# Meson scattering in a non-minimally Lorentz-violating scalar QED at finite
temperature

M. C. Araújo<EMAIL_ADDRESS>Universidade Federal do Ceará (UFC),
Departamento de Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.  R. V. Maluf
<EMAIL_ADDRESS>Universidade Federal do Ceará (UFC), Departamento de
Física,
Campus do Pici, Fortaleza - CE, C.P. 6030, 60455-760 - Brazil.  J. Furtado
<EMAIL_ADDRESS>Universidade Federal do Cariri (UFCA), Av. Tenente
Raimundo Rocha,
Cidade Universitária, Juazeiro do Norte, Ceará, CEP 63048-080, Brasil
Department of Physics, Faculty of Science, Gazi University, 06500 Ankara,
Turkey

(August 28, 2024)

###### Abstract

In this paper we study meson scattering in a non-minimally Lorentz-violating
scalar QED at finite temperature. The meson scatterings were investigated in
tree level and the finite temperature effects were addressed by using the
thermofield dynamics formalism. We have considered three types of scattering,
namely, meson-antimeson of $a$-type into meson-antimeson of $b$-type, meson-
antimeson of $a$-type into meson-antimeson of $a$-type and meson-meson of
$a$-type into meson-meson of $a$-type. For each scattering we have computed
the cross section in order to investigate the influence of the finite
temperature effects.

Mesons scattering at finite temperature, Lorentz symmetry breaking, Lorentz-
violating scalar QED, Thermo Field Dynamics.

##  I Introduction

In recent years, there has been a focus on exploring potential expansions of
the Standard Model (SM). Within this framework, the discussion has shifted
towards the breaking of Lorentz and CPT symmetries, considered as a
significant subject Kos1 ; Kos2 ; Kos3 ; Kos4 ; Colladay:1996iz ;
Colladay:1998fq . Typically, Lorentz symmetry is disrupted by introducing
specific directions in space-time, represented by additive terms that are
proportionate to small constant vectors or tensors. The Standard Model
Extension (SME) Colladay:1996iz ; Colladay:1998fq  stands out as the most
established model addressing the consequences of Lorentz and CPT symmetry
violation. SME, being an effective field theory, incorporates all conceivable
terms violating Lorentz and CPT symmetries in its Lagrangian.

Kostelecky Edwards:2018lsn . proposed an extension of the scalar sector
incorporating Lorentz-violating effects. This model presents a comprehensive
effective scalar field theory across any spacetime dimension, featuring
explicit perturbative spin-independent Lorentz-violating operators of
arbitrary mass dimension. The significance of this development lies in the
predominance of spin in most fundamental particles of the SM, with the Higgs
boson being the sole example of a fundamental spinless particle. Despite the
relatively minor role of the scalar sector of QED (sQED) compared to strong
interactions in describing meson coupling, it has been suggested
Edwards:2019lfb  that a Lorentz-violating extension of sQED could effectively
address slight CPT deviations in neutral-meson oscillations.

The minimal extension of SME was extensively studied in the last years in
several contexts, such as radiative corrections Jackiw:1999yp ;
Kostelecky:2001jc ; Ferrari:2021eam , neutrinos oscillations MINOS:2008fnv ;
MINOS:2010kat ; MiniBooNE:2011pix , Euler-Heisenberg effective action
Ferrari:2021eam ; Furtado:2014cja , gravitational context Kostelecky:2016kkn ;
Assuncao:2018jkq , finite temperature field theory Assuncao:2018jkq ;
Leite:2013pca ; Leite:2011jg ; Assuncao:2016fko ; Araujo:2023izx  among others
(for a more complete review, see f.e. Mariz:2022oib ). The non-minimal
extension, despite its posses only nonrenormalizable terms, have been
receiving attention in the literature Myers:2003fd ; Kostelecky:2009zp ;
Kostelecky:2011gq ; Mariz:2010fm ; Rubtsov:2012kb . A strong motivation lies
in the fact that some relevant astrophysical processes impose severe
restrictions on the coefficients associated with the operators with dimension
$d\geq 5$, being such contributions comparable or even dominant when compared
with the ones that arise from dimension $d\leq 4$ operators. Within the
nonminimal LV framework, an important role was played by the paper
Kostelecky:2009zp , in which the simplest cases of such operators were, for
the very first time, introduced in the LV scenario. The first studies of their
perturbative impact were investigated in Gomes:2009ch ; Mariz:2010fm ;
Gazzola:2010vr ; Mariz:2011ed ; BaetaScarpelli:2013rmt . And finally, all
possible LV extensions of SME with dimensions up to 6 for fermion-dependent
operators were listed in Kostelecky:2018yfa .

The topological structure of the thermofield dynamics (TFD) formalism allows
the study of thermal effects in several different systems, ranging from
particle physics to quantum computing and black hole physics Araujo:2022qke ;
Ulhoa:2023opw ; quantumcomp1 . The central points of the TFD formalism are the
duplication of the Hilbert space and the use of Bogoliubov transformation
Umezawa1 ; Umezawa2 . The thermal Hilbert space (or doubled Hilbert space)
allows the construction of a so called thermal ground state. Such ingredients
are related to an important result in which it is shown that for any arbitrary
operator, its statistical avarage is equal to its vacuum expectation value. In
the context of Lorentz symmetry violation, the TFD formalism is being widely
and successfully used for computing the finite temperature effects of tree
level (while finite temperature loop corrections are usully computed using the
Matsubara formalism Araujo:2023izx ; Assuncao:2020cdo ) scatterings for QED
processes in both spinorial Cabral:2024tqa ; Santos:2021hed ; Souza:2021uuf
and scalar sectors Araujo:2022qke . Moreover, the thermal effects of Lorentz-
violating extension of non-Abelian sectors were also investigated in the TFD
formalism Santos:2022zfz .

In this paper we study meson scattering in a non-minimally Lorentz-violating
scalar QED at finite temperature. The meson scatterings were investigated in
tree level and the finite temperature effects were addressed by using the
thermofield dynamics formalism. We have considered three types of scattering,
namely, meson-antimeson of $a$-type into meson-antimeson of $b$-type, meson-
antimeson of $a$-type into meson-antimeson of $a$-type and meson-meson of
$a$-type into meson-meson of $a$-type. For each scattering we have computed
the cross section in order to investigate the influence of the finite
temperature effects.

This paper is organized as follows. In Sec. II, a revision of the thermofield
dynamics formalism is presented. In Sec. III, we discuss the non-minimally
Lorentz-violating scalar sector of QED we are considering in both zero and
finite temperature. In Sec. IV we calculate the differential cross sections
for mesons at finite temperature by employing the TFD formalism. Finally, in
Sec. V, the main results of the paper are summarized. Throughout the paper, we
have used the natural units $\hslash=c=k_{B}=1$, with $k_{B}$ being the
Boltzmann constant, and assumed $\eta^{\mu\nu}=diag(1,-1,-1,-1)$ as the
Minkowski metric.

##  II Thermofield Dynamics Formalism

The Thermofield Dynamics (TFD) is a real-time formalism that allows the
investigation of temperature effects on systems described by a quantum field
theory. Its main feature is the definition of a thermal vacuum state
$|\,0(\beta)\,\rangle$ through the duplication of the degrees of freedom of
the system, characterized by the double-space
$\hat{\mathcal{S}}=\mathcal{S}\otimes\tilde{\mathcal{S}}$. Here, $\tilde{S}$
is to be thought of as a copy of the original space $S$, where all the
physical observables are actually defined, and $\beta=T^{-1}$, being $T$ the
system temperature. Due to this doubled structure, for each operator
$\mathcal{O}$ in the space $\mathcal{S}$, there exists an operator
$\hat{\mathcal{O}}$ in $\hat{\mathcal{S}}$ defined by

$\displaystyle\hat{\mathcal{O}}\equiv\mathcal{O}-\tilde{\mathcal{O}},$ (1)

where $\tilde{\mathcal{O}}$ is the analogous operator in the tilde-space
$\tilde{\mathcal{S}}$. Tilde and non-tilde operators are connected by the
following conjugation rules:

$\displaystyle\widetilde{(\mathcal{O}_{i}\mathcal{O}_{j})}$ $\displaystyle=$
$\displaystyle\tilde{\mathcal{O}_{i}}\tilde{\mathcal{O}_{j}},$ (2)
$\displaystyle\widetilde{(a\mathcal{O}_{i}+b\mathcal{O}_{j})}$
$\displaystyle=$ $\displaystyle
a^{\ast}\tilde{\mathcal{O}_{i}}+b^{\ast}\tilde{\mathcal{O}_{j}},$ (3)
$\displaystyle(\tilde{\mathcal{O}_{i}})^{\dagger}$ $\displaystyle=$
$\displaystyle\widetilde{(\mathcal{O}_{i}^{\dagger})},$ (4)
$\displaystyle\widetilde{(\tilde{\mathcal{O}}_{i})}$ $\displaystyle=$
$\displaystyle\kappa\mathcal{O}_{i},$ (5)

where $\kappa=1\,(-1)$ for bosons (fermions). As a consequence, thermal
operators can be defined through Bogoliubov transformations, which consist of
a rotation in the tilde and non-tilde variables. For bosons, for example, the
relations between thermal and non-thermal operators are given by

$\displaystyle a(\mathbf{k},\beta)$ $\displaystyle=$ $\displaystyle
u_{B}(\mathbf{k},\beta)a(\mathbf{k})-v_{B}(\mathbf{k},\beta)\tilde{a}^{\dagger}(\mathbf{k}),$
(6) $\displaystyle a^{\dagger}(\mathbf{k},\beta)$ $\displaystyle=$
$\displaystyle
u_{B}(\mathbf{k},\beta)a^{\dagger}(\mathbf{k})-v_{B}(\mathbf{k},\beta)\tilde{a}(\mathbf{k}),$
(7) $\displaystyle\tilde{a}(\mathbf{k},\beta)$ $\displaystyle=$ $\displaystyle
u_{B}(\mathbf{k},\beta)\tilde{a}(\mathbf{k})-v_{B}(\mathbf{k},\beta)a^{\dagger}(\mathbf{k}),$
(8) $\displaystyle\tilde{a}^{\dagger}(\mathbf{k},\beta)$ $\displaystyle=$
$\displaystyle
u_{B}(\mathbf{k},\beta)\tilde{a}^{\dagger}(\mathbf{k})-v_{B}(\mathbf{k},\beta)a(\mathbf{k}),$
(9)

where $a\,(\tilde{a})$ and $a^{\dagger}\,(\tilde{a}^{\dagger})$ stand for an
annihilation and creation operators in the space $\mathcal{S}$
($\tilde{\mathcal{S}}$), respectivally. Above,

$\displaystyle u_{B}(\mathbf{k},\beta)$ $\displaystyle=$ $\displaystyle
e^{\beta\,|E(\mathbf{k})|/2}v_{B}(\mathbf{k},\beta),$ (10)

and

$\displaystyle v_{B}(\mathbf{k},\beta)$ $\displaystyle=$
$\displaystyle(e^{\beta\,|E(\mathbf{k})|}-1)^{-1/2},$ (11)

are the functions that carry the temperature dependence of the thermal
operators. Here, $E(\mathbf{k})$ is the energy of oscillation associated with
the mode $\mathbf{k}$. Also, it can be shown that thermal operators must
satisfy

$\displaystyle\left[a(\mathbf{k},\beta),a^{\dagger}(\mathbf{k^{\prime}},\beta)\right]$
$\displaystyle=$
$\displaystyle(2\pi)^{3}\delta^{3}(\mathbf{k}-\mathbf{k^{\prime}}),$ (12)

with the same conventional algebra for tilde operators and all the other
commutation relations being zero.

In the TFD formalism, when written in the center-of-mass reference frame, the
differential cross section takes the form of

$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{\beta,cm}=\frac{1}{2E^{2}_{cm}}\frac{|\mathbf{k}|}{16\pi^{2}E_{cm}}|\hat{\mathcal{M}}(\beta)|^{2},$
(13)

where

$\hat{\mathcal{M}}(\beta)\equiv\sum_{n=0}^{\infty}\frac{(-i)^{n}}{n!}\int\,d^{4}z_{1}\cdots
d^{4}z_{n}\,{}_{\beta}\langle\,f\,|\mathcal{T}\left[\hat{\mathcal{L}}_{int}(z_{1})\cdots\hat{\mathcal{L}}_{int}(z_{n})\right]|\,i\,\rangle_{\beta},$
(14)

defines the temperature-dependent transition amplitude for a scattering of
interest. In expressions (13) and (14), $E_{cm}$ is the center-of-mass energy,
$\mathbf{k}$ is the $3$-momentum of an outgoing particle,
$\hat{\mathcal{L}}_{int}(z)$ is the interaction Lagrangian density in the
space $\hat{\mathcal{S}}$, and $\mathcal{T}$ is the time-ordering operator.
From now on, the procedure to be taken is quite standard in quantum field
theory, however, we must make use of the Bogoliubov transformations (6)-(9) in
order to first write Eq. (14) in terms of the thermal creation and
annihilation operators for some Hamiltonian of interest. This will allow us to
deal properly with the initial and final states of particles respectively
defined by

$\displaystyle|\,i\,\rangle_{\beta}$ $\displaystyle\equiv$
$\displaystyle\sqrt{2E(\mathbf{k}_{1})2E(\mathbf{k}_{2})}\,a^{\dagger}(\mathbf{k}_{1},\beta)a^{\dagger}(\mathbf{k}_{2},\beta)\,|\,0(\beta)\,\rangle,$
(15)

and

$\displaystyle{}_{\beta}\langle\,f\,|$ $\displaystyle\equiv$
$\displaystyle\langle\,0(\beta)\,|\,a(\mathbf{k}_{3},\beta)a(\mathbf{k}_{4},\beta)\sqrt{2E(\mathbf{k}_{3})2E(\mathbf{k}_{4})},$
(16)

where $E(\mathbf{k}_{i})=\sqrt{\mathbf{k}_{i}^{2}+m^{2}}$ is the energy
associated with the mode $\mathbf{k}_{i}$.

##  III Non-minimally Lorentz-violating scalar QED

The model we are interested in, which will allow us to investigate possible
effects of Lorentz symmetry violation in meson scattering, is given by the
Lagrangian density

$\displaystyle\mathcal{L}=(D_{\mu}\phi)^{*}(D^{\mu}\phi)-m^{2}\phi^{*}\phi-\frac{1}{4}F^{\mu\nu}F_{\mu\nu},$
(17)

where

$\displaystyle
D_{\mu}=\partial_{\mu}-ieA_{\mu}-i\frac{g}{2}\epsilon_{\mu\nu\alpha\beta}\omega^{\nu}F^{\alpha\beta}$
(18)

is a covariant derivative allowing for non-minimal coupling between the
complex scalar field and the gauge field. Supposed to be constant, the vector
$\omega^{\mu}$ sets a preferential direction in spacetime and thus breaks the
equivalence between particle and observer transformations, thereby promoting
the violation of Lorentz symmetry. By substituting Eq. (18) into (17), we can
conclude that the only modifications to the usual Feynman’s rules of the
scalar QED come from the terms that make up the Lagrangian density of
interaction $\mathcal{L}_{int}$:

$\displaystyle\mathcal{L}_{int}=ieA_{\mu}(\phi^{\ast}\partial^{\mu}\phi-\partial^{\mu}\phi^{\ast}\phi)+ig\epsilon^{\mu\nu\alpha\beta}\omega_{\nu}\partial_{\alpha}A_{\beta}(\phi^{\ast}\partial_{\mu}\phi-\partial_{\mu}\phi^{\ast}\phi)+\cdots,$
(19)

where we have omitted interaction terms that are not relevant to our analysis.
The first term of the equation above is already present in the usual scalar
QED and leads to the vertex

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1);
\vertex[right=0.9 of a1] (a2); \vertex[right=0.9 of a2] (a3);
\vertex[below=0.9 of a1] (b1); \vertex[right=0.9 of b1] (b2);
\vertex[right=1.3 of b2] (b3); \vertex[below=0.9 of b1] (c1);
\vertex[right=0.9 of c1] (c2); \vertex[right=0.9 of c2] (c3); \diagram* { (a1)
-- [charged scalar, momentum=$p$] (b2), (b2) -- [photon,] (b3), (c1) -- [anti
charged scalar, momentum'=$p^{\prime}$] (b2), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=ie\eta^{\mu\nu}(p_{\mu}-p^{\prime}_{\mu}).$
(20)

