# QED Fermions in a noisy magnetic field background

Jorge David Castaño-Yepes<EMAIL_ADDRESS>Facultad de Física, Pontificia
Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago, Chile  Marcelo
Loewe<EMAIL_ADDRESS>Facultad de Física, Pontificia Universidad Católica
de Chile, Vicuña Mackenna 4860, Santiago, Chile  Centre for Theoretical and
Mathematical Physics, and Department of Physics, University of Cape Town,
Rondebosch 7700, South Africa  Centro Científico Tecnológico de Valparaíso
CCTVAL, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso,
Chile  Facultad de Ingeniería, Arquitectura y Diseño, Universidad San
Sebastián, Santiago, Chile  Enrique Muñoz<EMAIL_ADDRESS>Facultad de
Física, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860,
Santiago, Chile  Center for Nanotechnology and Advanced Materials CIEN-UC,
Avenida Vicuña Mackenna 4860, Santiago, Chile  Juan Cristóbal Rojas
<EMAIL_ADDRESS>Departamento de Física, Universidad Católica del Norte, Angamos
610, Antofagasta, Chile  Renato Zamora<EMAIL_ADDRESS>Centro de
Investigación y Desarrollo en Ciencias Aeroespaciales (CIDCA), Fuerza Aérea de
Chile, Casilla 8020744, Santiago, Chile.  Instituto de Ciencias Básicas,
Universidad Diego Portales, Casilla 298-V, Santiago, Chile.

###### Abstract

We consider the effects of a noisy magnetic field background over the fermion
propagator in QED, as an approximation to the spatial inhomogeneities that
would naturally arise in certain physical scenarios, such as heavy-ion
collisions or the quark-gluon plasma in the early stages of the evolution of
the Universe. We considered a classical, finite and uniform average magnetic
field background $\langle\mathbf{B}(\mathbf{x})\rangle=\mathbf{B}$, subject to
white-noise spatial fluctuations with auto-correlation of magnitude
$\Delta_{B}$. By means of the Schwinger representation of the propagator in
the average magnetic field as a reference system, we used the replica
formalism to study the effects of the magnetic noise in the form of
renormalized quasi-particle parameters, leading to an effective charge and an
effective refraction index, that depend not only on the energy scale, as
usual, but also on the magnitude of the noise $\Delta_{B}$ and the average
field $\mathbf{B}$.

##  I Introduction

High-energy physics under the presence of strong magnetic fields is an
important subject of research in many scenarios, such as heavy-ion collisions
Alam  _et al._ (2021); Ayala  _et al._ (2022); Inghirami, Gabriele  _et al._
(2020); Ayala  _et al._ (2020a, 2017), the quark-gluon plasma Busza  _et al._
(2018); Hattori and Satow (2016, 2018); Buballa (2005) and the early-universe
evolution Inghirami, Gabriele  _et al._ (2020); Blaschke  _et al._ (2020). In
such systems, rather strong magnetic fields can emerge in comparatively small
regions of space and, moreover, strong spatial anisotropies and fluctuations
can develop in the magnitude of such fields Inghirami, Gabriele  _et al._
(2020); Alam  _et al._ (2021).

Remarkably, magnetic fields can influence the physical properties of both
charged as well as neutral particles, the later due to the quantum mechanical
fluctuations of the vacuum, that lead to the creation of virtual charged
fermion-antifermion pairs. In the context of high-energy physics, the effect
of a constant and “classical” magnetic field background has been studied since
the seminal work of Schwinger Schwinger (1951), followed with extensive
discussions in the literature in the context of semi-classical effective
Lagrangians Dittrich and Reuter (1985); Dittrich and Gies (2000). More
recently, the effect of magnetic fields on nucleon parameters have been
discussed in the context of QCD Dominguez  _et al._ (2020). In addition,
several studies have been reported concerning the effects of a classical,
static and uniform background magnetic field on the charged vacuum
fluctuations leading to the gluon polarization tensor Hattori and Itakura
(2013); Hattori and Satow (2016); Ayala  _et al._ (2020b); Hattori and Satow
(2018), in particular the role of the field in the breaking of the Lorentz
invariance, thus predicting the emergence of the vacuum birefringence
phenomena Hattori and Itakura (2013); Ayala  _et al._ (2020b). On the other
hand, vacuum fluctuations also affect the propagation of fermions in such
magnetized background Ayala, Alejandro  _et al._ (2021), as expressed by the
self-energy, that leads to the definition of a magnetic mass and, according to
recent studies in QED Ayala  _et al._ (2021), to an spectral width involving
the contribution of all the magnetic Landau levels. Moreover, non-perturbative
theoretical approaches Miransky and Shovkovy (2015) reveal the magnetic
catalysis effect, where the presence of a strong uniform magnetic field leads
to the emergence of effective masses for the fermion species, regardless of
their bare mass.

Interestingly, in most of these studies, the background magnetic field is
always idealized as static and uniform, and hence the presence of spatial
anisotropies or fluctuations in its magnitude are disregarded in the state of
the art of such calculations. A non-uniform but deterministic background
magnetic field has been studied in QED by means of a path-integral formulation
Gies and Roessler (2011). On the other hand, statistical fluctuations near a
zero average magnetic field have been studied in the context of QED2+1 Zhao
_et al._ (2017); Gusynin  _et al._ (2001), as it arises as an effective
continuum theory for certain low-dimensional Dirac materials such as Graphene
Miransky and Shovkovy (2015). In the later, however, the average background
field is assumed to be zero, and hence the reference system is characterized
by a free fermion propagator rather than by a Schwinger propagator as we
consider in the present study. Since spatial fluctuations with respect to a
finite background magnetic field may indeed exist in the different
aforementioned physical scenarios Inghirami, Gabriele  _et al._ (2020), in the
present work we shall study their effect over the renormalization of the
fermion propagator itself in QED. As we shall discuss in this article, a
perturbative treatment of such fluctuations in the framework of the replica
method Mèzard and Parisi (1991); Kardar  _et al._ (1986) allows us to show
that their effect can be captured in terms of a renormalization of the charge
$e\rightarrow z_{3}e$ and an effective refraction index $v^{\prime}/c=z^{-1}$.
Moreover, we show that $z_{3}$ and $z$ depend not only on the energy scale, as
usual, but also on the magnitude of the average magnetic field $|\mathbf{B}|$,
as well as on the strength of its spatial fluctuations, that we define as
$\Delta_{B}$.

##  II The model

We shall consider a physical scenario where a classical and static magnetic
field background, possessing random spatial fluctuations, modifies the quantum
dynamics of a system of fermions. For this purpose, we shall assume the
standard QED theory involving fermionic fields $\psi(x)$, as well as gauge
fields $A^{\mu}(x)$. In the later, we shall distinguish three physically
different contributions

$\displaystyle A^{\mu}(x)\rightarrow A^{\mu}(x)+A^{\mu}_{\text{BG}}(x)+\delta
A^{\mu}_{\text{BG}}(\mathbf{x}).$ (1)

Here, $A^{\mu}(x)$ represents the dynamical photonic quantum field, while BG
stands for “background”, representing the presence of a classical external
field imposed by the experimental conditions. Moreover, for this BG
contribution, we consider the effect of static (quenched) white noise spatial
fluctuations $\delta A^{\mu}_{\text{BG}}(\mathbf{x})$ with respect to the mean
value $A_{\text{BG}}^{\mu}(x)$, satisfying the statistical properties

$\displaystyle\langle\delta A^{j}_{\text{BG}}(\mathbf{x})\delta
A^{k}_{\text{BG}}(\mathbf{x}^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\Delta_{B}\delta_{j,k}\delta^{3}(\mathbf{x}-\mathbf{x}^{\prime}),$
$\displaystyle\langle\delta A^{\mu}_{\text{BG}}(\mathbf{x})\rangle$
$\displaystyle=$ $\displaystyle 0.$ (2)

These statistical properties are represented by a Gaussian functional
distribution of the form

$\displaystyle dP\left[\delta A^{\mu}_{\text{BG}}\right]=\mathcal{N}e^{-\int
d^{3}x\,\frac{\left[\delta
A_{\text{BG}}^{\mu}(\mathbf{x})\right]^{2}}{2\Delta_{B}}}\mathcal{D}\left[\delta
A_{\text{BG}}^{\mu}(\mathbf{x})\right].$ (3)

Therefore, we write the Lagrangian for this model as a superposition of two
terms

$\displaystyle\mathcal{L}=\mathcal{L}_{\text{FBG}}+\mathcal{L}_{\text{NBG}},$
(4)

where the first represents the system of Fermions (and photons) immersed in
the deterministic background field (FBG)

$\displaystyle\mathcal{L}_{\text{FBG}}=\bar{\psi}\left(\mathrm{i}\not{\partial}-e\not{A}_{\text{BG}}-e\not{A}-m\right)\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu},$
(5)

while the second therm represents the interaction between the Fermions and the
classical noise (NBG)

$\displaystyle\mathcal{L}_{\text{NBG}}=\bar{\psi}\left(-e\delta\not{A}_{\text{BG}}\right)\psi.$
(6)

The generating functional (in the absence of sources) for a given realization
of the noisy fields is given by

$\displaystyle Z[A]=\int\mathcal{D}[\bar{\psi},\psi]e^{\mathrm{i}\int
d^{4}x\left[\mathcal{L}_{\text{FBG}}+\mathcal{L}_{\text{NBG}}\right]}.$ (7)

To study the physics of this system, we need to calculate the statistical
average over the magnetic background noise $\delta A_{\text{BG}}^{\mu}$ of the
$\overline{\ln Z}$. For this purpose, we apply the replica method, which is
based on the following identity Mèzard and Parisi (1991)

$\displaystyle\overline{\ln Z[A]}=\lim_{n\rightarrow
0}\frac{\overline{Z^{n}[A]}-1}{n}.$ (8)

Here, we defined the statistical average according to the Gaussian functional
measure of Eq. (3), and $Z^{n}$ is obtained by incorporating an additional
“replica” component for each of the Fermion fields, i.e.
$\psi(x)\rightarrow\psi^{a}(x)$, for $1\leq a\leq n$. The “replicated”
Lagrangian has the same form as Eqs. (5) and (6), but with an additional sum
over the replica components of the Fermion fields. Therefore, the averaging
procedure leads to

$\displaystyle\overline{Z^{n}[A]}$ $\displaystyle=$
$\displaystyle\int\prod_{a=1}^{n}\mathcal{D}[\bar{\psi}^{a},\psi^{a}]\int\mathcal{D}\left[\delta
A_{\text{BG}}^{\mu}\right]e^{-\int d^{3}x\,\frac{\left[\delta
A_{\text{BG}}^{\mu}(\mathbf{x})\right]^{2}}{2\Delta_{B}}}$ (9)
$\displaystyle\times e^{\mathrm{i}\int
d^{4}x\sum_{a=1}^{n}\left(\mathcal{L}_{\text{FBG}}[\bar{\psi}^{a},\psi^{a}]+\mathcal{L}_{DBG}[\bar{\psi}^{a},\psi^{a}]\right)}$
$\displaystyle=$
$\displaystyle\int\prod_{a=1}^{n}\mathcal{D}[\bar{\psi}^{a},\psi^{a}]e^{\mathrm{i}\bar{S}\left[\bar{\psi}^{a},\psi^{a};A\right]},$

where in the last step we explicitly performed the Gaussian integral over the
background noise, leading to the definition of the effective averaged action
for the replica system

$\displaystyle\bar{S}\left[\bar{\psi}^{a},\psi^{a};A\right]$ $\displaystyle=$
$\displaystyle\int
d^{4}x\left(\sum_{a}\bar{\psi}^{a}\left(\mathrm{i}\not{\partial}-e\not{A}_{\text{BG}}-e\not{A}-m\right)\psi^{a}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right)$
(10) $\displaystyle+$ $\displaystyle\mathrm{i}\frac{e^{2}\Delta_{B}}{2}\int
d^{4}x\int
d^{4}y\sum_{a,b}\sum_{j=1}^{3}\bar{\psi}^{a}(x)\gamma^{j}\psi^{a}(x)\bar{\psi}^{b}(y)\gamma_{j}\psi^{b}(y)\delta^{3}(\mathbf{x}-\mathbf{y}).$

Clearly, we end up with an effective interacting theory, with an instantaneous
local interaction proportional to the fluctuation amplitude $\Delta_{B}$ that
characterizes the magnetic noise, as defined in Eq. (2). The “free” part of
the action corresponds to Fermions in the average background classical field
$A_{\text{BG}}^{\mu}(x)$. We choose this background to represent a uniform,
static magnetic field along the $z$-direction $\mathbf{B}=\hat{e}_{3}B$, using
the gauge Dittrich and Reuter (1985)

$\displaystyle A_{\text{BG}}^{\mu}(x)=\frac{1}{2}(0,-Bx^{2},Bx^{1},0).$ (11)

Therefore, this allows us to use directly the Schwinger proper-time
representation of the free-Fermion propagator dressed by the background field,
as follows Schwinger (1951); Dittrich and Reuter (1985)

$\displaystyle\left[S_{\text{F}}(k)\right]_{a,b}$
$\displaystyle=-\mathrm{i}\delta_{a,b}\int_{0}^{\infty}\frac{d\tau}{\cos(eB\tau)}e^{\mathrm{i}\tau\left(k_{\parallel}^{2}-\mathbf{k}_{\perp}^{2}\frac{\tan(eB\tau)}{eB\tau}-m^{2}+\mathrm{i}\epsilon\right)}$
(12)
$\displaystyle\times\left\\{\left[\cos(eB\tau)+\mathrm{i}\gamma^{1}\gamma^{2}\sin(eB\tau)\right](m+\not{k}_{\parallel})\right.$
$\displaystyle\left.+\frac{\not{k}_{\perp}}{\cos(eB\tau)}\right\\},$

which is clearly diagonal in replica space. Here, as usual, we separated the
parallel from the perpendicular directions with respect to the background
external magnetic field by splitting the metric tensor as
$g^{\mu\nu}=g_{\parallel}^{\mu\nu}+g_{\perp}^{\mu\nu}$, with

$\displaystyle g_{\parallel}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\text{diag}(1,0,0,-1),$ $\displaystyle g_{\perp}^{\mu\nu}$
$\displaystyle=$ $\displaystyle\text{diag}(0,-1,-1,0),$ (13)

thus implying that for any 4-vector, such as the momentum $k^{\mu}$, we write

$\displaystyle\not{k}=\not{k}_{\perp}+\not{k}_{\parallel},$ (14)

and

$\displaystyle k^{2}=k_{\parallel}^{2}-\mathbf{k}_{\perp}^{2}.$ (15)

In particular, $k_{\parallel}^{2}=k_{0}^{2}-k_{3}^{2}$, while
$\mathbf{k}_{\perp}=(k^{1},k^{2})$ is the Euclidean 2-vector lying in the
plane perpendicular to the field, such that its square-norm is
$\mathbf{k}_{\perp}^{2}=k_{1}^{2}+k_{2}^{2}$. The Schwinger propagator can be
expressed as

$\displaystyle\left[S_{\text{F}}(k)\right]_{a,b}=-\mathrm{i}\delta_{a,b}\left[\left(m+\not{k}\right)\mathcal{A}_{1}\right.$
$\displaystyle\left.+(\mathrm{i}eB)\mathrm{i}\gamma^{1}\gamma^{2}\left(m+\not{k}_{\parallel}\right)\frac{\partial\mathcal{A}_{1}}{\partial\mathbf{k}_{\perp}^{2}}+\left(\mathrm{i}eB\right)^{2}\not{k}_{\perp}\frac{\partial^{2}\mathcal{A}_{1}}{\partial(\mathbf{k}_{\perp}^{2})^{2}}\right]$
$\displaystyle=-\mathrm{i}\delta_{a,b}\left[\left(m+\not{k}_{\parallel}\right)\mathcal{A}_{1}+\mathrm{i}\gamma^{1}\gamma^{2}\left(m+\not{k}_{\parallel}\right)\mathcal{A}_{2}+\mathcal{A}_{3}\not{k}_{\perp}\right]$
(16)

Here, we defined the function

$\displaystyle\mathcal{A}_{1}(k,B)=\int_{0}^{\infty}d\tau
e^{\mathrm{i}\tau\left(k_{\parallel}^{2}-m^{2}+\mathrm{i}\epsilon\right)-\mathrm{i}\frac{\mathbf{k}_{\perp}^{2}}{eB}\tan(eB\tau)},$
(17)

that clearly reproduces the inverse scalar propagator (with Feynman
prescription) in the zero-field limit

$\displaystyle\lim_{B\rightarrow
0}\mathcal{A}_{1}(k,B)=\frac{\mathrm{i}}{k^{2}-m^{2}+\mathrm{i}\epsilon}\equiv\frac{\mathrm{i}}{\mathcal{D}_{0}(k)},$
(18)

with

$\displaystyle\mathcal{D}_{0}(k)=k^{2}-m^{2}+\mathrm{i}\epsilon,$ (19)

and its derivatives

$\displaystyle\mathcal{A}_{2}(k,B)$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{\infty}d\tau~{}\tan(eB\tau)e^{\mathrm{i}\tau\left(k_{\parallel}^{2}-t_{B}({\tau})\mathbf{k}_{\perp}^{2}-m^{2}+\mathrm{i}\epsilon\right)}$
(20a) $\displaystyle=$
$\displaystyle\mathrm{i}eB\frac{\partial\mathcal{A}_{1}}{\partial(\mathbf{k}_{\perp}^{2})},$
$\displaystyle\mathcal{A}_{3}(k,B)$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{\infty}\frac{d\tau}{\cos^{2}(eB\tau)}e^{\mathrm{i}\tau\left(k_{\parallel}^{2}-t_{B}({\tau})\mathbf{k}_{\perp}^{2}-m^{2}+\mathrm{i}\epsilon\right)}$
(20b) $\displaystyle=$
$\displaystyle\mathcal{A}_{1}+(\mathrm{i}eB)^{2}\frac{\partial^{2}\mathcal{A}_{1}}{\partial(\mathbf{k}_{\perp}^{2})^{2}}.$

Moreover, with these definitions it is straightforward to verify that the
inverse of the Schwinger propagator Eq. (16) is given by:

$\displaystyle\hat{S}_{\text{F}}^{-1}(k)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}}{\mathcal{D}(k)}\left[\left(m-\not{k}_{\parallel}\right)\mathcal{A}_{1}-\mathrm{i}\gamma^{1}\gamma^{2}\left(m-\not{k}_{\parallel}\right)\mathcal{A}_{2}\right.$
(21) $\displaystyle\left.-\mathcal{A}_{3}\not{k}_{\perp}\right],$

where

$\displaystyle\mathcal{D}(k)=\mathcal{A}_{3}^{2}\mathbf{k}_{\perp}^{2}-\left(\mathcal{A}_{1}^{2}-\mathcal{A}_{2}^{2}\right)\left(k_{\parallel}^{2}-m^{2}\right).$
(22)

Then, all the relevant expressions will be given in terms of
$\mathcal{A}_{1}$.

