with ${{\bf k}}$ running over $\mathbb{N}_0^d$ are linearly independent over $gr^\cdot K$. By Prop. A.21 in [@SZ] we may permute the factors in $\sigma(b_\alpha^{-l}{{\bf b}}^{{\bf k}})$ arbitrarily. Hence $gr^\cdot
D_{[\rho_1,\rho_2]}(N_0,K,\alpha)$ is a free $gr^\cdot
K$-module with basis $\{ \sigma({{\bf b}}^{{\bf k}}) : {{\bf k}}\in
\mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}} \}$.
On the other hand, we of course have $$\begin{aligned}
\|f({{\bf b}})\|_{\rho_1,\rho_2} & \leq \max_{{{\bf k}}\in \mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}}
|d_{{\bf k}}||{{\bf b}}^{{\bf k}}|_{\rho_1,\rho_2} \\
& = \max_{{{\bf k}}\in \mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}} |d_{{\bf k}}|
\max(\rho_1^{{\bf k}},\rho_2^{{\bf k}}) \\
& = \max( \max_{{{\bf k}}\in \mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}}
|d_{{\bf k}}|\rho_1^{{\bf k}}, \max_{{{\bf k}}\in \mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}}
|d_{{\bf k}}|\rho_2^{{\bf k}}) \\
& = |f|_{A_\alpha[\rho_1,\rho_2]} \ .\end{aligned}$$ This means that if we introduce on $\mathcal{O}_K(A_\alpha[\rho_1,\rho_2])$ the filtration defined by the spectral norm then the asserted map respects the filtrations, and by the above reasoning it induces an isomorphism between the associated graded rings. Hence, by completeness of these filtrations, it is an isometric bijection.
Now we turn to the construction of $\mathcal{R}(N_1,\ell)$. The problem with (naïve) microlocalization is that the ring $\mathfrak{R}_0$ is not finitely generated over $\varphi(\mathfrak{R}_0)$. The reason for this is that $\varphi$ improves the order of convergence for a power series in $\mathfrak{R}_0$. In the case $G\neq {\mathrm{GL}}_2(\mathbb{Q}_p)$ the operator $\varphi=\varphi_s$ acts by conjugation on $N_{\beta}$ by raising to the $\beta(s)$-th power. Whenever $\beta\in \Phi^+\setminus
\Delta$ is not a simple root then $\beta(s)=p^{m_\beta}>\alpha(s)=p$ where $m_\beta$ is the degree of the map $\beta\circ\xi\colon\mathbb{G}_m\to\mathbb{G}_m$.
\[trivest\] We have $\|b_\beta\|_\rho=\|b_\alpha\|_\rho=\rho$ and $$\|\varphi(b_\beta)\|_\rho=\max_{0\leq j\leq
m_{\beta}}(\rho^{p^j}p^{j-m_\beta})<\max(\rho^p,p^{-1}\rho)=\|\varphi(b_\alpha)\|_\rho$$ for any $p^{-1}<\rho<1$. In general, we have $\|\varphi_t(b_\beta)\|_\rho=\max\limits_{0\leq j\leq \mathrm{val}_p(\beta(t))}(\rho^{p^j}p^{j-\mathrm{val}_p(\beta(t))})$.
We compute $$\|\varphi_t(b_\beta)\|_\rho=\|(1+b_\beta)^{\beta(t)}-1\|_\rho=\|\sum_{i=1}^\infty \binom{\beta(t)}{i}b_\beta^{i}\|_\rho=\max_{0\leq j\leq {\mathrm{val}}_p(\beta(t))}(\rho^{p^j}p^{j-\mathrm{val}_p(\beta(t))})\
.$$ Here we use the trivial estimate $\mathrm{val}_p\binom{n}{k}={\mathrm{val}}_p(\frac{n}{k}\binom{n-1}{k-1})\geq
\mathrm{val}_p(n)-{\mathrm{val}}_p(k)$ for $n:=\beta(t)\in\mathbb{Z}_p$ and $k\in
\mathbb{N}$. We see immediately that whenever $m_\beta>1$ then $\rho^{p^{j}}p^{j-m_\beta}<\rho^p$ for $1\leq j\leq m_\beta$ and $p^{-m_\beta}\rho<p^{-1}\rho$.
Now choose an ordering $<$ on $\Phi^+$ such that $(i)$ $m_{\beta_1}<m_{\beta_2}$ implies $\beta_1>\beta_2$ and $(ii)$ $\alpha>\beta$ for any $\alpha\neq
\beta,\beta_1,\beta_2\in\Phi^+$. Then by Prop. \[4\] any element in $\mathfrak{R}_0$ has a skew Laurent-series expansion $$f({\bf b})=\sum_{{\bf
k}\in\mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}}c_{\bf k}{\bf b}^{\bf k}$$ such that there exists $p^{-1}<\rho<1$ such that for all $\rho<\rho_1<1$ we have $|c_{\bf k}|_p\rho_1^{\sum k_\beta}\to0$ as $\sum |k_\beta|\to\infty$. By Lemma \[trivest\] and the discussion above we clearly have the following
\[ex\] Let $\beta\in\Phi^+\setminus\Delta$ be a non-simple root. Then the series $\sum_{n=1}^\infty b_{\beta}^nb_\alpha^{-n}$ does not belong to $\mathfrak{R}_0(N_0)$. However, the series $\sum_{n=1}^\infty\varphi(b_\beta^nb_\alpha^{-n})$ converges in each $D_{[\rho_1,\rho_2]}(N_0,\alpha)$ (for arbitrary $p^{-1}<\rho_1<\rho_2<1$) hence defines an element in $\mathfrak{R}_0(N_0)$. Therefore we cannot have a continuous left inverse $\psi$ to $\varphi$ on $\mathfrak{R}_0(N_0)$ as otherwise $\psi(\sum_{n=1}^{\infty}\varphi(b_{\beta}^nb_\alpha^{-n}))=\sum_{n=1}^\infty b_{\beta}^nb_\alpha^{-n}$ would converge. In particular, we cannot write $\mathfrak{R}_0(N_0)$ as the topological direct sum $ \bigoplus_{u\in N_0/\varphi(N_0)}u\varphi(\mathfrak{R}_0(N_0))$ of closed subspaces in $\mathfrak{R}_0(N_0)$ as otherwise the operator $$\begin{aligned}
\psi\colon \mathfrak{R}_0(N_0)&\to& \mathfrak{R}_0(N_0)\\
\sum_{u\in J(N_0/\varphi(N_0))}u\varphi(f_u)&\mapsto& \varphi^{-1}(u_0)f_{u_0}\end{aligned}$$ for the unique $u_0\in J(N_0/\varphi(N_0))\cap \varphi(N_0)$ would be a continuous left inverse to $\varphi$. In fact, we even have $\mathfrak{R}_0(N_0)\neq \bigoplus_{u\in N_0/\varphi(N_0)}u\varphi(\mathfrak{R}_0(N_0))$ algebraically, however, the proof of this requires the forthcoming machinery (see Remark \[notetale\]).
In order to overcome the above counter-example we are going to consider the ring $\mathcal{R}(N_1,\ell)$ of all the skew power series of the form $f({\bf b})$ such that $f(\varphi_t({\bf b}))$ is convergent in $\mathfrak{R}_0$ for some $t\in T_+$. A priori it is not clear that these series form a ring, so we are going to give a more conceptual construction.
Take an arbitrary element $t\in T_+$. The conjugation by $t$ on $N_0$ gives an isomorphism $\varphi_t\colon
N_0\to\varphi_t(N_0)$ of pro-$p$ groups (since it is injective). Hence $\varphi_t(N_0)$ is also a uniform pro-$p$ group with minimal set of generators $\{\varphi_t(n_\beta)\}_{\beta\in\Phi^+}$. So we may define the distribution algebra $D(\varphi_t(N_0)):=D(\varphi_t(N_0),K)$. The inclusion $\varphi_t(N_0)\hookrightarrow N_0$ induces an injective homomorphism of Fréchet-algebras $\iota_{1,t}\colon D(\varphi_t(N_0))\hookrightarrow
D(N_0)$. It is well-known [@ST1] that we have $$D(N_0)=\bigoplus_{n\in J(N_0/\varphi_t(N_0))}n\iota_{1,t}(D(\varphi_t(N_0)))$$ as right $D(\varphi_t(N_0))$-modules. Moreover, the direct summands are closed in $D(N_0)$. For each real number $p^{-1}<\rho<1$ the $\rho$-norm on $D(N_0)$ defines a norm $r_t(\rho)$ on $D(\varphi_t(N_0))$ by restriction. Note that this is different from the $\rho$-norm on $D(\varphi_t(N_0))$ (using the uniform structure on $\varphi_t(N_0)$). However, the family $(r_t(\rho))_\rho$ of norms defines the Fréchet topology on $D(\varphi_t(N_0))$. On the other hand, whenever $r$ is a norm on $D(\varphi_t(N_0))$ then we may extend $r$ to a norm $q_t(r)$ on $D(N_0)$ by putting $$\|\sum_{n\in J(N_0/\varphi_t(N_0))}n\iota_{1,t}(x_n)\|_{q_t(r)}:=\max(\|x_n\|_r).$$ These norms define the Fréchet topology on $D(N_0)$. More precisely, if $\beta(t)=p^{m(\beta,t)}u(\beta,t)$ with $m(\beta,t):={\mathrm{val}}_p(\beta(t))\geq 0$ integer and $u(\beta,t)\in\mathbb{Z}_p^{\times}$, then we have
\[equivnorm\] $$\|x\|_\rho\leq \|x\|_{q_t(r_t(\rho))}\leq \rho^{-\sum_{\beta\in\Phi^+}(p^{m(\beta,t)}-1)}\|x\|_\rho$$ for any $p^{-\frac{1}{\max_{\beta\in\Phi^+}p^{m(\beta,t)}}}<\rho<1$ and $x\in D(N_0)$. In particular, the norms $\rho$ and $q_t(r_t(\rho))$ define the same topology.
The inequality on the left is clear from the triangle inequality. For the other inequality note that our assumption on $\rho$ implies in particular that $$\rho^{p^{m(\beta,t)}}= \rho^{p^j}\rho^{p^{m(\beta,t)}-p^j}>\rho^{p^j}p^{-\frac{p^{m(\beta,t)}-p^j}{p^{m(\beta,t)}}}>\rho^{p^j}p^{j-m(\beta,t)}$$ for all $0\leq j<m(\beta,t)$. Hence by Lemma \[trivest\], we have $\rho^{p^{m(\beta,t)}}=\|\binom{\beta(t)}{p^{m(\beta,t)}}b_\beta^{p^{m(\beta,t)}}\|_\rho=\|\varphi_t(b_\beta)\|_\rho$. Moreover, there exists an *invertible* element $y$ in the Iwasawa algebra $\Lambda(N_{0,\beta})$ such that $y\varphi_t(b_\beta)\equiv\binom{\beta(t)}{p^{m(\beta,t)}}b_\beta^{p^{m(\beta,t)}}\pmod{p}$ (as both sides have the same principal term). However, by the choice of $\rho$, $|p|=1/p<\rho^{p^{m(\beta,t)}}=\|\varphi_t(b_\beta)\|_\rho=\|\varphi_t(b_\beta)\|_{q_t(r_t(\rho))}$. Therefore we also have $$\begin{aligned}
\rho^{p^{m(\beta,t)}}=\|b_{\beta}^{p^{m(\beta,t)}}\|_\rho=\|\binom{\beta(t)}{p^{m(\beta,t)}}b_\beta^{p^{m(\beta,t)}}\|_\rho=\|\varphi_t(b_{\beta})\|_\rho=
\|\varphi_t(b_\beta)\|_{q_t(r_t(\rho))}=\\
=\|y\varphi_t(b_\beta)\|_{q_t(r_t(\rho))}=\|\binom{\beta(t)}{p^{m(\beta,t)}}b_\beta^{p^{m(\beta,t)}}\|_{q_t(r_t(\rho))}=\|b_{\beta}^{p^{m(\beta,t)}}\|_{q_t(r_t(\rho))}\end{aligned}$$ whence $$\|b_\beta^{k_\beta}\|_{q_t(r_t(\rho))}=\|b_\beta^{p^{m(\beta,t)}}\|_{q_t(r_t(\rho))}^{k_{1,\beta}}\|b_\beta^{k_{2,\beta}}\|_{q_t(r_t(\rho))}\leq
\rho^{k_{1,\beta}p^{m(\beta,t)}}\leq\rho^{-p^{m(\beta,t)}+1}\|b_\beta^{k_\beta}\|_\rho\label{est}$$ where $k_\beta=p^{m(\beta,t)}k_{1,\beta}+k_{2,\beta}$ with $0\leq
k_{2,\beta}\leq p^{m(\beta,t)}-1$ and $k_{1,\beta}$ nonnegative integers.
Now consider an element of $D(N_0)$ of the form $$x=\sum_{{\bf k}=(k_\beta)\in\mathbb{N}^{\Phi^+}}c_{\bf k}\prod_{\beta\in\Phi^+}b_\beta^{k_\beta}\ .$$ We may assume without loss of generality that $J(N_0/\varphi_t(N_0))=\{\prod_{\beta\in\Phi^+}n_\beta^{j_\beta}\mid 0\leq j_\beta\leq p^{m(\beta,t)}-1\}$ where the product is taken in the reversed order. Let $\eta\in\Phi^+$ be the largest root (with respect to the ordering $<$ defined after Lemma \[trivest\]) such that there exists a ${\bf k}\in\mathbb{N}^{\Phi^+}$ with $c_{\bf k}\neq 0$ and $k_{\eta}\neq0$. We are going to show the estimate $$\|x\|_{q_t(r_t(\rho))}\leq \rho^{-\sum_{\beta\leq\eta}(p^{m(\beta,t)}-1)}\|x\|_\rho$$ by induction on $\eta$. This induction has in fact finitely many steps since $|\Phi^+|<\infty$. At first we write $b_{\eta}^{k_{\eta}}=\sum_{j_{\eta}=0}^{p^{m(\eta,t)}-1}n_\eta^{j_\eta}f_{{\bf
k},j_\eta}(\varphi_t(b_\eta))$ for each ${\bf k}\in\mathbb{N}^{\Phi^+}$. Note that—by the choice of the ordering on $\Phi^+$—for any fixed $\eta$ the set $\prod_{\beta<\eta}N_{0,\beta}$ is a normal subgroup of $N_0$. Moreover, the conjugation by any element of $N_0$ preserves the $\rho$-norm on $D(N_0)$. Therefore we may write $$\prod_{\beta\leq\eta}b_\beta^{k_\beta}=\sum_{j_\eta=0}^{p^{m(\eta,t)}-1}n_{\eta}^{j_\eta}x_{{\bf
k},j_{\eta}}f_{{\bf k},j_\eta}(\varphi_t(b_\eta))$$ such that $$x_{{\bf
k},j_\eta}:=n_\eta^{-j_\eta}\left(\prod_{\beta<\eta}b_\beta^{k_\beta}\right)
n_\eta^{j_\eta}\in D(\prod_{\beta<\eta}N_{0,\beta})\ .$$ By we have $$\|f_{{\bf k},j_\eta}(\varphi_t(b_\eta))\|_{q_t(r_t(\rho))}=\|f_{{\bf k},j_\eta}(\varphi_t(b_\eta))\|_\rho
\leq \|b_\eta^{k_\eta}\|_{q_t(r_t(\rho))}\leq
\rho^{-p^{m(\eta,t)}+1}\|b_\eta^{k_\eta}\|_\rho\ .$$ Since the $r_t(\rho)$-norm is multiplicative on $D(\varphi_t(N_0))$, for any $a\in D(N_0)$ and $b\in D(\varphi_t(N_0))$ we also have $\|a\iota_{1,t}(b)\|_{q_t(r_t(\rho))}=\|a\|_{q_t(r_t(\rho))}\|b\|_{r_t(\rho)}$. Indeed, if we decompose $a$ as $a=\sum_{n\in J(N_0/\varphi_t(N_0))}n\iota_{1,t}(a_n)$ then we have $a\iota_{1,t}(b)=\sum_{n\in
J(N_0/\varphi_t(N_0))}n\iota_{1,t}(a_nb)$. Now $f_{{\bf
k},j_\eta}(\varphi_t(b_\eta))$ lies in $\iota_{1,t}(D(\varphi_t(N_0)))$, so we see that $$\|x_{{\bf k},j_\eta}f_{{\bf
k},j_\eta}(\varphi_t(b_\eta))\|_{q_t(r_t(\rho))}=\|x_{{\bf
k},j_\eta}\|_{q_t(r_t(\rho))}\|f_{{\bf
k},j_\eta}(\varphi_t(b_\eta))\|_{q_t(r_t(\rho))}\ .$$ On the other hand, the inductional hypothesis tells us that $$\|x_{{\bf k},j_\eta}\|_{q_t(r_t(\rho))}\leq
\rho^{-\sum_{\beta<\eta}(p^{m(\beta,t)}-1)}\|x_{{\bf
k},j_\eta}\|_\rho=\rho^{-\sum_{\beta<\eta}(p^{m(\beta,t)}-1)}\|\prod_{\beta<\eta}b_\beta^{k_\beta}\|_\rho\ .$$ Hence we compute $$\begin{aligned}
\|x\|_{q_t(r_t(\rho))}=\|\sum_{\bf k}c_{\bf k}\sum_{j_\eta=0}^{p^{m(\eta,t)}-1}n_{\eta}^{j_\eta}x_{{\bf
k},j_{\eta}}f_{{\bf
k},j_\eta}(\varphi_t(b_\eta))\|_{q_t(r_t(\rho))}\leq\\
\leq\max_{{\bf k},j_\eta}\left(|c_{\bf
k}|\|x_{{\bf
k},j_{\eta}}f_{{\bf k},j_\eta}(\varphi_t(b_\eta))\|_{q_t(r_t(\rho))}\right)\leq\\
\leq \max_{\bf k}\left(|c_{\bf
k}|\rho^{-\sum_{\beta<\eta}(p^{m(\beta,t)}-1)}\|\prod_{\beta<\eta}b_\beta^{k_\beta}\|_\rho\rho^{-p^{m(\eta,t)}+1}\|b_\eta^{k_\eta}\|_\rho\right)
=\rho^{-\sum_{\beta\leq\eta}(p^{m(\beta,t)}-1)}\|x\|_\rho\ .\end{aligned}$$
In particular, for each $p^{-\frac{1}{\max_{\beta}p^{m(\beta,t)}}}<\rho<1$ the completion of $D(N_0)$ with respect to the topology defined by $\|\cdot\|_\rho$ and by $\|\cdot\|_{q_t(r_t(\rho))}$ are the same, ie.$$\label{dec0}
D_{[0,\rho]}(N_0)= \bigoplus_{n\in
J(N_0/\varphi_t(N_0))}n\iota_{1,t}(D_{r_t([0,\rho])}(\varphi_t(N_0)))$$ where $D_{r_t([0,\rho])}(\varphi_t(N_0))$ denotes the completion of $D(\varphi_t(N_0))$ with respect to the norm $r_t(\rho)$.
Now we turn to the microlocalization and first of all note that $\varphi_t(b_\alpha)=(b_\alpha+1)^{\alpha(t)}-1$ is divisible by $b_\alpha$. So if $\varphi_t(b_\alpha)$ is invertible in a ring then so is $b_\alpha$. On the other hand, if $p^{-\frac{1}{p^{m(\alpha,t)}}}<\rho<1$ then by Lemma \[trivest\] we have $$\|\varphi_t(b_\alpha)-\binom{\alpha(t)}{p^{m(\alpha,t)}}b_\alpha^{p^{m(\alpha,t)}}\|_\rho<\|\binom{\alpha(t)}{p^{m(\alpha,t)}}b_\alpha^{p^{m(\alpha,t)}}\|_\rho\ .$$ Hence $\varphi_t(b_\alpha)$ is invertible in the Banach algebra $D_{[\rho_1,\rho_2]}(N_0,\alpha)$ for any $p^{-\frac{1}{p^{m(\alpha,t)}}}<\rho_1<\rho_2<1$ since it is close to the invertible element $\binom{\alpha(t)}{p^{m(\alpha,t)}}b_\alpha^{p^{m(\alpha,t)}}$ (as the binomial coefficient $\binom{\alpha(t)}{p^{m(\alpha,t)}}$ is not divisible by $p$). This shows that the microlocalisation of $D_{[0,\rho_2]}(N_0)$ with respect to the multiplicative set $\varphi_t(b_\alpha)^{\mathbb{N}}$ and norm $\max(\rho_1,\rho_2)$ equals $D_{[\rho_1,\rho_2]}(N_0,\alpha)$. Therefore for each $p^{-\frac{1}{p^{m(\alpha,t)}}}<\rho_1<\rho_2<1$ we obtain $$D_{[\rho_1,\rho_2]}(N_0,\alpha)= \bigoplus_{n\in
J(N_0/\varphi_t(N_0))}n\iota_{0,1}(D_{r_t([\rho_1,\rho_2])}(\varphi_t(N_0),\alpha))$$ by microlocalizing both sides of . Now letting $\rho_2$ tend to $1$ and then also $\rho_1\to 1$ we get $$\mathfrak{R}_0(N_0,\alpha)=\bigoplus_{n\in
J(N_0/\varphi_t(N_0))}n\iota_{1,t}(\mathfrak{R}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha))\label{dec1}$$ for all $t\in T_+$. Here we define $$\mathfrak{R}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha):=\varinjlim_{\rho_1\to
1}\varprojlim_{\rho_2\to 1}D_{r_t([\rho_1,r(\rho_2)])}(\varphi_t(N_0),\alpha)$$ which is in general different from $\mathfrak{R}_0(\varphi_t(N_0),\alpha)$ (in which by definition we use norms $\rho$ such that $\|\varphi_t(b_\beta)\|_\rho=\|\varphi_t(b_\alpha)\|_\rho$) by Example \[ex\]. Indeed, for $t=s$ the sum $\sum_{n=1}^{\infty}\varphi(b_\beta^nb_\alpha^{-n})$ converges in $\mathfrak{R}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha)$, but not in $\mathfrak{R}_0(\varphi_t(N_0),\alpha)$.
By the entirely same proof we also obtain $$\mathfrak{R}_{0,r_{t_1}(\cdot)}(\varphi_{t_1}(N_0),\alpha)=\bigoplus_{n\in
J(\varphi_{t_1}(N_0)/\varphi_{t_1t_2}(N_0))}n\iota_{t_1,t_1t_2}(\mathfrak{R}_{0,r_{t_1t_2}(\cdot)}(\varphi_{t_1t_2}(N_0),\alpha))\label{dec2}$$ for each pair $t_1,t_2\in T_+$ where $\iota_{t_1,t_1t_2}$ is the inclusion of the rings above induced by the natural inclusion $\varphi_{t_1t_2}(N_0)\hookrightarrow \varphi_{t_1}(N_0)$.
Now we would like to define continuous homomorphisms $$\begin{aligned}
\varphi_{t_2t_1,t_1}\colon
\mathfrak{R}_{0,r_{t_1}(\cdot)}(\varphi_{t_1}(N_0),\alpha)&\to&
\mathfrak{R}_{0,r_{t_1t_2}(\cdot)}(\varphi_{t_1t_2}(N_0),\alpha)\\
\varphi_{t_1}(b_\beta)&\mapsto&\varphi_{t_1t_2}(b_\beta)\end{aligned}$$ induced by the group isomorphism $\varphi_{t_2}\colon \varphi_{t_1}(N_0)\to
\varphi_{t_1t_2}(N_0)$ so that we can take the injective limit $$\mathfrak{R}(N_1,\ell):=\varinjlim_{t}\mathfrak{R}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha)$$ with respect to the maps $\varphi_{t_2t_1,t_1}$. This is not possible for all $t_2$ since the map $\varphi_{t_2}$ will not always be norm-decreasing on monomials ${{\bf b}}^{{{\bf k}}}$ for ${{\bf k}}\in\mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}$. To overcome this we define the pre-ordering $\leq_\alpha$ (depending on the choice of the simple root $\alpha$) on $T_+$ the following way: $t_1\leq_\alpha t_2$ if and only if $|\beta(t_2t_1^{-1})|\leq |\alpha(t_2t_1^{-1})|\leq 1$ for all $\beta\in\Phi^+$. (In other words if and only if we have $m(\beta,t_2t_1^{-1})\geq m(\alpha,t_2t_1^{-1})\geq 0$.) In particular, $t_1\leq_\alpha t_2$ implies $t_2t_1^{-1}\in
T_+$ and it is equivalent to $1\leq_\alpha t_2t_1^{-1}$. We also have $1\leq_\alpha s$ for any $\alpha\in \Delta$. It is clear that $\leq_\alpha$ is transitive and reflexive. Moreover, if $t_2\leq_\alpha t_1\leq_\alpha t_2$ then $|\beta(t_2t_1^{-1})|=1$ for all $\beta\in\Phi^+$ whence $t_2t_1^{-1}$ lies in $T_0$. Therefore $\leq_\alpha$ defines a partial ordering on the quotient monoid $T_+/T_0$.
\[rightfilt\] The partial ordering $\leq_\alpha$ on $T_+/T_0$ is right filtered, ie. any finite subset of $T_+/T_0$ has a common upper bound with respect to $\leq_\alpha$.
Let $t_1,t_2\in T_+$ be arbitrary with $|\alpha(t_1)|\leq |\alpha(t_2)|$. Since the simple roots $\beta\in\Delta$ are linearly independent in $X^*(T)={\mathrm{Hom}}_{alg}(T,\mathbb{G}_m)$, and the pairing $X^*(T)\times X_*(T)\to\mathbb{Z}$ is perfect, we may choose $s_{\overline{\alpha}}\in T$ so that $|\beta(s_{\overline{\alpha}})|<|\alpha(s_{\overline{\alpha}})|=1$ for all $\alpha\neq\beta\in\Delta$. Since all the positive roots are positive linear combinations of the simple roots, we see immediately that $s_{\overline{\alpha}}\in T_+$. Moreover, if $\alpha\neq\gamma\in\Phi^+$ then $\gamma$ is not a scalar multiple of $\alpha$ hence writing $\gamma=\sum_{\beta\in\Delta}m_{\beta,\gamma}\beta$ there is a $\alpha\neq\beta\in\Delta$ with $m_{\beta,\gamma}>0$ whence $|\gamma(s_{\overline{\alpha}})|<1$. So we have $t_1\leq_\alpha t_1s_{\overline{\alpha}}^k$ for any $k\geq 0$ and $t_2\leq_\alpha t_1s_{\overline{\alpha}}^k$ for $k$ big enough.
Fix an element $1\leq_\alpha t\in T_+$ and let $p^{-\frac{1}{\max_{\beta\in \Phi^+} p^{m(\beta,t)+m(\alpha,t)}}}<\rho_1<\rho_2<1$ be a real numbers in $p^{\mathbb{Q}}$. Note that $\varphi_t\colon
N_0\to\varphi_t(N_0)$ is an isomorphism of pro-$p$ groups. Hence it induces an isometric isomorphism $$\begin{aligned}
\varphi_t\colon D_{[0,\rho_2^{p^{m(\alpha,t)}}]}(N_0)&\to& D_{[0,\rho_2^{p^{m(\alpha,t)}}]}(\varphi_t(N_0))\\
\sum_{\bf k} c_{\bf k}\prod_\beta b_\beta^{k_\beta}&\mapsto& \sum_{\bf k}
c_{\bf k}\prod_\beta \varphi_t(b_\beta)^{k_\beta}\end{aligned}$$ of Banach algebras where $D_{[0,\rho_2^{p^{m(\alpha,t)}}]}(\varphi_t(N_0))$ denotes the completion of $D(\varphi_t(N_0))$ with respect to the $\rho_2^{p^{m(\alpha,t)}}$-norm defined by the set of generators $\{\varphi_t(n_\beta)\}_{\beta\in\Phi^+}$ of $\varphi_t(N_0)$. To avoid confusion, from now on we denote by the subscript $\rho,N_0$ the $\rho$-norm (as before) on $D(N_0)$ and by the subscript $\rho,\varphi_t(N_0)$ the $\rho$-norm on $D(\varphi_t(N_0))$. By Lemma \[trivest\] we have $$\|\varphi_t(b_\beta)\|_{\rho,N_0}=\rho^{p^{m(\beta,t)}}\leq\rho^{p^{m(\alpha,t)}}=\|\varphi_t(b_\alpha)\|_{\rho,N_0}$$ for any $\beta\in\Phi^+$ and $\rho=\rho_1$ or $\rho=\rho_2$ because of our assumption $1\leq_\alpha t$. This shows that for any monomial $\prod_{\beta\in\Phi^+}\varphi_t(b_{\beta})^{k_\beta}$ (with $k_\beta\geq 0$ for all $\beta\in\Phi^+$) we have $$\|\prod_{\beta\in\Phi^+}\varphi_t(b_{\beta})^{k_\beta}\|_{r_t(\rho)}=\|\prod_{\beta\in\Phi^+}\varphi_t(b_{\beta})^{k_\beta}\|_{\rho,N_0}\leq\rho^{p^{m(\alpha,t)}{{\bf k}}}=\|\prod_{\beta\in\Phi^+}\varphi_t(b_{\beta})^{k_\beta}\|_{\rho^{p^{m(\alpha,t)}},\varphi_t(N_0)}$$ since both norms are multiplicative on $D(\varphi_t(N_0))$. Hence we obtain a norm decreasing homomorphism $$D_{[0,\rho_2^{p^{m(\alpha,t)}}]}(N_0)\overset{\sim}{\rightarrow}
D_{[0,\rho_2^{p^{m(\alpha,t)}}]}(\varphi_t(N_0))\rightarrow
D_{r_t([0,\rho_2])}(\varphi(N_0))\hookrightarrow
D_{r_t([\rho_1,\rho_2])}(\varphi(N_0),\alpha)\ .$$
Moreover, the element $\varphi_t(b_\alpha)$ is invertible in $D_{r_t([\rho_1,\rho_2])}(\varphi_t(N_0),\alpha)$ and for each $\rho_1\leq
\rho\leq\rho_2$ and $x\in D_{[0,\rho_2]}(N_0)$ we have $$\|\varphi_t(x)\varphi_t(b_\alpha)^{-k}\|_{r_t(\rho)}\leq\|x\|_{\rho^{p^{m(\alpha,t)}},N_0}\|b_\alpha^{-k}\|_{\rho^{p^{m(\alpha,t)}},N_0}\ .$$ Therefore by the universal property of microlocalisation (Prop. A.18 in [@SZ]) we obtain a norm decreasing homomorphism $$\begin{aligned}
\varphi_{t,1}\colon D_{[\rho_1^{p^{m(\alpha,t)}},\rho_2^{p^{m(\alpha,t)}}]}(N_0,\alpha)&\to&
D_{r_t([\rho_1,\rho_2])}(\varphi_t(N_0),\alpha)\notag\\
b_\beta &\mapsto& \varphi_t(b_\beta)\ .\label{phit1}\end{aligned}$$ Note that this above map is not surjective in general by Example \[ex\].
The map is injective.
Take an element $f({{\bf b}})=\sum_{{{\bf k}}}d_{{{\bf k}}}{{\bf b}}^{{\bf k}}\in D_{[\rho_1^{p^{m(\alpha,t)}},\rho_2^{p^{m(\alpha,t)}}]}(N_0,\alpha)$ and pairwise distinct ${{\bf k}}_1,\dots,{{\bf k}}_r\in\mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}$. Note that $\|\varphi_t(b_\beta)\|_{\rho,N_0}>\|\varphi_t(b_\beta)-\binom{\beta(t)}{p^{m(\beta,t)}}b_\beta^{p^{m(\beta,t)}}\|_{\rho,N_0}$ hence we obtain $$\begin{aligned}
\|\sum_{j=1}^r d_{{{\bf k}}_j}\varphi_t({{\bf b}})^{{{\bf k}}_j}\|_{r_t(\rho_1),r_t(\rho_2)}=\|\sum_{j=1}^r d_{{{\bf k}}_j}\prod_{\beta\in\Phi^+}\left(\binom{\beta(t)}{p^{m(\beta,t)}}b_\beta^{p^{m(\beta,t)}}\right)^{k_{j,\beta}}\|_{\rho_1,\rho_2}=\\
\max_j\| d_{{{\bf k}}_j}\prod_{\beta\in\Phi^+}\left(\binom{\beta(t)}{p^{m(\beta,t)}}b_\beta^{p^{m(\beta,t)}}\right)^{k_{j,\beta}}\|_{\rho_1,\rho_2}=\max_j \|d_{{{\bf k}}_j}\varphi_t({{\bf b}})^{{{\bf k}}_j}\|_{r_t(\rho_1),r_t(\rho_2)}\end{aligned}$$ using Prop. \[4\] as we have $\prod_{\beta\in\Phi^+}b_\beta^{p^{m(\beta,t)}k_{j_1,\beta}}\neq \prod_{\beta\in\Phi^+}b_\beta^{p^{m(\beta,t)}k_{j_2,\beta}}$ for $1\leq j_1\neq j_2 \leq r$.
Since the map $\varphi_{t,1}$ is norm decreasing, we have $\|d_{{\bf k}}\varphi_t({{\bf b}})^{{\bf k}}\|_{r_t(\rho_1),r_t(\rho_2)}\to 0$ as ${{\bf k}}\to\infty$. Therefore we also have $\|\sum_{{{\bf k}}}d_{{{\bf k}}}\varphi_t({{\bf b}})^{{\bf k}}\| _{r_t(\rho_1),r_t(\rho_2)}=\max_{{{\bf k}}} \|d_{{{\bf k}}}\varphi_t({{\bf b}})^{{{\bf k}}}\|_{r_t(\rho_1),r_t(\rho_2)}$ which is nonzero if there exists a ${{\bf k}}$ with $d_{{{\bf k}}}\neq 0$. Therefore the injectivity.
Taking projective and injective limits we obtain an injective ring homomorphism $$\varphi_{t,1}\colon\mathfrak{R}_0(N_0,\alpha)\hookrightarrow\mathfrak{R}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha)$$ for any $1\leq_\alpha t\in T_+$.
Note that $\mathfrak{R}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha)$ is a subring of $\mathfrak{R}_0(N_0,\alpha)$ via the map $\iota_{1,t}$ (for all $t\in T_+$). Hence for $1\leq_\alpha t$ we obtain a ring homomorphism $\varphi_t=\iota_{1,t}\circ\varphi_{t,1}\colon \mathfrak{R}_0(N_0,\alpha)\to \mathfrak{R}_0(N_0,\alpha)$. However, if $1\not\leq_\alpha t$ for some $t\in T_+$ then we in fact do not have a continuous ring homomorphism $\varphi_t\colon
\mathfrak{R}_0(N_0,\alpha)\to\mathfrak{R}_0(N_0,\alpha)$. Indeed, in this case there exists a $\beta\in \Phi^+$ such that $|\beta(t)|>|\alpha(t)|$ so there exist integers $k_\beta>k_\alpha$ such that $\|\varphi_t(b_\beta^{k_\beta}b_\alpha^{-k_\alpha})\|_{\rho}=\rho^{k_\beta/|\beta(t)|-k_\alpha/|\alpha(t)|}>1$ for any $p^{-|\alpha(t)|}<\rho<1$ therefore $\sum_{n=1}^{\infty}\varphi_t(b_\beta^{nk_\beta}b_\alpha^{-nk_\alpha})$ does not converge in $\mathfrak{R}_0(N_0,\alpha)$ even though $\sum_{n=1}^{\infty}b_\beta^{nk_\beta}b_\alpha^{-nk_\alpha}$ does.
\[notetale\] If $\Phi^+\neq \Delta$ (e.g. if $G={\mathrm{GL}}_n(\mathbb{Q}_p)$, $n>2$) then we have $$\mathfrak{R}_0(N_0,\alpha)=\bigoplus_{u\in
J(N_0/\varphi_t(N_0))}u\iota_{1,s}(\mathfrak{R}_{0,r_s(\cdot)}(\varphi(N_0),\alpha))\supsetneq \bigoplus_{n\in
J(N_0/\varphi(N_0))}u\varphi(\mathfrak{R}_{0}(N_0,\alpha))$$ by (with the choice $t=s$) and Example \[ex\] (which shows that $\varphi_{s,1}$ is not surjective).
In a similar fashion we get for $t_1\in T_+$ (and $1\leq_\alpha t\in T_+$) an injective homomorphism $$\varphi_{tt_1,t_1}\colon\mathfrak{R}_{0,r_{t_1}(\cdot)}(\varphi_{t_1}(N_0),\alpha)\to\mathfrak{R}_{0,r_{tt_1}(\cdot)}(\varphi_{tt_1}(N_0),\alpha)\ .$$ In view of Lemma \[rightfilt\] we define $$\mathcal{R}(N_1,\ell):=\varinjlim_{t\in T_+}\mathfrak{R}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha)$$ with respect to the maps $\varphi_{t_1,t_2}$ for $t_2\leq_\alpha t_1$.
Now take any $t\in T_+$ (not necessarily satisfying $1\leq_\alpha t$). The map $$\varphi_t:=\varinjlim_{t_1}\iota_{t_1,tt_1}\colon\mathcal{R}(N_1,\ell)\to\mathcal{R}(N_1,\ell)$$ is defined as the direct limit of the inclusion maps $$\iota_{t_1,tt_1}\colon \mathfrak{R}_{0,r_{tt_1}(\cdot)}(\varphi_{tt_1}(N_0),\alpha)\hookrightarrow
\mathfrak{R}_{0,r_{t_1}(\cdot)}(\varphi_{t_1}(N_0),\alpha)$$ induced by $\varphi_{tt_1}(N_0)\subseteq \varphi_{t_1}(N_0)$. By definition, for any $t\in T_+$ the ring $\mathfrak{R}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha)$ consists of formal power series $\sum_{\bf k}c_{\bf k}\varphi_t({\bf b})^{\bf k}$ that converge in $\mathfrak{R}_0(N_0,\alpha)$. Therefore the map $$\begin{aligned}
\bigcup_{t\in T_+}\left\{
\sum_{{\bf k}\in\mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}}c_{\bf
k}{\bf b}^{\bf k}
\text{ }{\Big |} \text{ }
\sum_{\bf k}c_{\bf k}\varphi_t({\bf b})^{\bf k}\text{
convergent in }\mathfrak{R}_0(N_0,\alpha)\right\}\to\mathcal{R}(N_1,\ell)\\
\sum_{\bf k}c_{\bf
k}{\bf b}^{\bf k}\mapsto\sum_{\bf k}c_{\bf k}\varphi_t({\bf b})^{\bf k}\in \mathfrak{R}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha)\hookrightarrow\mathcal{R}(N_1,\ell)\end{aligned}$$ is well-defined and bijective since $\sum_{\bf k}c_{\bf k}\varphi_t({\bf b})^{\bf k}$ converges for some $t\in T_+$ and the connecting homomorphisms in the injective limit defining $\mathcal{R}(N_1,\ell)$ are injective and given by $\varphi_{t_1,t_2}$ for $t_2\leq_\alpha t_1$.
Hence we may identify $$\mathcal{R}(N_1,\ell)=\bigcup_{t\in T_+}\left\{
\sum_{{\bf k}\in\mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}}c_{\bf
k}{\bf b}^{\bf k}
\text{ }{\Big |} \text{ }
\sum_{\bf k}c_{\bf k}\varphi_t({\bf b})^{\bf k}\text{
convergent in }\mathfrak{R}_0(N_0,\alpha)\right\}
\label{expandR}$$ and obtain
The natural map $\varphi_t\colon\mathcal{R}(N_1,\ell)\to \mathcal{R}(N_1,\ell)$ is injective for all $t\in T_+$ and we have the decomposition $$\mathcal{R}(N_1,\ell)=\bigoplus_{n\in J(N_0/\varphi_t(N_0))}n\varphi_t(\mathcal{R}(N_1,\ell))\ .$$ In particular, $\mathcal{R}(N_1,\ell)$ is a free (right) module over itself via $\varphi_t$ and it is a $\varphi$-ring over $N_0$ with $\varphi=\varphi_s$ in the sense of Definition \[phiringH\_0\].
By we have $$\mathfrak{R}_{0,r_{t_1}(\cdot)}(\varphi_{t_1}(N_0),\alpha)=\bigoplus_{n\in J(N_0/\varphi_t(N_0))}\varphi_{t_1}(n)\iota_{t_1,tt_1}(\mathfrak{R}_{0,r_{tt_1}(\cdot)}(\varphi_{tt_1}(N_0),\alpha))$$ for any $t_1\in T_+$. The statement follows by taking the injective limit of both sides (with respect to $t_1$) and noting that $\varphi_{t_1,1}(n)=\varphi_{t_1}(n)\in\varphi_{t_1}(N_0)\subseteq \mathfrak{R}_{0,r_{t_1}(\cdot)}(\varphi_{t_1}(N_0),\alpha)$ for $n\in N_0\subseteq \mathfrak{R}_0(N_0)$ and $1\leq_\alpha t_1$ therefore $n$ corresponds to $\varinjlim_{1\leq_\alpha t_1}(\varphi_{t_1}(n))_{t_1}$ via the identification .
\[expandRell\] The ring $\mathcal{R}(N_1,\ell)$ via the description consists of exactly those Laurent-series $$x=\sum_{{\bf k}\in\mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}}c_{\bf
k}{\bf b}^{\bf k}$$ that converge on the open annulus of the form $$\left\{\rho_2<|z_\alpha|<1,\ |z_{\beta}|\leq |z_\alpha|^r\text{ for
}\beta\in\Phi^+\setminus\{\alpha\}\right\}\ .\label{annulus}$$ for some $p^{-1}<\rho_2<1$ and $1\leq r\in\mathbb{Z}$.
If $x\in \mathcal{R}(N_1,\ell)$ then there exists a $t\in T_+$ such that $\varphi_t(x)$ converges in $\mathfrak{R}_0(N_0)$, ie. it converges in the norm $\|b_\beta\|_\rho=\rho$ for all $\beta\in \Phi^+$ for some fixed $p^{-1}<\rho_0<1$ and all $\rho\in(\rho_0,1)$. By Lemma \[rightfilt\] we may assume that $|\alpha(t)|=1$ whence $\|\varphi_t(b_\alpha)\|_\rho=\rho$ for all $\rho<1$ as we may take $t=s_{\overline{\alpha}}^k$ for $k$ large enough. Now let $\rho_2:=\rho_0$ and $r:=\max_{\beta\in\Phi^+}([|1/\beta(t)|]+1)\in\mathbb{Z}$. Then $x$ converges on the annulus as we have $\rho^r\leq \rho^{1/|\beta(t)|}\leq\|\varphi_t(b_{\beta})\|_\rho$ for all $\beta\in\Phi^+\setminus\{\alpha\}$ by Lemma \[trivest\].
Conversely for any fixed $p^{-1}<\rho_2<1$ and integer $r\geq 1$ we need to find a $t\in T_+$ and a $\rho_0\in (p^{-1},1)$ such that for all $\rho\in (\rho_0,1)$ we have $\rho_2<\|\varphi_t(b_\alpha)\|_{\rho}<1$ and $\|\varphi_t(b_\beta)\|_{\rho}\leq \|\varphi_t(b_\alpha)\|_{\rho}^r$. We take $t:=s_{\overline{\alpha}}^k$ and $\rho_0:=\max(\rho_2,p^{-|\beta(t)|}\mid \beta\in\Phi^+\setminus\{\alpha\})$ where $k:=\max_{\beta\in\Phi^+\setminus\{\alpha\}}([-\frac{\log r}{\log |\beta(s_{\overline{\alpha}})|}]+1)$ (for the definition of $s_{\overline{\alpha}}$ see the proof of Lemma \[rightfilt\]). Indeed, since $|\alpha(s_{\overline{\alpha}}^k)|$ equals $1$, we have $\rho_2<\rho=\|\varphi_{s_{\overline{\alpha}}^k}(b_\alpha)\|_{\rho}<1$ (for any $k$). On the other hand, we have $|\beta(s_{\overline{\alpha}})|<1$ for all $\alpha\neq\beta\in\Phi^+$ (whence, in particular, the definition of $k$ makes sense), so we obtain $$\|\varphi_{t}(b_\beta)\|_{\rho}=\max\limits_{0\leq j\leq \mathrm{val}_p(\beta(t))}(\rho^{p^j}p^{j-\mathrm{val}_p(\beta(t))})=\rho^{p^{\mathrm{val}_p(\beta(t))}}=\rho^{1/|\beta(s_{\overline{\alpha}})|^k}\leq\rho^r$$ for all $\beta\in\Phi^+\setminus\{\alpha\}$ by Lemma \[trivest\], the choice of $k$ so that we have $r\leq 1/|\beta(s_{\overline{\alpha}})|^k$, and the choice of $p^{-|\beta(t)|}<\rho$ so that we have $\max\limits_{0\leq j\leq \mathrm{val}_p(\beta(t))}(\rho^{p^j}p^{j-\mathrm{val}_p(\beta(t))})=\rho^{p^{\mathrm{val}_p(\beta(t))}}$.
Bounded rings
-------------
Let us denote by $\mathfrak{R}^b_0$ (resp. by $\mathfrak{R}^{int}_0$) the set of elements $x\in\mathfrak{R}_0$ such that $\lim_{\rho\to1}\|x\|_{\rho,\rho}$ exists (resp. exists and is at most $1$). These are subrings of $\mathfrak{R}_0$ by Prop. A.28 in [@SZ]. Moreover, since $\varphi_t$ is norm-decreasing for any $1\leq_\alpha t$ (see ), these subrings are stable under the action of $\varphi_t$ ($1\leq_\alpha t\in T_+$). We put $$\begin{aligned}
\mathfrak{R}_{0,r_t(\cdot)}^b(\varphi_t(N_0),\alpha)&:=&\mathfrak{R}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha)\cap\mathfrak{R}_{0}^b(N_0,\alpha)\ ,\quad\text{and}\\
\mathfrak{R}_{0,r_t(\cdot)}^{int}(\varphi_t(N_0),\alpha)&:=&\mathfrak{R}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha)\cap\mathfrak{R}_{0}^{int}(N_0,\alpha) \end{aligned}$$ where the intersection is taken inside $\mathfrak{R}_0$ under the inclusion $\iota_{1,t}\colon\mathfrak{R}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha)\hookrightarrow\mathfrak{R}_0$. Hence $$\mathcal{R}^b(N_1,\ell):=\varinjlim_t\mathfrak{R}^b_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha)\text{ and }\mathcal{R}^{int}(N_1,\ell):=\varinjlim_t\mathfrak{R}^{int}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha)$$ are $T_+$-stable subrings of $\mathcal{R}(N_1,\ell)$ (the injective limit is taken with repsect to the maps $\varphi_{t_1,t_2}$ for $t_1\leq_\alpha t_2\in T_+$ as in the construction of $\mathcal{R}(N_1,\ell)$). Further, Lemma \[equivnorm\] shows that for any $t\in T_+$ and $x\in \mathfrak{R}_0$ we have $$\label{lims}
\lim_{\rho\to1}\|x\|_{\rho}=\lim_{\rho\to1}\|x\|_{q_t(r_t(\rho))}\ .$$ Indeed, we may use Lemma \[equivnorm\] in the context of $\mathfrak{R}_0$ the following way. The elements of $D_{[\rho_1,\rho_2]}(N_0,\alpha)$ are Cauchy sequences $(a_n\varphi_t(b_\alpha)^{-k_n})_{n\in\mathbb{N}}$ (in the norm $\max(\|\cdot\|_{\rho_1},\|\cdot\|_{\rho_2})$) with $a_n\in D_{[0,\rho_2]}(N_0)$ and $k_n\geq 0$. Since $\|\cdot\|_\rho$ is multiplicative for any $\rho_1\leq\rho\leq\rho_2$ in $p^{\mathbb{Q}}$ and so is its restriction to $D(\varphi_t(N_0))$ we compute $$\begin{aligned}
\|a_n\varphi_t(b_\alpha)^{-k_n}\|_{\rho}\rho^{\sum_{\beta\in\Phi^+}(p^{m(\beta,t)}-1)}=\frac{\|a_n\|_\rho}{\|\varphi_t(b_\alpha)^{k_n}\|_{\rho}\rho^{-\sum_{\beta\in\Phi^+}(p^{m(\beta,t)}-1)}}\leq\frac{\|a_n\|_{q_t(r_t(\rho))}}{\|\varphi_t(b_\alpha)^{k_n}\|_{q_t(r_t(\rho))}}=\\
=\|a_n\varphi_t(b_\alpha)^{-k_n}\|_{q_t(r_t(\rho))}\leq \frac{\|a_n\|_{\rho}\rho^{-\sum_{\beta\in\Phi^+}(p^{m(\beta,t)}-1)}}{\|\varphi_t(b_\alpha)^{k_n}\|_{\rho}}
\leq \|a_n\varphi_t(b_\alpha)^{-k_n}\|_{\rho}\rho^{-\sum_{\beta\in\Phi^+}(p^{m(\beta,t)}-1)}\ .\end{aligned}$$ If $\rho\to 1$ and $n\to\infty$ we obtain . Combining this observation with we obtain $$\begin{aligned}
\mathfrak{R}^b_0(N_0,\alpha)=\bigoplus_{n\in
J(N_0/\varphi_t(N_0))}n\iota_{1,t}(\mathfrak{R}^b_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha))\ ;\\
\mathfrak{R}^{int}_0(N_0,\alpha)=\bigoplus_{n\in
J(N_0/\varphi_t(N_0))}n\iota_{1,t}(\mathfrak{R}^{int}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha))\ .\end{aligned}$$ So by a similar argument as for $\mathcal{R}(N_1,\ell)$ we also obtain $$\begin{aligned}
\mathcal{R}^b(N_1,\ell)=\bigoplus_{n\in J(N_0/\varphi_t(N_0))}n\varphi_t(\mathcal{R}^b(N_1,\ell))\ ;\\
\mathcal{R}^{int}(N_1,\ell)=\bigoplus_{n\in J(N_0/\varphi_t(N_0))}n\varphi_t(\mathcal{R}^{int}(N_1,\ell)),\end{aligned}$$ in other words these are $\varphi$-rings over $N_0$ in the sense of Definition \[phiringH\_0\].
\[int\] Note that by Lemma A.27 in [@SZ] an element $\sum_{{\bf k}\in\mathbb{N}^{\Phi^+\setminus\{\alpha\}}\times\mathbb{Z}}c_{\bf
k}{\bf b}^{\bf k}\in \mathcal{R}(N_1,\ell)$ (under the parametrization ) lies in $\mathcal{R}^b(N_1,\ell)$ (resp. in $\mathcal{R}^{int}(N_1,\ell)$) if and only if $|c_{\bf k}|$ is bounded (resp. $\leq 1$) for ${\bf k}\in\mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}$.
Relation with the completed Robba ring and overconvergent ring {#relate}
--------------------------------------------------------------
\[intmult\] There exists a continuous (in the weak topology of $\Lambda_\ell(N_0)$) injective ring homomorphism $j_{int}\colon\mathcal{R}^{int}(N_1,\ell)\to\Lambda_\ell(N_0)$ respecting Laurent series expansions. The image of $j_{int}$ is contained in $\mathcal{O}_{\mathcal{E}}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\subset \Lambda_\ell(N_0)$.
We proceed in 3 steps. In Step 1 we construct a map $j_{int,0}= j_{int\mid\mathfrak{R}_0^{int}}\colon\mathfrak{R}^{int}_0\to \Lambda_\ell(N_0)$ which is a priori continuous and $o_K$-linear. In Step 2 we show that $j_{int,0}$ is multiplicative hence a ring homomorphism. In Step 3 we extend it to $\mathcal{R}^{int}(N_1,\ell)$ and show that the image lies in $\mathcal{O}_\mathcal{E}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\subset\mathcal{O_E}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}=\Lambda_\ell(N_0)$.
*Step 1.* By Lemma \[4\] and Remark \[int\] we may write any element in $\mathfrak{R}^{int}_0$ in a Laurent series expansion $\sum_{{\bf k}\in\mathbb{N}^{\Phi^+\setminus\{\alpha\}}\times\mathbb{Z}}c_{\bf
k}{\bf b}^{\bf k}$ with coefficients $c_{\bf k}$ in $o_K$. So we may collect all the terms containing $b_\alpha^{k_\alpha}$ for some fixed $k_\alpha$ into an element of the Iwasawa algebra $\Lambda(N_1)$ to obtain an expansion $\sum_{n\in\mathbb{Z}}b_\alpha^nf_n$ with $f_n\in\Lambda(N_1)$. These power series satisfy the convergence property that there exists a real number $p^{-1}<\rho_1<1$ such that $\rho^n\|f_n\|_{\rho}\to0$ as $|n|\to\infty$ for all $\rho_1<\rho<1$. In particular, if $n\to-\infty$ then $f_n\to 0$ in the compact topology of $\Lambda(N_1)$. Hence the sum $\sum_nb_\alpha^nf_n$ also converges in $\Lambda_\ell(N_0)$. This way we obtained a right $\Lambda(N_0)$-linear injective map $j_{int,0}\colon\mathfrak{R}_0^{int}\to\Lambda_\ell(N_0)$.
Recall that the weak topology (see [@SVe], [@SVi], [@SVZ] for instance) on $\Lambda_\ell(N_0)$ is defined by the open neighbourhoods of $0$ of the form $\mathcal{M}(r)=\mathcal{M}_\ell(N_0)^r+\mathcal{M}(N_0)^r$ where $\mathcal{M}_\ell(N_0)=\Lambda_\ell(N_0)\mathcal{M}(N_1)$ denotes the maximal ideal of $\Lambda_\ell(N_0)$ and $\mathcal{M}(N_i)$ denotes the maximal ideal of $\Lambda(N_i)\subseteq \Lambda_\ell(N_0)$ ($i=0,1$). For any fixed $p^{-1}<\rho_1<\rho<1$ the preimage of $\mathcal{M}(r)$ in $\mathfrak{R}^{int}_{0}\cap D_{[\rho_1,1)}(N_1,\alpha)$ contains the open ball $\{x\mid \|x\|_\rho<p^{-r}\}$. Indeed, if $x=\sum_{n\in\mathbb{Z}}b_\alpha^nf_n$ then for any $n<0$ we have $\|f_n\|_\rho<p^{-r}$ hence $f_n\in \mathcal{M}(N_1)^r$ and $b_\alpha^nf_n\in\mathcal{M}_\ell(N_0)$. On the other hand, the positive part $\sum_{n\geq 0}b_\alpha^nf_n$ lies in $\Lambda(N_0)$ and has $\rho$-norm smaller than $p^{-r}$ therefore lies in $\mathcal{M}(N_0)^r$. Hence the continuity.
*Step 2.* Now by the continuity and linearity of $j_{int,0}$ it suffices to show that it is multiplicative on monomials ${\bf b}^{\bf k}$. Moreover, each monomial is a linear combination of elements of the form $b_\alpha^ng$ with $g\in N_0$. In order to expand the product $(b_\alpha^{n_1}g_1)(b_\alpha^{n_2}g_2)$ into a skew Laurent series it suffices to expand $g_1b_\alpha^{n_2}$ with $n_2<0$. However, if $g_1b_\alpha^{n_2}=\sum_n b_\alpha^nh_n$ is the expansion in $\mathcal{R}^{int}(N_1,\ell)$ then $\sum_{|n|<n_0}b_\alpha^nh_nb_\alpha^{-n_2}$ tends to $g_1$ (as $n_0\to+\infty$) in the topology of $\mathfrak{R}^{int}_0$ (induced by the norms) hence also in the weak topology. Therefore the expansion in $\Lambda_\ell(N_0)$ is also $g_1b_\alpha^{n_2}=\sum_n b_\alpha^nh_n$. So the above constructed map $j_{int,0}$ is indeed a ring homomorphism as claimed.
*Step 3.* Finally, take an element $x\in\mathcal{R}^{int}(N_1,\ell)$. There exists an element $1\leq_\alpha
t\in T_+$ such that $\varphi_t(x)$ lies in the image of the composite map $$\mathfrak{R}^{int}_{0,r_t(\cdot)}(\varphi_t(N_0),\alpha)\hookrightarrow
\mathfrak{R}^{int}_0(N_0,\alpha)\hookrightarrow \mathcal{R}^{int}(N_1,\ell)$$ where the first arrow is induced by the inclusion $\varphi_t(N_0)\subseteq
N_0$. Now if we reduce $j_{int,0}(\varphi_t(x))\in\Lambda_\ell(N_0)$ modulo the ideal generated by $N_l-1$ for some integer $l\geq 1$ then we obtain an element in $\varphi_t(\mathcal{O}_\mathcal{E}^\dagger[N_1/N_l,\ell])$. Indeed, $\varphi_t(\mathcal{O_E}[N_1/N_l,\ell])$ is a closed subspace in $\mathcal{O_E}[N_1/N_l,\ell]$ and all the monomials $j_{int,0}(\varphi_t({\bf b}^{\bf k}))$ map into this subspace under the reduction modulo $(N_l-1)$. Hence the image lies in $\varphi_t(\mathcal{O_E}[N_1/N_l,\ell]$. Moreover, by the convergence property of elements in $\mathfrak{R}^{int}_0$, we may expand $$\varphi_t(x)=\sum_{n\in\mathbb{Z}}b_\alpha^nf_n$$ with $f_n\in\Lambda(N_1)$ and $\rho^n\|f_n\|_\rho\to0$ as $n\to\infty$ for all $\rho_1<\rho<1$ and a fixed $p^{-1}<\rho_1<1$ depending on $x$. Since the reduction map $\Lambda(N_1)\to o[N_1/N_l]$ is continuous in the $\rho$-norm, we obtain that the reduction of $j_{int,0}(\varphi_t(x))$ modulo $(N_l-1)$ also lies in $\mathcal{O}_\mathcal{E}^\dagger[N_1/N_l,\ell]$. Hence we have $j_{int,0}(\varphi_t(x))\pmod{N_l-1}\in\varphi_t(\mathcal{O}_\mathcal{E}^\dagger[N_1/N_l,\ell])=\varphi_t(\mathcal{O_E}[N_1/N_l,\ell])\cap \mathcal{O}_\mathcal{E}^\dagger[N_1/N_l,\ell]$. Taking the limit we see (using ) that $j_{int,0}(\varphi_t(x))$ lies in $$\varprojlim_l\varphi_t(\mathcal{O}_\mathcal{E}^\dagger[N_1/N_l,\ell])=\varphi_t(\mathcal{O}_\mathcal{E}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]})\ .$$ So we put $j_{int}(x):=\varphi_t^{-1}(j_{int,0}(\varphi_t(x)))$. This extends the ring homomorphism $j_{int,0}$ to a continuous ring homomorphism $j_{int}\colon\mathcal{R}^{int}(N_1,\ell)\hookrightarrow\mathcal{O}_\mathcal{E}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\subset\Lambda_\ell(N_0)$ by Lemma \[rightfilt\]. Moreover, $j_{int}$ is $T_+$-equivariant as it respects power series expansions.
Now the following proposition compares $\mathcal{R}(N_1,\ell)$ with the previous construction $\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$.
\[j\] There exists a natural $T_{+}$-equivariant ring homomorphism $$j\colon\mathcal{R}(N_1,\ell)\to \mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$$ with dense image.
At first we construct the map $j_0=j_{\mid\mathfrak{R}_0}$ on $\mathfrak{R}_0\subset \mathcal{R}(N_1,\ell)$ with dense image. We are going to show that for any open characteristic subgroup $H\leq N_1$ we have an isomorphism $\mathfrak{R}_0/\mathfrak{R}_0(H-1)\cong\mathcal{R}[N_1/H,\ell]$. Note that $N_1$ being a compact $p$-adic Lie group, $N_1$ has a system of neighbourhoods of $1$ consisting of open uniform characteristic subgroups (in fact $N_1$ is uniform—since so is $N_0$ by assumption—and one can take repeatedly the Frattini subgroups of $N_1$ which are characteristic subgroups, ie. stable under all the continuous automorphisms of $N_1$). So we may assume without loss of generality that $H$ is uniform with topological generators $h_1,h_2,\dots,h_d$ with $d=\dim N_1$ as a $p$-adic Lie group.
Under the parametrization in Prop. \[4\] the elements of $\mathfrak{R}_0$ can be written as power series $\sum_{n\in\mathbb{Z}}b_\alpha^nf_n$ with $f_n\in D(N_1,K)$ and the convergence property that there exists a real number $\rho_1<1$ such that $\rho^n\|f_n\|_\rho\rightarrow 0$ (as $|n|\rightarrow\infty$) for all $\rho_1\leq\rho<1$. Now note that we have $D(N_1,K)=\bigoplus_{u\in J(N_1/H)}uD(H,K)$. Hence the right ideal $D(N_1,K)(H-1)$ in $D(N_1,K)$ is generated by the elements $h_i-1$ for $1\leq i\leq d$ and it is the kernel of the natural projection $\pi_H\colon D(N_1,K)\to D(N_1/H)=K[N_1/H]$. Moreover, this quotient map factors through the inclusion $D(N_1,K)\hookrightarrow D_{[0,\rho]}(N_1,K)$ for any $p^{-1}<\rho<1$. Hence $\rho^n\|\pi_H(f_n)\|\to 0$ where $\|x\|:=\max_u |x_u|$ with $x=\sum_{u\in N_1/H}x_uu$, $x_u\in K$. Therefore we obtain a map $$\begin{aligned}
\pi_H\quad\colon\quad\mathfrak{R}_0&\to&\mathcal{R}[N_1/H,\ell]=\bigoplus_{u\in N_1/H}\mathcal{R}u\\
\sum_{n\in\mathbb{Z}}b_\alpha^nf_n&\mapsto&\sum_{u\in N_1/H}\sum_{n\in\mathbb{Z}}\pi_H(f_n)_uT^nu\ .\end{aligned}$$ A priori this map is only known to be $K$-linear, continuous, and surjective between topological $K$-vectorspaces. So for the multiplicativity it suffices to show that $\pi_H({\bf b}^{{\bf k}_1}{\bf b}^{{\bf k}_2})=\pi_H({\bf b}^{{\bf k}_1})\pi_H({\bf b}^{{\bf k}_2})$ for monomials ${\bf b}^{{\bf k}_i}$ with ${\bf k}_i\in\mathbb{N}\times\mathbb{Z}^{d}$ ($i=1,2$). On the other hand, these monomials are contained in the subring $\mathfrak{R}_0^{int}$. By Lemma \[intmult\] we have a commutative diagram $$\xymatrix{
\mathfrak{R}_0^{int}\ar[r] \ar[d]_{\pi_{H,\mathcal{O}_\mathcal{E}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}}} &\mathfrak{R}_0 \ar[d]_{\pi_{H}}\\
\mathcal{O}_\mathcal{E}^\dagger [N_1/H,\ell]\ar[r] & \mathcal{R}[N_1/H,\ell]
}$$ of $o$-modules such that all the maps are ring homomorphisms except possibly for $\pi_H$. However, from the commutativity of the diagram it follows that also $\pi_H$ is multiplicative on monomials therefore a ring homomorphism. By taking the projective limit of maps $\pi_H$ we obtain a ring homomorphism $j_0\colon\mathfrak{R}_0\to \mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ with dense image and extending $j_{int,0}\colon\mathfrak{R}_0^{int}\hookrightarrow \mathcal{O}_\mathcal{E}^\dagger$.
Finally, the homomorphism $j_0$ is extended to $\mathcal{R}(N_1,\ell)$ as in the proof of Lemma \[intmult\]. The $T_+$-equivariance is clear on monomials by Lemma \[intmult\] and follows in general from the continuity and linearity.
\[log\] The above constructed map $j\colon\mathcal{R}(N_1,\ell)\to\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ is not injective in general. Indeed, for any root $\beta\neq\alpha$ in $\Phi^+$ the element $\log(n_\beta)=\log(1+b_\beta)$ lies in $D(N_1)\subset \mathcal{R}(N_1,\ell)$. It is easy to see that $\log(1+b_\beta)$ is divisible by $\varphi^r(b_\beta)$ for any nonnegative integer $r$. Indeed, we clearly have $b_\beta\mid\log(1+b_\beta)$. Applying $\varphi^r$ on the both sides of the divisibility we obtain $$\varphi^r(b_\beta)\mid\varphi^r(\log(1+b_\beta))=\log(1+b_\beta)^{p^{rm_\beta}}=p^{rm_\beta}\log(1+b_\beta)\mid\log(1+b_\beta)$$ as $p^{rm_\beta}$ is invertible in $\mathcal{R}$. Therefore $\log(1+b_\beta)$ lies in the kernel of $\pi_H$ for all $H=N_r$ hence also in the kernel of $j$.
Note that via the inclusion $\mathcal{O}_{\mathcal{E}}^\dagger\subseteq
\mathcal{R}$ we also have $\mathcal{O}_{\mathcal{E}}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\subseteq \mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$. However, if $N_1\neq 1$ then we have $j_{int}(\mathcal{R}^{int}(N_1,\ell))\neq j(\mathcal{R}(N_1,\ell))\cap \mathcal{O}_\mathcal{E}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\subset\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$.
Assume $N_1\neq 1$, so we have a positive root $\beta\neq\alpha\in\Phi^+$. We proceed in 3 steps. In Step 1 we are going to construct an element $x\in\mathcal{R}(N_1,\ell)$ with several properties. In Step 2 we are going to show that $j(x)$ lies in $\mathcal{O}_\mathcal{E}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\subset\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$. In Step 3 we prove that $j(x)$ does not lie in $j_{int}(\mathcal{R}^{int}(N_1,\ell))$. Note that the other inclusion $j_{int}(\mathcal{R}^{int}(N_1,\ell))\subset j(\mathcal{R}(N_1,\ell))\cap \mathcal{O}_\mathcal{E}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ is obvious.
*Step 1.* We denote by $s_n:=\sum_{i=1}^n(-1)^{i+1}\frac{b_\beta^i}{i}$ the $n$th estimating sum of $\log(1+b_\beta)\in\mathcal{R}(N_1,\ell)$. Note that $k_n:=[\log_pn]$ is the smallest positive integer such that $$\label{k_n}
p^{k_n}s_n\in\mathbb{Z}_p[N_{\beta,0}]\subseteq \mathcal{R}(N_1,\ell)$$ where $[\cdot]$ denotes the integer part of a real number. We further choose a sequence of real numbers $p^{-1}<\rho_1<\dots<\rho_n<\dots<1$ in $p^{\mathbb{Q}}$ such that $\lim_{n\to\infty}\rho_n=1$. Now for any fixed positive integer $n$ let $i_n$ be the smallest positive integer satisfying the following properties $$\begin{aligned}
\log_{\rho_{n-1}}(\|p^{k_{i_{n-1}}}\log(1+b_\beta)\|_{\rho_{n-1}})+1<\log_{\rho_n}(\|p^{k_{i_n}}\log(1+b_\beta)\|_{\rho_n})\ ;\notag\\
\frac{\|\log(1+b_\beta)\|_{\rho_n}}{p^{n}}>\|\log(1+b_\beta)-s_{i_n}\|_{\rho_n}\ ;\label{i_n}\\
p^{k_{i_n}/2}>\|\log(1+b_\beta)\|_{\rho_n}\ ;\notag\\
\|\varphi^i(\log(1+b_\beta))\|_{\rho_j}>\|\varphi^i(\log(1+b_\beta)-s_{i_n})\|_{\rho_j}\notag\end{aligned}$$ for all $1\leq i,j\leq n$. Such an $i_n$ exists as for any fixed $1\leq i,j\leq n$ we have $\lim_{k\to\infty}\|\varphi^i(\log(1+b_\beta)-s_{k})\|_{\rho_j}=0$. The first condition in makes the definition of $i_n$ inductive. As a consequence, we have $\|\log(1+b_\beta)\|_{\rho_n}=\|s_{i_n}\|_{\rho_n}$ by the ultrametric inequality. Now define $j_n\in\mathbb{Z}$ so that $$\label{j_n}
\rho_n^{j_n+1}<\frac{\|s_{i_n}\|_{\rho_n}}{p^{k_{i_n}}}=\|p^{k_{i_n}}s_{i_n}\|_{\rho_n}=\|p^{k_{i_n}}\log(1+b_\beta)\|_{\rho_n}\leq \rho_n^{j_n}\ .$$ (In other words $j_n=[\log_{\rho_n}(\|p^{k_{i_n}}\log(1+b_\beta)\|_{\rho_n})]$.) By we have $j_n\geq 0$. Moreover, by the first condition in the sequence $(j_n)_n$ is strictly increasing: $j_{n-1}<j_n$ for all $n>1$. On the other hand, $(-1)^{p^{k_{i_n}}}b_\beta^{p^{k_{i_n}}}$ is a summand in $p^{k_{i_n}}s_{i_n}$, therefore we have $\rho_n^{p^{k_{i_n}}}\leq \|p^{k_{i_n}}s_{i_n}\|_{\rho_n}\leq \rho_n^{j_n}$ whence $$\label{j_ni_n}
j_n\leq p^{k_{i_n}}\leq i_n\ .$$ Put $x:=\sum_{n=1}^{\infty}p^{k_{i_n}}(\log(1+b_\beta)-s_{i_n})b_\alpha^{-j_n}$. Our goal in this step is to show that the sum $x$ converges in $\mathfrak{R}_0(N_0,\alpha)\subset \mathcal{R}(N_1,\ell)$. For this it suffices to verify that for any fixed $k\geq 1$ we have $\|p^{k_{i_n}}(\log(1+b_\beta)-s_{i_n})b_\alpha^{-j_n}\|_{\rho_k}\to 0$ as $n\to\infty$. Note that in the power series expansion of $\log(1+b_\beta)-s_{i_n}$ all the terms have degree $>i_n\geq j_n$ by . Therefore in the power series expansion of $x$ all the terms have positive degree. In particular, for $k<n$ we have $\|y\|_{\rho_k}\leq \|y\|_{\rho_n}$ whenever $y$ is a monomial in the expansion of $x$. By and we obtain $$\begin{aligned}
\|p^{k_{i_n}}(\log(1+b_\beta)-s_{i_n})b_\alpha^{-j_n}\|_{\rho_k}\leq \|p^{k_{i_n}}(\log(1+b_\beta)-s_{i_n})b_\alpha^{-j_n}\|_{\rho_n}<\\
<\frac{\|p^{k_{i_n}}\log(1+b_\beta)\|_{\rho_n}}{p^n}\rho_n^{-j_n}\leq \frac{1}{p^n}\end{aligned}$$ for $k<n$. Hence we have $x\in\mathfrak{R}_0(N_0,\alpha)\subset \mathcal{R}(N_1,\ell)$.
*Step 2.* Note that by Remark \[log\] $\log(1+b_\beta)$ lies in the kernel of $\pi_H$ for all open normal subgroup $H\leq N_1$. Hence by the continuity of $\pi_H$ we obtain $\pi_H(x)=\sum_{n=1}^{\infty}\pi_H(-p^{k_{i_n}}s_{i_n}b_\alpha^{-j_n})\in\mathcal{O}_{\mathcal{E}}^\dagger[N_1/H,\ell]\subseteq \mathcal{R}[N_1/H,\ell]$ as we have $-p^{k_{i_n}}s_{i_n}\in\mathbb{Z}_p[N_1]$ and $\mathcal{O}_{\mathcal{E}}^\dagger$ is closed in $\mathcal{R}$.
*Step 3.* Assume finally that $j_{int}(z)=j(x)$ for some $z\in \mathcal{R}^{int}(N_1,\ell)$. Note that both $z$ and $j(x)\in\mathcal{O}_{\mathcal{E}}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\subset \mathcal{O_E}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ have a power series expansion. By the injectivity of $j_{int}$ these expansions are equal. Hence put $z=\sum_{{\bf k}\in\mathbb{Z}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}}d_{\bf k}{\bf b}^{\bf k}$ with $d_{\bf k}\in\mathbb{Z}_p$. By the definition of $\mathcal{R}^{int}(N_1,\ell)$ there exists an element $t\in T_+$ such that $\varphi_t(z)$ lies in $\mathfrak{R}_0^{int}$. This means that there exists a positive integer $K_0$ such that for all fixed $k\geq K_0$ and $\varepsilon>0$ we have $\|\varphi_t(d_{\bf k}{\bf b}^{\bf k})\|_{\rho_k}<\varepsilon$ for all but finitely many ${\bf k}\in\mathbb{Z}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}$. In particular, for any fixed $k\geq K_0$ we have $$\|\varphi_t(-p^{k_{i_n}}s_{i_n}b_\alpha^{-j_n})\|_{\rho_k}<\varepsilon$$ for all but finitely many positive integers $n$ since the sequence $j_n$ is strictly increasing by construction therefore the terms in $x=\sum_{n=1}^{\infty}p^{k_{i_n}}(\log(1+b_\beta)-s_{i_n})b_\alpha^{-j_n}$ cannot cancel each other. Now we clearly have $\|\varphi_t(b_\alpha)\|_{\rho_k}\leq\rho_k$. On the other hand, we compute (for $n>\max(k,m(\beta,t))$ large enough) $$\|\varphi_t(-p^{k_{i_n}}s_{i_n})\|_{\rho_k}=\frac{\|\varphi^{m(\beta,t)}(s_{i_n})\|_{\rho_k}}{p^{k_{i_n}}}=\frac{\|\varphi^{m(\beta,t)}(\log(1+b_\beta))\|_{\rho_k}}{p^{k_{i_n}}}=\frac{\|\log(1+b_\beta)\|_{\rho_k}}{p^{m(\beta,t)+k_{i_n}}}\ .$$ Hence we obtain $$\begin{aligned}
\varepsilon>\|\varphi_t(-p^{k_{i_n}}s_{i_n}b_\alpha^{-j_n})\|_{\rho_k}\geq\frac{\|\log(1+b_\beta)\|_{\rho_k}}{p^{m(\beta,t)+k_{i_n}}\rho_k^{j_n}}>\frac{\rho_k\|\log(1+b_\beta)\|_{\rho_k}}{p^{m(\beta,t)+k_{i_n}}\|p^{k_{i_n}}\log(1+b_\beta)\|_{\rho_n}^{\log_{\rho_n}\rho_k}}=\\
=\frac{\rho_k\|\log(1+b_\beta)\|_{\rho_k}}{p^{m(\beta,t)}}\frac{p^{k_{i_n}(\log_{\rho_n}\rho_k-1)}}{\|\log(1+b_\beta)\|_{\rho_n}^{\log_{\rho_n}\rho_k}}>\frac{\rho_k\|\log(1+b_\beta)\|_{\rho_k}}{p^{m(\beta,t)}}p^{k_{i_n}(1/2\cdot\log_{\rho_n}\rho_k-1)}\end{aligned}$$ using and . This is a contradiction as the right hand side above tends to $\infty$ as $n\to\infty$. Therefore $j(x)$ is not in the image of $j_{int}$ as claimed.
The elements of $\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ cannot be expanded as a skew Laurent series of the form $\sum_{{\bf k}\in\mathbb{Z}^{\Phi^+}}d_{\bf k}{\bf b^k}$ in general. Indeed, the sum $\sum_{n=1}^{\infty}\varphi^n(b_\beta)/p^{2^n}=\sum_{n=1}^{\infty}((b_\beta+1)^{p^n}-1)/p^{2^n}$ converges in $\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ for any simple root $\beta\neq\alpha$ but does not have a skew Laurent-series expansion as the coefficient of $b_\beta$ in its expansion would be the non-convergent sum $\sum_{n=1}^\infty p^{n-2^n}$.
We end this section by a diagram showing all the rings constructed. $$\xymatrix{
\mathcal{O_E}\ar@{^{(}->}[rrr] & & &\mathcal{O_E}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}=\Lambda_\ell(N_0)\\
\mathcal{O}_\mathcal{E}^\dagger\ar@{^{(}->}[u]\ar@{^{(}->}[r]\ar@{^{(}->}[d] & \mathfrak{R}_0^{int}(N_0,\alpha)\ar@{^{(}->}[r]\ar@{^{(}->}[d] & \mathcal{R}^{int}(N_1,\ell)\ar@{^{(}->}[r]^{j_{int}}\ar@{^{(}->}[d] & \mathcal{O}_\mathcal{E}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\ar@{^{(}->}[u]\ar@{^{(}->}[d]\\
\mathcal{E}^\dagger\ar@{^{(}->}[r]\ar@{^{(}->}[d] & \mathfrak{R}_0^{bd}(N_0,\alpha)\ar@{^{(}->}[r]\ar@{^{(}->}[d] & \mathcal{R}^{bd}(N_1,\ell)\ar@{^{(}->}[r]^{j_{int}\otimes_{\mathbb{Z}_p}\mathbb{Q}_p}\ar@{^{(}->}[d] & \mathcal{E}^{\dagger}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\ar@{^{(}->}[d]\\
\mathcal{R}\ar@{^{(}->}[r] & \mathfrak{R}_0(N_0,\alpha)\ar@{^{(}->}[r] & \mathcal{R}(N_1,\ell)\ar[r]^{j} & \mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}}$$
Here $ \mathcal{R}(N_1,\ell)$ consists of Laurent series $\sum_{\bf k}c_{\bf
k}{\bf b}^{\bf k}$ with $c_{{\bf k}}\in K$ that converge on the open annulus of the form $$\left\{\rho_2<|z_\alpha|<1,\ |z_{\beta}|\leq |z_\alpha|^r\text{ for
}\beta\in\Phi^+\setminus\{\alpha\}\right\}$$ for some $0<\rho_2<1$ and $1\leq r\in\mathbb{Z}$. The elements of $\mathfrak{R}_0(N_0,\alpha)$ are exactly those for which we can take $r=1$. Their analogous integral (resp. bounded) versions consist of those Laurent series having the same convergence condition for which $c_{{\bf k}}\in o_K$ for all ${{\bf k}}\in\mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}$ (resp. for which $\{c_{{\bf k}}\mid {{\bf k}}\in\mathbb{Z}^{\{\alpha\}}\times\mathbb{N}^{\Phi^+\setminus\{\alpha\}}\}\subset K$ bounded).
Towards an equivalence of categories for overconvergent and Robba rings {#overconv}
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Note that Propositions \[equivcat\] and \[T\_+equiv\] apply in both the cases $R=\mathcal{O_E}$ and $R=\mathcal{O}^\dagger_{\mathcal{E}}$. In both cases the category $\mathfrak{M}(R,\varphi)$ is the category of *étale* $\varphi$-modules over $R$. Moreover, by the main result of [@CC] (see also [@Ke2]), we also have an equivalence of categories between finite free étale $(\varphi,\Gamma)$-modules over $\mathcal{O}_{\mathcal{E}}^\dagger$ and finite free étale $(\varphi,\Gamma)$-modules over $\mathcal{O_E}$ given by the base change $\mathcal{O}_{\mathcal{E}}\otimes_{\mathcal{O}_{\mathcal{E}}^\dagger}\cdot$. On the other hand, $T_\ell$ acts by automorphisms on an object $D$ in $\mathfrak{M}(\mathcal{O}_{\mathcal{E}},T_+)$ and also on an object $D^\dagger$ in $\mathfrak{M}(\mathcal{O}_{\mathcal{E}}^\dagger,T_+)$. Since automorphisms correspond to automorphism in an equivalence of categories, we obtain
The functors $$\begin{aligned}
\mathcal{O}_{\mathcal{E}}\otimes_{\mathcal{O}_{\mathcal{E}}^\dagger}\cdot\colon
\mathfrak{M}(\mathcal{O}_{\mathcal{E}}^\dagger,T_+)&\rightarrow&
\mathfrak{M}(\mathcal{O}_{\mathcal{E}},T_+)\\
\cdot^{\dagger}\colon \mathfrak{M}(\mathcal{O}_{\mathcal{E}},T_+)&\rightarrow&\mathfrak{M}(\mathcal{O}_{\mathcal{E}}^\dagger,T_+)\end{aligned}$$ are quasi-inverse equivalences of categories.
Note that for the Robba ring $\mathcal{R}$ étaleness is stronger than what we assumed for a module $D_{rig}^{\dagger}$ to belong to $\mathfrak{M}(\mathcal{R},\varphi)$. The category $\mathfrak{M}(\mathcal{R},\varphi)$ is just the category of $\varphi$-modules over the Robba ring. Recall that an object $D^{\dagger}_{rig}$ in $\mathfrak{M}(\mathcal{R},\varphi)$ is *étale* (or unit-root, or pure of slope zero) whenever it comes from an overconvergent étale $\varphi$-module $D^{\dagger}$ over the ring of “overconvergent” power series $\mathcal{O}_{\mathcal{E}}^\dagger$ by base extension. We denote by $\mathfrak{M}^0(\mathcal{R},\varphi)$ the category of étale $\varphi$-modules over the Robba ring $\mathcal{R}$. We consequently define the categories $\mathfrak{M}^0(\mathcal{R},T_{+})$, $\mathfrak{M}^0(\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},\varphi)$, and $\mathfrak{M}^0(\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},T_{+})$ as the full subcategory of étale objects in the corresponding categories without superscript $0$. Note that via the equivalence of categories \[T\_+equiv\], étale objects correspond to each other. Combining this observation with the main result of [@B] leads to
\[etaleRobba\] We have a commutative diagram of equivalences of categories $$\begin{matrix}
\mathfrak{M}^0(\mathcal{R},T_+)&\leftarrow&\mathfrak{M}(\mathcal{E}^\dagger,T_+)&\rightarrow&\mathfrak{M}(\mathcal{E},T_+)\\
\downarrow&&\downarrow&&\downarrow\\
\mathfrak{M}^0(\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},T_+)&\leftarrow&\mathfrak{M}(\mathcal{E}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},T_+)&\rightarrow&\mathfrak{M}(\mathcal{E}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},T_+).
\end{matrix}$$
The left horizontal arrows are also equivalences of categories by [@B] noting that $T_\ell$ acts via automorphisms on both types of objects in the upper row.
The category $\mathfrak{M}^0(\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},T_+)$ of étale $T_+$-modules is embedded into the bigger category $\mathfrak{M}(\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},T_+)$. So we may speak of *trianguline* objects in $\mathfrak{M}^0(\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},T_+)$ as in the classical case (see for instance [@B1]). Indeed, we call an object $M_{rig}^{\dagger}$ in $\mathfrak{M}^0(\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},T_+)$ trianguline if it becomes a successive extension of objects in $\mathfrak{M}(\mathcal{R}{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]},T_+)$ of rank $1$ after a finite base extension $L\otimes_K\cdot$. It is clear that trianguline objects correspond to trianguline objects via the first vertical arrow in Corollary \[etaleRobba\].
It would be interesting to construct a noncommutative version of the “big” rings $\tilde{\bf{A}}_{\mathbb{Q}_p}$ and $\tilde{\bf{A}}_{\mathbb{Q}_p}^\dagger$ in [@Ke2] and generalize (the proofs of) Theorems 2.3.5, 2.4.5, and 2.6.2 to this noncommutative setting. For this, one would need a generalization for results in the present paper to base fields other than $\mathbb{Q}_p$.
Since we have the natural inclusions $\mathcal{O}_{\mathcal{E}}^\dagger\hookrightarrow
\mathcal{R}^{int}(N_1,\ell)\hookrightarrow
\mathcal{O}_{\mathcal{E}}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$, we have a fully faithful functor $$\Theta:=\left(\mathcal{R}^{int}(N_1,\ell)\otimes_{\mathcal{O}_{\mathcal{E}}^\dagger}\cdot\right)\circ
\left(\mathcal{O}_{\mathcal{E}}^\dagger\otimes_{\ell,\mathcal{O}_{\mathcal{E}}^\dagger[[
N_1,\ell]]}\cdot\right)\circ
\left(\mathcal{O}_{\mathcal{E}}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}\otimes_{\mathcal{R}^{int}(N_1,\ell)}\cdot\right)$$ from the category $\mathfrak{M}(\mathcal{R}^{int}(N_1,\ell),T_+)$ to itself. Whether or not it is essentially surjective (or equivalently that it is naturally isomorphic to the identity functor) is not clear. However, we have $\Theta\cong\Theta\circ\Theta$ naturally.
The faithfulness is clear since the objects in the category $\mathfrak{M}(\mathcal{R}^{int}(N_1,\ell),T_+)$ are free modules, the maps $\mathcal{O}_{\mathcal{E}}^\dagger\hookrightarrow
\mathcal{R}^{int}(N_1,\ell)\hookrightarrow
\mathcal{O}_{\mathcal{E}}^\dagger{[\hspace{-0.04cm}[}N_1,\ell{]\hspace{-0.04cm}]}$ are injective, and the functor $\mathcal{O}_{\mathcal{E}}^\dagger\otimes_{\ell,\mathcal{O}_{\mathcal{E}}^\dagger[[
N_1,\ell]]}\cdot$ in the middle is an equivalence of categories by Prop. \[T\_+equiv\]. The assertion $\Theta\cong\Theta\circ\Theta$ is also clear by Prop. \[T\_+equiv\]. For the fullyness let $f\colon\Theta(\mathcal{M}_1)\to\Theta(\mathcal{M}_2)$ be a morphism in $\mathfrak{M}(\mathcal{R}^{int}(N_1,\ell))$. Then we have $\Theta(f-\Theta(f))=0$ and by the faithfulness of $\Theta$ obtain $f=\Theta(f)$.
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abstract: 'We describe the positive cone generated by bigraded Betti diagrams of artinian modules of codimension two, whose resolutions become pure of a given type when taking total degrees. If the differences of these total degrees, $p$ and $q$, are relatively prime, the extremal rays are parametrised by order ideals in ${\mathbb N}^2$ contained in the region $px + qy < (p-1)(q-1)$. We also consider some examples concerning artinian modules of codimension three.'
address:
- |
Institusjonen for Matematik, KTH\
S-100 44 Stockholm\
Sweden
- |
Matematisk Institutt\
Johs. Brunsgt. 12\
5008 Bergen\
Norway
author:
- Mats Boij
- 'Gunnar Fl[ø]{}ystad'
title: The cone of Betti diagrams of bigraded artinian modules of codimension two
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\[section\] \[theorem\][Corollary]{} \[theorem\][Lemma]{} \[theorem\][Proposition]{} \[theorem\][Conjecture]{}
\[theorem\][Definition]{}
\[theorem\][Notation]{} \[theorem\][Remark]{} \[theorem\][Example]{}
Å[[A]{}]{}
Introduction {#introduction .unnumbered}
============
In [@EFW], D.Eisenbud, J.Weyman, and the second author gave for every sequence of integers ${{\bf d}}: d_0 < d_1 < \cdots < d_n$ a construction of pure resolutions of graded artinian modules over a polynomial ring $S = {{\Bbbk}}[x_1, \ldots, x_n]$ (char ${{\Bbbk}}= 0$) $$S(-d_0)^{\beta_0} {\leftarrow}S(-d_1)^{\beta_1}
{\leftarrow}\ldots {\leftarrow}S(-d_n)^{\beta_n}.$$ Moreover these resolutions were ${{GL}}(n)$-equivariant, and so in particular invariant for the diagonal matrices and hence ${{\mathbb Z}}^n$-graded.
In the case when $S = {{\Bbbk}}[x_1,x_2]$, the first author and J.Söderberg in [@BS Remark 3.2] gave a different construction of pure resolutions of artinian bigraded modules. It had a bigraded Betti diagram distinct from that of the equivariant resolution.
Suppose $d_1 - d_0 = 2$ and $d_2 - d_1 = 3$. The equivariant resolution has the following form where we have written the bidegrees of the generators below the terms. $$\label{IntroLigEkvi}
\underset{\scriptsize{\begin{matrix} (2,0) \\ (1,1) \\ (0,2) \end{matrix}}} {S^3} {\leftarrow}\underset{\scriptsize{\begin{matrix} (4,0) \\ (3,1) \\ (2,2) \\ (1,3) \\ (0,4) \end{matrix}}}
{S^5} {\leftarrow}\underset{\scriptsize{\begin{matrix} (4,3) \\ (3,4) \end{matrix}}} {S^2}.$$ Let $\beta_1$ be its bigraded Betti table. The resolution in [@BS] is of a quotient of a pair of monomial ideals. For the type above the resolution has the following bidegrees. $$\label{IntroLigBS}
\underset{\scriptsize{\begin{matrix} (4,0) \\ (2,2) \\ (0,4) \end{matrix}}}
{S^3} {\leftarrow}\underset{\scriptsize{\begin{matrix} (6,0) \\ (4,2) \\ (3,3) \\ (2,4) \\ (0,6)
\end{matrix}}} {S^5} {\leftarrow}\underset{\scriptsize{\begin{matrix} (6,3) \\ (3,6) \end{matrix}}} {S^2}.$$ Denote by $\beta_2$ be its Betti diagram.
This indicated that there may be many types of multigraded Betti diagrams of ${{{\mathbb Z}}}^n$-graded artinian modules of codimension $n$ whose resolutions become pure of a given type when taking total degrees. In [@Fl] the second author showed that the multigraded Betti diagram of the equivariant resolution has a fundamental position. This diagram and its twists with ${{\bf a}}\in {{\mathbb Z}}^n$ form a basis for the linear space generated by multigraded Betti diagrams of artinian ${{\mathbb Z}}^n$-graded modules whose resolutions become pure of the given type when taking total degrees. Even more natural it is to describe the positive cone generated by the multigraded Betti diagrams.
In this paper we to this in the case when $S = {{\Bbbk}}[x_1, x_2]$. Let $e_1 = d_1 - d_0$ and $e_2 = d_2 - d_1$. We describe all the extremal rays of the positive cone $P(e_1,e_2)$ generated by bigraded Betti diagrams of artinian bigraded modules of codimension two whose resolutions become pure when taking total degrees, and where the differences of these total degrees are $e_1$ and $e_2$. In the example above the two resolutions, or rather their Betti diagrams, are essentially the full story in the sense that the extremal rays in $P(2,3)$ are exactly the rays generated by $\beta_1({{\bf a}})$ and $\beta_2({{\bf a}})$ for ${{\bf a}}\in {{\mathbb Z}}^2$. To explain the general situation assume here for simplicity that $e_1$ and $e_2$ are relatively prime. Let $R(e_1,e_2)$ be the integer coordinate points in the region of the first quadrant of the coordinate plane bounded by the line $e_1 x + e_2 y < (e_1 - 1)(e_2 - 1)$. There is a partial order on ${{\mathbb N}}^2$ given by $(a_1,a_2) \leq (b_1, b_2)$ if $a_1 \leq a_2$ and $b_1 \leq b_2$, and the region $R(e_1,e_2)$ inherits this. An order ideal in $R(e_1,e_2)$ corresponds to a partition ${\lambda}$. We give a construction which to each partition ${\lambda}$ in $R(e_1,e_2)$ associates a bigraded resolution $$S^{e_2} {\leftarrow}S^{e_2 + e_1} {\leftarrow}S^{e_1}.$$ Let $\beta_{\lambda}$ be the bigraded Betti diagram of this complex. The following is our main result in the case that $e_1$ and $e_2$ are relatively prime.
The extremal rays in the cone $P(e_1,e_2)$ are the $\beta_{\lambda}({{\bf a}})$ where ${{\bf a}}$ varies over ${{\mathbb Z}}^2$ and ${\lambda}$ ranges over partitions contained in the region $R(e_1,e_2)$.
The general case is formulated in Theorems \[PosconeTheMain\] and \[EksisTheMain\]. In the region $R(e_1,e_2)$ there are two distinguished partitions, the maximal one and the empty one. It turns out that the maximal one corresponds to the equivariant complex and the empty one corresponds to the bigraded resolution of a quotient of monomial ideals constructed in [@BS].
The organisation of the paper is as follows. Section 1 contains preliminaries. First we give the multigraded Herzog-Kühl equations which give strong restrictions on Betti diagrams of multigraded artinian modules. We recall the equivariant resolution, and the result of [@Fl] that its twists generate the linear space of multigraded Betti diagrams of artinian ${{{\mathbb Z}}}^n$-graded modules of codimension $n$ whose resolution becomes pure when taking total degrees. This give us a very simple alternative description of the positive cone $P(e_1,e_2)$. This is used in Section 2 where we show that the extremal rays of the positive cone $P(e_1,e_2)$ are generated by the Betti diagrams $\beta_{\lambda}({{\bf a}})$ for ${{\bf a}}\in {{\mathbb Z}}^2$, provided these diagrams really come from resolutions. And that such resolutions really exist is established in Section 3. In Section 4 we briefly discuss the positive cone in the case of three variables, providing an example.
Preliminaries
=============
Let $S = {{\Bbbk}}[x_1, \ldots, x_n]$ be the polynomial ring over a field ${{\Bbbk}}$. We shall study ${{\mathbb Z}}^ n$-graded free resolutions of artinian ${{\mathbb Z}}^ n$-graded $S$-modules $$F_0 {\leftarrow}F_1 {\leftarrow}\cdots {\leftarrow}F_n.$$ For a multidegree ${{\bf a}}= (a_1, a_2, \ldots, a_n)$ in ${{\mathbb Z}}^ n$ let $|{{\bf a}}| = \sum a_i$ be its total degree. We shall be interested in the case that these resolutions become pure resolutions if we make them singly graded by taking total degrees. That is there is a sequence $d_0 < d_1 < \cdots < d_n$ such that $$F_i = \oplus_{|{{\bf a}}| = d_i} S(- {{\bf a}})^ {\beta_{i, {{\bf a}}}}.$$
Betti diagrams and the multigraded Herzog-Kühl equations
--------------------------------------------------------
The [*multigraded Betti diagram*]{} of such a resolution is the element $$\{\beta_{i,{{\bf a}}}\}_{\scriptsize \underset{}
{\begin{matrix} i=0, \ldots, n \\ {{\bf a}}\in {{\mathbb Z}}^n \end{matrix} }}
\in \oplus_{{{\mathbb Z}}^n} {{\mathbb N}}^{n+1}.$$
A way of representing a multigraded Betti table which will be very convenient for us is to represent $\beta = \{ \beta_{i, {{\bf a}}} \}$ where $i = 0,\ldots,n$ and ${{\bf a}}\in {{\mathbb Z}}^{n}$ by Laurent polynomials $$B_i(t) = \sum_{{{\bf a}}\in {{\mathbb Z}}^n} \beta_{i,{{\bf a}}} \cdot t^{{\bf a}}.$$ We thus get an $(n+1)$-tuple of Laurent polynomials $$B = (B_0, B_1, \ldots, B_n).$$ Also the module $\oplus_{{\bf a}}S(-{{\bf a}})^{\beta_{i,{{\bf a}}}}$ may be conveniently denoted as $S.B_i$.
Let $e_i = d_i - d_{i-1}$, so we get the differences ${{\bf e}}=
(e_1, \ldots, e_n)$. Now let $L({{\bf e}})$ be the linear subspace of $\oplus_{{{\bf a}}\in {{\mathbb Z}}^n}
{\mathbb Q}^{n+1}$ generated by multigraded Betti diagrams of ${{\mathbb Z}}^n$-graded artinian $S$-modules whose resolutions become pure when taking total degrees, and where the difference sequence of these total degrees is ${{\bf e}}$. Similarly let $P({{\bf e}})$ be the positive cone in $\oplus_{{{\bf a}}\in {{\mathbb Z}}^n}
{\mathbb Q}^{n+1}$ generated by such Betti diagrams.
There are some natural restrictions on $L({{\bf e}})$ coming from the multigraded Herzog-Kühl equations. If the resolution resolves the artinian module $M$, the multigraded Hilbert series of $M$ is the polynomial $$h_M(t) = \frac{\sum_{i,{{\bf a}}} (-1)^i \beta_{i, {{\bf a}}} \cdot t^ {{{\bf a}}}}
{\Pi_{k=1}^ n (1-t_i)},$$ which gives $$\label{SetLigBeta}
\sum_{i,{{\bf a}}} (-1)^ i \beta_{i, {{\bf a}}} t^ {{{\bf a}}} = h_M(t) \cdot \Pi_{k=1}^n(1-t_i).$$ For each multigrade ${{\bf a}}\in {{\mathbb Z}}^ {n}$ and integer $k = 1, \ldots, n$, let the projection $\pi_k({{\bf a}})$ be $(a_1,\ldots, \hat{a}_k, \ldots, a_n)$, the $n-1$-tuple where we omit $a_k$.
Now we have the multigraded analogs of the Herzog-Kühl (HK) equations. We obtain these by setting $t_k = 1$ in (\[SetLigBeta\]) for each $k$. This gives for every $\hat{{{\bf a}}}$ in ${{\mathbb Z}}^{n-1}$ and $k = 1, \ldots, n$ an equation $$\label{SetLigHK}
\sum_{i,\pi_k({{\bf a}})= \hat{{{\bf a}}}} (-1)^i \beta_{i, {{\bf a}}} = 0.$$
Let $L^\prime({{\bf e}})$ be the linear space of elements in $\oplus_{{{\bf a}}\in {{\mathbb Z}}^{n}}
{\mathbb Q}^{n+1}$ which fulfil the multigraded HK-equations above, and which become pure diagrams when taking total degrees with the difference sequence of these total degrees equal to ${{\bf e}}$. Also let $P^\prime({{\bf e}})$ be the cone in $L^\prime({{\bf e}})$ consisting of the elements with nonnegative coordinates. There are natural injections $L({{\bf e}}) {\rightarrow}L^\prime({{\bf e}})$ and $P({{\bf e}}) {\rightarrow}P^\prime({{\bf e}})$. In [@Fl] the second author showed that the first injection is an isomorphism and moreover gave an explicit basis for $L({{\bf e}})$ which we now describe.
The equivariant resolution
--------------------------
In [@EFW] the second author together with D.Eisenbud and J.Weyman constructed ${{GL}}(n)$-equivariant pure resolutions of artinian modules. For a partition $\lambda = ({\lambda}_1, \ldots,
{\lambda}_n)$ let $S_\lambda$ be the associated Schur module, it is an irreducible representation of ${{GL}}(n)$ (see for instance [@FuH]). The action of the diagonal matrices in ${{GL}}(n)$ gives a decomposition of $S_\lambda$ as a ${{\mathbb Z}}^n$-graded vector space. The basis elements are given by semi-standard Young tableau of shape ${\lambda}$ with entries from $1,2, \ldots, n$. All the nonzero graded pieces in this decomposition have total degree $|{\lambda}| = \sum_{i=1}^n {\lambda}_i$. The free module $S {\otimes}_k S_{\lambda}$ then becomes a free multigraded module where the generators all have total degree $|{\lambda}|$.
Now given the difference vector ${{\bf e}}$, let $${\lambda}_i = \Sigma_{j = i+1}^n e_j - 1$$ and define a sequence of partitions for $i=0, \ldots, n$ by $$\alpha({{\bf e}},i) = ({\lambda}_1 + e_1, {\lambda}_2 + e_2, \ldots, {\lambda}_i + e_i, {\lambda}_{i+1},
\ldots, {\lambda}_n).$$ The construction in [@EFW] then gives a ${{GL}}(n)$-equivariant resolution $$\label{SetLigEe}
E({{\bf e}}) : S {\otimes}_k S_{\alpha({{\bf e}},0)} {\leftarrow}S {\otimes}_k S_{\alpha({{\bf e}},1)} {\leftarrow}\cdots {\leftarrow}S {\otimes}_k S_{\alpha({{\bf e}},n)}$$ of an artinian $S$-module.
In the case of two variables $S = {{\Bbbk}}[x_1, x_2]$ the resolution takes the form $$\label{SetLigEqui2}
E(e_1,e_2): S {\otimes}_k S_{e_2 - 1,0} {\leftarrow}S {\otimes}_k S_{e_1 + e_2 -1, 0}
{\leftarrow}S {\otimes}_k S_{e_1 + e_2 - 1, e_2}.$$
The linear space of Betti diagrams of multigraded artinian modules
------------------------------------------------------------------
For a multigraded Betti diagram $\beta = \{ \beta_{i, {{\bf a}}} \}$ and a multidegree ${{\bf t}}$ in ${{\mathbb Z}}^ {n}$, we get the twisted Betti diagram $\beta(-{{\bf t}})$ which in homological degree $i$ and multidegree ${{\bf a}}$ is given by $\beta_{i,{{\bf a}}-{{\bf t}}}$. If ${F_{{\displaystyle \cdot}}}$ is a resolution with Betti diagram $\beta$, then ${F_{{\displaystyle \cdot}}}(-{{\bf t}})$ is a resolution with Betti diagram $\beta(-{{\bf t}})$.
Also let $F_r : S {\rightarrow}S$ be the map sending $x_i \mapsto x_i^r$. Denote by $S^{(r)}$ the ring $S$ with the $S$-module structure given by $F_r$. Given any complex ${F_{{\displaystyle \cdot}}}$ we may tensor it with $- {\otimes}_S S^{(r)}$ and get a complex we denote by ${F_{{\displaystyle \cdot}}}^{(r)}$. Note that if ${F_{{\displaystyle \cdot}}}$ is pure with degrees ${{\bf d}}$, then ${F_{{\displaystyle \cdot}}}^{(r)}$ is pure with degrees $r \cdot {{\bf d}}$.
In [@Fl] we showed the following.
\[LinbettiTheMain\] Let $m = \gcd(e_1, \ldots, e_n)$ and let ${{\bf e}}= m \cdot {{\bf e}}^\prime$. The space $L({{\bf e}})$ is equal to the space $L^\prime({{\bf e}})$ of diagrams fulfilling the HK-equations, and the $\beta_{E({{\bf e}}^\prime)^{(m)}}({{\bf a}})$ where ${{\bf a}}$ varies over ${{\mathbb Z}}^n$, form a basis for $L({{\bf e}})$.
Moreover if $E^\prime$ is another resolution such that the $\beta_{E^\prime}({{\bf a}})$ form a basis, then $\beta_{E^\prime}$ is an integer multiple of $\beta_{E({{\bf e}}^\prime)^{(m)}}({{\bf a}})$ for some ${{\bf a}}$.
This may also be formulated in terms of the associated $(n+1)$-tuple of Betti polynomials.
\[SetCorLinpoly\] Let $s = (s_0, \ldots, s_n)$ be the $(n+1)$-tuple of Betti polynomials of $E({{\bf e}}^\prime)^{(m)}$. If $B = (B_0, \ldots, B_n)$ is any $(n+1)$-tuple of Betti polynomials of an artinian ${{\mathbb Z}}^n$-graded module whose resolution becomes pure when taking total degrees and with difference vector ${{\bf e}}$ of the total degrees, then $B = p \cdot s$ for some homogeneous Laurent polynomial $p$.
The linear space in the case of two variables {#SetSubsecLinto}
---------------------------------------------
Now assume $S = {{\Bbbk}}[x_1, x_2]$. Let $\xi_d(t,u) = t^{d-1} + t^{d-2}u + \cdots + u^{d-1}$ be the cyclotomic polynomial. The first and last Betti polynomials of the equivariant resolution (\[SetLigEqui2\]) are then respectively $$\xi_{e_2}(t,u), \quad (tu)^{e_2} \xi_{e_1}(t,u)$$ and the middle Betti polynomial is $$\label{SetLigXi}
\xi_{e_1 + e_2} = t^{e_2}\xi_{e_1}(t,u) + u^{e_1}\xi_{e_2}(t,u) =
u^{e_2}\xi_{e_1}(t,u) + t^{e_1} \xi_{e_2}(t,u).$$ By Corollary \[SetCorLinpoly\] the space $L(e_1, e_2)$ may now be described as follows.
\[SetLemB\] Let $e_1 = {m}{q}$ and $e_2 = {m}{p}$ where $m$ is the greatest common divisor of $e_1$ and $e_2$. A triple of homogeneous Laurent polynomials $B_0, B_1, B_2$ whose degrees have $e_1$ and $e_2$ as differences, is in $L(e_1, e_2)$ if and only if the following two equations hold: $$\begin{aligned}
\label{SetLigB02}
B_2(t,u) \cdot \xi_{p}(t^{m}, u^{m}) &=& (tu)^{{m}{p}}
B_0(t,u) \cdot
\xi_{{q}}(t^{m}, u^{m}), \\ \label{SetLigB03}
B_1(t,u) &=& u^{-{p}{m}} B_2(t,u) + u^{{q}{m}} B_0(t,u) \\ \notag
&=& t^{-{p}{m}} B_2(t,u) + t^{{q}{m}} B_0(t,u) .\end{aligned}$$
By Corollary \[SetCorLinpoly\] we have $$\notag (B_0, B_1, B_2) = f(t,u) \cdot
(\xi_{q}(t^m, u^m), \xi_{{p}+ {q}}(t^m, u^m), (tu)^{{p}+ {q}}
\xi_{{q}}(t^m, u^m)).$$ This gives (\[SetLigB02\]). Also (\[SetLigB03\]) follows by (\[SetLigXi\]). Conversely, if (\[SetLigB02\]) and (\[SetLigB03\]) hold, we may deduce that the equation above holds, so $(B_0,B_1,B_2)$ is in $L(e_1, e_2)$.
For a homogeneous Laurent polynomial $f(t,u)$ denote by $f^{dh}(t)$ its dehomogenisation with respect to $u$. If we now dehomogenise equation (\[SetLigB02\]) we get an equation $$B_2^{dh}/t^{{p}{m}} \cdot \xi_{p}(t^{m}) = B_0^{dh} \cdot \xi_{q}(t^{m}).$$ Each of the first factors are uniquely determined by the other, and if the triple comes from an actual complex, the coefficients are non-negative.
With some abuse of notation we also identify the cone $P^\prime = P^\prime(e_1, e_2)$ with the positive cone of pairs of Laurent polynomials $(A(t),B(t))$ in one variable $t$ and with non-negative coefficients, such that $$B(t) \xi_{p}(t^{m}) = A(t) \xi_{q}(t^{m}).$$ We shall in the next section describe the cone $P^\prime$ completely. Recall that we have an injective map $P(e_1,e_2) {\rightarrow}P^\prime(e_1,e_2)$. In Section \[EksisSec\] we show that this map is an isomorphism.
The positive cone of bigraded Betti diagrams {#PosSec}
============================================
In this section we describe completely the positive cone $P^\prime(e_1,e_2)$ of diagrams fulfilling the HK-equations (\[SetLigHK\]). We shall show that there is a finite number of diagrams $\beta_\lambda$ parametrised by certain partitions ${\lambda}$ such that extremal rays in the positive cone are the one-dimensional rays generated by $\beta_{\lambda}({{\bf a}})$ for ${{\bf a}}\in {{\mathbb Z}}$.
In the following we let $e_1 = {m}{q}$ and $e_2 = {m}{p}$ where ${m}$ is the greatest common divisor of $e_1$ and $e_2$.
Partitions
----------
Let ${\mathbb N}^2$ have the partial ordering where $(a_1,a_1) \leq (b_1,b_2)$ if $a_1 \leq b_1$ and $a_2 \leq b_2$. An order ideal $T$ in ${\mathbb N}^2$ (a set closed under taking smaller elements) gives rise to two partitions. The first is given by $$\lambda_j = 1 + \max \{ i \,| \, (i,j) \in T \}, \,\, j \geq 0.$$ The second is the dual partition $$\mu_i = 1 + \max \{ j \, | \, (i,j) \in T \}, \,\, i \geq 0.$$ (If for a given $j$ no $(i,j)$ is in $T$, we set $\lambda_j =0$ and correspondingly for $\mu_i$.) Note that $\lambda$ and $\mu$ are dual partitions. So $\mu_i$ is the cardinality of $\{ j\,| \, \lambda_j > i \}$.
We shall be interested in order ideals $T$ which are contained in the region $R({p},{q})$ in the first quadrant bounded by the following strict inequality $$\notag
{p}x+{q}y < ({p}-1)({q}-1).$$
Let the order ideal $T$ correspond to the partition $\lambda$. Then $T$ is contained in the region above if and only if every $a{q}-{p}\lambda_{{p}-1-a}$ is nonnegative for $0 \leq a < {p}$. Correspondingly for the dual partition $\mu$.
First note that $a{q}- {p}\lambda_{{p}-1-a} \geq 0$ if and only if $$({p}-1-a){q}+ (\lambda_{{p}-1-a}-1){p}\leq {p}{q}-{p}-{q}.$$ Assume $0 \leq a < p$. If $T$ is contained in $R(p,q)$ then if $\lambda_{p-1-a} \geq 1$ it fulfils the second equation above and therefore the first. If $\lambda_{p-1-1} = 0$ the first equation is also fulfilled. Suppose now that $T$ fulfils the first equation. Then when $\lambda_{p-1-a} \geq1$ the point $(p-1-a, \lambda_{p-1-a} - 1)$ is in $R(p,q)$, so $T$ is contained in $R(p,q)$.
The following easy lemma will be useful.
\[PosLemHopp\] Let $P(t) = \sum c_i t^i$ be a polynomial with positive coefficients. Write $P(t) \xi_d(t) = \sum_{j \in {{\mathbb Z}}} \alpha_jt^j$. Then $\alpha_j - \alpha_{j-1} = c_j - c_{j-d}$.
This is clear from $\alpha_j = \sum_{i = j-d+1}^{j} c_i$.
The following result will essentially describe the extremal rays.
Suppose ${p}$ and ${q}$ are relatively prime and let $T$ be an order ideal in $R({p},{q})$. Write $$A_T(t) = \sum_{a=0}^{{p}-1} t^{a{q}-{p}\lambda_{{p}-1-a}},
\quad B_T(t) = \sum_{a=0}^{{q}-1} t^{a{p}-{q}\mu_{{q}-1-a}}.$$ Then $$A_T(t) \xi_{q}(t) = B_T(t) \xi_{p}(t).$$
Note that since ${p}$ and ${q}$ are relatively prime, the coefficient of each power $t^j$ in $A_T$ or $B_T$ is $0$ or $1$. Writing $\sum \alpha_j t^j$ for the product $A_T(t) \xi_{q}(t)$ we see that when $\alpha_j > \alpha_{j-1}$ we have $\alpha_j = \alpha_{j-1} + 1$. We shall show that the indices $j$ for which this happens are exactly when $j = 0$ or $j = {p}{q}-{p}-{q}-{q}u-{p}v$ where $(u,v)$ is a maximal element in the poset $T$, i.e. $(u,v)$ is in $T$, but neither $(u+1,v)$ nor $(u,v+1)$ is in $T$. Since the analog holds for the product $B_T(t) \xi_{p}(t)$, these products must increase exactly at the same indices. An analog argument also show that they decrease at exactly the same indices, namely $\alpha_j < \alpha_{j-1}$ iff $j = {p}{q}-{q}u -{p}v$ where $(u,v)$ is not in $T$ but $(u-1,v)$ and $(u,v-1)$ are either in $T$ or have $-1$ as a coordinate. Hence the products are equal.
Now $\alpha_j > \alpha_{j-1}$ when $j = a{q}- {p}\lambda_{{p}-1-a}$ for some $a$ but $(a-1){q}- {p}\lambda_{{p}-1-a}$ is not a power in $A(t)$. Thus either $a = 0$ or $\lambda_{{p}-a}
< \lambda_{{p}-1-a}$. But this means that $j = 0$ or $(u,v) = (\lambda_{{p}-1-a} - 1,{p}-1-a)$ is a maximal element in $T$. We easily compute that $$j = a{q}-{p}\lambda_{{p}-1-a} = {p}{q}-{p}-{q}-{q}u-{p}v.$$
\[PosRemMin\] The empty poset $T = \emptyset$ corresponds to the polynomials $$\begin{aligned}
\notag A_{\emptyset}(t) & = & \xi_{p}(t^{q}) \\
\notag B_{\emptyset}(t) & = & \xi_{q}(t^{p}).\end{aligned}$$ Via the correspondence at the end of Subsection \[SetSubsecLinto\] these corresponds to the Betti diagram of a resolution of an artin module. This is the module described in [@BS Remark 3.2] which is the quotient $I/J$ of two monomial ideals in $k[x,y]$: the ideal $I = (x^{({p}-1){q}}, x^{({p}-2){q}}y^{q},
\ldots, y^{({p}-1){q}})$ and the ideal $J = (x^{{p}{q}}, x^{{p}({q}-1)}y^{p},
\ldots, y^{{p}{q}})$
\[PosRemMax\] There is also a maximal order ideal $\hat T$ in the region $R({p},{q})$ and this corresponds to the polynomials $$\begin{aligned}
\notag A_{\hat T}(t) & = & \xi_{p}(t) \\
\notag B_{\hat T}(t) & = & \xi_{q}(t)\end{aligned}$$ which again via the correspondence at the end of Subsection \[SetSubsecLinto\] corresponds to the Betti diagrams of the $GL(2)$-equivariant resolutions $E(p,q)$ constructed in [@EFW].
Decomposing
-----------
Now any polynomial $A(t)$ may be written $$A(t) = \sum_{a=0}^{{p}-1} \sum_{b \in {{\mathbb Z}}} \alpha_{a,b} t^{a{q}-b{p}}.$$ For each $a$ let $\lambda_{{p}-1-a}$ be the maximum of the set $\{ b \, | \, \alpha_{a,b} \neq 0 \}$. We may then write $$A(t) = A_{\min}(t) + A_+(t)$$ where $$A_{\min}(t) = \sum_{a=0}^{{p}-1}
\alpha_{a,\lambda_{{p}-1-a}} t^{a{q}-\lambda_{{p}-1-a}{p}}.$$ Correspondingly we may write $$B(t) = \sum_{a=0}^{{q}-1} \sum_{b \in {{\mathbb Z}}} \beta_{a,b} t^{a{p}-b{q}}.$$ For each $a$ let $\mu_{{q}-1-a}$ be the maximum of the set $\{ b \, | \, \beta_{a,b} \neq 0 \}$. We may then write $$B(t) = B_{\min}(t) + B_+(t)$$ where $$B_{\min}(t) = \sum_{a=0}^{{q}-1} \beta_{a,\mu_{{q}-1-a}} t^{a{p}-\mu_{{q}-1-a}{p}}.$$
\[PosconePropABmin\] Let ${p}$ and ${q}$ be relatively prime. Assume $A(t)$ and $B(t)$ are polynomials with nonnegative coefficients and nonzero constant terms. Suppose $$A(t) \xi_{q}(t) = B(t) \xi_{p}(t).$$ Let $\lambda$ and $\mu$ be the sequences corresponding to $A_{\min}(t)$ and $B_{\min}(t)$. Then these sequences are partitions which are dual.
Write the product above as $\sum \alpha_j t^j$. Let $0 \leq b < {p}-1$ and assume $b{q}- {p}\lambda$ occurs as a power in $A_{\min}(t)$, so $\lambda = \lambda_{{p}-1-b}$. We want to show that $\lambda_{{p}-b-2} \geq \lambda_{{p}-1-b}$. If $(b+1){q}- {p}\lambda$ occurs as a power in $A(t)$ then clearly $\lambda_{{p}-2-b} \geq \lambda = \lambda_{{p}-1-b}$. So assume $(b+1){q}- {p}\lambda$ does not occur in $A(t)$. By Lemma \[PosLemHopp\] applied to $A(t) \xi_q(t)$: $$\alpha_{(b+1){q}-{p}\lambda} < \alpha_{(b+1){q}-{p}\lambda-1}$$ so $(b+1){q}-{p}(\lambda+1)$ is a power in $B(t)$. We may now write $$(b+1){q}-{p}(\lambda+1) = ({q}-\lambda-1){p}- {q}({p}-b-1).$$ There will then be an $a^\prime \leq {q}-\lambda-1$ such that $a^\prime {p}- {q}({p}-b-1)$ is in $B(t)$ but $(a^\prime -1) {p}- {q}({p}-b-1)$ is not. By Lemma \[PosLemHopp\] applied to $B(t) \xi_p(t)$: $$\alpha_{a^\prime {p}- {q}({p}-b-1)} >
\alpha_{a^\prime {p}- {q}({p}-b-1)-1}.$$ Now we may write $$a^\prime {p}- {q}({p}-b-1) = (b+1){q}- {p}({q}-a^\prime)$$ and recall that ${q}-a^\prime \geq \lambda + 1$. Again by Lemma \[PosLemHopp\] we get that the number in this equation will occur as a power in $A(t)$. But this means that $$\lambda_{{p}-2-b} \geq {q}- a^\prime > \lambda
= \lambda_{{p}-1-b}.$$ Since $A(t)$ and equivalently $B(t)$ has nonzero constant term, we have $\lambda_{{p}-1} = 0$ so we get a partition $\lambda$.
An analog argument gives that the sequence of $\mu_i$’s also form a partition.
Now let $T$ be the order ideal corresponding to $\lambda$ and $T^\prime$ the order ideal corresponding to $\mu$. We show that they are equal and so $\lambda$ and $\mu$ will be dual partitions. Suppose $\lambda_{{p}-b-1} < \lambda_{{p}-b-2}$. Then $b{q}- {p}\lambda_{p-b-2}$ is not in $A(t)$. By Lemma \[PosLemHopp\] $$\alpha_{(b+1){q}-{p}\lambda_{{p}-b-2}} >
\alpha_{(b+1){q}-{p}\lambda_{{p}-b-1}-1}.$$ And this implies again by Lemma \[PosLemHopp\] that $(b+1){q}-{p}\lambda_{{p}-b-2}$ occurs as a power in $B(t)$, Rewriting, this is $({q}-1 - (\lambda_{{p}-b-2}-1)){p}- {q}({p}-b-1)$. And this means that $(\lambda_{{p}-b-2}-1, r)$ is in $T^\prime$ for some $r \geq {p}-b-2$. The upshot is that $T^\prime$ contains $T$. Analogously we could show the opposite inclusion so these are in fact equal.
\[PosconeCorABmin\] The polynomials $A(t) = \sum_{T,i} \gamma_{T,i} t^{c_{T,i}} A_T(t)$ and $B(t) = \sum_{T,i} \gamma_{T,i} t^{c_{T,i}} B_T(t)$ where the sum is over order ideals $T$ in $R(p,q)$ and a running index $i$ for each $T$.
Let $\alpha$ be the minimal positive coefficient of $A_{\min}(t)$ and $B_{\min}(t)$ and suppose these correspond to the order ideal $T$. Then we can subtract off $\alpha A_T(t)$ from $A(t)$ getting $A^\prime(t)$ and similarly subtract off $\alpha B_T(t)$ and get $B^\prime(t)$. Then also $$A^\prime(t) \xi_{q}(t) = B^\prime(t) \xi_{p}(t)$$ and we may proceed inductively, since then new polynomials have no more terms than the original ones, and one of them strictly less.
From this we obtain our goal of describing the extremal rays of the cone $P^\prime$ described at the end of Subsection \[SetSubsecLinto\].
\[PosconeTheMain\] Let $e_1 = m{q}$ and $e_2 = m {p}$ where ${p}$ and ${q}$ are relatively prime. The rays generated by $(t^cA_T(t^m), t^cB_T(t^m))$ where $T$ is an order ideal in $R({p},{q})$ and $c \in {{\mathbb Z}}$, are the extremal rays in the cone $P^\prime(e_1, e_2)$. In particular any element in this cone may be written as a positive linear combination of these elements.
In the case $m=1$ this follows immediately from Proposition \[PosconeCorABmin\]. Suppose then $m > 1$. If we have $$A(t) \xi_{q}(t^m) = B(t) \xi_{p}(t^m)$$ we may write $A(t) = \sum_{i = 0}^{m-1} t^i A_i(t^m)$ and correspondingly for $B(t)$. We must then have the equations $$A_i(t^m) \xi_{q}(t^m) = B_i(t^m) \xi_{p}(t^m)$$ for each $i$. By Corollary \[PosconeCorABmin\] we may then conclude.
Such a positive linear combination is in general not unique.
We see that the extremal rays fall into classes, one for each order ideal $T$ in $R({p},{q})$. These form a poset with a minimal element $T = \emptyset$ and a maximal element $\hat T$. In Remarks \[PosRemMin\] and \[PosRemMax\] we showed that these correspond to Betti diagrams of well known resolutions.
Existence of resolutions {#EksisSec}
========================
We will now show that for any extremal ray in $P^\prime (e_1,e_2)$ there is a resolution whose Betti diagram is in this extremal ray. This will show that $P^\prime(e_1,e_2) = P(e_1,e_2)$.
Given an order ideal $T$ in $R(p,q)$ where $p$ and $q$ are relatively prime. If $e_1 = mp$ and $e_2 = mq$, we have the two polynomials $A_T(t^m)$ and $B_T(t^m)$. Homogenising these we may construct an associated triple $B_0,B_1,B_2$ fulfilling the equations of Lemma \[SetLemB\], with positive integer coefficients. These lie on an extremal ray in $P^\prime(e_1,e_1)$. Note that in $B_0$ and $B_2$ each monomial occurs with coefficient $0$ or $1$ and similarly for $B_2$. We may therefore apply the following proposition whose proof will occupy this section.
\[EksisPropTreB\] Let $(B_0,B_1,B_2)$ be a triple of homogeneous Laurent polynomials of increasing degrees, fulfilling the HK-equations (\[SetLigHK\]). If the coefficients of each monomial of $B_0$ and $B_2$ is $0$ or $1$, there is a resolution $$\label{EksiLigCx}
S.B_0(t,u) {\overset{\tiny{\alpha}}{\longleftarrow}} S.B_1(t,u) {\overset{\tiny{\beta}}{\longleftarrow}} S.B_2(t,u)$$ of an artinian $S$-module.
As a consequence we get the following.
\[EksisTheMain\] Let $e_1 = m {q}$ and $e_2 = m {p}$ where $p$ and $q$ are relatively prime. Let $(B_0, B_1, B_2)$ be the triple of homogeneous Laurent polynomials associated to an order ideal $T$ in $R(p,q)$, with $t^m$ as argument. Then this is a triple of Betti polynomials associated to a bigraded artinian module. Hence the cone $P(e_1, e_2) = P^\prime(e_1,e_2)$.
Proposition \[EksisPropTreB\] holds for any $B_0$ and $B_2$ with nonnegative integer coefficients. But for ease of demonstration we make the above assumptions.
In the case of three variables it is not true that $P(e_1,e_2,e_2)$ is equal to $P^\prime(e_1,e_2,e_3)$. We provide an example where this is not so in the last section.
We shall prove the above proposition towards the end of this section. But the following outlines what we need to show. Since $\ker \alpha$ is a free module, $\ker \alpha/ {\text{im}\,}\beta$ will be either $0$ or nonzero of codimension one or zero. But the latter is equivalent to ${\text{coker}\,}\beta^{\vee}$ being of codimension one or zero. Hence we need to show the following.
- ${\text{coker}\,}\alpha$ is of codimension two.
- ${\text{coker}\,}\beta^\vee$ is of codimension two.
- The composition $\alpha \circ \beta = 0$.
First we have the following.
\[EksisLemBigrad\] Given a bidegree $(i,j)$ with $i+j \geq \deg B_2(t,u)-1$. Then the dimension of the bigraded part $S.B_1(t,u)_{i,j}$ is the sum of the dimensions of $S.B_0(t,u)_{i,j}$ and $S.B_2(t,u)_{i,j}$.
The bigraded Hilbert function is $$h(t,u) = \frac{\sum_{i,{{\bf a}}} (-1)^i \beta_{i, {{\bf a}}} \cdot t^{a_1}t^{a_2}}
{(1-t)(1-u)}$$ for some polynomial $h$. Writing $h(t,u)$ as $\sum \alpha_{i,j} t^iu^j$, the coefficient $\alpha_{i,j}$ will be the alternating sum of the dimensions of the $S.B_p(t,u)_{i,j}$. We will show that $\alpha_{i,j} = 0$ for $i+j \geq \deg B_2(t,u)-1$. But if such a coefficient is nonzero, the pair $(i+1,j+1)$ must occur as a power in the numerator in the fraction above. But this implies in turn that $i+j+2$ is less or equal to the degree of $B_2(t,u)$.
To facilitate the discussion we now introduce some notation. Let $s, e: [1, \ldots, n] {\rightarrow}[1, \ldots, m]$ be two weakly increasing functions such that $s(i) \leq e(i)$. The subset $D = \{ (i,j) \, | \, s(i) \leq j \leq e(i) \}$ of $[1, \ldots, n] \times [1, \ldots, m]$ is a [*thick diagonal*]{}. We then write $s = s_D$ and $e = e_D$. If $s(1) = 1$ and $e(n) = m$ and $s$ and $e$ are strictly increasing we call $D$ a [*strict*]{} thick diagonal. If $s$ is only strictly increasing as soon as $s(i) > 1$ and $e$ is only strictly increasing as long as $e(i) < m$, we call $D$ [*semi-strict*]{}.
Let $B_0(1,1) = p$ and $B_2(1,1) = q$ and write $B_0(t,u) =
\sum_{i = 1}^{p} t^{a^1_i} u^{a^2_i}$ where $\{a^1_i\}$ is strictly increasing and $\{a^2_i\}$ is strictly decreasing. Similarly for $B_2(t,u)$ with pairs $(c^1_k, c^2_k)$ and for $B_1(t,u)$ with pairs $(b^1_j, b^2_j)$ but now with the $\{b^1_j\}$ only weakly increasing and the $\{b^2_j\}$ only weakly decreasing.
We may now note that the positions where $\alpha$ may have nonzero entries, i.e. those pairs $(i,j)$ such that $(a^1_i, a^2_i) \leq (b^1_j, b^2_j)$, form a thick diagonal $D_\alpha$ of $[1, \ldots, p] \times [1, \ldots, p+q]$. It has no zero rows because of the HK-equations (\[SetLigHK\]) : for each $(a^1_i,a^2_i)$ there is a $(b^1_j, b^2_j)$ with $a^1_i = b^1_j$. Similarly we have a thick diagonal $D_{\beta^\vee}$ in $[1,\ldots,q] \times [1, \ldots, p+q]$.
\[EksisLemSe\] a. $s_{D_\alpha}(i) = j$ if and only if $j$ is the smallest index such that $a^1_i = b^1_j$.
b\. $e_{D_\alpha}(i) = j$ if and only if $j$ is the largest index for which $a^2_i = b^2_j$.
The analog result holds for $D_{\beta^\vee}$.
Let $s_{D_\alpha}(i) = j$ and let $\tilde{j}$ be the smallest index such that $a^1_i = b^1_{\tilde{j}}$. Such an index exists by the HK-equations. Clearly $j \leq \tilde{j}$. But if $j < \tilde{j}$ then $b^1_{j} < b^1_{\tilde{j}}$ and so we could not have $(a^1_i, a^2_i) \leq (b^1_j, b^2_j)$ The other arguments are analogous.
\[EksisCorStrict\] The thick diagonal $D_\alpha$ is strict. Similarly $D_{\beta^\vee}$ is strict.
Since the $a^1_i$ are strictly increasing, we get that $s_{D_\alpha}$ is strictly increasing. We thus need to show that $s_{D_\alpha}(1) = 1$. Let $s_{D_\alpha}(1) = j$. Suppose $j$ is not $1$. Then $b^1_1 < a^1_1$. By the HK-equations there will be $(c^1_k, c^2_k)$ with $c^1_k = b^1_1$. But then again there will be a $(b^1_{j^\prime}, b^2_{j^\prime})$ with $b^2_{j^\prime} = c^2_k$ and this would have $b^1_{j^\prime} < c^1_k = b^1_1$ which is impossible. Thus $s_{D_\alpha}(1) = j = 1$.
\[EksisLemRekt\] Let $D$ be a semi-strict diagonal of $[1, \ldots, n] \times [1, \ldots, n+1]$ with $e_D(1) > 1$ and $s_D(n) < n+1$. Let $A$ be a general matrix of type $D$. Then there is a vector in the null space of $A$ with nonzero first and last coordinates.
If we omit the first column we get an $n \times n$-matrix of semi-strict diagonal type. But a general such matrix is easily seen to be non-singular. Hence a null vector must have nonzero first coordinate. Similarly for the last coordinate.
\[EksisLemCoker\] If $\alpha$ is nonzero in positions $(i, s_{D_\alpha}(i))$ and $(i, e_{D_\alpha}(i))$ for $i = 1, \ldots, p$, then ${\text{coker}\,}\alpha $ has codimension two. Similarly for the map $\beta^\vee$.
By Lemma \[EksisLemSe\], in the first position of each row there is a power of $y$. Hence for the matrix to degenerate we must have $y = 0$. Similarly there is a power of $x$ in the last position, and so $x = 0$ when the matrix degenerates.
Now when $\alpha$ and $\beta$ are composed, columns in $\beta$ are multiplied with the rows of $\alpha$. Motivated by this we have the following.
\[EksisLemAB\] Let $k$ be a column in $D_\beta$ which starts in position $(j_0,k)$ and ends in $(j_1,k)$. Then $D_\alpha$ restricted to $[1, \ldots, p] \times [j_0, j_1]$ has $j_1 - j_0$ nonzero rows, say the interval $[i_0, i_1]$ where $j_1 - j_0 = i_1 - i_0 + 1$, and $D_\alpha$ restricted to $[i_0, i_1] \times [j_0, j_1]$ is semi-strict with $e_{D_\alpha}(i_0) > j_0$ and $s_{D_\alpha}(i_1) < j_1$.
1\. That $j_1 - j_0 = i_1 - i_0 +1$ follows from Lemma \[EksisLemBigrad\] by restricting to the bidegree $(c^1_k,c^2_k)$.
2\. Now we show $e_{D_\alpha}(i_0) > j_0$. By the HK-equations there is a $(b^1_j, b^2_j)$ with $b^2_j = c^2_k$. Since the $b^2_j$ are decreasing, this must happen for $j = j_0$. (This is the analog of Lemma \[EksisLemSe\] for $\beta^\vee$.) Clearly $e_{D_\alpha}(i_0) \geq j_0$. If we have equality, by Lemma \[EksisLemSe\] $a^2_{i_0} = b^2_{j_0}$. But then $a^2_{i_0} = c^2_k$ and by the HK-equations there must then be two $(b^1_j, b^2_j)$ with $b^2_j = a^2_{i_0} = c^2_k$. But this would again give $s_D(i_0) > j_0$. Similarly we can argue that $s_{D_\alpha}(i_1) < j_1$.
3\. That the restriction is semi-strict follows from i) $s_{D_\alpha}$ and $e_{D_\alpha}$ are strictly increasing, ii) $s_{D_\alpha}(i_o) \leq j_0$, and iii) $e_{D_\alpha}(i_1) \geq j_1$. To show ii) note that if $s_{D_\alpha}(i_0) > j_0$ then clearly $s_{D_\alpha}(i_1) > j_0 + i_1 - i_0 = j_1 - 1$. But this is not possible since $e_{D_\alpha} (i_1) \leq j_1 - 1$. Similarly we can show iii).
We choose $\alpha$ to be a general matrix, homogeneous with respect to the multidegrees. It will be of type $D_\alpha$ and it degenerates in codimension two by Lemma \[EksisLemCoker\].
By Lemma \[EksisLemAB\] we get for each column $k$ in $D_\beta$ a vector in the kernel of $\alpha$ which is nonzero in positions $s_{D_{\beta^\vee}}(k)$ and $e_{D_{\beta^\vee}}(k)$. Hence these kernel vectors make up the columns of a map $\beta$ such that $\beta^\vee$ degenerates in codimension two by Lemma \[EksisLemCoker\]. Also the composition $\alpha \circ \beta = 0$, and this is what we needed to show.
Resolutions of trigraded artinian modules of codimension three
==============================================================
In the case of trigraded artinian modules over the polynomial ring ${{\Bbbk}}[x,y,z]$ where the resolution has pure total degrees, we do not know much. The following are natural questions.
- For Betti diagrams with given total degrees, are there, up to translation, only a finite number of extremal rays in the positive cone of such Betti diagrams?
- Suppose the above property 2. holds. From Section \[PosSec\] we know that the translation classes of extremal rays form a poset with a unique minimal member and a unique maximal member. Is there a maximal member in the translation classes in the three variable case also?
We do not know the answer to these questions. A general fact we do know is that $L({{\bf e}}) = L^\prime({{\bf e}})$. However in three variables it is not the case that the injection $P({{\bf e}}) {\rightarrow}P^\prime({{\bf e}})$ is an isomorphism. Let us consider as example the case of resolutions of type $0,1,2,1$. The equivariant resolution of this type has the form (we have listed the tridegrees of the generators below each free module) $$\label{TrigLigETE}
\underset{ \begin{matrix} 100 \\010 \\ 001 \end{matrix}}{S^3} {\leftarrow}\underset{ \begin{matrix} 200 \\020 \\002\\110 \\ 101 \\ 011 \end{matrix}}{S^6} {\leftarrow}\underset{ \begin{matrix} 211 \\121\\ 112\\220 \\ 202 \\ 022 \end{matrix}}{S^6} {\leftarrow}\underset{ \begin{matrix} 221 \\212 \\ 122 \end{matrix}}{S^3}.$$
To facilitate notation write $\sum_i k_i\beta(a_i,b_i,c_i)$ as $\sum_i [k_i(a_i,b_i,c_i)] \beta$. Let $\beta$ be the Betti diagram of the complex (\[TrigLigETE\]). One may check that $$[(2,1,0) + (0,2,1) + (1,0,2) - (1,1,1)] \beta$$ gives a diagram with no negative entries (and it fulfils the HK-equations). But no multiple of this is the Betti diagram of a module. If $F_{\bullet}$ is a complex with this diagram, then $S(-3,-1,0)$ is a term in $F_0$. But there is no term $S(-3,-1,*)$ in $F_1$ (but there is one in $F_3$), and so the cokernel of $F_1 {\rightarrow}F_0$ cannot have codimension three. In particular this diagram is in $P^\prime(1,2,1)$ but not in $P(1,2,1)$. However let $\alpha$ be the diagram $$[(2,1,0) + (2,0,1) + (1,2,0) + (0,2,1) + (1,0,2) +
(0,1,2) - (1,1,1)] \beta.$$
The diagrams $\beta$ and $\alpha$ are Betti diagrams of resolutions of indecomposable artinian trigraded modules of codimension three, and they generate rays which are extremal rays in the cone $P(1,2,1)$.
That $\beta$ is a Betti diagram is clear and that it resolves an indecomposable module is also immediate to see from the resolution. That $\alpha$ is a Betti diagram of a resolution of an indecomposable module, may be checked on Macaulay 2 by filling in general monomial matrices with the tridegrees of $\alpha$. Now the only way $\alpha$ can decompose into nonnegative diagrams which are not on its ray, may be worked out to be as follows. $$\begin{aligned}
\notag &[ & c_1( (2,1,0) + (0,2,1) + (1,0,2) - (1,1,1)) \\
\notag &+& c_2 ((1,2,0) + (0,1,2) +
(2,0,1) - (1,1,1)) + c_3 (1,1,1)] \beta \end{aligned}$$ where $c_1 = c_2 = c_3$. But the same argument used to show that the diagram corresponding to the first term is not a resolution may be used to show that a linear combination as above is the diagram of a resolution only if $c_1 = c_2 = c_3$ is a positive integer.
It would be interesting to know if there are other extremal rays in the cone $P$ apart from the translates of $\alpha$ and $\beta$.
[99]{}
M.Boij, J.Söderberg, [*Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture*]{}, Journal of the London Mathematical Society, [**78**]{} no.1, (2008), p.78-101.
D.Eisenbud, G.Fløystad, J.Weyman, [*The existence of pure free resolutions*]{}, arXiv:0709.1529, to appear in Annales de l’institut Fourier.
D.Eisenbud, F.-O. Schreyer, [*Betti numbers of graded modules and cohomology of vector bundles*]{}, Journal of the American Mathematical Society [**22**]{} (2009), p.859-888.
G.Fløystad, [*The linear space of Betti diagrams of multigraded artinian modules*]{}, arXiv:1001.3235.
W.Fulton, J.Harris. [*Representation theory*]{}, GTM 129, Springer Verlag 1991.
E.Miller, B.Sturmfels, [*Combinatorial commutative algebra*]{} GTM 227, Springer Verlag 2005.
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---
abstract: 'This work presents an end-to-end trainable deep bidirectional LSTM (Long-Short Term Memory) model for image captioning. Our model builds on a deep convolutional neural network (CNN) and two separate LSTM networks. It is capable of learning long term visual-language interactions by making use of history and future context information at high level semantic space. Two novel deep bidirectional variant models, in which we increase the depth of nonlinearity transition in different way, are proposed to learn hierarchical visual-language embeddings. Data augmentation techniques such as multi-crop, multi-scale and vertical mirror are proposed to prevent overfitting in training deep models. We visualize the evolution of bidirectional LSTM internal states over time and qualitatively analyze how our models “translate” image to sentence. Our proposed models are evaluated on caption generation and image-sentence retrieval tasks with three benchmark datasets: Flickr8K, Flickr30K and MSCOCO datasets. We demonstrate that bidirectional LSTM models achieve highly competitive performance to the state-of-the-art results on caption generation even without integrating additional mechanism (e.g. object detection, attention model etc.) and significantly outperform recent methods on retrieval task.'
author:
- |
Cheng Wang, Haojin Yang, Christian Bartz, Christoph Meinel\
\
\
@hpi.de\
bibliography:
- 'references.bib'
title: Image Captioning with Deep Bidirectional LSTMs
---
<ccs2012> <concept> <concept\_id>10010147.10010178.10010179.10010182</concept\_id> <concept\_desc>Computing methodologies Natural language generation</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010293.10010294</concept\_id> <concept\_desc>Computing methodologies Neural networks</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010178.10010224.10010240</concept\_id> <concept\_desc>Computing methodologies Computer vision representations</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
Introduction
============
Automatically describe an image using sentence-level captions has been receiving much attention recent years [@karpathy2015deep; @karpathy2014deep; @kiros2014unifying; @kuznetsova2014treetalk; @kuznetsova2012collective; @mao2014deep; @socher2014grounded; @vinyals2015show]. It is a challenging task integrating visual and language understanding. It requires not only the recognition of visual objects in an image and the semantic interactions between objects, but the ability to capture visual-language interactions and learn how to “translate” the visual understanding to sensible sentence descriptions. The most important part of this visual-language modeling is to capture the semantic correlations across image and sentence by learning a multimodal joint model. While some previous models [@li2011composing; @kulkarni2013babytalk; @mitchell2012midge; @kuznetsova2014treetalk; @kuznetsova2012collective] have been proposed to address the problem of image captioning, they rely on either use sentence templates, or treat it as retrieval task through ranking the best matching sentence in database as caption. Those approaches usually suffer difficulty in generating variable-length and novel sentences. Recent work [@karpathy2015deep; @karpathy2014deep; @kiros2014unifying; @mao2014deep; @socher2014grounded; @vinyals2015show] has indicated that embedding visual and language to common semantic space with relatively shallow recurrent neural network (RNN) can yield promising results. In this work, we propose novel architectures to the problem of image captioning. Different to previous models, we learn a visual-language space where sentence embeddings are encoded using bidirectional Long-Short Term Memory (Bi-LSTM) and visual embeddings are encoded with CNN. Bi-LSTM is able to summarize long range visual-language interactions from forward and backward directions. Inspired by the architectural depth of human brain, we also explore the deep bidirectional LSTM architectures to learn higher level visual-language embeddings. All proposed models can be trained in end-to-end by optimizing a joint loss.
*Why bidirectional LSTMs?* In unidirectional sentence generation, one general way of predicting next word $w_t$ with visual context $I$ and history textual context $w_{1:t-1}$ is maximize $\log P(w_t|I,w_{1:t-1})$. While unidirectional model includes past context, it is still limited to retain future context $w_{t+1:T}$ that can be used for reasoning previous word $w_t$ by maximizing $\log P(w_t|I,w_{t+1:T})$. Bidirectional model tries to overcome the shortcomings that each unidirectional (forward and backward direction) model suffers on its own and exploits the past and future dependence to give a prediction. As shown in Figure \[illustration\_bidirectional\_examples\], two example images with bidirectionally generated sentences intuitively support our assumption that bidirectional captions are complementary, combining them can generate more sensible captions.
*Why deeper LSTMs?* The recent success of deep CNN in image classification and object detection [@krizhevsky2012imagenet; @simonyan2014very] demonstrates that deep, hierarchical models can be more efficient at learning representation than shallower ones. This motivated our work to explore deeper LSTM architectures in the context of learning bidirectional visual-language embeddings. As claimed in [@pascanu2013construct], if we consider LSTM as a composition of multiple hidden layers that unfolded in time, LSTM is already deep network. But this is the way of increasing “horizontal depth” in which network weights $W$ are reused at each time step and limited to learn more representative features as increasing the “vertical depth” of network. To design deep LSTM, one straightforward way is to stack multiple LSTM layers as hidden to hidden transition. Alternatively, instead of stacking multiple LSTM layers, we propose to add multilayer perception (MLP) as intermediate transition between LSTM layers. This can not only increase LSTM network depth, but can also prevent the parameter size from growing dramatically.
The core contributions of this work are threefold:
- We propose an end-to-end trainable multimodal bidirectional LSTM (see Sec.\[sec.bi-lstm\]) and its deeper variant models (see Sec.\[sec.deep-lstm\]) that embed image and sentence into a high level semantic space by exploiting both long term history and future context.
- We visualize the evolution of hidden states of bidirectional LSTM units to qualitatively analyze and understand how to generate sentence that conditioned by visual context information over time (see Sec.\[sec.vis\]).
- We demonstrate the effectiveness of proposed models on three benchmark datasets: Flickr8K, Flickr30K and MSCOCO. Our experimental results show that bidirectional LSTM models achieve highly competitive performance to the state-of-the-art on caption generation (see Sec.\[sec.com\_generation\]) and perform significantly better than recent methods on retrieval task (see Sec.\[sec.retrieval\]).
Related Work
============
Multimodal representation learning [@ngiam2011multimodal; @srivastava2012multimodal] has significant value in multimedia understanding and retrieval. The shared concept across modalities plays an important role in bridging the “semantic gap” of multimodal data. Image captioning falls into this general category of learning multimodal representations.
Recently, several approaches have been proposed for image captioning. We can roughly classify those methods into three categories. The first category is template based approaches that generate caption templates based on detecting objects and discovering attributes within image. For example, the work [@li2011composing] was proposed to parse a whole sentence into several phrases, and learn the relationships between phrases and objects within an image. In [@kulkarni2013babytalk], conditional random field (CRF) was used to correspond objects, attributes and prepositions of image content and predict the best label. Other similar methods were presented in [@mitchell2012midge; @kuznetsova2014treetalk; @kuznetsova2012collective]. These methods are typically hard-designed and rely on fixed template, which mostly lead to poor performance in generating variable-length sentences. The second category is retrieval based approach, this sort of methods treat image captioning as retrieval task. By leveraging distance metric to retrieve similar captioned images, then modify and combine retrieved captions to generate caption [@kuznetsova2014treetalk]. But these approaches generally need additional procedures such as modification and generalization process to fit image query.
Inspired by the success use of CNN [@krizhevsky2012imagenet; @zeiler2014visualizing] and Recurrent Neural Network [@mikolov2010recurrent; @mikolov2011extensions; @bahdanau2014neural]. The third category is emerged as neural network based methods [@vinyals2015show; @xu2015show; @kiros2014unifying; @karpathy2014deep; @karpathy2015deep]. Our work also belongs to this category. Among those work, Kiro *et al.*[@kiros2014multimodal] can been as pioneer work to use neural network for image captioning with multimodal neural language model. In their follow up work [@kiros2014unifying], Kiro *et al.* introduced an encoder-decoder pipeline where sentence was encoded by LSTM and decoded with structure-content neural language model (SC-NLM). Socher *et al*.[@socher2014grounded] presented a DT-RNN (Dependency Tree-Recursive Neural Network) to embed sentence into a vector space in order to retrieve images. Later on, Mao *et al*.[@mao2014deep] proposed m-RNN which replaces feed-forward neural language model in [@kiros2014unifying]. Similar architectures were introduced in NIC [@vinyals2015show] and LRCN [@donahue2015long], both approaches use LSTM to learn text context. But NIC only feed visual information at first time step while Mao *et al*.[@mao2014deep] and LRCN [@donahue2015long]’s model consider image context at each time step. Another group of neural network based approaches has been introduced in [@karpathy2014deep; @karpathy2015deep] where image captions generated by integrating object detection with R-CNN (region-CNN) and inferring the alignment between image regions and descriptions.
Most recently, Fang *et al*.[@fang2015captions] used multi-instance learning and traditional maximum-entropy language model for description generation. Chen *et al*.[@chen2015mind] proposed to learn visual representation with RNN for generating image caption. In [@xu2015show], Xu *et al*. introduced attention mechanism of human visual system into encoder-decoder framework. It is shown that attention model can visualize what the model “see” and yields significant improvements on image caption generation. Unlike those models, our deep LSTM model directly assumes the mapping relationship between visual-language is antisymmetric and dynamically learns long term bidirectional and hierarchical visual-language interactions. This is proved to be very effective in generation and retrieval tasks as we demonstrate in Sec.\[sec.com\_generation\] and Sec.\[sec.retrieval\].
Model
=====
In this section, we describe our multimodal bidirectional LSTM model (Bi-LSTM for short) and explore its deeper variants. We first briefly introduce LSTM which is at the center of model. The LSTM we used is described in [@zaremba2014learning].
Long Short Term Memory
----------------------
Our model builds on LSTM cell, which is a particular form of traditional recurrent neural network (RNN). It has been successfully applied to machine translation [@cho2014learning], speech recognition [@graves2013speech] and sequence learning [@sutskever2014sequence]. As shown in Figure \[fig:lstm\], the reading and writing memory cell $c$ is controlled by a group of sigmoid gates. At given time step $t$, LSTM receives inputs from different sources: current input $\mathbf{x}_t$, the previous hidden state of all LSTM units $\mathbf{h}_{t-1}$ as well as previous memory cell state $\mathbf{c}_{t-1}$. The updating of those gates at time step $t$ for given inputs $\mathbf{x}_t$, $\mathbf{h}_{t-1}$ and $\mathbf{c}_{t-1}$ as follows. $$\begin{aligned}
\mathbf{i}_t=\sigma (\mathbf{W}_{xi}\mathbf{x}_t+\mathbf{W}_{hi}\mathbf{h}_{t-1}+\mathbf{b}_i)\\
\mathbf{f}_t=\sigma (\mathbf{W}_{xf}\mathbf{x}_t+\mathbf{W}_{hf}\mathbf{h}_{t-1}+\mathbf{b}_f)\\
\mathbf{o}_t=\sigma (\mathbf{W}_{xo}\mathbf{x}_t+\mathbf{W}_{ho}\mathbf{h}_{t-1}+\mathbf{b}_o)\\
\mathbf{g}_t=\phi (\mathbf{W}_{xc}\mathbf{x}_t+\mathbf{W}_{hc}\mathbf{h}_{t-1}+\mathbf{b}_c)\\
\mathbf{c}_t=\mathbf{f}_t\odot\mathbf{c}_{t-1}+\mathbf{i}_t\odot\mathbf{g}_t\\
\mathbf{h}_t=\mathbf{o}_t\odot\phi(\mathbf{c}_t)\end{aligned}$$ where $\mathbf{W}$ are the weight matrices learned from the network and $\mathbf{b}$ are bias vectors. $\sigma$ is the sigmoid activation function $\sigma(x)=1/(1+\exp(-x))$ and $\phi$ presents hyperbolic tangent $\phi(x)=(\exp(x)-\exp(-x))/(\exp(x)+\exp(-x))$. $\odot$ denotes the products with a gate value. The LSTM hidden output $\mathbf{h}_t$=$\{\mathbf{h}_{tk}\}_{k=0}^K$, $\mathbf{h}_t\in \mathbf{R}^K$ will be used to predict the next word by Softmax function with parameters $\mathbf{W}_s$ and $\mathbf{b}_s$: $$\mathcal{F}(\mathbf{p}_{ti};\mathbf{W}_s,\mathbf{b}_s)=\frac{\exp(\mathbf{W}_s\mathbf{h}_{ti}+\mathbf{b}_s)}{\sum_{j=1}^{K}\exp(\mathbf{W}_s\mathbf{h}_{tj}+\mathbf{b}_s)}$$ where $\mathbf{p}_{ti}$ is the probability distribution for predicted word.
![\[fig:lstm\] Long Short Term Memory (LSTM) cell. It is consist of an input gate $i$, a forget gate $f$, a memory cell $c$ and an output gate $o$. The input gate decides let incoming signal go through to memory cell or block it. The output gate can allow new output or prevent it. The forget gate decides to remember or forget cell’s previous state. Updating cell states is performed by feeding previous cell output to itself by recurrent connections in two consecutive time steps.](lstm_cell.png){width="22.00000%"}
Our key motivation of chosen LSTM is that it can learn long-term temporal activities and avoid quick exploding and vanishing problems that traditional RNN suffers from during back propagation optimization.
![\[fig:framework\] Multimodal Bidirectional LSTM. L1: sentence embedding layer. L2: T-LSTM layer. L3: M-LSTM layer. L4: Softmax layer. We feed sentence in both forward (blue arrows) and backward (red arrows) order which allows our model summarizes context information from both left and right side for generating sentence word by word over time. Our model is end-to-end trainable by minimize a joint loss.](framework.png){width="46.00000%"}
{width="100.00000%"}
Bidirectional LSTM {#sec.bi-lstm}
------------------
In order to make use of both the past and future context information of a sentence in predicting word, we propose a bidirectional model by feeding sentence to LSTM from forward and backward order. Figure \[fig:framework\] presents the overview of our model, it is comprised of three modules: a CNN for encoding image inputs, a Text-LSTM (T-LSTM) for encoding sentence inputs, a Multimodal LSTM (M-LSTM) for embedding visual and textual vectors to a common semantic space and decoding to sentence. The bidirectional LSTM is implemented with two separate LSTM layers for computing forward hidden sequences $\overrightarrow{\mathbf{h}}$ and backward hidden sequences $\overleftarrow{\mathbf{h}}$. The forward LSTM starts at time $t=1$ and the backward LSTM starts at time $t=T$. Formally, our model works as follows, for raw image input $\widetilde{I}$, forward order sentence$\overrightarrow{S}$ and backward order sentence $\overleftarrow{S}$, the encoding performs as $$\begin{aligned}
\mathbf{I}_t=\mathcal{C}(\widetilde{I};\mathbf{\Theta}_v)~~
\overrightarrow{\mathbf{h}}_t^1=\mathcal{T}(\overrightarrow{\mathbf{E}}\overrightarrow{\mathbf{S}};\overrightarrow{\mathbf{\Theta}}_l)~~ \overleftarrow{\mathbf{h}}_t^1=\mathcal{T}(\overleftarrow{\mathbf{E}}\overleftarrow{\mathbf{S}};\overleftarrow{\mathbf{\Theta}}_l)
\end{aligned}$$ where $\mathcal{C}$, $\mathcal{T}$ represent CNN, T-LSTM respectively and $\mathbf{\Theta}_v$, $\mathbf{\Theta}_l$ are their corresponding weights. $\overrightarrow{\mathbf{E}}$ and $\overleftarrow{\mathbf{E}}$ are bidirectional embedding matrices learned from network. Encoded visual and textual representations are then embedded to multimodal LSTM by: $$\begin{aligned}
\overrightarrow{\mathbf{h}}_t^2=\mathcal{M}(\overrightarrow{\mathbf{h}}_t^1,\mathbf{I}_t;\overrightarrow{\mathbf{\Theta}}_m)~~~~ \overleftarrow{\mathbf{h}}_t^2=\mathcal{M}(\overleftarrow{\mathbf{h}}_t^1,\mathbf{I}_t;\overleftarrow{\mathbf{\Theta}}_m)\\
\end{aligned}$$ where $\mathcal{M}$ presents M-LSTM and its weight $\mathbf{\Theta}_m$. $\mathcal{M}$ aims to capture the correlation of visual context and words at different time steps. We feed visual vector $\mathbf{I}$ to model at each time step for capturing strong visual-word correlation. On the top of M-LSTM are Softmax layers which compute the probability distribution of next predicted word by $$\begin{aligned}
\overrightarrow{\mathbf{p}}_{t+1}=\mathcal{F}(\overrightarrow{\mathbf{h}}_t^2;\mathbf{W}_s,\mathbf{b}_s)~~~~ \overleftarrow{\mathbf{p}}_{t+1}=\mathcal{F}(\overleftarrow{\mathbf{h}}_t^2;\mathbf{W}_s,\mathbf{b}_s)\\
\end{aligned}$$ where $\mathbf{p}\in \mathbf{R}^K$ and $K$ is the vocabulary size.
Deeper LSTM architecture {#sec.deep-lstm}
------------------------
To design deeper LSTM architectures, in addition to directly stack multiple LSTMs on each other that we named as Bi-S-LSTM (Figure \[fig:extension\_LSTM1\](c)), we propose to use a fully connected layer as intermediate transition layer. Our motivation comes from the finding of [@pascanu2013construct], in which DT(S)-RNN (deep transition RNN with shortcut) is designed by adding hidden to hidden multilayer perception (MLP) transition. It is arguably easier to train such network. Inspired by this, we extend Bi-LSTM (Figure \[fig:extension\_LSTM1\](b)) with a fully connected layer that we called Bi-F-LSTM (Figure \[fig:extension\_LSTM1\](d)), shortcut connection between input and hidden states is introduced to make it easier to train model. The aim of extension models is to learn an extra hidden transition function $F_h$. In Bi-S-LSTM $$\mathbf{h}_t^{l+1}=F_h(\mathbf{h}_t^{l-1},\mathbf{h}_{t-1}^l)=\mathbf{U}\mathbf{h}_t^{l-1}+\mathbf{V}\mathbf{h}_{t-1}^l$$ where $\mathbf{h}_t^l$ presents the hidden states of $l$-th layer at time $t$, $\mathbf{U}$ and $\mathbf{V}$ are matrices connect to transition layer (also see Figure \[fig:transition\_function\](L)). For readability, we consider one direction training and suppress bias terms. Similarly, in Bi-F-LSTM, to learn a hidden transition function $F_h$ by $$\mathbf{h}_t^{l+1}=F_h(\mathbf{h}_t^{l-1})=\phi_r(\mathbf{W}\mathbf{h}_t^{l-1}\oplus(\mathbf{V}(\mathbf{U}\mathbf{h}_t^{l-1}))$$ where $\oplus$ is the operator that concatenates $\mathbf{h}_t^{l-1}$ and its abstractions to a long hidden states (also see Figure \[fig:transition\_function\](R)). $\phi_r$ presents rectified linear unit (Relu) activation function for transition layer, which performs $\phi_r(x)=\max(0,x)$.
![Transition for Bi-S-LSTM(L) and Bi-F-LSTM(R)[]{data-label="fig:transition_function"}](transition_function.png){width="35.00000%"}
Data Augmentation
-----------------
One of the most challenging aspects of training deep bidirectional LSTM models is preventing overfitting. Since our largest dataset has only 80K images [@lin2014microsoft] which might cause overfitting easily, we adopted several techniques such as fine-tuning on pre-trained visual model, weight decay, dropout and early stopping that commonly used in the literature. Additionally, it has been proved that data augmentation such as randomly cropping and horizontal mirror [@simonyan2014two; @lu2014rapid], adding noise, blur and rotation [@Wang:2015:DIY:2733373.2806219] can effectively alleviate over-fitting and other. Inspired by this, we designed new data augmentation techniques to increase the number of image-sentence pairs. Our implementation performs on visual model, as follows:
- **Multi-Corp**: Instead of randomly cropping on input image, we crop at the four corners and center region. Because we found random cropping is more tend to select center region and cause overfitting easily. By cropping four corners and center, the variations of network input can be increased to alleviate overfitting.
- **Multi-Scale**: To further increase the number of image-sentence pairs, we rescale input image to multiple scales. For each input image $\widetilde{I}$ with size $H\times W$, it is resized to 256 $\times$ 256, then we randomly select a region with size of $s*H\times s*W$, where $s\in[1,0.925,0.875,0.85]$ is scale ratio. $s=1$ means we do not multi-scale operation on given image. Finally we resize it to AlexNet input size 227 $\times$227 or VggNet input size 224 $\times$ 224.
- **Vertical Mirror**: Motivated by the effectiveness of widely used horizontal mirror, it is natural to also consider the vertical mirror of image for same purpose.
Those augmentation techniques are implemented in real-time fashion. Each input image is randomly transformed using one of augmentations to network input for training. In principle, our data augmentation can increase image-sentence training pairs by roughly 40 times (5$\times$4$\times$2).
Training and Inference
----------------------
Our model is end-to-end trainable by using Stochastic Gradient Descent (SGD). The joint loss function $L=\overrightarrow{L}+\overleftarrow{L}$ is computed by accumulating the Softmax losses of forward and backward directions. Our objective is to minimize $L$, which is equivalent to maximize the probabilities of correctly generated sentences. We compute the gradient $\triangledown L$ with Back-Propagation Through Time (BPTT) algorithm.
The trained model is used to predict a word $w_t$ with given image context $I$ and previous word context $w_{1:{t-1}}$ by $P(w_t|w_{1:{t-1}},I)$ in forward order, or by $P(w_t|w_{{t+1:T}},I)$ in backward order. We set $w_1$=$w_T$=$\mathbf{0}$ at start point respectively for forward and backward directions. Ultimately, with generated sentences from two directions, we decide the final sentence for given image $p(w_{1:T}|I)$ according to the summation of word probability within sentence $$\begin{aligned}
\label{equ:final_caption}
&p(w_{1:T}|I)=\max({\sum\nolimits_{t=1}^{T}}(\overrightarrow{p}(w_{t}|I)), \sum\nolimits_{t=1}^T(\overleftarrow{p}(w_{t}|I))) \\
&\overrightarrow{p}(w_{t}|I)=\prod\nolimits_{t=1}^Tp(w_t|w_{1},w_{2},...,w_{t-1},I) \\
&\overleftarrow{p}(w_{t}|I)=\prod\nolimits_{t=1}^Tp(w_t|w_{t+1},w_{t+2},...,w_{T},I) \end{aligned}$$
Follow previous work, we adopted beam search to consider the best $k$ candidate sentences at time $t$ to infer the sentence at next time step. In our work, we fix $k=1$ in all experiments although the average of 2 BLEU [@papineni2002bleu] points better results can be achieved with $k=20$ compare to $k=1$ as reported in [@vinyals2015show].
Experiments
===========
In this section, we design several groups of experiments to accomplish following objectives:
- Qualitatively analyze and understand how bidirectional multimodal LSTM learns to generate sentence conditioned by visual context information over time.
- Measure the benefits and performance of proposed bidirectional model and its deeper variant models that we increase their nonlinearity depth from different ways.
- Compare our approach with state-of-the-art methods in terms of sentence generation and image-sentence retrieval tasks on popular benchmark datasets.
Datasets
--------
To validate the effectiveness, generality and robustness of our models, we conduct experiments on three benchmark datasets: Flickr8K [@rashtchian2010collecting], Flickr30K [@young2014image] and MSCOCO [@lin2014microsoft].
**Flickr8K**. It consists of 8,000 images and each of them has 5 sentence-level captions. We follow the standard dataset divisions provided by authors, 6,000/1,000/1,000 images for training/validation/testing respectively.
**Flickr30K**. An extension version of Flickr8K. It has 31,783 images and each of them has 5 captions. We follow the public accessible[^1] dataset divisions by Karpathy *et al.* [@karpathy2015deep]. In this dataset splits, 29,000/1,000/1,000 images are used for training/validation/testing respectively.
**MSCOCO**. This is a recent released dataset that covers 82,783 images for training and 40,504 images for validation. Each of images has 5 sentence annotations. Since there is lack of standard splits, we also follow the splits provided by Karpathy *et al*. [@karpathy2015deep]. Namely, 80,000 training images and 5,000 images for both validation and testing.
Implementation Details
----------------------
**Visual feature**. We use two visual models for encoding image: Caffe [@jia2014caffe] reference model which is pre-trained with AlexNet [@krizhevsky2012imagenet] and 16-layer VggNet model [@simonyan2014very]. We extract features from last fully connected layer and feed to train visual-language model with LSTM. Previous work [@vinyals2015show; @mao2014deep] have demonstrated that with more powerful image models such as GoogleNet [@szegedy2015going] and VggNet [@simonyan2014very] can achieve promising improvements. To make a fair comparison with recent works, we select the widely used two models for experiments.
**Textual feature**. We first represent each word $w$ within sentence as one-hot vector, $w\in \mathbf{R}^K$, $K$ is vocabulary size built on training sentences and different for different datasets. By performing basic tokenization and removing the words that occurs less than 5 times in the training set, we have 2028, 7400 and 8801 words for Flickr8K, Flickr30K and MSCOCO dataset vocabularies respectively.
----------------------------------------------------------------------------------- ---------------------------------------------------
A man in a black jacket is walking down the street Street the on walking is suit a in man a
2 7 3 2 23 76 8 41 38 4 36 36 4 5 41 8 193 2 3 7 2
----------------------------------------------------------------------------------- ---------------------------------------------------
+:---------------:+:---------------:+:---------------:+:---------------:+
| \ | \ | \ | \ |
+-----------------+-----------------+-----------------+-----------------+
| | | | |
+-----------------+-----------------+-----------------+-----------------+
Our work uses the LSTM implementation of [@donahue2015long] on Caffe framework. All of our experiments were conducted on Ubuntu 14.04, 16G RAM and single Titan X GPU with 12G memory. Our LSTMs use 1000 hidden units and weights initialized uniformly from \[-0.08, 0.08\]. The batch sizes are 150, 100, 100, 32 for Bi-LSTM, Bi-S-LSTM, Bi-F-LSTM and Bi-LSTM (VGG) models respectively. Models were trained with learning rate $\eta=0.01$ (except $\eta=0.005$ for Bi-LSTM (VGG)), weight decay $\lambda$ is 0.0005 and we used momentum 0.9. Each model is trained for 18$\sim$35 epochs with early stopping. The code for this work can be found at *[ https://github.com/deepsemantic/image\_captioning]( https://github.com/deepsemantic/image_captioning)*.
Evaluation Metrics
------------------
We evaluate our models on two tasks: caption generation and image-sentence retrieval. In caption generation, we follow previous work to use BLEU-N (N=1,2,3,4) scores [@papineni2002bleu]: $$B_N=\min(1,e^{1-\frac{r}{c}})\cdot e^{\frac{1}{N}\sum_{n=1}^N\log p_n}$$ where $r$, $c$ are the length of reference sentence and generated sentence, $p_n$ is the modified *$n$-gram* precisions. We also report METETOR [@lavie2014meteor] and CIDEr [@vedantam2015cider] scores for further comparison. In image-sentence retrieval (image query sentence and vice versa), we adopt R@K (K=1,5,10) and Med $r$ as evaluation metrics. R@K is the recall rate R at top K candidates and Med $r$ is the median rank of the first retrieved ground-truth image and sentence. All mentioned metric scores are computed by MSCOCO caption evaluation server[^2], which is commonly used for image captioning challenge[^3].
Visualization and Qualitative Analysis {#sec.vis}
--------------------------------------
The aim of this set experiment is to visualize the properties of proposed bidirectional LSTM model and explain how it works in generating sentence word by word over time.
First, we examine the temporal evolution of internal gate states and understand how bidirectional LSTM units retain valuable context information and attenuate unimportant information. Figure \[fig:gates\] shows input and output data, the pattern of three sigmoid gates (input, forget and output) as well as cell states. We can clearly see that dynamic states are periodically distilled to units from time step $t=0$ to $t=11$. At $t=0$, the input data are sigmoid modulated to input gate $\mathbf{i}(t)$ where values lie within in \[0,1\]. At this step, the values of forget gates $\mathbf{f}(t)$ of different LSTM units are zeros. Along with the increasing of time step, forget gate starts to decide which unimportant information should be forgotten, meanwhile, to retain those useful information. Then the memory cell states $\mathbf{c}(t)$ and output gate $\mathbf{o}(t)$ gradually absorb the valuable context information over time and make a rich representation $\mathbf{h}(t)$ of the output data.
Next, we examine how visual and textual features are embedded to common semantic space and used to predict word over time. Figure \[fig:T-M-LSTM\] shows the evolution of hidden units at different layers. For T-LSTM layer, units are conditioned by textual context from the past and future. It performs as the encoder of forward and backward sentences. At M-LSTM layer, LSTM units are conditioned by both visual and textual context. It learns the correlations between input word sequence and visual information that encoded by CNN. At given time step, by removing unimportant information that make less contribution to correlate input word and visual context, the units tend to appear sparsity pattern and learn more discriminative representations from inputs. At higher layer, embedded multimodal representations are used to compute the probability distribution of next predict word with Softmax. It should be noted, for given image, the number of words in generated sentence from forward and backward direction can be different.
Figure \[fig:generated\_example\_mscoco\] presents some example images with generated captions. From it we found some interesting patterns of bidirectional captions: (1) **Cover different semantics**, for example, in (b) forward sentence captures “*couch*” and “*table*” while backward one describes “*chairs*” and “*table*”. (2) **Describe static scenario and infer dynamics**, in (a) and (d), one caption describes the static scene, and the other one presents the potential action or motion that possibly happen in the next time step. (3) **Generate novel sentences**, from generated captions, we found that a significant proportion (88% by randomly select 1000 images on MSCOCO validation set) of generated sentences are novel (not appear in training set). But generated sentences are highly similar to ground-truth captions, for example in (d), forward caption is similar to one of ground-truth captions (*“A passenger train that is pulling into a station”*) and backward caption is similar to ground-truth caption (*“a train is in a tunnel by a station”*). It illustrates that our model has strong capability in learning visual-language correlation and generating novel sentences.
--------------------------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- ------------------
NIC[@vinyals2015show]$^{G,\ddagger}$ 63 41 27.2 - $\underline{66.3}$ 42.3 27.7 18.3 66.6 46.1 32.9 24.6
LRCN[@donahue2015long]$^{A,\ddagger}$ - - - - 58.8 39.1 25.1 16.5 62.8 44.2 30.4 -
DeepVS[@karpathy2015deep]$^{V}$ 57.9 38.3 24.5 16 57.3 36.9 24.0 15.7 62.5 45 32.1 23
m-RNN[@mao2014deep]$^{A,\ddagger}$ 56.5 38.6 25.6 17.0 54 36 23 15 - - - -
m-RNN[@mao2014deep]$^{V,\ddagger}$ - - - - 60 41 28 19 67 49 35 $\underline{25}$
Hard-Attention[@xu2015show]$^{V}$ $\mathbf{67}$ $\underline{45.7}$ $\underline{31.4}$ $\underline{21.3}$ $\mathbf{66.9}$ $\mathbf{43.9}$ $\mathbf{29.6}$ $\mathbf{19.9}$ $\mathbf{71.8}$ $\mathbf{50.4}$ $\mathbf{35.7}$ $\mathbf{25}$
Bi-LSTM$^{A}$ 61.9 43.3 29.7 20.0 58.9 39.3 25.9 17.1 63.4 44.7 30.6 20.6
Bi-S-LSTM$^{A}$ 64.2 44.3 29.2 18.6 59.5 40.3 26.9 17.9 63.7 45.7 31.8 21.9
Bi-F-LSTM$^{A}$ 63.0 43.7 29.2 19.1 58.6 39.2 26.0 17.4 63.5 44.8 30.7 20.6
Bi-LSTM$^{V}$ $\underline{65.5}$ $\mathbf{46.8}$ $\mathbf{32.0}$ $\mathbf{21.5}$ 62.1 $\underline{42.6}$ $\underline{28.1}$ $\underline{19.3}$ $\underline{67.2}$ $\underline{49.2}$ $\underline{35.2}$ 24.4
--------------------------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- ------------------
\[tab:BLEU\]
Results on Caption Generation {#sec.com_generation}
-----------------------------
Now, we compare with state-of-the-art methods. Table \[tab:BLEU\] presents the comparison results in terms of BLEU-N. Our approach achieves very competitive performance on evaluated datasets although with less powerful AlexNet visual model. We can see that increase the depth of LSTM is beneficial on generation task. Deeper variant models mostly obtain better performance compare to Bi-LSTM, but they are inferior to latter one in B-3 and B-4 on Flickr8K. We conjecture it should be the reason that Flick8K is a relatively small dataset which suffers difficulty in training deep models with limited data. One of interesting facts we found is that by stacking multiple LSTM layers is generally superior to LSTM with fully connected transition layer although Bi-S-LSTM needs more training time. By replacing AlexNet with VggNet brings significant improvements on all BLEU evaluation metrics. We should be aware of that a recent interesting work [@xu2015show] achieves the best results by integrating attention mechanism [@lecun2015deep; @xu2015show] on this task. Although we believe incorporating such powerful mechanism into our framework can make further improvements, note that our current model Bi-LSTM$^V$ achieves the best or second best results on most of metrics while the small gap in performance between our model and Hard-Attention$^V$ [@xu2015show] is existed.
The further comparison on METEOR and CIDEr scores is plotted in Figure \[fig:METEOR\_CIDEr\]. Without integrating object detection and more powerful vision model, our model (Bi-LSTM$^A$) outperforms DeepVS$^V$[@karpathy2015deep] in a certain margin. It achieves 19.4/49.6 on Flickr 8K (compare to 16.7/31.8 of DeepVS$^V$) and 16.2/28.2 on Flickr30K (15.3/24.7 of DeepVS$^V$). On MSCOCO, our Bi-S-LSTM$^A$ obtains 20.8/66.6 for METEOR/CIDEr, which exceeds 19.5/66.0 in DeepVS$^V$.
---------- ----------------------------------------- ----------------- ----------------- ------------------ -------------- ----------------- ----------------- ----------------- ---------------
Datasets Methods R@5 R@10 R@1 R@5 R@10
DeViSE[@frome2013devise] 4.8 16.5 27.3 28 5.9 20.1 29.6 29
SDT-RNN[@socher2014grounded] 4.5 18.0 28.6 32 6.1 18.5 29.0 29
DeFrag[@karpathy2014deep]$^{+O}$ 12.6 32.9 44.0 14 9.7 29.6 42.5 15
Kiros et al. [@kiros2014unifying]$^{A}$ 13.5 36.2 45.7 13 10.4 31.0 43.7 14
Kiros et al. [@kiros2014unifying]$^{V}$ 18 40.9 55 8 12.5 37 51.5 10
m-RNN[@mao2014deep]$^{A}$ 14.5 37.2 48.5 11 11.5 31.0 42.4 15
Mind’s Eye[@chen2015mind]$^{V}$ 17.3 42.5 57.4 7 15.4 40.6 50.1 8
DeepVS[@karpathy2015deep]$^{+O,V}$ 16.5 40.6 54.2 7.6 11.8 32.1 44.7 12.4
NIC[@vinyals2015show]$^{G}$ 20 - 60 6 19 - $\mathbf{64}$ 5
Bi-LSTM$^{A}$ 21.3 44.7 56.5 6.5 15.1 37.8 50.1 9
Bi-S-LSTM$^{A}$ 19.6 43.7 55.7 7 14.5 36.4 48.3 10.5
Bi-F-LSTM$^{A}$ 19.9 44.0 56.0 7 14.9 37.4 49.8 10
Bi-LSTM$^{V}$ $\mathbf{29.3}$ $\mathbf{58.2}$ $\mathbf{69.6}$ $\mathbf{3}$ $\mathbf{19.7}$ $\mathbf{47.0}$ 60.6 $\mathbf{5}$
DeViSE[@frome2013devise] 4.5 18.1 29.2 26 6.7 21.9 32.7 25
SDT-RNN[@socher2014grounded] 9.6 29.8 41.1 16 8.9 29.8 41.1 16
Kiros et al. [@kiros2014unifying]$^{A}$ 14.8 39.2 50.9 10 11.8 34.0 46.3 13
Kiros et al. [@kiros2014unifying]$^{V}$ 23.0 50.7 62.9 5 16.8 42.0 56.5 8
LRCN[@donahue2015long]$^{A}$ 14 34.9 47 11 - - - -
NIC[@vinyals2015show]$^{G}$ 17 - 56 7 17 - 57 8
m-RNN[@mao2014deep]$^{A}$ 18.4 40.2 50.9 10 12.6 31.2 41.5 16
Mind’s Eye[@chen2015mind]$^{V}$ 18.5 45.7 58.1 7 16.6 42.5 $\mathbf{58.9}$ 8
DeFrag [@karpathy2014deep]$^{+O}$ 16.4 40.2 54.7 8 10.3 31.4 44.5 13
DeepVS[@karpathy2015deep]$^{+O,V}$ 22.2 48.2 61.4 $4.8$ 15.2 37.7 50.5 9.2
Bi-LSTM$^{A}$ 18.7 41.2 52.6 8 14.0 34.0 44.0 14
Bi-S-LSTM$^{A}$ 21 43.0 54.1 7 15.1 35.3 46.0 12
Bi-F-LSTM$^{A}$ 20 44.4 55.2 7 15.1 35.8 46.8 12
Bi-LSTM$^{V}$ $\mathbf{28.1}$ $\mathbf{53.1}$ $\mathbf{64.2}$ $\mathbf{4}$ $\mathbf{19.6}$ $\mathbf{43.8}$ 55.8 $\mathbf{7}$
DeepVS[@karpathy2015deep]$^{+O,V}$ 39.2 52.0 10.7 29.6 42.2 14.0
Bi-LSTM$^{A}$ 10.8 28.1 38.9 18 7.8 22.4 32.8 24
Bi-S-LSTM$^{A}$ 13.4 33.1 44.7 13 9.4 26.5 37.7 19
Bi-F-LSTM$^{A}$ 11.2 30 41.2 16 8.3 24.9 35.1 22
Bi-LSTM$^{V}$ $\mathbf{16.6}$ $\mathbf{39.4}$ $\mathbf{52.4} $ $\mathbf{9}$ $\mathbf{11.6}$ $\mathbf{30.9}$ $\mathbf{43.4}$ $\mathbf{13}$
---------- ----------------------------------------- ----------------- ----------------- ------------------ -------------- ----------------- ----------------- ----------------- ---------------
Results on Image-Sentence Retrieval {#sec.retrieval}
-----------------------------------
For retrieval evaluation, we focus on image to sentence retrieval and vice versa. This is an instance of cross-modal retrieval [@feng2014cross; @costa2014role; @jiang2015deep] which has been a hot research subject in multimedia field. Table \[tab:recall\] illustrates our results on different datasets. The performance of our models exceeds those compared methods on most of metrics or matching existing results. In a few metrics, our model didn’t show better result than Mind’s Eye [@chen2015mind] which combined image and text features in ranking (it makes this task more like multimodal retrieval) and NIC [@vinyals2015show] which employed more powerful vision model, large beam size and model ensemble. While adopting more powerful visual model VggNet results in significant improvements across all metrics, with less powerful AlexNet model, our results are still competitive on some metrics, e.g. R@1, R@5 on Flickr8K and Flickr30K. We also note that on relatively small dataset Filckr8K, shallow model performs slightly better than deeper ones on retrieval task, which in contrast with the results on the other two datasets. As we explained before, we think deeper LSTM architectures are better suited for ranking task on large datasets which provides enough training data for more complicate model training, otherwise, overfitting occurs. By increasing data variations with our implemented data augmentation techniques can alleviate it in a certain degree. But we foresee further significant improvement gains as training example grows, by reducing reliance on augmentation with fresh data. Figure \[fig:retr\_example\] presents some examples of retrieval experiments. For each caption (image) query, sensible images and descriptive captions are retrieved. It shows our models captured the visual-textual correlation for image and sentence ranking.
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Discussion
----------
**Efficiency**. In addition to showing superior performance, our models also possess high computational efficiency. Table \[tab:time\_costs\] presents the computational costs of proposed models. We randomly select 10 images from Flickr8K validation set, and perform caption generation and image to sentence retrieval test for 5 times respectively. The table shows the averaged time costs across 5 test results. The time cost of network initialization is excluded. The costs of caption generation includes: computing image feature, sampling bidirectional captions, computing the final caption. The time costs for retrieval considers: computing image-sentence pair scores (totally 10 $\times$ 50 pairs), ranking sentences for each image query. As can be seen from Table \[tab:BLEU\], \[tab:recall\] and \[tab:time\_costs\], deep models have only slightly higher time consumption but yield significant improvements and our proposed Bi-F-LSTM can strike the balance between performance and efficiency.
Bi-LSTM Bi-S-LSTM Bi-F-LSTM
------------ --------- ----------- -----------
Generation 0.93s 1.1s 0.97s
Retrieval 5.62s 7.46s 5.69s
: Time costs for testing 10 images on Flickr8K[]{data-label="tab:time_costs"}
**Challenges in exact comparison**. It is challenging to make a direct, extract comparison with related methods due to the differences in dataset division on MSCOCO. In principle, testing on smaller validation set can lead to better results, particularly in retrieval task. Since we strictly follow dataset splits as in [@karpathy2015deep], we compare to it in most cases. Another challenge is the visual model that utilized for encoding image inputs. Different models are employed in different works, to make a fair and comprehensive comparison, we select commonly used AlexNet and VggNet in our work.
Conclusions
===========
We proposed a bidirectional LSTM model that generates descriptive sentence for image by taking both history and future context into account. We further designed deep bidirectional LSTM architectures to embed image and sentence at high semantic space for learning visual-language models. We also qualitatively visualized internal states of proposed model to understand how multimodal LSTM generates word at consecutive time steps. The effectiveness, generality and robustness of proposed models were evaluated on numerous datasets. Our models achieve highly completive or state-of-the-art results on both generation and retrieval tasks. Our future work will focus on exploring more sophisticated language representation (e.g. word2vec) and incorporating multitask learning and attention mechanism into our model. We also plan to apply our model to other sequence learning tasks such as text recognition and video captioning.
[^1]: <http://cs.stanford.edu/people/karpathy/deepimagesent/>
[^2]: https://github.com/tylin/coco-caption
[^3]: http://mscoco.org/home/
|
---
abstract: 'Iterative imputation, in which variables are imputed one at a time each given a model predicting from all the others, is a popular technique that can be convenient and flexible, as it replaces a potentially difficult multivariate modeling problem with relatively simple univariate regressions. In this paper, we begin to characterize the stationary distributions of iterative imputations and their statistical properties. More precisely, when the conditional models are compatible (defined in the text), we give a set of sufficient conditions under which the imputation distribution converges in total variation to the posterior distribution of a Bayesian model. When the conditional models are incompatible but are valid, we show that the combined imputation estimator is consistent.'
author:
- |
Jingchen Liu, Andrew Gelman, Jennifer Hill, and Yu-Sung Su\
\
Columbia University, Columbia University, New York University,\
Tsinghua University
bibliography:
- 'bibstat.bib'
title: On the Stationary Distribution of Iterative Imputations
---
Introduction
============
Iterative imputation is a widely used approach for imputing multivariate missing data. The procedure starts by randomly imputing missing values using some simple stochastic algorithm. Missing values are then imputed one variable at a time, each conditionally on all the others using a model fit to the current iteration of the completed data. The variables are looped through until approximate convergence (as measured, for example, by the mixing of multiple chains).
Iterative imputation can be an appealing way to express uncertainty about missing data. There is no need to explicitly construct a joint multivariate model of all types of variables: continuous, ordinal, categorical, and so forth. Instead, one need only specify a sequence of families of conditional models such as linear regression, logistic regression, and other standard and already programmed forms. The distribution of the resulting imputations is implicitly defined as the invariant (stationary) distribution of the Markov chain corresponding to the iterative fitting and imputation process.
Iterative, or chained, imputation is convenient and flexible and has been implemented in various ways in several statistical software packages, including [mice]{} [@MICE10] and [mi]{} [@MI10] in `R`, [IVEware]{} [@IVEware10] in `SAS`, and [ice]{} in `Stata` [@ICE04; @ICE05]. The popularity of these programs suggests that the resulting imputations are believed to be of practical value. However, the theoretical properties of iterative imputation algorithms are not well understood. Even if, as we would prefer, the fitting of each imputation model and the imputations themselves are performed using conditional Bayesian inference, the stationary distribution of the algorithm (if it exists) does not in general correspond to Bayesian inference on any specified multivariate distribution.
Key questions are: (1) Under what conditions does the algorithm converge to a stationary distribution? (2) What statistical properties does the procedure admit given that a unique stationary distribution exists?
Regarding the first question, researchers have long known that the Markov chain may be non-recurrent (“blowing up” to infinity or drifting like a nonstationary random walk), even if each of the conditional models is fitted using a proper prior distribution. In this paper, we focus mostly on the second question—the characterization of the stationary distributions of the iterative imputation conditional on its existence. Unlike usual MCMC algorithms, which are designed in such a way that the invariant distribution and target distribution are identical, the invariant distribution of iterative imputation (even if it exists) is largely unknown.
The analysis of iterative imputation is challenging for at least two reasons. First, the range of choices of conditional models is wide, and it would be difficult to provide a solution applicable to all situations. Second, the literature on Markov chains focuses on known transition distributions. With iterative imputation, the distributions for the imputations are known only within specified parametric families. For example, if a particular variable is to be updated conditional on all the others using logistic regression, the actual updating distribution depends on the logistic regression coefficients which are themselves estimated given the latest update of the missing values.
The main contribution of this paper is to develop a mathematical framework under which the asymptotic properties of iterative imputation can be discussed. In particular, we demonstrate the following results.
1. \[ContribComp\]Given the existence of a unique invariant (stationary) distribution of the iterative imputation Markov chain, we provide a set of conditions under which this distribution converges in total variation to the posterior distribution of a joint Bayesian model, as the sample size tends to infinity. Under these conditions, iterative imputation is asymptotically equivalent to full Bayesian imputation using some joint model. Among these conditions, the most important is that the conditional models are *compatible*—that there exists a joint model whose conditional distributions are identical to the conditional models specified by the iterative imputation (Definition \[DefCompatible\]). We discuss in Section \[SecCompatible\].
2. \[ContribCompNec\]We consider model compatibility as a typically necessary condition for the iterative imputation distribution to converge to the posterior distribution of some Bayesian model (Section \[SecNec\]).
3. \[ContribIncomp\]For *incompatible* models whose imputation distributions are generally different from any Bayesian model, we show that the combined completed-data maximum likelihood estimate of the iterative imputation is a consistent estimator if the set of conditional models is valid, that is, if each conditional family contains the true probability distribution (Definition \[DefValid\] in Section \[SecIncomp\].).
The analysis presented in this paper connects to the existing separate literatures on missing data imputation and Markov chain convergence. Standard textbooks on imputation inference are [@LiRu02; @Rubin1987], and some key papers are [@Li1991; @BaRu99; @Meng1994; @Meng1992; @Rubin77; @Rubin96; @Sch97]. Large sample properties are studied by [@RoWa00; @ScWe88; @RoWa98], small samples are by [@BaRu99], and the issue of congeniality between the imputer’s and analyst’s models is considered by [@Meng1994].
Our asymptotic findings for compatible and incompatible models use results on convergence of Markov chains, a subject on which there is a vast literature on stability and rate of convergence ([@GeGe84; @Amit91; @AmGr91]). In addition, empirical diagnostics of Markov chains have been suggested by many authors, for instance, [@rhat]. For the analysis of compatible models, we need to construct a bound for convergence rate using renewal theory [@Bax05; @MT93; @Ros95], which has the advantage of not assuming the existence of an invariant distribution, which is naturally yielded by the minorization and drift conditions.
In Section \[SecBack\] of this article, we lay out our notation and assumptions. We then briefly review the framework of iterative imputation and the Gibbs sampler. In Section \[SecCompatible\], we investigate compatible conditional models. In Section \[SecIncomp\], the discussion focuses on incompatible models. Section \[SecLinear\] includes several simulation examples. An appendix is attached containing the technical developments and a brief review of the literature for Markov chain convergence via renewal theory.
Background {#SecBack}
==========
Consider a data set with $n$ cases and $p$ variables, where $\xx = (\xx_1,...,\xx_p)$ represents the complete data and $\xx_i = (x_{1,i},...,x_{n,i})^\top$ is the $i$-th variable. Let $\rr_i$ be the vector of observed data indicators for variable $i$, equaling $1$ for observed variables and $0$ for missing, and let $\xx_i^{obs}$ and $\xx_i^{mis}$ denote the observed and missing subsets of variable $i$: $$\xx^{obs}=\{\xx_i^{obs}: i=1,...,p\},\quad
\xx^{mis}=\{\xx_i^{mis}: i=1,...,p\}, \quad \rr = \{\rr_i: i =1,...,p\}.$$ To facilitate our description of the procedures, we define $$\xx^{obs}_{-j}=\{\xx_i^{obs}: i=1,...,j-1,j+1,...,p\},\quad
\xx^{mis}_{-j}=\{\xx_i^{mis}: i=1,...,j-1,j+1,...,p\}.$$ We use boldface $\xx$ to denote the entire data set and $x$ to denote individual observations. Therefore, $x_j$ denotes the $j$-th variable of one observation and $x_{-j}$ denotes all the variables except for the $j$-th one.
Throughout, we assume that the missing data process is ignorable. One set of sufficient conditions for ignorability is that the $\rr_i$ process is missing at random and the parameter spaces for $\rr_i$ and $\xx$ are distinct, with independent prior distributions [@LiRu02; @Rubin1987].
Inference using multiple imputations
------------------------------------
Multiple imputation is a convenient tool to handle incomplete data set by means of complete-data procedures. The framework consists of producing $m$ copies of the imputed data and applying the users’ complete data procedures to each of the multiply imputed data sets. Suppose that $m$ copies of point estimates and variance estimates are obtained, denoted by $(\hat\theta^{(i)},U^{(i)})$, $i=1,...,m$. The next step is to combine them into a single point estimate and a single variance estimate $(\hat \theta_m, \hat T_m)$ [@LiRu02]. If the imputed data are drawn from the joint posterior distribution of the missing data under a Bayesian model, under appropriate congeniality conditions, $\hat \theta_m$ is asymptotically equal to the posterior mean of $\theta$ and $\hat T_m$ is asymptotically equal to the posterior variance of $\theta$ ([@Rubin1987; @Meng1994]). The large sample theory of Bayesian inference ensures that the posterior mean and variance are asymptotically equivalent to the maximum likelihood estimate and its variance based on the observed data alone (see [@CoxHin74]). Therefore, the combined estimator from imputed samples is efficient. Imputations can also be constructed and used under other inferential frameworks; for example, Robins and Wang [@RoWa00; @RoWa98] propose estimates based on estimating equations and derive corresponding combining rules. For our purposes here, what is relevant is that the multiple imputations are being used to represent uncertainty about the joint distribution of missing values in a multivariate dataset.
Bayesian modeling, imputation, and Gibbs sampling {#SecFull}
-------------------------------------------------
In Bayesian inference, multiply imputed data sets are treated as samples from the posterior distribution of the full (incompletely-observed) data matrix. In the parametric Bayesian approach, one specifies a family of distributions $f(\xx|\theta)$ and a prior $\pi(\theta)$ and then performs inference using i.i.d. samples from the posterior predictive distribution, $$\label{Imp}p(\xx^{mis}|\xx^{obs}) = \int_{\Theta}
f(\xx^{mis}|\xx^{obs},\theta)p(\theta|\xx^{obs})d\theta,$$ where $p(\theta|\xx)\propto \pi(\theta) f(\xx|\theta)$. Direct simulation from (\[Imp\]) is generally difficult. One standard solution is to draw approximate samples using the Gibbs sampler or some more complicated Markov chain Monte Carlo (MCMC) algorithm. In the scenario of missing data, one can use the “data augmentation” strategy to iteratively draw $\theta$ given $(\xx^{obs},\xx^{mis})$ and $\xx^{mis}$ given $(\xx^{obs},\theta)$. Under regularity conditions (positive recurrence, irreducibility, and aperiodicity; see [@GeGe84]), the Markov process is ergodic with limiting distribution $p(\xx^{mis},\theta|\xx^{obs})$.
In order to connect these results to the iterative imputation that is the subject of the present article, we consider a slightly different Gibbs scheme which consists of indefinite iteration of following $p$ steps:,
Step 1.
: Draw $\theta\sim p(\theta | \xx^{obs}_1,\xx_{-1})$ and $\xx^{miss}_1\sim f(\xx^{miss}_1| \xx^{obs}_1,\xx_{-1},\theta)$;
Step 2.
: Draw $\theta \sim p(\theta|\xx^{obs}_2,\xx_{-2})$ and $\xx^{miss}_2\sim f(\xx^{miss}_2| \xx^{obs}_2,\xx_{-2},\theta)$;
Step $p$.
: Draw $\theta \sim p(\theta| \xx^{obs}_p,\xx_{-p})$ and $\xx^{miss}_p\sim f(\xx^{miss}_p| \xx^{obs}_p,\xx_{-p},\theta)$.
At each step, the posterior distribution is based on the updated values of the parameters and imputed data. It is not hard to verify that the Markov chain evolving according to steps 1 to $p$ (under mild regularity conditions) converges to the posterior distribution of the corresponding Bayesian model.
Iterative imputation and compatibility {#SecIter}
--------------------------------------
For iterative imputation, we need to specify $p$ conditional models, $$g_j(\xx_j|\xx_{-j},\theta_j),$$ for $\theta_j \in \Theta_j$ with prior distributions $\pi_j (\theta_j)$ for $j=1,...,p$. Iterative imputation adopts the following scheme to construct a Markov chain,
Step 1.
: Draw $\theta_1$ from $p_1 (\theta_1 | \xx^{obs}_1,\xx_{-1})$, which is the posterior distribution associated with $g_1$ and $\pi_1$; draw $\xx^{miss}_1$ from $g_1(\xx^{miss}_1|
\xx^{obs}_1,\xx_{-1},\theta_1)$;
Step 2.
: Draw $\theta_2$ from $p_2 (\theta_2 | \xx^{obs}_2,\xx_{-2})$, which is the posterior distribution associated with $g_2$ and $\pi_2$; draw $\xx^{miss}_2$ from $g_2(\xx^{miss}_2|
\xx^{obs}_2,\xx_{-2},\theta_2)$;
Step $p$.
: Draw $\theta_p$ from $p_p (\theta_p | \xx^{obs}_p,\xx_{-p})$, which is the posterior distribution associated with $g_p$ and $\pi_p$; draw $\xx^{miss}_p$ from $g_p(\xx^{miss}_p|
\xx^{obs}_p,\xx_{-p},\theta_p)$.
Iterative imputation has the practical advantage that, at each step, one only needs to set up a sensible regression model of $\xx_{j}$ given $\xx_{-j}$. This substantially reduces the modeling task, given that there are usually standard linear or generalized linear models for univariate responses of different variable types. In contrast, full Bayesian (or likelihood) modeling requires the more difficult task of constructing a joint model for $\xx$. Whether it is preferable to perform $p$ easy task or one difficult task, depends on the problem at hand. All that is needed here is the recognition that, in [*some*]{} settings, users prefer the $p$ easy steps of iterative imputation.
But iterative imputation has conceptual problems. Except in some special cases, there will not in general exist a joint distribution of $\xx$ such that $f(\xx_j |
\xx_{-j},\theta) = g_j (\xx_j|\xx_{-j},\theta_j)$ for each $j$. In addition, it is unclear whether the Markov process has a probability invariant distribution; if there is such a distribution, it lacks characterization.
In this paper, we discuss the properties of the stationary distribution of the iterative imputation process by first classifying the set of conditional models as compatible (defined as there existing a joint model $f$ which is consistent with all the conditional models) or incompatible.
We refer to the Markov chain generated by the scheme in Section \[SecFull\] as the *Gibbs chain* and that generated by the scheme in Section \[SecIter\] as the *iterative chain*. Our central analysis works by coupling the two.
Compatible conditional models {#SecCompatible}
=============================
Model compatibility
-------------------
Analysis of iterative imputation is particularly challenging partly because of the large collection of possible choices of conditional models. We begin by considering a restricted class, *compatible conditional models*, defined as follows:
\[DefCompatible\] A set of conditional models $\{g_j(x_j|x_{-j},\theta_j):
\theta_j \in \Theta_j, j = 1,...,p\}$ is said to be *compatible* if there exists a joint model $\{f(x|\theta): \theta\in \Theta\}$ and a collection of surjective maps, $\{t_j : \Theta \rightarrow \Theta_j
: j=1,...,p\}$ such that for each $j$, $\theta_j \in \Theta_j$, and $\theta \in t_j^{-1}(\theta_j)= \{\theta: t_j(\theta) = \theta_j\}$, $$g_j(x_j | x_{-j},\theta_j)=f(x_j | x_{-j},\theta).$$ Otherwise, $\{g_j: j =1,...,p\}$ is said to be *incompatible*.
Though imposing certain restrictions, compatible models do include quite a collection of procedures practically in use (e.g. [ice]{} in `Stata`). In what follows, we give a few examples of compatible and incompatible conditional models.
We begin with a simple linear model, which we shall revisit in Section \[SecLinear\].
\[ExToy\] Consider a binary continuous variable $(x,y)$ and conditional models $$x|y \sim N(\alpha_{x|y}+\beta_{x|y}y, \tau^2_x),\quad y|x \sim N(\alpha_{y|x}+\beta_{x|y}x, \tau^2_y).$$ These two conditional models are compatible if and only if $(\beta_{x|y},\beta_{x|y},\tau_x,\tau_y)$ lie on a subspace determined from the joint model, $$\left(
\begin{array}{c}
x \\
y\end{array}\right) \sim N\left( \left(
\begin{array}{c}
\mu_{x} \\
\mu_{y}\end{array}\right) ,\Sigma \right) ,\quad
$$ =(
[cc]{} \_[x]{}\^[2]{} & \_[x]{}\_[y]{}\
\_[x]{}\_[y]{} & \_[y]{}\^[2]{}
), with $\sigma_x, \sigma_y >0$ and $\rho \in [-1,1]$. The reparameterization from $(\mu_x,\mu_y,\sigma_x,\sigma_y,\rho)$ to the parameters of the conditional models is: $$\begin{aligned}
t_{1}(\mu_{x},\sigma_{x}^{2},\mu_{y},\sigma_{y}^{2},\rho
)=(\alpha_{x|y},\beta_{x|y},\tau_x^2)=\Big(\mu_{x}-\frac{\rho \sigma
_{x}}{\sigma_{y}}\mu_{y},\frac{\rho \sigma_{x}}{\sigma_{y}},(1-\rho
^{2})\sigma_{x}^{2}\Big)\\
t_{2}(\mu_{x},\sigma_{x}^{2},\mu_{y},\sigma_{y}^{2},\rho
)=(\alpha_{y|x},\beta_{y|x},\tau_y^2)=\Big(\mu_{y}-\frac{\rho \sigma
_{y}}{\sigma_{x}}\mu_{x},\frac{\rho \sigma_{y}}{\sigma_{x}},(1-\rho
^{2})\sigma_{y}^{2}\Big ).\end{aligned}$$
The following example is a natural extension.
\[ExLinear\] Consider a set of conditional linear models: for each $j$, $$x_j | x_{-j}, \beta_j, \sigma^2_j \sim N\left((\mathbf
1,x_{-j})\beta_j,\sigma^2_j \right),$$ where $\beta_j$ is a $p\times 1$ vector, $\mathbf 1 = (1,...,1)^\top$. Consider the joint model of $(x_1,...,x_p)\overset {i.i.d.}\sim N(\mu,
\Sigma)$. Then the conditional distribution of each $x_j$ given $x_{-j}$ is Gaussian. The maps $t_j$’s can be derived by conditional multivariate Gaussian calculations.
\[ExLogit\] Let $x_1$ be a Bernoulli random variable and $x_2$ be a continuous random variable. The conditional models are as follows: $$x_1|x_2 \sim Bernoulli \left(\frac{e^{\alpha + \beta
x_2}}{1+e^{\alpha + \beta x_2}}\right),\quad x_2|x_1
\sim N(\beta_0 + \beta_1 x_1 ,\sigma^2).$$ The above conditional models are compatible with the following joint model: $$x_1 \sim Bernoulli (p), \quad x_2|x_1
\sim N(\beta_0 + \beta_1 x_1 ,\sigma^2).$$ If we let $$\begin{aligned}
t_1 (p,\beta_0,\beta_1,\sigma^2)&=&\left(\log\frac p {1-p}
-\frac{\beta_1^2}{2\sigma^2}, \frac {\beta_1}{2
\sigma^2}\right)=(\alpha,\beta)\\
t_2 (p,\beta_0,\beta_1,\sigma^2)&=& (\beta_0,\beta_1),\end{aligned}$$ the conditional models and this joint model are compatible with each other. Similarly compatible models can be defined for other natural exponential families. See [@Efron75; @MN98].
\[ExCounter\] There are many incompatible conditional models. For instance, $$x|y \sim N(\beta_1 y+ \beta_2 y^2, 1),\quad y|x
\sim N(\lambda_1 x, 1),$$ are compatible only if $\beta_2 = 0$.
Total variation distance between two transition kernels {#SecPriorImp}
-------------------------------------------------------
Let $\{\xx^{mis,1}(k): k \in \mathbb Z^+\}$ be the Gibbs chain and $\{\xx^{mis,2}(k): k \in \mathbb Z^+\}$ be the iterative chain. Both chains live on the space of the missing data. We write the completed data as $\xx^{i}(k) = (\xx^{mis,i}(k),\xx^{obs})$ for the Gibbs chain ($i=1$) and the iterative chain ($i=2$). The transition kernels are $$\label{kernel}
K_i (w,dw')=P(\xx^{mis,i}(k+1) \in d w'|\xx^{mis,i}(k)=w), \mbox{ for } i=1,2.$$ where $w$ is a generic notation for the state of the processes. The transition kernels ($K_1$ and $K_2$) depend on $\xx^{obs}$. For simplicity, we omit the index of $\xx^{obs}$ in the notation of $K_i$. Also, we let $$K_i^{(k)} (\nu, A)\triangleq P_\nu(\xx^{mis,i}(k)\in A),$$ for $\xx^{mis,i}(0)\sim \nu$, $\nu$ being some starting distribution. The probability measure $P_{\nu}$ also depends on $\xx^{obs}$. Let $d_{TV}$ denote the total variation distance between two measures, that is, for two measures, $\nu_1$ and $\nu_2$, defined on the same probability space $$d_{TV}(\nu_1,\nu_2) = \sup_{A\in \mathcal F} |\nu_1 (A) - \nu_2 (A)|.$$ We further define $$\Vert\nu\Vert_V = \sup_{|h|\leq V}\int h(x) \nu (dx)$$ and $\Vert\nu\Vert_{\mathbf 1} = \Vert\nu\Vert_{V} $ for $V\equiv 1$. Let $\nu^{\xx^{obs}}_{i}$ be the stationary distribution of $K_{i}$. We intend to establish conditions under which $$d_{TV}(\nu^{\xx^{obs}}_{1}, \nu^{\xx^{obs}}_{2})\rightarrow 0$$ in probability as $n\rightarrow \infty$ and thus the iterative imputation and the joint Bayesian imputation are asymptotically the same.
Our basic strategy for analyzing the compatible conditional models is to first establish that the transition kernels $K_{1}$ and $K_{2}$ are close to each other in a large region $A_{n}$ (depending on the observed data $\xx^{obs}$), that is, $\Vert K_{1}(w,\cdot) - K_{2}(w,\cdot)\Vert_{\mathbf 1}\rightarrow 0$ as $n\rightarrow \infty$ for $w\in A_{n}$; and, second, to show that the two stationary distributions are close to each other in total variation in that the stationary distributions are completely determined by the transition kernels. In this subsection, we start with the first step, that is, to show that $K_{1}$ converges to $K_{2}$.
Both the Gibbs chain and the iterative chain evolve by updating each missing variable from the corresponding posterior predictive distributions. Upon comparing the difference between the two transition kernels associated with the simulation schemes in Sections \[SecFull\] and \[SecIter\], it suffices to compare the following posterior predictive distributions (for each $j=1,...,p$), $$\begin{aligned}
f(\xx_{j}^{mis}|\xx^{obs}_j,\xx_{-j}) &=&\int f(\xx_{j}^{mis}|\xx^{obs}_j,\xx_{-j},\theta )p(\theta |\xx^{obs}_j,\xx_{-j})d\theta \label{f} \\
g_{j}(\xx_{j}^{mis}|\xx^{obs}_j,\xx_{-j}) &=&\int g_{j}(\xx_{j}^{mis}|\xx^{obs}_j,\xx_{-j},\theta_{j})p_{j}(\theta_{j}|
\xx^{obs}_j,\xx_{-j})d\theta_{j},
\label{g}\end{aligned}$$where $p$ and $p_j$ denote the posterior distributions under $f$ and $g_j$ respectively. Due to compatibility, the distributions of the missing data given the parameters are the same for the joint Bayesian model and the iterative imputation model: $$f(\xx_{j}^{mis}|\xx^{obs}_j,\xx_{-j},\theta )=g_{j}(\xx_{j}^{mis}|\xx^{obs}_j,\xx_{-j},\theta_{j}),$$ if $t_{j}(\theta )=\theta_{j}$. The only difference lies in their posterior distributions. In fact, the $\Vert \cdot \Vert _{\mathbf 1}$ distance between two posterior predictive distributions is bounded by the distance between the posterior distributions of parameters. Therefore, we move to comparing $p(\theta|\xx^{obs}_j,\xx_{-j})$ and $p_j(\theta_j|\xx^{obs}_j,\xx_{-j})$.
#### Parameter augmentation.
Upon comparing the posterior distributions of $\theta$ and $\theta_{j}$, the first disparity to reconcile is that the dimensions are usually different. Typically $\theta_{j}$ is of a lower dimension. Consider the linear model in Example \[ExToy\]. The conditional models include three parameters (two regression coefficients and variance of the errors), while the joint model has five parameters $(\mu_{x},\mu_{y},\sigma_{x},\sigma_{y},\rho )$. This is because the (conditional) regression models are usually conditional on the covariates. The joint model not only parameterizes the conditional distributions of $\xx_{j}$ given $\xx_{-j}$ but also the marginal distribution of $\xx_{-j}$. Therefore, it includes extra parameters, although the distributions of the missing data is independent of these parameters. We augment the parameter space of the iterative imputation to $(\theta_{j},\theta_{j}^{\ast })$ with the corresponding map $\theta
_{j}^{\ast }=t_{j}^{\ast }(\theta )$. The augmented parameter $(\theta_{j},\theta_{j}^{\ast })$ is a non-degenerated reparameterization of $\theta $, that is, $T_{j}(\theta
)=(t_{j}(\theta ),t_{j}^{\ast }(\theta ))$ is a one-to-one (invertible) map. To illustrate this parameter augmentation, we consider the linear model in Example \[ExToy\] in which $\theta =(\mu_{x},\sigma_{x}^{2},\mu
_{y},\sigma_{y}^{2},\rho )$, where we use $\mu_{x}$ and $\sigma_{x}^{2}$ to denote mean and variance of the first variable, $\mu_{y}$ and $\sigma
_{y}^{2}$ to denote the mean and variance of the second, and $\rho $ to denote the correlation. The reparameterization is, $$\begin{aligned}
\theta_{2} &=&t_{2}(\mu_{x},\sigma_{x}^{2},\mu_{y},\sigma_{y}^{2},\rho
)=(\alpha_{y|x},\beta_{y|x},\tau_y^2)=(\mu_{y}-\frac{\rho \sigma
_{y}}{\sigma_{x}}\mu_{x},\frac{\rho \sigma_{y}}{\sigma_{x}},(1-\rho
^{2})\sigma_{y}^{2}),\\
\theta_{2}^{\ast } &=&t_{2}^{\ast }(\mu_{x},\sigma_{x}^{2},\mu
_{y},\sigma_{y}^{2},\rho )=(\mu_{x},\sigma_{x}^{2}).\end{aligned}$$The function $t_{2}$ maps to the regression coefficients and the variance of the residuals; $t_{2}^{\ast }$ maps to the marginal mean and variance of $x$. Similarly, we can define the map of $t_{1}$ and $t_{1}^{\ast
}$.
#### Impact of the prior distribution.
Because we are assuming compatibility, we can drop the notation $g_{j}$ for conditional model of the $j$-th variable. Instead, we unify the notation to that of the joint Bayesian model $f(\xx_{j}|\xx_{-j},\theta )$. In addition, we abuse the notation and write $f(\xx_{j}|\xx_{-j},\theta_{j})=f(\xx_{j}|\xx_{-j},\theta )$ for $t_{j}(\theta )=\theta_{j}$. For instance, in Example \[ExToy\], we write $f(y|x,\alpha_{y|x},\beta_{y|x},\sigma_{y|x})=f(y|x,\mu_{x},\mu_{y},\sigma_{x},\sigma_{y},\rho )$ as long as $\alpha_{y|x}=\mu_{y}-\frac{\rho \sigma_{y}}{\sigma_{x}}\mu
_{x}$, $\beta_{y|x}=\frac{\rho \sigma_{y}}{\sigma_{x}}$, and $\sigma
_{y|x}^{2}=(1-\rho ^{2})\sigma_{y}^{2}$.
The prior distribution $\pi$ on $\theta$ for the joint Bayesian model implies a prior on $(\theta_{j},\theta_{j}^{\ast })$, denoted by $$\pi_{j}^{\ast }(\theta_{j},\theta_{j}^{\ast })=|\det (\partial
T_{j}/\partial \theta )|^{-1}\pi (T_{j}^{-1}(\theta_{j},\theta_{j}^{\ast
})).$$For the full Bayesian model, the posterior distribution of $\theta_{j}$ is $$p(\theta_{j}|\xx^{obs}_j,\xx_{-j})=\int
p(\theta_{j},\theta_{j}^{\ast }|\xx^{obs}_j,\xx_{-j})d\theta_{j}^{\ast }\propto \int f(\xx^{obs}_j,\xx_{-j}|\theta_{j},\theta_{j}^{\ast })\pi
_{j}^{\ast }(\theta_{j},\theta_{j}^{\ast })d\theta_{j}^{\ast }.$$Because $f(\xx_{j}^{obs}|\xx_{-j},\theta
_{j},\theta_{j}^{\ast })=f(\xx_{j}^{obs}|\xx_{-j},\theta_{j})$, the above posterior distribution can be further reduced to $$p(\theta_{j}|\xx^{obs}_j,\xx_{-j})\propto f(\xx_{j}^{obs}|\xx_{-j},\theta_{j})\int f(\xx_{-j}|\theta_{j},\theta
_{j}^{\ast })\pi_{j}^{\ast }(\theta_{j},\theta_{j}^{\ast })d\theta
_{j}^{\ast }.$$If we write $$\pi_{j,\xx_{-j}}(\theta_{j})\triangleq \int f(\xx_{-j}|\theta_{j},\theta_{j}^{\ast })\pi_{j}^{\ast }(\theta_{j},\theta
_{j}^{\ast })d\theta_{j}^{\ast },$$then the posterior distribution of $\theta_j$ under the joint Bayesian model is $$p(\theta_{j}|\xx^{obs}_j,\xx_{-j})\propto f(\xx_{j}^{obs}|\xx_{-j},\theta_{j})\pi_{j,\xx_{-j}}(\theta_{j}).$$Compared with the posterior distribution of the iterative imputation procedure, which is proportional to $$p_{j}(\theta_{j}|\xx^{obs}_j,\xx_{-j})\propto g_{j}(\xx_{j}^{obs}|\xx_{-j},\theta_{j})\pi_{j}(\theta_{j})=f(\xx_{j}^{obs}|\xx_{-j},\theta_{j})\pi_{j}(\theta_{j}),$$the difference lies in the prior distributions, $\pi_{j}(\theta_{j})$ and $\pi_{j,\xx_{-j}}(\theta_{j})$.
#### Controlling the distance between the posterior predictive distributions.
We put forward tools to control the distance between the two posterior predictive distributions in and . Let $\xx$ be the generic notation the observed data, and let $f_{\xx}(\theta)$ and $g_{\xx}(\theta)$ be two posterior densities of $\theta$. Let $h(\tilde{x}|\theta )$ be the density function for future observations given the parameter $\theta $, and let $\tilde{f}_{\xx}(\tilde{x})$ and $\tilde{g}_{\xx}(\tilde{x})$ be the posterior predictive distributions: $$\tilde{f}_{\xx}(\tilde{x})=\int h(\tilde{x}|\theta )f_{\xx}(\theta )d\theta ,\quad \tilde{g}_{\xx}(\tilde{x})=\int h(\tilde{x}|\theta )g_{\xx}(\theta )d\theta .$$It is straightforward to obtain that $$\label{PropPred}
\Vert\tilde{f}_{\xx}-\tilde{g}_{\xx}\Vert_{\mathbf 1}\leq
\Vert f_{\xx}-g_{\xx}\Vert_{\mathbf 1}.$$ The next proposition provides sufficient conditions that $\Vert f_{\xx}-g_{\xx}\Vert_{\mathbf 1}$ vanishes.
\[PropPrior\] Let $n$ be the sample size. Let $f_{\xx}(\theta )$ and $g_{\xx}(\theta )$ be two posterior density functions that share the same likelihood but have two different prior distributions $\pi_{f}$ and $\pi_{g}$. Let $$L(\theta )=\frac{\pi_{g}(\theta)}{\pi_{f}(\theta )},\qquad r(\theta) = \frac{g_{\xx}(\theta)}{f_{\xx}(\theta)} = \frac{L(\theta)}{\int L(\theta) f_{\xx}(\theta)d \theta},$$ and $n$ denote sample size. Let $\partial L(\theta)$ be the partial derivative with respect to $\theta$ and let $\xi $ be a random variable such that $$L(\theta )=L(\mu_{\theta })+\partial L(\xi )\cdot (\theta -\mu_{\theta }),$$where “ $\cdot$” denotes inner product and $\mu_{\theta} = \int \theta f_{\xx}(\theta)d\theta$. If there exists a random variable $Z(\theta)$ with finite variance under $f_{\xx}$, such that $$\left\vert \sqrt n \partial L(\xi )\cdot (\theta -\mu
_{\theta})\right\vert\leq |\partial L(\mu_{\theta})|Z(\theta), \label{Dom}$$ then there exists a constant $\kappa>0$ such that for $n$ sufficiently large $$\label{pbd}\Vert\tilde{f}_{\xx}-\tilde{g}_{\xx}\Vert_{\mathbf 1}\leq \frac{\kappa \sqrt {|\partial \log L(\mu_{\theta})|}}{n^{1/4}}.
$$
We prove this proposition in Appendix \[Apdbd\].
\[RemLocal\] We adapt Proposition \[PropPrior\] to the analysis of the conditional models. Expresion implies that the posterior standard deviation of $\theta$ is $O(n^{-1/2})$. For most parametric models, is satisfied as long as the observed Fisher information is bounded from below by $\varepsilon n$ for some $\varepsilon >0$. In particular, we let $\hat \theta(\xx)$ be the complete-data MLE and $A_{n} = \{\xx: |\hat \theta(\xx)| \leq \gamma\}$. Then, is satisfied on the set $A_{n}$ for any fixed $\gamma$.
In order to verify that $\partial\log L(\theta)$ is bounded, one only needs to know $\pi_f$ and $\pi_g$ up to a normalizing constant. This is because the bound is in terms of $\partial L(\theta)/L(\theta)$. This helps to handle the situation when improper priors are used and it is not feasible to obtain a normalized prior distribution. In the current context, the prior likelihood ratio is $L(\theta_{j})= \pi_{j}(\theta_{j})/\pi_{j,\xx_{-j}}(\theta_{j})$.
If $L(\theta)$ is twice differentiable, the convergence rate in can be improved to $O(n^{-1/2})$. However, $O(n^{-1/4})$ is sufficient for the current analysis.
Convergence of the invariant distributions {#SecCgt}
------------------------------------------
With Proposition \[PropPrior\] and Remark \[RemLocal\], we have established that the transition kernels of the Gibbs chain and the iterative chain are close to each other in a large region $A_{n}$. The subsequent analysis falls into several steps. First, we slightly modify the processes by conditioning them on the set $A_{n}$ with stationary distributions $\tilde \nu^{\xx^{obs}}_{i}$ (details provided below). The stationary distributions of the conditional processes and the original processes ($\tilde \nu^{\xx^{obs}}_{i}$ and $\nu^{\xx^{obs}}_{i}$) are close in total variation. Second, we show (in Lemma \[ThmComp\]) that, with a bound on the convergence rate, $\tilde \nu^{\xx^{obs}}_{1}$ and $\tilde \nu^{\xx^{obs}}_{2}$ are close in total variation and so it is with $\nu^{\xx^{obs}}_{1}$ and $ \nu^{\xx^{obs}}_{2}$. The bound of convergence rate can be established by Proposition \[PropDrift\].
To proceed, we consider the chains conditional on the set $A_{n}$ where the two transition kernels are uniformly close to each other. In particular, for each set $B$, we let $$\label{condK}
\tilde K_{i} (w,B) = \frac { K_{i}(w,B\cap A_{n})}{ K_{i}(w,A_{n})}.$$ That is, we create another two processes, for which we update the missing data conditional on $\xx \in A_{n}$. The next lemma shows that we only need to consider the chains conditional on the set $A_{n}$.
\[LemCond\] Suppose that both $K_{1}$ and $K_{2}$ are positive Harris recurrent. We can choose $A_{n}$ as in the form of Remark \[RemLocal\] and $\gamma$ sufficiently large so that $$\label{condcgt}
\nu^{\xx^{obs}}_{i}(A_{n})\rightarrow 1$$ in probability as $n\rightarrow \infty$. Let $\tilde {\xx}^{mis,i}(k)$ be the Markov chains following $\tilde K_{i}$, defined as in , with invariant distribution $\tilde\nu^{\xx^{obs}}_{i}$. Then, $$d_{TV}(\nu^{\xx^{obs}}_{i},\tilde \nu^{\xx^{obs}}_{i})\rightarrow 0,$$ as $n\rightarrow \infty$.
The proof is elementary by the representation of $\nu_{i}^{\xx^{obs}}$ through the renewal theory and therefore is omitted. Based on the above lemma, we only need to show that $d_{TV}( \tilde \nu_{1}^{\xx^{obs}},\tilde \nu^{\xx^{obs}}_{2})\rightarrow 0$. The expression $\Vert K_{1}(w,\cdot)-K_{2}(w,\cdot)\Vert_{\mathbf 1}$ approaches 0 uniformly for $w\in A_{n}$. This implies that $$\Vert\tilde K_{1}(w,\cdot),\tilde K_{2}(w,\cdot)\Vert_{\mathbf 1}\rightarrow 0$$ as $n\rightarrow\infty$ uniformly for $w\in A_{n}$. With the above convergence, we use the following lemma to establish the convergence between $\tilde \nu_{1}^{\xx^{obs}}$ and $\tilde \nu^{\xx^{obs}}_{2}$.
\[ThmComp\] Let $\tilde{\xx}^{mis,i}(k)$ admit data-dependent transition kernels $\tilde K_i$ for $i=1,2$. We use $n$ to denote sample size. Suppose that each $\tilde K_i$ admits a data-dependent unique invariant distribution, denoted by $\tilde \nu^{\xx^{obs}}_i$, and that the following two conditions hold:
1. The convergence of the two transition kernels $$d(A_{n})\triangleq\sup_{w\in A_{n}}\Vert\tilde K_{1}(w,\cdot )-\tilde K_{2}(w,\cdot )\Vert_{V}\rightarrow
0, \label{BdKernal}$$in probability as $n\rightarrow \infty $. The function $V$ is either a geometric drift function for $ \tilde K_2$ or a constant, i.e., $V=1$.
2. Furthermore, there exists a monotone decreasing sequence $r_k \rightarrow 0$ (independent of data) and a starting measure $\nu$ (depending on data) such that $$\label{BdProb}
P\left[\Vert \tilde K_i^{(k)}(\nu,\cdot)-\tilde \nu^{\xx^{obs}}_i(\cdot)\Vert_V\leq r_k,\forall
k>0\right] \rightarrow 1,$$ as $n\rightarrow \infty$.
Then, $$\label{CgtMea}
\Vert\tilde \nu^{\xx^{obs}}_1-\tilde \nu^{\xx^{obs}}_2\Vert_V\rightarrow 0,$$ in probability as $n\rightarrow \infty$.
The above lemma holds if $V=1$ or $V$ is a drift function. For the analysis of convergence in total variation, we only need that $V=1$. The results when $V$ is a drift function is prepared for the analysis of incompatible models.
The first condition in the above lemma has been obtained by the result of Proposition \[PropPrior\]. Condition is more difficult to establish. According to the standard results in [@Ros95] (see also in the appendix), one set of sufficient conditions for is that the chains $\tilde K_{1}$ and $\tilde K_{2}$ admit a common small set, $C$; in addition, each of them admits their own drift functions associated with the small set $C$ (c.f. Appendix \[SecMC\]).
Gibbs chains typically admit a small set $C$ and a drift function $V$, that is, for some positive measure $\mu$ $$\label{small}
\tilde K_1(w,A)\geq q_{1} \mu_{1}(A),$$ for $w\in C$, $q_{1}\in (0,1)$; for some $\lambda_{1}\in (0,1)$ and for all $w\notin C$ $$\label{drift1}
\lambda_1 V(w)\geq\int V(w') \tilde K_{1}(w,dw').$$ With the existence of $C$ and $V$ a bound of convergence $r_{k}$ (with starting point $w\in C$) can be established for the Gibbs chain by standard results (see, for instance, [@Ros95]), and $r_{k}$ only depends on $\lambda_{1}$ and $q_{1}$. Therefore, it is necessary to require that $\lambda_{1}$ and $q_{1}$ are independent of $\xx^{obs}$. In contrast, the small set $C$ and drift function $V$ could be data-dependent.
Given that $\tilde K_{1}$ and $\tilde K_{2}$ are close in “$\Vert \cdot \Vert_{\mathbf 1}$”, the set $C$ is also a small set for $\tilde K_{2}$, that is $\tilde K_2(w,A)\geq q_{2} \mu_{2}(A),$ for some $q_{2}\in (0,1)$, all $w\in C$, and all measurable set $A$. The following proposition, whose proof is given in the appendix, establishes the conditions under which $V$ is also a drift function for $\tilde K_{2}$.
\[PropDrift\] Assume the following conditions hold.
1. The transition kernel $\tilde K_{1}$ admits a small set $C$ and a drift function $V$ satisfying .
2. Let $L_{j}(\theta_{j}) = \pi_j(\theta_j)/\pi_{j,\xx_{-j}}(\theta_j)$ ($j=1,...,p$) be the the ratio of prior distributions for each conditional model (possibly depending on the data) so that on the set $A_{n}$ $\sup_{|\theta_{j}|<\gamma}\partial L_{j}(\theta_{j})/L_{j}(\theta_{j}) <\infty.$
3. For each $j$ and $1\leq k \leq p-j$, there exists a $Z_{j}(\theta_{j})$ serving as the bound in for each $L_{j}$. In addition, $Z_{j}$ satisfies the following moment condition $$\label{cond2}
\tilde E_{1}\left[~ Z_{j+1}^{2}(\theta_{j+1}) V^{2}(w_{j+k})~|~w_{j} ~\right]= o( n)V^{2}(w_{j}), \qquad$$ where $ \tilde E_{1}$ is the expectation associated with the updating distribution of $\tilde K_{1}$ and $w_{j}$ is the state of the chain when the $j$-th variable is just updated. The convergence $o(n)/n \rightarrow 0$ is uniform in $w_{j}\in A_{n}$.
Then, there exits $\lambda_2\in (0,1)$ such that as $n$ tends to infinity with probability converging to one the following inequality holds $$\label{Lyp}\lambda_2 V(w)\geq\int V(w') \tilde K_{2}(w,dw').$$
The intuition of the above proposition is as follows. $V$ satisfying inequality is a drift function of $\tilde K_{1}$ to $C$. Since the $\tilde K_{1}$ and $\tilde K_{2}$ are close to each other, we may expect that $\int V(w') \tilde K_{1}(w,dw') \approx \int V(w') \tilde K_{2}(w,dw')$. The above proposition basically states the conditions under which this approximation is indeed true and suggests that $V$ be a drift function of $\tilde K_{2}$ if it is a drift function of $\tilde K_{1}$. Condition is imposed for a technical purpose. In particular, we allow the expectation of $ Z^{2}_{j+1}(\theta_{j+1}) V^{2}(w_{j+k})$ to grow to infinity but at a slower rate than $ n$. Therefore, it is a mild condition.
We now summarize the analysis and the results of the compatible conditional models in the following theorem.
\[ThmCompRev\] Suppose that a set of conditional models $\{g_{j}(x_{j}|x_{-j},\theta_{j}): \theta_{j}\in \Theta_{j},j=1,...,p\}$ is compatible with a joint model $\{f(x|\theta): \theta \in \Theta\}$. The Gibbs chain and the iterative chain then admit transition kernels $K_{i}$ and unique stationary distributions $\nu^{\xx^{obs}}_{i}$. Suppose the following conditions are satisfied:
- Let $A_{n} = \{\xx: |\hat \theta(\xx)| \leq \gamma\}$. One can choose $\gamma$ sufficiently large so that $\nu^{\xx^{obs}}_{i}(A_{n})\rightarrow 0,$ in probability as $n\rightarrow \infty$.
- The conditions in Proposition \[PropDrift\] hold.
Then, $$d_{TV}(\nu_{1}^{\xx^{obs}}, \nu_{2}^{\xx^{obs}})\rightarrow 0$$ in probability as $n\rightarrow \infty$.
One sufficient condition for A1 is that the stationary distributions of $\hat \theta(\xx)$ under $\nu^{\xx^{obs}}_{i}$ converge to a value $\theta^{i}$, where $\theta^{1}$ and $\theta^{2}$ are not necessarily the same.
In addition to the conditions of Proposition \[PropPrior\], Proposition \[ThmComp\] also requires that one constructs a drift function towards a small set for the Gibbs chain. One can usually construct $q_{1}$ and $\lambda_{1}$ free of data if the proportion of missing data is bounded from the above by $1-\varepsilon$. The most difficult task usually lies in constructing a drift function. For illustration purpose, we construct a drift function (in the supplement material) for the linear example in Section \[SecLinear\].
We summarize the analysis of compatible models in this proof. If $g_{j}$’s are compatible with $f$, then the conditional posterior predictive distributions of the Gibbs chain and the iterative chain are given in and . Thanks to compatibility, the “$\Vert \cdot\Vert_{\mathbf 1}$” distance between the posterior predictive distributions are bounded by the distance between the posterior distributions of their own parameters as in .
On the set $A_{n}$, the Fisher information of the likelihood has a lower bound of $\varepsilon n$ for some $\varepsilon$. Then, by Proposition \[PropPrior\] and the second condition in Proposition \[PropDrift\], the distance between the two posterior distributions is of order $O(n^{-1/4})$ uniformly on set $A_{n}$. Similar convergence result holds for the conditional transition kernels, that is, $\Vert\tilde K_{1}(w,\cdot)-\tilde K_{2}(w,\cdot)\Vert_{\mathbf 1}\rightarrow 0$. Thus, the first condition in Lemma \[ThmComp\] has been satisfied.
To verify the conditions of Proposition \[PropDrift\], one needs to construct a small set $C$ such that holds for both chains and a drift function $V$ for one of the two chains such that holds. Based on the results of Proposition \[PropDrift\], $\tilde K_{1}$ and $\tilde K_{2}$ share a common data-dependent small set $C$ with $q_{i}$ independent of data and a drift function $V$ (possibly with different rate $\lambda_1$ and $\lambda_2$).
According to the standard bound of Markov chain rate of convergence (for instance, [@Ros95] and in the appendix), there exists a common starting value $w\in C$ and a bound $r_{k}$ such that the bound in Lemma \[ThmComp\] is satisfied. Thus both conditions in Lemma \[ThmComp\] have been satisfied and further $$d_{TV}(\tilde\nu_{1}^{\xx^{obs}},\tilde\nu_{2}^{\xx^{obs}})\rightarrow 0,$$ in probability as $n\rightarrow \infty$. According to condition A1 and Lemma \[LemCond\], the above convergence implies that $$d_{TV}(\nu_{1}^{\xx^{obs}},\nu_{2}^{\xx^{obs}})\rightarrow 0.$$ Thereby, we conclude the analysis.
On the necessity of model compatibility {#SecNec}
---------------------------------------
Theorem \[ThmCompRev\] shows that for compatible models and under suitable technical conditions, iterative imputation is asymptotically equivalent to Bayesian imputation. The following theorem suggests that model compatibility is typically necessary for this convergence.
Let $P^{f}$ denote the probability measure induced by the posterior predictive distribution of the joint Bayesian model and $P^{g}_j$ denote those induced by the iterative imputation’s conditional models. That is, $$\begin{aligned}
P^{f}(\xx_j^{mis}\in A|\xx^{mis}_{-j},\xx^{obs})&=& \int_A f(\xx_{j}^{mis}|\xx^{mis}_{-j},\xx^{obs},\theta )p(\theta |\xx^{mis}_{-j},\xx^{obs})d\theta\\
P^{g}_j(\xx_j^{mis}\in A|\xx^{mis}_{-j},\xx^{obs})&=& \int_A g_j(\xx_{j}^{mis}|\xx^{mis}_{-j},\xx^{obs},\theta )p_j(\theta |\xx^{mis}_{-j},\xx^{obs})d\theta.\end{aligned}$$ Furthermore, denote the stationary distributions of the Gibbs chain and the iterative chain by $\nu_1^{\xx^{obs}}$ and $\nu_2^{\xx^{obs}}$.
\[PropNece\] Suppose that for some $j\in \mathbb Z^+$, sets $A$ and $C$, and $\varepsilon\in(0,1/2)$ $$\inf_{\xx^{mis}_{-j}\in C}P^{g}_j(\xx_j^{mis}\in A|\xx^{mis}_{-j},\xx^{obs})>
\sup_{\xx^{mis}_{-j}\in C}P^{f}(\xx_j^{mis}\in A|\xx^{mis}_{-j},\xx^{obs})+\varepsilon$$ or $$\sup_{\xx^{mis}_{-j}\in C}P^{g}_j(\xx_j^{mis}\in A|\xx^{mis}_{-j},\xx^{obs})<
\inf_{\xx^{mis}_{-j}\in C}P^{f}(\xx_j^{mis}\in A|\xx^{mis}_{-j},\xx^{obs})-\varepsilon$$ and $\nu_1^{\xx^{obs}}(\xx_{-j}^{mis}\in
C)>q \in (0,1)$.
Then there exists a set $B$ such that $$\left|\nu_2^{\xx^{obs}}(\xx^{mis}\in B) - \nu_1^{\xx^{obs}}(\xx^{mis}\in
B)\right|> q \varepsilon /4.$$
Suppose that $$\inf_{\xx^{mis}_{-j}\in C}P^{g}_j(\xx_j^{mis}\in A|\xx^{mis}_{-j},\xx^{obs})>
\sup_{\xx^{mis}_{-j}\in C}P^{f}(\xx_j^{mis}\in A|\xx^{mis}_{-j},\xx^{obs})+\varepsilon,$$ The “less than” case is completely analogous. Consider the set $B=\{\xx^{mis}: \xx_{-j}^{mis}\in C,
\xx_{j}^{mis}\in A\}$. If $$\label{CC}|\nu_2^{\xx^{obs}}(\xx^{mis}_{-j}\in C) -
\nu_1^{\xx^{obs}}(\xx^{mis}_{-j}\in C)| \leq q\varepsilon/2 ,$$then, by the fact that $$\begin{aligned}
\nu_1^{\xx^{obs}}(\xx^{mis}\in B)=\nu_1^{\xx^{obs}}(\xx^{mis}_{-j}\in C)\int P^{f}(\xx_j^{mis}\in
A|\xx^{mis}_{-j},\xx^{obs}) \nu_1^{\xx^{obs}}(d\xx^{mis}_{-j}|\xx^{mis}_{-j}\in C),\\
\nu_2^{\xx^{obs}}(\xx^{mis}\in B)=\nu_2^{\xx^{obs}}(\xx^{mis}_{-j}\in C)\int P^{g}(\xx_j^{mis}\in
A|\xx^{mis}_{-j},\xx^{obs}) \nu_2^{\xx^{obs}}(d\xx^{mis}_{-j}|\xx^{mis}_{-j}\in C),\end{aligned}$$ we obtain $$|\nu_2^{\xx^{obs}}(\xx^{mis}\in B) - \nu_1^{\xx^{obs}}(\xx^{mis}\in
B)|> q \varepsilon /4.$$ Otherwise, if does not hold, let $B=\{\xx^{mis}: \xx^{mis}_{-j}\in
C\}$.
For two models with different likelihood functions, one can construct sets $A$ and $C$ such that the conditions in the above theorem hold. Therefore, if among the predictive distributions of all the $p$ conditional models there is one $g_j$ that is different from $f$ as stated in Theorem \[PropNece\], then the stationary distribution of the iterative imputation is different from the posterior distribution of the Bayesian model in total variation by a fixed amount. For a set of incompatible models and any joint model $f$, there exists at least one $j$ such that the conditional likelihood functions of $\xx_j$ given $\xx_{-j}$ are different for $f$ and $g_j$. Their predictive distributions have to be different for $\xx_j$. Therefore, such an iterative imputation using incompatible conditional models typically does not correspond to Bayesian imputation under any joint model.
Incompatible conditional models {#SecIncomp}
===============================
In this section, we proceed to the discussion of incompatible conditional models. We first extend the concept of model compatibility to semi-compatibility which includes the regression models we have generally seen in practical uses of iterative imputation. We then introduce the validity of semi-compatible models. Finally, we show that if the conditional models are semi-compatible and valid (together with a few mild technical conditions) the combined imputation estimator is consistent.
Semi-compatibility and model validity
-------------------------------------
As in the previous section, we assume that the invariant distribution exists. For compatible conditional models, we used the posterior distribution of the corresponding Bayesian model as the natural benchmark and show that the two imputation distributions converge to each other. We can use this idea for the analysis of incompatible models. In this setting, the first issue is to find a natural Bayesian model associated with a set of incompatible conditional models. Naturally, we introduce the concept of semi-compatibility.
\[defSemiComp\] A set of conditional models $\{h_j(x_j|x_{-j},\theta_j,\varphi_j): j=1,...,p\}$, each of which is indexed by two sets of parameters $(\theta_j,\varphi_j)$, is said to be *semi-compatible*, if there exists a set of compatible conditional models $$\label{SemiComp}
g_j(x_j|x_{-j},\theta_j)=h_j(x_j|x_{-j},\theta_j,\varphi_j=0),$$ for $j=1,...,p$. We call $\{g_j:j=1,...,p\}$ a *compatible element* of $\{h_j: j=1,...,p\}$.
By definition, every set of compatible conditional models is semi-compatible. A simple and uninteresting class of semi-compatible models arises with iterative regression imputation. As typically parameterized, these models include complete independence as a special case. A *trivial* compatible element, then, is the one in which $x_j$ is independent of $x_{-j}$ under $g_j$ for all $j$. Throughout the discussion of this section, we use $\{g_j: j=1,...,p\}$ to denote the compatible element of $\{h_j:j=1,...,p\}$ and $f$ to denote the joint model compatible with $\{g_j:j=1,...,p\}$.
Semi-compatibility is a natural concept connecting a joint probability model to a set of conditionals. One foundation of almost all statistical theories is that data are generated according to some (unknown) probability law. When setting up each conditional model, the imputer chooses a rich family that is intended to include distributions that are close to the true conditional distribution. For instance, as recommended by [@Meng1994], the imputer should try to include as many predictors as possible (using regularization as necessary to keep the estimates stable). Sometimes, the degrees of flexibility among the conditional models are different. For instance, some includes quadratic terms or interactions. This richness usually results in incompatibility. Semi-compatibility includes such cases in which the conditional models are rich enough to include the true model but may not be always compatible among themselves. To proceed, we introduce the following definition.
\[DefValid\] Let $\{h_j:j=1,...,p\}$ be semi-compatible, $\{g_j:j=1,...,p\}$ be its compatible element, and $f$ be the joint model compatible with $g_j$. If the joint model $f(x|\theta)$ includes the true probability distribution, we say $\{h_j:j=1,...,p\}$ is a set of *valid semi-compatible models*.
In order to obtain good prediction, we need the validity of the semi-compatible models. A natural issue is the performance of valid semi-compatible models. Given that we have given up compatibility, we should not expect the iterative imputation to be equivalent to any joint Bayesian imputation. Nevertheless, under mild conditions, we are able to show the consistency of the combined imputation estimator.
Main theorem of incompatible conditional models
-----------------------------------------------
Now, we list a set of conditions:
1. Both the Gibbs and iterative chains admit their unique invariant distributions, $\nu_1^{\xx^{obs}}$ and $\nu_2^{\xx^{obs}}$.
2. The posterior distributions of $\theta $ (based on $f$) and $(\theta_j, \varphi_j)$ (based on $h_j$) given a complete data set $\xx$ have the representation $|\theta -\tilde \theta| \leq \xi n^{-1/2}, |(\theta_{j}-\tilde \theta_{j},\varphi_j - \tilde \varphi_j)|\leq \xi_{j} n^{-1/2},$ where $\tilde \theta$ is the maximum likelihood estimate of $f(\xx|\theta)$, $(\tilde \theta_j, \tilde \varphi_j)$ is the maximum likelihood estimate of $h_j(\xx_j|\xx_{j},\theta_j,\varphi_j)$, and $Ee^{|\xi_{j}|}\leq \kappa $, $E e^{|\xi|} \leq \kappa$ for some $\kappa >0$.
3. All the score functions have finite moment generating functions under $f(\xx^{mis} |\xx^{obs}, \theta)$.
4. For each variable $j$, there exists a subset of observations $\iota_j$ so that for each $i \in\iota_{j}$ $x_{i,j}$ is missing and $x_{i,-j}$ is fully observed. In addition, the cardinality $\#(\iota_j) \rightarrow \infty$ as $n\rightarrow \infty$.
Conditions B2 and B3 impose moment conditions on the posterior distribution and the score functions. They are satisfied by most parametric families. Condition B4 rules out certain boundary cases of missingness patterns and is imposed for technical purposes. The condition it is not very restrictive because it only requires that the cardinality of $\iota_{j}$ tends to infinity, not necessarily even of order of $O(n)$.
We now express the fifth and final condition which requires the following construction. Assume the conditional models are valid and that the data $\xx$ is generated from $f(\xx|\theta^0)$. We use $\theta_{j}^{0}=t_{j}(\theta ^{0})$ and $\varphi _{j}^{0}=0$ to denote the true parameters under $h_{j}$. We define $$\label{EE}
\hat{\theta}=\sup_{\theta }f(\xx^{obs}|\theta ),\qquad\hat{\theta}_{j}=t_{j}(\hat{\theta}),$$ be the observed-data MLE and $$\label{note}
\hat\theta^{(2)} = \arg\sup_\theta \int \log f(\xx|\theta) \nu^{\xx^{obs}}_2 (d\xx^{mis}),\quad
(\hat\theta_j^{(2)},\hat\varphi_j^{(2)}) = \arg\sup_{\theta_j,\varphi_j} \int \log h_j(\xx_j|\xx_{-j},\theta_j,\varphi_j) \nu^{\xx^{obs}}_2 (d\xx^{mis})$$ where $\xx= (\xx^{obs},\xx^{mis})$.
Consider a Markov chain $x^{\ast }(k)$ corresponding to one observation—one row of the data matrix—living on $R^{p}$. The chain evolves as follows. Within each iteration, each dimension $j$ is updated conditional on the others according to the conditional distribution $$h_{j}(x_{j}|x_{-j},\theta _{j},\varphi _{j}),$$where $(\theta _{j},\varphi _{j})=(\hat{\theta}_{j},0)+\varepsilon \xi _{j}$ and $\xi_j$ is a random vector with finite MGF (independent of everything at every step). Alternatively, one may consider $(\theta_{j},\varphi_{j})$ as a sample from the posterior distribution corresponding to the conditional model $h_{j}$. Thus, $x^*(k)$ is the marginal chain of one observation in iterative chain. Given that $\xx^{mis,2}(k)$ admits a unique invariant distribution, $x^{\ast }(k)$ admits its unique stationary distribution for $\varepsilon$ sufficiently small. Furthermore, consider that $x(k)$ is a Gibbs sampler and it admits stationary distribution $f(x|\hat{\theta})$, that is, each component is updated according to the conditional distribution $f(x_j|x_{-j},\hat \theta)$ and the parameters of the updating distribution are set at the observed data maximum likelihood estimate, $\hat \theta$. The last condition is stated as follows.
1. $x^{\ast }(k)$ and $x(k)$ satisfy conditions in Lemma \[ThmComp\] as $\varepsilon \rightarrow 0$, that is, the invariant distributions of $x^{*}(k)$ and $x(k)$ converges in $\Vert\cdot \Vert_{V}$ norm, where $V$ is a drift function for $x^*(k)$. There exists a constant $\kappa$ such that all the score functions are bounded by $$\partial\log f(x| \theta^{0})\leq \kappa V(x),\qquad \partial\log h_j(x_j|x_{-j},t_{j}(\theta^{0}),\varphi_j=0)\leq \kappa V(x) .$$
By choosing $\varepsilon$ small, the transition kernels of $x^*(k)$ and $x(k)$ converge to each other. Condition B5 requires that Lemma \[ThmComp\] applies in this setting, that their invariant distributions are close in the sense stated in the Lemma. This condition does not suggest that Lemma \[ThmComp\] applies to $\nu^{\xx^{obs}}_1$ and $\nu^{\xx^{obs}}_2$, which represents the joint distribution of many such $x^*(k)$’s and $x(k)$’s.
We can now state the main theorem in this section.
\[ThmIncomp\] Consider a set of valid semi-compatible models $\{h_{j}:j=1,...,p\}$, and assume conditions B1–5 are in force. Then, following the notations in , the following limits hold: $$\label{consist}
\hat\theta^{(2)}\rightarrow \theta ^{0},\qquad
\hat\theta_j^{(2)}\rightarrow t_j(\theta^0),
\quad \hat\varphi_j^{(2)}\rightarrow 0,$$ in probability as sample size $n\rightarrow \infty$ for all $j$.
The expression $\hat \theta^{(2)}$ corresponds to the following estimator. Impute the missing data from distribution $\nu^{\xx^{obs}}_{2}$ $m$ times to obtain $m$ complete datasets. Stack the $m$ datasets to one big dataset. Let $\hat \theta^{(2)}_{m}$ be the maximum likelihood estimator based on the big dataset. Then, $\hat \theta^{(2)}_{m}$ converges to $\hat \theta^{(2)}$ as $m\rightarrow \infty$. Furthermore, $\hat \theta^{(2)}$ is asymptotically equivalent to the combined point estimator of $\theta$ according to Rubin’s combining rule (with infinitely many imputations). Similarly, $(\hat \theta_j^{(2)}, \hat \varphi_j^{(2)})$ is asymptotically equivalent to the combined estimator of the conditional model. Therefore, Theorem \[ThmIncomp\] suggests that the combined imputation estimators are consistent under conditions B1–5.
Linear example {#SecLinear}
==============
A simple set of compatible conditional models
---------------------------------------------
In this subsection, we study a linear model as an illustration. Consider $n$ i.i.d. bivariate observations $(\xx,{\mbox{$\mathbf y$}})=\{(x_i,y_i): i=1,...,n\}$ and a set of conditional models $$\label{IterLinear}x_i | y_i
\sim N(\beta_{x|y} y_i, \tau_x^2), \quad y_i |x_i \sim N(\beta_{y|x}
x_i, \tau_y^2).$$ To simplify the discussion, we set the intercepts to zero. As discussed previously, the joint compatible model assumes that $(x,y)$ is a bivariate normal random variable with mean zero, variances $\sigma_x^2$ and $\sigma_y^2$, and correlation $\rho$. The reparameterization from the joint model to the conditional model of $y$ given $x$ is $$\beta_{y|x}= \frac{\sigma_y}{\sigma_x}\rho, \quad \tau_y^2 = (1-\rho^2) \sigma_y^2.$$ Figure \[FigData\] displays the missingness pattern we are assuming for this simple example, with $a$ denoting the set of observations for which both $x$ and $y$ are observed, $b$ denote those with missing $y$’s, and $c$ denoting those with missing $x$’s; $n_a$, $n_b$, and $n_c$ denote their respective sample sizes, and $n= n_a + n_b + n_c$. To keep the example simple, we assume that there are no cases for which both $x$ and $y$ are missing.
![Missingness pattern for our simple example with two variables. Gray and white areas indicate observed and missing data, respectively. This example is constructed so that there are no cases for which both variables are missing.[]{data-label="FigData"}](Linear-bak.png){height="2in"}
#### Positive recurrence and limiting distributions.
The Gibbs chain and the iterative chain admit a common small set containing the observed-data maximum likelihood estimate. The construction of the drift functions is tedious and is not particularly relevant to the current discussion, and so we leave their detailed derivations to the supplemental materials available at http://stat.columbia.edu/$\sim$jcliu/paper/driftsupp.pdf. We proceed here by assuming that they are in force.
#### Total variation distance between the kernels.
The results for incompatible models apply here. Thus, condition A1 in Theorem \[ThmCompRev\] has been satisfied. We now check the boundedness of $\partial L(\theta)$. The posterior distribution of the full Bayes model is $$\begin{aligned}
p(\sigma_x^2,\tau_y^2,\beta_{y|x}|\xx,{\mbox{$\mathbf y$}})&\propto& f( \xx,{\mbox{$\mathbf y$}}|\sigma_x^2,\tau_y^2, \beta_{y|x})
\pi^*(\sigma_x^2,\tau_y^2, \beta_{y|x}) \\
&=&f({\mbox{$\mathbf y$}}|\tau_y^2,\beta_{y|x},\xx)
f(\xx | \sigma_x^2)\pi^*(\sigma_x^2,\tau_y^2, \beta_{y|x}).\end{aligned}$$ The posterior distribution of $(\tau_y^2,\beta_{y|x})$ with $\sigma_x^2$ integrated out is $$p(\tau_y^2,\beta_{y|x}|\xx,{\mbox{$\mathbf y$}})\propto f({\mbox{$\mathbf y$}}|\tau_y^2,\beta_{y|x},\xx)\pi_{\xx}(\beta_{y|x},\tau_y^2),$$ where $$\pi_{\xx}(\beta_{y|x},\tau_y^2) \propto \int f(\xx|
\sigma_x^2)\pi^*(\sigma_x^2,\tau_y^2, \beta_{y|x}) d\sigma_x^2.$$ The next task is to show that $\pi_{\xx}(\beta_{y|x},\tau_y^2)$ is a diffuse prior satisfying the conditions in Proposition \[PropPrior\]. We impose the following independent prior distributions on $\sigma_x^2$, $\sigma_y^2$, and $\rho$: $$\label{prior}
\pi(\sigma_x^2,\sigma_y^2,\rho )\propto \sigma_x\sigma_y I_{[-1,1]}(\rho).$$ The distribution of $\xx$ does not depend on $(\sigma_y^2,\rho)$. Therefore, under the posterior distribution given $\xx$, $\sigma_x^2$ and $(\sigma_y^2,\rho)$ are independent. Conditional on $\xx $, $\sigma_x^2$ is inverse-gamma. Now we proceed to develop the conditional/posterior distribution of $(\tau_y^2, \beta_{y|x})$ given $\xx$. Consider the following change of variables $$\sigma_y^2 = \tau_y^2 + \beta_{y|x}^2 \sigma_x^2, \quad \rho = \beta_{y|x}\sqrt{\frac{
\sigma_x^2}{\tau_y^2 + \beta_{y|x}^2 \sigma_x^2}}.$$Then, $$\det \left(\frac{\partial (\sigma_y^2,\rho,\sigma_x^2)}{\partial
(\tau_y^2, \beta_{y|x},\sigma_x^2)}\right)= \frac{\sigma_x }{\sqrt{\tau_y^2 + \beta_{y|x}^2 \sigma_x^2}}.$$ Together with $$\pi(\sigma_y^2, \rho^2)\propto \sigma_y^{},$$ we have $$\begin{aligned}
\pi_{\xx}(\tau_y^2, \beta_{y|x})&\propto& \int \det \left(\frac{\partial (\sigma_y^2,\rho,\sigma_x^2)}{\partial
(\tau_y^2, \beta_{y|x},\sigma_x^2)}\right) \pi(\sigma^2_y, \rho) p(\sigma_x^2|\xx )d\sigma_x^2 \\&=&\int \sigma_x p(\sigma_x^2|\xx )d\sigma_x^2= C(\xx).\end{aligned}$$
If one chooses $\pi_2 (\tau_y^2,\beta_{y|x})\propto 1$ for the iterative imputation and (\[prior\]) for the joint Bayesian model, the iterative chain and the Gibbs chain happen to have identical transition kernels and, therefore, identical invariant distributions. This is one of the rare occasions that these two procedures yield identical imputation distributions.
If one chooses Jeffreys’ prior, $\pi_2 (\tau_y^2,\beta_{y|x})\propto \tau_y^{-2}$, then $$L(\tau_y^2,\beta_{y|x})=\frac{\pi_{\xx}(\tau^2_y,\beta_{y|x})}{\pi_2 (\tau_y^2,\beta_{y|x})}\propto \tau_y^2,$$ and $\partial L$ is bounded in a suitably chosen compact set containing the true parameters. Thus, Theorem \[ThmCompRev\] applies.
#### Empirical convergence check.
![Quantile-quantile plots demonstrating the closeness of the posterior distribution of the Bayesian model and the compatible iterative imputation distributions for $\beta_{x}$ and $\beta_{y}$ with sample size $n_a=200$.[]{data-label="FigPost200"}](postbeta200.png){height="2in"}
To numerically confirm the convergence of the two distributions, we generate the following data sets. To simplify analysis, let $(x_i,y_i)$’s be bivariate Gaussian random vectors with mean zero, variance one, and correlation zero. We set $n_a = 200$, $n_b = 80$, and $n_c = 80$. For the iterative imputation we use Jeffreys’ prior $p(\tau_y^2,\beta_{y|x})\propto \tau_y^{-2}$ and $p(\tau_x^2,\beta_{x|y})\propto \tau_x^{-2}$. For the full Bayesian model, the prior distribution is chosen as in (\[prior\]).
We monitor the posterior distributions of the following statistics: $$\label{monitor}\beta_{x}= \frac{\sum_{i\in b}x_iy_i}{\sum_{i\in
b}y_i^2},\quad \beta_{y}= \frac{\sum_{i\in
c}x_iy_i}{\sum_{i\in c}x_i^2}.$$ Figures \[FigPost200\] shows the quantile-quantile plots of the distributions of $\beta_x$ and $\beta_y$ under $\nu_1^{\xx^{obs}}$ and $\nu_2^{\xx^{obs}}$ based on $1$ million MCMC iterations. The differences between these two distributions are tiny.
Higher-dimensional linear models
--------------------------------
We next consider a more complicated and realistic situation, in which there are $p$ continuous variables, $x_{1},...,x_{p}$. Each conditional model is linear in the sense that, for each $j$, $$x_{j}|x_{-j}\sim N((1, x_{-j}^{\top})\beta_{j},\sigma_{j}^{2}),$$ which is the set of compatible models presented in Example \[ExLinear\].
In the simulation, we generate 1000 samples of $(x_{1},...,x_7)$ from a 7-dimensional multivariate normal distribution with mean 0 and covariance matrix that equals 1 on the diagonals and 0.4 on the off-diagonal elements. We then generate another variable $y\sim N( -2 + x_1 + x_2 + x_3 + x_4 - x_5 - x_6 - x_7, 1)$. Hence the dataset contains $y, x_1, x_2, \ldots, x_7$. For each variable, we randomly select $30\%$ of the observations and set them to be missing. Thus, the missing pattern of the dataset is missing completely at random (MCAR). We impute the missing values in two ways: iterative imputation and a multivariate Gaussian joint Bayesian model. After imputation, we use the imputed datasets and regress $y$ on all $x$’s to obtain the regression coefficients. The quantile-quantile plots in Figure \[fig:normbymi\] compare the imputation distribution of the least-square estimates of the regression coefficients of the iterative imputation procedure and the multivariate Gaussian joint model.
![Quantile-quantile plots of the imputation distributions of the regression coefficients ($y$ on $x$’s) from the joint Bayesian imputation and the iterative imputation.[]{data-label="fig:normbymi"}](normbymi.pdf "fig:"){width="\textwidth"}\
Simulation study for incompatible models
----------------------------------------
We next consider conditional models that are incompatible and valid. To study the frequency properties of the iterative imputation algorithm, we generate 1000 datasets independently each with a sample size of 10,000. For each dataset, $y_1\sim\mbox{Bernouli}(0.45)$, $y_2\sim\mbox{Bernouli}(0.65)$, $y_{1}$ and $y_{2}$ are independent, and the remaining variables come from this conditional distribution: $x_1,\ldots, x_5|y_1,y_2\sim N(\mu_1 y _1 + \mu_2 y_2, \Sigma)$, where $\mu_1$ is a vector of 1’s and $\mu_2$ is a vector of $0.5$’s and $\Sigma$ is a $5\times 5$ matrix that is 1 on the diagonals and 0.2 on the off-diagonal elements.
As before, we remove 30% of the data completely at random and then impute the dataset using iterative imputation. We impute $y_1$ and $y_2$ using logistic regressions and $x_1, \ldots, x_5$ using linear regressions. In particular, $y_1$ is conditionally imputed given $y_{2}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$, and the interactions $x_1y_2$ and $x_2y_2$; $y_2$ is conditionally imputed given $y_{1}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$, and the interactions $x_1y_1$ and $x_2y_1$; and each $x_j$, $j=1,\dots,5$, is conditionally imputed given $y_1,y_2,$ and the other four $x_j$’s. The conditional models for the $x_{j}$’s are simple linear models, whereas the logistic regressions for $y_{i}$ also include interactions. As a result, the set of conditional models is no longer compatible but is still valid. To check whether or not the incompatible models result in reasonable estimates, we impute the missing values using these conditional models. For each dataset, we obtain combined estimates of the regression coefficients of $x_{1}$ given the others by averaging the least-square estimates over 50 imputed datasets. That is, for each dataset, we have 50 imputations, for each of which we obtain the estimated regression coefficients of $x_1|y_1, y_2, x_2, x_3, x_4, x_5$. Next, we average over 50 sets of coefficients to obtain a single set of coefficients. We repeat the whole procedure on 1000 datasets to get 1000 sets of estimated coefficients. Figure \[fig:histglm\_hi\] shows the distribution of the estimated coefficients of $x_1$ regressing on $y_1, y_2, x_2, x_3, x_4, x_5$ based on the 1000 independent datasets. The frequentist distributions of the combined estimate are centered around their true values indicated by the dashed line. This is consistent with Theorem \[ThmIncomp\].
![Distributions of coefficients of $x_1$ regressing on $y_1, y_2, x_2, x_3, x_4, x_5$ from 1000 imputed datasets using an iterative imputation routine [@Su2011]. The dashed vertical lines represents the true value of the regression coefficients of the simulation setting, which are $0.5, 0.25, 0.125, 0.125, 0.125, 0.125$.[]{data-label="fig:histglm_hi"}](histglm_hi.pdf "fig:"){width="\textwidth"}\
Discussion
==========
Iterative imputation is appealing in that it promises to solve the difficult task of multivariate modeling and imputation using the flexible and simple tools of regression modeling. But two key concerns arise: does the algorithm converge to a stationary distribution and, if it converges, how to interpret the resulting joint distribution of the imputations, given that in general it will not correspond exactly to the fitted conditionals from the regression. In this article, we have taken steps in that direction.
There are several natural directions for future research. From one direction, it should be possible to obtain exact results for some particular classes of models such as linear regressions with Gaussian errors and Gaussian prior distributions, in which case convergence can be expressed in terms of simple matrix operations. In the more general case of arbitrary families of regression models, it would be desirable to develop diagnostics for stationarity (along with proofs of the effectiveness of such diagnostics, under some conditions) and empirical measures of the magnitude of discrepancies between fitted and stationary conditional distributions.
Another open problem here is how to consistently estimate the variance of the combined imputation estimator. Given that the imputation distribution of incompatible models is asymptotically different from that of any joint Bayesian imputation, there is no guarantee that Rubin’s combined variance estimator is asymptotically consistent. We acknowledge that this is a challenging problem. Even for joint Bayesian imputation, estimating the variance of the combined estimator is still a nontrivial task under specific situations; see, for instance, [@Kim04; @Meng1994]. Therefore, we leave this issue to future studies.
We conclude with some brief notes.
#### A special case of the compatible models.
In the analysis of the conditional models, suppose that the parameter spaces of the conditional distribution and the covariates are separable, that is, $f(x_{j},x_{-j}|\theta_{j},\theta_{j}^{*}) = f(x_{j}|x_{-j},\theta_{j}) f(x_{-j}|\theta_{j}^{*})$ and there exists a prior $\pi$ for the joint model $f$ such that $\theta_{j}$ and $\theta_{j}^{*}$ are a priori independent for all $j$. The, the boundedness of $\partial\log L({\theta_{j}})$ becomes straightforward to obtain. Note that $L(\theta_{j}) = \pi(\theta_{j})/ \pi_{j,\xx_{-j}}(\theta_{j})$ and $$\pi_{j,\xx_{-j}}(\theta_{j})\triangleq \pi_{j}^{\ast }(\theta_{j}) \int f(\xx_{-j}|,\theta_{j}^{\ast })\pi_{j}^{\ast }(\theta_{j}^{*})d\theta_{j}^{\ast }.$$Thus, $L(\theta_{j}) = \pi(\theta_{j})/ \pi_{j}^{*}(\theta_{j})$ is independent of the data.
#### A example when Theorem \[ThmCompRev\] does not apply.
The expression $\partial\log L(\theta_{j})$ in Proposition \[PropPrior\] is not always bounded. For example, suppose that $\pi_{j,\xx_{-j}}(\theta_{j})$ is an informative prior for which the covariates of the regression model $\xx_{-j}$ provide strong information on the regression coefficients $\theta_{j}$ according to the joint model $f$. For instance, in Example \[ExLogit\] on page , the marginal distribution of the covariate $x_{2}$ provides $\sqrt n$ amount of information on the coefficients of the logistic regression. Under this situation, Proposition \[PropPrior\] still holds, but the right-hand side of does not necessarily converge to zero. Consequently, Theorem \[ThmCompRev\] (the main result in this section, presented later) does not apply. For the properties of iterative imputation under such situations, we can apply the consistency result for incompatible models, as discussed in Section \[SecIncomp\]. That is, the combined estimator is still consistent though it is not equivalent to any Bayesian model.
Proofs in Section \[SecCompatible\] {#Apdbd}
===================================
\[LemTV1\]Let $Q_{0}$ and $Q_{1}$ be probability measures defined on the same $\sigma$-field $\mathcal{F}$ and such that $dQ_{1}=r^{-1}dQ_{0}$ for a positive r.v. $r>0$. Suppose that for some $\varepsilon>0$, $E^{Q_{1}}\left(
r^{2}\right) =E^{Q_{0}}r\leq1+\varepsilon$. Then, $$\sup_{|f|\leq 1}\left\vert E ^{Q_{1}}( f(X)) - E^{Q_{0}}(
f(X)) \right\vert \leq\varepsilon^{1/2}.$$
$$\begin{aligned}
\left\vert E ^{Q_{1}}( f(X)) - E^{Q_{0}}(
f(X)) \right\vert &
=\left\vert E^{Q_{1}}\left[ (1-r) f(X)\right] \right\vert \\
& \leq E^{Q_{1}}\left( \left\vert r-1\right\vert \right) \leq [E^{Q_{1}}( r-1) ^{2}]^{1/2}=\left( E^{Q_{1}}r^{2}-1\right) ^{1/2}\leq\varepsilon^{1/2}.\end{aligned}$$
From Lemma \[LemTV1\], we need to show that $$\int r^{2}(\theta )f_{\xx}(\theta )d\theta \leq 1+\frac{\kappa^2|\partial L(\mu _{\theta })|}{\sqrt n L(\mu
_{\theta })}.$$Let $\mu _{L}=E^{f}L(\theta )$. $$r(\theta )=\frac{L(\theta )}{\mu _{L}}=\frac{L(\mu _{\theta })+\partial
L(\xi )(\theta -\mu _{\theta })}{\mu _{L}}.$$Then$$\begin{aligned}
E^{f}(r^{2}(\theta ))\frac{\mu _{L}^{2}}{L^{2}(\mu _{\theta })} &=&1+2E\frac{\partial L(\xi )(\theta -\mu _{\theta })}{L(\mu _{\theta })}+E\frac{(\partial
L(\xi ))^{2}(\theta -\mu _{\theta })^{2}}{L^{2}(\mu _{\theta })} \\
&\leq &1+2\frac{|\partial L(\mu_{\theta} )|E Z}{L(\mu _{\theta })\sqrt{n}}+\frac{(\partial L(\mu _{\theta }))^{2}}{L^{2}(\mu _{\theta })}\frac{EZ^{2}}{n}.\end{aligned}$$With a similar argument, there exists a constant $\kappa_{1}$ such that$$\left\vert \frac{\mu _{L}^{2}}{L^{2}(\mu _{\theta })}-1\right\vert \leq
\frac{|\partial L(\mu _{\theta })|}{L(\mu _{\theta })}\frac{\kappa^2_1 }{\sqrt{n}}.$$Therefore, there exists some $\kappa>0$ such that$$\begin{aligned}
E^{f}r^{2}(\theta ) &\leq &\left( 1+2E\frac{|\partial L(\mu_{\theta} )|Z}{L(\mu
_{\theta })\sqrt{n}}+\frac{(\partial L(\mu _{\theta }))^{2}}{L^{2}(\mu
_{\theta })}\frac{EZ^{2}}{n}\right) \frac{L^{2}(\mu _{\theta })}{\mu _{L}^{2}} \\
&\leq &1+\frac{|\partial L(\mu _{\theta })|}{L(\mu _{\theta })}\frac{\kappa }{\sqrt{n}}.\end{aligned}$$Using Lemma \[LemTV1\], we conclude the proof.
For any $\varepsilon ,\delta >0$, let $k_{\varepsilon }=\inf \{j:\forall
k>j,r_{k}\leq \varepsilon \}$. Then, for any $m>k_{\varepsilon }$ $$\begin{aligned}
\Vert\tilde \nu^{\xx^{obs}} _{1}-\tilde \nu^{\xx^{obs}} _{2}\Vert_{V} &\leq &\left\Vert \tilde \nu^{\xx^{obs}}
_{1}-\frac{1}{m}\sum_{k=1}^{m}\tilde K_{1}^{(k)}(\nu ,\cdot
)\right\Vert _{V}+\left\Vert \tilde \nu^{\xx^{obs}} _{2}-\frac{1}{m}
\sum_{k=1}^{m}\tilde K_{2}^{(k)}(\nu ,\cdot )\right\Vert _{V} \notag \\
&&+\left\Vert \frac{1}{m}\sum_{k=1}^{m}\tilde K_{1}^{(k)}(\nu ,\cdot )-\frac{1}{m}
\sum_{k=1}^{m}\tilde K_{2}^{(k)}(\nu ,\cdot )\right\Vert _{V}. \notag\end{aligned}$$By the definition of $k_{\varepsilon }$, each of the first two terms is bounded by $\varepsilon +k_{\varepsilon }C_{\varepsilon}/m$, where $C_{\varepsilon}= \max\{\int V(w)\tilde K_i^{(k)}(\nu,dw): i = 1,2, ~~ 1\leq k\leq k_{\varepsilon} \}$. For the last term, for each $k\leq m$ and $|f|\leq V$, $$\begin{aligned}
&&\left\vert \int f(w)[\tilde K_{1}^{(k+1)}(\nu ,dw)-\tilde K_{2}^{(k+1)}(\nu
,dw)]\right\vert \\
&\leq &\left\vert \int \left( \tilde K_{1}^{(k)}(\nu ,dw)-\tilde K_{2}^{(k)}(\nu
,dw)\right) \int f(w')\tilde K_{2}(w,dw')\right\vert +\int \tilde K_{1}^{(k)}(\nu
,dw)\Vert \tilde K_{1}(w,\cdot)-K_{2}(w,\cdot)\Vert_V \\
&\leq &\left\Vert \tilde K_{1}^{(k)}(\nu ,\cdot )-\tilde K_{2}^{(k)}(\nu ,\cdot
)\right\Vert _{V}+d(A_n)
\\
&=&\left\Vert \tilde K_{1}^{(k)}(\nu ,\cdot )-\tilde K_{2}^{(k)}(\nu ,\cdot
)\right\Vert _{V} + o(1),\end{aligned}$$as $n\rightarrow \infty$. The second inequality in the above display holds, if $V\equiv 1$; if $V$ is a drift function of $\tilde K_{2}$, we replace $f$ by $V$ and the inequality hold by noticing that $\int V(w')\tilde K_{2}(w,dw')\leq \lambda_{2} V(w)$. Then, by induction, for all $k\leq m$,$$\left\Vert \tilde K_{1}^{(k)}(\nu ,\cdot )-\tilde K_{2}^{(k)}(\nu ,\cdot )\right\Vert
_{V}\leq o(1)$$Therefore, the last term is$$\left\Vert \frac{1}{m}\sum_{k=1}^{m}\tilde K_{1}^{(k)}(\nu ,\cdot )-\frac{1}{m}\sum_{k=1}^{m}\tilde K_{2}^{(k)}(\nu ,\cdot )\right\Vert _{V}= o(1), $$as $n\rightarrow \infty$. Thus, for each $\varepsilon > 0$, we first choose $\kappa_{\varepsilon}$ and $C_{\varepsilon}$, then choose $m$ large such that $2C_{\varepsilon}k_{\varepsilon }/m < \varepsilon $, lastly choose $n$ large such that $\left\Vert \tilde K_{1}^{(k)}(\nu ,\cdot )-\tilde K_{2}^{(k)}(\nu ,\cdot )\right\Vert
_{V} < \varepsilon$. Therefore,$$\left\Vert \tilde \nu^{\xx^{obs}} _{1}-\tilde \nu^{\xx^{obs}} _{2}\right\Vert _{V}\leq
5\varepsilon .$$
The proof uses a similar idea as that of Proposition \[PropPrior\]: $\tilde K_1$ is equivalent to updating the missing values from the posterior predictive distribution of $f$ condition on that $\xx\in A_n$. Similarly, $\tilde K_2$ corresponds to the posterior predictive distributions of $g_j$’s. By Proposition \[PropPrior\], for all $\xx\in A_{n}$, $|K_{1}(\xx,B) - K_{2}(\xx,B)|= O(n^{-1/4})$, which implies that $$\frac{K_{1}(w,A_{n})}{K_{2}(w,A_{n})}=1+O(n^{-1/4}).$$The posterior distribution is a joint distribution of the parameter and the missing values. Therefore, $\theta_{j}$ is part of the vector $w$. Let$$R= \frac{\tilde K_2(w,dw')}{\tilde K_1(w,dw')}=\prod_{j=1}^{p}r_{j}(\theta _{j})\frac{K_{1}(w,A_{n})}{K_{2}(w,A_{n})},$$where $r_{j}(\theta_{j})$ is the normalized prior ratio corresponding to the imputation model of the $j$-th variable, whose definition is given in Proposition \[PropPrior\].
For the verification of the drift function, $$\begin{aligned}
\int V(w')\tilde{K}_{2}(w,dw^{\prime }) &=&\int R\times V(w^{\prime })\tilde{K}_{1}(w,dw^{\prime }) \notag \\
&\leq &(1+O(n^{-1/4}))^{p}\int V(w^{\prime })\prod_{j=1}^{p}\left( 1+2\frac{\partial L(\mu _{\theta _{j}})Z_{j}}{L(\mu _{\theta _{j}})\sqrt{n}}\right)
\tilde{K}_{1}(w,dw^{\prime }). \label{seq}\end{aligned}$$Let $w_{j}$ be the state of the chain when the $j$-th variable is just updated (then, $w' = w_{p}$). Then, according to the condition in , we have that for each $j+k\leq p$ $$\tilde E_{1}\left[~ V(w_{j+k})\Big( 1+2\frac{\partial L(\mu _{\theta
_{j+k}})Z_{j+k}}{L(\mu _{\theta _{j+k}})\sqrt{n}}\Big)~ \Big |~ w_{j}~\right]
=\tilde E_{1}\left( V(w_{j+k})| w_{j}\right) +o(1)V(w_{j})$$Since the $o(1)$ is uniform in $w_{j}\in A_{n}$, we can apply induction on the product in (\[seq\]) by conditioning on $\mathcal F_{j}=\sigma (w_{1},...,w_{j})$ sequentially for $j=1,...,n$. Therefore, we have $$\begin{aligned}
\int V(w') \tilde{K}_{2}(w,dw') &=& (1+o(1))\int V(w')\tilde K_{1}(w,dw')+o(1)V(w) \\
&\leq &(\lambda _{1}+o(1))V(w).\end{aligned}$$Then, we can find another $\lambda _{2}\in (0,1)$ such that the above display is bounded by $\lambda _{2}V(w)$. Thus, $V(w)$ is also a drift function for $\tilde{K}_{2}$.
Proof of Theorem \[ThmIncomp\]
==============================
Throughout this proof we use the following notation for asymptotic behavior. We say that $0\leq g(n)=O(h(n))$ if $g(n)\leq ch(n)$ for some constant $c\in (0,\infty )$ and all $n\geq 1$. We also write $g(n)=o(h(n))$ as $n\nearrow \infty $ if $g(n)/h(n)\rightarrow 0$ as $n\rightarrow \infty $. Finally, we write $X_{n} = O_{p}(g(n))$ if $|X_{n }/g(n)|$ is stochastically dominated by some distribution with finite exponential moment.
Let $\xx(k)$ be the iterative chain starting from its stationary distribution $\nu _{2}^{\xx^{obs}}$. Furthermore, let $\nu _{2,j}^{\xx^{obs}}$ be the distribution of $\xx(k)$ when the $j$-th variable is just updated. Due to incompatibility, $\nu _{2,j}^{\xx^{obs}}$’s are not necessarily identical. Thanks to stationarity, $\nu _{2,j}^{\xx^{obs}}$ does not depend on $k$ and $\nu _{2}^{\xx^{obs}}=\nu
_{2,0}^{\xx^{obs}}=\nu _{2,p}^{\xx^{obs}}$. Let $$(\tilde{\theta}_{j},\tilde{\varphi}_{j})=\arg \sup_{\theta_{j},\varphi_{j}} \int \log
h_{j}(\xx_{j}|\xx_{-j},\theta_j ,\varphi_j )\nu _{2,j-1}^{\xx^{obs}}(d\xx^{mis}).$$
The proof consists of two steps. Step 1, we show that for all $j$, $\tilde{\varphi}_{j}\rightarrow 0,\tilde{\theta}_{j}-\hat{\theta}_{j}\rightarrow 0$, as $n\rightarrow \infty$, where $\hat \theta_{j}$ is the observed-data maximum likelihood estimate based on the joint model $f$ (defined as in ). That is, each variable is updated approximately from the conditional distribution $f(x_j|x_{-j},\hat \theta)$. Step 2, we establish the statement of the theorem.
#### Step 1.
We prove this step by contradiction. Suppose that there exist $\varepsilon
_{0}$ and $j_{0}$ such that $|\tilde{\varphi}_{j_{0}}|>\varepsilon _{0}$ or $|\tilde{\theta}_{j_{0}}-\hat{\theta}_{j_{0}}|>\varepsilon _{0}$. Let $\xx^{\ast }(k)$ be the Gibbs chain whose stationary distribution ($\nu _{1}^{\xx^{obs}}$) is the posterior predictive distribution associated with the joint model $f$ (c.f. Definition \[DefValid\]). In addition, $\xx^{*}$ starts from its stationary distribution $\nu _{1}^{\xx^{obs}}$. We now consider the KL divergence$$D(\nu _{1}^{\xx^{obs}}||\nu _{2,j}^{\xx^{obs}}).$$Let $\xx(k,j)$ and $\xx^{\ast }(k,j)$ be the state at iteration $k+1$ and the $j$-th variable is just updated. Since both chains are stationary, the distributions of $\xx(k,j)$ and $\xx^{*}(k,j)$ are free of $k$. To simplify notation, we let $$\begin{aligned}
u &=&\xx_{j}^{mis}(0,j-1),\quad v=\xx_{-j}(0,j-1)=\xx_{-j}(0,j),\quad w=\xx_{j}^{mis}(0,j), \\
u^{\ast } &=&\xx_{j}^{\ast mis}(0,j-1),\quad v^{\ast }=\xx_{-j}^{\ast
}(0,j-1)=\xx_{-j}^{\ast }(0,j),\quad w^{\ast }=\xx_{j_{0}}^{\ast mis}(0,j).\end{aligned}$$That is, $u$ is the missing value of variable $j$ from the previous step and $w$ is the updated missing value of variable $j$. $v$ stands for the variables that do not change in this update. Let $p_{j}(\cdot )$ be a generic notation for density functions of $(u,v,w)$ and $p_{j}^{\ast }(\cdot )$ for $(u^{\ast },v^{\ast },w^{\ast })$. By the chain rule, we have that $$\begin{aligned}
&&\int \log \frac{p_{j}^{\ast }(u,v,w)}{p_{j}(u,v,w)}p_{j}^{\ast
}(u,v,w)dudvdw \notag\\
&=&\int \log \frac{p_{j}^{\ast }(u,v)}{p_{j}(u,v)}p_{j}^{\ast }(u,v)dudv+\int \log \frac{p_{j}^{\ast }(w|u,v)}{p_{j}(w|u,v)}p_{j}^{\ast }(u,v,w)dudvdw \notag\\
&=&\int \log \frac{p_{j}^{\ast }(v,w)}{p_{j}(v,w)}p_{j}^{\ast
}(v,w)dvdw+\int \log \frac{p_{j}^{\ast }(u|v,w)}{p_{j}(u|v,w)}p_{j}^{\ast
}(u,v,w)dudvdw. \label{A}\end{aligned}$$By construction,$$\label{B}
\int \log \frac{p_{j}^{\ast }(u,v)}{p_{j}(u,v)}p_{j}^{\ast }(u,v)dudv=D(\nu
_{1}^{\xx^{obs}}||\nu _{2,j-1}^{\xx^{obs}})$$ and$$\label{C}
\int \log \frac{p_{j}^{\ast }(v,w)}{p_{j}(v,w)}p_{j}^{\ast }(v,w)dvdw=D(\nu
_{1}^{\xx^{obs}}||\nu _{2,j}^{\xx^{obs}}).$$ Furthermore, $p_{j}^{\ast }(w|u,v)$ is the posterior predictive distribution according to $f$ and $p_{j}(w|u,v)$ is the posterior predictive according to $h_{j}$. Note that $f$ is a sub-family of $h_{j}$ for the prediction of variable $j$. Under $p_{j}^{\ast}(u,v,w)$ that is the posterior distribution of $f$, we have $$\log \frac{p_{j}^{\ast }(w|u,v)}{f(\xx_{j}^{mis}|\xx_{-j},\hat \theta)}=O_{p}(1),\qquad \log \frac{p_{j}(w|u,v)}{f(\xx_{j}^{mis}|\xx_{-j},\hat \theta)}=O_{p}(1).$$ To understand the above estimate, for the posterior predictive distribution of the Gibbs chain, one first draw $\theta$ from the posterior distribution which is $\hat\theta+O_{p}(n^{-1/2})$ and then draw each $x_{j}$ from $f(x_{j}|x_{-j},\theta)$. Thus, we may consider that $p_{j}^{\ast }(w|u,v)\approx f(\xx_{j}^{mis}|\xx_{-j},\hat \theta + \xi/\sqrt n)$. Together with the fact that $\partial \log f(\xx_{j}^{mis}|\xx_{-j},\hat \theta) = O_{p}(n^{-1/2})$ we obtain the above approximation. The same argument applies to the second estimate for $p_{j}(w|u,v)$ too. With these two estimates, we have $$\log \frac{p_{j}^{\ast }(w|u,v)}{p_{j}(w|u,v)}=O_{p}(1).$$ According to condition B3 that all the score functions has exponential moments, then we have $$\label{D}
\int \log \frac{p_{j}^{\ast }(w|u,v)}{p_{j}(w|u,v)}p_{j}^{\ast
}(u,v,w)dudvdw=O(1).$$ We insert , , and back to . For all $1\leq j\leq p$, we have that $$D(\nu _{1}^{\xx^{obs}}||\nu _{2,j-1}^{\xx^{obs}})=D(\nu _{1}^{\xx^{obs}}||\nu _{2,j}^{\xx^{obs}})+O(1)+\int \log \frac{p_{j}^{\ast }(u|v,w)}{p_{j}(u|v,w)}p_{j}^{\ast }(u,v,w)dudvdw.$$We denote the last piece by $$A_{j}=\int \log \frac{p_{j}^{\ast }(u|v,w)}{p_{j}(u|v,w)}p_{j}^{\ast
}(u,v,w)dudvdw.$$ Note that $\nu _{2}^{\xx^{obs}}=\nu
_{2,0}^{\xx^{obs}}=\nu _{2,p}^{\xx^{obs}}$. Then, we have that $$\begin{aligned}
D(\nu _{1}^{\xx^{obs}}||\nu _{2}^{\xx^{obs}}) &=&D(\nu _{1}^{\xx^{obs}}||\nu
_{2,0}^{\xx^{obs}}) \\
&=&D(\nu _{1}^{\xx^{obs}}||\nu _{2,1}^{\xx^{obs}})+A_{1}+O(1) \\
&=&... \\
&=&D(\nu _{1}^{\xx^{obs}}||\nu _{2,p}^{\xx^{obs}})+\sum_{j=1}^{p}A_{p}+O(1)
\\
&=&D(\nu _{1}^{\xx^{obs}}||\nu _{2}^{\xx^{obs}})+\sum_{j=1}^{p}A_{p}+O(1).\end{aligned}$$Thus, $\sum_{j=1}^{p}A_{p}=O(1)$. Note that each $A_{j}$ is non-negative. Thus, $A_{j}$ must be bounded for all $j$, that is$$A_{j}=O(1). \label{contra}$$In what follows, we establish contradiction by showing that $A_{j_{0}}\rightarrow \infty $ if $|\tilde{\varphi}_{j_{0}}|+|\tilde{\theta}_{j_{0}}-\hat{\theta}_{j_{0}}|>\varepsilon _{0}$.
Now, we change all the $j$’s to $j_{0}$, that is, $$\begin{aligned}
u &=&\xx_{j_{0}}^{mis}(0,j_{0}-1),\quad v=\xx_{-j_{0}}(0,j_{0}-1)=\xx_{-j_{0}}(0,j_{0}),\quad w=\xx_{j_{0}}^{mis}(0,j_{0}), \\
u^{\ast } &=&\xx_{j_{0}}^{\ast mis}(0,j_{0}-1),\quad v^{\ast }=\xx_{-j_{0}}^{\ast }(0,j_{0}-1)=\xx_{-j_{0}\ast }(0,j_{0}),\quad w^{\ast }=\xx_{j_{0}}^{\ast mis}(0,j_{0}).\end{aligned}$$Note that $u$ is the missing values of $\xx_{j_{0}}$ from the previous step and $w$ is the missing value for the next step. In addition, the update of $\xx_{j_{0}}^{mis}$ does not depend on the previously imputed values. Therefore, $u$ and $w$ are independent conditional on $v$. Thus, $A_{j_{0}}$ is reduced to$$\begin{aligned}
\int \log \frac{p_{j_{0}}^{\ast }(u|v,w)}{p_{j_{0}}(u|v,w)}p_{j_{0}}^{\ast
}(u,v,w)dudvdw &=&\int \log \frac{p_{j_{0}}^{\ast }(u|v)}{p_{j_{0}}(u|v)}p_{j_{0}}^{\ast }(u,v)dudv \\
&=&\int \log \frac{d\nu _{1}^{\xx^{obs}}(\xx_{j_{0}}^{mis}|\xx_{-j_{0}})}{d\nu _{2,j_{0}-1}^{\xx^{obs}}(\xx_{j_{0}}^{mis}|\xx_{-j_{0}})}\nu _{1}^{\xx^{obs}}(d\xx^{mis}).\end{aligned}$$We further let $\iota $ be the set of observations where $x_{j_{0}}$ is missing and $x_{-j_{0}}$ are observed. Use $\xx_{\iota ,j_{0}}^{mis}$ to denote the missing $x_{j_{0}}$’s of the subset $\iota $. Then$$\int \log \frac{p_{j_{0}}^{\ast }(u|v,w)}{p_{j_{0}}(u|v,w)}p_{j_{0}}^{\ast
}(u,v,w)dudvdw\geq \int \log \frac{d\nu _{1}^{\xx^{obs}}(\xx_{\iota
,j_{0}}^{mis}|\xx_{-j_{0}})}{d\nu _{2,j_{0}-1}^{\xx^{obs}}(\xx_{\iota
,j_{0}}^{mis}|\xx_{-j_{0}})}\nu _{1}^{\xx^{obs}}(d\xx^{mis}),$$that is the joint K-L divergence is bounded from below by the marginal of K-L divergence on the subset $\iota $. Note that $\xx_{\iota ,j_{0}}^{\ast
mis}(0,j_{0}-1)$ is the starting value of $\xx^{\ast }$ and was sampled from the conditional stationary distribution $\nu _{1}^{\xx^{obs}}(\xx_{\iota
,j_{0}}^{mis}|\xx_{-j_{0}})$. Equivalently, $\xx_{\iota
,j_{0}}^{\ast mis}(0,j_{0}-1)$ is sampled from $f(\xx_{\iota ,j_{0}}|\xx_{\iota ,-j_{0}},\theta )$ where $\theta = \hat \theta + O_{p}(n^{-1/2}) $ is a posterior sample and $\xx_{\iota ,-j_{0}}$ is fully observed (by the construction of set $\iota $).
On the other hand, $\xx_{\iota ,j_{0}}^{mis}(0,j_{0}-1)$ follows the stationary distribution of the iterative chain and is sampled from the previous step (step $k-1$) according to the conditional model $h_{j_{0}}(x_{j_{0}}|x_{i,-j_{0}},\theta _{j_{0}},\varphi _{j_{0}})$ where $(\theta _{j_{0}},\varphi _{j_{0}})= (\tilde{\theta}_{j_{0}},\tilde{\varphi}_{j_{0}}) +O_{p}(n^{-1/2})$ is a draw from the posterior distribution. In addition, by assumption, the parameters are different by at least $\varepsilon _{0}$, that is, $|\tilde{\varphi}_{j_{0}}|+|\tilde{\theta}_{j_{0}}-\hat{\theta}_{j_{0}}|>\varepsilon _{0}$. Thus, the conditional distributions $h_{j_{0}}(\cdot |x_{i,-j_{0}},\tilde \theta_{j_{0}},\tilde \varphi_{j_{0}})$ and $f(\cdot |x_{i,-j_{0}},\hat \theta)$ are different. For some $\lambda _{0}>0$ (depending on $\varepsilon _{0}$), the KL divergence between the two updating distributions of $x_{i,j_{0}}$ is bounded below by some $\lambda _{0}>0$, that is, for $i\in
\iota $ $$D(~f(\cdot |x_{i,-j_{0}},\hat \theta) ~\Vert ~ h_{j_{0}}(\cdot |x_{i,-j_{0}}, \tilde \theta_{j_{0}},\tilde \varphi_{j_{0}}) ~)\geq
\lambda _{0}.$$This provides a lower bound of the KL divergence of one observation. The posterior predictive distributions for the observations in $\iota $ are conditionally independent given $(\theta _{j_{0}},\varphi _{j_{0}})$. Thus, the KL divergence of the joint distributions is approximately the sum of the individual KL divergence of all the observations in $\iota $. Then, we obtain that for some $\lambda_{1}>0$ $$\begin{aligned}
A_{j_{0}}=\int \log \frac{p_{j_{0}}^{\ast }(u|v,w)}{p_{j_{0}}(u|v,w)}p(u,v,w)dudvdw
&\geq &\int \log \frac{d\nu _{1}^{\xx^{obs}}(\xx_{\iota ,j_{0}}^{mis}|\xx_{-j_{0}})}{d\nu _{2,j_{0}-1}^{\xx^{obs}}(\xx_{\iota ,j_{0}}^{mis}|\xx_{-j_{0}})}\nu _{1}^{\xx^{obs}}(d\xx^{mis}) \label{far} \\
&\geq &\lambda _{1}\#(\iota ). \nonumber\end{aligned}$$Since the number of observations $\#(\iota )\rightarrow \infty $ (condition B5), we reached a contradiction to (\[contra\]). Thus, $|\tilde{\varphi}_{j}|+|\tilde{\theta}_{j}-\hat{\theta}_{j}|=o(1)$ as $n\rightarrow \infty $. Thereby, we conclude Step 1.
#### Step 2.
We first show the consistency of $\hat \theta^{(2)}$. It is sufficient to show that $|\hat{\theta}^{(2)}-\hat{\theta}|\rightarrow 0$. $\hat{\theta}^{(2)}$ solves equation $$\int \partial \log f(\xx^{mis},\xx^{obs}|\hat\theta^{(2)} )\nu _{2}^{\xx^{obs}}(d\xx^{mis})=0.$$By Taylor expansion, the MLE has the representation that $$\hat \theta^{(2)} - \hat\theta = O(n^{-1})\int \partial \log f(\xx^{mis},\xx^{obs}~|~\hat\theta)\nu _{2}^{\xx^{obs}}(d\xx^{mis}).$$ Thus, it is sufficient to show that $$\int \partial \log f(\xx^{mis},\xx^{obs}|\hat\theta)\nu _{2}^{\xx^{obs}}(d\xx^{mis})=o_{p}(n).$$ Given that $\hat \theta - \theta^{0} = O_{p}(n^{-1/2})$. It is sufficient to show that $$\label{cd}
\int \partial \log f(\xx^{mis},\xx^{obs}|\theta^{0})\nu _{2}^{\xx^{obs}}(d\xx^{mis})=o(n).$$ Notice that the observed data MLE $\hat \theta$ satisfies $$\int \partial \log f(\xx^{mis},\xx^{obs}|\hat \theta)f(d\xx^{mis}|\xx^{obs},\hat{\theta})=0$$and further $$\int \partial \log f(\xx^{mis},\xx^{obs}|\theta^{0})f(d\xx^{mis}|\xx^{obs},\hat{\theta})=O_{p}(1).$$Then, we only need to show that $$\label{van}
\int \partial \log f(\xx^{mis},\xx^{obs}|\theta^{0})\Big [\nu _{2}^{\xx^{obs}}(d\xx^{mis})-f(d\xx^{mis}|\xx^{obs},\hat{\theta})\Big ] = o_{p}(n).$$ Consider a single observation $x^{mis}(k)$. Without loss of generality, suppose that $x^{mis}(k)=(x_{1}(k),...,x_{j}(k))$ and $x^{obs}=(x_{j+1},...,x_{p})$. The result of Step 1 suggests that each coordinate of $x^{mis}$ is updated from $$h_{j}(x_{j}|x_{-j},\hat \theta_{j}+o_{p}(1), \varphi_{j}= o_{p}(1)).$$ Thus, $x^{mis}(k)$ follows precisely the transition kernel of $x^*(k)$ described in condition B5. Therefore, we apply Lemma \[ThmComp\] and have that $$\begin{aligned}
&&\int \partial \log f(x^{mis},x^{obs}|\theta^{0})\nu _{2}^{\xx^{obs}}(dx^{mis}) \\
&=&\int \partial \log f(x^{mis},x^{obs}|\theta^{0})f(x^{mis}|x^{obs},\hat{\theta})dx^{mis}+o(1).\end{aligned}$$Then, (\[van\]) is satisfied immediately by adding up the above integral for all observations. Therefore, is satisfied and further $\hat
\theta^{(2)} - \hat \theta \rightarrow 0$. The proof for $\hat \theta_j^{(2)}
$ and $\hat \varphi_j^{(2)}$ are completely analogous and therefore is omitted. Thereby, we conclude the proof.
Markov chain stability and rates of convergence {#SecMC}
===============================================
In this section, we discuss the pending topic of the Markov chain’s convergence. A bound on the convergence rate $q_k$ is required for both Lemma \[ThmComp\] and \[ThmIncomp\]. In this section, we review strategies in existing literature to check the convergence. We first provide a brief summary of methods to control the rate of convergence via renewal theory.
#### Markov chain stability by renewal theory.
We first list a few conditions (cf. [@Bax05]), which we will refer to later.
1. Minorization condition: A homogeneous Markov process $W(n)$ with state space in $\mathcal X$ and transition kernel $K(w,dw')= P(W(n+1) \in dw' | W(n) =w)$ is said to satisfy a *minorization condition* if for a subset $C\subset \mathcal X$, there exists a probability measure $\nu$ on $\mathcal X$, $l\in \mathbb Z^+$, and $q\in (0,1]$ such that $$K^{(l)}(w,A) \geq q \nu(A)$$ for all $w\in C$ and measurable $A\subset\mathcal X$. $C$ is called a *small set*.
2. Strong aperiodicity condition: There exists $\delta >0$ such that $q \nu(C) > \delta$.
3. Geometric drift condition: there exists a non-negative and finite drift function, $V$ and scalar $\lambda \in (0,1)$ such that for all $w\bar\in C$, $$\lambda V(w)\geq\int V(w') K(w,dw'),$$ and for all $w\in C$, $\int V(w') K(w,dw')\leq b$.
Chains satisfying A1–3 are ergodic and admit a unique stationary distribution $$\pi(\cdot) =\lim_{n\rightarrow \infty} \frac 1 n \sum_{l=1}^n
K^{(l)}(w,\cdot)$$ for all $w$. Moreover, there exists $\rho<1$ depending only (and explicitly) on $q$, $\delta$, $\lambda$, and $b$ such that whenever $\rho <
\gamma <1$, there exists $M<\infty$ depending only (and explicitly) on $q$, $\delta$, $\lambda$, and $b$ such that $$\label{cgtbd}\sup_{|g|\leq V} |\int g(w') K^{(k)}(w,dw')- \int g(w') \pi(dw')|\leq MV(w)\gamma^k,$$ for all $w$ and $k\geq 0$, where the supremum is taken over all measurable $g$ satisfying $g(w) \leq V(w)$. See [@Ros95] and more recently [@Bax05] for a proof via the coupling of two Markov processes.
#### A practical alternative.
In practice, one can check for convergence empirically. There are many diagnostic tools for the convergence of MCMC; see [@rhat] and the associated discussion. Such empirical studies can show stability within the range of observed simulations. This can be important in that we would like our imputations to be coherent even if we cannot assure they are correct. In addition, most theoretical bounds are conservative in the sense that the chain usually converges much faster than what it is implied by the bounds. On the other hand, purely empirically checking supplies no theoretical guarantee that the chain converges to any distribution. Therefore, a theoretical development of the convergence is recommended when it is feasible given available resources (for instance, time constraint).
|
---
abstract: 'The web ecosystem is rapidly evolving with changing business and functional models. Cloud platforms are available in a SaaS, PaaS and IaaS model designed around commoditized Linux based servers. 10 billion users will be online and accessing the web and its various content. The industry has seen a convergence around IP based technology. Additionally, Linux based designs allow for a system wide profiling of application characteristics. The customer is an OEM who provides Linux based servers for telecom solutions. The end customer will develop business applications on the server. Customers are interested in a latency profiling mechanism which helps them to understand how the application behaves at run time. The latency profiler is supposed to find the code path which makes an application block on I/O, and other synchronization primitives. This will allow the customer to understand the performance bottleneck and tune the system and application parameters.'
author:
- 'Shaun C. D’Souza'
bibliography:
- 'bibliography-file.bib'
title: Evolving system bottlenecks in the as a service cloud
---
Introduction
============
[**Web N.**]{} Open source has enabled the development of more efficient internet systems. As application performance is a top constraint, profiling is used to verify the performance of a multi-process and multi-threaded workloads. Additionally, a good developer always makes the optimal use of the platform architecture resources. Applications are deployed today in cloud environment and as a developer it is key to ensure that the code is well documented including maintenance of architectural UML diagrams to ensure minimal errors. Cloud computing combined with service-oriented architecture (SOA) and parallel computing have influenced application software development. Code is implemented in a variety of frameworks to ensure performance including OpenMP, MPI, MapReduce [@dean2008mapreduce] and Hadoop. Parallel programs present a variety of challenges including race conditions, synchronization and load balancing overhead.
[**Virtualization.**]{} This hosted infrastructure is complemented by a set of virtualization utilities that enable rapid provisioning and deployment of the web infrastructure in a distributed environment. Virtualization abstracts the underlying platform from the OS enabling a flexible infrastructure. Developers can partition resources across multiple running applications to ensure maximum utilization of resources. Additionally, systems can be ported across platforms with ease. Virtualization enables isolation of application run-time environment including kernel libraries ensuring that multi-threaded and multi-process workloads run in a scalable manner without errors. It decouples product development platforms from the conventional SDLC models enabling existing infrastructure to scale in available resources. The virtualization layer enables application profiling and development.
[**Open Source.**]{} Additionally, the cloud ecosystem is supported by the Open Source community enabling an accelerated scale of development and collaboration [@mahmood2013software]. This has been enabled by the internet and version control systems.
OOP and Java have enabled enterprise system architecture. Java is an algorithms, web and enterprise centric programming language. It allows for deployment of applications on a host of platforms running a virtual machine. 3 billion mobile devices run Java. Enterprise applications provide the business logic for an enterprise. Architectures have evolved from monolithic systems, to distributed tiered systems, to Internet connected cloud systems today.
Computing and the internet are more accessible and available to the larger community. Machine learning [@d2018system] has made extensive advances with the availability of modern computing. It is used widely in Natural Language Processing, Speech Recognition and Web Search.
- Web N, 10 Billion users, Intelligent machines, Turing test
- Social media, enterprise mobility, data analytics and cloud
- Technology and enterprise
- Virtualization, Open Source
- Machine learning, compilers, algorithms [@d2017parser], systems
Queries
=======
We evaluate the customer application in the internet. We developed a full-system simulator to evaluate web workloads in a server client environment including network. We found opportunity for the use of virtualization technology to efficiently utilize cloud resources. Additionally, use the virtualization layers for performance related benchmarking functionality.
Review the use of profiling tools with the customer including COTS commercial solutions and alternative open source solutions. Both tools present a set of tradeoffs which would vary in the customer application. A commercial solution like Intel VTune, VMware vmkperf would be suitable for a range of applications and provide more accurate profiling data on a host platform running Windows.
However, as part of our solution implementation we will be reviewing the use of an open source solution in Linux perf and Xenoprof (OProfile). We would review the nature of the customer application including availability of source code. Additionally, we would assess the availability of a high level software architecture specification including UML, use case diagrams. These would allow us to evaluate a first-order survey of the system application bottlenecks on the target platform. This would include the choice of application UI (presentation layer), middle layer (business logic layer) and data access (data layer).
We would evaluate the application run time environment. If the application is implemented in C++ it would support the symbol tagged Linux kernel libraries. A java application would run with a compatible Java Runtime Environment (JRE) to ensure detection of class binary symbol information. We would evaluate the possibility of using the application source code vs. benchmarking the application binary. A binary would allow us to profile application performance information. Availability of source code would allow us to profile the application in a simulation environment including addition of custom flags and debug messages to benchmark the application. It would further allow us to customize the application to improve performance on the host platform.
As internet platforms evolve towards a cloud service model, we would evaluate the application runtime environment and opportunities for hosting the application in a private, public cloud. This would ensure application performance and scalability in a deployed setting. The application would scale in the usage model and number of users on a cloud platform.
Ensure customer is aligned with current business and technology environment. Architect and design cloud applications to support multitenancy, concurrency management, parallel processing and service-oriented architecture supporting rest services.
- Law of accelerating returns
- Prices and margins, competition, converging global supply and demand, evolving business models
- Tier vs. batch processing, open source (availability of source code), language – C++, Java, security and reliability.
- Public cloud - SaaS, PaaS, IaaS, in-house
- Numbers of users, usage model
- Structured data
- Web N, Availability of cloud computing platforms – SaaS, PaaS, IaaS. Use of virtualization infrastructure. Rapid provisioning and performance profiling of available resources
- Simulation. Full system simulation of end-end internet. Order of magnitude (slower)
- Type of application. Web based application hosted on a cloud computing platform.
Assumptions related to functionality / requirements
===================================================
There is a host of cloud computing infrastructure deployed on Linux based platforms. Linux is open source and supports benchmarking and profiling of various applications. Additionally, it supports the use of Virtualization like Xen and VMware. We use OOP languages including Java, C++ and python. A NoSQL database MongoDB to store the profiling results data. This data is read and output to a web browser using Meteor web server. Kibana is used to store the profiling data in Elasticsearch. A dashboard is created to analyze the data in a user viewable histogram and pie chart.
- [**Web N.**]{} The internet is increasingly accessible to more than 10 billion users. It has been designed around Internet protocols and standards. The next generation of the web will use various Semantic web technologies.
- [**Cloud computing.**]{} Rapidly commoditized infrastructure and Linux servers
- Linux, Python, NoSQL Database MongoDB, Meteor web server, Kibana
- [**Knowledge systems.**]{} Vast repositories of structured, unstructured data
- [**Efficient programming languages.**]{} github
With the exponential growth in technology development in the recent years we find that developers have increased access to commoditized compute technology. Platforms based on commodity Linux solution are widely deployed in the enterprise. Application developers are concerned about application performance and scalability in the cloud. Application performance bottlenecks are constantly evolving in a tiered internet. They vary around system constraints limitations in the kernel functionality. However, application scalability is bounded in fundamental constraints of application development arising from a producer consumer model. The Producer Consumer or Bounded buffer problem is an example of a multi-process synchronization challenge. It forms a central role in any Operating system design that allows concurrent process activity.
semaphore empty, mutex, full
function PRODUCER
while (true) do
WAIT(empty);
WAIT(mutex);
// add item
// increment "head"
SIGNAL(mutex);
SIGNAL (full);
function CONSUMER
while (true) do
WAIT(full);
WAIT(mutex);
// remove item
// increment "tail"
SIGNAL(mutex);
SIGNAL(empty);
![Producer Consumer.[]{data-label="fig:prodcons"}](prodcons.pdf){width="0.6\columnwidth"}
As we have N producers, N consumers and N queues in the application – Figure \[fig:prodcons\] we can see that there are opportunities for the synchronization through the use of semaphores, deadlock avoidance and starvation. If we imagine infinite resources then the producer continues writing to the queue and the consumer has only to wait till there is data in the queue. The dining philosopher’s problem is another demonstration of the challenges in concurrency and synchronization.
Application workloads today
---------------------------
![Semantic web stack.[]{data-label="fig:semantic"}](semantic.pdf){width="0.2\columnwidth"}
In the broader context of the internet it is always beneficial to host resources close to the client consumption including providing a larger bandwidth to the consumer. Additionally, open platforms and standards enable for a balanced distribution of available bandwidth resources allowing for a scalable platform for 10 billion consumers. Innovation and advancement is enabled through open source and open platforms around internet based wireless technology. The protocol stacks comprising the future semantic web data are as – Figure \[fig:semantic\].
Technical Risks & Mitigation plan
=================================
Development environment Availability of compilers and code generators. Availability of full-system Linux simulator supporting customer application functionality.
------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------
Technology Custom application binary support for user profiling tools including symbol tag information. Generation of dependency graphs. Recompilation of driver libraries.
Availability of source code Annotated source code. Proprietary system applications.
Customer application environment Support for virtualization technology for Host OS and application compatibility.
Product size, Business impact, Customer related Code reuse, Open source, Revenue, delivery deadline, New customer, customer reviews
: Top 5 Risk and mitigation
Modeling Notation
=================
1. Unified Modeling Language (UML)
1. Class diagram
2. Use case
3. Sequence
2. Eclipse
3. JDeveloper
4. JavaScript wavi
See Figure \[fig:tomcat\].
![Tomcat MongoDB server.[]{data-label="fig:tomcat"}](tomcat){width="0.8\columnwidth"}
Comparative Analysis
====================
Alternative options considered
------------------------------
There are a wide range of profiling tools for the Linux kernel.
1. Data collection
1. sysstat package – iostat, pidstat
2. sar, atop
2. Online data - top
1. iotop, iftop
3. Tracing – strace, perf\_events, mutrace, ftrace
1. perf-tools, gprof
Virtualization technology is now common in the internet datacenter. It is also used for performance gathering. Will the customer be deploying the application in a private / public cloud? Evaluate the application in the cloud infrastructure. Increasingly vendors are adopting an as a service model for hosting applications on a platform. Evaluate opportunities for improving the application performance using native and hosted virtualization techniques. As per the type of application there are a host of performance profiling environments including perf, Intel VTune, VMware vmkperf.
System level application analysis is supported in Unix through the use of top, ps, vmstat, sar. However, these applications only allow for visibility and performance metrics at the application level. Additionally, Unix top allows the user to measure application compute and memory resource utilization. It provides information on processes running in the application.
Simulators are used to benchmark application systems. M5 [@binkert2011gem5] is a full-system simulator and supports modeling of compute, memory and network components. We implement an accurate view of the user system in a simulation environment. The simulator enables full system visibility of the technology stack allowing the user to configure application and kernel OS. Additionally, it is an open source simulator and can be extended to implement customer functionality to benchmark the application. It supports tracing of application and kernel function calls.
There are a host of cloud platforms available to deploy the customer application in a web environment. Cloud environment enables scalability of the application. As per the vendor environment there are a number of utilities to evaluate application performance in the cloud.
System bottlenecks are constantly evolving. As infrastructure is increasingly being commoditized with a growth of development around open source technologies. It is essential that adequate bandwidth is provisioned in the cloud to allow for application scalability. Virtualization technology enables efficient partitioning of additional resources. As a metric, it is key to replicate scale the infrastructure maintaining redundancy to ensure quality of service in the end-to-end internet.
At the same time, it is good to ensure efficient code is developed and maintained. There are a host of projects on github that allow for efficient development of user application code.
Additionally, availability of source code enables for maintenance of code counters to measure performance. We developed a full-system simulator M5. It enables application profiling in a simulation environment including server client.
The simulator is available as open source and allows you to model compute, memory and network functionality in the system. It is an event driven simulator and supports benchmarking of key workload characteristics, check pointing, forwarding and replay. Additionally, these capabilities can be extended in the simulator ecosystem including updation of the Linux kernel to support the application code functionality and profiling. The simulator and Linux environment are accessible online. However, as we are enabling the performance profiling in simulation we do have the disadvantage that metrics gathering is at 10X slower as the entire system is running in a simulation environment.
Recommended option
==================
As a solution we would take a multi-step approach in resolving the performance profile. As per our initial set of queries we would evaluate the availability of a high-level software architecture specification. We would evaluate opportunities for improving performance utilization and bottlenecks in the application per the specifics of the architecture. We would evaluate the application deployment in a cloud environment and support for virtualization technology. This would allow for application portability on a host of platforms and ensure scalability.
We would evaluate the application throughput requirements in the end-to-end internet. This would include all memory and network bandwidth in the system that would be utilized in a finite amount of time. We would ensure that there is no saturation of resources on all the data pipes in the platform. Data intensive workloads are usually writing large amounts of data to and from memory including serving the information to a client on the network. Consequently, as per the application requirements we would evaluate the average and peak bandwidth requirements for all the interfaces on the platform. This would ensure a first order evaluation of the application requirements.
We would then evaluate the application in Linux perf [@de2010new]. For the purposes of our performance profiling we will be using the Linux perf utility. Additionally, we would benchmark the application in the M5 simulator. Simulation gives us the ability to evaluate the application stack in a full system environment. We are able to view the function call trace and kernel system calls. Utilities like doxygen and gprof output a call graph for the source code statically at compile time enabling us to view function calls to locking / synchronization primitives. It outputs the control flow graph to view the structure of the program.
