# Photon polarization tensor in a magnetized plasma: Absorptive part

Xinyang Wang<EMAIL_ADDRESS>Department of Physics, Jiangsu University,
Zhenjiang 212013, People’s Republic of China  Igor Shovkovy
<EMAIL_ADDRESS>College of Integrative Sciences and Arts, Arizona State
University, Mesa, Arizona 85212, USA  Department of Physics, Arizona State
University, Tempe, Arizona 85287, USA

###### Abstract

We calculate the absorptive part of the photon polarization tensor in a hot
magnetized relativistic plasma. In the derivation, we utilize a Landau-level
representation for the fermion Green’s function in a mixed coordinate-momentum
space and obtain a closed-form expression for the one-loop polarization
tensor. At the leading order in the coupling, its absorptive part is
determined by particle and antiparticle splitting processes
($e^{-}\leftrightarrow e^{-}+\gamma$ and $e^{+}\leftrightarrow e^{+}+\gamma$,
respectively), as well as by particle-antiparticle annihilation processes
($e^{-}+e^{+}\leftrightarrow\gamma$). The interpretation in terms of quantum
transitions between Landau levels is also given. By making use of the photon
polarization tensor, we study the differential photon emission rate in the
quantum limit of magnetized relativistic plasma. At low energies, the photon
emission has a prolate profile with the symmetry axis along the line of the
magnetic field. At high energies, on the other hand, the photon emission has
an oblate profile. The underlying reasons for such emission profiles are given
in both regimes. The general result for the photon polarization tensor is also
used to calculate the longitudinal and transverse components of magneto-
optical conductivity.

###### pacs:

12.38.Mh,25.75.-q,11.10.Wx,13.88+e

##  I Introduction

Relativistic plasmas appear in a wide range of high-energy systems. The
archetypal examples include the hot plasma of the early Universe and the
quark-gluon plasma created in heavy-ion collisions. One also finds various
forms of relativistic plasmas in astrophysics, ranging from cold dense matter
inside compact stars to electromagnetic plasmas in astrophysics jets powered
by black holes. Even in condensed matter physics, pseudorelativistic electron
plasmas can be realized in Dirac and Weyl semimetals. In many cases, the
corresponding plasmas could be also strongly magnetized.

The properties of relativistic plasmas in weak magnetic fields have been
studied extensively over the years and are well understood. On the other hand,
the regime of strongly magnetized relativistic plasma is less explored. One of
the specific long-standing and largely unresolved problems is photon
polarization. Formally, the photon polarization tensor can be calculated by
making use of the well-known Schwinger’s result for the fermion Green’s
function in a background magnetic field Schwinger:1951nm . While this appears
to be a straightforward task conceptually, the calculation of the relevant
Feynman diagrams is plugged with serious technical difficulties, stemming from
an elaborate structure of the Green’s function. From a physics viewpoint,
there are several reasons for complications. One of them is the absence of the
Lorentz symmetry, which is broken explicitly by the background magnetic field.
At a nonzero temperature, the symmetry is further reduced by the thermal bath
effects. Additionally, the quantum states of charged fermions in a magnetic
field are much more complicated than the simple plane waves. They are
classified by a discrete Landau level index that replaces the two transverse
components of the particle momentum. The matters are made worse also because
the Landau level states are highly degenerate.

Before addressing the photon polarization effects in a hot magnetized plasma,
it should be noted that even the case of a strongly magnetized vacuum is very
challenging. The first calculations of the vacuum polarization in a
homogeneous magnetic field were done in the 1970s Tsai:1974ap ; Shabad:1975ik
, but their physical meaning was not always clear because of the complicated
structure. Beyond the weak field case Tsai:1974fa ; Urrutia:1977xb , only the
limit of an ultrastrong magnetic field, where the lowest Landau approximation
becomes reliable, was understood well Loskutov:1976aa ; Calucci:1993fi . More
recently, some progress in solving the problem of QED vacuum in a strong
magnetic field beyond the lowest Landau approximation was reported in Refs.
Hattori:2012je ; Hattori:2012ny ; Karbstein:2013ufa ; Ishikawa:2013fxa ;
Hattori:2020htm , where the combination of analytical and numerical methods
was employed.

In the vacuum, the regime of a quantizing magnetic field is achieved when the
field strength is comparable to the fermion mass or larger. In the case of
quantum electrodynamics, this means that the magnetic field should be
comparable to or larger than $B_{c}\simeq m_{e}^{2}/e\approx 4.414\times
10^{13}~{}\mbox{G}$. In hot (dense) matter, besides, the Landau energy scale
should be comparable to or exceed the value of temperature (chemical
potential). While such relativistic systems are not common, sufficiently
strong magnetic fields are realized naturally under certain conditions. One
example is the magnetars that host a strong enough field to turn the vacuum
into a birefringent medium. The corresponding theoretical prediction stems
from the well-known Heisenberg-Euler effective action Heisenberg:1935qt .
Tentatively, it has been supported by the observation of thermal emission with
a large degree of polarization coming from the isolated neutron star RX
J1856.5–3754 Mignani:2016fwz .

Superstrong magnetic fields can also appear in the early Universe and possibly
affect its evolution Vachaspati:1991nm ; Brandenburg:1996fc ; Grasso:2000wj ;
Banerjee:2004df . Moreover, some observable signatures of the primordial
fields can be detected in present-day astronomy Neronov:1900zz ;
Tavecchio:2010mk ; Dolag:2010ni ; Taylor:2011bn ; Tashiro:2013bxa ;
Tashiro:2013ita . Similar conditions, albeit on much smaller spatial scales,
could be also created in ultrarelativistic heavy-ion collisions. Indeed, such
collisions produce not only a hot quark-gluon plasma but also strong magnetic
fields. Theoretical predictions suggest that the strength of the magnetic
field in noncentral heavy-ion collisions could be comparable to the QCD energy
scales, i.e., $B\gtrsim m_{\pi}^{2}/e\approx 3.1\times 10^{18}~{}\mbox{G}$
Skokov:2009qp ; Deng:2012pc . The presence of such a strong field is likely to
have numerous observable effects that lead to distinct features in various
multi-particle correlators of charged particles Kharzeev:2015znc ;
Miransky:2015ava .

To make further progress in studies of strongly magnetized plasmas, it is
beneficial to have a more detailed knowledge of the photon polarization
effects. Despite promising attempts reported by several research groups,
however, they remain largely unknown in the regime of nonzero temperature and
chemical potential. At nonzero temperature, for example, the polarization
tensor was obtained in the lowest Landau level approximation in Ref.
Bandyopadhyay:2016fyd  and the weak-field limit in Ref. Das:2019nzv ;
Ghosh:2019kmf . Some calculations were also done within the framework of the
Ritus method and the real-time formalism in Refs. Sadooghi:2016jyf  and
Ghosh:2018xhh ; Ghosh:2020xwp .

Here we report a partial solution to the general problem in the regime of a
strongly magnetized thermal plasma. In particular, by making use of the Landau
level representation for the fermion Green’s function, we derive a relatively
simple closed-form analytical expression for the absorptive part of the
polarization tensor foot1  at nonzero energy and momentum. It should be
emphasized that we consider a magnetized relativistic plasma in the quantizing
limit when the effects of Landau level quantization play an important role.
This is in contrast to the semiclassical regime that assumes a weak magnetic
field when the quantization effects play no role. The absorptive part of the
polarization function gets contributions from the following particle-particle,
antiparticle-antiparticle, and particle-antiparticle processes: (i)
$e^{-}\rightarrow e^{-}+\gamma$, (ii) $e^{+}\rightarrow e^{+}+\gamma$, and
(iii) $e^{-}+e^{+}\rightarrow\gamma$. By extending this work in the future, we
hope to obtain also the real part of the polarization tensor within a similar
framework.

In this study, we will determine the photon polarization function in a
magnetized plasma by direct calculation. We will find that it contains 4
symmetric and 2 antisymmetric tensor structures. As required by the gauge
symmetry, all tensor structures are transverse with respect to the external
photon momentum. Our results may have direct applications to several
observable effects, including the photon emission and absorption, optical
conductivity, and Landau damping in strongly magnetized relativistic plasmas.
We expect that the polarization effects in this regime could be useful also
for illuminating the anomalous properties of chiral plasmas Miransky:2015ava .

To demonstrate possible applications of the main result, we use the absorptive
part of the polarization tensor to calculate the differential photon emission
rate in a hot QED plasma in a strong magnetic field. As we will argue below,
one of the contributions is analogous to the synchrotron emission in the
quantum limit Sokolov:1986nk , while the other to the one-photon particle-
antiparticle annihilation. This particular result was applied in Ref.
Wang:2020dsr  to a study of the hot quark-gluon plasma, produced in
relativistic heavy-ion collisions. In particular, it was argued that the
angular dependence of the photon emission in a strong magnetic field can
resolve the so-called $v_{2}$-puzzle in the observed photon spectra. (For
earlier studies, see also Refs. Tuchin:2012mf ; Basar:2012bp ; Yee:2013qma ;
Tuchin:2014pka ; Zakharov:2016mmc .) To further verify the validity of our
result, here we also calculate the optical conductivity of the magnetized hot
QED plasma and compare it with similar results obtained for Dirac semimetals.

This paper is organized as follows. In Sec. II, we start from the definition
of the one-loop photon polarization tensor in terms of the fermion Green’s
functions in a magnetic field. In the same section, we also obtain a general
expression for the absorptive part of the polarization tensor. The photon
emission in a magnetized hot QED plasma is studied in detail in Sec. III. The
derivation of the magneto-optical conductivity and the corresponding numerical
results are presented in Sec. IV. The summary of the main results and
conclusions are given in Sec. V. Some technical details and calculations are
provided in several appendices at the end of the paper.

##  II Polarization function

In a hot magnetized plasma, each charged particle species (or flavor) affects
the photon polarization function. At leading order in the coupling, the
individual contributions to the polarization are represented by the same one-
loop Feynman diagram in Fig. 1. This implies that the effects of different
particle species are additive in this approximation. Therefore, without loss
of generality, it is sufficient to analyze a partial contribution of a single
fermion species of mass $m$ and charge $q$. In the case of QED plasma, for
example, the values of the corresponding model parameters are
$m=0.511~{}\mbox{MeV}$ and $q=-e$, where $e>0$ is the absolute value of the
electron charge. Such a QED plasma, which is made of equal number densities of
electrons $e^{-}$ and positrons $e^{+}$, can be viewed as a default choice in
the analysis below. However, by changing the values of $m$ and $q$, the same
results will also apply to other relativistic plasmas made of charged fermions
(e.g., the quark-gluon plasma, where the polarization effects are determined
primarily by the lightest up, down, and perhaps strange quarks).

Let us briefly comment on the validity of the one-loop approximation used
here. In the context of QED, which is the main emphasis in the current study,
it is an excellent approximation because the fine structure constant is very
small. Of course, the same approximation is not necessarily as good in QCD.
Nevertheless, as argued in Ref. Wang:2020dsr , it may work with some caveats
in the regime of a very strong magnetic field. Possible ways of improving the
approximation by going beyond the leading order were also outlined in Ref.
Wang:2020dsr , but the prospects of real progress in calculating subleading
corrections are not clear at this point.

Figure 1: The leading-order one-loop Feynman diagram for the photon
polarization tensor in a magnetic field.

Without loss of generality, we will assume that the magnetic field
$\mathbf{B}$ points in the $+z$ direction, see Fig. 2. For the vector
potential, we will use the simplest Landau gauge: $\mathbf{A}=(-By,0,0)$,
where $B$ is the magnetic field strength. As is easy to verify, the
corresponding field strength tensor reads $F^{\mu\nu}=-\varepsilon^{0\mu\nu
3}B$. In such a constant background field, the fermion Green’s function takes
the following form Miransky:2015ava :

$G(t-t^{\prime};\mathbf{r},\mathbf{r}^{\prime})=e^{i\Phi(\mathbf{r}_{\perp},\mathbf{r}_{\perp}^{\prime})}\bar{G}(t-t^{\prime};\mathbf{r}-\mathbf{r}^{\prime}),$
(1)

where $\mathbf{r}=(x,y,z)$ is the position vector and
$\mathbf{r}_{\perp}=(x,y)$ is its projection on the plane perpendicular to the
magnetic field. The explicit expression for the Schwinger phase is given by
$\Phi(\mathbf{r}_{\perp},\mathbf{r}_{\perp}^{\prime})=-qB(x-x^{\prime})(y+y^{\prime})/2$.
As one can see, the Schwinger phase is the only part of the Green’s function
that breaks the translation invariance. Note that this phase does not affect
the calculation of the photon polarization tensor at the leading one-loop
order. The same will not remain true at higher loops, however.

Figure 2: A schematic illustration of the coordinate system used. The photon
momentum and its projection on the plane perpendicular to the magnetic field
are denoted by $\mathbf{k}$ and $\mathbf{k}_{\perp}$, respectively.