##  III Perturbation theory: Self-energy and vertex corrections

Our goal is to develop a perturbation theory in powers of $\Delta$, where as
described in Section II and particularly in Eq. (10), the effective fermion-
fermion interaction arises as a result of averaging over the background
magnetic noise. Starting from a free Fermion propagator, as defined by Eq.
(12), we include the magnetic noise-induced interaction effects by “dressing”
the propagator with a self-energy, as shown diagrammatically in the Dyson
equation depicted in Fig. 1. We remark that for this theory, the skeleton
diagram for the self-energy is represented in Fig. 2.

Figure 1: Dyson equation for the ”dressed” propagator (double-line), in terms
of the free propagator (single-line) and the self-energy $\Sigma$. Figure 2:
Skeleton diagram representing the self-energy for the effective interacting
theory. The dashed line is the disorder-induced interaction $\Delta_{B}$,
while the box $\hat{\Gamma}$ represents the 4-point vertex function.

##  IV Self-energy at order $\Delta$

Figure 3: Self-energy diagram at first order in $\Delta=e^{2}\Delta_{B}$.

It is possible to express the background-noise contribution to the self-energy
(where for notational simplicity, we define the parameter $\Delta\equiv
e^{2}\Delta_{B}$), as depicted in the Feynman diagram in Fig. (3), by the
integral expression

$\displaystyle\hat{\Sigma}_{\Delta}(q)$ $\displaystyle=$
$\displaystyle(\mathrm{i}\Delta)\int\frac{d^{3}p}{(2\pi)^{3}}\gamma^{j}\hat{S}_{F}(p+q;p_{0}=0)\gamma_{j}$
$\displaystyle=$ $\displaystyle\frac{i(\mathrm{i}\Delta)}{(2\pi)^{3}}\int
d^{3}p\left\\{3\left(\gamma^{0}q_{0}-m\right)\mathcal{A}_{1}(q_{0},p_{3};\mathbf{p}_{\perp})\right.$
$\displaystyle\left.+\mathrm{i}\gamma^{1}\gamma^{2}\left(m-q_{0}\gamma^{0}\right)(\mathrm{i}eB)\frac{\partial}{\partial\mathbf{p}_{\perp}^{2}}\mathcal{A}_{1}(q_{0},p_{3};\mathbf{p}_{\perp})\right\\}.$

The derivative term in the last expression can be integrated in cylindrical
coordinates $d^{3}p=\pi dp_{3}d(\mathbf{p}_{\perp}^{2})$, as follows

$\displaystyle\int
d^{3}p\,\frac{\partial}{\partial\mathbf{p}_{\perp}^{2}}\mathcal{A}_{1}(q_{0},p_{3};\mathbf{p}_{\perp})$
$\displaystyle=\pi\int_{-\infty}^{+\infty}dp_{3}\int_{0}^{\infty}d(\mathbf{p}_{\perp}^{2})\frac{\partial}{\partial\mathbf{p}_{\perp}^{2}}\mathcal{A}_{1}(q_{0},p_{3};\mathbf{p}_{\perp})$
$\displaystyle=-\pi\int_{-\infty}^{+\infty}dp_{3}\mathcal{A}_{1}(q_{0},p_{3};\mathbf{p}_{\perp}=0),$
(24)

where the identity
$\lim_{\mathbf{p}_{\perp}^{2}\rightarrow\infty}\mathcal{A}_{1}(q_{0},p_{3};\mathbf{p}_{\perp})=0$
was applied. Substituting this result into Eq. (IV), we finally obtain the
exact expression (valid at all orders in the background average magnetic field
$B$)

$\displaystyle\hat{\Sigma}_{\Delta}(q)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}(\mathrm{i}\Delta)}{(2\pi)^{3}}\left[3\left(\gamma^{0}q_{0}-m\right)\widetilde{\mathcal{A}}_{1}(q_{0})\right.$
(25) $\displaystyle\left.-\mathrm{i}\gamma^{1}\gamma^{2}(\mathrm{i}\pi
eB)\left(m-q_{0}\gamma^{0}\right)\widetilde{\mathcal{A}}_{2}(q_{0})\right],$

where we have defined:

$\displaystyle\widetilde{\mathcal{A}}_{1}(q_{0})$ $\displaystyle\equiv$
$\displaystyle\int d^{3}p\mathcal{A}_{1}(q_{0},p_{3};\mathbf{p}_{\perp}),$
$\displaystyle\widetilde{\mathcal{A}}_{2}(q_{0})$ $\displaystyle\equiv$
$\displaystyle\int_{-\infty}^{+\infty}dp_{3}\mathcal{A}_{1}(q_{0},p_{3};\mathbf{p}_{\perp}=0).$
(26)

Inserting the first-order in $\Delta$ expression for the self-energy Eq. (25)
into the Dyson equation, as depicted diagrammatically in Fig. 1, we obtain the
dressed inverse propagator at first-order in $\Delta$

$\displaystyle\hat{S}_{\Delta}^{-1}(k)=\hat{S}_{\text{F}}^{-1}(k)-\hat{\Sigma}_{\Delta},$
(27)

so that by using Eqs. (21) and (25) we explicitly obtain:

$\displaystyle\hat{S}_{\Delta}^{-1}(q)$ $\displaystyle=$
$\displaystyle\mathrm{i}\left[\frac{m\mathcal{A}_{1}(q)}{\mathcal{D}(q)}+\frac{3m(\mathrm{i}\Delta)}{(2\pi)^{3}}\widetilde{\mathcal{A}}_{1}(q_{0})\right]$
$\displaystyle-$
$\displaystyle\mathrm{i}\left[\frac{\mathcal{A}_{1}(q)}{\mathcal{D}(q)}+\frac{3(\mathrm{i}\Delta)}{(2\pi)^{3}}\widetilde{\mathcal{A}}_{1}(q_{0})\right]\left(q_{0}\gamma^{0}\right)$
$\displaystyle-$
$\displaystyle\mathrm{i}\left[\frac{m\mathcal{A}_{2}(q)}{\mathcal{D}(q)}-\mathrm{i}\frac{m\pi(\mathrm{i}\Delta)(eB)}{(2\pi)^{3}}\widetilde{\mathcal{A}}_{2}(q_{0})\right]\left(\mathrm{i}\gamma^{1}\gamma^{2}\right)$
$\displaystyle+$
$\displaystyle\mathrm{i}\left[\frac{\mathcal{A}_{2}(q)}{\mathcal{D}(q)}-\frac{\mathrm{i}\pi(\mathrm{i}\Delta)(eB)}{(2\pi)^{3}}\widetilde{\mathcal{A}}_{2}(q_{0})\right]\left(\mathrm{i}\gamma^{1}\gamma^{2}q_{0}\gamma^{0}\right)$
$\displaystyle-$
$\displaystyle\mathrm{i}\frac{\mathcal{A}_{1}(q)}{\mathcal{D}(q)}\left(q_{3}\gamma^{3}\right)+\mathrm{i}\frac{\mathcal{A}_{2}}{\mathcal{D}(q)}\left(\mathrm{i}\gamma^{1}\gamma^{2}q_{3}\gamma^{3}\right)-\mathrm{i}\frac{\mathcal{A}_{3}}{\mathcal{D}(q)}\not{q}_{\perp}.$

##  V Renormalization of the propagator

Let us define by $m^{\prime}$, $z$ and $z_{3}$ as the renormalization factors
for the mass, the wave function and the charge, respectively. While $z$ will
emerge as a global factor in the dressed propagator, the factor $z_{3}$ will
only be associated to the tensor structures involving the spin-magnetic field
interaction $e\sigma_{\mu\nu}F^{\mu\nu}_{\text{BG}}=i\gamma_{1}\gamma_{2}eB$.
Therefore, we can compare Eq. (21) with Eq. (LABEL:Scorrected), in order to
identify the corresponding scalar factors for each tensor structure in both
expressions, thus leading to the definition of the renormalized coefficients
as follows

  *   * •

For $\mathbb{1}$:

$\displaystyle\frac{m\mathcal{A}_{1}(q)}{\mathcal{D}(q)}+\frac{3m(\mathrm{i}\Delta)}{(2\pi)^{3}}\widetilde{\mathcal{A}}_{1}(q_{0})\equiv
z\frac{m^{\prime}\mathcal{A}_{1}(q)}{\mathcal{D}(q)}$ (29a)

  * •

For $\gamma^{1}\gamma^{2}\gamma^{0}$:

$\displaystyle\frac{\mathcal{A}_{2}(q)}{\mathcal{D}(q)}-\frac{\mathrm{i}\pi(\mathrm{i}\Delta)(eB)}{(2\pi)^{3}}\widetilde{\mathcal{A}}_{2}(q_{0})\equiv
z\cdot z_{3}\frac{\mathcal{A}_{2}(q)}{\mathcal{D}(q)}$ (29b)

  * •

For $\gamma^{1}\gamma^{2}$:

$\displaystyle\frac{m\mathcal{A}_{2}(q)}{\mathcal{D}(q)}-\frac{\mathrm{i}m\pi(\mathrm{i}\Delta)(eB)}{(2\pi)^{3}}\widetilde{\mathcal{A}}_{2}(q_{0})\equiv
z\cdot z_{3}\frac{m^{\prime}\mathcal{A}_{2}(q)}{\mathcal{D}(q)}$

Then, solving the system of equations we obtain

$\displaystyle
z=1+\frac{3\mathrm{i}\Delta}{(2\pi)^{3}}\frac{\widetilde{\mathcal{A}}_{1}(q_{0})}{\mathcal{A}_{1}(q)}\mathcal{D}(q),$
(30a) $\displaystyle
z_{3}=\frac{1-\frac{\mathrm{i}\pi(\mathrm{i}\Delta)(eB)}{(2\pi)^{3}}\frac{\widetilde{\mathcal{A}}_{2}(q_{0})}{\mathcal{A}_{2}(q)}\mathcal{D}(q)}{1+\frac{3\mathrm{i}\Delta}{(2\pi)^{3}}\frac{\widetilde{\mathcal{A}}_{1}(q_{0})}{\mathcal{A}_{1}(q)}\mathcal{D}(q)},$
(30b) $\displaystyle m^{\prime}=m,$ (30c) and
$\displaystyle\frac{v^{\prime}}{c}=z^{-1}=\left(1+\frac{3\mathrm{i}\Delta}{(2\pi)^{3}}\frac{\widetilde{\mathcal{A}}_{1}(q_{0})}{\mathcal{A}_{1}(q)}\mathcal{D}(q)\right)^{-1}$
(30d)

With these definitions into the “magnetic noise-dressed” inverse propagator
Eq. (LABEL:Scorrected), after organizing the different tensor structures, we
obtain the expression

$\displaystyle
S_{\Delta}^{-1}(q)=\frac{\mathrm{i}z}{\mathcal{D}(q)}\left[\left(m-q_{0}\gamma^{0}-z^{-1}q_{3}\gamma^{3}\right)\mathcal{A}_{1}(q)\right.$
(31)
$\displaystyle\left.-z_{3}\left(\mathrm{i}\gamma^{1}\gamma^{2}\right)\left(m-q_{0}\gamma^{0}-z^{-1}q_{3}\gamma^{3}\right)\mathcal{A}_{2}(q)\right.$
$\displaystyle\left.-\mathrm{i}\mathcal{A}_{3}(q)z^{-1}\not{q}_{\perp}\right]$
$\displaystyle=$
$\displaystyle\frac{\mathrm{i}z}{\mathcal{D}(q)}\left[\left(m-\tilde{\not{q}}_{\parallel}\right)\mathcal{A}_{1}(q)-z_{3}\left(\mathrm{i}\gamma^{1}\gamma^{2}\right)\left(m-\tilde{\not{q}}_{\parallel}\right)\mathcal{A}_{2}(q)\right.$
$\displaystyle\left.-\mathrm{i}\mathcal{A}_{3}(q)\tilde{\not{q}}_{\perp}\right],$

where in the last line we defined the four-vector
$\tilde{q}^{\mu}=(q^{0},z^{-1}\mathbf{q})$ that incorporates the definition of
the effective refraction index $v^{\prime}/c=z^{-1}$ due to the random
magnetic fluctuations. By comparing Eq. (31) with Eq. (21), it is clear that
they possess the same tensor structure. Therefore, by means of the elementary
properties of the Dirac matrices, this expression can be readily inverted to
obtain the “magnetic noise-dressed” fermion propagator

$\displaystyle
S_{\Delta}(q)=-\mathrm{i}z^{-1}\frac{\mathcal{D}(q)}{\tilde{\mathcal{D}}(q)}\left[\left(m+\tilde{\not{q}}_{\parallel}\right)\mathcal{A}_{1}(q)\right.$
$\displaystyle\left.+\mathrm{i}z_{3}\gamma^{1}\gamma^{2}\left(m+\tilde{\not{q}}_{\parallel}\right)\mathcal{A}_{2}(q)+\mathcal{A}_{3}(q)\tilde{\not{q}}_{\perp}\right],$
(32)

where $\mathcal{D}(q)$ was defined in Eq. (22), and

$\displaystyle\tilde{\mathcal{D}}(q)=\mathcal{A}_{3}^{2}z^{-2}\mathbf{q}_{\perp}^{2}-\left(\mathcal{A}_{1}^{2}-\mathcal{A}_{2}^{2}\right)\left(z^{-2}q_{\parallel}^{2}-m^{2}\right)$
(33)

Let us now discuss the explicit magnetic field and magnetic noise dependence
of the renormalized parameters defined in Eqs. (30a) and (30b), i.e $z$ and
$z_{3}$. For this purpose, we shall distinguish three different regimes,
corresponding to the very weak, the intermediate and the ultra-intense
magnetic field, respectively.

###  V.1 Very weak field $eB/m^{2}\ll 1$

As shown in detail in the Appendix A, for very weak fields $eB/m^{2}\ll 1$ the
function $\mathcal{A}_{1}(k,B)$ can be expanded in terms of the power series

$\displaystyle\mathcal{A}_{1}(k,B)=\frac{\mathrm{i}}{\mathcal{D}_{\parallel}}\left(1+\sum_{j=1}^{\infty}\left(\frac{\mathrm{i}eB}{\mathcal{D}_{\parallel}}\right)^{j}\mathcal{E}_{j}(x)\right),$
(34)

where for notational simplicity, we defined the “parallel” inverse scalar
propagator

$\displaystyle\mathcal{D}_{\parallel}=k_{\parallel}^{2}-m^{2}+\mathrm{i}\epsilon,$
(35)

and the dimensionless variable $x=\mathbf{k}_{\perp}^{2}/eB$. We also defined
the polynomials $\mathcal{E}_{j}(x)$, as those generated by the function
$e^{-\mathrm{i}x\tan v}$, i.e.

$\displaystyle\mathcal{E}_{j}(x)=\lim_{v\rightarrow
0}\frac{\partial^{j}}{\partial v^{j}}\left(e^{-\mathrm{i}x\tan v}\right).$
(36)

For instance, the explicit analytical expressions for the first three
polynomials ($j=1,2,3$) are as follows

$\displaystyle\mathcal{E}_{1}(x)$ $\displaystyle=$
$\displaystyle-\mathrm{i}x,$ $\displaystyle\mathcal{E}_{2}(x)$
$\displaystyle=$ $\displaystyle-x^{2},$ $\displaystyle\mathcal{E}_{3}(x)$
$\displaystyle=$ $\displaystyle-2\mathrm{i}x+\mathrm{i}x^{3}.$

At the lowest order, after subtracting the divergent vacuum contribution from
Eq. (34), we have after Eq. (LABEL:AA1_low) (see Appendix A for details),

$\displaystyle\mathcal{A}_{1}(k,B)-\mathcal{A}_{1}(k,0)=\frac{-2\mathrm{i}\left(eB\right)^{2}\mathbf{k}_{\perp}^{2}}{\left[k^{2}-m^{2}+\mathrm{i}\epsilon\right]^{4}}+O((eB)^{4}).$
(37)

Therefore, using this weak field expansion of the propagator, we calculate the
integral (details in Appendix C)

$\displaystyle\widetilde{\mathcal{A}}_{1}(q_{0})$ $\displaystyle=$
$\displaystyle-2\mathrm{i}(eB)^{2}\int
d^{3}p\frac{\mathbf{p_{\perp}}^{2}}{(q_{0}^{2}-p_{3}^{2}-\mathbf{p_{\perp}}^{2}-m^{2}+\mathrm{i}\epsilon)^{4}}$
(38) $\displaystyle=$
$\displaystyle-\frac{\pi^{2}}{6}\frac{(eB)^{2}}{(q_{0}^{2}-m^{2})^{3/2}}.$

In addition, at order $\mathcal{O}((eB)^{2})$ we also need to evaluate the
integral (details in Appendix C)

$\displaystyle\widetilde{\mathcal{A}}_{2}(q_{0})$ $\displaystyle=$
$\displaystyle\mathrm{i}\int_{-\infty}^{+\infty}\frac{dp_{3}}{q_{0}^{2}-(p^{3})^{2}-m^{2}+\mathrm{i}\epsilon}$
(39) $\displaystyle=$ $\displaystyle\frac{\pi}{\sqrt{q_{0}^{2}-m^{2}}}.$

Therefore, from Eqs. (30a), (37), (38), and (39), we can directly evaluate the
renormalization parameters to obtain

$\displaystyle z$ $\displaystyle=$ $\displaystyle
1+\frac{3\mathrm{i}\Delta}{(2\pi)^{3}}\frac{\widetilde{\mathcal{A}}_{1}(q_{0})}{\mathcal{A}_{1}(q)}\mathcal{D}(q)$
(40) $\displaystyle=$ $\displaystyle
1+\frac{\Delta(eB)^{4}}{8\pi}\frac{\mathbf{q}_{\perp}^{2}}{(q^{2}-m^{2}+\mathrm{i}\epsilon)^{3}(q_{0}^{2}-m^{2}+\mathrm{i}\epsilon)^{3/2}}$
$\displaystyle=$ $\displaystyle 1+O((eB)^{4}),$

and similarly from Eq. (30b)