Rationale behind suggested solution
===================================
Perf supports performance modeling of a variety of events in the user application and kernel code. We are able to capture this data in the OS and output it to the customer in a web interface. The other set of utilities do not provide sufficient information profiling other than compute utilization and availability of memory resources. There is no capability to view thread resources in the system. Linux pthreads are scheduled and provisioned in the OS to enable performance scalability.
Utilities like gprof and doxygen support creation of UML and call graph. These enable visualization of the application function call and libraries used. They show percentage utilization of application code and opportunities for optimization.
Estimation
==========
- Java KLOC per week = 0.85
- Python KLOC per week = 0.75
{width="0.8\columnwidth"}
\[tab:wbs\]
----- --------------------------------- --------------- ----- ------------- -------- -------------- --------- -------
Sl. Modules Estimated LOC
CUT Requirement Design Architecture Testing Total
1 Compilation and library support 2000 12 3 4 1 9 29
2 Code annotation 1000 6 1 2 1 4 14
3 Simulator integration 1000 6 1 2 1 4 14
4 Operating System integration 2000 12 3 4 1 9 29
5 Virtualization 1500 9 2 3 1 7 22
6 Performance analysis 2000 12 3 4 1 9 29
7 Input data to Elasticsearch 1000 6 1 2 1 4 14
8 Kibana visualization 2000 12 3 4 1 9 29
----- --------------------------------- --------------- ----- ------------- -------- -------------- --------- -------
: WBS
SDLC
====
SDLC model to be used
---------------------
We use an Agile, continuous development SDLC model. Incremental development is used to deliver the product in short iterations of 1 to 4 weeks. Incremental delivery includes functions and features that have been developed. Continuous integration is used to integrate work frequently. Each integration is verified and tested in an automated build to detect errors. This enables the rapid development of cohesive software.
Rationale
---------
Agile development and continuous integration. Enabled early detection and integration of defect and ensures code quality. Development happens in short iterations with fully automated regression tests. High level functional requirements are documented as user stories. Software development follows a model of Figure \[fig:sdlc\].
![Agile continuous development.[]{data-label="fig:sdlc"}](sdlc.pdf){width="0.4\columnwidth"}
- Requirements – Design – Coding – Testing – Deployment
Agile focuses on individuals and interactions, working software, customer collaboration and responding to change. Quality is maintained through the use of automated unit testing, test driven development, design patterns and code refactoring.
Suggested Customizations proposed
---------------------------------
We would use Lean techniques and Kanban. These ensure standardization and balance of the development environment.
Architectural (non-functional) requirements
===========================================
- Interoperability
- Scalability
- Quality of Service
- Time, Cost and Productivity
- Distributed Complex Sourcing
- Faster Delivery of Innovation
- Increasing Complexity
- SaaS vs. PaaS
- Number of users, usage model
- Reuse of Web Services
- Agile, continuous development
- QOS is supported in tiered cloud service provider
Critical success factors
========================
- Compatibility of development environments.
- Availability of source code and High-level architecture specification.
- Service Oriented Architecture (SOA)
- Linux based application hosted to run in a cloud platform.
Platforms while constantly updating themselves there is a standardization of functionality around the Linux open source movement. There has been a proliferation of development on the Linux platform and enabled commoditization of computing technology. Enterprises are now able to host a distributed platform at a marginal expense. Availability of open source software has accelerated the development of applications. Access to compiler technology in the GNU compiler collection has driven availability of shared libraries and the Linux file system. Applications can share run-time environments. Additionally, availability of version control systems has enabled collaboration across the conventional barriers.
We consequently choose an open source solution for our performance profiler. Availability of symbol tagged application libraries enables us to view run-time statistics including kernel system calls. Applications like top and perf provide system visibility in the Linux kernel. System performance information is captured using performance counters.
- Quality of service
- User base
- Open Source, Intellectual property, Revenue, Products and services
- Licensing, R&D, Open innovation, open source. Strategy vary in business needs
Architectural overview of the requirement and the feasibility in Linux operating system
=======================================================================================
![Host architecture.[]{data-label="fig:host"}](host.pdf){width="0.3\columnwidth"}
Linux supports a host of virtualization and performance profiling tools [@du2011performance]. A virtualized architecture consists of – Figure \[fig:host\].
1. Data collection
1. sysstat package – iostat, pidstat
2. sar, atop
2. Online data - top
1. iotop, iftop
3. Tracing – strace, perf\_events, mutrace
4. Application profiling. perf. gprof
5. Virtualization performance. VMware
6. Simulator. M5
![N-tier, SOA.[]{data-label="fig:soa"}](soa){width="0.6\columnwidth"}
The customer is interested in profiling the application on a host platform. Figure \[fig:soa\] shows the characteristics of a host platform and n-tier application stack. As architects we are able to evaluate the solution using the high-level specification. An architecture use case, sequence diagram would provide us with insight on the specific usage model. We are able to investigate bottlenecks in the application. Availability of performance profiling utilities enable us to evaluate the application latency profiling requirements.
Linux Perf
----------
Perf is a performance profiling tool for Linux. It supports trace functionalities for various file system, block layer and syscall events. Tracepoints and instrumentation points are placed at logical locations in the application and kernel code. These have negligible overhead and are used by the perf command to collect information on timestamps and stack traces. It uses a set of performance counters to record and output a report for the application code.
Intel VTune
-----------
VTune is a proprietary performance profiling utility [@malladi2009using]. It supports the implementation of performance counters that are used to profile the application. It supports a GUI and command line interface. It is capable of monitoring thread level and process level performance. It supports compute performance, threading, scalability and bandwidth monitoring - Figure \[fig:vtune\].
![Intel VTune.[]{data-label="fig:vtune"}](vtune){width="0.8\columnwidth"}
Simulation M5
-------------
Simulator design and development explores a high-level architecture design space. Simulation enables the user to evaluate various deployment topologies are varying level of abstraction. It examines the architectural building blocks in the context of performance optimization. We use the M5 architectural simulator developed at the University of Michigan, Ann Arbor. It enables us to model an end-to-end client server simulation with the native OS and application stack. There is a minimal overhead in modeling the various system components. However, the flexibility in modeling performance coupled with the high level design knowledge enable for an efficient resolution of application performance constraints.
In our research on PicoServer [@kgil2006picoserver], we were able to evaluate a generic architecture for the next generation internet based on internet protocols and open standards. We proposed an architecture for Web N with 10 billion users online.
We extend this research further in proposing a generic architectural framework for performance modeling. Given an architecture specification we are able to upfront evaluate dependency graphs and bottlenecks in the design. We take various approaches from compiler development in implementing efficient scheduling algorithms for scaling application in the cloud.
![3-tier use case diagram.[]{data-label="fig:usecase"}](usecase){width="0.6\columnwidth"}
Figure \[fig:usecase\] shows the architecture of an n-tier application in the cloud. Service Oriented Architecture (SOA) applications consist of a discrete function unit that provides services to other components using a web-based protocol. A typical 3-tier application consists of a presentation layer which provides the user interface functionality. It accesses a business logic layer (middle layer) and data layer. The application is typically consolidated in a tiered cloud service provider in a SaaS, PaaS and IaaS model using a private, public cloud.
In Figure \[fig:webserv\], we demonstrate the functionality of a server in the cloud which consists of compute, memory and network. With the proliferation of the internet and globalization we are seeing a commoditization of the infrastructure layer. Increasingly the server components are available as pluggable building blocks that scale in the internet. We see that a server comprises of compute and network blocks that are used to access memory. Additionally, some architectures enable direct access to network elements to improve performance and reduce wait and synchronization times.
![Web server.[]{data-label="fig:webserv"}](webserv){width="0.6\columnwidth"}
We evaluate the opportunities for building a set of heuristics for performance profiling including ad hoc and statistical techniques and algorithms such as shortest path, directed / un-directed graphs and minimum spanning tree. Evaluate the dynamic call tree.
Types of process wait state used to achieve the profiling
=========================================================
Unix POSIX supports a variety of multi-threading and multi-process synchronization primitives. Pthread library is used to implement multi-threading. It supports the creation of threads using pthread\_create and synchronization wait using pthread\_join. Pthread supports the use of mutually exclusive locks through the use of pthread\_mutex\_t.
Unix supports multiprocessing capabilities through the use of fork, join. Multi-processor synchronization is implemented using wait and semaphore locks semget, semop. Additionally, there are a host of programming practices that help in the implementation of efficient parallel code with reduced wait times, starvation and deadlocks / livelocks. Linux top enables viewing of various system statistics including user / kernel mode, idle and I/O wait. Perf supports the profiling of sleep and wait time in a program. These are obtained using the perf profiling events sched.
>> perf record -e sched:* -g -o perf.data.raw <<application>>
>> perf inject -v -s -i perf.data.raw -o perf.data
>> perf report --stdio --show-total-period -i perf.data
Latency is derived from multiple sources including scheduler, I/O and context switch. A waiting process can be blocked on events including availability of network, access to memory [@ghemawat2003google]. Multi-process systems enable optimization of the wait time by scheduling priority processes while a concurrent process is waiting on an external resource. There are various sources of latency in the Linux kernel
- Call to disk
- Memory management
- Fork and exit of a process
For an application blocked on I/O, we will see a latency on a read / write response to memory mapped input output (MMIO). Application driver code accesses a reserved section of the system memory map to communicate with external devices. Delayed access to a memory location results in an increased execution time of the driver library. MMIO’s are implemented using kernel ioread and iowrite. Address information is configured using ioremap.
perf supports profile gathering for various types of events sched, cpu-clock, syscalls, ext4, block.
>> perf record -e sched:* -e cpu-clock -e syscalls:* -e ext4:* -e block:* <<application>>
We investigate the opportunities for reducing wait times in a multi-threaded, multi-process workload. We see that a large amount of time is used in sched events including pthread mutex and join - Figure \[fig:multithread\].
![Multi-threaded / process use case.[]{data-label="fig:multithread"}](multithread.pdf){width="0.8\columnwidth"}
Xenoprof architecture
---------------------
Xenoprof is an open source profiling utility base on OProfile \[fig:xenoprofarch\]. It is developed on the Xen virtual machine environment and enables the user to gather system wide data. Xen is an open source virtual machine monitor (VMM). OProfile can be used to profile kernel and user level applications and libraries. It enables profiling of applications running in a virtualized environment.
``` {#xenoprof caption="xenoprof" output="" label="xenoprof"}
Function %Instructions Module
e1000_intr 13 .32 e1000
tcp_v4_rcv 8 .23 vmlinux
main 5 .47 rcv22
```
Virtualization profiling mechanisms provide real time capabilities vs. a simulator. As all the application system calls are serviced in the virtualization engine it is suitable to extend the VMM to support profiling. This functionality enables gathering of system wide data.
![Xenoprof architecture.[]{data-label="fig:xenoprofarch"}](xenoprofarch.pdf){width="0.6\columnwidth"}
High level architecture specification enables us to define a dependency graph for the application. This is facilitated by the availability of source code. A compiler is a sequential batch architecture - Figure \[fig:v8js\]. AST represents the structure of the source code. Parser turns flat text into a tree. AST is a kind of a program intermediate form (IR).
![V8 JS, g++ compiler.[]{data-label="fig:v8js"}](v8js.pdf){width="0.4\columnwidth"}
[**Compilers and translators.**]{} Compilers translate information from one representation to another. Most commonly, the information is a program. Compilers translate from high-level source code to low-level code. Translators transform representations at the same level of abstraction.
- Windows – 50 million LOC
- Google internet services – 2 billion LOC
Compilers enables us to evaluate the application dependency graphs at compile time. It allows us to find cyclical dependencies in the program that might cause a deadlock. At runtime we would find extremely large idle wait time consumed by a blocking thread. In such scenarios the OS scheduler would stop the process and release its resources and restart.
A number of IDE’s support the generation of a dependency graph for software profiling. MS Visual Studio generates a dependency for the Project Solution. A Dynamic call tree can be used to evaluate application critical paths at architecture design time.
Additionally, programs like strace output library information and can be used to evaluate program execution paths. strace provides system wide profiling including accesses to kernel system calls. Trace utilities like strace, ltrace and dtrace can be used to profile system performance.
>> strace -rT
<<strace output>>
Mechanism used to get the profiling data
========================================
Linux perf
----------
Performance data metrics will be gathered using the perf utility. There are a large number of Unix programs that are available for performance profiling including top, iostat and sar. We run the perf utility in user mode to collect data for a specific application. The call-graph option is used to obtain function sub-tree information
>> perf record -g <<application>>
We can profile system wide metrics using perf sleep. This enables the perf utility to run in the background while the application is running on the host platform. This approach can be used in validating system state for network and web server applications like tomcat and apache web server. perf captures performance data for various shared libraries including system kernel drivers. Users can profile drivers loaded using insmod, lsmod.
/lib/modules/kernel/drivers/ .ko
/lib/ .so
>> perf record -a sleep <<N>> &
<<application>>
Recorded data is saved in a file perf.data for offline use. There a variety of utilities that enable viewing the data. We use the report implementation to view a percentage utilization of the shared libraries and functions in the application - Listing \[perf\].
>> perf report
``` {#perf caption="perf" output="" label="perf"}
29.88% a.out libstdc++.so.6.0.19 [.] std::basic_ostream<char, std::char_traits<char> >& std::__ostream_insert<char, std::char_traits<char>>(std::basic_ostream<char, std::char_traits<char> >&, char const*, long)
17.53% a.out libstdc++.so.6.0.19 [.] std::basic_filebuf<char, std::char_traits<char> >::xsputn(char const*, long)
10.09% a.out libstdc++.so.6.0.19 [.] std::basic_streambuf<char, std::char_traits<char> >::xsputn(char const*, long)
7.60% a.out libstdc++.so.6.0.19 [.] std::ostream::sentry::sentry(std::ostream&)
5.86% a.out libc-2.19.so [.] __memcpy_sse2_unaligned
5.50% a.out libstdc++.so.6.0.19 [.] std::basic_ostream<char, std::char_traits<char> >& std::operator<< <std::char_traits<char>>(std::basic_ostream<char, std::char_traits<char> >&, char const*)
4.67% a.out [kernel.kallsyms] [k] 0xffffffff811ee670
4.31% a.out libc-2.19.so [.] strlen
3.05% a.out libstdc++.so.6.0.19 [.] std::codecvt<char, char, __mbstate_t>::do_always_noconv() const
2.26% a.out a.out [.] _ZStlsISt11char_traitsIcEERSt13basic_ostreamIcT_ES5_PKc@plt
1.66% a.out libstdc++.so.6.0.19 [.] strlen@plt
1.66% a.out libstdc++.so.6.0.19 [.] _ZSt16__ostream_insertIcSt11char_traitsIcEERSt13basic_ostreamIT_T0_ES6_PKS3_l@plt
1.54% a.out libstdc++.so.6.0.19 [.] memcpy@plt
1.07% a.out a.out [.] foo()
0.99% a.out libstdc++.so.6.0.19 [.] _ZNSo6sentryC1ERSo@plt
0.91% a.out libstdc++.so.6.0.19 [.] _ZNSt15basic_streambufIcSt11char_traitsIcEE6xsputnEPKcl@plt
0.83% a.out a.out [.] main
0.55% a.out a.out [.] bar()
0.04% a.out libstdc++.so.6.0.19 [.] std::__basic_file<char>::xsputn_2(char const*, long, char const*, long)
```
Network drivers
---------------
We evaluate the performance of the Linux network drivers using perf events sock, net and skb. scsi information is obtained using the scsi event.
>> perf record -e sock:* -e net:* -e skb:* -e scsi:* -e cpu-clock <<application>>
gprof
-----
gprof is used to profile data using compiler annotated binary format. Application is compiled using the –pg option.
>> g++ -pg <<application.cpp>>
After the program has been compiled it will output a gmon.out file on run containing the application performance information. This can be viewed in the gprof application.
>> gprof <<application>>
``` {caption="gprof" output=""}
% cumulative self self total
time seconds seconds calls ms/call ms/call name
41.64 0.12 0.12 main
31.23 0.21 0.09 1 90.56 90.56 foo()
26.02 0.29 0.08 1 75.47 166.02 bar()
0.00 0.29 0.00 3 0.00 0.00 std::operator|(std::_Ios_Openmode, std::_Ios_Openmode)
0.00 0.29 0.00 1 0.00 0.00 _GLOBAL__sub_I__Z3foov
0.00 0.29 0.00 1 0.00 0.00 __static_initialization_and_destruction_0(int, int)
```
Simulators enables full-system visibility including application and OS. Simulation provides the user with a range of metrics related to application performance. Additionally, with the availability of simulator and application OS source code, the user can customize debug messages to be outputted in the simulator and kernel dmesg logs. This enables us to evaluate the metrics in an offline mode and profile application characteristics including function bottlenecks.
Unix mutrace is used to obtain thread mutex debug information including contention and wait time - Listing \[mutrace\]. It provides information on the number of mutex locks used in the design, the number of times the lock changed and the average wait time for each lock.
``` {#mutrace caption="mutrace" output="" label="mutrace"}
Mutex # Locked Changed Cont. tot.Time[ms] avg.Time[ms] max.Time[ms] Flags
0 8 4 4 45381.448 5672.681 6303.132 M-.?-.
```
There are a host of utilities that enable measuring network packet latency. Systemtap provides the user with the ability to define event handlers for the Linux kernel system calls. The user is able to log system calls and event timing information.
Availability of driver source code and build environment enables the user to add printk KERN\_DEBUG messages in the kernel logs. These provide timing information for the driver execution. However, the driver libraries have to be rebuilt.
{width="0.5\columnwidth"} \[fig:ast\]
![Control flow graph.[]{data-label="fig:cfg"}](cfg){width="0.2\columnwidth"}
Compilers support the generation of an Abstract Syntax Tree (AST). This can be used to traverse multi-threaded, multi-process libraries to obtain critical path information. Availability of output from gprof and doxygen enables us to view the frequently used libraries and function calls. Compilers enable us to evaluate a static compile time profile for the application code. A control flow graph (CFG) is a graph representation of the user code. They are built recursively at compile time and are a high-level representation of the source code.
- CFG = (N, E)
- N – Set of nodes
- E $\subseteq$ N X N – Set of edges
Nodes in the graph constitute the basic blocks in application code including the function call and libraries accessed. These are used to evaluate critical paths in the design and dependency chains. Application profiling at compile time enables the user to profile the application during architecture design. The CFG is essential to compiler optimizations and static analysis. It is used in various directed approaches to latency reduction including scheduling and parallelism.
Compilers support a variety of optimizations including dead code elimination, common sub-expression elimination, loop invariant code motion, loop unrolling and parallelization. Detection of independent sub-graphs in the application code is used in application parallelizing. Function call and for loops can be partitioned by the compiler at compile time to optimize the application code for a multi-process / distributed cloud platform. Addition of user directives in the application code enables auto-partitioning of the code to ensure performance scalability.
Data storage, retrieval and presentation
========================================
We explore the usage of various visualization features in presenting the perf report data to the customer. We use the script command to output a raw report for the performance data. This includes information on the application command, time stamp and shared library related to the event. This data is exported into an Excel csv file.
Perf supports a set of command line utilities that can be used to visualize the report data information.
>> perf report --call-graph --stdio
Script is used to display and store a trace output for the test run.
>> perf script
Logstash
--------
Logstash is used to input the user data into Kibana. A conf file is used to setup the Logstash pipeline – Figure \[fig:logstash\].
![Logstash pipeline.[]{data-label="fig:logstash"}](logstash){width="0.6\columnwidth"}
We use the csv input filter to read the csv data file. An output filter is used to add the data to the Elasticsearch index. A template format is specified for reading the csv column values in the date time and string format. Time values of ss.SSS are used to read the perf event logs.
>> cat linuxperf.csv | logstash -f linuxperf.conf
Elasticsearch
-------------
Elasticsearch is a distributed and scalable data store. It provides search and analytics capabilities with Rest services. It supports indexing of a JSON document. Input is through Logstash.
Kibana
------
The data set is loaded in Kibana and a visualization dashboard is created to display the performance data to the customer.
- Data output in json format
- Stored in Kibana Elasticsearch
- Input using Logstash
<!-- -->
>> curl 'localhost:9200/_cat/indices?v'
>> curl -XDELETE 'http://localhost:9200/linuxperf'
>> curl -XPUT localhost:9200/_bulk --data-binary @linuxperf.json
Availability of user data in a csv format enables us to store the data in a NoSQL database like MongoDB. Documents are stored in the database in the json format consisting of field value pairs. MongoDB provides high performance data persistence. A collection is created for the perf data. Data is then input to the database using the pymongo library. Users can insert, update documents in the database using json dictionary values.
Alternatively, the Linux perf data is stored in the run folder in the perf.data file. This is used alongside a host of perf report utilities to view the application call graph and utilization information.
gprof similary supports output in the gmon.out format. Applications are compiled in g++ for profiling using the symbol table information in the application binary. Run-time application call graph information is stored in the gmon.out file for profiling. gprof profiles function execution utilization information.
Characteristics of the user interface and the details about its functionality
=============================================================================
Kibana is an open source framework for data visualization [@gupta2015kibana]. It enables the user to analyze and explore structured and unstructured data in an intuitive interface. It supports the use of graphs, histogram and pie charts on large amounts of data. Users can view and discover the data in the UI. They can create various visualizations that can be integrated into a comprehensive dashboard. Figure \[fig:dashboard\] shows a custom dashboard implemented for the Linux perf data.
![Kibana dashboard.[]{data-label="fig:dashboard"}](dashboard.pdf){width="0.8\columnwidth"}
Kibana user interface visualization
-----------------------------------
We evaluate a network workload consisting of a secure file transfer and zip compression of the data. As can be seen in the data a large amount of time is utilized in the libcrypto and gzip application. The histogram plots the total number of events per second as recorded / reported in perf. We additionally output a pie chart of the percentage of utilization for each application and the shared libraries accessed at runtime. We capture the data for all resources on the system for the duration of the application.
As the data is input in Kibana using Logstash, we create an index for the loaded data. The data can now be viewed as a list of input values for discovery. We use this data to create a set of visualizations. We create an area plot for the performance data using the logged time values in ss.SSS. We use the date histogram feature on the X-axis and split the plot using the run time command information. We create a pie chart using the command and corresponding shared library information in a sub-plot. Time series values are encoded in the date number format. Other input data is input in the string format with the option ‘not\_analyzed’ set.
![Kibana visualization.[]{data-label="fig:visual"}](visual){width="0.8\columnwidth"}
Python, MongoDB, Tomcat server
------------------------------
We additionally investigate the use of a NoSQL database in storing the results data. This is an architecture deployed in the Holmes Helpdesk platform. Python is used to update the performance profiling data in MongoDB code. This is then output to the customer in a HTML file in a DataTable - Figure \[fig:html\].
![HTML DataTable.[]{data-label="fig:html"}](html){width="0.8\columnwidth"}
[**HTML DataTable.**]{} JavaScript DataTables code runs natively on the HTML file allowing the user to browse, sort the performance data. A Tomcat server is enabled to output the MongoDB performance data to a Java Servlet so the customer can access the information remotely - Figure \[fig:tomcat\].
Functional and nonfunctional requirements of the profiler
=========================================================
The performance profiler has been developed for a Linux platform and supports the Unix POSIX function calls. It should support the C++, Java development libraries and run-time environments. There should be a compatible JRE running on the system. Additionally, a lot of the UI features will be supported in a browser based application. The platform features like Kibana and DataTables will use a browser.
- The application must support the linux-common-tools corresponding to the kernel release.
- Availability of High-level architecture documents.
- Requirements engineering – Implementation – Testing – Evaluation
We will be exploring the design space of a Linux performance profiler that would allow us to evaluate bottlenecks arising from an inefficiently designed application. The application would support an intuitive user interface and present a metrics dashboard to the user highlighting the compute utilization for the application code. The user would be able to view the total run time for the application and library wise break down of run-time utilization. We would design the user interface such that any user would be able to evaluate the performance characteristics of an application.
The profiler would run in a Linux based environment enabling the reuse of existing Unix kernel and application libraries. There are a number of Unix utilities that support application and system level performance monitoring. For the purposes of our investigation we will explore a hybrid of open source and COTS implementation including a simulator and compiler design. Design of the simulator and compiler are highly intensive projects with simulator designs running into 100K+ LOC and compilers at 50K+ LOC. As we will be exploring open source alternatives in our design. We will look at the use of the M5 simulator which is a full-system architectural simulator and the Unix g++ and java compilers.
Unix standards enable us to run a host of application on a shared platform architecture. Additionally, with the availability of open libraries and kernel binaries the user is able to observe the performance of an application in the Linux environment. Utilities like top and iostat enable us to view system compute utilization and I/O wait times. We will assume the functionality of an SOA application running on a web-based platform. The application would run on the OS natively enabling us to benchmark the application and gather data for offline viewing. We would provide all the application utilities on the test platform. The user would launch the application using a script which would gather the data and input it to Kibana using Logstash. We would run Kibana and Elasticsearch on a user machine for viewing the results data. It is possible that running the data viewer utility on the same test platform could corrupt the data. Alternatively, we could run the Kibana UI on a VMM on the same physical machine.
- Non-functional requirements include data requirements, constraints and quality requirements.
- Product requirements – portability, reliability, usability, efficiency, performance
- Organization – delivery, implementation, standards
- Availability of documented code.
Eclipse is an open source IDE and supports a variety of programming languages including plugin functionality. Eclipse supports the standard GNU environment for compiling, building and debugging applications. The CDT is a plugin which enables development of C/C++ applications in Eclipse. It enables functionality including code browsing, syntax highlighting and code completion.
Eclipse supports a number of programming languages including C/C++, Java, PHP, XML, and HTML. It is an open source IDE and can be used on multiple platforms including Windows, Linux. It supports plugins to extend the functionality of the IDE for source code language modeling and analysis.
![Stages of compilation parser.[]{data-label="fig:compiler"}](compiler.pdf){width="0.6\columnwidth"}
We use the CDT to function as a compiler frontend - Figure \[fig:compiler\]. The CDT uses a translation unit to represent a source file cpp and h. The CDT core supports a Visitor API which is used to traverse the AST. AST rewrite API is used to update the source code. We access the code AST using the Eclipse CDT API.
We evaluate the generation of a dependency graph for software profiling. The AST is used to profile critical application paths including access to lock and synchronization primitives. These enable us to evaluate a latency profile at design time. Eclipse supports a plugin to analyze library dependencies. MS Visual Studio generates a dependency graph for all the drivers and files in the solution architecture.
Conclusion
==========
We have looked at architectures for the next generation enterprise including end to end solutions for the web infrastructure. These highlight the challenges in bringing billions of users online on a commodity platform. There is a large opportunity in enabling technology consumption for more than a billion users. Technologies like social media, enterprise mobility, data analytics and cloud are disrupting the enterprise. Research and Development (R&D) is an enabler for the enterprise. We are in the midst of a modern day information revolution enabled by increased availability of technology. I have highlighted the technology direction and challenges that we as architects foresee and resolve. These include researching machine learning, compilers, algorithms and developing efficient language models to enable applications and systems in the enterprise.
|
---
abstract: 'Motivated by the difference between the dynamics of magnetization textures in ferromagnets and antiferromagnets, the Landau-Lifshitz equation of motion is explored. A typical one-dimensional domain wall in a bulk ferromagnet with biaxial magnetic anisotropy is considered. In the framework of Walker-type of solutions of steady-state ferromagnetic domain wall motion, the reduction of the non-linear Landau-Lifshitz equation to a Lorentz-invariant sine-Gordon equation typical for antiferromagnets is formally possible for velocities lower than a critical velocity of the topological soliton. The velocity dependence of the domain wall energy and the domain wall width are expressed in the relativistic-like form in the limit of large ratio of the easy-plane/easy-axis anisotropy constants. It is shown that the mapping of the Landau-Lifshitz equation of motion to the sine-Gordon equation can be performed only by going beyond the steady-motion Walker-type of solutions.'
author:
- 'R. Rama-Eiroa'
- 'R. M. Otxoa'
- 'P. E. Roy'
- 'K. Y. Guslienko'
bibliography:
- 'SineGordonBibtex.bib'
title: 'Steady one-dimensional domain wall motion in biaxial ferromagnets: mapping of the Landau-Lifshitz equation to the sine-Gordon equation'
---
Introduction {#sec:intro}
============
The exchange integral in the Heisenberg Hamiltonian determines the relative orientation between neighboring spins. When it is positive, the favored magnetization orientation between neighboring atomic sites is parallel, being this type of media known as ferromagnets (FM), while when it is negative, an antiparallel orientation is preferred, which are denominated as antiferromagnets (AFM). Both types of systems have similar types of spin excitations, such as spin waves (SW) and domain walls (DW), and the magnetization dynamics can be described, in continuum field approximation, by the Landau-Lifshitz (LL) equation of motion. [@LandauLifshitz1935] Interestingly, the dynamics in FM and AFM result in different SW frequency modes with a natural frequency of the order of GHz and THz, respectively. [@GomonayBaltzBrataasEtAl2018] Likewise, the dynamics of magnetic textures present not only quantitative differences between FM and AFM, but also qualitative ones. [@KosevichIvanovKovalev1990] The stable DW dynamics in FM is possible up to a limiting velocity, from which intrinsic instabilities appear in the propagating magnetic texture due to the combination of internal translational and oscillatory modes, which is known as Walker breakdown (WB). [@SchryerWalker1974] On the other hand, in AFM it is possible to reach higher velocities in a stable steady-state-like motion existing, however, a limit that cannot be exceeded, which is given by the maximum magnon group velocity in the medium. This is because the DW dynamics in AFM can be described, in the framework of the non-linear $\sigma$-model, through a Lorentz-invariant relativistic-like expression known as sine-Gordon (SG) equation. [@Fradkin2013] As a consequence, while the dynamics of DW in FM are described in terms of conventional Galilean dynamics, in the case of AFM the movement of the aforementioned magnetic textures will follow the precepts of special relativity. All of this due to a single change in the Heisenberg Hamiltonian.
The magnetization switching in FM and AFM is typically related to the nucleation and propagation of an inhomogeneous magnetization reversal mode. To implement spintronic devices whose functionality is based on the propagation of inhomogeneous magnetization textures, ultrafast and controllable dynamics are essential in order to reduce switching time. For FM, the fundamental problem in this context lies in the difficulty of reaching high speeds for the reversal mode propagation while preserving stability. Thus, AFM have been erected as a solid alternative, at least theoretically, because DW in these media can reach speeds of the order of tens of km/s without entering an irregular regime. [@shiino2016antiferromagnetic] However, the usefulness of AFM in the field of spintronics has been rather directed so far to a passive role, such as a necessary element to convert a FM free layer into a pinning one through the exchange field bias generated by it. This has been mainly due to the difficulty of exciting and tracking the dynamics of magnetic textures in this type of systems, as opposed to FM, at least until very recently in a very particular type of structures. [@gomonay2016high] Accordingly, many efforts have been invested in trying to obtain higher velocities in FM while ensuring the integrity of magnetic textures. This could be possible if the appearance of instabilities induced by the WB could be avoided, or at least delayed. Some results obtained through micromagnetic simulations show that, in fact, this is possible. In this direction it has been observed that it is possible to eradicate the WB for two-dimensional (2D) DW in FM nanowires. [@YanHertelPRL2010] However, the speed of the magnetic texture will be limited in this case by the minimum phase velocity of the SW of the medium. If this threshold is exceeded, the DW will begin to emit SW, which is known as the spin Cherenkov effect, [@YanAndreasKakayEtAl2011] and is the kind of phenomenon that is theoretically foreseen in AFM too. Also, in the context of one-dimensional (1D) DW, the inclusion of a Dzyaloshinskii-Moriya exchange interaction in ultrathin films with perpendicular anisotropy results in the stabilization of the magnetic texture, making it possible to delay the appearance of the instabilities and to increase the maximum DW velocity before this phenomenon begins. [@ThiavilleRohartJueEtAl2012] However, these approaches present challenges from an experimental point of view for their implementation.
Given this background, an alternative would be to consider analytically the dynamics of magnetic textures in a FM system as simple as possible and try to reduce the LL equations in a SG-like expression as in AFM. If it were possible to find a situation in which this happened, perhaps its experimental implementation could be addressed, and even more complicated systems could be considered. The 1D motion of a DW is the simplest case, in which the magnetization configuration can be considered as a function of only one spatial coordinate. The next step would be to reduce the non-linear LL equation into a simpler non-linear expression. For this, there are two main approaches: i) the method of collective coordinates, and ii) the asymptotic method. The first method is based on the inclusion of the DW center position and the azimuthal angle of the DW magnetization as the generalized coordinates of the system. This allows, ultimately, to reduce the LL equation in a system of coupled differential equations. [@SchryerWalker1974; @ThiavilleRohartJueEtAl2012] The second method aims to describe the dynamics of the magnetic texture in terms of a dimensionless parameter that allows to apply perturbation theory when it can be considered small. In this context, this condition can be transferred to one of the angles that describe the magnetization. [@KosevichIvanovKovalev1990] Our work will be framed within this second approach.
One of the simplest systems that can be evaluated analytically is a FM system with biaxial anisotropy in which the dynamics of a DW is 1D. Within the Walker approximation, where solutions to dynamic equations are sought under the assumption that the azimuthal angle is independent of the spatial coordinate, [@SchryerWalker1974] it is possible, in fact, to find an exact solution for the aforementioned system, which was demonstrated by Schlömann. [@Schloemann1971] In the more realistic context in which one works beyond the Walker-type of solutions, it was heuristically demonstrated by Enz that it is possible to reduce the LL equation into a SG-like expression. [@Enz1964; @DoddMorrisEilbeckEtAl1982] However, this type of approach contradicts the steady-state DW motion regime in an infinite medium, where the spatial and temporal derivatives of magnetization cannot be considered as independent. Also, within the context of the mapping proposed by Enz, it has not been found yet what is the expression for the DW energy, or if this solution is stable or not. If it were the case that it was stable, and that the associated DW energy is lower than that in the case of steady-state Walker-type of solutions, it could be confirmed that there is, in fact, a case in which the dynamics of a DW in a biaxial FM can be described by SG-like expression for at least a restricted range of velocities for which this solution tends to the exact Schlömann solution.
Therefore, in this article we consider a 1D DW in a bulk FM with biaxial magnetic anisotropy. The underlying physical basic principles of the 1D magnetic soliton theory are presented in Sec. \[sec:model\]. The approach introduced by Schlömann in which the DW dynamics are parameterized, without dissipation, through the dispersion relation of the linear SW with complex wave vector and frequency, which reside in the tails of the moving soliton, [@Schloemann1971] is introduced in Sec. \[section:limit\]. It is shown that in the case of a biaxial magnetic anisotropy of the easy-plane/easy-axis type, the maximum speed of the steady DW motion cannot exceed the maximum phase velocity of the linear SW with imaginary wave vector for the Walker-type of solutions. In Sec. \[section:gordon\] we consider the mapping of the LL equation of motion to the more simple SG equation within the Walker approximation and show that the mapping can be performed only by going beyond the steady-motion Walker-type of solutions assuming a constant magnetization azimuthal angle. In order to corroborate that in the case of large easy-plane anisotropy the DW dynamics obey the precepts of special relativity, this situation was explored using atomistic spin dynamics simulations, which is exposed in Sec. \[section:simulations\]. Finally, conclusions are set out in Sec. \[section:conclusions\].
Theoretical basis {#sec:model}
=================
We consider a 1D DW in a bulk anisotropic FM, as sketched in Fig. \[im:1\]. The DW at rest is located in the $yz$ anisotropy easy-plane (see Fig. \[im:1\] (a)), and moves along the $x$-[*th*]{} direction (see Fig. \[im:1\] (b)). The total magnetic energy of the system per unit DW square (per unit area) is $E \left[ \boldsymbol{m} \right]= \int {\mathop{}\!\mathrm{d}}x \, e \left( \boldsymbol{m} \right)$, being $e$ the energy density which, in continuum approximation, is given by $e \left( \boldsymbol{m} \right)=A \, {\left( \partial_x \boldsymbol{m} \right)}^2+e_a \left( \boldsymbol{m} \right) + e_m \left( \boldsymbol{m} \right)$. Here, $A$ represents the exchange stiffness constant, $\boldsymbol{m} \left( x, t \right) = \boldsymbol{M} \left( x, t \right) / M_s$ denotes the unit magnetization vector, $M_s$ is the saturation magnetization, $e_a$ is the anisotropy energy density, and $e_m$ stands for the magnetostatic energy density. We consider a general quadratic form for $ e_a \left( \boldsymbol{m} \right)$ that, accounting for the restriction ${\boldsymbol{m}}^2=1$, can be expressed in the form of a biaxial anisotropy, $e_a \left( \boldsymbol{m} \right) = K_x m^2_x-K_z m^2_z$ (see Fig. \[im:1\] (c) for the particular case $\lambda=10$, where $ \lambda = K_x / K_z $, being the anisotropy energy in $2 K_z$ units). Assuming that $ K_x, K_z> 0 $, it is possible to define the uniform “vacuum" state of magnetization far from the DW center, $ x \rightarrow \pm \infty $, and that the anisotropy $K_x$ is of an easy-plane type. The magnetostatic energy density for the case of a bulk FM with the magnetization varying along the $x$-[*th*]{} direction is local, $e_m \left( \boldsymbol{m} \right)= 2 \pi m^2_x$, and results in the renormalization of the anisotropy constant $ K_x $. The same expression for $e_a \left( \boldsymbol{m} \right)$ can be applied for thin magnetic stripes with $e_m \left( \boldsymbol{m} \right)= 2 \pi m^2_z$ (being in this case absorbed by $ K_z $), when the $y$-[*th*]{} component of the demagnetizing field is neglected. [@ThiavilleRohartJueEtAl2012]
We parameterize the unit magnetization vector using the spherical angles, $ \boldsymbol{m} =\boldsymbol{m} \left( \theta, \phi \right)$. The angles $ \theta, \phi $ are functions of the spatial coordinate $ x $ and time $ t $. Writing the DW energy density in units of $ 2K_z $ and lengths in units of the static DW width $ \Delta_0 = \sqrt{A / K_z} $, an expression for the energy that depends on a single dimensionless parameter $ \lambda $ can be written $$e \left( \theta , \phi \right) = \frac{1}{2} \left[ {\left( \theta_x \right)}^2 + \left( 1+ \lambda \cos^2 \phi +{\left( \phi_x \right)}^2 \right) \sin^2 \theta \right],
\label{eq:1}$$ where the spatial variable subscript $x$ indicates derivative with respect to it. The corresponding effective Lagrangian density for FM is given by $\mathcal{L}\left( \theta, \phi \right) = e \left( \theta, \phi \right) +\dot{\phi} \cos \theta $, [@Doering1948] where overdot means derivative with respect to time. Henceforth time is expressed in units of $ t_0=1/ \gamma H_a $, where $ \gamma $ is the gyromagnetic ratio, and $ H_a = 2 K_z / M_s $. The LL equations of motion in the angular representation, $\dot{\theta} \, \sin \theta=-\delta e / \delta \phi$ and $\dot{\phi} \, \sin \theta= \delta e / \delta \theta$, can be found from a first variation of the Lagrangian taking into account the energy density given by Eq. , which results in $$\begin{aligned}
\dot{\theta} \, \sin \theta= \left[ \lambda \cos \phi \sin \phi +\phi_{xx} \right] \sin^2 \theta + \theta_x \, \phi_x \sin 2 \theta , \, \\
\dot{\phi} \, \sin \theta=\left[ 1+\lambda \cos^2 \phi + {\left( \phi_x \right)}^2 \right] \cos \theta \sin \theta -\theta_{xx}.
\end{aligned}
\label{eq:2}$$
The system of Eqs. has been intensively investigated in literature along with its integrals of motion. [@KosevichIvanovKovalev1990] For the particular case of a moving DW, the focus is usually on the steady-state motion Walker-type of solutions assuming $\phi_x = 0$. [@SchryerWalker1974] We will work within the framework of the main assumption of the theory of 1D topological magnetic solitons, [@KosevichIvanovKovalev1990] which is known as the “travelling wave" [*ansatz*]{}, i.e., that the solutions of Eqs. can be written in the form $\theta= \theta \left( \xi \right)$, $\phi= \tilde{\omega} t+ \phi_0 \left( \xi \right)$, where $\xi=x-vt$, being $v$ the soliton velocity, and $\tilde{\omega}$ the soliton precession frequency in the moving frame with velocity $v$. The moving soliton is treated as a bounded state of many SW (magnons), and the steady velocity of the soliton is interpreted as the group velocity of the SW packet, $v=v_g$. The frequencies in the laboratory frame, $\omega$, and moving frame, $\tilde{\omega}$, are related by $\tilde{\omega}=\omega - \boldsymbol{k} \cdot \boldsymbol{v}_g$. Here, $\boldsymbol{v}_g= \partial \omega / \partial \boldsymbol{k}$ denotes the group velocity of the linear SW, and $\boldsymbol{k}= k \boldsymbol{\hat{x}}$. Therefore, it is natural that $\omega=\tilde{\omega}+vk$. Those magnetic solitons with $\tilde{\omega} \neq 0$ are known as precession or dynamic solitons. [@KosevichIvanovKovalev1990]
Moving domain wall energy and critical velocities {#section:limit}
=================================================
The current approach assumes that the calculation of the DW energy and the limiting DW velocities is done through the spectra of the linear SW that reside in the DW tails in the case of a saturated FM. Far from the center of the moving DW, the magnetization can be considered as uniform and parallel to the $z$-[*th*]{} anisotropy easy-axis, see Fig. \[im:1\] (b). The magnetization dynamics outside the DW can be described in terms of small amplitude (linear) SW assuming a complex wave vector and frequency. [@Schloemann1971] Therefore, the DW dynamics can be described considering the SW of its tails as long as the DW magnetization configuration does not change. The linear SW dispersion relation for a biaxial FM, $\omega \left( k \right) $, is well known, and can be deduced from the linearization of Eqs. with respect to the ground state at the tails of the DW. It is explicitly given by the dispersion equation $\omega^2= \left( 1+k^2 \right) \left( 1+ \lambda + k^2 \right)$. The generalization to the complex wave numbers and frequencies is straightforward $$\Omega^2=\left( 1+K^2 \right) \left( 1+ \lambda + K^2 \right),
\label{eq:3}$$ where $\Omega=\omega+{{i\mkern1mu}}\kappa v$, and $K=k+{{i\mkern1mu}}\kappa$.
![1D DW magnetization configuration. (a) Definition of the magnetization vector $\boldsymbol{M}$ in terms of the polar $\theta$ and azimuthal $\phi$ angles relative to a Cartesian coordinate system. The angle $\varepsilon = \pi /2- \phi$ describes deviation of the magnetization from the static $yz$ DW plane. (b) Sketch of the DW magnetization configuration along the $ x $-[*th*]{} direction of motion. (c) Spatial distribution of the anisotropy energy density $ e_a \left( \boldsymbol{m} \right) $, in $2K_z$ units, for $ \lambda = 10 $.[]{data-label="im:1"}](Sine_Gordon_Physical_System_V){width="8cm"}
From now on, we only consider stationary soliton motion, which assume $\tilde{\omega}=0$. This allows to write the expression $\Omega = v K$. One can find from Eq. that the velocity is real in two regions disconnected from each other, $ \left[ 0, v_- \right] $ and $ \left[ \left. v_+, \infty \right) \right. $, where $ v_{\pm} = \sqrt{1+ \lambda} \pm 1 $.[@Schloemann1971; @EleonskiKirovaKulagin1978] We note that velocities are in units of $\Delta_0 / t_0 = 2 \gamma \sqrt{ A K_z}/M_s$. The first critical velocity, $v_-$, possesses physical sense of the maximum phase velocity of SW with imaginary wave vector $K= {{i\mkern1mu}}\kappa$ and imaginary frequency $\Omega= {{i\mkern1mu}}\kappa v$. In fact, this velocity corresponds to the maximum DW velocity for the case of the steady-state motion regime. [@SobolevHuangChen1995] The velocity $v_-$ is higher than the critical Walker velocity $v_W$ in a uniaxial FM assuming a driving force due to external magnetic field or spin polarized current. However, the ratio $v_-$/$v_W$ is not very large, $v_-$/$v_W$ =$ \sqrt{2 \, (2+ \lambda)} \, (\sqrt{1+ \lambda} - 1)/ \lambda$ and approaches $ \sqrt{2}$ at $\lambda \gg 1$. [@SchryerWalker1974] The second critical velocity, $v_+$, can be interpreted as the minimal phase velocity of SW with real frequency $\Omega = \omega$ and real wave vector $K=k$. The steady motion of DW (topological solitons satisfying the boundary conditions $\theta \left( \pm \infty \right) = 0,\pi$) is possible only within the interval $\left[ 0, v_- \right]$. These DW solutions satisfy the condition $\phi\left( \xi \right) = \mathrm{const}$, and describe Bloch DW ($\phi = \pm \pi /2 $), Néel DW ($ \phi =0, \pi$), or a hybrid DW (other values of $\phi$). The complicated solutions existing within the velocity interval $\left( v_-, v_+ \right)$ accounts for solitary magnetization waves, not topological solitons, satisfying the boundary condition $\theta \left( \pm \infty \right) =0$. The region $ \left[ \left. v_+, \infty \right) \right. $ accommodates non-linear SW with real wave vector $k$. [@EleonskiKirovaKulagin1978]
It is possible to find an explicit form of the complex wave vector dependence on the soliton velocity, $ K \left (v \right) $, from Eq. , which is given by $$K^2 \left( v \right) = \frac{1}{2} \left( v^2 - \lambda \right)-1 \pm \frac{1}{2} \sqrt{\left( v^2-v^2_- \right) \left( v^2-v^2_+ \right)} \, .
\label{eq:4}$$
The minus sign in Eq. corresponds to an unstable DW solution at $v < v_{-}$ (Néel DW at $v=0$). [@Schloemann1971] The plus sign in Eq. holds for a stable solution (Bloch DW at $v=0$). [@MagyariThomas1985]
The first integral of Eqs. , ${\left( \theta_x \right)}^2+{\left( \phi_x \right)}^2 \, \sin^2 \theta=\sin^2 \theta \left[ 1+ \lambda \cos^2 \phi \right]$, [@EleonskiiKirovaKulagin1976] allows to calculate the energy, $E_{\mathrm{DW}}$, for the stable solution as the doubled exchange energy. The DW energy for the Walker-type solution $\phi = \mathrm{const}$, $E_{\mathrm{DW}} \left( v \right) = E_0 \, \kappa \left( v \right)$ in units of $2K_z \Delta_0$, increases with velocity up to a maximal value $E_{\mathrm{DW}}=E_0{\left( 1+\lambda \right)}^{1/4}$, where $E_0=2$ is the static DW energy. The DW width, $\Delta \left( v \right) = 1/ \kappa \left( v \right)$, decreases with velocity (dynamical contraction) reaching the finite minimal value $\Delta \left( v_- \right) = {\left( 1+\lambda \right)}^{-1/4}$. In addition, it is possible to verify that in the $\left[ 0, v_- \right]$ region the DW plane orientation angle, $\phi \left( v \right)$, decreases as the velocity $v$ increases from $\pi /2$ to $\phi \left( v_- \right) = \arccos \sqrt{v_- / \lambda } \,$. The decomposition of $E_{\mathrm{DW}}\left( v \right)$ in series on small velocities ($v \ll v_-$), $E_{\mathrm{DW}}=E_0+m_{\mathrm{DW}}v^2/2$, allows to find the DW D[ö]{}ring mass for the stable Bloch-like DW (in absolute units) $m_{\mathrm{DW}}=1/ 2 \pi \gamma^2 \Delta_0$. [@Doering1948; @HubertSchaefer2008] Therefore, Eqs. (\[eq:2\], \[eq:3\]) lead to correct results for relatively small SW velocities lying within the interval $\left[ 0, v_- \right]$. The second, unstable, solution of Eq. yields a negative D[ö]{}ring mass. The instability arises with respect to a inhomogeneous perturbation localized at the DW plane (corrugation mode). [@MagyariThomas1985; @Khodenkov2003]
As it has been pointed out, the critical velocities $ v_{\pm} $ separate regions that hold different moving magnetization textures. Taking advantage of the parameterization of the DW dynamics in the region $ \left [0, v_- \right] $ through Eq. , it is possible to discuss how well this generalization adapts to the rest of the velocity regions. This has been done in Fig. \[im:2\]. In the interval $ \left[ 0, v_- \right] $, being $ K = {{i\mkern1mu}}\kappa $ and $ \Omega = {{i\mkern1mu}}\kappa v $, the magnetization waves are localized spatially forming a DW. The DW width contracts as $ v $ increases but its structure remains unchanged and its non-zero topological charge is conserved, at least until reaching the limiting velocity $ v_- $. In the domain $ \left( v_-, v_+ \right) $, where both $ K $ and $ \Omega $ are complex, the real component of the wave vector $k$ appears above the velocity $ v_- $ and increases with $v$ increasing, and the imaginary component ${\kappa}$ decreases to be zero at the critical velocity $ v_+ $. According to Ref. \[[ -‘ [@KosevichIvanovKovalev1990] ]{}\], the magnetization profile within the region $ \left( v_-, v_+ \right) $ can be described as a localized envelope of the soliton (the area of localization is $1/{\kappa}$) modulated by a periodic pattern with the wavelength about of $1/k$ (that is, some SW oscillations appear along the soliton envelope). Finally, in the last region $\left[ \left. v_+, \infty \right) \right.$, only non-linear SW would be expected, obtaining the logical analogue in our case within the linear SW approximation.
![Real $ k $ and imaginary $ \kappa $ wave vector components calculated by the generalization of the linear SW dispersion relation of biaxial FM as functions of the soliton velocity $v$ according to Eq. for $ \lambda = 10 $. The region $0<v <v_-$ corresponds to a moving DW.[]{data-label="im:2"}](Sine_Gordon_Plot_Wave_Number_Regions_Lambda=10){width="8cm"}
Mapping to a sine-Gordon equation {#section:gordon}
=================================
The Lorentz-invariant SG equation, having the exact $N$-soliton solutions, [@Faddeev1974] is one of the simplest non-linear equations. The dynamics of FM cannot be described, in general, by Lorentz-invariant equations, unlike in the case of AFM dynamics, where it is naturally described by a relativistic SG equation within the non-linear $\sigma$-model if the Zeeman and Dzyaloshinskii-Moriya interactions are absent. [@IvanovKolezhuk1995] Is it possible to reduce the non-linear LL equations, given by Eqs. , to the simple non-linear SG equation? Is it really necessary to consider an extremely large easy-plane anisotropy to achieve this mapping? [@BraunLoss1996] To answer these questions we introduce a new variable $ \eta $ defined as $ \tan \theta / 2 = \mathrm{exp} \left(- \eta \right)$ (such substitution is often used in the theory of magnetic solitons within the Hirota representation). [@KosevichIvanovKovalev1990] This allows to rewrite Eqs. as $$\begin{aligned}
v \eta_{\xi}= \lambda \cos \phi \sin \phi + \phi_{\xi \xi} -2 \phi_{\xi} \eta_{\xi} \tanh \eta , \\
-v \phi_{\xi}= \eta_{\xi \xi}+ \left[ 1+\lambda \cos^2 \phi +{\left( \phi_{\xi} \right)}^2-{\left( \eta_{\xi} \right)}^2 \right] \tanh \eta .
\end{aligned}
\label{eq:5}$$
The standard approach to this problem is to consider steady motion Walker-type of solutions, assuming $ \phi_{\xi} = 0 $. [@SchryerWalker1974] This simplification leads to the system of equations given by $v \eta_{\xi}=\lambda \cos \phi_0 \, \sin \phi_0$, and $1+\lambda \cos^2 \phi_0 ={\left( \eta_{\xi} \right)}^2$. A stable solution of the form $\eta \left( \xi \right)= \xi / \Delta \left( v \right)$ exists when the DW velocity $v$ does not exceed $v_-$.
The linear dependence $\eta \left( \xi \right)$ leads to the equation $\theta_{\xi \xi}= \sin 2 \theta / \, 2 \Delta^2 $, the same as in the static case except for the velocity-dependent DW width, $\Delta \left( v \right) = 1/ \kappa \left( v \right)$. Although the exact solution of the LL equation exists for the Walker case $ \phi_{\xi} = 0 $, the equation for the angle $\theta$ can be formally written as a SG equation without any assumptions about the value of the parameter $\lambda$, giving rise to $$\theta_{xx}-\frac{1}{v^2_-} \, \ddot{\theta}=\frac{1}{2 \Delta^2_e} \sin 2 \theta ,
\label{eq:6}$$ where $\Delta_e=\Delta / \sqrt{1-{\left( v /v_- \right)}^2}$, being $v_-$ the maximum DW velocity. Although the Walker-type solution is stable, it is not possible to reproduce this result through an effective relativistic Lagrangian because the kinetic term is canceled due to the simplified type of solution considered. The condition $\phi \left( \xi \right) = \phi_0$ results in the nullification of the kinetic part of the Lagrangian density, $\mathcal{L}_{\mathrm{kin}}= - \phi \, \dot{\theta} \sin \theta$. The energy density defined in Eq. also does not lead to Eq. . Therefore, the formal SG equation given by Eq. is nonphysical and another mapping should be found.
On the other hand, the limit $ \lambda \gg 1$ can be studied. Such limit can be realized even in soft magnetic materials like permalloy (NiFe alloy) or YIG with an induced uniaxial magnetic anisotropy $K_z$, where $\lambda = 2 \pi M^2_s / K_z >>1$, $K_x = 0$. For instance, a ratio $ \lambda = 21$ was used by Schryer [*et al*]{}. for YIG. [@SchryerWalker1974] We assume that within the limit $ \lambda \gg 1$ the magnetization component $m_x$, perpendicular to the “easy”-plane, is small, $m_x\ll 1$, and develop a perturbation theory with respect to it. A new scalar field, $\psi \left( x,t \right)$, is introduced for convenience through the equations $m_y=\sqrt{1-m^2_x} \, \sin \psi$, and $m_z= \sqrt{1-m^2_x} \, \cos \psi$. There are different ways to define $m_x$. We choose the [*ansatz*]{} $m_x= \varepsilon \sin \psi$, being $\varepsilon=\pi/2-\phi$ as sketched in Fig. \[im:1\], and we assume that $\varepsilon \left( \lambda \right) \rightarrow 0$ at $\lambda \gg 1$. In general, $\varepsilon$ does not have to be small and may be a function of $x$ and $t$. The initial variables ($\theta, \phi$) are related to new ones ($\psi, \varepsilon$) by the expressions $\cos \theta =\sqrt{1-\varepsilon^2 \sin^2 \psi} \, \cos \psi$, and $\cos \phi = \varepsilon / \sqrt{1+\varepsilon^2 \cos^2 \psi}$. If $\varepsilon \rightarrow 0$, $\cos \phi = \varepsilon +\mathcal{O} \left( \varepsilon^3 \right)$, and $\sin \phi =1+ \mathcal{O} \left( \varepsilon^2 \right)$. Substituting these expressions in Eqs. and assuming that $\varepsilon \left( \xi \right) =\mathrm{const}$, we obtain $v \eta_{\xi}= \lambda \varepsilon$, $\eta_{\xi \xi}=0$, and $1+ \lambda \varepsilon^2= {\left( \lambda \varepsilon / v \right)}^2$. The last equation has the solution $\varepsilon \left( \lambda, v \right)=v/ \lambda \sqrt{1-v^2 / \lambda}$. At this point, a new critical velocity can be introduced, $c= \sqrt{\lambda}$. It is straightforward to show that $\displaystyle c= \lim_{\lambda \gg 1} v_- \left( \lambda \right)$. The variable $\psi \left( x, t \right)$ coincides with the polar angle, $\theta \left( x, t \right)$, in the limit $\varepsilon \ll 1 $, if the terms of order $\mathcal{O} \left( \varepsilon^2 \right)$ are neglected. In this context, accounting for the approximate solution $\eta \left( \xi \right) = \xi / \Delta^{\prime} \left( v \right) = \xi / \sqrt{1-v^2/c^2}$, the SG equation for the DW profile angle $\theta \left( x, t \right) = 2 \arctan \mathrm{exp} \left(- \eta \left( \xi \right) \right)$ can be deduced $$\theta_{xx}- \frac{1}{c^2} \ddot{\theta}=\frac{1}{2} \sin 2 \theta,
\label{eq:7}$$ which corresponds to the particular case of Eq. in which $\Delta=\Delta^{\prime}$ and $v_-=c$.
Therefore, in the limit $ \lambda \gg 1$ the DW energy and the DW width can be represented in a relativistic-like form $$E^{\prime}_{\mathrm{DW}}\left( v \right)= \frac{E_0}{\sqrt{1-v^2/c^2}}, \, \Delta^{\prime} \left( v \right)= \sqrt{1-v^2/c^2}.
\label{eq:8}$$ The DW energy found by Eq. smoothly increases with velocity increasing. However, this increasing behavior is far from being that of the normalized relativistic-like form, $E^{\prime}_{\mathrm{DW}}/E_0$, exposed in Eqs. , which accentuates the difference of the discussed dynamic equations with a SG equation for any finite $\lambda$, as it can be seen in Fig. \[im:3\]. The correct energy decomposition can be obtained from Eqs. only when the DW velocity is small ($v^2/c^2 \ll 1$), $E^{\prime}_{\mathrm{DW}}\left( v \right)=E_0+E_0v^2/2c^2$, leading to the DW D[ö]{}ring mass $m_{\mathrm{DW}}=E_0 / c^2$, which coincides with the one defined above. However, in the limit $v \rightarrow c $, the energy is singular, $E^{\prime}_{\mathrm{DW}}\left( v \right) \rightarrow \infty$, and the DW width matches the ultimate case of the Lorentz contraction, $\Delta^{\prime} \left( v \right) \rightarrow 0$. These results are nonphysical because the parameter $\varepsilon \left( \lambda, v \right)=v/ \lambda \sqrt{1-v^2 / \lambda}$ diverges at $v \rightarrow c$, and cannot be considered as a small parameter anymore. The exact solutions of Eqs. (\[eq:2\], \[eq:3\]) predict the finite DW energy $E_{\mathrm{DW}} \left( v_- \right) = E_0 \, {\left( 1 + \lambda \right)}^{1/4}$ and finite DW width $\Delta \left( v_- \right) ={\left( 1+ \lambda \right)}^{-1/4}$ at $v \rightarrow v_-$. Thus, although one can write a SG-like expression by means of Eq. and the relativistic-like Eqs. , they are strictly valid only in the limit of small DW velocity $v^2/c^2 \ll 1$. However, Eqs. are very good approximation at $ \lambda \gg 1$ if the DW velocity is far enough from the maximal velocity, $c$. The subsequent conclusion is that the approximate solution obtained for the case in which $\varepsilon=\textrm{const}$, given by Eqs. , is asymptotically exact when $\lambda \gg 1$, provided that $v^2/c^2 \ll 1$, being far from the singularity that this solution presents, becoming virtually indistinguishable from the exact solution in biaxial FM obtained through Eq. .
There is another approximate solution of the system of Eqs. or in the limit $\lambda \gg 1$ assuming Walker-type of solutions which was proposed by Sklyanin, [@Sklyanin1979] $\phi \left( \xi \right) = \mathrm{const}$. The solution was found to have the form $m_x \left( \xi^{\prime} \right)= \varepsilon_s \sin \Psi \left( \xi^{\prime} \right)$, being $\varepsilon_s= V / R \, \Delta_s$, $\Psi \left( \xi^{\prime} \right)= 2 \arctan \mathrm{exp} \left( \xi^{\prime} / \Delta_s \right)$, and $\xi^{\prime}=x-Vt$, $V=\sqrt{R} \, v$, $\Delta_s = \sqrt{R / \gamma^{\prime}} \, \sqrt{1-{\left( V / V_0 \right)}^2}$, and $V_0= \sqrt{R} \,$, in the aforementioned limit $R^2 / \gamma^{\prime}= \lambda \rightarrow \infty$. The function $\Psi \left( x, t \right)$ satisfies a SG equation of the form $$\Psi_{xx}- \frac{1}{V_0^2} \, \ddot{\Psi}= \frac{\gamma^{\prime}}{2R} \sin 2 \Psi,
\label{eq:9}$$ which constitutes a particular case of Eq. at $\Delta_e = \sqrt{R / \gamma^{\prime}}$. The DW width $\Delta_s$ goes to infinity for any finite DW velocity $V < V_0$. The maximum DW velocity, $V_0 ={\left( \gamma^{\prime} \lambda \right)}^{1/4}$, is not correct (it should be equal to $v_-= \sqrt{\lambda}$). A redefinition of the value of the parameter $\gamma^{\prime}$ was proposed by Kivshar [*et al*]{}., being $\gamma^{\prime} =1$, [@KivsharMalomed1989] keeping in this way the same form of the SG-like expression given by Eq. .
![Comparison between the moving DW energy obtained by using Eq. , $ E_{\mathrm{DW}} / E_0 $, for $ \lambda = 10 $, and a relativistic Lorentz-like energy, $E^{\prime}_{\mathrm{DW}} /E_0$, for which the maximum speed is given by the maximum SW phase velocity $ v_- $.[]{data-label="im:3"}](Sine_Gordon_Plot_Energy_Lambda=10){width="8cm"}
The small parameter $\varepsilon_s$ is similar to the previously defined one, $\varepsilon = v / \lambda \, \Delta^{\prime} \left( v \right) $. Both of them are singular as the velocity $v$ approaches the critical value $V_0 ={\lambda}^{1/4}$ or $v_-= \sqrt{\lambda}$. That is a peculiarity of all solutions of Eq. with $\phi(\xi)=\mathrm{const}$. However, it must be faced that the solution studied, proposed by Sklyanin, [@Sklyanin1979] is nonphysical, since it corresponds to the vanishing right-hand side in Eq. and infinitely wide DW at $V<V_0$. Moreover, the Sklyanin’s solution $\Psi (x, t)$ does not satisfy the LL equation, which in the limit $\lambda \rightarrow \infty$ is reduced to the SG-like expression given by Eq. . Therefore, Eq. is not correct. This is due to that the Sklyanin’s assumption that $K_x$ is proportional to $R$ and that $K_z$ is proportional to $1/R$ and goes to $0$ at $R \rightarrow \infty$ is physically incorrect. The correct limit corresponds to consider the ratio $K_x/K_z = \lambda \gg 1$ keeping a finite value of the anisotropy constant $K_z$. Otherwise, the DW separating two domains with opposite magnetizations directed along $z$-[*th*]{} axis disappears. A finite value of $K_z$ allows the proper normalization of the DW energy in the units of $2K_z \Delta_0$ and DW width in the units of $\Delta_0$.
Therefore, accounting the drawbacks of the approach with $\varepsilon \left( \xi \right) =\mathrm{const}$, to properly calculate the limit $\lambda \gg 1$, $m_x \ll 1$ at finite DW velocity $v < v_{-}$ and get a SG equation for the polar angle $\theta$, we need to assume that the azimuthal angle $\varepsilon \left( \xi \right) \ll 1$ is a function of the coordinate and time, and solve Eqs. . Following this approach the Walker-type solutions $\varepsilon \left( \xi \right) = \mathrm{const}$ considered above can be refined, and the high velocity singularity of the steady DW solution, $\varepsilon \left( \xi , v \right)$ at $v \rightarrow c$, disappears. However, any solution with variable angle $\varepsilon \left( \xi \right)$ is beyond the current theory of 1D topological magnetic solitons and will be considered elsewhere.
We note that the SG equation given by Eq. requires a kinetic Lagrangian density term of the form $\mathcal{L}_{\mathrm{kin}} \propto \dot{\theta}^2$, which is similar to the one for AFM within the non-linear $\sigma$-model giving rise to relativistic DW dynamics in this kind of systems. [@IvanovKolezhuk1995] To understand the appearance of such term, the kinetic Lagrangian density term for FM, $\mathcal{L}_{\mathrm{kin}}= -\dot{\phi} \cos \theta$, can be rewritten in the equivalent form $\mathcal{L}_{\mathrm{kin}}=- \phi \, \dot{\theta} \sin \theta$. Evaluating Eqs. at $\phi \left( \xi \right)= \pi /2 - \varepsilon \left( \xi \right)$, we find that the term $ \sin \theta $ is proportional to the time derivative $ \dot{\theta} $, namely $ \sin \theta = \dot{\theta} / \lambda \varepsilon $, which means that the kinetic Lagrangian density term can be expressed in a mass-like form, $\mathcal{L}_{\mathrm{kin}}= \dot{\theta}^2/ \lambda$. The effective Lagrangian density, $\mathcal{L}=e \left( \theta, \phi \right) - \mathcal{L}_{\mathrm{kin}}$, within the limit $m_x \ll 1$ is given by the expression $$\mathcal{L}= {\left( \theta_x \right)}^2 +\sin^2 \theta - \frac{1}{c^2} \, \dot{\theta}^2,
\label{eq:10}$$ which is compatible with the SG equation.
On the other hand, for an uniaxial AFM within the non-linear $\sigma$-model, the SW dispersion relation is $\omega^2=\omega^2_0+c^2k^2$, where $\omega_0$ is a frequency gap due to the uniaxial magnetic anisotropy. Employing the formalism of the complex wave vectors, $K=k+ {{i\mkern1mu}}\kappa$, it can be proved that $v_-=v_+=c$, and that the dependence $\kappa \left( v \right)$, in units of $\kappa_0=\omega_0 /c$, is expressed as $\kappa \left( v \right) = 1/ \sqrt{1-v^2/c^2}$. This immediately leads to relativistic-like expressions for the DW energy and DW width of AFM, $\Delta \left( v \right) = 1/ \kappa \left( v \right)$, similarly as to Eqs. , being $$E_{\mathrm{DW}}\left( v \right)= \frac{E_0}{\sqrt{1-v^2/c^2}}, \, \Delta \left( v \right)=\Delta_0 \sqrt{1-v^2/c^2}.
\label{eq:11}$$
However, in comparison to the approximate Eqs. for biaxial FM, Eqs. are exact within the non-linear $\sigma$-model of an uniaxial AFM. [@Fradkin2013] The SW velocity $c$ in AFM, limiting the DW velocity, is essentially higher than the one in FM due to the exchange enhancement.
We consider that the mapping of the LL equation into a SG-like equation for biaxial FM results in more deep understanding of the DW dynamics. In particular, the DW velocity increase along with the Lorentz contraction of DW width (as a result of the SG equation of DW motion) can lead to considerable spin Peltier effect not only in specific AFM-like $\textrm{Mn}_2 \textrm{Au}$, [@OtxoaAtxitiaRoyEtAl2019] but also in traditional FM metals with biaxial magnetic anisotropy.
Relativistic-like signatures in atomistic simulations {#section:simulations}
=====================================================
As it was previously introduced in Sec. \[section:limit\], there is a dynamical contraction of the DW width as it travels through a biaxial FM. Also, the higher the value of the magnetic anisotropy constants ratio $\lambda $, the greater this process will be. With this in mind, and in the spirit of the search for signatures of relativistic-like behaviors of the DW energy and DW width as was obtained in Eqs. for FM, we investigate this process numerically. To do this, we studied, through atomistic spin dynamic simulations (fifth order Runge-Kutta method to solve numerically the Landau-Lifshitz-Gilbert equation site by site), how the velocity of the magnetic texture, $ v $, and the DW width, $ \Delta $, behaves, as the applied magnetic field, $ H $, directed along the anisotropy easy-axis, is increased. We exploited atomistic spin dynamics simulations because the DW width for high velocities is expected to be about of $1$ nm and the continuous approach fails. With this goal in mind, the simulated system will be given by a 1D FM spin chain consisting of $ 60000$ atomic sites. We use the typical magnetic parameters for the FM layers that make up the layered AFM $\textrm{Mn}_2 \textrm{Au} $, [@BarthemColinMayaffreEtAl2013] being the exchange integral $ I= 115$ K, the atomic moment $\mu = 4 \, \mu_B$, and the lattice period $a_0=0.3328$ nm. In addition, the hard-axis anisotropy constant is given by $K_{x} \, a_0^3 = I$, the easy-axis anisotropy constant possess the value $K_z \, a_0^3=1.302 \cdot 10^{-24}$ J, the gyromagnetic ratio is equal to $\gamma=2.21 \cdot 10^5$ m/(A$\cdot$s), and the Gilbert damping constant is expressed by $\alpha =0.001$. These particular values of the anisotropy constants were chosen to secure the limit $ m_x \ll 1 $, $\varepsilon \left( \xi \right) \ll 1$.
To find moving DW solutions for the case of non-zero magnetic field and non-zero damping we include the corresponding terms to Eqs. . We consider only the Walker-type of solution $\phi \left( \xi \right) = \mathrm{const}$, which gives rise to $$\begin{aligned}
\dot{\theta}= \lambda \cos \phi \sin \phi \sin \theta ,
\\
\alpha \, \dot{\theta} - h \sin \theta=\left( 1+\lambda \cos^2 \phi \right) \cos \theta \sin \theta -\theta_{xx},
\end{aligned}
\label{eq:12}$$ where $h=H/H_a$ is the reduced external magnetic field.
We want to keep the kink solution of Eqs. (\[eq:2\], \[eq:5\]) unchanged and search for a specific Walker-type solution assuming that the damping- and field-terms cancel each other, i.e., $\alpha \, \dot{\theta} = h \, \sin \theta$. Accounting that for the kink solution, $v / \Delta (v)=\lambda \cos \phi_0 \sin \phi_0$, the equation connecting the DW velocity and magnetic field for any value of $\lambda$ is given by $$\frac{ v } {\Delta (v)} = \frac {h} {\alpha},
\label{eq:13}$$ where the velocity-dependent DW width $\Delta(v) = 1/ \kappa(v)$ can be determined from Eq. .
Now, to evaluate how well the data thrown by the simulations fits into a relativistic-like behavior which is a result of the SG equation of motion, it is necessary to take into account the DW contraction $\Delta(v) $ as the velocity increases. The DW width dependence on the DW velocity for the limiting case $ \lambda \gg 1$ can be approximately described by Eqs. as $ \Delta(v) = \Delta_0 \sqrt{1-v^2 / c^2} $, where $ \Delta_0 $ is the DW width at rest, which in the simulations is $ \Delta_0 = 11.62 $ nm, and $c$ is the maximal DW velocity, which was simulated to be $c=4.981$ km/s. Recalculating the atomistic parameters to the micromagnetic ones, we get $M_s=1006$ kA/m, $A=4.77$ pJ/m, $K_x=43.08$ $\textrm{MJ/m}^3$, $K_z= 0.0353$ $\textrm{MJ/m}^3$, and extremely large $\lambda=1220$. The maximal steady DW velocity then is $v_-=4.869$ km/s. This value is very close to the simulated maximal value of the DW velocity, $c$.
The equation for the DW velocity $v(h)$ can be easily solved for the ultimate case $ \lambda \gg 1$. The expression of velocity as a function of the magnetic field is explicitly given by $$v(h)= \frac{h/ \alpha}{\sqrt{1+(h/ \alpha c)^2}}.
\label{eq:14}$$
As it can be seen in Fig. \[im:4\], there is a very good match between the simulations and what is predicted by the theory in the limit $ \lambda \gg 1$. We note that for small fields (velocities) the expression for the DW velocity coincides with the standard Walker expression derived in the limit $\lambda =0$. [@SchryerWalker1974] On the other side, the expression for $v(h)$ has the same form as the expression for DW velocity in weak FM-like ${\textrm{YFeO}}_3$. [@bar2006dynamics] Only the DW mobility $v/H$ is different. In the weak FM it is determined by the exchange and Dzyaloshinskii-Moriya interactions. Therefore, this endorses the idea that, in extreme case of biaxial FM, it is possible to obtain traces of the behavior that characterizes weak FM (AFM), since the DW velocity saturates as the applied magnetic field increases and the DW width contracts drastically as the velocity of the magnetic texture increases. The DW steady motion velocity $v(h)$ is not an arbitrary parameter as it was assumed in previous sections. The steady-state DW motion of this new type of Walker solutions is possible only for a definite value of the velocity which is determined by the given magnetic field and damping parameter according to the equation for $v(h)$. Good agreement between the simulated dependencies $ \Delta(v) $, $v(h)$ and the ones calculated within the Walker approximation $\phi(\xi)=\textrm{const}$ means that the effect of the variable $\varepsilon \left( \xi \right) \ll 1$ on the DW dynamics is small for the very large $\lambda \gg 1$ in a wide DW velocity region up to $v_-$.
In fact, small values of the DW widths at high velocities give rise to a fundamental question. In micromagnetic simulations, which are usually used to evaluate the dynamics of magnetic textures in FM, a certain cell size must be chosen at the beginning of the numerical process. However, if the DW width contracts, it could be the case that, at a certain moment, the cell size chosen is insufficient to capture all the physics present in the problem, which could lead to artificial results that do not correspond to reality. Therefore, it is important to draw attention to this fact, since it could have an enormous impact in some cases. A perhaps more precise way to work in the context of micromagnetic simulations would be to apply a correction to the cell size according to the effect that a Lorentz-like factor could have on the DW width in the medium.
![Atomistic spin dynamics simulations for a FM simple cubic lattice using the parameters listed in the text, from which a relativistic-like behavior can be extracted for the DW width $ \Delta $ (which contracts as the velocity increases) and for the DW velocity $ v $ (which saturates as the external magnetic field increases).[]{data-label="im:4"}](Atomistic_Simulations_FM_with_Mn2Au_Parameters_Domain_Wall_Width_vs_Velocities_Paper_Reduced_Region){width="8cm"}
Conclusions {#section:conclusions}
===========
We addressed the problem of the reduction of the LL equation of motion to the SG equation in a bulk biaxial FM. The investigation on how to increase the DW velocity of anisotropic FM and whether such velocity increase is related to the SG equation of magnetization motion is of considerable importance. In the framework of steady-state Walker-type of solutions, it is formally possible to obtain the aforementioned Lorentz-invariant SG equation for the general case of arbitrary $\lambda$ for any DW velocity $ v < v_- $. However, the mapping has physical sense only in the limit of extremely large ratio of easy-plane and easy-axis magnetic anisotropy constants $ \lambda \gg 1 $ for DW velocities $ v \ll v_- $. The singularities were found for the later case in which the magnetization angle $ \varepsilon = \pi / 2 - \phi $ was operated in the limit $ \varepsilon \ll 1 $. We believe that accounting for the spatial and time dependence of the variable $ \varepsilon $ is sufficient to avoid the singularities in the limit $\lambda \gg 1$ and get the Lorentz-invariant Lagrangian along with SG equation for the magnetization polar angle $\theta (\xi)$. The possibility of mapping the LL equation into the SG equation does not imply an increase of the maximum DW velocity over the Walker-type solution velocity limit $v_-$. The case of the variable azimuthal magnetization angle $\phi (\xi)$ or even the essentially simpler case of the variable $ \varepsilon (\xi) \ll 1 $ are beyond of the theory of 1D magnetic solitons and will be considered elsewhere. The 1D DW dynamics considered above can be used not only for the description of bulk anisotropic FM and 1D spin chains, but also for 2D magnetic systems such as nanowires and nanostripes with small cross-section, if the conditions of applicability of 1D DW model discussed in Refs. \[[ -‘ [@ThiavilleRohartJueEtAl2012; @ThiavilleNakatani2006] ]{}\] are satisfied. Moreover, as it has been proven through atomistic spin dynamics simulations, it is in fact possible to replicate through the precepts of special relativity the behavior of the dynamics of a DW in a biaxial FM for the case $ \lambda \gg 1 $. Such a remarkable result supports what is analytically predicted throughout this text, and shows that, if there is a FM material for which $ \lambda $ was large enough, a system could be experimentally implemented in which WB-induced instabilities disappear without using complicated geometries. Furthermore, as mentioned, the selection of cell size in micromagnetic simulations requires a deeper reconsideration. In the high-speed regime, due to the DW width contraction, the chosen cell size may be insufficient to consider that the continuum approximation continues to be satisfied, which could cause the exchange interaction between the spins that make up the DW to be wrong, due to that the angle between spins would actually be smaller than the micromagnetic simulations would predict, leading to a more abrupt unreal transition through the DW. Because of this, it would be necessary to introduce a dynamic exchange length that took into account relativistic effects at high speeds. As a result, atomistic spin dynamics simulations would generally yield a more precise result even in the case of FM.
Acknowledgements {#section:acknowledgements}
================
K. G. acknowledges support by IKERBASQUE (the Basque Foundation for Science) and Spanish MINECO project FIS2016-78591-C3-3-R. The work of R. O. and K. G. was partially supported by the STSM Grants from the COST Action CA17123 “Ultrafast opto-magneto-electronics for non-dissipative information technology".
|
---
abstract: 'We present results of the one-loop corrections originating from the penguin diagrams for the improved staggered fermion operators constructed using various fat links such as Fat7, Fat7+Lepage, $\overline{\rm Fat7}$, HYP (I) and HYP (II). The main results include the diagonal/off-diagonal mixing coefficients and the matching formula between the continuum and lattice operators.'
address: 'School of Physics, Seoul National University, Seoul, 151-747, South Korea'
author:
- Keunsu Choi and Weonjong Lee
title: |
-0.9 cm SNUTP-03-021
-0.4 cm Penguin diagrams for the HYP staggered fermions[^1]
---
INTRODUCTION {#sec:intr}
============
The low energy effective Hamiltonian of the standard model includes $\Delta S = 1$ four-fermion operators with corresponding Wilson coefficients, which contains all the short-distance physics. The low energy effects of the electroweak and strong interactions can be expressed in terms of matrix elements of the four-fermion operators between hadronic states. Lattice QCD is well-suited to calculate these matrix elements non-perturbatively at low energy. One essential step in using lattice QCD is to find the relationship between the continuum and lattice operators, which is often called “matching formula”. There are two classes of Feynman diagrams at the one-loop level: (1) current-current diagrams and (2) penguin diagrams. At the one loop level, it is possible to treat the penguin contribution and the current-current contribution separately. In the case of the current-current diagrams, the matching formula at the one-loop level is given in [@ref:wlee:1].
Here, we focus on penguin diagrams in which one of the quarks in the four-fermion operator is contracted with one of the anti-quarks to form a closed loop. Hence, the main goal is to calculate the penguin diagrams for improved staggered operators constructed using various fat links and to provide the corresponding matching formula.
Here, we adopt the same notation and Feynman rules outlined in [@ref:wlee:1].
PENGUIN DIAGRAMS
================
Here, we study penguin diagrams. On the lattice, the gauge non-invariant four fermion operators such as Landau gauge operators mix with lower dimension operators, which are gauge non-invariant [@ref:sharpe:1]. It is required to subtract these contributions non-perturbatively. However, it is significantly harder to extract the divergent mixing coefficients in a completely non-perturbative way. Therefore, it is impractical to use gauge non-invariant operators for the numerical study of the CP violations. Hence, it is prerequisite to use gauge invariant operators in order to avoid unwanted mixing with lower dimension operators. For this reason, we choose gauge invariant operators in this study.
In the staggered fermion formalism, there are four penguin diagrams at the one loop level as shown in Fig. \[fig:1\].
These diagrams allow mixing with lower dimension operators as well as four fermion operators of the same dimension or higher. The mixing coefficients with lower dimension operators are divergent ([*i.e.*]{} proportional to inverse power of the lattice spacing). The perturbation is, however, not reliable with divergent coefficients. Hence, we must use non-perturbative method to determine them and subtract away the lower dimension operators. In the case of mixing with operators of the same dimension, the perturbation is expected to be reliable as long as the size of one-loop correction is small enough, which can be achieved by using improved staggered fermions.
In Fig. \[fig:1\], diagrams (a) and (b) have their correspondence in the continuum and diagrams (c) and (d) are pure lattice artifacts. However, diagrams (c) and (d) play an essential role to keep the gauge invariance. Basically, the contribution from diagrams (c) and (d) can be re-expressed as a sum of diagrams (e) and (f) as shown in Fig. \[fig:2\]. The first key point is that the sum of diagrams (a) and (e) generates bilinear operators in a gauge invariant form. The main key point is that the contributions from diagrams (b) and (f) ,as shown in Fig. \[fig:4\], leads to four fermion operators of our interests in a gauge invariant form.
The details of bilinear mixing (diagrams (a) and (e)) in Fig. \[fig:3\] will be presented in [@ref:wlee:2] and here we skip them. Here we focus on the diagrams (b) and (f) and present the final result. The final result is $$\begin{aligned}
G_{(b+f)} &=& \bigg( - \frac{1}{N_f} \bigg)
\frac{g^2}{ (4\pi)^2 } \bigg(\sum_I T^I_{ab} T^I_{cd} \bigg) I_c
\nonumber \\
& & \cdot \sum_{\mu}
\overline{\overline{ (\gamma_{S'} \otimes \xi_{F'}) }}_{C'D'}
\overline{\overline{ (\gamma_{\mu} \otimes 1 ) }}_{CD}
\nonumber \\
& & \cdot \delta_{S,\mu} \delta_{F,1} \Big[ h_{\mu\mu}(k) \Big]^2
\label{eq:b+f:1}\end{aligned}$$ where $k = q-p$ is strictly on shell. $$\begin{aligned}
I_c &=& \frac{16}{3} \bigg(
-\ln( 4 m^2 a^2 ) - \gamma_E + F_{0000} \bigg) - 9.5147
\nonumber \\
& & + {\cal O}(m^2 a^2)\end{aligned}$$ $I_c$ is also given in [@ref:sharpe:1]. The details of deriving Eq. (\[eq:b+f:1\]) will be presented in [@ref:wlee:2]. From Eq. (\[eq:b+f:1\]) we can derive the following theorem: [**Theorem 1 (Equivalence)**]{}\
[*At the one loop level, the diagonal mixing coefficients of penguin diagrams are identical between (a) the unimproved (naive) staggered operators constructed using the thin links and (b) the improved staggered operators constructed using the fat links such as HYP (I), HYP (II), Fat7, Fat7+Lepage, and $\overline{\rm
Fat7}$*]{}.[^2] The details on the proof of this theorem will be given in [@ref:wlee:2].
By construction, gluons carrying a momentum close to $k \sim \pi/a$ are physical in staggered fermions and lead to taste changing interactions, which is a pure lattice artifact. In the case of unimproved staggered fermions, it is allowed to mix with wrong taste ($\ne 1$) and the mixing coefficient is substantial. In contrast, in the case of improved staggered fermions using fat links of our interest such as Fat7, $\overline{\rm Fat7}$ and HYP (II), the off-diagonal mixing with wrong taste vanishes and is absent. In the case of the improvement using HYP (I) and Fat7 + Lepage, the off-diagonal mixing with wrong taste is significantly suppressed. The details of this off-diagonal mixing will be given in [@ref:wlee:2].
In summary, the diagonal mixing occurs only when the original operator has the spin and taste structure of $S=\mu$ and $F=1$ regardless of that of the spectator bilinear. The diagonal mixing coefficient is identical between the unimproved staggered operators and the improved staggered operators constructed using fat links such as Fat7, Fat7+Lepage, $\overline{\rm Fat7}$, HYP (I) and HYP (II). This is a direct consequence of the fact that the contribution from the improvement changes only the mixing with higher dimension operators and off-diagonal mixing, which are unphysical.
The result of this paper, combined with that of [@ref:wlee:1] provides a complete set of one-loop matching formula.
[99]{} W. Lee and S. Sharpe, hep-lat/0306016. S. Sharpe and A. Patel, Nucl. Phys. [**B417**]{} (1994) 307. W. Lee [*et al.*]{}, in preparation.
[^1]: Presented by K. Choi. Research supported in part by BK21, by the SNU foundation & Overhead Research fund, by KRF contract KRF-2002-003-C00033 and by KOSEF contract R01-2003-000-10229-0.
[^2]: Note that AsqTad is NOT included on the list. In this case, by construction the operators are made of the fat links which are not the same as those used in the action due to the Naik term. In addition, the choice of the fat links are open and not unique.
|
---
abstract: 'Further progress in hadron spectroscopy necessitates the phenomenological description of three particle reactions. We consider the isobar approximation, where the connected part of the $\3\to\3$ amplitude is first expressed as a sum over initial and final pairs, and then expanded into a truncated partial wave series. The resulting unitarity equation is automatically fulfilled by the $B$-matrix solution, which is an integral equation for the partial wave amplitudes, analogous to the $K$-matrix parameterization used to describe $\2\to\2$ amplitudes. We study the one particle exchange and how its analytic structure impacts rescattering solutions such as the triangle diagram. The analytic structure is compared to other parameterizations discussed in the literature. We briefly discuss the analogies with recent formalisms for extracting $\3\to\3$ scattering amplitudes in lattice QCD.'
author:
- 'A. Jackura'
- 'C. Fernández-Ramírez'
- 'V. Mathieu'
- 'M. Mikhasenko'
- 'J. Nys'
- 'A. Pilloni'
- 'K. Saldaña'
- 'N. Sherrill'
- 'A. P. Szczepaniak'
bibliography:
- 'bibliography.bib'
title: |
Phenomenology of Relativistic $\3\to\3$ Reaction Amplitudes\
within the Isobar Approximation
---
Introduction {#sec:Introduction}
============
Modern high-energy experiments are accumulating high quality data on three-hadron final states, that are expected to be the main decay channels of several poorly known or missing resonances. These include, for example, the enigmatic $a_1$, $\omega_2$, and the exotic $\pi_1$ resonances that can be studied in peripheral production at COMPASS, GlueX, and CLAS12 [@Adolph:2014rpp; @Adolph:2015pws; @Adolph:2015tqa; @Akhunzyanov:2018pnr; @Bookwalter:2011cu; @Ghoul:2015ifw; @AlGhoul:2017nbp]. In addition to conventional hadrons, many of the exotic $XYZ$ and pentaquark states observed in the heavy quarkonium sector [@Esposito:2016noz; @Lebed:2016hpi; @Olsen:2017bmm], are also found in three particle final states.
Many of these newly observed or anticipated states lie close to thresholds of their decay products. For example, the mass difference between the $X(3872)$ [@Choi:2003ue] and the $D^{0}\bar{D}^0 \pi^0$ threshold is only $6 \text{ MeV}$. The proximity of the three particle threshold together with the possibility of long-range interactions mediated by a single pion exchange can significantly influence the $X(3872)$ line-shape [@Braaten:2010mg] and one needs to carefully analyze the role of pion exchange and whether it is able to bind $D^{*0}$ and $\bar D^{0}$ [@Thomas:2008ja; @Baru:2011rs; @Kalashnikova:2012qf; @Guo:2017jvc]. In the light meson sector, the recently observed $a_1(1420)$ [@Adolph:2015pws] is yet another candidate for a state not expected in the quark model that can be influenced by meson exchange interactions and thresholds [@Basdevant:2015wma; @Ketzer:2015tqa].
On the theory side, lattice QCD has made substantial progress in extracting the resonance spectrum from simulations of $\2\to\2$ reactions [@Wilson:2014cna; @Lang:2015sba; @Dudek:2016cru; @Briceno:2016mjc; @Moir:2016srx; @Briceno:2017qmb; @Briceno:2017max; @Woss:2018irj; @Brett:2018jqw], and recently, the formalism for $\3\to\3$ scattering has been developed [@Hansen:2014eka; @Hansen:2015zga; @Hansen:2016ync; @Briceno:2017tce; @Briceno:2018mlh; @Mai:2017bge; @Mai:2018djl; @Polejaeva:2012ut; @Hammer:2017kms; @Hammer:2017uqm; @Doring:2018xxx]. Analysis and interpretation of both experimental data and lattice simulations require input in the form of reaction amplitudes that can be analytically continued into the complex energy plane. For example, in partial waves, resonances appear as pole singularities, while particle exchanges lead to logarithmic branch points. Fortunately, reaction amplitudes are constrained by unitarity, which can be used to determine the discontinuities of partial waves in the near threshold region.
The problem of constraining $\3\to\3$ reactions from the $S$-matrix principles of unitarity and analyticity has been studied previously in Refs. [@Fleming:1964zz; @Holman:1965; @Aitchison:1966lpz; @Grisaru:1966; @Ascoli:1975mn; @Mai:2017vot]. In this paper we extend these earlier works and clarify some of the results. Moreover, we present the $\3 \to \3$ reaction amplitudes in a way that can be directly translated to the finite volume.
Our description relies on the isobar approximation, where the amplitude is constructed as a sum of truncated partial wave expansions. This provides a good description of three-particle final states in the resonance region, where analyses of Dalitz plots indicate that they are dominated by intermediate two-body resonances. For example, the decay of the $a_1(1260)$ resonance into three pions occurs primarily via a decay to the $\rho\pi$ intermediate state with the subsequent decay of $\rho$ to two pions [@Adolph:2015tqa; @Akhunzyanov:2018pnr]. The isobar approximation can be regarded as an effective way to incorporate the relevant singularities in all Mandelstam variables, and will be discussed in detail later.
The rest of the paper is organized as follows. In Sec. \[sec:Kin\_Inv\_Amp\] we define the $\3\to\3$ amplitude for three spinless particles and discuss the relevant kinematics. In Sec. \[sec:Isobar\_Model\] we introduce the isobar approximation and investigate the consequences of unitarity. We explain the difference between isobar and the partial wave amplitudes , which are often confused. In short, we use the isobar representation to describe the $\3\to\3$ amplitude, $\A = \sum \A_{kj}$, where the indices $k$ and $j$ label the spectator particle in the final and initial state, respectively. We refer to the $ \A_{kj}$’s as isobar-spectator amplitudes, since they can be pictured as scattering of a quasi-particle, the isobar, and a stable spectator. The latter are expanded in partial waves of the three-particle system. Unitarity constrains the $\3\to\3$ amplitudes on the real energy axis, which results in specific relations involving the imaginary parts of the partial-wave-projected isobar-spectator amplitudes. Unitarity alone does not uniquely specify partial wave amplitudes, as evident, for example, in the $K$-matrix parametrization of $\2\to\2$ scattering amplitudes [@Castillejo:1955ed; @Gribov:2009zz]. In Sec. \[sec:B-Matrix\] we discuss a specific parameterization for the isobar-spectator amplitudes which satisfies the three-body and two-body unitarity. It is given as a solution of a set of linear integral equations that involve, among others, the one particle exchange (OPE) as a driving term. We call this the $B$-matrix parameterization and it satisfies, $$\A_{kj} = \B_{kj} + \B_{kn} \tau_{n} \A_{nj}, \label{1}$$ where $\B$ is the driving term that contains the OPE, $\tau$ is a known function of the phase space and of the $\2\to\2$ amplitudes. The product formally represents an integration over the intermediate isobar mass. In contrast to Ref. [@Mai:2017vot], we restrict the domain of the integrals to physical values of energies. This enables us to use the experimentally accessible subchannel amplitudes and we also discuss the consequences of this restriction. We derive Eq. (\[1\]) for isobars with arbitrary spin $s$, and for any value of the isobar-spectator orbital angular momentum $\ell$.
The $B$-matrix parameterization can be analytically continued to the complex energy plane and in Sec. \[sec:AnalyticProperties\] we discuss aspects of its analytic properties. Specifically, the one particle exchange process has some unique features, as it contains a kinematic singularity due to the exchange of a real particle, which can be isolated from the full $\3\to\3$ scattering amplitude. In addition, we also study the triangle amplitude that emerges from the $B$-matrix parameterization, and the relation to the Bethe-Salpeter solution. We summarize our results in Sec. \[sec:Conclusion\].
Kinematics, Invariants, & Amplitudes {#sec:Kin_Inv_Amp}
====================================
We consider elastic scattering of three distinguishable, spinless particle, $D\pi \bar D$, $K\pi \bar K$, or $\pi^+\pi^-\pi^0$. The particles have mass $m_j$ , where $j = 1, 2, $ or $3$ labels the individual particles. A single particle state, with four-momentum $p_j = (\omega_j,\p_j)$, where $\omega_j = \sqrt{m_j^2 + \lvert \p_j\rvert^2}$ is the energy and $\p_j$ is the three-momentum, is denoted $\ket{\p_j}$ and has relativistic normalization $\braket{\p_k'|\p_j} = (2\pi)^{3}\,2\omega_j \delta^{(3)}(\p_k'-\p_j)\delta_{kj}$. We are interested in the $S$-matrix element of the elastic $\3\to\3$ scattering process. We can decompose the $S$-matrix as $S = \1 + i T$. The $T$-matrix contains two terms, $T = T_d + T_c$, where the disconnected part, $T_d$, involves interactions of two particles at the time with the third one being a spectator, while the connected part, $T_c$, contains interactions of all three particles. The disconnected part can always be identified kinematically by the spectator momentum conserving delta function [@Itzykson:1980rh]. The disconnected part is written as $T_d = \sum_{j}\1_{j} \otimes T^{(j)}$, where $\1_{j}$ is the identity operator in the single particle space of the spectator, $j$ and $T^{(j)}$ describes $\2 \to \2$ scattering between the other two particles. The amplitudes associated with the matrix elements of scattering operators $T_d$ and $T_c$ are defined as ${\F}$ and ${\A}$, respectively. Specifically, the connected amplitude $\A$ is given by $$\label{eq:3to3c_Amp}
\bra{\p'}T_c\ket{\p} = (2\pi)^4\delta^{(4)}(P' - P) \A(\p';\p),$$ where $\ket{\p} \equiv \ket{\p_1\p_2\p_3}$ and $\ket{\p'} \equiv \ket{\p_1'\p_2'\p_3'}$ denote the initial and final states of the three particles, and $P = p_1 + p_2 + p_3$ and $P' = p_1' + p_2' + p_3'$ are the initial and final total four-momenta, respectively, as illustrated in Fig. \[fig:3to3\_Symbols\]. Time-Reversal symmetry implies that the amplitude is symmetric in the initial-final state variables, $\A(\p';\p) = \A(\p;\p')$. The chosen normalization implies that the amplitude $\A(\p';\p)$ has mass dimension $-2$. The disconnected amplitudes $\F_{j}$ are defined by $$\label{eq:2to2d_Amp}
\begin{split}
\bra{\p'} T_d \ket{\p} & = (2\pi)^4\delta^{(4)}(P' - P)
\\ & \times
\sum_{j=1}^{3} (2\pi)^{3}\,2\omega_j \delta^{(3)}(\p_j'-\p_j) \F_{j}(\p';\p),
\end{split}$$ where the delta function enforces that the spectator $j$ does not interact. We remark that the $\F_{j}$ is the genuine $\2\to\2$ scattering amplitude, as required by the LSZ construction [@Itzykson:1980rh]. We also define $P_{j} \equiv P - p_{j}$ and $P_{j}' \equiv P' - p_{j}'$ as the initial and final total four-momenta of the interacting pair recoiling against spectator $j$, Fig. \[fig:3to3\_Kinematics\].
In this paper we adopt the so-called [*[spectator notation]{}*]{} or [*[odd-one-out notation]{}*]{} [@Giebink:1985zz], where the $\2\to \2$ amplitudes associated with the spectator $j$ are labeled by the spectator index. The spectator notation requires additional information specifying the first particle in the two-particle system. There are two conventions which are useful for our discussions: the two-pair convention, and the cyclic convention. The two-pair convention is more practical when interaction in one of the three pairs is negligible. An example of such a system is $\pi^+\pi^+\pi^-$, where the $\pi^+\pi^+$ system interacts weakly. In this case it is convenient to choose the noninteracting system as, say, particles $(13)$ and designate particle $2$ as the second particle for both the interacting sub-systems. Therefore, the spectator index $j=1$ and $j=3$ uniquely identifies the two orderings in the pairs to be $(32)$ and $(12)$, respectively. If the interactions in all three subchannels are important, one can define the ordering through cyclical permutation, the spectator label $j=1,2,3$ corresponds to ordering of the two particles subsystems as $(23)$, $(31)$, and $(12)$, respectively. For simplicity, in the following we assume only two relevant subchannels, and use the former convention. Generalization to the latter case is straightforward. The type of applications we have in mind are systems like $M\bar M \pi$ elastic scattering, where $M$ is an open-flavor meson, such as $K$, $D$, and $B$. The interacting pairs will be assumed in the $M\pi$ and $\bar{M}\pi$ channels only, and pion being designated as particle $j=2$.
The $\3\to\3$ amplitude depends on eight independent kinematic variables. The choice of variables largely depends on the kinematical range of interest, the low vs high total energy region. Here we are interested in the low-energy region and use the following redundant set of Mandelstam variables,
$$\begin{aligned}
s & = (p_1 + p_2 + p_3)^2 = (p_1'+p_2'+p_3')^2, \\
t_{jk} & = (p_j - p_k')^2 = (P_{j} - P_{k}')^2, \\
u_{jk} & = (P_{j} - p_k')^2 = ((P - p_j) - p_k')^2, \\
\sigma_{j} & = P_{j}^2 = (P - p_{j})^2, \\
\sigma_{k}' & = P_{k}'^2 = (P - p_{k}')^2.\end{aligned}$$
where $s$, $\sigma_j$, and $\sigma_k'$ are the invariant mass squares of the total three particle system, the initial pair, and the final pair, respectively. The transferred momenta, $t_{jk}$ and $u_{jk}$, are between the initial and final spectators and the initial pair and final spectator, respectively. The Mandelstam invariants are related by energy-momentum conservation,
$$\begin{aligned}
s + t_{jk} + u_{jk} & = \sigma_{j} + \sigma_{k}' + m_j^2 + m_k^2, \label{eq:NRG_const} \\
\sum_{j=1}^{3}\sigma_{j} & = s + \sum_{j=1}^{3}m_{j}^2, \label{eq:subNRG_const1} \\
\sum_{k=1}^{3}\sigma_{k}' & = s + \sum_{k=1}^{3}m_{k}^2. \label{eq:subNRG_const2}\end{aligned}$$
In the physical region of the $\3\to\3$ reaction, $s$ can take any value above the three particle threshold, $s \ge s_{\mathrm{th}} = (m_1 + m_2 + m_3)^2$, while the subchannel invariant masses $\sigma_{j}$ and $\sigma_{k}'$ are bounded by ${\sigma_j^{(\mathrm{th})}} \le \sigma_{j} \le (\sqrt{s} - m_j)^2$ and ${\sigma_k^{(\mathrm{th})}} \le \sigma_{k}' \le (\sqrt{s} - m_k)^2$, where ${\sigma_j^{(\mathrm{th})}}$ are the sub-energy thresholds, ${\sigma_1^{(\mathrm{th})}} = (m_{2} + m_{3})^2$. We will need the relations between the invariants and energies and scattering angles, in three reference frames. The frames of interest will be the overall center-of-momentum frame (CMF) and the isobar rest frame (IRF). There are two IRFs corresponding to the initial and final states: the initial IRF$_j$, labeled by the spectator $j$, and the final IRF$'_k$, labeled with spectator $k$ and a prime.
To distinguish momenta in the CMF we denote them by a $\star$, $\P^{\star} = \P'^{\star} = \0$. In the CMF the scattering angle, $\Theta_{kj}^{\star}$, is defined as the angle between the initial and final state spectator momenta, $\cos\Theta_{kj}^{\star} \equiv \wh{\p}_k'^{\star} \cdot \wh{\p}_j^{\star}$, where $\wh{\p}_j^{\star}$ and $\wh{\p}_k'^{\star}$ denote the CMF orientations of the initial and final spectators, respectively. The kinematic variables in the other frames, IRF$_j$ ($\P_j=\0$) and IRF$'_k$ ($\P_k'=\0$) are obtained from the CMF by a Lorentz boost in the direction opposite to momentum of the corresponding spectator. The momentum of the first particle in the pair is denoted by $\q_j$, and $\q_k'$ in IRF$_j$ and IRF$'_k$, respectively. Orientation of these momenta are given by solid angles, $\wh{\q}_j=(\gamma_j,\chi_j)$ and $\wh{\q}_k' = (\gamma_k',\chi_k')$, respectively. Here, $\gamma_j$ and $\gamma_k'$ are the azimuthal angles between the decay plane of the isobar and the isobar-spectator scattering plane and $\chi_j$ and $\chi_k'$ are helicity angles, Fig. \[fig:3bodykins\] for the specific scheme $(23)1 \to (12)3$. The relations between all relevant kinematical variables and the Mandelstam invariants are given in Appendix \[sec:app\_A\]. In the following, we will use the set $(\wh{\q}'_k, \sigma_k, s, t_{jk}, \sigma_j, \wh{\q}_j)$ to describe the isobar-spectator amplitude.
Unitarity Relations {#sec:UnitarityRelations}
-------------------
We consider elastic unitarity in the physical region of the $\3 \to \3$ reaction below inelastic thresholds. It yields two relations [@Eden:1966dnq], one for the disconnected $\2\to\2$ amplitude $\F_j$ and one for the connected $\3\to\3$ amplitude $\A$. For $\F_{j}$, one finds $$\label{eq:2to2discUnit}
\im{\F_{j}(\p';\p)} = \rho_2(\sigma_j) \int d\wh{\q}_j'' \, \F_{j}^{*}(\p'';\p') \F_{j}(\p'';\p)$$ where $$\label{eq:2bodyPS}
\rho_2(\sigma_j) = \frac{1}{64\pi^2} \frac{2\lvert \q_j \rvert}{\sqrt{\sigma_j}}$$ is the phase space for the two particle system, and $\q''_j$ is the intermediate state relative momentum. The IRFs are defined with their $z$-axes defined along the opposite direction of the spectator and their $x$-axes defined by their azimuthal angles w.r.t. the total CMF plane spanned by the initial and final spectator momenta, Fig. \[fig:3bodykins\]. Note that from energy-momentum conservation, $ \lvert \q_j \rvert = \lvert \q_j' \rvert = \lvert \q_j'' \rvert$. Figure \[fig:2to2\_disc\_unitarity\] is a diagrammatic representations of Eq. .
![Diagrammatic representation for the $\2\to\2$ disconnected amplitude unitarity relation in Eq. . The red vertical dashed line indicates the intermediate particles are put on-shell. []{data-label="fig:2to2_disc_unitarity"}](figures/2to2_connected_unitarity){width=".8\columnwidth"}
Elastic unitarity yields the following condition for the connected $\3\to\3$ amplitude,
$$\label{eq:3to3connUnit} \
\begin{split}
\im{\A(\p';\p)} & = \frac{1}{2(2\pi)^{5}} \int \frac{d^{3}\p_1''}{2\omega_1''} \frac{d^{3}\p_2''}{2\omega_2''} \frac{d^{3}\p_3''}{2\omega_3''} \, \delta^{(4)}(P'' - P) \,\A^{*}(\p'';\p') \A(\p'';\p) \\
\
& + \sum_{k}\,
\rho_2(\sigma_k')
\int d\wh{\q}_k'' \,\F_k^{*}(\p'';\p') \A(\p'';\p)\rvert_{\p''_k = \p_k'} \Theta(\sigma_k' - \sigma_{k}^{(\mathrm{th})}) \\
\
& + \sum_{j}\,
\rho_2(\sigma_j)
\int d\wh{\q}_j'' \,\A^{*}(\p'';\p') \rvert_{\p''_j = \p_j} \F_{j}(\p'';\p)\Theta(\sigma_j - \sigma_{j}^{(\mathrm{th})}) \\
\
& + \sum_{\substack{j,k \\ j\ne k}} \pi \, \delta(u_{jk} - \mu_{jk}^2)\,\F_{k}^{*}(\p'';\p')\rvert_{\p''_j = \p_j} \F_{j}(\p'';\p) \rvert_{\p''_k = \p_k'},
\end{split}$$
where $\mu_{jk}$ is the mass of the exchanged particle that is neither $j$ nor $k$, if $j=1$, and $k=3$, then the exchanged mass is $\mu_{13} = m_2$. Note that the evaluations $\p_k'' = \p_k'$ in the second and fourth lines enforce that $\sigma_k' = \sigma_k''$, and similarly in lines three and four, $\p_j'' = \p_j$ implies that $\sigma_j'' = \sigma_j$. Figure \[fig:3to3\_conn\_unitarity\] is a diagrammatic representation of Eq. and its derivation is given in Appendix \[sec:app\_B\].
The implications of unitarity for the $\F_j$ are summarized below. The unitarity relation for the connected, $\A$ amplitude is more complicated. The first term in Eq. is analogous to the $\2\to\2$ case, in the sense that it is given by the product of the same connected amplitude $\A$. The next two terms originate from the contribution to $S^{\dag}S$ given by the product of $T_c$ and $T_d$, and represents the situation when only two of the three particles rescatter. The last term is the contribution to the imaginary part of the connected amplitude from the product of two disconnect amplitudes and reflects the real one particle exchange process. Since the unitarity relation deals with physical, on-shell amplitudes, this last contribution is non-vanishing only when the exchanged particle is on-shell, where it is singular and proportional to $\delta(u_{jk} - \mu_{jk}^2)$.
The implications of unitarity for the analytic properties of the $\2\to\2$ amplitude are well known [@Gribov:2009zz]. In the physical region the partial wave expansion $$\label{eq:2to2PW}
\F_{j}(\p';\p) = \sum_{s_j=0}^{\infty} \N_{s_j}^{2}
f_{s_j}(\sigma_j) P_{s_j}(\wh{\q}'_{j} \cdot \wh{\q}_j ),$$ converges and reduces the integral relation given by Eq. to a countable set of algebraic ones. Here $s_j$ is the angular momentum of the two-particle system $j$ defined in the IRF$_j$-frame, $\N_{s_j}^2 = ( 2s_j + 1) / 4\pi$ is a normalization constant, $f_{s_j}(\sigma_j)$ is the partial wave amplitude, and $P_{s_j}(\wh{\q}'_{j} \cdot \wh{\q}_j )$ is the Legendre polynomial describing the rotation dependence in terms of the cosine of the $\2\to\2$ scattering angle. The unitarity relation is diagonalized to the partial wave unitarity relation, $$\label{eq:2to2UnitPW}
\im f_{s_j}(\sigma_j) = \rho_2(\sigma_j) \lvert f_{s_j} (\sigma_j)\rvert^{2} \Theta(\sigma_j - \sigma_j^{\sss (\mathrm{th})}).$$ This equation is automatically satisfied by $$\label{eq:AmpKmat}
f_{s_{j}}^{-1}(\sigma_{j}) = K_{s_j}^{-1}(\sigma_j) - \frac{1}{\pi} \int_{\sigma_{j}^{(\mathrm{th})}}^{\infty} d\wh{\sigma} \, \frac{\rho_2(\wh{\sigma})}{\wh{\sigma} - \sigma_j}$$ where the $K$-matrix is a real function along the unitarity cut.
The $\3\to\3$ amplitude in the physical region can be expanded in partial waves in any of the $(12)$, $(13)$, $(23)$ subsystems. We refer to a subchannel of choice, $(12)$ as the direct channel and to the others as the cross channels. Since each term in the partial wave expansion is analytic in the angular variables, and therefore in the $(13)$ and $(23)$ invariant masses, singularities in the latter variables can happen only when the series diverges. In contrast to the $\2\to\2$ case, the unitarity equations for each partial wave would not decouple, and would contain an infinite number of terms. Since in practice one must truncate the series, the amplitude would be regular in the $(13)$ and $(23)$ invariant masses, and the information about the cross channels dynamics would be lost. Instead, we will represent $\3\to\3$ amplitude in an isobar approximation, where only a finite number of terms in the direct and cross channels are included.
The Isobar Representation {#sec:Isobar_Model}
=========================
To be concrete, the partial wave expansion of the connected $\3\to\3$ amplitude reads $$\label{eq:PW_expansion}
\begin{split}
\A(\p';\p) & = \sum_{J} \sum_{\ell_k',s_k'} \sum_{\ell_j,s_j} \mathcal{M}_{\ell_k' s_k'; \ell_j s_j}^{J}(\sigma_k',s,\sigma_j) \\
& \times \sum_{M}Z_{\ell_k' s_k'}^{JM\,*}(\wh{\P}_k'^{\star},\wh{\q}'_{k}) Z_{\ell_j s_j}^{J M}(\wh{\P}_j^{\star},\wh{\q}_j) ,
\end{split}$$ where we project the amplitude onto the chosen $j$ and $k$ initial and final channels. Here $s_j$ ($s_k'$) is the angular momentum of the initial (final) pair, $\ell_j$ ($\ell_k'$) is the angular momentum between the pair and the spectator, $J$ and $M$ are the total angular momentum of the three particles and its projection, and $\mathcal{M}_{\ell_k' s_k';\ell_j s_j}^{J}$ is the partial wave amplitude. The angles $\wh{\P}_j^{\star}$ and $\wh{\P}_k'^{\star}$ are the orientations of the initial and final pair, which are related to the CMF scattering angle via $\cos{\Theta_{kj}^{\star}}=\wh{\P}_k'^{\star}\cdot\wh{\P}_j^{\star}$. The functions $Z_{\ell s}^{JM}$ contain the rotational dependence of the amplitude $\A$, which are defined as $$\label{eq:Zfcn}
Z_{\ell s}^{JM} (\wh{\P},\wh{\q}) = \N_{\ell} \N_{s} \sum_{\lambda=-s}^{s} \braket{J \lambda | \ell 0 s \lambda} \D_{M \lambda}^{(J)}(\wh{\P}) \D_{\lambda 0}^{(s)}(\wh{\q}).$$ The $Z$-functions contain all the angular dependence, and they fulfill the orthonormality condition $$\label{eq:ZfcnOrtho}
\int d\wh{\P} \int d\wh{\q} \, Z_{\ell' s'}^{J'M' *}(\wh{\P},\wh{\q}) Z_{\ell s}^{JM}(\wh{\P},\wh{\q}) = \delta_{JJ'}\delta_{MM'}\delta_{\ell\ell'}\delta_{ss'}.$$ More details are in Appendix \[sec:app\_C\].
We next discuss the relation between partial wave expansion, isobar representation, and finally the isobar approximation. The partial wave expansion given by Eq. is in principle an exact representation of the amplitude in the physical region of $\3 \to \3$ scattering. However, unlike the analogous expansion in $\2 \to \2$ scattering, the partial wave expansion cannot be used in practice in the $\3 \to \3$ case.
In practice, one needs to restrict the series to a finite number of partial waves. In the physical region of $\2 \to \2$ scattering, the low-energy behavior of the partial waves is determined by barrier factors due to the finite range of interactions. This suppresses the strength of higher partial waves at threshold, provided the latter are regular in the cross channel Mandelstam variables. Cross channel exchanges generate singularities that spoil the convergence of the partial wave series. However, in the $\2 \to \2$ kinematics, these singularities do not overlap with the direct channel physical region. Therefore, the partial wave series can be safely truncated in a finite domain of CMF energies above the two particle threshold.
This is not the case, for example, when one of the particles can decay to the other three, and similarly it is never the case for $\3\to\3$ scattering. If we consider indistinguishable particles, explicit Bose symmetry is lost for the $\mathcal{M}_{\ell_k' s_k';\ell_j s_j}^{J}$ partial waves, since the partial wave expansion in the initial and final states singles out specific two-body channels. The symmetry is only recovered upon resummation. The isobar representation, in principle, takes care of this problem. One writes the connected $\3\to\3$ amplitude as a redundant sum of expansions in all the initial and final pairs to make the symmetry explicit. Bose symmetry is thus preserved upon truncation.
As discussed above, one can manage only a finite number of terms in the sums over the subchannel spins. Therefore one reduces the isobar representation $$\label{eq:Isobar_Expand}
\A(\p';\p) = \sum_{j,k} \A_{kj}(\p';\p),$$ to the isobar approximation, by representing the connected $\3\to\3$ amplitude as a sum over a finite number of isobar-spectator amplitudes, $$\label{eq:PWIS_expansion}
\begin{split}
\A_{kj}(\p';\p) & = \sum_{J} \sum_{\ell_k',s_k'}^{\text{max}'} \sum_{\ell_j,s_j}^{\text{max}} \A_{\ell_k' s_k'; \ell_j s_j}^{J}(\sigma_k',s,\sigma_j) \\
& \times \sum_{M}Z_{\ell_k' s_k'}^{JM\,*}(\wh{\P}_k'^{\star},\wh{\q}'_{k}) Z_{\ell_j s_j}^{J M}(\wh{\P}_j^{\star},\wh{\q}_j) ,
\end{split}$$ as shown in Fig. \[fig:3to3\_Isobar\_Expand\]. The truncation is reflected by “max" in the sums. We projected the isobar-spectator amplitudes onto the total angular momentum $J$ of the three particle system. In the following, we refer to $\A_{\ell_k' s_k'; \ell_j s_j}^{J}(\sigma_k',s,\sigma_j)$ as the partial wave isobar spectator (PWIS) amplitudes. We emphasize that, while truncation in $s_k'$ and $s_j$ cannot be avoided in practice, unitarity is diagonal in the total angular momentum. Amplitudes for each $J$ are thus independent and can in principle be resummed.
We also stress that the PWIS amplitudes $\A_{\ell_k' s_k'; \ell_j s_j}^{J}$ are not the genuine $\3\to\3$ partial wave amplitudes $\mathcal{M}_{\ell_k' s_k'; \ell_j s_j}^{J}$ in Eq. : $$\label{eq:3to3_recover}
\begin{split}
\mathcal{M}_{\ell_k' s_k' ; \ell_j s_j}^{J}(\sigma_k',s,\sigma_j) & = \A_{\ell_k' s_k' ; \ell_j s_j}^{J}(\sigma_k',s,\sigma_j) \\
&+ X_{\ell_k' s_k' ; \ell_j s_j}^{J}(\sigma_k',s,\sigma_j),
\end{split}$$ where $X_{\ell_k' s_k' ; \ell_j s_j}^{J}$ contains all the cross channel terms which recouple to the direct channel amplitude, $$\label{eq:cross_channel}
\begin{split}
& X_{\ell_k' s_k' ; \ell_j s_j}^{J}(\sigma_k',s,\sigma_j) \\
& = \sum_{\substack{\, a\ne j,\\ b\ne k} } \sum_{\ell_b' , s_b'} \sum_{\ell_a, s_a} \int d\wh{\P}_k'^{\star} \int d\wh{\q}_k' \, \int d\wh{\P}_j^{\star} \int d\wh{\q}_j \, \\
& \times Z_{\ell_k' s_k' }^{J M }(\wh{\P}_k'^{\star},\wh{\q}_k') Z_{\ell_b' s_b' }^{JM\,*}(\wh{\P}_b'^{\star},\wh{\q}_b') \\
& \times Z_{\ell_j s_j}^{JM\,*}(\wh{\P}_j^{\star},\wh{\q}_j) Z_{\ell_a s_a}^{JM}(\wh{\P}_a^{\star},\wh{\q}_a) \\
& \times \A_{\ell_b' s_b' ; \ell_a s_a}^{J}(\sigma_b',s,\sigma_a) .
\end{split}$$ The kinematic relations given in Appendix \[sec:app\_A\] can be used to write the cross channel variables in terms of the direct channel variables.
![Diagrammatic representation of the isobar approximation amplitude in Eq. . The double lines with the black disk represents the isobar amplitude $f_{s_j}(\sigma_j)$, while the gray disk represents the isobar-spectator amplitude $\A_{kj}(\p';\p)$. []{data-label="fig:3to3_Isobar_Expand"}](figures/isobar_expansion_v2){width=".85\columnwidth"}
Often in the literature, Bose symmetry is considered as a motivation for Eq. . However, this is completely independent: the representation can be applied to the distinguishable particle case (in this case the various $\A_{kj}(\p';\p)$ contain different physics and have different functional forms), and Bose symmetry can be imposed to the expansion in Eq. without requiring an explicit sum over channels. For example we consider the $\pi^+\pi^-\pi^0 \to \pi^+\pi^-\pi^0$ process in the isoscalar vector channel, where the $\omega$ is observed. Thinking in isospin basis, where the three pions are indistinguishable, and in the charge basis, where they are distinguishable, leads to the same form of the amplitude, showing that Bose symmetry plays no role in defining the representation.
Isobars parameterize the $\2\to\2$ dynamics in a given subchannel and angular momentum state. Contrary to the $\2\to\2$ partial waves, they have only right hand singularities constrained by unitarity. In the $N/D$ formalism, the isobars can be identified with the denominator function, where the left hand cuts are removed via a dispersive integral [@Bjorken:1960zz]. In the following, we will ignore all left hand singularities of the $\2\to\2$ amplitudes, and identify their partial waves with the isobars. Although we do not need to assume any resonant content for the isobars (we could use an isobar to describe the $\pi^+\pi^+$ dynamics), it is a popular picture to think of them as a quasi-particle, and to identify the invariant mass and angular momentum of the pair with the isobar mass and spin. Isobars are customarily labeled with the name of the dominant resonance, if any. Isobars can be parameterized as in Eq. . For example, the $a_1(1260)$ decays into three pions dominantly in the $\rho\pi$ and $\sigma\pi$ channels [@Tanabashi:2018oca]. If one chooses to perform a truncated partial wave expansion of the $\3\to\3$ amplitude in only the $\rho\pi \to a_{1}(1260) \to \rho\pi$ channel, rescattering effects between the $\rho\pi \to \sigma\pi$ isobars are ignored. The isobar approximation corrects this by including amplitudes for $\sigma\pi \to a_{1}(1260) \to \rho\pi$, $\rho\pi \to a_{1}(1260) \to \sigma\pi$, and $\sigma\pi \to a_{1}(1260) \to \sigma\pi$.
The approximation is expected to be valid at low values of energy, where a finite number of singularities dominate the amplitude. Moving to higher energies, the left hand cuts controlling the crossed $\2\to\mathbf{4}$ processes will become relevant, and the behavior of the amplitude will be controlled by analyticity in angular momentum, rather than direct-channel unitarity.
Since the isobar approximation includes the cross channel effects in the summation, the isobar-spectator amplitudes contain only normal threshold singularities determined by unitarity. Therefore, the analytic structure of each isobar-spectator amplitude in the energy variables, $s$, $\sigma_j$, and $\sigma_k'$, are determined by unitarity.
The problem of convergence in $J$ is more severe. The $\3 \to \3$ amplitude contains an OPE process (see the last diagram in Fig. \[fig:3to3\_conn\_unitarity\]), which can go on-shell in the direct channel, and results in an interaction of infinite range. In this case the cross channel singularities overlap with the physical region and project onto an infinite number of partial waves. The analytic properties of the projected amplitude are highly nontrivial. We discuss them in detail detail in Section \[sec:AnalyticProperties\]. However, since the main goal in this and similar studies of three particle scattering is to identify the spectrum, ultimately one needs to deal with amplitudes of well defined total angular momentum $J$. In other words, these amplitudes diagonalize unitarity, which is the basis for analytic continuation and identification of complex singularities as resonance poles. For this reason, in the following we will not address the problem of convergence in $J$.
Unitarity Relations {#sec:IM_unitarity .unnumbered}
-------------------
It is advantageous to introduce an amputated PWIS amplitude $\wt{\A}_{\ell_k' s_k';\ell_j s_j}^{J}$, in which the isobar amplitudes are factorized, $$\label{eq:amputation}
\A_{\ell_k' s_k'; \ell_j s_j}^{J}= f_{s_k'}(\sigma_k') \wt{\A}_{\ell_k' s_k'; \ell_j s_j}^{J}(\sigma_k',s,\sigma_j) f_{s_j}(\sigma_j).$$ The amputation reduces the number of terms in the isobar-spectator unitarity relation by making use of subchannel unitarity in Eq. . However, the amputated PWIS amplitudes still have a non-trivial dependence on the subchannel energies due to rescattering effects. As shown in detail in Appendix \[sec:app\_C\], combining Eqs. , , , , , and results in the amputated PWIS unitarity relation
$$\label{eq:PWIS_unitarity}
\begin{split}
\im \, & \wt{\A}^{J}_{ \ell_k' s_k' ; \ell_j s_j }(\sigma_k',s,\sigma_j) \\
& = \frac{1}{\pi(32\pi^2)^2} \sum_{n} \sum_{\ell_n'' , s_n''} \int_{\sigma_{n}^{(\mathrm{th})}}^{(\sqrt{s} - m_n)^2} d\sigma_n'' \, \frac{\lvert \q_n'' \rvert \lvert \p_n''^{\star} \rvert }{ \sqrt{\sigma_n''} \sqrt{s}} \lvert f_{s_n''}(\sigma_n'') \rvert^2 \, \wt{\A}^{J\,*}_{\ell_n'' s_n'' ; \ell_k' s_k' }(\sigma_n'',s,\sigma_k') \wt{\A}^{J}_{\ell_n'' s_n'' ; \ell_j s_j }(\sigma_n'',s,\sigma_j) \Theta(s - s_{\mathrm{th}}) \\
\
& + \frac{1}{2\pi s(32\pi^2)^2} \sum_{\substack{n,r \\ n \ne r}} \sum_{\ell_n'' , s_n''}\sum_{\ell_r'' , s_r''} \int_{\sigma_{n}^{(\mathrm{th})}}^{(\sqrt{s} - m_n)^2} d\sigma_n'' \,\int_{\sigma_r^{(-)}(\sigma_n'')}^{\sigma_r^{(+)}(\sigma_n'')} d\sigma_r'' \, f_{s_r''}^{*}(\sigma_r'')f_{s_n''}(\sigma_n'') \\
\
& \qquad\qquad \times \C_{\ell_n'' s_n'' ; \ell_r'' s_r''}^{J}(\sigma_n'',s,\sigma_r'')\, \wt{\A}^{J\,*}_{\ell_r'' s_r''; \ell_k' s_k' }(\sigma_r'',s,\sigma_k') \wt{\A}^{J}_{\ell_n'' s_n'' ; \ell_j s_j }(\sigma_n'',s,\sigma_j) \Theta(s - s_{\mathrm{th}}) \\
\
& + \frac{1}{64 \pi^2 \sqrt{s}} \frac{1}{\lvert \p_k^{\prime\,\star} \rvert} \sum_{\substack{r \ne k}} \sum_{\ell_r'', s_r''}\, \int_{\sigma_r^{(\mathrm{th})}}^{(\sqrt{s}-m_r)^2} d\sigma_r'' \, \C_{\ell_k' s_k' ; \ell_r'' s_r''}^{J}(\sigma_k',s,\sigma_r'') \, f_{s_r''}(\sigma_r'') \, \wt{\A}_{\ell_r'' s_r'' ; \ell_j s_j }^{J}(\sigma_r'',s,\sigma_j) \Theta(\sigma_k' - \sigma_{k}^{(\mathrm{th})}) \\
\
& + \frac{ 1 }{64 \pi^2 \sqrt{s}} \frac{1}{\lvert \p_j^{\star} \rvert} \sum_{\substack{n \ne j}} \sum_{\ell_n'' , s_n''} \, \int_{\sigma_n^{(\mathrm{th})}}^{(\sqrt{s}-m_n)^2} d\sigma_n'' \, \C_{\ell_n'' s_n'' ; \ell_j s_j}^{J}(\sigma_n'',s,\sigma_j) \, f_{s_n''}^{*}(\sigma_n'') \, \wt{\A}_{ \ell_k' s_k'; \ell_n'' s_n''}^{J\,*}(\sigma_k',s,\sigma_n'') \Theta(\sigma_j - \sigma_{j}^{(\mathrm{th})}) \\
\
& + \frac{\pi}{2\lvert \p_j^{\star} \rvert \lvert \p_k'^{\star} \rvert} \, \C_{\ell_k' s_k' ; \ell_j s_j}^{J}(\sigma_k',s,\sigma_j)
(1 - \delta_{jk}) \Theta(1 - \cos^2\theta_{kj}^{\star}) ,
\end{split}$$
where $\C_{\ell_n'' s_n'' ; \ell_r'' s_r''}^{J}(\sigma_n'',s,\sigma_r'') $ is a purely kinematical recoupling coefficient between different intermediate state isobars, $$\label{eq:recoupling_coef}
\begin{split}
\C_{\ell_k s_k ; \ell_j s_j}^{J}(\sigma_k,s,\sigma_j) & = 2\pi \, \N_{s_j} \N_{s_k} \N_{\ell_j}\N_{\ell_k} \N_{J}^{-2} \sum_{\lambda_k,\lambda_k} \braket{J \lambda_k | \ell_k 0 s_k \lambda_k} \braket{J \lambda_j | \ell_j 0 s_j \lambda_j}\,\\
& \times d_{\lambda_k 0}^{(s_k)}(\cos\chi_k
)\,d_{\lambda_k \lambda_j}^{(J)}(\cos\theta_{kj}^{\star}
) \,d_{\lambda_j 0}^{(s_j)}(\cos\chi_j
) .
\end{split}$$ The recoupling coefficients relate two different orientations of three particles in the same frame [@Ascoli:1974sp; @Ascoli:1975mn; @Giebink:1985zz]. Appendix \[sec:app\_C\] contains details on the derivation of the recoupling coefficients from the rotational matrices in Eq. . The helicity angles and the CMF angle between particles $j$ and $k$, $\theta_{kj}^{\star}$, are functions of the invariants (Appendix \[sec:app\_A\]).
The second term contains two integrals over the Dalitz region of the three-particles in the intermediate state, where the physical region is bounded by $\sigma_n^{(\mathrm{th})} \le \sigma_n'' \le (\sqrt{s} - m_n)^2$ and $\sigma_r^{(-)} \le \sigma_r'' \le \sigma_r^{(+)}$, where $\sigma_r^{\,(\pm)}$ is a function of $\sigma_n''$ and gives the physical boundary $\cos\chi_{n}'' = \pm 1$, for $n = 1$ and $r=3$, $$\begin{split}
\sigma_3^{(\pm)}(\sigma_1'') & = m_1^2 + m_2^2 - \frac{1}{2\sigma_1''} (\sigma_1'' - s + m_1^2)(\sigma_1'' + m_2^2 - m_3^2) \\
& \pm \frac{1}{2\sigma_1''} \lambda^{1/2}(s,\sigma_1'',m_1^2)\lambda^{1/2}(\sigma_1'',m_2^2,m_3^2).
\end{split}$$ Eq. is illustrated in Fig. \[fig:PWIS\_unitarity\]. Appendix \[sec:app\_C\] contains a sketch of the derivation of the amputated PWIS unitarity relations. The first term of Eq. involves the direct propagation of an isobar in the intermediate state, whereas the second, third, and fourth term involve the exchange of a particle between isobars. The rescattering between isobars modifies the line shape of the isobar amplitudes [@Niecknig:2012sj; @Danilkin:2014cra]. The final term is the contribution from the OPE process, which gives and additional imaginary part to the amplitude in the physical region.
At this stage we have not factored out the threshold factors from partial waves. This is straightforward to implement, however we do not do it here as we consider angular momenta in $S$-wave in further sections.
The B-Matrix Parameterization {#sec:B-Matrix}
=============================
Motivated by $S$-matrix theory, we present a parameterization for the PWIS amplitudes that satisfies real axis unitarity given by Eq. . In the $\2\to\2$ case, the $K$-matrix, $f^{-1} = K^{-1} - i\rho_2$, is an example of a parameterization satisfying unitarity. For the $\3\to\3$ case, we present the $B$-matrix parameterization for the PWIS amplitudes. The $B$-matrix parameterization is a linear integral equation for the amputated PWIS amplitudes that satisfy the unitarity relations Eq. : $$\label{eq:BMatrixParam}
\begin{split}
& \wt{\A}_{\ell_k' s_k' ; \ell_j s_j}^{J}(\sigma_k',s,\sigma_j) = \wt{\B}_{\ell_k' s_k' ; \ell_j s_j}^{J}(\sigma_k',s,\sigma_j) \\
& + \sum_{n} \sum_{\ell_n'',s_n''} \int_{\sigma_n^{(\mathrm{th})}}^{(\sqrt{s} - m_n)^2} d\sigma_n'' \, \wt{\B}_{\ell_k' s_k' ; \ell_n'' s_n''}^{J}(\sigma_k',s,\sigma_n'') \\
& \qquad\qquad \times \tau_n(s,\sigma_n'') \wt{\A}_{\ell_n'' s_n'' ; \ell_j s_j}^{J}(\sigma_n'',s,\sigma_j),
\end{split}$$ where the $B$-matrix $\wt{\B}_{\ell_k' s_k' ; \ell_j s_j}^{J}$ contains two terms, $$\label{eq:Bmatrix}
\wt{\B}_{\ell_k' s_k' ; \ell_j s_j}^{J} = \wt{\R}_{\ell_k' s_k' ; \ell_j s_j}^{J} + \wt{\E}_{\ell_k' s_k' ; \ell_j s_j}^{J}.$$ The function $\wt{\E}_{\ell_k' s_k' ; \ell_j s_j}^{J}$ is the amputated partial wave OPE amplitude, $\wt{\R}_{\ell_k' s_k' ; \ell_j s_j}^{J}$ is a real function that represents the short-distance three-body interactions unconstrained by unitarity, and $\tau_n$ is the product of the isobar-spectator phase space between and of the isobar amplitude $$\label{eq:tau}
\tau_{n}(s,\sigma_n) = \rho_{3}(s,\sigma_n) f_{s_n}(\sigma_n),$$ with $$\label{eq:3bodyPhaseSpace}
\rho_{3}(s,\sigma_n) = \frac{1}{64\pi^3 } \frac{2\lvert \p_n^{\star} \rvert }{\sqrt{s}}.$$ The parameterization is diagrammatically represented in Fig. \[fig:B-Matrix\_param\]. The OPE amplitude is defined as $$\label{eq:OPE}
\begin{split}
\E_{kj}(\p';\p) & = \F_k(\p';\p) \frac{1}{ \mu_{jk}^2 - u_{jk} -i\epsilon } \F_j(\p';\p),
\end{split}$$ where we note that the OPE only contributes to off-diagonal amplitudes, $j\ne k$. In principle, the OPE could contain a regular function of the energy in addition to the pole term, however unitarity only constrains the pole, and we assume all other real functions to be absorbed by $\R$. The amputated partial wave projected OPE amplitude $\wt{\E}_{\ell_k' s_k' ; \ell_j s_j}^{J}$ can be constructed using Eqs. and . The $\R$ represents the freedom of short-distance physics for the scattering of three particles, and can be any real function. In an effective field theory approach, it represents a low order polynomial of contact interactions. For simplicity, in the following we assume the latter for $\R$. Appendix \[sec:app\_E\] illustrates how the $B$-matrix parameterizations satisfies the amputated PWIS unitarity relations. Aspects of its analytical properties are examined in Sec. \[sec:AnalyticProperties\].
The $B$-matrix parameterization in Eq. differs from Mai et al. [@Mai:2017vot] in the lower limit of the integral: the latter is derived using Lippman-Schwinger equations with a relativistic potential model, and includes contributions from the unphysical subthreshold region, $\sigma_n < \sigma_n^{(\text{th})}$. Obviously, both parameterization have the same imaginary part in the physical region, since both satisfy unitarity.
![(a) Diagrammatic representation of the $B$-matrix parameterization in Eq. . The gray disk represent the amputated PWIS, and the gray box the $B$-matrix. (b) The $B$-matrix is composed of a short-range real $\R$ amplitude, and the OPE $\E$, see Eq. . []{data-label="fig:B-Matrix_param"}](figures/B-Matrix_Param3 "fig:"){width="0.95\columnwidth"} (-120,55)[[(a)]{}]{} (-120,-5)[[(b)]{}]{}
For notational simplicity, let $\wt{\A}_{kj}(s) \equiv \wt{\A}_{\ell_k' , s_k' ; \ell_j, s_j }^{J }(\sigma_k',s,\sigma_j)$, so that the amplitudes are matrices in the isobar sub-energies and angular momenta, which are indicated by the spectator indices. Equation is then a matrix relation with the integrations over intermediate isobars formally represented as matrix multiplications. Recalling that we work with the convention that isobars exists only in the (12) and (32) channels, we write the $B$-matrix parameterization as the set of coupled equations
\[eq:Bmat\_coupled\_12\] $$\begin{aligned}
\wt{\A}_{13}(s) & = \wt{\B}_{13}(s) + \wt{\B}_{13}(s)\tau_{3}(s) \wt{\A}_{33}(s) ,\label{eq:Bmat_coupled_1} \\
\wt{\A}_{33}(s) & = \wt{\B}_{31}(s)\tau_{1}(s)\wt{\A}_{13}(s) \label{eq:Bmat_coupled_2} ,\end{aligned}$$
with the other two amplitudes given by a similar set of equations,
\[eq:Bmat\_coupled\_34\] $$\begin{aligned}
\wt{\A}_{31}(s) & = \wt{\B}_{31}(s) + \wt{\B}_{31}(s)\tau_{1}(s) \wt{\A}_{11}(s) ,\label{eq:Bmat_coupled_3} \\
\wt{\A}_{11}(s) & = \wt{\B}_{13}(s)\tau_{3}(s)\wt{\A}_{31}(s) \label{eq:Bmat_coupled_4} .\end{aligned}$$
The Eqs. can be combined into one integral equation for $\wt{\A}_{13}$, $$\label{eq:BMat_integral1}
\wt{\A}_{13}(s) = \wt{\B}_{13}(s) + \Kc_{11}(s) \tau_1(s) \wt{\A}_{13}(s),$$ where the kernel $\Kc_{11}$ is $$\label{eq:kernel}
\Kc_{11}(s) = \wt{\B}_{13}(s) \tau_{3}(s) \wt{\B}_{31}(s).$$ Similarly, Eqs. give $$\label{eq:BMat_integral2}
\wt{\A}_{31}(s) = \wt{\B}_{31}(s) + \Kc_{33}(s) \tau_3(s) \wt{\A}_{31}(s),$$ where the kernel $\Kc_{33}$ is given by exchanging the $1 \leftrightarrow 3$ indices in Eq. . Eqs. and can be formally inverted to yield the solutions,
\[eq:Bmat1234\] $$\begin{aligned}
\wt{\A}_{13}(s) & = \left[ \mathbbm{1} - \Kc_{11}(s) \tau_1(s) \right]^{-1} \wt{\B}_{13}(s), \label{eq:Bmat1} \\
\wt{\A}_{33}(s) & = \left[ \mathbbm{1} - \Kc_{33}(s) \tau_{3}(s) \right]^{-1} \Kc_{33}(s), \label{eq:Bmat2} \\
\wt{\A}_{31}(s) & = \left[ \mathbbm{1} - \Kc_{33}(s) \tau_3(s) \right]^{-1} \wt{\B}_{31}(s), \label{eq:Bmat3} \\
\wt{\A}_{11}(s) & = \left[ \mathbbm{1} - \Kc_{11}(s) \tau_{1}(s) \right]^{-1} \Kc_{11}(s), \label{eq:Bmat4}\end{aligned}$$
Several terms can be identified in the kernels, $\Kc_{kj}(s) = \mathcal{G}_{kj}(s) + \mathcal{H}_{kj}(s) + \mathcal{T}_{kj}^{(1)}(s) + \mathcal{T}_{kj}^{(2)}(s)$, where $\mathcal{G}$ is a bubble diagram, $\mathcal{H}$ is a box diagram, and the $\mathcal{T}$’s are triangle diagrams, generated by integrals over OPE and contact terms in Eq. . Explicitly,
\[eq:kernels\_all\] $$\begin{aligned}
\mathcal{G}_{kj}(s) & = \sum_{n}\wt{\R}_{kn}(s)\tau_{n}(s) \wt{\R}_{nj}(s), \label{eq:kernel_bubble} \\
\mathcal{T}_{kj}^{(1)}(s) & = \sum_{n}\wt{\E}_{kn}(s) \tau_{n}(s) \wt{\R}_{nj}(s), \label{eq:kernel_tri1}\\
\mathcal{T}_{kj}^{(2)}(s) & = \sum_{n}\wt{\R}_{kn}(s) \tau_{n}(s) \wt{\E}_{nj}(s), \label{eq:kernel_tri2} \\
\mathcal{H}_{kj}(s) & = \sum_{n}\wt{\E}_{kn}(s) \tau_n(s) \wt{\E}_{nj}(s). \label{eq:kernel_box}\end{aligned}$$
These diagrams occur in the denominators of the amplitudes in Eqs. , Fig. \[fig:denom\]. They differ to the Feynman diagrams obtained in a perturbative QFT since the integrations are only over the physical region, changing the analytic structure below threshold (see Sec. \[sec:AnalyticProperties\]).
(-330,0)[[(a)]{}]{} (-240,0)[[(b)]{}]{} (-150,0)[[(c)]{}]{} (-80,0)[[(d)]{}]{}
The solutions can be interpreted as an infinite series of exchange and bubble diagrams. For example, expanding the solution for $\wt{\A}_{13}$, $$\begin{split}
\wt{\A}_{13}(s) & = \wt{\B}_{13}(s) + \Kc_{11}(s)\tau_1(s) \wt{\B}_{13}(s) \\
& + \Kc_{11}(s)\tau_1(s)\Kc_{11}(s)\tau_1(s)\wt{\B}_{13}(s) + \cdots.
\end{split}$$ The first term is the OPE and contact interaction, the second term is a ladder diagram with three exchanges, and various combinations of bubbles and OPE, and so on. The unitarization of bubble diagrams has been considered in quasi-two-body models [@Basdevant:1978tx; @Bhandari:1982ck; @Jackura:2016llm; @Mikhasenko:2017jtg]. In these models it is easy to show how additional cuts appear in the unphysical sheets due to the isobar decay.
Three-body resonances manifest as poles in the complex $s$-plane of the scattering amplitude. Rearranging the constituents of the kernel relates the two denominators $$\label{eq:denom_shift}
\begin{split}
\wt{\B}_{13}(s)\tau_{3}(s) & \left[\mathbbm{1} - \Kc_{33}(s)\tau_3(s)\right]^{-1} \\
& \,\, = \left[\mathbbm{1} - \Kc_{11}(s)\tau_1(s)\right]^{-1} \wt{\B}_{13}(s) \tau_3(s) .
\end{split}$$ Thus, we can write the full $\3\to\3$ amplitude in terms of a single Fredholm determinant. The determinants are independent of the external isobar energies, and the intermediate integrations will modify the phase space factors to incorporate rescattering effects. Resonance poles can be determined by solving $$\label{eq:det}
\det{\left[ \mathbbm{1} - \Kc_{11}(s) \tau_{1}(s) \right]} = 0.$$ The $B$-matrix solutions are real-boundary values of analytic functions in the complex $s$-plane. The physical amplitudes are defined by $s \to s + i\epsilon$, $\sigma_j \to \sigma_j + i\epsilon$, and $\sigma_k' \to \sigma_k' + i\epsilon$. Aspects of its analytic properties are discussed in the following section.
Relation to the finite volume formalism {#relation-to-the-finite-volume-formalism .unnumbered}
----------------------------------------
In finite volume studies for lattice QCD, substantial progress has been made to understand the connection between discrete energy levels and properties of hadron scattering amplitudes [@Hansen:2014eka; @Hansen:2015zga; @Hansen:2016ync; @Briceno:2017tce; @Briceno:2018mlh; @Polejaeva:2012ut; @Mai:2017bge; @Hammer:2017uqm; @Hammer:2017kms; @Doring:2018xxx; @Mai:2018djl]. In the case of $\2\to\2$ scattering the two-particle finite volume spectrum constrains the values of the infinite volume partial wave amplitudes via the Lüscher quantization condition [@Luscher:1986pf]. The multi-variable nature of $\3\to\3$ scattering amplitudes makes the derivation of the finite volume quantization condition much more complicated and different groups have approached the problem from a different angle. For example, in Refs. [@Hansen:2014eka; @Hansen:2015zga; @Hansen:2016ync; @Briceno:2017tce; @Briceno:2018mlh] the authors introduce amplitudes labeled by subchannel spins and the spectator 3-momenta. Furthermore, ladder diagrams generated by OPE are considered independently from other interactions. This implies that partial wave projection to total spin, which is necessary if one is interested in extracting properties of three-body resonances, would be performed after resummation of the OPE ladder. On the other hand, in Refs. [@Mai:2017bge; @Mai:2018djl], the quantization conditions are derived starting from a set of amplitudes projected onto the total and subchannel spins from the start, before OPE resummation, in a spirit close to our work. In Refs. [@Polejaeva:2012ut; @Hammer:2017uqm; @Hammer:2017kms] the quantization conditions are derived in a nonrelativistic EFT framework, and the direct comparison with our $S$-matrix approach is more complicated. It is more interesting to discuss the differences with Refs. [@Hansen:2014eka; @Hansen:2015zga; @Hansen:2016ync; @Briceno:2017tce]. Since we do not aim to address the subtleties of the finite volume here, we compare with the infinite volume equations derived in there on the basis of the finite volume formalism. For simplicity we ignore coupling to the $\2$-body channel. In Refs. [@Hansen:2014eka; @Hansen:2015zga; @Hansen:2016ync; @Briceno:2017tce], the $\3 \to \3$ connected amplitude is denoted by $\mathcal{M}_{33}(\vec k,\vec k')$ (see Eq. (112) in Ref. [@Briceno:2017tce]). It contains the resummed OPE ladder and the amputated amplitude $\mathcal{T}_{33}(\vec k,\vec k')$ that is generated by the kernel $\mathcal{K}_{\text{df},33}(\vec k,\vec k')$, which is analogous to our driving term $\wt{\R}_{\ell_k' s_k' ; \ell_j s_j}^{J}(\sigma_k',s,\sigma_j)$. Both the OPE ladder and the amplitude $\mathcal{T}_{33}(\vec k,\vec k')$ are solutions of linear integral equations (see Eqs. (87) and (106) in Ref. [@Briceno:2017tce]), which are analogous to our Eq. .
To further illustrate the connection between our amplitudes and those of Ref. [@Hansen:2014eka; @Briceno:2017tce] we shall consider the case of three identical particles in $S$-waves. The phase space $\rho_2$ (Eq. ) of the two particle subsystem gains a factor $1/2!$ to account for their identical nature. The resulting unitarity relations have a similar form as in Eq. . In matrix notation, one finds $$\label{imd}
\begin{split}
\im{\wt{\A}} & = \im{\wt{\E}} + \im{\wt{\E}} \rho_3 f \wt{\A} + \wt{\A}^{*} f^{*} \rho_3 \im{\wt{\E}} \\
& + \wt{\A}^* \rho_3 \im f \wt{\A} + \wt{\A}^* f^* \rho_3 \im\wt{\E} \rho_3 f \wt{\A},
\end{split}$$ where the matrices are in the $\sigma'$, $\sigma$ space with $f$ and $\rho_3$ diagonal matrices. The $S$-wave projection of the OPE is given by the symmetric matrix $\wt{\E}$, and is found by the inverse relation of Eq. on the OPE amplitude in Eq. . It is straightforward to show that the $B$-matrix parameterization (Eq. ), $\wt{\A} = \wt{\B} + \wt{\B} \rho_3 f \wt{\A}$, $$\wh{\A} = [1 - (\wt{\E} + \wt{\R}) \rho_3 f ]^{-1} (\wt{\E} + \wt{\R}) \label{alat}$$ satisfies the unitarity relation, Eq. . Moreover, after simple manipulations, Eq. can be rewritten as $$\A = f \wt{\A} f
= \mathcal{D} +
\mathcal{L} [1 -
\wt{R} \rho_3 \mathcal{L} ]^{-1} \wt{R}
\mathcal{L}^{\top}$$ where the $\mathcal{D}$ amplitude is the ladder sum of OPE, given by $$\mathcal{D} = f \wt{\E} f + f \wt{\E}\rho_3 \mathcal{D}$$ and $\mathcal{L} \equiv f + \mathcal{D} \rho_3$. Finally we introduce the amplitude $\mathcal{T}$ satisfying $$\mathcal{T} = \wt{\R} +
\wt{\R} \rho_3 \mathcal{L}
\mathcal{T},$$ and obtain an expression closely resembling that in Refs. [@Hansen:2014eka; @Briceno:2017tce], $$\A = \mathcal{D} + \mathcal{L} \mathcal{T}\mathcal{L}^{\top}.\label{af}$$ The difference between Eq. and the corresponding expression for ${\cal M}_{33}^{(u,u)}$ in Refs. [@Hansen:2014eka; @Briceno:2017tce] is in the definition of $\mathcal{L}$. In our notation, the $\mathcal{L}$ of Refs. [@Hansen:2014eka; @Briceno:2017tce] contains an additional $1/3$ constant, and the $f$ and $\mathcal{D}$ matrices contain an extra factor of $\rho_2$.
Although these analogies should be verified with care, two main differences appear. One is in the treatment of the OPE dynamics, which in Ref. [@Briceno:2017tce] is resummed before projection onto the total spin and in our case the projection is done first. It is likely that these approaches will ultimately prove to be equivalent, since in practical applications only a finite number of partial waves in total spin or spectator momentum components can be kept. The other difference is in the $\mathcal{L}$ function, which could possibly be related to a discrepancy in the definitions of $\wt{\R}$ and $\mathcal{K}_{\textrm{df},33}$. It would be interesting to see if our approach and the corresponding equation of Refs. [@Hansen:2014eka; @Briceno:2017tce] provide the same result, and to determine the origin of the difference.
Aspects of Analytic Properties {#sec:AnalyticProperties}
==============================
In this section, we examine the singularities of the OPE amplitude and the triangle diagram from the $B$-matrix parameterization. We numerically evaluate an amplitude where all external particles have unit mass ($m_1 = m_2 = m_3 = 1$) and coupling. In these studies, the units are arbitrary. For simplicity, we consider waves only, $J(\ell' s')_{k} (\ell s)_{j} = 0(00)_{k}(00)_{j}$. Generalizing to nonzero angular momenta does not change the analytic properties.
One Particle Exchange {#sec:ope}
---------------------
As seen in Eqs. , the building block for the $B$-matrix kernels is the OPE amplitude. Projecting Eq. using Eq. gives the $S$-wave OPE amplitude, $$\label{eq:OPE_Swave}
\begin{split}
\wt{\E}^S_{kj}(\sigma_k',s,\sigma_j) & = \frac{1}{4 \lvert \p_k'^{\star}\rvert \lvert \p_j^{\star}\rvert} \log{ \left( \frac{z_{kj} - 1 }{z_{kj} + 1} \right) }.
\end{split}$$ where $z_{kj}$ is given as $$\label{eq:z_mu}
\begin{split}
z_{kj} = \frac{2s(\sigma_{j} + m_{k}^2 - \mu_{jk}^2) - (s + \sigma_{j} - m_j^2) (s + m_k^2 - \sigma_{k}' ) }{ \lambda^{1/2}(s,\sigma_{j},m_j^2)\lambda^{1/2}(s,\sigma_{k}',m_k^2) },
\end{split}$$ where $\lambda(a,b,c) = a^2 + b^2 + c^2 - 2(ab+bc+ca)$ is the Källén triangle function. Eq. . We investigate the OPE as a function of $s$ for fixed real $\sigma_j$ and $\sigma_k'$. The imaginary part of the OPE is $$\label{eq:OPE_Swave_imag}
\im {\wt{\E}^S_{kj}(\sigma_k',s,\sigma_j) } = \frac{\pi }{4\lvert \p_j^{\star} \rvert \lvert \p_k'^{\star} \rvert} \Theta(1 - \lvert z_{kj} \rvert^2),$$ which is given by the unitarity relations in Eq. . The OPE has four branch points in $s$, one at zero, one at infinity, and two which we label $s_{kj}^{(\pm)}$, $$\label{eq:spm}
\begin{split}
s_{kj}^{(\pm)} & = \frac{1}{2\mu_{jk}^2} \bigg[ (m_k^2 - \sigma_j)(m_j^2 - \sigma_k') - \mu_{jk}^4 \\
& + \mu_{jk}^2(m_k^2 + m_j^2 + \sigma_j + \sigma_k') \\
& \pm \lambda^{1/2}(\mu_{jk}^2,m_k^2,\sigma_j) \lambda^{1/2}(\mu_{jk}^2,m_j^2,\sigma_k') \bigg],
\end{split}$$ which depend on the isobar invariant masses. The momenta in the denominator do not contribute additional branch points, because the logarithm vanishes and cancel the singularity, as expected from the -wave threshold behavior. The $s_{kj}^{(\pm)}$ branch points are in general complex. There are then two branch cuts: one where $s\in (-\infty,0]$, called the virtual particle exchange (VPE) cut, and one connecting $s_{kj}^{(-)}$ to $s_{kj}^{(+)}$, called the real particle exchange (RPE) cut. The VPE cut is associated with the exchange of a virtual particle, generating long-range forces. Historically, the RPE cut is associated with the exchange of a real particle between isobars, when it is kinematically allowed for an isobar to decay. This corresponds to when the RPE branch points lie on the real axis above the isobar-spectator threshold. If the isobar invariant masses are below the decay threshold, then the RPE branch points move in the complex plane below the isobar-spectator threshold. For convenience, however, we will always call this the RPE cut, and emphasize that a real particle exchange occurs only if it is kinematically accessible. Note that although the value of the isobar mass dictates the physics of the OPE, the OPE is blind to the decay products of the isobar and the physical threshold in $s$ is $\max\{(\sqrt{\sigma_j}+m_j)^2,(\sqrt{\sigma_k'}+m_k)^2\}$.
We can understand the analytic structure of the OPE by writing a dispersive representation in $s$. Eq. is nonzero in two regions, leading to the relation $$\label{eq:OPE_disp}
\begin{split}
\wt{\E}^S_{kj}(\sigma_k',s,\sigma_j) & = \int_{\Gamma_V} ds' \frac{1}{s' - s - i\epsilon} \frac{1}{4 \lvert \p_k'^{\star} \rvert \lvert \p_j^{\star} \rvert} \\
& + \int_{\Gamma_R} ds' \frac{1}{s' - s- i\epsilon} \frac{1}{4\lvert \p_k'^{\star} \rvert \lvert \p_j^{\star} \rvert},
\end{split}$$ where $\Gamma_V$ is the contour over the VPE cut and $\Gamma_R$ is the contour over the RPE cut. The integrand has four branch points associated with the thresholds and pseudo-thresholds of the initial and final momenta. We choose to orient the branch cuts such that the lowest branch point ($\min\{(\sqrt{\sigma_k'}-m_k)^2, (\sqrt{\sigma_j}-m_j)^2\}$) has a cut running to $-\infty$, the highest branch point ($\max\{(\sqrt{\sigma_k'}+m_k)^2, (\sqrt{\sigma_j}+m_j)^2\}$) has a cut running to $+\infty$, and the other two branch points have a branch cut joining them. The contour $\Gamma_V$ is always taken above the real axis, whereas the contour $\Gamma_R$ depends on the external masses. The physical amplitude is defined as the boundary value when $s \to s + i\epsilon$ , below the RPE cut.
(-470,150)[[(a)]{}]{} (-220,150)[[(b)]{}]{} (-470,30)[[(c)]{}]{} (-220,30)[[(d)]{}]{} \[fig:OPE\_cuts\]
{width="1.\textwidth"}
For fixed $\sigma_j > \sigma_j^{(\mathrm{th})}$, the RPE cut can be categorized by different regions in $\sigma_k'$. In Fig. \[fig:OPE\_integrand\_cuts\], we illustrate how the analytic structure of the integrand and the integration contours change in these regions. Assuming a small imaginary part $\mu_{jk}^2 \to \mu_{jk}^2 - i\epsilon$, the RPE branch points have a finite imaginary part for $\sigma_k' < \sigma_k^{(\mathrm{th})}$, with opposite signs. When $\sigma_k' > \sigma_k^{(\mathrm{th})}$, the branch points are infinitesimally close to the real axis. In the physical region, $s_{kj}^{(-)}$ has inflection points at two locations of $\sigma_k'$:
\[eq:inflect\] $$\begin{aligned}
\sigma_k^{(a)} & = m_j^2 + \mu_{jk}^2 + \frac{m_j (\sigma_j + \mu_{jk}^2 - m_k^2)}{\sqrt{ \sigma_j }}, \label{eq:inflectA}\\
\
\sigma_k^{(b)} & = -\frac{1}{2 m_k^2} \bigg[ 2 m_k^2 ( \sigma_j - m_j^2 ) - m_k^4 \label{eq:inflectB}\\
& - (\sigma_j - \mu_{jk}^2)^2 + (m_k^2 + \mu_{jk}^2 - \sigma_j) \nn \\
& \times \sqrt{4 m_j^2 m_k^2 + \lambda(\sigma_j,\mu_{jk}^2,m_k^2) } \bigg], \nn\end{aligned}$$
which follow from $ds_{kj}^{(-)}/d\sigma_k' = 0$ corresponding to $\im s_{kj}^{(-)} = 0$. Both $\sigma_k^{(a)}$ and $\sigma_k^{(b)}$ correspond to when $s_{kj}^{(-)}$ crosses the real $s$-axis. The $s_{kj}^{(+)}$ branch point always lies in the upper-half plane.
We can therefore classify the regions according to when the RPE branch points are both in the upper-half plane or when they approach the real axis.
(a) $\sigma_k' \ge \sigma_k^{(b)}$, see region (a) in Fig. \[fig:spm\]. Here $s_{kj}^{(-)}$ is below the real axis, and $s_{kj}^{(+)}$ is above the real axis. The RPE cut connects these two points by crossing the real axis below the threshold $(\sqrt{\sigma_k'}+m_k)^2$. Real particle exchange in this case has consequences when considering the OPE processes embedded in the triangle diagram, which is discussed in the next section. For $k=1$ and $j = 3$, Fig. \[fig:OPE\_integrand\_cuts\](a) shows the RPE and VPE contours, $\Gamma_R$ and $\Gamma_V$, respectively. Note that in the logarithmic representation, the RPE cut is circular, whereas in the dispersive representation, one can define the cut in any chosen manner as long as singularities are not crossed.
(b) $ \sigma_k^{(a)} \le \sigma_k' \le \sigma_k^{(b)}$, see region (b) in Fig. \[fig:spm\]. When $\sigma'_k$ decreases below the inversion point $\sigma^{(b)}_k$, $s_{kj}^{(-)}$ wanders above the real axis. The RPE cut directly connects $s_{kj}^{(-)}$ to $s_{kj}^{(+)}$ without crossing the real axis. This is the typical case when considering the exchange of a real particle, illustrated in Fig. \[fig:OPE\_integrand\_cuts\](b). Note that when $\sigma_k' = \sigma_j$ for equal masses $m_j = m_k$, then the integrand branch points merge into pole singularities.
(c) $\sigma_k^{(\mathrm{th})} \le \sigma_k' \le \sigma_k^{(a)}$, see region (c) in Fig. \[fig:spm\]. The RPE branch points again wrap around the real axis, Fig. \[fig:OPE\_integrand\_cuts\](c), with the cut crossing the real axis below the threshold $(\sqrt{\sigma_j}+m_j)^2$.
(d) $\sigma_k' \le \sigma_k^{(\mathrm{th})}$, see region (d) in Fig. \[fig:spm\]. The branch points $s_{kj}^{(\pm)}$ move deep into the complex plane, as shown in Fig. \[fig:OPE\_integrand\_cuts\](d). In this region, the isobar cannot decay, and therefore it is unphysical for the $\3\to\3$ elastic scattering. Real particle exchange cannot occur, leaving only the virtual contributions.
If we evaluate the OPE along the real $s$-axis in regions (a) or (c), we find that the real part of the OPE has a jump due to crossing the RPE cut. In the logarithmic representation, this crossing occurs when $z_{kj} = 0$, that is, when $$\begin{split}
s_{kj}^{(0)} & = \frac{1}{2} \bigg[ m_j^2 + m_k^2 - 2 \mu_{jk}^2 + \sigma_j + \sigma_k' \\
& + \bigg(
4 (m_j^2 - \sigma_j) (m_k^2 - \sigma_k') \\
&+ (m_j^2 + m_k^2 -
2 \mu_{jk}^2 + \sigma_j + \sigma_k')^2 \bigg)^{1/2} \bigg].
\end{split}$$ When choosing a different contour for $\Gamma_R$ in the dispersive representation, the location of this crossing depends on where real axis crosses the chosen contour.
(-35,30)[[(a)]{}]{} (-35,30)[[(b)]{}]{} (-35,30)[[(c)]{}]{}
(-35,30)[[(d)]{}]{} (-35,10)[[(e)]{}]{} (-35,10)[[(f)]{}]{}
These cases are illustrated in Fig. \[fig:OPE\_integrand\_cuts\] for spectators $k=1$ and $j=3$. We plot the OPE amplitudes, Eq. , as a function of $s$ for fixed $\sigma_3$ and $\sigma_1'$ in Fig. \[fig:OPE\_figs\]. Figure \[fig:OPE1\] shows the OPE computed at $\sigma_1'$ in region (a). At this energy, the $s_{kj}^{(-)}$ is below the real axis, and the RPE cut wraps around the real axis, passing below the threshold $(\sqrt{\sigma_1'} + m_1)^2$. The jump in the real part at $s_{13}^{(0)}$ is due to crossing the RPE cut. Figure \[fig:OPE2\], is evaluated at $\sigma_1'$ in region (b), where both branch points are above the real axis. Here, we illustrate that as $\sigma_1'$ decreases, the width of the imaginary part decreases and the peak increases. The narrowing imaginary region physically represents that less phase space is available for real particle propagation in the intermediate state. Figure \[fig:OPE3\] is computed for $\sigma_1'$ in region (c), right above the two-particle threshold. There is a jump in the real part at $s_{13}^{(0)}$ from crossing the RPE cut. The final case is illustrated in Fig. \[fig:OPE4\], where the OPE computed in the unphysical region (d). There is an imaginary part due to the VPE cut only, as it is kinematically inaccessible for the exchange of a real particle. The jump in the real part at $s_{13}^{(0)}$ comes from crossing the RPE cut. Figures \[fig:OPE5\] and \[fig:OPE6\] shows a 3-dimensional plot of the real and imaginary part of the logarithmic representation of the OPE, Eq. . The circular cut is clearly visible connecting the RPE branch points. The physical region is taken as the region approaching the real axis, below the RPE cut.
To summarize, the analytic structure of the OPE is given by two branch cuts, the VPE and RPE cuts. The VPE cut is present for $-\infty < s \le 0$, and is associated with the exchange of an off-shell particle. For physical isobars, the RPE cut is in the physical region. We have shown different scenarios, identified by the isobar masses, in which the RPE branch points can approach the physical region, which impact the structure of the $B$-matrix kernels.
Triangle Diagrams {#sec:triangle}
-----------------
To understand resonance poles of $\3\to\3$ systems, the analytic structure of the $B$-matrix parameterization, Eq. must be understood in the complex $s$-plane. This means understanding the properties of the $B$-matrix kernels. Here, we investigate the triangle diagram, and leave the box diagram for future studies. Let us work with the triangle $\mathcal{T}_B \equiv \mathcal{T}_{11}^{(2)}$ introduced in Eq. , where all angular momenta are in $S$-wave. For convenience, let $\wt{\R} = 1$, thus the amplitudes are independent of $\sigma_1'$, and given by $$\begin{split}
\mathcal{T}_B(s) = \int_{\sigma_3^{(\mathrm{th})}}^{(\sqrt{s} - m_3)^2} d\sigma_3'' \, \tau_{3}(s,\sigma_3'') \wt{\E}^S_{31}(\sigma_3'',s,\sigma_1) ,
\end{split}$$ where $\tau_3(s,\sigma_3'') = \rho_3(s,\sigma_3'')D_{3}^{-1}(\sigma_3'')$ , and the dependence of $\mathcal{T}_B$ on $\sigma_1$ has been understood. To ensure the correct analytic properties of the isobar amplitude, we introduce its dispersive representation $$\begin{split}
f_3(\sigma_3'') = \frac{1}{\pi}\int_{\sigma_3^{(\mathrm{th})}}^{\infty} d\wh{\sigma} \, \frac{\im{f_3(\wh{\sigma})}}{\wh{\sigma} - \sigma_3'' - i\epsilon},
\end{split}$$ giving the form for $\mathcal{T}_B$ $$\label{eq:TB_1}
\begin{split}
\mathcal{T}_B(s) & = \frac{1}{\pi} \int_{\sigma_{3}^{(\mathrm{th})}}^{\infty} d\wh{\sigma}\, \im{f_3(\wh{\sigma})}\, \\
& \times \int_{\sigma_3^{(\mathrm{th})}}^{(\sqrt{s} - m_3)^2} d\sigma_3'' \frac{\rho_{3}(s,\sigma_3'') \wt{\E}_{31}^{S}(\sigma_3'',s,\sigma_1) }{\wh{\sigma} - \sigma_3'' - i\epsilon}.
\end{split}$$ We see the $\sigma_3''$-integral does not depend on $f_3(\wh{\sigma})$, so for simplicity we take the narrow width limit $\im{f_3(\wh{\sigma})} = \pi \delta(\wh{\sigma} - M^2)$ , where $M$ is the mass of the isobar. The narrow width limit shifts the unitarity cut in the triangle diagram to begin at the threshold . However, for a general isobar shape, Eq. can be used to sum over its distribution, recovering the correct unitarity branch cut starting at $s_{\textrm{th}}$. Therefore, the triangle diagram has the form $$\label{eq:TB}
\begin{split}
\mathcal{T}_B(s) = \int_{\sigma_3^{(\mathrm{th})}}^{(\sqrt{s} - m_3)^2} d\sigma_3'' \frac{\rho_{3}(s,\sigma_3'') \wt{\E}_{31}^{S}(\sigma_3'',s,\sigma_1) }{M^2 - \sigma_3'' - i\epsilon}.
\end{split}$$ Figure \[fig:tri\_label\] shows the triangle diagram in consideration. The $B$-matrix triangle contains singularities on the physical sheet. These are due to the $s$-singularities in $\rho_3$ and $\wt{\E}_{13}^S$, and to endpoint singularities when the integration limits hit the $\sigma_3''$-singularities of the integrand. The upper integration limit gives a branch cut for $s<0$. Since $\rho_3(s,\sigma_3'') \propto \{[\sigma_3'' - (\sqrt{s} + m_3)^2][\sigma_3'' - (\sqrt{s} - m_3)^2]\}^{1/2} / s$, there is a pole at $s = 0$. When the integration variable hits the lower limit $\sigma_3'' = \sigma_3^{(\mathrm{th})}$, there are two branch point singularities at $s = (\sqrt{\sigma_3^{(\mathrm{th})}} \pm m_3)^2 = (m_1 + m_2 \pm m_3)^2$. In the narrow width limit, the unitarity cut opens when the upper limit of the integral hits the pole in the isobar propagator, for $s = (M + m_3)^2$. On the real $s$-axis, the OPE has a discontinuity in the real part when $z_{31} = 0$, at $\sigma_3'' = \sigma_3^{(\mathrm{th})}$. This discontinuity from crossing the RPE cut is present in $\mathcal{T}_B$. Two more singularities occur when $\sigma_3''$ hits the two inflection points $\sigma_3^{(a)}$ and $\sigma_3^{(b)}$, which are defined in Eqs. . The OPE pinches the real axis at $\sigma_3'' = \sigma_3^{(a)}$, and generates a singularity in $\mathcal{T}_B$ at the initial state threshold $s = (\sqrt{\sigma_1} + m_1)^2$. This can be understood by realizing that the OPE branch points, Eq. , can alternatively be written in terms of $\sigma_3''$ as a function of $s$ for fixed $\sigma_1$. The branch points are then $\sigma_3''^{(\pm)}$, where $$\begin{split}
\sigma_3''^{(\pm)} & = \frac{1}{2\sigma_1} \bigg[ \sigma_1(s + m_1^2 + m_3^2 + \mu_{13}^2) \\
& + (m_1^2 - s)(m_3^2 - \mu_{13}^2) - \sigma_1^2 \\
& \pm \lambda^{1/2}(s,\sigma_1,m_1^2)\lambda^{1/2}(\sigma_1,\mu_{13}^2,m_3^2) \bigg],
\end{split}$$ and $\sigma_3^{\prime\prime(-)}$ lies infinitesimally below the real axis in the physical region. Figure \[fig:sigpm\] shows the motion of $\sigma_3''^{(\pm)}$ in the complex $\sigma_3''$-plane as a function of $s$ for fixed $\sigma_1$. At the three particle threshold, $s = (m_1 + m_2 + m_3)^2$, the branch points have finite imaginary part and are on opposite sides in the $\sigma_3''$-plane. As $s$ approaches the initial state threshold $s = (\sqrt{\sigma_1} + m_1)^2 $, the $\sigma_3''^{(\pm)}$ branch points pinch the real axis at $\sigma_3'' = \sigma_3^{(a)}$. Since the $\mathcal{T}_B$ integration is on the real axis starting from $\sigma_3^{(\mathrm{th})}$, the integration path is pinched, causing a singularity in $\mathcal{T}_B$ at $s = (\sqrt{\sigma_1} + m_1)$. At $\sigma_3'' = \sigma_3^{(b)}$, the branch point migrates back below the real axis at a value greater than the threshold $\sigma_3^{(\textrm{th})}$ close to the real axis. When $M^2 > \sigma_3^{(b)}$, this effect generates the triangle singularity [@Peierls:1961zz; @Aitchison:1966; @Eden:1966dnq]. The triangle singularity has been studied as a possible mechanism to explain anomalous structures observed in heavy flavor experiments [@Guo:2015umn; @Guo:2016bkl; @Ketzer:2015tqa; @Szczepaniak:2015eza]. The peak of the triangle singularity coincides with the $s_{31}^{(-)}$ branch point, $s_{\textrm{tr}} = s_{31}^{(-)}$.
![The triangle diagram $\mathcal{T}_B$ contribution to the kernel $\Kc_{11}$. We take the isobar to have a narrow width with mass $M$. For numerical evaluations, $m_1 = m_3 = \mu_{31} = 1$. []{data-label="fig:tri_label"}](figures/triangle_labels_v2){width="0.8\columnwidth"}
![Motion of the $\sigma_3''^{(\pm)}$ in the $\sigma_3''$-plane as a function of $s$ for fixed $\sigma_1$. Shown in red is $\sigma_3''^{(+)}$ and in blue $\sigma_3''^{(-)}$. The points indicate various $s$ values starting from the three particle threshold, $s=(m_1+m_2+m_3)^2 = 9$. The inset shows that the branch points pinch the real $\sigma_1'$ axis at $s = (\sqrt{\sigma_1} + m_1)^2 = 16$, which is responsible for a pinch singularity in $\mathcal{T}_B$. []{data-label="fig:sigpm"}](figures/sigpm){width="1.\columnwidth"}
Aside for the unitarity branch cut starting at $s = (M + m_3)^2$ and the triangle singularity, these additional singularities in the physical $s$-plane are not allowed by analyticity. The extra singularities are moved to the second sheet when we consider the integration over the isobar shape, Eq. , leaving only the unitarity cut starting at $s = (m_1 + m_2 + m_3)^2$ and the triangle singularity.
{width="1.\columnwidth"} {width="1.\columnwidth"}
![The $B$-matrix triangle Eq. in with the triangle singularity. Shown in black are the real (solid) and imaginary (dashed) parts evaluated at $\sigma_1 = 4.41$ and $M^2 = 4.41$. Shown in red are the real (solid) and imaginary (dashed) parts of the OPE piece of the triangle, Eq. , where $\sigma_3'' = M^2$. The blue dashed lines indicate the threshold $(M+m_3)^2 = 13.7641$ and the lower RPE branch point at $s_{31}^{(-)} = s_{\textrm{tr}} = 13.8619$. The normal threshold accounts for the first peak in the triangle diagram, while the second peak is caused by the triangle singularity. Note we scaled the triangle diagram to account for the phase space normalization of the triangle.[]{data-label="fig:triOPE"}](figures/ftriOPE){width="1.\columnwidth"}
(-35,0)[[(a)]{}]{}
(-35,0)[[(b)]{}]{}
(-80,-20)[[(c)]{}]{}
We compare the structure of Eq. with that of a Feynman diagram triangle in a QFT (see Appendix \[sec:app\_F\] for a review of the Feynman triangle), which can be written as $$\label{eq:TF}
\mathcal{T}_{F}(s) = \int_{\Gamma_{T}} ds' \frac{\rho_3(s',M^2)\wt{\E}_{13}^{S}(M^2,s',\sigma_1)}{s' - s - i\epsilon},$$ where $\Gamma_T$ is the path from the threshold $(M + m_3)^2$ to $\infty$, and the $S$-wave amplitudes are normalized according to Eq. . Figure \[fig:triReal\] shows the real and imaginary parts, respectively, of the two triangle diagrams $\mathcal{T}_F$ and $\mathcal{T}_B$, below the region of the triangle singularity. Notice that the Feynman triangle has only a normal threshold singularity at $s = (M + m_3)^2$, and is smooth everywhere else. The imaginary parts of both triangles are identical above threshold, as required by unitarity. The $B$-matrix triangle has noticeable kinks in both the real and imaginary parts below threshold, corresponding to the singularities discussed above. The black dashed lines indicate the location of the singularities. Starting from low energy, the first additional singularity is the $s=0$ singularity from the phase space. The next two singularities occur at $s = (\sqrt{\sigma_3^{(\mathrm{th})}} - m_3)^2 $ and $(\sqrt{\sigma_3^{(\mathrm{th})}} + m_3)^2 $, which are from the phase space evaluated at the lower integration limit. The real part contains a singularity from evaluating the OPE across the RPE cut. Note that the imaginary part does not contain this jump, consistent with the OPE description in the previous section. The next singularity occurs at the initial state threshold $s = (\sqrt{\sigma_1} + m_1)^2 $, which is due to the pinching of the $\sigma_3''$ contours by the OPE branch points. Finally the normal threshold at $s = (M + m_3)^2$. Figure \[fig:triOPE\] shows $\mathcal{T}_B$ and the OPE in the region where the triangle singularity develops. The line shape shows the production threshold at $(M+m_3)^2$, and the peak at $s = s_{\textrm{tr}}$. The OPE branch point $s_{31}^{(-)}$ clearly coincides with the triangle peak.
Figure \[fig:tri\_compare\_figsB\] shows the $\mathcal{T}_B$ and $\mathcal{T}_F$ as a function of $s$ at fixed $\sigma_1$ and varying $M^2$ in the region below and above $M^2 = \sigma_3^{(b)}$. Figure \[fig:TriAllReal\] shows the real parts, and Fig. \[fig:TriAllImag\] shows the imaginary parts. At $M^2 = \sigma_3^{(b)}$, the triangle singularity develops, corresponding to when $s = s_{31}^{(-)} = s_{\textrm{tr}}$ . One can see a second threshold in the line shape above threshold $(M + m_3)^2$. Figure \[fig:Tricontours\] shows the RPE cut of the OPE in the $s$-plane at the corresponding values for the triangle amplitude.
We also compare the $B$-matrix triangle with the analogous one from Mai [@Mai:2017vot], that we denote as $\mathcal{T}_M$, $$\label{eq:TM}
\mathcal{T}_M(s) = \int_{-\infty}^{(\sqrt{s} - m_3)^2} d\sigma_3'' \frac{\rho_{3}(s,\sigma_3'') \wt{\E}_{31}^{S}(\sigma_3'',s,\sigma_1) }{M^2 - \sigma_3'' - i\epsilon},$$ where we take their contact term equal to unity, and the lower integration limit in their model accounts for the physics in the unphysical region. As $\sigma_3'' \to -\infty$, the OPE amplitude goes like $1/\sigma_3''$, while the phase space grows as $\sigma_3''$, thus the integrand goes like $1/\sigma_3''$ and the function is logarithmically divergent. Numerically, we choose to cut off the integral at some large value, $-200$ to investigate the behavior. For $\mathcal{T}_M$, all lower limit endpoint singularities in $s$ from the phase space and OPE are moved toward $-\infty$. The $s=0$ pole from the phase space persist, and the normal threshold singularity at $s = (M+m_3)^2 $ is present since it is from the upper limit. The pinch singularity at $s = (\sqrt{\sigma_1} + m_1)^2 $ is also present, as well as the pinch singularity at $s = (\sqrt{\sigma_1} - m_1)^2$, Fig. \[fig:sigpm\]. The second pinch singularity occurs when the integration over $\sigma_3''$ hits $\sigma_3'' = \sigma_3^{(c)}$, where $$\sigma_3^{(c)} = m_1^2 + \mu_{31}^2 - \frac{m_1 (\sigma_1 + \mu_{31}^2 - m_3^2)}{\sqrt{\sigma_1}},$$ is a third inflection point in the unphysical region, occurring at $s$ is at the threshold $s = (\sqrt{\sigma_1} - m_1)^2$ (when $\im{s_{kj}^{(-)}} = 0$). This pinch singularity is absent in the $B$-matrix triangle, as the integral is only over the physical region. Figure \[fig:triReal\] compares the line shapes of all three triangles, $\mathcal{T}_B$, $\mathcal{T}_F$, and $\mathcal{T}_M$. Although $\mathcal{T}_M$ has a logarithmic divergence, we fix the lower integration limit to $-200$. We see how the line shape below threshold smooths out except at the remaining singularities, shown with the black dashed lines. The red dashed line indicates the second pinch singularity in $\mathcal{T}_M$.
The Feynman triangle can be recovered from $\mathcal{T}_M$ with the method discussed by Aitchison and Pasquier [@Aitchison:1966lpz], where the isobar approximation for $\mathbf{1} \to \3$ decays was studied. Using their inversion technique, it was found that the Feynman triangle can be written as a dispersive integral over the isobar invariant mass as in Ref. [@Mai:2017vot], plus additional terms. The latter are real in the physical region, but cure the below threshold singularities shown in the $B$-matrix. The additional terms also cancel the logarithmic divergence, leaving a finite amplitude.
Removal of Unphysical Singularities
-----------------------------------
As shown, the $B$-matrix parameterization contains additional singularities which do not match the expected analytic behavior of the amplitudes. This happens in both our formulation and Mai [@Mai:2017vot]. One possible venue for improving that is to substitute the $B$-matrix kernels, Eq. , with the Feynman one. This is in the same spirit of the Chew-Mandelstam phase space in the $\2\to\2$ parameterizations, which removes the unphysical singularities of the phase space. Although the kernels will now have the proper analytic structure (no physical sheet singularities except for the unitarity cut), the resulting amplitude will still contain singularities from iterating the kernel. Consider the solution for $\wt{\A}_{33}$ in Eq. , where the kernel is replaced by the Feynman one, $\Kc_{33} \to \Kc_{33}^{F}$, where $$\Kc_{33}^{F}(s) = \int_{{\textrm{th}}}^{\infty} ds' \frac{\wt{\B}_{31}(s') \rho_3(s') \wt{\B}_{13}(s')}{s' - s - i\epsilon}.$$ Now expand the solution Eq. in an infinite series, $$\label{eq:FeynmanSwitch}
\wt{\A}_{33} = \Kc_{33}^{F}(s) + \Kc_{33}^{F}(s)\tau_{3}(s)\Kc_{33}^{F}(s) + \cdots.$$ The first term is the kernel, composed of Feynman diagrams which have the correct analytic properties. Let the kernel consist only of the triangle diagram, then the second term is two Feynman triangles joined with a $\tau$-function. The equivalent Feynman diagram would have two exchanges integrated over the four-momenta, which is not equivalent to what is shown in Eq. due to the $\tau$-function. This diagram, as well as the higher-order ones, contain non-analyticities in a similar manner to what was shown for the triangle diagram. The unintegrated singularities from the phase space are always present. Therefore the simple kernel substitution does not produce the correct analytic behavior in the $B$-matrix solution. However, it can still be advantageous, as it corrects some of the unphysical singularities in the present $B$-matrix solution.
The remaining singularities should disappear if one was to solve the proper Bethe-Salpeter equations of the underlying QFT. The $B$-matrix parameterization is indeed reminiscent of that for $\2\to\2$ scattering. We examine some differences between these formalisms. The $B$-matrix parameterization is a covariant integral equation for the on-shell isobar-spectator amplitudes. It satisfies unitarity relations and does not have additional imposed constraints from analyticity. Thus, for complex energies on the physical Riemann sheet, the $B$-matrix parameterization contains the unitarity cut, and has additional $s$-singularities from the $\tau$ and OPE.
The Bethe-Salpeter equation is a covariant integral equations that incorporate an infinite number of exchanges for any given QFT [@Itzykson:1980rh]. Solving it amounts to summation of exchange diagrams, similarly to the $B$-matrix. The resulting amplitudes are analytic functions in the complex $s$-plane, as the QFT amplitudes inherently obey analyticity constraints. The physical sheet thus has only the allowed singularities, such as the unitarity cut and possible bound state poles. Lippmann-Scwhinger equations are nonrelativistic equations for the scattering amplitude in a given potential model. The $B$-matrix has similarities to the Lippmann-Schwinger equations in that both involve in a three-dimensional integral over the momenta [@Mai:2017vot]. In this work however, we focus on the physical region, and truncate the isobar mass integration appropriately. Conversely, the Bethe-Salpeter equation contains integrations over four-momenta, which results in integrating over the off-shell behavior of the amplitude. Introducing dispersive integrations in the $B$-matrix amounts to the same procedure, and would remove the unphysical singularities.
Conclusions {#sec:Conclusion}
===========
In summary, we have discussed the phenomenological description of $\3\to\3$ elastic scattering of spinless particles. The $\3\to\3$ amplitude was described in the isobar representation. We constructed the unitarity relations for the isobar-spectator amplitudes for general partial wave quantum numbers. For a practical use, the infinite sums are truncated, leading to the standard isobar approximation. We parameterize the isobar-spectator partial wave amplitudes with the $B$-matrix formalism, which automatically satisfies the unitarity. The $B$-matrix parameterization explicitly includes the one pion exchange as a long-range contribution required by unitarity. The short-range part is not constrained by unitarity, and it can be incorporated by a specific (model-dependent) choice of the parameterization. This gives to the framework enough freedom to incorporate QCD resonances. The approach here differs from Mai [@Mai:2017vot] in that the $\2\to\2$ amplitudes required as input are only needed to be known in the physical energy regions. The singularities of the OPE directly impact the analytic structure of the $B$-matrix kernels, and are discussed explicitly for the triangle-like diagram. The singularities in the unphysical region of our solution differ from the Maiones, and from the Feynman diagram triangle. This results in a different value for the real part of the amplitudes in the physical region. Further studies are needed to understand how to remove unexpected singularities from the $B$-matrix. We also compare our formalism to the most recent ones discussed in the literature to extract three-body scattering amplitudes from lattice QCD. In particular, the main difference with Refs. [@Hansen:2014eka; @Hansen:2015zga; @Hansen:2016ync; @Briceno:2017tce; @Briceno:2018mlh] consists in the order of how the partial wave expansion and the one particle exchange ladder summation is performed. It remains to be seen whether the two operations commute, and whether the resulting amplitudes coincide.
Future studies will investigate the continuation to the unphysical energy sheets. This venue will allow us to constrain the role of one particle exchange in generating resonant structures, as it is assumed in some molecular models for the $XYZ$ states.
We thank Ian Aitchison, Raúl Briceño, Michael Döring, and Maxim Mai for many useful discussions, and Arkaitz Rodas for useful comments on the manuscript. We thank the Institute for Nuclear Theory at the University of Washington for its hospitality during the completion of this work. KS was supported by the U.S. National Science Foundation REU grant PHY-1757646. This work was supported by the U.S. Department of Energy under grants No. DE-AC05-06OR23177 and No. DE-FG02-87ER40365, PAPIIT-DGAPA (UNAM, Mexico) grants No. IA101717 and No. IA101819, CONACYT (Mexico) grant No. 251817, Research Foundation – Flanders (FWO), U.S. National Science Foundation under award No. PHY-1415459, and German Bundesministerium für Bildung und Forschung.
Kinematics for $\3\to\3$ Reactions {#sec:app_A}
==================================
In this Appendix, we discuss some of the technical details of the kinematics for $\3\to\3$ processes. We first consider the system in the CMF, $\P^{\star} = \P'^{\star} = \0$. The momenta in terms of invariants are $$\lvert \p_{j}^{\star} \rvert = \frac{\lambda^{1/2}(s,m_{j}^2,\sigma_{j})}{2\sqrt{s}}, \quad \lvert \p_{k}'^{\star} \rvert = \frac{\lambda^{1/2}(s,m_{k}^2,\sigma_{k}')}{2\sqrt{s}}.$$ Considering the particles $j$ and $k$ as spectators, then the recoiling two particles has a total momentum $\P_{j}^{\star} = -\p_{j}^{\star}$ and $\P_{k}^{\star\,\prime} = -\p_k'^{\star}$, for the initial and final system, respectively. The invariants $t_{jk}$ and $u_{jk}$ are related to the CMF scattering angle between spectators via
$$\begin{aligned}
t_{jk} & = (p_j - p_k')^2 \\
& = m_{j}^2 + m_{k}^2 -\frac{1}{2s} (s + m_j^2 - \sigma_{j} ) (s + m_k^2 - \sigma_{k}' ) \nonumber \\ &
+ \frac{1}{2s} \lambda^{1/2}(s,\sigma_{j},m_j^2)\lambda^{1/2}(s,\sigma_{k}',m_k^2)z_{jk}^{\star}, \nonumber \\
u_{jk} & = ((P - p_j)- p_k')^2 \\
& = \sigma_{j} + m_{k}^2 -\frac{1}{2s} (s + \sigma_{j} - m_j^2) (s + m_k^2 - \sigma_{k}' ) \nonumber \\ &
- \frac{1}{2s} \lambda^{1/2}(s,\sigma_{j},m_j^2)\lambda^{1/2}(s,\sigma_{k}',m_k^2)z_{jk}^{\star},\nonumber\end{aligned}$$
where $z_{jk}^{\star} = \cos\Theta_{jk}^{\star}$. The cosine of the CMF angle between particles $j$ and $k$ is $$\label{eq:angle_appA}
\begin{split}
\cos\theta_{kj}^{\star} = \frac{2s(\sigma_{j} + m_{k}^2 - \mu_{jk}^2) - (s + \sigma_{j} - m_j^2) (s + m_k^2 - \sigma_{k} ) }{ \lambda^{1/2}(s,\sigma_{j},m_j^2)\lambda^{1/2}(s,\sigma_{k},m_k^2) },
\end{split}$$ where $\mu_{jk}$ is the mass of the particle that is neither $j$ nor $k$.
The remaining variables needed to completely describe the $\3\to\3$ process are found by examining the IRFs. The initial IRF$_j$ and final IRF$'_k$ are defined when $\P_j = \0$ and $\P_k' = \0$, respectively. We use the convention that initial and final state variables are evaluated in their own respective IRF. The momentum of the first particle in the initial pair is denoted as $\q_j$ in the IRF$_j$. Similarly, the first particle in the final pair is $\q_k'$ in the IRF$'_k$. For example, for the final spectator $3$ in the IRF$'_3$, $\q_3'$ is the final momentum of particle 1, and in the IRF$_1$ of spectator 1, $\q_1$ is the initial momentum of particle 3. In terms of invariants, these momenta are $$\lvert \q_3' \rvert = \frac{\lambda^{1/2}(\sigma_3',m_1^2,m_2^2)}{2\sqrt{\sigma_3'}}, \,\, \lvert \q_1 \rvert = \frac{\lambda^{1/2}(\sigma_1,m_3^2,m_2^2)}{2\sqrt{\sigma_1}}.$$ The spectator momenta in these frames are $$\lvert \p_3' \rvert = \frac{\lambda^{1/2}(s,\sigma_3',m_3^2)}{2\sqrt{\sigma_3'}},$$ for the final state and $$\lvert \p_1 \rvert = \frac{\lambda^{1/2}(s,\sigma_1,m_1^2)}{2\sqrt{\sigma_1}},$$ for the initial state. The helicity angles of the first particle in the IRFs are given by $\chi_j$ and $\chi_k'$, for the initial and final states, respectively. The helicity angles are defined w.r.t. the opposite line-of-flight of the spectator. The azimuthal angles for the initial and final state are $\gamma_j$ and $\gamma_k'$, respectively. The azimuthal angles are defined as the angle between the plane of the two particles in the CMF, and the IRFs, Fig. \[fig:3bodykins\]. Note that the azimuthal angles $\gamma_j$ and $\gamma_k'$ are invariant with respect to the Lorentz boost between the IRFs and CMF, so $\gamma_j = \gamma_j^{\star}$ and $\gamma_k' = \gamma_k'^{\star}$.
The invariant masses of the other two pairs in their respective frames are related to the helicity angles. For example, in the IRF$'_3$, $$\begin{split}
\sigma_1' & = (P - p_1')^2 \\
& = s + m_1^2 - \frac{1}{2\sigma_3'}(s+\sigma_3' - m_3^2)(\sigma_3' + m_1^2 - m_2^2) \\ &
+ \frac{1}{2 \sigma_3'} \lambda^{1/2}(s,\sigma_3',m_3^2)\lambda^{1/2}(\sigma_3',m_1^2,m_2^2) \cos\chi_3' , \\
\sigma_2' & = (P - p_2')^2 \\
& = s + m_2^2 - \frac{1}{2\sigma_3'} (s+\sigma_3' - m_3^2)(\sigma_3' + m_2^2 - m_1^2) \\ &
- \frac{1}{2\sigma_3'} \lambda^{1/2}(s,\sigma_3',m_3^2) \lambda^{1/2}(\sigma_3',m_1^2,m_2^2) \cos\chi_3'.
\end{split}$$
Derivation of $\3\to\3$ Unitarity Relations {#sec:app_B}
===========================================
In this appendix we derive the general elastic unitarity relations for the $\3\to\3$ elastic scattering of distinguishable spinless particles [@Fleming:1964zz; @Holman:1965; @Eden:1966dnq]. For convenience, in this section we adopt the notation that the normalization of a single particle state is $\braket{\p_k'|\p_j} = (2\pi)^{3}\,2\omega_j \delta^{(3)}(\p'_k - \p_j)\delta_{jk} \equiv \wt{\delta}(p'_k - p_j)\delta_{jk}$, and the invariant measure is $\wt{d}p_j \equiv d^{3}\p_j / (2\pi)^{3}\,2\omega_j $. The $S$-matrix is a unitary operator, $S^{\dag}S = \1$, which implies that $T - T^{\dag} = iT^{\dag}T$, where $S = \1 + iT$. We consider the system in an energy range above the three particle threshold, but below the first inelastic threshold, $s_{\mathrm{th}} \le s < s_{\mathrm{inel}}$ . Taking matrix elements of this operator between initial and final states $\ket{\p}$ and $\ket{\p'}$, and inserting the completeness relation $\1= \int \wt{d}p_1''\wt{d}p_2'' \wt{d}p_3''\,\ket{\p''}\bra{\p''}$, gives the unitarity relation $$\label{eq:unitarity_app}
\begin{split}
& \bra{\p'}T\ket{\p} -\bra{\p'} T^{\dag}\ket{\p} \\
& \qquad = i \int \wt{d}p_1'' \wt{d}p_2'' \wt{d}p_3'' \bra{\p'}T^{\dag}\ket{\p''}\bra{\p''}T\ket{\p}.
\end{split}$$ Since $T = T_d + T_c$, where $T_d = \sum_{j} \1_j \otimes T^{(j)}$, then the matrix element is $$\label{eq:Tc+Td_app}
\begin{split}
\bra{\p'}T\ket{\p} & = \bra{\p'}T_{c}\ket{\p} \\
& \, + \sum_{j} \wt{\delta}(p_j' - p_j) \bra{\p'}T^{(j)} \ket{\p} .
\end{split}$$ The matrix elements $\bra{\p'} T^{\dag}\ket{\p}$ are equal to $\bra{\p'}T\ket{\p}^{*}$ by the property of Hermitian analyticity [@Olive:1962; @Eden:1966dnq]. Thus the left hand side of Eq. gives the imaginary part of the matrix element, $$\begin{split}
\text{LHS} & = 2i\,\im\bra{\p'}T_c\ket{\p} \\
& + \sum_{j}\wt{\delta}(p_j'-p_j) 2i\,\im \bra{\p'}T^{(j)}\ket{\p}.
\end{split}$$ The right hand side of Eq. is evaluated by substituting Eq. and expanding the product into four terms,
$$\begin{split}
\text{RHS} & = i \int \widetilde{d}p_1'' \wt{d}p_2'' \wt{d}p_3'' \bigg[ \bra{\p'}T_c^{\dag}\ket{\p''} \bra{\p''}T_c\ket{\p} \\
& \qquad \qquad \qquad \quad + \sum_{k} \wt{\delta}(p_k''-p_k')\bra{\p'}T^{(k)\,\dag}\ket{\p''}\bra{\p''}T_c\ket{\p} \\
& \qquad \qquad \qquad \quad + \sum_{j} \wt{\delta}(p_j''-p_j)\bra{\p'}T_c^{\dag}\ket{\p''}\bra{\p''}T^{(j)}\ket{\p} \\
& \qquad \qquad \qquad \quad + \sum_{j,k} \wt{\delta}(p_k''-p_k')\wt{\delta}(p_j''-p_j) \bra{\p'}T^{(k)\,\dag}\ket{\p''} \bra{\p''}T^{(j)}\ket{\p} \bigg].
\end{split}$$
The fourth term contains two cases, one where $j=k$, and one where $j\ne k$, so we split the sum into the two distinct terms $$\label{eq:DiscSplit_app}
\begin{split}
\sum_{j,k} \widetilde{\delta}(p_k''-p_k')\wt{\delta}(p_j''-p_j) & \bra{\p'}T^{(k)\,\dag}\ket{\p''} \bra{\p''}T^{(j)}\ket{\p} \\
& = \sum_{j} \wt{\delta}(p_j''-p_j')\wt{\delta}(p_j''-p_j) \bra{\p'}T^{(j)\,\dag}\ket{\p''} \bra{\p''}T^{(j)}\ket{\p} \\
& \quad + \sum_{\substack{ j,k \\ j\ne k}} \wt{\delta}(p_k''-p_k')\wt{\delta}(p_j''-p_j) \bra{\p'}T^{(k)\,\dag}\ket{\p''} \bra{\p''}T^{(j)}\ket{\p}.
\end{split}$$ We can write $\wt{\delta}(p_j''-p_j')\wt{\delta}(p_j''-p_j) = \wt{\delta}(p_j' - p_j)\wt{\delta}(p_j'' - p_j)$ in the first term in Eq. , thus we can identify the disconnected unitarity relation as being proportional to the spectator singularity $\wt{\delta}(p_j' - p_j)$, $$\label{eq:2to2DiscUnit_app}
2\im \bra{\p'}T^{(j)}\ket{\p} = \int \wt{d}p_{j_1}'' \wt{d}p_{j_2}'' \bra{\p'}T^{(j)\,\dag}\ket{\p''} \bra{\p''}T^{(j)}\ket{\p},$$ and the connected unitarity relation $$\label{eq:3to3ConnUnit_app}
\begin{split}
2\im \bra{\p'}T_c\ket{\p} & = \int \widetilde{d}p_1'' \widetilde{d}p_2'' \widetilde{d}p_3'' \bigg[ \bra{\p'}T_c^{\dag}\ket{\p''} \bra{\p''}T_c\ket{\p} \\
& + \sum_{k} \wt{\delta}(p_k''-p_k')\bra{\p'}T^{(k)\,\dag}\ket{\p''}\bra{\p''}T_c\ket{\p} \\
& + \sum_{j} \wt{\delta}(p_j''-p_j)\bra{\p'}T_c^{\dag}\ket{\p''}\bra{\p''}T^{(j)}\ket{\p} \\
& + \sum_{\substack{ j,k \\ j\ne k}} \wt{\delta}(p_k''-p_k')\wt{\delta}(p_j''-p_j) \bra{\p'}T^{(k)\,\dag}\ket{\p''} \bra{\p''}T^{(j)}\ket{\p} \bigg].
\end{split}$$ The momenta with $j_1$ and $j_2$ in Eq. identify the first and second particle in the pair. Substituting Eqs. and into Eqs. and , and evaluating the phase space integrals yield the unitarity relations Eqs. and .
Derivation of PWIS Unitarity Relations {#sec:app_C}
======================================
Using the assumption of the isobar model Eq. , we derive a set of unitarity relations for the amputated PWIS amplitudes. Inserting Eq. into the unitarity relations Eq. leads to a unitarity relation for the $\A_{kj}$ isobar-spectator amplitude, $$\label{eq:app_C_3to3UnitIsobar}
\begin{split}
\im{\A_{kj}(\p';\p)} & = \frac{1}{\pi(32\pi^2)^2} \sum_{n} \int_{\sigma_{n}^{(\mathrm{th})}}^{(\sqrt{s} - m_n)^2} d\sigma_n'' \, \frac{\lvert \q_n'' \rvert \lvert \p_n''^{\star} \rvert }{ \sqrt{\sigma_n''} \sqrt{s}} \int d\wh{\P}_n''^{\star} \int d\wh{\q}_n'' \, \A_{nk}^{*}(\p'';\p') \A_{nj}(\p'';\p) \Theta(s - s_{\mathrm{th}}) \\
\
& + \frac{1}{\pi(32\pi^2)^2} \sum_{\substack{n,r \\ n \ne r}} \int_{\sigma_{n}^{(\mathrm{th})}}^{(\sqrt{s} - m_n)^2} d\sigma_n'' \, \frac{\lvert \q_n''\rvert \lvert \p_n''^{\star} \rvert }{ \sqrt{\sigma_n''} \sqrt{s}} \int d\wh{\P}_n''^{\star} \int d\wh{\q}_{n}'' \,\A_{rk}^{*}(\p'';\p') \A_{nj}(\p'';\p) \Theta(s - s_{\mathrm{th}}) \\
\
& +
\rho_2(\sigma_k')\int d\wh{\q}_k'' \,\F_k^{*}(\p'';\p') \A_{kj}(\p'';\p)\rvert_{\p''_k = \p_k'} \Theta(\sigma_k' - \sigma_{k}^{(\mathrm{th})}) \\
\
& +
\rho_2(\sigma_j)\int d\wh{\q}_j'' \,\A_{kj}^{*}(\p'';\p') \rvert_{\p''_j = \p_j} \F_{j}(\p'';\p)\Theta(\sigma_j - \sigma_{j}^{(\mathrm{th})}) \\
\
& +
\rho_2(\sigma_k')\sum_{\substack{r \\ k \ne r}}\, \int d\wh{\q}_k''\,\F_k^{*}(\p'';\p') \A_{rj}(\p'';\p)\rvert_{\p''_k = \p_k'} \Theta(\sigma_k' - \sigma_{k}^{(\mathrm{th})}) \\
\
& +
\rho_2(\sigma_j)\sum_{\substack{n \\ n \ne j}}\, \int d\wh{\q}_j'' \,\A_{kn}^{*}(\p'';\p') \rvert_{\p''_j = \p_j} \F_{j}(\p'';\p)\Theta(\sigma_j - \sigma_{j}^{(\mathrm{th})}) \\
\
& + \pi \, \delta(u_{jk} - \mu_{jk}^2)\,\F_{k}^{*}(\p'';\p')\rvert_{\p''_j = \p_j} \F_{j}(\p'';\p) \rvert_{\p''_k = \p_k'} (1 - \delta_{jk}) ,
\end{split}$$
where we wrote the three-body phase space factor in the first two terms, $$\label{eq:app_C_3bodyPhaseSpace1}
\begin{split}
& \frac{1}{2(2\pi)^5}\int \frac{d^{3}\p_1''}{2\omega_1''}\frac{d^{3}\p_2''}{2\omega_2''}\frac{d^{3}\p_3''}{2\omega_3''} \delta^{(4)}(P'' - P) \\
\
& = \frac{1}{\pi (32\pi^2)^2} \int_{\sigma_n^{(\mathrm{th})}}^{(\sqrt{s} - m_n)^2} d\sigma_n'' \, \frac{\lvert \q_n''\rvert \lvert \p_n^{\prime\prime\,\star} \rvert }{ \sqrt{\sigma_n''} \sqrt{s}} \int d\wh{\P}_n''^{\star} \int d\wh{\q}_n'' ,
\end{split}$$
where $\wh{\P}_n''^{\star} $ is the orientation of the isobar associated with spectator $n$ in the intermediate state, and $\wh{\q}_n'' $ is the orientation of the first particle in the intermediate isobar in its rest frame. The terms in the intermediate state have been split up into two groups, diagonal and off-diagonal. The diagonal terms in Eq. (first, third, and fourth line) are terms such that the isobar propagates in the intermediate state without breaking up. The off-diagonal terms (second, fifth, and sixth line in Eq. ) are ones where the isobar breaks up in the intermediate state, and combines with the spectator to form a new isobar. Figure \[fig:isobar\_spectator\_unitarity\] shows the diagonal and off-diagonal topologies in the intermediate state.
The partial wave expansion Eq. can be derived by considering the expansion in three steps. The first step is to perform the expansion of the isobars into definite spin and helicity, $$\label{eq:app_C_isobar_strip}
\begin{split}
\A_{kj} = \sum_{s_k',\lambda_k'}\sum_{s_j,\lambda_j} Y_{\lambda_k'}^{s_k' }(\wh{\q}_k') \A_{s_k' \lambda_k' ; s_j \lambda_j} Y_{\lambda_j}^{s_j \, *}(\wh{\q}_j),
\end{split}$$ where $\lambda_j$, $\lambda_k'$ are defined along the direction of the isobar in the CMF. The expansion removes the $\wh{\q}_k'$ and $\wh{\q}_j$ dependence in the helicity amplitude $\A_{s_k' \lambda_k' ; s_j \lambda_j}$. Second, the helicity amplitude can be expanded in partial waves, $$\label{eq:app_C_hel_expansion}
\begin{split}
\A_{s_k' \lambda_k' ; s_j \lambda_j} & = \sum_{J} \N_{J}^2 \, \A_{s_k' \lambda_k' ; s_j \lambda_j}^{J}(\sigma_k',s,\sigma_j) \\
& \times \sum_{M} \D_{M \lambda_k'}^{(J)\,*}(\wh{\P}_{k}'^{\star}) \D_{M \lambda_j}^{(J)}(\wh{\P}_j^{\star}),
\end{split}$$ where $\N_J^2 = (2J + 1) / 4\pi$. Finally, since parity is not a good quantum number in the helicity basis, we convert the helicity partial wave amplitudes into $LS$ partial wave amplitudes, $$\label{eq:app_C_LS_hel}
\A_{\ell_k' s_k' ; \ell_j s_j}^{J} = \sum_{\lambda_k',\lambda_j} C^{J}_{\ell_k' s_k' \lambda_k'} C^{J}_{\ell_j s_j \lambda_j} \A_{s_k' \lambda_k' ; s_j \lambda_j}^{J} ,$$ where $C_{\ell s \lambda}^{J} = \sqrt{ (2\ell + 1) / (2 J + 1)} \braket{J \lambda | \ell 0 s \lambda}$, and has the completeness relation $$\label{eq:app_C_rec_ortho}
\sum_{\lambda} C_{\ell s \lambda}^{J} C_{\ell' s' \lambda}^{J} = \delta_{JJ'}\delta_{\ell\ell'} \delta_{ss'}.$$ Combining Eqs. , , and yields the partial wave expansion Eq. . We apply the expansion to Eq. to obtain the PWIS unitarity relations. The diagonal terms are most directly evaluated using the orthonormality condition, Eq. . Since $\p_k'' = \p_k'$ in the third term, and $\p_j'' = \p_j$ in the fourth term, then the intermediate isobar orientation is identical to that of the final and initial state isobar, respectively. The integrations over $\wh{\q}_k'$ and $\wh{\q}_j$ can be performed by writing Eq. using the spherical harmonic addition theorem.
The off-diagonal terms are more challenging, as they involve two different angles in the intermediate state, thus the rotational functions will not directly integrate out. We can make use of the group properties of rotations to simplify the intermediate rotational functions to a recoupling coefficient. The off-diagonal terms on the right-hand side of Eq. under the expansion Eq. will contain terms of the form $$\begin{split}
& \D_{\lambda_n 0}^{(s_n)\,*} (\wh{\q}_n) \D_{M\lambda_n}^{(J)\,*}(\wh{\P}_n^{\star}) \D_{M\lambda_r}^{(J)}(\wh{\P}_r^{\star}) \D_{\lambda_r 0}^{(s_r)}(\wh{\q}_r) \\
& \quad = d_{\lambda_n 0}^{(s_n)} (\cos\chi_n) \D_{M \lambda_n}^{(J)\,*}(R_n^{\star}) \D_{M \lambda_r}^{(J)}(R_r^{\star}) d_{\lambda_r 0 }^{(s_r)}(\cos\chi_r),
\end{split}$$ where $n\ne r$, and we combined the terms with $\gamma_n$ and the orientation of the isobar to the set of angles $R_n^{\star} = (\alpha_n^{\star},\beta_n^{\star},\gamma_n^{\star})$, where $\alpha_n^{\star}$ is the azimuthal angle of the isobar and $\beta_n^{\star}$ is the polar angle, w.r.t. some fixed coordinate system. Note that since we boost along the direction of the isobar to go between CMF and IRFs, $\gamma_n = \gamma_n^{\star}$. The angles $R^{\star}$ are the Euler angles describing the orientation of the three particles in their CMF. Since these two sets of angles describe the same configuration of three particles, with the only difference being which particle is the spectator, the angles $R_n^{\star}$ and $R_r^{\star}$ must be related by a rotation.
Each set of angles can be found by rotating from some initial standard configuration. We define the standard configuration such that the three particle system lies in the $xz$-plane, where the spectator momenta is along the negative $z$-axis, Fig. \[fig:SC\]. Then, the difference in the Euler angles is a rotation about the $y$-axis, $$R_r^{\star} = R_n^{\star} r_{nr}^{\star},$$ where $r_{nr}^{\star}$ is the rotation relating the two standard configurations [@Ascoli:1975mn; @Giebink:1985zz]. Here, the rotation is about the $y$-axis, $r_{nr}^{\star} = R_{y}(\theta_{nr}^{\star})$, where $\theta_{nr}^{\star}$ is given in Eq. . Thus, the rotation is a function of the invariant masses of the isobars, $\theta_{nr}^{\star} = \theta_{nr}^{\star}(\sigma_n,s,\sigma_r)$. Note that the inverse rotation is given by $r_{rn }^{\star} = r_{nr}^{\star\,-1}$.
(-145,10) (-30,10)
Therefore, we can relate the two Wigner $\D$-matrices using the group addition property $$\begin{split}
\D_{M\lambda_r}^{(J)}(R_{r}^{\star}) = \sum_{\lambda_n} \D_{M \lambda_n}^{(J)}(R_{n}^{\star}) \D_{\lambda_n \lambda_r}^{(J)}(r_{nr}^{\star}).
\end{split}$$ The integration over the Euler angles in the intermediate state can be performed, leaving one rotation that recouples the isobars, $$\label{eq:recouple_property}
\begin{split}
\int d R_{n}^{\star} \,& \D_{M\lambda_n}^{(J)\,*}(R_{n}^{\star}) \D_{M \lambda_r}^{(J)}(R_{r}^{\star}) \\
& = \sum_{\lambda} \int d R_n^{\star}\, \D_{M\lambda_n}^{(J)\,*}(R_{n}^{\star})\D_{M \lambda}^{(J)}(R_{n}^{\star})\D_{\lambda \lambda_r}^{(J)}(r_{nr}^{\star}) \\
& = \frac{8\pi^2}{2J+1}
d_{\lambda_n \lambda_r}^{(J)}(\cos{\theta_{nr}^{\star}}).
\end{split}$$ where $dR_n^{\star} = d\alpha_n^{\star} d\cos\beta_n^{\star} d\gamma_n^{\star}$.
Since the angle $\chi$ can be written in terms of the invariant masses of the isobars, it is advantages to write the phase space in terms of the Dalitz variables, $$\label{eq:app_C_3bodyPhaseSpace2}
\begin{split}
& \frac{1}{(2\pi)^5}\int \frac{d^{3}\p_1''}{2\omega_1''}\frac{d^{3}\p_2''}{2\omega_2''}\frac{d^{3}\p_3''}{2\omega_3''} \delta^{(4)}(P'' - P) \\
\
& = \frac{1}{\pi s \, (32\pi^2)^{2}} \int_{\sigma_n^{(\mathrm{th})}}^{(\sqrt{s} - m_n^2)^2} d\sigma_n'' \int_{\sigma_r^{(-)}}^{\sigma_r^{(+)}} d\sigma_r''\, \int d R_{n}''^{\star} ,
\end{split}$$ where we used $$\label{eq:app_C_3bodyPhaseSpace3}
d\sigma_r'' = 2 \frac{\sqrt{s}}{\sqrt{\sigma_n''}} \lvert \q_n'' \rvert \lvert \p_n''^{\star} \rvert \,d\cos\chi_n'',$$ to rewrite Eq. . The Dalitz region is bounded by $\sigma_n^{(\mathrm{th})} \le \sigma_n'' \le (\sqrt{s} - m_n)^2$ and $\sigma_r^{(-)} \le \sigma_r'' \le \sigma_r^{(+)}$, where $\sigma_r^{(\pm)} = \sigma_r^{(\pm)}(\sigma_n'')$ is found by the physical boundary $\cos\chi_n'' = \pm 1$, for $n = 1$ and $r=3$, then $$\begin{split}
\sigma_3^{(\pm)} & = m_1^2 + m_2^2 - \frac{1}{2\sigma_1''} (\sigma_1'' - s + m_1^2)(\sigma_1'' + m_2^2 - m_3^2) \\
\
& \pm \frac{1}{2\sigma_1''} \lambda^{1/2}(s,\sigma_1'',m_1^2)\lambda^{1/2}(\sigma_1'',m_2^2,m_3^2).
\end{split}$$
The last piece needed is the partial wave projection of the OPE term. To evaluate the partial wave projection, we write the delta-function as $$\label{eq:app_C_ope_term1}
\delta( u_{jk} - \mu_{jk}^2) = \frac{1}{2\lvert \p_k'^{\star} \rvert \lvert \p_j^{\star} \rvert} \delta(z_{kj}^{\star} - z_{kj}),$$ where $z_{kj}$ is defined in Eq. . Then, the completeness relation for the delta-function allows us to write Eq. as $$\delta(z_{kj}^{\star} - z_{kj}) = \sum_{J}\left( \frac{2J + 1}{2}\right) d_{\lambda' \lambda}^{(J)}(z_{kj}^{\star}) d_{\lambda'\lambda}^{(J)}(z_{kj}),$$ where $\lambda$ and $\lambda'$ are arbitary, and thus we may choose them to align with $\lambda_j$ and $\lambda_k'$, respectively. Then, $d_{\lambda_j \lambda_k'}^{(J)}(z_{kj}^{\star})$ is written in terms of the angles $\wh{\P}_j^{\star}$ and $\wh{\P}_k'^{\star}$, via the group addition property, $$d_{\lambda'\lambda}^{(J)}(z_{kj}^{\star}) = \sum_{M} \D_{M \lambda'}^{(J)\,*}(\wh{\P}_k'^{\star})\D_{M\lambda}^{(J)}(\wh{\P}_j^{\star}).$$
Finally, the $\2\to\2$ amplitudes are written using Eq. $$\begin{split}
F_{k}^{*}F_j & = f^{*}_{s_k'}(\sigma_k') f_{s_j}(\sigma_j) P_{s_k'}(\wh{\ov{\q}}_j \cdot \wh{\q}_k') P_{s_j}(\wh{\q}_j \cdot \wh{\ov{\q}}_k') \\
\
& =f^{*}_{s_k'}(\sigma_k') f_{s_j}(\sigma_j) \sum_{\lambda_k'} \D_{\lambda_k'0}^{(s_k')\,*}(\wh{\q}_k')\D_{\lambda_k'0}^{(s_k')}(\wh{\ov{\q}}_{j}) \\
& \qquad \times \sum_{\lambda_j} \D_{\lambda_j 0}^{(s_j)\,*}(\wh{\ov{\q}}_k') \D_{\lambda_j 0}^{(s_j)}(\wh{\q}_j),
\end{split}$$ where $\bar{\q}_j$ is the momentum of the first particle in the final state in the IRF$_j$, and $\bar{\q}_k'$ is the momentum of the first particle in the initial state in the IRF$'_k$. Since $\p_k'' = \p_k'$ and $\p_j'' = \p_j$, then the azimuthal angles are identical, $\gamma_k' = \gamma_j$. The helicity angles of the first particle in the opposite frames are defined as $\ov{\chi}_j$ and $\ov{\chi}_k'$, Fig. \[fig:ope\_planes\]. However, we can easily identify that $\chi_k' = \ov{\chi}_k'$ and $\chi_j = \ov{\chi_k'}$ since the intermediate spectator is aligned for the OPE and the IRFs merely differ by the rotation about $z$.
Combining all of this together yields the PWIS unitarity relations,
$$\label{eq:app_C_PWIS_unitarity}
\begin{split}
\im\, & \A^{J}_{ \ell_k' s_k' ; \ell_j s_j }(\sigma_k',s,\sigma_j) \\
& = \frac{1}{\pi(32\pi^2)^2} \sum_{n} \sum_{\ell_n'' , s_n''} \int_{\sigma_{n}^{(\mathrm{th})}}^{(\sqrt{s} - m_n)^2} d\sigma_n'' \, \frac{\lvert \q_n'' \rvert \lvert \p_n''^{\star} \rvert }{ \sqrt{\sigma_n''} \sqrt{s}} \, \A^{J\,*}_{\ell_n'' s_n'' ; \ell_k' s_k' }(\sigma_n'',s,\sigma_k') \A^{J}_{\ell_n'' s_n'' ; \ell_j s_j }(\sigma_n'',s,\sigma_j) \Theta(s - s_{\mathrm{th}}) \\
& + \frac{1}{2\pi s(32\pi^2)^2} \sum_{\substack{n,r \\ n \ne r}} \sum_{\ell_n'' , s_n''}\sum_{\ell_r'' , s_r''} \int_{\sigma_{n}^{(\mathrm{th})}}^{(\sqrt{s} - m_n)^2} d\sigma_n'' \,\int_{\sigma_r^{(-)}(\sigma_n'')}^{\sigma_r^{(+)}(\sigma_n'')} d\sigma_r'' \, \\
& \qquad\qquad \times \C_{\ell_n'' s_n'' ; \ell_r'' s_r''}^{J}(\sigma_n'',s,\sigma_r'')\, \A^{J\,*}_{\ell_r'' s_r''; \ell_k' s_k' }(\sigma_r'',s,\sigma_k') \A^{J}_{\ell_n'' s_n'' ; \ell_j s_j }(\sigma_n'',s,\sigma_j) \Theta(s - s_{\mathrm{th}}) \\
& +
\rho_2(\sigma_k')\, f_{s_k'}^{*}(\sigma_k') \A_{\ell_k' s_k' ; \ell_j s_j}^{J}(\sigma_k',s,\sigma_j) \\
& +
\rho_2(\sigma_j)\, \A_{\ell_k' s_k' ; \ell_j s_j}^{J \, *}(\sigma_k',s,\sigma_j) f_{s_j}(\sigma_j) \\
& + \frac{1}{64 \pi^2 \sqrt{s}} \frac{1}{\lvert \p_k^{\prime\,\star} \rvert} \sum_{\substack{r \\ r \ne k}} \sum_{\ell_r'', s_r''}\, \int_{\sigma_r^{(\mathrm{th})}}^{(\sqrt{s}-m_r)^2} d\sigma_r'' \, \C_{\ell_k' s_k' ; \ell_r'' s_r''}^{J}(\sigma_k',s,\sigma_r'') \, f_{s_k'}^{*}(\sigma_k') \, \A_{\ell_r'' s_r'' ; \ell_j s_j }^{J}(\sigma_r'',s,\sigma_j) \Theta(\sigma_k' - \sigma_{k}^{(\mathrm{th})}) \\
& + \frac{ 1 }{64 \pi^2 \sqrt{s}} \frac{1}{\lvert \p_j^{\star} \rvert} \sum_{\substack{n \\ n \ne j}} \sum_{\ell_n'' , s_n''} \, \int_{\sigma_n^{(\mathrm{th})}}^{(\sqrt{s}-m_n)^2} d\sigma_n'' \, \C_{\ell_n'' s_n'' ; \ell_j s_j}^{J}(\sigma_n'',s,\sigma_j) \, f_{s_j}(\sigma_j) \, \A_{ \ell_k' s_k'; \ell_n'' s_n''}^{J\,*}(\sigma_k',s,\sigma_n'') \Theta(\sigma_j - \sigma_{j}^{(\mathrm{th})}) \\
& + \frac{\pi}{2\lvert \p_j^{\star} \rvert \lvert \p_k'^{\star} \rvert} \, f_{s_k'}^{*}(\sigma_k') f_{s_j}(\sigma_j)\,\C_{\ell_k' s_k';\ell_j s_j}^{J}(\sigma_k',s,\sigma_j)
(1 - \delta_{jk}) \Theta(1 - \lvert z_{kj} \rvert^2) ,
\end{split}$$
where the recoupling coefficients are defined in Eq. . Notice that the first, third, and fourth term involve direct channel exchanges in the intermediate state, while the others involve rescattering between cross channels.
Finally, we introduce the amputated amplitude $\wt{\A}_{\ell_k' s_k' ; \ell_j s_j}^{J}$, defined in Eq. . The amputation eliminates the unitarity cut from the $\2\to\2$ amplitude in the two particle subsystem in the third and fourth term of Eq. . Taking the imaginary part of Eq. , $$\label{eq:ImagFact}
\begin{split}
& \im{\big[ f_{s_{k}'}(\sigma_{k}')\wt{\A}_{\ell_k' s_k' ; \ell_j s_j}(\sigma_{k}',s,\sigma_{j}) f_{s_{j}}(\sigma_{j}) \big]} \\
& \, = \im{\big[f_{s_{k}'}(\sigma_{k}')\big]}\wt{\A}_{\ell_k' s_k' ; \ell_j s_j}(\sigma_{k}',s,\sigma_{j}) f_{s_{j}}(\sigma_{j}) \\
&\quad + f_{s_{k}'}^{*}(\sigma_{k}') \im{\big[\wt{\A}_{\ell_k' s_k' ; \ell_j s_j}(\sigma_{k}',s,\sigma_{j}) \big]} f_{s_{j}}(\sigma_{j}) \\
&\quad + f_{s_k'}^{*}(\sigma_k')\wt{\A}_{\ell_k' s_k' ; \ell_j s_j}^{*}(\sigma_k',s,\sigma_j) \im{\big[ f_{s_j}(\sigma_j)\big]}.
\end{split}$$ The amputation removes the contribution from the isobar amplitude unitarity cut using Eq. , leaving only rescattering corrections to the isobar shape. We then arrive at the amputated PWIS unitarity relations Eq. .
The $B$-matrix and Unitarity {#sec:app_E}
============================
In this appendix, we demonstrate that the $B$-matrix parameterization satisfies the unitarity relations Eq. , specifically for $\wt{\A}_{13}$. Recall that the $B$-matrix parameterization for $\wt{\A}_{13}$ is $$\label{eq:app_Bmat1}
\begin{split}
\wt{\A}_{13}(s) & = \wt{\B}_{13}(s) + \wt{\B}_{13}(s)\tau_{3}(s)\wt{\A}_{33}(s),
\end{split}$$ which has an imaginary part $$\label{eq:app_E_derive1}
\begin{split}
\im{\wt{\A}_{13}(s)} & = \im{\wt{\B}_{13}}(s) \\
& + \im{\wt{\B}_{13}(s)}\tau_{3}(s)\wt{\A}_{33}(s) \\
& + \wt{\B}_{13}^{*}(s)\im{\tau_{3}(s)}\wt{\A}_{33}(s) \\
& + \wt{\B}_{13}^{*}(s)\tau_{3}^{*}(s)\im{\wt{\A}_{33}(s)}.
\end{split}$$ From Eqs. and , $$\im{\tau_n(s,\sigma_n)} = \rho_3(s,\sigma_n)\rho_2(\sigma_n) \lvert f_{s_n}(\sigma_n) \rvert^{2},$$ and since $\wt{\R}_{kj}$ is real, $\im\wt{\B}_{kj} = \im{\wt{\E}_{kj}}$, which is known from projecting Eq. into partial waves.
The imaginary part of $\wt{\A}_{33}$ is found by using Eq. , $$\label{eq:app_E_derive2}
\begin{split}
\im{\wt{\A}_{33}(s)} & = \im{\left[\mathbbm{1} - \Kc_{33}(s)\tau_{3}(s) \right]^{-1}}\Kc_{33}(s) \\
& + \left[\mathbbm{1} - \Kc_{33}^{*}(s)\tau_{3}^{*}(s) \right]^{-1} \im{\Kc_{33}(s)},
\end{split}$$ where the kernel $\Kc_{33}(s) = \wt{\B}_{31}(s)\tau_1(s) \wt{\B}_{13}(s)$ . The imaginary part of $\left[\mathbbm{1} - \Kc_{33}(s)\tau_3(s)\right]^{-1}$ is found by the identity $\im\left[A^{-1}A\right] = \im A^{-1} A + A^{*\,-1}\im A = 0$, giving $$\label{eq:app_E_derive3}
\begin{split}
\im{\left[ \mathbbm{1} - \Kc_{33}(s)\tau_3(s) \right]^{-1}} & =\left[ \mathbbm{1} - \Kc_{33}^{*}(s) \tau_{3}^{*}(s) \right]^{-1} \\
& \times \im\left[ \Kc_{33}(s) \tau_{3}(s)\right] \\
& \times \left[ \mathbbm{1} - \Kc_{33}(s) \tau_{3}(s) \right]^{-1},
\end{split}$$ with $\im\left[ \Kc_{33}(s) \tau_{3}(s)\right] = \im\Kc_{33}(s) \tau_3(s) + \Kc_{33}^{*}(s) \im \tau_3(s)$. Combining Eqs. , , and give
$$\label{eq:app_E_derive4}
\begin{split}
\im{\wt{\A}_{13}(s)} & = \im{\wt{\B}_{13}}(s) \\
& + \im{\wt{\B}_{13}(s)}\tau_{3}(s)\wt{\A}_{33}(s) \\
\
& + \wt{\B}_{13}^{*}(s)\im{\tau_{3}(s)}\wt{\A}_{33}(s) \\
\
& + \wt{\B}_{13}^{*}(s)\tau_{3}^{*}(s)\left[\mathbbm{1} - \Kc_{33}^{*}(s)\tau_{3}^{*}(s) \right]^{-1} \im{\Kc_{33}(s)}\tau_3(s)\left[\mathbbm{1}-\Kc_{33}(s)\tau_3(s) \right]^{-1}\Kc_{33}(s) \\
\
& + \wt{\B}_{13}^{*}(s)\tau_{3}^{*}(s)\left[\mathbbm{1} - \Kc_{33}^{*}(s)\tau_{3}^{*}(s) \right]^{-1} \Kc_{33}^{*}(s)\im{\tau_3}(s)\left[\mathbbm{1}-\Kc_{33}(s)\tau_3(s) \right]^{-1}\Kc_{33}(s) \\
\
& + \wt{\B}_{13}^{*}(s)\tau_{3}^{*} \left[\mathbbm{1} - \Kc_{33}^{*}(s)\tau_{3}^{*}(s) \right]^{-1} \im{\Kc_{33}(s)}.
\end{split}$$
The imaginary part of the kernel is $$\begin{split}
\im{\Kc_{33}(s)} & = \im{\wt{\B}_{31}(s)} \tau_1(s) \wt{\B}_{13}(s) \\
& + \wt{\B}_{31}^{*}(s) \im{\tau_1(s)} \wt{\B}_{13}(s) \\
& + \wt{\B}_{31}^{*}(s) \tau_1^{*}(s) \im{\wt{\B}_{13}(s)}.
\end{split}$$ We use Eq. to shift the last three lines of Eq. in terms $\Kc_{11}=\wt{\B}_{13}(s)\tau_3(s)\wt{\B}_{31}(s)$, $$\label{eq:app_E_derive5}
\begin{split}
\im{\wt{\A}_{13}(s)} & = \im{\wt{\B}_{13}}(s) \\
& + \im{\wt{\B}_{13}(s)}\tau_{3}(s)\wt{\A}_{33}(s) \\
& + \wt{\B}_{13}^{*}(s)\im{\tau_{3}(s)}\wt{\A}_{33}(s) \\
\
& + \left[\mathbbm{1} - \Kc_{11}^{*}(s)\tau_{1}^{*}(s) \right]^{-1}\wt{\B}_{13}^{*}(s)\tau_{3}^{*} \im{\wt{\B}_{31}(s)}\tau_1(s)\wt{\B}_{13}(s) \tau_3(s)\left[\mathbbm{1}-\Kc_{33}(s)\tau_3(s) \right]^{-1}\Kc_{33}(s) \\
\
& + \left[\mathbbm{1} - \Kc_{11}^{*}(s)\tau_{1}^{*}(s) \right]^{-1}\Kc_{11}^{*}(s)\im{\tau_1(s)}\wt{\B}_{13}(s)\tau_3(s)\left[\mathbbm{1}-\Kc_{33}(s)\tau_3(s) \right]^{-1}\Kc_{33}(s) \\
\
& + \left[\mathbbm{1} - \Kc_{11}^{*}(s)\tau_{1}^{*}(s) \right]^{-1}\Kc_{11}^{*}(s)\tau_1^{*}(s) \im{\wt{\B}_{13}(s) }\tau_3(s)\left[\mathbbm{1}-\Kc_{33}(s)\tau_3(s) \right]^{-1}\Kc_{33}(s) \\
\
& + \wt{\B}_{13}^{*}(s)\tau_{3}^{*}(s) \left[\mathbbm{1} - \Kc_{33}^{*}(s)\tau_{3}^{*}(s) \right]^{-1}\Kc_{33}^{*}(s)\im{\tau_3}(s)\left[\mathbbm{1}-\Kc_{33}(s)\tau_3(s) \right]^{-1}\Kc_{33}(s) \\
\
& + \left[\mathbbm{1} - \Kc_{11}^{*}(s)\tau_{1}^{*}(s) \right]^{-1}\wt{\B}_{13}^{*}(s)\tau_{3}^{*}(s) \im{\wt{\B}_{31}(s)}\tau_1(s)\wt{\B}_{13}(s) \\
\
& + \left[\mathbbm{1} - \Kc_{11}^{*}(s)\tau_{1}^{*}(s) \right]^{-1}\Kc_{11}^{*}(s)\im{\tau_1(s)}\wt{\B}_{13}(s) \\
\
& + \left[\mathbbm{1} - \Kc_{11}^{*}(s)\tau_{1}^{*}(s) \right]^{-1}\Kc_{11}^{*}(s)\tau_1^{*}(s) \im{\wt{\B}_{13}(s) }.
\end{split}$$ Grouping common terms in $\im\tau_n$ and $\im{\wt{\B}_{kj}}$, and identifying the forms of the amplitudes from Sec. \[sec:B-Matrix\], yields $$\label{eq:app_E_derive6}
\begin{split}
\im{\wt{\A}_{13}(s)} & = \im{\wt{\B}_{13}}(s) \\
& + \im{\wt{\B}_{13}(s)}\tau_{3}(s)\wt{\A}_{33}(s) \\
\
& + \wt{\A}_{11}^{*}(s)\tau_1^{*}(s) \im{\wt{\B}_{13}(s) }\\
\
& + \wt{\A}_{13}^{*}(s)\im{\tau_{3}(s)}\wt{\A}_{33}(s) \\
\
& + \wt{\A}_{11}^{*}(s)\im{\tau_1(s)}\wt{\A}_{13}(s)\\
\
& + \wt{\A}_{11}^{*}(s)\tau_1^{*}(s) \im{\wt{\B}_{13}(s) }\tau_3(s)\wt{\A}_{33}(s) \\
\
& + \wt{\A}_{13}^{*}(s)\tau_{3}^{*}(s) \im{\wt{\B}_{31}(s)}\tau_1(s)\wt{\A}_{13}(s).
\end{split}$$ Substituting for the imaginary parts of $\tau_n$ and $\wt{\B}_{kj}$ gives the PWIS unitarity relation for $\wt{\A}_{13}$. The unitarity relations for the other amplitudes can be found in a similar manner.
The Feynman Triangle Diagram {#sec:app_F}
============================
For reference, we state the basic formulae for computing the Feynman triangle diagram, Ref. [@Itzykson:1980rh]. The perturbative Feynman diagram has the form $$\mathcal{T}_{F}(s) = i\int \frac{d^{4}k}{(2\pi)^4}\frac{1}{D_1D_2D_3},$$ shown in Fig. \[fig:tri\_label\_app\], where the denominator is the product of internal propagators, $$\begin{split}
D_1D_2D_3 & = \left[k^2 - \mu_{jk}^2 + i\epsilon\right] \\
& \times \left[(k+P_1)^2 - m_3^2 + i\epsilon\right] \\
& \times \left[(k-p_1)^2 - M^2 + i\epsilon\right].
\end{split}$$
![The triangle diagram with loop momentum labels.[]{data-label="fig:tri_label_app"}](figures/triangle_labels_app){width="0.8\columnwidth"}
Using the Feynman parameterization and standard loop integration techniques, the Feynman diagram has the form $$\mathcal{T}_{F}(s) = \frac{1}{16\pi^2} \int_{0}^{1} d\alpha_1 \int_{0}^{1-\alpha_1} d\alpha_2 \, F(s;\alpha_1,\alpha_2),$$ where $$\begin{split}
F^{-1}(s;\alpha_1,\alpha_2) & = M^2 \alpha_1 + m_3^2 \alpha_2 + \mu{}^2 (1 - \alpha_1 - \alpha_2) \\
& + m_1^2 \alpha_1( \alpha_1 - 1) + \sigma_1 \alpha_2 (\alpha_2 - 1) \\
& - (s - \sigma_1 - m_1^2) \alpha_1 \alpha_2 - i\epsilon.
\end{split}$$ The remaining integrals over the Feynman parameters can be computed either numerically, or by analytically performing the integral over $\alpha_2$, then numerically computing the remaining integral over $\alpha_1$.
![Contours for dispersive triangle Eq. shown in red, and the integrand cuts. The three cases are (a) $M^2 \ge \sigma_3^{(b)}$, (b) $\sigma_3^{(a)} \le M^2 \le \sigma_3^{(b)}$, and (c) $\sigma_3^{(\mathrm{th})} \le M^2 \le \sigma_3^{(a)}$. Case (a) corresponds to the usual triangle singularity, which occurs since the OPE branch points pinch the integration region. Case (c) happens when the initial state of the OPE has a higher threshold then the intermediate state. The blue region indicates the physical region from the initial threshold. Note that a triangle singularity does not occur in this case and the integration is not pinched.[]{data-label="fig:tri_contours"}](figures/cuts_triangle "fig:"){width="0.8\columnwidth"} (-150,10)[[(c)]{}]{} (-150,110)[[(b)]{}]{} (-150,190)[[(a)]{}]{}
Alternatively, the Feynman triangle can be written with a dispersive representation in $s$ using the Cutkosky rules [@Itzykson:1980rh], $$\label{eq:TF_app}
\mathcal{T}_{F}(s) = \int_{\Gamma_{T}} ds' \frac{\rho_3(s',M^2)\wt{\E}_{13}(M^2,s',\sigma_1)}{s' - s - i\epsilon},$$ where $\Gamma_T$ is the path from the threshold $(M + m_3)^2$ to $\infty$, $\rho_3(s,M^2)$ is given by Eq. , $\wt{\E}_{13}$ is given by Eq. , and the $S$-wave amplitudes are normalized according to Eq. . The phase space contributes branch point singularities from the threshold and pseudothreshold, $(M \pm m_3)^2$, and a pole at $s=0$. The OPE has branch points $s = s_{31}^{(\pm)}$ near the integration region. Following the discussion in Section \[sec:ope\], the OPE branch points give us the following scenarios:
(a) $M^2 \ge \sigma_3^{(b)}$. The RPE branch points pinch the integration region which starts at $s = (M+m_3)^2$. Figure \[fig:tri\_contours\](a) shows the integrand branch cuts and the dispersive contour. The RPE branch point $s_{31}^{(-)}$ lie in the unphysical sheet close to threshold, causing the known as the triangle singularity [@Peierls:1961zz; @Aitchison:1966; @Eden:1966dnq; @Szczepaniak:2015eza]. The triangle singularity produces an extra threshold in the physical region above the threshold $s = (M+m_3)^2$, and is associated with real particle exchange in the intermediate state. The location of the triangle singularity occurs at $$\begin{split}
s_{\textrm{tri}} & = \frac{1}{2 m_2^2} \bigg[ (m_3^2 - \sigma_1)(m_1^2 - M^2 ) - m_2^4 \\
& + m_2^2(m_3^2 + m_1^2 + \sigma_1 + M^2) \\
& \pm \lambda^{1/2}(m_2^2,m_3^2,\sigma_1) \lambda^{1/2}(m_2^2,m_1^2,M^2) \bigg].
\end{split}$$
(b) $\sigma_3^{(a)} \le M^2 \le \sigma_3'^{(b)}$. The RPE branch points are both above the real axis, and cause no additional singular behavior. Figure \[fig:tri\_contours\](b) illustrates this case.
(c) $\sigma_3^{(\mathrm{th})} \le M^2 \le \sigma_3^{(a)}$. The RPE branch points are again on opposite sides of the real axis. However, the integration region begins below the location where the RPE cut crosses the real axis. This is due to the fact that the initial state has a higher threshold then the intermediate state, so the physical region is above the RPE crossing location. This is illustrated in Fig. \[fig:tri\_contours\] where the blue region indicates the physical region starting at the initial state threshold, and the integration contour is a path around the RPE branch point $s_{31}^{(-)}$. No singularity occurs in this region as the RPE branch points do not pinch the integration region.
Notice that in contrast to the $B$-matrix triangle, Eq. , the dispersive triangle moves all the singularities from the phase space and OPE to the unphysical sheet. Thus, the only singularity present on the physical sheet is the unitarity cut starting at $s = (M + m_3)^2$.
|
---
abstract: 'We present detailed magnetometry and muon-spin rotation data on polycrystalline samples of overdoped, non-superconducting LiFe$_{1-x}$Ni$_x$As ($x = 0.1,\,0.2$) and Li$_{1-y}$Fe$_{1+y}$As ($0\leq y\leq 0.04$) as well as superconducting LiFeAs. While LiFe$_{1-x}$Ni$_x$As exhibits weak antiferromagnetic fluctuations down to $1.5\,{\rm K}$, Li$_{1-y}$Fe$_{1+y}$As samples, which have a much smaller deviation from the $1:1:1$ stoichiometry, show a crossover from ferromagnetic to antiferromagnetic fluctuations on cooling and a freezing of dynamically fluctuating moments at low temperatures. We do not find any signatures of time-reversal symmetry breaking in stoichiometric LiFeAs that would support recent predictions of triplet pairing.'
author:
- 'J. D. Wright'
- 'M. J. Pitcher'
- 'W. Trevelyan-Thomas'
- 'T. Lancaster'
- 'P. J. Baker'
- 'F. L. Pratt'
- 'S. J. Clarke'
- 'S. J. Blundell'
title: 'Magnetic fluctuations and spin freezing in non-superconducting LiFeAs derivatives'
---
Of all the known Fe-based superconductors, LiFeAs remains one of the most intriguing: unlike other pnictides, such as BaFe$_2$As$_2$ [@avci2012] and NaFeAs [@parker2010], LiFeAs is a superconductor in its stoichiometric form [@pitcher2008; @tapp2008] and any chemical substitution on the Fe-site (with Co or Ni for instance) causes a reduction in the transition temperature, $T_{\rm c}$ [@pitcher2010]. In contrast with other systems, no ordered magnetic phase or structural transition has yet been observed in LiFeAs, a fact that has provoked much debate given the tendency for band-structure calculations to predict similar magnetic ground states to those seen in other pnictides [@zhang2010; @li-liu2009; @li2009]. Applied pressure suppresses superconductivity, but does not induce magnetism [@mito2009].
Magnetic fluctuations, however, have been observed. Inelastic neutron scattering (INS) experiments uncover incommensurate fluctuations close to the wavevector $Q = (0.5, 0.5, 0)$ in both superconducting [@taylor2011; @qureshi2012] and non-superconducting (apparently Li-deficient) [@wang2011] forms of LiFeAs. This $Q$-vector is the same as that which gives rise to the striped antiferromagnetic ground state seen in other pnictides and which is predicted by the aforementioned theoretical studies [@zhang2010; @li-liu2009; @li2009]. This suggests that there is a degree of commonality between LiFeAs and other pnictides but that some crucial difference prevents it from ordering magnetically as they do. ARPES measurements of the Fermi surface [@borisenko2010] suggest that this difference may be the comparatively poor nesting between electron and hole pockets.
To account for this, and to fit LiFeAs into a unified scheme for the pnictides, a recent report [@wang2012] suggested that LiFeAs behaves analogously to the electronically overdoped versions of other systems. Common features in INS data support this idea, and it would explain why further electron doping, such as with Co or Ni, only reduces $T_{\rm c}$. However, it would also suggest that removing electrons (hole doping) may induce an ordered magnetic state, and no evidence for this has yet been reported. An alternative approach to account for the special status of LiFeAs is centred around the suggestion that it may exhibit triplet pairing [@brydon2011; @aperis2013], which has gathered some experimental support [@baek2012; @baek2013; @hanke2012].
In this paper we present studies of two series of non-superconducting LiFeAs derivatives, LiFe$_{1-x}$Ni$_x$As ($x = 0.1$ and $0.2$) and Li$_{1-y}$Fe$_{1+y}$As ($y = 0.01, 0.018$ and $0.04$), as well as a stoichiometric (superconducting) compound. Superconductivity is known to be suppressed when $x\geq 0.1$ and $y\geq 0.01$ [@pitcher2010]. For the Fe-rich series, the concentration of Fe on the Li site was obtained from a Rietveld refinement of the structure against both synchrotron x-ray and neutron powder diffraction data. All samples were found to be of very high purity: no extra phases were observed in the diffraction data and magnetisation data taken at room temperature ruled out the presence of any magnetic impurities the diffraction experiments may have missed. Details of these analyses, along with synthesis procedures, can be found in Ref. [@pitcher2010].
![Susceptibility data (zero-field cooled and field-cooled in $50\,{\rm Oe}$) and analysis for all samples. The data for a) the LiFe$_{1-x}$Ni$_{x}$As series and b) the Li$_{1-y}$Fe$_{1+y}$As series are presented, with the superconducting response of stoichiometric LiFeAs shown in the inset to (b). Panel (c) compares the inverse susceptibility for LiFe$_{0.8}$Ni$_{0.2}$As and Li$_{0.96}$Fe$_{1.04}$As; the former is demonstrates only antiferromagnetic behaviour \[see inset to (c)\], whereas correlations in the latter seem to cross over from antiferromagnetic (AFM) to ferromagnetic (FM) on warming. A comparison of the variation in moment size with both $x$ and $y$ is given in (d); the lines through the points are a guide to the eye. All values for calculated moment sizes are given in Table \[mu\_eff\].[]{data-label="squid"}](SQUID_full.pdf){width="\columnwidth"}
Figures \[squid\](a) and (b) show the magnetic susceptibility data for the Ni-doped and Fe-rich series respectively, and at first sight they seem similar: both series produce a paramagnetic response and a divergence of field-cooled and zero-field cooled signals at low temperatures, which may indicate a spin-glass transition. Significant differences are revealed, however, upon fitting to a Curie-Weiss dependence \[$\chi = C/(T-\theta)$\]. Shown most clearly in the inverse susceptibility plots of Fig.\[squid\](c), it is found that the Fe-rich Li$_{0.96}$Fe$_{1.04}$As sample exhibits both ferromagnetic ($\theta>0$) and antiferromagnetic ($\theta<0$) correlations, whereas only antiferromagnetic behaviour is found in the Ni-doped sample LiFe$_{0.8}$Ni$_{0.2}$As. Additionally, the size of these moments is generally larger across the Fe-rich series compared to the Ni-doped series, as Fig.\[squid\](d) shows. The values for all extracted moment sizes are given in Table \[mu\_eff\].
-------------- -------------- ------------------------------------- ------------- -------------------------------------
$\theta$(K) $\mu_{\rm eff}$($\mu_{\rm B}$/f.u.) $\theta$(K) $\mu_{\rm eff}$($\mu_{\rm B}$/f.u.)
Fe$_{1.01}$ $-21\,(1)$ $1.27\,(2)$ - -
Fe$_{1.018}$ $-32\,(2)$ $2.35\,(2)$ $30\,(2)$ $1.47\,(2)$
Fe$_{1.04}$ $-12\,(1)$ $3.77\,(4)$ $78\,(4)$ $1.75\,(5)$
Ni$_{0.1}$ $-9.3\,(5)$ $0.78\,(1)$ - -
Ni$_{0.2}$ $-4.7\,(10)$ $1.11\,(6)$ - -
-------------- -------------- ------------------------------------- ------------- -------------------------------------
: Comparison of effective moment sizes and nature of correlations observed in all samples studied.[]{data-label="mu_eff"}
In the Li$_{1-y}$Fe$_{1+y}$As series, one might expect these moments to be associated solely with the Fe ions sitting on the Li site, which may act as impurity spins like those in dilute alloys. However, such a scheme cannot account for several features of our data; namely, the sizes of the moments, the fact that these change with Fe concentration and their unusual correlations. Nickel is widely believed to act primarily as an electron donor [@pitcher2010; @parker2010], rather than an impurity scatterer, and as such is driving the destruction of superconductivity in LiFe$_{1-x}$Ni$_{x}$As by altering the band filling. We therefore suggest that, as for the Ni-doped series, the moments present in Li$_{1-y}$Fe$_{1+y}$As have an itinerant character.
To help illuminate the low-temperature phases in these systems, be they glassy or otherwise, we used muon-spin rotation ($\mu$SR). This technique uses the asymmetric emission of positrons during muon decay to track the depolarisation of muons implanted within a sample, thus probing the local field distribution. The details of such experiments can be found in Ref. [@blundell1999]. This technique has been useful in studying the rich variety of magnetic states in Fe-based superconductors [@klaussLaFeAsO; @bakerSrFeAsF; @aczelBaFe2As2; @jescheSrFe2As2], particularly when the moment sizes are too small to be detected by other techniques [@parker2010; @wright2012].
{width="18cm"}
Figure \[compare\] summarises the data for all samples, taken in zero applied field (ZF). For stoichiometric LiFeAs \[Fig.\[compare\](a)\] the data are best described by a single Gaussian Kubo-Toyabe function, suggesting that the muons experience a field solely due to randomly orientated, quasistatic nuclear dipole moments [@blundell1999]. The fit remains unaltered across the entire temperature range; Fig.\[compare\](a) shows the data taken at $1.5\,{\rm K}$ with the fit from $80\,{\rm K}$ superimposed. Together with the observation of antiferromagnetic fluctuations in INS experiments [@taylor2011; @qureshi2012; @wang2011], our data cast doubt on the triplet-pairing predictions in Refs. [@brydon2011; @aperis2013]. Zero-field (ZF-) $\mu$SR is known to be sensitive to the small magnetic fields induced under spontaneous time-reversal symmetry breaking (which may be due to triplet pairing) in Sr$_2$RuO$_4$ [@luke1998] and LaNiC$_2$ [@hillier2009]. On crossing $T_{\rm c}$ no such fields can be resolved here, so we find no evidence to support a triplet pairing hypothesis in LiFeAs.
We find that the $\mu$SR data for our two members of the LiFe$_{1-x}$Ni$_x$As series can also be described by temperature-independent Kubo-Toyabe functions which are almost identical to that which describes the LiFeAs data. The only difference is a slight increase of the second moment of the local magnetic field distribution, $\Delta$, observed as the Ni concentration, $x$, increases \[see Fig.\[compare\](a)([*Inset*]{})\]. This is a consequence of the Ni nuclear moment ($-0.75\,\mu_{\rm N}$) being significantly larger than that of Fe ($0.09\,\mu_{\rm N}$). These data suggest that the LiFe$_{1-x}$Ni$_x$As samples exhibit a dynamically fluctuating state at all measured temperatures and do not exhibit a spin-frozen state, despite some apparently glassy behaviour observed in the susceptibility data.
By contrast, an emergent magnetic phase is identified in the Li$_{1-y}$Fe$_{1+y}$As series: Figure \[compare\](b), (c) and (d) show the ZF data for the $y\,=\,0.01, 0.018$ and $0.04$ members of the Li$_{1-y}$Fe$_{1+y}$As series respectively. In all three samples we observe Kubo-Toyabe functions at high temperatures, but the relaxation becomes more exponential on cooling. These data are consistent with a freezing of the induced moments and the effect is clearly stronger for samples with higher Fe concentrations.
To analyse the $\mu$SR data, we assumed the existence of two distinct muon sites in the unit cell, as found in related materials [@maeter2009]. In the ordered phases of NaFeAs, one observes two frequencies related by a constant factor of $\sim 10$ which indicates the relative coupling strengths between each of these two muon sites and the ordered Fe moments. For our isostructural Li$_{1-y}$Fe$_{1+y}$As series, we therefore fitted our spectra to the two-component function
$$A(t) = G_{\rm KT}(\Delta, t)\,[\alpha\,{\rm e}^{-\lambda t}+(1-\alpha)\,{\rm e}^{-\lambda Rt}],
\label{asym}$$
where the relaxation rate ratio, $R$, was fixed throughout the fitting. The fractional amplitude of each contribution, $\alpha$, did not vary with temperature but did scale with $y$: going from $0.35$ for $y\,=\,0.01$ to $0.9$ for $y\,=\,0.04$ \[this is shown in Fig.\[zf\_full\](c)\]. The free parameter, $\lambda$, is the relaxation rate describing fluctuations in the local magnetic field caused by the dynamics of the electronic moments. Both muon sites will experience relaxation due to nuclear moments and so a Gaussian Kubo-Toyabe function, with a fixed width ($\Delta$), was included as an overall multiplicative component.
![(a) The evolution of $\lambda(T)$ for the Li$_{1-y}$Fe$_{1+y}$As series. The spin-freezing temperature, $T_{\rm f}$, is defined as the onset of the power law increase in $\lambda(T)$. [*Inset:*]{} spectra at $10\,{\rm K}$ in both zero-field and a longitudinal field (LF) of $1000\,{\rm G}$. The weak relaxation still present in the LF spectrum indicates dynamic behaviour, pointing to a spin-freezing picture, as opposed to static local order (see text). (b) The values of the $T_{\rm f}$ extracted from the behaviour of $\lambda(T)$. (c) The variation of the fast relaxing amplitude, $\alpha$ (defined in Eq.\[asym\]), with Fe concentration. \[zf\_full\]](zf_full.pdf){width="\columnwidth"}
Figure \[zf\_full\](a) plots the temperature variation of the larger relaxation rate $\lambda$ for all samples, along with a comparison of spectra taken in zero-field and a longitudinal field (LF) of $1000\,{\rm G}$ for Li$_{0.96}$Fe$_{1.04}$As at $10\,{\rm K}$. The weak relaxation observed in the LF spectra suggests that these moments are dynamically fluctuating, and that the increase in $\lambda$ seen at low temperatures corresponds to a slowing-down of these fluctuations. The best fit lines in Fig.\[zf\_full\](a) assume that both power-law and temperature-independent relaxation processes contribute to $\lambda(T)$ in quadrature. We can define the spin freezing temperature, $T_{\rm f}$, for each sample as the onset of the power-law increase in $\lambda(T)$ (defined as the temperature at which the two contributions to $\lambda(T)$ are equal), as shown by the arrows in Fig.\[zf\_full\](a). These values are plotted in Fig.\[zf\_full\](b) and are used to compose the phase diagram in Fig.\[phase\].
It was difficult to resolve the second (smaller) relaxation rate, so in fact the plots of $\lambda(T)$ in Fig.\[zf\_full\](a) were obtained with the ratio $R$ set to zero. This is unsurprising if we can assume the observed dynamics operate in the fast-fluctation limit, such that $\lambda = 2\Delta_{\rm site}^2/\nu$; where $\Delta_{\rm site}/\gamma_{\mu}$ is the rms value of the local field at a given muon site, and $\nu$ is the fluctuation rate. Because the strength of the dipolar coupling to moments on the Fe site probably differs between the two muon sites by a factor of $\sim 10$ (based on the frequencies seen in NaFeAs), we would expect the relative relaxation rates to differ by a factor of $\sim 100$. This explains our inability to fit the smaller relaxation rate explicitly.
An alternative explanation would be to assume a single relaxation rate and interpret Eq.\[asym\] as describing mesoscopic phase separation where a fraction of muons sit in isolated regions of local order. However, the dynamic behaviour observed in LF spectra, the glassy behaviour seen in the susceptibility data and the overwhelming evidence for a two-site model from isostructural systems leads us to suggest the picture outlined above is the most plausible explanation for what we observe.
![Phase diagram for the Li$_{1-y}$Fe$_{1+y}$As and LiFe$_{1-x}$Ni$_x$As series, showing regions of superconductivity (SC), spin freezing (SF) as well as (anti)ferromagnetic fluctuations. \[phase\]](phaseplot.pdf){width="\columnwidth"}
In conclusion, we have demonstrated that suppressing superconductivity by subsituting Fe onto the Li site induces fluctuating, correlated, itinerant moments that freeze at low temperatures. On warming, the SQUID data show that magnetic correlations are most likely antiferromagnetic in nature, but there appears to be a cross-over to ferromagnetic correlations for samples with the largest Fe concentration (see Fig.\[phase\]). The changing size of these moments and the strength of their correlation cannot be explained as simply being the result of incorporating dilute Fe moments onto the Li site, and demonstrates emergent itinerant magnetic behaviour. The effects of an induced moment were proposed in relation to an anomalous result from earlier work on a supposedly Li-deficient sample [@pratt2009]; we now believe that this sample may have also contained a small amount of Fe on the Li site ($y\,<\,0.01$) which could result in a similar spin freezing effect. No evidence of any such behaviour is observed if superconductivity is suppressed by Ni subsitution onto the Fe site, so this state is not associated with a general suppression of superconductivity but is unique to samples with Fe substituted for Li. Despite a detailed search, we find no evidence of spontaneous fields below $T_{\rm c}$ in stoichiometric LiFeAs and thus no evidence of triplet pairing.
Part of this work was carried out using the GPS Spectrometer at the Paul Scherrer Institut, Villigen, CH. We acknowledge the financial support of EPSRC (UK).
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|
---
address: |
Nikhef, Science Park 105,\
1098 XG Amsterdam, Netherlands
author:
- |
M. MULDER\
on behalf of the LHCb Collaboration
title: ' THE BRANCHING FRACTION AND EFFECTIVE LIFETIME OF [${B^0_{(s)}\to \mu^+\mu^-}$ ]{}AT LHCb WITH RUN 1 AND RUN 2 DATA'
---
Introduction
============
In the SM, Flavour Changing Neutral Currents (FCNCs) occur at loop level and are suppressed by the GIM mechanism, and sometimes helicity suppression. As New Physics (NP) is not necessarily suppressed, FCNCs probe physics at energies beyond the LHC centre-of-mass energy.
One such FCNC is the transition. Several theory groups have performed global fits to observables in the Effective Field Theory (EFT) framework, and find that the data deviate by $\sim 4$ standard deviations with respect to the SM [@Altmannshofer:2014rta; @Hurth:2016fbr; @Descotes-Genon:2015uva].
One example of a transition is the decay of a [$B^0_{s}$]{}meson into two muons ([${B^0_{s}\to \mu^+\mu^-}$]{}). For [${B^0_{(s)}\to \mu^+\mu^-}$ ]{}decays, the decay amplitude can be written as A([${B^0_{(s)}\to \mu^+\mu^-}$ ]{}) = = \_[i]{} , where $G_F$ is the Fermi constant, and are CKM elements, $\Cppi$ are perturbative Wilson coefficients and $\Oppepi$ are non-perturbative Wilson operators. These operators describe axial vector (10 ), scalar () and pseudo-scalar () contributions respectively, and a prime indicates a right-handed contribution. In the SM, only 10 contributes. Interestingly, contributions from 10 are helicity suppressed, while (pseudo-)scalar contributions are not. Therefore, [${B^0_{(s)}\to \mu^+\mu^-}$ ]{}observables are very sensitive to (pseudo-)scalar NP.
As [${B^0_{(s)}\to \mu^+\mu^-}$ ]{}decays are purely leptonic, the hadronic part of the amplitude reduces to the decay constant, which is well determined using lattice QCD [@Aoki:2016frl]. The branching fractions of [${B^0\to \mu^+\mu^-}$]{}and [${B^0_{s}\to \mu^+\mu^-}$]{}are consequently well predicted in the SM [@Bobeth:2013uxa; @Fleischer:2017ltw]:
[$\mathcal{B}({B^0_{s}\to \mu^+\mu^-})$]{}&=& (3.57 0.18) 10\^[-9]{}\
[$\mathcal{B}({B^0\to \mu^+\mu^-})$]{}&=& (1.06 0.06) 10\^[-10]{}
A second observable in the [${B^0_{(s)}\to \mu^+\mu^-}$ ]{}system is the effective lifetime of [${B^0_{s}\to \mu^+\mu^-}$]{} [@DeBruyn:2012wk]. [$B^0_{(s)}$]{}mesons mix with their anti-particles and form mass (and lifetime) eigenstates. The mass-eigenstate rate asymmetry is given by
\_=
In the SM, $\mathcal{A}_{\Delta\Gamma}=1$, which means that only the heavy [$B^0_{(s)}$]{}eigenstate decays to two muons. For [$B^0_{s}$]{}mesons, the lifetime difference between their two eigenstates is measured to be $\Delta\Gamma/\Gamma = 0.12$ [^1] By measuring the effective lifetime from the untagged decay distribution, the mass-eigenstate rate asymmetry is probed. The effective lifetime of [${B^0_{s}\to \mu^+\mu^-}$]{}is an independent and complementary observable to the branching fraction.
Here, the new LHCb analysis of [${B^0_{(s)}\to \mu^+\mu^-}$ ]{}observables is presented [@Aaij:2017vad]. It is based on data corresponding to an integrated luminosity of 1of $pp$ collisions at a centre-of-mass energy of = 7[$\mathrm{\,Te\kern -0.1em V}$]{}, 2at = 8[$\mathrm{\,Te\kern -0.1em V}$]{}and 1.4at = 13[$\mathrm{\,Te\kern -0.1em V}$]{}. The first two datasets are referred to as Run 1, the latter as Run 2.
In Section \[sec:BF\], the branching fraction measurement is presented. In Section \[sec:tau\], the effective lifetime measurement is presented. As there is a large overlap in selection and analysis strategy, only differences with the branching fraction measurement are mentioned in the latter Section.
[${B^0_{s}\to \mu^+\mu^-}$]{}and [${B^0\to \mu^+\mu^-}$]{}branching fraction measurement {#sec:BF}
========================================================================================
Backgrounds and selection {#sec:bkg_sel}
-------------------------
The analysis strategy is to search for muon pairs with opposite charges, $m_{\mu^+\mu^-} \in [4900,6000] {\ensuremath{\mathrm{\,Me\kern -0.1em V}}\xspace}$ [^2] and a well reconstructed, clearly displaced decay vertex.
Four background categories exist for the branching fraction measurement. First, decays, where both muons combine to fake a signal. This is called combinatorial background, and all other backgrounds are hereafter referred to as peaking backgrounds. Second, [$B^0_{(s)}\to h^+h^{(')-}$]{}decays, where $h \in [K,\pi]$ and both hadrons are misidentified as muons. Third, [$\ensuremath{B^0}\to \pi^- \mu^+
\nu_\mu$]{}, [$\ensuremath{B^0_s}\to K^- \mu^+
\nu_\mu$]{}, and , where one hadron is misidentified as a muon and the neutrino is not reconstructed. Fourth, [$B^+_c\to J/\psi(\to \mu^+\mu^-)\mu^+\nu_{\mu}$]{}and decays, where two real muons are combined and one particle is not reconstructed.
To reject both combinatorial and peaking backgrounds, two multivariate classifiers are used.
For the separation of [${B^0_{(s)}\to \mu^+\mu^-}$ ]{}from combinatorial background, a Boosted Decision Tree (BDT) is trained. This BDT is used to divide the sample based on signal purity. The main discriminating variables in this BDT are based on track isolation, as decays have extra tracks close to the muon tracks.
To reject peaking backgrounds, neural networks that discriminate between muons and hadrons are trained. The neural network selection is optimised for the [$\mathcal{B}({B^0\to \mu^+\mu^-})$]{}measurement, as due to its lower mass it is more affected by peaking backgrounds.
Simulation is used for each peaking background to estimate the number of events after selection, as well as the mass and BDT shape. The main backgrounds ([$B^0_{(s)}\to h^+h^{(')-}$]{}, [$\ensuremath{B^0}\to \pi^- \mu^+
\nu_\mu$]{}, [$\ensuremath{B^0_s}\to K^- \mu^+
\nu_\mu$]{}) are also controlled using a fit to data with the $h\mu$ mass hypothesis.
Signal calibration and normalisation {#sec:sig_calib}
------------------------------------
To convert a yield after selection into a branching fraction, other B decays with known branching fractions are used for normalisation. In addition, the mass and BDT shape of the signal have to be calibrated.
The normalisation is performed with [$B^+\to J/\psi(\to \mu^+\mu^-)K^+$]{}and [${\ensuremath{B^0}\xspace}\to K^+\pi^-$]{}decays. The two muons in the final state for the [$B^+\to J/\psi(\to \mu^+\mu^-)K^+$]{}decay are detected similarly to the muons from [${B^0_{(s)}\to \mu^+\mu^-}$ ]{}. The efficiency to detect the extra kaon track in the final state is corrected for using simulation. For [${\ensuremath{B^0}\xspace}\to K^+\pi^-$]{}, the kinematics are very similar to [${B^0_{(s)}\to \mu^+\mu^-}$ ]{}, which means the same BDT can be applied. However, the trigger and PID selection for hadrons and muons are very different, and are corrected for using a data-driven approach.
For the BDT calibration, fits to [${\ensuremath{B^0}\xspace}\to K^+\pi^-$]{}decays per BDT bin are used to determine the [${B^0_{(s)}\to \mu^+\mu^-}$ ]{}BDT distributions. The same corrections as for the normalisation are applied, and the BDT distribution is found to be consistent with simulation.
![ (left) Mass distribution of selected [${B^0_{(s)}\to \mu^+\mu^-}$ ]{}candidates in the four most sensitive BDT bins. The result of the fit is overlaid. (right) The 2D confidence interval in ${\ensuremath{\mathcal{B}({B^0_{s}\to \mu^+\mu^-})}\xspace},{\ensuremath{\mathcal{B}({B^0\to \mu^+\mu^-})}\xspace}$ from this measurement. []{data-label="fig:bf_fit"}](Fig1){width="1.0\linewidth"}
![ (left) Mass distribution of selected [${B^0_{(s)}\to \mu^+\mu^-}$ ]{}candidates in the four most sensitive BDT bins. The result of the fit is overlaid. (right) The 2D confidence interval in ${\ensuremath{\mathcal{B}({B^0_{s}\to \mu^+\mu^-})}\xspace},{\ensuremath{\mathcal{B}({B^0\to \mu^+\mu^-})}\xspace}$ from this measurement. []{data-label="fig:bf_fit"}](Fig22){width="1.0\linewidth"}
Branching fraction fit and results
----------------------------------
To measure both branching fractions, a simultaneous maximum likelihood fit for [$\mathcal{B}({B^0_{s}\to \mu^+\mu^-})$]{}and [$\mathcal{B}({B^0\to \mu^+\mu^-})$]{}is performed to $m_{\mu^+\mu^-}$ in bins of BDT per dataset (Run 1 and Run 2). The only free nuisance parameters are the combinatorial background shape parameters and yields. The shape is the same for all BDT bins in a dataset, the yields are free for each BDT bin.
From the fit, the [${B^0_{s}\to \mu^+\mu^-}$]{}decay is observed with a statistical significance of 7.8 standard deviations, and its branching fraction is measured to be ${\ensuremath{\mathcal{B}({B^0_{s}\to \mu^+\mu^-})}\xspace}= (3.0 \pm 0.6 ^{+0.3}_{-0.2}) \times 10^{-9}$ , where the first uncertainty is statistical and the second systematic. The statistical uncertainty is obtained by repeating the fit with all nuisance parameters fixed. No significant excess of [${B^0\to \mu^+\mu^-}$]{}decays is observed, and a 95% confidence level upper limit is determined, ${\ensuremath{\mathcal{B}({B^0\to \mu^+\mu^-})}\xspace}< 3.4 \times 10^{-10}$.
Effective lifetime measurement of [${B^0_{s}\to \mu^+\mu^-}$]{} {#sec:tau}
================================================================
In contrast to the branching fraction, the effective lifetime is measured for the [${B^0_{s}\to \mu^+\mu^-}$]{}decay only. Therefore, the selection and fitting procedure are optimised independently.
First, events with $m_{\mu^+\mu^-} < 5320 {\ensuremath{\mathrm{\,Me\kern -0.1em V}}\xspace}$, and thus all peaking backgrounds, as well as [${B^0\to \mu^+\mu^-}$]{}events, are cut away. Second, the PID selection is loosened, as the mass cut makes the measurement less sensitive to peaking backgrounds. Third, a cut is applied on the BDT, instead of using the BDT to divide the sample in bins.
With the method [@Pivk:2004ty], a mass fit is used to background-subtract the decay time distribution. The mass fit contains just two shapes ([${B^0_{s}\to \mu^+\mu^-}$]{}and combinatorial background). The background-subtracted decay time distribution is fitted with an exponential, convoluted with an acceptance function.
![(left) Mass distribution of [${B^0_{s}\to \mu^+\mu^-}$]{}candidates. The result of the fit is overlaid. (right) Background-subtracted decay-time distribution with the fit result superimposed.[]{data-label="fig:tau_eff"}](Fig2top){width="1.0\linewidth"}
![(left) Mass distribution of [${B^0_{s}\to \mu^+\mu^-}$]{}candidates. The result of the fit is overlaid. (right) Background-subtracted decay-time distribution with the fit result superimposed.[]{data-label="fig:tau_eff"}](Fig2bot){width="1.0\linewidth"}
The effective lifetime is measured for the first time and found to be $\tau ({\ensuremath{{B^0_{s}\to \mu^+\mu^-}}\xspace}) = 2.04 \pm 0.44 \pm 0.05$ ps. This measurement is consistent with the $\mathcal{A}_{\Delta\Gamma}=+1 (-1)$ hypothesis at 1.0 (1.4) standard deviations. The current experimental precision does not set a strong constraint, but shows the potential LHCb will have to constrain NP using the effective lifetime.
Conclusions
===========
The [${B^0_{s}\to \mu^+\mu^-}$]{}decay is observed with a significance of 7.8 standard deviations, and its branching fraction is measured to be ${\ensuremath{\mathcal{B}({B^0_{s}\to \mu^+\mu^-})}\xspace}= (3.0 \pm 0.6 ^{+0.3}_{-0.2}) \times 10^{-9}$ , where the first uncertainty is statistical and the second systematic. In addition, the first measurement of the [${B^0_{s}\to \mu^+\mu^-}$]{}effective lifetime is performed: $\tau({\ensuremath{{B^0_{s}\to \mu^+\mu^-}}\xspace}) = 2.04 \pm 0.44 \pm 0.05$ ps. No significant excess of [${B^0\to \mu^+\mu^-}$]{}decays is observed, and a 95% confidence level upper limit is determined, ${\ensuremath{\mathcal{B}({B^0\to \mu^+\mu^-})}\xspace}< 3.4 \times 10^{-10}$. All results are consistent with the SM and constrain New Physics in processes. Already, these results have sparked interesting discussions [@Fleischer:2017ltw; @Altmannshofer:2017wqy; @Bobeth:2017xry; @Chiang:2017etj].
Acknowledgments {#acknowledgments .unnumbered}
===============
The author wishes to acknowledge the financial support from NWO and thank the organisers of Moriond EW 2017 for an interesting and captivating conference.
References {#references .unnumbered}
==========
[10]{}
W. Altmannshofer and D. Straub, [*Eur. Phys. J.*]{} [**[C 75]{}**]{}, 382, (2015).
T. Hurth [*[et al.]{}*]{}, [*Nucl. Phys.*]{} [**[B 909]{}**]{}, 737-777, (2016).
S. Descotes-Genon [*[et al.]{}*]{}, [*JHEP*]{} [**[06]{}**]{}, 092, (2016).
S. Aoki [*[et al.]{}*]{}, [*Eur. Phys. J.*]{} [**[C 77]{}**]{}, 112, (2017).
C. Bobeth [*[et al.]{}*]{}, [**[112]{}**]{}, 101801, (2014).
R. Fleischer [*[et al.]{}*]{}, arXiv:1703.10160 (2017).
K. De Bruyn [*[et al.]{}*]{}, [**[109]{}**]{}, 041801, (2012).
R. Aaij [*[et al.]{}*]{}, arXiv:1703.05747 (2017).
M. Pivk and F. Le Diberder, , 356-369,(2005).
W. Altmannshofer [*[et al.]{}*]{}, arXiv:1702.05498 (2017).
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C. Chiang [*[et al.]{}*]{}, arXiv:1703.06289 (2017).
[^1]: For [$B^0$]{}mesons, no lifetime difference has been observed. It is expected to be $\mathcal{O}(10^{-3})$. The effective lifetime of the [${B^0\to \mu^+\mu^-}$]{}decay is not measured due to the small lifetime difference and its small branching fraction.
[^2]: For reference, the [$B^0_{s}$]{}mass is 5367 [${\mathrm{\,Me\kern -0.1em V\!/}c^2}$]{}and the [$B^0$]{}mass is 5280 [${\mathrm{\,Me\kern -0.1em V\!/}c^2}$]{}.
|
---
abstract: 'Re-identification of individual animals in images can be ambiguous due to subtle variations in body markings between different individuals and no constraints on the poses of animals in the wild. Person re-identification is a similar task and it has been approached with a deep convolutional neural network (CNN) that learns discriminative embeddings for images of people. However, learning discriminative features for an individual animal is more challenging than for a person’s appearance due to the relatively small size of ecological datasets compared to labelled datasets of person’s identities. We propose to improve embedding learning by exploiting body landmarks information explicitly. Body landmarks are provided to the input of a CNN as confidence heatmaps that can be obtained from a separate body landmark predictor. The model is encouraged to use heatmaps by learning an auxiliary task of reconstructing input heatmaps. Body landmarks guide a feature extraction network to learn the representation of a distinctive pattern and its position on the body. We evaluate the proposed method on a large synthetic dataset and a small real dataset. Our method outperforms the same model without body landmarks input by 26% and 18% on the synthetic and the real datasets respectively. The method is robust to noise in input coordinates and can tolerate an error in coordinates up to 10% of the image size.'
author:
- 'Olga Moskvyak, Frederic Maire, Feras Dayoub and Mahsa Baktashmotlagh [^1][^2][^3]'
bibliography:
- 'egbib.bib'
title: 'Learning landmark guided embeddings for animal re-identification'
---
Introduction
============
Animal re-identification in images is an instance level recognition and retrieval problem which aims to distinguish between individual animals and find matching examples in an image database. Individual animals can be told apart by subtle variations in natural markings on their body such as belly patterns on manta rays, stripes on tigers and zebras.
Automatic re-identification of animals in photos is of high importance for wildlife monitoring and conservation because it is less time consuming than manual visual inspection and more efficient than collecting biology samples or attaching and tracking microchips [@into-photo-id-rays-sharks].
![(a) The spot pattern on these two different manta rays is the same (consists of one black dot). (b) Localised images of the spot pattern are ambiguous. (c) To distinguish between individuals like these we propose to exploit body landmark coordinates (e.g., eyes, gills, a tail) in the re-identification system. Photo credit: David Biddulph, John Gransbury.[]{data-label="fig:intro_image"}](intro_image.pdf){width="\linewidth"}
{width="0.85\linewidth"}
The task is similar to person recognition that has been approached with deep convolutional neural networks [@person-review]. A network learns embeddings for images of people’s appearances in such a way that the distance between embeddings of the same person is smaller than the distance between embeddings of different people. However, visual animal re-identification is mainly based on body markings which are more ambiguous than a person’s appearance because of the similarities among different individuals. For example, different manta rays can have a very similar spot pattern but located at different positions on the belly. Figure \[fig:intro\_image\] shows two manta rays with only one black dot on the belly and the only difference is the location of the spot with respect to the landmarks (e.g., eyes, a base of the tail and gills).
There are limitations in transferring face and person re-identification methods to images of animals:
- faces are usually normalised to an upright frontal pose thanks to robust methods to detect facial landmarks and body postures are aligned vertically;
- warping to a canonical position or alignment is not always possible for animals due to the sensitivity of these methods to errors in coordinates of body landmarks;
- wildlife datasets have limited data compared to large public datasets for face and person re-identification so there is less chance to learn the relation between the body landmarks and unique markings from the data itself.
Previous work on the manta ray re-identification system [@manta-reid] uses cropped images of spot patterns to focus the model’s attention on the pattern itself and avoid distraction from the background. However, the cropped patch of a sport pattern loses information about its relative position on the body so it is not likely to correctly identify individuals with similar patterns that differ only in a position like in Figure \[fig:intro\_image\].
We build on a strong model for person re-identification [@person-reid-baseline] and propose to improve embedding learning for animal re-identification by adding locations of body landmarks to the model input. The new model explicitly receives information about the position of distinctive features with respect to the body. The motivation of using body landmarks is the scarcity of annotated datasets with animal identities compared to large labelled datasets for person re-identification. Identification based only on the pattern itself without the knowledge about the position of a specific mark can be error prone. We favor heatmaps over exact coordinates to encode the estimated body landmark location because heatmaps can represent uncertainty. The key contributions of this paper are:
- a novel method to exploit body landmark locations to improve the performance of re-identification system;
- a novel heatmap augmentation method to train the model to handle missing or not visible landmarks;
- robustness to uncertainty in body landmark coordinates up to 10% of the image size.
Related work
============
{width="0.9\linewidth"}
There are multiple approaches to re-identification and some of them use only pixel intensities and some leverage additional information such as the semantic structure of the object (e.g., body parts of a person). We discuss re-identification methods that include some degree of pose or body landmark information. Pose information can be exploited to align the object of interest to a standard pose or crop patches from the image to obtain local features.
The body and face alignment based on keypoints is used to eliminate pose variance and improve recognition performance. Zheng et al. [@person-posebox] introduce the PoseBox structure to align pedestrians to a standard pose. The alignment is used extensively for face recognition [@parkhi2015deep; @schroff2015facenet; @sun2014deep]. Normalizing the head orientation of right whales improves the re-identification performance [@whale-re-id]. However, accurate body landmark information is hard to obtain and alignment methods are sensitive to precise coordinates [@multy-view-geometry]. It is not feasible to transfer alignment methods directly from person to animal re-identification. Landmark coordinates are used to extract local features from various patches cropped from the image. Recognition and fine-grained classification methods use these local features to complement a global representation. Guo and Farrell [@birds-pose] construct object representation as the concatenation of hierarchical pose-aligned regions features extracted from patches around pairs of body landmarks. Tiger re-identification has been improved by concatenating global features extracted from the whole image with local features from limb’s patches [@tiger-part-pose-guided]. Su et. al [@person-posedriven] explicitly leverage human body part cues to detect and normalize body parts to extract local features and combine them in a pose driven feature weighting subnetwork.
Pose information has also been used to enhance re-identification by generating new data samples in a pose-transferable person re-id framework [@person-transferrable-reid]. However, training image generation models requires a large amount of data so it cannot be transferred directly to smaller datasets of animal identities.
Several works include pose information to guide feature extraction. Sarfraz et. al [@pose-guided-person] improve person re-identification by incorporating both fine and coarse pose information into learning discriminative embeddings. Fine pose information is confidence maps from off-the-shelf body landmark predictor. Coarse pose information is the quantization (‘front’, ‘back’, ‘side’) of a person’s orientation to the camera. Liu et. al [@tiger-pose-guided] simplifies the tiger body pose into two categories according to the heading direction of the tigers to reduce pose variations. However, these approaches are not transferable to other objects due to coarse pose labels are specific to the task. In this paper, we introduce a generic method of exploiting body landmark information to improve learning of discriminative embeddings.
Learning landmark guided embeddings
===================================
A pose of an animal’s body with respect to the camera greatly affects the appearance of natural markings and the visibility of body landmarks in the photo. It is hard to obtain accurate locations of body landmarks because images are taken in the wild environment with an unknown pose of the animal in front of the camera, complex natural backgrounds and changing lighting conditions. Information about the pose has the potential to improve re-identification performance.
Baseline re-identification model
--------------------------------
As a baseline re-identification model, we use the second best model developed for person re-identification [@person-reid-baseline] that is generic enough to be transferred from people to animals. The state-of-the-art for person re-identification requires spatial-temporal information [@person-reid-sota] and cannot be transferred to our task. The backbone of the baseline model is ResNet50 [@resnet] that is initialized with pre-trained parameters on ImageNet. The model outputs ReID features $f$ and ID prediction logits $p$. ReID features $f$ are used to calculate a triplet loss [@triplet-loss] and a center loss [@center-loss]. Triplet loss pulls embeddings of images from the same individual closer together while pushing embeddings of images of different individuals above a specified margin. Center loss penalizes the distance between embeddings and their corresponding class centers where each individual is a class. ID prediction logits $p$ are used to calculated a smoothed cross entropy loss [@smooth-ce] over training classes to facilitate learning of discriminative features and are discarded at inference. The training process and all hyperparameters are inherited from the original work [@person-reid-baseline]. The baseline model is optimized with a weighted combination of three losses: the smoothed cross-entropy $L_{\textnormal{ID}}$ over training classes, the triplet loss $L_{\textnormal{Triplet}}$ and the center loss $L_{\textnormal{Center}}$:
$$L_{\textnormal{ReId}} = L_{\textnormal{ID}} + L_{\textnormal{Triplet}} + \beta L_{\textnormal{Center}}$$
where $\beta = 0.0005$ as in [@person-reid-baseline].
Landmark aware re-identification model
--------------------------------------
We add the body landmark information to the model input by concatenating $k$ extra channels with three RGB image channels ($k$ is a number of landmarks). Each channel is a grayscale heatmap representing the likelihood of a landmark location. Figure \[fig:manta\_heatmaps\] shows two images of manta rays and corresponding heatmaps. These additional channels guide feature extraction to learn embeddings that are aware of the location of distinctive features with respect to body parts. Information about landmarks can be obtained from another model that predicts landmarks based on the image (this task is out of the scope of the current work). Landmarks can also be annotated manually.
The model is encouraged to use landmark information by learning the auxiliary reconstruction task of input heatmaps from embeddings, see Figure \[fig:model\_architecture\]. We also experimented with the heatmap reconstruction block branched off after the third dimensionality reduction step and the results were similar to the reconstruction from the final features. We call this model a *Landmark-Id* model.
The *Landmark-Id* model is optimised with the following loss:
$$L = L_{\textnormal{ReId}} + \alpha L_{\textnormal{HR}}$$
where heatmap reconstruction loss $L_{\textnormal{HR}}$ is a binary cross-entropy. We experimented with $\alpha$ equal to 0.1, 1 and 10 and observed no difference in accuracy so we set $\alpha = 1$.
The *Landmark-Id* model is trained in two stages. At the first stage only randomly initialised weights in the first layer and in the final classification layer are trained while all other weights remain fixed. Once these layers are adapted to the rest of the network, the whole network is fine-tuned. The parameters in the first layer of ResNet50 are initialized randomly because the number of input channels differs from the number of channels in ImageNet due to the additional heatmap input. The rest of the parameters in ResNet50 model are initialized with ImageNet pretrained weights. Heatmap reconstruction block is not trained at this stage.
At the second stage, the heatmap reconstruction branch is added with randomly initialised weights. Only the heatmap reconstruction block is trained for the first ten epochs to tune random weights. Then the whole model is fine-tuned with a ten times smaller learning rate than in the first stage.
Heatmap augmentation
--------------------
![Noisy landmark augmentation (NLA) on heatmaps with different levels of uncertainty about landmark locations. The true coordinate is located inside the bright blob but not necessary in the middle.[]{data-label="fig:heatmaps_size_noise"}](heatmaps_size_noise.pdf){width="\linewidth"}
We introduce two augmentation techniques for heatmaps to improve the generalization ability of the *Landmark-Id*: noisy landmark augmentation (NLA) and missing landmark augmentation (MLA). Locations of body landmarks cannot always be annotated correctly especially when these are obtained from an automated landmark detection method. Due to large variations in animal poses some landmarks may not be visible in the image. NLA and MLA address these two problems.
NLA randomly shifts the blob in each heatmap (by default the center of the blob is the landmark) by a number of pixels less or equal than the radius of the blob, see Figure \[fig:heatmaps\_size\_noise\]. This way the landmark location is still contained within a blob but not always in the middle.
MLA has two parameters: a minimum number $M$ of visible landmarks (specific to the dataset) and a probability $p_{\textnormal{mla}}$. If there are more than $M$ landmarks visible in the image, than some of them may be set to missing with probability $p_{\textnormal{mla}}$. In practice, a missing landmark means that the corresponding heatmap is set to all zeros. The motivation for this augmentation is imbalanced data when there are not enough examples for the model to learn to reconstruct black heatmaps for not visible landmarks. We list the hyperparameters used for MLA in the Experiments section.
Experiments
===========
Datasets
--------
### Synthetic dataset
We verify ideas on a synthetic dataset first as it gives the ability to control the number and variety of examples. The design of synthetic images is inspired by manta rays belly patterns but does not aim to replicate it. Consider a collection of seed patterns $ \mathcal{P} = \{P_1, \ldots, P_n\}$ where each pattern $ P_{i} $ is a unique pattern of black filled ellipses on a white background inside at triangle area in the center as illustrated in Figure \[fig:synthetic\_patterns\] (first column). The corners of the triangle play the role of body landmarks. The image itself does not have any information about landmark locations. Landmark coordinates are recorded in a separate array.
![Example of three identities from the synthetic dataset. Each row shows a seed pattern and three generated examples for one identity. Coloured points represent locations of three landmarks and are plotted over images for illustration only and do not appear on images in the generated dataset.[]{data-label="fig:synthetic_patterns"}](synthetic_patterns.pdf){width="\linewidth"}
Seed patterns represent a canonical view from a camera placed directly in front of it. When the camera moves, the projection of the pattern on the camera plane will be related to the canonical view by a homography. We call an *identity* $I_i$ a unique pattern of ellipses where examples belonging to this identity are generated by applying random projective transformations to the canonical pattern $P_i$ and adding a random background (see examples in Figure \[fig:synthetic\_patterns\]). To randomise textures of a background and a pattern we use patches from images showing underwater scenes without any salient objects [@unsplash]. The pixel intensities in a background image are rescaled to be lighter than pixel intensities of a pattern texture to avoid merging of the pattern with the background. Finally, images are converted to grayscale and Gaussian noise is added. Landmark coordinates are warped the same way as a pattern and are recorded in a corresponding array.
The dataset consists of 3 subsets: a train, a gallery and a query. Each subset has images for 750 identities with the resolution $128\times128$. The gallery and the query subsets share the same identities while having disjoint identities with the training set. The training and the gallery sets have only 3 examples for each individual to simulate a limited data scenario. More examples per individual would make the re-identification task easier. The query set has 5 images per individual.
### Dataset of real images
As a real dataset, we use images of manta rays collected by Project Manta (a multidisciplinary research program based at the University of Queensland, Brisbane, Australia). The dataset is challenging as images are captured underwater at oblique angles in different illumination conditions and with small occlusions (fish, water bubbles). Each image has been manually annotated with five most distinctive landmarks: right eye, left eye, outer corner of the fifth right gill, outer corner of the fifth left gill, tail (see examples in Figure \[fig:manta\_identities\]). We select eyes and a tail as landmarks as these are easy to identify in images. Bottom gill slits on both sides have distal black marks that are salient and visible most times [@biology-manta]. Only around half of the images have all 5 landmarks visible, 30% of the images have 4 visible landmarks and the rest have 3 and less visible landmarks.
![Example of three identities from a manta rays dataset (real images). Each row shows four examples of one identity. Coloured points are landmark locations and are plotted over images for illustration only and do not appear on images in the dataset.[]{data-label="fig:manta_identities"}](manta_identities.pdf){width="\linewidth"}
The training set has 110 identities with 1422 images in total. The test set consists of 18 identities (different from the training) with 321 examples in total. Images are taken by a large number of researchers and photographers so we assume that each image comes from a different camera. Due to the limited size of the data we use one test set instead of a separate gallery and query sets. The gallery set is created by combining the training set with two random images of each individual from the test set. The rest of the images from the test set are used as a query set. This way each query image has two matching examples in the gallery.
Landmarks input as heatmaps
---------------------------
Landmark coordinates are converted to heatmaps with one grayscale channel per a landmark, see Figure \[fig:manta\_heatmaps\]. The heatmap is created by running a Gaussian filter over a white disk on a black background to smooth the edges. The center of the heatmap has equal intensity so there is no additional clue where the landmark is located. Heatmaps are used as an input to the model instead of exact coordinates to accommodate different levels of uncertainty in landmark locations. If the landmark is not visible the heatmap is all zeros.
Heatmaps for the synthetic dataset are generated with three settings for the radius of the blob (5%, 10% and 20% of the image size) to evaluate the sensitivity of the model to the uncertainty in landmark locations. Heatmaps for the manta ray dataset have the radius 5% of the image size.
Model architecture
------------------
We use ResNet50 model as a core feature extractor with the output feature maps pooled globally to produce a vector of size 2048. Then one fully connected layer is used to reduce the dimension to 256.
The heatmap reconstruction block decodes heatmaps from an embedding using three blocks consisting of bi-linear upsampling with a factor of 2, a convolutional layer with the kernel $3\times3$, a batch normalization layer and a relu activation function. Reconstructed heatmaps have resolution $64\times64$ for any input size. This does not affect the network’s ability to reconstruct locations of body landmarks and allows us to minimise the number of parameters in the heatmap reconstruction branch.
Training and evaluation
-----------------------
Data augmentation is applied on the fly to images and corresponding heatmaps in the same way. We use rotations up to 360 degrees, zooming up to 20% of image size and translations up to 20%. The same augmentation is applied when training the baseline model. Heatmap augmentation NLA shifts the blob in heatmaps to imitate noise in landmark coordinates. The minimal visible landmarks in MLA is set to 2 (out of possible 3) for the synthetic dataset and 3 (out of 5) for the manta ray dataset. The probability of missing a landmark is 50%.
The model is trained on the training subset. The test accuracy is obtained on new identities never seen during training. The test accuracy is computed by retrieving predictions from the gallery set for each image in the query set. We use top-1, top-5 and top-10 test accuracy for model evaluation.
Results
-------
### Landmark-Id vs baseline
The baseline results are obtained with only RGB images as input. *Landmark-Id* model outperforms the baseline model on both synthetic and real datasets that demonstrates that additional pose information is beneficial for learning discriminative embeddings, see Tables \[tab:poseid\_synthetic\_accuracy\] and \[tab:poseid\_manta\_accuracy\].
![Progress of top-1 accuracy on the test set during training evaluated each 10 epochs on the synthetic dataset. *Landmark-Id* Stage 2 continues training from Stage 1. Models are trained until convergence of the loss.[]{data-label="fig:training_plot"}](training_plot.pdf){width="\linewidth"}
Model Top-1 Top-5 Top-10
--------------------- ------------ ------------ ------------
Baseline Reid 63.81% 85.35% 90.94%
Landmark-Id Stage 1 78.10% 91.82% 94.41%
Landmark-Id Stage 2 **89.53%** **95.96%** **96.98%**
: *Landmark-Id* outperforms the baseline re-identification model on the synthetic dataset. Stage 1 is the model trained with additional heatmap input and Stage 2 is the model with the heatmap reconstruction block.[]{data-label="tab:poseid_synthetic_accuracy"}
Model Top-1 Top-5 Top-10
--------------------- ------------ ------------ ------------
Baseline Reid 44.00% 78.60% 84.70%
Landmark-Id Stage 1 52.67% 80.11% 86.18%
Landmark-Id Stage 2 **62.04%** **89.82%** **91.96%**
: Accuracy of re-identification on manta ray dataset. *Landmark-Id* outperforms the baseline re-identification model. Stage 1 is the model trained with additional heatmap input and Stage 2 is the model with the heatmap reconstruction block.[]{data-label="tab:poseid_manta_accuracy"}
*Landmark-Id* model reaches 89.53% top-1 accuracy versus 63.81% top-1 accuracy of baseline model on the synthetic data (Table \[tab:poseid\_synthetic\_accuracy\]). The real data is more challenging. Baseline model demonstrates 44.00% top-1 accuracy while *Landmark-Id* model goes up to 62.04% (Table \[tab:poseid\_manta\_accuracy\]).
We evaluate top-1 accuracy on the test set during training every 10 epochs on the synthetic dataset, see Figure \[fig:training\_plot\]. *Landmark-Id* model at Stage 1 (heatmap input with no reconstruction) shows higher accuracy than the baseline model. Stage 2 with auxiliary heatmap reconstruction further boosts the performance. *Landmark-Id* model without reconstruction block outperforms the baseline model. Adding the heatmap reconstruction block is useful as it promotes usage of pose information during feature extraction and improves accuracy on both synthetic and real data.
The above results are obtained with no noise in landmark coordinates and heatmaps of 5% of the image size. We analyse the sensitivity of the model to uncertainty and noise in the next section.
### Sensitivity to uncertainty in landmark locations
Model Top-1 Top-5 Top-10
--------------------- -------- -------- --------
Landmark-Id, hm 5% 86.13% 93.12% 95.82%
Landmark-Id, hm 10% 84.72% 92.84% 95.31%
Landmark-Id, hm 20% 66.62% 83.18% 88.85%
: Sensitivity of *Landmark-Id* to uncertainty in landmark locations is analysed with three sizes of heatmaps: 5%, 10% and 20% of the image size. The model shows almost equal performance for heatmaps with the blob radius up to $\pm10\%$ of the image size.[]{data-label="tab:synth-sensitivity"}
We investigate the sensitivity of the model to uncertainty in landmark locations by training and evaluating the model on the synthetic dataset with different settings for the size of the bright blob in heatmaps. Three experiments are conducted with the radius of the blob 5%, 10% and 20% of the image size (see Figure \[fig:heatmaps\_size\_noise\]). NLA adds noise to heatmaps shifting the center from the actual landmark location. The blob with a radius of $r\%$ means that the average noise in a landmark location is $\pm r\%$ of the image size.
Noise of 5% and 10% in landmark locations slightly decreases the accuracy (Table \[tab:synth-sensitivity\]). The noise of $\pm20\%$ decreases the top-1 accuracy to 66.62%. This is a high level of uncertainty because the blob with the radius 20% of the image size covers almost a quarter of the image. We conclude that a landmark detection model should have at most 10% error to predict landmark coordinates useful for re-identification.
### Sensitivity to missing landmarks
Model Top-1 Top-5 Top-10
----------------------- ------------ ------------ ------------
Landmark-Id, with MLA **62.04%** **89.82%** **91.96%**
Landmark-Id, no MLA 55.30% 87.47% 89.12%
: MLA (missing landmark augmentation) improves robustness of *Landmark-Id* to not visible landmarks. Evaluated on real dataset of manta ray images.[]{data-label="tab:real-sensitivity-missing"}
To evaluate the sensitivity of *Landmark-Id* model to missing landmarks, we train the model without the MLA augmentation. The synthetic dataset has most of the landmarks visible at all times so we use real data in this experiment. The manta ray dataset has around 50% images with all five landmarks visible, 30% of images with four landmarks visible and the rest with three and less visible landmarks.
Without MLA augmentation top-1 accuracy drops to 55.30% from 62.04% on manta ray dataset (Table \[tab:real-sensitivity-missing\]).
Conclusion
==========
We demonstrated that the additional input of body landmarks improves learning of discriminative embeddings. This method is robust to uncertainty in landmark locations and tolerates errors in landmark coordinates up to 10% of the image size.
We will conduct experiments on other real datasets (e.g., ATRW [@tiger-dataset], ELPephants [@elephants-dataset]). In the future, we plan to investigate how to train an accurate body landmark predictor on a small dataset and integrate it with the re-identification model.
[^1]: O. Moskvyak, F. Maire and F. Dayoub are with the School of Electrical Engineering and Computer Science, Queensland University of Technology, Brisbane, QLD 4000, Australia.
[^2]: M. Baktashmotlagh is with the School of Information Technology and Electrical Engineering, The University of Queensland, St Lucia, QLD 4072, Australia.
[^3]: Corresponding author O. Moskvyak: [email protected]
|
---
abstract: 'This paper investigates ideal-theoretic as well as homological extensions of the Prüfer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal. The new results both compare and contrast with recent results on trivial ring extensions (and pullbacks) as well as yield original families of examples issued from amalgamated duplications subject to various Prüfer conditions.'
address:
- 'Department of Mathematics, FST, University S. M. Ben Abdellah, Fez 30000, Morocco'
- 'Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran 31261, KSA'
- 'Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran 31261, KSA'
- 'Department of Mathematics, FST, University S. M. Ben Abdellah, Fez 30000, Morocco'
author:
- 'M. Chhiti'
- 'M. Jarrar'
- 'S. Kabbaj $^{(1)}$'
- 'N. Mahdou'
title: |
Prüfer conditions in an amalgamated duplication\
of a ring along an ideal $^{(\star)}$
---
[^1]
[^2]
Introduction
============
All rings considered in this paper are commutative with unity and all modules are unital. Let $A$ be a ring, $I$ an ideal of $A$, and $\pi:A\rightarrow \frac{A}{I}$ the canonical surjection. The amalgamated duplication of $A$ along $I$, denoted by $A\bowtie I$, is the special pullback (or fiber product) of $\pi$ and $\pi$; i.e., the subring of $A \times A$ given by $$A\bowtie I:=\pi\times_{\frac{A}{I}}\pi=\{(a,a+i)\mid a\in A, i\in I\}.$$ This construction was introduced and its basic properties were studied by D’Anna and Fontana in [@DF1; @DF2] and then it was investigated by D’Anna in [@D] with the aim of applying it to curve singularities (over algebraic closed fields) where he proved that the amalgamated duplication of an algebroid curve along a regular canonical ideal yields a Gorenstein algebroid curve [@D Theorem 14 and Corollary 17]. In [@DFF1; @DFF2], with Finocchiaro, they have considered the more general context of amalgamated algebra $A\bowtie^{f} J:=\{(a,f(a)+j)\mid a\in A, j\in J\}$ for a given homomorphism of rings $f: A\rightarrow B$ and ideal $J$ of $B$. In particular, they have studied amalgamations in the frame of pullbacks which allowed them to establish numerous (prime) ideal and ring-theoretic basic properties for this new construction. Two more recent works on amalgamated duplications are [@MY; @Sh]. The interest of amalgamation resides, partly, in its ability to cover several basic constructions in commutative algebra, including pullbacks and trivial ring extensions (also called Nagata’s idealizations).
A domain is Prüfer if all its non-zero finitely generated ideals are invertible [@K; @P]. There are well-known extensions of this notion to arbitrary rings (with zero divisors). Namely, for a ring $R$,\
$R$ is semihereditary, i.e., every finitely generated ideal of $R$ is projective [@CE];\
$R$ has weak global dimension $\operatorname{w.gl.dim}(R)\leq1$ [@G1; @G2];\
$R$ is arithmetical, i.e., every finitely generated ideal of $R$ is locally principal [@Fu; @J];\
$R$ is an fqp-ring, i.e., every finitely generated ideal of $R$ is quasi-projective [@AJK].\
$R$ is Gaussian, i.e., $c(fg)=c(f)c(g), \forall f,g\in R[x]$, where $c(f)$ is the content of $f$ [@T];\
$R$ is Prüfer, i.e., every finitely generated regular ideal of $R$ is projective [@BS; @Gr].\
The following diagram summarizes the relations between these Prüfer-like conditions where the implications cannot be reversed in general [@AJK; @BG; @BG2; @G2; @G3]:
$R$ is semihereditary\
$\Downarrow$\
$\operatorname{w.gl.dim}(R)\leq 1$\
$\Downarrow$\
$R$ is arithmetical\
$\Downarrow$\
$R$ is an fqp-ring\
$\Downarrow$\
$R$ is Gaussian\
$\Downarrow$\
$R$ is Prüfer
All these forms coincide in the context of domains [@AJK; @G3]. Glaz [@G3] and Bazzoni & Glaz [@BG2] constructed examples which show that all these notions are distinct in the context of arbitrary rings. It is notable that original examples, marking the distinction of each of the above classes of Prüfer-like rings are rare in the literature. New examples, in this regard, were provided via the study of these notions in diverse settings of trivial ring extensions [@AJK; @BKM]. Pullbacks issued from rings with zero divisors were also considered for a similar study; namely, let $T$ be an arbitrary ring (possibly, with zero divisors), $I$ a (regular) ideal of $T$, $\pi:T\rightarrow \frac{T}{I}$ the canonical surjection, and $i:D\hookrightarrow \frac{T}{I}$ an inclusion of rings. Let $R:=i\times_{\frac{T}{I}}\pi$ be the pullback of $i$ and $\pi$. In [@Bo1; @Bo2], the author examined the transfer of the Prüfer conditions (except the fqp property) from $D$ and $T$ to $R$. At this point, it is worthwhile noticing that an amalgamated duplication along an ideal $I$ collapses to a trivial ring extension $A\ltimes I$ for $I^{2}=0$ and overlaps with the above pullbacks for $I=0$ (i.e., $A\bowtie I\cong A$).
This paper investigates necessary and sufficient conditions for an amalgamated duplication of a ring along an ideal to inherit the six aforementioned Prüfer notions, and hence provides new families of examples subject to these conditions. In this vein, we shall omit the case $A\bowtie A=A\times A$ since all these notions are stable under finite products by [@B Theorem 3.4] and Remark \[fqp1\]. That is, in all main results, the ideal $I$ of the amalgamation will be assumed to be proper. Section 2 examines the transfer of the notions of local Prüfer ring and total ring of quotients. Section 3 deals with the arithmetical, Gaussian, and fqp conditions. Section 4 is devoted to the weak global dimension and the transfer of the semihereditary condition.
Throughout, $A\bowtie I$ will denote the amalgamated duplication of a ring $A$ along an ideal $I$ of $A$. If $J$ is an ideal of $A$, then $J\bowtie I:=\{(j,j+i)\mid j\in J, i\in I\}$ is an ideal of $A\bowtie I$ with $\frac{A\bowtie I}{J\bowtie I}\cong \frac{A}{J}$ [@DFF1 Proposition 5.1]. Under the natural injection $A\hookrightarrow A\bowtie I$ defined by $i(a)=(a,a)$, we identify $A$ with its respective image in $A\bowtie I$; and the natural surjection $A\bowtie I\twoheadrightarrow A$ yields the isomorphism $\frac{A\bowtie I}{(0)\bowtie I}\cong A$ [@D Remark 1]. Also, for a ring $R$, $Q(R)$ will denote the total ring of quotients and $\operatorname{Z}(R)$, $\operatorname{U}(R)$, $\operatorname{Nil}(R)$, and $\operatorname{J}(R)$ will denote, respectively, the set of zero divisors, set of invertible elements, nilradical, and Jacobson radical of $R$. Finally, $\operatorname{Max}(R)$ shall denote the set of maximal ideals of $R$, $\operatorname{Max}(R, I):=\{\m\in\operatorname{Max}(R)\mid I\subseteq \m\}$, and $\operatorname{Ann}(I)$ the annihilator of $I$ for any ideal $I$ of $R$.
Transfer of the Prüfer condition {#P}
================================
This section handles the notion of Prüfer ring. An ideal $I$ of a ring $R$ is invertible if $II^{-1}=R$, where $I^{-1}:=\{x\in Q(R)\mid xI\subseteq R\}$; and $R$ is Prüfer if every finitely generated regular ideal of $R$ is invertible (or, equivalently, projective) [@BS; @Gr]. We refer the reader to [@BG Theorem 2.13] which collects fifteen conditions equivalent to this definition. Finally, recall that the class of Prüfer rings contains strictly the class of total rings of quotients.
Next, before we announce the main result of this section (Theorem \[P1\]), we make the following useful remark.
\[P5\] Let $A$ be a ring, $I$ an ideal of $A$, and $P$ a prime ideal of $A$. In [@D Propositions 5 & 7], D’Anna proved that if $I\nsubseteqq P$, then $\tilde{P}:=\{(p+i,p)\mid p\in P,\ i\in I\}$ and $P\bowtie I$ are the only prime ideals of $A\bowtie I$ lying over $P$ and we have $$\dfrac{A\bowtie I}{\tilde{P}}\cong\dfrac{A\bowtie I}{P\bowtie I}\cong \dfrac{A}{P}\ \mbox{ and }\
(A\bowtie I)_{\tilde{P}}\cong(A\bowtie I)_{P\bowtie I}\cong A_{P}.$$ Notice that $P\bowtie I$ and $\tilde{P}$ are incomparable. However, if $I\subseteq P$, then $P\bowtie I=\tilde{P}$ is the unique prime ideal of $A\bowtie I$ lying over $P$ and we have $$\dfrac{A\bowtie I}{P\bowtie I}\cong \dfrac{A}{P}\ \mbox{ and }\
(A\bowtie I)_{P\bowtie I}\cong A_{P}\bowtie I_{P}.$$ As a consequence, $(A,\m)$ is local with $I\subseteq \m$ if and only if $A\bowtie I$ is local with maximal ideal $\m\bowtie I$. This basic fact will be used throughout this paper without explicit mention.
Now, to the main result:
\[P1\] Let $(A, \m)$ be a local ring and $I$ a proper ideal of $A$. Then $A\bowtie I$ is a Prüfer ring if and only if $A$ is a Prüfer ring and $I=aI$ for every $a\in \m\setminus \operatorname{Z}(A)$.
\[P5.1\]
Let $(A,\m)$ be a local Prüfer ring. One can easily check that:
$aI=a^{2}I,\ \forall\ a\in \m$ (i.e., $\forall\ a\in A$)\
$\Downarrow$\
$I=aI,\ \forall\ a\in \m\setminus \operatorname{Z}(A)$ (i.e., $\forall\ a\in A\setminus \operatorname{Z}(A)$)\
$\Downarrow$\
$I\subseteq \operatorname{Z}(A)\subseteq \m$.
So, by Theorem \[P1\], if $I$ is a proper regular ideal of $A$ (i.e., $I\nsubseteq\operatorname{Z}(A)$), then the amalgamation $A\bowtie I$ is never a Prüfer ring. The first assumption “$aI=a^{2}I,\ \forall\ a\in \m$" will be used later in Theorem \[AGfqp1\] to characterize amalgamations subject to the Gaussian and fqp conditions.
Notice that, in the setting $0\not=I\subseteq \operatorname{Z}(A)$, the assumption “$ I=aI,\ \forall\ a\in \m\setminus \operatorname{Z}(A)$" is not necessarily embedded in the (local) Prüfer condition. For instance, let $A:=\Z_{(2)}\ltimes \Q$ and $I:=0\ltimes \Z_{(2)}$. Then $A$ is a chained ring [@BKM Theorem 2.1(2)] with maximal ideal $2\Z_{(2)}\ltimes \Q$ and $I^{2}=0$; whereas $I\not= (4,0)I$. So, by Theorem \[P1\], $A\bowtie I = A\ltimes I$ is not a Prüfer ring.
The proof of the theorem relies on the following lemmas which are of independent interest. Recall at this point that a polynomial $f$ over a ring $R$ is Gaussian if the content ideal equation $c(fg) = c(f)c(g)$ holds for any polynomial $g$ over $R$ [@T].
\[P2\] Let $A$ be a ring and $I$ an ideal of $A$. If the polynomial $F(x):=\sum_{i=0}^{n}(a_{i},a_{i})x^{i}$ is Gaussian over $A\bowtie I$, then $f(x):=\sum_{i=0}^{n}a_{i}x^{i}$ is Gaussian over $A$.
Straightforward.
\[P3\] Let $A$ be a ring and $I$ an ideal of $A$. If $A\bowtie I$ is Prüfer, then $A$ is Prüfer.
Assume that $A\bowtie I$ is a Prüfer ring. Let $J:=\sum_{i=0}^{n}a_{i}A$ be a finitely generated regular ideal of $A$ and $a$ a regular element of $J$. Clearly, $G:=\sum_{i=0}^{n}(a_{i},a_{i})A\bowtie I$ is a finitely generated regular ideal of $A\bowtie I$ since $(a,a)\in G$. Then $G$ is invertible and hence the polynomial $F(x):=\sum_{i=0}^{n}(a_{i},a_{i})x^{i}$ is Gaussian over $A\bowtie
I $. By Lemma \[P2\], $f(x):=\sum_{i=0}^{n}a_{i}x^{i}$ is Gaussian over $A$. Therefore $J=c(f)$ is invertible in $A$ by [@BG Theorem 4.2], making $A$ a Prüfer ring, as desired.
Next we recall a nice result by Maimani and Yassemi which provides a full description for the set of zero divisors of $A\bowtie I$ for any arbitrary commutative ring. In the sequel, the subset $\{(a,a+i)\mid a\in\operatorname{Z}(A), i\in I\}$ of $A\bowtie I$ will be denoted by $\operatorname{Z}(A)\bowtie I$.
\[P3.1\] Let $A$ be a ring and $I$ an ideal of $A$. Then $$\begin{array}{rl}
\operatorname{Z}(A\bowtie I) &=\operatorname{Z}(A)\bowtie I\ \cup\ \big\{(i,0)\mid i\in I\big\}\\
&\cup\ \big\{(a, a+i)\mid a\ \mbox{regular and}\ j(a+i)=0\ \mbox{for some}\ 0\not=j\in I\big\}.
\end{array}$$
\[P3.2\] Let $R$ be a local Prüfer ring and let $x$ be a regular element of $R$. Then $xR$ is comparable with every principal ideal of $R$.
The proof follows immediately from [@AJK Lemma 3.8].
\[P4\] Let $A$ be a local Prüfer ring and $I$ an ideal of $A$. Then: $$I\subseteq \operatorname{Z}(A) \Leftrightarrow\operatorname{Z}(A\bowtie I)=\operatorname{Z}(A)\bowtie I.$$
Assume $I\subseteq \operatorname{Z}(A)$. Let $a$ be a regular element of $A$ and let $i\in I$. We claim that $a+i$ is regular in $A$. Indeed, the ideals $aA$ and $iA$ are comparable by Lemma \[P3.2\]. It follows that $i=ka$ for some non-unit $k\in A$ since $I\subseteq \operatorname{Z}(A)$. Thus $a+i=(1+k)a\in A\setminus\operatorname{Z}(A)$, as claimed. Consequently, the set $\big\{(a, a+i)\mid a\ \mbox{regular and}\ j(a+i)=0\ \mbox{for some}\ 0\not=j\in I\big\}$ is empty. In view of the description of $\operatorname{Z}(A\bowtie I)$ in Lemma \[P3.1\], it merely collapses to $\operatorname{Z}(A)\bowtie I$, as desired. The converse is trivial by the same lemma.
\(1) $(A,\m)$ is assumed to be local and $I\subseteq \m$. This is equivalent to saying that $A\bowtie I$ is local. Suppose that $A\bowtie I$ is Prüfer. By Lemma \[P3\], $A$ is Prüfer. Note that $\operatorname{Z}(A)\subseteq\m$. We claim that $I\subseteq \operatorname{Z}(A)$. Deny and let $i\in I\setminus \operatorname{Z}(A)$. Clearly, $(i,i)$ is regular in $A\bowtie I$. By Lemma \[P3.2\], the ideals $\big((0,i)\big)$ and $\big((i,i)\big)$ must be comparable in $A\bowtie I$ and, necessarily, $(0,i)=(i,i)(b,b+j)$ for some $b\in A$ and $j\in I$. So that $b=0$ and $i=ij$, whence $j=1$, the desired contradiction. Next, let $a\in A\setminus \operatorname{Z}(A)$ and $i\in I$. By Lemma \[P4\], $(a,a+i)$ is regular in $A\bowtie I$. As above, via Lemma \[P3.2\], we get $(0,i)=(a,a+i)(b,b+j)$ for some $b\in A$ and $j\in I\subseteq \m$. Therefore, $b=0$ and thus $i=aj(1-j)^{-1}\in aI$, as desired.
