# Dyson-Schwinger equation approach to Lorentz symmetry breaking with finite
temperature and chemical potential

Y.M.P. Gomes<EMAIL_ADDRESS>Instituto de Física - Universidade do
Estado do Rio de Janeiro (UERJ), 20550-900, Rio de Janeiro, Rio de Janeiro,
Brazil.

###### Abstract

In this work, we investigate the dynamical breakdown of Lorentz symmetry in 4
dimensions by the condensation of a fermionic field described by a Dirac
lagrangean with a four-fermion interaction. Using the Keldysh formalism we
show that the Lorentz symmetry breaking modifies the Dyson-Schwinger equations
of the fermionic propagator. We analyze the non-perturbative solutions for the
Dyson-Schwinger equations using the combination of the rainbow and quenched
approximations and show that, in equilibrium, the Lorentz symmetry breakdown
can occur in the strong coupling regime and new features arise from this
approach. Finally, we analyze the contributions of temperature and chemical
potential and find the respective phase diagram of the model and analyze the
dependence of the critical temperature and chemical potential as functions of
the coupling constant.

###### pacs:

11.30. Cp, 11.15. Ex, 14.70. - e;

##  I Introduction

Based on the fact that the standard model of particle physics (SM) is, despite
its great success, unable to fully explain the phenomenology of particle
physics, several models have been proposed to improve the knowledge about the
quantum realm of fundamental particles, as supersymmetry, extra-dimensions,
among others intro1 ; intro2 ; intro3 .

An attempt to extend the SM is to analyze extensions of the standard model
with the introduction of Lorentz symmetry violation. In ref. string  the
authors shown that in string theory could be possible that a spontaneous
Lorentz symmetry breaking could occur and some phenomenological implications
could arise in particle physics. This approach became a paradigm in Lorentz
symmetry breaking. In particular, the simpler extension of the SM can be done
by introducing new sectors where a vector field acquires a non-null
expectation value in the vacuum. This background vector can be coupled with
the standard model currents minimally or non-minimally and, in principle,
could make contributions to the Charge-Parity (CP) symmetry violation or even
to CPT symmetry violation lsv1 ; lsv2 ; lsv3 ; lsv4 ; lsv5 ; lsv6 ; lsv7 ;
lsv8 .

Beyond the well-known works searching for LSV in particle physics, the
approach of dynamical (instead of spontaneous) Lorentz symmetry breaking was
implemented. In particular, a perturbative approach of dynamical Lorentz
symmetry breaking (DLSB) is used to shed light on how this breakdown could
occur in 4 and 3 dimensions DLSB1 ; DLSB2 ; DLSB3 ; DLSB4 ; DLSB5 ; jenkins .
The main idea is that in analogy with the models of chiral symmetry breaking
miranski , the breakdown of the Lorentz symmetry appears from a fermion
condensation through a four-fermion interaction.

An example of this subject can be seen in ref. jenkins , where a model with a
vector four-fermion interaction $(\bar{\psi}\gamma_{\mu}\psi)^{2}$ is proposed
and the dynamical symmetry breaking is analyzed. When the DLSB occurs, the
vector acquires a non-null vacuum expectation value
$\langle\bar{\psi}\gamma_{\mu}\psi\rangle\neq 0$, such way, if this DLSB
generates a time-like constant vector, we can interpret this result as an
effective chemical potential and its implications are well discussed in the
paper.

In the present work, we analyze the DLSB of an axial (or pseudo-vector)
fermionic current starting from the following action proposed firstly by ref.
DLSB1 :

$S=\int
dx\left[\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi-\frac{G}{2}(\bar{\psi}\gamma_{\mu}\gamma_{5}\psi)^{2}\right]~{},$
(1)

where $G$ is the coupling constant and $G>0$ in order to maintain an
attractive interaction and provide the fermionic condensation. We can
introduce an auxiliary field $A_{\mu}$ such that the action can be rewritten
as:

$S=\int
dx\left[\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-g\gamma^{\mu}A_{\mu}\gamma_{5}-m)\psi+\frac{g^{2}}{2G}A_{\mu}A^{\mu}\right]~{}.$
(2)

In general, the $g$ parameter is introduced to expand the physical quantities
and find quantum corrections. Although this is not our intention, we maintain
$g$ explicitly to compare with the perturbative implementation.

As can be seen in DLSB1 ; DLSB2 , in this perturbative approach the fermionic
condensation can be calculated by a gap equation and it gives us the result
$\alpha_{\mu}=\langle A_{\mu}\rangle\propto G^{-1/2}$ showing that for a
strong coupling constant $G$ we find a small value of DLSB parameter, and
vice-versa. For instance, if $G\propto M^{-2}$, with $M$ representing some
mass scale, $\alpha_{\mu}\propto M$. As we will see in this paper, the non-
perturbative approach gives us a very different result, and a phase transition
will occur. Although we will discuss this result in our conclusions, we can
already affirm that instead of a contribution for the chemical potential such
can be seen in ref. jenkins , our results can be interpreted as a chiral
chemical potential $\mu_{5}$ above some critical coupling constant. The
validity of calling $\mu_{5}$ a true chemical potential will be discussed in
the conclusions.

This work is organized as follows: In Section II we introduce the Keldysh
formalism and through the generating functional we find the proper Dyson-
Schwinger (DS) equations with the inclusion of Lorentz symmetry breaking. In
Section III we focus on the DS equations of the full fermion propagator and
show the approximations made. In Section IV we look for equilibrium solutions
in some physical limits. In Section V we present our final comments and
perspectives.

