#  $\displaystyle e^{+}e^{-}\to l^{+}l^{-}$ scattering at finite temperature
in the presence of a classical background magnetic field

D. S. Cabral<EMAIL_ADDRESS>Instituto de Física, Universidade
Federal de Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil  A. F. Santos
<EMAIL_ADDRESS>Instituto de Física, Universidade Federal de
Mato Grosso,
78060-900, Cuiabá, Mato Grosso, Brazil

###### Abstract

In this work the $\displaystyle e^{+}e^{-}\to l^{+}l^{-}$ scattering process
is investigated. The cross-section is calculated considering three different
effects: temperature, external magnetic field and chemical potential. The
effect due to an external field is inserted into the problem through a
redefinition of the fermionic field operator. Effects due to temperature and
chemical potential are introduced using the Thermo Field Dynamics formalism.

##  I Introduction

The study of processes in quantum field theory at finite temperatures is very
important. It serves not only to determine the well-known quantities in a more
real situation, adding the thermal effects, but also to obtain constraints and
limits that show a description of the primordial world, which are systems
under conditions similar to those at the beginning of the universe. In
addition to temperature, strong magnetic fields and finite chemical potential
play highly important roles, particularly in phase transition regimes [1, 2].
In order to examine these conditions and features a particular scattering
process is considered, namely, electron-positron annihilation decaying in a
lepton-antilepton pair, $\displaystyle e^{+}e^{-}\to l^{+}l^{-}$ scattering.

To investigate the influence of an external background field on the problem,
it is added to the quantum photonic field in the QED Lagrangian. The procedure
here is similar to that used by [3], however, it is adapted to its thermal
approach by the Thermo Field Dynamics (TFD) formalism. This, in turn, is a
thermal field theory that consists of doubling the Hilbert space by making a
copy of it and introducing the temperature-dependent states and operators by a
rotation of this doubled space [4, 5]. There are many works studying processes
at finite temperature by different approaches [6, 7]. There are also some
investigations using the TFD formalism [8, 9, 10, 11, 12, 4]. However, despite
its importance, in most cases, the contribution of the chemical potential has
not been considered.

The connection with thermodynamics, however, is made through the grand
canonical partition function instead of the canonical one, which, in addition
to the temperature, also introduces the chemical potential into the problem.
The presence of the chemical potential ($\displaystyle\mu$) is very important
in a scattering process since it is the energy required to add a particle to a
system without changing entropy and volume [13]. Then, the contribution of
this term will be considered and calculated. An alternative way to put the
chemical potential in the system, which not will be done here, is to change
the Lagrangian [14].

For the investigation carried out in this work, it is important to observe
that the classical background field leads to a modification in the wave
function for the fermionic field, as well as in the energy expression.
Strictly speaking, this feature modifies the particle and antiparticle spinors
by adding to them a new coordinate dependency and a new quantum number that
emerges. Therefore, for the first there is a new completeness relation
depending on the special Landau functions, and the second gives the possible
energies that the system can occupy as a function of the field strength.

This paper is organized as following. In Sec. II, it is presented how to
describe the Dirac field focusing on the new forms of the energy, as well as
on the spinors and their modified completeness relations and, consequently,
introducing new terms called Landau functions. Then, to work with the thermal
effects of this theory, the formalism and the structure of TFD are presented.
In addition, the differential cross-section is calculated for a $\displaystyle
a+b\to c+d$ reaction, looking at the dependence on Landau levels
$\displaystyle n$. In Sec. III, a review of the meaning of chemical potential
in a physical system is made. Its functional dependence on temperature and
lepton density is obtained by two different methods: the first is a Sommerfeld
approximation and the second is a Taylor series expansion. Then, the
interpretation of these results in a QED scattering process and the role of
this quantity in a quantum field theory are discussed. Finally, in Sec. IV the
thermal $\displaystyle e^{+}e^{-}\to l^{+}l^{-}$ scattering in the presence of
an external classical background magnetic field is investigated, showing the
calculations of its cross-section in the limit of very strong fields. Some
conclusions remarks are presented in the Sec. V.

In this work, natural units of particle physics, $\displaystyle\hbar=c=1$,
have been used. The scattering diagrams are made with the Jaxodraw application
[15]. The calculations are performed with Mathematica software [16].

##  II The Dirac field

Let us consider a charged massive fermion field $\displaystyle\psi$ whose
charge and mass are $\displaystyle q$ and $\displaystyle M$, respectively. The
Hamiltonian describing this particle in the presence of a magnetic field is
written as

$\displaystyle\displaystyle
H=\vec{\alpha}\cdot\left(-i\vec{\nabla}-q\vec{A}^{(B)}\right)+\mathbb{T}M,$
(1)

where $\displaystyle\vec{A}^{(B)}$ is the background classical potential
vector [17] and

$\displaystyle\displaystyle\vec{\alpha}=\begin{pmatrix}0&\vec{\sigma}\\\
\vec{\sigma}&0\end{pmatrix},\quad\quad\quad\mathbb{T}=\begin{pmatrix}I&0\\\
0&-I\end{pmatrix},$ (2)

with $\displaystyle\vec{\sigma}$ and $\displaystyle I$ being the Pauli and
identity $\displaystyle 2\times 2$ matrices, respectively.

In this way, the Lagrangian of the system is given as

$\displaystyle\displaystyle\mathcal{L}=\bar{\psi}\left(i\gamma^{\nu}\Pi_{\nu}-M\right)\psi,$
(3)

where $\displaystyle\gamma^{\nu}$ are the Dirac matrices and
$\displaystyle\Pi_{\nu}=\partial_{\nu}+iqA_{\nu}^{(B)}$, which looks like a
covariant derivative, but with the background field playing the role of the
gauge field. From Eq. (1) the motion equations, i.e. the modified Dirac
equation, can be extracted, which becomes the eigenvalue relation
$\displaystyle H\psi=E\psi$. Then the equation to be solved is

$\displaystyle\displaystyle i\gamma^{\nu}\Pi_{\nu}\psi=M\psi,$ (4)

whose solutions are given by

$\displaystyle\displaystyle\psi=e^{-iEt}\begin{pmatrix}\psi_{R}(\vec{r})\\\
\psi_{L}(\vec{r})\end{pmatrix},$ (5)

with $\displaystyle(t,\vec{r})=(t,x,y,z)$ being the position in space-time.
The spinorial field $\displaystyle\psi$ is a column vector represented by
$\displaystyle\psi_{R}$ and $\displaystyle\psi_{L}$.

It is important to note that, choosing a constant magnetic field in the
z-direction, the potential is given by

$\displaystyle\displaystyle A^{\nu}_{(B)}=\left(0,-yB,0,0\right),$ (6)

such that, in Cartesian coordinates,
$\displaystyle\vec{B}=\vec{\nabla}\times\vec{A}=B\hat{z}$, with $\displaystyle
B>0$ being the amplitude of the field.

Therefore, under these conditions, taking $\displaystyle q=-e$ as the electron
charge, the form of Eq. (5) in addition to Eq. (4) gives us a differential
equation whose solutions are

$\displaystyle\displaystyle\psi_{R,L}(x^{\nu})=C_{R,L}^{(1)}u_{R,L}e^{-ip_{\nu}x^{\nu}_{\cancel{y}}}+C_{R,L}^{(2)}v_{R,L}e^{ip_{\nu}x^{\nu}_{\cancel{y}}},$
(7)

where the notation $\displaystyle x^{\nu}_{\cancel{y}}$ is adopted and
represents a space-time position without $\displaystyle y$-component, this
implies $\displaystyle p_{\nu}x^{\nu}_{\cancel{y}}=p_{0}t-p_{x}x-p_{z}z$. The
indices $\displaystyle R$ and $\displaystyle L$ represent the right- and left-
handed spinor solutions while $\displaystyle C^{(j)}_{R,L}$ are constants with
$\displaystyle j=1,2$ representing positive and negative energy eigenstates,
respectively.

Note in Eq. (7) that the plane wave parts of the solution have the
$\displaystyle y$-component equal to zero, such that this dependence is shown
in the $\displaystyle u_{s}$ and $\displaystyle v_{s}$ spinors. Doing this,
the problem to be solved becomes a special Hermite problem [18] and for the
positive energy solutions it has the following

$\displaystyle\displaystyle
u_{1}(y,n,\vec{p}_{\cancel{y}})=\begin{pmatrix}I_{n-1}(\xi^{+})\\\ 0\\\
\frac{p_{z}}{E_{n}+M}I_{n-1}(\xi^{+})\\\
-\frac{\sqrt{2neB}}{E_{n}+M}I_{n}(\xi^{+})\end{pmatrix},\quad\quad\quad
u_{2}(y,n,\vec{p}_{\cancel{y}})=\begin{pmatrix}0\\\ I_{n}(\xi^{+})\\\
-\frac{\sqrt{2neB}}{E_{n}+M}I_{n-1}(\xi^{+})\\\
-\frac{p_{z}}{E_{n}+M}I_{n}(\xi^{+})\end{pmatrix},$ (8)

and for the negative energy solutions,

$\displaystyle\displaystyle
v_{1}(y,n,\vec{p}_{\cancel{y}})=\begin{pmatrix}\frac{p_{z}}{E_{n}+M}I_{n-1}(\xi^{-})\\\
\frac{\sqrt{2neB}}{E_{n}+M}I_{n}(\xi^{-})\\\ I_{n-1}(\xi^{-})\\\
0\end{pmatrix},\quad\quad\quad
v_{2}(y,n,\vec{p}_{\cancel{y}})=\begin{pmatrix}\frac{\sqrt{2neB}}{E_{n}+M}I_{n-1}(\xi^{-})\\\
-\frac{p_{z}}{E_{n}+M}I_{n}(\xi^{-})\\\ 0\\\ I_{n}(\xi^{-})\end{pmatrix}.$ (9)

The $\displaystyle s=1,2$ index represents the spin-up and spin-down states
and

$\displaystyle\displaystyle E_{n}=\pm\sqrt{M^{2}+p_{z}^{2}+2neB},$ (10)

is the energy spectrum, obtained by the dispersion relation coming from the
Hermite equation, with $\displaystyle n=0,1,2,...$.

The term $\displaystyle I_{n}$ appearing in the expressions (8) and (9) is the
$\displaystyle n$-th Landau function given by

$\displaystyle\displaystyle
I_{n}(\xi)=\left(\frac{\sqrt{eB}}{2^{n}n!\sqrt{\pi}}\right)^{1/2}e^{-\xi^{2}/2}H_{n}(\xi)$
(11)

with $\displaystyle H_{n}(\xi)$ being the Hermite polynomials of order
$\displaystyle n$. The functions $\displaystyle I_{n-1}$ are related to spin-
up solutions, while $\displaystyle I_{n}$ performs the same for spin-down
solutions. Furthermore, for $\displaystyle n=0$ there is no solution for the
spin-up state, then $\displaystyle I_{-1}\equiv 0$. In addition, these
functions obey the following completeness relations

$\displaystyle\displaystyle\sum_{n}I_{n}(\xi)I_{n}(\xi^{\prime})=\sqrt{eB}\delta(\xi-\xi^{\prime})=\delta(y-y^{\prime})$
(12)

and

$\displaystyle\displaystyle\int_{-\infty}^{+\infty}I_{n}(\xi)I_{j}(\xi)d\xi=\sqrt{eB}\delta_{n,j}.$
(13)

The variable $\displaystyle\xi$ present in the spinor solutions is given by

$\displaystyle\displaystyle\xi^{\pm}=\sqrt{eB}\left(y\mp\frac{p_{x}}{eB}\right),$
(14)

which is a term that makes a connection between $\displaystyle y$-coordinate
and $\displaystyle p_{x}$-momentum.