It is convenient to write the translation invariant part of Green’s function
$\bar{G}(t;\mathbf{r})$ in Eq. (1) by using a mixed coordinate-momentum space
representation Miransky:2015ava ,

$\bar{G}(t;\mathbf{r})=\int\frac{d\omega dp_{z}}{(2\pi)^{2}}e^{-i\omega
t+ip_{z}z}\bar{G}(\omega;p_{z};\mathbf{r}_{\perp}),$ (2)

where the Fourier transform is given as a sum over the Landau levels, i.e.,

$\bar{G}(\omega,p_{z};\mathbf{r}_{\perp})=i\frac{e^{-\mathbf{r}_{\perp}^{2}/(4\ell^{2})}}{2\pi\ell^{2}}\sum_{n=0}^{\infty}\frac{\tilde{D}_{n}(\omega,p_{z};\mathbf{r}_{\perp})}{\omega^{2}-p_{z}^{2}-m^{2}-2n|qB|}.$
(3)

The numerator in the $n$th Landau level contribution has the following
explicit form:

$\tilde{D}_{n}(\omega,p_{z};\mathbf{r}_{\perp})=\left(\omega\gamma^{0}-p_{z}\gamma^{3}+m\right)\left[\mathcal{P}_{+}L_{n}\left(\frac{\mathbf{r}_{\perp}^{2}}{2\ell^{2}}\right)+\mathcal{P}_{-}L_{n-1}\left(\frac{\mathbf{r}_{\perp}^{2}}{2\ell^{2}}\right)\right]-\frac{i}{\ell^{2}}(\mathbf{r}_{\perp}\cdot\bm{\gamma}_{\perp})L_{n-1}^{1}\left(\frac{\mathbf{r}_{\perp}^{2}}{2\ell^{2}}\right),$
(4)

where $L_{n}^{\alpha}(z)$ are the generalized Laguerre polynomials Gradshtein
, $\mathcal{P}_{\pm}\equiv\frac{1}{2}\left(1\pm
is_{\perp}\gamma^{1}\gamma^{2}\right)$ are the spin projectors, and
$\ell=1/\sqrt{|qB|}$ is the magnetic length. By definition,
$s_{\perp}=\,\mbox{sign}(qB)$ and $L_{-1}^{\alpha}(z)\equiv 0$.

By making use of the fermion Green’s function in the Landau-level
representation, it is straightforward to write down the momentum-space
expression for the polarization function in a magnetized plasma. Within the
Matsubara finite-temperature formalism, the corresponding one-loop result
reads

$\Pi^{\mu\nu}(i\Omega_{m};\mathbf{k})=4\pi N_{f}\alpha
T\sum_{k=-\infty}^{\infty}\int\frac{dp_{z}}{2\pi}\int
d^{2}\mathbf{r}_{\perp}e^{-i\mathbf{r}_{\perp}\cdot\mathbf{k}_{\perp}}\,\mbox{tr}\left[\gamma^{\mu}\bar{G}(i\omega_{k},p_{z};\mathbf{r}_{\perp})\gamma^{\nu}\bar{G}(i\omega_{k}-i\Omega_{m},p_{z}-k_{z};-\mathbf{r}_{\perp})\right],$
(5)

where $\alpha=q^{2}/(4\pi)$ is the coupling constant, $N_{f}$ is the number of
active fermion flavors in the plasma, and the trace in the integrand runs over
the Dirac indices. As per standard convention, the fermionic and bosonic
Matsubara frequencies are $\omega_{k}=(2k+1)\pi T$ and $\Omega_{m}=2n\pi T$,
respectively. By substituting Green’s function (3) into the definition for
$\Pi^{\mu\nu}$ and performing the Matsubara summation with the help of Eq.
(48) in Appendix A, we derive the following expression for the polarization
tensor:

$\Pi^{\mu\nu}(i\Omega_{m};\mathbf{k})=-\frac{\alpha
N_{f}}{\pi\ell^{4}}\sum_{n,n^{\prime}=0}^{\infty}\int\frac{dp_{z}}{2\pi}\sum_{\lambda=\pm
1}\frac{(E_{n,p_{z}}-\lambda
E_{n^{\prime},p_{z}-k_{z}})\left[n_{F}(E_{n,p_{z}})-n_{F}(\lambda
E_{n^{\prime},p_{z}-k_{z}})\right]}{2\lambda
E_{n,p_{z}}E_{n^{\prime},p_{z}-k_{z}}\left[(E_{n,p_{z}}-\lambda
E_{n^{\prime},p_{z}-k_{z}})^{2}+\Omega_{m}^{2}\right]}\sum_{i=1}^{4}I_{i}^{\mu\nu},$
(6)

where $E_{n}=\sqrt{m^{2}+p_{z}^{2}+2n|qB|}$ is the fermion energy in the $n$th
Landau level. Note that, in the last expression, we also calculated the Dirac
traces and integrated over the transverse spatial coordinates
($\mathbf{r}_{\perp}$) by using the results in Appendices B and C. The
explicit expressions for functions $I_{i}^{\mu\nu}$ can be found in Appendix
C.

By imposing a suitable regularization for the Landau-level sum in Eq. (6), in
principle, one could use the brute-force numerical methods to evaluate the
(retarded) photon polarization function in a hot magnetized plasma. Moreover,
the partial contributions in the sum have a clear physical interpretation in
terms of quantum transitions between Landau-level states. In practice,
however, the corresponding calculation is not easy. Additional complications
come from the need to perform the analytic continuation to real values of
photon energy, $i\Omega_{m}\to\Omega+i\epsilon$. A partial resolution of the
problem is to extract the real and imaginary parts of the retarded
polarization function and study them separately. As we will discuss below, the
structure of the imaginary part is much simpler than that of the real one.
Thus, in this paper, we will concentrate primarily on the imaginary part and
leave the real part of the polarization tensor for future studies.

After replacing $i\Omega_{m}\to\Omega+i\epsilon$, it is straightforward to
extract the imaginary part of the retarded polarization tensor. The result
reads

$\mbox{Im}\left[\Pi_{R}^{\mu\nu}(\Omega;\mathbf{k})\right]=\frac{\alpha
N_{f}}{2\ell^{4}}\sum_{n,n^{\prime}=0}^{\infty}\int\frac{dp_{z}}{2\pi}\sum_{\lambda,\eta=\pm
1}\frac{n_{F}(E_{n,p_{z}})-n_{F}(\lambda
E_{n^{\prime},p_{z}-k_{z}})}{2\eta\lambda
E_{n,p_{z}}E_{n^{\prime},p_{z}-k_{z}}}\sum_{i=1}^{4}I_{i}^{\mu\nu}\delta\left(E_{n,p_{z}}-\lambda
E_{n^{\prime},p_{z}-k_{z}}+\eta\Omega\right).$ (7)

(Strictly speaking, this is the absorptive rather then imaginary part of the
polarization tensor foot1 .) The $\delta$ function has a nonvanishing support
when the following energy conservation equation

$E_{n,p_{z}}-\lambda E_{n^{\prime},p_{z}-k_{z}}+\eta\Omega=0$ (8)

is satisfied. Therefore, by finding the values of $p_{z}$ that solve the
corresponding equation, the integration over $p_{z}$ can be performed
analytically.

Before proceeding further with the analysis, let us note that the imaginary
part of the polarization tensor in Eq. (7) is an odd function of the photon
frequency $\Omega$. Taking this into account, we will assume without loss of
generality that $\Omega>0$ in the rest of the paper. From a physics viewpoint,
the case of $\Omega>0$ could be associated with the photon emission processes
while $\Omega<0$ with the photon absorption processes. In equilibrium, of
course, the rates should be the same as required by the principle of detailed
balance.

Depending on the choice of the signs of $\lambda$ and $\eta$, the energy
conservation equation (8) represents one of the three possible processes
involving particle and/or antiparticle states with the Landau level indices
$n$ and $n^{\prime}$. Schematically, the corresponding processes are shown in
Fig. 3. Two of them are the particle and antiparticle splitting processes
processes, $e^{-}\to e^{-}+\gamma$ and $e^{+}\to e^{+}+\gamma$, shown in
panels (a) and (b), respectively. They correspond to transitions between
Landau levels with the energies of the same signs, as shown schematically in
panels (d) and (e), respectively. Note that these processes represent the
quantum version of the synchrotron radiation. Unlike the quasiclassical
version, the current description in terms of the Landau level transitions
captures all quantum effects of the synchrotron radiation. The latter includes
the quantization of photon energies and the fermion recoil effects due to
photon emission.

The third process type is the particle-antiparticle annihilation, shown
diagrammatically in panel (c) of Fig. 3. In terms of Landau levels, it
corresponds to transitions between the states with energies of the opposite
signs, shown in panel (f).

Figure 3: Three types of processes involving fermion states with the Landau
level indices $n$ and $n^{\prime}$: (a) $e^{-}\to e^{-}+\gamma$, (b) $e^{+}\to
e^{+}+\gamma$, (c) $e^{-}+e^{+}\to\gamma$. The corresponding Landau level
transitions are shown schematically in panels (d), (e) and (f), respectively.

By examining the energy conservation equation (8), it is easy to check that
the two splitting processes are described by setting $\lambda=+1$. The sign of
$\eta$ further specifies whether the splitting process is $e^{-}\to
e^{-}+\gamma$ ($\eta=-1$) or $e^{+}\to e^{+}+\gamma$ ($\eta=+1$). In these two
cases, real solutions for $p_{z}$ exist only when the following necessary
conditions are satisfied:

$\displaystyle\sqrt{\Omega^{2}-k_{z}^{2}}\leq k_{-}$ and $\displaystyle
n>n^{\prime},$ (9)

for $e^{-}\to e^{-}+\gamma$ ($\lambda=+1$, $\eta=-1$), and

$\displaystyle\sqrt{\Omega^{2}-k_{z}^{2}}\leq k_{-}$ and $\displaystyle
n<n^{\prime},\quad$ (10)

for $e^{+}\to e^{+}+\gamma$ ($\lambda=+1$, $\eta=+1$). In Eqs. (9) and (10),
we introduced the shorthand notation for the transverse momentum threshold

$k_{-}=\left|\sqrt{m^{2}+2n|qB|}-\sqrt{m^{2}+2n^{\prime}|qB|}\right|,$ (11)

which depends on the Landau level indices $n$ and $n^{\prime}$ of the initial
and final quantum states, respectively.

The annihilation process $e^{-}+e^{+}\to\gamma$ is realized when $\lambda=-1$
and $\eta=-1$. In this case, the necessary condition for the existence of
real-valued solutions for $p_{z}$ is

$\sqrt{\Omega^{2}-k_{z}^{2}}\geq k_{+},$ (12)

where, by definition, the transverse momentum threshold is

$k_{+}=\left|\sqrt{m^{2}+2n|qB|}+\sqrt{m^{2}+2n^{\prime}|qB|}\right|.$ (13)

When $\Omega>0$, there are no physical processes that correspond to
$\lambda=-1$ and $\eta=+1$. Accordingly, there are no real solutions to the
energy conservation equation (8) in this case.

In all cases, when real solutions exist, they are given by the following
explicit expressions:

$\displaystyle p_{z}^{(\pm)}$ $\displaystyle=$
$\displaystyle\frac{k_{z}}{2}\left[1+\frac{2(n-n^{\prime})|qB|}{\Omega^{2}-k_{z}^{2}}\pm\frac{\Omega}{|k_{z}|}\sqrt{\left(1-\frac{k_{-}^{2}}{\Omega^{2}-k_{z}^{2}}\right)\left(1-\frac{k_{+}^{2}}{\Omega^{2}-k_{z}^{2}}\right)}\right].$
(14)

Note that there are two solutions labeled by the plus and minus signs in the
superscript. By using this result and the energy conservation equation,
written in the form
$(E_{n,p_{z}}+\eta\Omega)^{2}=E_{n^{\prime},p_{z}-k_{z}}^{2}$, it is
straightforward to derive the following explicit expressions for the fermions
energies:

$\displaystyle\left.E_{n,p_{z}}\right|_{p_{z}=p_{z}^{(\pm)}}$ $\displaystyle=$
$\displaystyle-\frac{\eta\Omega}{2}\left[1+\frac{2(n-n^{\prime})|qB|}{\Omega^{2}-k_{z}^{2}}\pm\frac{|k_{z}|}{\Omega}\sqrt{\left(1-\frac{k_{-}^{2}}{\Omega^{2}-k_{z}^{2}}\right)\left(1-\frac{k_{+}^{2}}{\Omega^{2}-k_{z}^{2}}\right)}\right],$
(15)
$\displaystyle\left.E_{n^{\prime},p_{z}-k_{z}}\right|_{p_{z}=p_{z}^{(\pm)}}$
$\displaystyle=$
$\displaystyle\frac{\lambda\eta\Omega}{2}\left[1-\frac{2(n-n^{\prime})|qB|}{\Omega^{2}-k_{z}^{2}}\mp\frac{|k_{z}|}{\Omega}\sqrt{\left(1-\frac{k_{-}^{2}}{\Omega^{2}-k_{z}^{2}}\right)\left(1-\frac{k_{+}^{2}}{\Omega^{2}-k_{z}^{2}}\right)}\right].$
(16)

As expected, these expressions define positive definite energies when the
necessary conditions in Eqs. (9), (10), and (12) are satisfied.