##  II Generating functional in Keldysh formalism

In this work we adopt the real-time formalism in order to evaluate the
contribution of the chemical potential and temperature. In this formalism,
known as the Keldysh formalism kamenevbook , we must double the fields
($\psi\rightarrow\\{\psi^{+},\psi^{-}\\}$,$\bar{\psi}\rightarrow\\{\bar{\psi}^{+},\bar{\psi}^{-}\\}$
, $A_{\mu}\rightarrow\\{A^{+},A^{-}\\}$), and the new action can be written as
$S=S^{+}-S^{-}$ and, applying the Keldysh rotations kamenevbook ; kamenev  we
can write the action
$S[\bar{\psi},\psi,A]=S_{0}[\bar{\psi},\psi,A]+S_{int}[\bar{\psi},\psi,A]$,
where:

$\displaystyle S_{0}[\bar{\psi},\psi,A]$ $\displaystyle=$ $\displaystyle\int
dxdy\Big{[}\bar{\psi}^{a}(x)(S_{0}^{-1})_{ab}(x,y)\psi^{b}(y)+$ (3)
$\displaystyle+\frac{1}{2}A_{\mu}^{a}(x)(D_{0}^{-1})_{ab}^{\mu\nu}(x,y)A_{\mu}^{b}(y)\Big{]},$

$S_{int}[\bar{\psi},\psi,A]=-g\int
dx\bar{\psi}^{a}(x)\gamma_{c,ab}^{\mu}\psi^{b}(x)A_{\mu}^{c}(x)$ (4)

with
$(S_{0}^{-1})_{ab}(\partial)=(i\gamma^{\mu}\partial_{\mu}-m)\hat{\sigma}^{cl}_{ab}$,
$\gamma^{\mu}_{c,ab}=\frac{1}{\sqrt{2}}\gamma^{\mu}\gamma_{5}(\hat{\sigma}_{c})_{ab}$,
$(D_{0}^{-1})^{\mu\nu}_{ab}(x,y)=\frac{g^{2}}{G}\delta(x-y)\hat{\sigma}^{q}_{ab}\eta^{\mu\nu}$,
with $\hat{\sigma}^{cl}=\begin{pmatrix}1&0\\\ 0&1\\\ \end{pmatrix}$ and
$\hat{\sigma}^{q}=\begin{pmatrix}0&1\\\ 1&0\\\ \end{pmatrix}$, where
$a,b,c=\\{cl,q\\}$. We also have $A^{cl}=\frac{1}{\sqrt{2}}(A^{+}+A^{-})$,
$A^{q}=\frac{1}{\sqrt{2}}(A^{+}-A^{-})$,
$\psi^{cl}=\frac{1}{\sqrt{2}}(\psi^{+}+\psi^{-})$,
$\psi^{q}=\frac{1}{\sqrt{2}}(\psi^{+}-\psi^{-})$,
$\bar{\psi}^{cl}=\frac{1}{\sqrt{2}}(\bar{\psi}^{+}-\bar{\psi}^{-})$ and
$\bar{\psi}^{q}=\frac{1}{\sqrt{2}}(\bar{\psi}^{+}+\bar{\psi}^{-})$. We
emphasize that the index structure (with Latin indexes $a,b,c$) is due to the
doubling of the fields in the Keldysh formalism, very similar to the closed-
time-path (CTP) formalism CTP , is necessary to construct the out-of-
equilibrium QFT action. A very instructive review of Keldysh formalism can be
seen in ref. kamenev . The generating functional is defined as follows Ryder :

$Z[\bar{\eta},\eta,J]=\int D[\bar{\psi},\psi,A]e^{iS[\bar{\psi},\psi,A]}$ (5)

where $D[\bar{\psi},\psi,A]=\Pi_{a,b,c}d\bar{\psi}^{a}d\psi^{b}dA_{\mu}^{c}$.
Since $A$ is not a gauge field, the necessity of the introduction of Faddeev-
Popov ghosts is absent. We can define the generating functional of connected
green functions as usual, i.e., $W=LnZ$. We can write the quantum version of
the field equations of motion (e.o.m.) of the fields as follows:

$\displaystyle\int dy\Bigg{\\{}(D_{0}^{-1})_{\mu\nu}^{ab}(x,y)\frac{\delta
W}{\delta J_{\nu}^{b}(y)}-g\delta(x-y)\frac{\delta
W}{\delta\eta^{b}(x)}\gamma^{\mu}_{c,ab}\frac{\delta
W}{\delta\bar{\eta}^{c}(y)}-g\delta(x-y)\frac{\delta}{\delta\eta^{b}(x)}\gamma^{\mu}_{c,ab}\frac{\delta
W}{\delta\bar{\eta}^{c}(y)}\Bigg{\\}}+J_{\mu}^{a}(x)=0$

from the $A_{\mu}^{a}(x)$ e.o.m., and:

$\int dy\Bigg{\\{}(S_{0}^{-1})_{cd}(x,y)\frac{\delta
W}{\delta\bar{\eta}^{d}(y)}-g\delta(x-y)\frac{\delta
W}{\delta\bar{\eta}^{d}(x)}\gamma^{\mu}_{a,cd}\frac{\delta W}{\delta
J^{\mu}_{a}(y)}-g\delta(x-y)\frac{\delta}{\delta\bar{\eta}^{d}(x)}\gamma^{\mu}_{a,cd}\frac{\delta
W}{\delta J^{\mu}_{a}(y)}\Bigg{\\}}+\eta^{c}(x)=0$ (7)

from $\bar{\psi}^{c}(x)$. Through (II) we can show an identity that will be
important to our analysis that is the following:

$\displaystyle\frac{\delta W}{\delta J_{\mu}^{a}(y)}\Big{|}_{J=0}$
$\displaystyle=$ $\displaystyle\langle A^{\mu}_{a}(x)\rangle=$
$\displaystyle=-ig\int dzTr[\gamma_{\nu
b,cd}S_{cd}(z,z)](D_{0})^{\mu\nu}_{ab}(z,x),$

where
$S_{bc}(x,y)=\frac{\delta^{2}W}{\delta\eta^{d}(x)\delta\bar{\eta}^{c}(y)}\Big{|}_{J=0}$
the full fermion propagator and we use $J=0$ meaning
$J_{\mu}^{a}=\bar{\eta}^{a}=\eta^{a}=0$. We also use $Tr$ as the trace over
the Dirac matrices space. Therefore, the $A_{a}^{\mu}$ field can assume a non-
vanishing vacuum expectation value $\alpha_{a}^{\mu}(x)=\langle
A^{\mu}_{a}(x)\rangle$ , i.e., a dynamical Lorentz symmetry breakdown can
occur. Is important to highlight that in the Keldysh formalism the Lorentz
symmetry breaking parameter acquire $x$-dependence. Going further, by the use
of the effective action $\Gamma[\bar{\psi},\psi,A]$ defined as follows:

$\displaystyle\Gamma[\bar{\psi},\psi,A]=W[\bar{\eta},\eta,J]+$
$\displaystyle+\int
dx\left[\bar{\eta}^{c}(x)\psi_{c}(x)+\bar{\psi}^{c}(x)\eta_{c}(x)+A_{\mu}^{a}(x)J^{\mu}_{a}(x)\right],$

such way that the DS equation for the full bosonic propagator is given by

$(D^{-1})_{ab}^{\mu\nu}(x,y)-(D^{-1}_{0})_{ab}^{\mu\nu}(x,y)=\Pi_{ab}^{\mu\nu}(x,y),$
(10)

where $(D^{-1})_{ab}^{\mu\nu}(x,y)=\frac{\delta^{2}\Gamma}{\delta
A^{a}_{\mu}(x)\delta A_{\nu}^{b}(y)}\Big{|}_{\Phi=0}$ is the inverse of the
full bosonic propagator, $\Phi=0$ means $A_{\mu}^{a}=\alpha_{\mu}^{a}$ and
$\bar{\psi}^{a}=\psi^{a}=0$. We also have that $\Pi_{ab}^{\mu\nu}(x,y)$, the
polarization tensor, is given by:

$\displaystyle\Pi_{ab}^{\mu\nu}(x,y)=-ig\int dx^{\prime}dy^{\prime}\times$
$\displaystyle
Tr\Big{[}\gamma^{\mu}_{a,cd}S_{dd^{\prime}}(x,y^{\prime})\Gamma^{\nu}_{b,c^{\prime}d^{\prime}}(y,x^{\prime},y^{\prime})S_{cc^{\prime}}(x,x^{\prime})\Big{]},$

where
$\Gamma^{\nu}_{b,c^{\prime}d^{\prime}}(y,x^{\prime},y^{\prime})=\frac{\delta^{3}\Gamma}{\delta
A_{b}^{\nu}(y)\delta\psi^{c^{\prime}}(x^{\prime})\delta\bar{\psi}^{d^{\prime}}(y^{\prime})}\Big{|}_{\Phi=0}$
is the full interaction vertex. The DS equation for the full fermionic
propagator is the following:

$\displaystyle S^{-1}_{cd}(x,y)-(S_{0}^{-1})_{cd}(x,y)$ $\displaystyle=$
$\displaystyle=\Xi_{cd}(x,y)-g\delta(x-y)\gamma^{\mu}_{a,cd}\alpha_{\mu}^{a}(x),$

where
$S^{-1}_{cd}(x,y)=\frac{\delta^{2}\Gamma}{\delta\psi^{c}(x)\delta\bar{\psi}^{d}(y)}\Big{|}_{\Phi=0}$,
and the fermionic self-energy is given by:

$\displaystyle\Xi_{cd}(x,y)$ $\displaystyle=$ $\displaystyle-ig\int
dzdw\Big{[}\gamma^{\mu}_{a,cf}S_{fg}(x,z)\times$ (13)
$\displaystyle\Gamma^{\nu}_{b,dg}(w,y,z)D_{\mu\nu}^{ab}(x,w)\Big{]},$

Finally, the $3$-point vertex respects the Bethe–Salpeter equation involving
the 4-point vertex miranski  and so on. As usual, we need to truncate the DS
equations in some $n$-point vertex, and our choice will be clarified in the
following section.

##  III Non-perturbative approach to DLSB

Starting from the DS equation for the retarded component of the full fermion
propagator:

$\displaystyle S^{-1}_{clcl}(x,y)$ $\displaystyle=$
$\displaystyle(S^{R})^{-1}(x,y)=$
$\displaystyle=(S_{0}^{R})^{-1}(x,y)+\Xi^{R}(x,y)-g\delta(x-y)\gamma^{\mu}\gamma_{5}\alpha_{\mu}(x),$

where $\alpha_{\mu}=\alpha^{cl}_{\mu}$ (and $\alpha^{q}_{\mu}=0$). Now we need
to choose the truncation of the vertex expansion, and we choose to use the so-
called rainbow approximation, given by :

$\Gamma^{\nu}_{b,dg}(w,y,z)\approx-g\gamma^{\nu}_{b,dg}\delta(w-y)\delta(z-y),$
(15)

which means that we are ignoring the contribution of the 4-point vertex. In
QED this truncation violates the Ward-Takahashi identity of gauge invariance.
But, due to the fact that our model does not have a gauge symmetry, this
approximation good enough to our proposal. Therefore, from eq. (13) the
retarded component of the fermionic self-energy can be rewritten as follows:

$\displaystyle\Xi^{R}(x,y)$ $\displaystyle=$ $\displaystyle
ig^{2}\gamma^{\mu}\gamma_{5}S^{K}(x,y)\gamma^{\nu}\gamma_{5}D_{\mu\nu}^{R}(y,x)+$
$\displaystyle+ig^{2}\gamma^{\mu}\gamma_{5}S^{R}(x,y)\gamma^{\nu}\gamma_{5}D_{\mu\nu}^{K}(x,y).$

Assuming
$D^{R}_{\mu\nu}(x,y)=\frac{G}{g^{2}}\eta_{\mu\nu}\delta(x-y)+O(g^{2})$,
$D_{\mu\nu}^{K}(x,y)=0+O(g^{2})$ (the so-called quenched approximation ref1 )
and applying the Wigner transformation (see appendix I) we can rewrite the
retarded component of the fermionic self-energy as:

$\Xi^{R}(X,p)\approx iG\int
dk\gamma^{\mu}\gamma_{5}S^{K}(X,k+p)\gamma_{\mu}\gamma_{5},$ (17)

with $dk=\frac{d^{4}k}{(2\pi)^{4}}$, $X=\frac{x+y}{2}$ and we assume a weak
dependence in $X$-coordinates for all functions of the model, which means a
small non-local character. Going further, we propose the following ansatz for
the inverse of the full propagator:

$(S^{R})^{-1}(X,p)=A(X,p)\not{p}-\Sigma(X,p)-\not{B}(X,p)\gamma_{5},$ (18)

where $A(X,p)$ is called renormalization function, $\Sigma(X,p)$ is the
position-dependent mass function and $B_{\mu}(X,p)$ is introduced here in
order to include the corrections to the parameter $\alpha_{\mu}$. Therefore,
can be shown that the exact full fermion propagator can be written as follows
fullprop :