Consequently, the fermionic field is written as

$\displaystyle\displaystyle\psi(x^{\nu})=\int\frac{dp_{x}dp_{z}}{(2\pi)^{2}}\sum_{s,n}\sqrt{\frac{E_{n}+M}{2E_{n}}}\left[a_{s}(n,\vec{p}_{\cancel{y}})u_{s}(y,n,\vec{p}_{\cancel{y}})e^{-ip_{\nu}x^{\nu}_{\cancel{y}}}+b_{s}^{\dagger}(n,\vec{p}_{\cancel{y}})v_{s}(y,n,\vec{p}_{\cancel{y}})e^{ip_{\nu}x^{\nu}_{\cancel{y}}}\right],$
(15)

whose anti-commutation relations are

$\displaystyle\displaystyle\left\\{\psi(x_{\nu}),\psi^{\dagger}(x^{\prime}_{\nu})\right\\}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\delta^{3}(\vec{x}-\vec{x^{\prime}}),$ (16)
$\displaystyle\displaystyle\left\\{a_{s}(n,\vec{p}_{\cancel{y}}),a^{\dagger}_{s^{\prime}}(n^{\prime},\vec{p^{\prime}}_{\cancel{y}})\right\\}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle(2\pi)^{2}\delta_{s,s^{\prime}}\delta_{n,n^{\prime}}\delta(p_{x}-p^{\prime}_{x})\delta(p_{z}-p^{\prime}_{z}),$
(17)

being the same for antiparticle $\displaystyle b$’s operators and equal to
zero for the others. The spinors obey the following modified completeness
relations

$\displaystyle\displaystyle\sum_{s}u_{s}(y,n,\vec{p}_{\cancel{y}})\bar{u}_{s}(y^{\prime},n,\vec{p}_{\cancel{y}})=\frac{1}{E_{n}+M}F_{u}(y,y^{\prime},n,\vec{p}_{\cancel{y}}),$
(18)

for the positive energy part and

$\displaystyle\displaystyle\sum_{s}v_{s}(y,n,\vec{p}_{\cancel{y}})\bar{v}_{s}(y^{\prime},n,\vec{p}_{\cancel{y}})=\frac{1}{E_{n}+M}F_{v}(y,y^{\prime},n,\vec{p}_{\cancel{y}}),$
(19)

for the negative energy part. Here $\displaystyle F_{u}$ and $\displaystyle
F_{v}$ are somewhat complicated functions depending on Landau terms
$\displaystyle I_{n}$ [3, 19].

In addition, due to the $\displaystyle y$-dependence, the spinors also satisfy
the following continuous properties

$\displaystyle\displaystyle\int_{-\infty}^{\infty}dy\left[u^{\dagger}_{s}(y,n,p_{\cancel{y}})u_{s^{\prime}}(y,n^{\prime},p_{\cancel{y}})\right]=\int_{-\infty}^{\infty}dy\left[v^{\dagger}_{s}(y,n,p_{\cancel{y}})v_{s^{\prime}}(y,n^{\prime},p_{\cancel{y}})\right]=\delta_{s,s^{\prime}}\delta_{n,n^{\prime}}\frac{2E_{n}}{E_{n}+M},$
(20)

and the other combinations as $\displaystyle u^{\dagger}v$ and $\displaystyle
v^{\dagger}u$ are equal to zero.

Some notes about Eq. (15) should be considered. (i) The field operator is a
superposition of plane waves, and the exponentials guarantee this fact, but it
is necessary to note that the $\displaystyle p_{y}$ component does not appear
on the expression because the $\displaystyle y$ component of space-time
vanishes. (ii) The spinors do not depend on $\displaystyle p_{y}$ since they
explicitly depend on Landau functions of the space. In other words, the
$\displaystyle y$-component of momentum is not zero, but undefined by the
uncertainly principle, and therefore the integral cannot be evaluated in this
direction because it will lead to divergence.

Finally, these $\displaystyle n$ indices of Landau functions, whose energy is
strongly dependent, although very large, do not go to infinity. This can be
seen from the spectra (10). For a fermionic incident particle with mass
$\displaystyle M$ and whose energy $\displaystyle E$ is given, from the
dispersion relation one can extract the limit $\displaystyle E^{2}-M^{2}\geq
2neB$. In other words, if $\displaystyle 2neB>E^{2}-M^{2}$ it would be
necessary for $\displaystyle p_{z}$ to be a complex number, which is
impossible because it is a measurable physical quantity [18].

Up to now, it has been described how to treat fermions in the presence of a
classical magnetic field. Where the new operator given by Eq. (15) and its
adjoint $\displaystyle\bar{\psi}=\psi^{\dagger}\gamma^{0}$ with the anti-
commutation rule Eq. (17) configures all Dirac mechanics, in this special
case, within a background potential, but still at zero temperature.

To make this field theory thermal, the Thermo Field Dynamics (TFD) formalism
can be used. This approach is based on the concepts of Lie algebra and
symmetries, in such a manner that it is characterized by doubling the Hilbert
space [5, 4]. In other words, in this formalism, the thermal effects are added
to the theory through the thermal space
$\displaystyle\mathbb{H}_{T}=\mathbb{H}\otimes\widetilde{\mathbb{H}}$, where
the first is the usual Hilbert space while the following is the copy of the
first one, in which, obey the commutation (or anti-commutation) relations

$\displaystyle\displaystyle[\mathcal{A},\mathcal{B}]=if_{ab}^{c}\mathcal{C},\quad\quad[\widetilde{\mathcal{A}},\widetilde{\mathcal{B}}]=-if_{ab}^{c}\widetilde{\mathcal{C}},\quad\quad[\widetilde{\mathcal{A}},\mathcal{B}]=0,$
(21)

where $\displaystyle\mathcal{A}$, $\displaystyle\mathcal{B}$ and
$\displaystyle\mathcal{C}$ are arbitrary operators in
$\displaystyle\mathbb{H}$ and $\displaystyle f_{ab}^{c}$ are structure
constants.

The tilde operators (which are in the $\displaystyle\widetilde{\mathbb{H}}$
space) are related to the non-tilde ones by

$\displaystyle\displaystyle(\mathcal{A}\mathcal{B})^{\widetilde{}}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\widetilde{\mathcal{A}}\widetilde{\mathcal{B}},\quad\quad(\widetilde{\mathcal{A}})^{\widetilde{}}=\varsigma\mathcal{A},$
$\displaystyle\displaystyle(a\mathcal{A}+\mathcal{B})^{\widetilde{}}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
a^{*}\widetilde{\mathcal{A}}+\widetilde{\mathcal{B}},\quad\quad(\mathcal{A}^{\dagger})^{\widetilde{}}=(\widetilde{\mathcal{A}})^{\dagger},$
(22)

where $\displaystyle a$ is a complex constant, $\displaystyle\varsigma=1$ for
bosons and $\displaystyle\varsigma=-1$ for fermions. Note that the
$\displaystyle\mathcal{A}$ operators are the true observables of the theory,
while $\displaystyle\widetilde{\mathcal{A}}$ exist only for symmetry purposes.

In this formalism, there are symmetry generators
$\displaystyle\widehat{\mathcal{A}}$ of the thermal space, related to the
physical transformations of the system, which are

$\displaystyle\displaystyle\widehat{\mathcal{A}}=\mathcal{A}-\widetilde{\mathcal{A}},$
(23)

in other words, although the tilde and non-tilde operators have the meaning
already discussed, it is often necessary to calculate the contribution given
by $\displaystyle\widehat{\mathcal{A}}$. Taking the Hamiltonian of the system
as an example, it is known that $\displaystyle H$ provides the possible energy
states of the system while $\displaystyle\widetilde{H}$ is the tilde-conjugate
of the first to duplicate the space and, consequently, composes the thermal
space. But only $\displaystyle\hat{H}$ is the operator that gives the time-
translational character, i.e., applying it in a certain state gives the
temporal evolution of the system.

In addition, by this approach, the average of a dynamical operator
$\displaystyle\mathcal{A}$ in thermal equilibrium is given by
$\displaystyle\left\langle\mathcal{A}\right\rangle=\bra{0(\beta,\mu)}\mathcal{A}\ket{0(\beta,\mu)}$,
where

$\displaystyle\displaystyle\ket{0(\beta,\mu)}=\frac{1}{\sqrt{Z(\beta,\mu)}}\sum_{n}e^{-\beta(E_{n}-\mu
N_{n})/2}\ket{n,\widetilde{n}}$ (24)

is the thermal vacuum state, with $\displaystyle Z(\beta,\mu)$ being the
grand-canonical partition function, $\displaystyle\mu$ is the chemical
potential and $\displaystyle E_{n}$ and $\displaystyle N_{n}$ are the energy
and occupation number eigenvalues of the $\displaystyle n$-th state,
respectively. The temperature $\displaystyle T$ is introduced by the factor
$\displaystyle\beta=1/k_{B}T$, where $\displaystyle k_{B}$ is the Boltzmann
constant.

It should be noted in Eq. (24) that, as the states $\displaystyle\ket{n}$ and
$\displaystyle\ket{\widetilde{n}}$ are in the non-tilde and tilde spaces,
respectively, the vector
$\displaystyle\ket{n,\widetilde{n}}=\ket{n}\otimes\ket{\widetilde{n}}$ is in
the thermal Hilbert space $\displaystyle\mathbb{H}_{T}$. Then, to make this
link between them, there is another important ingredient of this formalism,
the Bogoliubov transformation, which acts by making the connection by uniting
thermal and non-thermal operators.

Such transformations permit writing the zero-temperature operators in terms of
the $\displaystyle\beta$-dependent ones and vice-versa. Then, looking at the
fermion field given by Eq. (15) these rules can be formulated for its creation
and annihilation operators as follows

$\displaystyle\displaystyle a_{s}(n,\vec{p}_{\cancel{y}})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle(1-f)^{1/2}a_{s}(\beta,n,\vec{p}_{\cancel{y}})+(f)^{1/2}\widetilde{a}^{\dagger}_{s}(\beta,n,\vec{p}_{\cancel{y}}),$
$\displaystyle\displaystyle a^{\dagger}_{s}(n,\vec{p}_{\cancel{y}})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle(1-f)^{1/2}a^{\dagger}_{s}(\beta,n,\vec{p}_{\cancel{y}})+(f)^{1/2}\widetilde{a}_{s}(\beta,n,\vec{p}_{\cancel{y}}),$
(25)

for particles, and

$\displaystyle\displaystyle b_{s}(n,\vec{p}_{\cancel{y}})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle(1-f)^{1/2}b_{s}(\beta,n,\vec{p}_{\cancel{y}})+(f)^{1/2}\widetilde{b}^{\dagger}_{s}(\beta,n,\vec{p}_{\cancel{y}}),$
$\displaystyle\displaystyle b^{\dagger}_{s}(n,\vec{p}_{\cancel{y}})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle(1-f)^{1/2}b^{\dagger}_{s}(\beta,n,\vec{p}_{\cancel{y}})+(f)^{1/2}\widetilde{b}_{s}(\beta,n,\vec{p}_{\cancel{y}}),$
(26)

for antiparticles. Here

$\displaystyle\displaystyle f=f(E_{n},\mu)=\frac{1}{e^{\beta(E_{n}-\mu)}+1},$
(27)

is the Fermi-Dirac distribution. It is important to note that the chemical
potential of the particle is different from that of the antiparticle.