$\displaystyle z_{3}$ $\displaystyle=$ $\displaystyle 1+O((eB)^{4}).$ (41)

###  V.2 Intermediate field

For intermediate magnetic field intensities, we can calculate the integral
$\mathcal{A}_{1}$ by means of an expansion in terms of Landau levels. For this
purpose, let us consider the generating function of the Laguerre
polynomialsGradshteyn and Rhyzik (2000a)

$\displaystyle
e^{-\frac{x}{2}\frac{1-t}{1+t}}=(1+t)e^{-x/2}\sum_{n=0}^{\infty}(-t)^{n}L_{n}^{0}(x),$
(42)

since

$\displaystyle e^{-\mathrm{i}x\tan v}$ $\displaystyle=$
$\displaystyle\exp\left[-x\left(1-e^{-2\mathrm{i}v}\right)/\left(1+e^{-2\mathrm{i}v}\right)\right]$
$\displaystyle=$
$\displaystyle\left(1+e^{-2\mathrm{i}v}\right)e^{-x}\sum_{n=0}^{\infty}(-1)^{n}e^{-2\mathrm{i}nv}L_{n}^{0}(2x)$

Therefore, we have (for $x=\mathbf{k}_{\perp}^{2}/eB$)

$\displaystyle\mathcal{A}_{1}(k)=e^{-x}\left(\sum_{n=0}^{\infty}(-1)^{n}L_{n}^{0}(2x)\int_{0}^{\infty}d\tau
e^{\mathrm{i}(\mathcal{D}_{\parallel}-2(n+1)eB)\tau}\right.$
$\displaystyle\left.+\sum_{n=0}^{\infty}(-1)^{n}L_{n}^{0}(2x)\int_{0}^{\infty}d\tau
e^{\mathrm{i}(\mathcal{D}_{\parallel}-2neB)\tau}\right)$ (44)

Evaluating the exponential integrals, we obtain

$\displaystyle\mathcal{A}_{1}(k)$ $\displaystyle=$
$\displaystyle\mathrm{i}e^{-x}\left(\sum_{n=0}^{\infty}(-1)^{n}\frac{L_{n}^{0}(2x)}{\mathcal{D}_{\parallel}-2(n+1)eB}\right.$
$\displaystyle\left.+\sum_{n=0}^{\infty}(-1)^{n}\frac{L_{n}^{0}(2x)}{\mathcal{D}_{\parallel}-2neB}\right)$
$\displaystyle=$
$\displaystyle\mathrm{i}\frac{e^{-x}}{\mathcal{D}_{\parallel}}\left[1+\sum_{n=1}^{\infty}\frac{(-1)^{n}\left[L_{n}^{0}(2x)-L_{n-1}^{0}(2x)\right]}{1-2n\frac{eB}{\mathcal{D}_{\parallel}}}\right]$

Inserting this expression into the definition of $\widetilde{\mathcal{A}}_{1}$
of Eq. (26), we obtain

$\displaystyle\widetilde{\mathcal{A}}_{1}(q_{0})$ $\displaystyle=$
$\displaystyle\int d^{3}p\mathcal{A}_{1}(q_{0},p_{3};\mathbf{p}_{\perp})$ (46)
$\displaystyle=$ $\displaystyle\mathrm{i}\int
d^{3}p\Bigg{[}\frac{e^{-\mathbf{p}_{\perp}^{2}/eB}}{q_{0}^{2}-p_{3}^{2}-m^{2}+\mathrm{i}\epsilon}$
$\displaystyle+$
$\displaystyle\sum_{n=1}^{\infty}(-1)^{n}e^{-\mathbf{p}_{\perp}^{2}/eB}\frac{L_{n}\left(\frac{2\mathbf{p}_{\perp}^{2}}{eB}\right)-L_{n-1}\left(\frac{2\mathbf{p}_{\perp}^{2}}{eB}\right)}{q_{0}^{2}-p_{3}^{2}-m^{2}-2neB+\mathrm{i}\epsilon}\Bigg{]}$
$\displaystyle=$
$\displaystyle\mathcal{I}_{1}+\sum_{n=1}^{\infty}(-1)^{n}\mathcal{I}_{2,n}$

Here, we defined

$\displaystyle\mathcal{I}_{1}$ $\displaystyle=$ $\displaystyle\mathrm{i}\int
d^{3}p\frac{e^{-\mathbf{p}_{\perp}^{2}/eB}}{q_{0}^{2}-p_{3}^{2}-m^{2}+\mathrm{i}\epsilon}$
(47) $\displaystyle=$
$\displaystyle\frac{\pi^{2}eB}{\sqrt{q_{0}^{2}-m^{2}+\mathrm{i}\epsilon}},$

and

$\displaystyle\mathcal{I}_{2,n}=\mathrm{i}\int
d^{3}p\,e^{-\mathbf{p}_{\perp}^{2}/eB}\frac{L_{n}\left(\frac{2\mathbf{p}_{\perp}^{2}}{eB}\right)-L_{n-1}\left(\frac{2\mathbf{p}_{\perp}^{2}}{eB}\right)}{q_{0}^{2}-p_{3}^{2}-m^{2}-2neB+\mathrm{i}\epsilon}$
(48)

We calculate the momentum integrals in cylindrical coordinates, making use of
the azimuthal symmetry, such that $d^{3}p=dp_{3}\pi
d(\mathbf{p}_{\perp}^{2})$. Moreover, in the integral over
$\mathbf{p}_{\perp}$, we define the auxiliary variable
$x=\frac{2\mathbf{p}_{\perp}^{2}}{eB}$, such that

$\displaystyle\mathcal{I}_{2,n}$ $\displaystyle=$ $\displaystyle\frac{\pi
eB}{2}\int_{-\infty}^{\infty}\frac{dp_{3}}{q_{0}^{2}-m^{2}-p_{3}^{2}-2nqB+\mathrm{i}\epsilon}$
(49)
$\displaystyle\times\int_{0}^{\infty}dx\,e^{-x/2}\left[L_{n}(x)-L_{n-1}(x)\right]$
$\displaystyle=$ $\displaystyle 2\pi
eB(-1)^{n}\int_{-\infty}^{\infty}\frac{dp_{3}}{q_{0}^{2}-m^{2}-p_{3}^{2}-2nqB+\mathrm{i}\epsilon}$

where we used the identity Gradshteyn and Rhyzik (2000b)

$\int_{0}^{\infty}dxe^{-bx}L_{n}(x)=\left(b-1\right)^{n}b^{-n-1}\,\,\,\text{Re}\,b>0.$
(50)

Figure 4: Spectral density for the Landau level spectrum $\rho(E)$, as a
function of the dimensionless energy scale $E/m$, for different values of the
average background magnetic field $eB/m^{2}$. The inset is shown in order to
appreciate in detail the staircase pattern produced by the discrete Landau
levels.

Inserting Eq. (47) and Eq. (49) into Eq. (46), we obtain (after shifting the
index $n\rightarrow n+1$)

$\displaystyle\tilde{\mathcal{A}}_{1}(q_{0})=\frac{\pi^{2}(eB)}{\sqrt{q_{0}^{2}-m^{2}+\mathrm{i}\epsilon}}$
(51) $\displaystyle+2\pi
eB\int_{-\infty}^{+\infty}dp_{3}\sum_{n=0}^{\infty}\frac{1}{q_{0}^{2}-m^{2}-p_{3}^{2}-2(n+1)eB+\mathrm{i}\epsilon}$

Let us introduce the density of states for Landau levels

$\displaystyle\rho(E)=\int_{-\infty}^{\infty}\frac{dp_{3}}{2\pi}\sum_{n=0}^{\infty}\delta\left(E-E_{n}(p_{3})\right),$
(52)

with the dispersion relation for the spectrum

$\displaystyle E_{n}(p_{3})=\sqrt{p_{3}^{2}+m^{2}+2(n+1)eB}.$ (53)

As shown in detail in Appendix B

$\displaystyle\rho(E)$ $\displaystyle=$
$\displaystyle\Theta(E-\sqrt{m^{2}+2eB})\frac{E}{\pi\sqrt{eB}}$ (54)
$\displaystyle\times\left[\zeta\left(\frac{1}{2},\frac{E^{2}-m^{2}-2eB}{2eB}-N_{max}(E)\right)\right.$
$\displaystyle\left.-\zeta\left(\frac{1}{2},\frac{E^{2}-m^{2}}{2eB}\right)\right],$

where we defined

$\displaystyle N_{max}(E)=\lfloor\frac{E^{2}-m^{2}}{2eB}-1\rfloor,$ (55)

with $\lfloor x\rfloor$ the integer part of $x$, and $\zeta(n,z)$ is the
Riemann zeta function. The spectral density Eq. (54) is represented in Fig. 4,
as a function of the dimensionless energy scale $E/m$. For large magnetic
fields $eB/m^{2}\gg 1$, the spectral density displays a clear staircase
pattern, where each step represents the contribution arising from a single
Landau level $n=0,1,\ldots$. On the other hand, for weak magnetic fields
$eB/m^{2}\ll 1$, the spectral density exhibits a denser, quasi-continuum
behavior.

With these definitions, we obtain from Eq. (V.2) the exact expression

$\displaystyle\tilde{\mathcal{A}}_{1}(q_{0})=\frac{\pi^{2}eB}{\sqrt{q_{0}^{2}-m^{2}+\mathrm{i}\epsilon}}+4\pi^{2}eB\int_{-\infty}^{+\infty}dE\frac{\rho(E)}{q_{0}^{2}-E^{2}+\mathrm{i}\epsilon},$

On the other hand, from Eq. (26) we have

$\displaystyle\widetilde{\mathcal{A}}_{2}(q_{0})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dp_{3}p\mathcal{A}_{1}(q_{0},p_{3};\mathbf{p}_{\perp}=0)$
$\displaystyle=$
$\displaystyle\mathrm{i}\int_{-\infty}^{\infty}dp_{3}\frac{1}{q_{0}^{2}-p_{3}^{2}-m^{2}+\mathrm{i}\epsilon}$
$\displaystyle+$
$\displaystyle\mathrm{i}\int_{-\infty}^{\infty}dp_{3}\sum_{n=1}^{\infty}(-1)^{n}\frac{L_{n}(0)-L_{n-1}(0)}{q_{0}^{2}-p_{3}^{2}-m^{2}-2neB+\mathrm{i}\epsilon},$

where the second term vanishes, given that $L_{n}(0)=1~{}\forall n$. Hence, we
end up with the simple expression

$\displaystyle\widetilde{\mathcal{A}}_{2}(q_{0})=\frac{\pi}{\sqrt{q_{0}^{2}-m^{2}+\mathrm{i}\epsilon}}.$
(58)

In order to evaluate the formulas for $z$ in Eq. (30a) and $z_{3}$ in Eq.
(30b), we also need to evaluate the following coefficients (for
$x=\mathbf{k}_{\perp}^{2}/(eB)$)

$\displaystyle\mathcal{A}_{2}$ $\displaystyle=$ $\displaystyle
ieB\frac{\partial\mathcal{A}_{1}}{\partial\mathbf{k}_{\perp}^{2}}=\mathrm{i}\frac{\partial\mathcal{A}_{1}}{\partial
x}$ $\displaystyle=$
$\displaystyle\frac{e^{-x}}{\mathcal{D}_{\parallel}}\left[1+\sum_{n=1}^{\infty}\frac{(-1)^{n}\left(L_{n}(2x)-2L_{n}^{{}^{\prime}}(2x)-L_{n-1}(2x)+2L_{n-1}^{{}^{\prime}}(2x)\right)}{1-2\frac{neB}{\mathcal{D}_{\parallel}}}\right]$
$\displaystyle\mathcal{A}_{3}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{1}+(\mathrm{i}eB)^{2}\frac{\partial^{2}\mathcal{A}_{1}}{\partial(\mathbf{k}_{\perp}^{2})^{2}}=\mathcal{A}_{1}-\frac{\partial^{2}\mathcal{A}_{1}}{\partial
x^{2}}$ (59) $\displaystyle=$ $\displaystyle
i\frac{e^{-x}}{\mathcal{D}_{\parallel}}\sum_{n=1}^{\infty}\frac{(-1)^{n}\left(L_{n}(2x)-L_{n-1}(2x)-2L_{n}^{{}^{\prime}}(2x)+4L_{n}^{{}^{\prime\prime}}(2x)+2L_{n-1}^{{}^{\prime}}(2x)-4L_{n-1}^{{}^{\prime\prime}}(2x)\right)}{1-2\frac{neB}{\mathcal{D}_{\parallel}}}$

Finally, in order to further simplify these expressions, it is convenient to
use the identities

$\displaystyle L_{n}^{{}^{\prime}}(2x)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}-L_{n-1}^{(1)}(2x),&n\geq 1\\\
0,&\text{otherwise}\end{array}\right.$ (62) $\displaystyle=$
$\displaystyle-\theta_{n-1}\cdot L_{n-1}^{(1)}(2x),$ (63) $\displaystyle
L_{n}^{{}^{\prime\prime}}(2x)$ $\displaystyle=$
$\displaystyle\theta_{n-2}L_{n-2}^{(2)}(2x),$ (64)

where $\theta_{n-k}$ is the Heaviside step function

$\displaystyle\theta_{n-k}=\left\\{\begin{array}[]{cc}1,&n\geq k\\\
0,&\text{otherwise}\end{array}\right.$ (67)

With these identities, we obtain the final expressions

$\displaystyle\mathcal{A}_{2}$ $\displaystyle=$
$\displaystyle\frac{e^{-x}}{\mathcal{D}_{\parallel}}\left[1+\sum_{n=1}^{\infty}\frac{(-1)^{n}\left(L_{n}(2x)+2L_{n-1}^{(1)}(2x)-L_{n-1}(2x)-2\theta_{n-2}\cdot
L_{n-2}^{(1)}(2x)\right)}{1-2\frac{neB}{\mathcal{D}_{\parallel}}}\right]$ (68)
$\displaystyle\mathcal{A}_{3}$ $\displaystyle=$ $\displaystyle
i\frac{e^{-x}}{\mathcal{D}_{\parallel}}\sum_{n=1}^{\infty}\frac{(-1)^{n}\left(L_{n}(2x)-L_{n-1}(2x)+2L_{n-1}^{(1)}(2x)+4\theta_{n-2}\cdot
L_{n-2}^{(2)}(2x)-2L_{n-1}^{(1)}(2x)-4\theta_{n-3}\cdot
L_{n-3}^{(2)}(2x)\right)}{1-2\frac{neB}{\mathcal{D}_{\parallel}}}$

With these expressions, we evaluate

$\mathcal{D}(k)=\mathcal{A}_{3}^{2}\mathbf{k}_{\perp}^{2}-\left(\mathcal{A}_{1}^{2}-\mathcal{A}_{2}^{2}\right)(k_{\parallel}^{2}-m^{2}),$
(69)

and finally evaluate $z$ and $z_{3}$ with Eq. (30a) and Eq. (30b),
respectively. These results can be appreciated in Figs.5–10, as a function of
the energy scale $q_{0}/m$, as well as the magnitude of the average background
magnetic field $eB/m^{2}$, respectively.

###  V.3 Ultra-intense (LLL) field $eB/m^{2}\gg 1$

Let us now analyze the asymptotic behaviour of the quasi-particle
renormalization parameters $z$, $z_{3}$, and $v^{\prime}/c$, respectively, in
the ultra-intense magnetic field regime $eB/m^{2}\gg 1$. Here, we obtain the
corresponding asymptotic expression for $\mathcal{A}_{1}(q)$ by considering
only the lowest-Landau level (LLL) $n=0$ in Eq. (V.2). Therefore, we have

$\displaystyle\mathcal{A}_{1}(q)\sim\mathrm{i}\frac{e^{-\mathbf{q}_{\perp}^{2}/eB}}{q_{\parallel}^{2}-m^{2}},$
(70)

and the corresponding expressions for its derivatives are

$\displaystyle\mathcal{A}_{2}(q)=\frac{e^{-\mathbf{q}_{\perp}^{2}/eB}}{q_{\parallel}^{2}-m^{2}},$
(71)

and

$\displaystyle\mathcal{A}_{3}(q)=0.$ (72)

Similarly, we also have

$\displaystyle\mathcal{D}(q)=2\frac{e^{-2\mathbf{q}_{\perp}^{2}/eB}}{q_{\parallel}^{2}-m^{2}}.$
(73)

Finally, the integrals of $\mathcal{A}_{1}(q)$ are given, in this
approximation, by the expressions

$\displaystyle\widetilde{\mathcal{A}}_{1}(q)=\frac{\pi^{2}eB}{\sqrt{q_{0}^{2}-m^{2}}},$
(74a)
$\displaystyle\widetilde{\mathcal{A}}_{2}(q_{0})=\frac{\pi}{\sqrt{q_{0}^{2}-m^{2}}}.$
(74b)

Applying these asymptotic results for the ultra-strong field regime, and
substituting into the general definitions Eq. (30a) and Eq. (30b), we obtain
explicit analytical expressions for the renormalization factors $z$ and
$z_{3}$, respectively, as follows

$\displaystyle z$ $\displaystyle=$ $\displaystyle
1+\frac{3}{4}\frac{\Delta(eB)e^{-\mathbf{q}_{\perp}^{2}/eB}}{\pi\sqrt{q_{0}^{2}-m^{2}}},$
(75)

and

$\displaystyle z_{3}$ $\displaystyle=$
$\displaystyle\left(1+\frac{\Delta(eB)e^{-\mathbf{q}_{\perp}^{2}/eB}}{4\pi\sqrt{q_{0}^{2}-m^{2}}}\right)z^{-1}$
(76) $\displaystyle=$
$\displaystyle\frac{1+\frac{\Delta(eB)e^{-\mathbf{q}_{\perp}^{2}/(eB)}}{4\pi\sqrt{q_{0}^{2}-m^{2}}}}{1+\frac{3}{4}\frac{\Delta(eB)e^{-\mathbf{q}_{\perp}^{2}/(eB)}}{\pi\sqrt{q_{0}^{2}-m^{2}}}}$

Remarkably, while for very large magnetic fields $z\sim eB/m^{2}$ grows
linearly, $z_{3}$ instead converges asymptotically to the constant limit

$\displaystyle\lim_{\frac{eB}{m^{2}}\rightarrow\infty}z_{3}=\frac{1}{3}.$ (77)