The second term of $\mathcal{L}_{int}$, in turn, will be responsible for the
corrections due to Lorentz violation for the scattering processes considered
here. The interaction vertex associated with it is given by

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1);
\vertex[right=0.9 of a1] (a2); \vertex[right=0.9 of a2] (a3);
\vertex[below=0.9 of a1] (b1); \vertex[dot, right=0.9 of b1] (b2){};
\vertex[right=1.3 of b2] (b3); \vertex[below=0.9 of b1] (c1);
\vertex[right=0.9 of c1] (c2); \vertex[right=0.9 of c2] (c3); \diagram* { (a1)
-- [charged scalar, momentum=$p$] (b2), (b2) -- [photon,momentum=$q$] (b3),
(c1) -- [anti charged scalar, momentum'=$p^{\prime}$] (b2), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=ig\epsilon^{\mu\nu\alpha\beta}\omega_{\nu}q_{\alpha}(p_{\mu}-p^{\prime}_{\mu}).$
(21)

Here is an observation. The vertex above takes into account the creation of a
photon from the annihilation of two particles. However, if the photon were
also being annihilated, an extra minus sign should be added to expression
(21). This is due to the factor $\partial_{\alpha}A_{\beta}$ present in the
second term of Eq. (19).

###  III.1 Finite Temperature

The temperature-dependent Feynman rules for the Lorentz-violating scalar QED
model described above directly follow from the expression for the scattering
matrix in TFD, that is, Eq. (14). The photon propagator in the non-tilde space
$\mathcal{S}$, which is defined as

$\displaystyle\langle\,0(\beta)\,|\mathcal{T}\left[A_{\mu}(x)A_{\nu}(y)\right]|\,0(\beta)\,\rangle,$
(22)

with

$\displaystyle
A_{\mu}(x)=\int\frac{d^{3}k}{(2\pi)^{3}}\,\frac{1}{\sqrt{2E_{\mathbf{k}}}}\,\sum_{m=0}^{3}\left[a^{m}(\mathbf{k})\epsilon^{m}_{\mu}(k)e^{-ik\cdot
x}+a^{m\dagger}(\mathbf{k})\epsilon^{\ast m}_{\mu}(k)e^{ik\cdot x}\right],$
(23)

being the usual field solution for photons and $\epsilon^{m}_{\mu}(k)$ being
the respective polarization vector, for exemple, can be diagrammatically read,
after some algebra, as

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(b1) {$\mu$};
\vertex[right= 2.0cm of b1] (b2) {$\nu$}; \par\diagram* { (b1) -- [photon,
momentum=$q$] (b2), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\frac{-i\eta_{\mu\nu}}{q^{2}}-2\pi\,\eta_{\mu\nu}v_{B}^{2}(\mathbf{q},\beta)\delta(q^{2}).$
(24)

On the same line, it can be shown that the propagator in the tilde space
$\tilde{\mathcal{S}}$, defined as in Eq. (22) by replacing $A_{\mu}$ with
$\tilde{A}_{\mu}$, can be derived simply by acting directly with the
conjugation rules (2)-(5) on Eq. (24). The same applies to the interaction
vertices, where, when in the space $\mathcal{S}$, they take the form of those
presented in equations (20) and (21).

##  IV Lorentz-violating Differential cross sections for mesons at finite
temperature

In this section, we will calculate the differential cross section at finite
temperature using the TFD formalism for three relevant scattering processes in
the context of a scalar QED non-minimally violating Lorentz symmetry. The
processes can be characterized as follows:

$\displaystyle M_{a}(p)+\bar{M}_{a}(p^{\prime})$ $\displaystyle\rightarrow$
$\displaystyle M_{b}(k)+\bar{M}_{b}(k^{\prime});$ (25) $\displaystyle
M_{a}(p)+\bar{M}_{a}(p^{\prime})$ $\displaystyle\rightarrow$ $\displaystyle
M_{a}(k)+\bar{M}_{a}(k^{\prime});$ (26) $\displaystyle
M_{a}(p)+M_{a}(p^{\prime})$ $\displaystyle\rightarrow$ $\displaystyle
M_{a}(k)+M_{a}(k^{\prime});$ (27)

where $M_{a}$ stands for an $a$-type meson, different from a $b$-type meson,
and $\bar{M}$ stands for the corresponding antiparticles.

From now on, the procedure to be followed is quite standard in quantum field
theory. First, we draw all the Feynman diagrams corresponding to the
scattering of interest. Then, we apply the rules presented in the previous
section to write the scattering matrix

$\displaystyle\hat{\mathcal{M}}(\beta)=\mathcal{M}(\beta)-\tilde{\mathcal{M}}(\beta),$
(28)

appropriately defined in Eq. (14). Having said that, let us examine each of
the processes in Eqs. (25)-(27) separately.

###  IV.1 Meson-antimeson of $a$-type into meson-antimeson of $b$-type

At the tree level, the Feynman diagrams, common to both tilde and non-tilde
spaces, for the scattering process represented in Eq. (25) are

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1);
\vertex[right=1.0 of a1] (a2); \vertex[right=1.0 of a2] (a3);
\vertex[right=1.0 of a3] (a4); \vertex[below=1.0 of a1] (b1);
\vertex[right=1.0 of b1] (b2); \vertex[right=1.0 of b2] (b3);
\vertex[right=1.0 of b3] (b4); \vertex[below=1.0 of b1] (c1);
\vertex[right=1.0 of c1] (c2); \vertex[right=1.0 of c2] (c3);
\vertex[right=1.0 of c3] (c4); \par\diagram* { (a1) -- [anti charged scalar,
edge label'=$p^{\prime}$] (b2), (b2) -- [photon, edge label=$q$] (b3), (b3) --
[anti charged scalar, edge label'=$k^{\prime}$] (a4), (c1) -- [charged scalar,
edge label=$p$] (b2), (b3) -- [charged scalar, edge label=$k$] (c4), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1);
\vertex[right=1.0 of a1] (a2); \vertex[right=1.0 of a2] (a3);
\vertex[right=1.0 of a3] (a4); \vertex[below=1.0 of a1] (b1); \vertex[dot,
right=1.0 of b1] (b2){}; \vertex[right=1.0 of b2] (b3); \vertex[right=1.0 of
b3] (b4); \vertex[below=1.0 of b1] (c1); \vertex[right=1.0 of c1] (c2);
\vertex[right=1.0 of c2] (c3); \vertex[right=1.0 of c3] (c4); \par\diagram* {
(a1) -- [anti charged scalar, edge label'=$p^{\prime}$] (b2), (b2) -- [photon,
edge label=$q$] (b3), (b3) -- [anti charged scalar, edge label'=$k^{\prime}$]
(a4), (c1) -- [charged scalar, edge label=$p$] (b2), (b3) -- [charged scalar,
edge label=$k$] (c4), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1);
\vertex[right=1.0 of a1] (a2); \vertex[right=1.0 of a2] (a3);
\vertex[right=1.0 of a3] (a4); \vertex[below=1.0 of a1] (b1);
\vertex[right=1.0 of b1] (b2); \vertex[dot, right=1.0 of b2] (b3){};
\vertex[right=1.0 of b3] (b4); \vertex[below=1.0 of b1] (c1);
\vertex[right=1.0 of c1] (c2); \vertex[right=1.0 of c2] (c3);
\vertex[right=1.0 of c3] (c4); \par\diagram* { (a1) -- [anti charged scalar,
edge label'=$p^{\prime}$] (b2), (b2) -- [photon, edge label=$q$] (b3), (b3) --
[anti charged scalar, edge label'=$k^{\prime}$] (a4), (c1) -- [charged scalar,
edge label=$p$] (b2), (b3) -- [charged scalar, edge label=$k$] (c4), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1);
\vertex[right=1.0 of a1] (a2); \vertex[right=1.0 of a2] (a3);
\vertex[right=1.0 of a3] (a4); \vertex[below=1.0 of a1] (b1); \vertex[dot,
right=1.0 of b1] (b2){}; \vertex[dot, right=1.0 of b2] (b3){};
\vertex[right=1.0 of b3] (b4); \vertex[below=1.0 of b1] (c1);
\vertex[right=1.0 of c1] (c2); \vertex[right=1.0 of c2] (c3);
\vertex[right=1.0 of c3] (c4); \par\diagram* { (a1) -- [anti charged scalar,
edge label'=$p^{\prime}$] (b2), (b2) -- [photon, edge label=$q$] (b3), (b3) --
[anti charged scalar, edge label'=$k^{\prime}$] (a4), (c1) -- [charged scalar,
edge label=$p$] (b2), (b3) -- [charged scalar, edge label=$k$] (c4), };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$ (29)

where we have considered up to second order in the Lorentz symmetry violation
parameter. By means of the Feynman rules discussed in the previous section, we
can express Eq. (28) as