In addition to Eqs. (25) and (26), we have the anti-commutation relations
given by

$\displaystyle\displaystyle\left\\{a_{s}(\beta,n,\vec{p}_{\cancel{y}}),a^{\dagger}_{s^{\prime}}(\beta,n^{\prime},\vec{p^{\prime}}_{\cancel{y}})\right\\}=(2\pi)^{2}\delta_{s,s^{\prime}}\delta_{n,n^{\prime}}\delta(p_{x}-p^{\prime}_{x})\delta(p_{z}-p^{\prime}_{z})$
(28)

for particles and analogously for antiparticles. The others are equal to zero.

Therefore, the interpretation of Eqs. (25), (26) and (28) is very simple,
[$\displaystyle b^{\dagger}_{s}$]$\displaystyle
a^{\dagger}_{s}(\beta,n,\vec{p}_{\cancel{y}})$ acts by creating a
[anti]fermion with temperature $\displaystyle T$, spin $\displaystyle s$,
3-momentum $\displaystyle\vec{p}=(p_{x},p_{z})$ and energy $\displaystyle
E_{n}$ at an almost delocalized space-time point (by the uncertainly
principle) in which there is a classical magnetic field. In “almost”, we mean
that we do not have information about $\displaystyle x$ and $\displaystyle z$
components but $\displaystyle y$ does, since it is given by spinor dependency
and, moreover, even if this operator acts in a situation where $\displaystyle
p_{y}=0$, this component cannot be defined. In this way, the description of
all fermionic mechanics in a presence of a background potential at finite
temperature becomes straightforward, since $\displaystyle\psi(x^{\nu},\beta)$
creates a Dirac particle with temperature $\displaystyle\beta$ at a space-time
point $\displaystyle x^{\nu}$ over a superposition of spin, energy and
momentum states. The same concepts can be used for an antiparticle using the
$\displaystyle\bar{\psi}(x^{\nu},\beta)$ operator.

Now, keeping it in mind, we can extract some important quantities like the
differential cross-section for any scattering process.

###  II.1 Calculation of the differential cross-section

In this subsection, the main objective is to calculate the differential cross-
section for a scattering process considering two specific cases of the
$\displaystyle n$ parameter which is related to the intensity of the external
magnetic field. In order to analyze these cases, let us consider the
dispersion relation given in Eq. (10). From this the following relation is
extracted

$\displaystyle\displaystyle
p^{\alpha}p_{\alpha}=M^{2}=E_{n}^{2}-|\vec{p}|^{2}=M^{2}+p_{z}^{2}+2neB-|\vec{p}|^{2}$
(29)

which implies on

$\displaystyle\displaystyle p_{x}^{2}+p_{y}^{2}=2neB.$ (30)

Furthermore, taking into consideration Eq. (30), we can see that this relation
forms a circle in the $\displaystyle p_{\perp}$ plane, with a radius of
$\displaystyle r=\sqrt{2neB}$. Therefore, when performing integrations over
the perpendicular momentum components, which may arise frequently in certain
calculations, we find that it is just

$\displaystyle\displaystyle\int d^{2}\vec{p^{\prime}}_{\perp}=\text{circle
area}=\pi r^{2}=(2\pi)neB.$ (31)

And, for Eq. (30) two different cases, $\displaystyle n=0$ and $\displaystyle
n\neq 0$, are considered. In this context, the differential cross-section is
investigated.

####  II.1.1 Calculations for $\displaystyle n\neq 0$

The differential cross-section is defined as

$\displaystyle\displaystyle d\sigma=\frac{\text{density of final
states}}{\text{flux of incoming particles}}W_{fi},$ (32)

where $\displaystyle W_{fi}$ is the probability density of the incoming state
$\displaystyle\ket{i}$ decay into outcome state $\displaystyle\ket{f}$. In a
presence of an external field, this quantity is given by

$\displaystyle\displaystyle
W_{fi}=\frac{(2\pi)^{4}}{V^{4}}\delta^{4}\left(\sum
p^{(\text{in})}_{\cancel{y}}-\sum
p^{(\text{out})}_{\cancel{y}}\right)\langle|\mathcal{M}_{fi}|^{2}\rangle,$
(33)

since the sums in $\displaystyle p^{(\text{in})}$ and $\displaystyle
p^{(\text{out})}$ guarantee the momentum conservation, but with their
$\displaystyle y-$components set to zero, and $\displaystyle V$ is a space-
time volume stands out for normalization purposes.

The density of final states, given in Eq. (32), for a scattering process
$\displaystyle p+k\to p^{\prime}+k^{\prime}$ in the center of mass frame,
where all particle energies are $\displaystyle E_{n}$, is

$\displaystyle\displaystyle\text{density}=g_{n}\frac{V}{(2\pi)^{3}}\frac{d^{3}\vec{p^{\prime}}}{2E_{n}}g_{n}\frac{V}{(2\pi)^{3}}\frac{d^{3}\vec{k^{\prime}}}{2E_{n}}=g_{n}^{2}\frac{V^{2}}{(2\pi)^{6}}d^{2}\vec{p^{\prime}}_{\perp}d^{2}\vec{k^{\prime}}_{\perp}\frac{dp_{z}^{\prime}dk_{z}^{\prime}}{(2E_{n})^{2}}$
(34)

in such a way that the phase space element, in the presence of a magnetic
field oriented in the $\displaystyle z$-direction, is given by

$\displaystyle\displaystyle\frac{d^{3}\vec{p}}{(2\pi)^{3}}=\frac{d^{2}\vec{p}_{\perp}dp_{z}}{(2\pi)^{3}}$
(35)

where $\displaystyle\vec{p}_{\perp}$ represents the momentum components
$\displaystyle p_{x}$ and $\displaystyle p_{y}$, which are orthogonal to the
field orientation, and $\displaystyle g_{n}$ is the spin degeneracy of the
energy state, with $\displaystyle g_{0}=1$ and $\displaystyle g_{n}=2$ for
$\displaystyle n=1,2,3,...$ [3].

For what follows, it is usual to write the Mandelstam variables which are
given as

$\displaystyle\displaystyle s=(p+k)^{2};\quad\quad
t=(p-p^{\prime})^{2};\quad\quad u=(p-k^{\prime})^{2}.$ (36)

Then, the flux written in Eq. (32), when the initial particles has mass
$\displaystyle M_{i}$, is

$\displaystyle\displaystyle\text{flux}=\frac{4}{V^{2}}\sqrt{\left(p\cdot
k\right)^{2}-M_{i}^{4}}=\frac{2}{V^{2}}\sqrt{s(s-4M_{i}^{2})}.$ (37)

Thus, the differential cross-section Eq. (32) becomes

$\displaystyle\displaystyle
d\sigma(\beta)=\langle|\mathcal{M}_{fi}(\beta)|^{2}\rangle
g_{n}^{2}\frac{\delta^{4}\left(p_{\cancel{y}}+k_{\cancel{y}}-p^{\prime}_{\cancel{y}}-k^{\prime}_{\cancel{y}}\right)}{2\sqrt{s(s-4M_{i}^{2})}}\frac{d^{2}\vec{p^{\prime}}d^{2}\vec{k^{\prime}}}{(2\pi)^{2}}\frac{dp^{\prime}_{z}dk^{\prime}_{z}}{(2E_{n})^{2}}.$
(38)

The cross-section is obtained from Eq. (38) by integration and by summing all
possibles Landau indices $\displaystyle n$. Performing the integration on
$\displaystyle\vec{k}^{\prime}$, the resulting delta function is only on
$\displaystyle E$, i.e.,

$\displaystyle\displaystyle\sigma(\beta)=\sum_{n}\int\langle|\mathcal{M}_{fi}(\beta)|^{2}\rangle
g_{n}^{2}\frac{\delta(E_{n}^{(p)}+E_{n}^{(k)}-E_{n}^{(p^{\prime})}-E_{n}^{(k^{\prime})})}{2\sqrt{s(s-4M_{i}^{2})}}\frac{d^{2}\vec{p^{\prime}_{\perp}}dp^{\prime}_{z}}{(2\pi)^{2}(2E_{n})^{2}}.$
(39)

However, due to the delta function evaluation, the transition amplitude
involves momentum conservation, given by $\displaystyle
k^{\prime}=p+k-p^{\prime}$. Therefore, Eq. (39) can be rewritten as

$\displaystyle\displaystyle\sigma(\beta)=\sum_{n}\sigma_{n}(\beta)$ (40)

with

$\displaystyle\displaystyle\sigma_{n}(\beta)=g_{n}^{2}\int\langle|\mathcal{M}_{fi}(\beta)|^{2}\rangle\frac{\delta(E_{n}^{(p)}+E_{n}^{(k)}-E_{n}^{(p^{\prime})}-E_{n}^{(k^{\prime})})}{2\sqrt{s(s-4M_{i}^{2})}}\frac{d^{2}\vec{p^{\prime}_{\perp}}dp^{\prime}_{z}}{(2\pi)^{2}(2E_{n})^{2}}.$
(41)

Making $\displaystyle E_{n}^{(p)}+E_{n}^{(k)}=\sqrt{s}$ and $\displaystyle
E_{n}^{(p^{\prime})}+E_{n}^{(k^{\prime})}=2\sqrt{M_{f}^{2}+p_{z}^{\prime
2}+2neB}=E^{\prime}_{n}$ in Eq. (41), and using

$\displaystyle\displaystyle
dE^{\prime}_{n}=2\frac{p^{\prime}_{z}dp^{\prime}_{z}}{\sqrt{M_{f}^{2}+p_{z}^{\prime
2}+2neB}}=\frac{E^{\prime}_{n}}{E_{n}^{2}}p^{\prime}_{z}dp^{\prime}_{z},$ (42)

the $\displaystyle n$-th cross-section is written as

$\displaystyle\displaystyle\sigma_{n}(\beta)=\frac{g_{n}^{2}}{(4\pi)^{2}}\int\frac{\langle|\mathcal{M}_{fi}(\beta)|^{2}\rangle}{2\sqrt{s(s-4M_{i}^{2})}}\delta(\sqrt{s}-E^{\prime}_{n})\frac{d^{2}\vec{p^{\prime}_{\perp}}dE^{\prime}_{n}}{p^{\prime}_{z}E^{\prime}_{n}}=\frac{g_{n}^{2}}{(4\pi)^{2}}\int\frac{\langle|\mathcal{M}_{fi}(\beta)|^{2}\rangle}{2sp^{\prime}_{z}\sqrt{s-(2M_{i})^{2}}}d^{2}\vec{p^{\prime}_{\perp}},$
(43)

where $\displaystyle M_{f}$ is the mass of the final fermion. As the field
strength reaches very high values, $\displaystyle\mathcal{M}_{fi}$ becomes
increasingly independent of the perpendicular momenta, allowing the direct
application of the integration in Eq. (31) to Eq. (43).