Figure 5: Wavefunction renormalization factor $z$ as a function of the
dimensionless energy scale $q_{0}/m$. Here
$\mathbf{p}_{\perp}^{2}=p_{\parallel}^{2}-m^{2}$. Figure 6: Wavefunction
renormalization factor $z$ as a function of the dimensionless magnetic field
scale $eB/m^{2}$. Here $\mathbf{p}_{\perp}^{2}=p_{\parallel}^{2}-m^{2}$, and
$eB/m^{2}\in[10,500]$ Figure 7: Charge renormalization factor $z_{3}$ as a
function of the dimensionless energy scale $q_{0}/m$. Here
$\mathbf{p}_{\perp}^{2}=p_{\parallel}^{2}-m^{2}$. Figure 8: Charge
renormalization factor $z_{3}$ as a function of the dimensionless magnetic
field scale $eB/m^{2}$. Here $\mathbf{p}_{\perp}^{2}=p_{\parallel}^{2}-m^{2}$,
and $eB/m^{2}\in[10,500]$ Figure 9: Effective refraction index $v^{\prime}/c$
as a function of the dimensionless energy scale $q_{0}/m$. Here
$\mathbf{p}_{\perp}^{2}=p_{\parallel}^{2}-m^{2}$. Figure 10: Effective
refraction index $v^{\prime}/c$ as a function of the dimensionless magnetic
field scale $eB/m^{2}$. Here $\mathbf{p}_{\perp}^{2}=p_{\parallel}^{2}-m^{2}$,
and $eB/m^{2}\in[10,500]$

Let us now summarize the behavior of the renormalization factors $z$ and
$z_{3}$ as a function of the energy scale $q_{0}/m$, and the average
background magnetic field $eB/m^{2}$, respectively, in the whole range of both
parameters, as displayed in Figs. 5–8, respectively. As can be appreciated in
Fig. 5, $z$ presents a monotonically decreasing behaviour as a function of the
energy $q_{0}/m$, that asymptotically reaches the limit $z\rightarrow 1$ as
$q_{0}/m\gg 1$, for all values of the average background magnetic field
$eB/m^{2}$. In physical terms, this shows that the quasi-particle
renormalization due to the random magnetic field fluctuations tends to be
negligible as the energy of the propagating fermions becomes very large, but
in contrast it can be quite significant at low energy scales. This trend is
also consistent with the effective refraction index $v^{\prime}/c=z^{-1}$, as
shown in Figs. 9,10. For low energy scales, $v^{\prime}/c<1$, indicating a
strong renormalization of the effective group velocity of the propagating
quasi-particles due to the presence of the magnetic background fluctuations.
In contrast, for larger energy scales the effect becomes weaker, thus
recovering the asymptotic limit $v^{\prime}/c\rightarrow 1$ as $q_{0}/m\gg 1$.
Since low energy and momentum components in the Fourier representation of the
propagator correspond to long-wavelength components in the space of
configurations, our results are consistent with the fact that such long-
wavelength components are more sensitive to the spatial distribution of the
magnetic fluctuations, and hence experience a higher degree of decoherence,
thus reducing the corresponding group velocity. In contrast, the high-energy
Fourier modes of the propagator, that correspond to short-wavelength
components in the configuration space, are less sensitive to the presence of
spatial fluctuations of the background magnetic field.

Concerning the charge renormalization factor $z_{3}$, as can be appreciated in
Fig.7 it experiences a strong effect $z_{3}<1$ at low quasi-particle energies
$q_{0}/m$, but this effect becomes negligible a large energy scales
$q_{0}/m\gg 1$, since $z_{3}\rightarrow 1$ as an asymptotic limit. This
behavior, that can be interpreted physically as a charge screening due to the
spatial magnetic fluctuations in the background, is consistent with the
aforementioned interpretation for the effective index of refraction as a
function of energy. On the other hand, as can be appreciated in Fig. 8,
$z_{3}$ tends to decrease as a function of the average background magnetic
field intensity, achieving an asymptotic limit $z_{3}\rightarrow 1/3$ as shown
in Eq. (77).

##  VI Vertex corrections at $O(\Delta^{2})$

Let us now consider the renormalization of the effective interaction term
$\Delta\rightarrow\tilde{\Delta}$, that characterizes the strength of the
effective interaction vertex in the effective, averaged action for the replica
system, Eq. (10). Following the skeleton diagrams for the perturbation theory,
as depicted in Fig. 2, the diagrams contributing at order $\Delta^{2}$ to the
4-point vertex $\hat{\Gamma}$ are depicted in Fig. 11.

Figure 11: Diagrams contributing to the 4-point vertex function $\hat{\Gamma}$
at order $\Delta^{2}$.

Therefore, the matrix elements corresponding to each diagram are given by the
following integral expressions

$\displaystyle\hat{\Gamma}_{\text{(a)}}=\int\frac{d^{3}q}{(2\pi)^{3}}\,\gamma^{i}S_{\text{F}}(p-q)\gamma^{j}\otimes\gamma_{i}S_{\text{F}}(p^{\prime}-q)\gamma_{j},$
(78a)
$\displaystyle\hat{\Gamma}_{\text{(b)}}=\int\frac{d^{3}q}{(2\pi)^{3}}\,\gamma^{i}S_{\text{F}}(p-q)\gamma^{j}\otimes\gamma_{i}S_{\text{F}}(p^{\prime}+q)\gamma_{j},$
(78b) and
$\displaystyle\hat{\Gamma}_{\text{(c)}}=\int\frac{d^{3}q}{(2\pi)^{3}}\,\gamma^{i}S_{\text{F}}(p+q)\gamma^{j}\otimes\gamma_{i}S_{\text{F}}(p^{\prime}-q)\gamma_{j}.$
(78c)

In order to compute the expressions above, it is convenient to introduce the
notation

$\displaystyle\hat{\Gamma}^{{}^{(\lambda,\sigma)}}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\,\gamma^{i}S_{\text{F}}(p+\lambda
q)\gamma^{j}\otimes\gamma_{i}S_{\text{F}}(p^{\prime}+\sigma q)\gamma_{j},$

where $\lambda,\sigma=\pm 1$. Then, we have the correspondence
$\hat{\Gamma}_{\text{(a)}}=\hat{\Gamma}^{(-,-)}$,
$\hat{\Gamma}_{\text{(b)}}=\hat{\Gamma}^{(-,+)}$, and
$\hat{\Gamma}_{\text{(c)}}=\hat{\Gamma}^{(+,-)}$, respectively.

Figure 12: real and imaginary parts of the correction coefficient from Eq.
(82) as a function of the fermion’s energy. Here, we take $Q=0$ and
$\mathbf{p}_{\perp}^{2}=p_{0}^{2}-m^{2}$.

By considering the tensor structure of the propagator, it is straightforward
to realize that the full vertex, taking into account the multiplicity and
symmetry factors for each diagram, is given by

$\displaystyle\hat{\Gamma}=2\hat{\Gamma}^{(-,-)}+2\hat{\Gamma}^{(-,+)}+4\hat{\Gamma}^{(+,-)}.$
(80)

The former leads to an effective interaction of the form

$\displaystyle\hat{\Gamma}=\tilde{\Delta}(\bar{\psi}\gamma^{i}\psi)(\bar{\psi}\gamma^{i}\psi)+{\rm{other\,\,tensor\,\,structures}},$
(81)

where the renormalized coefficient $\tilde{\Delta}$ is given, up to second
order in $\Delta$ (see Appendix D for details) by the expression

$\displaystyle\tilde{\Delta}=\Delta+2\Delta^{2}\left(\mathcal{J}_{2}^{(-,-)}+\mathcal{J}_{2}^{(-,+)}+2\mathcal{J}_{2}^{(+,-)}\right.$
$\displaystyle\left.+\left(1-\partial_{x}^{2}\right)(1-\partial_{y}^{2})\mathcal{J}_{3}^{(-,-)}+\left(1-\partial_{x}^{2}\right)(1-\partial_{y}^{2})\mathcal{J}_{3}^{(-,+)}\right.$
$\displaystyle\left.+2\left(1-\partial_{x}^{2}\right)(1-\partial_{y}^{2})\mathcal{J}_{3}^{(+,-)}\right).$
(82)

Here, we defined the integrals

$\displaystyle\mathcal{J}_{1}^{(\lambda,\sigma)}(p,p^{\prime})\equiv\int\frac{d^{3}q}{(2\pi)^{3}}\mathcal{A}_{1}(p+\lambda
q)\mathcal{A}_{1}(p^{\prime}+\sigma q),$
$\displaystyle\mathcal{J}_{2}^{(\lambda,\sigma)}(p,p^{\prime})\equiv\int\frac{d^{3}q}{(2\pi)^{3}}q_{\parallel}^{2}\mathcal{A}_{1}(p+\lambda
q)\mathcal{A}_{1}(p^{\prime}+\sigma q),$ and
$\displaystyle\mathcal{J}_{3}^{(\lambda,\sigma)}(p,p^{\prime})\equiv\int\frac{d^{3}q}{(2\pi)^{3}}\mathbf{q}_{\perp}^{2}\mathcal{A}_{1}(p+\lambda
q)\mathcal{A}_{1}(p^{\prime}+\sigma q).$ (83c)

In order to calculate the integrals $\mathcal{J}_{i}$, we shall use the
analytical expression for $\mathcal{A}_{1}(k)$ Eq. (LABEL:eq_A1_U) (details in
Appendix A):

$\displaystyle\mathcal{A}_{1}(k)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}e^{-\mathbf{k}_{\perp}^{2}/eB}}{2eB}\exp\left[-\frac{\mathrm{i}\pi\left(k_{\parallel}^{2}-m^{2}\right)}{2eB}\right]$
$\displaystyle\times$
$\displaystyle\Gamma\left(-\frac{k_{\parallel}^{2}-m^{2}}{2eB}\right)U\left(-\frac{k_{\parallel}^{2}-m^{2}}{2eB},0,\frac{2\mathbf{k}_{\perp}^{2}}{eB}\right).$

###  VI.1 The integral $\mathcal{J}_{1}$

Let us first consider the integral

$\displaystyle\mathcal{J}_{1}^{(\lambda,\sigma)}(p,p^{\prime})=\int\frac{d^{3}q}{(2\pi)^{3}}\mathcal{A}_{1}(p+\lambda
q)\mathcal{A}_{1}(p^{\prime}+\sigma q).$ (85)

For the case $(\lambda,\sigma)=(-1,-1)$ we change the integration variables as
follows

$\displaystyle p^{\prime}-q$ $\displaystyle=$ $\displaystyle q^{\prime}+Q,$
$\displaystyle p-q$ $\displaystyle=$ $\displaystyle q^{\prime}-Q.$ (86)

For notational simplicity, in what follows we shall use $q$ instead of
$q^{\prime}$, and we shall define the parameters

$\displaystyle a$ $\displaystyle=$
$\displaystyle-\frac{\mathcal{D}_{\parallel}(q_{3}+Q_{\parallel})}{2eB},$
$\displaystyle a^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{\mathcal{D}_{\parallel}(q_{3}-Q_{\parallel})}{2eB}.$ (87)

Furthermore, we shall use the identity (for $z=2\mathbf{q}_{\perp}^{2}/eB$)

$\displaystyle\Gamma(a)U(a,\epsilon,z)=\frac{1}{a}M(a,\epsilon,z)+\Gamma(-1+\epsilon)zM(1+a,2,z),$

along with the singular expansion for $\epsilon\rightarrow 0^{+}$

$\displaystyle\Gamma(-1+\epsilon)=\frac{-1}{\epsilon}+\gamma_{e}-1+O(\epsilon),$
(89)

where $\gamma_{e}=0.577$ is the Euler-Mascheroni constant. In addition, given
that there is a strong exponential damping in the integral, we consider the
expansion of the Kummer function for small values of its argument, that is
given by

$\displaystyle M(a,b,z)=1+\frac{a}{b}z+O(z^{2}),$ (90)

so that, after regularization by removing the divergences in $1/\epsilon$, we
end up with the integral

$\displaystyle\mathcal{J}_{1}^{(-,-)}(p,p^{\prime})$ $\displaystyle=$
$\displaystyle\left(\frac{\mathrm{i}}{2eB}\right)^{2}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}\int_{-\infty}^{\infty}\frac{dq_{3}}{2\pi}~{}e^{\mathrm{i}\pi\left(a+a^{\prime}\right)}\int_{0}^{\infty}\frac{d^{2}q_{\perp}}{(2\pi)^{2}}e^{-\frac{2\mathbf{q}^{2}_{\perp}}{eB}}~{}\Gamma\left(a\right)U\left(a,\epsilon,\frac{(q+Q)_{\perp}^{2}}{eB}\right)\Gamma\left(a^{\prime}\right)U\left(a^{\prime},\epsilon,\frac{(q-Q)_{\perp}^{2}}{eB}\right)$
(91) $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\left(\frac{\mathrm{i}}{2eB}\right)^{2}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}\int_{-\infty}^{\infty}dq_{3}~{}e^{\mathrm{i}\pi\left(a+a^{\prime}\right)}$
$\displaystyle\times$
$\displaystyle\int_{0}^{\infty}d^{2}q_{\perp}e^{-\frac{2\mathbf{q}^{2}_{\perp}}{eB}}\left[\frac{1}{a}+\frac{\gamma_{e}-1}{eB}\left(\mathbf{q}_{\perp}^{2}+2\mathbf{Q}_{\perp}\cdot\mathbf{q}_{\perp}+\mathbf{Q}_{\perp}^{2}\right)\right]\left[\frac{1}{a^{\prime}}+\frac{\gamma_{e}-1}{eB}\left(\mathbf{q}_{\perp}^{2}-2\mathbf{Q}_{\perp}\cdot\mathbf{q}_{\perp}+\mathbf{Q}_{\perp}^{2}\right)\right].$

Furthermore, we shall set the external 3-momenta to zero, except for the
presence of the $Q_{\perp}$ factors, that we shall keep as finite in order to
use this expression as a generating function. Therefore, the integral reduces
to the simpler expression

$\displaystyle\mathcal{J}_{1}^{(-,-)}(p,p^{\prime})=-\frac{eB}{4\pi^{2}}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}$
$\displaystyle\times$
$\displaystyle\int_{-\infty}^{\infty}\frac{dq_{3}}{\left(q_{3}^{2}+m^{2}+\mathrm{i}\epsilon\right)^{2}}\exp\left[\frac{\mathrm{i}\pi}{2eB}\left(q_{3}^{2}+m^{2}\right)\right]$
$\displaystyle\times$
$\displaystyle\Bigg{[}1+\frac{(\gamma_{e}-1)\mathbf{Q}_{\perp}^{2}\left(q_{3}^{2}+m^{2}\right)}{(eB)^{2}}+\frac{(\gamma_{e}-1)\left(q_{3}^{2}+m^{2}\right)}{eB}\Bigg{]}$

Performing the last integral explicitly, we obtain (details in Appendix D)

$\displaystyle\mathcal{J}_{1}^{(-,-)}(p,p^{\prime})=-\frac{eB}{4\pi^{2}}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}\Bigg{\\{}\frac{(1-\mathrm{i})\pi}{2\sqrt{eB}m^{2}}e^{\frac{\mathrm{i}\pi
m^{2}}{2eB}}$ $\displaystyle+$ $\displaystyle\frac{(eB+\mathrm{i}\pi
m^{2})\pi}{2(eB)m^{3}}\left[1-\sqrt{\frac{\pi}{2}}\text{erf}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\frac{m}{\sqrt{eB}}\right)\right]$
$\displaystyle+$
$\displaystyle\frac{\pi(\gamma_{e}-1)}{(eB)m}\left(1+\frac{\mathbf{Q}_{\perp}^{2}}{eB}\right)\left[1-\sqrt{\frac{\pi}{2}}\text{erf}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\frac{m}{\sqrt{eB}}\right)\right]\Bigg{\\}},$

where $\text{erf}(x)$ is the error function.

###  VI.2 The integral $\mathcal{J}_{2}$

For this second integral, we notice that $q_{\parallel}^{2}=-q_{3}^{2}$, so
that after integration over $z=2\mathbf{q}_{\perp}^{2}/(eB)$ we obtain
(details in Appendix D)

$\displaystyle\mathcal{J}_{2}^{(-,-)}(p,p^{\prime})=\frac{eB}{4\pi^{2}}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}$
$\displaystyle\times$
$\displaystyle\int_{-\infty}^{\infty}dq_{3}\frac{q_{3}^{2}}{\left(q_{3}^{2}+m^{2}+\mathrm{i}\epsilon\right)^{2}}\exp\left[\frac{\mathrm{i}\pi}{2eB}\left(q_{3}^{2}+m^{2}\right)\right]$
$\displaystyle\times$
$\displaystyle\Bigg{[}1+\frac{(\gamma_{e}-1)\mathbf{Q}_{\perp}^{2}\left(q_{3}^{2}+m^{2}\right)}{(eB)^{2}}+\frac{(\gamma_{e}-1)\left(q_{3}^{2}+m^{2}\right)}{eB}\Bigg{]}$

After performing the remaining momentum integral over $q_{3}$, as shown in
detail in Appendix D, we finally obtain

$\displaystyle\mathcal{J}_{2}^{(-,-)}(p,p^{\prime})=\frac{eB}{4\pi^{2}}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}$
$\displaystyle\times$
$\displaystyle\Bigg{\\{}\frac{(\mathrm{i}-1)\pi}{\sqrt{eB}}e^{\frac{\mathrm{i}\pi
m^{2}}{2eB}}$ $\displaystyle+$ $\displaystyle\frac{\left(eB-\mathrm{i}\pi
m^{2}\right)\pi}{2(eB)m}\left[1-\sqrt{\frac{\pi}{2}}\text{erf}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\frac{m}{\sqrt{eB}}\right)\right]$
$\displaystyle-$
$\displaystyle\frac{m\pi(\gamma_{e}-1)}{(eB)}\left(1+\frac{\mathbf{Q}_{\perp}^{2}}{eB}\right)\left[1-\sqrt{\frac{\pi}{2}}\text{erf}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\frac{m}{\sqrt{eB}}\right)\right]\Bigg{\\}}.$

###  VI.3 The integral $\mathcal{J}_{3}$

By following the same procedure as in the previous two cases, as shown in
Appendix D, it is straightforward to obtain the analytical expression

$\displaystyle\mathcal{J}_{3}^{(-,-)}(p,p^{\prime})=-\frac{eB^{2}}{8\pi^{2}}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}\Bigg{\\{}\frac{(1-\mathrm{i})\pi}{2\sqrt{eB}m^{2}}e^{\frac{\mathrm{i}\pi
m^{2}}{2eB}}$ $\displaystyle+$ $\displaystyle\frac{(eB+\mathrm{i}\pi
m^{2})\pi}{2(eB)m^{3}}\left[1-\sqrt{\frac{\pi}{2}}\text{erf}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\frac{m}{\sqrt{eB}}\right)\right]$
$\displaystyle+$
$\displaystyle\frac{\pi(\gamma_{e}-1)\mathbf{Q}_{\perp}^{2}}{(eB)^{2}m}\left[1-\sqrt{\frac{\pi}{2}}\text{erf}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\frac{m}{\sqrt{eB}}\right)\right]\Bigg{\\}}.$

Moreover, it is straightforward to verify the following relations

$\displaystyle\mathcal{J}_{n}^{(+,+)}(p,p^{\prime})$ $\displaystyle=$
$\displaystyle\mathcal{J}_{n}^{(-,-)}(p,p^{\prime})$
$\displaystyle\mathcal{J}_{n}^{(+,-)}(p,p^{\prime})$ $\displaystyle=$
$\displaystyle\mathcal{J}_{n}^{(-,+)}(p,p^{\prime})=\mathcal{J}_{n}^{(-,-)}(p,p^{\prime})\mid_{Q\rightarrow
P},$

for $n=1,2,3$, that allows us to generate all the remaining expressions from
these three explicit analytical results.