By making use of the explicit solutions, the $\delta$ function in Eq. (7) can
be rewritten in a much simpler form, i.e.,

$\displaystyle\delta\left(E_{n,p_{z}}-\lambda
E_{n^{\prime},p_{z}-k_{z}}+\eta\Omega\right)=\sum_{s=\pm}\frac{2E_{n,p_{z}}E_{n^{\prime},p_{z}-k_{z}}\delta\left(p_{z}-p_{z}^{(s)}\right)}{\sqrt{\left(\Omega^{2}-k_{z}^{2}-k_{-}^{2}\right)\left(\Omega^{2}-k_{z}^{2}-k_{+}^{2}\right)}}.$
(17)

Indeed, this can be derived by using the following result:

$\displaystyle\left|\frac{\partial\left(E_{n,p_{z}}-\lambda
E_{n^{\prime},p_{z}-k_{z}}\right)}{\partial
p_{z}}\right|_{p_{z}=p_{z}^{(\pm)}}=\frac{\left|k_{z}E_{n,p_{z}}+\eta
p_{z}^{(\pm)}\Omega\right|}{E_{n,p_{z}}E_{n^{\prime},p_{z}-k_{z}}}\Bigg{|}_{p_{z}=p_{z}^{(\pm)}}=\frac{\sqrt{\left(\Omega^{2}-k_{z}^{2}-k_{-}^{2}\right)\left(\Omega^{2}-k_{z}^{2}-k_{+}^{2}\right)}}{2E_{n,p_{z}}E_{n^{\prime},p_{z}-k_{z}}}.$
(18)

To obtain the last form of the expression, we used Eqs. (14) and (15).

By substituting Eq. (17) into Eq. (7), we finally derive the expression for
the imaginary part of the polarization tensor:

$\mbox{Im}\left[\Pi_{R}^{\mu\nu}(\Omega;\mathbf{k})\right]=\frac{\alpha
N_{f}}{4\pi\ell^{4}}\sum_{n,n^{\prime}=0}^{\infty}\sum_{\lambda,\eta=\pm
1}\sum_{s=\pm
1}\Theta_{\lambda,\eta}^{n,n^{\prime}}(\Omega,k_{z})\frac{n_{F}(E_{n,p_{z}})-n_{F}(\lambda
E_{n^{\prime},p_{z}-k_{z}})}{\eta\lambda\sqrt{\left(\Omega^{2}-k_{z}^{2}-k_{-}^{2}\right)\left(\Omega^{2}-k_{z}^{2}-k_{+}^{2}\right)}}\sum_{i=1}^{4}I_{i}^{\mu\nu}\Bigg{|}_{p_{z}=p_{z}^{(s)}}.$
(19)

where threshold function $\Theta_{\lambda,\eta}^{n,n^{\prime}}(\Omega,k_{z})$
is defined as follows:

$\Theta_{\lambda,\eta}^{n,n^{\prime}}(\Omega,k_{z})=\left\\{\begin{array}[]{lll}\theta\left(k_{-}^{2}+k_{z}^{2}-\Omega^{2}\right)&\mbox{for}&\lambda=1,\eta=-1,n>n^{\prime},\\\
\theta\left(k_{-}^{2}+k_{z}^{2}-\Omega^{2}\right)&\mbox{for}&\lambda=1,\eta=1,n<n^{\prime},\\\
\theta\left(\Omega^{2}-k_{z}^{2}-k_{+}^{2}\right)&\mbox{for}&\lambda=-1,\eta=-1,\end{array}\right.$
(20)

and $\Theta_{\lambda,\eta}^{n,n^{\prime}}(\Omega,k_{z})=0$ otherwise. This
function defines the windows of the parameter space where the necessary
conditions (9), (10), and (12) are satisfied.

This analytical expression for
$\mbox{Im}\left[\Pi_{R}^{\mu\nu}(\Omega;\mathbf{k})\right]$ is one of the main
results of the paper. We will use it in the next two sections to calculate the
photon emission rate and the optical conductivity of a strongly magnetized hot
relativistic plasma. Before proceeding to the applications, however, it is
instructive to verify that the polarization tensor is transverse (i.e.,
$k_{\mu}\Pi^{\mu\nu}=0$ and $\Pi^{\mu\nu}k_{\nu}=0$), as required by the gauge
invariance of the theory.

Since both thermal bath and background magnetic field break the Lorentz
symmetry, the final tensor structure of the polarization tensor is rather
complicated. Its explicit form is determined by tensors $I_{i}^{\mu\nu}$,
defined in Appendix C. As is clear from Eq. (6) and the explicit expressions
for $I_{i}^{\mu\nu}$ in Appendix C, the general tensor structure is same for
both real and imaginary parts of the polarization function. Thus, by analyzing
all similar terms of the absorptive part of
$\Pi_{R}^{\mu\nu}(\Omega;\mathbf{k})$ in Appendix D, we find that the
polarization tensor takes following form:

$\displaystyle\Pi_{R}^{\mu\nu}(\Omega;\mathbf{k})$ $\displaystyle=$
$\displaystyle\left(\frac{k_{\parallel}^{\mu}k_{\parallel}^{\nu}}{k_{\parallel}^{2}}-g_{\parallel}^{\mu\nu}\right)\Pi_{1}+\left(g_{\perp}^{\mu\nu}+\frac{k_{\perp}^{\mu}k_{\perp}^{\nu}}{k_{\perp}^{2}}\right)\Pi_{2}+\left(\frac{k_{\parallel}^{\mu}\tilde{k}_{\parallel}^{\nu}+\tilde{k}_{\parallel}^{\mu}k_{\parallel}^{\nu}}{k_{\parallel}^{2}}+\frac{\tilde{k}_{\parallel}^{\mu}k_{\perp}^{\nu}+k_{\perp}^{\mu}\tilde{k}_{\parallel}^{\nu}}{k_{\perp}^{2}}\right)\Pi_{3}$
(21) $\displaystyle+$
$\displaystyle\left(\frac{k_{\parallel}^{\mu}k_{\perp}^{\nu}+k_{\perp}^{\mu}k_{\parallel}^{\nu}}{k_{\parallel}^{2}}+\frac{k_{\perp}^{2}}{k_{\parallel}^{2}}g_{\parallel}^{\mu\nu}-g_{\perp}^{\mu\nu}\right)\Pi_{4}+\left(\frac{F^{\mu\nu}}{B}+\frac{k_{\parallel}^{\mu}\tilde{k}_{\perp}^{\nu}-\tilde{k}_{\perp}^{\mu}k_{\parallel}^{\nu}}{k_{\parallel}^{2}}\right)\tilde{\Pi}_{5}+\frac{\tilde{k}_{\parallel}^{\mu}\tilde{k}_{\perp}^{\nu}-\tilde{k}_{\perp}^{\mu}\tilde{k}_{\parallel}^{\nu}}{k_{\parallel}^{2}}\tilde{\Pi}_{6},$

where we used the following notation:

$\begin{array}[]{lll}g_{\parallel}^{\mu\nu}=\mbox{diag}(1,0,0,-1),&k_{\parallel}^{\mu}=g_{\parallel}^{\mu\nu}k_{\nu}=k_{0}\delta^{\mu}_{0}+k_{z}\delta^{\mu}_{3},&\tilde{k}_{\parallel}^{\mu}=-\varepsilon^{\mu
12\nu}k_{\nu}=k_{z}\delta^{\mu}_{0}+k_{0}\delta^{\mu}_{3},\\\
g_{\perp}^{\mu\nu}=\mbox{diag}(0,-1,-1,0),&k_{\perp}^{\mu}=g_{\perp}^{\mu\nu}k_{\nu}=k_{x}\delta^{\mu}_{1}+k_{y}\delta^{\mu}_{2},&\tilde{k}_{\perp}^{\mu}=-\varepsilon^{0\mu\nu
3}k_{\nu}=k_{y}\delta^{\mu}_{1}-k_{x}\delta^{\mu}_{2}.\\\ \end{array}$ (22)

Note that
$\tilde{k}_{\perp,\mu}\tilde{k}_{\perp}^{\mu}=k_{\perp,\mu}k_{\perp}^{\mu}=-k_{\perp}^{2}$,
$\tilde{k}_{\parallel,\mu}\tilde{k}_{\parallel}^{\mu}=-k_{\parallel,\mu}k_{\parallel}^{\mu}=-k_{\parallel}^{2}$,
and $k_{\mu}\tilde{k}_{\perp}^{\mu}=k_{\mu}\tilde{k}_{\parallel}^{\mu}=0$. As
we see the polarization tensor in Eq. (21) contains four symmetric tensor
structures and two antisymmetric ones. Naively, it may appear that another
symmetric tensor, i.e.,
$\tilde{k}_{\parallel}^{\mu}\tilde{k}_{\parallel}^{\nu}/k_{\parallel}^{2}$,
can be constructed from $\tilde{k}_{\parallel}^{\mu}$. As one can verify,
however, it is exactly the same as
$k_{\parallel}^{\mu}k_{\parallel}^{\nu}/k_{\parallel}^{2}-g_{\parallel}^{\mu\nu}$.
Note that the antisymmetric contributions, defined by the component functions
$\tilde{\Pi}_{5}$ and $\tilde{\Pi}_{6}$, appear because the time-reversal
symmetry is broken by a nonzero magnetic field.

By using the definitions in Eq. (22), one can verify that all six tensor
structures in Eq. (21) are transverse. Therefore, the polarization tensor is
transverse as well, i.e., $k_{\mu}\Pi^{\mu\nu}=0$ and $\Pi^{\mu\nu}k_{\nu}=0$,
which is consistent with the gauge invariance. The explicit expressions for
the (absorptive part of) component functions are given in Appendix D.

##  III Photon emission from QED plasma

Here we will use the general result for
$\mbox{Im}[\Pi_{R}^{\mu\nu}(\Omega;\mathbf{k})]$ which is obtained in the
previous section to study the photon emission from a strongly magnetized hot
QED plasma. The corresponding emission is directly observable when the plasma
is optically thin, i.e., its size is small compared to the photon mean free
path in the plasma. For optically thick systems, the photon emission will come
only from the surface layer of a depth comparable to the mean free path.

###  III.1 Photon emission: Analytical results

In quantum field theory, the photon production rate from a thermally
equilibrated charged plasma can be expressed in terms of the imaginary part of
the retarded polarization tensor $\Pi^{\mu\nu}_{R}(\Omega,\mathbf{k})$ as
follows Kapusta:2006pm :

$\Omega\frac{d^{3}R}{d^{3}\mathbf{k}}=-\frac{1}{(2\pi)^{3}}\frac{\mbox{Im}\left[\Pi^{\mu}_{R,\mu}(\Omega,\mathbf{k})\right]}{\exp\left(\frac{\Omega}{T}\right)-1},$
(23)

where $\Omega$ and $\mathbf{k}$ are the photon frequency and momentum,
respectively, and $T$ is the temperature of the plasma.

Because of the rotation symmetry about the $z$ axis, it is convenient to use
the spherical coordinates and write $d^{3}\mathbf{k}=k^{2}dk\sin\theta d\theta
d\phi$. Note that the production rate is independent of the polar angle
$\phi$. Thus, without loss of generality, the differential rate is fully
characterized by

$\frac{d^{2}R}{k\,dk\,d(\cos\theta)}=-\frac{1}{(2\pi)^{2}}\frac{\mbox{Im}\left[\Pi^{\mu}_{R,\mu}(k,\mathbf{k})\right]}{\exp\left(\frac{k}{T}\right)-1},$
(24)

where we substituted $\Omega=k$ for the on-shell photons. The photon momentum
is given by $k=\sqrt{k_{\perp}^{2}+k_{z}^{2}}$, where $k_{\perp}=k\sin\theta$
and $k_{z}=k\cos\theta$ are the components perpendicular and parallel to the
magnetic field, respectively.