As already discussed, for the treated problem to be physical it is necessary
that $\displaystyle E^{2}-M^{2}\geq 2neB$. In other words, a limit for
$\displaystyle n$ can be set, i.e.,

$\displaystyle\displaystyle
n_{\text{max}}=\text{int}\left\\{\frac{E^{2}-M^{2}}{2eB}\right\\},$ (44)

in such a manner that in the limit of the strongest possible magnetic field,
we get $\displaystyle n_{\text{max}}=0$. Then, the total cross-section Eq.
(40) is written as

$\displaystyle\displaystyle\sigma(\beta)=\sum_{n=0}^{n_{\text{max}}}\sigma_{n}(\beta).$
(45)

In the limit without field, $\displaystyle B\to 0$, the spacing between each
$\displaystyle E_{n}$ and $\displaystyle E_{n+1}$ becomes smaller and smaller.
Therefore, the energy levels become a continuum spectrum and $\displaystyle
g_{n}\equiv 2$ for all $\displaystyle n$. Then, Eq. (45) reads

$\sigma_{B\to 0}(\beta)=\int_{0}^{n_{\text{max}}}\sigma_{n}(\beta)dn,$ (46)

and using Eq. (43), the total cross-section will be

$\displaystyle\displaystyle\sigma_{B\to 0}(\beta)$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\lim_{B\to
0}\frac{g_{n}^{2}}{(4\pi)^{2}}\frac{1}{2s\sqrt{s-(2M_{i})^{2}}}\int\left[\int_{0}^{n_{\text{max}}}\langle|\mathcal{M}_{fi}(\beta)|^{2}\rangle\frac{n}{\sqrt{E_{n}^{2}-M_{f}^{2}-2neB}}dn\right]d^{2}\vec{p^{\prime}_{\perp}}$
(47)

which is strongly dependent on the form of
$\displaystyle\langle|\mathcal{M}_{fi}|^{2}\rangle$, i.e., the functional
dependence of $\displaystyle B$ and $\displaystyle n$. The last dependence is
given uniquely by the form of spin completeness relations Eq. (18) and Eq.
(19).

####  II.1.2 Calculations for $\displaystyle n=0$

When $\displaystyle n=0$ it is necessary to look for a special case. From Eq.
(30) it is extracted $\displaystyle p_{x}=p_{y}=0$. In other words, if the
situation $\displaystyle n_{\text{max}}=0$ is taken, the momentum of the
resulting particles can only be in $\displaystyle p_{z}$ one-dimensional real
space. Therefore, the density of the final states coming from the phase space
element is

$\displaystyle\displaystyle\text{density}=g_{0}\frac{V}{(2\pi)}\frac{dp^{\prime}_{z}}{2E_{0}}g_{0}\frac{V}{(2\pi)}\frac{dk^{\prime}_{z}}{2E_{0}}=\frac{V^{2}}{(2\pi)^{2}}\frac{dp^{\prime}_{z}dk^{\prime}_{z}}{(2E_{0})^{2}},$
(48)

and the probability density of the scattering process becomes

$\displaystyle\displaystyle W_{fi}=\frac{(2\pi)^{2}}{V^{4}}\delta\left(\sum
p_{z}^{\text{(in)}}-\sum
p_{z}^{\text{(out)}}\right)\delta\left(E_{n}^{(p)}+E_{n}^{(k)}-E_{n}^{(p^{\prime})}-E_{n}^{(k^{\prime})}\right)\langle|\mathcal{M}_{fi}|^{2}\rangle.$
(49)

The flux of incoming particles is the same as given in Eq. (37). Therefore, in
this present case, we get

$\displaystyle\displaystyle\sigma_{0}(\beta)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int\frac{\langle|\mathcal{M}_{fi}(\beta)|^{2}\rangle}{2\sqrt{s(s-4m_{e}^{2})}}\delta\left(E_{n}^{(p)}+E_{n}^{(k)}-E_{n}^{(p^{\prime})}-E_{n}^{(k^{\prime})}\right)\frac{dp^{\prime}_{z}}{(2E_{0})^{2}}$
(50) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int\frac{\langle|\mathcal{M}_{fi}(\beta)|^{2}\rangle}{8\sqrt{s(s-4m_{e}^{2})}}\delta(\sqrt{s}-E^{\prime}_{n})\frac{dE^{\prime}_{n}}{p^{\prime}_{z}E^{\prime}_{n}}=\frac{\langle|\mathcal{M}_{fi}(\beta)|^{2}\rangle}{8p^{\prime}_{z}s\sqrt{s-4m_{e}^{2}}}.$

In short, any $\displaystyle 1+2\to 3+4$ scattering process carried out at
finite temperature in the center of mass frame subject to an external magnetic
field will have the total cross-section given by Eq. (45), where
$\displaystyle\sigma_{n}(\beta)$ is given by Eq. (43) for $\displaystyle n>1$
and $\displaystyle\sigma_{0}(\beta)$ is given in Eq. (50). When the field
strength $\displaystyle B$ becomes negligible, $\displaystyle
n_{\text{max}}\to\infty$, then the energy levels become a continuum and
$\displaystyle\sigma_{n}$ needs to be integrated.

On the other hand, when the scattering process is analyzed over a very strong
field the only allowed level, with respect to Eq. (44), is the one where
$\displaystyle n_{\text{max}}=0$. Then the only quantity needed to be taken
into account is $\displaystyle\sigma_{0}(\beta)$.

In the next section, the meaning of the chemical potential and its importance
for the scattering process are investigated.

##  III The Chemical Potential

Starting from the TFD formalism, one can observe the dependence of a physical
system on temperature and chemical potential, as can be seen in Eq. (24). Note
that temperature is a well-known quantity, whereas chemical potential does not
have a well-understood physical picture either in basic physics or in more
complex studies of quantum field theory [20]. In this way, first we need to
explain the meaning of this quantity and, later, what it represents in QED
processes.

###  III.1 The meaning of the chemical potential

In Eqs. (25) and (26) the chemical potential $\displaystyle\mu$ appears due to
Eq. (27). Although almost never discussed, the role of chemical potential in
QED processes is very important [7]. This quantity comes from the free energy
related to particle changes. Since the vacuum is not empty, such as real
particles, there are a very large number of liquid particles present in any
real scattering process that are created and subsequently annihilated without
violating the uncertainly principle [21].

The chemical potential in a system of fermionic particles is equal to the
Fermi energy at zero temperature. Where is the highest energy level of the
system at the fundamental state [22]. In addition, the study of the
contribution of this term is very important in several cases where it is
necessary to consider the macroscopic behavior of the problem, i.e., the
thermodynamic connection.

As the grand-canonical formalism of statistical physics is used, it is known
that the connection with thermodynamics is made by the link $\displaystyle
Z(\beta,\mu)\to\Phi(T,\mu)$, where $\displaystyle\Phi$ is the thermodynamic
grand-potential [23]. In this way, it is known that the chemical potential is
the free energy related to the changes of the particles. As in the present
discussion the Dirac field is treated, there are two types of particles, with
positive and negative energy. Then, can be defined

$\displaystyle\displaystyle
N_{+}=\frac{\partial\Phi}{\partial\mu_{+}},\quad\quad\quad
N_{-}=\frac{\partial\Phi}{\partial\mu_{-}}.$ (51)

It is important to note that the annihilation of particles implies the
creation of antiparticles and vice-versa, such that $\displaystyle
dN_{+}=-dN_{-}$. Therefore defining $\displaystyle\mu$ as the chemical
potential of the system we get $\displaystyle\mu_{+}\equiv\mu$ and
$\displaystyle\mu_{-}=-\mu$, for fermions and anti-fermions, respectively.

(a) The distribution function of a particle at close-to-zero temperature
(black solid line) and $\displaystyle T>0$ (red dashed line) cases.

(b) The distribution function of an antiparticle at close-to-zero temperature
(black solid line) and $\displaystyle T>0$ (red dashed line) cases.

(c) The distribution function of an antiparticle with the change
$\displaystyle-E\to\epsilon$, at close-to-zero temperature (black solid line)
and $\displaystyle T>0$ (red dashed line) cases.

Figure 1: Distribution functions in terms of the Energy.

The distribution function $\displaystyle f(E,\mu)$ given in Eq. (27) is
plotted in Figure 1. The Figure 1(a) is the well-known Fermi-Dirac function
that describes the population of energy states of a given system with chemical
potential $\displaystyle\mu$ in two different temperature cases. At zero
temperature, all states with energy less than $\displaystyle\mu$ are filled
and empty otherwise. When the temperature increases, the forward states become
filled. Looking at Figure 1(b), we notice that for antiparticles, although the
step is moved to $\displaystyle-\mu$, the profile is similar to the previous
graph and the same analysis can be applied to this case. The last function
shown in Figure 1(c) is the same as the previous one, but with the
redefinition of the energy by $\displaystyle\epsilon=-E$, and this approach
displays an electron distribution like the first figure.

In a quantum field theory, such as QED, it is possible to associate a chemical
potential with each conserved quantum number such as the lepton [2]. For our
present purposes, the lepton number density is written as

$\displaystyle\displaystyle\rho=N_{+}-N_{-}=\sum_{k}g_{k}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\left[f(E_{k}(p_{z}),\mu)-f(E_{k}(p_{z}),-\mu)\right],$
(52)

where the Fermi-Dirac distribution given in Eq. (27) is used [7, 22]. The
$\displaystyle k$-sum and the $\displaystyle\vec{p}$-integration ensure that
all energy eigenstates, Eq. (10), need to be taken into account. Then, Eq.
(52) can be written as

$\displaystyle\displaystyle\rho=\sum_{n}g_{n}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\left[\frac{1}{\zeta^{-1}e^{\beta\sqrt{M^{2}+p_{z}^{2}+2neB}}+1}-\frac{1}{\zeta
e^{\beta\sqrt{M^{2}+p_{z}^{2}+2neB}}+1}\right],$ (53)

with $\displaystyle\zeta=e^{\beta\mu}$ being the fugacity of the system. At
this moment, it is again necessary to make a distinction between the cases
$\displaystyle n\neq 0$ and $\displaystyle n=0$. For the former, looking at
Eq. (35) and Eq. (31), we get from Eq. (53)

$\displaystyle\displaystyle\rho=\sum_{n}(2neB)g_{n}\int_{-\infty}^{\infty}\frac{dp_{z}}{(2\pi)^{2}}\left[\frac{1}{\zeta^{-1}e^{\beta\sqrt{M^{2}+p_{z}^{2}+2neB}}+1}-\frac{1}{\zeta
e^{\beta\sqrt{M^{2}+p_{z}^{2}+2neB}}+1}\right]\equiv\sum_{n}\rho_{n}.$ (54)

Then, it is necessary to calculate all $\displaystyle n$-states densities and
add them together.

For the case $\displaystyle n=0$, Eq. (53) is written as

$\displaystyle\displaystyle\rho=\int_{-\infty}^{\infty}\frac{dp_{z}}{2\pi}\left[\frac{1}{\zeta^{-1}e^{\beta\sqrt{M^{2}+p_{z}^{2}}}+1}-\frac{1}{\zeta
e^{\beta\sqrt{M^{2}+p_{z}^{2}}}+1}\right].$ (55)

Due to the constraint imposed by the intense magnetic field on the particle’s
motion in the $\displaystyle z$-direction, the integration in Eq. (55) is one-
dimensional, unlike Eq. (54). In addition, both cases of lepton densities have
very different forms, and this is due to the strong dependence of the state
density on the Landau levels, which as already discussed are finite numbers
determined by the amplitude of the external magnetic field. Furthermore, it is
interesting to reinforce that, in the case of very strong magnetic fields,
only the $\displaystyle n=0$ level is allowed. Consequently, in this regime,
the energy does not depend on the magnetic field $\displaystyle B$, and, as a
result, neither does the fermion density $\displaystyle\rho$. The chemical
potential $\displaystyle\mu$ is determined by inverting the relation between
$\displaystyle\rho$ and the parameters $\displaystyle\beta$ and
$\displaystyle\zeta$ as given in Eqs. (54) and (55) resulting in
$\displaystyle\mu=\mu(\beta,\rho)$. Consequently, when considering
$\displaystyle n=0$, $\displaystyle\mu$ is independent of the magnetic field.
However, for $\displaystyle n\neq 0$, $\displaystyle\mu$ becomes dependent on
the magnetic field. In other words, in the regime where the energy does not
depend on the external field, the chemical potential behaves similarly.
Conceptually, we can view $\displaystyle\mu$ as the energy required to add a
particle to the fermionic system at zero temperature. Although Eqs. (54) and
(55) cannot be solved analytically due to the form of the dispersion relation,
they can be used to determine the chemical potential as a function of
temperature and lepton density. But in the following, we only look at the
latter case. Furthermore, it is interesting to do some analysis on Eq. (55)
looking at the limiting cases, i.e., extracting information when the system is
subject to high and low temperatures. For high temperatures (or small values
of $\displaystyle\beta$), Eq. (55) will be strongly dependent on the value of
$\displaystyle\mu$. When the charges involved are symmetrically related, we
get $\displaystyle\rho=0$ and therefore $\displaystyle\mu=0$.