An explicit numerical evaluation of our analytical expressions for the
renormalized effective interaction, expressed by the combination
$(\tilde{\Delta}-\Delta)/(2\Delta^{2})$, is displayed in Fig. 12. Clearly,
this effective coupling develops both a real as well as an imaginary part
(left and right panels in Fig. 12, respectively). In particular, the emergence
of an imaginary component implies an imaginary contribution to the self-
energy, corresponding to a relaxation time (spectral broadening) of the quasi-
particle spectrum. This is a natural consequence of the decoherence mechanism
induced by the random fluctuating magnetic environment. On the other hand, as
can be appreciated in Fig. 12, both the real and imaginary contributions to
the effective interaction $\tilde{\Delta}$ display a large enhancement (in
absolute value) at low energy scales $p_{0}/m<1$, while asymptotically
$\tilde{\Delta}\rightarrow\Delta$ at higher energies $p_{0}/m\gg 1$. This
strong renormalization effect at low-energies is consistent with the effect
observed in the previous section for the charge $z_{3}$ and refraction index
$v^{\prime}/c$, respectively, and can be explained in similar terms due to the
short-range spatial distribution of the magnetic noise, that therefore
renormalizes mainly the long-wavelength components of the propagator,
corresponding to the small energy-momentum components in Fourier space.

##  VII Conclusions

We have studied the effects of quenched, white noise spatial fluctuations in
an otherwise uniform background magnetic field, over the properties of the QED
fermion propagator. This configuration is important in different physical
scenarios, including heavy-ion collisions and the quark-gluon plasma, where
spatial anisotropies of the background magnetic field may be present. We
developed explicit results, that we carried over by combining the replica
method to average over spatial fluctuations, with a perturbation theory based
on the Schwinger propagator for the average background field. Upon averaging
over magnetic fluctuations, we obtained an effective action in the replica
fields, with an effective particle-particle interaction proportional to the
strength $\Delta$ of the spatial auto-correlation function of the background
noise. Our perturbative results show that, up to first order in $\Delta$, the
propagator retains its form, thus representing renormalized quasi-particles
with the same mass $m^{\prime}=m$, but propagating in the medium with a
magnetic field and noise-dependent index of refraction $v^{\prime}/c=z^{-1}$,
and effective charge $e^{\prime}=z_{3}e$, where $z$ and $z_{3}$ are
renormalization factors. We showed that $z$ presents a monotonically
decreasing behaviour as a function of the energy $q_{0}/m$, that reaches the
asymptotic limit $z\rightarrow 1$ as $q_{0}/m\gg 1$, for all values of the
average background magnetic field $eB/m^{2}$. In physical terms, this shows
that the quasi-particle renormalization due to the random magnetic field
fluctuations, while being quite significant at low energy scales, tends to be
negligible as the energy of the propagating fermions becomes very large. This
trend is also observed in the effective refraction index
$v^{\prime}/c=z^{-1}$, since at low energy scales $v^{\prime}/c<1$, indicating
a strong renormalization of the effective group velocity of the propagating
quasi-particles due to the presence of the magnetic background fluctuations.
In contrast, for larger energy scales the effect becomes weaker, thus
recovering the asymptotic limit $v^{\prime}/c\rightarrow 1$ as $q_{0}/m\gg 1$.

Our results show that the effective quasi-particle charge experiences a strong
renormalization $z_{3}<1$ at low energies $q_{0}/m$, while the effect becomes
negligible a large energy scales $q_{0}/m\gg 1$, since $z_{3}\rightarrow 1$ as
an asymptotic limit. We interpret this as a charge screening due to the
spatial magnetic fluctuations in the background. On the other hand, $z_{3}$
tends to decrease as a function of the average background magnetic field
intensity, achieving an asymptotic limit $z_{3}\rightarrow 1/3$ as shown in
Eq. (77).

We remark that both the effective refraction index $v^{\prime}/c=z^{-1}$ and
charge screening $z_{3}$ display a similar, and therefore consistent,
renormalization behaviour of the quasi-particle properties in the magnetically
fluctuating environment. In order to understand such effects in physical
terms, we remark that the low energy and momentum components in the Fourier
representation of the propagator correspond to long-wavelength components in
the space of configurations. Therefore, our results are consistent with the
fact that such long-wavelength components are more sensitive to the spatial
distribution of the background magnetic noise, and hence experience a higher
degree of decoherence, thus reducing the corresponding group velocity and
enhancing the charge screening. In contrast, the high-energy Fourier
components of the propagator, that correspond to short-wavelength components
in the configuration space, are less sensitive to the presence of spatial
fluctuations of the background magnetic field. In addition, we remark that the
intensity of the average background magnetic field defines a characteristic
length-scale known as the Landau radius $l_{B}=1/\sqrt{eB}$, that determines
the support of the quasi-particle propagator in configuration space. Moreover,
in the semi-classical picture this length-scale represents the typical size of
the “cyclotron radius” of the helycoidal trajectories that propagate along the
magnetic field axis. Therefore, the stronger the magnetic field, the smaller
the Landau radius, and hence the quasi-particle propagator is modulated
towards higher momentum and energy components that, as previously discussed,
are more sensitive to the magnetic noise renormalization effects, as is
verified by the trend observed both in $z_{3}$ and in $v^{\prime}/c$, that
strongly decrease as the average magnetic field intensity increases
$eB/m^{2}\gg 1$.

Moreover, we also showed that 4-point vertex corrections at the second order
in $\Delta^{2}$ lead to a renormalized $\tilde{\Delta}=\Delta+O(\Delta^{2})$,
whose relative magnitude grows with the average magnetic field intensity
$eB/m^{2}$, and tend to decrease with the quasi-particle energy scale
$p_{0}/m$, in agreement with the behavior of $v^{\prime}/c$ and $z_{3}$ and
the physical interpretation previously discussed.

The analysis and results presented in this work only concern the study of the
quasi-particle fermion propagator in the noisy magnetic field background.
However, the effective model obtained via the replica method and its
consequences can be extended towards the study of other physical quantities,
such as the photon polarization tensor. We are currently investigating this
and it will be communicated in a separate article.

###### Acknowledgements.

J.D.C.-Y. and E.M. acknowledge financial support from ANID PIA Anillo
ACT/192023. E.M. also acknowledges financial support from Fondecyt 1190361. M.
L. acknowledges support from ANID/CONICYT FONDECYT Regular (Chile) under
grants No. 1200483, 1190192 and 1220035. M. L. also acknowledges support from
ANID/PIA/APOYO AFB180002 (Chile).

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##  Appendix A The coefficient $\mathcal{A}_{1}$

We can calculate the integral $\mathcal{A}_{1}$ in terms of Landau levels by
means of the generating function of the Laguerre polynomials

$\displaystyle
e^{-\frac{x}{2}\frac{1-t}{1+t}}=(1+t)e^{-x/2}\sum_{n=0}^{\infty}(-t)^{n}L_{n}^{0}(x),$
(97)

since

$\displaystyle e^{-\mathrm{i}x\tan v}$ $\displaystyle=$ $\displaystyle
e^{-x\left(1-e^{-2\mathrm{i}v}\right)/\left(1+e^{-2\mathrm{i}v}\right)}$
$\displaystyle=$
$\displaystyle\left(1+e^{-2\mathrm{i}v}\right)e^{-x}\sum_{n=0}^{\infty}(-1)^{n}e^{-2\mathrm{i}nv}L_{n}^{0}(2x)$

Therefore, we have (for $x=\mathbf{k}_{\perp}^{2}/eB$)

$\displaystyle\mathcal{A}_{1}=e^{-x}\left(\sum_{n=0}^{\infty}(-1)^{n}L_{n}^{0}(2x)\int_{0}^{\infty}d\tau
e^{\mathrm{i}(\mathcal{D}_{\parallel}-2(n+1)eB)\tau}\right.$
$\displaystyle\left.+\sum_{n=0}^{\infty}(-1)^{n}L_{n}^{0}(2x)\int_{0}^{\infty}d\tau
e^{\mathrm{i}(\mathcal{D}_{\parallel}-2neB)\tau}\right)$ (99)

Evaluating the exponential integrals,

$\displaystyle\mathcal{A}_{1}$ $\displaystyle=$
$\displaystyle\mathrm{i}e^{-x}\left(\sum_{n=0}^{\infty}(-1)^{n}\frac{L_{n}^{0}(2x)}{\mathcal{D}_{\parallel}-2(n+1)eB}\right.$
(100)
$\displaystyle\left.+\sum_{n=0}^{\infty}(-1)^{n}\frac{L_{n}^{0}(2x)}{\mathcal{D}_{\parallel}-2neB}\right)$
$\displaystyle=$
$\displaystyle\mathrm{i}\frac{e^{-x}}{\mathcal{D}_{\parallel}}\left[1+\sum_{n=1}^{\infty}\frac{(-1)^{n}\left[L_{n}^{0}(2x)-L_{n-1}^{0}(2x)\right]}{1-2n\frac{eB}{\mathcal{D}_{\parallel}}}\right]$

This expansion seems is fair for $eB>0$. However, we would like to inspect if
it is possible to use it to generate a valid expansion near $eB=0$.

###  A.1 Expansion for low magnetic fields

Notice that, from Eq. (V.2) we have the identities

$\displaystyle
e^{-x}\sum_{n=0}^{\infty}(-1)^{n}e^{-2\mathrm{i}nv}L_{n}^{0}(2x)=\frac{e^{-\mathrm{i}x\tan
v}}{1+e^{-2\mathrm{i}v}}$ (101) $\displaystyle
e^{-x}\sum_{n=0}^{\infty}(-1)^{n}e^{-2\mathrm{i}(n+1)v}L_{n}^{0}(2x)=\frac{e^{-2\mathrm{i}v}e^{-\mathrm{i}x\tan
v}}{1+e^{-2\mathrm{i}v}}$ (102)

Therefore, we can calculate the following expansions

$\displaystyle
e^{-x}\sum_{n=0}^{\infty}\frac{(-1)^{n}L_{n}^{0}(2x)}{\mathcal{D}_{\parallel}-2(n+1)eB}=\frac{1}{\mathcal{D}_{\parallel}}\sum_{k=0}^{\infty}\left(\frac{2eB}{\mathcal{D}_{\parallel}}\right)^{k}$
$\displaystyle\times\left[e^{-x}\sum_{n=0}^{\infty}(-1)^{n}(n+1)^{k}L_{n}^{0}(2x)\right]$
$\displaystyle=\frac{1}{\mathcal{D}_{\parallel}}\sum_{k=0}^{\infty}\left(\frac{2eB}{\mathcal{D}_{\parallel}}\right)^{k}\frac{1}{(-2\mathrm{i})^{k}}\left.\frac{\partial^{k}}{\partial
v^{k}}\left(\frac{e^{-2\mathrm{i}v}e^{-\mathrm{i}x\tan
v}}{1+e^{-2\mathrm{i}v}}\right)\right|_{v\rightarrow 0}$ (103)

and similarly

$\displaystyle
e^{-x}\sum_{n=0}^{\infty}\frac{(-1)^{n}L_{n}^{0}(2x)}{\mathcal{D}_{\parallel}-2neB}=\frac{1}{\mathcal{D}_{\parallel}}\sum_{k=0}^{\infty}\left(\frac{2eB}{\mathcal{D}_{\parallel}}\right)^{k}$
$\displaystyle\times\left[e^{-x}\sum_{n=0}^{\infty}(-1)^{n}n^{k}L_{n}^{0}(2x)\right]$
$\displaystyle=\frac{1}{\mathcal{D}_{\parallel}}\sum_{k=0}^{\infty}\left(\frac{2eB}{\mathcal{D}_{\parallel}}\right)^{k}\frac{1}{(-2\mathrm{i})^{k}}\left.\frac{\partial^{k}}{\partial
v^{k}}\left(\frac{e^{-\mathrm{i}x\tan
v}}{1+e^{-2\mathrm{i}v}}\right)\right|_{v\rightarrow 0}$ (104)

Substituting both expressions into Eq.(100), we obtain the infinite series

$\displaystyle\mathcal{A}_{1}=\frac{\mathrm{i}}{\mathcal{D}_{\parallel}}\left(1+\sum_{k=1}^{\infty}\left(\frac{\mathrm{i}eB}{\mathcal{D}_{\parallel}}\right)^{k}\mathcal{E}_{k}(x)\right)$
(105)

Here, we have defined the polynomials $\mathcal{E}_{k}(x)$ (for
$x=k_{\perp}^{2}/eB$) as generated by the function $e^{-\mathrm{i}x\tan v}$,

$\displaystyle\mathcal{E}_{k}(x)=\lim_{v\rightarrow
0}\frac{\partial^{k}}{\partial v^{k}}\left(e^{-\mathrm{i}x\tan v}\right)$
(106)

The first few cases

$\displaystyle\mathcal{E}_{1}(x)$ $\displaystyle=$ $\displaystyle-\mathrm{i}x$
$\displaystyle\mathcal{E}_{2}(x)$ $\displaystyle=$ $\displaystyle-x^{2}$
$\displaystyle\mathcal{E}_{3}(x)$ $\displaystyle=$
$\displaystyle-2\mathrm{i}x+\mathrm{i}x^{3}$ $\displaystyle\mathcal{E}_{4}(x)$
$\displaystyle=$ $\displaystyle x^{4}-8x^{2}$
$\displaystyle\mathcal{E}_{5}(x)$ $\displaystyle=$
$\displaystyle-\mathrm{i}x^{5}+20\mathrm{i}x^{3}-16\mathrm{i}x$
$\displaystyle\mathcal{E}_{6}(x)$ $\displaystyle=$
$\displaystyle-x^{6}+40x^{4}-136x^{2}$ (107) $\displaystyle\vdots$ (108)

Substituting these expressions into the series Eq. (34), it can be reorganized
as an expansion in terms of the variable
$y=\mathbf{k}_{\perp}^{2}/\mathcal{D}_{\parallel}$, as follows

$\displaystyle\mathcal{A}_{1}$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}}{\mathcal{D}_{\parallel}}\left[(1+y+y^{2}+y^{3}+\ldots)\right.$
(109)
$\displaystyle\left.-2\left(\frac{eB}{\mathcal{D}_{\parallel}}\right)^{2}y\left(1+4y+10y^{2}+20y^{4}+\ldots\right)\right.$
$\displaystyle\left.+8\left(\frac{eB}{\mathcal{D}_{\parallel}}\right)^{4}y\left(2+17y+77y^{2}+\ldots\right)\right]+O\left(\frac{eB}{\mathcal{D}_{\parallel}}\right)^{6}$
$\displaystyle=$
$\displaystyle\frac{\mathrm{i}}{\mathcal{D}_{\parallel}}\left[\frac{1}{1-y}-2\left(\frac{eB}{\mathcal{D}_{\parallel}}\right)^{2}\frac{y}{(1-y)^{4}}\right.$
$\displaystyle\left.+8\left(\frac{eB}{\mathcal{D}_{\parallel}}\right)^{4}\frac{y(3y+2)}{(1-y)^{7}}\right]+O\left(\frac{eB}{\mathcal{D}_{\parallel}}\right)^{6}$

Finally, using the simple identity

$\displaystyle\mathcal{D}_{\parallel}(1-y)=\mathcal{D}_{\parallel}(1-\frac{\mathbf{k}_{\perp}^{2}}{\mathcal{D}_{\parallel}})=k^{2}-m^{2}+\mathrm{i}\epsilon$
(110)

we obtain the final form

$\displaystyle\mathcal{A}_{1}$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}}{k^{2}-m^{2}+\mathrm{i}\epsilon}\left[1-2\left(\frac{eB}{\mathcal{D}_{\parallel}}\right)^{2}\frac{y}{(1-y)^{3}}\right.$
$\displaystyle\left.+8\left(\frac{eB}{\mathcal{D}_{\parallel}}\right)^{4}\frac{y(3y+2)}{(1-y)^{6}}\right]+O(\frac{eB}{\mathcal{D}_{\parallel}})^{6}$
$\displaystyle=$
$\displaystyle\frac{\mathrm{i}}{k^{2}-m^{2}+\mathrm{i}\epsilon}+\frac{-2\mathrm{i}(eB)^{2}\mathbf{k}_{\perp}^{2}}{\left[k^{2}-m^{2}+\mathrm{i}\epsilon\right]^{4}}+O((eB)^{4}).$

###  A.2 A closed expression in terms of Hypergeometric functions

In order to simplify the integrals $\mathcal{J}_{i}$, let us to provide an
analytical expression for $\mathcal{A}_{1}(k)$. From the generating function
of the Laguerre’s polynomials:

$\displaystyle\sum_{n=0}^{\infty}(-1)^{n}t^{n}L_{n}^{\alpha}(x)=\frac{1}{(1+t)^{\alpha}}\exp\left(\frac{t}{1+t}x\right),$
(112)

then, by defining:

$\displaystyle b\equiv\frac{t}{1+t},$ (113)

we get:

$\displaystyle(1-b)^{t+\alpha}e^{bx}=\sum_{n=0}^{\infty}(-1)^{n}\frac{b^{n}}{(1-b)^{n}}L_{n}^{\alpha}(x).$
(114)

Multiplying by $b^{\beta}/(1-b)^{\beta+2}$:

$\displaystyle
b^{\beta}(1+b)^{\alpha-\beta-1}e^{-bx}=\sum_{n=0}^{\infty}(-1)^{n}b^{n+\beta}(1-b)^{-n-\beta-2}L_{n}^{\alpha}(x),$

so that:

$\displaystyle\int_{0}^{-\infty}db~{}b^{\beta}(1+b)^{\alpha-\beta-1}e^{-bx}=-\sum_{n=0}^{\infty}\frac{(-1)^{n}L_{n}^{\alpha}(x)}{n+\beta+1}.$

Now by setting $b\to-b$:

$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}L_{n}^{\alpha}(x)}{n+\beta+1}$
$\displaystyle=$
$\displaystyle(-1)^{\beta}\int_{0}^{\infty}db~{}b^{\beta}(1+b)^{\alpha-\beta-1}e^{-bx}$
$\displaystyle=$ $\displaystyle
e^{\mathrm{i}\pi\beta}\Gamma(1+\beta)U(1+\beta,1+\alpha,x),$

where $\Gamma(z)$, and $U(a,b,z)$ are the gamma-function and the confluent
hypergeometric function, respectively.