By performing the Lorentz contraction in expression (19), we derive the
following result:

$\displaystyle\mbox{Im}\left[\Pi^{\mu}_{R,\mu}(\Omega,\mathbf{k})\right]$
$\displaystyle=$ $\displaystyle\frac{\alpha
N_{f}}{4\pi\ell^{4}}\sum_{n>n^{\prime}}^{\infty}\theta\left(k_{-}^{2}+k_{z}^{2}-\Omega^{2}\right)\sum_{s=\pm
1}\frac{n_{F}(E_{n^{\prime},p_{z}-k_{z}})-n_{F}(E_{n,p_{z}})}{\sqrt{\left(k_{-}^{2}+k_{z}^{2}-\Omega^{2}\right)\left(k_{+}^{2}+k_{z}^{2}-\Omega^{2}\right)}}\sum_{i=1}^{4}\mathcal{F}_{i}\Bigg{|}_{p_{z}=p_{z}^{(s)},\lambda=1,\eta=-1}$
(25) $\displaystyle+$ $\displaystyle\frac{\alpha
N_{f}}{4\pi\ell^{4}}\sum_{n<n^{\prime}}^{\infty}\theta\left(k_{-}^{2}+k_{z}^{2}-\Omega^{2}\right)\sum_{s=\pm
1}\frac{n_{F}(E_{n,p_{z}})-n_{F}(E_{n^{\prime},p_{z}-k_{z}})}{\sqrt{\left(k_{-}^{2}+k_{z}^{2}-\Omega^{2}\right)\left(k_{+}^{2}+k_{z}^{2}-\Omega^{2}\right)}}\sum_{i=1}^{4}\mathcal{F}_{i}\Bigg{|}_{p_{z}=p_{z}^{(s)},\lambda=\eta=1}$
$\displaystyle+$ $\displaystyle\frac{\alpha
N_{f}}{4\pi\ell^{4}}\sum_{n,n^{\prime}=0}^{\infty}\theta\left(\Omega^{2}-k_{z}^{2}-k_{+}^{2}\right)\sum_{s=\pm
1}\frac{n_{F}(E_{n,p_{z}})+n_{F}(E_{n^{\prime},p_{z}-k_{z}})-1}{\sqrt{\left(\Omega^{2}-k_{z}^{2}-k_{-}^{2}\right)\left(\Omega^{2}-k_{z}^{2}-k_{+}^{2}\right)}}\sum_{i=1}^{4}\mathcal{F}_{i}\Bigg{|}_{p_{z}=p_{z}^{(s)},\lambda=\eta=-1},$

where, by definition, $\mathcal{F}_{i}=g_{\mu\nu}I_{i}^{\mu\nu}$ (with
$i=1,2,3,4$). As shown in Appendix C, $\mathcal{F}_{2}=\mathcal{F}_{3}=0$,
while $\mathcal{F}_{1}$ and $\mathcal{F}_{4}$ are nonzero. The explicit
expressions for the latter two functions are given by

$\displaystyle\mathcal{F}_{1}$ $\displaystyle=$ $\displaystyle
4\pi\left[\left(\Omega^{2}-k_{z}^{2}\right)\ell^{2}-2(n+n^{\prime})\right]\left(\mathcal{I}_{0}^{n-1,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n,n^{\prime}-1}(\xi)\right)+8\pi
m^{2}\ell^{2}\left(\mathcal{I}_{0}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}-1}(\xi)\right),$
(26) $\displaystyle\mathcal{F}_{4}$ $\displaystyle=$ $\displaystyle
16\pi\,\mathcal{I}_{2}^{n-1,n^{\prime}-1}(\xi),$ (27)

where $\xi=k_{\perp}^{2}\ell^{2}/2$. For the definition of functions
$\mathcal{I}_{i}^{n,n^{\prime}}(\xi)$ in terms of the Laguerre polynomials,
see Appendix B.

By making use of the explicit expressions for the fermion energies in Eqs.
(15) and (16), it is straightforward to show that the first two contributions
in Eq. (25), which describe the photon emission due to the particle splitting
$e^{-}\to e^{-}+\gamma$ ($\lambda=+1$, $\eta=-1$) and the antiparticles
splitting $e^{+}\to e^{+}+\gamma$ ($\lambda=+1$, $\eta=1$) processes, are
equal to each other. This is not surprising, of course, because the two
processes are related by the charge conjugation symmetry. (Note, however, that
the same is not true in a plasma at a nonzero chemical potential Wang:2021eud
.) By taking this symmetry into account and substituting the result into Eq.
(24), we obtain the final expression for the rate:

$\displaystyle\frac{d^{2}R}{k\,dk\,d(\cos\theta)}$ $\displaystyle=$
$\displaystyle\frac{\alpha
N_{f}}{(2\pi)^{3}\ell^{4}\left[\exp\left(\frac{k}{T}\right)-1\right]}\sum_{n>n^{\prime}}^{\infty}\frac{g(n,n^{\prime})\left[\theta\left(\Omega^{2}-k_{z}^{2}-k_{+}^{2}\right)-\theta\left(k_{-}^{2}+k_{z}^{2}-\Omega^{2}\right)\right]}{\sqrt{\left(k_{-}^{2}+k_{z}^{2}-\Omega^{2}\right)\left(k_{+}^{2}+k_{z}^{2}-\Omega^{2}\right)}}\left(\mathcal{F}_{1}+\mathcal{F}_{4}\right)$
(28) $\displaystyle+$ $\displaystyle\frac{\alpha
N_{f}}{2(2\pi)^{3}\ell^{4}\left[\exp\left(\frac{k}{T}\right)-1\right]}\sum_{n=0}^{\infty}\frac{g_{0}(n)\theta\left(\Omega^{2}-k_{z}^{2}-k_{+}^{2}\right)}{\sqrt{\left(\Omega^{2}-k_{z}^{2}\right)\left(\Omega^{2}-k_{z}^{2}-k_{+}^{2}\right)}}\left(\mathcal{F}_{1}+\mathcal{F}_{4}\right),$

where

$\displaystyle g(n,n^{\prime})$ $\displaystyle=$ $\displaystyle
2-\sum_{s_{1},s_{2}=\pm}n_{F}\left(\frac{\Omega}{2}+s_{1}\frac{\Omega(n-n^{\prime})|qB|}{\Omega^{2}-k_{z}^{2}}+s_{2}\frac{|k_{z}|}{2(\Omega^{2}-k_{z}^{2})}\sqrt{\left(\Omega^{2}-k_{z}^{2}-k_{-}^{2}\right)\left(\Omega^{2}-k_{z}^{2}-k_{+}^{2}\right)}\right),$
(29) $\displaystyle g_{0}(n)$ $\displaystyle=$ $\displaystyle
g(n,n)=2-2\sum_{s=\pm}n_{F}\left(\frac{\Omega}{2}+s\frac{|k_{z}|}{2}\sqrt{1-\frac{4(m^{2}+2n|qB|)}{\Omega^{2}-k_{z}^{2}}}\right).$
(30)

While the analytical expression in Eq. (28) for the photon production rate is
relatively simple, its predictions are not obvious. As is clear, each term in
the sum has a simple interpretation as a partial contribution from a splitting
or annihilation process that involves a quantum transition between Landau
levels $n$ and $n^{\prime}$. Schematically, the relevant low-energy
transitions for the two types of processes ($e^{-}\to e^{-}+\gamma$ and
$e^{-}+e^{+}\to\gamma$, respectively) are shown in Figs. 4 and 5 for several
fixed values of the photon energy and several fixed directions of emission.
Because of the superposition of a large (infinite) set of relevant
transitions, the final energy and angular dependence of the differential rate
is hard to grasp from Eq. (28). Therefore, in the next subsection, we use
numerical methods to study the photon production rate in more detail.

Figure 4: Schematic representation of Landau level transitions for the
splitting processes $e^{-}\to e^{-}+\gamma$ with a fixed direction of the
photon emission, i.e., $\theta=0$ (blue), $\theta=\pi/3$ (green), and
$\theta=\pi/2$ (red), allowed by the energy conservation. Note that, at
$\theta=0$, only the transitions between the adjacent Landau levels ($n+1\to
n$) contribute nontrivially. The individual panels show the results for
different photon energies: (a) $\Omega=m$, (b) $\Omega=2m$, (c) $\Omega=3m$.

Figure 5: Schematic representation of Landau level transitions for the
annihilation processes $e^{-}+e^{+}\to\gamma$ with a fixed direction of the
photon emission, i.e., $\theta=\pi/6$ (blue), $\theta=\pi/3$ (green), and
$\theta=\pi/2$ (red), allowed by the energy conservation. Note that, the
emission is forbidden at $\theta=0$. The individual panels show the results
for different photon energies: (a) $\Omega=5m$, (b) $\Omega=10m$, (c)
$\Omega=15m$.

Let us note in passing that the general result in Eq. (28), derived in the
strong-field case, should also have a well-defined zero-field limit. To obtain
such a limit, one has to perform an infinite sum over the Landau levels
analytically. While such a task is beyond the scope of this study, it may not
be hopeless if the resumming technique of Ref. Sokolov:1986nk  is utilized.

###  III.2 Photon emission: Numerical results

To get an insight into the underlying physics of the photon emission from a
magnetized relativistic plasma, here we study the corresponding differential
rate by numerical methods. To stay as general as possible, we will express all
dimensionful parameters (e.g., the magnetic field strength and temperature) in
units of the fermion mass $m$. In particular, we will consider the following
three values of temperature: $T=5m$, $T=15m$, and $T=50m$, which cover a range
of different regimes (from moderately relativistic to ultrarelativistic). To
explore the interplay of the magnetic field and thermal effects in each
regime, we will calculate the photon production rate for two different values
of the magnetic field, i.e., $\sqrt{|qB|}=5m$ and $\sqrt{|qB|}=15m$.

For the QED plasma, the mass of particles (electrons) is
$m=m_{e}=0.511~{}\mbox{MeV}$ and the charge is $q=-e$. In this case, as is
easy to check, the two choices of the magnetic field $\sqrt{|qB|}=5m_{e}$ and
$\sqrt{|qB|}=15m_{e}$ correspond to $B\approx 1.1\times 10^{15}~{}\mbox{G}$
and $B\approx 9.9\times 10^{15}~{}\mbox{G}$, respectively. Note that the
approximate conversion formula reads $B\approx 4.414\times
10^{13}(\sqrt{|qB|}/m_{e})^{2}~{}\mbox{G}$, where both $\sqrt{|qB|}$ and
$m_{e}$ are given in the energy units. On the other hand, if the charged
particles are quarks with charge $q$ and mass $m=5~{}\mbox{MeV}$, the same
numerical results, rescaled by a factor of $(q/e)^{2}$, will also describe the
partial contribution of photon emission from the given quark species in a
magnetized quark-gluon plasma at $B\approx 1.1\times 10^{17}|e/q|~{}\mbox{G}$
and $B\approx 9.5\times 10^{17}|e/q|~{}\mbox{G}$, respectively.

When calculating the rate in Eq. (28) numerically, the sum over Landau levels
has to be truncated at some finite value of $n_{\rm max}$. Qualitatively, such
truncation is equivalent to setting an ultraviolet energy cutoff at about
$\Lambda_{UV}=\sqrt{2n_{\rm max}|qB|}$. In application to the photon emission,
this implies that the photon energy should be limited from above by the
cutoff, i.e., $\Omega\lesssim\Lambda_{UV}$. Indeed, the photon emission is
meaningful within this framework only when the energies of both initial and
final Landau-level states lie below $\Lambda_{UV}$. In general, it is expected
that the emission rate in the plasma is a rapidly decreasing function of the
photon energy at large $\Omega$. However, its value can be calculated reliably
from Eq. (28) only if sufficiently many Landau levels are included in the sum,
i.e., $n_{\rm max}\gtrsim\Omega^{2}/(2|qB|)$.

As the numerical analysis will show, the photon emission is dominated by the
particle and antiparticle splitting processes ($e^{-}\to e^{-}+\gamma$ and
$e^{+}\to e^{+}+\gamma$), provided the temperature is comparable to or higher
than the magnetic energy scale $\sqrt{|qB|}$ and the photon energy $\Omega$ is
not much larger than $\sqrt{|qB|}$. Nevertheless, the particle-antiparticle
annihilation processes ($e^{-}+e^{+}\to\gamma$) also contribute, especially at
small temperatures and large $\Omega$. For a fixed value of $\Omega$, we find
that the maximum Landau-level indices are constrained by Eq. (12), namely,
$\mbox{max}(n,n^{\prime})\leq
n^{*}_{\Omega}=\left[\Omega(\Omega-2m)/(2|qB|)\right]$. Therefore, the
corresponding sum of the annihilation contributions can be limited to $n_{\rm
max}=n^{*}_{\Omega}$. Note that $n^{*}_{\Omega}$ grows quadratically with the
photon energy $\Omega$ and decreases with the magnetic field as the inverse
power law $1/B$. Numerically, one finds that $n^{*}_{\Omega}=196$ for
$\Omega=100m$ and $|qB|=25m^{2}$. On the other hand, $n^{*}_{\Omega}=21$ for
$\Omega=100m$ and $|qB|=225m^{2}$.

As we will discuss later in more detail, the low-energy quantization of Landau
levels also sets a limit for the photon energy from below, namely
$\Omega\gtrsim\Lambda_{IR}$, where $\Lambda_{IR}=\sqrt{|qB|}/\sqrt{2n_{\rm
max}}$. The restriction comes from the fact that the separation between the
fermionic Landau levels is of the order of $|qB|$ at low energies. As a
result, the emission of photons with small energies, $\Omega\ll\sqrt{|qB|}$,
can come only from the quantum transitions between the states with
sufficiently large longitudinal momenta and/or Landau-level indices, where the
vertical separation between the levels is equal to or smaller than $\Omega$,
see Fig. 4(a). From a physics viewpoint, this implies that the rate is
strongly suppressed when $\Omega\ll\sqrt{|qB|}$. On a technical side, however,
this means that a reliable calculation requires the inclusion of a
sufficiently large number of Landau levels, i.e., $n_{\rm
max}\gtrsim|qB|/(2\Omega^{2})$. In the numerical calculation below, we will
set $n_{\rm max}\simeq 450$. Even with such large $n_{\rm max}$, one finds
that the range of photon energies $\Omega$ is strongly constrained, i.e.,
$\sqrt{|qB|}/30\lesssim\Omega\lesssim 30\sqrt{|qB|}$. To remain in the range
of validity, therefore, we will consider the following range of photon
energies: $m\leq\Omega\leq 100m$, which lies within the required limits for
both choices of the magnetic field.