$\displaystyle\hat{\mathcal{M}}(\beta)$ $\displaystyle=$
$\displaystyle\frac{1}{q^{2}}\left(\prod
v_{B}\right)\left(1+\prod\frac{u_{B}}{v_{B}}\right)\left\\{1+2\pi
i\,v_{B}^{2}(\mathbf{q},\beta)\,q^{2}\,\delta(q^{2})\left[\frac{1-\prod\frac{u_{B}}{v_{B}}}{1+\prod\frac{u_{B}}{v_{B}}}\right]\right\\}$
(30) $\displaystyle\times$ $\displaystyle\left\\{e^{2}\left(k^{\prime}\cdot
p^{\prime}-p\cdot k^{\prime}-k\cdot p^{\prime}+k\cdot
p\right)+2eg\,\epsilon^{\mu\nu\alpha\beta}\,\omega_{\nu}\,q_{\alpha}\,(k_{\beta}-k^{\prime}_{\beta})(p_{\mu}-p^{\prime}_{\mu})\right.$
$\displaystyle+$
$\displaystyle\left.g^{2}\,\epsilon^{\mu\nu\alpha\beta}\,\epsilon^{\rho\sigma\gamma\eta}\,\eta_{\beta\rho}\,\omega_{\nu}\,\omega_{\gamma}\,q_{\alpha}\,q_{\eta}\,(k_{\sigma}-k^{\prime}_{\sigma})(p_{\mu}-p^{\prime}_{\mu})\right\\},$

where

$\displaystyle\prod v_{B}\equiv
v_{B}(\vec{p},\beta)\,v_{B}(\vec{p^{\prime}},\beta)\,v_{B}(\vec{k},\beta)\,v_{B}(\vec{k^{\prime}},\beta),$
(31)

and

$\displaystyle\prod\frac{u_{B}}{v_{B}}\equiv\frac{u_{B}(\vec{p},\beta)\,u_{B}(\vec{p^{\prime}},\beta)\,u_{B}(\vec{k},\beta)\,u_{B}(\vec{k^{\prime}},\beta)}{v_{B}(\vec{p},\beta)\,v_{B}(\vec{p^{\prime}},\beta)\,v_{B}(\vec{k},\beta)\,v_{B}(\vec{k^{\prime}},\beta)}.$
(32)

Note that $\mathcal{\hat{M}}=\mathcal{\hat{M}}_{s}$, where the subscript ‘$s$’
indicates that we are only dealing with s-channel diagrams. From Eq. (30), we
can directly compute $|\hat{\mathcal{M}}(\beta)|^{2}$ and evaluate the result
from the center-of-mass reference frame of the system, where

$\displaystyle p=(E,E\,\hat{\mathbf{z}});$ $\displaystyle k=(E,\mathbf{k});$
$\displaystyle p^{\prime}=(E,-E\,\hat{\mathbf{z}});$ $\displaystyle
k^{\prime}=(E,-\mathbf{k});$ (33)

with

$\displaystyle\hat{\mathbf{x}}\cdot\mathbf{k}$ $\displaystyle=$ $\displaystyle
E\sin\theta\cos\phi,$ (34) $\displaystyle\hat{\mathbf{y}}\cdot\mathbf{k}$
$\displaystyle=$ $\displaystyle E\sin\theta\sin\phi,$ (35)
$\displaystyle\hat{\mathbf{z}}\cdot\mathbf{k}$ $\displaystyle=$ $\displaystyle
E\cos\theta,$ (36)

and $E_{cm}=2E$. This evaluation allows us to calculate the cross section
using Eq. (13). Note that when writing Eqs. (IV.1), we are implicitly assuming
that our analysis is conducted in the high-energy limit, where we can take
$m_{a}=m_{b}=0$, with $m_{a}$ and $m_{b}$ representing the masses of the
particles of type $a$ and $b$, respectively. In addition, concerning the
Lorentz violation parameter, we will separately investigate two distinct
possibilities based on whether $\omega^{\mu}$ is a timelike or spacelike four-
vector.

####  IV.1.1 Timelike case

In this case, we must consider $\omega^{\nu}=(\omega_{0},\mathbf{0})$, where
$\omega_{0}$ is a constant parameter that sets the Lorentz violation effects.
In accordance with what was previously established, after some algebraic
manipulation, we can write

$\displaystyle|\hat{\mathcal{M}}(\beta)|^{2}=B(\beta)\,e^{4}\,\cos^{2}\theta,$
(37)

where

$\displaystyle
B(\beta)=\frac{1}{4}\left\\{C(\beta)+\pi^{2}\,\delta\left(E_{cm}^{2}\right)^{2}\,D(\beta)\right\\},$
(38)

is the thermal correction factor for a meson and antimeson of an $a$-type
scattering off a meson and antimeson of a $b$-type. The temperature-dependent
functions $C(\beta)$ and $D(\beta)$ are given by

$\displaystyle
C(\beta)=\cosh^{2}\left(\frac{\beta\,E_{cm}}{2}\right)\text{csch}^{4}\left(\frac{\beta\,E_{cm}}{4}\right),$
(39)

and

$\displaystyle D(\beta)=\left(\frac{2\,E_{cm}}{e^{\frac{\beta
E_{cm}}{2}}-1}\right)^{4},$ (40)

respectively. Figure (1) illustrates the behavior of these functions in terms
of the parameter $\beta$.

Figure 1: Graph of temperature-dependent functions $C$ and $D$ as a function
of $\beta$. We have considered $E_{cm}=1$.

The temperature-dependent differential cross section for this specific
configuration can be written from Eq. (13) as

$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{\beta,cm}$
$\displaystyle=$ $\displaystyle
B(\beta)\left(\frac{d\sigma_{0}}{d\Omega}\right)_{cm}$ (41)

where

$\displaystyle\left(\frac{d\sigma_{0}}{d\Omega}\right)_{cm}=\frac{e^{4}}{64\,\pi^{2}\,E_{cm}^{2}}\cos^{2}\theta,$
(42)

represents the corresponding differential cross section at zero temperature.

The above result reveals that, although the cross section is corrected for
temperature through a factor $B(\beta)$, no Lorentz violation effect will be
observed if the vector $\omega^{\nu}$ characterizing it is timelike. In fact,
the expression in Eq. (42) represents the usual outcome for the scattering of
distinct mesons at zero temperature, thus aligning completely with what was
found in reference Araujo:2022qke .

In terms of the dependence of the cross section on temperature, we observe
that it is entirely governed by the function in Eq. (38). As we can notice
(see Figure (1)), the temperature corrections become highly relevant in the
limit of very high temperatures, where $C(\beta)$ and $D(\beta)$, and
consequently $B(\beta)$, assumes large values. Conversely, as the temperature
approaches zero, Eq. (38) tends towards unity, allowing us to recover the
usual result. Figure (2) shows the behavior of the differential thermal cross
section in terms of the temperature and the angle $\theta$. As we can also
observe, there still exists in $B(\beta)$ a product of delta function with
equal arguments, namely $\delta\left(E_{cm}^{2}\right)^{2}$. Such a term is
highly characteristic in the TFD formalism and represents a kind of apparent
"pathology" that are actually called pinch singularity Landsman:1986uw .
However, as mentioned in Refs. Landsman:1986uw ; van2001introduction , all
these problems are avoided by working with the regularized form of delta-
functions and their derivatives, namely

$\displaystyle 2\pi i\frac{1}{n!}\frac{\partial^{n}}{\partial
x^{n}}\delta(x)=\left(-\frac{1}{x+i\epsilon}\right)^{n+1}-\left(-\frac{1}{x-i\epsilon}\right)^{n+1}.$
(43)

Figure 2: Angular and temperature dependencies of the differential cross
section for $a$-type meson-antimeson scattering in $b$-type meson-antimeson in
the timelike case. For this plot we have considered $e=1$ and $E_{cm}=1$.