In the low-temperature limit ($\displaystyle\beta\to\infty$), the first term
in square brackets in Eq. (55) is dominant upon its subsequent one because of
the exponential present in the denominator (this fact can be seen in Figures
1(a) and 1(b)). Then, one can write

$\displaystyle\displaystyle\rho=\frac{1}{\pi}\int_{0}^{\infty}\frac{dp_{z}}{e^{\beta(E-\mu)}+1}.$
(56)

It is useful to write it in terms of energy, i.e., putting $\displaystyle
dp_{z}$ in terms of $\displaystyle dE$ and shifting the displacement
$\displaystyle E\to E+M$. Thus

$\displaystyle\displaystyle\rho=\frac{1}{\pi}\int_{0}^{\infty}dE\frac{E+M}{\sqrt{E^{2}+2EM}}\frac{1}{e^{\beta(E+M-\mu)}+1}\equiv\int_{0}^{\infty}G(E)f(E,\mu-M)dE,$
(57)

with $\displaystyle G(x)=(x+M)/(\pi\sqrt{x^{2}+2Mx})$. Integrating Eq. (57) by
parts, we get

$\displaystyle\displaystyle\int_{0}^{\infty}G(E)f(E,\mu-M)dE=-\int_{0}^{\infty}\left[\int_{0}^{E}G(\varepsilon)d\varepsilon\right]\left(\frac{\partial
f}{\partial E}\right)dE,$ (58)

and expanding the term in the integrand around $\displaystyle E=\mu-M$, we
have

$\displaystyle\displaystyle\int_{0}^{E}G(\varepsilon)d\varepsilon=\int_{0}^{\mu-M}G(\varepsilon)d\varepsilon+\sum_{k=1}^{\infty}\frac{(E-\mu+M)^{k}}{k!}\frac{d^{k}}{dE^{k}}\left[\int_{0}^{E}G(\varepsilon)d\varepsilon\right]_{E=\mu-M}.$
(59)

In this temperature regime, the Sommerfeld expansion can be used,

$\displaystyle\displaystyle\int_{0}^{\infty}G(E)f(E,\mu-M)dE=\int_{0}^{\mu-M}G(\varepsilon)d\varepsilon+\frac{\pi^{2}}{6}\left(\frac{1}{\beta}\right)^{2}G^{\prime}(\mu-M)+\mathcal{O}\left[\left(\frac{1}{\beta}\right)^{4}\right],$
(60)

where $\displaystyle G^{\prime}(x)$ is the derivative of $\displaystyle
G(\varepsilon)$ applied to $\displaystyle\varepsilon=x$ [24]. Then, Eq. (57)
up to second order becomes

$\displaystyle\displaystyle\rho=\frac{1}{\pi}\left[\sqrt{\mu^{2}-M^{2}}-\frac{\pi^{2}}{6}\left(\frac{1}{\beta}\right)^{2}\frac{M^{2}}{\left(\mu^{2}-M^{2}\right)^{3/2}}\right].$
(61)

Note that Eq. (61) is a way to determine the chemical potential of this system
in terms of the $\displaystyle\rho_{N}$ and $\displaystyle T$ over a sum of
all possible energy states $\displaystyle E_{n}$. At zero temperature, we get

$\displaystyle\displaystyle\mu_{0}=M\cosh{\theta_{0}},\quad\quad\text{where}\quad\quad\rho_{0}=\frac{M}{\pi}\sinh{\theta_{0}},$
(62)

similarly to that obtained by [7], but in the context of the presence of a
background field. Furthermore, to obtain an expression for the temperature-
dependent chemical potential it is necessary to take the following
thermodynamical relation [25]

$\displaystyle\displaystyle\frac{\partial\mu}{\partial
T}=\left.-\left({\frac{\partial\rho}{\partial
T}}\right/{\frac{\partial\rho}{\partial\mu}}\right),$ (63)

which implies (up to second order on temperature) in

$\displaystyle\displaystyle\frac{\partial\mu}{\partial
T}=\frac{k_{B}^{2}M^{2}\pi^{2}}{3}\frac{T}{\mu(\mu^{2}-M^{2})}.$ (64)

Then, the temperature-dependent chemical potential, up to second order, is
given by

$\displaystyle\displaystyle\mu(T)=\left[M^{2}+\sqrt{(\mu_{0}^{2}-M^{2})^{2}+\frac{2}{3}\left(k_{B}M\pi\right)^{2}T^{2}}\right]^{1/2}$
(65)

or, expanding once more in the Taylor series,

$\displaystyle\displaystyle\mu(T)=\mu_{0}+\frac{k_{B}^{2}M^{2}\pi^{2}}{6\mu_{0}(\mu_{0}^{2}-M^{2})}T^{2}.$
(66)

On the other hand, evidencing the Boltzmann factor $\displaystyle e^{-\beta
E}$, Eq. (55) becomes

$\displaystyle\displaystyle\rho=\int_{0}^{\infty}\frac{dp_{z}}{\pi}e^{-\beta\sqrt{M^{2}+p_{z}^{2}}}\left[\frac{\zeta}{\zeta
e^{-\beta\sqrt{M^{2}+p_{z}^{2}}}+1}-\frac{\zeta^{-1}}{\zeta^{-1}e^{-\beta\sqrt{M^{2}+p_{z}^{2}}}+1}\right]$
(67)

and, since $\displaystyle e^{-x}<1$ for $\displaystyle x>0$ the term in square
brackets can be expanded into power series, then

$\displaystyle\displaystyle\rho=\sum_{k=1}^{\infty}(-1)^{k+1}\int_{0}^{\infty}\frac{dp_{z}}{\pi}e^{-\beta
k\sqrt{M^{2}+p_{z}^{2}}}\left(\zeta^{k}-\zeta^{-k}\right).$ (68)

As the integrand of Eq. (68) is exponentially damped, it is only necessary to
take a few first values of $\displaystyle k$. Then, doing just $\displaystyle
k=1$, we get

$\displaystyle\displaystyle\rho=\frac{2}{\pi}\sinh{\beta\mu}\int_{0}^{\infty}dp_{z}e^{-\beta\sqrt{M^{2}+p_{z}^{2}}},$
(69)

which, through an expansion in temperature, can be summarized in two different
regimes,

$\displaystyle\displaystyle\rho=\frac{2}{\pi\beta}\sinh{\beta\mu}\quad\quad(\beta
M<<1)$ (70)

and

$\displaystyle\displaystyle\rho=\sqrt{\frac{2M}{\pi\beta}}e^{-\beta
M}\sinh{\beta\mu}\quad\quad(\beta M>>1).$ (71)

These results are similar to those obtained in [26]. The comparison between
$\displaystyle\beta M$ and the unity shows us how greater the factor
$\displaystyle k_{B}T$ is in terms of the rest energy $\displaystyle mc^{2}$.

Expanding Eq. (70) and Eq. (71) we get

$\displaystyle\displaystyle\mu=\frac{\pi}{2}\rho\quad\quad(\beta M<<1)$ (72)

and

$\displaystyle\displaystyle\mu(T)=M-\frac{1}{2}k_{B}T\ln{\left(\frac{k_{B}T}{M}\right)}+\frac{\alpha}{2}k_{B}T\quad\quad(\beta
M>>1)$ (73)

where $\displaystyle\alpha$ is a constant, which depends on the leptonic
density of the system.

At very high temperatures, $\displaystyle\beta\to 0$ or $\displaystyle\beta
M<<1$, as discussed $\displaystyle\rho=0$, therefore $\displaystyle\mu=0$. The
result obtained in Eq. (73) is very interesting and is similar to the one
obtained without external field by [7]. In addition, as Eq. (73) can be taken
in the low temperatures regime, i.e. $\displaystyle\beta M>>1$, we get a
behavior similar to those obtained in Eq. (65) obtained in Sommerfeld
formalism, but making $\displaystyle\mu_{0}=M$ and taking very low values of
$\displaystyle T$. When the temperature increases, both expressions become
very different and this is because to obtain Eq. (65) we assume many
approximations that do not happen in the calculation of Eq. (73).

Figure 2: Chemical potential function described by Eq. (74).

From this, the functional form of the lepton density chemical potential can be
summarized as

$\displaystyle\displaystyle\mu(T)=\begin{cases}M-\frac{1}{2}k_{B}T\ln{\frac{k_{B}T}{M}}+\frac{\alpha}{2}k_{B}T,\quad&\text{if}\quad
T<T_{\text{cut}}\\\ 0,\quad&\text{if}\quad T\geq T_{\text{cut}}\end{cases}$
(74)

whose maximum value occurs for

$\displaystyle\displaystyle T_{0}=\frac{M}{k_{B}}e^{\alpha-1}.$ (75)

This function is plotted in Figure 2.

In the next subsection, an interpretation of the chemical potential in a QED
scattering process is presented.

###  III.2 The interpretation of the chemical potential in a QED scattering
reaction

In ordinary thermodynamics, the chemical potential is defined as

$\displaystyle\displaystyle\mu=\left(\frac{\partial U}{\partial
N}\right)_{S,V}\approx U(N+1)-U(N),$ (76)

where $\displaystyle U$ is the internal energy of the system. This quantity is
the energy needed to add a particle to the system keeping entropy and volume
constant [23].

In a QED system surrounded by thermal and particle reservoirs, the particle
population can be described by the functions given in Figures 1(a) and 1(c),
i.e., there are two types of fermions, with positive and negative energy, as
shown in Figure 3(a). The existence of particles with negative energy can be
understood with the Fermi sea formalism [27]. In this way, the antiparticle is
understood as if there were “holes” in the distribution of negative-energy
fermions. The processes of annihilation and pair creation can be explained by
the hole-fermion interaction. Pair creation occurs when a negative energy
fermion absorbs an external photon, it jumps to an empty state leaving an
empty space where it was, that is, a hole with positive energy and charge, a
anti-fermion. On the other hand, the pair annihilation process happens when
the positive energy fermion decays into an empty space in the negative energy
spectrum emitting a photon with energy equal to the sum of the energies of the
fermion and the hole.

(a) The particle population of the system where the energy range of the
fermions with the highest positive and negative energies is
$\displaystyle\Delta E=2\mu_{0}$.

(b) The particle population of the system when the highest negative fermion
absorbs an external photon with energy $\displaystyle E_{\gamma}>2\mu_{0}$.

Figure 3: Schemes for energy spectra of electrons at $\displaystyle T=0$
temperature.