Figure 6: The differential photon production rate as a function of the angle
$\theta$ for $|qB|=25m^{2}$ and $T=5m$.

Figure 7: The differential photon production rate as a function of the angle
$\theta$ for $|qB|=25m^{2}$ and $T=15m$.

Figure 8: The differential photon production rate as a function of the angle
$\theta$ for $|qB|=25m^{2}$ and $T=50m$.

Figure 9: The differential photon production rate as a function of the angle
$\theta$ for $|qB|=225m^{2}$ and $T=5m$.

Figure 10: The differential photon production rate as a function of the angle
$\theta$ for $|qB|=225m^{2}$ and $T=15m$.

Figure 11: The differential photon production rate as a function of the angle
$\theta$ for $|qB|=225m^{2}$ and $T=50m$.

###  III.3 Photon emission at $|qB|=(5m)^{2}$

Let us start by presenting the numerical results for the differential photon
emission rate in the case the magnetic field $|qB|=(5m)^{2}$ and temperature
$T=5m$. The corresponding rates as functions of the angular coordinate
$\theta$ are shown in Fig. 6 for several representative sets of photon
energies. The results for six smaller energies ($\Omega\lesssim\sqrt{|qB|}$)
are shown in panel (a), while the results for six larger energies
($\Omega\gtrsim\sqrt{|qB|}$) are shown in panel (b). By comparing the results,
we see that the differential rate has a qualitatively different overall
behavior in the two regimes. While at small photon energies the emission tends
to be strongest along the magnetic field direction ($\theta\approx 0$), at
large energies the emission tends to be strongest in the directions
perpendicular to the magnetic field ($\theta\approx\pi/2$). In other words,
the photon emission has a nonzero ellipticity Wang:2020dsr , with the overall
profile changing from a prolate shape at $\Omega\lesssim\sqrt{|qB|}$ to an
oblate shape at $\Omega\gtrsim\sqrt{|qB|}$. The corresponding qualitative
change of the emission profile is visualized by panels (c), (d), and (e) in
Fig. 6, where the polar plots of the differential rates are presented. As one
can see from panel (a), the overall photon emission has a well-pronounced
prolate shape at small energies. The profile gradually changes from prolate to
oblate at intermediate energies, see panel (b). Finally, as seen from panel
(c), it becomes oblate at high photon energies.

As is evident from Fig. 6, the photon emission rate is not a smooth function
of the angular coordinate $\theta$. It has numerous spikes that come from the
Landau-level quantization in the spectrum of charged fermions in a magnetic
field. In essence, the spikes result from a near-threshold photon production
when $k_{\perp}=\Omega\sin\theta\approx k_{\pm}$. As one can see from Eq.
(28), the rate near the Landau-level thresholds is described by a singular
inverse-square-root dependence on the energy (at fixed $\theta$) or the
angular coordinate $\theta$ (at fixed energy). Of course, the actual
singularity is a consequence of the clean limit approximation that neglects
interaction effects of charged particles. In a more refined approximation, as
we will demonstrate in Sec. III.7, interactions lead to a nonzero
quasiparticle width that smoothes out the dependence of the rate on the energy
and the angular coordinate $\theta$. For simplicity, however, we will ignore
the interaction effects since they are not expected to change the overall
qualitative behavior of the emission rate. Moreover, since the inverse-square-
root singularities are integrable, they have little effect on the net
integrated rate (see also Sec. III.7).

As is clear from Eq. (28), there are two types of thresholds in the photon
production. One of them is associated with the splitting (i.e.,
$e^{-}\rightarrow e^{-}+\gamma$ and $e^{+}\rightarrow e^{+}+\gamma$)
processes, while the other with the annihilation (i.e.,
$e^{-}+e^{+}\rightarrow\gamma$) processes. They are characterized by a
qualitatively different dependence of the rate on the angular coordinate
$\theta$ in Fig. 6. For the particle splitting processes, the rate grows
gradually as $\theta$ approaches the threshold value from below and then drops
suddenly at the threshold. For the annihilation processes, on the other hand,
the rate first jumps to a large value (infinite value in the clean limit) at
the threshold and then decreases gradually with $\theta$. Both types could be
easily identified in the differential rates shown in Fig. 6.

For example, one can readily identify the first annihilation peak at
$\sin\theta=2m/\Omega$ (provided $\Omega\geq 2m$), which is connected with the
threshold for transitions between the lowest Landau-level states with positive
and negative energies. For example, the rates for the photon energies
$\Omega=3m$, $\Omega=4m$, $\Omega=5m$, and $\Omega=6m$, presented in Fig.
6(a), have the corresponding peaks at $\theta\approx 0.232\pi$,
$\theta=\pi/6$, $\theta\approx 0.131\pi$, and $\theta\approx 0.108\pi$,
respectively.

Similarly, one can identify some thresholds for the splitting processes. The
thresholds for the transitions between the $n=1$ and $n=0$ Landau levels
appear when $\sin\theta=(\sqrt{m^{2}+2|qB|}-m)/\Omega$ (provided
$\Omega\gtrsim 6m$). For example, the rates for the photon energies
$\Omega=10m$, $\Omega=15m$, $\Omega=20m$, $\Omega=25m$, and $\Omega=30m$,
presented in Fig. 6(b), have the corresponding peaks at $\theta\approx
0.210\pi$, $\theta=0.134\pi$, $\theta\approx 0.099\pi$, $\theta\approx
0.079\pi$, and $\theta\approx 0.066\pi$, respectively. In the rates at some
smaller photon energies, shown in Fig. 6(a), one can also identify the
thresholds for the transitions between the $n=2$ and $n=1$ Landau levels which
occur when $\sin\theta=(\sqrt{m^{2}+4|qB|}-\sqrt{m^{2}+2|qB|})/\Omega$
(provided $\Omega\gtrsim 3m$). For $\Omega=3m$, $\Omega=4m$, $\Omega=5m$, and
$\Omega=6m$, the corresponding peaks appear at $\theta\approx 0.421\pi$,
$\theta=0.259\pi$, $\theta\approx 0.198\pi$, and $\theta\approx 0.161\pi$,
respectively.

By comparing the results at different photon energies $\Omega$ presented in
Fig. 6, we see that the rates tend to decrease overall with increasing
$\Omega$. This qualitative conclusion is not surprising and will be
reconfirmed more rigorously later by analyzing the dependence of the
integrated rates on the photon energy. It is explained largely by the strong
energy dependence of the Fermi-Dirac distribution functions that appear in the
definition of the polarization tensor, see Eq. (25). From a physics viewpoint,
the emission of high-energy photons is possible only when there exist occupied
Landau-level states with sufficiently large positive energies or empty Landau-
level states with sufficiently large negative energies. In a thermal plasma,
however, the availability of such states rapidly decreases with increasing the
energy.

For the most part, the same qualitative behavior of the differential photon
emission rate is obtained at higher temperatures, $T=15m$ and $T=50m$. The
corresponding numerical results are presented in Figs. 7 and 8, respectively.
Naturally, the photon rate grows as a whole with increasing the temperature of
plasma. Also, as one might expect, it tends to become a smoother function of
the angular coordinate $\theta$ at higher temperatures. Most importantly, we
find that, with increasing the temperature, the ellipticity of photon emission
remains qualitatively the same. In particular, the emission profile starts
from a prolate shape at low energies ($\Omega\lesssim\sqrt{|qB|}$) and
gradually evolves to an oblate shape at high energies
($\Omega\gtrsim\sqrt{|qB|}$). As one might check, even the degree of the
ellipticity at high energies remains about the same (i.e., $\sim 20\%$) .

###  III.4 Photon emission at $|qB|=(15m)^{2}$

The results for the stronger magnetic field $qB=(15m)^{2}$ are shown in Figs.
9, 10 and 11 for the same three default choices of temperature, $T=5m$,
$T=15m$, and $T=50m$, respectively. Qualitatively, the functional dependence
of the differential rate has many features similar to those at the smaller
magnetic field. However, some differences can be clearly seen as well. For
example, the quantization effects at the lowest temperature $T=5m$ are much
more pronounced. While the overall rates are small in the low-temperature
regime, the partial contribution of the annihilation processes is substantial.
For $\Omega\leq 6m$, as one can see from Fig. 9(a), there is only one well
pronounced singularity in the angular dependence and it is associated with the
first singularity at $\sin\theta=2m/\Omega$, provided the photon energy
exceeds the threshold, i.e., $\Omega>2m$. Below the threshold energy
($\Omega<2m$), on the other hand, the rate is strongly suppressed. The latter
is explained by the fact that the charge particles of a low-temperature plasma
($T\ll\sqrt{|qB|}$) reside primarily in the lowest Landau level. Since the
occupation numbers of the higher levels are negligible, the phase space for
the quark and antiquark splitting processes is highly restricted and, in turn,
the corresponding contributions to the rate are negligible.

From the polar representation of the differential rates, shown in panels (c)
through (e) of Fig. 9, the ellipticity of the emission profile may not be
obvious at the lowest temperature, $T=5m$. The main complication comes from a
very nonsmooth dependence of the rate on the angular coordinate $\theta$. By
utilizing a Fourier transform, however, one can still verify that the profile
evolves from a largely prolate shape at small energies
($\Omega\lesssim\sqrt{|qB|}$) to an oblate shape at large energies
($\Omega\gtrsim\sqrt{|qB|}$).

By comparing the results in Figs. 9, 10 and 11, we see that the differential
rates not only grow with increasing the temperature, but also their angular
dependence gets smoother. One can also verify that the relative contribution
of the splitting (annihilation) processes gets larger (smaller) with at
temperatures. Because of the smoother rates at $T=15m$ and $T=50m$, the
ellipticity of the emission profiles can be easily identified from the polar
representation of the differential rates, see (c) through (e) of Figs. 10 and
11. They are qualitatively the same as in the case of the weaker
$qB=(5m)^{2}$.

###  III.5 Integrated photon production rate

For both values of the magnetic field, $qB=(5m)^{2}$ and $qB=(15m)^{2}$, we
see that the differential rates tend to grow overall with increasing the
temperature. In order to see this clearly, it is instructive to calculate the
total integrated photon production rate, i.e.,

$\frac{dR}{k\,dk}=\int_{0}^{\pi}\frac{d^{2}R}{k\,dk\,d(\cos\theta)}\sin\theta
d\theta=-\frac{1}{(2\pi)^{2}}\int_{0}^{\pi}\frac{\mbox{Im}\left[\Pi^{\mu}_{R,\mu}(k,\mathbf{k})\right]}{\exp\left(\frac{k}{T}\right)-1}\sin\theta
d\theta.$ (31)

The corresponding numerical results for the integrated rates as a function of
the photon energy are presented in Fig. 12 for each of the three fixed values
of temperature, $T=5m$, $T=15m$, and $T=50m$. The two separate panels contains
the results for $qB=(5m)^{2}$ and $qB=(15m)^{2}$, respectively. Overall, as we
see, the photon rates grow with the temperature. This is not surprising since
hotter plasmas should shine brighter.

Figure 12: Panel (a): The energy dependence of the integrated photon
production for $|qB|=25m^{2}$ and three different temperatures $T=5m$ (blue
line) and $T=15m$ (green line), and $T=50m$ (red line). Panel (b): Same as
panel (a) but for $|qB|=225m^{2}$.

The energy dependence in both panels of Fig. 12 shows that the integrated
rates fall rapidly at large values of $\Omega$. This is expected, of course,
since the emission of high-energy photons is possible only when there are some
occupied (empty) charged particle (antiparticle) states with sufficiently high
positive (negative) energies. In a thermal plasma at fixed temperature $T$,
however, the number densities of such states are controlled by the Fermi-Dirac
distributions, which drop quickly with increasing particle energies.

The dependence of the integrated rates in the region of small photon energies
is more subtle. Our analysis reveals that the rate has a local maximum at some
value of $\Omega$, determined by the magnetic field strength and the
temperature. When $\Omega$ decreases further, the rate gets strongly
suppressed. This is seen in the case of the stronger magnetic field
$|qB|=225m^{2}$ in panel (b) of Fig. 12 for the rates at $T=5m$ and $T=15m$.
As we will explain in Sec. III.6 below, the same should happen also in the
case of the weaker magnetic field, $|qB|=25m^{2}$, but at smaller energies.
The corresponding suppression of the rate at small photon energies is caused
by the Landau-level quantization.

###  III.6 Landau-level quantization at small $\Omega$

As one might expect, the Landau-level quantization has a strong effect on the
photon emission at small energies. This is clear already from the illustration
of the relevant quantum transitions in Figs. 4 and 5. The corresponding
processes are possible only when the photon energy is greater than or equal to
the separation between the neighboring Landau levels. This is a necessary
condition imposed by the energy conservation.