Conversely, suppose $A$ is a (local) Prüfer ring with $I=aI$ for every $a\in \m\setminus \operatorname{Z}(A)$ (i.e., for every $a\in A\setminus \operatorname{Z}(A)$). Let $F:=\big((a,a+i),(b,b+j)\big)$ be a regular ideal of $A\bowtie I$. Assume one, at least, of the two generators of $F$ is regular. By Lemma \[P4\], $a$ or $b$ is regular in $A$. So, by Lemma \[P3.2\], $(a)$ and $(b)$ are comparable in $A$; say, $a$ is regular in $A$ and $b=ac$ for some $c\in A$. By hypothesis, there is $k\in I$ such that $j-ic=(a+i)k$. So one can easily check that $(b,b+j)=(a,a+i)(c,c+k)$; i.e., $F:=\big((a,a+i)\big)$. Now, assume both generators of $F$ are zero divisors and let $(r, r+h)$ be a regular element of $F$. Similar arguments as above yield $a=ra'$, $i-ha'=(r+h)k_{1}$, $b=rb'$, and $j-hb'=(r+h)k_{2}$, for some $a',b'\in A$ and $k_{1}, k_{2}\in I$; leading to $F:=\big((r,r+h)\big)$. So in both cases $F$ is principal (and, a fortiori, invertible) making $A\bowtie I$ a Prüfer ring [@BG Theorem 2.13(2)]. This completes the proof of the local case.
As an application of Theorem \[P1\] (combined with Theorem \[AGfqp1\]), one can construct new examples of (non-Gaussian) Prüfer rings as shown below.
\[P5.2\] Let $R:=\frac{\Z}{8\Z}\bowtie \frac{2\Z}{8\Z}$. We have $\operatorname{Z}(\frac{\Z}{8\Z})=\frac{2\Z}{8\Z}$ and hence, by Theorem \[P1\], $R$ is a local Prüfer ring (which is not Gaussian by Theorem \[AGfqp1\](2)). Further, $R$ is neither a trivial ring extension nor a pullback of the type studied in [@Bo1; @Bo2; @Bo3].
Total rings of quotients are important source of Prüfer rings. Next, we study the transfer of this notion to an amalgamation.
\[P6\] Let $A$ be a ring and $I$ an ideal of $A$ such that $I\subseteq \operatorname{J}(A)$. Then $A$ is a total ring of quotients if and only if $A\bowtie I$ is a total ring of quotients.
Assume $A$ is a total ring of quotients and let $(x,x+i)\in A\bowtie I$. If $x$ is a zero divisor in $A$, then so is (x,x+i) in $A\bowtie I$ since $\operatorname{Z}(A)\bowtie I\subseteq \operatorname{Z}(A\bowtie I)$ always holds. Now suppose that $x$ is invertible in $A$ and let $y:=x^{-1}$ and $j:=-iy^{2}(1+yi)^{-1}$. Since $I\subseteq \operatorname{J}(A)$, then $j\in I$. Further, we have $(x,x+i)(y,y+j)=(1,1)$. So $(x,x+i)$ is invertible in $A\bowtie I$. Conversely, assume $A\bowtie I$ is a total ring of quotients and let $x\in A$. Then $(x,x)$ is either a zero divisor or invertible in $A\bowtie I$. Clearly, this forces $x$ to be either a zero divisor or invertible in $A$, completing the proof.
Let $(A,\m)$ be a local ring and let $n$ be an integer $\geq2$. By Proposition \[P6\], $\frac{A}{\m^{n}}\bowtie \frac{\m^{n-1}}{\m^{n}}$ $\left(=\ \frac{A}{\m^{n}}\ltimes \frac{\m^{n-1}}{\m^{n}}\right)$ is a local total ring of quotients and, a fortiori, a local Prüfer ring.
Recall that the notion of Prüfer ring is not stable under factor rings [@BSh Example 3.3] (also [@L Example 3.6] and [@BKM Example 2.8]). A ring $R$ is locally Prüfer if $R_{p}$ is Prüfer $\forall\ p\in\operatorname{Spec}(R)$ [@Bo3 Definition 2.1]. Lucas proved that if $R_{\m}$ is Prüfer $\forall\ \m\in\operatorname{Max}(R)$ (a fortiori, if $R$ is locally Prüfer), then $R$ is Prüfer [@L Proposition 2.10]; and constructed a non-local Prüfer ring which is not locally Prüfer [@L Example 2.11]. Recently, Boynton provided an example of a local Prüfer ring which is not locally Prüfer [@Bo3 Example 2.4].
Is Theorem \[P1\] valid in the global case? i.e., when $A$ is Prüfer (not necessarily local) or locally Prüfer. One, particularly, needs to find the *right and natural* globalization for the assumption “$I=aI,\ \forall\ a\in \m\setminus \operatorname{Z}(A)$."
Transfer of the arithmetical, Gaussian, and fqp conditions {#AGfqp}
==========================================================
A ring $R$ is arithmetical if the ideals of any localization of $R$ are linearly ordered; equivalently, if every finitely generated ideal of $R$ is locally principal [@Fu; @J]. A local arithmetical ring is also called a chained or valuation ring. The ring $R$ is Gaussian if for every $f, g$ in the polynomial ring $R[x]$, one has the content ideal equation $c(fg) = c(f)c(g)$ [@T]. Both arithmetical and Gaussian notions are local; i.e., a ring is arithmetical (resp., Gaussian) if and only if its localizations with respect to maximal ideals are arithmetical (resp., Gaussian). We will make frequent use of an important characterization of a local Gaussian ring; namely, “*for any two elements $a, b$ in the ring, we have $(a,b)^{2}=(a^{2})$ or $(b^{2})$; moreover, if $ab=0$ and, say, $(a,b)^{2}=(a^{2})$, then $b^{2}=0$*" [@BG2 Theorem 2.2].
An ideal $I$ of a ring $R$ is quasi-projective if the natural map $\operatorname{Hom}_{R}(I,I)\rightarrow\operatorname{Hom}_{R}(I,I/J)$, defined by $f\mapsto \overline{f}$, is surjective for every subideal $J$ of $I$. A ring $R$ is an fqp-ring if every finitely generated ideal of $R$ is quasi-projective [@AJK]. An arithmetical ring is an fqp-ring and an fqp-ring is Gaussian, where the implications are irreversible in general [@AJK Theorem 3.2]. It is worthwhile recalling, at this point, that the fqp condition is stable under formation of rings of fractions [@AJK Lemma 3.6]; though, the question of whether it is a local property is still elusively open [@AJK]. As mentioned in the introduction, we shall omit the case $A\bowtie A=A\times A$ since the fqp property, too, is stable under finite products as shown below.
\[fqp1\] Let $R_{1}$ and $R_{2}$ be two fqp-rings, $R:=R_{1}\times R_{2}$, and $I:=I_{1}\times I_{2}$, where $I_{i}$ is a finitely generated ideal of $R_{i}$ for $i=1,2$. Let $f: I\rightarrow I/K$ be an $R$-map, where $K$ is a subideal of $I$ and write $K=K_{1}\times K_{2}$ and $f=f_{1}\times f_{2}$, where $K_{i}$ is a subideal of $I_{i}$ and $f_{i}\in\operatorname{Hom}_{R}(I_{i},I_{i}/K_{i})$ defined by $f_{1}(x):=a$ such that $f(x,0)=(a,b)$ and similarly for $f_{2}$. Therefore, there is $g_{i}\in\operatorname{Hom}_{R}(I_{i},I_{i})$ such that $\overline{g_{i}}=f_{i}$. It is clear that $\overline{g}=\overline{g_{1}}\times \overline{g_{2}}=f $. It follows that $R$ is an fqp-ring. The converse is more straightforward.
The main result of this section examines necessary and sufficient conditions for amalgamations issued from local rings to inherit the notions of arithmetical, Gaussian, and fqp-ring, respectively. In particular, it turns out that, among amalgamated duplications of local rings, only trivial extensions can inherit the Gaussian or fqp properties. Thereby, a second result examines the global case.
\[AGfqp1\] Let $(A, \m)$ be a local ring and $I$ a proper ideal of $A$. Then:
1. $A\bowtie I$ is arithmetical if and only if $A$ is arithmetical and $I=0$.
2. $A\bowtie I$ is Gaussian if and only if $A$ is Gaussian, $I^{2}=0$, and $aI=a^{2}I\ \forall\ a\in \m$.
3. $A\bowtie I$ is an fqp-ring if and only if $A$ is an fqp-ring (resp., a Prüfer ring), $\big(\operatorname{Z}(A)\big)^{2}=0$, and $aI=a^{2}I\ \forall\ a\in \m$.
The proof of this theorem draws on the following results.
\[fqp2\] Let $A$ be a ring and $I$ a proper ideal of $A$. Let $J$ be an ideal of $A$ and $K$ a subideal of $I$. Then $J\bowtie K$ is an ideal of $A\bowtie I$ if and only if $JI\subseteq K$.
The proof is straightforward and may be left to the reader.
\[fqp3\] Let $A$ be a ring and $I$ a proper ideal of $A$. If $A\bowtie I$ is an fqp-ring, then $A$ is an fqp-ring.
Assume that $A\bowtie I$ is an fqp-ring and let $J:=(a_{1},...,a_{n})$ be a finitely generated ideal of $A$, $K$ a subideal of $J$, and $f\in \operatorname{Hom}_{A}(J,J/K)$. We need to prove the existence of $g\in \operatorname{Hom}_{A}(J,J)$ such that $f=\overline{g}$ (mod $K$). For this purpose, consider the ideal of $A\bowtie I$ given by $U:=J\bowtie JI$ (Lemma \[fqp2\]). We claim that $U=\big((a_{1},a_{1}),...,(a_{n},a_{n})\big)$. Obviously, $(a_{i},a_{i})\in U\ \forall\ i=1, ..., n$. Next, let $(x,x+h)\in U$. We have $$\begin{array}{rcl}
(x,x+h) &= &(x,x)+(0,\sum e_jf_j)\ \big(\mbox{for some } x\in J \mbox{ and } (e_j,f_j)\in J\times I, 1\leq j\leq m\big)\\
&= &\sum_{i} (r_i,r_i)(a_i,a_i)+ (0,\sum_j(\sum_{i} s_{ij} a_i)f_j)\ \big(\mbox{for some } r_i, s_{ij} \in A, 1\leq i\leq n\big)\\
&= &\sum_{i} (r_i,r_i)(a_i,a_i)+\sum_j(\sum_{i} s_{ij}a_i,\sum s_{ij}a_i)(0,f_j)\\
&= &\sum_{i} (r_i,r_i)(a_i,a_i)+\sum_j(0,f_j)\sum_{i} (s_{ij},s_{ij})(a_i,a_i)\\
&= &\sum_{i} (r_i,r_i+\sum_{j}f_js_{ij})(a_i,a_i), \mbox{ as desired.}
\end{array}$$ Let $V:=K\bowtie KI$, a subideal of $U$ by Lemma \[fqp2\], and consider the function $$\begin{array}{cccc}
F: &U &\longrightarrow & U/V \cong J/K \bowtie JI/KI\\
&\sum_{i=1}^{n} \lambda_{i}(a_i,a_i) &\longrightarrow &\sum_{i=1}^{n} \lambda_{i}\big(f(a_i),f(a_i)\big).
\end{array}$$ One can check that $F$ is well-defined and hence an $A\bowtie I$-map. Since $U$ is quasi-projective, there exists $G\in \operatorname{Hom}_{A\bowtie I}(U,U)$ such that $F=\overline{G}$ (mod $V$). Now, let $a\in J$ and let $g(a)$ equal the first coordinate of $G(a,a)$. Clearly, $g\in \operatorname{Hom}_{A}(J,J)$. Moreover, $\overline{G(a,a)}=F(a,a)=(f(a),f(a))$ yields $f=\overline{g}$.
\[fqp4\] Let $R$ be a local fqp-ring which is not a chained ring. Then $\big(\operatorname{Nil}(R)\big)^{2}=0$.
\[fqp5\] Let $R$ be a local fqp-ring which is not a chained ring. Then $\operatorname{Z}(R)=\operatorname{Nil}(R)$.
\(1) Assume that $A\bowtie I$ is (local) arithmetical (i.e., chained ring). Then $A$, too, is (local) arithmetical since the arithmetical property is stable under factor rings. Moreover, for each $i\in I$, the ideals $(i,0)A\bowtie I$ and $(0,i)A\bowtie I$ are comparable. In case $(i,0)\in (0,i)A\bowtie I$, there is an element $(a,j)\in A\times I$ such that $(0,i)=(a,a+j)(i,0)=(ai,0)$; so that $i=0$. Similarly, the other case yields $i=0$. So, we conclude that $I=0$, as desired. The converse is trivial since $A\bowtie (0)\cong A$.
\(2) Assume $A\bowtie I$ is (local) Gaussian. Then so is $A$ since the Gaussian property is stable under factor rings. Next, we prove $I^{2}=0$. Let $a, b \in I$. In $A\bowtie I$, we have $((a,a), (0,a))^{2}=((0,a)^{2})$ or $((a,a)^{2})$. The two cases yield, respectively, $a^{2}=0$ or $a^{2}(1-i)=0$ for some $i\in I\subseteq \m$. It follows that $a^{2}=0$. Likewise, $b^{2}=0$. Now appeal to the Gaussian property in $A$ to get $ab=0$. To prove the last statement, let $a\in A$ and $i\in I$. In $A\bowtie I$, we have $((a,a), (0,i))^{2}=((a,a)^{2})$ since $I^{2}=0$. It follows that $ai=a^{2}j$ for some $j\in I$. That is, $aI=a^{2}I$, as desired.
Conversely, let $(a,a+i), (b,b+j)\in A\bowtie I$. Since $A$ is local Gaussian, then, say, $(a, b)^{2}=(a^{2})$. Hence $b^{2}=a^{2}x$ and $ab=a^{2}y$ for some $x,y\in A$. Moreover, $ab=0$ implies $b^{2}=0$. By assumption, there exist $i_{1},i_{2},i_{3},j_{1},j_{2}\in I$ such that $2bj=a^{2}xj_{1}$, $2axi=a^{2}i_{1}$, $aj=a^{2}j_{2}$, $bi=a^{2}xi_{2}$, and $2ayi=a^{2}i_{3}$. Using the fact $I^{2}=0$, simple calculations show that $(b,b+j)^{2}=(a,a+i)^{2}(x,x+xj_{1}-i_{1})$ and $(a,a+i)(b,b+j)=(a,a+i)^{2}(y,y+xi_{2}+j_{2}-i_{3})$. Further, assume $(a,a+i)(b,b+j)=0$. Hence $ab=0$, whence $b^{2}=0$ and $2bj=0$ since $bI=0$. So that $(b,b+j)^{2}=0$. Therefore, $A\bowtie I$ is (local) Gaussian, completing the proof of (2).
\(3) Without loss of generality, we assume that $A\bowtie I$ is a local fqp-ring that is not a chained ring (i.e., $I\not= 0$). By Lemma \[fqp3\], $A$ is an fqp-ring (and hence a Prüfer ring). So $\operatorname{Z}(A)$ is a (prime) ideal of $A$. Moreover, by (2), $aI=a^{2}I$ for every $a\in A$. In particular, $I=aI$ for every regular element $a$ of $A$ and, hence, $I\subseteq \operatorname{Z}(A)$ by Remark \[P5.1\]. So, by Lemma \[P4\], $\operatorname{Z}(A\bowtie I)=\operatorname{Z}(A)\bowtie I$. Finally, (1) combined with Lemmas \[fqp4\] & \[fqp5\] yield $\big(\operatorname{Z}(A\bowtie I)\big)^{2}=0$. Consequently, $\big(\operatorname{Z}(A)\big)^{2}=0$.
Conversely, assume that $A$ is a Prüfer ring, $\big(\operatorname{Z}(A)\big)^{2}=0$, and $aI=a^{2}I$ for every element $a$ in $A$. We aim to prove that $A\bowtie I$ is an fqp-ring. Throughout the proof, we will also be using the basic facts $I\subseteq \operatorname{Z}(A)$, $I^2=0$, and $I=aI\ \forall\ a\in A\setminus\operatorname{Z}(A)$.
Let $(a,a+i)$ and $(b,b+j)$ be two nonzero *incomparable* elements of $A\bowtie I$.
$a,b \in \operatorname{Z}(A)$
Indeed, assume, by way of contradiction, that $a$ is regular in $A$. By Lemma \[P3.2\], $(a)$ and $(b)$ are comparable. Suppose $b=ac$ for some $c\in A$. There is $k\in I\ (=aI)$ such that $j-ci=ak$ which leads to $(b,b+j)=(a,a+i)(c,c+k)$, absurd. Now, if $a=bc$ for some $c\in A$, necessarily, $b$ is regular and hence similar arguments lead to the same absurdity, proving the claim.
$\operatorname{Ann}(a,a+i)=\operatorname{Ann}(b,b+j)$ and $\big((a,a+i)\big)\cap\big((b,b+j)\big)=0$.
Clearly, $\operatorname{Ann}(a,a+i)\subseteq \operatorname{Z}(A\bowtie I)=\operatorname{Z}(A)\bowtie I$ by Lemma \[P4\]. The reverse inclusion is straight in view of Claim 1. So $\operatorname{Ann}(a,a+i)=\operatorname{Z}(A)\bowtie I=\operatorname{Ann}(b,b+j)$, as desired. It remains to show that $\big((a,a+i)\big)\cap\big((b,b+j)\big)=0$. For this purpose, let $(x,x+h)$ and $(y,y+k)$ be two elements of $A\bowtie I$ such that $$(a,a+i)(x,x+h)=(b,b+j)(y,y+k).$$ We get via Claim 1 $$ax=by\ \text{ and }\ xi=yj.$$ We claim that $x$ or $y \in \operatorname{Z}(A)$. Deny and assume, by way of contradiction, that both $x$ and $y$ are regular in $A$. By Lemma \[P3.2\], $xA$ and $yA$ are comparable; say, $x=ry$ for some $r\in A$. So $ary=by$ and $ryi=yj$ yield $b=ra$ and $j=ri$. It follows that $(b,b+j)=(a,a+i)(r,r)$, the desired contradiction. Consequently, $x$ or $y \in \operatorname{Z}(A)$. This forces $ax=by=xi=yj=0$, completing the proof of the claim.
Finally, let $J$ be a finitely generated ideal of $A\bowtie I$ with a minimal generating set $\big\{(a_{1},a_{1}+i_{1}), \ldots, (a_{n},a_{n}+i_{n})\big\}$. By Claim 2, we obtain $$\operatorname{Ann}(a_{h},a_{h}+i_{h})=\operatorname{Ann}(a_{k},a_{k}+i_{k}),\ \forall\ h\not=k\in\{1, \ldots, n\};$$ $$J=\big((a_{1},a_{1}+i_{1})\big)\oplus \big((a_{2},a_{2}+i_{2})\big)\oplus \ldots \oplus\big((a_{n},a_{n}+i_{n})\big).$$ Therefore $\big((a_{h},a_{h}+i_{h})\big)\cong \big((a_{k},a_{k}+i_{k})\big)$ and so $\big((a_{h},a_{h}+i_{h})\big)$ is $\big((a_{k},a_{k}+i_{k})\big)$-projective, for all $h, k$. By [@FH Corollary 1.2], $J$ is quasi-projective, completing the proof of the theorem.
\[fqp6\] It is worthwhile noting that, in Theorem \[AGfqp1\](2-3), the two facts $I^{2}=0$ and $(\operatorname{Z}(A))^{2}=0$ are independent of the assumption “$aI=a^{2}I,\ \forall\ a\in A$." For instance, this latter does not hold for the chained ring $A$ and ideal $I$ given in Remark \[P5.1\](2); though $I^{2}=(\operatorname{Z}(A))^{2}=0$ since $I\subset\operatorname{Z}(A)=0\ltimes \Q$. Conversely, let $A:=\frac{\Z}{8\Z}$ and $I:=\frac{4\Z}{8\Z}$. One can verify that $(\operatorname{Z}(A))^{2}=I\not=0$ and $aI=a^{2}I=0,\ \forall\ a\in \frac{2\Z}{8\Z}$.
The next corollary handles the global case.
\[AGfqp2\] Let $A$ be a ring and $I$ a proper ideal of $A$. Then:
1. $A\bowtie I$ is arithmetical if and only if $A$ is arithmetical and $I_{\m}=0,\ \forall\ \m\in\operatorname{Max}(A,I)$.
2. $A\bowtie I$ is locally an fqp-ring if and only if $A$ is locally an fqp-ring, $\big(\operatorname{Z}(A_{\m})\big)^{2}=0$, and $aI_{\m}=a^{2}I_{\m},\ \forall\ \m\in\operatorname{Max}(A,I)$ and $\forall\ a\in \m$.
3. $A\bowtie I$ is Gaussian if and only if $A$ is Gaussian, $I_{\m}^{2}=0$, and $aI_{\m}=a^{2}I_{\m},\ \forall\ \m\in\operatorname{Max}(A,I)$ and $\forall\ a\in \m$.
Let $\m\in\operatorname{Max}(A)$. By Remark \[P5\], $(A\bowtie I)_{\m\bowtie I}\cong A_{\m}\bowtie I_{\m}$ if $I\subseteq \m$ and $(A\bowtie I)_{\m\bowtie I}\cong A_{\m}$ if $I\nsubseteq \m$. So Theorem \[AGfqp1\] combined with the known facts that the arithmetical and Gaussian properties are local leads to the conclusion.
As an application of Corollary \[AGfqp2\], one can easily construct new examples of non-local arithmetical, fqp, or Gaussian rings as shown below. The “non-local" assumption here is meant to discriminate against the family of local Prüfer-like rings which can be built via [@AJK Theorem 4.4] or [@BKM Theorem 3.1].
\[fqp7\] Let $A:=\frac{\Z}{12\Z}$, $\m_{1}:=2A$, $\m_{2}:=3A$, and $I:=\m_{1}\m_{2}$. Then $A\bowtie I=A\ltimes I$ is locally an fqp ring which is not arithmetical. Indeed, $\left(\operatorname{Z}(A_{\m_{i}})\right)^{2}=\m_{i}^{2}A\m_{i}=0$ (for $i=1,2$); $I_{\m_{1}}=6A_{\m_{1}}\not=0$; $I_{\m_{2}}=0$; and readily $aI_{\m_{1}}=a^{2}I_{\m_{1}}=0,\ \forall\ a\in\m_{1}$.
\[fqp8\] Is there a satisfactory global analogue of Theorem \[AGfqp1\](3) for the fqp property (i.e, $A$ not necessarily local)?
The weak global dimension and transfer of the semihereditary condition {#WS}
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A ring $R$ is semihereditary if every finitely generated ideal of $R$ is projective [@CE]. Recall for convenience that a ring $R$ has weak global dimension at most 1 (denoted $\operatorname{w.gl.dim}(R)\leq1$) if and only if every finitely generated ideal of $R$ is flat if and only if $R_{p}$ is a valuation domain for every prime ideal $p$ of $R$ [@BG Theorem 3.4]. Therefore, if $\operatorname{w.gl.dim}(R)\leq1$, then $R$ is arithmetical. Also, if $R$ is semihereditary, then $\operatorname{w.gl.dim}(R)\leq1$; and the converse holds if $R$ is coherent (i.e., every finitely generated ideal is finitely presented) [@BG Theorem 3.3].
The main result of this section investigates the weak global dimension of an amalgamation and its possible inheritance of the semihereditary condition.
\[WS1\] Let $A$ be a ring and $I$ a proper ideal of $A$. Then:
1. $\operatorname{w.gl.dim}(A\bowtie I)\leq 1$ if and only if $\operatorname{w.gl.dim}(A)\leq 1$ and $I_{\m}=0,\ \forall\ \m\in\operatorname{Max}(A,I)$.
2. Assume $I$ is finitely generated. Then $A\bowtie I$ is semihereditary if and only if $A$ is semihereditary and $I_{\m}=0,\ \forall\ \m\in\operatorname{Max}(A,I)$.
The proof requires the following lemma which examines the transfer of coherence to amalgamations.
\[WS2\] Let $A$ be a ring and $I$ a proper ideal of $A$. If $A\bowtie I $ is a coherent ring, then so is $A$. The converse holds when $I$ is finitely generated.
If $A\bowtie I$ is coherent, then $A$ is coherent, by [@G1 Theorem 4.1.5], since $A$ is a retract of $A\bowtie I$ via the retraction $\psi : A\bowtie I\longrightarrow A$ defined by $\psi(a,a+i)=a$. Conversely, assume that $A$ is coherent and $I$ is finitely generated. Recall that $I\times 0$ is an ideal of $A\bowtie I$ with $\frac{A\bowtie I}{I\times 0}\cong A$ [@D Remark 1(b)]. We claim that $I\times 0$ is $A\bowtie I$-coherent. Indeed, let $H$ be a finitely generated subideal of $I\times 0$. We will show that $H$ is finitely presented. Clearly, $H:=\sum_{i=1}^{n}A\bowtie I(a_{i},0)$, for some positive integer $n$ and $a_i \in I$. Consider the exact sequence of $A\bowtie I -$modules $$0\rightarrow \operatorname{Ker}(u)\rightarrow (A\bowtie I)^{n}\stackrel{u}\rightarrow H\rightarrow 0$$ where $u(r_{i},r_{i}+e_{i})_{1\leq i\leq n}=\sum_{i=1}^{n}(r_{i},r_{i}+e_{i})(a_{i},0)=(\sum_{i=1}^{n}r_{i}a_{i},0)$. So that $\operatorname{Ker}(u)=\{(r_{i},r_{i}+e_{i})_{1\leq i\leq n}\in(A\bowtie I)^{n}\mid\sum_{i=1}^{n}r_{i}a_{i}=0\}$. Now, set $J :=\sum_{i=1}^{n}Ra_{i}$, a finitely generated subideal of $I$, and consider the exact sequence of $A$-modules $$0\rightarrow \operatorname{Ker}(v)\rightarrow A^{n}\stackrel{v}\rightarrow J\rightarrow 0$$ where $v(b_{i})_{1\leq i\leq n}=\sum_{i=1}^{n}b_{i}a_{i}$. So, under the $A\bowtie I -$module identification $(A\bowtie I)^{n}=A^{n}\bowtie I^{n}$, we have $\operatorname{Ker}(u) = \operatorname{Ker}( v)\bowtie I^{n}$. But $J$ is finitely presented since $A$ is coherent. Hence, $\operatorname{Ker}(v) $ is a finitely generated $A$-module. Whence $\operatorname{Ker}(u)$ is a finitely generated $A\bowtie I$-module (recall $I$ is finitely generated). It follows that $H$ is finitely presented and thus $I\times 0$ is $A\bowtie I$-coherent, as claimed. By [@G1 Theorem 2.4.1(2)], $A\bowtie I$ is coherent, proving the lemma.
\(1) If $I_{\m}=0$ for every $\m\in\operatorname{Max}(A,I)$, then Remark \[P5\] yields $(A\bowtie I)_{\tilde{\m}}\cong(A\bowtie I)_{\m\bowtie I}\cong A_{\m}, \ \forall\ \m\in\operatorname{Max}(A)$, where $\tilde{\m}:=\{(x+i,x)\mid x\in \m,\ i\in I\}$. This fact combined with Corollary \[AGfqp2\](1) leads to the conclusion.
\(2) Combine Lemma \[WS2\] with (1) and the known fact that a ring is semihereditary if and only if it is coherent and has weak global dimension at most 1 [@BG Theorem 3.3].
\[WS3\] By Theorem \[WS1\], $\frac{\Z}{12\Z}\bowtie \frac{4\Z}{12\Z}=\frac{\Z}{12\Z}\ltimes \frac{4\Z}{12\Z}$ is a semihereditary ring since $(\frac{4\Z}{12\Z})_{\frac{2\Z}{12\Z}}=0$. Notice, however, that the above results do not allow for a discrimination among the classes of arithmetical rings, rings with weak global dimension at most 1, and semihereditary rings.
Recall at this point that Osofsky in 1969 (resp., Glaz in 2005) proved that the weak global dimension of an arithmetical (resp., a coherent Gaussian) ring is 0, 1, or $\infty$ [@Os] (resp., [@G2 Theorem 3.3]). One can use amalgamations to build new examples of non-arithmetical non-Coherent Gaussian rings with infinite weak global dimension, as shown in the next example.
\[WS4\] Let $A$ be a local non-coherent Gaussian ring and $0\not=I$ a proper ideal of $A$ with $I^{2}=0$ and $aI=a^{2}I\ \forall\ a\in A$. Assume $0\times I$ is not flat in $A\bowtie I$ (in particular, if $I$ is finitely generated or not flat in $A$). Then the amalgamation $R:=A\ltimes I$ is a local non-arithmetical non-coherent Gaussian ring with $\operatorname{w.gl.dim}(R)=\infty$. For an explicit example, one may take $A:=\Z_{(2)}\ltimes\Q$ and $I:=0\ltimes\Q$.
By Theorem \[AGfqp1\], Theorem \[WS1\], and Lemma \[WS2\], $R$ is a local non-arithmetical non-coherent Gaussian ring with $\operatorname{w.gl.dim}(R)\geq2$. Next, assume $0\times I$ is not flat in $R$. Let $\{f_{i}\}_{i\in \Delta}$ be a set of generators of $I$ and consider the $R$-map $u: R^{(\Delta)}\rightarrow 0\times I$ defined by $u(a_{i},e_{i})_{i\in \Delta}=\sum_{i\in \Delta} (a_{i},e_{i})(0,f_{i})=(0,\sum_{i\in \Delta}a_{i}f_{i})$. Clearly, $\operatorname{Ker}(u)=V\bowtie I^{(\Delta)}$, where $V:=\{(a_{i})_{i}\in A^{(\Delta)}/\sum_{i\in \Delta}a_{i}f_{i}=0 \}$. Here we are identifying $R^{(\Delta)}$ with $A^{(\Delta)}\bowtie I^{(\Delta)}$ as $R$-modules. We have the exact sequence of $R$ modules $$0\rightarrow V\bowtie I^{(\Delta)}\rightarrow R^{(\Delta)}\stackrel{u}\rightarrow 0\times I\rightarrow 0.$$ On the other hand, $V\bowtie I^{(\Delta)}=V^{\star}\oplus (0\times I)^{(\Delta)}$, where $V^{\star}=\{(a,a)/ r\in V\}$. Since $0\times I$ is not flat, the above sequence yields $$\operatorname{fd}(0\times I)\leq \operatorname{fd}\left(V^{\star}\oplus (0\times I)^{(\Delta)}\right)\leq \operatorname{fd}(0\times I)-1.$$ Therefore $\operatorname{fd}(0\times I)=\operatorname{w.gl.dim}(R)=\infty$, as desired.
Now, if $I$ is finitely generated, then $0\times I$ is not $R$-flat since $R$ is local and $(a,0)(0\times I)= 0$ for any $0\not=a\in I$. Also, using the interpretation of flatness in terms of relations [@Bour Ch. I, §2, Corollary 1], one can easily verify that if $0\times I$ is $R$-flat, then $I$ is $A$-flat.
For the explicit example, it is readily seen that $I^{2}=0$ and $aI=a^{2}I\ \forall\ a\in 2\Z_{(2)}\ltimes\Q$. Moreover, $A$ is an arithmetical (hence Gaussian) ring by [@BKM Theorem 2.1] and not coherent by [@KM Theorem 2.8]. Finally, we claim that $I:=0\ltimes\Q$ is not flat in $A:=\Z_{(2)}\ltimes\Q$ since it is not flat in $\Q\ltimes\Q$. The evidence for this fact is already included in the proof of [@BKM Lemma 2.3].
Is Theorem \[WS1\](2) true if $I$ is not assumed to be finitely generated?
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[^1]: $^{(\star)}$ Supported by King Fahd University of Petroleum & Minerals under Research Project \#: RG1208-1/2.
[^2]: $^{(1)}$ Corresponding author.
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abstract: 'In this paper, we solve a generalized Klein-Gordon oscillator in the cosmic string space-time with a scalar potential of Cornell-type within the Kaluza-Klein theory and obtain the relativistic energy eigenvalues and eigenfunctions. We extend this analysis by replacing the Cornell-type with Coulomb-type potential in the magnetic cosmic string space-time and analyze a relativistic analogue of the Aharonov-Bohm effect for bound states.'
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=18 pt
[**Faizuddin Ahmed**]{}[^1]\
[**Ajmal College of Arts and Science, Dhubri-783324, Assam, India**]{}
[**keywords:**]{} Klein-Gordon oscillator, topological defects, Kaluza-Klein theory, special functions, Aharonov-Bohm effect.
[**PACS Number(s):**]{} 03.65.Pm, 03.65.Ge, 04.50.Cd, 04.50.+h, 11.10.Kk
Introduction
============
A unified formulation of Einstein’s theory of gravitation and theory of electromagnetism in four-dimensional space-time was first proposed by Kaluza [@bb12] by assuming a pure gravitational theory in five-dimensional space-time. The so called cylinder condition was later explained by Klein when the extra dimension was compactified on a circle $S^1$ with a microscopic radius [@bb13], where the spatial dimension becomes five-dimensional. The idea behind introducing additional space-time dimensions has found application in quantum field theory, for instance, in string theory [@bb26]. There are studies on Kaluza-Klein theory with torsion [@bb31; @bb32], in the Grassmannian context [@ss1; @ss2; @ss3], in Kähler fields [@ss4], in the presence of fermions [@bb33; @bb34; @bb35], and in the Lorentz-symmetry violation (LSV) [@bb36; @bb37; @bb38]. Also, there are investigations in space-times with topological defect in the context of Kaluza-Klein theory, for example, the magnetic cosmic string [@bb14] (see also, [@ss5]), and magnetic chiral cosmic string [@bb28] in five-dimensions.
Aharonov-Bohm effect [@bb39; @bb40; @bb50] is a quantum mechanical phenomena that has been investigated in several branches of physics, such as in, graphene [@RJ], Newtonian theory [@MAA], bound states of massive fermions [@VRK], scattering of dislocated wave-fronts [@CC], torsion effect on a relativistic position-dependent mass system [@ff3; @AHEP], non-minimal Lorentz-violating coupling [@HB]. In addition, Aharonov-Bohm effect has been investigated in the context of Kaluza-Klein theory by several authors [@bb28; @bb15; @bb16; @aa6; @EVBL; @EVBL2; @EPJC], and the geometric quantum phase in graphene [@KB]. It is well-known in condensed matter [@NB; @WCT; @LD; @MBu; @ACB] and in the relativistic quantum systems [@Bakke; @Bakke2] that when there exist dependence of the energy eigenvalues on geometric quantum phase [@bb50], then, persistent current arise in the systems. The studies of persistent currents have explored systems that deal with the Berry phase [@DL; @DL2], the Aharonov-Anandan quantum phase [@XCG; @TZQ], and the Aharonov-Casher geometric quantum phase [@AVB; @SO; @HM2; @HM3]. Investigation of magnetization and persistent currents of mass-less Dirac Fermions confined in a quantum dot in a graphene layer with topological defects was studied in [@cc17].
Klein-Gordon oscillator theory [@bb1; @bb2] was inspired by the Dirac oscillator [@bb3]. This oscillator field is used to study the spectral distribution of energy eigenvalues and eigenfunctions in $1-d$ version of Minkowski space-times [@bb4]. Klein-Gordon oscillator was studied by several authors, such as, in the cosmic string space-time with an external fields [@bb5], with Coulomb-type potential by two ways : (i) modifying the mass term $m \rightarrow m+S(r)$ [@bb6], and (ii) via the minimal coupling [@bb7] in addition to a linear scalar potential, in the background space-time generated by a cosmic string [@bb8], in the Gödel-type space-times under the influence of gravitational fields produced by topological defects [@bb9], in the Som-Raychaudhuri space-time with a disclination parameter [@ff2], in non-commutative (NC) phase space [@bb10], in $(1+2)$-dimensional Gürses space-time background [@ff4], and in $(1+2)$-dimensional Gürses space-time background subject to a Coulomb-type potential [@ff5]. The relativistic quantum effects on oscillator field with a linear confining potential was investigated in [@ff6].
We consider a generalization of the oscillator as described in Refs. [@EPJC; @ff5] for the Klein-Gordon. This generalization is introduced through a generalized momentum operator where the radial coordinate $r$ is replaced by a general function $f (r)$. To author best knowledge, such a new coupling was first introduced by K. Bakke [*et al.*]{} in Ref. [@Bakke] and led to a generalization of the Tan-Inkson model of a two-dimensional quantum ring for systems whose energy levels depend on the coupling’s control parameters. Based on this, a generalized Dirac oscillator in the cosmic string space-time was studied by F. Deng [*et al.*]{} in Ref. [@cc9] where the four-momentum $p_{\mu}$ is replaced with its alternative $p_{\mu}+m\,\omega\,\beta\,f_{\mu} ( x_{\mu} )$. In the literature, $f_{\mu} (x_{\mu})$ has chosen similar to potentials encountered in quantum mechanics (Cornell-type, exponential-type, singular, Morse-type, Yukawa-like etc.). A generalized Dirac oscillator in $(2+1)$-dimensional world was studied in [@cc10]. Very recently, the generalized K-G oscillator in the cosmic string space-time in [@FD], and non-inertial effects on a generalized DKP oscillator in the cosmic string space-time in [@SZ] was studied.
The relativistic quantum dynamics of a scalar particle of mass $m$ with a scalar potential $S (r)$ [@bb41; @WG] is described by the following Klein-Gordon equation: $$\left [\frac{1}{\sqrt{-g}}\,\partial_{\mu} (\sqrt{-g}\,g^{\mu\nu}\,\partial_{\nu})-(m + S)^2 \right]\,\Psi=0,
\label{1}$$ with $g$ is the determinant of metric tensor with $g^{\mu\nu}$ its inverse. To couple Klein-Gordon field with oscillator [@bb1; @bb2], following change in the momentum operator is considered as in [@dd2; @bb5]: $$\vec{p}\rightarrow \vec{p}+i\,m\,\omega\,\vec{r},
\label{2}$$ where $\omega$ is the oscillatory frequency of the particle and $\vec{r}=r\,\hat{r}$ where, $r$ being distance from the particle to the string. To generalized the Klein-Gordon oscillator, we adopted the idea considered in Refs. [@EPJC; @ff5; @cc9; @FD; @SZ] by replacing $r \rightarrow f (r)$ as $$X_{\mu}=(0, f(r), 0, 0, 0).
\label{3}$$ So we can write $\vec{p}\rightarrow \vec{p}+i\,m\,\omega\,f (r)\,\hat{r}$, and we have, $p^2 \rightarrow (\vec{p}+i\,m\,\omega\, f (r)\,\hat{r})(\vec{p}-i\,m\,\omega\,f (r)\,\hat{r})$. Therefore, the generalized Klein-Gordon oscillator equation: $$\left[\frac{1}{\sqrt{-g}}\,(\partial_{\mu}+m\,\omega\,X_{\mu})\{\sqrt{-g}\,g^{\mu\nu}\,(\partial_{\nu}-m\,\omega\,X_{\nu})\}-(m+ S)^2 \right]\,\Psi=0,
\label{4}$$ where $X_{\mu}$ is given by Eq. (\[3\]).
Various potentials have been used to investigate the bound state solutions to the relativistic wave-equations. Among them, much attention has given on Coulomb-type potential. This kind of potential has widely used to study various physical phenomena, such as in, the propagation of gravitational waves [@gg], the confinement of quark models [@gg1], molecular models [@gg2], position-dependent mass systems [@gg3; @gg4; @gg5], and relativistic quantum mechanics [@bb7; @bb8; @bb6]. The Coulomb-type potential is given by $$S(r)=\frac{\eta_{c}}{r}.
\label{5}$$ where $\eta_{c}$ is Coulombic confining parameter.
Another potential that we are interest here is the Cornell-type potential. The Cornell potential, which consists of a linear potential plus a Coulomb potential, is a particular case of the quark-antiquark interaction, one more harmonic type term [@gg6]. The Coulomb potential is responsible by the interaction at small distances and the linear potential leads to the confinement. Recently, the Cornell potential has been studied in the ground state of three quarks [@CA]. However, this type of potential is worked on spherical symmetry; in cylindrical symmetry, which is our case, this type of potential is known as Cornell-type potential [@bb9]. This type of interaction has been studied in [@bb9; @ff6; @bb41; @RLLV]. Given this, let us consider this type of potential $$S(r)=\eta_{L}\,r+\frac{\eta_c}{r},
\label{6}$$ where $\eta_{L}, \eta_{c}$ are the confining potential parameters.
The aim of the present work is to analyze a relativistic analogue of the Aharonov-Bohm effect for bound states [@bb39; @bb40; @bb50] for a relativistic scalar particle with potential in the context of Kaluza-Klein theory. First, we study a relativistic scalar particle by solving the generalized Klein-Gordon oscillator with a Cornell-type potential in the five-dimensional cosmic string space-time. Secondly, by using the Kaluza-Klein theory [@bb12; @bb13; @bb26] a magnetic flux through the line-element of the cosmic string space-time is introduced, and thus write the generalized Klein-Gordon oscillator in the five-dimensional space-time. In the later case, a Coulomb-type potential by modifying the mass term $m \rightarrow m + S(r)$ is introduced which was not study earlier. Then, we show that the relativistic bound states solutions can be achieved, where the relativistic energy eigenvalues depend on the geometric quantum phase [@bb50]. Due to this dependence of the relativistic energy eigenvalue on the geometric quantum phase, we calculate the persistent currents [@NB; @WCT] that arise in the relativistic system.
This paper comprises as follow : In [*section 2*]{}, we study a generalized Klein-Gordon oscillator in the cosmic string background within the Kaluza-Klein theory with a Cornell-type scalar potential; in [*section 3*]{}, a generalized Klein-Gordon oscillator in the magnetic cosmic string in the Kaluza-Klein theory subject to a Coulomb-type scalar potential and obtain the energy eigenvalues and eigenfunctions; and the conclusion one in [*section 4*]{}.
Generalized Klein-Gordon oscillator in cosmic string space-time with a Cornell-type potential in Kaluza-Klein theory
====================================================================================================================
The purpose of this section is to study the Klein-Gordon equation in cosmic string space-time with the use of Kaluza-Klein theory with interactions. The first study of the topological defects within the Kaluza-Klein theory was carried out in [@bb14]. The metric corresponding to this geometry can be written as, $$ds^2=-dt^2+dr^2+\alpha^2\,r^2\,d\phi^2+dz^2+dx^2,
\label{7}$$ where $t$ is the time coordinate, $x$ is the coordinate associated with the fifth additional dimension and $(r, \phi, z)$ are cylindrical coordinates. These coordinates assume the ranges $-\infty < (t, z) < \infty$, $0 \leq r < \infty$, $0 \leq \phi \leq 2\,\pi$, $0 < x < 2\,\pi\,a$, where $a$ is the radius of the compact dimension $x$. The $\alpha$ parameter characterizing the cosmic string, and in terms of mass density $\mu$ given by $\alpha=1-4\,\mu$ [@bb21]. The cosmology and gravitation imposes limits to the range of the $\alpha$ parameter which is restricted to $\alpha <1$ [@bb21].
By considering the line element (\[7\]) into the Eq. (\[4\]), we obtain the following differential equation : $$\begin{aligned}
&&[-\frac{\partial^2}{\partial t^2}+\frac{1}{r}\,\left(\frac{\partial}{\partial r} + m\,\omega\,f (r) \right)\,\left (r\,\frac{\partial}{\partial r}-m\,\omega\,r\,f (r) \right)+\frac{1}{\alpha^2\,r^2}\,\frac{\partial^2}{\partial \phi^2}+\frac{\partial^2}{\partial z^2}\nonumber\\&&+\frac{\partial}{\partial x^2}
-(m+ S(r))^2]\,\Psi (t, r, \phi, z, x)=0,\nonumber\\
&&[-\frac{\partial^2}{\partial t^2}+\frac{\partial^2}{\partial r^2}+\frac{1}{r}\,\frac{\partial}{\partial r}-m\,\omega\,\left (f'(r)+\frac{f (r)}{r} \right)-m^2\,\omega^2\,f^{2} (r)+\frac{1}{\alpha^2\,r^2}\,\frac{\partial^2}{\partial \phi^2}\nonumber\\
&&+\frac{\partial^2}{\partial z^2}+\frac{\partial}{\partial x^2}-(m+ S(r))^2]\,\Psi (t, r, \phi, z, x)=0.
\label{8}\end{aligned}$$ Since the metric is independent of $t, \phi ,z, x$. One can choose the following ansatz for the function $\Psi$ $$\Psi (t, r, \phi, z, x)=e^{i\,(-E\,t+l\,\phi+k\,z+q\,x)}\,\psi(r),
\label{9}$$ where $E$ is the total energy, $l=0,\pm\,1,\pm\,2..$, and $k, q$ are constants.
Substituting the above ansatz into the Eq. (\[8\]), we get the following equation for $\psi (r)$ : $$\begin{aligned}
&&[ \frac{d^2}{dr^2} + \frac{1}{r}\,\frac{d}{dr} + E^2-k^2-q^2-\frac{l^2}{\alpha^2\,r^2}-m\,\omega\,\left (f'(r)+\frac{f (r)}{r} \right)\nonumber\\
&&-m^2\,\omega^2\,f^{2}(r)-\left(m+S(r) \right)^2]\,\psi(r)=0.
\label{10}\end{aligned}$$
We choose the function $f(r)$ a Cornell-type given by [@EPJC; @ff5; @cc9; @SZ] $$f(r)=a\,r+\frac{b}{r}\quad,\quad a, b>0.
\label{11}$$
Substituting the function (\[11\]) and Cornell potential (\[6\]) into the Eq. (\[9\]), we obtain the following equation: $$\left [\frac{d^2}{dr^2} + \frac{1}{r}\,\frac{d}{dr} + \lambda-\Omega^2\,r^2-\frac{j^2}{r^2}-\frac{2\,m\,\eta_{c}}{r}-2\,m\,\eta_{L}\,r \right]\,\psi(r)=0,
\label{12}$$ where $$\begin{aligned}
&&\lambda=E^2-k^2-q^2-m^2-2\,m\,\omega\,a-2\,m^2\,\omega^2\,a\,b-2\,\eta_{L}\,\eta_{c},\nonumber\\
&&\Omega=\sqrt{m^2\,\omega^2\,a^2+\eta^2_{L}},\nonumber\\
&&j=\sqrt{\frac{l^2}{\alpha^2}+m^2\,\omega^2\,b^2+\eta^{2}_{c}}.
\label{13}\end{aligned}$$ Transforming $\rho=\sqrt{\Omega}\,r$ into the equation (\[12\]), we get $$\left [\frac{d^2}{d\rho^2} + \frac{1}{\rho}\,\frac{d}{d\rho} + \zeta-\rho^2-\frac{j^2}{\rho^2}-\frac{\eta}{\rho}-\theta\,\rho \right]\,\psi (\rho)=0,
\label{14}$$ where $$\zeta=\frac{\lambda}{\Omega}\quad,\quad \eta=\frac{2\,m\,\eta_c}{\sqrt{\Omega}}\quad,\quad \theta=\frac{2\,m\,\eta_L}{\Omega^{\frac{3}{2}}}.
\label{15}$$
Let us impose that $\psi (\rho) \rightarrow 0$ when $\rho \rightarrow 0$ and $\rho \rightarrow \infty$. Suppose the possible solution to the Eq. (\[14\]) is $$\psi (\rho)=\rho^{j}\,e^{-\frac{1}{2}\,(\rho+\theta)\,\rho}\,H (\rho).
\label{16}$$ Substituting the solution Eq. (\[16\]) into the Eq. (\[14\]), we obtain $$H''(\rho)+\left [\frac{\gamma}{\rho}-\theta-2\,\rho \right ]\,H'(\rho)+\left [-\frac{\beta}{\rho}+\Theta \right]\,H (\rho)=0,
\label{17}$$ where $$\begin{aligned}
&&\gamma=1+2\,j,\nonumber\\
&&\Theta=\zeta+\frac{\theta^2}{4}-2\,(1+j),\nonumber\\
&&\beta=\eta+\frac{\theta}{2}\,(1+2\,j).
\label{18}\end{aligned}$$ Equation (\[17\]) is the biconfluent Heun’s differential equation [@ff3; @AHEP; @bb15; @bb16; @aa6; @EVBL; @EVBL2; @EPJC; @bb7; @bb8; @bb9; @ff2; @ff5; @ff6; @bb41; @bb42; @bb46; @bb47; @dd51; @dd52] and $H (\rho)$ is the Heun polynomials.
The above equation (\[17\]) can be solved by the Frobenius method. We consider the power series solution around the origin [@bb43] $$H (\rho)=\sum_{i=0}^{\infty}\,c_{i}\,\rho^{i}
\label{19}$$ Substituting the above power series solution into the Eq. (\[17\]), we obtain the following recurrence relation for the coefficients: $$c_{n+2}=\frac{1}{(n+2)(n+2+2\,j)}\,\left[\left\{\beta+\theta\,(n+1) \right\}\,c_{n+1}-(\Theta-2\,n)\,c_{n} \right].
\label{20}$$ And the various coefficients are $$\begin{aligned}
&&c_1=\left(\frac{\eta}{\gamma}-\frac{\theta}{2} \right)\,c_0,\nonumber\\
&&c_2=\frac{1}{4\,(1+j)}\,[\left(\beta+\theta \right)\,c_{1}-\Theta\,c_{0}].
\label{21}\end{aligned}$$
The quantum theory requires that the wave function $\Psi$ must be normalized. The bound state solutions $\psi (\rho)$ can be obtained because there is no divergence of the wave function at $\rho \rightarrow 0$ and $\rho \rightarrow \infty$. Since we have written the function $H (\rho)$ as a power series expansion around the origin in Eq. (\[19\]). Thereby, bound state solutions can be achieved by imposing that the power series expansion (\[19\]) becomes a polynomial of degree $n$. Through the recurrence relation (\[20\]), we can see that the power series expansion (\[19\]) becomes a polynomial of degree $n$ by imposing two conditions [@ff3; @AHEP; @bb15; @bb16; @aa6; @EVBL; @EVBL2; @EPJC; @bb7; @bb8; @bb9; @ff2; @ff5; @ff6; @bb41; @bb42; @bb46; @bb47]: $$\begin{aligned}
\Theta&=&2\,n \quad (n=1,2,...),\nonumber\\
c_{n+1}&=&0
\label{23}\end{aligned}$$
By analyzing the condition $\Theta=2\,n$, we get expression of the energy eigenvalues $E_{n,l}$: $$\begin{aligned}
&&\frac{\lambda}{\Omega}+\frac{\theta^2}{4}-2\,(1+j)=2\,n\nonumber\\\Rightarrow
&&E^{2}_{n,l}=k^2+q^2+m^2+2\,\Omega\,\left(n+1+\sqrt{\frac{l^2}{\alpha^2}+m^2\,\omega^2\,b^2+\eta^{2}_{c}} \right)\nonumber\\
&&+2\,m^2\,\omega^2\,a\,b+2\,m\,\omega\,a+2\,\eta_{L}\,\eta_c-\frac{m^2\,\eta^2_{L}}{\Omega^2}.
\label{24}\end{aligned}$$
We plot graphs of the above energy eigenvalues w. r. t. different parameters. In fig. 1, the energy eigenvalues $E_{1,1}$ against the parameter $\eta_c$. In fig. 2, the energy eigenvalues $E_{1,1}$ against the parameter $\eta_L$. In fig. 3, the energy eigenvalues $E_{1,1}$ against the parameter $M$. In fig. 4, the energy eigenvalues $E_{1,1}$ against the parameter $\omega$. In fig. 5, the energy eigenvalues $E_{1,1}$ against the parameter $\Omega$.
Now we impose additional condition $c_{n+1}=0$ to find the individual energy levels and corresponding wave functions one by one as done in [@bb44; @bb45]. As example, for $n=1$, we have $\Theta=2$ and $c_2=0$ which implies $$\begin{aligned}
&&c_1=\frac{2}{\beta+\theta}\,c_0\Rightarrow\left(\frac{\eta}{1+2\,j}-\frac{\theta}{2} \right)=\frac{2}{\beta+\theta}\nonumber\\
&&\Omega^3_{1,l}-\frac{\eta^2}{2\,(1+2\,j)}\Omega^2_{1,l}-\eta\,\theta\,(\frac{1+j}{1+2\,j})\,\Omega_{1,l}-\frac{\theta^2}{8}\,(3+2\,j)=0
\label{25}\end{aligned}$$ a constraint on the parameter $\Omega_{1,l}$. The relation given in Eq. (\[25\]) gives the possible values of the parameter $\Omega_{1,l}$ that permit us to construct first degree polynomial to H(x) for $n=1$. Note that its values changes for each quantum number $n$ and $l$, so we have labeled $\Omega \rightarrow \Omega_{n,l}$. Besides, since this parameter is determined by the frequency, hence, the frequency $\omega_{1,l}$ is so adjusted that the Eq. (\[25\]) can be satisfied, where we have simplified our notation by labeling: $$\omega_{1,l}=\frac{1}{m\,a}\sqrt{\Omega^2_{1,l}-\eta^2_{L}}.
\label{26}$$ It is noteworthy that a third-degree algebraic equation (\[25\]) has at least one real solution and it is exactly this solution that gives us the allowed values of the frequency for the lowest state of the system, which we do not write because its expression is very long. We can note, from Eq. (\[25\]) that the possible values of the frequency depend on the quantum numbers and the potential parameter. In addition, for each relativistic energy level, we have a different relation of the magnetic field associated to the Cornell-type potential and quantum numbers of the system $\{l, n \}$. For this reason, we have labeled the parameters $\Omega$ and $\omega$ in Eqs. (\[25\]) and (\[26\]).
Therefore, the ground state energy level and corresponding wave-function for $n=1$ are given by $$\begin{aligned}
&&E^{2}_{1,l}=k^2+q^2+m^2+2\,\Omega_{1,l}\,\left(2+\sqrt{\frac{l^2}{\alpha^2}+m^2\,\omega^2\,b^2+\eta^{2}_{c}} \right)\nonumber\\
&&+2\,m^2\,\omega^2_{1,l}\,a\,b+2\,m\,\omega_{1,l}\,a+2\,\eta_{L}\,\eta_c-\frac{m^2\,\eta^2_{L}}{\Omega^2_{1,l}},\nonumber\\
&&\psi_{1,l}=\rho^{\sqrt{\frac{l^2}{\alpha^2}+m^2\,\omega^2_{1,l}\,b^2+\eta^{2}_{c}}}\,e^{-\frac{1}{2}\,\left (\frac{2\,m\,\eta_L}{\Omega^{\frac{3}{2}}_{1,l}}+\rho \right)\,\rho}\,\left(c_0+c_1\,\rho\right),
\label{27}\end{aligned}$$ where $$\begin{aligned}
c_1&=&\frac{1}{\Omega^{\frac{1}{2}}_{1,l}}\,\left [\frac{2\,m\,\eta_c}{\left(1+2\,\sqrt{\frac{l^2}{\alpha^2}+m^2\,\omega^2_{1,l}\,b^2+\eta^{2}_{c}} \right)}-\frac{m\,\eta_L}{\Omega_{1,l}} \right]\,c_0.
\label{28}\end{aligned}$$ Then, by substituting the real solution of Eq. (\[26\]) into the Eqs. (\[27\])-(\[28\]) it is possible to obtain the allowed values of the relativistic energy for the radial mode $n=1$ of a position dependent mass system. We can see that the lowest energy state defined by the real solution of the algebraic equation given in Eq. (\[26\]) plus the expression given in Eq. (\[27\]) is defined by the radial mode $n=1$, instead of $n=0$. This effect arises due to the presence of the Cornell-type potential in the system.
For $\alpha \rightarrow 1$, the relativistic energy eigenvalue (\[25\]) becomes $$\begin{aligned}
&&E^{2}_{n,l}=k^2+q^2+m^2+2\,\Omega\,\left(n+1+\sqrt{l^2+m^2\,\omega^2\,b^2+\eta^{2}_{c}} \right)\nonumber\\
&&+2\,m^2\,\omega^2\,a\,b+2\,m\,\omega\,a+2\,\eta_{L}\,\eta_c-\frac{m^2\,\eta^2_{L}}{\Omega^2}.
\label{29}\end{aligned}$$ Equation (\[29\]) is the relativistic energy eigenvalue of a scalar particles via the generalized Klein-Gordon oscillator subject to a Cornell-type potential in the Minkowski space-time in the Kaluza-Klein theory.
We discuss bellow a very special case of the above relativistic system.
: Considering $\eta_{L}=0$, that is, only Coulomb-type potential $S(r)=\frac{\eta_c}{r}$.
We want to investigate the effect of Coulomb-type potential on a scalar particle in the background of cosmic string space-time in the Kaluza-Klein theory. In that case, the radial wave-equation Eq. (\[12\]) becomes $$\left [\frac{d^2}{dr^2}+\frac{1}{r}\,\frac{d}{dr}+\lambda_0-m^2\,\omega^2\,a^2\,r^2-\frac{j^2}{r^2}-\frac{2\,m\,\eta_{c}}{r} \right]\,\psi(r)=0,
\label{aa1}$$ where $$\lambda_0=E^2-k^2-q^2-m^2-2\,m\,\omega\,a-2\,m^2\,\omega^2\,a\,b
\label{aa2}$$
Transforming $\rho=\sqrt{m\,\omega\,a}\,r$ into the Eq. (\[aa1\]), we get $$\left [\frac{d^2}{d\rho^2}+\frac{1}{\rho}\,\frac{d}{d\rho}+\frac{\lambda_0}{m\,\omega\,a}-\rho^2-\frac{j^2}{\rho^2}-\frac{2\,m\,\eta_c}{\sqrt{m\,\omega\,a}}\,\frac{1}{\rho} \right]\,\psi(\rho)=0.
\label{aa6}$$ Suppose the possible solution to Eq. (\[aa6\]) is $$\psi (\rho)=\rho^{j}\,E^{-\frac{\rho^2}{2}}\,H (\rho).
\label{aa7}$$ Substituting the solution Eq. (\[aa7\]) into the Eq. (\[aa6\]), we obtain $$H ''(\rho)+\left [\frac{1+2\,j}{\rho}-2\,\rho \right]\, H' (\rho)+\left[-\frac{\tilde{\eta}}{\rho}+\frac{\lambda_0}{m\,\omega\,a}-2\,(1+j) \right]\, H (\rho),
\label{aa8}$$ where $\tilde{\eta}=\frac{2\,m\,\eta_c}{\sqrt{m\,\omega\,a}}$. Equation (\[aa8\]) is the Heun’s differential equation [@ff3; @AHEP; @bb15; @bb16; @aa6; @EVBL; @EVBL2; @EPJC; @bb7; @bb8; @bb9; @ff2; @ff5; @ff6; @bb41; @bb42; @bb46; @bb47; @dd51; @dd52] with $H (\rho)$ is the Heun polynomial.
Substituting the power series solution Eq. (\[19\]) into the Eq. (\[aa8\]), we obtain the following recurrence relation for coefficients $$c_{n+2}=\frac{1}{(n+2)(n+2+2\,j)}\,\left [\tilde{\eta}\,c_{n+1}-\{\frac{\lambda_0}{m\,\omega\,a}-2\,(1+j)-2\,n \}\,c_n \right]
\label{aa9}$$ The power series solution becomes a polynomial of degree $n$ provided [@ff3; @AHEP; @bb15; @bb16; @aa6; @EVBL; @EVBL2; @EPJC; @bb7; @bb8; @bb9; @ff2; @ff5; @ff6; @bb41; @bb42; @bb46; @bb47] $$\begin{aligned}
\frac{\lambda_0}{m\,\omega\,a}-2\,(1+j)&=&2\,n\quad (n=1,2,...)\nonumber\\
c_{n+1}&=&0.
\label{aa10}\end{aligned}$$
Using the first condition, one will get the following energy eigenvalues of the relativistic system : $$\begin{aligned}
&&E_{n,l}=\pm\{k^2+q^2+m^2+2\,m\,\omega\,a\,\left(n+2+\sqrt{\frac{l^2}{\alpha^2}+m^2\,\omega^2\,b^2+\eta^{2}_c} \right)\nonumber\\
&&+2\,m^2\,\omega^2\,a\,b\}^{\frac{1}{2}}.
\label{aa3}\end{aligned}$$
The ground state energy levels and corresponding wave-function for $n=1$ are given by $$\begin{aligned}
&&E_{1,l}=\pm\{k^2+q^2+m^2+2\,m\,\omega_{1,l}\,a\,\left(3+\sqrt{\frac{l^2}{\alpha^2}+m^2\,\omega^2\,b^2+\eta^{2}_{c}} \right)\nonumber\\
&&+2\,m^2\,\omega^2\,a\,b \}^{\frac{1}{2}},\nonumber\\
&&\psi_{1,l} (\rho)=\rho^{\sqrt{\frac{l^2}{\alpha^2}+m^2\,\omega^2_{1,l}\,b^2+\eta^{2}_{c}}}\,e^{-\frac{\rho^2}{2}}\,\left(c_0+c_1\,\rho \right),
\label{aa4}\end{aligned}$$ where $$\begin{aligned}
c_1&=&\frac{2\,m\,\eta_{c}}{\sqrt{m\,\omega_{1,l}\,a}\,\left(1+2\,\sqrt{\frac{l^2}{\alpha^2}+m^2\,\omega^2_{1,l}\,b^2+\eta^{2}_{c}}\right)}\nonumber\\
&=&\left(\frac{2}{1+2\,\sqrt{\frac{l^2}{\alpha^2}+m^2\,\omega^2_{1,l}\,b^2+\eta^{2}_{c}}}\right)^{\frac{1}{2}}\,c_0,\nonumber\\
\omega_{1,l}&=&\frac{2\,m\,\eta^2_{c}}{a\,\left(1+2\,\sqrt{\frac{l^2}{\alpha^2}+m^2\,\omega^2_{1,l}\,b^2+\eta^{2}_c} \right)}.
\label{aa5}\end{aligned}$$ a constraint on the frequency parameter $\omega_{1,l}$.
:
We consider another case corresponds to $a \rightarrow 0$, $b \rightarrow 0$ and $\eta_{L}=0$, that is, a scalar quantum particle in the cosmic string background subject to a Coulomb-type scalar potential within the Kaluza-Klein theory. In that case, from Eq. (\[12\]) we obtain the following equation: $$\psi''(r)+\frac{1}{r}\,\psi'(r)+[\tilde{\lambda}-\frac{\tilde{j}^2}{r^2}-\frac{2\,m\,\eta_{c}}{r}]\,\psi(r)=0.
\label{bb1}$$ Equation (\[bb1\]) can be written as $$\psi''(r)+\frac{1}{r}\,\psi'(r)+\frac{1}{r^2}\,(-\xi_1\,r^2+\xi_2\,r-\xi_3)\,\psi(r)=0,
\label{bb2}$$ where $$\xi_1=-\tilde{\lambda}=-(E^2-k^2-q^2-m^2),\quad \xi_2=-2\,m\,\eta_{c},\quad \xi_3=\tilde{j}^2=\frac{l^2}{\alpha^2}+\eta^2_{c}.
\label{bb3}$$
Compairing the Eq (\[bb2\]) with Equation (\[A.1\]) in appendix A, we get $$\begin{aligned}
&&\alpha_1=1,\quad \alpha_2=0,\quad \alpha_3=0,\quad \alpha_4=0,\quad \alpha_5=0,\quad \alpha_6=\xi_1,\nonumber\\
&&\alpha_7=-\xi_2,\quad \alpha_8=\xi_3,\quad \alpha_9=\xi_1,\quad \alpha_{10}=1+2\,\sqrt{\xi_3},\nonumber\\
&&\alpha_{11}=2\,\sqrt{\xi_1},\quad \alpha_{12}=\sqrt{\xi_3},\quad \alpha_{13}=-\sqrt{\xi_1}.
\label{bb6}\end{aligned}$$
The energy eigenvalues using Eqs. (\[bb3\])-(\[bb6\]) into the Eq. (\[A.8\]) in appendix A is given by $$E_{n,l}=\pm\,m\,\sqrt{1-\frac{\eta^2_{c}}{(n+\sqrt{\frac{l^2}{\alpha^2}+\eta^{2}_{c}}+\frac{1}{2})^2}+\frac{k^2}{m^2}+\frac{q^2}{m^2}},
\label{bb4}$$ where $n=0,1,2,..$ is the quantum number associated with the radial modes, $l=0,\pm\,1,\pm\,2,.$ are the quantum number associated with the angular momentum operator, $k$ and $q$ are arbitrary constants. Equation (\[bb4\]) corresponds to the relativistic energy eigenvalues of a free-scalar particle subject to a Coulomb-type scalar potential in the background of cosmic string within the Kaluza-Klein theory.
The corresponding radial wave-function is given by $$\begin{aligned}
\psi_{n,l} (r)&=&|N|\,r^{\frac{\tilde{j}}{2}}\,{\sf e}^{-\frac{r}{2}}\,{\sf L}^{(\tilde{j})}_{n} (r)\nonumber\\
&=&|N|\,r^{\frac{1}{2}\sqrt{\frac{l^2}{\alpha^2}+\eta^2_{c}}}\,{\sf e}^{-\frac{r}{2}}\,{\sf L}^{(\sqrt{\frac{l^2}{\alpha^2}+\eta^2_{c}})}_{n} (r).
\label{bb7}\end{aligned}$$ Here $|N|$ is the normalization constant and ${\sf L}^{(\sqrt{\frac{l^2}{\alpha^2}+\eta^2_{c}})}_{n} (r) $ is the generalized Laguerre polynomial.
For $\alpha \rightarrow 1$, the relativistic energy eigenvalues Eq. (\[bb4\]) becomes $$E_{n,l}=\pm\,m\,\sqrt{1-\frac{\eta^2_{c}}{(n+\sqrt{l^2+\eta^{2}_{c}}+\frac{1}{2})^2}+\frac{k^2}{m^2}+\frac{q^2}{m^2}}.
\label{bb5}$$ Equation (\[bb5\]) correspond to the relativistic energy eigenvalue of a scalar particle subject to a Coulomb-type scalar potential in the Minkowski space-time within the Kaluza-Klein theory.
Generalized Klein-Gordon oscillator in the magnetic cosmic string with a Coulomb-type potential in Kaluza-Klein theory
======================================================================================================================
Let us consider the quantum dynamics of a particle moving in the magnetic cosmic string background. In the Kaluza-Klein theory [@bb12; @bb13; @bb28], the corresponding metrics with Aharonov-Bohm magnetic flux $\Phi$ passing along the symmetry axis of the string assumes the following form $$ds^2=-dt^2+dr^2+\alpha^2\,r^2\,d\phi^2+dz^2+(dx+\frac{\Phi}{2\,\pi}\,d\phi)^2
\label{30}$$ with cylindrical coordinates are used. The quantum dynamics is described by the equation (\[4\]) with the following change in the inverse matrix tensor $g^{\mu\nu}$, $$g^{\mu\nu}=\left (\begin{array}{lllll}
-1
& 0 & \quad 0 & 0 & \quad 0 \\
\quad 0 & 1 & \quad 0 & 0 & \quad 0 \\
\quad 0 & 0 & \quad \frac{1}{\alpha^2\,r^2} & 0 & -\frac{\Phi}{2\,\pi\,\alpha^2\,r^2} \\
\quad 0 & 0 & \quad 0 & 1 & \quad 0 \\
\quad 0 & 0 & -\frac{\Phi}{2\,\pi\,\alpha^2\,r^2} & 0 & 1+\frac{\Phi^2}{4\,\pi^2\,\alpha^2\,r^2}
\end{array} \right).
\label{31}$$ By considering the line element (\[30\]) into the Eq. (\[4\]), we obtain the following differential equation : $$\begin{aligned}
&&[-\partial_{t}^2+\partial_{r}^2+\frac{1}{r}\,\partial_{r}+\frac{1}{\alpha^2\,r^2}\,(\partial_{\phi}-\frac{\Phi}{2\,\pi}\,\partial_{x})^2+\partial_{z}^2+\partial_{x}^2\nonumber\\
&&-m\,\omega\,\left(f' (r)+\frac{f(r)}{r} \right)-m^2\,\omega^2\,f^{2}(r)-\left(m + S(r) \right)^2]\,\Psi=0.
\label{32}\end{aligned}$$ Since the space-time is independent of $t, \phi, z, x$, substituting the ansatz (\[9\]) into the Eq. (\[32\]), we get the following equation : $$\begin{aligned}
&&\psi ''(r)+\frac{1}{r}\,\psi'(r)+[E^2-k^2-q^2-\frac{l^2_{eff}}{r^2}-m\,\omega\,\left(f'(r)+\frac{f(r)}{r} \right)\nonumber\\
&-&m^2\,\omega^2\,f^{2}(r)-\left(m+S (r) \right)^2]\,\psi (r)=0,
\label{33}\end{aligned}$$ where the effective angular quantum number $$l_{eff}=\frac{1}{\alpha}\,(l-\frac{q\,\Phi}{2\,\pi}).
\label{34}$$
Substituting the function (\[11\]) into the Eq. (\[33\]) and using Coulomb-type potential (\[5\]), the radial wave-equation becomes $$\left [\frac{d^2}{dr^2} + \frac{1}{r}\,\frac{d}{dr} + \lambda_0-m^2\,\omega^2\,a^2\,r^2-\frac{\chi^2}{r^2}-\frac{2\,m\,\eta_{c}}{r} \right]\,\psi(r)=0,
\label{35}$$ where $$\begin{aligned}
&&\lambda_0=E^2-k^2-q^2-m^2-2\,m\,\omega\,a-2\,m^2\,\omega^2\,a\,b,\nonumber\\
&&\chi=\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+m^2\,\omega^2\,b^2+\eta^{2}_{c}}.
\label{36}\end{aligned}$$ Transforming $\rho=\sqrt{m\,\omega\,a}\,r$ into the equation (\[35\]), we get $$\left [\frac{d^2}{d\rho^2}+\frac{1}{\rho}\,\frac{d}{d\rho}+ \frac{\lambda_0}{m\,\omega\,a}-\rho^2-\frac{\chi^2}{\rho^2}-\frac{\tilde{\eta}}{\rho} \right]\,\psi (\rho)=0,
\label{37}$$ where $\tilde{\eta}=\frac{2\,m\,\eta_c}{\sqrt{m\,\omega\,a}}$.
Suppose the possible solution to Eq. (\[37\]) is $$\psi (\rho)=\rho^{\chi}\,e^{-\frac{\rho^2}{2}}\,H (\rho)
\label{42}$$ Substituting the solution Eq. (\[42\]) into the Eq. (\[37\]), we obtain $$H'' (\rho)+\left[\frac{1+2\,\chi}{\rho}-2\,\rho \right]\,H' (\rho)+\left [-\frac{\tilde{\eta}}{\rho}+\frac{\lambda_0}{m\,\omega\,a}-2\,(1+\chi) \right]\,H (\rho).
\label{43}$$ Equation (\[43\]) is the second order Heun’s differential equation [@ff3; @AHEP; @bb15; @bb16; @aa6; @EVBL; @EVBL2; @EPJC; @bb7; @bb8; @bb9; @ff2; @ff5; @ff6; @bb41; @bb42; @bb46; @bb47; @dd51; @dd52] with $H (\rho)$ is the Heun polynomial.
Substituting the power series solution Eq. (\[19\]) into the Eq. (\[43\]), we obtain the following recurrence relation for the coefficients: $$c_{n+2}=\frac{1}{(n+2)\,(n+2+2\,\chi)}\,\left[\tilde{\eta}\,c_{n+1}-\left\{ \frac{\lambda_0}{m\,\omega\,a}-2-2\,\chi-2\,n \right\}\,c_n \right].
\label{44}$$ The power series becomes a polynomial of degree $n$ by imposing the following conditions [@ff3; @AHEP; @bb15; @bb16; @aa6; @EVBL; @EVBL2; @EPJC; @bb7; @bb8; @bb9; @ff2; @ff5; @ff6; @bb41; @bb42; @bb46; @bb47] $$c_{n+1}=0\quad,\quad \frac{\lambda_0}{m\,\omega\,a}-2-2\,\chi=2\,n\quad (n=1,2,...)
\label{45}$$
By analyzing the second condition, we get the following energy eigenvalues $E_{n,l}$: $$\begin{aligned}
&&E^{2}_{n,l}=k^2+q^2+m^2+2\,m\,\omega\,a\,\left(n+2+\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+m^2\,\omega^2\,b^2+\eta^{2}_c}\right)\nonumber\\
&&+2\,m^2\,\omega^2\,a\,b.
\label{38}\end{aligned}$$ Equation (\[38\]) is the energy eigenvalues of a generalized Klein-Gordon oscillator in the magnetic cosmic string with a Coulomb-type scalar potential in the Kaluza-Klein theory. Observed that the relativistic energy eigenvalues Eq. (\[38\]) depend on the Aharonov-Bohm geometric quantum phase [@bb50]. Thus, we have that $E_{n, l} (\Phi+\Phi_0)=E_{n, l \mp \tau} (\Phi)$ where, $\Phi_0=\pm\,\frac{2\,\pi}{q}\,\tau$ with $\tau=0,1,2..$. This dependence of the relativistic energy eigenvalue on the geometric quantum phase $\Phi$ gives rise to a relativistic analogue of the Aharonov-Bohm effect for bound states [@ff3; @bb15; @bb28; @bb39; @bb40; @bb50].
We plot graphs of the above energy eigenvalues w. r. t. different parameters. In fig. 6, the energy eigenvalues $E_{1,1}$ against the parameter $\eta_c$. In fig. 7, the energy eigenvalues $E_{1,1}$ against the parameter $M$. In fig. 8, the energy eigenvalues $E_{1,1}$ against the parameter $\omega$. In fig. 9, the energy eigenvalues $E_{1,1}$ against the parameter $\Phi$.
The ground state energy levels and corresponding wave-function for $n=1$ are given by $$\begin{aligned}
&&E^{2}_{1,l}=k^2+q^2+m^2+2\,m\,\omega_{1,l}\,a\,\left(3+\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+m^2\,\omega^2\,b^2+\eta^{2}_c} \right)\nonumber\\
&&+2\,m^2\,\omega^2_{1,l}\,a\,b\quad,\nonumber\\
&&\psi_{1,l} (\rho)=\rho^{\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+m^2\,\omega^2_{1,l}\,b^2+\eta^{2}_c}}\,e^{-\frac{\rho^2}{2}}\,\left(c_0+c_1\,\rho \right),
\label{39}\end{aligned}$$ where $$\begin{aligned}
c_1&=&\frac{2\,m\,\eta_{c}}{\sqrt{m\,\omega_{1,l}\,a}\,(1+2\,\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+m^2\,\omega^2_{1,l}\,b^2+\eta^{2}_{c}})}\nonumber\\
&=&\left(\frac{2}{1+2\,\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+m^2\,\omega^2_{1,l}\,b^2+\eta^{2}_c}}\right)^{\frac{1}{2}}\,c_0. \nonumber\\
\omega_{1,l}&=&\frac{2\,m\,\eta^2_{c}}{a\,\left (1+2\,\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+m^2\,\omega^2_{1,l}\,b^2+\eta^{2}_c}\right)}
\label{40}\end{aligned}$$ a constraint on the physical parameter $\omega_{1,l}$.
Equation Eq. (\[39\]) is the ground states energy eigenvalues and corresponding eigenfunctions of a generalized Klein-Gordon oscillator in the presence of Coulomb-type scalar potential in a magnetic cosmic string space-time in the Kaluza-Klein theory.
For $\alpha \rightarrow 1$, the energy eigenvalues (\[38\]) becomes $$\begin{aligned}
&&E^{2}_{n,l}=k^2+m^2+q^2+2\,m\,\omega\,a\,\left(n+2+\sqrt{(l-\frac{q\,\Phi}{2\,\pi})^2+m^2\,\omega^2\,b^2+\eta^{2}_{c}}\right)\nonumber\\
&&+2\,m^2\,\omega^2\,a\,b.
\label{41}\end{aligned}$$ Equation (\[41\]) is the relativistic energy eigenvalue of the generalized Klein-Gordon oscillator field with a Coulomb-type scalar potential with a magnetic flux in the Kaluza-Klein theory. Observed that the relativistic energy eigenvalue Eq. (\[41\]) depend on the geometric quantum phase [@bb50]. Thus, we have that $E_{n,l} (\Phi+\Phi_0)=E_{n,l \mp \tau} (\Phi)$ where, $\Phi_0=\pm\,\frac{2\,\pi}{q}\,\tau$ with $\tau=0,1,2..$. This dependence of the relativistic energy eigenvalue on the geometric quantum phase gives rise to an analogous effect to Aharonov-Bohm effect for bound states [@ff3; @bb15; @bb28; @bb39; @bb40; @bb50].
:
We discuss below a special case corresponds to $b \rightarrow 0$, $a \rightarrow 0$, that is, a scalar quantum particle in a magnetic cosmic string background subject to a Coulomb-type scalar potential in the Kaluza-Klein theory. In that case, from Eq. (\[35\]) we obtain the following equation: $$\psi''(r)+\frac{1}{r}\,\psi'(r)+[\tilde{\lambda}-\frac{\tilde{\chi}^2}{r^2}-\frac{2\,m\,\eta_{c}}{r}]\,\psi(r)=0,
\label{cc1}$$ where $$\begin{aligned}
&&\tilde{\lambda}=E^2-k^2-q^2-m^2,\nonumber\\
&&\tilde{\chi}_0=\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+\eta^{2}_c}.
\label{cc2}\end{aligned}$$ The above Eq. (\[cc1\]) can be written as $$\psi''(r)+\frac{1}{r}\,\psi'(r)+\frac{1}{r^2}\,\left(-\xi_1\,r^2+\xi_2\,r-\xi_3 \right)\,\psi(r)=0,
\label{cc3}$$ where $$\xi_1=-\tilde{\lambda}\quad,\quad \xi_2=-2\,m\,\eta_{c}\quad,\quad \xi_3=\tilde{\chi}^{2}_0.
\label{Bakke}$$
Following the similar technique as done earlier, we get the following energy eigenvalues $E_{n,l}$: $$E_{n,l}=\pm\,m\,\sqrt{1-\frac{\eta^2_{c}}{\left (n+\sqrt{\frac{1}{\alpha^2}\,(l-\frac{q\,\Phi}{2\,\pi})^2+\eta^{2}_{c}}+\frac{1}{2}\right)^2}+\frac{k^2}{m^2}+\frac{q^2}{m^2}},
\label{cc5}$$ where $n=0,1,2,..$ is the quantum number associated with radial modes, $l=0,\pm\,1,\pm\,2,....$ are the quantum number associated with the angular momentum, $k$ and $q$ are constants. Equation (\[cc5\]) corresponds to the relativistic energy levels for a free-scalar particle subject to Coulomb-type scalar potential in the background of magnetic cosmic string in a Kaluza-Klein theory.
The radial wave-function is given by $$\begin{aligned}
&&\psi_{n,l} (r)=|N|\,r^{\frac{\tilde{\chi}_0}{2}}\,{\sf e}^{-\frac{r}{2}}\,{\sf L}^{(\tilde{j})}_{n} (r)\nonumber\\
&=&|N|\,r^{\frac{1}{2},\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+\eta^{2}_c}}\,{\sf e}^{-\frac{r}{2}}\,{\sf L}^{(\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+\eta^{2}_c})}_{n} (r).
\label{cc7}\end{aligned}$$ Here $|N|$ is the normalization constant and ${\sf L}^{(\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+\eta^{2}_c})}_{n} (r) $ is the generalized Laguerre polynomial.
For $\alpha \rightarrow 1$, the energy eigenvalues (\[cc5\]) becomes $$E_{n,l}=\pm\,m\,\sqrt{1-\frac{\eta^2_{c}}{\left (n+\sqrt{(l-\frac{q\,\Phi}{2\,\pi})^2+k^{2}_{c}}+\frac{1}{2}\right)^2}+\frac{k^2}{m^2}+\frac{q^2}{m^2}},
\label{cc6}$$ which is similar to the energy eigenvalue obtained in [@bb16] (see Eq. (12) in [@bb16]). Thus we can see that the cosmic string $\alpha$ modify the relativistic energy eigenvalue (\[cc5\]) in comparison to those results obtained in [@bb16].
Observe that the relativistic energy eigenvalues Eq. (\[cc5\]) depend on the cosmic string parameter $\alpha$, the magnetic quantum flux $\Phi$, and potential parameter $\eta_c$. We can see that $E_{n, l} (\Phi+\Phi_0)=E_{n, l \mp \tau} (\Phi)$ where, $\Phi_0=\pm\,\frac{2\,\pi}{q}\,\tau$ with $\tau=0,1,..$. This dependence of the relativistic energy eigenvalues on the geometric quantum phase gives rise to a relativistic analogue of the Aharonov-Bohm effect for bound states [@ff3; @bb15; @bb28; @bb39; @bb40; @bb50].
Persistent currents of the Relativistic System
----------------------------------------------
By following [@NB; @WCT; @LD], the expression for the total persistent currents is given by $$I=\sum_{n,l}\,I_{n,l},
\label{dd1}$$ where $$I_{n,l}=-\frac{\partial E_{n,l}}{\partial \Phi}
\label{dd2}$$ is called the Byers-Yang relation [@NB].
Therefore, the persistent current that arises in this relativistic system using Eq. (\[38\]) is given by $$\begin{aligned}
&&I_{n,l}=-\frac{\partial E_{n,l}}{\partial \Phi}\nonumber\\
&&=\mp\frac{m\,\omega\, a\,(\frac{\partial\,\chi}{\partial\,\Phi})}{\sqrt{k^2+q^2+m^2+2m^2\omega^2 a b+2 m \omega a\left(n+2+\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+m^2\omega^2 b^2+\eta^{2}_c}\right)}},\quad\quad
\label{bb55}\end{aligned}$$ where $$\begin{aligned}
\frac{\partial\,\chi}{\partial\,\Phi}=-\frac{q\,(l-\frac{q\,\Phi}{2\,\pi})}{2\,\alpha^2\,\pi\,\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+m^2\,\omega^2\,b^2+\eta^{2}_c}}.
\label{dd4}\end{aligned}$$
Similarly, for the relativistic system discussed in [**case A**]{} in this section, this current using Eq. (\[cc5\]) is given by $$\begin{aligned}
I_{n,l}&=&\pm\,\frac{m\,q\,\eta^{2}_c\,(l-\frac{q\,\Phi}{2\,\pi})}{2\,\pi\,\alpha^2\,\left(n+\frac{1}{2}+\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+\eta^2_{c}} \right)^3\,\sqrt{\frac{(l-\frac{q\,\Phi}{2\,\pi})^2}{\alpha^2}+\eta^2_{c}}}\nonumber\\
&&\times\frac{1}{\sqrt{1-\frac{\eta^2_{c}}{\left(n+\sqrt{\frac{1}{\alpha^2}\,(l-\frac{q\,\Phi}{2\,\pi})^2+\eta^{2}_c}+\frac{1}{2}\right)^2}+\frac{k^2}{m^2}+\frac{q^2}{m^2}}}.
\label{dd5}\end{aligned}$$ For $\alpha \rightarrow 1$, the persistent currents expression given by Eq. (\[dd5\]) reduces to the result obtained in Ref. [@bb16]. Thus we can see that the presence of the cosmic string parameter modify the persistent currents Eq. (\[dd5\]) in comparison to those results in Ref. [@bb16].
By introducing a magnetic flux through the line element of the cosmic string space-time in five dimensions, we see that the relativistic energy eigenvalue Eq. (\[38\]) depend on the geometric quantum phase [@bb50] which gives rise to a relativistic analogue of the Aharonov-Bohm effect for bound states [@ff3; @bb15; @bb28; @bb39; @bb40; @bb50]. Moreover, this dependence of the relativistic energy eigenvalues on the geometric quantum phase has yielded persistent currents in this relativistic quantum system.
Conclusions
===========
In Ref. [@bb16], Aharonov-Bohm effects for bound states of a relativistic scalar particle by solving the Klein-Gordon equation subject to a Coulomb-type potential in the Minkowski space-time within the Kaluza-Klein theory were studied. They obtained the relativistic bound states solutions and calculated the persistent currents. In Ref. [@bb14], it is shown that the cosmic string space-time and the magnetic cosmic string space-time can have analogue in five dimensions. In Ref. [@bb28], quantum mechanics of a scalar particle in the background of a chiral cosmic string using the Kaluza-Klein theory was studied. They shown that the wave functions, the phase shifts, and scattering amplitudes associated with the particle depend on the global features of those space-times. These dependence represent the gravitational analogues of the well-known Aharonov-Bohm effect. In addition, they discussed the Landau levels in the presence of a cosmic string within the framework of Kaluza-Klein theory. In Ref. [@aa6], the Klein-Gordon oscillator on the curved background within the Kaluza-Klein theory were studied. The problem of the interaction between particles coupled harmonically with topological defects in the Kaluza-Klein theory were studied. They considered a series of topological defects and then treated the Klein-Gordon oscillator coupled to this background, and obtained the energy eigenvalue and corresponding eigenfunctions in this cases. They have shown that the energy eigenvalue depend on the global parameters characterizing these space-times. In Ref. [@EVBL], a scalar particle with position-dependent mass subject to a uniform magnetic field and a quantum magnetic flux, both coming from the background which is governed by the Kaluza-Klein theory were investigated. They inserted a Cornell-type scalar potential into this relativistic systems and determined the relativistic energy eigenvalue of the system in this background of extra dimension. They analyzed particular cases of this system and a quantum effect were observed: the dependence of the magnetic field on the quantum numbers of the solutions. In Ref. [@EPJC], the relativistic quantum dynamics of a scalar particle subject to linear potential on the curved background within the Kaluza-Klein theory was studied. We have solved the generalized Klein-Gordon oscillator in the cosmic string and magnetic cosmic string space-time with a linear potential within the Kaluza-Klein theory. We have shown that the energy eigenvalues obtained there depend on the global parameters characterizing these space-times and the gravitational analogue to the Aharonov-Bohm effect for bound states [@ff3; @bb15; @bb28; @bb39; @bb40; @bb50] of a scalar particle was analyzed.
In this work, we have investigated the relativistic quantum dynamics of a scalar particle interacting with gravitational fields produced by topological defects via the Klein-Gordon oscillator of the Klein-Gordon equation in the presence of cosmic string and magnetic cosmic string within the Kaluza-Klein theory with scalar potential. We have determined the manner in which the non-trivial topology due to the topological defects and a quantum magnetic flux modifies the energy spectrum and wave-functions of a scalar particle. We then have studied the quantum dynamics of a scalar particle interacting with fields by introducing a magnetic flux through the line element of a cosmic string space-time using the five-dimensional version of the General Relativity. The quantum dynamics in the usual as well as magnetic cosmic string cases allow us to obtain the energy eigenvalues and corresponding wave-functions that depend on the external parameters characterize the background space-time, a result known by gravitational analogue of the well studied Aharonov-Bohm effect.
In [*section 2*]{}, we have chosen a Cornell-type function $f(r)=a\,r+\frac{b}{r}$ and Cornell-type potential $S(r)=\eta_{L}\,r+\frac{\eta_c}{r}$ into the relativistic systems. We have solved the generalized Klein-Gordon oscillator in the cosmic string background within the Kaluza-Klein theory and obtained the energy eigenvalues Eq. (\[24\]). We have plotted graphs of the energy eigenvalues Eq. (\[24\]) w. r. t. different parameters by figs. 1–5. By imposing the additional recurrence condition $c_{n+1}=0$ on the relativistic eigenvalue problem, for example $n=1$, we have obtained the ground state energy levels and wave-functions by Eqs. (\[27\])–(\[28\]). We have discussed a special case corresponds to $\eta_{L} \rightarrow 0$ and obtained the relativistic energy eigenvalues Eq. (\[aa3\]) of a generalized Klein-Gordon oscillator in the cosmic string space-time within the Kaluza-Klein theory. We have also obtained the relativistic energy eigenvalues Eq. (\[bb4\]) of a free-scalar particle by solving the Klein-Gordon equation with a Coulomb-type scalar potential in the background of cosmic string space-time in the Kaluza-Klein theory.
In [*section 3*]{}, we have studied the relativistic quantum dynamics of a scalar particle in the background of magnetic cosmic string in the Kaluza-Klein theory with a scalar potential. By choosing the same function $f(r)=a\,r+\frac{b}{r}$ and a Coulomb-type scalar potential $S(r)=\frac{\eta_c}{r}$, we have solved the radial wave-equation in the considered system and obtained the bound states energy eigenvalues Eq. (\[38\]). We have plotted graphs of the energy eigenvalues Eq. (\[38\]) w. r. t. different parameters by figs. 6–9. Subsequently, the ground state energy levels Eq. (\[39\]) and corresponding wave-functions Eq. (\[40\]) for the radial mode $n=1$ by imposing the additional condition $c_{n+1}=0$ on the eigenvalue problem is obtained. Furthermore, a special case corresponds to $a\rightarrow 0$, $b\rightarrow 0$ is discussed and obtained the relativistic energy eigenvalues Eq. (\[cc5\]) of a scalar particle by solving the Klein-Gordon equation with a Coulomb-type scalar potential in the magnetic cosmic string space-time in the Kaluza-Klein theory. For $\alpha \rightarrow 1$, we have seen that the energy eigenvalues Eq. (\[cc5\]) reduces to the result obtained in Ref. [@bb16]. As there is an effective angular momentum quantum number, $l \rightarrow l_{eff}=\frac{1}{\alpha}\,(l-\frac{q\,\Phi}{2\pi})$, thus the relativistic energy eigenvalues Eqs. (\[38\]) and (\[cc5\]) depend on the geometric quantum phase [@bb50]. Hence, we have that $E_{n, l} (\Phi+\Phi_0)=E_{n, l \mp \tau} (\Phi)$ where, $\Phi_0=\pm\,\frac{2\,\pi}{q}\,\tau$ with $\tau=0,1,2,.$. This dependence of the relativistic energy eigenvalues on the geometric quantum phase gives rise to a relativistic analogue of the Aharonov-Bohm effect for bound states [@bb15; @bb39; @bb40; @bb50]. Finally, we have obtained the persistent currents by Eqs. (\[bb55\])–(\[dd5\]) for this relativistic quantum system because of the dependence of the relativistic energy eigenvalues on the geometric quantum phase.
So in this paper, we have shown some results which are in addition to those results obtained in Refs. [@bb28; @bb15; @bb16; @aa6; @EVBL; @EVBL2; @EPJC] presents many interesting effects.
Data Availability {#data-availability .unnumbered}
=================
No data has been used to prepare this paper.
Conflict of Interest {#conflict-of-interest .unnumbered}
====================
Author declares that there is no conflict of interest regarding publication this paper.
Acknowledgement {#acknowledgement .unnumbered}
===============
Author sincerely acknowledge the anonymous kind referee(s) for their valuable comments and suggestions and thanks the editor.
Appendix A : Brief review of the Nikiforov-Uvarov (NU) method {#appendix-a-brief-review-of-the-nikiforov-uvarov-nu-method .unnumbered}
=============================================================
The Nikiforov-Uvarov method is helpful in order to find eigenvalues and eigenfunctions of the Schrödinger like equation, as well as other second-order differential equations of physical interest. According to this method, the eigenfunctions of a second-order differential equation [@bb49] $$\frac{d^2 \psi (s)}{ds^2}+\frac{(\alpha_1-\alpha_2\,s)}{s\,(1-\alpha_3\,s)}\,\frac{d \psi (s)}{ds}+\frac{(-\xi_1\,s^2+\xi_2\,s-\xi_3)}{s^2\,(1-\alpha_3\,s)^2}\,\psi (s)=0.
\label{A.1}$$ are given by $$\psi (s)=s^{\alpha_{12}}\,(1-\alpha_3\,s)^{-\alpha_{12}-\frac{\alpha_{13}}{\alpha_3}}\,P^{(\alpha_{10}-1,\frac{\alpha_{11}}{\alpha_3}-\alpha_{10}-1)}_{n}\,(1-2\,\alpha_3\,s).
\label{A.2}$$ And that the energy eigenvalues equation $$\begin{aligned}
&&\alpha_2\,n-(2\,n+1)\,\alpha_5+(2\,n+1)\,(\sqrt{\alpha_9}+\alpha_3\,\sqrt{\alpha_8})+n\,(n-1)\,\alpha_3+\alpha_7\nonumber\\
&&+2\,\alpha_3\,\alpha_8+2\,\sqrt{\alpha_8\,\alpha_9}=0.
\label{A.3}\end{aligned}$$ The parameters $\alpha_4,\ldots,\alpha_{13}$ are obtained from the six parameters $\alpha_1,\ldots,\alpha_3$ and $\xi_1,\ldots,\xi_3$ as follows: $$\begin{aligned}
&&\alpha_4=\frac{1}{2}\,(1-\alpha_1)\quad,\quad \alpha_5=\frac{1}{2}\,(\alpha_2-2\,\alpha_3),\nonumber\\
&&\alpha_6=\alpha^2_{5}+\xi_1\quad,\quad \alpha_7=2\,\alpha_4\,\alpha_{5}-\xi_2,\nonumber\\
&&\alpha_8=\alpha^2_{4}+\xi_3\quad,\quad \alpha_9=\alpha_6+\alpha_3\,\alpha_7+\alpha^{2}_3\,\alpha_8,\nonumber\\
&&\alpha_{10}=\alpha_1+2\,\alpha_4+2\,\sqrt{\alpha_8}\quad,\quad \alpha_{11}=\alpha_2-2\,\alpha_5+2\,(\sqrt{\alpha_9}+\alpha_3\,\sqrt{\alpha_8}),\nonumber\\
&&\alpha_{12}=\alpha_4+\sqrt{\alpha_8}\quad,\quad \alpha_{13}=\alpha_5-(\sqrt{\alpha_9}+\alpha_3\,\sqrt{\alpha_8}).
\label{A.4}\end{aligned}$$
A special case where $\alpha_3=0$, as in our case, we find $$\lim_{\alpha_3\rightarrow 0} P^{(\alpha_{10}-1,\frac{\alpha_{11}}{\alpha_3}-\alpha_{10}-1)}_{n}\,(1-2\,\alpha_3\,s)=L^{\alpha_{10}-1}_{n} (\alpha_{11}\,s),
\label{A.5}$$ and $$\lim_{\alpha_3\rightarrow 0} (1-\alpha_3\,s)^{-\alpha_{12}-\frac{\alpha_{13}}{\alpha_3}}=e^{\alpha_{13}\,s}.
\label{A.6}$$ Therefore the wave-function from (\[A.2\]) becomes $$\psi (s)=s^{\alpha_{12}}\,e^{\alpha_{13}\,s}\,L^{\alpha_{10}-1}_{n} (\alpha_{11}\,s),
\label{A.7}$$ where $L^{(\alpha)}_{n} (x)$ denotes the generalized Laguerre polynomial.
The energy eigenvalues equation reduces to $$n\,\alpha_2-(2\,n+1)\,\alpha_5+(2\,n+1)\,\sqrt{\alpha_9}+\alpha_7+2\,\sqrt{\alpha_8\,\alpha_9}=0.
\label{A.8}$$ Noted that the simple Laguerre polynomial is the special case $\alpha=0$ of the generalized Laguerre polynomial: $$L^{(0)}_{n} (x)=L_{n} (x).
\label{A.9}$$
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[^1]: [email protected] ; [email protected]
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---
abstract: 'We discuss the prospects of studying lepton number violating processes in order to identify Majorana neutrinos from low scale seesaw mechanisms at lepton-proton colliders. In particular, we consider the scenarios of colliding electrons with LHC energy protons and, motivated by the efforts towards the construction of a muon collider, the prospects of muon-proton collisions. We find that present constraints on the mixing of the Majorana neutrinos still allow for a detectable signal at these kind of facilities given the smallness of the Standard Model background. We discuss possible cuts in order to further increase the signal over background ratio and the prospects of reconstructing the neutrino mass from the kinematics of the final state particles.'
author:
- Carl Blaksley
- Mattias Blennow
- Florian Bonnet
- Pilar Coloma
-
title: 'Heavy Neutrinos and Lepton Number Violation in $\boldsymbol{\ell p}$ Colliders'
---
Introduction
============
The first light of the Large Hadron Collider (LHC) has marked the beginning of a new era of high-energy collider physics. This actualizes the concept of not only proton-proton collisions, but also what kind of physics could be probed in other high-energy collisions exploiting proton energies similar to the LHC ones. At the same time, it is the hope that new colliders will not only probe the mechanism behind electroweak symmetry breaking and the stabilization of the electroweak scale, but also trace and identify the source of other known shortcomings of the Standard Model (SM), such as the existence of neutrino masses and dark matter. In the case of neutrino masses, the most popular scenario is generally considered to be the type-I seesaw models [@Minkowski:1977sc; @Mohapatra:1979ia; @Yanagida:1979as; @GellMann:1980vs] and it has already been shown that low-scale realizations [@Mohapatra:1986bd; @Bernabeu:1987gr; @Branco:1988ex; @Buchmuller:1990du; @Buchmuller:1991tu; @Datta:1991mf; @Ingelman:1993ve] of such models can give observable signals in collider experiments [@Datta:1993nm; @Almeida:2000pz; @Panella:2001wq; @delAguila:2005mf; @delAguila:2005pf; @Bray:2005wv; @Han:2006ip; @delAguila:2006dx; @Atwood:2007zza; @Bray:2007ru; @deAlmeida:2007gc; @delAguila:2007em; @delAguila:2008cj].
One of the most striking predictions of the type-I seesaw is that neutrinos are Majorana particles and thus, that lepton number is broken through their Majorana mass terms. One of the most promising probes of the Majorana nature of neutrinos is the search for neutrinoless double beta decay ([${0\nu\beta\beta }$]{}), which would produce a distinct peak at the end of the double beta decay spectrum (for a recent review see [Ref.]{} [@Bilenky:2010zz]). Any search for processes violating lepton number will typically have a signal suppressed by the tiny neutrino masses [@Kersten:2007vk; @Ibarra:2010xw]. While in low scale seesaw mechanisms this smallness is due to a cancellation of different large contributions, this very same cancellation will suppress any lepton number violating signal unless the energy probed in the process is such that some particular contributions are enhanced or suppressed due to kinematics. In [${0\nu\beta\beta }$]{} searches, contributions from neutrinos with masses above the nuclear scale of $\sim 100$ MeV are suppressed with respect to those below (see [*e.g.*]{} [Ref.]{} [@Blennow:2010th] for a discussion) and the cancellation can be avoided. Here we will discuss the prospects of observing the manifestly lepton number violating processes $p\ell^\mp \to \ell^\pm jjj$. Similar to [${0\nu\beta\beta }$]{}, the contributions of mass eigenstates above the energy probed by the colliders will be suppressed, evading the cancellation behind the small neutrino masses. Moreover, collider experiments, and proton-lepton colliders in particular can, unlike [${0\nu\beta\beta }$]{} searches, produce these neutrinos on-shell via s-channel contributions with the consequent enhancement of the signal.
The rest of this paper is organized as follows: in [Sec.]{} \[sec:low-scale\], we review the low-scale seesaw model and summarize the current bounds. We then go on to discuss in [Sec.]{} \[sec:LNV\], within the low-scale seesaw model, the lepton number violating processes that constitute our signal. Section \[sec:numerics\] is dedicated to providing the details of our numerical computations, while in [Sec.]{} \[sec:results\] we present the results. Finally, in [Sec.]{} \[sec:summary\], we summarize our results and give our conclusions.
The low-scale seesaw model and present constraints {#sec:low-scale}
==================================================
Let us consider the Lagrangian of the standard [type-I]{} seesaw model, which consists of the Standard Model (SM) Lagrangian plus an extra piece containing the allowed couplings between the SM fields and additional gauge singlet fermions ([*i.e.*]{}, right-handed neutrinos) $N_\mathrm{R}^i$: $$\label{eq:The3FormsOfNuMassOp}
\mathscr{L} = \mathscr{L}_\mathrm{SM} - \left[\frac{1}{2} \overline{N_\mathrm{R}^i} M_{ij} N^{c j}_\mathrm{R} + (Y_{N})_{i\alpha}\overline{N_\mathrm{R}^i} \phi^\dagger
L^\alpha +{\mathrm{H.c.}}\right]\; .$$ Here, $\phi$ denotes the SM Higgs field, which breaks the electroweak (EW) symmetry after acquiring its vacuum expectation value, $v_{\mathrm{EW}}$. In this work, we will use the basis in which $M$ is diagonal with real positive entries. After electroweak symmetry breaking, this produces a $6 \times 6$ mass matrix $$\mathcal M = {\left(\begin{array}{cc} 0 & m_D^T \\ m_D & M \end{array}\right)},$$ where $m_D = v_{EW} Y_N$, in the basis $[\nu^\alpha_L, (N_\mathrm{R}^{i})^c]^T$. In terms of the light mass eigenstates $\nu^i$ and the heavy eigenstates $N^i$, the neutrino flavor eigenstates can be written as $$\nu^\alpha_L \simeq (\delta_{\alpha\beta}-\frac{1}{2} \theta_{\alpha j}\theta_{\beta j}^*)U_{\beta i} \nu^i + \theta_{\alpha i} N^i,$$ where $\theta = m_D^\dagger M^{-1}$. In the cases where the heavy neutrinos can be integrated out, this implies a non-unitary mixing matrix $\mathcal N = (1 - \eta/2) U$, where $\eta = \theta \theta^\dagger$, for the light neutrinos [@Langacker:1988ur; @Nardi:1994iv; @Tommasini:1995ii; @Broncano:2002rw]. Inserting this relation into the electroweak interaction Lagrangian, we obtain the following coupling between the heavy neutrinos and the charged leptons via the $W$ bosons $$\mathscr L_{\rm int} = \frac{g}{\sqrt 2} W_\mu \overline{\ell^\alpha_L} \gamma^\mu \theta_{\alpha i} N^i + {\mathrm{H.c.}}.$$
The unitarity deviations implied by the heavy neutrinos can thus be used to constrain the elements of $\theta$ through the relation $\eta = \theta \theta^\dagger$ which implies $$|\theta_{\alpha i}|^2 \leq |\eta_{\alpha\alpha}| = 0.005 \pm 0.005$$ at the 90 % Confidence Level (CL). This constraint stems from the effects that such terms would have in universality tests of the weak interactions as well as the invisble width of the $Z$ [@Antusch:2006vwa]. Notice that somewhat stronger bounds $|\eta_{e e}| < 4.0 \cdot 10^{-3}$, $|\eta_{\mu \mu}| < 1.6 \cdot 10^{-3}$ and $|\eta_{\tau \tau}| < 5.3 \cdot 10^{-3}$ are obtained when measurements of $G_F$ from muon decay and the CKM unitarity are assumed [@Antusch:2008tz] instead of the invisible width of the $Z$. The combination of both data sets would, however, result in a larger allowed region since their respective best fits do not coincide. Indeed, while the invisible width of the $Z$ prefers non-zero values of $\eta$ in order to accommodate the present $2 \sigma$ deviation from the SM, such a result is not favored by the measuremets of $G_F$. For definiteness we will fix here the size of the signal by accommodating the data on the invisble width of the $Z$, setting $|\eta_{\alpha \alpha}| = 5.0 \cdot 10^{-3}$. However, the rescaling of the signal under study to smaller mixing angles is trivial through an overall quadratic dependence, as we will discuss later.
If the masses of the heavy neutrinos display a moderate hierarchy, the signal will mainly be dominated by the contribution of the mass eigenstate within the reach of the collider energy. Thus, in order to simplify the discussion, we will assume there is only one heavy neutrino contributing to the signal, while the other neutrinos, necessary for the generation of the observed pattern of masses and mixings at low energies, are heavier and do not contribute significantly. The mixing of this neutrino with the SM fermions will be set to $ |\theta_{\alpha 1}|^2 \equiv |\theta_{\alpha}|^2 = 0.005 $ as discussed above.
Note that, if we also impose the condition that neutrino masses are small while maintaining large Yukawa couplings, then an additional constraint $|\eta_{\alpha\alpha}|\cdot|\eta_{\beta\beta}| \simeq |\eta_{\alpha\beta}|^2$ applies [@Buchmuller:1990du; @Datta:1991mf; @Ohlsson:2010ca]. In particular, strong constraints on the product will then stem from processes such as $\mu \to e\gamma$. Therefore, we will consider scenarios with only one $\theta_{\alpha} \neq 0$, where the mixing is to only one flavour. Thus, we will not consider lepton flavour violation in the signal but only lepton number violation.
Finally, it should be noted that [${0\nu\beta\beta }$]{} decay searches in general provide strong bounds on $\theta_{ei}$ [@Ibarra:2010xw]. However, it is possible to satisfy both the constraints from [${0\nu\beta\beta }$]{} and bounds on neutrino masses while still maintaining large Yukawa couplings [@Ingelman:1993ve].
Lepton number violating processes {#sec:LNV}
=================================
Here we will consider the process $p \ell^- \to \ell^+ jjj$ to investigate the possibility of producing the heavy right-handed neutrinos of the type-I Seesaw mechanism and observe the lepton-number-violating signals associated to their Majorana nature at colliders. The advantage of lepton-proton collisions with respect to purely hadron or lepton colliders is the cleanness of the signal. Indeed, since the lepton number in the initial state is one, the observation of a charged antilepton in the final state is a clear signal of violation of lepton number by two units, as long as there is no missing energy. This last requisite translates in the presence of three jets in the final state, two of them reconstructing an invariant mass equal to the $W$ mass. Conversely, lepton number violation in hadronic or $\ell^- \ell^+$ colliders implies tagging two leptons of the same charge in the final state together with either the presence of missing energy in the form of neutrinos or a higher number of jets, making the signal more challenging to search for.
In the [type-I]{} seesaw, two diagrams contribute to the signal under study (see [Fig.]{} \[fig:diagrams\]).
(220,140)(0,0) (10,10)(60,10) (8,10)\[r\][$u$]{} (60,10)(210,10) (212,10)\[l\][$j$]{} (210,40)(160,70) (212,42)\[lt\][$j$]{} (160,70)(210,90) (212,88)\[lb\][$j$]{} (160,70)(110,90)[2]{}[5.5]{} (133,78)\[tr\][$W$]{} (210,130)(110,90) (212,130)\[lb\][$\ell^+$]{} (10,90)(60,90) (8,90)\[r\][$\ell^-$]{} (60,90)(110,90) (85,92)\[b\][$N_R$]{} (60,90)(60,10)[-2]{}[6.5]{} (64,50)\[l\][$W$]{}
(170,140)(0,0) (10,10)(60,10) (8,10)\[r\][$u$]{} (10,90)(60,90) (8,90)\[r\][$\ell^-$]{} (60,90)(110,50) (87,72)\[lb\][$N_R$]{} (110,50)(60,10)[2]{}[5]{} (83,32)\[rb\][$W$]{} (60,90)(110,110)[2]{}[4.5]{} (83,102)\[rb\][$W$]{} (110,110)(160,130) (162,128)\[bl\][$j$]{} (160,90)(110,110) (162,92)\[tl\][$j$]{} (160,50)(110,50) (162,50)\[l\][$\ell^+$]{} (60,10)(160,10) (162,10)\[l\][$j$]{}
The diagram with the Majorana neutrino in $s$-channel will be enhanced and dominate the signal if the collider energy is high enough to produce it on shell. We will therefore explore this possibility, which can help overcome the expected suppression of the signal by the smallness of neutrino masses. Thus, the process basically corresponds to the on shell production of the heavy neutrino, via the exchange of a $W$ boson, and its subsequent decay to $\ell^+ W^-$ with the $W$ decaying hadronically. The branching ratio for the first decay is roughly 25 % [@Buchmuller:1990vh]. On the other hand, the production process is weighted by the mixing $|\theta_{\alpha}|^2$ of the heavy neutrino $N$ with the charged lepton $\ell_\alpha$. Thus, it is trivial to extend the analysis from the assumed values of the mixing angles of $|\theta_{\alpha}|^2 = 5 \cdot 10^{-3}$ to smaller mixings by a general rescaling.
The electron-proton process we discuss here was already studied in [@Buchmuller:1990vh; @Buchmuller:1991tu; @Buchmuller:1992wm; @Ingelman:1993ve], mainly focusing on the DESY experiment but also extended to a combination of LEP and LHC. The production cross section and the decay rate of the heavy neutrino were computed in order to estimate the rate of the signal that could be expected in a electron-proton collider. In [Ref.]{} [@Ingelman:1993ve] a first numerical simulation of the signal was also performed. The analysis presented here is motivated by the improved simulation techniques presently available, in particular regarding hadronization processes, and better knowledge of the parton distribution functions (PDFs) of the proton. We also extend their analysis to the higher collider energies discussed in the LHCeC proposal [@LHCeC], which results in an enhancement of the signal and sensitivity to higher neutrino masses. We have therefore extended the original analysis to a wider range of neutrino masses than initially considered and showed explicitly the dependence of the signal on the mediating neutrino mass.
Furthermore, we also explore for the first time more sophisticated collider technologies, such as muon-proton colliders, inspired by the recent efforts towards a muon collider (see [*e.g.*]{} [Ref.]{} [@Shiltsev:2010qg] for a recent status review). This setup would allow to explore different matrix elements with respect to the electron-proton option, providing a completely independent search channel also with respect to [${0\nu\beta\beta }$]{} searches. Moreover, the energies that could be expected from such a facility imply a huge gain in sensitivity allowing to explore much smaller mixings or larger sterile neutrino masses than their electron counterparts. In both scenarios we will assume a detector setup based on the capabilities of ATLAS and CMS.
Regarding possible sources of background, the manifest lepton number violating nature of the signal in absence of missing energy makes it very clean and difficult to mimic by SM processes. Several sources of background were already discussed in [Ref.]{} [@Ingelman:1993ve] where it was found that, for the lepton number violating signal under study, the dominant background stems from $W$ production with its subsequent decay into the $\ell^+$ required for the signal[^1]. The original $\ell^-$ can be missed for example if it becomes a neutrino or is lost in the beamline. We have simulated and studied these two sources of background. In the second case the contribution to $W$ production is dominated by the exchange of an almost real photon with a very collinear outgoing electron. In order to compute this process, we have simulated a proton-photon collision and convoluted it with the photon PDF in a charged lepton. This turns out to be the dominant source of background, in particular when the two final state jets are produced by QCD processes. In [Ref.]{} [@Ingelman:1993ve], this background was reduced by means of a cut in the invariant mass of the two jets in the final state $m_{jj}$ compatible with that of a $W$ boson $M_W$, complemented with a cut in the minimum transverse momentum of the outgoing $\ell^+$ ($p_{T,\ell^+}$). However, as we will show in Sect. \[sec:results\], the presence of neutrinos in the final state, necessary to produce the final state positron through SM processes, suggests also the use of cuts in missing transverse energy ($E_{T,\rm miss}$) to suppress the background. We have found that cuts in maximum $E_{T,\rm miss}$ and minimum $p_{T,\ell^+}$ actually provide a better signal/background ratio than those in $m_{jj}$. Moreover, in the case of the $p \ell^- \to \ell^+ jjj \nu_l \nu_l$ background, two of the jets typically originate from a $W$ boson decay. Thus, the invariant mass cut around $M_W$ will not avail to reduce this source of background. On the other hand, the cuts in maximum $E_{T,\rm miss}$ and minimum $p_{T,\ell^+}$ prove to be useful for this other background too. This extra cut in $E_{T,\rm miss}$ thus represents a very useful complementary tool to the cut in $p_{T,\ell^+}$ in order to reduce all sources of SM background.
Numerical analysis {#sec:numerics}
==================
We base our numerical study of the signal and background on the use of the MadGraph/MadAnalysis [@Alwall:2007st] software tools supplemented by Pythia [@Sjostrand:2006za] and PGS [@PGS] to process the resulting events. The type-I seesaw description for MadGraph/MadAnalysis was obtained via the FeynRules [@Christensen:2008py] software. For the protons we assume an LHC-like beam with an energy of 7 TeV, while several different choices are examined for the lepton beam. In the case of an electron beam we study both the conservative setup with a beam energy of 50 GeV and a more optimistic setup where the beam energy is 150 GeV [@LHCeC]. For muons, on the other hand, we consider the beam energies that have been discussed for a future muon collider. These vary between 500 GeV for the more conservative case and 2 TeV for the more optimistic proposals [@Shiltsev:2010qg]. In all cases under study, we employ a detector with capabilities similar to the ATLAS and CMS detectors. When generating events, we impose the acceptance cuts specified in [Tab.]{} \[tab:gcuts\].
[**Variable**]{} [**Jets**]{} [**Leptons**]{} [**Photons**]{}
------------------ -------------- ----------------- -----------------
$\; p_T$ $> 20$ GeV $> 10$ GeV $> 10$ GeV
$\; \eta$ $< 5$ $< 2.5$ $< 2.5$
: Acceptance cuts used in MadGraph for our simulations.[]{data-label="tab:gcuts"}
For the simulation of the SM background process $p \ell^- \to \ell^+ j j j \nu \nu$, we use the default SM implementation provided by the MadGraph distribution. In the case of the $p \ell^- \to \ell^- \ell^+ j j j \nu $, which is dominated by the exchange of an almost real photon with a very collinear outgoing electron, we simulate instead the process $p \gamma \to \ell^+ j j j \nu $ convoluted with the PDF of a photon inside the charged lepton.
For each experimental setup, we simulate the signal for the case of a heavy neutrino mass of $M_N = 250$, 500, and 750 GeV while setting the mixing to the value required to accommodate the invisible width of the $Z$: $|\theta_{\alpha}|^2 = 0.005$. This provides a quite optimistic signal, but the scaling of the results is trivial with $|\theta_{\alpha}|^2$, as previously discussed. For all cases, we generate a total of $8 \cdot 10^4$ events satisfying the acceptance cuts. At a later stage, after presenting the bare results of the simulation, we will implement additional cuts to reduce the background. The implementation of these cuts will be made at the MadAnalysis level.
Results {#sec:results}
=======
As we are discussing new physics searches, it is fundamental to distinguish between the signal and the SM background. Thus, in order to increase the sensitivity of these searches, cuts which can suppress the background while not affecting the signal too adversely should be investigated. Naturally, an expected background lower than one event would not be very worrisome and the constraining factor would then be one of the signal cross section. However, as the achievable luminosities for these speculative facilities are uncertain, we will present cuts designed to enhance the signal-to-background ratio regardless of the smallness of the cross section. For definiteness, we will assume a baseline integrated luminosity of $\sim100$ fb$^{-1}$ and include this value in the plots for reference. Indeed, this value is in the ballpark of the $10^{33}$ cm$^{-2}$s$^{-1}$ being discussed for the LHCeC proposal [@LHCeC] for its lower energy version. For a 150 GeV electron beam the luminosity could be around an order of magnitude smaller. As for a prospective muon-proton collider the luminosities under discussion for a muon collider range between $10^{33}$ cm$^{-2}$s$^{-1}$ and $10^{35}$ cm$^{-2}$s$^{-1}$ [@Geer:2009zz].
The major question is thus what separates the signal events from the background ones. A cut that has been previously studied in the literature is to reject events where the outgoing $\ell^+$ does not have a minimum transverse momentum. The reason for this cut to reject the background more strongly than the signal is that, while the signal is mainly due to on-shell heavy neutrinos, the background kinematics are such that the positron is predominantly emitted in the beam direction. Since all of the SM backgrounds include at least one neutrino, which will constitute missing energy, while the signal only includes particles visible to the detector, another viable cut could be to impose an upper bound on the missing transverse energy of the accepted events.
In [Fig.]{} \[fig:nocut\_e50\], we show our numerical results for the signal and background distributions in both of these variables at an electron beam energy of 50 GeV in order to illustrate how the signal and background differ.
![Results for the differential cross sections for the 50 GeV electron beam facility. Here, $E_T$ is the missing transverse energy and $p_{T,e^+}$ is the transverse momentum of the positron. Blue, red, and green lines lines correspond to the signal simulated for heavy neutrino masses of $M_N = 250$, 500 and 750 GeV, respectively. The shaded area shows the main backgrounds, the smaller $p e^- \to e^+ j j j \nu_e \nu_e$ in light gray and the larger $p \gamma \to e^+ j j j \nu_e$ in green.[]{data-label="fig:nocut_e50"}](Nocut_e50_ET "fig:"){width="48.00000%"} ![Results for the differential cross sections for the 50 GeV electron beam facility. Here, $E_T$ is the missing transverse energy and $p_{T,e^+}$ is the transverse momentum of the positron. Blue, red, and green lines lines correspond to the signal simulated for heavy neutrino masses of $M_N = 250$, 500 and 750 GeV, respectively. The shaded area shows the main backgrounds, the smaller $p e^- \to e^+ j j j \nu_e \nu_e$ in light gray and the larger $p \gamma \to e^+ j j j \nu_e$ in green.[]{data-label="fig:nocut_e50"}](Nocut_e50_ptPositron "fig:"){width="48.00000%"}
The behavior for different beam energies, and for the muon beams, is similar. In order to illustrate the impact a given cut would have, we show in [Fig.]{} \[fig:cutse\] how the signal and backgrounds for an electron beam would be affected as a function of the value at which the cut is implemented.
![Total cross section for the 50 GeV (upper) and 150 GeV (lower) electron beam facilities as a function of a cut on the maximum missing transverse energy (left) and the minimum transverse momentum of the positron (right). The blue, red, and green lines lines correspond to simulated heavy neutrino masses of $M_N = 250$, 500 and 750 GeV, respectively. The shaded area shows the main backgrounds, the smaller $p e^- \to e^+ j j j \nu_e \nu_e$ in light gray and the larger $p \gamma \to e^+ j j j \nu_e$ in green. The horizontal line represents a cross section of $10^{-2}$ fb, which is the required cross section to have one expected event at an integrated luminosity of 100 fb$^{-1}$.[]{data-label="fig:cutse"}](e50_ETvar "fig:"){width=".48\textwidth"} ![Total cross section for the 50 GeV (upper) and 150 GeV (lower) electron beam facilities as a function of a cut on the maximum missing transverse energy (left) and the minimum transverse momentum of the positron (right). The blue, red, and green lines lines correspond to simulated heavy neutrino masses of $M_N = 250$, 500 and 750 GeV, respectively. The shaded area shows the main backgrounds, the smaller $p e^- \to e^+ j j j \nu_e \nu_e$ in light gray and the larger $p \gamma \to e^+ j j j \nu_e$ in green. The horizontal line represents a cross section of $10^{-2}$ fb, which is the required cross section to have one expected event at an integrated luminosity of 100 fb$^{-1}$.[]{data-label="fig:cutse"}](e50_PTvar "fig:"){width=".48\textwidth"}\
![Total cross section for the 50 GeV (upper) and 150 GeV (lower) electron beam facilities as a function of a cut on the maximum missing transverse energy (left) and the minimum transverse momentum of the positron (right). The blue, red, and green lines lines correspond to simulated heavy neutrino masses of $M_N = 250$, 500 and 750 GeV, respectively. The shaded area shows the main backgrounds, the smaller $p e^- \to e^+ j j j \nu_e \nu_e$ in light gray and the larger $p \gamma \to e^+ j j j \nu_e$ in green. The horizontal line represents a cross section of $10^{-2}$ fb, which is the required cross section to have one expected event at an integrated luminosity of 100 fb$^{-1}$.[]{data-label="fig:cutse"}](e150_ETvar "fig:"){width=".48\textwidth"} ![Total cross section for the 50 GeV (upper) and 150 GeV (lower) electron beam facilities as a function of a cut on the maximum missing transverse energy (left) and the minimum transverse momentum of the positron (right). The blue, red, and green lines lines correspond to simulated heavy neutrino masses of $M_N = 250$, 500 and 750 GeV, respectively. The shaded area shows the main backgrounds, the smaller $p e^- \to e^+ j j j \nu_e \nu_e$ in light gray and the larger $p \gamma \to e^+ j j j \nu_e$ in green. The horizontal line represents a cross section of $10^{-2}$ fb, which is the required cross section to have one expected event at an integrated luminosity of 100 fb$^{-1}$.[]{data-label="fig:cutse"}](e150_PTvar "fig:"){width=".48\textwidth"}
As can be seen from these figures, both cuts are indeed effective in increasing the signal-to-background ratio, although there are some qualitative differences between them. In particular, due to the peaked nature of the signal in the transverse momentum of the outgoing lepton, the cut on this variable is starting to deteriorate also the signal quite early in the cases of low $M_N$. On the other hand, the shape of the transverse energy distribution is similar for all $M_N$ in the area close to zero missing transverse energy, thus leading to a uniform behavior for the signal suppression. It should also be noted that even a very mild cut on the missing transverse energy would be enough to reduce the background by a factor of a few without noticeably affecting the signal and thus, at least a modest cut should be implemented in any analysis. For illustration, we will implement cuts of $p_{T,\ell^+} > 70$ GeV and $E_{T,\rm miss} < 10$ GeV. This roughly corresponds to the regions where the signals start to be noticeably reduced.
The impact of the different cuts can be seen from the remaining total cross sections, which are presented in [Tab.]{} \[tab:xsects\]. In the table we also show the cut in the jets invariant mass $m_{jj}$ around the mass of the $W$ performed in [Ref.]{} [@Ingelman:1993ve] for comparison. It can be seen that this cut strongly reduces the signal while similar backgrounds suppressions can be achieved through the cuts in $p_{T,\ell^+}$ and $E_{T,\rm miss}$ instead.
As can be seen from the table the two cuts are fairly independent and complementary in nature, since their combination provides a significant background reduction with respect to the implementation of only one of them. Moreover, depending on the facility, luminosity and part of the parameter space that is being explored, some combination of the two would provide an optimal cut. Another interesting kinematic variable is the reconstructed invariant mass of two jets and the outgoing lepton $M_{jj\ell}$. In the case of the signal, this should provide a measurement of the mass of the on-shell intermediate heavy neutrino, and thus, the majority of all signal events cluster around this value. A bump of events at a certain value would not only be in favor of signal over background, but also indicate the actual mass of the intermediate particle. Even if a search of this sort is performed, the cuts previously discussed still provide a complementary suppression of the background. This is illustrated in [Fig.]{} \[fig:Mnjje\], where we show the differential cross section with respect to $M_{jj\ell}$ at the 50 GeV electron-proton collider as an example.
![The differential cross section with respect to the reconstructed mass $M_{jj\ell}$ at the 50 GeV electron beam facility. The results are shown both without (left) and with (right) cuts on $p_{T,e^+}$ and $E_{T,\rm miss}$. Blue, red and green lines correspond to the signals for heavy neutrino masses of $M_N = 250$, 500 and 750 GeV, respectively. The shaded area shows the main backgrounds, the smaller $p e^- \to e^+ j j j \nu_e \nu_e$ in light gray and the larger $p \gamma \to e^+ j j j \nu_e$ in green.[]{data-label="fig:Mnjje"}](Nocut_e50_Mnjje "fig:"){width="48.00000%"} ![The differential cross section with respect to the reconstructed mass $M_{jj\ell}$ at the 50 GeV electron beam facility. The results are shown both without (left) and with (right) cuts on $p_{T,e^+}$ and $E_{T,\rm miss}$. Blue, red and green lines correspond to the signals for heavy neutrino masses of $M_N = 250$, 500 and 750 GeV, respectively. The shaded area shows the main backgrounds, the smaller $p e^- \to e^+ j j j \nu_e \nu_e$ in light gray and the larger $p \gamma \to e^+ j j j \nu_e$ in green.[]{data-label="fig:Mnjje"}](ETPTcut_e50_Mnjje "fig:"){width="48.00000%"}\
An additional cut (not discussed in detail here) which could in principle be exploited to enhance the signal over possible backgrounds is the isolation of the final state antilepton.
Finally, in [Fig.]{} \[fig:sens\] we show an estimate of the achievable $90~\%$ CL sensitivity to the mixing between the heavy neutrino and the charged lepton as a function of the luminosity and for different values of the neutrino mass. The sensitivity has been defined as the minimum value of the mixing angle that can be excluded with at least 50 % probability at the $90~\%$ CL in absence of signal. Given the low statistics expected for small mixings, a Poisson distribution was used to compute the sensitivity. The discrete nature of the Poisson distribution is the reason of the abrupt jumps in sensitivity depicted. The sensitivity estimate is conservative and based purely on a counting experiment without exploiting kinematic distributions beyond the $E_T$ and $p_T$ cuts such as the reconstructed mass $M_{jj\ell}$ of [Fig.]{} \[fig:Mnjje\].
![The $90~\%$ CL sensitivity to the mixing between the heavy neutrino and electrons (upper panels) as a function of the luminosity at the 50 GeV (left) and 150 GeV (right) electron beam facilities. The lower panels are the corresponding plots for muons at the 500 GeV (left) and 2 TeV (right) muon beam facilities. Red, blue and black lines correspond to heavy neutrino masses of $M_N = 250$, 500 and 750 GeV, respectively. The solid/dotted lines are the sensitivities with/without cuts. []{data-label="fig:sens"}](ep50-90log "fig:"){width="48.00000%"} ![The $90~\%$ CL sensitivity to the mixing between the heavy neutrino and electrons (upper panels) as a function of the luminosity at the 50 GeV (left) and 150 GeV (right) electron beam facilities. The lower panels are the corresponding plots for muons at the 500 GeV (left) and 2 TeV (right) muon beam facilities. Red, blue and black lines correspond to heavy neutrino masses of $M_N = 250$, 500 and 750 GeV, respectively. The solid/dotted lines are the sensitivities with/without cuts. []{data-label="fig:sens"}](ep150-90log "fig:"){width="48.00000%"}\
![The $90~\%$ CL sensitivity to the mixing between the heavy neutrino and electrons (upper panels) as a function of the luminosity at the 50 GeV (left) and 150 GeV (right) electron beam facilities. The lower panels are the corresponding plots for muons at the 500 GeV (left) and 2 TeV (right) muon beam facilities. Red, blue and black lines correspond to heavy neutrino masses of $M_N = 250$, 500 and 750 GeV, respectively. The solid/dotted lines are the sensitivities with/without cuts. []{data-label="fig:sens"}](mp500-90log "fig:"){width="48.00000%"} ![The $90~\%$ CL sensitivity to the mixing between the heavy neutrino and electrons (upper panels) as a function of the luminosity at the 50 GeV (left) and 150 GeV (right) electron beam facilities. The lower panels are the corresponding plots for muons at the 500 GeV (left) and 2 TeV (right) muon beam facilities. Red, blue and black lines correspond to heavy neutrino masses of $M_N = 250$, 500 and 750 GeV, respectively. The solid/dotted lines are the sensitivities with/without cuts. []{data-label="fig:sens"}](mp2000-90log "fig:"){width="48.00000%"}
As can be seen from the figure, just based on a counting analysis sensitivities more than two orders of magnitude better than the current best fit from the invisible width of the $Z$ would be allowed in these facilities for luminosities of $100$ fb$^{-1}$.
Summary and conclusions {#sec:summary}
=======================
We have discussed the prospects of testing the existence of heavy Majorana neutrinos of a low scale type-I seesaw mechanism in lepton-proton collisions. The question of the Majorana nature of neutrinos together with the origin of neutrino masses is one of the most fundamental issues still unsolved in particle physics. The quest for answers to these unknowns is particularly challenging since all probes testing the Majorana character of neutrinos typically face a strong suppression of the signal due to the smallness of the active neutrino masses. In low scale seesaw mechanisms, however, this smallness originates from a cancellation between different large contributions that could, individually, lead to observable signals at searches such as [${0\nu\beta\beta }$]{} decay or collider experiments. Thus, if the mass spectrum of the heavy Majorana neutrinos is such that some contributions are enhanced or suppressed by the kinematics of the process, the cancellation responsible for the smallness of neutrino masses will not take place. This is particularly true in collider searches, as the one discussed here, given the fact that the contribution of a given neutrino can be enhanced via its s-channel production if the collider has enough energy to produce it on-shell.
We have simulated the lepton number violating signal that the contribution of a heavy Majorana neutrino within the reach of the lepton-proton collider would provide. The simulations were performed for two types of prospective facilities. We have extended previous analyses of electron-proton colliders and updated them to the recent LHCeC proposals in which 7 TeV protons collide against electrons with energies between 50 and 150 GeV. As a more ambitious facility we also considered a muon-proton collider inspired by the efforts towards a muon collider. We simulated muons with energies between 0.5 and 2 TeV colliding against 7 TeV protons. Such a facility would not only allow to probe smaller mixings and higher neutrino masses through higher cross sections of the signal at higher energies, but also provide a complementary search channel, since it would probe areas of the parameter space completely independent to the electron-proton collisions or [${0\nu\beta\beta }$]{} searches.
In order to simulate the signal, we have set the mixing between the new heavy neutrinos and the charged leptons to the value required to accommodate the $2 \sigma$ deviation of the invisible width of the $Z$ with respect to the Standard Model prediction. Even this optimistic assumption leads to a small signal, as expected from the challenging task of testing Lepton number violation. We have shown that, despite the smallness of the expected cross section, the facilities under study would allow for an observable signal in areas of the parameter space still allowed by present data. Moreover, the signal from these facilities is particularly clean, with a very small Standard Model background to obscure it. The dominant background is originated from $W$ production, with its subsequent decay into an $\ell^+$ and the original $\ell^-$ missed by the detector. Nevertheless, we have studied how the signal over background ratio could be improved further through different kinematic cuts so as to allow the search of even smaller values of the mixing angle, provided sufficiently high collider luminosities. A conservative estimate of the sensitivity to the mixing between the heavy neutrinos and charged leptons yields an improvement of more than two orders of magnitude with respect to the present constraints for luminosities of $100$ fb$^{-1}$.
We conclude that lepton-proton colliders provide a particularly clean probe of the elusive Majorana nature of neutrinos and the type-I seesaw mechanism and would complement other ongoing searches by exploring different parts of the allowed parameter space.
We acknowledge illuminating discussions with Paolo Checchia, Andrea Donini, Vicent Mateu and Juan Terron. MB is supported by the European Union through the European Commission Marie Curie Actions Framework Programme 7 Intra-European Fellowship: Neutrino Evolution and. PC acknowledges financial support from the Comunidad Autónoma de Madrid, project HEPHACOS S2009/ESP-1473 and the Spanish Government under the Consolider-Ingenio 2010 programme CUP, “Canfranc Underground Physics”, Project Number CSD00C-08-44022, and under project FPA2009-09017 (DGI del MCyT, Spain). PC would also like to thank the Institute for Particle Physics Phenomenology at Durham where part of this work was completed. MB, PC and EFM acknowledge financial support from the European Community under the European Comission Framework Programme 7, Design Study: EUROnu, Project Number 212372 (the EU is not liable for any use that may be made of the information contained herein).
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[^1]: Other sources of background discussed in [Ref.]{} [@Ingelman:1993ve] such as boson-gluon fusion and DIS are negligible because of the lepton number violating nature of the signal considered.
|
---
abstract: 'We show from the action integral that in the special environment of a flux tube, QCD$_4$ in (3+1) dimensional space-time can be approximately compactified into QCD$_2$ in (1+1) dimensional space-time. In such a process, we find out how the coupling constant $g_{2D}$ in QCD$_2$ is related to the coupling constant $g_{4D}$ in QCD$_4$. We show how the quark and the gluon in QCD$_2$ acquire contributions to their masses arising from their confinement within the tube, and how all these quantities depend on the excitation of the partons in the transverse degrees of freedom. The compactification facilitates the investigation of some dynamical problems in QCD$_4$ in the simpler dynamics of QCD$_2$ where the variation of the gluon fields leads to a bound state.'
author:
- 'Andrew V. Koshelkin'
- 'Cheuk-Yin Wong'
title: ' The Compactification of QCD$_4$ to QCD$_2$ in a Flux Tube'
---
\#1
Introduction
============
Previously, t’Hooft showed that in the limit of large $N_c$ with fixed $g^2N_c$ in single-flavor QCD$_4$, planar diagrams with quarks at the edges dominate, whereas diagrams with the topology of a fermion loop or a wormhole are associated with suppressing factors of $1/N_c$ and $1/N_c^2$, respectively [@tho74a]. In this case a simple-minded perturbation expansion with respect to the coupling constant $g$ cannot describe the spectrum, while the $1/N_c$ expansion may be a reasonable concept, in spite of the fact that $N_c$ is equal to 3 and is not very big. The dominance of the planar diagram allows one to consider QCD in one space and one time dimensions (QCD$_2$) and the physics resembles those of the dual string or a flux tube, with the physical spectrum of a straight Regge trajectory [@tho74b]. Since the pioneering work of t’Hooft, the properties of QCD in two-dimensional space-time have been investigated by many workers [@tho74a; @tho74b; @Fri93; @Dal93; @Fri94; @Arm95; @Kut95; @Gro96; @Abd96; @Dal98; @Arm99; @Eng01; @Tri02; @Abr04; @Li87; @Wit84].
The flux tube picture of longitudinal dynamics is phenomenologically supported in hadron spectroscopy [@Isg85], in hadron collisions, and in $e^+e^-$ annihilations at high energies [@Cas74; @And83; @Won91; @Gat92; @Won94; @Won09; @Won10]. In these high-energy processes, the average transverse momenta of produced hadrons are observed to be limited, of the order of a few hundred MeV. In contrast, the longitudinal momenta of the produced hadrons can be very large, as described by a rapidity plateau with a large average longitudinal momentum. This average longitudinal momentum increases with the collision energy. The limitation of the average transverse momenta of the produced hadrons means that the average momenta of partons in produced hadrons are also limited,[^1] consistent with the picture that the produced partons as constituents of the produced hadrons are transversely confined in a flux tube. Further idealization of the three-dimensional flux tube as a one-dimensional string leads to the picture of the particle production process as a string fragmentation in (1+1) space-time dimensions. The particle production description of Casher, Kogut, and Susskind [@Cas74] in (1+1) dimensional Abelian gauge theory led to results that mimics the dynamics of particle production in hadron collisions and in the annihilation of $e^+e^-$ pairs at high energies. Furthermore, the Lund model of classical string fragmentation has been quite successful in describing quantitatively the process of particle production in these high energy processes [@And83; @Won94].
With the successes of the theoretical description of Casher $et~al.$ and the Lund model of string fragmentation, it should be possible to compactify quantum chromodynamics in (3+1) dimensional space-time (QCD$_4$) approximately to quantum chromodynamics in (1+1) dimensional space-time (QCD$_2$), in the special environment appropriate for particle production at high-energies. It is useful to examine the circumstances under which such a compactification is possible. Such a link was given earlier in [@Won09; @Won10] and reported briefly in [@And12]. Here, we would like to examine the problem from the more general viewpoint of the action integral.
We note that the process of string fragmentation occurs when a valence quark-antiquark pair pull part from each other at high energies, as described in [@Cas74; @And83]. It is therefore reasonable to examine the QCD$_4$ compactification under the dominance of longitudinal dynamics in the center-of-mass frame of the receding valence $q \bar q$ pair. Under such a longitudinal dominance in this frame, not only are the magnitudes of the longitudinal momenta of the leading valence quark and antiquark dominant over their transverse momenta, so too are the magnitudes of longitudinal momenta of the produced $q\bar q$ parton pairs. The spatially one-dimensional string is an idealization of a more realistic three-dimensional flux-tube. The description of produced $q\bar q$ parton pairs residing within the string or flux tube presumes the confinement of these produced partons in the string. Hence, it is reasonable to examine further the QCD$_4$ compactification under transverse confinement. As transverse confinement is a nonperturbative process and is beyond the realm of perturbative QCD, we can describe the transverse confinement property in terms of a confining scalar interaction $S({\bb
r}_\perp)$ in transverse coordinates $\bb r_\perp$, with the quark mass function described by $m(\bb r_\perp)$=$m_0$+$S({\bb
r}_\perp)$ where $m_0$ is the quark rest mass.
Having spelled out explicitly the circumstances under which the QCD$_4$ compactification may occur, we proceed to start with the QCD$_4$ action integral and begin our process of compactification. We need to find out how we can relate the field variables in four-dimensional space-time to those in 2-dimensional space-time in such a way that the four-dimensional action integral can be simplified to contain only field quantities in two-dimensional space-time. What is the form of the two-dimensional action integral after compactification? How are the coupling constant $g_{2D}$ in the two-dimensional action integral related to the coupling constant $g$=$g_{4D}$ in QCD$_4$ in four-dimensional space-time? Are there additional terms in the two-dimensional action integral that arise from the compactification? How do all these quantities depend on the excitation of the partons in the transverse degrees of freedom?
We shall show that the compactification for QCD$_4$ in a flux tube leads to an action integral of a QCD gauge field coupled to the quark field in two-dimensional space time, which can be appropriately called QCD$_2$. The QCD gauge field coupling constant is found to depend on the quark transverse wave function in the flux tube. There are additional quark- and gluon-mass terms that arise from the confinement of the quark and the gluon within the tube.
The success of the compactification program facilitates the examination of some problems in QCD$_4$ in the simpler dynamics of QCD$_2$. The QCD$_2$ action integral allows one to obtain the equations of motion for the quark field and the gauge field. We find self-consistent solution of a boson state with a mass in the flux tube environment, similar to Schwinger’s solution of a massive boson in two-dimensional Abelian gauge field theory.
It should be noted that the occurrence of a massive composite bound state in gauge field theories has been known in many previous investigations [@xxx]. While the basic principles of the massive bound state as arising from interactions of the gauge fields in these theories are the same as in the present investigation in a flux tube, the physical environments and the constraints are quite different. How the massive boson in a flux tube environment examined here can be related to the massive boson formed by purely gluons as a pole in the three gluon vertex in 4-dimensional space-time [@Pap11; @Pap12] is a subject worthy of further investigation.
This paper is organized as follows. In Sec. II, we show how the action integral in QCD$_4$ can be compactified into QCD$_2$, under the assumption of longitudinal dominance and transverse confinement. The relationship between the 4-dimensional (4D) quantities and those two-dimensional counterpart are expressed explicitly. The fermions and gauge bosons acquire contributions to their masses that arises from the confinement. In Sec. III, we solve the Dirac field equation in (1+1) space-time, and obtain the relation between the current and the gauge field. In Sec. IV, we examine the gauge field degrees of freedom in two-dimensional (2D) space-time. In Sec. V, we determine the equation of transverse motion for fermions in a tube. In Sec. VI, we present our conclusions and discussions.
$4D \to 2D$ Compactification in the Action Integral
====================================================
We employ the convention that before compactification is achieved, all field quantities and gamma matrices are in four-dimensional space-time unless specified otherwise. With fermions interacting with an SU(N) gauge field and a scalar field $m(x)$ in the (3+1) Minkowski space-time, the SU(N) gauge invariant action integral $\cal A$ is given by [@Pes95] $$\begin{aligned}
&& {\cal A} = \int d^4 x \Bigg\{ Tr \Bigg[ {1\over 2} \left[
{\bar \Psi} \gamma^\mu \Pi_\mu \ \Psi
- {\bar \Psi} ~ m (x)~ \Psi \right]
- {1\over 2} \left[ {\bar \Psi} \gamma^\mu
{\overleftarrow \Pi_\mu } \ \Psi + {\bar \Psi} ~ m (x) ~
\Psi \right] \Bigg] -
{1\over 4 } F^a_{\mu \nu} \ F_a^{\mu \nu} \Bigg\}, \label{1}\end{aligned}$$ where $A^a_\mu $ and $\Psi $ are the gauge and fermion fields respectively in the Minkowski (3+1)-dimensional space-time with coordinates $x\equiv x^\mu = (x^0 , {\bb x})=(x^0 , x^1, x^2,
x^3)$ and transverse coordinates $\bb r_\perp=(x^1, x^2)$. Here in Eqs. (\[1\]) , $$\begin{aligned}
\label{eq2}
\Pi_\mu &=& i \partial_\mu + g_{4D} ~ T_a A^a_\mu,
\nonumber \\
F^a_{\mu \nu} &=&
\partial_\mu A^a_\nu - \partial_\nu A^a_\mu + g_{4D} ~ f^a_{~bc } ~ A^b_\mu ~ A^c_\nu ,
\\&\equiv& \partial_\mu
A^a_\nu - \partial_\nu A^a_\mu - i g_{4D} [ A^b_\mu ,~ A^c_\nu]^a
, \label{4}\end{aligned}$$ $\gamma^{\nu}$ are the standard Dirac matrices, $\partial_{\mu} =
(\partial /\partial t , \nabla )$, and $a,b,c = 1 \dots N^2 -1$ are SU(N) group indices. We use the signature of (1, -1, -1, -1) for the diagonal elements of metric tensor $ {g}_{\mu \nu}$. We should note that the action integral is gauge invariant since $
m(x)$ is independent of the SU(N) group generators $T_a$.
Fermion part of the action integral
-----------------------------------
The $4D$-action integral ${\cal A } $ resides in four-dimensional (3+1) space-time. There are environments in which the full four-dimensional space-time is necessary, as for example in the discussion of the phase transition in a hot quark-gluon plasma [@Kar09; @Fod04; @Phi11]. There are however environments which are susceptible for compactification to two-dimensional (1+1) space-time, in which the dynamics can be greatly simplified.
A proper environment for the compactification of QCD$_4$ can be found in the special case in which a valence quark and antiquark pull part from each other at high energies, as in the case examined by Casher, Kugut, and Susskind [@Cas74]. It is convenient to work in the center-of-mass frame of the receding quark-antiquark pair in which the magnitudes of the longitudinal momenta of the valence quark pairs are very large, much larger than the magnitudes of their transverse momenta. Under such a dominance of longitudinal dynamics, not only are the magnitudes of the longitudinal momenta of the leading valence quark and antiquark pair large, so are also those of the produced $q$ and $\bar q$ partons. It is then convenient to choose the Lorentz gauge $$\begin{aligned}
\partial ^\nu A_\nu^a=0.
\label{7A}\end{aligned}$$ In this Lorentz gauge, $A_\nu^a$ is given by an integral of the current $J_\nu^a$. For a system with longitudinal dominance, the magnitudes of the transverse currents are much smaller than the magnitudes of the longitudinal currents. As a consequence, the magnitudes of the gauge field transverse components, $A_1^a$ and $A_2^a$, along the transverse directions are small in comparison with those of $A_0^a$ and $A_3^a$. The gauge field components $A_1^a$ and $A_2^a$ can be neglected. The absence of the transverse components of the gauge fields in the Lorentz gauge provides a needed simplification for compactification. However, both $A_0^a$ and $A_3^a$ still depend on the $4D$ space-time variables, $A_0^a ( x^0 , {\bb x} ), ~ A_3^a ( x^0 , {\bb x} )$.
The dominance of the longitudinal motion implies that the valence leading quark and anti-quark lie inside a longitudinal tube. The limiting average transverse momentum suggests further that the produced quarks reside within the longitudinal tube with a radius inversely proportional to this limiting average transverse momentum. As the confinement of the produced quarks within the tube is a nonperturbative process that is beyond the realm of perturbative QCD, we can represent the confinement property in terms of a confining scalar interaction $S({\bb r}_\perp)$ in transverse coordinates $\bb r_\perp$, with the quark mass function $m(\bb r_\perp)=m_0+S({\bb
r}_\perp)$. The origin of ${\bb r}_\perp$ coordinates lies along the longitudinal axis of the receding valence quark pair. Because of the presence of a scalar interaction $m(\bb r_\perp)$, our dynamical problem does not maintain general Lorentz in all directions. There remains however approximate Lorentz invariance with respect to a finite boost along the longitudinal axis and the range of this finite boost increases as the energy of the receding quark pair increase.
Under such circumstances, we can carry out the compactification of QCD$_4$ in (3+1) dimensions as follows. The fermion part of the $4D$-action ${\cal A } _F $ in (\[1\]) is given by $$\begin{aligned}
\label{eq7}
{\cal A }_F \!=\! Tr \!\! \int\! d^4 x \Biggl\{ \!{1\over 2}
{\bar \Psi} \gamma^\mu\Pi_\mu \Psi \! -\! {1\over 2} {\bar
\Psi} \gamma^\mu{\overleftarrow \Pi_\mu } \Psi \!-\! {\bar
\Psi} m ({\bb r}_\perp )
\Psi \!\Biggr\},~
\label{7}\end{aligned}$$ where $\mu = 0,1,2,3$ and $ \gamma^\mu$ is the $4D$-Dirac matrices, $$\begin{aligned}
\label{eq8}
&& \gamma^0 = \left( \begin{array}{cccc} 0 \
\ \ \ \ \ I \\ \\
I \ \ \ \ \ \ 0 \\
\end{array} \right) , \ \ \ {\bb \gamma } = \left( \begin{array}{cccc} 0 \
\ \ \ \ \ - {\bb \sigma} \\ \\
{\bb \sigma} \ \ \ \ \ \ 0
\end{array} \right).\end{aligned}$$ To relate the field variables in four-dimensional space-time to those in 2-dimensional space-time in such a way that the four-dimensional action integral can be simplified, we write the Dirac fermion field $\Psi ( x )$ in terms of the following bispinor with transverse functions $G_\pm({\bb r}_\perp)$ and $x^0$-$x^3$ functions $f_\pm(x^0,x^3)$ [@Won91], $$\begin{aligned}
\label{eq9}
&& \Psi ( x ) \equiv \left( \begin{array}{cccc} \varphi ( x^0 ;
{\bb x} )
\\
\chi ( x^0 ; {\bb x} )
\end{array} \right) \equiv
\left( \begin{array}{cccc} \varphi_1 ( x^0 ; {\bb x} )
\\ \varphi_2 ( x^0 ; {\bb x} ) \\
\chi_1 ( x^0 ; {\bb x} ) \\ \chi_2 ( x^0 ; {\bb x} )
\end{array} \right)
= \frac {1}{\sqrt{2}}\left( \begin{array}{cccc} G_1 ({\bb
r}_\bot ) \left( f_+ (x^0 ; x^3 ) + f_- (x^0 ; x^3 ) \right) \
\\
- G_2 ({\bb r}_\perp) \left( f_+ (x^0 ; x^3 ) - f_- (x^0 ; x^3 )
\right)
\\
G_1 ({\bb r}_\perp) \left( f_+ (x^0 ; x^3 ) - f_- (x^0 ; x^3 )
\right)
\\
G_2 ({\bb r}_\perp) \left( f_+ (x^0 ; x^3 ) + f_- (x^0 ; x^3 )
\right)
\end{array} \right) ,\end{aligned}$$ where ${\bb r}_\perp$ is a vector in the plane perpendicular to the $x^3$ axis. Using this explicit form of the Dirac bispinor $\Psi$, we can carry out simplifications (with detailed derivation given in Appendix A) that lead from Eq. (\[7\]) eventually to $$\begin{aligned}
{\cal A }_F &=& Tr~ \int d^2 X \Biggl\{ {1\over 2}
{\bar \Psi} (2D, X)\left[ i \gamma^\mu (2D) \partial_\mu + g_{2D}
\gamma^\mu T_a A^a_\mu (2D, X)\
\right] \Psi (2D, X) \nonumber \\
&-& {1\over 2} {\bar \Psi} (2D, X) \left[ i \gamma^\mu (2D)
\overleftarrow{\partial}_\mu - g_{2D} \gamma^\mu (2D) T_a A^a_\mu
(2D, X)\ \right]\Psi (2D, X)\nonumber\\
&-& {\bar \Psi} (2D, X) ~ m_{qT} ~
\Psi (2D, X) \Biggr\}\equiv {\cal A }_F (2D) ,~~~~~ \mu=0,3,
\label{10}\end{aligned}$$ where we have introduced in the Dirac fermion field $\Psi(2D,X)$, $\gamma$-matrices, and metric tensor $g_{\mu \nu}$, according to the following specifications in the (1+1)-dimensional QCD$_2$ space-time $$\begin{aligned}
&& \Psi ( 2D,X )= \left( \begin{array}{cccc} f_+ ( X ) \\
f_- ( X )
\end{array} \right) , \ \ \ \ \ \ \ X = ( x^0 ; x^3 ) ,
\label{11a}\\
&&
\gamma^0 (2D)= \left( \begin{array}{cccc} 1\ \ \ \ \ \ 0 \\ \\
0
\ \ \ \ \ -1 \\
\end{array} \right) , \ \ \ \gamma^3 (2D) = \left( \begin{array}{cccc} 0 \
\ \ \ \ \ 1 \\ \\ -1 \ \ \ \ \ \ 0
\end{array} \right), ~~~~~ g_{\mu \nu }(2D)= \left( \begin{array}{cccc} 1 \ \ \ \ \ \ 0 \\ \\
0
\ \ \ \ \ \ -1 \\
\end{array} \right).
\label{11}\end{aligned}$$ The $2D$ coupling constant, $g_{2D}$, is defined by the following equation (see Appendix A) $$\begin{aligned}
g_{2D}=\int dx^1 dx^2 g_{4D} [{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{3/2},
\label{13}\end{aligned}$$ where the transverse wave functions $G_{1,2}({\bb r}_\perp)$ are normalized according to $$\begin{aligned}
\label{norm} \int d x^1 d x^2 \left( \vert G_1 ({\bb r}_\perp)
\vert^2 +
\vert G_2 ({\bb r}_\perp) \vert^2 \right) = 1 .
\label{14}\end{aligned}$$ In the special case of the transverse ground state, we can approximate the transverse density by a uniform distribution with a sharp transverse radius $R_{T{\rm sharp}}$, $$\begin{aligned}
\label{eq15}
\left( \vert G_1 ({\bb r}_\perp)
\vert^2 +
\vert G_2 ({\bb r}_\perp) \vert^2 \right) \sim \frac{1}{\pi R_{T{\rm sharp}}^2}\Theta(R_{T{\rm sharp}}-|\bb r_T|),\end{aligned}$$ we then obtain for a sharp distribution in the transverse ground state the approximate relation [@Won09] $$\begin{aligned}
\label{eq16}
g_{2D}\sim \frac{g_{4D}}{\sqrt{\pi} R_{T{\rm sharp}}}.\end{aligned}$$
If we characterize the transverse ground state with a Gaussian profile and a root-mean-square transverse radius $R_T=\sqrt{2}\sigma_T$ as $$\begin{aligned}
\label{Gauss} \left( \vert G_1 ({\vec r}_\bot ) \vert^2 +
\vert G_2 ({\vec r}_\bot ) \vert^2 \right)&=&\frac{1}{2\pi \sigma^2} \exp\{ - \frac{r^2}{2\sigma_T^2}
\},
\label{16a}\end{aligned}$$ then the corresponding $g_{2D}$ coupling constant becomes $$\begin{aligned}
\label{g2D}
g_{2D} &=& \frac{g_{4D}}{R_T}~ \sqrt{\frac{2}{9\pi}}.\end{aligned}$$ The transverse quark mass $m_{qT}$ in Eq. (\[10\]) is given by (see Appendix A) $$\begin{aligned}
m_{qT} \!=\!\! \int\!\! d x^1 d x^2 \left\{
m ({\bb r}_\perp) \left( \vert G_1
({\bb r}_\perp) \vert^2 - \vert G_2 ({\bb r}_\perp) \vert^2
\right) + \left( G^\ast_1 ({\bb r}_\perp) ( p_1 - ip_2 ) G_2 ({\bb r}_\bot ) \right) - \left( G_1 ({\bb r}_\perp) ( p_1 + ip_2 )
G^\ast_2 ({\bb r}_\perp) \right) \right\}\!.
\label{mqT}\end{aligned}$$ The transverse quark mass $m_{qT}$ contains a contribution from the quark rest mass (through $m(r)$), in addition to a contribution arising from the confinement of the quark in the flux tube (through the confining wave functions $G_{1,2}({\bb
r}_\perp)$). In obtaining these results, we have considered $2D$ gauge fields $A_\mu^a(2D,x^0,x^3)$ to be related to the $4D$-field gauge fields $A_\mu^a(x^0,x^3,{\bb r}_\perp)$ by $$\begin{aligned}
\label{AA} A_\mu^a(x^0,x^3,{\bb r}_\perp)&=& \sqrt{ |G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2} A_\mu^a(2D,x^0,x^3),~~~\mu=0,3.
\label{20}\end{aligned}$$ The above equation means that along with the confinement of the fermions, for which the wave function $G_{1,2}({\bb r}_\perp)$ is confined within a finite region of transverse coordinates ${\bb
r}_{\perp}$, the gauge field $A_\mu^a ( x)$, $\mu=0,3$, is also considered to be confined within the same finite region of transverse coordinates, as in the case for a flux tube. Note that because of the longitudinal dominance, we have assumed that $
A_\mu^a(x^0,x^3,{\bb
r}_\perp)= 0$ for $\mu=1,2$.
Gauge field part of the action integral
---------------------------------------
Having reduced the fermion part of the action integral ${\cal
A}_F$, we come to examine the gauge field part of the action integral ${\cal
A}_A$, $$\begin{aligned}
&& {\cal A}_A = -\frac{1}{4}\int d^4 x
F^a_{\mu \nu} \ F_a^{\mu \nu} .
\label{19}\end{aligned}$$ Our task is to find out what will be the form of ${\cal A}_A$ involving the gauge fields $A_\mu(2D)$ the two-dimensional space-time, when $A_\mu(2D)$ the $A_\mu$ in four-dimensional space-time are related by Eq. (\[20\]).
In Eq. (\[19\]) the summation over $\mu,\nu$ includes terms with $\mu,\nu=1,2$. Previously, in going from ${\cal A}_F$ in Eq. (\[7\]) to ${\cal A}_F(2D)$ in the action integral of Eq.(\[10\]), we have assumed that the currents in the $x^0$ and $x^3$ directions are much greater in magnitude than the currents in the transverse directions so that $A_1^a$ and $A_2^a$ are small in comparison and can be neglected. As a consequence, $F_{12}(4D)=0$ (we omit the superscript color index $a$ for simplicity).
We consider now the contribution one of the terms, $F_{03}F^{03}$, in Eq. (\[19\]). Equation (\[20\]) gives $F_{03}(x^0,x^3,{\bb
r}_\perp)$ in four-dimensional space-time as $$\begin{aligned}
F_{03}(x^0,x^3,{\bb r}_\perp) &=& [{ |G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2}
[\partial_0 A_3(2D,x^0,x^3)-\partial_3 A_0(2D,x^0,x^3)]
\nonumber\\
& & - i g_{4D} [{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}] [A_0(2D,x^0,x^3),A_3(2D,x^0,x^3)]. \label{eq21}\end{aligned}$$ On the other hand, the gauge field $F_{03}(2D,x^0,x^3)$ in two dimensional space-time is given by definition as $$\begin{aligned}
\label{eq22}
F_{03}(2D,x^0,x^3)&=&\partial_0 A_3(2D)-\partial_3 A_0(2D)
-ig_{2D} [A_0(2D),A_3(2D)],
\end{aligned}$$ where for brevity of notation, the coordinates $(x^0,x^3)$ in $A_\mu (2D,x^0,x^3)$ will be understood. As a consequence, $F_{03}(2D,x^0,x^3)$ in two-dimensional space-time and $F_{03}(x^0,x^3,{\bb r}_\perp)$ in four-dimensional space-time are related by $$\begin{aligned}
\label{eq23}
F_{03}(x^0,x^3,{\bb r}_\perp)&=& [{ |G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2}\{ F_{03}(2D,x^0,x^3)+ig_{2D} [A_0(2D),A_3(2D)]\}
\nonumber\\
& & - i g_{4D} [{|G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}] [A_0(2D),A_3(2D)].\end{aligned}$$ The above equation can be re-written as $$\begin{aligned}
\label{eq24}
F_{03}(x^0,x^3,{\bb r}_\perp) &=& [{ |G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2}
\biggl \{ F_{03}(2D,x^0,x^3)
\nonumber\\
& &+\biggl [ i g_{2D} - i g_{4D} [{ |G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2}\biggr ] [A_0(2D),A_3(2D)] \biggr \}.\end{aligned}$$ The product $F_{03} (x)F^{03}(x)$ in eq. (\[19\]) becomes $$\begin{aligned}
\label{eq25}
& &F_{03}(x^0,x^3,{\bb r}_\perp) F^{03}(x^0,x^3,{\bb r}_\perp) =
[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]\biggl \{ F_{03}(2D,x^0,x^3)F^{03}(2D,x^0,x^3)
\nonumber\\
&+& \biggl [ i g_{2D} - i g_{4D} \{{( |G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2})\}^{1/2}\biggr ] \biggl ( F_{03}(2D) [A^0(2D),A^3(2D)]
+ [A_0(2D),A_3(2D)]F^{03}(2D) \biggr )
\nonumber\\
& & +\biggl [ i g_{2D} - i g_{4D} [{( |G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2})]^{1/2}\biggr ]^2\biggl ( [A_0(2D),A_3(2D)]
[A^0(2D),A^3(2D)] \biggr )
\biggr \}. \label{26}\end{aligned}$$ The action integral ${\cal A}_A$ in Eq. (\[19\]) involves the integration of the above quantity over $x_1$ and $x_2$. Upon integration over $x^1$ and $x^2$, the second term inside the curly bracket of the above equation, is zero, $$\begin{aligned}
\label{second} & &\int dx^1 dx^2 [{( |G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2})]\biggl [ig_{2D} - ig_{4D} [{( |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2})]^{1/2}\biggr ]=0,
\label{25}\end{aligned}$$ where we have used the relation between $g_{2D}$ and $g_{4D}$ as given by Eq.(\[13\]) and the normalization condition of (\[norm\]). As a consequence, the integral of $F_{03}
(x)F^{03}(x)$ in Eq. (\[19\]) becomes $$\begin{aligned}
& &\int dx F_{03}(x^0,x^3,{\bb r}_\perp) F^{03}(x^0,x^3,{\bb
r}_\perp) =\int dx [{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]\biggl \{ F_{03}(2D,x^0,x^3)F^{03}(2D,x^0,x^3)
\nonumber\\
& & +\biggl [ i g_{2D} - i g_{4D} [{( |G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2})]^{1/2}\biggr ]^2\biggl ( [A_0(2D),A_3(2D)]
[A^0(2D),A^3(2D)] \biggr )
\biggr \}.\end{aligned}$$ For the second term in the curly bracket, the integral over $dx^1$ and $dx^2$ is $$\begin{aligned}
\int dx^1 dx&^2 [{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]\biggl [ i g_{2D} - i g_{4D} [{( |G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2})]^{1/2}\biggr ]^2\end{aligned}$$ which can be considered as an integral over $g_{2D}$ in the form $$\begin{aligned}
2 i\int dg_{2D}\int dx^1 dx&^2 [{ |G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]\biggl [ i g_{2D} - i g_{4D} [{( |G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2})]^{1/2}\biggr ].\end{aligned}$$ Because of Eq. (\[25\]), the above integral gives an irrelevant constant which we can set to zero. After these manipulations, we obtain $$\begin{aligned}
\label{eq29}
\int d x^1 d x^2 F_{03}(x^0,x^3,{\bb r}_\perp) F^{03}(x^0,x^3,{\bb
r}_\perp) &=& \int d x^1 d x^2
[{( |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2})] F_{03}(2D,x^0,x^3)F^{03}(2D,x^0,x^3)
\nonumber\\
&=& F_{03}(2D,x^0,x^3)F^{03}(2D,x^0,x^3).\end{aligned}$$ Following the same way (see Appendix B), we calculate terms containing $F_{01}(4D)$, $F_{02} (4D)$, $F_{31} (4D)$, and $F_{32}
(4D)$. For the gauge field part of the action integral, we obtain $$\begin{aligned}
\frac{1}{4} \int dx
F^a_{\mu \nu} F_a^{\mu
\nu} &=&\frac{1}{4} \int {dx^0 dx^3} F^a_{03 } (2D,x^0,x^3 )
F_a^{03}(2Dx^0,x^3) \nonumber\\
&-& \int \frac{dx^0 dx^3}{ 4 } \int dx^1 dx^2 \biggl (
\{\partial_1[{ |G_1({\bb r}_\perp)|^2
+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2
\nonumber\\
& &\hspace*{2.0cm}+\{\partial_2[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2\biggr )
\nonumber\\
& &\times [A_0(2D,x^0,x^3)A^0(2D,x^0,x^3)
+A_3(2D,x^0,x^3)A^3(2D,x^0,x^3)]. \label{28a}\end{aligned}$$ It is useful to introduce the gluon mass $m_{gT}$ that arises from the confinement of the gluons in the transverse direction $$\begin{aligned}
\label{mgT}
m_{gT}^2= \frac{1}{2} \int dx^1 dx^2
\biggl [ \{\partial_1[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2+\{\partial_2[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2\biggr ].\end{aligned}$$ Equation (\[28a\]) becomes $$\begin{aligned}
\label{eq32}
\frac{1}{4} \int {d x}
F^a_{\mu \nu} \ F_a^{\mu
\nu}&=& \frac{1}{4} \int {dx^0 dx^3}
\biggl \{ F^a_{03 } (2D) F_a^{03}(2D) - 2 m_{gT}^2 [A_0^a (2D) A^0_a(2D)
+A_3^a (2D)A^3_a(2D)] \biggr \}.\end{aligned}$$ We collect all the fermion and gauge field parts of the action in ${\cal A} (4D )$ in Eq. (\[1\]). The action integral ${\cal
A}={\cal A}_F+{\cal A}_A$ that was an integral in four-dimensional space-time now turns into an integral only in two-dimensional space-time. All quantities in ${\cal A} (4D )$ are completely defined in (1+1) dimensional space-time coordinates, we rename this action integral ${\cal A} (2D )$ that is given explicitly by $$\begin{aligned}
{\cal A} (2D ) = \int d^2 X &\Bigg\{& Tr \Bigg[ {1\over 2}
\left[ {\bar \Psi}
(2D,X) \gamma^k(2D) \Pi_k(2D) \Psi (2D,X) - {\bar \Psi} (2D,X) m_{qT} \Psi (2D,X)
\right]
\nonumber \\
& &- {1\over 2} \left[ {\bar \Psi} (2D,X) \gamma^k(2D)
{\overleftarrow \Pi_k }(2D) \ \Psi (2D,X) + {\bar \Psi} (2D,X) m_{qT} \Psi (2D,X)
\right] \Bigg] \nonumber \\ && -{1\over 4 } F^a_{\mu \nu} (2D)
\ F_a^{\mu \nu} (2D) +{1\over 2 }m_{gT}^2 [A_a^\mu(2D)A^a_\mu(2D)] \Bigg\},
\label{33}\end{aligned}$$ where $\{\mu,\nu\}$=0,3, and $$\begin{aligned}
\label{eq34,35}
\Pi_\mu (2D) &=& i \partial_\mu + g_{2D} ~ T_a A^a_\mu (2D, X)
= p_\mu + g_{2D} ~ T_a A^a_\mu (2D, X) .\end{aligned}$$ Here, all terms (including matrices and coefficients) in the action integral of Eq. (\[33\]) are in the $(1+1) $ Minkowski space-time. Thus, in the environment of longitudinal dominance and transverse confinement, we succeed in compactifying the action integral in four-dimensional space-time to two-dimensional space-time by judiciously relating the field operators in four-dimensional space-time to the corresponding field operators in two-dimensional space-time.
The result in this subsection indicates that the compactified two-dimensional action integral has the same form as QCD in two-dimensional space-time, and the compactified field theory can be appropriately call QCD$_2$. It has the feature that the coupling constant $g_{2D}$ in QCD$_2$ acquires the dimension of a mass, and is related to $g_{4D}$ and the wave functions of the confined fermions in the flux tube. Fermions in different excited states inside the tube will have different coupling constants as indicated in Eq. (\[13\]). Furthermore, the action integral gains additional transverse mass terms with an effective quark mass $m_{qT}$ and gluon mass $m_{gT}$ that also depend on the transverse fermion wave functions, as given in Eqs. (\[mqT\]) and (\[mgT\]) respectively. The transverse quark mass includes a contribution from the quark rest mass, in addition to a contribution due to the confinement of the flux tube. In the lower two-dimensional space-time, fermions in excited transverse states have a quark transverse mass different from those in the ground transverse states. All the transverse flux tube information is subsumed under these quantities.
Provided that the fields $A_\mu^a(4D, x^1,x^2,x^0,x^3)$ are governed by the standard gauge tarnsformation [@Pes95], the 2D gauged fields $A_\mu^a(2D,x^0,x^3)$ introduced according to Eq.(\[AA\]) are found to transform under a gauge tarnsformation as follows (see Appendix C) : $$\begin{aligned}
\label{2DDtrans} &&A_\mu^a(2D,x^0,x^3) \to {\tilde
A}_\mu^a(2D,x^0,x^3) = A_\mu^a(2D,x^0,x^3) +
f^{a}_{~b c} \varepsilon^b (x^0, x^3)
A_\mu^c (2D, x^0 ,x^3)\end{aligned}$$ As a consequence, the term $A_\mu^a(2D)A^{\mu}_{ a} (2D)$ transforms under a gauge transformation as $$\begin{aligned}
A_\mu^a(2D,x^0,x^3) A^\mu_a(2D,x^0,x^3) & \to & {\tilde
A}_\mu^a(2D,x^0,x^3) {\tilde A}^\mu_a(2D,x^0,x^3) ,\end{aligned}$$ which indicates that the mass term in Eq. (\[33\]) does not violate gauge invariance and does not violate the Slavnov-Taylor [@Slavnov] identities (see Appendix C) due to the 2D gauge transformations given by Eqs.(\[2DDtrans\]), (\[2Dtrans\]).
Solution of the Dirac fields in (1+1) space-time
=================================================
Having completed the program of compactification of QCD$_4$ to QCD$_2$, we shall employ the new notation henceforth that all field quantities and gamma matrices are in two-dimensional space-time with $\mu=0,3$, unless specified otherwise. We can use the QCD$_2$ action integral to get the equation of motion for the field. Varying the action integral ${\cal A}(2D)$ given by Eq.(\[33\]) with respect to $ {\bar \Psi}$, we derive the 2D Dirac equation, $$\begin{aligned}
&& \left\{ i \gamma^{\mu} \left( \partial_{\mu} - i g_{2D} \cdot
A_{\mu}^a (X)~ T_a \right) - m_{qT} \right\} \Psi (X) = 0 ,
\label{28}\end{aligned}$$ where $2D$ Dirac matrices are those given in Eq. (\[11\]). The gauge field $A_{\mu}^a$ written in component form is $$\begin{aligned}
\label{eq37}
&& A_{\mu}^a (X) = \left( A^a_{0} , - A^a_{3} \right) ,
~~~~~~~ X = ( x^0 ; x^3 ) .\end{aligned}$$ We express the fermion field $\Psi $ in terms of $ f_+ (X)$ and $f_- (X)$ as in Eq. (\[11a\]), $$\begin{aligned}
\label{eq38}
&& \Psi = \left(
\begin{array}{cccc}
1 \\
0
\end{array} \right) ~f_+ (X) + \left(
\begin{array}{cccc}
0 \\
1
\end{array} \right) ~f_- (X).\end{aligned}$$ Then, the Dirac equation (37) becomes $$\begin{aligned}
\label{eq39}
&& i { \partial f_+ (t, z) \over \partial t} + i {
\partial f_- (t, z) \over \partial z} + g_{2D}\ T_a \ A_0^a f_+
(t, z) - g_{2D}\ T_a \ A_3^a \ f_- (t, z) =
m_{qT} f_+ (t, z), \\
&& - i { \partial f_- (t, z) \over \partial t} - i {
\partial f_+ (t, z) \over \partial z} - g_{2D}\ T^a \ A^0_a \
f_- (z) + g_{2D}\ T_a \ A_3^a \ f_+ (t, z) = m_{qT} f_- (t, z) ,
\nonumber \\ && t\equiv x^0 ; ~~~ z\equiv x^3.
\nonumber \end{aligned}$$ We introduce new functions as sum and difference of $f_+$ and $f_-$: $$\begin{aligned}
\label{eq40}
&& \eta (t, z)= f_+ (t, z) - f_- (t, z) , \nonumber \\
&& \zeta (t, z)= f_+ (t, z) + f_- (t, z).
\end{aligned}$$ As a result, we obtain $$\begin{aligned}
&& i { \partial \eta (t, z) \over \partial t} - i {
\partial \eta (t, z) \over \partial z} + g_{2D} {\hat A}_1 \eta (t, z)
= m_{qT} \zeta(t, z), \nonumber \\
&& i { \partial \zeta (t, z) \over \partial t} + i {
\partial \zeta (t, z) \over \partial z} + g_{2D} {\hat A}_2 \zeta (t, z) =
m_{qT} \eta(t, z) , \label{42}
\end{aligned}$$ where $$\begin{aligned}
\label{eq42}
&& {\hat A}_1 = T_a ( A_0^a + A_3^a ); \ \ \ {\hat A}_2 = T_a (
A_0^a - A_3^a ).
\end{aligned}$$ We look for a solution of Eq. (\[42\]) in the form $$\begin{aligned}
\label{eq43}
&& \eta (t, z) = F (t, z ) \chi (t, z),
\nonumber \\
&& \zeta (t, z) = G ( t, z ) \chi (t, z),\end{aligned}$$ where the functions $\chi (t, z )$, $F (t, z )$ and $G (t, z )$ satisfy the following equations: $$\begin{aligned}
&& i { \partial \chi (t, z) \over \partial t} - i {
\partial \chi (t, z) \over \partial z} + g_{2D} {\hat A}_1 \eta (t, z)
= 0, \nonumber \\
&& i { \partial \chi (t, z) \over \partial t} + i {
\partial \chi (t, z) \over \partial z} + g_{2D} {\hat A}_2 \zeta (t, z) =
0 , \label{36}
\end{aligned}$$ while $$\begin{aligned}
&& i { \partial F (t, z) \over \partial t} - i {
\partial F (t, z) \over \partial z} = m_{qT} G (t, z),
\nonumber \\
&& i { \partial G (t, z) \over \partial t} + i {
\partial G (t, z) \over \partial z} =
m_{qT} F (t, z). \label{37}
\end{aligned}$$ The solution of Eq. (\[36\]) can be formally written in the operator form as follows: $$\begin{aligned}
\label{eq46}
&&\chi (t, z) = \{T_{l (M_0 ; M) }\exp\} \left\{ i g_{2D} T_a
\int dx^\mu A^a_\mu \right\},\end{aligned}$$ where the symbol $\{T_{l (M_0 ; M) }\exp\}$ means that the integration is to be carried out along the line on the light cone from the point $M_0$ to the point $M$ such that the factors in exponent expansion are chronologically ordered from $M_0$ to $M$. Eq. (\[37\]) are the free 2D Dirac equations. When $m_{qT}$ is a constant, the solution can be found as the superposition of 2D plane waves: $$\begin{aligned}
\label{eq47}
&& F (t, z) = \int {d^2 P\over 2\pi} F (P) e^{\left( -i P X
\right)}= \int {d^2 P\over 2\pi} F (P) e^{ -i (\omega t
-p_z z) },\nonumber \\
&& G (t, z) = \int {d^2 P\over 2\pi} G (P)e^{ \left( -i P X \right)}= \int {d^2 P\over 2\pi} G (P)e^{-i (\omega t
-p_z z) }.
\end{aligned}$$ Substituting the last expansion into Eq. (\[37\]), we obtain $$\begin{aligned}
\label{eq48}
&& F (P) \left( \omega + p \right) - m_{qT} G (P) = 0 ,
\nonumber \\
&& \left( \omega - p \right) G (P)- m_{qT} \ F (P) = 0 ,
\nonumber \\
&& P\equiv P^\mu = (\omega; {\bb p})= (\omega; p_z)\equiv(\omega; p).\label{48}
\end{aligned}$$ As a result, we have $$\begin{aligned}
\label{eq49}
f_+ =\frac{\zeta+\eta}{2}\propto G(P)+F(p),
\nonumber\\
f_- =\frac{\zeta-\eta}{2}\propto G(P)-F(p).\end{aligned}$$ Taking $F(P)=1$ and $G(P)=(\epsilon+p)/m_{qT}$, we derive $$\begin{aligned}
\label{eq50}
f_+ \propto \frac{\omega +p}{m_{qT}}+1,
\nonumber\\
f_- \propto \frac{\omega +p}{m_{qT}}-1.\end{aligned}$$ The solution of Eq. (\[28\]) becomes $$\begin{aligned}
\label{eq51}
\Psi ( 2D, X ) &&= \Psi ( x^0 ; x^3 ) = f_+ (
x^0 ; x^3 ) \left(
\begin{array}{c}
1 \\
0
\end{array} \right) + f_- ( x^0 ; x^3 ) \left(
\begin{array}{c}
0 \\
1
\end{array} \right)
\nonumber\\
&&= \int {d^2 P\over 2\pi} e^{ -i (\omega t -p_z z) } N(\omega,p)
\begin{pmatrix}
\frac{\omega +p}{m_{qT}}+1 \cr
\frac{\omega +p}{m_{qT}}-1\cr
\end{pmatrix} \cdot \chi (X) ,\end{aligned}$$ where $N(\omega,p)$ is some normalization multiplier. We can take the normalization condition $$\begin{aligned}
\label{eq52}
\left ( N(\omega,p)
\begin{pmatrix}
\frac{\omega +p}{m_{qT}}+1 \cr
\frac{\omega +p}{m_{qT}}-1\cr
\end{pmatrix}
\right )^\dag
N(\omega,p)
\begin{pmatrix}
\frac{\omega +p}{m_{qT}}+1 \cr
\frac{\omega +p}{m_{qT}}-1\cr
\end{pmatrix}=\frac{1}{L},\end{aligned}$$ where $L$ is the flux tube length. Then, we obtain $$\begin{aligned}
\label{eq53}
N(\omega,p)
=\frac{m_{qT}}{2\sqrt{L(\omega^2+p\omega})}.\end{aligned}$$ As a result, the general solution is $$\begin{aligned}
\Psi (X) &=&\int\limits_{-\infty}^{+\infty} \frac {d\omega}{2\sqrt L} \sum_p {m_{qT}\ \over \sqrt{\omega^2 + p \omega}} \
\exp (-i P_\mu x^\mu ) \ a(p, \omega) \left[ \delta (\omega - \varepsilon (p)) + \delta (\omega + \varepsilon
(p))\right] ~
\left(
\begin{array}{cccc}
{\omega + p \over m_{qT}} + 1 \\ \\ {\omega + p \over m_{qT}} - 1
\end{array} \right)
\nonumber\\
& \times & \{T_{l (M_0 ; M) }\exp\} \left\{ i g_{2D} T_a \int
dx^\mu A^a_\mu \right\}, \label{46a}
\end{aligned}$$ where $a(p, \omega)$ are coefficients related to either particles or anti-particles under the field quantization. We have not deliberately separated out positive and negative frequency terms in Eq. (\[46a\]) because the structure of the fermion vacuum is strongly dependent on the explicit form of the external field $A^a_\mu (X)$. Furthermore, when the external field depends on time, there will be no stationary particles and antiparticles states.
Fermion current and gauge fields
--------------------------------
We envisage that a perturbative gauge field is introduced inside the flux tube, such a field will generate a current, and the current in turn will produce a gauge field self-consistently. How do these quantities relate to each other? We therefore need to obtain a relationship between the fermion current and the gauge field.
The fermion field solution in Eq. (\[46a\]) leads to a fermion current $J_a^\mu$ $$\begin{aligned}
&& J_a^\mu (2D) = g_{2D}~Tr \left\{ {\bar \Psi ( X)} \gamma^\mu T_a \Psi
(X' )\right\}, \ \ \ \ \ X^\prime \to X.
\label{47a}
\end{aligned}$$ Owing to the operation of trace calculation in the last formula, the current (\[47a\]) contains the factor $$\begin{aligned}
&&
(T\exp) \left\{ i g_{2D}~ T_a \int\limits_{X}^{X'} A^a_\mu
dX^\mu \right\}.\end{aligned}$$ We expand the operator exponent in the last equation as a series with respect to $( X' - X ) \to 0 $, $$\begin{aligned}
\label{eq57}
(T\exp) \left\{ i g_{2D}~ T_a \int\limits_{X}^{X'} A^a_\mu
dX^\mu \right\}&=& 1 + i g_{2D}~ T_a (X' - X )^\mu A^a_\mu (\xi) +
{i\over 2} g_{2D}~ T_a (X' - X )^\mu (X' - X )^\nu
\partial_\nu
A^a_\mu (\xi) \nonumber \\
& &~- g^2_{2D}~ ( T_a T_b ) ( {\tilde X}' - {\tilde X} )^\mu ( X'
- X )^\nu A^a_\mu ({\tilde \xi } ) A^b_\nu (\xi) \theta({\tilde
\xi } - \xi ), \label{49a}\end{aligned}$$ where ${\tilde \xi } \in [ {\tilde X} , {\tilde X}' ],$ and $\xi
\in [ X ,
X' ];~ \ X' \to X$. We take the limits $( {\tilde X}' - {\tilde X})\to 0 $ and $( X' -
X )\to 0 $ such that $$\begin{aligned}
\label{eq58}
&& {( {\tilde X}' - {\tilde X} )\over ( X' -
X )} \to 0.\end{aligned}$$ Then, the last term in the expansion in Eq. (\[49a\]) is equal to zero. Substituting $(T\exp) \{ i g_{2D}~ T_a
\int\limits_{X}^{X'} A^a_\mu dx^\mu \}$ into Eq.(\[47a\]), we obtain for $(X'-X)\to 0$ $$\begin{aligned}
\label{eq59}
J^\mu_a &&= \frac {g_{2D}}{ L} Tr \int d\omega \sum_{{ p}, f }
\Biggl\{ {<a^\dag_f ( p, \omega) a_f (p, \omega)
> }( P^\mu T_a ) \left( - {\partial \over \partial P_\nu } \exp
\left( - i P (X' - X ) \right) \right) ~ \left[ \delta (\omega +
\varepsilon (p)) ~ + ~ \delta (\omega - \varepsilon (p))\right]
\nonumber \\
&&\times \left( g_{2D}~ T_b A^b_\nu (\xi) + {1\over 2} g_{2D}~ T_b
(X' - X )^\lambda
\partial_\nu
A^b_\lambda (\xi) \right)
\Biggr\} ,
\end{aligned}$$ where $f$ denotes flavor states. In reaching the last equation, we have successively calculated a trace, gone from summation to integration, introduced the additional integration with respect to the $p$ variable, and integrated by parts. We note that upon taking the partial derivative $\partial^2\equiv \partial_{(X)}^2$ on $(X'-X)^\lambda \partial_{\nu{(X)}} A^b_\lambda (\xi)$, we get $$\begin{aligned}
\partial_{(X)}^2 \lim \limits_{X' \to~
X} \left\{ (X' - X )^\lambda \partial_{\nu(X)} A^b_\lambda (\xi )
\right\} &=&\lim \limits_{X' \to~ X}
\partial_{(X)}^2 \left\{ (X' - X )^\lambda
\partial_{\nu(X)} A^b_\lambda(X)
\right\}
\nonumber\\
&=& \lim \limits_{X' \to~ X} \left\{ - 2
\partial_{(X)}^\lambda
\partial_{\nu(X)} A^b_\lambda (X) + (X' - X)^\lambda
\partial_{\kappa(X)} ~ \partial_{(X)}^\kappa \{ \partial_{ \nu (X)} A^b_\lambda (X) \}\right\}.
\end{aligned}$$ Upon taking the limit $X' \to X$, the second term vanishes . Therefore, we have in the limit of $X' \to X$, $$\begin{aligned}
\label{cc1}
\lim \limits_{X' \to~ X} \left( (X' - X )^\lambda
\partial_\nu A^b_\lambda (\xi) \right) &=& - 2 \frac{\partial^\lambda
\partial_\nu}{
\partial^2} A^b_\lambda (X).\end{aligned}$$
It should be noted that as QCD$_4$ in the (3+1) dimensional space-time is gauge invariant, and we have chosen the Lorentz gauge (\[7A\]) in QCD$_4$ to simplify the compactified action integral in QCD$_2$. We need to continue to use the Lorentz gauge in QCD$_2$ for consistency. In the Lorentz gauge, the last term in the second circular brackets in Eq.(\[eq59\]) is equal to zero because of Eq.(\[cc1\]). Calculating a trace with respect to the color variables according to Eq.(6), we can represent the current $J_a^\mu$ in the following form $$\begin{aligned}
J_a^\mu(2D, X) &=& \frac { g^2_{2D}~ {\cal S}}{4}\ A_a^\mu
(2D, X),
\nonumber \\
{\cal S} &=& \frac {1}{2\pi } \sum_f \int d^2 P \frac
{\partial}{\partial P^\mu} \left\{ \left[ \delta (\omega +
\varepsilon (p)) ~ + ~ \delta (\omega - \varepsilon (p))\right] ~
P^\mu ~ < a^\dag(p, \omega )_f ~a_f(p, \omega) > \right\}; ~~~ d^2
P = d\omega ~dp,
\label{60}
\end{aligned}$$ where $P^\mu$ is the momentum introduced in Eq. (\[48\]). We can introduce a boson mass $m_{gf T}$ by $$\begin{aligned}
m_{gf\,T}^2= \frac{ g^2_{2D}{\cal S}}{4}.\end{aligned}$$ Then, the current in Eq.(\[60\]) can be written as $$\begin{aligned}
\label{SchCur }
J_a^\mu(2D , X) &=& m_{gf\,T}^2 ~
A_a^\mu(2D,X).
\label{63}
\end{aligned}$$ We can calculate the quantity ${\cal S}$. Changing $p$ by $- p$ in the term corresponding to the negative $\omega$, we have $$\begin{aligned}
{\cal S} = \frac {1}{2\pi } \sum_f \int d^2 P \frac
{\partial}{\partial P^\mu} \left\{ P^\mu ~ \left( \delta
(\omega - \varepsilon (p)) < a^\dag(p, \omega)_f ~a_f(p, \omega) >
+ ~ \delta (\omega + \varepsilon (p)) < a^\dag(- p, \omega)_f
~a_f( -p, \omega) > \right) \right\}.
\label{61}
\end{aligned}$$ Integrating out the $\delta$-functions, we obtain $$\begin{aligned}
&& {\cal S} = \frac {1}{\pi } \sum_f \int d p ~ \left( <
a^\dag_f ( p, \varepsilon(p) ) ~a_f( p, \varepsilon(p)) > + <
a^\dag_f (- p, - \varepsilon(p)) ~a_f( -p, - \varepsilon(p))
>\right) \left( 1 - \frac{m^2_{qT}}{2\varepsilon^2 (p)}\right).
\label{62}
\end{aligned}$$ Since a fermion moves either along or opposite to the only spatial axis, we have $$\begin{aligned}
&& \int d p ~ < a^\dag_f ( p, \varepsilon(p) ) ~a_f( p,
\varepsilon(p)) > = \int d p ~ < a^\dag_f (- p, - \varepsilon(p))
~a_f( -p, - \varepsilon(p))
> = 1.
\label{62a}
\end{aligned}$$ In the case of a flux tube for which $m_{qT} \gg p$, we obtain $$\begin{aligned}
{\cal S} &=& \frac{2}{\pi}~N_f ,
\end{aligned}$$ where $N_f$ is the number of flavors. We note in passing that in the special case of the massless QED$_2$ [@Sch62], we obtain after summing over all spin states of a fermion $$\begin{aligned}
&& {\cal S}_{\rm QED_2} = \frac {4}{\pi }, ~~(N_f =1),
\label{62b}
\end{aligned}$$ and $$\begin{aligned}
&& m_{{gf~T}}^2 ({\rm QED_2})= \ \frac{g^2}{\pi},
\end{aligned}$$ which agrees with the Schwinger massless QED$_2$ result [@Sch62].
Finally, we note that under the gauge transformation $$\begin{aligned}
\label{eq64}
&& \delta A_a^\mu = \varepsilon_b f_a^{\ bc} A_c^\mu
\end{aligned}$$ the current (\[60\]) satisfies the gauge relation $$\begin{aligned}
\label{eq65}
&& \delta J_a^\mu = \varepsilon_b f^{~ b c }_a J_c^\mu.
\end{aligned}$$
Equation of motion for the 2D Gauge Fields
==========================================
The action integral ${\cal A}$ allows us to obtain the equation of motion for the 2D gauge field. We rewrite the action integral (\[33\]) by expressing explicitly the term corresponding to the interaction between the fermion and the gauge field, $$\begin{aligned}
\label{QPA} {\cal A} (2D ) = \int d^2 X &\Bigg\{& {i\over 2}
\left[ {\bar \Psi}
\gamma^k \partial_k \Psi - {\bar \Psi} m_{qT} \Psi
\right]
- {i\over 2} \left[ {\bar \Psi} \gamma^k
{\overleftarrow \partial_k } \ \Psi + {\bar \Psi} m_{qT} \Psi
\right] + J_a^\mu A_\mu^a \nonumber \\ && -{1\over 4 } F^a_{\mu \nu}
\ F_a^{\mu \nu} +{1\over 2 }m_{gT}^2 A^a_\nu A_a^\nu
\Bigg\},\end{aligned}$$ where $J_a^\mu(x)$ is the fermion current governed by Eq.(\[63\]). Substituting $J_a^\mu(x)$ given by Eq.(\[63\]) into the 2D action integral (\[QPA\]), we obtain $$\begin{aligned}
\label{QPA1} {\cal A} (2D ) = \int d^2 X &\Bigg\{& {i\over 2}
\left[ {\bar \Psi}
\gamma^k \partial_k \Psi - {\bar \Psi} m_{qT} \Psi
\right]
- {i\over 2} \left[ {\bar \Psi} \gamma^k
{\overleftarrow \partial_k } \ \Psi + {\bar \Psi} m_{qT} \Psi
\right] \nonumber \\ && -{1\over 4 } F^a_{\mu \nu}
\ F_a^{\mu \nu} +{1\over 2 }M_{gT}^2 A^a_\nu A_a^\nu
\Bigg\}.\end{aligned}$$ Here the constant $M_{gT}$ is given by $$\begin{aligned}
\label{M_gT} M_{gT}^2& =& \frac{1}{2} \int
dx^1 dx^2 \biggl [ \{\partial_1[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2+\{\partial_2[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2\biggr ] + \frac{ g^2_{2D} {\cal S}}{2} \nonumber \\
& \equiv & m_{gT}^2 + m_{gf\,T}^2 \ge 0.\end{aligned}$$
To find out the meaning of $M_{gT}$, we consider the variation of the action integral (\[QPA1\]) with respect to a variation of the gauge field $ A_a^\nu (x)$. We obtain equation of motion for the variation $A_a^\nu (x)$. As a result, we derive the Klein-Gordon-like equation: $$\begin{aligned}
\label{KG} \square ~ A_a^\nu &=& M_{gT}^2 ~ A_a^\nu.\end{aligned}$$ We look for a solution for the variation of the gauge field in Eq. (\[KG\]) of the form $$\begin{aligned}
\label{solKG} && A_a^\nu = b_a (k, \nu )~ e_a^\nu (k)
\exp (- i k_\mu X^\mu ) , \nonumber \\ \nonumber \\
&& k^\mu = (k^0 ; \bb k),~~ e_a^0 = \frac{\vert\bb k
\vert}{M_{gT}}~ (1, 0 ) , ~~~ e_a^3 = \frac{\vert k^0
\vert}{M_{gT}} (0, 1),\end{aligned}$$ where $e_a^\nu$ denotes a pair orthogonal vectors; $b_a (k, \nu )$ are some coefficients being independent on $X$. Substituting $
A_a^\nu $ given by Eq. (\[solKG\]) into Eq. (\[KG\]), we obtain: $$\begin{aligned}
\label{eq74}
&& (k^0 )^2 = \bb k^2 + M^2_{gT}.
\end{aligned}$$ Because of both the positivity of $M^2_{gT}$ and Eq. (\[eq74\]), $M_{gT}$ can be interpreted as a mass of the particle whose energy is $$\begin{aligned}
\label{eq75}
&& E(k) \equiv k^0 = + \sqrt{ \bb k^2 + M^2_{gT}} \label{spect}.
\end{aligned}$$ Eqs. (\[solKG\]) and (\[spect\]) allow us to write down the general solution of Eq. (\[KG\]). Following the standard way [@Pes95], and separating the negative and positive frequency terms, we obtain $$\begin{aligned}
A_a^\nu (2D, X) &=&\sum_k {e_a^\nu ~ M_{gT} \over \sqrt{
\left( \bb k^2+ M^2_{gT}\right)^3 }}
\Biggl\{
\exp (-i k X) \ b_a(k, \nu) ~ + \exp ( + i k X) \ {\bar b}_a^\dag (k, \nu)
\Biggr\} , \label{gensol-g}\end{aligned}$$ where the symbols $b_a(k, \nu)$ and ${\bar b}_a^\dag (k, \nu)$ are the operators of annihilation and creation of a boson with the mass $M_{gT}$. In this way, $M_{gT}$ corresponds to the mass of the boson responding to the space-time variation of the gauge field variation.
The boson mass $M_{gT}$ in the action integral Eq.(\[QPA1\]) arises from the compactification of $4D \to 2D$ and from the interaction of the compactified fermions and gauge field. We should also note that all masses (boson and fermion field) as well as $2D$ coupling constant are governed by the functions of the transverse motion of a fermion, $G_{1, 2} ({\vec r}_\bot )$. It is independent of the color index $a$. Out of the gauge field variations of different color components $A_a^\mu$, one can construct a colorless variation of the type $$\begin{aligned}
A_{\rm color-singlet}^\nu = \frac{1}{\sqrt{8}}\sum_a A_a^\mu
|8,a\rangle,\end{aligned}$$ where $|8,a\rangle$ is the color-octet state with component $a$. Eq. (\[KG\]) gives $$\begin{aligned}
\square ~ A_{\rm color-singlet}^\nu &=& M_{gT}^2 ~
A_{\rm color-singlet}^\nu .\end{aligned}$$ Thus, we find that $M_{gT}$ is also the mass corresponding to a colorless variation of the gauge field of different color components in a flux tube. Such a colorless variation should lead to an observable quantity. If one considers pion as the colorless dynamical response of the variations of the gage fields in a string, then $M_{gT}$ may be presumed to be the mass of the pion within the environment of a flux tube under consideration.
Equations of transverse motion in a tube and the Fermion effective mass
=======================================================================
To obtain the equations of motion for the functions $G_1 ({\vec
r}_\bot )$ and $G_2 ({\vec r}_\bot )$, we vary the action integral ${\cal A} (4D)$ in Eq. (\[1\]) with the fermion fields $\Psi
(4D , x)$ given by Eq.(\[eq9\]), under the constraint of the normalization condition Eq. (\[norm\]). To do this we construct a new functional ${\cal F}$,
$$\begin{aligned}
\label{eq80}
&& {\cal F} = {\cal A }(4 D ) + \frac{ \lambda}{2} ~ \int d x^1 d
x^2 \left( \vert G_1 ({\vec r}_\bot ) \vert^2 + \vert G_2 ({\vec
r}_\bot ) \vert^2 \right) ~\int d x^0 d x^3 \left( {\bar \Psi}
(x^0 , x^3 )
~ { \Psi} (x^0 , x^3 ) \right),\end{aligned}$$
where $\lambda$ is the Lagrange multiplier. The last term in Eq.(\[eq80\]) takes into account the unitarity of a fermion field in the 4D space-time. Varying the last equation with respect to the functions $G_1 ({\vec r}_\bot )$ and $G_2 ({\vec
r}_\bot )$, we obtain $$\begin{aligned}
\label{eq81}
&&
( p_1 + ip_2 ) G_1 ({\vec r}_\bot ) = ( m ({\vec r}_\bot ) +\lambda ) G_2 ({\vec r}_\bot ), \nonumber \\
&& ( p_1 - ip_2 ) G_2 ({\vec r}_\bot ) = (\lambda - m ({\vec
r}_\bot ) )
G_1 ({\vec r}) ,\nonumber \\
&&( p_1 + ip_2 ) G^\ast_2 ({\vec r}_\bot ) = ( m ({\vec r}_\bot ) - \lambda ) G^\ast_1 ({\vec r}_\bot ), \nonumber \\
&& ( p_1 - ip_2 ) G^\ast_1 ({\vec r}_\bot ) = - ( m ({\vec
r}_\bot ) + \lambda ) G^\ast_2 ({\vec r}).\end{aligned}$$ Carrying out complex conjugation in the last two equations, we obtain $$\begin{aligned}
\label{eq82}
&& \lambda = \lambda^\ast.\end{aligned}$$ Combining Eq. (\[eq81\]), we get $$\begin{aligned}
\label{eq83}
&& \left( p^2_1 + p^2_2 - \lambda^2 + m^2 ({\vec r}_\bot )
\right) G_1 ({\vec r}_\bot ) = G_2 ({\vec r}_\bot ) ( p_1 - ip_2
) m ({\vec r}_\bot )
\nonumber \\
&& \left( p^2_1 + p^2_2 - \lambda^2 + m^2 ({\vec r}_\bot )
\right) G_2 ({\vec r}_\bot ) = - G_1 ({\vec r}_\bot ) ( p_1 +
ip_2 ) m ({\vec r}_\bot ).\end{aligned}$$ Substituting the equations (\[eq81\]) for $G_{1,2} ({\vec
r}_\bot )$ functions into the formula (\[mqT\]) for $m_{qT}$, we find that $$\begin{aligned}
\label{eq84}
m_{qT}& = & \lambda .\end{aligned}$$ Thus, the effective mass of the compactified 2D fermion field is equal to the energy eigenvalue for the transverse motion of the 4D fermion as described in Eqs. (\[eq81\]). We should note here that the 2D fermion can generally gain a mass even when the initial 4D fermion appears to be massless. The compactification effectively leads to a constraint in moving a fermion from one point of a space-time to another point due to decreasing the number of trajectories in the 2D space-time as compared with the 4D situation. This contraint leads to the presence of an effective mass.
Conclusions and Discussions
===========================
Encouraged by the successes of the particle production model of Casher, Kogut, and Susskind using the Abelian gauge field theory in two-dimension space-time [@Cas74] and the Lund model of string fragmentation [@And83], we seek a compactification of QCD$_4$ to QCD$_2$ in the environment of a flux tube. Under the assumption of longitudinal dominance and transverse confinement, the SU(N) gauge invariant field theory of QCD$_4$ can be compactified in the (1+1) Minkowski space-time, from the consideration of the action integral. This is achieved by finding a way to relate the field variables in 2-dimensional space-time to those in four-dimensional space-time.
The compactified 2D action integral ${\cal A} (2D)$ depends only on fields that are defined in two-dimensional space-time. It has the same structure as those in QCD in four-dimensional space-time and can therefore be appropriately called QCD$_2$. In the compactified QCD$_2$ quantum field theory, the coupling constant is found to be dimensional, and there are additional terms in the action associated with an effective quark mass and effective gauge field mass as a result of the flux tube confinement. These quantities depends on the transverse profile and the transverse state of the quarks in the flux tube.
On a basis of the derived QCD$_2$ action integral, the equations of motion for the fields can be obtained for both the fermion field and the gauge field. The solution of 2D Dirac equation can then be formally obtained. The structure of the solution allows one to consider the effects of the fermion-gluon coupling. As a result, the 2D action integral can be re-written in the form such that the gauge field acquires an additional effective mass due to interaction with fermions. The structure of the derived mass term appears to be identical to the one obtained by Schwinger [@Sch62] in the special case of massless QED$_2$.
The occurrence of a massive composite bound state in gauge field theories has been known in many previous investigations [@xxx]. How the massive bosons as a pole in the three gluon vertex in 4-dimensional space-time [@Pap11; @Pap12] can be produced in the flux tube environment in high-energy collisions will be an interesting subject worthy of further investigations.
**Acknowledgment**
The research was supported in part by the Division of Nuclear Physics, U.S. Department of Energy.
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Substituting $\Psi (4D, x)\equiv \Psi ( x)$ given by Eq. (7) into the first term in Eq. (5), we obtain $$\begin{aligned}
\label{a1}
&& {\bar \Psi (x)} \gamma^\mu \Pi_\mu \ \Psi (x) = \chi^\dag
\left( \Pi_0 - ( p_1 \sigma_1 + p_2 \sigma_2 ) - \Pi_3 \sigma_3
\right) \chi + \varphi^\dag \left( \Pi_0 + ( p_1 \sigma_1 + p_2
\sigma_2 ) + \Pi_3 \sigma_3 \right) \varphi \nonumber \\ \nonumber \\
&& = \chi^\dag \left[ \left( \begin{array}{cccc} \Pi_0 - \Pi_3 \
\ \ \ \ \ 0 \\ \\
0 \ \ \ \ \ \ \Pi_0 + \Pi_3 \\
\end{array} \right) - \left( \begin{array}{cccc} 0 \
\ \ \ \ \ p_1 - i p_2 \\ \\
p_1 + i p_2 \ \ \ \ \ \ 0
\end{array} \right)
\right] \chi + \varphi^\dag \left[ \left( \begin{array}{cccc}
\Pi_0 + \Pi_3 \
\ \ \ \ \ 0 \\ \\
0 \ \ \ \ \ \ \Pi_0 - \Pi_3 \\
\end{array} \right) + \left( \begin{array}{cccc} 0 \
\ \ \ \ \ p_1 - i p_2 \\ \\
p_1 + i p_2 \ \ \ \ \ \ 0
\end{array} \right)
\right] \varphi \nonumber \\
&&= \chi_1^\ast (\Pi_0 - \Pi_3 ) \chi_1 + \chi_2^\ast (\Pi_0 +
\Pi_3 ) \chi_2 - \chi_1^\ast ( p_1 - ip_2 ) \chi_2 - \chi_2^\ast (
p_1 + ip_2 ) \chi_1 + \varphi_1^\ast (\Pi_0 + \Pi_3 ) \varphi_1 +
\varphi_2^\ast (\Pi_0 - \Pi_3 ) \varphi_2 \nonumber \\
&&~~~~~+ \varphi_1^\ast ( p_1 - ip_2 ) \varphi_2 + \varphi_2^\ast
( p_1
+ ip_2 ) \varphi_1 \nonumber \\
&&= \chi_1^\ast (\Pi_0 - \Pi_3 ) \chi_1 + \chi_2^\ast (\Pi_0 +
\Pi_3 ) \chi_2 + \varphi_1^\ast (\Pi_0 + \Pi_3 ) \varphi_1 +
\varphi_2^\ast (\Pi_0 - \Pi_3 ) \varphi_2 \nonumber \\
&&~~~~~ - \chi_1^\ast ( p_1 - ip_2 ) \chi_2 - \chi_2^\ast ( p_1 +
ip_2 ) \chi_1 + \varphi_1^\ast ( p_1 - ip_2 ) \varphi_2 +
\varphi_2^\ast ( p_1 + ip_2 ) \varphi_1.\end{aligned}$$ Integration of the last equation gives $$\begin{aligned}
\label{a2}
\int d^4 x \ {\bar \Psi (x)} \gamma^k \Pi_k \ \Psi (x) &=&
\int d^4 x \left\{ \chi_1^\ast (\Pi_0 - \Pi_3 ) \chi_1 +
\chi_2^\ast (\Pi_0 + \Pi_3 ) \chi_2 + \varphi_1^\ast (\Pi_0 +
\Pi_3 ) \varphi_1 + \varphi_2^\ast (\Pi_0 - \Pi_3 ) \varphi_2
\right\} \nonumber\\
& & +
\int d^4 x \left\{ - \chi_1^\ast ( p_1 - ip_2 ) \chi_2 -
\chi_2^\ast ( p_1 + ip_2 ) \chi_1 + \varphi_1^\ast ( p_1 - ip_2 )
\varphi_2 + \varphi_2^\ast ( p_1 + ip_2 ) \varphi_1 \right\}
\nonumber \\
&=& \int d^4 x \left\{ \chi_1^\ast (\Pi_0 - \Pi_3 ) \chi_1 +
\chi_2^\ast (\Pi_0 + \Pi_3 ) \chi_2 + \varphi_1^\ast (\Pi_0 +
\Pi_3 ) \varphi_1 + \varphi_2^\ast (\Pi_0 - \Pi_3 ) \varphi_2
\right\} \nonumber \\
&& - \int d^4 x \left\{ G^\ast_1 ({\vec r}_\bot ) ( p_1 - ip_2
) G_2 ({\vec r}_\bot ) \right\} \left( \vert f_+ \vert^2 - \vert
f_- \vert^2 \right) \nonumber \\
&=& \int d^4 x \left( \vert G_1 ({\vec r}_\bot ) \vert^2 +
\vert G_2 ({\vec r}_\bot ) \vert^2 \right) \left[ \ f^\ast_+
\Pi_0 f_+ + \ f^\ast_- \Pi_0 f_- + \ f^\ast_+ \Pi_3 f_- + \
f^\ast_- \Pi_3 f_+ \right] \nonumber \\
&& - \int d^4 x \left\{ G^\ast_1 ({\vec r}_\bot ) ( p_1 - ip_2 )
G_2 ({\vec r}_\bot ) \right\} \left( \vert f_+ \vert^2 - \vert f_-
\vert^2 \right).\end{aligned}$$ Following the same way, we derive for the term ${\bar \Psi} (x)
\gamma^k {\overleftarrow \Pi_k } \ \Psi (x) $ $$\begin{aligned}
\label{a3}
\int d^4 x \ {\bar \Psi (x)} \gamma^k {\overleftarrow\Pi}_k \ \Psi
(x) &=& \int d^4 x \left( \vert G_1 ({\vec r}_\bot ) \vert^2 +
\vert G_2 ({\vec r}_\bot ) \vert^2 \right) \left[ \ f^\ast_+
{\overleftarrow\Pi}_0 f_+ + \ f^\ast_- {\overleftarrow\Pi}_0 f_- +
\ f^\ast_+ {\overleftarrow\Pi}_3 f_ - +f^\ast_-
{\overleftarrow\Pi}_3 f_+ \right] \nonumber \\ & & - \int d^4
x \left\{ G_1 ({\vec r}_\bot ) ( p_1 + ip_2 ) G^\ast_2 ({\vec r}_\bot
) \right\} \left( \vert f_+ \vert^2 - \vert f_- \vert^2 \right).\end{aligned}$$ We substitute $ \Psi (x)$ of Eq. (7) into the last term in Eq.(5), and we obtain $$\begin{aligned}
\label{a4}
&& {\bar \Psi} (x) m ({\vec r}_\bot ) \Psi (x) = m ({\vec
r}_\bot
) \ \left( \vert G_1 ({\vec r}_\bot ) \vert^2 - \vert G_2 ({\vec
r}_\bot ) \vert^2 \right) \left[ \vert f_+ \vert^2 - \vert f_-
\vert^2 \right].\end{aligned}$$ Collecting the above results and introducing the $2D$-fermion wave function $\Psi (X)$, $2D$-gamma matrices $\gamma^\mu$, and the metric tensor $g_{\mu \nu}(2D)$ as given in Eqs. (\[11a\]) and (\[11\]), we obtain the Fermion part of the action integral in Eq. (8).
To compactify the gauge field parts of the (3+1) dimensional space-time to (1+1) dimensional space-time, we need to evaluate $F_{01}$, $F_{02}$, $F_{31}$, and $F_{32}$. Direct calculations give (color indexes are omitted for simplicity) $$\begin{aligned}
\label{b1}
F_{01}(x^0,x^3,{\bb r}_\perp)&=& -\partial_1 A_0( x^0,x^3,{\bb
r}_\perp)
\nonumber\\
&=&-\partial_1[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} A_0(2D,x^0,x^3),\end{aligned}$$ $$\begin{aligned}
\label{b2}
F_{01}( x^0,x^3,{\bb r}_\perp) F^{01}( x^0,x^3,{\bb r}_\perp)
&=&\{-\partial_1[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}\{-\partial^1[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}
\nonumber\\
& & \times A_0(2D,x^0,x^3)A^0(2D,x^0,x^3),~~~
\nonumber\\
&=&- \{\partial_1 [{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2
A_0(2D,x^0,x^3)A^0(2D,x^0,x^3),\end{aligned}$$ which contribute a gauge field mass in the $A_0(2D,x^0,x^3)$ gauge field. Similarly, we can calculate $$\begin{aligned}
\label{b3}
&&F_{02}( x^0,x^3,{\bb r}_\perp) F^{02}( x^0,x^3,{\bb r}_\perp) =
-\{\partial_2[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2 A_0(2D,x^0,x^3)A^0(2D,x^0,x^3),
\nonumber \\ \nonumber \\
&& F_{31}( x^0,x^3,{\bb r}_\perp) F^{31}( x^0,x^3,{\bb r}_\perp) =
-\{\partial_1[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2 A_3(2D,x^0,x^3)A^3(2D,x^0,x^3),\end{aligned}$$ which contribute a gauge field mass in the $A_3(2D,x^0,x^3)$ gauge field. Similarly, we have also $$\begin{aligned}
F_{32}( x^0,x^3,{\bb r}_\perp) F^{32}( x^0,x^3,{\bb r}_\perp)
&=&-\{\partial_2[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2 A_3(2D,x^0,x^3)A^3(2D,x^0,x^3).\end{aligned}$$ Combining all similar terms, we get $$\begin{aligned}
\label{b4}
& &[F_{01}F^{01}+F_{02}F^{02}+F_{31}F^{31} +F_{32}F^{32}](
x^0,x^3,{\bb r}_\perp)
\nonumber\\
&=&-\biggl ( \{\partial_1[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2+\{\partial_2[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2\biggr )
\nonumber\\
& &\times [A_0(2D,x^0,x^3)A^0(2D,x^0,x^3)
+A_3(2D,x^0,x^3)A^3(2D,x^0,x^3)]
\nonumber\\
&=& -\biggl ( \{\partial_1[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2+\{\partial_2[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2\biggr )
\nonumber\\
& &\times [A_0(2D,x^0,x^3)A^0(2D,x^0,x^3)
+A_3(2D,x^0,x^3)A^3(2D,x^0,x^3)].\end{aligned}$$ Then, due to the normalization relation (\[norm\]) we have the following (see Eq. (26) and (27)) for the gauge field part: $$\begin{aligned}
\label{b5}
& &{1\over 4 } \int d^4 x F^a_{\mu \nu}(4D) \ F_a^{\mu \nu}(4D)
={1\over 4 } \int dx^0 dx^3 \int dx^1 dx^2 ( |G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2) F^a_{03 } (2D)
F_a^{03}(2D) \nonumber\\
& & -{1\over 4 } \int dx^0 dx^3 \int dx^1 dx^2 \biggl (
\{\partial_1[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2+\{\partial_2[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2\biggr )
\nonumber\\
& &~~~~~~~~~~~\times [A_0(2D,x^0,x^3)A^0(2D,x^0,x^3)
+A_3(2D,x^0,x^3)A^3(2D,x^0,x^3)]
\nonumber\\
&=& {1\over 4 } \int dx^0 dx^3 F^a_{0
3} (2D) \ F_a^{0 3}(2D) - {1\over 2 } \int dx^0 dx^3
m_{gT}^2[A_0(2D)A^0(2D) +A_3(2D)A^3(2D)],\end{aligned}$$ where $m_{gT}^2$ is the mass term that arises from the confinement of the gluons in the transverse direction $$\begin{aligned}
\label{b6}
m_{gT}^2= \frac{1}{2} \int dx^1 dx^2
\biggl [ \{\partial_1[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2+\{\partial_2[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2\biggr ].\end{aligned}$$ Note that using integration by parts, we get $$\begin{aligned}
\label{b7}
& & -\int dx^1 dx^2
\{\partial_1[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2
\nonumber\\
&=& \int dx^1 dx^2 [{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \partial_1^2
[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2}.\end{aligned}$$ Therefore, adding the terms together, we obtain: $$\begin{aligned}
\label{b8}
& & m_{gT}^2= \frac{1}{2} \int dx^1 dx^2 \biggl [ \{\partial_1[{
|G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2
+\{\partial_2[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \}^2 \biggr ]
\nonumber\\
&=& - \frac{1}{2} \int dx^1 dx^2 [{ |G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2}( \partial_1^2 + \partial_2^2)
[{2 |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2}
\nonumber\\
&=& \int dx^1 dx^2 [{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} \left( - \frac{1}{2} \nabla_T^2 \right)
[{ |G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}]^{1/2} .\end{aligned}$$
Transformation of a gauge field in the 2D space-time
----------------------------------------------------
We would like to write down the gauge transformation properties for $A_\mu^a (2D, x^0 , x^3)$. For the corresponding gauge field $A_\mu^a (x)$ in the 4D space-time $x=(x^0, x^3,{\bb r}_\perp)$, it transforms under a gauge transformation as [@Pes95] $$\begin{aligned}
\label{2Dtrans} && A_\mu^a(x) \to {\tilde A}_\mu^a(x) =
A_\mu^a(x) +\delta A_\mu^a ( x).\end{aligned}$$ where $$\begin{aligned}
&& \delta A_\mu^a ( x) = f^{a}_{~b c} \varepsilon^b (x) A_\mu^c (
x) - \frac{1}{g(4D)} \partial_\mu \varepsilon^a (x)
\label{cgauge}\end{aligned}$$ According to Eq. (\[AA\]) the gauge fields in the 2D and 4D space-time are related to each other as follows: $$\begin{aligned}
\label{2} && A_\mu^a(2D,x^0,x^3) = \frac{ A_\mu^a(x^0,x^3,{\bb
r}_\perp)}{\sqrt {|G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}},~~~\mu=0,3,\nonumber \\ \nonumber \\
&& A_\mu^a(x^0,x^3,{\bb r}_\perp) =0 , ~~ \mu =1,2\end{aligned}$$ From the last equation we have $$\begin{aligned}
\label{3} &&\delta A_\mu^a(x^0,x^3,{\bb r}_\perp) =0 , ~~ \mu
=1,2. ~~~~ \Rightarrow ~~ \partial_\mu \varepsilon^a (x) = 0 , ~~
\mu =1,2.\end{aligned}$$ Then, we have $$\begin{aligned}
&& \varepsilon^a (x) = \varepsilon^a (x^0 ,x^3)\end{aligned}$$ Next, we would like to transform $A_\mu^a(2D,x^0,x^3)$ by using the first relation in Eq.(\[2\]). $$\begin{aligned}
\label{5} &&\delta A_\mu^a(2D,x^0,x^3) = \frac{\delta
A_\mu^a(x^0,x^3,{\bb r}_\perp)}{\sqrt {|G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}} + A_\mu^a(x^0,x^3,{\bb r}_\perp) \delta \left(\frac{1
}{\sqrt {|G_1({\bb r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}}\right)\end{aligned}$$ The last term in Eq.(\[5\]) is equal to zero since $\varepsilon^a = \varepsilon^a (x^0 ,x^3)$. Then, substituting Eq. (\[cgauge\]) into Eq. (\[5\]) we obtain $$\begin{aligned}
\label{c6} &&\delta A_\mu^a(2D,x^0,x^3) =
f^{a}_{~b c} \varepsilon^b (x^0, x^3)
A_\mu^c (2D, x^0 ,x^3) - \frac{1}{g(4D)~\sqrt {|G_1({\bb
r}_\perp)|^2+|G_2({\bb
r}_\perp)|^2}} \partial_\mu
\varepsilon^a (x^0,x^3)\end{aligned}$$ Since the left-hand side of Eq. (\[c6\]) depends on $(x^0,x^3)$ the same must be for the right-hand side of this equation. This means that $\varepsilon^a (x^0,x^3)$= constant and the transformation relation for $ A_\mu^a(2D,x^0,x^3)$ is $$\begin{aligned}
&&\delta A_\mu^a(2D,x^0,x^3) =
f^{a}_{~b c} \varepsilon^b (x^0, x^3)
A_\mu^c (2D, x^0 ,x^3)\end{aligned}$$ Varying the mass term in the 2D Lagrangian with respect to the group variables, we obtain $$\begin{aligned}
\delta {\cal L}_{m_{gT}} &=& {1\over 2 }m_{gT}^2 \delta
[A_a^\mu(2D)A^a_\mu(2D)] = m_{gT}^2 \delta
[A_a^\mu(2D)]~[A^a_\mu(2D)]
\nonumber\\
&=& m_{gT}^2
f^{a}_{~b c} \varepsilon^b (x^0, x^3)
[A_\mu^c (2D)] [A_a^\mu(2D)]= 0,\end{aligned}$$ due to the anti-symmetry of the structure constants $ f^{a}_{~b
c}$. Using Eq. (\[cgauge\]) for the infinitesimal transformation of the gauge of the field $A_\mu^a(2D,x^0,x^3)$ we calculate the $n$-th variation of $A_\mu^a(2D,x^0,x^3)$. After such calculations we derive that the gauge transformation of the 2D gauge field has the form $$\begin{aligned}
\label{8} \delta^{(n)} { A}_\mu^a(2D,x^0,x^3) &=& f^{a}_{~b c}
\varepsilon^b (x^0, x^3) f^{c}_{~~b_1 c_1} \varepsilon^{b_1} (x^0,
x^3) A_\mu^{c_1} (2D, x^0 ,x^3) \dots f^{c_{n-2}}_{~~~b_{n-1}
c_{n-1}} \varepsilon^{b_{n-1}} (x^0, x^3) A_\mu^{c_{n-1}} (2D,
x^0 ,x^3), \nonumber \\ {\tilde A}_\mu^a(2D,x^0,x^3) &=&
e^{f^{a}_{~b c} \varepsilon^b (x^0, x^3)}~
A_\mu^c (2D, x^0 ,x^3), \nonumber \\
{\tilde A}^\mu_a(2D,x^0,x^3) &=& A_\mu^c (2D, x^0 ,x^3)~ e^{-
f^{a}_{~b c} \varepsilon^b (x^0, x^3)}.\end{aligned}$$ As a consequence, $$\begin{aligned}
{\tilde A}_\mu^a (2D, x^0 ,x^3) ~ {\tilde A}^\mu_a(2D,x^0,x^3) = {
A}_\mu^a (2D, x^0 ,x^3)~ {A}^\mu_a(2D,x^0,x^3),\end{aligned}$$ which maintains the 2D gauge invariance of the derived 2D action integral Eq. (\[33\]).
The Slavnov-Taylor identities in the 2D space-time in the Lorentz gauge
-----------------------------------------------------------------------
The Slavnov-Taylor identities in the standard 4D space-time in the Lorentz gauge has the form [@Slavnov]: $$\begin{aligned}
\label{10AB} \int \exp \left( i {\cal A} (A) \right) \cdot \int
dz \left( J^\nu_b (z)
\partial_\nu (M^{-1})^{ba} (z,y) + g_{4D} f_{d~c}^{~b} ~ J_\nu^d (z) ~
A_b^\nu (z) ~ (M^{-1})^{ca} (z,y) \right) \Delta (A) d A = 0 ,\end{aligned}$$ where $J_\nu^a (z)$ is a fermion current, $(M^{-1})^{ca} (z,y)$ is the propagator of a scalar field, and $\Delta (A)$ is the Faddeev-Popov determinate. After the compactification with respect to Eqs. (\[eq9\]), (\[norm\]), and (\[AA\]), the action integral ${\cal A} (A)$ becomes ${\cal A}[2D, A(2D)]$. Integrating the first term in the circular brackets by parts with respect to the $z$ variable and using Eqs.(\[eq9\]) and (\[AA\]), we obtain $$\begin{aligned}
\label{10c} \int dz \left( J^\nu_b (z)
\partial_\nu (M^{-1})^{ba} (z,y) \right)& =& - \int dz (M^{-1})^{ba} (z,y)
\nonumber \\
& & \times \Biggl( [{ |G_1({\bb z}_\perp)|^2+|G_2({\bb
z}_\perp)|^2}]^{2}(\partial_0 J^0_b (z^0 ,z^3)
+ \partial_3 J^3_b (z^0 ,z^3))
\nonumber \\
& & ~~~-
i Tr \left\{(p_1 - ip_2 ) G_1^\ast ({\bb
z}_\perp) G_2 ({\bb z}_\perp) {\bar \Psi }(z^0 ,z^3) T_b \Psi (z^0
,z^3) \right\} \Biggr).\end{aligned}$$ The integral involving the first term inside the above curly bracket is equal to zero because of the Lorentz gauge for the $A^\mu_a$ field and Eq.(\[SchCur \]) while the second one is found to be the same due to the trace calculation. Thus the first term in the circular brackets in Eq. (\[10AB\]) is equal to zero.
As for the second term in the circular brackets in Eq.(\[10AB\]), by using Eqs. (\[norm\]), (\[AA\]), and (\[SchCur \]) it can be written as $$\begin{aligned}
\int dz & &\left( g_{4D} f_{d~c}^{~b} ~ J_\nu^d (z) ~
A_b^\nu (z) ~ (M^{-1})^{ca} (z,y) \right) \nonumber \\
&=& g_{4D} f_{d~c}^{~b} \int dz [{ |G_1({\bb
z}_\perp)|^2+|G_2({\bb
z}_\perp)|^2}]^{3/2}
J_\nu^d (2D, z^0 ,z^3) ~ A_b^\nu (2D, z^0 ,z^3) ~ (M^{-1})^{ca}
(z,y) \nonumber \\
& = & m_{gf\,T}^2 ~ g_{4D} f_{d~c}^{~b} \int dz [{ |G_1({\bb
z}_\perp)|^2+|G_2({\bb
z}_\perp)|^2}]^{3/2} A^d_\nu (2D, z^0 ,z^3)
~ A_b^\nu (2D, z^0 ,z^3) ~ (M^{-1})^{ca}(z,y) = 0.\end{aligned}$$ The last expression is equal to zero because of the anti-symmetry of the structure constant. Thus, the pre-exponent in Eq.(\[10AB\]) is found to be equal to zero after the $4D \to 2D $ compactification. This means that the Slavnov-Taylor identities are not violated in the 2D space-time we have considered.
[^1]: Even though the average transverse momenta of the partons are limited, the tails of the parton transverse momentum distribution of partons in the produced hadrons can still extend to the high $p_T$ region, but with small probabilities.
|
---
abstract: 'The unsupervised training of GANs and VAEs has enabled them to generate realistic images mimicking real-world distributions and perform image-based unsupervised clustering or semi-supervised classification. Combining the power of these two generative models, we introduce Multi-Adversarial Variational autoEncoder Networks (MAVENs), a novel network architecture that incorporates an ensemble of discriminators in a VAE-GAN network, with simultaneous adversarial learning and variational inference. We apply MAVENs to the generation of synthetic images and propose a new distribution measure to quantify the quality of the generated images. Our experimental results using datasets from the computer vision and medical imaging domains—Street View House Numbers, CIFAR-10, and Chest X-Ray datasets—demonstrate competitive performance against state-of-the-art semi-supervised models both in image generation and classification tasks.'
author:
- |
Abdullah-Al-Zubaer Imran\
`[email protected]` Demetri Terzopoulos\
`[email protected]`\
Computer Science Department\
University of California, Los Angeles
bibliography:
- 'references.bib'
title: 'Multi-Adversarial Variational Autoencoder Networks'
---
Introduction
============
Training deep neural networks usually requires a large pool of labeled data, yet obtaining large datasets for tasks such as image classification remains a fundamental challenge. Although there has been explosive progress in the production of vast quantities of high resolution images, large collections of labeled data required for supervised learning remain scarce. Especially in domains such as medical imaging, datasets are limited in size due to privacy issues, and manual annotation by medical experts is expensive, time-consuming, and prone to subjectivity, human error, and variance across different experts. Even when large labeled datasets become available, they are often highly imbalanced and nonuniformly distributed. For instance, in an imbalanced medical dataset there will be an over-representation of common medical problems and an under-representation of rare conditions. Such biases make the training of neural networks across multiple classes with similar effectiveness very challenging.
The small-training-data problem is traditionally mitigated through simplistic and cumbersome data augmentation, often by creating new training examples through translation, rotation, flipping, etc. The missing or mismatched label problem can be addressed by evaluating similarity measures over the training examples. This is not always robust and the efficiency largely depends on the performance of the similarity measuring algorithms.
Generative models, such as VAEs [@kingma2013auto] and GANs [@goodfellow2014generative], have recently become popular because of their ability to learn underlying data distributions from training samples. This has made generative models more practical in ever-frequent scenarios where there is an abundance of unlabeled data. With minimal annotation, an efficient semi-supervised learning model could be a go-to approach. More specifically, based on small quantities of annotation, generative models could be utilized to learn real-data distributions and synthesize realistic new training images. Both VAEs and GANs can be employed for this purpose.
VAEs can learn the dimensionality-reduced representation of training data and, with an explicit density estimation, can generate new samples. However VAE-generated samples are usually blurry (Fig. \[fig:gan\_collapse\]b). On the other hand, despite the successes in generating images and semi-supervised classifications, GAN frameworks are still very difficult to train and there are challenges in using GAN models, such as non-convergence due to unstable training, mode collapsed image generation (Fig. \[fig:gan\_collapse\]c), diminished gradient, overfitting, and high sensitivity to hyper-parameters.
To stabilize GAN training and combat mode collapse, several variants have been proposed. @nguyen2017dual proposed a model, where a single generator is used alongside dual discriminators. @durugkar2016generative proposed a model with a single generator and feedback aggregated over several discriminators considering either the average loss of all discriminators or by picking only the discriminator with the maximum loss in relation to the generator’s output. @neyshabur2017stabilizing proposed a framework where a single generator simultaneously trains against an array of discriminators, each of which operates on a different low-dimensional projection of the data. @mordido2018dropout, arguing that all the previous approaches restrict the discriminator’s architecture, which compromises the extensibility of the framework, instead proposed a Dropout-GAN, where a single generator is trained against a dynamically changing ensemble of discriminators. However, there could be a risk of dropping out all the discriminators. Feature matching and minibatch discrimination techniques have been proposed [@salimans2016improvedT] for eliminating mode collapsing and preventing overfitting in GAN training.
Although there have been wide ranging efforts in high quality image generation with GANs and VAEs, accuracy and image quality are usually not ensured in the same model, especially in multi-class image classification. To tackle this issue, we propose a novel method that can learn joint image generation and multi-class image classification. Our specific contribution is the Multi-Adversarial Variational autoEncoder Network, or MAVEN, a novel multi-class image classification model incorporating an ensemble of discriminators in a combined VAE-GAN network. An ensemble layer combines the feedback from multiple discriminators at the end of each batch. With the inclusion of ensemble learning at the end of a VAE-GAN, both generated image quality and classification accuracy are improved simultaneously. We also introduce a simplified version of the Descriptive Distribution Distance (DDD) measure for evaluating any generative model, which better represents the distribution of the generated data and quantifies its closeness to the real data. Our experimental results on a number of different datasets in both the computer vision and medical imaging domains indicate that our MAVEN model improves upon the joint image generation and classification performance of a GAN and a VAE-GAN with the same set of hyper-parameters.
Related Work
============
Generative modeling has attracted much attention in the computer vision and medical imaging research communities. In particular, realistic image generation greatly helps address many problems involving the scarcity of labeled data. GANs and their variants have been applied in different architectures in continuing efforts to improve the accuracy and effectiveness of image classification. The GAN framework has been utilized in numerous works as a more generic approach to generating realistic training images that synthetically augment datasets in order to combat overfitting; e.g., for synthetic data augmentation in liver lesions [@frid2018gan], retinal fundi [@guibas2017synthetic], histopathology [@hou2017unsupervised], and chest X-rays [@salehinejad2018generalization]. @calimeri2017biomedical employed a LAPGAN [@Denton2015DeepGI] and @han2018gan used a WGAN [@arjovsky2017wasserstein] to generate synthetic brain MR images. @bermudez2018learning used a DCGAN [@radford2015unsupervised] to generate 2D brain MR images followed by an autoencoder for image denoising. @chuquicusma2018fool utilized a DCGAN to generate lung nodules and then conducted a Turing test to evaluate the quality of the generated samples. GAN frameworks were also shown to improve accuracy of image classification via generation of new synthetic training images. @frid2018gan used a DCGAN and a ACGAN [@Odena2017Conditional] to generate images of three liver lesion classes to synthetically augment the limited dataset and improve the performance of CNN for liver lesion classification. Similarly, @salehinejad2018generalization employed a DCGAN to artificially simulate pathology across five classes of chest X-rays in order to augment the original imbalanced dataset and improve the performance of a CNN model in chest pathology classification.
The GAN framework has also been utilized in semi-supervised learning architectures to help leverage the vast number of unlabeled data alongside limited labeled data. The following efforts demonstrate how incorporating unlabeled data in the GAN framework has led to significant improvements in the accuracy of image-level classification: @madani2018semi used an order of magnitude less labeled data with a DCGAN in semi-supervised learning and showed comparable performance to a traditional supervised CNN classifier. Furthermore, their study also demonstrated reduced domain over-fitting by simply supplying unlabeled test domain images. @springenberg2015unsupervised combined a WGAN and CatGAN [@wang2017catgan] for unsupervised and semi-supervised learning of feature representation of dermoscopy images.
Despite these successes, GAN frameworks are very difficult to train, as was discussed in the previous section. Our work mitigates the limitations of training the GAN framework; it enables training on a limited number of labeled data, prevents overfitting to a specific data domain source, prevents mode collapse, and enables multi-class image classification.
MAVEN Architecture
==================
![Our MAVEN architecture compared to those of VAE, GAN, and VAE-GAN. In the MAVEN, inputs to $D$ can be real data $X$, generated data $\hat{X}$, or $\tilde{X}$. An ensemble ensures the combined feedback from the discriminators to the generator.[]{data-label="fig:archs"}](maven_archs){width="\linewidth"}
Fig. \[fig:archs\] illustrates the preliminary models building up to our MAVEN architecture.
The VAE is an explicit generative model that uses two neural nets—an encoder $E$ and decoder $D^\prime$. Network $E$ learns an efficient compression of the real data point $x$ into a lower dimensional latent representation space $z(x)$; i.e., $q_\lambda(z\vert x)$. With neural network likelihoods, computing the gradient becomes intractable. However via differentiable, non-centered re-parameterization, sampling is performed from an approximate function $q_{\lambda}(z\vert x) = N(z;
\mu_{\lambda}, \sigma_{\lambda}^2)$, where $z = \mu_\lambda +
\sigma_\lambda \odot \hat{\varepsilon}$ with $\hat{\varepsilon} \sim
N(0, 1)$. Encoder $E$ results in $\mu$ and $\sigma$, and with the re-parameterization trick, $z$ is sampled from a Gaussian distribution. Then with $D^\prime$, new samples are generated or real data samples are reconstructed. So, $D^\prime$ provides parameters for the real data distribution; i.e., $p_\lambda(x\vert z)$. Later, a sample drawn from $p_\phi(x\vert z)$ may be used to reconstruct the real data by marginalizing out $z$.
The GAN is an implicit generative model where a generator $G$ and a discriminator $D$ compete in a mini-max game over the training data to improve their performance. Generator $G$ tries to mimic the underlying distribution of the training data and generates fake samples while discriminator $D$ learns to discriminate fake generated samples from real samples. The GAN model is trained on the following objectives: $$\begin{aligned}
\label{eqn:discriminator}
\max_{D}V(D) &= E_{x\sim p_\text{data}(x)}[\log D(x)] + E_{x\sim p_g(z)}[\log(1 - D(G(z))];\\
\label{eqn:generator}
\min_{G}V(G) &= E_{x \sim p_z(z)}[\log(1 - D(G(z))].\end{aligned}$$ $G$ takes a noise sample $z\sim p_g(z)$ and learns to map into image space as if they are coming from the original data distribution $p_\text{data}(x)$. The discriminator $D$ takes either real image data or fake image data as the input and provides feedback to the generator $G$, regarding whether the input to $D$ is real or fake. $D$ wants to maximize the likelihood for real samples and minimize the likelihood of generated samples. On the other hand, $G$ wants $D$ to maximize the likelihood of generated samples. A Nash equilibrium state is possible when $D$ can no longer distinguish real and generated samples meaning that the model distribution will be the same as the data distribution.
![The three convolutional neural networks, $E$, $G$, and $D$, in the MAVEN.[]{data-label="fig:arch_details"}](arch_EGD){width="0.75\linewidth"}
@makhzani2015adversarial proposed the adversarial training of VAEs; i.e., VAE-GANs. Although they kept both $D^\prime$ and $G$, one can merge $D^\prime$ and $G$ since both can generate data samples from the noise samples of the representation $z$. In this case, $D$ either receives generated samples $\tilde{x}$ via $G$ or fake samples $\hat{x}$, and real data samples $x$. Although $G$ and $D$ compete against each other, at some point the feedback from $D$ becomes predictable for $G$ and it keeps generating samples from the same class. At that time, the generated samples lack variety. Fig. \[fig:gan\_collapse\]c shows an example where all the generated images are of the same class. @durugkar2016generative proposed that using multiple discriminators in a GAN model helps improve performance, especially resolving the mode collapse issue. Moreover, a dynamic ensemble of multiple discriminators has recently been proposed, addressing the same issue [@mordido2018dropout].
In our MAVEN, the VAE-GAN combination is extended to have multiple discriminators aggregated in an ensemble layer. As in a VAE-GAN, the MAVEN has three components $E$, $G$, and $D$; all are convolutional neural networks with convolutional or transposed convolutional layers (Fig. \[fig:arch\_details\]). $E$ takes real samples and generates a dimensionality-reduced representation $z(x)$. $G$ can take samples from noise distribution $z\sim p_g(z)$ or sampled noise $z(x)\sim q_\lambda(x)$, and it generates fake or completely new samples. $D$ takes inputs from distributions of real labeled data, real unlabeled data, and fake generated data. Fractionally strided convolutions are performed in $G$ to obtain the image dimension from the latent code. The goal of an autoencoder is to maximize the Evidence Lower Bound (ELBO). The intuition here is to show the network more real data. The more real data that it sees, the more evidence is available to it and, as a result, the ELBO can be maximized faster. $K$ discriminators are collected in an ensemble layer and the combined feedback $$\label{eqn: mean_ensemble}
V(D) = \frac{1}{K}\sum_{i=1}^K w_iD_i$$ is passed to $G$. In order to randomize the feedback from multiple discriminators, a single discriminator is randomly selected.
$steps \leftarrow \frac{m}{B}$ Sample minibatch $z_i;{z^{(1)},\dots,z^{(m)}},z_i\sim p_g(z)$ Sample minibatch $x_i; {x^{(1)},\dots,x^{(m)}}, x_i\sim p_\text{data}(x)$ Update discriminator $D_k$ by ascending along its gradient: $$\nabla_{\theta_{D_k}} \frac{1}{m}\sum_{i=1}^m[\log D_k(x_i) + \log(1 - D_k(G(z_i)))]$$ Sample minibatch $z_{k_i}, i=1,\dots, m, k=1,\dots, K, z_{k_i}\sim p_g(z)$
Assign weights $w_k$ for each of the discriminators $D_k$ Determine the mean discriminator $D_\mu$ of the discriminators $D_1,...,D_k$ $$D_\mu = \frac{1}{K}\sum_i^K w_iD_i$$
Update the generator $G$ by descending along its gradient from the ensemble of discriminator $D_{\mu}$: $$\nabla_{\theta_{G}} \frac{1}{m}\sum_{i=1}^m[\log(1 - D_{\mu}(G(z_i)))]$$ Sample minibatch $x_i; {x^{(1)},\dots,x^{(m)}}, x_i\sim p_\text{data}(x)$ Update encoder along its expectation function: $$\nabla_{\theta_{ E_{q_\lambda(z|x)}}} \left[\log \frac{p(z)}{q_\lambda (z|x)} \right]$$
Semi-Supervised Learning
========================
The overall training procedure of the proposed MAVEN model is presented in Algorithm \[alg:MAVEN\]. In the forward pass, the real samples to $E$ and noise samples to $G$ are presented multiple times for the presence of multiple discriminators. In the backward pass, the combined feedback from the $D$s is determined and passed to $G$ and $E$.
In the original image generator GAN, $D$ works as a binary classifier—it classifies the input image as real or synthetic. In order to facilitate the training for a $n$-class classifier, the role of $D$ is changed to an $(n+1)$-classifier. For multiple logit generation, the sigmoid function is replaced by a softmax function. Now, it can receive an image $x$ as input and outputs an $(n+1)$-dimensional vector of logits $\{{l}_1, {l}_2,\dots,{l}_{n+1}\}$. These logits are finally transformed into class probabilities for the final classification. Class ${(n+1)}$ is for the fake data and the remaining $n$ are for the multiple labels in the real data. The probability of $x$ being fake is $$\label{eqn:fake_prob}
p(y = n+1 | x) = \frac{\exp(l_{n+1})}{\sum_{j=1}^{n+1}\exp(l_j)},$$ and the probability that $x$ is real and belongs to class $i$ is $$\label{eqn:real_prob}
p(y= i|x, i< n+1) = \frac{\exp(l_i)}{\sum_{j=1}^{n+1}\exp(l_j)}.$$ As a semi-supervised classifier, the model only takes labels for a small portion of training data. For the labeled data, it is then like supervised learning, while it learns in an unsupervised manner for the unlabeled data. The advantage comes from generating new samples. The model learns the classifier by generating samples from different classes.
Losses
------
Three networks $E$, $G$, and $D$ are trained on different objectives. $E$ is trained on maximizing the ELBO, $G$ is trained on generating realistic samples, and $D$ is trained to learn a classifier that classifies fake generated samples or particular classes for the real data samples.
#### D Loss:
Since the model is trained on both labeled and unlabeled training data, the loss function of $D$ includes both supervised and unsupervised losses. When the model receives real labeled data, it is just the standard supervised learning loss $$\label{eqn:supervised_loss}
L_{D_\text{supervised}} =
- \mathbb{E}_{\mathbf{x},y\sim p_\text{data}} \log[p(y = i|\mathbf{x}, i< n+1)].$$ When it receives unlabeled data from three different sources, the unsupervised loss contains the original GAN loss for real and fake data from two different sources: fake1 directly from $G$ and fake2 from $E$ via $G$. The three losses $$\label{eqn:D_real}
L_{D_\text{real}} = - \mathbb{E}_{x \sim p_\text{data}} \log [ 1 - p(y = n+1 | \mathbf{x})],$$ $$\label{eqn:D_fake}
L_{D_\text{fake1}} = - \mathbb{E}_{\hat{x} \sim G} \log [p(y = n+1 | \hat{\mathbf{x}})],$$ and $$\label{eqn:D_recon}
L_{D_\text{fake2}}=-\mathbb{E}_{\tilde{x} \sim G} \log [p(y = n+1 | \tilde{x})],$$ are combined as the unsupervised loss in $D$: $$\label{eqn:D_unsupervised}
L_{D_\text{unsupervised}} = L_{D_\text{real}} + L_{D_\text{fake1}} + L_{D_\text{fake2}}.$$
#### G Loss:
For $G$, the feature loss is used along with the original GAN loss. Activation $f(x)$ from an intermediate layer of $D$ is used to match the feature between real and fake samples. Feature matching has shown a lot of potential in semi-supervised learning [@salimans2016improvedT]. The goal of feature matching is to push the generator to generate data that matches real data statistics. The discriminator specifies those statistics; it is natural that $D$ can find the most discriminative features in real data against data generated by the model: $$\label{eqn:G_feature}
L_{G_\text{feature}} = || \mathbb{E}_{x \sim p_\text{data}} f(x) - \mathbb{E}_{\hat{x} \sim G}f(\hat{x}) ||^2_2.$$ The total $G$ loss becomes the combined feature loss and $G$ costs maximizing the log-probability of $D$ making a mistake for generated data (fake1/fake2). Therefore, the $G$ loss $$\label{eqn:G_loss}
L_G = L_{G_\text{feature}} + L_{G_\text{fake1}} + L_{G_\text{fake2}}.$$ is the combination of three losses, (\[eqn:G\_feature\]), $$\label{eqn:G_fake}
L_{G_\text{fake1}} = - \mathbb{E}_{\hat{x} \sim G} \log [ 1 - p(y = n+1 | \hat{x})],$$ and $$\label{eqn:G_recon}
L_{G_\text{fake2}} = - \mathbb{E}_{\tilde{\mathbf{x}} \sim G} \log [ 1 - p(y = n+1 | \tilde{x}].$$
#### E Loss:
In the encoder $E$, the maximization of ELBO is equivalent to minimization of KL-divergence, allowing approximate posterior inferences. Therefore the loss function includes the KL-divergence and also a feature loss to match the features in the fake2 data with the real data distribution. The loss for the encoder is $$\label{eqn:E_loss}
L_E = L_{E_\text{KL}} + L_{E_\text{feature}},$$ where $$L_{E_\text{KL}} = -KL [q_{\lambda}(z|x)|| p(z)]
= \mathbb{E}_{q_\lambda(z|x)} \left[\log \frac{p(z)}{q_\lambda (z|x)} \right] \approx \mathbb{E}_{q_\lambda(z|x)}$$ and $$L_{E_\text{feature}} = || \mathbb{E}_{x \sim p_\text{data}} f(x) - \mathbb{E}_{\tilde{x} \sim G}f(\tilde{x}) ||^2_2.$$
Experiments and Results
=======================
Data
----
We used three datasets to evaluate our MAVEN model for image generation and automatic image classification from 2D images in a semi-supervised learning scheme, and we constrained the experiments to limited labeled training data, considering that a large portion of annotation is missing; specifically:
1. The Street View House Numbers (SVHN) dataset [@netzer2011reading]. There are 73,257 digit images for training and 26,032 digit images for testing in the SVHN dataset. Out of two versions of the images, we used the version which has MNIST-like $32\times32$ pixel images centered around a single character, in RGB channels. Each of the training and test images are labeled as one of the ten digits (0–9).
2. The CIFAR-10 dataset [@krizhevsky2009learning], which consists of 60,000 $32\times32$ pixel color images in 10 classes. There are 50,000 training images and 10,000 test images in the CIFAR-10 dataset. This is a 10-class classification with classes airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck.
3. The anterior-posterior Chest X-Ray (CXR) dataset [@kermany2018identifying] for the classification of pneumonia and normal images. We performed 3-class classification: normal, bacterial pneumonia, and virus pneumonia. The dataset contains 5,216 training and 624 test images.
Implementation Details
----------------------
To compare the image generation and multi-class classification performance of our MAVEN model, we used two baselines: DC-GAN and VAE-GAN. The same generator and discriminator architectures were used for DC-GAN and MAVEN models and the same encoder was used for the VAE-GAN and MAVEN models. For our MAVENs, we experimented with 2, 3, and 5 discriminators. In addition to using the proposed mean feedback of the multiple discriminators, we also experimented with feedback from a randomly selected discriminator. All the models were implemented in TensorFlow and run on a single Nvidia Titan GTX (12GB) GPU. For the CXR dataset, the images were normalized and resized to $128\times128$ pixels before passing them to the models, while for the SVHN and CIFAR-10 datasets, the normalized images were passed to the models in their original $(32\times32\times3)$ pixel sizes. For the discriminator, after every convolutional layer, a dropout layer was added with a dropout rate of 0.4. For all the models, we consistently used the Adam optimizer with a learning rate of $2e-4$ for $G$ and $D$, and $1e-5$ for $E$ with a momentum of 0.5. All the convolutional layers were followed by batch normalizations. Leaky ReLU activations were used with $\alpha = 0.2$. For all the experiments, only 10% training data were used along with the corresponding labels. The classification performance was measured with cross-validation and average scores were reported after running each model 10 times.
Evaluation
----------
#### Image Generation Performance:
There are no perfect performance metrics for the unsupervised learning in measuring the quality of generated samples. However, to assess the quality of the generated images, we employed the widely used Fréchet Inception Distance (FID) [@heusel2017gans] and a simplified version of the Descriptive Distribution Distance (DDD) [@imran2017optimization]. To measure the Fréchet distance between two multivariate Gaussians, the generated samples and real data samples are compared through their distribution statistics: $$\label{eqn:fid}
\text{FID} = ||\mu_\text{data} - \mu_\text{fake}||^2 + Tr(\Sigma_\text{data} + \Sigma_\text{fake} - 2\sqrt{\Sigma_\text{data}\Sigma_\text{fake}}).$$ Two distribution samples are calculated from the 2048-dimensional activations of the pool3 layer of Inception-v3 [@salimans2016improvedT]. DDD measures the closeness of a synthetic data distribution to a real data distribution by comparing descriptive parameters from the two distributions. We propose a simplified version based on the first four moments of the distributions, computed as the weighted sum of normalized differences of moments, as follows: $$\label{eqn:ddd}
\text{DDD} = - \sum_{i=1}^{i=4} \log{w_i}|\mu_{\text{data}_i} - \mu_{\text{fake}_i}|.$$ The higher-order moments are weighted more, as the stability of a distribution can be better represented by them. For both FID and DDD, lower scores are better.
#### Image Classification Performance:
To evaluate model performance in classification, we used two measures: image-level classification accuracy and class-wise F1 scoring. The F1 score is $$\label{eqn:f1-score}
\text{F1} = \frac{2\times \text{precision} \times \text{recall}}{\text{precision} + \text{recall}},$$ with $$\text{precision} = \frac{\text{TP}}{\text{TP} + \text{FP}} \textrm{\qquad and \qquad} \text{recall} = \frac{\text{TP}}{\text{TP} + \text{FN}},$$ where TP, FP, and FN are the number of true positives, false positives, and false negatives, respectively.
Results
-------
### SVHN
For the SVHN dataset, we trained the network on $32\times32$ pixel images. From the training set, we randomly picked 7,326 labeled images and the remaining unlabeled images were passed to the network. All the models were trained for 150 epochs and then evaluated. We generated an equal number of new images as the training set size. Fig. \[fig:svhn\_images\] presents a qualitative comparison of the generated digit images from the DC-GAN, VAE-GAN, and ALEAN models relative to the real training images, suggesting that our MAVEN-generated images are more realistic.
This was further confirmed by the FID and DDD scoring. FID and DDD measurement was performed by drawing 10,000 samples from the generated images and 10,000 samples from the real training images. The generated image quality measurement was performed for eight different models, and the resultant FID and DDD scores are reported in Table \[table:fid-ddd\]. For FID score calculation, the FID score is reported after running the pre-trained Inception-v3 network for 20 epochs for each model. Per the scores, the MAVEN-rand model with 3 discriminators achieved the best FID and the best DDD was achieved for the MAVEN-mean model with 5 discriminators.
For the semi-supervised classification, both image-level accuracy and class-wise F1 scores were calculated. Table \[table:svhn-accuracy\] compares the classification performance of all the models for the SVHN dataset. The MAVEN model consistently outperformed the DC-GAN and VAE-GAN classifiers both in classification accuracy and class-wise F1 scores. Among all the models, our MAVEN-mean model with 2 and 3 discriminators were found to be the most accurate.
### CIFAR-10
For the CIFAR-10 dataset, all the models were trained for 300 epochs and then evaluated. We generated an equal number of new images as the training set size. Fig. \[fig:cifar\_images\] visually compares the generated images from the GAN, VAE-GAN, and ALEAN models relative to the real training images.
The FID and DDD measurements were performed with the distribution of 10,000 samples drawn from the generated images and 10,000 samples from the real training images. For the FID score calculation, the pre-trained Inception-v3 network was run for 20 epochs and the FID score was recorded. The FID and DDD scores are reported in Table \[table:fid-ddd\]. As the tabulated results suggest, our proposed MAVEN models achieved better FID scores than some of the recently published models. Note that, those models were implemented in a different settings. As for the visual comparison, the FID and DDD scores confirmed more realistic image generation with our ALELAN models than the DC-GAN and VAE-GAN models. Except for MAVEN-mean with 2 discriminators, all other MAVEN models have smaller FID scores; MAVEN-rand with 3 discriminators has the smallest FID score among all the models.
For the semi-supervised classification, both image-level accuracy and class-wise F1 scores were calculated. Table \[table:cifar10-accuracy\] compares the performance of all the models for the CIFAR-10 dataset.
### CXR
For the CXR dataset, all the models were trained for 150 epochs and then evaluated. We generated an equal number of new images as the training set size. Fig. \[fig:chex\_images\] presents a visual comparison of synthesized and real image samples.
The FID and DDD measurements were performed for distribution of generated and real training samples, indicating that more realistic images were generated by the MAVEN models than by the GAN and VAE-GAN models. The FID and DDD scores presented in Table \[table:fid-ddd\] show that the mean MAVEN model with 3 discriminators (MAVEN-mean3D) has the smallest FID and DDD scores.
The classification performance reported in Table \[table:chex-accuracy\] suggests our proposed MAVEN model-based classifers are more accurate than the basline GAN and VAE-GAN classifiers. Among all the models, MAVEN-mean classifier with 3 discriminators found to be the most accurate in classifying the B-pneumonia and V-pneumonia from normal. However, the overall performance is not so good for the CXR dataset compared to the natural image datasets. A possible reason could be the shortage of data and the omission of a larger portion of the labels. The main issue in the medical image dataset is that, unlike natural images, every case is different than others, even though they are labeled as the same class. It may be possible to resolve this by augmenting the training set with the generated images from each of the models. However, the goal of our present work was to devise a generative model architecture that could be equally competitive as a generator and a classifier. Even with the relatively smaller dataset, the proposed MAVEN models perform better than the baseline models.
Conclusions
===========
We have demonstrated the advantages of an ensemble of discriminators in the adversarial learning of variational autoencoders and the application of this idea to semi-supervised classification from limited labeled data. Training our new MAVEN models on a small, labeled dataset and leveraging a large number of unlabeled examples, we have shown superior performance relative to prior GAN and VAE-GAN based classifiers, suggesting that our MAVEN models can be very effective in concurrently generating high-quality realistic images and improving multi-class classification performance. However, it remains an open problem to find the optimal number of discriminators that can perform consistently. Our future work will consider more complex image analysis tasks beyond classification and include more extensive experimentation spanning additional domains.
Comparison of Distributions
===========================
Through histogram-density diagrams, Fig. \[fig:density\] compares the distributions of each of the models against the real distribution, showing that the distributions of images synthesized by our MAVENs are generally closer to the real image distributions for the SVHN, CIFAR-10, and CXR datasets.
Comparison of Images
====================
Figs. \[fig:svhn\_images\], \[fig:cifar\_images\], and \[fig:chex\_images\] present visual comparisons of image samples from the SVHN, CIFAR-10, and CXR datasets, respectively, relative to those generated by the different models.
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---
abstract: 'This paper is based on a formulation of the Navier-Stokes equations developed by Iyer and Constantin [@Cont] , where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. Our contribution is to establish this probabilistic representation formula for mild solutions of the Navier-Stokes equations on ${\mathbb R}^{d} $.'
author:
- 'C. Olivera[^1]'
title: ' Probabilistic representation for mild solution of the Navier-Stokes equations '
---
*Key words and phrases. Navier-Stokes equations, Stochastic differential equation, Iyer-Cosntantin representation formula, Mild Solution.*
Introduction {#Intro}
============
We consider the classical Cauchy problem for the Navier-Stokes system, describing the evolution of a velocity field $u$ of an incompressible fluid with kinematic viscosity $\nu$
$$\label{Navier}
\left \{
\begin{aligned}
&\partial_t u(t, x) =\nu \Delta u(t, x) - [u(t, x)\cdot \nabla] u(t, x) - \nabla \pi(t, x)
\\& \text{ div } u(t, x)=0
\\ &u(0,x)= u_{0}(x)
\end{aligned}
\right .$$
The unknown quantities are the velocity $u(t, x) = (u_{1}(t, x), . . . , u_{d}(t, x))$ of the fluid element at time $t$ and position $x$ and the pressure $\pi(t, x)$. Such equations always attract the attention of many researchers, with an enormous quantity of publications in the literature. Concerning classical results about (1.1), we refer to the book by Teman [@Tema]. In the 1960s, mild solutions were first constructed by Kato and Fujita ([@Kato3] and [@Kato2]) that are continuous in time and take values in the Sobolev spaces $u\in C([0,T], H^{s}({\mathbb R}^{d}))$ ($s\geq \frac{d}{2}-1$). Results on the existence of mild solutions with value in $L^{p}$, were established by Fabes, Jones and Riviere [@Fabes] and by Giga [@Giga2]. In 1992, a modern treatment for mild solutions in $H^{s}$ was given by Chemin [@Che]. For recent developments see Lemarie-Rieusset [@Lema]. One of the (still open) million dollar problems posed by the Clay Institute is to show that given a smooth initial data $u_0$ the solution to (\[Navier\]) in three dimensions remains smooth for all time. We are interested in developing probabilistic techniques, that could help solve this problem
Probabilistic representations of solutions of partial differential equations as the expected value of functionals of stochastic processes date back to the work of Einstein, Feynman, Kac, and Kolmogorov in physics and mathematics. The Feynman-Kac formula is the most well-known example, which has provided a link between linear parabolic partial differential equations and probability theory, see [@Kara]. In 2008 Constantin and Iyer [@Cont](see also [@Cont2] and [@Iyer]) established a probabilistic Lagrangian representation formula by making use of stochastic flows. They show that $u$ is classical solution to the Navier-Stokes equation (\[Navier\]) if an only if $u$ satisfies the stochastic systems
$$\label{itoass}
X_{t}(x)= x + \int_{0}^{t} u(r, X_{r}(x)) \ dr + B_{t},$$
$$\label{repre}
u(t,x)= \mathbb{P} {\mathbb E}[ (\nabla X_{t}^{-1})^{\ast} u_{0}(X_{t}^{-1}) ]$$
where $B_{t}$ denoting the standard Brownian motion, $\mathbb{P}$ is the Leray-Hodge projection and $\ast$ denotes the transposition of matrix. We mention that Fang, D Luo [@Fang] obtained formula (\[repre\]) on a compact manifolds, Rezakhanlou [@Reza] wrote the representation (\[repre\]) in the context of symplectic geometry and Zhang [@Zhang3] extended the formula (\[repre\]) for non-local operators. Different probabilistic representations of the solution of the the Navier-Stokes equations were studied by S. Albeverio, Y. Belopolskaya [@Albe], Busnello [@Bu], Busnello, Flandoli, Romito [@Bu2] Cipriano, Cruzeiro [@Cip], Cruzeiro, Shamarova [@Cru] and Zhang [@Zhang3].
Strong solutions to the equation (\[itoass\]) are known for irregular $u$ , the best result (after previous investigations of Zvonkin [@Zo], Veretennikov[@Ve], among others) being proved by Krylov, Röckner in [@Krylov]. More recently Flandoli, Gubinelli, Priola, see [@FGP2] and [@FGP], proved that if the drift term is Hölder continuous then $x \rightarrow X_{s,t} $ is a $C^{1}$- stochastic flow.
The contribution of this paper is to show that the unique mild solution of the equation (\[Navier\]) with values in $ C([0,T], H^{s}({\mathbb R}^{d}))$ has the stochastic representation (\[repre\]). The proof is simple and it is based in stability properties of the mild solution and in the flow properties associated to the equation (\[itoass\]). The result is the following theorem.
In fact, through of this paper, we fix a stochastic basis with a $d$-dimensional Brownian motion $\big( \Omega, \mathcal{F}, \{
\mathcal{F}_t: t \in [0,T] \}, \mathbb{P}, (B_{t}) \big)$. We denoted $M$ a generic constant.
Preliminaries {#Intro}
=============
Mild Solution.
--------------
In this subsection we recall some results on the Stokes operator.
$$L_{\sigma}^{2}= \ the \ closure \ in \ [L^{2}({\mathbb R}^{d})]^d \ of \{ u\in [C_{0}^{\infty}({\mathbb R}^{d})]^d, \ div u=0\}$$ and $$G^{2}= \{ \nabla q, q\in W^{1,2}({\mathbb R}) \}.$$
We then have the following Helmholtz decomposition $$[L^{2}({\mathbb R})]^d= L_{\sigma}^{2} \oplus G^{2},$$ the sum above reduces to the orthogonal decomposition and $L_{\sigma}^2$ is a separable Hilbert space, whose scalar product is denoted by $(\cdot,\cdot)$.
Let $\mathbb{P}$ be the continuous projection from $L^{2}({\mathbb R}^{d})$ to $L_{\sigma}^{2}$ associated with this decomposition and let $\Delta$ be the Laplace operator. Now, we define the Stokes operator $A$ in $L_{\sigma}^{2}$ by $A=-\mathbb{P}\Delta$. The operator $-A$ generates a bounded analytic semigroup $\{S(t)\}_{t\ge 0}$, see [@Soh].
The potential space $H^{s}({\mathbb R}^{d})$ is defined as the space $(\mathit{I}-\Delta)^{-s/2}L^{2}$ equipped with the norm $\| f\|_{H^{s}}:=\| (\mathit{I}-\Delta)^{s/2}\|_{L^{2}}$. It is well know that
$$\| fg\|_{H^{s}}\leq M_{s} \ \| f\|_{H^{s}} \ \| g\|_{H^{s}} \ \ \ \ if \ s>\frac{d}{2}$$
We also recall that
$$\label{stimaSpr}
\| S(t) u \|_{H^{s}} \le \ M_{s} \| u \|_{H^{s}} $$
$$\label{stimaASp}
\|A^{\alpha} S(t) u \|_{H^{s}}\le \ \frac{M_{s}}{t^{\alpha}} \ \| u \|_{H^{s}} .$$
We consider the Navier -Stokes initial problem in the space $H^{s}({\mathbb R}^{d})$. Applying the projection operator $\mathbb{P}$ to we get rid of the pressure term; setting $\nu=1$, equation becomes $$\label{eq-abst}
\begin{cases}
du(t) + Au(t)\ dt= B(u(t)) \ dt, & t>0
\\ u(0)=u_{0}
\end{cases}$$ where the non linear term $B$ is defined by $B(u)=-\mathbb{P}[(u\cdot \nabla)u]$. Since $u$ is divergence free, we also have the representation $B(u)= -\mathbb{P}[ \text{div}\ (u\otimes u )]$ which will be useful later on. We consider the mild solution
\[defimild\] A measurable function $u: [0,T]\rightarrow H^{s}({\mathbb R}^{d})$ is a mild solution of the equation (\[eq-abst\]) if
1. $u \in C(0,T; H^{s}({\mathbb R}^{d}))$,
2. for all $t \in (0,T]$, we have $$\label{mild}
u(t)= S(t)u_{0}+ \ \int_{0}^{t} S(t-s)B(u(s)) \ ds$$
We assume that there exist $T>0$ such that satisfies $u$ satisfies items 1 and 2 in definition \[defimild\]. Then we called $u$ of local mild solution.
Stochastic Flows.
-----------------
In this subsection we follow the seminar paper by Flandoli, Gubinelli and Priola in [@FGP2]. We consider the SDE
$$\label{sde3}
dX_{t} (x)= b(t,X_{s,t}(x) ) dt + d B_{t}, \ X_{s}=x\in{\mathbb R}^{d},$$
where $X_{s,t}(x)= X(s,t,x)$, also $X_{t}(x)= X(0,t,x)$. Moreover, the inverse $Y_{s,t}(x):=X_{s,t}^{-1}(x)$ satisfies the following backward stochastic differential equation $$\label{itoassBac}Y_{s,t}(x)= x - \int_{s}^{t} b(r, Y_{r,t}(x)) \ dr - (B_{t}-B_{s}).$$
We denote by $\phi_{s,t}$ the flow associated to $X_{s,t}$ and $\psi_{s,t}$ its inverse. Let $T> 0 $ be be fixed. For any $\alpha\in (0,1)$, we denoted by $L^{\infty}([0,T],C_{b}^{\alpha}({\mathbb R}^{d}))$ the space bounded Borel functions $f : [0,T]\times {\mathbb R}^{d}\rightarrow {\mathbb R}$ such that
$$\begin{aligned}
\|f\|_{\alpha,T}:= \sup_{t\in [0,T]} \sup_{x\neq y, |x-y|\leq 1} \frac{|f(t,x)-f(t,y)|}{|x-y|^\theta} < \infty\,.\end{aligned}$$
We also recall the important results in [@FGP2].
\[difH\] We assume that $b \in L^{\infty}([0,\infty),C_{b}^{\alpha}({\mathbb R}^{d}))$. Then
- There exists a unique solution of the SDE (\[sde3\]).
- There exists a stochastic flow $\phi_{s,t}$ of diffeomorphisms associated to equation (\[sde3\]). The flow is the class $C^{1+\alpha^{\prime}}$ for every $0<\alpha^{\prime}< \alpha$.
- Let $b^{n}\in L^{\infty}([0,\infty),C_{b}^{\alpha}({\mathbb R}^{d}))$ be a sequence of the vector field and $\phi^{n}$ be the corresponding stochastic flow. If $b^{n}\rightarrow b$ in $L^{\infty}([0,\infty),C_{b}^{\alpha}({\mathbb R}^{d}))$, then for any $p\geq 1$ we have
$$\label{es1}
\lim_{n\rightarrow \infty} \sup_{x\in {\mathbb R}^{d}} \sup_{s\in[0,T]} {\mathbb E}[\sup_{r\in[s,T]}|\phi^{n}_{s,r}-\phi_{s,r}|^{p}]=0,$$
$$\label{es2}
\sup_{n}\sup_{x\in {\mathbb R}^{d}} \sup_{s\in[0,T]} {\mathbb E}[\sup_{r\in[s,T]}|D\phi_{s,r}^{n}|^{p}]< \infty,$$
$$\label{es3}
\lim_{n\rightarrow \infty} \sup_{x\in {\mathbb R}^{d}} \sup_{s\in[0,T]} {\mathbb E}[\sup_{r\in[s,T]}|D\phi_{s,r}^{n}-D\phi_{s,t}|^{p}]=0.$$
The same results are valid for the backward flows $\psi_{s,t}^{n}$ and $\psi_{s,t}$ since are solutions of the same SDE driven by the drifts $-b_{n}$ and $-b$.
Result
======
Let $\{\rho_n\}_n$ be a family of standard symmetric mollifiers. We define the family of regularized initial data as $u_{0}^{n}(x) = (u \ast \rho_\varepsilon) (x) $. Let $T>0$. Now, we assume that for all $n$ there exist $u^{n}$ a classical solution in $[0,T]\times {\mathbb R}^{d}$ of
$$\label{eq-abstreg}
\begin{cases}
du^{n}(t) + Au^{n}(t)\ dt= B(u^{n}(t)) \ dt, & t>0
\\ u(0)=u_{0}^{n}.
\end{cases}$$
\[T1\] We fix $T>0$ and we assume $u_{0}\in H^{s}({\mathbb R}^{d})$. Let be $u \in C([0,T], H^{s}({\mathbb R}^{d}))$ a local mild solution with $s> \frac{d}{2}$ such that $u^{n}$ converge to $u$ in $C([0,T], H^{s}({\mathbb R}^{d}))$. Then we have that
$$\label{repr2}
u(t,x)= \mathbb{P} {\mathbb E}[ (\nabla X_{t}^{-1})^{\ast} u_{0}(X_{t}^{-1}) ].$$
[*Step 1 : Regular initial data.*]{}
By Itô formula or by Constantin-Iyer [@Cont2] we have
$$u^{n}(t,x)= \mathbb{P} {\mathbb E}[ (\nabla Y_{t}^{n})^{\ast} u_{0}^{n}(Y_{t}^{n}) ].$$
where $Y_{t}^{n}$ is the inverse of
$$dX_{t}^{n}= u^{n}(t,X_{t}^{n}) dt + d B_{t}, \ X_{0}=x\in{\mathbb R}^{d}.$$
[*Step 2 : Convergence II.*]{}
From $H^{s}({\mathbb R}^{d})\subset C_{b}^{\alpha}({\mathbb R}^{d})$ with $\alpha=s-\frac{d}{2}$, hypothesis and theorem \[difH\] we have
$$\label{c1}
\lim_{n\rightarrow \infty} \sup_{x\in {\mathbb R}^{d}} {\mathbb E}[\sup_{t\in[0,T]}|Y^{n}_{t}-Y_{t}|^{p}]=0,$$
$$\label{c3}
\lim_{n\rightarrow \infty} \sup_{x\in {\mathbb R}^{d}} {\mathbb E}[\sup_{t\in[0,T]}|DY_{t}^{n}-DY_{t}|^{p}]=0,$$
$$\label{c4}
\sup_{x\in {\mathbb R}^{d}} {\mathbb E}[\sup_{t\in[0,T]}|DY_{t}^{n}|^{p}]< \infty,$$
where $Y_{t}$ is the inverse of $X_{t}$ and it verifies (\[sde3\]) with drift $u(t,x)$.
[*Step 3 : Convergence III.*]{}
We observe that
$$\begin{aligned}
& {\mathbb E}[ (\nabla Y_{t}^{n})^{\ast} u_{0}^{n}(Y_{t}^{n}) ]- {\mathbb E}[ (\nabla Y_{t})^{\ast} u_{0}(Y_{t}) ]|
\\ &
\leq{\mathbb E}[ (\nabla Y_{t}^{n})^{\ast} u_{0}^{n}(Y_{t}^{n})-(\nabla Y_{t}^{n})^{\ast} u_{0}(Y_{t}^{n})] |
\\ &
+ \ {\mathbb E}[ (\nabla Y_{t}^{n})^{\ast} u_{0}(Y_{t}^{n})-(\nabla Y_{t}^{n})^{\ast} u_{0}(Y_{t})] |
\\ &
+ \ {\mathbb E}[ (\nabla Y_{t}^{n})^{\ast} u_{0}(Y_{t})-(\nabla Y_{t})^{\ast} u_{0}(Y_{t})] | \
\\ & = I_{1}+ I_{2} + I_{3}. \end{aligned}$$
By Hölder inequality and (\[c4\]) we have
$$\begin{aligned}
& \int_{{\mathbb R}^{d}} |I_{1}|^{2} \ dx \\ &
\leq
\int_{{\mathbb R}^{d}} {\mathbb E}|\nabla Y_{t}^{n}|^{2} \ {\mathbb E}|u_{0}^{n}(Y_{t}^{n})- u_{0}(Y_{t}^{n})|^{2} \ dx
\\ & \leq \sup_{x,t} {\mathbb E}|\nabla Y_{t}^{n}|^{2} \ \int_{{\mathbb R}^{d}} {\mathbb E}|u_{0}^{n}(Y_{t}^{n})- u_{0}(Y_{t}^{n})|^{2} \ dx
\\ & = C \ \int_{{\mathbb R}^{d}} {\mathbb E}|u_{0}^{n}(x)- u_{0}(x)|^{2} \ dx \end{aligned}$$
it follows that $I_{1} \rightarrow 0 $ in $C([0,T], L^2({{\mathbb R}^{d}}))$.
By Hölder inequality and (\[c4\]) we obtain
$$\begin{aligned}
& \int_{{\mathbb R}^{d}} |I_{2}|^{2} \ dx \\ & \leq \ \int_{{\mathbb R}^{d}} {\mathbb E}|\nabla Y_{t}^{n}|^{2} \ {\mathbb E}|u_{0}(Y_{t}^{n})- u_{0}(Y_{t}) |^{2} \ dx \\ & \leq \sup_{x,t} {\mathbb E}|\nabla Y_{t}^{n}|^{2} \ \int_{{\mathbb R}^{d}}{\mathbb E}|u_{0}(Y_{t}^{n})- u_{0}(Y_{t}) |^{2} \ dx, \end{aligned}$$
from (\[c1\]) and dominated convergence we get that $I_{2}\rightarrow 0 $ in $C([0,T], L^2({{\mathbb R}^{d}}))$.
We observe that
$$\begin{aligned}
& \int_{{\mathbb R}^{d}} |I_{3}|^{2} \ dx \\ &
\leq \ \int_{{\mathbb R}^{d}} {\mathbb E}| (\nabla Y_{t}^{n})^{\ast}- (\nabla Y_{t})^{\ast}|^{2} {\mathbb E}|u_{0}(Y_{t}) |^{2} \ dx
\\ & \leq C \sup_{x,t}{\mathbb E}| (\nabla Y_{t}^{n})^{\ast}- (\nabla Y_{t})^{\ast}|^{2} \ dx \end{aligned}$$
from (\[c3\]) we deduce that $I_{3}\rightarrow 0 $ in $C([0,T], L^2({{\mathbb R}^{d}}))$.
Thus we conclude that $ {\mathbb E}[ (\nabla Y_{t}^{n})^{\ast} u_{0}^{n}(Y_{t}^{n})]\rightarrow {\mathbb E}[(\nabla Y_{t})^{\ast} u_{0}(Y_{t})]$ strong in $C([0,T], L^2({{\mathbb R}^{d}}))$. This implies that $u^{n}= \mathbb{P} {\mathbb E}[(\nabla Y_{t}^{n})^{\ast} u_{0}^{n}(Y_{t}^{n})]$converge to $ \mathbb{P} {\mathbb E}[(\nabla Y_{t})^{\ast} u_{0}(Y_{t})]$ in $C([0,T], L^{2}({\mathbb R}^{d}))$.
[*Step 4 : Conclusion.*]{}
From step I and hypothesis we conclude that $u(t,x)= \mathbb{P} {\mathbb E}[(\nabla Y_{t})^{\ast} u_{0}(Y_{t})]$.
We observed that by Kato construction of the mild solution we can take $T$ sufficiently small such that there exists an unique mild solution in $C(0,T, ;H^{s}({\mathbb R}^{d}))$ with initial conditions $u_{0}$ and $u_{0}^{n}$, see for instance [@Lema].
We assume $u_{0}\in H^{s}({\mathbb R}^{d})$ and $T$ is small enough. Let be $u \in C([0,T], H^{s}({\mathbb R}^{d}))$ the unique local mild solution with $s> \frac{d}{2}$. Then we have that
$$\label{repr3}
u(t,x)= \mathbb{P} {\mathbb E}[ (\nabla X_{t}^{-1})^{\ast} u_{0}(X_{t}^{-1}) ].$$
Now, we consider $u^{n}$ the unique local mild solution of
$$\label{eq-abstreg2}
\begin{cases}
du^{n}(t) + Au^{n}(t)\ dt= B(u^{n}(t)) \ dt, & t>0
\\ u(0)=u_{0}^{n}.
\end{cases}$$
in $[0,T]$. We have
$$u(t)- u^{n}(t)=u_{0}- u_{0}^{n} + \ \int_{0}^{t} S(t-s) \big(B(u(s))-B( u^{n}(s)) \big) \ ds .$$
By classical estimations we obtain
$$\begin{aligned}
& \| u(t)-\ u^{n}(t)\|_{H_{s}} \\ & \leq \|u_{0}- u_{0}^{n}\|_{H_{s}} \\ &
+ \int_{0}^{t} \|S(t-s) \big(B(u(s))-B( u^{n}(s)) \big)\|_{H_{s}} \ ds
\\ & \le \|u_{0}- u_{0}^{n}\|_{H_{s}} \\ & + \int_{0}^{t} \frac{M}{(t-s)^{\frac{1}{2}}} (\| u^{n}(s)\|_{H_{s}}+ \| u(s)\|_{H_{s}}) \|u(s)-u^{n}(s)\|_{H_{s}} \ ds. \end{aligned}$$
It is well know that
$$\| u(t)\|_{H^{s}}\leq M \|u_0\|_{H^{s}}$$
and
$$\| u^{n}(t)\|_{H^{s}}\leq M \|u_0^{n}\|_{H^{s}}\leq M\|u_0\|_{H^{s}}.$$
Thus we have
$$\begin{aligned}
& \sup_{t\in [0,T]}\| u(t)-\ u^{n}(t)\|_{H_{s}}\\ &
\leq \|u_{0}- u_{0}^{n}\|_{H_{s}} M T^{\frac{1}{2}} (\sup_{t\in [0,T]}\| u^{n}(t)\|_{H_{s}}+
\sup_{t\in [0,T]}\| u(t)\|_{H_{s}}) \\ & \times \sup_{t\in [0,T]} \|u(s)-u^{n}(s)\|_{H_{s}}
\\ & \leq \|u_{0}- u_{0}^{n}\|_{H_{s}}
+ M T^{\frac{1}{2}} \| u_{0}\|_{H_{s}} \sup_{t\in [0,T]} \|u(s)-u^{n}(s)\|_{H_{s}}.\end{aligned}$$
If $M T^{\frac{1}{2}} \| u_{0}\|_{H_{s}} < 1$ we get
$$\sup_{t\in [0,T]}\| u(t)-\ u^{n}(t)\|_{H_{s}}\leq C \|u_{0}- u_{0}^{n}\|_{H_{s}}.$$
Then we deduce
$$\sup_{t\in [0,T]}\| u(t)-\ u^{n}(t)\|_{H_{s}}\rightarrow 0 \ as \ n\rightarrow\infty.$$
The representation (\[repr3\]) we follow from the theorem \[T1\].
Acknowledgements {#acknowledgements .unnumbered}
================
Christian Olivera is partially supported by FAPESP by the grants 2017/17670-0 and 2015/07278-0 . Also supported by CNPq by the grant 426747/2018-6.
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[^1]: Departamento de Matemática, Universidade Estadual de Campinas, Brazil. E-mail: *[email protected]*.
|
---
abstract: 'Today, many different probabilistic programming languages exist and even more inference mechanisms for these languages. Still, most logic programming based languages use backward reasoning based on SLD resolution for inference. While these methods are typically computationally efficient, they often can neither handle infinite and/or continuous distributions, nor evidence. To overcome these limitations, we introduce distributional clauses, a variation and extension of Sato’s distribution semantics. We also contribute a novel approximate inference method that integrates forward reasoning with importance sampling, a well-known technique for probabilistic inference. To achieve efficiency, we integrate two logic programming techniques to direct forward sampling. Magic sets are used to focus on relevant parts of the program, while the integration of backward reasoning allows one to identify and avoid regions of the sample space that are inconsistent with the evidence.'
author:
- |
Bernd Gutmann, Ingo Thon, Angelika Kimmig, Maurice Bruynooghe and Luc De Raedt\
Department of Computer Science, Katholieke Universiteit Leuven,\
Celestijnenlaan 200A - bus 2402, 3001 Heverlee, Belgium\
{firstname.lastname}@cs.kuleuven.be
bibliography:
- 'bibtex.bib'
title: The Magic of Logical Inference in Probabilistic Programming
---
Introduction
============
The advent of statistical relational learning [@Getoor07; @DeRaedtAPRIL08] and probabilistic programming [@NIPSWorkshop] has resulted in a vast number of different languages and systems such as PRISM [@SatoKameya:01], ICL [@Poole08], ProbLog [@DeRaedt07-IJCAIa], Dyna [@Eisner05], BLPs [@Kersting08], CLP($\mathcal{BN}$) [@clpbn], BLOG [@Milch05], Church [@Goodman08], IBAL [@Pfeffer01], and MLNs [@Richardson:06]. While inference in these languages generally involves evaluating the probability distribution defined by the model, often conditioned on evidence in the form of known truth values for some atoms, this diversity of systems has led to a variety of inference approaches. Languages such as IBAL, BLPs, MLNs and CLP($\mathcal{BN}$) combine knowledge-based model construction to generate a graphical model with standard inference techniques for such models. Some probabilistic programming languages, for instance BLOG and Church, use sampling for approximate inference in generative models, that is, they estimate probabilities from a large number of randomly generated program traces. Finally, probabilistic logic programming frameworks such as ICL, PRISM and ProbLog, combine SLD-resolution with probability calculations.
So far, the second approach based on sampling has received little attention in logic programming based systems. In this paper, we investigate the integration of sampling-based approaches into probabilistic logic programming frameworks to broaden the applicability of these. Particularly relevant in this regard are the ability of Church and BLOG to sample from continuous distributions and to answer conditional queries of the form $p(q |e )$ where $e$ is the evidence. To accommodate (continuous and discrete) distributions, we introduce *distributional clauses*, which define random variables together with their associated distributions, conditional upon logical predicates. Random variables can be passed around in the logic program and the outcome of a random variable can be compared with other values by means of special built-ins. To formally establish the semantics of this new construct, we show that these random variables define a basic distribution over facts (using the comparison built-ins) as required in Sato’s distribution semantics [@Sato:95], and thus induces a distribution over least Herbrand models of the program. This contrasts with previous instances of the distribution semantics in that we no longer enumerate the probabilities of alternatives, but instead use arbitrary densities and distributions.
From a logic programming perspective, BLOG [@milch:aistats2005] and related approaches perform *forward reasoning*, that is, the samples needed for probability estimation are generated starting from known facts and deriving additional facts, thus generating a *possible world*. PRISM and related approaches follow the opposite approach of *backward reasoning*, where inference starts from a query and follows a chain of rules backwards to the basic facts, thus generating *proofs*. This difference is one of the reasons for using sampling in the first approach: exact forward inference would require that all possible worlds be generated, which is infeasible in most cases. Based on this observation, we contribute a new inference method for probabilistic logic programming that combines sampling-based inference techniques with forward reasoning. On the probabilistic side, the approach uses rejection sampling [@KollerFriedman09], a well-known sampling technique that rejects samples that are inconsistent with the evidence. On the logic programming side, we adapt the *magic set* technique [@bancilhon] towards the probabilistic setting, thereby combining the advantages of forward and backward reasoning. Furthermore, the inference algorithm is improved along the lines of the *SampleSearch* algorithm [@Gogate09samplesearch:importance], which avoids choices leading to a sample that cannot be used in the probability estimation due to inconsistency with the evidence. We realize this using a heuristic based on backward reasoning with limited proof length, the benefit of which is experimentally confirmed. This novel approach to inference creates a number of new possibilities for applications of probabilistic logic programming systems, including continuous distributions and Bayesian inference.
This paper is organized as follows: we start by reviewing the basic concepts in Section \[sec:prelim\]. Section \[sec:semantics\] introduces the new language and its semantics, Section \[sec:algorithms\] a novel forward sampling algorithm for probabilistic logic programs. Before concluding, we evaluate our approach in Section \[sec:experiments\].
Preliminaries {#sec:prelim}
=============
Probabilistic Inference {#sec:probinf}
-----------------------
A discrete probabilistic model defines a probability distribution $p(\cdot)$ over a set $\Omega$ of basic outcomes, that is, value assignments to the model’s random variables. This distribution can then be used to evaluate a conditional probability distribution $p(q|e) =\frac{p(q\wedge e)}{p(e)}$, also called *target distribution*. Here, $q$ is a query involving random variables, and $e$ is the *evidence*, that is, a partial value assignment of the random variables[^1]. Evaluating this target distribution is called *probabilistic inference* [@KollerFriedman09]. In probabilistic logic programming, random variables often correspond to ground atoms, and $p(\cdot)$ thus defines a distribution over truth value assignments, as we will see in more detail in Sec. \[sec:ds\] (but see also ). Probabilistic inference then asks for the probability of a logical query being true given truth value assignments for a number of such ground atoms.
In general, the probability $p(\cdot)$ of a query $q$ is in the discrete case the sum over those outcomes $\omega\in \Omega$ that are consistent with the query. In the continuous case, the sum is replaced by an (multidimensional) integral and the distribution $p(\cdot)$ by a (product of) densities $\mathbf{F}(\cdot)$ That is, $$\label{eq:exact}
p(q)= \sum_{\omega\in \Omega} p(\omega) \mymathbb{1}_{q}(\omega), \quad \quad and\quad \quad p(q)=\idotsint\limits_\Omega \mymathbb{1}_q(\omega) d\mathbf{F}(\omega)$$ where $\mymathbb{1}_{q}(\omega)=1$ if $\omega\models q$ and $0$ otherwise. As common (e.g. [@wasserman04allof]) we will use for convenience the notation $\int xdF(x)$ as unifying notation for both discrete and continuous distributions.
As $\Omega$ is often very large or even infinite, exact inference based on the summation in quickly becomes infeasible, and inference has to resort to approximation techniques based on *samples*, that is, randomly drawn outcomes $\omega\in
\Omega$. Given a large set of such samples $\{s_1,\ldots,s_N\}$ drawn from $p(\cdot)$, the probability $p(q)$ can be estimated as the fraction of samples where $q$ is true. If samples are instead drawn from the target distribution $p(\cdot|e)$, the latter can directly be estimated as $$\hat{p}(q|e) := \frac{1}{N}\sum_{i=1}^N \mymathbb{1}_q(s_i) \enspace.$$
However, sampling from $p(\cdot|e)$ is often highly inefficient or infeasible in practice, as the evidence needs to be taken into account. For instance, if one would use the standard definition of conditional probability to generate samples from $p(\cdot)$, all samples that are not consistent with the evidence do not contribute to the estimate and would thus have to be discarded or, in sampling terminology, *rejected*.
More advanced sampling methods therefore often resort to a so-called *proposal distribution* which allows for easier sampling. The error introduced by this simplification then needs to be accounted for when generating the estimate from the set of samples. An example for such a method is *importance sampling*, where each sample $s_i$ has an associated *weight* $w_i$. Samples are drawn from an *importance distribution* $\pi(\cdot |e)$, and weights are defined as $w_i=\frac{p(s_i|e)}{\pi(s_i|e)}$. The true target distribution can then be estimated as $$\hat{p}(q|e) = \frac{1}{W} \sum_{i=1}^Nw_i\cdot \mymathbb{1}_q(s_i)$$ where $W=\sum_i w_i$ is a normalization constant. The simplest instance of this algorithm is *rejection sampling* as already sketched above, where the samples are drawn from the prior distribution $p(\cdot)$ and weights are $1$ for those samples consistent with the evidence, and $0$ for the others. Especially for evidence with low probability, rejection sampling suffers from a very high rejection rate, that is, many samples are generated, but do not contribute to the final estimate. This is known as the *rejection problem*. One way to address this problem is *likelihood weighted sampling*, which dynamically adapts the proposal distribution during sampling to avoid choosing values for random variables that cause the sample to become inconsistent with the evidence. Again, this requires corresponding modifications of the associated weights in order to produce correct estimates.
Logical Inference {#sec:forward}
-----------------
A (definite) clause is an expression of the form $\mathtt{h{\mbox{ :- }}b_1, \ldots ,b_n}$, where $\mathtt{h}$ is called head and $\mathtt{b_1,\ldots,b_n}$ is the body. A program consists of a set of clauses and its semantics is given by its least Herbrand model. There are at least two ways of using a definite clause in a logical derivation. First, there is *backward chaining*, which states that to prove a goal $\mathtt{h}$ with the clause it suffices to prove $\mathtt{b_1, \ldots,b_n}$; second, there is *forward chaining*, which starts from a set of known facts $\mathtt{b_1, \ldots ,b_n}$ and the clause and concludes that $\mathtt{h}$ also holds (cf. [@nilsson:book]). Prolog employs backward chaining (SLD-resolution) to answer queries. SLD-resolution is very efficient both in terms of time and space. However, similar subgoals may be derived multiple times if the query contains recursive calls. Moreover, SLD-resolution is not guaranteed to always terminate (when searching depth-first). Using forward reasoning, on the other hand, one starts with what is known and employs the immediate consequence operator $T_P$ until a fixpoint is reached. This fixpoint is identical to the least Herbrand model.
\[def:tpop\] Let $P$ be a logic program containing a set of definite clauses and $ground(P)$ the set of all ground instances of these clauses. Starting from a set of ground facts $S$ the $T_P$ operator returns $$T_P(S) = \{ \mathtt{h} \mid \mathtt{h {\mbox{ :- }}b_1,\ldots,b_n} \in
ground(P) \mbox{ and } \{ \mathtt{b_1}, \ldots , \mathtt{b_n}\}
\subseteq S\}$$
Distribution Semantics {#sec:ds}
----------------------
Sato’s distribution semantics [@Sato:95] extends logic programming to the probabilistic setting by choosing truth values of basic facts randomly. The core of this semantics lies in splitting the logic program into a set $F$ of *facts* and a set $R$ of *rules*. Given a probability distribution $P_F$ over the facts, the rules then allow one to extend $P_F$ into a distribution over least Herbrand models of the logic program. Such a Herbrand model is called a *possible world*.
More precisely, it is assumed that $DB = F\cup R$ is ground and denumerable, and that no atom in $F$ unifies with the head of a rule in $R$. Each truth value assignment to $F$ gives rise to a unique least Herbrand model of $DB$. Thus, a probability distribution $P_F$ over $F$ can directly be extended into a distribution $P_{DB}$ over these models. Furthermore, Sato shows that, given an enumeration $f_1,f_2,\ldots$ of facts in $F$, $P_F$ can be constructed from a series of finite distributions $P_F^{(n)}(f_1=x_1,\ldots, f_n=x_n)$ provided that the series fulfills the so-called compatibility condition, that is, $$\label{eq:compat}
P_F^{(n)}(f_1=x_1,\ldots, f_n=x_n) =
\sum_{x_{n+1}}P_F^{(n+1)}(f_1=x_1,\ldots, f_{n+1}=x_{n+1})$$
Syntax and Semantics {#sec:semantics}
====================
Sato’s distribution semantics, as summarized in Sec. \[sec:ds\], provides the basis for most probabilistic logic programming languages including PRISM [@SatoKameya:01], ICL [@Poole08], CP-logic [@Vennekens09] and ProbLog [@DeRaedt07-IJCAIa]. The precise way of defining the basic distribution $P_F$ differs among languages, though the theoretical foundations are essentially the same. The most basic instance of the distribution semantics, employed by ProbLog, uses so-called *probabilistic facts*. Each ground instance of a *probabilistic fact* directly corresponds to an independent random variable that takes either the value “true” or “false”. These probabilistic facts can also be seen as binary switches, cf. [@Sato:95], which again can be extended to multi-ary switches or choices as used by PRISM and ICL. For switches, at most one of the probabilistic facts belonging to the switch is “true” according to the specified distribution. Finally, in CP-logic, such choices are used in the head of rules leading to the so-called *annotated disjunction*. Hybrid ProbLog [@gutmann10ilp] extends the distribution semantics with continuous distributions. To allow for exact inference, Hybrid ProbLog imposes severe restrictions on the distributions and their further use in the program. Two sampled values, for instance, cannot be compared against each other. Only comparisons that involve one sampled value and one number constant are allowed. Sampled values may not be used in arithmetic expressions or as parameters for other distributions, for instance, it is not possible to sample a value and use it as the mean of a Gaussian distribution. It is also not possible to reason over an unknown number of objects as BLOG [@Milch05] does, though this is the case mainly for algorithmic reasons.
Here, we alleviate these restrictions by defining the basic distribution $P_F$ over probabilistic facts based on both discrete and continuous random variables. We use a three-step approach to define this distribution. First, we introduce explicit random variables and corresponding distributions over their domains, both denoted by terms. Second, we use a mapping from these terms to terms denoting (sampled) outcomes, which, then, are used to define the basic distribution $P_F$ on the level of probabilistic facts. For instance, assume that an urn contains an unknown number of balls where the number is drawn from a Poisson distribution and we say that this urn contains many balls if it contains at least $10$ balls. We introduce a random variable $\mathtt{number}$, and we define $\mathtt{many} {\mbox{ :- }}\mathtt{dist\_gt({\simeq\!\!}(number), 9).}$ Here, ${\simeq\!\!}(\mathtt{number})$ is the Herbrand term denoting the sampled value of $\mathtt{number}$, and $\mathtt{dist\_gt({\simeq\!\!}(number), 9)}$ is a probabilistic fact whose probability of being true is the expectation that this value is actually greater than $9$. This probability then carries over to the derived atom $\mathtt{many}$ as well. We will elaborate on the details in the following.
Syntax {#sec:syntax}
------
In a logic program, following Sato, we distinguish between probabilistic facts, which are used to define the basic distribution, and rules, which are used to derive additional atoms.[^2] Probabilistic facts are not allowed to unify with any rule head. The distribution over facts is based on random variables, whose distributions we define through so called distributional clauses.
\[def:disclause\] A *distributional clause* is a definite clause with an atom $\mathtt{h} \sim \mathcal{D}$ in the head where $\sim$ is a binary predicate used in infix notation.
For each ground instance $(\mathtt{h}\sim \mathcal{D} {\mbox{ :- }}\mathtt{b_1,\ldots,b_n})\theta$ with $\theta$ being a substitution over the Herbrand universe of the logic program, the distributional clause defines a random variable $\mathtt{h}\theta$ and an associated distribution $\mathcal{D}\theta$. In fact, the distribution is only defined when $(\mathtt{b_1,\ldots,b_n})\theta$ is true in the semantics of the logic program. These random variables are terms of the Herbrand universe and can be used as any other term in the logic program. Furthermore, a term ${\simeq\!\!}(d)$ constructed from the reserved functor ${\simeq\!\!}/1$ represents the outcome of the random variable $d$. These functors can be used inside calls to special predicates in $dist\_rel =\{ dist\_eq/2, dist\_lt/2, dist\_leq/2, dist\_gt/2,
dist\_geq/2\}$. We assume that there is a fact for each of the ground instances of these predicate calls. These facts are the *probabilistic facts* of Sato’s distribution semantics. Note that the set of probabilistic facts is enumerable as the Herbrand universe of the program is enumerable. A term ${\simeq\!\!}(d)$ links the random variable $d$ with its outcome. The probabilistic facts compare the outcome of a random variable with a constant or the outcome of another random variable and succeed or fail according to the probability distribution(s) of the random variable(s).
\[ex:probrules\]
$$\begin{aligned}
\mathtt{nballs \sim poisson(6)}.
& \label{lbl:ex1}\\
\mathtt{color(B)\sim\ [0.7:b, 0.3:g]}
&{\mbox{ :- }}\mathtt{between(1,{\simeq\!\!}(nballs),B).} \label{lbl:ex2}\\
\mathtt{ diameter(B,MD)\sim gamma(MD/20,20) }
&\mathtt{{\mbox{ :- }}between(1,{\simeq\!\!}(nballs),B),}\nonumber \\
&\quad\ \mathtt{mean\_diameter({\simeq\!\!}(color(B)),MD).}\label{lbl:ex3}
\end{aligned}$$
The defined distributions depend on the following logical clauses: $$\begin{aligned}
\mathtt{mean\_diameter(C,5) } & \mathtt{{\mbox{ :- }}dist\_eq(C,b).}\\
\mathtt{mean\_diameter(C,10) } & \mathtt{{\mbox{ :- }}dist\_eq(C,g). }\\
\mathtt{ between(I,J,I)} & \mathtt{{\mbox{ :- }}dist\_leq(I,J).} \\
\mathtt{between(I, J, K)} & \mathtt{{\mbox{ :- }}dist\_lt(I,J), \ I1\ is\ I + 1, between(I1,J,K)}.
\end{aligned}$$
The distributional clause models the number of balls as a Poisson distribution with mean 6. The distributional clause models a discrete distribution for the random variable color(B). With probability 0.7 the ball is blue and green otherwise. Note that the distribution is defined only for the values B for which $between(1, {\simeq\!\!}(nballs), B)$ succeeds. Execution of calls to the latter give rise to calls to probabilistic facts that are instances of $dist\_leq(I,{\simeq\!\!}(nballs))$ and $dist\_lt(I,{\simeq\!\!}(nballs))$. Similarly, the distributional clause defines a gamma distribution that is also conditionally defined. Note that the conditions in the distribution depend on calls of the form $mean\_diameter({\simeq\!\!}(color(n)), MD)$ with $n$ a value returned by between/3. Execution of this call finally leads to calls $dist\_eq({\simeq\!\!}(color(n)),b)$ and $dist\_eq({\simeq\!\!}(color(n)),g)$.
It looks feasible, to allow ${\simeq\!\!}(d)$ terms everywhere and to have a simple program analysis insert the special predicates in the appropriate places by replacing $</2$, $>/2$, $\leq/2$, $\geq/2$ predicates by $dist\_rel/2$ facts. Though extending unification is a bit harder: as long as a ${\simeq\!\!}(h)$ term is unified with a free variable, standard unification can be performed; only when the other term is bound an extension is required. In this paper, we assume that the special predicates $\mathtt{dist\_eq/2}$, $\mathtt{dist\_lt/2}$, $\mathtt{dist\_leq/2}$, $\mathtt{dist\_gt/2}$, and $\mathtt{dist\_geq/2}$ are used whenever the outcome of a random variable need to be compared with another value and that it is safe to use standard unification whenever a $\mathtt{{\simeq\!\!}(h)}$ term is used in another predicate.
For the basic distribution on facts to be well-defined, a program has to fulfill a set of validity criteria that have to be enforced by the programmer.
\[def:validprog\] A program $P$ is called *valid* if:
(V1)
: In the relation $\mathtt{h} \sim \mathcal{D}$ that holds in the least fixpoint of a program, there is a functional dependency from $\mathtt{h}$ to $ \mathcal{D}$, so there is a unique ground distribution $ \mathcal{D}$ for each ground random variable $\mathtt{h}$.
(V2)
: The program is *distribution-stratified*, that is, there exists a function $rank(\cdot)$ that maps ground atoms to $\mathbb{N}$ and that satisfies the following properties: (1) for each ground instance of a distribution clause $\mathtt{h}\sim
\mathcal{D} {\mbox{ :- }}\mathtt{b_1, \ldots b_n}$ holds $rank(\mathtt{h}\sim \mathcal{D} > rank(\mathtt{b_i})$ (for all $i$). (2) for each ground instance of another program clause: $\mathtt{h} {\mbox{ :- }}\mathtt{b_1, \ldots b_n}$ holds $rank(\mathtt{h})\geq rank(\mathtt{b_i})$ (for all $i$). (3) for each ground atom $\mathtt{b}$ that contains (the name of) a random variable $\mathtt{h}$, $rank(\mathtt{b}) \geq rank(\mathtt{h} \sim
\mathcal{D})$ (with $\mathtt{h} \sim \mathcal{D}$ the head of the distribution clause defining $\mathtt{h}$).
(V3)
: All ground probabilistic facts or, to be more precise, the corresponding indicator functions are *Lebesgue-measurable*.
(V4)
: Each atom in the least fixpoint can be derived from a finite number of probabilistic facts (*finite support condition* [@Sato:95]).
Together, (V1) and (V2) ensure that a single basic distribution $P_F$ over the probabilistic facts can be obtained from the distributions of individual random variables defined in $P$. The requirement (V3) is crucial. It ensures that the series of distributions $P_F^{(n)}$ needed to construct this basic distribution is well-defined. Finally, the number of facts over which the basic distribution is defined needs to be countable. This is true, as we have a finite number of constants and functors: those appearing in the program.
Distribution Semantics {#sec:distributionsemantics}
----------------------
We now define the series of distributions $P_F^{(n)}$ where we fix an enumeration $f_1,f_2,\ldots$ of probabilistic facts such that $i < j
\implies rank(f_i) \leq rank(f_j)$ where $rank(\cdot)$ is a *ranking function* showing that the program is distribution-stratified. For each predicate $\mathtt{rel/2}\in
dist\_rel$, we define an *indicator function* as follows: $$\begin{aligned}
I^1_{rel}(X_1,X_2) & =
\begin{cases}
1 & \text{if } \mathtt{rel}(X_1,X_2) \text{ is true} \\
0 & \text{if } \mathtt{rel}(X_1,X_2) \text{ is false}
\end{cases}\end{aligned}$$
Furthermore, we set $I^0_{rel}(X_1,X_2) = 1.0 - I^1_{rel}(X_1,X_2)$. We then use the expected value of the indicator function to define probability distributions $P_F^{(n)}$ over finite sets of ground facts $f_1,\ldots,f_n$. Let $\{rv_1,\ldots rv_m\}$ be the set of random variables these $n$ facts depend on, ordered such that if $rank(rv_i)<rank(rv_j)$, then $i<j$ (cf. (V2) in Definition \[def:validprog\]). Furthermore, let $f_i =
rel_i(t_{i1},t_{i2})$, $x_j\in\{1,0\}$, and $\theta^{-1} =
\{{\simeq\!\!}(rv_1)/V_1,\ldots, {\simeq\!\!}(rv_m)/V_m\}$. The latter replaces all evaluations of random variables on which the $f_i$ depend by variables for integration. $$\begin{aligned}
\lefteqn{P_F^{(n)}(f_1 = x_1, \ldots , f_n = x_n)
= \mathbb{E} [
I^{x_1}_{rel_1}(t_{11},t_{12}), \ldots ,
I^{x_n}_{rel_n}(t_{n1},t_{n2}) ]}\label{eq:series}\\
& = \idotsint \left( I^{x_1}_{rel_1}(t_{11}\theta^{-1}, t_{12}\theta^{-1})\cdots
I^{x_n}_{rel_n}(t_{n1}\theta^{-1}, t_{n2})\theta^{-1}\right) d\mathcal{D}_{rv_1}(V_1)\cdots d\mathcal{D}_{rv_m}(V_m) \nonumber\end{aligned}$$
Let $f_1,f_2,\ldots = dist\_lt({\simeq\!\!}(b1), 3), dist\_lt({\simeq\!\!}(b2), {\simeq\!\!}(b1)), \ldots$. The second distribution in the series then is $$\begin{aligned}
\lefteqn{P_F^{(2)}(dist\_lt({\simeq\!\!}(b1), 3)=x_1 ,dist\_lt({\simeq\!\!}(b2), {\simeq\!\!}(b1))=x_2)}\\
&= \mathbb{E} [ I_{\mathtt{dist\_lt}}^{x_1} ({\simeq\!\!}(b1),3),I_{\mathtt{dist\_lt}}^{x_2} ({\simeq\!\!}(b2),{\simeq\!\!}(b1))]\\
&= \int\int \left(I_{\mathtt{dist\_lt} }^{x_1} (V1,3) ,I_{\mathtt{dist\_lt} }^{x_2} (V2,V1) \right)d\mathcal{D}_{b1}(V1)d\mathcal{D}_{b2}(V2)\end{aligned}$$
By now we are able to prove the following proposition.
\[prop:adm\] Let $P$ be a valid program. $P$ defines a probability measure $P_P$ over the set of fixpoints of the $T_P$ operator. Hence, $P$ also defines for an arbitrary formula $\mathtt{q}$ over atoms in its Herbrand base the probability that $\mathtt{q}$ is true.
It suffices to show that the series of distributions $P_F^{(n)}$ over facts (cf. ) is of the form that is required in the distribution semantics, that is, these are well-defined probability distributions that satisfy the compatibility condition, cf. . This is a direct consequence of the definition in terms of indicator functions and the measurability of the underlying facts required for valid programs.
$T_P$ Semantics {#sec:tp}
---------------
In the following, we give a procedural view onto the semantics by extending the $T_P$ operator of Definition \[def:tpop\] to deal with probabilistic facts $\mathtt{dist\_rel(t_1,t_2)}$. To do so, we introduce a function <span style="font-variant:small-caps;">ReadTable</span>$(\cdot)$ that keeps track of the sampled values of random variables to evaluate probabilistic facts. This is required because interpretations of a program only contain such probabilistic facts, but not the underlying outcomes of random variables. Given a probabilistic fact $\mathtt{dist\_rel(t1,t2)}$, <span style="font-variant:small-caps;">ReadTable</span> returns the truth value of the fact based on the values of the random variables $h$ in the arguments, which are either retrieved from the table or sampled according to their definition $\mathtt{h}\sim \mathcal{D}$ as included in the interpretation and stored in case they are not yet available.
\[def:stp\] Let $P$ be a valid program and $ground(P)$ the set of all ground instances of clauses in $P$. Starting from a set of ground facts $S$ the $ST_P$ operator returns $$\begin{aligned}
ST_P(S) := \Big\{ \mathtt{h}\ \Big|\ &
\mathtt{h {\mbox{ :- }}b_1,\ldots,b_n} \in ground(P) \mbox{ and }
\forall\ \mathtt{b_i}: \mbox{ either } \mathtt{b_i}\in S \mbox{
or }\\
& \big(\mathtt{b_i}=dist\_rel(t1,t2) \wedge
(t_j={\simeq\!\!}(h)\rightarrow (\mathtt{h}\sim\mathcal{D})\in S) \wedge\\
& \ \mbox{\textsc{ReadTable}}(b_i)=true \big) \Big\}\end{aligned}$$
<span style="font-variant:small-caps;">ReadTable</span> ensures that the basic facts are sampled from their joint distribution as defined in Sec. \[sec:distributionsemantics\] during the construction of the standard fixpoint of the logic program. Thus, each fixpoint of the $ST_P$ operator corresponds to a possible world whose probability is given by the distribution semantics.
Forward sampling using Magic Sets and backward reasoning {#sec:algorithms}
========================================================
In this section we introduce our new method for probabilistic forward inference. To this aim, we first extend the magic set transformation to distributional clauses. We then develop a rejection sampling scheme using this transformation. This scheme also incorporates backward reasoning to check for consistency with the evidence during sampling and thus to reduce the rejection rate.
Probabilistic magic set transformation {#sec:magic}
--------------------------------------
The disadvantage of forward reasoning in logic programming is that the search is not goal-driven, which might generate irrelevant atoms. The *magic set* transformation [@bancilhon; @nilsson:book] focuses forward reasoning in logic programs towards a goal to avoid the generation of uninteresting facts. It thus combines the advantages of both reasoning directions.
\[def:magicsets\] If $P$ is a logic program, then we use $\textsc{Magic}(P)$ to denote the smallest program such that if $\mathtt{A_0 {\mbox{ :- }}A_1,\ldots, A_n}
\in P$ then
- $\mathtt{A_0 {\mbox{ :- }}c(A_0), A_1, \ldots, A_n }\in \textsc{Magic}(P)$ and
- for each $1\le i \le n$: $\mathtt{c(A_i) {\mbox{ :- }}c(A_0), A_1,\ldots, A_{i-1} }\in \textsc{Magic}(P)$
The meaning of the additional $\mathtt{c/1}$ atoms (c=call) is that they “switch on” clauses when they are needed to prove a particular goal. If the corresponding switch for the head atom is not true, the body is not true and thus cannot be proven. The magic transformation is both sound and complete. Furthermore, if the SLD-tree of a goal is finite, forward reasoning in the transformed program terminates. The same holds if forward reasoning on the original program terminates.
We now extend this transformation to distributional clauses. The idea is that the distributional clause for a random variable $h$ is activated when there is a call to a probabilistic fact $dist\_rel(t_1, t_2)$ depending on $h$.
\[def:pmagicsets\] For program $P$, let $P_L$ be $P$ without distributional clauses. $\textsc{M}(P)$ is the smallest program s.t. $\textsc{Magic}(P_L)
\subseteq \textsc{M}(P)$ and for each $\mathtt{h} \sim \mathcal{D}
\mathtt{ {\mbox{ :- }}b_1,\ldots,b_n} \in P$ and $\mathtt{rel} \in
\{\mathtt{eq, lt, leq, gt, geq}\}$:
- $\mathtt{h} \sim \mathcal{D}{\mbox{ :- }}\mathtt{(c(dist\_rel({\simeq\!\!}(h), X)); c(dist\_rel(X ,
{\simeq\!\!}(h))),b_1,\ldots,b_n.} \in \textsc{M}(P)$.
- $\mathtt{c(b_i) {\mbox{ :- }}(c(dist\_rel({\simeq\!\!}(h) , X));
c(dist\_rel(X, {\simeq\!\!}(h))), b_1,\ldots, b_{i-1}. } \in \textsc{M}(P)$.
Then $\textsc{PMagic}(P)$ consists of:
- a clause $\mathtt{a\_p(t_1,\ldots,t_n)} {\mbox{ :- }}\mathtt{c(p(t_1,\ldots,t_n)), p(t_1,\ldots,t_n)}$ for each built-in predicate (including $\mathtt{dist\_rel/2}$ for $\mathtt{rel} \in \{\mathtt{eq, lt, leq, gt, geq} \}$) used in $\textsc{M}(P)$.