$S^{R}=\frac{(A\not{p}-\not{B}\gamma_{5}+\Sigma)(p^{2}A^{2}-\Sigma^{2}-B^{2}+A[\not{p},\not{B}]\gamma_{5})}{(p^{2}A^{2}-\Sigma^{2}-B^{2})^{2}+4A^{2}(p^{2}B^{2}-(p\cdot
B)^{2})+i\epsilon},$ (19)

where we omit that $S^{R},A,B$ and $\Sigma$ are functions of $X$ and $p$ for
the sake of clarity. From the DS equation for the full fermionic propagator we
reach:

$(S^{R})^{-1}(X,p)=\not{p}-m+\Xi^{R}(X,p)-g\gamma^{\mu}\gamma_{5}\alpha_{\mu}(X),$
(20)

where

$\displaystyle g\alpha_{\mu}(X)$ $\displaystyle=$ $\displaystyle-iG~{}\int
dpTr[\gamma_{\mu}\gamma_{5}S^{K}(X,p)]=$ $\displaystyle=2GIm\int
dpTr[\gamma_{\mu}\gamma_{5}S^{R}(X,p)F(X,p)],$

where we use that $S^{K}$ is the Keldysh component of the full propagator and
is given by $S^{K}(X,p)=2iIm[S^{R}(X,p)F(X,p)]$ up to some $\partial_{X}$
contribution which we discard due to the small non-local approximation. The
function $F(X,p)$ is called Wigner function (or Wigner quasiprobability
distribution) and in our case is related to a generalization of the Fermi-
Dirac distribution in out-of-equilibrium systems (see appendix II for
details).

With the appropriate traces over the Dirac matrices, we can find equations for
the components of the full fermionic propagator and we can see that due to the
quenched approximation the $p$-dependence of $B_{\mu}$ and $\Sigma$
disappears. Lastly, assuming that the fermionic Wigner function $F(X,p)$ is a
real function and a scalar in the Dirac space and therefore $S^{K}(X,p)\approx
2iIm[S^{R}(X,p)]F(X,p)$, we can write the equations for the components of the
full fermion propagator as follows:

$\displaystyle A(p,X)-1$ $\displaystyle=$
$\displaystyle-\frac{G}{2p^{2}}\times$ $\displaystyle Im\int
dk\Big{\\{}Tr[\not{p}\gamma^{\mu}\gamma_{5}S^{R}(X,k)\gamma_{\mu}\gamma_{5}]F(X,k)\Big{\\}},$
$\displaystyle B^{\lambda}(X)-g\alpha^{\lambda}(X)$ $\displaystyle=$
$\displaystyle-\frac{G}{2}\times$ $\displaystyle Im\int
dk\Big{\\{}Tr[\gamma^{\lambda}\gamma_{5}\gamma^{\mu}\gamma_{5}S^{R}(X,k)\gamma_{\mu}\gamma_{5}]F(X,k)\Big{\\}},$

and

$\displaystyle\Sigma(X)-m=\frac{G}{2}\times$ $\displaystyle Im\int
dk\Big{\\{}Tr[\gamma^{\mu}\gamma_{5}S^{R}(X,k)\gamma_{\mu}\gamma_{5}]F(X,k)\Big{\\}},$

##  IV Equilibrium solutions:

The results we achieve in the previous section are general and can be applied
in a variety of physical contexts. The out-of-equilibrium QFT can be analyzed
in a temperature gradient or chemical potential environment, or the presence
of external fields and until in the cosmological context. The advantage of
this approach is that it contains naturally the equilibrium as a solution and
it will be our background from now on.

The equations which we find in the quenched and rainbow approximations are
simplified since $A$, $B_{\mu}$ and $\Sigma$ are independent of $p$ and in a
special case can also be local configurations (i.e., independent of $X$). This
is already the case of equilibrium systems. Therefore, in this regime we find
that:

$A(p)-1=-\frac{4G}{p^{2}}Im\int
dk\frac{A(k)\left(k.p\left(A(k)^{2}k^{2}-3B^{2}-\Sigma^{2}\right)+2B.kB.p\right)}{(A(k)^{2}k^{2}-\Sigma^{2}-B^{2})^{2}+4A(k)^{2}(k^{2}B^{2}-(k\cdot
B)^{2})+i\epsilon}F(k),$ (25)

where in equilibrium $F(k)=F_{eq}(k)=tanh\left((k_{0}-\mu)/2T\right)$ (see
appendix II). Since $A(p)-1\propto\frac{1}{p}$ and using the fact that the
integrand is an odd function of $k_{\mu}$ we will assume $A(p)=1$ from now on.
Using this assumption we reach:

$\displaystyle B_{\mu}$ $\displaystyle=$ $\displaystyle 4G\times$
$\displaystyle Im\int
dk\frac{B_{\mu}\left(B^{2}-3k^{2}+\Sigma^{2}\right)+2k_{\mu}(k\cdot
B)}{D(k,\Sigma,B)+i\epsilon}F(k),$ $\Sigma-m=8\Sigma GIm\int
dk\frac{\left(B^{2}-k^{2}+\Sigma^{2}\right)}{D(k,\Sigma,B)+i\epsilon}F(k),$
(27)

where $D(k,\Sigma,B)=(k^{2}-\Sigma^{2}-B^{2})^{2}+4(k^{2}B^{2}-(k\cdot
B)^{2})$. Important to note that the DLSB parameter respects the identity
$g\alpha_{\mu}=2B_{\mu}$. Going further, we study in the following subsections
some special cases where the above equations have manageable solutions.