Figure 3(a) shows the energy spectra of the system when the distribution is
given by 1(a) and 1(b). In this scheme, we can see that in the energy range
$\displaystyle(0,\mu_{0})$ such as $\displaystyle(-\infty,-\mu_{0})$ are fully
filled while the interval $\displaystyle(-\mu_{0},0)$ is a forbidden zone.
Furthermore, the interpretation of Eq. (76) shows us that at zero temperature
$\displaystyle\mu_{0}$ is the energy required to add a fermion to the system.
It is important to notice that, to remove the higher energy particle it is
necessary to remove a fermion with energy $\displaystyle\mu_{0}$ or, in other
words, add an $\displaystyle E=-\mu_{0}$ to the system. Therefore, the latter
implies the addition of a negative energy fermion to the system.

In terms of the pair creation and annihilation processes, since the
antiparticle is a hole in the negative energy fermion distribution, it can
only exist if the excited negative energy particle jumps to a state where
$\displaystyle E>\mu_{0}$, where are only the free states. In this point of
view, it is necessary for the fermion to absorb an external photon with energy
greater than the separation interval between the states, as shown in Figure
3(b). In short, anti-fermions cannot be created if the received energy is not
greater than $\displaystyle 2\mu_{0}$.

The idea behind the use of the grand-canonical formalism is that the vacuum
can behave like a reservoir of particles. This approach lies in the fact that
infinite virtual processes happen in a vacuum, that is, quanta of particles
are created and annihilated all the time without violating the energy-time
uncertainly principle [21]. But when a virtual pair absorbs an external
photon, it becomes real, so this fact configures the vacuum as a reservoir.

Figure 4 shows a pictorial scheme of the QED system connected with both
reservoirs. Special attention is required to the particle reservoir. Note that
it is impossible to conceive an antiparticle, with energy $\displaystyle
E<\mu_{0}$, in a physical reaction, since there are no negative energy
fermions in the forbidden zone. In addition, it is not possible to create
another particle where $\displaystyle E<\mu_{0}$, since all states are filled.
Therefore, for any particle-antiparticle reaction, it is necessary that the
condition $\displaystyle E>\mu_{0}$ is satisfied.

Figure 4: Scheme of a connected system with two reservoirs, one for particles
and the other for a thermal bath. In the former, the energy spectrum of the
particles is represented by the energy levels placed on the left.

Furthermore, following the results given in Eq. (62) or Eq. (73), the chemical
potential of a QED system at zero temperature is just the mass of the fermion
that composes it. This fact can be easily verified in a practical manner. It
would be reasonable to think that the minimum energy needed to create a
particle is that which keeps the particle at rest.

On the other hand, in a system composed of fermions of different flavors, we
get from Eq. (76),

$\displaystyle\displaystyle\mu_{i}=\left(\frac{\partial U}{\partial
N_{i}}\right)_{S,V,N_{j}}\approx U(N_{i}+1)-U(N_{i}),$ (77)

where the indices $\displaystyle i$ and $\displaystyle j$ ranges cover all
flavors with $\displaystyle i\neq j$. Then, looking at the $\displaystyle
e^{+}e^{-}\to l^{+}l^{-}$ reaction, it is necessary to consider the electron
and lepton distribution functions. That is, there are two reservoirs, one for
each particle.

For bosons, the discussion is a little different, because at zero temperature
the energy necessary to add a bosonic particle to the system is that in which
the Bose-Einstein condensate are. Especially for photons, which are massless
and non-interacting particles, we can return to the black-body radiation
problem, where we have $\displaystyle\mu=0$ for all temperatures [22].

Finally, we can say that a QED reaction, at zero temperature, can only occur
if the total energy of the initial state is greater than $\displaystyle
2\mu_{0}$. When the temperature assumes very high values, i.e., greater than
$\displaystyle T_{0}$, given by Eq. (75), the interpretation of the chemical
potential no longer makes sense, since that $\displaystyle\mu$ can be defined
as zero. Therefore, all important quantities in the limit
$\displaystyle\beta\to 0$ do not depend on a comparison between $\displaystyle
E$ and $\displaystyle\mu$.

Up to now, we have seen how to describe a fermion under the effect of a
classical background magnetic field at finite temperature, what is chemical
potential, and what it represents in a reaction. In the next section, the QED
reactions are revised and it is shown which features become relevant when we
take these phenomena into account.

##  IV The scattering process

The QED Lagrangian for an arbitrary lepton with mass $\displaystyle m_{l}$ is
given by

$\displaystyle\displaystyle\mathcal{L}=\bar{\psi}_{l}\left(iD_{\nu}\gamma^{\nu}-m_{l}\right)\psi_{l}-\frac{1}{4}F_{\alpha\nu}F^{\alpha\nu},$
(78)

with $\displaystyle D_{\nu}=\partial_{\nu}-ieA_{\nu}$ being the covariant
gauge derivative. It is important to note that, in the presence of a
background field, the potential is written as

$\displaystyle\displaystyle A_{\nu}(x)=A_{\nu}^{(D)}(x)+A_{\nu}^{(B)}(x),$
(79)

where $\displaystyle A_{\nu}^{(D)}$ and $\displaystyle A_{\nu}^{(B)}$ are the
dynamical and background fields, respectively. In other words, while the first
quantity is the free field of a propagating photon, the second is the
potential vector of classical background magnetic field, as in Eq. (6).

Note from Eq. (79) that $\displaystyle
F_{\alpha\nu}=F_{\alpha\nu}^{(D)}+F_{\alpha\nu}^{(B)}$, with

$\displaystyle\displaystyle
F_{\alpha\nu}^{(D)}=\partial_{\alpha}A_{\nu}^{(D)}-\partial_{\nu}A_{\alpha}^{(D)}\quad\quad\mathrm{and}\quad\quad\quad
F_{\alpha\nu}^{(B)}=\partial_{\alpha}A_{\nu}^{(B)}-\partial_{\nu}A_{\alpha}^{(B)},$
(80)

where due to the structure of the potential field, $\displaystyle
F_{\alpha\nu}^{(B)}$ is constant. Then the contribution of this term is a
constant of motion and does not affect the field equations [28]. Therefore,
the Lagrangian given in Eq. (78) becomes

$\displaystyle\displaystyle\mathcal{L}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\bar{\psi}_{l}\left[i(\partial_{\nu}-ieA_{\nu}^{(B)})\gamma^{\nu}-m_{l}\right]\psi_{l}+e\bar{\psi}_{l}A_{\nu}^{(D)}\gamma^{\nu}\psi_{l}-\frac{1}{4}F_{\alpha\nu}^{(D)}F^{\alpha\nu}_{(D)}$
(81) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\bar{\psi}_{l}\left(iD_{\nu}\gamma^{\nu}-m_{l}\right)\psi_{l}-\frac{1}{4}F_{\alpha\nu}^{(D)}F^{\alpha\nu}_{(D)},$

where $\displaystyle D_{\nu}=-i\Pi_{\nu}-ieA_{\nu}^{(D)}$ and
$\displaystyle\Pi_{\nu}$ is the same as given by Eq. (3).

Therefore, from Eq. (3), Eq. (81) can be understood as the theory describing
the usual QED, but the dynamics of a Dirac field is now formulated for a
lepton with electric charge $\displaystyle-e$ and mass $\displaystyle m_{l}$
in the presence of a constant magnetic field. Thus, the interaction with the
gauge photon can be described by a minimal coupling. In other words, the
Feynman rules can be applied to the present case.

From Eq. (81) the interaction Lagrangian is given as

$\displaystyle\displaystyle\mathcal{L}_{\text{int}}(x)=e\bar{\psi}_{l}(x)\gamma^{\nu}\psi_{l}(x)A_{\nu}^{(D)}(x).$
(82)

Using this Lagrangian the term of second order of the scattering matrix is
written as

$\displaystyle\displaystyle\widehat{S}^{(2)}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{(-i)^{2}}{2}\int
d^{4}x_{1}d^{4}x_{2}:\widehat{\mathcal{L}}(x_{1})\widehat{\mathcal{L}}(x_{2}):,$
(83) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{(-i)^{2}}{2}\int
d^{4}x_{1}d^{4}x_{2}:\left[\mathcal{L}(x_{1})-\widetilde{\mathcal{L}}(x_{1})\right]\left[\mathcal{L}(x_{2})-\widetilde{\mathcal{L}}(x_{2})\right]:,$

where $\displaystyle:(\quad):$ is the ordering operator. It is important to
note that only non-tilde operators are related to measurable physical
quantities, therefore, after performing the Bogoliubov transformations to make
this theory thermal, one can cut the tilde creation and annihilation operators
in Eq. (83).

For the scattering process $\displaystyle e^{+}e^{-}\to l^{+}l^{-}$ in the
presence of a background magnetic field within the TFD formalism, the initial
and final states are described as

$\displaystyle\displaystyle\ket{i(\beta,\mu)}=\sqrt{2E_{n}}\sqrt{2E_{j}}a^{\dagger}_{s}(\beta,n,\vec{p}_{\cancel{y}})b^{\dagger}_{\lambda}(\beta,j,\vec{k}_{\cancel{y}})\ket{0(\beta,\mu)},$
$\displaystyle\displaystyle\ket{f(\beta,\mu)}=\sqrt{2E_{n^{\prime}}}\sqrt{2E_{j^{\prime}}}a^{\dagger}_{s^{\prime}}(\beta,n^{\prime},\vec{p^{\prime}}_{\cancel{y}})b^{\dagger}_{\lambda^{\prime}}(\beta,j^{\prime},\vec{k^{\prime}}_{\cancel{y}})\ket{0(\beta,\mu)},$
(84)

where the pre-factors $\displaystyle\sqrt{2E}$ are present for normalization
purposes [29].

Therefore, the probability amplitude of this scattering process to occur is
given by

$\displaystyle\displaystyle\mathcal{M}_{fi}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\bra{f(\beta,\mu)}\widehat{S}^{(2)}\ket{i(\beta,\mu)}$
(85) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\frac{e^{2}}{2}\int
d^{4}x_{1}d^{4}x_{2}\bra{f}:\left\\{\left[\bar{\psi}_{e}(x_{1})\gamma^{\alpha}\psi_{e}(x_{1})A_{\alpha}^{(D)}(x_{1})\right]^{\widetilde{}}-\bar{\psi}_{e}(x_{1})\gamma^{\alpha}\psi_{e}(x_{1})A_{\alpha}^{(D)}(x_{1})\right\\}$
$\displaystyle\displaystyle\times$
$\displaystyle\displaystyle\left\\{\left[\bar{\psi}_{l}(x_{2})\gamma^{\nu}\psi_{l}(x_{2})A_{\nu}^{(D)}(x_{2})\right]^{\widetilde{}}-\bar{\psi}_{l}(x_{2})\gamma^{\nu}\psi_{l}(x_{2})A_{\nu}^{(D)}(x_{2})\right\\}:\ket{i}.$

This scattering process at tree level is described by the Feynman diagram
given in Figure 5. The thermal propagator of the photon with momentum
$\displaystyle\kappa$ is
$\displaystyle\Delta_{\alpha\nu}(\kappa)=\Delta^{(0)}_{\alpha\nu}(\kappa)+\Delta^{(\beta)}_{\alpha\nu}(\kappa)$,
where

$\displaystyle\displaystyle\Delta^{(0)}_{\alpha\nu}(\kappa)=\frac{\eta_{\alpha\nu}}{\kappa^{2}}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}$ (86)

is the temperature-independent part, and

$\displaystyle\displaystyle\Delta^{(\beta)}_{\alpha\nu}(\kappa)=-\frac{2\pi
i\delta(\kappa^{2})}{e^{\beta q_{0}}-1}\begin{pmatrix}1&e^{\beta q_{0}/2}\\\
e^{\beta q_{0}/2}&1\end{pmatrix}\eta_{\alpha\nu}$ (87)

is the temperature-dependent part of the propagator [10]. The
$\displaystyle\eta_{\alpha\nu}$ is the Minkowski metric with
$\displaystyle\det{\eta}=-2$. Observe that, in the expression for the thermal
propagator given in Eq. (87) the chemical potential of the photon does not
appear and this fact occurs because $\displaystyle\mu_{\text{photon}}=0$ as
discussed earlier.