From general considerations, the Landau quantization has the strongest effect
when $\Omega\ll\sqrt{|qB|}$. In this case, transitions between particle states
are highly restricted because the typical separation between the low-lying
Landau levels is of the order of $\sqrt{|qB|}$, while the photon energy is
much smaller. Strictly speaking, this does not apply to the annihilation
processes that involve only the (positive- and negative-energy) states in the
lowest Landau level. They still contribute when $\Omega>2m$, see Appendix E.
Nevertheless, the partial contribution of such annihilation processes to the
rate decreases rapidly when $\Omega$ goes down and approaches the threshold.
Furthermore, for $\Omega<2m$, it turns off completely. Thus, to understand the
universal suppression mechanism for the photon emission rate at small
$\Omega$, we will concentrate on the splitting processes and ignore the
annihilation channel.

The kinematic restrictions for the splitting processes at small $\Omega$ can
be understood from the visualization of the quantum transitions in Fig. 4(a).
In particular, the transitions between the neighboring Landau levels with
small indices $n$ are possible only at large enough longitudinal momenta,
where the energy levels are sufficiently close to each other. The quantitative
estimate for the momenta can be obtained from the solution of the energy
conservation condition in Eq. (14). In particular, one finds that
$|p_{z}|\sim|qB|/\left[\Omega(1+|\cos\theta|)\right]$. Since the particle
states with such large $|p_{z}|$ also have large energies, i.e.,
$E_{p_{z}}\gtrsim|qB|/\left[\Omega(1+|\cos\theta|)\right]$, their occupation
numbers and, in turn, their contributions to the photon emission rate are
exponentially suppressed when $\Omega\to 0$. One should add that there are
also numerous transitions involving states with large Landau indices (with $n$
up to about $|qB|/\Omega^{2}$) that contribute to the rate too. However, all
relevant transitions occur between states with comparable or larger energies,
implying that their contributions are subject to the same exponential
suppression by the Fermi-Dirac distributions.

As the value of $\Omega$ grows and becomes comparable to the Landau energy
scale, $\Omega\sim\sqrt{|qB|}$, the quantization effects relax gradually. This
is seen from the visualization of quantum transitions in panels (b) and (c) of
Fig. 4(a). When the photon energy $\Omega$ grows, the transitions start to
involve particle states with lower energies. The larger occupation numbers of
such states lead to a higher photon production rate. With further increasing
of $\Omega$, however, the role of quantization gradually diminishes.
Eventually, when $\Omega\gg\sqrt{|qB|}$, the rates start to decrease again
with increasing $\Omega$. Therefore, the photon production rate as a function
of the photon energy has the following universal behavior. It starts with a
vanishingly small value when $\Omega\simeq 0$, reaches a maximum at an
intermediate energy, and then decreases at large $\Omega$.

As argued in Ref. Wang:2020dsr , the quantization of Landau levels also
explains the prolate shape of the photon emission profile in the region of
small $\Omega$. The latter follows from the angular dependence of the particle
momenta of the lowest energy states participating in the splitting processes,
i.e., $|p_{z}|\sim|qB|/\left[\Omega(1+|\cos\theta|)\right]$. The smallest
values of $|p_{z}|$ (and, thus, the particle energies) are achieved when
$\theta=0$ and $\theta=\pi$ (along the line of the magnetic field). These
determine the directions of the strongest photon emission. In contrast, the
largest values of $|p_{z}|$ correspond to $\theta=\pi/2$, implying that the
photons emitted perpendicular to the magnetic field have the lowest rate.

###  III.7 Interaction effects in the vicinity of Landau-level thresholds

In this study we used the clean limit approximation that ignores all particle
interactions except for the interaction with a strong background magnetic
field. One of the consequences of such an approximation was the appearance of
numerous singular spikes in the differential photon rate as a function of the
angular coordinate $\theta$ or energy $\Omega$. The corresponding inverse-
square-root singularities were connected with Landau-level thresholds in the
three processes responsible for the photon production (i.e., $e^{-}\rightarrow
e^{-}+\gamma$, $e^{+}\rightarrow e^{+}+\gamma$, and
$e^{-}+e^{+}\rightarrow\gamma$). As we explain below, the singularities will
be removed and the rate will have smooth dependence when interaction effects
are taken into account. Qualitatively, this can be understood as follows.

In an interacting plasma, charged particles should have a nonzero
quasiparticle width $\Gamma$, which is determined by the imaginary part of the
self-energy. When the quasiparticle width is small, its effect can be captured
in the calculation of the photon emission rate by replacing the $\delta$
function in the particle spectral weight with a Gaussian profile of width
$\Gamma$. The inclusion of $\Gamma$ will modify the on-shell condition for
interacting particles and, in turn, relax slightly the energy conservation
relation in the processes responsible for the emission. From a technical point
of view, one can replace the $\delta$ function, that enforces the energy
conservation condition on the right-hand side of Eq. (19), with a Gaussian
profile, i.e.,

$\displaystyle\delta\left(E_{n,p_{z}}-\lambda
E_{n^{\prime},p_{z}-k_{z}}+\eta\Omega\right)$ $\displaystyle\to$
$\displaystyle\frac{1}{\sqrt{2\pi}\Gamma}\int_{-\infty}^{\infty}d\varepsilon
e^{-\frac{\varepsilon^{2}}{2\Gamma^{2}}}\delta\left(E_{n,p_{z}}-\lambda
E_{n^{\prime},p_{z}-k_{z}}+\eta\Omega+\varepsilon\right)$ (32)
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}\Gamma}\int_{-\infty}^{\infty}d\Omega^{\prime}e^{-\frac{(\Omega^{\prime}-\Omega)^{2}}{2\Gamma^{2}}}\delta\left(E_{n,p_{z}}-\lambda
E_{n^{\prime},p_{z}-k_{z}}+\eta\Omega^{\prime}\right),$

where, in the second line, we changed the integration variable from
$\varepsilon$ to $\Omega^{\prime}=\Omega+\eta\varepsilon$. Such an
introduction of the quasiparticle width effects leads to a simple modification
of the expression for the imaginary part of the polarization tensor:

$\displaystyle\mbox{Im}\left[\Pi^{\mu}_{R,\mu}(\Omega,\mathbf{k})\right]\to\frac{1}{\sqrt{2\pi}\Gamma}\int_{-\infty}^{\infty}d\Omega^{\prime}e^{-\frac{(\Omega^{\prime}-\Omega)^{2}}{2\Gamma^{2}}}\mbox{Im}\left[\Pi^{\mu}_{R,\mu}(\Omega^{\prime},\mathbf{k})\right].$
(33)

Note that the result is a convolution of the result obtained in the clean
limit and a Gaussian distribution. Let us verify now that the modification has
the predicted smearing effect on the inverse-square-root singularities in the
near-threshold production of photons. It suffices to consider an isolated
singularity at $\Omega=\Omega_{\rm thr}$, e.g.,

$\displaystyle\frac{\theta(\Omega-\Omega_{\rm thr})}{\sqrt{\Omega-\Omega_{\rm
thr}}}f(\Omega)$ $\displaystyle\to$
$\displaystyle\frac{1}{\sqrt{2\pi}\Gamma}\int_{-\infty}^{\infty}d\Omega^{\prime}e^{-\frac{(\Omega^{\prime}-\Omega)^{2}}{2\Gamma^{2}}}\frac{\theta(\Omega^{\prime}-\Omega_{\rm
thr})}{\sqrt{\Omega^{\prime}-\Omega_{\rm thr}}}f(\Omega^{\prime})$ (34)
$\displaystyle\approx$
$\displaystyle\sqrt{\frac{\pi|\omega|}{8\Gamma}}e^{-\frac{\omega^{2}}{4}}\left\\{\theta(\omega)\left[I_{-1/4}\left(\frac{\omega^{2}}{4}\right)+I_{1/4}\left(\frac{\omega^{2}}{4}\right)\right]+\frac{\sqrt{2}}{\pi}\theta(-\omega)K_{1/4}\left(\frac{\omega^{2}}{4}\right)\right\\}f(\Omega_{\rm
thr}),$

where $\omega=(\Omega-\Omega_{\rm thr})/\Gamma$ is a dimensionless deviation
from the threshold energy and $f(\Omega)$ is a smooth function. The effect of
smearing is visualized in Fig. 13, where we show the partial contribution of
an isolated threshold singularity over the background value $R_{0}$. Note that
the local peak of the near-threshold rate is shifted from $\Omega_{\rm thr}$
to $\Omega^{*}_{\rm thr}\approx\Omega_{\rm thr}+0.765\Gamma$. The maximum
value of the function is approximately $1.021f(\Omega_{\rm
thr})/\sqrt{\Gamma}$.

Figure 13: The effect of quasiparticle width $\Gamma$ on the near-threshold
behavior of the rate as a function of the photon energy. A singular dependence
$R-R_{0}\propto\theta(\omega)/\sqrt{\omega}$, where
$\omega=(\Omega-\Omega_{\rm thr})/\Gamma$ and $R_{0}$ is the background rate,
is replaced by a smooth function defined in Eq. (34).

As we see, the quasiparticle width $\Gamma$ indeed smoothes out the threshold
singularities in the photon rate. In the weakly interacting regime, when
$\Gamma$ is small, the corresponding modifications occur only in small windows
of energies near the thresholds, i.e., $|\Omega-\Omega_{\rm
thr}|\lesssim\Gamma$. Thus, the overall features of the photon rate obtained
above in the clean limit of a strongly magnetized plasma will remain
approximately the same even after the interaction effects are accounted for.
Moreover, one can also verify that the integrated rates will not change much
either. Indeed, by integrating both singular and smoothed functions in Eq.
(34), one can verify that the relative error is less than $0.84\%$ when the
integration range is as small as $|\Omega-\Omega_{\rm thr}|\leq 4\Gamma$. With
expanding the integration range further, the error rapidly decreases too.

##  IV Magneto-optical conductivity of relativistic plasma

In this section, we demonstrate another use of the general result for the
imaginary part of the photon polarization function obtained in Sec. II. In
particular, below we calculate the absorptive part of the magneto-optical
conductivity of a hot relativistic plasma. The corresponding frequency-
dependent transport coefficient is a valuable characteristics in a plasma
under time dependent perturbations. The general theoretical predictions could
be also applied to pseudorelativistic electron plasmas of Dirac and Weyl
semimetals Ashby:2013aa ; Long:2018aa ; PhysRevB.102 , where it can be tested
experimentally PRL115.176404 .

In the linear response theory, the magneto-optical conductivity can be
expressed in terms of the imaginary part of the photon polarization tensor as
follows:

$\sigma^{ij}(\Omega)=\frac{\mbox{Im}\left[\Pi_{R}^{ij}(\Omega;\bm{0})\right]}{\Omega}.$
(35)

Note that only the diagonal components of the conductivity tensor are nonzero
in the case of the vanishing chemical potential, which is studied in this
paper. Indeed, the off-diagonal (Hall) conductivity $\sigma_{H}(\Omega)$ is
trivial because of the charge conjugation symmetry.

Compared to the general result for the polarization tensor obtained in Sec.
II, the conductivity tensor has a much simpler structure. This is the result
of taking the limit $\mathbf{k}\to\bm{0}$. Indeed, by making use of the
expression in Eq. (21), supplemented by the definitions of individual
component functions in Appendix D, we find that the magneto-optical
conductivity takes the following form:

$\sigma^{ij}(\Omega)=\delta^{i}_{3}\delta^{j}_{3}\sigma_{\parallel}(\Omega)+\left(\delta^{i}_{1}\delta^{j}_{1}+\delta^{i}_{2}\delta^{j}_{2}\right)\sigma_{\perp}(\Omega)+\varepsilon^{0ij3}\sigma_{H}(\Omega),$
(36)

where the longitudinal and transverse Hall conductivities are given by

$\displaystyle\sigma_{\parallel}(\Omega)$ $\displaystyle=$
$\displaystyle\frac{\mbox{Im}\left[\Pi_{1}(\Omega;\bm{0})\right]}{\Omega}$
(37) $\displaystyle=$ $\displaystyle-\frac{\alpha
N_{f}}{\Omega\ell^{2}}\sum_{n,n^{\prime}=0}^{\infty}\sum_{\lambda,\eta=\pm
1}\Theta_{\lambda,\eta}^{n,n^{\prime}}(\Omega,0)\frac{n_{F}(E_{n,p_{z}^{(+)}})-n_{F}(\lambda
E_{n^{\prime},p_{z}^{(+)}})}{\eta\lambda\sqrt{\Omega^{2}\left(\Omega^{2}-k_{+}^{2}\right)}}k_{+}^{2}\left(\delta_{n,n^{\prime}}+\delta_{n-1,n^{\prime}-1}\right),$
$\displaystyle\sigma_{\perp}(\Omega)$ $\displaystyle=$
$\displaystyle\frac{\mbox{Im}\left[\Pi_{4}(\Omega;\bm{0})\right]}{\Omega}$
(38) $\displaystyle=$ $\displaystyle-\frac{\alpha
N_{f}}{\Omega\ell^{4}}\sum_{n,n^{\prime}=0}^{\infty}\sum_{\lambda,\eta=\pm
1}\Theta_{\lambda,\eta}^{n,n^{\prime}}(\Omega,0)\frac{n_{F}(E_{n,p_{z}^{(+)}})-n_{F}(\lambda
E_{n^{\prime},p_{z}^{(+)}})}{\eta\lambda\sqrt{\left(\Omega^{2}-k_{-}^{2}\right)\left(\Omega^{2}-k_{+}^{2}\right)}}\left[\Omega^{2}\ell^{2}-2(n+n^{\prime})\right]\left(\delta_{n,n^{\prime}-1}+\delta_{n-1,n^{\prime}}\right).$

Here we used explicit expressions for the component functions
$\mbox{Im}\left[\Pi_{i}(\Omega;\bm{0})\right]$, which are obtained from the
definitions in Appendix D by taking the limit $\mathbf{k}\to\bm{0}$. In the
derivation, we also took into account that the solutions of the energy
conservation equation at $k_{z}=0$ are given by

$\displaystyle p_{z}^{(\pm)}$ $\displaystyle=$
$\displaystyle\pm\frac{1}{2\Omega}\sqrt{\left(\Omega^{2}-k_{-}^{2}\right)\left(\Omega^{2}-k_{+}^{2}\right)},$
(39)

and the corresponding quasiparticle energies are

$\displaystyle E_{n,p_{z}^{(\pm)}}$ $\displaystyle=$
$\displaystyle-\frac{\eta}{2\Omega}\left[\Omega^{2}+2(n-n^{\prime})|qB|\right],$
(40) $\displaystyle E_{n^{\prime},p_{z}^{(\pm)}}$ $\displaystyle=$
$\displaystyle\frac{\lambda\eta}{2\Omega}\left[\Omega^{2}-2(n-n^{\prime})|qB|\right],$
(41)

which are the same for $p_{z}^{(-)}$ and $p_{z}^{(+)}$.