###  IV.1 $T=0$ and $\mu=0$:

In the limit $T,\mu\rightarrow 0$, $F\rightarrow 1$ and is useful to apply the
Euclidean rotation given by $k_{0}=ik_{4}$, $B_{0}=iB_{4}$ such that
$k^{2}=-k_{E}^{2}$, $B^{2}=-B_{E}^{2}$ and $k.B=-k_{E}.B_{E}$. The gap
equations can be rewritten as:

$\displaystyle(B_{E})^{2}+4GRe\int\frac{d^{4}k_{E}}{(2\pi)^{4}}\times$
$\displaystyle~{}~{}\Big{[}\frac{(B_{E}^{2})\left(B_{E}^{2}-3k_{E}^{2}-\Sigma^{2}\right)+2(k_{E}\cdot
B_{E})^{2}}{D(k_{E},\Sigma,B_{E})+i\epsilon}\Big{]}=0,$

and

$\displaystyle\Sigma-m+8\Sigma GRe\int\frac{d^{4}k_{E}}{(2\pi)^{4}}\times$
$\displaystyle~{}~{}~{}~{}~{}~{}\times\Big{[}\frac{\left(k_{E}^{2}-B_{E}^{2}+\Sigma^{2}\right)}{D(k_{E},\Sigma,B_{E})+i\epsilon}\Big{]}=0,$

where
$D(k_{E},\Sigma,B_{E})=(k_{E}^{2}+\Sigma^{2}-B_{E}^{2})^{2}+4(k_{E}^{2}B_{E}^{2}-(k_{E}\cdot
B_{E})^{2})$ and we contract eq. (IV) with $B_{\mu}$ after the Euclidean
rotation. Using spherical coordinates in 4D Euclidean space as
$k_{1}=|k_{E}|\cos\beta$, $k_{2}=|k_{E}|\sin\beta\cos\theta$ ,
$k_{3}=|k_{E}|\sin\beta\sin\theta\cos\phi$ and
$k_{4}=|k_{E}|\sin\beta\sin\theta\sin\phi$, fixing $B_{E}$ such that
$B_{E}\cdot k_{E}=|B_{E}||k_{E}|\cos\beta$ and integrating over
$\beta\in[0,\pi]$, $\theta\in[0,\pi]$, $\phi\in[0,2\pi]$,
$|k_{E}|\in[0,\Lambda]$ , where we introduce the cutoff parameter $\Lambda$,
and $d^{4}k_{E}=d\theta\sin\theta d\phi d\beta\sin^{2}\beta
d|k_{E}||k_{E}|^{3}$, we find the following set of gap equations:

$\tilde{b}^{2}+\tilde{G}F_{1}(\tilde{b},\tilde{\Sigma})=0$ (30)

and

$\tilde{\Sigma}-\tilde{m}+\tilde{G}\tilde{\Sigma}F_{2}(\tilde{b},\tilde{\Sigma})=0$
(31)

where $b=|B_{E}|$, and $\tilde{G}=G\Lambda^{2}$,
$\tilde{b}=|B_{E}|\Lambda^{-1}$ ,$\tilde{\Sigma}=\Sigma\Lambda^{-1}$,
$\tilde{m}=m\Lambda^{-1}$ and omit the structure of $F_{1,2}$ for the sake of
simplicity. In fig. (1) we can see the solution of (30) for some values of
$\tilde{\Sigma}$ and in fig. (2) we can see the solution of (31) for some
values of $\tilde{b}$.

Figure 1: Solution for $\tilde{b}$ in equation (30) for $\tilde{b}^{2}>0$; for
$\tilde{\Sigma}=0$ (black curve) and for $\tilde{\Sigma}=0.7$ (dashed curve),
in units of $b$.  Figure 2: Normalized solution of $\tilde{\Sigma}$ in
equation (31) and $\tilde{b}=0$ (black curve) and for $\tilde{b}=0.7$ (dashed
curve), in units of $m$.

If we assume $\tilde{b}=0$, eq. (31) reduces to
$\tilde{\Sigma}-\tilde{m}+\frac{G}{2\pi^{2}}\tilde{\Sigma}\left(\tilde{\Sigma}^{2}\ln\left(\frac{\tilde{\Sigma}^{2}}{\tilde{\Sigma}^{2}+1}\right)+1\right)=0$,
and it’s solutions is shown by the black curve of figure (2). In the case of
massless fermions ($m=0$), the solution $\tilde{\Sigma}=0$ is possible and in
this case we find the following gap equation for $\tilde{b}$ :

$\displaystyle\frac{1}{\tilde{G}}-\frac{\left(15+\tilde{b}^{2}\left(48\ln
2-31+24\ln\left(\frac{\tilde{b}^{2}}{\tilde{b}^{2}+1}\right)\right)\right)}{24\pi^{2}}=0~{}.$

As we can see in fig. (3), the solution is very different of the (massless)
perturbative result DLSB1 ; DLSB2  given by $b=\sqrt{\frac{3\pi^{2}}{|G|}}$,
which goes to infinity where $G$ approaches to zero, and goes to zero in the
limit $G\rightarrow\infty$. Our result has (in the massless case) a critical
coupling $\tilde{G}_{c}=\frac{24\pi^{2}}{15}\approx 15.8$, and for values of
$\tilde{G}>\tilde{G}_{c}$ the Lorentz symmetry breaking occurs. Is important
to highlight that we only find time-like solutions, i.e., solutions where
$\tilde{b}^{2}>0$.

Figure 3: Plot of the gap equation solution for $m=0$ given by eq. (IV.1)
(black curve) and the perturbative solution $\tilde{b}=\sqrt{3\pi^{2}/G}$
DLSB1 ; DLSB2  (dashed curve).

We can improve our result for $\tilde{b}$ assuming that $\Sigma\approx m$.
This approximation is valid for $\tilde{G}<1$. In this approximation the gap
equation is given by $\tilde{b}^{2}+\tilde{G}F_{1}(\tilde{b},\tilde{m})=0$ and
we can see in fig. (4) the plot of $\tilde{b}$ in function of $\tilde{m}$ for
$\tilde{G}=10^{-2}$ and the perturbative result
$b^{2}=3\pi^{2}\left(\frac{1}{|G|}+\frac{m^{2}}{2\pi^{2}}\ln\left(\frac{m^{2}}{\mu^{2}}\right)\right)$
from ref. DLSB1 . In the limit $\tilde{m}\rightarrow 0$ the DLSB parameter $b$
vanishes, whereas the perturbative result is given by
$b^{2}=\frac{3\pi^{2}}{|G|}$.