Now, from Eq. (100):

$\displaystyle\mathcal{A}_{1}$ $\displaystyle=$
$\displaystyle\mathrm{i}e^{-x}\sum_{n=0}^{\infty}\left(\frac{(-1)^{n}L_{n}^{0}(2x)}{\mathcal{D}_{\parallel}-2(n+1)eB}+\frac{(-1)^{n}L_{n}^{0}(2x)}{\mathcal{D}_{\parallel}-2neB}\right)$
$\displaystyle=$
$\displaystyle-\frac{\mathrm{i}e^{-x}}{2eB}\sum_{n=0}^{\infty}\left[\frac{(-1)^{n}L_{n}^{0}(2x)}{n+\left(1-\frac{\mathcal{D}_{\parallel}}{2eB}\right)}+\frac{(-1)^{n}L_{n}^{0}(2x)}{n-\frac{\mathcal{D}_{\parallel}}{2eB}}\right].$

Therefore, by using Eq. (LABEL:ResultSuma):

$\displaystyle\mathcal{A}_{1}$ $\displaystyle=$
$\displaystyle-\frac{\mathrm{i}e^{-x}}{2eB}$ $\displaystyle\times$
$\displaystyle\Bigg{[}e^{-\mathrm{i}\pi\frac{\mathcal{D}_{\parallel}}{2eB}}\Gamma\left(1-\frac{\mathcal{D}_{\parallel}}{2eB}\right)U\left(1-\frac{\mathcal{D}_{\parallel}}{2eB},1,2x\right)$
$\displaystyle+$ $\displaystyle
e^{-\mathrm{i}\pi\left(1+\frac{\mathcal{D}_{\parallel}}{2eB}\right)}\Gamma\left(-\frac{\mathcal{D}_{\parallel}}{2eB}\right)U\left(-\frac{\mathcal{D}_{\parallel}}{2eB},1,2x\right)\Bigg{]}.$

The latter can be simplified with the properties of the gamma and the
hypergeometric functions. First, from the identity $\Gamma(z+1)=z\Gamma(z)$:

$\displaystyle\mathcal{A}_{1}=\frac{\mathrm{i}e^{-x}}{2eB}\exp\left(-\frac{\mathrm{i}\pi\mathcal{D}_{\parallel}}{2eB}\right)\Gamma\left(-\frac{\mathcal{D}_{\parallel}}{2eB}\right)$
$\displaystyle\times$
$\displaystyle\Bigg{[}\frac{\mathcal{D}_{\parallel}}{2eB}U\left(1-\frac{\mathcal{D}_{\parallel}}{2eB},1,2x\right)+U\left(-\frac{\mathcal{D}_{\parallel}}{2eB},1,2x\right)\Bigg{]}.$

Moreover, given that:

$\displaystyle U(a,b,z)-aU(a+1,b,z)=U(a,b-1,z),$ (121)

we finally arrive to:

$\displaystyle\mathcal{A}_{1}(k)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}e^{-\mathbf{k}_{\perp}^{2}/eB}}{2eB}\exp\left[-\frac{\mathrm{i}\pi\left(k_{\parallel}^{2}-m^{2}\right)}{2eB}\right]$
$\displaystyle\times$
$\displaystyle\Gamma\left(-\frac{k_{\parallel}^{2}-m^{2}}{2eB}\right)U\left(-\frac{k_{\parallel}^{2}-m^{2}}{2eB},0,\frac{2\mathbf{k}_{\perp}^{2}}{eB}\right).$

##  Appendix B The density of states $\rho(E)$

In this appendix, we show the details for the calculation of the density of
states for the Landau level spectrum $\rho(E)$ defined in the main text. We
start from the definition of the density of states

$\displaystyle\rho(E)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{dp_{3}}{2\pi}\sum_{n=0}^{\infty}\delta\left(E-\sqrt{p_{3}^{2}+m^{2}+2(n+1)eB}\right)$
(123) $\displaystyle=$ $\displaystyle
2\sum_{n=0}^{\infty}\int_{0}^{\infty}\frac{dp_{3}}{2\pi}\frac{\delta\left(p^{3}-\sqrt{E^{2}}-m^{2}-2(n+1)eB\right)}{p^{3}/E}$
$\displaystyle=$
$\displaystyle\frac{E}{\pi}\sum_{n=0}^{\infty}\frac{\Theta\left(E-\sqrt{m^{2}+2(n+1)eB}\right)}{\sqrt{E^{2}-m^{2}-2(n+1)eB}}$

Since this function, and hence the corresponding sum, is defined for each
fixed value of the energy $E$, there is a maximum integer $n=N_{max}(E)$ at
which the sum is truncated by the condition imposed on the Heaviside step
function

$\displaystyle E-\sqrt{m^{2}+2(N_{max}+1)eB}=0,$ (124)

that leads to the definition (with $\lfloor z\rfloor$ the lowest integer part)

$\displaystyle N_{max}(E)=\lfloor\frac{E^{2}-m^{2}}{2eB}-1\rfloor.$ (125)

Hence, we have

$\displaystyle\rho(E)$ $\displaystyle=$
$\displaystyle\Theta(E-\sqrt{m^{2}+2eB})\frac{E}{\pi}$ (126)
$\displaystyle\times\sum_{n=0}^{N_{max}(E)}\frac{1}{\sqrt{E^{2}-m^{2}-2(n+1)eB}}$

In this finite sum, $0\leq n\leq N_{max}(E)$, we can redefine the index by

$\displaystyle\ell\equiv N_{max}(E)-n\Longrightarrow 0\leq\ell\leq
N_{max}(E),$ (127)

and hence we have the equivalent expression

$\displaystyle\rho(E)$ $\displaystyle=$
$\displaystyle\Theta(E-\sqrt{m^{2}+2eB})\frac{E}{\pi\sqrt{eB}}$ (128)
$\displaystyle\times\sum_{\ell=0}^{N_{max}(E)}\frac{1}{\sqrt{\frac{E^{2}-m^{2}-2(N_{max}(E)+1)eB}{2eB}}+\ell}$

Finally, using the property of the Riemann Zeta function

$\displaystyle\sum_{\ell=0}^{N}\left(z+\ell\right)^{-s}=\zeta(s,z)-\zeta(s,z+N+1),$
(129)

we obtain

$\displaystyle\rho(E)$ $\displaystyle=$
$\displaystyle\Theta(E-\sqrt{m^{2}+2eB})\frac{E}{\pi\sqrt{eB}}$ (130)
$\displaystyle\times\left[\zeta\left(\frac{1}{2},\frac{E^{2}-m^{2}-2eB}{2eB}-N_{max}(E)\right)\right.$
$\displaystyle\left.-\zeta\left(\frac{1}{2},\frac{E^{2}-m^{2}}{2eB}\right)\right]$

##  Appendix C Computing $\tilde{A}_{1}$ and $\tilde{A}_{2}$

###  C.1 Weak magnetic field limit

We need to compute:

$\displaystyle\widetilde{\mathcal{A}}_{1}(q_{0})=-2\mathrm{i}(eB)^{2}\int
d^{3}p\frac{p_{\perp}^{2}}{(q_{0}^{2}-p_{3}^{2}-p_{\perp}^{2}-m^{2}+\mathrm{i}\epsilon)^{4}}$

so that in spherical coordinates, with $p_{3}=p\cos\theta$ and
$p_{\perp}=p\sin\theta$:

$\displaystyle\widetilde{\mathcal{A}}_{1}(q_{0})$ (132) $\displaystyle=$
$\displaystyle-4\pi\mathrm{i}(eB)^{2}\int_{0}^{\pi}d\theta\sin^{3}\theta\int_{0}^{\infty}dp\frac{p^{4}}{(q_{0}^{2}-p^{2}-m^{2}+\mathrm{i}\epsilon)^{4}}$
$\displaystyle=$
$\displaystyle-4\pi\mathrm{i}(eB)^{2}\frac{4}{3}\int_{0}^{\infty}dp\frac{p^{4}}{(q_{0}^{2}-p^{2}-m^{2}+\mathrm{i}\epsilon)^{4}}$
$\displaystyle=$
$\displaystyle-4\pi\mathrm{i}(eB)^{2}\frac{4}{3}\int_{0}^{\infty}dp\frac{p^{4}}{(q_{0}^{2}-p^{2}-m^{2}+\mathrm{i}\epsilon)^{4}}$

By defining

$\displaystyle a^{2}\equiv q_{0}^{2}-m^{2}+\mathrm{i}\epsilon,$ (133)

and

$\displaystyle a_{\pm}=\sqrt{q_{0}^{2}-m^{2}}\pm\mathrm{i}\epsilon,$ (134)

we can use complex integration:

$\displaystyle\widetilde{\mathcal{A}}_{1}(q_{0})$ $\displaystyle=$
$\displaystyle-4\pi\mathrm{i}(eB)^{2}\frac{4}{3}\frac{1}{2}\int_{-\infty}^{\infty}dz\frac{z^{4}}{(z^{2}-a^{2})^{4}}$
(135) $\displaystyle=$
$\displaystyle-4\pi\mathrm{i}(eB)^{2}\frac{4}{3}\frac{1}{2}\frac{2\pi\mathrm{i}}{3!}\lim_{z\to
a_{+}}\frac{d^{3}}{dz^{3}}\left[\frac{z^{4}}{(z-a_{-})^{4}}\right]$
$\displaystyle=$
$\displaystyle-4\pi\mathrm{i}(eB)^{2}\frac{4}{3}\frac{1}{2}\frac{2\pi\mathrm{i}}{3!}\frac{(-3)}{16(q_{0}^{2}-m^{2})^{3/2}}$
$\displaystyle=$
$\displaystyle-\frac{\pi^{2}}{6}\frac{(eB)^{2}}{(q_{0}^{2}-m^{2})^{3/2}}.$

On the other hand, at order $\mathcal{O}((eB)^{2})$:

$\displaystyle\widetilde{\mathcal{A}}_{2}(q_{0})$ $\displaystyle=$
$\displaystyle\mathrm{i}\int_{-\infty}^{+\infty}\frac{dp_{3}}{q_{0}^{2}-(p^{3})^{2}-m^{2}+\mathrm{i}\epsilon}$
(136) $\displaystyle=$
$\displaystyle-\mathrm{i}\int_{-\infty}^{+\infty}\frac{dz}{z^{2}-a^{2}}$
$\displaystyle=$
$\displaystyle-\mathrm{i}\int_{-\infty}^{+\infty}\frac{dz}{(z-a_{+})(z+a_{-})}$

By choosing a contour closing upside, we get from the residue theorem:

$\displaystyle-\mathrm{i}\int_{-\infty}^{+\infty}\frac{dz}{(z-a_{+})(z+a_{-})}=\frac{\pi}{a_{+}}.$
(137)

Then:

$\displaystyle\widetilde{\mathcal{A}}_{2}(q_{0})=\frac{\pi}{\sqrt{q_{0}^{2}-m^{2}}}$
(138)

###  C.2 Arbitrary magnetic field

From the definition of $\widetilde{\mathcal{A}}_{1}$:

$\displaystyle\widetilde{\mathcal{A}}_{1}(q_{0})$ $\displaystyle=$
$\displaystyle\int d^{3}p\mathcal{A}_{1}(q_{0},p_{3};\mathbf{p}_{\perp})$
(139) $\displaystyle=$ $\displaystyle\mathrm{i}\int
d^{3}p\Bigg{[}\frac{e^{-\mathbf{p}_{\perp}^{2}/eB}}{q_{0}^{2}-p_{3}^{2}-m^{2}+\mathrm{i}\epsilon}$
$\displaystyle+$
$\displaystyle\sum_{n=1}^{\infty}(-1)^{n}e^{-\mathbf{p}_{\perp}^{2}/eB}\frac{L_{n}\left(\frac{2\mathbf{p}_{\perp}^{2}}{eB}\right)-L_{n-1}\left(\frac{2\mathbf{p}_{\perp}^{2}}{eB}\right)}{q_{0}^{2}-p_{3}^{2}-m^{2}-2neB+\mathrm{i}\epsilon}\Bigg{]}$
$\displaystyle=$
$\displaystyle\mathcal{I}_{1}+\sum_{n=1}^{\infty}(-1)^{n}\mathcal{I}_{2,n}$

Here, we defined

$\displaystyle\mathcal{I}_{1}$ $\displaystyle=$ $\displaystyle\mathrm{i}\int
d^{3}p\frac{e^{-\mathbf{p}_{\perp}^{2}/eB}}{q_{0}^{2}-p_{3}^{2}-m^{2}+\mathrm{i}\epsilon}$
(140) $\displaystyle=$ $\displaystyle-\mathrm{i}\int
d^{2}p_{\perp}e^{-\mathbf{p}_{\perp}^{2}/eB}\int_{-\infty}^{\infty}\frac{dz}{(z-a-\mathrm{i}\epsilon)(z+a+\mathrm{i}\epsilon)}$
$\displaystyle=$
$\displaystyle-\mathrm{i}\pi(eB)\frac{2\pi\mathrm{i}}{2(a+\mathrm{i}\epsilon)}=\frac{\pi^{2}eB}{\sqrt{q_{0}^{2}-m^{2}+\mathrm{i}\epsilon}},$

and

$\displaystyle\mathcal{I}_{2,n}=\mathrm{i}\int
d^{3}p\,e^{-\mathbf{p}_{\perp}^{2}/eB}\frac{L_{n}\left(\frac{2\mathbf{p}_{\perp}^{2}}{eB}\right)-L_{n-1}\left(\frac{2\mathbf{p}_{\perp}^{2}}{eB}\right)}{q_{0}^{2}-p_{3}^{2}-m^{2}-2neB+\mathrm{i}\epsilon}$
(141)

In cylindrical coordinates, with azimuthal symmetry, $d^{3}p=dp_{3}\pi
d(\mathbf{p}_{\perp}^{2})$. Moreover, in the integral over
$\mathbf{p}_{\perp}$, define $x=\frac{2\mathbf{p}_{\perp}^{2}}{eB}$, such that

$\displaystyle\mathcal{I}_{2,n}$ $\displaystyle=$ $\displaystyle\frac{\pi
qB}{2}\int_{-\infty}^{\infty}\frac{dp_{3}}{q_{0}^{2}-m^{2}-p_{3}^{2}-2nqB+\mathrm{i}\epsilon}$
(142)
$\displaystyle\times\int_{0}^{\infty}dx\,e^{-x/2}\left[L_{n}(x)-L_{n-1}(x)\right]$
$\displaystyle=$ $\displaystyle 2\pi
qB(-1)^{n}\int_{-\infty}^{\infty}\frac{dp_{3}}{q_{0}^{2}-m^{2}-p_{3}^{2}-2nqB+\mathrm{i}\epsilon}$

where we used the identity (Gradshteyn-Ryzhik)

$\int_{0}^{\infty}dxe^{-bx}L_{n}(x)=\left(b-1\right)^{n}b^{-n-1}\,\,\,\text{Re}\,b>0$
(143)

Inserting Eq. (47) and Eq. (49) into Eq. (46), we obtain (after shifting the
index $n\rightarrow n+1$)

$\displaystyle\tilde{\mathcal{A}}_{1}(q_{0})=\frac{\pi^{2}(eB)}{\sqrt{q_{0}^{2}-m^{2}+\mathrm{i}\epsilon}}$
(144) $\displaystyle+2\pi
qB\int_{-\infty}^{+\infty}dp_{3}\sum_{n=0}^{\infty}\frac{1}{q_{0}^{2}-m^{2}-p_{3}^{2}-2(n+1)eB+\mathrm{i}\epsilon}$

Let us introduce the density of states for Landau levels

$\displaystyle\rho(E)=\int_{-\infty}^{\infty}\frac{dp_{3}}{2\pi}\sum_{n=0}^{\infty}\delta\left(E-E_{n}(p_{3})\right),$
(145)

with the dispersion relation for the spectrum

$\displaystyle E_{n}(p_{3})=\sqrt{p_{3}^{2}+m^{2}+2(n+1)eB}.$ (146)

With these definitions, we obtain from Eq. (V.2) the exact expression

$\displaystyle\tilde{\mathcal{A}}_{1}(q_{0})=\frac{\pi^{2}eB}{\sqrt{q_{0}^{2}-m^{2}+\mathrm{i}\epsilon}}+4\pi^{2}eB\int_{-\infty}^{+\infty}dE\frac{\rho(E)}{q_{0}^{2}-E^{2}+\mathrm{i}\epsilon},$

where, as shown in detail in Appendix B

$\displaystyle\rho(E)$ $\displaystyle=$
$\displaystyle\Theta(E-\sqrt{m^{2}+2eB})\frac{E}{\pi\sqrt{eB}}$ (148)
$\displaystyle\times\left[\zeta\left(\frac{1}{2},\frac{E^{2}-m^{2}-2eB}{2eB}-N_{max}(E)\right)\right.$
$\displaystyle\left.-\zeta\left(\frac{1}{2},\frac{E^{2}-m^{2}}{2eB}\right)\right],$

where we defined

$\displaystyle N_{max}(E)=\lfloor\frac{E^{2}-m^{2}}{2eB}-1\rfloor,$ (149)

with $\lfloor x\rfloor$ the integer part of $x$, and $\zeta(n,z)$ is the
Riemann zeta function.