- a clause $\mathtt{h} {\mbox{ :- }}\mathtt{b_1',\ldots,b_n'}$ for each clause $\mathtt{h} {\mbox{ :- }}\mathtt{b_1,\ldots,b_n} \in
\textsc{M}(P)$ where $\mathtt{b_i'=a\_b_i}$ if $\mathtt{b_i}$ uses a built-in predicate and else $\mathtt{b_i'=b_i}$.
Note that every call to a built-in $\mathtt{b}$ is replaced by a call to $\mathtt{a\_b}$; the latter predicate is defined by a clause that is activated when there is a call to the built-in ($\mathtt{c(b)}$) and that effectively calls the built-in. The transformed program computes the distributions only for random variables whose value is relevant to the query. These distributions are the same as those obtained in a forward computation of the original program. Hence we can show:
Let $P$ be a program and $\textsc{PMagic}(P)$ its probabilistic magic set transformation extended with a seed $c(q)$. The distribution over $q$ defined by $P$ and by $\textsc{PMagic}(P)$ is the same.
In both programs, the distribution over $q$ is determined by the distributions of the atoms $dist\_eq(t_1,t_2)$, $dist\_leq(t_1,t_2)$, $dist\_lt(t_1,t_2)$, $dist\_geq(t_1,t_2)$, and $dist\_gt(t_1,t_2)$ on which $q$ depends in a forward computation of the program $P$. The magic set transformation ensures that these atoms are called in the forward execution of $\textsc{PMagic}(P)$. In $\textsc{PMagic}(P)$, a call to such an atom activates the distributional clause for the involved random variable. As this distributional clause is a logic program clause, soundness and completeness of the magic set transformation ensures that the distribution obtained for that random variable is the same as in $P$. Hence also the distribution over $q$ is the same for both programs.
Rejection sampling with heuristic lookahead
-------------------------------------------
As discussed in Section \[sec:probinf\], sampling-based approaches to probabilistic inference estimate the conditional probability $p(q|e)$ of a query $q$ given evidence $e$ by randomly generating a large number of samples or possible worlds (cf. Algorithm \[alg:sampling\]). The algorithm starts by preparing the program $L$ for sampling by applying the <span style="font-variant:small-caps;">PMagic</span> transformation. In the following, we discuss our choice of subroutine <span style="font-variant:small-caps;">STPMagic</span> (cf. Algorithm \[alg:rejection\]) which realizes likelihood weighted sampling. It is used in Algorithm \[alg:sampling\], line \[line:callWeightedsample\], to generate individual samples. It iterates the stochastic consequence operator of Definition \[def:stp\] until either a fixpoint is reached or the current sample is inconsistent with the evidence. Finally, if the sample is inconsistent with the evidence, it receives weight 0.
$L^*:=$<span style="font-variant:small-caps;">PMagic</span>$(L)\cup \{c(a) | a \in e \cup{q} \}$ $n^+ :=0$ $n^- := 0$ $(I,w):=$<span style="font-variant:small-caps;">STPMagic</span>$(L^*,L,e,Depth)$\[line:callWeightedsample\] **if** $q\in I$ **then** $n^+:= n^+ + w$ **else** $n^-:= n^- + w$ $n^+ / (n^+ + n^-)$
$T_{pf}:=\varnothing$, $T_{dis}:=\varnothing$, $w:=1$, $I_{old}:=\varnothing$, $I_{new}:=\varnothing$ $I_{old}:=I_{new}$ split body in $B_{PF}$ (prob. facts) and $B_L$ (the rest) $s := true$, $w_d := 1$ select and remove $pf$ from $B_{PF}$ $(b_{pf},w_{pf}) := $<span style="font-variant:small-caps;">ReadTable</span>$(pf\theta, I_{old},
T_{pf}, T_{dis}, L, e, Depth)$ \[line:samplehead\] $s := s \wedge b_{pf}$ $w_d := w_d \cdot w_{pf}$ $(I_{new}, 0)$ $I_{new} := I_{new} \cup \{h\theta\}$ $w := w \cdot w_d$ $(I_{new}, w)$ $(I_{new}, 0)$
$(false,0)$ $(false,0)$ \[line:sample\] $(b,w) :=$ $(false,0)$ extend $T_{pf}$ with $(pf,b,w)$ $(b,w)$ as stored in $T_{pf}$ for $pf$ $w_h:=1$, $\mathcal{D}' := \mathcal{D}$ $\mathcal{D}' := \Call{Norm}{\mathcal{D}' \setminus
\{p_j:\mathtt{a_j}\}}$, $w_h:=w_h\times (1-p_j)$ $\mathcal{D}' := [1 : v]$, $w_h:=w_h\times
p$
$\mathcal{D}' := \Call{Norm}{\mathcal{D}' \setminus
\{p_j:\mathtt{a_j}\}}$, $w_h:=w_h\times (1-p_j)$ **if** [$\mathcal{D}'=\varnothing$]{} false Sample $x$ according to $\mathcal{D'}$, extend $T_{dis}$ with $(h,x)$ and true
Algorithm \[alg:likelihoodweighting\] details the procedure used in line \[line:samplehead\] of Algorithm \[alg:rejection\] to sample from a given distributional clause. The function <span style="font-variant:small-caps;">ReadTable</span> returns the truth value of the probabilistic fact, together with its weight. If the outcome is not yet tabled, it is computed. Note that `false` is returned when the outcome is not consistent with the evidence. Involved distributions, if not yet tabled, are sampled in line \[line:sample\]. In the infinite case, <span style="font-variant:small-caps;">Sample</span> simply returns the sampled value. In the finite case, it is directed towards generating samples that are consistent with the evidence. Firstly, all possible choices that are inconsistent with the negative evidence are removed. Secondly, when there is positive evidence for a particular value, only that value is left in the distribution. Thirdly, it is checked whether each left value is consistent with all other evidence. This consistency check is performed by a simple depth-bounded meta-interpreter. For positive evidence, it attempts a top-down proof of the evidence atom in the original program using the function <span style="font-variant:small-caps;">MaybeProof</span>. Subgoals for which the depth-bound is reached, as well as probabilistic facts that are not yet tabled are assumed to succeed. If this results in a proof, the value is consistent, otherwise it is removed. Similarly for negative evidence: in <span style="font-variant:small-caps;">MaybeFail</span>, subgoals for which the depth-bound is reached, as well as probabilistic facts that are not yet tabled are assumed to fail. If this results in failure, the value is consistent, otherwise it is removed. The $Depth$ parameter allows one to trade the computational cost associated with this consistency check for a reduced rejection rate.
Note that the modified distribution is normalized and the weight is adjusted in each of these three cases. The weight adjustment takes into account that removed elements cannot be sampled and is necessary as it can depend on the distributions sampled so far which elements are removed from the distribution sampled in <span style="font-variant:small-caps;">Sample</span> (the clause bodies of the distribution clause are instantiating the distribution).
Experiments {#sec:experiments}
===========
We implemented our algorithm in YAP Prolog and set up experiments to answer the questions
- Does the lookahead-based sampling improve the performance?
- How do rejection sampling and likelihood weighting compare?
To answer the first question, we used the distributional program in Figure \[fig:experimentnogreen\], which models an urn containing a random number of balls. The number of balls is uniformly distributed between 1 and 10 and each ball is either red or green with equal probability. We draw 8 times a ball with replacement from the urn and observe its color. We also define the atom $\mathtt{nogreen(}D\mathtt{)}$ to be true if and only if we did not draw any green ball in draw 1 to $D$. We evaluated the query $P(\mathtt{dist\_eq(\simeq(color(\simeq(drawnball(1)))),red)}\
|\mathtt{nogreen(}D\mathtt{)})$ for $D=1,2,\ldots, 8$. Note that the evidence implies that the first drawn ball is red, hence that the probability of the query is 1; however, the number of steps required to proof that the evidence is inconsistent with drawing a green first ball increases with D, so the larger is D, the larger Depth is required to reach a 100% acceptance rate for the sample as illustrated in Figure \[fig:experimentnogreen\]. It is clear that by increasing the depth limit, each sample will take longer to be generated. Thus, the $Depth$ parameter allows one to trade off convergence speed of the sampling and the time each sample needs to be generated. Depending on the program, the query, and the evidence there is an optimal depth for the lookahead.
$$\begin{aligned}
&{ }\\
&{ }\\
&{ }\\
&\mathtt{numballs \sim uniform([1,2,3,4,5,6,7,8,9,10]).}\\
&\mathtt{ball(M) {\mbox{ :- }}between(1,numballs,M).}\\
&\mathtt{color(B)\sim uniform([red,green]) {\mbox{ :- }}ball(B).}\\
&\mathtt{draw(N) {\mbox{ :- }}between(1,8,N).}\\
&\mathtt{nogreen(0).}\\
\end{aligned}$$
![ The program modeling the urn (left); rate of accepted samples (right) for evaluating the query $P(\mathtt{dist\_eq(\simeq(color(\simeq(drawnball(1)))),red)}\ |$ $\mathtt{nogreen(}D\mathtt{)})$ for $D=1,2,\ldots, 10$ and for $Depth=1,2,\ldots,8$ using Algorithm \[alg:sampling\]. The acceptance rate is calculated by generating 200 samples using our algorithm and counting the number of sample with weight larger than 0.[]{data-label="fig:experimentnogreen"}](pics/experimentgreen)
$$\begin{aligned}
&\mathtt{nogreen(D) {\mbox{ :- }}dist\_eq(\simeq(color(\simeq(drawnball(D)))),red),\ D2\ is\
D-1,\
nogreen(D2).}\\
&\mathtt{drawnball(D) \sim uniform(L) {\mbox{ :- }}draw(D),}\mathtt{findall(B,ball(B),L).}
\end{aligned}$$
To answer **Q2**, we used the standard example for BLOG [@Milch05]. An urn contains an unknown number of balls where every ball can be either green or blue with $p=0.5$. When drawing a ball from the urn, we observe its color but do not know which ball it is. When we observe the color of a particular ball, there is a $20\%$ chance to observe the wrong one, e.g. green instead of blue. We have some prior belief over the number of balls in the urn. If 10 balls are drawn with replacement from the urn and we saw 10 times the color green, what is the probability that there are $n$ balls in the urn? We consider two different prior distributions: in the first case, the number of balls is uniformly distributed between 1 and 8, in the second case, it is Poisson-distributed with mean $\lambda=6$.
![Results of urn experiment with forward reasoning. 10 balls with replacement were drawn and each time green was observed. Left: Uniform prior over \# balls, right: Poisson prior $(\lambda=6)$.[]{data-label="fig:urnproblog"}](pics/uniform_std "fig:") ![Results of urn experiment with forward reasoning. 10 balls with replacement were drawn and each time green was observed. Left: Uniform prior over \# balls, right: Poisson prior $(\lambda=6)$.[]{data-label="fig:urnproblog"}](pics/poisson_std "fig:")
One run of the experiment corresponds to sampling the number $N$ of balls in the urn, the color for each of the $N$ balls, and for each of the ten draws both the ball drawn and whether or not the color is observed correctly in this draw. Once these values are fixed, the sequence of colors observed is determined. This implies that for a fixed number $N$ of balls, there are $2^N\cdot N^{10}$ possible proofs. In case of the uniform distribution, exact PRISM inference can be used to calculate the probability for each given number of balls, with a total runtime of $0.16$ seconds for all eight cases. In the case of the Poisson distribution, this is only possible up to 13 balls, with more balls, PRISM runs out of memory. For inference using sampling, we generate 20,000 samples with the uniform prior, and 100,000 with Poisson prior. We report average results over five repetitions. For these priors, PRISM generates 8,015 and 7,507 samples per second respectively, ProbLog backward sampling 708 and 510, BLOG 3,008 and 2,900, and our new forward sampling (with rejection sampling) 760 and 731. The results using our algorithm for both rejection sampling and likelihood weighting with $Depth=0$ are shown in Figure \[fig:urnproblog\]. As the graphs show, the standard deviation for rejection sampling is much larger than for likelihood weighting.
Conclusions and related work {#sec:conclusions}
============================
We have contributed a novel construct for probabilistic logic programming, the distributional clauses, and defined its semantics. Distributional clauses allow one to represent continuous variables and to reason about an unknown number of objects. In this regard this construct is similar in spirit to languages such as BLOG and Church, but it is strongly embedded in a logic programming context. This embedding allowed us to propose also a novel inference method based on a combination of importance sampling and forward reasoning. This contrasts with the majority of probabilistic logic programming languages which are based on backward reasoning (possibly enhanced with tabling [@SatoKameya:01; @Mantadelis10iclp]). Furthermore, only few of these techniques employ sampling, but see [@Kimmig11] for a Monte Carlo approach using backward reasoning. Another key difference with the existing probabilistic logic programming approaches is that the described inference method can handle evidence. This is due to the magic set transformation that targets the generative process towards the query and evidence and instantiates only relevant random variables.
P-log [@Baral09] is a probabilistic language based on Answer Set Prolog (ASP). It uses a standard ASP solver for inference and thus is based on forward reasoning, but without the use of sampling. Magic sets are also used in probabilistic Datalog [@Fuhr00], as well as in Dyna, a probabilistic logic programming language [@Eisner05] based on rewrite rules that uses forward reasoning. However, neither of them uses sampling. Furthermore, Dyna and PRISM require that the exclusive-explanation assumption. This assumption states that no two different proofs for the same goal can be true simultaneously, that is, they have to rely on at least one basic random variable with different outcome. Distributional clauses (and the ProbLog language) do not impose such a restriction. Other related work includes MCMC-based sampling algorithms such as the approach for SLP [@cussensmc]. Church’s inference algorithm is based on MCMC too, and also BLOG is able to employ MCMC. At least for BLOG it seems to be required to define a domain-specific proposal distribution for fast convergence. With regard to future work, it would be interesting to consider evidence on continuous distributions as it is currently restricted to finite distribution. Program analysis and transformation techniques to further optimize the program w.r.t. the evidence and query could be used to increase the sampling speed. Finally, the implementation could be optimized by memoizing some information from previous runs and then use it to more rapidly prune as well as sample.
Acknowledgements {#acknowledgements .unnumbered}
================
Angelika Kimmig and Bernd Gutmann are supported by the Research Foundation-Flanders (FWO-Vlaanderen). This work is supported by the GOA project 2008/08 Probabilistic Logic Learning and by the European Community’s Seventh Framework Programme under grant agreement First-MM-248258.
[^1]: If $e$ contains assignments to continuous variables, $p(e)$ is zero. Hence, evidence on continuous values has to be defined via a probability density function, also called a sensor model.
[^2]: A rule can have an empty body, in which case it represents a deterministic fact.
|
---
bibliography:
- 'comment.bib'
---
**Comment on “General solution to the $U(1)$ anomaly equations" \[PRL 123, 151601 (2019)\]**
Ref. [@Costa_Dobrescu_Fox_2019] finds solutions to the anomaly equations $$\begin{aligned}
\label{anomalyeq}
\sum_{i=1}^n z_i=0,\; \sum_{i=1}^n z_i^3=0\end{aligned}$$ for $n$ charges $z_i
\in {\mathbb{Z}}$ in a $U(1)$ gauge theory. The solutions obtained there are general, except they omit solutions which do not correspond to chiral representations of $U(1)$, or which are obtainable by permuting charges, or concatenating solutions of lower $n$, or by adding zeros.
Here, we show that the ingenious methods of [@Costa_Dobrescu_Fox_2019] have a simple geometric interpretation, corresponding to elementary constructions long known to number theorists [@Mordell_1969]. Viewing them in this context allows a fully general solution (i.e. [*without*]{} the exceptions above) to be written down in one fell swoop. It also allows us to give a variety of other, qualitatively similar, parameterisations of the general solution, as well as a qualitatively different (arguably simpler) form of general solution for even $n$.
To put the method on a geometric footing, we first observe that, by clearing denominators, every solution in integers of (\[anomalyeq\]) is equivalent to a solution in the rational number field ${\mathbb{Q}}$. Eliminating $z_n$, we get the homogeneous cubic equation $$\begin{gathered}
\label{cubichyper}
\sum_{i=1}^{n-1} z_i^3-\left(\sum_{i=1}^{n-1} z_i\right)^3=0,\end{gathered}$$ defining a hypersurface in the projective space $\mathrm{P}\mathbb{Q}^{n-2}$. As well as identifying solutions differing by a common multiple, working projectively allows us to sidestep annoyances associated with points at infinity in affine space (such as the fact that that affine lines can be variously intersecting, parallel, or skew, while projective lines either intersect or are disjoint).
A result of antiquity tells us that a chord through two given rational points on a cubic hypersurface intersects the hypersurface in a third rational point, giving us a way of generating new solutions from old ones. (This is precisely the obscure construction called a ‘merger’ in [@Costa_Dobrescu_Fox_2019].) Further, a rather more recent (though equally elementary) result of Mordell [@Mordell_1969] shows that all rational points in a cubic [*surface*]{} can be constructed in this way, starting from two given disjoint lines in the surface. The generalisation to an arbitrary cubic [*hyper*]{}surface $X$ is immediate and gives the following : Let $\Gamma_1, \Gamma_2\subset X$ be disjoint hyperplanes of dimensions $d_1,d_2=m_o:= (n-3)/2$, if $n$ is odd and of dimensions $d_1=m_e := (n-2)/2$ and $d_2=m_e-1$ if $n$ is even. Every rational point $p\in \mathrm{P}\mathbb{Q}^{n-2}$ ([*ergo*]{} every $p \in X$) lies on a chord joining a point in $\Gamma_1$ to a point in $\Gamma_2$. : The result is obvious if $p \in \Gamma_2$. If $p\notin
\Gamma_2$, then $p$ and $\Gamma_2$ define a ($d_2+1$)-d hyperplane, which intersects $\Gamma_1$ in a point $p^1$. The line through $p$ and $p^1$ intersects $\Gamma_2$ in a point $p^2$, yielding a chord. QED.
In the case of interest, the (projective) line $L=\alpha_1 p^1+\alpha_2 p^2$ through $p^{1,2}$ with homogeneous parameter $[\alpha_1:\alpha_2] \in \mathrm{P}\mathbb{Q}^1$ intersects the cubic hypersurface $X$ defined by (\[cubichyper\]) when $$\begin{aligned}
\label{lineincube}
3\alpha_1 \alpha_2\sum_{i=1}^{n-1}\left(\alpha_1 p^2_i P^1_i+\alpha_2 p^1_i P^2_i\right)=0,\;
P^a_i:= (p^a_i)^2-\left(\sum_{j=1}^{n-1} p^a_j\right)^2. \nonumber\end{aligned}$$ Thus, along with the points $p^{1,2}$ (corresponding to $\alpha_{2,1}=0$) we get either a third rational point on $X$ at $$\begin{aligned}
[\alpha_1:\alpha_2]=\Bigg[\sum_{i=1}^{n-1}p^1_i P^2_i:-\sum_{i=1}^{n-1}p^2_iP^1_i\Bigg], \nonumber\end{aligned}$$ or, if the terms on the RHS both vanish, we have that every rational point on $L$ is on $X$.
To get a fully general solution, we just need to find suitable $\Gamma_1,\Gamma_2$. Many choices are possible; let us choose the one that makes closest contact with [@Costa_Dobrescu_Fox_2019]. To wit, $$\begin{aligned}
\Gamma^{e}_{1}&=[k_1: \cdots: k_{m_e}:-\tilde
l_1:-k_{1}:\cdots:-k_{m_e}]\nonumber \\
\Gamma^e_{2}&=[0:l_1: \cdots: l_{m_e}:-l_1:\cdots:-l_{m_e}] \nonumber \\
\Gamma^o_1 &=[k_1:\cdots:k_{m_o+1}:-k_1:\cdots:- k_{m_o+1}]\nonumber \\
\Gamma^o_{2}&=[l_2:\cdots: l_{m_o}:\tilde
k_1:0:-l_1:\cdots:-l_{m_0}:\tilde k_1]. \nonumber\end{aligned}$$ These parameterisations cover $\Gamma_{1,2}$, so by the Theorem yield all solutions of (\[anomalyeq\]). The parameterisations of [@Costa_Dobrescu_Fox_2019], in contrast, have $\tilde l_1=l_1$ and $\tilde k_1=k_1$; as a result they are unable to reproduce solutions in which one of the points $p^{1,2}$ in the construction is at infinity in affine space, e.g. $[0:1:-1:0]$ for $n=4$, which isn’t chiral and has zeros.
Two further remarks are in order. Firstly, as we have seen, our parameterization of the general solution is somewhat distasteful, in that occasionally the chord joining points on $\Gamma_{1,2}$ lies in $X$, and so yields not one, but infinitely many solutions. By repeating the construction with 2 different choices of $\Gamma_{1,2}$, one can ensure that every solution arises as a unique third point of intersection of a chord with $X$. One can even choose the original $\Gamma_{1,2}$ such that this can be achieved by simply permuting the co-ordinates $z_i$. Secondly, in the case where $n$ is even, a completely different, and arguably even simpler, construction of a general solution is possible. Indeed, in such cases, the cubic hypersurface has double points, where both the LHS of (\[cubichyper\]) and its partial derivatives vanish (e.g. the rational point \[+1:-1:…:+1\]). A line through such a double point intersects the cubic in one other rational point (or the line lies entirely in $X$) and thus all solutions can be obtained by constructing all lines through just a single double point, as it were.
This work was supported by STFC consolidated grants ST/P000681/1 and ST/S505316/1.\
\
\
B.C. Allanach$^{a,1}$, Ben Gripaios$^{b,2}$ and Joseph Tooby-Smith$^{b,3}$\
\
$^{a}$ DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom\
$^{b}$ Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge, CB3 0HE, United Kingdom\
\
$^{1}$ [email protected]\
$^{2}$ [email protected]\
$^{3}$ [email protected]
|
---
abstract: 'For the past 10 years there has been an active debate over whether fast shocks play an important role in ionizing emission line regions in Seyfert galaxies. To investigate this claim, we have studied the Seyfert 2 galaxy Mkn 78, using HST UV/optical images and spectroscopy. Since Mkn 78 provides the archetypal jet-driven bipolar velocity field, if shocks are important anywhere they should be important in this object. Having mapped the emission line fluxes and velocity field, we first compare the ionization conditions to standard photoionization and shock models. We find coherent variations of ionization consistent with photoionization model sequences which combine optically thick and thin gas, but are inconsistent with either autoionizing shock models or photoionization models of just optically thick gas. Furthermore, we find absolutely no link between the ionization of the gas and its kinematic state, while we do find a simple decline of ionization degree with radius. We feel this object provides the strongest case to date against the importance of shock related ionization in Seyferts.'
title: 'Ionization Mechanisms in Jet-Dominated Seyferts: A Detailed Case Study'
---
Introduction
==============
Ionization studies of Seyferts have a long history. Early work led to the establishment of nuclear photoionization as the favored NLR ionizing mechanism. But in the past decade or so, standard models have been called into question because, among other reasons, they strongly underestimate the strengths of many of the weaker high-ionization and high-excitation lines (see [@binette96; @robinson00] for a more complete discussion). This led to the development of alternative models, as well as refinements to standard nuclear photoionization. In particular, photoionizing shocks, driven by AGN jets and outflows, have emerged as a viable ionizing source, following work by [@viegas89] and [@dopita96].
We try to resolve this debate by taking the following approach: we choose a Seyfert with strong, NLR-wide jet-gas interactions. If shocks are important in providing the ionizing power in Seyferts, we should expect to see unambiguous signs of their presence in this object’s spectrum. If not, current refinements to nuclear photoionization can be tested.
Mkn 78 : A Jet-Gas Interaction Archetype
========================================
The Seyfert 2 Mkn 78 was selected as a target because it lies well off the virial correlation for Seyferts ([@whittle92 Whittle 1992]), indicating the presence of widespread non-gravitational motions in the ionized gas. This makes it one of the best candidates for a strong radio jet/ISM interaction among the sample of nearby Seyferts. [@paper1] discuss the structural aspects of the interaction in detail.
We use a extensive dataset consisting of HST-STIS longslit spectra from four slits sampling all the major emission line features in the NLR at high spatial resolution ($\sim 0.05$ arcsec). Our spectra give us almost complete FUV and optical wavelength coverage, allowing the measurement of many lines of different ionization state and excitation level. In addition, medium resolution ($\sim 30$ km/s) spectra allow us to accurately estimate the kinematics of the line emitting gas.
Ionization Mechanisms
=======================
We consider three types of ionization models to compare to the observations.
Standard Nuclear Photoionization
----------------------------------
Early photoionization models (e.g., [@davidson79 Davidson & Netzer 1979]) invoked a population of Lyman thick (ionization-bounded or IB) clouds illuminated by a power-law AGN ionizing continuum (of the form $F_{\nu}\propto \nu^{\alpha}$). Using CLOUDY ([@ferland96 Version 94.0, Ferland 1996]), we ran a set of constant density $\alpha=-1.0$ and $\alpha=-1.4$ models, with sequences in the ionization parameter $U=n_{i}/n_{e}=10^{-1}-10^{-3}$. In all cases, the $\alpha=-1.4$ model was the better match to the data.
Multi-Component Nuclear Photoionization : the $A_{m/i}$ Sequence
------------------------------------------------------------------
The $U$ models can be generalized by the introdution of matter-bounded or MB clouds that are optically thin to the Lyman continuum ([@binette96 Binette et al. 1996]). By varying $A_{m/i}$, the relative contribution of the two components to the spectrum, the range of observed emission line ratios can be reproduced for average Seyfert NLRs. Mkn 78, however, has unusually weak high-ionization lines and so we used CLOUDY to generate an $A_{m/i}$ sequence with the ionization parameter of the MB component reduced by a factor of 4 compared to the Binette et al. value, which allowed us to adequately match \[NeV\] and other high-ionization lines.
Shock models
--------------
We use the [@dopita96] shock models, in which the hot postshock gas generates a photoionized quiescent “precursor” region. In this way, shock models are inherently two-component in nature, but with a self-consistent prescription for the relative line contribution from postshock and precursor material. We use models with shock velocity $V_{sh}=200-500\;$km/s and magnetic parameter $B/\sqrt{n} = 0-2\;\mu G\;\textrm{cm}^{\frac{3}{2}}$. We have also included a more recent sequence of equipartition magnetic shocks from [@allen04], with $V_{sh}=200-1000$ km/s.
Methods of Analysis
===================
We employ a series of tests to search for signs of shock excited gas and/or check the consistency of the AGN photoionization paradigm.
Line-ratio vs. Line-ratio Diagrams
----------------------------------
Here, we use the well-known method from [@baldwin81] of plotting sets of line-ratios vs. other line-ratios and comparing the data to the predictions of models. We divide these diagrams into three basic types:
1. Excitation diagrams, such as \[OIII\]$\lambda 5007$/\[OII\]$\lambda
3727$ vs. \[OIII\]/H$\beta$ \[Fig. \[fig1\](*a*)\]. As expected, the data points lie in the intersection space of all the models, with implied $U\sim10^{-2}-10^{-3}$, $A_{m/i}\sim0.2-10.0$ and $V_{sh}\sim300-500$ km/s. However, trends in the data marginally support nuclear photoionization over shocks.
2. Shock discriminators, such as \[SII\]$\lambda 6720$/H$\alpha$ vs. \[OIII\]/H$\beta$ \[Fig. \[fig1\](*b*)\]. The shock sequences lie almost perpendicular to the photoionization sequences and the trends in the data clearly follow the $U$ and $A_{m/i}$ model loci.
3. U discriminators, which use line-ratios known to be troublesome for standard photoionization. Fig. \[fig1\](*c*) shows \[NeIII\]$\lambda 3868$/\[OII\] vs. \[NeV\]/\[NeIII\]. The U models predict far weaker \[NeV\] than is measured, even though this line is already unusually weak in Mkn 78.
We conclude that nuclear photoionization probably dominates the line emission processes in the NLR of Mkn 78, with a mixture of optically thin and optically thick gas needed to explain the range of excitation.
Line-ratios vs. Other Quantities
--------------------------------
We can measure a host of physical and dynamical properties from our spectra, which we then compare to model predictions. For example, the [@dopita96] shock models predict strong correlations between shock velocity and excitation tracing line-ratios like \[OIII\]/H$\beta$. On the other hand, the data seems to show absolutely no correlation between excitation and either bulk line velocity or FWHM. There does appear to be, however, a strong, significant drop in ionization state with distance from the nucleus, with \[OIII\]/\[OII\] $\propto r^{-1}$ \[Fig. \[fig1\](*d*)\]. A proper interpretation of this trend is, nevertheless, quite complicated. Here, we can only note that the radial trend is evidence for a nuclear ionizing field in the NLR.
Profile Comparisons
-------------------
A hallmark of two-component models is that the line spectrum of each component is radically different. In the case of shocks, most of the flux in high ionization lines is produced in the kinematically quiescent precursor, while the strongly disturbed postshock cooling region generates a low-ionization spectrum. Thus, it is reasonable to expect a shock-excited spectrum to show significantly different line kinematics between low-ionization lines, like \[NII\]$\lambda 6584$ and high ionization lines like \[OIII\]. This is applicable to the MB/IB scenario as well and can be used to set constraints on the level of co-spatiality of the two components.
To look for profile differences, we compared the \[OIII\] line profile to the profiles of a number of lines of different ionization species, after correcting for the wavelength shift and instrumental broadening. Only very minor differences were found, even in areas with strong signs of jet-gas interaction. We conclude that strong shocks are unlikely and components of different optical depths share a common velocity field.
Estimates of Ionizing Luminosity
--------------------------------
We can test for nuclear photoionization by comparing the UV ionizing flux of the AGN with the total emission line luminosity. Since the direct UV flux of the AGN is obscured, we use the FIR luminosity as a surrogate for the dust-reprocessed UV luminosity. From IRAS FSC measurements and estimates of geometrical and covering factors \[see [@paper2]\], we derive $L_{UV}\sim L_{FIR}\sim 10^{43.5}\textrm{ erg s}^{-1}$, which is approximately equal to the total emission line luminosity, taken to be about $10\times L_{5007}$. Clearly, an ionizing field from the AGN is sufficient to power the observed line emission in Mkn 78 and there is no need for any additional source of ionizing photons, such as fast shocks.
To conclude, all the evidence suggests that the principal ionizing source of Mkn 78, and possibly most Seyferts, is the central AGN, coupled with a realistic multi-component ionized gas distribution. The role of ionizing shocks are negligible. This is borne out in detail in the dynamical analysis of [@paper3].
Allen, M.G. 2004, private communication
Binette, L., Wilson, A.S., & Storchi-Bergmann, T. 1996, A&A, 312, 365
Baldwin, J.A., Phillips, M.M., & Terlevich, R. 1981, PASP, 93, 5
Davidson, K., & Netzer, H. 1979, Rev. Mod. Phys. 51, 715
Dopita, M.A., & Sutherland, R.S. 1996, ApJS, 102, 161
Ferland, G.J. 1996, [*Hazy, a Brief Introduction to Cloudy*]{}, University of Kentucky Department of Physics and Astronomy Internal Report.
Robinson, T.G., Tadhunter, C.N., Axon, D.J., & Robinson, A. 2000, MNRAS, 317, 922
Viegas-Aldrovandi, S.M., & Contini, M. 1989, ApJ, 339, 689
Whittle, M. ApJ, 387, 109
Whittle, M., & Wilson, A.S. 2004, AJ, 127, 606 (Paper I)
Whittle, M., Rosario, D.J., Silverman, J.D., Nelson, C.H., & Wilson, A.S. 2004a, AJ, submitted (Paper II).
Whittle, M., Silverman, J.D., Rosario, D.J., Nelson, C.H., & Wilson, A.S. 2004b, AJ, submitted (Paper III).
|
<span style="font-variant:small-caps;">Two-dimensional DTQWs in electromagnetic fields\
(Publications [@AD15] and [@AD16])</span> \[Chap:Electromagnetic\_2D\_DTQWs\]
=======================================================================================
Higher-dimensional (discrete-time) quantum walks {#sec:higher}
------------------------------------------------
\
There are several ways to extend DTQWs to higher-dimensional coin and/or node spaces. Historically, the first extensions to higher dimensions that we encounter in the litterature under the name of ‘quantum walks’ are those developed by quantum-algorithmics authors, in the early 2000’s, but particular cases of such extensions had actually already appeared before in the community of physics simulation, and more precisely, of cellular automata, in the 90’s, as we will detail below. Let us first come back a bit in time.
### From the Turing machine to Meyer’s and then Ambainis’s one-dimensional quantum walks
The principle of modern computers is the Turing machine, proposed by Turing, mathematician and one of the first computer scientists, in his seminal paper of 1936 [@Turing1937]. Cellular automata (CA) were then introduced by both the mathematician Ulam and the mathematician and physicist Von Neumann [@VonNeumann66], to simulate, with computing machines, (classical) physics governed by local rules. Feynamn, physicist, added his contribution to the computing era by proposing in 1982 a model to quantize these CA [@Feynman1982], which was formally developed by Deutsch [@Deutsch1985]. The term ‘quantum cellular automaton’ (QCA) appeared in a paper by Grössing and Zeilinger [@Grossing_Zeilinger_88] in 1988, although their model differs from that of Feynman and Deutsch. Meyer, mathematical physicist, entered quantum computing and quantum simulation with the aforementioned background. The main part of his work is nowadays still at this crossing between quantum simulation and quantum computing. In his seminal paper “From quantum cellular automata to quantum lattice gases” [@Meyer96a], Meyer shows how to obtain what are now called 1D DTQWs as a particular family of 1D reversible (i.e. unitary[^1]) QCA[^2]. In particular, he explains that imposing unitarity to homogeneous 1D QCA, results in a trivial dynamics. That is why [@Meyer96a] (i) Grössing and Zeilinger relax the unitarity constraint in their paper [@Grossing_Zeilinger_88], and (ii) Meyer weakens the homogeneity constraint, as follows. First, (ii.a) Meyer weakens the strict homogeneity[^3] to a one-every-two-cells homogeneity. To preserve unitarity, the only case that can be retained is that of ‘non-interacting’ pairs of cells, i.e. each pair evolves independently from the other pairs, with no quantum superpositions of pairs. This can be viewed as creating the spin degree of freedom, but each spin is still independent from the others. Second, (ii.b) Meyer adds to the model the alternate evolution (sometimes called partitioning or staggered rule), characteristic of DTQWs, and performed by what we refer to nowadays as the spin-dependent shift, which makes the pairs of cells (i.e. the spins) interact (i.e. superpose) and allow propagation to occur. He thus connects his work to Feynman’s checkerboard, recalling the link with the Dirac equation. Computational work on 1D DTQWs is then continued by computer scientists, the denomination ‘quantum walk’ appearing for the first time in a paper title in the paper “One-dimensional quantum walk” by Ambainis et al. [@Ambainis2001], in 2001.
### Higher dimensions in, essentially, a quantum-computing perspective: exponentional speedup of this research field in the 2000’s?
#### Higher-dimensional coin space in a combinatorial abstract node space {#subsubsec:higher_dim_node_space}
This 1D-DTQW model [@Ambainis2001] is then quickly extended to arbitrary $d$-regular graphs, by Dorit Aharonov, Ambainis, Kempe and Vazirani [@Aharonov2001]. In this model, the coin operation acts, at each of the $n$ nodes of the graph, on qudits (generalization of qubits i.e. spins-1/2), belonging to a $d$-dimensional coin space, and the qudit-dependent shift transports each component of the qudit to one of the $d$ different neighbors. In this work, the vertex (i.e. position) space is abstract, and is not connected to physical dimensions. The authors (i) define the basic tools to study the spreading properties of DTQWs on such graphs, among which the mixing time, and (ii) prove the existence of a disappointing lower bound in this mixing time: DTQWs can *mix* at most ‘only’ (almost) quadratically faster than their classical counterparts, classical random walks (CRWs) (mixing time of $O(n \log n)$ for DTQWs, vs. $O(n^2)$ for CRWs). This ‘bad news’ is then reexamined on the particular case of an hypercube[^4] by Moore and Russell [@Moore2002], who show that the same bound exists for continuous-time quantum walks (CTQWs). These results are however quickly followed by better news in a paper by Kempe [@Kempe2003]: DTQWs *hit* exponentially faster than CRWs (first evidence of exponential speedup enabled by DTQWs). Previous and similar but somehow weaker results were given for CTQWs by Fahri and Guttman in their seminal paper [@FG98a], and then reviewed by Childs and the two previous authors [@Childs2002]: these authors found an exponentially-smaller ‘local’ (*I* use this word) hitting time on some particular graphs.
#### Higher-dimensional physical node space
Shortly after the publication of this generalization of DTQWs by computer scientists, a different extension to higher dimensions was proposed by the physicists Mackay, Bartley, Stephenson and Sanders [@Mackay2002]. This extension is explicitly viewed as a way to move a discrete-time quantum walker in higher-dimensional physical spaces. For a $D$-dimensional physical space, the coin space is built as the tensorial product of $D$ two-dimensional coin spaces, and has thus dimension $2^D$. The spin-dependent shift then moves a walker located at some node, not to its $2D$ nearest neighbors in the ‘natural’ $D$-dimensional hypercubic lattice induced by the $D$ elementary qubit-dependent shifts, but to its $2^D$ $D$-th nearest neighbors, and the walker does thus not explore this whole ‘natural’ hypercubic lattice, but only one of the $D$ independent sublattices generated by such a transport (which are generically not hypercubic). For example in two dimensions, the $2$- (i.e. next-) nearest neighbors of the node $(0,0)$, are $(1,1)$, $(-1,-1)$, $(1,-1)$ and $(-1,1)$. In $D=2$, the number of nearest beighbors and of $D$-th nearest neighbors happen to be equal (to 4) so that, apart from the fact that there is a (unique) useless sublattice, the coin space has a dimension which *is* the minimal one to move a walker on a square lattice, which is the natural hypercubic lattice in $2D$ (square lattice); but this is just a coincidence: the lattice explored, although square, is *not* the original one induced by the $D=2$ elementary qubit-dependent shifts. This coincidence dissapears for $D>2$. In $D=3$, for example, the number of nearest neighbors is $2D=6 < 2^D=8$, the number of next-next-nearest neighbors. To fully explore a given cubic lattice, one could use the generalization described in the previous subsubsection, with a coin space having dimension $2D=6$. The sublattice explored by the present generalization is not cubic, but *body-centered cubic*, and this scheme is natural to explore such a lattice. Note that this body-centered cubic sublattice can only be $d=2^D$ regular either if it is infinite, i.e., in practice, in the bulk, or if we consider periodic boundary conditions, which makes the $D$-dimensional lattice a $D$-dimensional torus.
After recalling the definition of two equally-weighting transformations, namely the discrete Fourier transform (DFT) and the Grover operator (GO), the authors show that the entanglement between the spatial dimensions, produced by these transformations (thanks to their alternate action with the spin-dependent shift), can serve to reduce the rate of spread of the DTQW, with respect to an evolution by the $D$-tensorized Hadamard gate, a separable transformation which thus produces no entanglement between the spatial dimensions. Since the extensions to higher dimensions are done in a systematic way, the rate of spread always increases with $D$ for a given transformation.
This seminal paper about the spreading properties of DTQWs in higher *physical* dimensions, was followed by another study by Tregenna, Flanagan, Mail and Kendon [@Tregenna2003]. In this paper are presented short but conceptually comprehensive and systematic studies of some phenomenal spreading properties of quantum walks on low-degree graphs, from two to four, as well as some comparisons with CTQWs; it is in particular shown how, thanks to the coin degree of freedom, one can better monitor the propagation in DTQWs than in CTQWs. A notable work of this team formed by Tregenna and Kendon is that “Decoherence can be useful in quantum walks” on the line, the cycle and the hypercube [@Kendon2003], which is underlined in Kendon’s review of 2006 on quantum walks [@Kendon2006].
#### Quantum walks for spatial search and universal computation
The results described in the two previous sections regarding the spreading properties of quantum walks, are bricks with which to elaborate more complex algorithms. Spatial search is one of those.
##### $\bullet$ Early quantum algorithms, Grover’s search and followings
The recent review of 2016 by Giri and Korepin [@GK16], gives a nice straight-to-the-point introduction to quantum search algorithms derived from Grover’s.
Before getting into the precise subject, the authors present, through some of the early quantum algorithms, some of the essential features of quantum algorithms that may be used to speedup computing power (provided these algorithms run on a quantum computer). First, they describe the very basic Deutsch algorithm, which enables to find, in single query to the quantum oracle (quantum query), whether some function of a single qubit and taking at most two different values (Boolean function) is constant or balanced, while one needs two queries to a classical oracle (classical query) for such a task to be fulfilled; this illustrates quantum parallelism. Second, they show that this number of quantum queries does not grow with the number $k$ of input qubits[^5] if one uses the Deutsch-Jozsa algorithm[^6], while one needs $2^{k-1} +1$ classical queries in the worst case. Rather than the classical equivalent to the Deustch algorithm, it is more efficient to use, to fulfill the proposed task with large inputs on a classical computer, an algorithm which is similar but fulfills the job in only $d$ classical queries; the authors show that the quantum version of this algorithm, variation of the Deutsch-Jozsa algorithm called the Bernstein-Vazirani algorithm, ensures the same performance of a single quantum query. The authors do not discuss Shor’s famous quantum algorithm of 1994 [@Shor94; @Shor97], which is (i) not a search algorithm, and (ii) widely studied in the litterature. Shor’s algorithm enables prime-number factorization of an integer $n$ in a time (i.e. number of steps, there is no oracle here) $O((\log n)^3)$; we say that the algorithm is polynomial, with exponent 3, in the physical input size, which is $\log n$. To be precise, the algorithm actually runs in a time $O((\log n)^2(\log \log n) (\log \log \log n))$ (which is better than $O((\log n)^3)$). This is a big improvement with respect to the best known classical algorithm performing such a task, “Gordon’s adaptation” [@Shor97] of the so-called number field sieve [@Gordon1993], which does the job in a sub-exponential time $O(\exp(c(\log n)^{1/3}(\log \log n )^{2/3}))$.
Giri and Korepin then present in depth many variants of Grover’s famous spatial search algorithm of 1996 [@Grover96]; there is in particular a distinction between full and partial database search, the latter being achieved by the Grover-Radhakrishnan-Korepin algorithm. The basic important result presented in Grover’s seminal paper is the following: instead of the $O(n)$ classical queries that are needed to find a given element in a database of $n$ elements, Grover’s quantum-mechanical algorithm only needs $O(\sqrt{n})$ quantum queries, which is a quadratic improvement. It can be shown, and this is an important restrictive result to have in mind, that $O(\sqrt{n})$ is a lower bound for *any* quantum search algorithm [@Bennett1997; @Boyer1998; @Zalka1999].
##### $\bullet$ Spatial search with DTQWs: AA, SKW, AKR, AKR-Tulsi, MNRS
We have seen that Grover’s algorithm finds a given item among $n$ in a time $O(\sqrt{n})$, which is a lower bound for any quantum search algorithm, and any DTQWs-based search algorithm will *not* go below this bound. One of the aims of DTQWs-based search algorithms is to search only with local operations, Grover’s algorithm being “highly non-local” [@Shenvi2003]. The first DTQW-based search algorithm to reach the aforementioned lower bound is the SKW algorithm, suggested by Shenvi, Kempe and Whaley [@Shenvi2003] for a search on the hypercube (abstract space). This paper was then followed by the well-known AKR algorithm, by Ambainis, Kempe and Rivosh [@AKR2005], that manages to adapt the previous algorithm to hypercubic lattices of dimension $D$ (physical space, explored by standard lattices, which *have much lower spreading capacities than the hypercube of dimension $d= \log n$*), containing a number of nodes $n=( \! \sqrt[D]{n} )^D$, the optimal performance of $O(\sqrt{n})$ being reached for $D\geq 3$. In $D=2$, the running time is $O(\sqrt{n} \log n)$, and the authors show that it is sufficient to have a two-dimensional (instead of a four-dimensional) coin space to reach this lower bound. The AKR algorithm combines (i) the monitoring capacities of coins in DTQWs [@Tregenna2003; @Shenvi2003], and (ii) the scheme developed by Aaronson and Ambainis [@Aaronson2005], in order to surmount the difficulty of implementing Grover’s algorithm on a 2D grid[^7] (they also add to this scheme ‘amplitude amplification’ to increase its probability of success). These results [@Aaronson2005; @Shenvi2003; @AKR2005] were compiled and phrased in a highly simplified framework by Szegedy [@Szegedy], as quantum counterparts to classical Markov chains.
Further developments have focused on increasing the probability of success of the search (without increasing the time search). All these developments were used by Magniez, Nayak, Roland and Santha, to construct their MNRS algorithm [@Magniez2007], a simplified version of which is given (i) in Santha’s review of 2008 [@Santha08quantumwalk], which deals with many computing problems other than search, and (ii) in an updated version of [@Magniez2007], namely [@Magniez2011]. Some of the latest developments in increasing the probability of success are the 100% chances of success achieved in the SKW algorithm [@Potoek2009], and the time $O(\sqrt{n \log n})$ reached in the AKR algorithm in 2011 [@Ambainis2013] (thus disproving Szegedy’s “probably optimal" [@Szegedy] referring to the $O(\sqrt{n} \log n)$); this last result is achieved by managing to get rid of the amplitude amplification included in the original AKR algorithm, which demands an $O(\sqrt{\log n})$ running time, but with no need of any other modification of the original algorithm, as previously done by Tulsi [@Tulsi2008] and later by Krovi et al. [@Krovi2010] to reach the same $O(\sqrt{n \log n})$. In the first three pragraphs of his paper [@Tulsi2008], Tulsi gives a clear and fast sum-up of the ‘previous episodes’.
##### $\bullet$ Parallel and/or joint research with CTQWs: search problem and universal computation, by Childs and others
Here we report some of the key steps reached by CTQWs in the run ‘between’ continuous- and discrete-time quantum walks in both the quest for the optimization of the search problem and universal quantum computation. Quickly after Kempe’s result on the exponential hitting-time speedup provided by DTQWs, Childs et al. [@Childs2003] published a paper showing that CTQWs can be exponentially fast in oracle-type problems, in particular, search (Kempe’s result dealt with a non-oracular problem). Two years later, in 2004, Childs and Goldstone [@Childs2004] show via CTQWs that the lower bound of $O(\sqrt{n})$ can be reached on the hypercube and in the *‘physical’* search problem, for lattices of dimension $D \geq 5$, while in lower dimensions only $O(\sqrt{n} \, \mathrm{polylog} \, n)$ is reached, a worse performance than the solution provided one year later by Ambainis et al. with DTQWs [@Aaronson2005] and then by the AKR algorithm [@AKR2005], already discussed above.
While many quantum-computing teams were busy improving the AKR algorithm, from 2005 to 2008, Childs kept working with CTQWs, “perhaps easier to define” [@Childs2009], and published in 2009 his “Universal computation by quantum walk” [@Childs2009], showing CTQWs can be used as computational primitives for *any* quantum algorithm, and not only search problems. This paper was followed in 2012 by an extensive generalization to multi-particle quantum walks [@CGW13].
In 2006, Strauch precisely connected the continuous- and the discrete-time quantum walks [@Strauch06b], underlying that, “at least in \[the\] simple case”[@Strauch06b] he treats, the coin degree of freedom is irrelevant to the speedup provided by DTQWs-based algorithms, and that the speedup is only due to a “simple interference process” [@Strauch06b]. This work was followed by an in-depth generalization to arbitrary graphs by Childs [@ChildsCD2009]. In 2010, the results of the first paper by Childs on universal quantum computation via CTQW, were recast in the framework of DTQW, by Lovett et al. [@Lovett2010].
##### $\bullet$ Recent developments in quantum computing
In 2014, a 3-nearest-neighbors CTQW-algorithm has been designed for spatial search on graphene lattices [@FG2014; @Foulger_thesis_2014]. A recent work by Chakraborty, Novo, Ambainis and Omar [@Chakraborty2016], show, using CTQWs, that “spatial search by quantum walk is optimal for almost all graphs”, meaning that “the fraction of graphs of $n$ vertices for which this optimality holds tends to one in the asymptotic limit” [@Chakraborty2016].
A recent paper by Wong [@Wong2016] proves with CTQWs that the Johnson graph $J(n,3)$ supports fast search (i.e. in a time $O(\sqrt{n})$), thus generalizing the same results obtained previously for (i) $J(n,1)$ with the CTQW version of Grover’s algorithm by Childs and Goldstone [@Childs2004], and (ii) $J(n,2)$. Note that Johnson graphs were already used as support by Ambainis to develp his “Quantum walk algorithm for element distinctness” [@Ambainis04quantumwalk; @AmbainisElementDistinctness14; @Amb07a].
### Higher dimensions in the perspective of simulating physical phenomena
#### The community of non-linear physics and kinetic theory of gases: FHP, LBE, QBE
##### $\bullet$ On the history of fluid dynamics, up to the Boltzmann equation
To learn about the history of fluid mechanics “From Newton’s principles to Euler’s equations”, see the paper by Darrigol and Frisch at <https://www-n.oca.eu/etc7/EE250/texts/darrigol-frisch.pdf>. In his *General Principles of the Motion of Fluids*[^8], published in 1757, Euler derives two out of the three equations which are nowadays known as the Euler equations for inviscid (i.e. non-viscous) fluids.
These two equations are (i) the momentum conservation with local pressure and generic external forces, and (ii) the mass conservation. These original equations are the current most general ones for inviscid fluids[^9]. In particular, they *are* valid for compressible fluids. The question of the solutions of these equations is a different one. We have (i) five scalar unknowns, namely the three components of the Eulerian speed field, the pressure, and the density, and (ii) four scalar equations: three given by the momentum conservation, and one by the mass conservation (also called continuity equation). This means that even if we manage to make Euler’s seminal system linear (with, for example, some ansatz for the unknowns), it will be under-determined[^10], except for incompressible fluids, whose density is a constant. Euler already mentions, after summing up his 3+1 equations, that a fifth equation should exist[^11] between pressure, density, and, in Euler’s words “the heat of the fluid particle”, which is nowadays called the internal specific (i.e. per-unit-volume) energy, essentially related to the temperature of the fluid (proportional, for ideal gases). Thermodynamics developed later, in the whole 19th century, and at Euler’s time the distinction between the concepts of ‘heat’ and ‘temperature’ was at least not properly formalized.
The equation between these three quantities (which also involves the Eulerian speed field), is the energy-balance equation. A first relation of this kind enabled Laplace to make the first correct calculation of the sound speed [@L1816]. This condition was later recognized as (i) an adiabatic condition [@CM13], in the context of a developing thermodynamics, and, more precisely, as (ii) an isentropic[^12] condition (which is an idealized adiabatic condition), after Clausius introduced the concept of entropy in 1865 [@C1865]: indeed, the equation can simply be stated as the conservation of the entropy of any fluid particle.
To focus on the convective term of the momentum-conservation equation, which is the non-linear term, the pressure term is sometimes dropped, which leads to the (inviscid) Burgers equation. In addition to be physically motivated in some cases, this approximation often simplifies the search for solutions for the speed field.
After deriving the two aforementioned equations, Euler shows that, when the fluid is incompressible, the integral form of the momentum-conservation equation yields Bernouilli’s original theorem, published in 1738, and which was only valid for incompressible fluids, although nowadays one can write down a compressible version. For a pre-Eulerian history of partial differentiation, see the paper by Cajori at <http://www.math.harvard.edu/archive/21a_fall_14/exhibits/cajori/cajori.pdf>.
In his *Memoir on the Laws of the Motion of Fluids* of 1822, Navier [@N1822] introduced the notion of viscosity, but “he did not developed the concept of shear”[^13]. The present form of the Navier-Stokes (momentum-conservation) equation was written by Stokes in 1845[^14]. The Navier-Stokes equations describe the fluid motion at the so-called mesoscopic scale, which is much greater than the microscopic scale, but still much smaller than the macroscopic scale[^15].
The Boltzmann equation, derived in 1872 [@Boltzmann1872; @Boltzmann2012], describes the evolution of an out-of-equilibrium fluid (gas or liquid), which emerges at mesoscopic and macroscopic scales from the collisional dynamics of the classical microscopic particles it is made of. It is an equation on the (one-point) probability distribution function of the fluid particles. From this Boltzmann equation, one can derive[^16] [@M1890; @E17; @CC52a] mesoscopic-scale equations (such as the standard Navier-Stokes equation), by taking the so-called hydrodynamical limit[^17] [@L09], which essentially consists in a “patching together of equilibria which are varying slowly in space and time” [@FHP86]. A recent paper (2016) by Chow et al. [@Chow2016] gives an analytical solution for the $N$-dimensional compressible Euler equations with damping, for a barotropic (pressure only function of the density) fluid, with the pressure as function of the density given by a power law.
##### $\bullet$ The lattice Boltzmann equation (LBE)
The following few lines on the history of the lattice Boltzmann equation are strongly inspired from Scholarpedia’s sum-up on the subject, see <http://www.scholarpedia.org/article/Lattice_Boltzmann_Method>. Experts on the discretization of the Boltzmann equation noted that the first *simplified* discretizations of the Boltzmann equation, from the 60’s to the mid 80’s, were rather used to find analytically-tractable solutions of the Boltzmann equation, and not as alternatives to the discretizations of the Navier-Stokes equation in a numerical-computation perspective. The idea of simulating fluid dynamics with lattice-gas models was introduced in 1986 by Frisch, Hasslacher and Pomeau, with their well-known FHP lattice-gas cellular automaton [@FHP86]. This model runs, in dimension 2, on a hexagonal lattice, to fulfill an isotropy condition, unsatisfied by the preceding HPP automaton developed by Hardy, de Pazzis and Pomeau [@Hardy1973; @Hardy1976] in the 70’s; in 3 dimensions, a suitable lattice is given by the projection of a face-centered-hypercube of dimension 4 on one of the coordinate axis [@FHHLPR87]. The collision rules at each vertex are inspired by a discrete Boltzmann model [@Harris1966]. Here is a 1989 review on “Cellular automaton fluids" [@RajLakshmi1989]. Several pitflaws of the FHP automaton lead to the development, in the late 80’s and 90’s, of the so-called lattice Boltzmann methods[^18] [@succi2001lattice].
##### $\bullet$ The quantum (lattice) Boltzmann equation (QBE): a DTQW!
In 1992, Succi and Benzi published their “Lattice Boltzmann equation for quantum mechanics” [@Succi1993], inspired (in part) by the FHP automaton [@FHP86]. The central idea of this important paper is that “the non-relativistic Schrödinger equation ensues from the relativistic Dirac equation under the same conditions which govern the passage from the LBE to the Navier Stokes equation” [@Succi1993]. To accept this statement without further clarifications, recall that the LBE is a *simplified* discrete Boltzmann equation, and this simplification procedure gets rid of some microscopical degrees of freedom of the original Boltzmann equation. Of course, one still has to find a suitable discretization of the Dirac equation, and this is done via an operator-splitting technique. It is straightforward to implement in this scheme a minimal coupling to an external Abelian potential. In 2015, the (1+1)D QBE is explicitly identified with a (1+1)D quantum walk by Succi [@Succi2015], the a straightforward generalization to (1+3)D, with steps taken alternatively in the 3 spatial directions of the lattice, and not at the same time as in [@MBSS2002], which enables to write these equations sticking to the ‘physical’ low-dimensional coin spaces associated to spinors, as opposed to the ‘algorithmical’ ones, built by tensorial products of qubits [@MBSS2002].
#### From particle physics to cellular automata: Wolfram, ’t Hooft, Bialynicki-Birula, Yepez
After a PhD in particle physics, obtained in 1979 at the age of 20, Wolfram turned into complex systems and cellular automata as a tool to model them. He developed the formal-computation software Mathematica and the search engine Wolfram Alpha. His ideas and findings on cellular automata are summarized in his book *A New Kind of Science* [@Wolfram2002]. In the mid 80’s, Wolfram had already developed many ideas on cellular automata, which inspired other theoretical physicists. His seminal paper on “Cellular automata as models of complexity” [@Wolfram1984] is cited in the FHP-automaton paper [@FHP86].
In 1988, ’t Hooft introduced a cellular automaton to describe quantum sytems [@Hooft1988]. In 2014, he wrote a book on the *Cellular Automaton interpretation of Quantum Mechanics* [@Hooft2014]; for an online free version, go to <https://arxiv.org/abs/1405.1548>. In 1994, Bialynicki-Birula wrote the “Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata” [@BB94a].
#### Note
More authors and works are evoked in the introduction, and discussed more lengthily in the conclusion. The authors include the teams Cirac-Zohar-Reznik, Zoller’s, D’Ariano-Perinotti-Bisio, Arrighi-Facchini-Di Molfetta, Chandrasekhar-Busch, to cite only some of the members of each team. All these works deal with relativistic aspects. A huge amount of works deal with non-relativistic quantum simulation, including, among those working with cold atoms, Zoller’s team, the collaboration Bloch-Cirac, the team Meschede-Alberti, and that of Dalibard. Among the teams working with photons, let us mention the team Silberhorn-Sansoni and the team Mataloni-Sciarrino, to cite only a few.
### Hamiltonian formulations of DTQWs
### Experimental implementation of quantum walks
Driven quantum systems: theory and experiments
----------------------------------------------
### Driven quantum walks
Publications [@AD15] and [@AD16]: DTQWs in electromagnetic fields {#sec:two_publications}
-----------------------------------------------------------------
\
I give below a compact presentation of these two publications [@AD15; @AD16].
Let us first give some background: first, an extended ‘recap’ of the 1D case [@DDMEF12a; @DMD14] (Subsection \[subsec:recap\_1D\]), and second, the expression of the 2D DTQW-operator with electromagnetic coupling (Subsection \[subsec:walk\_2D\]). This 2D walk operator is the central ingredient of the two publications, whose initial purpose and first achievement is to extend the 1D results to dimension 2.
Recall that the the one-step evolution is given by Eq. (\[eq:protocol\]), that is (I make explicit the time-dependence of the walk operator, which is in order in the cases we are going to deal with): $$\label{eq:protocol_2}
\ket {\Psi_{j+1}} = \hat{W_j} \ket{\Psi_j}.$$
### State-of-the-art ‘recap’ on the 1D electric DTQW {#subsec:recap_1D}
In dimension 1, there can be no magnetic field[^19], and the 1D DTQW-operator with electric coupling reads (see Eqs. (\[eq:electric\_walk\_intro\])): $$\label{eq:1D_walk_operator}
W_j^{\text{1D}} = e^{i\Delta \alpha_j} U(\Delta \theta, \Delta \xi_j) \, S \, ,$$ where $\Delta \alpha_{j,p}$ is an overall (i.e. spin-independent) local (i.e. time- and position-dependent) phase shift, the spin-dependent shift $S$ is given by Eq. (\[eq:shift\_compact\]), and where I have introduced a 1D[^20] coin operation with mixing angle $\Delta\theta$ and spin-dependent local phase shift $\pm\Delta \xi_{j,p}$, $$U(\Delta\theta,\Delta\xi_{j,p}) = C(\Delta\theta) \, F(\Delta\xi_{j,p}) \, ,$$ with the standard coin operation $C(\theta)$ and the spin-dependent phase-shift operator $F(\xi)$, given respectively by Eqs. (\[eq:standard\_coin\]) and (\[eq:phase\_shift\]). We write the mixing angle and the phases as (see below why)
$$\begin{aligned}
\Delta\theta &= - \epsilon_m \, m \\
\label{eq:deltaalpha}
\Delta\alpha_{j,p} &= \epsilon_A \, (A_0)_{j,p} \\
\label{eq:deltaxi}
\Delta\xi_{j,p} &= \epsilon_A \, (A_1)_{j,p} \, . \end{aligned}$$
Note that $\Delta\theta$ could be chosen $(j,p)$-dependent.
The electric[^21] interpretation of this walk is threefold. The first interpretation is explained in depth in Subsection \[subsec:continuum\_limit\_inhomogeneous\][^22], the second is introduced in Subsection \[subsec:gauge\_inv\], and the third is a generalization, to an arbitrary electric field, of the interpretation adopted in Refs. [@Bauls2006; @mesch13a; @ced13] in the case of a non-generic constant and uniform electric field[^23]:
- First, considering $\Psi_{j,p}=\Psi(\epsilon_l \, \! j, \epsilon_l \, \! p)$, where $\epsilon_l=\epsilon_x=\epsilon_t$ is the spacetime-lattice step (the subscript ‘$l$’ is for ‘lattice’), choosing $\epsilon_m=\epsilon_A=\epsilon_l=\epsilon$[^24], and taking the continuum limit $\epsilon \rightarrow 0$, the one-step evolution equation (\[eq:protocol\_2\]) with walk operator (\[eq:1D\_walk\_operator\]) yields the (1+1)D Dirac equation for a relativistic spin-1/2 fermion of mass $m$ and charge $-1$, coupled to an electric potential with covariant time and spatial components $A_0$ and $A_1$, see Eq. (\[eq:Dirac\_coupled\]).
- Second, the walk satisfies the following gauge invariance [@DMD14] on the spacetime lattice with nodes labelled by $(j,p)$ (we have recast the results of Subsection \[subsec:gauge\_inv\]): Eq. (\[eq:protocol\_2\]) with the walk operator (\[eq:1D\_walk\_operator\]) is invariant under the following substitutions:
\[eq:discrete\_gauge\_inv\] $$\begin{aligned}
\Psi &\rightarrow \Psi' = \Psi e^{-i\phi} \\
\label{eq:discrete_gauge_inv_gauge_field}
A_{\mu} & \rightarrow A'_{\mu} = A_{\mu} - d_{\mu} \phi \ ,\end{aligned}$$
where $\phi_{j,p}$ is an arbitrary local phase change, and where we have introduced the following discrete derivatives (finite difference operators): $$\label{eq:finite_diff}
d_0 = (L - \Sigma_1)/\epsilon_A \ , \ \ \ \ \ \ \
d_1 = \Delta_1/ \epsilon_A ,$$ where, for any $(j,p)$-dependent quantity $Q$,
$$\begin{aligned}
(L Q)_{j,p} &= Q_{j+1,p} \\
(\Sigma_1 Q)_{j,p} &= \frac{Q_{j,p+1}+Q_{j,p-1}}{2} \\
(\Delta_1 Q)_{j,p} &= \frac{Q_{j,p+1}-Q_{j,p-1}}{2} \ .\end{aligned}$$
Note that $d_1$ is a standard finite difference in the spatial direction (defined over two sites), while $d_0$ can be viewed as the mean between two standard finite-differentiated convective derivatives (with Eulerian speed equal to $1$), one going forward, and the other backwards[^25]: $$(d_0Q)_{j,p} = \frac{1}{2 \epsilon_A} \Big[ \big\{(Q_{j+1,p}-Q_{j,p}) + (Q_{j,p}-Q_{j,p-1}) \big\} + \big\{ (Q_{j+1,p}-Q_{j,p}) + (Q_{j,p}-Q_{j,p+1}) \big\} \Big] \, .$$ In the continuum limit, the finite-difference operators $d_0$ and $d_1$ yield the standard partial derivatives $\partial_0$ and $\partial_1$, and the lattice gauge invariance (\[eq:discrete\_gauge\_inv\]) yields the standard gauge invariance of the Dirac equation in (1+1)D continuous spacetime.
Eventually, one can define the following lattice equivalent of the electric[^26] tensor in 1+1 dimensions, $$\label{eq:lattice_1D_strength_tensor}
(F_{\mu\nu})_{j,p} = (d_{\mu} A_{\nu})_{j,p} - (d_{\nu} A_{\mu})_{j,p} \ ,$$ where $(\mu,\nu) \in \{0,1\}$. This quantity is (i) antisymmetric by construction, (ii) invariant under (\[eq:discrete\_gauge\_inv\]), and (iii) its continuum limit yields the usual electric tensor. Moreover, we can check the following conservation equation on the lattice: $$\label{eq:current_cons}
d_0 J^0 + d_1 J^1 = 0 \ ,$$ where $J^{\mu}$ is invariant under (\[eq:discrete\_gauge\_inv\]) and has the same expression as the Dirac current in (1+1)D continuous spacetime (but is defined on the lattice).
- Third, the walk operator (\[eq:1D\_walk\_operator\]) can be obtained from the standard walk having only an angle $\Delta\theta\neq0$, i.e. with $\Delta\alpha_{j,p}=\Delta \xi_{j,p}=0$, by implementing a ‘Bloch phase’ $e^{i(\Delta\alpha_{j,p}\pm\Delta\xi_{j,p})}$ [^27], which reminds of that acquired by tight-binding electrons driven, in dimension 1, by a superimposed arbitrary electric field, where the $+$ is for the electron tunneling in one direction, and the $-$ for the other direction. To be precise on the tight-binding procedure in the presence of electromagnetic fields: one mathematically *adds* the *scalar* potential $\sim \Delta\alpha_{j,p}$ to the tight-binding Hamiltonian, and performs, to account for the presence of the *vector* potential, a Peierls substitution on the hopping (off-diagonal) matrix elements, namely $t \rightarrow t \, e^{\pm\Delta\xi_{j,p}}$, see Eq. (12) in [@Graf1995]. In the case of a constant and uniform electric field, this condensed-matter system yields the well-known Bloch oscillations of the electrons, with period inversely proportional to the electric field. Despite the phenomenal similarities between such a DTQW-scheme and the 1D tight-binding Hamiltonian with superimposed electric field, there are two *fundamental* differences. First: in the DTQW, the particle, which has a spin (or pseudo-spin), undergoes a spin-dependent transport, while in the tight-binding Hamiltonian, the left-right jump has *nothing* to do with spin, it is due to *tunneling*[^28]. Second, here (i) the Peierls flux exponential is implemented, not on the Hamiltonian, as standardly done with tight-binding Hamiltonians, but on the (one-step) evolution operator, and (ii) the implementation of the scalar potential is not equivalent either, although there is a closer matching[^29], because of the generic non-commutativity of the one-step evolution operators. Eventually, there is a third difference whose ‘importance’ (to define) I haven’t completely evaluated yet: in the DTQW-scheme, time is discrete, while it is continuous in tight-binding Hamiltonians.
### Walk operator for the 2D DTQW with electromagnetic (i.e. Abelian) coupling {#subsec:walk_2D}
Now the walker $\ket{\Psi_j}$ lives, in real space, on a 2D lattice with nodes $(p_1,p_2) \in \mathbb{Z}^2$.
We build the 2D DTQW operator by doing a 1D walk in the first spatial direction, with spin-dependent phase shift $\Delta\xi^{(1)}_{j,p_1,p_2}$, followed by another 1D walk but in the other spatial direction, with spin-dependent phase shift $\Delta\xi^{(2)}_{j,p_1,p_2}$[^30]: $$\label{eq:walk_operator_2D}
W_j^{\text{2D}} = e^{i\Delta \alpha_j} \, \, \left[ U \! \left(f^-(\Delta\theta), \Delta \xi^{(2)}_j\right) \, S^{(2)} \right] \, \,\left[ U \! \left(f^+(\Delta \theta), \Delta \xi^{(1)}_j\right) \, S^{(1)} \right] \, ,$$ where the mixing angle $\Delta \theta$ is encoded through $$f^{\pm}(\Delta\theta) = \pm \frac{\pi}{4} + \frac{\Delta \theta}{2} \, .$$
### Publication [@AD15]: Landau levels for discrete-time quantum walks in artificial magnetic fields {#subsec:B}
\
#### Short but detailed review and comments
In Section 2 of the paper, we introduce the above walk, Eq. (\[eq:walk\_operator\_2D\]), but *only* in the particular case $$\Delta\alpha_{j,p_1,p_2}=0 \ , \ \ \ \ \ \ \Delta\xi^{(1)}_{j,p_1,p_2} = 0 \ , \ \ \ \ \ \ \Delta\xi^{(1)}_{j,p_1,p_2} = B p_1 \epsilon \ ,$$ where $B \in \mathbb{R}$ is the parameter that is interpreted as a constant and uniform magnetic field in the continuum limit[^31], derived in Section 3 of the paper.
Note this technical formal point: the two coin matrices introduced in the paper do *not* correspond to the two $U$’s of the above equation, (\[eq:walk\_operator\_2D\]), but after expanding the matrix products in the paper, one can recast the equations exactly as (\[eq:walk\_operator\_2D\]). This is so simply because the correspondence between the DTQW-equations and the two matrices chosen to write these equations in a compact form is not one to one, but one to many[^32].
Section 4 is devoted to a study of how the non-vanishing step of the spacetime lattice modifies the eigenstates of the continuum-limit Hamiltonian, which are the well-known relativistic Landau levels. The study is perturbative in the step $\epsilon$, the zeroth order being the continuum situation. The zeroth-order eigen-energies are well-known; they vary as the square root of the level[^33]. There are two usual choices for the eigenstate basis of the Landau levels. We choose the so-called Hermite basis, for which the momentum along $y$ is a good quantum number[^34] (this is possible in the chosen Landau gauge); this choice is natural because translations are symmetries of the mesh on which the walker lives, and the walker ‘sees’ this mesh at higher orders. In Appendix B of the paper, we compute analytically the first-order corrections to (i) the eigen-energies (which vanish) and to (ii) the Hermite eigenvectors, and we numerically check in the body of the paper that these corrections are correct[^35] (see Figs. 4 and 5 of the paper). We also provide, in Fig. 3, a typical (relative) change induced by the corrections on the probabilities of presence of the five first Landau levels. These plots may be useful for, e.g., a comparison with future experimental data.
Section 5 aims at giving a flavor of the phenomena produced by our scheme well beyond the continuum limit (the perturbative computation can intrinsically only go, a priori, slightly beyond). We show that the parameter $B$ has the same qualitative confinement properties as a standard magnetic field in continuous spacetime.
In the conclusion (Section 6), we stress that, if we view the DTQW as a way to discretize the Dirac equation, (i) this discretization is a naive symmetric one, and suffers from fermion doubling, which happens at the boundaries of the Brillouin zone, but that (ii) this does not preclude the use of this discretization around the Dirac cone, located at the center of the Brillouin zone, i.e. for small vavectors (in comparison with the size of the Brillouin zone), which is the only limiting case where the Dirac-fermion interpretation, i.e. the continuum limit, makes sense.
### Publication [@AD16]: Quantum walks and discrete gauge theories {#subsec:EB}
\
#### Short but detailed review and comments
This is the paper where, in Section II, we give the proper compact form of the 2D DTQW-operator with generic electromagnetic coupling, Eq. (\[eq:walk\_operator\_2D\]). The continuum electromagnetic interpretation is given in Section II: the continuum limit of the scheme is that of a spin-1/2 Dirac fermion, with mass $m$ and charge $-1$, coupled to an electromagnetic potential with contravariant components $A_0 \sim \Delta\alpha$, $A_1 \sim -\Delta\xi^{(1)}$ and $A_2 \sim - \Delta\xi^{(2)}$, where, as for the 1D case, the symbol ‘$\sim$’ means that between the $A_{\mu}$’s and the phase shifts, there is a factor $\epsilon_A$ (see Eqs. (\[eq:deltaalpha\]) and (\[eq:deltaxi\])) which is taken equal to the spacetime-lattice step $\epsilon_l$ and goes to zero in the continuum, where the discrete coordinate $p_1$ ($p$ in the paper) becomes the continuous one $x$ ($X$ in the paper), and $p_2$ ($q$ in the paper) becomes $y$ ($Y$ in the paper). In addition to this formal continuum limit, shown on the evolution equation for the quantum particle, we show numerically that the limit is correct, as follows: we run the MQTD scheme with an initial condition whose time evolution through the Dirac equation (for constant and uniform crossed electric and magnetic fields) can be computed in closed form, and we compare this explicit continuum time evolution to the MQTD time evolution for decreasing spacetime-lattice steps, see Figs. 1 and 2 of the paper.
After a generalization, in Section III, of the 1D results on the lattice gauge invariance (Eq. (8) of the paper generalizes Eq. (\[eq:discrete\_gauge\_inv\])), and gauge invariants (Eq. (11) of the paper generalizes Eq. (\[eq:lattice\_1D\_strength\_tensor\])), we present, in Section 4, a lattice version of (the inhomogeneous[^36]) Maxwell’s equations (Eq. (20) of the paper), which ensure the current conservation on the lattice (Eq. (17) of the paper). These lattice Maxwell equations yield the standard ones in the continuum limit. This means we can theoretically quantum simulate, in the continuum limit, a first-quantized Abelian gauge theory: the spin-1 bosons[^37] evolve through Maxwell’s equations, and the fermions through the Dirac equation.
Section 5 is devoted to the phenomenal properties of the 2D electromagnetic DTQW beyond the continuum limit. A detailed discussion of the two different ‘small parameters’ is now in order. The first small parameter, $\epsilon_l$, is the spacetime-lattice step. The second small parameter, $\epsilon_A$, is that associated to the lattice gauge field. The continuum limit requires that both $\epsilon_l$ and $\epsilon_A $ tend to zero. But one can envisage taking only one of the two parameters as actually small, either (i) large wavefunctions, i.e. $\epsilon_l \simeq 0$, or (ii) a weak gauge field[^38]^,^[^39]:
1. If we are in situation (ii), we recover known phenomena. First, we recover the classical $\boldsymbol{E} \times \boldsymbol{B}$ drift [@book_Jackson], even if (i) is not satisfied – i.e. even if the walker ‘sees’ the edge of the Brillouin zone –, which is a priori non-obvious; see Figs. 4 (bottom propagating front) and 5 of the paper. Second, we recover tight-binding-like phenomena: Bloch oscillations for a vanishing magnetic field in Fig. 3, and top propagating front [@Kolovski03; @Kolovsky04; @KolovskyMantica14] for a non-vanishing magnetic field in Fig. 4. In contrast with the classical $\boldsymbol{E} \times \boldsymbol{B}$ drift, these tight-binding-like phenomena *demand* that (i) be unsatisfied at some point in the dynamics, i.e. that the walker ‘see’ the edge of the Brillouin zone[^40]. These tight-binding phenomena are qualitatively not that unexpected (see the third item in Subsection \[subsec:recap\_1D\]).
2. If (ii) is not satisfied, i.e. if the gauge-field components ‘see’ the edge of the Brillouin zone, i.e. reach sizeable fractions of $2\pi$, then the dynamics is strongly dependent on whether the electric and magnetic field are *exact* rational fractions of $2\pi$, see Figs. 6 and 7 of the paper. These features have already been studied in full mathematical depth in dimension 1, for a sole electric field, by Cedzich et al. (2013) [@ced13], and have been studied more recently in dimension 2 for a sole magnetic field by Yal[ç]{}[i]{}nkaya et al. (2015) [@Yalcinkaya2015], and, for a sole electric field, by Bru et al. (2016) [@Bru2016]. The results given by Cedzich et al. [@ced13] for the 1D DTQW in a constant and uniform electric field, are summarized in Table. of the paper, and very briefly summarized in Appendix .
[^1]: Unitarity implies reversibility, and the converse also holds in standard quantum mechanics, see the following discussion, <https://physics.stackexchange.com/questions/150733/unitarity-of-a-transformation-and-reversibility-imply-one-another>. The relaxation of the unitarity constraint is usually explicitly mentioned.
[^2]: Again (see above the definition I give for classical cellular automata), and as commonly done, the locality of the evolution rules is implied in what I call QCA.
[^3]: This means (I use Meyer’s notations) that at time $t+1$ and discrete position $x$ on the line, the state $\phi(t+1,x)$ of the QCA is obtained from the states which are at $t$ on positions $x-1$, $x$ and $x+1$, namely $\phi(t,x-1)$, $\phi(t,x)$ and $\phi(t,x+1)$, in a manner which is the same for each $\phi(t+1,x)$ on the space line; to be more precise, $\phi(t,x) = w_{-1}\phi(t-1,x-1) + w_{0}\phi(t-1,x) + w_{+1}\phi(t-1,x+1) $, with $w$’s that do not depend on $x$ at $t+1$ (for any $t$).
[^4]: The hypercube is widely used in algorithmics, to handle computations on bit strings. It is one of the regular graphs for which the number of nodes satisfies $n=2^d$. For any $d$-regular graph with $n$ nodes, the number of edges is $e=nd/2$, but this is still not sufficient to characterise the hypercube.
[^5]: More precisely, the input is a series of $k$ bits, namely a $k$-bit, and this $k$-bit is treated as a $k$-qubit by the algorithm, i.e. it is viewed as a basis vector of a $2^k$-dimensional Hilbert space.
[^6]: The funtion is said balanced if it takes one of the two possible values for exactly half of all the possible inputs, i.e. half of the possible $k$-bits.
[^7]: As he explicitly stated in the abstract of his paper, Benioff [@Benioff2002] considered null the possibility that the implementation of Grover’s algorithm on a 2D grid could outperform classical search on such a grid. As explicitly stated in the abstract of their paper, Aaronson and Ambainis [@Aaronson2005] disproved Benioff’s claim.
[^8]: A scan of the original manuscript, written in French and published by the Prussian Royal Academy of Sciences and Belles-Lettres, in Berlin, is available on The Euler Achive at <http://eulerarchive.maa.org//docs/originals/E226.pdf>. Ref. [@U08] is an English translation adapted from Burton’s by Frisch.
[^9]: This is true in the hytohesis of local thermodynamical equilibrium; otherwise, one can use, in some situations, the more general framework of so-called extended thermodynamics, a branch of non-equilibrium thermodynamics.
[^10]: For a general definition of over- and under-determination of a system of PDEs, see <https://www.ljll.math.upmc.fr/frey/cours/UdC/ma691/ma691_ch3.pdf>. The author of the following notes on the subject, <https://www.researchgate.net/file.PostFileLoader.html?id=56498d8b5cd9e3774f8b4569&assetKey=AS:296325611573251@1447660939976>, feels “we should not attach too much importance to this terminology”. One can put the Euler equations and, more generally, the Navier-Stokes equations, into the form of a quasilinear system.
[^11]: I wonder whether such a statement is made by thinking rather ‘physically’ or ‘mathematically’, thinking of the determination of linear systems; probably both. I should investigate on the thermodynamics of Euler’s time.
[^12]: Note that Kelvin’s circulation theorem holds for isentropic or barotropic fluids.
[^13]: See Huilier’s note on Navier at <http://www.daniel-huilier.fr/Enseignement/Histoire_Sciences/Histoire.pdf>.
[^14]: A scan of the original paper [@S1845] can be found at <http://www.chem.mtu.edu/~fmorriso/cm310/StokesLaw1845.pdf>.
[^15]: See <http://s2.e-monsite.com/2010/03/16/02/meca_flu.pdf>, Chap. 1, for a quick discussion on these three scales.
[^16]: Maxwell first realized this, see the following online scan of his *Scientific papers*, <http://strangebeautiful.com/other-texts/maxwell-scientificpapers-vol-ii-dover.pdf>.
[^17]: See the following notes by, respectively, Golse, <http://www.cmls.polytechnique.fr/perso/golse/Surveys/FGEcm04.pdf>, and Villani, <http://archive.numdam.org/article/SB_2000-2001__43__365_0.pdf>, which presents results obtained by Bardos, Golse, Levermore, Lions, Masmoudi and Saint-Raymond.
[^18]: A short review of the book by Succi [@succi2001lattice], written by Yeomans, is available at <http://physicstoday.scitation.org/doi/pdf/10.1063/1.1537916>.
[^19]: Unless one embeds this single dimension into a higher-dimensional space, which is sometimes called ‘effective 1D’, i.e. the motion is constrained to be one-dimensional but the actual physical space has a higher dimension. This is not the case here, the scheme is strictly 1D.
[^20]: In contrast with coin operations acting on coin spaces greater than 2 and are associated to multidimensional spin-dependent shifts.
[^21]: There is no magnetic field in this 1D scheme.
[^22]: This first interpretation is only valid in the limit of both (i) wavefunctions whose width is large with respect to the spatial-lattice step, and which vary over time periods which are large with respect to the time step, and (ii) a weak gauge field, i.e. $\epsilon_A|A^{\mu}_{j,p}| \ll 2\pi$ for $\mu = 0,1$ and for all $j$’s and $p$’s.
[^23]: The paper of 2006 by Bañuls et al. [@Bauls2006] shows that this constant and uniform electric field can be obtained through either a spatial or a temporal discrete derivative, but the discrete gauge invariance is not exhibited (which is linked to the fact that the case of an arbitrary electric field is not treated). In the ‘Electric-quantum-walk’ experiment with neutral atoms in a 1D optical lattice, by Genske et al. [@mesch13a], the electric field is obtained with the (discrete) *spatial* derivative; the paper by Cedzich et al. [@ced13] provides theoretical interpretations of the experimental results, which are far beyond the continuum situation, and deal with revivals of the initial state, due to the periodicity of the system (this periodicity dissapears in the continuum limit).
[^24]: If these small parameters differ from each other by proportionality factors only, it changes nothing to the equation obtained in the continuum limit except from such proportionality factors in front of either $A$, $m$, or the lattice coordinates. However, one of these small parameters can be, for instance, a power of another small parameter, possibly non-integer, which makes the continuum limit change; see Subsection \[subsec:continuum\].
[^25]: This was pointed out to me by D. Meschede.
[^26]: It contains no magnetic field.
[^27]: I use the same denomination as in [@mesch13a]: the so-called Bloch phase is that which implements the electric potential, namely, with contravariant components, $\sim (\Delta\alpha_{j,p}, \Delta\xi_{j,p})$.
[^28]: The DTQW-scheme is intrinsically chiral, and this has nothing to do with the presence or absence of a magnetic field. In the case of tight-binding Hamiltonians, one can include spin-orbit corrections [@Jaffe1987; @Barreteau2016] when a magnetic field is applied, but again the left/right jump has *nothing* to do with that, it is due to tunneling. (Note that the probabilities of going left or right by tunneling can be modified by a superimposed magnetic field *only* if the latter breaks the translational symmetry of the system; this might be a way of simulating a biased coin operation with such a condensed-matter system, or, rather, a simulation of this condensed-matter system.)
[^29]: To the exponential implemented on the DTQW one-step evolution operator, $e^{i\Delta\alpha_{j,p}}$, corresponds a scalar potential $\sim\Delta\alpha_{j,p}$, which is precisely that *added* (no multiplication like for the Peierls substitution) to the tight-binding Hamiltonian.
[^30]: Similar constructions can be found, in a quantum-computing perspective (if we are to categorize), in the AKR paper (2005) [@AKR2005] and in the paper by Di Franco et al. (2011) [@DiFranco2011], and, in the perspective of simulating quantum physics with cellular automata, in the papers by Succi et al. (1993) [@Succi1993], Bialynicki-Birula (1994) [@BB94a], Yepez (2002) [@Yepez2002Dirac], Strauch (2006) [@Strauch06a], and, more recently, Arrighi et al. (2015, 2016) [@Arrighi_higher_dim_2014; @AF16] and Succi et al. (2015) [@Succi2015]. Note the following papers by Yepez: these two rather old ones (2002) [@Yepez2002Schro; @Yepez2002Burgers], and this recent proceeding of 2016 [@Yepez2016].
[^31]: Outside the continuum limit, the confinement properties of $B$, exhibited in Section 5 of the paper, also endow $B$ with a phenomenal *qualitative* magnetic interpretation.
[^32]: When writing this paper we had not figured out the suitable compact form of Eq. (\[eq:walk\_operator\_2D\]) because we had not figured out the minimal possible gauge invariance on the lattice. That compact form is given in our next paper [@AD16]; it enables, in particular, a straightforward correspondence between the lattice gauge fields and those in the continuum.
[^33]: Have in mind that this square-root dependence is that expected for relativistic Dirac fermions or gapped graphene-like quasiparticles (which behave effectively exactly as the former), in contrast with the linear dependence expected for non-relativistic particles.
[^34]: The other eigenvector basis is the Laguerre basis, for which the good quantum number is a certain gauge-invariant angular momentum, see [@HRR93] and Appendix \[app:gauge\_invariant\_gen\] for an expression of the gauge-invariant generator of an arbitrary symmetry of the electromagnetic field.
[^35]: I have checked that, for the Laguerre basis at zeroth order, the DTQW-scheme converges when the lattice step goes to zero, but the convergence is not improved when I add the first-order corrections to the eigenvectors, maybe because these corrections only make sense if the rotational symmetry of space is not broken for a non-vanishing step, while this is *not* the case in our scheme, which uses a square lattice.
[^36]: The homogeneous ones are satisfied *by construction* on the lattice.
[^37]: The spin-1 boson is, in this (1+2)D case, $(A_0,A_1,A_2)$, and thus has three internal states.
[^38]: The weak-gauge-field condition is that needed to Taylor expand $\exp(i\epsilon_A A^{\mu}_{j,p_1,p_2})$ around zero, i.e. $\epsilon_A|A^{\mu}_{j,p_1,p_2}| \ll 2\pi$ for $\mu = 0,1,2$ and for any $(j,p_1,p_2)$. This condition thus enforces the variations of the $\epsilon_A A^{\mu}_{j,p_1,p_2}$’s between two points of the spacetime lattice to be much smaller than $2\pi$ as well. Conversely, the picture is different: choosing small variations, i.e., in our case, $\epsilon_A E$ and $\epsilon_A B$ $\ll 2\pi$, ensures the weak-gauge-field condition *only for some time*.
[^39]: One can obviously refine this picture by ‘dezooming’ only on either the temporal or the spatial lattice. Such a distinction is necessary for a mapping to CTQWs.
[^40]: The $\boldsymbol{E} \times \boldsymbol{B}$ drift is ‘blind’ to (i), while the tight-binding-like phenomena demand non-(i).
|
---
abstract: 'We solve the Kakeya needle problem and construct a Besicovitch and a Nikodym set for rectifiable sets.'
address: 'Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA'
author:
- 'Alan Chang, Marianna Csörnyei'
title: The Kakeya needle problem and the existence of Besicovitch and Nikodym sets for rectifiable sets
---
Introduction
============
Let $E\subset{\mathbb{R}}^2$ be a rectifiable set. Our aim in this paper is to show that the classical results about rotating a line segment in arbitrarily small area, and the existence of a Besicovitch and a Nikodym set hold if we replace the line by the set $E$. We will explain our results in more details below, but first we present two illustrative examples.
1. If $E$ is the graph of a convex function $f:\,{\mathbb{R}}\to{\mathbb{R}}$, our results imply the following: *E can be rotated continuously by $360^\circ$ covering only a set of zero Lebesgue measure, if at each time moment $t$ we are allowed to delete just one point from the rotated copy of $E$.*
2. If $f$ is not just convex but strictly convex, then: *E can be moved, using only translations, to any other shifted position, if at each time moment $t$ we are allowed to delete just one point from the translated copy of $E$.*
In the two examples given above, our movement $t\mapsto E_t $ is continuous, but the point $x_t \in E_t$ we delete cannot be chosen continuously. However, all our constructions in this paper are Borel.
In the first example, if we take $E$ to be a general rectifiable set, the result still holds, but instead of a single point, we need to delete an ${\mathcal{H}}^1$-null subset of $E$ (see ). For the generalization of the second example to rectifiable sets, see .
In the first example above, $\bigcup_t (E_t\setminus \{x_t\})$ is Lebesgue null. Therefore, $\bigcup_t (E_t\setminus \{x_t\})$ is a *Besicovitch set*: in each direction it contains not just a “unit line segment of the line $E$” but a whole copy of the set $E$ except for one of its points.
On the other hand, since $\bigcup_t E_t$ has non-empty interior, we can cover ${\mathbb{R}}^2$ by taking a countable union of copies of $\bigcup_t E_t$. Therefore, the countable union of copies of $\bigcup_t(E_t\setminus \{x_t\})$ is a *Nikodym set*: it has measure zero, and through each point $x\in {\mathbb{R}}^2$, it contains a copy of the set $E$ with one point removed.
For the case when $E$ is a line, see, e.g., [@mattila15] for both classical and recent results.
The Kakeya needle problem for general sets has been studied before. In [@chl] the authors introduced the following definitions: a planar set $E$ has the *Kakeya property* if there exist two different positions of $E$ such that $E$ can be moved continuously from the first position to the second in such a way that the area covered by $E$ along the movement is arbitrarily small. A planar set $E$ has the *strong Kakeya property* if it can be moved in the plane continuously to any other shifted or rotated position in a set of arbitrarily small area.
In [@chl] it is shown that if $E$ is a closed connected set that has the Kakeya property, then $E$ must be a subset of a line or of a circle. Moreover, if $E$ is an arbitrary closed set that has the Kakeya property, then the union of the non-trivial connected components of $E$ must be a subset of parallel lines or of concentric circles. In [@hl] the authors show that short enough circular arcs of the unit circle possess the strong Kakeya property.
Let us consider a related question for circular arcs: can we translate a full circle continuously to any other position covering arbitrarily small area, if at each point of the translation, we are allowed to delete an arc of the circle of a given length? How long must the deleted arc be? Because of rotational symmetry, the question of which circular arcs have the strong Kakeya property is equivalent to this one, as long as we choose the deleted arc piecewise continuously.
In this paper, we will answer the “piecewise continuous question” for an arbitrary rectifiable set $E$ of finite ${\mathcal{H}}^1$-measure in the following way: we only need to delete points whose tangent directions lie in a small interval.
Let us state our results precisely. We will use the following notation and terminology.
We let ${\mathbb{P}}^1 \simeq {\mathbb{R}}/\pi{\mathbb{Z}}$ denote the set of all directions in ${\mathbb{R}}^2$. We will use the standard embedding of ${\mathbb{R}}^2$ into the projective plane ${\mathbb{P}}^2$, so that ${\mathbb{P}}^2 = {\mathbb{R}}^2 \cup {\mathbb{P}}^1$. The arc-length metric on the unit sphere $S^2$ together with the quotient map $S^2 \to {\mathbb{P}}^2$ gives us a metric on ${\mathbb{P}}^2$. Let $({\mathbb{P}}^2)^*$ denote all the lines in ${\mathbb{P}}^2$. We denote by $|\cdot|$ the Lebesgue measure on ${\mathbb{R}}^2$ or ${\mathbb{P}}^1$, and by ${\mathcal{H}}^1$ the $1$-dimensional Hausdorff measure on ${\mathbb{R}}^2$. As usual, $B(x,r)$ denotes the open ball centered at $x$ of radius $r$, and $B(S, r)$ denotes the open $r$-neighborhood of a set $S$. We denote by $\operatorname{cl}S$ the closure of $S$. We write $A \lesssim B$ to mean $A \leq CB$ for some absolute constant $C > 0$.
Recall that every rectifiable set $E$ has a tangent field, which is defined for ${\mathcal{H}}^1$-almost every $x \in E$ (see ). We let $\theta_x \in {\mathbb{P}}^1$ denote the tangent of $E$ at $x$, and we let $\nu_x \in ({\mathbb{P}}^2)^*$ denote the normal line of $E$ at $x$. (The direction of $\nu_x$ is the one orthogonal to $\theta_x$.) Note that $\nu_x$ is the normal *line* passing through the point $x$, and not just a normal vector.
We will start by proving the following theorem:
\[theorem:translations\] Let $E \subset {\mathbb{R}}^2$ be a rectifiable set of finite ${\mathcal{H}}^1$-measure. Let ${{\varepsilon}}> 0$ be arbitrary. Then between the origin and any prescribed point in ${\mathbb{R}}^2$, there exists a polygonal path $P = \bigcup_{i=1}^n L_i$ with each $L_i$ a line segment, and for each $i$ there exists a direction $\theta_i\in{\mathbb{P}}^1$, such that $$\label{eq:theorem-translations}
|\bigcup_{i}\bigcup_{p\in L_i} (p+\{x\in E:\,\theta_x\not\in B(\theta_i,{{\varepsilon}})\})|< {{\varepsilon}}.$$
Although the tangent field of a rectifiable set is defined only ${\mathcal{H}}^1$-almost everywhere, for the statement of (and for all other results in this paper), we need to define it pointwise. We will show that regardless of which pointwise representation we choose, the results remain true (see ).
has an immediate corollary:
\[corollary:circle\] If we remove an arbitrary neighborhood of two diametrically opposite points from a circle, the resulting set can be moved continuously to any other position in the plane in arbitrarily small area.
This strengthens the previously known result [@hl] that sufficiently short circular arcs have the strong Kakeya property.
We note that does not handle the classical Kakeya needle problem: clearly it is not possible to translate a line segment to every other position in small area. We can still apply with $E$ a line segment, but since every point of $E$ has the same tangent direction, it allows us to delete the entire line segment at every point $p \in P$. To obtain a more meaningful statement for line segments, we need to consider what happens if we allow rotations as well as translations.
In order to unify translations and rotations, it is helpful to consider the projective plane ${\mathbb{P}}^2$. We can consider a translation in direction $\theta \in {\mathbb{P}}^1$ to be a “rotation” around the infinite point $\theta^\perp \in {\mathbb{P}}^1 \subset {\mathbb{P}}^2$ (see ).
We need to generalize the notion of a polygonal path from a path in ${\mathbb{R}}^2$ to one in $\operatorname{Isom}^+({\mathbb{R}}^2)$, the space of all orientation-preserving isometries of ${\mathbb{R}}^2$. (This space is also known as the *special Euclidean group* $SE(2)$.) The polygonal path in can be viewed as a sequence of vectors, each indicating in which direction and how far to translate. Then, a polygonal path of rotations should be a sequence of rotations, indicating around which point and how much to rotate.
Specifying a sequence of rotations is slightly trickier than a sequence of translations: when we rotate a set around a point, the centers of all the other rotations move. To avoid this problem, we will find it much more convenient to specify our sequence in the *intrinsic coordinate system*. That is, with $\rho_i$ denoting rotations around $z_i\in{\mathbb{R}}^2$, our continuous movement will be to rotate first with center $z_1$, then with center $\rho_1(z_2)$, and so on.
Our *polygonal path* $P$ will be specified by the *intrinsic sequence* $\rho_i$, but it will still lie in the space $\operatorname{Isom}^+({\mathbb{R}}^2)$, and its points will be isometries not in the intrinsic but in the standard coordinate system. For each sequence $\{\rho_i\}$ we obtain a $P=\bigcup_i L_i$. For each “line segment” $L_i$ in $P$, the rotations in $\{p'\circ p^{-1} : p, p' \in L_i\}$ all have the same center. (It is important to remember that this center depends not only on $z_i$ but also on the previous rotations.)
Also, we find it much more convenient to specify a rotation not by the point that we rotate around, but by the image of this point in the projective plane when we embed ${\mathbb{R}}^2$ into ${\mathbb{P}}^2$. We will call this the *projective center* of $\rho$ (both for translations and rotations). First we will prove a preliminary result (see ). The exact statement is quite technical, but essentially says that instead of using translations, we can move our set $E$ using rotations whose projective centers are almost aligned: if we want to connect $\rho\in\operatorname{Isom}^+({\mathbb{R}}^2)$ to the identity map by a polygonal path, we can choose a line $\ell\in({\mathbb{P}}^2)^*$ that passes through the center of $\rho$, and choose the (intrinsic) rotations so that their projective centers lie in $B(\ell, {{\varepsilon}})$. We also obtain, for each $i$, a $u_i \in \ell$ such that: $$\label{eq:throw-out-belt}
|\bigcup_i\bigcup_{p\in L_i}p(\{x\in E:\, \nu_x \cap \ell \cap B(u_i, {{\varepsilon}})=\emptyset\})|<{{\varepsilon}}.$$
can be viewed as a special case of by taking $\rho$ to be a translation and then taking $\ell$ to be ${\mathbb{P}}^1$. In this case, the centers lie on $\ell$, not just in an ${{\varepsilon}}$-neighborhood of $\ell$. The reason we need an ${{\varepsilon}}$-neighborhood for rotations is that, unlike for translations, the composition of a rotation around $z_1$ and a rotation around $z_2$ does not equal a rotation around a point on the line through $z_1,z_2$. (Recall the centers are specified with intrinsic coordinates.) This makes the statement and the proof of more complicated than those of . We will need careful error estimates on how far the centers move, and, consequently, how large area the set $E$ covers during its movement. The classical Kakeya needle problem, i.e., rotating a line segment in arbitrarily small area, is an immediate corollary of .
and provide a new insight into the result mentioned at the beginning of the introduction, that the non-trivial connected components of a closed set with the Kakeya property can be covered by parallel lines or by concentric circles [@chl]. It turns out that the key property of the line and the circle is that they are homogeneous: by rotating around the center of the circle, any sub-arc can be mapped onto any other sub-arc of the same length, by a continuous movement that covers only zero area. The same is true for lines with shifts. Therefore our piecewise continuous deletion of the line segments/sub-arcs in / can be replaced by a continuous one. No rectifiable set except for the union of parallel lines or concentric circles has this property.
\[unbounded\] The set $E$ in needs to be bounded. Take, for example, $E$ to be a union of countably many circles with centers $z_i$ and radius $r_i$, such that $\sum {r_i} < \infty$ and $\sum r_i |z_i| = \infty$. Then it is a rectifiable set with finite ${\mathcal{H}}^1$-measure, but every continuous rotation with a fixed center covers infinite area, even with a normal line removed. However, for the limit version (explained below), in which the centers of the rotations no longer need to be piecewise constant along the path, we can drop the boundedness condition.
Next, in , we study what happens in the limit as ${{\varepsilon}}\to 0$. By taking a sequence of ${{\varepsilon}}$ tending to zero, the balls $B(u_i, {{\varepsilon}})$ shrink to a single point in ${\mathbb{P}}^2$, and the area covered shrinks to zero. We obtain in the limit a continuous movement $P \subset \operatorname{Isom}^+({\mathbb{R}}^2)$ such that the set $E$ covers only *zero* area, where at each time moment we only need to delete a subset of ${\mathcal{H}}^1$-measure zero (see ). The resulting set of zero area is an analogue of a Besicovitch set for $E$.
Consider, e.g., the special case where there is a line $\ell \in ({\mathbb{P}}^2)^*$ such that there is a neighborhood of $\ell$ in which no two normal lines of $E$ intersect. Then says that we can rotate $E$ continuously by $360^\circ$, covering a set of zero Lebesgue measure, where at each time moment, we only need to delete *one point*. This happens, e.g., in the special case when $E$ is the graph of a convex function; by choosing the line $\ell$ to lie below the graph, there is a neighborhood of $\ell$ where no two normal lines meet.
If $E$ is strictly convex, then we can apply with $\ell={\mathbb{P}}^1$ and hence translate $E$ to any arbitrary position in the plane in a set of zero Lebesgue measure, deleting one point at each time moment. For moving a circle, we can choose any line $\ell$. In this case we need to delete, at each time moment, not just one but two diametrically opposite points of the circle, since they have the same normal line.
By the continuity of $P$, we can construct, from these Besicovitch sets, analogues of Nikodym sets, using the technique outlined at the beginning of this introduction. We state these more precisely in .
There is not only one continuous $P$, but residually many, in the sense of Baire category (see ). For results of similar nature when $E$ is a line, see, e.g., [@korner] and [@cchk].
It is well-known that there are no sets in ${\mathbb{R}}^n$ ($n\geq 2$) which have measure zero and contain a circle centered at every point. Stein first proved this for $n \geq 3$ by his estimates on spherical maximal functions [@stein]. Bourgain and Marstrand independently showed the same non-existence result holds for $n = 2$ around the same time [@bourgain; @marstrand]. Bourgain’s paper actually treats smooth curves with non-vanishing curvature. More work has been done on such curves, e.g., [@mitsis; @wolff97; @wolff00].
The non-existence results concern placing a copy of $E$ around every point in ${\mathbb{R}}^2$. For our Nikodym result, we instead place a copy of $E$ *through* every point of ${\mathbb{R}}^2$. With this change, such a construction is now possible.
We do not know whether the sizes of the sets we delete are sharp. While tells us that we only need to delete an ${\mathcal{H}}^1$-nullset of $E$ at each time moment, perhaps it is possible to delete much fewer points than specified by the theorem. (For more precise information on the size of the sets we delete, see also , , and .)
For example, it would be interesting to know whether it is possible to translate a circle in a set of Lebesgue measure zero, deleting only one point at every time moment.
Main ideas of the proof of Theorems \[theorem:translations\] and \[theorem:rotations-on-line\]
==============================================================================================
Our proof of relies on two key ideas.
The first key idea
------------------
Our first key idea is the “small neighborhood lemma”: suppose we move a compact set $E \subset {\mathbb{R}}^2$ along a path of isometries $P \subset \operatorname{Isom}^+({\mathbb{R}}^2)$. If we perturb $P$ by a small amount, the area covered by the perturbed movement will not increase very much because the new region covered is contained in a small neighborhood of the original. This simple and obvious fact turns out to be extremely useful.
\[lemma:small-nbhd-small\] Let $E \subset {\mathbb{R}}^2$ be any compact set, and let $P$ be an arbitrary path in $\operatorname{Isom}^+({\mathbb{R}}^2)$. Then for every ${{\varepsilon}}> 0$ there exists a neighborhood $U \subset \operatorname{Isom}^+({\mathbb{R}}^2)$ of $P$ such that $$|\bigcup_{p \in U} p(E)| \leq |\bigcup_{p \in P} p(E)| + {{\varepsilon}}.
$$
The second key idea
-------------------
The second key idea is more technical, so we give only an informal presentation here and defer the precise details to and . First, note that:
\[lemma:luzin\] For any polygonal path $P\subset {\mathbb{R}}^2$ and for an arbitrary $E\subset{\mathbb{R}}^2$, if we translate $E$ along $P$, then the area covered is $\lesssim{\mathcal{H}}^1(E){\mathcal{H}}^1(P)$.
If $B$ is a ball of radius $r$, where $r$ is smaller than the line segments in the polygonal path, then for each line segment $L\subset P$, by translating $B$ along $L$ we cover a set of area $\lesssim r{\mathcal{H}}^1(L)$. Adding these up for all line segments $L$ and approximating $E$ by a union of small balls, we obtain .
\[remark:luzinrem\] shows that in our proof of we can ignore small subsets of $E$, since in the movements these will cover only small area. Also, we can ignore small subsets of $P$.
Our second key idea is the simple observation that the estimate in can be improved if we also take into account the directions of the tangents of $E$. For simplicity, suppose that $E$ is a $C^1$ curve. Then we can cover $E$ with thin rectangles that approximate the curve. Each thin rectangle $R$ has the property that translating $E \cap R$ along a line segment $L$ in the direction of the long side of $R$ covers area $\lesssim \delta{\mathcal{H}}^1(L)$, where $\delta$ is the length of the short side of the rectangle. If the rectangle is thin enough, then this is a much better estimate than the estimate ${\mathcal{H}}^1(E\cap R){\mathcal{H}}^1(L)$ that we obtain from .
For general rectifiable sets $E$, instead of thin rectangles, we will choose $R\subset E$ such that $\theta_x$ is almost constant on $R$. The key idea remains the same (see ).
Combining the key ideas
-----------------------
We combine these two “key ideas” to construct polygonal paths in a Venetian blind-type construction. (For Venetian blinds, see, e.g., [@falconer2014 Theorem 6.9] or [@mattila15 Lemma 11.8].) Again, we give an informal presentation. See for the precise details.
The method is as follows. Suppose that $E$ is a $C^1$ curve, which we cover by thin rectangles. Suppose our initial path is a translation along a horizontal segment. Let $R, R'$ be two rectangles from our cover and let $\theta, \theta'$ be the directions of their long sides, with $\theta \neq \theta'$.
1. First, we replace our horizontal segment by a zigzag so that every other segment has direction $\theta$. Then $R \cap E$ will cover small area when translated along these segments.
2. Now we repeat the previous step, replacing each segment in direction $\theta$ with a new zigzag such that every other segment has direction $\theta'$. Then $R' \cap E$ will cover small area when translated along the segments in direction $\theta'$. Furthermore, if we make these new zigzags sufficiently “fine” (many turns and small enough segments), then these zigzags will remain close to the segments of direction $\theta$ that we just replaced. Then by the small neighborhood lemma, $R \cap E$ also covers small area when moved along the segments in direction $\theta'$.
By the end of step (2), we now have a “Venetian blind.” The line segments in direction $\theta'$ are the “good” segments, because translations along these segments cover small area for both rectangles $R$ and $R'$. By iterating with more angles, we can increase the number of rectangles for which translations along the good segments cover small area.
We also need the total length of the remaining “bad” segments to be strictly smaller than the initial segment, so that the size of the bad segments tends to zero when we iterate the Venetian blind construction. (For this reason, we cannot deviate too far from the initial horizontal direction. This leads to condition .) After sufficiently many iterations, we can ignore the bad segments by .
The main ideas of the proof of , where we use rotations, are similar. As in the proof for translations, we combine the small neighborhood lemma with the covering of $E$ by sets $R$ such that rotating $R$ around an appropriate point $z$ covers only a small area. We still use a Venetian blind construction, but now our zigzags will be in $\operatorname{Isom}^+({\mathbb{R}}^2)$. The general ideas of the argument are the same, but, as we explained in the introduction, they will require more delicate estimates than for translations.
Preliminaries
=============
Tangents of rectifiable sets {#sec:rectifiable}
----------------------------
Recall that a set $E\subset{\mathbb{R}}^2$ is called *rectifiable* if ${\mathcal{H}}^1$-a.e. point of $E$ can be covered by countably many $C^1$ curves. For any two $C^1$ curves, their tangent directions agree at ${\mathcal{H}}^1$-a.e. point of their intersection. Therefore, there exists a tangent field to $E$, i.e., a map $x \mapsto \theta_x$ from $E$ to ${\mathbb{P}}^1$ such that for any $C^1$ curve $\Gamma$, the tangent direction to $\Gamma$ at $x$ agrees with $\theta_x$ for ${\mathcal{H}}^1$-a.e. $x \in \Gamma \cap E$. This gives one of the (many equivalent) descriptions of a tangent field of a rectifiable set.
Of course, the tangent field is uniquely defined only up to an ${\mathcal{H}}^1$-null subset of $E$. That is, if we change the tangent field along a set that meets each $C^1$ curve in a set of ${\mathcal{H}}^1$-measure zero, it is still a tangent field.
In order to prove and also our other results, we fix a particular tangent field $x \mapsto \theta_x$ on $E$ as follows: first we fix a subset $E'\subset E$ of full ${\mathcal{H}}^1$-measure and a cover of $E'$ by countably many $C^1$ curves $\{ \Gamma_i \}$. Next, for each $x \in E$, if all the curves $\Gamma_i$ that go through $x$ have the same tangent direction at that point, then we let $\theta_x \in {\mathbb{P}}^1$ be that direction. (This also defines the normal line $\nu_x$ at $x$.)
Consider the set of those $x \in E$ where our $\theta_x,\nu_x$ either (1) are not defined, or (2) are defined but do not agree with the $\theta_x$, $\nu_x$ from the statements of our theorems. This is a set of zero ${\mathcal{H}}^1$-measure; hence we can ignore it by when we work with translations, and we will be able to ignore it by (below) when we work with rotations.
Rotations {#rott}
---------
We denote by $\operatorname{Isom}^+({\mathbb{R}}^2)$ the space of all orientation preserving isometries of ${\mathbb{R}}^2$. Each element of $\operatorname{Isom}^+({\mathbb{R}}^2)$ is either a translation by a vector $v$, or a rotation around a point $z\in{\mathbb{R}}^2$ by angle $\phi$. Using complex notation, such a rotation is the map $u\mapsto e^{i\phi}(u-z)+z$. The image of $0$ under this mapping is $z(1-e^{i\phi})$, so it is natural to denote $$\label{v}
v:=z(1-e^{i\phi}).$$ We can see from that $v=z(-i\phi+O(\phi^2))$. We denote $$w=\begin{cases}
z\phi & \phi\neq 0\\
iv& \phi=0.\\
\end{cases}$$ The motivation behind our notation is that, for small $\phi$ and near the origin, the rotation acts, to first order, like translation by $-iw$.
Both translations and rotations can now be specified by an ordered pair $(w, \phi) \in {\mathbb{R}}^2 \times {\mathbb{R}}$. (We have $\phi=0$ for translations.) From now on, we will refer to translations as rotations as well.
For $\rho\neq\operatorname{id}$, we define the *projective center* of $\rho$ to be the image of $(w, \phi)$ under the quotient map ${\mathbb{R}}^3 \setminus \{0\}\to{\mathbb{P}}^2$. We still use $(w, \phi)$ to denote the image in ${\mathbb{P}}^2$. If $\phi \neq 0$, then, using homogeneous coordinates, this reduces to $(z\phi,\phi)=(z, 1)$, as expected. If $\phi = 0$, then $(w, 0) = (iv, 0)=(v^\perp,0)$, which is indeed the point at infinity orthogonal to the direction of $v$.
Even though we now view translations as rotations around infinite points, translations and rotations are still different, even when viewed in ${\mathbb{P}}^2$. For example, a rotation with angle $\phi\neq 0$ fixes just one point in ${\mathbb{P}}^2$ (its projective center) whereas a translation fixes an entire line (${\mathbb{P}}^1\subset{\mathbb{P}}^2$).
We will use the notation $\rho(w,\phi)$ for a rotation whose projective center is $(w,\phi)\in{\mathbb{P}}^2$ and whose angle is $\phi$. That is, we assign to each point $x=(x_1,x_2,x_3)\in {\mathbb{R}}^3\setminus\{0\}$ the rotation $\rho(x)$ whose:
- projective center is the image of $(x_1,x_2,x_3)$ under the projection ${\mathbb{R}}^3\setminus\{0\} \to{\mathbb{P}}^2$;
- angle is the last coordinate $x_3$.
We will use the same notation $\rho=\rho(w,\phi)$ for the mapping $\rho : {\mathbb{R}}^2\to {\mathbb{R}}^2$ and for the continuous movement that rotates ${\mathbb{R}}^2$ around a point. For example, if $\phi=2\pi$ then the former is the identity mapping and the latter is not. It will be always clear from the context which one we mean.
The “second key idea” for translations {#sec:second-key-idea-lines}
--------------------------------------
We fix a small $\delta>0$, and a direction $\theta$. Let $R$ be a subset of $E$ such that $|\theta_x-\theta|\lesssim\delta$ for every $x\in R$. Our aim is to estimate how large area we cover if we translate $R$ by a vector $v$ of direction $\theta$.
For each $x\in R$ there is a $C^1$ curve $\Gamma_i$ from that goes through the point $x$. We choose a decomposition $R=\bigcup R_i$ such that $R_i\subset\Gamma_i$ for each $i$. Then locally, i.e., in a neighborhood of $x\in R_i$, $\Gamma_i$ is the graph of a Lipschitz function $f$ in the $(\theta,\theta^\perp)$ coordinate system, with Lipschitz constant $\lesssim\delta$. Without loss of generality we can assume that $\theta=0$.
Now, when we translate $R_i$ by the horizontal vector $v$, for each fixed $t\in{\mathbb{R}}$ we obtain $\#\{x\in{\mathbb{R}}:\,f(x)=t,\,(x,f(x))\in R_i\}$ many (not necessary disjoint) horizontal line segments on the line $y=t$, each of length $|v|$. Therefore, by Fubini’s theorem, the area covered is $$\leq |v|\int \#\{x\in{\mathbb{R}}:\,f(x)=t,\,(x,f(x))\in R_i\}\,dt\lesssim \delta{\mathcal{H}}^1(R_i)|v|,$$ where the second inequality follows from the co-area formula (see [@federer Theorem 3.2.22]) and the fact that $f$ has Lipschitz constant $\lesssim \delta$. Summing over $i$, we obtain the following:
\[lemma:variation\] Let $\delta>0$ be sufficiently small, and let $\theta$ be an arbitrary direction. Let $R$ be a subset of $E$ such that $|\theta_x-\theta|\lesssim \delta$ for every $x\in R$. Then if we translate $R$ by a vector $v$ of direction $\theta$, the total area covered is $\lesssim \delta{\mathcal{H}}^1(R)|v|$.
Note that $|\theta_x-\theta|\lesssim \delta$ if and only if $\nu_x$ meets a $\lesssim \delta$-neighborhood of $\theta^\perp$ in ${\mathbb{P}}^1$. Using this observation, we generalize to rotations in the next section.
The “second key idea” for rotations {#sec:second-key-idea-rot}
-----------------------------------
Let $z\in{\mathbb{R}}^2$, and let $\phi$ be an arbitrary angle. If we rotate the set $E$ around the center $z$ by angle $\phi$, then each point $x\in E$ moves along a circular arc of length $|x-z||\phi|$. Therefore, the trivial estimate we get is that by rotating $E$, the area covered is $$\label{trivirot}
\leq|\phi|\int_{\mathbb{R}}r\,\#\{x\in E:\,|x-z|=r\}\,dr\le |\phi|\int_E |x-z|\,d\,{\mathcal{H}}^1(x).$$ Here we used the fact that if we parametrize the curve $\Gamma_i$ by arc-length, the mapping $t\mapsto |x(t)-z|$ is Lipschitz, with Lipschitz constant at most 1.
For a general rectifiable set, the right-hand side of can be infinite (cf. ). From now on, in this section we assume that $E$ is bounded. More precisely, we assume that $E\subset B(0,r)\subset{\mathbb{R}}^2$ (here, we used the Euclidean metric). We will show that there is a constant $c$ (that depends only on $r$) such that the following two lemmas hold.
\[lemma:luzin-rot\] Let $\rho=\rho(y)$ be a rotation and let $R\subset E$ be arbitrary. Then, if we rotate $R$ by $\rho$, the area covered is $\lesssim c {\mathcal{H}}^1(R)|y|$.
\[var2\] Let $\delta>0$ be sufficiently small (depending on $r$). Let $\rho=\rho(y)$ be a rotation with projective center $z$. Let $R\subset E$ be such that, for each $x\in R$, $\nu_x\cap B(z,\delta)\neq \emptyset$. (Here, the ball $B(z, \delta)$ is defined with respect to the metric on ${\mathbb{P}}^2$.) Then, when we rotate $R$ by $\rho$, the area covered is $$\lesssim c\delta{\mathcal{H}}^1(R)|y|.$$
Let $y = (w, \phi)$. By , we know that holds (with $c=1$) when $\rho$ is a translation. Now suppose that $z\in{\mathbb{R}}^2$ and $\phi\neq 0$. Then there is a constant $c_1$ (that depends only on $r$) such that $|x-z|\le r+|z|\le c_1\sqrt{1+|z|^2}$ for every $x\in E$. Since $|y|=\sqrt{|z|^2\phi^2+\phi^2}=|\phi|\sqrt{1+|z|^2}$, therefore follows from the trivial estimate , with $E$ replaced by $R$, and $|x-z|$ replaced by $c_1\sqrt{1+|z|^2}$.
Let $y = (w, \phi)$. First, suppose that $z \in {\mathbb{R}}^2$ and $\phi \neq 0$. We note that we can improve the estimate by noticing that the derivative of $t\mapsto |x(t)-z|$ is $\langle \dot x(t),\frac{x(t)-z}{|x(t)-z|}\rangle = \frac{1}{|x(t)-z|}\operatorname{dist}(\nu_{x(t)},z)$. (Here, $\operatorname{dist}$ denotes the Euclidean distance.) Therefore, rotating the set $R$ covers area $$\label{eq:rotate-R-bound}
\leq |\phi|\int_R \operatorname{dist}(\nu_x,z)\,d {\mathcal{H}}^1(x).$$
Thus, it suffices to show that if $\nu_x$ intersects the $\delta$-neighbourhood of $z$ in ${\mathbb{P}}^2$, then $\operatorname{dist}(\nu_x,z) \leq c\delta\sqrt{1+|z|^2}$ in ${\mathbb{R}}^2$. If $|z| \leq 2r$ and $\delta$ is sufficiently small, then the Euclidean and projective distances are comparable, and $\sqrt{1+|z|^2}$ is comparable to 1 (where the implied constants depend only on $r$), so there is nothing to prove. Now, suppose $|z| > 2r$. Since $E\subset B(0,r)$, therefore, for $\delta$ sufficiently small, the projective ball $B(z, \delta)$ is bounded away from $x\in E$. Let $\pi$ denote the quotient map $\pi:\,S^2\to{\mathbb{P}}^2$. Then there is a constant $c_2$ (that depends only on $r$) such that the angle between any two great circles through $\pi^{-1}x$ that meet $\pi^{-1}(B(z,\delta))$ is $\le c_2\delta$. Then there is a constant $c_3$ (that depends only on $r$) such that the angle between $x-z$ and $\nu_x$ in ${\mathbb{R}}^2$ is $\le c_2c_3\delta$. Therefore $\operatorname{dist}(\nu_x,z)\le c_2c_3\delta|x-z|\le c_1c_2c_3\delta\sqrt{1+|z|^2}$, as desired.
Now, we prove when $z \in {\mathbb{P}}^1$ (i.e., when $\rho$ is a translation). Again, for $\delta$ sufficiently small, the projective ball $B(z,\delta)$ is bounded away from $E$. Thus, if $\nu_x$ intersects $B(z, \delta)$, then the angle between $\nu_x$ and ${\mathbb{P}}^1$ is bounded away from zero. Therefore, there is a constant $c_4$ (that depends only on $r$) such that if $\nu_x$ intersects $B(z, \delta)$ for some $x \in E$, then the projective distance between $z$ and $\nu_x \cap {\mathbb{P}}^1$ is $\leq c_4 \delta$. Hence, we can apply to obtain our desired result.
Kakeya needle problem for translations {#sec:translations}
======================================
In this section $E$ is an arbitrary rectifiable set of finite ${\mathcal{H}}^1$-measure. Without loss of generality we assume that ${\mathcal{H}}^1(E)=1$, and that $\theta_x$ is defined for each $x\in E$, as in .
Notation
--------
We say that a subset of ${\mathbb{P}}^1$ is an *interval* if it is connected. For $\theta_1,\theta_2\in {\mathbb{P}}^1$ with $|\theta_1-\theta_2|<\pi/2$, we denote by $[\theta_1,\theta_2]$ the interval in ${\mathbb{P}}^1$ whose endpoints are $\theta_1,\theta_2$ and has length less than $\pi/2$. (When we use this notation, we do not specify which one is the left and which one is the right endpoint.) The symbol ${{\bf i}}$ will always denote a finite binary sequence, i.e., a sequence $i_1i_2\dots i_k$, where $k \geq 0$ and each term $i_j$ is 0 or 1. The length of ${{\bf i}}$ is denoted by $|{{\bf i}}|$. We denote ${{\bf i}}'=i_1i_2\dots i_{k-1}$ (note that $\emptyset'$ is not defined). We will say that ${{\bf i}}'$ is the *parent* of ${{\bf i}}$, and ${{\bf i}}$ is a *child* of ${{\bf i}}'$, respectively. The *ancestors* and the *descendants* of an ${{\bf i}}$ are defined in the obvious way. We will also say that a sequence is *bad* if it ends with a $0$ and *good* if it ends with a $1$. (The empty sequence of length 0 is also good.)
Basic zigzag {#sec:basic-zigzag-lines}
------------
A *basic zigzag* is a polygonal path which is made up of $N$ congruent and equally spaced segments in direction $\theta_0$ interlaced with $N$ congruent segments in direction $\theta_1$.
The fundamental procedure in our construction is taking a line segment $L$ and replacing it with a basic zigzag with the same endpoints. The key properties of basic zigzags are the following two geometrically obvious facts:
- With $L, \theta_0, \theta_1$ fixed, we can ensure that the basic zigzag lies in an arbitrarily small neighborhood of $L$ by making the zigzag sufficiently “fine,” i.e., making $N$ sufficiently large.
- The total length of each of the two parallel pieces of the basic zigzag depends only on $L, \theta_0, \theta_1$ and not on the fineness of the zigzag.
Venetian blind {#sec:venetian}
--------------
Like the basic zigzag, a Venetian blind is a polygonal path of line segments of two fixed directions. These segments are constructed by iterating the basic zigzag construction. We fix a line segment $L$, small parameters $\beta \geq \gamma > 0$ and a sign $\pm$. The Venetian blind construction is as follows.
Let $\theta_L\in{\mathbb{P}}^1$ denote the direction of $L$. In our first step, we replace $L$ by a basic zigzag with directions $\theta_L\mp\beta$, $\theta_L\pm\gamma$. Let $G_1$ denote the union of the line segments of the basic zigzag in direction $\theta_L\pm\gamma$. Iteratively, in our $i^{th}$ step for $i\ge 2$, we replace each line segment in $G_{i-1}$ by a basic zigzag of directions $\theta_L\mp\beta$, $\theta_L\pm i\gamma$, and let $G_i$ denote the union of the line segments in direction $\theta_L\pm i\gamma$. We stop this procedure after $k$ steps, where $k$ is defined by $$\label{rat}
k\gamma\in[\pi/2-2\beta-\gamma,\pi/2-2\beta).$$
The zigzag we end up with is what we call our *Venetian blind*. We denote by $L_1$ the final set $G_k$ obtained by this construction, and we denote by $L_0$ the rest of the Venetian blind. That is, the Venetian blind is the polygonal path $L_0\cup L_1$ where $L_0$ and $L_1$ are unions of line segments of directions $\theta_L\mp \beta$ and $\theta_L\pm k\gamma$ respectively. We call $L_0$ the *bad* part of the Venetian blind and $L_1$ the *good* part.
Usually, in the literature, a Venetian blind consists only of the “good” line segments. In our definition of a Venetian blind, it contains both $L_0$ and $L_1$.
\[remark:Venetian-blind-lengths\] The lengths of $L_0$ and $L_1$ depend only on ${\mathcal{H}}^1(L)$ and $\beta,\gamma$. They *do not* depend on the fineness of the zigzags. Furthermore, condition ensures that there is a constant $c(\beta)<1$ such that $$\label{ratt}
{\mathcal{H}}^1(G_j)\le {\mathcal{H}}^1(L),\ {\mathcal{H}}^1(L_i)\le c(\beta){\mathcal{H}}^1(L)$$ for each $j=1,2,\dots,k$ and $i=0,1$.
Main construction {#section:proof-of-translations}
-----------------
Our strategy of proving is to iterate the Venetian blind construction. Given a point in ${\mathbb{R}}^2$, we construct a polygonal path from the origin to this point. We start with the line segment $L_\emptyset$ joining the origin to this point, and then, iteratively, for each finite sequence ${{\bf i}}$, we apply the Venetian blind construction to each segment in $L_{{{\bf i}}}$ with some parameters $\beta = \beta_{{\bf i}},\gamma=\gamma_{{\bf i}}, \pm = \pm_{{\bf i}}$. We let $L_{{{\bf i}}0}$ be the union of all the bad parts of the Venetian blinds and $L_{{{\bf i}}1}$ be the union of all the good parts (as defined in ). Since we use the same $\beta_{{\bf i}}, \gamma_{{\bf i}}, \pm_{{\bf i}}$ on each line segment in $L_{{{\bf i}}}$, it follows by induction that every $L_{{\bf i}}$ is a union of parallel line segments of some direction $\theta_{{\bf i}}$.
We also iteratively assign, to each ${\bf i}$, an interval $I_{{\bf i}}\subset{\mathbb{P}}^1$ by the following simple method. We put $I_\emptyset=\emptyset$. Then, for each finite sequence ${{\bf i}}\neq{\emptyset}$, we define $I_{{{\bf i}}}:=I_{{{\bf i}}'}\cup[\theta_{{\bf i}},\theta_{{{\bf i}}'}]$. Then clearly, by induction, we can see that for every ${{\bf i}}\neq\emptyset$, $I_{{{\bf i}}}$ is an interval and $\theta_{{{\bf i}}}\in I_{{\bf i}}$.
Choosing the parameters $\beta_{{\bf i}},\gamma_{{\bf i}}$ {#sec:choose}
----------------------------------------------------------
For each ${{\bf i}}$, we fix a small ${{\varepsilon}}_{{\bf i}}$ that we will specify later. They will depend only on ${{\varepsilon}}$ (where ${{\varepsilon}}$ is from the statement of ). We denote the number of 1’s in the sequence ${{\bf i}}$ by $n_{{\bf i}}$. Then we can choose our parameters $\beta_{{\bf i}}\ge \gamma_{{\bf i}}>0$ in our Venetian blind constructions such that they satisfy:
1. $\beta_{{{\bf i}}}\le\beta_{{{\bf i}}'}$ for every ${{\bf i}}$;
2. $\beta_{{{\bf i}}}=\beta_{{{\bf k}}}$, where ${{\bf k}}$ is the last (i.e., youngest) good sequence among ${{\bf i}}$ and its ancestors;
3. $\beta_{{\bf i}}\le 1/n_{{\bf i}}$ for every ${{\bf i}}$;
4. $\beta_{{{\bf i}}}{\mathcal{H}}^1(L_{{{\bf i}}})\le {{\varepsilon}}_{{{\bf i}}}$ if ${{\bf i}}$ is good;
5. $\gamma_{{\bf i}}{\mathcal{H}}^1(L_{{{\bf i}}})\le {{\varepsilon}}_{{{\bf i}}}$ for every ${{\bf i}}$.
We can indeed make these choices, since ${\mathcal{H}}^1(L_{{{\bf i}}})$ is determined by the $\beta$’s and $\gamma$’s of its ancestors.
We will also use the notation:
1. $\alpha_{{\bf i}}=\beta_{{{\bf i}}'}$ if ${{\bf i}}$ is bad, and $\alpha_{{\bf i}}=\gamma_{{{\bf i}}'}$ if ${{\bf i}}$ is good.
Choosing the signs $\pm_{{\bf i}}$
----------------------------------
We choose each sign $\pm_{{\bf i}}$ such that it makes $I_{{{\bf i}}1} = I_{{\bf i}}\cup [\theta_{{\bf i}}, \theta_{{{\bf i}}1}]$ as large as possible. That is, if $\theta_{{\bf i}}$ is in the right half of the interval $I_{{\bf i}}$ (where we embed $I_{{\bf i}}$ into ${\mathbb{R}}$), then we choose the $(+)$ sign; otherwise, we choose the $(-)$ sign. (If ${{\bf i}}=\emptyset$, or if $\theta_{{\bf i}}$ is in the middle of the interval $I_{{\bf i}}$, or if $I_{{\bf i}}={\mathbb{P}}^1$, then we can choose the sign arbitrarily.) Our choice of $\pm$ ensures that the length of the interval $I_{{{\bf i}}1}$ can be estimated by $$|I_{{{\bf i}}1}|\ge |I_{{{\bf i}}}|/2+|\theta_{{\bf i}}-\theta_{{{\bf i}}1}|\ge |I_{{\bf i}}|/2+\pi/2-2\beta_{{\bf i}}.$$ The second inequality follows from . This can be re-written as: $$\pi-|I_{{{\bf i}}1}|\le(\pi-|I_{{\bf i}}|)/2+2\beta_{{\bf i}}.$$ Suppose ${{\bf k}}$ is the last good sequence among ${{\bf i}}$ and its ancestors, and ${{{\bf m}}}$ is the second-to-last one. Then since the intervals $I_{{\bf i}}$ are increasing and the parameters $\beta_{{\bf i}}$ are decreasing along each family line, we have $\pi-|I_{{{\bf i}}}|\le \pi-|I_{{{\bf k}}}|$ and hence $$\label{I}
\pi-|I_{{{\bf i}}}|\le(\pi-|I_{{{\bf m}}}|)/2+2\beta_{{{\bf m}}}.$$
Stopping time {#sec:stopping-time}
-------------
We need to define when we stop our Venetian blind constructions on various family lines. In order to construct our polygonal path $P$ (for ), we need to ensure that ultimate extinction occurs. We will, of course, define the polygonal path $P$ as the union of those $L_{{\bf i}}$ where the construction stops.
First of all, we stop our Venetian blind construction at $L_{{\bf i}}$ if ${\mathcal{H}}^1(L_{{{\bf i}}})\le{{\varepsilon}}_{{\bf k}}$, where ${{\bf k}}$ is the last good sequence among ${{\bf i}}$ and its ancestors. By and condition (2) in , $${\mathcal{H}}^1(L_{{\bf i}})\ge c(\beta_{{\bf i}})^{-1}{\mathcal{H}}^1(L_{{{\bf i}}0})\ge c(\beta_{{\bf i}})^{-2}{\mathcal{H}}^1(L_{{{\bf i}}00})\ge\dots$$ This ensures that, for each ${{\bf i}}$, the family line ${{\bf i}},{{{\bf i}}0},{{{\bf i}}00},\dots$ dies out after finitely many generations, where the number of generations depends only on ${{\bf i}}$. Therefore $\min\{n_{{\bf i}}:\,|{{\bf i}}|=k\}\to\infty$ as $k\to\infty$, and consequently, by condition (3), $\max\{\beta_{{\bf i}}:\,|{{\bf i}}|=k\}\to 0$. Using this and , if $k$ is large enough, then $$\label{k}
\max\{\pi-|I_{{\bf i}}|:\,|{{\bf i}}|=k\}<{{\varepsilon}}.$$ We stop our whole construction after $k$ generations, where $k$ is so large that holds.
By choosing the parameters ${{\varepsilon}}_{{\bf i}}$ such that $\sum_{{\bf i}}{{\varepsilon}}_{{\bf i}}$ is small enough, by , and our assumption ${\mathcal{H}}^1(E)=1$, we can ignore those $L_{{\bf i}}$ for which ${\mathcal{H}}^1(L_{{{\bf i}}})\le{{\varepsilon}}_{{\bf k}}$, where ${{\bf k}}$ is the last good sequence among ${{\bf i}}$ and its ancestors. (This is because each ${{\bf i}}$ at which we stop our construction has a different “last good sequence among ${{\bf i}}$ and its ancestors.”) For the line segments that belong to the remaining part of the polygonal path, we have $|I_{{\bf i}}|>\pi-{{\varepsilon}}$ by .
Using the previous paragraph, we choose our balls $B(\theta_i, {{\varepsilon}})$ for as follows. If we ignore $L_{{\bf i}}$ (as described in the previous paragraph), then for each line segment $L_i$ in $L_{{{\bf i}}}$, we let $\theta_i$ to be any point we like. If we do not ignore $L_{{\bf i}}$, then for each $L_i$ in $L_{{\bf i}}$, we choose $\theta_i$ so that ${\mathbb{P}}^1 \setminus B(\theta_i, {{\varepsilon}}) \subset I_{{\bf i}}$. We can do this because $|I_{{\bf i}}|>\pi-{{\varepsilon}}$.
In order to finish the proof of , it suffices to show that $$\label{4.6}
A:=\bigcup_{L_{{\bf i}}\subset P} (L_{{\bf i}}+\{x\in E:\,\theta_x\in I_{{\bf i}}\})$$ has small measure.
So far, our definition of the polygonal path $P$ did not depend on the set $E$. In what follows, we will show that if the zigzags we use are sufficiently fine (depending on the set $E$) then indeed the set $A$ in has small measure. Note that the fineness of the zigzags is the only remaining parameter we need to specify. The parameters $\beta_{{\bf i}}, \gamma_{{\bf i}}, \pm_{{\bf i}}$, the lengths ${\mathcal{H}}^1(L_{{\bf i}})$, the stopping time, and the intervals $I_{{\bf i}}$ are all independent of the fineness of the zigzags and of $E$.
Fineness of the zigzags, and the small neighborhood lemma
---------------------------------------------------------
We have already chosen all the directions we use in all the basic zigzags to construct $P$. These directions divide ${\mathbb{P}}^1$ into finitely many intervals, which we call *elementary intervals*. By an elementary interval we mean a *closed* interval $I\subset{\mathbb{P}}^1$ such that its endpoints are directions used in our construction, and such that $I$ does not contain any other such direction.
Since we already know the length ${\mathcal{H}}^1(P)$, we also know how large subset of $E$ we may ignore by and . Therefore, by throwing away a sufficiently small subset of $E$ if necessary, we can assume that $E$ is compact and also that $x\mapsto\theta_x$ is a continuous function on $E$.
For an elementary interval $I$, we denote $$E_I=\{x\in E:\,\theta_x\in I\}.$$ Because of our assumptions above, $E_I$ is also compact.
Here is our strategy for choosing the fineness of the zigzags. Suppose that for some $E_I$ and for some line segment $L$ in our construction, we have obtained the estimate $|L + E_I| < \eta$ for some $\eta$. Then, we require all zigzags descending from $L$ to be fine enough so that they stay in a sufficiently small neighborhood of $L$. This ensures that by the small neighborhood lemma, translating $E_I$ along the descendants of $L$ still covers area $< \eta$. In the next section, we obtain finitely many estimates of the form $|L + E_I| < \eta$. We make the zigzags sufficiently fine at each step so that these estimates are preserved by the descendants of $L$, as explained above.
Area estimate
-------------
We fix an elementary interval $I$ and the corresponding set $E_I$, and revisit the Venetian blind construction. Our aim is to estimate the measure of the set $$\label{4.7}
A_I:=\bigcup_{L_{{\bf i}}\subset P \text{ s.t. } I \subset I_{{\bf i}}} (L_{{\bf i}}+E_I).$$ Our final goal is to show $|\bigcup_I A_I| < {{\varepsilon}}$. In the next two paragraphs, we will use the same notations as in .
First assume that the elementary interval $I$ is contained in the interval $[\theta_L\pm (j-1)\gamma$, $\theta_L\pm j\gamma]$ for some $j=1,2,\dots,k$. Since the line segments of $G_j$ are of direction $\theta_L\pm j\gamma$, it follows from (and the estimate ) that translating $E_I$ along the line segments of $G_j$ covers area $\lesssim\gamma{\mathcal{H}}^1(E_I){\mathcal{H}}^1(G_j)
\le\gamma{\mathcal{H}}^1(E_I){\mathcal{H}}^1(L)$. By our remarks in the previous section about choosing the fineness of the zigzags, the same estimate $\gamma {\mathcal{H}}^1(E_I){\mathcal{H}}^1(L)$ remains true if we translate $E_I$ along the line segments of $G_k$.
We can argue similarly when the elementary interval is contained in $[\theta_L,\theta_L\mp\beta]$. Therefore, for an $L_{{\bf i}}$, by adding up these estimates for all the line segments $L$ in $L_{{\bf i}}$, we proved the following lemma:
\[lemma:lemm\] Suppose that an elementary interval $I$ is contained in $[\theta_{{\bf i}},\theta_{{{\bf i}}'}]$. Then $|L_{{\bf i}}+ E_I|\lesssim\alpha_{{\bf i}}{\mathcal{H}}^1(E_I){\mathcal{H}}^1(L_{{{\bf i}}'})$.
Now consider an $L_{{\bf i}}\subset P$ with $I\subset I_{{\bf i}}$. Since the intervals $I_{{\bf i}},I_{{{\bf i}}'},I_{{{\bf i}}''},\dots$ are decreasing, there is a ${{\bf k}}$ among ${{\bf i}}$ and its ancestors such that $I\subset I_{{\bf k}}\setminus I_{{{\bf k}}'}\subset[\theta_{{\bf k}},\theta_{{{\bf k}}'}]$. By , $$\label{l''}
|L_{{{\bf k}}} + E_I| \lesssim \alpha_{{\bf k}}{\mathcal{H}}^1(E_I){\mathcal{H}}^1(L_{{{\bf k}}'}).$$
The estimates are precisely those that we would like to maintain when we replace the set $L_{{\bf k}}$ by all of its final descendants in $P$, as described in the previous section. Thus, we make the zigzags sufficiently fine so that these estimates are preserved.
Therefore, instead of taking the sum of the estimates for *all* finite sequences ${{\bf k}}$, it is sufficient to take the sum for *some* ${{\bf k}}$, each of which belongs to a different family line. Let ${{\bf k}}_1,{{\bf k}}_2,\dots$ be arbitrary sequences from different family lines.
We distinguish two cases: if ${{\bf k}}_m$ is good, then by (5) and (6) in , $$\label{1}
\alpha_{{{\bf k}}_m}{\mathcal{H}}^1(E_I){\mathcal{H}}^1(L_{{{{\bf k}}_m}'})
\le{{\varepsilon}}_{{{{\bf k}}_m}'}{\mathcal{H}}^1(E_I).$$ With the bad ${{\bf k}}_m$, the same trivial bound does not work. Nonetheless, because of the “different family lines condition,” each bad ${{\bf k}}_m$ has a different “last good among ${{\bf k}}_m$ and its ancestors.” Therefore by (2), (4), and (6) in , we have $$\label{2}
\sum_{{{\bf k}}_m \text{ bad}} \alpha_{{{\bf k}}_m}{\mathcal{H}}^1(E_I){\mathcal{H}}^1(L_{{{{\bf k}}_m}'})
\le\sum_{{{\bf k}}}{{\varepsilon}}_{{{\bf k}}}{\mathcal{H}}^1(E_I),$$ where the summation on the right is taken over all ${{{\bf k}}}$. Adding together the estimates for all good ${{\bf k}}_m$ and , we have $$\label{eq:AI-bound}
|A_I|\le 2\sum_{{{\bf k}}}{{\varepsilon}}_{{{\bf k}}}{\mathcal{H}}^1(E_I).$$ Since each $x\in E$ belongs to at most two of the sets $E_I$, by summing over $I$ and choosing $\sum_{{{\bf k}}}{{\varepsilon}}_{{{\bf k}}}$ small enough, the proof of is finished.
Kakeya needle problem for rotations {#sec:rotations}
===================================
Our aim in this section is to prove the following theorem, which can be thought of as a direct analogue of . Recall when we create a polygonal path $P \subset \operatorname{Isom}^+({\mathbb{R}}^2)$ from a sequence of rotations $\{\rho_i\}$, we always interpret the rotations in the *intrinsic* coordinate system. We will occasionally use the phrase *intrinsic rotation* to remind ourselves of this convention.
\[theorem:rotations-on-line\] Let $E \subset {\mathbb{R}}^2$ be a bounded rectifiable set of finite ${\mathcal{H}}^1$-measure. Let ${{\varepsilon}}> 0$, and let $\rho\in\operatorname{Isom}^+({\mathbb{R}}^2)$ be arbitrary. Let $\ell\subset{\mathbb{P}}^2$ be a line through the projective center $z$ of $\rho$.
Then there are intrinsic rotations $\rho_i=\rho(x_i)$ with projective centers $z_i \in B(\ell, {{\varepsilon}}) \subset {\mathbb{P}}^2$ such that the corresponding polygonal path $P=\bigcup_i L_i\subset\operatorname{Isom}^+({\mathbb{R}}^2)$ connects the identity and $\rho$, and for each $i$, there exists a $u_i \in \ell$ such that $$\label{eq:theorem-rotations}
|\bigcup_{i}\bigcup_{p\in L_i} p(\{x\in E: \nu_x \cap \ell \cap B(u_i, {{\varepsilon}}) = \emptyset\})|< {{\varepsilon}}.$$
Basic zigzags, deconstructed {#sec:basic-zigzag}
----------------------------
The heart of the matter in our proof of was that we repeatedly replaced line segments by basic zigzags. Each line segment $L$ represented a translation. In our proof of we will do an analogue construction with rotations instead of translations. However, this is a bit more delicate.
The first step of the basic zigzag construction for translations divides a line segment $L$ into $N$ equal parts. This step is easy to understand for rotations: we replace a rotation $\rho=\rho(x)$ by $N$ copies of $\rho(x/N)$, which are rotations around the same projective center as $\rho$ but with angle reduced by a factor of $N$. In the intrinsic coordinate system, if we apply $\rho(x/N)$ repeatedly $N$ times, then indeed we obtain $\rho(x)$.
In the second step, for translations, we replace a line segment $L$ by two line segments of given directions. We can represent this by the vector sum $v = v_1+ v_2$. The analogue of this would be to replace $\rho(x)$ by $\rho(x_1)$ and $\rho(x_2)$, for some $x_1, x_2$. We need to determine the necessary condition on $x,x_1, x_2$, i.e., the analogue of $v = v_1 + v_2$. It is not as simple as $x = x_1 + x_2$; the composition of $\rho(x_1)$ followed by $\rho(x_2)$ is not necessarily $\rho(x_1+x_2)$. Therefore, first we need to understand which rotations a given $\rho$ can be replaced by. We do this in the next section.
The structure of intrinsic compositions
---------------------------------------
Using the notation $\rho=\rho(w,\phi)$ and $v$ from , we can see that $\rho_3$ can be replaced by $\rho_1,\rho_2$ if $$\label{eq:phi123}
\phi_1+\phi_2=\phi_3$$ and $$\label{v12}
v_1+e^{i\phi_1}v_2=v_3.$$ Indeed, says that by applying $\rho_1$ and $\rho_2$, we rotate ${\mathbb{R}}^2$ by angle $\phi_1+\phi_2$. And says that the image of $0$ after applying $\rho_1$ and $\rho_2$ will be $v_1+e^{i\phi_1}v_2$; this is true since the path of $0$ can be obtained by integrating its velocity, and when we apply $\rho_2$, its velocity is rotated by $\phi_1$. If two rotations have the same angle and they map 0 to the same point, then they are the same rotation.
For $x_j\in{\mathbb{R}}^3\setminus\{0\}$, we will use the notation $x_3=x_1\star x_2$ if and hold for $\rho_j=\rho(x_j)$.
We do not need the following fact in this paper, but the conditions and imply that $\star$ is a group operation on ${\mathbb{R}}^3$. The group $({\mathbb{R}}^3, \star)$ has the structure of the semidirect product ${\mathbb{R}}^2 \rtimes {\mathbb{R}}$, where $\phi \in {\mathbb{R}}$ acts on $v \in {\mathbb{R}}^2$ by $v \mapsto e^{i\phi} v$.
The extra difficulty in our proof for rotations is essentially due to the failure of $\star$ to agree with $+$. Nonetheless, we can modify the proof for translations to obtain a proof for rotations because for small $x_1, x_2$, $\star$ is “close enough” to $+$, as we show in the next section.
Our main estimate is the following:
\[lemma:star-approx-add\] Let $x_j=(w_j,\phi_j)\in {\mathbb{R}}^3\setminus\{0\}$ with $x_3=x_1\star x_2$ and $|\phi_j|\lesssim 1$ for each $j$. Then $$\label{eq:star-approx-add}
|w_1 + w_2 - w_3| \lesssim |w_2\phi_1| + |w_1\phi_1| + |w_2\phi_2| + |w_3\phi_3|.$$
Observe that $|v_j| \leq |w_j|$ and $|v_j + iw_j| \lesssim |z_j \phi_j^2| = |w_j \phi_j|$. (For the second inequality, we used $|\phi_j| \lesssim 1$.) By , $|v_1 + v_2 - v_3| = |v_2(1-e^{i\phi_1})| \leq |w_2 \phi_1|$. Thus indeed, $$\begin{aligned}
|w_1 + w_2 - w_3|
&\leq
|v_1 + v_2 - v_3| + |v_1 + iw_1| + |v_2 + iw_2| + |v_3 + iw_3|
\\
&\lesssim
|w_2 \phi_1| + |w_1 \phi_1| + |w_2 \phi_2| + |w_3\phi_3|.
\qedhere\end{aligned}$$
Basic zigzag construction for rotations
---------------------------------------
Now we are ready to define our basic zigzag construction in general. This construction, for given $x,x_0,x_1\in{\mathbb{R}}^3\setminus\{0\}$ with $x=x_0+x_1$ and a given $N$, replaces the rotation $\rho(x)$ by the sequence of intrinsic rotations $\rho(y_0),\rho(y_1),\dots,\rho(y_0),\rho(y_1)$. We define $y_0=x_0/N$, and then $y_1=\tilde x_1/N$ is defined by $y_0 \star y_1 = x/N$.
The key properties of the construction are the following.
\[lemma:tilde-x1\] For any given ${{\varepsilon}}>0$, if $N$ is sufficiently large, then:
1. $|y_j|<{{\varepsilon}}$ for $j = 0, 1$;
2. $|\tilde x_1-x_1|<{{\varepsilon}}$.
Since $y_0 = x_0/N$, property $(1)$ for $j = 0$ is obvious. For $j = 1$, this property follows from $y_1 = \tilde x_1/N$ and from (2).
Let $x = (w, \phi)$, $x_j = (w_j, \phi_j)$, and $\tilde x_j = (\tilde w_j, \tilde \phi_j)$. To prove (2), it suffices to show that $\tilde w_1\to w_1$ as $N\to\infty$, since $\tilde\phi_1=\phi_1$. If $N$ is large enough, then $\frac{\phi}{N},\frac{\phi_j}{N}\lesssim 1$, so we can apply for $x/N= (x_0/N) \star (\tilde x_1/N)$ to obtain: $$|\tilde w_1/N+w_0/N-w/N|\lesssim
\frac{1}{N^2}(|w_0\phi_1|+|\tilde w_1\phi_1|+|w_0\phi_0|+|w\phi|).$$ Therefore $$|\tilde w_1-w_1|=|\tilde w_1+w_0-w|\lesssim \frac{1}{N}(|w_0\phi_1|+|\tilde w_1\phi_1|+|w_0\phi_0|+|w\phi|).$$ Note that $w, w_0, w_1, \phi, \phi_0, \phi_1$ do not depend on $N$. Therefore, as $N\to\infty$, $|\tilde w_1-w_1|=o(|\tilde w_1|+o(1))$ and hence indeed $\tilde w_1\to w_1$.
Property (1) allows the polygonal path for $y_0 \star y_1 \star \cdots \star y_0 \star y_1$ to stay within an arbitrarily small neighborhood of the line segment defined by $x$. This is because by decomposing $x$ into $(x/N)\star\dots\star(x/N)$, we divide the line segment into $N$ equal segments. When we replace each segment by $y_0\star y_1$, we stay in a small neighborhood of it.
Iterating the basic zigzag {#sec:iterating-basic-zigzag}
--------------------------
In our proof of , we started from a line segment $L$ and then, iteratively, we replaced each line segment by a Venetian blind; the indices ${{\bf i}}$ indexed the Venetian blinds. However, in this section, we need to focus also on basic zigzags, hence we introduce a new set of indices ${{\bf j}}$ (finite binary sequences) to index the basic zigzags. For ${{\bf j}}= j_1 \cdots j_k$, we denote ${{\bf j}}' = j_1 \cdots j_{k-1}$.
Suppose we start our construction from with a vector $v$. In our first basic zigzag, we chose two directions $\theta_0$, $\theta_1$. Then we can uniquely decompose $v = v_0 + v_1$, where $v_j$ is in direction $\theta_j$. If the fineness is $N$, we can represent the basic zigzag as $$v = (v_0/N) + (v_1/N) + \cdots + (v_0/N) + (v_1/N).$$ This gives us $N$ copies of the segments $v_0/N$ and $v_1/N$. We set $M_0 = M_1 = N$.
Now suppose we have $M_{{\bf j}}$ copies of $v_{{\bf j}}/M_{{\bf j}}$. To apply a basic zigzag on every copy, we write $v_{{\bf j}}=v_{{{\bf j}}0}+v_{{{\bf j}}1}$ and choose a fineness $N_{{\bf j}}$. Then our basic zigzag is $$\frac{v_{{\bf j}}}{M_{{\bf j}}}
=
\frac{v_{{{\bf j}}0}}{M_{{\bf j}}N_{{\bf j}}}+\frac{v_{{{\bf j}}1}}{M_{{\bf j}}N_{{\bf j}}}
+
\cdots
+
\frac{v_{{{\bf j}}0}}{M_{{\bf j}}N_{{\bf j}}}+\frac{v_{{{\bf j}}1}}{M_{{\bf j}}N_{{\bf j}}}.$$ Here we have $N_{{\bf j}}$ copies of $\frac{v_{{{\bf j}}0}}{M_{{\bf j}}N_{{\bf j}}}+\frac{v_{{{\bf j}}1}}{M_{{\bf j}}N_{{\bf j}}}$ and $M_{{{\bf j}}0}=M_{{{\bf j}}1}=M_{{\bf j}}N_{{\bf j}}$.
We let $L_{{\bf j}}\subset {\mathbb{R}}^2$ be the union of the $M_{{\bf j}}$ congruent and parallel line segments corresponding to the $M_{{\bf j}}$ copies of $v_{{\bf j}}/M_{{\bf j}}$. We let $\theta_{{\bf j}}$ be the direction of these segments.
The vectors $v_{{\bf j}}$ do not depend on the fineness of the zigzags. Note also that $|v_{{\bf j}}| = {\mathcal{H}}^1(L_{{\bf j}})$.
\[belonging\] As noted earlier, the indices ${{\bf j}}$ index the basic zigzag constructions from , *not* the Venetian blinds. However, each ${{\bf j}}$ lies between an index ${{\bf i}}$ and its parent ${{\bf i}}'$ in the following way. The segments $v_{{\bf j}}/M_{{\bf j}}$ were created in the Venetian blind construction starting from $L_{{{\bf i}}'}$. If $L_{{\bf j}}$ is in $L_{{{\bf i}}' 0}$ then we will say that ${{\bf j}}$ is between ${{\bf i}}'$ and ${{\bf i}}$, where ${{\bf i}}:= {{\bf i}}' 0$ is the bad child of ${{\bf i}}'$. Otherwise, ${{\bf j}}$ is between ${{\bf i}}'$ and ${{\bf i}}$, where ${{\bf i}}$ is the good child.
In the paragraphs above, we showed how to construct $\{v_{{\bf j}}\}$ given an iteration of basic zigzags. Conversely, we could start with a collection $\{v_{{\bf j}}\}$ satisfying $v_{{\bf j}}= v_{{{\bf j}}0} + v_{{{\bf j}}1}$ and turn this into instructions for iterating the basic zigzags. (We would also need to specify the fineness $N_{{\bf j}}$ at each step.)
The analogue of the above scheme for rotations is the following. Suppose that we are given some points $x_{\bf j}\in{\mathbb{R}}^3\setminus\{0\}$, where the ${{\bf j}}$ are finite binary sequences, such that $x_{{{\bf j}}}=x_{{{\bf j}}0}+x_{{{\bf j}}1}$ for each ${{\bf j}}$. We also fix a small $r>0$.
In our first step of the construction, we choose a sufficiently large $N$ and choose $y_0$, $y_1$ as in the previous section. That is, we replace $x$ by $N$ copies of $(x_0/N)\star(\tilde x_1/N)$: $$x=(x_0/N)\star(\tilde x_1/N)\star\dots\star(x_0/N)\star(\tilde x_1/N).$$ We choose $N$ so large that $|\tilde x_1-x_1|<r$. (We can do this by (2).) We also put $\tilde x=x$, $\tilde x_0=x_0$, and $M_0=M_1=N$.
Now suppose that we have already chosen $\tilde x_{{\bf j}}$ and an $M_{{\bf j}}$ for some sequence ${{\bf j}}$, and $|\tilde x_{{\bf j}}-x_{{\bf j}}|<r$. Then we apply a basic zigzag construction with $x$ replaced by $\tilde x_{{\bf j}}/M_{{\bf j}}$, $x_0$ replaced by $x_{{{\bf j}}0}/M_{{{\bf j}}}$ and $x_1$ replaced by $(x_{{{\bf j}}1}+\tilde x_{{\bf j}}-x_{{\bf j}})/M_{{\bf j}}$, and with fineness $N_{{\bf j}}$. That is, we replace $\tilde x_{{\bf j}}/M_{{\bf j}}N_{{\bf j}}=(x_{{{\bf j}}0}/M_{{\bf j}}N_{{\bf j}})\star(\tilde x_{{{\bf j}}1}/M_{{\bf j}}N_{{\bf j}})$, giving us $$\frac{\tilde x_{{\bf j}}}{M_{{\bf j}}}
=
\left(\frac{x_{{{\bf j}}0}}{M_{{\bf j}}N_{{\bf j}}}\right)\star\left(\frac{\tilde x_{{{\bf j}}1}}{M_{{\bf j}}N_{{\bf j}}}\right)
\star\dots\star
\left(\frac{x_{{{\bf j}}0}}{M_{{\bf j}}N_{{\bf j}}}\right)\star\left(\frac{\tilde x_{{{\bf j}}1}}{M_{{\bf j}}N_{{\bf j}}}\right).$$ If $N_{{\bf j}}$ is very large, then $\tilde x_{{{\bf j}}1}/M_{{\bf j}}$ will be very close to $(x_{{{\bf j}}1}+\tilde x_{{\bf j}}-x_{{\bf j}})/M_{{\bf j}}$, which means that $\tilde x_{{{\bf j}}1}$ will be very close to $x_{{{\bf j}}1}+\tilde x_{{\bf j}}-x_{{\bf j}}$. Therefore, by choosing $N_{{\bf j}}$ large enough, $|\tilde x_{{{\bf j}}1}-x_{{{\bf j}}1}|<r$ holds. We put $\tilde x_{{{\bf j}}0}=x_{{{\bf j}}0}$ and $M_{{{\bf j}}0}=M_{{{\bf j}}1}=M_{{\bf j}}N_{{\bf j}}$.
Using this procedure, we obtain an $\tilde x_{{\bf j}}$ and an $M_{{\bf j}}$ for each ${{\bf j}}$, such that $|\tilde x_{{\bf j}}-x_{{\bf j}}|<r$, and for $y_{{\bf j}}:=\tilde x_{{\bf j}}/M_{{\bf j}}$: $$y_{{\bf j}}=y_{{{\bf j}}0}\star y_{{{\bf j}}1}\star\dots\star y_{{{\bf j}}0}\star y_{{{\bf j}}1}$$ (where we have $N_{{\bf j}}$ copies of $y_{{{\bf j}}0}\star y_{{{\bf j}}1}$).
In this way, we have shown how to take a collection $\{x_{{\bf j}}\}$ with $x_{{\bf j}}= x_{{{\bf j}}0} + x_{{{\bf j}}1}$, together with fineness $N_{{\bf j}}$, and turn this data into a sequence of rotations, the composition of which is the original rotation $\rho(x)$.
For translations, the sequence $\{v_{{\bf j}}\}$ tells us every direction we will translate in, even before the fineness $N_{{\bf j}}$ are chosen. However, for rotations, the sequence $\{x_{{\bf j}}\}$ alone does not tell us the projective centers of the rotations we will use. The centers are given by $\{\tilde x_{{\bf j}}\}$, which depend on $N_{{\bf j}}$. The $N_{{\bf j}}$ in turn depend on $\{x_{{\bf j}}\}$ and $r$ (in the way explained above) as well as on the area estimates in the following sections.
Turning the translations into rotations {#sec:translations-into-rotations}
---------------------------------------
In the previous section, we showed how to turn a collection $\{x_{{\bf j}}\}$ into a sequence of rotations, but we did not say which sequence $\{x_{{\bf j}}\}$ to start with. We specify that now. The construction of $\{x_{{\bf j}}\}$ is actually very simple: we use a rotation in ${\mathbb{R}}^3$ to “transform” a sequence of vectors $\{v_{{\bf j}}\}$ in ${\mathbb{R}}^2$ into our desired sequence $\{x_{{\bf j}}\}$.
Let $\rho(x)$ and ${{\varepsilon}}$ be as in the statement of . Then we can apply the results of to $E$; let $c$ be the constant in and . Without loss of generality we can assume that ${{\varepsilon}}$ is small enough, so that the conclusion of holds for every $\delta<{{\varepsilon}}$.
Let $v \in {\mathbb{R}}^2$ be an arbitrary vector with $|v|=|x|$. We can follow the steps in to construct the vectors $v_{{\bf j}}$ with $v_\emptyset = v$ as well as the stopping time.
Our aim is to “turn” the sequence $\{v_{{\bf j}}\}$ into a sequence of rotations. Let $Q:{\mathbb{R}}^3 \to {\mathbb{R}}^3$ be a linear rotation that maps $(v,0)$ to $x$, and that maps the plane $\phi=0$ (i.e., those $x=(w,\phi) \in {\mathbb{R}}^3$ for which $\rho(x)$ is a translation) onto the plane of $\ell$. We define $x_{{\bf j}}:=Q(v_{{\bf j}},0)$ for each ${{\bf j}}$. Since $Q$ is linear, we do indeed have $x_{{\bf j}}=x_{{{\bf j}}0}+x_{{{\bf j}}1}$.
We denote by $z_{{\bf j}}$ the projective image of $x_{{\bf j}}$ onto ${\mathbb{P}}^2$. Then $z_{{\bf j}}\in\ell$. A trivial but very important property we have is this: since $Q$ is an isometry, the distance between any two $z_{{\bf j}}$ is the same as the angle between the corresponding vectors $v_{{\bf j}}$. If ${{\bf j}}$ is between ${{\bf i}}'$ and ${{\bf i}}$ (see ), we denote $\alpha_{{\bf j}}:= \alpha_{{\bf i}}$ and let $$\label{eq:def-B-jj}
B_{{\bf j}}= B(z_{{\bf j}}, 2\alpha_{{\bf j}}) \subset {\mathbb{P}}^2.$$
\[remark:ball-contains-interval\] Suppose ${{\bf j}}$ is between ${{\bf i}}$ and ${{\bf i}}'$. If ${{\bf i}}$ is good, then the ball $B_{{\bf j}}$ contains $[z_{{\bf j}},z_{{{\bf j}}'}]\subset \ell$, which is the image of $[\theta_{{\bf j}}, \theta_{{{\bf j}}'}] \subset {\mathbb{P}}^1$ under the rotation $Q$. If ${{\bf i}}$ is bad, then $B_{{\bf j}}$ contains $[z_{{\bf j}}, z_{{{\bf i}}'}] = [z_{{\bf i}}, z_{{{\bf i}}'}] \subset \ell$.
So far, none of the objects we defined depend on the fineness of the zigzags; they depend only on $E$, $\ell$, $\rho(x)$ and ${{\varepsilon}}$.
Now we use the basic zigzag iteration process in to obtain $\{\tilde x_{{\bf j}}\}$ with $N_{{\bf j}}$ large enough (that we will specify in the next section). We denote the projective center of the rotations by $\tilde z_{{\bf j}}$. That is, $\tilde z_{{\bf j}}$ is the image of $y_{{\bf j}}$ (which is the same as the image of $\tilde x_{{\bf j}}$) under the projection ${\mathbb{R}}^3\setminus\{0\}\to{\mathbb{P}}^2$. We will also denote $x_{{\bf j}}=(w_{{\bf j}},\phi_{{\bf j}})$ and $\tilde x_{{\bf j}}=(\tilde w_{{\bf j}},\phi_{{\bf j}})$. (Caution: we do not use the notation $v_{{\bf j}}$ as in . Instead, the $v_{{\bf j}}$ satisfy $x_{{\bf j}}= Q(v_{{\bf j}}, 0)$.)
Recall from that $\tilde x_{{\bf j}}\in B(x_{{\bf j}}, r) \subset {\mathbb{R}}^3$, where we can choose $r$ as small as we wish. We choose $r$ small enough so that $r \leq \frac{1}{2} \min_{{{\bf j}}} |x_{{\bf j}}|$ and so that for each ${{\bf j}}$, the image of $B(x_{{\bf j}},r)$ under the projection ${\mathbb{R}}^3\setminus\{0\}\to{\mathbb{P}}^2$ is contained in $B_{{\bf j}}\cap B(\ell, {{\varepsilon}})$. It follows that $|\tilde x_{{\bf j}}| \lesssim |x_{{\bf j}}|$ and $\tilde z_{{\bf j}}\in B(z_{{\bf j}},2\alpha_{{\bf j}}) \cap B(\ell, {{\varepsilon}})$ for each ${{\bf j}}$.
In the end, we have two polygonal paths. One is $P = \bigcup_{{\bf j}}L_{{\bf j}}\subset {\mathbb{R}}^2$, corresponding to $\{v_{{\bf j}}\}$; the other is $\tilde P = \bigcup_{{\bf j}}\tilde L_{{\bf j}}\subset \operatorname{Isom}^+({\mathbb{R}}^2)$, corresponding to $\{x_{{\bf j}}\}$. In both cases, we use the same fineness $N_{{\bf j}}$ (still to be specified). (We also have the same stopping time since that is encoded in the sequences $\{v_{{\bf j}}\}$, $\{x_{{\bf j}}\}$.)
Thus, $Q$ “transforms” a polygonal path $P = \bigcup_{{\bf j}}L_{{\bf j}}\subset {\mathbb{R}}^2$ into a polygonal path $\tilde P = \bigcup_{{\bf j}}\tilde L_{{\bf j}}\subset \operatorname{Isom}^+({\mathbb{R}}^2)$ by “transforming” $L_{{\bf j}}$ into $\tilde L_{{\bf j}}$. Our next aim is to turn the estimates for $P$ we obtained in into estimates for $\tilde P$.
Ignoring small parts of $E$ and of $\tilde P$
---------------------------------------------
Recall the definition of the intervals $I_{{\bf i}}$, the elementary intervals $I$, and the sets $E_I$ from . Because of the rotation $Q$, the relevant objects are now $J_{{\bf i}}:=QI_{{\bf i}}$, $J:=QI\subset \ell$, and $E_J:=\{x\in E:\,\nu_x \cap J \neq \emptyset\}$.
We made the sets $E_I$ compact by “ignoring” a sufficiently small subset of $E$. Since we knew the length of the final polygon $P$ (this depended on the stopping time, but *not* on the fineness of the zigzags) we also knew from that during our movement, small enough subsets of $E$ will automatically cover small area. By the same reason, we could also “ignore” those $L_{{\bf i}}$ for which ${\mathcal{H}}^1(L_{{{\bf i}}})\le{{\varepsilon}}_{{\bf k}}$, where ${{\bf k}}$ is the last good sequence among ${{\bf i}}$ and its ancestors.
We now obtain the analogue estimates for rotations, by applying in place of . Indeed, since $|\tilde x_{{\bf j}}| \lesssim |x_{{\bf j}}| = |v_{{\bf j}}| = {\mathcal{H}}^1(L_{{\bf j}})$ for each ${{\bf j}}$, therefore every subset $R\subset E$ will cover, during the movement by $\tilde P$, an area $\lesssim c{\mathcal{H}}^1(R)\sum_{{\bf j}}|\tilde x_{{\bf j}}|\lesssim c{\mathcal{H}}^1(R)\sum_{{\bf j}}{\mathcal{H}}^1(L_{{\bf j}})= c{\mathcal{H}}^1(R){\mathcal{H}}^1(P)$, where the sums are over all ${{\bf j}}$ with $L_{{\bf j}}\subset P$ (or, equivalently, $\tilde L_{{\bf j}}\subset \tilde P$). That is, we obtain a $c$ times larger estimate than in . Similarly, when we move any $R$ by $\tilde L_{{\bf i}}$, we cover an area at most $c{\mathcal{H}}^1(R)|\tilde x_{{\bf i}}|\lesssim c {\mathcal{H}}^1(R){\mathcal{H}}^1(L_{{\bf i}})$ instead of ${\mathcal{H}}^1(R){\mathcal{H}}^1(L_{{\bf i}})$.
Since $Q$ is a rotation, $|J_{{\bf i}}| = |I_{{\bf i}}|$. Similarly as in , for each line segment $\tilde L \subset \tilde L_{{{\bf i}}}$ appearing in the final polygon $\tilde P$, we choose $B(u_i, {{\varepsilon}})$ of so that $\ell \setminus B(u_i, {{\varepsilon}})\subset J_{{\bf i}}$ whenever $|J_{{\bf i}}|=|I_{{\bf i}}|\ge \pi-{{\varepsilon}}$. If $|J_{{\bf i}}|< \pi-{{\varepsilon}}$, we can choose $B(u_i, {{\varepsilon}})$ arbitrarily.
Area estimates
--------------
For each $J$, let $A_J$ denote the set covered by moving $E_J$ along those $\tilde L_{{\bf i}}\subset \tilde P$ for which $J\subset J_{{\bf i}}$ (cf. ). Our final goal is to show $\sum_J |A_J| < \tilde c {{\varepsilon}}$, for some $\tilde c$ independent of ${{\varepsilon}}$. This would imply that holds with ${{\varepsilon}}$ replaced by $\tilde c{{\varepsilon}}$ in its right hand side.
First we prove the following analogue of .
\[lemma:lemm-rot\] By making the basic zigzags sufficiently fine, we can achieve the following: if $J$ is an elementary interval contained in $[z_{{\bf i}}, z_{{{\bf i}}'}]$, then the area covered by moving $E_{J}$ along $\tilde L_{{\bf i}}$ is $\lesssim c\alpha_{{\bf i}}{\mathcal{H}}^1(E_{J}) {\mathcal{H}}^1(L_{{{\bf i}}'})$.
Suppose $J$ is an elementary interval contained in $[z_{{\bf i}}, z_{{{\bf i}}'}]$. If ${{\bf i}}$ is good, then there is a ${{\bf j}}$ between ${{\bf i}}$ and ${{\bf i}}'$ such that $ J \subset [z_{{\bf j}}, z_{{{\bf j}}'}]$. If ${{\bf i}}$ is bad, then for all ${{\bf j}}$ between ${{\bf i}}$ and ${{\bf i}}'$, $J \subset [z_{{\bf j}}, z_{{{\bf j}}'}]$.
Suppose that $J \subset [z_{{\bf j}}, z_{{{\bf j}}'}]$. Applying with $\rho=\rho(y_{{\bf j}})$ and $R = E_{J}$ (noting ), we see that if we move $E_{J}$ by the rotation $\rho(y_{{\bf j}})$, the area covered is $\lesssim c\alpha_{{\bf j}}{\mathcal{H}}^1(E_J)|\tilde x_{{\bf j}}|/M_{{\bf j}}$. Hence the total area covered by moving $E_{J}$ by all $M_{{\bf j}}$ copies of $\rho(y_{{\bf j}})$ is $\lesssim c\alpha_{{\bf j}}{\mathcal{H}}^1(E_J)|\tilde x_{{\bf j}}| \lesssim c\alpha_{{\bf j}}{\mathcal{H}}^1(E_J) {\mathcal{H}}^1(L_{{\bf j}})$. We make the zigzags so fine in our constructions that the same estimate $$\label{eq:EQI-Ljj}\lesssim c\alpha_{{\bf j}}{\mathcal{H}}^1(E_J){\mathcal{H}}^1(L_{{\bf j}})$$ remains true when we rotate the set $E_J$ by the descendants of the $M_{{\bf j}}$ copies of $y_{{\bf j}}$.
Now, we break into two cases. If ${{\bf i}}$ is good, then $L_{{\bf i}}$ descends from $L_{{\bf j}}$, so the statement of the lemma follows from ${\mathcal{H}}^1(L_{{\bf j}}) \leq {\mathcal{H}}^1(L_{{{\bf i}}'})$.
If ${{\bf i}}$ is bad, we use the fact that $L_{{{\bf i}}} = \bigcup_{{\bf j}}L_{{\bf j}}$ and $\tilde L_{{{\bf i}}} = \bigcup_{{\bf j}}\tilde L_{{\bf j}}$, where the unions are over all ${{\bf j}}$ between ${{\bf i}}$ and ${{\bf i}}'$. Then summing over the estimate for each such ${{\bf j}}$, we have that moving along $\tilde L_{{{\bf i}}}$, the area is $$\lesssim
c \alpha_{{\bf i}}{\mathcal{H}}^1(E_{J})
\sum_{{\bf j}}{\mathcal{H}}^1(L_{{\bf j}})
=
c \alpha_{{\bf i}}{\mathcal{H}}^1(E_{J})
{\mathcal{H}}^1(L_{{\bf i}})
\leq
c \alpha_{{\bf i}}{\mathcal{H}}^1(E_{J})
{\mathcal{H}}^1(L_{{{\bf i}}'})$$ which completes the proof.
Having established this estimate, the proof continues in the same way as in , to obtain $|A_{J}| \lesssim 2c \sum_{{{\bf i}}} {{\varepsilon}}_{{\bf i}}{\mathcal{H}}^1(E_{J})$, the analogue of . We explain some details below.
Consider an $\tilde L_{{\bf i}}\subset \tilde P$ with $J\subset J_{{\bf i}}$. Since the intervals $J_{{\bf i}},J_{{{\bf i}}'},J_{{{\bf i}}''},\dots$ are decreasing, there is a ${{\bf k}}$ among ${{\bf i}}$ and its ancestors such that $J\subset J_{{\bf k}}\setminus J_{{{\bf k}}'}\subset[z_{{\bf k}},z_{{{\bf k}}'}]$. By , the total area covered when we move $E_J$ along $\tilde L_{{{\bf k}}}$ is $$\label{l''-rot}
\lesssim c\alpha_{{\bf k}}{\mathcal{H}}^1(E_J){\mathcal{H}}^1(L_{{{\bf k}}'}).$$
By making the zigzags sufficiently fine, the same estimate remains true when we move $E_J$ along all the descendants of $\tilde L_{{\bf k}}$ in $\tilde P$.
Therefore, similarly as in section 4, the area of $A_J$ can be estimated by summing the estimate for those ancestors that are on different family lines. Let ${{\bf k}}_1,{{\bf k}}_2,\dots$ be arbitrary sequences from different family lines.
We distinguish two cases: if ${{\bf k}}_m$ is good, then $$\label{1-rot}
\alpha_{{{\bf k}}_m}{\mathcal{H}}^1(E_J){\mathcal{H}}^1(L_{{{{\bf k}}_m}'})
\le{{\varepsilon}}_{{{{\bf k}}_m}'}{\mathcal{H}}^1(E_J).$$ With the bad ${{\bf k}}_m$, because of the different family lines condition, each bad ${{\bf k}}_m$ has a different “last good among ${{\bf k}}_m$ and its ancestors” so $$\label{2-rot}
\sum_{{{\bf k}}_m \text{ is bad}} \alpha_{{{\bf k}}_m}{\mathcal{H}}^1(E_J){\mathcal{H}}^1(L_{{{{\bf k}}_m}'})
\le\sum_{{{\bf k}}}{{\varepsilon}}_{{{\bf k}}}{\mathcal{H}}^1(E_J),$$ where the summation on the right is taken over all ${{{\bf k}}}$. Adding together the estimates for all good ${{\bf k}}_m$ and , we proved that $$\label{eq:AJ-bound}
|A_J|\lesssim 2c\sum_{{{\bf k}}}{{\varepsilon}}_{{{\bf k}}}{\mathcal{H}}^1(E_J).$$
Let $c'$ be the implied constant in . Since each $x\in E$ belongs to at most two of the sets $E_J$, we proved that $\sum_J|A_J|$ is at most $cc'$ times larger than the bound of ${{\varepsilon}}$ for $\sum_I |A_I|$ that we obtained in . In other words, we showed $\sum_J|A_J| < cc'{{\varepsilon}}$. The constant $cc'$ depends only $\ell$ and $E$ (and not on ${{\varepsilon}}$). This completes the proof.
Further remarks
---------------
\[remark:small-nbhd-initial-mvmt\] In both and , we constructed a polygonal path that replaced a continuous movement with a fixed intrinsic projective center by a sequence of intrinsic rotations. By choosing all the zigzags sufficiently fine in our constructions, we can stay in an arbitrarily small neighborhood of the initial movement in $\operatorname{Isom}^+({\mathbb{R}}^2)$.
\[remark:z-not-in-closure-ball\] It is possible to choose the $u_i$ in so that $z$, the initial center of rotation, is not in any of the closed balls $\operatorname{cl}B(u_i, {{\varepsilon}})$.
By applying to the initial rotation $\rho$ and a sufficiently small ball $B(z,\eta)$, we see that rotating the set $R = \{ x \in E : \nu_x \cap \ell \cap B(z, \eta)\}$ by $\rho$ covers small area. By making the zigzags sufficiently fine and using the small neighborhood lemma, the set $R$ still covers small area when moved by the final polygonal path. Thus, holds with $B(u_i,{{\varepsilon}})$ replaced by $B(u_i,{{\varepsilon}})\setminus B(z,\eta)$, so we can reselect the $u_i$ so that $z \not\in \operatorname{cl}B(u_i, {{\varepsilon}})$.
This property will be used in the proof of .
Besicovitch and Nikodym sets {#section:besi-niko}
============================
We conclude this paper by showing that when we iterate the polygonal constructions in and and “take the limit,” we obtain the analogues of Besicovitch and Nikodym sets for rectifiable sets.
Construction of a Besicovitch set for translations {#subsec:besicovitch-translations}
--------------------------------------------------
We start with the following, somewhat technical conditions. Afterwards, we will discuss some interesting special cases.
Suppose that we are given some rectifiable sets $E_1\subset E_2\subset\dots$, and a tangent field $x\mapsto\theta_x$ of $\bigcup E_n$, satisfying the following:
- each $E_n$ is compact, and has finite ${\mathcal{H}}^1$-measure;
- each $E_n$ has a subset $E_n'$ of full ${\mathcal{H}}^1$-measure, such that the restriction of the tangent $\theta$ to $E_n'$ is continuous, and for each $y\in E_n$, $$\label{thetay}
\theta_y\in\bigcap_{r>0}\operatorname{cl}(\theta(B(y,r)\cap E_n')).$$
We will prove the following proposition:
\[propo\] Suppose that the sets $E_n$ satisfy the assumptions above. Let $P_0$ be an arbitrary path in ${\mathbb{R}}^2$. Then for any neighborhood of $P_0$, there is a path $P$ in this neighborhood with the same endpoints as $P_0$, and there is a Borel mapping $p\mapsto\theta_p\in{\mathbb{P}}^1$ such that $$\label{pep}
|\bigcup_{p\in P} (p+\{x\in \bigcup E_n:\,\theta_x\neq \theta_p\}|=0.$$
Given any neighborhood of $P_0$, let $P^0$ be a polygonal path in this neighborhood with the same endpoints as $P_0$. For each $n$, we choose an ${{\varepsilon}}_n>0$ with $\sum{{\varepsilon}}_n<\infty$. Then iteratively, for each $n\ge 1$ we apply to each segment $L \subset P^{n-1}$ with $E$ replaced by $E_n'$ and ${{\varepsilon}}$ replaced by some ${{\varepsilon}}_L > 0$ such that $\sum_{L \subset P^{n-1}} {{\varepsilon}}_L < {{\varepsilon}}_n$. This gives us a polygonal path $P^n=\bigcup_i L_i^n$ and directions $\theta_i^n$ such that $$\label{pn}
|\bigcup_i\bigcup_{p\in L_i^n}(p+E_i^n)|<{{\varepsilon}}_n,$$ where $$\label{Ei}
E_i^n:= \operatorname{cl}{\{x\in E_n':\,\theta_x\not\in B(\theta_i^n,{{\varepsilon}}_n)\}}.$$ Although gives us the sets $E_i^n$ without their closure, we can take the closure in since, by our assumptions, doing so does not increases their measure. We know that moving an ${\mathcal{H}}^1$-null set along a polygon covers only zero area, so indeed, holds.
We construct $P^{n+1}$ by replacing each line segment $L_i^n$ of $P^n$ by a polygonal path that stays in such a small neighborhood of $L_i^n$ that the area estimate in remains true when, instead of $L_i^n$, we shift the sets $E_i^n$ along the line segments that we replace $L_i^n$ with. (Here we used and that the sets $E_i^n$ are compact.)
Also, we choose the neighborhoods small enough so that the polygonal paths $P^n$ converge to a continuous limit curve $P$. For each $p\in P$, and for each fixed $n$, we have an $i=i(p,n)$ such that $$\bigcup_{p\in P}(p+E_{i(p,n)}^n)|<{{\varepsilon}}_n$$ holds. We denote $$\label{eq:def-Ep}
E_p:=\limsup_{n\to\infty}E_{i(p,n)}^n.$$ Then $$|\bigcup_{p\in P}(p+E_p)|
\le
|\bigcup_{p\in P}(p+\bigcup_{m\ge n}E_{i(p,m)}^m)|
=
|\bigcup_{m\ge n}\bigcup_{p\in P}(p+E_{i(p,m)}^m)|
\le
\sum_{m\ge n}{{\varepsilon}}_m.$$ Since this is true for every $n$, it follows that $\bigcup_{p\in P}(p+E_p)$ is Lebesgue null.
By the definition , if a point $y\in\bigcup E_n$ does not belong to $E_p$, then for every large enough $n$, it has a neighborhood disjoint from $\{x\in E_n':\,\theta_x\not\in B(\theta_i^n,{{\varepsilon}}_n)\}$. That is, there is an $r>0$ such that $\theta_x\in B(\theta_i^n,{{\varepsilon}}_n)$ for every $x\in B(y,r)\cap E_n'$. Hence, by our assumption , $\theta_y\in\operatorname{cl}(\theta(B(y,r)\cap E_n'))\subset \operatorname{cl}B(\theta_i^n,{{\varepsilon}}_n)$. That is, $\theta_y$ is in $\liminf_{n\to\infty}\operatorname{cl}B(\theta_{i(p,n)}^n,{{\varepsilon}}_n)$, which has at most one point. For $p \in P$, if this set has one point, then we let $\theta_p$ denote that point. Otherwise, we let $\theta_p$ be arbitrary.
Then for each $p$, $\{x\in \bigcup E_n:\,\theta_x \neq \theta_p\} \subset E_p$, and the proof is finished.
For every rectifiable set $E$, we can choose the sets $E_n = E_n'$ such that they satisfy the requirements at the beginning of this section, and such that $\bigcup E_n$ is a subset of $E$ of full ${\mathcal{H}}^1$-measure. Therefore we obtain the following theorem:
\[theorem:besicovitch\] Let $E$ be an arbitrary rectifiable set, and let $x\mapsto\theta_x$ be an arbitrary tangent field of $E$. Then there is an $E_0\subset E$ of full ${\mathcal{H}}^1$-measure in $E$ for which the following holds.
For every path $P_0$ in ${\mathbb{R}}^2$, and for any neighborhood of $P_0$, there is a path $P$ in this neighborhood with the same endpoints as $P_0$, and there is a Borel mapping $p\mapsto\theta_p\in{\mathbb{P}}^1$ such that $$|\bigcup_{p\in P} (p+\{x\in E_0:\,\theta_x\neq \theta_p\}|=0.$$
\[rempre1\] Another interesting corollary of is the following. Suppose that $E$ can be covered by a finite union of (not necessarily disjoint) $C^1$ curves, or $E$ is the graph of a convex function. In these cases there is an $E_0\subset E$ of full measure so that the tangent is continuous on $E_0$. Moreover, we can define the tangent on $E\setminus E_0$ (in a natural way) and find the sets $E_n, E_n'$ so that they satisfy our requirements and so that $\bigcup_n E_n$ covers $E$. Therefore the statement of holds with $E_0$ replaced by $E$.
For example, if $E$ is the graph of a strictly convex function, then it is enough to delete at most one point for each $p\in P$, as we claimed in the introduction.
Construction of a Besicovitch set for rotations
-----------------------------------------------
The main ideas for rotations are the same as for translations.
\[propo-rot\] Suppose that the sets $E_n$ satisfy the assumptions as in the beginning of . Let $P_0$ be an arbitrary path in $\operatorname{Isom}^+({\mathbb{R}}^2)$. Then for any neighborhood of $P_0$, there is a path $P$ in the neighborhood of $P_0$ with the same endpoints as $P_0$, and there is a Borel mapping $p \mapsto x_p \in {\mathbb{P}}^2$ such that $$|\bigcup_{p\in P} p(\{x\in \bigcup E_n :\, x_p \not\in \nu_x \})|=0.$$
We begin with choosing $P^0$ to be an arbitrary polygonal path in the neighborhood of $P_0$ with the same endpoints as $P_0$. We iterate to construct the polygonal paths $P^n$ in $\operatorname{Isom}^+({\mathbb{R}}^2)$, each lying in a small neighborhood of the previous one. Here, the details are now a bit more technical, and we need to be careful when we specify our parameters for .
As before, we choose an ${{\varepsilon}}_n>0$ for each $n$ such that $\sum_n{{\varepsilon}}_n<\infty$. Each line segment $L_i^n \subset P^n$ corresponds to a rotation $\rho_i^n$ with projective center $z_i^n$. We choose a line $\ell_i^n$ containing $z_i^n$ and a $0<\delta_i^n<{{\varepsilon}}_n$. (We will impose additional conditions on $\ell_i^n,\delta_i^n$ in .) Then we replace $\rho_i^n$ by a sequence of intrinsic rotations by applying and with $E$ replaced by $E_{n+1}'$, $\ell$ replaced by $\ell_i^n$, and ${{\varepsilon}}$ replaced by $\delta_i^n$.
Choosing each of the parameters $\delta_i^n$ sufficiently small, we obtain the balls $B(u_i^n,\delta_i^n)$ and: $$\label{6.8}
|\bigcup_i\bigcup_{p\in L_i^n}p(E_i^n)|<{{\varepsilon}}_n,$$ where $$E_i^n:=\operatorname{cl}\{x\in E_n' : \nu_x\cap B(u_i^n,\delta_i^n)={\emptyset}\}.$$
We define $i(p, n)$ as in the previous section, and again take $E_p:=\limsup_{n\to\infty}E_{i(p,n)}^n$. Then as in the previous section, the movement $\bigcup_{p\in P} p(E_p)$ covers only a null set.
Since $\delta_{i(p,n)}^n \to 0$, we know $\liminf_{n\to\infty}\operatorname{cl}B(u_{i(p,n)}^n,\delta_{i(p,n)}^n)$ can have at most one point. If it has one point, let $x_p$ be that point. Otherwise, let $x_p$ be arbitrary.
Now suppose that $y\in E_n$ and $y \not\in E^n_{i(p,n)}$. Then $\nu_y \cap \operatorname{cl}B(u_{i(p,n)}^n,\delta_{i(p,n)}^n) \neq \emptyset$. Therefore indeed $\{x \in \bigcup E_n :\, x_p \not\in \nu_x\} \subset E_p$, and the proof is finished.
The main theorem {#hide}
----------------
In , the points on $E$ that we hide at each $p \in P$ are those whose normal line passes through a particular point $x_p$. Since we would like to hide as little of $E$ as possible, it would be undesirable if an $x_p$ from our construction has the property that the normal line of positively many points of $E$ pass through $x_p$.
Fortunately such points are very rare:
There are at most countably many points with the property that the normal line of positively many points of $E$ pass through this point.
Note that for any two such points there is only one common line, and there can be only an ${\mathcal{H}}^1$-nullset of points of $E$ which have a given normal line. Since $E$ has $\sigma$-finite ${\mathcal{H}}^1$-measure, it cannot have more than countably many subsets of positive measure such that their pairwise intersections are null.
We denote the exceptional points above by $x_1,x_2,\dots$. In what follows, we show how to choose the parameters in our construction more carefully to avoid these points, i.e., so that $x_p \not\in \{x_1, x_2, \ldots \}$ for any $p \in P$.
We use the notation from the previous section. For each $n\ge 1$ and for each $L_i^n\subset P^n$, let $S_i^n$ denote the strip $B(\ell,\delta)$ assigned to *the parent* of $L_i^n$, i.e., to the line segment in $P^{n-1}$ that we replaced by a polygon in the construction of $L_i^n$. Then we choose $\ell_i^n$, $\delta_i^n$ such that $B(\ell_i^n,\delta_i^n)\subset S_i^n$ and such that $\operatorname{cl}B(\ell_i^n,\delta_i^n)\setminus\{z_i^n\}$ does not contain any of the points $x_m$ with $m \leq n$.
Then $\liminf_{n\to \infty} \operatorname{cl}B(u_{i(p, n)}^n, \delta_{i(p,n)}^n)$ is either empty, or contains one point. Suppose it contains a point $x_p$. Since $u_{i(p, n)}^n \in \ell_{i(p, n)}^n$ and the strips $\{B(\ell_{i(p, n)}^n,\delta_{i(p, n)}^n)\}_n$ are nested, it follows that $x_p \in \bigcap_n \operatorname{cl}B(\ell_{i(p, n)}^n,\delta_{i(p, n)}^n)$. By , $z_{i(p,n)}^n\not\in \operatorname{cl}B(u_{i(p, n+1)}^{n+1}, \delta_{i(p, n+1)}^{n+1})$, so $x_p \neq z_{i(p,n)}^n$ for all $n$.
Thus we have shown the following. This is the main theorem in our paper.
\[theorem:besicovitch-rot\] Let $E$ be an arbitrary rectifiable set, and let $x\mapsto\theta_x$ be an arbitrary tangent field of $E$. Then there is an $E_0\subset E$ of full ${\mathcal{H}}^1$-measure in $E$ for which the following holds.
For every path $P_0$ in $\operatorname{Isom}^+({\mathbb{R}}^2)$, and for any neighborhood of $P_0$, there is a path $P$ in the neighborhood of $P_0$ with the same endpoints as $P_0$, and there is a Borel mapping $p \mapsto x_p \in {\mathbb{P}}^2$ such that $$\label{pep-rot}
|\bigcup_{p\in P} p(\{x\in E_0:\, x_p \not \in \nu_x\})|=0.$$ Furthermore, for each $p$, the set $\{x\in E_0:\, x_p \not \in \nu_x\}$ has full ${\mathcal{H}}^1$-measure in $E$.
\[rempre2\] By the same argument as at the end of the previous section, we can get a stronger statement if the set $E$ has nice geometric properties. For instance, if it is covered by finitely many $C^1$ curves, or if it is the graph of a convex function, then the statement holds with $E_0$ replaced by $E$.
As mentioned in the introduction, consider the special case where there is a line $\ell \in ({\mathbb{P}}^2)^*$ such that there is a neighborhood of $\ell$ in which no two normal lines of $E$ intersect. Then by choosing all the lines $\ell_i^n$ to lie inside this neighborhood, we can ensure that all the $x_p$ do as well. Hence, says that we can rotate $E$ continuously by $360^\circ$, covering a set of zero Lebesgue measure, where at each time moment, we only need to delete *one point*.
\[remark:residual\] By the small neighborhood lemma, we can see that holds (with the same sets $E_i^n$) not only for the path $P^n$ but for every continuous path $P$ sufficiently close to $P^n$. Using this observation, we obtain a dense open set of curves, and then, by taking the limit, a residual set of continuous paths $P$ connecting the endpoints of $P_0$, for which the statement of holds.
Construction of a Nikodym set
-----------------------------
We conclude this paper by explaining how the continuous Besicovitch sets can be used to construct Nikodym sets for rectifiable curves.
Let $E \subset {\mathbb{R}}^2$ be an arbitrary rectifiable set. We fix an arbitrary (continuous) rectifiable curve $\Gamma \subset {\mathbb{R}}^2$ (if $E$ contains such a curve, we can choose $\Gamma$ to be that curve). By “putting a copy of $E$ onto a point $y$,” we mean that the corresponding copy of $\Gamma$ (i.e., the same isometry applied to $\Gamma$) goes through $y$.
For every continuous rectifiable curve $\Gamma$, there is a path $P_0 \subset \operatorname{Isom}^+({\mathbb{R}}^2)$ and a neighborhood of $P_0$ such that $\Gamma$ covers a set of non-empty interior along any path $P$ which lies in this neighborhood and has the same endpoints as $P_0$. (For example, if $\Gamma$ is a circle, we make sure that it is not possible for $P$ to be a rotation around the circle’s center.)
We apply with $E$ and with this neighborhood of $P_0$ to obtain a path $P$, and for each $p\in P$ to obtain a subset $E_p\subset E$ of full ${\mathcal{H}}^1$-measure so that $|\bigcup_{p \in P} p(E_p)| = 0$.
By our choice of $P_0$, we know that $\bigcup_{p \in P} p(\Gamma)$ has nonempty interior. Thus, $\bigcup_{q \in {\mathbb{Q}}^2} \bigcup_{p \in P} (q + p(\Gamma)) = {\mathbb{R}}^2$, whereas $$\label{eq:nikodym}
A
:=
\bigcup_{q \in {\mathbb{Q}}^2} \bigcup_{p \in P} (q + p(E_p))$$ has measure zero. Thus, we have shown the following.
\[theorem:nikodym\] Let $E$ be a rectifiable set and $\Gamma$ a rectifiable curve. Then the set $A$ defined by is a *Nikodym set* for $E$:
1. $A$ has Lebesgue measure zero;
2. Through each point $y \in {\mathbb{R}}^2$, $A$ contains a copy of ${\mathcal{H}}^1$-a.e. point of $E$. That is, for all $y \in {\mathbb{R}}^2$, there is an $E_y \subset E$ and a $p_y \in \operatorname{Isom}^+({\mathbb{R}}^2)$ such that ${\mathcal{H}}^1(E \setminus E_y) = 0$, $y \in p_y(\Gamma)$, and $p_y(E_y) \subset A$.
With in place of we can prove a result about placing *translated* copies of $E$ at each point $y \in {\mathbb{R}}^2$. By essentially the same arguments as above, we now obtain a path $P \subset {\mathbb{R}}^2$ and $\theta_p \in {\mathbb{P}}^1$ such that $\bigcup_{p \in P} (p + E_p)$ has Lebesgue measure zero, where $E_p = \{x \in E_0 : \theta_x \neq \theta_p\}$, and such that $\bigcup_{p \in P} (p + \Gamma)$ has nonempty interior. Thus, $\bigcup_{q \in {\mathbb{Q}}^2} \bigcup_{p \in P} (q + p + \Gamma) = {\mathbb{R}}^2$, whereas $$\label{eq:nikodym-translations}
A
:=
\bigcup_{q \in {\mathbb{Q}}^2} \bigcup_{p \in P} (q + p + E_p)$$ has Lebesgue measure zero. To ensure that $E_p$ has full ${\mathcal{H}}^1$-measure in $E$, it is sufficient to assume that $\{x \in E : \theta_x = \theta\}$ is ${\mathcal{H}}^1$-null for every $\theta \in {\mathbb{P}}^1$.
\[theorem:nikodym-translations\] Let $E$ be a rectifiable set and $\Gamma$ a rectifiable curve. Suppose that for every direction $\theta \in {\mathbb{P}}^1$, the set $\{x \in E : \theta_x = \theta\}$ is ${\mathcal{H}}^1$-null. Then the set $A$ defined by satisfies the following:
1. $A$ has Lebesgue measure zero;
2. Through each point $y \in {\mathbb{R}}^2$, $A$ contains a translated copy of ${\mathcal{H}}^1$-a.e. point of $E$. That is, for all $y \in {\mathbb{R}}^2$, there is an $E_y \subset E$ and a $p_y \in {\mathbb{R}}^2$ such that ${\mathcal{H}}^1(E \setminus E_y) = 0$, $y \in p_y + \Gamma $, and $p_y + E_y \subset A$.
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|
---
abstract: 'Quantum state tomography is an important tool in quantum information science for complete characterization of multi-qubit states and their correlations. Here we report a method to perform a joint simultaneous read-out of two superconducting qubits dispersively coupled to the same mode of a microwave transmission line resonator. The non-linear dependence of the resonator transmission on the qubit state dependent cavity frequency allows us to extract the full two-qubit correlations without the need for single shot read-out of individual qubits. We employ standard tomographic techniques to reconstruct the density matrix of two-qubit quantum states.'
author:
- 'S. Filipp'
- 'P. Maurer'
- 'P. J. Leek'
- 'M. Baur'
- 'R. Bianchetti'
- 'J. M. Fink'
- 'M. Göppl'
- 'L. Steffen'
- 'J. M. Gambetta'
- 'A. Blais'
- 'A. Wallraff'
title: 'Two-Qubit State Tomography using a Joint Dispersive Read-Out'
---
[^1]
Quantum state tomography allows for the reconstruction of an a-priori unknown state of a quantum system by measuring a complete set of observables [@Paris2004]. It is an essential tool in the development of quantum information processing [@Nielsen2000] and has first been used to reconstruct the Wigner-function [@Schleich2001] of a light mode [@Smithey1993] by homodyne measurements, as suggested in a seminal paper by Vogel and Risken [@Vogel1989]. Subsequently, state tomography has been applied to other systems with a continuous spectrum, for instance, to determine vibrational states of molecules [@Dunn1995], ions [@Leibfried1996] and atoms [@Kurtsiefer1997]. Later, techniques have been adapted to systems with a discrete spectrum, for example nuclear spins [@Chuang1998], polarization entangled photon pairs [@White1999], electronic states of trapped ions [@Roos2004], states of hybrid atom-photon systems [@Volz2006], and spin-path entangled single neutrons [@Hasegawa2007].
Recent advances have enabled the coherent control of individual quantum two-level systems embedded in a solid-state environment. Numerous experiments have been performed with superconducting quantum devices [@Clarke2008], manifesting the rapid progress and the promising future of this approach to quantum information processing. In particular, the strong coupling of superconducting qubits to a coplanar waveguide resonator can be exploited to perform cavity quantum electrodynamics (QED) experiments on a chip [@Wallraff2004b; @Blais2004; @Schoelkopf2008] in an architecture known as circuit QED. The high level of control over the dynamics of this coupled quantum system has been demonstrated, e. g., in [@Leek2007; @Hofheinz2008]. State tomographic methods have already been used in superconducting circuits to verify the entanglement between two phase qubits [@Steffen2006a]. There, the state is determined for each individual qubit with single-shot read-out such that two-qubit correlations can be evaluated by correlating the single measurement outcomes. In contrast, in this letter we extract two-qubit correlations from the simultaneous averaged measurement of two qubits dispersively coupled to a common resonator. This possibility has also been pointed out in Ref. .
In the setup shown in Fig. \[fig:setup\], two superconducting qubits are coupled to a transmission line resonator operating in the microwave regime [@Majer2007].
![Schematic of the experimental setup with two qubits coupled via the capacitances $\rm{C}_{\rm{g}}$ to a microwave resonator operated at a temperature of about $20~\rm{mK}$. The transition frequencies of the qubits are adjusted by external fluxes $\Phi_1$ and $\Phi_2$. The resonator-qubit system is probed through the input and output capacitances $\rm{C}_{\rm{in}}$ and $\rm{C}_{\rm{out}}$ by a microwave signal at frequency $\omega_{m}$. Additionally, local control of the qubits is implemented by capacitively coupled signals $\omega_{d1}$ and $\omega_{d2}$, which are phase and amplitude modulated using IQ mixers. The output signal is detected in a homodyne measurement at room temperature.[]{data-label="fig:setup"}](setupst){width="86mm"}
Due to the large dipole moment of the qubits and the large vacuum field of the resonator the strong coupling regime with $g_{1,2}\gg
\kappa,\gamma_1$ is reached. $g_1/2\pi \approx g_2/2\pi = 133~\rm{MHz}$ denotes the similar coupling strenghts of both qubits and $\kappa/2\pi\approx 1.65~\rm{MHz}$, $\gamma_1/2\pi \approx 0.25~\rm{MHz}$ the photon and the qubit decay rates, respectively. The qubits are realized as transmons [@Koch2007], a variant of a split Cooper pair box [@Bouchiat1998] with exponentially suppressed sensitivy to 1/f charge noise [@Schreier2008]. The transition frequencies $\omega_{aj}$ ($j=1,2$) of the qubits are tuned separately by external flux bias coils. Both qubits can be addressed individually through local gate-lines using amplitude and phase modulated microwaves at frequencies $\omega_{d1}$ and $\omega_{d2}$. Read-out is accomplished by measuring the transmission of microwaves applied to the resonator input at frequency $\omega_m$ close to the fundamental resonator mode $\omega_r$. At large detunings $\Delta_{j} \equiv \omega_{aj}-\omega_r$ of both qubits from the resonator, the dispersive qubit-resonator interaction gives rise to a qubit state dependent shift of the resonator frequency. In this dispersive limit and in a frame rotating at $\omega_m$ the relevant Hamiltonian reads [@Blais2007] $$\begin{aligned}
\label{eq:hdisp}
H &=& \hbar\big(\Delta_{rm} + \chi_1{\hat{\sigma}}_{z1}+\chi_2
{\hat{\sigma}}_{z2}\big){\hat{a}}^\dagger {\hat{a}}\\\nonumber
&&+\frac{\hbar}{2}\sum_{j=1,2}\left(\omega_{aj}+\chi_j\right){\hat{\sigma}}_{zj}
+ \hbar\epsilon(t) ({\hat{a}}^\dagger + {\hat{a}}),\end{aligned}$$ where $\Delta_{rm}\equiv \omega_r - \omega_m$ is the detuning of the measurement drive from the resonator frequency. The coefficients $\chi_{1,2}$ are determined by the detuning $\Delta_{1,2}$, the coupling strength $g_{1,2}$ and the design parameters of the qubit [@Koch2007]. The last term in Eq. (\[eq:hdisp\]) models the measurement drive with amplitude $\epsilon(t)$.
The operator ${\hat{\chi}}\equiv \chi_1{\hat{\sigma}}_{z1}+\chi_2
{\hat{\sigma}}_{z2} $, which describes the dispersive shift of the resonator frequency, is linear in both qubit states. It does not contain two-qubit terms like ${\hat{\sigma}}_{z1}{\hat{\sigma}}_{z2}$ from which information about the qubit-qubit correlations could be obtained. However, in circuit QED instead of measuring frequency shifts directly, we record quadrature amplitudes of microwave transmission through the resonator which depend nonlinearly on these shifts. The average values of the field quadratures $\langle
{\hat{I}}(t)\rangle = \operatorname{Tr}[{\hat{\rho}}(t)({\hat{a}}^\dagger +
{\hat{a}})]$ and $\langle {\hat{Q}} \rangle =
{{\mathrm i}}\operatorname{Tr}[{\hat{\rho}}(t)({\hat{a}}^\dagger - {\hat{a}})]$ are determined from the amplified voltage signal at the resonator output in a homodyne measurement. Here, ${\hat{\rho}}(t) = U_m(t)
{\hat{\rho}}(0)U_m(t)^\dagger$ denotes the time evolved state of both resonator and qubit under measurement. In the dispersive approximation we can safely assume this state to be separable before the measurement, which is taken to start at time $t_m$, ${\hat{\rho}}(t_m) = {|0\rangle\langle0|}\otimes
{\hat{\rho}}_q(t_m)$. Using these expressions, we find $\langle {\hat{Q}}(t) \rangle = {{\mathrm i}}\operatorname{Tr}_q[{\hat{\rho}}_q(t_m){\langle 0|}{\hat{U}}^\dagger_m(t)
({\hat{a}}^\dagger - {\hat{a}}) {\hat{U}}_m(t){|0\rangle}]$ (and similarly for $\langle {\hat{I}}(t) \rangle$), where $\operatorname{Tr}_q$ denotes the partial trace over the qubit. This expression is evaluated using the input-ouput formalism [@Gardiner1992] including cavity decay $\kappa$. In the steady-state, this yields $\langle
{\hat{I}} \rangle_s, \langle Q \rangle_s = -\epsilon \operatorname{Tr}_q
[\rho_q(t_m){\hat{M}}_{I,Q}]$ with $$\begin{aligned}
\label{eq:it}
{\hat{M}}_I &=& \frac{2(\Delta_{rm} + {\hat{\chi}})}{(\Delta_{rm} +{\hat{\chi}})^2 +
(\kappa/2)^2},\\ {\hat{M}}_Q &=&\frac{\kappa}{(\Delta_{rm} + {\hat{\chi}})^2 +
(\kappa/2)^2}.
\end{aligned}$$ We note that the measurement operators are nonlinear functions of ${\hat{\chi}}$. Thus, ${\hat{M}}_{I,Q}$ comprises in general also two-qubit correlation terms proportional to $\sigma_{z1} \sigma_{z2}$, which allow to reconstruct the full two-qubit state.
In our experiments the phase of the measurement microwave at frequency $\Delta_{rm} = (\chi_1+\chi_2)$ is adjusted such that the $Q$-quadrature of the transmitted signal carries most of the signal when both qubits are in the ground state. The corresponding measurement operator can be expressed as $$\label{eq:M}
{\hat{M}} =\frac{1}{4}\left( \beta_{00} {\hat{{\text{id}}}} + \beta_{10} {\hat{\sigma}}_{z1} + \beta_{01} {\hat{\sigma}}_{z2} + \beta_{11} {\hat{\sigma}}_{z1} {\hat{\sigma}}_{z2}\right),$$ where $\beta_{ij} = \alpha_{\text{\scriptsize -\,-}} + (-1)^{j}
\alpha_{\text{\scriptsize -+}} + (-1)^i \alpha_{\text{\scriptsize
+\,-}} + (-1)^{i+j}\alpha_{\text{\scriptsize ++}}$, with the coefficients $$\label{eq:alpha}
\alpha_{\text{\scriptsize $\pm\pm$}} = -\epsilon\kappa \{(\kappa/2)^2 + (\Delta_{rm} \pm \chi_1 \pm
\chi_2)^2\}^{-1/2}$$ representing the qubit state dependent $Q$-quadrature amplitudes of the resonator field in the steady-state limit and for infinite qubit-lifetime (Fig. \[fig:phaseshift\](a)).
Since we operate in a regime, where the qubit relaxation cannot be neglected, the steady-state expression is of limited practical use. The decay of a qubit to its ground state changes the resonance frequency of the resonator and consequently limits the read-out time to $\sim 1/\gamma_1$. A typical averaged time-trace of the resonator response for pulsed measurements is shown in Fig. \[fig:phaseshift\](b), similar to the data presented in [@Majer2007]. The qubits are prepared initially in the states ${|ee\rangle}$, ${|eg\rangle}$, ${|ge\rangle}$ and ${|gg\rangle}$, respectively, using the local gate lines. The time dependence of the measurement signal is determined by the rise time of the resonator and the decay time of the qubits. It is in excellent agreement with calculations (solid lines in Fig. \[fig:phaseshift\](b)) of the dynamics of the dispersive Jaynes-Cummings Hamiltonian [@Blais2007; @Bianchetti2008].
![(a) $Q$-quadrature of the resonator field for the qubits in states $gg$, $eg$, $ge$ and $ee$ as a function of the detuning $\Delta_{rm}$. Tomography measurements have been performed at $\Delta_{rm}=(\chi_1+\chi_2)$ indicated by an arrow. (b) Measured (data points) time evolution of the $Q$-quadrature for the indicated initial states compared to numerically calculated responses (solid lines). All parameters have been determined in independent measurements.[]{data-label="fig:phaseshift"}](responseQ){width="86mm"}
Due to the quantum non-demolition nature of the measurement [@Blais2004], ${\hat{M}}$ remains diagonal in the instantaneous qubit eigenbasis during the measurement process. Therefore, a suitable realistic measurement operator ${\hat{M}}'$ can be defined by replacing the $\alpha_{\pm\pm}$ in Eq. (\[eq:alpha\]) with the integrated signal from $t_m$ to the final time $T$, $\alpha_{\pm\pm}' = 1/N \int_{t_m}^T
(\langle {\hat{M}}(t) \rangle_{\pm\pm} - \langle{\hat{M}}(t)
\rangle_{\text{\scriptsize -\,-}}) dt$ with the ground state response $\langle{\hat{M}}(t)\rangle_{\text{\scriptsize -\,-}}$ subtracted. The normalization constant $N$ is chosen such that $\alpha_{\text{\scriptsize
+\,-}}'=1$.
To reconstruct the combined state ${\hat{\rho}}_q$ of both qubits, a suitable set of measurements has to be found to determine unambiguously the 16 coefficients $r_{ij}$ of the density matrix ${\hat{\rho}}_q =
\sum_{i,j=0}^3 r_{ij}\, {\hat{\sigma}}_{i} \otimes {\hat{\sigma}}_{j}$ with the identity ${\hat{\sigma}}_0={\hat{{\text{id}}}}$ and $\{{\hat{\sigma}}_{1},{\hat{\sigma}}_{2},{\hat{\sigma}}_{3}\} =
\{{\hat{\sigma}}_{x},{\hat{\sigma}}_{y},{\hat{\sigma}}_{z}\}$. Such a complete set of measurements is constructed by applying appropriate single qubit rotations ${\hat{U}}_k \in SU(2)\otimes SU(2)$ before the measurement in order to measure the expectation values $\langle {\hat{M}}_k\rangle = \operatorname{Tr}[{\hat{M}} {\hat{U}}_k{\hat{\rho}}_q
{\hat{U}}_k^\dagger] = \operatorname{Tr}[{\hat{U}}_k^\dagger {\hat{M}}
{\hat{U}}_k{\hat{\rho}}_q]$. The latter equality defines the set of measurement operators ${\hat{M}}_k\equiv
{\hat{U}}_k^\dagger {\hat{M}} U_k$. This illustrates again that a measurement operator ${\hat{M}}$ involving non-trivial two-qubit terms $\sigma_{i1} \sigma_{j2}$ is necessary for state tomography. In fact, single-qubit operations $U_k = U_{k1} \otimes U_{k2}$ alone cannot be used to generate correlation terms since $U_k^\dagger ({\hat{{\text{id}}}} \otimes \sigma_{z}) U_k = {\hat{{\text{id}}}} \otimes
(U_{k2}^\dagger\sigma_{z} U_{k2})$, for instance. As $\operatorname{Tr}[(\sigma_{k}\otimes\sigma_{l})(\sigma_{m}\otimes\sigma_{n})]=\delta_{km}\delta_{ln}$, some coefficients $r_{ij}$ of the density matrix ${\hat{\rho}}_q$ would not be determined in an averaged measurement.
To identify the coefficients $r_{ij}$ we perform 16 linearly-independent measurements. The condition for the completeness of the set of tomographic measurements is the non-singularity of the matrix $A$ defined by the relation $\langle {\hat{M}}_k\rangle =
\sum_{l=0}^{15} A_{kl} r_l$ between the the expectation values $\langle {\hat{M}}_k\rangle$ and the coefficients of the density matrix $r_l$ with $l\equiv
i+4j$. This condition is only violated if one of the coefficients $\beta_{ij}$ of ${\hat{M}}$ in Eq. (\[eq:M\]) vanishes. For instance, $\beta_{01}=\beta_{10}=0$ for $\Delta_{rm}=0$, which reflects the fact that we cannot distinguish two identical qubits due to symmetry reasons as apparent from Fig. \[fig:phaseshift\](a).
Our pulse scheme for the state tomography is shown in Fig. \[fig:pulsescheme\]. The transition frequencies of the qubits are adjusted to $\omega_{a1}/2\pi = 4.5~\rm{GHz}$ and $\omega_{a2}/2\pi = 4.85~\rm{GHz}$. At this detuning from the resonator frequency $\omega_r/2\pi = 6.442~\rm{GHz}$ the cavity pulls are $\chi_1 = - 1~\rm{MHz}$ and $\chi_2=-1.5~\rm{MHz}$ [@Koch2007]. First, a given two-qubit state is prepared. Then a complete set of tomography measurements is formed by applying the combination of $\{(\pi/2)_x,\,(\pi/2)_y,(\pi),{\text{id}}\}$ pulses to both qubits over their individual gate lines using amplitude and phase controlled microwave signals. The wanted rotation angles are realized with an accuracy better than $4^\circ$. Finally, the measurement drive is applied at $\omega_m = 6.445~\rm{GHz}$ corresponding to the maximum transmission frequency of the resonator with both qubits in the ground state.
![Pulse scheme for state tomography, see text.[]{data-label="fig:pulsescheme"}](pulsescheme){width="86mm"}
To determine the measurement operator ${\hat{M}}'$, $\pi$-pulses are alternately applied to both qubits to yield signals as shown in Fig. \[fig:phaseshift\](b). From this data the coefficients $(\beta_{00}',\beta_{01}',\beta_{10}',\beta_{11}')=(0.8,-0.3,-0.4,-0.1)$ of ${\hat{M}}'$ are deduced, where the non-vanishing $\beta_{11}'$ allows for a measurement of arbitrary quantum states. As an example of this state reconstruction, in Fig. \[fig:statetomprod\](a) the extracted density matrix ${\hat{\rho}}_{q}$ of the product state ${|\Psi_{\rm{sep}}\rangle}=1/\sqrt{2}\left({|g\rangle}+{|e\rangle}\right)\otimes
1/\sqrt{2}\left({|g\rangle}+{{\mathrm i}}{|e\rangle}\right)$ is shown. In Fig. \[fig:statetomprod\](b) the Bell state ${|\Phi\rangle} =
1/\sqrt{2}\left({|g\rangle}\otimes{|g\rangle}-{{\mathrm i}}{|e\rangle}\otimes{|e\rangle}\right)$ prepared by a sequence of sideband pulses [@Wallraff2007; @Blais2007; @Leek2008] is reconstructed. $6.6\times 10^4$ and $6.6\times 10^5$ records have been averaged, respectively, for each of the 16 tomographic measurement pulses to determine the expectation values $\langle {\hat{M}}_k'\rangle$ for the two states. The corresponding ideal state tomograms are depicted in Fig. \[fig:statetomprod\](c) and (d). To avoid unphysical, non positive-semidefinite, density matrices originating from statistical uncertainties, all tomography data has been processed by a maximum likelihood method [@Hradil1997; @James2001]. The corresponding fidelities $\mathcal{F}_{\psi} \equiv ({\langle \psi|}{\hat{\rho}}_{q}{|\psi\rangle})^{1/2}$ are $\mathcal{F}_{\rm{sep}} = 95\%$ and $\mathcal{F}_{\Phi^-}=74\%$. These results are in close agreement with theoretically expected fidelities when taking finite photon and qubit-lifetimes into account. As a result, the loss in fidelity and concurrence of the Bell state are not due to measurement errors but to the long preparation sequence [@Leek2008].
{width="172mm"}
In conclusion, we have presented a method to jointly and simultaneously read-out the full quantum state of two qubits dispersively coupled to a microwave resonator. In a measurement of the field quadrature amplitudes of microwaves transmitted through the resonator each photon carries information about the state of both qubits. In this way the two-qubit correlations can be extracted from an averaged measurement of the transmission amplitude without the need for single shot or single qubit read-out. This method can also be extended to multi-qubit systems coupled to the same resonator mode.
This work was supported by Swiss National Science Foundation (SNF) and ETH Zurich. P. J. L. was supported by the EC with a MC-EIF, J. M. G. by CIFAR, MITACS and ORDCF and A. B. by NSERC and CIFAR.
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[^1]: The first two authors contributed equally to this work.
|
---
abstract: 'We established a new algorithm for correlation process in radio astronomy. This scheme consists of the 1st-stage Fourier Transform as a filter and the 2nd-stage Fourier Transform for spectroscopy. The “FFX” correlator stands for Filter and FX architecture, since the 1st-stage Fourier Transform is performed as a digital filter, and the 2nd-stage Fourier Transform is performed as a conventional FX scheme. We developed the FFX correlator hardware not only for the verification of the FFX scheme algorithm but also for the application to the Atacama Submillimeter Telescope Experiment (ASTE) telescope toward high-dispersion and wideband radio observation at submillimeter wavelengths. In this paper, we present the principle of the FFX correlator and its properties, as well as the evaluation results with the production version.'
author:
- 'Satoru <span style="font-variant:small-caps;">Iguchi</span> and Takeshi <span style="font-variant:small-caps;">Okuda</span>'
title: The FFX Correlator
---
Introduction
============
The signals received by the antennas obey the stationary stochastic process and then ergodic process. The ergodic theory can be applied to the auto-correlation function for a spectrometer and the cross-correlation function for radio interferometer. Under such conditions, Weinreb (1963) developed the first digital spectrometer. This digital spectrometer is called the XF correlator in which the correlation is calculated before Fourier Transform. Meanwhile, Chikada et al. (1987) developed the first the FX correlator of an another design, in which Fourier Transform is performed before cross multiplication. Although there is a difference of property between two basic designs, the obtained astronomical spectra of them were confirmed to be identical.
Determining the number of correlation lags in the XF scheme or of Fourier Transform points in the FX scheme is essential for the realization of high-dispersion and wideband observation, because the frequency resolution is derived as $$\Delta f = 1/(\Delta t_\mathrm{s} N) = 2B/N,
\label{eq:it}$$ where $\Delta t_\mathrm{s}$ is the sampling period, $N$ is the number of correlation lags or Fourier Transform points, and the bandwidth of B is equal to $1/(2 \Delta t_\mathrm{s})$. The material size and cost of the correlator strongly depend on the sampling period, $\Delta t_\mathrm{s}$, and the number of correlation lags or Fourier Transform points, $N$.
The new XF architecture with the digital Tunable Filter Bank that is designed with the Finite Impulse Response (FIR) has been proposed and developed for the next generation radio interferometers, the Expanded Very Large Array (EVLA) and the Atacama Large Millimeter/submillimeter Array (ALMA) ([@EVLA], [@ALMA1]). This is called the “FXF correlator”. The architecture of the FXF scheme can make the material size smaller in comparison with that of the conventional XF scheme. Since the digital filter allows a variety of observation modes \[scientific and observational availability were shown in Iguchi et al. (2004)\], the FXF scheme will provide us with the most appropriate specifications which meet the scientific requirements. This will lower the risk of over-engineering of the correlator.
The improved FX architecture with DFT filterbank was developed by Bunton (2000). The use of polyphase filter banks allows arbitrary filter responses to be implemented in the FX scheme (Bunton 2003). This is called the “Polyphase FX Correlator”. This scheme has a possibility to achieve the spectral leakage of about -120 dB. In particular, this performance is significant to suppress the leakage from the spurious lines mixed in receiving, down-converting or digitizing.
The FFX Correlator is a new algorithm for correlation process in radio astronomy. The FFX scheme consists of 2-stage Fourier Transform blocks, which perform the 1st-stage Fourier Transform as a digital filter, and the 2nd-stage Fourier Transform to achieve higher dispersion. The first ’F’ of the FFX is the initial letter of the word “Filter”. In this paper, we present a new FFX architecture. The principle of the FFX scheme in section 2, the properties of the FFX scheme in section 3, the algorithm verification and performance evaluation with the developed FFX correlator in sections 4 and 5, and the summary of this paper in section 6 are presented.
Principle of the FFX Correlator
===============================
This section shows the algorithm and the data flow diagram of the signal processing in the Fourier Transform of the FFX scheme (see figure \[fig:ffx\]). Suppose that $x_\mathrm{n}$ are the digital waveforms at the correlator input from the astronomical radio signals that are received by the telescope. The inputs, $x_\mathrm{n}$, are real digital signals at sampling period of $\Delta t_\mathrm{s}$, and obey the zero-mean Gaussian random variable. The suffix $n$ is an integer for time.
Fig1
(160mm,200mm)[fig1.eps]{}
\[Step 1\] The correlator receives the time-domain digital sampling signals from the Analog-to-Digital Converter (ADC), and accumulate them up to $N_\mathrm{n}$ points.
\[Step 2\] The time-domain $N_\mathrm{n}$-point data are transferred to the frequency-domain by using the $N_\mathrm{n}$-point Discrete Complex Fourier Transform as follows: $$X_\mathrm{p} = \Delta t_\mathrm{s} \sum^{N_\mathrm{n}-1}_{n=0} x_\mathrm{n}
\exp\left(- j \frac{2 \pi p n}{N_\mathrm{n}}\right),$$ where $X$ is the spectrum after the 1st Fourier Transform, the suffix $p$ is an integer for frequency, and $\Delta t_\mathrm{s}$ is equal to $1/(2B_\mathrm{1st})$ at the bandwidth of $B_\mathrm{1st}$. The $\Delta f_\mathrm{1st}$ is the minimum frequency resolution of the 1st Fourier Transform, which is equal to $1/(\Delta t_\mathrm{s} N_\mathrm{n})$.
\[Step 3\] The extraction of the $N_\mathrm{k}$ points from the frequency domain $N_\mathrm{n}/2$-point data after the 1st Fourier Transform is conducted as if filter and frequency conversion are performed simultaneously: $$X'_\mathrm{k} = X_{\mathrm{p}} \hspace{5mm} (p=k+k_0, \hspace{2mm} k = 0, \cdot \cdot \cdot, N_\mathrm{k}-1),
\label{eq:df}$$ where $k_0$ is the minimum frequency channel in the extraction, and the suffix $k$ is an integer for frequency.
\[Step 4\] The $N_\mathrm{k}$-point data after Inverse Fourier Transform is written by $$x'_\mathrm{l} = \frac{1}{\Delta t_\mathrm{f} N_\mathrm{k}} \sum^{N_\mathrm{k}-1}_{k=0} X'_\mathrm{k} \exp\left[j \frac{2 \pi (k-N_\mathrm{k}/2) l}{N_\mathrm{k}}\right],$$ where $x'$ is the time-domain signal after inverse Fourier Transform, the suffix $l$ is an integer for time, and $\Delta t_\mathrm{f}$ is the sampling period after filtering at the bandwidth of $B_\mathrm{2nd}$ $(\Delta t_\mathrm{f}=1/B_\mathrm{2nd}=1/\Delta f_\mathrm{1st} N_\mathrm{k})$.
\[Step 5\] By repeating the procedure from Step 1 to Step 4, the data are gathered up to $N_\mathrm{m}$ points as follows; $$x'_\mathrm{m} = x'_\mathrm{l + d N_\mathrm{k}}$$ where $m$ is $l+d N_\mathrm{k}$, and $d$ is the number of repeating times of the procedure from Step 1 to Step 4.
\[Step 6\] The time-domain $N_\mathrm{m}$-point data after gathering are transferred to the frequency-domain by using the $N_\mathrm{m}$-point Discrete Complex Fourier Transform as follows: $$X'_\mathrm{q} = \Delta t_\mathrm{f} \sum^{N_\mathrm{m}-1}_{m=0} x'_\mathrm{m}
\exp\left(- j \frac{2 \pi q m}{N_\mathrm{m}}\right),$$ where $X'$ is the spectrum after the 2nd Fourier Transform, and the suffix $q$ is an integer for frequency. The $\Delta f_\mathrm{2nd}$ is the minimum frequency resolution after the 2nd Fourier Transform, which is equal to $1/(\Delta t_\mathrm{f} N_\mathrm{m})$ (=$\Delta f_\mathrm{1st} N_\mathrm{k}/N_\mathrm{m}$).
Mark Explanation
------------------------- --------------------------------------------
$B_\mathrm{1st}$ Bandwidth of the input signals
$\Delta t_\mathrm{s}$ Sampling period of the input signals
$N_\mathrm{n}$ Number of the points of 1st FT
$\Delta f_\mathrm{1st}$ Minimum frequency resolution of 1st FT
$N_\mathrm{k}$ Number of the extraction times as a filter
$B_\mathrm{2nd}$ Bandwidth after extraction
$\Delta t_\mathrm{f}$ Sampling period after filtering
$N_\mathrm{m}$ Number of the points of 2nd FT
$\Delta f_\mathrm{2nd}$ Minimum frequency resolution of 2nd FT
: Definition of functions.
\[table:word\]
Note that FT is Fourier Transform.
Equation
----- ------------------------- ------------------------------------------------------------------
(a) $\Delta t_\mathrm{s}$ = $1/(2B_\mathrm{1st}) $
(b) $\Delta f_\mathrm{1st}$ = $2B_\mathrm{1st}/N_\mathrm{n} $
= $1/(\Delta t_\mathrm{s} N_\mathrm{n})$
(c) $\Delta t_\mathrm{f}$ = $1/B_\mathrm{2nd}$
= $1/(\Delta f_\mathrm{1st} N_\mathrm{k}) $
= $\Delta t_\mathrm{s} N_\mathrm{n}/N_\mathrm{k} $
(d) $\Delta f_\mathrm{2nd}$ = $B_\mathrm{2nd} / N_\mathrm{m}$
= $1/(\Delta t_\mathrm{f} N_\mathrm{m})$
= $\Delta f_\mathrm{1st} N_\mathrm{k}/N_\mathrm{m}$
= $2B_\mathrm{1st}/N_\mathrm{n} \cdot N_\mathrm{k}/N_\mathrm{m}$
: Relationship among the functions (see table \[table:word\]).
\[table:summary\]
The definition of all functions used in this section is summarized in table \[table:word\]. Also, the summary of the relationship among the functions (see table \[table:word\]) is listed in table \[table:summary\]. The following relations are derived: $$\begin{aligned}
\Delta f_\mathrm{1st} &=& 2B_\mathrm{1st}/N_\mathrm{n}, \\
\Delta f_\mathrm{2nd} &=& 2B_\mathrm{1st}/N_\mathrm{n} \cdot N_\mathrm{k}/N_\mathrm{m}. \end{aligned}$$ The frequency resolution of the FFX scheme is determined by the number of of the 1st Fourier Transform points($N_\mathrm{n}$), the number of the extractions as a filter ($N_\mathrm{k}$), and the number of the 2nd Fourier Transform points($N_\mathrm{m}$).
Properties of the FFX Correlator
================================
The frequency responses for spectroscopic observations are finally derived by Fourier Transform in all schemes. For finite length of Fourier Transform, the responses are multiplied by a rectangular window function, which corresponds to convolving sinc function in the frequency domain. The frequency profile of the XF scheme becomes the shape of sinc function profile, while that of the FX scheme is sinc squared function profile. This indicates the FX scheme (including the FFX and polyphase FX schemes) is better than the XF scheme (including the FXF scheme) from the view points of the frequency profile and sharpness of individual frequency channels, and the spectral leakage.
For the realization of the high frequency resolution in the conventional FX scheme, the $N$-point complex FFT may be divided into $N/2$-point FFTs, $N$-point twiddle factor multiplications, and $N/2$-point second FFTs, because the circuit size of the LSI (Large-Scale Integration) is limited ([@IG02]). The memory and circuit for the twiddle factor multiplications are critical. However, the high frequency resolution can be realized in the FFX scheme without the twiddle factor multiplications.
The FFX scheme has advantages of selectivity for frequency resolution and bandwidth in comparison with other schemes. The comparable functions are also realized by implementing a digital LO circuits ([@ALMA1]). The digital LO circuits need to be delicately designed to avoid the spurious in mixing the digital LO signals due to the rounding errors in the calculation process.
(147mm,100mm)[fig2.eps]{}
For the investigation of aliasing or foldover from frequencies over the bandedge in the FFX scheme, it is necessary and important to estimate the frequency response. As shown in \[Step 3\] in figure \[fig:ffx\], the desired frequency response, equation (\[eq:df\]) can be rewritten as $$\begin{aligned}
X'_\mathrm{k} &=&
X'_\mathrm{p} \hspace{2mm} (p=k+k_0, \hspace{2mm} k = 0, \cdots, N_\mathrm{k}-1), \\
X'_\mathrm{p} &=& H_\mathrm{p} \cdot X_\mathrm{p}, \\
H_\mathrm{p} &=&
\left\{
\begin{array}{l@{}l}
1 & \hspace{2mm} (p = k_0, \cdots, k_0+N_\mathrm{k}-1) \\
0 & \hspace{2mm} (p < k_0, \hspace{2mm} p > k_0+N_\mathrm{k}-1).
\end{array}
\right.
\label{eq:desiredFSF}\end{aligned}$$ However, “$H_\mathrm{k_0}$” should be replaced by zero to reduce the aliasing or foldover of noise from frequencies over the bandedge. Thus, in the FFX scheme, equation (\[eq:desiredFSF\]) need to be replaced as $$\begin{aligned}
H_\mathrm{p} &=&
\left\{
\begin{array}{l@{}l}
1 & \hspace{2mm} (p = k_0+1, \cdots, k_0+N_\mathrm{k}-1) \\
0 & \hspace{2mm} (p < k_0+1, \hspace{2mm} p > k_0+N_\mathrm{k}-1).
\end{array}
\right.
\label{eq:desiredFFX}\end{aligned}$$ According to the two (1st and inverse) Fourier Transforms at the same resolution, the actual designed transfer function of the impulse response, which is derived with the square of sinc function, is represented as follows: $$\begin{aligned}
|H(f)| &=& \sum_{p=0}^{N_\mathrm{n}-1} H_\mathrm{p}^2 \left\{ \frac{\sin \left[\pi \left(N_\mathrm{n} \Delta t_\mathrm{s}f - p \right)\right]}{\pi \left(N_\mathrm{n} \Delta t_\mathrm{s} f - p \right) } \right\}^2, \\
P(f) &=& \left\{ \sum_{p=0}^{N_\mathrm{n}-1} H_\mathrm{p}^2 \left\{ \frac{\sin \left[\pi \left(N_\mathrm{n} \Delta t_\mathrm{s} f - p \right)\right]}{\pi \left(N_\mathrm{n} \Delta t_\mathrm{s} f - p \right) } \right\}^2 \right\}^2,
\label{eq:theory}\end{aligned}$$ where $f$ is an arbitrary frequency, and also the response of a sinc function is caused by one Fourier Transform. It can be confirmed that the designed transfer function approaches the desired frequency response by increasing $N_\mathrm{n}$ while keeping $N_\mathrm{n}/N_\mathrm{k}$ constant. The filtering process in the FFX scheme is similar to that used in the design method of a Frequency Sampling Filter, FSF ([@FSF]).
Also, to improve the frequency response, the window function before 1st Fourier Transform can be multiplied. In that case, equation (\[eq:theory\]) should be written as $$\begin{aligned}
&&|H(f)| = \nonumber \\
&&\sum_{p=0}^{N_\mathrm{n}-1} H_\mathrm{p}^2 W(N_\mathrm{n} \Delta t_\mathrm{s} f - p) \frac{\sin \left[\pi \left(N_\mathrm{n} \Delta t_\mathrm{s} f - p \right)\right]}{\pi \left(N_\mathrm{n} \Delta t_\mathrm{s} f - p \right) } , \\
&&P(f) = \nonumber \\
&&\left\{ \sum_{p=0}^{N_\mathrm{n}-1} H_\mathrm{p}^2 W(N_\mathrm{n} \Delta t_\mathrm{s} f - p) \frac{\sin \left[\pi \left(N_\mathrm{n} \Delta t_\mathrm{s} f - p \right)\right]}{\pi \left(N_\mathrm{n} \Delta t_\mathrm{s} f - p \right) } \right\}^2,
\label{eq:theory2}\end{aligned}$$ where $W(f)$ is the response after the window function $w(n)$ is transferred to the frequency-domain by Fourier Transform. If the window is a rectangular window function, W(f) becomes a sinc function. In that case, equation (\[eq:theory2\]) consists with equation (\[eq:theory\]). There are the following famous window functions: Hanning, Hamming, Blackman, and Kaiser. By well making a choice of the window function, the first sidelobe of stopband in the frequency response will be improved. It is well known that the first sidelobe levels with Backman and Kaiser as the window are better than those Hanning and Hamming. For the FFX scheme, it is found that the Bessel function of zeroth order $J_0$ is better than others. This frequency response is shown in figure \[fig:bessel\]. The first and second sidelobe levels is about -34 dB, the fifth and sixth sidelobe levels is about -50 dB, and higher-order sidelobe levels will be better than -60 dB (see figure \[fig:bessel\]a). The stopband response of the FFX scheme is not better than that of the polyphase FX scheme (Bunton 2000) and that of the FXF scheme for EVLA ([@EVLA]). On the other hand, the ripple response in the passband is less than 0.4 dB peak-to-peak (see figure \[fig:bessel\]b). This performance is better than that of the polyphase FX scheme.
Note that the stopband performance to suppress the spurious lines can be improved by installing “the detection and cancellation techniques of high and low frequency spurious lines” into the FFX scheme (Chikada found this algorithm in the development of ALMA/ACA correlator).
Requirements and Specifications for the Development of the FFX Correlator
=========================================================================
The development of the FFX correlator hardware is significant for the verification of the FFX scheme algorithm. It is necessary to define the requirements and specifications of the hardware of the FFX correlator. The hardware size can be optimized for the verification of algorithm. On the other hand, there were the scientific requests for the application to the Atacama Submillimeter Telescope Experiment (ASTE), which is a new project to install and operate a 10-m submillimeter telescope at a high latitude site (4,800 m) in the Atacama desert in northern Chile ([@ASTE]). Under these situations, the FFX correlator hardware for the algorithm verification was specified by also considering the scientific requirements for submillimeter astronomy including the application to ASTE.
In case of spectroscopic observations of atomic / molecular line emissions, their line width are extended by the Doppler shift along a line of sight with movement of inter stellar matter (ISM). In nuclear regions of external galaxies and Ultra Luminous Infrared Galaxies (ULIRGs), the molecular clouds which have various velocity components can be observed simultaneously, and line widths of the observed atomic / molecular emission lines are extended. For example, the line width of CO line emission of external galaxy is sometimes extended to more than 800 km s$^{-1}$ (i.e. [@nar05]), which corresponds to about 920 MHz in $^{12}$CO(J=3–2) ($\nu_{\rm rest}\sim 345.796$ GHz) and about 2.2 GHz in $^{12}$CO(J=7–6) ($\nu_{\rm rest}\sim 806.652$ GHz), by rotating around its nuclear region. On the other hand, in order to evaluate the kinematics of protoplanetary disks and the internal structure of molecular clouds in the Milky Way, it is also necessary to resolve their thermal line widths using spectrometer with high frequency dispersion; for instance, frequency resolution of 32 kHz corresponds to velocity resolution of 0.032 km s$^{-1}$ at 1 mm wave length.
FFT stage FFT segment length
------------------ --------------------
First stage FFT 1024 points
Inverse FFT 8 points
Second stage FFT 4096 points
: FFT segment length.
\[table:NumFFT\]
As the full bandwidth of more than 3 GHz is required, the FFX correlator need to achieve the processing speed of 8192 Mega sample per second (Msps). In that case, for the realization of two-type spectral resolutions of about 5 MHz and less than 32 kHz, the FFX correlator must meet at least the specifications in table \[table:NumFFT\].
(170mm,70mm)[fig3.eps]{}
Development and Evaluation of the FFX Correlator
================================================
The correlation processing block diagram of the FFX correlator is shown in figure \[fig:blockdiagram\]. The FFX correlator consists of DTS-R (Data Transmission System Receiver) module, Correlation Module, and the Monitor $\&$ Control Computer. In the DTS-R module, the DTS-R Board or the ADC (Analog-to-Digital Converter) Board is implemented as an EIB (Electrical input Interface Board). The input data rate of the FFX correlator is about 48 Giga bit per second (Gbps) with 3-bit quantization at the sampling frequency of 8192 or 4096 Msps, which is 8192 Msps x 3 bits x 2 IF or 4096 Msps x 3 bits x 4 IFs. The DCDCB (Delay Correction and Data Configuration Board) effectively distributes the input signals to the next boards for the parallel correlation processing. For the data processing at the throughput of 8192 Msps, the correlation is performed with 16 parallels, and both of the real and imaginary parts are used in FFT. The data is sent to each 16-parallel CORB (Correlation Board) per one-segment length. In the correlation mode of the FFX scheme, the data is sent per total segment length which is determined considering the signal process including the second stage of FFT. Final correlation output is obtained by adding 16-parallel correlation results. The correlation output is sent to the Monitor $\&$ Control Computer via LAN cable. Switching between the FX processing and the FFX processing is normally operated by setting the command into the Monitor $\&$ Control Computer.
The correlation processing flow of the FFX correlator is shown in figure \[fig:corrflow\]. All main logics are implemented in FPGAs (Field Programmable Gate Array).
(85mm,100mm)[fig4.eps]{}
Delay Correction and Data Configuration Board (DCDCB)
-----------------------------------------------------
### Delay Correction
The FFX correlator has a delay correction circuit per bit for every 3-bit sampling signal. Delay tracking for every single bit is realized by extracting 64-sample length from time-sequential 128 samples that are produced by splitting the output from FIFO (First In, First Out) memory in two, and shifting one side or the other in one-clock phase (see figure \[fig:DelayCorr\]). Delay correction circuit has a FIFO memory of 1.024 Mega samples to each bit for delay tracking. In case of 8192 Msps, the delay correction range is $\pm$ 512 kilo samples, that is $\pm$ 62.5 $\mu$sec (=$\pm$ 18.75 km).
(75mm,80mm)[fig5.eps]{}
### Data Distribution
(85mm,120mm)[fig6.eps]{}
Given the operation speed of the device (FPGA etc.), 16-parallel correlation processing is essential to achieve the throughput of 8192 Msps. To do this, the input signals are divided by one segment length of FFT and sent to the Correlation Board (CORB) so that each parallel processing is performed separately.
In the FFX processing, the data is divided by single segment length of the second-stage FFT. From table \[table:NumFFT\], the input signals are divided into: $$1024 \times 4096/8 = 512 \hspace{2mm} \hat{k},$$ where $\hat{k}$ is 1024 ($2^{10}$). This value determines the one segment length of this FFT processing. 64-parallel signals are converted to 4-parallel after being output to each distribution buffer (see figure \[fig:DataDist\]).
Correlation Board (CORB)
------------------------
### Operation Format
(85mm,120mm)[fig7.eps]{}
The operations of the correlation processing are performed using 16-bit floating point format. FFT calculation is expressed as $$-1^s \cdot 2^{e\cdot 16} \cdot (1+ma/1024),$$ and the range is from $\pm$ 1/32768 to 131008. When the index ($e$) is 0, it is regarded as 0 instead of 2$^{-16}$ irrespective of mantissa.
This not only simplifies the processing but also improves the compatibility with IEEE single precision floating point data. IEEE single precision floating point has 32-bit representing sign ($S$): 1 bit, exponential ($E$): 8 bits, and mantissa ($Ma$): 23 bits, and is expressed as $$-1^S \cdot 2^{E\cdot 127} \cdot (1+M/223),$$ and they have following relations $$\begin{aligned}
S &=& s, \\
E &=& e + 111, \\
Ma &=& ma \times 8192 (2^{13}). \end{aligned}$$ Conversion between 16-bit floating point and IEEE single precision floating point is also shown in figure \[fig:bitformat\].
### Parallel processing of two datasets {#section:twodata}
(85mm,120mm)[fig8.eps]{}
Since FFT is the linear response, the real and imaginary parts of the input can be used separately. Since the signals received by an antenna are real part only, two datasets are combined into one complex dataset for the FFT processing (see figure \[fig:twodatasets\]). One dataset is inserted into a real part, and then next dataset is inserted into an imaginary part. These two datasets are shifted with one-FFT segment length ($M$), and then processed as one complex data. Combination of two datasets is performed by Read/Write sequence control of the received buffer (=input buffer in FPGA). This method can reduce the material size of the correlator.
### F-Block (FX mode)
(85mm,120mm)[fig9.eps]{}
(85mm,120mm)[fig10.eps]{}
The FX processing is performed with 16$\hat{k}$-point FFT. In terms of the throughrate, pipeline processing is applied to each FFT stage processing. To reduce the memory size in the pipeline processing, 16$\hat{k}$-point FFT is realized by being divided into three parts: 32-point FFT, 32-point FFT, and 16-point FFT (see figure \[fig:pipeline16kFFT\]). The input signals are multiplied by the window function between the input buffer and FFT, so that the input buffer size is based on the 3-bit signals. FFT is realized by changing the twiddle factor according to the processing stages. This helps minimize the ROM of the twiddle factors.
### F-Block (FFX mode)
The FFX mode is shown in figure \[fig:pipelineF-FX\]. The first-stage FFT (1$\hat{k}$-point FFT) is performed in the same manner as 32 points $\times$ 32 points of FX mode.
The second-stage FFT (4$\hat{k}$-point FFT) is serially processed with one butterfly computing unit. Since 2$\times$8-point data is obtained every single process of 1k-point first-stage FFT, 4096-point data is obtained by repeating the process of the first-stage FFT 256 times. The required time for this process is: 512 x 256 = 128$\hat{k}$ \[CLK\]. On the other hand, the required time for the process of 4096-point FFT using one butterfly computing unit is: $$4096 \times \log_2 (4096) / 2 = 24\hat{k} \hspace{2mm} \mathrm{[CLK]}.$$ Compared with these processing times, the processing time of 4096-point FFT with one butterfly computing unit is much shorter.
### Window function processing
After re-allocated to two datasets, the signals are multiplied by window function. Any of the following window functions are selected: None (rectangular window), Hanning, Hamming, and Blackman. Window functions are generated in the CPU according to the selected command (“WINDOW”) from the Monitor $\&$ Control Computer before the correlation process starts.
### Data conversion
(30mm,40mm)[fig11.eps]{}
(80mm,120mm)[fig12.eps]{}
The input signals go through 3-bit processing. By setting a command (“SMPLBIT”) from the Monitor $\&$ Control Computer, the input signals are processed as 1-bit or 2-bit data. Also, conversion to 1-bit data with (M) middle bit only is available. Assumed threshold value for each bit of sampling data is given as shown in figure \[fig:samplingbitlevel\]. Based on the above threshold value, converted value of the input data is calculated as shown in figure \[fig:samplingbitlevel2\]. The input signals are converted into 16-bit floating point operation format before FFT.
### X-block (FX mode)
(80mm,120mm)[fig13.eps]{}
(80mm,120mm)[fig14.eps]{}
The composition of X-block in normal FX processing is shown in figure \[fig:xblockFX\]. Correlation processing is performed in 8$\hat{k}$-channel either of USB or LSB. For improving the sensitivity loss (see table 6 of [@oku01]), 2-channel frequency binning is performed to generate 4$\hat{k}$-channel data (Frq. bin). Integration of 100 milliseconds is conducted in time integration (S.T.Ing). If the integration takes longer than 100 milliseconds, it is performed in the Long-Term Accumulation and Output Board (LTAOB) at the time of 16-parallel data synthesis.
In OTF mode, the data is compressed from 8$\hat{k}$ frequency channels to 32 frequency channels by frequency binning. With time component of 1 millisecond, the correlation buffer area is changed every 1 millisecond. The output to the Long-Term Accumulation and Output Board (LTAOB) is performed every 100 millisecond.
Prior to the correlation processing, the data from the F-block needs some pre-processing such as split of two datasets, USB/LSB (upper and lower sideband) selection, and 90-degree and 180-degree phase switching demodulations. In splitting process of two datasets, the input data are added and subtracted, and then the two-data components combined in the F-block are split. When LSB is selected, the sign of the split Imag component is inversed. On the other hand, when USB is selected, the split data is sent to the correlation process without any additional processing. In 90-degree and 180-degree phase-switching demodulations, switching between Real and Imag, and sign inversion are performed according to the two phase-switching signals.
### X-block (FFX mode)
X-block in the FFX processing is shown in figure \[fig:xblockFFX\]. In the FFX processing, X-block has a dual structure considering the processing speed of the device. The first-stage FFT is the same as that of FX mode, however frequency binning is not performed. Split of two datasets in the second-stage FFT is not performed in X-block, since the process is performed in F-block. The results of the second stage FFT are output together with USB and LSB. Two-channel frequency binning is performed for improving the sensitivity loss (see table 6 of [@oku01]).
### $\Delta W$ correction
(80mm,120mm)[fig15.eps]{}
(80mm,120mm)[fig16.eps]{}
----------- ---------- ------------ ----------------- -------------
Bandwidth Spectral Spectral Velocity Correlation
points resolution resolution
at 1 mm
4096 MHz 4096 1 MHz 1.0 km s$^{-1}$ 2AC–1CC
2048 MHz 4096 0.5 MHz 0.5 km s$^{-1}$ 4AC–1CC
----------- ---------- ------------ ----------------- -------------
\[table:FXopmode\]
[llllll]{} Stage & Bandwidth & Spectral & Spectral & Velocity & Correlation\
& & points & resolution & resolution &\
& & & & at 1 mm &\
1$^{\rm st}$ & 4096 MHz & 512 & 8 MHz & 8.0 km s$^{-1}$ & 2AC–1CC\
2$^{\rm nd}$ & 64 MHz & 2048 & 31.25 kHz & 0.031 km s$^{-1}$ & 2AC–1CC\
\
1$^{\rm st}$ & 4096 MHz & 512 & 8 MHz & 8.0 km s$^{-1}$ & 2AC–1CC\
2$^{\rm nd}$ & 128 MHz & 4096 & 31.25 kHz & 0.031 km s$^{-1}$ & 1AC\
\
1$^{\rm st}$ & 2048 MHz & 512 & 4 MHz & 4.0 km s$^{-1}$ & 4AC–2CC\
2$^{\rm nd}$ & 32 MHz & 2048 & 15.625 kHz & 0.016 km s$^{-1}$ & 4AC–2CC\
\
1$^{\rm st}$ & 2048 MHz & 512 & 4 MHz & 4.0 km s$^{-1}$ & 2AC–1CC\
2$^{\rm nd}$ & 64 MHz & 4096 & 15.625 kHz & 0.016 km s$^{-1}$ & 2AC–1CC\
\
1$^{\rm st}$ & 2048 MHz & 512 & 4 MHz & 4.0 km s$^{-1}$ & 2AC–1CC\
2$^{\rm nd}$ & 128 MHz & 8192 & 15.625 kHz & 0.016 km s$^{-1}$ & 1AC\
\
1$^{\rm st}$ & 2048 MHz & 512 & 4 MHz & 4.0 km s$^{-1}$ & 2AC–1CC\
2$^{\rm nd}$ & 96 MHz & 6144 & 15.625 kHz & 0.016 km s$^{-1}$ & 1AC\
2$^{\rm nd}$ & 32 MHz & 2048 & 15.625 kHz & 0.016 km s$^{-1}$ & 1AC\
\
\
\
\[table:FFXopmode\]
$\Delta W$ correction is performed based on the baseline. $\Delta W$ correction coefficient is generated within FPGA. Firstly, $\Delta W_0(t)$, the gradient of $\Delta W$ to time change, is evaluated and then the value of each frequency channel is calculated (see figure \[fig:deltaW\]). Circuit diagram of the line graphs above is shown in figure \[fig:deltaWcir\].
For every 1 millisecond, the initial value (Init) is provided by the Monitor $\&$ Control Computer, and read as a initial gradient \[$\Delta W_0(0)$\]. In the border between segments, the previous value is multiplied by the gradient (Grad) data to generate a gradient of a new segment \[$\Delta W_0(t)$\]. In the arbitrary time $(t)$, initial value (the value of DC) is set to 0. $\Delta W$ is set for every 128 channel in the full bandwidth of 8$\hat{k}$ channel, which means the full bandwidth (8$\hat{k}$ channel) is corrected with 64 steps. The initial value of $\Delta W_0(t)$ can be set for every 1 millisecond, however, a given value is set within 128 channel. The gradient of $\Delta W_0(t)$ is the gradient variation of $\Delta W$ per one segment length for FFT. The Monitor $\&$ Control Computer specifies a set of the initial value and gradient approximately every 1 second.
Long-Term Accumulation/Output Board (LTAOB)
-------------------------------------------
(80mm,120mm)[fig17.eps]{}
(130mm,120mm)[fig18.eps]{}
Final correlation value is calculated by adding the output of 16 correlators. The composition of the Long-Term Accumulation/Output Board (LTAOB) is shown in figure \[fig:LTA\]. When the correlation result is output, the data is converted into the format of IEEE single-precision floating point.
In the FX processing, the number of the output frequency channels is normally 4 $\hat{k}$ (=4096). Thus the output data size per one correlation is $$4 \hat{k} \times 32
\times 2 \times
3 = 768 \hat{k} \hspace{1mm} \mathrm{bits},$$ where $32$ is the number of single-precision floating bits, $2$ is the complex, and $3$ is the number of correlations; auto-correlations of $x$ and $y$, and a cross-correlation between them. Assuming the minimum integration time is 0.1 second, the estimated output speed is approximately 7.7 Mbps. In the FFX processing, the number of the output frequency channels is 512 + 2 $\hat{k}$, thus the data size per one correlation is 480 $\hat{k}$ bits. Consequently, the estimated output speed with 0.1-second integration time is approximately 4.8 Mbps. Correlation results are sent to the Monitor $\&$ Control Computer using TCP/IP protocol of 100 BaseT-Ether.
Operation mode
--------------
The FFX correlator are divided into four F parts in principle (not physically). Each F part in a FX mode and the first FFT stage of FFX mode are operated at 2048 MHz, and each F part at the second FFT stage of FFX mode are operated at 32 MHz. Also, the digital signals input from EIB are distributed to arbitrary F parts by using the command (“FCHSEL”) from the Monitor $\&$ Control Computer. In FX mode and the first FFT stage of FFX mode, the digital signals of 8192 Msps are operated by combining two F parts (= 2$\times$2048 MHz), while the signals of 4096 Msps are operated by using one parts (=1$\times$2048 MHz). At the second FFT stage of FFX mode, the digital signals of 8192 Msps are operated by combining two F parts (=2$\times$32 MHz), while the signals of 4096 Msps are operated using only one parts (=1$\times$32 MHz). By using the command (“FCHSEL”), the effective bandwidth after the second FFT stage can be changed.
All of the operation modes available in this FFX correlator are listed in table \[table:FXopmode\] and \[table:FFXopmode\].
Hardware of the FX Correlator
-----------------------------
The Hardware of the FX Correlator is shown in figure \[fig:farm\]. The DTS-R module consists of two Electrical input Interface Boards (EIBs), two Delay Correction and Data Configuration Boards (DCDCBs), and one DTS-R Monitor $\&$ Control Boards (DRMCBs). The Correlation module consists of eight Correlation Boards (CORBs), Long-Term Accumulation/Output Board (LTAOB), and one Correlation Monitor $\&$ Control Board(CORMCB). Each module is connected to the independent Power module.
The power consumption of the filter module is 400 W, while that of the output module is 600 W. The total AC power is 750 W at 1-phase 100-220 VAC 50/60 Hz (100 V $\pm 10\%$ or 220 V $\pm 10\%$). The total weight is 71.3 kg.
Evaluation and Discussion
-------------------------
(80mm,65mm)[fig19.eps]{}
Figure \[fig:measure\] shows a block diagram of the measurement setup of the frequency response of the FFX Correlator. To investigate the frequency response, it is useful to use the CW signal, which can measure the folding effects by sweeping the frequency range of 0 to 4096 MHz. The white noise is important to measure the frequency response, because the astronomical signals obey the Gaussian random variable. To obtain the input signals that are approximated to the zero-mean Gaussian probability, mixing of the CW signal with the white noise is necessary.
(147mm,100mm)[fig20.eps]{}
The frequency response of the FFX correlator when CW is included is written as $$\begin{aligned}
P_{\mathrm{on}}(f) &=& a_{\mathrm{on}} \cdot \left[|C(f)|^2 +|N(f)|^2 \right] \nonumber \\
&&\cdot H_{1}(f) H_{2}^{\ast}(f) \cdot |H_{\mathrm{D}}(f)|^2,
\label{eq:oncor}\end{aligned}$$ where $C(f)$ is the frequency response of the CW signal, $N(f)$ is the frequency response of the white noise from the ASTE analog backend subsystem, $H_{1}$ and $H_{2}$ are the frequency response by different transmission paths, in which $H_1 =H_{\mathrm{t}1} H_{\mathrm{a}1}$ and $H_2 =H_{\mathrm{t}2} H_{\mathrm{a}2}$ (see figure \[fig:measure\]), and $H_{\mathrm{D}}$ is the frequency response of the FFX correlator, including the effects of requantization and the folding noise after downsampling. The bandpass calibration is essential for estimating the CW power accurately, because the bandpass response becomes a time-variable due to outdoor air temperature. The frequency response without the CW signal is written as $$P_{\mathrm{off}}(f) = a_{\mathrm{off}} \cdot |N(f)|^2 \cdot H_{1}(f) H_{2}^{\ast}(f) \cdot |H_{\mathrm{D}}(f)|^2.
\label{eq:coroff}$$ The ADCs work as 1-bit performance ([@oku07]). In that case, it is important to adjust the power as precisely as possible so as to avoid the high-order spurious effects. The frequency responses of $P_{\mathrm{on}}(f)$ and $P_{\mathrm{off}}(f)$ depend on the relative power of the CW signal to the white noise, and also the threshold levels in quantization. To correct these effects, we need to calibrate the bandpass by sensitively adjusting the continuum floor level of $P_{\mathrm{off}}$ to that of $P_{\mathrm{on}}$ in the data analysis. These values are $a_{\mathrm{on}}$ and $a_{\mathrm{off}}$.
From equations (\[eq:oncor\]) and (\[eq:coroff\]), the frequency response in a FFX mode is written as $$\begin{aligned}
&&P^{\mathrm{F}}_{\mathrm{on}}(f) \nonumber \\
&&= a_{\mathrm{on}} \cdot \left[|C(f)|^2 +|N(f)|^2 \right] \cdot H_{1}(f) H_{2}^{\ast}(f) \cdot |H^{\mathrm{F}}_{\mathrm{D}}(f)|^2,
\label{eq:FFXon}\end{aligned}$$ while the frequency response without the CW signal can be written as $$P^{\mathrm{F}}_{\mathrm{off}}(f) = a_{\mathrm{off}} \cdot |N(f)|^2 \cdot H_{1}(f) H_{2}^{\ast}(f) \cdot |H^{\mathrm{F}}_\mathrm{D}(f)|^2,
\label{eq:FFXoff}$$ and then the CW frequency response including the response of the measurement system is derived as $$\begin{aligned}
P^{\mathrm{F}}(f) &=& P^{\mathrm{F}}_{\mathrm{on}}-P^{\mathrm{F}}_{\mathrm{off}} \cdot \frac{a_{\mathrm{on}}}{a_{\mathrm{off}}} \nonumber \\
&=& a_{\mathrm{on}} \cdot |C(f)|^2 \cdot H_{1}(f) H_{2}^{\ast}(f) \cdot |H^{\mathrm{F}}_\mathrm{D}(f)|^2.
\label{eq:FFXpwr}\end{aligned}$$ Similarly, the frequency response in a FX mode is written as $$\begin{aligned}
&&P^{\mathrm{N}}_{\mathrm{on}}(f) \nonumber \\
&&= a_{\mathrm{on}} \cdot \left[|C(f)|^2 +|N(f)|^2 \right] \cdot H_{1}(f) H_{2}^{\ast}(f) \cdot |H^{\mathrm{N}}_{\mathrm{D}}(f)|^2,
\label{eq:FXon}\end{aligned}$$ while the frequency response without the CW signal can be written as $$P^{\mathrm{N}}_{\mathrm{off}}(f) = a_{\mathrm{off}} \cdot |N(f)|^2 \cdot H_{1}(f) H_{2}^{\ast}(f) \cdot |H^{\mathrm{N}}_\mathrm{D}(f)|^2,
\label{eq:FXoff}$$ and then the CW frequency response including the response of the measurement system is derived as $$\begin{aligned}
P^{\mathrm{N}}(f) &=& P^{\mathrm{N}}_{\mathrm{on}}-P^{\mathrm{N}}_{\mathrm{off}} \cdot \frac{a_{\mathrm{on}}}{a_{\mathrm{off}}} \nonumber \\
&=& a_{\mathrm{on}} \cdot |C(f)|^2 \cdot H_{1}(f) H_{2}^{\ast}(f) \cdot |H^{\mathrm{N}}_\mathrm{D}(f)|^2.
\label{eq:FXpwr}\end{aligned}$$
From equations (\[eq:FFXpwr\]) and (\[eq:FXpwr\]), we can derive the frequency response in a FFX mode from the correlated spectra obtained in FFX and FX modes as $$|H^{\mathrm{F}}_\mathrm{D}(f)|^2 = \frac{P^{\mathrm{F}}(f)}{P^{\mathrm{N}}(f)} \cdot |H^{\mathrm{N}}_\mathrm{D}(f)|^2.$$ Since the frequency response of $H^{\mathrm{N}}_\mathrm{D}(f)$ in the FX mode is well known ([@TH01]), the frequency response in a FFX mode can be derived finally.
Figure \[fig:response\] shows the frequency response of the FFX correlator at a bandwidth of 64 MHz in the range of 2048 to 2112 MHz, and that the lower limits of about -33 dB to -40 dB are successfully measured with this method. The measurement frequency resolution was 31.25 kHz. The measurement results show that the effective bandwidth is about 59.28 MHz, which was obtained by the passband responses of about $-1$ dB at 2046.40625 and 2106.28125 MHz, $-3$ dB at 2045.3125 and 2107.28125 MHz, and by a stopband response with the first sidelobe of about $-20$ dB. The measurement results are well consistent with the theoretical curve in the passband, both bandedges (sharpness), and first sidelobe levels. In the stopband response except these responses, however, it is shown that there are differences between theoretical curve and measured results. This problem is probably due to the non-linear response of 1-bit ADC and the precision of the data reduction process in this measurement method. The cross-modulation distortion is strongly generated in digitizing the CW signals at 1 bit. This character complicates the data reduction method, and will reduce the measurement precision. If the ADCs with 3 bits or more are feasible, this problem will be relaxed.
Finally, the measurement results show that the theory of the FFX scheme can be confirmed, and the development of the FFX Correlator was successfully realized.
Summary
=======
There are two basic designs of a digital correlator: the XF-type in which the cross-correlation is calculated before Fourier transformation, and the FX-type in which Fourier transformation is performed before cross multiplication. To improve the FX-type correlator, we established a new algorithm for correlation process, that is called the FFX scheme. The FFX scheme demonstrates that the realization of a stopband response with first and second sidelobes of $-34$ dB and higher-order sidelobes of $-60$ dB is technically feasible. The FFX scheme consists of 2-stage Fourier Transform blocks, which perform the 1st-stage Fourier Transform as a digital filter, and the 2nd-stage Fourier Transform to achieve higher dispersion. The FFX scheme provides flexibility in the setting of bandwidth within the sampling frequency.
The input data rate of the developed FFX correlator is about 48 Giga bit per second (Gbps) with 3-bit quantization at the sampling frequency of 8192 or 4096 Msps, which is 8192 Msps x 3 bits x 2 IF or 4096 Msps x 3 bits x 4 IFs. We have successfully evaluated the feasibilities of the FFX correlator hardware. Also, this hardware will be installed and operated as a new spectrometer for ASTE.
We successfully developed the FFX correlator, measured its performances, and demonstrated the capability of a wide-frequency coverage and high-frequency resolution of the correlation systems. Our development and measurement results will also be useful and helpful in designing and developing the next generation correlator.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to acknowledge Yoshihiro Chikada for his helpful technical discussions. The author would like to express gratitude to Brent Carlson who provided constructive comments and suggestions on this paper. This research was partially supported by the Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Young Scientists (B), 17740114, 2005.
Bunton, J, 2000, ALMA Memo 342, (Charlottesville: NRAO) Bunton, J, 2003, ALMA Memo 447, (Charlottesville: NRAO) Carlson, B. 2001, NRC-EVLA Memo 014 Chikada, Y., Ishiguro, M., Hirabayashi, H., Morimoto, M., Morita, K., Kanzawa, T., Iwashita, H., Nakazima, K., et al. 1987, Proc. IEEE, 75, 1203 Escoffier, R. P., Comoretto, G. Webber, J. C., Baudry, A., Broadwell, C. M., Greenberg, J. H., R. R. Treacy, R. R., Cais, P., et al. 2007, A$\&$A 462, 801 Ezawa, H., Kawabe, R., Kohno, K., Yamamoto, S. 2004, SPIE 5489, 763 Iguchi, S., Okuramu, S. K., Okiura, M., Momose, M., Chikada, Y. URSI General Assembly (J6:RECENT SCIENTIFIC DEVELOPMENTS), Maastricht, 2002. Iguchi, S., Kurayama, T., Kawaguchi, N., $\&$ Kawakami, K. 2005, PASJ 57, 259 Narayanan, D., Groppi, C. E., Kulesa, C. A., & Walker, C. K. 2005, , 630, 269 Okuda, T., Iguchi, S. 2008, PASJ 60, 315 Okumura, S. K., Chikada, Y., Momose, M., Iguchi, S. 2001, ALMA Memo No.350 (Charlottesville: NRAO) Rabiner, L. R., Schafer, R. W. 1971, IEEE Trans. Audio Electroacoust., AU-19, 200 Thompson, A. R., Moran, J. M., $\&$ Swenson, G. W.Jr. 2001, Interferometry and Synthesis in Radio Astronomy, 2nd Ed., (New York:John Wiley $\&$ Sons), 289 Weinreb, S. 1963, Digital Spectral Analysis Technique and Its Application to Radio Astronomy, (R. L. E., MIT, Cambridge, Mass.), Tech. Rep. No. 412
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---
abstract: 'Heteronuclear alkali-metal dimers represent the class of molecules of choice for creating samples of ultracold molecules exhibiting an intrinsic large permanent electric dipole moment. Among them, the KCs molecule, with a permanent dipole moment of 1.92 Debye still remains to be observed in ultracold conditions. Based on spectroscopic studies available in the literature completed by accurate quantum chemistry calculations, we propose several optical coherent schemes to create ultracold bosonic and fermionic KCs molecules in their absolute rovibrational ground level, starting from a weakly bound level of their electronic ground state manifold. The processes rely on the existence of convenient electronically excited states allowing an efficient stimulated Raman adiabatic transfer of the level population.'
address:
- '$^1$Laboratoire Aim$\acute{e}$ Cotton, CNRS/Universit$\acute{e}$ Paris-Sud/ENS Cachan, Orsay Cedex, France'
- '$^1$Laboratoire Aim$\acute{e}$ Cotton, CNRS/Universit$\acute{e}$ Paris-Sud/ENS Cachan, Orsay Cedex, France'
- '$^1$Laboratoire Aim$\acute{e}$ Cotton, CNRS/Universit$\acute{e}$ Paris-Sud/ENS Cachan, Orsay Cedex, France'
- '$^1$Laboratoire Aim$\acute{e}$ Cotton, CNRS/Universit$\acute{e}$ Paris-Sud/ENS Cachan, Orsay Cedex, France'
- '$^1$Laboratoire Aim$\acute{e}$ Cotton, CNRS/Universit$\acute{e}$ Paris-Sud/ENS Cachan, Orsay Cedex, France'
- '$^1$Laboratoire Aim$\acute{e}$ Cotton, CNRS/Universit$\acute{e}$ Paris-Sud/ENS Cachan, Orsay Cedex, France'
- '$^{3}$ UFR de Physique, Université de Cergy-Pontoise, France'
author:
- 'D. Borsalino$^{1}$'
- 'R. Vexiau$^{1}$'
- 'M. Aymar$^{1}$'
- 'E. Luc-Koenig$^{1}$'
- 'O. Dulieu$^{1}$'
- 'N. Bouloufa-Maafa$^{1,3}$'
title: Prospects for the formation of ultracold polar ground state KCs molecules via an optical process
---
Introduction {#sec:intro}
============
Dilute atomic and molecular gases at ultracold temperatures ($T=E/k_B \ll 1$ millikelvin) offer the fascinating opportunity of long observation time allowing for measurements with unprecedented accuracy. For instance, due to their extremely low relative velocity in such gases, particles have their maximal presence probability at large mutual distances $R$, well beyond the range of electron exchange. Therefore the dynamics of ultracold gases is dominated by their weak long-range (van der Waals) interaction varying as $R^{-6}$, which is isotropic for identical particles in free space with spherical symmetry (like atoms, or molecules with vanishing total angular momentum). Once the particles are immersed in an external magnetic or electric field, their intrinsic properties (permanent magnetic or electric dipole moment) induce the anisotropy of their long-range interaction, which can now vary as $R^{-3}$ and depends on the relative orientation of their molecular axis, or of their angular momentum [@lepers2013; @zuchowski2013]. Manifestations of anisotropic interactions have already been observed experimentally with ultracold quantum degenerate gases of magnetic atoms [@stuhler2005; @bismut2012; @aikawa2012; @lu2011], and during ultracold collisions between KRb polar molecules (*i.e.* possessing a permanent electric dipole moment in their own frame) [@ni2010; @yan2013]. Such so-called ultracold dipolar gases are expected to reveal novel physical phenomena for instance in the context of quantum degeneracy where hamiltonians involved in condensed matter physics could be simulated with the opportunity for controlling the interaction between particles with external fields (see for instance the review papers of Refs.[@baranov2012; @lahaye2009]). As for molecules, the recent review articles of Refs.[@quemener2012; @stuhl2014] provide in-depth presentations of theory and experiments of collisions and reactions with ultracold molecules, emphasizing on their implications in the development of the new research area of ultracold chemistry dominated by quantum mechanical effects [@nesbitt2012].
Despite amazing experimental progress, the main challenge for experimentalists dealing with ultracold molecules is still their formation as a gaseous sample with sufficient number density and with a good control of their internal state. An overview of the various methodologies to create ultracold neutral molecules and of their potential opportunities and applications is available in several review articles [@carr2009; @dulieu2009; @jin2012], and we will not cover them here. In brief, there are two classes of approaches to obtain ultracold ground state molecules: (i) manipulating pre-existing polar diatomic or polyatomic molecules with external magnetic or electric fields to design slow molecular beams [@vandemeerakker2012; @hutzler2012; @narevicius2012], and for some specific polar species, cooling diatomic molecules with laser [@dirosa2004; @hummon2013; @zhelyazkova2014; @barry2014; @kobayashi2014]; (ii) associating a pair of ultracold atoms into an ultracold molecule using laser photoassociation (PA) toward an electronic excited state followed by radiative emission (RE) to the ground state [@jones2006; @ulmanis2012], or magnetoasssociation (MA) [@kohler2006] in a weakly bound level of the electronic ground state manifold via magnetically tunable Feshbach resonances [@chin2010], with a subsequent stimulated radiative transfer (SRT) process to populate the lowest molecular bound level of the ground state.
We focus for the rest of this paper on the latter option (MA+SRT), which has been successfully demonstrated experimentally in a still limited number of cases with homonuclear molecules like Cs$_2$ [@danzl2010] and Rb$_2$ [@lang2008], and heteronuclear molecules like KRb [@ni2008] and RbCs [@takekoshi2014; @gregory2015]. This is a quite general method for the class of alkali-metal diatomic molecules which all possess a wealth of Feshbach resonances (see Ref.[@patel2014] for KCs isotopologues). The population transfer toward the lowest bound level of the ground state is achieved via the well-known coherent process of stimulated Raman Adiabatic Passage (STIRAP, [@bergmann1998; @vitanov2001; @koch2012]). The efficiency of the transfer relies on the identification of a pair of so-called pump and dump electric dipole allowed transitions with comparable Rabi frequencies, and thus on the detailed knowledge of the spectroscopy of the molecule of interest. In a previous paper [@borsalino2014], hereafter referred to as paper I, we modeled the STIRAP approach for the bosonic and fermionic KRb molecules. Using up-to-date spectroscopic data, we analyzed the efficiency of several transition schemes over the entire range of accessible laser frequencies determined by the excited electronic states, and we confirmed the suitability of the experimentally chosen scheme for KRb. We also investigated STIRAP efficiency on a limited range of frequencies for RbCs [@debatin2011; @takekoshi2014].
Our study concerns the formation of ultracold bosonic $^{39}$K$^{133}$Cs and fermionic $^{40}$K$^{133}$Cs polar molecules in their absolute ground state by STIRAP, which has not yet been achieved experimentally. The present investigation aims at guiding ongoing experiments in the choice of their laser set-up to implement the STIRAP scheme. Its main outcome is that the most efficient STIRAP schemes do not generally correspond to the intuitive picture delivered by the Franck-Condon principle describing the strongest molecular transitions in terms limited to the spatial overlap of vibrational wave functions. The KCs species exhibits several specific properties which contrast with the KRb one studied in paper I. It possesses an intrinsic permanent electric dipole moment (PEDM) of 1.92 D almost 4 times larger than the KRb one [@aymar2005]. The potential well depth of the KCs molecular ground state being larger than the Cs$_2$ one, but smaller than the K$_2$ one, the KCs molecule is stable in its lowest ground state level against ultracold collisions with surrounding Cs atoms and other KCs molecules [@zuchowski2010], but not with surrounding K atoms. These features will represent a decisive advantage for their further manipulation to create for instance a quantum degenerate molecular gas without the request of trapping molecules inside an optical lattice. Moreover, the KCs spectroscopy has been already quite well investigated experimentally for the X$^1\Sigma^+$ ground state and the lowest triplet state a$^3\Sigma^+$ [@ferber2009; @ferber2013], and for several excited states [@tamanis2010; @kruzins2010; @busevica2011; @birzniece2012; @klincare2012; @kruzins2013]. The spectrum of KCs Feshbach resonances based on these results has also been modeled in detail [@patel2014].
The present paper is organized as follows. We first recall in Section \[sec:model\] the basic principle of STIRAP, and we characterize the initial, and final molecular electronic states chosen for the implementation of STIRAP in $^{39}$KCs and in $^{40}$KCs (the index for the Cs mass will be omitted in the rest of the paper). The choice of the intermediate state for STIRAP is discussed in Section \[sec:interm\], emphasizing on the necessary knowledge of the requested molecular structure data, *i.e.* potential energy curves (PECs), transition electric dipole moments (TEDMs) and spin-orbit couplings (SOCs). Three transition paths for STIRAP in KCs relying on different kinds of couplings between molecular levels are identified and their efficiencies are compared among each other (Section \[sec:results\]). Experimental prospects are discussed in Section \[sec:prospects\] in the perspective of the extension of such studies to other alkali-metal polar diatomic species.
When appropriate, the atomic unit of length ($1\,a_0= 0.052 917 721 092$ nm) and of dipole moment (1 a.u.$\equiv ea_0=$ 2.541 580 59 D) will be used.
Model for STIRAP with KC molecules {#sec:model}
==================================
The STIRAP principle has been proposed by Bergmann *et al.*[@bergmann1998; @vitanov2001] and further discussed for instance in Ref.[@koch2012] in the context of coherent photoassociation of ultracold atoms. We also presented a summary in paper I and we only recall here a few aspects which are relevant for the present study.
The central idea of STIRAP is to adiabatically transfer the population of a quantum system from an initial state [$|\,i\,\rangle$]{} to a well-defined final state [$|\,g\,\rangle$]{} via an intermediate excited state [$|\,e\,\rangle$]{}, in such a way that the [$|\,e\,\rangle$]{} state is actually not populated. This is achieved by cleverly shaping two laser pulses overlapping in time inducing the [$|\,i\,\rangle$]{} $\rightarrow$ [$|\,e\,\rangle$]{} (pump) transition and the [$|\,e\,\rangle$]{} $\rightarrow$ [$|\,g\,\rangle$]{} (dump) transition. The Hamiltonian of the system dressed by the pulses admits a “dark” eigenstate which cannot radiatively decay and insures the transfer from [$|\,i\,\rangle$]{} to [$|\,g\,\rangle$]{} without loss of population. One can show that the optimal efficiency of the transfer is reached when one finds a level [$|\,e\,\rangle$]{} such as the amplitude of the time-dependent Rabi frequencies $$\overline{\Omega}_{ei} = {\ensuremath{\langle\,e\,|\,\vec{d} \cdot \vec{E}_{\mathrm{pump}}\,|\,i\,\rangle}} \, / \, \hbar \,\mathrm{;}\, \overline{\Omega}_{ge} = {\ensuremath{\langle\,g\,|\,\vec{d} \cdot \vec{E}_{\mathrm{dump}}\,|\,e\,\rangle}}\, / \, \hbar
\label{eq:STIRAP-1}$$ for the pump and dump transitions are equal. In Eq.(\[eq:STIRAP-1\]) $\vec{d}(R)$ is the electronic transition dipole moment function (TEDM), and $E_{\mathrm{pump}}$ (resp. $E_{\mathrm{dump}}$) is the amplitude of the laser field driving the pump (resp. dump) transition, associated to an intensity $I_{\mathrm{pump}}$ (resp $I_{\mathrm{dump}}$). This equality is achieved either with equal TEDM matrix elements for the pump and dump transitions assuming equal intensities for both pulses which is often convenient in experimental setups, or by slight adjustments of the laser intensities within the experimental feasibility to approach the strict equality of the Rabi frequencies.
Here [$|\,g\,\rangle$]{} is the lowest level $v_X=0$ of the X$^1\Sigma^+$ electronic ground state of KCs, and [$|\,i\,\rangle$]{} is a weakly-bound level of the X$^1\Sigma^+$ and a$^3\Sigma^+$ state manifolds (hereafter referred to as the X and a states) coupled by the hyperfine interaction. Figures \[fig:PECs-0\] and \[fig:PECs-1\] display two possible choices of excited electronic state for the intermediate state [$|\,e\,\rangle$]{} that are discussed below.
The weakly-bound initial state [$|\,i\,\rangle$]{} results from the magnetoassociation of a pair of ultracold $^{39}$K (or $^{40}$K) and Cs atoms by tuning an external magnetic field onto a Feshbach resonance of the pair. Such a so-called Feshbach molecule is populated in a high-lying rovibrational level with a combination of triplet ($S=1$) and singlet ($S=0$) characters. The mixing coefficients depend on the choice of the Feshbach resonance, experimentally investigated in Ref.[@ferber2013] and accurately modeled in Ref.[@patel2014]. For a molecular system, there is an additional requirement for STIRAP to work, that the radial wavefunctions of the [$|\,i\,\rangle$]{} and [$|\,e\,\rangle$]{} on one hand, and of the [$|\,e\,\rangle$]{} and [$|\,g\,\rangle$]{} levels on the other hand should overlap each other in a region where the TEDMs are not vanishingly small. As it can be seen from Figs.\[fig:PECs-0\]a and \[fig:PECs-1\]a, this is well achieved if the [$|\,i\,\rangle$]{} levels contains a significant component on the a$^3\Sigma^+$ state which has an inner classical turning point in the suitable range of $R$. Just like in paper I, the first hypothesis of our model is to choose [$|\,i\,\rangle$]{} with a pure triplet character, so that one can describe it with a radial wavefunction belonging to the single a$^3\Sigma^+$ PEC with approximately the same binding energy as the one of the Feshbach molecule bound level. This is the case of the uppermost a$^3\Sigma^+$ level assigned to $v_a=35$ with a binding energy of about 0.05 GHz at zero magnetic field, which possesses a triplet character up to 95% [@ferber2013]. Note, however, that a pump transition starting from a pure singlet level would, in principle, be possible (see the dashed arrows in Fig. \[fig:PECs-0\]a) but is not discussed further here.
Such an assumption is reasonable, as even if there is a significant singlet component in the chosen level, the nodal structure of the triplet component of the radial wavefunction will not be affected in the region of the inner turning point of the a$^3\Sigma^+$ PEC, while its amplitude may be changed. Therefore the matrix elements in Eq. (\[eq:STIRAP-1\]) would be affected only through a global scaling factor. An example of such a wave function concerns a level resulting from the mixture of two a$^3\Sigma^+$ levels ($v_a = 32$ and $v_a = 33$) and two singlet levels ($v_X = 102$ and $v_a = 103$) (see Fig.6 of Ref.[@ferber2013]). In a recent proposal, Klincare *et al.* [@klincare2012] proposed a STIRAP implementation based on a pump transition mainly acting around the outer turning point of the X PEC, using a similar hypothesis of a pure singlet weakly-bound level as the [$|\,i\,\rangle$]{} state.
The choice of the intermediate STIRAP levels in KCs {#sec:interm}
===================================================
The [$|\,e\,\rangle$]{} state must be optically coupled to both [$|\,i\,\rangle$]{} $\equiv$[$|\,{a\ensuremath{^3\Sigma^+}}\: v_a = 35\,\rangle$]{} and [$|\,g\,\rangle$]{} $\equiv$ [$|\,{X\ensuremath{^1\Sigma^+}}\: v_X = 0\,\rangle$]{} and thus must exhibit favorable transition dipole moments and good spatial overlaps with [$|\,i\,\rangle$]{} and [$|\,g\,\rangle$]{} vibrational wavefunctions. As in paper I, such a mixed singlet/triplet character is offered by the excited electronic states converging to the K($4s$) + Cs($6p$) dissociation limit, namely [b$^3\Pi$]{}, [A$^1\Sigma^+$]{}, [c$^3\Sigma^+$]{} and [B$^1\Pi$]{}, hereafter referred to as b, A, c and B states respectively (see Figs.\[fig:PECs-0\]a and \[fig:PECs-1\]a). These states are significantly affected by the spin-orbit (SO) interaction, resulting in coupled states with both spin characters, labeled with the Hund’s case $(c)$ quantum number $\Omega=0^+, 1$ for the projection of the total electronic angular momentum on the molecular axis. In order to provide predictions to the experimentalists, it is crucial to rely on all available spectroscopic information about these states, so that the corresponding PECs are built piecewise, combining spectroscopic and quantum chemistry determinations. Note that the implementation proposed in Ref.[@klincare2012] relies on a higher electronic excited state, the (4)$^1\Sigma^+$ state correlated to the K($4s$)+Cs($5d$) dissociation limit, perturbed by neighboring triplet states.
Unlike the KRb case, our study of the $\Omega = 0^+$ symmetry has been greatly facilitated by the extensive spectroscopic study from Refs.[@kruzins2010; @tamanis2010; @kruzins2013], providing the relevant PECs (Fig. \[fig:PECs-0\]a) and $R$-dependent spin-orbit couplings (SOC) (Fig. \[fig:PECs-0\]b). Following these authors, we use a four-coupled-channel model which accounts for the dominant SO interaction between the [A$^1\Sigma^+$]{} state and the $\Omega = 0^+$ component b$_0$ of the [b$^3\Pi$]{} state, as well as the interaction with other molecular states responsible for the asymmetric splitting between the $\Omega=0,1,2$ components of the b$^3\Pi$ state. In addition it includes the weak rotational interactions with the other $\Omega=1, 2$ components b$_1$ and b$_2$ of the [b$^3\Pi$]{} state, scaling with $\beta = \hbar^2/(2\mu R^2)$ where $\mu$ is the KCs reduced mass. The resulting $R$-dependent potential energy matrix is expressed, for a given total angular momentum $J$ (including the total electronic angular momentum and the rotation of the molecule, but not the nuclear spins, and $X=J(J+1)$), as the sum of Born-Oppenheimer (BO) PEC matrix $V_{\mathrm{BO}}$ and SOC matrix $W_{\mathrm{so}}^{(0^+)}$ [$$\begin{aligned}
\label{eq:WSO-0}
V_{\mathrm{BO}}+W_{\mathrm{so}}^{(0^+)} = \\ \nonumber
\bordermatrix{~ & {\ensuremath{|\,b2\,\rangle}} &{\ensuremath{|\,b1\,\rangle}} & {\ensuremath{|\,b0\,\rangle}} & {\ensuremath{|\,A \,\rangle}} \cr
& V'_{b2}(R)+A_{\mathrm{so}}^+(R)&-\beta(1-\gamma_b)\sqrt{2(X-2)}& 0 & 0 \cr
& -\beta(1-\gamma_b)\sqrt{2(X-2)} &V'_{b1}(R) &-\beta(1-\gamma_b)\sqrt{2X} &-\beta\zeta_{Ab1}\sqrt{2X}\cr
& 0 &-\beta(1-\gamma_b)\sqrt{2X} &V'_{b0}(R)-A_{\mathrm{so}}^-(R)&-\sqrt{2}\xi^{Ab0}_{\mathrm{so}}(R)\cr
& 0 &-\beta\zeta_{Ab1}\sqrt{2X}&-\sqrt{2}\xi^{Ab0}_{\mathrm{so}}(R)& V'_A(R) \cr
}\end{aligned}$$ ]{} For compactness purpose we used the notations for the potential energies including the centrifugal term: $V'_{b2}(R)=V_b(R)+\beta (1-\gamma_b) (X-2)$, $V'_{b1}(R)=V_b(R)+\beta(1-\gamma_b)(X+2)$, $V'_{b0}(R)=V_b(R)+\beta(1-\gamma_b)(X+2)$, and $V'_A(R)=V_A(R)+\beta(1-\gamma_b)(X+2)$. This matrix reduces to a $3\times 3$ form if $J=1$. Note that, strictly speaking, the labels “so” and $0^+$ in $W_{\mathrm{so}}^{(0^+)}(R)$ are approximate, referring to the dominant SO interaction. We will keep them in the following, for convenience.
For the bound level calculations, we extend the experimental PECS of the A and b states at large distances with a $C_n / R^n$ ($n=6,8$) expansion using the $C_n$ coefficients from Ref.[@marinescu1999]. At large distances the functions $A_{\mathrm{so}}^+(R)$, $A_{\mathrm{so}}^-(R)$ and $\xi^{Ab0}_{\mathrm{so}}(R)$ reach the SO constant of the Cs atom, i.e. $\xi^{\mathrm{Cs}}_{\mathrm{so}} = {{\ensuremath{184.68 \: \mathrm{cm^{-1}}}}}$. The constants $\zeta_{Ab1}$ and $\gamma_b$ are empirically adjusted to take the relevant off-diagonal interaction into account. All those terms are obtained using the analytical formulas given in Refs.[@kruzins2010; @tamanis2010; @kruzins2013]. For completeness, we also report in Fig. \[fig:PECs-0\]b the SOC functions which were computed in Ref. [@kim2009] prior to the spectroscopic analysis of Ref. [@ferber2013], showing a remarkable agreement between the two determinations.
The amazing quality of the spectroscopic data for the {b,A} complex allows calculating the energies of the $\Omega = 0^+$ levels with experimental precision. There are only a few observed levels assigned to levels with [b$^3\Pi$]{}$_1$ and [b$^3\Pi$]{}$_2$ character, so that the prediction of the energies of unobserved levels is not as accurate. However, as it has been experimentally demonstrated for RbCs [@docenko2010; @takekoshi2014], such a model indeed provides reliable information about these [b$^3\Pi$]{}$_1$ levels.
For the $\Omega=1$ case, no full spectroscopic analysis for the b, c, and B coupled molecular states exists in the literature. Kim *et al.* computed the relevant $R$-dependent SOC functions $W_{bc}$, $W_{bB}$, and $W_{Bc}$ with a quantum chemistry approach [@kim2009], and we used them in our model. The SO Hamiltonian matrix is expressed in a way similar to Eq. (\[eq:WSO-0\]) as $$\label{eq:WSO-1}
V_{\mathrm{BO}}+W_{\mathrm{so}}^{(1)} =
\bordermatrix{ ~ & {\ensuremath{|\,b\,\rangle}} & {\ensuremath{|\,c\,\rangle}} & {\ensuremath{|\,B\,\rangle}} \cr
& V'_b(R) & W_{bc}(R) & -W_{bB}(R) \cr
& W_{bc}(R) & V'_c(R) & W_{Bc}(R) \cr
& -W_{bB}(R) & W_{Bc}(R) & V'_B(R) \cr
}$$ with $V'_{\alpha}(R)=V_{\alpha}(R)+\beta X$ for $\alpha=$b, c, B. The coupling functions are reported in Fig. \[fig:PECs-1\]b, showing that they all converge toward $\xi^{\mathrm{Cs}}_{\mathrm{so}}$. For the bound level calculations, we used as above the experimental $b$ PEC [@tamanis2010; @kruzins2013]. The c PEC is obtained from our own quantum chemistry calculations based on semi-empirical effective core potentials (ECP), following the procedure described in Refs.[@aymar2005; @aymar2006a], completed by the ECP parameters reported in Ref. [@guerout2010]. The spectroscopy of the bottom of the B PEC has been achieved in Ref.[@birzniece2012], which thus accounts for SOC in an effective way. Therefore we shifted it in energy before its connection to our own computed PEC curve in order to ensure that the three-coupled-channel calculation actually delivers the correct energies for the measured levels. As above, these PECs are connected at large distances to an asymptotic expansion using coefficients from Ref.[@marinescu1999].
The treatment of the [$\Omega=1$]{} complex is not as accurate as the [$\Omega=0^+$]{} one, as there is no spectroscopic analysis available in the literature. However, the few spectroscopically observed vibrational levels at the bottom of the B state provide useful information for predicting a good STIRAP transfer using an [$\Omega=1$]{} intermediate level.
In addition, TEDM functions connecting the X and a states with the A, b, B, and c states from our own quantum chemistry calculations are drawn in Fig. \[fig:PECs-0\]c and Fig. \[fig:PECs-1\]c. Note that as stated by Kim *et al.* in their article [@kim2009], both our own PECs and TEDMs are in excellent agreement with their results. It is also worthwhile to remark that the TEDMs are quite similar to the KRb ones, with an important difference however. The magnitude of $d_{ba}(R)$ is larger than its KRb counterpart [@borsalino2014] at the inner turning point of the a PEC: $d_{ba}$($R$=9.5a.u.) $\sim 0.21 ea_0$ in KCs, whereas in KRb $d_{ba}$($R$=9.3a.u.) $\sim 0.03 ea_0$. As discussed later, this order of magnitude difference will have important consequences for the experimental realization of the STIRAP approach based on $\Omega = 0^+$ states.
Our quantum chemistry data for PECs and TEDMs and the piecewise PECs elaborated above are provided in the Supplemental Material for convenience.
Finally, according to the authors, the semiempirical curves and parameters derived in Refs.[@ferber2009; @ferber2013; @kruzins2010; @tamanis2010; @kruzins2013] are correctly mass-invariant, so that it is possible to use them to model the levels of the other isotopologues $^{40}$KCs and $^{41}$KCs. This is partly verified for the {b,A} complex by the ability of the semiempirical curves to reproduce a few measured levels [@kruzins2010] of the $^{41}$K$^{133}$Cs molecule. For X and a PECs, the derived curves yield reliable scattering lengths for elastic collisions of each isotopic combination, thanks to the good quality of the long range part [@ferber2009; @ferber2013; @kruzins2013].
Three possible implementations of STIRAP in KCs {#sec:results}
===============================================
In order to evaluate the relevant transition matrix elements (TMEs) involved in the Rabi frequencies in Eq.(\[eq:STIRAP-1\]), vibrational energies and radial wave functions are computed with the Mapped Fourier Grid Hamiltonian (MFGH) method [@kokoouline1999; @kokoouline2000] as described in paper I. The Hamiltonian operator governing the nuclear motion is $\hat{H} = \hat{T} + \hat{V}_{\mathrm{BO}} + \hat{W}_{\mathrm{so}}^{(\Omega)}$, where $\hat{T}$ refers to the kinetic energy operator. It is represented as a $Nq \times Nq$ matrix where $N$ is the number of coupled channels and $q$ is the number of grid points for the $R$ coordinate determined such that the diagonalization yields eigenenergies reproducing the bound states energies at the experimental accuracy. The variable grid step at the heart of the MFGH method allows for accurately calculating bound levels with energies very close to the dissociation limit (and thus with a large vibrational amplitude) while limiting the number of grid points to about $q=590$. The resulting vibrational wave functions [$|\,i\,\rangle$]{} $\equiv$ [$|\,{a\ensuremath{^3\Sigma^+}}\: v_a = 35\,\rangle$]{}, and [$|\,g\,\rangle$]{} $\equiv$ [$|\,{X\ensuremath{^1\Sigma^+}}\: v_X = 0\,\rangle$]{}, are then used in our calculations.
The vibrational levels resulting from the diagonalization are labeled with an index $v'_{\Omega}$ referring to the global numbering of the increasing eigenenergies. The corresponding radial wavefunctions $\left|\Omega; \,v'_{\Omega}\right\rangle$ are expressed as linear combinations of the $N$ coupled electronic states $$\left|\Omega; \,v'_{\Omega}\right\rangle = \sum_{\alpha=1}^N \frac{1}{R} \psi_{\alpha}^{\Omega v'_{\Omega}}(R) \left|\alpha\right\rangle .
\label{eq:wf}$$ The weight $w_{\alpha}^{\Omega v'_{\Omega}}$ on each electronic state is defined by the squared radial components $\left|\psi_{\alpha}^{\Omega v'_{\Omega}}(R)\right|^2$ such as $$\sum_{\alpha=1}^N w_{\alpha}^{\Omega v'_{\Omega}} \equiv \sum_{\alpha=1}^N \int_0^\infty \left|\psi_{\alpha}^{\Omega v'_{\Omega}}(R)\right|^2 dR =1
\label{eq:norm}$$ The TMEs for the pump (resp. dump) transition involve the vibrational functions $\varphi_a^{v_a}$ (resp. $\varphi_X^{v_X}$) of the [a$^3\Sigma^+$]{} (resp. [X$^1\Sigma^+$]{}) state and the triplet part $\psi_{\alpha_t}^{\Omega v'_{\Omega}}(R)$ (resp. the singlet part $\psi_{\alpha_s}^{\Omega v'_{\Omega}}(R)$) of the coupled wave function of the intermediate level [$|\,\Omega; v'_{\Omega}\,\rangle$]{} (eq. (\[eq:wf\])) $$\begin{aligned}
{d}_{\alpha_t a}^{v'_{\Omega} v_a} &=& \left\langle \Omega; v'_{\Omega}\right| \hat{d}_{\alpha_t a} \left| a; v_a \right\rangle \nonumber \\
&=& \int_0^\infty \psi_{\alpha_t}^{\Omega v'_{\Omega}}(R) \, d_{\alpha_t a}(R) \, \varphi_a^{v_a}(R) dR \\
{d}_{X \alpha_s}^{v_X v'_{\Omega}} &=& \left\langle X; v_X\right|\hat{d}_{X \alpha_s}\left| \Omega; v'_{\Omega} \right\rangle \nonumber \\
&=& \int_0^\infty \varphi_X^{v_X}(R) \, d_{X \alpha_s}(R) \, \psi_{\alpha_s}^{\Omega v'_{\Omega}}(R) dR
\label{eq:d-matrix}
\end{aligned}$$ The squared matrix elements $\left|{d}_{\alpha_t a}^{v'_{\Omega} v_a}\right|^2$ and $\left|{d}_{X \alpha_s}^{v_X v'_{\Omega}}\right|^2$ determine the efficiency of the STIRAP process and are systematically calculated in the following. Note that these TMEs will have to be multiplied by the appropriate Höln-London factors to take in account the experimentally chosen polarizations of the pump and dump lasers.
In the next sections, graphs for TMEs will be drawn for the energy region where they are of comparable magnitude for the pump and dump transitions, for clarity. The full list of TMEs are given in the Supplementary Material attached to the present paper.
STIRAP via the A – b$_0$ spin-orbit coupled states {#ssec:b0-A}
--------------------------------------------------
We considered the lowest allowed $J=1$ value for which the matrix of Eq. (\[eq:WSO-0\]) reduces to a $N=3$ dimension. We display in Fig. \[fig:TME-0\_39\] for $^{39}$KCs and in Fig. \[fig:TME-0\_40\] for $^{40}$KCs, the relevant TMEs ${d}_{b_0 a}^{v'_{0^+} v_a}$ (closed squares) and ${d}_{X A}^{v_X v'_{0^+}}$ (closed circles). They are extracted from the calculations above, involving the coupling matrix of Eq.(\[eq:WSO-0\]). As expected, their global behavior is very similar for the two isotopologues, but of course the recommended levels for the optimal transfer are slightly different, as summarized in Table \[tab:0+\].
The data points for $\left|{d}_{X A}^{v_X v'_{0^+}}\right|^2$ (closed circles in Figs. \[fig:TME-0\_39\] and \[fig:TME-0\_40\]) are associated to *all* eigenvectors yielded by the diagonalization, and of course related to the magnitude of their A component. The TMEs present strong variations associated to levels with main mixed A-b$_0$ character (upper zone of the data points, associated to states with strong A component, alternating with states with strong b$_0$ component), and to levels with main weight on the b$_1$ state (lower zone of the data, corresponding to states with very weak A component). The TMEs reported for $\left|{d}_{b_0 a}^{v'_{0^+} v_a}\right|^2$ (closed squares in Figs. \[fig:TME-0\_39\] and \[fig:TME-0\_40\]) correspond to states with either a main component on b$_0$ or on A, disregarding those with main b$_1$ character for clarity. Thus this data is complementary to the upper part of the data for $\left|{d}_{X A}^{v_X v'_{0^+}}\right|^2$.
Thus the optimal STIRAP region for equal pump and dump transitions, exemplified in the figures by the selected $v'_0=202$ level in $^{39}$KCs and by $v'_0=198$ and $v'_0=207$ in $^{40}$KCs, is located where the upper part of the A $\rightarrow$ X data crosses the a $\rightarrow$ b$_0$ data (with closed squares). Note that the picture is qualitatively similar than the one obtained in paper I for KRb, except that the b$_1$ and b$_2$ states were not included in the spin-orbit coupling matrix. However, the large $R$-dependent TEDM around the inner turning point of the a PEC in KCs compared to KRb taken in similar conditions, namely starting from the uppermost level $v=31$ of the a state (see Table \[tab:0+\]) induces a much larger TME than in KRb, which makes this STIRAP scheme attractive for a future experimental implementation in KCs.
$v'$ $E_{bind}$ $E_{pump}$ $E_{dump}$ $w_{b_0}$ $w_{b_1}$ $w_{A}$ $|d_{ab}|^2$ $|d_{AX}|^2$
------------------- ------ ------------ ------------ ------------ ----------- ----------- --------- -------------- --------------
$^{39}$KCs 202 -3418.866 8128.764 12163.843 0.79 0.20(-6) 0.21 1.21(-6) 1.57(-6)
$^{40}$KCs 207 -3380.794 8166.841 12202.246 0.31 0.13(-6) 0.69 1.40(-6) 1.40(-6)
198 -3500.188 8047.448 12082.852 0.60 0.17(-6) 0.40 2.00(-6) 7.36(-6)
$^{39}$K$^{87}$Rb 103 -3450.1 9287.297 13467.396 0.09 - 0.91 9.74(-9) 2.0 (-8)
\[tab:0+\]
STIRAP via the A – b$_1$ rotationnally coupled states {#ssec:b1-A}
-----------------------------------------------------
The relevant TMEs are ${d}_{b_1 a}^{v'_{1} v_a}$ (open triangles in Figs. \[fig:TME-0\_39\] and \[fig:TME-0\_40\]) and ${d}_{X A}^{v_X v'_{0^+}}$ (the low set of closed circles in Figs. \[fig:TME-0\_39\] and \[fig:TME-0\_40\]). The data points correspond to levels with main b$_1$ character *and* with the largest possible component on the A state (typically $\approx 10^{-4}$). Due to the weak rotational coupling (the constant $\beta$ in Eq.(\[eq:WSO-0\]) amounts $\approx$0.04 cm$^{-1}$ around $R=$10 a.u.), vibrational levels of the unperturbed A and b$_1$ PECs must be quite close in energy to be effectively coupled. As the available KCs spectroscopic data is of good quality, we identified one level in $^{39}$KCs and two levels in $^{40}$KCs of main b$_1$ character with such characteristics (Table \[tab:0+-1\]). Despite small TMEs and quite unbalanced transition matrix elements for the pump and dump transitions, we predict a situation which is comparable to the one modeled and already observed in RbCs, which characteristics are recalled in Table \[tab:0+-1\]. This is actually such a circumstance which recently allowed for an efficient STIRAP implementation to create a dense sample of ultracold $^{87}$RbCs molecules [@debatin2011; @takekoshi2014]. This mechanism is expected to be even more favorable if a more deeply-bound level is chosen for [$|\,i\,\rangle$]{} (as done in RbCs [@debatin2011]) since the amplitude of its assumed pure triplet wave function around the a$^3\Sigma^+$ inner turning point grows up.
------------- ------ ------------ ------------ ------------ ----------- ----------- --------- -------------- --------------
$v'$ $E_{bind}$ $E_{pump}$ $E_{dump}$ $w_{b_1}$ $w_{b_0}$ $w_{A}$ $|d_{ab}|^2$ $|d_{AX}|^2$
$^{39}$KCs 122 -4500.405 7047.225 11082.304 0.99987 0.00003 0.00010 4.42(-7) 9.19(-5)
$^{40}$KCs 127 -4459.692 7087.944 11123.348 0.99138 0.00191 0.00671 1.37(-6) 2.05(-3)
162 -3982.269 7565.367 11600.771 0.99826 0.00055 0.00118 1.01(-6) 9.24(-6)
$^{87}$RbCs 68 -5124.586 6423.042 10234.613 0.99603 0.00083 0.00314 9.49(-7) 1.71(-4)
------------- ------ ------------ ------------ ------------ ----------- ----------- --------- -------------- --------------
\[tab:0+-1\]
STIRAP via the B–b–c spin-orbit coupled states {#ssec:B-c}
----------------------------------------------
The relevant TMEs ${d}_{b_1 a}^{v'_{1} v_a}$, ${d}_{c a}^{v'_{1} v_a}$, and ${d}_{X B}^{v_X v'_{1}}$ are extracted from the calculations involving the coupling matrix of Eq.(\[eq:WSO-1\]). They are presented in Fig. \[fig:TME-1\] for $^{39}$KCs as a representative isotopologue, as the full spectroscopy of the $\Omega=1$ states is not yet available. The recommended levels for an optimal STIRAP implementation are displayed in Table \[tab:1-Bc\]. As in KRb [@borsalino2014], due to the larger TEDM for the a $\rightarrow$ c than for the a $\rightarrow$ b transition, these levels are characterized by TMEs of comparable magnitude for the pump $\rightarrow$ c transition and the dump B $\rightarrow$ X transition. Thus we qualify this case as being induced by the B–c spin-orbit coupling. But in contrast with KRb, it is likely to reach quite high-lying B vibrational levels ($v_B=$20, 23, while for KRb we had $v_B=$8) to ensure an optimal STIRAP. These levels are located at an energy corresponding to the quantum chemistry part of the B PEC, while its spectroscopic determination is yielded only up to the energy of $v_B=5$ in Ref.[@birzniece2012]. In the $v_B \leq 5$ energy range a couple of $\Omega=1$ levels with main b character are also expected to be interesting for STIRAP and are reported in Table \[tab:1-Bc\]. For instance the $v' = 125$ level energy is predicted close to the location of $v_B=0$ which is well known experimentally. Due to their noticeable weight on the B state, such levels are most likely present in the recorded data of Ref.[@birzniece2012], even if not yet assigned.
Finally, this B–b–c STIRAP option is based on TMEs which are larger than those for the A–b$_0$ and A–b$_1$ options by about two orders of magnitude only, in strong contrast with KRb where the difference was at least of four orders of magnitude. As anticipated above, this is due to the large a–b TEDM in KCs compared to the KRb one.
------------ ------------- ------------ ------------ ------------ --------- --------- --------- -------------- -------------- --------------
$v'$ $E_{bind}$ $E_{pump}$ $E_{dump}$ $w_{b}$ $w_{c}$ $w_{B}$ $|d_{ba}|^2$ $|d_{ca}|^2$ $|d_{XB}|^2$
$^{39}$KCs 188 -814.4 10733.3 14768.4 0.193 0.275 0.532 1.32(-5) 4.54(-5) 3.72(-4)
($v_B=20$)
195 -751.5 10796.2 14831.2 0.138 0.282 0.579 7.18(-7) 6.64(-5) 8.71(-5)
($v_B=23$)
125 -1543.3 10004.3 14039.4 0.630 0.357 0.013 3.44(-6) 2.32(-4) 1.08(-3)
($v_b=90$)
163 -1067.5 10480.1 14515.2 0.528 0.440 0.032 1.60(-6) 2.83(-4) 6.17(-4)
($v_b=100$)
------------ ------------- ------------ ------------ ------------ --------- --------- --------- -------------- -------------- --------------
\[tab:1-Bc\]
Prospects for experimental implementation {#sec:prospects}
=========================================
The present investigation of the possible pathways for the formation of ultracold KCs molecules in their absolute ground state is a follow-up of our previous study on KRb. But the current situation on the experimental side is very different for the two molecules.
The formation of ultracold KRb molecules in their absolute ground state via a STIRAP scheme has been undoubtedly boosted in part by the wealth of spectroscopic data available for the B$^1\Pi$ state in the region of its PEC minimum, with an accurate modeling of the perturbations by the b$^3\Pi$ and c$^3\Sigma^+$ states induced by spin-orbit interaction. Our complete analysis in paper I confirmed the choice of the experimental groups for the implementation of STIRAP based on the B–b–c scheme. Indeed, the weak TEDM for the a$^3\Sigma^+ \rightarrow$ b$^3\Pi$ pump transition does not favor the implementation via the other A–b$_0$ scheme relying on the spin-orbit coupled A$^1\Sigma^+$ and b$^3\Pi$ states.
The situation is reversed for the KCs species which has not yet been observed in the ultracold regime. The spectroscopy of the coupled A$^1\Sigma^+$ and b$^3\Pi$ states is much better known than the one of the B$^1\Pi$, b$^3\Pi$ and c$^3\Sigma^+$ states. Moreover, the large $R$-dependent TEDM around the inner turning point of the a PEC in KCs compared to KRb induces a much larger TMEs for the A–b$_0$ and A–b$_1$ schemes in KCs, which makes this STIRAP scheme attractive for a future experimental implementation. Thus our study shows that there are more possible options than in KRb to implement efficient STIRAP in KCs, namely, using either the A–b$_0$ spin-orbit coupled states, the A–b$_1$ rotationnally coupled states, or the B–b–c spin-orbit coupled states. Note that the A–b$_1$ case was not analyzed in paper I, but it has been successfully implemented in the RbCs experiment of Ref.[@debatin2011; @takekoshi2014]. The magnitudes of the relevant RbCs transition matrix elements are recalled in Table \[tab:0+-1\], which provide a reference for the experimental feasibility of such a scheme in the KCs case. In this respect the STIRAP scheme based on A–b$_1$ coupled states is particularly attractive: it relies on the accurately known spectroscopy of the A–b coupled states. Moreover, despite the very weak hyperfine structure expected for $\Omega=0^+$ levels, their coupling with closeby $\Omega=1$ levels identified in KCs, -which are expected to possess a large hyperfine structure- allows for controlling the hyperfine level during STIRAP, as achieved for RbCs. We also predict that the STIRAP scheme yielding the largest TMEs for both the pump and dump transitions is the one based on the B–b–c spin-orbit coupled states, as observed in KRb. However, its implementation would require further spectroscopic investigations, which may not be the priority of the interested experimental groups.
Generally, the presented study, just like the previous one on KRb, demonstrates that the choice of an efficient STIRAP scheme to create ultracold molecules in their absolute ground state level cannot be determined by invoking the Franck-Condon (FC) principle which only involves the spatial overlap of the vibrational wave functions. The variation of the relevant TEDMs along the internuclear distance plays a central role. Moreover a reasonable balance of the TMEs for the pump and dump transitions should be achieved, which generally corresponds to transitions departing from the most favorable ones identified by the FC principle. In this respect, the provided Supplementary material should be a great help for setting up an experiment. We anticipate that the present study comes at the appropriate time to guide future experiments aiming at creating ultracold samples of KCs molecules. The hyperfine structure not taken into account in this work will be modeled in an upcoming study and the STIRAP schemes will reexamined in this framework.
Acknowledgments {#acknowledgments .unnumbered}
===============
This project is supported in part by the contract BLUESHIELD of Agence Nationale de la Recherche (ANR, contract ANR-14-CE34-0006). R.V. acknowledges partial support from ANR under the project COPOMOL (contract ANR-13-IS04-0004). Stimulating discussions with Hanns-Christoph Nägerl and Emil Kirilov are gratefully acknowledged.
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|
---
author:
- |
M. Williams,${}^1$ C.P. Burgess,${}^{1,2}$ Anshuman Maharana${}^3$ and F. Quevedo${}^{3,4}$\
${}^1$Department of Physics & Astronomy, McMaster University\
1280 Main Street West, Hamilton ON, Canada. ${}^2$Perimeter Institute for Theoretical Physics\
31 Caroline Street North, Waterloo ON, Canada. $^3$ DAMTP/CMS, University of Cambridge, Cambridge CB3 0WA, UK. $^4$ Abdus Salam ICTP, Strada Costiera 11, Trieste 34014, Italy.
title: |
New Constraints (and Motivations) for Abelian\
Gauge Bosons in the MeV – TeV Mass Range
---
Introduction and summary of results
===================================
New particles need not have very large masses in order to have evaded discovery; they can also be quite light provided they couple weakly enough to the other particles we [*do*]{} see. This unremarkable observation has been reinforced by recent dark matter models, many of which introduce new particles at GeV or lower scales in order to provide dark-matter interpretations for various astrophysical anomalies [@DarkSector]. This model-building exercise has emphasized how comparatively small experimental efforts might close off a wide range of at-present allowed couplings and masses for putative new light particles [@toroschuster; @ovanesyan].
[*Light spin-one bosons*]{}
Spin-one gauge bosons are particularly natural kinds of particles to seek at low energies, since (unlike most scalars) these can have light masses in a technically natural way. Furthermore, their couplings are reasonably restrictive, allowing only two kinds of dimensionless interactions with ordinary Standard Model particles: direct gauge couplings to ordinary matter and kinetic mixing [@holdom2u1s] with Standard Model gauge bosons. Most extant surveys of constraints on particles of this type assume the existence of one or the other of these couplings, with older studies studying only direct gauge-fermion interactions [@carlson; @bonly] and later studies (particularly for dark-matter motivated models) [@millichargebounds; @ChangNgWu; @HiddenU1Bounds; @maximsec; @cmb] usually allowing only kinetic mixing.