Figure 4: Plot of the gap equation solution
$\tilde{b}^{2}+\tilde{G}F_{1}(\tilde{b},\tilde{\Sigma})=0$ for $\Sigma\approx
m$ and for $\log_{10}\tilde{G}=-2$ (black), and the perturbative solution for
$\log_{10}\tilde{G}=-2$ , normalized in units of $\mu$ (black and dashed).

###  IV.2 $T=0$ and $\mu\neq 0$:

As we have shown in the previous section, the vector $B_{\mu}$ can acquire a
non-null vacuum expectation value $\langle B\rangle=b$ at zero temperature and
chemical potential, even in the non-perturbative regime. In the case where we
consider the global U(1) symmetry of the action in eq. (1), a chemical
potential must be added and this is achieved applying the shift in the
equilibrium fermionic function $F(k_{0})\rightarrow F(k_{0}-\mu)$ where $\mu$
is the chemical potential. Explicitly, the fermionic function will be given as
follows:

$F(k_{0}-\mu)=1+\left(\frac{1}{e^{\beta(k_{0}-\mu)}+1}+\frac{1}{e^{\beta(k_{0}+\mu)}+1}\right)\Theta(k_{0})$
(33)

where $\beta=T^{-1}$ and $\Theta(x)$ is defined as $\Theta(x)=1$ for $x>0$ and
$\Theta(x)=0$ for $x<0$. In the regime where $T$ is negligible ($T<<\mu$) we
can use the Sommerfeld expansion as follows. Let $H$ a function of $k_{0}$,
therefore:

$\displaystyle I(\mu)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dk_{0}H(k_{0})F(k_{0}-\mu)$ (34)
$\displaystyle=$ $\displaystyle I(0)+2\int_{0}^{\mu}dk_{0}H(k_{0}).$

Therefore, the gap equation for $B$ can be rewritten as follows:

$B^{2}(\mu)=b^{2}-8\pi gG\left(2\int_{0}^{\mu}dk_{0}H(k_{0})\right),$ (35)

where

$\displaystyle H(k_{0})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}\vec{k}}{(2\pi)^{4}}\Big{[}$ $\displaystyle
B^{2}\left(B^{2}-3k^{2}+\Sigma^{2}\right)+2(k\cdot B)^{2}\Big{]}\delta(D(k)),$

with $D(k)=(k^{2}-\Sigma^{2}-B^{2})^{2}+4(k^{2}B^{2}-(k\cdot B)^{2})$ 111We
use the identity $Im\frac{1}{D(k)+i\epsilon}=\pi\delta(D(k))$. In the case of
time-like DLSB parameter we can choose a reference frame such that
$B_{\nu}=\delta_{\nu 0}B$ (also $b_{\nu}=\delta_{\nu 0}b$ ) , and in the
massless case we can assume $\Sigma=0$, and using the minimal subtraction
scheme (MS) in the divergent component $B(\mu=0)=b+(\text{divergent terms})$,
with $b$ finite, we find:

$B^{2}-b^{2}+\frac{BG\mu\left(15B^{2}+\mu^{2}\right)}{6\pi^{2}}=0\text{~{}~{},
if~{}~{}}B>\mu,$ (37) $\displaystyle
B^{2}-b^{2}+\frac{B^{2}G\left(3B^{2}\ln\left(\frac{\mu}{B}\right)+5B^{2}+3\mu^{2}\right)}{3\pi^{2}}=0,$
$\displaystyle\text{~{}~{}if~{}~{}}B<\mu$ (38)

The solution of the gap equation is given by $B=F(\mu,G)$( with $F(0,G)=b$),
can be find numerically and is shown in fig. (5). As can be seen in fig. (5)
and fig. (6) the effect of the chemical potential is to decrease the value of
$B$.

Figure 5: Normalized gap equation solutions in the massless case for:
$\mu=0.0$ (black curve), $\mu=1.0$ (dashed curve), and $\mu=10.0$, in units of
$b$.  Figure 6: Normalized gap equation solutions in the massless as a
function of $\mu$ for: $\tilde{G}=10^{3}$ (black curve), $\tilde{G}=10^{8}$
(dashed curve), in units of b.

###  IV.3 $T\neq 0$ and $\mu\neq 0$:

Finally, from eq. (IV) we can explicitly find the phase diagram through
$B(\mu,T,G)=0$ in the massless case, and the solutions are shown in fig. (7).
We can see that the critical temperature depends upon the magnitude of the
DLSB parameter at zero temperature $b$. Particularly, at zero chemical
potential the critical temperature can be written as
$T_{c}\approx\tilde{b}^{2}T_{0}$ with $T_{0}=1.73G^{-1/2}$. In the zero-
temperature limit, the critical chemical potential can be written as
$\mu_{c}\approx\tilde{b}^{2}\mu_{0}$ with $\mu_{0}=2.55G^{-1/2}$. Since $b$ is
itself a function of $G$ we should combine these expressions with the results
of fig. (1) and this combination gives us the fig. (8). As we can see, the
critical temperature and critical chemical potential have the same dependence
in $\tilde{G}$, up to a multiplicative constant.

Figure 7: Phase transition diagram for a time-like DLSB with $m=0$;
$b=\sqrt{0.5/G}$ (black curve), for $b=\sqrt{G}$ (black and dashed) and for
$b=\sqrt{2.0/G}$. Here $T_{0}\approx 1.73G^{-1/2}$ and $\mu_{0}\approx
2.55G^{-1/2}$.  Figure 8: Plot of the $\log_{10}$ of the critical temperature
$\tilde{T}_{c}$ (black curve) and critical chemical potential
$\tilde{\mu}_{c}$ (black and dashed) as a function of $\tilde{G}$ and for
$m=0$.

##  V Final comments and perspectives

In this work, we investigate aspects of dynamical Lorentz symmetry breaking.
Firstly, by use of the Keldysh formalism, we find the DS equations for non-
equilibrium fermionic systems with interactions given by eq. (1). We also show
that through the Keldysh formalism the vacuum expectation value (v.e.v) of the
composite field $A_{\mu}$ given by $g\langle A_{\mu}\rangle=g\alpha_{\mu}=2B$
will be in general dependent on the non-local coordinate X. This shows that
the violation of the Lorentz symmetry and non-equilibrium can be naturally
combined by the use of Keldysh formalism.