####  IV.1.2 Spacelike case

For this case, we must consider $\omega^{\nu}=(0,\mathbf{w})$, where
$\mathbf{w}$ is a constant vector in an arbitrary space direction. Let
$\theta_{\omega z}$ be the angle that the vector $\mathbf{w}$ makes with the
z-axis, and $\phi_{\omega x}$ the angle that the projection of $\mathbf{w}$
onto the xy-plane makes with the x-axis. Then, following the same steps as in
the previous case, after some algebra, we can express the temperature-
dependent differential cross-section as

$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{\beta,cm}$
$\displaystyle=$ $\displaystyle
B(\beta)\left(\frac{d\sigma}{d\Omega}\right)_{cm},$ (44)

where

$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{cm}$ $\displaystyle=$
$\displaystyle\frac{e^{4}}{64\,\pi^{2}\,E_{cm}^{2}}\cos^{2}\theta+g\,|\mathbf{w}|\,\frac{e^{3}}{32\,\pi^{2}\,E_{cm}}\,\sin(2\theta)\,\sin(\theta_{\omega
z})\,\sin\left(\phi-\phi_{\omega x}\right)$ (45) $\displaystyle+$
$\displaystyle
g^{2}\,|\mathbf{w}|^{2}\,\frac{e^{2}}{32\,\pi^{2}}\,\sin^{2}(\theta_{\omega
z})\Big{[}\sin(\theta)\,\cos(\theta)\,\cot(\theta_{\omega
z})\,\cos\left(\phi-\phi_{\omega x}\right)$ $\displaystyle+$ $\displaystyle
2\sin^{2}(\theta)\,\sin^{2}\left(\phi-\phi_{\omega
x}\right)-\cos^{2}(\theta)\Big{]}$ $\displaystyle-$ $\displaystyle
g^{3}\,|\mathbf{w}|^{3}\,\frac{e\,E_{cm}}{16\,\pi^{2}}\,\sin^{2}(\theta)\,\sin^{3}(\theta_{\omega
z})\,\sin\left(\phi-\phi_{\omega
x}\right)\Big{[}\cot(\theta)-\cot(\theta_{\omega
z})\,\cos\left(\phi-\phi_{\omega x}\right)\Big{]}$ $\displaystyle-$
$\displaystyle
g^{4}\,|\mathbf{w}|^{4}\,\frac{E_{cm}^{2}}{1024\,\pi^{2}}\,\sin^{2}(\theta_{\omega
z})\,\Big{\\{}\cos(2\theta)\,\big{[}2\cos(2\theta_{\omega
z})\big{(}3+\cos(2\left(\phi-\phi_{\omega x}\right))\big{)}+1\big{]}$
$\displaystyle+$ $\displaystyle 4\cos(2\theta_{\omega
z})\,\sin^{2}\left(\phi-\phi_{\omega
x}\right)+8\sin(2\theta)\,\sin(2\theta_{\omega
z})\,\cos\left(\phi-\phi_{\omega x}\right)$ $\displaystyle+$ $\displaystyle
2\sin^{2}(\theta)\,\cos(2\left(\phi-\phi_{\omega x}\right))-9\Big{\\}},$

corresponds to the result at zero temperature, taking into account terms up to
fourth order in the Lorentz violation parameter. From Eq. (44), we can observe
once again that the behavior of the differential cross section with
temperature is entirely governed by the function $B(\beta)$, such that
corrections become highly relevant in the limit of very high temperatures, and
the usual result ($T=0$) is promptly recovered. However, unlike the timelike
case, Lorentz violation effects are now observed, suggesting that the
scattering in question is more sensitive in scenarios where $\omega^{\mu}$ is
a spacelike four-vector. Note also that such effects are completely canceled
for situations where $\mathbf{w}$ lies in the plane containing the line
defining the collision trajectory of the particles at the beginning of the
process. Figure (3) illustrates the angular dependencies of the differential
cross section at zero temperature for some values of $\theta_{\omega z}$ and
$\phi_{\omega x}$. It is evident that the behavior changes dramatically from
one situation to another, particularly concerning the region where the
differential cross section reaches its maximum value.

(a)

(b)

(c)

Figure 3: Angular dependencies of the zero-temperature differential cross
section for $a$-type meson-antimeson scattering in $b$-type meson-antimeson in
the spacelike case. For this plot we took $e=1$, $E_{cm}=1$. In (3(a)), we
have the usual case ($g|\mathbf{w}|=0$). In (3(b)) and (3(c)), we have
considered $g|\mathbf{w}|=0.9$, but for the former $\theta_{\omega
z}=\frac{\pi}{2}$ and $\phi_{\omega x}=0$, while for the latter
$\theta_{\omega z}=\phi_{\omega x}=\frac{\pi}{4}$.

###  IV.2 Meson-antimeson of $a$-type into meson-antimeson of $a$-type

When considering the scattering process represented by Eq. (26) at the tree
level, we must also take into account four more Feynman diagrams, namely

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1);
\vertex[below=1.0 of a1] (a2); \vertex[below=1.0 of a2] (a3);
\vertex[below=1.0 of a3] (a4); \vertex[right=1.0 of a1] (b1);
\vertex[below=1.0 of b1] (b2); \vertex[below=1.0 of b2] (b3);
\vertex[below=1.0 of b3] (b4); \vertex[right=1.0 of b1] (c1);
\vertex[below=1.0 of c1] (c2); \vertex[below=1.0 of c2] (c3);
\vertex[below=1.0 of c3] (c4); \par\diagram* { (a1) -- [anti charged scalar,
edge label'=$p^{\prime}$] (b2), (b2) -- [photon, edge label=$q^{\prime}$]
(b3), (b3) -- [anti charged scalar, edge label'=$p$] (a4), (c1) -- [charged
scalar, edge label=$k^{\prime}$] (b2), (b3) -- [charged scalar, edge
label=$k$] (c4), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1);
\vertex[below=1.0 of a1] (a2); \vertex[below=1.0 of a2] (a3);
\vertex[below=1.0 of a3] (a4); \vertex[right=1.0 of a1] (b1); \vertex[dot,
below=1.0 of b1] (b2){}; \vertex[below=1.0 of b2] (b3); \vertex[below=1.0 of
b3] (b4); \vertex[right=1.0 of b1] (c1); \vertex[below=1.0 of c1] (c2);
\vertex[below=1.0 of c2] (c3); \vertex[below=1.0 of c3] (c4); \par\diagram* {
(a1) -- [anti charged scalar, edge label'=$p^{\prime}$] (b2), (b2) -- [photon,
edge label=$q^{\prime}$] (b3), (b3) -- [anti charged scalar, edge label'=$p$]
(a4), (c1) -- [charged scalar, edge label=$k^{\prime}$] (b2), (b3) -- [charged
scalar, edge label=$k$] (c4), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1);
\vertex[below=1.0 of a1] (a2); \vertex[below=1.0 of a2] (a3);
\vertex[below=1.0 of a3] (a4); \vertex[right=1.0 of a1] (b1);
\vertex[below=1.0 of b1] (b2); \vertex[dot, below=1.0 of b2] (b3){};
\vertex[below=1.0 of b3] (b4); \vertex[right=1.0 of b1] (c1);
\vertex[below=1.0 of c1] (c2); \vertex[below=1.0 of c2] (c3);
\vertex[below=1.0 of c3] (c4); \par\diagram* { (a1) -- [anti charged scalar,
edge label'=$p^{\prime}$] (b2), (b2) -- [photon, edge label=$q^{\prime}$]
(b3), (b3) -- [anti charged scalar, edge label'=$p$] (a4), (c1) -- [charged
scalar, edge label=$k^{\prime}$] (b2), (b3) -- [charged scalar, edge
label=$k$] (c4), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(a1);
\vertex[below=1.0 of a1] (a2); \vertex[below=1.0 of a2] (a3);
\vertex[below=1.0 of a3] (a4); \vertex[right=1.0 of a1] (b1); \vertex[dot,
below=1.0 of b1] (b2){}; \vertex[dot, below=1.0 of b2] (b3){};
\vertex[below=1.0 of b3] (b4); \vertex[right=1.0 of b1] (c1);
\vertex[below=1.0 of c1] (c2); \vertex[below=1.0 of c2] (c3);
\vertex[below=1.0 of c3] (c4); \par\diagram* { (a1) -- [anti charged scalar,
edge label'=$p^{\prime}$] (b2), (b2) -- [photon, edge label=$q^{\prime}$]
(b3), (b3) -- [anti charged scalar, edge label'=$p$] (a4), (c1) -- [charged
scalar, edge label=$k^{\prime}$] (b2), (b3) -- [charged scalar, edge
label=$k$] (c4), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,$ (46)

in addition to those depicted in Eq. (29). These new diagrams correspond to
the t-channel of the interaction process in question, such that the total
thermal scattering matrix $\hat{\mathcal{M}}(\beta)$ now becomes