In this paper we have both motivational and phenomenological goals. On the phenomenological side, we analyze the constraints on new (abelian) gauge bosons, including both direct gauge-fermion couplings and gauge-boson kinetic mixing. In this way we include all of their dimensionless couplings, which (if all other things are equal) should dominate their behaviour at low energies. We can follow the interplay of these couplings with one another, and how this changes the bounds that can be inferred concerning the allowed parameter space. In particular we find in some cases (such as beam dump experiments) that bounds derived under the assumption of the absence of the other coupling can sometimes weaken, rather than strengthen, once the most general couplings are present.
Our motivational goal in this paper is twofold. First, we argue that the existence of gauge bosons directly coupled to ordinary fermions is very likely to be a generic and robust property of any phenomenologically successful theory for which the gravity scale is much smaller than the GUT scale [@StringLow; @ADD; @RS]. Next, we argue that these gauge bosons often very naturally have extremely weak gauge interactions within reasonable UV extensions of the low-energy theory, such as extra-dimensional models [@6Dflux] and low-energy string vacua [@IQ]. Besides motivating the otherwise potentially repulsive feature of having very small couplings, the smallness of these couplings (together with the low value for the fundamental gravity scale) also naturally tends to make the corresponding gauge bosons unusually light.
The remainder of this paper is organized as follows. The rest of this section, §1, briefly summarizes the basic motivational arguments and phenomenological results. §2 then provides a more detailed theoretical background that motivates the sizes and kinds of couplings we consider, which may be skipped for those interested only in the bounds themselves. In §3 we briefly summarize the basic properties of the new gauge boson, with details given in an Appendix. By diagonalizing all kinetic terms and masses we identify the physical combination of couplings that are bounded in the subsequent sections. The next three sections, §4, §5 and §6, then explore the bounds on these couplings that are most restrictive for successively lighter bosons, starting at the weak scale and working down to MeV scales.
Motivational summary {#motivational-summary .unnumbered}
--------------------
Why consider light gauge bosons that couple directly to ordinary fermions? And why should their couplings be so small? We here briefly summarize the more lengthy motivations given below, in §2.
[*Low-scale gravity and proton decay*]{}
Weakly coupled gauge bosons are likely to be generic features of any (phenomenologically viable) UV physics for which the fundamental gravity scale is systematically small relative to the GUT scale, $M_{\rm GUT} \sim 10^{15}$ GeV. Such bosons arise because of the difficulty of reconciling a low gravity scale with the observed stability of the proton. After all, higher-dimension baryon- and lepton-violating interactions that generically cause proton decay are not adequately suppressed if they arise accompanied by a gravity scale that is much smaller than $M_{\rm GUT}$. Similarly, global symmetries cannot themselves stop proton decay if the present lore about the absence of global symmetries in quantum gravity [@NoGlobinGrav] should prove to be true (as happens in string theory, in particular [@NoGlobSinST; @hyper]).
This leaves low-energy gauge symmetries as the remaining generic mechanism for suppressing proton decay. Indeed, extra gauge bosons are often found in string vacua, and when the string scale is much smaller than the GUT scale, $M_s \ll M_{\rm GUT}$, these bosons typically play a crucial role in protecting protons from decaying. Furthermore, very weak gauge couplings appear naturally in such string compactifications, once modulus stabilization is included. In these systems the gauge couplings can be small because they are often inversely proportional to the volume of some higher-dimensional cycle, whose volume gets stabilized at very large values [@LV]. Similar things can also occur in non-stringy extra-dimensional models [@BraneBR].
[*Unbroken gauge symmetry without unbroken gauge symmetry*]{}
We believe there is a generic low-energy lesson to be drawn from how proton decay is avoided in phenomenological string constructions. This is because in these models, even though proton decay is forbidden by conservation of a gauged charge, the gauge boson that gauges this symmetry is not massless [@IQ]. This combines the virtues of an unbroken symmetry (no proton decay), with the virtues of a broken symmetry (no new forces mediated by a massless gauge boson).[^1] Usually this happy situation arises in the string examples because the gauge symmetry in question is anomalous, if judged solely by the light fermion content, with anomaly freedom restored through Green-Schwarz anomaly cancellation. But in four dimensions Green-Schwarz anomaly cancellation relies on the existence of a Goldstone boson, whose presence also ensures that the gauge boson acquires a nonzero mass.
For these constructions the effective lagrangian obtained just below the string scale from matching to the stringy UV completion is invariant under the symmetry apart from an anomaly-cancelling term that breaks the symmetry in just the way required to cancel the fermion loop anomalies. §2 argues that this property remains true (to all orders in perturbation theory) as one integrates out modes down to low energies. Leading symmetry breaking contributions arise non-perturbatively, exponentially suppressed by the relevant gauge couplings. Consequently they remain negligibly small provided only that the gauge groups involved in the anomalies are weakly coupled. Although supersymmetry also plays a role in the explicit string examples usually examined, our point here is that this is not required for the basic mechanism that allows massive gauge bosons to coexist with conservation of the corresponding gauge charge.
Phenomenological summary {#phenomenological-summary .unnumbered}
------------------------
We next summarize, for convenience of reference, the combined bounds obtained from the constraints examined throughout the following sections.
[*Mass vs coupling*]{}
Fig. \[allofem\] presents a series of exclusion plots in the $\alpha_\ssX-M_\ssX$ plane, where $\alpha_\ssX = g^2_\ssX/4\pi$ is the gauge-fermion coupling and $M_\ssX$ is the gauge boson mass. Each panel shows these bounds for different fixed values of the kinetic mixing parameter, $\sh\,\eta$ (for details on the definition of variables, see §\[Xprops\]). The figure shows the collective exclusion area of all of the different bounds considered in this paper. For concreteness they are calculated for a vector-like charge assignment, $X_{f\ssL} = X_{f\ssR}$, with the choice $X = B-L$ denoted by a lighter shading and the choice $X=B$ denoted with a heavier shading. Comparison of the cases $X = B$ and $X = B-L$ shows how much the bounds strengthen once direct couplings to leptons are allowed.
For $\eta =0$, the dominant bounds are from neutrino scattering, upsilon decay, anomalous magnetic moments, beam-dump experiments, neutron-nucleus scattering and nucleosynthesis. Once kinetic mixing is introduced, many of these bounds improve, with the exception of the beam-dump bounds. Once $\sh\,\eta\gtrsim0.06$, kinetic mixing becomes sufficiently strong that the $W$-mass bound prevails over any other bounds in the $M_\ssX {{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}M_\ssZ$ region. For $\sh\,\eta=1$, we discard the region where the oblique $T$ parameter is large (for details, see §\[e+e-annihil\]), and focus on the region where $M_\ssX>385$ GeV. In this region, it is the neutrino-electron scattering bound and the $W$-mass bound that dominate.
[*Mass vs Mixing angle*]{}
It is useful to show these same bounds as exclusion plots in the mixing-angle/boson-mass plane, for fixed choices of the gauge-fermion coupling, $\alpha_\ssX$. This allows contact to be made with similar bounds obtained in the context of dark matter-inspired $U(1)$ models [@HiddenU1Bounds; @toroschuster; @ovanesyan; @maximsec], which correspond to the $\alpha_\ssX \to 0$ limit of the bounds we find here. This version of the plots is shown in Figure \[alphasummary\], restricted to the MeV-GeV mass range (in order to facilitate the comparison with earlier work).
For small, but non-zero, gauge coupling ($\alpha_\ssX \sim 10^{-10}$) the bounds from beam dump experiments weaken significantly. However, another strong bound from neutrino-electron scattering also begins to take effect. This bound dominates for larger $\alpha_\ssX$, and once $\alpha_\ssX {{\mathrel{\raise.3ex\hbox{$>$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}10^{-7}$ the entire MeV$-$GeV mass range is excluded.
Since the bounds in Figure \[alphasummary\] all rely on coupling to leptons, in the case where $X=B$ the constraints arise through the kinetic mixing and are independent of $\alpha_\ssX$. The resulting plot for $X=B$ is therefore the same as is shown in the figure for $\alpha_\ssX=0$. However, as the gauge coupling is increased the neutron-nucleus scattering bound — discussed in §\[neutnuclsec\] — eventually becomes important, first being visible as an exclusion in the $\sh\,\eta$ – $M_\ssX$ plane in the panel for $\alpha_\ssX \simeq 10^{-8}$ in Fig. \[alphasummary\].
Theoretical motivation
======================
This section elaborates the motivations for weakly coupled, very light gauge bosons alluded to above. This is done both by summarizing the consistency conditions they must satisfy within the low-energy effective theory relevant to experiments, and by describing how such bosons actually arise from several representative UV completions in string theory and extra-dimensional models.
Low-energy gauge symmetries, consistency and anomaly cancellation
-----------------------------------------------------------------
Very general arguments [@AnySpin; @MasslessGauge] indicate that the interplay between unitarity and Lorentz invariance require massless gauge bosons only to couple to conserved charges that generate exact symmetries of the matter action. Consequently we normally expect the direct couplings of very light gauge bosons to be similarly restricted. This section reviews these arguments, emphasizing how they can break down [@GaugeUnitarity; @AnomalyScale; @NonlinIsBroken] if the energy scale, $\Lambda$, of any UV completion is sufficiently small compared with the gauge boson mass, $M$, and coupling, $g$: $\Lambda {{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}4\pi M/g$. For the present purposes it suffices to restrict our attention to abelian gauge bosons (see however [@NonlinIsBroken] for some discussion of the nonabelian case).
The upshot of the arguments summarized here is that massive spin-one bosons can couple in an essentially arbitrary way if their mass, $M$, lies within a factor $g/4\pi$ of the scale of UV completion. But once $M$ becomes smaller than $g\Lambda/4\pi$, then the corresponding boson must gauge an honest-to-God, linearly realized exact symmetry. In particular this symmetry must be anomaly free. However any anomalies that Standard Model fermions give a putative new gauge charge needn’t be cancelled by adding new, exotic low-energy fermions; they can instead be cancelled by the Goldstone boson whose presence is in any case required if the gauge boson has a mass. But this latter sort of cancellation also requires the UV completion scale to satisfy $\Lambda {{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}4\pi M/g$.
Notice that for any given $M$ the condition $\Lambda {{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}4\pi M/g$ need not require $\Lambda$ to lie below the TeV scale if the coupling $g$ is small enough. For instance, if $M \simeq 1$ MeV then $\Lambda$ lies above the TeV scale provided $g {{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}10^{-5}$ (an upper limit often required in any case by the strong phenomenological bounds we find below). And, as subsequent sections argue, such small couplings can actually arise in a natural way from reasonable UV completions.
### Massless spin-one bosons {#massless-spin-one-bosons .unnumbered}
What goes wrong if a spin-one particle is not coupled to matter by gauging an exact symmetry of the matter action? If the spin-one particle is massless, then the problem is that one must give up either Lorentz invariance or unitarity (provided the particle has non-derivative, Coulomb-like couplings that survive in the far infrared). Lorentz invariance and unitarity fight one another because the basic field, $A_\mu(x)$, cannot transform as a Lorentz 4-vector if $A_\mu$ creates and destroys massless spin-one particles [@AnySpin; @MasslessGauge]. Instead it transforms as a 4-vector [*up to a gauge transformation*]{}, $A_\mu \to A_\mu + \partial_\mu \omega$, and so interactions must be kept gauge invariant in order to be Lorentz invariant [@WbgBook].
Massive spin-one bosons {#massive-spin-one-bosons .unnumbered}
-----------------------
For massive spin-one particles the argument proceeds differently, as is now described. The difference arises because a 4-vector field, $A_\mu$, can represent a [*massive*]{} spin-one particle [@AnySpin].
To examine the relevance of symmetries, it is worth first considering coupling massive spin-one particles to other matter fields, $\psi$, in some arbitrary non-gauge-invariant way, with lagrangian density $\cL(A_\mu,\psi)$. The first observation to make is that any such a lagrangian can be made gauge invariant for free, by introducing a Stückelberg field, $\phi$, according to the replacements $A_\mu \to \cA_\mu := A_\mu - \partial_\mu \phi$ and $\psi \to \Psi := \exp[-i \phi \, Q] \psi$, where $Q$ is a hermitian matrix acting on the fields $\psi$. With this replacement the lagrangian $\cL(\cA_\mu, \Psi)$ is automatically invariant under the symmetry $A_\mu \to A_\mu + \partial_\mu \omega$, $\phi \to \phi + \omega$ and $\psi \to \exp[i \omega \, Q] \psi$, since both $\cA_\mu$ and $\Psi$ are themselves invariant under these replacements. The original non-symmetric formulation corresponds to the specific gauge $\phi = 0$. For gauge symmetry, absence of gauge invariance is evidently equivalent to nonlinearly realized gauge invariance (similar arguments can also be made in the nonabelian case [@NonlinIsBroken]).
But this gauge invariance is obtained at the expense of introducing a new scale. Since $\phi$ is dimensionless, its kinetic term involves a scale, $v$, \_[kin]{} = - 1[4g\^2]{} F\_ F\^ - 2 (\_- A\_) (\^ - A\^) . In $\phi = 0$ gauge the scale $v$ is seen to be related to the gauge boson mass by the relation $M = g v$. In a general gauge the scale $v$ controls the size of couplings between the canonically normalized field, $\varphi = \phi \, v$, and other particles. For instance the coupling \[psipsiphicoupling\] \_[coupling]{} = - i (\^Q) (A\_- ) , shows that the $(\overline\psi \gamma^\mu Q\psi) \partial_\mu\varphi$ coupling is dimension-five, being suppressed by the scale $v = M/g$.
Lagrangians with nonrenormalizable couplings like this must be interpreted as effective field theories, whose predictive power relies on performing a low-energy expansion in powers of $E/\Lambda$, for some UV scale $\Lambda$. The interpretation of the scale $v$ then generically depends on the how high $\Lambda$ is relative to $4\pi M/g = 4\pi v$. We consider each case in turn.
[*Light spin-one bosons: $M \ll g \Lambda/4\pi$*]{}
If the gauge boson is very light compared with the UV scale, then its low-energy interactions should be describable by some renormalizable theory. But renormalizability is only consistent with a dimension-five interaction[^2] like the $(\overline\psi \gamma^\mu Q\psi) \partial_\mu \varphi$ coupling of eq. if this coupling is a [*redundant*]{} interaction, such as it would be if it could be removed by a field redefinition. A sufficient condition for an interaction of the form $J^\mu \partial_\mu \varphi$ to be redundant in this way is if the field equations for $\psi$ were to imply the quantity $J^\mu(\psi)$ satisfies $\partial_\mu J^\mu = 0$ [@EFTrev]. This shows that if the gauge boson is to be arbitrarily light relative to $\Lambda$, its low-energy, renormalizable couplings must be to a (dimension-three) conserved current. This is the usual prescription for obtaining these couplings by gauging a linearly realized matter symmetry, for which $J^\mu$ is the usual Noether current.
[*More generic massive spin-one bosons: $M {{\mathrel{\raise.3ex\hbox{$>$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}g \Lambda/4\pi$*]{}
If, on the other hand, the dimension-five coupling $(\overline\psi \gamma^\mu Q\psi) \partial_\mu\varphi$ is not redundant, then there must be an upper bound on the UV scale: $\Lambda {{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}4\pi v = 4\pi M/g$. Sometimes this may be seen from the energy-dependence predicted for the cross section of reactions in the low energy theory: if $\sigma(E) \propto 1/(4 \pi v)^2$ then this would be larger than the unitarity bound $\sigma {{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}1/E^2$ for energies $E {{\mathrel{\raise.3ex\hbox{$>$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}\Lambda \simeq 4 \pi v$, indicating the failure at these energies of the low-energy approximation. If so, the full UV completion must intervene at or below these energies to keep the theory unitary.
The upshot is that spin-one particles can couple fairly arbitrarily to matter provided they are massive, and provided the energy scale, $\Lambda$, of any UV completion satisfies $\Lambda {{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}4\pi M/g$, where $M$ is the gauge boson mass and $g$ is its coupling strength. (Everyday examples of spin-one particles of this type include the $\rho$ meson or spin-one nuclei.) It is only spin-one particles with $M < g\Lambda/4\pi$ that must gauge linearly realized symmetries.
Anomaly cancellation
--------------------
Any new gauge symmetry — henceforth denoted $U(1)_\ssX$ — must be an exact symmetry (though possibly spontaneously broken), and in particular must be anomaly free. This is true regardless of whether the symmetry is the linearly realized symmetry of a light gauge boson, or the nonlinearly realized symmetry of a massive gauge boson.
Of particular interest in this paper are models where the new symmetry acts on ordinary fermions, because a robust motivation for thinking about light gauge bosons is the avoidance of proton decay in models with a low gravity scale (more about which below). In this case these ordinary fermions usually contribute gauge anomalies for the new symmetry, and an important issue is how these anomalies are ultimately cancelled. The two main anomaly-cancellation scenarios then divide according to whether or not anomalies cancel among the SM fields themselves, or require the addition of new particles.
### Anomaly cancellation using only SM fields {#anomaly-cancellation-using-only-sm-fields .unnumbered}
The simplest situation is where the new gauge symmetry is simply a linear combination of one or more of the SM’s four classical global symmetries — baryon number $B$, electron number $L_e$, muon number $L_\mu$ and tau number $L_\tau$. In this situation there are only two independent combinations of these symmetries that are anomaly free[^3] [@SM], corresponding to arbitrary linear combinations of the anomaly-free symmetries $L_e - L_\mu$ and $L_\mu - L_\tau$: X = a (L\_e - L\_) + b (L\_- L\_) . Of course, evidence for neutrino oscillations [@Numassev] make it unlikely that these symmetries are unbroken in whatever replaces the Standard Model in our ultimate understanding of Nature.
### Anomaly cancellation using the Green-Schwarz mechanism {#anomaly-cancellation-using-the-green-schwarz-mechanism .unnumbered}
If more general combinations of $B$, $L_e$, $L_\mu$ and $L_\tau$ are to be gauged, it is necessary to introduce new particles that can cancel their Standard Model anomalies. For a new $U(1)_\ssX$ symmetry the minimal way to do this is to add only the Goldstone boson, which must in any case be present if the corresponding gauge boson has a mass (as it typically must to avoid mediating a macroscopic, long-range new force, whose presence is strongly disfavoured by observations [@NewForce]). For a $U(1)_\ssX$ symmetry this can always be done using the 4D version [@4DGS] of the Green-Schwarz mechanism [@GS]. Besides its intrinsic interest, this is a way of cancelling anomalies that actually arises from plausible UV physics, such as low energy string models.
In principle, there are four types of new anomalies that can arise in 4D once the SM is supplemented by a new gauge symmetry, $U(1)_\ssX$. These are proportional to Tr\[$XXX$\], Tr\[$XXY$\], Tr\[$XYY$\] and Tr\[$X G^a G^a$\], where the trace is over all left-handed fermions and $X$ denotes the new symmetry generator, $Y$ is Standard Model hypercharge, and $G^a$ represents the generators of the Standard Model nonabelian gauge groups, $SU(2)_\ssL \times SU(3)_c$, as well as the generators of Lorentz transformations. In four dimensions CPT invariance implies the absence of pure gravitational anomalies, and anomaly cancellation within the Standard Model ensures the absence of anomalies of the form Tr\[$YYY$\] and Tr\[$G^a G^b G^c$\].
It is always possible to redefine the new symmetry generator, $V := X + \zeta \, Y$, to remove one of the two mixed anomalies. For instance, Tr$[VVY] = \hbox{Tr}[XXY] + 2 \,\zeta \, \hbox{Tr}[XYY]$ can be made to vanish by choosing $\zeta$ appropriately (provided Tr\[$XYY$\] does not vanish). It suffices then to consider only the case of nonzero anomalies of the form Tr\[$VVV$\] and Tr\[$V G^a G^a$\], where $G^a$ now includes also the generator $Y$. The anomaly then can be written in the $G^a$- and Lorentz-invariant form[^4] \[anomaly\] &=& - \^4 x { c\_F\_F\_+ c\_a \[ F\_a F\_a \] - c\_\[ R R \] }\
&=& - \^4 x { c\_( F\_+ F\_) (F\_+ F\_) + c\_a \[ G\^a G\^a \] - c\_ \[ R R \]} ,where $\Gamma$ is the ‘quantum action’ (generator of 1PI correlations), the symmetry parameter is normalized by $\delta X_\mu = \partial_\mu \omega$ and the coefficients, $c_\ssX$, $c_a$ and $c_\ssL$, are calculable. Here $F_\ssV = \exd V = F_\ssX + \zeta \, F_\ssY$ is the gauge-boson field strength for the generator $X + \zeta \, Y$, while $F_a$ is the same for the Standard Model gauge bosons and $R$ is the gravitational curvature 2-form.
Given the coefficients $c_\ssX$, $c_a$ and $c_\ssL$, here is how 4D Green-Schwarz anomaly cancellation works [@4DGS]. Consider the gauge kinetic lagrangian, including the Stückelberg field $\phi$, \[stract\] [L]{} &=& \_[inv]{} - [1 4g\^[2]{} ]{} F\^\_ F\_\^ - [1 4 g\_[a]{}\^[2]{} ]{} [Tr]{} \[G\^[a]{}\_ G\_[a]{}\^ \] - [v\^2 2]{} ( \^ - X\^ ) (\_- X\_)\
&& + { c\_( F\_+ F\_) (F\_+ F\_) + c\_a \[ G\^a G\^a \] - c\_ \[ R R \]} . Here $\cL_{\rm inv}$ denotes those parts of the lagrangian that are invariant under all of the gauge symmetries that are not written explicitly. The second line is not invariant under gauge transformations because $\phi$ is not; its variation precisely cancels the fermion anomaly, eq. .
An important observation is that the anomaly cancelling term is dimension-five, and so is not renormalizable. For instance, in terms of the canonically normalized field, $\varphi = \phi v$, the first anomaly cancelling term is $\cL_{\rm anom} = (\varphi/f) F_\ssX \wedge F_\ssX + \cdots$, where $f = v/c_\ssX$. As before, this implies the existence of a UV-completion scale, $\Lambda$, above which the low-energy effective description breaks down [@AnomalyScale]. For weakly coupled theories typically $\Lambda {{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}4\pi v \simeq 4\pi M/g \simeq 4 \pi c_\ssX f$ marks the scale where the fields arise that are required to extend the Goldstone boson to a linear representation of the symmetry.
Perhaps the most interesting feature of cancelling anomalies with the Green-Schwarz mechanism in this way is that the lagrangian remains invariant under the $U(1)$ symmetry, apart from the anomaly-cancelling term. This is interesting because it means that the corresponding charge still appears to be conserved in the low-energy theory, [*despite the gauge field being massive*]{}. This opens up interesting phenomenological possibilities for the gauging of symmetries like $U(1)_{B}$ and $U(1)_{B-L}$, which appear to be conserved in Nature but which are also ruled out as sources of the new long-range force that a massless gauge boson would imply.
One might worry that arbitrary symmetry-breaking interactions might be generated by embedding the anomaly cancelling interactions (or the fermion triangle anomaly graph) into a quantum fluctuation. For instance if $X = B$, so the new gauge boson couples to baryon number, then why can’t some complicated loop generate a $\Delta B = \pm 1$ interaction, $\cO_{\pm 1}$, that can mediate proton decay? After all, this can be $U(1)_B$ invariant if it arises multiplied by a factor $e^{\mp i\phi}$, which carries baryon number $\Delta B = \mp 1$.
The difficulty with generating this kind of interaction is that it must involve $\phi$ undifferentiated. But if we restrict $\cL_{\rm anom}$ to constant $\phi$ configurations, it becomes a total derivative. For constant $\phi$, the dependence of observables on $\phi$ is similar to the dependence of observables on the vacuum angle, $\theta$. Consequently it arises at best only non-perturbatively, proportional at weak coupling to a power of $\sim \exp[- 8\pi^2/g^2]$, where $g$ is the anomalous gauge coupling. As a result the only potentially dangerous contribution of this type comes from the mixed $X$-QCD-QCD anomaly, which can generate nontrivial $\phi$-dependence once we integrate down to scales ${{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}\Lambda_{\scriptscriptstyle QCD}$. This is not dangerous in particular for the classical symmetries, $B$, $L_e$, $L_\mu$ and $L_\tau$, since these do not have mixed QCD anomalies [@SM].
### Anomaly cancellation using new fermions {#anomaly-cancellation-using-new-fermions .unnumbered}
More complicated possibilities for new gauge bosons emerge if new, light exotic fermions are allowed that also carry the new $X$ charge (and so can also take part in the anomaly cancellation). We briefly describe some features involving such new exotic particles, although they do not play any role in our later phenomenological studies.
The simplest example along these lines is $X = B-L$, which is anomaly-free provided only that the SM spectrum is supplemented by three right-handed neutrinos (one for each generation). Furthermore, conservation of $L$ is consistent with all evidence for neutrino oscillations, although it would be ruled out should neutrinoless double-beta decay ever be witnessed.
A practical way in which such new fermions can arise at TeV scales is if the UV theory at these scales is supersymmetric. In this case the plethora of new superpartners can change anomaly cancellation in one of two ways (or both). They can either directly contribute to the anomalies themselves, and possibly help anomalies cancel without recourse to the Green-Schwarz mechanism. Alternatively, they can modify the details of how the Green-Schwarz mechanism operates if the UV scale, $v$, associated with it is larger than the supersymmetry breaking scale, $M_{\rm susy}$.
In particular, supersymmetry typically relates the kinetic term for the Stückelberg field, ([\[stract\]]{}), with a Fayet-Iliopoulos term in the scalar potential [@GS4Dhet], S\_[FI]{} = - [1 g\^[2]{} ]{} d\^4 x( - \_[i]{} q\_[i]{} \_[i]{}\^ \_[i]{} )\^[2]{} , where $\tau$ is a dynamical field whose vev acts as the low-energy Fayet-Iliopoulos parameter; the $q_{i}$ are the charges of the fields $\phi_{i}$ under the $U(1)$ in question. In string examples the field $\tau$ corresponds to a modulus of the compactification, which controls the size of a cycle in the internal geometry on which some branes wrap. We note that the vanishing of the D-term is consistent with vanishing vevs of the charged fields if $\tau = 0$, [*i.e.*]{} the symmetry survives as an [*exact*]{} global symmetry when the cycle size vanishes (the singular locus). Small values of the vev are obtained if the cycle size is small.
Motivations from UV physics
---------------------------
The above summary outlines some of the theoretical constraints on coupling ordinary fermions to very light gauge bosons. This section shows how very small couplings can naturally appear in well-motivated ultraviolet physics, such as extra-dimensional models or string vacua. In particular, they often arise due to considerations of proton stability in constructions for which the gravity scale is small compared with the Planck scale, as we now explain.
### Proton decay in low-scale gravity models {#proton-decay-in-low-scale-gravity-models .unnumbered}
One of the surprises of the late 20th century was the discovery that the scale, $M_g$, of quantum gravity could be much smaller than the Planck scale, $M_p = (8\pi G)^{-1/2} \simeq 10^{18}$ GeV [@StringLow]. From the point of view of particle physics this possibility is remarkable for several reasons. Most obvious is the potential it allows for experimental detection if it should happen that $M_g$ is in the vicinity of the TeV scale [@ADD; @RS].
But there is a potentially more wide-reaching consequence that $M_g \ll M_p$ has for the low-energy sector: the suppression by powers of $M_g/M_p$ it allows for otherwise UV-sensitive radiative corrections [@Ubernat]. This suppression arises because the contribution of short-wavelength degrees of freedom can saturate at $M_g$, allowing their effects to be suppressed by powers of the gravitational coupling.
The most precise examples of this are provided by string theory, in the regime where the string scale is low, $M_g := M_s \ll M_p$ [@StringLow]. String theory makes the suppression of UV-sensitive contributions precise by providing an explicit stringy ultraviolet completion within which the effects of the full UV sector can be explored. Large-volume (LV) models [@LV] are particularly useful laboratories for these purposes, since these systematically exploit the expansion of inverse powers of the extra-dimensional volume (in string units), $\cV := (\hbox{Vol})/\ell_s^6 \gg 1$, and it is ultimately these kinds of powers that enforce the suppressions of interest since $M_s/M_p \propto \cV^{-1/2}$.
Proton decay — that is, its experimental absence — turns out to impose a very general constraint on any fundamental theory of this type, with $M_g \ll M_p$. It does so because having $M_g$ very small removes two of the standard ways of keeping the proton stable in specific models. On one hand quantum gravity, and string theory in particular [@NoGlobSinST; @hyper], seems to preclude the existence of global symmetries, and this forbids ensuring proton stability by simply using a conserved global charge (such as baryon number).
If $M_g$ is too small then it is also unlikely that such a symmetry simply emerges by accident for the lowest-dimension interactions in the low-energy effective theory. The problem in this case is that we know that generic higher-dimensional interactions, \_[eff]{} = \_i , eventually do arise in the low-energy effective theory, such as the standard baryon-number violating 4-quark operators arising at dimension $d_i = 6$ [@BVdim6] in the low-energy limit of grand-unified theories (GUTs) [@GUTFirst; @GUTRevs]. But a dimension-six interaction of the form $\cO/M^2$ generically contributes a proton-decay rate of order $\Gamma \simeq m_p^5/M^4$, where $m_p$ is the proton mass, which is too large to agree with observations once $M$ falls below $M_\GUT \simeq 10^{16}$ GeV.
The way theories with $M_g \ll M_p$ usually evade proton decay is through the appearance of a [*gauged*]{} $U(1)$, whose conservation forbids the decay. Of course, to be useful the gauged $U(1)$ that appears must couple to the proton or its decay products in order to forbid its decay. But because this means ordinary particles couple to the new gauge boson, it potentially introduces other phenomenological issues. If the gauge symmetry is conserved, why isn’t the gauge boson massless? If the gauge boson is light, why isn’t the new boson seen in low-energy observations? If the gauge boson is heavy, the corresponding symmetry must be badly broken and so how can it help with proton decay? Interestingly, extant models can naturally address both of these issues, and often the low-energy mechanism that is used is Green-Schwarz anomaly cancellation with gauge boson mass generated through the Stückelberg mechanism described above. Sometimes this mechanism is also combined with supersymmetry to suppress the dangerous decays.
The existence of these gauge bosons, their properties, and the way they evade the above issues, may be among the few generic low-energy consequences of viable theories with a low gravity scale: $M_g \ll M_\GUT$.
[*Sample symmetries:*]{}
The simplest proposals for new low-energy gauge groups that forbid proton decay are either baryon or lepton number, $X = B$ or $X = L$. If the anomalies for these symmetries due to Standard Model fermions are cancelled through the Green-Schwarz mechanism, then no new light particles are required besides the massive gauge boson itself.
More complicated examples are possible if the low-energy theory at TeV scales is supersymmetric. In this case symmetries like $B-L$, that in themselves cannot forbid proton decay, can help suppress proton decay if taken together with supersymmetry [@IQ]. (For instance, the parity $R = (-)^{F + 3(B-L)}$ that is usually used to suppress proton decay in the MSSM is a combination of fermion number and $B-L$.)
More general combinations of $B$ and $L$ can also suppress proton decay in supersymmetric theories. Ref. [@IQ] provides a list of the kinds of symmetries of this type that can be relevant to proton decay, as well as the conditions they must satisfy in order to have their anomalies be cancelled through the Green-Schwarz mechanism. The general form for the low-energy charge may be written X = m T\_+ n A + p L , where $T_\ssR$ is right-handed isospin; $A$ is an axionic PQ symmetry; and $L$ is lepton number, with the charge assignments given in the Table. The coefficients $m$, $n$ and $p$ are subject to (but not over-constrained by) several anomaly cancellation conditions [@IQ]. In particular $B$ and $L$ violating interactions can be forbidden up to and including dimension six for some choices of these symmetries in the supersymmetric limit, as can the $\mu$-terms of the superpotential – $W \simeq \mu_\ssL L \ol{H}$ and $W \simeq \mu H \ol{H}$ – if $n \ne 0$.
$Q$ $U$ $D$ $L$ $E$ $H$ $\ol{H}$
---------- ----- ----- ------- ------- -------- ------- ----------
$T_\ssR$ 0 1 $-1$ 0 $-1$ 1 $-1$
$A$ 0 0 1 1 0 $-1$ 0
$L$ 0 0 0 1 $-1$ 0 0
$X$ 0 $m$ $n-m$ $n+p$ $-m-p$ $m-n$ $-m$
### Very light and weakly coupled gauge bosons from extra-dimensional models {#very-light-and-weakly-coupled-gauge-bosons-from-extra-dimensional-models .unnumbered}
For the phenomenological discussions of later sections we consider gauge bosons in the MeV to TeV mass range, whose direct couplings to Standard Model fermions are much smaller than those arising within the Standard Model itself. This section and the next one describe several way that very light and weakly coupled bosons can arise from reasonable UV physics.
Extra-dimensional supergravity provides a simple way to obtain very light gauge bosons that are very weakly coupled. A concrete example is six-dimensional chiral gauged supergravity [@NS], for which the bosonic part of the gravity multiplet contains the metric, $g_{\ssM\ssN}$, a Kalb-Ramond 2-form potential, $B_{\ssM\ssN}$, and a scalar, $\phi$. Because it is chiral this supergravity potentially has anomalies, whose cancellation imposes demands on the matter content. In six dimensions Green-Schwarz anomaly cancellation is not automatic, because cancellation of the pure gravitational anomalies requires the existence of a specific number of gauge multiplets [@6Danom]. Given these multiplets, mixed gauge-gravity anomalies can be cancelled through the Green-Schwarz mechanism using the couplings of the field $B_{\ssM\ssN}$.
The resulting supergravity admits simple solutions for which the extra dimensions are a sphere [@SSsoln], whose moduli can be stabilized by a combination of background fluxes in the extra dimensions [@6Dflux], and branes coupling to the 6D dilaton [@6DGWmech; @BraneBR]. An important feature of this stabilization is that the value of the dilaton field becomes related by the field equations to the size of the extra dimensions: \[6Dmodstab\] e\^= , where $M_6$ denotes the 6D Planck scale. This ensures these models are a rich source of $U(1)$ gauge bosons, some of whom can have massless modes that survive to low energies below the Kaluza-Klein scale. Some of these gauge modes also naturally acquire masses through the Stückelberg mechanism [@6Dflux] (with the Stückelberg field arising as a component of the Kalb-Ramond field, $B_{\ssM\ssN}$).
Besides having light gauge bosons, these models also naturally furnish them with very small coupling constants. This is because the loop-counting parameter for all bulk interactions turns out to be the value of the 6D dilaton, $\phi$, with $g^2 \simeq e^\phi$. But modulus stabilization, eq. , ensures that this coupling can be extremely small because it scales inversely with the size of the extra dimensions (measured in 6D Planck units).
### Very light and weakly coupled gauge bosons from low-energy string vacua {#very-light-and-weakly-coupled-gauge-bosons-from-low-energy-string-vacua .unnumbered}
A related mechanism also often arises in low-scale string models. In early heterotic models the role of the Goldstone boson is played by a member of the dilaton super-multiplet: $a \simeq \hbox{Im} \, S$ [@GS4Dhet], while in later Type I and Type II models it is twisted closed string multiplets that instead play this role [@GS4DII; @IQ]. Although the universal couplings of the dilaton restrict the kinds of symmetries that can arise in heterotic constructions of this type, the same is not true for Type I and II models.
There is a simple reason why additional $U(1)$ gauge groups often arise. The basic building blocks for constructing models of particle physics in type IIB and IIA string theory are D-branes. Generically, the gauge group associated with a stack of $N$ D-branes is $U(N)$, but the Standard Model gauge group involves special unitary groups, $SU(3) \times SU(2) \times U(1)$. Typical GUT models also involve special unitary groups, like $SU(5)$, $SU(3) \times SU(2) \times SU(2) \times U(1)$ (Left-Right symmetric models) or $SU(4) \times SU(2) \times SU(2)$ (Pati-Salam models). It is the additional $U(1)$s that distinguish the Standard Model $SU(N)$ factors from the $U(N)$ factors arising from the D-branes, that give new low-energy gauge symmetries.
Furthermore, anomaly cancellation in string theory typically demands the presence of additional D-brane stacks, in addition to those providing the Standard Model gauge group factors. These stacks also lead to extra $U(1)$s under which Standard Model particles are charged. Extra $U(1)$s also appear naturally in F-theory models (for a recent discussion see [@timo]). In many concrete examples these additional gauge fields correspond to $U(1)_{B}$ or $U(1)_{B-L}$, hence can be relevant for the stability of the proton [@fqib; @IQ] (see also [@cmq] for a recent discussion).
[*Masses and couplings*]{}
For string vacua the masses and couplings of any gauged $U(1)$s can be computed, as we now briefly describe.
Consider first the $U(1)$s associated with the same stack of D-branes as gives rise to the Standard Model gauge group. As discussed earlier, such gauge bosons often acquire masses from the Stückelberg mechanism. The size of the mass generated in this way is the string scale when the $U(1)$ is anomalous [@wijn; @cmq], but it is the smaller Kaluza-Klein scale for non-anomalous $U(1)$s.
For models with the compactification volume not too much larger than the string scale, these $U(1)$ gauge bosons are very heavy. On the other hand, for large-volume models the string scale can be quite low, leading to additional $U(1)$s potentially as light as the TeV scale. The latter can have interesting low energy phenomenology (see for instance [@fqib; @david; @zwo; @zwt; @kr]). In these models the strength of the gauge coupling for the additional $U(1)$s is roughly the same as for the Standard Model gauge couplings (evaluated at the string scale), because both have the same origin: the world-volume theory of the stack. Hence they cannot be extremely small.
The masses and couplings of the extra $U(1)$ gauge bosons vary more widely when they arise from D-brane stacks whose $SU(N)$ factors are not part of the Standard Model gauge group. For instance, the case of additional $U(1)$s associated with D7 branes wrapping bulk four cycles of the compactification is discussed in detail in [@hyper]. The value of the gauge coupling in this case is inversely proportional to the volume (in string units) of the cycle, $\Sigma$, that the D7 brane wraps, g\^[2]{} . In the context of the large volume scenario (LVS) of modulus stabilization [[@LV]]{}, the size of the bulk cycle associated with the overall volume of compactification can easily be approximately $\cV_\Sigma {{\mathrel{\raise.3ex\hbox{$>$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}10^{9}$ in string units, set by the requirement that one generate TeV-scale soft terms. Thus one can obtain gauge couplings as low as $g {{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}2 \times 10^{-4}$ [@hyper; @LVSU1] (couplings larger than this can be obtained if the D7 brane wraps a cycle different from the one associated with the overall volume). With couplings this small, the gauge boson mass can be $M_\ssX \simeq g v {{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}100$ MeV even if $v$ is a TeV.
Gauge boson properties {#Xprops}
======================
With the above motivation, our goal in the remainder of the paper is to work out various constraints on the parameters of a massive (yet comparatively light) gauge boson, the $X$ boson, that couples to a new $U(1)_\ssX$ symmetry. Since the lowest dimension interactions dominate in principle at low energies, we include in our analysis all of the dimensionless couplings that such a boson could have with Standard Model particles: [*i.e.*]{} both direct fermion-gauge couplings and gauge kinetic mixing. We see how these are constrained by present data as a function of the gauge boson mass.
More specifically, we consider an effective lagrangian density below the supersymmetry breaking scale of the form $$\cL = \cL_\SM + \cL_\ssX +
\cL_{\rm mix}$$ where $\cL_\SM$ is the usual Standard Model lagrangian; $\cL_\ssX$ describes the $X$ boson, including its couplings to the SM fermions; and $\cL_{\rm mix}$ is the kinetic-mixing interaction between the $X$ boson and that of the SM gauge factor $U(1)_\ssY$ [@holdom2u1s]. Explicitly \_= - 14 X\_ X\^ - X\_X\^+ i J\^\_X\_, where $X_{\mu \nu } := \partial_\mu X_\nu - \partial_\nu X_\mu$ is the curl of the appropriate gauge potential, $X_\mu$, and $J^\mu_\ssX$ is the current for the $U(1)_\ssX$ gauge symmetry involving the SM fermions. Similarly, $\cL_{\rm mix}$ has the form, \_[mix]{} = B\_ X\^ where $B_\mu$ is the SM gauge boson for the gauge factor $U(1)_\ssY$.
The analysis we provide complements and extends earlier studies of extra gauge boson phenomenology. In the lower part of the mass range we may compare with [@carlson], who some time ago considered the special cases $X = B-L$ and $\chi = 0$. Contact is also possible in this mass range with more recent Dark Matter models [@HiddenU1Bounds; @toroschuster; @ovanesyan] in the absence of direct matter couplings, $g_\ssX J^\mu = 0$. At masses much lower than those considered here other constraints on kinetic mixing have also been studied, from the cosmic microwave background [@cmb], and from the absence of new long-range forces [@NewForce] or milli-charged particles [@holdom2u1s; @millichargebounds].
There is also a broad literature on the phenomenology of gauge bosons at the upper end of the mass range, largely done in the context of a $Z^{\prime}$ field and often motivated by GUTs [@GUTFirst; @so10; @e6; @hewettrizzo; @leftright; @atthetev]. Until recently, most did not include the kinetic mixing term. Constraints including kinetic mixing arising from precision electroweak experiments are considered in [@preseweakmix; @BKM; @Leike]; more recent bounds are found in [@HIW; @langacker; @wells; @ChangNgWu]. Many of these analyses overlap parts of our parameter space. For instance $Z^\prime$ searches, such as [@d0], give bounds on the mass of the $Z^\prime$ that apply in the regime that the couplings to fermions are identical to that of the $Z$. Others [@bonly; @CMbarnum] derive bounds for a $Z^{\prime }$ coupled only to baryon number.
One difference between the models examined here and those usually considered for $Z'$ phenomenology at the weak scale, such as those of ref. [@massmixing], is the absence in $\cL$ of mixing between the $X$ and the $Z$ bosons in the mass matrix ([*i.e.*]{} a term of the form $\cL_{\rm mix} = \delta m^2 Z_\mu X^\mu$). We do not consider this type of mixing because we imagine the models of interest here to break the $X$ symmetry with a SM singlet. Notice that because the SM Higgs is uncharged under the $X$ symmetry, the strong bounds as found, for example, in [@CGR] don’t apply.
The mixed lagrangian
--------------------
In this section we diagonalize the gauge boson kinetic mixing terms (and SM mass terms) and identify the physical combination of parameters relevant for phenomenology within the accuracy to which we work. Our goal in so doing is to follow ref. [@stuvwx; @bigfit] and identify once and for all how the gauge boson mixing contributes to fermion couplings and to oblique parameters [@oblique] modified by the gauge-boson mixing. This allows an efficient identification of how observables depend on the mixing parameters.
We begin by writing the lagrangian of interest more explicitly, after spontaneous symmetry breaking. Because it is the $Z$ and photon that potentially mix with the $X$ boson, we also focus on these sectors of the SM lagrangian. In order to distinguish the fields before and after mixing, where appropriate we denote the still-mixed fields with carets, [*e.g.*]{} $\hat X_\mu$, reserving variables like $X_\mu$ for the final, diagonalized fields.
With this notation, the lagrangian of interest is $$\cL = \cL_{\rm gauge} + \cL_{f}
+\cL_{\rm int} \,,$$ where \_[gauge]{} = \_[kin]{} + \_[mass]{} with $$\begin{aligned}
\cL_{\rm kin} &=& - \frac14 \hat{W}_{\mu\nu}^3
\hat{W}_3^{\mu\nu} - \frac14 \hat{B}_{\mu\nu}
\hat{B}^{\mu\nu} - \frac14 \hat{X}_{\mu\nu}
\hat{X}^{\mu\nu} + \frac{\chi}{2} \hat{B}_{\mu\nu}
\hat{X}^{\mu \nu } \\
\cL_{\rm mass} &=& -\frac12 \left( m_3 \hat{W}_\mu^3
-m_0 \hat{B}_\mu \right) \left( m_3 \hat{W}_3^\mu
-m_{0} \hat{B}^\mu \right) - \frac{m_\ssX^2}{2}
\, \hat{X}_\mu \hat{X}^\mu \,,\end{aligned}$$ and $$\begin{aligned}
\cL_{f} &=& - \sum_f \ol{f} \left( \dsl
+ m_f \right) f \\
\cL_{\rm int} &=& i \sum_f \left\{ g_2 \left( \ol{f}
\gamma^\mu T_{3f} \gamma_\ssL f \right) \hat{W}_\mu^3
+ g_1 \left[ \ol{f} \gamma^\mu
\left( Y_{f \ssL} \gamma_\ssL + Y_{f \ssR} \gamma_\ssR \right)
f \right] \hat{B}_\mu \right. \\
&& \quad \left. + g_\ssX \left[ \ol{f} \gamma^\mu
\left( X_{f \ssL} \gamma_\ssL + X_{f \ssR} \gamma_\ssR \right)
f \right]
\hat{X}_\mu \right\} \,.\end{aligned}$$ Here $T_{3f}$, $Y_{f\ssL}$ and $Y_{f\ssR}$ denote the usual SM charge assignments, while $X_{f\ssL}$ and $X_{f\ssR}$ are the fermion charges under the new $U(1)_\ssX$ symmetry. The SM masses, $m_3$ and $m_0$, are defined as usual [@SM] in terms of the standard model gauge couplings, $g_1$ and $g_2$, and the Higgs VEV, $v$: $m_3 = \frac12 \, g_2 v$ and $m_0 = \frac12 \, g_1 v$. $\gamma_\ssL$ and $\gamma_\ssR$ are the usual left- and right-handed Dirac projectors.
Defining the gauge-field-valued vector $\hat{\bf V}$ to be $$\hat{\bf V}=\begin{bmatrix}
\hat{W}^{3} \\
\hat{B} \\
\hat{X}
\end{bmatrix} \,,$$ the above lagrangian can be written in matrix form $$\label{startLeq}
\cL_{\rm gauge} + \cL_{\rm int} = -\frac14 \hat{\bf V}_{\mu\nu}^\ssT \hat K
\hat{\bf V}^{\mu\nu} - \frac12 \hat{\bf V}_\mu^\ssT
\hat M \hat{\bf V}^\mu + i \hat{\bf J}_\mu^\ssT
\hat{\bf V}^\mu$$ where $$\label{startKMeq}
\hat K :=
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & -\chi \\
0 & -\chi & 1
\end{bmatrix}
\quad \hbox{and} \quad
\hat M :=
\begin{bmatrix}
m_{3}^2 & -m_{3}m_{0} & 0 \\
-m_{3}m_{0} & m_{0}^2 & 0 \\
0 & 0 & m_\ssX^2
\end{bmatrix} \,,$$ and $$\hat{\bf J}_\mu :=
\begin{bmatrix}
J_\mu^3 \\
J_\mu^\ssY \\
\hat{J}_\mu^\ssX
\end{bmatrix}
= \sum_f \begin{bmatrix}
g_2 \left[ \ol{f} \gamma_\mu T_{3f} \gamma_\ssL f \right] \\
g_1 \left[ \ol{f} \gamma_\mu \left( Y_{f\ssL} \gamma_\ssL
+ Y_{f\ssR} \gamma_\ssR \right) f \right] \\
g_\ssX \left[ \ol{f} \gamma_\mu \left( X_{f\ssL}
\gamma_\ssL + X_{f\ssR} \gamma_\ssR \right) f \right]
\end{bmatrix} \,.$$ The off-diagonal elements of $\hat M$ ensure it has a zero eigenvalue, and the condition that the matrix $\hat K$ be positive definite requires $\chi^2 < 1$.
Physical couplings {#physcouplings}
------------------
In order to put this lagrangian into a more useful form we must diagonalize the kinetic and mass terms, and then eliminate the SM electroweak parameters in terms of physically measured input quantities like the $Z$ mass, $M_\ssZ$, the fine-structure constant, $\alpha = e^2/4\pi$, and Fermi’s constant, $G_\ssF$, as measured in muon decay.
The diagonalization is performed explicitly in the Appendix, leading to the diagonalized form $$\cL = -\frac14 \, {\bf V}_{\mu\nu}^\ssT
{\bf V}^{\mu\nu} - \frac{M_\ssZ^2}{2} \, Z_\mu Z^\mu
-\frac{M_\ssX^2}{2} \, X_\mu X^\mu +
i {\bf J}_\mu^\ssT {\bf V}^\mu \,,$$ where the physical masses are M\_\^2 = ( 1 + s\_\^2 \^2 + r\_\^2 \^2 + \_ ) \[masseigs1\] and M\_\^2 = ( 1 + s\_\^2 \^2 + r\_\^2 \^2 - \_ ) . \[masseigs\] In these expressions $m_\ssZ^2 := \frac14 \left( g_1^2 + g_2^2 \right) v^2$, \_:= \_:= \_:= \_ := , while := := := := . Finally, the quantities $r_\ssX$ and $\vartheta_\ssX$ are defined by r\_:= \_ := {
[c]{} +1 r\_ > 1\
-1 r\_ < 1
. , which ensures $M_\ssZ \rightarrow m_\ssZ$ and $M_\ssX \rightarrow m_\ssX$ as $\eta \rightarrow 0$.
The currents in the physical basis are similarly read off as $${\bf J}_\mu
:=
\begin{bmatrix}
J_\mu^\ssZ \\
J_\mu^\ssA \\
J_\mu^\ssX
\end{bmatrix} =
\begin{bmatrix}
\check J_\mu^\ssZ c_\xi +
\left( -\check J_\mu^\ssZ \hat s_\ssW \sh \, \eta
+ \check J_\mu^\ssA \hat c_\ssW \sh \, \eta
+ \check J_\mu^\ssX \ch \, \eta \right) s_\xi \\
\check J_\mu^\ssA \\
- \check J_\mu^\ssZ s_\xi
+ \left( - \check J_\mu^\ssZ \hat s_\ssW \sh \, \eta
+ \check J_\mu^\ssA \hat c_\ssW \sh \, \eta
+ \check J_\mu^\ssX \ch \, \eta \right) c_\xi
\end{bmatrix} \,,$$ where
J\_\^\
J\_\^\
J\_\^
:=
J\_\^3 c\_- J\_\^s\_\
J\_\^3 s\_+ J\_\^c\_\
\_\^
= \_f
i e\_ \_f\
ie \_Q\_f f\
ig\_ \_f
, and $e := g_2 \hat s_\ssW = g_1 \hat c_\ssW$, $\hat e_\ssZ := e/(\hat s_\ssW \hat c_\ssW)$ and $Q_{f} = T_{3f} + Y_{f\ssL} = Y_{f\ssR}$. Finally, $c_\xi := \cos \xi$ and $s_\xi := \sin \xi$ with the angle $\xi $ given by $$\tan 2\xi =\frac{-2 \hat s_\ssW \sh \eta }{1-\hat s_\ssW^2 \sh^2 \eta-r_\ssX^2 \ch^2\eta} \,.
\label{tan2alpha}$$ Writing the resulting lagrangian as $$\cL = \cL_\SM + \delta \cL_\SM +
\cL_\ssX \,, \label{Leff1}$$ shows that the $X$ boson has two kinds of physical implications: $(i)$ direct new couplings between the $X$ boson and SM particles; $(ii)$ modifications (due to mixing) of the couplings among the SM particles themselves.
### Modification of SM couplings {#modification-of-sm-couplings .unnumbered}
The modification to the SM self-couplings caused by $Z-X$ mixing are given by \[Leff2\] \_= - m\_\^2 Z\_Z\^ + i e\_\_f Z\_, with [@bigfit] $z := (M_\ssZ^2 - m_\ssZ^2)/m_\ssZ^2$ and g\_[f()]{} = (c\_-1) g\_[f()]{}+ s\_( s\_(Q\_f c\_\^2 -g\_[f()]{}) + X\_[f()]{} ) .
The last step before comparing these expressions with observations is to eliminate the parameters $\hat s_\ssW$ and $m_\ssZ$ (the second of which enters the interactions through $r_\ssX$) from the lagrangian in favour of a physically defined weak mixing angle, $s_\ssW$, and the physical mass, $M_\ssZ$. This process reveals the physical combination of new-physics parameters that is relevant to observables, and thereby provides a derivation [@bigfit] of the $X$-boson contributions to the oblique electroweak parameters [@oblique].
To this end define the physical weak mixing angle, $s_\ssW$, so that the Fermi constant, $G_\ssF$, measured in muon decay is given by the SM formula, $$\frac{G_\ssF}{\sqrt{2}} := \frac{e^2}{8 s_\ssW^2 c_\ssW^2 M_\ssZ^2} \,.
\label{G_fdef}$$ But this can be compared with the tree-level calculation of the Fermi constant obtained from $W$-exchange using the above lagrangian, giving (see Appendix) $$\hat s_\ssW^2 = s_\ssW^2 \left[
1 + \frac{z \, c_\ssW^2}{c_\ssW^2 - s_\ssW^2}
\right] \,,$$ to linear order in $z$ (which we assume is small — as is justified shortly by the phenomenological bounds).
Eliminating $\hat s_\ssW$ in favour of $s_\ssW$ in the fermionic weak interactions introduces a further shift in these couplings, leading to our final form for the neutral-current lagrangian: \_ = i e\_\_f \^f Z\_, \[dLNC\] where $e_\ssZ := e/s_\ssW c_\ssW$ and $$\begin{aligned}
\label{DgfLR}
\Delta g_{f \ssL(\ssR)} &=& - \frac{z}{2} g_{f \ssL(\ssR)}^{\SM} - z \left(\frac{s_\ssW^2 c_\ssW^2}{c_\ssW^2-s_\ssW^2}\right) Q_f
+\delta g_{f\ssL(\ssR)} \nn\\
&=& \frac{\alpha T}{2} \, g_{f\ssL(\ssR)}^{\SM}
+\alpha T \left( \frac{s_\ssW^2 c_\ssW^2}{c_\ssW^2
-s_\ssW^2} \right) Q_f + \delta g_{f\ssL(\ssR)} \,.\end{aligned}$$ The SM couplings are (as usual) $g_{f\ssL}^\SM := T_{3f} - Q_f s_\ssW^2$ and $g_{f\ssR}^\SM := - Q_f s_\ssW^2$, while the oblique parameters [@oblique] $S$, $T$ and $U$ are given by \[obliqueresult0\] S = U = 0 , and $$\begin{aligned}
\label{obliqueresult}
\alpha T = - z \,.\end{aligned}$$
### Direct $X$-boson couplings {#direct-x-boson-couplings .unnumbered}
The terms explicitly involving the $X$ boson similarly are \_= - 14 X\_ X\^ - X\_X\^ + i \_f \_( k\_[f]{} \_ + k\_[f]{} \_ ) f X\^, with k\_[f()]{} = c\_ g\_X\_[f()]{} + c\_ (Q\_f c\_\^2-g\_[f()]{}\^)-s\_e\_g\_[f()]{}\^. \[DkfLR\]
We are now in a position to compute how observables depend on the underlying parameters, and so bound their size. When doing so we follow [@bigfit] and work to linear order in the deviations, $\Delta g_{f\ssL(\ssR)}$, of the SM couplings, since we know these are observationally constrained to be small.
High-energy constraints\[e+e-annihil\]
======================================
This section considers the constraints on the $X$ boson coming from its influence on various precision electroweak observables measured at high-energy colliders. There are two main types of observables to consider: those that test the changes that $X$-boson mixing induces in SM couplings; and those sensitive to the direct couplings of the $X$ boson to SM fermions. We consider each type in turn.
We begin with two well-measured observables that are sensitive only to changes to the SM self-couplings: the $W$ boson mass, $M_\ssW$, and the $Z$-boson branching fraction into leptons, $\Gamma(Z\rightarrow \ell^+ \ell^-)$. Later subsections then consider reactions to which direct $X$ exchange can contribute, such as the cross section, $\sigma_{\rm res}(e^+ e^- \to h)$, for electron-positron annihilation into hadrons evaluated at the $Z$ resonance.
### Consistency limits on accessible parameter space {#consistency-limits-on-accessible-parameter-space .unnumbered}
Since the SM is in such good agreement with experiment [@ewwg], it is useful to linearize corrections to the SM parameters as we have done in the previous section. To be consistent, we limit ourselves to considering the subset of parameter space which is consistent with this linearization procedure. In practice, we require that the following two conditions of $z$ be satisfied:
1. $z$ must be real (see the discussion in the Appendix), which amounts to demanding that it is obtained by a physically allowed choice for the initial parameters $m_\ssX$ and $\chi$. This implies that \_\^2 - R\_\^2 s\_\^2 \^20 , where $\Delta_\ssX$ is defined in eq. . This simplifies to |\_-| \[zrealconstr\] where :=s\_\^2\^2.
2. $z$ must be small: $z\ll1$. To quantify this statement, we assume that $z$ (or, equivalently, $\alpha T$) will be at most within $2\sigma$ from its global fit value [@bigfit]: |z|0.014 . This bound has been considered in [@CPS] in the context of hidden sector dark matter models.
In figure \[zbound\], we show the regions in the $M_\ssX-\sh\,\eta$ parameter space that are excluded by each of these bounds. From this, we see that the first condition is dominant when $\sh\,\eta < 3\times10^{-2}$, whereas for greater values of $\sh\,\eta$, it is the second condition that is dominant.
Effects due to modified $W,Z$ couplings
---------------------------------------
We start with several examples of constraints that probe the induced changes to the SM self-couplings.
### The $W$ mass\[wmasssect\] {#the-w-masswmasssect .unnumbered}
Mixing with the $X$ boson modifies the SM prediction for the $W$ mass due to its contribution to the electroweak oblique parameter $T$, as follows [@bigfit; @oblique]: $$\begin{aligned}
M_\ssW^2 &=& m_\ssW^2 = m_\ssZ^2 \left( 1- \hat s_\ssW^2\right) \\
&=& \Bigl[ M_\ssZ^2 \left( 1+\alpha T\right) \Bigr]
\left[ 1-s_\ssW^2 \left( 1-\frac{c_\ssW^2 \alpha T }{c_\ssW^2
-s_\ssW^2} \right) \right] \\
&\simeq & (M_\ssW^2 )_\SM \left[
1 + \alpha T \left( 1+\frac{s_\ssW^2}{c_\ssW^2
-s_\ssW^2}\right) \right] \,,\end{aligned}$$ where $(M_\ssW^2)_\SM$ is the full SM prediction, including radiative corrections: $(M_\ssW^2)_\SM = M_\ssZ^2 (1 - s_\ssW^2) +$ loops. Because both the SM radiative corrections and the oblique corrections are known to be small, we can neglect their product in the above expression.
At this point one might ask why bother examine the $W$ mass correction separately, since the $W$ mass is one of the observables included in the global fits to oblique parameters, and we have already assumed that $z$ must be small enough to ensure that the oblique parameter $T$ lies within its 2-$\sigma$ range obtained from global electroweak fits (as in Figure E.2 of [@ewwg]). The reason we re-examine the $W$ mass is that it leads to a slightly stronger constraint, because the mixing between the $Z$ and $X$ bosons does not contribute to the $S$ parameter, and this prior information leads to a slightly stronger limit on $T$ (as is shown in Figure \[STellipse\]). Using the result, eq. , $\alpha T = - z$ together with eq. for $z$ as a function of $\eta$ and $M_\ssX$ gives the desired expression for $\Delta M_\ssW$ as a function of $\eta$ and $M_\ssX$. In the limit when the $Z$ and $X$ masses are very close to one another — [*i.e.*]{} when $\Delta_\ssX$ is such that the equality in eq. holds — the expression for $z$ becomes z = = - \_s\_|| + s\_\^2 \^2 + (\^3) , and so |M\_| M\_c\_ 2.75 ( ) . \[deltamw1\]
Moving away from degeneracy, we find the expression for $z$ can be simplified as follows: z = = - + (\^2) = + (\^4) , where $R_\ssX := M_\ssX/M_\ssZ$ ([*c.f.*]{} eq. ). So when $M_\ssX$ and $M_\ssZ$ are very different, M\_ ( ) 1.1010\^[5]{}( ) \^[3]{} . \[deltamw\] The large-$M_\ssX$ limit of eq. agrees with the result given in [@wells], which finds $$\Delta M_\ssW \simeq \left( 17\ \text{MeV}\right)
\left( \frac{\eta }{0.1}\right)^2
\left( \frac{250\text{ GeV}}{M_{\ssX}} \right)^2 \,.$$
The experimental agreement of the measured $W$ mass with the SM prediction implies $\Delta M_\ssW \leq 0.05$ GeV [@pdg] ($2\sigma$ uncertainty), and the constraint this imposes on $\sh\,\eta$ as a function of $M_\ssX$ is shown in Figure \[wmass\]. Several points about the comparison given in the figure are of note:
- The $W$-mass bound on $\eta$ is model-independent inasmuch as it relies only on the kinetic mixing and does not depend at all on the fermion quantum numbers to which $X$ couples;
- The strongest constraints on $\eta$ occur for $M_\ssX$ nearest to the $Z$ pole, where $|\eta_{\rm pole}| \leq 1.8 \times 10^{-3}$;
- When $M_\ssX \ll M_\ssZ,$ the bound on $\eta$ becomes approximately $M_\ssX$-independent: $|\eta| \leq 6.2\times 10^{-2}$. This behaviour is also visible in the analytic expression, eq. ;
- When $M_\ssX \gg M_\ssZ,$ the $W$ mass bounds the ratio $\eta/M_\ssX$: giving $M_\ssX/\eta {{\mathrel{\raise.3ex\hbox{$>$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}1.5$ TeV.
### $Z$ Decay {#z-decay .unnumbered}
The $Z$ decay rate has been measured with great accuracy at LEP and SLC (for details regarding their analysis, see [@ewwg]). The experimental value [@pdg] for the decay $Z\rightarrow \ell^+ \ell^-$, where $\ell$ can be any of the charged leptons, is $\Gamma_{\ell^+ \ell^-} = 83.984 \pm 0.086$ MeV (1 $\sigma$), and agrees well with the SM result [@pdg] $83.988\pm 0.016$ MeV. The modified $Z$-fermion couplings change the tree-level decay rate, $$\Gamma_{\ell^+ \ell^-} = \frac{M_\ssZ e_\ssZ^2}{24\pi }
\left( g_{\ell \ssL}^2 + g_{\ell \ssR}^2 \right) \,,$$ where the couplings $g_{\ell \ssI} = g_{\ell \ssI}^\SM + \Delta g_{\ell \ssI}$ (with $I = L, R$) are defined by the interaction . The deviation from the SM prediction therefore is $$\Delta \Gamma_{\ell^+ \ell^-} :=
\Gamma_{\ell^+ \ell^-} - \Gamma_{\ell^+ \ell^-}^\SM
\simeq \frac{M_\ssZ e_\ssZ^2}{24\pi} \sum_{\ssI = \ssL,\ssR}
\left[ 2 g_{\ell \ssI}^\SM + \Delta g_{\ell \ssI} \right]
\Delta g_{\ell \ssI} \,.$$ Notice that this vanishes if $\Delta g_{\ell \ssI} = 0$ or when $\Delta g_{\ell \ssI} = - 2 g_{\ell \ssI}^\SM$. It can therefore happen that $\Delta \Gamma_{\ell^+ \ell^-}$ vanishes for two separate regions as one varies through parameter space.
To obtain bounds on $\eta$ and $M_\ssX$ we use eq. to eliminate $\Delta g_{f\ssL(\ssR)}$, giving \[Dgell\] g\_ = (c\_-)(-12 \_ + s\_\^2) + -s\_Here $-\frac12 \, \delta_{\ssI \ssL} + s_\ssW^2$ is the SM contribution, $g_{\ell \ssI}^\SM$, where $\delta_{\ssI \ssL}$ denotes a Kronecker delta function. Requiring $\Delta \Gamma_{\ell^+ \ell^-}$ to be smaller than the experimental (2 $\sigma$) experimental error gives the desired bound on the parameters $g_\ssX$, $\eta$ and $M_\ssX$. Figure \[zdecay\] shows the excluded values in the $\alpha_\ssX = g_\ssX^2/4\pi$ vs $M_\ssX$ plane, with the leptonic $X$-boson charge assumed to be $X_{\ell \ssL} = X_{\ell \ssR} = -1$ (such as would apply if $X = B-L$). Each panel of the figure corresponds to a different choice for $\sh\,\eta$. For the panel in which $\sh\,\eta=1$ bounds at lower mass scales than roughly $385$ GeV are not plotted, since these would conflict with a $z=-\alpha T$ satisfying the global electroweak fit, as outlined in figure \[zbound\].
In order to understand the features present in the plots it is useful to consider the small-$\eta$ limit of $z$ and $\xi$. As discussed above for the $W$ mass bound, the small-$\eta$ limit when $M_\ssX$ and $M_\ssZ$ are very similar or very different must be considered separately. The expressions when $M_\ssX$ and $M_\ssZ$ are very different are z . As might be expected, all terms in $\Delta g_{\ell \ssI}$ are suppressed by a factor of $1/R_\ssX^2 \simeq M_\ssZ^2/M_\ssX^2$ and so go to zero when $M_\ssX \gg M_\ssZ$. In the opposite limit, $R_\ssX \to 0$, $\Delta g_{\ell \ssI} \simeq X_{\ell \ssI} \eta s_\ssW (g_\ssX/e_\ssZ) + \eta^2 s_\ssW^2 c_\ssW^4/(c_\ssW^2 - s_\ssW^2)$, which can pass through zero (if $X_{\ell \ssI}\eta < 0$) when $|X_{\ell \ssI}| (g_\ssX/e_\ssZ) \simeq \cO(\eta)$.
Several features of these plots should be highlighted:
- The best bounds come for $M_\ssX \simeq M_\ssZ$, even for small couplings $g_\ssX$, because in this limit the $Z-X$ mixing parameter $\xi$ becomes maximal ($\tan 2 \xi \, \raro \, \infty$), leading to strong constraints.
- For a similar reason, once $\eta$ is sufficiently large ($\sh\,\eta \simeq 0.1$ — see also figure \[zdecaylowmcut\]) the regime of vanishingly small $\alpha_\ssX$ remains excluded because $\Delta g_{\ell \ssL(\ssR)}$ is dominated by the oblique corrections to the weak mixing angle.
- For $M_\ssX \gg M_\ssZ$ the excluded area approaches a straight line, corresponding to a bound on the ratio $g_\ssX/M_\ssX^2$, as expected from the form of $\Delta g_{\ell \ssL(\ssR)}$.
- The graph is more intricate for $M_\ssX \ll M_\ssZ$, with slivers of allowed parameter space emerging for a narrow, $\eta$-dependent but $M_\ssX$-independent, value of $\alpha_\ssX$. This happens (for sufficiently large $\eta$) because $\Delta \Gamma = 0$ is a multiple-valued condition on the parameters, as discussed above.
Figure \[zdecaylowmcut\] provides a view of the bounds taken on a different slice through the three-dimensional parameter space ($\eta$, $\alpha_\ssX$, $M_\ssX$). This figure plots the constraints on $\alpha_\ssX$ vs $\sh\,\eta$, in the regime where $M_\ssX \ll M_\ssZ$, showing how a wider range of $\alpha_\ssX$ is allowed as $\sh\,\eta$ shrinks. Note that bounds are only shown for the region where $z \ll 1$.
Processes involving $X$-boson exchange
--------------------------------------
In this section we consider precision electroweak observables, like the resonant cross section for $e^+ e^- \to$ hadrons, that receive direct contributions from $X$-boson exchange, in addition to the modifications to SM $Z$-boson couplings.
### The annihilation cross section {#the-annihilation-cross-section .unnumbered}
We again proceed by computing the leading change to the tree-level cross section for $e^+ e^- \to f \ol{f}$ at leading order in the new interactions. Interference terms between SM loops and $X$-boson contributions may be neglected under the assumption that their product is negligible [@bigfit]. The relevant Feynman diagrams are shown in Figure \[schannel\], where the exchanged boson is either a photon, $Z$ or $X$ boson.
Neglecting fermion masses the relevant spin-averaged squared matrix element for this process is (see, [*e.g.*]{} [@SM] for a treatment of SM scatterings using similar conventions) $$\frac14 \sum \left\vert \mathcal{M} \right\vert^2
= N_c \left[ \left( \left\vert A_{\ssL\ssL}
\left( s\right) \right\vert ^2 + \left\vert A_{\ssR\ssR}
\left( s\right) \right\vert ^2\right) u^2
+\left( \left\vert A_{\ssL \ssR} \left(
s\right) \right\vert ^2 + \left\vert A_{\ssR \ssL}
\left( s\right) \right\vert^2\right) t^2\right] \,,$$ where $s$, $t$ and $u$ are the usual Mandelstam variables and $$A_{\ssI\ssJ} \left( s\right) := \frac{e^2 Q_e Q_f }{s}
+ \frac{e_\ssZ^2 g_{e\ssI}g_{f\ssJ}}{s - M_\ssZ^2
+i\Gamma_\ssZ M_\ssZ} + \frac{k_{e\ssI} k_{f\ssJ}}{s
-M_\ssX^2 + i\Gamma_\ssX M_\ssX} \,.$$ The total unpolarized cross section that follows from this is $$\sigma \left( e^+ e^- \rightarrow f \overline{f} \right)
= \frac{N_c s}{48\pi } \left( \left\vert A_{\ssL\ssL}
\right\vert^2 + \left\vert A_{\ssR\ssR} \right\vert^2
+\left\vert A_{\ssL\ssR} \right\vert^2
+\left\vert A_{\ssR\ssL} \right\vert^2\right) \,.$$ The couplings $g_{f\ssI}$ and $k_{f\ssI}$ in these expressions are defined in terms of $\eta$, $g_\ssX$ and $M_\ssX$ by eqs. and . The quantities $\Gamma_\ssZ$ and $\Gamma_\ssX$ are only important near resonance, and denote the full decay widths for the $Z$ and $X$ boson, respectively: $$\begin{aligned}
\Gamma_\ssZ &=& \frac{e_\ssZ^2 M_\ssZ}{24\pi }
\sum_{2m_f \leq M_\ssZ}
\left[ g_{f\ssL}^2 + g_{f\ssR}^2 \right] N_{c} \\
\hbox{and} \quad
\Gamma_\ssX &=& \frac{M_\ssX}{24\pi } \sum_{2m_f \leq M_\ssX}
\left[ k_{f\ssL}^2 + k_{f\ssR}^2 \right] N_c \,,\end{aligned}$$ where $N_c$ is the colour degeneracy for fermion $f$.
### The Hadronic Cross Section at the $Z$ Pole {#the-hadronic-cross-section-at-the-z-pole .unnumbered}
Summing the above over all quarks lighter than $M_\ssZ$ and evaluating at $\sqrt{s} = M_\ssZ$ gives the leading correction to the resonant cross section into hadrons, $\sigma_{had} \left( s = M_\ssZ^2 \right)$, which is well-measured to be $41.541 \pm 0.037$ nb [@pdg]. Requiring the deviation from the SM to be smaller than the 2$\sigma$ error gives the desired constraints. Figure \[sigmahad\] shows a number of exclusion limits for the coupling $\alpha_\ssX$ vs the $X$-boson mass (for $X_{f\ssL} = X_{f\ssR} = (B - L)_f$, and $M_\ssX$ in the range of $10 - 10^3$ GeV), with each panel corresponding to a different choice for $\eta$.
These plots reflect several features seen in the analytic expressions for the couplings:
- For $M_\ssX \ll M_\ssZ$ and when $\eta$ is small enough, the mass dependence of the bound on $\alpha_\ssX$ completely drops out, leaving $\alpha_\ssX {{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}10^{-2}$ in this limit. For larger $\eta$ small values of $\alpha_\ssX$ can still be ruled out because the contributions of mixing are already too large. This mixing also ensures that the region near $M_\ssX = M_\ssZ$ tends to give the strongest bounds.
- The regime $M_\ssX \gg M_\ssZ$ similarly constrains only the combination $M_\ssX^2/\alpha_\ssX {{\mathrel{\raise.3ex\hbox{$>$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}800$ GeV (when $\eta$ is small).
- For $\eta$ not too small and $M_\ssX$ smaller than $M_\ssZ$, figure \[sigmahad\] shows a window of unconstrained couplings, for the same kinds of reasons discussed above for $\Gamma_{\ell^+ \ell^-}$.
Figure \[shadlowmcut\] shows a sample slice of the constraint region in the $\alpha_\ssX$ vs $\sh\,\eta$ plane, in the limit $M_\ssX \ll M_\ssZ$. Once again, bounds are not plotted within regions of parameter space for which $z$ is not $\ll 1$. This plot shows that the smallest $\eta$ for which small $\alpha_\ssX$ can be ruled out is $\sh\,\eta {{\mathrel{\raise.3ex\hbox{$>$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}0.06$. Once $\eta$ is larger than this, mixing rules out the $X$ boson even with arbitrarily small gauge couplings.
Constraints at intermediate energies
====================================
Better constraints on lower-mass $X$ bosons can be obtained from low-energy scattering of muon neutrinos with electrons and nuclei. The purpose of this section is to quantify these bounds by identifying how the cross section depends on the parameters $g_\ssX$, $\eta$ and $M_\ssX$. We consider electron and nuclear scattering in turn.
Neutrino-electron scattering
----------------------------
The Feynman graphs relevant for $\nu_\mu e^-$ scattering are those of Fig. \[schannel\], with three changes: ($i$) the gauge bosons are exchanged in the $t$-channel rather than $s$-channel; ($ii$) there is no photon-exchange graph and ($iii$) omission of right-handed neutrino polarizations.
Crossing to $t$-channel can be obtained by performing the following substitution $$\begin{array}{ccc}
s\rightarrow t, & t\rightarrow u, & u\rightarrow s
\end{array}$$ among the Mandelstam variables in the invariant amplitude $\frac12 \sum \left\vert \mathcal{M} \right\vert^2$. With these replacements, the differential cross section for the process $\nu_\mu e^- \rightarrow \nu_\mu e^-$ is ( \_e\^- \_e\^- ) = - , where $$\label{Aijnu}
A_{\ssI \ssJ} (t) = e_\ssZ^2 \frac{ g_{e\ssI} g_{\nu \ssJ}}{
t - M_\ssZ^2} + \frac{ k_{e\ssI} k_{\nu \ssJ}}{t - M_\ssX^2} \,.$$
In the rest frame of the initial electron $s \simeq 2 \,m_e E_\nu$ and $t \simeq -2 y \,m_e E_\nu$, where $E_\nu$ is the incoming neutrino energy and $y$ is the fractional neutrino energy loss, $y := E^f_e/E_\nu$ where $E^f_e$ is the energy of the outgoing electron. (In such experiments [@CHARM2; @radelbeyer], $E_\nu,\,E^f_e \sim 1-10$ GeV so ratios of the form $m_e/ E_\nu$ and $m_e / E^f_e$ can be neglected.) In terms of these new variables the differential cross section is $$\label{nuediffcross}
\frac{\exd\sigma }{\exd y} \left( \nu_\mu e^-
\rightarrow \nu_\mu e^- \right) = \frac{m_e E_\nu}{4\pi}
\left[ \Bigl\vert A_{\ssL\ssL} [t(y E_\nu)] \Bigr\vert^2
+ \Bigl\vert A_{\ssR \ssL} [t(y E_\nu)] \Bigr\vert^2 (1 - y)^2
\right] \,.$$
The cross section for anti-neutrino scattering is easily found from the above by interchanging $A_{\ssL \ssL} \leftrightarrow A_{\ssR \ssL}$.
### Special case: Low-energy limit with $\eta = 0$ {#special-case-low-energy-limit-with-eta-0 .unnumbered}
One case of practical interest is when the boson masses, $M_\ssZ$ and $M_\ssX$, are much greater than the invariant energy exchange in the process of interest ([*i.e.*]{} $\sqrt{|t|} \ll M_\ssX, M_\ssZ)$. When this holds the amplitudes, $A_{\ssI \ssL}$, can be simplified to $$\begin{aligned}
A_{\ssI \ssL} &\simeq & - \frac{e_\ssZ^2 g_{e\ssI}
g_{\nu \ssL}}{M_\ssZ^2} - \frac{k_{e \ssI}
k_{\nu \ssL} }{M_\ssX^2} \notag \\
&=& - \frac{e_\ssZ^2 g_{\nu \ssL}}{M_\ssZ^2}
\left[ g_{e \ssI} + \frac{M_\ssZ^2}{M_\ssX^2} \left(
\frac{k_{\nu \ssL} k_{e \ssI}}{e^2_\ssZ g_{\nu \ssL}}
\right) \right] \,, \label{Aneuij}\end{aligned}$$ allowing the effects of $X$-boson exchange be interpreted as an effective shift in the electron’s electroweak couplings. For $E_\nu \simeq 1$ GeV and $y$ order unity this approximation remains good down to $M_\ssX \simeq 30$ MeV.
The resulting cross section is particularly simple in the case of no kinetic mixing, for which we can substitute the SM values $g_{e\ssI} = - \frac12 \delta_{\ssI\ssL} + s_\ssW^2$ and $g_{\nu \ssL} = \frac12$ and the $X$-boson couplings $k_{e\ssI} = g_\ssX X_{e \ssI}$ and $k_{\nu \ssJ} = g_\ssX X_{\nu \ssJ}$, and obtain $$A_{\ssI \ssL} \simeq -2 \sqrt{2} \,G_{\ssF}
\left( -\frac12 \, \delta_{\ssI \ssL} + s_\ssW^2
+\frac{g_\ssX^2 X_{e\ssI} X_{\nu \ssL}}{2\sqrt{2} \,
G_{\ssF} M_\ssX^2} \right) \,,$$ using the SM result $2\sqrt{2} \,G_{\ssF} \simeq e_\ssZ^2/2M_\ssZ^2$. We see that the $X$-boson contribution can be regarded as an additional contribution to $s_\ssW^2$ in this limit. This is convenient because it allows the simple use of constraints on $s_\ssW^2$ to directly constrain the ratio $g_\ssX^2/M_\ssX^2$.
The bounds are usually taken from the following ratio [@radelbeyer] of total cross sections, R := . Given the differential cross section $$\frac{\exd\sigma }{\exd y} \left( \nu_\mu e^-
\rightarrow \nu_\mu e^- \right) =\frac{2G_\ssF^2
m_e E_\nu}{\pi } \left[ g_{e\ssL}^2 + g_{e\ssR}^2
(1 - y)^2\right] \,,
\label{dsigneuelect}$$ the total cross section becomes $$\sigma \left( \nu_\mu e^- \rightarrow
\nu_\mu e^- \right) = \frac{2G_\ssF^2 m_e
E_\nu}{\pi } \left( g_{e\ssL}^2 +\frac{g_{e\ssR}^2}{3}
\right) \,,$$ and so $$\sigma \left( \overline\nu_\mu e^-
\rightarrow \overline\nu_\mu e^- \right)
= \frac{2G_\ssF^2 m_e E_\nu }{\pi }
\left( \frac{g_{e\ssL}^2}{3} + g_{e\ssR}^2 \right) \,.$$ Specializing to SM couplings the result depends only on $s_\ssW$: $$R = \frac{3 g_{e\ssL}^2 + g_{e\ssR}^2}{g_{e\ssL}^2
+3g_{e \ssR}^2} = \frac{3 - 12 s_\ssW^2
+ 16s_\ssW^4}{1-4s_\ssW^2+16s_\ssW^4}
= \frac{1+\kappa +\kappa ^2}{1-\kappa +\kappa ^2} \,,$$ where $\kappa \equiv 1-4s_\ssW^2 \ll 1$.
Using the experimental limit [@CHARM2] $\Delta s_\ssW^2 = 0.0166$ ($2\sigma$ error) with $G_{\ssF} = 1.1664\times 10^{-5} \text{ GeV}^{-2}$ [@pdg] to constrain $\Delta s_\ssW^2 = {g_\ssX^2}/{2 \sqrt{2} \, G_{\ssF} M_\ssX^2}$ (assuming the choice $X_{e\ssI} X_{\nu \ssL} = 1$, as would be true for $X = B-L$ for example), gives [@gjneuelect] $$\frac{M_\ssX}{g_\ssX}\gtrsim 4\ \text{TeV} \,.$$
### General Case: $\protect\eta \neq 0$ {#general-case-protecteta-neq-0 .unnumbered}
More generally, the couplings $k_{f \ssI}$ also acquire contributions from $Z-X$ mixing even when $g_\ssX = 0$, as the above calculations show. In this case the more general bounds on $g_\ssX$, $\eta$ and $M_\ssX$ can be extracted by demanding that these contribute within the experimental limit $\Delta R$. Since the experimental limit is often quoted in terms of $s_\ssW^2$ [@CHARM2], we translate using $\Delta R = \left\vert {\exd R_{SM}}/ {\exd s_\ssW^2 } \right\vert \Delta s_\ssW^2$. In obtaining $R$, we integrate over $y$ using $\sqrt{|t|}\ll M_\ssZ^2$, but [*without*]{} assuming that $\sqrt{|t|} \ll M_\ssX^2$. When evaluating $R$, we set $E_\nu$ to a nominal value of 1 GeV. Figure \[nuelect\] shows the resulting bound in the $\alpha_\ssX - M_\ssX$ plane, for several choices for $\eta$ assuming $X_{e \ssI} X_{\nu \ssL} = 1$.
The resulting curves inspire a few comments:
- For large $M_\ssX$ the bound is independent of $\eta$ due to the $M_\ssX / M_\ssZ$ suppression of the mixing in $\Delta g$ and $\Delta k$. This allows the direct $g_\ssX^2/M_\ssX^2$ term to dominate. The bounds in this regime are relatively strong, and compete with those found in direct searches (e.g., by CDF [@CDF] in the case of a SM-like $Z'$).
- For smaller $M_\ssX$, it is the terms in the couplings that are linear in $\eta$ that influence the deviation from the $\eta=0$ result. To see this, note that g\_=- g\_X\_+ (\^2)\
, and so there will be a term in $|A_{\ssI \ssL}|^2$ that is linear in $\eta$ with the parametric dependence $g_\ssX/M_\ssX^2$. When $g_\ssX \ll 0$, it is this term that is dominant compared to the $g_\ssX^2/M_\ssX^2$ term from $X$-boson exchange. However, when $g_\ssX\sim\eta$, this new term is no longer dominant and the bound regresses back to its original slope from the $\eta=0$ case at high masses.
- Once $1\leq M_\ssX \leq 10$ MeV and $M_\ssX {{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}\sqrt{|t|}$, the bound loses its dependence on $M_\ssX$ and levels out to some fixed value. This is expected from the form of eq. .
- When $\sh \, \eta =1$, much of the parameter space is excluded due to the requirement that $z \ll 1$. Therefore, only a small region with $M_\ssX > 385$ GeV is bounded by electron-neutrino scattering in this case.
Neutrino-nucleon scattering
---------------------------
For the bounds from neutrino-nucleon scattering it is worth first recalling how the standard analysis is performed. In terms of the neutral-current quark couplings, the quark-level cross sections for neutral-current muon-neutrino scattering are $$\begin{aligned}
\label{neuxsects1}
\sigma \left( \nu_\mu u \rightarrow \nu_\mu u\right)
&=& \sigma_0 \left( g_{u\ssL}^2 + \frac{g_{u\ssR}^2}{3} \right)
\,, \qquad
\sigma \left( \nu_\mu d \rightarrow \nu_\mu d\right)
= \sigma_0 \left( g_{d\ssL}^2 + \frac{g_{d\ssR}^2}{3} \right) \\
\sigma \left( \ol\nu_\mu u \rightarrow \ol\nu_\mu u \right)
&=& \sigma_0 \left( \frac{g_{u\ssL}^2}{3} + g_{u \ssR}^2 \right)
\,, \qquad
\sigma \left( \ol\nu_\mu d \rightarrow \ol\nu_\mu d \right)
= \sigma_0 \left( \frac{g_{d\ssL}^2}{3} + g_{d \ssR}^2 \right) \nn\end{aligned}$$ while those for charged currents are \[neuxsects2\] ( \_d \^- u) = \_0 ( \_u \^+ d) = , where $\sigma_0 := 2 N_c G_{\ssF}^2 m_e E_\nu/\pi$ and $N_{c}=3$.
These show that the quark neutral-current and charged-current cross sections are all proportional to one another. The resulting cross section for neutrino-nucleon scattering in the deep-inelastic limit is obtained by summing incoherently over the quark contributions, giving $$\begin{aligned}
\sigma \left( \nu_\mu N \rightarrow \nu_\mu X\right) &=&
\varepsilon_\ssL^2 \, \sigma \left( \nu_\mu N \rightarrow
\mu^- X\right) + \varepsilon_\ssR^2 \, \sigma \left( \ol\nu_\mu
N\rightarrow \mu^+ X \right) \nn\\
\sigma \left( \ol\nu_\mu N \rightarrow \ol\nu_\mu X \right)
&=& \varepsilon_\ssL^2 \sigma \left( \ol\nu_\mu N
\rightarrow \mu^+ X \right) + \varepsilon_\ssR^2
\sigma \left( \nu_\mu N \rightarrow \mu^- X \right) \,,\end{aligned}$$ where \_[( ) ]{}\^2 := g\_[u ( ) ]{}\^2 + g\_[d( ) ]{}\^2 .
The experimental bounds come from the following ratios: $$\begin{aligned}
R^{\nu } &:= & \frac{\sigma \left( \nu_{\mu } N
\rightarrow \nu_{\mu} X\right) }{\sigma \left( \nu_{\mu }
N\rightarrow \mu ^{-} X\right) }
= \varepsilon_\ssL^2 + r \, \varepsilon_\ssR^2 \nn\\
R^{\bar{\nu}} &:=& \frac{\sigma \left( \ol{\nu}_{\mu }
N\rightarrow \ol{\nu}_{\mu } X\right) }{\sigma
\left( \ol{\nu}_{\mu } N \rightarrow \mu^{+} X\right) }
=\varepsilon_{\ssL}^2 + \frac{\varepsilon_{\ssR}^2}{r}\end{aligned}$$ where $r := \sigma \left( \bar{\nu}_{\mu }N \rightarrow \mu^{+} X\right) /\sigma \left( \nu_{\mu } N\rightarrow \mu^{-}X\right)$. Most useful is the Paschos-Wolfenstein ratio [@pw], from which the comparatively uncertain ratio $r$ cancels: \[PWratio\] R\^[-]{} := = = \_\^2-\_\^2 . Experiments measure the following values [@nutev] $$\begin{aligned}
\label{nutevbound}
\varepsilon _{\ssL}^2 &=&0.30005\pm 0.00137 \\
\varepsilon _{\ssR}^2 &=&0.03076\pm 0.00110 \,.\nn\end{aligned}$$
To constrain the $X$-boson coupling parameters we work in the regime with $\sqrt{-t} \ll M_\ssX$, for which the effects of $X$-boson mixing and exchange can both be rolled into a set of effective neutral-current couplings. The cross sections for quark-level scattering are then given by integrating eqs. using , leading to expressions identical with eqs. but with g\_[q ]{} g\_[q ]{}\^[eff]{} := 2 , where $q = u, d$ and $I = L, R$. Using these in eq. gives constraints on $\varepsilon_{\ssI}^2 = \left( g_{u\ssI}^{\SM} + \Delta g_{u \ssI}^{\rm eff} \right)^2 + \left( g_{d\ssI}^{\SM} + \Delta g_{d \ssI}^{\rm eff} \right)^2$.
Figure \[nuhad\] plots the constraint found by requiring $\Delta \varepsilon_{\ssL}^2 \leq 0.00137$, assuming that $X = B-L$. The plots are cut off at low mass where the condition $\left\vert t/M_{\ssX}^2 \right\vert \leq 0.01$ breaks down. Notice that for $\eta =0$ the bound is similar to that found for neutrino-electron scattering, with stronger bounds on $\alpha_\ssX$ at smaller $M_\ssX$. For nontrivial $\eta$ the strength of $X-Z$ mixing eventually provides the strongest constraint, leading to strong bounds even for small $g_\ssX$ at sufficiently low $M_\ssX$.
Low-energy constraints
======================
We finally turn to constraints coming from lower-energy processes.
Anomalous magnetic moments\[amm-sec\]
-------------------------------------
The accuracy of anomalous magnetic moment (AMM) measurements [@pdg] produce a strong constraint on the parameters of an extra gauge boson. We consider the bound arising from both the electron and muon AMM on the $X$ gauge coupling as a function of the mass $M_\ssX$, for various values of the kinetic mixing parameter $\sh\,\eta$.
The correction to the AMM of a lepton, $\ell$, is given by [@ovanesyan] a\_= \_0\^1 dz , where the vector and axial couplings to the $X$ boson are of the form k\_ &:=&= c\_-s\_e\_(-14+s\_\^2 )\
k\_ &:=&= c\_-s\_e\_(-14) .
There is, however, some subtlety in comparing this shift with experiment [@maximsec]: since the electron AMM, $\delta a_e$, is used to determine the fine-structure constant, $\alpha$. The best bound on $X$ boson couplings therefore comes from the next most precise experiment that measures $\alpha$, and not the errors from the $(g-2)$ experiments themselves. Following [@maximsec] this leads to the constraints $\delta a_e < 1.59 \times 10^{-10}$ and $\delta a_\mu < 7.4 \times 10^{-9}$, which when compared with the above expression gives the bounds shown in Figure \[amm\]. These plots reproduce the results found in [@ovanesyan] when $\sh\,\eta = 0$. In particular, the $M_\ssX$ values below which any gauge coupling is excluded are consistent with the bounds shown in [@maximsec].
Since these bounds are often considered (e.g. in [@maximsec], [@HiddenU1Bounds; @toroschuster]) in the context of a constraint on kinetic mixing, we also plot the constraint on $\sh\,\eta$ as a function of the $X$ boson mass, for various values of the gauge coupling. This is shown in Figure \[alphaamm\].
Upsilon decay
-------------
The bound we present here is an extension of the result found in [@toroschuster]. By looking at the decay rate of the $\Upsilon(3 s)$ $b\ol b$ bound state, researchers from the BABAR collaboration were able to place a bound on the occurrence of a particular channel involving a light pseudoscalar $A_0$ [@dimuon]: e\^+ + e\^- (3 s) + A\_0 + \^+ + \^- . Their upper limit on the number of events N = (e\^+ + e\^- (3 s)) ((3 s) + A\_0) (A\_0 \^+ + \^-) , places a bound on the quantity $Q:=\mathrm{Br} (\Upsilon(3 s) \rightarrow \gamma + A_0) \times \mathrm{Br} (A_0 \rightarrow \mu^+ + \mu^-)$.
However, the reaction of interest to us is e\^+ + e\^- + X + \^+ + \^- , which would have an identical signature. So the measured bound can also be reinterpreted as applying to the quantity Q\_:= (X \^+ + \^-) The experimental limit [@dimuon] $Q_\ssX < 3 \times 10^{-6}$ gives the plots found in Figure \[upsilon\] over the range $2 m_\mu < M_\ssX < E_{cm} ( = 10.355$ GeV). This bound is quite strong, as it eliminates the entire region for $\sh\,\eta \gtrsim 0.002$ (as is shown in [@toroschuster]). For smaller $\sh\,\eta$, the bound is roughly constant when $M_\ssX \ll E_{cm}$.
As in the case of the AMM bounds, we also plot the constraint on $\sh\,\eta$ as a function of the $X$ boson mass for various values of the gauge coupling — as shown in Figure \[alphaupsilon\].
Beam-dump experiments {#bdexpts}
---------------------
In the MeV$-$GeV mass range, small $g_\ssX$ and $\eta$ are constrained by several beam dump experiments. These bounds are considered in detail in [@toroschuster]; we apply a simplified version of their analysis here.
In these experiments, a large number $N_e$ of electrons with initial energy $E$ are collided with a fixed target made of either aluminum or tungsten. Many of the resulting collision products are absorbed either by the target or by some secondary shielding. (Here, we use $t$ to denote the total thickness of both the target and the shielding.) The remaining products continue along an evacuated tube to the detector, located at some distance $D$ away from the target. For a summary of values for these parameters, see Table \[beamdumpvals\].
The bound arises from the non-observation of $X$ decay products. The incoming electron emits an $X$ boson as bremsstrahlung during photon exchange with the nucleon ($N$): $e^- + N \rightarrow e^- +N+X$. The $X$ can then decay into either an $e^+ e^-$ or $\mu^+ \mu^-$ pair. However, a decay that occurs too soon is absorbed by the shield while a decay that occurs too late occurs past the detector. Therefore, the number of lepton anti-lepton pairs observed at the detector can be computed by multiplying the number of $X$ bosons produced, $N_\ssX$, by the probability for the $X$ to decay between $z=t$ to $z=D$: N\_[obs]{} = N\_\_t\^D dz ( e\^[-z/\_0]{}) . Here, we write the lab frame decay length as $\ell_0:=\gamma c \tau$, where $\gamma = (1-v^2)^{-1/2}$ is the relativistic time-dilation factor and $\tau$ is the inverse of the $X$ rest-frame decay rate: $\tau:=1/\Gamma_\ssX$.
In estimating the number of $X$’s produced, we use the following result from [@toroschuster]: N\_\~N\_e \^2 , where $\epsilon = \chi c_\ssW$ and $\mu^2 \simeq 2.5 \,\, \textrm{MeV}^2$ is an overall factor that contains information regarding the details of the nuclear interaction, and is shown in [@toroschuster] to be roughly constant for $M_\ssX$ between 1 and 100 MeV. There is, however, an obstacle in applying this result directly to our analysis: it was derived without including any coupling to $J_\ssX^\mu$. In order to introduce the $k_{e \ssL(\ssR)}$-dependence in this expression, we note from [@toroschuster] that the $\epsilon$-dependence above arises from the cross section $\sigma(e^- \gamma \rightarrow e^- X)$ under the assumption that the electron is massless. This means that the left- and right-handed helicity $X - e$ interactions contribute equally to the cross section, allowing the substitution \^2 ( ), with the normalization chosen so that the above expression reduces to $\chi^2 c_\ssW^2$ in the case where $X_{e\ssL(\ssR)}=0$, $\sh\,\eta \ll 1$ and $M_\ssX \ll M_\ssZ$.
All in all, we find that the number of $X$’s we expect to observe is given by N\_[obs]{} \~ ( ) ( e\^[-t/\_0]{} - e\^[-D/\_0]{} ) . Applying the experimental exclusions [@toroschuster] $N_{obs} < 17$ events (E774), $N_{obs} < 1000$ events (E141), and $N_{obs} < 10$ events (E137) gives the bounds shown in Figure \[beamdump\].
The plot for $\sh\,\eta=0$ gives good agreement with a similar plot in [@ovanesyan] (in the region over which these results overlap). The lower bounds for each experiment are approximately flat because, in the region where $t \ll D \ll \ell_0$, the fraction of $X$’s that decay is just $D/\ell_0$, which gives N\_[obs]{} \~ ( ) . The leading $M_\ssX$-dependence then cancels since $\ell_0 \sim 1/M_\ssX^2$. The upper bound results from the situation where the $X$ bosons decay too quickly, and the decay products do not escape the shielding.
We have only included plots for the cases where $\sh\,\eta=0$ and 0.001 because the bounds become too weak to constrain any region of this parameter space whenever $\sh \, \eta > 0.007$.
An interesting feature of these bounds is that, at any given value of $\sh\,\eta$, the gauge coupling can be increased such that the bounds are evaded. This occurs because a stronger gauge coupling causes the $X$ bosons to decay within the shielding. Therefore, any bound on kinetic mixing which results from these experiments can weaken if the direct coupling of electrons to the $X$ is taken to be non-zero. To demonstrate this, consider the bounds shown in Figure \[alphabeamdump\], which plots the bound on kinetic mixing as a function of the $X$-boson mass, for various values of $\alpha_\ssX$. Note that, for $\alpha_\ssX {{\mathrel{\raise.3ex\hbox{$>$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}1\times 10^{-6}$, these bounds are satisfied for all values of $\sh\,\eta$ in the relevant mass range.
Neutron-nucleus scattering\[neutnuclsec\]
-----------------------------------------
Low-energy neutron-nucleus scattering is important because most of the other low-energy bounds evaporate if the new boson doesn‘t couple to leptons (such as if $X = B$). For neutron-nucleus scattering a bound is obtained by considering the effects of the new Yukawa-type potential that would arise from a non-zero vector coupling of the $X$ to neutrons. For light $X$ bosons this can be seen over the strong nuclear force because it has a longer range, and can affect the angular dependence of the differential cross section for elastic scattering, $d\sigma(n N \to n N)/d\Omega$. This bound is discussed in the context of a scalar boson in [@barbericson] and more generally in [@psizeanom].
Following these authors we parameterize the differential cross section as = ( 1+ E cos) , where $\sigma_0$ and $\omega$ are to be taken from experiments. Then an interaction of the form V\_[nN]{}(r) = ( ) leads to a correction to the expected value of $\omega$, which is measured experimentally in the energy range $E\sim 1$–$10$ keV for neutrons scattering with ${}^{208}$Pb.
Agreement with observations leads to the bound [@barbericson; @psizeanom] < 3.410\^[-11]{} , where $k_{n\ssV} = k_{u\ssV} + 2 k_{d\ssV}$ with $k_{f\ssV} := \frac12 (k_{f\ssL}+k_{f\ssR})$, as above. Figure \[neutnucl\] shows a plot of this bound for the nominal case $\sh\,\eta=0$.
For this combination of couplings, an interesting cancellation occurs. For small kinetic mixing, the correction $\Delta k_{f\ssL(\ssR)}$ has the form $$\begin{aligned}
\Delta k_{f\ssL(\ssR)} &= \eta \frac{e}{c_\ssW} (Q_f c_\ssW^2 - g^\SM_{f\ssL(\ssR)})+\eta s_\ssW e_\ssZ g^\SM_{f\ssL(\ssR)} +\cO(\eta^2) \nn\\
&= \eta e c_\ssW Q_f + \cO(\eta^2) \,,\end{aligned}$$ and so the leading correction in $\eta$ vanishes for any electrically neutral particle, like a neutron. This makes this bound relatively insensitive to changes in kinetic mixing, not varying appreciably over the range $0\leq \sh\,\eta \leq 1$. A similar cancellation occurs in the case of nucleosynthesis, considered in §\[bbn\].
Atomic parity violation\[weakcharge\]
-------------------------------------
The Standard Model predicts a low-energy effective coupling between the electron axial current and the vector currents within a given nucleus. The so-called weak charge of a nucleus with $Z$ protons and $N$ neutrons is defined (up to an overall constant) as the coherent sum of the $Z$-boson vector couplings over the constituents of that nucleus [@bouchfayet]: Q\_(Z,N):=4 . where g\_[ff]{}:= g\_[f]{}:= .
In terms of these the leading parity-violating effective electron-nuclear interaction generated by $Z$ boson exchange is \_[eff]{}=- G\_g\_[e]{} Q\_(e \_\_5 e) (\^) . where $\Psi$ is the field describing the nucleus. $X$-boson exchange adds an additional term to this effective lagrangian of the form \_[eff]{}\^=- (e \_\_5 e) (\^) where Q\_:= Z ( 2 k\_[u]{} + k\_[d]{}) + N ( k\_[u]{} + 2 k\_[d]{}) . Therefore, the total shift in the $Q_\ssW$ due to the $X$ boson is \[deltaqw\] Q\_= - where Q\_\^(Z,N) &=& 4\
&=& Z (1- 4 s\_\^2 ) -N . Notice that the bracketed term in $\Delta Q_\ssW$ goes to $0$ as $\eta\rightarrow0$, whereas the second term does not as long as $k_{\ssA e}$ does not vanish in the same limit. The total effective lagrangian for this system can then be written as \_[eff]{}+\_[eff]{}\^= ( Q\_\^+ Q\_) (e \_\_5 e) (\^) .
It is expected that the second term in eq. will be dominant, so it is useful to consider the form of $k_{\ssA e}/M_\ssX^2$ in the limit where $M_\ssX \ll M_\ssZ$ and $\eta \ll 1$: = - . Therefore, if $X_{\ssL e} = X_{\ssR e}$, then the constraint becomes significantly less stringent at low masses since, instead of bounding the ratio $g_\ssX^2/M_\ssX^2$, it is now the combination $g_\ssX \eta$ that is bounded. In order to emphasize the strength of this bound when $X_{\ssA e}\neq0$, we use the charge assignments as shown in Table \[apvcharges\].
If the $X$ boson is light enough the above effective interaction eventually becomes inaccurate in describing the electron-nucleus interactions. In this case, rather than pursuing a detailed analysis of the microscopic lagrangian, we follow ref. [@bouchfayet] and introduce a corrective factor $K(M_\ssX)$ to account for the non-locality caused by the small mass of the $X$ boson. This modifies our expression for $\Delta Q_\ssW$ as follows: Q\_= - K(M\_) . In [@bouchfayet] a table is given for $K$ for various values of $M_\ssX$ in the range $0.1$ MeV $< M_\ssX < 100$ MeV. In order to render the graphs shown here, we have interpolated values of $K$ by doing a least squares fit to the values in [@bouchfayet].
As with the neutrino-electron scattering bounds, the slope of the bound changes for $\eta\neq0$ due to the production of a new dominant term through cancellation with the modified $Z$-fermion coupling. Once again, we exclude the region below $385$ GeV for the $\sh\,\eta=1$ plot in order to avoid conflict with the electroweak oblique fits that require $z\ll1$.
Since this bound relies crucially on there being an axial vector coupling to the electron, we did not include it when compiling the summary of bounds given in figures in §1.
Primordial nucleosynthesis\[bbn\]
---------------------------------
We close with the study of constraints coming from cosmology, which for the mass range of interest in this paper consists dominantly of Big Bang Nucleosynthesis.
Any $X$ bosons light enough to be present in the primordial soup at temperatures below $T \sim 1$ MeV can destroy the success of Big Bang Nucleosynthesis (BBN) if they make up a sufficiently large fraction (${{\mathrel{\raise.3ex\hbox{$<$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}10\%$) of the universal energy density, leading to potentially strong constraints. In particular, such a boson poses a problem if it is in thermal equilibrium at these temperatures.
Quantitatively, measurements of primordial nuclear abundances forbid the existence of the number of additional neutrino species (beyond the usual 3 of the SM) to be [@4hebbn] $\delta N_{\nu } \leq 1.44$ (at $95\%$ C.L.). But since each boson in equilibrium counts $\frac87$ times more strongly in the equilibrium abundance, and since a massive $X$ boson carries 3 independent spin states, the corresponding bound on the number, $N_\ssX$, of new species of spin-1 particles in equilibrium at BBN is N\_0.84 . Even just one additional massive spin-1 boson into relativistic equilibrium is excluded at the $95\%$ confidence level.
In a universe containing only the $X$ boson and ordinary SM particles at energies of order 1 MeV, this leads to two kinds of constraints: either the $X$ boson’s couplings are weak enough that it does not ever reach equilibrium; or if the $X$ boson is in equilibrium it must be heavy enough (${{\mathrel{\raise.3ex\hbox{$>$\kern-0.85em
\lower1ex\hbox{$\sim$}}}}}1$ MeV) to have a Boltzmann-suppressed abundance. Figure \[bbnweta\] sketches the regions in the coupling-mass plane that are excluded by these conditions. The vertical line corresponds to the situation where abundance is suppressed by Boltzmann factors.
The constraints on couplings arise only for sufficiently light particles, and express the condition that the couplings be weak enough to avoid equilibrium, at least up until the freeze-out temperature $T_\ssF$. There are two curves of this type drawn, which differ by whether it is collision or decay processes that are the dominant equilibration mechanisms. Qualitatively, the requirement that reactions like $X \nu \leftrightarrow X \nu$ not equilibrate the $X$ bosons leads to a constraint on the couplings that is $M_\ssX$-independent in the limit where $M_\ssX \ll T_\ssF$, because then the size of both the reaction rate and Hubble scale is set by the temperature. The same is not true for decay reactions, $X \leftrightarrow \ol\nu \, \nu$, since the rate for this also depends on the $X$-boson mass.
A few other comments are appropriate for Figure \[bbnweta\]. First, because they are outside the main scope of this study, the bounds shown are derived assuming that $M_\ssX \ll T$ (rather than being evaluated numerically as a function of $M_\ssX$) and so are drawn only up to the mass range within 0.5 MeV of the freeze-out temperature. Second, the resulting expressions depend only weakly on $\eta$, showing little difference over the range $0<\sh\,\eta<1$. As discussed in earlier sections, this is a consequence of the neutrino’s electrical neutrality, which ensures that the leading small-$\eta$ limit of the kinetic mixing first arises at $\cO(\eta^2)$ rather than $\cO(\eta)$.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Brian Batell, Joseph Conlon, Rouven Essig, Sven Krippendorf, David Poland, Maxim Pospelov, Philip Schuster, Natalia Toro and Michael Trott for helpful discussions. AM and FQ thank McMaster University and Perimeter Institute for hospitality. CB and AM thank the Abdus Salam International Centre for Theoretical Physics (ICTP) for its kind hospitality while part of this work was done. The work of AM was supported by the EU through the Seventh Framework Programme and Cambridge University. CB and MW’s research was supported in part by funds from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Research at the Perimeter Institute is supported in part by the Government of Canada through Industry Canada, and by the Province of Ontario through the Ministry of Research and Information (MRI).
Note Added {#note-added .unnumbered}
==========
Since posting, we have learned of a beam dump analysis [@newBeamDumpBB] that has enlarged[^5] the exclusion regions discussed in §\[bdexpts\].
Diagonalizing the gauge action
==============================
This appendix provides the details of the diagonalization of the gauge boson kinetic and mass mixings. The starting point is eq. , $$\label{AppstartLeq}
\cL = -\frac14 \hat{\bf V}_{\mu\nu}^\ssT \hat K
\hat{\bf V}^{\mu\nu} - \frac12 \hat{\bf V}_\mu^\ssT
\hat M \hat{\bf V}^\mu + \hat{\bf J}_\mu^\ssT
\hat{\bf V}^\mu \,,$$ with $\hat K$ and $\hat M$ given in eqs. .
### Diagonalization {#diagonalization .unnumbered}
We begin by performing the usual weak-mixing rotation to diagonalize the mass term: $$\hat{\bf V} = R_1 \check{\bf V} :=
\begin{bmatrix}
\hat{c}_\ssW & \hat{s}_\ssW
& 0 \\
-\hat{s}_\ssW & \hat{c}_\ssW
& 0 \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
\check{Z} \\
\check{A} \\
\check{X}\end{bmatrix}$$ where \_:= \_:= \_:= \_ := . The lagrangian then becomes $$\cL = - \frac14 \check{\bf V}_{\mu\nu}^\ssT
\check K \check{\bf V}^{\mu\nu} - \frac12
\check{\bf V}_\mu^\ssT \check M \check{\bf V}^\mu
+ \check{\bf J}_\mu^\ssT \check{\bf V}^\mu \,,$$ with new matrices $$\check{K} = R_1^\ssT \hat K R_1 =
\begin{bmatrix}
1 & 0 & \chi \hat s_\ssW \\
0 & 1 & -\chi \hat c_\ssW \\
\chi \hat s_\ssW & -\chi \hat c_\ssW & 1 \end{bmatrix}
\quad \hbox{and} \quad
\check{M} = R_1^\ssT \hat M R_1 =
\begin{bmatrix}
m_\ssZ^2 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & m_\ssX^2 \end{bmatrix}$$ where $m_\ssZ^2 := \frac14 \left( g_1^2 + g_2^2 \right) v^2$. Under the same transformation the currents become $$\begin{aligned}
\check{\bf J}_\mu &=& R_1^\ssT \hat{\bf J}_\mu =
\begin{bmatrix}
\hat J_\mu^3 \,\hat c_\ssW - \hat J_\mu^\ssY \,\hat s_\ssW \\
\hat J_\mu^3 \,\hat s_\ssW + \hat J_\mu^\ssY \,\hat c_\ssW \\
\hat{J}_\mu^\ssX
\end{bmatrix} \\
&=& \sum_f
\begin{bmatrix}
\hat ie_\ssZ \ol{f} \gamma_\mu \left[ T_{3f} \gamma_\ssL
- Q_f \hat s_\ssW^2 \right] f \\
ie \, \ol{f} \gamma_\mu Q_f f \\
ig_\ssX \ol{f} \gamma_\mu \left[ X_{f\ssL} \gamma_\ssL
+ X_{f \ssR} \gamma_\ssR \right] f
\end{bmatrix} :=
\begin{bmatrix}
\check J_\mu^\ssZ \\
\check J_\mu^\ssA \\
\check J_\mu^\ssX
\end{bmatrix} \,, \notag\end{aligned}$$ which defines $\hat e_\ssZ := e/(\hat s_\ssW \hat c_\ssW)$ and uses the standard SM relations $g_2 \hat s_\ssW = g_1 \hat c_\ssW := e$ and $Q_{f} = T_{3f} + Y_{f\ssL} = Y_{f\ssR}$.
The kinetic term is diagonalized by letting $$\check{\bf V} := L \tilde{\bf V} :=
\begin{bmatrix}
1 & 0 & - \hat s_\ssW \, \sh \, \eta \\
0 & 1 & \hat c_\ssW \, \sh \, \eta \\
0 & 0 & \ch \, \eta
\end{bmatrix} \begin{bmatrix}
\tilde Z \\
\tilde A \\
\tilde X
\end{bmatrix}$$ with := := := := . This gives, by construction $$\tilde K = L^\ssT \check K L =
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{bmatrix}$$ and $$\tilde M = L^\ssT \check M L=
\begin{bmatrix}
m_\ssZ^2 & 0 & - m_\ssZ^2 \hat s_\ssW \sh \, \eta \\
0 & 0 & 0 \\
- m_\ssZ^2 \hat s_\ssW \sh \, \eta & 0 & m_\ssX^2 \ch^2 \, \eta
+ m_\ssZ^2 \hat s_\ssW^2 \sh^2 \, \eta
\end{bmatrix}\,,$$ while the currents become $$\tilde{\bf J}_\mu := L^\ssT \check{\bf J}_\mu =
\begin{bmatrix}
\check J_\mu^\ssZ \\
\check J_\mu^\ssA \\
- \check J_\mu^\ssZ \hat s_\ssW \sh \, \eta
+ \check J_\mu^\ssA \hat c_\ssW \sh \, \eta
+ \check J_\mu^\ssX \ch \, \eta
\end{bmatrix} \,.$$ (Notice that $L$ and $R_1$ satisfy $L R_1 = R_1 L$, so it is immaterial whether we first diagonalize the SM mass or the kinetic terms.)
Finally, the mass matrix is diagonalized by letting $$\tilde{\bf V} = R_2 {\bf V} :=
\begin{bmatrix}
c_\xi & 0 & -s_\xi \\
0 & 1 & 0 \\
s_\xi & 0 & c_\xi \end{bmatrix} \begin{bmatrix}
Z \\
A \\
X
\end{bmatrix}$$ where $c_\xi := \cos \xi$ and $s_\xi := \sin \xi$ with the angle $\xi $ given by $$\tan 2\xi =\frac{-2 \hat s_\ssW \sh \eta }{1-\hat s_\ssW^2 \sh^2 \eta-r_\ssX^2 \ch^2\eta} \,,
\label{Apptan2alpha}$$ where we define for convenience r\_:= .
The diagonalized lagrangian then is $$\cL = -\frac14 \, {\bf V}_{\mu\nu}^\ssT
{\bf V}^{\mu\nu} - \frac{M_\ssZ^2}{2} \, Z_\mu Z^\mu
-\frac{M_\ssX^2}{2} \, X_\mu X^\mu +
{\bf J}_\mu^\ssT {\bf V}^\mu \,,$$ where the physical masses are M\_\^2 &=& ( 1 + s\_\^2 \^2 + r\_\^2 \^2 + \_ )\
M\_\^2 &=& ( 1 + s\_\^2 \^2 + r\_\^2 \^2 - \_ ) \[Appmasseigs\] with $\vartheta_{\ssX}$ defined such that $M_\ssZ \rightarrow m_\ssZ$ and $M_\ssX \rightarrow m_\ssX$ as $\eta \rightarrow 0$: \_ := {
[c]{} +1 r\_ > 1\
-1 r\_ < 1
. . The currents in the physical basis are similarly read off as $${\bf J}_\mu =
\begin{bmatrix}
\check J_\mu^\ssZ c_\xi +
\left( -\check J_\mu^\ssZ \hat s_\ssW \sh \, \eta
+ \check J_\mu^\ssA \hat c_\ssW \sh \, \eta
+ \check J_\mu^\ssX \ch \, \eta \right) s_\xi \\
\check J_\mu^\ssA \\
- \check J_\mu^\ssZ s_\xi
+ \left( - \check J_\mu^\ssZ \hat s_\ssW \sh \, \eta
+ \check J_\mu^\ssA \hat c_\ssW \sh \, \eta
+ \check J_\mu^\ssX \ch \, \eta \right) c_\xi
\end{bmatrix} :=
\begin{bmatrix}
J_\mu^\ssZ \\
J_\mu^\ssA \\
J_\mu^\ssX
\end{bmatrix} \,.$$
Since we are eventually interested in obtaining bounds in terms of the physical masses $M_\ssZ$ and $M_\ssX$, it is useful to invert these mass equations to find the input parameters $m^2_\ssZ$ and $m^2_\ssX$ as a function of the physical masses and $\eta$. This gives m\_\^2 &=& (1+R\_\^2 +\_)\
m\_\^2 &=& (1+R\_\^2 -\_) where $R_\ssX$ is used to denote the ratio of the physical masses: \[AppRXeqn\] R\_:= . Also, the sign $\vartheta_\ssX$ is now $+1$ for $R_\ssX>1$ and $-1$ for $R_\ssX<1$.
Given this inversion, the angle $\xi$ can now be written as a function of $R_\ssX$ and $\eta$ only: $$\tan 2\xi(R_\ssX,\eta) =- \left(\frac{2 \hat s_\ssW \sh \eta }{1-\hat s_\ssW^2 \sh^2\eta - r_\ssX^2(R_\ssX,\eta) \ch^2\eta}\right) \,,$$ where r\_\^2(R\_,) = .
Physical couplings {#Appphyscouplings .unnumbered}
------------------
We are now in a position to read off the physical implications of the $X$ boson. That is, we may write $$\cL = \cL_\SM + \delta \cL_\SM +
\cL_\ssX \,, \label{AppLeff1}$$ where the modification to the SM self-couplings are given by \[AppLeff2\] \_= - m\_\^2 Z\_Z\^ + i e\_\_f Z\_, with [@bigfit] z (R\_,) := = , \[Appzeqn\] where \[AppDeltaeqn\] \_:=(R\_\^2-1) . (Note that the $\eta \rightarrow 0$ limit of $z$ is easily verified by implementing the identity $\Delta_\ssX=\vartheta_\ssX \sqrt{\Delta_\ssX^2}$.)
Given the form of $z$, one might worry that, for some choice of the parameters $M_\ssX$ and $\sh\,\eta$, $z$ would yield a complex value. However, any such choice does not correspond to a choice of real values for the original parameters of the lagrangian, $m_\ssX$, $m_\ssZ$, and $\chi$. This happens because sufficiently large kinetic mixing tends to preclude the existence of mass eigenvalues, $M_\ssX$ and $M_\ssZ$, that are too close to one another. This is why this region of parameter space is excluded from the plots of §\[e+e-annihil\].
The fermion couplings are similarly g\_[f()]{} = (c\_-1) g\_[f()]{}+ s\_( s\_(Q\_f c\_\^2 -g\_[f()]{}) + X\_[f()]{} ) . The terms explicitly involving the $X$ boson are $$\begin{aligned}
\cL_\ssX &=& - \frac14 \, X_{\mu \nu } X^{\mu \nu } -
\frac{M_\ssX^2}{2} \, X_\mu X^\mu \\
&& \qquad + i \sum_f \ol{f} \gamma_\mu \left( k_{f\ssL} \gamma_{\ssL}
+ k_{f\ssR} \gamma_{\ssR} \right) f X^\mu \,, \notag\end{aligned}$$ with k\_[f()]{} = c\_( g\_X\_[f()]{} + (Q\_f c\_\^2-g\_[f()]{})) -s\_e\_g\_[f()]{} . Notice that in this basis $X_{\mu }$ does not couple directly to the electroweak gauge bosons at tree-level, but has acquired modified fermion couplings due to the mixing.
### Oblique parameters {#oblique-parameters .unnumbered}
The only remaining step is to eliminate parameters like $\hat s_\ssW$ and $m_\ssZ$ in the lagrangian in favour of a physically defined weak mixing angle, $s_\ssW$, and mass $M_\ssZ$. This process reveals the physical combination of new-physics parameters that is relevant to observables, and thereby provides a derivation [@bigfit] of the $X$-boson contributions to the oblique electroweak parameters [@oblique].
We have already seen how to do this for the $Z$ mass, for which $$m_\ssZ \simeq M_\ssZ \left( 1 - \frac{z}{2} \right) \,.$$ For the weak mixing angle it is convenient to define $s_\ssW$ so that the Fermi constant, $G_\ssF$, measured in muon decay is given by the SM formula, $$\frac{G_\ssF}{\sqrt{2}} := \frac{e^2}{8 s_\ssW^2 c_\ssW^2 M_\ssZ^2} \,.
\label{AppG_fdef}$$ But this can be compared with the tree-level calculation of the Fermi constant obtained in our model from $W$-exchange, $$\frac{G_\ssF}{\sqrt{2}} = \frac{g_2^2}{8 m_\ssW^2}
= \frac{e^2}{8 \hat s_\ssW^2 \hat c_\ssW^2 m_\ssZ^2} \,,
\label{AppG_f}$$ to infer $$\hat s_\ssW^2 \hat c_\ssW^2 = s_\ssW^2 c_\ssW^2
\left( 1+z\right) \,,$$ which, to linear order in $z$, implies that $$\hat s_\ssW^2 = s_\ssW^2 \left[
1 + \frac{z \, c_\ssW^2}{c_\ssW^2 - s_\ssW^2}
\right] \,.$$
Eliminating $\hat s_\ssW$ in favour of $s_\ssW$ in the fermionic weak interactions introduces a further shift in these couplings, leading to our final form for the neutral-current lagrangian: $$\begin{aligned}
\cL_{\NC} &=& \frac{ie}{\hat s_\ssW \hat c_\ssW}
\sum_f \left[ \ol{f} \gamma^\mu \left( T_{3f} \gamma_{\ssL}
-Q_f \hat s_\ssW^2 \right) f \right. \nn\\
&&\qquad\qquad\qquad \left. +
\ol{f} \gamma^\mu \left( \delta g_{f\ssL} \gamma_{\ssL}
+\delta g_{f \ssR} \gamma_{\ssR} \right) f \right] Z_\mu \notag \\
&\simeq& \frac{ie}{s_\ssW c_\ssW} \left(
1 - \frac{z}{2} \right) \sum_f \left\{ \ol{f} \gamma^\mu
\left[ T_{3f} \gamma_{\ssL} - Q_f s_\ssW^2 \left(
1 + \frac{z \, c_\ssW^2}{c_\ssW^2 - s_\ssW} \right)
\right] f \right. \nn\\
&& \qquad\qquad \left. + \phantom{\frac12} \overline{f} \gamma^\mu
\left( \delta g_{f\ssL} \gamma_{\ssL}
+\delta g_{f\ssR} \gamma_{\ssR}\right)
f \right\} Z_\mu \notag \\
&:=& i e_\ssZ \sum_f \overline{f}
\gamma^\mu \left[ \left( g_{f\ssL}^\SM + \Delta g_{f\ssL}
\right) \gamma_\ssL + \left( g_{f\ssR}^\SM + \Delta g_{f\ssR}
\right) \gamma_\ssR \right] f \, Z_\mu \,, \label{AppdLNC}\end{aligned}$$ where $e_\ssZ := e/s_\ssW c_\ssW$ and $$\label{AppDgfLR}
\Delta g_{f \ssL(\ssR)} = - \frac{z}{2} g_{f \ssL(\ssR)}^{\SM} - z \left(\frac{s_\ssW^2 c_\ssW^2}{c_\ssW^2-s_\ssW^2}\right) Q_f
+\delta g_{f\ssL(\ssR)} \,,$$ where (as usual) $g_{f\ssL}^\SM := T_{3f} - Q_f s_\ssW^2$ and $g_{f\ssR}^\SM := - Q_f s_\ssW^2$. It is assumed throughout that the corrections $z$, $\delta g_{f\ssL(\ssR)}$, and \[AppdefDkfLR\] k\_[f()]{}:=k\_[f()]{}-g\_X\_[f()]{} are small, so that any expression can be linearized in these variables. In particular, this means that one can replace hatted electroweak parameters (i.e. $\hat s_\ssW$, etc...) with unhatted ones in our previous expressions to give: z (R\_,) &=&\
g\_[f()]{} &=& (c\_-1) g\_[f()]{}\^+ s\_( s\_(Q\_f c\_\^2 -g\_[f()]{}\^) + X\_[f()]{} )\
k\_[f()]{} &=& (c\_-1) g\_X\_[f()]{} + c\_ (Q\_f c\_\^2-g\_[f()]{}\^)-s\_e\_g\_[f()]{}\^. \[AppDkfLR\] Alternatively, one can use the relationship between $z$ and $\eta$ to determine the contribution to the oblique parameters [@oblique] $S = U = 0$ and $\alpha T = - z$, where (as usual) $\alpha := e^2/4\pi$. In this case, $\Delta g_{f \ssL(\ssR)}$ can be written as in [@bigfit] $$\Delta g_{f\ssL(\ssR)} = \frac{\alpha T}{2} \, g_{f\ssL(\ssR)}^{\SM}
+\alpha T \left( \frac{s_\ssW^2 c_\ssW^2}{c_\ssW^2
-s_\ssW^2} \right) Q_f + \delta g_{f\ssL(\ssR)} \,.$$
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[^1]: Superconductors are similar in this regard: the photon acquires a mass without implying gross violations of charge conservation.
[^2]: The careful reader will recognize that this argument assumes negligible anomalous dimensions, and so needs re-examination for strongly coupled theories.
[^3]: Notice that $B-L$ carries a Standard Model anomaly in the absence of sterile right-handed neutrinos (see below).
[^4]: A similar formulation can be made using the anomaly in its ‘consistent’ form, rather than the ‘covariant’ form used in the text.
[^5]: We thank Johannes Blümlein for bringing this to our attention
|
---
abstract: |
In this paper, we study a class of generalized intersection matrix Lie algebras $\gim(M_n)$, and prove that its every finite-dimensional semi-simple quotient is of type $M(n,{\bf a}, {\bf c},{\bf d})$. Particularly, any finite dimensional irreducible $\gim(M_n)$ module must be an irreducible module of $M(n,{\bf a}, {\bf c},{\bf d})$ and any finite dimensional irreducible $M(n,{\bf a}, {\bf c},{\bf d})$ module must be an irreducible module of $\gim(M_n)$.
[*Key Words*]{}: intersection matrix algebras; irreducible modules; quotient algebras; affine Lie algebras.
author:
- Yun Gao$^1$
- 'Li-meng Xia$^{2,\dag}$'
title: 'Finite-dimensional representations for a class of generalized intersection matrix Lie algebras'
---
$^1$Department of Mathematics and Statistics, York University, Canada
$^2$Faculty of Science, Jiangsu University, P.R. China
\[section\] \[theo\][Theorem]{} \[theo\][Definition]{} \[theo\][Lemma]{} \[theo\][Corollary]{} \[theo\][Proposition]{} \[theo\][Remark]{}
Introduction
============
In the early to mid-1980s, Peter Slodowy discovered that matrices like $$M =\left[\begin{array}{cccc}
2&-1&0&1\\
-1&2&-1&1\\
0& -2&2&-2\\
1&1&-1&2
\end{array}\right]$$ were encoding the intersection form on the second homology group of Milnor fibres for germs of holomorphic maps with an isolated singularity at the origin \[S1\], \[S2\]. These matrices were like the generalized Cartan matrices of Kac-Moody theory in that they had integer entries, $2$’s along the diagonal, and $m_{i,j}$ was negative if and only if $m_{j,i}$ was negative. What was new, however, was the presence of positive entries off the diagonal. Slodowy called such matrices generalized intersection matrices:
[([@S1])]{} An $n\times n$ integer-valued matrix $M=(m_{i,j})_{1\leq i,j\leq n}$ is called a generalized intersection matrix () if
$m_{i,i}=2$,
$m_{i,j}<0$ if and only if $m_{j,i}<0$, and
$m_{i,j}>0$ if and only if $m_{j,i}>0$
[for $1\leq i, j\leq n$ with $i\not= j$.]{}
Slodowy used these matrices to define a class of Lie algebras that encompassed all the Kac-Moody Lie algebras:
Given an $n\times n$ generalized intersection matrix $M = (m_{i,j})$, define a Lie algebra over $\mc$, called a generalized intersection matrix (GIM) algebra and denoted by $\gim(M)$, with:
generators: $e_1, . . . , e_n, f_1, . . . , f_n, h_1, . . . h_n$,
relations: (R1) for $1\leq i, j \leq n$, = m\_[i,j]{}e\_j,&[\[h\_i, f\_j \]]{} =-m\_[i,j]{}f\_j,&[\[e\_i, f\_i\]]{} = h\_i,(R2) for $m_{i,j}\leq 0$, = 0 = \[f\_i, e\_j \],&&(e\_i)\^[-m\_[i,j]{}+1]{}e\_j = 0 = (f\_i)\^[-m\_[i,j]{}+1]{}f\_j,(R3) for $m_{i,j} > 0$, $i\not= j$, = 0 = \[f\_i, f\_j \],&&(e\_i)\^[m\_[i,j]{}+1]{}f\_j = 0 = (f\_i)\^[m\_[i,j]{}+1]{}e\_j.
If the $M$ that we begin with is a generalized Cartan matrix, then the $3n$ generators and the first two groups of axioms, (R1) and (R2), provide a presentation of the Kac-Moody Lie algebras \[GbK\], \[C\], \[K\].
Slodowy and, later, Berman showed that the GIM algebras are also isomorphic to fixed point subalgebras of involutions on larger Kac- Moody algebras \[S1\], \[Br\]. So, in their words, the GIM algebras lie both “beyond and inside” Kac-Moody algebras.
Further progress came in the 1990s as a byproduct of the work of Berman and Moody, Benkart and Zelmanov, and Neher on the classification of root-graded Lie algebras \[BrM\], \[BnZ\], \[N\]. Their work revealed that some families of intersection matrix ($\emph{\textbf{im}}$) algebras, were universal covering algebras of well understood Lie algebras. An $\emph{\textbf{im}}$ algebra generally is a quotient algebra of a GIM algebra associated to the ideal generated by homogeneous vectors those have long roots (i.e., $(\alpha,\alpha)>2$).
A handful of other researchers also began engaging these new algebras. For example, Eswara Rao, Moody, and Yokonuma used vertex operator representations to show that $\emph{\textbf{im}}$ algebras were nontrivial \[EMY\]. Analogous compact forms of $\emph{\textbf{im}}$ algebras arising from conjugations over the complex field were considered in \[G\]. Peng found relations between $\emph{\textbf{im}}$ algebras and the representations of tilted algebras via Ringel-Hall algebras \[P\]. Berman, Jurisich, and Tan showed that the presentation of GIM algebras could be put into a broader framework that incorporated Borcherds algebras \[BrJT\].
In present paper, we study the GIM algebra $\gim(M_n)$ constructed through intersection matrix M\_n=(m\_[i,j]{})\_[nn]{}=\_[nn]{}where $n\geq 3$, and we build the representation theory of finite-dimensional modules for $\gim(M_n)$.
Generally, $\gim(M_n)$ can be illustrated by the following diagram (also called Dynkin diagram):
where the numbered circles present the indices of $e_i, f_i$ ($1\leq i\leq n$), the solid line between two circles $i,i+1$ means that $m_{i,i+1}=m_{i+1,i}=-1$ and the unique dotted line means that $m_{1,n}=m_{n,1}=1$. There is no line between any other pair $(i,j)$, it means that $m_{i,j}=m_{j,i}=0$.
Construction of epimorphisms
============================
Suppose that $\cl$ is a non-trivial semi-simple Lie algebra and ${\bf a}\in\mz_{\geq0}, {\bf c, d}\in\{0,1\}$. If $n\geq 3$ and $\cl$ is isomorphic to a direct sum of ${\bf a}$ copies of $sl_{2n}$, ${\bf c}$ copies of $sp_{2n}$ and ${\bf d}$ copies of $so_{2n}$, then we say that $\cl$ is of $M(n,{\bf a}, {\bf c}, {\bf d})$ type.
Let $\cl=\oplus_{k=1}^K\cl_k$ be of type $M(n,{\bf a}, {\bf c}, {\bf d})$, in this section we construct an epimorphism from $\gim(M_n)$ to $\cl$.
For convenience, we fix a Chevalley generators $\{e_{\alpha_i}, f_{\alpha_i}|1\leq i\leq n\}$ for simple Lie algebra of type $C_n$ or $D_n$, and $\{e_{\alpha_i}, f_{\alpha_i}|1\leq i\leq 2n-1\}$ for $A_{2n-1}$. The associative Dynkin diagrams are
\[L2.1\] Let $a\in\mc^\times, a\not=\pm1$. If $\{e_{\alpha_i},f_{\alpha_i}|1\leq i\leq 2n-1\}$ is the Chevalley generators of $A_{2n-1}$, let e\_[\_[2n]{}]{}&=&a\[f\_[\_[2n-1]{}]{},\],\
f\_[\_[2n]{}]{}&=&a\^[-1]{}\[, e\_[\_[2n-1]{}]{}\],then e\_ie\_[\_i]{}-f\_[\_[n+i]{}]{},&&f\_if\_[\_i]{}-e\_[\_[n+i]{}]{},defines a Lie algebra homomorphism from $\gim(M_n)$ to $A_{2n-1}$, where $1\leq i\leq n$.
Let X\_i&=&e\_[\_i]{}-f\_[\_[n+i]{}]{},\
Y\_i&=&f\_[\_i]{}-e\_[\_[n+i]{}]{},\
H\_i&=&\[X\_i,Y\_i\],for all $1\leq i\leq n$.
Note that the subalgebra generated by $e_{\alpha_i},f_{\alpha_i} (1\leq i\leq 2n-1, i\not=n)$ is isomorphic to $sl_n\oplus sl_n$. So the map e\_ie\_[\_i]{}-f\_[\_[n+i]{}]{},&&f\_if\_[\_i]{}-e\_[\_[n+i]{}]{},restricted to $1\leq i\leq n-1$ induces a diagonal injective map $\Phi_1\oplus\Phi_2$, where \_1: &e\_ie\_[\_i]{},&f\_if\_[\_i]{},\
\_2:& e\_i-f\_[\_[n+i]{}]{},&f\_i-e\_[\_[n+i]{}]{}.Particularly, $\Phi_1$ can be viewed as an inclusion and $\Phi_2$ can be viewed the composition of Chevalley involution and inclusion.
It is sufficient to check the relations involving elements $X_n,Y_n,H_n$. H\_i&=&\[e\_[\_i]{}-f\_[\_[n+i]{}]{}, f\_[\_i]{}-e\_[\_[n+i]{}]{}\]=h\_[\_i]{}-h\_[\_[n+i]{}]{}, 1in-1,\
H\_n&=&\[e\_[\_n]{}-f\_[\_[2n]{}]{}, f\_[\_n]{}-e\_[\_[2n]{}]{}\]=h\_[\_n]{}+(h\_[\_1]{}++h\_[\_[2n-1]{}]{}),where $h_{\alpha_j}=[e_{\alpha_j}, f_{\alpha_j}]$ for all $1\leq j\leq 2n-1$. ([*For the computation of $H_n$, see Remark 1.*]{}) Then &=&\[h\_[\_n]{}+(h\_[\_1]{}++h\_[\_[2n-1]{}]{}),e\_[\_n]{}-f\_[\_[2n]{}]{}\]\
&=&\_n(h\_[\_[n-1]{}]{}+2h\_[\_n]{}+h\_[\_[n+1]{}]{})e\_[\_n]{}-(\_1++\_[2n-1]{})(h\_[\_n]{}+(h\_[\_1]{}++h\_[\_[2n-1]{}]{}))f\_[\_[2n]{}]{}\
&=&\_n(h\_[\_n]{})e\_[\_n]{}-(\_1++\_[2n-1]{})(h\_[\_1]{}+h\_[\_[2n-1]{}]{})f\_[\_[2n]{}]{}\
&=&2X\_n,\
[\[H\_n, Y\_n\]]{}&=&-\_n(h\_[\_n]{})f\_[\_n]{}+(\_1++\_[2n-1]{})(h\_[\_1]{}+h\_[\_[2n-1]{}]{})e\_[\_[2n]{}]{}\
&=&-2Y\_n, and &=&\_1(h\_[\_n]{}+(h\_[\_1]{}++h\_[\_[2n-1]{}]{}))e\_[\_1]{}-(-\_[n+1]{})(h\_[\_n]{}+(h\_[\_1]{}++h\_[\_[2n-1]{}]{}))f\_[\_[n+1]{}]{}\
&=&\_1(h\_[\_1]{}+h\_[\_2]{})e\_[\_1]{}+\_[n+1]{}(2h\_[\_n]{}+h\_[\_[n+1]{}]{}+h\_[\_[n+2]{}]{})f\_[\_[n+1]{}]{}\
&=&X\_1,\
[\[H\_n, Y\_1\]]{}&=&-\_1(h\_[\_1]{}+h\_[\_2]{})f\_[\_1]{}-\_[n+1]{}(2h\_[\_n]{}+h\_[\_[n+1]{}]{}+h\_[\_[n+2]{}]{})e\_[\_[n+1]{}]{}\
&=&-Y\_1,\
[\[H\_n, X\_[n-1]{}\]]{}&=&\_[n-1]{}(h\_[\_n]{}+(h\_[\_1]{}++h\_[\_[2n-1]{}]{}))e\_[\_[n-1]{}]{}-(-\_[2n-1]{})(h\_[\_n]{}+(h\_[\_1]{}++h\_[\_[2n-1]{}]{}))f\_[\_[2n-1]{}]{}\
&=&\_[n-1]{}(h\_[\_[n-2]{}]{}+h\_[\_[n-1]{}]{}+2h\_[\_n]{})e\_[\_[n-1]{}]{}+\_[2n-1]{}(h\_[\_[2n-2]{}]{}+h\_[\_[2n-1]{}]{})f\_[\_[2n-1]{}]{}\
&=&-X\_[n-1]{},\
[\[H\_n, Y\_[n-1]{}\]]{}&=&-\_[n-1]{}(h\_[\_[n-2]{}]{}+h\_[\_[n-1]{}]{}+2h\_[\_n]{})f\_[\_[n-1]{}]{}-\_[2n-1]{}(h\_[\_[2n-2]{}]{}+h\_[\_[2n-1]{}]{})e\_[\_[2n-1]{}]{}\
&=&Y\_[n-1]{}.Similar argument implies that &&\[H\_1, X\_n\]=X\_n,\[H\_1, Y\_n\]=-Y\_n,\[H\_[n-1]{}, X\_n\]=-X\_n,\[H\_[n-1]{}, Y\_n\]=Y\_n,\
&&[\[X\_1, X\_n\]]{}=\[Y\_1, Y\_n\]=\[Y\_[n-1]{},X\_n\]=\[X\_[n-1]{},Y\_n\]=0,\
&&\[X\_i,\[X\_i,Y\_j\]\]=\[Y\_i,\[Y\_i,X\_j\]\]=0,{i,j}={1,n},\
&&\[X\_i,\[X\_i,X\_j\]\]=\[Y\_i,\[Y\_i,Y\_j\]\]=0,{i,j}={n-1,n}. Finally, the following relation is clear: =\[X\_n,Y\_i\]=\[Y\_n, X\_i\]=\[Y\_n,Y\_i\]=0for all $2\leq i\leq n-2$.
[**Remark 1.**]{}
*Suppose that $\alpha, \beta, \alpha+\beta\in\Delta^+$ in $A_{2n-1}$ and $[e_\alpha, f_\alpha]=\alpha^\vee, [e_\beta, f_\beta]=\beta^\vee$. Then it holds that $[[e_\alpha,e_\beta],[f_\beta, f_\alpha]]=\alpha^\vee+\beta^\vee$. Repeatedly using this formula, we infer that =h\_[\_1]{}++h\_[\_[2n-1]{}]{}.Together with that $\alpha_1+\cdots+\alpha_{2n-1}\pm\alpha_n$ is not a root, we obtain the computation of $H_n$.*
One can also understand it in an easy way: Let $A_{2n-1}$ be the matrix Lie algebra $sl_{2n}$ and $e_{\alpha_i}=E_{i,i+1}, f_{\alpha_i}=E_{i,i+1}$ for $i<2n$. Then $e_{\alpha_{2n}}=aE_{2n,1}, f_{\alpha_{2n}}=a^{-1}E_{1,2n}$, which implies that =E\_[1,1]{}-E\_[2n,2n]{}=\_[i=1]{}\^[2n-1]{}(E\_[i,i]{}-E\_[i+1,i+1]{})=\_[i=1]{}\^[2n-1]{}h\_[\_i]{}.
\[L2.2\] If $\{e_{\alpha_i},f_{\alpha_i}|1\leq i\leq n\}$ is the Chevalley generators of $C_n$, let E&=&\[f\_[\_[n-1]{}]{},\],\
F&=&\[, e\_[\_[n-1]{}]{}\],then e\_ie\_[\_i]{},&&f\_if\_[\_i]{},1in-1,\
e\_nE,&&f\_nF,defines a Lie algebra homomorphism from $\gim(M_n)$ to $C_{n}$.
It is sufficient to check the relation involving elements $E,F$. Actually, $F$ (respectively $E$) is a root vector of highest short root (respectively lowest short root). Then =\[E, f\_[\_i]{}\]=0&& 2in-1,\
[\[F, f\_[\_i]{}\]]{}=[\[E, e\_[\_i]{}\]]{}=0&& 1in-2,and =\[e\_[\_1]{},\[e\_[\_1]{},F\]\]=0,&&\[E,\[E,f\_[\_1]{}\]\]=\[f\_[\_1]{},\[f\_[\_1]{},E\]\]=0,\
[\[F,\[F,f\_[\_[n-1]{}]{}\]\]]{}=\[f\_[\_[n-1]{}]{},\[f\_[\_[n-1]{}]{},F\]\]=0,&&\[E,\[E,e\_[\_[n-1]{}]{}\]\]=\[e\_[\_[n-1]{}]{},\[e\_[\_[n-1]{}]{},E\]\]=0. Moreover, $H:=[E,F]=-(2h_{\alpha_n}+h_{\alpha_1}+\cdots+h_{\alpha_{n-1}})$, where $h_{\alpha_i}=[e_{\alpha_i},f_{\alpha_i}]$ for all $1\leq i\leq n$. Hence, we have &=&(-\_1--\_n)(-(2h\_[\_n]{}+h\_[\_1]{}++h\_[\_[n-1]{}]{}))E=2E,\
[\[H,F\]]{}&=&(\_1++\_n)(-(2h\_[\_n]{}+h\_[\_1]{}++h\_[\_[n-1]{}]{}))F=-2F,and =e\_[\_1]{},&&\[H,f\_[\_1]{}\]=-f\_[\_1]{},\
[\[H,e\_[\_[n-1]{}]{}\]]{}=-e\_[\_[n-1]{}]{},&&\[H,f\_[\_[n-1]{}]{}\]=f\_[\_[n-1]{}]{},\
[\[h\_[\_1]{}]{},E\]=E,&& [\[h\_[\_[n-1]{}]{}]{},E\]=-E,\
[\[h\_[\_1]{}]{},F\]=-F,&& [\[h\_[\_[n-1]{}]{}]{},F\]=F.The above calculation implies our statement and the proof is completed.
\[L2.3\] If $\{e_{\alpha_i},f_{\alpha_i}|1\leq i\leq n\}$ is the Chevalley generators of $D_n$, let E&=&\[f\_[\_[n-1]{}]{},\],\
F&=&\[, e\_[\_[n-1]{}]{}\],then e\_ie\_[\_i]{},&&f\_if\_[\_i]{},1in-1,\
e\_nE,&&f\_nF,defines a Lie algebra homomorphism from $\gim(M_n)$ to $D_{n}$.
It is sufficient to check the relation involving elements $E,F$. Actually, $F$ (respectively $E$) is a root vector of highest root (respectively lowest root) of subalgebra generated by $e_{\alpha_i},f_{\alpha_i} (2\leq i\leq n)$. Then =\[E, f\_[\_i]{}\]=0&& 2in-1,\
[\[F, f\_[\_i]{}\]]{}=[\[E, e\_[\_i]{}\]]{}=0&& 2in-2,and =\[f\_[\_[n-1]{}]{},\[f\_[\_[n-1]{}]{},F\]\]=0,&&\[E,\[E,e\_[\_[n-1]{}]{}\]\]=\[e\_[\_[n-1]{}]{},\[e\_[\_[n-1]{}]{},E\]\]=0. Notice that $\alpha_1+\cdots+\alpha_n$ is a root, and neither of $-\alpha_1+\alpha_2+\cdots+\alpha_n$, $2\alpha_1+\cdots+\alpha_n$ and $\alpha_1+2(\alpha_2+\cdots+\alpha_n)$ is a root, hence we have =\[e\_[\_1]{},\[e\_[\_1]{},F\]\]=0,&&\[E,\[E,f\_[\_1]{}\]\]=\[f\_[\_1]{},\[f\_[\_1]{},E\]\]=0,\
[\[F, f\_[\_1]{}\]]{}=0,&&[\[E, e\_[\_1]{}\]]{}=0.
Moreover, $H:=[E,F]=-(h_{\alpha_2}+\cdots+h_{\alpha_{n}})$, where $h_{\alpha_i}=[e_{\alpha_i},f_{\alpha_i}]$ for all $1\leq i\leq n$. Hence, we have &=&(-\_2--\_n)(-(h\_[\_2]{}++h\_[\_[n]{}]{}))E=2E,\
[\[H,F\]]{}&=&(\_2++\_n)(-(h\_[\_2]{}++h\_[\_[n]{}]{}))F=-2F,and =e\_[\_1]{},&&\[H,f\_[\_1]{}\]=-f\_[\_1]{},\
[\[H,e\_[\_[n-1]{}]{}\]]{}=-e\_[\_[n-1]{}]{},&&\[H,f\_[\_[n-1]{}]{}\]=f\_[\_[n-1]{}]{},\
[\[h\_[\_1]{}]{},E\]=E,&& [\[h\_[\_[n-1]{}]{}]{},E\]=-E,\
[\[h\_[\_1]{}]{},F\]=-F,&& [\[h\_[\_[n-1]{}]{}]{},F\]=F.The above calculation implies our statement and the proof is completed.
Next we begin to explicitly construct an epimorphism from $\gim(M_n)$ to $\cl$. The construction is divided into four distinguish cases.
[**Case 1.**]{} $\cl_k\cong sl_{2n}$ for all $1\leq k\leq K$.
Let $\{e_{\alpha_i}^{[k]}, f_{\alpha_i}^{[k]}|1\leq i\leq 2n-1\}$ be the analogue of Chevalley generators of $A_{2n-1}$ in $\cl_k$ for all $k$. Choose a $K$-tuple $\underline{a}:=(a_1,\cdots, a_K)\in(\mc^\times)^K$ such that $a_k\not=\pm1$ and $a_k\not=a_{j}^{\pm1}$ for all $k\not=j$. Set e\_[\_[2n]{}]{}\^[\[k\]]{}&=&a\_k\[f\_[\_[2n-1]{}]{}\^[\[k\]]{},\],\
f\_[\_[2n]{}]{}\^[\[k\]]{}&=&a\_k\^[-1]{}\[, e\_[\_[2n-1]{}]{}\^[\[k\]]{}\],for all $1\leq k\leq K$, where $e_{\alpha_{2n}}^{[k]}$ and $f_{\alpha_{2n}}^{[k]}$ are root vectors of lowest root and highest root, respectively.
[**Case 2.**]{} $\cl_1\cong sp_{2n}$ and $\cl_k\cong sl_{2n}$ for all $2\leq k\leq K$.
Let $\{e_{\alpha_i}^{[1]}, f_{\alpha_i}^{[1]}|1\leq i\leq n\}$ be the analogue of Chevalley generators of $C_n$ in $\cl_1$. and $\{e_{\alpha_i}^{[k]}, f_{\alpha_i}^{[k]}|1\leq i\leq 2n-1\}$ be the analogue of Chevalley generators of $A_{2n-1}$ in $\cl_k$ for $k\geq 2$. Choose a $K$-tuple ${\underline{a}}:=(1,a_2,\cdots, a_K)\in(\mc^\times)^K$ such that $a_k\not=\pm1$ and $a_k\not=a_{j}^{\pm1}$ for all $k\not=j$. Set e\_[\_[2n]{}]{}\^[\[k\]]{}&=&a\_k\[f\_[\_[2n-1]{}]{}\^[\[k\]]{},\],\
f\_[\_[2n]{}]{}\^[\[k\]]{}&=&a\_k\^[-1]{}\[, e\_[\_[2n-1]{}]{}\^[\[k\]]{}\],for all $2\leq k\leq K$, and e\_[\_[2n]{}]{}\^[\[1\]]{}&=&f\_[\_n]{}\^[\[1\]]{}-\[, e\_[\_[n-1]{}]{}\^[\[1\]]{}\],\
f\_[\_[2n]{}]{}\^[\[1\]]{}&=&e\_[\_n]{}\^[\[1\]]{}-\[f\_[\_[n-1]{}]{}\^[\[1\]]{},\],\
&&e\_[\_[n+1]{}]{}\^[\[1\]]{}==e\_[\_[2n-1]{}]{}\^[\[1\]]{}=0,\
&&f\_[\_[n+1]{}]{}\^[\[1\]]{}==f\_[\_[2n-1]{}]{}\^[\[1\]]{}=0,where $f_{\alpha_{n}}^{[1]}-e_{\alpha_{2n}}^{[1]}$ and $e_{\alpha_{n}}^{[1]}-f_{\alpha_{2n}}^{[1]}$ are root vectors of highest short root and lowest short root, respectively.
[**Case 3.**]{} $\cl_1\cong so_{2n}$ and $\cl_k\cong sl_{2n}$ for all $2\leq k\leq K$.
Let $\{e_{\alpha_i}^{[1]}, f_{\alpha_i}^{[1]}|1\leq i\leq n\}$ be the analogue of Chevalley generators of $C_n$ in $\cl_1$. and $\{e_{\alpha_i}^{[k]}, f_{\alpha_i}^{[k]}|1\leq i\leq 2n-1\}$ be the analogue of Chevalley generators of $A_{2n-1}$ in $\cl_k$ for $k\geq 2$. Choose a $K$-tuple ${\underline{a}}:=(-1,a_2,\cdots, a_K)\in(\mc^\times)^K$ such that $a_k\not=\pm1$ and $a_k\not=a_{j}^{\pm1}$ for all $k\not=j$. Set e\_[\_[2n]{}]{}\^[\[k\]]{}&=&a\_k\[f\_[\_[2n-1]{}]{}\^[\[k\]]{},\],\
f\_[\_[2n]{}]{}\^[\[k\]]{}&=&a\_k\^[-1]{}\[, e\_[\_[2n-1]{}]{}\^[\[k\]]{}\],for all $2\leq k\leq K$, and e\_[\_[2n]{}]{}\^[\[1\]]{}&=&f\_[\_n]{}\^[\[1\]]{}-\[, e\_[\_[n-1]{}]{}\^[\[1\]]{}\],\
f\_[\_[2n]{}]{}\^[\[1\]]{}&=&e\_[\_n]{}\^[\[1\]]{}-\[f\_[\_[n-1]{}]{}\^[\[1\]]{},\],\
&&e\_[\_[n+1]{}]{}\^[\[1\]]{}==e\_[\_[2n-1]{}]{}\^[\[1\]]{}=0,\
&&f\_[\_[n+1]{}]{}\^[\[1\]]{}==f\_[\_[2n-1]{}]{}\^[\[1\]]{}=0.
[**Case 4.**]{} $\cl_1\cong so_{2n}$, $\cl_2\cong sp_{2n}$ and $\cl_k\cong sl_{2n}$ for all $3\leq k\leq K$.
Let $\{e_{\alpha_i}^{[1]}, f_{\alpha_i}^{[1]}|1\leq i\leq n\}$ be the analogue of Chevalley generators of $D_n$ in $\cl_1$, $\{e_{\alpha_i}^{[2]}, f_{\alpha_i}^{[2]}|1\leq i\leq n\}$ be the analogue of Chevalley generators of $C_n$ in $\cl_2$, and $\{e_{\alpha_i}^{[k]}, f_{\alpha_i}^{[k]}|1\leq i\leq 2n-1\}$ be the analogue of Chevalley generators of $A_{2n-1}$ in $\cl_k$ for $k\geq 3$. Choose a $K$-tuple ${\underline{a}}:=(-1,1, a_3,\cdots, a_K)\in(\mc^\times)^K$ such that $a_k\not=\pm1$ and $a_k\not=a_{j}^{\pm1}$ for all $k\not=j$. Set e\_[\_[2n]{}]{}\^[\[k\]]{}&=&a\_k\[f\_[\_[2n-1]{}]{}\^[\[k\]]{},\],\
f\_[\_[2n]{}]{}\^[\[k\]]{}&=&a\_k\^[-1]{}\[, e\_[\_[2n-1]{}]{}\^[\[k\]]{}\],for all $3\leq k\leq K$, and e\_[\_[2n]{}]{}\^[\[1\]]{}&=&f\_[\_n]{}\^[\[1\]]{}-\[, e\_[\_[n-1]{}]{}\^[\[1\]]{}\],\
f\_[\_[2n]{}]{}\^[\[1\]]{}&=&e\_[\_n]{}\^[\[1\]]{}-\[f\_[\_[n-1]{}]{}\^[\[1\]]{},\],\
&&e\_[\_[n+1]{}]{}\^[\[1\]]{}==e\_[\_[2n-1]{}]{}\^[\[1\]]{}=0,\
&&f\_[\_[n+1]{}]{}\^[\[1\]]{}==f\_[\_[2n-1]{}]{}\^[\[1\]]{}=0,\
e\_[\_[2n]{}]{}\^[\[2\]]{}&=&f\_[\_n]{}\^[\[2\]]{}-\[, e\_[\_[n-1]{}]{}\^[\[2\]]{}\],\
f\_[\_[2n]{}]{}\^[\[2\]]{}&=&e\_[\_n]{}\^[\[2\]]{}-\[f\_[\_[n-1]{}]{}\^[\[2\]]{},\],\
&&e\_[\_[n+1]{}]{}\^[\[2\]]{}==e\_[\_[2n-1]{}]{}\^[\[2\]]{}=0,\
&&f\_[\_[n+1]{}]{}\^[\[2\]]{}==f\_[\_[2n-1]{}]{}\^[\[2\]]{}=0.
In all above four cases, we define a homomorphism from $\gim(M_n)\rightarrow\cl$ via \_:&& e\_i\_[k=1]{}\^K(e\_[\_i]{}\^[\[k\]]{}-f\_[\_[n+i]{}]{}\^[\[k\]]{}), f\_i\_[k=1]{}\^K(f\_[\_i]{}\^[\[k\]]{}-e\_[\_[n+i]{}]{}\^[\[k\]]{}),for all $1\leq i\leq n$.
$\Psi_{\underline{a}}$ is a Lie algebra epimorphism.
Let $P_k$ be the projection from $\cl$ to $\cl_k$. By Lemmas \[L2.1\]-\[L2.3\], we infer that $P_k\circ \Psi_{\underline{a}}$ is a homomorphism for all $1\leq k\leq K$, then $\Psi_{\underline{a}}=\oplus_{k=1}^K P_k\circ\Psi_{\underline{a}}$ is a homomorphism of Lie algebras. The detailed proof for that $\Psi_{\underline{a}}$ is an epimorphism, is very similar to the proof for Lemmas 5.1-5.4 in Section 5.
Main results
============
Our main result can be stated by the following three theorems.
If $\cl$ is of type $M(n, {\bf a}, {\bf c}, {\bf d})$, then there exists an ideal $I$ of $\gim(M_n)$, such that (M\_n)/I.
If there exists an ideal $I$ of $\gim(M_n)$, such that $\gim(M_n)/I$ is finite-dimensional and semi-simple, then $\gim(M_n)/I$ is of type $M(n, {\bf a}, {\bf c}, {\bf d})$.
The above two theorems will be proved in below sections.
\(1) If $V$ is an irreducible finite-dimensional module of an $M(n, {\bf a}, {\bf c}, {\bf d})$ type Lie algebra, then $V$ is an irreducible $\gim(M_n)$-module.
\(2) If $V$ is an irreducible $\gim(M_n)$-module with finite dimension, then $V$ is an irreducible module of some $M(n, {\bf a}, {\bf c}, {\bf d})$ type Lie algebra.
\(1) Suppose that $V$ is an irreducible finite-dimensional module of $M(n, {\bf a}, {\bf c}, {\bf d})$ type Lie algebra $\cl$, and that $\zeta:\cl\rightarrow End(V)$ is the representation map. By Proposition 2.5, $V$ is an irreducible $\gim(M_n)$-module with representation map $\zeta\circ\Psi_{\underline{a}}$.
\(2) Suppose that $V$ is non-trivial and the representation is define by $\kappa: \gim(M_n)\rightarrow End(V)$ and the image of $\kappa$ has Levi decomposition $$Im(\kappa)=S\dot+H\dot+W,$$ where $S$ is semi-simple, $H$ is diagonal and $W$ is nilpotent. Undoubtedly, it holds that $[S, H]=0$.
Because $V$ is non-trivial and there exists a positive integer $k$ such that $W^k$ trivially acts on $V$, we infer that $W\cdot V$ is a proper submodule and we may assume that $W=0$.
Suppose that $\sh$ is a Cartan subalgebra of $S$. Then $V$ has a basis such that both $\kappa(\sh)$ and $\kappa(H)$ are diagonal, then any $S$-module must be an $H$-module. So $V$ is irreducible as $\gim(M_n)$-module if and only if $V$ is irreducible as $S$-module. By Theorem 3.2, $S$ has to be isomorphic to some $M(n, {\bf a}, {\bf c}, {\bf d})$ type Lie algebra, and thus the statement holds.
Evaluation representations of $A_{2n-1}^{(1)}$
==============================================
Let $\sg$ be the simple Lie algebra $sl_{2n}(\mc)$. Then the affine Lie algebra of type $A_{2n-1}^{(1)}$ has a realization: $$\hat{\sg}=\sg\otimes\mc[t,t^{-1}]\oplus\mc c,$$ with bracket: $$[c,\hat{\sg}]=0,\quad [x\otimes t^m, y\otimes t^k]=[x,y]\otimes t^{m+k}+m\delta_{m,-k}(x,y)c,$$ where $x,y\in\sg, m,k\in\mz$ and $(,)$ is a non-degenerate invariant form on $\sg$, which is a scalar of the Killing form.
The following is the Dynkin diagram of $\sg$:
Let $\Pi=\{\alpha_1,\cdots, \alpha_{2n-1}\}$ be the associated prime root system, and $\dot\Delta$ be its root system. Set $\{E_\alpha, H_i|\alpha\in\dot\Delta, 1\leq i\leq 2n-1\}$ be the Chevalley basis of $\sg$. Then elements $E_{\pm\alpha_i}\otimes 1(1\leq i\leq 2n-1)$ and $E_{\pm\alpha^\flat}\otimes t^\mp$ are Chevalley generators of $\hat{\sg}$, where $\alpha^\flat=\alpha_1+\cdots+\alpha_{2n-1}$.
Set &e\_1=E\_[\_1]{}1-E\_[-\_[n+1]{}]{}1,f\_1=E\_[-\_1]{}1-E\_[\_[n+1]{}]{}1, h\_1=H\_11-H\_[n+1]{}1,\
&e\_2=E\_[\_2]{}1-E\_[-\_[n+2]{}]{}1,f\_2=E\_[-\_2]{}1-E\_[\_[n+2]{}]{}1, h\_2=H\_21-H\_[n+2]{}1,\
&\
&e\_n=E\_[\_n]{}1+E\_[\^]{}t\^[-1]{}, f\_n=E\_[-\_n]{}1+E\_[-\^]{}t,h\_n=H\_n1+(H\_1++H\_[2n-1]{})1-c, and let $\hat{\sg}_{fp}$ be the subalgebra generated by $\{\hat e_i,\hat f_i|1\leq i\leq n\}$. $\hat{\sg}_{fp}$ is called a fixed point subalgebra of $\hat{\sg}$.
\[L4.1\] There exists an algebra homomorphism $\phi: \cu(\gim(M_n))\rightarrow \cu(\hat{\sg}_{fp})$ via the action on generators: (e\_i)=e\_i,(f\_i)=f\_i,1in. More over, it induces a Lie algebra homomorphism (also denoted by $\phi$) from $\gim(M_n)$ to $\hat{\sg}_{fp}$.
This is a consequence of the following equations: \[e\_[i]{}, f\_[i]{}\]=h\_i, &&i=1,,n,\
[\[h\_i, e\_[j]{}\]]{}=m\_[i,j]{}e\_[j]{}, \[h\_i, f\_[j]{}\]=-m\_[i,j]{}f\_[j]{}, &&1i,jn,\
[\[e\_i,f\_j,\]]{}=0,&&i=j, {i,j}={1,n},\
[\[e\_i, \[e\_i, e\_[j]{}\]\]]{}=0, [\[f\_i, \[f\_i, f\_[j]{}\]\]]{}=0,&& |i-j|=1,\
[\[e\_i, e\_[j]{}\]]{}=0, [\[f\_i, f\_[j]{}\]]{}=0,&& |i-j|>1,\
[\[e\_i,\[e\_i,f\_j,\]\]]{}=\[f\_i,\[f\_i,e\_j,\]\]=0,&& {i,j}={1,n}.
[**Remark 2.**]{} [*In fact, the map $\phi$ is an isomorphism (see [@Br]). More over, if we set e\_n=E\_[\_n]{}1-E\_[\^]{}t\^[-1]{}, f\_n=E\_[-\_n]{}1-E\_[-\^]{}t, then Proposition 4.1 still holds. However, these two maps are equivalent under the automorphism of $\hat{sl}_{2n}$ induced by $x\otimes t^m\mapsto x\otimes (-t)^m$.* ]{}
Let $a\in\mc^\times$, an irreducible evaluation $\hat{\sg}$-module $V_{\lambda,a}$ is an irreducible highest weight $\sg$-module with highest weight $\lambda$, and the action is defined by: (xt\^m)v=a\^mxv, &cv=0.for all $x\in\sg, m\in\mz$ and $v\in V_{\lambda,a}$.
For any $V_{\lambda,a}$, the map $\phi$ induces a representation for $\gim({M_n})$. Particularly, for any dominate weight $\lambda$, $V_{\lambda,a}$ is finite-dimensional.
Let $V_{\lambda,a}$ be the $2n$-dimensional irreducible module, i.e., the natural representation of $\sg$. Particularly, we may assume that $\sg=sl_{2n}$. The following proposition is an important tool of the proof for our main results.
\[P4.2\] The following map induces a representation for $\gim(M_n)$: &\_a: &(M\_n)sl\_[2n]{}\
&&\_a(e\_1)=E\_[1,2]{}-E\_[n+2,n+1]{},\_a(f\_1)=E\_[2,1]{}-E\_[n+1,n+2]{}\
&&\_a(e\_2)=E\_[2,3]{}-E\_[n+3,n+2]{},\_a(f\_2)=E\_[3,2]{}-E\_[n+2,n+3]{}\
&&\
&&\_a(e\_[n-1]{})=E\_[n-1,n]{}-E\_[2n,2n-1]{},\_a(f\_[n-1]{})=E\_[n,n-1]{}-E\_[2n-1,2n]{}\
&&\_a(e\_n)=E\_[n,n+1]{}+a\^[-1]{}E\_[1,2n]{},\_a(f\_[n]{})=E\_[n+1,n]{}+aE\_[2n,1]{}.Particularly, if $a\not=\pm1$, the image of $\psi_a$ is $sl_{2n}$, if $a=1$, the image of $\psi_a$ is $sp_{2n}$, and if $a=-1$, then the image of $\psi_a$ is $so_{2n}$.
[**Remark 3.**]{} [*In case of assuming that e\_n=E\_[\_n]{}1-E\_[\^]{}t\^[-1]{}, f\_n=E\_[-\_n]{}1-E\_[-\^]{}t, Proposition 4.2 still holds after exchanging the positions of results when $a=1$ and $a=-1$, respectively.* ]{}
Set $x_i=\psi_a(e_i), y_i=\psi_a(f_i)$ for $1\leq i\leq n$, and $h_i=E_{i,i}-E_{i+1,i+1}-E_{i+n,i+n}+E_{i+1+n,i+1+n}$ for $1\leq i\leq n-1$. Then it is easy to know that the elements $x_i,y_i (1\leq i\leq n-1)$ generate the Lie algebra $$S=\left\{X=\left[\begin{array}{cc}A&0\\0&-A^t\end{array}\right]\Big| A\in sl_n\right\},$$ which is isomorphic to $sl_n$.
More over, &=&E\_[n-1, n+1]{}+a\^[-1]{}E\_[1, 2n-1]{},\
[\[x\_[n-2]{},\[x\_[n-1]{},x\_n\]\]]{}&=&E\_[n-2, n+1]{}+a\^[-1]{}E\_[1, 2n-2]{},\
[\[x\_[n-3]{},\[x\_[n-2]{},\[x\_[n-1]{},x\_n\]\]\]]{}&=&E\_[n-3, n+1]{}+a\^[-1]{}E\_[1, 2n-3]{},\
&&\
[\[x\_2,\[x\_[n-1]{},x\_n\]\]]{}&=&E\_[2, n+1]{}+a\^[-1]{}E\_[1, n+2]{},\
[\[x\_1,\[x\_2,\[x\_[n-1]{},x\_n\]\]\]]{}&=&(1+a\^[-1]{})E\_[1,n+1]{},&=&E\_[n+1, n-1]{}+aE\_[2n-1,1]{},\
[\[\[y\_[n]{},y\_[n-1]{}\],y\_[n-2]{}\]]{}&=&E\_[n+1, n-2]{}+aE\_[2n-2,1]{},\
[\[\[\[y\_[n]{},y\_[n-1]{}\],y\_[n-2]{}\], y\_[n-3]{}\]]{}&=&E\_[n+1, n-3]{}+aE\_[2n-3,1]{},\
&&\
[\[,y\_2\]]{}&=&E\_[n+1, 2]{}+aE\_[n+2,1]{},\
[\[\[,y\_2\],y\_1\]]{}&=&(1+a)E\_[n+1,1]{},then we divide the proof into three cases according to the value of $a$.
\(1) [*Case 1.*]{}$a\not=\pm1$.
By equations &=&(1+a\^[-1]{})E\_[1,n+1]{},\
[\[\[,y\_2\],y\_1\]]{}&=&(1+a)E\_[n+1,1]{},we have $E_{1,n+1}, E_{n+1,1}\in Im(\psi_a)$, and hence, for every $1\leq i\leq n-1$, 2E\_[i+1,n+i+1]{}=\[y\_i,\[y\_i, E\_[i,n+i]{}\]\],\
2E\_[n+i+1,i+1]{}=\[x\_i,\[x\_i, E\_[n+i,i]{}\]\],thus it holds that $$E_{i,n+i}, E_{n+i,i}\in Im(\psi_a), \quad \forall 1\leq i\leq n.$$
Set \_k=E\_[n+k,k]{}&&[**y**]{}\_k=E\_[k,n+k]{},then $$[x_{n}, {\bf x}_n]=E_{2n,n+1}-a^{-1}E_{1,n},\; [y_{n}, {\bf y}_n]=E_{n+1,2n}-aE_{n,1}$$ this implies that $E_{2n, n+1}, E_{n+1,2n}\in Im(\psi_a)$, and hence, $E_{1,n}, E_{n,1}\in Im(\psi_a)$.
Further, the iteration formula =E\_[i+1,n]{}, &&\[E\_[n,i]{}, x\_i\]=E\_[n,i+1]{}, implies that $E_{n,i}, E_{i,n}\in Im(\psi_a)$ for all $i\leq n$, and thus $E_{n+i,2n}, E_{2n,n+i}\in Im(\psi_a)$ for all $i\leq n$.
We also have iteration formula =E\_[i,n-1]{}, \[x\_[n-1]{}, E\_[n,i]{}\]=E\_[n-1,i]{},\
[\[E\_[i,k+1]{}, y\_[k]{}\]]{}=E\_[i,k]{}, \[x\_[k]{}, E\_[k+1,i]{}\]=E\_[k,i]{}, i<k, and this implies that $E_{i,j}, E_{j,i}\in Im(\psi_a)$ for all $i<j\leq n$. Finally, we infer that $E_{i,j}, E_{i+n,j+n}\in Im(\psi_a)$ for all $1\leq i,j\leq n$ and $i\not=j$. Notice that $$[[E_{2,n}, E_{n,2}], x_n]=-E_{n,n+1},\; [[E_{2,n}, E_{n,2}], y_n]=E_{n+1,n},$$ so we have $E_{i,i+1}, E_{i+1,i}\in Im(\psi_a)$ for all $1\leq i\leq 2n-1$, this is a set of Chevalley generators of $sl_{2n}$, and then $Im(\psi_a)=sl_{2n}$.
\(2) [*Case 2.*]{}$a=1$.
It is easy to check that =\[[**y**]{}\_1, x\_i\]=0&&i=1,,n-1,\
[\[[**x**]{}\_1, x\_i\]]{}=\[[**y**]{}\_1, y\_i\]=0&&i=2,,n-1,\
[\[\[[**x**]{}\_1, x\_1\],x\_1\]]{}=2E\_[n+2,2]{}=2[**x**]{}\_2,&& \[\[[**y**]{}\_1, y\_1\],y\_1\]=2E\_[2,n+2]{}=2[**y**]{}\_2,\
[\[[**x**]{}\_2, x\_1\]]{}=\[[**y**]{}\_2, y\_1\]=0, &&[\[[**x**]{}\_1,\[[**x**]{}\_1, x\_1\]\]]{}=\[[**y**]{}\_1, \[[**y**]{}\_1, y\_1\]\]=0, then the Lie algebra generated by $S$ and ${\bf x}_1, {\bf y}_1$ is isomorphic to $sp_{2n}$. In particular, $\{x_i, y_i, {\bf x}_1, {\bf y}_1|1\leq i\leq n-1\}$ is a set of Chevalley generators. The associated Dynkin diagram is
Notice that $x_i, y_i\in sp_{2n}$ for all $1\leq i\leq n$, we have S, [**x**]{}\_1, [**y**]{}\_1Im(\_a)=x\_i, y\_i |1insp\_[2n]{},the dimension relation $\dim\langle S, {\bf x}_1, {\bf y}_1\rangle=\dim sp_{2n}$ implies that $Im(\psi_a)=sp_{2n}$.
\(3) [*Case 3.*]{}$a=-1$. Then $x_n, y_n$ have the form $\left[\begin{array}{cc}0&B\\0&0\end{array}\right]$ and $\left[\begin{array}{cc}0&0\\B&0\end{array}\right]$, respectively, and where $B=-B^t\in gl_n$ is anti-symmetric.
Set x\^&=&\[,y\_2\]=E\_[n+1,2]{}-E\_[n+2,1]{},\
y\^&=&\[x\_2,\]=E\_[2, n+1]{}-E\_[1, n+2]{},we have =\[[**y**]{}\^, x\_i\]=0,&&i=1,,n-1,\
[\[[**x**]{}\^, x\_i\]]{}=\[[**y**]{}\^, y\_i\]=0,&&i=1,3,,n-1,and the set $\{{\bf x}^\sharp, {\bf y}^\sharp, x_2, y_2\}$ generates a simple Lie algebra of type $A_2$.
Then the Lie algebra generated by $S$ and ${\bf x}^\sharp, {\bf y}^\sharp$ is isomorphic to $so_{2n}$. In particular, $\{x_i, y_i, {\bf x}^\sharp, {\bf y}^\sharp|1\leq i\leq n-1\}$ is a set of Chevalley generators. Notice that $x_i, y_i\in so_{2n}$ for all $1\leq i\leq n$, we have S, [**x**]{}\^, [**y**]{}\^Im(\_a)=x\_i, y\_i |1inso\_[2n]{},the dimension relation $\dim\langle S, {\bf x}^\sharp, {\bf y}^\sharp\rangle=\dim so_{2n}$ implies that $Im(\psi_a)=so_{2n}$.
Proof for Theorem 3.1
=====================
Let $\ca=\oplus_{k=1}^K\sg_i=\sg^{\oplus n}$ be the direct sum of $K$ copies of $\sg=sl_{2n}$. Fix a $K$-tuple $\underline{a}=(a_1,\cdots,a_K)\in(\mc^\times)^K$ such that $a_k\not=a_j^{\pm1}$ for all $k\not=j$. Define the map \_=\_[k=1]{}\^K\_[a\_k]{}:&&(M\_n),where $\psi_{a_k}:\gim(M_n)\rightarrow\sg_k$ is an analogue of the evaluation map defined in Proposition \[P4.2\]..
[*Theorem 3.1 can be proved by the following Lemmas \[L5.1\]-\[L5.4\].* ]{}
\[L5.1\] If $a_k\not=\pm1$ for every $k$, then $\psi_{\underline{a}}$ is an epimorphism.
For convenience, we let $\psi(x)^{[k]}$ denote the image of $x$ under $\psi_{a_k}$. By the definition of $\psi$, we have that &&\_[a\_k]{}(e\_1)=(E\_[1,2]{}-E\_[n+2,n+1]{})\^[\[k\]]{},\_[a\_k]{}(f\_1)=(E\_[2,1]{}-E\_[n+1,n+2]{})\^[\[k\]]{}\
&&\_[a\_k]{}(e\_2)=(E\_[2,3]{}-E\_[n+3,n+2]{})\^[\[k\]]{},\_[a\_k]{}(f\_2)=(E\_[3,2]{}-E\_[n+2,n+3]{})\^[\[k\]]{}\
&&\
&&\_[a\_k]{}(e\_[n-1]{})=(E\_[n-1,n]{}-E\_[2n,2n-1]{})\^[\[k\]]{},\_[a\_k]{}(f\_[n-1]{})=(E\_[n,n-1]{}-E\_[2n-1,2n]{})\^[\[k\]]{}\
&&\_[a\_k]{}(e\_n)=(E\_[n,n+1]{}+a\_k\^[-1]{}E\_[1,2n]{})\^[\[k\]]{},\_[a\_k]{}(f\_[n]{})=(E\_[n+1,n]{}+a\_kE\_[2n,1]{})\^[\[k\]]{}.Set x\_i\^[\[k\]]{}=\_[a\_k]{}(e\_i),&& y\_i\^[\[k\]]{}=\_[a\_k]{}(f\_i).Then &=&(E\_[n-1, n+1]{}+a\_k\^[-1]{}E\_[1, 2n-1]{})\^[\[k\]]{},\
[\[x\^[\[k\]]{}\_[n-2]{},\[x\^[\[k\]]{}\_[n-1]{},x\^[\[k\]]{}\_n\]\]]{}&=&(E\_[n-2, n+1]{}+a\_k\^[-1]{}E\_[1, 2n-2]{})\^[\[k\]]{},\
&&\
[\[x\_1\^[\[k\]]{},\[x\_2\^[\[k\]]{},\[x\_[n-1]{}\^[\[k\]]{},x\_n\^[\[k\]]{}\]\]\]]{}&=&(1+a\_k\^[-1]{})(E\_[1,n+1]{})\^[\[k\]]{},&=&(E\_[n+1, n-1]{}+a\_k E\_[2n-1,1]{})\^[\[k\]]{},\
[\[\[y\_[n]{}\^[\[k\]]{},y\_[n-1]{}\^[\[k\]]{}\],y\_[n-2]{}\^[\[k\]]{}\]\]]{}&=&(E\_[n+1, n-2]{}+a\_kE\_[2n-2,1]{})\^[\[k\]]{},\
&&\
[\[\[,y\_2\^[\[k\]]{}\],y\_1\^[\[k\]]{}\]]{}&=&(1+a\_k)(E\_[n+1,1]{})\^[\[k\]]{},we infer that H\_[1,n+1]{}&:=&\
&=&\_[k=1]{}\^K\[(1+a\_k\^[-1]{})(E\_[1,1+n]{})\^[\[k\]]{},(1+a\_k)(E\_[n+1,1]{})\^[\[k\]]{}\]\
&=&\_[k=1]{}\^K(2+a\_k+a\_k\^[-1]{})(E\_[1,1]{}-E\_[n+1,n+1]{})\^[\[k\]]{}. Consider its action on $\psi_{\underline{a}}(e_1)=\sum_{k=1}^Kx_1^{[k]}$ and $\psi_{\underline{a}}(f_1)=\sum_{k=1}^Ky_1^{[k]}$, we have that (H\_[1,n]{})(\_(e\_1))&=&\_[k=1]{}\^K(2+a\_k+a\_k\^[-1]{})x\_1\^[\[k\]]{},\
(-H\_[1,n]{})(\_(f\_1))&=&\_[k=1]{}\^K(2+a\_k+a\_k\^[-1]{})y\_1\^[\[k\]]{},\
(H\_[1,n]{})\^2(\_(e\_1))&=&\_[k=1]{}\^K(2+a\_k+a\_k\^[-1]{})\^2x\_1\^[\[k\]]{},\
(-H\_[1,n]{})\^2(\_(f\_1))&=&\_[k=1]{}\^K(2+a\_k+a\_k\^[-1]{})\^2y\_1\^[\[k\]]{},\
&&\
(H\_[1,n]{})\^K(\_(e\_1))&=&\_[k=1]{}\^K(2+a\_k+a\_k\^[-1]{})\^Kx\_1\^[\[k\]]{},\
(-H\_[1,n]{})\^K(\_(f\_1))&=&\_[k=1]{}\^K(2+a\_k+a\_k\^[-1]{})\^Ky\_1\^[\[k\]]{}.
If $2+a_k+a_k^{-1}=2+a_j+a_j^{-1}$, then $(a_ka_j-1)(a_k-a_j)=0$. By our assumption, every $2+a_k+a_k^{-1}$ is non-zero and $2+a_k+a_k^{-1}\not=2+a_j+a_j^{-1}$ for all $k\not=j$. By the invertibility of Vandermonde matrix, we infer that x\_1\^[\[1\]]{}, y\_1\^[\[1\]]{}, x\_1\^[\[2\]]{},y\_1\^[\[2\]]{},x\_1\^[\[K\]]{},y\_1\^[\[K\]]{}Im(\_).
Let $H_{1,2}^{[k]}=[x_1^{[k]},y_1^{[k]}]=(E_{1,1}-E_{2,2})^{[k]}$, and consider its action on $\psi_{\underline{a}}(e_2), \psi_{\underline{a}}(f_2)$, we infer that x\_2\^[\[1\]]{}, y\_2\^[\[1\]]{}, x\_2\^[\[2\]]{},y\_2\^[\[2\]]{},x\_2\^[\[K\]]{},y\_2\^[\[K\]]{}Im(\_).Repeat this process, similar argument implies that x\_i\^[\[k\]]{},y\_i\^[\[k\]]{}Im(\_)for every $k$. By Proposition \[P4.2\], we can infer that $\sg_k\in Im(\psi_{\underline{a}})$ for all $k$, and thus the map $\psi_{\underline{a}}$ is surjective.
\[L5.2\] If $a_1=1$ and $a_k\not=-1$ for every $k\geq 2$, then $Im(\psi_{\underline{a}})$ is a direct sum of one copy of $sp_{2n}$ and $K-1$ copies of $sl_{2n}$.
We may repeat the proof for Lemma \[L5.1\] until we obtain the result x\_i\^[\[k\]]{},y\_i\^[\[k\]]{}Im(\_), 1kK, 1in,then $Im(\psi_{\underline{a}})=\bigoplus_{k=1}^K Im(\psi_{a_k})$, by Proposition \[P4.2\], $Im(\psi_{a_1})$ is isomorphic to $sp_{2n}$, and $Im(\psi_{a_k})$ is isomorphic to $sl_{2n}$ for $k\geq 2$.
\[L5.3\] If $a_1=-1, a_2=1$, then $Im(\psi_{\underline{a}})$ is a direct sum of one copy of $so_{2n}$, one copy of $sp_{2n}$ and $K-2$ copies of $sl_{2n}$.
We may repeat the proof for Lemma \[L5.1\] until we obtain $H_{1,n+1}$. Notice that, in this case H\_[1,n+1]{}&:=&\_[k=2]{}\^K(2+a\_k+a\_k\^[-1]{})(E\_[1,1]{}-E\_[n+1,n+1]{})\^[\[k\]]{}.Then go on with the process in Lemma \[L5.2\], we get that x\_i\^[\[k\]]{},y\_i\^[\[k\]]{}Im(\_), 2kK, 1in,then x\_i\^[\[1\]]{}=\_(e\_i)-(\_[k=2]{}\^Kx\_i\^[\[k\]]{})Im(\_),&&y\_i\^[\[1\]]{}=\_(f\_i)-(\_[k=2]{}\^Ky\_i\^[\[k\]]{})Im(\_),it also holds that $Im(\psi_{\underline{a}})=\bigoplus_{k=1}^K Im(\psi_{a_k})$. By Proposition \[P4.2\], $Im(\psi_{a_1})$ is isomorphic to $so_{2n}$, $Im(\psi_{a_2})$ is isomorphic to $sp_{2n}$, and $Im(\psi_{a_k})$ is isomorphic to $sl_{2n}$ for $k\geq 3$.
\[L5.4\] If $a_1=-1$ and $a_k\not=1$ for $k\geq 2$, then $Im(\psi_{\underline{a}})$ is a direct sum of one copy of $so_{2n}$ and $K-1$ copies of $sl_{2n}$.
The proof is very similar to that for Lemma \[L5.3\].
Proof for Theorem 3.2
=====================
If $G(t)$ is a polynomial with non-zero constant term, we may define the quotient algebra /G(t),and denoted by $\hat{\sg}/G(t)$. Note that, we may regard the central element $c$ as $0$ in the quotient algebra $\hat{\sg}/G(t)$ for all non-trivial polynomial $G(t)$.
If $G(t)|F(t)$, then $\hat{\sg}/G(t)$ is a quotient of $\hat{\sg}/F(t)$. Particularly, /G(t).
This result follows from the homomorphism fundamental theorem.
If $(x-a)^r|G(t)$ for some $r>1$, then $\hat{\sg}/G(t)$ has a nilpotent ideal $$I=\sg\otimes\frac{G(t)}{(t-a)}\mc[t,t^{-1}].$$
Obviously, for any $x\in\hat{\sg}$ and $y\in I$, we have $[x,y]\in I$. Suppose that $r^*\geq r$ is the multiple of prime factor $t-a$, then for any $u\otimes f(t), v\otimes g(t)\in I$, we have $(t-a)^2|G(t)$, and $$G(t)\big|\frac{G(t)^2}{(t-a)^2},\quad \frac{G(t)^2}{(t-a)^2}\big|f(t)g(t),$$and thus $[I,I]=0$.
If $G(t)$ has degree $K$, then it is well-known that $\hat{\sg}/G(t)$ is the direct sum of $K$ copies of $\sg$ if $G(t)$ has $K$ many different factors $t-a_i$, where $a_i\not=0 (1\leq i\leq K)$.
Let $G(t)=\prod_{i=1}^k(t-a_i)$, then there exists $c_i\not=0$, such that \_[i=1]{}\^k c\_i=1,then /G(t)=\_[i=1]{}\^k/G(t)\_[i=1]{}\^k.
Let $\phi$ be as defined in Section 4. Then &=&E\_[\_1++\_n]{}(1+t\^[-1]{}),\
[\[,\_1\]]{}&=&E\_[-\_1--\_[n-1]{}]{}1-E\_[\_[n+1]{}++\_[2n-1]{}]{}1,\
[\[\[,\_1\],E\_[\_1++\_n]{}(1+t\^[-1]{})]{}\]&=&(E\_[\_n]{}-E\_[\_[1]{}++\_[2n-1]{}]{})(1+t\^[-1]{}),\
[\[(E\_[\_n]{}-E\_[\_[1]{}++\_[2n-1]{}]{})(1+t\^[-1]{}), \_n\]]{}&=&H\_n(1+t\^[-1]{})+(H\_1+H\_[2n-1]{})(1+t)-c, this implies that :=H\_nt\^[-1]{}+(H\_1++H\_[2n-1]{})t((M\_n)),then &=&H\_nt\^[-m]{}+(H\_1++H\_[2n-1]{})t\^m,\
[\[()\^m(\_1),\_1\]]{}&=&H\_1t\^[m]{}-H\_[n+1]{}t\^[-m]{},for all $m>0$. Similarly, we infer that H\_it\^[-m]{}-H\_[n+i]{}t\^m((M\_n)),for all $1\leq i\leq n$ and $m\in\mz\setminus\{0\}$, where $H_{2n}=-H_1-\cdots-H_{2n-1}$.
Moreover, (H\_it\^[-m]{}-H\_[n+i]{}t\^m)(H\_it\^[m]{}-H\_[n+i]{}t\^[-m]{})&=&(H\_iH\_[n+i]{})(t\^mt\^[-m]{}).
Suppose that $I$ is an ideal of $\gim(M_n)$ such that the quotient algebra $\gim(M_n)/I$ is finite-dimensional and semi-simple. Since $\phi$ is an isomorphism, we have that $\phi(I)$ is an ideal of $\hat{\sg}_{fp}=\phi(\gim(M_n))\cong\gim(M_n)$ and $\hat{\sg}_{fp}/\phi(I)\cong\gim(M_n)/I$. Note that every polynomial in variable $t+t^{-1}$ is a linear combination of $t^{m}+t^{-m} (m\geq 0)$. Then there must hold that (H\_i-H\_[n+i]{})(t)(I),&&1in,where $\theta(t)=\eta(t+t^{-1})$ for some polynomial $\eta$. Otherwise, the elements {H(t\^j+t\^[-j]{})| j\_[0]{}}are linearly independent in $\hat{\sg}_{fp}/\phi(I)$ for some $H\in span\{H_i-H_{n+i}|1\leq i\leq n\}$, which contradicts to the assumption of the finite-dimension.
Actually, $\hat{h}_i=H_i-H_{n+i}$ for all $1\leq i\leq n$. Let $\sh_{fp}=span\{\hat{h}_i|1\leq i\leq n\}$. Then $\hat{\sg}_{fp}$ has a root decomposition \_[fp]{}=(\_[fp]{})\_0+\_(\_[fp]{})\_,where $(\hat{\sg}_{fp})_0=\{x\in \hat{\sg}_{fp}| [\sh_{fp},x]=0\}$ and $(\hat{\sg}_{fp})_\alpha=\{x\in \hat{\sg}_{fp}|[h,x]=\alpha(h)x,\forall h\in\sh_{fp}\}$.
By Berman’s result ([@Br], Proposition 1.12), it holds that $\hat{\sg}=\hat{\sg}_{fp}\oplus M$, where $M$ is a $\hat{\sg}_{fp}$-module and $[M,M]\subseteq \hat{\sg}_{fp}$. If $x=\sum_{j=1}^d x_j\otimes p_j(t)\theta(t)\in (\hat{\sg}_{fp})_\alpha, \alpha\not=0$, then there exists $H\in\sh_{fp}$ such that $\alpha(H)\not=0$ and x=\_[j=1]{}\^d \[H(t), x\_jp\_j(t)\](I).Otherwise, we infer that $\sum_{j=1}^d x_j\otimes p_j(t)\not\in\hat{\sg}_{fp}$ and thus $x\not\in \hat{\sg}_{fp}$, which is a contradiction.
Moreover, $(\hat{\sg}_{fp})_0\subseteq \sh\otimes\mc[t,t^{-1}]\oplus\mc c$ and (c)\_[fp]{}&=&\_[m=0]{}\^\_[i=1]{}\^n((H\_i-H\_[n+i]{})(t\^m+t\^[-m]{})-\_[i,n]{}c),then we have the following result:
We let $J$ be the ideal of $\hat{\sg}$ generated by $\{H_i\otimes\theta(t) |1\leq i\leq 2n-1\}$, then ((M\_n))J(I),and naturally there exists an epimorphism (M\_n)/I.
For convenience, we let $\theta^*(t)=t^r\eta(t+t^{-1})$, where $r=\deg\eta$, then $\theta^*$ is a polynomial with $\theta^*(0)\not=0$ and it holds that /J=/\^\*(t).Moreover, $\theta^*(t)$ and $\eta(t+t^{-1})$ have same roots.
Suppose that $\hat{\sg}/\theta^*(t)$ has a Levi decomposition /\^\*(t)=S+U,where $S$ is semi-simple and $U$ is the solvable radical. Let ${\bf p}: S+U\rightarrow (S+U)/U$ be the canonical map. If ${\bf i}:\gim(M_n)/I\rightarrow\hat{\sg}/\theta^*(t)\subseteq S$ is the injective map, then ${\bf p}\circ{\bf i}$ is injective. Hence we may assume that $$\theta^*(t)=\prod_{i=1}^K(t-a_i),$$ where $a_i\not=a_j$ if $i\not=j$.
Out of questions, $\gim(M_n)/I$ is a subalgebra of $\hat{\sg}/\theta^*(t)$ through isomorphism $\phi$.
We may assume that $c_i, d_i\in\mc^\times$ such that \_[i=1]{}\^Kc\_i=1, d\_i ( t-a\_i),we also assume that $\hat{\sg}/\theta^*(t)=\bigoplus_{i=1}^K \sg_i$, and $\psi^\clubsuit$ is the canonical map from $\hat{\sg}$ to $\hat{\sg}/\theta^*(t)$.
Let $$\psi_{a_i}:\hat{\sg}\rightarrow\hat{\sg}_i=\frac{\sg\otimes \theta^*(t)/(t-a_i)\mc[t,t^{-1}]}{(t-a_i)}\cong\hat{\sg}/(t-a_i)$$ be the evaluation map, where $\psi_{a_i}(x\otimes t^m\frac{\theta^*(t)}{(t-a_i)})=d_ia_i^mx^{[i]}$.
\[L6.4\] For any $x\otimes t^m$, we have $$\psi^\clubsuit(x\otimes t^m)=\sum_{i=1}^Kc_id_i\psi_{a_i}(x\otimes t^m)=\sum_{i=1}^Kc_id_ia_i^mx^{[i]}.$$ Particularly, $c_id_i=1$ for any $1\leq i\leq K$, then $\psi^\clubsuit=\psi_{\underline{a}}$, where $\underline{a}=(a_1,\cdots,a_K)$.
Since $\sum_{i=1}^Kc_i\frac{\theta^*(t)}{t-a_i}=1$, evaluate $t=a_i$ in both sides, we have $c_id_i=1$. For any $x\otimes t^m$, we have \^(xt\^m)&=&\_[i=1]{}\^Kc\_ixt\^m\
&=&\_[i=1]{}\^Kc\_id\_i a\_i\^mx\^[\[i\]]{}=\_[i=1]{}\^K a\_i\^mx\^[\[i\]]{}.
\[L6.5\] Let $a\not=0, \pm1$, then associated to the map $$\psi_{a}\oplus\psi_{a^{-1}}:\gim(M_n)/I\rightarrow \hat{\sg}/(t-a)\oplus \hat{\sg}/(t-a^{-1}),$$ we have that $$Im(\psi_{a}\oplus\psi_{a^{-1}})\cong Im(\psi_a).$$
As we know that : &&\
&&xt\^mxt\^[-m]{},is an isomorphism. Moreover, $\psi_a\circ \sigma=\psi_{a^{-1}}$, hence we have that \_[a]{}\_[a\^[-1]{}]{}=\_a(1),notice that $1\oplus\sigma$ is diagonally injective, then the image of $\psi_{a}+\psi_{a^{-1}}$ is isomorphic to $Im(\psi_a)$.
[**Proof for Theorem 2.3:**]{}Undoubtedly, since $\eta$ is a polynomial in variable $t+t^{-1}$ and $\theta^*(0)\not=0$, we have that $(t-a)|\theta^*(t)$ if and only if $(t-a^{-1})|\theta^*(t)$. By Lemma \[L6.4\], $\psi^\clubsuit=\psi_{\underline{a}}$. Here, $a_k\in {\underline{a}}$ if and only if $a_k^{-1}\in {\underline{a}}$. However, the approach of proof for Lemma \[L5.1\] still play a major role in this case. Follow the process of the proof for Lemma \[L5.1\] and Lemma \[L5.3\], we can successfully reach that: x\_i\^[\[k\]]{}, y\_i\^[\[k\]]{}Im(\_),&& a\_k=1, 1in\
x\_i\^[\[k\]]{}+x\_i\^[\[j\]]{}, y\_i\^[\[k\]]{}+y\_i\^[\[j\]]{}Im(\_),&& a\_k=1, a\_ka\_j=1, 1in.
By Proposition \[P4.2\] and Lemma \[L6.5\], the subalgebra generated by $x_i^{[k]}, y_i^{[k]} (a_k=\pm1, 1\leq i\leq n)$ is one of $so_{2n}, sp_{2n}$, and the subalgebra generated by $x_i^{[k]}+x_i^{[j]}, y_i^{[k]}+y_i^{[j]} (a_ka_j=1, 1\leq i\leq n)$ is $sl_{2n}$. Since $t-1$ or $t+1$ appears in $\theta(t)$ at most one time, each of $so_{2n}$ and $sp_{2n}$ appears at most one time. Then $\gim(M_n)/I$ is of type $M(n, {\bf a}, {\bf c}, {\bf d})$.
The proof is completed.
.5cm
[**Acknowledgements.** ]{}
The first author is partially supported by NSERC of Canada. The second author is supported by the NNSF of China (Grant No. 11001110, 11271131) and [*Jiangsu Government Scholarship for Overseas Studies*]{}.
.3cm
[ABC]{} B. Allison, G. Benkart and Y. Gao, [*Lie algebras graded by the root system $BC_r$, $r\geq2$*]{}, Mem. A.M.S.(751)158(2002) S. Bhargava and Y. Gao, [*Realizations of BCr-graded intersection matrix algebras with grading subalgebras of type $B_r, r\geq3$*]{}, Pacific Journal of Mathematics, Vol. 263 (2013), No. 2, 257-281. G. Benkart and E. Zelmanov, [*Lie algebras graded by finite root systems and intersection matrix algebras*]{}, Invent. Math. 126 (1996), 1-45. S. Berman, [*On generators and relations for certain involutory subalgebras of Kac-Moody Lie Algebras*]{}, Comm. Alg. 17 (1989), 3165-3185. S. Berman, E. Jurisich and S. Tan, [*Beyond Borcherds Lie Algebras and Inside*]{}, Trans. AMS 353 (2001), 1183-1219. S. Berman and R.V. Moody, [*Lie algebras graded by finite root systems and the intersection matrix algebras of Slowdowy*]{}, Invent. Math. 108 (1992), 323-347. R. Carter, [*Lie Algebras of Finite and Affine Type*]{}, Cambridge Univ. Press, Cambridge, 2005. V. Chari, A. A. Moura, [*Spectral characters of finite-dimensional representations of affine algebras*]{}, Journal of Algebra 279 (2004) 820-839. S. Eswara Rao, R.V. Moody and T. Yokonuma, [*Lie algebras and Weyl groups arising from vertex operator representations*]{}, Nova J. of Algebra and Geometry 1 (1992), 15-57. Y. Gao, [*Involutive Lie algebras graded by finite root systems and compact forms of IM algebras*]{}, Math. Zeitschrift 223 (1996), 651- 672. O. Gabber and V.G. Kac, [*On Defining Relations of Certain Infinite-Dimensional Lie Algebras*]{}, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 185-189. J.E. Humphreys, [*Introduction to Lie Algebras and Representation Theory*]{}, Springer, New York, 1972. N. Jacobson, [*Lie Algebras*]{}, Interscience, New York, 1962. V.G. Kac, [*Infinite Dimensional Lie Algebras*]{}, 3rd edition, Cambridge Univ. Press, Cambridge, 1990. R. V. Moody,[*Euclidean Lie algebras*]{}, Canad. J. Math. 21 (1969), 1432¨C1454. R. V. Moody,[*Simple quotients of Euclidean Lie algebras*]{}, Canad. J. Math. 22 (1970), 839¨C846. E. Neher, [*Lie algebras graded by 3-graded root systems and Jordan pairs covered by grids*]{}, Amer. J. Math. 118 (1996), no. 2, 439-491. L. Peng, [*Intersection matrix Lie algebras and Ringel-Hall Lie algebras of tilted algebras*]{}, Representations of Algebra Vol. I, II, 98-108, Beijing Normal Univ. Press, Beijing, 2002. P. Slodowy, [*Beyond Kac-Moody algebras and inside*]{}, Can. Math. Soc. Conf. Proc. 5 (1986), 361-371. P. Slodowy, [*Singularitäten, Kac-Moody Lie-Algebren, assoziierte Gruppen und Verallgemeinerungen*]{}, Habilitationsschrift, Universität Bonn, March 1984.
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abstract: 'The jet compositions, central engines, and progenitors of gamma-ray bursts (GRBs) remain open questions in GRB physics. Applying broadband observations, including GRB prompt emission and afterglow properties derived from [*Fermi*]{} and [*Swift*]{} data, as well as from Keck host-galaxy observations, we address these questions for the peculiar, bright GRB 110731A. By using the pair-opacity method, we derive $\Gamma_{0}>190$ during the prompt emission phase. Alternatively, we derive $\Gamma_{0} \approx 580$ and $\Gamma_{0} \approx 154$ by invoking the early-afterglow phase within the homogeneous density and wind cases, respectively. On the other hand, nondetection of a thermal component in the spectra suggests that the prompt emission is likely powered by dissipation of a Poynting-flux-dominated jet leading to synchrotron radiation in an optically thin region. The nondetection of a jet break in the X-ray and optical bands allows us to place a lower limit on the jet opening angle $\theta_j>5.5^{\circ}$. Within a millisecond magnetar central engine scenario, we derive the period $P_{0}$ and polar magnetic field strength $B_{\rm p}$, which have extreme (but still allowed) values. The moderately short observed duration (7.3s) and relatively large redshift ($z=2.83$) place the burst as a “rest-frame short” GRB, so the progenitor of the burst is subject to debate. Its relatively large $f_{{\rm eff}, z}$ parameter (ratio of the 1s peak flux of a pseudo-GRB and the background flux) and a large physical offset from a potential host galaxy suggest that the progenitor of GRB 110731A may be a compact-star merger.'
author:
- 'HouJun Lü, XiangGao Wang, RuiJing Lu, Lin Lan, He Gao, EnWei Liang, Melissa L. Graham, WeiKang Zheng, Alexei V. Filippenko, and Bing Zhang'
title: 'The peculiar GRB 110731A: Lorentz factor, jet composition, central engine, and progenitor'
---
Introduction {#sec:intro}
============
Despite decades of investigation, there still exist several open questions in gamma-ray burst (GRB) physics, particularly regarding their progenitors, central engines, and jet compositions (e.g., Zhang 2011; Kumar & Zhang 2015).
Traditionally, GRBs are classified into long/soft and short/hard categories based on their distributions in the $T_{90}$ vs. hardness-ratio plane (Kouveliotou et al. 1993). However, the measurement of $T_{90}$ is energy and instrument dependent (Qin et al. 2013). Observations of the host-galaxy properties and supernova associations of GRBs suggest that the majority of long-duration GRBs originate from core collapse of massive stars (“collapsars”; Woosley 1993), while the majority of short-duration GRBs originate from coalescence of two compact stars (Paczýnski 1986; Eichler et al. 1989). But with only duration information, the physical category of a GRB is sometimes misclassified (e.g., Gehrels et al. 2006; Levesque et al. 2010); multiwavelength data are needed to make correct classifications (Zhang et al. 2009). Correctly identifying compact-star merger systems is of great interest, since they are promising gravitational wave sources to be detected by advanced LIGO/Virgo (e.g., Chu et al. 2016).
Within both the collapsar and compact-star merger models, another interesting question is what central engine launches the relativistic outflow. A widely discussed scenario invokes a hyperaccreting stellar-mass black hole with an accretion rate of 0.1-1$M_{\odot}$s$^{-1}$ (e.g., Popham et al. 1999; Narayan et al. 2001; Chen & Beloborodov 2007; Liu et al. 2008; Kumar et al. 2008; Lei et al. 2013). On the other hand, some GRBs, both long and short, have been discovered to have a plateau emission component in their X-ray afterglows (Zhang et al. 2006; O’Brien et al. 2006; Liang et al. 2007), some of them having an extremely steep drop following the plateau (known as internal plateaus; Troja et al. 2007; Liang et al. 2007; Lyons et al. 2010; Rawlinson et al. 2010, 2013; Lü & Zhang 2014; Lü et al. 2015). Such behavior is consistent with a millisecond magnetar central engine (Usov 1992; Dai & Lu 1998; Zhang & Mészáros 2001; Gao & Fan 2006; Fan & Xu 2006; Metzger et al. 2010). The steep drop at the end of the plateau may be consistent with the collapse of a supramassive neutron star (NS) to a black hole (e.g., Zhang 2014), which has profound implications for the inferred NS equation of state (Fan et al. 2013a; Lasky et al. 2014; Ravi & Lasky 2014; Lü et al. 2015; Gao et al. 2016). The existence of a supramassive NS as a compact-star merger product also enhances electromagnetic signals of gravitational wave sources (Zhang 2013; Gao et al. 2013a; Yu et al. 2013; Metzger & Piro 2014; Fan et al. 2013b), which gives encouraging prospects of confirming the astrophysical nature of gravitational wave sources detected by advanced LIGO/Virgo (e.g., Abbott et al. 2016). One challenging task is to distinguish millisecond magnetars (possibly supramassive) from hyperaccreting black holes based on data.
The next open question is regarding the composition of the relativistic jet launched from the central engine, as well as how energy is dissipated to give rise to prompt emission. Competing models include the fireball internal shock model (Rees & Mészáros 1994), dissipative photosphere models (Thompson 1994; Rees & Mészáros 2005; Pe’er et al. 2006), and the internal-collision-induced magnetic reconnection and turbulence (ICMART) model (Zhang & Yan 2011; Deng et al. 2015). The first two models have the magnetization parameter $\sigma$ much less than unity at the GRB emission site, while the ICMART model has a moderately large $\sigma>1$ at the emission site, with the GRB emission powered by directly dissipating the magnetic energy to radiation. These models have distinct predictions for GRB spectra, light curves, and other properties. Observations can be used to differentiate among them.
Finally, it is well known that most of the broadband afterglow emission is produced from forward and reverse external shocks (Mészáros & Rees 1997; Sari et al. 1998; Kobayashi 2000; Mészáros 2002; Zhang & Mészáros 2004; Gao et al. 2013b), but the properties of the ambient medium as well as the shock microphysics parameters remain poorly constrained (Santana et al. 2014; Wang et al. 2015).
GRB 110731A is a bright GRB jointly detected by the [*Swift*]{} and [*Fermi*]{} satellites. The abundant data collected from the burst make it a good target to address the open questions discussed above (e.g., Ackermann et al. 2013; Lemoine et al. 2013; Fraija 2015; Hascoet et al. 2015). In this paper, we reduce the available high-energy data and present new observations of the host galaxy of the GRB (§2). We use the broadband data to constrain the properties of the GRB, including its bulk Lorentz factor (§3), jet composition (§4), central engine (§5), and progenitor (§6). Our conclusions and discussion are given in §7.
Data Reduction and Analysis
===========================
Data Reduction
--------------
Both the [*Swift*]{} Burst Alert Telescope (BAT) and the [*Fermi*]{} Gamma-ray Burst Monitor (GBM) triggered GRB 110731A. The burst was also detected by the Large Area Telescope (LAT), so high-energy photons above 100MeV were detected with $>10\sigma$ significance (Ackermann et al. 2013). The [*Swift*]{} X-Ray Telescope (XRT) and Ultraviolet/Optical Telescope (UVOT) promptly slewed to the source 56s and 62s after the BAT trigger, respectively. Bright X-ray and optical afterglows were detected with a spectroscopic redshift $z=2.83$ identified (Tanvir et al. 2011).
Prompt Emission
---------------
The [*Fermi*]{} Gamma-ray Space Telescope comprises two science instruments, the GBM (Meegan et al. 2009) and the LAT (Atwood et al. 2009). The GBM has 12 sodium iodide (NaI) detectors covering an energy range from 8keV to 1MeV, and two bismuth germanate (BGO) scintillation detectors sensitive to higher energies between 200keV and 40MeV (Meegan et al. 2009). The signals from each of the 14 GBM detectors have three different types: CTIME, CSPEC, and TTE. The TTE event data files contain individual photons with time and energy tags. The LAT observes the energy of photons from 20MeV to 300GeV (Atwood et al. 2009). The standard LAT analysis is performed with the latest Pass 8 release data ($>100$MeV). However, the LAT Low Energy (LLE) data are produced by increasing the effective area of the LAT at low energy ($\sim30$ MeV) and with very loose event selection. It required only minimal information, such as the existence of a reconstructed direction. Also, it is suitable for studying transient phenomena, such as GRBs. The LLE analysis also has been updated to the latest Pass 8 event reconstruction; for more information, refer to the official [*Fermi*]{} website[^1]. As suggested by the [*Fermi*]{} team,[^2], it is suitable to use standard LAT event data at high energies and LLE data at low energies.
Based on the standard `heasoft` tools (v. 6.19) and the [*Fermi*]{} `ScienceTool` (v10r0p5), the [*PYTHON*]{} source package $gtBurst$[^3] is designed to analyze the GBM and LAT data, as well as the LLE data. A step-by-step guide to the $gtBurst$ can be found on the website[^4]. We downloaded GBM, LLE, and LAT data for GRB 110731A from the public science support center at the [*Fermi*]{} website. Then, we extracted the light curves and performed spectral analysis based on the package [*gtBurst*]{}. By invoking the `heasoft` command `fselect` and the `ScienceTool` command `gtbin`, we extracted light curves with a bin size of 0.064 s. However, for the standard LAT data, we employed an unblinded likelihood analysis method to build the LAT light curve based on [*gtBurst*]{}. By invoking the `Standardcut` function in $gtBurst$, we adopted the photons above 100MeV in a region of interest of $12^\circ$ and excluded the photons with zenith angle $>
100^\circ$ to avoid the contribution of Earth’s limb. Then, we extracted the light curve with a bin size of 0.1 s by using the command `gtbin`. The resulting light curves of the GBM, LAT, and LLE data for GRB 110731A are shown in Figure \[fig:GBMLC\]. The high-energy photons above 500MeV from the LLE and LAT are also overplotted at the bottom of Figure \[fig:GBMLC\].
By invoking “Tasks$\rightarrow$Make spectra for XSPEC” in $gtBurst$, we extracted the source spectra of the GBM and LLE data. The background spectra are extracted from the time intervals before and after the prompt emission phase and modeled with a polynomial function. Then, we extracted the source spectra by applying the background model to the prompt emission phase. We derived the LAT spectrum files and response files by invoking “Tasks$\rightarrow$Make likelihood analysis” in $gtBurst$. First, a standard cut was performed as done above. The Galactic interstellar emission model (`gll_iem_v06.fits`) and the isotropic spectral template (`iso_P8R2_SOURCE_V6_V06.txt`) will be used to reduce the Galactic diffuse and isotropic emission background contamination. A simple three-component model (an isotropic diffuse component, a Galactic diffuse component, and a point source with a power-law spectrum) and the IRF P8R2\_TRANSIENT020\_V6 were used for unbinned likelihood analysis. Second, the corresponding response file and background spectrum file of the GRB’s *PHA1* spectrum file were obtained by using the tools of `gtbin`, `gtrspgen`, and `gtbkg`. Finally, we adopted XSPEC to conduct a joint spectral fit with a CSTAT statistic method with the GBM, LLE, and LAT spectra. The results for three time-resolved spectra and a time-integrated spectrum are shown in Figure \[fig:GBMSP\].
The duration of GRB 110731A is $T_{90}=(7.3\pm0.3)$s in the energy band 50–300keV (Gruber 2011), and the fluence is $\sim (2.22\pm0.01)\times 10^{-5}$ergcm$^{-2}$ in the energy band 10–1000keV. The time-averaged (from $T_0$s to $T_0+8.6$s, where $T_0$ is the BAT trigger time) spectrum is shown in Figure 2, which can be well fit by a Band function with $E_{\rm
peak}=299^{+54}_{-44}$keV, $\hat{\alpha}=-0.92^{+0.07}_{-0.06}$, and $\hat{\beta}=-2.34^{+0.02}_{-0.03}$ without the need for an additional thermal component[^5]. According to the concordance cosmology with parameters $H_0 = 71$kms$^{-1}$Mpc$^{-1}$, $\Omega_M=0.30$, and $\Omega_{\Lambda}=0.70$, the total isotropic-equivalent energy in the 10keV–10GeV band is $E_{\gamma,{\rm iso}}=(6.8\pm 0.1)\times 10^{53}$erg with $z=2.83$.
We also extracted light curves obtained with [*Swift*]{}/BAT. For this process, we developed an IDL script to automatically download and maintain the [*Swift*]{}/BAT data, and then we used standard HEASOFT tools (v. 6.12) to process the data. By running [*bateconvert*]{} from the HEASOFT software release, we obtain the energy scale for the BAT events. The light curves are extracted by running [*batbinevt*]{} in the 15–150keV energy range (bottom panel in Fig. \[fig:BATLAT\]). The time-averaged spectrum can be best fit by a simple power-law model with photon index $\Gamma_{\rm ph}=1.24\pm 0.08$ owing to the narrow energy range. No obvious spectral evolution was seen in the time-resolved spectral BAT data. The spectral index within the range 6.5–8.6s is $\Gamma_{\rm ph}\approx 1.5$.
![GBM and LAT light curves of GRB 110731A $\gamma$-ray emission in different energy bands. The stars indicate the LLE and LAT high-energy photons above 500MeV. The vertical dotted lines indicate the time intervals for the spectral fitting.[]{data-label="fig:GBMLC"}](GBMlightcurve.eps)
GeV Flare
---------
Figure \[fig:BATLAT\]) shows the minute-scale structure of the LAT light curves. We fit the light curves with a model of multiple power-law components: a power-law function $F=F_{0}t^{\alpha}$, or a broken power-law function $F=F_{0}[(t/t_{\rm p})^{\alpha_{1}s}+(t/t_{\rm
p})^{\alpha_{2}s}]^{1/s}$, where $\alpha$, $\alpha_1$, and $\alpha_2$ are the temporal slopes, $t_{\rm p}$ is the peak time, and $s$ measures the sharpness of a peak of the light-curve component. One has a fast-rising ($t^{2.5\pm
0.12}$) phase initially within the time interval 3.2–7.5s, followed by a very rapid decay ($t^{-7.4\pm 0.23}$) phase within the time interval 7.5–16.5s (see Figs. \[fig:BATLAT\] and \[fig:BATXRT\]). Finally, one flat component follows the rapid decay phase. The peak of the GeV emission is at around 8s. Two highest-energy photons ($\sim2$GeV) were detected in the time interval of 8–9s after the BAT trigger (see Fig. \[fig:GBMSP\], which shows the arrival time for high-energy photons above 500MeV). The LAT spectrum is fitted well by a single power-law model, $N(E)\propto E^{-\Gamma_{\rm LAT}}$, and a clear hard-to-soft spectral evolution during the decay phase is present.
Ackermann et al. (2013) and Fraija (2015) suggested that this GeV emission is the afterglow onset from the external shock. However, the steep temporal indices of both the rising and decaying phases, as well as the hard-to-soft spectral evolution, disfavor the afterglow onset scenario. Rather, it is more like flare emission, which is similar to the X-ray flare emission commonly observed in X-ray afterglows (Burrows et al. 2005; Chincarini et al. 2007; Margutti et al. 2010). We therefore define this GeV emission as a GeV flare and discuss its physical origin. The isotropic energy and luminosity of this GeV flare are $E_{\rm flare, iso} \approx 1.4\times
10^{53}$erg and $L_{\rm flare, iso} \approx 1.1\times
10^{53}$ergs$^{-1}$ with $z=2.83$. Figure \[fig:BATLAT\] shows a comparison between the light curves in the LAT and BAT bands.
X-Ray Afterglow
---------------
We take the XRT data from the [*Swift*]{} UK XRT team website[^6]. The X-ray afterglow light curve shows a rapid increase ($t^{7.0\pm0.82}$) and then a normal decay ($t^{-1.18\pm0.08}$), with a peak time at $65\pm8$s in the observed frame[^7]. No jet break feature was detected up to $\sim 7.5$days. The photon index during the X-ray afterglow phase is $\Gamma_{\rm XRT}\approx 2.0\pm 0.1$, which is similar to the photon index in the LAT band. Figure 4 shows the BAT and XRT light curves from the GRB trigger to $\sim 10^6$s later. One finds that the prompt emission phase is essentially a very short plateau with a decay index $t^{-0.15\pm0.09}$ on a logarithmic scale. Then, it is followed by a sharp drop $\propto t^{-8.6\pm1.12}$ within the time interval 6.5–8.6s[^8], and the break time is $t_{p} \approx 6.5\pm0.8$s.
![Top: [*Fermi*]{}/LAT light curve with energy range 100MeV–100GeV. Bottom: [*Swift*]{}/BAT light curve with energy range 15keV–150keV.[]{data-label="fig:BATLAT"}](BATLAT.EPS)
![Light curves of GRB 110731A observed with [*Swift*]{}/BAT (blue triangles; 15keV–150keV), XRT (black filled circles; 0.3keV–10keV), and [*Fermi*]{}/LAT (red filled circles; 100MeV–100GeV). The empirical fit with the power-law model is shown as a solid line. The evolution of the photon index in the LAT band is also shown in the inset. The vertical line marks the peak time of the GeV flares observed with LAT.[]{data-label="fig:BATXRT"}](BATXRT.EPS)
Keck Observation of a Potential Host Galaxy
-------------------------------------------
We took images of GRB 110731A using the Low Resolution Imaging Spectrometer (Oke et al. 1995) with the Keck I 10m telescope on 15 June 2015 UT in the $I$ and $V$ filters with $3
\times 400$s exposures (see Fig. \[fig:Host\]). We use the optical image from GROND to double-check the position of GRB 110731A, which is at (J2000) $\alpha = 18^h42^m01.011^s$, $\delta = -28^\circ32'13.43''$ (Ackermann et al. 2013). The foreground stars are very saturated and crowded; thus, in order to calibrate the magnitudes of stars in the field, we took an image with the 1.0m Nickel telescope at Lick Observatory on 2015 June 30 UT. At the exact afterglow position, we did not find an apparent host galaxy of GRB 110731A, with a limiting magnitude of $m_{I}\approx 24.9$.
On the other hand, we found a potential extended source to the northeast of the afterglow position, which might be the host galaxy of GRB 110731A. The source has an $I$-band magnitude of $m_I \approx 23.5$ (as shown in Fig. 5 with circle B). Following Bloom et al. (2002) and Berger (2010), we calculate the probability of association for a given galaxy of brightness $m$ at a separation $\delta R$ from a GRB position: $$\begin{aligned}
P=1-e^{-\pi(\delta R)^2\sum(\leq m)},
\label{depth}\end{aligned}$$ where the galaxy number counts are given by $\sum(\leq m)= 1.3
\times 10^{0.33(m-24)-2.44}$arcsec$^{-2}$ (Hogg et al. 1997). Assuming a physical association, the offset between this putative host galaxy and the GRB afterglow is $1.63''$, which corresponds to $\sim 13.0$kpc at $z=2.83$. We find that the possibility that GRB 110731A resides in this host galaxy is about 3%.
![Slightly trailed Keck I image of the field of GRB 110731A in the $I$ band. The lower circle (circle A) marks the position of the GRB 110731A optical afterglow with $0.3''$ uncertainty and a magnitude limit $m_{I}\approx 24.9$. The upper circle (circle B) marks the position of its potential host galaxy, with $0.5''$ uncertainty and $m_{I}= 23.5$mag. The separation between circles A and B is $1.63''$, which corresponds to $\sim 13.0$kpc at $z=2.83$.[]{data-label="fig:Host"}](Hostgalaxy.eps)
Constraints on the Bulk Lorentz Factor
======================================
The broadband data allow us to constrain the bulk Lorentz factor $\Gamma$ of the GRB with two different methods.
The first is the pair-opacity method. The broadband featureless Band-function spectra extending to very high energies ($>100$MeV) pose a lower limit on $\Gamma$ to avoid two-photon pair production ($\gamma \gamma\rightarrow
e^{+}e^{-}$). If a cutoff energy is detected in the spectrum, one may constrain $\Gamma$ and the radius of the emission region ($R_\gamma$) by requiring the optical depth (Zhang & Pe’er 2009) to be $$\begin{aligned}
\tau_{\gamma\gamma}(E_{\rm cut}) & = &
\frac{F(\hat{\beta})\sigma_{T}D^{2}_{L}f_0}{-1-\hat{\beta}}
\left(\frac{E_{\rm cut}}{m^{2}_{e}c^{4}}\right)^{-1-\hat{\beta}} \nonumber \\
& \times & R^{-2}_{\gamma} \left(\frac{\Gamma}{1+z}\right)^{2+2\hat{\beta}} = 1,
\label{depth}\end{aligned}$$ where $D_{\rm L}$ and $z$ are the luminosity distance and redshift, respectively, and $m_{e\rm }$, $c$, and $\sigma_{\rm T}$ are the fundamental constants of electron mass, speed of light, and Thomson cross section, respectively. The parameter $f_0$ is related to the Band-function parameters as $$\begin{aligned}
f_0= A \cdot \Delta T
\left[\frac{E_{\rm p}(\hat{\alpha}-\hat{\beta})}{2+\hat{\alpha}}\right]^{\hat{\alpha}-\hat{\beta}}
\rm
exp(\hat{\beta}-\hat{\alpha})(100\,{\rm keV})^{-\hat{\beta}},
\label{f0}\end{aligned}$$ where $\Delta T$ is the time interval for spectral fitting, 8.6s. Also, $\hat{\alpha}$, $\hat{\beta}$, and $E_{\rm p}$ are the parameters of the time-integrated spectrum within 0–8.6s by invoking a Band-function fit, and the fitting results are presented in Table 1. The Band-function normalization is $A =
0.044$phcm$^{-2}$s$^{-1}$keV$^{-1}$. The coefficient $F(\hat{\beta})$ can be expressed as (Svensson 1987) $$\begin{aligned}
F(\hat{\beta})= \frac{7}{6(1-\hat{\beta})(-\hat{\beta})^{5/3}}.
\label{Fbeta}\end{aligned}$$
For the spectra of GRB 110731A, a cutoff power-law model is not consistent with the data. However, a Band function combined with an extra power-law component provides a good fit from 7.3s to 8.6s. There is no cutoff feature in the spectra when this extra power-law component extends to the GeV energy band. The highest-energy observed photon had $\sim1.9$GeV. Hence, we can use this maximum-energy photon $E_{\rm max}$ to replace $E_{\rm cut}$ in Eq. 2, and we estimate the lower limit of the bulk Lorentz factor ($\Gamma_{\rm min}$) by assuming that both $>100$MeV and sub-MeV photons are from the same zone. By using the $\tau_{\gamma\gamma}(E_{\rm max})\leq 1$ condition, one derives $\Gamma_{\rm min} \approx 190$ within the internal-shock model, $R_{\gamma}\approx \Gamma^2 c \delta
t/(1+z)$, where $\delta t$ is the minimum variability timescale. But $\delta t$ is subject to large uncertainties because GRB light curves are chaotic, without a characteristic timescale, and it also depends on the bin size and energy. In our calculation, $\delta t=0.5$s is adopted to get a higher signal-to-noise ratio for the GBM-LAT light curve. The constraints on the range of $\Gamma$ and $R_\gamma$ are shown in Figure \[fig:Gammar\]. However, if the $>100$MeV and sub-MeV photons are from different regions, then the estimated bulk Lorentz factor is more complex (Zhao et al. 2011; Zou et al. 2011).
![The $R_{\gamma}$–$\Gamma$ diagram of GRB 110731A. The constraint on $R_{\gamma}$ is displayed with a black solid line, above which is the allowed parameter space (gray shadow). The internal shock radius, $R_{\rm IS}\approx \Gamma^{2} c \delta t/(1+z)$, is plotted as the dotted line.[]{data-label="fig:Gammar"}](gamma_r.EPS)
![Comparison of GRB 110731A estimated from the early afterglow onset time with other typical long GRBs in the $\Gamma_0$–$E_{\gamma, {\rm iso}}$ plane. The filled star is GRB 110731A assuming a homogeneous density profile (ISM). GRBs marked with dots are taken from Liang et al. (2010). Two dashed lines mark the $2\sigma$ region of the correlation, and $\sigma$ is the standard deviation of the best fit.[]{data-label="fig:GammaEiso"}](gamma_eiso.eps)
The second method of estimating the initial Lorentz factor is to use the onset time of the early afterglow. According to a broken power-law fit, the early X-ray afterglow light curve of GRB 110731A peaks at $t_{\rm peak} \approx 65$s. The deceleration time is $t_{\rm dec}=t_{\rm peak}/(1+z)$. We apply the standard afterglow model with a constant-density medium (i.e., the interstellar medium \[ISM\]) to derive the initial Lorentz factor, which reads $$\begin{aligned}
\Gamma_0 &
\approx & 170\ t_{\rm dec,2}^{-3/8}\left(\frac{1+z}{2}\right)^{3/8}E_{\rm iso,52}^{1/8}
(n\eta)^{-1/8}.
\label{gamma}\end{aligned}$$ We take $n \approx 0.1$ cm$^{-3}$ and radiative efficiency $\eta=\frac{E_{\gamma, {\rm iso}}}{E_{\gamma, {\rm iso}}+E_{\rm
K, iso}}\approx 90\%$ in this analysis and derive[^9]$\Gamma_0 \approx 580$. Liang et al. (2010) discovered a tight relation between $\Gamma_0$ and $E_{\gamma, {\rm iso}}$. We test to compare whether GRB 110731A is consistent with this correlation. The values $n \approx 1$cm$^{-3}$ and $\eta
\approx 20$% (performed by Liang et al. 2010) are adopted to recalculate the Lorentz factor of GRB 110731A. One has $\Gamma_0 \approx 525$, which is consistent with the correlation between $\Gamma_0$ and $E_{\gamma, {\rm iso}}$ discovered by Liang et al. (2010). Figure \[fig:GammaEiso\] shows the $\Gamma_0$–$E_{\gamma, {\rm iso}}$ plot and compares GRB 110731A in a constant-density ISM (filled star) with other typical long GRBs taken from Liang et al (2010). Alternatively, if the medium has a density that decays with radius (wind model), then $$\begin{aligned}
\Gamma_0 &
= & 1.44[\frac{E_{\rm iso}(1+z)}{8\pi A_{\ast}m_p c^{3}\eta t_{\rm dec}}]^{1/4}
\nonumber \\
&\approx& 40(\eta E_{\rm iso, 52})^{1/4}(\frac{1+z}{2})^{1/4}(t_{\rm dec, 2})^{-1/4},
\label{gamma}\end{aligned}$$ where $A_{\ast}=3\times 10^{35}$ cm$^{-1}$ is the wind parameter (Yi et al. 2015) and $\eta \approx 90$% is adopted. One has $\Gamma_0\approx 154$, which is much lower than in the ISM case.
Jet Composition
===============
Different jet energy dissipation models of prompt emission predict different properties in the emission region, such as the magnetization parameter $\sigma$, the spectral shape, and the $R_{\gamma}$–$\Gamma$ relation. As shown in Figure 6, the $\tau_{\gamma\gamma}(E_{\rm max})\leq 1$ condition places a tight lower limit on $R_\gamma$ for the burst, which is much larger than the photosphere radius (typically at $R_{\rm
ph}\approx 10^{11}$–$10^{12}$cm). Also, the nondetection of a thermal component in the spectrum also suggests that the photosphere component is suppressed, suggesting that the outflow is Poynting flux dominated (e.g., Zhang & Pe’er 2009; Gao & Zhang 2015).
The rapid decay of X-ray emission at the end of prompt emission ($t^{-(8.6\pm 1.12)}$) is also consistent with such a picture. The standard model predicts that the decay slope cannot be steeper than a decay index $\alpha = 2+\beta$, where $\beta$ is the spectral index (Kumar & Panaitescu 2000). Such a “curvature effect” prediction is valid if the outflow moves with a constant Lorentz factor. Uhm & Zhang (2015) pointed out that the decay slope can be steeper than this prediction if the outflow is undergoing acceleration. Applying the theory to X-ray flares by properly correcting the zero-time effect (Liang et al. 2006), Uhm & Zhang (2016) suggested that the X-ray flare emission region is undergoing rapid acceleration; see also Jia et al. (2016) for an extended analysis of a larger sample of X-ray flares. For GRB 110731A, the rapid decay of X-ray emission at the end of prompt emission is $\alpha \approx
8.6$. However, this decay slope is dependent on the zero-time ($T_{\rm z}$), which is uncertain.
Owing to the short duration of the prompt emission, we reanalyze the temporal behavior of the X-ray emission at the end of the prompt emission; one has $\alpha \approx 4$ if $T_{\rm z}=4$s is adopted. On the other hand, the photon index of spectra ($\Gamma_{\rm ph}$) at the end of the prompt emission is about 1.5 by invoking the power-law model fit, so $\beta= \Gamma_{\rm ph}-1
\approx 0.5$. Within the curvature-effect scenario, the temporal index $\alpha$ and spectral index $\beta$ should satisfy $\alpha= 2+\beta$. However, $\alpha > 2+\beta = 2.5$ if $T_{\rm z}=4$s is adopted, so the curvature effect is unlikely. Instead, the data seem to be consistent with a model that invokes dissipation of a moderately high $\sigma$ Poynting flux in the emission region (e.g., ICMART model; Zhang & Yan 2011). Alternatively, if $T_{\rm z}=5$s is adopted, then $\alpha\approx2.8$, only a little larger than $2+\beta = 2.5$. In this case, the curvature effect cannot be ruled out.
To summarize, the constraint on the emission region $R_\gamma$, the nondetection of a thermal component in the spectrum, and the possibility of bulk acceleration in the emission region all point toward a consistent picture regarding the jet composition of GRB 110731A: it is very likely Poynting flux dominated.
Central Engine
==============
Two types of GRB central engine models have been discussed in the literature (see, e.g., Kumar & Zhang 2015, for a review). One type invokes a hyperaccreting stellar-mass black hole (e.g., Popham et al. 1999; Narayan et al. 2001; Lei et al. 2013). The second type invokes a rapidly spinning, strongly magnetized NS called a millisecond magnetar, which has been invoked to interpret the shallow-decay, long-lasting, early-afterglow phase (Dai & Lu 1998; Zhang & Meszaros 2001) in both long and short GRBs (Fan & Xu 2006; Troja et al. 2007; Rowlinson et al. 2010, 2013; Lü et al. 2015). Within the black hole central engine, the plateau and subsequent steep decay are more difficult to interpret. Here, we test whether the magnetar central engine can power GRB 110731A based on the observed properties of GRB 110731A — i.e., the plateau phase produced by energy injection from a magnetar wind, and the sharp drop thereafter being due to the collapse of the magnetar forming a black hole.
According to Zhang & Mészáros (2001), the energy reservoir is the total rotation energy of the millisecond magnetar, $$\label{Erotaion}
E_{\rm rot} = \frac{1}{2} I \Omega_{0}^{2}
\approx 2 \times 10^{52}~{\rm erg}~
M_{1.4} R_6^2 P_{0,-3}^{-2},
\label{Erot}$$ where $I$ is the moment of inertia, $\Omega_0 = 2\pi/P_0$ is the initial angular frequency of the NS, $M_{1.4} = M/1.4~{\rm
M}_\odot$, and the convention $Q = 10^x Q_x$ is adopted in cgs units for all other parameters throughout the paper. The characteristic spin-down luminosity and spin-down timescale are related to the magnetar initial parameters $$L_0 = (1.0 \times 10^{49})\,(B_{p,15}^2 P_{0,-3}^{-4} R_6^6)~{\rm erg~s^{-1}},
\label{L0}$$ $$\tau = (2.05 \times 10^3)\,(I_{45} B_{p,15}^{-2} P_{0,-3}^2 R_6^{-6})~{\rm s},
\label{tau}$$ where $B_{\rm p}$ and $P_0$ correspond to the surface polar cap magnetic field and initial spin period, respectively.
Using Eq. \[L0\] and Eq. \[tau\], one can derive $B_{\rm
p}$ and $P_0$ as $$\begin{aligned}
B_{p,15} = 2.05\,(I_{45} R_6^{-3} L_{0,49}^{-1/2} \tau_{3}^{-1})~{\rm G},
\label{Bp}\end{aligned}$$ $$\begin{aligned}
P_{0,-3} = 1.42\,(I_{45}^{1/2} L_{0,49}^{-1/2} \tau_{3}^{-1/2})~{\rm s}.
\label{P0}\end{aligned}$$ Through light-curve fitting, one can derive the break-time luminosity as $$\begin{aligned}
L_b = 4\pi D^2 F_b,\end{aligned}$$ where $F_{\rm b}$ is flux at break time $t_{\rm b}$. For a plateau, the characteristic spin-down luminosity can be estimated as $$\begin{aligned}
L_0 \approx L_b.
\label{L0=Lb}\end{aligned}$$
The spin-down timescale can be generally identified as $$\begin{aligned}
\tau \geq t_b/(1+z).
\label{tautb}\end{aligned}$$ The “greater than” sign takes into account that the supramassive magnetar collapses to a black hole before it is significantly spun down. One can therefore derive $B_{\rm p} \leq
9.9\times 10^{15}$G and $P_0 \leq 0.56$ms if we assume that the magnetar wind is isotropic (see Fig. \[fig:magnetar\]a). Since this value of $P_0$ is shorter than the breakup limit of an NS ($P_0=0.96$ms; Lattimer & Prakash 2004), we also consider a possible beaming factor ($f_{\rm b}$) of the GRB outflow, $$\begin{aligned}
\label{fb}
f_b = 1-\cos \theta_j \approx (1/2) \theta_j^2,\end{aligned}$$ where $\theta_{\rm j}$ is jet opening angle. The jet opening angle can be estimated as $$\begin{aligned}
\label{theta_j}
\theta_j &=& 0.07\,\left(\frac{t_j}{1~{\rm
day}}\right)^{3/8}\left(\frac{1+z}{2}\right)^{-3/8} \nonumber \\& \times &
\left(\frac{E_{\rm K,iso}}{10^{53}~{\rm erg}}\right)^{-1/8}\left(\frac{n}{0.1 ~\rm
cm^{-3}}\right)^{-1/8}~{\rm rad},\end{aligned}$$ where $n \approx 0.1$cm$^{-3}$ is the ambient-medium density and $E_{\rm K,iso}$ is the kinetic energy of the outflow. The temporal index of the normal decay segment is $\alpha_2=1.18\pm0.01$, and the spectral index is $\beta_X=0.85\pm0.13$. They satisfy $2\alpha_2 \approx
3\beta_X$, suggesting a spectral regime $\nu_m < \nu < \nu_c$. Following Zhang et al. (2007), we derive $$\begin{aligned}
E_{\rm K,iso,52} & = & \left[\frac{\nu F_\nu (\nu=10^{18}~{\rm Hz})}{6.5\times
10^{-13} ~{\rm erg\,s^{-1}\,cm^{-2}} }\right]^{4/(p+3)} \nonumber \\& \times &
D_{28}^{8/(p+3)}(1+z)^{-1}
t_d^{3(p-1)/(p+3)}\nonumber \\
& \times &f_p^{-4/(p+3)} \epsilon_{B,-2}^{-(p+1)/(p+3)}
\epsilon_{e,-1}^{4(1-p)/(p+3)} \nonumber \\ & \times & n^{-2/(p+3)}
\nu_{18}{^{2(p-3)/(p+3)}},
\nonumber \\\end{aligned}$$ where $p=2\beta+1$. With standard values of microphysics parameters (e.g., Panaitescu & Kumar 2002; Yost et al. 2003), such as $\epsilon_{\rm e} = 0.1$, $\epsilon_B = 0.01$, and $n
\approx 0.1$cm$^{-3}$, we derive $E_{\rm K,iso} \approx
7.5\times10^{52}$erg. No jet break feature was detected up to $\sim 7.5$days of afterglow emission; we therefore set a lower limit on the jet opening angle, $\theta_{\rm
j}>5.5^{\circ}$.
![Inferred magnetar parameters, initial spin period $P_0$ vs. surface polar cap magnetic field strength $B_p$, of GRB 110731A (red star). (a)the case of isotropic winds; (b) The case of beaming correction with jet opening angle $\theta_{\rm j}$ in the range 5.5$^\circ$–12.2$^\circ$. The vertical solid line is the breakup spin period for an NS (Lattimer & Prakash 2004).[]{data-label="fig:magnetar"}](magnetar_a.EPS "fig:") ![Inferred magnetar parameters, initial spin period $P_0$ vs. surface polar cap magnetic field strength $B_p$, of GRB 110731A (red star). (a)the case of isotropic winds; (b) The case of beaming correction with jet opening angle $\theta_{\rm j}$ in the range 5.5$^\circ$–12.2$^\circ$. The vertical solid line is the breakup spin period for an NS (Lattimer & Prakash 2004).[]{data-label="fig:magnetar"}](magnetar_b.EPS "fig:")
Within the magnetar model, one can also set an upper limit of $\theta_j$ by requiring that the total energy does not exceed the total spin energy of the magnetar: $E_{\rm rot}\geq
E_\gamma+E_{\rm K}+E_{\rm flare}$, where $E_\gamma=E_{\gamma,
{\rm iso}} f_b$, $E_{\rm K} = E_{\rm K, iso} f_b$, and $E_{\rm
flare} = E_{\rm flare, iso} f_b$ is the energy of the GeV flare. This gives $\theta_{\rm j} < 12.2^{\circ}$. By including the range of the opening angle $5.5^{\circ}< \theta_{\rm j} <
12.2^{\circ}$, we rederive the magnetar parameters for GRB 110731A (red stars in Fig. \[fig:magnetar\]b). We find that the $P_0$ upper limit is in the allowed range (longer than the breakup limit), but the inferred $B_{\rm p}$ upper limit is significantly larger than the $B_{\rm p}$ values inferred for other GRBs. In any case, since $B_{\rm p}$ is an upper limit, the magnetar model is possible given that the spin-down time $\tau$ is much longer than $t_{\rm b}/(1+z)$.
Within this supramassive NS collapsing framework, there is another energy budget, the total magnetic field energy after the magnetosphere is expelled following the collapse of the NS. It may be estimated as (Zhang 2014) $$\begin{aligned}
E_{\rm B, iso} &\approx& \int^{R_{\rm LC}}_{R}4\pi r^2
\frac{B^2_p}{8\pi}\left(\frac{r}{R}\right)^{-6} dr \nonumber \\& \approx &
(1/6)B^2_p R^3 \approx (1.7\times 10^{47})\, B^2_{p, 15} R^3_{6}~ {\rm erg},
\label{EB}\end{aligned}$$ where $R_{\rm LC}\gg R$ is the light-cylinder radius. This is a relatively small energy for typical parameters, but can be important if $B_{\rm p}$ is large (close to the upper limit inferred above).
Figure \[fig:energy\] gives all the energy components as a function of jet opening angle, with the allowed $\theta_{\rm
j}$ range marked.
![(a)Energy $E_{\gamma}$ (green line), $E_{\rm K}$ (blue line), $E_{\rm
flare}$ (red line), $E_{\rm B}$ (cyan line), and $E_{\rm total}$ (black line) as a function of jet opening angle. The horizontal line is the total energy budget of the magnetar ($E_{\rm rot}\approx 2\times 10^{52}$erg); (b)Inferred upper limit of $P_0$ as a function of $\theta_{\rm j}$.[]{data-label="fig:energy"}](energy_a.EPS "fig:") ![(a)Energy $E_{\gamma}$ (green line), $E_{\rm K}$ (blue line), $E_{\rm
flare}$ (red line), $E_{\rm B}$ (cyan line), and $E_{\rm total}$ (black line) as a function of jet opening angle. The horizontal line is the total energy budget of the magnetar ($E_{\rm rot}\approx 2\times 10^{52}$erg); (b)Inferred upper limit of $P_0$ as a function of $\theta_{\rm j}$.[]{data-label="fig:energy"}](energy_b.EPS "fig:")
Progenitors
===========
With the measured $z = 2.83$, GRB 110731A has a rest-frame duration $[T_{90}/(1 + z)]$ shorter than 2s. This naturally raises the interesting question regarding the progenitor system of this burst (Type I vs. Type II; Zhang et al. 2009). In the past, there have been quite a few GRBs that are rest-frame short, including several high-$z$ GRBs such as GRB 080913 at $z=6.7$ with $T_{90} =8$s (Greiner et al. 2009), GRB 090423 at $z = 8.2$ with $T_{90} = 10.3$s (Salvaterra et al. 2009; Tanvir et al. 2009), and GRB 090429B at $z = 9.4$ with $T_{90}
= 5.5$s (Cucchiara et al. 2011). Multiwavelength observed properties suggested that those three high-$z$ GRBs are likely of a Type II (massive star) origin (Zhang et al. 2009).
Lü et al. (2014) proposed a method to judge whether a rest-frame short GRB is more likely the “tip of iceberg” of a long-duration GRB. They move a specific long GRB to progressively larger redshifts until the rest-frame duration is shorter than 2s and then define the ratio of the peak flux of this pseudo-GRB to the background flux as $f_{{\rm eff},z}$. The $f_{{\rm eff},z}$ value of long GRBs is typically smaller than 2. The three high-$z$ GRBs all have $f_{{\rm eff},z}$ smaller than 2, suggesting that they are consistent with being a long GRB as observed at high redshift. We perform the same analysis for GRB 110731A following Lü et al. (2014), and find that the value is $f_{{\rm eff},z}=2.67$. As shown in Figure \[fig:ft90\], this value (red star) is higher than that of typical long GRBs (gray), but is consistent with short GRBs (blue triangles). Following Lü et al. (2014), we also calculate the probability ($p$) of this being a disguised short GRB: $p\approx 0.03$. In general, these arguments suggest that the GRB is consistent with having a Type I (compact-star merger) origin.
The host-galaxy information seems to also support a Type I origin for GRB 110731A. As discussed in § 2.5, our Keck observation revealed an extended source northeast of the afterglow position, which might be the host galaxy of GRB 110731A. The source has an $I$-band magnitude $m_I \approx
23.5$. The offset between this source and the GRB afterglow is $1.63''$, which corresponds to $\sim13.0$kpc, with the possibility of a physical association being $\sim 3$%. First, we compare the distribution of long GRB physical offsets with those of short GRBs; the K-S test yields $P_{\rm K-S}=0.31$, suggesting that the objects are not drawn from the same population. Then, we compare the offset of GRB 110721A with the distributions of the physical offsets of both long and short GRBs observed by the [*Hubble Space Telescope*]{} (Fong et al. 2010; Blanchard et al. 2016). We find that the physical offset of GRB 110731A is larger than that of almost all long GRBs, but is statistically consistent with typical short GRBs (see Fig. \[fig:offset\]). This also supports the Type I origin of the burst, if the extended source is indeed the host galaxy of GRB 110731A.
Empirically, most long GRBs are found to satisfy a relationship between $E_{\rm p}(1+z)$ and $E_{\gamma, {\rm iso}}$ ($E_{\rm
p}(1+z)\propto E^{1/2}_{\gamma, {\rm iso}}$; Amati et al. 2002), but outliers do exist (e.g., long GRBs 980425, 031203, and 050826). In contrast, most short GRBs are inconsistent with this empirical relation (Zhang et al. 2009) and seem to follow a different correlation with a larger dispersion. However, outliers to this relation also exist (e.g., short GRB 060121). In order to compare GRB 110731A with long and short GRBs, we calculate $E_{\gamma, {\rm iso}}\approx 4.5\times 10^{53}$erg from 1keV to $10^{4}$keV based on the spectral properties and then plot it in the $E_{\rm p}(1+z)$–$E_{\gamma, {\rm
iso}}$ diagram (see Fig. \[fig:Amati\]). GRB 110731A falls in the 3$\sigma$ confidence band of power-law fitting of long GRBs. However, this empirical relation does not provide persuasive evidence that GRB 110731A is from massive star core collapse (outliers do exist), and it may be caused by some selection effects.
![$T_{90}/(1 + z)-f_{{\rm eff},z}$ diagram of both long and short GRBs taken from Lü et al (2014). The red star denotes GRB 110731A, and the vertical line is $T_{90}/(1+z)=2$s.[]{data-label="fig:ft90"}](feffectz.EPS)
![Bottom panel: the distribution of physical offsets for short GRBs (blue; Fong et al. 2010) and long GRBs (gray; Blanchard et al. 2016); the value of the K-S test is about 0.31. The blue dotted line and gray dotted line indicate the best Gaussian fits. Top panel: a cumulative distribution of long (gray) and short (blue) GRBs. The red vertical line corresponds to GRB 110731A.[]{data-label="fig:offset"}](offset.eps)
![$E_{\rm p}$-$E_{\rm \gamma,iso}$ diagram of both long (blue dots) and short (green diamonds) GRBs. The data points for long and short GRBs are taken from Amati et al. (2002) and Zhang et al. (2009), respectively. The solid lines are the best fit with a power-law model, and dashed lines mark 3$\sigma$ confidence bands. The red star corresponds to GRB 110731A.[]{data-label="fig:Amati"}](Amati.eps)
The Origin of the GeV Flare
===========================
The origin of the GeV flare is mysterious. As shown in Figure 3, it rises during the prompt emission phase but peaks after the BAT-band emission finished. The time-integrated spectral energy distribution is fitted by a Band function alone, and it seems that the GeV emission has the same origin. However, when we analyze the time-resolved spectra during the GeV flare (from $T_0+7.3$s to $T_0+8.6$s), they are well fitted by a Band function with an additional power-law component (Table 1 and Fig. \[fig:GBMSP\]). This suggests that the GeV flare may have a distinct origin from the sub-MeV emission.
If this is the case, one possible scenario is to interpret the GeV flare within the framework of a supramassive NS collapsing into a black hole. Zhang (2014) suggested that the ejection of the magnetosphere may be accompanied by a fast radio burst. It is possible that such an ejection may power a GeV flare. However, the total amount of magnetospheric energy (Eq. 17) is typically smaller than the GeV flare energy, so one runs into an energy budget problem. Nonetheless, the uncertainty in the beaming factor $f_{\rm b}$ makes it possible that $E_{\rm B}> E_{\rm
flare}$ in a certain range of jet opening angles. From Figure 9a, one can derive that the condition is $5.5^{\circ}<\theta_{\rm j}
< 8.5^{\circ}$. The specific emission mechanism depends on the particle acceleration details within the ejected magnetosphere, but the sudden acceleration of the magnetosphere makes it plausible to have a GeV peak right after the rapid decline in sub-MeV emission.
Alternatively, the GeV emission may be produced from the external shock (e.g., Kumar & Barniol-Duran 2009, 2010; Ghisellini et al. 2010; Zhang 2011; Maxham et al. 2011; He et al. 2012; Liu et al. 2012). The initial steep decay may suggest a reverse-shock component (e.g., Wang et al. 2001, 2002). However, this interpretation is in conflict with our suggestion of the X-ray peak at 65s as the deceleration time.
Conclusions and Discussion
==========================
GRB 110731A is a peculiar GRB with a duration of $\sim7.3$s detected by [*Swift*]{} and [*Fermi*]{}, and a measured redshift of $z=2.83$. The total isotropic-equivalent energy in the 10keV–10GeV range is $E_{\gamma, {\rm iso}}\approx (6.8\pm
0.1)\times 10^{53}$erg for the prompt emission. One GeV flare was detected by LAT with its highest photon energy $\sim
2$GeV, and the peak time of the GeV flare corresponds to the transition break time from the prompt emission (plateau) to a steeper decay. The total isotropic-equivalent energy of the GeV flare is $E_{\rm iso, flare}\approx 1.4\times 10^{53}$erg. Our Keck $I$-band image of the field placed an upper limit to the host-galaxy brightness ($m_I \approx 24.9$mag) at the afterglow position and identified a putative galaxy with a physical offset of $\sim13$kpc. We analyze the broadband data and compare them with GRB theoretical models, finding the following results.
- We apply both the pair-opacity method and afterglow deceleration time method to constrain the Lorentz factor of the ejecta. The former gives $\Gamma>190$, while the latter gives $\Gamma \approx 580$ and $\Gamma\approx 154$ within the homogeneous and wind density profiles, respectively.
- The broadband featureless Band-function spectra cover 5–6 orders of magnitude in energy, as well as the very high energy gamma-ray emission ($\geq
1$GeV). Nondetection of a thermal component may be consistent with a Poynting-flux-dominated flow as the jet composition of the burst. On the other hand, during the rapid decay phase following prompt emission, the temporal decay index ($\alpha$) is steeper than the curvature-effect prediction $\alpha=2+\beta$, which supports possible acceleration of the emission region. However, this decay slope is dependent on the selected zero-time, which is uncertain.
- The central engine of the GRB may be a millisecond magnetar, but with a relatively large upper limit of both $B_p$ and $P_0$. With a beaming correction, the total observed energy (sum of $E_{\rm \gamma}$, $E_{\rm K}$, and $E_{\rm flare}$) is within the energy budget provided by the spin energy of the millisecond magnetar ($E_{\rm rot}\approx 2\times 10^{52}$erg) when the jet opening angle satisfies $5.5^{\circ}<
\theta_{\rm j} < 12.2^{\circ}$.
- The burst has a rest-frame duration shorter than 2s. A relatively large $f_{{\rm eff},z}$ value and a substantial physical offset from the putative host galaxy suggest that the progenitor of GRB 110731A is likely to come from a compact-star merger.
- The GeV flare is mysterious. However, within the magnetar central scenario, the GeV flare may be produced during the ejection of the magnetosphere when the magnetar collapses to form a black hole.
We acknowledge the use of public data from the [*Swift*]{} and [*Fermi*]{} data archives and the UK [*Swift*]{} Science Data Center. We thank Xue-Feng Wu and Wei-Hua Lei for helpful comments and discussions, as well as D. Alexander Kann for observation suggestions. This work is supported by the National Basic Research Program (973 Programme) of China 2014CB845800, the National Natural Science Foundation of China (grant nos. 11603006, 11533003, 11673006, 11603003, 11543005, U1331202, 11303005, 11363002), the One-Hundred-Talents Program of Guangxi colleges, the high-level innovation team and outstanding scholar program in Guangxi colleges, the Guangxi Science Foundation (2016GXNSFCB380005, 2016GXNSFFA380006, 2013GXNSFFA019001, 2014GXNSFAA118011), and the Scientific Research Foundation of Guangxi University (grant no. XGZ150299). A.V.F.’s group at UC Berkeley has been supported by Gary and Cynthia Bengier, the Richard & Rhoda Goldman Fund, the Christopher R. Redlich Fund, the TABASGO Foundation, and US NSF grant AST-1211916. Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and NASA; the observatory was made possible by the generous financial support of the W. M. Keck Foundation. Research at Lick Observatory is partially supported by a generous gift from Google.
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[lllllllllllll]{}
& $\hat{\alpha}$ &$\hat{\beta}$ &$E_{\rm p}(keV)$ & $\lambda\tablenotemark{a}$ & $\chi^2$/dof\
$0--8.6$ & $-0.89\pm 0.06$ &$-2.32\pm0.03$ &$285\pm 41$ & – & 373/315\
$0--2\tablenotemark{b}$ & $-0.87\pm 0.12$ &– &$145\pm 21$ & – & 228/242\
$2--7.3$ & $-0.74\pm 0.09$ &$-2.32\pm 0.03$ &$277\pm 59$ & – & 254/315\
$7.3--8.6$ & $-1.08\pm 0.29$ &$-2.15\pm 0.16$ &$2793\pm 673$ & $1.57\pm 0.47$ & 263/321\
[^1]: http://fermi.gsfc.nasa.gov/ssc/data/
[^2]: http://fermi.gsfc.nasa.gov/ssc/data/analysis/LAT\_caveats.html
[^3]: http://sourceforge.net/projects/gtburst/
[^4]: fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/gtburst.html/
[^5]: In order to avoid confusion with temporal ($\alpha$) and spectral ($\beta$) indices, we use $\hat{\alpha}$ and $\hat{\beta}$ to indicate the low-energy and high-energy photon spectral indices of the Band function.
[^6]: http://www.swift.ac.uk
[^7]: Ackermann et al. (2013) explained the rapid flux increase in the XRT band around 65s as an X-ray flare. However, since both the X-ray and optical light curves after the peak decay as a power-law function with a typical index, the initial rapid increase of the XRT light curve would be more like the onset of the afterglow with flickering contamination, as observed in GRB 061007 (Mundell et al. 2007).
[^8]: However, the temporal decay index is dependent on the zero-time ($T_{\rm z}$), which is more difficult to determine. In our calculation, we assume that the BAT trigger time $T_0$ is also the zero-time.
[^9]: Sari & Piran (1999) derived $\Gamma_0 =
[{3E_{\rm iso}(1+z)^{3}} / {32\pi n \eta m_p c^5 t_{\rm dec}^3}
]^{1/8}$. Lü et al. (2012) introduced a factor of 1.4 through numerical integration. A more precise treatment by including pressure in the energy-momentum tensor gives our numerical coefficient.
|
---
abstract: 'We present the SPICA Coronagraphic Instrument (SCI), which has been designed for a concentrated study of extra-solar planets (exoplanets). SPICA mission provides us with a unique opportunity to make high contrast observations because of its large telescope aperture, the simple pupil shape, and the capability for making infrared observations from space. The primary objectives for the SCI are the direct coronagraphic detection and spectroscopy of Jovian exoplanets in infrared, while the monitoring of transiting planets is another important target. The specification and an overview of the design of the instrument are shown. In the SCI, coronagraphic and non-coronagraphic modes are applicable for both an imaging and a spectroscopy. The core wavelength range and the goal contrast of the coronagraphic mode are 3.5–27$\mu$m, and 10$^{-6}$, respectively. Two complemental designs of binary shaped pupil mask coronagraph are presented. The SCI has capability of simultaneous observations of one target using two channels, a short channel with an InSb detector and a long wavelength channel with a Si:As detector. We also give a report on the current progress in the development of key technologies for the SCI.'
address:
- |
Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency,\
3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan
- 'The Graduate University for Advanced Studies, 3-1-1 Yoshinodai, Sagamihara, Tyuou-ku, Kanagawa 252-5210, Japan'
- 'Department of Physics, Graduate School of Science, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan'
- 'Aerospace Research and Development Directorate, Japan Aerospace Exploration Agency, Tsukuba Spcae Center, 2-1-1 Sengen, Tsukuba-shi, Ibaraki 305-8505, Japan'
- 'Institute of Astronomy, School of Science, University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015,Japan'
- 'National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan'
- 'Subaru Telescope, National Astronomical Observatory of Japan, 650 North A’ohoku Place, Hilo, Hawaii96720, U.S.A.'
- 'Department of Earth and Space Science, Graduate School of Science, Osaka University, 1-1Machikaneyama, Toyonaka, Osaka 560-0043, Japan'
- 'Division of Applied Physics, Faculty of Engineering, Hokkaido University, Sapporo 060-8628, Japan'
- 'Graduate School of Science and Technology, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan'
- 'Department of Information Sciences, Kanagawa University, 2946 Tsuchiya, Hiratsuka, Kanagawa, 259-1293, Japan'
- 'Institute of Astrophysics and Planetary Sciences, Faculty of Science, Ibaraki University, 2-1-1 Bunkyo,Mito, Ibaraki 310-8512, Japan'
- 'Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8551, Japan'
- 'Institute of Astronomy and Astrophysics, Academia Sinica. P.O. Box 23-141, Taipei 10617, Taiwan, R.O.C'
- 'Laboratoire Fizeau, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France'
- 'Subaru Telescope, National Astronomical Observatory of Japan, 650 North A’ohoku Place, Hilo, Hawaii96720, U.S.A.'
- 'Boston Micromachines Corporation, 30 Spinelli Place, Cambridge, MA 02138, U.S.A'
- 'Optcraft Corporation, 3-16-8 Higashi-Hashimoto, Sagamihara, Midori-ku, Kanagawa 252-0144, Japan'
author:
- 'K. Enya,'
- 'T. Kotani,'
- 'K. Haze,'
- 'K. Aono,'
- 'T. Nakagawa,'
- 'H. Matsuhara,'
- 'H. Kataza,'
- 'T. Wada,'
- 'M. Kawada,'
- 'K. Fujiwara,'
- 'M. Mita,'
- 'S. Takeuchi,'
- 'K. Komatsu,'
- 'S. Sakai,'
- 'H. Uchida,'
- 'S. Mitani,'
- 'T. Yamawaki,'
- 'T. Miyata,'
- 'S. Sako,'
- 'T. Nakamura,'
- 'K. Asano,'
- 'T. Yamashita,'
- 'N. Narita,'
- 'T. Matsuo,'
- 'M. Tamura,'
- 'J. Nishikawa,'
- 'E. Kokubo,'
- 'Y. Hayano,'
- 'S. Oya,'
- 'M. Fukagawa,'
- 'H. Shibai,'
- 'N. Baba,'
- 'N. Murakami,'
- 'Y. Itoh,'
- 'M. Honda,'
- 'B. Okamoto,'
- 'S. Ida,'
- 'M. Takami,'
- 'L. Abe,'
- 'O. Guyon,'
- 'P. Bierden,'
- 'T. Yamamuro'
title: ' The SPICA coronagraphic instrument (SCI) for the study of exoplanets '
---
\
SPICA ,coronagraph ,instrument ,SCI ,exoplanet ,infrared
=0.5 cm
Background and Scientific Objective
===================================
We regard the systematic study of extra-solar planets (exoplanets) to be one of the most important tasks to be undertaken in space science in the near future. The enormous contrast between the parent star and the planet presents us with a serious problem. Therefore, there is a requirement for special instruments using techniques specifically designed to improve the contrast in order to perform a systematic observation of exoplanets. There are a number of different techniques currently used to observe exolanets. Since the first report by @Mayor1995 using the radial velocity method, more than 450 exoplanets have been discovered. It has also been shown that observations monitoring the transits of exoplanets provide a valuable means for studying them [@Charbonneau2000]. Not only detection but also spectroscopic studies of some transiting exoplanets have been carried out (e.g., @Deming2005; @Tinetti2007; @Swain2010). Recently, the spatially resolved direct detection of an exoplanet by coronagraphic imaging was finally reported (e.g., @Marois2008; @Kalas2008). While these methods are quite valuable, the current targets of these methods tend to be strongly biased. Many of the targets for radial velocity observations and the monitoring of transiting planets are ”Hot-Jupiters” that are close to their parent stars. On the other hand, exoplanets observed by current coronagraphic imaging tend to be strongly limited to young, giant outer planets. So the coronagraphic observation of mature, outer planets is still missing from the population. Considering this situation, we believe that the next important step in this field is a systematic study of exoplanets of various ages, masses, and distances from their parent stars. A spectroscopic survey of known exoplanets is especially important for characterizing planetary atmospheres.
The Space Infrared telescope for Cosmology and Astrophysics (SPICA) is an astronomical mission optimized for mid-infrared (MIR) and far-infrared astronomy with a 3m class ($\sim$3.0m effective pupil diameter/$\sim$3.2m physical diameter in the current design) on-axis telescope cooled to $<$6K [@Nakagawa2010]. We have proposed to develop a Coronagraphic Instrument for SPICA (SCI) and to carry out essential study in exoplanet science with the SCI (P.I.: K. Enya; @Enya2010 and its references).
SPICA has advantages as the platform for the SCI: it is perfectly free from band pass limitations and wavefront turbulence caused by the atmosphere. The cryogenic telescope provides high sensitivity in the infrared region. High stability is expected as the cryogenic telescope is to be launched into deep space, the Sun-Earth L2 Halo orbit. In addition, the structure of the SPICA telescope, adopting a monolithic primary mirror and carefully designed secondary support, yields a clean point spread function (PSF).
The major scientific tasks are described below. We regard that the instrument should be designed to carry out two “critical tasks”; one is the coronagraphic detection and characterization of exoplanets, and, the other is the monitoring of transiting planets. There are other important scientific tasks which can be carried out with the “given” performance of the designed instrument.
1. :\
One of the two critical scientific tasks for the SCI is to pursue the direct detection and spectroscopy of exoplanets. The SCI suppresses light from the parent star, and in spectroscopy mode working with coronagraph reveals essentially important spectral features in the MIR region, CH$_4$, H$_2$O, CO$_2$, CO, NH$_3$. Jovian exoplanets around (1) nearby stars($\leq$10pc) and (2) young and moderately old stars ($\leq$1Gyr old) are the primary targets for coronagraphic observations with the SCI (Fig.\[figure1\]). The former are suitable for detailed spatially-resolved observations, and the latter for gaining an understanding of the history of the formation of the planetary system. With the SCI we expect to be able to create an atlas of the various spectra of exoplanets by making observations on $\sim$100s of targets. More detail is given in @Fukagawa2009, @Matsuo2011.\
2. :\
The other critical scientific task is precise monitoring of transiting planets in order to characterize their spectral features. Coronagraphic observations by the SCI cover the “outer planets”, i.e., those at $\sim$10AU or further from the parent star. Therefore, the observation of transiting planets and coronagraphic observations are complementary techniques. It should be noted that the method for monitoring transiting planets requires the planets to be in edge-on orbits, and therefore discovery before observation with SPICA. The SCI has many advantages as a transit monitoring instrument. It has the capability for simultaneous observations with two detectors, having filters optimized for the spectral features of planetary atmospheres. Of the focal plane instruments on the SPICA mission, only the SCI covers the spectrum down to wavelengths of $\sim$1$\mu$m (coronagraphic high contrast images are guaranteed only for wavelengths down to 3.5$\mu$m; however the instrument has an InSb detector which is sensitive at shorter wavelengths). Another advantages of the SCI for this task are the superior pointing stability in the SPICA instruments, 0.05arcsec can be realized by an internal sensor, and potential capability for defocusing to avoid saturation by using an internal deformable mirror. Additionally, partial readout of the detectors can be adopted to improve the exposure/readout duty cycle when imaging.\
3. :\
Some other important observations relating to extra-solar planetary systems are planned with the given instrument design. Color Differential Astrometry (CDA) is being considered for the observation of planetary systems [@Abe2009]. CDA is a challenging method, but it has the advantage of having the capability of observing “inner planets” and does not require planets in edge-on orbits. Observations of the “snow line” (e.g., @Honda2009) and other features in circumstellar disks are other important targets for the SCI (@Tamura2009; @Takami2010).\
By using the spectral data sets of various exoplanets obtained by the SCI, we expect that our understanding of the whole planetary system will be significantly improved.
Instrument
==========
Specification
-------------
The specification of the instrument is summarized in Table\[table1\]. An overview of the current optical design of the SCI is shown in Fig.\[figure2\]. The requirement that gives us the limiting short wavelength (3.5$\mu$m) is derived for the direct detection and spectroscopy of Jovian exoplanets. As shown in Fig.\[figure1\], it is expected that the Spectral Energy Distribution (SED) of Jovian exoplanets has a peak in the 3.5–5$\mu$m wavelength region (Burrows2003). So this wavelength region is one of the most appropriate regions for the direct detection of Jovian planets. Wavelength coverage from MIR to 3.5$\mu$m allows us to study interesting molecular features of the planetary atmospheres, e.g., H$_2$O, CH$_4$, NH$_3$. The goal contrast (10$^{-6}$ at PSF) is derived for the systematic study of Jovian planets, including not only extremely massive, young planets but lower mass planets (down to $\sim$1M$_J$ with 1Gyr old targets) and older planets (up to 5Gyr old massive targets). It should be noted that the PSF subtraction technique has the potential to improve the final contrast after image processing, e.g., by one order or more ([@Trauger2007]; [@Haze2009]). Although the diffraction limited image at a wavelength of 5$\mu$m is a specification of the SPCIA telescope, wavefront correction using a deformable mirror (DM), make observations at shorter wavelengths possible with the SCI. The SCI is primarily used for coronagraphic imaging and in coronagraphic spectroscopy mode. On the other hand, non-coronagraphic imaging and spectroscopy are also possible because the coronagraph mask can be removed. In non-coronagraphic mode the SCI is useful as a general purpose fine-pixel camera and a spectrometer.
The high contrast in the coronagraphic image is guaranteed only within a field determined by the inner working angle (IWA), the outer working angle (OWA), and the number of actuators in the DM, whereas the SCI has a FoV of 1’$\times$1’. The number of DM actuators is 32$\times$32 which provides the capability to control the wavefront over an area of $16\lambda/D \times 16\lambda/D$ in the coronagraphic image. This high contrast area, $16\lambda/D \times 16\lambda/D$, determines the limit to the outer working angle (OWA) in the design of the coronagraph shown in Table.\[table1\].
Two channels, a short and a long wavelength channel, have been adopted, together with a beam splitter. The short and long wavelength channels have an InSb and a Si:As detector, respectively, and cover the wavelength regions $\leq$5$\mu$m and $\geq$5$\mu$m. This two channel design has some advantages: First, a higher sensitivity for the 3.5–27$\mu$m wavelength region is obtained compared to a single channel design. Secondary, the simultaneous observation of one target with two channels is possible. Thirdly, an appropriate pixel scale can be determined for each channel. In the current design, 1.5$\times$ Nyquist sampling (i.e., over sampling) at 3.5$\mu$m and 5$\mu$m has been adopted for the short and long channels, respectively. The simultaneous observation of one target with two channels is possible.
The IWA of the SCI, 3.3$\lambda/D$, corresponds to $\sim$1 and $\sim$2arcsec at wavelengths of 5$\mu$m and 10$\mu$m, respectively. These values are no smaller than those of the next generation of ground based coronagraphs, e.g.,the Gemini Planet Imager (GPI) or coronagraphs for 30m class telescopes. However, such coronagraphs used with ground based telescopes mainly work only in the window of atmospheric transmittance in infrared. This suggests that ground based coronagraphs are useful for imaging young planets, and are complementary to the SCI’s capability for wide band spectroscopy for cooler exoplanets. In comparison with the JWST coronagraphs, the most important advantage of the SCI is its capability for spectroscopic measurements which JWST does not have. Furthermore, the contrast of the SCI is significantly higher owing to the active wavefront control. The capability of the simultaneous use of two channels in the SCI is useful for image subtraction of coronagraphic observations, and monitoring transiting planets with wide wavelength coverage, which is not possible with JWST. [@Matsuo2011] presents more on the comparison between the SCI and the JWST coronagraphs, especially on the point of detectability of outer cooled planets.
Overview of the Instrument
---------------------------
The optics of the SCI is compact with the optical axis in one plane (Fig.\[figure2\]). The total mass is $\sim$30kg and the whole instrument, together with the telescope, is cooled to 5K before operation to enable the SPICA to achieve ultra-high sensitivity in the far- and mid-infrared regions. The SCI uses the center of the field of view in the focal plane of the telescope to obtain the best wavefront accuracy and symmetricity of the obscuration by the secondary mirror and its support structure(Fig.\[figure3\]). All the mirrors for collimation and focusing in the SCI are off-axis parabolas. Aberration is minimized at the center of the FoV in order to obtain the best coronagraphic PSF of the parent star. The pre-coronagraphic optics and the coronagraphic optics are common for both wavelength channels, while the post-coronagraphic optics is split into two channels.
The pre-coronagraphic optics consists of reflective optical devices in order to avoid ghosts and/or a wavelength dependent refractive indices which are sometimes issues in lens optics. In contrast to the other focal plane instruments on the SPICA mission, the SCI has no pick-off mirror, with the beam from the telescope secondary mirror arriving directly on the collimating mirror (Fig.\[figure2\]). This configuration makes the SCI compact under the constraint presupposing no lens optics. Minimizing the number of mirrors reduces the total light scattering at the surfaces of the mirrors. The first mirror of the SCI is placed after the focal point of the telescope. This is convenient for optically testing the SCI and the telescope. A DM is included in the SCI to correct the wavefront errors of the telescope.
As shown in Fig.\[figure4\], the baseline solution for the coronagraphic method is a binary-shaped pupil mask (see Sec.\[sec\_crg\]). With this method, only one pupil mask modifies the PSF and provides the required contrast. A focal plane mask is used to obscure the bright core in the coronagraphic PSF and to prevent scattered light from the PSF core polluting the dark region of the coronagraphic PSF in the post-coronagraphic optics. Interchangeable focal plane masks will be used to realize different observing modes, including a slit that can be used both with and without the pupil mask to provide spectroscopic capability in coronagraphic and non-coronagraphic observing modes. Fig.\[figure5\] shows the concept of spectroscopy working together with coronagraphy in the SCI.
The post-coronagraphic optics uses a beam splitter to split the optical path into two channels, the short wavelength channel with an InSb detector and the long wavelength channel with a Si:As detector. Each channel has filter wheels, and simultaneous observation of the same target with the two channels is possible. Each filter wheel contains a transmissive disperser (e.g., a grism) for spectroscopy.
In order to reduce technical risk in the development, another design of the instrument without the use of a DM is also under consideration, together with the full-equipped solution described above. As the result of this simplification, the contrast at the PSF is limited to be $\sim$10$^{-4}$. In this case, the advantage of high contrast over JWST is basically lost. However, the spectroscopy working with coronagraph remain a unique capability of the SCI. It is already know that there are young outer planets observed by direct imaging (e.g., @Marois2008). These planets are enough bright in infrared for the spectroscopic observation using the simplified design of the SCI. We believe that the spectrum data of such targets for the wide infrared wavelength range is essentially unique and important.
Key technologies
================
The SCI requires challenging technologies to realize high contrast for both imaging and spectroscopy over a wide MIR wavelength range. One of the critical technologies is the design, development and manufacture a coronagraph which can yield a high contrast PSF. We focused on a coronagraph using a multiple 1-dimensional barcode mask [@EnyaAbe2010] which is a type of binary shaped pupil mask (e.g., [@Kasdin2005]; [@Vanderbei2004]; [@Tanaka2006]). Another key technology is the adaptive optics, which improves the wavefront and stability of the PSF because the specification for these is more critical for the SCI than for the SPICA. Optical ghosts and scattering in the SCI should be enough nullified practically to avoid polluting the high contrast image. With this in mind, the development of a cryogenic MIR chamber for end-to-end tests of the coronagraph are ongoing. More details are presented in the following sections.
Coronagraph {#sec_crg}
-----------
A coronagraph to produce high contrast images is the core function of the SCI. The coronagraph for SPICA has to work over a wide MIR wavelength region in a cryogenic environment, so a coronagraph without transmissive optical devices is preferable. The coronagraph should be insensitive to telescope pointing errors caused by vibration from the cryogenic cooling system and the attitude control system. Achromatism (except the scaling effect for the size of the PSF) is also an important property. The coronagraph should be applicable for the pupil of the SPICA, which is partly obscured by the secondary mirror and its support structure. After taking these points into consideration, a coronagraph using a binary shaped pupil mask was selected as the primary method for our study. Experiments were carried out to confirm the feasibility of this strategy.
- :\
First we carried out experiments to validate the performance of the coronagraph with a checkerboard mask, which is a type of binary shaped pupil mask, using a visible laser in an air ambient. Checkerboard masks consisting of 100nm thick aluminum patterns on BK7 glass substrates were constructed using nano-fabrication technology. Electron beam lithography and a lift-off process were used in the fabrication process. Optimization of the checkerboard masks was executed using the LOCO software presented by @Vanderbei1999. The contrast, derived by averaging over the dark region using a linear scale and comparing this with the peak intensity, was 6.7$\times$10$^{-8}$ (Fig.\[figure6\]-(c),(d)).\
- :\
We installed multi-band Super Luminescent Diode Light Sources (SLED) for this experiment, and confirmed that the binary shaped pupil mask coronagraph works for light sources with center wavelengths ($\lambda$)of 650, 750, 800, 850nm with bandwidths ($\Delta\lambda$) of 8, 21, 25, and 55nm, respectively. The results of this experiment are shown in Fig.\[figure7\]. More details are given in @Haze2010.\
- :\
We designed some binary shaped pupil masks for the SPICA pupil mask having the obscuration shown in Fig.\[figure4\]. These designs consist of multi-barcode masks and have coronagraphic power in one dimension only. As a result, large opening angle is realized with keeping to satisfy the specification for the IWA. In the masks shown in Fig.\[figure4\], mask-1 is the baseline design. Additionally, mask-2 provides a small IWA to explore the field closer to the parent star. On the other hand, the contrast of mask-2 is not as high as mask-1 because there is a trade-off between the IWA and the contrast. These two masks are complementary, and can be changed by a mechanical mask changer. It should be noted that the principle of the barcode mask was presented by @Kasdin2005. More details are presented in @EnyaAbe2010\
- \
The trial fabrication of free standing masks (i.e., masks without a substrate) was carried out using various manufacturing methods. We tested the coronagraphic performance using a manufactured free standing mask made of thin copper plate and a visible He-Ne laser (Fig.\[figure8\]). This mask was also used in an experiment to demonstrate wavefront correction, and this was found to work successfully with this mask (see Sec.\[sec\_wfc\]).\
Combining these results implies that it is reasonable to assume that our binary shaped pupil mask coronagraph will work at MIR wavelengths. The development of a cryogenic chamber for end-to-end demonstration of the MIR coronagraph is ongoing (see Sec.\[sec\_milct\]). Toughness tests with vibration and acoustic load is to be undertaken for the mask in future work.\
Adaptive optics
---------------
The specification for the wavefront quality of the SPICA telescope is 350nm rms, while the requirement for the SCI is higher by a factor of 10 or more. Therefore, the SCI needs an internal active wavefront correction system which works at the temperature of the SCI (i.e., 5K). Therefore, we began the development of a cryogenic DM, which is one of the critical components needed to realize the SCI.
- :\
We developed a prototype DM with 32 channels consisting of a Micro Electro Mechanical Systems (MEMS) chip fabricated by the Boston Micromachines Corporation (BMC) and a special silicon substrate designed to minimize the thermal stress caused by cooling. Fig.\[figure9\] shows the result of a demonstration with this prototype in a cryogenic environment. For the first step of this work, a liquid nitrogen cooled chamber was used (i.e., without liquid helium) for convenience. As a result, the lowest temperature of the DM was limited to 95K. More than 80% of the linear thermal shrinkage of the silicon when cooling from room temperature to 5K occurs in the temperature range from 293 to 95K [@Okaji1999]. This fact suggests that the development strategy starting is reasonable. For the full demonstration of the cryogenic DM, development of a cryogenic 32$\times$32 channel DM and a 5K chamber is ongoing. More details are given in @Enya2009.\
- :\
From the studies with the prototype, we found that the thermal stress is the critical issue for our cryogenic DM if we intend to go to a larger format, e.g., a 1000 channel DM. Therefore, we carried out simulation studies of the thermal deformation and stress induced by cooling in order to obtain design solutions that would maintain the flatness of a 1000 channel DM cooled to 5K. We obtained an initial design, and further improvement is ongoing.\
- :\
Parasitic heat passing through the cables between the cooled DM and the warm electronics is another important issue. In the SPICA mission, the allocated parasitic heat for all focal plane instruments is limited to only 10mW. We attempted the manufacture of film print cables in order to reduce the parasitic heat to an ultra-low level ( $<$1mW via 1000 cables). We confirmed that the first version of our film print cable is sufficiently thin and provides electrical insulation up to 200V (Fig.\[figure9\]).\
These results suggest that the development of a cryogenic DM with 1000 channels is viable and that it will work at 5K. We are also considering a cryogenic multiplexer as a backup solution to the film print cables. Testing of toughness of the cryogenic DM will be also undertaken in future work.
Wavefront correction {#sec_wfc}
--------------------
Another important issue is how to operate a DM in order to correct wavefront errors. To develop an algorithm for SPICA/SCI, we initiated experiments using a visible laser, a commercially-available DM from BMC, a free standing coronagraph mask having the checkerboard design, and a CCD camera. Coronagraphic images were taken, and the speckle nulling method was applied to cancel the speckle in a dark region of the PSF [@Malbet1995]. A remarkable improvement of the coronagraphic contrast, to become much better than 10$^{-6}$ , was confirmed for an area just outside the inner working angle, which is an important area in the search for exoplanets (Fig.\[figure10\]). The algorithm used in this experiment included hundreds of iterations to produce anti-speckles. We plan in future work to try a more sophisticated algorithm which requires less number of images with different phases as presented in [@Borde2006], [@Giveon2007]. For the SCI, the phase shift will be achieved by operating an internal device (i.e., a deformable mirror) or moving the secondary mirror of the SPICA telescope. More detail is given in [@Kotani2010].
Cryogenic MIR Testbed {#sec_milct}
---------------------
Our previous laboratory experiments on the coronagraph were performed at room temperature and atmospheric pressure, and at visible wavelengths, whereas the SPICA coronagraph will have to be evaluated at cryogenic temperatures, in a vacuum, and at infrared wavelengths. In order to complete end-to-end testing of the MIR coronagraph at 5K, we are developing a cryogenic vacuum chamber in the Institute of Space and Astronautical Science (ISAS) of the Japan Aerospace Exploration Agency (JAXA). The cryogenic DM and free-standing mask described above will be included in this chamber.
Internal optics structure
-------------------------
It is planned that the internal off-axis mirror, its support structure, and the optical bench in the SCI will be made of the same material in order to avoid deformation of the optics due to mismatching of the thermal expansion coefficients. The primary candidate for this material is aluminum. Determination of the specification for the mirror quality is ongoing. It is expected that our requirements for the quality of the mirror surface are more relaxed than those for optical coronagraph missions targeting terrestrial planets [@Shaklan2006] because of the differences in target contrast and the observation wavelength region.
As presented above, key technologies required to realize a coronagraph with spectroscopic capabilities in the MIR for SPICA are being assessed. The indications are that all the major issues, including the binary shaped pupil mask coronagraph for the MIR and the cryogenic adaptive optics, can be resolved.
Acknowledgments {#acknowledgments .unnumbered}
===============
We deeply thank and pay our respects to all the pioneers in this field, especially R. Vanderbei and J. Kasdin. This work is supported by JAXA.
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--------------------------- -------------------------------------------------------------------------
Wavelength ($\lambda$)\* Core wavelength lambda = 3.5-27$\mu$m
Short wavelength channel: $\lambda$ = 3.5-5$\mu$m
Long wavelength channel: $\lambda \ge 5 \mu$m
Coronagraph method Binary shaped pupil mask
Observation mode Coronagraphic imaging
Coronagraphic spectroscopy
Non-coronagraphic imaging
Non-coronagraphic spectroscopy
contrast $10^{-6}$ @PSF
Inner working angle (IWA) 3.3$\lambda/D$ \*\*
Outer working angle (OWA) 16$\lambda/D$
Filter Band-pass filters for each channel
Detector a 1K $\times$ 1K Si:As array for the long wavelength channel
a 1K $\times$ 1K InSb array for the short wavelength channel
FoV High-contrast coronagraphic FoV:16$\lambda/D$
(FoV of $1' \times 1'$ is available but high-contrast is not guaranteed
out of 16$\lambda/D$)
Spectral resolution $\sim$20 and $\sim$200 (realized by transmissive dispersion devices,
e.g. grisms)
--------------------------- -------------------------------------------------------------------------
: Specification for the SCI
\[table1\]
\* At $\lambda < 3.5\mu$m, high contrast imaging is not guaranteed, but the instrument has sensitivity through the InSb detector.\
\*\* $D$ is telescope aperture diameter.
![ Calculated SEDs of 1Gyrs (left) and 5Gyrs (right) old Jovian planets with various masses presented as Fig.13 in @Burrows2003, and properties relating to SPICA observations. 10pc is assumed as the distance to the planetary system. The gray solid curve shows the sensitivity limit of imaging with SPICA. The orange and purple dashed lines show the scaled SEDs of G2 and M0 type stars, respectively. These figures are from @Fukagawa2009. []{data-label="figure1"}](fig1.eps){width="16cm"}
![Overview of the optical design of the SCI. []{data-label="figure2"}](fig2.eps){width="16cm"}
![Current design of the distribution of the FoV for the SPICA instruments. []{data-label="figure3"}](fig3.eps){width="9cm"}
![ Design of a multi-barcode 1D coronagraph for the SPICA pupil. Top and bottom panels show current baseline and an optional design, respectively. The mask design (left) and the simulated PSF using this pupil (right). Transmissivity of the mask is 1 and 0 at white and black area, respectively. []{data-label="figure4"}](fig4.eps){width="9.5cm"}
![Spectroscopy working together with coronagraphy in the SCI. []{data-label="figure5"}](fig5.eps){width="14cm"}
![ (a): Configuration used in demonstrating the operation of a binary shaped pupil mask. (b): Checkerboard mask manufactured on a glass substrate by nano-fabrication technology using electron beam lithography and a lift-off process. Top and bottom panels show whole of the device, and microscopic image, respectively. This mask is designed to produce four dark region of square shape, DR1-DR4, around the core of the PSF. (c): Experimental PSF (top) and high sensitivity image of a dark region, DR2, of the PSF (bottom). (d): The observed and theoretical coronagraphic profiles as well as the theoretical Airy profile. Each profile is normalized to the peak intensity in each image. []{data-label="figure6"}](fig6.eps){width="16cm"}
![ Results of a multi-color/broadband experiment using a checkerboard mask. Only images of the dark region are shown. Ghosting caused by the lens puts a practical limit on the contrast at longer wavelengths. []{data-label="figure7"}](fig7.eps){width="16cm"}
![ Left: a free standing checkerboard mask made of a thin copper plate installed in the test setup using a visible laser in air. Middle and right: microscope pictures of the free standing checkerboard mask made of thin copper plate. []{data-label="figure8"}](fig8.eps){width="16cm"}
![ (a)-(c): schematic views and a photograph of the prototype cryogenic DM unit. (d): results of a demonstration of the prototype cryogenic DM. (e): A sample of the film print cable. []{data-label="figure9"}](fig9.eps){width="16cm"}
![ Results of the speckle nulling experiment. Left and middle: images of a dark region of the PSF obtained with a square hole mask, before and after speckle nulling, respectively. Right: Diagonal cut through the PSF. []{data-label="figure10"}](fig10.eps){width="16cm"}
|
---
abstract: 'We consider the problem of approximating a linear cocycle (or, more generally, a vector bundle automorphism) over a fixed base dynamics by another cocycle admitting a dominated splitting. We prove that the possibility of doing so depends only on the homotopy class of the cocycle, provided that the base dynamics is a minimal diffeomorphism and the fiber dimension is least $3$. This result is obtained by means of a general theorem on the existence of almost invariant sections for fiberwise isometries of bundles with compact fibers and finite fundamental group. The main novelty of the proofs is the use of a quantitative homotopy result due to Calder, Siegel, and Williams.'
address: 'Facultad de Matemáticas, Pontificia Universidad Católica de Chile'
author:
- Jairo Bochi
date: 'August 26, 2014'
title: Cocycles of isometries and denseness of domination
---
[The author was partially supported by project Fondecyt 1140202 and by the Center of Dynamical Systems and Related Fields ACT1103.]{}
Introduction
============
A dynamical interplay
---------------------
This paper deals with the dynamics of certain classes of fiber bundle automorphisms. In particular, these include skew-products $g(x,y) = (f(x),g_x(y))$ acting on trivial bundles $X \times Y$. If the map $x \mapsto g_x$ takes values in a specific group $G$ of transformations of the fiber $Y$, it is called a *$G$-cocycle*.
The first class we consider consists of vector bundle automorphisms, which in particular include linear cocycles. To investigate them, it is often useful to consider induced automorphisms on other (not necessarily linear) fiber bundles. The simplest example is the projectivization of a vector bundle automorphism. A related linearly-induced automorphism of a fiber bundle with ${\mathrm{SO}}(m)$ fibers was used in an essential way by V.I. Oseledets in the proof of his celebrated theorem: see [@Oseledets p. 229]. Quoting [@Selgrade], compactness of the fibers “allows the use of techniques not available for the vector bundle”.
Nevertheless, it is also useful to consider linearly-induced automorphisms on bundles with non-compact fibers, especially if these have some extra structure. For example, a ${\mathrm{SL}}(2,{\mathbb{R}})$-cocycle induces a cocycle of Möbius transformations of the complex half-plane, which are isometries with respect to the hyperbolic metric, and many linear-algebraic properties of the former cocycle can be understood in terms of geometric properties of the latter. A far reaching extension of this interplay is revealed by the Karlsson–Margulis theorem [@KM] on cocycles of isometries of spaces of nonpositive curvature, which yields Oseledets theorem as a corollary. The remarkable generality and simplicity of the Karlsson–Margulis theorem have instant appeal and justify the study of cocycles of isometries for its own sake.
The notion of *dominated splittings* is central to the dynamics of vector bundle automorphisms and is a major motivation for this paper. It basically consists on a projective form of hyperbolicity, and it is equivalent to ordinary uniform hyperbolicity in the case of ${\mathrm{SL}}(2,{\mathbb{R}})$-cocycles. The term “domination” was coined by R. Mañé in the 1970’s, although the concept was actually introduced earlier in differential equations theory under the name “exponential separation”: see [@Sambarino; @Palmer] and references therein. Dominated splittings are intrinsically related to chain recurrence properties of the induced projectivized automorphism [@Selgrade; @CK], and can also be characterized in terms of separation between singular values [@BG; @Morris]. Since Mañé, dominated splittings continue to play a important role in differentiable dynamics on compact manifolds: see [@BDV; @Sambarino].
The issue we are concerned with here is denseness of domination: when can a given vector bundle automorphism be approximated by another having a dominated splitting? Under reasonable assumptions, we reduce this question to a problem about the existence of almost invariant sections for fiberwise isometries, which we them solve in a much greater generality. That general result is the core of this paper. It turns out to have other applications: we use it to characterize almost coboundaries on compact Lie groups with finite center.
Other very general constructions of invariant and almost invariant sections for cocycles of isometries appear in the works [@CNP; @BN_geometric]; these rely on nonpositive curvature and are highly geometrical. By contrast, the isometries considered in this paper act on compact fibers, whose geometries are less favorable: for example, shortest geodesics between pairs of points are not necessarily unique. Actually the arguments developed here are much more topological than geometrical, and use as a crucial ingredient beautiful results on *quantitative homotopy* by Calder, Siegel, and Williams [@CS80; @SW89].
Let us proceed with precise statements.
Domination and the problem of denseness
---------------------------------------
Let $X$ be a compact Hausdorff space. Let $m \ge 2$ be an integer, and let ${\mathbb{E}}$ be a $m$-plane bundle over $X$, that is, a real vector bundle with base space $X$ and fibers of dimension $m$. We endow ${\mathbb{E}}$ with a Riemannian norm.
If $f \colon X \to X$ is a homeomorphism, we let ${\mathrm{Aut}}({\mathbb{E}},f)$ denote the space of automorphisms of ${\mathbb{E}}$ fibering over $f$, endowed with the uniform (i.e. $C^0$) topology.
When the vector bundle is trivial, that is ${\mathbb{E}}= X \times {\mathbb{R}}^m$, there is an identification ${\mathrm{Aut}}({\mathbb{E}},f) = C(X,{\mathrm{GL}}(m,{\mathbb{R}}))$; indeed every automorphism is of the form $(x,v) \mapsto (f(x),A(x)v)$ for some continuous map $A \colon X \to {\mathrm{GL}}(m,{\mathbb{R}})$, which is called a *linear cocycle*.
Consider a splitting ${\mathbb{E}}= {\mathbb{E}}^1 \oplus {\mathbb{E}}^2 \oplus \cdots \oplus {\mathbb{E}}^k$ of the bundle ${\mathbb{E}}$ as a sum of proper nontrivial subbundles ${\mathbb{E}}^i$. This splitting is called *dominated* with respect to an automorphism $A \in {\mathrm{Aut}}({\mathbb{E}},f)$ if each subbundle ${\mathbb{E}}^i$ is $A$-invariant and moreover there is a constant integer $\ell \in {\mathbb{N}}$ such that for all $x \in X$, all $i \in \{1,\dots,k-1\}$, and all unit vectors $v_i \in {\mathbb{E}}^i_x$, $v_{i+1} \in {\mathbb{E}}^{i+1}_x$, we have $$\| A^\ell (v_i) \| > \|A^\ell (v_{i+1})\| \, .$$ That is, up to replacing $A$ by a power, any vector in ${\mathbb{E}}^i_x$ is relatively more expanded than any vector in ${\mathbb{E}}^{i+1}_x$. We also say that ${\mathbb{E}}^i$ *dominates* ${\mathbb{E}}^{i+1}$.
Let us consider the base dynamics $f$ as fixed. An important feature of domination is *openness*: the set of automorphisms admitting a dominated splitting is open in ${\mathrm{Aut}}({\mathbb{E}},f)$. On the other hand, domination is not dense in general. If $f$ has a periodic point $x$ of period $p$ such that the restriction of the power $A^p$ to the fiber ${\mathbb{E}}_x$ has exactly two eigenvalues of maximum absolute value, and these eigenvalues are non-real, then the automorphism $A$ cannot admit a dominated splitting whose top subbundle ${\mathbb{E}}^1$ is one-dimensional. Such a condition is open in ${\mathrm{Aut}}({\mathbb{E}},f)$. With this kind of reasoning we can exhibit nonempty open subsets of ${\mathrm{Aut}}({\mathbb{E}},f)$ formed by automorphisms that admit no dominated splitting at all, provided $f$ has sufficiently many periodic points.
Different obstructions to domination may be due to topological reasons: sometimes the homotopy type of $A$ forbids the existence of an invariant splitting, and in particular, of a dominated one. (See § \[ss.two\_types\] for an example.)
Suppose that the base dynamics $f$ is minimal (and the base space is infinite), and so periodic orbit obstructions do not arise. Our first main result basically states that all robust obstructions to domination are topological, provided the (linear) dimension is at least $3$. The precise statement is as follows:
\[t.densedom\] Let $f \colon X \to X$ be a minimal diffeomorphism of a compact manifold $X$ of positive dimension. Let ${\mathbb{E}}$ be a $m$-plane bundle over $X$, where $m \ge 3$. Then for each fibered homotopy class ${\mathcal{C}}\subset {\mathrm{Aut}}({\mathbb{E}},f)$,
1. \[i.obstruction\] either no automorphism in ${\mathcal{C}}$ has a proper nontrivial invariant subbundle;
2. \[i.densedom\] or there is an open and dense subset ${\mathcal{D}}\subset {\mathcal{C}}$ such that all automorphisms in ${\mathcal{D}}$ have a dominated splitting.
Here the *fibered homotopy class* of an automorphism is its path-connected component in ${\mathrm{Aut}}({\mathbb{E}},f)$; the corresponding paths are called *fibered homotopies*. So the theorem states that if an automorphism $A$ is fibered homotopic to another having a nontrivial continuous invariant field of planes then a perturbation of $A$ has a dominated splitting. In particular, domination is either empty or dense inside each fibered homotopy class.
Notice that Theorem \[t.densedom\] requires $f$ to be a diffeomorphism. (In this paper, we assume all manifolds to be $C^\infty$, without boundary, and paracompact, and all diffeomorphisms to be $C^\infty$.) Although this assumption should be stronger than necessary, it is technically very convenient for certain parts of the construction (especially those in Appendix \[s.appendix\]), and so we have not tried to optimize it.
More information about the classes ${\mathcal{C}}$ of type (\[i.obstruction\]) in Theorem \[t.densedom\] is available: generically in ${\mathcal{C}}$ the automorphism is uniformly subexponentially quasiconformal (by a result of [@B_Studia]), and densely in ${\mathcal{C}}$ there is an invariant conformal structure (by a result of [@BN_elementary]); see § \[ss.dom\_vs\_conf\] for details.
Theorem \[t.densedom\] does not hold in dimension $m=2$, because in this case there exists another obstruction to domination related to the rotation number: see § \[ss.dim2\].
For examples, we refer the reader to §§ \[ss.two\_types\] and \[ss.indices\], where we show that ${\mathrm{Aut}}({\mathbb{E}},f)$ can indeed contain classes of both types (\[i.obstruction\]) and (\[i.densedom\]), and that a class of type (\[i.densedom\]) can contain different types of domination.
Other results on denseness of domination may be found in the papers [@Million] (for autonomous linear differential equations), [@Cong] (for bounded measurable cocycles), [@ABD1; @ABD2] (for continuous ${\mathrm{SL}}(2,{\mathbb{R}})$-cocycles over uniquely ergodic dynamics), [@FJZ] (for Hölder-continuous ${\mathrm{SL}}(2,{\mathbb{R}})$-cocycles over generic irrational flows on the two-torus), and [@AJS] (for analytic complex-valued cocycles over rotations).
We next describe the setting of fiberwise isometries, which we will later relate to Theorem \[t.densedom\].
Fiberwise isometries and almost invariant sections {#ss.fib_isom_sections}
--------------------------------------------------
Let $X$ be a compact Hausdorff space, and let $Y$ be a manifold. A fiber bundle ${{Y} \hookrightarrow {Z} \overset{{p}}{\to} {X}}$ is called *fiberwise smooth* if its structural group is formed by diffeomorphisms of $Y$. Then each fiber $Z_x \coloneqq p^{-1}(x)$ has a manifold structure and is diffeomorphic to $Y$. If $g \colon Z \to Z$ is a bundle automorphism then there exists a homeomorphism $f \colon X \to X$ such that $g$ diffeomorphically maps the fiber $Z_x$ to the fiber $Z_{f(x)}$. We say that $g$ fibers over $f$. The set of all automorphisms of $Z$ is denoted by ${\mathrm{Aut}}(Z)$, and the set of automorphisms fibering over a given $f \in{\mathrm{Homeo}}(X)$ is denoted by ${\mathrm{Aut}}(Z,f)$.
The *vertical tangent bundle* is the union $\bigsqcup_{x\in X} TZ_x$ of the tangent bundles of the fibers, endowed with the obvious vector bundle structure. A *fibered Riemannian structure* on the fiberwise smooth bundle ${{Y} \hookrightarrow {Z} \overset{{p}}{\to} {X}}$ is a continuous field of positive definite quadratic forms on the vertical tangent bundle whose restriction to each $TZ_x$ is a (smooth) Riemannian metric on the manifold $Z_x$. Such structures always exist. An automorphism $g \in {\mathrm{Aut}}(Z)$ is called a *fiberwise isometry* if it preserves a given fibered Riemannian structure.
Let ${\mathrm{Sec}}(Z)$ denote the space of all sections of $Z$, that is, all continuous maps $\sigma \colon X \to Z$ such that $p \circ \sigma = {\mathrm{id}}_X$. The *distance* between $\sigma$, $\sigma' \in {\mathrm{Sec}}(Z)$ is defined as $$\label{e.def_distance}
\mathrm{d}(\sigma, \sigma') \coloneqq \sup_{x \in X} \mathrm{d}_x( \sigma(x), \sigma'(x)),$$ where $\mathrm{d}_x$ denotes Riemannian distance on the fiber $Z_x$. This makes ${\mathrm{Sec}}(Z)$ a metric space.
Throughout this paper, we denote the unit interval as $$I \coloneqq [0,1] \, .$$ We say that $\sigma$, $\sigma' \in {\mathrm{Sec}}(Z)$ are *fibered homotopic* if they are homotopic through sections, that is, there exists a continuous curve $t \in I \mapsto \sigma_t \in {\mathrm{Sec}}(Z)$ (called a *fibered homotopy*) from $\sigma_0 = \sigma$ to $\sigma_1 = \sigma'$. If the fibered homotopy is of the form $\sigma_t = (g_t)_* \sigma$ for some continuous curve $t \in I \mapsto g_t \in {\mathrm{Aut}}(Z,{\mathrm{id}})$ starting from $g_0 = {\mathrm{id}}$ then we say that $\sigma$ and $\sigma'$ are *isotopic*, and that $\{g_t\}_{t \in I}$ is an *ambient isotopy* that moves the section $\sigma$ to the section $\sigma'$.
Given $g \in {\mathrm{Aut}}(Z)$ and $\sigma \in {\mathrm{Sec}}(Z)$, we define a new section $g_*\sigma \in {\mathrm{Sec}}(Z)$ by $$\label{e.push_section}
(g_* \sigma)(x) \coloneqq g(\sigma(f^{-1}(x))) \, ,$$ where $f$ is the homeomorphism over which $g$ fibers. A section $\sigma$ is called:
- *$g$-invariant* if $g_* \sigma = \sigma$;
- *${\varepsilon}$-almost $g$-invariant* for some ${\varepsilon}> 0$ if $\mathrm{d}(g_* \sigma, \sigma) < {\varepsilon}$;
- *$g$-invariant up to homotopy* if $\sigma$ and $g_* \sigma$ are fibered homotopic;
- *$g$-invariant up to isotopy* if $\sigma$ and $g_* \sigma$ are isotopic.
We can now state the second main result of this paper:
\[t.sections\] Let $f \colon X \to X$ be a minimal diffeomorphism of a compact manifold $X$ of positive dimension. Let $Y$ be a compact connected manifold with finite fundamental group. Consider a fiberwise smooth bundle ${{Y} \hookrightarrow {Z} \overset{{p}}{\to} {X}}$ endowed with a fibered Riemannian structure, and let $g \in {\mathrm{Aut}}(Z,f)$ be a fiberwise isometry. Suppose that $\sigma \in {\mathrm{Sec}}(Z)$ is $g$-invariant up to isotopy. Then for any ${\varepsilon}>0$ there exists an ${\varepsilon}$-almost $g$-invariant section $\omega \in {\mathrm{Sec}}(Z)$ that is fibered homotopic to $\sigma$.
Actually it is equivalent to suppose that $\sigma$ is $g$-invariant up to homotopy. This equivalence can be proven by using vector bundle neighborhoods [@Palais Thrm. 12.10] (a tool generally used to endow ${\mathrm{Sec}}(Z)$ with a Banach manifold structure), but we will not provide the technical details.
As a corollary of Theorem \[t.sections\], we will show in § \[ss.coboundaries\] that a cocycle on a compact Lie group with finite center is an almost coboundary if and only if it is homotopic to a coboundary.
Theorem A from [@BN_geometric] also constructs almost invariant sections for fiberwise isometries, but under hypotheses very different from those of Theorem \[t.sections\]. See § \[ss.unification\] for a discussion of possible connections between these two results.
Comments on the proofs and organization of the paper
----------------------------------------------------
The broad strategy that we follow to prove Theorem \[t.densedom\] is the same used in [@ABD1; @ABD2] in a more restricted setting. Absence of domination allows us to mix Lyapunov exponents and make the dynamics conformal, after a suitable perturbation. Conformality allows us to induce certain fiberwise isometries, and using almost invariant sections we introduce some weak domination with further perturbations. (See §§ \[ss.dom\_vs\_conf\]–\[ss.proof\_densedom\] for details.)
Let us comment how the construction of almost invariant sections presented in this paper relates to previous ones. The papers [@ABD1; @ABD2] use “dynamical stratifications” (see § \[ss.strat\]) and towers to construct almost-invariant sections. Actually these constructions, which only form part of these papers, can be considerably simplified by the geometric methods of [@BN_geometric] or the more specific linear-algebraic methods of [@BN_elementary]. Unfortunately, these “cleaner” methods require a convenient geometry and do not apply to the situation considered here. Thus our constructions are closer to those of [@ABD1; @ABD2] (though we do not directly use results from these papers).
The generality of Theorem \[t.sections\] creates new topological problems, and we need two novel tools: One tool is a certain regularity property of the dynamical stratifications which, despite being natural, is not straightforward to obtain. The other tool is actually not new, but this is perhaps the first time it is used for dynamical applications: it is a “quantitative homotopy” result from [@CS80; @SW89].
The rest of this paper is organized as follows: In Section \[s.consequences\] we explain a result from [@BN_elementary] which is then combined with Theorem \[t.sections\] to deduce Theorem \[t.densedom\]; we also explain an independent application of Theorem \[t.sections\] to almost coboundaries. In Section \[s.ingredients\] we explain the two new tools mentioned above, which we then employ in Section \[s.proof\_sections\] to prove Theorem \[t.sections\]. Section \[s.examples\] contains examples and remarks on the necessity of the various hypotheses in our theorems, as well as questions for future research. The more technical construction of regular dynamical stratifications is given in Appendix \[s.appendix\].
Discussions with Carlos Tomei (PUC–Rio) lead me to believe that slow homotopies should exist under mild conditions, and so propelled me to search the literature until I found the papers by Calder, Siegel, and Williams.
Consequences of Theorem \[t.sections\] {#s.consequences}
======================================
Domination versus conformality {#ss.dom_vs_conf}
------------------------------
The link between our two main Theorems \[t.densedom\] and \[t.sections\] is made by means of the following result:
\[t.BN\] Let $f \colon X \to X$ be a minimal homeomorphism of a compact space $X$ of finite dimension. Let ${\mathbb{E}}$ be a vector bundle over $X$. Then there exists a dense subset ${\mathcal{I}}\subset {\mathrm{Aut}}({\mathbb{E}},f)$ such that for every $A \in {\mathcal{I}}$,
1. \[i.BN\_noDS\] either $A$ has a dominated splitting;
2. \[i.BN\_conformal\] or $A$ is conformal with respect to some Riemannian metric on ${\mathbb{E}}$.
Domination evidently fails in the second alternative, and it does so in the most extreme of ways: all vectors in the same fiber are expanded (at time $1$) at exactly the same rate.
Let us summarize what is involved in the proof of this result. The first part of the proof is to apply a theorem from [@B_Studia] which, extending previous results of [@BV; @AB], states that generic elements of ${\mathrm{Aut}}({\mathbb{E}},f)$ either admit dominated splittings or are uniformly subexponentially quasiconformal (i.e., such that its Oseledets decompositions are all trivial). The one-phrase rationale behind it is this: absence of domination allows Lyapunov exponents to be mixed by suitable perturbations. The second part of the proof of Theorem \[t.BN\] is to construct a Riemannian metric with respect to which the quasiconformal distortion is small. The third and final part is to perturb the automorphism to become conformal with respect to this metric – which is not obvious, because the new metric is usually very distorted as compared to the initial one. The second and third parts of the proof can be carried out by using elementary linear-algebraic tools, as it is done in [@BN_elementary], or by geometric constructions on fiberwise isometries of suitable spaces, as in [@BN_geometric]. Finally, let us mention that the three parts of the proof can be refined in order to yield similar conformality properties inside the subbundles of the finest dominated splitting: the result is Theorem 2.4 from [@BN_elementary], and Theorem \[t.BN\] is actually a corollary of it.
The Grassmannian bundle and deduction of Theorem \[t.densedom\] {#ss.proof_densedom}
---------------------------------------------------------------
Let us prepare the ground for the use of Theorem \[t.sections\]. A general procedure for obtaining fiberwise isometries is as follows:
\[p.fibered\_riem\] Consider a fiberwise smooth bundle ${{Y} \hookrightarrow {Z} \overset{{p}}{\to} {X}}$ whose structural group $H \subset {\mathrm{Diff}}(Y)$ is compact. Then the bundle admits a fibered Riemannian structure with respect to which any $H$-automorphism of $Z$ is a fiberwise isometry.
Start with any Riemannian metric on $Y$. By averaging with respect to the Haar measure of $H$, we obtain a Riemannian metric on $Y$ that is preserved by $H$. Therefore we can pull it back by bundle charts and obtain a well defined fibered Riemannian structure on $Z$. This structure is obviously preserved by any $H$-automorphism of $Z$.
Given integers $1 \le k < m$, let ${\mathrm{Gr}}(k,m)$ denote the *Grassmannian* whose elements are the $k$-planes in ${\mathbb{R}}^m$; these are compact connected manifolds, and their fundamental groups are (see e.g. [@Arkowitz p. 189]): $$\label{e.pi1grass}
\pi_1 ({\mathrm{Gr}}(k,m)) =
\begin{cases}
{\mathbb{Z}}&\text{if $m=2$,} \\
{\mathbb{Z}}_2 &\text{if $m\ge 3$.}
\end{cases}$$ Each linear automorphism of ${\mathbb{R}}^m$ induces a diffeomorphism of ${\mathrm{Gr}}(k,m)$ in the obvious way. This defines a homomorphism $\iota \colon {\mathrm{GL}}(d,{\mathbb{R}}) \to {\mathrm{Diff}}({\mathrm{Gr}}(k,m))$ whose kernel is formed by the nonzero multiples of the identity matrix.
If ${\mathbb{E}}$ is a $m$-plane bundle over a compact Hausdorff space $X$, let $G_k({\mathbb{E}})$ denote the set of all $k$-planes contained in the fibers of ${\mathbb{E}}$. This set can be given the structure of a fiber bundle with base space $X$, typical fiber ${\mathrm{Gr}}(k,m)$, and structural group $\iota({\mathrm{GL}}(d,{\mathbb{R}}))$. Any automorphism $A$ of ${\mathbb{E}}$ induces an automorphism $\bar{A}$ of $G_k({\mathbb{E}})$.
Let us explain how a Riemannian metric on the vector bundle ${\mathbb{E}}$ induces a fibered Riemannian structure on the fiber $G_k({\mathbb{E}})$. Given such a Riemannian metric, we use it to stiffen the fiber bundle structure of ${\mathbb{E}}$ so that the structural group is the orthogonal group ${\mathrm{O}}(m)$. We also stiffen the fiber bundle $G_k({\mathbb{E}})$ so that the structural group is $\iota({\mathrm{O}}(m))$. Since this group is compact, Proposition \[p.fibered\_riem\] provides us with a fibered Riemannian structure on $G_k({\mathbb{E}})$ that has the following property: for any automorphism $A$ of ${\mathbb{E}}$ that is conformal with respect to the Riemannian metric, the induced automorphism $\bar{A} \in {\mathrm{Aut}}(G_k({\mathbb{E}}))$ is a fiberwise isometry.
Let $f \colon X \to X$ be a minimal diffeomorphism. Let ${\mathbb{E}}$ be a $m$-plane bundle over $X$, where $m \ge 3$. Fix a fibered-homotopy class ${\mathcal{C}}\subset {\mathrm{Aut}}({\mathbb{E}},f)$. Let ${\mathcal{D}}$ be the open subset of ${\mathcal{C}}$ formed by the automorphisms that have a dominated splitting. Suppose that we are not in case (\[i.obstruction\]) in the statement of the theorem, that is, there exist $A_0 \in {\mathcal{C}}$ and $k \in \{1,2,\dots,m-1\}$ with a continuous invariant field of $k$-planes. This means that the induced automorphism $\bar{A}_0 \colon G_k({\mathbb{E}}) \to G_k({\mathbb{E}})$ has an invariant section $\sigma \colon X \to G_k({\mathbb{E}})$.
Take an arbitrary open set ${\mathcal{V}}\subset {\mathcal{C}}$; we will show that ${\mathcal{D}}\cap {\mathcal{V}}\neq {\varnothing}$, so concluding that property (\[i.densedom\]) holds and therefore proving the theorem.
Let ${\mathcal{I}}$ be the dense subset of ${\mathrm{Aut}}({\mathbb{E}},f)$ provided by Theorem \[t.BN\]. Fix $A_1 \in {\mathcal{I}}\cap {\mathcal{V}}$. If $A_1 \in {\mathcal{D}}$ then we have nothing to show, so assume that $A_1 \not\in {\mathcal{D}}$. Then we are in case (\[i.BN\_conformal\]) in Theorem \[t.BN\], that is, there is a Riemannian metric on ${\mathbb{E}}$ with respect to which $A_1$ is conformal. As explained above, we can endow the bundle $G_k({\mathbb{E}})$ with a fibered Riemannian structure with respect to which the automorphism $\bar{A}_1$ is a fiberwise isometry.