Going further, analyzing the DS equations in the equilibrium limit and we see
that, through the solution of the DS equations with meaningful approximations,
we can access new sectors of the model proposed in DLSB1 ; DLSB2 , we reach
the solutions of the gap equations for both DLSB and self-energy and we find
that system has two special sectors, the first is the weak coupling limit,
where the DLSB is dependent of the bare mass $m$. The second sector is the
strong coupling limit, where the self-energy vanishes and the DLSB became
intense and rules the dynamics of the system (see fig. (2) and fig. (3)).
Between these two sectors, we have a transition region where both the DLSB
parameter and the self-energy are important to the dynamic of the system. In
fig. (4) we plot the behavior of the DLSB parameter in the small coupling
regime ($\tilde{G}<1$) as a function of the mass (assuming the self-energy
$\Sigma\approx m$) for $\tilde{G}=10^{-2}$ and the perturbative solution found
in ref. DLSB1 . As we can see, the behavior of both curves is similar, and the
non-perturbative solution can be seen as an improvement of the perturbative
one, without any abrupt change in the character of the DLSB parameter. An
important difference is that in the limit $m\rightarrow 0$ the DLSB vanishes,
whereas the perturbative result is finite ($b\propto G^{-1/2}$).

In particular, we show that in the massless case the DLSB is a possibility of
strongly coupled systems with $\tilde{G}>\tilde{G}_{c}\approx 15.8$. As can be
seen in fig. (1) and fig. (2), there are two different regions concerning the
strength of the coupling constant $G$. The first, $\tilde{G}<<\tilde{G}_{c}$
is characterized by a massless fermionic theory with a null DLSB parameter,
and the second, with $\tilde{G}>\tilde{G}_{c}$ we find a massless fermionic
theory and with non-null DLSB parameter. We show that assuming $m=0$ the DS
equations shows that for $\tilde{G}>\tilde{G}_{c}$, $T_{c}\approx
1.73\tilde{b}^{2}G^{-1/2}$ at zero chemical potential and $\mu_{c}\approx
2.55\tilde{b}^{2}G^{-1/2}$ at zero temperature, and the behavior of this
critical parameters can be shown in fig. (8).

In this context, we can discuss the interpretation of the (time-like) DLSB
parameter as a true chiral chemical potential. Due to the existence of a mass
in the self-energy gap equation solution, there is no chiral symmetry in the
model, unless $m=0$. Therefore, the model will have a true chemical potential
only if $m=0$ or in the limit $\tilde{G}\rightarrow\infty$.

In general a chiral chemical potential is related to a conservation of the
chiral current, i.e., $\partial_{\mu}j_{5}^{\mu}=0$, where
$j_{5}^{\mu}=\bar{\psi}\gamma^{\mu}\gamma_{5}\psi$. Since our model has not a
gauge symmetry, there is no anomalies and the symmetry is not broken if $m=0$.
Therefore, in the massless case we have a conserved charge given by
$Q_{5}=\int d^{3}xj_{5}^{0}=\int
d^{3}x\left(\psi_{L}^{\dagger}\psi_{L}-\psi_{R}^{\dagger}\psi_{R}\right)=N_{L}-N_{R}$,
with $L,R$ representing the chiral components of the Dirac fermion. In a
thermodynamical perspective, a chiral chemical potential generates a chiral
charge density $\rho_{5}=\rho_{L}-\rho_{R}$ which is called a chiral imbalance
and is given by
$\rho_{5}=\frac{\mu_{5}^{3}}{3\pi^{2}}+\frac{\mu_{5}}{3}\left(T^{2}+\mu^{2}\right)$
chiraleff , and in our case $\mu_{5}=g\langle A_{0}\rangle=2B$. This effect
violates CP and CPT symmetries, such way it can produce a barion number
violation and can be a source for Baryogenesis baryo ; baryo2 .

Even if we assume $m\neq 0$, in the strong coupling limit
$\tilde{G}>>\tilde{G}_{c}$ we can approximate $\Sigma\approx 0$ and in this
case, the DLSB parameter is not a true chemical potential and it still
contributes to a “chiral” imbalance, but in this case, we must take into count
that the equality between chirality and helicity is not exact. From the
perspective of Keldysh formalism, a way to implement the chiral chemical
potential is to expand the Wigner function $F$ character in Dirac space
through projectors $P_{L,R}=\frac{1}{2}(1\pm\gamma_{5})$ to accommodate this
new feature. This implementation is beyond the scope of this work and will be
investigated in the future.

Going further, a way to improve our results can occur by extending the rainbow
approximation through the use of the Bethe-Salpeter equation. We also can
improve our result by extending the quenched approximation, taking into
account the polarization tensor contribution, and taking into account the non-
locality of the renormalization function $A(X,p)$. These modifications will
introduce new and more complex non-localities which difficult the extract
information, but numerical analysis such as ref2  can be made.

New features can also appear if we take a more complex structure of the Wigner
function $F(X,p)$, particularly the spinorial character. Since the DLSB
parameter is proportional to $S^{K}=2iIm[S^{R}F]$ if we give up the scalar
property of the Wigner function a much richer tensor structure arose. In
special, $S^{K}$ could have off-shell contributions which is a feature of non-
equilibrium physics and can give us new sources of Lorentz symmetry breaking.
Although the model represented by the action (3) and analyzed with the non-
perturbative tools of the DS equations can be called a toy model, some
applications can be thought of in condensed matter, e.g. in the context of 3D
Weyl semi-metals models weyl1 . The study of these other applications will be
the theme of a forthcoming paper.

###### Acknowledgements.

This work was supported by FAPERJ. The author is grateful to P.C. Malta, P. de
Fabritiis, and the referee for interesting discussions and relevant criticism.