The expressions for the longitudinal and transverse conductivities, defined by
Eqs. (37) and (38), can be further simplified, i.e.,

$\displaystyle\sigma_{\parallel}(\Omega)$ $\displaystyle=$
$\displaystyle\frac{\alpha
N_{f}}{\ell^{2}}\sum_{n=0}^{\infty}\left(2-\delta_{n,0}\right)\frac{4M_{n}^{2}\theta\left(\Omega^{2}-4M_{n}^{2}\right)}{\Omega^{2}\sqrt{\Omega^{2}-4M_{n}^{2}}}\tanh\left(\frac{|\Omega|}{4T}\right),$
(42) $\displaystyle\sigma_{\perp}(\Omega)$ $\displaystyle=$
$\displaystyle\frac{2\alpha
N_{f}\sinh\left(\frac{\Omega}{2T}\right)}{\Omega\ell^{4}\left[\cosh\left(\frac{\Omega}{2T}\right)+\cosh\left(\frac{|qB|}{T\Omega}\right)\right]}\sum_{n=1}^{\infty}\frac{\left[2(2n-1)-\Omega^{2}\ell^{2}\right]\theta\left[(M_{n}-M_{n-1})^{2}-\Omega^{2}\right]}{\sqrt{\left[(M_{n}-M_{n-1})^{2}-\Omega^{2}\right]\left[(M_{n}+M_{n-1})^{2}-\Omega^{2}\right]}}$
(43) $\displaystyle+$ $\displaystyle\frac{2\alpha
N_{f}\sinh\left(\frac{|qB|}{T\Omega}\right)}{\Omega\ell^{4}\left[\cosh\left(\frac{\Omega}{2T}\right)+\cosh\left(\frac{|qB|}{T\Omega}\right)\right]}\sum_{n=1}^{\infty}\frac{\left[\Omega^{2}\ell^{2}-2(2n-1)\right]\theta\left[\Omega^{2}-(M_{n}+M_{n-1})^{2}\right]}{\sqrt{\left[\Omega^{2}-(M_{n}-M_{n-1})^{2}\right]\left[\Omega^{2}-(M_{n}+M_{n-1})^{2}\right]}},$

where we used the shorthand notation $M_{n}=E_{n,0}=\sqrt{2n|qB|+m^{2}}$.

Note that the longitudinal conductivity $\sigma_{\parallel}(\Omega)$ in Eq.
(42) comes exclusively from the annihilation (or “interband”) transitions.
Thus, while the conductivity is nonzero for $\Omega\geq 2m$, it vanishes below
the lowest frequency annihilation threshold, i.e., $\Omega<2m$. (Needless to
say that the threshold goes away in the limit of massless fermions.)
Numerically, the longitudinal conductivity as a function of frequency is
relatively easy to calculate even in the limit of a weak magnetic field. This
is because the sum in Eq. (42) terminates at a finite Landau-level index
$n_{\rm max}=\left[(\Omega^{2}-4m^{2})/(4|qB|)\right]$. The corresponding
numerical results for the two choices of the magnetic field and three
different values of temperature are shown in Fig. 14. By comparing the energy
dependence in the two panels, we see that the longitudinal conductivities for
the weaker and stronger magnetic fields differ primarily by the separation
scale between the neighboring peaks.

It is interesting to note that the longitudinal part of conductivity, shown in
Fig. 14, tends to decrease with increasing the temperature of plasma. Such
decrease should not be too surprising however. Mathematically, it comes from
$\tanh[\Omega/(4T)]$ in Eq. (42). From a physics viewpoint, it is explained by
the gradual depletion of the occupied low-energy states when the temperature
grows.

Figure 14: Longitudinal magneto-optical conductivity as a function of
frequency $\Omega$ for $|qB|=25m^{2}$ (panel a) and $|qB|=225m^{2}$ (panel b).

Unlike the longitudinal conductivity, the transverse conductivity
$\sigma_{\perp}(\Omega)$ contains two different types of contributions,
represented by the two sums in Eq. (43). The first sum represents the
contributions due to particle-particle and antiparticle-antiparticle (or
“intraband”) transitions. As one can verify, they are nonzero only in the
region of sufficiently small frequencies, i.e., $\Omega<\sqrt{2|qB|+m^{2}}-m$.
Note that the quoted upper limit is determined by the largest vertical
separation between neighboring Landau levels, i.e., the energy difference
between the $0$th and $1$th levels at $p_{z}=0$. The second sum in Eq. (43)
captures the contributions due to annihilation (“interband”) transitions. This
one is nonvanishing only at sufficiently large frequencies, i.e.,
$\Omega>\sqrt{2|qB|+m^{2}}+m$. As one can see, the frequency windows for the
two types of processes do not overlap and are separated by a gap of width
$2m$. Interestingly, for both types of contributions, the sums over Landau
levels terminate at $n_{\rm
max}=\left[((2|qB|+\Omega)^{2}-4m^{2}\Omega^{2})/(8|qB|\Omega^{2})\right]$.

The numerical results for the transverse conductivity $\sigma_{\perp}(\Omega)$
are presented in Fig. 15. Panels (a) and (b) display the overall dependence of
the conductivity on frequency for the two fixed values of the magnetic field,
$|qB|=(5m)^{2}$ and $|qB|=(15m)^{2}$, respectively. As predicted, only the
intraband transitions contribute to $\sigma_{\perp}(\Omega)$ in the low-
frequency region ($\Omega<\sqrt{2|qB|+m^{2}}-m$) and only the interband
transitions contribute in the high-frequency region
($\Omega>\sqrt{2|qB|+m^{2}}+m$). In the intermediate window of frequencies,
i.e., $\sqrt{2|qB|+m^{2}}-m<\Omega<\sqrt{2|qB|+m^{2}}+m$, the transverse
conductivity vanishes identically.

Figure 15: Transverse magneto-optical conductivity as a function of frequency
$\Omega$ for $|qB|=25m^{2}$ (panel a) and $|qB|=225m^{2}$ (panel b). Panels
(c) and (d) show the same results at small frequencies, where only the
“intraband” transitions contribute.

While it is easy to resolve and identify the annihilation peaks in the high-
frequency regime in panels (a) and (b) of Fig. 15, the low-frequency region
appears much more convoluted due to many nearly overlapping thresholds for
particle-particle (antiparticle-antiparticle) transitions. Thus, to resolve
the details of the transverse conductivity better, we show the low-frequency
results separately in panels (c) and (d) for $|qB|=(5m)^{2}$ and
$|qB|=(15m)^{2}$, respectively.

The temperature dependence of the transverse conductivity in the high-
frequency region can be understood qualitatively from the results in panels
(a) and (b) of Fig. 15. Just above the annihilation threshold, the
conductivities have an inverse ordering with the largest (smallest) value of
conductivity obtained for the lowest (highest) temperature. At sufficiently
high frequencies, however, the ordering gradually becomes normal, i.e., the
largest (smallest) value of conductivity is obtained for the (lowest)
temperature. The transition from the inverse-ordering
($\sigma_{\perp}(\Omega)$ increasing with $T$) to the normal-ordering regime
($\sigma_{\perp}(\Omega)$ decreasing with $T$) depends both on the magnetic
field and the ratio $T/\sqrt{|qB|}$.

As one can see from panels (c) and (d) of Fig. 15, the transverse conductivity
mostly grows with temperature in the region of low frequencies. This is
expected, of course, since the main part of the result comes from many
transitions between neighboring Landau levels with sufficiently close energies
and, thus, large indices $n$ (running up to about $n\sim|qB|/\Omega^{2}$).
When the temperature increases, the occupation numbers of the relevant high-
energy states grow and, as a result, $\sigma_{\perp}(\Omega)$ becomes larger.
As we see from Figs. 15(c) and 15(d), this trend gradually changes as the
frequency increases and approaches the largest particle-particle threshold
$\sqrt{2|qB|+m^{2}}-m$ from below. Near the highest particle-particle
threshold, the ordering of results for the three values of temperature
($T=5m$, $T=15m$, and $T=50m$) is opposite. The width of the corresponding
near-threshold region is not universal however. It changes depending on the
ratio of energy scales set by the magnetic field and temperature. As one can
see from the results shown in panels (c) and (d) of Fig. 15, the near-
threshold region with inverse ordering of conductivities for the three fixed
temperatures widens when $T/\sqrt{|qB|}$ grows.

Similarly to the photon emission discussed in Sec. III, both longitudinal and
transverse magneto-optical conductivities have numerous singular peaks,
associated with the threshold effects. From the analytical expressions in Eqs.
(42) and (43), we see that the singularities are also of the same inverse-
square-root type. As argued in Sec. III.7, the peaks are expected to smooth
out when one takes quasiparticle interaction effects into account.

Our results for the magneto-optical conductivity agree with those obtained in
the quantum limit for Dirac semimetals in Refs. Ashby:2013aa ; Long:2018aa ;
PhysRevB.102 . Indirectly, such an agreement provides a valuable cross-check
for the derivation of the photon polarization tensor in Sec. II. Unlike the
conductivity, though, the polarization tensor extends to the case of nonzero
photon momenta and has a considerably more complicated structure.

##  V Summary and Conclusions

In this study, we utilized a coordinate representation for the fermion Green’s
function in a constant magnetic field to derive a closed-form analytical
expression for the absorptive part of the photon polarization tensor in a
relativistic plasma. We found that the polarization tensor contains 4
different symmetric structures and 2 antisymmetric ones. Such an abundance of
tensor structures is the result of the Lorentz symmetry breaking by the
magnetic field and the thermal bath. As required by the gauge symmetry, all 6
tensor structures are transverse. We found that the antisymmetric parts of the
tensor vanish at zero chemical potential ($\mu=0$), which was the main focus
of this study. This is the consequence of the charge conjugation symmetry. It
is expected, however, that both antisymmetric structures will be nonzero when
$\mu\neq 0$ Wang:2021eud .

The derivation in this study was focused on the absorptive part of the photon
polarization. The latter includes the imaginary parts of symmetric tensor
structures and, in principle, the real parts of antisymmetric ones. As we
argued before Wang:2020dsr , the absorptive part of the tensor is determined
by the following three types of processes: (i) $e^{-}\rightarrow
e^{-}+\gamma$, (ii) $e^{+}\rightarrow e^{+}+\gamma$, and (iii)
$e^{-}+e^{+}\rightarrow\gamma$. In a quantum description, they correspond to
particle-particle, antiparticle-antiparticle, and particle-antiparticle
transitions between Landau levels, respectively. At the vanishing chemical
potential, both particle-particle and antiparticle-antiparticle transitions
contribute equally to the polarization tensor.

By utilizing the photon polarization tensor, we studied in detail the
differential photon emission rate for a strongly magnetized thermal
relativistic plasma. In this context, the particle-particle and antiparticle-
antiparticle transitions can be viewed as splitting ($1\to 2$) processes,
while the particle-antiparticle ones as annihilation ($2\to 1$) processes. All
of them are affected by the Landau-level quantization. In particular, the
photon rate as a function of the photon energy (for a fixed direction of
emission) or as a function of the angular coordinate (for fixed photon energy)
contains numerous inverse-square-root singularities. They are associated with
the threshold effects in photon production. Since the inverse-square-root
singularities are integrable, they cause no complications in studying rates
integrated over the angular coordinates or the energy. Conceptually, the
actual singularities are an artifact of the clean-limit approximation that
ignores all particle interactions except for the interaction with the
background magnetic field. As we showed, taking interactions into account will
smooth out the threshold singularities. Quantitative effects of interactions
can be captured by the quasiparticle width $\Gamma$. It the value of $\Gamma$
that also determines the width of an energy window near the threshold, where
the singular behavior will be smeared out.