On the other hand:

$\displaystyle\widetilde{\mathcal{A}}_{2}(q_{0})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dp_{3}p\mathcal{A}_{1}(q_{0},p_{3};\mathbf{p}_{\perp}=0)$
$\displaystyle=$
$\displaystyle\mathrm{i}\int_{-\infty}^{\infty}dp_{3}\frac{1}{q_{0}^{2}-p_{3}^{2}-m^{2}+\mathrm{i}\epsilon}$
$\displaystyle+$
$\displaystyle\mathrm{i}\int_{-\infty}^{\infty}dp_{3}\sum_{n=1}^{\infty}(-1)^{n}\frac{L_{n}(0)-L_{n-1}(0)}{q_{0}^{2}-p_{3}^{2}-m^{2}-2neB+\mathrm{i}\epsilon},$

where the second term vanishes, given that $L_{n}(0)=1~{}\forall n$. Hence

$\displaystyle\widetilde{\mathcal{A}}_{2}(q_{0})=\mathcal{I}_{1}=\frac{\pi^{2}eB}{\sqrt{q_{0}^{2}-m^{2}+\mathrm{i}\epsilon}}.$
(151)

###  C.3 Strong magnetic field limit

In this limit:

$\displaystyle\mathcal{A}_{1}(q)=\mathrm{i}\frac{e^{-\mathbf{q}_{\perp}^{2}/eB}}{q_{\parallel}^{2}-m^{2}},$
$\displaystyle\mathcal{A}_{2}(q)=\frac{e^{-\mathbf{q}_{\perp}^{2}/eB}}{q_{\parallel}^{2}-m^{2}},$
(152b) $\displaystyle\mathcal{A}_{3}(q)=0,$ (152c)
$\displaystyle\mathcal{D}(q)=2\frac{e^{-2\mathbf{q}_{\perp}^{2}/eB}}{q_{\parallel}^{2}-m^{2}}$
(152d)

and

$\displaystyle\widetilde{\mathcal{A}}_{1}(q)=\frac{\pi^{2}eB}{\sqrt{q_{0}^{2}-m^{2}}}$
(153a)
$\displaystyle\widetilde{\mathcal{A}}_{2}(q_{0})=\frac{\pi}{\sqrt{q_{0}^{2}-m^{2}}}.$
(153b)

##  Appendix D Vertex corrections at $O(\Delta^{2})$

The diagrams contributing at order $\Delta^{2}$ to the 4-point vertex are
depicted in Fig. 11, and hence their corresponding matrix elements are given
by the following integral expressions

$\displaystyle\hat{\Gamma}_{\text{(a)}}=\int\frac{d^{3}q}{(2\pi)^{3}}\,\gamma^{i}S_{\text{F}}(p-q)\gamma^{j}\otimes\gamma_{i}S_{\text{F}}(p^{\prime}-q)\gamma_{j},$
$\displaystyle\hat{\Gamma}_{\text{(b)}}=\int\frac{d^{3}q}{(2\pi)^{3}}\,\gamma^{i}S_{\text{F}}(p-q)\gamma^{j}\otimes\gamma_{i}S_{\text{F}}(p^{\prime}+q)\gamma_{j},$
and
$\displaystyle\hat{\Gamma}_{\text{(c)}}=\int\frac{d^{3}q}{(2\pi)^{3}}\,\gamma^{i}S_{\text{F}}(p+q)\gamma^{j}\otimes\gamma_{i}S_{\text{F}}(p^{\prime}-q)\gamma_{j}$

In order to compute the former expressions, it is convenient to introduce the
notation

$\displaystyle\hat{\Gamma}^{{}^{(\lambda,\sigma)}}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\,\gamma^{i}S_{\text{F}}(p+\lambda
q)\gamma^{j}\otimes\gamma_{i}S_{\text{F}}(p^{\prime}+\sigma q)\gamma_{j}.$

where $\lambda,\sigma=\pm 1$. Then, we have the correspondence
$\hat{\Gamma}_{\text{(a)}}=\hat{\Gamma}^{(-,-)}$,
$\hat{\Gamma}_{\text{(b)}}=\hat{\Gamma}^{(-,+)}$, and
$\hat{\Gamma}_{\text{(c)}}=\hat{\Gamma}^{(+,-)}$, respectively. By considering
the tensor structure of the propagator, it is straightforward to realize that
the full vertex, taking into account the multiplicity factors for each
diagram,

$\displaystyle\hat{\Gamma}=2\hat{\Gamma}^{(-,-)}+2\hat{\Gamma}^{(-,+)}+4\hat{\Gamma}^{(+,-)},$
(156)

leads to an effective interaction of the form

$\displaystyle\hat{\Gamma}=\tilde{\Delta}(\bar{\psi}\gamma^{i}\psi)(\bar{\psi}\gamma^{i}\psi)+{\rm{other\,\,tensor\,\,structures}},$
(157)

where the renormalized coefficient $\tilde{\Delta}$ is given up to second
order in $\Delta$ by

$\displaystyle\tilde{\Delta}=\Delta+2(\Delta)^{2}\left(\mathcal{J}_{2}^{(-,-)}+\mathcal{J}_{2}^{(-,+)}+2\mathcal{J}_{2}^{(+,-)}\right.$
$\displaystyle\left.+\left(1-\partial_{x}^{2}\right)(1-\partial_{y}^{2})\mathcal{J}_{3}^{(-,-)}+\left(1-\partial_{x}^{2}\right)(1-\partial_{y}^{2})\mathcal{J}_{3}^{(-,+)}\right.$
$\displaystyle\left.+2\left(1-\partial_{x}^{2}\right)(1-\partial_{y}^{2})\mathcal{J}_{3}^{(+,-)}\right)$
(158)

Now, from Eq. (16):

$\displaystyle\hat{\Gamma}^{{}^{(\lambda,\sigma)}}$ $\displaystyle=$
$\displaystyle-\left(\hat{\Gamma}_{11}^{{}^{(\lambda,\sigma)}}+\hat{\Gamma}_{12}^{{}^{(\lambda,\sigma)}}+\hat{\Gamma}_{13}^{{}^{(\lambda,\sigma)}}+\hat{\Gamma}_{21}^{{}^{(\lambda,\sigma)}}\right.$
$\displaystyle+$
$\displaystyle\left.\hat{\Gamma}_{22}^{{}^{(\lambda,\sigma)}}+\hat{\Gamma}_{23}^{{}^{(\lambda,\sigma)}}+\hat{\Gamma}_{31}^{{}^{(\lambda,\sigma)}}+\hat{\Gamma}_{32}^{{}^{(\lambda,\sigma)}}+\hat{\Gamma}_{33}^{{}^{(\lambda,\sigma)}}\right),$

where

$\displaystyle\hat{\Gamma}_{11}^{{}^{(\lambda,\sigma)}}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\left(m+\not{p}_{\parallel}+\lambda\not{q}_{\parallel}\right)\left(m+\not{p}_{\parallel}^{\prime}+\sigma\not{q}_{\parallel}\right)$
(160a) $\displaystyle\times\mathcal{A}_{1}(p+\lambda
q)\mathcal{A}_{1}(p^{\prime}+\sigma q),$
$\displaystyle\hat{\Gamma}_{12}^{{}^{(\lambda,\sigma)}}$ $\displaystyle=$
$\displaystyle\mathrm{i}\gamma^{1}\gamma^{2}\int\frac{d^{3}q}{(2\pi)^{3}}\left(m+\not{p}_{\parallel}+\lambda\not{q}_{\parallel}\right)\left(m+\not{p}_{\parallel}^{\prime}+\sigma\not{q}_{\parallel}\right)$
(160b) $\displaystyle\times\mathcal{A}_{1}(p+\lambda
q)\mathcal{A}_{2}(p^{\prime}+\sigma q),$
$\displaystyle\hat{\Gamma}_{13}^{{}^{(\lambda,\sigma)}}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\left(m+\not{p}_{\parallel}+\lambda\not{q}_{\parallel}\right)\left(\not{p}^{\prime}_{\perp}+\sigma\not{q}_{\perp}\right)$
(160c) $\displaystyle\times$ $\displaystyle\mathcal{A}_{1}(p+\lambda
q)\mathcal{A}_{3}(p^{\prime}+\sigma q),$
$\displaystyle\hat{\Gamma}_{21}^{{}^{(\lambda,\sigma)}}$ $\displaystyle=$
$\displaystyle\mathrm{i}\gamma^{1}\gamma^{2}\int\frac{d^{3}q}{(2\pi)^{3}}\left(m+\not{p}_{\parallel}+\lambda\not{q}_{\parallel}\right)\left(m+\not{p}_{\parallel}^{\prime}+\sigma\not{q}_{\parallel}\right)$
(160d) $\displaystyle\times$ $\displaystyle\mathcal{A}_{2}(p+\lambda
q)\mathcal{A}_{1}(p^{\prime}+\sigma q),$
$\displaystyle\hat{\Gamma}_{22}^{{}^{(\lambda,\sigma)}}$ $\displaystyle=$
$\displaystyle-\int\frac{d^{3}q}{(2\pi)^{3}}\left(m+\not{p}_{\parallel}+\lambda\not{q}_{\parallel}\right)\left(m+\not{p}_{\parallel}^{\prime}+\sigma\not{q}_{\parallel}\right)$
(160e) $\displaystyle\times$ $\displaystyle\mathcal{A}_{2}(p+\lambda
q)\mathcal{A}_{2}(p^{\prime}+\sigma q),$
$\displaystyle\hat{\Gamma}_{23}^{{}^{(\lambda,\sigma)}}$ $\displaystyle=$
$\displaystyle\mathrm{i}\gamma^{1}\gamma^{2}\int\frac{d^{3}q}{(2\pi)^{3}}\left(m+\not{p}_{\parallel}+\lambda\not{q}_{\parallel}\right)\left(\not{p}^{\prime}_{\perp}+\sigma\not{q}_{\perp}\right)$
(160f) $\displaystyle\times$ $\displaystyle\mathcal{A}_{2}(p+\lambda
q)\mathcal{A}_{3}(p^{\prime}+\sigma q),$
$\displaystyle\hat{\Gamma}_{31}^{{}^{(\lambda,\sigma)}}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\left(\not{p}_{\perp}+\lambda\not{q}_{\perp}\right)\left(m+\not{p}_{\parallel}^{\prime}+\sigma\not{q}_{\parallel}\right)$
(160g) $\displaystyle\times$ $\displaystyle\mathcal{A}_{3}(p+\lambda
q)\mathcal{A}_{1}(p^{\prime}+\sigma q),$
$\displaystyle\hat{\Gamma}_{32}^{{}^{(\lambda,\sigma)}}$ $\displaystyle=$
$\displaystyle-\mathrm{i}\gamma^{1}\gamma^{2}\int\frac{d^{3}q}{(2\pi)^{3}}\left(\not{p}_{\perp}+\lambda\not{q}_{\perp}\right)\left(m+\not{p}_{\parallel}^{\prime}+\sigma\not{q}_{\parallel}\right)$
(160h) $\displaystyle\times$ $\displaystyle\mathcal{A}_{3}(p+\lambda
q)\mathcal{A}_{2}(p^{\prime}+\sigma q),$
$\displaystyle\hat{\Gamma}_{33}^{{}^{(\lambda,\sigma)}}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\left(\not{p}_{\perp}+\lambda\not{q}_{\perp}\right)\left(\not{p}_{\perp}^{\prime}+\sigma\not{q}_{\perp}\right)$
(160i) $\displaystyle\times$ $\displaystyle\mathcal{A}_{3}(p+\lambda
q)\mathcal{A}_{3}(p^{\prime}+\sigma q).$

The latter equations can be condensed by defining a single master integral in
terms of $\mathcal{A}_{1}$ and its derivatives. To do so, note that:

$\displaystyle\left(m+\not{p}_{\parallel}+\lambda\not{q}_{\parallel}\right)\left(m+\not{p}_{\parallel}^{\prime}+\sigma\not{q}_{\parallel}\right)$
(161) $\displaystyle=$ $\displaystyle
m^{2}+m\left[\not{p}_{\parallel}+\not{p}_{\parallel}^{\prime}+(\sigma+\lambda)\not{q}_{\parallel}\right]+\not{p}_{\parallel}\not{p}_{\parallel}^{\prime}$
$\displaystyle+$
$\displaystyle\sigma\not{p}_{\parallel}\not{q}_{\parallel}+\lambda\not{q}_{\parallel}\not{p}_{\parallel}^{\prime}+\lambda\sigma\left(\not{q}_{\parallel}\right)^{2},$

and given that $\mathcal{A}_{i}$ are even functions of $q$, the linear terms
can be ignored. Then:

$\displaystyle\left(m+\not{p}_{\parallel}+\lambda\not{q}_{\parallel}\right)\left(m+\not{p}_{\parallel}^{\prime}+\sigma\not{q}_{\parallel}\right)$
(162) $\displaystyle\to$ $\displaystyle
m^{2}+m\left(\not{p}_{\parallel}+\not{p}_{\parallel}^{\prime}\right)+\not{p}_{\parallel}\not{p}_{\parallel}^{\prime}+\lambda\sigma
q_{\parallel}^{2}.$

Now, it is convenient to define:

$\displaystyle P$ $\displaystyle\equiv$ $\displaystyle\frac{p^{\prime}+p}{2},$
$\displaystyle Q$ $\displaystyle\equiv$ $\displaystyle\frac{p^{\prime}-p}{2},$
(163)

from which:

$\displaystyle\left(m+\not{p}_{\parallel}+\lambda\not{q}_{\parallel}\right)\left(m+\not{p}_{\parallel}^{\prime}+\sigma\not{q}_{\parallel}\right)$
$\displaystyle\to$ $\displaystyle
m^{2}+2m\not{P}_{\parallel}+\left(\not{P}_{\parallel}-\not{Q}_{\parallel}\right)\left(\not{P}_{\parallel}+\not{Q}_{\parallel}\right)+\lambda\sigma
q_{\parallel}^{2}.$

Similarly:

$\displaystyle\left(m+\not{p}_{\parallel}+\lambda\not{q}_{\parallel}\right)\left(\not{p}^{\prime}_{\perp}+\sigma\not{q}_{\perp}\right)$
(165) $\displaystyle\to$ $\displaystyle
m\left(\not{P}_{\perp}+\not{Q}_{\perp}\right)+\left(\not{P}_{\parallel}-\not{Q}_{\parallel}\right)\left(\not{P}_{\perp}+\not{Q}_{\perp}\right),$
$\displaystyle\left(\not{p}_{\perp}+\lambda\not{q}_{\perp}\right)\left(m+\not{p}_{\parallel}^{\prime}+\sigma\not{q}_{\parallel}\right)$
(166) $\displaystyle\to$ $\displaystyle
m\left(\not{P}_{\perp}-\not{Q}_{\perp}\right)+\left(\not{P}_{\perp}-\not{Q}_{\perp}\right)\left(\not{P}_{\parallel}+\not{Q}_{\parallel}\right),$

and

$\displaystyle\left(\not{p}_{\perp}+\lambda\not{q}_{\perp}\right)\left(\not{p}_{\perp}^{\prime}+\sigma\not{q}_{\perp}\right)$
(167) $\displaystyle\to$
$\displaystyle\left(\not{P}_{\perp}-\not{Q}_{\perp}\right)\left(\not{P}_{\perp}+\not{Q}_{\perp}\right)-\lambda\sigma\mathbf{q}_{\perp}^{2}.$

Moreover, Eqs. (20) provide relations between $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$ and $\mathcal{A}_{3}$ so that by introducing the variables

$\displaystyle
x=\frac{\mathbf{p}_{\perp}^{2}}{eB},~{}~{}y=\frac{\mathbf{p}_{\perp}^{{}^{\prime}2}}{eB},$
(168)

so that

$\displaystyle\hat{\Gamma}^{{}^{(\lambda,\sigma)}}$ $\displaystyle=$
$\displaystyle-\left[\left(\not{P}_{\parallel}-\not{Q}_{\parallel}\right)\left(\not{P}_{\parallel}+\not{Q}_{\parallel}\right)+2m\not{P}_{\parallel}+m^{2}\right]\left[1+\partial_{x}\partial_{y}-\gamma^{1}\gamma^{2}\left(\partial_{x}-\partial_{y}\right)\right]\mathcal{J}_{1}^{(\lambda,\sigma)}(p,p^{\prime})$
(169)
$\displaystyle-\left[\left(\not{P}_{\parallel}-\not{Q}_{\parallel}\right)\left(\not{P}_{\perp}+\not{Q}_{\perp}\right)+m\left(\not{P}_{\perp}+\not{Q}_{\perp}\right)\right]\left(1-\gamma^{1}\gamma^{2}\partial_{x}\right)\left(1-\partial^{2}_{y}\right)\mathcal{J}_{1}^{(\lambda,\sigma)}(p,p^{\prime})$
$\displaystyle-\left[\left(\not{P}_{\perp}-\not{Q}_{\perp}\right)\left(\not{P}_{\parallel}+\not{Q}_{\parallel}\right)+m\left(\not{P}_{\perp}-\not{Q}_{\perp}\right)\right]\left(1+\gamma^{1}\gamma^{2}\partial_{y}\right)\left(1-\partial^{2}_{x}\right)\mathcal{J}_{1}^{(\lambda,\sigma)}(p,p^{\prime})$
$\displaystyle-\left(\not{P}_{\perp}-\not{Q}_{\perp}\right)\left(\not{P}_{\perp}+\not{Q}_{\perp}\right)\left(1-\partial_{x}^{2}\right)\left(1-\partial_{y}^{2}\right)\mathcal{J}_{1}^{(\lambda,\sigma)}(p,p^{\prime})$
$\displaystyle-\lambda\sigma\left[1+\partial_{x}\partial_{y}-\gamma^{1}\gamma^{2}\left(\partial_{x}-\partial_{y}\right)\right]\mathcal{J}_{2}^{(\lambda,\sigma)}(p,p^{\prime})-\lambda\sigma\left(1-\partial_{x}^{2}\right)\left(1-\partial_{y}^{2}\right)\mathcal{J}_{3}^{(\lambda,\sigma)}(p,p^{\prime}),$

where

$\displaystyle\mathcal{J}_{1}^{(\lambda,\sigma)}(p,p^{\prime})\equiv\int\frac{d^{3}q}{(2\pi)^{3}}\mathcal{A}_{1}(p+\lambda
q)\mathcal{A}_{1}(p^{\prime}+\sigma q),$
$\displaystyle\mathcal{J}_{2}^{(\lambda,\sigma)}(p,p^{\prime})\equiv\int\frac{d^{3}q}{(2\pi)^{3}}q_{\parallel}^{2}\mathcal{A}_{1}(p+\lambda
q)\mathcal{A}_{1}(p^{\prime}+\sigma q),$ and
$\displaystyle\mathcal{J}_{3}^{(\lambda,\sigma)}(p,p^{\prime})\equiv\int\frac{d^{3}q}{(2\pi)^{3}}\mathbf{q}_{\perp}^{2}\mathcal{A}_{1}(p+\lambda
q)\mathcal{A}_{1}(p^{\prime}+\sigma q),$