Since $A_0$ and $A_1$ belong to the class ${\mathcal{C}}$, there exists a fibered homotopy $(A_t)_{t \in I}$ in ${\mathrm{Aut}}(E,f)$ between $A_0$ and $A_1$. Then $(\bar{A}_t \circ \bar{A}_0^{-1})_{t\in I}$ is an ambient isotopy that moves the section $\sigma$ to the section $(\bar{A}_1)_* \sigma$. In particular, $\sigma$ is $\bar{A}_1$-invariant up to isotopy.
The fibers of the bundle $G_k({\mathbb{E}})$ satisfy the hypotheses of Theorem \[t.sections\]: they are compact connected manifolds which by have finite fundamental groups. Therefore for each $i \in {\mathbb{N}}$ we can apply Theorem \[t.sections\] and obtain an $1/i$-almost $\bar{A}_1$-invariant section $\omega_i \colon X \to G_k({\mathbb{E}})$. This means that $\omega_i$ is uniformly $1/i$-close to the section $\omega_i'$ defined by $\omega_i'(x) \coloneqq A_{1\star}(f^{-1}x) (\omega_i(x))$.
For each $i \in {\mathbb{N}}$, we can find an automorphism $R_i \in {\mathrm{Aut}}({\mathbb{E}},{\mathrm{id}})$ such that for each $x \in X$, $R_i(x)$ is an orthogonal linear map (with respect to the Riemannian metric on the fiber ${\mathbb{E}}_x$) and sends the $k$-plane $\omega_i(x)$ to the $k$-plane $\omega_i'(x)$. Moreover, it is possible to choose the sequence $(R_i)$ converging to the identity autormorphism.
For each $i \in {\mathbb{N}}$ and $x \in X$, let $D_i(x)$ be the isomorphisms of ${\mathbb{E}}_x$ that preserves the $k$-plane $\omega_i(x)$ and its orthogonal complement $\omega_i^\perp(x)$, and whose restriction to $\omega_i(x)$ (resp. $\omega_i^\perp(x)$) is $e^{1/i}$ (resp. $e^{-1/i}$) times the identity. This defines a sequence of automorphisms $D_i \in {\mathrm{Aut}}({\mathbb{E}},{\mathrm{id}})$ that converges to the identity.
The sequence $(B_i)$ on ${\mathrm{Aut}}({\mathbb{E}},f)$ defined by $B_i \coloneqq D_i \circ R_i \circ A_1$ converges to $A_1$. Moreover each $B_i$ belongs to ${\mathcal{D}}$, since it admits the dominated splitting $\omega_i \oplus \omega_i^\perp$. Therefore ${\mathcal{D}}\cap {\mathcal{V}}\neq {\varnothing}$, as we wanted to prove.
Almost coboundaries {#ss.coboundaries}
-------------------
We will describe another application of Theorem \[t.sections\].
Let $f \colon X \to X$ be a homeomorphism of a compact Hausdorff space, let $G$ be a topological group, and let $A \colon X \to G$ be continuous cocycle over $f$. We say that $A$ is a *coboundary* if there exists a continuous map $B \colon X \to G$ such that $$A(x) = B(f(x)) B(x)^{-1} \, .$$ A cocycle is called an *almost coboundary* if it is the limit of a sequence of coboundaries.
\[c.coboundaries\] Let $f \colon X \to X$ be a minimal diffeomorphism of a compact manifold $X$ of positive dimension. Let $G$ be a compact Lie group with finite center. Then a cocycle $A \colon X \to G$ is an almost coboundary if and only if it is homotopic to a coboundary.
Endow $G$ with a bi-invariant metric, and consider the product bundle $X \times G$ over $X$. Then each cocycle $A \colon X \to G$ induces a fiberwise isometry $g(x,y) = (f(x), A(x) y)$; moreover $A$ is a coboundary if and only if $g$ has an invariant section, and $A$ is homotopic to a coboundary if and only if $g$ has a section that is invariant up to isotopy.
First consider the case of connected $G$. Since the Lie algebra of $G$ has trivial center, by a theorem of Weyl (see e.g. [@Hsiang p. 82]), the fundamental group of $G$ is finite. Therefore the corollary follows from Theorem \[t.sections\].
In general, if a cocycle $A$ is homotopic to a coboundary $x \mapsto B(f(x)) B(x)^{-1}$, then the cocycle $x \mapsto B(f(x))^{-1} A(x) B(x)$ takes values in the identity component of $G$, and therefore the corollary follows from the previous case.
Ingredients for the proof of Theorem \[t.sections\] {#s.ingredients}
===================================================
Speed control for homotopies
----------------------------
We begin with an informal motivation for quantitative homotopy problems. Let $X$ be any topological space, and let $Y$ be a Riemannian manifold. A homotopy $F \colon X \times I \to Y$ is called *$c$-Lipschitz* if $$d(F(x,t), F(x,s)) \le c |t - s| , \quad \text{for all $t$, $s\in I$ and $x\in X$.}$$ Assume that $X$ is compact; then it is not difficult to see that any two homotopic maps $X \to Y$ are Lipschitz homotopic. Calder, Siegel and Williams have dealt with this kind of question: *If $Y$ is compact, can we choose the homotopy above with a “small” Lipschitz constant?* More precisely, *is there a finite constant $b = b(X,Y)$ such that one can always find a $b$-Lipschitz homotopy between any given pair of homotopic maps $X \to Y$?*
The answer is clearly negative in general. For example, it is easy to see that $b(I,S^1) = \infty$: despite all maps $I \to S^1$ being homotopic, an homotopy that has to unwind many turns will necessarily have large Lipschitz constant.
Let us see some situations where the answer is positive. First, $b(I,S^2)<\infty$; this can be shown by using a deformation retraction of the punctured sphere to a point. The same trick shows that $b(I^n, S^{n+1}) < \infty$ for each $n$. What about $b(I^2,S^2)$? Since any map $I^2 \to S^2$ can be lifted with respect to the Hopf fibration to a map $I^2 \to S^3$, we can perform a controlled Lipschitz homotopy on $S^3$ and then project back to $S^2$. Thus $b(I^2,S^2) < \infty$. The same argument shows that $b(X,S^2) < \infty$ for any $2$-dimensional manifold $X$.
After considering the examples above, one may guess that the topology of $X$ is not very important, and it is the topology of $Y$ which determines the finiteness of $b(X,Y)$. This is indeed true; in fact, the following very general result holds:
\[t.CSW\] Let $d \in {\mathbb{N}}$ and let $Y$ be a compact Riemannian manifold with finite fundamental group. Then there exists $b = b(d,Y) > 0$ with the following properties. Let $X$ be a compact CW-complex of dimension $d$, and let $A \subset X$ be a subcomplex. Let $f_0$, $f_1 \colon X \to Y$ be homotopic relative to $A$. There there exists a $b$-Lipschitz homotopy relative to $A$ between the two maps.
Theorem \[t.CSW\] is contained in Corollary 2.6 from [@SW89]. In the case $A={\varnothing}$, Theorem \[t.CSW\] was obtained previously in [@CS80]: see Theorem 0.2 and Corollary 3.6 in that paper.
Let us give a brief sketch of the proofs, which, similarly to the informal discussion above, involve lifting to a convenient larger space. For simplicity we discuss only the case $A = {\varnothing}$. The pair of maps $f_0$, $f_1$ can be seen as a single map $(f_0,f_1) \colon X \to Y \times Y$. Each homotopy between them corresponds to a lift of $(f_0,f_1)$ with respect to the fibration $p \colon C(I,Y) \to Y \times Y$ that sends a free path to its endpoints. Finiteness of $\pi_1(Y)$ actually implies that the (infinite dimensional) fibers of $p$ have the homotopy type of a CW-complex with finitely many cells in each dimension. Calder and Siegel use this property to deform maps $X \to C(I,Y)$ along the fibers of $p$ so the image becomes contained in a compact subset $C_d \subset C(I,Y)$ depending not on $X$, but only on its dimension $d$. From compactness it is relatively simple to conclude the existence of a uniform Lipschitz constant.
If $F \colon X \times I \to Y$ is a given homotopy between $f_0$ and $f_1$ relative to $A$ then the $b$-Lipschitz homotopy $G \colon X \times I \to Y$ provided by Theorem \[t.CSW\] is itself homotopic to $F$, relative to $A \times I \cup X \times \partial I$. We will not use this fact, however.
Dynamical stratifications {#ss.strat}
-------------------------
In all this subsection, we assume that $X$ is a compact manifold of dimension $d > 0$, and $f \colon X \to X$ is a minimal diffeomorphism. Our aim here is to describe certain decompositions of the space $X$ with good dynamical and topological properties.
Fix a compact set $K \subset X$ with nonempty interior. Then, for each $x \in X$, let: $$\begin{aligned}
\ell^+(x) &\coloneqq \min \{j \ge 0 ;\; f^j(x) \in \operatorname{int}K\} \, , \label{e.ell_plus}\\
\ell^-(x) &\coloneqq \min \{j > 0 ;\; f^{-j}(x) \in \operatorname{int}K\} \, , \label{e.ell_minus}\\
{\mathcal{L}}(x) &\coloneqq \{ j \in {\mathbb{Z}};\; -\ell^-(x) < j < \ell^+(x) , \ f^j(x) \in \partial K \} \, . \label{e.L}\end{aligned}$$ By minimality, all these numbers are finite and uniformly bounded. Following [@ABD2 p. 75], define closed sets: $$X_i \coloneqq \{ x \in X ;\; \#{\mathcal{L}}(x) \ge i\} \, ,
\quad i = 0,1,2,\dots$$ The sequence $$X = X_0 \supset X_1 \supset \cdots$$ is called the *dynamical stratification of $X$ associated to $K$.* An example is pictured in Fig. \[f.strat\].
(-.5,-.5) rectangle (.5,.5); (2\*[.6571]{}-2,2\*[.2317]{}-1) circle \[radius=[.457]{}\]; plot([2\*[.6571]{}-2+[.457]{}\*cos()]{},[2\*[.2317]{}-1+[.457]{}\*sin()]{});
(2\*[.6571]{}-1,2\*[.2317]{}-1) circle \[radius=[.457]{}\]; plot([2\*[.6571]{}-1+[.457]{}\*cos()]{},[2\*[.2317]{}-1+[.457]{}\*sin()]{});
(2\*[.6571]{}-2,2\*[.2317]{}) circle \[radius=[.457]{}\]; (2\*[.6571]{}-1,2\*[.2317]{}) circle \[radius=[.457]{}\]; plot([2\*[.6571]{}-1+[.457]{}\*cos()]{},[2\*[.2317]{}+[.457]{}\*sin()]{});
(-[.6571]{},-[.2317]{}) circle \[radius=[.457]{}\]; plot([-[.6571]{}+[.457]{}\*cos()]{},[-[.2317]{}+[.457]{}\*sin()]{}); plot([-[.6571]{}+[.457]{}\*cos()]{},[-[.2317]{}+[.457]{}\*sin()]{}); plot([-[.6571]{}+[.457]{}\*cos()]{},[-[.2317]{}+[.457]{}\*sin()]{});
(-[.6571]{}+1,-[.2317]{}) circle \[radius=[.457]{}\]; plot([-[.6571]{}+1+[.457]{}\*cos()]{},[-[.2317]{}+[.457]{}\*sin()]{}); plot([-[.6571]{}+1+[.457]{}\*cos()]{},[-[.2317]{}+[.457]{}\*sin()]{}); plot([-[.6571]{}+1+[.457]{}\*cos()]{},[-[.2317]{}+[.457]{}\*sin()]{});
(-[.6571]{},-[.2317]{}+1) circle \[radius=[.457]{}\]; plot([-[.6571]{}+[.457]{}\*cos()]{},[-[.2317]{}+1+[.457]{}\*sin()]{}); plot([-[.6571]{}+[.457]{}\*cos()]{},[-[.2317]{}+1+[.457]{}\*sin()]{}); plot([-[.6571]{}+[.457]{}\*cos()]{},[-[.2317]{}+1+[.457]{}\*sin()]{});
(-[.6571]{}+1,-[.2317]{}+1) circle \[radius=[.457]{}\]; plot([-[.6571]{}+1+[.457]{}\*cos()]{},[-[.2317]{}+1+[.457]{}\*sin()]{}); plot([-[.6571]{}+1+[.457]{}\*cos()]{},[-[.2317]{}+1+[.457]{}\*sin()]{}); plot([-[.6571]{}+1+[.457]{}\*cos()]{},[-[.2317]{}+1+[.457]{}\*sin()]{});
([.6571]{}-1,[.2317]{}-1) circle \[radius=[.457]{}\]; ([.6571]{},[.2317]{}-1) circle \[radius=[.457]{}\]; ([.6571]{}-1,[.2317]{}) circle \[radius=[.457]{}\]; ([.6571]{},[.2317]{}) circle \[radius=[.457]{}\]; ( 0, 0) circle \[radius=[.457]{}\];
\(a) circle (.2pt) (b) circle (.2pt); (a) circle (.2pt) (b) circle (.2pt); (a) circle (.2pt) (b) circle (.2pt); (a) circle (.2pt) (b) circle (.2pt); (a) circle (.2pt) (b) circle (.2pt); (a) circle (.2pt) (b) circle (.2pt); (a) circle (.2pt) (b) circle (.2pt); in [29.03,42.95]{}[([2\*[.6571]{}-2+[.457]{}\*cos()]{},[2\*[.2317]{}-1+[.457]{}\*sin()]{}) circle\[radius=.2pt\];]{} in [88.95,159.09]{}[([2\*[.6571]{}-1+[.457]{}\*cos()]{},[2\*[.2317]{}-1+[.457]{}\*sin()]{}) circle\[radius=.2pt\];]{} in [239.75,262.88]{}[([2\*[.6571]{}-1+[.457]{}\*cos()]{},[2\*[.2317]{}+[.457]{}\*sin()]{}) circle\[radius=.2pt\];]{}
(-.5,-.5) rectangle (.5,.5); (zzero) at (.55,.12) [$\partial K$]{}; (zzero) – (.455,.12);
(menosum) at (.55,.27) [$f^{-1}(\partial K)$]{}; (menosum) – (.51,.34); (menosum) – (.51,.20);
(dois) at (.55,-.04) [$f^{2}(\partial K)$]{}; (dois) – (.51,.04); (dois) – (.51,-.12);
(um) at (.55,-.27) [$f(\partial K)$]{}; (um) – (.51,-.34); (um) – (.51,-.20);
The usual definition of stratification also asks that each $X_i {\smallsetminus}X_{i+1}$ is either empty or a submanifold of codimension $i$. We do not want to impose this requirement, although the stratifications we will actually use satisfy it.
Recall that $I \coloneqq [0,1]$. If $Y$ is a topological space and $A \subset Y$ is a closed set, then the pair $(Y,A)$ is said to have the *homotopy extension property* if $Y \times \{0\} \cup A \times I$ is a retract of $Y \times I$. See either [@Arkowitz p. 25] or [@Hatcher p. 15] for a discussion of this property, including the equivalent characterization that explains its name.
We say that a compact set $K\subset X$ of nonempty interior is *regular*, or equivalently that the dynamical stratification $X_0 \supset X_1 \supset \cdots$ associated to it is *regular* if the following two properties hold:
- $X_{d+1} = {\varnothing}$ (where $d \coloneqq \dim X$);
- for each $i\in \{0,1,\dots,d\}$, the pair $(X_i \cap K, X_{i+1} \cap K)$ has the homotopy extension property.
It is not difficult to check that that the stratification of Fig. \[f.strat\]. is regular. In general, regularity will be obtained by means of the following:
\[t.regularity\] Let $f \colon X \to X$ be a minimal diffeomorphism of a compact manifold $X$ of positive dimension. Then every point in $X$ has a basis of neighborhoods consisting on regular embedded $d$-dimensional disks.
The precise proof of the theorem is somewhat laborious, and since its arguments are independent of the rest of the paper, we present it separately in Appendix \[s.appendix\].
We will also need the following simple fact from [@ABD2 p. 75], whose proof we include for the reader’s convenience:
\[l.loc\_const\] The function $\ell^+$ is locally constant on each set $X_i {\smallsetminus}X_{i+1}$.
It is easy to see that ${\mathcal{L}}$ and $\ell^+$ are upper semicontinuous on $X$, that is, for any $x \in X$, if $y$ is sufficiently close to $x$ then ${\mathcal{L}}(y) \subset {\mathcal{L}}(x)$ and $\ell^+(y) \le \ell^+(x)$. Assume for a contradiction that there exists a sequence $(x_n)$ in $X_i {\smallsetminus}X_{i+1}$ converging to some $x \in X_i {\smallsetminus}X_{i+1}$ such that $\ell^+(x_n) < \ell^+(x)$ for each $n$. By passing to a subsequence we can assume that $\ell^+(x_n) = k$ is independent of $n$. On one hand, $f^k(x) \not\in \operatorname{int}K$, and in the other hand $f^k(x) = \lim f^k(x_n) \in \overline{\operatorname{int}(K)} \subset K$, showing that $f^k(x) \in \partial K$. In particular, $k \in {\mathcal{L}}(x) {\smallsetminus}{\mathcal{L}}(x_n)$ for each $n$. However, both ${\mathcal{L}}(x)$ and ${\mathcal{L}}(x_n)$ have cardinality $i$. This contradiction proves the lemma.
Proof of Theorem \[t.sections\] {#s.proof_sections}
===============================
Concentrating non-invariance
----------------------------
Using regular dynamical stratifications, we will construct sections that are invariant except on a small set. These kind of sections were used in [@ABD1; @ABD2]. Here, for topological reasons, we need to keep track of not only one such section, but a whole family of them, one for each automorphism along an isotopy.
\[l.predom\_inv\] Let $f \colon X \to X$ be a minimal diffeomorphism of a compact manifold $X$ of positive dimension. Consider a fiberwise smooth bundle ${{Y} \hookrightarrow {Z} \overset{{p}}{\to} {X}}$, an automorphism $g \in {\mathrm{Aut}}(Z,f)$, a section $\sigma \in {\mathrm{Sec}}(Z)$ that is $g$-invariant up to isotopy, and a regular set $K \subset X$ contained in a trivializing domain of the bundle. Then there exist:
- a continuous family of automorphisms $\{g_t\}_{t\in I} \subset {\mathrm{Aut}}(X, f)$ with $g_1 = g$;
- a continuous family of sections $\{{\varphi}_t\}_{t\in I} \subset {\mathrm{Sec}}(Z)$ with ${\varphi}_0 = \sigma$;
such that: $$\label{e.predom_inv}
(x,t) \in (X {\smallsetminus}\operatorname{int}K) \times I {\enspace{\cup}\enspace}X \times \{0\}
\ \Rightarrow \ g_t ({\varphi}_t(x)) = {\varphi}_t (f(x)) \, .$$
Let $\{\hat{g}_t\}_{t \in I} \subset {\mathrm{Aut}}(X, {\mathrm{id}})$ be an ambient isotopy that moves the section $\sigma$ to the section $g_* \sigma$, that is, $$\hat{g}_0 = {\mathrm{id}}\quad \text{and} \quad (\hat{g}_1)_*(g_*(\sigma)) = \sigma \, .$$ Define a continuous family $\{g_t\}_{t\in I}$ in ${\mathrm{Aut}}(X, f)$ by $g_t \coloneqq \hat{g}_{1-t} \circ g$. Then $$g_1 = g \quad \text{and} \quad (g_0)_*(\sigma) = \sigma \, .$$
We want to define a map ${\varphi}\colon X \times I \to Z$ such that ${\varphi}_t = {\varphi}(\mathord{\cdot},t)$ are sections satisfying . Since we also want ${\varphi}_0 = \sigma$, we start defining ${\varphi}$ on $X \times \{0\}$ by ${\varphi}(x,0) \coloneqq \sigma(x)$. The extension to $X \times I$ will be made by a inductive procedure with $d+1$ steps, where $d \coloneqq \dim X$. Consider the regular dynamical stratification associated to the set $K$: $$X = X_0 \supset X_1 \supset \cdots \supset X_{d+1} = {\varnothing}\, .$$ Let $i \in \{0,\dots,d\}$ and assume that ${\varphi}$ is already continuously defined on $X \times \{0\} {\enspace{\cup}\enspace}X_{i+1} \times I$, and that it satisfies condition where it makes sense. (Notice that this assumption is already met for $i=d$, which is the starting point of the induction.) We will explain how to extend ${\varphi}$ to $X \times \{0\}{\enspace{\cup}\enspace}X_i \times I$.
By regularity of the stratification, the pair $(X_i \cap K, X_{i+1} \cap K)$ has the homotopy extension property. This means that there exists a retraction $$r \colon (X_i\cap K) \times I \to L , \quad \text{where }
L\coloneqq (X_i\cap K) \times \{0\} {\enspace{\cup}\enspace}(X_{i+1}\cap K) \times I \, .$$ Notice that $L$ is the intersection of the domain of $r$ and the current domain of ${\varphi}$. Since $K$ is contained in a trivializing domain of the fiber bundle, there exists a homeomorphism $h \colon K \times Y \to p^{-1}(K)$ such that $p \circ h$ equals the projection on the first factor. Then there is a unique map $$\eta \colon L \to Y \quad \text{such that} \quad
{\varphi}(x,t) = h(x,\eta(x,t)) \text{ for every } (x,t) \in L.$$ We extend ${\varphi}$ to an intermediate domain $$\label{e.intermediate}
X \times \{0\} {\enspace{\cup}\enspace}\big(X_{i+1} \cup (X_i \cap \operatorname{int}K) \big) \times I$$ by setting $${\varphi}(x,t) \coloneqq h \big( x , \eta \circ r(x,t) \big)
\quad \text{for } (x,t) \in (X_i \cap \operatorname{int}K) \times I \, .$$ Notice that this is coherent with the previously defined values of ${\varphi}$, and the new map ${\varphi}$ is continuous. For the new points in the domain of ${\varphi}$, condition is vacuously verified.
To complete the induction step, we extend ${\varphi}$ from the intermediate domain to $X \times \{0\} \cup X_i \times I$ by letting $${\varphi}(x,t) \coloneqq g_t^{-\ell^+(x)} \Big( {\varphi}\big( f^{\ell^+(x)}(x) \big) , t \Big) \quad \text{for every }
(x,t) \in (X_i {\smallsetminus}X_{i+1}) \times I ;$$ note that this map is well-defined because ${\mathcal{L}}(f^{\ell^+(x)}(x))$ always has the same cardinality as ${\mathcal{L}}(x)$, and extends the previous ${\varphi}$ because $\ell^+$ vanishes on $\operatorname{int}K$ and ${\varphi}_0$ is $g_0$-invariant. This extension evidently keeps property true where it makes sense. Let us check that this new ${\varphi}$ is continuous. It is sufficient to show that ${\varphi}| X_i \times I$ is continuous. Actually, it is sufficient to prove continuity on points $(x,t) \in X_{i+1} \times I$, since $\ell^+$ is continuous on $X_i {\smallsetminus}X_{i+1}$ by Lemma \[l.loc\_const\].
Take a sequence $(x_n,t_n)$ in $X_i \times I$ converging to $(x,t)$. We can break the sequence $(x_n)$ into finitely many subsequences, where each subsequence is either contained in $X_{i+1}$, or is contained in $X_i {\smallsetminus}X_{i+1}$ and has a constant value of $\ell^+$. Using that ${\varphi}$ is continuous on the domain , it follows that ${\varphi}(x_{n_k},t_{n_k}) \to {\varphi}(x,t)$ for each of those subsequences $(x_{n_k})$. Therefore the new map ${\varphi}$ is continuous.
The induction stops after $d+1$ steps, when the map ${\varphi}$ is defined on the set $X \times \{0\} {\enspace{\cup}\enspace}X_0 \times I = X \times I$. The lemma is proved.
Dissipating non-invariance along a tower
----------------------------------------
We have seen in Lemma \[l.predom\_inv\] how to find sections whose non-invariant part is concentrated in a small set $K$. Next we want to “dissipate” this non-invariant part to a high tower $K \sqcup f(K) \sqcup \cdots \sqcup f^n(K)$ and in this way obtain an almost invariant section. A major problem is that the sections we are working with may be extremely twisted, and here is where the quantitative homotopy Theorem \[t.CSW\] comes in handy.
Fix the diffeomorphism $f$, the bundle ${{Y} \hookrightarrow {Z} \overset{{p}}{\to} {X}}$, the automorphism $g$, and the section $\sigma$ satisfying the hypotheses of the theorem. Let ${\varepsilon}>0$ be arbitrary.
We endow $Y$ with a Riemannian structure. Let $d = \dim X$, and let $b = b(d,Y)$ be given by Theorem \[t.CSW\].
Let $U_0 \subset X$ be a trivializing domain for the fiber bundle. Then there exists a homeomorphism $h \colon U_0 \times Y \to p^{-1}(U_0)$ such that for each $x \in U_0$, the map $h (x, \mathord{\cdot})$ is a diffeomorphism from $Y$ to the fiber $Z_x = p^{-1}(x)$. Take an open $U \neq {\varnothing}$ such that $\overline{U} \subset U_0$. Let $c>0$ be an upper bound for the Lipschitz constants of all maps $h (x, \mathord{\cdot})$ with $x \in U$. Fix an integer $$n > \frac{c b}{{\varepsilon}} \, .$$
By Theorem \[t.regularity\], we can choose a regular embedded closed $d$-dimensional disk $K \subset U$ sufficiently small so that it is disjoint from its $n$ first iterates. We apply Lemma \[l.predom\_inv\] to this set $K$, and so obtain continuous families $\{g_t\}_{t\in I} \subset {\mathrm{Aut}}(X, f)$ and $\{{\varphi}_t\}_{t\in I} \subset {\mathrm{Sec}}(Z)$ such that $g_1 = g$, ${\varphi}_0 = \sigma$, and $$\label{e.predom_inv_again}
(x,t) \in (X {\smallsetminus}\operatorname{int}K) \times I {\enspace{\cup}\enspace}X \times \{0\}
\ \Rightarrow \ g_t ({\varphi}_t(x)) = {\varphi}_t (f(x)) \, .$$ It follows that the sections $((g^{-n}_t)_* {\varphi}_t)(x) = g_t^{-n}({\varphi}_t(f^n(x)))$ satisfy: $$\label{e.tower}
((g^{-n}_t)_* {\varphi}_t)(x) = {\varphi}_t(x) \quad \text{if} \quad x \not\in\bigcup_{j=0}^{n-1} f^{-j}(\operatorname{int}K)
\quad \text{or} \quad t = 0.$$
\[l.cube\] The restrictions of the sections ${\varphi}_1$ and $g^{-n}_* {\varphi}_1$ to the disk $K$ are fibered homotopic relative to $\partial K$. In other words, there exists a continuous map $\eta \colon K \times I \to Y$ such that $${\varphi}_1(x) = h(x, \eta(x,0)) , \quad
(g^{-n}_* {\varphi}_1)(x) = h(x, \eta(x,1)) \quad \text{for all } x \in K ,$$ and moreover if $x \in \partial K$ then $\eta(x,s)$ does not depend on $s \in I$.
The pair $(K,\partial K)$ is homeomorphic to $(D^d, S^{d-1})$ – the unit disk and the unit sphere in ${\mathbb{R}}^d$. Therefore the pair $$\big( K \times I, \ \partial(K\times I) \big) =
\big( K \times I, \ \partial K \times I {\enspace{\cup}\enspace}K \times\{0,1\} \big)$$ is homeomorphic to $(D^{d+1}, S^d)$. It is well-known that the latter pair has the homotopy extension property (see e.g. [@Arkowitz p. 9] or [@Hatcher p. 15]), and thus so has the former. This means that there exists a retraction $r \colon K \times I \times I \to C$, where $$C \enspace{\coloneqq}\enspace K \times I \times \{0\} {\enspace{\cup}\enspace}\partial K \times I \times I {\enspace{\cup}\enspace}K \times\{0,1\} \times I \, .$$ Define continuous maps $\xi_0$, $\xi_1 \colon K \times I \to Y$ by: $${\varphi}_t(x) = h(x, \xi_0(x,t)) \, , \quad
((g^{-n}_t)_* {\varphi}_t)(x) = h(x, \xi_1(x,t)) \, .$$ It follows from that $$\xi_0 = \xi_1 \quad \text{on} \quad K \times \{0\} {\enspace{\cup}\enspace}\partial K \times I \, .$$ Define a map $\xi \colon C \to Y$ by $$\xi(x,s,t) \coloneqq
\begin{cases}
\xi_0(x,t) &\quad\text{if } (x,t) \in K \times \{0\} {\enspace{\cup}\enspace}\partial K \times I \, , \\
\xi_s(x,t) &\quad\text{if } s\in \{0,1\} \, .
\end{cases}$$ Notice that $\xi$ is well-defined and continuous. Extend $\xi$ to $K \times I \times I$ by imposing $\xi = \xi \circ r$. The announced map $\eta$ is $\eta(x,s) \coloneqq \xi(x,s,1)$.
Applying the Calder–Siegel–Williams Theorem \[t.CSW\], we find another homotopy $\zeta \colon K \times I \to Y$ relative to $\partial K$ between the maps $\eta(\mathord{\cdot},0)$ and $\eta(\mathord{\cdot},1)$ with the additional property of being $b$-Lipschitz.
The desired section $\omega \colon X \to Z$ is defined as follows: $$\omega(x) \coloneqq
\begin{cases}
g^j \left( h \left( f^{-j}(x), \zeta \left( f^{-j}(x), \frac{j}{n} \right) \right)\right)
&\text{if } x \in f^j(\operatorname{int}K), \ 0 \le j \le n. \\
{\varphi}_1(x)
&\text{if } x \not\in\bigcup_{j=0}^{n} f^j(\operatorname{int}K) \, .
\end{cases}$$ Continuity follows from the fact that the homotopy $\zeta$ is relative to $\partial K$. Notice that $\omega$ also coincides with ${\varphi}_1$ on $K \cup f^n(K)$.
Let us check almost-invariance of $\omega$. If $x \not\in\bigcup_{j=0}^{n-1} f^j(\operatorname{int}K)$ then by , ${\varphi}_1(f(x)) = g({\varphi}_1(x))$, that is, $$\omega(f(x)) = g(\omega(x)).$$ If, on the other hand, $x \in f^j(\operatorname{int}K)$ for some $j \in \{0,1,\dots,n-1\}$ then $$\begin{aligned}
\omega(f(x)) &\coloneqq g^{j+1} \left( h \left( f^{-j}(x), \zeta \left( f^{-j}(x), \tfrac{j+1}{n} \right) \right)\right) \, ,
\\
g(\omega(x)) &\coloneqq g^{j+1} \left( h \left( f^{-j}(x), \zeta \left( f^{-j}(x), \tfrac{j}{n} \right) \right)\right) \, .\end{aligned}$$ Using that the maps $\zeta(f^{-j}(x), \mathord{\cdot})$ and $h(f^{-j}(x) , \mathord{\cdot})$ are respectively $b$-Lipschitz and $c$-Lipschitz, and that the map $g^{j+1}$ is isometric along fibers of $Z$, it follows that $$\mathrm{d}_x \big( \omega(f(x)) , g(\omega(x)) \big) \le
\frac{cb}{n} < {\varepsilon}\, .$$ This shows that the section $\omega$ is ${\varepsilon}$-almost invariant under $g$.
Finally, let us check that $\omega$ and $\sigma = {\varphi}_0$ are fibered homotopic. Notice that if we replace the fraction $j/n$ that appears in the definition of $\omega$ by $1$ we obtain the following section, which is fibered homotopic to $\omega$: $$\tilde\omega(x) \coloneqq
\begin{cases}
(g_*^{-(n-j)} {\varphi}_1)(x)
&\text{if } x \in f^j(\operatorname{int}K), \ 0 \le j \le n. \\
{\varphi}_1(x)
&\text{if } x \not\in\bigcup_{j=0}^{n} f^j(\operatorname{int}K) \, .
\end{cases}$$ For each $j$ with $0 \le j \le n$, the restrictions of the sections ${\varphi}_1$ and $g^{-(n-j)}_* {\varphi}_1$ to the disk $f^j(K)$ are fibered homotopic relative to $\partial f^j(K)$, by the exact same argument of the proof of Lemma \[l.cube\]. It follows that the sections $\tilde \omega$ and ${\varphi}_1$ are fibered homotopic. Since the latter is obviously fibered homotopic to ${\varphi}_0 = \sigma$, we conclude that $\omega$ and $\sigma$ are fibered homotopic, as announced. This ends the proof of Theorem \[t.sections\].
Further comments and questions {#s.examples}
==============================
In this section we collect a number of examples that illustrate and test the sharpness of Theorems \[t.densedom\] and \[t.sections\]. Some questions are posed along the way.
A space of cocycles containing classes of both types {#ss.two_types}
----------------------------------------------------
We will give an example where ${\mathrm{Aut}}({\mathbb{E}},f)$ contains fibered homotopy classes of both types (\[i.obstruction\]) and (\[i.densedom\]) in Theorem \[t.densedom\], and therefore such that domination is neither empty nor dense.
Since the sphere $S^3$ is a Lie group containing circle subgroups, by a theorem of Fathi and Herman [@FH], there exists a minimal diffeomorphism $f \colon S^3 \to S^3$ that is homotopic to the identity. Let ${\mathbb{E}}$ be the trivial bundle $S^3 \times {\mathbb{R}}^3$. Then ${\mathrm{Aut}}({\mathbb{E}},f) = C(S^3, {\mathrm{GL}}(3,{\mathbb{R}}))$.
We regard the sphere $S^3$ as the group of unit quaternions in $\mathbb{H} = {\mathbb{R}}^4$, the space ${\mathbb{R}}^3$ as the set of purely imaginary quaternions, and $S^2$ as $S^3 \cap {\mathbb{R}}^3$. Let $\rho \colon S^3 \to {\mathrm{SO}}(3)$ be the homomorphism (and also a double covering) that associates to each unit quaternion $q \in S^3$ the orthogonal linear map $v \in {\mathbb{R}}^3 \mapsto q^{-1}vq \in {\mathbb{R}}^3$ (see [@Thurston p. 105], [@GHV p. 75]). For each $n \in {\mathbb{Z}}$, let $P_n \colon S^3 \to S^3$ be the power map $q \mapsto q^n$, and let $A_n$ be the composition of the following maps: $$S^3 \xrightarrow{P_n} S^3 \xrightarrow{\rho} {\mathrm{SO}}(3) \hookrightarrow {\mathrm{GL}}(3,{\mathbb{R}}) \, .$$ Let ${\mathcal{C}}_n$ be the fibered homotopy class of $A_n$.
\[fa.two\_types\] The class ${\mathcal{C}}_n$ is of type (\[i.obstruction\]) if $n \neq 0$, and of type (\[i.densedom\]) if $n = 0$.
Before proving this, we need to establish the following:
\[fa.Hopf\] For each continuous map $h \colon S^3 \to S^2$ there exists a unique $n \in {\mathbb{Z}}$ such that $h$ is homotopic to the map $h_n(q) \coloneqq (A_n(q))(\mathbf{i})$, where $\mathbf{i} = (0,1,0,0)$ is the first imaginary unit in quaternionic space $\mathbb{H} = {\mathbb{R}}^4$.
Since the Hopf invariant $H \colon \pi_2(S^3) \to {\mathbb{Z}}$ is an isomorphism it is sufficient to show that each map $h_n$ has Hopf invariant $H(h_n) = n$. Since $h_1$ is the projection map of the Hopf fibration (see [@Thurston p. 106]), its Hopf invariant is $1$. It follows that the Hopf invariant of $h_n = h_1 \circ P_n$ is the degree of the map $P_n$ (see [@Hatcher p. 428]), which is exactly $n$ (see [@GHV p. 104]).
Since ${\mathcal{C}}_0$ is the class of cocycles homotopic to constant, it is indeed of type (\[i.densedom\]). Conversely, suppose that the class ${\mathcal{C}}_n$ is of type (\[i.densedom\]), that is, some $A \in {\mathcal{C}}_n$ has a continuous invariant $k$-plane field $\sigma \colon S^3 \to {\mathrm{Gr}}(k,3)$ for some $k \in \{1,2\}$. We will only discuss the case $k=1$, since the case $k=2$ is entirely analogous. Since $S^3$ is simply connected, the line field $\sigma$ is orientable, that is, it lifts to a map $h \colon S^3 \to S^2$. Since $f \simeq {\mathrm{id}}$ and $A \simeq A_n$, the map $h$ is homotopic to the map $h'(q) \coloneqq (A_n(q))(h(q)) = q^{-n} h(q) q^n$. By Fact \[fa.Hopf\], there exists $m \in {\mathbb{Z}}$ be such that $h \simeq h_m$. Then $h' \simeq h_{m+n}$, and by Fact \[fa.Hopf\] again we have $n = 0$.
Different types of domination inside the same class {#ss.indices}
---------------------------------------------------
The *indices* of a dominated splitting ${\mathbb{E}}= {\mathbb{E}}^1 \oplus {\mathbb{E}}^2 \oplus \cdots \oplus {\mathbb{E}}^k$ (where ${\mathbb{E}}^i$ dominates ${\mathbb{E}}^{i+1}$) are the dimensions of the subbundles ${\mathbb{E}}^1$, ${\mathbb{E}}^1\oplus {\mathbb{E}}^2$, …, ${\mathbb{E}}^1 \oplus \cdots \oplus {\mathbb{E}}^{k-1}$.
Let us show that if ${\mathcal{C}}$ is a fibered homotopy class of type (\[i.densedom\]) in Theorem \[t.densedom\], it is not necessarily true that there is a common index of domination that appears open and densely in ${\mathcal{C}}$.
Let $X = S^3 \times S^2 \times S^6$. Let ${\mathbb{E}}'$, ${\mathbb{E}}''$, and ${\mathbb{E}}$ be the vector bundles with base space $X$ whose fibers over $(x_1,x_2,x_3) \in S^3 \times S^2 \times S^6$ are respectively ${\mathbb{E}}'(x) \coloneqq T_{x_2} S^2$, ${\mathbb{E}}''(x) \coloneqq T_{x_3} S^6$, and ${\mathbb{E}}(x) \coloneqq {\mathbb{E}}'(x) \oplus {\mathbb{E}}''(x)$.
${\mathbb{E}}$ has no subbundle of fiber dimension $4$.
Let ${\mathbb{F}}$ be a subbundle of fiber dimension $4$. Since $X$ is simply connected, ${\mathbb{F}}$ is orientable. Then ${\mathbb{E}}= {\mathbb{F}}\oplus {\mathbb{F}}^\perp$ (oriented Whitney sum). Let us consider cohomology groups with integer coefficients. Since $H^m(S^n) \neq 0$ iff $m = 0$ or $n$, and $4$ is not a sum of different numbers in $\{3,2,6\}$, by the Künneth formula the cohomology group $H^4(X)$ vanishes. In particular, the Euler classes $e({\mathbb{F}})$, $e({\mathbb{F}}^\perp)$ vanish, and therefore so does $e({\mathbb{E}}) = e({\mathbb{F}}) \smallsmile e({\mathbb{F}}^\perp)$ (see [@MS p. 100]). On the other hand, the vector bundle ${{{\mathbb{R}}^8} \hookrightarrow {{\mathbb{E}}} \overset{{}}{\to} {X}}$ is the cartesian product (see [@MS p. 27]) of the following three vector bundles $${{\{0\}} \hookrightarrow {{\mathbb{G}}} \overset{{}}{\to} {S^3}}, \quad
{{{\mathbb{R}}^2} \hookrightarrow {TS^2} \overset{{}}{\to} {S^2}}, \quad
{{{\mathbb{R}}^6} \hookrightarrow {TS^6} \overset{{}}{\to} {S^6}}, \quad$$ Therefore (see [@MS p. 100]), $$e({\mathbb{E}}) = \underbrace{e({\mathbb{G}})}_{\in H^0(S^3)={\mathbb{Z}}} \times \underbrace{e(TS^2)}_{\in H^2(S^2)={\mathbb{Z}}}
\times \underbrace{e(TS^6)}_{\in H^6(S^6)={\mathbb{Z}}} \, ,$$ where $\times$ is the cross product. But $e({\mathbb{G}}) = 1$ and $e(TS^{2k}) = \chi(S^k) = 2$, which imply that $e({\mathbb{E}}) \neq 0$. Contradiction.
There is a minimal homeomorphism $f \colon X \to X$ and there is a homotopy class ${\mathcal{C}}\subset {\mathrm{Aut}}({\mathbb{E}},f)$ containing an automorphism having a dominated splitting with index $2$ and another automorphism having a dominated splitting with index $6$.
Consider a free action of $S^1$ on $S^3$ by diffeomorphisms. By multiplying by the identity on $S^2 \times S^6$ we obtain a free action $t \in S^1 \mapsto {\varphi}_t \in {\mathrm{Diff}}(X)$. Choose some ${\varphi}_t \neq {\mathrm{id}}_X$ and call it $g$.
For each $\lambda > 0$, let $B_\lambda \in {\mathrm{Aut}}({\mathbb{E}}, g)$ be the automorphism that preserves the subbundles ${\mathbb{E}}'$ and ${\mathbb{E}}''$, whose restriction to ${\mathbb{E}}'$ is an homothecy of factor $\lambda$, and whose restriction to ${\mathbb{E}}''$ is an homothecy of factor $\lambda^{-1}$.
Since $\{{\varphi}_t\}$ is a free action of the circle, by a theorem of Fathi and Herman [@FH], minimal diffeomorphisms form a residual subset of the $C^\infty$-closure of the union of the $C^\infty$-conjugacy classes of the maps ${\varphi}_t$. In particular, there is a sequence of diffeomorphisms $h_n \to {\mathrm{id}}_X$ such that each $f_n = h_n \circ g$ is minimal. Taking the partial derivative of the $S^2 \times S^6$ component of $h_n$ with respect to itself, we obtain a sequence of automorphisms $D_n \in {\mathrm{Aut}}({\mathbb{E}}, h_n)$ that converges to ${\mathrm{id}}_{\mathbb{E}}$.
Let $A_{\lambda,n} \coloneqq D_n \circ B_\lambda \in {\mathrm{Aut}}({\mathbb{E}}, f_n)$. For each $t$, we have $A_{\lambda,n} \to B_\lambda$ as $n \to \infty$. Since $B_2$ and $B_{1/2}$ have dominated splittings of respective indices $2$ and $6$, if $n$ is large enough then $A_{2,n}$ and $A_{1/2,n}$ also have dominated splittings of indices $2$ and $6$. These two automorphisms are homotopic in ${\mathrm{Aut}}({\mathbb{E}}, f_n)$. So, taking $f = f_n$, the fact is proved.
However, no element of ${\mathcal{C}}$ can have simultaneously domination indices $2$ and $6$, because it would then have an invariant “middle” bundle of dimension $4$, which we have seem that is impossible.
Failure of Theorem \[t.densedom\] in dimension two {#ss.dim2}
--------------------------------------------------
The statement of Theorem \[t.densedom\] is false for $m=2$, even for trivial bundles. For example, if $f$ is minimal but not uniquely ergodic and $A$ is a cocycle of rotations homotopic to constant whose fibered rotation numbers are not the same for all invariant measures then $A$ cannot be approximated by dominated cocycles. For similar reasons, the theorem also fails for uniquely ergodic homeomorphisms such that the range of the Schwartzman asymptotic cycle is not dense. See [@ABD2] for details.
For ${\mathrm{SL}}(2,{\mathbb{R}})$-cocycles (or, slightly more generally, orientation-preserving linear cocycles), these are basically all the possible counterexamples, as it follows from the results of [@ABD2].
In the lack of orientability or when the vector bundle is nontrivial, new topological and dynamical problems appear. We hope to address these in a later paper.
Going to vector bundles ${\mathbb{E}}$ of arbitrary fiber dimension, another interesting problem is to describe the domination types that appear openly inside a given fibered homotopy class in ${\mathrm{Aut}}({\mathbb{E}},f)$.
Theorem \[t.sections\] fails if $g$ is not a fiberwise isometry
---------------------------------------------------------------
Consider the product bundle ${{S^2} \hookrightarrow {S^3 \times S^2} \overset{{}}{\to} {S^3}}$. Its sections can be identified with continuous maps $S^3 \to S^2$. As in § \[ss.two\_types\], consider a minimal diffeomorphism $f \colon S^3 \to S^3$ homotopic to the identity. Define $g \in {\mathrm{Aut}}(Z,f)$ by $g(x,y) \coloneqq (f(x),h(y))$, where $h \colon S^2 \to S^2$ is a diffeomorphism homotopic to the identity with a single attracting fixed point and a single repelling fixed point (e.g., a “north pole – south pole” map). Then any section is $g$-invariant up to homotopy. On the other hand, if ${\varepsilon}>0$ is sufficiently small, then every ${\varepsilon}$-almost $g$-invariant section is homotopic to a constant. Since there exist sections that are not homotopic to constants (for example, the Hopf map $S^3 \to S^2$), we conclude that Theorem \[t.sections\] does not apply to $g$.
More general fibers {#ss.unification}
-------------------
For a fiberwise isometry $g$ of a bundle having simply connected fibers of nonpositive curvature, Theorem A (with Remark 2.15) from [@BN_geometric] characterizes the value $$\inf \big\{{\varepsilon}>0 ; \; \text{$g$ has an ${\varepsilon}$-invariant section}\}$$ as the *maximal drift* of the cocycle, defined as the linear rate of growth of the distances between the iterates of an arbitrary section and itself.
If we want to extend such a result to bundles whose fibers are non contractible, it seems natural to measure distances between sections by using the *homotopy distance* [@CS80] instead. If the fiber $Y$ is compact with finite fundamental group then the Calder–Siegel theorem states that these homotopy distances are uniformly bounded, and therefore the “maximal homotopy drift” would always vanish when it is finite.
These remarks indicate that Theorem A from [@BN_geometric] and Theorem \[t.sections\] may be manifestations of a more general phenomenon.
Construction of regular dynamical stratifications {#s.appendix}
=================================================
In this appendix we prove Theorem \[t.regularity\]. The proof has basically three steps: First we define a transversality property concerning the iterates of the boundary $\partial K$ of an embedded $d$-dimensional disk $K$, and show that this property can always be obtained by perturbation. Second, we show that this transversality property implies the regularity of a certain auxiliary stratification. Third, we deduce the regularity of the dynamical stratification.
Before going into the details, let us highlight the basic ideas. To check that a pair $(Y,A)$ has the homotopy extension property we only need to understand how a neighborhood of $A$ fits inside $Y$: see [@Hatcher Example 0.15, p. 15]. In our situation, this local topology is controlled using transversality between $\partial K$ and its iterates. If the dimension $d$ equals $2$, a typical situation is shown in Fig. \[f.strat\]. In dimension $3$, a typical neighborhood of a point in $X_3 \cap \operatorname{int}K$ is shown in Fig. \[f.d3\]. A direct construction of the necessary retractions in order to prove regularity of dynamical stratifications would be messy, so we use as a convenient technical device some auxiliary stratifications with a simpler local topology.
(3,2) – (2,2) – (0,0) – (6,0) – (8,2) – (7,2); (7,2) – (3,2); (4,1) – (3,0) – (3,3) – (5,5) – (5,4); (5,4) – (5,2) – (4,1); (4,1) – (4,4) – (7,4) – (7,1) – cycle;
Transverse hits {#ss.transv}
---------------
Let $E$ be a finite-dimensional real vector space. Let $H_1$, …, $H_k$ be a finite family of hyperplanes of $E$ (i.e., codimension $1$ subspaces). Let $\lambda_1$, …, $\lambda_k \in E^*$ be linear functionals such that $\operatorname{Ker}\lambda_i = H_i$ for each $i$; they are unique up to nonzero factors. We say that the hyperplanes $H_1$, …, $H_k$ are *independent* if the linear functionals $\lambda_1$, …, $\lambda_k$ are linearly independent. An empty family of hyperplanes is also considered independent.
\[l.transv\_equivalence\] A family of hyperplanes $H_1$, …, $H_k$ of $E$ is independent iff their cartesian product $H_1 \times \cdots \times H_k$ is transverse to the diagonal $\Delta_k(E)$ of the cartesian power $E^k$.
For each hyperplane $H_i$, fix a functional $\lambda_i \in E^*$ whose kernel is $H_i$.
Assume that the functionals $\lambda_1$, …, $\lambda_k$ are linearly independent. Take vectors $u_1$, …, $u_k \in E$ such that $\lambda_i(u_j) = \delta_{ij}$. Given $(v_1, \dots, v_k) \in E^k$, let $w = \sum_{i=1}^k \lambda_i(v_i) u_i$. Then $v_j - w \in H_j$ for each $j$, which shows that $(v_1, \dots, v_k)$ is spanned by $H_1 \times \cdots \times H_k$ and $\Delta_k(E)$. This proves that these two spaces are transverse.
Conversely, assume that $H_1 \times \cdots \times H_k {\;\;\makebox[0pt]{$\top$}\makebox[0pt]{\small $\cap$}\;\;}\Delta_k(E)$. Suppose $a_1$, …, $a_k \in {\mathbb{R}}$ are such that $\sum_i a_i \lambda_i = 0$. Then the linear map $\Lambda \colon E^k \to {\mathbb{R}}$ defined by $\Lambda(v_1, \dots, v_k) \coloneqq \sum_{i=1}^k a_i \lambda_i(v_i)$. vanishes both on the product $H_1 \times \cdots \times H_k$ and on the diagonal $\Delta_k E$, and so it must be zero. Hence $a_i = 0$ for every $i$, showing that $\lambda_1$, …, $\lambda_k$ are linearly independent.
Let $K \subset X$ be a embedded $d$-dimensional (closed) disk. Let $N$ be a finite set of integers, and let $x \in X$ be a point. We say that *the $N$-hits of $x$ at $\partial K$ are transverse* if the hyperplanes $Df^{-j} \left(T_{f^j(x)} (\partial K) \right) \subset T_x X$, where $j$ runs on the elements of $N$ such that $f^j(x) \in \partial K$, form an independent family. If this condition is satisfied for every $x\in X$ then we say that $K$ has the *transverse $N$-hits property*.
\[l.transverse\_hits\] Let $K \subset X$ be an embedded $d$-dimensional disk, and let $U$ be a neighborhood of $\partial K$. Then, for any finite set $N \subset {\mathbb{Z}}$, there exists an embedded $d$-dimensional disk $\tilde K$ with the transverse $N$-hits property and such that $\tilde K \mathbin{\vartriangle} K \subset U$.
Let $K \subset X$ be the image of an embedding $h$ of the closed unit disk $\bar{B}(0,1) \subset {\mathbb{R}}^d$. We can extend $h$ to a diffeomorphism between the open disk $B(0,2)$ and a neighborhood of $K$. Let $U$ be any given neighborhood of $\partial K$. Reducing $U$ if necessary, we can assume it is the image under $h$ of a spherical shell $B(0,1+\delta) {\smallsetminus}\bar{B}(0,1-\delta)$, for some $\delta \in (0,1)$.
Fix a finite set $N \subset {\mathbb{Z}}$, and let $n$ be its diameter. Since $f$ has no periodic points, we can cover the unit sphere $S^{d-1}$ by open disks $B_1$, …, $B_m$ on ${\mathbb{R}}^d$ of radii less than $1/2$ and such that $$\label{e.disjoint_images}
h(B_\ell) \cap f^j(h(B_\ell)) = {\varnothing}\quad \text{for all $j\in\{1,\dots,n\}$ and $\ell\in \{1,\dots,m\}$.}$$ Let $\{\rho_1, \dots, \rho_m\}$ be a $C^\infty$ partition of unity subordinate to this cover.
For each $k \in \{1,2,\dots,n\}$, define a map $$\Psi_k \colon (B(0,{\varepsilon}))^m \times (S^{d-1})^{k} \to X^{k}$$ (where ${\varepsilon}\in (0,\delta)$ will be determined later) by $$\Psi_k (y_1, \dots, y_m, z_1, \dots, z_k) \coloneqq
\left( h \left( z_i + \sum_{\ell=1}^m \rho_\ell(z_i) y_\ell \right) \right)_{i=1,\dots,k}$$ For each subset $J \subset N$ of cardinality $k$, say $J = \{j_1 < \cdots < j_k\}$, let $f_J \colon X^{k} \to X^{k}$ be the diffeomorphism $$f_J(x_1,\dots,x_k) = \big(f^{j_1}(x_1), f^{j_2}(x_2), \dots, f^{j_k}(x_k) \big) \, .$$ Let $\Delta_{k} X$ be the diagonal of $X^{k}$, and let $G_J \coloneqq f_J(\Delta_{k} X)$; both are closed submanifolds of $X^{k}$.
\[fa.transv\] If ${\varepsilon}\in (0,\delta)$ is chosen sufficiently small then for every nonempty $J \subset N$, if $k \coloneqq \# J$ then the map $\Psi_k$ is transverse to the submanifold $G_J$.
Let $J = \{j_1 < \cdots < j_k\} \subset N$ and let $(z_1, \dots, z_k) \in (S^{d-1})^{k}$ be such that $\Psi_k (0, \dots, 0, z_1, \dots, z_k) \in G_J$, that is, there exists $x_1 \in \partial K$ such that $$\label{e.hit}
h(z_i) = f^{j_i}(x_1) \quad \text{for every $i \in \{1,\dots, k\}$.}$$ We will actually prove that the derivative of $D\Psi_k$ at the point $(0, \dots, 0, z_1, \dots, z_k)$ is onto, which implies the fact.
Indeed, for each $i \in \{1,\dots,k\}$, we can choose $\ell_i$ such that $\rho_{\ell_i}(z_i) \neq 0$, and in particular $z_i \in B_{\ell_i}$. By , the points $h(z_i)$ belong to a common segment of orbit of $f$ of length at most $n+1$, and so using we conclude that the indices $\ell_i$ are pairwise distinct. In particular, $\rho_{\ell_i}(z_t) = 0$ whenever $t \neq i$. Suppose that $(y_1, \dots, y_m)$ is such that $y_\ell = 0$ if $\ell \not\in \{\ell_1, \dots, \ell_k\}$; then $$\Psi_k (y_1, \dots, y_m, z_1, \dots, z_k) \coloneqq
\left( h \left( z_i + \rho_{\ell_i}(z_i) y_{\ell_i} \right) \right)_{i=1,\dots,k}$$ Taking the derivative with respect to $(y_{\ell_1}, \dots, y_{\ell_k})$, we obtain an onto linear map; in particular the derivative $D\Psi_k(0, \dots, 0, z_1, \dots, z_k)$ is also onto, as we wanted to show.
Fix some ${\varepsilon}\in (0,\delta)$ with the property given by Fact \[fa.transv\]. It follows from the parametric transversality theorem [@Hirsch p. 79] that for each nonempty $J \subset N$ there exists a residual subset of $R_J \subset (B(0,{\varepsilon}))^m$ such that if $(y_1, \dots, y_m) \in R_J$ and $k = \# J$ then $$\label{e.I_like_cake}
\Psi_k (y_1, \dots, y_m, \ \cdot \ ) \colon (S^{d-1})^{k} \to X^{k} \text{ is transverse to } G_J \, .$$ Choose and fix a point $(y_{1}^*, \dots, y_{m}^*)$ in the residual set $\bigcap_{{\varnothing}\neq J\subset N} R_J \subset (B(0,{\varepsilon}))^m$ close enough to $(0,\dots,0)$ so that the map $\tilde{h} \colon \bar{B}(0,1) \to X$ defined by $$\tilde{h} (z) \coloneqq h \left( z + \sum_{\ell=1}^m \rho_\ell(z) y_{\ell}^* \right)$$ is a diffeomorphism. We define an embedded disk $\tilde{K} \coloneqq \tilde{h}(\bar{B}(0,1))$. Notice that the boundary $\partial \tilde{K}$ is contained in $V$, and thus so is the symmetric difference $\tilde K \mathbin{\vartriangle} K$.
Let us check that $\partial \tilde{K}$ has the transverse $N$-hits property. Fix any $x \in X$ and consider $$J = \big\{ j_1 < j_2 < \cdots < j_k \big\} \coloneqq \big\{ j \in N ; \; f^j(x) \in \partial \tilde{K} \big\} \, .$$ We need to show that the hyperplanes $$H_i \coloneqq Df^{-j_i} \left(T_{f^{j_i}(x)} (\partial \tilde{K}) \right) , \quad (i=1,2,\dots,k)$$ are independent. Assume that $k > 1$, otherwise there is nothing to prove. Let $\psi$ be the restriction of $\tilde{h}$ to the unit sphere, so $\partial \tilde{K} = \psi (S^{d-1})$. For each $i \in \{1,\dots,k\}$, let $z_{i}^* \coloneqq \psi^{-1} (f^{j_i}(x)) \in S^{d-1}$. Condition specialized to the point $(y_{1}^*,\dots,y_{m}^*)$ means that the map $$\psi_k \colon (S^{d-1})^{k} \to X^{k}
\quad\text{defined by }
\psi(z_1, \dots, z_k) \coloneqq (\psi(z_1), \dots, \psi(z_k))$$ is transverse to $G_J = f_J(\Delta_{k} X)$. Equivalently, $f_J^{-1} \circ \psi_k {\;\;\makebox[0pt]{$\top$}\makebox[0pt]{\small $\cap$}\;\;}\Delta_{k} X$. In particular, since $f_J^{-1} \circ \psi_k (z_{1}^*,\dots, z_{k}^*) = (x,\dots,x)$, the spaces $${\mathop{\mathrm{Im}}}D (f_J^{-1} \circ \psi_k) (z_{1}^*, \dots, z_{k}^*) = H_1 \times \cdots \times H_k
\quad \text{and} \quad
T_{(x,\dots,x)}(\Delta_{k} X) = \Delta_{k} (T_{x} X)$$ are transverse in $(T_{x} X)^{k}$. By Lemma \[l.transv\_equivalence\], this means that the hyperplanes $H_1$, …, $H_k$ are independent. The proof of Lemma \[l.transverse\_hits\] is concluded.
Fine stratifications
--------------------
Before proving Theorem \[t.regularity\], we need to establish analogous regularity properties for certain auxiliary finer stratifications that have a simpler local structure.
Suppose $K\subset X$ is a closed set with nonempty interior. Define $m(K)$ as the least positive integer $m$ such that $\bigcup_{j=0}^{m-1} f^{-j}(\operatorname{int}K) = X$. Notice that the numbers defined in and satisfy the bounds: $$\ell^+(x) \le m(K)-1 \quad \text{and} \quad
\ell^-(x) \le m(K) \quad \text{for every } x \in X.$$ Let $$N(K) \coloneqq \{ j \in {\mathbb{Z}}\; ; -m(K) \le j \le m(K)-1 \} \, ,$$ and for each $x \in X$, let $${\mathcal{M}}(x) \coloneqq \{j\in N(K) ; \; f^j(x) \in \partial K \} \, .$$ Therefore the set ${\mathcal{L}}(x)$ defined in equals ${\mathcal{M}}(x) \cap (-\ell^-(x) , \ell^+(x))$.
Recall that the dynamical stratification associated to $K$ is the sequence $$X = X_0 \supset X_1 \supset \cdots \, , \quad \text{where }
X_i \coloneqq \{x \in X ;\; \#{\mathcal{L}}(x) \ge i\} \, .$$ The *strata* of the stratification $(X_i)$ are defined as the connected components of the nonempty sets $X_i {\smallsetminus}X_{i+1}$, and form a partition of $X$.
We now define the *fine dynamical stratification associated to $K$* as the sequence $$X = W_0 \supset W_1 \supset \cdots \, , \quad \text{where }
W_i \coloneqq \{x \in X ;\; \#{\mathcal{M}}(x) \ge i\} \, .$$ The corresponding strata are the connected components of the nonempty sets $W_i {\smallsetminus}W_{i+1}$. Note that $W_i \supset X_i$ for every $i$. In other words, the strata of $(W_i)$ form a finer partition than those of $(X_i)$. Also note that $W_{2m(K)+1} = {\varnothing}$.
The following lemma relates locally the two stratifications:
\[l.frontier\] Letting $K_i \coloneqq K \cap X_i$ for each $i \ge 0$, we have $\overline{W_i {\smallsetminus}K_j} \cap K_j \subset W_{i+1}$ for any $i$, $j \ge 0$.
The proof of this lemma is somewhat similar to that of Lemma \[l.loc\_const\]:
If $i<j$ then $K_j \subset W_{i+1}$ and the assertion becomes trivial. So assume that $i \ge j$. Consider a point $x \in \overline{W_i {\smallsetminus}K_j} \cap K_j$; let us show that $x \in W_{i+1}$. Choose a sequence $(x_n)$ in $W_i {\smallsetminus}K_j$ converging to $x$. By passing to a subsequence, we can assume that ${\mathcal{M}}(x_n)$, $\ell^+(x_n)$ and $\ell^-(x_n)$ are all independent of $n$. By continuity of $f$ we have $$\label{e.3_cases}
{\mathcal{M}}(x_n) \subset {\mathcal{M}}(x) \, , \quad
\ell^+(x_n) \le \ell^+(x) \quad \text{and} \quad
\ell^-(x_n) \le \ell^-(x) \, .$$ Notice that the sets $${\mathcal{L}}(x) = {\mathcal{M}}(x) \cap (-\ell^-(x), \ell^+(x)) \quad \text{and} \quad
{\mathcal{L}}(x_n) = {\mathcal{M}}(x_n) \cap (-\ell^-(x_n), \ell^+(x_n))$$ are different, because $\# {\mathcal{L}}(x) \ge j > \# {\mathcal{L}}(x_n)$. Therefore at least one of the relations in is strict. We consider the three possible cases:
- If ${\mathcal{M}}(x_n) \subsetneqq {\mathcal{M}}(x)$ then $\# {\mathcal{M}}(x) > \# {\mathcal{M}}(x_n) \ge i$, so $x \in W_{i+1}$.
- If $\ell^+(x_n) < \ell^+(x)$ then $z: = f^{\ell^+(x_n)}(x) \not \in \operatorname{int}K$; on the other hand, $$z = \lim_{n \to \infty} f^{\ell^+(x_n)}(x_n) \in \overline{\operatorname{int}(K)} \subset K \, ,$$ so $z \in \partial K$. This shows that $\ell^+(x_n) \in {\mathcal{M}}(x) {\smallsetminus}{\mathcal{M}}(x_n)$, and so by the previous case, $x \in W_{i+1}$.
- If $\ell^+(x_n) < \ell^+(x)$ then $x \in W_{i+1}$ analogously.
This proves Lemma \[l.frontier\].
The following important lemma yields regularity properties for the stratification $(W_i)$:
\[l.reg\_strat\_fine\] If $K$ is an embedded $d$-dimensional disk with the transverse $N(K)$-hits property then the fine dynamical stratification $(W_i)$ associated to $K$ satisfies the following properties:
- $W_{d+1} = {\varnothing}$;
- for each $i\in \{0,1,\dots,d\}$, the pair $(W_i, W_{i+1})$ has the homotopy extension property.
Let us summarize what is involved in the proof of this lemma. It follows from transversality that $W_{d+1} = {\varnothing}$ and moreover for each $i\in \{0,1,\dots,d\}$, every stratum $S \subset W_i {\smallsetminus}W_{i+1}$ is a submanifold of codimension $i$ such that $\overline{S} {\smallsetminus}S \subset W_{i+1}$. We construct a vector field $\mathbf{v}$ on $X$ that is tangent to each strata and “points to” strata of higher codimension. Despite being discontinuous, this vector field can be integrated to a flow whose restriction to each strata is continuous and moreover has the property that if a point in $W_i {\smallsetminus}W_{i+1}$ is close to $W_{i+1}$ then its flow hits $W_{i+1}$ in small positive time. Using this flow we construct the desired retractions in order to conclude that each pair $(W_i, W_{i+1})$ has the homotopy extension property.
Let $K$ be an embedded $d$-dimensional disk with the transverse $N$-hits property, where $N=N(K)$. Let $(W_i)$ be the associated fine dynamical stratification. Since no $d+1$ hyperplanes in a $d$-dimensional space can be independent, we have $W_{d+1} = {\varnothing}$.
Fix a smooth map $\lambda_0 \colon X \to {\mathbb{R}}$ having $0$ as a regular value and such that the submanifold $\lambda_0^{-1}(0)$ is precisely $\partial K$. Let $\lambda_j \coloneqq \lambda_0 \circ f^j$ for each $j \in N$.
Given $x \in X$, let $$\sigma_1(x) \le \sigma_2(x) \le \cdots \le \sigma_{2m}(x)$$ be the ordered list of the values $|\lambda_j(x)|$, where $j$ runs on the set $N$, with possible repetitions. Notice that for each $i$, the function $\sigma_i$ is continuous and its zero set is $W_i$.
Fixed $x \in X$, let $i = i(x)$ be the least nonnegative integer such that $x \in W_i {\smallsetminus}W_{i+1}$. Let $$\big \{j_1 = j_1(x) < j_2 = j_2(x) < \cdots < j_i = j_{i(x)}(x) \big\}$$ be the set of times $j \in N$ such that $\lambda_j(x) = 0$, that is, $f^j(x) \in \partial K$. By the transverse $N$-hits property, the functionals $D\lambda_{j_1}(x)$, …, $D\lambda_{j_i}(x)$ are linearly independent. Therefore we can choose smooth functions $\chi_1$, …, $\chi_{d-i}$ on a neighborhood $U(x)$ of $x$ such that $$\psi_x \coloneqq \big( \lambda_{j_1} , \dots , \lambda_{j_i} , \chi_1, \dots, \chi_{d-i})$$ is a diffeomorphism from $U(x)$ onto a subset of ${\mathbb{R}}^d$. (If $i=0$ then $\psi_x$ is an arbitrary chart around $x$.) Let $$\delta(x) \coloneqq \frac{\sigma_{i+1}(x)}{2} \, .$$ Reducing the neighborhood $U(x)$ if necessary, we ensure that for every point $y \in U(x)$ we have $$\label{e.delta}
\max \big( |\lambda_{j_1}(y)|, |\lambda_{j_2}(y)| , \dots , |\lambda_{j_i}(y)| \big) < \delta(x) < \sigma_{i+1}(y) \, ,$$ where $\max {\varnothing}\coloneqq 0$. In particular, $U(x) \subset W_i {\smallsetminus}W_{i+1}$.
Next, take a finite subcover of the manifold $X$ by these neighborhoods: $$X = \bigcup_{\alpha=1}^\nu U_\alpha, \qquad U_\alpha \coloneqq U(x_\alpha).$$
For each $i \in \{1,\dots, d\}$, define the following (discontinuous) vector field on the euclidian space ${\mathbb{R}}^d$: $$\mathbf{u}_i(z_1, \dots, z_d) \coloneqq \big( -\operatorname{sgn}(z_1) ,\dots, -\operatorname{sgn}(z_i), 0, \dots, 0 \big),$$ where $\operatorname{sgn}(t)$ is defined as $1$, $0$ or $-1$, depending on whether $t$ is positive, zero or negative, respectively. Also let $\mathbf{u}_0 \coloneqq 0$.
For each $\alpha \in \{1,\dots, \nu\}$, let $i_\alpha \coloneqq i(x_\alpha)$ and let $\mathbf{v}_\alpha$ be the pull-back of the vector field $\mathbf{u}_{i_\alpha}$ under the diffeomorphism $\psi_\alpha \coloneqq \psi_{x_\alpha}$, that is, $$\mathbf{v}_\alpha(y) \coloneqq [D\psi_\alpha(y)]^{-1} \big( \mathbf{u}_{i_\alpha}( \psi_\alpha(y) ) \big) \quad \text{for all } y \in U_\alpha \, .$$ See Fig. \[f.fields\]. Notice that for each stratum $S$ intersecting $U_\alpha$, the vector field $\mathbf{v}_\alpha$ restricted to the submanifold $S\cap U_\alpha$ is smooth and tangent to it.
plot([u]{},[[.06]{}\*ŭ]{}); plot([[-.2]{}\*v]{},[v]{});
ĭn [-.75,.75]{}[ plot([+[-.2]{}\*v]{},[+[.06]{}\*ŭ]{}); ]{} ǐn [-.75,.75]{}[ plot([+[-.2]{}\*v]{},[+[.06]{}\*ŭ]{}); ]{} ĭn [1.5,3]{}[ plot([+[-.2]{}\*v]{},[+[.06]{}\*ŭ]{}); ]{} ǐn [-.75,.75]{}[ plot([+[-.2]{}\*v]{},[+[.06]{}\*ŭ]{}); ]{}
ĭn [-.5,0,.5]{}[ ǐn [-.5,0,.5]{}[ ([+[-.2]{}\*v]{},[+[.06]{}\*ŭ]{}) circle(.03); ]{} ]{}
ĭn [1.75,2.25,2.75]{}[ ǐn [-.5,0,.5]{}[ ([+[-.2]{}\*v]{},[+[.06]{}\*ŭ]{}) circle(.03); ]{} ]{}
at (0,0) [$x_\alpha$]{}; at (2.25,[.06]{}\*2.25\*2.25) [$x_\beta$]{}; at (-.75+[-.2]{}\*.75\*.75, .75+[.06]{}\*.75\*.75) [$U_\alpha$]{}; at (3+[-.2]{}\*.75\*.75, .75+[.06]{}\*3\*3) [$U_\beta$]{};
Let $(\rho_\alpha)$ be a smooth partition of unity subordinate the open cover $(U_\alpha)$. Define a (discontinuous) vector field $\mathbf{v}$ on $X$ by the formula: $$\mathbf{v} (y) \coloneqq \sum_{\alpha ; \; U_\alpha \ni y} \rho_\alpha(y) \mathbf{v}_\alpha(y) \, .$$ Then the restriction of $\mathbf{v}$ to each stratum is smooth (and in particular, locally Lipschitz) and tangent to it.
Let $\delta_\alpha \coloneqq \delta(x_\alpha)$ and $\delta_* \coloneqq \min_\alpha \delta_\alpha$. We claim that the vector field $\mathbf{v}$ has following property: $$\label{e.it_s_a_trap}
D\lambda_j(y)(\mathbf{v} (y)) = -\operatorname{sgn}(\lambda_j(y))
\text{ for every } j \in N \text{ such that } |\lambda_j(y)| < \delta_* \, .$$ Indeed, assume that $j \in N$ and $|\lambda_j(y)| \le \delta_*$. Fix any $\alpha$ such that $U_\alpha \ni y$. Let $i = i_\alpha$ and $j_1 = j_1(x_\alpha) < \cdots < j_i = j_{i_\alpha}(x_\alpha)$. Then, by , $$\{ k \in N ;\; |\lambda_k(y)| < \delta_\alpha \} = \{ j_1, \dots, j_i \} \, .$$ The index $j$ belongs to this set, because $\delta_* \le \delta_\alpha$. It now follows from the definition of $\mathbf{v}_\alpha$ that $$D\lambda_j(y)(\mathbf{v}_\alpha (y)) = -\operatorname{sgn}(\lambda_j(y)) \, .$$ Since this equality holds for every $\alpha$ such that $U_\alpha \ni y$, we conclude that holds.
From now on, let $i \in \{0,\dots,d\}$ be fixed. We need to find a retraction $$\label{e.g_i}
g_i \colon W_i \times I \to W_i \times \{0\} {\enspace{\cup}\enspace}W_{i+1} \times I.$$ If $i=d$ then $g_d(x,s)$ is simply $(x,0)$. Let us assume that $i<d$.
For each $x \in W_i {\smallsetminus}W_{i+1}$, the ODE $x'(t) = \mathbf{v}( x(t) )$ with initial value $x(0)=x$ has a unique solution $x(t) = {\varphi}_i(x, t)$ taking values in $W_i {\smallsetminus}W_{i+1}$. Let $\tau_i(x) \in (0,\infty]$ be the supremum of the maximal interval where this solution is defined. Note that: $$\label{e.hitting_time}
x \in W_i {\smallsetminus}W_{i+1}, \
\sigma_{i+1}(x) < \delta_* \ \Rightarrow \ \tau(x) = \sigma_{i+1}(x);$$ indeed once the quantity $\sigma_{i+1}$ is less than $\delta_*$, by it decreases with unit speed until $W_{i+1}$ is hit.
Property has the following consequences:
- The map $\tau_i$ is continuous, and can be continuously extended to a map $\bar{\tau}_i$ on $W_i$ which vanishes on $W_{i+1}$.
- If $x\in W_i {\smallsetminus}W_{i+1}$ is such that $\tau_i(x) < \infty$ then $$\pi_i(x) \coloneqq {\varphi}_i(x, \tau_i(x)-) = \lim_{t \to \tau_i(x)-} {\varphi}_i(x, t)$$ exists and belongs to $W_{i+1}$. Moreover, the map $\pi_i$ is continuous.
It follows that the map $\bar{{\varphi}}_i \colon W_i \times [0, \infty) \to W_i$ defined by $$\bar{{\varphi}}_i (x,t) \coloneqq
\begin{cases}
x &\text{if $x \in W_{i+1}$ or $t=0$,} \\
{\varphi}_i(x,t) &\text{if $x \not\in W_{i+1}$ and $0 < t < \tau_i(x)$,} \\
\pi_i(x) &\text{if $x \not\in W_{i+1}$ and $t \ge \tau_i(x)$}
\end{cases}$$ is continuous. (This map can be viewed as the positive-time flow generated by the vector field $\mathbf{v}$ on $W_i$ and with $W_{i+1}$ as an absorbing barrier.)
Define a continuous map $\bar{g}_i \colon W_i \times I \times I \to W_i \times I$ by $$\label{e.deformation}
\bar{g}_i (x,s,t) \coloneqq \left( \bar{{\varphi}}_i \big(x, \min\{\bar{\tau}_i(x),s,t\} \big), s - \min\{\bar{\tau}_i(x),s,t\} \right) \, .$$ (This map can be viewed as the flow generated by the vector field $(\mathbf{v},-1)$ on $W_i \times I$ and with $W_i \times \{0\} \cup W_{i+1}\times I$ as an absorbing barrier.) Then the map $g_i \coloneqq \bar{g}_i (\mathord{\cdot},\mathord{\cdot},1)$ is the desired retraction . This proves that $(W_i, W_{i+1})$ has the homotopy extension property.
\[scholium\] The retractions $g_i \colon W_i \times I \to W_i \times \{0\} {\enspace{\cup}\enspace}W_{i+1} \times I$ constructed in the proof of Lemma \[l.reg\_strat\_fine\] have the following property: if $g_i(x,s) = (x',s')$ then ${\mathcal{L}}(x') \supset {\mathcal{L}}(x)$.
We will make use of the map $\bar{g}_i$ defined by , which is actually a deformation retraction. Let $(x,s) \in W_i \times I$ and let $(x',s') = g_i(x,s)$. Assume that $i<d$ and $(x,s) \not\in W_i \times \{0\} {\enspace{\cup}\enspace}W_{i+1} \times I$, otherwise there is nothing to prove. For $t\in I$, let $\xi_i(x,s,t)$ be first coordinate of $\bar{g}_i(x,s,t)$. Then $\xi_i(x,s,t) \in W_i {\smallsetminus}W_{i+1}$ for every $t \in [0, t_*)$, where $t_*\coloneqq\min\{\bar{\tau}_i(x),s\} > 0$. By connectedness we conclude that ${\mathcal{M}}(\xi_i(x,s,t))$ is independent of $t \in [0, t_*)$. It follows that $${\mathcal{M}}(\xi_i(x,s,t_*) ) \supset {\mathcal{M}}( \xi_i(x,s,0) ) \, , \quad \text{that is,}\quad
{\mathcal{M}}(x') \supset {\mathcal{M}}(x) \, .$$ A similar argument shows that $\ell^+(x') \ge \ell^+(x)$ and $\ell^-(x') \ge \ell^-(x)$. Therefore ${\mathcal{L}}(x') \supset {\mathcal{L}}(x)$, as announced.
End of the proof
----------------
\[l.reg\_strat\_2\] If $K$ is an embedded $d$-dimensional disk with the transverse $N(K)$-hits property then it is regular.
Apply Lemma \[l.reg\_strat\_fine\] to the disk $K$ and the associated fine dynamical stratification $(W_i)$. Then $X_{d+1} \subset W_{d+1} = {\varnothing}$, which is the first regularity property that we need to check.
For each $i \in \{0, \dots, d\}$, Lemma \[l.reg\_strat\_fine\] gives us a retraction $$g_i \colon W_i \times I \to W_i \times \{0\} {\enspace{\cup}\enspace}W_{i+1} \times I \, .$$ Denote $K_i \coloneqq K \cap X_i$.
Let $j \in \{0,1,\dots, d\}$ be fixed. We will explain how to find a retraction $$\label{e.wanted}
r \colon K_j \times I \to K_j \times \{0\} {\enspace{\cup}\enspace}K_{j+1} \times I \, .$$ First notice that, for any $i$, $$\label{e.fit}
g_i \left( (K_j \cap W_i) \times I \right) \subset K_j \times \{0\} {\enspace{\cup}\enspace}(K_j \cap W_i) \times I \, .$$ Indeed this follows from the property of $g_i$ provided by the Scholium \[scholium\]. Define a nested sequence of closed sets $$K_j \times I = Q_j \supset Q_{j+1} \supset \cdots \supset Q_d \supset Q_{d+1} = K_j \times \{0\} {\enspace{\cup}\enspace}K_{j+1} \times I \, .$$ by $$Q_i \coloneqq K_j \times \{0\} {\enspace{\cup}\enspace}\left( K_{j+1} \cup (K_j \cap W_i) \right) \times I \, .$$ For each $i \ge j$, the set $$Q_i {\smallsetminus}Q_{d+1} = (K_j \cap W_i {\smallsetminus}K_{j+1}) \times (I {\smallsetminus}\{0\});$$ is contained in $(K_j \cap W_i) \times I$; therefore by we can define a map $$h_i \colon Q_i \to Q_{i+1} \quad \text{by} \quad
h_i =
\begin{cases}
{\mathrm{id}}&\text{on } Q_{d+1} \, , \\
g_i &\text{on } Q_i {\smallsetminus}Q_{d+1} \, .
\end{cases}$$ Notice that $$\begin{aligned}
\overline{Q_i {\smallsetminus}Q_{d+1}} {\enspace{\cap}\enspace}Q_{d+1}
&\enspace{\subset}\enspace \left( \, \overline{W_i {\smallsetminus}K_{j+1}} \times I \right) {\enspace{\cap}\enspace}Q_{d+1} \\
&\enspace{\subset}\enspace W_i \times \{0\} {\enspace{\cup}\enspace}\left( \, \overline{W_i {\smallsetminus}K_{j+1}} \cap K_{j+1} \right) \times I \\
&\enspace{\subset}\enspace W_i \times \{0\} {\enspace{\cup}\enspace}W_{i+1} \times I \quad \text{(by Lemma~\ref{l.frontier}),}\end{aligned}$$ and so $g_i = {\mathrm{id}}$ on these sets. It follows that $h_i$ is continuous.
The desired retraction is $$r \coloneqq h_{d+1} \circ h_d \circ \cdots \circ h_{i+1} \circ h_i \colon Q_i \to Q_{d+1} \, .$$ This shows the regularity of the dynamical stratification $(X_i)$.
Given any point $x \in X$ and any open neighborhood $V \ni x$, let $K_1$, $K_2 \subset V$ be embedded $d$-dimensional disks containing $x$ in its interior, with $K_1 \subset \operatorname{int}K_2$. Let $N = N(K_1)$. By Lemma \[l.transverse\_hits\], there exists a disk $K$ with the transverse $N$-hits property such that $K \mathbin{\vartriangle} K_2 \subset V {\smallsetminus}K_1$. Note that $x \in K \subset V$ and so the sets $K$ obtained in this way form a basis of neighborhoods of $x$. Moreover, $K \supset K_1$ and in particular $N(K) \subset N(K_1)$. So $K$ also has the transverse $N(K)$-hits property, and therefore it is a regular set by Lemma \[l.reg\_strat\_2\].
[ABD2]{}
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abstract: 'We show global existence backwards from scattering data for models of Einstein’s equations in wave coordinates satisfying the weak null condition. The data is in the form of the radiation field at null infinity recently shown to exist for the forward problem in Lindblad [@L17]. Our results are sharp in the sense that we show that the solution has the same spacial decay as the radiation field does at null infinity, as for the forward problem.'
author:
- Hans Lindblad and Volker Schlue
title: 'Scattering from infinity for semilinear models of Einstein’s equations satisfying the weak null condition'
---
[*Address:*]{} [<span style="font-variant:small-caps;">Johns Hopkins University, 404 Krieger Hall, 3400 N. Charles Street, Baltimore, MD 21218, US</span>]{}\
[*Email:*]{} [[email protected]]{}
[*Address:*]{} [<span style="font-variant:small-caps;">LJLL, Université Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France</span>]{}\
[*Email:*]{} [[email protected]]{}
|
---
author:
- |
[Christian Bayer^a^, Raúl Tempone^b,c^, Sören Wolfers^c,^[^1]]{}\
[^a^ Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany]{}\
[^b^ RWTH Aachen University, Germany]{}\
[^c^ King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia]{}\
title: Pricing American Options by Exercise Rate Optimization
---
**Abstract** We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal *exercise regions*, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the *exercise rates* of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Numerical experiments on vanilla put options in the multivariate Black–Scholes model and a preliminary theoretical analysis underline the efficiency of our method, both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of the exercise rates. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black–Scholes model, and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model.\
**Keywords** Computational finance, American option pricing, stochastic optimization problem, Monte Carlo, multivariate approximation, rough volatility
**2010 Mathematics Subject Classification** 91G60, 91G20, 49M20, 90C90, 65K10, 65C05
Introduction
============
American options on $d\geq 1$ underlying assets $S_{t}=(S_{1,t},\dots,S_{d,t})$ may be exercised by their holder at any time $t$ before a given expiration time $T\in\R_+:=[0,\infty)$, upon which the holder receives the payoff ${g}(t,S_t)$ for some previously agreed function ${g}\colon [0,T] \times \R_{+}^{d} \to \R_{+}$.
If the underlying market is Markovian and has a security with interest rate $r>0$, then the arbitrage-free value of an American option under a risk-neutral measure $\mathbb{Q}$ is determined solely by the current asset values. The value function $V\colon \R_{+}^{d}\to\R_{+}$ satisfies $$\label{fundstop}
V(s_0)=\sup_{\tau \in \mathcal{S}}\E_{\mathbb{Q}} [Y_{\tau\wedge
T}|{S}_{0}=s_0],\quad s_0\in \R_{+}^{d},$$ where $Y_{t}:=\exp(-rt){g}(t, S_{t})$, $t\geq 0$ is the [discounted payoff process]{} and $\mathcal{S}$ denotes the set of all stopping times with respect to the filtration generated by $(S_t)_{0\leq t\leq T}$ [@karatzas1998methods Theorem 5.3]. In the remainder of this work, all expectations are taken with respect to the same risk-neutral measure $\mathbb{Q}$ and denoted by $\E$.
Most state-of-the-art methods for American option pricing – including all variants of the Longstaff–Schwartz [@longstaff2001valuing], PDE [@achdou2005computational], binomial tree [@cox1979option], and stochastic mesh [@broadie1997pricing] methods – exploit the dynamic programming principle to determine the value function using a backwards-iteration scheme. Further approaches are based on dual problems [@rogers2002montecarlo; @andersen2004primal], policy iteration [@belomestny2018advanced], or (quasi-)analytic solutions [@barone1987efficient; @kuske1998optimal]. The computational cost of many methods grows exponentially with respect to the number of dimensions, thus making them prohibitively expensive for options on many underlying assets. This phenomenon has been coined the curse of dimensionality [@reisinger2007efficient; @bellman2015adaptive].
In this work, we propose a method that is based on the following variation of , which states that the optimization may be restricted to *hitting times* instead of general stopping times: $$\label{fund}
V(s_0)=\sup_{E\in \mathcal{B}([0,T]\times \R_{+}^{d})}\E [Y_{\tau_{E}\wedge T}|{S}_{0}=s_0],\quad s_0\in \R_{+}^{d}.$$ Here, the supremum is taken over Borel-measurable subsets of $E\subset[0,T]\times \R_{+}^{d}$, whose [hitting times]{} are given by $\tau_{E}:=\inf\{t\geq 0 :
(t,S_t)\in E\}$. To be precise, both and require some technical conditions on the processes $(Y_{t})_{0\leq t\leq T}$ and $(S_t)_{0\leq t \leq T}$ [@shiryaev2007optimal Corollary 2, Section 3.3.1]. Throughout this work, we assume that such conditions hold and restrict our attention to the solution of .
To the best of our knowledge, optimization of the exercise region in was first proposed in [@grant1997path] and developed in [@andersen1999simple; @garcia2003convergence; @ibanez2004monte; @belomestny2011on; @gemmrich2012master], but it has not yet found its way into the canon of numerical algorithms for American option pricing. In [@grant1997path], separate exercise regions were determined for each exercise date of an American Asian option in a backwards iteration. The optimization at each step was performed in a brute force fashion, which explains why only two parameters were allowed in the parametrization of the exercise regions. In [@garcia2003convergence; @gemmrich2012master], ad hoc parametrizations that exploit known behavior of the optimal exercise regions were used to optimize exercise regions as subsets of time-space without applying a backwards iteration.
In general, optimization of the exercise region faces two challenges. First, as mentioned in [@gemmrich2012master], it is not obvious how to parametrize the possible exercise regions in a multi-dimensional setting, or even in a one-dimensional setting that goes beyond vanilla options in the Black–Scholes model. Second, once a parametrization has been found, it is not obvious how to find the global optimum [@garcia2003convergence; @gemmrich2012master]. Indeed, when the expectation in is replaced by an empirical average for the purpose of numerical approximations of the expected payoff, the quantity to be maximized depends highly irregularly on the exercise region $E$ (see below). Furthermore, even if a large number of sample paths is used to reduce the small scale oscillatory behavior, the resulting surface may still be non-concave and exhibit isolated local optima, as reported in [@garcia2003convergence].
To address these challenges, we introduce, in , a relaxation of the optimization problem in wherein the exercise regions $E\subset
[0,T] \times \R_{+}^{d}$ are replaced by *exercise rates* $f\colon[0,T] \times \R_{+}^{d}\to\R_{+}$, which define *randomized exercise strategies* where options are exercised with an infinitesimal probability depending on the current time and asset values.[^2] The space of exercise rates can easily be parametrized even in high dimensions using a finite-dimensional spaces of polynomials on $[0,T]\times \R^{d}$. The resulting optimization problem exhibits the same maximum as the original optimization problem over deterministic strategies but has the advantage of a differentiable objective function and a lower risk of getting stuck in local minima because of a richer search space. Indeed, by integrating analytically with respect to the exponential distribution that underlies the random exercise decision, we obtain an objective function that is smooth even when finitely many sample paths are used in the computations. We may then use gradient-based optimization routines to determine an optimal coefficient vector. Furthermore, we may start this optimization from an exercise rate that has a constant non-zero value across time and space and let the optimization routine gradually refine this neutral strategy towards an optimal one with marked variations in the exercise rate. This facilitates the search for a global optimum without requiring an informed initial guess that is already close to the optimum. Details of the numerical implementation are discussed in . There, we also briefly discuss how the accuracy of our method depends on the various discretization parameters. In particular, we provide heuristic bounds on the number of degrees of freedom in the exercise rate that are required for satisfactory randomized exercise strategies. These bounds are given in terms of the smoothness of the optimal exercise boundary as a manifold, not as a function of time.
Finally, presents numerical experiments for various market models and options. In , we consider vanilla put options in the classical Black–Scholes model. In the case of a single underlying, the exercise boundary of an American put option, whose payoff function is given by ${g}(t,s):={g}(s):=(K-s)^{+}$ for some [strike]{} $K>0$, can be written as a function of time with asymptotic behavior $s(t)\approx K-C_1\sqrt{(T-t)\log (T-t)}$ for some $C_1>0$ as $t\to T$. Despite the square-root singularity near the expiration time, the experiments presented in show that low-degree polynomials suffice to capture the optimal exercise boundary well. In fact, we obtain a relative error of less than $0.1\%$ with quadratic polynomials. This can be explained by the fact that the graph of the similar function $\tilde{s}(t)=K-C_1\sqrt{(T-t)}$ is smooth as a one-dimensional manifold in $\R^2$ and, indeed, coincides with the zero level set (intersected with $x<K$) of the quadratic polynomial $f(t,s):=(K-s)^2-C_1^2(T-t)$, whose scalar multiples therefore constitute close-to-optimal exercise rates.
Although we solve non-concave maximization problems, we are able to find global optima starting from a constant exercise rate. Furthermore, in we show that our algorithm outperforms the Longstaff–Schwartz algorithm with respect to the required polynomial degree for the pricing of basket put options, which is crucial when the number of underlying asset is large.
In , we consider call options on the maximum of a number of underlying assets, $g(s)=\max_{i=1}^{d}(s_i-K)^{+}$. Numerical algorithms for the pricing of such max call options were previously discussed in [@andersen2004primal; @ludkovski2018kriging]. Max call options pose a challenge to the direct determination of exercise regions because the optimal exercise regions are disconnected [@broadie1997valuation]. Still, our results show that polynomials of low degree suffice to obtain highly accurate estimates despite the nontrivial topology of the optimal exercise region.
In , we consider the [Heston]{} model, in which the underlying asset and its stochastic volatility form a joint Markov process. Since our method involves the market model for the generation of random sample paths only, its application in this scenario is straightforward. Finally, we consider the non-Markovian [rough Bergomi]{} model [@bayer2016pricing] in . To recover Markovianity, we must extend our process by its past values. In practice, using a large but finite number of past values leads to very high-dimensional approximation problems. However, our experiments indicate that exercise strategies depending only on the spot values of the underlying asset and its volatility achieve near-optimal performance.
Exercise rate optimization {#sec:ero}
==========================
We let ${\mathcal{T}}:=[0,T]$ and assume throughout that $(S_{t})_{t\in {\mathcal{T}}}$ is conditioned on $S_{0}=s_0$.
For any $f\colon{\mathcal{T}}\times \R_{+}^{d}\to\R_{+}$, the *randomized exercise strategy with exercise rate* $f$ is given by early exercise at the time $$\label{randomstop}
\tau_{f}:=\inf\{t\geq 0:\int_{0}^{t}\lambda_u{\mathop{}\!\mathrm{d}}u \geq X\},$$ where $
\lambda_{t}:=f(t,S_t)
$, $t\in{\mathcal{T}}$, and $X$ is a standard exponential random variable that is independent of $(S_t)_{t\in {\mathcal{T}}}$.
The exercise time $\tau_f$ equals the first jump time of a Poisson process with rate $(\lambda_{t})_{t\in {\mathcal{T}}}$. In other words, the exercise rate $f$ determines the time- and space-dependent infinitesimal probability with which the American option is exercised in a infinitesimal time interval ${\mathop{}\!\mathrm{d}}t$.
With in mind, we are interested in the expected payoff under a randomized exercise strategy with early exercise time $\tau_{f}$, which we denote by $$\label{psi}
\psi(f):=\E [Y_{\tau_{f}\wedge T}].$$ Since $\int_{0}^{t}\lambda_u{\mathop{}\!\mathrm{d}}u$ is a deterministic function of the asset path until $t$, and $X$ is independent of $(S_u)_{u\in {\mathcal{T}}}$, we have $$\mathbb{P}(\tau_f \geq t\mid (S_{u})_{u\in {\mathcal{T}}}) =\mathbb{P}(X>\int_{0}^{t}\lambda_u{\mathop{}\!\mathrm{d}}u\mid (S_{u})_{u\in {\mathcal{T}}})=\exp\left(-\int_{0}^{t}\lambda_u {\mathop{}\!\mathrm{d}}u\right)=:U_{t}$$ and $$\mathbb{P}(\tau_f \in {\mathop{}\!\mathrm{d}}t\mid (S_{u})_{u\in{\mathcal{T}}})= -{\mathop{}\!\mathrm{d}}U_t
= \lambda_t U_t {\mathop{}\!\mathrm{d}}t .$$ Hence, we obtain $$\phi(f,(S_{u})_{u\in{\mathcal{T}}}):=\E [Y_{\tau_{f}\wedge T}\mid (S_{u})_{u\in{\mathcal{T}}}] = \int_{0}^T Y_t \lambda_t U_t {\mathop{}\!\mathrm{d}}t +
Y_T U_T.$$ By the law of total expectation, which we may apply because all the random variables involved are nonnegative, we deduce the formula $$\label{fund1}
\psi(f)= \E[\phi(f,(S_{u})_{u\in{\mathcal{T}}})] = \E\left[\int_{0}^T Y_t \lambda_t U_t {\mathop{}\!\mathrm{d}}t +
Y_T U_T\right].$$ It is advisable to replace $\lambda U_t {\mathop{}\!\mathrm{d}}t$ by $-{\mathop{}\!\mathrm{d}}U_t$ in numerical implementations of this formula to avoid cancellations. The following proposition shows that, in theory, exercise rate optimization yields the correct option value. It is a special case of Theorem 2.2 in [@gyongy2008randomized].
\[equality\] We have $$\label{fund3}
V(s_0)=\sup_{f\colon [0,T]\times \R_{+}^{d}\to \R_{+}}\psi(f).$$
For any $E\in \mathcal{B}({\mathcal{T}}\times \R_{+}^{d})$, we may formally insert the indicator function $$f_{E}(t,s):=\begin{cases}
+\infty, & (t,s)\in E\\
0, &(t,s)\not\in E
\end{cases}$$ into to obtain $\tau_{f_E}=\tau_E$. After replacing $+\infty$ with large numbers that diverge to $+\infty$ and applying Fatou’s lemma, we may take the supremum over $E$ to conclude from that $\sup_{f\colon [0,T] \times \R_{+}^{d}\to \R_{+}}\psi(f)\geq V(s_0)$.
Conversely, the law of total expectation shows, for any $f\colon [0,T]
\times \R_{+}^{d}\to\R_{+}$, that $$\begin{aligned}
\psi(f)=\E[Y_{\tau_{f}\wedge T}]=\E\Big[\E\left[Y_{\tau_f\wedge T}\mid X\right]\Big].
\end{aligned}$$ Because $\tau_{f}$ conditioned on $X$ is a stopping time and $(S_{t})_{t\in{\mathcal{T}}}$ is independent of $X$, implies that $\E\left[Y_{\tau_{f}\wedge T}\mid X\right]\leq V(s_0)$ almost surely; hence, $\psi(f)\leq V(s_0)$.
Numerical algorithm {#sec:algo}
-------------------
To determine optimal exercise rates numerically, we
(i) replace the time-continuous model of the stochastic process $(S_{t})_{t\in{\mathcal{T}}}$ with a discretization with $N<\infty$ time steps, such as the the Euler–Maruyama scheme;
(ii) replace the expectation in with an average over $M<\infty$ fixed sample paths $(S^{(m)}_{n})_{1\leq n\leq N,1\leq m \leq M}$;
(iii) \[pparam\] introduce a $B$-dimensional, $B<\infty$ parametrization $\R^{B}\ni\bm{c}\mapsto f_{\bm{c}}$ of the space of exercise rates;
(iv) maximize the surrogate function $$\begin{aligned}
\overline{\psi}\colon &\R^{B}\to \R\\
&\bm{c}\mapsto \frac{1}{M}\sum_{m=1}^{M} \phi(f_{\bm{c}},(S^{(m)}_{t})_{1\leq n\leq N}).\footnotemark
\end{aligned}$$
#### Parametrization
To address step (iii), we work with the logarithmic asset values $x_i:=\log(s_i)$, ${1\leq i\leq d}$ and let $$F_{\mathcal{P}}:=\left\{f_{p}(t,x):=1_{{g}(t,s)>0}\exp(p(t,x))\;\big|\;p\in \mathcal{P}\right\}$$ for any finite-dimensional linear space $\mathcal{P}$ of functions on ${\mathcal{T}}\times\R^{d}$. After choosing a basis of $\mathcal{P}$, we obtain the desired parametrization $\bm{c}\mapsto f_{\bm{c}}$. Throughout the remainder of this manuscript, we work with spaces $\mathcal{P}_{k}$ of polynomials of degree less than or equal to $k\geq 0$ in $d+1$ variables, and we use an orthonormal basis with respect to the inner product $\|f\|^2:=\frac{1}{NM}\sum_{n=1}^{N}\sum_{m=1}^{M} f(t_n,x_{n,m})$ induced by the time-space samples $(t_{n},x_{n,m}:=\log(S_{n}^{m}))_{1\leq n\leq N, 1\leq m\leq M}$.
#### Optimization
Concerning step (iv), it is not clear that globally optimal coefficients, which may even lie at infinity, can be found numerically because $\overline{\psi}$ is not concave. However, in our numerical experiments, we found that the Quasi-Newton L-BFGS-B algorithm [@byrd1995limited], as implemented in Python’s SciPy library[^3], performs well and does not get stuck in local maxima when started from a constant exercise rate.
The advantage of exercise rate optimization over exercise region optimization is illustrated by . Even a simple gradient ascent algorithm could be used to maximize $\overline{\psi}$ in , where we show the dependence on the coefficient $c_{(0,0)}$ of the constant polynomial $p_{(0,0)}\equiv1$ for a one-dimensional put option. For comparison, this is not possible for the function shown in , which arises from the optimization of deterministic exercise regions and requires the use of finite-difference stochastic-gradient algorithms.
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Differentiability of $\phi$, $\psi$, and $\overline{\psi}$ with respect to $f$ is easy to show. Using the fact that $\lambda_t U_t{\mathop{}\!\mathrm{d}}t=-{\mathop{}\!\mathrm{d}}U_t$, we obtain the simple gradient formula $$\begin{aligned}
\langle \nabla_{f}\phi(f,(S_{t})_{t\in{\mathcal{T}}}),h\rangle &= -\int_{0}^{T}Y_t{\mathop{}\!\mathrm{d}}\langle \nabla_f U_t,h\rangle+\langle \nabla_f U_{T},h\rangle Y_T,\quad h\colon{\mathcal{T}}\times\R_{+}^{d}\to\R,\end{aligned}$$ where $$\begin{aligned}
\langle \nabla_f U_{t},h\rangle&=-U_t\int_{0}^{t}{h(u,{S}_u)}{\mathop{}\!\mathrm{d}}u, \quad t\in {\mathcal{T}}.\end{aligned}$$
shows four snapshots of the search for an optimal exercise rate for max call options on two underlying securities.
[0.49]{} ![Four iterations of the exercise rate optimization for a max call option (all figures show a slice of the exercise rate at $t=T/2$). High color intensities represent high exercise rates. The white region in the bottom left contains the points with zero payoff, $\{{g}=0\}$. Random sample values of the two underlying securities at $T/2$ are shown in blue.[]{data-label="fig:anim"}](figures/animation/animation0_10.png "fig:")
[0.49]{} ![Four iterations of the exercise rate optimization for a max call option (all figures show a slice of the exercise rate at $t=T/2$). High color intensities represent high exercise rates. The white region in the bottom left contains the points with zero payoff, $\{{g}=0\}$. Random sample values of the two underlying securities at $T/2$ are shown in blue.[]{data-label="fig:anim"}](figures/animation/animation0_20.png "fig:")
\
[0.49]{} ![Four iterations of the exercise rate optimization for a max call option (all figures show a slice of the exercise rate at $t=T/2$). High color intensities represent high exercise rates. The white region in the bottom left contains the points with zero payoff, $\{{g}=0\}$. Random sample values of the two underlying securities at $T/2$ are shown in blue.[]{data-label="fig:anim"}](figures/animation/animation0_30.png "fig:")
[0.49]{} ![Four iterations of the exercise rate optimization for a max call option (all figures show a slice of the exercise rate at $t=T/2$). High color intensities represent high exercise rates. The white region in the bottom left contains the points with zero payoff, $\{{g}=0\}$. Random sample values of the two underlying securities at $T/2$ are shown in blue.[]{data-label="fig:anim"}](figures/animation/animation0_40.png "fig:")
#### Accuracy
To obtain accurate results, we must choose large enough values for the number of samples, $M$, the number of time steps, $N$, the number of iterations of the optimization routine, $\ell$, and the polynomial degree, $k$.
For a fixed exercise rate and a fixed number of time steps, convergence with respect to the number of sample paths, $M$, occurs asymptotically at the Monte Carlo rate $M^{-1/2}$. Pre-asymptotically, the number of Monte Carlo samples has to be larger than a threshold depending on the dimension of the polynomial subspace to avoid overfitting, see the next paragraph.
For a fixed, smooth exercise rate, the expected payoff converges at the weak convergence rate of the discretization scheme with respect to the number of time steps (e.g., $N^{-1}$ for the Euler–Maruyama scheme). In the limit of increasingly steep exercise rates approaching the optimal deterministic exercise regions, the weak convergence rate is expected to deteriorate to $N^{-1/2}$. However, this effect does not become noticeable in our numerical experiments (see ).
With everything else held fixed, we expect exponential or faster convergence with respect to $\ell$, depending on what type of deterministic optimization routine is used. in provides numerical evidence of exponential convergence using the L-BFGS-B algorithm.
To characterize the convergence of the optimal exercise rate with respect to $k$ under the simplifying assumptions $M=\infty$ and $N=\infty$, we note that for any polynomial $0\neq p_k\in \mathcal{P}_{k}$ the randomized exercise strategies with exercise rates $f_{L}:=\exp(L p_k)\in
F_{k}$ converge to a deterministic strategy with early exercise region $E_k:=\{p_k\geq 0\}$ as $L\to \infty$. Therefore, it suffices to study the approximability of the optimal exercise region $E_*$ by polynomial superlevel sets, and the sensitivity of the expected payoff on the right-hand side of with respect to perturbations of the exercise region. Regarding the approximability of $E_*$, we observe that if $E_*$ is a bounded $C^m$-submanifold, $m\geq 2$, of $(0,T)\times \{{g}>0\}$, then there exists a sequence of polynomials $p_k$ such that the boundaries $B_k:=\partial E_k$ of the corresponding exercise regions $E_k:=\{p_k\geq 0\}$ satisfy $$\label{perturb}
B_k=\{(t,s)+\Theta(t,s) : (t,s) \in B_{*}\}$$ for some $\Theta\colon B_{*}\to\R^{1+d}$ such that $$\sup_{(t,s)\in B_{*}}|\Theta(t,s)|<C k^{-m}.$$ This follows from a combination of the multi-dimensional Jackson theorem [@BagbyBosLevenberg2002] with a partition of unity and elementary geometry. Regarding the sensitivity of the expected payoff, [@gobet2006sensitivities] showed differentiability with respect to perturbations of the exercise region in spatial directions under the assumption that $(0,s_0)\not\in E_*$ and that the payoff function lies in some Hölder space $C^{1,\alpha}$, $\alpha>0$. Unfortunately, this result is not quite general enough for our purposes, since we require bounds with respect to general, spatio-temporal perturbations of the domain (as in ) and for payoff functions that are only Lipschitz.
A rigorous analysis of the interplay of the various discretizations will be the topic of future work; some numerical results are presented in below.
#### Overfitting
Choosing a subspace with a large number of degrees of freedom, $B\gg 1$, to improve the flexibility of the candidate exercise rates increases the cost of computations and the risk of *overfitting*. This means that the value of $\overline{\psi}(\bm{c}^*)$ at the optimized coefficients $\bm{c}^{*}$ may overestimate the true value $\psi(f_{\bm{c}^*})$ unless a correspondingly large number $M=M(B)$ of sample paths is used. Numerical experiments indicate that $M(B)\approx CB^2$ for some $C>0$ but we were not able to prove such a formula. In practice, we can simply compute an unbiased estimate of $\psi(f_{\bm{c}^{*}})$ using a new set of sample paths $(\tilde{S}^{(m)}_{t})_{t\in{\mathcal{T}}}$, $1\leq m\leq {M}$; similar techniques are used in classical regression-based methods such as the Longstaff–Schwartz algorithm. Following statistical learning terminology, we refer to the biased and unbiased estimators of $\psi(f_{\bm{c}^{*}})$ as *training* and *test* values, respectively. One way to avoid overfitting is to recompute the test value at each step of the optimization and to terminate as soon as the test value decreases. Note that, as in the case of the Longstaff–Schwartz algorithm, the test values are biased low, i.e., are Monte Carlo estimates of *lower bounds* of the option price.
Numerical experiments {#sec:practice}
=====================
Throughout this section, we use the L-BFGS-B algorithm with initial coefficients $\bm{c}\equiv 0$ to maximize $\overline{\psi}$.
Convergence with respect to discretization parameters {#sec:practice0}
-----------------------------------------------------
In this subsection, we study the convergence of our method with respect to the discretization parameters $M$, $N$, $k$, and $\ell$ by pricing the vanilla put option from with strike $K=100$ and expiry $T=1$ in the Black–Scholes model with volatility $\sigma=0.3$, risk-free interest rate $r=0.05$, and spot price $s_0=100$. Using a binomial tree algorithm with $\num{50000}$ levels (i.e., $\num{50000}$ time steps and $\num{50000}$ spatial discretization nodes at $T=1$), we obtain the reference value $V^*=9.8701$. show that the prices found through exercise rate optimization with polynomial degree $k=2$ and $M_n:=200\times 4^{n}$ sample paths with $N_n:=2^{n}$ time-steps converge towards this reference value as $n\to\infty$.
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In particular, our maximization does not get stuck in local optima of $\overline{\psi}$. Furthermore, shows that test and training values converge at roughly the same speed, which means that we do not suffer from overfitting. This is not surprising, since the space of bivariate quadratic polynomials is only 6-dimensional. We restrict the following plots to the test value, which constitutes an unbiased estimate of the quality of a given exercise rate.
In the logarithmic scale of , we see that our approximations converge to the reference value at roughly the rate $2^{-n}=\mathcal{O}(N_n^{-1}+M_n^{-1/2})$. We obtain an accuracy of about four significant digits, despite using only quadratic polynomials for the exercise boundary approximation. This confirms that singularities of the exercise boundary as a function of time do not pose a problem for our polynomial approximation scheme. For comparison, show results for $k\in \{0,1\}$, that is, for constant exercise rates and for exercise rates that depend only linearly on space and time, respectively. For $k=1$, the results are astoundingly similar to the case $k=2$, though closer inspection on a logarithmic scale reveals stagnation at a relative error of $0.5\%$. For $k=0$, our method stagnates around the value $9.35$, which is roughly the price of a European option with the same parameters.
To study the effects of $M$, $N$, and $k$, we performed experiments in this and the following subsection with the tolerance of the L-BFGS-B optimization set to machine precision, which required between $70$ and $200$ function evaluations to achieve. However, an error comparable to that of the remaining discretization errors can already be achieved with significantly fewer evaluations. Indeed, for $n=4$ and $k=2$ the relative error between $\overline{\psi}(\bm{c}_{\ell})$ and the final value is already below $0.1\%$ when $\ell=20$ (). For this reason, we limit the number of iterations below to $20$.
Comparison with Longstaff–Schwartz algorithm {#sec:practice1}
--------------------------------------------
In this subsection, we consider *basket put options* on linear combinations of $d\in\{2,5\}$ underlying assets. The payoff function of such options is given by ${g}(s):=(K-c\cdot {s})^{+}$ for $K>0$ and $c\in\R^{d}$. In our experiments, we use $K:=100$ and $c_i:=1/d$, $1\leq i\leq d$.
We compare our method to the Longstaff–Schwartz algorithm, as implemented in the freely available version 16 of the derivative pricing software Premia[^4]. Like our method, the Longstaff–Schwartz algorithm requires specification of the number of sample paths, the number of time-steps used for their simulation, and the polynomial degree, which controls the accuracy of approximations of the value function. For simplicity, we restrict the simulations in this section to $N=8$ time steps. To prevent our comparison being skewed by the fact that the two algorithms use different sample paths, we use the same large number of $M=3.2\times 10^6$ samples for both. Finally, we use a risk-free interest rate $r=0.05$ and a diagonal volatility matrix $\Sigma_{ij}=0.3^2\delta_{ij}$, $1\leq i,j\leq d$ in the underlying Black–Scholes model with $s_0=(100,\dots,100)$.
To emphasize the efficiency of exercise rate optimization with respect to the polynomial degree, we compute reference values $V^*=6.5479$ and $V^*=3.6606$ using exercise rate optimization with polynomial degree $k_{\mathrm{ERO}}=2$ for $d=2$ and $d=5$, respectively. shows that the Longstaff–Schwartz algorithm converges to these values as $k_{\mathrm{LS}}\to\infty$, but only achieves a comparable performance for $k\approx 6$. We show $95\%$ confidence bands around our reference value, which are based on the empirical variance in the evaluation of our test value. From these we see that the remaining difference between the two methods can be explained by the random sampling error.
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#### Runtime comparison
To obtain a fair runtime comparison, we created a Python package[^5] with straightforward implementations of both algorithms, which we ran on a 12 core Intel Xeon X5650 CPU.
For the same polynomial degree, exercise rate optimization is slower than the Longstaff–Schwartz algorithm. However, as we have seen above, the latter requires larger polynomial degrees for accurate results. Since the ratio between the dimensions of polynomial subspaces with degrees $k=2$ and $k>2$ grows with respect to the dimension of the domain, exercise rate optimization returns accurate results faster than the Longstaff–Schwartz algorithm in high-dimensional examples.
For example, for a basket put option as above with $d=10$, the Longstaff–Schwartz algorithm returns $2.235$ with $k=2$ after $530$ seconds and $2.237$ with $k=4$ after $7437$ seconds. Exercise rate optimization, on the other hand, returns $2.240$ with $k=2$ after $2493$ seconds. All these results were obtained with the same $3.2\times 10^6$ Brownian motion samples.
Max call options {#sec:practice1b}
----------------
In this subsection, we consider max call options on two underlying assets, for which $g(s):=\max\{(s_1-K)^{+},(s_2-K)^{+}\}$. These max call options present an interesting challenge for our method, since the optimal exercise region at any time before expiry has two connected components [@broadie1997valuation]. Lower and upper bounds for the option prices in the Black–Scholes model with $r=0.05$, $\Sigma_{ij}=0.2^2\delta_{ij}$, $K=100$, $N=8$ and dividend $\delta=0.1$ are taken from [@andersen2004primal] and provided in alongside the results of our method for $k\in\{1,2,3\}$ and $M=\SI{1000000}{}$.
------- ----- -------------------- -------- -------- --------
$1$ $2$ $3$
90 \[8.053,8.082\] 7.126 8.009 8.039
$s_0$ 100 \[13.892,13.934\] 12.311 13.821 13.865
110 \[21.316, 21.359\] 19.133 21.220 21.256
------- ----- -------------------- -------- -------- --------
: Prices of max call option. 95% confidence intervals (CI) taken from [@andersen2004primal].[]{data-label="table:maxcall"}
The optimized exercise rates with $k\in\{2,3\}$ are shown in .
As expected, they are almost deterministic, which means that they exhibit steep slopes from values close to zero to values close to infinity. Since the specific values are irrelevant, we restrict our plots to the level sets of exercise rate $0.001$ and $1000$. The results in this subsection were obtained using a maximal number of $20$ optimization steps. Performing more steps would further reduce the distance between these level sets without a noticeable difference in the resulting option price. As predicted by theory, there are two disjoint regions of high exercise rates. Furthermore, due to the symmetry of the underlying model and the payoff, the optimized exercise rate is almost axisymmetric even though we do not enforce this symmetry. While modeling the disconnected regions is not possible with log-linear exercise rates available for $k=1$, the hyperbolic conic sections available with $k=2$ already provide satisfactory approximations.
Stochastic volatility {#sec:practice2}
---------------------
In this subsection, we apply our method to pricing in a stochastic volatility model.
For this purpose, we consider the basic Heston model as described in [@heston1993closed], which models the evolution of a single underlying asset $X_t$ and its instantaneous variance ${v}_t$ using the coupled system of stochastic differential equations $$\begin{aligned}
\label{hestonunderlying}
{\mathop{}\!\mathrm{d}}X_t &= \mu X_t{\mathop{}\!\mathrm{d}}t +\sqrt{{v}_t} X_t {\mathop{}\!\mathrm{d}}W^{X}_t,\\
\label{hestonvolatility}
{\mathop{}\!\mathrm{d}}{v}_t &= \kappa(\theta-{v}_t){\mathop{}\!\mathrm{d}}t+\xi \sqrt{{v}_t}{\mathop{}\!\mathrm{d}}W^{{v}}_{t},\end{aligned}$$ where $\mu>0, \kappa>0$, $\theta>0$, $\xi>0$ with $2\kappa\theta>\xi^2$, and $W^{X}_t$ and $W^{{v}}_t$ are Wiener processes with correlation $-1\leq\rho\leq 1$.
Since our method requires Markovian markets, we must include the volatility and define $S_t:=(X_t,{v}_t)$, $t\in{\mathcal{T}}$. This means that knowledge of the current volatility is required to make optimal exercise decisions in stochastic volatility models.
To obtain a risk neutral measure, we replace $\mu$ with the risk-free rate $r=0.05$ in . We choose the remaining parameters $\kappa=3$, $\theta=0.05$, $\xi=0.5$, $\rho=-0.5$ and compute estimates of $v_K(s_0)$ for a put option with $s_0=(100,0.15)$ and $25$ different values of the strike $K\in[90,150]$. For this purpose, we use polynomials of degree $k\in\{0,1,2\}$ and $M=\SI{100000}{}$ samples with $N=32$ time steps.
For comparison, we also show the results of the finite difference method `FD_Hout_Heston` implemented in Premia, with $32$ time steps and a grid of $100\times 100$ nodes in the discretization of the stock-volatility plane. The results are shown in .
The maximal relative difference between the two methods is $1\%$ and occurs around $K^*=130$. Up to roundoff error, the prices computed by our method are equal to $K-100$ for all $K\geq K^*$. This behavior is expected, since for large enough $K$ the initial point $(100,0.15)$ lies within the optimal exercise region and the option is thus exercised immediately.
shows the numerically optimized exercise rates (with $k=2$) at $t=0.5$ for $K\in\{100,110\}$.
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Finally, we consider a 10-dimensional portfolio where each underlying $(X_t^{i})_{t\in{\mathcal{T}}}, 1\leq i\leq 10$ follows with the same volatility process $({v}_t)_{t\in{\mathcal{T}}}$ (and the same parameter values as in the one-dimensional case) but different Wiener processes $(W^{X^i})_{t\in{\mathcal{T}}}$, $1\leq i\leq 10$ such that the $11$-dimensional Wiener process $(W^{X^1}_t,\dots,W^{X^{10}}_t,W^{{v}}_t)$ has the covariance matrix $$\Sigma=\begin{pmatrix}
1. & 0.2 & 0.2 & 0.35 & 0.2 & 0.25 & 0.2 & 0.2 & 0.3 & 0.2 & -0.5 \\
0.2 & 1. & 0.2 & 0.2 & 0.2 & 0.125& 0.45 & 0.2 & 0.2 & 0.45 & -0.5 \\
0.2 & 0.2 & 1. & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0.45 & 0.2 & -0.5 \\
0.35 & 0.2 & 0.2 & 1. & 0.2 & 0.2 & 0.2 & 0.2 & 0.425& 0.2 & -0.5 \\
0.2 & 0.2 & 0.2 & 0.2 & 1. & 0.1 & 0.2 & 0.2 & 0.5 & 0.2 & -0.5 \\
0.25 & 0.125& 0.2 & 0.2 & 0.1 & 1. & 0.2 & 0.2 & 0.35 & 0.2 & -0.5 \\
0.2 & 0.45 & 0.2 & 0.2 & 0.2 & 0.2 & 1. & 0.2 & 0.2 & 0.2 & -0.5 \\
0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & 1. & 0.2 & -0.1 & -0.5 \\
0.3 & 0.2 & 0.45 & 0.425& 0.5 & 0.35 & 0.2 & 0.2 & 1. & 0.2 & -0.5 \\
0.2 & 0.45 & 0.2 & 0.2 & 0.2 & 0.2 & 0.2 & -0.1 & 0.2 & 1. & -0.5 \\
-0.5 & -0.5 & -0.5 & -0.5 & -0.5 & -0.5 & -0.5 & -0.5 & -0.5 & -0.5 & 1 \\
\end{pmatrix}$$ shows estimates of the values of American basket put options (with coefficients $c\equiv 1/10$) that were obtained by exercise rate optimization for the corresponding $11$-dimensional process $S_t:=(X^{1}_t,\dots,X^{10}_t,{v}_t)$ using the same discretization parameters as before.
Rough volatility {#sec:practice3}
----------------
To illustrate the wide applicability of our method, we conclude this section with the non-Markovian *rough Bergomi* model, which was previously applied to explain implied volatility smiles and other phenomena in the pricing of European options [@bayer2016pricing]. In non-Markovian models, does not hold because optimal exercise strategies may be based on the entire history of the path $(S_t)_{t\in{\mathcal{T}}}$, which we again assume to include the underlying asset $(X_t)_{t\in{\mathcal{T}}}$ as well as the volatility $({v}_t)_{t\in{\mathcal{T}}}$. Therefore, we consider the infinite-dimensional Markovian extension $$\tilde{S}_t:=(S_u)_{u\in [0,t]},\quad t\in{\mathcal{T}},$$ for which formally holds with subsets of ${\mathcal{T}}\times \R_{+}^{d}$ replaced by subsets of ${\mathcal{T}}\times \Gamma$, where $\Gamma:=\bigcup_{t\in{\mathcal{T}}}\{s:[0,t]\to\R_{+}^{d}\}$.
For numerical purposes, we subsample realizations of $S_t$ (with the convention that $S_t:=S_{0}$ for $t<0$) and define $$\tilde{\bm{S}}_t:=(S_{t},S_{t-\Delta_1},\dots,S_{t-\Delta_J})\in \R^{d_{\text{eff}}}:=\R^{2\times (1+J)},\;\quad t\in {\mathcal{T}}$$ for some $J<\infty$ and $0<\Delta_1<\dots<\Delta_J$. We apply the algorithm described in to the resulting problem of finding exercise rates on the extended space ${\mathcal{T}}\times \R^{d_{\text{eff}}}$.
Following [@bayer2016pricing Section 4], we generate samples from the risk-neutral measure induced by $$\begin{aligned}
\label{rbunderlying}
{\mathop{}\!\mathrm{d}}X_t &= rX_t{\mathop{}\!\mathrm{d}}t+ X_t \sqrt{{v}_{t}} {\mathop{}\!\mathrm{d}}W^{X}_t, \quad X_0 = x_0,\\
\label{rbvolatility}
{v}_{t}&:={v}_0\mathcal{E}\left(\eta\sqrt{2H}\int_{0}^{t}\frac{1}{(t-u)^{1/2-H}}{\mathop{}\!\mathrm{d}}W^{{v}}_{u}\right),\end{aligned}$$ where $\mathcal{E}$ is the stochastic exponential in the Wick sense, $H = 0.07$, $r=0.05$, $\eta = 1.9$, and $W^{X}$, $W^{{v}}$ are Wiener processes with correlation $\rho=-0.9$. Since the asset price process $X_t$ is a continuous local martingale, standard no arbitrage theory applies even though ${v}_t$ is not a semi-martingale.
shows the American option prices for $x_0=100$, ${v}_0=0.09$, $T=1$, and different strikes, which we computed using the discretization parameters $M=\SI{100000}{}$, $N=128$, $k=2$, and $\Delta_j:=j/8$, $1\leq j\leq J$, $J\in\{0,1,3,7\}$. For comparison, we include the European prices computed by simple Monte Carlo simulation. The difference between our estimates for $J=0$ and $J=7$ is not consistently larger than the Monte Carlo sampling error, indicating that the exploitation of non-Markovian features does not yield significantly improved exercise strategies. This is not to say, however, that American option prices in non-Markovian and Markovian models are similar. The non-Markovianity of the samples of $(S_t)_{t\in{\mathcal{T}}}$ plays an important role in the evaluation of any given strategy, even when the strategy only depends on the spot values.
-------------- ------ ------ ------ ------ ------- ------- ------- ------- -------
$70$ $80$ $90$ $100$ $110$ $120$ $130$ $140$
1.83 3.13 5.06 7.98 12.21 17.99 25.35 33.88
0 1.88 3.23 5.32 8.51 13.24 20 30 40
1 1.88 3.23 5.31 8.50 13.22 20 30 40
\[0pt\][J]{} 3 1.88 3.21 5.31 8.50 13.22 20 30 40
7 1.88 3.22 5.30 8.50 13.23 20 30 40
-------------- ------ ------ ------ ------ ------- ------- ------- ------- -------
: Prices of put options in the rough Bergomi model.[]{data-label="table:bergomi"}
The numerically optimized exercise rates at $t=0.5$ for $J=0$ and $K\in\{100,110\}$ are shown in .
[0.49]{}
[0.5]{}
Conclusion
==========
We have introduced a method of pricing American options by optimization of randomized exercise strategies, in which deterministic exercise regions are replaced by probabilistic exercise rates.
Since the objective function of the corresponding relaxed optimization problem is smooth, optimal exercise rates can be found using simple deterministic optimization routines. Our numerical experiments show that exercise rates based on quadratic polynomials are sufficient to obtain remarkably accurate price estimates and that the resulting non-concave objective functions can be globally maximized using only a few iterations. Since the market model only appears in the simulation of sample paths, our method is quite flexible and easy to implement. We demonstrated its practical applicability in uni- and multivariate Black–Scholes, Heston and rough Bergomi models.
In even higher-dimensional situations than those considered in this work, already the space of quadratic polynomials may be prohibitively large. In that case, the polynomial subspace $\mathcal{P}$ could be designed in an anisotropic way to exploit, for example, the fact that the exercise decision of basket put options with coefficients $c$ is most sensitive to the coordinate $\tilde{s}_1:=c\cdot s$. For situations where large polynomial subspaces are unavoidable, a rigorous analysis of the number of samples that are required to determine a given number of degrees of freedom without significant overfitting would be of interest; similar but not directly transferable results were established in [@belomestny2011rates; @zanger2018convergence].
To accelerate numerical implementations, multilevel Monte Carlo methods [@Giles2015] could be used for evaluations of the expected payoff and its gradient.
It is an open question whether efficiently computable upper bounds on the option price [@belomestny2013solving] can be constructed using exercise rates as well.
[ **Acknowledgments** This work was supported by the KAUST Office of Sponsored Research (OSR, award URF/1/2584-01-01), the German Research Foundation (DFG, grant BA5484/1) and the Alexander von Humboldt Foundation. R. Tempone and S. Wolfers are members of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering. ]{}
[^1]: Corresponding author. Email address: `[email protected]`
[^2]: We were informed after the initial submission of this manuscript that randomized stopping was previously studied from a theoretical perspective [@gyongy2008randomized; @krylov2008controlled]. These references do not contain discussions of numerical solution of the resulting stochastic optimization problem, however.
[^3]: <https://docs.scipy.org/doc/scipy/reference/optimize.minimize-lbfgsb.html>
[^4]: <https://www.rocq.inria.fr/mathfi/Premia>
[^5]: <https://pypi.org/project/pryce/>
|
---
abstract: 'We investigate the possibility that the dark matter consists of clusters of the heavy family quarks and leptons with zero Yukawa couplings to the lower families. Such a family is predicted by the [*approach unifying spin and charges*]{} as the fifth family. We make a rough estimation of properties of baryons of this new family members, of their behaviour during the evolution of the universe and when scattering on the ordinary matter and study possible limitations on the family properties due to the cosmological and direct experimental evidences.'
address: 'Department of Physics, FMF, University of Ljubljana, Jadranska 19, 1000 Ljubljana'
author:
- 'G. Bregar, N.S. Mankoč Borštnik'
title: 'Does dark matter consist of baryons of new stable family quarks?\'
---
\#1)\#2 \#1\]\#2 \#1[\_[1.5pt ]{}]{}
Introduction
============
Although the origin of the dark matter is unknown, its gravitational interaction with the known matter and other cosmological observations require from the candidate for the dark matter constituent that: i. The scattering amplitude of a cluster of constituents with the ordinary matter and among the dark matter clusters themselves must be small enough, so that no effect of such scattering has been observed, except possibly in the DAMA/NaI [@rita0708] and not (yet?) in the CDMS and other experiments [@cdms]. ii. Its density distribution (obviously different from the ordinary matter density distribution) causes that all the stars within a galaxy rotate approximately with the same velocity (suggesting that the density is approximately spherically symmetrically distributed, descending with the second power of the distance from the center, it is extended also far out of the galaxy, manifesting the gravitational lensing by galaxy clusters). iii. The dark matter constituents must be stable in comparison with the age of our universe, having obviously for many orders of magnitude different time scale for forming (if at all) solid matter than the ordinary matter. iv. The dark matter constituents had to be formed during the evolution of our universe so that they contribute today the main part of the matter ((5-7) times as much as the ordinary matter).
There are several candidates for the massive dark matter constituents in the literature, like, for example, WIMPs (weakly interacting massive particles), the references can be found in [@dodelson; @rita0708]. In this paper we discuss the possibility that the dark matter constituents are clusters of a stable (from the point of view of the age of the universe) family of quarks and leptons. Such a family is predicted by the approach unifying spin and charges [@pn06; @n92; @gmdn07], proposed by one of the authors of this paper: N.S.M.B. This approach is showing a new way beyond the standard model of the electroweak and colour interactions by answering the open questions of this model like: Where do the families originate?, Why do only the left handed quarks and leptons carry the weak charge, while the right handed ones do not? Why do particles carry the observed $SU(2), U(1)$ and $SU(3)$ charges? Where does the Higgs field originate from?, and others.
There are several attempts in the literature trying to understand the origin of families. All of them, however, in one or another way (for example through choices of appropriate groups) simply postulate that there are at least three families, as does the standard model of the electroweak and colour interactions. Proposing the (right) mechanism for generating families is to our understanding the most promising guide to physics beyond the standard model.
[*The approach unifying spin and charges is offering the mechanism for the appearance of families.*]{} It introduces the [*second kind*]{} [@pn06; @n92; @n93; @hn02hn03] of the Clifford algebra objects, which generates families as the [*equivalent representations to the Dirac spinor representation*]{}. The references [@n93; @hn02hn03] show that there are two, only two, kinds of the Clifford algebra objects, one used by Dirac to describe the spin of fermions. The second kind forms the equivalent representations with respect to the Lorentz group for spinors [@pn06] and the families do form the equivalent representations with respect to the Lorentz group. The approach, in which fermions carry two kinds of spins (no charges), predicts from the simple starting action more than the observed three families. It predicts two times four families with masses several orders of magnitude bellow the unification scale of the three observed charges.
Since due to the approach (after assuming a particular, but to our opinion trustable, way of a nonperturbative breaking of the starting symmetry) the fifth family decouples in the Yukawa couplings from the lower four families (whose the fourth family quark’s mass is predicted to be at around $250$ GeV or above [@pn06; @gmdn07]), the fifth family quarks and leptons are stable as required by the condition iii.. Since the masses of all the members of the fifth family lie, due to the approach, much above the known three and the predicted fourth family masses, the baryons made out of the fifth family form small enough clusters (as we shall see in section \[properties\]) so that their scattering amplitude among themselves and with the ordinary matter is small enough and also the number of clusters (as we shall see in section \[evolution\]) is low enough to fulfil the conditions i. and iii.. Our study of the behaviour of the fifth family quarks in the cosmological evolution (section \[evolution\]) shows that also the condition iv. is fulfilled, if the fifth family masses are large enough. Let us add that there are several assessments about masses of a possible (non stable) fourth family of quarks and leptons, which follow from the analyses of the existing experimental data and the cosmological observations. Although most of physicists have doubts about the existence of any more than the three observed families, the analyses clearly show that neither the experimental electroweak data [@okun; @pdg], nor the cosmological observations [@pdg] forbid the existence of more than three families, as long as the masses of the fourth family quarks are higher than a few hundred GeV and the masses of the fourth family leptons above one hundred GeV ($\nu_4$ could be above $50$ GeV). We studied in the ¨references [@pn06; @gmdn07; @n09] possible (non perturbative) breaks of the symmetries of the simple starting Lagrangean which, by predicting the Yukawa couplings, leads at low energies first to twice four families with no Yukawa couplings between these two groups of families. One group obtains at the last break masses of several hundred TeV or higher, while the lower four families stay massless and mass protected [@n09]. For one choice of the next break [@gmdn07] the fourth family members ($u_{4}, d_{4}, \nu_4, e_4$) obtain the masses at ($224$ GeV (285 GeV), $285$ GeV (224 GeV), $84$ GeV, $170$ GeV), respectively. For the other choice of the next break we could not determine the fourth family masses, but when assuming the values for these masses we predicted mixing matrices in dependence on the masses. All these studies were done on the tree level. We are studying now symmetries of the Yukawa couplings if we go beyond the tree level. Let us add that the last experimental data [@moscow09] from the HERA experiments require that there is no $d_4$ quark with the mass lower than $250$ GeV.
Our stable fifth family baryons, which might form the dark matter, also do not contradict the so far observed experimental data—as it is the measured (first family) baryon number and its ratio to the photon energy density, as long as the fifth family quarks are heavy enough ($> 1$ TeV). (This would be true for any stable heavy family.) Namely, all the measurements, which connect the baryon and the photon energy density, relate to the moment(s) in the history of the universe, when baryons of the first family where formed ($k_b T$ bellow the binding energy of the three first family quarks dressed into constituent mass of $m_{q_1}c^2 \approx 300$ MeV, that is bellow $10$ MeV) and the electrons and nuclei formed atoms ($k_b \,T
\approx 1$ eV). The chargeless (with respect to the colour and electromagnetic charges) clusters of the fifth family were formed long before (at $ k_b T\approx E_{c_5}$ (see Table \[TableI.\])), contributing the equal amount of the fifth family baryons and anti-baryons to the dark matter, provided that there is no fifth family baryon—anti-baryon asymmetry (if the asymmetry is nonzero the colourless baryons or anti-baryons are formed also at the early stage of the colour phase transition at around $1$ GeV). They manifest after decoupling from the plasma (with their small number density and small cross section) (almost) only their gravitational interaction.
In this paper we estimate the properties of the fifth family members ($u_5,d_5,\nu_5,e_5$), as well as of the clusters of these members, in particular the fifth family neutrons, under the assumptions that:\
I. Neutron is the lightest fifth family baryon.\
II. There is no fifth family baryon—anti-baryon asymmetry.\
The assumptions are made since we are not yet able to derive the properties of the family from the starting Lagrange density of the approach. The results of the present paper’s study are helpful to better understand steps needed to come from the approach’s starting Lagrange density to the low energy effective one.
From the approach unifying spin and charges we learn:\
i. The stable fifth family members have masses higher than $\approx 1$ TeV and smaller than $ \approx 10^6$ TeV.\
ii. The stable fifth family members have the properties of the lower four families; that is the same family members with the same (electromagnetic, weak and colour) charges and interacting correspondingly with the same gauge fields.
We estimate the masses of the fifth family quarks by studying their behaviour in the evolution of the universe, their formation of chargeless (with respect to the electromagnetic and colour interaction) clusters and the properties of these clusters when scattering on the ordinary (made mostly of the first family members) matter and among themselves. We use a simple (the hydrogen-like) model [@gnBled07] to estimate the size and the binding energy of the fifth family baryons, assuming that the fifth family quarks are heavy enough to interact mostly by exchanging one gluon. We solve the Boltzmann equations for the fifth family quarks (and anti-quarks) forming the colourless clusters in the expanding universe, starting in the energy region when the fifth family members are ultrarelativistic, up to $\approx 1$ GeV when the colour phase transition starts. In this energy interval the one gluon exchange is the dominant interaction among quarks and the plasma. We conclude that the quarks and anti-quarks, which succeed to form neutral (colourless and electromagnetic chargeless) clusters, have the properties of the dark matter constituents if their masses are within the interval of a few TeV $< m_{q_5} c^2< $ a few hundred TeV, while the rest of the coloured fifth family objects annihilate within the colour phase transition period with their anti-particles for the zero fifth family baryon number asymmetry. We estimate also the behaviour of our fifth family clusters if hitting the DAMA/NaI—DAMA-LIBRA [@rita0708] and CDMS [@cdms] experiments presenting the limitations the DAMA/NaI experiments put on our fifth family quarks when recognizing that CDMS has not found any event (yet). The fifth family baryons are not the objects (WIMPS), which would interact with only the weak interaction, since their decoupling from the rest of the plasma in the expanding universe is determined by the colour force and their interaction with the ordinary matter is determined with the fifth family “nuclear force” (this is the force among clusters of the fifth family quarks, manifesting much smaller cross section than does the ordinary, mostly first family, “nuclear force”) as long as their mass is not higher than $10^{4} $ TeV, when the weak interaction starts to dominate as commented in the last paragraph of section \[dynamics\].
Properties of clusters of the heavy family {#properties}
==========================================
Let us study the properties of the fifth family of quarks and leptons as predicted by the approach unifying spin and charges, with masses several orders of magnitude greater than those of the known three families, decoupled in the Yukawa couplings from the lower mass families and with the charges and their couplings to the gauge fields of the known families (which all seems, due to our estimate predictions of the approach, reasonable assumptions). Families distinguish among themselves (besides in masses) in the family index (in the quantum number, which in the approach is determined by the second kind of the Clifford algebra objects’ operators [@pn06; @n92; @n93] $\tilde{S}^{ab}=\frac{i}{4}(\tilde{\gamma}^a \tilde{\gamma}^b - \tilde{\gamma}^b
\tilde{\gamma}^a)$, anti-commuting with the Dirac $\gamma^a$’s), and (due to the Yukawa couplings) in their masses.
For a heavy enough family the properties of baryons (protons $p_5$ $(u_5 u_5 d_5)$, neutrons $n_5$ $(u_5 d_5 d_5)$, $\Delta_{5}^{-}$, $\Delta_{5}^{++}$) made out of quarks $u_5$ and $d_5$ can be estimated by using the non relativistic Bohr-like model with the $\frac{1}{r}$ dependence of the potential between a pair of quarks $V= - \frac{2}{3} \frac{\hbar c \,\alpha_c}{r}$, where $\alpha_c$ is in this case the colour coupling constant. Equivalently goes for anti-quarks. This is a meaningful approximation as long as the one gluon exchange is the dominant contribution to the interaction among quarks, that is as long as excitations of a cluster are not influenced by the linearly rising part of the potential [^1]. The electromagnetic and weak interaction contributions are of the order of $10^{-2}$ times smaller. Which one of $p_5$, $n_5$, or maybe $\Delta_{5}^-$ or $\Delta_{5}^{++}$, is a stable fifth family baryon, depends on the ratio of the bare masses $m_{u_5}$ and $m_{d_5}$, as well as on the weak and the electromagnetic interactions among quarks. If $m_{d_5}$ is appropriately smaller than $m_{u_5}$ so that the weak and electromagnetic interactions favor the neutron $n_5$, then $n_5$ is a colour singlet electromagnetic chargeless stable cluster of quarks, with the weak charge $-1/2$. If $m_{d_5}$ is larger (enough, due to the stronger electromagnetic repulsion among the two $u_5$ than among the two $d_5$) than $m_{u_5}$, the proton $p_5$ which is a colour singlet stable nucleon with the weak charge $1/2$, needs the electron $e_5$ or $e_1$ or $\bar{p}_1$ to form a stable electromagnetic chargeless cluster (in the last case it could also be the weak singlet and would accordingly manifest the ordinary nuclear force only). An atom made out of only fifth family members might be lighter or not than $n_5$, depending on the masses of the fifth family members.
Neutral (with respect to the electromagnetic and colour charge) fifth family particles that constitute the dark matter can be $n_5,\nu_5$ or charged baryons like $p_5, \Delta^{++}_5$, $\Delta^{-}_5$, forming neutral atoms with $e^{-}_5$ or $\bar{e}^{+}_5$, correspondingly, or (as said above) $p_{5} \bar{p}_1$ . We treat the case that $n_5$ as well as $\bar{n}_5$ form the major part of the dark matter, assuming that $n_5$ (and $\bar{n}_5$) are stable baryons (anti-baryons). Taking $m_{\nu_5}< m_{e_5}$ also $\nu_5$ contributes to the dark matter. We shall comment this in section \[directmeasurements\].
In the Bohr-like model we obtain if neglecting more than one gluon exchange contribution $$\begin{aligned}
\label{bohr}
E_{c_{5}}\approx -3\; \frac{1}{2}\; \left( \frac{2}{3}\, \alpha_c \right)^2\; \frac{m_{q_5}}{2} c^2,
\quad r_{c_{5}} \approx \frac{\hbar c}{ \frac{2}{3}\;\alpha_c \frac{m_{q_5}}{2} c^2}. \end{aligned}$$ The mass of the cluster is approximately $m_{c_5}\, c^2 \approx
3 m_{q_5}\, c^2(1- (\frac{1}{3}\, \alpha_c)^2)$. We use the factor of $\frac{2}{3}$ for a two quark pair potential and of $\frac{4}{3}$ for a quark and an anti-quark pair potential. If treating correctly the three quarks’ (or anti-quarks’) center of mass motion in the hydrogen-like model, allowing the hydrogen-like functions to adapt the width as presented in Appendix I, the factor $-3\; \frac{1}{2}\; (\frac{2}{3})^2\; \frac{1}{2}$ in Eq. \[bohr\] is replaced by $0.66$, and the mass of the cluster is accordingly $3 m_{q_5} c^2(1-0.22\, \alpha_{c}^2)$, while the average radius takes the values as presented in Table \[TableI.\].
Assuming that the coupling constant of the colour charge $\alpha_c$ runs with the kinetic energy $- E_{c_{5}}/3$ and taking into account the number of families which contribute to the running coupling constant in dependence on the kinetic energy (and correspondingly on the mass of the fifth family quarks) we estimate the properties of a baryon as presented on Table \[TableI.\] (the table is calculated from the hydrogen-like model presented in Appendix I),
$\frac{m_{q_5} c^2}{{\rm TeV}}$ $\alpha_c$ $\frac{E_{c_5}}{m_{q_5} c^2}$ $\frac{r_{c_5}}{10^{-6}{\rm fm}}$ $\frac{\Delta m_{ud} c^2}{{\rm GeV}}$
--------------------------------- ------------ ------------------------------- ----------------------------------- ---------------------------------------
$1 $ 0.16 -0.016 $3.2\, \cdot 10^3$ 0.05
$10 $ 0.12 -0.009 $4.2\, \cdot 10^2$ 0.5
$10^2$ 0.10 -0.006 $52$ 5
$10^3$ 0.08 -0.004 $6.0$ 50
$10^4$ 0.07 -0.003 $0.7$ $5 \cdot 10^2$
$10^5$ 0.06 -0.003 $0.08$ $5 \cdot 10^3$
: \[TableI.\] The properties of a cluster of the fifth family quarks within the extended Bohr-like (hydrogen-like) model from Appendix I. $m_{q_5}$ in TeV/c$^2$ is the assumed fifth family quark mass, $\alpha_c$ is the coupling constant of the colour interaction at $E\approx (- E_{c_{5}}/3)\;$ (Eq.\[bohr\]) which is the kinetic energy of quarks in the baryon, $r_{c_5}$ is the corresponding average radius. Then $\sigma_{c_5}=\pi r_{c_5}^2 $ is the corresponding scattering cross section.
The binding energy is approximately $\frac{1}{ 100}$ of the mass of the cluster (it is $\approx \frac{\alpha_{c}^2}{3}$). The baryon $n_5$ ($u_{5} d_{5} d_{5}$) is lighter than the baryon $p_{5}$, ($u_{q_5} d_{q_5} d_{q_5}$) if $\Delta m_{ud}=(m_{u_5}-m_{d_5})$ is smaller than $\approx (0.05,0.5,5, 50, 500, 5000)$ GeV for the six values of the $m_{q_5} c^2$ on Table \[TableI.\], respectively. We see from Table \[TableI.\] that the ”nucleon-nucleon” force among the fifth family baryons leads to many orders of magnitude smaller cross section than in the case of the first family nucleons ($\sigma_{c_5}= \pi r_{c_5}^2$ is from $10^{-5}\,{\rm fm}^2$ for $m_{q_5} c^2 = 1$ TeV to $10^{-14}\, {\rm fm}^2$ for $m_{q_5} c^2 = 10^5$ TeV). Accordingly is the scattering cross section between two fifth family baryons determined by the weak interaction as soon as the mass exceeds several GeV.
If a cluster of the heavy (fifth family) quarks and leptons and of the ordinary (the lightest) family is made, then, since ordinary family dictates the radius and the excitation energies of a cluster, its properties are not far from the properties of the ordinary hadrons and atoms, except that such a cluster has the mass dictated by the heavy family members.
Evolution of the abundance of the fifth family members in the universe {#evolution}
======================================================================
We assume that there is no fifth family baryon—anti-baryon asymmetry and that the neutron is the lightest baryon made out of the fifth family quarks. Under these assumptions and with the knowledge from our rough estimations [@gmdn07] that the fifth family masses are within the interval from $1$ TeV to $10^6$ TeV we study the behaviour of our fifth family quarks and anti-quarks in the expanding (and accordingly cooling down [@dodelson]) universe in the plasma of all other fields (fermionic and bosonic) from the period, when the fifth family members carrying all the three charges (the colour, weak and electromagnetic) are ultra relativistic and is their number (as there are the numbers of all the other fermions and bosons in the ultra relativistic regime) determined by the temperature. We follow the fifth family quarks and anti-quarks first through the freezing out period, when the fifth family quarks and anti-quarks start to have too large mass to be formed out of the plasma (due to the plasma’s too low temperature), then through the period when first the clusters of di-quarks and di-anti-quarks and then the colourless neutrons and anti-neutrons ($n_5$ and $\bar{n}_5$) are formed. The fifth family neutrons being tightly bound into the colourless objects do not feel the colour phase transition when it starts bellow $ k_b T\approx 1$ GeV ($k_b$ is the Boltzmann constant) and decouple accordingly from the rest of quarks and anti-quarks and gluons and manifest today as the dark matter constituents. We take the quark mass as a free parameter in the interval from $1$ TeV to $10^6$ TeV and determine the mass from the observed dark matter density. At the colour phase transition, however, the coloured fifth family quarks and anti-quarks annihilate to the today’s unmeasurable density: Heaving much larger mass (of the order of $10^{5} $ times larger), and correspondingly much larger momentum (of the order of $10^3$ times larger) as well as much larger binding energy (of the order of $10^5$ times larger) than the first family quarks when they are “dressed” into constituent mass, the coloured fifth family quarks succeed in the colour phase transition region to annihilate with the corresponding anti-quarks to the non measurable extend, if it is no fifth family baryon asymmetry. In the freezing out period almost up to the colour phase transition the kinetic energy of quarks is high enough so that the one gluon exchange dominates in the colour interaction of quarks with the plasma, while the (hundred times) weaker weak and electromagnetic interaction can be neglected.
The quarks and anti-quarks start to freeze out when the temperature of the plasma falls close to $m_{q_5}\,c^2/k_b $. They are forming clusters (bound states) when the temperature falls close to the binding energy (which is due to Table \[TableI.\] $\approx \frac{1}{100} m_{q_5} c^2$). When the three quarks (or three anti-quarks) of the fifth family form a colourless baryon (or anti-baryon), they decouple from the rest of plasma due to small scattering cross section manifested by the average radius presented in Table \[TableI.\].
Recognizing that at the temperatures ($ 10^6$ TeV $> k_b T >1 $ GeV) the one gluon exchange gives the dominant contribution to the interaction among quarks of any family, it is not difficult to estimate the thermally averaged scattering cross sections (as the function of the temperature) for the fifth family quarks and anti-quarks to scatter:\
$\;\;\;\;$ i. into all the relativistic quarks and anti-quarks of lower mass families ($<\sigma v>_{q\bar{q}}$),\
$\;\;\;$ ii. into gluons ($<\sigma v>_{gg}$),\
$\;\;$ iii. into (annihilating) bound states of a fifth family quark and an anti-quark mesons $\;\;\;$ ($<\sigma v>_{(q\bar{q})_b}$),\
$\;\;$ iv. into bound states of two fifth family quarks and into the fifth family baryons ($<\sigma v>_{c_5}$) (and equivalently into two anti-quarks and into anti-baryons).
The one gluon exchange scattering cross sections are namely (up to the strength of the coupling constants and up to the numbers of the order one determined by the corresponding groups) equivalent to the corresponding cross sections for the one photon exchange scattering cross sections, and we use correspondingly also the expression for scattering of an electron and a proton into the bound state of a hydrogen when treating the scattering of two quarks into the bound states. We take the roughness of such estimations into account by two parameters: The parameter $\eta_{c_5}$ takes care of scattering of two quarks (anti-quarks) into three colourless quarks (or anti-quarks), which are the fifth family baryons (anti-baryons) and about the uncertainty with which this cross section is estimated. $\eta_{(q \bar{q})_b}$ takes care of the roughness of the used formula for $<\sigma v>_{(q\bar{q})_b}$.
The following expressions for the thermally averaged cross sections are used$$\begin{aligned}
\label{sigmasq}
< \sigma v>_{q\bar{q}} &=& \frac{16 \,\pi}{9}
\;\left( \frac{\alpha_{c} \hbar c}{m_{q_{5}}\,c^2}\right)^2 \, c ,\nonumber\\
< \sigma v>_{gg} &=& \frac{37 \,\pi}{108}\;\left( \frac{\alpha_{c}
\hbar c}{m_{q_{5}}\,c^2}\right)^2\, c, \nonumber\\
< \sigma v>_{c_5} &=& \eta_{c_5}\; 10 \;\left( \frac{\alpha_{c} \hbar c}{m_{q_5}\,c^2} \right)^2\, c\;
\sqrt{\frac{ E_{c_5}}{ k_b T}} \ln{\frac{E_{c_5}}{ k_b T}}, \nonumber\\
<\sigma v>_{(q \bar{q})_b}&=& \eta_{(q \bar{q})_b} \;10 \;\left(
\frac{\alpha_{c} \hbar c}{m_{q_5}\,c^2}\right)^2\, c\;
\sqrt{\frac{ E_{c_5}}{ k_b T}} \ln{\frac{E_{c_5}}{ k_b T}}, \nonumber\\
\sigma_{T } &=& \frac{8 \pi}{3} \left(\frac{\alpha_{c} \hbar c }{m_{q_5} \, c^2}\right)^2,\end{aligned}$$ where $v$ is the relative velocity between the fifth family quark and its anti-quark, or between two quarks and $E_{c_5}$ is the binding energy for a cluster (Eq. \[bohr\]). $\sigma_{T }$ is the Thompson-like scattering cross section of gluons on quarks (or anti-quarks). To see how many fifth family quarks and anti-quarks of a chosen mass form the fifth family baryons and anti-baryons today we solve the coupled systems of Boltzmann equations presented bellow as a function of time (or temperature). The value of the fifth family quark mass which predicts the today observed dark matter is the mass we are looking for. Due to the inaccuracy of the estimated scattering cross sections entering into the Boltzmann equations we tell the interval within which the mass lies. We follow in our derivation of the Boltzmann equations (as much as possible) the ref. [@dodelson], chapter 3. Let $T_0$ be the today’s black body radiation temperature, $T(t)$ the actual (studied) temperature, $a^2(T^0) =1$ and $a^2(T)= a^2(T(t))$ is the metric tensor component in the expanding flat universe—the Friedmann-Robertson-Walker metric: ${\rm diag}\, g_{\mu \nu} =
(1, - a(t)^2, - a(t)^2, - a(t)^2),\;$ $(\frac{\dot{a}}{a})^2= \frac{8 \pi G}{3} \rho$, with $\rho= \frac{\pi^2}{15} \, g^*\, T^4$, $ \, T=T(t)$, $g^*$ measures the number of degrees of freedom of those of the four family members (f) and gauge bosons (b), which are at the treated temperature $T$ ultra-relativistic ($g^*= \sum_{i\in {\rm b}} \,g_i + \frac{7}{8} \sum_{i\in {\rm f}} \,g_i$). $H_0 \, \approx 1.5\,\cdot 10^{-42} \,\frac{{\rm GeV} c}{\hbar c} $ is the present Hubble constant and $G = \frac{\hbar c }{ (m_{pl}^2)}$, $m_{pl} c^2 =
1.2 \cdot 10^{19}$ GeV.
Let us write down the Boltzmann equation, which treats in the expanding universe the number density of all the fifth family quarks as a function of time $t$. The fifth family quarks scatter with anti-quarks into all the other relativistic quarks (with the number density $n_{q}$) and anti-quarks ($n_{\bar{q}}$ ($< \sigma v>_{q\bar{q}}$) and into gluons ($< \sigma v>_{gg}$). At the beginning, when the quarks are becoming non-relativistic and start to freeze out, the formation of bound states is negligible. One finds [@dodelson] the Boltzmann equation for the fifth family quarks $n_{q_5}$ (and equivalently for anti-quarks $n_{\bar{q}_5}$) $$\begin{aligned}
\label{boltzq1}
a^{-3}\frac{d( a^3 n_{q_5})}{dt} &=& < \sigma v>_{q\bar{q}}\; n^{(0)}_{q_5} n^{(0)}_{\bar{q}_5}\,
\left( - \frac{n_{q_5} n_{\bar{q}_5}}{n^{(0)}_{q_5} n^{(0)}_{\bar{q}_5}} +
\frac{n_{q} n_{\bar{q}}}{n^{(0)}_{q} n^{(0)}_{\bar{q}}} \right) + \nonumber\\
&&< \sigma v>_{gg} \;
n^{(0)}_{q_5} n^{(0)}_{\bar{q}_5}\,
\left( - \frac{n_{q_5} n_{\bar{q}_5}}{n^{(0)}_{q_5} n^{(0)}_{\bar{q}_5}} +
\frac{n_{g} n_{g}}{n^{(0)}_{g} n^{(0)}_{g}} \right). \end{aligned}$$ Let us tell that $n^{(0)}_{i} = g_i\, (\frac{m_i c^2 k_b T}{(\hbar c)^2})^{\frac{3}{2}}
e^{-\frac{m_i c^2}{ k_b T}}$ for $m_i c^2 >> k_b T$ and $\frac{g_i}{\pi^2}\, (\frac{ k_b T}{\hbar c})^3$ for $m_i c^2 << k_b T$. Since the ultra-relativistic quarks and anti-quarks of the lower families are in the thermal equilibrium with the plasma and so are gluons, it follows $\frac{n_{q} n_{\bar{q}}}{n^{(0)}_{q} n^{(0)}_{\bar{q}}}=1=
\frac{n_{g} n_{g}}{n^{(0)}_{g} n^{(0)}_{g}}$. Taking into account that $(a\, T)^3 \, g^*(T)$ is a constant it is appropriate [@dodelson] to introduce a new parameter $x=\frac{m_{q_5}c^2}{k_b T}$ and the quantity $Y_{q_5}=
n_{q_5}\, (\frac{\hbar c}{k_b T})^3$, $Y^{(0)}_{q_5}=
n^{(0)}_{q_5}\, (\frac{\hbar c}{k_b T})^3$. When taking into account that the number of quarks is the same as the number of anti-quarks, and that $\frac{dx}{dt} = \frac{h_m \,m_{q_5}c^2}{x} $, with $h_m = \sqrt{\frac{4 \pi^3 g^*}{45}}\,
\frac{c}{\hbar c \, m_{pl} c^2}$, Eq. \[boltzq1\] transforms into $\frac{dY_{q_5}}{dx} =
\frac{\lambda_{q_5}}{x^2}\, (Y^{(0)2}_{q_5} - Y^{2}_{q_5}), $ with $\lambda_{q_5} = \frac{(<\sigma v>_{q\bar{q}} +
<\sigma v>_{gg}) \, m_{q_5} c^2}{h_{m}\, (\hbar c)^3}$. It is this equation which we are solving (up to the region of $x$ when the clusters of quarks and anti-quarks start to be formed) to see the behaviour of the fifth family quarks as a function of the temperature.
When the temperature of the expanding universe falls close enough to the binding energy of the cluster of the fifth family quarks (and anti-quarks), the bound states of quarks (and anti-quarks) and the clusters of fifth family baryons (in our case neutrons $n_{5}$) (and anti-baryons $\bar{n}_{5}$—anti-neutrons) start to form. To a fifth family di-quark ($q_5 + q_5 \rightarrow $ di-quark + gluon) a third quark clusters ( di-quark $+ q_5 \rightarrow c_5 +$ gluon) to form the colourless fifth family neutron (anti-neutron), in an excited state (contributing gluons back into the plasma in the thermal bath when going into the ground state), all in thermal equilibrium. Similarly goes with the anti-quarks clusters. We take into account both processes approximately within the same equation of motion by correcting the averaged amplitude $< \sigma v>_{c_5} $ for quarks to scatter into a bound state of di-quarks with the parameter $\eta_{c_5}$, as explained above. The corresponding Boltzmann equation for the number of baryons $n_{c_5}$ then reads $$\begin{aligned}
\label{boltzc}
a^{-3}\frac{d( a^3 n_{c_5})}{dt} &=& < \sigma v>_{c_5}\; n^{(0)^2}_{q_5}\,
\left( \left( \frac{n_{q_5}}{n^{(0)}_{q_5}} \right)^2 -
\frac{n_{c_5}}{ n^{(0)}_{c_5}} \right).\end{aligned}$$ Introducing again $Y_{c_5}= n_{c_5}\, (\frac{\hbar c}{k_b T})^3$, $Y^{(0)}_{c_5}= n^{(0)}_{c_5}\, (\frac{\hbar c}{k_b T})^3$ and $\lambda_{c_5} = \frac{<\sigma v>_{c_5} \, m_{q_5} c^2}{h_m\, (\hbar c)^3}$, with the same $x$ and $h_m$ as above, we obtain the equation $\frac{dY_{c_5}}{dx} =
\frac{\lambda_{c_5}}{x^2}\, (Y^{2}_{q_5} - Y_{c_5} \,Y^{(0)}_{q_5}\,
\frac{Y^{(0)}_{q_5}}{Y^{(0)}_{c_5}} )$.
The number density of the fifth family quarks $n_{q_5}$ (and correspondingly $Y_{q_5}$), which has above the temperature of the binding energy of the clusters of the fifth family quarks (almost) reached the decoupled value, starts to decrease again due to the formation of the clusters of the fifth family quarks (and anti-quarks) as well as due to forming the bound state of the fifth family quark with an anti-quark, which annihilates into gluons. It follows $$\begin{aligned}
\label{boltzq2}
a^{-3}\frac{d( a^3 n_{q_5})}{dt} &=&
< \sigma v>_{c_5}\; n^{(0)}_{q_5}\, n^{(0)}_{q_5}
\left[ -\left( \frac{n_{q_5}}{n^{(0)}_{q_5}} \right)^2 + \frac{n_{c_5}}{ n^{(0)}_{c_5}} -
\frac{\eta_{(q\bar{q})_b}}{\eta_{c_5}} \;\left( \frac{n_{q_5}}{n^{(0)}_{q_5}} \right)^2 \right] + \nonumber\\
&& < \sigma v>_{q\bar{q}}\; n^{(0)}_{q_5} n^{(0)}_{\bar{q}_5}\,
\left(- \frac{n_{q_5} n_{\bar{q}_5}}{n^{(0)}_{q_5} n^{(0)}_{\bar{q}_5}} +
\frac{n_{q} n_{\bar{q}}}{n^{(0)}_{q} n^{(0)}_{\bar{q}}} \right) + \nonumber\\
&&< \sigma v>_{gg} \;
n^{(0)}_{q_5} n^{(0)}_{\bar{q}_5}\,
\left(- \frac{n_{q_5} n_{\bar{q}_5}}{n^{(0)}_{q_5} n^{(0)}_{\bar{q}_5}} +
\frac{n_{g} n_{g}}{n^{(0)}_{g} n^{(0)}_{g}} \right), \end{aligned}$$ with $\eta_{(q\bar{q})_b}$ and $\eta_{c_5}$ defined in Eq. \[sigmasq\]. Introducing the above defined $Y_{q_5}$ and $Y_{c_5}$ the Eq. \[boltzq2\] transforms into $\frac{dY_{q_5}}{dx} =
\frac{\lambda_{c_5}}{x^2}\, (- Y^{2}_{q_5} + Y_{c_5} \,Y^{(0)}_{q_5}\,
\frac{Y^{(0)}_{q_5}}{Y^{(0)}_{c_5}} ) + \frac{\lambda_{(q\bar{q})_b}}{x^2}\, (- Y^{2}_{q_5})
+ \frac{\lambda_{q_5}}{x^2}\, (Y^{(0)2}_{q_5} - Y^{2}_{q_5})$, with $\lambda_{(q\bar{q})_b} = \frac{<\sigma v>_{(q\bar{q})_b} \, m_{q_5} c^2}{h_m\, (\hbar c)^3}$ (and with the same $x$ and $h_m$ as well as $\lambda_{c_5}$ and $\lambda_{q_5}$ as defined above). We solve this equation together with the above equation for $Y_{c_5} $.
Solving the Boltzmann equations (Eqs. \[boltzq1\], \[boltzc\], \[boltzq2\]) we obtain the number density of the fifth family quarks $n_{q_5}$ (and anti-quarks) and the number density of the fifth family baryons $n_{c_5}$ (and anti-baryons) as a function of the parameter $x=\frac{m_{q_5} c^2}{k_b T}$ and the two parameters $\eta_{c_5}$ and $\eta_{(q\bar{q})_b}$. The evaluations are made, as we explained above, with the approximate expressions for the thermally averaged cross sections from Eq.( \[sigmasq\]), corrected by the parameters $\eta_{c_5}$ and $\eta_{(q \bar{q})_b}$ (Eq. \[sigmasq\]). We made a rough estimation of the two intervals, within which the parameters $\eta_{c_5}$ and $\eta_{(q \bar{q})_b}$ (Eq. \[sigmasq\]) seem to be acceptable. More accurate evaluations of the cross sections are under consideration. In fig. \[DiagramI.\] both number densities (multiplied by $(\frac{\hbar \, c}{ k_b T})^3$, which is $Y_{q_5}$ and $Y_{c_5}$, respectively for the quarks and the clusters of quarks) as a function of $ \frac{m_{q_5} \, c^2}{ k_b T}$ for $\eta_{(q\bar{q})_3}=1$ and $\eta_{c_5}=\frac{1}{50}$ are presented. The particular choice of the parameters $\eta_{(q\bar{q})_3}$ and $\eta_{c_5}$ in fig. \[DiagramI.\] is made as a typical example. The calculation is performed up to $ k_b T=1$ GeV (when the colour phase transition starts and the one gluon exchange stops to be the acceptable approximation).
![The dependence of the two number densities $n_{q_5}$ (of the fifth family quarks) and $n_{c_5}$ (of the fifth family clusters) as function of $\frac{m_{q_5} \, c^2}{ k_b \, T}$ is presented for the special values $m_{q_5} c^2= 71 \,{\rm TeV}$, $\eta_{c_5} = \frac{1}{50}$ and $\eta_{(q\bar{q})_b}=1$. We take $g^*=91.5$. In the treated energy (temperature $ k_b T$) interval the one gluon exchange gives the main contribution to the scattering cross sections of Eq.(\[sigmasq\]) entering into the Boltzmann equations for $n_{q_5}$ and $n_{c_5}$. In the figure we make a choice of the parameters within the estimated intervals.](slika.png){width="15cm"}
\[DiagramI.\]
Let us repeat how the $n_5$ and $\bar{n}_5
$ evolve in the evolution of our universe. The quarks and anti-quarks are at high temperature ($\frac{m_{q_5} c^2}{k_b T}<< 1$) in thermal equilibrium with the plasma (as are also all the other families and bosons of lower masses). As the temperature of the plasma (due to the expansion of the universe) drops close to the mass of the fifth family quarks, quarks and anti-quarks scatter into all the other (ultra) relativistic fermions and bosons, but can not be created any longer from the plasma (in the average). At the temperature close to the binding energy of the quarks in a cluster, the clusters of the fifth family ($n_{c_5}, n_{\bar{c}_5}$) baryons start to be formed. We evaluated the number density $n_{q_5} (T) \,
(\frac{\hbar c}{ k_b T})^3 = Y_{q_5} $ of the fifth family quarks (and anti-quarks) and the number density of the fifth family baryons $n_{c_5} (T) \,(\frac{\hbar c}{ k_b T})^3 = Y_{c_5} $ for several choices of $m_{q_5}, \eta_{c_5}$ and $\eta_{(q \bar{q})_b}$ up to $ k_b T_{lim}= 1$ GeV $=\frac{m_{q_5} c^2}{x_{lim}} $. From the calculated decoupled number density of baryons and anti-baryons of the fifth family quarks (and anti-quarks) $n_{c_5}(T_1)$ at temperature $ k_b T_1=1$ GeV, where we stopped our calculations as a function of the quark mass and of the two parameters $\eta_{c_5}$ and $\eta_{(q\bar{q})_b}$, the today’s mass density of the dark matter follows (after taking into account that when once the $n_{5}$ and $\bar{n}_{5}$ decouple, their number stays unchanged but due to the expansion of the universe their density decreases according to $a^{3}_1 n_{c_5}(T_1)=
a^{3}_2 n_{c_5}(T_2)$, with the today’s $a_0=1$ and the temperature $T_0=2.725^0 $ K) leading to [@dodelson] $$\begin{aligned}
\label{dm}
\rho_{dm} &=& \Omega_{dm} \rho_{cr}= 2 \, m_{c_5}\, n_{c_5}(T_1) \,
\left(\frac{T_0}{T_1} \right)^3 \frac{g^*(T_1)}{g^*(T_0)},\end{aligned}$$ where we take into account that $g^*(T_1) (a_1 T_1)^3= g^*(T_0)(a_0 T_0)^3$, with $T_0 = 2.5 \,\cdot 10^{-4}\,\frac{{\rm eV}}{k_b}$, $g^*(T_0)= 2 + \frac{7}{8}\,\cdot3 \,\cdot \,
(\frac{4}{11})^{4/3}$, $g^*(T_1)= 2 + 2\,\cdot 8 + \frac{7}{8}\, (5\cdot3\cdot2\cdot2 + 6\cdot
2\cdot2)$ and $\rho_{cr} \,c^2 \,\approx \frac{3\, H^{2}_{0}\, c^2}{8 \pi G} \approx 5.7\, \cdot 10^3 \frac{{\rm eV}}{{\rm cm}^3}$, factor $2$ counts baryons and anti-baryons.
The intervals for the acceptable parameters $\eta_{c_5}$ and $\eta_{(q \bar{q})_b}$ (determining the inaccuracy, with which the scattering cross sections were evaluated) influence the value of $n_{c_5}$ and determine the interval, within which one expects the fifth family mass.
$\frac{m_{q_5} c^2}{{\rm TeV}}$ $\eta_{(q\bar{q})_b}=\frac{1}{10}$ $\eta_{(q\bar{q})_b}=\frac{1}{3}$ $\eta_{(q\bar{q})_b}=1$ $\eta_{(q\bar{q})_b}=3$ $\eta_{(q\bar{q})_b}=10$
--------------------------------- ------------------------------------ ----------------------------------- ------------------------- ------------------------- --------------------------
$\eta_{c_5}=\frac{1}{50}$ 21 36 71 159 417
$\eta_{c_5}=\frac{1}{10} $ 12 20 39 84 215
$\eta_{c_5}=\frac{1}{3} $ 9 14 25 54 134
$\eta_{c_5}=1 $ 8 11 19 37 88
$\eta_{c_5}= 3$ 7 10 15 27 60
$\eta_{c_5}=10$ 7\* 8\* 13 22 43
: \[TableII.\] The fifth family quark mass is presented (Eq.(\[dm\])), calculated for different choices of $\eta_{c_5}$ (which takes care of the inaccuracy with which a colourless cluster of three quarks (anti-quarks) cross section was estimated and of $\eta_{(q\bar{q})_b}$ (which takes care of the inaccuracy with which the cross section for the annihilation of a bound state of quark—anti-quark was taken into account) from Eqs. (\[dm\], \[boltzc\], \[boltzq1\]). \* denotes non stable calculations.
We read from Table \[TableII.\] the mass interval for the fifth family quarks’ mass, which fits Eqs. (\[dm\], \[boltzc\], \[boltzq1\]): $$\begin{aligned}
\label{massinterval}
10 \;\; {\rm TeV} < m_{q_5}\, c^2 < {\rm a\, few} \cdot 10^2 {\rm TeV}.\end{aligned}$$ From this mass interval we estimate from Table \[TableI.\] the cross section for the fifth family neutrons $\pi (r_{c_5})^2$: $$\begin{aligned}
\label{sigma}
10^{-8} {\rm fm}^2 \, < \sigma_{c_5} < \, 10^{-6} {\rm fm}^2.\end{aligned}$$ (It is at least $10^{-6} $ smaller than the cross section for the first family neutrons.)
Let us comment on the fifth family quark—anti-quark annihilation at the colour phase transition, which starts at approximately $1$ GeV. When the colour phase transition starts, the quarks start to “dress” into constituent mass, which brings to them $\approx 300$ MeV/$c^2$, since to the force many gluon exchanges start to contribute. The scattering cross sections, which were up to the phase transition dominated by one gluon exchange, rise now to the value of a few ${\rm fm}^2$ and more, say $(50 {\rm fm})^2$. Although the colour phase transition is not yet well understood even for the first family quarks, the evaluation of what happens to the fifth family quarks and anti-quarks and coloured clusters of the fifth family quarks or anti-quarks can still be done as follows. At the interval, when the temperature $ k_b T$ is considerably above the binding energy of the “dressed” first family quarks and anti-quarks into mesons or of the binding energy of the three first family quarks or anti-quarks into the first family baryons or anti-baryons, which is $\approx $ a few MeV (one must be more careful with the mesons), the first family quarks and anti-quarks move in the plasma like being free. (Let us remind the reader that the nuclear interaction can be derived as the interaction among the clusters of quarks [@bmmn].) 25 years ago there were several proposals to treat nuclei as clusters of dressed quarks instead of as clusters of baryons. Although this idea was not very fruitful (since even models with nuclei as bound states of $\alpha$ particles work many a time reasonably) it also was not far from the reality. Accordingly it is meaningful to accept the description of plasma at temperatures above a few $ 10$ MeV/$k_b$ as the plasma of less or more “dressed” quarks with the very large scattering amplitude (of $\approx (50{\rm fm})^2$). The fifth family quarks and anti-quarks, heaving much higher mass (several ten thousands GeV/$c^2$ to be compared with $\approx 300$ MeV/$c^2$) than the first family quarks and accordingly much higher momentum, “see” the first family quarks as a “medium” in which they (the fifth family quarks) scatter among themselves. The fifth family quarks and anti-quarks, having much higher binding energy when forming a meson among themselves than when forming mesons with the first family quarks and anti-quarks (few thousand GeV to be compared with few MeV or few $10$ MeV) and correspondingly very high annihilation probability and also pretty low velocities ($\approx 10^{-3} c$), have during the scattering enough time to annihilate with their anti-particles. The ratio of the scattering time between two coloured quarks (of any kind) and the Hubble time is of the order of $\approx 10^{-18}$ and therefore although the number of the fifth family quarks and anti-quarks is of the order of $10^{-13}$ smaller than the number of the quarks and anti-quarks of the first family (as show the solutions of the Boltzmann equations presented in fig. \[DiagramI.\]), the fifth family quarks and anti-quarks have in the first period of the colour phase transition (from $\approx$ GeV to $\approx 10$ MeV) enough opportunity to scatter often enough among themselves to deplete (their annihilation time is for several orders of magnitude smaller than the time needed to pass by). More detailed calculations, which are certainly needed, are under considerations. Let us still do rough estimation about the number of the coloured fifth family quarks (and anti-quarks). Using the expression for the thermally averaged cross section for scattering of a quark and an anti-quark and annihilating ($<\sigma v>_{(q \bar{q})_b} $ from Eq.(\[sigmasq\])) and correcting the part which determines the scattering cross section by replacing it with $ \eta\, ( 50 {\rm fm})^2 c\; $ (which takes into account the scattering in the plasma during the colour phase transition in the expanding universe) we obtain the expression $ <\sigma v>_{(q \bar{q})_b}= \eta_{(q \bar{q})_b} \, \eta\, ( 50 {\rm fm})^2 c\;
\sqrt{\frac{ E_{c_5}}{ k_b T}} \ln{\frac{E_{c_5}}{ k_b T}}$, which is almost independent of the velocity of the fifth family quarks (which slow down when the temperature lowers). We shall assume that the temperature is lowering as it would be no phase transition and correct this fact with the parameter $\eta$, which could for a few orders of magnitude (say $10^2$) enlarge the depleting probability. Using this expression for $<\sigma v>_{(q \bar{q})_b}$ in the expression for $\lambda= \frac{<\sigma v>_{(q \bar{q})_b} \; m_{q_5} c^2}{h_m (\hbar c)^3}$, we obtain for a factor up to $10^{19}$ larger $\lambda$ than it was the one dictating the freeze out procedure of $q_5$ and $\bar{q}_{5}$ before the phase transition. Using then the equation $\frac{dY_{q_5}}{dx} =
\frac{\lambda_{c_5}}{x^2}\, (- Y^{2}_{q_5})$ and integrating it from $Y_1$ which is the value from the fig. \[DiagramI.\] at $1$ GeV up to the value when $ k_b T\approx 20 $ MeV, when the first family quarks start to bindd into baryons, we obtain in the approximation that $\lambda $ is independent of $x$ (which is not really the case) that $\frac{1}{Y(20 {\rm MeV})}= 10^{32} \frac{1}{2\cdot 10^5} $ or $Y(20 {\rm MeV}) = 10^{-27}$ and correspondingly $n_{q_5}(T_0)= \eta^{-1} 10^{-24} cm^{-3}$. Some of these fifth family quarks can form the mesons or baryons and anti-baryons with the first family quarks $q_1$ when they start to form baryons and mesons. They would form the anomalous hydrogen in the ratio: $\frac{n_{ah}}{n_{h}} \approx \eta^{-1} \cdot 10^{-12} $, where $n_{ah}$ determines the number of the anomalous (heavy) hydrogen atoms and ${n_{h}}$ the number of the hydrogen atoms, with $\eta$ which might be bellow $10^{2}$. The best measurements in the context of such baryons with the masses of a few hundred TeV/${\rm c}^2$ which we were able to find were done $25$ years ago [@superheavy]. The authors declare that their measurements manifest that such a ratio should be $\frac{n_{ah}}{n_{h}}< 10^{-14}$ for the mass interval between $10$ TeV/$c^{2}$ to $10^{4}$ TeV/$c^{2}$. Our evaluation presented above is very rough and more careful treating the problem might easily lead to lower values than required. On the other side we can not say how trustable is the value for the above ratio for the masses of a few hundreds TeV. Our evaluations are very approximate and if $\eta= 10^{2}$ we conclude that the evaluation agrees with measurements.
Dynamics of a heavy family baryons in our galaxy {#dynamics}
================================================
There are experiments [@rita0708; @cdms] which are trying to directly measure the dark matter clusters. Let us make a short introduction into these measurements, treating our fifth family clusters in particular. The density of the dark matter $\rho_{dm}$ in the Milky way can be evaluated from the measured rotation velocity of stars and gas in our galaxy, which appears to be approximately independent of the distance $r$ from the center of our galaxy. For our Sun this velocity is $v_S \approx (170 - 270)$ km/s. $\rho_{dm}$ is approximately spherically symmetric distributed and proportional to $\frac{1}{r^2}$. Locally (at the position of our Sun) $\rho_{dm}$ is known within a factor of 10 to be $\rho_0 \approx 0.3 \,{\rm GeV} /(c^2 \,{\rm cm}^3)$, we put $\rho_{dm}= \rho_0\, \varepsilon_{\rho},$ with $\frac{1}{3} < \varepsilon_{\rho} < 3$. The local velocity distribution of the dark matter cluster $\vec{v}_{dm\, i}$, in the velocity class $i$ of clusters, can only be estimated, results depend strongly on the model. Let us illustrate this dependence. In a simple model that all the clusters at any radius $r$ from the center of our galaxy travel in all possible circles around the center so that the paths are spherically symmetrically distributed, the velocity of a cluster at the position of the Earth is equal to $v_{S}$, the velocity of our Sun in the absolute value, but has all possible orientations perpendicular to the radius $r$ with equal probability. In the model that the clusters only oscillate through the center of the galaxy, the velocities of the dark matter clusters at the Earth position have values from zero to the escape velocity, each one weighted so that all the contributions give $ \rho_{dm} $. Many other possibilities are presented in the references cited in [@rita0708].
The velocity of the Earth around the center of the galaxy is equal to: $\vec{v}_{E}= \vec{v}_{S} + \vec{v}_{ES} $, with $v_{ES}= 30$ km/s and $\frac{\vec{v}_{S}\cdot \vec{v}_{ES}}{v_S v_{ES}}\approx \cos \theta \, \sin \omega t, \theta = 60^0$. Then the velocity with which the dark matter cluster of the $i$- th velocity class hits the Earth is equal to: $\vec{v}_{dmE\,i}= \vec{v}_{dm\,i} - \vec{v}_{E}$. $\omega $ determines the rotation of our Earth around the Sun.
One finds for the flux of the dark matter clusters hitting the Earth: $\Phi_{dm} = \sum_i \,\frac{\rho_{dm \,i}}{m_{c_5}} \,
|\vec{v}_{dm \,i} - \vec{v}_{E}| $ to be approximately (as long as $\frac{v_{ES}}{|\vec{v}_{dm \,i}- \vec{v}_S|}$ is small) equal to $$\begin{aligned}
\label{flux}
\Phi_{dm}\approx \sum_i \,\frac{\rho_{dm \,i}}{m_{c_5}} \,
\{|\vec{v}_{dm \,i} - \vec{v}_{S}| - \vec{v}_{ES} \cdot \frac{\vec{v}_{dm\, i}- \vec{v}_S}{
|\vec{v}_{dm \,i}- \vec{v}_S|} \}.\end{aligned}$$ Further terms are neglected. We shall approximately take that $\sum_i \, |\vec{v_{dm \,i}}- \vec{v_S}| \,\rho_{dm \,i} \approx \varepsilon_{v_{dmS}}
\, \varepsilon_{\rho}\, v_S\, \rho_0 $, and correspondingly $ \sum_i \, \vec{v}_{ES} \cdot \frac{\vec{v}_{dm \,i}- \vec{v}_S}{
|\vec{v}_{dm \,i}- \vec{v}_S|} \approx v_{ES} \varepsilon_{v_{dmS}}
\cos \theta \, \sin \omega t $, (determining the annual modulations observed by DAMA [@rita0708]). Here $\frac{1}{3} < \varepsilon_{v_{dmS}} <
3$ and $\frac{1}{3} < \frac{\varepsilon_{v_{dmES}}}{\varepsilon_{v_{dmS}}} < 3$ are estimated with respect to experimental and (our) theoretical evaluations.
Let us evaluate the cross section for our heavy dark matter baryon to elastically (the excited states of nuclei, which we shall treat, I and Ge, are at $\approx 50$ keV or higher and are very narrow, while the average recoil energy of Iodine is expected to be $30$ keV) scatter on an ordinary nucleus with $A$ nucleons $\sigma_{A} =
\frac{1}{\pi \hbar^2} <|M_{c_5 A}|>^2 \, m_{A}^2$. For our heavy dark matter cluster is $m_{A} $ approximately the mass of the ordinary nucleus [^2]. In the case of a coherent scattering (if recognizing that $\lambda= \frac{h}{p_A}$ is for a nucleus large enough to make scattering coherent when the mass of the cluster is $1$ TeV or more and its velocity $\approx v_{S}$), the cross section is almost independent of the recoil velocity of the nucleus. For the case that the ”nuclear force” as manifesting in the cross section $\pi\, (r_{c_5})^2$ in Eq.(\[bohr\]) brings the main contribution [^3] the cross section is proportional to $(3A)^2$ (due to the square of the matrix element) times $(A)^2$ (due to the mass of the nuclei $m_A\approx 3 A \,m_{q_1}$, with $m_{q_1}\, c^2 \approx \frac{1 {\rm GeV}}{3}$). When $m_{q_5}$ is heavier than $10^4 \, {\rm TeV}/c^2$ (Table \[TableI.\]), the weak interaction dominates and $\sigma_{A}$ is proportional to $(A-Z)^2 \, A^2$, since to $Z^0$ boson exchange only neutron gives an appreciable contribution. Accordingly we have, when the ”nuclear force” dominates, $\sigma_A \approx \sigma_{0} \, A^4 \, \varepsilon_{\sigma}$, with $\sigma_{0}\, \varepsilon_{\sigma}$, which is $\pi r_{c_5}^2 \,
\varepsilon_{\sigma_{nucl}} $ and with $\frac{1}{30} < \varepsilon_{\sigma_{nucl}} < 30$. $\varepsilon_{\sigma_{nucl}}$ takes into account the roughness with which we treat our heavy baryon’s properties and the scattering procedure. When the weak interaction dominates, $ \varepsilon_{\sigma}$ is smaller and we have $
\sigma_{0}\, \varepsilon_{\sigma}=(\frac{m_{n_1} G_F}{\sqrt{2 \pi}}
\frac{A-Z}{A})^2 \,\varepsilon_{\sigma_{weak}} $ ($=( 10^{-6} \,\frac{A-Z}{ A} \, {\rm fm} )^2 \,\varepsilon_{\sigma_{weak}} $), $ \frac{1}{10}\, <\, \varepsilon_{\sigma_{weak}} \,< 1$. The weak force is pretty accurately evaluated, but the way how we are averaging is not.
Direct measurements of the fifth family baryons as dark matter constituents {#directmeasurements}
===========================================================================
We are making very rough estimations of what the DAMA [@rita0708] and CDMS [@cdms] experiments are measuring, provided that the dark matter clusters are made out of our (any) heavy family quarks as discussed above. We are looking for limitations these two experiments might put on properties of our heavy family members. We discussed about our estimations and their relations to the measurements with R. Bernabei [@privatecommRBJF] and J. Filippini [@privatecommRBJF]. Both pointed out (R.B. in particular) that the two experiments can hardly be compared, and that our very approximate estimations may be right only within the orders of magnitude. We are completely aware of how rough our estimation is, yet we conclude that, since the number of measured events is proportional to $(m_{c_5})^{-3}$ for masses $\approx 10^4$ TeV or smaller (while for higher masses, when the weak interaction dominates, it is proportional to $(m_{c_5})^{-1}$) that even such rough estimations may in the case of our heavy baryons say whether both experiments do at all measure our (any) heavy family clusters, if one experiment clearly sees the dark matter signals and the other does not (yet?) and we accordingly estimate the mass of our cluster. Let $N_A$ be the number of nuclei of a type $A$ in the apparatus (of either DAMA [@rita0708], which has $4\cdot 10^{24}$ nuclei per kg of $I$, with $A_I=127$, and $Na$, with $A_{Na}= 23$ (we shall neglect $Na$), or of CDMS [@cdms], which has $8.3 \cdot 10^{24}$ of $Ge$ nuclei per kg, with $A_{Ge}\approx 73$). At velocities of a dark matter cluster $v_{dmE}$ $\approx$ $200$ km/s are the $3A$ scatterers strongly bound in the nucleus, so that the whole nucleus with $A$ nucleons elastically scatters on a heavy dark matter cluster. Then the number of events per second ($R_A$) taking place in $N_A$ nuclei is due to the flux $\Phi_{dm}$ and the recognition that the cross section is at these energies almost independent of the velocity equal to $$\begin{aligned}
\label{ra}
R_A = \, N_A \, \frac{\rho_{0}}{m_{c_5}} \;
\sigma(A) \, v_S \, \varepsilon_{v_{dmS}}\, \varepsilon_{\rho} \, ( 1 +
\frac{\varepsilon_{v_{dmES}}}{\varepsilon_{v_{dmS}}} \, \frac{v_{ES}}{v_S}\, \cos \theta
\, \sin \omega t).\end{aligned}$$ Let $\Delta R_A$ mean the amplitude of the annual modulation of $R_A$ $$\begin{aligned}
\label{anmod}
\Delta R_A &=& R_A(\omega t = \frac{\pi}{2}) - R_A(\omega t = 0) = N_A \, R_0 \, A^4\,
\frac{\varepsilon_{v_{dmES}}}{\varepsilon_{v_{dmS}}}\, \frac{v_{ES}}{v_S}\, \cos \theta,\end{aligned}$$ where $ R_0 = \sigma_{0} \, \frac{\rho_0}{m_{c_5}} \, v_S\, \varepsilon$, $R_0$ is for the case that the ”nuclear force” dominates $R_0 \approx \pi\, (\frac{3\, \hbar\, c}{\alpha_c \, m_{q_5}\, c^2})^2\,
\frac{\rho_0}{m_{q_5}} \, v_S\, \varepsilon$, with $\varepsilon =
\varepsilon_{\rho} \, \varepsilon_{v_{dmES}} \varepsilon_{\sigma_{nucl}} $. $R_0$ is therefore proportional to $m_{q_5}^{-3}$. We estimated $10^{-4} < \varepsilon < 10$, which demonstrates both, the uncertainties in the knowledge about the dark matter dynamics in our galaxy and our approximate treating of the dark matter properties. (When for $m_{q_5} \, c^2 > 10^4$ TeV the weak interaction determines the cross section $R_0 $ is in this case proportional to $m_{q_5}^{-1}$.) We estimate that an experiment with $N_A$ scatterers should measure the amplitude $R_A \varepsilon_{cut\, A}$, with $\varepsilon_{cut \, A}$ determining the efficiency of a particular experiment to detect a dark matter cluster collision. For small enough $\frac{\varepsilon_{v_{dmES}}}{\varepsilon_{v_{dmS}}}\,
\frac{v_{ES}}{v_S}\, \cos \theta$ we have $$\begin{aligned}
R_A \, \varepsilon_{cut \, A} \approx N_{A}\, R_0\, A^4\,
\varepsilon_{cut\, A} = \Delta R_A \varepsilon_{cut\, A} \,
\frac{\varepsilon_{v_{dmS}}}{\varepsilon_{v_{dmES}}} \, \frac{v_{S}}{v_{ES}\, \cos \theta}.
\label{measure}\end{aligned}$$ If DAMA [@rita0708] is measuring our heavy family baryons scattering mostly on $I$ (we neglect the same number of $Na$, with $A =23$), then the average $R_I$ is $$\begin{aligned}
\label{ridama}
R_{I} \varepsilon_{cut\, dama} \approx \Delta R_{dama} \frac{\varepsilon_{v_{dmS}}}{\varepsilon_{v_{dmES}}}\,
\frac{v_{S} }{v_{ES}\, \cos 60^0 } ,\end{aligned}$$ with $\Delta R_{dama}\approx
\Delta R_{I} \, \varepsilon_{cut\, dama}$, this is what we read from their papers [@rita0708]. In this rough estimation most of unknowns about the dark matter properties, except the local velocity of our Sun, the cut off procedure ($\varepsilon_{cut\, dama}$) and $\frac{\varepsilon_{v_{dmS}}}{\varepsilon_{v_{dmES}}}$, (estimated to be $\frac{1}{3} < \frac{\varepsilon_{v_{dmS}}}{\varepsilon_{v_{dmES}}} < 3$), are hidden in $\Delta R_{dama}$. If we assume that the Sun’s velocity is $v_{S}=100, 170, 220, 270$ km/s, we find $\frac{v_S}{v_{ES} \cos \theta}= 7,10,14,18, $ respectively. (The recoil energy of the nucleus $A=I$ changes correspondingly with the square of $v_S $.) DAMA/NaI, DAMA/LIBRA [@rita0708] publishes $ \Delta R_{dama}= 0.052 $ counts per day and per kg of NaI. Correspondingly is $R_I \, \varepsilon_{cut\, dama} =
0,052 \, \frac{\varepsilon_{v_{dmS}}}{\varepsilon_{v_{dmES}}}\, \frac{v_S}{v_{SE} \cos \theta} $ counts per day and per kg. CDMS should then in $121$ days with 1 kg of Ge ($A=73$) detect $R_{Ge}\, \varepsilon_{cut\, cdms}$ $\approx \frac{8.3}{4.0} \,
(\frac{73}{127})^4 \; \frac{\varepsilon_{cut\,cdms}}{\varepsilon_{cut \,dama}}\,
\frac{\varepsilon_{v_{dmS}}}{\varepsilon_{v_{dmES}}}\;
\frac{v_S}{v_{SE} \cos \theta} \; 0.052 \cdot
121 \;$ events, which is for the above measured velocities equal to $(10,16,21,25)
\, \frac{\varepsilon_{cut\, cdms}}{\varepsilon_{cut\,dama}}\;
\frac{\varepsilon_{v_{dmS}}}{\varepsilon_{v_{dmES}}}$. CDMS [@cdms] has found no event.
The approximations we made might cause that the expected numbers $(10,16,21,25)$ multiplied by $\frac{\varepsilon_{cut\,Ge}}{\varepsilon_{cut\,I}}\;
\frac{\varepsilon_{v_{dmS}}}{\varepsilon_{v_{dmES}}}$ are too high (or too low!!) for a factor let us say $4$ or $10$. If in the near future CDMS (or some other experiment) will measure the above predicted events, then there might be heavy family clusters which form the dark matter. In this case the DAMA experiment puts the limit on our heavy family masses (Eq.(\[measure\])).
Taking into account all the uncertainties mentioned above, with the uncertainty with the ”nuclear force” cross section included (we evaluate these uncertainties to be $10^{-4} <\,\varepsilon^{"}\,< 3\cdot 10^3$), we can estimate the mass range of the fifth family quarks from the DAMA experiments: $(m_{q_5}\, c^2)^3= \frac{1}{\Delta R_{dama}}
N_I\,A^4\, \pi \,(\frac{3 \,\hbar c}{\alpha_c})^2 \,
\rho_0\, c^2\, v_{ES} \,\cos \theta\, \varepsilon^{"}= (0.3\, \cdot 10^7)^3 \,
\varepsilon^{"} (\frac{0.1}{\alpha_c})^{2}
$ GeV. The lower mass limit, which follows from the DAMA experiment, is accordingly $m_{q_5}\, c^2> 200$ TeV. Observing that for $m_{q_5} \, c^2> 10^4$ TeV the weak force starts to dominate, we estimate the upper limit $m_{q_5}\, c^2< 10^5$ TeV. Then $200 {\rm\; TeV} < m_{q_5} \, c^2 < 10^5$ TeV.
Let us at the end evaluate the total number of our fifth family neutrons ($n_5$) which in $\delta t= 121$ days strike $1$ kg of Ge and which CDMS experiment could detect, that is $R_{Ge} \delta t
\varepsilon_{cut_{Ge}}= N_{Ge} \sigma_0 \frac{\rho_0}{m_{c_5}}\,v_{S}\, A^{4}_{Ge}\, \varepsilon \varepsilon_{cut+{Ge}}$ (Eq. \[measure\]), with $N_{Ge} = 8.3 \cdot 10^{24}$/kg, with the cross section from Table \[TableI.\], with $A_{Ge} = 73$ and $1$ kg of Ge, while $10^{-5} < \varepsilon \varepsilon_{cut_{Ge}}< 5\cdot 10$. The coefficient $\varepsilon \varepsilon_{cut_{Ge}}$ determines all the uncertainties: about the scattering amplitudes of the fifth family neutrons on the Ge nuclei (about the scattering amplitude of one $n_5$ on the first family quark, about the degree of coherence when scattering on the nuclei, about the local density of the dark matter, about the local velocity of the dark matter and about the efficiency of the experiment). Quite a part of these uncertainties were hidden in the number of events the DAMA/LIBRA experiments measure, when we compare both experiments. If we assume that the fifth family quark mass ($m_{q_5}$) is several hundreds TeV, as evaluated (as the upper bound (Eq. \[massinterval\])) when considering the cosmological history of our fifth family neutrons, we get for the number of events the CDMS experiment should measure: $\varepsilon \varepsilon_{cut_{Ge}} \cdot 10^{4}$. If we take $\varepsilon \varepsilon_{cut_{Ge}}= 10^{-5}$, the CDMS experiment should continue to measure 10 times as long as they did.
Let us see how many events CDMS should measure if the dark matter clusters would interact weakly with the Ge nuclei and if the weak interaction would determine also their freezing out procedure, that is if any kind of WIMP would form the dark matter. One easily sees from the Boltzmann equations for the freezing out procedure for $q_5$ that since the weak massless boson exchange is approximately hundred times weaker than the one gluon exchange which determines the freeze out procedure of the fifth family quarks, the mass of such an object should be hundred times smaller, which means a few TeV. Taking into account the expression for the weak interaction of such an object with Ge nuclei, which leads to $10^{-2}$ smaller cross section for scattering of one such weakly interacting particle on one proton (see derivations in the previous section), we end up with the number of events which the CDMS experiment should measure: $\varepsilon \varepsilon_{cut_{Ge}} 5 \cdot 10^3$. Since the weak interaction with the matter is much better known that the (”fifth family nuclear force”) interaction of the colourless clusters of $q_5$ ($n_5$), the $\varepsilon $ is smaller. Let us say $\varepsilon$ is $5 \cdot10^{-4}$. Accordingly, even in the case of weakly interacting dark matter particles the CDMS should continue to measure to see some events.
Concluding remarks {#conclusion}
===================
We estimated in this paper the possibility that a new stable family, predicted by the approach unifying spin and charges [@pn06; @n92; @gmdn07] to have the same charges and the same couplings to the corresponding gauge fields as the known families, forms baryons which are the dark matter constituents. The approach (proposed by S.N.M.B.) is to our knowledge the only proposal in the literature so far which offers the mechanism for generating families, if we do not count those which in one or another way just assume more than three families. Not being able so far to derive from the approach precisely enough the fifth family masses and also not (yet) the baryon asymmetry, we assume that the neutron is the lightest fifth family baryon and that there is no baryon—anti-baryon asymmetry. We comment what changes if the asymmetry exists. We evaluated under these assumptions the properties of the fifth family members in the expanding universe, their clustering into the fifth family neutrons, the scattering of these neutrons on ordinary matter and find the limit on the properties of the stable fifth family quarks due to the cosmological observations and the direct experiments provided that these neutrons constitute the dark matter.
We use the simple hydrogen-like model to evaluate the properties of these heavy baryons and their interaction among themselves and with the ordinary nuclei. We take into account that for masses of the order of $1$ TeV/$c^2$ or larger the one gluon exchange determines the force among the constituents of the fifth family baryons. Studying the interaction of these baryons with the ordinary matter we find out that for massive enough fifth family quarks ($m_{q_5}> 10^4$ TeV) the weak interaction starts to dominate over the ”nuclear interaction” which the fifth family neutron manifests. The non relativistic fifth family baryons interact among themselves with the weak force only.
We study the freeze out procedure of the fifth family quarks and anti-quarks and the formation of baryons and anti-baryons up to the temperature $ k_b T= 1$ GeV, when the colour phase transition starts which to our estimations depletes almost all the fifth family quarks and anti-quarks while the colourless fifth family neutrons with very small scattering cross section decouples long before (at $ k_b T= 100$ GeV).
The cosmological evolution suggests for the mass limits the range $10$ TeV $< m_{q_5} \, c^2 < {\rm a \, few} \cdot 10^2$ TeV and for the scattering cross sections $ 10^{-8}\, {\rm fm}^2\, < \sigma_{c_5}\, < 10^{-6} \,{\rm fm}^2 $. The measured density of the dark matter does not put much limitation on the properties of heavy enough clusters.
The DAMA experiments [@rita0708] limit (provided that they measure our heavy fifth family clusters) the quark mass to: $ 200 \,{\rm TeV} < m_{q_{5}}c^2 < 10^5\, {\rm TeV}$. The estimated cross section for the dark matter cluster to (elastically, coherently and nonrelativisically) scatter on the (first family) nucleus is in this case determined on the lower mass limit by the ”fifth family nuclear force” of the fifth family clusters ($ (3\cdot 10^{-5}\,A^2\, {\rm fm} )^2$) and on the higher mass limit by the weak force ($ ( A (A-Z)\, 10^{-6} \, {\rm fm} )^2 $). Accordingly we conclude that if the DAMA experiments are measuring our fifth family neutrons, the mass of the fifth family quarks is a few hundred TeV $/c^2$.
Taking into account all the uncertainties in connection with the dark matter clusters (the local density of the dark matter and its local velocity) including the scattering cross sections of our fifth family neutrons on the ordinary nuclei as well as the experimental errors, we do expect that CDMS will in a few years measure our fifth family baryons.
Let us point out that the stable fifth family neutrons are not the WIMPS, which would interact with the weak force only: the cosmological behaviour (the freezing out procedure) of these clusters are dictated by the colour force, while their interaction with the ordinary matter is determined by the “fifth family nuclear force” if they have masses smaller than $10^4$ TeV/$c^2$.
In the ref. [@mbb] [^4] the authors study the limits on a scattering cross section of a heavy dark matter cluster of particles and anti-particles (both of approximately the same amount) with the ordinary matter, estimating the energy flux produced by the annihilation of such pairs of clusters. They treat the conditions under which would the heat flow following from the annihilation of dark matter particles and anti-particles in the Earth core start to be noticeable. Using their limits we conclude that our fifth family baryons of the mass of a few hundreds TeV/${c^2} $ have for a factor more than $100$ too small scattering amplitude with the ordinary matter to cause a measurable heat flux on the Earth’s surface. On the other hand could the measurements [@superheavy] tell whether the fifth family members do deplete at the colour phase transition of our universe enough to be in agreement with them. Our very rough estimation show that the fifth family members are on the allowed limit, but they are too rough to be taken as a real limit.
Our estimations predict that, if the DAMA experiments observe the events due to our (any) heavy family members, (or any heavy enough family clusters with small enough cross section), the CDMS experiments [@cdms] will in the near future observe a few events as well. If CDMS will not confirm the heavy family events, then we must conclude, trusting the DAMA experiments, that either our fifth family clusters have much higher cross section due to the possibility that $u_5$ is lighter than $d_5$ so that their velocity slows down when scattering on nuclei of the earth above the measuring apparatus bellow the threshold of the CDMS experiment (and that there must be in this case the fifth family quarks—anti-quarks asymmetry) [@maxim]) while the DAMA experiment still observes them, or the fifth family clusters (any heavy stable family clusters) are not what forms the dark matter.
Let us comment again the question whether it is at all possible (due to electroweak experimental data) that there exist more than three up to now observed families, that is, whether the approach unifying spin and charges by predicting the fourth and the stable fifth family (with neutrinos included) contradict the observations. In the ref. [@mdnbled06] the properties of all the members of the fourth family were studied (for one particular choice of breaking the starting symmetry). The predicted fourth family neutrino mass is at around $100$ GeV/$c^2$ or higher, therefore it does not due to the detailed analyses of the electroweak data done by the Russian group [@okun] contradict any experimental data. The stable fifth family neutrino has due to our calculations considerably higher mass. Accordingly none of these two neutrinos contradict the electroweak data. They also do not contradict the nucleosynthesis, since to the nucleosynthesis only the neutrinos with masses bellow the electron mass contribute. The fact that the fifth family baryons might form the dark matter does not contradict the measured (first family) baryon number and its ratio to the photon energy density as well, as long as the fifth family quarks are heavy enough ($>$1 TeV). All the measurements, which connect the baryon and the photon energy density, relate to the moment(s) in the history of the universe, when the baryons (of the first family) where formed ($m_1 c^2 \approx
k_b T = 1$ GeV and lower) and the electrons and nuclei were forming atoms ($k_b \,T
\approx 1$ eV). The chargeless (with respect to the colour and electromagnetic charges, not with respect to the weak charge) clusters of the fifth family were formed long before (at $ k_b T\approx E_{c_5}$ (Table \[TableI.\])). They manifest after decoupling from the plasma (with their small number density and small cross section) (almost) only their gravitational interaction.
Let the reader recognize that the fifth family baryons are not the objects—WIMPS—which would interact with only the weak interaction, since their decoupling from the rest of the plasma in the expanding universe is determined by the colour force and their interaction with the ordinary matter is determined with the fifth family “nuclear force” (the force among the fifth family nucleons, manifesting much smaller cross section than does the ordinary “nuclear force”) as long as their mass is not higher than $10^{4} $ TeV, when the weak interaction starts to dominate as commented in section \[dynamics\].
Let us conclude this paper with the recognition: If the approach unifying spin and charges is the right way beyond the standard model of the electroweak and colour interaction, then more than three families of quarks and leptons do exist, and the stable (with respect to the age of the universe) fifth family of quarks and leptons is the candidate to form the dark matter. The assumptions we made (i. The fifth family neutron is the lightest fifth family baryon, ii. There is no fifth family baryon asymmetry), could be derived from the approach unifying spins and charges and we are working on these problems. The fifth family baryon anti-baryon asymmetry does not very much change the conclusions of this paper as long as the fifth family quarks’s mass is a few hundreds TeV or higher.
Appendix I. Three fifth family quarks’ bound states {#betterhf}
====================================================
We look for the ground state solution of the Hamilton equation $H\,
|\psi\rangle= E_{c_5}\,|\psi\rangle $ for a cluster of three heavy quarks with $$\begin{aligned}
H=\sum_{i=1}^3 \,\frac{p_{i}^2}{2 \,m_{q_5}}
-\frac{2}{3}\, \, \sum_{i<j=1}^3
\frac{\hbar c \; \alpha_c}{|\vec{x}_i-\vec{x}_j|},
\end{aligned}$$ in the center of mass motion $$\begin{aligned}
\vec{x}=\vec{x}_2 - \vec{x}_1,\quad \vec{y}=\vec{x}_3 - \frac{\vec{x}_1+\vec{x}_2}{2},\quad
\vec{R}=\frac{\vec{x}_1+\vec{x}_2+\vec{x}_3}{3},
\end{aligned}$$ assuming the anti-symmetric colour part ($|\psi\rangle_{c,\, \cal{A} }$), symmetric spin and weak charge part ($|\psi\rangle_{w \, {\rm spin},\, \cal{S} }$) and symmetric space part ($|\psi\rangle_{{\rm space}, \, \cal{S}}$). For the space part we take the hydrogen-like wave functions $ \psi_a(\vec{x})=$$\frac{1}{\sqrt{\pi a^3}} \; e^{-|\vec{x}|/a}$ and $\psi_b(\vec{y})=$$\frac{1}{\sqrt{\pi b^3}} \; e^{-|\vec{y}|/b}$, allowing $a$ and $b$ to adapt variationally. Accordingly $\langle\, \vec{x}_1, \vec{x}_2, \vec{x}_3|\psi\rangle_{{\rm space}\, \cal{S}}=
\mathcal{N}
\left( \psi_a(\vec{x}) \psi_{b}(\vec{y}) + \textrm{symmetric permutations} \right)$. It follows $\langle\, \vec{x}_1, \vec{x}_2, \vec{x}_3|\psi\rangle_{{\rm space}\, \cal{S}}=
\mathcal{N} \, \left( 2 \psi_a(\vec{x}) \psi_{b}(\vec{y}) +
2 \psi_a(\vec{y}-\frac{\vec{x}}{2}) \psi_{b}(\frac{\vec{y}}{2}+\frac{3 \vec{x}}{4})) +
2 \psi_a(\vec{y}+\frac{\vec{x}}{2}) \psi_{b}(\frac{\vec{y}}{2}-\frac{3 \vec{x}}{4}) \right) $.
The Hamiltonian in the center of mass motion reads $H=\frac{p_x^2}{2 (\frac{m_{q_5}}{2})}+\frac{p_y^2}{2 (\frac{2m_{q_5}}{3})}+\frac{p_R^2}{2
\cdot 3 m_{q_5}}
-\frac{2}{3} \hbar c \; \alpha_c \left(\frac{1}{x}+\frac{1}{|\vec{y}+\frac{\vec{x}}{2}|}+
\frac{1}{|\vec{y}-\frac{\vec{x}}{2}|} \right).
$ Varying the expectation value of the Hamiltonian with respect to $a$ and $b$ it follows: $\frac{a}{b}=1.03, \, \frac{a\, \alpha_c\, m_{q_5}\, c^2}{\hbar c} = 1.6$.
Accordingly we get for the binding energy $ E_{c_5}=0.66\; m_{q_5}\, c^2 \alpha_{c}^2$ and for the size of the cluster $\sqrt{\langle |\vec{x}_2-\vec{x}_1|^2 \rangle} = 2.5\, \frac{\hbar c}{\alpha_c m_{q_5 \, c^2}}
$.
To estimate the mass difference between $u_5$ and $d_5$ for which $u_5 d_5 d_5$ is stable we treat the electromagnetic ($\alpha_{elm}$) and weak ($\alpha_w $) interaction as a small correction to the above calculated binding energy: $H'= \alpha_{elm\,w} \; \hbar c \,
\left(\frac{1}{x}+\frac{1}{|\vec{y}+\frac{\vec{x}}{2}|}+
\frac{1}{|\vec{y}-\frac{\vec{x}}{2}|} \right)$. $\alpha_{elm\,w} $ stays for electromagnetic and weak coupling constants. For $m_{q_5}= 200$ TeV we take $\alpha_{elm\,w} =\frac{1}{100}$, then $|m_{u_5}- m_{d_5}|< \frac{1}{3}\, E_{c_5} \frac{(\frac{3}{2} \alpha_{elm\,w})^2}{\alpha_c^2}
= 0.5\,\cdot 10^{-4} \; m_{q_5}\, c^2 $.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank all the participants of the workshops entitled “What comes beyond the Standard models”, taking place at Bled annually (usually) in July, starting in 1998, and in particular H. B. Nielsen, since all the open problems were there very openly discussed.
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[^1]: Let us tell that a simple bag model evaluation does not contradict such a simple Bohr-like model.
[^2]: Let us illustrate what is happening when a very heavy ($10^4$ times or more heavier than the ordinary nucleon) cluster hits the nucleon. Having the “nuclear force” cross section of $10^{-8}$ ${\rm fm}^2$ or smaller, it “sees” with this cross section a particular quark, which starts to move. But since at this velocities the quark is tightly bound into a nucleon and nucleon into the nucleus, the hole nucleus is forced to move with the moving quark.
[^3]: The very heavy colourless cluster of three quarks, hitting with the relative velocity $\approx 200 \,{\rm km}/{\rm s}$ the nucleus of the first family quarks, ”sees” the (light) quark $q_1$ of the nucleus through the cross section $\pi\, (r_{c_5})^2$. But since the quark $q_{1}$ is at these velocities strongly bound to the proton and the proton to the nucleus, the hole nucleus takes the momentum.
[^4]: The referee of PRL suggested that we should comment on the paper [@mbb].
|
---
abstract: 'Most of known RR Lyraes are type ab RR Lyraes (RRLab), and they are the excellent tool to map the Milky Way and its substructures. We find that 1148 RRLab stars determined by Drake et al.(2013) have been observed by spectroscopic surveys of SDSS and LAMOST. We derived radial velocity dispersion, circular velocity and mass profile from 860 halo tracers in our paper I. Here, we present the stellar densities and radial velocity distributions of thick disk and halo of the Milky Way. The 288 RRLab stars located in the thick disk have the mean metallicity of \[Fe/H\]$=-1.02$. Three thick disk tracers have the radial velocity lower than 215 km $\rm s^{-1}$. With 860 halo tracers which have a mean metallicity of \[Fe/H\]$=-1.33$, we find a double power-law of $n(r) \propto r^{-2.8}$ and $n(r) \propto r^{-4.8}$ with a break distance of 21 kpc to express the halo stellar density profile. The radial velocity dispersion at 50 kpc is around 78 km $\rm s^{-1}$.'
author:
- Iminhaji Ablimit and Gang Zhao
title: The Density Profile and Kinematics of the Milky Way with RR Lyrae Stars
---
Introduction
============
The recent photometric and spectroscopic surveys have been providing information of stars to measure the stellar number density profile and kinematics of the Galactic disk and halo (e.g., Bland-Hawthorn & Gerhard 2016), and the stellar density profile is the one of the key issues for knowing the nature of the Milky Way. The Galactic thick disk is different from the Galactic thin disk by its unique chemistry, older age and higher elevation (e.g. Bensby 2014; Hawkins et al. 2015). It has been showed that an edge in the stellar disk could be in the range $R_{\rm GC} = 10-15$ kpc from the surveys data (Habing 1988; Minniti et al. 2011).
The Milky Way’s stellar halo is an important component to understand our Galaxy’s formation history. Many observational studies have claimed the broken power-law slope of space density distributions with RR Lyraes (RRLs), globular clusters and blue horizontal branch stars so on (e.g., Saha 1985; Wetterer & McGraw 1996; Miceli et al. 2008; Bell et al. 2008; Watkins et al. 2009; Sear et al. 2010, 2011; Deason et al. 2011). The theoretical study of Font et al. (2011) predict the broken power-law slope of the mass-density profile. The study of Deason et al. (2013) always found the broken stellar halo profile of the Galaxy, and they theoretically explained the origin of the break radius in the RRL profiles in terms of the apocentric distance of the satellites that were accreted. More recently, Iorio et al. (2017) demonstrated the properties of the Galactic stellar halo with RRL in the Gaia data, and they showed that the inner halo ($R_{\rm GC} < 28$ kpc) stellar density profile is well approximated by a single power-law with exponent $\alpha = -2.96$. Cohen et al. (2017) presented RRLs determined in the Palomar Transient Facility database, and derived the stellar density profile with the power-law index of -4 for the outer halo of the Milky Way.
RRL is a very good standard candle because of the narrow luminosity-metallicity-relation in the visual band and period-luminosity-metallicity relations in the near-infrared wavelengths. Besides, most of RRL stars are type ab RRLs (RRLab), and they are old ($>10$ Gyr) with low metallicity, so that they are mainly distributed in the bulge, thick disk and halo (Smith 1995; Demarque et al. 2000).
In this paper, we investigate density profiles of the Galactic thick disk and halo by using 1148 RRLab variable stars presented in our previous paper (Ablimit & Zhao 2017, hereafter paper I). We briefly describe our RRL sample and methods in §2. The kinematic and density information of the thick disk and halo are discussed in §3. In the last section (§4), the conclusions are given.
The sample and methods {#sec:model}
======================
The RRLab sample of this work were selected from 12227 RRLab stars of Catalina Surveys Data Release 1 (Drake et al. 2013a), and the cross matches were made with Sloan Digital Sky Survey (SDSS) spectroscopic data release 8 (DR8) and Large sky Area Multi-Object fiber Spectroscopic Telescope (LAMOST) DR4 within an angular distance of 3 arcsecond. LAMOST is a Chinese national scientific research facility operated by National Astronomical Observatories, Chinese Academy of Sciences (Zhao et al. 2006, 2012; Cui et al. 2012). We found 797 LAMOST matched RRLab stars and 351 SDSS matched RRLab stars with the corrected radial velocities (uncertainty $<15$ $\rm km\,\rm s^{-1}$) and reliable metallicities (in the region of Galactic centric distance $\leq 50$ kpc, for more details see paper I).
An absolute magnitude-metallicity relation has been adopted for distance determination (Sandage 1981). The method of Chaboyer (1999) and Cacciari & Clementini (2003) is used in this work as, $$M_{\rm V} = (0.23 \pm 0.04)([\rm Fe/H] + 1.5) + (0.59 \pm 0.03),$$ where \[Fe/H\] is the metallicity of an RR Lyrae star. We find our overall uncertainties is around 0.15 mag from the uncertainties from the photometric calibration and the variations in metallicity and uncertainty in RRab absolute magnitudes (also see Dambis et al. (2013) for the uncertainty in absolute magnitude). The uncertainties of $\sim 7\%$ in distances are expected by our overall uncertainties of $M_{\rm V}$. The heliocentric $d$ and Galactocentric distances $R_{\rm GC}$ can be derived from the equations, $$d = 10^{(<V> - M_{\rm V} + 5)/5}\, {\rm kpc},$$ where $<V>$ average magnitudes were corrected for interstellar medium extinction using Schlegel et al. (1998) reddening maps, and $$R_{\rm GC} = (R_\odot - d{\rm cos}\,b\, {\rm cos}\,l)^2 + d^2{\rm cos}^2\,b\, {\rm sin}^2\,l +
d^2{\rm sin}^2\,b \, {\rm kpc},$$ where $R_\odot$, $l$ and $b$ are the distance from the sun to the Galactic center (8.33 kpc in this work, see Gillessen et al. 2009), Galactic longitude and latitude of the stars, respectively.
We obtain the fundamental (Galactic) standard of rest (FSR) of stars by using the heliocentric radial velocities ($V_{\rm h}$, the corrected ones:see our paper one) and the solar peculiar motion of (U, V, W) = (11.1, 12, 7.2) km ${\rm s}^{-1}$ (Binney & Dehnen 2010) which are defined in a right-handed Galactic system with U pointing toward the Galactic center, V in the direction of rotation, and W toward the north Galactic pole. The value of $235\pm7$ km ${\rm s}^{-1}$ is taken for the local standard of rest ($\rm{V}_{\rm lsr}$, Reid at al. 2014) in the equation below, $$V_{\rm FRS} = V_{\rm h} + {\rm U} {\rm cos}\, b\, {\rm cos}\, l + ({\rm V} + {\rm V}_{\rm lsr})
{\rm cos}\, b\, {\rm sin}\, l + {\rm W} {\rm sin}\, b.$$
For deriving the spatial density of our RRLab sample, we followed the density calculation as a function of Galactocentric distance described by Wetterer & McGraw (1996), as $${\rho}({R_{\rm GC}}) = \frac{1}{4\pi {R^2_{\rm GC}} f({R_{\rm GC}})}\frac{dN}{dR}$$ $N$ is the number of RRLab as a function of distance and $f({R_{\rm GC}})$ is the fraction of the total halo volume at $R_{\rm GC}$ sampled by the survey. The efficiency or completeness of sampling (selection process) is a way to achieve each $f({R_{\rm GC}})$ for each individual field. Drake et al. (2013a) discussed the Catalina Surveys efficiency of RRL sampling, and their Figure 13 showed the detection completeness as a function of magnitude. We followed Drake et al. (2013a) for $f({R_{\rm GC}})$ by adopting completeness as 70% for $V < 17.5$ mag and it is gradually reach to 0% from $V = 17.5$ to 20 mag.
Results
========
Our sample of 1148 RRLab stars contains 288 thick disc stars with $1
< |z| < 4$ kpc, and also 860 halo stars with $|z| > 4$ kpc (see Figure 3 of paper I). We use the equations introduced above section to demonstrate the stellar number density and velocity distributions of the thick disk and halo.
The thick disk profile
----------------------
Figure 1 demonstrates the stellar density map in the ${R_{\rm GC}}$–Z plane for the thick disk, the bin size is $2\times0.5$ kpc. We assume the thick disk has a shape of cylinder, and get the volume from the ${R_{\rm GC}}$ and Z (Z as a height). The thick disk has a range of $1< |z| < 4$ kpc in the vertical direction, and there is a gap between -1 and 1 kpc because of thin disk, RRLab stars are old and metal-poor stars. The tomographic map distributed in a Galactocentric distance range of 4.5–14.5 kpc. There are two ring areas in the Figure 1, the high density ring showed at the region of ${R_{\rm GC}}= 8$ – $10.5$ kpc and $Z= -1.5$ – $-2.5$ kpc, while the relatively lower density ring showed up at the region of ${R_{\rm GC}}= 8$ – $ 9.5$ kpc and $Z= 2.2$ – $ 2.8$ kpc (for the similar results see Newberg et al (2003), Juri$\acute{\rm c}$ et al. (2008) and Ivezi$\acute{\rm c}$ et al. (2008)). LAMOST survey covers different areas at different longitudes. Thus, these two density regions might be caused by the selection effect of LAMOST. From $Z= -1.5$ – $- 4$ kpc the density distribution yield a heart shape between ${R_{\rm GC}}= 7$ – $ 13$ kpc, and gradually decreases until ${R_{\rm GC}}= 14.5$ kpc (the outer edge). Combining our results of the thick disk and halo, we agree the conclusion of Liu et al. (2017) which is that the disk smoothly transit to the halo without any truncation (also see Liu et al. 2017).
The metallicity and velocity profiles of the thick disk RRLab stars are given in the upper and lower panels of Figure 2. Most of the RRLab are metal-poor stars which have \[Fe/H\] around or lower than -1 dex while few of them with around 0. The red line in the upper panel of the figure is the mean value (\[Fe/H\]$=-1.02$) of all the thick disk tracers. There are three stars which have the radial velocity $<$ -215 km ${\rm s}^{-1}$ around 7.5 kpc, while other the RRLab have the radial velocities higher than -210 km ${\rm s}^{-1}$ (see $V_{\rm FSR}$ distribution in the lower panel of Figure 2).
The halo profile
----------------
The number density of RRL has been on debate such as break or no break in the density distribution. Ivezi$\acute{\rm c}$ et al. (2000) claimed the existence of a break in the density distribution in the halo at ${R_{\rm GC}}\sim 50$ kpc by using 148 SDSS RRLs. However, Ivezi$\acute{\rm c}$ et al. (2004) and Vivas & Zinn (2006) have found no break until $\sim$ 60 or 70 kpc. A broken-power law has been considered as a better number density profile for the RRLs by a number of works (e.g., Saha 1985; Sesar et al. 2007; Keller et al. 2008; Watkins et al. 2009; Akhter et al. 2012; Faccioli et al. 2014). Drake et al. (2013b) presented 1207 RRLs taken by the Caltalina Survey’s Mount Lemmon telescopes, and found the number density out to 100 kpc with $\sim$70% detection efficiency and a break appeared around 50 kpc (see the Figure 12 of Drake et al. (2013b)). They claimed their density profile is good agreement with the Watkins et al.(2009), and the different break is caused by the density enhanced by RRLs in the Sagittarius stream leading and trailing arms. We further analyze 860 RRLab stars from Drake et al. (2013a) by combining LAMOST DR4 and SDSS DR8 data to show the density profile in the 9–50 kpc range. We consider a spherical averaged number density and fit by following formula,
$$n(R_{\rm GC}) = n_0 \left\{ \begin{array}{ll}
(\frac{R_0}{R_{\rm GC}})^{\alpha} & \textrm{if $R_{\rm min}<{R_{\rm GC}} < R_0$}\\
\\
(\frac{R_0}{R_{\rm GC}})^{\beta} & \textrm{if $R_0<{R_{\rm GC}} < R_{\rm max}$},
\end{array} \right.$$
and derive the following values for the halo parameters : $n_0 = 0.35\pm0.18$ ${\rm kpc}^{-3}$, $R_0=21\pm2$ kpc, $\alpha=2.8\pm0.4$ and $\beta=4.8\pm0.4$ (see Figure 3). Watkins et al. (2009) gave their best results as: $(n_0,\,R_0,\,\alpha,\,\beta)=(0.26\,{\rm kpc}^{-3},\,23\,{\rm kpc},\,2.4\,,4.5)$. Faccioli et al. (2014) demonstrated 318 RRLs observed by Xuyi Schmidt telescope photometric survey, and obtained the density profiles by including and removing the possible Sagittarius RRLs. There is not a significant difference between their results with and without Sagittarius RRLs. Their spherical double-power model results are shown in their table 2, and the result with all RRLs are $n_0 = 0.42\pm0.16$ ${\rm kpc}^{-3}$, $R_0=21.5\pm2.2$ kpc, $\alpha=2.3\pm0.5$ and $\beta=4.8\pm0.5$. It is obvious that our result are strongly supports the results of Watkins et al. (2009) and Faccioli et al. (2014).
Figure 4 shows the velocity $V_{\rm FSR}$ distributions of 860 RRLab stars. In general, the radial velocities of RRLab stars are smoothly distributed with the distance, and a small part has a higher radial velocities at inner Galaxy, this may caused by the stream effect. We find the radial velocity dispersion at 50 kpc is $\sim$78 km ${\rm s}^{-1}$ (for more details also see paper I), which is smaller than $\sim$90 km ${\rm s}^{-1}$ derived by Cohen et al. (2017). We used 860 RRLab stars to measure the radial velocity dispersion for the halo based on the SDSS and LAMOST spectroscopic surveys, and Cohen et al. (2017) determined the radial velocity dispersion by only using 112 RRLs based on the moderate resolution spectra with Deimos on the Keck 2 Telescope.
Conclusions
===========
In this work, we have investigated the density profiles and velocity distributions of the thick disk and halo of the Milky Way, based on 1148 RRLab variables with precise distances (7% uncertainty) and reliable radial velocities (uncertainty $<$ 15 km $\rm s^{-1}$) presented in paper I. The 288 thick disk RRLab stars have the mean metallicity of \[Fe/H\]$=-1.02$. Despite three RRLab with the radial velocity lower than -215 km $\rm s^{-1}$, other disk tracers distributed in a region $>$ -210 km $\rm s^{-1}$. Our result shows that the edge of the thick disk is around 14.5 kpc. The halo of the Milky Way have been studying by using RRLab variables. Comparing to previous works, we present a larger sample (860) of halo RRLab variables with the mean metallicity of \[Fe/H\]$=-1.33$. The stellar density distribution of the halo tracers can be well fitted by a broken power-law, and power law index of -2.8 for $<$ 21 kpc & the index of -4.8 for $\geq$ 21 kpc. This density distribution is agreed by most of other works, especially by the works which used RRLs as the tracer. The radial velocity dispersion at 50 kpc is $\sim$78 km $\rm s^{-1}$, and few halo tracers show high radial velocities while that of others smoothly distributed.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by JSPS International Postdoctoral Fellowship of Japan (P17022, JSPS KAKENHI grant no. 17F17022), and also supported by National Natural Science Foundation of China under grant number 11390371 and 11233004. Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.
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![The mean spatial density for RRLab stars. The color codes for each $R_{\rm GC}$–Z bin. There is gap between -1 and 1 kpc, because the thick disk is considered as a disk of $1<|Z|<4$ kpc.[]{data-label="fig:1"}](f1){width="4.5in"}
![The upper panel shows the metallicity distribution of the thick disk tracers (the red line represents the mean value), and the velocity distribution of the fundamental standard rests are shown in the lower panel.[]{data-label="fig:1"}](f2a "fig:"){width="4.1in"} ![The upper panel shows the metallicity distribution of the thick disk tracers (the red line represents the mean value), and the velocity distribution of the fundamental standard rests are shown in the lower panel.[]{data-label="fig:1"}](f2b "fig:"){width="4.7in"}
![The number density of the halo RRLab stars as a function of Galactocentric distance in a range of 9–50 kpc. The blue line is the fit result of Watkins et al. (2009), and the red line is the result derived in this work.[]{data-label="fig:1"}](f3){width="4.5in"}
![The velocity distribution of the fundamental standard rests $V_{\rm FSR}$ of the halo RRLab stars.[]{data-label="fig:1"}](f4){width="4.7in"}
|
---
abstract: 'The incidence correspondence in the grassmannian which determines the tautological bundle defines a map between cycle spaces on grassmannians. These cycle spaces decompose canonically into a product of Eilenberg-MacLane spaces. These decompositions and the associated maps are calculated up to homotopy.'
address: |
Max-Planck-Institut für Mathematik\
Vivatsgasse 7\
D-53111 Bonn\
Germany
author:
- 'Luis E. Lopez'
title: The Incidence Correspondence and its associated maps in Homotopy
---
Introduction
============
Let ${{\mathcal C}}^p_d({\mathbb{P}}^n)$ denote the space of algebraic cycles of codimension $p$ and degree $d$ in ${\mathbb{P}}^n$. This set can be given the structure of an algebraic variety via the Cayley-Chow-Van der Waerden embedding which takes an irreducible cycle $X$ into $\Psi(X)$ where the Chow Form $\Psi$ is obtained from the following incidence correspondence: $$\xymatrix{ & \{ (x,\Lambda) \in {\mathbb{P}}^n \times {\mathbb{G}}_{p-1}({\mathbb{P}}^n) \mid x \in \Lambda \} \ar[dl]_{\pi_1} \ar[dr]^{\pi_2} \\
{\mathbb{P}}^n & & {\mathbb{G}}_{p-1}({\mathbb{P}}^n) \\
& \Psi(X) = \pi_2\pi_1^{-1}(X)&
}$$ This map is then extended additively to the topological monoid ${{\mathcal C}}^p({\mathbb{P}}^n)$ of all cycles and to its naive group completion ${{\mathcal Z}}^p({\mathbb{P}}^n)$. The cycle $\Psi(X)$ has codimension $1$ in the grassmannian, therefore $\Psi$ defines a map $$\Psi: {{\mathcal Z}}^p({\mathbb{P}}^n) \rightarrow {{\mathcal Z}}^1\left({\mathbb{G}}_{p-1}({\mathbb{P}}^n)\right)$$ Moreover, if $B = \bigoplus B_d$ denotes the coordinate ring of the grassmannian in the Plücker embedding, then every irreducible hypersurface $Z$ of degree $d$ in the grassmannian is given by an element $f \in B_d$ defined uniquely up to a constant factor (see [@GKZ]), this defines a grading on the space of hypersurfaces in the grassmannian and with respect to this grading the map above preserves the degree.\
The topology thus inherited defines in turn a topology in the space ${{\mathcal C}}^p({\mathbb{P}}^n)$ of all codimension $p$ algebraic cycles in ${\mathbb{P}}^n$ (c.f. [@FM] for this and other equivalent definitions) The results presented here are of the following type:
The Chow Form map $$\Psi: {{\mathcal Z}}^p({\mathbb{P}}^n) \rightarrow {{\mathcal Z}}^1\left({\mathbb{G}}_{p-1}({\mathbb{P}}^n)\right)$$ can be represented with respect to canonical decompositions of the corresponding spaces in the following way: $$\xymatrix{ \prod_{j=0}^p K({\mathbb{Z}},2j) \ar[r]^-{p} & K({\mathbb{Z}},0) \times K({\mathbb{Z}},2)}$$ where $p$ is the projection into the first two factors of the product
hence, the Chow Form map can be interpreted as the classifying map of a (non trivial) line bundle in the space of algebraic cycles in ${\mathbb{P}}^n$, the class of this line bundle generates $H^2\left({{\mathcal Z}}^1\left({\mathbb{G}}_{p-1}({\mathbb{P}}^n)\right)\right)$.\
An equivalent way of looking at the theorem is via the chain of inclusions $${\mathbb{G}}^p({\mathbb{P}}^n) \subset {{\mathcal Z}}^p({\mathbb{P}}^n) \subset {{\mathcal Z}}^1\left({\mathbb{G}}_{p-1}({\mathbb{P}}^n)\right)$$
The first space classifies $p$-dimensional bundles (not all of them), the second space classifies all total integer cohomology classes in $\bigoplus_{j=0}^p H^{2j}(-;{\mathbb{Z}})$ and the third space classifies all integer cohomology classes in $H^0(-;{\mathbb{Z}}) \times H^{2}(-;{\mathbb{Z}})$. The corresponding maps associated to the inclusions are the total chern class map (see [@LM]) and the projection into the first two factors.\
The Chow Form Map
=================
In this section we prove the theorem regarding the chow form map using an explicit description of the map. In order to state the results and its proofs we recall some definitions and facts about Lawson Homology. A survey of these and other related results is given in [@lawsurvey].
Let $X$ be a projective variety. The [*Lawson Homology groups*]{} $L_pH_n(X)$ of $X$ are defined by $$L_pH_n(X) := \pi_{n-2p}\left({{\mathcal Z}}_p(X)\right)$$ where ${{\mathcal Z}}_p(X)$ is the naive group completion of the Chow monoid ${{\mathcal C}}_p(X)$ of all $p$-dimensional effective algebraic cycles in $X$
These groups stand between the group of algebraic cycles modulo algebraic equivalence ${\mathcal A}_p(X) = L_pH_{2p}(X)$ and the singular homology group $H_{n}(X) = L_0H_n(X)$. Friedlander and Mazur defined a [*cycle map*]{} between the Lawson Homology groups and the singular homology groups $$s^{(p)}_X:L_pH_n(X) \rightarrow H_n(X;{\mathbb{Z}})$$ this map will be referred to as the F-M map.\
The following results of Lima-Filho [@L-FCyclemap] will be used throughout the section.
\[smaps\] The F-M cycle map coincides with the composition $$\xymatrix{
L_pH_n(X) = \pi_{n-2p}({{\mathcal Z}}_{n-p}(X)) \ar[r]^-{e_{*}} & \pi_{n-2p}({\mathfrak Z}_{2p}(X)) \ar[r]^-{{\mathcal A}} & H_n(X; {\mathbb{Z}})
}$$ where $e_{*}$ is the map induced by the inclusion $$e: {{\mathcal Z}}_m(X) \rightarrow \mathfrak{Z}_{m}(X)$$ of the space of algebraic cycles into the space of all integral currents and ${\mathcal A}$ is the Almgren isomorphism defined in [@Almgren]. In particular, the F-M is functorial and is compatible with proper push-forwards and flat-pullbacks of cycles.
\[inclusion\] If $X$ is a projective variety with a cellular decomposition in the sense of Fulton, (i.e., $X$ is an algebraic cellular extension of $\emptyset$) then the inclusion $${{\mathcal Z}}_p(X) \hookrightarrow \mathfrak{Z}_{2p}(X)$$ into the space $\mathfrak{Z}_{2p}(X)$ of integral currents is a homotopy equivalence
We also recall some facts about the cohomology of the Grassmannian. We follow the notation of [@Fulton] Chapter $14$.
\[schubert\] Let ${\mathbb{G}}_k({\mathbb{P}}^n)$ be the Grassmann variety of $k$-dimensional linear spaces in ${\mathbb{P}}^n$. ${\mathbb{G}}_k({\mathbb{P}}^n)$ is a smooth algebraic variety of complex dimension $d :=(k+1)(n-k)$. The [*special Schubert classes*]{} are the homology classes $\sigma_m \in H_{2(d-m)}({\mathbb{G}}_k({\mathbb{P}}^n))$ defined by the cycle class of $$\sigma_m := \{ L \in {\mathbb{G}}_k({\mathbb{P}}^n) \mid L \cap A \neq \emptyset \}$$ where $A$ is any linear subspace of ${\mathbb{P}}^n$ of codimension $k+m$.
The integral cohomology of the grassmannian ${\mathbb{G}}_k({\mathbb{P}}^n)$ is generated by the Poincare duals $c_m(Q)$ of the special Schubert classes $\sigma_m$. These Poincare duals are the chern classes of the universal quotient bundle $Q$.
\[chowform\] The Chow Form map $$\Psi: {{\mathcal Z}}^p({\mathbb{P}}^n) \rightarrow {{\mathcal Z}}^1\left({\mathbb{G}}_{p-1}({\mathbb{P}}^n)\right)$$ can be represented with respect to canonical decompositions of the corresponding spaces in the following way: $$\xymatrix{ \prod_{j=0}^p K({\mathbb{Z}},2j) \ar[r]^-{p} & K({\mathbb{Z}},0) \times K({\mathbb{Z}},2)}$$ where $p$ is the projection into the first two factors of the product
The homotopy equivalences $${{\mathcal Z}}^p({\mathbb{P}}^n) \simeq \prod_{j=1}^p K({\mathbb{Z}},2j)$$ and $${{\mathcal Z}}^1\left({\mathbb{G}}_{p-1}({\mathbb{P}}^n)\right) \simeq K({\mathbb{Z}},0) \times K({\mathbb{Z}},2)$$ are a consequence of theorem \[inclusion\] and Almgren’s theorem which asserts that $$\pi_i(\mathfrak{Z}_k{X}) \cong H_{i+k}(X)$$ for the second homotopy equivalence, if $d=\dim_{{\mathbb{C}}} {\mathbb{G}}_{p-1}({\mathbb{P}}^n)$ then $$\pi_i\left({{\mathcal Z}}_{d-1}{\mathbb{G}}_{p-1}({\mathbb{P}}^n)\right) \cong
\begin{cases}
H_{2d-1}\left({\mathbb{G}}_{p-1}({\mathbb{P}}^n)\right) \cong {\mathbb{Z}}& \text{if $i=0$}\\
H_{2d}\left({\mathbb{G}}_{p-1}({\mathbb{P}}^n)\right) \cong {\mathbb{Z}}& \text{if $i=2$}\\
H_{d-1+i}\left({\mathbb{G}}_{p-1}({\mathbb{P}}^n)\right) = 0 & \text{otherwise}
\end{cases}$$ this last calculation being well known for grassmannians (see [@BottandTu]). Now we use theorem \[smaps\] to calculate the induced maps in homotopy. We have the following commutative diagram: $$\xymatrix{
\pi_{m}({{\mathcal Z}}_{n-p}({\mathbb{P}}^n))=L_{n-p}H_{m+2(n-p)}({\mathbb{P}}^n) \ar[r]^-{\Psi} \ar[d]_-s & \pi_{m}({{\mathcal Z}}_{d-1}{\mathbb{G}}_{p-1}({\mathbb{P}}^n)) \cong L_{d-1}H_{m+2(g-1)}({\mathbb{G}}_{p-1}({\mathbb{P}}^n)) \ar[d]^-s \\
H_{m+2(n-p)}({\mathbb{P}}^n) \ar[r]^-{\tilde{\Psi}} & H_{m+2(d-1)}({\mathbb{G}}_{p-1}({\mathbb{P}}^n))
}$$ where $\Psi$ is the Chow Form map and $\tilde{\Psi}$ is the corresponding map in the space of integral currents.
Theorem \[inclusion\] implies that the vertical maps are isomorphisms. Since we know the homology of the grassmannian, the only non-zero dimensions in the lower right correspond to the cases $m=0$ and $m=2$. We will describe explicitly the morphism in these two cases.
- In this case $H_{2(n-p)}({\mathbb{P}}^n)$ is generated by the class of an $(n-p)$-plane $\Lambda$ in ${\mathbb{P}}^n$. The cycle $\tilde{\Psi}(\Lambda)$ is then $$\tilde{\Psi}(\Lambda)=\{ P \in {\mathbb{G}}_{p-1}({\mathbb{P}}^n) \mid \Lambda \cap P \neq \emptyset \}$$ this is precisely definition \[schubert\] of the special Schubert cycle $\sigma_1$.
- In this case $H_{2 + 2(n-p)}({\mathbb{P}}^n)$ is generated by the class of an $(n-(p-1))$-plane $\Lambda$ in ${\mathbb{P}}^n$. The cycle $\tilde{\Psi}(\Lambda)$ is then $$\tilde{\Psi}(\Lambda)=\{ P \in {\mathbb{G}}_{p-1}({\mathbb{P}}^n) \mid \Lambda \cap P \neq \emptyset \}$$ but by a dimension count this is the whole variety ${\mathbb{G}}_{p-1}({\mathbb{P}}^n)$ again, this corresponds to the special Schubert cycle $\sigma_0$
The results of Lima-Filho about the F-M cycle class map may also be used to prove the following theorem of Lawson and Michelsohn which appeared in [@LM].
The complex join pairing in the cycle spaces $$\#:{{\mathcal Z}}^q({\mathbb{P}}^n) \wedge {{\mathcal Z}}^{q'}({\mathbb{P}}^m) \rightarrow {{\mathcal Z}}^{q+q'}({\mathbb{P}}^{n+m+1})$$ represents the cup product pairing in the canonical decompositions $$\cup: \prod_{s=0}^{q}K({\mathbb{Z}},2s) \wedge \prod_{t=0}^{q'}K({\mathbb{Z}},2t) \rightarrow \prod_{r=0}^{q+q'}K({\mathbb{Z}},2r)$$ i.e. if $i_{2c}$ represents the generator of $H^{2c}(K({\mathbb{Z}},2c);{\mathbb{Z}}) \cong {\mathbb{Z}}$ then $$(\#)^{*}(i_{2c}) = \sum_{\begin{aligned} a+b = c \\ a > 0 \\ b > 0 \end{aligned}}i_{2a} \otimes i_{2b}$$
The linear join of two irreducible varieties $X \subset {\mathbb{P}}^n$ and $Y \subset {\mathbb{P}}^m$ with defining ideals $\langle F_i \rangle \in {\mathbb{C}}[x_0,\ldots,x_n]$ and $\langle \G_j \rangle \in {\mathbb{C}}[y_0,\ldots,y_m]$ is the variety in ${\mathbb{P}}^{n+m+1}$ defined by the ideal $\langle F_i,G_j \rangle \in {\mathbb{C}}[x_0,\ldots,x_n,y_0,\ldots,y_m]$. A synthetic construction of this pairing is given in [@LM]. The join pairing $\#$ is obtained by extending bi-additively to all cycles and taking the induced pairing in the smash product. The join pairing induces a bilinear pairing $$\pi_s({{\mathcal Z}}^q({\mathbb{P}}^n)) \times \pi_r({{\mathcal Z}}^{q'}({\mathbb{P}}^m)) \rightarrow \pi_{r+s}({{\mathcal Z}}^{q+q'}({\mathbb{P}}^{n+m+1}))$$ which corresponds to a homomorphism $$\#_*:L_{n-q}H_{s+2(n-q)}({\mathbb{P}}^n) \otimes L_{m-q'}H_{r+2(m-q')}({\mathbb{P}}^m) \rightarrow L_{(n-q)+(m-q')+1}H_{s+2(n-q)+r+2(m-q')+2}({\mathbb{P}}^{n+m+1})$$ This homomorphism is non-zero only when $s$ and $r$ are even, so we may assume $s=2a$ and $r= 2b$. Now we take the F-M map and we get the following commutative diagram $$\xymatrix{
L_{n-q}H_{s+2(n-q)}({\mathbb{P}}^n) \otimes L_{m-q'}H_{r+2(m-q')}({\mathbb{P}}^m) \ar[r]^-{\#_*} \ar[d] & L_{(n-q)+(m-q')+1}H_{s+2(n-q)+r+2(m-q')+2}({\mathbb{P}}^{n+m+1}) \ar[d] \\
H_{s+2(n-q)}({\mathbb{P}}^n) \otimes H_{r+2(m-q')}({\mathbb{P}}^m) \ar[r]^-{\phi} & H_{s+2(n-q)+r+2(m-q')+2}({\mathbb{P}}^{n+m+1})
}$$ where we define $\phi$ on the generators and we extend it bilinearly, namely, if the cycle classes of the planes $\Lambda_1 \in H_{2a+2(n-q)}({\mathbb{P}}^n)$ and $\Lambda_2 \in H_{2b+2(m-q')}({\mathbb{P}}^m)$ are generators of the corresponding groups, then $\phi([\Lambda_1] \otimes [\Lambda_2])=[\Lambda_1 \# \Lambda_2]$, this last class is again a generator of $H_{2(a+b)+2(n-q)+2(m-q')+2}({\mathbb{P}}^{n+m+1})$. The vertical F-M maps are isomorphisms by theorem \[inclusion\], so we get that the generator $i_{2c}$ gets pulled back precisely to the sum of the generators $i_{2a} \otimes i_{2b}$ with $a+b=c$.
Generalizations
===============
The proof of theorem \[chowform\] suggests a generalization of the result. Instead of taking the Chow form map we will look at general correspondences.
Let $\Sigma_k$ denote the [*incidence correspondence*]{} defined by $$\xymatrix{ & \Sigma_k := \{ (x,\Lambda) \in {\mathbb{P}}^n \times {\mathbb{G}}_{k}({\mathbb{P}}^n) \mid x \in \Lambda \} \ar[dl]_{\pi_1} \ar[dr]^{\pi_2} \\
{\mathbb{P}}^n & & {\mathbb{G}}_{k}({\mathbb{P}}^n) \\
& \Psi_k(X) = \pi_2\pi_1^{-1}(X)&
}$$
The map $\pi_1$ is flat and the map $\pi_2$ is proper (see [@harris]). Therefore $\Psi_k$ is a well defined map of cycle spaces $$\Psi_k: {{\mathcal C}}^p({\mathbb{P}}^n) \rightarrow {{\mathcal C}}^{p-k}({\mathbb{G}}_k({\mathbb{P}}^n)).$$
Notice that the incidence correspondence which defines the universal quotient bundle is $\Sigma_{p-1}$.\
With this notation we have the following
The map $$\Psi_k: {{\mathcal Z}}^p({\mathbb{P}}^n) \rightarrow {{\mathcal Z}}^{p-k}\left({\mathbb{G}}_{k}({\mathbb{P}}^n)\right)$$ can be represented with respect to canonical decompositions of the corresponding spaces in the following way $$\xymatrix{\prod_{j=0}^p K({\mathbb{Z}},2j) \ar[r]^-{p} & \prod_{r=0}^{p-k} \prod_{\alpha \in H_{2r}({\mathbb{G}}_k({\mathbb{P}}^n))}K({\mathbb{Z}},2r)_{\alpha}}$$ where $p$ is the projection into the factors corresponding to the classes $\sigma_0,\sigma_1,\ldots,\sigma_{(p-k)}$
The argument follows the proof of theorem \[chowform\]. We take the homotopy groups and then we use the cycle map. In this case we have the following commutative diagram: $$\xymatrix{
\pi_{m}({{\mathcal Z}}_{n-p}({\mathbb{P}}^n))=L_{n-p}H_{m+2(n-p)}({\mathbb{P}}^n) \ar[r]^-{\Psi_k} \ar[d]_-s & \pi_{m}({{\mathcal Z}}_{m}{\mathbb{G}}_{k}({\mathbb{P}}^n)) \cong L_{d-(p-k)}H_{d+2[d-(p-k)]}({\mathbb{G}}_{k}({\mathbb{P}}^n)) \ar[d]^-s \\
H_{m+2(n-p)}({\mathbb{P}}^n) \ar[r]^-{\tilde{\Psi_k}} & H_{m+2[d-(p-k)]}({\mathbb{G}}_{k}({\mathbb{P}}^n))
}$$
Since the homology of the grassmannian is zero in odd degrees we are only concerned with the case $m=2r$. The lower right corner of the diagram imposes the condition $0 \leq 2r \leq 2(p-k)$. Once again, the definition of the special Schubert cycles implies that if $\Lambda_{2r + 2(n-p)}$ is a plane of dimension $2r + 2(n-p)$ (i.e. the class of a generator of $H_{2r + 2(n-p)}({\mathbb{P}}^n)$) then $\tilde{\Psi_k}(\Lambda_{2r + 2(n-p)}) = \sigma_{p-(k+r)}$
|
---
abstract: 'We provide a first-principles, perturbative derivation of the $AdS_5$/$CFT_4$ Y-system that has been proposed to solve the spectrum problem of $\mathcal{N}=4$ SYM. The proof relies on the computation of quantum effects in the fusion of some loop operators, namely the transfer matrices. More precisely we show that the leading quantum corrections in the fusion of transfer matrices induce the correct shifts of the spectral parameter in the T-system. As intermediate steps we study UV divergences in line operators up to first order and compute the fusion of line operators up to second order for the pure spinor string in $AdS_5\times S^5$. We also argue that the derivation can be easily extended to other integrable models, some of which describe string theory on $AdS_4$, $AdS_3$ and $AdS_2$ spacetimes.'
author:
- Raphael Benichou
---
Theoretische Natuurkunde, Vrije Universiteit Brussel and\
The International Solvay Institutes,\
Pleinlaan 2, B-1050 Brussels, Belgium\
*[email protected]*
Introduction
============
The AdS/CFT correspondence [@Maldacena:1997re][@Gubser:1998bc][@Witten:1998qj] implies that type IIB string theory in $AdS_5 \times S^5$ is equivalent to $\mathcal{N}=4$ Super-Yang-Mills in four dimensions. In the classical string theory limit, or equivalently in the planar gauge theory limit, integrable structures appear. This has lead to impressive progress in the understanding of this system (see [@Beisert:2010jr] for a review).
The AdS/CFT dictionary relates the energy of string states in the bulk to the conformal dimensions of operators on the boundary. A set of equations known as the Y-system has been put forward in [@Gromov:2009tv] to solve the spectrum problem of planar $\mathcal{N}=4$ SYM, or equivalently of string theory in $AdS_5 \times S^5$. The goal of the present article is to make a new step towards a definite proof of the validity of this set of equations.
There is by now solid evidence in favor of the validity of the Y-system. It reproduces the results of the Asymptotic Bethe Ansatz [@Beisert:2006ez], but it does not suffer from the same limitations. For instance it contains [@Gromov:2009tq][@Gromov:2010vb] the spectrum of the quasi-classical string at large ’t Hooft coupling (see e.g. [@Gromov:2007ky]). Even more impressively, it lead to correct predictions for the dimension of the Konishi operator both at large [@Gromov:2009zb] and at small [@Arutyunov:2010gb] ’t Hooft coupling.
In order to claim that the spectrum problem for $\mathcal{N}=4$ SYM has been definitively solved, it would be comfortable to have a proof of the validity of the Y-system. At that point the only known derivation of the Y-system relies on the Thermodynamic Bethe Ansatz [@Zamolodchikov:1989cf] (see e.g. [@Bajnok:2010ke] for a review). This approach was studied in [@Gromov:2009bc][@Bombardelli:2009ns][@Arutyunov:2009ur]. The Thermodynamic Bethe Ansatz has been very successful and lead to numerous remarkable results. However this method relies on several crucial assumptions. In the first place, one has to assume quantum integrability of the theory. Then one needs the “string hypothesis”: the spectrum of excitations that contribute to the thermodynamic limit of the theory essentially has to be guessed. Most importantly, this method only gives the ground state energy. The spectrum of excited state can be obtained by analytic continuation, but the reason why this works is not understood.
In this paper, we will initiate a different approach to derive the Y-system from first principles. We will use only elementary tools of two dimensional conformal field theory; in this aspect this article can be related to the seminal work of [@Bazhanov:1994ft] where the Y-system was derived for the minimal models. We will be able to prove the validity of the Y-system up to first non-trivial order at large ’t Hooft coupling.
#### The idea of the proof.
Up to a change of variables, the Y-system can be rewritten as a T-system, also known as the Hirota equation: \[Tsystem\] \_[a,s]{}(u + 1) \_[a,s]{}(u - 1) = \_[a+1,s]{}(u+1)\_[a-1,s]{}(u-1) + \_[a,s+1]{}(u-1)\_[a,s-1]{}(u+1) In the above equation, $u$ is a spectral parameter and the indices $(a,s)$ are integers that label representations of the global symmetry group of the system. In the case at hand this group is $PSU(2,2|4)$. These labels take values in a T-shaped lattice. More details are given in section \[proofTsys\].
The T-functions are expected to be related to special line operators of the string worldsheet theory, namely the transfer matrices (see e.g. [@Gromov:2010vb]). The transfer matrices play a central role in the study of classical integrability. Indeed these operators code an infinite number of conserved charges. By definition, the transfer matrix is the supertrace of the monodromy matrix. The monodromy matrix itself is an element of the supergroup $PSU(2,2|4)$. Thus the classical transfer matrix is a supercharacter. If the shifts of the spectral parameter are neglected, the T-system reduces to a character identity that is known to hold (see e.g. [@Gromov:2010vb][@Kazakov:2007na]). We deduce that the shifts of the spectral parameter come from some kind of quantum effects.
In this paper we take the identification the T-functions to the transfer matrices seriously. The T-system is promoted to an operator identity between quantum transfer matrices. We postulate that the product appearing in the T-system is the fusion of line operators. The process of fusion of line operators involve quantum effects, that are responsible for the appearance of the shifts in the T-system. We will show that this picture is indeed correct up to first non-trivial order in the large ’t Hooft coupling expansion. More precisely, we will compute the leading quantum correction in the fusion of two transfer matrices, and show that it correctly gives the shifts of the T-system at first order. The same strategy was previously successfully applied for the sigma-model on the supergroup $PSl(n|n)$ in [@Benichou:2010ts].
#### Organization of the paper.
In section \[worldsheetTheory\] we describe the features of the pure spinor string on $AdS_5\times S^5$ that are relevant for our purposes. We also introduce the relevant line operators. In section \[fusion\] we present the central computation of this work: we study the fusion of line operators up to second order in perturbation theory. In section \[proofTsys\] we make good use of this computation to deduce the validity of the T-system up to first order in perturbation theory. Section \[extensions\] contains a discussion of the extension of this method to other integrable models. Eventually final remarks are gathered in section \[conclusion\].
In order to keep the bulk of the paper as readable as possible, most of the details of the computations are gathered in the appendices. Appendix \[conventions\] contains the conventions. In Appendix \[covCurrents\] we revisit the computation of the current-current OPEs in the pure spinor formalism using a novel and efficient method. In Appendix \[computationDiv\] we study the UV divergences in line operators. Eventually Appendix \[comFusion\] contains the computations relevant for the fusion of line operators.
The pure spinor string on $AdS_5 \times S^5$ {#worldsheetTheory}
============================================
To describe superstring theory in $AdS_5 \times S^5$ we will use the pure spinor formalism. This choice is a matter of convenience. Indeed the computations of section \[fusion\] are simpler in the conformal gauge, where target-space covariance is preserved.
In this section we introduce the pure spinor string on $AdS_5 \times S^5$. We only discuss the features of this formalism that are relevant for the purpose of this paper. A more detailed discussion can be found for instance in [@Berkovits:2000fe][@Berkovits:2004xu][@Mazzucato:2011jt]. We introduce the flat connection, and show that the commutator of equal-time connections can be written in the canonical form of a $(r,s)$ system. We also introduce the line operators that are defined as the path-ordered exponential of the line integral of the flat connection. Finally we discuss the UV divergences that appear in these line operators because of quantum effects. Most of the results discussed in this section have appeared before in the literature. Some new results are presented concerning the current algebra, the commutator of equal-time connections and the renormalization of the line operators.
Generalities
------------
The target-space $AdS_5\times S^5$ is embedded in a superspace with 32 supercharges. It is realized as the supercoset $PSU(2,2|4)/SO(4,1)\times SO(5)$. The Lie superalgebra $\mathcal{G}=psu(2,2|4)$ admits the action of a $\mathbb{Z}_4$ automorphism. This automorphism induces a $\mathbb{Z}_4$ grading on the elements of the Lie superalgebra. We can decompose the Lie algebra $\mathcal{G}$ according to this grading: \[Z4decompo\] = \_0 \_1 \_2 \_3 where the subscript gives the $\mathbb{Z}_4$ grade. Bosonic (respectively fermionic) generators of the Lie superalgebra have an even (respectively odd) grade.
#### The action.
Let us introduce the currents $J$ and $\bar J$ defined in terms of the group element $g \in PSU(2,2|4)$ as: \[J=gdg\] J = g\^[-1]{} g ; |J = g\^[-1]{} |g They take values in the Lie superalgebra $\mathcal{G}$. We decompose the current $J$ according to the $\mathbb{Z}_4$ grading of the Lie superalgebra: J = J\_[0]{} + J\_[1]{} + J\_[2]{} + J\_[3]{} and similarly for $\bar J$. Let us also introduce the bosonic pure spinor ghosts $\lambda, \hat \lambda$ as well as their conjugate momenta $w, \hat w$. They expand on the fermionic generators of the superalgebra with the following gradings: $\lambda, \hat w \in \mathcal{H}_1$ and $\hat \lambda, w \in \mathcal{H}_3$. The ghosts satisfy the pure spinor constraint: $ \lambda \gamma^\mu \lambda = 0 = \hat \lambda \gamma^\mu \hat \lambda$, where the $\gamma^\mu$’s are the $SO(9,1)$ gamma matrices. The pure spinor Lorentz currents are: N=-{w,} ; N = -{w, } The action reads: $$\begin{aligned}
\label{action} S =& \frac{R^2}{4\pi } STr \int d^2 z \left( J_2 \bar J_2 + \frac{3}{2} J_3 \bar J_1 + \frac{1}{2} \bar J_3 J_1 \right) \cr
&+ \frac{R^2}{2 \pi} STr \int d^2 z \left( N \bar J_0 + \hat N J_0 - N \hat N + w \bar \p \lambda + \hat w \p \hat \lambda \right)\end{aligned}$$ The first line of the action contains a kinetic term both for the bosonic and fermionic target space coordinates. This implies in particular that the model does not exhibit kappa-symmetry, contrary to the Green-Schwarz string. The radius of the target space is denoted by $R$ in units of the string length. Later on we will work perturbatively in a large radius expansion: the small parameter is $R^{-2}$.
#### Gauge symmetry.
The action admits a $\mathcal{H}_{0}$ gauge symmetry: g = g h\_0, = \[,h\_0\], = \[,h\_0\], w = \[w,h\_0\], w = \[w,h\_0\] The holomorphic currents transform as: i0: J\_[i]{} = \[J\_[i]{} ,h\_0\]; J\_[0]{} = h\_0 + \[J\_[0]{} ,h\_0\]; N = \[N,h\_0\] and similarly for the anti-holomorphic currents. We introduce the associated covariant derivative: = + \[J\_[0]{},\] ; |= |+ \[|J\_[0]{},\]
#### Parity.
The model enjoys a $\mathbb{Z}_2$ symmetry that exchanges holomorphic and anti-holomorphic worldsheet coordinates. It also flips the grade of the fermionic elements of the Lie superalgebra: the subalgebras $\mathcal{H}_1$ and $\mathcal{H}_3$ are exchanged.
#### The Maurer-Cartan equation.
A consequence of is that the current satisfies the Maurer-Cartan equation: \[MC\] |J - |J + \[J, |J\] = 0 This is a crucial equation that is essentially responsible for the integrable properties of the model.
#### Equations of motion.
The equations of motion combined with the Maurer-Cartan equation lead to: \[MC+EOM\]
[lll]{} |J\_1 = \[J\_3,|J\_2\] + \[J\_2,|J\_3\] + \[N,|J\_1\] - \[J\_1,N\] &&|J\_1 = \[N,|J\_1\] - \[J\_1,N\] |J\_2 = \[J\_3,|J\_3\] + \[N,|J\_2\] - \[J\_2,N\] &&|J\_2 = - \[J\_1,|J\_1\] + \[N,|J\_2\] - \[J\_2,N\] |J\_3 = \[N,|J\_3\] - \[J\_3,N\] &&|J\_3 =- \[J\_1,|J\_2\] - \[J\_2,|J\_1\] + \[N,|J\_3\] - \[J\_3,N\] |N = -\[N,N\] &&N = \[N,N\]
The current algebra {#subKalgebra}
-------------------
In this section we discuss the current-current OPEs that are the elementary input needed for the computations of section \[fusion\]. The set of currents we consider are the currents $J_0$, $J_1$, $J_2$, $J_3$ as well as the ghost Lorentz current $N$, together with their anti-holomorphic partners $\bar J_0$, $\bar J_1$, $\bar J_2$, $\bar J_3$ and $\hat N$. In order to simplify the expressions in the following computations, we introduce the generic notation $K_m$, $\bar K_m$ for the currents. The index $m$ takes the values in the set $\{ 0,1,2,3,g \}$. For $m=0,1,2,3$ we define $K_m\equiv J_m$, $\bar K_m \equiv \bar J_m$. For the particular value $m=g$ we define $K_g\equiv N$, $\bar K_g \equiv \hat N$, and $g$ stands for “ghost". The index $m$ codes the $\mathbb{Z}_4$-grade of the current. The ghost currents have grade zero.
The OPEs of the gauge covariant currents have been discussed in various papers. The OPEs at first-order in the $R^{-2}$ expansion have been analyzed in [@Bianchi:2006im][@Puletti:2006vb][@Puletti:2008ym][@Mikhailov:2007mr]. The $R^{-4}$ corrections to the second-order poles have been computed in [@Bedoya:2010av]. For the purpose of the present article, the knowledge of the current algebra at order $R^{-2}$ is enough.
In appendix \[covCurrents\] we present a new and rather efficient way of computing the current algebra. The idea is to demand compatibility with the Maurer-Cartan equation and with the equations of motion (more precisely with the reparametrization invariance of the path integral). Notice that the OPEs involving $J_0$ and $\bar J_0$ generically suffer from some ambiguities because of the gauge freedom. At the end of appendix \[covCurrents\] we compare the version of the current algebra we compute with the ones that appeared previously in the literature.
It is convenient to expand the current on a basis of the Lie superalgebra that is compatible with the $\mathbb{Z}_4$ grading. We write: K\_m = K\_m\^[A\_m]{} t\_[A\_m]{} The indices $A_m$ are adjoint indices[^1] restricted to the subspace of the Lie superalgebra of grade $m$. The generators $t_{A_m}$ form a basis of the subspace $\mathcal{H}_m$.
The current algebra takes the form: $$\begin{aligned}
\label{Kalgebra}
K_m^{A_m}(z) K_n^{B_n}(w) = &
R^{-2} C_{mn} \frac{\kappa^{B_n A_m}}{(z-w)^2}
+ R^{-2} \sum_p C_{mn}^p \frac{{f_{C_p}}^{B_n A_m} K_p^{C_p}}{z-w} \cr
& + R^{-2} \sum_p C_{mn}^{\bar p} {f_{C_p}}^{B_n A_m} \bar K_p^{C_p} \frac{\bar z - \bar w}{(z-w)^2} +... \cr
%%
K_m^{A_m}(z) \bar K_n^{B_n}(w) = &
R^{-2} C_{m \bar n} \kappa^{B_n A_m} 2\pi \delta^{(2)}(z-w)
+ R^{-2} \sum_p C_{m\bar n}^p \frac{{f_{C_p}}^{B_n A_m} K_p^{C_p}}{\bar z-\bar w} \cr
& + R^{-2} \sum_p C_{m\bar n}^{\bar p} \frac{{f_{C_p}}^{B_n A_m} \bar K_p^{C_p}}{z-w} +...\cr
%%
\bar K_m^{A_m}(z) \bar K_n^{B_n}(w) = &
R^{-2} C_{\bar m \bar n} \frac{\kappa^{B_n A_m} }{(\bar z - \bar w)^2}
+ R^{-2} \sum_p C_{\bar m\bar n}^p {f_{C_p}}^{B_n A_m} K_p^{C_p}\frac{z-w}{(\bar z-\bar w)^2} \cr
& + R^{-2} \sum_p C_{\bar m\bar n}^{\bar p} \frac{{f_{C_p}}^{B_n A_m} \bar K_p^{C_p}}{\bar z-\bar w} +...\end{aligned}$$ The tensors $\kappa^{AB}$ and ${f_{C}}^{BA}$ are respectively the metric and the structure constants (see appendix \[conventions\] for conventions). The non-trivial data in the current algebra is coded in the coefficients $C_{**}$, $C_{**}^{*}$. These coefficients should be read as follows: the coefficient $C_{13}$ give the coefficient of the identity operator in the OPE between the currents $J_1$ and $J_3$, the coefficient $C_{2\bar 2}^g$ give the coefficient of the ghost current $N$ in the OPE between $J_2$ and $\bar J_2$, and so on. We introduced a sum over all currents in the first-order poles in order to simplify the writing, but many of the $C$’s are clearly zero since they do not respect the $\mathbb{Z}_4$ grading. The coefficients $C_{**}$, $C_{**}^{*}$ are symmetric in their two lower indices. Also $\sum_p C_{mn}^{\bar p}$ should be understood as $C_{mn}^{\bar 1} + C_{mn}^{\bar 2} +... $. The non-zero coefficients are given below. The non-vanishing second-order poles are: C\_[13]{} = -1 ,C\_[1 |3]{} = 1 ,C\_[|1 3]{} = 1 ,C\_[|1 |3]{} = -1 ,C\_[22]{} = -1 ,C\_[2|2]{} = 1 ,C\_[|2 |2]{} = -1 The first-order poles involving only the currents of non-zero grade are:
[c]{} C\_[11]{}\^2 = 2 ,C\_[11]{}\^[|2]{} = 1 ,C\_[1 |1]{}\^2 = 1 ,C\_[|1 |1]{}\^[|2]{} = 1 C\_[33]{}\^2 = 1 ,C\_[3|3]{}\^[|2]{} = 1 ,C\_[|3 |3]{}\^2 = 1 ,C\_[|3 |3]{}\^[|2]{} = 2 C\_[12]{}\^3 = 2 ,C\_[12]{}\^[|3]{} = 1 ,C\_[|1 2]{}\^3 = 1 ,C\_[1|2]{}\^3 = 1 ,C\_[|1 |2]{}\^[|3]{} = 1 C\_[32]{}\^1 = 1 ,C\_[|3 2]{}\^[|1]{} = 1 ,C\_[3 |2]{}\^[|1]{} = 1 ,C\_[|3 |2]{}\^1 = 1 ,C\_[|3 |2]{}\^[|1]{} =2
The first-order poles involving the ghosts currents are:
[c]{} C\_[22]{}\^g = 1 ,C\_[22]{}\^[|g]{} = -1 ,C\_[2 |2]{}\^g = 1 ,C\_[2 |2]{}\^[|g]{} = 1 ,C\_[|2 |2]{}\^g = -1 ,C\_[|2 |2]{}\^[|g]{} = 1 C\_[13]{}\^g = 1 ,C\_[13]{}\^[|g]{} = -1 ,C\_[1 |3]{}\^g = 1 ,C\_[1 |3]{}\^[|g]{}= 1 ,C\_[|1 3]{}\^g = 1 ,C\_[|1 3]{}\^[|g]{} = 1 ,C\_[|1 |3]{}\^g = -1 ,C\_[|1 |3]{}\^[|g]{} = 1 C\_[gg]{}\^g = -1 ,C\_[|g |g]{}\^[|g]{} = -1
Eventually the first-order poles involving the currents $J_0$, $\bar J_0$ are:
[c]{} C\_[10]{}\^1 = 1 ,C\_[0 |1]{}\^1 = 1 ,C\_[|0 1]{}\^[|1]{} = 1 ,C\_[|0 |1]{}\^[|1]{} = 1 C\_[02]{}\^2 = 1 ,C\_[0 |2]{}\^2 = 1 ,C\_[|0 2]{}\^[|2]{} = 1 ,C\_[|0 |2]{}\^[|2]{} = 1 C\_[30]{}\^3 = 1 ,C\_[0 |3]{}\^3 = 1 ,C\_[|0 3]{}\^[|3]{} = 1 ,C\_[|0 |3]{}\^[|3]{} = 1 C\_[13]{}\^0 = 1 ,C\_[13]{}\^[|0]{} = 1 ,C\_[|1 |3]{}\^0 = 1 ,C\_[|1 |3]{}\^[|0]{} = 1 C\_[22]{}\^0 = 1 ,C\_[22]{}\^[|0]{} = 1 ,C\_[|2 |2]{}\^0 = 1 ,C\_[|2 |2]{}\^[|0]{} = 1
The method we are using to compute the current algebra does not fix completely the self-OPEs of $J_0$ and $\bar J_0$. We only obtain the following constraints: $$\begin{aligned}
\label{constraintsC00*}
C_{00}^0 = C_{0\bar 0}^0 = -C_{\bar 0 \bar 0}^0 \quad ; \quad
- C_{00}^{\bar 0} = C_{0\bar 0}^{\bar 0} = C_{\bar 0 \bar 0}^{\bar 0} \cr
C_{00}^g = C_{0\bar 0}^g = -C_{\bar 0 \bar 0}^g \quad ; \quad
- C_{00}^{\bar g} = C_{0\bar 0}^{\bar g} = C_{\bar 0 \bar 0}^{\bar g} \end{aligned}$$ It turns out that these constraints are enough to perform explicitly the computations presented in this paper[^2].
The flat connection and the $(r,s)$ system
------------------------------------------
Similarly to the Green-Schwarz string [@Metsaev:1998it][@Bena:2003wd], the pure spinor string on $AdS_5 \times S^5$ admits a one-parameter family of flat connections [@Vallilo:2003nx]. This implies that the classical theory admits an infinite number of conserved charges. The flat connection $A(y)$ is defined as: $$\begin{aligned}
\label{defA} A(y) = & (J_0 + y J_1 + y^2 J_2 + y^3 J_3 + (y^4-1) N)dz \cr
& + (\bar J_0 + y^{-3} \bar J_1 + y^{-2} \bar J_2 + y^{-1} \bar J_3 + (y^{-4}-1)\hat N) d \bar z \end{aligned}$$ The flat connection is invariant under parity combined with the exchange of $y$ and $y^{-1}$. The equations of motion together with the Maurer-Cartan equation imply that the previous connection is flat for all values of the spectral parameter $y$: d A(y) + A(y) A(y) = 0 In the following we study line operators that are the path-ordered exponential of the integral of the flat connection along a given contour. We will only consider integration contours that lie at constant time. Consequently only the spacelike component of the flat connection will appear. For simplicity, we use the same notation $A(y)$ for the connection and for its spacelike component. The advantage of the version of the current algebra we are working with is that the commutator of two equal-time space-component of the flat connection can be written as a $(r,s)$ system: $$\begin{aligned}
\label{r,sSyst}
[ A_R(y;\sigma), A_{R'}(y';\sigma')] = &2s \p_\sigma\delta^{(2)}(\sigma-\sigma') + [ A_R(y;\sigma)+ A_{R'}(y';\sigma'),r] \delta^{(2)}(\sigma-\sigma') \cr
& + [ A_R(y;\sigma)- A_{R'}(y';\sigma'),s] \delta^{(2)}(\sigma-\sigma')\end{aligned}$$ where $R$ and $R'$ denote the representations the two connections are transforming in. The commutator transforms in the tensor product $R \otimes R'$. Only the terms explicitly written down in the OPEs contribute to the commutator . The infinite number of subleading singularities contained in the ellipses of do not contribute to the commutator of equal-time currents (see e.g. [@Benichou:2010ts]). As shown in appendix \[appRSmatrices\], the constant matrices $r$ and $s$ are given by: $$\begin{aligned}
\label{rMatrix}
r = & i\pi R^{-2} \left( r_{13}t_{A_1}^R \otimes t_{B_3}^{R'} \kappa^{B_3 A_1} + r_{22}t_{A_2}^R \otimes t_{B_2}^{R'} \kappa^{B_2 A_2} + r_{31}t_{A_3}^R \otimes t_{B_1}^{R'} \kappa^{B_1 A_3} + r_{00}t_{A_0}^R \otimes t_{B_0}^{R'} \kappa^{B_0 A_0} \right) \cr \cr
%
&r_{13} = \frac{(y^2-y^{-2})^2 + (y'^2-y'^{-2})^2}{y^4-y'^4}y y'^3 \quad ; \quad
r_{22} = \frac{(y^2-y^{-2})^2 + (y'^2-y'^{-2})^2}{y^4-y'^4}y^2 y'^2 \cr\cr
&r_{31} = \frac{(y^2-y^{-2})^2 + (y'^2-y'^{-2})^2}{y^4-y'^4}y^3 y' \quad ; \quad
r_{00} = 2 \frac{(y^4-1)(y'^4-1)}{y^4-y'^4} \end{aligned}$$ and: $$\begin{aligned}
\label{sMatrix}
s = & i\pi R^{-2} \left( s_{13}t_{A_1}^R \otimes t_{B_3}^{R'} \kappa^{B_3 A_1} + s_{22}t_{A_2}^R \otimes t_{B_2}^{R'} \kappa^{B_2 A_2} + s_{31}t_{A_3}^R \otimes t_{B_1}^{R'} \kappa^{B_1 A_3} + s_{00}t_{A_0}^R \otimes t_{B_0}^{R'} \kappa^{B_0 A_0} \right) \cr\cr
%
& s_{13} = \frac{1}{y^3 y'} - y y'^3 \quad ; \quad
s_{22} = \frac{1}{y^2 y'^2} - y^2 y'^2 \quad ; \quad
s_{31} = \frac{1}{y y'^3} - y^3 y' \quad ; \quad
s_{00} = 0\end{aligned}$$ Later it will be important that the $r$ matrix simplifies in the limit where the difference between the spectral parameters $y$ and $y'$ is small: \[rLim\] y-y’ 0r \~ y(y\^2+y\^[-2]{})\^2 t\_A\^R t\_B\^[R’]{} \^[BA]{} The $(r,s)$ matrices , first appeared in [@Mikhailov:2007eg]. A detailed study of the $(r,s)$ system for string theory in $AdS_5\times S^5$ and its properties can be found in [@Magro:2008dv][@Vicedo:2009sn][@Vicedo:2010qd] (see appendix \[appCompKAlgebras\] for more details).
The line operators
------------------
#### Definitions.
We are interested in studying line operators that are the path-ordered exponential of the integral of the flat connection on a given contour. When the contour is an interval $[a,b]$, the line operator is called the transition matrix. We denote it as $T^{b,a}_R(y)$: T\^[b,a]{}\_R(y) = P ( - \_a\^b A\_R(y)) The transition matrix is labelled by the representation $R$ in which the flat connection transforms. Flatness of the connection implies that the classical transition matrix does not depend on the integration path chosen. This property has been argued to extend to the quantum theory in [@Puletti:2008ym]. For simplicity we consider only constant-time contours.
For string theory purposes we are lead to define the theory on a cylinder. Then we can define the monodromy matrix which is the line operator associated with a closed contour winding once around the cylinder: \_R(y) = P (-A\_R(y) ) Flatness of the connection implies that the eigenvalues of the monodromy matrix are independent on time. Consequently they code an infinite number of conserved charges. Eventually the transfer matrix is the supertrace of the monodromy matrix: \[defTransfer\] \_R(y) = STr P (-A\_R(y) )
#### Regularization of UV divergences.
In a quantum theory the line operators are generically ill-defined since the collisions of integrated connections lead to divergences. To properly define line operators one has to regularize these divergences, and then renormalize the line operators. In order to study the UV divergences, we first have to expand the exponentials in the line operators. We write the transition matrix as: T\^[b,a]{}\_R(y) = \_[M=0]{}\^(-1)\^M T\_[R,(M)]{}\^[b,a]{}(y) where the $M$-th term is the path-ordered integral of $M$ connections: T\_[R,(M)]{}\^[b,a]{}(y) = P(\_a\^b A\_R(y) )\^M = \_[b>\_1>...>\_M>a]{}d\_1...d\_M A\_R(y;\_1)...A\_R(y;\_M) and similarly for the monodromy and transfer matrices. Divergences occur when two integrated connections collide. It is clear from the current algebra that the collision of two connections leads to second- and first-order poles[^3]. In order to regularize these divergences, we introduce a UV cut-off $\epsilon$. We use a principal-value regularization scheme as suggested in [@Mikhailov:2007eg]. The OPE between two equal-time connections $A(\sigma)$ and $A(\sigma')$ is regularized by a small shift in time, in a symmetric way: \[defReg\] A() A(’) ( A(+i) A(’) + A() A(’+i) )For instance, a first-order pole is regularized as: \[reg1pole\] P.V. = ( + ) = This regularization scheme turns out to be very convenient to discuss the fusion of line operators, as explained in section \[fusion\].
#### Divergences at order $R^{-2}$.
The first-order divergences in line operators in the pure spinor string on $AdS_5 \times S^5$ were first studied in [@Mikhailov:2007mr]. In this paper the authors used a different regularization scheme: the OPEs were regularized by imposing that the distance between two connections cannot be smaller than the UV cut-off. The authors of [@Mikhailov:2007mr] also used a slightly different version of the current algebra. In appendix \[computationDiv\] we revisit the analysis of [@Mikhailov:2007mr] using the regularization scheme and the current algebra . The main difference we obtain with respect to [@Mikhailov:2007mr] is that the linear divergences do cancel thanks to our choice of regularization scheme. Below we summarize the results derived in appendix \[computationDiv\] .
The transition matrices contain logarithmic divergences. Schematically, these divergences read: \~\_[i=0]{}\^3 \# { t\^[A\_i]{} t\_[A\_i]{}, T\^[b,a]{}(y) } The precise expression for these divergences is given in equation . Consequently the transition matrices need to be renormalized. These divergences are cancelled by a simple wave-function renormalization.
The monodromy matrix also contains logarithmic divergences. These are given in . Schematically, these divergences read: \[divOmega\] \~\_[i=0]{}\^3 \# ( t\^[A\_i]{} t\_[A\_i]{} (y) + (y) t\^[A\_i]{} t\_[A\_i]{} - 2 t\^[A\_i]{} (y) t\_[A\_i]{} ) The new divergences with respect to the transition matrices come from collisions between connections sitting on both sides of the starting point of the integration contour. These divergences are cancelled by a simple wave-function renormalization of the monodromy matrix.
The most important result for the purpose of this paper is that the transfer matrix is completely free of divergences at order $R^{-2}$. This follows simply by taking the supertrace of equation . This remarkable property strongly relies on the vanishing of the dual Coxeter number of the global symmetry group $PSU(2,2|4)$. For instance in generic WZW models, cancellation of divergences in the transfer matrices require both a wave-function renormalization and a renormalization of the spectral parameter [@Bachas:2004sy]. The divergences identified in [@Bachas:2004sy] also vanish if the dual Coxeter number of the group is zero.
Fusion of line operators {#fusion}
========================
In this section we study the fusion of two line operators. The fusion is the process of bringing the integration contours of two line operators on top of each other. We are interested in the quantum effects that occur in this process. The fusion of line operators for the pure-spinor string in $AdS_5 \times S^5$ was studied at first-order in perturbation theory in [@Mikhailov:2007eg]. In this section we will revisit and extend the first-order computations of [@Mikhailov:2007eg]. Then we will further extend the computation of fusion up to second order in perturbation theory.
The structure of the computations is similar to the ones presented in [@Benichou:2010ts], where more details can be found. In [@Benichou:2010ts] the computations were performed in the sigma model on the supergroup $PSl(n|n)$. This theory is a good toy model for the pure spinor string on $AdS_5 \times S^5$. Indeed the complications coming from the coset structure and the pure spinor ghosts are absent.
Setting up the computation {#recipeFusion}
--------------------------
Let us consider two transition matrices $T^{b,a}_R(y)$ and $T^{d,c}_{R'}(y')$ that transform respectively in the representations $R$ and $R'$. The fusion of these two matrices transforms in the tensor product $R\otimes R'$. In the following we will omit the symbol $\otimes$ to lighten the formulas. We represent the fusion of these two transition matrices with the symbol $\F$. This process is defined as: \[defFusion\] T\^[b,a]{}\_R(y)T\^[d,c]{}\_[R’]{}(y’) = \_[0\^+]{} T\^[b+i,a+i]{}\_R(y)T\^[d,c]{}\_[R’]{}(y’) Assuming the integration contour of the transition matrices $T^{d,c}_{R'}(y')$ lies at constant time $\tau$, then the integration contour of $T^{b+i\epsilon,a+i\epsilon}_R(y)$ lies at constant time $\tau + \epsilon$. If the intervals $[a,b]$ and $[c,d]$ do not overlap, the process of fusion is trivial. In the following we assume that the overlap of these intervals is non-zero. As the distance between the two contours goes to zero, the OPEs between integrated connections sitting on the two contours produce quantum corrections to the classical process of fusion. These are the corrections we will evaluate.
Let us consider the OPE between two connections $A_R(y;\sigma+i\epsilon)$ and $A_{R'}(y';\sigma')$ integrated respectively on the first and on the second contour. We write this OPE as: $$\begin{aligned}
\label{splitAA'}
A_R(y;\sigma+i\epsilon)A_{R'}(y';\sigma') = &\frac{1}{2}\left( A_R(y;\sigma+i\epsilon)A_{R'}(y';\sigma') +A_R(y;\sigma)A_{R'}(y';\sigma'+i\epsilon) \right) \cr &+ \frac{1}{2}\left( A_R(y;\sigma+i\epsilon)A_{R'}(y';\sigma') -A_R(y;\sigma)A_{R'}(y';\sigma'+i\epsilon) \right) \end{aligned}$$ Comparing with equation , we notice that the first term in the previous equation should be understood as regularized OPE in the quantum line operator obtained after the fusion has been completed. On the other hand, the second term in should be understood as producing a quantum correction proper to the process of fusion. This are the corrections we want to compute.
In order to understand better the meaning of , let us isolate a first-order pole in the OPE between the two connections. Under the decomposition , it is rewritten as: $$\begin{aligned}
\label{splitPole} \frac{1}{\sigma+i \epsilon - \sigma'}
& = \frac{1}{2} \left( \frac{1}{\sigma+i \epsilon - \sigma'} + \frac{1}{\sigma-i \epsilon - \sigma'} \right)
+ \frac{1}{2} \left( \frac{1}{\sigma+i \epsilon - \sigma'} - \frac{1}{\sigma-i \epsilon - \sigma'} \right) \cr
& = P.V.\ \frac{1}{\sigma - \sigma'} - i \pi \delta_\epsilon(\sigma-\sigma')\end{aligned}$$ The term we focus on is the second term on the right-hand side. As the notation suggests, it is actually a regularization of the delta-function. Once we integrate upon the free coordinates, it produces a finite quantum corrections to the fusion of line operators. We can perform a similar manipulation for all first- and second-order poles appearing in the OPE between the two connections $A_R(y;\sigma+i\epsilon)$ and $A_{R'}(y';\sigma')$. In order to isolate the contribution to the quantum corrections associated with fusion, we subtract the principal value from the singularities. We obtain that all the terms that contribute to the quantum corrections from fusion come with (derivatives of) regularized delta functions.
The upshot is the following: in order to compute the quantum corrections in the process of fusion, we have to subtract the “principal value” piece from the OPE between connections. What we are left with is essentially the commutator between connections that we computed in : \[1-PVAA’\] (1-P.V.) A\_R(y;+i) A\_[R’]{}(y’;’) \[A\_[R]{}(y;), A\_[R’]{}(y’;’) \]
Fusion at first-order
---------------------
We begin with the corrections of order $R^{-2}$. Since all terms in the current algebra are of order $R^{-2}$, it is enough to perform one OPE. The computation of the first-order corrections in the fusion of two transition matrices was performed in [@Benichou:2010ts] using OPE techniques. This computation holds provided the commutator of connections can be written as a $(r,s)$ system, which is the case for the pure spinor string on $AdS_5\times S^5$ (see ). The result obtained in [@Benichou:2010ts] matches the hamiltonian analysis of [@Maillet:1985ek]: $$\begin{aligned}
\label{fusionOrder1}
& T^{b,a}_R(y)\F T^{d,c}_{R'}(y')
= T^{b,a}_R(y)T^{d,c}_{R'}(y') \cr
&+ \chi(b;c,d) T^{d,b}_{R'}(y') \frac{r+s}{2} T^{b,a}_R(y)T^{b,c}_{R'}(y') - \chi(a;c,d) T^{b,a}_R(y) T^{d,a}_{R'}(y') \frac{r+s}{2} T^{b,c}_{R'}(y') \cr
& + \chi(d;a,b) T^{b,d}_R(y) \frac{r-s}{2} T^{d,a}_R(y) T^{d,c}_{R'}(y') - \chi(c;a,b) T^{b,c}_R(y) T^{d,c}_{R'}(y') \frac{r-s}{2} T^{c,a}_R(y) \cr
& + \mathcal{O}(R^{-4})\end{aligned}$$ The first term on the right-hand side is the zeroth-order result. The remaining terms are the first-order corrections. The function $\chi(a;b,c)$ is the characteristic function of the interval $[b,c]$ which takes the value $1$ if $b>a>c$ and $0$ if $a>b$ or $a<c$. For the special case where the integration intervals of the line operators have coinciding endpoints, that is for $a=b$ or $a=c$, then the characteristic function $\chi(a;b,c)$ has to be evaluated as $\frac{1}{2}$. This prescription essentially had to be guessed in the hamiltonian formalism [@Maillet:1985fn]. In the OPE formalism it is a consequence of the definition [@Benichou:2010ts].
Notice that the first-order corrections in are anti-symmetric in the exchange of the two line operators. So they contribute only to the commutator of the line operators. This follows from the fact that the quantum corrections associated with fusion come from the anti-symmetric part in the OPEs (see ).
From equation we can deduce the fusion of transfer matrices [@Benichou:2010ts]. We obtain that the fusion of transfer matrices at first order is trivial: $$\begin{aligned}
\T_R(y) \F \T_{R'}(y') = \T_R(y) \T_{R'}(y') + \mathcal{O}(R^{-4})\end{aligned}$$ Indeed the first-order corrections associated with fusion in take the form of constant matrices inserted at the endpoints of the overlap of the integration intervals. Since the transfer matrices have no endpoints, it is not surprising that these corrections vanish. This implies in particular that the commutator of the transfer matrices is zero at first order.
Fusion at second-order {#subFusion2nd}
----------------------
Remember that our main goal is to show that the leading quantum correction in the fusion of two transfer matrices gives the shifts in the T-system at first order. Since the fusion of transfer matrices is trivial at first-order, we need to study the fusion of line operators at second-order.
There are two different ways we can obtain $R^{-4}$ corrections in the process of fusion:
- The first way is to take one single OPE between two integrated connections, and include $R^{-4}$ corrections to the current algebra . As argued in section \[recipeFusion\], only the anti-symmetric part of the current-current OPEs contributes to the process of fusion. Consequently at this order the $R^{-4}$ corrections to the current algebra lead to $R^{-4}$ corrections to the commutator of line operators. We postpone the computation of these corrections for future work since the current algebra is only partially known at order $R^{-4}$ [@Bedoya:2010av].
- The second way is to perform two OPEs between integrated connections and use the current algebra at order $R^{-2}$. Following the logic explained in section \[recipeFusion\], we again consider only the anti-symmetric part in each OPE. Since we perform an even number of OPEs this time the result will be symmetric under the exchange of the two line operators. More precisely we obtain a contribution to the symmetric product of the line operators of order $R^{-4}$. This are the terms that we will compute in this paper.
More generally, the arguments of section \[recipeFusion\] imply that any quantum correction in the process of fusion that involve an even (respectively odd) number of OPEs would contribute only to the symmetric product (respectively commutator) of the line operators. Consequently the $R^{-4}$ corrections to the current algebra would contribute to the symmetric product of line operators at order $R^{-6}$ and higher.
Below we describe the different steps in the computation of the symmetric fusion of line operators at order $R^{-4}$. In an attempt to keep this section readable, the technical details of the computation have been gathered in appendix \[comFusion\]. More details can also be found in [@Benichou:2010ts].
Let us consider two line operators with contours separated in time by a small distance $\epsilon$. We have to take two OPEs between the connections integrated on the contours. The simplest way is to take two OPEs between two distinct pairs of connections. But we can also take one OPE between two connections, and then take the OPE of the resulting currents with a third connections. We will call this latter process a triple collision (see figure \[tripleCollision\]).
![A triple collision. The first OPE is taken between two connections sitting on different contours. The second OPE is taken between a third connection and the currents resulting from the first OPE.\[tripleCollision\]](tripleCollision)
The computation is conveniently decomposed in two steps.
- A first part of the total answer is obtained in the way depicted in Figure \[fusO2step1\]. We start from the result of the fusion at first order. We pull the contours away, and then re-fuse the line operators. The result of this procedure was computed in [@Benichou:2010ts] for an arbitrary $(r,s)$ system. We obtain new insertion of constant matrices at the endpoints of the overlap of the contours of the line operators. Roughly speaking, the first-order result exponentiate[^4]. What is important for our purposes is that this procedure gives once again a vanishing result for the fusion of transfer matrices. This simply follows from the fact that there is no first-order correction in the fusion of transfer matrices.
- The procedure described above does not capture correctly the quantum corrections coming from triple collisions. Indeed in this procedure, the intermediate currents in the triple collisions are distributed in an arbitrary way on the two integration contours so that they recombine into connections. This induces a source of errors. So we have to compute separately additional corrections coming from the triple collisions. This is done in appendix \[comFusion\].
![We can compute one piece of the result for fusion at second order in the following way. In step we compute the first-order corrections from fusion. In step we separate the contours again. Finally in step we perform a second time the fusion at first order. This computation does not give the full result since the triple collisions are not properly accounted for. For transfer matrices the fusion at first order is trivial, so the quantum corrections obtained in this way actually vanish. \[fusO2step1\]](fusion2times1)
The analysis of appendix \[comFusion\] shows that there are two types of corrections that we need to add on top of the result obtained by the procedure described in figure \[fusO2step1\]. The first type of corrective terms contain the integration of an operator $\tilde K$ on the overlap of the contours: $$\begin{aligned}
\label{FusionTT1}
& R^{-4} \sum_{M,M'=0}^\infty (-)^{M+M'} \sum_{i=0}^M \sum_{i'=0}^{M'} \int_{[a,b]\cap[c,d]}d\sigma
\left \lfloor \int_\sigma^b A \right\rceil^i \left \lfloor \int_\sigma^d A' \right\rceil^{i'} \tilde K (\sigma) \left \lfloor \int_a^\sigma A \right\rceil^{M-i} \left \lfloor \int_c^\sigma A' \right\rceil^{M'-i'} \end{aligned}$$ where we introduced the convenient notation $\lfloor \int_a^b A \rceil^M$ to describe the path-ordered integral of $M$ connections on the interval $[a,b]$. The precise expression for the operator $\tilde K$ is explicitly given in . Schematically, the operator $\tilde K$ is a linear combination of the currents multiplied by three generators of the Lie superalgebra in the representations $R$ or $R'$, and contracted with structure constants. The second type of corrective terms contain a constant matrix $\tilde{tt}$ inserted in between the integrated connections on the overlap of the integration contours: $$\begin{aligned}
\label{FusionTT2}
& R^{-4} \sum_{M,M'=0}^\infty (-)^{M+M'} \sum_{i=0}^M \sum_{i'=0}^{M'} \int_a^b d\sigma \int_c^d d \sigma'
\left \lfloor \int_\sigma^b A \right\rceil^i \left \lfloor \int_\sigma^d A' \right\rceil^{i'} \cr
& \quad \times \delta^2_{\epsilon}(\sigma-\sigma') \tilde{tt} \left \lfloor \int_a^\sigma A \right\rceil^{M-i} \left \lfloor \int_c^\sigma A' \right\rceil^{M'-i'} \end{aligned}$$ The precise expression for the matrix $\tilde{tt}$ is given in . Schematically, it is a linear combination of the tensor product of two generators taken in the representation $R$ and $R'$ and contracted with structure constants. Notice that the integration over the regularized delta function squared produce a linear divergence when the UV regulator $\epsilon$ is sent to zero. It would be interesting to perform a complete analysis of the second-order divergences in the line operators along the lines of section 3 in [@Benichou:2010ts], to see whether the transfer matrices as well as the result of the fusion of transfer matrices are free of divergences up to second-order.
#### Fusion of transfer matrices at second order: upshot.
The detailed expression for the symmetric fusion of transition and monodromy matrices at second order is quite indigestible, so we refrain from giving an explicit formula for those. On the other hand the symmetric fusion of transfer matrices, that is crucial for the purposes of this paper, turns out to be rather simple: $$\begin{aligned}
\label{summaryFusionTT}
&\T_R(y)\ \SF \ \T_{R'}(y') = \frac{1}{2}\{ \T_R(y), \T_{R'}(y')\} \cr
& + R^{-4} STr \left( \int_0^{2\pi} d\sigma \
T_R^{2\pi, \sigma}(y) T_{R'}^{2\pi, \sigma}(y') \ \tilde K (\sigma) \ T_R^{ \sigma,0}(y) T_{R'}^{\sigma,0}(y') \right)\cr
& + R^{-4} STr \left( \int_0^{2\pi} d\sigma \int_0^{2\pi} d\sigma' \
T_R^{2\pi, \sigma}(y) T_{R'}^{2\pi, \sigma}(y') \delta^2_{\epsilon}(\sigma-\sigma')\ \tilde{tt} \ T_R^{ \sigma,0}(y) T_{R'}^{\sigma,0}(y') \right)\cr
&+ \mathcal{O}(R^{-6})\end{aligned}$$ where we denoted by $\SF$ the symmetrized fusion product. This result is schematically represented in figure \[fusionTs\]. The detailed expressions for the operator $\tilde K$ and the constant matrix $\tilde{tt}$ can be read from equations and .
![Schematic representation of the symmetric fusion of transfer matrices at second order . The first term is the classical result. In the second term, an additional operator $\tilde{K}$ is integrated in between the connections. In the third term, a constant matrix $\tilde{tt}$ is inserted in between the integrated connections.\[fusionTs\]](fusionTs)
The $AdS_5$/$CFT_4$ T-system {#proofTsys}
============================
In this section we use the previous computations to obtain a first-principle perturbative derivation of the T-system. As explained in the introduction, the idea is to promote the T-system to an operator identity, where the product between transfer matrices is understood as the fusion product: \[QTsystem\] \_[a,s]{}(u + 1) \_[a,s]{}(u - 1) = \_[a+1,s]{}(u+1)\_[a-1,s]{}(u-1) + \_[a,s-1]{}(u+1) \_[a,s+1]{}(u-1) We expect the transfer matrices to commute in the quantum theory. This has been proven in section \[fusion\] at order $R^{-2}$. Consequently we can equivalently use the symmetric fusion product to define the T-system .
The integer label $a,s$ label unitary irreducible representation of $PSU(2,2|4)$. These labels take value in a T-shaped lattice (see e.g. [@Gromov:2010kf][@Volin:2010xz]). The corresponding representations are associated with rectangular Young tableaux which size is given by the value of the labels $a,s$. It is known that these representations satisfy the following supercharacter identity (see e.g. [@Gromov:2010vb][@Kazakov:2007na]): \[charId1\] (a,s)\^2 = (a+1,s)(a-1,s)+(a,s+1)(a,s-1) In the limit where we neglect both the shifts of the spectral parameter as well as the quantum effects associated with fusion, the T-system reduces to the character identity .
Next we want to show that the T-system holds at first order. We consider: $$\begin{aligned}
\label{Tsys1}
0 \stackrel{?}{=} & \mathcal{T}_{a,s}(y+\delta) \F \mathcal{T}_{a,s}(y-\delta) -
\mathcal{T}_{a+1,s}(y+\delta)\F \mathcal{T}_{a-1,s}(y-\delta) - \mathcal{T}_{a,s-1}(y+\delta) \F \mathcal{T}_{a,s+1}(y-\delta) \cr & \equiv \sum_{R,R'} \mathcal{T}_{R}(y+\delta) \F \mathcal{T}_{R'}(y-\delta) \end{aligned}$$ where we use the shorthand $\sum_{R,R'}$ to denote the sum over representations that appears in the T-system. We look for a value of the shift of the spectral parameters $\delta$ such that the previous quantity does indeed vanish, and the T-system holds. Then we can deduce the relationship between the spectral parameter $y$ used to define the flat connection , and the spectral parameter $u$ that appears in the T-system .
We assume that $\delta$ is of order $R^{-2}$. We will now show that the terms of order $R^{-2}$ in do vanish. More precisely, the terms of order $R^{-2}$ coming from the derivative expansion of the transfer matrices cancel against the leading quantum correction coming from the process of fusion.
#### Fusion of transfer matrices when the difference of spectral parameter is small.
In section \[fusion\] we obtained that the leading quantum correction in the fusion of transfer matrices is of order $R^{-4}$. However when the difference of the spectral parameter is of order $R^{-2}$, a piece of the leading quantum correction actually becomes of order $R^{-2}$. So it has the right order of magnitude to cancel the first term in the derivative expansion of the transfer matrices in . More details can be found at the end of appendix \[comFusion\].
The interesting term comes from the second line in . We assume $y-y' = \mathcal{O}(R^{-2})$. From equations we obtain that simplifies to: $$\begin{aligned}
\label{fusTTSmallDif}
&\T_R(y) \F \T_{R'}(y') = \T_R(y) \T_{R'}(y') \cr
& + R^{-4} \frac{\pi^2 }{32} \frac{y^2(y^2-y^{-2})^4}{y-y'}
STr \left( \int_0^{2\pi} d\sigma
T_R^{2\pi, \sigma}(y) T_{R'}^{2\pi, \sigma}(y')
\right. \cr &
\quad \times \left( -\sum_{m,n,p,q,r=0}^3\p_y A^{E_r}(y)
\left({f_{C_p}}^{B_n A_m} {f_{E_r}}^{C_p D_q} \{ t_{D_q}^R , t_{A_m}^R] t_{B_n}^{R'} + {f_{C_p}}^{B_n A_m} {f_{E_r}}^{ D_q C_p} t_{A_m}^R \{ t_{B_n}^{R'}, t_{D_q}^{R'}]\right)\right)\cr
&\quad \times \left. T_R^{ \sigma,0}(y) T_{R'}^{\sigma,0}(y') \right) + \mathcal{O}(R^{-4})\end{aligned}$$ We observe that the linear combination of the currents that appears is now proportional to the derivative of the flat connection.
#### Useful character identities.
To further simplify the quantum corrections that appear in , we need to use some character identities that holds for the particular combination of representations that appear in the T-system. In [@Kazakov:2007na] the validity of the T-system was proven for transfer matrices associated to $Gl(k|m)$ spin chains. Then a large family of character identities was deduced by expanding the T-system as an infinite series. Here we go the other way: we try to reconstruct the T-system from a perturbative expansion. Thus it makes sense that we need to use the character identities of [@Kazakov:2007na]. In the appendix E of [@Benichou:2010ts], it was shown that some of these character identities imply in particular that for any group element $g$ and for any function $K^E$: $$\begin{aligned}
\label{charIdKV1}
\sum_{R,R'} K^E {f_C}^{BA} {f_E}^{CD} & STr(\{ t_D^R, t_A^R] g^R \otimes t_B^{R'} g^{R'}) \cr
& = 2 \sum_{R,R'} STr( g^R \otimes K^E t_E^{R'} g^{R'}) - STr( K^E t_E^{R} g^R \otimes g^{R'}) \end{aligned}$$ and similarly: $$\begin{aligned}
\label{charIdKV2}
\sum_{R,R'} K^E {f_C}^{BA} {f_E}^{DC} & STr(t_A^R g^R \otimes \{ t_B^{R'},t_D] g^{R'}) \cr
& = 2 \sum_{R,R'} STr( g^R \otimes K^E t_E^{R'} g^{R'}) - STr( K^E t_E^{R} g^R \otimes g^{R'}) \end{aligned}$$ Essentially, these character identities allow to replace the complicated combination of structure constants and generators appearing in by single generators, assuming we consider the sum of representation that appear in the T-system. So we obtain: $$\begin{aligned}
\label{Tsys2}
\sum_{R,R'} & \T_R(y+\delta) \F \T_{R'}(y-\delta) = \sum_{R,R'} \T_R(y+\delta) \T_{R'}(y-\delta) %\cr &
+ \sum_{R,R'} R^{-4}\frac{\pi^2 }{16} \frac{y^2(y^2-y^{-2})^4}{\delta} \cr
& \times STr \left( \int_0^{2\pi} d\sigma
T_R^{2\pi, \sigma}(y) T_{R'}^{2\pi, \sigma}(y) \left( - \p_y A^{E}(y;\sigma) (-t_E^R + t_E^{R'}) \right)
T_R^{ \sigma,0}(y) T_{R'}^{\sigma,0}(y) \right)
+ \mathcal{O}(R^{-4})\cr
%
=& \sum_{R,R'} \T_R(y+\delta) \T_{R'}(y-\delta)
- R^{-4} \frac{\pi^2 }{16} \frac{y^2(y^2-y^{-2})^4}{\delta} \left( \p_y \T_R(y) \T_{R'}(y) - \T_R(y) \p_y \T_{R'}(y) \right) \cr
&\quad + \mathcal{O}(R^{-4})\end{aligned}$$
#### The T-system at first order.
Performing a Taylor expansion for the $\T$’s in , we deduce: $$\begin{aligned}
\label{Tsys3}
\sum_{R,R'} & \T_R(y+\delta) \F \T_{R'}(y-\delta) =
\sum_{R,R'} \T_R(y) \T_{R'}(y) \cr
& +\left(\delta - R^{-4} \frac{\pi^2 }{16} \frac{y^2(y^2-y^{-2})^4}{\delta} \right) \left( \p_y \T_R(y) \T_{R'}(y) - \T_R(y) \p_y \T_{R'}(y) \right) + \mathcal{O}(R^{-4})\end{aligned}$$ The first term in the previous expression vanishes because of the classical identity . In order for the first-order corrections to vanish as well, we have to take: \[defDelta\] = R\^[-2]{} y(y\^2-y\^[-2]{})\^2 Thus have have shown that: \[Tsys4\] \_[R,R’]{} \_R(y+) \_[R’]{}(y-) = 0 + (R\^[-4]{})
#### Redefinition of the spectral parameter.
In order to write the T-system in the canonical form we define: \[defU\] u = + cst such that $u(y\pm \delta)=u(y) \pm 1$. Then equation is rewritten as: \_[a,s]{}(u + 1) \_[a,s]{}(u - 1) = \_[a+1,s]{}(u+1)\_[a-1,s]{}(u-1) + \_[a,s-1]{}(u+1) \_[a,s+1]{}(u-1) + ... This is the canonical form of the T-system .
#### Comparison with the T-system obtained via the Thermodynamic Bethe Ansatz.
We can now perform a consistency check with the Thermodynamic Bethe Ansatz derivation of the $AdS_5\times S^5$ T-system [@Gromov:2009bc][@Bombardelli:2009ns][@Arutyunov:2009ur]. In this context the T-system is obtained in a slightly different form: only the T-functions on the left-hand side have a shifted spectral parameter: \[TsystemTBA\] \^[TBA]{}\_[a,s]{}(u + 1) \^[TBA]{}\_[a,s]{}(u - 1) = \^[TBA]{}\_[a+1,s]{}(u) \^[TBA]{}\_[a-1,s]{}(u) + \^[TBA]{}\_[a,s-1]{}(u) \^[TBA]{}\_[a,s+1]{}(u) This mismatch is easily cured by a redefinition of the T-functions. Let us define the functions $\T^{TBA}(u)$ as: \^[TBA]{}\_[a,s]{}(u) = \_[a,s]{}(u+a-s)Then the functions $\T^{TBA}$’s satisfy if and only if the functions $\T$’s satisfy the T-system .
However the previous redefinition does not change the magnitude of the shift on the left-hand side of the T-system. Thus the matching of the shifts gives a quantitative check of the consistency between the approach taken in this paper, and the TBA approach[^5]. Next we perform this matching using the conventions of [@Gromov:2009tv][@Gromov:2009bc]. The flat connection is written in terms of a spectral parameter $x$ as $A(x) = J_0 dz + (x-1)/(x+1)J_2 dz +...$. Comparing with we deduce that the spectral parameter $y$ that we use is related to $x$ as: $y^2=(x-1)/(x+1)$. The variable $u^{TBA}$ that enters the TBA T-system is linked to the spectral parameter $x$ via the Zhukowsky map: $u^{TBA}/g=x+1/x$, where $g$ is related to the ’t Hooft coupling $\lambda$ as $g=\sqrt{\lambda}/4\pi$. The parameter $R$ can be linked to the ’t Hooft coupling $\lambda$ by identification of the prefactor of the worldsheet action. This gives $\sqrt{\lambda}/2\pi = R^2/4\pi$. Consequently the parameter $u$ that we obtained is related to the parameter $u^{TBA}$ as: $ u = 2 u^{TBA}$, assuming the free constant in takes the value $R^2/2\pi$. The analysis of [@Gromov:2009bc] gives a T-system where the parameter $u^{TBA}$ is shifted by $\pm i/2$. Given that $u$ is shifted in our case by $\pm 1$, there is an apparent mismatch by a factor of $i$. This comes from the fact that we have been working on an euclidean worldsheet. If we Wick-rotate the worldsheet to a minkowskian signature, then our analysis produce the T-system with imaginary shifts[^6] in perfect agreement with the TBA analysis.
#### Upshot.
As claimed previously, we have derived the T-system up to first order in the large radius expansion. More precisely, we have sown that the shifts of the spectral parameter in the T-system come from quantum effects in the fusion of transfer matrices. Moreover we have checked that the shifts are the same than the ones obtained in the Thermodynamic Bethe Ansatz derivation of the T-system.
Notice that the vanishing of the divergences in the transfer matrices is important. Indeed if the transfer matrices would need to be renormalized, then the renormalization factor would most likely depend on the representation in which the transfer matrix is taken (see e.g. [@Bachas:2004sy]). It implies that the different terms in the T-system would be renormalized differently, which would most likely destroy the balance needed for the previous computation to work.
Generalization to other integrable theories {#extensions}
===========================================
In this paper we have proven that the T-system is realized in the pure spinor string on $AdS_5 \times S^5$ up to first order in the large radius expansion. In [@Benichou:2010ts] a similar proof was given for the non-linear sigma model on the supergroup $PSl(n|n)$. It is natural to look for other theories where this derivation can be easily generalized. A close look at the computation leads to the following conclusion: there are only a few necessary and sufficient conditions that a given model has to fulfill in order for the derivation to apply. Obviously the theory has to exhibit a one-parameter family of flat connections. We assume that the connection takes value in a Lie algebra. The other conditions are the following:
- The equal-time commutator of the spacelike component of the connection can be written as a $(r,s)$ system. This guarantees that formula can be directly reproduced. Moreover the $r$ matrix must satisfy a property similar to equation : in the limit where the difference between the spectral parameter is small, the $r$ matrix needs to be proportional to the Casimir $\kappa^{BA} t_A \otimes t_B$. This condition is necessary for the simplification leading to equation to occur.
- The Lie group needs to possess an equivalent of the character identities , . Given the role played by the results of [@Kazakov:2007na], it is tempting to speculate that a sufficient condition is that there exists a spin chain with the same symmetry group that realizes the T-system.
- Eventually the transfer matrix has to be free of divergences at first order in perturbation theory[^7]. A crucial condition here is that the dual Coxeter number of the symmetry group vanishes. This condition prevents the renormalization of the transfer matrices from destroying the balance needed for the computation to work.
Next we discuss several candidate theories that may fulfill these requirements.
#### Candidate theories relevant for the AdS/CFT correspondence.
Let us begin with the theories that describe superstrings in Anti-de Sitter backgrounds. These theories are built on sigma models on (coset of) supergroups, see e.g. [@Zarembo:2010sg] for a classification of the relevant $\mathbb{Z}_4$ cosets. Notice that all the supergroups involved have a vanishing dual Coxeter number. This should not come as a surprise, since the vanishing of the dual Coxeter number is tightly related to the vanishing of the spacetime supercurvature, and thus to the fact that the equations of motion of supergravity are satisfied.
Spacetime covariance was helpful in the previous analysis. Consequently we will mostly discuss theories of the pure-spinor type that allow for a covariant quantization. Obviously it would be interesting to reproduce the previous computations in Green-Schwarz-like theories that realize kappa-symmetry. The structure of the computations would be identical, but the computations themselves would be more tedious because the gauge-fixing of kappa symmetry usually comes with a breaking of the target space isometries.
The first obvious candidate is string theory on $AdS_4 \times CP^3$. Indeed in [@Gromov:2009tv] a Y-system was conjectured to hold in this theory. The TBA derivation of the Y-system was performed in [@Bombardelli:2009xz][@Gromov:2009at]. In order to actually reproduce the computations described in the present paper, the pure spinor formulation of superstring theory in $AdS_4 \times CP^3$ developed in [@Fre:2008qc] is a natural starting point (see also [@Stefanski:2008ik]).
The second candidate is string theory on $AdS_3 \times S^3$. The analysis of [@Benichou:2010ts] applies to the sigma-model on $PSU(1,1|2)$. In [@Berkovits:1999im] the hybrid formalism was developed to describe superstrings in $AdS_3 \times S^3$ in a superspace with eight supercharges. In this formalism the worldsheet theory is the sigma model on $PSU(1,1|2)$, coupled to ghosts. This theory admits a consistent expansion in the ghosts. Thus the analysis of [@Benichou:2010ts] implies that the hybrid string in $AdS_3 \times S^3$ realizes the T-system up to first order in the large radius expansion, and up to zeroth order in the ghosts expansion. This is valid for $AdS_3 \times S^3$ supported by RR fluxes, NS fluxes or by any mixing of these fluxes. It would be instructive to dress up the computation of [@Benichou:2010ts] with the hybrid ghosts. There has been some interest in the question of integrability for string theory in $AdS_3 \times S^3$, see e.g. [@Pakman:2009mi]. However the progress have been rather slower than in the case of $AdS_5 \times S^5$, mostly because the dual Conformal Field Theory is not as well understood. Presumably the approach presented in [@Benichou:2010ts] and in the present article can lead to a faster road to the solution of this problem.