## Appendix I: Wigner transformation

Since our results are dependent on the knowledge of the fermionic function, we
need to find the solution of the kinetic equation for F(x,y). We can achieve
this with the use of the Wigner transformation (WT) which is defined as
follows. Let $A(x,x^{\prime})$ a function and $X=\frac{x+x^{\prime}}{2}$ and
$\tilde{x}=x-x^{\prime}$, thus:

$A(X,p)=\int
d\tilde{x}e^{-ip\tilde{x}}A\left(X+\frac{\tilde{x}}{2},X-\frac{\tilde{x}}{2}\right).$
(39)

The inverse of the Wigner transformation is given by:

$A(x,x^{\prime})=\int
dpe^{ip(x-x^{\prime})}A\left(\frac{x+x^{\prime}}{2},p\right).$ (40)

with $dp=\frac{d^{d}p}{(2\pi)^{d}}$ for any dimension $d$. Let $C=A\circ B$ a
convolution, the WT of $C$ reads:

$\displaystyle C(X,p)$ $\displaystyle=$ $\displaystyle
A(X,p)e^{\frac{i}{2}(\overleftarrow{\partial}_{X}\overrightarrow{\partial}_{p}-\overleftarrow{\partial}_{p}\overrightarrow{\partial}_{X})}B(X,p)$
(41) $\displaystyle\approx
A(X,p)B(X,p)+\frac{i}{2}\Big{(}\overrightarrow{\partial}_{X}A(X,p)\overrightarrow{\partial}_{p}B(X,p)-$
$\displaystyle-\overrightarrow{\partial}_{p}A(X,p)\overrightarrow{\partial}_{X}B(X,p)\Big{)}+...$

When $A(X,p)$ and $B(X,p)$ is weakly dependent on $X$ coordinate the
approximation is valid up to first order in $\partial_{X}$. Other important
property is the WT of a product of two functions. Let
$C(x,x^{\prime})=A(x,x^{\prime})B(x,x^{\prime})$. Thus:

$C(X,p)=\int dqA(X,p-q)B(X,q)$ (42)

If the function $A(x,x^{\prime})$ are invariant under translations, thus
$A(x,x^{\prime})=A(x-x^{\prime})$ implying
$\overrightarrow{\partial}_{X}A(X,p)=0$, therefore $A(X,p)=A(p)$ and the WT is
equivalent to the Fourier transformation.

## Appendix II: kinetic equations

The full fermionic propagator $S_{cd}(x,y)$, as can be reach from eq. (II),
obey the following equation:

$\int
dy\left[(S_{0}^{-1})_{cd}(x,y)+g\delta(x-y)\gamma^{\mu}_{a,cd}\alpha_{\mu}^{a}(x)+\Xi_{cd}(x,y)\right]S_{df}(y,z)=\delta_{cf}\delta(x-z)$
(43)

Due to the causality structure the full fermionic propagator in the Keldysh
formalism it has the following structure in the Keldysh space:

$S_{df}(y,z)=\begin{pmatrix}S^{R}(y,z)&S^{K}(y,z)\\\
0&S^{A}(y,z)\end{pmatrix}$ (44)

The self-energy contribution obeys a similar structure, i.e.:

$\Xi_{df}(y,z)=\begin{pmatrix}\Xi^{R}(y,z)&\Xi^{K}(y,z)\\\
0&\Xi^{A}(y,z)\end{pmatrix}$ (45)

Therefore the retarded (advanced) component of $S_{df}(y,z)$ obey the
following (Kadanoff-baym-like) nonlinear equations:

$\displaystyle\left(i\gamma^{\mu}\partial_{\mu}+g\gamma^{\mu}\gamma_{5}\alpha^{cl}_{\mu}(x)-m\right)S^{R(A)}(x,y)+$
$\displaystyle+\int dz\Xi^{R(A)}(x,z)S^{R(A)}(z,y)=\delta(x-y),$

where

$g\alpha_{\mu}^{cl}(x)=iG~{}Tr[\gamma^{\mu}\gamma_{5}S^{K}(x,x)]$ (47)

Important to highlight that the causal structure of the full fermionic
propagator fix $\alpha_{\mu}^{q}(x)=0$ unambiguously. Therefore, we use
$\alpha_{\mu}^{cl}(x)=\alpha_{\mu}(x)$ for short. Since $S^{K}$ is skew-
Hermitian, by parametrization we can write $S^{K}=S^{R}\circ F-F\circ S^{A}$
with $F=F(x,y)$ a generic hermitian tensor in Dirac matrices space and we use
the symbol “$\circ$” as the convolution operator. Therefore, $F$ obey the
follow equation:

$\displaystyle\int
dy\left[\Delta(x,y)F(y,z)-F(x,y)\Delta(y,z)\right]=\Xi^{K}(x,z)+$
$\displaystyle\hskip 14.22636pt+\int
dy\left[F(x,y)\Xi^{R}(y,z)-\Xi^{A}(x,y)F(y,z)\right],$

where
$\Delta(x,y)=\left(i\gamma^{\mu}\partial_{\mu}+g\gamma^{\mu}\gamma_{5}\alpha_{\mu}(x)-m\right)\delta(x-y)$.
Applying the Wigner transformation in eq. (Appendix II: kinetic equations) we
find:

$Re\left[\left(\gamma^{\mu}+\frac{\partial}{\partial
p^{\mu}}\tilde{\Xi}^{R}(X,p)\right)\frac{\partial}{\partial
X_{\mu}}F(X,p)-\frac{\partial}{\partial
X^{\mu}}\tilde{\Xi}^{R}(X,p)\frac{\partial}{\partial
p_{\mu}}F(X,p)\right]=I_{coll}[F]$ (49)

where
$\tilde{\Xi}^{R}(X,p)=\Xi^{R}(X,p)+g\gamma^{\nu}\gamma_{5}\alpha_{\nu}(X)$ ,
$\alpha_{\mu}(X)$ is the WT of $\alpha_{\mu}(x)$ and is given by
$\alpha_{\mu}(X)=iG~{}\int
dp^{\prime}Tr[\gamma_{\mu}\gamma_{5}S^{K}(X,p^{\prime})]$ and
$I_{coll}[F]=-i\Xi^{K}(X,p)+2Im[\Xi^{R}(X,p)F(X,p)]$ is the collision
integral. The equilibrium solution is given by $I_{coll}[F]=0$ and can be
shown that for fermionic fields in equilibrium
$F_{eq}(k)=tanh(\frac{k_{0}-\mu}{2T})$ and its connected with the fluctuation
– dissipation theorem (FDT) kamenevbook .

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