Figure 5: Tree-level Feynman diagram of $\displaystyle e^{+}e^{-}\to
l^{+}l^{-}$ scattering process.

At this point, from Eq. (85), it is necessary to make a distinction between
electronic and leptonic fields, since their masses are different. Looking at
Eq. (15), $\displaystyle\psi_{e}$ and $\displaystyle\psi_{l}$ are obtained by
putting $\displaystyle M=m_{e}$ and $\displaystyle M=m_{l}$, respectively.
Then

$\displaystyle\displaystyle\psi_{e}(x^{\nu})=\int\frac{dp_{x}dp_{z}}{(2\pi)^{2}}\sum_{s,n}\sqrt{\frac{E_{n}+m_{e}}{2E_{n}}}\left[a_{s}(n,\vec{p}_{\cancel{y}})u_{s}(y,n,\vec{p}_{\cancel{y}})e^{-ip_{\nu}x^{\nu}_{\cancel{y}}}+b_{s}^{\dagger}(n,\vec{p}_{\cancel{y}})v_{s}(y,n,\vec{p}_{\cancel{y}})e^{ip_{\nu}x^{\nu}_{\cancel{y}}}\right]$
(88)

for electrons, and

$\displaystyle\displaystyle\psi_{l}(x^{\nu})=\int\frac{dp_{x}dp_{z}}{(2\pi)^{2}}\sum_{s,n}\sqrt{\frac{E_{n}+m_{l}}{2E_{n}}}\left[a_{s}(n,\vec{p}_{\cancel{y}})u_{s}(y,n,\vec{p}_{\cancel{y}})e^{-ip_{\nu}x^{\nu}_{\cancel{y}}}+b_{s}^{\dagger}(n,\vec{p}_{\cancel{y}})v_{s}(y,n,\vec{p}_{\cancel{y}})e^{ip_{\nu}x^{\nu}_{\cancel{y}}}\right]$
(89)

for leptons.

Note from Eqs. (86) and (87) that both equations do not depend on the Landau
index $\displaystyle n$, and this is because the photon has no charge and
therefore does not couple to the external magnetic field [18, 3].

###  IV.1 Limit of very strong magnetic fields

Here a special case is addressed, a very strong magnetic field limit. This
occurs when the amplitude $\displaystyle B$ is very large compared to
$\displaystyle E^{2}-m_{l}^{2}$. In this regime, the only physical condition
of interest is one in which the Landau index is zero ($\displaystyle n=0$),
where the movement of the electron and lepton is restricted to a one-
dimensional $\displaystyle z$-axis.

In this situation, the completeness relations given by Eqs. (18) and (19) take
the form

$\displaystyle\displaystyle\sum_{s}u_{s}(y,\vec{p}_{\cancel{y}})\bar{u}_{s}(y^{\prime},\vec{p}_{\cancel{y}})=\frac{1}{2(E_{0}+m_{l})}\left[m_{l}(1-\Sigma_{z})+\not{p}_{||}+\bar{\not{p}}_{||}\gamma^{5}\right]I_{0}(\xi^{+})I_{0}(\xi^{+\prime}),$
(90)

for the positive energy part and

$\displaystyle\displaystyle\sum_{s}v_{s}(y,\vec{p}_{\cancel{y}})\bar{v}_{s}(y^{\prime},\vec{p}_{\cancel{y}})=-\frac{1}{2(E_{0}+m_{l})}\left[m_{l}(1-\Sigma_{z})+\not{p}_{||}+\bar{\not{p}}_{||}\gamma^{5}\right]I_{0}(\xi^{-})I_{0}(\xi^{-\prime}),$
(91)

for the negative energy part. Here have been used
$\displaystyle\Sigma_{z}=i\gamma^{1}\gamma^{2}$,
$\displaystyle\not{p}_{||}=\gamma^{0}p_{0}-\gamma^{3}p_{3}$,
$\displaystyle\bar{\not{p}_{||}}=\gamma^{0}p_{3}-\gamma^{3}p_{0}$ and
$\displaystyle\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$.

Note that $\displaystyle\mathcal{M}_{fi}$ is an average over thermal vacuum
state. Then it is necessary to make the connection between finite and zero-
temperature operators. The transition amplitude can be written using the
Bogoliubov transformations Eqs. (25) and (26) and the tilde conjugation rules
Eq. (22). Thus, Eq. (85) becomes

$\displaystyle\displaystyle\mathcal{M}_{fi}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-e^{2}F(\beta)\left(E_{0}+m_{e}\right)\left(E_{0}+m_{l}\right)\int
d^{4}x_{1}d^{4}x_{2}\left[\bar{v}_{\lambda}(y_{1},\vec{k}_{\cancel{y}})\gamma^{\alpha}u_{s}(y_{1},\vec{p}_{\cancel{y}})e^{-i(p+k)\cdot
x_{1\cancel{y}}}\right.$ (92) $\displaystyle\displaystyle\times$
$\displaystyle\displaystyle\left.\bar{u}_{s^{\prime}}(y_{2},\vec{p^{\prime}}_{\cancel{y}})\gamma^{\nu}v_{\lambda^{\prime}}(y_{2},\vec{k^{\prime}}_{\cancel{y}})e^{i(p^{\prime}+k^{\prime})\cdot
x_{2\cancel{y}}}\Delta_{\alpha\nu}(x_{1}-x_{2})\right],$

where $\displaystyle F(\beta)$ is a thermal function that associates the
Fermi-Dirac distribution of all particles in the following functional form

$\displaystyle\displaystyle
F(\beta)=\left[f(E_{e^{-}},\mu_{e})+f(E_{e^{+}},-\mu_{e})-1\right]\left[f(E_{l^{-}},\mu_{l})+f(E_{l^{+}},-\mu_{l})-1\right]$
(93)

whose analytical appearance is very dependent on the choice of the reference
frame. In the center of mass we have

$\displaystyle\displaystyle p=\left(E,\vec{p}\right),\quad\quad
k=\left(E,-\vec{p}\right),\quad\quad
p^{\prime}=\left(E,\vec{p^{\prime}}\right),\quad\quad
k^{\prime}=\left(E,-\vec{p^{\prime}}\right),$ (94)

such that $\displaystyle m_{e}^{2}=E^{2}-|\vec{p}|^{2}$ and $\displaystyle
m_{l}^{2}=E^{2}-|\vec{p^{\prime}}|^{2}$. Thus, the thermal function Eq. (93)
becomes

$\displaystyle\displaystyle F(\beta)=\frac{\sinh^{2}{\beta
E}}{\left(\cosh{\beta\mu_{e}}+\cosh{\beta
E}\right)\left(\cosh{\beta\mu_{l}}+\cosh{\beta E}\right)}.$ (95)

The photon propagator $\displaystyle\Delta_{\alpha\nu}(x_{1}-x_{2})$ in the
coordinate-space is given as

$\displaystyle\displaystyle\Delta_{\alpha\nu}(x_{1}-x_{2})=i\int\frac{d^{4}\kappa}{(2\pi)^{4}}e^{-i\kappa\cdot(x_{2}-x_{1})}\Delta_{\alpha\nu}(\kappa).$
(96)

Performing the integration on $\displaystyle x_{1}$ and $\displaystyle x_{2}$
in Eq. (92), we get

$\displaystyle\displaystyle\mathcal{M}_{fi}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-
ie^{2}F(\beta)\left(E_{0}+m_{e}\right)\left(E_{0}+m_{l}\right)\int\frac{d^{4}\kappa}{2\pi}\int
dy_{1}dy_{2}\left[e^{i\kappa_{y}y_{1}}\bar{v}_{\lambda}(y_{1},\vec{k}_{\cancel{y}})\gamma^{\alpha}u_{s}(y_{1},\vec{p}_{\cancel{y}})\right.$
(97) $\displaystyle\displaystyle\times$
$\displaystyle\displaystyle\left.\delta^{4}_{\cancel{y}}(p+k-\kappa)e^{-i\kappa_{y}y_{2}}\bar{u}_{s^{\prime}}(y_{2},\vec{p^{\prime}}_{\cancel{y}})\gamma^{\nu}v_{\lambda^{\prime}}(y_{2},\vec{k^{\prime}}_{\cancel{y}})\delta^{4}_{\cancel{y}}(p^{\prime}+k^{\prime}-\kappa)\Delta_{\alpha\nu}(\kappa)\right],$

where the definition of the Dirac delta function

$\displaystyle\displaystyle\int\frac{d^{4}x}{(2\pi)^{4}}e^{\pm(a+b)\cdot
x_{\cancel{y}}}=\int\frac{dy}{2\pi}\delta^{4}_{\cancel{y}}(a+b)$ (98)

has been used.

Looking at the integration in $\displaystyle\kappa$, the Eq. (97) becomes

$\displaystyle\displaystyle\mathcal{M}_{fi}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-
ie^{2}F(\beta)\left(E_{0}+m_{e}\right)\left(E_{0}+m_{l}\right)\int
dy_{1}dy_{2}\left[\delta(y_{1}-y_{2})\bar{v}_{\lambda}(y_{1},\vec{k}_{\cancel{y}})\gamma^{\alpha}u_{s}(y_{1},\vec{p}_{\cancel{y}})\right.$
(99) $\displaystyle\displaystyle\times$
$\displaystyle\displaystyle\left.\bar{u}_{s^{\prime}}(y_{2},\vec{p^{\prime}}_{\cancel{y}})\gamma^{\nu}v_{\lambda^{\prime}}(y_{2},\vec{k^{\prime}}_{\cancel{y}})\delta^{4}_{\cancel{y}}(p^{\prime}+k^{\prime}-\kappa)\Delta_{\alpha\nu}(p+k)\right].$

Since the delta function imposes conservation of the total momentum, the last
equation can be written as

$\displaystyle\displaystyle\mathcal{M}_{fi}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-
ie^{2}F(\beta)\left(E_{0}+m_{e}\right)\left(E_{0}+m_{l}\right)$ (100)
$\displaystyle\displaystyle\times$ $\displaystyle\displaystyle\int
dy\left[\bar{v}_{\lambda}(y,\vec{k}_{\cancel{y}})\gamma^{\alpha}u_{s}(y,\vec{p}_{\cancel{y}})\Delta_{\alpha\nu}(p+k)\bar{u}_{s^{\prime}}(y,\vec{p^{\prime}}_{\cancel{y}})\gamma^{\nu}v_{\lambda^{\prime}}(y,\vec{k^{\prime}}_{\cancel{y}})\right].$

For an initial unpolarized beam, the final and initial spin states are
averaged. Then

$\displaystyle\displaystyle\langle|\mathcal{M}_{fi}|^{2}\rangle=\frac{1}{4}\sum_{\begin{subarray}{c}s,s^{\prime}\\\
\lambda,\lambda^{\prime}\end{subarray}}|\mathcal{M}_{fi}|^{2}.$ (101)

Using Eqs. (90) and (91) we obtain

$\displaystyle\displaystyle\langle|\mathcal{M}_{fi}|^{2}\rangle=2e^{4}H(\beta)\left(\frac{eB}{2\pi}\right)\left[m_{e}^{2}m_{l}^{2}+\left(\text{E}^{2}+|p|^{2}\right)\left(\text{E}^{2}+|p^{\prime}|^{2}\cos^{2}\theta\right)\right],$
(102)

with

$\displaystyle\displaystyle
H(\beta)=F^{2}(\beta)\left[\frac{1}{s^{2}}+\frac{(2\pi)^{2}\delta^{2}(s)}{(e^{2\beta
E}-1)^{2}}\right].$ (103)

Here has been used

$\displaystyle\displaystyle\int
dy\left[I_{0}^{2}(\xi^{+})I_{0}^{2}(\xi^{-})\right]=\sqrt{\frac{eB}{2\pi}}.$
(104)

Remember that we are dealing with the case where the background field is very
strong, so this approach is considered $\displaystyle p_{x}=0$, which implies
$\displaystyle p=(E,0,0,p_{z})$. In addition, the angle that appears in Eq.
(102) is defined between $\displaystyle p$ and $\displaystyle p^{\prime}$, and
since the motion is one-dimensional, it can assume only two values:
$\displaystyle\theta=0$ or $\displaystyle\theta=\pi$ [30].