$\displaystyle\hat{\mathcal{M}}(\beta)=\hat{\mathcal{M}}_{s}(\beta)+\hat{\mathcal{M}}_{t}(\beta),$
(47)

with

$\displaystyle\hat{\mathcal{M}}_{t}(\beta)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{q^{\prime 2}}\left(\prod
v_{B}\right)\left(1+\prod\frac{u_{B}}{v_{B}}\right)\left\\{1+2\pi
i\,v_{B}^{2}(\mathbf{q^{\prime}},\beta)\,q^{\prime 2}\,\delta(q^{\prime
2})\left[\frac{1-\prod\frac{u_{B}}{v_{B}}}{1+\prod\frac{u_{B}}{v_{B}}}\right]\right\\}$
(48) $\displaystyle\times$ $\displaystyle\left\\{-\left(p\cdot
k^{\prime}+k\cdot k^{\prime}+k\cdot p^{\prime}+p\cdot
p^{\prime}\right)+2\frac{g}{e}\,\epsilon^{\mu\nu\alpha\beta}\,\omega_{\nu}\,q^{\prime}_{\alpha}\,(k_{\beta}+p_{\beta})(k^{\prime}_{\mu}+p^{\prime}_{\mu})\right.$
$\displaystyle+$
$\displaystyle\left.\frac{g^{2}}{e^{2}}\,\epsilon^{\mu\nu\alpha\beta}\,\epsilon^{\sigma\gamma\eta\rho}\,\eta_{\beta\rho}\,\omega_{\nu}\,\omega_{\gamma}\,q^{\prime}_{\alpha}\,q^{\prime}_{\eta}\,(k^{\prime}_{\sigma}+p^{\prime}_{\sigma})(k_{\mu}+p_{\mu})\right\\},$

representing the contribution from the $t$-channel and
$\hat{\mathcal{M}}_{s}(\beta)$ (still given by Eq. (30)) the contribution from
the $s$-channel, both constructed according to the Feynman rules discussed in
section (III). As before, by knowing $\hat{\mathcal{M}}(\beta)$, we can
evaluate its squared matrix in the center-of-mass reference frame (see Eq.
(IV.1)) and write, through Eq. (13), an expression for the thermal
differential cross section in the two cases of interest: when $\omega^{\mu}$
is a timelike four-vector and when $\omega^{\mu}$ is a spacelike four-vector.

####  IV.2.1 Timelike case

(a)

(b)

Figure 4: Angular and temperature dependencies of the differential cross
section for $a$-type meson-antimeson scattering in $a$-type meson-antimeson in
the timelike case. For this plot we took $e=1$ and $E_{cm}=1$. In (4(a)), no
Lorentz violation has been considered ($g\omega_{0}=0$). In (4(b)), we have
set $g\omega_{0}=3$.

Since $\omega^{\nu}=(\omega_{0},\mathbf{0})$ for this case, after some
algebraic manipulation, the temperature-dependent differential cross section
for the scattering process characterized in Eq. (26) takes the form

$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{\beta,cm}$
$\displaystyle=$
$\displaystyle\frac{e^{4}}{4096\,\pi^{2}\,E_{cm}^{2}}\Biggl{\\{}(7+\cos(2\theta))^{2}\csc^{4}\left(\frac{\theta}{2}\right)C(\beta)$
(49) $\displaystyle+$ $\displaystyle
16\,\pi^{2}\,D(\beta)\Big{[}\cos(\theta)\,\delta\left(E_{cm}^{2}\right)+\left(3+\cos(\theta)\right)\delta\left(E_{cm}^{2}(\cos(\theta)-1)\right)\Big{]}^{2}\Biggr{\\}}$
$\displaystyle-$
$\displaystyle\frac{1}{512}\left(\frac{e\,g\,\omega_{0}}{\pi}\right)^{2}\Biggl{\\{}(7+\cos(2\theta))\cot^{2}\left(\frac{\theta}{2}\right)C(\beta)$
$\displaystyle+$
$\displaystyle\,\pi^{2}\,D(\beta)\sin(\theta)\,\delta\left(E_{cm}^{2}(\cos(\theta)-1)\right)\big{[}\sin(2\theta)\,\delta\left(E_{cm}^{2}\right)$
$\displaystyle+$
$\displaystyle\left(6\sin(\theta)+\sin(2\theta)\right)\delta\left(E_{cm}^{2}(\cos(\theta)-1)\right)\big{]}\Biggr{\\}}$
$\displaystyle+$
$\displaystyle\frac{\pi^{2}\,E_{cm}^{2}}{256}\left(\frac{g\,\omega_{0}}{\pi}\right)^{4}\Biggl{\\{}\cos^{4}\left(\frac{\theta}{2}\right)C(\beta)$
$\displaystyle+$
$\displaystyle\frac{\pi^{2}}{4}D(\beta)\sin^{4}(\theta)\Big{[}\delta\left(E_{cm}^{2}(\cos\theta-1)\right)\Big{]}^{2}\Biggr{\\}},$

where $C(\beta)$ and $D(\beta)$ are again the temperature-dependent functions
given respectively in Eqs. (39) and (40). Note that, in comparison with what
was found in the previous case, the result in Eq. (49) suggests that the
scattering process described by Eq. (26) is more sensitive to Lorentz
violation effects when these are characterized by the parameter $\omega_{0}$
of the timelike four-vector $\omega^{\mu}$. However, observe that neither
first-order nor third-order terms in the coupling constant $g$ are present in
the above result. As we will see, this seems to be a characteristic of
scattering involving mesons and antimesons of the same species within the
context of the non-minimally Lorentz-violating scalar QED framework presented
in this work. Figure (4) illustrates the behavior of the differential cross
section with the temperature ($\beta=1/T$) and the angle $\theta$ for the
cases with and without Lorentz symmetry violation. As we can observe, the
effects of anisotropy characterized by the parameter $\omega_{0}$ become more
pronounced in the limit of very high temperatures, suggesting a new
perspective (or path) in the search for Lorentz-violating measurable effects.

####  IV.2.2 Spacelike case

(a)

(b)

(c)

Figure 5: Angular dependencies of the zero-temperature differential cross
section for $a$-type meson-antimeson scattering in $a$-type meson-antimeson in
the spacelike case. For this plot we took $e=1$, $E_{cm}=1$. In (5(a)), we
have the usual case ($g|\mathbf{w}|=0$). In (5(b)) and (5(c)), we have
considered $g|\mathbf{w}|=2$, but for the former $\theta_{\omega
z}=\frac{\pi}{2}$ and $\phi_{\omega x}=0$, while for the latter
$\theta_{\omega z}=\frac{\pi}{2}$ and $\phi_{\omega x}=\pi$.