Other formulations of string theory on $AdS_3 \times S^3$ involve a $\mathbb{Z}_4$ coset of the supergroup $PSU(1,1|2)\times PSU(1,1|2)$. It is the case of the hybrid string with sixteen manifest supercharges [@Berkovits:1999du]. It is reasonable to expect that the T-system is also realized in this formalism.
Another natural candidate is the hybrid description of superstrings in $AdS_2 \times S^2$ discussed in [@Berkovits:1999zq]. It is also based on a $\mathbb{Z}_4$ coset of the supergroup $PSU(1,1|2)$, and it is integrable [@Young:2005jv][@Adam:2007ws].
In the classification of [@Zarembo:2010sg] we find other candidate string backgrounds that are $\mathbb{Z}_4$ coset of supergroups with vanishing dual Coxeter number: $AdS_3\times S^3 \times S^3$, $AdS_2\times S^2 \times S^2$, $AdS_2 \times S^3$ and $AdS_2$. Quantum integrability is likely to show up at least in some of these examples. No formalism has been proposed to covariantly quantize string theory in these backgrounds yet.
#### Other candidates.
Other theories that may not be directly relevant for string theory presumably also realize the T-system in the way described in this paper. These are the sigma models on (cosets of) supergroups with vanishing dual Coxeter number, some of which play a role in condensed matter (see e.g. [@PS][@Efetov:1983xg][@Zirnbauer:1999ua]).
A first example is the sigma model on the supergroup $OSp(2n+2|2n)$. This model shares many of the remarkable properties of the sigma model on $PSl(n|n)$ [@Bershadsky:1999hk], see e.g. [@Babichenko:2006uc].
Next $\mathbb{Z}_2$ cosets of supergroups with vanishing dual Coxeter number are classically integrable [@Bena:2003wd][@Babichenko:2006uc]. Some of them also display nice quantum features [@Candu:2010yg]. In these model there is a current that is both flat and conserved. Consequently the current algebra is very similar to the one found in sigma models on supergroups [@Ashok:2009xx]. This follows from the generic method introduced in [@Benichou:2010rk] and used in appendix \[covCurrents\] to compute the current algebra.
Classical integrability extends to $\mathbb{Z}_4$ and more generally to $\mathbb{Z}_m$ cosets [@Young:2005jv]. It would be interesting to understand these models better. In particular it may shed some new light on the question of the role of the pure spinor ghosts for quantum integrability of the pure spinor string in $AdS_5 \times S^5$.
Conclusion
==========
#### Summary of the results.
We have studied the fusion of line operators in the pure spinor string on $AdS_5 \times S^5$ up to second order in perturbation theory. We deduced that the pure spinor string on $AdS_5 \times S^5$ realizes the T-system as an operator identity, with the fusion product, up to first order in the large ’t Hooft coupling expansion. The quantum effects in the fusion of the transfer matrices give the shifts in the T-system.
#### Comparison with the Thermodynamic Bethe Ansatz.
The T-system was previously derived using the TBA machinery [@Gromov:2009bc][@Bombardelli:2009ns][@Arutyunov:2009ur]. Here we will compare the advantages of both approaches.
A weakness of the TBA is that it relies on several assumptions that are notoriously difficult to check. In particular one has to assume quantum integrability to start with. Moreover the spectrum of excitations that contribute in the thermodynamic limit essentially has to be guessed through the string hypothesis. The approach of the current article has the big advantage of starting from first principles.
The other drawback of the TBA is that the derivation of the T-system only applies to the ground state. The fact that the same set of equations also codes the spectrum of excited states upon analytic continuation is essentially an empirical observation. In this paper we have derived the T-system as an operator identity. Thus there is no doubt that all states of the theory satisfy the T-system.
On the other hand, the approach we are using here is intrinsically perturbative. The computations needed to derive the T-system at first order were already quite heavy. It would take a lot of efforts to go to the next order. The TBA approach is free of this limitation since it produces the full T-system in one go.
There are also some by-products of the TBA approach that were not reproduced in the present work. In particular the TBA gives a explicit formula to extract the spectrum from the T-functions. It also gives some informations about the analytic properties of these functions. It would be interesting to investigate these questions with the elementary techniques used in the present paper. We hope to come back to these questions in future work.
The computation presented in this paper gives a very strong argument in favor of the validity of the T-system. It is not a definite proof since it is perturbative. However the previous discussion shows that the approach presented here is complementary with the TBA analysis. Indeed the weak points of the TBA are the the strong points in our approach, and vice-versa. So the combination of both methods leaves little room for doubts.
Moreover this works sheds a new light on the T-system. The fact that it should be understood as an operator identity where the product is the fusion of line operators, may be helpful to understand better the integrable structures that appear in the AdS/CFT correspondence.
#### The role of the pure spinor ghosts.
The pure spinor ghosts are expected to play an important role in the quantum worldsheet theory. We can ask the question of the role of the pure spinor ghosts in the computation described in this paper. Interestingly, the same results would be obtained if we would set the pure spinor ghosts to zero from the start. The reason is that at tree level, the ghosts form a closed subsector. More precisely the OPE of a ghost current with any other current can only produce ghost current. In other words, all the coefficients of the type $C_{g*}^m$ and $C_{\bar g*}^m$ in the current algebra are zero if the index $m$ is not $g$ or $\bar g$.
The fact that the computation does not relies on the pure spinor ghosts can be tracked back to the fact that we only needed the tree-level current algebra to compute the crucial term that produces the shifts in the T-system. Presumably this is not going to be the case at higher order. Continuing the computation of [@Bedoya:2010av] to get the full current algebra at second order would be interesting. Already the pure spinor should play an important role. Hopefully they allow for the cancellation of second-order divergences in the transfer matrices, and they also insure that transfer matrices commute up to second order in perturbation theory.
In this paper we used the pure spinor formalism that allows for a covariant quantization. It may also be instructive to reproduce this computation in the Green-Schwarz formulation of [@Metsaev:1998it].
#### The algebra of transfer matrices.
The computations we performed allow to address the question of the algebra of transfer matrices for the pure spinor string on $AdS_5 \times S^5$. Naively, the fusion of two transfer matrices is rather complicated. It seems from that it does not even close on transfer matrices. However by selecting a particular combination of representations we managed to close the algebra. For these representations, the algebra of transfer matrices is nothing but the T-system.
There might exists a generalization of the T-system that applies for other representations. It would be interesting to further explore this issue.
#### Generalization to other integrable field theories.
It would be also interesting to try the approach advocated here in other integrable theories that play a role in the AdS/CFT correspondence. Some examples were listed in section \[extensions\]. This approach may be more efficient than trying to reproduce the historical steps that were performed for $AdS_5 \times S^5$. More generally, developing worldsheet technology for strings in RR background is certainly worthwhile. Even if the progress in that direction have been rather slow, the results presented here together with other recent works (see e.g. [@Ashok:2009jw][@Vallilo:2011fj]) demonstrate that quantum string theory in some RR backgrounds can be studied with the tools that are currently available.
The results presented here also suggest that the integrable models relevant for the AdS/CFT correspondence may belong to a special family. It is not clear that the interpretation of the T-system advocated here applies straightforwardly to generic integrable field theories. Indeed generically the transfer matrices have to be renormalized when the dual Coxeter number of the symmetry group is non-zero (see e.g. [@Bachas:2004sy]). This would complicate a tentative derivation of the T-system from the fusion of transfer matrices.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to thank Gleb Arutyunov, Oscar Bedoya, Denis Bernard, Nikolay Gromov, Volodya Kazakov, Marc Magro, Valentina Puletti, Sakura Schafer-Nameki, Joerg Teschner, Jan Troost, Benoit Vicedo, Dmitro Volin and in particular Pedro Vieira for useful discussions and correspondence. The author is a Postdoctoral researcher of FWO-Vlaanderen. This research is supported in part by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole IAP VI/11 and by FWO-Vlaanderen through project G011410N.
Conventions
===========
Let ${t_A}$ be a basis of the generators of the Lie superalgebra. The metric is defined as: \_[AB]{} = STr(t\_A t\_B) where the supertrace $STr$ is a non-degenerate graded-symmetric inner product. We define the inverse metric as: \^[AB]{} \_[AC]{} = \^B\_C The metric and its inverse are graded-symmetric: \_[AB]{} = (-)\^[AB]{}\_[BA]{} where $(-)^{AB}$ is a minus sign if and only if both indices $A$ and $B$ are fermionic. An element $X$ of the Lie superalgebra is expanded as: J = J\^A t\_A We adopt “NE-SW" conventions for the contraction of indices. Indices are raised and lowered with the metric in the following way: X\^A = \^[AB]{}X\_B ; X\_A = X\^B \_[BA]{}
#### Structure constants.
The graded commutator for the generators is defined as: \[t\_A,t\_B} = t\_A t\_B - (-)\^[AB]{} t\_B t\_A We define the structure constants ${f_{AB}}^C$ as: \[t\_A,t\_B} = [f\_[AB]{}]{}\^C t\_C The identity $STr(t_A[t_B,t_C\}) = Str([t_A,t_B\}t_C)$ follows from the graded-symmetry of the supertrace. It implies for the structure constants: \^D \_[AD]{} = [f\_[AB]{}]{}\^D \_[DC]{} In agreement with our conventions we define: f\_[ABC]{} = [f\_[AB]{}]{}\^D \_[DC]{} and so on. The structure constants are graded-antisymmetric in the 1-2 and 2-3 indices[^8]: f\_[ABC]{} = -(-)\^[AB]{}f\_[BAC]{} ; f\_[ABC]{} = -(-)\^[BC]{} f\_[ACB]{} Under the exchange of the first and third indices, we have: f\_[ABC]{} = -(-)\^f\_[CBA]{} Under cyclic permutation of their indices, the structure constant also satisfy: f\_[ABC]{} = (-)\^A f\_[BCA]{}
#### Tensor product.
The tensor product $t_A^R \otimes t_B^{R'}$ of two generators taken in different representation $R$ and $R'$ is graded: t\_A\^R t\_B\^[R’]{} = (-)\^[AB]{} (1\^R t\_B\^[R’]{})(t\_A\^R 1\^[R’]{}) In order to lighten the expressions in the bulk of the paper we often get rid of the $\otimes$ symbol: t\_A\^R t\_B\^[R’]{} t\_A\^R t\_B\^[R’]{} = (-)\^[AB]{} t\_B\^[R’]{} t\_A\^R
A new look at the gauge covariant current algebra {#covCurrents}
=================================================
Derivation of the current-current OPEs
--------------------------------------
In this appendix we give a new derivation of the tree-level gauge-covariant current algebra for the pure spinor string on $AdS_5 \times S^5$. The method we use is inspired by the analysis of [@Benichou:2010rk] for the current algebra in sigma-models on supergroups. The different steps are the following. We make a natural ansatz for the current-current OPEs. Then we demand that this ansatz is compatible with reparametrization invariance of the path integral, and the with Maurer-Cartan equation. This typically gives more constraints than the number of free coefficients in the ansatz. Finally we solve these constraints to get the current algebra.
This method can be generalized to compute the quantum corrections to the current algebra. This computation can be efficiently organized recursively [@Benichou:2010rk]. Here we will only compute the tree-level coefficients since this is sufficient for the purpose of this article.
We choose the ansatz for the current algebra, that we reproduce here for clarity: $$\begin{aligned}
\label{ansatzKK}
K_m^{A_m}(z) K_n^{B_n}(w) = &
R^{-2} C_{mn} \frac{\kappa^{B_n A_m}}{(z-w)^2}
+ R^{-2} \sum_p C_{mn}^p \frac{{f_{C_p}}^{B_n A_m} K_p^{C_p}}{z-w} \cr
& + R^{-2} \sum_p C_{mn}^{\bar p} {f_{C_p}}^{B_n A_m} \bar K_p^{C_p} \frac{\bar z - \bar w}{(z-w)^2} +... \cr
%%
K_m^{A_m}(z) \bar K_n^{B_n}(w) = &
R^{-2} C_{m \bar n} \kappa^{B_n A_m} 2\pi \delta^{(2)}(z-w)
+ R^{-2} \sum_p C_{m\bar n}^p \frac{{f_{C_p}}^{B_n A_m} K_p^{C_p}}{\bar z-\bar w} \cr
& + R^{-2} \sum_p C_{m\bar n}^{\bar p} \frac{{f_{C_p}}^{B_n A_m} \bar K_p^{C_p}}{z-w} +...\cr
%%
\bar K_m^{A_m}(z) \bar K_n^{B_n}(w) = &
R^{-2} C_{\bar m \bar n} \frac{\kappa^{B_n A_m} }{(\bar z - \bar w)^2}
+ R^{-2} \sum_p C_{\bar m\bar n}^p {f_{C_p}}^{B_n A_m} K_p^{C_p}\frac{z-w}{(\bar z-\bar w)^2} \cr
& + R^{-2} \sum_p C_{\bar m\bar n}^{\bar p} \frac{{f_{C_p}}^{B_n A_m} \bar K_p^{C_p}}{\bar z-\bar w} +...\end{aligned}$$ This ansatz is based on dimensional analysis and symmetry. We only wrote down the second- and first-order poles, but there is an infinite series of less and less singular terms that come with operators of (classical) dimension greater or equal to two. Notice also that this ansatz is suitable for the tree-level current algebra, but it should be slightly modified if one is to take into account quantum corrections [@Bedoya:2010av]. In the following we will compute the coefficients $C$’s. Many of these coefficient vanish trivially because the current algebra has to be compatible with the $\mathbb{Z}_4$ grading. Parity also induces some redundancy in the remaining coefficients. These two symmetries leave 57 independent coefficients that we need to compute.
#### Equations of motion and path-integral reparametrization invariance.
In this subsection we demand that the current algebra is compatible with the reparametrization invariance of the path integral. In particular this guarantees that the current algebra is compatible with the equations of motion.
Let us consider the action for the pure spinor string in $AdS_5 \times S^5$ . We consider a small variation of the group element $g$ parametrized by a element of the Lie superalgebra $X$: \[shiftG\] g = g X The variation of the currents is given by: $$\begin{aligned}
& \delta J = \p X + [J,X] && \delta N = 0\cr
&\delta \bar J = \bar \p X + [\bar J,X] & &\delta \hat N = 0 \end{aligned}$$ We can decompose the infinitesimal shift $X$ on the $\mathbb{Z}_4$ subspaces of the Lie superalgebra as $X=X_0+X_1+X_2+X_3$. Since $X_0$ generates gauge transformations that leave the action invariant, we set $X_0 = 0$. We obtain the variation of the $J_i$’s: \[deltaJ\] J\_[0]{} &=& \[J\_[1]{},X\_3\] + \[J\_[2]{},X\_2\] + \[J\_[3]{},X\_3\] J\_[1]{} &=& X\_1 + \[J\_[0]{},X\_1\] + \[J\_[2]{},X\_3\] + \[J\_[3]{},X\_2\] J\_[2]{} &=& X\_2 + \[J\_[0]{},X\_2\] + \[J\_[1]{},X\_1\] + \[J\_[3]{},X\_3\] J\_[3]{} &=& X\_3 + \[J\_[0]{},X\_3\] + \[J\_[1]{},X\_2\] + \[J\_[2]{},X\_1\] and similarly for the $\bar J_i$’s. We deduce the variation of the action under the infinitesimal shift of the group element : $$\begin{aligned}
\label{deltaS} \delta S = \frac{R^2}{4\pi} STr \int d^2 z & \left\{
X_1 \left( - \frac{3}{2} \bar \nabla J_{3} - \frac{1}{2} \nabla \bar J_{ 3} - \frac{1}{2} [J_{1},\bar J_{2}]- \frac{1}{2} [J_{2}, \bar J_{1}] + 2[N,\bar J_3] - 2 [J_3, \hat N] \right) \right.\cr
& + X_2 \left(- \bar \nabla J_{2} - \nabla \bar J_{ 2} - [J_{1},\bar J_{1}]+ [J_{3},\bar J_{3}] + 2[N,\bar J_2] - 2 [J_2, \hat N] \right) \cr
& \left.+ X_3 \left( -\frac{1}{2} \bar \nabla J_{1} - \frac{3}{2} \nabla \bar J_{ 1} + \frac{1}{2} [J_{2},\bar J_{3}]+ \frac{1}{2} [J_{3},\bar J_{2}]+ 2[N,\bar J_1] - 2 [J_1, \hat N]\right) \right\}
\end{aligned}$$ Now we consider the following quantity: J\_1(z) = J\_1(z) e\^[-S]{} where $ \mathcal{D}\Phi $ is the path integral measure over the fields. The previous one-point function, whatever its value is, is invariant under the reparametrization of the path integral . We further assume that the path-integral measure is also invariant under . Let us mention at that point that we are simply following the method that would provide a path integral derivation of the Ward identity for a global symmetry, if were indeed a global symmetry. We obtain: \[dJ=JdS\] J\_1(z) - J\_1(z) S = 0 It is convenient to rewrite the variation of the current as an integral over the worldsheet: J\_1(z) = d\^2 w ( X\_1(w) ’(z-w) + (\[J\_0(w),X\_1(w)\]+\[J\_2(w),X\_3(w)\]+\[J\_3(w),X\_2(w)\]) (z-w) ) Projecting equation on the $\mathbb{Z}_4$ subspaces, we obtain three operator identities: $$\begin{aligned}
\label{appJ1EOM} J_1^{A_1}(z) & \left(-\frac{3}{2} \bar \nabla J_3^{B_3}(w) - \frac{1}{2} \nabla \bar J_3^{B_3}(w) - \frac{1}{2}{f_{C_1 D_2}}^{B_3} :J_1^{C_1} \bar J_2^{D_2}:(w) - \frac{1}{2} {f_{C_2 D_1}}^{B_3} :J_2^{C_2} \bar J_1^{D_1}:(w) \right. \cr
& \left. \qquad + 2 {f_{C_0 D_3}}^{B_3} :N^{C_0} \bar J_3^{D_3}:(w) - 2 {f_{C_3 D_0}}^{B_3} :J_3^{C_3} \hat N^{D_0}:(w) \right) \cr
& = 4 \pi R^{-2} \kappa^{B_3 A_1} \p_z \delta(z-w) \cr
%
J_1^{A_1}(z) & \left(- \bar \nabla J_2^{B_2}(w) - \nabla \bar J_2^{B_2}(w) - {f_{C_1 D_1}}^{B_2} :J_1^{C_1} \bar J_1^{D_1}:(w) + {f_{C_3 D_3}}^{B_2} :J_3^{C_3} \bar J_3^{D_3}:(w) \right. \cr
& \left.\qquad + 2 {f_{C_0 D_2}}^{B_2} :N^{C_0} \bar J_2^{D_2}:(w) - 2 {f_{C_2 D_0}}^{B_2} :J_2^{C_2} \hat N^{D_0}:(w) \right) \cr
& = 4 \pi R^{-2} {f_{C_3}}^{B_2 A_1} J_3^{C_3}(w) \delta(z-w) \cr
%
J_1^{A_1}(z) & \left(-\frac{1}{2} \bar \nabla J_1^{B_1}(w) - \frac{3}{2} \nabla \bar J_1^{B_1}(w) + \frac{1}{2}{f_{C_2 D_3}}^{B_1} :J_2^{C_2} \bar J_3^{D_3}:(w) + \frac{1}{2} {f_{C_3 D_2}}^{B_1} :J_3^{C_3} \bar J_2^{D_2}:(w) \right. \cr
& \left. \qquad + 2 {f_{C_0 D_1}}^{B_1} :N^{C_0} \bar J_1^{D_1}:(w) - 2 {f_{C_1 D_0}}^{B_1} :J_1^{C_1} \hat N^{D_0}:(w) \right) \cr
& = 4 \pi R^{-2} {f_{C_2}}^{B_1 A_1} J_2^{C_2}(w) \delta(z-w)\end{aligned}$$ where the colons stand for normal ordering. Next we plug the ansatz into these equations. More precisely, we use the ansatz to perform the OPEs on the left-hand side of the identities . Since we are working at first-order in $R^{-2}$, the OPEs involving composite operators are easily dealt with: a single OPE has to be taken with one or the other of the components of the composite operator. We use the equalities: \_[|w]{} = -2(z-w) = \_w ; \_[|w]{} = 2’(z-w) = \_w We are left with some identities between operators multiplied by functions that are singular when $z-w \to 0$. We demand that the operator identities do hold for the singular terms of order two and three: all the terms multiplying either a derivative of a delta function, a delta function, or a second-order pole shall cancel against each other[^9]. We obtain a set of linear equations that the free coefficients in the ansatz have to satisfy: $$\begin{aligned}
& 2 = -\frac{3}{2} C_{13} + \frac{1}{2} C_{1 \bar 3}
& 0 = \frac{3}{2} C_{13}^{\bar g} - \frac{1}{2} C_{1 \bar 3}^{\bar g} - 2 C_{13}
& \qquad 0 = \frac{3}{2} C_{13}^g + \frac{1}{2} C_{1 \bar 3}^g - 2C_{1 \bar 3} \cr
%
& 0 = \frac{3}{2} C_{13}^{\bar 0} - \frac{1}{2} C_{1 \bar 3}^{\bar 0} + \frac{3}{2} C_{13}
& 2 = \frac{3}{2} C_{13}^0 + \frac{1}{2} C_{1 \bar 3}^0 + \frac{1}{2} C_{1 \bar 3}
& \qquad 0 = C_{12}^{\bar 3} - C_{1 \bar 2}^{\bar 3} + C_{13} \cr
%
& 2 = C_{12}^3 + C_{1 \bar 2}^3 - C_{1 \bar 3}
& 0 = \frac{1}{2} C_{11}^{\bar 2} - \frac{3}{2} C_{1 \bar 1}^{\bar 2} + \frac{1}{2} C_{13}
& \qquad 2 = \frac{1}{2} C_{11}^2 + \frac{3}{2} C_{1 \bar 1}^{2} - \frac{1}{2} C_{1 \bar 3}\end{aligned}$$ We can play the same game replacing in equation $J_1$ by another current. For each current we obtain a new set of equations. To get more constraints for the OPEs involving the ghosts, we can also vary the ghosts variable instead of . In total we get 39 linear equations that the coefficients $C$’s in the ansatz have to satisfy.
#### The Maurer-Cartan equation.
We can further constraint the coefficients in the ansatz by demanding compatibility with the Maurer-Cartan equation . Projecting this equation according to the $\mathbb{Z}_4$ grading we obtain: $$\begin{aligned}
\label{MCcomp}
& \p \bar J_0 - \bar \p J_0 + [J_0, \bar J_0] + [J_1, \bar J_3] + [J_2, \bar J_2] + [J_3, \bar J_1] = 0 \cr
& \p \bar J_1- \bar \p J_1 + [J_0, \bar J_1] + [J_1, \bar J_0] + [J_2, \bar J_3] + [J_3, \bar J_2] = 0 \cr
& \p \bar J_2 - \bar \p J_2 + [J_0, \bar J_2] + [J_1, \bar J_1] + [J_2, \bar J_0] + [J_3, \bar J_3] = 0 \cr
& \p \bar J_3 - \bar \p J_3 + [J_0, \bar J_3] + [J_1, \bar J_2] + [J_2, \bar J_1] + [J_3, \bar J_0] = 0
\end{aligned}$$ The strategy is to demand that the Maurer-Cartan equation does hold as an operator identity. More precisely, we demand that the OPE between a current and the left-hand side of does vanish. Let us consider one example for illustrative purposes: we take the OPE between the current $J_1$ and the left-hand side of the first line in equation : $$\begin{aligned}
0 = J_1^{A_1}(z) & \left( \p \bar J_0^{B_0}(w) - \bar\p J_0^{B_0}(w) + {f_{C_0 D_0}}^{B_0} :J_0^{C_0} \bar J_0^{D_0}:(w) + {f_{C_1 D_3}}^{B_0} :J_1^{C_1} \bar J_3^{D_3}:(w) \right. \cr
& \left. \qquad + {f_{C_2 D_2}}^{B_0} :J_2^{C_2} \bar J_2^{D_2}:(w) + {f_{C_3 D_1}}^{B_0} :J_3^{C_3} \bar J_1^{D_1}:(w) \right) \end{aligned}$$ As previously we plug the ansatz in the previous equation, and demand that the singular terms of order three and two do vanish. We obtain the following equations: 0 = C\_[10]{}\^[|1]{} + C\_[1 |0]{}\^[|1]{} + C\_[13]{} 0 = C\_[10]{}\^1 - C\_[1 |0]{}\^1 - C\_[1 |3]{} Repeating the same procedure for the different current and the different lines of equation , we obtain in total 43 linear equations that the coefficients $C$’s in the ansatz have to satisfy.
Let us make a side remark here. There is no doubt that the Maurer-Cartan identity does hold at tree level. However it may get quantum corrections. In order to generalize the method described here to compute quantum corrections to the current algebra, one needs to assume that the Maurer-Cartan identity holds in the quantum theory as well. This may be interpreted as postulating quantum integrability of the model. This provides an efficient way to use quantum integrability of the model to compute the quantum current algebra. We leave it for future work.
#### The coefficients of the current algebra.
Using the Maurer-Cartan identity and reparametrization invariance of the path integral, we find in total 82 equations that constrain the 57 independent coefficients of the current algebra . This system of equation can be easily decomposed in subsystems of eight equations or less. It is remarkable that there exists a solution to this set of equations. The non-zero coefficients are given in section \[subKalgebra\].
There is however one exception for the OPEs between two of the currents $J_0$ and $\bar J_0$. In that case the equations we obtain only provide the constraints .
#### Associativity.
The current algebra has to be associative. Associativity of the current algebra can be tested in the following way. Let us consider a 3-points function, for instance: J\_1(x) J\_1(y) J\_2(z) It can be computed by taking first the OPE between $J_1(x)$ and $ J_1(y)$, and then take the OPE between the resulting current and $J_2(z)$. But one can also start by taking the OPE between $J_1(y)$ and $J_2(z)$, and then take the OPE of the result with $J_1(x)$. The two methods lead to the same result if the coefficients of the current algebra satisfy: C\_[11]{}\^2 C\_[22]{} = C\_[12]{}\^3 C\_[13]{} We can play the same game with any three-points functions. We find a large set of constraints that are all satisfied by the current algebra obtained previously.
The $(r,s)$ system. {#appRSmatrices}
-------------------
In this section we give some details on the derivations of equations , and . We want to compute the commutator of two equal-time connections $A_R(y;\sigma)$ and $A_{R'}(y';\sigma')$ evaluated for different values of the spectral parameter $y$ and $y'$ and taken in possibly different representation $R$ and $R'$. We can deduce this commutator from the current algebra. We define the commutator of equal-time operators as: = \_[0\^+]{} ( A(+i )B(0) - B(i )A() ) From this definition we extract an operative dictionary between OPEs and commutators. Let us consider for instance the following OPE: \[ABOPE\] A(z) B(0) = + + E \^[(2)]{}(z) + + + + + ... We deduce the commutator: \[ABcom\] \[ A(), B(0) \] = C ’() - D ’() - F(0) () - G(0)() + H(0)() + I(0)() This dictionary shows that the first-order computation of fusion presented in section \[fusionOrder1\] is equivalent to the computation of the Poisson bracket of line operators in the Hamiltonian formalism. Notice that the OPE generically contains additional sub-leading singularities, for instance $\frac{\bar z}{z}$, or even logarithms. They do not contribute to the commutator [@Benichou:2010ts]. Generically, the OPE contains more information than the commutator.
In order to simplify the following expressions, we write the (spacelike component of the) flat connection $A(y)$ defined in as: $$\begin{aligned}
\label{defAwithFs} A(y) =& \sum_m F_m(y) K_m + \bar F_m(y) \bar K_m \end{aligned}$$ Using the previous dictionary, we obtain for the commutator of two connections: $$\begin{aligned}
\label{[A,A']}
[A_R&(y;\sigma), A_{R'}(y';\sigma')] = \cr
& 2\pi i R^{-2} \p_\sigma \delta(\sigma- \sigma') \sum_{m,n} \kappa^{B_n A_m} t_{A_m}^R t_{B_n}^{R'} ( F_m(y)F_n(y')C_{mn} -
\bar F_m(y)\bar F_n(y')C_{\bar m \bar n}) \cr
& + 2\pi i R^{-2} \delta(\sigma- \sigma') \sum_{m,n,p} {f_{C_p}}^{B_n A_m} t_{A_m}^R t_{B_n}^{R'}
K_p^{C_p} ( -F_m(y)F_n(y') C_{mn}^p \cr
& \hspace{2.5 cm} + F_m(y)\bar F_n(y') C_{m\bar n}^p + \bar F_m(y)F_n(y') C_{\bar m n}^p + \bar F_m(y) \bar F_n(y') C_{\bar m \bar n}^p) \cr
& + 2\pi i R^{-2} \delta(\sigma- \sigma') \sum_{m,n,p} {f_{C_p}}^{B_n A_m} t_{A_m}^R t_{B_n}^{R'}
\bar K_p^{C_p} ( -F_m(y)F_n(y') C_{mn}^{\bar p} \cr
& \hspace{2.5 cm} - F_m(y)\bar F_n(y') C_{m\bar n}^{\bar p} - \bar F_m(y)F_n(y') C_{\bar m n}^{\bar p} + \bar F_m(y) \bar F_n(y') C_{\bar m \bar n}^{\bar p}) \end{aligned}$$ We wish to write this commutator as a $(r,s)$ system . From the terms coming with a derivative of the delta function in the commutator , we can read directly the $s$-matrix. We obtain: s = i R\^[-2]{} \_[m,n]{} \^[B\_n A\_m]{} t\_[A\_m]{}\^R t\_[B\_n]{}\^[R’]{} ( F\_m(y)F\_n(y’)C\_[mn]{} - |F\_m(y)|F\_n(y’)C\_[|m |n]{}) Plugging in the value of the coefficients, we obtain . To obtain the $r$ matrix, we have to compare the terms coming with a delta function in and . This leads to the following equations for the components of the $r$ and $s$ matrices: $$\begin{aligned}
\label{eqsForRMatrix} \forall\ m,n,p: \quad
F_p(y) & r_{4-n,n} - F_p(y') r_{m,4-n} = - F_p(y) s_{4-n,n} - F_p(y') s_{m,4-n} - 2 F_m(y) F_n(y') C_{m n}^{p} \cr
&+ 2 F_m(y) \bar F_n(y') C_{m \bar n}^{p}+ 2 \bar F_m(y) F_n(y') C_{\bar m n}^{p} + 2 \bar F_m(y) \bar F_n(y') C_{\bar m \bar n}^{p} \cr \cr
\bar F_p(y) & r_{4-n,n} - \bar F_p(y') r_{m,4-n} = - \bar F_p(y) s_{4-n,n} - \bar F_p(y') s_{m,4-n} - 2 F_m(y) F_n(y') C_{m n}^{\bar p} \cr
&- 2 F_m(y) \bar F_n(y') C_{m \bar n}^{\bar p}- 2 \bar F_m(y) F_n(y') C_{\bar m n}^{\bar p} + 2 \bar F_m(y) \bar F_n(y') C_{\bar m \bar n}^{\bar p}\end{aligned}$$ In the previous equations, when the indices $m,n$ take the value $0$ or $g$, one should understand“$r_{4,0}$" and “$r_{4-g,g}$" as being $r_{0,0}$, etc. Remarkably, this largely over-constrained system is solved by the $r$ and $s$ matrices and .
#### Comparison with previous analyses. {#appCompKAlgebras}
The current-current OPEs were previously discussed in the literature. In [@Puletti:2006vb] the OPEs for the currents of non-zero grade were computed using the background field methods. Some of the OPEs involving the grade zero currents were further given in [@Puletti:2008ym]. The results we obtained here agree with these papers.
In [@Mikhailov:2007mr] the current algebra was also computed using Feynman diagram technology. The OPEs do match the ones we derived here except for those involving the currents $J_0$, $\bar J_0$. This is not surprising given the gauge choice that was explicitly made for the coset element in [@Mikhailov:2007mr]. A consequence of this discrepancy is that the commutator of equal-time connections can not be written as a $(r,s)$ system with the OPEs of [@Mikhailov:2007mr]. However one should keep in mind that it is only an issue of gauge fixing. Indeed in [@Mikhailov:2007eg] the OPEs of [@Mikhailov:2007mr] were used to compute the fusion of line operators at first order. Then the $r$ and $s$ matrices were deduced by comparison with the expectations from the Hamiltonian formalism. These matrices agree with the ones that we derived in this paper.
In [@Magro:2008dv] the hamiltonian formalism was used to compute the commutator of equal time connections. A careful treatment of the constraints was performed. It was argued that in the Hamiltonian formalism, the flat connection realizes a $(r,s)$ system up to constraints generating gauge transformations. It was shown that one should add to the flat connection a term proportional to the constraints so that the commutator of connections take exactly the form of a $(r,s)$ system. The resulting $(r,s)$ system is slightly different than the one used here and in [@Mikhailov:2007mr]. The flat connection obtained in [@Magro:2008dv], including the additional term proportional to the constraints, was derived from first principles in [@Vicedo:2009sn] in the Green-Schwarz formalism. It is remarkable that the analysis of [@Vicedo:2009sn] leads to the pure spinor-like flat connections of [@Vallilo:2003nx] (without the pure spinor ghosts contribution) and not to the Bena-Polchinski-Roiban flat connections [@Bena:2003wd]. This provides some evidence for the equivalence of the pure spinor and Green-Schwarz formulations of string theory on $AdS_5 \times S^5$. In [@Vicedo:2010qd] it was shown that the $(r,s)$ system of [@Magro:2008dv] has a nice algebraic interpretation.
For the purposes of this paper it is important that the $r$ matrix found in [@Magro:2008dv] is identical to the one we worked with in the limit where the difference of spectral parameter is small . This guarantees that the results derived in the present paper would also hold if one were to work with the $(r,s)$ system of [@Magro:2008dv]. In order to reproduce the $(r,s)$ system found in [@Magro:2008dv] using OPEs technology, the first step would be to gauge-fix the $\mathcal{H}_0$ gauge symmetry via a BRST procedure. Then one should generalize the analysis of [@Vallilo:2003nx] by including in the flat connections additional terms written in terms of the ghosts resulting from the $\mathcal{H}_0$ gauge-fixing. These new connections should realize the $(r,s)$ system of [@Magro:2008dv][^10].
Divergences in line operators {#computationDiv}
=============================
In this appendix we give some details about the computations of the first-order divergences in line operators. As explained in section \[worldsheetTheory\], we use a principal-value regularization scheme. A first-order pole is regularized as: P.V. = ( + ) = and a second-order pole is regularized as: P.V. = ( + ) =
Divergences in transition matrices
----------------------------------
![The three individual sources of first-order divergences in line operators. In the first-order poles in the OPE between two connections are considered. In and the second-order poles are considered.\[divergences\]](divergences)
There are three sources of divergences in the transition matrices. They are depicted in figure \[divergences\]. The first divergences come from the first order poles in the OPE of two neighboring connections, say $A(y;\sigma_1)$ and $A(y;\sigma_2)$ (case in figure[divergences]{}). We evaluate the resulting currents at the point $\sigma_2$ and perform the integration over $\sigma_1$. We obtain a logarithmic divergences: $$\begin{aligned}
\label{divAA1}
& (-\log \epsilon) \sum_{m,n,p} \left(
K_p^{C_p}(\sigma_2)(F_m F_n C_{mn}^p + \bar F_m F_n C_{\bar m n}^p + F_m \bar F_n C_{m \bar n}^p + \bar F_m \bar F_n C_{\bar m \bar n}^p) \right. \cr
& \left. + \bar K_p^{C_p}(\sigma_2)(F_m F_n C_{mn}^{\bar p} + \bar F_m F_n C_{\bar m n}^{\bar p} + F_m \bar F_n C_{m \bar n}^{\bar p} + \bar F_m \bar F_n C_{\bar m \bar n}^{\bar p}) \right)
{f_{C_p}}^{B_n A_m} t_{A_m} t_{B_n}\end{aligned}$$ where the functions $F$’s were defined in .
The second type of divergences come from the second-order poles in the OPE between two neighboring connections (case in figure[divergences]{}). After performing the integration over the positions of the two connections, we obtain a logarithmic divergence: $$\begin{aligned}
\label{divAA2}
& \log \epsilon \sum_{m,n}
(F_m F_n C_{mn} + \bar F_m \bar F_n C_{\bar m \bar n}) \kappa^{B_n A_m} t_{A_m} t_{B_n}\end{aligned}$$ Notice that there is no linear divergences. This is a pleasant feature of the regularization scheme that we are using. Eventually the third type of divergences come from the second-order poles in the OPE between two connections that are separated by a third one sitting in between (case in figure[divergences]{}). Let us denote this third connection by $A(y;\sigma)$. After performing the integrations, we obtain another logarithmic divergence: $$\begin{aligned}
\label{divAAA2}
& (-\log \epsilon) \sum_{m,n}
(F_m F_n C_{mn} + \bar F_m \bar F_n C_{\bar m \bar n}) %\cr &\times
\left( \frac{1}{2}\left\{ \kappa^{B_n A_m} t_{A_m} t_{B_n} ,A(y;\sigma) \right\} \right. \cr
& \quad \left. -\frac{1}{2} \sum_{p,q} \left({f_{C_p}}^{D_q A_m} t_{A_m} t_{D_q} + {f_{C_p}}^{B_n D_q}t_{D_q} t_{B_n}\right)\left(F_p K_p^{C_p}(\sigma) + \bar F_p \bar K_p^{C_p}(\sigma)\right) \right)\end{aligned}$$ There is some freedom in how we write the last expression since we can commute the generators in different ways. We choose a writing that is symmetric with respect to the central connection $A(y;\sigma)$.
Starting from a transition matrix, we compute all the different OPEs of the types described previously that lead to divergences. Next we sum all these terms. Most of the terms of the type cancel against the first terms in . We obtain: $$\begin{aligned}
\label{divTot1}
\sum_{M=0}^\infty&\left((-)^M \log \epsilon \frac{1}{2} \sum_{m,n} (F_m F_n C_{mn} + \bar F_m \bar F_n C_{\bar m \bar n})
\left\{ \kappa^{B_n A_m} t_{A_m} t_{B_n}, \left \lfloor \int_a^b A \right \rceil^M \right\}\right.\cr
& + (-)^{M+1} \log \epsilon \sum_{i=0}^{M-1} \int_a^b d \sigma \left \lfloor \int_\sigma^b A \right \rceil^i
\sum_{m,n,p} {f_{C_p}}^{B_n A_m} t_{A_m} t_{B_n} \cr
& \qquad \times ( K_p^{C_p}(\sigma) (F_m F_n C_{mn}^p + \bar F_m F_n C_{\bar m n}^p + F_m \bar F_n C_{m \bar n}^p + \bar F_m \bar F_n C_{\bar m \bar n}^p \cr
& \qquad \qquad +\frac{1}{2}F_p \sum_q(F_m F_q C_{mq} + \bar F_m \bar F_q C_{\bar m \bar q} + F_n F_q C_{nq} + \bar F_n \bar F_q C_{\bar n \bar q})) \cr
& \qquad \quad + \bar K_p^{C_p}(\sigma)(F_m F_n C_{mn}^{\bar p} + \bar F_m F_n C_{\bar m n}^{\bar p} + F_m \bar F_n C_{m \bar n}^{\bar p} + \bar F_m \bar F_n C_{\bar m \bar n}^{\bar p} \\
& \left.\qquad \qquad +\frac{1}{2}\bar F_p \sum_q(F_m F_q C_{mq} + \bar F_m \bar F_q C_{\bar m \bar q} + F_n F_q C_{nq} + \bar F_n \bar F_q C_{\bar n \bar q})))%\cr & \times
\left \lfloor \int_a^\sigma A \right \rceil^{M-i-1} \right) \nonumber\end{aligned}$$
#### Consequences of the vanishing of the dual Coxeter number.
Here we derive some identities that are useful to show the vanishing of some divergences in the line operators. These identities were first derived in [@Mikhailov:2007mr]. The vanishing of the dual Coxeter number can be written as: \[ftt=0\] [f\_C]{}\^[BA]{}\[t\_A,t\_B} = 0 The super-Jacobi identity together with the fact that $\kappa^{A_1 B_3} \{t_{A_1},t_{B_3}\} = 0$ implies: \[13C=31C\] \^[B\_3 A\_1]{}\[t\_[A\_1]{},\[t\_[B\_3]{},t\_C}} = \^[A\_1 B\_3]{}\[t\_[B\_3]{},\[t\_[A\_1]{},t\_C}} The identities and further imply: $$\begin{aligned}
\label{C=1}
{f_{C_1}}^{D_2 B_3}[t_{B_3},t_{D_2}\} = {f_{C_1}}^{D_0 A_1}[t_{A_1},t_{D_0}\} = 0 \end{aligned}$$ $$\begin{aligned}
\label{C=3}
{f_{C_3}}^{D_2 B_1}[t_{B_1},t_{D_2}\} = {f_{C_3}}^{D_0 A_3}[t_{A_3},t_{D_0}\} = 0 \end{aligned}$$ $$\begin{aligned}
\label{C=2}
{f_{C_2}}^{D_1 A_1}[t_{A_1},t_{D_1}\} = {f_{C_2}}^{D_3 A_3}[t_{A_3},t_{D_3}\} = - {f_{C_2}}^{D_0 B_2}[t_{B_2},t_{D_0}\} \end{aligned}$$
#### Cancellation of divergences.
Let us come back to the expression . We will now argue that the second piece of vanishes, as first shown in [@Mikhailov:2007mr]. Using the identities and , we observe that the terms proportional to $J_1$, $\bar J_1$, $J_3$ and $\bar J_3$ vanish straight away. Then using the identities together with the actual value of the coefficients of the current algebra, it is straightforward to check that the terms proportional to $J_2$ and $\bar J_2$ also drop out.
The vanishing of the terms proportional to $J_0$, $\bar J_0$, $N$ and $\hat N$ depends on the value of the simple poles in the OPEs $J_0.J_0$, $J_0. \bar J_0$ and $\bar J_0 . \bar J_0$. The method explained in appendix \[covCurrents\] to compute the current algebra does not fix completely these OPEs, but only gives the constraint . The identity combined with the value of the other coefficients of the current algebra implies the vanishing of all terms proportional to $J_0$, $\bar J_0$, $N$ and $\hat N$ provided we have: $$\begin{aligned}
\label{possibleValueC00*}
C_{00}^0 = C_{0\bar 0}^0 = -C_{\bar 0 \bar 0}^0 = 0 \quad ; \quad
- C_{00}^{\bar 0} = C_{0\bar 0}^{\bar 0} = C_{\bar 0 \bar 0}^{\bar 0} = 0\cr
C_{00}^g = C_{0\bar 0}^g = -C_{\bar 0 \bar 0}^g = 2 \quad ; \quad
- C_{00}^{\bar g} = C_{0\bar 0}^{\bar g} = C_{\bar 0 \bar 0}^{\bar g} = 2\end{aligned}$$ Demanding consistency with the analysis of [@Mikhailov:2007mr] implies the previous equations. We deduce that only the first term in survives. We can write it as: $$\begin{aligned}
&(-)^{M+1} \log \epsilon \frac{y^4 + y^{-4}}{2}
\left\{ \kappa^{B_3 A_1} t_{A_1} t_{B_3} + \kappa^{B_2 A_2} t_{A_2} t_{B_2} + \kappa^{B_1 A_3} t_{A_3} t_{B_1}, \left \lfloor \int_a^b A \right \rceil^M \right\}\cr\end{aligned}$$ So the first-order divergences in the transition matrix can be rewritten as: \[divTransition\] - { \^[B\_3 A\_1]{} t\_[A\_1]{} t\_[B\_3]{} + \^[B\_2 A\_2]{} t\_[A\_2]{} t\_[B\_2]{} + \^[B\_1 A\_3]{} t\_[A\_3]{} t\_[B\_1]{}, T\^[b,a]{}(x) }
Divergences in monodromy and transfer matrices
----------------------------------------------
In loop operators we have additional divergences coming from the collisions between two connections sitting on either side of the starting point of the integration contour. Only the second-order pole in such a collision lead to a divergence. These divergences read: $$\begin{aligned}
\sum_{M=0}^\infty & (-)^M(-\log \epsilon) \int_{2\pi>\sigma_1>...>\sigma_M>0} d\sigma_1...d\sigma_M A^{B^{(1)}}(\sigma_1)...A^{B^{(M)}}(\sigma_M)\cr
& \qquad \times \sum_{m,n}(F_m F_n C_{mn} + \bar F_m \bar F_n C_{\bar m \bar n})
\kappa^{C_n D_m} t_{D_m} t_{B^{(1)}}...t_{B^{(M)}} t_{C_n} \prod_{i=1}^M(-)^{C_n B^{(i)}}\end{aligned}$$ So the first-order divergences in the monodromy matrix add up to: $$\begin{aligned}
\label{divMonodromy} & \log \epsilon \frac{y^4 + y^{-4}}{2}
\sum_{M=0}^{\infty} (-)^M \int_{2\pi>\sigma_1>...>\sigma_M>0} d\sigma_1...d\sigma_M A^{B^{(1)}}(\sigma_1)...A^{B^{(M)}}(\sigma_M)\cr
& \qquad \times ( -(\kappa^{C_3 D_1} t_{D_1} t_{C_3} + \kappa^{C_2 D_2} t_{D_2} t_{C_2} + \kappa^{C_1 D_3} t_{D_3} t_{C_1}) t_{B^{(1)}}...t_{B^{(M)}} \cr
& \qquad \qquad - t_{B^{(1)}}...t_{B^{(M)}} (\kappa^{C_3 D_1} t_{D_1} t_{C_3} + \kappa^{C_2 D_2} t_{D_2} t_{C_2} + \kappa^{C_1 D_3} t_{D_3} t_{C_1}) \cr
& \qquad \qquad + 2\kappa^{C_3 D_1} t_{D_1}t_{B^{(1)}}...t_{B^{(M)}} t_{C_3}\prod_{i=1}^M(-)^{D_1 B^{(i)}}
+ 2\kappa^{C_2 D_2} t_{D_2}t_{B^{(1)}}...t_{B^{(M)}} t_{C_2}\prod_{i=1}^M(-)^{D_2 B^{(i)}}\cr
& \qquad \qquad
+ 2\kappa^{C_1 D_3} t_{D_3}t_{B^{(1)}}...t_{B^{(M)}} t_{C_1}\prod_{i=1}^M(-)^{D_3 B^{(i)}}
)\end{aligned}$$ Taking the supertrace, we see that the transfer matrix is free of divergences at first order.
Fusion at second order: computations {#comFusion}
====================================
In this appendix we give some details concerning the computation of the fusion of line operators at second order. In particular we describe the computation that leads to and . These terms are produced by triple collisions of connections. A triple collision means that we take one OPE between two connections, and then take the OPE of the resulting currents with a third connection.
#### Treatment of the OPEs.
As explain in section \[recipeFusion\], one needs to disentangle two contributions from the OPEs. On one hand there is the contribution that gives a quantum correction associated with fusion. On the other hand there is the contribution that is interpreted as a regularized OPE in the double line operator resulting from the process of fusion. In order to isolate the interesting part associated with fusion, we subtract the principal value from the singularities. We obtain[^11]: $$\begin{aligned}
\label{regPoles}
&\frac{1}{(\sigma \pm i \epsilon - \sigma')^2} -P.V. \frac{1}{(\sigma-\sigma')^2} = \pm i \pi \delta'_\epsilon(\sigma-\sigma') \cr
&\frac{1}{\sigma \pm i \epsilon - \sigma'} - P.V. \frac{1}{\sigma-\sigma'} = \mp i \pi \delta_\epsilon(\sigma-\sigma') \cr
&\frac{\sigma \mp i \epsilon - \sigma'}{(\sigma \pm i \epsilon - \sigma')^2} - P.V. \frac{1}{\sigma-\sigma'} = \mp i \pi \delta_\epsilon(\sigma-\sigma') \end{aligned}$$
#### Computation of the individual terms.
Let us now face the computation of the individual quantum corrections that add up to and . In the first step of the computation of a triple collision we perform an OPE between two connections sitting on different contours. We obtain intermediate currents that we evaluate on one of the two contours[^12]. The contour on which these intermediate currents are evaluated matters for the second step of the computation. The relevant OPE for the first step of the computation is thus: $$\begin{aligned}
\label{firstOPE} (1&-P.V.) A_R(y;\sigma+i\epsilon)A_{R'}(y';\sigma') \supset \pi i R^{-2} \delta_\epsilon(\sigma-\sigma')\sum_{m,n=0}^3 \sum_{p} {f_{C_p}}^{b_n A_m} t_{A_m}^R t_{B_n}^{R'} \cr
& \times (D_{m n}^{p} K_p^{C_p}(\sigma+i\epsilon) + {D'}_{m n}^{p} K_p^{C_p}(\sigma') +
D_{m n}^{\bar p} \bar K_p^{C_p}(\sigma+i\epsilon) + {D'}_{m n}^{\bar p} \bar K_p^{C_p}(\sigma') )\end{aligned}$$ where the index $p$ can take the values $\{0,1,2,3,g\}$. In the following when the range of the sum for some index is not specified, it is understood that the sum runs over the set $\{0,1,2,3,g\}$. The coefficients $D_{**}^{*}$ depends on the precise location where the currents are evaluated in the simple poles of the current algebra . From the coefficient given in section \[subKalgebra\] we can only deduce the sums $ D_{m n}^{p} + {D'}_{m n}^{p} $ and $ D_{m n}^{\bar p} + {D'}_{m n}^{\bar p} $. It turns out that we don’t need more information about the coefficients $D_{**}^{*}$ for the purpose of this article (see equation ).
Actually what we want to compute here is not exactly the contribution of the triple collisions to the fusion of line operators. It is rather the part of this contribution that has not been taken into account by the first part of the computation described in section \[subFusion2nd\] and depicted in figure \[fusO2step1\]. Removing the piece of the triple collisions already taken into account amounts to perform the following replacement in the first OPE : $$\begin{aligned}
\label{defDtilde}
&D_{mn}^p \to \tilde D_{mn}^p = D_{mn}^p - \frac{1}{2}F_p(y)(s_{4-n,n}+r_{4-n,n}) \cr
&D_{mn}^{\bar p} \to \tilde D_{mn}^{\bar p} = D_{mn}^{\bar p} - \frac{1}{2}\bar F_p(y)(s_{4-n,n}+r_{4-n,n}) \cr
&{D'}_{mn}^p \to \tilde {D'}_{mn}^p = {D'}_{mn}^p - \frac{1}{2}F_p(y')(s_{m,4-m}-r_{m,4-m}) \cr
&{D'}_{mn}^{\bar p} \to \tilde {D'}_{mn}^{\bar p} = {D'}_{mn}^{\bar p} - \frac{1}{2}\bar F_p(y')(s_{m,4-m}-r_{m,4-m})\end{aligned}$$
![The different types of triple collisions that contribute to the fusion at second order. The first OPE produces currents denoted $K's$. In and the contribution from the first-order poles in the second OPE is singled out. In the four other cases the contribution from the second-order poles in the second OPE is considered.\[allTripleCollisions\]](allTripleCollisions)
Now we can compute the individual terms that add up to and . These different terms are schematically depicted in figure \[allTripleCollisions\]. We compute separately the contribution of the first- and second-order poles in the second OPE (obviously only the first order poles have to be taken into account in the first OPE). Let us begin with the first-order poles in the second OPE. The triple collision involves three connections, two of which are integrated on the same contour. There are two different cases. For a triple collision involving two neighboring connections on the first contour (case in figure \[allTripleCollisions\]), we obtain: $$\begin{aligned}
\label{RRR'1}
(i\pi R^{-2})^2 & \delta_{\epsilon}(\sigma_1-\sigma')\delta_{\epsilon}(\sigma_2-\sigma') \sum_{m,n=0}^3\sum_{p,q,r}
{f_{C_p}}^{B_n A_m} {f_{E_r}}^{C_p D_q} \{ t_{D_q}^R , t_{A_m}^R] t_{B_n}^{R'} \cr
& \times (K_r^{E_r}({\tilde{D'}}_{mn}^p F_q C_{pq}^r - {\tilde{D'}}_{mn}^p \bar F_q C_{p\bar q}^r - {\tilde{D'}}_{mn}^{\bar p} F_q C_{\bar pq}^r - {\tilde{D'}}_{mn}^{\bar p} \bar F_q C_{\bar p \bar q}^r ) \cr
& \quad + \bar K_r^{E_r}({\tilde{D'}}_{mn}^p F_q C_{pq}^{\bar r} + {\tilde{D'}}_{mn}^p \bar F_q C_{p\bar q}^{\bar r} + {\tilde{D'}}_{mn}^{\bar p} F_q C_{\bar pq}^{\bar r} - {\tilde{D'}}_{mn}^{\bar p} \bar F_q C_{\bar p \bar q}^{\bar r} ))\end{aligned}$$ where we wrote $F_q$ as a shorthand for $F_q(y)$. Similarly we will write $F'_q$ for $F_q(y')$. For a triple collision involving two neighboring connections on the second contour (case in figure \[allTripleCollisions\]), we obtain: $$\begin{aligned}
\label{RR'R'1}
(i\pi R^{-2})^2 & \delta_{\epsilon}(\sigma'_1-\sigma)\delta_{\epsilon}(\sigma'_2-\sigma) \sum_{m,n=0}^3\sum_{p,q,r}
{f_{C_p}}^{B_n A_m} {f_{E_r}}^{ D_q C_p} t_{A_m}^R \{ t_{B_n}^{R'}, t_{D_q}^{R'}] \cr
& \times (K_r^{E_r}(-{\tilde{D}}_{mn}^p F'_q C_{pq}^r + {\tilde{D}}_{mn}^p \bar F'_q C_{p\bar q}^r + {\tilde{D}}_{mn}^{\bar p} F'_q C_{\bar pq}^r + {\tilde{D}}_{mn}^{\bar p} \bar F'_q C_{\bar p \bar q}^r ) \cr
& \quad + \bar K_r^{E_r}(-{\tilde{D}}_{mn}^p F'_q C_{pq}^{\bar r} - {\tilde{D}}_{mn}^p \bar F'_q C_{p\bar q}^{\bar r} - {\tilde{D}}_{mn}^{\bar p} F'_q C_{\bar pq}^{\bar r} + {\tilde{D}}_{mn}^{\bar p} \bar F'_q C_{\bar p \bar q}^{\bar r} ))\end{aligned}$$ Next we consider the second-order poles in the second OPE. There are now four different cases. For a triple collision involving two neighboring connections on the first contour (case in figure \[allTripleCollisions\]), we obtain: $$\begin{aligned}
\label{RRR'2}
& (i\pi R^{-2})^2 \delta'_{\epsilon}(\sigma_1-\sigma')\delta_{\epsilon}(\sigma_2-\sigma') \sum_{m,n=0}^3\sum_{p,q}
{f}^{B_n A_m D_q} t_{D_q}^R t_{A_m}^R t_{B_n}^{R'} (-{\tilde{D'}}_{mn}^p F_q C_{pq} + {\tilde{D'}}_{mn}^{\bar p} \bar F_q C_{\bar p\bar q}) \cr
& + (i\pi R^{-2})^2 \delta_{\epsilon}(\sigma_1-\sigma')\delta'_{\epsilon}(\sigma_2-\sigma') \sum_{m,n=0}^3\sum_{p,q}
{f}^{B_n A_m D_q} (-)^{A_m D_q} t_{A_m}^R t_{D_q}^R t_{B_n}^{R'} (-{\tilde{D'}}_{mn}^p F_q C_{pq} + {\tilde{D'}}_{mn}^{\bar p} \bar F_q C_{\bar p\bar q})\end{aligned}$$ and for a triple collision involving two neighboring connections on the second contour (case in figure \[allTripleCollisions\]), we obtain: $$\begin{aligned}
\label{RR'R'2}
& -(i\pi R^{-2})^2 \delta_{\epsilon}(\sigma'_1-\sigma)\delta'_{\epsilon}(\sigma'_2-\sigma) \sum_{m,n=0}^3\sum_{p,q}
{f}^{ D_q B_n A_m} t_{A_m}^R t_{B_n}^{R'} t_{D_q}^{R'} (-{\tilde{D}}_{mn}^p F'_q C_{pq} + {\tilde{D}}_{mn}^{\bar p} \bar F'_q C_{\bar p\bar q}) \cr
& - (i\pi R^{-2})^2 \delta'_{\epsilon}(\sigma'_1-\sigma)\delta_{\epsilon}(\sigma'_2-\sigma) \sum_{m,n=0}^3\sum_{p,q}
{f}^{ D_q B_n A_m} t_{A_m}^R (-)^{D_q B_n} t_{D_q}^{R'} t_{B_n}^{R'} (-{\tilde{D}}_{mn}^p F'_q C_{pq} + {\tilde{D}}_{mn}^{\bar p} \bar F'_q C_{\bar p\bar q})\end{aligned}$$ We also obtain a non-zero contribution if the two connections that are on the same contour are separated by a third one. When this happens on the first contour (case in figure \[allTripleCollisions\]), we obtain: $$\begin{aligned}
\label{RARR'}
(i\pi R^{-2})^2 & \delta'_{\epsilon}(\sigma_1-\sigma')\delta_{\epsilon}(\sigma_2-\sigma') \frac{1}{2} \sum_{m,n=0}^3\sum_{p,q}
\{ A_R(\sigma), {f}^{B_n A_m D_q} [ t_{D_q}^R , t_{A_m}^R \} t_{B_n}^{R'} \} \cr
& \quad \times (-{\tilde{D'}}_{mn}^p F_q C_{pq} + {\tilde{D'}}_{mn}^{\bar p} \bar F_q C_{\bar p\bar q})\cr
+ (i\pi R^{-2})^2 & \delta'_{\epsilon}(\sigma_1-\sigma')\delta_{\epsilon}(\sigma_2-\sigma') \frac{1}{2} \sum_{m,n,q=0}^3 \sum_{p,r,s}
{f_{C_p}}^{B_n A_m} {f_{E_r}}^{C_p D_q} \{ t_{D_q}^R , t_{A_m}^R ] t_{B_n}^{R'} \cr
& \quad \times ({\tilde{D'}}_{mn}^s F_p C_{sp} - {\tilde{D'}}_{mn}^{\bar s} \bar F_p C_{\bar s\bar p})
(F_r K_r^{E_r} + \bar F_r \bar K_r^{E_r})\cr
+ (i\pi R^{-2})^2 & \delta'_{\epsilon}(\sigma_1-\sigma')\delta_{\epsilon}(\sigma_2-\sigma') \frac{1}{2} \sum_{n,p,q=0}^3\sum_{m,r,s}
{f_{C_p}}^{B_n A_m} {f_{E_r}}^{C_p D_q} \{ t_{D_q}^R , t_{A_m}^R ] t_{B_n}^{R'} \cr
& \quad \times ({\tilde{D'}}_{pn}^s F_m C_{sm} - {\tilde{D'}}_{pn}^{\bar s} \bar F_m C_{\bar s\bar m})
(F_r K_r^{E_r} + \bar F_r \bar K_r^{E_r})\end{aligned}$$ Finally for two connections separated by a third one in the second contour (case in figure \[allTripleCollisions\]), we obtain: $$\begin{aligned}
\label{RR'A'R'}
(i\pi R^{-2})^2 & \delta'_{\epsilon}(\sigma'_1-\sigma)\delta_{\epsilon}(\sigma'_2-\sigma) \frac{1}{2} \sum_{m,n=0}^3\sum_{p,q}
\{ A_{R'}(\sigma'), {f}^{D_q B_n A_m} t_{A_m}^R [ t_{B_n}^{R'},t_{D_q}^{R'}\} \} \cr
& \quad \times (-{\tilde{D}}_{mn}^p F'_q C_{pq} + {\tilde{D}}_{mn}^{\bar p} \bar F'_q C_{\bar p\bar q})\cr
+ (i\pi R^{-2})^2 & \delta'_{\epsilon}(\sigma'_1-\sigma)\delta_{\epsilon}(\sigma'_2-\sigma) \frac{1}{2} \sum_{m,n,q=0}^3\sum_{p,r,s}
{f_{C_p}}^{B_n A_m} {f_{E_r}}^{D_q C_p} t_{A_m}^R \{ t_{B_n}^{R'},t_{D_q}^{R'}] \cr
& \quad \times ({\tilde{D}}_{mn}^s F'_p C_{sp} - {\tilde{D}}_{mn}^{\bar s} \bar F'_p C_{\bar s\bar p} )
(F'_r K_r^{E_r} + \bar F'_r \bar K_r^{E_r})\cr
+ (i\pi R^{-2})^2 & \delta'_{\epsilon}(\sigma'_1-\sigma)\delta_{\epsilon}(\sigma'_2-\sigma) \frac{1}{2} \sum_{n,p,q=0}^3 \sum_{m,r,s}
{f_{C_p}}^{B_n A_m} {f_{E_r}}^{D_q C_p} t_{A_m}^R \{ t_{B_n}^{R'},t_{D_q}^{R'}] \cr
& \quad \times ({\tilde{D}}_{pn}^s F'_m C_{sm} - {\tilde{D}}_{pn}^{\bar s} \bar F'_m C_{\bar s\bar m})
(F'_r K_r^{E_r} + \bar F'_r \bar K_r^{E_r})\\end{aligned}$$
#### Performing the integration.
Next we have to perform the integration over the free coordinates in the previous results. The integrals over the regularized delta functions provide a well-defined answer. This is an advantage of the OPE formalism with respect to the Hamiltonian formalism. The integrals needed are given below. The results are given in the limit $\epsilon \to 0$. $$\begin{aligned}
\int_{b>\sigma_1>\sigma_2>a}d\sigma_1 d\sigma_2 \delta_\epsilon(\sigma_1-\sigma')\delta_\epsilon(\sigma_2-\sigma') = \frac{1}{2} \chi(\sigma';a,b)\end{aligned}$$ $$\begin{aligned}
\int_{b>\sigma_1>\sigma>\sigma_2>a}d\sigma_1 d\sigma_2 \int_c^d d\sigma' \delta'_\epsilon(\sigma_1-\sigma')\delta_\epsilon(\sigma_2-\sigma') = -\frac{1}{2} \chi(\sigma;c,d)\end{aligned}$$ $$\begin{aligned}
& \int_{b>\sigma_1>\sigma_2>a}d\sigma_1 d\sigma_2 \int_c^d d\sigma' \delta'_\epsilon(\sigma_1-\sigma')\delta_\epsilon(\sigma_2-\sigma') \cr
&\qquad = \frac{1}{2} \chi(b;c,d) - \int_{a}^b d\sigma_2 \int_c^d d\sigma' \delta^2_\epsilon(\sigma_2-\sigma')\end{aligned}$$ It is sometimes convenient to write $\chi(b;c,d)$ as $\frac{1}{2}( \chi(b;c,d) + \chi(a;c,d) + \chi(c;a,b) - \chi(d;a,b) )$. Notice that the integral over the squared regularized delta function is divergent in the limit $\epsilon \to 0$: \_[a]{}\^b d\_2 \_c\^d d’ \^2\_(\_2-’) = |\[a,b\]| where we denoted by $|[a,b]\cap[c,d]|$ the length of the overlap of the intervals $[a,b]$ and $[c,d]$. Similarly we have: $$\begin{aligned}
& \int_{b>\sigma_1>\sigma_2>a}d\sigma_1 d\sigma_2 \int_c^d d\sigma' \delta_\epsilon(\sigma_1-\sigma')\delta'_\epsilon(\sigma_2-\sigma') = -\frac{1}{2} \chi(a;c,d) + \int_{a}^b d\sigma_2 \int_c^d d\sigma' \delta^2_\epsilon(\sigma_2-\sigma')\end{aligned}$$ where we can also replace $\chi(a;c,d)$ by $\frac{1}{2}( \chi(b;c,d) + \chi(a;c,d) - \chi(c;a,b) + \chi(d;a,b) ) $.
#### Summing the terms.
Finally we can sum the various contributions from triple collisions. The terms of the form combined with the second and third terms of lead to: $$\begin{aligned}
\label{fusionttt'}
\sum_{M=0}^\infty& (-)^{M+M'+3} \frac{1}{2} \sum_{i=0}^M \sum_{i'=0}^{M'} \int_{[a,b]\cap[c,d]}d\sigma
\left \lfloor \int_\sigma^b A \right\rceil^i \left \lfloor \int_\sigma^d A' \right\rceil^{i'} \cr
& \times (i\pi R^{-2})^2 \tilde{\sum_{m,n,p,q,r}}
{f_{C_p}}^{B_n A_m} {f_{E_r}}^{C_p D_q} \{ t_{D_q}^R , t_{A_m}^R] t_{B_n}^{R'} \cr
&\quad \times (K_r^{E_r}({\tilde{D'}}_{mn}^p F_q C_{pq}^r - {\tilde{D'}}_{mn}^p \bar F_q C_{p\bar q}^r - {\tilde{D'}}_{mn}^{\bar p} F_q C_{\bar pq}^r - {\tilde{D'}}_{mn}^{\bar p} \bar F_q C_{\bar p \bar q}^r \cr
&\quad \quad\quad+ \frac{1}{2} F_r \sum_s ({\tilde{D'}}_{mn}^s F_p C_{sp} + {\tilde{D'}}_{pn}^s F_m C_{sm} - {\tilde{D'}}_{mn}^{\bar s} \bar F_p C_{\bar s\bar p} - {\tilde{D'}}_{pn}^{\bar s} \bar F_m C_{\bar s\bar m})) \cr
& \quad\quad + \bar K_r^{E_r}({\tilde{D'}}_{mn}^p F_q C_{pq}^{\bar r} + {\tilde{D'}}_{mn}^p \bar F_q C_{p\bar q}^{\bar r} + {\tilde{D'}}_{mn}^{\bar p} F_q C_{\bar pq}^{\bar r} - {\tilde{D'}}_{mn}^{\bar p} \bar F_q C_{\bar p \bar q}^{\bar r} \cr
& \quad\quad\quad+ \frac{1}{2} \bar F_r \sum_s ({\tilde{D'}}_{mn}^s F_p C_{sp} + {\tilde{D'}}_{pn}^s F_m C_{sm} - {\tilde{D'}}_{mn}^{\bar s} \bar F_p C_{\bar s\bar p} - {\tilde{D'}}_{pn}^{\bar s} \bar F_m C_{\bar s\bar m}))) \cr
& \times \left \lfloor \int_a^\sigma A \right\rceil^{M-i} \left \lfloor \int_c^\sigma A' \right\rceil^{M'-i'} \end{aligned}$$ In order to shorten the previous expression we introduce the symbol $\tilde{\sum}$ with the following meaning: for each term in the expression, the lower indices of a coefficient $D$ have to be summed over the values $\{0,1,2,3\}$, while all other indices have to be summed over the values $\{0,1,2,3,g\}$. Similarly the terms of the form combined with the second and third terms of lead to: $$\begin{aligned}
\label{fusiontt't'}
\sum_{M=0}^\infty& (-)^{M+M'+3} \frac{1}{2} \sum_{i=0}^M \sum_{i'=0}^{M'} \int_{[a,b]\cap[c,d]}d\sigma
\left \lfloor \int_\sigma^b A \right\rceil^i \left \lfloor \int_\sigma^d A' \right\rceil^{i'} \cr
& \times (i\pi R^{-2})^2 \tilde{\sum_{m,n,p,q,r}}
{f_{C_p}}^{B_n A_m} {f_{E_r}}^{ D_q C_p} t_{A_m}^R \{ t_{B_n}^{R'}, t_{D_q}^{R'}] \cr
&\quad \times (K_r^{E_r}(-{\tilde{D}}_{mn}^p F'_q C_{pq}^r + {\tilde{D}}_{mn}^p \bar F'_q C_{p\bar q}^r + {\tilde{D}}_{mn}^{\bar p} F'_q C_{\bar pq}^r + {\tilde{D}}_{mn}^{\bar p} \bar F'_q C_{\bar p \bar q}^r \cr
&\quad \quad\quad- \frac{1}{2} F'_r \sum_s ({\tilde{D}}_{mn}^s F'_p C_{sp} + {\tilde{D}}_{pn}^s F'_m C_{sm} - {\tilde{D}}_{mn}^{\bar s} \bar F'_p C_{\bar s\bar p} - {\tilde{D}}_{pn}^{\bar s} \bar F'_m C_{\bar s\bar m})) \cr
& \quad\quad + \bar K_r^{E_r}(-{\tilde{D}}_{mn}^p F'_q C_{pq}^{\bar r} - {\tilde{D}}_{mn}^p \bar F'_q C_{p\bar q}^{\bar r} - {\tilde{D}}_{mn}^{\bar p} F'_q C_{\bar pq}^{\bar r} + {\tilde{D}}_{mn}^{\bar p} \bar F'_q C_{\bar p \bar q}^{\bar r} ) \cr
& \quad\quad\quad- \frac{1}{2} \bar F'_r \sum_s ({\tilde{D}}_{mn}^s F'_p C_{sp} + {\tilde{D}}_{pn}^s F'_m C_{sm} - {\tilde{D}}_{mn}^{\bar s} \bar F'_p C_{\bar s\bar p} - {\tilde{D}}_{pn}^{\bar s} \bar F'_m C_{\bar s\bar m}))) \cr
& \times \left \lfloor \int_a^\sigma A \right\rceil^{M-i} \left \lfloor \int_c^\sigma A' \right\rceil^{M'-i'} \end{aligned}$$ The sum of and leads to where the operator $\tilde K$ is given by: $$\begin{aligned}
\label{Ktilde}
\tilde K = \frac{\pi^2}{2} &\tilde{\sum_{m,n,p,q,r}}
{f_{C_p}}^{B_n A_m} {f_{E_r}}^{C_p D_q} \{ t_{D_q}^R , t_{A_m}^R] t_{B_n}^{R'} \cr
&\quad \times (K_r^{E_r}({\tilde{D'}}_{mn}^p F_q C_{pq}^r - {\tilde{D'}}_{mn}^p \bar F_q C_{p\bar q}^r - {\tilde{D'}}_{mn}^{\bar p} F_q C_{\bar pq}^r - {\tilde{D'}}_{mn}^{\bar p} \bar F_q C_{\bar p \bar q}^r \cr
&\quad \quad\quad+ \frac{1}{2} F_r \sum_s ({\tilde{D'}}_{mn}^s F_p C_{sp} + {\tilde{D'}}_{pn}^s F_m C_{sm} - {\tilde{D'}}_{mn}^{\bar s} \bar F_p C_{\bar s\bar p} - {\tilde{D'}}_{pn}^{\bar s} \bar F_m C_{\bar s\bar m})) \cr
& \quad\quad + \bar K_r^{E_r}({\tilde{D'}}_{mn}^p F_q C_{pq}^{\bar r} + {\tilde{D'}}_{mn}^p \bar F_q C_{p\bar q}^{\bar r} + {\tilde{D'}}_{mn}^{\bar p} F_q C_{\bar pq}^{\bar r} - {\tilde{D'}}_{mn}^{\bar p} \bar F_q C_{\bar p \bar q}^{\bar r} \cr
& \quad\quad\quad+ \frac{1}{2} \bar F_r \sum_s ({\tilde{D'}}_{mn}^s F_p C_{sp} + {\tilde{D'}}_{pn}^s F_m C_{sm} - {\tilde{D'}}_{mn}^{\bar s} \bar F_p C_{\bar s\bar p} - {\tilde{D'}}_{pn}^{\bar s} \bar F_m C_{\bar s\bar m})))\cr
%
&+{f_{C_p}}^{B_n A_m} {f_{E_r}}^{ D_q C_p} t_{A_m}^R \{ t_{B_n}^{R'}, t_{D_q}^{R'}] \cr
&\quad \times (K_r^{E_r}(-{\tilde{D}}_{mn}^p F'_q C_{pq}^r + {\tilde{D}}_{mn}^p \bar F'_q C_{p\bar q}^r + {\tilde{D}}_{mn}^{\bar p} F'_q C_{\bar pq}^r + {\tilde{D}}_{mn}^{\bar p} \bar F'_q C_{\bar p \bar q}^r \cr
&\quad \quad\quad- \frac{1}{2} F'_r \sum_s ({\tilde{D}}_{mn}^s F'_p C_{sp} + {\tilde{D}}_{pn}^s F'_m C_{sm} - {\tilde{D}}_{mn}^{\bar s} \bar F'_p C_{\bar s\bar p} - {\tilde{D}}_{pn}^{\bar s} \bar F'_m C_{\bar s\bar m})) \cr
& \quad\quad + \bar K_r^{E_r}(-{\tilde{D}}_{mn}^p F'_q C_{pq}^{\bar r} - {\tilde{D}}_{mn}^p \bar F'_q C_{p\bar q}^{\bar r} - {\tilde{D}}_{mn}^{\bar p} F'_q C_{\bar pq}^{\bar r} + {\tilde{D}}_{mn}^{\bar p} \bar F'_q C_{\bar p \bar q}^{\bar r} ) \cr
& \quad\quad\quad- \frac{1}{2} \bar F'_r \sum_s ({\tilde{D}}_{mn}^s F'_p C_{sp} + {\tilde{D}}_{pn}^s F'_m C_{sm} - {\tilde{D}}_{mn}^{\bar s} \bar F'_p C_{\bar s\bar p} - {\tilde{D}}_{pn}^{\bar s} \bar F'_m C_{\bar s\bar m})))\end{aligned}$$ The combination of the terms and together with the first terms of and simplifies to: $$\begin{aligned}
\label{fusiontt'}
\sum_{M=0}^\infty& (-)^{M+M'+3} \frac{1}{2} \sum_{i=0}^M \sum_{i'=0}^{M'} \int_{a}^b d\sigma \int_{c}^d d\sigma'
\left \lfloor \int_{\sigma}^b A \right\rceil^i \left \lfloor \int_{\sigma'}^d A' \right\rceil^{i'} \cr
& \times (i\pi R^{-2})^2 \delta_\epsilon^2(\sigma-\sigma') \tilde{\sum_{m,n,p,q,r}} f^{B_n A_m D_q} {f_{D_q A_m}}^{E_r} t_{E_r}^R t_{B_n}^{R'}\cr
& \quad (-{\tilde{D'}}_{mn}^p F_q C_{pq} + {\tilde{D'}}_{mn}^{\bar p} \bar F_q C_{\bar p\bar q}
-{\tilde{D}}_{rm}^p F'_q C_{pq} + {\tilde{D}}_{rm}^{\bar p} \bar F'_q C_{\bar p\bar q})\cr
& \times \left \lfloor \int_a^{\sigma} A \right\rceil^{M-i} \left \lfloor \int_c^{\sigma'} A' \right\rceil^{M'-i'} \end{aligned}$$ This can be rewritten as where the matrix $\tilde{tt}$ is given by: $$\begin{aligned}
\label{tttilde}
\tilde{tt}=&\frac{\pi^2}{2} \tilde{\sum_{m,n,p,q,r}} f^{B_n A_m D_q} {f_{D_q A_m}}^{E_r} t_{E_r}^R t_{B_n}^{R'}
\cr& \quad
\times (-{\tilde{D'}}_{mn}^p F_q C_{pq} + {\tilde{D'}}_{mn}^{\bar p} \bar F_q C_{\bar p\bar q}
-{\tilde{D}}_{rm}^p F'_q C_{pq} + {\tilde{D}}_{rm}^{\bar p} \bar F'_q C_{\bar p\bar q})\end{aligned}$$
#### Simplifications in the limit $y-y'=\mathcal{O}(R^{-2})$.
For the purposes of this paper it is interesting to take the limit where the difference of spectral parameters is small. More precisely we assume that the difference $y-y'$ is of order $\mathcal{O}(R^{-2})$. In this limit one term dominates the previous result. This follows essentially from the observation that the $r$ matrix satisfies . In the limit $y-y'=\mathcal{O}(R^{-2})$, the $r$ matrix is no longer of order $R^{-2}$ but rather of order $R^0$. Consequently the coefficients $\tilde{D}_{**}^*$ introduced in behave like: $$\begin{aligned}
\label{DsLim}
& \tilde{D}_{mn}^p = -\frac{1}{4}F_p(y)\frac{y(y^2+y^{-2})^2}{y-y'} + \mathcal{O}(R^0) \cr
& \tilde{D}_{mn}^{\bar p} = -\frac{1}{4}\bar F_p(y)\frac{y(y^2+y^{-2})^2}{y-y'} + \mathcal{O}(R^0) \cr
& \tilde{D'}_{mn}^p = +\frac{1}{4}F_p(y)\frac{y(y^2+y^{-2})^2}{y-y'} + \mathcal{O}(R^0) \cr
& \tilde{D}_{mn}^{\bar p} = +\frac{1}{4}\bar F_p(y)\frac{y(y^2+y^{-2})^2}{y-y'} + \mathcal{O}(R^0) \end{aligned}$$ Using equation , we deduce that simplifies to: $$\begin{aligned}
\label{fusionttt'Lim}
& (-)^{M+M'+3} \frac{1}{2} \sum_{i=0}^M \sum_{i'=0}^{M'} \int_{[a,b]\cap[c,d]}d\sigma
\left \lfloor \int_\sigma^b A \right\rceil^i \left \lfloor \int_\sigma^d A' \right\rceil^{i'} \cr
& \times (i\pi R^{-2})^2 \sum_{m,n,p,q=0}^3\sum_{r}
{f_{C_p}}^{B_n A_m} {f_{E_r}}^{C_p D_q} \{ t_{D_q}^R , t_{A_m}^R] t_{B_n}^{R'} \cr
&\quad \times \left( -\frac{1}{16}\frac{y^2(y^2+y^{-2})^4}{y-y'}\right) \left(\p_y F_r(y) K_r^{E_r} + \p_y \bar F_r(y) \bar K_r^{E_r}\right) \cr
& \times \left \lfloor \int_a^\sigma A \right\rceil^{M-i} \left \lfloor \int_c^\sigma A' \right\rceil^{M'-i'} +\mathcal{O}(R^{-4})\end{aligned}$$ Remarkably the combination of currents that factors out is the derivative of the components of the flat connection with respect to the spectral parameter. Similarly simplifies to: $$\begin{aligned}
\label{fusiontt't'Lim}
& (-)^{M+M'+3} \frac{1}{2} \sum_{i=0}^M \sum_{i'=0}^{M'} \int_{[a,b]\cap[c,d]}d\sigma
\left \lfloor \int_\sigma^b A \right\rceil^i \left \lfloor \int_\sigma^d A' \right\rceil^{i'} \cr
& \times (i\pi R^{-2})^2 \sum_{m,n,p,q=0}^3\sum_{r}
{f_{C_p}}^{B_n A_m} {f_{E_r}}^{ D_q C_p} t_{A_m}^R \{ t_{B_n}^{R'}, t_{D_q}^{R'}] \cr
&\quad \times \left( -\frac{1}{16}\frac{y^2(y^2+y^{-2})^4}{y-y'}\right) \left(\p_y F_r(y) K_r^{E_r} + \p_y \bar F_r(y) \bar K_r^{E_r}\right) \cr
& \times \left \lfloor \int_a^\sigma A \right\rceil^{M-i} \left \lfloor \int_c^\sigma A' \right\rceil^{M'-i'} +\mathcal{O}(R^{-4})\end{aligned}$$ So the operator $\tilde K$ becomes: $$\begin{aligned}
\label{KtildeLim}
\tilde{K} &= - \frac{\pi^2}{32}\frac{y^2(y^2+y^{-2})^4}{y-y'}
\sum_{m,n,p,q=0}^3\sum_{r} \p_y A^{E_r}(y)\cr
& \times \left( {f_{C_p}}^{B_n A_m} {f_{E_r}}^{C_p D_q} \{ t_{D_q}^R , t_{A_m}^R] t_{B_n}^{R'}
+{f_{C_p}}^{B_n A_m} {f_{E_r}}^{ D_q C_p} t_{A_m}^R \{ t_{B_n}^{R'}, t_{D_q}^{R'}] \right)
+\mathcal{O}(R^{0})\end{aligned}$$ Eventually the term remains of order $R^{-4}$, since: \[tttildeLim\] -\_[mn]{}\^p F\_q C\_[pq]{} + \_[mn]{}\^[|p]{} |F\_q C\_[|p|q]{} -\_[rm]{}\^p F’\_q C\_[pq]{} + \_[rm]{}\^[|p]{} |F’\_q C\_[|p|q]{} = 0 +(R\^[-4]{}) Consequently the matrix $\tilde{tt}$ defined in remains of order one. Actually it might be that the term and thus the matrix $\tilde{tt}$ cancel exactly. This could be decided by computing the coefficients of the derivatives of the currents in the current-current OPEs.
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[^1]: In most of the literature the notation are $A_0\to [ab]$, $A_1 \to \alpha$, $A_2 \to a$, $A_3 \to \hat \alpha$. Although slightly less explicit, the notations used here allow for a much more compact writing.
[^2]: More precisely, the constraints implies that the coefficients cancel against each other in the computation of the commutator of equal-times connections . The reason is essentially that the currents $J_0$, $\bar J_0$ appear in the flat connection with no dependence on the spectral parameter. This in turns implies that these coefficients cancel against each other in the computation of the fusion line operators presented in section \[fusion\]. Notice however that the cancellation of some divergences in the line operators depends on the value of the coefficients , see .
[^3]: The ellipses in the current algebra contain subleading singularities, including possible logarithmic singularities. Such terms do not lead to any UV divergences in the line operators. Indeed the integral of these subleading singularities gives a finite result.
[^4]: The precise expression is slightly more complicated than the one given in [@Benichou:2010ts], since the formula (4.15) in [@Benichou:2010ts] does not generalizes to the coset. This implies that the exponentiation observed in [@Benichou:2010ts] is not exact in the case at hand.
[^5]: The author would like to thank N. Gromov for stressing this point.
[^6]: Let us be a bit more precise on this point. On a minkowskian worldsheet, the imaginary shift $i \epsilon$ in e.g. would be replaced by a real shift $\epsilon$. This implies that in the computation of fusion all OPEs would come with an additional factor of $i$. Consequently the second-order corrections from fusion would come with an additional minus sign. This would in turn induce a factor of $i$ in the shift of the T-system.
[^7]: Actually a weaker condition is that all combinations of transfer matrices that enters the T-system are renormalized with the same coefficient.
[^8]: With different conventions (SE-NW), it would be the 1-2 and 1-3 indices.
[^9]: Demanding that the singular terms of order one or less do also vanish is not consistent with the ansatz , since the subleading terms in the current algebra that we did not write in would contribute [@Benichou:2010rk].
[^10]: The author would like to thank B. Vicedo for illuminating discussions on this point.
[^11]: The regularized delta-function in the third line of is not exactly the same one as in the first two lines. However to keep the formulas simple we will adopt the same notations for both regularizations of the delta-function.
[^12]: We can also choose to evaluate these intermediary currents in between the two contours. This would not change equations and .
|
---
abstract: 'A non-local gravity model, which includes a function $f(\Box^{-1} R)$, where $\Box$ is the d’Alembert operator, is considered. For the model with an exponential $f(\Box^{-1} R)$ de Sitter solutions are explored, without any restrictions on the parameters. Using Hubble-normalized variables, the stability of the de Sitter solutions is investigated, with respect to perturbations in the Bianchi I metric, in the case of zero cosmological constant, and sufficient conditions for stability are obtained.'
author:
- |
E. Elizalde$^{1}$[^1] , E.O. Pozdeeva$^{2}$[^2] , and S.Yu. Vernov$^{1,2}$[^3]\
$^1$Instituto de Ciencias del Espacio (ICE/CSIC) and\
Institut d’Estudis Espacials de Catalunya (IEEC)\
Campus UAB, Facultat de Ciències, Torre C5-Parell-2a planta,\
E-08193, Bellaterra (Barcelona), Spain\
$^2$Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,\
Leninskie Gory 1, 119991, Moscow, Russia
date:
title: '**Stability of de Sitter Solutions in Non-local Cosmological Models**'
---
Introduction
============
Modern cosmological observations allow to obtain joint constraints on cosmological parameters (see, for example, [@Kilbinger:2008gk]) and indicate that the current expansion of the Universe is accelerating. The simplest model able to reproduce this late-time cosmic acceleration is general relativity with a cosmological constant. Other models involve modifications of gravity, as for instance $F(R)$ gravity, with $F(R)$ an (in principle) arbitrary function of the scalar curvature (for reviews see [@Review-Nojiri-Odintsov; @Book-Capozziello-Faraoni]).
Higher-derivative corrections to the Einstein–Hilbert action are being actively studied in the context of quantum gravity (as one of the first papers we can mention [@Stelle]). A non-local gravity theory obtained by taking into account quantum effects has been proposed in [@Deser:2007jk]. Also, the string/M-theory is usually considered as a possible theory for all fundamental interactions, including gravity. The appearance of nonlocality within string field theory is a good motivation for studying non-local cosmological models. Most of the non-local cosmological models explicitly include a function of the d’Alembert operator, $\Box$, and either define a non-local modified gravity [@Non-local-gravity-Refs; @Odintsov0708; @Jhingan:2008ym; @Non-local-FR; @Koivisto:2008; @Nojiri:2010pw; @Bambu1104; @ZS; @EPV2011] or add a non-local scalar field, minimally coupled to gravity [@Non-local_scalar].
In this paper, we consider a modification that includes a function of the $\Box^{-1}$ operator. This modification does not assume the existence of a new dimensional parameter in the action and the ensuing non-local model has a local scalar-tensor formulation. The most currently studied example [@Odintsov0708; @Jhingan:2008ym; @Koivisto:2008; @Nojiri:2010pw; @Bambu1104; @ZS; @EPV2011] of a model of this kind with de Sitter solutions is characterized by a function $f(\Box^{-1}R)=f_0 e^{(\Box^{-1}\!R)/\beta}$, where $f_0$ and $\beta$ are real parameters. It has been shown in [@Odintsov0708] that a theory of this kind, being consistent with Solar System tests, may actually lead to the known Universe history sequence: inflation, radiation/matter dominance and a dark epoch. Expanding universe solutions $a\sim t^n$ have been found in [@Odintsov0708; @ZS]. In [@Koivisto:2008] the ensuing cosmology at the four basic epochs: radiation dominated, matter dominated, accelerating, and a general scaling has been studied for non-local models involving, in particular, an exponential form of $f(\eta)$. An explicit mechanism to screen the cosmological constant in non-local gravity was discussed in [@Nojiri:2010pw; @Bambu1104; @ZS].
De Sitter solutions play a very important role in cosmological models, because both inflation and the late-time Universe acceleration can be described as a de Sitter solution with perturbations. A few de Sitter solutions for this model have been found in [@Odintsov0708] and also analyzed in [@Bambu1104]. In [@EPV2011] de Sitter solutions have been obtained without any restriction and it has been shown that the model can have de Sitter solutions only if the function $f(\Box^{-1}R)$ satisfies a given second order linear differential equation. The simplest solution of this equation is an exponential function.
The Bianchi I metric can be considered as a minimal generalization of Friedmann–Lemaître–Robertson–Walker (FLRW) spatially flat metric. Considering the stability of de Sitter solutions in Bianchi I metric we include anisotropic perturbations in our consideration. For the model with the exponential function $f(\Box^{-1}R)$ and nonzero cosmological constant $\Lambda$ the stability of de Sitter solutions in the Bianchi I metric has been analysed in [@EPV2011].
In the case $\Lambda=0$, the stability of the fixed point for the system of equations in terms of Hubble-normalized variables has been discussed in [@Odintsov0708] and further investigated in [@Jhingan:2008ym; @EPV2011]. In all these papers the stability of solutions has been analysed only with respect to isotropic perturbations of the initial conditions, in other words, in the FLRW metric. Here we investigate the stability of de Sitter solutions at $\Lambda=0$ in the Bianchi I metric, and show that the stability conditions, in the Bianchi I metric and in the FLRW metric, are the same.
Non-local gravitational models in the Bianchi I metric
======================================================
Consider the following action for non-local gravity $$\label{nl1} S=\int d^4 x \sqrt{-g}\left\{ \frac{1}{2\kappa^2}\left[
R\Bigl(1 + f\left(\Box^{-1}R \right)\Bigr) -2 \Lambda \right] +
\mathcal{L}_\mathrm{matter} \right\} \, ,$$ where ${\kappa}^2 \equiv 8\pi/{M_{\mathrm{Pl}}}^2$, the Planck mass being $M_{\mathrm{Pl}} = 1.2 \times 10^{19}$ GeV. The determinant of the metric tensor $g_{\mu\nu}$ is $g$, $\Lambda$ is the cosmological constant, $f$ is a differentiable function, and $\mathcal{L}_\mathrm{matter}$ is the matter Lagrangian. We use the signature $(-,+,+,+)$.
Note that the modified gravity action (\[nl1\]) does not include a new dimensional parameter. This non-local model has a local scalar-tensor formulation. Introducing two scalar fields, $\eta$ and $\xi$, we can rewrite action (\[nl1\]) in the following local form: $$\label{anl2} S = \int d^4 x \sqrt{-g}\left\{
\frac{1}{2\kappa^2}\left[R\left(1 + f(\eta)-\xi\right) + \xi\Box\eta -
2 \Lambda \right] + \mathcal{L}_\mathrm{matter} \right\} \, .$$ By varying the action (\[anl2\]) over $\xi$, we get $\Box\eta=R$. Substituting $\eta=\Box^{-1}R$ into action (\[anl2\]), one reobtains action (\[nl1\]).
Variation of action (\[anl2\]) with respect to $\eta$ yields $\Box\xi+ f'(\eta) R=0$, where the prime denotes derivative with respect to $\eta$. Varying action (\[anl2\]) with respect to the metric tensor $g_{\mu\nu}$ yields $$\label{nl4}
\begin{split}
& \frac{1}{2}g_{\mu\nu} \left[R\left(1 + f(\eta) -
\xi\right)
- \partial_\rho \xi \partial^\rho \eta - 2 \Lambda \right]
- R_{\mu\nu}\left(1 + f(\eta) - \xi\right)+{} \\ &+ \frac{1}{2}\left(\partial_\mu \xi \partial_\nu \eta
+ \partial_\mu \eta \partial_\nu \xi \right)
-\left(g_{\mu\nu}\Box - \nabla_\mu \partial_\nu\right)\left( f(\eta) -
\xi\right) + \kappa^2T_{\mathrm{matter}\, \mu\nu}=0\, ,
\end{split}$$ where $\nabla_\mu$ is the covariant derivative and $T_{\mathrm{matter}\,\mu\nu}$ the energy–momentum tensor of matter.
Let us consider the Bianchi I metric with the interval $$\label{Bianchi}
{ds}^{2}={}-{dt}^2+a_1^2(t)dx_1^2+a_2^2(t)dx_2^2+a_3^2(t)dx_3^2.$$ The Bianchi universe models are spatially homogeneous anisotropic cosmological models. Interpreting the solutions of the Friedmann equations as isotropic solutions in the Bianchi I metric, we include anisotropic perturbations in our consideration. A similar stability analysis has been made for cosmological models with scalar fields and phantom scalar fields in [@ABJV0903]. It is convenient to express $a_i$ in terms of new variables $a$ and $\beta_i$ (we use the notation of [@Pereira]): $$a_i(t)= a(t) e^{\beta_i(t)}.$$ Imposing the constraint $\beta_1(t)+\beta_2(t)+\beta_3(t)=0$, at any $t$, one has the following relations $$a(t)=\left[a_1(t)a_2(t)a_3(t)\right]^{1/3},
\quad
H_i\equiv \frac{\dot a_i}{a_i}= H+\dot\beta_i, \quad\mbox{and}\quad
H\equiv \frac{\dot a}{a}=\frac{1}{3}(H_1+H_2+H_3).$$ In the case of the FLRW spatially flat metric we have $a_1=a_2=a_3=a$, all $\beta_i=0$, and $H$ is the Hubble parameter. Following [@Pereira], we introduce the shear $$\sigma^2\equiv \dot\beta_1^2+\dot\beta_2^2+\dot\beta_3^2.$$
In the Bianchi I metric $R=12H^2+6\dot H+\sigma^2$. The equations of motion for the scalar fields are as follows: $$\label{equ3} \ddot\eta={}-3H\dot\eta-12H^2-6\dot H-\sigma^2.$$ $$\label{equ4}
\ddot \xi={}-3H \dot\xi+\left(12H^2+6\dot H+\sigma^2\right)f'(\eta),$$ For a perfect matter fluid, we have $T_{\mathrm{matter}\, 0 0} =
\rho_{\mathrm{m}}$ and $T_{\mathrm{matter}\, i j} = P_{\mathrm{m}} g_{i
j}$. The equation of state is $$\label{equ_rho} \dot\rho_{\mathrm{m}}={}-
3H(P_{\mathrm{m}}+\rho_{\mathrm{m}}).$$ The Einstein equations have the form: $$\label{equ1B} {}\left[\frac{\sigma^2}{2}- 3 H^2\right]\!\left(1 + \phi
- \xi\right) + \frac{1}{2}\dot\xi \dot\eta
- 3H\left( \dot\phi -\dot\xi \right) + \Lambda
+ \kappa^2 \rho_{\mathrm{m}}=0\, ,$$ $$\label{equ2B} \left[2\dot H +
3H^2+\frac{\sigma^2}{2}-\ddot\beta_j-3H\dot\beta_j\right]\! \left(1 +
\phi - \xi\right) + \frac{1}{2}\dot\xi\dot\eta + \ddot\phi -\ddot\xi +
(2H-\dot\beta_j)(\dot\phi -\dot\xi) = \Lambda - \kappa^2
P_{\mathrm{m}},$$ where $\phi\equiv f(\eta)$. Summing Eqs. (\[equ2B\]) for $j=1,2,3$, we get $$\label{equ2Bsum} \left[2\dot H + 3H^2+\frac{\sigma^2}{2}\right]\!
\left(1 + \phi - \xi\right) + \frac{1}{2}\dot\xi \dot\eta +
\ddot\phi-\ddot\xi + 2H\left(\dot\phi -\dot\xi\right) = \Lambda -
\kappa^2 P_{\mathrm{m}}.$$ From equations (\[equ2B\]) it is easy to get $$\left[\ddot\beta_j+3H\dot\beta_j\right]\! \left(1 + \phi -
\xi\right)+\dot\beta_j\left(\dot\phi -\dot\xi\right)=0,
\label{equbeta}$$ $$\label{equvartheta}
\left[\frac{d}{dt}\left(\sigma^2\right)+6H\sigma^2\right]\! \left(1 +
\phi - \xi\right)+2\sigma^2\left(\dot\phi -\dot\xi\right)=0.$$ The functions $H(t)$, $\sigma^2(t)$, $\xi(t)$, $\eta(t)$, and $\rho_m(t)$ can be obtained from equations (\[equ3\])–(\[equ1B\]), (\[equ2Bsum\]) and (\[equvartheta\]). If $H(t)$ and the scalar fields are known, then $\beta_j(t)$ can be found from (\[equbeta\]).
Following [@Bambu1104], we consider matter with a state parameter $w_{\mathrm{m}}\equiv P_{\mathrm{m}}/\rho_{\mathrm{m}}$ to be a constant not equal to $-1$. Thus, Eq. (\[equ\_rho\]) has the following general solution: $$\rho_{\mathrm{m}}=\rho_0\,e^{{}-3(1+w_{\mathrm{m}})H_0t},$$ where $\rho_0$ is an arbitrary constant.
It has been shown in [@EPV2011] that the model (\[anl2\]) can have de Sitter solutions for functions $f$ of the following forms: $$\label{f1}
f_1(\eta)
=\frac{C_2}{4}e^{\eta/2}+C_3e^{3\eta/2}+C_4-\frac{\kappa^2\rho_0}{3(1+3w_{\mathrm{m}})H_0^2}
e^{3(w_{\mathrm{m}}+1)\eta/4}\,,\quad \mbox{for}\quad w_{\mathrm{m}}\neq {}-\frac{1}{3}\,,$$ $$\label{fw13} \tilde{f}_1(\eta)
=\frac{C_2}{4}e^{\eta/2}+C_3e^{3\eta/2}+C_4+\frac{\kappa^2\rho_0}{4H_0^2}\left(1-\frac{1}{3}\eta\right)
e^{\eta/2}, \quad \mbox{for}\quad w_{\mathrm{m}}= {}-\frac{1}{3}\,,$$ where $C_2$, $C_3$, and $C_4$ are arbitrary constants. One can see that the key ingredient common to all these functions $f_i(\eta)$ is the exponential form. For the models with $f(\eta)$ equal to a simple exponential function or to a sum of exponential functions, particular de Sitter solutions have been found in [@Odintsov0708; @Bambu1104]. In the most general form, de Sitter solutions for the case of the exponential function $f(\eta)$ have been obtained in [@EPV2011].
De Sitter solutions and their stability
=======================================
Let us consider the action (\[anl2\]), with $$\label{f} f(\eta)=f_0 e^{\eta/{\beta}}\, ,$$ where $f_0$ and $\beta$ are real constants. This form of $f(\eta)$ is the simplest function which belongs to the set of functions described by (\[f1\]). De Sitter solutions with a constant nonzero $H=H_0$ have the following expression [@EPV2011] $$\begin{aligned}
\label{eta} \eta(t) &=& {}-4H_0(t-t_0),\\
\label{sol_eta}
\xi(t)&=&{}-\frac{3f_0\beta}{3\beta-4}e^{-4H_0(t-t_0)/\beta}+\frac{c_0}{3H_0}e^{-3H_0(t-t_0)}-\xi_0,
\quad \mbox{at}\quad \beta\neq 4/3,\label{sol_xi}\\
\xi(t) &=& {}-f_0(c_0+3H_0(t-t_0))e^{-3H_0(t-t_0)}- \xi_0, \quad \mbox{at}\quad \beta= 4/3,\end{aligned}$$ where $c_0$ and $t_0$ are arbitrary constants, $$\xi_0 ={}-1 -\frac{\Lambda}{3H_0^2},\qquad \rho_{0} = \frac{6
\left(\beta-2\right) H_0^2f_0}{\kappa^2\beta}\ ,\qquad w_{\mathrm{m}} =
{}-1+\frac{4}{3\beta}.
\label{CondeSit}$$
The case $\beta=2$ corresponds to $\rho_0=0$. Thus, the model with exponential $f(\eta)$ has no de Sitter solution if we add matter with $w_{\mathrm{m}} =-1/3$. The type of function $f(\eta)$, which can have such solutions, is given by (\[fw13\]). The case $\beta=4/3$ corresponds to dark matter, because $w_{\mathrm{m}}=0$.
Using (\[equ3\]) and (\[equ4\]), we get equation (\[equ2Bsum\]) in the form $$\label{equ12n2B}
2\left[1 + \frac{\beta-6}{\beta}\phi-
\xi\right]\dot H =4H\left[\frac{\phi\dot\eta}{\beta}-\dot\xi\right]-\frac{\phi\dot\eta^2}{\beta^2}
+\frac{24}{\beta}H^2\phi - \dot\xi \dot\eta - \frac{4\kappa^2}{3\beta}
\rho_{\mathrm{m}}-\left[1 + \frac{\beta-2}{\beta}\phi-
\xi\right]\sigma^2.$$ For $H_0>0$ and $\beta>0$, $$\phi\rightarrow 0,\qquad\xi\rightarrow {}-\xi_0, \quad\mbox{at}\quad t \rightarrow +\infty.$$ Therefore, the coefficient of $\dot H$ in (\[equ12n2B\]) tends to $\Lambda/(3H_0^2)$. In the case of nonzero $\Lambda$, the stability of de Sitter solutions at late times can be analysed without using of the Hubble-normalized variables. It has been found in [@EPV2011] that for $H_0>0$ and $\beta>0$, the de Sitter solutions are stable with respect to fluctuations of the initial conditions in the Bianchi I metric at any nonzero value of $\Lambda$.
Here we consider the stability of de Sitter solutions with respect to fluctuations of the initial conditions in the Bianchi I metric, in the case $\Lambda=0$. To analyze the stability of the de Sitter solutions at $\Lambda=0$, we transform the system of equations using the Hubble-normalized variables $$X={}-\frac{\dot{\eta}}{4H},\qquad
W= \frac{\dot{\xi}}{6Hf},\qquad
Y=\frac{1-\xi}{3f},\qquad
Z=\frac{\kappa^2\rho_m}{3H^2f}, \qquad K=\frac{\sigma^2}{2H^2}$$ and the independent variable, $N$, $$\frac{d}{dN}\equiv a\frac{d}{da}= \frac{1}{H}\frac{d}{dt}\ .$$ The use of these variables makes the equation of motion dimensionless. Equations (\[equ3\]), (\[equ4\]), (\[equ\_rho\]), and (\[equvartheta\]) are equivalent to the following ones, in terms of the new variables, $$\begin{aligned}
\frac{dX}{dN}&=&3(1-X)+\frac{1}{H}\left(\frac32-X\right)\frac{dH}{dN}+\frac{K}{2}\,,\label{dXsys} \\
\frac{dW}{dN}&=&\frac{2}{\beta}\left(1+2WX\right)-3W+\frac{1}{H}\left(\frac{1}{\beta}
-W\right)\frac{dH}{dN}+\frac{K}{3\beta}\,,\label{dWsys}\\
\frac{dZ}{dN}&=&\frac{4}{\beta}(X-1)Z-2\frac{Z}{H}\frac{dH}{dN}\,,\label{dZsys}\\
\left(\frac{dK}{dN}\right.&+&\left.\frac{2K}{H}\frac{dH}{dN}+6K\right)(3Y+1)=4K\left(\frac{2X}{\beta}+3W\right)\label{eq5}.\end{aligned}$$
To get the full system of the first order differential equations we need to get one for $\frac{dH}{dN}$ and to eliminate $Y$. To do this, we use Eq. (\[equ1B\]), which can be written in terms of the new variables as $$Y={}-\frac{1}{3}+\frac{2\beta(2X-3)W-4X-\beta Z}{\beta(K-3)}. \label{equ1nv}$$ Differentiating (\[equ1nv\]), substituting (\[dXsys\])–(\[eq5\]), and using $$\frac{dY}{dN}=2\left(\frac{2XY}{\beta}-W\right)=\frac{4X}{3\beta^2(3-K)}\Bigl(\beta(K-3)+6\beta(3-2X)W+12X+3\beta Z\Bigr)-2W,\label{dY}$$ one gets $$\label{dH}
\begin{split}
\Bigl(2(2X-3)(\beta W-1)-\beta Z-2K\Bigr)\frac{1}{H}\frac{dH}{dN}&=\frac{8(3-K)X^2}{3\beta}+{}\\
{}+\frac{4}{3}(6-9\beta W+K)X+2Z+12(\beta W-1)&+\left(2-\frac{2}{3}Z+(2W+Z)\beta\right)K+\frac{2}{3}K^2,
\end{split}$$
In terms of the new variables, the system (\[dXsys\])–(\[eq5\]), (\[dH\]) has the following fixed point $$\label{FixPoi}
H=H_0,\qquad X_0=1,\qquad Z_0=\frac{2(\beta-2)}{\beta}, \qquad W_0=\frac{2}{3\beta-4}, \qquad K_0=0,$$ which corresponds to de Sitter solution for $\beta\neq 4/3$, with $c_{0}=0$. In the case of an arbitrary $c_0$, for the de Sitter solution, we get $$W=\frac{2}{3\beta-4}-\frac{c_0}{6H_0f_0}e^{-(3-4/\beta)(N-N_0)},$$ where $N_0=H_0t_0$. The function $W$ tends to infinity at large $N$ for $\beta<4/3$ and $\lim\limits_{N\rightarrow\infty} W=W_0$ at $\beta>4/3$. So, the fixed point can be stable only at $\beta>4/3$. Under this condition all de Sitter solutions tend to a fixed point, what means that, for any $\varepsilon>0$, there exists a number, $N_1$, such that the de Sitter solution is in the $\varepsilon/2$ neighborhood of the fixed point, for all $N>N_1$. Therefore, the stability of the fixed point guarantees the stability of all de Sitter solutions.
For $\beta=4/3$ the function $W$, corresponding to de Sitter solutions, depends on $N$ for any value of parameters. Thus, this choice of dimensionless variable is not suitable to analyse stability of the de Sitter solutions for $\beta=4/3$. Here we will deal with the case $\beta\neq 4/3$, only.
Let us consider perturbations in the neighborhood of (\[FixPoi\]): $$X=1+\varepsilon x_1,\quad Z=Z_0(1+\varepsilon z_1),\quad
W=W_0(1+\varepsilon w_1),\quad H=H_0(1+\varepsilon h_1),\quad K=\varepsilon k_1,$$ where $\varepsilon$ is a small parameter. To first order in $\varepsilon$, after some work we obtain the system of linear equations: $$\label{equx1z1} \frac{dx_1}{dN}={}-3x_1+\frac{1}{2}\frac{dh_1}{dN}+\frac{1}{2} k_1,\qquad \frac{dz_1}{dN}=\frac{4}{\beta}x_1-2\frac{dh_1}{dN},$$ $$\label{equw1}
\frac{dw_1}{dN}=\frac{4}{\beta}x_1+\frac{\beta-4}{2\beta}\frac{dh_1}{dN}+\left(\frac{4}{\beta}-3\right)w_1
+\frac{3\beta-4}{6\beta}k_1,$$ $$\label{equh1}
\frac{dh_1}{dN}=\frac{8(4-\beta)}{\beta(3\beta^2-11\beta+12)}x_1
-\frac{2(3\beta-4)(\beta-2)}{\beta(3\beta^2-11\beta+12)}z_1-\frac{3\beta^2-5\beta+4}{3\beta^2-11\beta+12} k_1,$$ $$\label{equk1}
\frac{dk_1}{dN}=\left(\frac{8}\beta-6\right)k_1.$$ Solving (\[equk1\]), we get $$k_1(N)=b_1e^{-(6-8/\beta)N},$$ where $b_1$ is an arbitrary constant and $k_1$ tends to zero for $\displaystyle N\rightarrow\infty$, if and only if $\beta>4/3$.
Substituting $k_1$ and (\[equh1\]) into (\[equx1z1\]), we get a system of two inhomogeneous differential equations. As known, the general solution of this system is a sum of the general solution of the corresponding homogeneous system and a particular solution of inhomogeneous one. The homogeneous system corresponds to the FLRW metric (the case $K=0$) and those general solution, which has been obtained in [@EPV2011], is bounded and tends to zero for $N\rightarrow\infty$, if $4/3<\beta\leqslant 2$. For any $\beta$ from this interval a particular solution of the inhomogeneous system tends to zero as well, because $k_1$ tends to zero at $\beta>4/3$. Therefore, the perturbations $x_1$ and $z_1$ decrease provided $4/3<\beta<2$. Substituting $x_1(N)$ and $z_1(N)$ into Eqs. (\[equw1\]) and (\[equh1\]) we get that $h_1(N)$ and $w_1(N)$ decrease as well. Note that $h_1(N)$ has a part, $H_{1}$, which does not depend on $N$ and, therefore, it can be considered as part of $H_0$. This result corresponds to the fact that, for $\Lambda=0$, the value of $H_0$ can be selected arbitrarily; thus, one can choose $\tilde{H}_0=H_0+H_{1}$ instead of $H_0$. We can summarize the above saying that the de Sitter solutions are stable with respect to perturbations of the Bianchi I metric, in the case $4/3<\beta\leqslant 2$. If $f_0>0$, then the stable de Sitter solution corresponds to $\rho_0\leqslant 0$.
Conclusions
===========
We have investigated de Sitter solutions in the non-local gravity model described by the action (\[nl1\]) (see [@Odintsov0708]). We have used the local formulation of the model (\[anl2\]), which includes two scalar fields. We have specifically considered the case of the exponential function $f(\eta)$, which is the simplest and most studied case, corresponding to the model (\[anl2\]), that admits de Sitter solutions.
In [@EPV2011], we have discussed the stability of de Sitter solutions in the Bianchi I metrics and obtained that, for $H_0>0$ and $\beta>0$, de Sitter solutions are stable, for all nonzero values of $\Lambda$. Here we have proved that in the case $\Lambda=0$ de Sitter solutions are stable for $H_0>0$ and $4/3<\beta\leqslant 2$. Thus, our conclusion is that de Sitter solutions, which are stable with respect to isotropic perturbations, are also stable with respect to anisotropic perturbations of the Bianchi I metric.
The authors thank Sergei D. Odintsov for useful discussions. E.E. was supported in part by MICINN (Spain), projects FIS2006-02842 and FIS2010-15640, by the CPAN Consolider Ingenio Project, and by AGAUR (Generalitat de Catalunya), contract 2009SGR-994. E.P. and S.V. are supported in part by the RFBR grant 11-01-00894, E.P. also by a state contract of the Russian Ministry of Education and Science 14.740.12.0846, and S.V. by a grant of the Russian Ministry of Education and Science NSh-4142.2010.2 and by CPAN10-PD12 (ICE, Barcelona, Spain).
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[^1]: E-mail: [email protected], [email protected]
[^2]: E-mail: [email protected]
[^3]: E-mail: [email protected], [email protected]
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---
abstract: 'In this paper we deal with verification of safety properties of parameterized systems with a tree topology. The verification problem is translated to a purely logical problem of finding a finite countermodel for a first-order formula, which further resolved by a generic finite model finding procedure. A finite countermodel method is shown is at least as powerful as regular tree model checking and as the methods based on monotonic abstraction and backwards symbolic reachability. The practical efficiency of the method is illustrated on a set of examples taken from the literature.'
author:
- Alexei Lisitsa
title: Finite countermodels for safety verification of parameterized tree systems
---
Finite Countermodel Method
==========================
The development of general automated methods for the verification of infinite-state and parameterized systems poses a major challenge. In general, such problems are undecidable, so one cannot hope for the ultimate solution and the development should focus on the restricted classes of systems and properties. In this paper we deal with a very general method for verification of *safety* properties of infinite-state systems which is based on a simple idea. If an evolution of a computational system is faithfully modeled by a derivation in a classical first-order logic then safety verification (non-reachability of unsafe states) can be reduced to the disproving of a first-order formula. The latter task can be (partially, at least) tackled by generic automated procedures searching for *finite* countermodels.
Such an approach to verification was originated in the research on formal verification of security protocols ([@Weid99; @S01; @GL08; @JW09; @Gut]) and later has been extended to the wider classes of infinite-state and parameterized verification tasks. Completeness of the approach for particular classes of systems (lossy channel systems) and relative completeness with respect to general method of regular model checking has been established in [@AL10] and [@AL10arxiv] respectively. The method has also been applied to the verification of safety properties of general term rewriting systems and its relative completeness with respect to the tree completion techniques has been shown in [@AL11].
In this paper we continue investigation of applicability of the method and show its power in the context of verification of safety properties of *parameterized tree-like systems*. We show the relative completeness of FMC methods with respect to *regular tree model checking* [@RTMC] and with respect to the methods based on monotonic abstraction and symbolic backwards reachability analysis [@PTS].
Preliminaries
-------------
\[sec:pre\] We assume that the reader is familiar with the basics of first-order logic. In particular, we use without definitions the following concepts: first-order predicate logic, first-order models, interpretations of relational, functional and constant symbols, satisfaction $M \models \varphi$ of a formula $\varphi$ in a model $M$, semantical consequence $ \varphi\models\psi$, deducibility (derivability) $\vdash$ in first-order logic. We denote interpretations by square brackets, so, for example, $[f]$ denotes an interpretation of a functional symbol $f$ in a model. We also use the existence of *complete* finite model finding procedures for the first-order predicate logic [@Model; @McCune], which given a first-order sentence $\varphi$ eventually produce a finite model for $\varphi$ if such a model exists.
Regular Tree Model Checking
===========================
Regular Tree Model Checking (RTMC) is a general method for the verification of parameterized systems that have tree topology [@RTMC; @ARTMC]. The definitions of this section are largely borrowed from [@RTMC].
Trees
-----
A *ranked alphabet* is a pair $(\Sigma,\rho)$, where $\Sigma$ is a finite set of symbols and $\rho : \Sigma \rightarrow Nat$ is an *arity mapping*. Let $\Sigma_{p}$ denote the set of symbols in $\Sigma$ of arity $p$. Intuitively, each node of a tree is a labeled with a symbol from $\Sigma$ and the out-degree of the node is the same as the arity of the symbol.
\[def:tree\] A tree $T$ over a ranked alphabet $(\Sigma, \rho)$ is a pair $(S,\lambda)$, where
- $S$, called tree structure, is a finite set of finite sequences over $Nat$. Each sequence $n$ in $S$ is called a *node* of $T$. $S$ is prefix-closed set, that is, if $S$ contains a node $n = b_{1}b_{2}\ldots b_{k}$, then $S$ also contains the node $n' = b_{1}b_{2}\ldots b_{k-1}$ and the nodes $n_{r} = b_{1}b_{2}\ldots b_{k-1}r$, for $r: 0 \le r < b_{k}$. We say that $n'$ is a parent of $n$, and that $n$ is a child of $n'$. A *leaf* of $T$ is a node $n$ which does not have any child.
- $\lambda$ is a a mapping from $S$ to $\Sigma$. the number of children of $n$ is equal to $\rho(\lambda(n))$. In particular, if $n$ is a leaf then $\lambda(n) \in \Sigma_{0}$.
We use $T(\Sigma)$ to denote the set of all trees over $\Sigma$. We write $n \in T$ when $n \in S$ and $f \in T$ denotes that $\lambda(n) = f$ for some $n \in T$. For a tree $T = (S, \lambda)$ and a node $n \in T$, a subtree of $T$ rooted at $n$ is a tree $T' = (S', \lambda_{n})$, where $S' \subseteq \{b \mid nb \in S\}$ and $\lambda_{n}(b) = \lambda(nb)$. Notice, that according to this definition a subtree of a tree T consists not necessarily all descendants of some node in T.
For a ranked alphabet $\Sigma$ let $\Sigma^{\bullet}(m)$ be the ranked alphabet which contains all tuples $(f_{1}, \ldots, f_{m})$ such that $m \ge 1$ and $f_{1}, \ldots, f_{m}\in \Sigma_{p}$ for some $p$. We put $\rho((f_{1}, \ldots, f_{m})) = \rho(f_{1})$.
For trees $T_{1} = (S_{1}, \lambda_{1})$ and $T_{2} = (S_{2}, \lambda_{2})$ we say that $T_{1}$ and $T_{2}$ are structurally equivalent, if $S_{1} = S_{2}$.
Let $T_{1} = (S, \lambda_{1}), \ldots, T_{m} = (S,\lambda_{m})$ are structurally equivalent trees. Then $T_{1} \times \ldots \times T_{m}$ denotes the tree $T = (S,\lambda)$ where $\lambda(n) = (\lambda_{1}(n), \ldots, \lambda_{m}(n)$).
Tree Automata and Transducers
-----------------------------
A *tree language* is a set of trees.
A tree automaton over a ranked alphabet $\Sigma$ is a triple $A = (Q,F,\delta)$, where $Q$ is a finite set of states, $F\subseteq Q$ is a set of final states, and $\delta$ is a transition relation, represented by a finite set of rules of the form $(q_{1}, \ldots, q_{p}) \rightarrow^{f} q$, where $f \in \Sigma_{p}$ and $q_{1}, \ldots q_{p},q \in Q$.
A *run* $r$ of $A$ on a tree $T = (S,\lambda) \in T(\Sigma)$ is a mapping from $S$ to $Q$ such that for each node $n \in T$ with children $n_{1}, \ldots, n_{k}$: $(r(n_{1}, \ldots, r(n_{k})) \rightarrow^{\lambda (n)} r(n)) \in \delta$.
For a state $q \in S$ we denote by $T \Rightarrow_{A}^{r} q$ that $r$ ia run of $A$ on $T$ such that $r(\epsilon) = q$. We say that $A$ accepts $T$ if $T \Rightarrow_{A}^{r} q$ for some run $r$ and some $q \in F$. The language of trees accepted by an automaton $A$ is defined as $L(A) = \{T \mid T \;\mbox{is accepted by}\; A\}$. The tree language $L$ is called *regular* iff there is a tree automaton $A$ such that $L = L(A)$.
A tree automaton over an alphabet $\Sigma^{\bullet}(2)$ is called *tree transducer*.
Let $D$ be a tree transducer over an alphabet $\Sigma^{\bullet}(2)$.
An one-step transition relation $R_{D} \subseteq T(\Sigma) \times T(\Sigma)$ is defined as $R^{D} = \{(T,T') \mid T \times T' \; \mbox{ is accepted by}\; D \}$. The reflexive and transitive closure of $R_{D}$ is denoted by $R_{D}^{\ast}$.
We use $\circ$ to denote the composition of two binary relations defined in the standard way. Let $R^{i}$ denote the $i$th power of $R$ i.e. $i$ compositions of $R$. Then we have $R^{\ast} = \cup_{i \ge 0} R^{i}$.
For any $L \subseteq T(\Sigma)$ and $R \subseteq T(\Sigma) \times T(\Sigma)$ we denote by $L \star R$ the set $\{y \mid \exists x (x,y) \in L \times T(\Sigma) \cap R \}$.
Regular Tree Model Checking deals with the following basic verification task.
\[problem:rtmc\] Given two tree automata $A_{I}$ and $A_{U}$ over an alphabet $\Sigma$ and a tree transducer $D$ over $\Sigma^{\bullet}(2)$. Does $(L(A_{I}) \star R^{\ast}_{D}) \cap L(A_{U}) = \emptyset$ hold?
In verification scenario, trees over $\Sigma$ denote states of the system to be verified, tree automata $A_{I}$ and $A_{U}$ define the sets of trees representing *initial*, respectively, *unsafe* states. Tree transducer $D$ defines the transitions of the system. Under such assumptions, the positive answer to an instance of Problem \[problem:rtmc\] means the *safety* property is established, namely, none of the unsafe states is reachable along the system transitions from any of the initial states.
The verification in RTMC proceeds by producing a tree transducer $TR$ approximating $R^{\ast}_{D}$ from above, that is $R^{\ast}_{D} \subseteq L(TR)$, and showing the emptiness of the set $(L(A_{I}) \star L(TR)) \cap L(A_{U})$
From RTMC to FMC
================
In this section we show that the generic regular tree model checking question posed in Problem \[problem:rtmc\] can be reduced to a purely logical problem of finding a finite countermodel for a first-order logic formula, which then can be resolved by application of generic model finding procedure. We show also the relative completeness of finite countermodel method with respect to RTMC.
Assume we are given an instance of the basic verification problem (over ranking alphabet $\Sigma$), that is
- a tree automaton $A_{I} = (Q_{I},F_{I}, \delta_{I})$ accepting a regular set of initial states;
- a tree automaton $A_{U} = (Q_{U}, F_{U}, \delta_{U} )$ accepting a regular set of unsafe states;
- a tree transducer $D = (Q_{D}, F_{D}, \delta_{D} )$ representation one-step transition relation $R_{D}$.
Now define a set formulae of first-order predicate logic as follows. The vocabulary consists of
- constants for all elements of $Q_{I} \sqcup Q_{U} \sqcup Q_{D} \sqcup \Sigma_{0}$;
- unary predicate symbols $Init^{(1)}$, $Unsafe^{(1)}$
- binary predicate symbols $Init^{2}$, $Unsafe^{2}$, $R$;
- a ternary predicate symbol $T$;
- a $p$-ary functional symbol $f_{\theta}$ for every $\theta \in \Sigma_{p}$
Given any tree $\tau$ from $T(\Sigma)$ define its term translation $t_{\tau}$ by induction:
- $t_{\tau} = c$ for a tree $\tau$ with one node labeled by $c \in \Sigma_{0}$;
- $t_{\tau} = f_{\theta}(t_{\tau_{1}}, \ldots, t_{\tau_{p}})$ for a tree $\tau$ with the root labeled by $\theta \in \Sigma_{p}$ and children $\tau_{1}, \ldots \tau_{p}$.
Let $\Phi$ be the set of the following formulae, which are all assumed to be universally closed:
1. $Init^{(2) }(a,q)$ for every $a \in \Sigma_{0}$, $q \in Q_{I}$ and $\rightarrow^{a} q$ in $\delta_{I}$;
2. $Init^{(2) }(x_{1},q_{1}) \land \ldots \land Init^{(2) }(x_{p},q_{p}) \rightarrow Init^{(2) }(f_{\theta}(x_{1},\ldots,x_{p}),q)$ for every $(q_{1},\ldots,q_{p}) \rightarrow^{\theta} q$ in $\delta_{I}$;
3. $\vee_{q \in F_{I}} Init^{(2) }(x,q) \rightarrow Init^{(1)}(x)$;
4. $Unsafe^{(2) }(a,q)$ for every $a \in \Sigma_{0}$, $q \in Q_{U}$ and $\rightarrow^{a} q$ in $\delta_{U}$;
5. $Unsafe^{(2) }(x_{1},q_{1}) \land \ldots \land Unsafe^{(2) }(x_{p},q_{p}) \rightarrow Unsafe^{(2) } (f_{\theta}(x_{1},\ldots,x_{p}),q)$ for every $(q_{1},\ldots,q_{p}) \rightarrow^{\theta} q$ in $\delta_{U}$;
6. $\vee_{q \in F_{U}} Unsafe^{(2) }(x,q) \rightarrow Unsafe^{(1)}(x)$;
7. $T(a,b,q)$ for every $\rightarrow^{(a,b)} q$ in $\delta_{D}$;
8. $T(x_{1},y_{1},q_{1}) \land \ldots \land T(x_{p},y_{p},q_{p}) \rightarrow T(f_{\theta_{1}}(x_{1}, \ldots, x_{p}),
f_{\theta_{2}}(y_{1}, \ldots, y_{p}),q)$ for every $(q_{1}, \ldots, q_{p})\rightarrow^{\theta_{1},\theta_{2}} q$ in $\delta_{D}$;
9. $\vee_{q \in F_{D}} T(x,y,q) \rightarrow R(x,y)$;
10. $R(x,x)$;
11. $R(x,y) \land R(y,z) \rightarrow R(x,z)$.
\[prop:init\_unsafe\](adequacy of Init and Unsafe translations)
If $\tau \in L(A_{I})$ then $\Phi \vdash Init^{(1)}(t_{\tau})$
If $\tau \in L(A_{U})$ then $\Phi \vdash Unsafe^{(1)}(t_{\tau})$
We prove only the first statement, the second one is dealt with in the same way.
\[lemma:\] For any tree $\tau$ and any run $r$ if $\tau \Rightarrow^{r}_{A_{I}} q$ then $\Phi \vdash Init^{2}(t_{\tau},q)$.
[*Proof of Lemma*]{}. By induction on the depth of the trees.
- *Induction Base Case*. Assume $\tau$ has a depth $0$, that is consists of one vertex labeled by some $a \in \Sigma_{0}$. Let $r$ be a run such that $\tau \Rightarrow^{r}_{A_{I}} q$. It follows (by the definition of run) that $\rightarrow^{a} q \in \delta_{I}$ and then $Init^{(2)}(a,q)$ is in $\Phi$ (by clause 1 of the definition of $\Phi$) and therefore $\Phi \vdash Init^{(2)}(a,q)$ Finally notice that term translation $t_{\tau}$ of $\tau$ is $a$.
- *Induction Step Case*. Assume $\tau$ has a root labeled by $\theta \in \Sigma_{p}$ and $\tau_{1}, \ldots, t_{p}$ are children of the root. For a run $r$ on $\tau$, assume $\tau \Rightarrow^{r}_{A_{I}} q$ and $\tau_{1} \Rightarrow^{r}_{A_{I}} q_{1},
\ldots, \tau_{p} \Rightarrow^{r}_{A_{I}} q_{n}$. By the definition of a run we have $(q_{1}, \ldots, q_{p}) \rightarrow^{\theta} q$ is in $\delta_{I}$. By induction assumption we have $\Phi \vdash Init^{(2)}(t_{\tau_{1}},q_{1}), \ldots, \Phi \vdash Init^{(2)}(t_{\tau_{p}},q_{p})$. By using clause 2 of the definition of $\Phi$ we get\
$\Phi \vdash Init^{(2)}(f_{\theta}(t_{\tau_{1}}, \ldots, t_{\tau_{p}}),q)$ and $\Phi \vdash Init^{2}(t_{\tau},q)$ $\Box$
Returning to the proof of the proposition we notice that if $\tau \in L(A_{I})$ then there is a run $r$ such that $\tau^{r}_{A_{I}} q$ for some $q \in F_{I}$. By Lemma \[lemma:\] $\Phi \vdash \vee_{q \in F_{I}}
Init^{(2)}(t_{\tau}, q)$. By using clause 3 of the definition of $\Phi$ we then get $\Phi \vdash Init^{(1)}(t_{\tau})$
\[prop:encoding\](adequacy of encoding)
If $\tau \in L(A_{I}) \star R^{\ast}_{D}$ then $\Phi \vdash \exists x \; Init^{(1)}(x) \land R(x,t_{\tau})$
Easy induction on the length of transition sequences.
- *Induction Base Case.* Let $\tau \in L(A_{I}) \subseteq L(A_{I}) \star R^{\ast}_{D}$. Then $\Phi \vdash Init^{(1)}(t_{\tau})$ (by Proposition \[prop:init\_unsafe\]) and, further $\Phi \vdash \exists x \; Init^{(1)}(x) \land R(x,t_{\tau})$ (using clause 10).
- *Induction Step Case.* Let $\tau \in L(A_{I}) \star R^{n+1}_{D}$. Then there exists $\tau'$ such that $\tau' \in L(A_{I}) \star R^{n}_{D}$ and $R(\tau',\tau)$ holds. Further, by the argument analogous to the proof of Proposition \[prop:init\_unsafe\], $R(\tau',\tau)$ entails $\Phi \vdash \vee_{q \in F_{D}}T(t_{\tau'},t_{\tau},q)$ and further $\Phi \vdash R(t_{\tau'},t_{\tau})$ (using clause 9). from this, the clause 11 and the induction assumption $\Phi \vdash \exists x \; Init^{(1)}(x) \land R(x,t_{\tau})$ follows.
Assume $\tau \in L(A_{I}) \star R^{\ast}_{D}$ then by definition of $\star$ there exists $\tau_{0} \in L(A_{I})$ such that $R_{D}^{\ast}(\tau_{0},\tau)$ holds.
\[cor:verification\](correctness of the verification method)
If $\Phi \not\vdash \exists x \exists y (Init^{(1)}(x) \land R(x,y) \land Unsafe^{(1)}(y)$ then $(L(A_{I}) \star R^{\ast}_{D}) \cap L(A_{U})=\emptyset$
The corollary \[cor:verification\] serves as a formal underpinning of the proposed FCM (finite countermodel) verification method. In order to prove safety, that is $(L(A_{I}) \star R^{\ast}_{D}) \cap L(A_{U})=\emptyset$ it is sufficient to demonstrate $\Phi \not\vdash \exists x \exists y (Init^{(1)}(x) \land R(x,y) \land Unsafe^{(1)}(y)$. In the FCM method we delegate this task to the generic finite model finding procedure, which searches for the finite countermodels for\
$\Phi \rightarrow \exists x \exists y (Init^{(1)}(x) \land R(x,y) \land Unsafe^{(1)}(y)$.
Relative completeness of FCM with respect to RTMC
-------------------------------------------------
In general, searching for finite countermodels to disprove non-valid first-order formulae may not always lead to success, because for some formulae countermodels are inevitably infinite. Here we show, however, this is not the case for the first-order encodings of the problems which can be positively answered by Regular Tree Model Checking. It follows then that FCM is at least as powerful in establishing safety as RTMC, provided a complete finite model finding procedure is used.
\[th:completeness\](relative completeness of FCM) Given an instance of the basic verification problem for RTMC, that is two tree automata $A_{I}$ and $A_{U}$ over an alphabet $\Sigma$ and a tree transducer $D = (Q_{D},F_{D},\delta_{D})$ over $\Sigma^{\bullet}(2)$. If there exists a regular tree language ${\cal R}$ such that $(L(A_{I}) \star R^{\ast}_{D}) \subseteq {\cal R}$ and ${\cal R} \cap L(A_{U}) = \emptyset$ then there is a finite countermodel for $\Phi \rightarrow \exists x \exists y (Init^{(1)}(x) \land R(x,y) \land Unsafe^{(1)}(y)$
Let $A = (Q,F,\delta)$ be a *deterministic* tree automaton recognizing the tree language ${\cal R}$, i.e. $L(A) = {\cal R}$. We take $Q \cup Q_{I} \cup Q_{U} \cup Q_{D} \cup \{ e \}$ to be domain of the required finite model. Here $e$ is a distinct element not in $Q \cup Q_{I} \cup Q_{U} \cup Q_{D}$.
Define interpretations as follows.
- For $a \in \Sigma_{0}$ $[a] = q \in Q$ such that $\rightarrow^{a} q$ is in $\delta$;
- For $\theta \in \Sigma_{p}$ $[f_{\theta}](q_{1}, \ldots, q_{p}) = q$ for any $(q_{1}, \ldots, q_{p}) \rightarrow^{\theta} q$ in $\delta$, and $[f_{\theta}](\ldots) = e$ otherwise;
- Interpretations of $Init^{2}$ and $Init^{1}$ are defined inductively, as the least subsets of pairs, respectively, elements of the domain, satisfying the formulae $(1)$ - $(3)$ (and assuming all interpretations above);
- Interpretations of $Unsafe^{2}$ and $Unsafe^{1}$ are defined inductively, as the least subsets of pairs, respectively, elements of the domain, satisfying the formulae $(4)$ - $(6)$ (and assuming all interpretations above);
- Interpretation of $T$ is defined inductively, as the least subsets of triples satisfying the formulae $(7)$ - $(8)$ (and assuming all interpretations above);
- Interpretation of $R$ and $Init^{1}$ is defined inductively, as the least subsets of pairs, satisfying the formulae $(9)$ - $(11)$ (and assuming all interpretations above);
Such defined a finite model satisfies $\Phi$ (by construction). Now we check that $\neg \exists x \exists y (Init^{(1)}(x) \land R(x,y) \land Unsafe^{(1)}(y)$ is satisfied in the model We have
1. $[Init^{(1)}] \star [R] \subseteq \{[t] \mid t \in L(A_{I}) \star R^{\ast}_{D}\}$ (by the minimality condition on interpretations of $Init^{(1)}$ and $R$);
2. $\{[t]\mid t \in L(A_{I}) \star R^{ast}_{D}\} \subseteq F \subseteq Q$ (by interpretations of terms and condition $(L(A_{I}) \star R^{\ast}_{D}) \subseteq {\cal R}$);
3. $[Init^{(1)}] \star [R] \subseteq F$ (by 1 and 2);
4. $[Unsafe^{(1)}] = \{[t] \mid t \in L(A_{U})\}$ (by definition of $[Unsafe^{1}]$, in particular by the minimality condition);
5. $\{[t] \mid t \in L(A_{U})\} \cap F = \emptyset$ (by condition ${\cal R} \cap L(A_{U}) = \emptyset$);
6. $Unsafe^{(1)} \cap F = \emptyset$ (by 4 and 5);
7. $[Init^{(1)}] \star [R] \cap Unsafe^{(1)} = \emptyset$ (by 3 and 6);
The case study
==============
In this section we illustrate FCM method by applying it to the verification of Two-way Token protocol. The system consists of finite-state processes connected to form a binary tree structure. Each process stores a single bit which represents the fact that the process has a token. During operation of the protocol the token can be passed up or down the tree. The correctness condition is that no two or more tokens ever appear. In parameterized verification we would like to establish correctness for all possible sizes of trees.
We take RTMC-style specification of Two-way Token from [@RTMC]. Let $\Sigma = \{t,n,T,N\}$ be the alphabet. Here $t,n \in \Sigma_{0}$ label processes on the leaves of a tree, and $T,N \in \Sigma_{2}$ label processes on the inner nodes of a tree. Further, $t,T$ label processes with a token and $n,N$ label processes without tokens. The automaton $A_{I} = (Q_{I},F_{I},\delta_{I})$ accepts the initial configurations of the protocol, that is the trees with exactly one token. Here $Q_{I} = \{q_{0}, q_{1}\}$, $F_{I} = \{q_{1}\}$ and $\delta_{I}$ consists of the following transition rules:
--------------------------------------- --------------------------------------- ---------------------------------------
$\rightarrow^{n} q_{0}$ $\rightarrow^{t} q_{1}$ $(q_{0},q_{0}) \rightarrow^{T} q_{1}$
$(q_{0},q_{0}) \rightarrow^{N} q_{0}$ $(q_{0},q_{1}) \rightarrow^{N} q_{1}$ $(q_{1},q_{0}) \rightarrow^{N} q_{1}$
--------------------------------------- --------------------------------------- ---------------------------------------
The tree transducer $D = (Q_{D},F_{D},\delta_{D})$ over $\Sigma^{\bullet}(2)$ represents the transitions of the protocol. Here $Q_{D} = \{q_{0},q_{1},q_{2},q_{3}\}$, $F = \{q_{2}\}$ and $\delta_{D}$ consists of the following transition rules:
------------------------------------------- -------------------------------------------
$\rightarrow^{(n,n)} q_{0}$ $\rightarrow^{(t,n)} q_{1}$
$\rightarrow^{(n,t)} q_{3}$ $(q_{0},q_{0}) \rightarrow^{(N,N)} q_{0}$
$(q_{0},q_{2}) \rightarrow^{(N,N)} q_{2}$ $(q_{2},q_{0}) \rightarrow^{(N,N)} q_{2}$
$(q_{0},q_{0}) \rightarrow^{(T,N)} q_{1}$ $(q_{3},q_{0}) \rightarrow^{(T,N)} q_{2}$
$(q_{0},q_{3}) \rightarrow^{(T,N)} q_{2}$ $(q_{0},q_{1}) \rightarrow^{(N,T)} q_{2}$
$(q_{1},q_{0}) \rightarrow^{(N,T)} q_{2}$ $(q_{0},q_{0}) \rightarrow^{(N,T)} q_{3}$
------------------------------------------- -------------------------------------------
The automaton $A_{U} = (Q_{U},F_{U},\delta_{U})$ accepts unsafe (bad) configurations of the protocol, that is the trees with at least two tokens. Here $Q_{U} = \{q_{0}, q_{1}, q_{2}\}$, $F_{U} = \{q_{2}\}$ and $\delta_{U}$ consists of the following transition rules:
--------------------------------------- --------------------------------------- ---------------------------------------
$\rightarrow^{n} q_{0}$ $\rightarrow^{t} q_{1}$ $(q_{0},q_{0}) \rightarrow^{N} q_{0}$
$(q_{0},q_{0}) \rightarrow^{T} q_{1}$ $(q_{0},q_{1}) \rightarrow^{N} q_{1}$ $(q_{1},q_{0}) \rightarrow^{N} q_{1}$
$(q_{0},q_{1}) \rightarrow^{T} q_{2}$ $(q_{1},q_{0}) \rightarrow^{T} q_{2}$ $(q_{1},q_{1}) \rightarrow^{T} q_{2}$
$(q_{0},q_{2}) \rightarrow^{T} q_{2}$ $(q_{2},q_{0}) \rightarrow^{T} q_{2}$ $(q_{1},q_{2}) \rightarrow^{T} q_{2}$
$(q_{2},q_{1}) \rightarrow^{T} q_{2}$ $(q_{2},q_{2}) \rightarrow^{T} q_{2}$ $(q_{1},q_{1}) \rightarrow^{N} q_{2}$
$(q_{0},q_{2}) \rightarrow^{N} q_{2}$ $(q_{2},q_{0}) \rightarrow^{N} q_{2}$ $(q_{1},q_{2}) \rightarrow^{N} q_{2}$
$(q_{2},q_{1}) \rightarrow^{N} q_{2}$ $(q_{2},q_{2}) \rightarrow^{N} q_{2}$
--------------------------------------- --------------------------------------- ---------------------------------------
The set $\Phi$ of the following formulae presents a translation of the verification problem. We use the syntax of first-order logic used in Mace4 finite model finder [@McCune].
T(n,n,q0).
T(t,n,q1).
T(n,t,q3).
T(x,z,q0) & T(y,v,q0) -> T(fT(x,y),fN(z,v),q1).
T(x,z,q1) & T(y,v,q0) -> T(fN(x,y),fT(z,v),q2).
T(x,z,q0) & T(y,v,q1) -> T(fN(x,y),fT(z,v),q2).
T(x,z,q0) & T(y,v,q0) -> T(fN(x,y),fN(z,v),q0).
T(x,z,q0) & T(y,v,q2) -> T(fN(x,y),fN(z,v),q2).
T(x,z,q2) & T(y,v,q0) -> T(fN(x,y),fN(z,v),q2).
T(x,z,q3) & T(y,v,q0) -> T(fT(x,y),fN(z,v),q2).
T(x,z,q0) & T(y,v,q3) -> T(fT(x,y),fN(z,v),q2).
T(x,z,q0) & T(y,v,q0) -> T(fN(x,y),fT(z,v),q3).
% Initial states automaton
Init(n,q0).
Init(t,q1).
Init(x,q0) & Init(y,q0) -> Init(fT(x,y),q1).
Init(x,q0) & Init(y,q1) -> Init(fN(x,y),q1).
Init(x,q0) & Init(y,q0) -> Init(fN(x,y),q0).
Init(x,q1) & Init(y,q0) -> Init(fN(x,y),q1).
% Bad states automaton
Bad(n,q0).
Bad(t,q1).
Bad(x,q0) & Bad(y,q0) -> Bad(fN(x,y),q0).
Bad(x,q0) & Bad(y,q0) -> Bad(fT(x,y),q1).
Bad(x,q0) & Bad(y,q1) -> Bad(fN(x,y),q1).
Bad(x,q1) & Bad(y,q0) -> Bad(fN(x,y),q0).
Bad(x,q0) & Bad(y,q1) -> Bad(fT(x,y),q2).
Bad(x,q1) & Bad(y,q0) -> Bad(fT(x,y),q2).
Bad(x,q1) & Bad(y,q1) -> Bad(fN(x,y),q2).
Bad(x,q1) & Bad(y,q2) -> Bad(fT(x,y),q2).
Bad(x,q2) & Bad(y,q1) -> Bad(fT(x,y),q2).
Bad(x,q2) & Bad(y,q2) -> Bad(fT(x,y),q2).
Bad(x,q1) & Bad(y,q1) -> Bad(fN(x,y),q2).
Bad(x,q0) & Bad(y,q2) -> Bad(fN(x,y),q2).
Bad(x,q2) & Bad(y,q0) -> Bad(fN(x,y),q2).
Bad(x,q1) & Bad(y,q2) -> Bad(fN(x,y),q2).
Bad(x,q2) & Bad(y,q1) -> Bad(fN(x,y),q2).
Bad(x,q2) & Bad(y,q2) -> Bad(fN(x,y),q2).
T(x,y,q2) -> R(x,y).
R(x,y) & R(y,z) -> R(x,z).
Init(x,q1) -> Init1(x).
Bad(x,q2) -> Bad1(x).
According to Proposition \[prop:encoding\] and Corollary \[cor:verification\] to establish safety for Two-way Token protocol it does suffice to show $\Phi \not\vdash \exists x \exists y ((Init1(x) \land R(x,y)) \land Bad1(y))$. We delegate this task to Mace4 finite model finder and it finds a countermodel for $\Phi \rightarrow \exists x \exists y ((Init1(x) \land R(x,y)) \land Bad1(y))$ in $0.03s$. The parameterized protocol is verified. Actual Mace4 input and output can be found in [@AL09].
Monotonic abstraction and symbolic reachability vc FCM
======================================================
Regular Tree Model Checking provides with a general method for the verification parameterized protocols for tree-shaped architectures. In [@PTS] a lightweight alternative to RTMC was proposed. It utilizes a generic approach to safety verification using *monotonic abstraction* and *symbolic reachability* applied to tree rewriting systems. This generic approach has previously been successfully applied to the verification of parameterized linear system [@Mon] (as an alternative to standard Regular Model Checking). In this section we demonstrate the flexibility of the FCM approach and show that one can translate safety verification problems for parameterized tree-shaped systems formulated using tree rewriting into the problem of disproving a first-order formulae using the same basic principles (reachability as FO derivability). For defined translation we show the relative completeness of the FCM with respect to monotonic abstraction and symbolic reachability and demonstrate its practical efficiency.
Parameterized Tree Systems
--------------------------
\[subsec:pts\]
The approach of [@PTS] to the verification of parameterized tree systems adopts the following viewpoint. A configuration of the system is represented by a tree over a finite alphabet, where elements of the alphabet represent the local states of the individual processes. The behaviors of the system is specified by a set of *tree rewriting* rules, which describe how the processes perform transitions. Transitions are enabled by the local states of the process together with the states of children and parent processes.
\[def:tree\_over-states\] A tree $T$ over a set of states $Q$ is a pair $(S,\lambda)$, where
- $S$ is a tree structure (cf. Definition \[def:tree\])
- $\lambda$ is a a mapping from $S$ to $Q$.
Notice that trees over a set of states are similar to the trees over ranked alphabets (Definition \[def:tree\]) with the only difference is that the same state can label the vertices with different number of children (e.g. leaves of the tree and internal vertices).
In what follows to assume for simplicity of presentation (after [@PTS]) that all trees are (no more than) binary, that is every node has either one or two children (internal node) or no children (leaf). It is straightforward to extend all constructions and results to the general case of not necessarily binary trees. Notice that configurations of the tree systems will be modeled by complete binary trees. Incomplete binary trees (which may contain nodes with one child) will appear only in the rewrite rules.
\[def:pts\] A parameterized tree system ${\cal P}$ is a tuple $(Q,R)$, where $Q$ is a finite set of states and $R \subseteq T(Q \times Q)$ is a finite set of rewrite rules.
For each rule $r = (S,\lambda) \in R$ we associate two trees, called *left* and *right* trees of $r$. We define $lhs(r) = (S,lhs(\lambda))$ and $rhs(r) = (S,rhs(\lambda))$, where $lhs(r)$ and $rhs(r)$ are left, respectively right projection of $\lambda$.
We will denote (labeled) binary trees by bracket expressions in a standard way.
Let $Q = \{q_{0}, q_{1}, q_{2}\}$ then $r = \langle q_{0},q_{1}\rangle(\langle q_{1},q_{1} \rangle, \langle q_{2},q_{0}\rangle) \in T(Q \times Q)$ is a rewriting rule. This rule has $\bullet (\bullet, \bullet)$ as it tree structure with one root and two leaves. The pairs of states $\langle q_{0},q_{1}\rangle$, $\langle q_{1},q_{1} \rangle$, $\langle q_{2},q_{0}\rangle$ label the root and two leaves respectively. We also have $lhs(r) = q_{0}(q_{1},q_{2})$ and $rhs(r) = q_{1}(q_{1},q_{0})$.
Let $Q$ be as above then $\langle q_{1},q_{2}\rangle (\langle q_{0},q_{1}\rangle)$ is a rewriting rule with the structure of incomplete binary tree $\bullet (\bullet)$
Given a parameterized tree system ${\cal P} = (Q,R)$ define one step transition relation $\Rightarrow_{\cal P} \subseteq
T(Q) \times T(Q)$ as follows: $\tau_{1} \Rightarrow_{\cal P} \tau_{2}$ iff for some $r \in R$ $\tau_{1}$ contains $lhs(r)$ as a subtree and $\tau_{2}$ obtained from $\tau_{1}$ by replacing this subtree with $rhs(r)$. Since $lhs(r)$ and $rhs(r)$ have the same tree structure, the operation of replacement and one step transition relation are well-defined.
Let ${\cal P} = (Q,R)$ with $Q = \{q_{0}, q_{1}, q_{2}\}$ and $R =\{\langle q_{0},q_{1}\rangle(\langle q_{1},q_{1} \rangle, \langle q_{2},q_{0}\rangle) \}$. Then we have (with the subtrees refered to in the definition of $\Rightarrow_{\cal P}$ inderlined):
- $\underline{q_{0}(q_{1},q_{2})} \Rightarrow_{\cal P} \underline{q_{1}(q_{1},q_{0})}$;
- $q_{2}(\underline{q_{0}(q_{1},q_{2})},q_{1}) \Rightarrow_{\cal P} q_{2}(\underline{q_{1}(q_{1},q_{0}}),q_{1})$;
- $\underline{q_{0}}(\underline{q_{1}}(q_{1},q_{0}),\underline{q_{2}}(q_{0},q_{2})) \Rightarrow_{\cal P}
\underline{q_{1}}(\underline{q_{1}}(q_{1},q_{0}),\underline{q_{0}}(q_{0},q_{2})) $;
Denote transitive and reflexive closure of $\Rightarrow_{\cal P}$ by $\Rightarrow_{\cal P}^{\ast}$.
\[def:embedding\](embedding) For $\tau_{1} = (S_{1},\lambda_{1})$ and $\tau_{2} = (S_{2},\lambda_{2})$ an injective function $f: S_{1} \rightarrow S_{2}$ is called embedding iff
- $s\cdot b \in S$ implies $f(s) \cdot b \le f(s \cdot b)$ for any $s \in S$
- $\lambda_{1}(s) = \lambda_{2}(f(s))$
We use $\tau_{1} \preceq_{f} \tau_{2}$ to denote that $f$ is embedding of $\tau_{1}$ into $\tau_{2}$ and write $\tau_{1} \preceq \tau_{2}$ iff there exists $f$ such that $\tau_{1} \preceq_{f} \tau_{2}$.
Using embeddability relation $\prec$ allows to describe infinite families of trees by *finitary* means.
We call a set of trees $T \subseteq T(Q)$ *finitely based* iff there is a finite set $B \subseteq T(Q)$ such that $T = \{ \tau \mid \exists \tau' \in B \tau' \preceq \tau \}$. Notice that finitely based set of trees are upwards closed with respect to $\preceq$, that is $\tau \in T$ and $\tau \preceq \tau'$ implies $\tau' \in T$.
Many safety verification problems for parameterized tree system can be reduced to the following coverability problem.
\[prob:mono\] Given a parameterized tree system ${\cal P} = (Q,R)$, a regulat tree language $Init \subseteq T(Q)$ of initial configurations and finitely based set of unsafe configurations $Unsafe \subseteq T(Q)$. Does $\tau \not\Rightarrow_{\cal P}^{\ast} \tau'$ hold for all $\tau \in Init$ and all $\tau' \in Unsafe$?
We formally defined regular tree languages over ranked alphabets. Regular tree languages over (unranked) states can be defined in a various ways. We will fix a particular convention in Assumption 1 below.
Now we briefly outline the monotonic abstraction approach [@PTS] to verification. Given the coverability problem above [@PTS] defines the monotonic abstraction $\Rightarrow_{\cal P}^{\cal A}$ of the transition relation $\Rightarrow_{\cal P}$ as follows. We have $\tau_{1} \Rightarrow_{\cal P}^{\cal A} \tau_{2}$ iff there exists a tree $\tau'$ such that $\tau' \preceq \tau_{1}$ and $\tau' \Rightarrow_{\cal P} \tau_{2}$. It is clear that such defined $\Rightarrow_{\cal P}^{\cal A}$ is an over-approximation of $\Rightarrow_{\cal P}$. To establish the safety property, i.e. to get a positive answer to the question of Problem \[prob:mono\], [@PTS] proposes using a symbolic backward reachability algorithm for monotonic abstraction. Starting with an *upwards* closed (wrt to $\prec$) set of unsafe configuration $Unsafe$ the algorithm proceeds iteratively with the computation of the set of configurations backwards reachable along $\Rightarrow_{\cal P}^{\cal A}$ from $Unsafe$:
- $U_{0} = Unsafe$
- $U_{i+1} = U_{i} \cup Pre(U_{i})$
where $Pre(U) = \{\tau \mid \exists \tau' \in U \land \tau \Rightarrow_{\cal P}^{\cal A} \tau' \}$. Since the relation $\preceq$ is a *well quasi-ordering* [@Kruskal:tree] this iterative process is guaranteed to stabilize, i.e. $U_{n+1} = U_{n} = U$ for some finite $n$. During the computation each $U_{i}$ is represented symbolically by a finite set of generators. Once the process stabilized on some $U$ the check is preformed on whether $Init \cap U = \emptyset$. If this condition is satisfied then the safety is established, for no bad configuration can be reached from initial configurations via $\Rightarrow-{\cal P}^{\cal A}$ and, a fortiori, via $\Rightarrow_{\cal P}$.
Parameterized Tree systems to FCM
---------------------------------
Here we show how to translate the coverability problem (Problem \[prob:mono\]) into the task of disproving a first-order formula and demonstrate the *relative completeness* of the FCM method with respect to monotonic abstraction approach.
Assume we are given an instance of the coverability problem, that is
- a parameterized tree system ${\cal P} = (Q,R)$,
- a regular tree language $Init$ of initial configurations, given by a tree automaton $A_{I} = (Q_{I},F,\delta)$, and
- finitely based set of unsafe configurations $Unsafe$ given by a finite set of generators $Un \subseteq T(Q)$.
For a set of states $Q$ let ${\cal F}_{Q} = \{ f_{q}^{(2)} \mid q \in Q\} \cup \{e\}$ be the set of corresponding binary functional symbols extended with a distinct functional symbol $e$ of arity $0$ (constant).
For any *complete* binary tree $\tau \in T(Q)$ define its term translation $t_{\tau}$ in vocabulary ${\cal F}_{Q}$ inductively:
- $t_{\tau} = f_{q}(e,e)$ if $\tau$ is a tree with one node labeled by a state $q$;
- $t_{\tau} = f_{q}(t_{\tau_{1}},t_{\tau_{2}})$ if the root of $\tau$ has two children and $\tau = q(\tau_{1},\tau_{2})$;
For any not necessarily complete binary tree $\tau \in T(Q \times Q)$ define inductively its translation $s_{\tau}$ as a set of pairs of terms in vocabulary ${\cal F}_{Q}$:
- $s_{\tau} = \{\langle f_{q_{1}}(e,e), f_{q_{2}}(e,e) \rangle \}$ if $\tau$ is a tree with one node labeled with states $(q_{1},q_{2})$;
- $s_{\tau} =\{\langle f_{q_1}(\rho_{1}, \rho_{2}),f_{q_2}(\rho_{3},\rho_{4}) \rangle \mid \langle \rho_{1},\rho_{3} \rangle \in s_{\tau_{1}}, \langle \rho_{2},\rho_{4} \rangle \in s_{\tau_{2}}\}$ if the root of $\tau$ is labeled by $(q_{1},q_{2})$ and it has two children $\tau_{1}$ and $\tau_{2}$, i.e. if $\tau = (q_1,q_2)(\tau_{1}, \tau_{2})$;
- $s_{\tau} = \{\langle f_{q_1}(\rho_{1},e),f_{q_2}(\rho_{2},e)\rangle \mid \langle \rho_{1},\rho_{2} \rangle \in s_{\tau_{1}}\} \cup \{\langle f_{q_1}(e,\rho_{1}),f_{q_2}(e,\rho_{2})\rangle \mid \langle \rho_{1},\rho_{2} \rangle \in s_{\tau_{1}} \}$ if the root of $\tau$ is labeled by $(q_{1},q_{2})$ and it has one child $\tau_{1}$, i.e. if $\tau = (q_{1},q_{2})(\tau_1)$.
For $\langle \rho_{1},\rho_{2} \rangle \in s_{\tau}$ we denote by $\rho_{1}^{gen}$ (by $\rho_{2}^{gen}$) a generalized term obtained by replacement of all occurences of constant $e$ in $\rho_{1}$ (in $\rho_{2}$, respectively,) with distinct variables.
Now we define first-order translation of the set of rules $R$ as the following set $\Phi_{R}$ of first-order formulae, which are all assumed to be universally closed:
1. $R(\rho_{1}^{gen},\rho_{2}^{gen})$ for all $r \in R$ and $\langle \rho_{1},\rho_{2} \rangle \in s_{r}$
2. $R(x,x)$
3. $R(x,y) \land R(y,z) \rightarrow R(x,z)$
4. $R(x,y) \land R(z,v) \rightarrow R(f_{q}(x,z),f_{q}(y,v))$ for all $q \in Q$\
In $1)$ we additionally require that generalizations $\rho_{1}^{gen}$ and $\rho_{2}^{gen}$ should be consistent, that means the variables used in the generalizations are the same in the same positions.
Now for simplicity we make the following
An automaton $A_{I} = (Q_{I},F_{I},\delta_{I})$ is given over ranked alphabet ${\cal F}_{Q}$.
We define the translation of $A_{I}$ as the set $\Phi_{I}$ of first-order formulae
5. $I_{\theta}(f_{q}(e,e))$ for all $\rightarrow^{e} \theta'$ and $(\theta',\theta') \rightarrow^{f_{q}} \theta$ in $\delta_{I}$;
6. $I_{\theta_{1}}(x) \land I_{\theta_{2}}(y) \rightarrow I_{\theta_{3}}(f_{q}(x,y))$ for all $(\theta_{1},\theta_{2}) \rightarrow^{f_{q}} \theta_{3})$ in $\delta_{I}$.
7. $\vee_{\theta \in F_{I}} I_{\theta}(x) \rightarrow Init(x)$
Let $A_{U} = (Q_{U},F_{U}, \delta_{U})$ is a tree automaton recognizing finitely based set $Unsafe$. Then its translation $\Phi_{U}$ defined analogously to the translation of $A_{I}$:
8. $U_{\theta}(f_{q}(e,e))$ for all $\rightarrow^{e} \theta'$ and $(\theta',\theta') \rightarrow^{f_{q}} \theta$ in $\delta_{U}$;
9. $U_{\theta_{1}}(x) \land U_{\theta_{2}}(y) \rightarrow U_{\theta_{3}}(f_{q}(x,y))$ for all $(\theta_{1},\theta_{2}) \rightarrow^{f_{q}} \theta_{3})$ in $\delta_{U}$.
10. $\vee_{\theta \in F_{U}} U_{\theta}(x) \rightarrow Unsafe(x)$
(Adequacy of encoding) For an instance of the coverability problem and the translation defined above the following holds true:
1. For any $\tau_{1}, \tau_{2} \in T(Q)$ if $\tau_{1} \Rightarrow_{\cal P}^{\ast} \tau_{2}$ then $\Phi_{R} \vdash R(t_{\tau_{1}},t_{\tau_{2}})$
2. For any $\tau \in Init$ $\Phi_{I} \vdash Init(t_{\tau})$;
3. For any $\tau \in Unsafe$ $\Phi_{U} \vdash Unsafe(t_{\tau})$
proceeds by straightforward inspection of definitions.
(safety verification) If $\Phi_{R} \cup \Phi_{I} \cup \Phi_{U} \not\vdash \exists x \exists y Init(x) \land Unsafe(y) \land R(x,y)$ then the coverability problem has a positive answer, that is $\tau \not\Rightarrow_{\cal P}^{\ast} \tau'$ holds for all $\tau \in Init$ and all $\tau' \in Unsafe$.
(relative completeness) Given a parameterized tree system ${\cal P} = (Q,R)$, the tree regular language of initial configurations $Init$, finitely based set of unsafe configurations $Unsafe$. Assume the backward symbolic reachability algorithm for monotonic abstraction described above terminates with the fixed-point $U = U_{n+1} = U_{n}$ for some $n$ and $Init \cap U = \emptyset$. Then there exists a finite model for $\Phi_{R} \land \Phi_{I} \land \Phi_{U}
\land \neg (\exists x \exists y Init(x) \land Unsafe(y) \land R(x,y))$.
First we observe that since the fixed-point $U$ has a finite set of generators it is a regular tree language. Let $A_{U^{\ast}} = (Q_{U^{\ast}},F_{U^{\ast}},\delta_{U^{\ast}})$ be a deterministic tree automaton recognizing $U$. We take $Q_{U^{\ast}}$ as a domain of the required model. Interpretations of all functional symbols from ${\cal F}_{Q}$ are given by $\delta_{U^{\ast}}$:
- $[f_{q}](\theta_{1},\theta_{2}) = \theta_{3}$ iff $(\theta_{1},\theta_{2}) \rightarrow^{f_{q}} \theta_{3}$ is in $\delta_{U^{\ast}}$
- $[e] = \theta$, where $\rightarrow^{e} \theta$ is in $\delta_{U^{\ast}}$.
Interpretations of predicates $R, I_{\theta}, Init, U_{\theta}, Unsafe$ are defined inductively as the least sets of tuples, or elements of the domains satisfying the axioms 1-4, 5-7, 8-10, respectively. That concludes the definition of the model which we denote by ${\cal M}$. We have ${\cal M} \models \Phi_{R} \land \Phi_{I} \land \Phi_{U}$ by construction. Now we check that ${\cal M} \models \neg (\exists x \exists y Init(x) \land Unsafe(y) \land R(x,y))$ is satisfied in the model. We have
1. $[Init]\star[R] \subseteq \{[\tau] \mid \exists \tau' \in Init \; \tau' \Rightarrow^{\ast}_{\cal P} \tau \}$ (by the minimality conditions on interpretations of $Init$ and $R$)
2. $\{[\tau] \mid \exists \tau' \in Init \; \tau'
\Rightarrow^{\ast}_{\cal P} \tau \} \subseteq \bar{F}_{U^{\ast}} = Q - F_{U^{\ast}}$ (by assumption $U \cap Init = \emptyset$)
3. $[Unsafe] \subseteq F_{U^{\ast}}$ (by $Unsafe \subseteq U$);
4. $([Init]\star[R]) \cap [Unsafe] = \emptyset$ (by 1-3).
The case study, II
------------------
In this section we illustrate the discussed variation of the FCM method by applying it again to the verification of Two-way Token Protocol, but specified differently. The specification of this protocol using trees over states and tree rewriting is taken from [@PTS]. The set of states $Q = \{n,t\}$, where $n$ and $t$ denote local states ‘no token’ and ‘token’, respectively. The set of $R$ of rewriting rules consists of the following rules:
- $\langle t,n \rangle(\langle n,t \rangle)$;
- $\langle n,t \rangle(\langle t,n \rangle)$;
The set $Init$ of initial configurations consists all complete binary trees over $Q$ with exactly one token. The set $Unsafe$ of unsafe configuration consists of all complete binary trees over $Q$ with at least two tokens. The set of the formulae $\Phi$ below is a first-order translation (in Mace4 syntax) of the verification task.
% rewriting rules
R(ft(fn(y,z),x),fn(ft(y,z),x)).
R(ft(x,fn(y,z)),fn(x,ft(y,z))).
R(fn(ft(y,z),x),ft(fn(y,z),x)).
R(fn(x,ft(y,z)),ft(x,fn(y,z))).
% reflexivity
R(x,x).
%congruence
(R(x,y) & R(z,v)) -> R(fn(x,z),fn(y,v)).
(R(x,y) & R(z,v)) -> R(ft(x,z),ft(y,v)).
% transitivity
(R(x,y) & R(y,z)) -> R(x,z).
% Initial states automaton
I1(fn(e,e)).
(I1(x) & I1(y)) -> I1(fn(x,y)).
(I1(x) & I1(y)) -> Init(ft(x,y)).
(Init(x) & I1(y)) -> Init(fn(x,y)).
(I1(x) & Init(y)) -> Init(fn(x,y)).
% Unsafe states automaton
B1(ft(x,y)).
B1(x) -> B1(fn(x,y)).
B1(y) -> B1(fn(x,y)).
B1(x) -> Unsafe(ft(x,y)).
B1(x) -> Unsafe(ft(y,x)).
B1(x) & B1(y) -> Unsafe(fn(x,y)).
B1(x) & B1(y) -> Unsafe(ft(x,y)).
Unsafe(x) -> Unsafe(fn(x,y)).
Unsafe(x) -> Unsafe(fn(y,x)).
Unsafe(x) -> Unsafe(ft(x,y)).
Unsafe(x) -> Unsafe(ft(y,x)).
Now, in order to establish safety, it is sufficient to show that $\Phi \not\vdash \exists x \exists y Init(x) \land R(x,y) \land Unsafe(y)$. Finite model finder Mace4 finds a model for $\Phi \land \neg(\exists x \exists y Init(x) \land R(x,y) \land Unsafe(y))$ in $0.04s$.
Experimental results
====================
We have applied both presented versions of FMC method to the verification of several parameterized tree-shaped systems. The tasks specified in RTMC tradition were taken from [@RTMC] and the first translation was used. To compare with monotonic abstraction based methods we used the second translation for the tasks from [@PTS].
In the experiments we used the finite model finder Mace4[@McCune] within the package Prover9-Mace4, Version 0.5, December 2007. The system configuration used in the experiments: Microsoft Windows XP Professional, Version 2002, Intel(R) Core(TM)2 Duo CPU, T7100 @ 1.8Ghz 1.79Ghz, 1.00 GB of RAM. The time measurements are done by Mace4 itself, upon completion of the model search it communicates the CPU time used. The table below lists the parameterized tree protocols and shows the time it took Mace4 to find a countermodel and verify a safety property. The time shown is an average of 10 attempts. We also show the time reported on the verification of the same protocols by alternative methods.
FCM vs RTMC
-----------
Protocol Time Time reported in [@ARTMC]$^{\ast}$
--------------- ------- ------------------------------------
Token 0.02s 0.06s
Two-way Token 0.03s 0.09s
$^{\ast}$ the system configuration used in [@ARTMC] was *Intel Centrino 1.6GHZ with 768MB of RAM*
Notice that [@ARTMC] discusses different methods for enhancement of RTMC within the abstract-check-refine paradigm and we included in the table the best times reported in [@ARTMC] for each verification problem.
FCM vs monotonic Abstraction
----------------------------
Protocol Time Time reported in [@PTS]$^{\ast}$
----------------- ------- ----------------------------------
Token 0.02s 1s
Two-way Token 0.03s 1s
Percolate 0.02s 1s
Leader Election 0.03s 1s
Tree-arbiter 0.02s 37s
IEEE 1394 0.04s 1h15m25s
$^{\ast}$ the system configuration used in [@PTS] was *dual Opteron 2.8 GHZ with 8 GB of RAM*
All specifications used in the experiments and Mace4 output can be found in [@AL09].
Related work
============
\[sec:rel\]
As mentioned Section 1 the approach to verification using the modeling of protocol executions by first-order derivations and together with countermodel finding for disproving was introduced within the research on the formal analysis of cryptographic protocols ([@Weid99],[@S01],[@GL08], [@JW09], [@Gut]).
This work continues the exploration of the FCM approach presented in [@AL09; @Avocs09; @ALWIng10; @AL10; @AL10arxiv; @AL11]. In [@AL10](which is an extended version of [@Avocs09]) it was shown that FCM provides a decision procedure for safety verification for lossy channel systems, and that FCM can be used for efficient verification of parameterised cache coherence protocols. The relative completeness of the FCM with respect to regular model checking and methods based on monotonic abstraction for linear parameterized systems was established in [@AL10arxiv](which is an extended version of the abstract [@ALWIng10]). The relative completeness of the FCM with respect to tree completion techniques for general term rewriting systems is shown in [@AL11]. Our treatment of tree rewriting in \[subsec:pts\] can be seen as a particular case of term rewriting considered in [@AL11] with slightly different translation of tree automata. Detailed comparison and/or unified treatment of FCM vs Tree Completion vs RTMC vs Monotonic Abstraction to be given elsewhere. Here we notice only that the reason for FCM to succeed in verification of safety of various classess of infinite-state and parameterized systems is the presence of *regular* sets of configurations (invariants) covering all reachable configurations and disjoint with the sets of unsafe configurations.
In a more general context, the work we present in this paper is related to the concepts of *proof by consistency* [@pbc], and *inductionless induction* [@ii] and can be seen as an investigation into the power of these concepts in the particular setting of the verification of parameterized tree systems via finite countermodel finding.
Conclusion
==========
We have shown how to apply generic finite model finders in the parameterized verification of tree-shaped systems, have demonstrated the relative completeness of the method with respect to regular tree model checking and to the methods based on monotonic abstraction and have illustrated its practical efficiency. Future work includes the investigation of scalability of FCM, and its applications to software verification.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author is grateful to anonymous referees of FMCAD 2011 conference who provided with many helpful comments on the previous version of this paper.
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abstract: |
Fringe fields in multipole magnets can have a variety of effects on the linear and nonlinear dynamics of particles moving along an accelerator beamline. An accurate model of an accelerator must include realistic models of the magnet fringe fields. Fringe fields for dipoles are well understood and can be modelled at an early stage of accelerator design in such codes as MAD8, MADX, GPT or ELEGANT. However, usually it is not until the final stages of a design project that it is possible to model fringe fields for quadrupoles or higher order multipoles. Even then, existing techniques rely on the use of a numerical field map, constructed either from magnetic modelling or from measurement, and which will not be available until the magnet design is well developed. Substitutes for the full field map exist but these are typically based on expansions about the origin and rely heavily on the assumption that the beam travels more or less on axis throughout the beam line. In some types of machine (for example, a non-scaling FFAG such as EMMA) this is not a good assumption. Some tracking codes (such as General Particle Tracer or GPT) use methods for including space charge effects that require fields to vary smoothly and continuously along a beamline: in such cases, realistic fringe field models are of significant importance.
In this paper, a method for calculating fringe fields based on analytical expressions is presented, which allows fringe field effects to be included at the start of an accelerator design project. The magnetostatic Maxwell equations are solved analytically and a solution that fits all orders of multipoles derived. Quadrupole fringe fields are considered in detail as these are the ones that give the strongest effects; the simplest specific solution is shown. Two examples of quadrupole fringe fields are also presented. The first example is a magnet in the LHC inner triplet, which consists of a set of four quadrupoles providing the final focus to the beam, just before the interaction point. Because of the very strong focus at the collision point, particles in the beam pass through the inner triplet quadrupoles a considerable distance from the axis and, as a result, fringe field effects are important. Quadrupoles in EMMA provide the second example. In both examples, the analytical expressions derived in this paper for quadrupole fringe fields provide a good approximation to the field maps obtained from a numerical magnet modelling code.
author:
- |
B.D. Muratori, J.K. Jones, ASTeC, STFC Daresbury Laboratory, and Cockcroft Institute, U.K.\
A. Wolski, Department of Physics, University of Liverpool, Liverpool, and Cockcroft Institute, U.K.
title: Analytical Expressions for Fringe Fields in Multipole Magnets
---
Introduction
============
Fringe fields represent regions that lie at the edges of a magnet where there is a transition from the nominal field to zero field, or to the nominal field in an adjacent magnet. In multipole magnets, the nominal field has no longitudinal component. However, in the fringe field region where the fields vary with longitudinal position, Maxwell’s equations require the presence of a non-zero longitudinal field component. For dipoles, the nominal field has only a single component. In dipole fringe fields, therefore, the field has only two components, and (assuming that the fields are independent of the horizontal transverse co-ordinate) analytical expressions for the field can be obtained by solving the two-dimensional Laplace equation. For quadrupoles and higher-order multipoles, however, the fields in the fringe region have three components, and analytical expressions for these fields must be obtained by solving the three-dimensional Laplace equation. Fringe fields can impact the motion of particles passing through magnets in a number of ways. For example, they can introduce nonlinearities in the equation of motion, or they can make a substantial contribution to the desired effects of the nominal field. The latter situation is the case in EMMA, for example, where the large aperture of the quadrupoles compared to their lengths means that the fringe fields dominate the focusing effects of these magnets. Nonlinearities from fringe fields can become important if the transverse size of the beam is large, or if the beam traverses a multipole at an angle and some distance from the magnetic axis: this is often the case in final focus quadrupoles in colliders. The implementation of fringe fields is also important in some tracking codes which include effects like space charge, as is the case in GPT (General Particle Tracer) [@GPT1], for example. This requires that all fields be continuous so that there are smooth regions where the fields transit from their maximal value to zero and vice versa.
There are several models available for the study of fringe fields for multipoles, see for example [@Etienne1; @Etienne2] and references therein. However, existing models are usually limited to on-axis and mid-plane approximations, meaning that the field in the full space of the multipole typically has to be computed with elliptic integrals. Therefore, for simple but accurate particle tracking, a significant amount of effort and computing power goes into just the calculation of the fields that particles see. The results obtained in this paper make possible an alternative method, based on analytical expressions for the fields as functions of position, that provide exact solutions to the static Maxwell equations in three dimensions. This allows for arbitrarily smooth fields to be constructed and used in tracking codes as well as the possible creation of transfer maps. The fringe fields considered here have associated scalar and vector potentials. From the scalar potential it is possible to inspect the shape of the pole-face for an iron-dominated magnet generating the given field. Using the vector potential it is possible to perform symplectic integration of the equations of motion for a particle in the field, leading to the construction of transfer maps for the fringe region. Further, it may be possible to improve the efficiency of the magnet design process by making some initial assumptions based on the formulae presented in this paper. However, a full three-dimensional numerical field map will ultimately be needed (obtained from a numerical magnet modelling code) in order to achieve the accuracy that is needed for validating the design of many accelerators.
This paper develops the mathematical framework that was initially presented in [@BM], gives further details and results, and presents two examples. Following the Introduction, fringe fields for dipoles are briefly reviewed. The formalism used for dipoles is extended to fully three-dimensional fields in Section 3. A complete solution to the static Maxwell equations, in a form suitable for application to multipole fringe fields, is derived and presented. Expressions for fringe fields in multipoles of arbitrary order are then given in Section 4. A particular case of a quadrupole fringe field, together with a of the simple fall-off in the form of an Enge function [@engecite], is then presented in Section 5. All the salient properties are described in order to demonstrate that the field behaves in the way expected of a quadrupole, first inside the magnet, then in the fringe field region, and finally at a large distance from the magnet (so that the field effectively falls to zero). In Section 6, the scalar and vector potentials for fringe fields in the case of a multipole of arbitrary order are discussed. In Section 7, two quadrupole examples are presented. The first is the HL-LHC inner triplet, where the beam size and trajectory in the quadrupoles providing strong focusing close to the interaction point make fringe field effects significant. The second example is based on the quadrupole magnets in the non-scaling FFAG, EMMA. The large aperture of these magnets compared to their length means that the fringe fields make a dominant contribution to the focusing effects. Conclusions are given in Section 8.
Fringe fields for dipoles
=========================
The goal is to derive expressions for multipole fringe fields that satisfy Maxwell’s equations. To ensure the validity of the solution and the corresponding assumptions, it is important to write all equations explicitly. For static fields in the absence of any electric current, the equations for the magnetic field $\vec{B}$ are: $$\nabla\times\vec{B} = \nabla\cdot\vec{B} = 0.$$ For dipole magnets, it is sufficient to consider a two dimensional version of the equations. Taking $B_x = 0$, we are left with: $$\partial_y B_y + \partial_z B_z = \partial_y B_z - \partial_z B_y = 0,
\label{maxwell}$$ together with: $$\partial_x B_z = \partial_x B_y = 0,$$ which excludes all dependence on $x$. Further, we seek fringe fields which have a possible fall-off along the axis of the magnet given by the six parameter Enge function [@engecite; @Berz]: $$\frac{1}{1 + e^{E(z)}},$$ with $E(z)$ given by: $$E(z) = \sum_{n=0}^5 a_n\left(\frac{z}{D}\right)^{\!\! n}.$$ All coefficients $a_n$ are constants determined by modelling and/or experiment. $D$ is the full aperture of the dipole. The main advantage of the Enge function is that it is analytic and can be made to tend to asymptotic values arbitrarily fast, if required. The main disadvantage of this function is that, if all coefficients $a_n$ with $n = 0, \ldots, 5$ are included, varying any one coefficient changes the effective length of the magnet (in this case, a dipole). Other functions which decay sufficiently rapidly may be used instead of the Enge function [@Kato1; @Kato2]. For simplicity, we consider in this paper only the case where $a_1 \neq 0$ and all other coefficients are set to zero. This has the further advantage that varying $a_1$ only changes the fringe field “hardness” without altering its length. Additionally, because we only have one non-zero coefficient, we can normalise it to $1$ without loss of generality.
Maxwell’s equations (\[maxwell\]) imply: $$\Delta_{y,z} B_y = \Delta_{y,z} B_z = 0,$$ where $\Delta_{y,z} = \partial_y^2 + \partial_z^2$. Both wave equations (for $B_y$ and $B_z$) can easily be solved: $$\begin{aligned}
B_y & = & e(z + iy) + f(z -iy), \nonumber \\
B_z & = & g(z + iy) + h(z - iy). \nonumber\end{aligned}$$ Requiring that equations (\[maxwell\]) be solved as well, we end up with: $$\begin{aligned}
B_y & = & e(z + iy) + f(z -iy), \nonumber \\
B_z &= & -ie(z + iy) + if(z - iy). \nonumber\end{aligned}$$ If we further restrict ourselves to real magnetic fields, we obtain: $$\begin{aligned}
B_y & = & e(z + iy) + \bar{e}(z -iy), \label{maxrel1} \\
B_z & = & - ie(z + iy) + i\bar{e}(z - iy). \label{maxrel2}\end{aligned}$$ $B_y$ and $B_z$ are given by twice the real and imaginary parts of the function $e(z + iy)$, respectively.
Traditionally, dipole fringe fields are described by: $$\begin{aligned}
B_y & = & \frac{B_0}{2\pi}\left( \pi - \arctan\!\left(\frac{z}{g + y}\right) - \arctan\!\left(\frac{z}{g - y}\right)\right), \nonumber \\
B_z & = & \frac{B_0}{4\pi}\left( \ln(z^2 + (g + y)^2) - \ln(z^2 + (g - y)^2)\right), \nonumber\end{aligned}$$ where $B_0$ is the nominal strength of the dipole field, and $g$ is a parameter (related to the aperture of the magnet) affecting the precise shape of the fringe field. Using the result: $$(z + i(g \pm y))(z - i(g \pm y)) = z^2 + (g \pm y)^2,$$ $B_z$ may be re-written as: $$B_z = \frac{B_0}{4\pi}\left[\ln(z + i(g + y)) + \ln(z - i(g + y)) - \ln(z + i(g - y)) - \ln(z - i(g - y))\right],$$ whence $e(z + iy)$ has the form: $$e(z + iy) = \frac{iB_0}{4\pi}\left[\ln(z + i(g + y)) - \ln(z - i(g - y))\right].$$ Then, using equation (\[maxrel1\]) we find $B_y$ to be given by: $$B_y = \frac{iB_0}{4\pi}\left[\ln(z + i(g + y)) - \ln(z - i(g + y)) + \ln(z + i(g - y)) - \ln(z - i(g - y))\right],$$ which may be converted into the same form as the $B_y$ component of the magnetic field used in GPT (General Particle Tracer) [@GPT2], for example. So, in summary, given a complex function $e(z + iy)$, Maxwell’s equations may be satisfied automatically by setting $B_y = 2\mathrm{Re}(e(z + iy))$ and $B_z = 2\mathrm{Im}(e(z + iy))$.
A possibility for having a magnetic field whose $B_y$ component falls off on-axis is given by the six parameter Enge function: $$B_y = \frac{1}{2(1 + e^{E(z + iy)})} + \frac{1}{2(1 + e^{E(z - iy)})}.
\label{engeby}$$ This would force $B_z$ to have the form: $$B_z = \frac{-i}{2(1 + e^{E(z + iy)})} + \frac{i}{2(1 + e^{E(z - iy)})},
\label{engebz}$$ for some complex function $E(z + iy)$. If we consider the simple case $E(z + iy) = z + iy$ (with aperture having unit diameter) then equations (\[engeby\]) and (\[engebz\]) simplify to: $$\begin{aligned}
B_y & = & \frac{(1 + e^z\cos y)}{1 + 2e^z\cos y + e^{2z}}, \nonumber \\
B_z & = & \frac{-e^z\sin y}{1 + 2e^z\cos y + e^{2z}}. \nonumber\end{aligned}$$ This may be extended to include as many parameters of the Enge function as desired, the only restriction being that $E = E(z + iy)$. So, if $E(z + iy) = b_1(z + iy) + b_2(z + iy)^2$, with $b_1$ and $b_2$ arbitrary constants, we have: $$\begin{aligned}
B_y & = & \frac{(1 + e^f\cos h)}{1 + 2e^f\cos h + e^{2f}}, \nonumber \\
B_z & = & \frac{-e^f\sin h}{1 + 2e^f\cos h + e^{2f}}, \nonumber\end{aligned}$$ where $f = b_1z + b_2(z^2 - y^2)$ and $h = y(b_1 + 2b_2z)$. This can be generalized to: $$E(z + iy) = \sum_{n = 1}^N b_n (z + iy)^n,$$ with $b_j$ constants if an $N$ parameter Enge function is desired. Indeed, any function can be used, provided it is a function of $z + iy$.
We can plot the fringe fields (\[engeby\]) and (\[engebz\]) in the simplest case ($E(z + iy) = z + iy$), as shown in Fig. \[engebybz\]. Each field component has singularities, of which two are visible in the plots. From the denominator of (\[engeby\]) and (\[engebz\]), it can be seen that the singularities appear when (re-introducing the arbitrary aperture diameter $D$) $1 + 2e^{z/D}\cos(y/D) + e^{2z/D} = 0$ or $\cos(y/D) = - 1 \rightarrow y/D = \pm n\pi$ where $n$ is an integer. This does not cause any problems for modelling the field as it is always possible to arrange that the singularities are located outside the region of interest. It is not possible to avoid the appearance of singularities since any solution to Laplace’s equation either has singularities or is constant. In practical terms, the singularities can be associated with the current in the windings of the dipole.
-- --
-- --
Note that the singularities have a $\tan$-like behaviour and that, at the exact point where $z = 0$, the value of the field is precisely half of the maximum and everything is smooth. This can be verified by setting $z = 0$ in the expressions for the dipole fringe fields above and using l’Hopital’s rule.
Locally, it is possible to find potentials for the dipole fringe fields. The usual vector potential $\vec{A}$ is related to the field by $\vec{B} = \nabla\times\vec{A}$. For magnetostatic fields in the absence of currents, the field can be derived from a scalar potential $\varphi$, by $\vec{B} = \nabla\varphi$. The existence of the vector potential is ensured by the Maxwell equation $\nabla\cdot\vec{B} = 0$, while the scalar potential exists because $\nabla\times\vec{B} = 0$. For the simple case discussed above, the scalar potential is given by: $$\varphi = y + i\ln\sqrt{\frac{1 + e^{z + iy}}{1 + e^{z - iy}}} + \mathrm{constant.}$$ with the only gauge freedom being given by the constant. From the scalar potential, it is possible to obtain a description of the pole-face geometry, since this is given by surfaces where $\varphi$ is constant. This is shown in Fig. \[dippf\] where two profiles of the pole-face can be seen. It should be remembered that there is a scale invariance in the expressions depending on the dimensions of the dipole. Figure \[dippf\] also shows that the pole-face profiles encompass the singularities: this is consistent with the assertion made earlier that the singularities are associated with the current in the coils of the dipole.
The vector potential has extensive gauge freedom and, in one of its simplest forms, can be given as: $$A_x = z - \ln\sqrt{(1+e^{z+iy})(1+e^{z-iy})},$$ with the other components of $\vec{A}$ set to zero.
General three dimensional solution in a fringe field {#sectiongeneralthreedsolution}
====================================================
General expressions for the fringe fields in a dipole followed from writing Maxwell’s equations in the form (\[maxwell\]). In a dipole, we only needed to consider two field components ($B_y$ and $B_z$) as functions of two co-ordinates ($y$ and $z$). For higher order multipoles, it is necessary to consider the dependence of all three field components on all three co-ordinates. One approach might be to look for solutions to the three dimensional Laplacian. A formal solution (due to Whittaker [@Piaggio; @Whit1; @Whit2]) is known, and can be expressed as: $$\varphi(x,y,z) \ = \ \int_0^{2\pi}f(z + ix\cos\vartheta + iy\sin\vartheta)\ {\rm d}\vartheta,$$ where: $$\Delta_{x, y, z}\varphi \equiv \partial_x^2\varphi + \partial_y^2\varphi + \partial_z^2\varphi \ = \ 0.$$ However, it is difficult to use this result to find expressions that are useful in practice. The only well-known practical solution is $f = (z + ix\cos\vartheta + iy\sin\vartheta)^{- 1}$, which gives the standard solution $\varphi = 2\pi/r$ with $r = \sqrt{x^2 + y^2 + z^2}$.
An alternative approach, which we develop here, is to define new variables in terms of which Maxwell’s equations can be written for a general three-dimensional magnetostatic field in a form very similar to (\[maxwell\]). This raises the possibility of finding expressions for fringe fields in higher order multipoles by applying similar techniques to those described in Section 2 for dipoles.
To write Maxwell’s equations for three-dimensional fields in a form similar to (\[maxwell\]), we define new variables: $$\begin{aligned}
u & = & \frac{1}{\sqrt{2}}(x + iy), \nonumber \\
v & = & \frac{1}{\sqrt{2}}(x - iy), \nonumber \\
\zeta & = & \sqrt{2}z. \nonumber\end{aligned}$$ We express the magnetic field in terms of components: $$\begin{aligned}
B_u & = & \frac{1}{\sqrt{2}}(B_x + i B_y), \nonumber \\
B_v & = & \frac{1}{\sqrt{2}}(B_x - i B_y), \nonumber \\
B_\zeta & = & \frac{1}{\sqrt{2}} B_z. \nonumber\end{aligned}$$ In terms of the new variables, Maxwell’s equations can be written: $$\begin{aligned}
\label{maxnew1}
\partial_u B_u + \partial_z B_z & = & 0, \\
\label{maxnew2}
\partial_v B_v + \partial_z B_z & = & 0, \\
\label{maxnew3}
\partial_z B_u - \partial_v B_z & = & 0, \\
\label{maxnew4}
\partial_z B_v - \partial_u B_z & = & 0.\end{aligned}$$
From (\[maxnew1\]) and (\[maxnew2\]), one can see immediately that, in the absence of any fringe fields, the general solution of Maxwell’s equations for any magnet, acting transversely only and without fringe ($B_\zeta = 0$) is given by: $$\begin{aligned}
B_u & = & f(v), \nonumber \\
B_v & = & h(u), \nonumber\end{aligned}$$ for any functions $f(v)$ and $h(u)$. The case of a multipole of order $n$ ($n=1$ for a quadrupole, $n=2$ for a sextupole, and so on) is given by: $$\begin{aligned}
B_u & = & iv^n, \nonumber \\
B_v & = & - iu^n, \nonumber \\
B_\zeta & = & 0. \nonumber\end{aligned}$$ For example, a quadrupole is described by $B_u = iv$, $B_v = - iu$ and $B_\zeta = 0$.
We now assume a very general form that we expect multipole fringe fields should take: $$\begin{aligned}
\label{ass1}
B_u & = & \frac{f_1(u,v,\zeta) + f_2(u,v,\zeta)e^\zeta}{(1 + 2f_3(u,v)e^\zeta + e^{2\zeta})}, \\
\label{ass2}
B_v & = & \frac{f_4(u,v,\zeta) + f_5(u,v,\zeta)e^\zeta}{(1 + 2f_3(u,v)e^\zeta + e^{2\zeta})}, \\
\label{ass3}
B_\zeta & = & \frac{f_6(u,v,\zeta) + f_7(u,v,\zeta)e^\zeta}{(1 + 2f_3(u,v)e^\zeta + e^{2\zeta})}.\end{aligned}$$ The form of this field is based on a generalisation of the fringe fields for the dipole case. Substituting (\[ass1\])–(\[ass3\]) into Maxwell’s equations (\[maxnew1\])–(\[maxnew4\]) gives a set of constraints on the possible forms of the functions $f_1$, $f_2$ etc. To obtain useful expressions for multipole fringe fields, we need to find solutions satisfying the various constraints: this is the task that we address in the remainder of this section.
Essentially, there are only two types of derivative that we need to consider. These are: $$\begin{aligned}
\partial_u B_u & = & \frac{\partial_u f_1 + \partial_u f_2 e^\zeta}{A} - \frac{2(f_1 + f_2 e^\zeta)e^\zeta \partial_u f_3}{A^2}, \nonumber \\
\partial_\zeta B_u & = & \frac{\partial_\zeta f_1 + \partial_\zeta f_2 e^\zeta + f_2 e^\zeta}{A}
- \frac{2(f_1 + f_2 e^\zeta)(e^\zeta f_3 + e^{2\zeta})}{A^2}, \nonumber\end{aligned}$$ where $A = 1 + 2f_3 e^\zeta + e^{2\zeta}$. For the remaining derivatives, we simply implement the following changes sequentially: $$\begin{aligned}
\partial_v B_u \equiv \partial_u B_u & {\rm under} & (u \leftrightarrow v), \nonumber \\
\partial_u B_v \equiv \partial_u B_u & {\rm under} & (f_1 \rightarrow f_4, \ f_2 \rightarrow f_5), \nonumber \\
\partial_v B_v \equiv \partial_u B_v & {\rm under} & (u \leftrightarrow v), \nonumber \\
\partial_\zeta B_v \equiv \partial_\zeta B_u & {\rm under} & (f_1 \rightarrow f_4, \ f_2 \rightarrow f_5), \nonumber \\
\partial_u B_\zeta \equiv \partial_u B_u & {\rm under} & (f_1 \rightarrow f_6, \ f_2 \rightarrow f_7), \nonumber \\
\partial_v B_\zeta \equiv \partial_u B_\zeta & {\rm under} & (u \leftrightarrow v), \nonumber \\
\partial_\zeta B_\zeta \equiv \partial_\zeta B_u & {\rm under} & (f_1 \rightarrow f_6, \ f_2 \rightarrow f_7). \nonumber \end{aligned}$$ As all equations (\[maxnew1\])–(\[maxnew4\]) are equal to zero, we can take out a factor of $A^2$ to give: $$\begin{aligned}
(\partial_x f_1 + \partial_x f_2 e^\zeta)(1 + 2f_3 e^\zeta + e^{2\zeta})
+ (\partial_y f_4 + \partial_y f_5 e^\zeta)(1 + 2f_3 e^\zeta + e^{2\zeta}) \quad & & \nonumber \\
+ (\partial_\zeta f_6 + \partial_\zeta f_7 e^\zeta + f_7 e^\zeta)(1 + 2f_3 e^\zeta + e^{2\zeta}) \quad & & \nonumber \\
- 2(f_1 + f_2 e^\zeta)e^\zeta \partial_x f_3 - 2(f_4 + f_5 e^\zeta)e^\zeta \partial_y f_3
- 2(f_6 + f_7 e^\zeta)(f_3 e^\zeta + e^{2\zeta}) & = & 0, \nonumber \\
& & \nonumber \\
(\partial_x f_4 + \partial_x f_5 e^\zeta)(1 + 2f_3 e^\zeta + e^{2\zeta})
- (\partial_y f_1 + \partial_y f_2 e^\zeta)(1 + 2f_3 e^\zeta + e^{2\zeta}) \quad & & \nonumber \\
- 2(f_4 + f_5 e^\zeta)e^\zeta \partial_x f_3 + 2(f_1 + f_2 e^\zeta)e^\zeta \partial_y f_3 & = & 0, \nonumber \\
& & \nonumber \\
(\partial_x f_6 + \partial_x f_7 e^\zeta)(1 + 2f_3 e^\zeta + e^{2\zeta})
- (\partial_\zeta f_1 + \partial_\zeta f_2 e^\zeta + f_2 e^\zeta)(1 + 2f_3 e^\zeta + e^{2\zeta}) \quad & & \nonumber \\
- 2(f_6 + f_7 e^\zeta)e^\zeta \partial_x f_3 + 2(f_1 + f_2 e^\zeta)(f_3 e^\zeta + e^{2\zeta}) & = & 0, \nonumber \\
& & \nonumber \\
(\partial_y f_6 + \partial_y f_7 e^\zeta)(1 + 2f_3 e^\zeta + e^{2\zeta})
- (\partial_\zeta f_4 + \partial_\zeta f_5 e^\zeta + f_5 e^\zeta)(1 + 2f_3 e^\zeta + e^{2\zeta}) \quad & & \nonumber \\
- 2(f_6 + f_7 e^\zeta)e^\zeta \partial_x f_3 + 2(f_4 + f_5 e^\zeta)(f_3 e^\zeta + e^{2\zeta}) & = & 0. \nonumber\end{aligned}$$ We can now equate coefficients of $e^\zeta$ giving: $$\begin{aligned}
\label{eq1}
e^{3\zeta}: & \ & \partial_u f_2 + \partial_\zeta f_7 - f_7 \ = \ 0 \\
\label{eq2}
& \ & \partial_v f_5 + \partial_\zeta f_7 - f_7 \ = \ 0 \\
\label{eq3}
& \ & \partial_u f_7 - \partial_\zeta f_5 + f_5 \ = \ 0 \\
\label{eq4}
& \ & \partial_v f_7 - \partial_\zeta f_2 + f_2 \ = \ 0 \\
\nonumber \\
\label{eq5}
e^{2\zeta}: & \ & f_2 \partial_u f_3 + f_6 - f_3 f_7 \ = \ 0 \\
\label{eq6}
& \ & f_5 \partial_v f_3 + f_6 - f_3 f_7 \ = \ 0 \\
\label{eq7}
& \ & f_7 \partial_u f_3 - f_4 + f_3 f_5 \ = \ 0 \\
\label{eq8}
& \ & f_7 \partial_v f_3 - f_1 + f_3 f_2 \ = \ 0 \\
\nonumber \\
\label{eq9}
e^\zeta: & \ & f_1 \partial_u f_3 + f_3 f_6 - f_7 \ = \ 0 \\
\label{eq10}
& \ & f_4 \partial_v f_3 + f_3 f_6 - f_7 \ = \ 0 \\
\label{eq11}
& \ & f_6 \partial_u f_3 + f_5 - f_3 f_4 \ = \ 0 \\
\label{eq12}
& \ & f_6 \partial_v f_3 + f_2 - f_3 f_1 \ = \ 0 \\
\nonumber \\
\label{eq13}
e^0: & \ & \partial_u f_1 + \partial_\zeta f_6 \ = \ 0 \\
\label{eq14}
& \ & \partial_v f_4 + \partial_\zeta f_6 \ = \ 0 \\
\label{eq15}
& \ & \partial_u f_6 - \partial_\zeta f_4 \ = \ 0 \\
\label{eq16}
& \ & \partial_v f_6 - \partial_\zeta f_1 \ = \ 0.\end{aligned}$$ Note that we have not included all the steps and the above equations represent the original set with all possible simplifications taking into account the set itself. Note also that equations (\[eq1\])–(\[eq4\]) and (\[eq13\])–(\[eq16\]) may be solved independently of the rest and can therefore be dealt with later. From equation (\[eq8\]), using (\[eq5\]) and (\[eq9\]), we see that: $$f_7(\partial_v f_3 \, \partial_u f_3 + f_3^2 - 1) = 0.$$ Had we looked at equations (\[eq7\]) and (\[eq11\]) instead, using (\[eq6\]) and (\[eq10\]), we would have had: $$f_6(\partial_v f_3 \, \partial_u f_3 + f_3^2 - 1) = 0,$$ with the same result from equation (\[eq12\]). Now, $f_6$ and $f_7$ cannot both be zero as this would mean $B_\zeta = 0$. Therefore, we must have: $$\partial_v f_3 \, \partial_u f_3 + f_3^2 - 1 = 0.$$ The general solution for $f_3$, using the method of characteristics [@Piaggio], is: $$f_3 = \sin h(u,v),$$ with $$h(u,v) = \frac{1}{b}u + bv + c, \label{hdefinition}$$ where $b$ and $c$ are constant. Without loss of generality and for simplicity, we set $c = 0$. Substituting this back into (\[eq5\])–(\[eq12\]) gives the relations: $$\begin{aligned}
f_2 & = & b^2 f_5, \nonumber \\
f_1 & = & b^2 f_4. \nonumber\end{aligned}$$ The equations reduce to just two independent equations that may be written as: $$\begin{aligned}
\label{eqn6}
\frac{1}{b}f_2 \cos h + f_6 - f_7 \sin h & = & 0, \\
\label{eqn11}
f_6 \cos h + \frac{1}{b}f_2 - \frac{1}{b}f_1 \sin h & = & 0.\end{aligned}$$ Using $f_1 = b^2 f_4$ and equations (\[eq13\]) and (\[eq14\]) we see that we require $$b^2 \partial_u f_4 = \partial_v f_4,$$ which can again be solved by the method of characteristics to give $f_4 \ = \ f_4(h, z)$. Using this with equations (\[eq15\]) and (\[eq16\]) we see that $f_6 \ = \ f_6(h, z)$. Similarly, $f_2 \ = \ b^2 f_5$ applied to (\[eq1\]) and (\[eq2\]) and, subsequently (\[eq3\]) and (\[eq4\]) gives $f_5 \ = \ f_5(h, z)$ and $f_7 \ = \ f_7(h, z)$. This leaves six equations to be satisfied from the original system (\[eq1\])–(\[eq16\]), namely (\[eqn6\]) and (\[eqn11\]) together with: $$\begin{aligned}
\label{eqn1}
\partial_u f_2 + \partial_\zeta f_7 - f_7 & = & 0 \\
\label{eqn4}
\partial_v f_7 - \partial_\zeta f_2 + f_2 & = & 0 \\
\label{eqn13}
\partial_u f_1 + \partial_\zeta f_6 & = & 0 \\
\label{eqn16}
\partial_v f_6 - \partial_\zeta f_1 & = & 0.\end{aligned}$$ After cross-differentiation, equations (\[eqn13\]) and (\[eqn16\]) give $$\begin{aligned}
\partial_{u,v}^2 f_6 + \partial^2_\zeta f_6 & = & 0, \nonumber \\
\partial_{u,v}^2 f_1 + \partial^2_\zeta f_1 & = & 0. \nonumber\end{aligned}$$
Now, we can re-express the partial derivatives in $u$ and $v$ in terms of $h$ only. The equations simplify to: $$\triangle f_1 = \triangle f_6 = 0,$$ where $\triangle \equiv \partial_h^2 + \partial_\zeta^2$. We introduce the co-ordinates: $$\begin{aligned}
w & = & h + i\zeta, \nonumber \\
\bar{w} & = & h - i\zeta. \nonumber\end{aligned}$$ Note that this operation is equivalent to complex conjugation in the $\zeta$ co-ordinate only and the function $h$ is untouched. Therefore, we have the solutions: $$\begin{aligned}
f_1 & = & p_1(w) + q_1(\bar{w}), \nonumber \\
f_6 & = & p_6 (w) + q_6(\bar{w}). \nonumber\end{aligned}$$ Substituting this back into (\[eqn13\]) and (\[eqn16\]), we see that the solutions are further constrained to: $$f_1 = - ibp_6 + ibq_6 + k,$$ with $k$ constant, from which we can get $f_4$ via $f_4 = \frac{1}{b^2}f_1$. Subsequently, we can get $f_2$ from (\[eqn11\]) and hence $f_5$ via $f_5 = \frac{1}{b^2}f_2$ and $f_7$ from (\[eqn6\]). The general result, in terms of $p_6$ and $q_6$ may be summarised as follows (with $h = \frac{1}{b}u + bv$): $$\begin{aligned}
\label{res1}
f_1 & = & - ibp_6 + ibq_6 + k, \\
\label{res2}
f_2 & = & (- ibp_6 + ibq_6 + k)\sin h - (bp_6 + bq_6)\cos h, \\
\label{res3}
f_3 & = & \sin h, \\
\label{res4}
f_4 & = & \frac{1}{b}\left(- ip_6 + iq_6 + \frac{k}{b}\right), \\
\label{res5}
f_5 & = & \frac{1}{b}\left(- ip_6 + iq_6 + \frac{k}{b}\right)\sin h - \frac{1}{b}\left(p_6 + q_6\right)\cos h, \\
\label{res6}
f_6 & = & p_6 + q_6, \\
\label{res7}
f_7 & = & \left(p_6 + q_6\right)\sin h + \left(- ip_6 + iq_6 + \frac{k}{b}\right)\cos h.\end{aligned}$$
We are left with equations (\[eqn1\]) and (\[eqn4\]) to be solved. Upon substitution of (\[res1\])–(\[res7\]), these are actually seen to be trivially satisfied with no further constraints on any of the functions $f_i$. In fact, the results constitute a Darboux transformation, where, given a solution to Maxwell’s equations expressed by (\[eq13\])–(\[eq16\]), a new solution may be created, given by (\[ass1\])–(\[ass3\]), provided (\[eq1\])–(\[eq12\]) are satisfied. Further, the results can be seen to imply the following solution to Maxwell’s equations: $$\begin{aligned}
\label{resbu}
B_u & = & - ibf(h + i\zeta) + ibg(h - i\zeta), \\
\label{resbv}
B_v & = & - \frac{i}{b}f(h + i\zeta) + \frac{i}{b}g(h - i\zeta), \\
\label{resbzp}
B_\zeta & = & f(h + i\zeta) + g(h - i\zeta).\end{aligned}$$ with $h$ defined by (\[hdefinition\]). This solution could have been arrived at by a shorter method, but the above workings show that, for the kind of functions we are interested in, it is the only solution that fits. We may have been forced to consider more complicated functions by introducing non-linearities in the coordinate dependance, for example. Note that the constant $b$ is purely a scaling constant of the coordinates, as well as giving a proportionality between $B_u$ and $B_v$. For the purposes of the remainder of this article, we shall refer to the solution (\[resbu\]), (\[resbv\]) and (\[resbzp\]) as an *elementary solution* of Maxwell’s equations in three dimensions. In the original (Cartesian) co-ordinates ($x = (u + v)/\sqrt{2}$, $y = (u - v)/\sqrt{2}i$, $z = \zeta/\sqrt{2}$) the elementary solution reads: $$\begin{aligned}
\label{resbx}
B_x & = & \frac{1}{\sqrt{2}}(B_u + B_v) = - idf(h + i\sqrt{2}z) + idg(h - i\sqrt{2}z), \\
\label{resby}
B_y & = & \frac{1}{\sqrt{2}i}(B_u - B_v) = ef(h + i\sqrt{2}z) - eg(h - i\sqrt{2}z), \\
\label{resbz}
B_z & = & \sqrt{2}B_\zeta = \sqrt{2}f(h + i\sqrt{2}z) + \sqrt{2}g(h - i\sqrt{2}z),\end{aligned}$$ where: $$\begin{aligned}
d & = & \frac{1}{\sqrt{2}} \left( \frac{1}{b} + b \right), \nonumber \\
e & = & \frac{1}{\sqrt{2}} \left( \frac{1}{b} - b \right), \nonumber \end{aligned}$$ and $h$ is now expressed as: $$h = dx + iey.$$
For some applications, it is useful to have expressions for the scalar potential $\varphi$ and vector potential $\vec{A}$, from which the field can be derived by: $$\vec{B} = \nabla \varphi, \label{scalarpotentialdefn}$$ or: $$\vec{B} = \nabla \times \vec{A}. \label{vectorpotentialdefn}$$ It can be readily verified by substitution into (\[scalarpotentialdefn\]) that an expression for the scalar potential for the elementary solution can be written: $$\varphi = -i \tilde{f}(h + i \sqrt{2}z) + i \tilde{g}(h - i \sqrt{2}z),$$ where: $$\begin{aligned}
\tilde{f}(\zeta) & = & \int_0^\zeta f(\zeta^\prime) \, \mathrm{d}\zeta^\prime, \nonumber \\
\tilde{g}(\zeta) & = & \int_0^\zeta g(\zeta^\prime) \, \mathrm{d}\zeta^\prime. \nonumber\end{aligned}$$ Setting the lower limit of each integral to zero fixes the gauge so that $\varphi = 0$ at the origin; a change of gauge can be made simply by adding a constant to $\varphi$. It can be verified by substitution into (\[vectorpotentialdefn\]) that the vector potential for the elementary solution may be written: $$\begin{aligned}
A_x & = & 0, \nonumber \\
A_y & = & \frac{\sqrt{2}}{d} \left( \tilde{f}(h + i \sqrt{2}z) + \tilde{g}(h - i \sqrt{2}z) \right), \nonumber \\
A_z & = & -\frac{e}{d} \left( \tilde{f}(h + i \sqrt{2}z) - \tilde{g}(h - i \sqrt{2}z) \right). \nonumber\end{aligned}$$ Here, we have chosen a gauge in which $A_x = 0$.
Note that any function of $h \pm i\zeta$ may also be expressed as a function of $\zeta \mp ih$: we shall use both forms in what follows in order to simplify the expressions as appropriate.
Multipole magnets with general fringe fields
============================================
The relations $f_2 = b^2 f_5$ and $f_1 = b^2 f_4$ found earlier imply that $B_u \propto B_v$: this means that no physical magnetic fields can be represented this way. However, because of the linearity of Maxwell’s equations, any linear combination of elementary solutions gives a solution to Maxwell’s equations. Physical solutions corresponding to multipole fields can be constructed by taking appropriate combinations of elementary solutions. Further (physical) constraints are that the field decays as $z \rightarrow \infty$ and that the field matches the nominal multipole field inside the magnet. In this section, we describe in detail the procedure for constructing an expression for the fringe field in a quadrupole, and then generalise our results to multipoles of any order. To obtain the correct behaviour of the field as a function of $z$, the functions $f(h + i\zeta)$ and $g(h - i\zeta)$ are each written as a product of two factors. The first factor, $(h + i\zeta)^n$ and $(h - i\zeta)^n$ respectively, represents the (nominal) multipole for some constant $n$. The second factor, $F(h + i\zeta)$ and $G(h - i\zeta)$ respectively, are multiplicative functions which are chosen to give the desired decay as $z$ becomes large ($z \rightarrow \infty$), and to give the correct field in the body of the magnet. Thus, we write: $$\begin{aligned}
f(h + i\zeta) & = & (h + i\zeta)^nF(h + i\zeta), \label{multipolefexpression} \\
g(h - i\zeta) & = & (h - i\zeta)^nG(h - i\zeta). \label{multipolegexpression}\end{aligned}$$ Following the form of (\[resbu\]), (\[resbv\]) and (\[resbzp\]), let $B_u$, $B_v$ and $B_\zeta$ be given by linear combinations of the elementary solution: $$\begin{aligned}
B_u & = & \sum_{j = 1}^{m}c_j \left( - ib_j(h_j + i\zeta)^nF_j(h_j + i\zeta) + ib_j(h_j - i\zeta)^nG_j(h_j - i\zeta) \right), \label{busummation} \\
B_v & = & \sum_{j = 1}^{m}c_j \left( - \frac{i}{b_j}(h_j + i\zeta)^nF_j(h_j + i\zeta) + \frac{i}{b_j}(h_j - i\zeta)^nG_j(h_j - i\zeta) \right),
\label{bvsummation} \\
B_\zeta & = & \sum_{j = 1}^{m}c_j \left( (h_j + i\zeta)^nF_j(h_j + i\zeta) + (h_j - i\zeta)^nG_j(h_j - i\zeta) \right). \label{bzetasummation}\end{aligned}$$ where $h_j = \frac{1}{b_j}u + b_jv$, and $m$ is a constant determined by the number of copies of the elementary solution needed to construct a physical multipole of the desired order. The constant $n$ determines the order of the multipole. Within the body of the magnet, $B_u = iv^n$, $B_v = -iu^n$, and $B_\zeta = 0$. We also expect to have to satisfy several relations between the constants $b_j$ and $c_j$.
Quadrupole magnets {#quads}
------------------
A quadrupole field is obtained by putting $n = 1$ in equations (\[multipolefexpression\]) and (\[multipolegexpression\]). Therefore, within the body of the magnet (far from any fringe field) $F(h_j + i\zeta) = G(h_j - i\zeta) = \pm 1$ for any $j$. The relative signs of $F(h_j + i\zeta)$ and $G(h_j - i\zeta)$ are to be determined by the required gradient inside the quadrupole and the fact that there should be no dependence on $z$ at the centre of the magnet. The required behaviour of the fields in the body of the magnet can be obtained by taking $m = 2$ (larger values of $m$ can be used, but are not required). Then, putting $B_u = iv$, $B_v = -iu$ and $B_\zeta = 0$ gives: $$\begin{aligned}
- ic_1b_1\bigg(\frac{u}{b_1} + b_1v + iz\bigg) - ic_1b_1\bigg(\frac{u}{b_1} + b_1v - iz\bigg)
- ic_2b_2\bigg(\frac{u}{b_2} + b_2v + iz\bigg) - ic_2b_2\bigg(\frac{u}{b_2} + b_2v - iz\bigg) & = & iv, \nonumber \\
- i\frac{c_1}{b_1}\bigg(\frac{u}{b_1} + b_1v + iz\bigg) - i\frac{c_1}{b_1}\bigg(\frac{u}{b_1} + b_1v - iz\bigg)
- i\frac{c_2}{b_2}\bigg(\frac{u}{b_2} + b_2v + iz\bigg) - i\frac{c_2}{b_2}\bigg(\frac{u}{b_2} + b_2v - iz\bigg) & = & - iu, \nonumber \\
- ic_1\bigg(\frac{u}{b_1} + b_1v + iz\bigg) - ic_1\bigg(\frac{u}{b_1} + b_1v - iz\bigg)
- ic_2\bigg(\frac{u}{b_2} + b_2v + iz\bigg) - ic_2\bigg(\frac{u}{b_2} + b_2v - iz\bigg) & = & 0. \nonumber\end{aligned}$$ Thus, we have three equations to satisfy: $$\begin{aligned}
2c_1b_1^2 + 2c_2b_2^2 & = & - 1, \nonumber \\
2\frac{c_1}{b_1^2} + 2\frac{c_2}{b_2^2} & = & 1, \nonumber \\
c_1 + c_2 & = & 0. \nonumber\end{aligned}$$ Therefore, inside the magnet, the coefficients $b_j$ and $c_j$ must satisfy the following constraints: $$\begin{aligned}
\label{bq}
b_1 & = & \pm \frac{1}{b_2}, \\
\label{cq}
c_1 & = & - c_2 = \frac{1}{2} \left(b_2^2 - \frac{1}{b_2^2}\right)^{-1}.\end{aligned}$$ We are left with the freedom of choosing one constant, which we take to be $b_2$. We shall consider the significance of this constant in more detail later, but for now we note that to avoid $b_1$, $c_1$ or $c_2$ becoming singular, $b_2$ must not be equal to $0$ or $1$. Also, the field is unchanged if we replace $b_2$ by $1/b_2$. Therefore, before implementing any simplifications, we have the following: $$\begin{aligned}
B_x & = & c_1d_1f(\zeta - ih_1) - c_1d_1g(\zeta + ih_1) + c_2d_2f(\zeta - ih_2) - c_2d_2g(\zeta + ih_2), \\
B_y & = & ic_1e_1f(\zeta - ih_1) - ic_1e_1g(\zeta + ih_1) + ic_2e_2f(\zeta - ih_2) - ic_2e_2g(\zeta + ih_2), \\
B_z & = & ic_1f(\zeta - ih_1) + ic_1g(\zeta + ih_1) + ic_2f(\zeta + ih_2) + ic_2g(\zeta + ih_2).\end{aligned}$$ Physically, it is necessary that our choices of constants should give a real field in the end. Now, since the two co-ordinates are related via $u = \bar{v}$ (where the bar denotes complex conjugation) $h_2 = \bar{h}_1$ to within a factor $\pm 1$. In detail, the first constraint above implies the following two cases: $$\begin{aligned}
b_1 = + \frac{1}{b_2}: & \Rightarrow & d_1 = d_2, e_1 = - e_2, h_1 = \bar{h}_2, \\
b_1 = - \frac{1}{b_2}: & \Rightarrow & d_1 = - d_2, e_1 = e_2, h_1 = - \bar{h}_2.\end{aligned}$$ Now, in either case, the second constraint ($c_1 = - c_2$) is the same and we factor the resulting constant out for clarity so, in terms of magnetic field components we are left with (for $b_1 = + 1/b_2$): $$\begin{aligned}
B_x & = & df(\zeta - ih) - dg(\zeta + ih) - df(\zeta - i\bar{h}) + dg(\zeta + i\bar{h}), \\
B_y & = & ief(\zeta - ih) - ieg(\zeta + ih) + ief(\zeta - i\bar{h}) - ieg(\zeta + i\bar{h}), \\
B_z & = & if(\zeta - ih) + ig(\zeta + ih) - if(\zeta - i\bar{h}) - ig(\zeta + i\bar{h}),\end{aligned}$$ where we have put $h_1 = h$. On the other hand, if we take $b_1 = - 1/b_2$, then the field components read: $$\begin{aligned}
B_x & = & df(\zeta - ih) - dg(\zeta + ih) + df(\zeta + i\bar{h}) - dg(\zeta - i\bar{h}), \\
B_y & = & ief(\zeta - ih) - ieg(\zeta + ih) - ief(\zeta + i\bar{h}) + ieg(\zeta - i\bar{h}), \\
B_z & = & if(\zeta - ih) + ig(\zeta + ih) - if(\zeta + i\bar{h}) - ig(\zeta - i\bar{h}).\end{aligned}$$ So we can see that, for both choices, $B_z$ is real, however, this is not the case for the other two components. In the first case ($b_1 = + 1/b_2$), it is not possible to say anything about $B_x$ or $B_y$ unless $f = g$, in which case both are real. However, in the second case ($b_1 = - 1/b_2$), all components are automatically real, irrespective of the choices of $f$ and $g$. Note that allowing the constant $b_1$ to take complex values only increases the complexity of the equations without bringing in any useful additional parameters; therefore, without loss of generality, we consider only cases where $b_1$ takes real values. Therefore, and so as to be able to keep $f$ and $g$ independent of each other, we choose $b_1 = - 1/b_2$.
The results for fields in quadrupole magnets can be extended to higher order multipoles in a straightforward way. However, the higher the order, the more copies of the elementary solution are needed to construct fields with the required properties. In the following subsections, we shall consider explicitly the cases of sextupole and octupole magnets, to establish a pattern from which the results for a general multipole magnet can be written down.
Sextupole magnets
-----------------
A sextupole field is obtained by putting $n = 2$ in equations (\[multipolefexpression\]) and (\[multipolegexpression\]). To obtain the constraints on the various coefficients $c_j$ and $b_j$, we can proceed in close analogy with the case of the quadrupole magnet. In particular, within the body of the magnet (far from any fringe field) $F(h_j + i\zeta) = G(h_j - i\zeta) = \pm 1$ for any $j$, and the field is given by: $$\begin{aligned}
B_u & = & iv^2, \nonumber \\
B_v & = & -iu^2, \nonumber \\
B_\zeta & = & 0. \nonumber\end{aligned}$$ The corresponding constraints are: $$\begin{aligned}
- ic_1b_1\bigg(\frac{u}{b_1} + b_1v + iz\bigg)^2 - ic_1b_1\bigg(\frac{u}{b_1} + b_1v - iz\bigg)^2
- ic_2b_2\bigg(\frac{u}{b_2} + b_2v + iz\bigg)^2 - ic_2b_2\bigg(\frac{u}{b_2} + b_2v - iz\bigg)^2 \quad & & \nonumber \\
- ic_3b_3\bigg(\frac{u}{b_3} + b_3v + iz\bigg)^2 - ic_3b_3\bigg(\frac{u}{b_3} + b_3v - iz\bigg)^2 & = & iv^2, \nonumber \\
- i\frac{c_1}{b_1}\bigg(\frac{u}{b_1} + b_1v + iz\bigg)^2 - i\frac{c_1}{b_1}\bigg(\frac{u}{b_1} + b_1v - iz\bigg)^2
- i\frac{c_2}{b_2}\bigg(\frac{u}{b_2} + b_2v + iz\bigg)^2 - i\frac{c_2}{b_2}\bigg(\frac{u}{b_2} + b_2v - iz\bigg)^2 \quad & & \nonumber \\
- i\frac{c_3}{b_3}\bigg(\frac{u}{b_3} + b_3v + iz\bigg)^2 - i\frac{c_3}{b_3}\bigg(\frac{u}{b_3} + b_3v - iz\bigg)^2 & = & - iu^2, \nonumber \\
- ic_1\bigg(\frac{u}{b_1} + b_1v + iz\bigg)^2 - ic_1\bigg(\frac{u}{b_1} + b_1v - iz\bigg)^2
- ic_2\bigg(\frac{u}{b_2} + b_2v + iz\bigg)^2 - ic_2\bigg(\frac{u}{b_2} + b_2v - iz\bigg)^2 \quad & & \nonumber \\
- ic_3\bigg(\frac{u}{b_3} + b_3v + iz\bigg)^2 - ic_3\bigg(\frac{u}{b_3} + b_3v - iz\bigg)^2 & = & 0. \nonumber\end{aligned}$$ There are four equations to satisfy, namely: $$\begin{aligned}
c_1b_1 + c_2b_2 + c_3b_3 & = & 0, \nonumber \\
\frac{c_1}{b_1} + \frac{c_2}{b_2} + \frac{c_3}{b_3} & = & 0, \nonumber \\
2c_1b_1^3 + 2c_2b_2^3 + 2c_3b_3^3 & = & - 1, \nonumber \\
2\frac{c_1}{b_1^3} + 2\frac{c_2}{b_2^3} + 2\frac{c_3}{b_3^3} & = & 1. \nonumber\end{aligned}$$ Finally, the constraints on the coefficients $b_j$ and $c_j$ may be written: $$\begin{aligned}
c_1 & = & - \frac{b_1}{2\left(b_1^2 - b_2^2\right)\left(b_1^2 - b_3^2\right)}, \nonumber \\
c_2 & = & - \frac{b_2}{2\left(b_2^2 - b_1^2\right)\left(b_2^2 - b_3^2\right)}, \nonumber \\
c_3 & = & - \frac{b_3}{2\left(b_3^2 - b_1^2\right)\left(b_3^2 - b_2^2\right)}, \nonumber \\
b_1 & = & \pm \frac{i}{b_2b_3}. \nonumber\end{aligned}$$ Note that, for sextupoles and higher order multipoles, it is unlikely that a particular choice of the associated constants results in the field components being automatically real. This is because of the additional number of constants involved. However, this should not result in any problem because the real part can just be taken at the end of the explicit construction given that any complex solution to Maxwell’s equations implies that, both the real and imaginary parts, are also solutions of the same equations.
Octupole magnets
----------------
An octupole field is obtained by putting $n = 3$ in equations (\[multipolefexpression\]) and (\[multipolegexpression\]). The field is given by: $$\begin{aligned}
B_u & = & iv^3, \nonumber \\
B_v & = & -iu^3, \nonumber \\
B_\zeta & = & 0. \nonumber\end{aligned}$$ Following the same procedure as for the quadrupole and the sextupole, the constraints on the coefficients $c_j$ and $b_j$ are: $$\begin{aligned}
- ic_1b_1\left(\frac{u}{b_1} + b_1v + iz\right)^3 - ic_1b_1\left(\frac{u}{b_1} + b_1v - iz\right)^3
- ic_2b_2\left(\frac{u}{b_2} + b_2v + iz\right)^3 - ic_2b_2\left(\frac{u}{b_2} + b_2v - iz\right)^3 \quad & & \nonumber \\
- ic_3b_3\left(\frac{u}{b_3} + b_3v + iz\right)^3 - ic_3b_3\left(\frac{u}{b_3} + b_3v - iz\right)^3
- ic_4b_4\left(\frac{u}{b_4} + b_4v + iz\right)^3 - ic_4b_4\left(\frac{u}{b_4} + b_4v - iz\right)^3 & = & iv^3, \nonumber \\
- i\frac{c_1}{b_1}\left(\frac{u}{b_1} + b_1v + iz\right)^3 - i\frac{c_1}{b_1}\left(\frac{u}{b_1} + b_1v - iz\right)^3
- i\frac{c_2}{b_2}\left(\frac{u}{b_2} + b_2v + iz\right)^3 - i\frac{c_2}{b_2}\left(\frac{u}{b_2} + b_2v - iz\right)^3 \quad & & \nonumber \\
- i\frac{c_3}{b_3}\left(\frac{u}{b_3} + b_3v + iz\right)^3 - i\frac{c_3}{b_3}\left(\frac{u}{b_3} + b_3v - iz\right)^3
- i\frac{c_4}{b_4}\left(\frac{u}{b_4} + b_4v + iz\right)^3 - i\frac{c_4}{b_4}\left(\frac{u}{b_4} + b_4v - iz\right)^3 & = & - iu^3, \nonumber \\
- ic_1\left(\frac{u}{b_1} + b_1v + iz\right)^3 - ic_1\left(\frac{u}{b_1} + b_1v - iz\right)^3
- ic_2\left(\frac{u}{b_2} + b_2v + iz\right)^3 - ic_2\left(\frac{u}{b_2} + b_2v - iz\right)^3 \quad & & \nonumber \\
- ic_3\left(\frac{u}{b_3} + b_3v + iz\right)^3 - ic_3\left(\frac{u}{b_3} + b_3v - iz\right)^3
- ic_4\left(\frac{u}{b_4} + b_4v + iz\right)^3 - ic_4\left(\frac{u}{b_4} + b_4v - iz\right)^3 & = & 0. \nonumber\end{aligned}$$ There are five equations to satisfy, namely: $$\begin{aligned}
c_1 + c_2 + c_3 + c_4 & = & 0, \nonumber \\
c_1b_1^2 + c_2b_2^2 + c_3b_3^2 + c_4b_4^2 & = & 0, \nonumber \\
\frac{c_1}{b_1^2} + \frac{c_2}{b_2^2} + \frac{c_3}{b_3^2} + \frac{c_4}{b_4^2} & = & 0, \nonumber \\
2c_1b_1^4 + 2c_2b_2^4 + 2c_3b_3^4 + 2c_4b_4^4 & = & - 1, \nonumber \\
2\frac{c_1}{b_1^4} + 2\frac{c_2}{b_2^4} + 2\frac{c_3}{b_3^4} + 2\frac{c_4}{b_4^4} & = & 1. \nonumber\end{aligned}$$ The constraints on the coefficients $b_j$ and $c_j$ can be written: $$\begin{aligned}
c_1 & = & - \frac{b_1^2}{2\left(b_1^2 - b_2^2\right)\left(b_1^2 - b_3^2\right)\left(b_1^2 - b_4^2\right)}, \nonumber \\
c_2 & = & - \frac{b_2^2}{2\left(b_2^2 - b_1^2\right)\left(b_2^2 - b_3^2\right)\left(b_2^2 - b_4^2\right)}, \nonumber \\
c_3 & = & - \frac{b_3^2}{2\left(b_3^2 - b_1^2\right)\left(b_3^2 - b_2^2\right)\left(b_3^2 - b_4^2\right)}, \nonumber \\
c_4 & = & - \frac{b_4^2}{2\left(b_4^2 - b_1^2\right)\left(b_4^2 - b_2^2\right)\left(b_4^2 - b_3^2\right)}, \nonumber \\
b_1 & = & \pm \frac{1}{b_2b_3b_4}. \nonumber\end{aligned}$$
General multipole magnets
-------------------------
The derivation of the constraints on the coefficients $c_j$ and $b_j$ in the case of quadrupoles, sextupoles and octupoles can be extended to any order of multipole. In the body of the multipole, the components of the field are: $$\begin{aligned}
B_u & = & iv^n, \nonumber \\
B_v & = & -iu^n, \nonumber \\
B_\zeta & = & 0. \nonumber\end{aligned}$$ Substituting from (\[busummation\]), (\[bvsummation\]) and (\[bzetasummation\]) gives: $$\begin{aligned}
- ic_1b_1\left(\frac{u}{b_1} + b_1v + iz\right)^n - ic_1b_1\left(\frac{u}{b_1} + b_1v - iz\right)^n
- ic_2b_2\left(\frac{u}{b_2} + b_2v + iz\right)^n - ic_2b_2\left(\frac{u}{b_2} + b_2v - iz\right)^n \quad & & \nonumber \\
+ \cdots - ic_{n+1}b_{n+1}\left(\frac{u}{b_{n+1}} + b_{n+1}v + iz\right)^n - ic_nb_{n+1}\left(\frac{u}{b_{n+1}} + b_{n+1}v - iz\right)^n & = & iv^n, \nonumber \\
- i\frac{c_1}{b_1}\left(\frac{u}{b_1} + b_1v + iz\right)^n - i\frac{c_1}{b_1}\left(\frac{u}{b_1} + b_1v - iz\right)^n
- i\frac{c_2}{b_2}\left(\frac{u}{b_2} + b_2v + iz\right)^n - i\frac{c_2}{b_2}\left(\frac{u}{b_2} + b_2v - iz\right)^n \quad & & \nonumber \\
+ \cdots - i\frac{c_{n+1}}{b_{n+1}}\left(\frac{u}{b_{n+1}} + b_{n+1}v + iz\right)^n
- i\frac{c_{n+1}}{b_{n+1}}\left(\frac{u}{b_{n+1}} + b_{n+1}v - iz\right)^n & = & - iu^n, \nonumber \\
- ic_1\left(\frac{u}{b_1} + b_1v + iz\right)^n - ic_1\left(\frac{u}{b_1} + b_1v - iz\right)^n
- ic_2\left(\frac{u}{b_2} + b_2v + iz\right)^n - ic_2\left(\frac{u}{b_2} + b_2v - iz\right)^n \quad & & \nonumber \\
+ \cdots - ic_{n+1}\left(\frac{u}{b_{n+1}} + b_{n+1}v + iz\right)^n - ic_{n+1}\left(\frac{u}{b_{n+1}} + b_{n+1}v - iz\right)^n & = & 0. \nonumber\end{aligned}$$ There are $n + 2$ equations to satisfy. Two of the equations can be written: $$\begin{aligned}
2c_1b_1^n + 2c_2b_2^n + \cdots + 2c_{n+1}b_{n+1}^n & = & - 1, \nonumber \\
2\frac{c_1}{b_1^n} + 2\frac{c_2}{b_2^n} + \cdots + 2\frac{c_{n+1}}{b_{n+1}^n} & = & 1. \nonumber\end{aligned}$$ The remaining equations have the form: $$\begin{aligned}
c_1b_1^\alpha + c_2b_2^\alpha + \cdots + c_{n+1}b_{n+1}^\alpha & = & 0, \nonumber \\
\frac{c_1}{b_1^\alpha} + \frac{c_2}{b_2^\alpha} + \cdots + \frac{c_{n+1}}{b_{n+1}^\alpha} & = & 0, \nonumber\end{aligned}$$ where, for convenience, we have defined $\alpha = n - 2p$ with $p$ a positive integer. The number of equations is determined by the order of the multipole and the restriction that $\alpha \ge 0$. Solving the equations leads to the following constraints on the coefficients $b_j$ and $c_j$: $$\begin{aligned}
b_1 & = & \pm \frac{i^{n-2}}{\prod_{j = 2}^{n+1}b_j}, \nonumber \\
c_1 & = & - \frac{b_1^{n-2}}{2\prod_{j = 2}^{n+1}\left(b_1^2 - b_j^2\right)}, \nonumber \\
c_2 & = & - \frac{b_2^{n-2}}{2\prod_{j = 1,j\neq2}^{n+1}\left(b_2^2 - b_j^2\right)}, \nonumber \\
& \vdots & \nonumber \\
c_n & = & - \frac{b_n^{n-2}}{2\prod_{j = 1,j\neq n}^{n+1}\left(b_n^2 - b_j^2\right)}. \nonumber\end{aligned}$$
Significance of parameters and scaling laws
-------------------------------------------
From equations (\[resbu\]), (\[resbv\]) and (\[resbzp\]), it is clear that in the case of the elementary solution, the constant $b$ acts as a scaling parameter. This remains the case when several elementary solutions are added together to represent the fringe field of a multipole magnet. We shall see later that the constant $b$ affects the rate of fall-off of the field strength with distance from the axis (keeping $z$ constant). This can be seen by inspecting the singularities present in the components of the field, as illustrated in the case of quadrupoles in the next section. The singularities are an inevitable consequence of the way in which the magnetostatic Maxwell equations in three dimensions were solved, by strict analogy with the two dimensional case. For quadrupole magnets, the locations of the singularities depend on $b$; it is expected that the same is true for higher order multipoles. The singularities depend on the inverse of $d$ or $e$; therefore, if $b$ is increased from zero towards one, the singularities move (transversely) away from the axis of the magnet. Looking at the individual components of the magnetic field in Cartesian co-ordinates (equations (\[resbx\]), (\[resby\]) and (\[resbz\])), it can be seen from the way that the scaling constant $b$ enters through the constants $d$, $e$ and $c$ (with $c = 4/de$), that the smaller the value of $b$, the larger the values of $d$ and $e$ and vice versa. These relationships are illustrated in Fig. \[d+e\].
![Behaviour of the factors $d$ and $e$.[]{data-label="d+e"}](d+e){width="55.00000%"}
Suppose that, for simplicity, we scale all the field components by a factor of $c$. Then, $B_x \sim d$, $B_y \sim e$ and $B_z$ is independent of $d$ or $e$. For small values of $b$, the transverse components $B_x$ and $B_y$ of the magnetic field dominate over the longitudinal component $B_z$. This agrees with the fact that, for a small value of $b$, the transverse fall-off of the field is very rapid, so the appearance in the fringe region of the longitudinal component of the magnetic field $B_z$ does not need to be as prolonged as for larger values of $b$. Ultimately, we shall see that the constant $b$, for quadrupoles, can be used as a parameter to fit the field behaviour with distance from the axis, in the same way that the Enge coefficients can be used as parameters to fit the field behaviour with distance along the axis. The constant $b$ and the Enge coefficients are ultimately dependent on the geometry of the magnet.
The number of constants increases as $(n - 2)/2$ with increasing order $n$ of the multipole. For a quadrupole, there are effectively two transverse directions in which the rate of the fall-off of the field in the fringe region can vary. The rate of fall-off in one direction can be adjusted by an overall re-scaling; to vary the rate of fall-off in the second direction (independently of the first direction) requires a separate parameter. Similarly, in a sextupole, there are three independent transverse directions in which the rate of fall-off of the field can vary, which means that there are two independent parameters to control the transverse behaviour. For octupoles, with four independent transverse directions, three parameters are required, and so on.
Multipole magnets with Enge-type fringe fields
==============================================
Having constructed general expressions for fringe fields in multipole magnets, it is worth investigating these expressions further, to show that the field has the appropriate behaviour. For a multipole magnet of arbitrary order, the expressions for the field can be rather complicated: therefore, we consider in detail only the case of a quadrupole. In this section, we shall discuss the behaviour of the fields in a quadrupole, first for the elementary solution, and then for the full solution (in which the fields are given by real numbers) obtained by adding two versions of the elementary solution. In order to plot the field, we need to make some assumption for the form of the roll-off of the field along the axis of the magnet: we shall consider the case that the roll-off is described by an Enge function. At the end of the section, we shall generalise the expressions for a full quadrupole solution with Enge roll-off of the gradient to higher order multipole magnets.
Of particular interest is the appearance of singularities in the magnetic field. Singularities are expected from the properties of Laplace’s equation: in two dimensions, solutions to Laplace’s equation are either constant everywhere, or have singularities somewhere. Our expressions for multipole fringe fields have been obtained by extending the two dimensional case to three dimensions. Mathematically, it is no surprise that singularities appear, but if the expressions we have derived for the fields are to be applied in physical situations, we should understand the position and nature of the singularities.
Elementary solution for a quadrupole with Enge-type fringe field {#elementaryquadsolution}
----------------------------------------------------------------
We take $n = 1$ for the elementary solution and examine its behaviour. For simplicity we assume that the fall-off functions are related by $F_1 = - G_1$. To simplify the notation we drop the subscript $j = 1$, so the field components (\[busummation\])–(\[bzetasummation\]) become: $$\begin{aligned}
B_u & = & b c \left( (\zeta + ih) G(\zeta + ih) - (\zeta - ih) G(\zeta - ih) \right), \nonumber \\
B_v & = & \frac{c}{b} \left( (\zeta + ih) G(\zeta + ih) - (\zeta - ih) G(\zeta - ih) \right), \nonumber \\
B_\zeta & = & - i c \left( (\zeta + ih) G(\zeta + ih) + (\zeta - ih) G(\zeta - ih) \right). \nonumber\end{aligned}$$ We shall see when we consider the full quadrupole solution in Section \[sectionfullquadrupoleenge\] that an Enge-type fall-off in the quadrupole gradient occurs when we choose the function $G(\zeta)$ as follows: $$G(\zeta) = \frac{\ln(1 + e^\zeta)}{\zeta} - \frac{\ln 2}{\zeta} - 1.$$ The components of the field (in a Cartesian basis) can be written in this case: $$\begin{aligned}
B_x & = & \frac{b}{2\sqrt{2}(1 - b^2)} \left( \ln(1 + e^{\zeta + i h}) - \ln(1 + e^{\zeta - i h}) - 2i h \right), \nonumber \\
B_y & = & \frac{i b}{2\sqrt{2}(1 + b^2)} \left( \ln(1 + e^{\zeta + i h}) - \ln(1 + e^{\zeta - i h}) - 2i h \right), \nonumber \\
B_z & = & \frac{i b^2}{\sqrt{2}(1 - b^4)} \left( 2\zeta + 2\ln 2 - \ln(1 + e^{\zeta + i h}) - \ln(1 + e^{\zeta - i h}) \right). \nonumber\end{aligned}$$ Along the axis of the magnet ($h = 0$), the transverse components of the field vanish and the longitudinal component of the field is purely imaginary. If we identify the real parts of the field with the physical magnetic field, then the longitudinal component of the field also vanishes along the axis. Singularities in the field occur when: $$\zeta \pm i h = i \ell \pi, \label{singularitycondition}$$ where $\ell$ is any odd integer. In terms of the Cartesian co-ordinates, the singularities occur at: $$(x, y) = \left( \frac{\sqrt{2} \ell \pi}{\frac{1}{b} + b} , \pm \frac{2 z}{\frac{1}{b} - b}\right).$$ Note, however, that the singularities in the different terms in the transverse components of the field cancel out when $z = 0$.
The behaviour of the field can be seen in Fig. \[engebxm4m2\], which shows the real parts of the $B_x$ and $B_z$ components of the magnetic field as functions of the transverse co-ordinates at various $z$ locations. For the elementary quadrupole-like solution, the constant $b$ amounts to a re-scaling in the coordinates and the field; therefore, we show plots for only a single value $b = 0.1$. Also, since no new features occur in the behaviour of $B_y$ compared to the behaviour of $B_x$, we only show plots for $B_x$ and $B_z$. The plots in Fig. \[engebxm4m2\] show that the field has the expected behaviour for a quadrupole: in particular, within the body of the magnet, $B_x$ is linear in the co-ordinate $y$, and independent of $x$. At increasing values of $z$, the slope of $B_x$ versus $y$ decreases (the gradient falls off); at $z = 0$, the quadrupole gradient is half the value at large negative $z$. As $z$ increases further, the gradient (and the field itself) falls to zero. The singularities in the field have the behaviour expected from (\[singularitycondition\]). In particular, we see that at $z = 0$, the singularities in the transverse field disappear completely.
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Note that it was not necessary to demand that $F = - G$ in the fall-off functions above. This was only done for simplicity and it could well be the case that a non-symmetric fall-off is desired and this would require $F$ and $G$ to be completely different. As they are just functions that dictate the behaviour the fringe fall-off, nothing changes inside the quadrupole and the magneto-static Maxwell equations are still satisfied.
Full solution for a quadrupole with Enge-type fringe field {#sectionfullquadrupoleenge}
----------------------------------------------------------
As shown above, we can construct physical (real) fields in a quadrupole by adding two versions of the elementary solution with $n = 1$: $$\begin{aligned}
B_u & = & \sum_{j = 1}^{2} b_j c_j \left( (\zeta + ih_j) G(\zeta + ih_j) - (\zeta - ih_j) G(\zeta - ih_j) \right), \label{fullsolutionquadbu} \\
B_v & = & \sum_{j = 1}^{2} \frac{c_j}{b_j} \left( (\zeta + ih_j) G(\zeta + ih_j) - (\zeta - ih_j) G(\zeta - ih_j) \right), \\
B_\zeta & = & - i \sum_{j = 1}^{2} c_j \left( (\zeta + ih_j) G(\zeta + ih_j) + (\zeta - ih_j) G(\zeta - ih_j) \right),\label{fullsolutionquadbzeta}\end{aligned}$$ where: $$h_j = d_jx + ie_jy,$$ and: $$\begin{aligned}
d_j & = & \frac{b_j}{\sqrt{2}} + \frac{1}{\sqrt{2}b_j}, \nonumber \\
e_j & = & \frac{1}{\sqrt{2}b_j} - \frac{b_j}{\sqrt{2}}, \nonumber\end{aligned}$$ together with the definitions of $b_j$ and $c_j$ given in (\[bq\]) and (\[cq\]), keeping in mind we took the negative solution in equation (\[bq\]). Working in cylindrical polar co-ordinates $r$, $\theta$, $z$, with the $x$ axis corresponding to the line $z = \theta = 0$, the radial field component can be expressed as a Taylor series in $r$: $$B_r = -r \sin(2\theta) \left( G(\zeta) + \zeta G^\prime(\zeta) \right) + O(r^2), \label{quadrupolegradient}$$ where $G^\prime$ is the derivative of $G$ with respect to its argument. Equation (\[quadrupolegradient\]) describes the behaviour we would expect from a magnet in which the quadrupole gradient $g(z)$ varies along the axis as: $$g(\zeta) = - G(\zeta) - \zeta G^\prime(\zeta). \label{gradientbigg}$$ A conventional model for the fringe field in a quadrupole magnet describes the gradient as an Enge function [@Brown]. Again to keep the analysis as simple as possible, we consider the case that the gradient varies as an Enge function with a single parameter: $$g(\zeta) = \frac{1}{1 + e^\zeta}.$$ Writing the gradient in this way determines $z = 0$ as the location along the axis at which the quadrupole gradient falls to half its nominal value within the body of the magnet. Integrating (\[gradientbigg\]) gives: $$G(\zeta) = \frac{\ln(1 + e^\zeta)}{\zeta} - \frac{\ln 2}{\zeta} - 1,$$ where the constant of integration has been chosen so that $G(\zeta)$ remains finite as $\zeta \to 0$.
The components of the field can now be written (in a Cartesian basis): $$\begin{aligned}
B_x & = & \sum_{j = 1}^{2}\frac{c_jd_j}{2}\left( - 2ih_j - \ln(1 + e^{\sqrt{2}z - ih_j}) + \ln(1 + e^{\sqrt{2}z + ih_j}) \right), \nonumber \\
B_y & = & i\sum_{j = 1}^{2}\frac{c_je_j}{2}\left( - 2ih_j - \ln(1 + e^{\sqrt{2}z - ih_j}) + \ln(1 + e^{\sqrt{2}z + ih_j}) \right), \nonumber \\
B_z & = & i\sum_{j = 1}^{2}\frac{c_j}{\sqrt{2}}\left( 2\ln(2) + 2\sqrt{2}z - \ln(1 + e^{\sqrt{2}z - ih_j}) - \ln(1 + e^{\sqrt{2}z + ih_j}) \right).
\nonumber\end{aligned}$$ The quadrupole gradient $g(\zeta)$ and the corresponding function $G(\zeta)$ are plotted as a function of distance along the axis of the quadrupole magnet in Fig. \[fall-offs\].
![Quadrupole gradient $g(\zeta)$ given by an Enge function (black line) and the corresponding function $-G(\zeta)$ (red line), as a function of distance along the axis of a quadrupole magnet.[]{data-label="fall-offs"}](falloff2){width="50.00000%"}
If we plot the two constants $d_j$ and $e_j$ as functions of $b_j$ (as shown in Fig. \[d+e\]) we see that the field components $B_x$ and $B_y$ will decay at different rates. The transverse field components only show the same behaviour when $b_2 = 0$; this value of $b_2$ is not allowed, since as $b_2 \to 0$, $e_2 \to \pm \infty$. Therefore, the symmetry $B_x \leftrightarrow B_y$ under $x \leftrightarrow y$ has to be imposed. This can be done by adding equivalent expressions to $B_x$, $B_y$ and $B_z$, with $x$ and $y$ interchanged but keeping $b_2$ the same, and dividing the result by $2$. Therefore, in what follows, the *full quadrupole solution* refers to the fully symmetric quadrupole solution. There may be cases where exact quadrupole symmetry is not required; however, we do not consider such cases here.
Figure \[engebxfm4m2\] shows the behaviour of the field components $B_x$, $B_y$ and $B_z$ for the full quadrupole solution, with the fixed value of the parameter $b_2 = 0.1$, and with the imposition of the symmetry $B_x \leftrightarrow B_y$ under $x \leftrightarrow y$. Within the body of the quadrupole (large negative $z$) we see that, as expected, $B_x \propto y$, $B_y \propto x$ and $B_z \approx 0$. At $z = 0$ the quadrupole gradient (the constant of proportionality between $B_x$ and $y$, or between $B_y$ and $x$) falls to half of its value within the body of the magnet. At a large distance from the quadrupole (large positive $z$), the field approaches zero.
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Full solution for a multipole magnet with Enge-type fringe field {#sectionfullmultipoleenge}
----------------------------------------------------------------
The full solution for a multipole of order $n$ can be written: $$\begin{aligned}
B_u & = & - \sum_{j = 1}^{n+1} b_j c_j \left( (-i)^{n+1} (\zeta + ih_j)^n G_n(\zeta + ih_j) - i^{n+1} (\zeta - ih_j)^n G_n(\zeta - ih_j) \right),
\nonumber \\
B_v & = & - \sum_{j = 1}^{n+1} \frac{c_j}{b_j} \left( (-i)^{n+1} (\zeta + ih_j)^n G_n(\zeta + ih_j) - i^{n+1} (\zeta - ih_j)^n G_n(\zeta - ih_j)
\right), \nonumber \\
B_\zeta & = & i \sum_{j = 1}^{n+1} c_j \left( (-i)^{n+1} (\zeta + ih_j)^n G_n(\zeta + ih_j) + i^{n+1} (\zeta - ih_j)^n G_n(\zeta - ih_j) \right).
\nonumber\end{aligned}$$ The function $G_0(\zeta)$ is an Enge function (with one coefficient): $$G_0(\zeta) = \frac{1}{1 + e^\zeta}. \label{engefunctionG0}$$ Functions $G_n(\zeta)$ for $n > 0$ are obtained by induction: $$\zeta^n \, G_n(\zeta) = \int_0^\zeta \zeta^{\prime n-1}\, G_{n-1}(\zeta^\prime) \, \mathrm{d}\zeta^\prime. \label{engefunctionGn}$$ It is then found that expanding the radial field component $B_r$ as a Taylor series in $r$ gives: $$B_r = \frac{r^n \sin ((n+1)\theta)}{2^\frac{n-1}{2} \, n! \, (1 + e^{\sqrt{2} z})} + O(r^{n+1}),$$ where $O(r^{n+1})$ represents terms in $r$ of order $n+1$. Thus, the field has the behaviour expected of a multipole magnet, with gradient that decays as an Enge function in the fringe field region.
The functions $G_n(\zeta)$ can also be written in the form: $$G_n(\zeta) = \frac{1}{n!} + \frac{\polylog_n(-e^\zeta)}{\zeta^n} - \sum_{j = 1}^n \frac{\polylog_j(-1)}{\zeta^j (n-j)!},$$ where $\polylog_n(\zeta)$ is the polylogarithm (or Jonquière function [@Jonquiere]) of order $n$.
Potentials for multipole magnet fringe fields
=============================================
For some applications it may be useful to have analytical expressions for scalar and vector potentials from which the fringe fields in multipole magnets can be derived. For example, in an iron dominated magnet the pole faces correspond to surfaces of constant scalar potential; this makes it possible to inspect the geometry of a magnet that would have a fringe field of a given form. The vector potential can be useful for symplectic tracking of particles through the field (see, for example, [@WFR]). General expressions for the scalar and vector potentials for the elementary solution were given in Section \[sectiongeneralthreedsolution\]. In this section, we consider in particular the case that the multipole gradient $g_n(\zeta)$ has a roll-off in the fringe field that can be represented by an Enge function.
Scalar potential and pole-face geometry
---------------------------------------
Consider a fringe field of the form: $$\begin{aligned}
B_u & = & - \sum_{j = 1}^{n+1} b_j c_j \left( (-i)^{n+1} g_n(\zeta + ih_j) - i^{n+1} g_n(\zeta - ih_j) \right), \label{generalfringebu} \\
B_v & = & - \sum_{j = 1}^{n+1} \frac{c_j}{b_j} \left( (-i)^{n+1} g_n(\zeta + ih_j) - i^{n+1} g_n(\zeta - ih_j) \right), \label{generalfringebv} \\
B_\zeta & = & i \sum_{j = 1}^{n+1} c_j \left( (-i)^{n+1} g_n(\zeta + ih_j) + i^{n+1} g_n(\zeta - ih_j) \right). \label{generalfringebzeta}\end{aligned}$$ In the case that: $$g_n(\zeta) = \zeta^n G_n(\zeta),$$ where the functions $G_n(\zeta)$ are defined in (\[engefunctionG0\]) and (\[engefunctionGn\]), this field represents the full solution for the fringe field in a multipole magnet of order $n$. For the moment, we do not assume any particular form for the function $g_n(\zeta)$. The field (\[generalfringebu\])–(\[generalfringebzeta\]) may be derived from a scalar potential given by: $$\varphi = \sum_{j = 1}^{n+1} (-i)^n c_j \tilde{g}_n(\zeta + i h_j) - i^n c_j \tilde{g}_n(\zeta - i h_j),$$ where: $$\tilde{g}_n(\zeta) = \int_0^\zeta g_n(\zeta^\prime)\, \mathrm{d}\zeta^\prime.$$ This expression for the scalar potential may be verified by substituting into $\vec{B} = \nabla \varphi$. If it is possible to integrate the function $g_n(\zeta)$ describing the fringe field, then it is possible to write down a scalar potential for the field. In the case of the fringe field of a multipole magnet of order $n$, with roll-off given by an Enge function, $\tilde{g}_n(\zeta)$ is given by: $$\tilde{g}_n(\zeta) = \int_0^\zeta g_n(\zeta^\prime)\, \mathrm{d}\zeta^\prime = \int_0^\zeta \zeta^{\prime n} G_n(\zeta^\prime)\,
\mathrm{d}\zeta^\prime = \zeta^{n+1}G_{n+1}(\zeta).$$ Hence, in a multipole magnet of order $n$ with fringe field falling off as an Enge function, the scalar potential can be written: $$\varphi = \sum_{j = 1}^{n+1} (-i)^n c_j (\zeta + i h_j)^{n+1}G_{n+1}(\zeta + i h_j) - i^n c_j (\zeta - i h_j)^{n+1}G_{n+1}(\zeta - i h_j).
\label{scalarpotentialmultipolefullsolution}$$
In an iron dominated magnet, the pole faces can be identified with surfaces of constant scalar potential: an example of such a surface (for an inner triplet quadrupole for the HL-LHC) is shown in Fig. \[hllhcitpolefaceshape\]. The surface has the geometry expected of the poles in an iron dominated quadrupole. Within the body of the magnet, the intersection of the pole face with a plane $z = $constant forms a hyperbola. As a function of position along the axis, the pole face has little dependence on $z$ for large negative $z$; but in the fringe field region the pole face abruptly “flattens” off to lie close to the plane $z = 0$.
Vector potential
----------------
It can be useful to know the vector potential for a given field for symplectic integration of the equations of motion for a charged particle moving through the field: see, for example, [@WFR]. It is possible to write down an expression for a vector potential $\vec{A}$ from which the field (\[generalfringebu\])–(\[generalfringebzeta\]) may be derived. In a gauge with $A_x = 0$, the components $A_y$ and $A_z$ are given by: $$\begin{aligned}
A_y & = & - \sqrt{2} i \sum_{j = 1}^{n+1} \frac{c_j}{d_j} \left( (-i)^n \tilde{g}_n(\zeta + ih_j) + i^n \tilde{g}_n(\zeta - ih_j) \right), \nonumber \\
A_z & = & - i \sum_{j = 1}^{n+1} \frac{c_j e_j}{d_j} \left( (-i)^n \tilde{g}_n(\zeta + ih_j) - i^n \tilde{g}_n(\zeta - ih_j) \right). \nonumber\end{aligned}$$ This form for the vector potential may be verified by substitution[^1] into $\vec{B} = \nabla \times \vec{A}$.
In the case that the field (\[generalfringebu\])–(\[generalfringebzeta\]) represents the fringe field in a multipole magnet: $$g_n(\zeta) = \zeta^n G_n(\zeta),$$ with $G_0(\zeta)$ an Enge function, it is possible to express the vector potential in terms of the functions $G_n(\zeta)$ (defined in (\[engefunctionG0\]) and (\[engefunctionGn\])) using: $$\tilde{g}_n(\zeta) = \zeta^{n+1}G_{n+1}(\zeta).$$
Examples
========
To illustrate the application of the methods and results described in the previous sections, we consider two examples: a quadrupole in a final focus or “inner triplet” region of the high luminosity upgrade of the Large Hadron Collider (HL-LHC), and a quadrupole in the EMMA non-scaling fixed-field alternating gradient accelerator. These magnets are chosen to provide two contrasting cases in terms of magnet technology and parameter regime. The HL-LHC inner triplet quadrupole is a superconducting magnet with large aperture and gradient (respectively, 150mm diameter and 140T/m). Design studies are still in progress; some field maps are available, but the impact of the fringe fields on the beam dynamics are still under investigation. The EMMA quadrupoles are normal-conducting electromagnets with more conventional gradient. Here, we consider one of the two types of EMMA magnet, namely the F quadrupoles [@emmaquads]. These magnets have large apertures (74mm diameter) given the length of the iron poles (73mm); as a result, the fringe fields make a dominant contribution to the focusing effects. The specified integrated gradient is 0.387T. Although it is possible to represent an EMMA quadrupole by a hard-edge model, beam dynamics studies (supported by experimental results) [@yoelthesis] indicate the need for a more realistic representation in order to give an accurate description of the longitudinal and transverse dynamics in the machine.
We should emphasise that the purpose of considering these illustrative cases is not to demonstrate close agreement between the numerical and analytical field models, but simply to show that it is possible, using analytical formulae from the previous sections, to construct a model of the fringe field in each case that satisfies Maxwell’s equations, and is closer to reality than a simple hard-edge model.
For the HL-LHC inner triplet quadrupole and for the EMMA F quadrupole we use the full solution for a quadrupole with Enge-type fringe field, presented in Section \[sectionfullquadrupoleenge\]. The numerical field data in each case are fitted using a roll-off function such that the quadrupole gradient has the form of an Enge function: $$g(z) = \frac{a_0}{1 + e^{a_1 + \sqrt{2} a_2 z}}.
\label{examplegradientenge}$$ The fit is optimised by varying the parameters $a_0$, $a_1$ and $a_2$. Following the same procedure as in Section \[sectionfullquadrupoleenge\], we find that the radial component of the magnetic field can be written to first order in the radial co-ordinate $r$: $$B_r = \frac{a_0 \sin(2\theta) r}{1 + e^{a_1 + \sqrt{2} a_2 z}} + O(r^3).$$ If the radial component of the field is known as a function of longitudinal position for some $\theta$ and (small) $r$, then $a_0$, $a_1$ and $a_2$ can be obtained from a nonlinear fit to the data. With the gradient described by an Enge function as in (\[examplegradientenge\]), the function $G(\zeta)$ in (\[fullsolutionquadbu\])–(\[fullsolutionquadbzeta\]) is given by: $$G(\zeta) = a_0 \left( 1 - \frac{\ln(1 + e^{a_1 + a_2 \zeta}) - \ln(1 + e^{a_1})}{a_2 \zeta} \right).
\label{exampleGfunction}$$ All field components at any point in the magnet (including the fringe field) can then be obtained using (\[fullsolutionquadbu\])–(\[fullsolutionquadbzeta\]).
HL-LHC inner triplet quadrupoles {#sectionhllhcitquads}
--------------------------------
As a first example, we show the results of a fit to the field in an HL-LHC inner triplet quadrupole. Field data are obtained from a magnetic model, and a fit to the field based on a function of the form (\[exampleGfunction\]) is then obtained using the method described above. The parameter values obtained by fitting the radial field component along a line at $\theta = \pi/4$ and various values of $r$ are shown in Table \[hllhcfitparams\]. If the chosen function provides a good description of the field, then we expect little variation in the parameters obtained by fitting along lines with different $\theta$ and $r$, at least for small $r$ (since the fit is based on the linear term in a series expansion of the field). For the HL-LHC inner triplet quadrupole, inspection of the values shown in Table \[hllhcfitparams\] shows changes in the fit parameters of less than 0.7%, comparing values obtained by fitting the field along lines with $r = r_\mathrm{max}/10$ and $r = r_\mathrm{max}/6$ (and fixed $\theta = \pi/4$).
radius of fit $a_0$ $a_1$ $a_2$
--------------------- ---------- ----------- ---------
$r_\mathrm{max}/10$ -55.9503 -0.520120 8.98913
$r_\mathrm{max}/8$ -55.9504 -0.521262 9.00549
$r_\mathrm{max}/6$ -55.9505 -0.523743 9.04082
$r_\mathrm{max}/4$ -55.9510 -0.530875 9.14139
$r_\mathrm{max}/3$ -55.9517 -0.540984 9.28116
$r_\mathrm{max}/2$ -55.9539 -0.570142 9.66955
: Parameters for a fit of an Enge function of the form (\[examplegradientenge\]), to the quadrupole gradient in an HL-LHC inner triplet quadrupole. The maximum radius $r_\mathrm{max}$ in the field data is 75mm.[]{data-label="hllhcfitparams"}
Having obtained values for the fit parameters, we can make direct comparisons of the field given by equations (\[fullsolutionquadbu\])–(\[fullsolutionquadbzeta\]) with the field obtained by the numerical magnetic model. An example of such a comparison, for the radial field component, is shown in Fig. \[examplefigbrvszvariousr\]. Although the analytical model does not match the numerical model exactly, the general behaviour of the field is reproduced quite closely. The impact of the residuals from the fit on the beam dynamics still needs to be studied; but the analytical model does include features of the field that would be completely omitted in a hard-edged magnet model.
Figure \[examplefigbrvszvarioustheta\] shows the radial field component as a function of position along lines of fixed radius and for different values of the polar co-ordinate $\theta$. Again, there is reasonable agreement between the numerical field data (black lines) and the analytical fit (red line). Finally, Fig. \[examplefigbzvszvariousr\] shows the longitudinal field component as a function of position along lines of given radius and with fixed $\theta = \pi/4$. Here, it appears that there are more significant discrepancies between the numerical field data and the analytical model; however, there is general agreement in the main features, especially for the region close to the axis of the magnet. The more detailed structure that appears at larger distances from the axis cannot be reproduced by the relatively simple (Enge) function that is used to describe the fall-off of the quadrupole gradient in the fringe field.
![Radial component of the magnetic field in an HL-LHC inner triplet quadrupole as a function of position along lines of given radius and with cylindrical polar co-ordinate $\theta = \pi/4$. Each pair of black and red lines shows the field at a different radius, from $r_\mathrm{max}/5$ to $4r_\mathrm{max}/5$, with $r_\mathrm{max} = 75$mm. The black lines show the field obtained from the (numerical) magnetic model; the red lines show the analytical model (\[fullsolutionquadbu\])–(\[fullsolutionquadbzeta\]), with fit parameters given in Table \[hllhcfitparams\] with radius of fit $r_\mathrm{max}/10$, and $b_1 = 2.5$.[]{data-label="examplefigbrvszvariousr"}](LHC_IT_Quad_Fringe_Br_vs_z-r_Enge){width="50.00000%"}
![Radial component of the magnetic field in an HL-LHC inner triplet quadrupole as a function of position along lines of fixed distance $r = r_\mathrm{max}/4$. Each pair of black and red lines shows the field at a different value of the cylindrical polar co-ordinate $\theta$, from $\pi/16$ to $\pi/4$ (in steps of $\pi/16$). The black lines show the field obtained from the (numerical) magnetic model; the red lines show the analytical model (\[fullsolutionquadbu\])–(\[fullsolutionquadbzeta\]), with fit parameters given in Table \[hllhcfitparams\] with radius of fit $r_\mathrm{max}/10$, and $b_1 = 2.5$.[]{data-label="examplefigbrvszvarioustheta"}](LHC_IT_Quad_Fringe_Br_vs_z-theta_Enge){width="50.00000%"}
![Longitudinal component of the magnetic field in an HL-LHC inner triplet quadrupole as a function of position along lines of given radius and with cylindrical polar co-ordinate $\theta = \pi/4$. Each pair of black and red lines shows the field at an increasing radius, from $r_\mathrm{max}/5$ to $4r_\mathrm{max}/5$, with $r_\mathrm{max} = 75$mm. The black lines show the field obtained from the (numerical) magnetic model; the red lines show the analytical model (\[fullsolutionquadbu\])–(\[fullsolutionquadbzeta\]), with fit parameters given in Table \[hllhcfitparams\] with radius of fit $r_\mathrm{max}/10$, and $b_1 = 2.5$.[]{data-label="examplefigbzvszvariousr"}](LHC_IT_Quad_Fringe_Bz_vs_z-r_Enge){width="50.00000%"}
It is worth considering the dependence of the field on the parameter $b_1 (= 1/b_2)$. Changing the value of $b_1$ has an effect on the way that the fringe field varies with distance from the magnetic axis. This can be seen by comparing the plots in Fig. \[examplefigbrvszvariousrb\], which show the radial field component as a function of position along lines with fixed polar co-ordinate $\theta = \pi/4$ and different distance $r$ from the axis of the magnet. The plot on the left in Fig. \[examplefigbrvszvariousrb\] shows a comparison between the numerical field map and the analytical model with $b_1 = 1.5$; the plot on the right compares the numerical field map and the analytical model with $b_1 = 3.5$. In regions close to the axis of the magnet, changes in the value of $b_1$ in the range 1.5 to 3.5 have little effect on the field; but differences are apparent at larger distances from the axis. The parameter $b_1$ therefore allows control over the field behaviour at large $r$, after the parameters $a_0$, $a_1$ and $a_2$ have been chosen to fit the field behaviour close to the axis of the magnet.
![As Fig. \[examplefigbrvszvariousr\], but with $b_1 = 1.5$ (left) and $b_1 = 3.5$ (right).[]{data-label="examplefigbrvszvariousrb"}](LHC_IT_Quad_Fringe_Br_vs_z-r_Enge_b1,5 "fig:"){width="45.00000%"} ![As Fig. \[examplefigbrvszvariousr\], but with $b_1 = 1.5$ (left) and $b_1 = 3.5$ (right).[]{data-label="examplefigbrvszvariousrb"}](LHC_IT_Quad_Fringe_Br_vs_z-r_Enge_b3,5 "fig:"){width="45.00000%"}
Although the HL-LHC inner triplet quadrupoles are superconducting magnets, we can nevertheless inspect the shape of the pole face that would be needed to give the same field. Assuming poles with infinite magnetic permeability, the shape of a pole face corresponds to a surface of constant magnetic scalar potential $\varphi$; in the case of a quadrupole with gradient falling off as an Enge function, the scalar potential is given by (\[scalarpotentialmultipolefullsolution\]). Figure \[hllhcitpolefaceshape\] shows an equipotential surface for the analytical field model fitted to the numerical HL-LHC inner triplet quadrupole field data. The surface has been chosen so that the magnetic scalar potential has the (arbitrary) value $\varphi = 0.25$Tm. We see that the equipotential surface has the shape that might be expected of an iron-dominated quadrupole, with the curved surace following an hyperbola in a plane of constant negative $z$, and the end of the pole being reasonably flat (close to a plane of constant $z \approx 0$).
![Surface of constant scalar potential $\varphi = 0.25$Tm in a representation of the HL-LHC inner triplet quadrupole, with gradient falling as an Enge function in the fringe field. The parameters of the Enge function (\[examplegradientenge\]) correspond to the first line in Table \[hllhcfitparams\].[]{data-label="hllhcitpolefaceshape"}](LHC_IT_Quad_PoleFace_phi-0,25_b2,5){width="60.00000%"}
EMMA quadrupoles
----------------
The quadrupole magnets in EMMA are iron-dominated, normal conducting magnets. The unusual feature of the EMMA quadrupoles is that the diameter of the aperture is comparable to the length of the magnet; this allows the accelerator to have a large transverse acceptance in a lattice consisting of magnets packed very close together. However, the relatively large aperture of the quadrupoles means that the gradient falls off rapidly from the centre of the magnet. There is no appreciable distance along the axis for which the gradient is constant, and a realistic model of the field must include some representation of the fringe fields.
Given numerical field data from a magnetic model of an EMMA quadrupole, we can repeat the analysis used for the HL-LHC inner triplet quadrupole in Section \[sectionhllhcitquads\]. We again use an Enge function of the form (\[examplegradientenge\]) to represent the fall-off of the gradient along the axis of the magnet. The field data cover an area closely approaching the pole tip (the field values are given on a rectangular grid, with transverse co-ordinates extending to 36mm). The results of fitting the parameters in the Enge function to field data along a line parallel to the axis (at different distances from the axis and fixed polar angle $\theta = \pi/4$) are shown in Table \[emmaquadfitparams\]. We see that there is much larger variation in the parameters if the fit is performed at different distances from the axis, compared to the case of the HL-LHC inner triplet quadrupole. This suggests that the quality of the fit will not be as good. Making a direct comparison of the field based on the analytical formula with the numerical field data confirms that this is the case: see Figs. \[examplefigbrvszvariousremma\], \[examplefigbrvszvariousthetaemma\] and \[examplefigbzvszvariousremma\]. Inspecting Fig. \[examplefigbrvszvariousremma\] suggests the reason for the poor quality of the fit. The gradient initially falls off quite rapidly along the axis from the centre of the magnet, but there is a long “tail” as the gradient approaches zero: this asymmetric behaviour cannot be represented accurately using an Enge function with a small number of coefficients. Using a larger number of Enge coefficients improves the quality of the fit for the field at a given radius (and polar angle $\theta$), but then performing the integral of the quadrupole gradient $g(z)$ to find the function $G(\zeta)$ becomes difficult.
It is of course possible to plot an equipotential surface to represent the shape of the pole in an EMMA quadrupole, in the same way that we did for a “normal conducting equivalent” HL-LHC inner triplet quadrupole. However, we find that the shape of the pole is much as expected for a normal conducting quadrupole, i.e. the plot is qualitatively very similar to that shown in Fig. \[hllhcitpolefaceshape\].
radius of fit $a_0$ $a_1$ $a_2$
--------------------- ----------- ----------- ---------
$r_\mathrm{max}/10$ 0.0138149 -0.162670 11.2914
$r_\mathrm{max}/8$ 0.0172756 -0.163023 11.3031
$r_\mathrm{max}/6$ 0.0230543 -0.163794 11.3286
$r_\mathrm{max}/4$ 0.0346675 -0.166075 11.4036
$r_\mathrm{max}/3$ 0.0463818 -0.169480 11.5140
$r_\mathrm{max}/2$ 0.0702287 -0.180796 11.8694
: Parameters for a fit of an Enge function of the form (\[examplegradientenge\]), to the quadrupole gradient in an EMMA F quadrupole. The maximum radius $r_\mathrm{max}$ in the field data is 36mm.[]{data-label="emmaquadfitparams"}
![Radial component of the magnetic field in an EMMA F quadrupole as a function of position along lines of given radius and with cylindrical polar co-ordinate $\theta = \pi/4$. Each pair of black and red lines shows the field at a different radius, from $r_\mathrm{max}/5$ to $4r_\mathrm{max}/5$, with $r_\mathrm{max} = 36$mm. The black lines show the field obtained from the (numerical) magnetic model; the red lines show the analytical model (\[fullsolutionquadbu\])–(\[fullsolutionquadbzeta\]), with fit parameters given in Table \[emmaquadfitparams\] with radius of fit $r_\mathrm{max}/10$, and $b_1 = 1.8$. The centre of the quadrupole is at the far left of the plot, $z = -53.6$mm.[]{data-label="examplefigbrvszvariousremma"}](EMMA_Quad_Fringe_Br_vs_z-r_Enge_b1,8){width="50.00000%"}
![Radial component of the magnetic field in an EMMA F quadrupole as a function of position along lines of fixed distance $r = r_\mathrm{max}/4$. Each pair of black and red lines shows the field at a different value of the cylindrical polar co-ordinate $\theta$, from $\pi/16$ to $\pi/4$ (in steps of $\pi/16$). The black lines show the field obtained from the (numerical) magnetic model; the red lines show the analytical model (\[fullsolutionquadbu\])–(\[fullsolutionquadbzeta\]), with fit parameters given in Table \[emmaquadfitparams\] with radius of fit $r_\mathrm{max}/10$, and $b_1 = 1.8$. The centre of the quadrupole is at the far left of the plot, $z = -53.6$mm.[]{data-label="examplefigbrvszvariousthetaemma"}](EMMA_Quad_Fringe_Br_vs_z-theta_Enge_b1,8){width="50.00000%"}
![Longitudinal component of the magnetic field in an EMMA F quadrupole as a function of position along lines of given radius and with cylindrical polar co-ordinate $\theta = \pi/4$. Each pair of black and red lines shows the field at an increasing radius, from $r_\mathrm{max}/5$ to $4r_\mathrm{max}/5$, with $r_\mathrm{max} = 36$mm. The black lines show the field obtained from the (numerical) magnetic model; the red lines show the analytical model (\[fullsolutionquadbu\])–(\[fullsolutionquadbzeta\]), with fit parameters given in Table \[emmaquadfitparams\] with radius of fit $r_\mathrm{max}/10$, and $b_1 = 1.8$. The centre of the quadrupole is at the far left of the plot, $z = -53.6$mm.[]{data-label="examplefigbzvszvariousremma"}](EMMA_Quad_Fringe_Bz_vs_z-r_Enge_b1,8){width="50.00000%"}
Conclusions
===========
A closed form analytic expression was presented for fringe fields in multipole magnets. For quadrupoles, the field described by the analytic expression was shown to have the expected properties. The expression can be extended to describe multipoles of any order. For ease of explanation and illustration, we looked in particular at fringe fields in which the multipole gradient had a roll-off along the axis of the magnet described by an Enge function with only a single parameter in the exponent. However, the technique can be applied to any function with the appropriate dependence on the co-ordinates (i.e. any function that depends on the co-ordinates combined in the form $\sqrt{2}z \pm i h$). Examples of other (non-Enge) functions that may be suitable for describing fringe fields may be found in [@Kato1; @Kato2].
Expressions were also given for scalar and vector potentials from which the multipole fringe fields presented here could be derived. Again, the expressions for the potentials can be extended to apply to multipole magnets of any order. The scalar potential is of interest since, in iron-dominated magnets, the pole faces form surfaces of constant scalar potential. This provides a connection between studies of the dynamics of particles moving through the fringe fields a particular magnet, and design studies of the magnet geometry. It is hoped that by having access to realistic analytical descriptions of fringe fields at an early stage in the design of an accelerator beamline, the design process (typically involving many iterations between beam dynamics studies and magnet design work) may be made more efficient.
The vector potential is of interest for particle tracking. In particular, some techniques for symplectic integration of the equations of motion for particles moving in magnetic fields are based on analytical expressions for the vector potential (see, for example, [@WFR]). Again, it is hoped that there will be benefits in being able to perform symplectic tracking through realistic fringe field models at an early stage in the design of an accelerator.
In some types of magnet, such as those used in non-scaling FFAGs, fringe fields dominate the effects of the magnet. In such cases, being able to study the impact of fringe fields at an early stage of the accelerator design is essential for making efficient progress with the design. It should be possible to implement the methods presented here in standard accelerator tracking codes; this will allow accurate modelling of fringe field effects in multipole magnets of arbitrary order, and enhance the range of tools available for accelerator design and simulation.
Acknowledgements
================
This work has been ongoing for many years and BDM thanks Deepa Angal-Kalinin for giving him the opportunity and required support to finish it. BDM is also pleased to acknowledge useful discussions on both form and content, as well as general encouragement over the years from the following: Jim Clarke, Chris Edmonds, Bas van der Geer, Fay Hannon, Werner Herr, David Holder, Marieke de Loos, Neil Marks, Giovanni Muratori Snr., Hywel Owen and Peter Williams. The authors would like to thank Susana Izquierdo Bermudez for the LHC IT quadrupole field data, and Chris Edmonds for the EMMA F quadrupole field data.
This work was partially supported by the FP7 HiLumi LHC Design Study http://hilumilhc.web.cern.ch.
[99]{} B. van der Geer and M. de Loos (2004) GPT (General Particle Tracer), http://www.pulsar.nl/ É. Forest (1998), “Beam dynamics”, The Physics and Technology of Particle and Photon Beams, Vol. [**8**]{}, Harwood Academic Publishers. É. Forest and J. Milutinović (1988), “Leading order hard edge fringe fields effects exact in $(1 + \delta)$ and consistent with Maxwell’s equations for rectilinear magnets”, Nucl. Inst. Meth. A [**269**]{}, pp. 474–482. B.D. Muratori *et al.* (2011), “Multipole fringe fields”, Proceedings of IPAC’11, San Sebastian, Spain (2011). H.A. Enge (1967), “Deflecting magnets”, in A. Septier (ed.) *Focusing of charged particles*, Vol. [**2**]{}, Academic Press, pp. 203–264. M. Berz, B. Erdélyi and K. Makino (2000), “Fringe field effects in small rings of large acceptance”, Physical Review Special Topics – Accelerators and Beams [**3**]{}, 124001 and references therein. S. Kato (2005), “An improved description of magnetic fringing field”, Nucl. Inst. Meth. A [**540**]{}, pp. 1–13. S. Kato (2009), “Three-dimensional magnetic field distribution function beyond the bore radius of quadrupole magnets”, Nucl. Inst. Meth. A [**611**]{}, pp. 1–13. B. van der Geer and M. de Loos (2004) GPT (General Particle Tracer) manual and documentation for dipole and rectangular magnets in GPT. H.T.H. Piaggio (1948), “An elementary treatise on differential equations and their applications”, G. Bell and Sons, Ltd. E.T. Whittaker (1902), “A course of modern analysis”, Cambridge University Press. E.T. Whittaker (1903), “On the partial differential equations of mathematical physics”, Mathematishe Annalen, Vol. [**57**]{}, pp. 333–355. K.L. Brown and J.E. Spencer (1981), “Non-linear optics for the final focus of the single-pass collider”, Proceedings of PAC’81, IEEE Trans. Nucl. Sci. [**28**]{} 2568–2570, SLAC-PUB-2678, PEP-0345. A. Jonquière (1889), “Note sur la série $\sum_{n=1}^{\infty} \frac{x^n}{n^s}$”, Bulletin de la S.M.F., tome [**17**]{}, pp. 142–152. Y.K. Wu, É. Forest, and D.S. Robin (2003), “Explicit symplectic integrator for s-dependent static magnetic field”, Phys. Rev. E [**68**]{}, 046502. N. Marks, B.J.A. Shepherd, B. Leigh, F. Goldie, M.J. Crawley, “Development and adjustment of the EMMA quadrupoles”, Proceedings of EPAC’08, Genoa, Italy (2008). Y. Giboudot (2011), “Study of beam dynamics in NS-FFAG EMMA with dynamical map”, PhD Thesis, Brunel University.
[^1]: The result $d_j^2 = e_j^2 + 2$ may be useful for verifying the expressions for the vector potential.
|
---
abstract: 'Given positive numbers $a$ and $b$, the function $\sqrt{at^2+b}$ is exponentially convex function of $t$ on the whole real axis. Three proofs of this result are presented.'
address: |
Department of Mathematics\
The Weizmann Institute\
76100, Rehovot\
Israel
author:
- Victor Katsnelson
title: 'The function $\boldsymbol{\cosh\big(\sqrt{a\,t^2+b}\big)}$ is exponentially convex.'
---
The exponential convexity result
================================
A function $f$ on $\mathbb{R}$, $f:\,\mathbb{R}\to[0,\infty)$, is said to be *exponentially convex* if
1. For every positive integer $N$, for every choice of real numbers $t_1,t_2,\,\ldots\,$, $t_{N}$, and complex numbers $\xi_1$, $\xi_2, \,\ldots\,, \xi_{N}$, the inequality holds $$\label{pqf}
\sum\limits_{r,s=1}^{N}f(t_r+t_s)\xi_r\overline{\xi_s}\geq 0;$$
2. The function $f$ is continuous on $\mathbb{R}$.
The class of exponentially convex functions was introduced by S.N.Bernstein, [@B], see §15 there. Russian translation of the paper [@B] can be found in [@B1 pp.370-425].
From it follows that the inequality $%\label{p2f}
f(t_1+t_2)\leq\sqrt{f(2t_1)f(2t_2)}$ holds for every $t_1\in\mathbb{R},t_2\in\mathbb{R}$. Thus the alternative takes place:\
*If $f$ is an exponentially convex function then either $f(t)\equiv 0$, or $f(t)>0$ for every $t\in\mathbb{R}$.*
**Properties of the class of exponentially convex functions.**
1. If $f(t$ if an exponentially convex function and $c\geq0$ is a nonnegative constant, then the function $cf(t)$ is exponentially convex.
2. If $f_1(t)$ and $f_2(t)$ are exponentially convex functions, then their sum $f_1(t)+f_2(t)$ is exponentially convex.
3. If $f_1(t)$ and $f_2(t)$ are exponentially convex functions, then their product $f_1(t)\cdot f_2(t)$ is exponentially convex.
4. Let $\lbrace f_{n}(t)\rbrace_{1\leq n<\infty}$ be a sequence of exponentially convex functions. We assume that for each $t\in\mathbb{R}$ there exists the limit $f(t)=\lim_{n\to\infty}f_{n}(t)$, and that $f(t)<\infty\ \forall t\in\mathbb{R}$. Then the limiting function $f(t)$ is exponentially convex.
From the functional equation for the exponential function it follows that for each real number $\lambda$, for every choice of real numbers $t_1,t_2,\,\ldots\,$, $t_{N}$ and complex numbers $\xi_1$, $\xi_2, \,\ldots\,, \xi_{N}$, the equality holds $$\label{ece}
\sum\limits_{r,s=1}^{N}e^{\lambda(t_r+t_s)}\xi_r\overline{\xi_s}=
\bigg|\sum\limits_{p=1}^{N}e^{\lambda t_p}\xi_p\,\bigg|^{\,2}\geq 0.$$ The relation can be formulated as
\[ECE\] For each real $\lambda$, the function $e^{\lambda t}$ of the variable $t$ is exponentially convex.
For $z\in\mathbb{C}$, the function $\cosh z$, which is called *the hyperbolic cosine of $z$*, is defined as $$\label{dch}
\cosh z =\frac{1}{2}\big(e^z+e^{-z}\big).$$ From Lemma \[ECE\] and property P2 we obtain
\[echc\] For each real $\mu$, the function $\cosh(\mu\, t)$ of the variable $t$ is exponentially convex.
The following result is well known.
[ ]{}\
1. Let $\sigma(d\lambda)$ be a nonnegative measure on the real axis, and let the function $f(t)$ be a two-sided Laplace transform of the measure $d\sigma(\lambda)$: $$\label{rep}
f(t)=\int\limits_{\lambda\in\mathbb{R}}e^{\lambda t}\,d\sigma(\lambda)$$ for any $t\in\mathbb{R}$. Then the function $f$ is exponentially convex.
2. Let $f(t)$ be an exponentially convex function. Then this function $f$ can be represented on $\mathbb{R}$ as a two-sided Laplace transform of a nonnegative measure $d\sigma(\lambda)$. In particular, the integral in the right hand side of is finite for any $t\in\mathbb{R}$. The representing measure $d\sigma(\lambda)$ is unique.
The assertion 1 of the representation theorem is an evident consequence of Lemma \[ECE\], of the properties P1,P2, P4, and of the definition of the integration.
The proof of the assertion 2 can be found in [@A],Theorem 5.5.4, and in [@Wi],Theorem 21.
Of course, Lemma \[echc\] is a special case of the representation theorem which corresponds to the representing measure $\sigma(d\lambda)=1/2\big(\delta(\lambda-\mu)+\delta(\lambda+\mu)\big)\,d\lambda$, where $\delta(\lambda\mp\mu)$ are Dirak’s $\delta$-functions supported at the points $\pm\mu$.\
\[ef\] The expression $$\label{mcf}
\varphi(t,a,b)=\cosh\big(\sqrt{at^2+b}\,\big).$$ is well defined for every complex numbers $t,a,b$. The function $\varphi(t,a,b)$ is an entire function of complex variables $(t,a,b)\in\mathbb{C}^3$. For each fixed $a>0$ and $b$, the function $\varphi(t,a,b)$, considered as a function of $t$, is an entire function of exponential type $\sqrt{a}$.
**. The function $\varphi(t,a,b)$ is a superposition of the entire function $\cosh\sqrt{\zeta}$ of variable $\zeta$ and the quadratic polynomial $\zeta(t,a,b)=at^2+b$. The assertion concerning the growth of this function is evident. $\Box$
In the paper [@BMV] a conjecture was formulated which now is commonly known as the BMV conjecture:\
**The BMV Conjecture.** Let $A$ and $B$ be Hermitian matrices of size $n\times{}n$. Then the function $$\label{TrF}
f_{A,B}(t)=
\textup{trace}\,\{\exp[tA+B]\}$$ of the variable $t$ is representable as a bilateral Laplace transform of a measure $d\sigma_{A,B}(\lambda)$ compactly supported on the real axis: $$\label{LaR}
f_{A,B}(t)=\!\!\int\limits_{\lambda\in(-\infty,\infty)}\!\!\exp(t\lambda)\,d\sigma_{A,B}(\lambda), \ \ \forall \,t\in(-\infty,\infty).$$
In general case, if the matrices $A$ and $B$ do not commute, the BMV conjecture remained an open question for longer than 35 years. In 2011, Herbert Stahl, [@St], gave an affirmative answer to the BMV conjecture.\
**Theorem**(H.Stahl) *Let $A$ and $B$ be $n\times{}n$ Hermitian matrices.* *Then the function $f_{A,B}(t)$ defined by is representable as the bilateral Laplace transform of a non-negative measure $d\sigma_{A,B}(\lambda)$ supported on the closed interval $[\lambda_{\min},\lambda_{\max}]$.*
The proof of Herbert Stahl is based on ingenious considerations related to Riemann surfaces of algebraic functions.
In the case $2\times 2$ matrices $A$ and $B$, the BMV conjecture is equivalent to the exponential convexity with respect to $t$ for each $a>0$ and $b\geq 0$ of the function $\varphi(t,a,b)=\cosh\big(\sqrt{at^2+b}\,\big)$ which was introduced in .
\[MT\] For each fixed $a>0$ and $b\geq0$, the function $\varphi(t,a,b)$ defined by is an exponentially convex function of variable $t$.
In what follow we present three different proofs of Theorem \[MT\]. The first and the second proofs are based on the representation theorem. We prove that the function $\widehat{d}(\lambda,a,b)$ which is defined by below takes positive values for $\lambda\in(-\sqrt{a},\sqrt{a})$. In the first proof we calculate the function $\widehat{d}(\lambda,a,b)$ explicitly expressing this function in terms of the modified Bessel function $I_{1}$. In the second proof, we prove the positivity of the function $\widehat{d}(\lambda,a,b)$ using the reasoning by Herbert Stahl in [@St]. (We use a very simple special case of this reasoning.) The third proof is based on the Taylor expansion of the function $\varphi(t,a,b)$, , with respect to parameter $b$. This proof does not use any integration in the complex plane. It based only on Lemma \[echc\] and on the properties P1 – P4 of the class of exponentially convex functions. As a by-product of this proof we obtain that all coefficients of this Taylor expansion are exponentially convex functions. However we can not conclude directly from this proof that the restriction of the representing measure on the *open* interval $(-\sqrt{a},\sqrt{a})$ is an absolutely continuous measure.
\[repl\] For each fixed $a>0,\,b\geq0$, the function $\varphi(t,a,b)$ defined by is representable in the form $$\label{rfv}
\varphi(t,a,b)=\cosh \sqrt{a}\,t+
\int\limits_{-\sqrt{a}}^{\sqrt{a}}\widehat{d}(\lambda,a,b)e^{\lambda t}\,d\lambda,
\ \ \ \forall t\in\mathbb{C},$$ where the function $\widehat{d}(\lambda,a,b)$ possesses the properties\
1. [ ]{} $$\label{L2}
\int\nolimits_{-\sqrt{a}}^{\sqrt{a}}|\widehat{d}(\lambda,a,b)|^2d\lambda<\infty;$$
2. The function $\widehat{d}(\lambda,a,b)$ is continuous with respect to $\lambda$ on the closed interval $[-\sqrt{a},\sqrt{a}]$, takes real values there, and is even.
3. The values of the function $\widehat{d}(\lambda,a,b)$ at the end points $\pm\sqrt{a}$ of the interval $[-\sqrt{a},\sqrt{a}]$ are: $$\label{vep}
\widehat{d}(\pm\sqrt{a},a,b)=\frac{b}{4\sqrt{a}}.$$
*Proof.* We introduce the function $$\label{df}
d(t,a,b)=\cosh \big(\sqrt{at^2+b}\big)-\cosh \sqrt{a}t$$ of variables $t,a,b$. Considered as a function of $t$ for fixed *positive* $a$ and $b$, $d(t,a,b)$ is entire function of exponential type $\sqrt{a}$. On the imaginary axis $d$ takes the form $$\label{dia}
d(i\tau,a,b)=\cos\sqrt{a\tau^2-b}-\cos\sqrt{a}\tau, \quad \tau\in\mathbb{R}.$$ From it follows that the function $d$ is a bounded and decaying on the imaginary axis: $|d(i\tau,a,b)\leq{}1+\cosh b,\,\tau\in\mathbb{R}$, $d(i\tau,a,b)=O(|\tau|^{-1}) \ \text{ as } \ \tau\to\pm\infty$. By the Wiener-Paley theorem, the function $d(i\tau,a,b)$ is representable in the form $$\label{pwr}
d(i\tau,a,b)=\int\limits_{-\sqrt{a}}^{\sqrt{a}}\widehat{d}(\lambda,a,b)e^{i\lambda \tau}\,d\lambda, \ \
\tau\in\mathbb{R},$$ where the function $\widehat{d}(\lambda,a,b)$ satisfies the condition . *The equality serves as a definition of the function $\widehat{d}(\lambda,a,b)$.* So, this function is defined only for $a>0,\,b>0,\,-\sqrt{a}\leq\lambda\leq\sqrt{a}$.
Since the function $d(i\tau,a,b)$ is even with respect to $\tau$ and real valued, its inverse Fourier transform $\widehat{d}(\lambda,a,b)$ is even with respect to $\lambda$ and real valued.
From we obtain that $$\label{ase}
d(i\tau,a,b)-\frac{b}{2}\,\frac{\sin\sqrt{a}\,\tau}{\sqrt{a}\,\tau}=O(\tau^{-2}) \text{ as } \tau\to\pm\infty.$$ Hence the function in the left hand side of is a Fourier transform of some function $r(\lambda)$ which is square summable and *continuous* at every $\lambda\in\mathbb{R}$. We remark that $$\frac{b}{2}\,\frac{\sin\sqrt{a}\,\tau}{\sqrt{a}\,\tau}=
\int\limits_{-\sqrt{a}}^{\sqrt{a}}\widehat{c}\,e^{i\lambda\tau}\,d\lambda, \ \ \tau\in\mathbb{R},$$ where $\widehat{c}=\frac{b}{4\sqrt{a}}$ is a constant function. Hence $$r(\lambda)=
\begin{cases}
\widehat{d}(\lambda,a,b)-\widehat{c},& \ \ \textup{for} \ |\lambda|<\sqrt{a},\\
0\phantom{( \ \ \ \ \ \lambda)-\widehat{c}},& \ \ \textup{for} \ |\lambda|>\sqrt{a}.
\end{cases}$$ Since $r(\lambda)=0$ for $|\lambda|>\sqrt{a}$, also $r(\pm\sqrt{a})=0$. Thus, holds. $\Box$
Representation the function $\boldsymbol{\widehat{d}(\lambda,a,b)}$\
by a contour integral.\[RCI\]
====================================================================
Let $S$ be a segment of the imaginary axis: $$\label{slit}
S=\left\lbrace\zeta=\xi+i\eta:\,\xi=0,-\sqrt{\tfrac{b}{a}}\leq\eta\leq\sqrt{\tfrac{b}{a}}\,\right\rbrace.$$ The function $\sqrt{a\zeta^{2}+b}$ is a single value function of $\zeta$ in the complex plane slitted along the vertical segment $S$. We choose the branch of this function which takes positive values for large real $\zeta$.
\[lir\] The function $\widehat{d}(\lambda,a,b)$, which was defined by , admits the integral representation $$\label{cir}
\widehat{d}(\lambda,a,b)=-\frac{1}{4\pi i}\ointctrclockwise\limits_{\Gamma}e^{-\sqrt{a\zeta^2+b}}e^{-\lambda\zeta}\,d\zeta, \ \
-\sqrt{a}<\lambda<\sqrt{a},$$ where $\Gamma$ is an arbitrary counterclockwise oriented closed Jordan curve which contains the slit $S$ inside.
*Proof.* According the inversion formula for the Fourier transform, $$\label{iff}
\widehat{d}(\lambda,a,b)=
\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\varphi(i\eta,a,b)e^{-i\lambda\eta}\,d\eta.$$ We interpret the integral in the right hand side of as the integral along the the vertical straight line $\lbrace\zeta:\, \operatorname{Re}\zeta =0\rbrace$: $$\label{iffv}
\widehat{d}(\lambda,a,b)=\frac{1}{2\pi i}\int\limits_{\operatorname{Re}\zeta=0}\varphi(\zeta,a,b)e^{-\lambda\zeta}\,d\zeta
=\lim_{R\to+\infty}\frac{1}{2\pi i}\int\limits_{-iR}^{+iR}\varphi(\zeta,a,b)e^{-\lambda\zeta}\,d\zeta.$$ Since the function $\varphi(\zeta,a,b)$ is bounded in each vertical strip $\lbrace\zeta:\,\alpha\leq\operatorname{Re}\zeta\leq\beta\rbrace$ and tends to zero as $\operatorname{Im}\zeta\to\pm\infty$ within this strip, the value of the integral in does not change if we integrate along any vertical line $\lbrace\zeta:\,\operatorname{Re}\zeta=\gamma\rbrace$, where $\gamma$ is an arbitrary real number: $$\label{iffvg}
\widehat{d}(\lambda,a,b)=\frac{1}{2\pi i}\int\limits_{\operatorname{Re}\zeta=
\gamma}\varphi(\zeta,a,b)e^{-\lambda\zeta}\,d\zeta,\ \ -\sqrt{a}\leq\lambda\leq\sqrt{a}\,.$$ Choosing $\gamma<0$ , we split the integral in into the sum $$\label{iffsp}
\widehat{d}(\lambda,a,b)=\frac{1}{2\pi i}\int\limits_{\operatorname{Re}\zeta=\gamma}\varphi_{+}(\zeta,a,b)e^{-\lambda\zeta}\,d\zeta+
\frac{1}{2\pi i}\int\limits_{\operatorname{Re}\zeta=\gamma}\varphi_{-}(\zeta,a,b)e^{-\lambda\zeta}\,d\zeta,$$ where $$\varphi_{+}(\zeta,a,b)=\frac{1}{2}\Big(e^{\sqrt{a\zeta^{2}+b}}-e^{\sqrt{a}\zeta}\Big),\quad
\varphi_{-}(\zeta,a,b)=\frac{1}{2}\Big(e^{-\sqrt{a\zeta^{2}+b}}-e^{-\sqrt{a}\zeta}\Big).
\label{ppm}$$
The function $\varphi_{+}(\zeta,a,b)$ is holomorphic in the halfplane $\lbrace\zeta:\,\operatorname{Re}\zeta\leq\gamma\rbrace$ and admits the estimate $$|\varphi_{+}(\zeta,a,b)|\leq c(\gamma)(1+|\zeta|)^{-1}e^{\sqrt{a}\operatorname{Re}\zeta}, \quad \forall \zeta: \operatorname{Re}\zeta\leq\gamma$$ there, where $c(\gamma)<\infty $ is a constant. Therefore $$\label{ipi+}
\frac{1}{2\pi i}\int\limits_{\operatorname{Re}\zeta=\gamma}\varphi_{+}(\zeta,a,b)e^{-\lambda\zeta}\,d\zeta=0,
\ \ \lambda<\sqrt{a} .$$ The function $\varphi_{-}(\zeta,a,b)$ is holomorphic in the slitted half plane $\lbrace\zeta:\,\operatorname{Re}\zeta\geq\gamma,$ $\zeta\not\in S\rbrace$ and admits the estimate $$|\varphi_{-}(\zeta,a,b)|\leq c(\gamma)(1+|\zeta|)^{-1}e^{-\sqrt{a}\operatorname{Re}\zeta}, \quad \forall \zeta: \operatorname{Re}\zeta\geq\gamma,\,\zeta\not\in S$$ there. Therefore $$\label{ipi-}
\frac{1}{2\pi i}\int\limits_{\operatorname{Re}\zeta=\gamma}\varphi_{-}(\zeta,a,b)e^{-\lambda\zeta}\,d\zeta=
\frac{1}{2\pi i}\ointclockwise\limits_{\Gamma}\varphi_{-}(\zeta,a,b)e^{-\lambda\zeta}e^{-\lambda\zeta}\,d\zeta,
\ \ \lambda>-\sqrt{a} ,$$ where $\Gamma$ is an arbitrary closed Jourdan curve which is oriented *clockwise* and contains the slit $S$ in its interior. Since the function $e^{-\sqrt{a}\zeta}$ is entire, $\oint\limits_{\Gamma}e^{-\sqrt{a}\zeta}\,d\zeta=0$. So $$\label{red}
\frac{1}{2\pi i}\ointclockwise\limits_{\Gamma}\varphi_{-}(\zeta,a,b)e^{-\lambda\zeta}\,d\zeta=
-\frac{1}{4\pi i}\ointctrclockwise\limits_{\Gamma}e^{-\sqrt{a\zeta^2+b}}e^{-\lambda\zeta}\,d\zeta,$$ where the integral in the right hand side is taken over the curve $\Gamma$ which is oriented *counterclockwise*. Comparing , , , and , we obtain . $\Box$
Explicite calculation of the function $\boldsymbol{\widehat{d}(\lambda,a,b)}$.
==============================================================================
\[lsir\] The function $\widehat{d}(\lambda,a,b)$ which was defined by admits the integral representation $$\label{ird}
\widehat{d}(\lambda,a,b)=\frac{1}{\pi}\sqrt{\frac{b}{a}}\int\limits_{0}^{1}\sinh\sqrt{b(1-\tau^2)}\cdot\cos\Big(\lambda\sqrt{\tfrac{b}{a}}\tau\Big)\,d\tau, \ \ -\sqrt{a}\leq\lambda\leq\sqrt{a}\,.$$
*Proof.* We derive Lemma \[lsir\] from Lemma \[lir\] showing that $$\begin{gathered}
\label{dll}
-\frac{1}{4\pi i}\ointctrclockwise\limits_{\Gamma}e^{-\sqrt{a\zeta^2+b}}e^{-\lambda\zeta}\,d\zeta=
\frac{1}{\pi}\sqrt{\frac{b}{a}}\int\limits_{0}^{1}\sinh\sqrt{b(1-\tau^2)}\cdot\cos\lambda\sqrt{\tfrac{b}{a}}\tau\,d\tau.\end{gathered}$$ The function $e^{-\sqrt{a\zeta^2+b}}e^{-\lambda\zeta}$ is holomorphic in the domain $\mathbb{C}\setminus S$ and continuous up to boundary $S=\partial(\mathbb{C}\setminus S)$ of this domain. Therefore the integral of this function over $\Gamma$ does not change if we shrink the original contour $\Gamma$ to the boundary $S$: $$\label{scr}
\frac{1}{4\pi i}\ointctrclockwise\limits_{\Gamma}e^{-\sqrt{a\zeta^2+b}}e^{-\lambda\zeta}\,d\zeta=
\frac{1}{4\pi i}\ointctrclockwise\limits_{S}e^{-\sqrt{a\zeta^2+b}}e^{-\lambda\zeta}\,d\zeta$$ To one “geometric” point $i\eta\in S$ there corresponds two topologically different “boundary” points $+0+i\eta$ and $-0+i\eta$ lying on the right edge $S^{+}$ and the left edge $S^{-}$ of the slit $S$ respectively. The chosen branch of the function $\sqrt{a\zeta^2+b}$ takes the following values on the boundary of the domain $\mathbb{C}\setminus S$: $$\label{onb}
\sqrt{a(+0+i\eta)^2+b}=-\sqrt{a(-0+i\eta)^2+b}=\sqrt{b-a\eta^2}, \ \ i\eta\in S.$$ If the point $\zeta=\pm0+i\eta$ runs over $S=\partial\, (\mathbb{C}\setminus S)$ counterclockwise, the $\eta$ increases from $-\sqrt{\tfrac{b}{a}}$ to $\sqrt{\tfrac{b}{a}}$ if $\zeta=\in S^{+}$ and $\eta$ decreases from $\sqrt{\tfrac{b}{a}}$ to $-\sqrt{\tfrac{b}{a}}$ if $\zeta=\in S^{-}$. Therefore $$\begin{aligned}
\sideset{}{\hspace*{-2.5ex}{\text{\raisebox{0.5ex}{\(\scriptstyle\curvearrowleft\)}}}}\int\limits_{S^{+}}e^{-\sqrt{a\zeta^2+b}}\,e^{-\lambda\zeta}\,d\zeta=&
+i\int\limits_{-\sqrt{b/a}}^{\sqrt{b/a}}e^{-\sqrt{b-a\eta^2}}e^{-i\lambda\eta}\,d\eta,\\
\sideset{}{\hspace*{-2.5ex}{\text{\raisebox{0.5ex}{\(\scriptstyle\curvearrowleft\)}}}}\int\limits_{S^{-}}e^{-\sqrt{a\zeta^2+b}}\,e^{-\lambda\zeta}\,d\zeta=&
-i\int\limits_{-\sqrt{b/a}}^{\sqrt{b/a}}e^{+\sqrt{b-a\eta^2}}e^{-i\lambda\eta}\,d\eta\end{aligned}$$ Thus $$\begin{gathered}
\label{ios}
\frac{1}{4\pi i}\ointctrclockwise\limits_{S}e^{-\sqrt{a\zeta^2+b}}e^{-\lambda\zeta}\,d\zeta=
\frac{1}{4\pi}\int\limits_{-\sqrt{b/a}}^{\sqrt{b/a}}(e^{-\sqrt{b-a\eta^2}}-e^{\sqrt{b-a\eta^2}})e^{-i\lambda\eta}\,d\eta=\\
=-\frac{1}{2\pi}\int\limits_{-\sqrt{b/a}}^{\sqrt{b/a}}\sinh\sqrt{b-a\eta^2}\,e^{-i\lambda\eta}\,d\eta=-\frac{1}{\pi}
\int\limits_{0}^{\sqrt{b/a}}\sinh\sqrt{b-a\eta^2}\,\cos\lambda\eta\,d\eta=\\
=-\frac{1}{\pi}\sqrt{\tfrac{b}{a}}\int\limits_{0}^{1}\sinh\sqrt{b(1-\eta^2)}
\cdot\cos\Big(\sqrt{\tfrac{b}{a}}\lambda\eta\Big)\,d\eta.\end{gathered}$$ Comparing with , we obtain .
\[lexex\] Let $a>0$ and $b>0$ be fixed positive numbers. Then
1. The function $\widehat{d}(\lambda,a,b)$ which was defined by can be expressed explicitly in terms of the modified Bessel function $I_1$: $$\label{exex}
\widehat{d}(\lambda,a,b)=\frac{\sqrt{b}}{2\sqrt{a-\lambda^2}}
I_1\Big({\textstyle\sqrt{\frac{(a-\lambda^2)b}{a}}}\,\Big), \ \ \ \ -\sqrt{a}\leq\lambda\leq\sqrt{a}.$$
2. The function $\widehat{d}(\lambda,a,b)$ is representable as the sum of the series $$\begin{gathered}
\label{dtb}
\widehat{d}(\lambda,a,b)=\frac{b}{4\sqrt{a}}\sum\limits_{k=0}^{\infty}\frac{1}{k!(k+1)!}
\bigg(\frac{(a-\lambda^2)b}{4a}\bigg)^k, \\ a>0, \ -\sqrt{a}\leq\lambda\leq\sqrt{a}, \ b\geq0.\end{gathered}$$
\[dwrb\] The expression in the right hand sides of is an entire function of three variables $(\lambda,\sqrt{a^{-1}},b)\in\mathbb{C}^3$. However the equalities , , hold only for $a>0,\,b>0,\, -\sqrt{a}\leq\lambda\leq\sqrt{a}$. We recall that the function $\widehat{d}(\lambda,a,b)$ was *defined* by only for $a>0,\,b>0,\, -\sqrt{a}\leq\lambda\leq\sqrt{a}$.
** We start from the formula . Using the Taylor expansion of the hyperbolic $\sinh$ function, we obtain $$\label{ehs}
\widehat{d}(\lambda,a,b)=\frac{1}{\pi}\sqrt{\frac{b}{a}}\sum\limits_{r=0}^{\infty}\frac{1}{(2r+1)!}
b^{\,r+\frac{1}{2}}\int\limits_{0}^{1}(1-\tau^2)^{r+\frac{1}{2}}\cos\Big(\lambda\sqrt{\tfrac{b}{a}}\tau\big)\,d\tau$$ The integral in the right hand side of can be expressed in terms of the Bessel function $J_{r+1}$, see [@AS **9.1.20**]: $$\begin{gathered}
\int\limits_{0}^{1}(1-\tau^2)^{r+\frac{1}{2}}\cos\Big(\lambda\sqrt{\tfrac{b}{a}}\tau\Big)\,d\tau=\\=\pi^{1/2}\,2^r\,\Gamma(r+3/2)a^{\frac{r+1}{2}}b^{-\frac{r+1}{2})}\,\lambda^{-(r+1)}\,
J_{r+1}\Big(\lambda\sqrt{\tfrac{b}{a}}\Big).\end{gathered}$$ Substituting the last equality into , we obtain the equality $$\widehat{d}(\lambda,a,b)=\pi^{-\frac{1}{2}}\sum\limits_{r=0}^{\infty}
\frac{\Gamma(r+3/2)}{(2r+1)!}2^ra^{\frac{r}{2}}b^{\frac{r+1}{2}}\lambda^{-(r+1)}
J_{r+1}\Big(\lambda\sqrt{\tfrac{b}{a}}\Big).$$ Taking into account the duplication formula for the Gamma-function, [@AS **6.1.18**]: $$\frac{\Gamma(r+\frac{3}{2})}{\Gamma(2r+2)}=\pi^{\frac{1}{2}}2^{-(2r+1)}\frac{1}{\Gamma(r+1)},$$ we transform the last equality to the form $$\label{pmt}
\widehat{d}(\lambda,a,b)=\sum\limits_{r=0}^{\infty}\frac{1}{r!}2^{-(r+1)}
a^{\frac{r}{2}}b^{\frac{r+1}{2}}\lambda^{-(r+1)}
J_{r+1}\Big(\lambda\sqrt{\tfrac{b}{a}}\Big).$$ Now we would like to reduce the equality to the form which occurs in the so called *Multiplication Theorem*[^1], see [@AS **9.1.74**]: $$\label{omt}
\widehat{d}(\lambda)=\tfrac{\sqrt{b}}{2\lambda}\sum\limits_{r=0}^{\infty}\frac{(-1)^r}{r!}
\big(-\tfrac{a}{\lambda^2}\big)^r\cdot
\Big(\tfrac{\lambda}{2}\sqrt{\tfrac{b}{a}}\,\Big)^r\,J_{r+1}\Big(\lambda\sqrt{\tfrac{b}{a}}\Big).$$ Let us introduce $\mu: \mu^2-1=-\dfrac{a}{\lambda^2}$, i.e. $$\label{lm}
\mu=i\frac{\sqrt{a-\lambda^2}}{\lambda}.$$ Then the equality takes the form $$\label{req}
\widehat{d}(\lambda,a,b)=\frac{\sqrt{b}}{2i\sqrt{a-\lambda^2}}\,\cdot\,\mu\sum\limits_{r=0}^{\infty}\frac{1}{r!}
(\mu^2-1)^r\Big(\tfrac{\lambda}{2}\sqrt{\tfrac{b}{a}}\,\Big)^r\,J_{r+1}\Big(\lambda\sqrt{\tfrac{b}{a}}\Big).$$ According to the Multiplication Theorem, $$\label{amt}
\mu\sum\limits_{r=0}^{\infty}\frac{1}{r!}
(\mu^2-1)^r\Big(\tfrac{\lambda}{2}\sqrt{\tfrac{b}{a}}\,\Big)^r\,
J_{r+1}\Big(\lambda\sqrt{\tfrac{b}{a}}\Big)=J_1\Big(i\sqrt{a-\lambda^2}\sqrt{\tfrac{b}{a}}\Big),
\ \ \forall \lambda,a,b.$$ Taking into account that $J_1(iz)=iI_1(z)$, we reduce the equality to the form .
Using the Taylor expansion of the modified Bessel function $I_1$, [@AS **9.6.10**], we represent the function $\widehat{d}(\lambda,a,b)$ as the sum of the series . $\Box$\
The first proof of Theorem \[MT\].
==================================
From the equality is evident that $$\label{pod}
\widehat{d}(\lambda,a,b)>0 \text{ for } \lambda\in[-\sqrt{a},\sqrt{a}]$$ Theorem \[MT\] follows from and . $\Box$\
\[ptc\] For each $a>0$, the function $\varphi(t,a,b)$ which was introduced in admits the Taylor expansion with respect to $b$:
\[tew\] $$\label{tewb}
\varphi(t,a,b)=\sum\limits_{k=0}^{\infty}\frac{1}{k!}\varphi_{k}(t,a)b^k, \ \ \forall\,t\in\mathbb{R},\
0\leq b<\infty.$$ For each $k\geq 0$, the function $\varphi_{k}(t,a)$, which is the $k$-th coefficient of the Taylor expansion , is exponentially convex: $$\begin{aligned}
\varphi_0(t,a)=&\cosh\sqrt{a}t,\label{k=0}\\
\varphi_k(t,a)=&\frac{1}{(k-1)!4^ka^{k-\frac{1}{2}}}\int\limits_{-\sqrt{a}}^{\sqrt{a}}
(a-\lambda^2)^{k-1}\,e^{\lambda t}\,d\lambda,\ \ \ k=1,2,3,\,\ldots\,.
\label{k>0}\end{aligned}$$
*.* The expansion can be presented as a Taylor expansion with respect to $b$:
\[ted\] $$\begin{gathered}
\widehat{d}(\lambda,a,b)=\sum\limits_{k=1}^{\infty}\frac{1}{k!}\widehat{d}_k(\lambda,a)\,b^k,
\label{ted1}\\
\intertext{where}
\widehat{d}_k(\lambda,a)=\frac{(a-\lambda^2)^{k-1}}{(k-1)!4^ka^{k-\frac{1}{2}}}, \ \ \ k=1,2,3,\,\ldots\,.