Finally, putting Eq. (102) into Eq. (50) the cross-section for the scattering
process reads

$\displaystyle\displaystyle\sigma_{0}(\beta)=\frac{\langle|\mathcal{M}_{fi}|^{2}\rangle}{8p^{\prime}_{z}s\sqrt{s-4m_{e}^{2}}}=\frac{e^{4}(\pi
eB)}{2(2\pi)^{2}p^{\prime}_{z}s\sqrt{s-4m_{e}^{2}}}H(\beta)\left[m_{e}^{2}m_{l}^{2}+\left(\text{E}^{2}+|p|^{2}\right)\left(\text{E}^{2}+|p^{\prime}|^{2}\right)\right].$
(105)

When temperatures are low the function $\displaystyle F(\beta)$ becomes a step
function around $\displaystyle E=\mu_{0}$, that is,

$\displaystyle\displaystyle\lim_{\beta\to\infty}F(\beta)=\begin{cases}0\quad\text{if}\quad
E<\mu_{0}\\\ 1\quad\text{if}\quad E>\mu_{0}\end{cases}$ (106)

which leads to

$\displaystyle\displaystyle\lim_{\beta\to\infty}H(\beta)=\frac{1}{s^{2}}.$
(107)

It is important to note that the cross-section vanishes for $\displaystyle
E<\mu_{0}$ and becomes

$\displaystyle\displaystyle\sigma_{0}(T=0)=\frac{e^{4}(\pi
eB)}{2(2\pi)^{2}p^{\prime}_{z}s^{3}\sqrt{s-4m_{e}^{2}}}\left[m_{e}^{2}m_{l}^{2}+\left(\text{E}^{2}+|p|^{2}\right)\left(\text{E}^{2}+|p^{\prime}|^{2}\right)\right],$
(108)

for $\displaystyle E>\mu_{0}$. Or, more elegantly, as

$\displaystyle\displaystyle\sigma_{0}(T=0)=\begin{cases}\frac{(\pi
eB)\alpha^{2}}{s^{3}}\frac{\left[s^{2}+8m_{e}^{2}m_{l}^{2}-s\left(m_{e}^{2}+m_{l}^{2}\right)\right]}{\sqrt{(s-4m_{e}^{2})(s-4m_{l}^{2})}}\quad\quad&\text{if}\quad\quad
E>\mu_{0}\\\ 0\quad\quad&\text{if}\quad\quad E<\mu_{0},\end{cases}$ (109)

where $\displaystyle\alpha=e^{2}/4\pi$ is the fine-structure constant. This
result is the same obtained by [31], without the EDM term (dipole term) that
comes from the Lorentz violation, when the high energy approximation
($\displaystyle m_{e}=m_{l}=0$) is taken.

The usual QED cross-section for the $\displaystyle e^{+}e^{-}\to l^{+}l^{-}$
scattering process is $\displaystyle\sigma_{\text{QED}}=4\pi\alpha^{2}/3s$
[29]. In Figure 6 is shown how $\displaystyle\sigma_{0}(T=0)$ behaves in terms
of $\displaystyle\sigma_{\text{QED}}$ for $\displaystyle E>\mu_{0}$.
Initially, it can be seen that there is an asymptotic limit at $\displaystyle
s=4m_{l}^{2}$ and the cross-section obtained here exhibits an energy
dependence $\displaystyle s^{-2}$ while the usual QED theory does for
$\displaystyle s^{-1}$. In addition, due to the type of dependence of the
$\displaystyle eB$ factor, stronger magnetic fields produce a linear increase
in cross-section value.

Figure 6: The ratio cross-section at zero temperature and
$\displaystyle\sigma_{\text{QED}}$ in terms of the centre of mass energy
($\displaystyle\sqrt{s}$) for the magnetic field amplitudes: $\displaystyle
B=2m_{l}^{2}/e$ (black solid line); $\displaystyle B=3m_{l}^{2}/e$ (red dashed
line) and $\displaystyle B=5m_{l}^{2}/e$ (blue dotted line).

When the temperatures are very high, we obtain from Eqs. (95) and (103) the
following approximation

$\displaystyle\displaystyle H(\beta)=\frac{(\beta
E)^{2}}{4s^{2}}+\frac{(2\pi)^{2}\delta^{2}(s)}{8}\left\\{\frac{1}{2}-\beta
E-\left[\left(\frac{\mu_{l}}{E}\right)^{2}+\left(\frac{\mu_{e}}{E}\right)^{2}-6\right]\frac{(\beta
E)^{2}}{8}\right\\}$ (110)

for any $\displaystyle E$ value. Then, the differential cross-section at this
regime becomes

$\displaystyle\displaystyle\lim_{T\to\infty}\sigma_{0}(\beta)=\frac{e^{4}(\pi
eB)}{2(2\pi)^{2}p^{\prime}_{z}s\sqrt{s-4m_{e}^{2}}}\frac{(2\pi)^{2}\delta^{2}(s)}{16}\left[m_{e}^{2}m_{l}^{2}+\left(\text{E}^{2}+|p|^{2}\right)\left(\text{E}^{2}+|p^{\prime}|^{2}\right)\right],$
(111)

which, as expected, does not depend on the chemical potential. The square
Dirac delta function that appears in the solutions can be treated and assumes
a real value considering its regularized forms [32].

Returning to Eq. (95), taking $\displaystyle\mu_{e}=\mu_{l}=0$ limit, we get

$\displaystyle\displaystyle F(\beta)=\frac{\sinh^{2}{\beta
E}}{\left(1+\cosh{\beta E}\right)^{2}}=\tanh^{2}{\frac{\beta E}{2}},$ (112)

and, in this way, the result presented in Eq. (105) using Eq. (112) is very
similar to those obtained by [11].

It is important to note that the weak magnetic field limit cannot be made by
doing $\displaystyle B\to 0$, and this fact comes from the strong field regime
($\displaystyle n=0$). To obtain the threshold of weaker background fields is
necessary to follow the steps present in Eq. (47). In other words, to make the
limit of $\displaystyle B\to 0$ it is necessary to take into account the
continuous sum over all $\displaystyle n$-states, which is completely contrary
to what we used initially, the very high field strength.

(a) The thermal function $\displaystyle F(\beta)$ in terms of inverse
temperature for $\displaystyle\mu\neq 0$ case (black solid line) given by Eq.
(95) and for $\displaystyle\mu=0$ case (red dashed line) given by Eq. (112) in
$\displaystyle E=2\mu$ environment.

(b) The thermal function $\displaystyle F(\beta)$ in terms of inverse
temperature for $\displaystyle\mu\neq 0$ case (black solid line) given by Eq.
(95) and for $\displaystyle\mu=0$ case (red dashed line) given by Eq. (112) in
$\displaystyle E=\mu/2$ environment.

Figure 7: The thermal functions in terms of the inverse temperature.

Regarding the thermal functions Eqs. (95) and (112), Figure 7 shows a
comparison between them. Note that, in Figure 7(a), a system with energy
$\displaystyle E>\mu$ shows similar behavior for both functions, although the
presence of the chemical potential changes some regimes by an amount. In
addition, the limits for high or lower temperatures are the same.

Now, looking at Figure 7(b) it can be seen that, for energies less than
$\displaystyle\mu$, the reaction cannot occur except for some short regime.
But, as discussed, for very high temperatures, or low
$\displaystyle\beta$-values, the interpretation of the chemical potential as
the energy required to add a particle to the system does not even make sense.

It is important to note that the masses of other leptons are larger than those
of electrons; hence, the effect of an external field on an electron is more
pronounced than on other leptons. Consequently, the field magnitudes that need
to be taken into account are those that can be felt by all leptons.
Furthermore, these magnitudes are so intense that the $\displaystyle n=0$
approximation can be applied. In simpler terms, when referring to a very
strong field, we mean magnitudes so large that they affect all leptons,
constraining them to motion only in the $\displaystyle z-$direction. In this
regime, the energy and chemical potential of all particles do not depend on
$\displaystyle B$, but only the spinors do. It is due to this dependence that
the final cross-section is directly influenced by the external field.

Finally, the thermal function Eq. (95) is strongly dependent on the lepton
flavor, since $\displaystyle\mu$ for $\displaystyle T=0$ lies around the
particle mass. For reactions where the final products are muon or tau
fermions, one can take $\displaystyle\mu_{e}=0$ since these masses are many
times greater than that of the electron. For the Bhabha scattering
$\displaystyle e^{+}e^{-}\to e^{+}e^{-}$, however, this assumption needs a
little bit more attention, since that $\displaystyle m_{l}=m_{e}$ and,
further, it is necessary to calculate another additional diagram in addition
to Figure 5, which will not be done in this paper.

##  V Conclusions

An electron-positron pair decaying into lepton-antileptons, subject to thermal
effects and a strong background classical magnetic field, has been presented.
The effect due to an external field is inserted into the problem through a
redefinition of the fermionic field operator that includes a dependence on
$\displaystyle y-$coordinate and Landau level $\displaystyle n$. It is because
in the quantization process, instead of writing the wave function as a well-
localized combination of plane waves with defined momenta, the $\displaystyle
y-$component is present in the spinor expression, leading to a $\displaystyle
n-$dependence of the energy relation. Furthermore, the energy function comes
from a new dispersion relation and the cross-section is calculated, where both
quantities involve these levels. These modifications are discussed. Since the
order of the external field in cosmological situations is very strong, the
assumption $\displaystyle n_{\text{max}}=0$ is valid and this guarantees our
choice of approach.

Temperature and chemical potential are introduced into the theory using the
TFD formalism. A functional relation between cross-section and temperature, as
well as for the chemical potential, is obtained. Although most published
papers do not consider the quantity $\displaystyle\mu$, we can see that it
makes a considerable difference in the final result. More explicitly, the
chemical potential at zero temperature determines when the reaction can or
cannot occur. Under these conditions, the reaction is analyzed for the very
strong magnetic field limit. Limits and estimates for temperature and chemical
potential effects are discussed. A temperature-dependent chemical potential
has been obtained. It is important to note that the cross-section at finite
temperature is not directly measurable in any many-body process. Such a
quantity, in this context, is an important input to treat a gas at low density
through a kinetic theory approach such as a Boltzmann equation. Then the
finite temperature cross-sections will form part of the collision term that
would lead to a distribution function by a solution of the Boltzmann equation.
Furthermore, the results obtained here are important for a better
understanding of the early Universe at finite temperature and density.

## Acknowledgments

This work by A. F. S. is partially supported by National Council for
Scientific and Technological Development - CNPq project No. 313400/2020-2. D.
S. C. thanks CAPES for financial support.

## Data Availability Statement

No Data associated in the manuscript.

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