For this case, we must again consider that $\omega^{\nu}=(0,\mathbf{w})$, with
$\mathbf{w}$ being the constant vector that establishes the effects of Lorentz
violation. Consequently, proceeding exactly as before, the temperature-
dependent differential cross section takes the form

$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{\beta,cm}$
$\displaystyle=\frac{e^{4}}{4096\,\pi^{2}\,E_{cm}^{2}}\Biggl{\\{}(7+\cos(2\theta))^{2}\csc^{4}\left(\frac{\theta}{2}\right)C(\beta)$
$\displaystyle+16\,\pi^{2}\,D(\beta)\Big{[}\cos(\theta)\,\delta\left(E_{cm}^{2}\right)+\left(3+\cos(\theta)\right)\delta\left(E_{cm}^{2}(\cos{\theta}-1)\right)\Big{]}^{2}\Biggr{\\}}$
$\displaystyle+g\,|\mathbf{w}|\,\frac{e^{3}}{256\,\pi^{2}\,E_{cm}}$
$\displaystyle\times\Biggl{\\{}(\cos(\theta)-3)(7+\cos(2\theta))\cot\left(\frac{\theta}{2}\right)\csc^{2}\left(\frac{\theta}{2}\right)\sin(\theta_{\omega
z})\sin(\phi-\phi_{\omega x})\,C(\beta)$
$\displaystyle+4\pi^{2}\,D(\beta)\,\sin(\theta)\,\sin(\theta_{\omega
z})\,\sin(\phi-\phi_{\omega
x})\,\Big{[}\,\delta\left(E_{cm}^{2}\right)-2\,\delta\left(E_{cm}^{2}(\cos(\theta)-1)\right)\Big{]}$
$\displaystyle\times\Big{[}\cos(\theta)\,\delta\left(E_{cm}^{2}\right)+\left(3+\cos(\theta)\right)\delta\left(E_{cm}^{2}(\cos{\theta}-1)\right)\Big{]}\Bigg{\\}}+\cdots,$
(50)

where the dots stands for higher-orders Lorentz-breaking terms. The exact form
of these corrections are lengthy and will not be displayed in detail here.
However, we must emphasize that all the terms up to fourth order in $g$ are
once again present in the above result. This suggests that the scattering
processes involving mesons and antimesons of the same type are more sensitive
to Lorentz violation effects characterized by a spacelike four-vector within
the context of the QED framework here discussed. From Eq. (IV.2.2), we again
see that the temperature dependence is shaped by functions $C(\beta)$ and
$D(\beta)$, indicating that at very high temperatures, thermal effects cannot
be neglected. Figure (5) illustrates the angular dependencies of the
differential cross section at zero temperature for some values of
$\theta_{\omega z}$ and $\phi_{\omega x}$. It is noteworthy that the effects
of anisotropy cancel out for $\theta=\pi$, irrespective of the azimuthal angle
$\phi$ and the orientation of $\mathbf{w}$.

###  IV.3 Meson-meson of $a$-type into meson-meson of $a$-type

Characterized by Eq. (27), the scattering process involving only mesons of the
same type should be evaluated at the tree level in terms of six Feynman
diagrams, namely

(51)

Here, the four upper diagrams correspond to the t-channel of the interaction,
while the four lower diagrams correspond to the u-channel. We must further
emphasize that the particle-flow arrows illustrated in Eq. (IV.3) are distinct
from those depicted in Eqs. (29) and (46). This distinction arises in this
section because we are dealing solely with particles (and not antiparticles).
Having said that, by standard procedure, we can write the thermal scattering
matrix for this case as

$\displaystyle\hat{\mathcal{M}}(\beta)=\hat{\mathcal{M}}_{u}(\beta)-\hat{\mathcal{M}}_{t}(\beta),$
(52)

with

$\displaystyle\hat{\mathcal{M}}_{u}(\beta)$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{q^{\prime\prime 2}}\left(\prod
v_{B}\right)\left(1+\prod\frac{u_{B}}{v_{B}}\right)\left\\{1+2\pi
i\,v_{B}^{2}(\mathbf{q^{\prime\prime}},\beta)\,q^{\prime\prime
2}\,\delta(q^{\prime\prime
2})\left[\frac{1-\prod\frac{u_{B}}{v_{B}}}{1+\prod\frac{u_{B}}{v_{B}}}\right]\right\\}$
(53) $\displaystyle\times$ $\displaystyle\left\\{\left(k^{\prime}\cdot
p^{\prime}+k\cdot k^{\prime}+k\cdot p+p\cdot
p^{\prime}\right)+2\frac{g}{e}\,\epsilon^{\mu\nu\alpha\beta}\,\omega_{\nu}\,q^{\prime\prime}_{\alpha}\,(k_{\beta}+p^{\prime}_{\beta})(k^{\prime}_{\mu}+p_{\mu})\right.$
$\displaystyle-$
$\displaystyle\left.\frac{g^{2}}{e^{2}}\,\epsilon^{\mu\nu\alpha\beta}\,\epsilon^{\gamma\rho\eta\sigma}\,\eta_{\beta\sigma}\,\omega_{\nu}\,\omega_{\rho}\,q^{\prime\prime}_{\alpha}\,q^{\prime\prime}_{\eta}\,(k_{\gamma}+p^{\prime}_{\gamma})(k^{\prime}_{\mu}+p_{\mu})\right\\},$

and $\hat{\mathcal{M}}_{t}(\beta)$ still given by Eq. (48). As in the previous
sections, once Eq. (52) is known, we must proceed by evaluating an expression
for the thermal differential cross section in the two cases of interest: when
$\omega^{\mu}$ is a timelike or spacelike four-vector.

####  IV.3.1 Timelike case

As before, $\omega^{\nu}=(\omega_{0},\mathbf{0})$ for this case. Thus, the
temperature-dependent differential cross section for mesons of $a$-type
scattering off mesons of $a$-type takes the form

$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{\beta,cm}$
$\displaystyle=$
$\displaystyle\frac{e^{4}}{256\,\pi^{2}\,E_{cm}^{2}}\Bigg{\\{}(\cos(2\theta)+7)^{2}\csc^{4}(\theta)\,C(\beta)$
(54) $\displaystyle+$
$\displaystyle\pi^{2}\,D(\beta)\,\big{[}(\cos(\theta)+3)^{2}\delta\left(E_{cm}^{2}(\cos(\theta)-1)\right)^{2}$
$\displaystyle-$
$\displaystyle(\cos(2\theta)-17)\delta\left(E_{cm}^{2}(\cos(\theta)+1)\right)\delta\left(E_{cm}^{2}(\cos(\theta)-1)\right)$
$\displaystyle+$
$\displaystyle(\cos(\theta)-3)^{2}\delta\left(E_{cm}^{2}(\cos(\theta)+1)\right)^{2}\big{]}\Bigg{\\}}$
$\displaystyle-$
$\displaystyle\frac{1}{128}\left(\frac{e\,g\,\omega_{0}}{\pi}\right)^{2}\Bigg{\\{}(\cos(2\theta)+7)\csc^{2}(\theta)\,C(\beta)$
$\displaystyle+$
$\displaystyle\frac{\pi^{2}}{2}\,D(\beta)\,\sin^{2}(\theta)\Big{[}(\cos(\theta)+3)\delta\left(E_{cm}^{2}(\cos(\theta)-1)\right)^{2}$
$\displaystyle-$
$\displaystyle(\cos(\theta)-3)\delta\left(E_{cm}^{2}(\cos(\theta)+1)\right)^{2}+6\delta\left(E_{cm}^{2}(\cos(\theta)-1)\right)\delta\left(E_{cm}^{2}(\cos(\theta)+1)\right)\Big{]}\Bigg{\\}}$
$\displaystyle+$
$\displaystyle\frac{\pi^{2}\,E_{cm}^{2}}{256}\,\left(\frac{g\,\omega_{0}}{\pi}\right)^{4}\Bigg{\\{}C(\beta)+\frac{\pi^{2}}{4}\,D(\beta)\,\sin^{4}(\theta)\,\Big{[}\delta\left(E_{cm}^{2}(\cos(\theta)-1)\right)^{2}$
$\displaystyle+$
$\displaystyle\delta\left(E_{cm}^{2}(\cos(\theta)+1)\right)^{2}+2\,\delta\left(E_{cm}^{2}(\cos(\theta)-1)\right)\delta\left(E_{cm}^{2}(\cos(\theta)+1)\right)\Big{]}\Bigg{\\}}.$

Once again, we observe that first and third-order terms in the coupling
constant $g$ are not present when Lorentz symmetry violation is characterized
by a spacetime quadrivector $\omega^{\mu}=(\omega_{0},\mathbf{0})$, completely
corroborating what was previously discussed in Sec. IV.2. The temperature
dependence is, as before, governed by functions $C(\beta)$ and $D(\beta)$.
Figure (6) illustrates the angular and temperature dependencies of the
differential cross section found in Eq. (54) for the cases with and without
Lorentz violation. Once again, it is observed that the effects of anisotropy
appear to be more pronounced in the limit of very high temperatures. This
corroborates findings in Sec. IV.2 and further emphasizes the need for
experimental investigations in this regime of temperature.