In order to simplify the integrals $\mathcal{J}_{i}$, we shall use the
analytical expression for $\mathcal{A}_{1}(k)$ Eq. (LABEL:eq_A1_U) (details in
Appendix A):

$\displaystyle\mathcal{A}_{1}(k)$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}e^{-\mathbf{k}_{\perp}^{2}/eB}}{2eB}\exp\left[-\frac{\mathrm{i}\pi\left(k_{\parallel}^{2}-m^{2}\right)}{2eB}\right]$
$\displaystyle\times$
$\displaystyle\Gamma\left(-\frac{k_{\parallel}^{2}-m^{2}}{2eB}\right)U\left(-\frac{k_{\parallel}^{2}-m^{2}}{2eB},0,\frac{2\mathbf{k}_{\perp}^{2}}{eB}\right).$

###  D.1 The integral $\mathcal{J}_{1}$

We shall consider the integral:

$\displaystyle\mathcal{J}_{1}^{(\lambda,\sigma)}(p,p^{\prime})=\int\frac{d^{3}q}{(2\pi)^{3}}\mathcal{A}_{1}(p+\lambda
q)\mathcal{A}_{1}(p^{\prime}+\sigma q)$ (172)

For the case $(\lambda,\sigma)=(-1,-1)$ we change the integration variables as
follows

$\displaystyle p^{\prime}-q$ $\displaystyle=$ $\displaystyle q^{\prime}+Q$
$\displaystyle p-q$ $\displaystyle=$ $\displaystyle q^{\prime}-Q$ (173)

in what follows, we shall use $q$ instead of $q^{\prime}$. For notation
simplicity, we shall define the parameters

$\displaystyle a$ $\displaystyle=$
$\displaystyle-\frac{\mathcal{D}_{\parallel}(q_{3}+Q_{\parallel})}{2eB},$
$\displaystyle a^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{\mathcal{D}_{\parallel}(q_{3}-Q_{\parallel})}{2eB},$
(174)

and we shall use the identity

$\displaystyle\Gamma(a)U(a,\epsilon,z)=\frac{1}{a}M(a,\epsilon,z)+\Gamma(-1+\epsilon)zM(1+a,2,z),$

together with for $\epsilon\rightarrow 0^{+}$

$\displaystyle\Gamma(-1+\epsilon)=\frac{-1}{\epsilon}+\gamma_{e}-1+O(\epsilon),$
(176)

where $\gamma_{e}=0.577$ is the Euler-Mascheroni constant. Also, given that
there is a strong exponential damping in the integral, we consider the
expansion of the Kummer function for small values of its argument, that is
given by

$\displaystyle M(a,b,z)=1+\frac{a}{b}z+O(z^{2}),$ (177)

so that, after removing the divergences, we end up with the integral

$\displaystyle\mathcal{J}_{1}^{(-,-)}(p,p^{\prime})$ $\displaystyle=$
$\displaystyle\left(\frac{\mathrm{i}}{2eB}\right)^{2}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}\int_{-\infty}^{\infty}\frac{dq_{3}}{2\pi}~{}e^{\mathrm{i}\pi\left(a+a^{\prime}\right)}\int_{0}^{\infty}\frac{d^{2}q_{\perp}}{(2\pi)^{2}}e^{-\frac{2\mathbf{q}^{2}_{\perp}}{eB}}~{}\Gamma\left(a\right)U\left(a,\epsilon,\frac{(q+Q)_{\perp}^{2}}{eB}\right)\Gamma\left(a^{\prime}\right)U\left(a^{\prime},\epsilon,\frac{(q-Q)_{\perp}^{2}}{eB}\right)$
(178) $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\left(\frac{\mathrm{i}}{2eB}\right)^{2}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}\int_{-\infty}^{\infty}dq_{3}~{}e^{\mathrm{i}\pi\left(a+a^{\prime}\right)}$
$\displaystyle\times$
$\displaystyle\int_{0}^{\infty}d^{2}q_{\perp}e^{-\frac{2\mathbf{q}^{2}_{\perp}}{eB}}\left[\frac{1}{a}+\frac{\gamma_{e}-1}{eB}\left(\mathbf{q_{\perp}^{2}}+2\mathbf{Q}_{\perp}\cdot\mathbf{q}_{\perp}+\mathbf{Q}_{\perp}^{2}\right)\right]\left[\frac{1}{a^{\prime}}+\frac{\gamma_{e}-1}{eB}\left(\mathbf{q_{\perp}^{2}}-2\mathbf{Q}_{\perp}\cdot\mathbf{q}_{\perp}+\mathbf{Q}_{\perp}^{2}\right)\right]$

Let us focus into the integrand. At order
$\mathcal{O}(\mathbf{q}_{\perp}^{2})$, we have:

$\displaystyle\left[\frac{1}{a}+\frac{\gamma_{e}-1}{eB}\left(\mathbf{q_{\perp}^{2}}+2\mathbf{Q}_{\perp}\cdot\mathbf{q}_{\perp}+\mathbf{Q}_{\perp}^{2}\right)\right]\left[\frac{1}{a^{\prime}}+\frac{\gamma_{e}-1}{eB}\left(\mathbf{q_{\perp}^{2}}-2\mathbf{Q}_{\perp}\cdot\mathbf{q}_{\perp}+\mathbf{Q}_{\perp}^{2}\right)\right]$
$\displaystyle\simeq$
$\displaystyle\frac{1}{aa^{\prime}}+\frac{(\gamma_{e}-1)\mathbf{q}_{\perp}^{2}}{eB}\left(\frac{1}{a}+\frac{1}{a^{\prime}}+\frac{2(\gamma_{e}-1)\mathbf{Q}_{\perp}^{2}}{eB}\right)+\frac{(\gamma_{e}-1)(2\mathbf{Q}_{\perp}\cdot\mathbf{q}_{\perp})}{eB}\left(\frac{1}{a^{\prime}}-\frac{1}{a}\right)$
$\displaystyle-$
$\displaystyle\frac{(\gamma_{e}-1)^{2}(2\mathbf{Q}_{\perp}\cdot\mathbf{q}_{\perp})^{2}}{eB^{2}}+\frac{(\gamma_{e}-1)\mathbf{Q}_{\perp}^{2}}{eB}\left(\frac{1}{a}+\frac{1}{a^{\prime}}\right).$

Then, by defining:

$\displaystyle z\equiv\frac{2\mathbf{q}_{\perp}^{2}}{eB},$ (180)

such that

$\displaystyle\mathbf{Q}_{\perp}\cdot\mathbf{q}_{\perp}$ $\displaystyle=$
$\displaystyle|\mathbf{Q}_{\perp}|~{}|\mathbf{q}_{\perp}|\cos\theta=\sqrt{\frac{eB}{2}}|\mathbf{Q}_{\perp}|z^{1/2}\cos\theta,$
$\displaystyle d^{3}q$ $\displaystyle=$ $\displaystyle
dq_{3}\,d^{2}q_{\perp}=\frac{eB}{4}dzd\theta dq_{3},$ (181)

where after angular integration we obtain:

$\displaystyle\mathcal{J}_{1}^{(-,-)}(p,p^{\prime})=\frac{1}{(2\pi)^{3}}\frac{\pi
eB}{2}\left(\frac{\mathrm{i}}{2eB}\right)^{2}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}$
(182) $\displaystyle\times$
$\displaystyle\int_{-\infty}^{\infty}dq_{3}e^{-\frac{\mathrm{i}\pi}{2eB}\left(\mathcal{D}_{\parallel}(q_{3}+Q_{\parallel})+\mathcal{D}_{\parallel}(q_{3}-Q_{\parallel})\right)}$
$\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dze^{-z}\Bigg{\\{}\frac{1}{aa^{\prime}}+\frac{(\gamma_{e}-1)}{2}\left(\frac{1}{a}+\frac{1}{a^{\prime}}\right)z$
$\displaystyle+$
$\displaystyle\frac{(\gamma_{e}-1)\mathbf{Q}_{\perp}^{2}}{eB}\left(\frac{1}{a}+\frac{1}{a^{\prime}}\right)\Bigg{\\}}.$

Performing the integration over $z$:

$\displaystyle\mathcal{J}_{1}^{(-,-)}(p,p^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{eB}{4\pi^{2}}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}e^{-\frac{\mathrm{i}\pi(Q_{\parallel}^{2}-m^{2})}{eB}}$
(183) $\displaystyle\times$
$\displaystyle\int_{-\infty}^{\infty}dq_{3}\frac{\exp\left(\frac{\mathrm{i}\pi}{2eB}q_{3}^{2}\right)}{\left(Q_{\parallel}^{2}-q_{3}^{2}-m^{2}+\mathrm{i}\epsilon\right)^{2}-4Q_{3}^{2}q_{3}^{2}}$
$\displaystyle\times$
$\displaystyle\Bigg{[}1-\frac{(\gamma_{e}-1)\mathbf{Q}_{\perp}^{2}\left(Q_{\parallel}^{2}-q_{3}^{2}-m^{2}\right)}{(eB)^{2}}$
$\displaystyle-$
$\displaystyle\frac{(\gamma_{e}-1)\left(Q_{\parallel}^{2}-q_{3}^{2}-m^{2}\right)}{eB}\Bigg{]}$

In what follows, we shall set the external 3-momenta to zero, except for the
presence of the $Q_{\perp}$ factors, those we shall keep in order to use this
expression as a generating function. Then:

$\displaystyle\mathcal{J}_{1}^{(-,-)}(p,p^{\prime})=-\frac{eB}{4\pi^{2}}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}$
$\displaystyle\times$
$\displaystyle\int_{-\infty}^{\infty}\frac{dq_{3}}{\left(q_{3}^{2}+m^{2}+\mathrm{i}\epsilon\right)^{2}}\exp\left[\frac{\mathrm{i}\pi}{2eB}\left(q_{3}^{2}+m^{2}\right)\right]$
$\displaystyle\times$
$\displaystyle\Bigg{[}1+\frac{(\gamma_{e}-1)\mathbf{Q}_{\perp}^{2}\left(q_{3}^{2}+m^{2}\right)}{(eB)^{2}}+\frac{(\gamma_{e}-1)\left(q_{3}^{2}+m^{2}\right)}{eB}\Bigg{]}$

Now, from the well-known results

$\displaystyle\int_{-\infty}^{\infty}\frac{dx}{x^{2}+m^{2}}\exp\left[\frac{\mathrm{i}\pi}{2b}(x^{2}+m^{2})\right]$
$\displaystyle=$
$\displaystyle\frac{\pi}{m}\left[1-(1-\mathrm{i})C\left(\frac{m}{\sqrt{b}}\right)-(1+\mathrm{i})S\left(\frac{m}{\sqrt{b}}\right)\right],$
and
$\displaystyle\int_{-\infty}^{\infty}\frac{dx}{(x^{2}+m^{2})^{2}}\exp\left[\frac{\mathrm{i}\pi}{2b}(x^{2}+m^{2})\right]$
$\displaystyle=$
$\displaystyle\frac{(1-\mathrm{i})\pi}{2\sqrt{b}m^{2}}e^{\frac{\mathrm{i}\pi
m^{2}}{2b}}$ $\displaystyle+$ $\displaystyle\frac{(b+\mathrm{i}\pi
m^{2})\pi}{2bm^{3}}\left[1-(1-\mathrm{i})C\left(\frac{m}{\sqrt{b}}\right)-(1+\mathrm{i})S\left(\frac{m}{\sqrt{b}}\right)\right],$

where $C(x)$ and $S(x)$ are the cosine and sine Fresnel integrals,
respectively. From the property:

$\displaystyle
C(x)+\mathrm{i}S(x)=\sqrt{\frac{\pi}{2}}\frac{1+\mathrm{i}}{2}\text{erf}\left(\frac{1-\mathrm{i}}{\sqrt{2}}x\right),$
(186)

we have:

$\displaystyle\mathcal{J}_{1}^{(-,-)}(p,p^{\prime})=-\frac{eB}{4\pi^{2}}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}\Bigg{\\{}\frac{(1-\mathrm{i})\pi}{2\sqrt{eB}m^{2}}e^{\frac{\mathrm{i}\pi
m^{2}}{2eB}}$ $\displaystyle+$ $\displaystyle\frac{(eB+\mathrm{i}\pi
m^{2})\pi}{2(eB)m^{3}}\left[1-\sqrt{\frac{\pi}{2}}\text{erf}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\frac{m}{\sqrt{eB}}\right)\right]$
$\displaystyle+$
$\displaystyle\frac{\pi(\gamma_{e}-1)}{(eB)m}\left(1+\frac{\mathbf{Q}_{\perp}^{2}}{eB}\right)\left[1-\sqrt{\frac{\pi}{2}}\text{erf}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\frac{m}{\sqrt{eB}}\right)\right]\Bigg{\\}},$

where $\text{erf}(x)$ is the error function.

###  D.2 The integral $\mathcal{J}_{2}$

For this integral, note that $q_{\parallel}^{2}=-q_{3}^{2}$, so that after
integration over $z$ we get:

$\displaystyle\mathcal{J}_{2}^{(-,-)}(p,p^{\prime})=\frac{eB}{4\pi^{2}}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}$
$\displaystyle\times$
$\displaystyle\int_{-\infty}^{\infty}dq_{3}\frac{q_{3}^{2}}{\left(q_{3}^{2}+m^{2}+\mathrm{i}\epsilon\right)^{2}}\exp\left[\frac{\mathrm{i}\pi}{2eB}\left(q_{3}^{2}+m^{2}\right)\right]$
$\displaystyle\times$
$\displaystyle\Bigg{[}1+\frac{(\gamma_{e}-1)\mathbf{Q}_{\perp}^{2}\left(q_{3}^{2}+m^{2}\right)}{(eB)^{2}}+\frac{(\gamma_{e}-1)\left(q_{3}^{2}+m^{2}\right)}{eB}\Bigg{]}$

By using:

$\displaystyle\int_{-\infty}^{\infty}dx\frac{x^{2}}{x^{2}+m^{2}}\exp\left[\frac{\mathrm{i}\pi}{2b}(x^{2}+m^{2})\right]$
$\displaystyle=$ $\displaystyle(1+\mathrm{i})e^{\frac{\mathrm{i}\pi
m^{2}}{2b}}\sqrt{b}$ $\displaystyle-$ $\displaystyle
m\pi\left[1-(1-\mathrm{i})C\left(\frac{m}{\sqrt{b}}\right)-(1+\mathrm{i})S\left(\frac{m}{\sqrt{b}}\right)\right],$
and
$\displaystyle\int_{-\infty}^{\infty}dx\frac{x^{2}}{(x^{2}+m^{2})^{2}}\exp\left[\frac{\mathrm{i}\pi}{2b}(x^{2}+m^{2})\right]$
$\displaystyle=$
$\displaystyle\frac{(\mathrm{i}-1)\pi}{\sqrt{b}}e^{\frac{\mathrm{i}\pi
m^{2}}{2b}}$ $\displaystyle+$ $\displaystyle\frac{(\mathrm{i}b+\pi
m^{2})\pi}{2bm}\left[-\mathrm{i}+(1+\mathrm{i})C\left(\frac{m}{\sqrt{b}}\right)-(1-\mathrm{i})S\left(\frac{m}{\sqrt{b}}\right)\right],$

we get:

$\displaystyle\mathcal{J}_{2}^{(-,-)}(p,p^{\prime})=\frac{eB}{4\pi^{2}}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}$
$\displaystyle\times$
$\displaystyle\Bigg{\\{}\frac{(\mathrm{i}-1)\pi}{\sqrt{eB}}e^{\frac{\mathrm{i}\pi
m^{2}}{2eB}}$ $\displaystyle+$ $\displaystyle\frac{\left(eB-\mathrm{i}\pi
m^{2}\right)\pi}{2(eB)m}\left[1-\sqrt{\frac{\pi}{2}}\text{erf}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\frac{m}{\sqrt{eB}}\right)\right]$
$\displaystyle-$
$\displaystyle\frac{m\pi(\gamma_{e}-1)}{(eB)}\left(1+\frac{\mathbf{Q}_{\perp}^{2}}{eB}\right)\left[1-\sqrt{\frac{\pi}{2}}\text{erf}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\frac{m}{\sqrt{eB}}\right)\right]\Bigg{\\}}$

###  D.3 The integral $\mathcal{J}_{3}$

By following the same procedure, it is straightforward to obtain at order
$\mathcal{O}(z)$:

$\displaystyle\mathcal{J}_{3}^{(-,-)}(p,p^{\prime})=-\frac{eB^{2}}{8\pi^{2}}e^{-\frac{2\mathbf{Q}_{\perp}^{2}}{eB}}\Bigg{\\{}\frac{(1-\mathrm{i})\pi}{2\sqrt{eB}m^{2}}e^{\frac{\mathrm{i}\pi
m^{2}}{2eB}}$ $\displaystyle+$ $\displaystyle\frac{(eB+\mathrm{i}\pi
m^{2})\pi}{2(eB)m^{3}}\left[1-\sqrt{\frac{\pi}{2}}\text{erf}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\frac{m}{\sqrt{eB}}\right)\right]$
$\displaystyle+$
$\displaystyle\frac{\pi(\gamma_{e}-1)\mathbf{Q}_{\perp}^{2}}{(eB)^{2}m}\left[1-\sqrt{\frac{\pi}{2}}\text{erf}\left(\frac{1-\mathrm{i}}{\sqrt{2}}\frac{m}{\sqrt{eB}}\right)\right]\Bigg{\\}},$

Moreover, it is easy to check that

$\displaystyle\mathcal{J}_{n}^{(+,+)}(p,p^{\prime})$ $\displaystyle=$
$\displaystyle\mathcal{J}_{n}^{(-,-)}(p,p^{\prime})$
$\displaystyle\mathcal{J}_{n}^{(+,-)}(p,p^{\prime})$ $\displaystyle=$
$\displaystyle\mathcal{J}_{n}^{(-,+)}(p,p^{\prime})=\mathcal{J}_{n}^{(-,-)}(p,p^{\prime})\mid_{Q\rightarrow
P},$

for $n=1,2,3$.