\label{ted2}\end{gathered}$$
Substituting the expansion into the integrand in , we obtain the expansion . It is evident that $\widehat{d}_k(\lambda,a)>0$ for $-\sqrt{a}<\lambda<\sqrt{a}$. The exponential convexity of the function $\varphi_k(t,a)$ follows from the representation .$\Box$\
\[rade\] The function $\varphi_k(t,a)$, , can be expressed in terms of the modified Bessel function $I_{k-\frac{1}{2}}$: $$\label{ade}
\varphi_k(t,a)=\pi^{\frac{1}{2}}2^{-(k+\frac{1}{2})}a^{-\frac{k}{2}+\frac{1}{4}}
t^{-(k-\frac{1}{2})}\,I_{k-\frac{1}{2}}(\sqrt{a}t)\,,\\
\ \ a>0,\ t\in\mathbb{R},
\ \ k=1,2,3,\,\ldots\,.$$ See .
\[prior1\] The formula appeared in , see formulas and there. In , the derivation of the expansion was done by a direct calculation, without any reference to the multiplication theorem for Bessel function. It should be mention that the series in the right hand side of appeared in as a perturbation series related to the BMV conjection for $2\times2$ matrices.
The second proof of Theorem \[MT\].
===================================
The starting point of the first as well as of the second is the representation of the value $\widehat{d}(\lambda,a,b)$ by the contour integral . See Lemma \[lir\].
In the first proof, we shrank the contour of integration over the slit $S$, so the contour of integration was the same for every $\lambda\in[-\sqrt{a},\sqrt{a}]$.
In contrast to this, in the second proof we choose the contour $\Gamma$ in such a way that the exponent $-\sqrt{a\zeta^2+b}-\lambda\zeta$ of the integrand $e^{-\sqrt{a\zeta^2+b}-\lambda\zeta}$ in takes real values on $\Gamma$. (So the contour $\Gamma$ depends on $\lambda$!). We denote this contour by $\Gamma_{\lambda}$
The function $\widehat{d}(\lambda,a,b)$ is even with respect to $\lambda$. Therefore to prove the exponential convexity of the function $\varphi(t,a,b)$, it is enough to prove that the value $\widehat{d}(\lambda,a,b)$ is positive for each $$\label{negl}
\lambda\in(-\sqrt{a},0).$$ We choose an arbitrary $\lambda$ satisfying the condition and fix this choice in the course of the proof.
Let us introduce the functions
\[Harf\] $$\begin{aligned}
\label{Harfr}
u(\zeta)=\operatorname{Re}(\sqrt{a\zeta^2+b}+\lambda\zeta),\ \ z\in\mathbb{C}\setminus S, \\
v(\zeta)=\operatorname{Im}(\sqrt{a\zeta^2+b}+\lambda\zeta),\ \ z\in\mathbb{C}\setminus S,
\label{Harfi}\end{aligned}$$
where $S$ is the vertical slit and the branch of the function $\sqrt{a\zeta^2+b}$ in $\mathbb{C}\setminus S$ is chosen which takes positive values for large real $\zeta$.
\[laze\] Let us assume that $a>0,b>0$ and $\lambda$ satisfies the condition . Then there exist $\varepsilon>0$ $R<\infty$, $\varepsilon=\varepsilon(a,b,\lambda),\,R=R(a,b,\lambda)$, such that
\[poda\] $$\begin{aligned}
\label{podal}
v(\zeta)/\operatorname{Im}\zeta>0,& \ \ \ \ \forall \zeta\in\mathbb{C}\phantom{\setminus S}:\,|\zeta|>R, \ \operatorname{Im}\zeta\neq0,\\
\label{podas}
v(\zeta)/\operatorname{Im}\zeta<0,& \ \ \ \ \forall \zeta\in\mathbb{C}\setminus S:\,|\zeta|<\varepsilon, \ \,\,\operatorname{Im}\zeta\neq0.\end{aligned}$$
From the identity $$\sqrt{a\zeta^2+b}-\sqrt{a}\zeta=\frac{b}{\sqrt{a\zeta^2+b}+\sqrt{a}\zeta}$$ we derive that $$\operatorname{Im}\sqrt{a\zeta^2+b}-\sqrt{a}\operatorname{Im}\zeta=-\big(\operatorname{Im}\sqrt{a\zeta^2+b}+\sqrt{a}\operatorname{Im}\zeta\big)\,\rho(\zeta)$$ where $\rho(\zeta)=b\big|\sqrt{a\zeta^2+b}+\sqrt{a}\zeta\big|^{-2}$. Thus $$\operatorname{Im}\sqrt{a\zeta^2+b}=\frac{1-\rho(\zeta)}{1+\rho(\zeta)}\cdot\sqrt{a}\, \operatorname{Im}\zeta$$ and $$\label{exprv}
v(\zeta)=\bigg(\sqrt{a}\,\frac{1-\rho(\zeta)}{1+\rho(\zeta)}+\lambda\bigg)\cdot\operatorname{Im}\zeta.$$ It is clear that[^2] $\rho(\zeta)\to0$ as $|\zeta|\to\infty$, $\rho(\zeta)\to1$ as $|\zeta|\to0,\zeta\not\in S$. Since $\sqrt{a}+\lambda>0$, the inequality holds if if $|\zeta|$ is large enough. Since $\lambda<0$, the inequality holds if $|\zeta|$ is small enough. $\Box$\
Let $N_{\lambda}$ be the set $$\label{zers}
N_{\lambda}=\lbrace\zeta\in\mathbb{C}\setminus S: v(\zeta)=0\rbrace,$$
[ ]{}\
\[szers\]
1. The set $N_{\lambda}$ is the union of the real axis and an ellipse $\Gamma_{\lambda}$ $$\label{szs}
N_{\lambda}=(\mathbb{R}\setminus0)+\Gamma_{\lambda}.$$ where the ellipse $\Gamma_{\lambda}$ is described by the equation: $$\label{eqel}
\frac{\xi^2}{A^2}+\frac{\eta^2}{B^2}=1, \ \ (\zeta=\xi+i\eta),$$ with $$\label{halfa}
A=\sqrt{\frac{b}{a}}\,\cdot\,\frac{|\lambda|}{\sqrt{a}}\bigg(1-\frac{\lambda^2}{a}\bigg)^{-\frac{1}{2}},\quad
B=\sqrt{\frac{b}{a}}\,\cdot\,\bigg(1-\frac{\lambda^2}{a}\bigg)^{-\frac{1}{2}}$$
2. The slit $S$ is contained in the interior of the ellipse $\Gamma_\lambda$.
\
Let $\zeta=\xi+i\eta,\,\,\sqrt{a\zeta^2+b}=p+iq$, where $\xi,\eta,p,q$ are real numbers. The equality $$\pm\sqrt{a\zeta^2+b}=p+iq$$ is equivalent to the system of equalities $$\label{syst}
\begin{cases}
a(\xi^2-\eta^2)+b&=p^2-q^2,\\
\ \ \ \ a\xi\eta&=\ \ pq.
\end{cases}$$ Here $p=p(\xi,\eta),\,q=q(\xi,\eta).$ Clearly $v(\xi,\eta)=q(\xi,\eta)+\lambda\eta$.
Let $\zeta\in N_{\lambda}$. This means that $v(\xi,\eta)=0$, i.e.
\[pq\] $$\label{q}
q=-\lambda\eta$$ Substituting this equality into the second equality of the system , we obtain the equality $a\xi\eta=-\lambda p\eta.$ Assuming that $\eta\neq0$, that is $\zeta\not\in\mathbb{R}$, we can cancel by $\eta$ and obtain $$\label{p}
p=-\frac{a\xi}{\lambda}$$
Substituting the equalities into the first equality of the system , we obtain that the equality holds for $\zeta=\xi+i\eta$. Thus we proved that $$\label{incl}
(N_{\lambda}\setminus\mathbb{R})\subseteq\Gamma_{\lambda}.$$ Let $$\label{hp}
\mathbb{H}^+=\lbrace\zeta:\,\operatorname{Im}\zeta>0\rbrace,\quad \mathbb{H}^-=\lbrace\zeta:\,\operatorname{Im}\zeta<0\rbrace$$ be the upper and the lower half-plane respectively.
According to Lemma \[laze\], there exist points $\zeta\in\mathbb{H}^{+}\setminus S$ where $v(\zeta)>0$ and points $\zeta\in\mathbb{H}^{+}\setminus S$ where $v(\zeta)<0$. This means that the set $N_{\lambda}$, , separates the domain $\mathbb{H}^{+}\setminus S$. In other words, the open set $(\mathbb{H}^{+}\setminus S)\setminus N_{\lambda}$ is disconnected. Since $v(\overline{\zeta})=-v(\zeta)$, the set $N_{\lambda}$ is symmetric with respect to the real axis. The set $\Gamma_{\lambda}\setminus N_{\lambda}$ also is symmetric with respect to the real axis. Since , the set $N_{\lambda}$ can not separate the domain $(\mathbb{H}^{+}\setminus S)$ if $\Gamma_{\lambda}\setminus N_{\lambda}\neq\emptyset$.\
. In view of , the inequality $0<A<B$ hold. So $A$ is the minor semiaxis of the ellips $\Gamma_{\lambda}$ and $B$ is its major semiaxis. Moreover, the inequality $\sqrt{\frac{b}{a}}<B$ holds. This means that the slit $S$ is contained inside the ellipse $\Gamma_\lambda$. $\Box$\
[ ]{}\
\[prhf\]
1. The functions $u(\zeta)$ and $v(\zeta)$ are conjugate harmonic function of $\zeta$ in the domain $\zeta\in\mathbb{C}\setminus S$.
2. The only critical points of the the functions $u$ and $v$ in the domain $\zeta\in\mathbb{C}\setminus S$ are the points $$\label{Crp}
\zeta_{+}(\lambda)=\sqrt{\frac{b}{a}}\,\cdot\,\frac{|\lambda|}{\sqrt{a}}\cdot\,
\bigg(1-\frac{\lambda^2}{a}\bigg)^{-\frac{1}{2}} \ \textup{ and } \
\zeta_{-}(\lambda)=-\sqrt{\frac{b}{a}}\,\cdot\,\frac{|\lambda|}{\sqrt{a}}\cdot\,
\bigg(1-\frac{\lambda^2}{a}\bigg)^{-\frac{1}{2}},$$ that is the points where the ellipse $\Gamma_{\lambda}$ and the real axis $\mathbb{R}$ intersect.
3. If $\zeta\in\mathbb{H}^{+}$ lies outside the contour $\Gamma_{\lambda}$, then $v(\zeta)>0$. If $\zeta\in\mathbb{H}^{+}\setminus S$ lies inside the contour $\Gamma_{\lambda}$, then $v(\zeta)<0$.
The functions $u$ and $v$ are the real and the imaginary parts of the holomorphic function $\sqrt{a\zeta^2+b}+\lambda\zeta$. From the Cauchy-Riemann equation it follows that the functions $u$ and $v$ have the same critical points. Moreover the point $\zeta$ is critical for $v$ if and only if $\zeta$ is a root of the derivative $a\zeta(a\zeta^2+b)^{-\frac{1}{2}}+\lambda$ of the function $\sqrt{a\zeta^2+b}+\lambda\zeta$. An explicit calculation shows that this derivative has only two roots $\zeta_{+}(\lambda)$ and $\zeta_{-}(\lambda)$, .
Let $\operatorname{Ext}(\Gamma_\lambda)$ and $\operatorname{Int}(\Gamma_{\lambda})$ be the exterior and the exterior of the contour $\Gamma_\lambda$ respectively. Each of the sets $\mathscr{E}^{Ext}_{\lambda}$ and $\mathscr{E}^{Int}_{\lambda}$, $$\label{extint}
\mathscr{E}^{Ext}_{\lambda}=\operatorname{Ext}(\Gamma_\lambda)\cap\mathbb{H}^{+},\quad
\mathscr{E}^{Int}_{\lambda}=\operatorname{Int}(\Gamma_\lambda)\cap(\mathbb{H}^{+}\setminus S)$$ is a connected open set. According to and , the continuous real valued function $v$ does not vanish on any of these two sets. Hence the values $v(\zeta)$ have the same sign, say $s^{Ext}$, at all points $\zeta$ of the set $\mathscr{E}^{Ext}_{\lambda}$, and the same same sign, say $s^{Int}$, at all points $\zeta$ of the set $\mathscr{E}^{Int}_{\lambda}$. Now the assertion 3 of Lemma \[prhf\] is a consequence of Lemma \[laze\].$\Box$\
. Let us chose the ellipse $\Gamma_{\lambda}$ as the contour of integration $\Gamma$ in the integral in the right hand side of . Since the imaginary part $v(\zeta)$ of the exponent of the integrand vanishes on $\Gamma_{\lambda}$, the integral representation takes the form $$\label{cirl}
\widehat{d}(\lambda,a,b)=-\frac{1}{4\pi i}\ointctrclockwise\limits_{\Gamma_\lambda}
e^{-u(\zeta)}\,d\zeta, \ \
-\sqrt{a}<\lambda<0.$$ Since $d\zeta=dx+idy$, we can split the integral in : $$\label{cirls}
\widehat{d}(\lambda,a,b)=-\frac{1}{4\pi i}\ointctrclockwise\limits_{\Gamma_\lambda}
e^{-u(\zeta)}\,dx(\zeta)-\frac{1}{4\pi}\ointctrclockwise\limits_{\Gamma_\lambda}
e^{-u(\zeta)}\,dy(\zeta).$$ Since the values $\widehat{d}(\lambda,a,b)$, $x(\zeta)$, $y(\zeta)$, and $e^{-u(\zeta)}$ are real, the first integral in the right hand side of vanishes. So the equality takes the form $$\label{cirlt}
\widehat{d}(\lambda,a,b)=-\frac{1}{4\pi}\ointctrclockwise\limits_{\Gamma_\lambda}
e^{-u(\zeta)}\,dy(\zeta).$$ Since the contour $\Gamma_{\lambda}$ is symmetric with respect to the real axis $\mathbb{R}$ and the function $u$ also is symmetric: $u(\zeta)=u(\overline{\zeta})$, the equality can be reduced to the form $$\label{cirlr}
\widehat{d}(\lambda,a,b)=-\frac{1}{2\pi}
\sideset{}{\hspace*{-2.5ex}{\text{\raisebox{0.5ex}{\(\scriptstyle\curvearrowleft\)}}}}\int\limits_{\Gamma_\lambda^{+}}
e^{-u(\zeta)}\,dy(\zeta),$$ where $\Gamma_\lambda^{+}=\Gamma_\lambda\cap\mathbb{H}^{+}$ is the upper half of the contour $\Gamma_{\lambda}$. Integrating by parts in , we obtain $$\label{cirlp}
\widehat{d}(\lambda,a,b)=\frac{1}{2\pi}
\sideset{}{\hspace*{-2.3ex}{\text{\raisebox{0.5ex}{\(\scriptstyle\curvearrowright\)}}}}\int\limits_{\Gamma_\lambda^{+}}
e^{-u(\zeta)}\,y(\zeta)\,du(\zeta),$$ (The values $y(\zeta_{\pm}(\lambda))$ at the end points $\zeta_{\pm}(\lambda)$, , of the integration path $\Gamma_{\lambda}^{+}$ vanish.)
The differential $du(\zeta)$ in can be represented as $$\label{redi}
du(\zeta)=\frac{du(\zeta(s))}{ds}\,ds,$$ where $s$ is a natural parameter[^3] on $\Gamma_{\lambda}^{+}$. In other words, the differential $du(\zeta)$ can be represented as $$\label{redv}
du(\zeta)=\frac{du}{d\vec{\tau}}(\zeta)\,ds(\zeta),$$ where $\vec{\tau}(\zeta)$ is the tangent vector to the curve $\Gamma_{\lambda}^{+}$ at the point $\zeta\in\Gamma_{\lambda}^{+}$. The direction of the vector $\vec{\tau}(\zeta)$ corresponds to the motion of the point $\zeta(s)$ along the path $\Gamma_{\lambda}^{+}$ from its left end point $\zeta_{-}(\lambda)$ to the right end point $\zeta_{+}(\lambda)$. If $\vec{n}(\zeta)$ is the vector of the *exterior* normal to $\Gamma_{\lambda}^{+}$ at the point $\zeta\in\Gamma_{\lambda}^{+}$, then the orientation of the frame $(\vec{\tau}(\zeta),\vec{n}(\zeta))$ coincides with the orientation of the natural frame of $\mathbb{R}^2$. According the Cauchy-Riemann equations, $$\label{CR}
\frac{du}{d\vec{\tau}}(\zeta)=\frac{dv}{d\vec{n}}(\zeta), \ \ \forall\,\zeta\in\Gamma_{\lambda}^{+}.$$ Thus the representation can be reduced to the form $$\label{cirlpr}
\widehat{d}(\lambda,a,b)=\frac{1}{2\pi}
\int\limits_{\Gamma_\lambda^{+}}
e^{-u(\zeta)}\,y(\zeta)\,\frac{dv}{d\vec{n}}(\zeta)\,ds(\zeta).$$ According the assertion 3 of Lemma \[prhf\], $$\label{crpo}
\frac{dv}{d\vec{n}}(\zeta)>0, \ \ \ \forall\,\zeta\in\Gamma_{\lambda}^{+}.$$ The inequality in is strict because $\frac{dv}{d\vec{n}}(\zeta)=
|\operatorname{grad}v(\zeta)|$ and the gradient $\operatorname{grad}v(\zeta)$ of the function $v$ vanishes only at the critical points $\zeta_{\pm}(\lambda)$ of the function $v$, which are the end points of the integration path $\Gamma_{\lambda}^{+}$. Evidently $y(\zeta)>0$ and $e^{-u(\zeta)}>0$ at every point $\zeta\in\Gamma_{\lambda}^{+}$. Thus the integrand in is strictly positive at every point $\zeta\in\Gamma_{\lambda}^{+}$. So the inequality $\widehat{d}(\lambda,a,b)>0$ holds. $\Box$
\[prior2\] The method which we use in the second proof of is the lite version of the method which Herbert Stahl, , used in his proof of the BMV conjecture.
The third proof of Theorem \[MT\].
==================================
For each fixed $\eta$, the function $\cosh\big(\eta\sqrt{t^2+\xi}\big)$ is an entire function of the variables $t,\xi$. Therefore, the Taylor expansion holds $$\label{ser}
\cosh\Big(\eta\sqrt{t^2+\xi}\Big)=\sum\limits_{0\leq k<\infty}\frac{1}{k!}\, \psi_{k}(t,\eta)\,\xi^{k},$$ where $$\psi_{k}(t,\eta)= \frac{d^k\cosh\Big(\eta\sqrt{t^2+\xi}\Big)}{d\xi^k}\raisebox{-1.0ex}{\(\big|_{\xi=0}\)},
\ \ \ k=0,1,2,\,\ldots\,\,.$$ It turns out that *for every fixed real $\eta$ and for every $k=0,1,2,\,\ldots$, the function $\psi_k(t,\eta)$ of the variable $t$ is exponentially convex.* We prove this by induction in $k$. Therefore for $\xi\geq0$, the sum of the series in is an exponentially convex function of $t$. To obtain Theorem \[MT\], we put $\eta=\sqrt{a}$, $\xi=b/a$ in . (For $a=0$, the statement of Theorem \[MT\] is trivially true.)
Our proof of the exponential convexity of the functions $\psi_{k}(t,\eta)$ is based on the identity $$\frac{\sinh\zeta}{\zeta}= \prod\limits_{1\leq m<\infty} \!\!\cosh\frac{\zeta}{2^m},$$ which holds for every $\zeta\in\mathbb{C}$. Substituting the expression $$\zeta= \eta\, \sqrt{t^2+\xi}$$ into this identity, we obtain the equality $$\frac{\sinh\big(\eta\sqrt{t^2+\xi}\big)}{\sqrt{t^2+\xi}} =\eta
\prod_{1\leq m<\infty} \cosh\Big(\frac{\eta}{2^m}\sqrt{t^2+\xi}\Big).$$ Using the equality $$\dfrac{d\cosh\big(\eta\sqrt{t^2+\xi}\big)}{d\xi}=\dfrac{\eta}{2}\,
\dfrac{\sinh\big(\eta\sqrt{t^2+\xi}\big)}{\sqrt{t^2+\xi}},$$ which holds for every $t,\xi,\eta$, we obtain the equality $$\label{Rec}
\dfrac{d\cosh\big(\eta\sqrt{t^2+\xi}\big)}{d\xi}=
\frac{\eta^2}{2}\prod_{1\leq m<\infty} \cosh\Big(\frac{\eta}{2^m}\sqrt{t^2+\xi}\Big).$$
By successive differentiation the equality with respect to $\xi$, we obtain the equality $$\label{sud}
\dfrac{d^{k+1}\cosh\big(\eta\sqrt{t^2+\xi}\big)}{d\xi^{k+1}}=
\frac{\eta^2}{2}\sum\limits_{|\mathfrak{l}|=k}\bigg(\prod_{1\leq m<\infty}
\frac{d^{\,l_{m}}\cosh\big(\frac{\eta}{2^m}\sqrt{t^2+\xi}\big)}{d\xi^{l_{m}}}\bigg),$$ where $k=0,1,2,3,\,\ldots\,\,$. In , the summation is extended over all sequences[^4] $\mathfrak{l}=(l_1,l_2,l_3,\,\ldots\,)$ of non-negative integers for which $|\mathfrak{l}|=l_1+l_2+l_3+\,\ldots\,=k.$
The equality holds for every $t,\xi,\eta$. Restricting this equality to the value $\xi=0$, we obtain the equality $$\label{sude}
\psi_{k+1}(t,\eta)=\frac{\eta^2}{2}\sum\limits_{|\mathfrak{l}|=k}\bigg(\prod_{1\leq m<\infty}
\psi_{l_{m}}\Big(t,\frac{\eta}{2^m}\Big)\bigg),$$ which holds for every $t$, $\eta$, and $k=0,1,2,3,\,\ldots\,\,$. In , the summation is extended over all sequences $\mathfrak{l}=(l_1,l_2,l_3,\,\ldots\,)$ of non-negative integers for which $|\mathfrak{l}|=l_1+l_2+l_3+\,\ldots\,=k.$
Let $\eta$ be an arbitrary real number. By Lemma \[echc\], the function $$\label{ne}
\psi_0(t,\eta)=\cosh \eta t$$ of $t$ is exponentially convex. Moreover, the function $\psi_0\Big(t,\dfrac{\eta}{2^m}\Big)$ is exponentially convex for every $m=1,2,3,\,\ldots\,.$ (The number $\frac{\eta}{2^m}$ here plays the same role as the number $\eta$ in : it is an arbitrary real number.)
Given $k\geq0$, assume that all functions $\psi_{l}(t,\frac{\eta}{2^m})$ with $0\leq l\leq k$ are exponentially convex functions of $t$. Then for each sequence $\mathfrak{l}=(l_1,l_2,l_3,\,\ldots\,)$ with $|\mathfrak{l}|=k$, the inequalities $0\leq l_m\leq k$ hold. Thus, all the factors $\psi_{l_{m}}\Big(t,\frac{\eta}{2^m}\Big)$ which appears in the product $\prod\limits_{1\leq m<\infty}
\psi_{l_{m}}\Big(t,\frac{\eta}{2^m}\Big)$ are exponentially convex functions of $t$. Hence the product itself is an exponentially convex function. Finally, the function $\psi_{k+1}(t,\eta$, , which is essentially equal to the sum of all such products with $|\mathfrak{l}|=k$, is exponentially convex. This finishes the proof.$\Box$
\[coex\] Comparing the expansions and , we see that $$\label{coeq}
\varphi_{k}(t,a)=\psi_k(t,\sqrt{a})a^{-k},\quad k=0,1,2,\,\ldots\,,\ \ t\in\mathbb{R}.$$ As a byproduct of the third proof of , we proved that each of the functions $\varphi_{k}(t,a)$ is exponentially convex. Thus we have given a second proof of .
\[Act\] Actually we proved more then we formulated in Theorem \[MT\]. Namely we proved that for any sequence $a_{k}(\eta)$ of non-negative numbers the sum of series $$\label{sos}
s(t)=\sum\limits_{0\leq k<\infty}a_k(\eta)\psi_{k}(t,\eta)$$ is an exponentially convex function if this series converges for every real $t$.
If $m$ is a positive integer and $\xi\geq0$, then the Taylor expansion $$\label{Tem}
\frac{d^{m}\cosh\big(\eta\sqrt{t^2+\xi})}{d\xi^{m}}=
\sum\limits_{m\leq k<\infty}\frac{1}{(k-m)!}\varphi_{k}(t,\eta)\,\xi^{k-m}$$ is of the form with $a_{k}(\eta)=0$ for $0\leq k <m$, $a_{k}(\eta)=\dfrac{1}{(k-m)!}\xi^{k-m}$ for $k=m,m+1,m+2,\,\ldots\,\,.$
In particular, for $m=1$ the following result holds:
\[meo\] For any $a>0$ and $b>0$, the function $\psi(t)=\dfrac{\sinh\big(\sqrt{at^2+b}\big)}{\sqrt{at^2+b}}$ is an exponentially convex function of the variable $t$.
[Mo]{} <span style="font-variant:small-caps;">Abramowitz,M.,Stegun,I.(Eds.)</span> *Handbook of Mathematical Functions.* National Bureau of Standarts Applied Mathematics Series$\cdot$55, Washington, D.C., Tenth Printing, 1972. Russian translation:\
<span style="font-variant:small-caps;">Абрамовиц, М., Стиган,И.</span>(Ред.) *Справочник по Специальным Функциям.* Физматгиз, Москва, 1979. <span style="font-variant:small-caps;">Ахиезер,Н.И.</span> *Классическая проблема моментов.* Физматгиз, Москва 1961. English translation:\
<span style="font-variant:small-caps;">Akhiezer,N.I.</span> *The classical moment problem*. Oliver&Boyd, Edinburg and London 1965. <span style="font-variant:small-caps;">Bernstein, S.N.</span> *Sur les fonctions absolument monotones.* Acta Math., **52** (1929), 1–66. <span style="font-variant:small-caps;">Бернштейн, С.Н.</span> *Собрание Сочинений, Том 1.* Издательство АН СССР, 1952. D.Bessis, P.Moussa,M.Villani. *Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics.* J. Mat. Phys., **16**:11 (1975), 2318 - 2325. <span style="font-variant:small-caps;">Mehta,M.L., Kumar,K.</span> *On an integral representation of the function $\textup{Tr}\,[e^{A-\lambda B}]$.* J.Phys.A: Math. Gen.,**9**, no.2 (1976), 197–206. <span style="font-variant:small-caps;">Schaftheitlin,P.</span> *Die Theorie der Besselschen Funktionen.* Teubner, Leipzig und Berlin, 1908. <span style="font-variant:small-caps;">Stahl,H.</span> *Proof of BMV conjecture.* Acta Math, **211** (2013), 255–290. <span style="font-variant:small-caps;">Watson,G.N.</span> *A Treatise on the Theory of Bessel Functions*. Cambridge Univ. Press, Cambridge 1922. Russian translation:\
<span style="font-variant:small-caps;">Ватсон, Г.Н.</span> *Теория Бесселевых Функций*. ИЛ, Москва, 1949. <span style="font-variant:small-caps;">Widder,D.</span> *The Laplace transform.*
[^1]: Proof of the Multiplication Theorem can be found in [@W Chapter V, sec.5.22], see formula (15) on page 142 of the English edition or on the page 156 of the Russian translation. See also [@Sc Chapter IV, sec.21].
[^2]: Here the choice of the branch of the function $\sqrt{a\zeta^2+b}$ is important.
[^3]: Length of arc.
[^4]: $\textup{For} \ l_m=0, \ \ \ \dfrac{d^{\,l_{m}}\cosh\big(\frac{\eta}{2^m}\sqrt{t^2+\xi}\big)}{d\xi^{l_{m}}}
\stackrel{\textup{\tiny def}}{=}\cosh\Big(\frac{\eta}{2^m}\sqrt{t^2+\xi}\Big).$
|
---
abstract: 'We present a method to approximate post-Hartree-Fock correlation energies by using approximate natural orbitals obtained by the random phase approximation (RPA). We demonstrate the method by applying it to the helium atom, the hydrogen and fluorine molecule, and to diamond as an example of a periodic system. For these benchmark systems we show that RPA natural orbitals converge the MP2 correlation energy rapidly. Additionally we calculated FCI energies for $\rm He$ and $\rm H_2$, which are in excellent agreement with literature and experimental values. We conclude that the proposed method may serve as a compromise to reach good approximations to correlation energies at moderate computational cost and we expect the method to be especially useful for theoretical studies on surface chemistry by providing an efficient basis to correlated wave function based methods.'
author:
- Benjamin Ramberger
- Zoran Sukurma
- Tobias Schäfer
- Georg Kresse
bibliography:
- 'literature.bib'
title: 'RPA natural orbitals and their application to post-Hartree-Fock electronic structure methods'
---
\[sec:intro\]Introduction
=========================
In *ab initio* quantum chemistry and computational materials physics there exists a well know trade off between accuracy and computational cost. Solving the many electron Schrödinger equation is an exponentially hard problem with respect to the number of electrons and therefore solving it exactly is out of scope for most practically relevant systems. Therefore approximations with varying degrees of accuracy are employed. While mean field methods like Hartree-Fock (HF) and density functional theory (DFT) possess computationally favorable scaling of $N^3$ with the number of electrons, they sometimes lack accuracy and fail to describe certain processes. For example describing the dissociation of simple molecules like $\rm H_2$ already poses a challenge for such methods. The properties that are not well described by the HF approximation are attributed to so called electron correlation and there exists a wide variety of electronic structure methods all of which attempt to take the correlation effects into account in an approximate manner. Most of them have in common that one can increase the accuracy systematically at the expense of computational resources, e.g. Møller-Plesset perturbation theory (MPn), coupled cluster (CCSD,CCSDT) and configuration interaction (CI) methods.
DFT and HF calculations can provide a useful single electron basis as a starting point for correlated methods. However the correlated methods have in common that they usually involve virtual orbitals, i.e. eigenstates of the single-electron Hamiltonian from the underlying HF or DFT calculation that are unoccupied in the respective underlying calculation. To obtain the energy exactly (within the approximation of the respective correlated method) a complete basis set of single particle states is required. The convergence behavior with respect to the number of single particle states depends of course on the specific set of orbitals employed. Therefore one can in principle obtain results that are converged up to machine precision with a finite number of virtual states and ideally reduce the computational costs significantly by using a very small number of appropriate single particle orbitals.
The essence of the above statement is that, by choosing an appropriate set of single particle orbitals, one can extract the relevant information resting in the full many body wave function from a small number of single particle wave functions. The question remains how to choose an appropriate set. One possible answer is to use natural orbitals as they turned out to capture much of the relevant information required to converge correlation energies quickly [@Davidson1972; @Gruneis2011].
In this article, we will show how we can approximate natural orbitals by employing the random phase approximation (RPA) and how to utilize the approximate natural orbitals to obtain accurate correlation energies at moderate computational cost. We will demonstrate the method by showing the convergence behavior of the MP2 energy with respect to the number of RPA natural orbitals (RPANOs) for the following prototypical systems: the helium atom, the hydrogen molecule at various internuclear distances, the fluorine molecule at equilibrium distance and diamond as an example for a periodic system. In the case of $\rm He$ and $\rm H_2$ we will also show results for FCI calculations using a RPANO basis and will furthermore compare the exact natural orbitals from FCI with the RPANOs.
\[sec:theory\]Theory
====================
In second quantization notation, the one particle reduced density matrix (1-RDM) is $$\begin{aligned}
n({\bf r}, {\bf r}')=\braket{\Psi|\hat{\psi^\dagger}({\bf r'})\hat{\psi}({\bf r})|\Psi} \label{eq:1-RDM}\end{aligned}$$ with the usual field operator $\hat{\psi}$. Natural orbitals $\lbrace \varphi_i \rbrace$ are just the eigenstates of the 1-RDM: $$\begin{aligned}
\int_{-\infty}^\infty {\rm d^3}{\bf r}' \; n({\bf r}, {\bf r}') \; \varphi_i({\bf r}') = f_i \; \varphi_i({\bf r}). \label{eq:NOs}\end{aligned}$$ The 1-RDM is the central object of reduced density matrix functional theory (RDMFT) [@Gilbert1975]. In comparison with conventional DFT, RDMFT has the advantage that besides the Hartree energy, also the kinetic and the exchange energy are obtained exactly. Finding an efficient way to approximate the 1-RDM as well as natural orbitals can therefore provide an interesting pathway for RDMFT [@Theophilou2015]. Though we did not pursuit it, we still want to note that the method described hereafter for approximating the 1-RDM within the RPA might also be interesting for research in the field of RDMFT.
We will now try to establish a connection between the 1-RDM and the RPA by using a Green’s function formalism. The 1-RDM is obtained from the Green’s function in the limit of small negative time: $$n({\bf r}, {\bf r}')=-{\operatorname{i}}\lim_{t \to 0^-} G({\bf r}, {\bf r}',t), \label{eq:1-RDM-G}$$ which can be easily seen from the definition of the Green’s function $$G({\bf r}, {\bf r}',t'-t)=-{\operatorname{i}}\braket{\Psi_0|\hat{T}\lbrace \hat{\psi}({\bf r},t)\hat{\psi^\dagger}({\bf r'},t') \rbrace|\Psi_0},$$ where $\hat{T}$ is the Wick time-ordering operator.
In a system with $N$ electrons, the independent particle Green’s function $G^0$ corresponds to a ground state with a single Slater determinant $\ket{\phi_1\cdots\phi_N}$ and the 1-RDM reduces to $$\label{eq:1-RDM-0th}
n^0({\bf r}, {\bf r}')=-{\operatorname{i}}\lim_{t \to 0^-} G^0({\bf r}, {\bf r}',t)=\sum_{i\in occ.} \phi_i({\bf r}) \phi_i^*({\bf r}').$$ The natural orbitals are then just the occupied independent particle orbitals $\lbrace\phi_i\rbrace$. This would be the situation for a DFT or HF calculation.
To improve on the independent particle 1-RDM and to include “information” on the unoccupied virtual space, the Dyson equation for the Green’s function $$G(2,1)=G^0(2,1)+G^0(2,3)\Sigma(3,4)G(4,1)$$ can be used to write a perturbation series for the 1-RDM as a functional of the self-energy $\Sigma$ and the the independent particle Green’s function $G^0$: $$\begin{aligned}
&n=n^0-{\operatorname{i}}\lim_{t \to 0^-}\left\lbrace G^0\Sigma G^0+G^0\Sigma G^0\Sigma G^0+... \right\rbrace.\end{aligned}$$ In the imaginary time/frequency domain, the first order term $n^{(1)}$ is explicitly obtained as [@Ramberger2017] $$\begin{aligned}
&n^{(1)}({\bf r}, {\bf r}')=\lim_{\tau \to 0^-} \frac{1}{2\pi} \int_{-\infty}^{\infty} {\rm d} \nu \; e^{-{\operatorname{i}}\nu \tau} \nonumber \\ &\int {\rm d}{\bf r}'' {\rm d}{\bf r}''' G^0({\bf r}, {\bf r}'',\nu) \Sigma({\bf r}'', {\bf r}''',\nu) G^0({\bf r}''', {\bf r}',\nu).\end{aligned}$$
In the method shown below we will use this first order correction to the density matrix. Furthermore, we will approximate the self-energy using the RPA so that we will ultimately obtain an approximate set of natural orbitals, the RPANOs, as eigenfunctions of the first order RPA density matrix $$n^{\rm RPA}({\bf r}, {\bf r}')= -{\operatorname{i}}\lim_{\tau \to 0^-} \left[G^0+G^0\Sigma^{\rm RPA}G^0\right]({\bf r}, {\bf r}',\tau). \label{eq:RPA-density}$$
\[sec:method\]Method
====================
Assuming that with only a few RPA natural orbitals one can span the relevant subspace of the one electron wave functions for advanced methods like MP2 or CI, we aim to accurately approximate electronic ground state energies of these methods using a small number of virtual states. Thereby we are hopefully able to reduce the computational cost in these calculations significantly. In the following subsection, we will give a step by step recipe for obtaining RPA natural orbitals in order to calculate approximate post-Hartree-Fock ground state energies (e.g. MP2, CC and CI) with a reduced number of virtual states.
\[subsec:RPANOs\] RPA natural orbitals
--------------------------------------
In the first step one calculates the electronic ground state at mean field level (HF or DFT) in order to obtain a well converged set of one electron wave functions for the occupied space. These occupied orbitals fix the respective mean field Hamiltonian for the subsequent steps. In the second step “all” unoccupied orbitals are calculated by diagonalizing the one particle Hamiltonian in the entire underlying basis set. “All” means that the number of eigenfunctions of the Hamiltonian (i.e. orbitals) that are calculated is equal to the number of basis functions used. Specifically for a plane wave basis with a cut-off energy $E_{\rm cut}$ this means that the sum of occupied orbitals $N_{\rm occ}$ and virtuals $N_{\rm virt}$ is equal to the number of plane waves below the cut-off, or in other words equal to the size of the underlying plane-wave basis set: $$N_{\rm occ}+N_{\rm virt}=\operatorname{card}\left\lbrace {\bf G} : \frac{\hbar^2 }{2m_e}|{\bf G}+{\bf k}|^2 < E_{\rm cut} \right\rbrace, \label{eq:NPW}$$ where ${\bf k}$ is the Bloch wave vector.
In the next step, the independent particle Green’s function $G^0$ is calculated from all orbitals to subsequently calculate the independent particle polarizability $\chi^0$, the RPA screened interaction $W^{\rm RPA}$, as well as the RPA self energy $\Sigma^{\rm RPA}$ [@Kaltak2014; @Kaltak2014a; @Liu2016]. Note that $\Sigma^{\rm RPA}$ is related to the $GW$-approximation as the self energy in the first self-consistency cycle [@Ramberger2017; @Ren2012] $$\Sigma^{\rm RPA}(\tau)=G^0(\tau)W^0(\tau).$$
Having the self energy at hand, one can use Eq. (\[eq:RPA-density\]) to calculate the RPA density matrix. The density matrix $n= n^0+n^{\rm RPA}$ is than diagonalized in the subspace of the virtual states, i.e. the occupied states from the mean field calculation are retained. The occupied orbitals together with the eigenstates obtained from that subspace diagonalization form the RPANOs basis set. Keeping the occupied states has practical reasons. In this way, the RPANOs reproduce the underlying HF or DFT ground state energy exactly, while at the same time the virtual states are rearranged to yield a more compact set for higher hierarchy methods when the orbital set is truncated. RPANOs obtained by this approach therefore provide a versatile basis for different applications.
The RPANOs are sorted by descending occupation number, i.e. eigenvalue with respect to the density matrix, and to obtain a truncated basis with $N$ basis functions, one simply uses the first $N$ RPANOs according to this order. In some cases, it is now necessary to diagonalize the mean-field Hamiltonian in the subspace spanned by the truncated RPANOs basis. For example, to calculate the MP2 energy with $256$ RPANOs, the HF Hamiltonian is diagonalized in the subspace spanned by the first $256$ RPANOs. The subspace HF orbitals from that diagonalization span the same space as the truncated RPANOs and allow at the same time to use the Brillouin theorem as required by most MP2 implementations.
The computational cost for all the above calculations scales cubically in system size [@Kaltak2014; @Kaltak2014a; @Liu2016], allowing the method to be used for relatively big systems ($\approx 100$ atoms) with moderate computing resources ($\approx 100$ CPUs).
\[subsec:FCI\] FCI
------------------
The full configuration interaction method (FCI) solves the non-relativistic Schrödinger equation within a given basis set exactly. Therefore it provides a useful benchmark for other correlation-consistent methods. Its computational cost scales exponentially with the number of orbitals and the number of electrons, limiting its range of application. In order to obtain the FCI solution, one starts with the HF orbitals and constructs all possible Slater determinants with $N_{\text{occ}}$ occupied orbitals. By separating each Slater determinant in its spin up and spin down part, and by using Slater-Condon rules, it is possible to efficiently evaluate contractions of the form $HC$, where $C$ is the vector of Slater determinant weights in the wave function. Following the method of Handy and Knowles [@Knowles1989] and using a Davidson-Liu algorithm to iteratively diagonalize the matrix $H$ [@Davidson1975], the FCI solution can be obtained. Using Eq. (\[eq:1-RDM\]), the FCI 1-RDM can be easily calculated once the FCI wave function is known. Finally, the diagonalization of this density matrix gives the FCI natural orbitals.
\[sec:application\]Application
==============================
All calculations regarding the RPANOs were performed using `VASP` [@Kresse1993; @Kresse1996; @Kresse1996a]. To obtain the RPANOs we tried two different approaches, namely (restricted) HF as well as PBE as underlying single determinant method. However, since the observations for these two approaches were by and large very similar, we will focus on presenting the results that we obtained from using the HF orbitals as the underlying approach for the RPA calculations. For large scale applications on many atom systems, one would most likely though revert to the more efficient DFT approximation. The RPANO calculations are based on the RPA implementation described in previous work [@Kaltak2014; @Kaltak2014a; @Liu2016]. For the systems investigated in this work, 8-12 points on the imaginary time/frequency axis were sufficient. For the MP2 calculations in this work, we used the standard available in `VASP` [@Marsman2009; @Schafer2017]. The FCI calculations were performed with a development version, briefly described above. Details can be found in the master thesis of Sukurma [@Sukurma2019]. Where not stated otherwise, the shown figures were obtained from calculations with a plane wave cut-off of $750$ eV. Note that the orbitals shown below are approximations and do not fulfil the exact cusp condition due to the use of a finite sized plane wave basis set \[compare Eq. (\[eq:NPW\])\].
\[subsec:He-atom\]Helium atom
-----------------------------
The first benchmark system that we studied is the helium atom. The helium atom is an ideal benchmark system for many body approaches for various reasons. First, unlike other model systems (e.g. Jellium) it is a real many body system that can be studied experimentally. Though it is a relatively simple system, its theoretical description faces already prototypical many-body challenges and it is not possible to write down a closed form for its exact solution. Furthermore a numerically exact solution is available, providing a perfect ground to compare to the results of approximate many body approaches [@Li2019].
The set-up was a $10\;{\rm \mathring{A}} \times 10\;{\rm \mathring{A}} \times 10\;{\rm \mathring{A}} $ unit cell with periodic boundary conditions. We used a plane-wave basis with different cut-offs up to $750$ eV and a $1/r$ potential.
First, we examined the convergence behavior of the MP2 correlation energy with respect to the RPANOs and compared it with that for canonical HF-orbitals. The results are shown in Fig. \[fig:He-MP2-convergence\]. While the correlation energy using canonical HF orbitals converges very slowly and would require almost the full basis set (for this set-up with a $750$ eV plane-wave cut-off, these are $4.7\cdot10^4$ orbitals), with only 128 RPANOs the correlation energy converged to $<1$ meV above the exact value. Note that in this case, converged results means converged for a specific fixed plane wave basis set size. To obtain accurate MP2 correlation energies one would also need to check convergence with respect to the plane wave cut-off. However to show the qualitative difference between RPANOs and HF orbitals it is sufficient to inspect the behavior at a fixed cut-off, where the error in the MP2 energy related to the cut-off is much smaller than the difference between the MP2 energy at a few hundred RPANOs and the same number of canonical HF orbitals.
![Convergence behavior of the MP2 correlation energy for the $\rm He$ atom. The convergence with respect to the number of RPANOs and HF-orbitals is compared. The black line shows the result for the full basis set. Note that we use two different ranges for the energy.[]{data-label="fig:He-MP2-convergence"}](He-MP2-convergence)
We repeated the procedure for FCI correlation energies using 10 to 60 orbitals, see Fig. \[fig:He-FCI-convergence\]. As was seen for the MP2 correlation energy, canonical HF-orbitals converge the energy very slowly. Again, to reach highly accurate results, one would also need to go to higher plane wave cut-offs, but for a qualitative comparison the chosen cut-off suffices for the same reasons as at the MP2 level. As a matter of fact, the ground state energy of an FCI calculation with a specific basis is always an upper bound for the real ground state energy. One can conclude from this, that the RPANOs are much better in spanning the relevant space of correlated single particle states than the HF-orbitals.
![Convergence behavior of the FCI correlation energy for the $\rm He$ atom. The convergence with respect to the number of RPANOs and HF-orbitals is compared. The FCI energy using 10-60 HF virtual orbitals is at least 1 eV above the converged value.[]{data-label="fig:He-FCI-convergence"}](He-FCIcor-convergence)
In order to obtain a more accurate approximation to the total energy of the helium atom, $E^{\rm tot}=E^{\rm HF}+E^{\rm cor}$, we first used a plane wave cut-off extrapolation for the FCI correlation and the HF energy [@Shepherd2012; @Klimes2014]. The correlation energy was extrapolated from FCI values at plane wave cut-offs of $474$ eV, $621$ eV and $750$ eV. The HF energy was extrapolated separately from plane wave cut-offs up to $6000$ eV in a $16\;{\rm \mathring{A}} \times 16\;{\rm \mathring{A}} \times 16\;{\rm \mathring{A}}$ unit cell in order to reach more accurate predictions. This procedure is justified, because the correlation energy is less than $1.5\%$ of the HF energy. For the plane wave extrapolation of the energies we used a linear regression with respect to the inverse number of plane waves, $N_{\rm PW}^{-1}$, which is itself proportional to $E_{\rm cut}^{-3/2}$, $$E^{\rm cor/HF}(E_{\rm cut}) \approx E^{\rm cor/HF}(\infty)+p \cdot E_{\rm cut}^{-3/2}. \label{eq:cut-off-extrapolation}$$ This value is the best approximation we can achieve for a fixed number of truncated RPANOs.
However, the value for the cut-off extrapolated correlation energy still depends on the number of RPANOs used in the FCI calculation. Therefore, to obtain a highly accurate correlation energy, we performed an orbital-set extrapolation on top of the cut-off extrapolation. To achieve this, we calculated cut-off extrapolated FCI correlation energies as described above with different numbers of RPANOs (10,20,...,60). These cut-off extrapolated values were extrapolated by another linear regression, this time with respect to the inverse number of RPANOs used in the respective FCI calculations, $$E^{\rm FCI-cor}(N_{\rm RPANOs}) \approx E^{\rm FCI-cor}(\infty)+q\cdot N_{\rm RPANOs}^{-1}. \label{eq:basis-set-extrapolation}$$ By doing so, we obtained an approximate value for the FCI total energy of $-79.019$ eV, which differs by less than $1$ meV from the literature value from the basis set extrapolation with Gaussian cc-pVxZ basis sets [@Li2019]. The extrapolated values for the two methods are therefore well within the error bars of each other. Another exact value from Hylleraas-like calculations is $-79.014$ eV [@Korobov2002]; our estimation is around $5$ meV bellow that. These results are summarized in Fig. \[fig:He-FCI-basis-set-convergence\].
![The cut-off extrapolated FCI total energies for the $\rm He$ atom are shown for different numbers of RPANOs. A basis set extrapolation with 1/RPANOs yields a $\rm He$ total energy that is $5$ meV below the value from Hylleraas-like calculations. The extrapolated value for RPANOs and cc-pVxZ are on top of each other.[]{data-label="fig:He-FCI-basis-set-convergence"}](He-FCI-basis-set-convergence)
Finally, we compared the shape of the natural orbitals along an axes through the nucleus of the atom. In Figs. \[fig:He-NOs-1s-all\]-\[fig:He-NOs-2s-FCI\] we show the 1s and 2s orbitals of helium for the different methods we employed. While the occupied 1s orbital is well described in the mean field methods, the 2s orbitals for the different methods vary significantly. This is not very surprising as the HF/DFT states are not computed as eigenstates of the density matrix, but stem from a mean field Hamiltonian. In fact, by construction, the virtual HF/DFT states are degenerate eigenstates to the HF/DFT density matrix with eigenvalue 0. This is easily seen by Eq. (\[eq:NOs\]) and (\[eq:1-RDM-0th\]), since the mean field single particle states are orthogonal to each other. Furthermore, from Fig. \[fig:He-NOs-2s-FCI\] we can deduce that the 2s FCI natural orbital is already converged with as little as 10 RPANOs, but far from converged with 60 HF orbitals. This substantiates the results from Figs. \[fig:He-MP2-convergence\] and \[fig:He-FCI-convergence\] that canonical HF orbitals calculated from a plane wave basis are impractical for correlated calculations, which is cured by introducing RPANOs.
![$\rm He$ 1s orbital. The occupied orbital is shown for PBE, HF, FCI@60RPANOs and FCI@60HF. RPANO and HF are by construction identical.[]{data-label="fig:He-NOs-1s-all"}](He-NOs-1s-all)
![$\rm He$ “2s orbital”. The first unoccupied mean-field or natural orbital is shown for PBE, HF, RPANO, FCI@60RPANOs and FCI@60HF.[]{data-label="fig:He-NOs-2s-all"}](He-NOs-2s-all)
![$\rm He$ “2s orbital”. The first unoccupied FCI natural orbital is shown for 10 and 60 RPANOs as well as for 10 and 60 HF orbitals.[]{data-label="fig:He-NOs-2s-FCI"}](He-NOs-2s-FCI)
\[subsec:H-molecule\]Hydrogen molecule
--------------------------------------
The second benchmark system investigated is the hydrogen molecule. Though $\rm H_2$ is the simplest molecule, it’s exact quantum mechanical description already shows prototypical challenges. Specifically the dissociation of the two hydrogen atoms is challenging from a theoretical point of view. In the dissociation limit an accurate wave function based method should yield a wave function corresponding to two separate hydrogen atoms. The restricted HF method, however, yields an inadequate dissociation limit with an energy substantially above the correct limit due to a flawed mixture of single electron states. Conversely, the spin-unrestricted HF (URHF) method yields a solution with an up electron on one site and a down electron on the other site, which is a spin contaminated singlet. The true singlet would have an equal expectation value for up and down electrons on both sites and is a mixture of (at least) two Slater determinants. The underlying problem is that a single determinant description is insufficient to describe the dissociation process. Similar problems exist in DFT [@Siegbahn2010]. Therefore the hydrogen molecule is, despite being seemingly simple, an interesting benchmark system for correlated electronic structure methods.
The computational set-up was again a $10\;{\rm \mathring{A}} \times 10\;{\rm \mathring{A}} \times 10\;{\rm \mathring{A}} $ unit cell with periodic boundary conditions. We used a plane-wave basis with different cut-offs up to $750$ eV. The internuclear distance was varied between the equilibrium distance of $0.74$ Å and $5.0$ Å.
As for the helium atom, we examined the convergence behaviors of the MP2 as well as the FCI correlation energy with respect to the RPANOs and compared them with convergence for HF-orbitals. The results of the calculations are shown in Figs. \[fig:H2-MP2-convergence\] and \[fig:H2-FCI-convergence\]. Similarly to the helium atom, the correlation energies converge very slowly with an increasing number of HF orbitals. Using RPANOs, we only need a few hundred basis functions to obtain converged results for the MP2-energy. The statements on the cut-off convergence in the previous section on the helium atom apply here as well.
![Convergence behavior of the MP2 correlation energy for the $\rm H_2$ molecule at equilibrium distance. The convergence with respect to the number of RPANOs and HF-orbitals is compared. The black line shows the result for the full basis set.[]{data-label="fig:H2-MP2-convergence"}](H2-MP2-convergence)
![Convergence behavior of the FCI correlation energy for the $\rm H_2$ molecule at equilibrium distance. The convergence with respect to the number of RPANOs and HF-orbitals is compared. The FCI energy using 10-60 HF virtual orbitals is at least 1 eV above the converged value.[]{data-label="fig:H2-FCI-convergence"}](H2-FCIcor-convergence)
For the FCI total energies, a cut-off extrapolation was done in the same fashion as for the helium atom according to Eq. (\[eq:cut-off-extrapolation\]) - from cut-offs up to $6000$ eV for HF, and from cut-offs of $474$ eV, $621$ and $750$ eV for the FCI correlation energy. Additionally we also performed an orbital set extrapolation with respect to the number of RPANOs, compare Eq. (\[eq:basis-set-extrapolation\]). In order to compare our results with experimental energies, we calculated the dissociation energy by subtracting twice the URHF energy for the isolated atom obtained with the same procedure (cut-off extrapolation), from the extrapolated FCI total energy at equilibrium distance ($0.74$ Å). An appropriate experimental value to compare our *ab-initio* Born-Oppenheimer approximation result with, was obtained by adding the experimental vibrational zero point energy ($E_{\rm ZPE}$) [@Irikura2007] to the experimental dissociation energy $D_0$ [@Zhang2004], $$D_{\rm e}=D_0+E_{\rm ZPE}. \label{eq:dissociation-energy}$$ Our extrapolated FCI binding energy ($-D_{\rm e}$) is $-4.749$ eV, which is $1$ meV below the corresponding experimental value of $-4.748$ eV. Additionally we performed FCI calculations with the Gaussian pp-VxZ basis sets, with $x=2,3,4$, and extrapolated in order to obtain an estimation for the basis set limit of the $\rm H_2$ binding energy in the Born-Oppenheimer approximation. The extrapolation was done along the lines of Ref. [@Li2019], [*i.e.*]{} for the correlation energy we extrapolated according to $$E^{\rm cor}(x)=E^{\rm cor}(\infty)+c \cdot x^{-3},$$ while for the HF energy we fitted $$E^{\rm HF}(x)=E^{\rm HF}(\infty)+a\cdot e^{-b\cdot x}.$$ The extrapolated energy from the pp-VxZ basis sets was $-4.755$ eV, $6$ meV below the value for RPANOs and $7$ meV below the experimental value. These results are summarized in Fig. \[fig:H2-FCI-basis-set-convergence\].
![The cut-off extrapolated FCI binding energies for the $\rm H_2$ molecule are shown at different numbers of RPANOs. A basis set extrapolation with 1/RPANOs yields a $\rm H_2$ binding energy that is 1meV below the corresponding experimental reference value, see Eq. \[eq:dissociation-energy\].[]{data-label="fig:H2-FCI-basis-set-convergence"}](H2-FCI-basis-set-convergence)
Again we also inspected the shape of the orbitals, this time along the bond-axis of the hydrogen molecule. In the equilibrium position, the first NO ([*i.e.*]{} the occupied 1$\sigma$ state) is well approximated by the mean field methods $1\sigma$ orbital, see Fig. \[fig:H2-EQ-NOs-1st-all\]. But the second natural orbital is already very different from DFT and even more so from the HF $1\sigma^*$, see Fig. \[fig:H2-EQ-NOs-2nd-all\]. This has the same reasons as for the $\rm He$ 2s orbital.
![$\rm H_2$ 1$\sigma$ orbital at equilibrium distance. The occupied orbital is shown for PBE, HF, FCI@60RPANOs and FCI@60HF. RPANO and HF are by construction identical.[]{data-label="fig:H2-EQ-NOs-1st-all"}](H2-EQ-NOs-1st-all)
![$\rm H_2$ 1$\sigma^*$ orbital at equilibrium distance. The first unoccupied orbital is shown for PBE, HF, RPANO, FCI@60RPANOs and FCI@60HF.[]{data-label="fig:H2-EQ-NOs-2nd-all"}](H2-EQ-NOs-2nd-all)
In Fig. \[fig:H2-EQ-NOs-2nd-all\], one can also see that the RPANO does not match the FCINO perfectly. But using just 10 RPANOs for the FCI calculation, the FCI natural orbital is already converged, see Fig. \[fig:H2-EQ-NOs-2nd-FCI\]. This is a hint that, at least in this case, the RPANOs span the relevant space efficiently. In contrast, the HF orbitals are not even close to convergence with 60 orbitals.
![$\rm H_2$ 1$\sigma^*$ orbital at equilibrium distance. The first unoccupied FCI orbital is shown for 10 and 60 RPANOs as well as for 10 and 60 HF orbtials.[]{data-label="fig:H2-EQ-NOs-2nd-FCI"}](H2-EQ-NOs-2nd-FCI)
Similar observations can be made for other cases, with varying internuclear distances and orbital levels as well as basis set sizes and underlying mean field methods. We have investigated a variety of cases, and we have picked a few illustrative examples in an effort to summarize our findings.
One of these examples is the third orbital (2$\sigma$) at an internuclear distance of 1.588 Å (=$\;3a_0$). This example is interesting since in this case the RPANO noticeably differs from the FCINO, see Fig. \[fig:H2-3au-NOs-3rd-all\]. However, inspecting Fig. \[fig:H2-3au-NOs-3rd-FCI\], one can see again that only very few RPANOs are necessary to describe the third FCINO correctly, even though the third RPANO does not match well with the third FCINO. Furthermore once again, the canonical HF-orbitals poorly describe the FCINO.
![$\rm H_2$ $2\sigma$ orbital at an internuclear distance of 1.588 Å (=$3\;a_0$). The second unoccupied FCI orbital ($\hat{=}2\sigma$) is shown for PBE, HF, RPANO, FCI@60RPANOs and FCI@60HF.[]{data-label="fig:H2-3au-NOs-3rd-all"}](H2-3au-NOs-3rd-all)
![$\rm H_2$ $2\sigma$ orbital at an internuclear distance of 1.588 Å (=$3\;a_0$). The second unoccupied FCI orbital ($\hat{=}2\sigma$) is shown for 10 and 60 RPANOs as well as for 10 and 60 HF orbtials.[]{data-label="fig:H2-3au-NOs-3rd-FCI"}](H2-3au-NOs-3rd-FCI)
Finally, we show the orbitals at dissociation (5 Å). This time we compare the first two orbitals again. At this distance, the occupied orbitals vary more between the employed methods than they did at equilibrium distance, compare Fig. \[fig:H2-EQ-NOs-1st-all\] and \[fig:H2-5A-NOs-1st-all\]. By construction the occupied RPANO is equal to the occupied HF orbitals. It is noteworthy that in this case, the PBE orbitals and hence the PBE electron density, is much closer to the FCINO than the HF orbital is.
![$\rm H_2$ 1$\sigma$ orbital at an internuclear distance of 5 Å. The first occupied FCI orbital ($\hat{=}1\sigma$) is shown for PBE, HF, and FCI@60RPANOs.[]{data-label="fig:H2-5A-NOs-1st-all"}](H2-5A-NOs-1st-all)
![$\rm H_2$ 1$\sigma^*$ orbital at an internuclear distance of 5 Å. The first unoccupied FCI orbital ($\hat{=}1\sigma^*$) is shown for PBE, HF, RPANO and FCI@60RPANOs.[]{data-label="fig:H2-5A-NOs-2nd-all"}](H2-5A-NOs-2nd-all)
\[subsec:fluor\] Fluorine molecule
----------------------------------
In the two electron systems $\rm H_2$ and $\rm He$, the direct and exchange contribution to the MP2 correlation energy are related by a simple factor of -2, i.e. $E^{(2)}_d=-2E^{(2)}_x$. As $\Sigma^{\rm RPA}$ consists of an infinite series of bubble diagrams ($\chi^0$), it intrinsically contains the direct part of the MP2 self energy $\Sigma^{(2)}_d$, but also accordingly models the exchange part $\Sigma^{(2)}_x$. For these two reasons, we expected the RPANOs to efficiently span the relevant subspace for MP2 calculations in $\rm H_2$ and $\rm He$. However, for a multi electron system like the fluorine molecule, the simple relation between direct and exchange part of the MP2 energy is no longer valid and it is therefore not clear from the outset that RPANOs will converge the MP2 energy quickly. Thus we investigated the convergence behavior for the direct (MP2D) and exchange (MP2X) contribution to the MP2 correlation energy separately and compared again the performance of RPANOs with HF orbitals. For these calculation, we employed the PAW method [@Kresse1994; @Kresse1999] and used again a $10\;{\rm \mathring{A}} \times 10\;{\rm \mathring{A}} \times 10\;{\rm \mathring{A}} $ unit cell and a plane wave cut-off of $750$ eV. We observed that for fluorine, the exchange contribution to the MP2 correlation energy converges rapidly and in the same manner as the direct contribution, see Fig. \[fig:F2-MP2XD-convergence\]. This is a strong indication that the proposed method, using RPANOs to construct orbitals for more expansive correlated methods, is straightforwardly applicable to complex, multi electron systems.
![Convergence behavior of the direct and the exchange contribution to the MP2 correlation energy for the $\rm F_2$ molecule. The convergence with respect to the number of RPANOs is shown. The black lines indicate the converged result for the full basis set. Note the different scales.[]{data-label="fig:F2-MP2XD-convergence"}](F2-MP2XD-convergence)
\[subsec:diamond\]Diamond
-------------------------
The fourth benchmark system, diamond, was chosen to demonstrate the method on a simple periodic multi electron system. Fig. \[fig:C-MP2-convergence\] shows that the convergence behavior of the MP2 energy is slightly improved by the use of RPANOs, but the effect is much smaller than in the molecular systems. Since we use the primitive cell, the number of plane waves in the full basis set is much smaller than it was for the molecules with large vacuum regions between the periodic images - in this case $\approx 400$ for diamond vs. $\approx 47000$ for $\rm H_2$. In this case, between 200-300 RPANOs were sufficient to reach convergence, similar to the numbers for the hydrogen molecule. Thus the improvement being small is not caused by an under-performance of RPANOs, but rather due to a comparably small number of orbitals in the full basis set.
![Convergence behavior of the MP2 correlation energy for diamond. The convergence with respect to the number of RPANOs and HF-orbitals is compared. The black line shows the result for the full basis set. The calculation was performed in the primitive cell with experimental lattice parameter $d=3.567\;{\rm \mathring{A}}$. We used the PAW method [@Kresse1994; @Kresse1999] with a $4\times4\times4$ k-mesh and a plane wave cut-off of $600$ eV.[]{data-label="fig:C-MP2-convergence"}](C-MP2-convergence)
Conclusion
==========
We have tested the efficiency of RPANOs for MP2 and FCI calculations and found that for atoms and molecules they can drastically reduce the number of orbitals that are necessary to reach converged correlation energies in comparison to canonical HF or PBE orbitals. In the present implementation, the calculation of RPANOs scales only cubically with system size. The compute cost of canonical MP2 or FCI calculations increases with the fifth power (MP2) or exponentially (CI) with respect to the number of orbitals. Thus the compute cost for the preceding compression of the space of orbitals will generally be small compared to the savings gained in the final accurate correlated calculations.
An important finding of the present work is that RPANOs yield a set of orbitals which allows to converge the correlation energy rapidly, even in cases, where the RPA itself is most likely not very accurate. Typical examples for such situations are bond breaking and bond making, in this study exemplified for the case of bond dissociation of H$_2$. Specifically, upon dissociation, we found cases where the first few RPANOs deviate significantly from the natural orbitals determined from the FCI correlated density matrix. Regardless of this difficulty, when the correlation energy is expanded in an increasing set of RPANOs, rapid convergence with the number of RPANOs is observed even in these “difficult” cases.
The present method is especially useful for systems comprised of large vacuum regions where the basis set size becomes unmanageable when plane waves are used (we are talking of ten thousands of plane wave orbitals). In fact, some of us have already applied a preliminary implementation of the algorithm presented here to calculate RPA natural orbitals and subsequently solve the BSE equations in this smaller basis. In this way, we were able to determine accurate quasiparticle energies in the GW$\Gamma$ approximation for molecules [@Maggio2017]. Apart from molecular systems, open structures (zeolites) and surface science studies with large vacuum regions are likely to be an interesting and promising field of application for the RPANO method. Recall that coupled cluster methods scale at least with the fourth order of the number of virtual orbitals. If one is capable to compress the number of virtual orbitals by say a factor 2-3, speed ups of one to two orders of magnitude can be expected. We also note that the RPA is expected to be already fairly accurate for the prediction of adsorption energies[@Schimka2010; @GarridoTorres2017], however, close to transition states we expect that methods beyond the RPA will be required. Our results for the bond dissociation in H$_2$ gives hope that the RPANOs will work also well in such challenging situations. Overall, the present implementation will greatly boost the applicability of plane wave basis sets allowing them to compete with the now omnipresent local Gaussian basis sets used in quantum chemistry.
Last not least, natural orbitals in periodic systems can be used as a stepping stone for “strongly” correlated calculations, such as dynamical mean field theory (DMFT) or density matrix renormalization group (DMRG) theory. There are already examples in literature indicating that such an approach might be at least competitive with the usual approach of maximally localized Wannier functions[@Rusakov2019; @Lan2015; @Rusakov2016], since natural orbitals with an occupancy far from zero or one, are likely to contribute most to the correlation energy. Thus, combining perturbative methods for weakly correlated orbitals with an accurate correlation method for the strongly correlated orbitals is an important future development to be pursued.\
\
[*Acknowledgment:*]{} Funding by the Austrian Science Fund (FWF) within the SFB ViCoM (F41) is gratefully acknowledged.
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[**Une étude asymptotique probabiliste des coefficients d’une série entière**]{}
Bernard Candelpergher, Michel Miniconi
[*Université Nice Sophia Antipolis, 06108 Nice Cedex 02, France*]{}
[[email protected], [email protected]]{}
(15 juillet 2013)
[*Résumé.* ]{} En partant des idées de Rosenbloom [@Rosen] et Hayman [@Hayman], Luis B' aez-Duarte donne dans [@LBD] une preuve probabiliste de la formule asymptotique de Hardy-Ramanujan pour les partitions d’un entier. Le principe général de la méthode repose sur la convergence en loi d’une famille de variables aléatoires vers la loi normale. Dans notre travail nous démontrons un théorème de type Liapounov (Chung [@Chung]) qui justifie cette convergence. L’obtention de formules asymptotiques simples nécessite une condition dite [*Gaussienne forte*]{} énoncée par Luis B' aez-Duarte, que nous démontrons dans une situation permettant d’obtenir une formule asymptotique classique pour les partitions d’un entier en entiers distincts (Erd" os-Lehner [@Erdos], Ingham [@Ingham]).
[*Abstract.* ]{} Following the ideas of Rosenbloom [@Rosen] and Hayman [@Hayman], Luis B' aez-Duarte gives in [@LBD] a probabilistic proof of Hardy-Ramanujan’s asymptotic formula for the partitions of an integer. The main principle of the method relies on the convergence in law of a family of random variables to a gaussian variable. In our work we prove a theorem of the Liapounov type (Chung [@Chung]) that justifies this convergence. To obtain simple asymptotic formulæ a condition of the so-called [*strong Gaussian* ]{} type defined by Luis B' aez-Duarte is required; we demonstrate this in a situation that make it possible to obtain a classical asymptotic formula for the partitions of an integer with distinct parts (Erdös-Lehner [@Erdos], Ingham [@Ingham]).
Introduction
============
[**Notation.**]{}
On désigne par ${{\cal O}}_{+}(D(0,1))$ l’ensemble des fonctions $f$ analytiques de rayon de convergence $1$ telles que $f(t)=\sum_{n\geq 0}a_{n}t^{n}$ avec $a_{n}$ réels positifs non tous nuls. En particulier on a $f(t)>0$ pour tout $t\in\ ]0,1[$.
**Définition.**
Soit $f\in\ {{\cal O}}_{+}(D(0,1))$ : pour tout $t\in\ ]0,1[$ on définit la mesure discrète sur $\mathbb{R}$ $$\mu _{t}(f)=\sum_{n\geq 0}\frac{a_{n}t^{n}}{f(t)}\delta _{n}$$
On associe à cette mesure une variable aléatoire $X_{t}$ définie sur l’espace probabilisé $\Omega =]0,1[,$ à valeurs dans $\mathbb{N},$ telle que $$P(X_{t}=n)=\frac{a_{n}t^{n}}{f(t)}$$ le but étant de démontrer des résultats sur le comportement asymptotique des $a_n$ en utilisant des méthodes probabilistes ([*voir*]{} Rosenbloom [@Rosen] qui attribue cette idée à Khinchin).
[**Moments et fonction caractéristique.**]{}
Les séries $\sum_{n\geq 0}n^{k}a_{n}t^{n}$ étant aussi convergentes dans le disque $D(0,1),$ on en déduit que $X_{t}$ possède un moment d’ordre $k$ pour tout $k\geq 1.$ En particulier pour $k=1$ on a, pour tout $t\in\ ]0,1[$ : $$E(X_{t})=\sum_{n\geq 0}n\frac{a_{n}t^{n}}{f(t)}=t\frac{f^{\prime }(t)}{f(t)}$$ On pose $m(t)=E(X_{t})$, $0<t<1$ ; la fonction $m$ est continue sur $]0,1[$.
La fonction caractéristique $\varphi _{X_{t}}$ (ou $\varphi _{\mu _{t}(f)}$) de $X_{t}$ est donnée par $$\varphi _{X_{t}}(\theta )=E(e^{i\theta X_{t}})=\sum_{n\geq
0}e^{in\theta }\frac{a_{n}t^{n}}{f(t)}=\frac{f(te^{i\theta })}{f(t)}$$
Les coefficients du développement de Taylor de $f$ sont liés à la fonction caractéristique $\varphi _{X_t}$ par $$\label{coeffTaylor}
a_{n}=\frac{f(t)}{2\pi t^{n}}\int_{-\pi }^{\pi }\varphi _{X_{t}}(\theta
)e^{-in\theta }d\theta$$ pour tout $t\in\ ]0,1[$.
[**Normalisation.**]{}
Soit $\sigma (t)=\sqrt{Var(X_{t})}$ et considérons la variable aléatoire centrée réduite $$Z_{t}=\frac{X_{t}-m(t)}{\sigma (t)}$$
On a $$\varphi _{Z_{t}}(x)=e^{-ix\frac{m(t)}{\sigma (t)}}\varphi _{X_{t}}\bigl(\frac{x}{\sigma (t)}\bigr)$$
donc en posant $\theta =\displaystyle\frac{x}{\sigma (t)}$ dans la formule (\[coeffTaylor\]) donnant $a_{n}$, on obtient pour tout $t\in\ ]0,1 \lbrack $ $$\begin{aligned}
a_{n} =\frac{f(t)}{2\pi \sigma (t)t^{n}}\int_{-\pi \sigma (t)}^{\pi \sigma
(t)}\varphi _{Z_{t}}(x)e^{i\frac{x}{\sigma (t)}(m(t)-n)}dx.\end{aligned}$$
Supposons qu’il existe une suite $t_{n}\rightarrow 1$ telle que $$m(t_{n})=n\text{ pour tout }n$$
alors on a $$a_{n}=\frac{f(t_{n})}{2\pi \sigma (t_{n})(t_{n})^{n}}\int_{-\pi \sigma
(t_{n})}^{\pi \sigma (t_{n})}\varphi _{Z_{t_{n}}}(x)dx$$
**Comportement asymptotique des** $a_{n}$ **lorsque** $n\rightarrow +\infty $
On suppose que la fonction $m$ est continue, strictement croissante et qu’elle tend vers $+\infty$ lorsque $t$ tend vers 1. Soit $(t_n)$ une suite dans $]0,1[$ tendant vers 1 et telle que
$$m(t_{n}) =n\quad \text{pour tout }n$$
et $$\sigma (t_{n}) \rightarrow +\infty$$
On a alors $$a_{n}=\frac{f(t_{n})}{2\pi \sigma (t_{n})(t_{n})^{n}}\int_{-\pi \sigma
(t_{n})}^{\pi \sigma (t_{n})}\varphi _{Z_{t_{n}}}(x)dx$$
Supposons en outre que l’on ait la convergence en loi de $Z_{t}$ vers une variable aléatoire $Z$ de loi N(0,1) quand $t\rightarrow 1$, ce qui veut dire que $$\varphi _{Z_{t}}(x)\rightarrow \varphi _{Z}(x)=e^{-\frac{x^2}{2}} \text{ pour tout } x\in \mathbb{R},$$ alors on peut espérer un résultat du type ([*voir*]{} Hayman [@Hayman]) $$a_{n}\backsim \frac{f(t_{n})}{2\pi \sigma (t_{n})(t_{n})^{n}}\int_{-\infty}^{+\infty }e^{-\frac{x^2}{2}}
dx=\frac{f(t_{n})}{\sqrt{2\pi} \sigma (t_{n})(t_{n})^{n}}$$
Pour que la formule ci-dessus donne un équivalent sous une forme analytique simple, il faudrait pouvoir résoudre explicitement l’équation $$m(t_{n})=n.$$ Quand ceci n’est pas possible, la stratégie consiste alors à utiliser un équivalent de la fonction $t\mapsto
m(t)$ lorsque $t\rightarrow 1$.
Soient $m_1$ et $\sigma_1$ des équivalents de $m$ et $\sigma$ respectivement lorsque $t\rightarrow 1$ : $$\begin{aligned}
m(t) &\thicksim &m_{1}(t) \\
\sigma (t) &\thicksim &\sigma _{1}(t)\end{aligned}$$
Soit alors $(\tau_{n})$ une suite dans $]0,1[$ tendant vers $1$ et telle que pour tout $n$ $$m_{1}(\tau _{n})=n$$
Posons $Z_{t}^{1}=\displaystyle\frac{X_{t}-m_{1}(t)}{\sigma _{1}(t)}$. On a comme précédemment
$$a_{n}=\frac{f(\tau _{n})}{2\pi \sigma _{1}(\tau _{n})\tau _{n}^{n}}\int_{-\sigma _{1}(\tau _{n})\pi }^{\sigma _{1}(\tau _{n})\pi
}E(e^{ixZ_{\tau _{n}}^{1}})dx$$
Sous l’hypothèse $$\frac{m(\tau _{n})-m_{1}(\tau _{n})}{\sigma _{1}(\tau _{n})}
\rightarrow 0\text{ quand }\tau _{n}\rightarrow 1$$ et la condition de [*convergence forte*]{} énoncée par B' aez-Duarte dans [@LBD] :
$$\int_{-\sigma (\tau _{n})\pi }^{\sigma (\tau _{n})\pi }\left| \varphi
_{Z_{\tau _{n}}}(x)-e^{-x^{2}/2}\right| dx\rightarrow 0$$
on peut alors obtenir $$a_{n}\backsim \frac{f(\tau_{n})}{\sqrt{2\pi} \sigma_1 (\tau_{n})(\tau_{n})^{n}}$$
On va appliquer la méthode que l’on vient de décrire à la fonction $$f(z)=\sum q(n)z^{n}$$
où $q(n)$ est le nombre de partitions restreintes de $n$, c’est-à-dire le nombre des décompositions $$n=n_1+\cdots+n_p$$ en entiers strictement positifs [*différents les uns des autres*]{} afin d’obtenir la formule asymptotique des partitions restreintes : $$q(n)\thicksim\frac{1}{4}\frac{e^{\frac{\pi \sqrt{n}}{\sqrt{3}}}}{3^{1/4}n^{3/4}}$$ ([*voir*]{} par exemple Erd" os [@Erdos] ou Ingham [@Ingham]).
Variable associée à un produit infini
=====================================
**Mesure associée à un produit**
---------------------------------
**Soient $f_{1}$ et $f_{2}$ deux fonctions dans ${{\cal O}}_{+}(D(0,1))$, alors le produit $f_{1}f_{2}$ est dans ${{\cal O}}_{+}(D(0,1))$ et $$\mu _{t}(f_{1}f_{2})=\mu _{t}(f_{1})\ast \mu _{t}(f_{2})$$**
Plus généralement, soient $f_{1}, f_{2},\ldots$ des fonctions dans ${{\cal O}}_{+}(D(0,1))$, alors le produit $f_{1}...f_{n}$ est dans ${{\cal O}}_{+}(D(0,1))$ : $$\mu _{t}(f_{1}...f_{n})=\mu _{t}(f_{1})\ast\cdots\ast \mu _{t}(f_{n})$$
[**Démonstration**]{}
Soient $f_{1}(t)=\sum_{n\geq 0}a_{n}t^{n}$ et $f_{2}(t)=\sum_{n\geq
0}b_{n}t^{n},$ on a $$f_{1}(t)f_{2}(t)=\sum_{n\geq 0}a_{n}t^{n}\sum_{n\geq
0}b_{n}t^{n}=\sum_{n\geq 0}\sum_{k+l=n}a_{k}b_{l}t^{n}$$ donc $$\mu _{t}(f_{1}f_{2})=\sum_{n\geq 0}\frac{\sum_{k+l=n}a_{k}b_{l}t^{n}}
{f_{1}(t)f_{2}(t)}\delta _{n}=\sum_{n\geq 0}\sum_{k+l=n}
\frac{a_{k}t^{k}b_{l}t^{l}}{f_{1}(t)f_{2}(t)}\delta _{k}\ast \delta _{l}
=\mu_{t}(f_{1})\ast \mu _{t}(f_{2})$$
Par récurrence on a $\mu _{t}(f_{1}...f_{n})=\mu _{t}(f_{1})\ast\cdots\ast
\mu _{t}(f_{n}).$
$\square $
**Soit $(f_{n})$ une suite de fonctions dans ${{\cal O}}_{+}(D(0,1))$ telle que le produit infini $\prod_{k\geq 1}^{+\infty }f_{k}$ converge uniformément sur tout compact de $D(0,1)$. Alors la fonction $f=\prod_{k\geq 1}^{+\infty }f_{k}$ est dans ${{\cal O}}_{+}(D(0,1))$ et la suite des mesures $\mu _{t}(f_{1}...f_{n})=\mu _{t}(f_{1})\ast\cdots\ast \mu _{t}(f_{n})$ converge en loi vers la mesure $\mu
_{t}(f)$ lorsque $n\rightarrow +\infty $.**
[**Démonstration.**]{} Comme le produit infini $\prod_{k\geq 1}^{+\infty }f_{k}$ converge uniformément sur tout compact de $D(0,1)$ on en déduit que la fonction $f=\prod_{k\geq 1}^{+\infty }f_{k}$ est analytique dans $D(0,1)$.
En outre on a $$f(z)=\prod_{k\geq 1}^{+\infty }f_{k}(z)=\prod_{k\geq 1}^{+\infty}\sum_{n\geq 0}a_{n,k}z^{n}
=\sum_{n\geq 0}z^{n}\!\!\!\sum_{n_{1}+...+n_{p}=n}a_{n_{1},1}...a_{n_{p},p}$$
donc $f(z)=\sum_{n\geq 0}a_{n}z^{n}$ avec $$a_{n}=\sum_{n_{1}+...+n_{p}=n}a_{n_{1},1}...a_{n_{p},p}$$
ce qui prouve que $f\in\ {{\cal O}}_{+}(D(0,1)).$
D’autre part, la fonction caractéristique de $\mu _{t}(f_{1}...f_{n})=\mu
_{t}(f_{1})\ast\cdots\ast \mu _{t}(f_{n})$ est égale au produit des fonctions caractéristiques de chacune des lois $$\varphi _{\mu _{t}(f_{1})\ast\cdots\ast \mu _{t}(f_{n})}(x)=\frac{f_{1}(te^{ix})}{f_{1}(t)}\cdots\frac{f_{n}(te^{ix})}{f_{n}(t)}$$
Comme le produit $f_{1}(z)...f_{n}(z)$ tend vers $f(z)$ pour tout $z$ dans $D(0,1)$, on a pour tout $t\in\ ]0,1[$ $$\lim_{n\rightarrow +\infty }\frac{f_{1}(te^{ix})}{f_{1}(t)}\cdots\frac{f_{n}(te^{ix})}{f_{n}(t)}=\frac{f(te^{ix})}{f(t)}$$
La suite des fonctions caractéristiques des mesures $\mu
_{t}(f_{1}...f_{n})$ converge donc simplement vers la fonction caractéristique de la mesure $\mu _{t}(f)$ associée à $f.$
$\square $
**La série des variables aléatoires associées** {#VAssoc}
-----------------------------------------------
Soit $(f_{n})$ une suite de fonctions dans ${{\cal O}}_{+}(D(0,1))$ telle que le produit infini $f=\prod_{k\geq 1}^{+\infty }f_{k}$ converge uniformément sur tout compact de $D(0,1)$. Par un théorème classique, à la suite des mesures de probabilité $(\mu _{t}(f_{n}))$ on peut associer une probabilité sur l’espace produit $\Omega =]0,1[^{\mathbb{N}}$ et une suite de variables aléatoires $(X_{n,t})$ à valeurs dans $\mathbb{N}$ *indépendantes* telle que pour tout $n\geq 1$ la variable aléatoire $X_{n,t}$ ait pour loi $\mu _{t}(f_{n})$. On peut alors affirmer que pour tout $n\geq 1,$ la mesure $\mu
_{t}(f_{1}...f_{n})$ est la loi de la somme $X_{1,t}+...+X_{n,t}.$
La convergence de la suite de mesures $\mu _{t}(f_{1}...f_{n})$ se traduit donc par la convergence en loi de la série $\sum_{n\geq 1}X_{n,t}.$ La loi de $\sum_{n\geq 1}X_{n,t}$ n’est autre que $\mu _{t}(f).$
D’après le théorème de Kolmogorov, si la série $$\sum_{n\geq 1}\sigma ^{2}(X_{n,t})=\sum_{n\geq 1}Var(X_{n,t}-E(X_{n,t}))$$ est convergente, alors la série $\sum_{n\geq 1}(X_{n,t}-E(X_{n,t}))$ converge presque sûrement sur $\Omega$. Si l’on suppose en outre que la série $\sum_{n\geq 1}E(X_{n,t})$ converge alors la série $\sum_{n\geq 1}X_{n,t}$ converge presque sûrement.
Les $X_{n,t}$ étant positives et indépendantes, on en déduit par le théorème de Beppo-Levi que
$$Var\bigl(\sum_{n\geq 1}X_{n,t}\bigr)=\sum_{n\geq 1}Var(X_{n,t})$$
Sous ces hypothèses on peut alors donner la définition suivante :
**Définition.**
Soit $X_t$ la variable aléatoire définie comme la somme de la série $\sum_{n\geq 1}X_{n,t}$ dont la loi sera notée $\mu_t(f)$. On pose $$m(t)=E(X_t)=\sum_{n\geq 1}E(X_{n,t})\quad \hbox{et}\quad \sigma (t)=\Bigl(Var(X_{t})\Bigr)^{1/2}=
\Bigl(\sum_{n\geq 1}Var(X_{n,t})\Bigr)^{1/2}$$
Application à l’étude des coefficients
--------------------------------------
On se donne une suite de fonctions $f_{n}=\sum_{n\geq 0}a_{k,n}z^{k}$ dans ${{\cal O}}_{+}(D(0,1))$ telle que le produit infini $\prod_{n\geq 1}^{+\infty }f_{n}$ converge uniformément sur tout compact de $D(0,1)$. La fonction $f=\prod_{n\geq 1}^{+\infty }f_{n}$ est dans ${{\cal O}}_{+}(D(0,1))$ et l’on peut écrire $f(z)=\sum_{n\geq 0}a_{n}z^{n}$ avec $$a_{n}=\sum_{n_{1}+...+n_{p}=n}a_{n_{1},1}...a_{n_{p},p}$$
Si les séries $\sum_{n\geq 1}E(X_{n,t})$ et $\sum_{n\geq 1}\sigma^2(X_{n,t})$ sont convergentes, considérons la variable aléatoire centrée réduite $$Z_{t}=\frac{\sum_{n\geq 1}X_{n,t}-m(t)}{\sigma (t)}$$
Supposons que $Z_{t}$ converge en loi quand $t\rightarrow 1$ vers une variable aléatoire $Z$ de loi N(0,1) et soit $(t_{n})$ une suite tendant vers 1 telle que $$m(t_{n})=n\text{ pour tout }n.$$
On a alors $$\sum_{n_{1}+...+n_{p}=n}a_{n_{1},1}...a_{n_{p},p}=\frac{f(t_{n})}{2\pi
\sigma (t_{n})(t_{n})^{n}}\int_{-\pi \sigma (t_{n})}^{\pi \sigma
(t_{n})}\varphi _{Z_{t_{n}}}(x)dx$$
Si $
\sigma (t_{n})\rightarrow +\infty \text{ quand }t_{n}\rightarrow 1
$ il est plausible que $$\lim_{t_{n}\rightarrow 1}\int_{-\sigma (t_{n})\pi }^{\sigma (t_{n})\pi }\varphi_{t_{n}}(x)dx
= \int_{-\infty }^{+\infty }\lim \varphi_{t_{n}}(x)dx=\int_{-\infty }^{+\infty }e^{-x^{2}/2}dx=\sqrt{2\pi }$$
et on en déduirait ainsi *la formule asymptotique des coefficients* $$\label{formuleAsymptGeneral}
\sum_{n_{1}+...+n_{p}=n}a_{n_{1},1}...a_{n_{p},p}\thicksim
\frac{f(t_{n})}{\sqrt{2\pi }\sigma (t_{n})(t_{n})^{n}}.$$
[**Passage par des équivalents.**]{}
Nous détaillons ici la méthode de Luis Báez-Duarte [@LBD].
Afin d’établir la formule (\[formuleAsymptGeneral\]) ci-dessus avec $\sigma_1$ à la place de $\sigma$ (et $\tau_n$ à la place de $t_n$), il reste à montrer que $$\lim_{n\rightarrow +\infty }\int_{-\sigma _{1}(\tau _{n})\pi }^{\sigma
_{1}(\tau _{n})\pi }E(e^{ixZ_{\tau _{n}}^{1}})dx=\sqrt{2\pi }.$$
On remarque que l’on peut écrire $$Z_{t}^{1}=\frac{X_{t}-m_{1}(t)}{\sigma _{1}(t)}
= Z_{t}\frac{\sigma (t)}{\sigma _{1}(t)}+\varepsilon(t)$$ où $\varepsilon(t)=\frac{m(t)-m_{1}(t)}{\sigma _{1}(t)}.$ On a alors $$\begin{aligned}
\int_{-\sigma _{1}(\tau _{n})\pi }^{\sigma _{1}(\tau _{n})\pi
}E(e^{ixZ_{\tau _{n}}^{1}})dx &=&\int_{-\sigma _{1}(\tau _{n})\pi }^{\sigma_{1}(\tau _{n})\pi }
E(e^{ix\frac{\sigma (\tau _{n})}{\sigma _{1}(\tau _{n})}
Z_{\tau _{n}}})e^{ix\varepsilon (\tau _{n})}dx \\
&=&\frac{\sigma _{1}(\tau _{n})}{\sigma (\tau _{n})}
\int_{-\sigma (\tau_{n})\pi }^{\sigma (\tau _{n})\pi }\varphi _{Z_{\tau _{n}}}(x)
e^{ix\frac{\sigma _{1}(\tau _{n})}{\sigma (\tau _{n})}\varepsilon (\tau _{n})}dx\end{aligned}$$
Pour justifier le remplacement par des équivalents on énonce deux hypothèses:
[**Hypothèse 1.**]{} Supposons que $$\frac{m(\tau _{n})-m_{1}(\tau _{n})}{\sigma _{1}(\tau _{n})}=\varepsilon
(\tau _{n})\rightarrow 0\text{ quand }\tau _{n}\rightarrow 1$$
[**Hypothèse 2.**]{} Supposons que $$\int_{-\sigma (\tau _{n})\pi }^{\sigma (\tau _{n})\pi }\left| \varphi
_{Z_{\tau _{n}}}(x)-e^{-x^{2}/2}\right| dx\rightarrow 0$$
Sous ces deux hypothèses il est facile de montrer que la suite $\displaystyle\int_{-\sigma (\tau _{n})\pi }^{\sigma (\tau _{n})\pi
}\varphi _{Z_{\tau _{n}}}(x)e^{ix\frac{\sigma _{1}(\tau _{n})}{\sigma (\tau
_{n})}\varepsilon (\tau _{n})}dx$ converge vers $\sqrt{2\pi }$.
En effet, on a
$$\begin{aligned}
&&\left| \int_{-\sigma (\tau _{n})\pi }^{\sigma (\tau _{n})\pi }\varphi_{Z_{\tau _{n}}}(x)
e^{ix\frac{\sigma _{1}(\tau _{n})}{\sigma (\tau _{n})}
\varepsilon (\tau _{n})}dx
-\int_{-\sigma (\tau _{n})\pi }^{\sigma (\tau_{n})\pi }
e^{ix\frac{\sigma _{1}(\tau _{n})}{\sigma (\tau _{n})}\varepsilon
(\tau _{n})}e^{-x^{2}/2}dx\right| \\
&\leq &\int_{-\sigma (\tau _{n})\pi }^{\sigma (\tau _{n})\pi }\left|
\varphi_{Z_{\tau _{n}}}(x)-e^{-x^{2}/2}\right| dx\end{aligned}$$
il suffit donc de montrer que
$$\lim_{\tau_n\rightarrow 1}\int_{-\sigma (\tau _{n})\pi }^{\sigma (\tau _{n})\pi }
e^{ix\frac{\sigma_{1}(\tau _{n})}{\sigma (\tau _{n})}\varepsilon (\tau_{n})}e^{-x^{2}/2}dx = \sqrt{2\pi }.$$
Ceci résulte du théorème de la convergence dominée, car on a $$\lim_{\tau_n\rightarrow 1}e^{ix\frac{\sigma
_{1}(\tau _{n})}{\sigma (\tau _{n})}\varepsilon (\tau
_{n})}e^{-x^{2}/2} = e^{-x^{2}/2}$$ et $$\left| e^{ix\frac{\sigma
_{1}(\tau _{n})}{\sigma (\tau _{n})}\varepsilon (\tau
_{n})}e^{-x^{2}/2}\right| \leq e^{-x^{2}/2}$$
Résumons ce qui précède dans le théorème suivant:
\[thmEquiv\] ****
(Théorème des équivalents) Soit $(f_{n}=\sum_{n\geq 0}a_{k,n}z^{k})$ une suite de fonctions dans ${{\cal O}}_{+}(D(0,1))$ telle que le produit infini $\prod_{n\geq 1}^{+\infty }f_{n}$ converge uniformément sur tout compact de $D(0,1)$. La fonction $f=\prod_{n\geq 1}^{+\infty }f_{n}$ est donc analytique dans $D(0,1)$ et $f(z)=\sum_{n\geq 0}a_{n}z^{n}$ avec $$a_{n}=\sum_{n_{1}+...+n_{p}=n}a_{n_{1},1}...a_{n_{p},p}$$
Si les séries $\sum_{n\geq 1}E(X_{n,t})$ et $\sum_{n\geq 1}\sigma^2(X_{n,t})$ sont convergentes, considérons la variable aléatoire centrée réduite $$Z_{t}=\frac{\sum_{n\geq 1}X_{n,t}-m(t)}{\sigma (t)}$$
Supposons que $Z_{t}$ converge en loi quand $t\rightarrow 1$ vers une variable aléatoire $Z$ de loi N(0,1) avec *la condition de convergence forte* ([*voir*]{} [@LBD]) : $$\label{condConvForte}
\lim_{t\rightarrow 1}\int_{-\sigma (t)\pi }^{\sigma (t)\pi }\left| \varphi
_{Z_{t}}(x)-e^{-x^{2}/2}\right| dx=0$$
Soient $m_1$ et $\sigma_1$ des équivalents de $m$ et $\sigma$ respectivement lorsque $t\rightarrow 1$ : $$\begin{aligned}
m(t) &\thicksim &m_{1}(t) \\
\sigma (t) &\thicksim &\sigma _{1}(t)\end{aligned}$$
Soit une suite $(\tau _{n})$ dans $]0,1[$ convergeant vers 1 et telle que pour tout $n$ on ait : $$m_{1}(\tau _{n})=n$$
avec $$\frac{m(\tau _{n})-m_{1}(\tau _{n})}{\sigma _{1}(\tau _{n})}\rightarrow 0\text{ quand }\tau _{n}\rightarrow 1.$$
Alors
$$\label{formuleAsymptGeneralTau}
a_{n}\thicksim \frac{f(\tau _{n})}{\sqrt{2\pi }\sigma _{1}(\tau _{n})\tau_{n}^{n}}$$
Un théorème de convergence
==========================
Nous énonçons et démontrons un théorème du type Liapounov ([*voir*]{} Chung [@Chung] p. 205 sq.) de convergence vers une loi normale concernant une famille continue de suites infinies de variables aléatoires.
\[thmConv\] **(Théorème de convergence) Soit une suite de variables aléatoires positives $(X_{n,t})_n$ dans $L^3(\Omega)$ telle que pour tout $t\in\ ]0,1[$ :**
[a)]{} les $X_{n,t}$ sont indépendantes
[b)]{} les séries $m(t) = \sum_{n\geq 1}E(X_{n,t})$, $\sigma^2(t) = \sum_{n\geq 1}Var(X_{n,t})$ et $\Gamma_{3}(t)=\sum_{n\geq 1}E(\left| X_{n,t}-E(X_{n,t})\right| ^{3})$ sont convergentes
[c)]{} la fonction $t\mapsto\displaystyle \frac{\Gamma_{3}(t)}{(\sigma (t))^{3}}$ tend vers $0$ quand $t\rightarrow 1$
[d)]{} ${\boldsymbol\lim_{t\rightarrow 1}}{\boldsymbol\sup}_{n\geq 1}\displaystyle\frac{Var(X_{n,t})}{\sigma ^{2}(t)}=0$
Alors la série $X_t=\sum_{n\geq 1}X_{n,t}$ est convergente presque sžrement et la fonction caractéristique $\varphi _{Z_{t}}$ de la variable aléatoire $$Z_{t}=\frac{X_t-m(t)}{\sigma (t)}$$
est telle que $$\varphi _{Z_{t}}(x)\rightarrow e^{-x^{2}/2}\quad\hbox{quand}\quad
t\rightarrow 1.$$
[**Démonstration.**]{}
Posons $Y_{n,t}=X_{n,t}-E(X_{n,t})$ on a $\displaystyle Z_{t}=\frac{\sum_{n\geq 1}Y_{n,t}}{\sigma (t)}.$
Par l’indépendance des $Y_{n,t}$ on voit que la variable aléatoire $Z_{t}$ a pour fonction caractéristique $$\varphi _{Z_{t}}(\theta )=E(e^{i\theta \sum \frac{Y_{n,t}}{\sigma (t)}})
= \prod_{n\geq 1}E(e^{i\theta \frac{Y_{n,t}}{\sigma (t)}})$$
On a $E(Y_{n,t})=0$ et $E(Y_{n,t}^{2})=\sigma _{n,t}^{2}=Var(X_{n,t}).$
**Sous les hypothèses du théorème (\[thmConv\]) ci-dessus, on a $$E\bigl(e^{i\theta \frac{Y_{n,t}}{\sigma (t)}}\bigr)=1-\frac{\theta ^{2}}{2}
\bigl(\frac{\sigma _{n,t}}{\sigma (t)}\bigr)^{2}+L_{n}(\theta ,t)$$**
avec $$\left| L_{n}(\theta ,t)\right| \leq \frac{\left| \theta \right| ^{3}}{6(\sigma (t))^{3}}E(\left| Y_{n,t}\right| ^{3}).$$
Ce lemme résulte de la formule de Taylor $$e^{ix}=1+ix-\frac{x^{2}}{2}-i\int_{0}^{1}\frac{(1-u)^{2}}{2}x^{3}e^{iux}du$$
qui nous donne$$e^{i\theta \frac{Y_{n,t}}{\sigma (t)}}=1+i\theta \frac{Y_{n,t}}{\sigma (t)}-
\frac{\theta ^{2}(\frac{Y_{n,t}}{\sigma (t)})^{2}}{2}-i\int_{0}^{1}\frac{(1-u)^{2}}{2}\theta ^{3}(\frac{Y_{n,t}}{\sigma (t)})^{3}
e^{iu\theta \frac{Y_{n,t}}{\sigma (t)}}du$$
Comme $E(Y_{n,t})=0$ on en déduit que $$\begin{aligned}
E\Bigl(e^{i\theta \frac{Y_{n,t}}{\sigma (t)}}\Bigr) &=&1-\frac{\theta ^{2}
(\frac{\sigma _{n,t}}{\sigma (t)})^{2}}{2}-i\int_{0}^{1}\frac{(1-u)^{2}}{2}\theta
^{3}E\Bigl(\bigl(\frac{Y_{n,t}}{\sigma (t)}
\bigr)^{3}e^{iu\theta \frac{Y_{n,t}}{\sigma (t)}}\Bigr)du \\
&=&1-\frac{\theta ^{2}}{2}(\frac{\sigma _{n,t}}{\sigma (t)})^{2}+L_{n}(\theta ,t)\end{aligned}$$
où $$L_{n}(\theta ,t)=-i\int_{0}^{1}\frac{(1-u)^{2}}{2}\theta ^{3}
E\Bigl(\bigl(\frac{Y_{n,t}}{\sigma (t)}\bigr)^{3}e^{iu\theta \frac{Y_{n,t}}{\sigma (t)}}\Bigr)du$$
On a ainsi la majoration $$\left| L_{n}(\theta ,t)\right| \leq |\theta ^{3}|E\Bigl(\bigl(\frac{|Y_{n,t}|}{\sigma (t)}
\bigr)^{3}\Bigr)\int_{0}^{1}\frac{(1-u)^{2}}{2}\, du\leq \frac{\left| \theta \right| ^{3}}{6(\sigma (t))^{3}}E(\left| Y_{n,t}\right| ^{3})$$
Ce qui termine la démonstration du lemme.
$\square $
Par l’indépendance des $Y_{n,t}$ la fonction caractéristique $\varphi _{Z_{t}}$ de la variable aléatoire $Z_{t}$ peut s’écrire $$\label{fonctCaract}
\varphi _{Z_{t}}(\theta )=\prod_{n\geq 1}\bigl(1-\frac{\theta ^{2}}{2}
\bigl(\frac{\sigma _{n,t}}{\sigma (t)}\bigr)^{2}+L_{n}(\theta ,t)\bigr)$$
Pour montrer que $\varphi _{Z_{t}}(\theta )\rightarrow e^{-\frac{\theta ^{2}}{2}}$ quand $t\rightarrow 1$ nous allons utiliser le lemme suivant (dont nous donnons la démonstration dans l’Appendice ([*voir*]{} section 5)) :
\[lemmeConv\] **Soit $(u_{n,t})_{n\geq 1}$ une famille de suites complexes indexées par $t\in\ ]0,1[$ telle que**
[($\imath$)]{} $\boldsymbol\sup_{n\geq 1}|u_{n,t}|\rightarrow 0$ quand $t\rightarrow 1.$
[($\imath\imath$)]{} il existe $M>0$ et $0<\alpha <1$ tel que $\sum_{n\geq 1}|u_{n,t}|\leq M$ pour tout $t\in\ ]\alpha ,1[.$
[($\imath\imath\imath$)]{} il existe $S\in\ \mathbb{C}$ tel que $\sum_{n\geq 1}u_{n,t}\rightarrow S
$ quand $t\rightarrow 1.$
Alors $$\lim_{t\rightarrow 1}\prod_{n\geq 1}(1+u_{n,t}) = e^{S}$$
On va appliquer ce lemme à la fonction caractéristique (\[fonctCaract\]) en posant $$u_{n,t}(\theta )=-\frac{\theta ^{2}}{2}(\frac{\sigma _{n,t}}{\sigma (t)}
)^{2}+L_{n}(\theta ,t)$$
Vérifions les trois conditions du lemme:
($\imath$) On a $$\begin{aligned}
\sup_{n\geq 1}|u_{n,t}(\theta )| &\leq &\frac{\theta ^{2}}{2}\sup_{n\geq 1}
(\frac{\sigma _{n,t}}{\sigma (t)})^{2}+\sup_{n\geq 1}L_{n}(\theta ,t) \\
&\leq &\frac{\theta ^{2}}{2}\sup_{n\geq 1}(\frac{\sigma _{n,t}}{\sigma (t)})^{2}+
\frac{\left| \theta \right| ^{3}}{6(\sigma (t))^{3}}\sum_{n\geq 1}E(\left| Y_{n,t}\right| ^{3})\end{aligned}$$
D’après les hypothèses c) et d) du théorème cette dernière quantité tend vers $0$ quand $t\rightarrow 1.$
($\imath\imath$) On a $$\begin{aligned}
\sum_{n\geq 1}|u_{n,t}(\theta )| &\leq &\frac{\theta ^{2}}{2}\sum_{n\geq 1}
(\frac{\sigma _{n,t}}{\sigma (t)})^{2}+\sum_{n\geq 1}L_{n}(\theta ,t) \\
&\leq &\frac{\theta ^{2}}{2}+\frac{\left| \theta \right| ^{3}}{6(\sigma(t))^{3}}\sum_{n\geq 1}E(\left| Y_{n,t}\right| ^{3})\end{aligned}$$
Or d’après c) la quantité $\frac{1}{(\sigma (t))^{3}}\sum_{n\geq
1}E(\left| Y_{n,t}\right| ^{3})$ est bornée au voisinage de 1.
($\imath\imath\imath$) On a $$\sum_{n\geq 1}u_{n,t}(\theta )=-\frac{\theta ^{2}}{2}\sum_{n\geq 1}
(\frac{\sigma _{n,t}}{\sigma (t)})^{2}+\sum_{n\geq 1}L_{n}(\theta ,t)=\frac{\theta
^{2}}{2}+\sum_{n\geq 1}L_{n}(\theta ,t)$$ et $\sum_{n\geq 1}L_{n}(\theta ,t)\rightarrow 0$ quand $t\rightarrow 1$ par c).
On a donc $\sum_{n\geq 1}u_{n,t}(\theta )\rightarrow -\frac{\theta ^{2}}{2}$ quand $t\rightarrow 1$ et par le lemme (\[lemmeConv\]) $$\varphi _{Z_{t}}(\theta )=\prod_{n\geq 1}\bigl(1-\frac{\theta ^{2}}{2}
\bigl(\frac{\sigma _{n,t}}{\sigma (t)}\bigr)^{2}+L_{n}(\theta ,t)\bigr)\rightarrow e^{-\frac{\theta ^{2}}{2}}$$
$\square $
Application aux partitions restreintes {#partRestr}
======================================
Considérons la fonction définie par le produit infini $$f(z)=\prod_{n\geq 1}^{+\infty }(1+z^{n})$$ Cette fonction est analytique dans height 0pt depth 8pt width 0pt $D(0,1)$ car la série $\sum_{n=1}^{+\infty }z^{n}$ converge uniformément sur tout compact de $D(0,1)$. On a $$f(z)=\sum q(n)z^{n}$$
où $q(n)$ est le nombre de partitions restreintes de $n,$ c’est-à-dire le nombre des décompositions $n=n_1+\cdots+n_p $ en entiers strictement positifs [*différents les uns des autres.*]{}
Le but de ce qui suit est d’appliquer la méthode décrite au début de cet article pour obtenir la formule asymptotique des partitions restreintes : $$q(n)\thicksim\frac{1}{4}\frac{e^{\frac{\pi \sqrt{n}}{\sqrt{3}}}}{3^{1/4}n^{3/4}}$$
Soit la mesure de probabilité associée à $f_{n}(t)=1+t^{n}$ $$\mu _{t}(f_{n})=\frac{1}{1+t^{n}}\delta _{0}+\frac{t^{n}}{1+t^{n}}\delta _{n}$$ où $\delta _0$ et $\delta _n$ représentent les mesures de Dirac en $0$ et $n$ respectivement.
On associe à ces mesures une suite de variables aléatoires indépendantes $(X_{n,t})$ ([*voir*]{} section (\[VAssoc\])). La variable $X_{n,t}$ prend les valeurs $0$ et $n$ et on a $$\begin{aligned}
E(X_{n,t}) &=&\frac{nt^{n}}{1+t^{n}} \\
Var(X_{n,t}) &=&\frac{n^{2}t^{n}}{\left( 1+t^{n}\right) ^{2}}\end{aligned}$$
**Dans ce qui suit on posera** $$t=e^{-r }$$ **où** $r >0,$ **de sorte que l’on a** $$t\rightarrow 1\Leftrightarrow r \rightarrow 0$$
Vérification des hypothèses du théorème de convergence
------------------------------------------------------
Les séries $$m(t)= \sum_{n\geq 1}E(X_{n,t}) = \sum_{n=1}^{+\infty }\frac{nt^{n}}{1+t^{n}}\quad\hbox{et}\quad
\sigma^{2}(t) = \sum_{n\geq 1}\sigma^2(X_{n,t}) = \sum_{k\geq 1}\frac{n^{2}t^{n}}{\left( 1+t^{n}\right) ^{2}}$$ sont clairement convergentes.
Examinons la série $\sum_{n\geq 1}E(\left| X_{n,t}-E(X_{n,t})\right| ^{3})$ : on a $$\begin{aligned}
E(\left| X_{n,t}-E(X_{n,t})\right| ^{3}) &=&(\frac{nt^{n}}{1+t^{n}})^{3}
\frac{1}{1+t^{n}}+(n-\frac{nt^{n}}{1+t^{n}})^{3}\frac{t^{n}}{1+t^{n}} \\
&=&n^{3}\frac{t^{3n}+t^{n}}{\left( 1+t^{n}\right) ^{4}}\end{aligned}$$
donc la série $\sum_{n\geq 1}E(\left| X_{n,t}-E(X_{n,t})\right| ^{3})$ est convergente.
Ainsi les hypothèses a) et b) du théorème de convergence (\[thmConv\]) sont bien vérifiées.
### Comportement asymptotique de $m$ et $\sigma^2$
Pour déterminer le comportement asymptotique quand $t\rightarrow 1$ des fonctions $m(t)$ et $\sigma^2(t)$ on va utiliser la formule d’Euler-McLaurin rappelée ci-dessous :
si $f\in\ C^{1}[0,+\infty \lbrack $ on a pour tout entier $n\geq 1$ $$\sum_{k=1}^{n}f(k)=\int_{1}^{n}f(x)dx+\frac{1}{2}(f(1)+f(n))+
\int_{1}^{n}b_{1}(x)f^{\prime }(x)dx$$ où $b_{1}(x)=x-[x]-\frac{1}{2}$. Si en outre $\sum_{k=1}^{+\infty }f(k)$ et $\int_{1}^{+\infty }f(x)dx$ sont convergentes alors $$\sum_{k=1}^{+\infty }f(k)=\int_{0}^{+\infty }f(x)dx+\int_{1}^{+\infty }b_{1}(x)f^{\prime }(x)dx+C$$ où $C=\frac12f(1)-\int_{0}^{1}f(x)dx.$
Les fonctions $f$ auxquelles on va appliquer cette formule seront du type $$f(x)=\frac{x^{p}e^{-arx}}{(1+e^{-rx})^{q}}$$ où $a,p,q$ sont des entiers supérieurs ou égaux à 1. Comme $ f(x)=\frac{1}{r^p}g(rx)$ où $g(u)=\frac{u^{p}e^{-au}}{(1+e^{-u})^{q}}$, on a $$\int_{0}^{+\infty }f(x)dx=\frac{1}{r^{p+1}}\int_{0}^{+\infty }g(u)du$$ et $$\Bigl|\int_{1}^{+\infty }b_{1}(x)f^{\prime }(x)dx\Bigr| =
\frac{1}{r^{p}}\Bigl|\int_{1}^{+\infty }b_{1}(\frac ur)g^{\prime }(u)du\Bigr|\leq \frac{1}{2r^{p}} \int_{1}^{+\infty }\vert g^{\prime }(u)\vert du$$ car la fonction $g^{\prime }$ est intégrable.
La formule d’Euler-MacLaurin nous donne pour $r\rightarrow 0+$ $$\sum_{k=1}^{+\infty}\frac{k^{p}e^{-akr}}{(1+e^{-kr})^{q}}=\frac{1}{r^{p+1}}\int_{0}^{+\infty }
\frac{u^{p}e^{-au}}{(1+e^{-u})^{q}}+O(\frac{1}{r^p}).$$
Pour $a=p=q=1$ on obtient $$\begin{aligned}
m(e^{-r })=\sum_{k=1}^{+\infty }\frac{ke^{-r k}}{1+e^{-r k}} =\frac{1}{r^2}\int_{0}^{+\infty
}\frac{ue^{-u}}{1+e^u}dx+O(\frac 1r)\end{aligned}$$
Notons que
$$\begin{aligned}
\frac{1}{r^2}\int_{0}^{+\infty
}\frac{ue^{-u}}{1+e^u}dx
=\frac{1}{r^2}\sum_{n=0}^{+\infty }(-1)^{n}\int_{0}^{+\infty }ue^{-u(n+1)}dx
=\frac{1}{r ^{2}}\sum_{n=0}^{+\infty }(-1)^{n}\frac{1}{\left(
n+1\right) ^{2}}=\frac{1}{r ^{2}}\frac{\pi ^{2}}{12}\end{aligned}$$
On a ainsi $$m(e^{-r })=m_{1}(e^{-r })+O(\frac 1r)\quad\hbox{avec}\quad m_{1}(e^{-r })=\frac{\pi ^{2}}{12}\frac{1}{r ^{2}}$$
Et de la même manière, on a
$$\label{sigma2}
\sigma ^{2}(e^{-r })\thicksim \int_{1}^{+\infty }
\frac{x^{2}e^{-r x}}{(1+e^{-r x})^{2}}dx
\thicksim _{r \rightarrow 0}\frac{\pi ^{2}}{6}\frac{1}{r ^{3}}=\sigma _{1}^{2}(e^{-r })$$
car
$$\int_{0}^{+\infty }\frac{x^{2}e^{-r x}}{(1+e^{-r x})^{2}}dx
=\sum_{n=1}^{+\infty }(-1)^{n-1}n\int_{0}^{+\infty }x^{2}e^{-r xn}dx
=\frac{2}{r ^{3}}\sum_{n=1}^{+\infty }(-1)^{n-1}\frac{1}{n^{2}}$$
### Les conditions c) et d)
Calculons $\Gamma _{3}(t)=\sum_{n\geq 1}E(\left| X_{n,t}-E(X_{n,t})\right|^{3})$ : $$\sum_{n\geq 1}E(\left| X_{n,t}-E(X_{n,t})\right| ^{3})=\sum_{n\geq 1}n^{3}
\frac{e^{-3nr }+e^{-nr }}{\left( 1+e^{-nr }\right) ^{4}}\thicksim
\int_{0}^{+\infty }\frac{x^{3}(e^{-3r x}+e^{-r x})}{(1+e^{-r x})^{4}}dx=\frac{C}{r ^{4}}$$ et donc $$\frac{\Gamma _{3}(t)}{\sigma (t)^{3}}\thicksim \frac{\frac{C}{r ^{4}}}{(\frac{\pi ^{2}}{6}\frac{1}{r ^{3}})^{3/2}}=C_{3}r ^{1/2}$$
On a donc bien $\frac{\Gamma _{3}(t)}{\sigma (t)^{3}}\rightarrow 0$ quand $t\rightarrow 1.$
Il reste à voir que $\lim_{t\rightarrow 1}\sup_{n\geq 1}\displaystyle\frac{Var(X_{n,t})}{\sigma ^{2}(t)}=0$. On a
$$\frac{Var(X_{n,t})}{\sigma ^{2}(t)}=\frac{1}{\sigma ^{2}(t)}n^{2}
\frac{t^{n}}{\left( 1+t^{n}\right) ^{2}}\leq \frac{1}{\sigma ^{2}(t)}n^{2}t^{n}$$
Or on a $n^{2}e^{-nr }\leq \frac{4}{r ^{2}}e^{-2}$ pour tout $n$ et $\sigma ^{2}(e^{-r })\thicksim \frac{\pi ^{2}}{6}\frac{1}{r ^{3}}$ d’après (\[sigma2\]) donc $$\lim_{t\rightarrow 1}\sup_{n\geq 1}\frac{Var(X_{n,t})}{\sigma ^{2}(t)}=0$$
Ainsi les hypothèses du théorème de convergence (\[thmConv\]) sont bien vérifiées. Par conséquent la fonction caractéristique $\varphi _{Z_{t}}$ de la variable aléatoire $$Z_{t}=\frac{\sum_{n\geq 1}X_{n,t}-m(t)}{\sigma (t)}$$ converge vers $e^{-x^{2}/2}$ quand $t\rightarrow 1$.
Vérification de la condition de convergence forte
-------------------------------------------------
Pour obtenir une formule asymptotique du nombre de partitions restreintes $q(n)$ défini au début de ce paragraphe (\[partRestr\]), on doit vérifier les hypothèses du théorème des équivalents, en particulier la condition de convergence forte :
$$\lim_{t\rightarrow 1}\int_{-\pi \sigma (t)}^{\pi \sigma (t)}\left| \varphi _{Z_t}(\theta)-e^{-\theta ^{2}/2}\right| d\theta = 0$$
Pour cela on va décomposer l’intégrale précédente en $$\int_{\left| \theta \right| \leq \frac{C}{r ^{1/2}}}\left| \varphi
_{Z_t}(\theta )-e^{-\theta ^{2}/2}\right| d\theta +
\int_{\frac{C}{r ^{1/2}}\leq \left| \theta \right| \leq \pi \sigma
(t)}\left| \varphi _{Z_t}(\theta )-e^{-\theta ^{2}/2}\right| d\theta$$ et majorer $\left| \varphi _{Z_t}(\theta )\right|$ sur chacun des domaines d’intégration.
\[lemmeA\] **Si $\left| \theta \right| \leq \displaystyle \frac{1}{4\frac{\Gamma
_{3}(t)}{\sigma ^{3}(t)}}\thicksim \frac{1}{4C_{3}}r ^{-1/2}$ alors $\left| \varphi _{Z_t}(\theta )\right| \leq e^{-\theta ^{2}/3}$.**
[**Démonstration.**]{}
Posons $Y_{n,t}=X_{n,t}-E(X_{n,t})$ on a $Z_{t}=\frac{\sum_{n\geq 1}Y_{n,t}}{\sigma (t)}.$ On a $$\varphi _{Z_t}(\theta )=\prod_{n\geq 1}\varphi _{Y_{n,t}}\bigl(\frac{\theta }{\sigma (t)}\bigr)$$
Pour majorer $\left| \varphi _{Z_t}(\theta )\right| $ on va utiliser le lemme suivant dont la démonstration est reportée à la section (\[preuveLemCramer\]) :
\[lemmeCramer\] **(Lemme de Cramér) Soit $Z$ une variable aléatoire centrée telle que $E(\left| Z\right| ^{3})<+\infty $ et $\varphi_{Z}$ sa fonction caractéristique. On a $$\left| \varphi _{Z}(\xi)\right| ^{2}\leq e^{-\xi^{2}E(Z^{2})+\frac{4}{3}\left|
\xi\right| ^{3}E(\left| Z\right| ^{3})}$$ En particulier si $\left| \xi\right| \leq \frac{1}{2}\frac{E(Z^{2})}{E(\left|
Z\right| ^{3})}$ alors $\left| \varphi _{Z}(\xi)\right| ^{2}\leq e^{-\frac{\xi^{2}}{3}E(Z^{2})}$.**
En appliquant ce lemme on obtient ainsi
$$\left| \varphi _{Y_{n,t}}\Bigl(\frac{\theta }{\sigma (t)}\Bigr)\right| ^{2}\leq
\exp\Bigl({-\frac{\sigma _{n,t}^{2}}{\sigma ^{2}(t)}\theta ^{2}+\frac{4}{3}\frac{\left|
\theta \right| ^{3}E(\left| Y_{n,t}\right| ^{3})}{\sigma ^{3}(t)}}\Bigr)$$
donc
$$\left| \varphi _{Z_t}(\theta )\right| ^{2}\leq \prod_{n\geq 1}
\exp\Bigl({-\frac{\sigma _{n,t}^{2}}{\sigma ^{2}(t)}\theta ^{2}+\frac{4}{3}\frac{\left| \theta
\right| ^{3}E(\left| Y_{n,t}\right| ^{3})}{\sigma ^{3}(t)}}\Bigr)
= \exp\Bigl({-\theta^{2}\bigl(1-\frac{4}{3}\left| \theta \right| \frac{\Gamma _{3}(t)}{\sigma ^{3}(t)}\bigr)}\Bigr)$$
Pour conclure, si $\left| \theta \right| \leq
\displaystyle\frac{1}{4\frac{\Gamma _{3}(t)}{\sigma ^{3}(t)}}$ alors $1-\frac{4}{3}\left| \theta \right|
\frac{\Gamma _{3}(t)}{\sigma^{3}(t)}\geq 2/3$ et par conséquent $\left| \varphi _{Z_t}(\theta )\right| ^{2}\leq e^{-2\theta ^{2}/3}$.
$\square $
Comme $\pi \sigma (t)\thicksim \displaystyle \sqrt{\frac{1}{6}}\frac{\pi ^{2}}{r ^{3/2}}$ lorsque $t$ tend vers 1 il suffit maintenant d’obtenir une majoration de la fonction caractéristique sur le domaine $\displaystyle\frac{1}{4C_{3}}r ^{-1/2}\leq \left| \theta \right| \leq \sqrt{\frac{1}{6}}\frac{\pi ^{2}}{r ^{3/2}}.$
\[lemmeB\] **Soit $C$ une constante positive. Sous l’hypothèse $\frac{C}{r ^{1/2}}\leq \left| \theta \right| <\pi\sigma (t)$ il existe un réel positif $B$ tel que l’on ait $\left| \varphi _{Z_t}(\theta )\right| \leq e^{-B/r }$.**
[**Démonstration.**]{}
La méthode consiste à écrire
$\ln(\left| \varphi _{Z_t}(\theta )\right| )=\sum_{k\geq 1}\ln(\left|
1+t^{k}e^{ik\theta /\sigma (t)}\right| )-\ln(1+t^{k})$.
On développe $$\left| 1+t^{k}e^{ik\theta /\sigma (t)}\right| ^{2}=1+t^{2k}+2t^{k}\cos
(k\theta /\sigma (t))$$ et on écrit $$\begin{aligned}
\ln(\left| \varphi _{Z_t}(\theta )\right| ) &=&\frac{1}{2}\sum_{k\geq
1}\ln(1+t^{2k}+2t^{k}\cos (k\theta /\sigma (t)))-\ln(1+t^{2k}+2t^{k}) \\
&=&\frac{1}{2}\sum_{k\geq 1}\ln(1+\frac{2t^{k}(\cos (k\theta /\sigma (t))-1)}{1+t^{2k}+2t^{k}}) \\
&\leq &\frac{1}{2}\sum_{k\geq 1}\frac{2t^{k}(\cos (k\theta /\sigma (t))-1)}{1+t^{2k}+2t^{k}} \\
&\leq &\frac{1}{4}\sum_{k\geq 1}t^{k}(\cos (k\theta /\sigma (t))-1)\end{aligned}$$
Or on a $$\sum_{k\geq 1}t^{k}(\cos (k\theta /\sigma (t))= {Re}\bigl(\frac{te^{i\theta
/\sigma (t)}}{1-te^{i\theta /\sigma (t)}}\bigr)=\frac{t\cos (\theta /\sigma
(t))-t^{2}}{1-2t\cos (\theta /\sigma (t))+t^{2}}$$
et puisque $Cr \leq \left| \theta \right| /\sigma (t)<\pi $ alors $\cos (\theta
/\sigma (t))\leq \cos (Cr )$. Par conséquent $$\sum_{k\geq 1}t^{k}(\cos (k\theta /\sigma (t))\leq \frac{t\cos (Cr )-t^{2}}{1-2t\cos (Cr )+t^{2}}$$ donc $$\ln(\left| \varphi _{Z_t}(\theta )\right| )\leq \frac{1}{4}\Bigl(\frac{t\cos(Cr )-t^{2}}{1-2t\cos (Cr )+t^{2}}
-\frac{t}{1-t}\Bigr)\backsim \left( \frac{1}{1+C^{2}}-1\right) r ^{-1}$$
On en déduit l’existence d’une constante $B>0$ telle que l’on ait : $$\left| \varphi _{Z_t}(\theta )\right| \leq e^{-B/r }$$
$\square $
**On a $$\lim_{t\rightarrow 1}\int_{-\pi \sigma (t)}^{\pi \sigma (t)}\left| \varphi _{Z_t}(\theta
)-e^{-\theta ^{2}/2}\right| d\theta = 0$$**
[**Démonstration.**]{}
D’après le lemme (\[lemmeB\]) : $$\left| \varphi _{Z_t}(\theta )\right| \leq e^{-B/r }\text{ si }
\frac{C}{r ^{1/2}}\leq \left| \theta \right| \leq \pi \sigma (t),\text{ avec }B>0$$
et clairement $$e^{-\theta ^{2}/2}\leq e^{-C^{2}/2r }$$ sous les mêmes conditions. On a donc, avec $D=\min (B,C^{2}/2)$ : $$\begin{aligned}
\int_{\frac{C}{r ^{1/2}}\leq \left| \theta \right| \leq \pi \sigma
(t)}\left| \varphi _{Z_t}(\theta )-e^{-\theta ^{2}/2}\right| d\theta &\leq
&\int_{\frac{C}{r ^{1/2}}\leq \left| \theta \right| \leq \pi \sigma
(t)}\left| \varphi _{Z_t}(\theta )\right| d\theta +\int_{\frac{C}{r
^{1/2}}\leq \left| \theta \right| \leq \pi \sigma (t)}e^{-\theta
^{2}/2}d\theta \\
&\leq &e^{-D/r }\Bigl(\pi \sigma (t)-\frac{C}{r ^{1/2}}\Bigr)\end{aligned}$$
et cette dernière quantité tend vers 0 lorsque $t$ tend vers 1 ([*i.e.*]{} lorsque $r$ tend vers 0).
Il reste à voir que $$\lim_{t\rightarrow 1}\int_{\left| \theta \right| \leq \frac{C}{r ^{1/2}}}\left| \varphi
_{Z_t}(\theta )-e^{-\theta ^{2}/2}\right| d\theta =0$$
D’après le lemme (\[lemmeA\]), sur cet intervalle on a $\left| \varphi
_{Z_t}(\theta )\right| \leq e^{-\theta ^{2}/3}$ donc $$\left| \varphi _{Z_t}(\theta )-e^{-\theta ^{2}/2}\right| \leq e^{-\theta
^{2}/3}+e^{-\theta ^{2}/2}$$
et on peut conclure par le théorème de la convergence dominée.
$\square $
Application du théorème des équivalents
---------------------------------------
On a choisi comme équivalent de la fonction $m$ lorsque $t$ tend vers 1 la fonction $m_1$ définie par
$$m_1(e^{-r})=\frac{\pi^2}{12}\frac{1}{r^2}.$$
La définition de $\tau_n$ par l’égalité $m_1(\tau_n)=n$ se traduit, en posant $\tau_n=e^{-\rho_n}$, par
$$m_1(e^{-\rho_n })=n\quad\hbox{ ce qui donne }\quad \rho _{n}=\frac{1}{2\sqrt{3}\sqrt{n}}\pi$$
et par conséquent $$\tau _{n}=e^{-\frac{1}{2\sqrt{3}\sqrt{n}}\pi}$$
On a aussi choisi comme équivalent de la fonction $\sigma$ lorsque $t$ tend vers 1 la fonction $\sigma_1$ définie par
$$\sigma_1(e^{-r })=\sqrt{\frac{\pi ^{2}}{6}\frac{1}{r^3}}$$
ce qui donne $$\sigma _{1}^{2}(\tau _{n})=\sigma_1^2(e^{-\rho_n })=\frac{\pi ^{2}}{6}\frac{1}{\rho_n^3}=\frac{\pi ^{2}}{6}\frac{1}{(\frac{1}{2\sqrt{3}\sqrt{n}}\pi )^{3}}
=\frac{4}{\pi }\left( n\right) ^{3/2}\sqrt{3}.$$
La condition du théorème des équivalents : $$\frac{m(\tau _{n})-m_{1}(\tau _{n})}{\sigma _{1}(\tau _{n})}\rightarrow 0 \text { quand } \tau _{n} \rightarrow 1$$
est bien satisfaite car on a vu en 4.1.1. que $m(e^{-r })=m_{1}(e^{-r })+O(\frac 1r),$ ce qui permet d’écrire
$$\frac{m(\tau _{n})-m_{1}(\tau _{n})}{\sigma _{1}(\tau _{n})}=\frac{m(e^{- \rho_{n}})-m_{1}(e^{- \rho_{n}})}{\sigma _{1}(e^{- \rho_{n}})}
=\frac{O(\frac{1}{\rho_n})}{\sigma_1(e^{-\rho_n })}=\frac{O(n^\frac 12)}
{3^{\frac 14}\sqrt{\frac{4}{\pi }} \ n^{3/4}}
\rightarrow 0$$
Le théorème des équivalents (\[formuleAsymptGeneralTau\]) nous permet donc d’obtenir la formule asymptotique : $$a_{n}\thicksim \displaystyle\frac{f(\tau _{n})}{\sqrt{2\pi }\sigma _{1}(\tau _{n})\tau _{n}^{n}}$$
Il reste à donner un équivalent de $$f(\tau _{n})=\prod_{k=1}^{+\infty }(1+e^{-\frac{1}{2\sqrt{3}\sqrt{n}}\pi k})$$
Passons au logarithme $$\ln (f(\tau _{n}))=\sum_{k\geq 1}\ln (1+e^{-\frac{1}{2\sqrt{3}\sqrt{n}}\pi k})$$
\[lemmeApplicEquiv\] **On a pour $\rho \rightarrow 0^{+}$ $$\sum_{k=1}^{+\infty }\ln(1+e^{-\rho k})=\frac{\pi ^{2}}{12\rho }-\frac{1}{2}\ln 2+O\left( \rho \right)$$**
[**Démonstration.**]{}
Appliquons la formule d’Euler-McLaurin: $$\sum_{k\geq 1}\ln(1+e^{-\rho k})=\int_{1}^{+\infty }\ln (1+e^{-\rho
x})dx+\frac{1}{2}\ln (1+e^{-\rho })-\rho \int_{1}^{+\infty }b_{1}(x)\frac{e^{-\rho x}}{1+e^{-\rho x}}dx$$
[**a) Le terme $\int_{1}^{+\infty }\ln (1+e^{-\rho x})dx$ :**]{}
On décompose l’intégrale:
$$\begin{aligned}
\int_{1}^{+\infty }\ln (1+e^{-\rho x})dx &=&\int_{0}^{\infty }\ln
(1+e^{-\rho x})dx-\int_{0}^{1}\ln (1+e^{-\rho x})dx \\
&=&\int_{0}^{+\infty }\sum_{n\geq 1}\frac{(-1)^{n-1}}{n}e^{-\rho
nx}dx-\ln 2+O\left( \rho \right) \\
&=&\sum_{n\geq 1}\frac{(-1)^{n-1}}{\rho n^{2}}-\ln 2+O\left(\rho\right)\end{aligned}$$
Donc $$\int_{1}^{+\infty }\ln (1+e^{-\rho x})dx=-\ln 2+\frac{\pi ^{2}}{12\rho }+O\left(\rho \right)$$
[**b) Le terme $\frac{1}{2}\ln (1+e^{-\rho})$ :**]{}
$$\frac{1}{2}\ln (1+e^{-\rho })=\frac{1}{2}\ln 2+O\left( \rho \right)$$
[**c) Le troisième terme :**]{}
On a $$-\rho \int_{1}^{+\infty }b_{1}(x)\frac{e^{-\rho x}}{1+e^{-\rho x}}dx=O\left(\rho e^{-\rho }\right)$$
En effet la fonction $x\mapsto \displaystyle\frac{e^{-\rho x}}{1+e^{-\rho x}}$ est positive décroissante et elle tend vers 0 à l’infini. Comme la fonction $b_{1}$ est périodique, par le lemme d’Abel on obtient la majoration $$\left| \int_{1}^{+\infty }b_{1}(x)\frac{e^{-\rho x}}{1+e^{-\rho x}}dx\right|
\leq C\frac{e^{-\rho }}{1+e^{-\rho }}\leq Ce^{-\rho }$$
$\square $
**Conclusion**
D’après le lemme (\[lemmeApplicEquiv\]) avec $\rho=\rho_n$ on a $$\sum_{k\geq 1}\ln (1+e^{-\frac{1}{2\sqrt{3}\sqrt{n}}\pi k})
= \frac{\pi \sqrt{n}}{2\sqrt{3}}-\frac{1}{2}\ln 2+O\left( \frac{1}{\sqrt{n}}\right)$$
donc
$$f(\tau _{n})=\prod_{k=1}^{+\infty }(1+e^{-\frac{1}{2\sqrt{3}\sqrt{n}}\pi
k})\thicksim \frac{1}{\sqrt{2}}e^{\frac{\pi \sqrt{n}}{2\sqrt{3}}}$$
lorsque $n$ tend vers l’infini, ce qui donne la formule asymptotique des partitions restreintes : $$q(n)\thicksim \frac{1}{\sqrt{2}}e^{\frac{\pi \sqrt{n}}{2\sqrt{3}}}
\frac{1}{\sqrt{2\pi }\sqrt{\frac{4}{\pi }\left( n\right) ^{3/2}\sqrt{3}}
e^{-\frac{\sqrt{n}}{2\sqrt{3}}\pi }}=\frac{1}{4}\frac{e^{\frac{\pi \sqrt{n}}{\sqrt{3}}}}{3^{1/4}n^{3/4}}$$
Appendice
=========
Démonstration du Lemme 3.3.
---------------------------
Comme $\sup_{n\geq 1}|u_{n,t}|\rightarrow 0$ quand $t\rightarrow 1,$ il existe $a<1$ tel que pour $t\in\ ]a,1[$ on a $|u_{n,t}|<1/2$ pour tout $n\geq 1.$ Donc $\ln(1+u_{n,t})$ est bien défini pour $t\in\ ]a,1[$ et $$\ln(1+u_{n,t})=\sum_{k=1}^{+\infty }\frac{(-1)^{k-1}}{k}(u_{n,t})^{k}$$
ce qui donne $$\left| \ln(1+u_{n,t})-u_{n,t}\right| \leq \left| u_{n,t}\right|
^{2}\sum_{k=2}^{+\infty }\frac{1}{k}\left| u_{n,t}\right| ^{k-2}\leq \left|
u_{n,t}\right| ^{2}\sum_{k=2}^{+\infty }(\frac{1}{2})^{k-2}=2\left|
u_{n,t}\right| ^{2}$$
D’autre part la série $\sum_{n\geq 1}|u_{n,t}|$ est supposée convergente pour tout $t\in\ ]\alpha ,1[,$ donc la série $\sum_{n\geq
1}\left| u_{n,t}\right| ^{2}$ est convergente si $t\in\ ]\sup (a,\alpha ),1[.$ On en déduit que la série $\sum_{n\geq 1}\ln(1+u_{n,t})$ est convergente si $t\in\ ]\sup (a,\alpha ),1[$ et il en est donc de même du produit infini $\prod_{n\geq 1}(1+u_{n,t}).$
D’autre part, pour tout $N\geq 1$ on a $$\left| \sum_{n=1}^{N}\ln(1+u_{n,t})-\sum_{n=1}^{N}u_{n,t}\right| \leq
\sum_{n=1}^{N}\left| \ln(1+u_{n,t})-u_{n,t}\right| \leq
2\sum_{n=1}^{N}\left| u_{n,t}\right| ^{2}$$
Comme $$\sum_{n=1}^{N}\left| u_{n,t}\right| ^{2}\leq \sup_{n\geq
1}|u_{n,t}|\sum_{n=1}^{N}\left| u_{n,t}\right| \leq M\sup_{n\geq
1}|u_{n,t}|$$
on en déduit que $\sum_{n=1}^{+\infty }\left| u_{n,t}\right| ^{2}\leq M\sup_{n\geq 1}|u_{n,t}|$ et que $$\lim_{N\rightarrow +\infty }\left|
\sum_{n=1}^{N}\ln(1+u_{n,t})-\sum_{n=1}^{N}u_{n,t}\right| \leq 2M\sup_{n\geq 1}|u_{n,t}|$$
Donc $$\left| \sum_{n=1}^{+\infty }\ln(1+u_{n,t})-\sum_{n=1}^{+\infty
}u_{n,t}\right| \leq 2M\sup_{n\geq 1}|u_{n,t}|.$$
Pour conclure il suffit de prendre la limite quand $t\rightarrow 1$, on obtient $$\lim_{t\rightarrow 1}\sum_{n=1}^{+\infty }\ln(1+u_{n,t})=\lim_{t\rightarrow
1}\sum_{n=1}^{+\infty }u_{n,t}=S$$
En passant à l’exponentielle on obtient $$\lim_{t\rightarrow 1}\prod_{n=1}^{+\infty }(1+u_{n,t})=e^{S}$$
$\square $
Démonstration du Lemme de Cramér (Lemme \[lemmeCramer\]) {#preuveLemCramer}
--------------------------------------------------------
[**Démonstration.**]{} ([*voir*]{} Cramér [@Cramer], Chung [@Chung] p. 210)
Soit $Y$ une variable aléatoire indépendante de $Z$ et de même loi. On a $$\left| \varphi _{Z}(\xi)\right| ^{2}=\varphi _{Z}(\xi)\overline{\varphi _{Y}(\xi)}
= E(e^{i\xi Z})\overline{E(e^{i\xi Y})}=E(e^{i\xi (Z-Y)})$$
ce qui permet d’écrire $$\left| \varphi _{Z}(\xi)\right| ^{2}=\int_{\mathbb{R}^{2}}e^{i\xi (z-y)}dP_{Z}(z)dP_{Y}(y)
=\int_{\mathbb{R}^{2}}\cos(\xi (z-y))dP_{Z}(z)dP_{Y}(y)$$
En utilisant la majoration $$\cos (u)\leq 1-\frac{u^{2}}{2}+\frac{\left| u\right| ^{3}}{6}$$ que l’on peut obtenir à l’aide de la formule de Taylor d’ordre deux avec reste intégral, on en déduit que $$\left| \varphi _{Z}(\xi)\right| ^{2}\leq 1-\frac{\xi^{2}}{2}\int_{\mathbb{R}^{2}}(z-y)^{2}dP_{Z}(z)dP_{Y}(y)
+\frac{\left| \xi\right| ^{3}}{6}\int_{\mathbb{R}^{2}}\left| z-y\right| ^{3}dP_{Z}(z)dP_{Y}(y)$$
La première intégrale n’est autre que $2E(Z^{2}).$ Pour la deuxième on utilise la majoration $$\left| z-y\right| ^{3}\leq 4\left| z\right| ^{3}+4\left| y\right| ^{3}$$
ce qui permet de majorer l’intégrale par $8E(\left| Z\right| ^{3}).$
On obtient finalement $$\left| \varphi _{Z}(\xi)\right| ^{2}\leq 1-\xi^{2}E(Z^{2})+\frac{4}{3}\left|
\xi\right| ^{3}E(\left| Z\right| ^{3})\leq e^{-\xi^{2}E(Z^{2})+\frac{4}{3}\left|
\xi\right| ^{3}E(\left| Z\right| ^{3})}$$
Pour la seconde partie du lemme, si $\left| \xi\right| \leq \frac{1}{2}\frac{E(X^{2})}{E(\left| X\right| ^{3})}$ il suffit de remarquer que $$-\xi^{2}E(Z^{2})+\frac{4}{3}\left| \xi\right| ^{3}E(\left| Z\right|
^{3})=-\xi^{2}[E(Z^{2})-\frac{4}{3}\left| \xi\right| E(\left| Z\right|
^{3})]\leq -\xi^{2}\frac{1}{3}E(Z^{2})$$
$\square$
**Remerciements**
Nous remercions Mesdames Cécile Fouilhé et No" emie El Qotbi pour l’intérêt qu’elles ont porté à l’étude de l’article de Luis B' aez-Duarte.
[9]{}
B' aez-Duarte, L., [*“Hardy-Ramanujan’s Asymptotic Formula for Partitions and the Central Limit Theorem,”*]{} Advances in Mathematics 125, 114-120 (1997). Chung, K.L., [*“A Course in Probability Theory, 3rd ed.”*]{} Academic Press, San Diego \[CA\] (2001). Cramér, H., [*“Random Variables and Probability Distributions,”*]{} 2nd ed., Cambridge Univ. Press, Cambridge (1963). Erdšs, P., Lehner, J., [*“The Distribution of the Number of Summands in the Partitions of a Positive Integer,”*]{} Duke Mathematical Journal Vol. 8, No.2 (June, 1941) Hayman, W.K., [*“A Generalisation of Stirling’s Formula,”*]{} J. Reine Angew. Mat. 196, Nos. 1/2, 67-95 (1956). Ingham, A.E., [*“A Tauberian Theorem for Partitions,”*]{} The Annals of Mathematics, Second Series, Vol. 42, No. 5, 1075-1090 (Dec., 1941) Rosenbloom, P.C., [*“Probability and Entire Functions,”*]{} Studies in Mathematical Analysis and Related Topics, Vol. 45, 325-332, Stanford Univ. Press, Palo Alto, CA (1962).
|
---
abstract: |
Superconductivity has been extensively studied since its discovery in 1911 [@citeulike:7386247]. However, the feasibility of room-temperature superconductivity is unknown. It is very difficult for both theory and computational methods to predict the superconducting transition temperatures ($T_\text{c}$) of superconductors for strongly correlated systems, in which high-temperature superconductivity emerges. Exploration of new superconductors still relies on the experience and intuition of experts, and is largely a process of experimental trial and error. In one study, only 3% of the candidate materials showed superconductivity [@1468-6996-16-3-033503]. Here we report the first deep learning model for finding new superconductors. We represented the periodic table in a way that allows a deep learning model to learn it. Although we used only the chemical composition of materials as information, we obtained an $R^{2}$ value of 0.92 for predicting $T_\text{c}$ for materials in a database of superconductors. We obtained three remarkable results. The deep learning method can predict superconductivity for a material with a precision of 62%, which shows the usefulness of the model; it found the recently discovered superconductor , which is not in the superconductor database; and it found Fe-based high-temperature superconductors (discovered in 2008) from the training data before 2008. These results open the way for the discovery of new high-temperature superconductor families.\
\
PHYSICAL SCIENCES: Applied Physical Sciences\
Keywords: Deep Learning, Superconductors, Materials Search, Machine Learning
author:
- Tomohiko Konno
- Hodaka Kurokawa
- Fuyuki Nabeshima
- Yuki Sakishita
- Ryo Ogawa
- Iwao Hosako
- Atsutaka Maeda
bibliography:
- 'bib\_deep\_matter.bib'
title: Deep Learning Model for Finding New Superconductors
---
Extensive research has been conducted on superconductors with a high superconducting transition temperature, $T_\text{c}$, because of their many promising applications, such as low-loss power cables, powerful electromagnets, and fast digital circuits. However, finding new superconductors is very difficult. In one study, it was reported [@1468-6996-16-3-033503] that only 3% of candidate materials showed superconductivity. Theoretical approaches have been proposed for predicting new superconducting materials. According to Bardeen-Cooper-Schrieffer (BCS) theory [@PhysRev.106.162], which explains phonon-mediated superconductivity in many materials, high $T_\text{c}$ is expected for compounds made of light elements. $T_\text{c}$ values of over 200 K have been reported for sulfur hydride [@drozdov2015conventional] and lanthanum hydride [@PhysRevLett.122.027001]. However, very high pressures (over 150 GPa) are required. Superconductivity with a rather high $T_\text{c}$ has been observed for cuprates [@bednorz1986possible] and iron-based materials [@kamihara2008iron] at ambient pressure, where unconventional superconductivity beyond the BCS framework is realized. However, the strong electron correlations in these materials make it very difficult to conduct first-principles calculations [@kohn1965self; @jain2013commentary; @kirklin2015open; @curtarolo2012aflowlib] to calculate their electronic structures and predict their $T_\text{c}$ values. Therefore, new approaches for finding superconductors are needed.
Materials informatics, which applies the principles of informatics to materials science, has attracted much interest [@butler2018machine; @ramprasad2017machine]. Among machine learning methods, deep learning has achieved great progress. Deep learning has been used to classify images [@krizhevsky2012imagenet], generate images [@goodfellow2014generative], play Go [@silver2016mastering], translate languages [@Vaswani2017AttentionIA], perform natural language tasks [@Peters:2018], and make its own network architecture [@2018arXiv180203268P; @Liu2018DARTSDA]. To predict the material properties of materials using the conventional methods in materials informatics, researchers must design the input features of the materials; this is called feature engineering. It is very difficult for a human to design the appropriate features. A deep learning method can design and optimize features, giving it higher representation capabilities and potential compared to those of conventional methods.
Reading periodic table
----------------------
Here, we report a deep learning model for the exploration of new superconductors. Using deep learning to discover new superconductor families from known ones is analogous to using one to recognize dogs from training data containing only cats. This form of learning, called zero-shot learning, is very difficult. However, the properties of elements, which can be learned by deep learning, can be applied to materials. Our strategy is to suitably represent these properties, use this representation as training data, and have the deep learning model learn these properties. We made the deep learning model learn how to read the periodic table as human experts do. Although humans cannot recall tens of thousands of data points, computers can. For this purpose, we represented the periodic table in a way that allows a deep learning model to learn it, as illustrated in Fig. \[fig: peridoc-table-neural-networks\]. This method, named reading periodic table, is our first contribution to deep learning. We considered inorganic crystal superconductors because the number of known organic superconductors is small. We used only the composition of materials because the applied superconductor database does not have sufficient spatial information.
[0.88]{} ![Proposed method named reading the period table. Top: the representation of a material by the method. The composition ratios of the material [@vesta3] are entered into the two-dimensional periodic table. We then divide the original table into four tables corresponding to s-, p-, d-, and f-blocks, which show the orbital characteristics of the valence electrons, to allow the deep learning model to learn the valence orbital blocks. The dimensions of the representation are $4\times32 \times 7$. The neural network learns the rules from the periodic table by convolutional layers. Bottom: the representation by the method and neural network.[]{data-label="fig: peridoc-table-neural-networks"}](./figures/schematic_periodic_table_2.png "fig:"){width="100.00000%"}
\
[0.88]{} ![Proposed method named reading the period table. Top: the representation of a material by the method. The composition ratios of the material [@vesta3] are entered into the two-dimensional periodic table. We then divide the original table into four tables corresponding to s-, p-, d-, and f-blocks, which show the orbital characteristics of the valence electrons, to allow the deep learning model to learn the valence orbital blocks. The dimensions of the representation are $4\times32 \times 7$. The neural network learns the rules from the periodic table by convolutional layers. Bottom: the representation by the method and neural network.[]{data-label="fig: peridoc-table-neural-networks"}](./figures/neural-networks-2.pdf "fig:"){width="100.00000%"}
We used the deep learning model to predict the critical temperatures, $T_\text{c}$, of superconductors in the SuperCon dataset [@supercon], which has the $T_\text{c}$ values of about 13,000 superconductors. We refer to the model trained with only SuperCon as the preliminary model. The train-test split was 0.05. A scatter plot of the predicted and actual $T_\text{c}$ values is shown in Fig. \[fig: scatter\_supercon\]. The $R^2$ value is 0.92, which is higher than that previously reported (0.88) for a random forest regression model [@stanev2018machine], where materials were restricted to those with $T_\text{c}> 10$ K (half of all materials). In contrast, our preliminary model does not have any restrictions regarding $T_\text{c}$ (see Supplementary Information). The random forest regression model requires many appropriate input features of the materials (e.g., atomic mass, band gap, atomic configuration, melting temperature) to be manually designed. Here, even without such feature engineering, we achieved much better results.
![Scatter plot of predicted and true (SuperCon) $T_\text{c}$ values.[]{data-label="fig: scatter_supercon"}](./figures/Scatter_supercon_3_within_1000_005.png){width=".7\hsize"}
The problem in using data of superconductors only and the method named garbage-in for overcoming it
---------------------------------------------------------------------------------------------------
We used the preliminary model trained with SuperCon to predict the $T_\text{c}$ values of 48,000 inorganic materials in the Crystallography Open Database (COD) to find new superconductors for experiments. However, for about 17,000 of the materials, the predicted $T_\text{c}$ was > 10 K, which is unreasonable. The failure to find new superconductors by this preliminary model seems to originate from the fact that the training data (SuperCon) included only 60 non-superconductors; the preliminary model was thus unable to learn non-superconductors. Data on non-superconductors are needed to differentiate superconductors from non-superconductors. However, no such dataset is available. Hence, we created synthetic data on non-superconductors, supposing that the $T_\text{c}$ values of the inorganic materials in COD that are not in SuperCon are 0 K under the assumption that most of these materials do not become superconductors with finite $T_\text{c}$. We used the synthetic data and SuperCon as the training data. We refer to this data generation method as garbage-in, which is our second contribution to deep learning.
As demonstrated by the above results for the preliminary model, scores of tests using only superconductor data, SuperCon, are not good for evaluating models. Usually, density functional theory is applied for evaluation in materials informatics; however, density functional theory cannot be used to evaluate models, because it is very difficult to calculate $T_\text{c}$ for strongly correlated systems. A database of non-superconductors is thus necessary.
Precision Recall f1 score
-------------------------- ----------- -------- ---------- --
Baseline (0 K) 32% – –
Our DL model Reg (0 K) 62% 67% 63%
Our DL model Cls (0 K) 72% 50% 59%
Random Forest Cls (0 K) 71% 27% 39%
Baseline (10 K) 10% – –
Our DL model Reg (10 K) 75% 76% 75%
Our DL model Cls (10 K) 76% 77% 77%
Random Forest Cls (10 K) 88% 26% 40%
: Scores for predictions of superconductivity for materials reported by Ref. [@1468-6996-16-3-033503]. Reg and Cls are abbreviations for regression and classification respectively.[]{data-label="tab: aprf_hosono"}
The prediction of superconductivity
-----------------------------------
We applied a list of materials reported by Ref. [@1468-6996-16-3-033503] to evaluate the models. The list has about 400 materials found since 2010; importantly, it includes 330 non-superconductors. To temporally separate the materials on the list from the training data, we used only the data added to SuperCon or COD before 2010 as training data. To compare the capability of a model with expert predictions, we evaluated whether the model could predict superconductivity for the given materials. Randomly selecting a material from the list with $T_\text{c}$ > 0 K yields a precision of 32%. This is considered the baseline because all the materials on the list were expected to be superconductors before the experiments. The model predicted $T_\text{c}$; with respect to whether the $T_\text{c}$ value will be higher than 0 K, the results had a precision of 62%, an accuracy of 76%, a recall of 67%, and an f1 score of 63%. The precision is about two times higher than the baseline (32%) and 10 sigma away from it. Another interesting threshold is 10 K because only a limited number of superconductors have a $T_\text{c}$ of > 10 K. The deep learning method predicted materials with this $T_\text{c}$ with a precision of 75%, which is about seven times higher than the baseline random precision (10%). The accuracy, recall, and f1 score were 95%, 76%, and 75%, respectively. In contrast, the preliminary model, trained with SuperCon only, predicted that all the materials would be superconductors, even though the training data were up to the year 2018 (i.e., not temporally separated). We also performed random forest binary classification with garbage-in and deep learning binary classification, which classify materials in terms of whether the $T_\text{c}$ is beyond 0 K or not. The results, summarized in Table \[tab: aprf\_hosono\], demonstrate that our deep learning model has good capability to predict superconductivity and outperformed random forest classification (see Methods).
The discovery of superconductor
--------------------------------
Next, we used the model to predict the $T_\text{c}$ values of the materials in COD. The number of materials predicted to be superconductors was different every time we trained the models from scratch, which is expected with deep learning. We made a search target list for the experiment. After we removed cuprates and Fe-based superconductors (FeSCs) from the list, we obtained 900 materials predicted to be superconductors with $T_\text{c} > 0$ K, 250 materials with $T_\text{c} > 4$ K, and 70 materials with $T_\text{c} > 10$ K, which is more reasonable compared to the results obtained using the preliminary model. These materials are candidates for new superconductors. Although the prediction results on materials reported by Hosono et al. show that the model is useful, experiments (currently under way) are required to validate the method. The list included , which was recently found to be a superconductor [@C6CP02856J] and is not listed in SuperCon. We had not known it was a superconductor beforehand. It can be concluded that the deep learning model found an actual superconductor.
The discovery of Fe-based superconductors (FeSCs)
-------------------------------------------------
To test the capability of our deep learning model of finding new types of superconductor, we investigated whether we could find high-$T_\text{c}$ FeSCs by using the model trained with data before 2008, the year FeSCs were discovered. We removed two materials, and , from the training data because their discovery in 2006 led to the discovery of high-$T_\text{c}$ FeSCs. We used the 1,399 FeSCs known as of 2018 in SuperCon as the test data. A total of about 130 training and test runs were used. Although the models were made stochastically, we found some FeSCs that were predicted to have finite $T_\text{c}$. A histogram of the number of predicted FeSCs with $T_\text{c}$ $> 0$ K is shown in Fig. \[fig: hist\_fe\_2007\_without-log-normal\]. We obtained the same results for high-$T_\text{c}$ cuprates (see Supplementary Information). When we used shallow 10-layer networks that had as good $R^2$, precision, etc., as the current large model, FeSCs were not found. This is not strange, because most iron compounds show magnetism, which is incompatible with superconductivity, and there are few superconductors including iron except for FeSCs. Indeed, few researchers had anticipated that FeSCs could have high Tc values. It is recognized that larger models have better generalization performances. The fact that the larger model found FeSCs can be explained by a larger model having an improved search capability for new superconductors. These results suggest that FeSCs and cuprate superconductors might have been found by our deep learning model.
![Histogram of the number of predicted FeSCs with $T_\text{c}$ > 0 K (log scale).[]{data-label="fig: hist_fe_2007_without-log-normal"}](./figures/Histogram_Predict_Fe_2007_without_two_body.pdf){width="70.00000%"}
Discussion {#discussion .unnumbered}
==========
If we had searched for FeSCs following the prediction, we would have discovered FeSCs. However, the predicted $T_\text{c}$ of the FeSCs was rather low in our attempt to *discover* FeSCs. FeSCs might thus have been a low-priority target depending on how the model prediction was used. This problem will be considered in future research. We will incorporate crystal structure information to enhance the capability of the model of finding new high-$T_\text{c}$ superconductor families. Nevertheless, the present model is still useful as an auxiliary tool. Furthermore, the present method could be applied to other problems where crystal structure is difficult to obtain. Even though our method does not require feature engineering, unlike conventional methods in materials informatics, it achieved much better results. Our deep learning method may replace existing methods, just as other deep learning methods have done in computer vision, natural language processing, and reinforcement learning. Deep learning requires failure data (e.g., non-superconductors) for accurate prediction. As many datasets in materials search are a random train-test split, we must prepare a temporally separated train-test datasets for the field to progress. Because our method does not use specific properties of superconductors and uses only chemical formulas, the method can be applied to other problems with, in particular, inorganic materials. We are almost ready for the paper on the application to other problems. We demonstrated the usefulness of our method and deep learning. Our results open the way to the discovery of new high-$T_\text{c}$ superconductor families.
methods {#methods .unnumbered}
=======
The method named reading periodic table: representation of periodic table
-------------------------------------------------------------------------
### Representation of elements as one-hot vectors {#sec: one-hot-1}
Any one of the $118$ elements of the periodic table can be represented by a one-hot vector. For example, He can be represented by a $118$-dimensional vector $(0,1,0,\cdots, 0)$ and H can be represented by $(1,0,\cdots,0$). The fictional compound would be represented by $(2,3,0,\cdots,0)$ or $(2/5,3/5,0,\cdots,0)$. There are two problems associated with representing materials by one-hot vectors. First, neural networks do not learn about elements and their combinations that do not appear in the training data. Second, one-hot vector representations do not reflect the properties of the elements, especially when data are scarce. Elements are treated as quite different entities in one-hot representations, even though the properties of the elements are known from quantum mechanics.
### Learning of periodic table
To overcome these problems, we introduce a method that enables the deep learning model to learn the periodic table. The information on elements is reflected by the data representation, which the deep learning model uses to learn the properties. The properties of the elements and their similarities are reflected in the periodic table. The composition ratios of materials are entered into the periodic table and we then divide the periodic table into four tables corresponding to the s-, p-, d-, and f-blocks because differences in the valence orbitals are important. The dimensions of the representation are $4\times32 \times 7$. The deep learning model learns the periodic table using its convolutional layers. With knowledge of the periodic table and element properties, the deep learning model can predict unknown materials from known ones.
Garbage-in: a method for creating synthetic data on non-superconductors
-----------------------------------------------------------------------
We have a database of superconductors. However, to explore new superconductors we also need a database of non-superconductors, which does not exist. Hence, we created synthetic data. Under the assumption that most of the inorganic materials in COD do not become superconductors with finite $T_\text{c}$, we input the inorganic materials in COD, with $T_c=0$, to the deep learning model as training data along with SuperCon. The method is illustrated in Fig. \[fig: garbage\_in\]. The overall scheme of training is illustrated in fig. \[fig: overall\].
![Synthetic data generation method named garbage-in.[]{data-label="fig: garbage_in"}](./figures/garbage-in.pdf){width="0.8\hsize"}
![Overall scheme. The data of superconductors from SuperCon and the data of non-superconductors synthesized by garbage-in are transformed into the representation by reading periodic table in order for neural networks to learn the rules, then the deep neural network is trained to output $T_\text{c}$ []{data-label="fig: overall"}](./figures/overall_scheme.pdf){width="0.9\hsize"}
List of candidate materials
---------------------------
We used the 48,000 inorganic materials in COD, 1,000 of which were used as test data. The remaining 47,000 materials and 12,000 materials in SuperCon were used as training data. Then, we obtained predicted $T_\text{c}$ values of the 1,000 materials in the test data. We repeated the procedure 48 times with different test data. This produced a candidate materials list. If we generate materials by generative models, which output chemical composition virtually, we do not yet know how to synthesize the generated materials. COD is thus used because it is a list of previously synthesized materials.
Data availability
-----------------
SuperCon [@supercon], COD [@gravzulis2011crystallography; @gravzulis2009crystallography; @Downs2003], and the materials reported by Ref. [@1468-6996-16-3-033503] are openly available and free to use. The materials reported by Hosono et al. have undetermined variables, such as $x$ in . We investigated related papers and input the values for such variables. We then made a list of materials for the evaluation of models. This list will be openly available for the community.
Code availability
-----------------
We are planning to open the code after some experiments on the candidate materials are completed. The code will be available on reasonable request.
Data handling
-------------
### Definitions of conventional, cuprate, and Fe-based superconductors
Cuprate superconductors are defined as materials that contain Cu, O, and more than one other element. The exceptions are , , and . FeSCs are defined as materials that contain Fe and either As, S, Se, or P. All other superconductors are considered to be conventional.
### Removal of problematic data
We removed materials whose composition ends with variables such as “$+$x”. About 7,000 materials were removed from SuperCon. If we know the accurate compositions of these materials and include them in the data, this should improve the deep learning model. We input appropriate values for variables such as “x” for the materials reported by Hosono et al. after reviewing the original studies, because there were only about 300 materials left after temporal separation.
### Treatment of materials with same composition but different $T_\text{c}$ values in SuperCon
SuperCon contains materials with the same composition but different $T_\text{c}$ values. We decided to use the median value of $T_\text{c}$.
### Treatment of materials without $T_\text{c}$ values in SuperCon
Of the 17,000 remaining materials, about 4,000 did not have $T_\text{c}$ values. We considered setting $T_\text{c}=0$ for these materials or just excluding them. A comparison of the regression results of the preliminary models with SuperCon indicated that excluding the materials without $T_\text{c}$ values was better, so this was done.
### Treatment of COD data
We use only the inorganic materials in COD. We remove duplicates, data with compositions difficult for machines to read, and overlap with SuperCon and the materials reported by Hosono et al. After this process, about 48,000 materials remained.
### Overlap among SuperCon, COD, and materials reported by Hosono et al.
The overlap with SuperCon was removed from COD and the materials reported by Hosono et al.
### Temporal separation of materials
Since the materials reported by Hosono et al. were collected starting from 2010, we used data from before 2010 as the training data. Using data from before 2008 as the training data and using the materials reported by Hosono et al. to check the reliability of the models also resulted in temporal separation.
Accuracy, precision, recall, and f1 score
-----------------------------------------
Consider disease detection. Suppose that we have 10,000 samples, 100 of which contain a disease. The task is to predict whether a given sample contains a disease. Accuracy is the rate of the prediction being right, irrespective of the prediction being positive or negative. Precision is the percentage of positive predictions that are correct. If a positive prediction is made for only one sample that is obviously positive, and all other samples are predicted to be negative, then you will get 100% precision, but you miss all the remaining $99$ samples with a disease, which is a problem for disease detection. Hence, we have recall. Recall is the percentage of identified samples with a disease out of all samples with a disease. If all 10,000 samples are predicted to be positive, recall will be 100% since all 100 samples with a disease were found, but accuracy and precision will be only 1%, which is unsatisfactory. The f1 score utilizes both precision and recall. It is given by the harmonic mean $2\times\frac{\text{precision}\times\text{recall}}{\text{precision}+\text{recall}}$. The best measure for evaluating a model depends on the specific problem.
Neural networks
---------------
A smooth L1 loss function was used. The optimizer was Adam [@Kingma2014AdamAM]. For the prediction of $T_\text{c}$ values for the materials in SuperCon by the preliminary model, the learning rate was $2\times 10^{-6}$, the batch size was 32, the number of epochs was 6,000, $T_\text{c}$ was in the linear scale, and the number of layers was 64. It took about 50 hours for training (see Supplementary Information). For the prediction of $T_\text{c}$ values for the materials in SuperCon by the model with garbage-in, the number of epochs was set to 1,000. It took about 45 hours for training because the training dataset was five times larger than the preliminary model. For the prediction of superconductivity for the materials reported by Hosono et al., the learning rate was $10^{-4}$, the batch size was 32, the number of epochs was 200, $T_\text{c}$ was in the linear scale, and the number of layers was 64. For the prediction of FeSCs, the learning rate was $10^{-4}$, the batch size was 32, the number of epochs was 200, $T_\text{c}$ was in the linear scale, and the number of layers was 64. For making the candidate material list for the experiment and the discovery of from the list, the learning rate was $10^{-4}$, the batch size was 32, the number of epochs was 500, $T_\text{c}$ was in the log scale after the addition of 0.1 to $T_\text{c}$, and the number of layers was 9. The network was different because these predictions were done at the start of our research. We also found the superconductor using the 64-layer network.
Random forest
-------------
Random forest analysis was performed by using the weighted average, weighted variance, maximum, minimum, range, mode, median, and mean absolute difference of the 32 basic features of elements in compositions (see Supplementary Information). The basic features were obtained from [Magpie](https://bitbucket.org/wolverton/magpie/src/master/). The results were averages over 10 models. Random forest analysis using only the data of superconductors, without garbage-in, encountered the same problem as our deep learning model. It predicted about 60% of the materials were superconductors. We performed classification regarding whether $T_\text{c}$ is beyond 0 K or not for materials reported by Hosono et al. because it is almost impossible for random forest regression to estimate $T_\text{c}=0$, due to random forest classification being an ensemble estimation. If even one tree estimates $T_\text{c}>0$, then random forest regression estimates $T_\text{c}>0$. Classification with respect to 10 K was also done.
Training and test data used for main results
--------------------------------------------
The training data and test data used for the main results are summarized in Table \[tab: summary-results\].
Main result Training data Test data
------------- -------------------------- -------------------------------------
Materials reported by Hosono et al.
SuperCon and COD in 2018 COD in 2018
: Summary of main results. []{data-label="tab: summary-results"}
\
Supplementary information {#supplementary-information .unnumbered}
=========================
Other hyper-parameters
----------------------
We obtained an $R^{2}$ value of 0.93 for the prediction of $T_\text{c}$ for the materials in SuperCon by the best preliminary model with the same train-test split (0.15) as that used in a previous study [@stanev2018machine], because in the previous study, only the $R^2$ value of the best model (to our understanding), 0.88, was reported, which is less than our $R^2$ value of 0.93. For a train-test split of 0.05, the median of $R^{2}$ was 0.92 for 56 models, which is presented in the main text. For the prediction of $T_\text{c}$ values for the materials in SuperCon by the model with garbage-in and a train-test split of 0.05, the median of $R^{2}$ was 0.85 for 55 models. For the prediction of superconductivity for the materials by deep learning regression reported by Ref. [@1468-6996-16-3-033503], the reported scores are the median values for 29 models. For deep learning classification, the scores are the averages over 32 models, and the hyper-parameters are the same as those of the regression models that output $T_\text{c}$ except for the binary classification and the binary classification entropy with the logit loss function.
Prediction of superconductivity in FeSCs by deep learning model
---------------------------------------------------------------
The number of predicted FeSCs varied with the model because models trained with the same training data can become different depending on their initial weights and the input order of the training data. The models were stochastically constructed. However, once a model is constructed, the output is deterministic unless stochastic layers are used. Of note, we always found some high-$T_\text{c}$ FeSCs that were predicted to have a finite $T_\text{c}$.
### Check of models based on materials reported by Hosono et al. {#sec: check-by-hosono}
A model that predicts superconductivity for all materials or makes random predictions will have a precision that is equal to the baseline random precision. The validity of the models was checked using the materials reported by Hosono et al., which can be used to reject the model. We used the models to predict whether the materials on the list had a superconducting transition temperature of above 0 K, and checked whether the precision was higher than the baseline random precision. The mean precision was 0.5 for about 130 training and test runs and the baseline random precision was 0.32. The precision was about two times higher than the baseline. We also checked whether each model satisfied the condition that the precision be sufficiently higher than the baseline.
### Predictions using various combinations of training data
To confirm the reliability of a model, we checked whether the model learned the feature of superconductivity by checking the effect of training data on the number of predicted FeSCs. We compared the predictions of five models trained using different data based on SuperCon and COD. The training data were as follows: (i) data before 2008 without LaFePFO and LaFePO; (ii) data before 2008 with LaFePFO and LaFePO; (iii) only conventional superconductors as of 2018; (iv) only cuprates as of 2018; and (v) both conventional superconductors and cuprates as of 2018. These models predicted the $T_\text{c}$ values of the FeSCs in SuperCon. In total, about 130 training and test runs were used for (i) and (ii) and about 170 training and test runs were used for (iii), (iv), and (v).
Figure \[fig: 2007-with-without\] shows the results for (i) and (ii). As shown, the number of predicted FeSCs in Fig. \[fig: 2007-with\] is higher than that in Fig. \[fig: 2007-without\]. The average number of FeSCs predicted to have finite $T_\text{c}$ increased from 80 to 129. The median increased from 44 to 130. We checked the validity of the models using the materials reported by Hosono et al. When evaluating model (i), we removed materials containing Fe from the list. This was not done when evaluating model (ii). The baseline random precision and model precision were 0.32 and 0.5, respectively, for model (i), and 0.32 and 0.5, respectively, for model (ii). The model precision was sufficiently higher than the baseline, indicating that the models were valid. These results show that these two materials had a large impact on the predictions. It is surprising, in view of deep learning, that 2 out of 60,000 training data points had such a significant influence on the model. However, this influence is reasonable because human experts can infer many FeSCs if they know that LaFePFO and LaFePO are superconductors.
[0.5]{} ![Comparison of histograms of predicted FeSCs by model trained with data before 2008 with and without LaFePFO and LaFePO (log scale).[]{data-label="fig: 2007-with-without"}](./figures/Histogram_Predict_Fe_2007_without_two.pdf "fig:"){width="110.00000%"}
[0.5]{} ![Comparison of histograms of predicted FeSCs by model trained with data before 2008 with and without LaFePFO and LaFePO (log scale).[]{data-label="fig: 2007-with-without"}](./figures/Histogram_Predict_Fe_2007_include_two.pdf "fig:"){width="110.00000%"}
The results for (iii) and (v) are shown in Fig. \[fig: fe-based-2018\]. Model (iv) (trained with only cuprates) could not predict any FeSCs. As shown in the results of (iii) and (v), the average number of the FeSCs predicted to have finite $T_\text{c}$ increased from 46 to 123 when cuprates were included in the training data. The median increased from 15 to 80. The model learned the feature of cuprates, and the number of predicted FeSCs was increased by the addition of cuprates. We checked the models using the materials reported by Hosono et al. We removed materials containing Fe from the materials on the list. The baseline random precision was 0.2. The mean precision values were 0.41 and 0.35, respectively, for models (iii) and (v), indicating that the models were valid since they are three times higher than the baseline value. The above results show that the addition of training data increased the number of FeSCs with finite $T_\text{c}$. We conclude that the model learned the feature of superconductivity. This confirms the reliability of the model, which was also confirmed by checking the precision of the prediction of superconductivities from the materials reported by Hosono et al.
[0.5]{} ![Histograms of predicted FeSCs (log scale).[]{data-label="fig: fe-based-2018"}](./figures/Histogram_Predict_Fe_from_conv.pdf "fig:"){width="110.00000%"}
[0.5]{} ![Histograms of predicted FeSCs (log scale).[]{data-label="fig: fe-based-2018"}](./figures/Histogram_Predict_Fe_from_conv_CuO.pdf "fig:"){width="110.00000%"}
Prediction of superconductivity in cuprates
-------------------------------------------
It was checked whether the deep learning model could predict cuprates when trained with data that did not include cuprates. We trained models with the following data combinations: (i) only conventional superconductors; (ii) only FeSCs; and (iii) both conventional superconductors and FeSCs. The data were as of 2018. We repeated the training and test approximately 130 times. The results for models (i) and (iii) are shown in Fig. \[fig: cuo-based-2018\]. Model (ii) (trained with only FeSCs) could not predict any cuprates with finite $T_\text{c}$, which is consistent with the prediction of FeSCs from cuprates. The average numbers of cuprates predicted to have finite $T_\text{c}$ were 12 and 17 for models (i) and (iii), respectively. The median values were 7 and 9, respectively. In contrast to the prediction of FeSCs, the mean and median values were almost unchanged by the addition of cuprates to the training data. In SuperCon, the number of FeSCs is less than a quarter of the number of cuprates, which might have led to the difference between the predictions of FeSCs and cuprates. The models were checked using the materials reported by Hosono et al. The baseline random precision was 0.20. The mean precision values were 0.41 and 0.52 for models (i) and (iii), respectively. The precision values are sufficiently higher than the baseline random precision. Although care should be taken when interpreting the results (i.e., deep learning predicted cuprates as candidate superconductors), the results show the possibility of exploring superconductors using deep learning.
[0.5]{} ![Histograms of predicted cuprate superconductors (log scale).[]{data-label="fig: cuo-based-2018"}](./figures/Histogram_Predict_CuO_from_conv.pdf "fig:"){width="110.00000%"}
[0.5]{} ![Histograms of predicted cuprate superconductors (log scale).[]{data-label="fig: cuo-based-2018"}](./figures/Histogram_Predict_CuO_from_conv_Fe.pdf "fig:"){width="110.00000%"}
Deep learning regression model and binary classification model
--------------------------------------------------------------
The difference between the deep learning regression model and binary classification model is that the regression model outputs $T_\text{C}$, whereas the binary classification model classifies materials with respect to whether the $T_\text{C}$ value is larger than a threshold value or not.
The features used in random forest classification
-------------------------------------------------
The 32 basic features of elements used for random forest classification are as follows.
> AtomicWeight, Column, DipolePolarizability, FirstIonizationEnergy, GSbandgap, GSenergy-pa, GSestBCClatcnt, GSestFCClatcnt, GSmagmom, GSvolume-pa, ICSDVolume, IsAlkali, IsDBlock, IsFBlock, IsMetal, IsMetalloid, IsNonmetal, MendeleevNumber, NdUnfilled, NdValence, NfUnfilled, NfValence, NpUnfilled, NpValence, NsUnfilled, NsValence, Number, NUnfilled, NValance, Polarizability, Row, FirstIonizationEnergies.
We used the weighted average, weighted variance, maximum, minimum, range, mode, median, and mean absolute difference of the basic features. Thus, in total, we used 256 features.\
**Author Contributions:** Tomohiko Konno conceived and supervised the research. Tomohiko Konno, Hodaka Kurokawa, and Fuyuki Nabeshima had the primary roles in the research, discussed the direction and interpretation of the analysis, and were the main writers of the manuscript. Tomohiko Konno made the deep learning models and the two methods (reading periodic table and garbage-in), and specified how to evaluate a model using the materials reported by Hosono et al. Hodaka Kurokawa checked the materials in the candidate materials list and found CaBi2 and investigated the corresponding original papers to determine the indefinite values in the materials reported by Hosono et al. Yuki Sakishita performed random forest analysis. Iwao Hosako brought together the experimenters and machine learning experts. All authors approved the final version of the manuscript for submission.
**Competing Interests:** The authors declare they have no competing financial interests.
**Correspondence:** Correspondence and requests for materials should be addressed to Tomohiko Konno ([email protected]) and Hodaka Kurokawa ([email protected]).
|
---
abstract: 'Emission mechanism of the magnetars is still controversial while various observational and theoretical studies have been made. In order to investigate mechanisms of both the persistent X-ray emission and the burst emission of the magnetars, we have proposed a model that the persistent X-ray emission consists of numerous micro-bursts of various sizes. If this model is correct, intensity Root Mean Square (RMS) variations of the persistent emission exceed the values expected from the Poisson distribution. Using [*Suzaku*]{} archive data of 11 magnetars (22 observations), the RMS intensity variations were calculated from 0.2keV to 70keV. As a result, we found significant excess RMS intensity variations from all the 11 magnetars. We suppose that numerous mircro-bursts constituting the persistent X-ray emission cause the observed variations, suggesting that the persistent X-ray emission and the burst emission have identical emission mechanisms. In addition, we found that the RMS intensity variations clearly increase toward higher energy bands for 4 magnetars (6 observations). The energy dependent RMS intensity variations imply that the soft thermal component and the hard X-ray component are emitted from different regions far apart from each other.'
author:
- 'Yujin <span style="font-variant:small-caps;">Nakagawa</span>, Ken <span style="font-variant:small-caps;">Ebisawa</span> and Teruaki <span style="font-variant:small-caps;">Enoto</span>'
title: 'Energy Dependent Intensity Variation of the Persistent X-ray Emission of Magnetars Observed with Suzaku'
---
Introduction {#sec:intro}
============
Magnetars are highly magnetized neutron stars [@duncan1992], and unique astrophysical objects to study physical phenomena under extremely high magnetic field strengths greater than the quantum critical level 4.4$\times$10$^{13}$G (e.g., [@lyne2006]). Among several classes of magnetars, soft gamma repeaters (SGRs) and anomalous X-ray pulsars (AXPs) are known to exhibit particularly intriguing X-ray emitting phenomena. While both exhibit persistent X-ray emission with typical luminosities of $\sim10^{34}$–$10^{35}$ergs$^{-1}$ in 2–10keV, the SGRs and some AXPs occasionally exhibit sporadic short bursts with typical durations of $\sim$100ms and luminosities of $\sim10^{39}$–$10^{40}$ergs$^{-1}$ in 2–100keV. These unusual phenomena are thought to be caused by extremely strong magnetic field dissipation [@duncan1992].
Models which reproduce spectra of the persistent X-ray emission of the magnetars were studied based on observations by RXTE [@kuiper2004], INTEGRAL [@molkov2005; @rea2009] and [*Suzaku*]{} [@esposito2007; @nakagawa2009b; @enoto2010a; @enoto2010b; @enoto2010c; @2017ApJEnoto]. These observational studies suggest that the magnetar persistent X-ray spectra consist of a soft thermal ($<$10keV) component and a hard X-ray ($>$10keV) component. The soft thermal component is reproduced by two blackbody functions (2BB) with typical temperatures of $\sim$0.5keV and $\sim$1.4keV, or a blackbody plus a power-law model (BB$+$PL) with a typical temperature of $\sim$0.5keV and a typical power-law photon index of $\sim$3 (e.g., [@nakagawa2009a]). The hard X-ray component is well reproduced by a power-law model (PL) with a typical power-law photon index of $\sim$1 (e.g., [@enoto2010c]). The hard X-ray component should have a cutoff in high energy greater than $\sim200$keV (e.g., [@enoto2010b]), otherwise the energy flux in the high energy goes to infinity. However, no clear evidence of the cutoff has been found ([@li2017] for upper limits of gamma-ray emission in 0.1–10GeV), while its hint has been reported [@yasuda2015].
Energy spectra of 50 short bursts from SGR1806$-$20 and 5 short bursts from SGR1900$+$14 with typical luminosities of $\sim10^{39}$–$10^{40}$ergs$^{-1}$ in 2–100keV were observed by High Energy Transient Explorer 2 (HETE-2) with a wide energy band of 2–400keV, and phenomenologically reproduced by the 2BB model [@nakagawa2007] or the 2BB$+$PL model [@nakagawa2011b]. In recent studies based on [*Suzaku*]{} observations, energy spectra of weak bursts with luminosities of $\sim10^{36}$–$10^{37}$ergs$^{-1}$ in 2–40keV from SGR0501$+$4516 [@nakagawa2011a] and AXP1E1547.0$-$5408 [@enoto2012], which have lower luminosities than the typical short bursts, are reproduced with the hard X-ray component (PL) and the soft thermal component (2BB). Thus the energy spectra of the persistent X-ray emission, the typical short bursts and the weak bursts are likely to be reproduced by the same spectral model.
Several physical models have been proposed to explain emission mechanisms of the short bursts and the persistent X-ray emission. One of the ideas for the bursts is that the bursts are caused by heating of the magnetic corona due to local magnetic reconnections [@lyutikov2003]. The soft thermal component of the persistent X-ray emission is explained, e.g., by the Surface Thermal Emission and Magnetospheric Scattering (STEMS) model [@guver2006], the Resonant Cyclotron Scattering (RCS) model [@lyutikov2006]. Meanwhile, the hard X-ray component is explained by, e.g., thermal bremsstrahlung at the neutron star surface [@thompson2005; @beloborodov2007], Compton scattering in high magnetic fields [@baring2007; @fernandez2007], synchrotron emission in the magnetosphere [@heyl2005b], a fallback disk model [@trumper2010] or a photon splitting model [@enoto2010c].
Based on the unified analysis of the persistent X-ray emission and the burst emission, two types of the correlation are reported: One is between the low and high temperatures of the 2BB components (2BB temperature correlation; [@nakagawa2009a]). The other is between the luminosities of the soft (2BB) and the hard (PL) components over five orders of magnitude (luminosity correlation; [@nakagawa2011a]).
Based on the unified spectral analysis, the 2BB temperature correlation, the luminosity correlation, and analogy with a relation between the solar microflare and the solar flare, we have proposed a new idea named ”micro-burst model” that the persistent X-ray emission is composed of numerous micro-bursts of various sizes [@nakagawa2009a; @nakagawa2011a]. The micro-bursts may have a duration much less than $\sim$100ms, a typical duration of the short bursts. If the persistent X-ray emission is composed of such numerous micro-bursts, a cumulative number-intensity distribution of the micro-bursts would show a power-law distribution which has been found for typical short bursts (e.g., [@nakagawa2007]). Such power-law distribution is often referred to as the Gutenberg-Richter law [@gutenberg1956].
We have calculated expected fluxes of the putative micro-bursts constituting the persistent X-ray emission by extrapolating the cumulative number-intensity distribution of typical bursts observed by HETE-2 for SGR1806$-$20 [@nakagawa2011b]. We found that the expected flux, accumulating the unresolved micro-bursts, is comparable to the observed persistent X-ray fluxes. A similar study was performed on an outburst of AXP1E1547.0$-$5408 with [*Suzaku*]{} [@enoto2012].
If the persistent X-ray emission is not static, but composed of numerous micro-bursts of various sizes following a particular cumulative number-intensity distribution, dispersion of the micro-burst intensities, as well as the persistent X-ray flux, should exceed the value expected from the Poisson distribution. In order to measure the dispersion quantitatively, in this paper, we calculate Root Mean Square (RMS) intensity variations in the persistent X-ray emission using the [*Suzaku*]{} data. [*Suzaku*]{} has great capabilities to estimate the RMS intensity variations, because the on-board narrow field instruments of X-ray imaging spectrometer (XIS; 0.2–12keV; [@koyama2007]) and the hard X-ray detector (HXD; 10–700keV; [@takahashi2007]) have high sensitivities and wide energy bands.
Observation and Data Reduction
==============================
The present studies are performed using [*Suzaku*]{} archive data of 11 magnetars (22 observations). Table \[tab:suzaku\_obs\_list\] shows a summary of the observations. Besides, [*Suzaku*]{} archive data include the magnetars AXPSwiftJ1834.9$-$0846 and AXP1E1841$-$045, which are not used in the present study due to no significant detection and contamination by a nearby supernova remnant, respectively. Since there are not enough photon counts in the HXD-GSO energy band, we focus on the XIS and HXD-PIN data. Although HXD-PIN event data have a time resolution of 61 $\mu$s, there are not enough photon counts to find any direct evidence of micro-bursts.
Reduction of the XIS and HXD-PIN event data were made using HEAsoft6.16 software. The latest calibration database (CALDB:20150312) was applied to unfiltered XIS event data using [*aepipeline*]{} (v1.1.0). We created light curves and spectra from the cleaned XIS event data using [*xselect*]{} (v2.4c). Response matrix files were generated by [*xisrmfgen*]{} (v2012-04-21), and ancillary response function files by [*xissimarfgen*]{} (v2010-11-05). The net exposures of the XIS data are summarized in table \[tab:suzaku\_obs\_list\].
The latest calibration database (CALDB:20110915) was applied to unfiltered HXD-PIN event data using [*aepipeline*]{} (v1.1.0). We created light curves and spectra using [*xselect*]{}. Dead time corrections were applied to the spectra using [*hxddtcor*]{} (v1.50) as well as to the light curves using the recipe published on a [*Suzaku*]{} website[^1]. Response matrix files were taken from CALDB[^2]. The net exposures of the HXD-PIN data are summarized in table \[tab:suzaku\_obs\_list\].
Data Analysis {#sec:data_analysis}
=============
For each observation, the XIS events were extracted from box regions centered on the objects, while the background events were extracted from box regions around the objects with the same area. The HXD-PIN background was subtracted using the background files supplied by Suzaku Guest Observer Facility[^3]. The quoted errors hereafter refer to 68% confidence levels.
In this paper, we define the RMS intensity variations $R$ as $$\label{rms_eq}
R = \frac{
\bigl[
\frac{1}{N-1}
\bigl\{
\sum_{i}(x_i - \overline{x})^2 - \sum_{i}{\delta_{x_i}}^2
\bigr\}
\bigr]^{\frac{1}{2}}
}{\overline{x}},$$ where $i$ is the bin number, $x_i$ is the background-subtracted counts per bin, $\overline{x}$ is the average of $x_i$, $\delta_{x_i}$ is the error of $x_i$ and $N$ is the number of bins. Here, $N$ is obtained by dividing the net exposures in table \[tab:suzaku\_obs\_list\] by the time bin-widths.
We calculated RMS intensity variations for each observation using the background-subtracted light curves in the 0.2–12keV (XIS) and 10–70keV (HXD-PIN) energy bands. Time resolutions of the light curves are 8s in 0.2–12keV and 128s in 10–70keV. The RMS intensity variations are found to be $R_{\mathrm{X}}=$1.3–135% in the 0.2–12keV energy band and $R_{\mathrm{P}}=$17–99% in the 10–70keV energy band, depending on sources and observations (table \[tab:suzaku\_obs\_list\]).
Based on Monte Carlo simulations, we confirmed that variations due to rotations of the magnetars ($\sim$2–12s; e.g., [@enoto2010c]), and long-term ($\sim$1day) flux variations do not affect the RMS intensity variations when using the 8s (0.2–12keV) light curves. We also estimated variations caused by background fluctuations using [*Suzaku*]{} data of hard and bright non-variable sources (table \[tab:suzaku\_obs\_list\_bkg\]), and confirmed that the background fluctuations are not significant for most cases.
Next, we estimate effects of obviously bright bursts, which would significantly affect RMS intensity variations. We performed burst search using the 0.2–12keV light curves with a 8s time resolution of the XIS. We searched for such bright bursts in the light curves that exceed $\lambda+5\sigma$, where $\lambda$ is the average and $\sigma$ is the standard deviation. Then we identified visually obvious bright bursts from 7 out of 22 observations as summarized in table \[tab:suzaku\_obs\_list\]. After removing the bright bursts from the light curves, we calculated the RMS variations as before. Consequently, the RMS intensity variations without influence of the bright burst emission are $R'_{\mathrm{X}}=$1.3–18.8% in the 0.2–12keV energy band and $R'_{\mathrm{P}}=$17–99% in the 10–70keV energy band as summarized in table \[tab:suzaku\_obs\_list\]. These variations are considered to be intrinsic variations of the persistent emission.
We also calculated the RMS intensity variations with finer energy bands using the background-subtracted light curves of the XIS with a time resolution of 32s. Figure \[fig:rms\_spc\_summary\] shows energy dependency of the RMS intensity variations with $E^2{f}(E)$ spectra where $f(E)$ is the photon spectrum for each observation. The $E^2{f}(E)$ spectra are consistent with @2017ApJEnoto. Among the observations, the RMS intensity variations clearly increase toward higher energy bands for 4 magnetars (6 observations) as shown in the panels (a), (b), (d), (h), (m) and (o) in figure \[fig:rms\_spc\_summary\].
Result
======
Summary of Data Analysis
------------------------
We calculated RMS intensity variations using the [*Suzaku*]{} data archive for 11 magnetars (22 observations). The RMS intensity variations are significantly greater than the values expected from the Poisson distribution for all the 22 observations of 11 magnetars in the 0.2–12keV energy band (XIS) and 5 magnetars in the 10–70keV energy band (HXD-PIN). For these 5 magnetars, there were 15 observations, and significant variation was detected from 8 out of them.
Mathematical Formulation of RMS Intensity Variations {#sec:math_rms}
----------------------------------------------------
In order to understand observed RMS intensity variations, we calculate expected RMS intensity variations with mathematical approach. We define the expected RMS intensity variations as $$\label{eq:rms_model}
R_{\mathrm{M}} = (\sigma_{\mathrm{c}}^2 - \sigma_{\mathrm{p}}^2)^{\frac{1}{2}}S_{\mathrm{a}}^{-1},$$ where $\sigma_{\mathrm{c}}^2$ is variance of the expected cumulative number-intensity distribution, $\sigma_{\mathrm{p}}^2$ is variance of Poisson distribution, and $S_{\mathrm{a}}$ is an average fluence of the micro-bursts. We assume that $\sigma_{\mathrm{c}}^2$, $\sigma_{\mathrm{p}}^2$ and $S_{\mathrm{a}}$ are defined as values in the 0.2–12keV energy band. The expected RMS intensity variations are independent of the observation exposure time exceeding 1ms, if we assume each micro-burst has 1ms duration.
The expected cumulative number-intensity distribution from a single magnetar is defined as $$\label{eq:logn_logs}
N_{\mathrm c}(>S_{\mathrm c}) = A_{\mathrm c}S_{\mathrm c}^{\alpha},$$ where $A_{\mathrm c}$ is a normalization, $S_{\mathrm c}$ is a fluence of micro-bursts, $\alpha$ is an index, and $N_{\mathrm c}(>S_{\mathrm c})$ is a cumulative number of micro-bursts whose fluences are greater than $S_{\mathrm c}$. The expected cumulative number-intensity distribution for a hypothetical magnetar is assumed to have $\alpha =-1.1$ [@nakagawa2007] and $A_{\mathrm c} = 7\times$10$^{-9}$burstsday$^{-1}$ at $S_{\mathrm c} = 1$ergcm$^{-2}$. A probability density function of the expected cumulative number-intensity distribution is defined as $$\begin{aligned}
\label{eq:pdf}
P(S_{\mathrm{c}}) &=& N'_{\mathrm c}(>S_{\mathrm c}) \left( \int_{S_1}^{S_2} N'_{\mathrm c}(>S_{\mathrm c}) dS_{\mathrm c} \nonumber \right)^{-1} \\
&=& \frac{{\alpha}S_{\mathrm c}^{\alpha-1}}{S_2^{\alpha} - S_1^{\alpha}},\end{aligned}$$ where $S_1$ and $S_2$ ($S_1 < S_2$) are minimum and maximum fluences of the interval of $S_{\mathrm c}$ which satisfy the probability density function. Using equation (\[eq:pdf\]), variance of the expected cumulative number-intensity distribution is calculated as $$\begin{aligned}
\label{eq:logn_logs_vari}
\sigma_{\mathrm{c}}^2 &=& E[S_{\mathrm c}^2] - (E[S_{\mathrm c}])^2 \nonumber \\
&=& \int_{S_1}^{S_2} S_{\mathrm c}^2 P(S_{\mathrm{c}}) dS_{\mathrm c}
- \left\{ \int_{S_1}^{S_2} S_{\mathrm c} P(S_{\mathrm{c}}) dS_{\mathrm c} \right\}^{2} \nonumber \\
%
&=&
-\alpha{A_{\mathrm c}}f_{\mathrm m}^{-1}
\int_{S_1}^{S_2} S_{\mathrm c}^{\alpha+1} dS_{\mathrm c}
-\left\{{\alpha{A_{\mathrm c}}}{f_{\mathrm m}^{-1}}
\int_{S_1}^{S_2} S_{\mathrm c}^{\alpha} dS_{\mathrm c} \right\}^{2}
\nonumber \\
%
&=&
\left\{ \begin{array}{ll}
\frac{\alpha}{\alpha+2}
{A_{\mathrm c}}f_{\mathrm m}^{-1}
\left(
S_1^{\alpha+2} - S_2^{\alpha+2}
%
\right)
-
\left\{
\frac{\alpha}{\alpha+1}
{A_{\mathrm c}}f_{\mathrm m}^{-1}\left(
S_1^{\alpha+1} - S_2^{\alpha+1}
%
\right)
\right\}^{2}
& (\alpha \ne -1 \land 0>\alpha>-2) \\
{A_{\mathrm c}}{f_{\mathrm m}^{-1}}\left(
S_1 - S_2
\right)
-
\left\{
{A_{\mathrm c}}{f_{\mathrm m}^{-1}}
\left(\log|S_1| - \log|S_2|\right)
\right\}^{2}
& (\alpha = -1)
\end{array} \right.,
\label{eq5}\end{aligned}$$ where $E[S_{\mathrm c}^2]$ is an expectation of $S_{\mathrm c}^2$, $E[S_{\mathrm c}]$ is an expectation of $S_{\mathrm c}$ and $f_{\mathrm m} = -A_{\mathrm c}(S_2^\alpha - S_1^\alpha)$ is a frequency of the micro-bursts with the fluences between $S_1$ and $S_2$. Variance of the expected cumulative number-intensity distribution $\sigma_{\mathrm{c}}^2$ depends on the frequency of micro-bursts $f_{\mathrm m}$ as well as the fluence distribution index $\alpha$. Hence, the values of $R_{\mathrm{M}}$ depend on choices of $S_1$, $S_2$ and $\alpha$. Figure \[fig:logn\_logs\_map\] (left) shows a two-dimentional contour graph of $R_{\mathrm{M}}$ with respect to $S_1$ and $S_2$, and figure \[fig:relation\_alpha\_rms\] (right) shows a relation between $\alpha$ and $R_{\mathrm{M}}$; other parameter values are fixed to those determined below.
Short bursts are known to have spiky structures in 0.5ms light curves, which may be caused by a rapid energy reinjection and cooling [@nakagawa2007]. We assume 1ms durations for the micro-bursts under the assumption that one cycle of the energy reinjection and cooling corresponds to a single micro-burst. Using the assumed duration of 1ms for the micro-bursts and typical flux of $10^{-11}$ergcm$^{-2}$s$^{-1}$ for the persistent X-ray emission, we assume $S_1 = 0.001$s$ \times $$10^{-11}$ergcm$^{-2}$s$^{-1} = 10^{-14}$ergcm$^{-2}$. We also assume $S_2 = 10^{-11}$ergcm$^{-2}$ which is substantially below a burst detection limit of [*Suzaku*]{} ($5\sigma \approx 10^{-10}$ergcm$^{-2}$ described in section \[sec:data\_analysis\]). We found $\sigma_{\mathrm{c}}^2 = 5.8\times10^{-26}$erg$^2$cm$^{-4}$ from equation (\[eq5\]).
The expected cumulative Poisson distribution is defined as $N_{\mathrm p}(>k') = A_{\mathrm p}(1-\mathrm{e}^{-\lambda}\sum^{k'}_{k=0}\lambda^{k}/k!)$, where $A_{\mathrm p}$ is a normalization, $\lambda$ is mean counts per bin of a light curve, $k'$ and $k$ are integer values, and $N_{\mathrm p}(>k')$ is a cumulative number of the bins corresponding to micro-bursts for which $k$ is greater than $k'$. The expected cumulative Poisson distribution for the hypothetical magnetar has $\lambda = 12.12$counts(2s)$^{-1}$ in the 0.2–12keV energy band [@nakagawa2011a] and an assumed normalization of $A_{\mathrm p} = 2\times$10$^{5}$burstsday$^{-1}$. We assume that $\lambda = 12.12$counts(2s)$^{-1}$ correspond to an average fluence of the micro-bursts in the 0.2–12keV energy band of $S_{\mathrm{a}} = 7.54\times$10$^{-13}$ergcm$^{-2}$. We also assume that $k'$ has a fluence $S_{\mathrm{p}}(k')$ where $S_{\mathrm{p}}(k'+1) - S_{\mathrm{p}}(k') = S_{\mathrm{p}}(1) = \lambda^{-1}S_{\mathrm{a}}$ is 6.22$\times$10$^{-14}$ergcm$^{-2}$. We found the variance of the expected Poisson distribution as $\sigma_{\mathrm{p}}^2 = {\lambda^{-1}}S_{\mathrm{a}}^2 = 4.7\times10^{-26}$erg$^2$cm$^{-4}$.
Finally, we obtain $R_{\mathrm{M}} = 14$% for the hypothetical magnetar by substituting $\sigma_{\mathrm{c}}^2$, $\sigma_{\mathrm{p}}^2$ and $S_{\mathrm{a}}$ shown above in equation (\[eq:rms\_model\]). This is consistent with the observed values of $R'_{\mathrm{X}} = $1.3–18.8%. Difference of the RMS intensity variations among magnetars may be explained by difference of the fluence distribution index $\alpha$.
Comparison of cumulative number-intensity distributions
-------------------------------------------------------
Figure \[fig:comp\_poisson\_cumulative\_ns\] shows comparison between the expected cumulative number-intensity distribution of the micro-bursts and that expected from Poisson distribution for a hypothetical magnetar. We see that the former clearly has a wider distribution than the latter. In particular, reduction of the bursts above $\sim2\times10^{-12}$ergcm$^{-2}$ makes the expected cumulative number-intensity distribution closer to that expected from Poisson distribution. We confirmed that the RMS intensity variations significantly exceed the values expected from the Poisson distribution even after removing bright bursts (section \[sec:data\_analysis\]). The wider cumulative number-intensity distribution can naturally explain the observed excess RMS intensity variations. Thus, the observation is consistent with the assumption that the persistent X-ray emission is composed of numerous mircro-bursts of various sizes subject to a particular cumulative number-intensity distribution.
Discussion {#sec:discussion}
==========
Expected Flux from Cumulative Number-Intensity Distribution
-----------------------------------------------------------
Integrating energies of the micro-bursts from $S_{\mathrm c} =
10^{-14}$ergcm$^{-2}$ to 10$^{-11}$ergcm$^{-2}$ using the expected cumulative number-intensity distribution in section \[sec:math\_rms\] [@nakagawa2007], we obtain the persistent X-ray flux of a hypothetical magnetar as $\sim$1.1$\times$10$^{-11}$ergcm$^{-2}$s$^{-1}$. This is comparable with a typical observed flux of $\sim$9.9$\times$10$^{-12}$ergcm$^{-2}$s$^{-1}$ for SGR1806$-$20 [@nakagawa2009b]. This result is consistent with the persistent X-ray fluxes estimated from observed cumulative number-intensity distributions obtained by @nakagawa2011b and @enoto2012.
Energy Dependent RMS Intensity Variations
-----------------------------------------
We discovered that energy dependencies of the RMS intensity variations for 4 magnetars (6 observations; panels (a), (b), (d), (h), (m) and (o) in figure \[fig:rms\_spc\_summary\]), clearly increase toward higher energy bands. Among these observations, the RMS intensity variations remarkably increase above $\sim$8keV and $\sim$4keV for SGR0501$+$4516 (OBSID=404078010; panel (a) in figure \[fig:rms\_spc\_summary\]) and AXP1E1547.0$-$5408 (OBSID=903006010; panel (m) in figure \[fig:rms\_spc\_summary\]), respectively. These energies correspond to the crossing points of the soft thermal components and the hard X-ray components, suggesting that the most variation is associated with the hard components.
The energy dependent RMS intensity variations may be explained by the micro-burst model presented in section \[sec:math\_rms\]. It is reported that indices of the cumulative number-intensity distribution of short bursts increase toward higher energy bands in SGR 1806–20 [@nakagawa2007]. Increase of the index causes a high dispersion of fluences, which leads to a large RMS intensity variation as shown in figure \[fig:relation\_alpha\_rms\] (right). If the same energy dependence of the index is applicable to the micro-bursts, smaller RMS intensity variation in lower energy bands is caused by smaller indices, and vice versa.
Comparison with Theoretical Models
----------------------------------
The present results will give constraints on persistent X-ray emission mechanisms of the magnetars. Significant RMS intensity variations in both the 0.2–12keV energy band and the 10–70keV energy band imply that neither the soft thermal component nor the hard X-ray component is from the stable neutron star surface in thermal equilibrium. In this context, thermal bremsstrahlung model at the neutron star surface [@thompson2005; @beloborodov2007] is unlikely.
Energy dependence of the RMS intensity variations suggests that the emission regions of the soft thermal component and the hard X-ray component are located separately. In the magnetospheric synchrotron model [@heyl2005b], the soft thermal component is emitted from a fireball near the neutron star and the hard X-ray component is emitted via synchrotron process far from the neutron star. Thus, this model seems consistent with the present result of the energy dependent RMS intensity variations. In fact, power-law indices of the hard X-ray components expected from the synchrotron model (0.5; [@heyl2005b]) is comparable to the observed indices (0.3–1.7; [@enoto2010c]). Furthermore, the synchrotron model is applicable not only to the persistent emission but also to the burst emissions [@heyl2005a]. This agrees with our idea that the persistent X-ray emission is composed of numerous micro-bursts, and that the persistent emission and the bust emission have the same origin.
A Unified View of the Magnetar X-ray Emission
---------------------------------------------
Figure \[fig:magnetar\_view\] shows a schematic illustration of our unified view (“Micro-Burst Model”) of both the persistent X-ray emission and the burst emission from magnetars, based on the present observation and theoretical models [@duncan1992; @thompson1995; @heyl2005a; @heyl2005b]. In this model, the burst emission is caused by a single energetic fireball, while the persistent X-ray emission consists of numerous micro-bursts caused by numerous small fireballs. In the unified view, both the persistent X-ray emission and the burst emission are explained under the same configuration, where only their luminosities are different. The persistent X-ray emission and the burst emission have typical luminosities of $\sim$10$^{35}$ergs$^{-1}$ and $\sim$10$^{37}$ – $\sim$10$^{40}$ergs$^{-1}$, respectively (e.g., [@nakagawa2011a]). Initially, a starquake occurs on the magnetar surface ([@duncan1992]; process 1 in figure \[fig:magnetar\_view\]). The starquake produces an electron-positron pair plasma fireball which has momenta to leave from the magnetar. The fireball travels in the magnetosphere and emits blackbody emissions (i.e., the soft-thermal components) at around $\sim100R_{\rm NS}$ where $R_{\rm NS}$ is a typical neutron star radius of $\sim$10km ([@heyl2005b]; process 2 in figure \[fig:magnetar\_view\]). The emissions is observed as two blackbody spectra, because of two different polarization modes in strong magnetic fields of $\sim10^{14}$G [@thompson1995]. Their typical temperatures are $\sim$0.5keV and $\sim$1.4keV for the persistent X-ray emission (e.g., [@nakagawa2009a]), or $\sim$4keV and $\sim$11keV for the burst emission (e.g., [@nakagawa2007]).
Eventually, the fireball turns to optically thin condition and emits synchrotron emissions (i.e., the hard X-ray components) at around $\sim1000R_{\rm NS}$ ([@heyl2005b]; process 3 in figure \[fig:magnetar\_view\]). The spatial scale of causality for the hard X-ray components is estimated to be $\sim30R_{\rm NS}$ in assuming that the micro-burst has 1ms duration. Therefore the size of each fireball should be less than $\sim30R_{\rm NS}$. Number of the electrons to produce the hard X-ray component for the persistent X-ray emission may be lower by 2–3 orders of magnitude than that for the burst emission [@nakagawa2011a]. Presumably, the magnetic disturbance is increased outward from the magnetars, so that the strong magnetic disturbance causes the larger RMS intensity variations in the hard X-ray components.
Conclusion
==========
Using the [*Suzaku*]{} archive data of 11 magnetars (22 observations), we found significant RMS intensity variations in the persistent X-ray emission from all the magnetars studied. In addition, we found that the RMS intensity variations increase toward higher energy band for 4 magnetars (6 observations). These RMS intensity variations are consistent with the micro-burst model, where the persistent X-ray emission is composed of numerous mircro-bursts of various sizes subject to a particular cumulative number-intensity distribution. We propose a unified view of the magnetar X-ray emission based on the present results.
Time scale of the flux variations expected in the micro-burst model is shorter than a few milliseconds [@nakagawa2007]. Future observations of magnetars with a higher sensitivity and better time resolution, which is expected to give more accurate measurements of the RMS intensity variations will allow us to strongly constrain magnetar emission mechanisms by directly comparing with theoretical models (e.g., [@duncan1992; @thompson1995; @heyl2005a; @heyl2005b]).
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![ Left: A two-dimentional contour graph of $R_{\mathrm{M}}$ with respect to $S_1$ and $S_2$. Contour lines are plotted for $R_{\mathrm{M}}$ ranging from 0 to 100. Right: A relation between $\alpha$ and $R_{\mathrm{M}}$. \[fig:logn\_logs\_map\] []{data-label="fig:relation_alpha_rms"}](logn_logs_map_v20170919.eps){height="7.19cm"}
![ Left: A two-dimentional contour graph of $R_{\mathrm{M}}$ with respect to $S_1$ and $S_2$. Contour lines are plotted for $R_{\mathrm{M}}$ ranging from 0 to 100. Right: A relation between $\alpha$ and $R_{\mathrm{M}}$. \[fig:logn\_logs\_map\] []{data-label="fig:relation_alpha_rms"}](relation_alpha_rms_v20170919.eps){height="7.19cm"}
![ An expected cumulative number-intensity distribution of the micro-bursts (solid line) and an assumed cumulative Poisson distribution (dashed line). See details in text. []{data-label="fig:comp_poisson_cumulative_ns"}](comp_poisson_cumulative_ns_v5.eps){width="8.19cm"}
![ A schematic illustration of our “Mircro-Burst Model” of both the persistent X-ray emission and the burst emission from magnetars. $R_{\rm NS}$ indicates a typical neutron star radius of $\sim$10km, $L$ indicates a typical luminosity, $kT_{\rm 1}$ and $kT_{\rm 2}$ indicate typical blackbody temperatures and $n_{\rm e}$ indicates number of electrons in emission regions. Magnetic disturbance may be strong at a distance of $\sim$1000$R_{\rm NS}$ and may cause the large RMS intensity variations. []{data-label="fig:magnetar_view"}](magnetar_view_v9.eps){width="16.38cm"}
[lllccccccc]{}
Object& OBSID& Date& $T_{\mathrm{X}}$ & $T_{\mathrm{P}}$ & $N_{b}$ & $R_{\mathrm{X}}$ & $R_{\mathrm{P}}$ & $R'_{\mathrm{X}}$ & $R'_{\mathrm{P}}$\
& & (UTC) & (ks) & (ks) & & (%) & (%) & (%) & (%)\
Object& OBSID& Date& $T_{\mathrm{X}}$ & $T_{\mathrm{P}}$ & $N_{b}$ & $R_{\mathrm{X}}$ & $R_{\mathrm{P}}$ & $R'_{\mathrm{X}}$ & $R'_{\mathrm{P}}$\
& & (UTC) & (ks) & (ks) & & (%) & (%) & (%) & (%)\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
SGR0501$+$4516 & 404078010 & 2009-08-17 & 43 & 25 & 0 & 9.2$\pm$1.1 & $50\pm6$ & 9.2$\pm$1.1 & $50\pm6$\
& 405075010 & 2010-09-20 & 60 & 49 & 0 & 7.2$\pm$2.5 & $99\pm86$ & 7.2$\pm$2.5 & $99\pm86$\
& 408013010& 2013-08-31 & 41 & 33 & 0 & 14$\pm$1 & $-$ & 14$\pm$1 & $-$\
& 903002010 & 2008-08-26 & 60 & 50 & 17 & 50.9$\pm$0.5 & 0 & 7.37$\pm$0.07 & 0\
SGR1833$-$0832 & 904006010 & 2010-03-27 & 42 & 10 & 2 & 27.5$\pm$0.7 & 0 & 18.8$\pm$1.1 & 0\
SGR1900$+$14 & 401022010 & 2006-04-01 & 22 & 13 & 0 & 6.8$\pm$2.7 & 0 & 6.8$\pm$2.7 & 0\
& 404077010 & 2009-04-26 & 53 & 39 & 0 & 9.6$\pm$3.5 & 0 & 9.6$\pm$3.5 & 0\
SGR1806$-$20 & 401021010 & 2007-03-30 & 19 & 16 & 2 & 51$\pm$1 & 26$\pm$24 & 6.4$\pm$4.8 & $31\pm19$\
& 401092010 & 2006-09-09 & 49 & 52 & 9 & 135$\pm$1 & 59$\pm$5 & 6.5$\pm$0.8 & $17\pm13$\
& 402094010 & 2007-10-14 & 52 & 46 & 6 & 82.7$\pm$0.9 & 75$\pm$7 & 4.2$\pm$0.5 & $<55$\
& 406069010 & 2012-03-24 & 71 & 60 & 0 & 16.4$\pm$0.4 & 0 & 16.4$\pm$0.4 & 0\
AXP1E1547.0$-$5408 & 405024010 & 2010-08-07 & 52 & 40 & 0 & 8.4$\pm$0.6 & $20\pm12$ & 8.4$\pm$0.6 & $20\pm12$\
& 903006010 & 2009-01-28 & 11 & 31 & 25 & 36.0$\pm$0.3 & 32$\pm$3 & 18.2$\pm$0.2 & $28\pm3$\
AXP4U0142$+$614 & 402013010 & 2007-08-13 & 100 & 95 & 0 & 2.0$\pm$0.1 & $<40$ & 2.0$\pm$0.1 & $<40$\
& 404079010 & 2009-08-10 & 107 & 92 & 0 & 1.3$\pm$0.3 & $33\pm20$ & 1.3$\pm$0.3 & $33\pm20$\
& 406031010 & 2011-09-07 & 39 & 39 & 0 & 1.8$\pm$0.3 & $<42$ & 1.8$\pm$0.3 & $<42$\
AXP1E1048.1$-$5937 & 403005010& 2008-11-30 & 100 & 63 & 1 & 3.0$\pm$1.0 & $-$ & 3.0$\pm$1.0 & $-$\
AXPSwiftJ1822.3$-$1606 & 906002010& 2011-09-13 & 41 & 34 & 0 & 6.0$\pm$0.3 & $-$ & 6.0$\pm$0.3 & $-$\
AXPCXOUJ164710.2$-$455216 & 901002010& 2006-09-23 & 39 & $-$ & 0 & 3.3$\pm$0.7 & $-$ & 3.3$\pm$0.7 & $-$\
AXP1RXSJ170849.0$-$400910 & 404080010 & 2009-08-23 & 61 & 48 & 0 & 7.85$\pm$0.07 & $23\pm16$ & 7.85$\pm$0.07 & $23\pm16$\
& 405076010 & 2010-09-27 & 63 & 55 & 0 & 9.25$\pm$0.06 & $<36$ & 9.25$\pm$0.06 & $<36$\
AXP1E2259$+$586 & 404076010& 2009-05-25 & 123 & 96 & 0 & 4.1$\pm$0.1 & $-$ & 4.1$\pm$0.1 & $-$\
This work was supported by JSPS KAKENHI Grant Numbers 24540309, 15K05117 and 16K05309.
RMS Intensity Variations Caused by Background Fluctuations {#rms-intensity-variations-caused-by-background-fluctuations .unnumbered}
==========================================================
Suzaku data of hard and bright non-variable sources (table \[tab:suzaku\_obs\_list\_bkg\]) are analyzed, in order to estimate the RMS intensity variations caused by background fluctuations and their time dependencies. The RMS intensity variations are found to be $R_{\mathrm{X}}=$0.8–2.5% in the 0.2–12keV energy band and $R_{\mathrm{P}}=$5–12% in the 10–70keV energy band, depending on observation periods. We found that the time dependency is marginal and confirmed that the background fluctuations are not significant for most cases.
[lllcccc]{}
Object& OBSID& Date& $T_{\mathrm{X}}$ & $T_{\mathrm{P}}$ & $R_{\mathrm{X}}$ & $R_{\mathrm{P}}$\
& & (UTC) & (ks) & (ks) & (%) & (%)\
Object& OBSID& Date& $T_{\mathrm{X}}$ & $T_{\mathrm{P}}$ & $R_{\mathrm{X}}$ & $R_{\mathrm{P}}$\
& & (UTC) & (ks) & (ks) & (%) & (%)\
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CasA& 100016010& 2005-09-01 & $-$ & 24 & $-$ & 12$\pm$5\
& 100043010& 2006-02-02 & $-$ & 10 & $-$ & $<40$\
& 100043020 & 2006-02-17 & 7 & 16 & 0.8$\pm$0.5 & $<26$\
& 507038010 & 2012-12-20 & 102 & 118 & 1.56$\pm$0.02 & $<28$\
ComaCluster & 801097010 & 2006-05-31 & 179 & 157 &0.8$\pm$0.3 & 5$\pm$4\
PerseusCluster& 101012020 & 2007-02-05 & 44 & 42 & $<1.6$ & 6$\pm$5\
& 102011010 & 2007-08-15 & 42 & 36 & 0.9$\pm$0.7 & 7$\pm$5\
& 102012010 & 2008-02-27 & 42 & 62 & 0.9$\pm$0.7 & 7$\pm$3\
& 103004010 & 2008-08-13 & 41 & 32 & 1.5$\pm$0.3 & 0\
& 103004020 & 2009-02-11 & 50 & 45 & 1.2$\pm$0.4 & $<16$\
& 103005010 & 2008-08-14 & 21 & 17 & $<12$ & $<386$\
& 103005020 & 2009-02-12 & 29 & 27 & $<6$ & 0\
& 104018010 & 2009-08-26 & 41 & 37 & 1.1$\pm$0.5 & $<46$\
& 104019010 & 2010-02-01 & 39 & 36 & 1.7$\pm$0.3 & $<173$\
& 104020010 & 2009-08-27 & 55 & 52 & 1.2$\pm$0.3 & $<20$\
& 104021010 & 2010-02-02 & 22 & 22 & $<4$ & $<31$\
& 105009010 & 2010-08-09 & 34 & 38 & 0.9$\pm$0.8 & 0\
& 105009020 & 2011-02-03 & 40 & 30 & 1.3$\pm$0.4 & $<16$\
& 105010010 & 2010-08-10 & 27 & 34 & $<2$ & $<20$\
& 105010020 & 2011-02-02 & 21 & 17 & $<3$ & $<21$\
& 105027010 & 2011-02-22 & 46 & 42 & 1.7$\pm$0.2 & 11$\pm$3\
& 105028010 & 2011-02-21 & 21 & 18 & 1.3$\pm$1.2 & 0\
& 106005010 & 2011-07-27 & 41 & 37 & $<6.4$ & $<412$\
& 106005020 & 2012-02-07 & 47 & 45 & 1.2$\pm$0.4 & 0\
& 106006010 & 2011-07-26 & 40 & 35 & 0.9$\pm$0.8 & 0\
& 106007010 & 2011-08-23 & 21 & 19 & 1.3$\pm$0.8 & 0\
& 106007020 & 2012-02-08 & 21 & 21 & $<3$ & 10$\pm$7\
& 106008010 & 2011-08-22 & 23 & 22 & $<2$ & $<19$\
& 107005010 & 2012-08-20 & 41 & 39 & 0.9$\pm$0.6 & 12$\pm$3\
& 107005020 & 2013-02-11 & 41 & 36 & 1.1$\pm$0.6 & 0\
& 107006010 & 2012-08-20 & 24 & 20 & $<2$ & 9$\pm$7\
& 107006020 & 2013-02-12 & 22 & 18 & 2.0$\pm$0.4 & 10$\pm$6\
& 108005010 & 2013-08-15 & 41 & 41 & 1.2$\pm$0.4 & $<38$\
& 108005020 & 2014-02-05 & 38 & $-$ & 1.2$\pm$0.5 & $-$\
& 108006010 & 2013-08-16 & 22 & 19 & 1.2$\pm$0.8 & $<20$\
& 108006020 & 2014-02-06 & 19 & $-$ & $<2$ & $-$\
& 109005010 & 2014-08-27 & 20 & $-$ & $<3.5$ & $-$\
& 109005020 & 2015-03-03 & 37 & $-$ & 1.4$\pm$0.3 & $-$\
& 109006010 & 2015-03-04 & 24 & $-$ & 2.5$\pm$0.4 & $-$\
& 109007010 & 2015-03-05 & 31 & $-$ & 1.2$\pm$0.8 & $-$\
Baring, M.G., & Harding, A.K. 2007, , 308, 109-118 Beloborodov, A.M., & Thompson, C. 2007, , 308, 631-639 Dennis, B. R. 1985, , 100, 465 Duncan, R.C., & Thompson, C. 1992, , 392, L9-L13 Enoto, T. 2010a, , 715, 665-670 Enoto, T. 2010b, , 62, 475-485 Enoto, T. 2010c, , 722, L162-L167 Enoto, T. 2012, , 427, 2824-2840 Enoto, T. 2017, , accepted Esposito, P. 2007, , 476, 321-330 Ferández, R., & Thompson, C. 2007, , 660, 615-640 Gavriil, F.P., & Kaspi, V.M. 2004, , 609, L67-L70 Götz, D. 2006, , 445, 313-321 Gutenberg, B., & Richter, C.F. 1956, Annals of Geophysics, 9, pp.1-15 Güver, T., Özel, F., & Lyutikov, M. 2006, arXiv:astro-ph/0611405 Heyl, J.S., & Hernquist, L. 2005a, , 618, 463-473 Heyl, J.S., & Hernquist, L. 2005b, , 362, 777-783 Kagan, Y. Y. 1999, Pure Appl. Geophys., 155, 537 Koyama, K. 2007, , 59, 23-33 Kuiper, L., Hermsen, W., & Mendez, M. 2004, , 613, 1173-1178 Li, J., Rea, N., Torres, F.D., & de O[ñ]{}a-Wilhelmi, E., , 835, 30 (10pp) Lin, L., Göğüş, E., Kaneko Y., & Kouveliotou C. 2013, , 778, 105 (11pp) Lyne, A., & Graham-Smith, F. 2006, Cambridge Astrophysics Series, 38, 149 Lyutikov, M. 2003, , 346, 540-554 Lyutkov, M., & Gavriil F.P. 2006, , 368, 690-706 Maeda, Y. 2009, , 61, 1217-1228 Molkov, S. 2005, , 433, L13-L16 Nakagawa, Y.E. 2007, , 59, 653-678 Nakagawa, Y.E., Yoshida, A., Yamaoka, K., & Shibazaki, N. 2009, , 61, 109-122 Nakagawa, Y.E. 2009, , 61, S387-S393 Nakagawa, Y.E., Makishima, K., & Enoto, T. 2011, , 63, S813-S820 Nakagawa, Y.E., Enoto, T., Makishima, K., Yoshida, A., Yamaoka, K., Sakamoto, T., Rea, N., & Hurley, K. 2011, High-Energy Emission from Pulsars and their Systems, Astrophysics and Space Science Proceedings (Berlin: Springer-Verlag Berlin Heidelberg), 323-327 Olive, J.-F., Hurley, K., Sakamoto, T., Atteia, J.-L., Crew, G., Ricker, G., Pizzichini, G., & Barraud, C. 2004, , 616, 1148-1158 Rea, N. 2009, , 396, 2419-2432 Takahashi, T. 2007, , 59, 35-51 Takahashi, T., Mitsuda, K., Kelley, R., Fabian, A., Mushotzky, R., Ohashi, T., & Petre, R. on behalf of the ASTRO-H Science Working Group 2014, arXiv: astro-ph/1412.2351 Thompson, C., & Beloborodov, A.M. 2005, , 634, 565-569 Thompson, C., & Duncan, R.C. 1995, , 275, 255-300 Trümper, J.E., Zezas, A., Ertan, Ü, & Kylafis, N.D. 2010, , 518, A46 (1-5) Yasuda, T. 2015, , 67, 41 (1-12)
[^1]: http://www.astro.isas.jaxa.jp/suzaku/analysis/hxd/hxdfaq/hxd\_dtcor\_lc.html (last accessed on 2017-10-16).
[^2]: We choose the response matrix files for each observation according to the instruction at http://www.astro.isas.jaxa.jp/suzaku/analysis/hxd/pinnxb/quick/index.html.
[^3]: https://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/pinbgd.html (last accessed Oct 16, 2017)
|
---
abstract: 'We calculate fragmentation functions for a b-quark to fragment into color-singlet P-wave bound states $\bar c b$ in the Heavy Quark Effective Theory with the exact account of $O(1/m_b)$ corrections. We demonstrate an agreement of the obtained results with the corresponding calculations carried out in quantum chromodynamics.'
---
=-10mm 44
[**HEAVY QUARK FRAGMENTATION FUNCTIONS IN THE HEAVY QUARK EFFECTIVE THEORY**]{}
[**Martynenko A.P., Saleev V.A.\
*Samara State University, Samara 443011, Russia***]{}
The study of heavy quarkonia properties is a subject of much current interest for understanding of quark-gluon interaction dynamics. $B_c$-mesons, consisting of b- and c-quarks, hold a unique position in the heavy quarkonia physics [@1]. In the first place, $B_c$ -mesons consist of two heavy quarks, so the predictions of potential models refer to $J/\Psi$- and $\Upsilon$ -mesons as well as to $B_c$ -mesons [@2]. In the second place, $B_c$ -mesons are constructed from the quarks of different flavors and masses, what essentially determine their decay characteristics [@3].
Production of mesons with heavy quarks in $e^+e^-$, $\gamma\gamma$ and $p\bar p$-interactions may be described in nonrelativistic perturbative quantum chromodynamics. At present, two mechanisms were investigated for the production of $B_c$-mesons: the recombination and the fragmentation. In the first case, $B_c$-mesons are formed from heavy quarks, produced independently in hard subprocess. The fragmentational mechanism demands the pair production of $b$- or $c$-quarks in hard subprocess with their subsequent fragmentation to $B_c$-mesons ($\bar b\to B_c \bar c, c\to B_c b$). The relative contributions of these mechanisms in the production cross-sections are different in various reactions. In $e^+e^-$ -annihilation only quark fragmentation is essential [@4]. In $p\bar p$ É $\gamma\gamma$ -interactions the fragmentational mechanism prevails also for $B_c$-mesons production with large transverse momenta [@5; @6]. In the range of small transverse momenta the recombination is dominant and determines the total cross section of $B_c$-meson production in $\gamma\gamma$ and $p\bar p$ interactions. But experimental conditions of $B_c$-meson discovery are more perspective in the large transverse momenta domain. So, the study of heavy quark fragmentation into $B_c$-mesons attracts considerable interest. An approach for the calculation of fragmentation functions $D_{\bar b\to B_c \bar c}, D_{c\to B_c b}$ in nonrelativistic perturbative quantum chromodynamics was suggested in [@7].
At the same time, in the last years there was suggested, based on QCD, the Heavy Quark Effective Theory (HQET) [@8; @9] for description of heavy hadrons properties. HQET make it possible to obtain the finite analytical result with some accuracy even for complicated processes of quark- gluon interaction. In this approach the matrix elements of different processes may be decomposed on degrees of two small parameters: the strong coupling constant $\alpha_s(m_Q)$ and $\Lambda_{QCD}/m_Q$, where $m_Q$ is the mass of the heavy quark. In the limit $m_Q\rightarrow\infty$ the effective lagrangian, which describes the strong interactions of heavy quarks has an exact spin-flavor symmetry [@8]. HQET is successfully used for investigation of exclusive and inclusive hadron decays [@9]. Recently it was shown [@10] that HQET may be used for study of b-quark fragmentation into S-wave pseudoscalar and vector mesons and the corresponding nonpolarized fragmentation functions were calculated. The HQET calculation of the b-quark fragmentation into the transverse and longitudinal polarized S-wave $B_c^*$-mesons have been made in [@10b]. In this work we have calculated the fragmentation functions of b-quark into P-wave color-singlet states ($\bar c b$) to the next to leading order in the heavy quark mass expansion using the methods of HQET.
Fragmentation functions into P-wave $\bar c b$-mesons.
======================================================
Heavy b-quark may fragment into bound states of two heavy quarks ($\bar c b$ - states) with orbital momentum l=1. There are four such states: $^1P_1,
^3P_J$ (J=0,1,2). Heavy quark fragmentation functions into P-wave $B_c$- mesons were calculated by Chen [@11] and Yuan [@12] in QCD, but the results of their calculations disagree. Let carry out the similar calculation of fragmentation functions in the HQET. Let $q=m_b v+k$ is 4-momentum of virtual heavy quark, $p_1=(1-r)Mv+\rho$ and $p_2=rMv-\rho$ are 4-momenta of $b-$ and $\bar c$-quarks correspondingly; $\rho$ is 4-momentum of relative motion. Let also $l=k-\rho$ is 4-momentum of the virtual gluon and k is the residual momentum of the fragmenting heavy quark. Fragmentation functions for the process $b\rightarrow B_c+c$ are determined by the next expression [@7]: $$D(z)=\frac{1}{16\pi^2}\int ds \theta\left(s-\frac{M^2}{z}-\frac{m_c^2}
{1-z}\right)\lim_{q_0\rightarrow\infty}\frac{\sum\vert T\vert^2}{\sum\vert
T_0\vert^2},$$ where $M=m_b+m_c$ is the mass of $B_c$-meson, T is the matrix element for production $B_c+\bar c$ from an off-shell $b^\ast$-quark with virtuality $s=q^2$, and $T_0$ is the matrix element for producing an on-shell b-quark with the same 3-momentum $\vec q$. The calculation can be greatly simplified by using the axial gauge with gauge parameter $n_\mu=(1,0,0,-1)$ in the frame where $q_\mu=(q_0,0,0,\sqrt{q_0^2-s})$:
$$D_{\sigma\lambda}(k)=\frac{1}{k^2+i0}\left[g_{\sigma\lambda}-\frac{k_\sigma
n_\lambda+
k_\lambda n_\sigma}{k\cdot n}+\frac{n^2 k_\sigma k_\lambda}{(k\cdot
n)^2}\right],$$
The part of amplitude T that involves production of the virtual $b^\ast$-quark can be treated as an unknown Dirac spinor $\Gamma$. In the limit $q_0\rightarrow
\infty$, the same spinor factor $\Gamma$ appears in the matrix element $T_0=\bar\Gamma v(q)$, what leads to cancellation of this factor $\Gamma$ in (1).
Let consider the fragmentation of b-quark into color-singlet bound state ($\bar c b$) $^1P_1$. The amplitude of such process involves the spinor factor $v(p_1)\bar u(p_2)$. To project the pair of quarks on $^1P_1$ bound state we have used the next substitution [@13]: $$v(p_2)\bar u(p_1)\rightarrow\sqrt{M}\frac{\hat p_2-m_c}{2m_c}\gamma_5\frac{\hat
p_1+m_b}{2m_b}.$$
The HQET Lagrangian, including the leading and the $1/m_b$ terms is given by [@8; @9]: $$L=\bar h_v\left\{iv\cdot D+\frac{1}{2m_b}\left[C_1(iD)^2-C_2(v\cdot iD)^2-
\frac{C_3}{2}g_s\sigma_{\mu\nu}G^{\mu\nu}\right]\right\}h_v,$$ where $$C_1=1,~~C_2=3\left(\frac{\alpha_s(\mu)}{\alpha_s(m_b)}\right)^{-\frac{8}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
{(33-2n_f)}}-2,~~C_3=\left(\frac{\alpha_s(\mu)}{\alpha_s(m_b)}\right)^{-\frac{9}
{(33-2n_f)}}.$$ All of these coefficients are equal to 1 at the heavy quark mass scale $\mu=m_b$. When we concerned the fragmentation into P-wave mesons, it is necessary to decompose the projecting operator (3) and the gluon propagator (2) on the relative motion momentum $\rho$. Using the Feynman rules, derived from HQET Lagrangian (4), we may write the full fragmentation amplitude into $^1P_1$ -state [@13]: $$iM(n^1P_1)=\frac{\sqrt{4\pi
M}\alpha_s}{3m_cm_b}R'_1(0)\epsilon_\alpha(L_z)\frac{\partial }
{\partial \rho_\alpha}\Big\{\frac{1}{l^2}(-g_{\mu\nu}+\frac{n_\nu l_\mu}{n\cdot
l})$$ $$\Bigl\{\bar u(p')\gamma^\nu(m_c \hat v-\hat\rho-m_c)\gamma_5(m_b\hat
v+\hat\rho+m_b)
[v^\mu+\frac{C_1}{2m_b}(\rho+k)^\mu-$$ $$-\frac{C_2}{2m_b}v\cdot(\rho+k)v^\mu+i\frac{C_3}{2m_b}\sigma^{\mu\lambda}
(\rho-k)_\lambda]\frac{1+\hat v}{2}\Gamma\frac{i}{v\cdot k+\frac{C_1}{2m_b}k^2-
\frac{C_2}{2m_b}(v\cdot k)^2}\Bigr\}\vert_{\rho=0},$$ where $\epsilon_\alpha(L_z)$ is the polarization vector of $^1P_1$ -state. To calculate amplitude (6) it is convenient to divide it into two parts on the degrees of small parameter $1/m_b$. When $m_b\rightarrow\infty$ in vertex function and in the propagator of heavy quark, we obtain the main contribution to the fragmentation amplitude of b-quark in the form: $$iM_1(n^1P_1)=\frac{\sqrt{4\pi M}\alpha_s
2R'_1(0)}{3r^2(s-m_b^2)^3}\epsilon^\ast_\alpha
(L_z)\bar u(p')W^\alpha\gamma_5\Gamma,$$ $$W_\alpha=(s-m_b^2)\left[(\hat v+1)\gamma_\alpha-\frac{v\cdot k}{n\cdot k}\hat n
(\hat v+1)\gamma_\alpha\right]+4mk_\alpha\left[1+\frac{k\cdot v}{k\cdot n}\hat
n
\right](\hat v-1)-$$ $$-2Mr(s-m_b^2)v_\alpha\frac{1}{n\cdot k}\hat n(\hat v-1)+2Mr(s-m_b^2)n_\alpha
\frac{v\cdot k}{(n\cdot k)^2}\hat n(\hat v-1).$$
All calculations of the fragmentation functions, which are rather complicated, were done by means of the system “REDUCE”. Substituting (7) into (1), we obtain: $$%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
D_1(n^1P_1)(y)=N_1\frac{(1-y)^2}{ry^8}(9y^4-4y^3+40y^2+96),~~~N_1=\frac{\alpha_s^2
\vert R'_{nP}(0)\vert^2}{54\pi r^5M^5}.$$ where $y=(1-z+rz)/rz$ is the so called Yaffe-Randall variable [@14], and $r=m_c/M$.
The amplitude of bound $\bar c b$ state $n^3P_J$ production may be derived from (6), changing $\gamma_5\rightarrow \hat\epsilon(S_z)$, where $\epsilon^\mu(S_z)$ is the spin wave function: $$iM(n^3P_J)=\frac{\sqrt{4\pi
M}\alpha_s}{3m_cm_b}R'_1(0)\sum_{S_z,L_z}<1,L_z;1,S_z\vert J,J_z>
\epsilon^\ast_\beta(S_z)\epsilon^\ast_\alpha(L_z)$$ $$\frac{\partial }
{\partial \rho_\alpha}\Bigl\{\frac{1}{l^2}(-g_{\mu\nu}+\frac{n_\nu
l_\mu}{n\cdot l})
\bar u(p')\gamma^\nu(m_c \hat v-\hat\rho-m_c)\gamma_\beta(m_b\hat
v+\hat\rho+m_b)
[v^\mu+\frac{C_1}{2m_b}(\rho+k)^\mu-$$ $$-\frac{C_2}{2m_b}v\cdot(\rho+k)v^\mu+i\frac{C_3}{2m_b}\sigma^{\mu\lambda}
(\rho-k)_\lambda]\frac{1+\hat v}{2}\Gamma\frac{i}{v\cdot k+\frac{C_1}{2m_b}k^2-
\frac{C_2}{2m_b}(v\cdot k)^2}\Bigr\}\vert_{\rho=0},$$ where we have expressed the Clebsch-Gordon coefficients and $\epsilon^\ast_
\beta(S_z), \epsilon^\ast_\alpha(L_z)$ by the bound state polarizations [@13]: $$\sum_{S_z,L_z}<1,L_z;1,S_z\vert J,J_z>\epsilon^\ast_\beta(S_z)
\epsilon^\ast_\alpha(L_z)=\Bigl\{
\begin{array}{l}
\frac{1}{\sqrt{3}}(g_{\alpha\beta}-v_\alpha v_\beta),~~~J=0\\
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
\frac{i}{\sqrt{2}}\epsilon_{\alpha\beta\lambda\rho}v_\lambda\epsilon^\ast_\rho(J_z),~~~J=1\\
\epsilon_{\alpha\beta}(J_z),~~~~~~~~~~~~~~J=2.
\end{array}$$
Taking in (9) only the terms of leading order on $1/m_b$ and doing necessary differentiation on $\rho_\alpha$, we have obtained: $$iM_1(n^3P_J)=\frac{\sqrt{4\pi
M}\alpha_s2R'_1(0)}{3r^2(s-m_b^2)^3}\sum_{S_z,L_z}<1,L_z;1,S_z\vert J,J_z>
\epsilon^\ast_\beta(S_z)\epsilon^\ast_\alpha(L_z)$$ $$\bar u(p')\Bigl\{(s-m_b^2)(1-\frac{k\cdot v}{k\cdot n}\hat
n)\gamma_\alpha\gamma_\beta-
4k_\alpha M(1+\frac{k\cdot v}{k\cdot n}\hat n)\gamma_\beta
-4rM^2\left(\frac{k\cdot v}{k\cdot n}\right)^2n_\alpha\hat n\gamma_\beta\Bigr\}
(1+\hat v)\Gamma.$$ This amplitude determines the basic contribution to fragmentation functions of b-quark into $^3P_J$- state: $$\begin{aligned}
D_1(n^3P_0)(y)=N_1\frac{(y-1)^2}{ry^8}(y^4-4y^3+8y^2+32),\\
D_1(n^3P_1)(y)=N_1\frac{2(y-1)^2}{ry^8}(3y^4-4y^3+16y^2+48),\\
D_1(n^3P_2)(y)=N_1\frac{20(y-1)^2}{ry^8}(y^4+4y^2+8).\end{aligned}$$
The heavy quark fragmentation functions into P-wave mesons were calculated in [@12] using the QCD. Our results (8), (12)-(14) coincide with the calculations of [@12] in the leading order on $1/m_b$. Let consider calculation of the next-to-leading order contributions $O(1/m_b)$ for $^1P_1$ -state. First of all, let observe, that the fragmentation functions, determined by the amplitudes (9) and (11), involves factor $1/[1+r(y-1)]$. Decompose it on degrees of r, we have the correction to (8): $$D_2(n^1P_1)(y)=N_1\frac{(1-y)^3}{y^8}(9y^4-4y^3+40y^2+96),$$
Without any calculations we can obtain the propagator correction $O(1/m_b)$ to (8), which arises as a result of following substitution: $$\frac{1}{v\cdot k}\rightarrow\frac{1}{v\cdot
k+\frac{C_1}{2m_b}k^2-\frac{C_2}{2m_b}
(v\cdot k)^2}\approx\frac{1}{v\cdot k}\left[1+r(-C_1+\frac{1}{2}C_2My)\right].$$ This correction has the next form: $$D_3(n^1P_1)(y)=N_1\frac{(y-1)^2}{y^8}(9y^4-4y^3+40y^2+96)(-2C_1+C_2My).$$ The vertex correction of necessary order in (6) is calculated in a more tedious way: $$iM_{vert.}(n^1P_1)=\frac{\sqrt{4\pi
M}\alpha_s}{3m_cm_b}R'_{nP}(0)\epsilon^\ast_
\alpha(L_z)\frac{\partial}{\partial\rho_\alpha}\Bigl\{\bar u(p')\gamma_\nu(m_c
\hat v-\hat\rho-m_c)\gamma_5$$ $$(m_b\hat v+\hat\rho+m_b)
\left[\frac{C_1}{2m_b}(\rho+k)_\mu-\frac{C_2}{2m_b}v\cdot(\rho+k)v_\mu+i
\frac{C_3}{2m_b}\sigma_{\mu\lambda}(k-\rho)_\lambda\right]$$ $$\frac{1+\hat v}{2}\Gamma\frac{i}{v\cdot k}\frac{1}{l^2}
\left(-g_{\mu\nu}+\frac{n_\nu l_\mu}{n\cdot l}
\right)\Bigr\}\vert_{\rho=0}.$$ It is natural to perform it as a sum of several terms. Putting $\rho=0$ in the square brackets, we obtain: $$\left[\frac{C_1}{2m_b}(\rho+k)_\mu-\frac{C_2}{2m_b}v\cdot(\rho+k)v_\mu-
\frac{C_3}{4m_b}[\gamma_\mu(\hat\rho-\hat k)-(\hat\rho-\hat
k)\gamma_\mu]\right]\vert_{\rho=0}=$$ $$=\left[\frac{C_1}{2m_b}k_\mu-\frac{C_2}{2m_b}(v\cdot k)v_\mu+\frac{C_3}
{4m_b}(\gamma_\mu\hat k-\hat k\gamma_\mu)\right].$$ An addendum $(-C_2(k\cdot v)v_\mu/2m_b)$ in (19) gives the contribution to fragmentation functions which differs only by sign from the similar quark propagator correction: $$D_4(n^1P_1)(y)=N_1\frac{(y-1)^2}{y^8}(9y^4-4y^3+40y^2+96)(-C_2My).$$ Two other terms in square brackets of (19) give rise the next fragmentation amplitudes: $$iM^{(1)}_{vert.}(n^1P_1)=\frac{\sqrt{4\pi M}2\alpha_s}{3m_cm_b}R'_{nP}(0)
\epsilon^\ast_\alpha(L_z)\frac{C_1}{2m_b}\frac
{1}{r(s-m_b^2)^2}
\bar u(p')\Bigl[2k_\alpha Mr\hat n\frac{(s-m_b^2)}{k\cdot n}-$$ $$-4k_\alpha m\hat k+
2n_\alpha\hat n Mr^2\frac{(s-m_b)^2}{(k\cdot n)^2}
+r\hat n\gamma_\alpha\frac{(s-m_b^2)^2}{k\cdot n}-\hat k\gamma_\alpha(s-m_b^2)
\Bigr]\gamma_5(1+\hat v)\Gamma,$$ $$iM^{(2)}_{vert.}(n^1P_1)=\frac{\sqrt{4\pi M}2\alpha_s}{3m_cm_b}R'_{nP}(0)
\epsilon^\ast_\alpha(L_z)\frac{C_3}{m_b}\frac{1}{r^2(s-m_b^2)^2}$$ $$\bar u(p')\Bigl\{-2k_\alpha Mr\hat n\frac{(s-m_b^2)}{k\cdot n}-8k_\alpha M\hat
k-
2(s-m_b^2)k_\alpha+2Mr\frac{(s-m_b^2)}{k\cdot n}\hat n\gamma_\alpha\hat k+$$ $$+r\hat n\gamma_\alpha\frac{(s-m_b^2)^2}{k\cdot n}+r(s-m_b^2)\hat k\gamma_\alpha
\Bigr\}\gamma_5(1+\hat v)\Gamma.$$ A further part of vertex correction $O(1/m_b)$ appears, when we differentiate the expression in square brackets of (18) on $\rho_\alpha$: $$iM^{(3)}_{vert.}(n^1P_1)=\frac{\sqrt{4\pi M}\alpha_s}{3m_cm_b}R'_{nP}(0)
\epsilon^\ast_\alpha(L_z)\frac{1}{2r(s-m_b^2)^2}$$ $$\bar u(p')\Bigl[C_1\left(k_\alpha\hat n\frac{1}{k\cdot n}-\gamma_\alpha\right)
+C_3\Bigl(k_\alpha\hat n\frac{1}{k\cdot n}+\frac{1}{2M}\frac{(s-m_b^2)}{k\cdot
n}
\hat n\gamma_\alpha-$$ $$\frac{1}{k\cdot n}\hat n\hat k\gamma_\alpha+2\gamma_\alpha\Bigr)
\Bigr]\gamma_5(1+\hat v)\Gamma.$$
The contributions of matrix elements (21)-(23) to fragmentation functions have the following form: $$D_5(n^1P_1)(y)=4N_1\frac{(y-1)^2y}{y^8}(3y^3-2y^2+6y+32)C_1,$$ $$D_6(n^1P_1)(y)=4N_1\frac{(y-1)y^2}{y^8}(-3y^3-4y^2+14y-16)C_3,$$ $$D_7(n^1P_1)(y)=2N_1\frac{(y-1)^2y}{y^8}\left[(3y^3+6y^2+4y+32)C_1+2y
(3y^2-2y-16)C_3\right].$$
The small component of heavy quark field also leads to the correction of type O(r) in function D(y). Substituting the small quark component propagator to (6) instead of heavy quark propagator $$\frac{i}{v\cdot k}\frac{1+\hat v}{2}\left[\frac{1}{2m_b}\sigma_{\mu\lambda}
k^\lambda\right]\frac{1-\hat v}{2},$$ we obtain the next amplitude of $B_c$- meson production: $$iM_{prop.}(n^1P_1)=-\frac{\sqrt{4\pi M}2\alpha_s}{3m_cm_b}R'_{nP}(0)
\epsilon^\ast_\alpha(L_z)\frac{4}{r^2(s-m_b^2)^3}$$ $$\bar u(p')\left[4k_\alpha M\hat k+2k_\alpha(s-m_b^2)+r\frac{(s-m_b^2)^2}
{k\cdot n}\hat n\gamma_\alpha+(s-m_b^2)\hat k\gamma_\alpha\right]
\gamma_5(1-\hat v)\Gamma.$$ This amplitude gives the contribution to fragmentation function of the kind: $$D_8(n^1P_1)(y)=6N_1\frac{(y-1)y}{y^8}(y^5+2y^4+3y^3+16y-16).$$
The calculation of b-quark fragmentation functions into $^3P_J$ states was done in a similar way. The results of $D(n^3P_J)$ calculations are represented in the next section.
Discussion of the results.
==========================
Let analyse b-quark fragmentation functions into P-wave mesons. As was mentioned earlier, in the leading order on $1/m_b$ the expressions (8), (12)- (14) coincide with QCD calculations of Yuan [@12]. The correction $O(r)$ to (8) is determined by the sum of terms (15), (17), (20), (24)-(26), (29). Setting $C_1=C_3=1$, we obtain the fragmentation function $D(n^1P_1)$ to the next-to-leading order on r: $$D(n^1P_1)(y)=\frac{\alpha_s^2\vert R'_{nP}(0)\vert^2}{54\pi r^5M^5}
\frac{(y-1)^2}{ry^8}\Bigl[(9y^4-4y^3+40y^2+96)-$$ $$-r(3y^5-31y^4+32y^3+8y^2-192y+96)\Bigr].$$ Similarly, for the $^3P_J$ states, we have: $$D(^3P_0)(y)=\frac{\alpha_s^2\vert R'_{nP}(0)\vert^2}{54\pi r^5M^5}
\frac{(y-1)^2}{ry^8}\Bigl[(y^4-4y^3+8y^2+32)+$$ $$+\frac{r}{3}(3y^5-11y^4+392y^2+192y-96)\Bigr],$$ $$D(^3P_1)(y)=\frac{\alpha_s^2\vert R'_{nP}(0)\vert^2}{27\pi r^5M^5}
\frac{(y-1)^2}{ry^8}\Bigl[(3y^4-4y^3+16y^2+48)+$$ $$+r(3y^5+5y^4+8y^3+32y^2+96y-48)\Bigr],$$ $$D(^3P_2)(y)=\frac{10\alpha_s^2\vert R'_{nP}(0)\vert^2}{27\pi r^5M^5}
\frac{(y-1)^2}{ry^8}\Bigl[(y^4+4y^2+8)+$$ $$+\frac{r}{15}(-3y^5-y^4+36y^3-164y^2+240y-120)\Bigr].$$ It follows immediately from (30)-(33), that our results coincide with the calculations of Yuan [@12] with the accuracy $O(r)$, if we take into account, that the nonperturbative factor of (30)-(33) in [@12] contains reduced mass $\mu$ contrary to our factor $rM$. Using obtained fragmentation functions (30)-(33), we may calculate the fragmentation probabilities of corresponding $(\bar c b)$-mesons [@7]: $$P_{b\rightarrow \bar c b}(n^1P_1)=\int_0^1 dz D_{b\rightarrow \bar c b}
(n^1P_1)(z,r)=$$ $$=\frac{\alpha_s^2\vert R'_{nP}(0)\vert^2}{54\pi r^5M^5}\Bigl[
\frac{r\ln r}{(1-r)^8}\left(21+32r+110r^2-184r^3-387r^4-168r^5\right)+$$ $$+\frac{1}{210(1-r)^7}\left(1032+4497r+21353r^2-2762r^3-65202r^4-76199r^5-
3679r^6\right)\Bigr].$$ $$P_{b\rightarrow \bar c b}(n^3P_0)
=\frac{\alpha_s^2\vert R'_{nP}(0)\vert^2}{54\pi r^5M^5}\Bigl[
\frac{r\ln r}{3(1-r)^8}\left( 3+8r+98r^2+1496r^3-1221r^4-960r^5\right)+$$ $$+\frac{1}{630(1-r)^7}\left(360+1767r-14977r^2+180778r^3+158658r^4-427697r^5-
19849r^6\right)\Bigr].$$ $$P_{b\rightarrow \bar c b}(n^3P_1)
=\frac{\alpha_s^2\vert R'_{nP}(0)\vert^2}{27\pi r^5M^5}\Bigl[
\frac{r\ln r}{(1-r)^8}\left(3+16r+10r^2+112r^3-237r^4-192r^5\right)+$$ $$+\frac{1}{210(1-r)^7}\left(348-177r+8419r^2+2714r^3+14334r^4-81769r^5-
4349r^6\right)\Bigr].$$ $$P_{b\rightarrow \bar c b}(n^3P_2)
=\frac{10\alpha_s^2\vert R'_{nP}(0)\vert^2}{27\pi r^5M^5}\Bigl[
\frac{r\ln r}{15(1-r)^8}\left(33-68r+130r^2-548r^3-291r^4+24r^5\right)+$$ $$+\frac{1}{3150(1-r)^7}\left(1710+2697r-2525r^2-10330r^3-146760r^4+3425r^5+
583r^6\right)\Bigr].$$ Putting here $\vert R'_{2P}(0)\vert^2=0.201$ GeV$^5$, $m_c=1.5$ GeV, $m_b=4.9$ GeV and $\alpha_s(2m_c)$=0.38 ($2m_c$ is a minimal energy of exchanged gluon), we have obtained the numerical value of the fragmentation probabilities, which are presented in table. We see, that our integral probabilities of P-wave $\bar c b$ meson production, founded by means of the b-quark fragmentation functions in the HQET are in good agreement with the results of QCD calculations of [@12].\
--------------------------------------------------------------------------------------------------------------------------------------------------
$P_{b\rightarrow \bar cb}(2^1P_1)$ $P_{b\rightarrow \bar cb}(2^3P_0)$ $P_{b\rightarrow\bar cb}(2^3P_1)$ $P_{b\rightarrow\bar c b}(2^3P_2)$
------------------------------------ ------------------------------------ ----------------------------------- ------------------------------------
$6.4\cdot 10^{-5}$ $2.5\cdot 10^{-5}$ $7.3\cdot 10^{-5}$ $10.5\cdot
10^{-5}$
--------------------------------------------------------------------------------------------------------------------------------------------------
So the performed calculations show that the Heavy Quark Effective Theory may be successfully used for the study of the heavy quark fragmentation. In this approach we may systematically take into account the $O(1/m_b)$ corrections in the amplitudes and the probabilities of the fragmentation, what increases the accuracy of HQET calculations. Moreover, it seems more important, that HQET leads to finite analytical answer, when we study complicated problems in the heavy quark physics. The approach, based on HQET, may be used for calculation of heavy quark fragmentation functions into D-wave mesons, and for the investigation of $B_c$ -meson hadroproduction.
We are grateful to Faustov R.N., Kiselev V.V., Likhoded A.K. for useful discussions of $B_c$ meson physics and the Heavy Quark Effective Theory, and to Braaten E., Cheung K., Fleming S. for the valuable information about obtained results.
This work was done under the financial support of the Russian Fund of Fundamental Researches (Grant 93-02-3545) and by State Committee on High Education of Russian Federation (Grant 94-6.7-2015).
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abstract: |
We consider foliations of the whole three dimensional hyperbolic space $\mathbb{H}^{3}$ by oriented geodesics. Let $\mathcal{L}$ be the space of all the oriented geodesics of $\mathbb{H}^{3}$, which is a four dimensional manifold carrying two canonical pseudo-Riemannian metrics of signature $\left( 2,2\right) $. We characterize, in terms of these geometries of $\mathcal{L}$, the subsets $\mathcal{M}$ in $\mathcal{L}$ that determine foliations of $\mathbb{H}^{3}$. We describe in a similar way some distinguished types of geodesic foliations of $\mathbb{H}^{3}$, regarding to which extent they are in some sense trivial in some directions: On the one hand, foliations whose leaves do not lie in a totally geodesic surface, not even at the infinitesimal level. On the other hand, those for which the forward and backward Gauss maps $\varphi ^{\pm }:\mathcal{M}\rightarrow \mathbb{H}^{3}\left( \infty \right) $ are local diffeomorphisms. Besides, we prove that for this kind of foliations, $\varphi ^{\pm }$ are global diffeomorphisms onto their images.
The subject of this article is within the framework of foliations by congruent submanifolds, and follows the spirit of the paper by Gluck and Warner where they understand the infinite dimensional manifold of all the great circle foliations of the three sphere.
author:
- |
Yamile Godoy and Marcos Salvai [^1]\
[, Ciudad Universitaria, 5000 Córdoba, Argentina]{}\
[email protected], [email protected]
bibliography:
- 'mybib.bib'
date:
title: |
Global smooth geodesic foliations\
of the hyperbolic space
---
*Key words and phrases:* geodesic foliation, hyperbolic space, space of oriented lines *Mathematics Subject Classification:* 53C12, 53C40, 53C50.
Introduction
============
Geodesic foliations
-------------------
A smooth geodesic foliation of a Riemannian manifold $N$ is given by a smooth unit vector field $V$ on $N$ all of whose integral curves, the leaves, are geodesics. Throughout the paper, smooth means of class $\text{C}^\infty$.
The standard examples of geodesic foliations of $\mathbb{R}^3$ are given by foliating the space by parallel planes which are in turn foliated by parallel lines, with smoothly varying directions. One can construct other examples by writing $\mathbb{R}^{3}$ smoothly as the disjoint union of the $z
$-axis and one-sheet hyperboloids of revolution around that axis, and considering on each one, coherently, one of the two ways of ruling it (the striction circles of the hyperboloids do not need to be at the same height). Applying a linear isomorphism one obtains new examples.
Notice that the foliations of the hyperbolic three space by totally geodesic surfaces, as well as the foliations of the hyperbolic plane by geodesics, are by far not as rigid as in the Euclidean case [@Ferus-1973; @Browne-1984]. So, the hyperbolic analogues of the standard examples of Euclidean geodesic foliations are richer. Recently, Nuchi studied the fiberwise homogeneous geodesic foliations of the three dimensional space forms [@Nuchi].
Global smooth geodesic foliations of the three dimensional Euclidean space were characterized in [@Salvai-2009] in terms of the geometry of the space of oriented lines. Now, we deal with the analogous problem in the hyperbolic context. The general basic theory for the Euclidean case is still useful, but some crucial definitions and arguments in the proofs must be adapted to the hyperbolic setting.
Let $\mathbb{H}^{3}$ be the three dimensional hyperbolic space of constant sectional curvature $-1$. Let $\mathcal{L}_{0}$ and $\mathcal{L}_{-}$ be the spaces of oriented geodesics of $\mathbb{R}^{3}$ and $\mathbb{H}^{3}$, respectively, which are manifolds of dimension four admitting canonical neutral pseudo-Riemannian metrics: $\mathcal{L}_{0}$ admits one (associated with the cross product) [@Guilfoyle-2005; @Salvai-2005], and $\mathcal{L}_{-}$ admits two of them, $g_{\times }$ and $g_{K}$, coming from the cross product and the Killing form on Iso$\,\left( \mathbb{H}^{3}\right) $, respectively [@Salvai-2007; @Georgiou-2010]. See the precise definitions below in the preliminaries. Distinguished geometries on spaces of oriented geodesics are also studied in [@Alekseevsky-2011] and [@Anciaux-2012].
While the geodesic foliations of $\mathbb{R}^{3}$ are described in terms of the canonical neutral metric on $\mathcal{L}_{0}$, the characterization of the geodesic foliations of $\mathbb{H}^{3}$ involves both $g_{\times }$ and $g_{K}$. This situation appears also in other problems in hyperbolic geometry; for instance, A. Honda needed both canonical neutral metrics on $\mathcal{L}_{-}$ in the study of the isometric immersions of the hyperbolic plane into $\mathbb{H}^{3}$ [@Honda-2012].
We will have two types of distinguished foliations. We call a geodesic foliation $\emph{nondegenerate}$ if the leaves do not lie in a totally geodesic surface, not even at the infinitesimal level. More precisely, if the only eigenvectors of $\nabla V$ are in $\mathbb{R}V$, where $V$ is the unit vector field that determines the foliation.
We have also another notion, which turns out to be weaker: a *semi-nondegenerate* foliation does not resemble, in any direction, a trivial foliation whose leaves are all orthogonal to a fixed horosphere. In the upper half space model of $\mathbb{H}^{3}$, these are foliations congruent to the those with vertical geodesics, with both orientations. See Definition \[nondeg\]. Both concepts generalize the Euclidean notion of nondegeneracy (see Corollary 4 in [@Salvai-2009]). For a higher dimensional (local) analogue, see the foliations of ${\mathbb{R}}^{n}$ by pairwise skew $p$-planes in [@Tabachnikov-2013].
We want to emphasize that the statements of the results are similar to those of [@Salvai-2009], but the technical meaning of the definitions involved, for instance, (almost) semidefinite submanifolds and (semi-)nondegenerate foliations are quite different in the Euclidean and the hyperbolic cases.
Foliations by congruent submanifolds
------------------------------------
The general setting of this article is the study of foliations of a smooth manifold $N$ by congruent submanifolds: Suppose that the Lie group $G$ acts on $N$, let $M$ be a closed submanifold of $N$, and let $\mathcal{C}$ be the set of all submanifolds of $N$ congruent to $M$ via $G$. Let $H$ be the subset of all points in $G$ that preserve $M$. Then $H$ is a closed Lie subgroup of $G$ since $M$ is closed in $N$ and we can identify $\mathcal{C}\cong G/H$. Sometimes $\mathcal{C}$ admits distinguished $G$-invariant geometries, which are useful to describe the foliations of $N$ by submanifolds congruent to $M$. More precisely, the problem is the following:
Describe geometrically which subsets $\mathcal{M}$ of $\mathcal{C}$ determine foliations of $N$.
The paradigm is the paper [@Gluck-1983], where foliations of $S^{3}$ by great circles are characterized in this way. See also [@Salvai-2002] (a partial generalization of [@Gluck-1983]) and [@Salvai-2009], with the global foliations of $\mathbb{R}^{3}$ by oriented lines, which includes a pseudo-Riemannian reformulation of the principal result of [Gluck-1983]{}. M. Czarnecki and R. Langevin are currently working, in this context, on the classification of codimension two totally geodesic foliations of the complex hyperbolic space.
Preliminaries
=============
A smooth geodesic foliation of $\mathbb{H}^{3}$ is given by a smooth unit vector field $V$ on $\mathbb{H}^{3}$ all of whose integral curves, the leaves, are geodesics. The set $\mathcal{M}$ of all the leaves admits a canonical differentiable structure. For the sake of completeness, we include its existence as a proposition.
\[est.dif\] The set $\mathcal{M}$ of all the leaves of a geodesic foliation of $\mathbb{H}^{3}$ admits a unique differentiable structure such that the canonical projection $P:\mathbb{H}^{3}\rightarrow \mathcal{M}$ is a smooth submersion.
Let $V$ be the smooth unit vector field on $\mathbb{H}^{3}$ associated with the geodesic foliation and consider the smooth distribution given by $\mathcal{D}=\mathbb{R}V$. By Theorem VIII in [@Palais-1957], it suffices to prove that $\mathcal{D}$ is regular, that is, if for each $p\in \mathbb{H}^{3}$ there is a cubical coordinate system $(U,(x_{1},x_{2},x_{3}))$ centered at $p$ such that $\left\{ \left. {(\partial /\partial x_{3}})\right\vert _{q}\right\} $ is a basis of $\mathcal{D}_{q}$ for all $q\in U$ and each leaf of $\mathcal{D}$ intersects $U$ in at most one 1-dimensional slice $\left( x_{1},x_{2}\right) =$ const.
Let us see that for each $p\in \mathbb{H}^{3}$ we have such a coordinate system. Let $u_{1},u_{2}\in T_{p}\mathbb{H}^{3}$ such that $\left\{
u_{1},u_{2},V(p)\right\} $ is an orthonormal basis of $T_{p}\mathbb{H}^{3}$ and let $F:\mathbb{R}^{2}\rightarrow \mathbb{H}^{3}$ be the totally geodesic submanifold given by $F\left( x,y\right) =$ Exp$_{p}(xu_{1}+yu_{2})$ (here $\text{Exp}$ is the geodesic exponential map). We consider the smooth map $$\alpha :\mathbb{R}^{3}\rightarrow \mathbb{H}^{3},\hspace{0.5cm}\alpha
(x,y,t)=\vartheta _{t}(F\left( x,y\right) )\text{,}$$where $\vartheta _{t}$ is the flow of $V$. Since $d\alpha _{0}$ is an isomorphism, there exist $\varepsilon >0$ and an open neighborhood $U\subset
\mathbb{H}^{3}$ of $p$ such that ${\alpha }:(-\varepsilon ,\varepsilon
)^{3}\rightarrow U$ is a diffeomorphism. Hence, $\left( U,\alpha
^{-1}=(x_{1},x_{2},x_{3})\right) $ is a cubical coordinate system centered at $p$ such that $\left. {(\partial /\partial x_{3}})\right\vert _{q}=V(q)$ for all $q\in U$. The 1-dimensional slices are clearly integral submanifolds od $\mathcal{D}$ and no leaf of $\mathcal{D}$ intersects two different slices of $U$, since geodesics in $\mathbb{H}^{3}$ transverse to a totally geodesic surface intersect it at most at one point.
The space $\mathcal{L}$ of all complete oriented geodesics of $\mathbb{H}^{3}
$ (up to orientation preserving reparametrizations) admits a unique differentiable structure such that the canonical projection $\Pi :T^{1}\mathbb{H}^{3}\rightarrow \mathcal{L}$ is a differentiable submersion (by [@Palais-1957], as above, with the spray as the vector field giving the foliation). We may think of $c\in \mathcal{L}$ as the equivalence class of unit speed geodesics $\gamma :\mathbb{R}\rightarrow \mathbb{H}^{3}$ with image $c$ such that $\{\dot{\gamma}(t)\}$ is a positive basis of $T_{\gamma
(t)}c$ for all $t$. If $\ell \in \mathcal{L}$, then by abuse of notation we sometimes write $z\in \ell $, meaning that $z$ is in the underlying line.
Fixing a point $o\in \mathbb{H}^{3}$, let $$H:T(T_{o}^{1}\mathbb{H}^{3})\rightarrow \mathcal{L} \label{H}$$be the map defined as follows: Let $u\in T_{o}^{1}\mathbb{H}^{3}$ and $v\in
T_{o}\mathbb{H}^{3}$ with $u\bot v$, then $H(u,v)$ is the oriented geodesic with initial point $\text{Exp}_{o}(v)$ and initial velocity the parallel transport of $u$ along the geodesic $t\mapsto \text{Exp}_{o}(tv)$ at $t=1$. Proposition 4.14 of [@Beem-1996] asserts that $H$ is a diffeomorphism.
Let $\gamma $ be a complete unit speed geodesic of $\mathbb{H}^{3}$ and let $\mathcal{J}_{\gamma }$ be the space of all Jacobi vector fields along $\gamma $ which are orthogonal to $\dot{\gamma}$. There exists a well-defined canonical isomorphism $$T_{\gamma }:\mathcal{J}_{\gamma }\rightarrow T_{[\gamma ]}\mathcal{\mathcal{L}}\text{,}\hspace{1cm}T_{\gamma }(J)=\left. {\frac{d}{dt}}\right\vert
_{0}[\gamma _{t}]\text{,} \label{isoT}$$where $\gamma _{t}$ is any variation of $\gamma $ by unit speed geodesics associated with $J$ (see [@Salvai-2007]).
Given a tangent vector $X$ to a pseudo-Riemannian manifold, we denote $\left\Vert X\right\Vert =\left\langle X,X\right\rangle $, the square norm associated with the metric $\left\langle .\,,.\right\rangle $, and $\left\vert
X\right\vert =\sqrt{\left\vert \left\langle X,X\right\rangle \right\vert }$. Also, given $v\in T\mathbb{H}^{3}$, we denote by $\gamma _{v}$ the unique geodesic in $\mathbb{H}^{3}$ with initial velocity $v$.
Now, we recall the definition of the two canonical pseudo-Riemannian metrics $g_{\times }$ and $g_{K}$ on $\mathcal{\mathcal{L}}$ given in [@Salvai-2007 Theorem 1]. In terms of the isomorphism (\[isoT\]) the square norms of these metrics may be written as follows [@Salvai-2007 page 362]: For $J\in \mathcal{J}_{\gamma }$, $$\begin{array}{lll}
\Vert T_{\gamma }(J)\Vert _{\times } & = & \langle \dot{\gamma}\times
J,J^{\prime }\rangle \text{,} \\
\Vert T_{\gamma }(J)\Vert _{K} & = & \left\vert J\right\vert
^{2}-\left\vert J^{\prime }\right\vert ^{2}\text{.}\end{array}
\label{n}$$The cross product $\times $ is induced by a fixed orientation of $\mathbb{H}^{3}$ and $J^{\prime }$ denotes the covariant derivative of $J$ along $\gamma $ (the right hand side in the expressions are constant functions, so they are well defined).
Let $\mathcal{M}$ be a submanifold of $\mathcal{L}$ and we take $[\gamma]\in {\mathcal{M}}$. Next we show that any tangent vector in $T_{[\gamma ]}\mathcal{M}$ corresponds (via $T_{\gamma }$) to a Jacobi vector field in $\mathcal{J}_{\gamma }$ associated with a variation of $\gamma$ by unit speed geodesics whose equivalence classes are in $\mathcal{M}$. In fact, given $X\in T_{[\gamma ]}\mathcal{M}$, there exists a smooth curve $c:(-\varepsilon ,\varepsilon )\rightarrow \mathcal{M}$ with $c(0)=[\gamma ]$ and $\dot{c}(0)=X$. By Proposition 3 in [@Salvai-2007], there exists a standard presentation of $c$, that is a function $\varphi :\mathbb{R}\times (-\varepsilon ,\varepsilon )
\rightarrow \mathbb{H}^{3}$ such that $s\mapsto \alpha _{t}(s):=\varphi (s,t)$ is a unit speed geodesic of $\mathbb{H}^{3}$ satisfying $c(t)=[\alpha _{t}]$, $\left\langle \dot{\beta}
(t),\dot{\alpha _{t}}(0)\right\rangle =0$ for all $t\in (-\varepsilon ,\varepsilon )$, where $\beta
(t)=\varphi (0,t)$, and $\varphi(0,0)=\gamma (0)$. It is easy to see that $$J(s)=\left. {\frac{d}{dt}}\right\vert _{0}\alpha _{t}(s)$$is a Jacobi field in $\mathcal{J}_{\gamma }$ and it satisfies $$T_{\gamma }(J)=\left. {\frac{d}{dt}}\right\vert _{0}[\alpha _{t}]=\left. {\frac{d}{dt}}\right\vert _{0}c(t)=X\text{.}$$
Global geodesic foliations of $\mathbb{H}^{3}$
==============================================
In this section we characterize, in terms of the canonical neutral metrics on $\mathcal{L}$, the subsets $\mathcal{M}$ in $\mathcal{L}$ that determine foliations of $\mathbb{H}^{3}$. To this end, it is convenient to give the following definition.
A submanifold $\mathcal{M}$ of $\mathcal{L}$ is said to be *almost semidefinite* if $\Vert X\Vert _{\times }=0$ for $X\in T\mathcal{M}$ only if $\Vert X\Vert _{K}\geq 0$.
\[adefinite\] Let $\mathcal{M}$ be a surface contained in $\mathcal{L}$ *(*the inclusion is a priori not even smooth*)*. Then the following statements are equivalent:
1. The surface $\mathcal{M}$ is the space of leaves of a smooth foliation of $\mathbb{H}^{3}$ by oriented geodesics, with the canonical differentiable structure.
2. The surface $\mathcal{M}$ is a closed almost semidefinite connected submanifold of $\mathcal{L}$.
Besides, if ${\mathcal{M}}$ satisfies *(a)* or *(b)*, ${\mathcal{M}}$ is diffeomorphic to ${\mathbb{R}}^2$.
Let $o$ be a fixed point in $\mathbb{H}^{3}$. We recall that $$T(T_{o}^{1}\mathbb{H}^{3})=\{(u,v)\in T_{o}^{1}\mathbb{H}^{3}\times T_{o}\mathbb{H}^{3} : \left\langle u,v\right\rangle =0\}\cong TS^{2}\text{.}$$Let $f:\mathcal{L}\rightarrow \mathbb{H}^{3}$ be the map that assigns to each oriented unit speed geodesic $\ell $ of the hyperbolic space its closest point to $o$. Considering the following diagram, $$\begin{array}{ccc}
T\left( T_{o}^{1}\mathbb{H}^{3}\right) & \overset{H}{\longrightarrow } &
\mathcal{L} \\
\pi _{2}\downarrow \ \ \ \ & & \ \ \downarrow f \\
T_{o}\mathbb{H}^{3} & \overset{\text{Exp}_{o}}{\longrightarrow } & \mathbb{H}^{3}\end{array}$$we have that $f$ is a smooth map, where $H$ is the diffeomorphism given in (\[H\]) and $\pi _{2}$ is the projection onto the second component.
Let $D:\mathcal{L}\rightarrow \mathbb{R}$ be the square distance from $o$. In particular, if $\ell =H\left( u,v\right) $, we have that $D(\ell
)=|v|^{2} $ and so $D$ is smooth.
For any unit speed geodesic $\gamma$, let $\psi _{\gamma }:T_{[\gamma] }
\mathcal{L}\simeq \mathcal{J}_{\gamma }\rightarrow \dot{\gamma}(0)^{\bot }$ be the linear map defined by $\psi _{\gamma }(J)=J(0)$.
\[distancia\] Let $\mathcal{M}$ be an almost semidefinite closed connected two-dimensional submanifold of $\mathcal{L}$.
1. For any $\ell=[\gamma] \in \mathcal{M}$, $\left.{\psi_{\gamma}}\right\vert_{T_{\ell}\mathcal{M}}$ is surjective.
2. Any critical point $\ell $ of $\left. D\right\vert _{\mathcal{M}}$ is a strict local minimum of $\left. D\right\vert _{\mathcal{M}}$ with $D(\ell
)=0$. Moreover, $D(\ell _{n})\rightarrow \infty $ as $n\rightarrow \infty $ for any sequence $\ell _{n}$ in $\mathcal{M}$ without cluster points.
\(a) It suffices to show that the map is injective ($T_{\ell }\mathcal{M}$ and $\dot{\gamma}(0)^{\bot }$ have the same dimension). If $\psi _{\gamma}(J)=0$, then $J(0)=0$ and from (\[n\]) we have that $\Vert T_{\gamma
}(J)\Vert _{\times}=0$. Since $\mathcal{M}$ is almost semidefinite, using (\[n\]) we obtain that $J^{\prime }(0)=0$, thus $J\equiv 0$.
\(b) Let $(u,v)\in T(T_{o}^{1}\mathbb{H}^{3})$ and let $H(u,v)=\ell$. In particular, $\ell =[\gamma _{U}]$ with $U=\left. \tau _{\gamma
_{v}}\right\vert _{0}^{1}(u)$, where $\tau _{\gamma _{v}}$ denotes the parallel transport along $\gamma _{v}$. By $(\text{a})$, there exists a Jacobi vector field $J\in \mathcal{J}_{\gamma _{U}}$, with $T_{\gamma
_{U}}\left( J\right) \in T_{\ell }\mathcal{M}$, such that $J(0)=\dot{\gamma}_{v}(1)$. We take a variation of $\gamma _{U}$ by unit speed geodesics $\Gamma (s,t)=\gamma _{t}(s)$ associated with $J$, with $[\gamma
_{t}]=H(u_{t},v_{t})\in \mathcal{M}$. We call $\alpha $ the smooth curve in $\mathcal{M}$ given by $\alpha (t)=[\gamma _{t}]$. We have $v_{t}=\pi
_{2}\circ \,H^{-1}\circ \,\alpha (t)$, thus $v_{t}$ is a smooth curve in $T_{o}\mathbb{H}^{3}$.
Suppose that $\ell \in \mathcal{M}$ is a critical point of $\left.
D\right\vert _{\mathcal{M}}$. First we verify that $D(\ell )=0$. We compute
$$0=\dot{\alpha}\left( 0\right) (D)=\left. \dfrac{d}{dt}\right\vert
_{0}D(\alpha (t))=\left. \frac{d}{dt}\right\vert _{0}|v_{t}|^{2}=2\langle
v,v_{0}^{\prime }\rangle \text{.} \label{pi}$$
Now, let $\lambda _{t}$ be the smooth curve in $\mathbb{R}$ such that $\text{Exp}_{o}(v_{t})=\gamma _{t}(\lambda _{t})$ (in particular, $\lambda _{0}=0$) and consider the Jacobi vector field $K$ associated with the geodesic variation $(s,t)\rightarrow \Delta (s,t)=\Gamma (s+\lambda _{t},t)$, that is, $$K(s)=\lambda _{0}^{\prime }\,\dot{\gamma}_{U}(s)+J(s)\text{.}$$Since $\text{Exp}_{o}(v_{t})=\Delta (0,t)$, we have $$(d\,\text{Exp}_{o})_{v}(v_{0}^{\prime })=K(0)=\lambda _{0}^{\prime }\,U+J(0)\text{.} \label{v'0}$$Besides, $(d\,\text{Exp}_{o})_{v}(v)=J(0)$. Then, by (\[pi\]), the Gauss Lemma and (\[v’0\]), we obtain $$0=\langle v,v_{0}^{\prime }\rangle =\langle \left( d\,\text{Exp}_{o}\right)
_{v}(v),\left( d\,\text{Exp}_{o}\right) _{v}(v_{0}^{\prime })\rangle
=|J(0)|^{2}=|v|^{2}=D(\ell ), \label{D}$$as desired. Next we see that $\ell $ is a strict local minimum. Let $X$ be a nonzero vector in $T_{\ell }\mathcal{M}$ and let $J\in \mathcal{J}_{\gamma _{u}}$ such that $X=T_{\gamma _{u}}(J)$. Since $J$ is not an identically zero Jacobi vector field, by (a) we have $J(0)\neq 0$. As above, we take a smooth curve $[\gamma _{t}]=H(u_{t},v_{t})$ in $\mathcal{M}$ such that its initial velocity is $X$ and $J(s)=\left. \frac{
d}{dt}\right\vert _{0}\gamma _{t}(s)$. Then, $$\left. \frac{d^{2}}{dt^{2}}\right\vert _{0}D([\gamma _{t}])=2(\langle
v_{0},v_{0}^{\prime \prime }\rangle +|v_{0}^{\prime }|^{2})=2|v_{0}^{\prime
}|^{2}>0,$$since $v_0 =v=0$ by (\[D\]) and $v_{0}^{\prime }\neq 0$ by (\[v’0\]).
The last statement is proved in a similar way as in Lemma 5(b) of [Salvai-2009]{}.
(a)$\Rightarrow $(b) Suppose that the foliation is given by a smooth unit vector field $V$ on $\mathbb{H}^{3}$ and let $P:\mathbb{H}^{3}\rightarrow \mathcal{M}$ be the smooth submersion induced by $V$ as in Proposition \[est.dif\]. Since $\mathcal{M}=P(\mathbb{H}^{3})$, we have that $\mathcal{M}$ is connected. The fact that the inclusion $i:\mathcal{M}\rightarrow \mathcal{L}$ is a submanifold is proved in the same way as in the Euclidean case (the beginning of (a)$\Rightarrow$(b) in the proof of Theorem 2 in [@Salvai-2009]).
Let us see that $\mathcal{M}$ is almost semidefinite. Let $X\in T_{[\gamma ]}\mathcal{M}$ with $\Vert X\Vert _{\times }=0$ and let $J\in \mathcal{J}_{\gamma }\mathcal{\ }$with $X=T_{\gamma }(J)$. We want to see that $\left\Vert X\right\Vert _{K}\geq 0$. First, we observe that if $\gamma _{t}$ is any variation of $\gamma $ by geodesics in the foliation, associated with $J $, we have that $\dot{\gamma}_{t}\left( s\right) =V\left( \gamma
_{t}\left( s\right) \right) $ and so we compute $$J^{\prime }(s)=\frac{D}{ds}\left. \frac{d}{dt}\right\vert _{0}\gamma
_{t}\left( s\right) =\left. \frac{D}{dt}\right\vert _{0}\frac{d}{ds}\gamma
_{t}\left( s\right) =\left. \frac{D}{dt}\right\vert _{0}\dot{\gamma}_{t}\left( s\right) =\nabla _{J(s)}V\text{. } \label{Jprima}$$By (\[n\]), $J\left( 0\right) $ and $J^{\prime }\left( 0\right) $ are linearly dependent. If $J(0)=0$, then $J^{\prime }(0)=0$ by (\[Jprima\]), and so $\Vert X\Vert _{K}=0$. If $J(0)\neq 0$, there exists $a\in \mathbb{R}$ such that $J^{\prime }(0)=aJ(0)$. So, $J(s)=(a\, \sinh s + \cosh s)Z(s)$, where $Z$ is a parallel vector field along $\gamma$ and orthogonal to $\dot{\gamma}$. If $|a|>1$ there exists $s_o=(\tanh)^{-1} (-1/a)$ such that $J(s_o)=0$. By (\[Jprima\]), we have that $J'(s_o)=0$. Hence, $J\equiv 0$, which is a contradiction. Therefore, $|a|\leq 1$ and consequently $\left\Vert X\right\Vert _{K}=(1-a^2)|J(0)|^2\geq 0$, as desired.
Next we show that $\mathcal{M}$ is closed. Let $[\gamma _{n}]=H(u_{n},v_{n})$ be a sequence in $\mathcal{M}$ with $\lim_{n\rightarrow \infty }[\gamma
_{n}]=[\gamma ]\in \mathcal{L}$. Let $(u,v)\in T(T_{o}^{1}\mathbb{H}^{3})$ such that $[\gamma ]=H(u,v)$. Since $H$ is a diffeomorphism we have that $(u_{n},v_{n})\rightarrow (u,v)$. Let $\bar{H}:T(T_{o}^{1}\mathbb{H}^{3})\rightarrow T^{1}\mathbb{H}^{3}$ be the smooth map defined by $\bar{H}(u,v)=\left. {\tau _{\gamma _{v}}}\right\vert _{0}^{1}(u)$ and recall that $H=\Pi \circ \bar{H}$ holds by definition of $H$, where $\Pi :T^{1}\mathbb{H}^{3}\rightarrow \mathcal{L}$ is the canonical projection. Since $[\gamma
_{n}]\in \mathcal{M}$, $\bar{H}(u_{n},v_{n})=V(\text{Exp}_{o}(v_{n}))$. So, to prove that $\mathcal{M}$ is closed we have to see that $\bar{H}(u,v)=V(\text{Exp}_{o}(v))$. Now, the assertion follows from the continuity of $\bar{H}\text{, Exp}_{o}$ and $V$.
(b)$\Rightarrow $(a) The facts that the union of all geodesics in $\mathcal{M}$ covers the whole space ${\mathbb{H}^3}$ and that two distinct geodesics in $\mathcal{M}$ do not intersect are proved in a similar way as in Theorem 2 of [@Salvai-2009], but using in this case Lemma [distancia]{} (b). As in that theorem, the hypotheses force the existence of only one critical point (cf. the second paragraph of Remark \[closed\]) and that ${\mathcal{M}}$ is diffeomorphic to ${\mathbb{R}}^2$.
Next, we define the vector field $V$ which determines the foliation. Given $z\in \mathbb{H}^{3}$, let $V(z)=\dot{\gamma}(t)$, where $[\gamma ]$ is the unique element in $\mathcal{M}$ such that $z$ is in the trajectory of $\gamma $ and $z=\gamma (t)$. Now, we verify that $V$ is smooth. The arguments differ from those in the Euclidean case only at the end, but we include the details for the sake of completeness. The image of $V$ coincides with $\Pi ^{-1}(\mathcal{M})$, and hence it is a smooth submanifold of $T^{1}\mathbb{H}^{3}$, since $\Pi $ is a fiber bundle. We have to check that zero is the only vertical (with respect to $p:T^{1}\mathbb{H}^{3}\rightarrow
\mathbb{H}^{3}$) tangent vector $\eta $ of the image of $V$. Suppose that $(dp)_{V(z)}(\eta )=0$ and let $t\mapsto V\circ c(t)$ be a smooth curve in $T^{1}\mathbb{H}^{3}$ such that $c(0)=z$ and with initial velocity equal to $\eta $. So, we have that $c^{\prime }(0)=0$. Let $\ell $ be the curve in $\mathcal{M}$ defined by $\ell (t)=\Pi (V(c(t)))$ and set $\ell ^{\prime
}(0)=X$. Let $\ell (0)=[\gamma ]$ with $\gamma (0)=c(0)$ and let $J(s)=\left. {\frac{d}{dt}}\right\vert _{0}\gamma _{V(c(t))}(s)$. We compute $J(0)=c^{\prime }(0)=0$ and we have that $J^{\prime }\left( 0\right) $ is orthogonal to $\dot{\gamma}\left( 0\right) $, since $V$ is a unit vector field. Hence, $X=T_{\gamma }(J)$ and $\Vert X\Vert _{\times }=0$ by (\[n\]) and so $\left\Vert X\right\Vert _{K}\geq 0$ since $\mathcal{M}$ is almost semidefinite. This implies, again by (\[n\]), that $J^{\prime }(0)=0$. Finally, if we consider the isomorphism $$(dp_{V(z)},\mathcal{K}_{V(z)}):T_{V(z)}T\mathbb{H}^{3}\rightarrow T_{z}\mathbb{H}^{3}\times T_{z}\mathbb{H}^{3}, \label{isophi_v}$$where $\mathcal{K}_{V(z)}$ is the connection operator, we obtain that $\eta $ is equal to zero, since $(dp_{V(z)},\mathcal{K}_{V(z)})(\eta
)=(J(0),J^{\prime }(0))=(0,0)$.
\[closed\] We construct in Proposition \[ejemplo\] below an example of a two dimensional submanifold $\mathcal{M}$ of $\mathcal{L}$ satisfying all conditions of part (b) in Theorem \[adefinite\], except to be closed. The geodesics in $\mathcal{M}$ not only fail to foliate the whole $\mathbb{H}^{3}$, as expected, but they do not even foliate the open set $\mathcal{U}$ in $\mathbb{H}^{3}$ given by the union of all their trajectories (there exist two geodesics in $\mathcal{M}$ intersecting at a point in $\mathcal{U}$).
The same proposition shows that if the hypothesis that $\mathcal{M}$ is closed in $\mathcal{L}$ is removed in Lemma \[distancia\], there might exist two different critical points $\ell _{1}$ and $\ell _{2}$ of $\left.
D\right\vert _{\mathcal{M}}$. One can take $o=f\left( 2,0\right) $ and as $\ell _{1}$ and $\ell _{2}$ the geodesics through $o$ with initial velocities $V_{\lambda}\left( 2,0\right) $ and $V_{\lambda}\left( 2,2\pi \right) $.
We begin by defining an immersion $f$ of an open set of the plane into a totally geodesic submanifold $S$ of $\mathbb{H}^{3}$ covering an annulus in $S$ in a non-injective way. Let $U$ be an open set in the half plane $\left\{ \left(
r,t\right) \mid r>0\right\} $ containing the rectangle $R=\left[ 1,3\right]
\times \left[ -\delta ,2\pi +\delta \right] $. Fix $o\in \mathbb{H}^{3}$ and define $f:U\rightarrow \mathbb{H}^{3}$ by $$f\left( r,t\right) =\text{Exp}_{o}\left( r\cos t~u_{o}+r\sin t~v_{o}\right)
\text{,}$$where $u_{o},v_{o}\in T_{o}\mathbb{H}^{3}$ are unit orthogonal vectors. We consider vector fields $u,v,w$ along $f$ forming an orthonormal basis of $T_{f\left( r,t\right) }\mathbb{H}^{3}$ for each $r,t$. They are given by $$u=\frac{\partial f}{\partial r},\ \ \ \ v=\frac{1}{\sinh r}\frac{\partial f}{\partial t},\ \ \ \ w=u\times v\text{.}$$Now, let $\alpha_{\lambda} \left( r,t\right) =\alpha _{0}+\lambda t-\lambda r$ for some $\lambda >0$ and $\alpha _{0}\in (0,\pi /2)$, and let $V_{\lambda}$ be the vector field along $f$ defined by $V_{\lambda}=\cos \alpha_{\lambda} ~v+\sin \alpha_{\lambda} ~w$. Let $R^{o}$ be the interior of $R$.
\[ejemplo\] For some $\lambda>0$, the map $$F_{\lambda}:R^{o}\rightarrow \mathcal{L}\text{, \ \ \ \ \ }F_{\lambda}\left( r,t\right) =\left[
\gamma _{V_{\lambda}\left( r,t\right) }\right]$$is an immersion and $g_{\times }$ induces a Riemannian metric on its image $F_{\lambda}\left( R^{o}\right) =\mathcal{M}$. Moreover, the trajectories of $\gamma
_{V_{\lambda}(r,0)}$ and $\gamma _{V_{\lambda}(r,2\pi )}$ intersect at $f(r,0)$, for each $r\in
(1,3)$.
We fix $\left( r,t\right) \in R^{o}$ and $0\neq \left( x,y\right) \in T_{\left( r,t\right) }R^{\circ}$. For the sake of simplicity we omit the subindex $\lambda$ and write $\alpha$ instead of $\alpha_{\lambda}(r,t)$. Let us see that $\left\Vert dF_{\left( r,t\right) }\left( x,y\right) \right\Vert _{\times }>0$. Let $J$ be the Jacobi field associated with the variation $s\mapsto \gamma _{V\left(
r+sx,t+sy\right) }$. We compute that $J\left( 0\right) =x\,u+y\sinh r\,v.$ Now, since$$\nabla _{u}u=0,\,\nabla _{u}v=0,\,\nabla _{u}w=0,\,\nabla _{v}u=\left( \coth
r\right) v,\,\nabla _{v}v=-\left( \coth r\right) u\,\,\text{and}\,\,\nabla
_{v}w=0\text{,}$$we obtain that $$J^{\prime }(0)=-\left( y\cosh r\cos \alpha \right) u+\lambda (x-y)\left(
\sin \alpha \right) v-\lambda (x-y)\left( \cos \alpha \right) w\text{.}$$On the other hand, calling $\gamma =\gamma _{V(r,t)}$, we have that $$\dot{\gamma}(0)\times J(0)=V(r,t)\times J(0)=-\left( y\sinh r\sin \alpha
\right) u+\left( x\sin \alpha \right) v-\left( x\cos \alpha \right) w\text{.}$$Then, since the expression for $g_{\times }$ in (\[n\]) is valid also if $J
$ is not orthogonal to $\dot{\gamma}$, we have that $$\left\Vert dF_{\left( r,t\right) }\left( x,y\right) \right\Vert _{\times
}=\langle \dot{\gamma}(0)\times J(0),J^{\prime }(0)\rangle =\lambda
x^{2}-\lambda xy+\tfrac{1}{4}\left( \sinh 2r\sin 2\alpha \right) \,y^{2}.$$Thus, for $\mathcal{M}$ to be Riemannian, it is enough that $\lambda$ makes this bilinear form positive definite for all $\left( r,t\right)
\in R$. Equivalently, that $h_{\lambda }\left( r,t\right) >0$ for all $\left( r,t\right) \in R$, where for each $\lambda >0$, $h_{\lambda
}:R\rightarrow \mathbb{R}$ is defined by $$h_{\lambda }\left( r,t\right) =\sinh \left( 2r\right) \sin \left( 2\left(
\alpha _{0}+\lambda t-\lambda r\right) \right) -\lambda \text{.}$$Now, $h_{\lambda }$ converges pointwise (and also uniformly, since $R$ is compact) to $\sinh \left( 2r\right) \sin \left( 2\alpha _{0}\right) $ as $\lambda \rightarrow 0^{+}$. Since the limit function is positive, for $\lambda >0$ small enough, $h_{\lambda }\left( r,t\right) >0$ for all $\left(
r,t\right) \in R$, as desired.
Global nondegenerate geodesic foliations of $\mathbb{H}^{3}$
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Two unit speed geodesics $\gamma $ and $\alpha $ of $\mathbb{H}^{3}$ are said to be *asymptotic* if there exists a positive constant $C$ such that $d(\gamma (s),\sigma (s))\leq C$, $\forall s\geq 0$ [@Eberlein-1996]. Two unit vectors $v,w\in T^{1}\mathbb{H}^{3}$ are said to be asymptotic if the corresponding geodesics $\gamma _{v}$ and $\gamma _{w}$ have this property.
A point at infinity for $\mathbb{H}^{3}$ is an equivalence class of asymptotic geodesics of $\mathbb{H}^{3}$. The set of all points at infinity for $\mathbb{H}^{3}$ is denoted by $\mathbb{H}^{3}(\infty )$ and has a canonical differentiable structure diffeomorphic to the $2$-sphere. The equivalence class represented by a geodesic $\gamma $ is denoted by $\gamma
(\infty )$ and the equivalence class represented by the oppositely oriented geodesic $s\mapsto \gamma (-s)$ is denoted by $\gamma (-\infty )$. Let $\varphi ^{\pm }:\mathcal{L}\rightarrow \mathbb{H}^{3}(\infty )$ be the forward Gauss map (for $+$) and the backward Gauss map (for $-$), defined by $\varphi ^{\pm }([\gamma ])=\gamma (\pm \infty )$, which are smooth.
In the introduction we commented on some distinguished types of geodesic foliations of $\mathbb{H}^{3}$, regarding to which extent they are in some sense trivial in some directions. That motivates the following precise definitions. Before we recall that by Theorem \[adefinite\], any smooth geodesic foliation of ${\mathbb{H}^3}$ has an associated submanifold ${\mathcal{M}}$ of ${\mathcal{L}}$.
\[nondeg\] We say that a smooth foliation by oriented geodesics of $\mathbb{H}^{3}$ is *semi-nondegenerate* if the Gauss maps $\varphi ^{\pm }:\mathcal{M}\rightarrow \mathbb{H}^{3}(\infty )$ are local diffeomorphisms, where $\mathcal{M}\subset \mathcal{L}$ is the space of leaves. And we say that it is *nondegenerate* if the only eigenvectors of $\nabla V$ are in ${\mathbb{R}}V$, where $V$ is the unit vector field that determines the foliation.
In order to characterize the semi-nondegenerate and nondegenerate global geodesic foliations of the hyperbolic space in terms of the geometry of $\mathcal{L}$, we have the next definition.
A submanifold $\mathcal{M}$ of $\mathcal{L}$ is said to be *semidefinite* if $\Vert X\Vert _{\times }=0$ for a nonzero $X\in T\mathcal{M}$ only if $\Vert X\Vert _{K}>0$.
\[SAD\] Let ${\mathcal{M}}$ be the submanifold of ${\mathcal{L}}$ associated with a foliation of ${\mathbb{H}^3}$ by oriented geodesics. Then
1. the foliation is semi-nondegenerate if and only if ${\mathcal{M}}$ is semidefinite.
2. the foliation is nondegenerate if and only if $g_\times $ induces on ${\mathcal{M}}$ a definite metric.
Some definitions and lemmas will be necessary to prove the theorem.
A Jacobi vector field $J$ along a geodesic $\gamma $ of $\mathbb{H}^{3}$ is said to be *stable* (*unstable*) if there exists a constant $c>0$ such that $$|J(s)|\leq c,\hspace{0.5cm}\forall s\geq 0\hspace{0.5cm}(\forall s\leq 0).$$It is well-known that a Jacobi vector field $J$ along a geodesic $\gamma$ of $\mathbb{H}^{3}$ and orthogonal to $\dot{\gamma}$ is stable (respectively, unstable) if and only if $J(s)=e^{-s}U(s)$ (respectively, $J(s)=e^{s}U(s)$) for some parallel vector field $U$ along $\gamma $ orthogonal to $\dot{\gamma}$.
We recall that given $v\in T^{1}\mathbb{H}^{3}$ and any point $p\in \mathbb{H}^{3}$, there exists a unique unit tangent vector at $p$ that is asymptotic to $v$ (see [@Eberlein-1996 Proposition 1.7.3]). A smooth vector field $W$ in $\mathbb{H}^{3}$ is called an asymptotic vector field if $W(p)$ and $W(q)$ are asymptotic for every $p,q\in \mathbb{H}^{3}$.
Let $c$ be a smooth curve in $\mathbb{H}^{3}$. Then an asymptotic vector field $W$ on $\mathbb{H}^{3}$ satisfies the following differential equation $$\nabla _{\dot{c}(t)}W=\langle \dot{c}(t),W_{c(t)}\rangle \,W_{c(t)}-\dot{c}(t)\text{.} \label{Ec.Asintotica}$$
After decomposing $\dot{c}(t)$ into its components tangent and orthogonal to $W_{c(t)}$, the statement follows directly from the following equations: $$\nabla _{X}\,W=-X,\,\text{if}\,\,X\bot \,W\hspace{0.5cm}\text{and}
\hspace{0.5cm}\nabla _{W}\,W=0. \label{campoasint.}$$ The first one is true (see (1.10.9) in [@Eberlein-1996]), since it is well known that the shape operator of a horosphere is the identity. The second one holds, since the integral curves of an asymptotic vector field are geodesics.
\[jacobi\] Let $\gamma $ be a geodesic of $\mathbb{H}^{3}$ and let $J\in
\mathcal{J}_{\gamma }$ be given by $J(s)=\left. {\frac{d}{dt}}\right\vert
_{0}\gamma _{u_{t}}(s)$, where $t\mapsto u_{t}$ is a smooth curve in $T^{1}\mathbb{H}^{3}$, with foot points $c(t)$ *(*in particular, $u_{0}=\dot{\gamma}\left( 0\right)
\bot \dot{c}\left( 0\right) $*)*. If $v_{t}\in T_{\gamma (0)}^{1}\mathbb{H}^{3}$ is the asymptotic vector to $u_{t}$ for each $t\in \mathbb{R}
$, then the Jacobi vector field $K$ along $\gamma$ associated with $v_{t}$ satisfies $$K^{\prime }(0)=J(0)+J^{\prime }(0).$$
For each $s,t\in \mathbb{R}$, let $V(s,t)$ be the unique unit vector at $c(s)$ that is asymptotic to $u_{t}$. In particular, $V(0,t)=v_{t}$ and $V(t,t)=u_{t}$. We compute $$\left. {\frac{D}{dt}}\right\vert _{0}u_{t}=\left. {\frac{D}{dt}}\right\vert
_{0}V(t,t)=\left. {\frac{D}{dt}}\right\vert _{0}V(t,0)+\left. {\frac{D}{dt}}\right\vert _{0}V(0,t)=\left. {\frac{D}{dt}}\right\vert _{0}V(t,0)+\left. {\frac{D}{dt}}\right\vert _{0}v_{t} \label{der.parc}$$The second equality follows from the well-known corresponding identity in the calculus of several variables (writing $V$ in coordinates). The vector field $V(t,0)$ is an asymptotic vector field along $t\mapsto c(t)$. So, using that $J$ is orthogonal to $\dot{\gamma}$ (in particular $\left\langle
\dot{c}\left( 0\right) ,V\left( 0,0\right) \right\rangle =0$) and ([Ec.Asintotica]{}) with $W_{ c\left( t\right) } =V\left( t,0\right) $, we have that $$\label{V(t,0)}
\left. {\frac{D}{dt}}\right\vert _{0}V(t,0)=-\dot{c}(0).$$ Finally, since $J^{\prime }(0)=\left. {\frac{D}{dt}}\right\vert _{0}u_{t}$ and $K^{\prime }(0)=\left. {\frac{D}{dt}}\right\vert _{0}v_{t}$, by ([der.parc]{}) and (\[V(t,0)\]) we obtain $K^{\prime }(0)=J(0)+J^{\prime }(0)$, as desired.
\(a) Suppose that the foliation is semi-nondegenerate. Let $[\gamma ]\in \mathcal{M}$ and let $0\neq X\in T_{[\gamma ]}\mathcal{M}$ such that $\Vert X\Vert _{\times }=0$. We want to see that $\left\Vert
X\right\Vert _{K}>0$. By (a) $\Rightarrow $ (b) in Theorem \[adefinite\] we have $\Vert X\Vert _{K}\geq 0$. Suppose now that $\Vert X\Vert _{K}=0$. Let $J\in \mathcal{J}_{\gamma }$ be the Jacobi vector field associated with $X$ via the isomorphism $T_{\gamma }$ given in (\[isoT\]). Hence, by (\[n\]), $\{J(0),J^{\prime }(0)\}$ is linearly dependent and $\left\vert J^{\prime }\left( 0\right) \right\vert ^{2}=|J(0)|^{2}$. Since $X\neq 0$, we have that $J$ is a stable or an unstable vector field. If $J$ is a stable Jacobi vector field, by Proposition 1.10.7 in [@Eberlein-1996], $J(s)=\left.
\frac{d}{dt}\right\vert _{0}\gamma _{u_{t}}(s)$ for some smooth curve $t\mapsto u_{t}\in T^{1}\mathbb{H}^{3}$ with $u_{0}=\dot{\gamma}(0)$ and $u_{t}$ asymptotic for all $t$. Then, $$(d\varphi ^{+})_{[\gamma ]}X=\left. \frac{d}{dt}\right\vert _{0}\varphi
^{+}([\gamma _{u_{t}}])=0,$$ which is a contradiction since $\varphi ^{+}$ is a local diffeomorphism. The case $J$ unstable is similar. Therefore, $\mathcal{M}$ is semidefinite.
Conversely, if ${\mathcal{M}}$ is semidefinite, we have to prove that the foliation is semi-nondegenerate, that is, that the backward and forward Gauss maps $\varphi ^{\pm }$ are local diffeomorphisms. So, let $[\gamma ]\in \mathcal{M}$ and $0\neq X\in T_{[\gamma ]}\mathcal{M}$. Let $J\in \mathcal{J}_{\gamma }$ be the Jacobi vector field associated with $X$ via (\[isoT\]) and consider a smooth curve $t\mapsto u_{t}\in T^{1}\mathbb{H}^{3}$ such that $[\gamma _{u_{t}}]\in \mathcal{M}$ and $J(s)=\left. {\frac{d}{dt}}\right\vert _{0}\gamma _{u_{t}}(s)$. Let us prove that $(d\varphi
^{+})_{[\gamma ]}X\neq 0$ (the proof of the corresponding assertion for $\varphi ^{-}$ instead of $\varphi $ $^{+}$ is similar). We have to see that the initial velocity of $t\mapsto \gamma _{u_{t}}(\infty )$ is different from zero. By the definition of the differentiable structure of $\mathbb{H}^{3}(\infty )$ we have that the map that assigns to each $v\in T_{\gamma
(0)}^{1}\mathbb{H}^{3}$ the equivalence class of $\gamma _{v}$ in $\mathbb{H}^{3}(\infty )$ is a diffeomorphism. So, we consider the smooth curve $t\mapsto v_{t}\in T_{\gamma (0)}^{1}\mathbb{H}^{3}$ such that $\gamma
_{u_{t}}(\infty )=\gamma _{v_{t}}(\infty )$ and we show that $\left. \frac{d}{dt}\right\vert _{0}v_{t}\neq 0$. Let $K$ be the Jacobi vector field along $\gamma $ given by $K(s)=\left. {\frac{d}{dt}}\right\vert _{0}\gamma
_{v_{t}}(s)$ with initial conditions $K(0)=0$ and $K^{\prime }(0)=\left. {\frac{D}{dt}}\right\vert _{0}v_{t}$. By the isomorphism given in ([isophi\_v]{}), it suffices to see that $K^{\prime }(0)\neq 0$. Now, by Lemma \[jacobi\], $K^{\prime }(0)=J(0)+J^{\prime }(0)$. If $\Vert X\Vert
_{\times }\neq 0$, the set $\{J(0),J^{\prime }(0)\}$ is linearly independent and so $K^{\prime }(0)\neq 0$. Now suppose that $\Vert X\Vert _{\times }=0$. Since $X\neq 0$ and $\mathcal{M}$ is semidefinite, then $\left\Vert X\right\Vert _{K}=|J(0)|^2-|J'(0)|^2 > 0$. By (\[n\]), the set $\{J(0),J^{\prime
}(0)\}$ is linearly dependent, and $J(0)\neq 0$. Hence, $J^{\prime
}(0)=\lambda J(0)$ with $\lambda \in \mathbb{R}-\{\pm 1\}$. Consequently, $K^{\prime }(0)=(1+\lambda )J(0)\neq 0$, as desired.
\(b) First, suppose that the foliation is nondegenerate, that is, the only eigenvectors of $\nabla V$ are in ${\mathbb{R}}V$. We want to see that $\|X\|_{\times}\neq 0$ for all $0\neq X\in T{\mathcal{M}}$. Suppose that there exists a nonzero vector $X\in T_{[\gamma]}{\mathcal{M}}$ such that $\|X\|_{\times}=0$. Let $J\in \mathcal{J}_{\gamma}$ the Jacobi vector field associated with $X$ via the isomorphism $T_{\gamma}$ defined in (\[isoT\]). By (\[Jprima\]), $\nabla_{J(0)}V=J'(0)$ and since $X\neq 0$ we obtain that $J(0)\neq 0$. Now, since $$0=\|X\|_{\times}=\|T_{\gamma}(J)\|_{\times}=\left\langle \dot{\gamma}(0)\times
J(0), J'(0)\right\rangle,$$ we have that $J(0)\times J'(0)$ is orthogonal to $\dot{\gamma}(0)=V(\gamma(0))$. Or equivalently, $J'(0)$ is a multiple of $J(0)$. Again by (\[Jprima\]), $J(0)$ is an eigenvector of $\nabla V$ orthogonal to $V(\gamma(0))$ (recall that $J\in \mathcal{J}_{\gamma}$), which is a contradiction.
Conversely, let $u\in T_{p}{\mathbb{H}^3}$ be an eigenvector of $\nabla V$ with eigenvalue $\lambda$. Let $c:(-\varepsilon, \varepsilon)\rightarrow {\mathbb{H}^3}$ be a smooth curve such that $\dot{c}(0)=u-\left\langle u, V(p)\right\rangle V(p) \bot V(p)$. So, the Jacobi vector field given by $J(s)=\left. \dfrac{d}{dt}\right\vert _{0}\gamma_{V(c(t))}\left( s\right)$ is in $\mathcal{J}_{\gamma}$, where $\gamma=\gamma_{V(p)}$. Since $J(0)=\dot{c}(0)$ and $\nabla _{V(p)}V=0$, we have that $\nabla_{J(0)} V=\lambda u$. As $\nabla_{J(0)} V=J'(0)$ by (\[Jprima\]), $T_{\gamma}(J)\in T{\mathcal{M}}$ satisfies $$\label{nulo}
\|T_{\gamma}(J)\|_{\times}=\left\langle \dot{\gamma}(0)\times J(0), J'(0)\right\rangle =
\left\langle V(p)\times \dot{c}(0), \lambda u\right\rangle=\left\langle V(p)\times u, \lambda u\right\rangle=0.$$ Since $g_{\times}$ induces on ${\mathcal{M}}$ a definite metric, we have that $T_{\gamma}(J)=0$ and so $\dot{c}(0)=J(0)=0$. Thus, $u$ is a multiple of $V(p)$, as desired.
The definition of semi-nondegenerate foliation says that geodesic varying smoothly within the foliation do not meet at infinity. The following theorem states that this local condition implies in fact the global property that geodesics in the foliation do not meet at infinity at all. In the proof we have to use coordinates in $\mathbb{H}^{3}$.
Let $\mathcal{M}$ be the space of leaves of a semi-nondegenerate smooth foliation of $\mathbb{H}^{3}$ by oriented geodesics. Then the forward and backward Gauss maps $\varphi ^{\pm }:\mathcal{M}\rightarrow \mathbb{H}^{3}\left( \infty \right) $ are one to one. In particular, they are diffeomorphisms onto their images.
Let $P:\mathbb{H}^{3}\rightarrow \mathcal{M}$ be the map assigning to each point $q$ in the hyperbolic space the oriented geodesic in the foliation containing $q$, that is, $P\left( q\right) =\left[ \gamma
_{V\left( q\right) }\right] $. This is a fiber bundle with typical fiber $\mathbb{R}$. Since $\mathcal{M}$ is diffeomorphic to $\mathbb{R}^{2}$ by Theorem \[adefinite\], there exists a global section $S:\mathcal{M}\rightarrow \mathbb{H}^{3}$. Let $F:\mathcal{M}^{\prime }\times \mathbb{R}\rightarrow \mathbb{H}^{3}$ be the diffeomorphism given by $F\left(
q,t\right) =\gamma _{V\left( q\right) }\left( t\right) $, where $\mathcal{M}^{\prime }=S\left( \mathcal{M}\right) \subset \mathbb{H}^{3}$. Let $$F^{\pm }:\mathcal{M}^{\prime }\rightarrow \mathbb{H}^{3}\left( \infty
\right) \text{,\ \ \ \ \ \ \ \ \ \ \ }F^{\pm }\left( q\right) =\gamma
_{V\left( q\right) }\left( \pm \infty \right) \text{,}$$ which satisfies $F^{\pm }\circ S=\varphi ^{\pm }$. Clearly, it suffices to prove that $F^{\pm }$ is one to one.
We consider the upper half space model of the hyperbolic space, that is,\
$\left\{ \left( x,y,z\right) \mid z>0\right\}$ with the Riemannian metric $ds^{2}=\frac{1}{z^{2}}\left( dx^{2}+dy^{2}+dz^{2}\right) $. Without loss of generality, we may suppose that $\left[ \gamma _{o}\right] \in \mathcal{M}$, where $\gamma _{o}\left( t\right) =\left( 0,0,e^{t}\right) $, and that $S\left( \left[ \gamma _{o}\right] \right) =\left( 0,0,1\right) $. In this model, $\mathbb{H}^{3}\left( \infty \right) =\left( \mathbb{R}^{2}\times
\left\{ 0\right\} \right) \cup \left\{ \infty \right\} $, $\varphi ^{+}\left[
\gamma _{o}\right] =\gamma _{o}\left( \infty \right) =\infty $ and $\varphi
^{-}\left[ \gamma _{o}\right] =\gamma _{o}\left( -\infty \right) =0$.
Since $\varphi ^{\pm }$ (or equivalently $F^{\pm }$) are local diffeomorphisms, there exists a neighborhood $A$ of $\left( 0,0,1\right) $ in $\mathcal{M}^{\prime }$, and neighborhoods $U_{+}$ and $U_{-}$ of $\infty
$ and $0$ in $\mathbb{H}^{3}\left( \infty \right) $, respectively, such that $F^{\pm }:A\rightarrow U_{\pm }$ is a diffeomorphism. Let $B_{+}\subset
U_{+} $ be the complement of a closed disk centered at $0$ of radius $R$ in $\mathbb{R}^{2}\times \left\{ 0\right\} $. Let $A^{\prime }\subset A$ and $B_{-}\subset U_{-}$ be such that $F^{\pm }:A^{\prime }\rightarrow B_{\pm }$ are diffeomorphisms. Taking, if necessary, a larger $R$, we may suppose that $B_{-}$ is contained in the disk of radius $\delta $ (also centered at $0$).
Let us see that $F\left( A^{\prime }\times \mathbb{R}\right) $ contains the horoball $\left\{ \left( x,y,z\right) \mid 2z\geq R+\delta \right\} $. If $\partial A^{\prime }$ is the border of $A^{\prime }$ in $\mathcal{M}^{\prime
}$, then $F\left( \partial A^{\prime }\times \mathbb{R}\right) $ is a cylinder in $\mathbb{H}^{3}$ separating the space in two connected components, in such a way that $F\left( A^{\prime }\times \mathbb{R}\right)$ is the component containing the trajectory of $\gamma _{o}$. The assertion follows from the fact that the cylinder is built up with trajectories of geodesics in $\mathbb{H}^{3}$ (vertical semicircles with center in $\mathbb{R
}^{2}\times \left\{ 0\right\} $) whose $z$-component is smaller than $\frac{1
}{2}\left( R+\delta \right) $.
Finally, given $\left[ \sigma \right] \in \mathcal{M}$ such that $\sigma
\left( \infty \right) =\gamma _{o}\left( \infty \right) $, we want to see that $\left[ \sigma \right] =\left[ \gamma _{o}\right] $. The geodesic $\sigma $ must have the form $\sigma \left( t\right) =\left(
x_{o},y_{o},z_{o}e^{t}\right) $ for some real numbers $x_{o},y_{o},z_{o}$, with $z_{o}>0$. For $t$ large enough, $\sigma \left( t\right) $ is in the horoball. In particular, there exists $t_{1}$ such that $\sigma \left(
t_{1}\right) =F\left( q,s\right) $ for some $\left( q,s\right) \in A^{\prime
}\times \mathbb{R}$. Hence $\sigma \left( t_{1}\right) =\gamma _{V\left(
q\right) }\left( s\right) $. Now, since for each point of $\mathbb{H}^{3}$ passes only one geodesic in $\mathcal{M}$, we have that $\left[ \sigma
\right] =\left[ \gamma _{V\left( q\right) }\right] $. Consequently, $\gamma
_{V\left( q\right) }\left( \infty \right) =\infty $ and so $q=\left(
0,0,1\right) $, since $F^{+}$ is one to one on $A^{\prime }$. Therefore $\left[ \sigma \right] =\left[ \gamma _{o}\right] $. The injectivity of $\varphi ^{-}$ is verified in a similar way.
[^1]: Partially supported by <span style="font-variant:small-caps;">CONICET, FONCyT, SECyT (UNC)</span>.
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