(a)

(b)

Figure 6: Angular and temperature dependencies of the differential cross
section for $a$-type meson-meson scattering in $a$-type meson-meson in the
timelike case. For this plot we took $e=1$ and $E_{cm}=1$. In (6(a)), no
Lorentz violation has been considered ($g\omega_{0}=0$). In (6(b)), we have
set $g\omega_{0}=5$.

####  IV.3.2 Spacelike case

(a)

(b)

(c)

Figure 7: Angular dependencies of the zero-temperature differential cross
section for $a$-type meson-meson scattering in $a$-type meson-meson in the
spacelike case. For this plot we took $e=1$, $E_{cm}=1$. In (7(a)), we have
the usual case ($g|\mathbf{w}|=0$). In (7(b)) and (7(c)), we have considered
$g|\mathbf{w}|=5$, but for the former $\theta_{\omega z}=\frac{\pi}{2}$ and
$\phi_{\omega x}=0$, while for the latter $\theta_{\omega z}=\frac{\pi}{2}$
and $\phi_{\omega x}=\frac{\pi}{2}$.

As we know, the violation of Lorentz symmetry is characterized by the
spacetime quadrivector $\omega^{\mu}=(0,\mathbf{w})$. Thus, for this case, the
thermal differential cross section can be written as

$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{\beta,cm}$
$\displaystyle=$
$\displaystyle\frac{e^{4}}{256\pi^{2}E_{cm}^{2}}\Bigg{\\{}(\cos(2\theta)+7)^{2}\csc^{4}(\theta)\,C(\beta)$
(55) $\displaystyle+$
$\displaystyle\pi^{2}D(\beta)\Big{[}(\cos(\theta)+3)^{2}\delta\left(E_{cm}^{2}(\cos(\theta)-1)\right)^{2}+(\cos(\theta)-3)^{2}\delta\left(E_{cm}^{2}(\cos(\theta)+1)\right)^{2}$
$\displaystyle-$
$\displaystyle(\cos(2\theta)-17)\delta\left(E_{cm}^{2}(\cos(\theta)-1)\right)\delta\left(E_{cm}^{2}(\cos(\theta)+1)\right)\Big{]}\Bigg{\\}}$
$\displaystyle-$ $\displaystyle
g\,|\mathbf{w}|\,\frac{e^{3}}{32\,\pi^{2}\,E_{cm}}\Bigg{\\{}(15\cos(\theta)+\cos(3\theta))\csc^{3}(\theta)\sin\left(\theta_{\omega
z}\right)\sin\left(\phi-\phi_{\omega x}\right)\,C(\beta)$ $\displaystyle+$
$\displaystyle\frac{\pi^{2}}{2}\,D(\beta)\Big{[}(6\sin(\theta)+\sin(2\theta))\sin\left(\theta_{\text{$\omega$z}}\right)\sin\left(\phi-\phi_{\omega
x}\right)\delta\left(E_{cm}^{2}(\cos(\theta)-1)\right)^{2}$ $\displaystyle+$
$\displaystyle(\sin(2\theta)-6\sin(\theta))\sin\left(\theta_{\omega
z}\right)\sin\left(\phi-\phi_{\text{$\omega$x}}\right)\delta\left(E_{cm}^{2}(\cos(\theta)+1)\right)^{2}$
$\displaystyle-$ $\displaystyle 2\sin(2\theta)\sin\left(\theta_{\omega
z}\right)\sin\left(\phi-\phi_{\omega
x}\right)\delta\left(E_{cm}^{2}(\cos(\theta)-1)\right)\delta\left(E_{cm}^{2}(\cos(\theta)+1)\right)\Big{]}\Bigg{\\}}$
$\displaystyle+$ $\displaystyle\cdots,$

where the dots stand for higher-order Lorentz-breaking terms, and once more,
since the exact form of these corrections is lengthy, they will not be
displayed in detail here. However, it is important to emphasize that all the
terms up to fourth order in $g$ are once again present, further corroborating
the suggestion that the scattering processes involving mesons and antimesons
of the same type are more sensitive to Lorentz violation effects characterized
by a spacelike four-vector within the context of the QED framework here
discussed. The thermal behavior is once again shaped by the functions
$C(\beta)$ and $D(\beta)$, resulting in a significant differential cross
section value at very high temperature limits. The zero-temperature result is
also promptly recovered, given that $C(\beta)\rightarrow 4$ and
$D(\beta)\rightarrow 0$ (see Figure (1)). The angular dependencies of the
zero-temperature differential cross section for some values of $\theta_{\omega
z}$ and $\phi_{\omega x}$ are illustrated in Figure (7), revealing that the
anisotropy effects are most pronounced at the boundaries of the interval to
which $\theta$ belongs.

##  V Conclusion

In this paper we study meson scattering in a non-minimally Lorentz-violating
scalar QED at finite temperature. The meson scatterings were investigated in
tree level and the finite temperature effects were addressed by using the
thermofield dynamics formalism. We have considered three types of scattering,
namely, meson-antimeson of $a$-type into meson-antimeson of $b$-type, meson-
antimeson of $a$-type into meson-antimeson of $a$-type and meson-meson of
$a$-type into meson-meson of $a$-type. All cases considered were addressed
similarly and for each scattering we have computed the cross section in order
to investigate the influence of the Lorentz symmetry violation and finite
temperature effects.

For the meson-antimeson of $a$-type into meson-antimeson of $b$-type, we have
considered contributions up to second order in the Lorentz-violating
parameter. Also, we have separately investigated two distinct possibilities
based on whether $\omega^{\mu}$ is a timelike or spacelike four-vector. For
the timelike case, no Lorentz violation effect in the cross section is
observed but the temperature influence becomes highly relevant in the limit of
high temperatures. For the spacelike case, unlike the timelike case, Lorentz
violation effects are observed, suggesting that the scattering in question is
more sensitive in scenarios where $\omega^{\mu}$ is a spacelike four-vector.
Also, such effects are completely canceled for situations where $\mathbf{w}$
lies in the plane containing the line defining the collision trajectory of the
particles at the beginning of the process.

For the meson-antimeson of a-type into meson-antimeson of a-type, the timelike
consideration shows us that the scattering process is more sensitive to
Lorentz violation effects since these are dependent of the parameter
$\omega_{0}$, i.e., the timelike four-vector $\omega^{\mu}$. It is important
to highlight that we observed that neither first-order nor third-order terms
in the coupling constant $g$ are present in the cross section. This seems to
be a characteristic of scattering involving mesons and antimesons of the same
species within the context of the non-minimally Lorentz-violating scalar QED
framework presented in this work. Also, the effects of anisotropy
characterized by the parameter $\omega_{0}$ become more pronounced in the
limit of very high temperatures, suggesting a new perspective (or path) in the
search for Lorentz-violating measurable effects. A similar behaviour occurs
for the spacelike consideration, but in this case all the terms up to fourth
order in $g$ are present.

Finally, for the meson-meson of $a$-type into meson-meson of $a$-type, we have
similar corrections due to the Lorentz symmetry violation and temperature in
comparison with the the meson-antimeson of a-type into meson-antimeson of
a-type process, being the coefficients of each order of contribution the sole
difference between the two cases. An interesting behaviour present in all
three processes is the angular dependency promoted by the Lorentz symmetry
violation regarding the vector $\mathbf{w}$ that sets the priviledge space
direction and the line defining the collision trajectory of the particles at
the beginning of the process. Such angular dependency can be seen as an
effective way of verifing the existence of a priviledge direction in
spacetime, by the measurement of two different cross sections for the same
process occuring in different directions.

## Acknowledgments

JF would like to thank the Fundação Cearense de Apoio ao Desenvolvimento
Científico e Tecnológico (FUNCAP) under the grant PRONEM
PNE0112-00085.01.00/16 for financial support, the National Council for
scientific and technological support (CNPq) under the grant 304485/2023-3 and
Gazi University for the kind hospitality. RVM would like to thank CNPq under
the Grant no. 200879/2022-7 for financial support. No Data associated in the
manuscript

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