In this study, we analyzed the angular dependence of the differential photon
emission rate in a wide range of model parameters. In all regimes, ranging
from a moderately relativistic to an ultrarelativistic plasma, the emission is
highly anisotropic. In addition to its spiky dependence on the angular
coordinate coming from the threshold effects, the emission has an overall
ellipsoidal profile that changes gradually with the photon energy. In general,
the profile is prolate at small energies ($\Omega\lesssim\sqrt{|qB|}$) and
oblate at large energies ($\Omega\gtrsim\sqrt{|qB|}$). In other words, the
dominant emission tends to be along the line of the magnetic field at small
energies and perpendicular to the magnetic field at large energies. The
prolate emission profile at small energies can be explained largely by the
Landau-level quantization that strongly restricts the kinematics of photon
emission. On the other hand, we speculate that the oblate profile at large
energies is a natural consequence of the synchrotron radiation in the
semiclassical limit Sokolov:1986nk , which goes predominantly in the
directions perpendicular to the magnetic field. Of course, the synchrotron
radiation comes only from the splitting ($1\to 2$) processes. However, there
are also contributions from the annihilation ($2\to 1$) processes. While the
latter are subdominant at sufficiently high temperatures, they become
important at low temperatures and high photon energies.

The combination of our results for several representative choices of the
magnetic field and temperature provides a rather detailed picture of the
photon emission. As one might expect, the photon rate tends to grow with
increasing the temperature of the plasma. Concerning the energy dependence,
however, we found that the rate has a nonmonotonic behavior. For small photon
energies, $\Omega\ll\sqrt{|qB|}$, it is strongly suppressed by the Landau-
level quantization. With increasing the energy, the quantization effects
gradually relax and the rate grows. After reaching a maximum value at an
intermediate value, however, the rate starts to decrease. The decreasing trend
also persists at very large energies, where the suppression comes from the
Fermi-Dirac distribution of charged particles with high energies.

The other application of the absorptive part of the photon polarization
tensor, presented in this paper, was the calculation of magneto-optical
conductivity in a quantum limit of relativistic plasma. By making use of the
explicit tensor structure, we calculated both longitudinal and transverse
parts of the conductivity. As in the case of photon emission, the Landau-level
quantization affects the main features of the frequency dependence of optical
conductivity. One of such characteristic features is the appearance of
singular spikes associated with the Landau-level threshold effects. Of course,
the singularities come from using the clean-limit approximation and should go
away when particle interactions are taking into account.

In the case of the longitudinal component of conductivity
$\sigma_{\parallel}(\Omega)$, we showed that only the annihilation
(“interband”) processes contribute. Accordingly, the result for
$\sigma_{\parallel}(\Omega)$ is nonvanishing only when $\Omega>2m$. The
transverse component of conductivity $\sigma_{\perp}(\Omega)$, on the other
hand, comes from the splitting (“intraband”) processes when
$\Omega<\sqrt{2|qB|+m^{2}}-m$ and the annihilation (“interband”) ones when
$\Omega>\sqrt{2|qB|+m^{2}}+m$. The two windows of frequencies are separated by
a gap of width $2m$, where the transverse conductivity vanishes.

While we carefully studied the imaginary part of the photon polarization at
$\mu=0$ in this paper, the case of nonzero $\mu$ was not addressed. It appears
that the corresponding problem poses only a technical complication that can be
overcome with a moderate effort. Such study is underway now Wang:2021eud . A
more difficult extension of the present work will be obtaining the real part
of the photon polarization. We are currently exploring several promising
approaches to address the problem.

###### Acknowledgements.

The work of X.W. was supported by the start-up funding No. 4111190010 of
Jiangsu University. The work of I.A.S. was supported by the U.S. National
Science Foundation under Grant No. PHY-1713950.

##  Appendix A Matsubara sums

In this Appendix, we present several general results for the fermionic
Matsubara sums needed in the calculation of the photon polarization function
in the main text.

Let us start by quoting the following standard sums

$\displaystyle
T\sum_{k=-\infty}^{\infty}\frac{1}{\left[\omega_{k}^{2}+a^{2}\right]\left[(\omega_{k}-\Omega_{m})^{2}+b^{2}\right]}$
$\displaystyle=$
$\displaystyle\frac{(a+b)\left[1-n_{F}(a)-n_{F}(b)\right]}{2ab\left[(a+b)^{2}+\Omega_{m}^{2}\right]}+\frac{(a-b)\left[n_{F}(a)-n_{F}(b)\right]}{2ab\left[(a-b)^{2}+\Omega_{m}^{2}\right]},$
(44) $\displaystyle
T\sum_{k=-\infty}^{\infty}\frac{\omega_{k}(\omega_{k}-\Omega_{m})}{\left[\omega_{k}^{2}+a^{2}\right]\left[(\omega_{k}-\Omega_{m})^{2}+b^{2}\right]}$
$\displaystyle=$
$\displaystyle\frac{(a+b)\left[1-n_{F}(a)-n_{F}(b)\right]}{2\left[(a+b)^{2}+\Omega_{m}^{2}\right]}-\frac{(a-b)\left[n_{F}(a)-n_{F}(b)\right]}{2\left[(a-b)^{2}+\Omega_{m}^{2}\right]},$
(45) $\displaystyle
T\sum_{k=-\infty}^{\infty}\frac{i\omega_{k}}{\left[\omega_{k}^{2}+a^{2}\right]\left[(\omega_{k}-\Omega_{m})^{2}+b^{2}\right]}$
$\displaystyle=$
$\displaystyle\frac{i\Omega_{m}\left[1-n_{F}(a)-n_{F}(b)\right]}{2b\left[(a+b)^{2}+\Omega_{m}^{2}\right]}+\frac{i\Omega_{m}\left[n_{F}(a)-n_{F}(b)\right]}{2b\left[(a-b)^{2}+\Omega_{m}^{2}\right]},$
(46) $\displaystyle
T\sum_{k=-\infty}^{\infty}\frac{i(\omega_{k}-\Omega_{m})}{\left[\omega_{k}^{2}+a^{2}\right]\left[(\omega_{k}-\Omega_{m})^{2}+b^{2}\right]}$
$\displaystyle=$
$\displaystyle-\frac{i\Omega_{m}\left[1-n_{F}(a)-n_{F}(b)\right]}{2a\left[(a+b)^{2}+\Omega_{m}^{2}\right]}+\frac{i\Omega_{m}\left[n_{F}(a)-n_{F}(b)\right]}{2a\left[(a-b)^{2}+\Omega_{m}^{2}\right]}.$
(47)

where $n_{F}(\epsilon)=1/\left[\exp(\epsilon/T)+1\right]$ is the Fermi-Dirac
distribution function, and $\omega_{k}=(2k+1)\pi T$ and $\Omega_{m}=2m\pi T$
are the fermionic and bosonic Matsubara frequencies, respectively. Note that
distribution function satisfies the following property:
$n_{F}(-x)=1-n_{F}(x)$.

By making use of Eqs. (44) through (47), it is straightforward to derive the
following Matsubara sum of a more general type:

$\displaystyle
T\sum_{k=-\infty}^{\infty}\frac{i\omega_{k}(i\omega_{k}-i\Omega_{m})X+i\omega_{k}Y_{1}+(i\omega_{k}-i\Omega_{m})Y_{2}+Z}{\left[(i\omega_{k})^{2}-a^{2}\right]\left[(i\omega_{k}-i\Omega_{m})^{2}-b^{2}\right]}$
$\displaystyle=\sum_{\lambda=\pm 1}\frac{(a-\lambda
b)\left[n_{F}(a)-n_{F}(\lambda b)\right]}{2\lambda ab\left[(a-\lambda
b)^{2}+\Omega_{m}^{2}\right]}\left[\lambda abX+i\Omega_{m}\frac{aY_{1}+\lambda
bY_{2}}{a-\lambda b}+Z\right]$ $\displaystyle=\sum_{\eta,\lambda=\pm
1}\frac{n_{F}(a)-n_{F}(\lambda b)}{4\lambda ab\left(a-\lambda b+\eta
i\Omega_{m}\right)}\left[\lambda abX-\eta\left(aY_{1}+\lambda
bY_{2}\right)+Z\right],$ (48)

where $X$, $Y_{1}$, $Y_{2}$, and $Z$ are arbitrary coefficient functions
independent of the fermion Matsubara frequency $\omega_{k}$.

In the calculation of the photon polarization function in the main text,
parameters $a$ and $b$ should be replaced with the Landau-level energies
$E_{n,p_{z}}=\sqrt{p_{z}^{2}+m^{2}+2n|qB|}$ and
$E_{n^{\prime},p_{z}-k_{z}}=\sqrt{(p_{z}-k_{z})^{2}+m^{2}+2n^{\prime}|qB|}$,
respectively.

##  Appendix B Integrals over transverse spatial coordinates

In this Appendix, we present the explicit results for all types of integrals
over the transverse spatial coordinates, $\mathbf{r}_{\perp}$, that appear in
the calculation of the photon polarization function.

Instead of the Cartesian coordinates $x$ and $y$, it is convenient to use the
polar coordinates $r_{\perp}\equiv|\mathbf{r}_{\perp}|$ and $\varphi_{\perp}$
in the $\mathbf{r}_{\perp}$ plane. Then, the results of the angular
integrations are given in terms of the Bessel functions. In particular, one
derives the following results for the integrals that appear in the
calculation:

$\displaystyle\int
d^{2}\mathbf{r}_{\perp}e^{-i\mathbf{r}_{\perp}\cdot\mathbf{k}_{\perp}}f(r_{\perp})$
$\displaystyle=$ $\displaystyle
2\pi\int_{0}^{\infty}r_{\perp}dr_{\perp}J_{0}\left(r_{\perp}k_{\perp}\right)f(r_{\perp}),$
(49) $\displaystyle\int
d^{2}\mathbf{r}_{\perp}e^{-i\mathbf{r}_{\perp}\cdot\mathbf{k}_{\perp}}r_{\perp}^{\mu}f(r_{\perp})$
$\displaystyle=$ $\displaystyle-2\pi
i\hat{k}_{\perp}^{\mu}\int_{0}^{\infty}r_{\perp}^{2}dr_{\perp}J_{1}\left(r_{\perp}k_{\perp}\right)f(r_{\perp})$
(50) $\displaystyle=$ $\displaystyle-\pi
ik_{\perp}^{\mu}\int_{0}^{\infty}r_{\perp}^{3}dr_{\perp}\left[J_{0}\left(r_{\perp}k_{\perp}\right)+J_{2}\left(r_{\perp}k_{\perp}\right)\right]f(r_{\perp}),$
$\displaystyle\int
d^{2}\mathbf{r}_{\perp}e^{-i\mathbf{r}_{\perp}\cdot\mathbf{k}_{\perp}}\mathbf{r}_{\perp}^{2}f(r_{\perp})$
$\displaystyle=$ $\displaystyle
2\pi\int_{0}^{\infty}r_{\perp}^{3}dr_{\perp}J_{0}\left(r_{\perp}k_{\perp}\right)f(r_{\perp}),$
(51) $\displaystyle\int
d^{2}\mathbf{r}_{\perp}e^{-i\mathbf{r}_{\perp}\cdot\mathbf{k}_{\perp}}r_{\perp}^{\mu}r_{\perp}^{\nu}f(r_{\perp})$
$\displaystyle=$ $\displaystyle
2\pi\int_{0}^{\infty}r_{\perp}^{3}dr_{\perp}\left[\left(-\hat{k}_{\perp}^{\mu}\hat{k}_{\perp}^{\nu}-\frac{1}{2}g_{\perp}^{\mu\nu}\right)J_{2}\left(r_{\perp}k_{\perp}\right)-\frac{1}{2}g_{\perp}^{\mu\nu}J_{0}\left(r_{\perp}k_{\perp}\right)\right]f(r_{\perp}),$
(52)

where we used the following relation for the Bessel functions:

$J_{1}\left(r_{\perp}k_{\perp}\right)=\frac{1}{2}r_{\perp}k_{\perp}\left[J_{0}\left(r_{\perp}k_{\perp}\right)+J_{2}\left(r_{\perp}k_{\perp}\right)\right].$
(53)

The remaining integrals over $r_{\perp}$ can be expressed in terms of the
following dimensionless functions:

$\displaystyle\mathcal{I}_{0}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\frac{1}{\ell^{2}}\int_{0}^{\infty}r_{\perp}dr_{\perp}e^{-\mathbf{r}_{\perp}^{2}/(2\ell^{2})}J_{0}\left(r_{\perp}k_{\perp}\right)L_{n}\left(\frac{\mathbf{r}_{\perp}^{2}}{2\ell^{2}}\right)L_{n^{\prime}}\left(\frac{\mathbf{r}_{\perp}^{2}}{2\ell^{2}}\right),$
(54) $\displaystyle\mathcal{I}_{1}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\frac{1}{\ell^{3}}\int_{0}^{\infty}r_{\perp}^{2}dr_{\perp}e^{-\mathbf{r}_{\perp}^{2}/(2\ell^{2})}J_{1}\left(r_{\perp}k_{\perp}\right)L_{n}\left(\frac{\mathbf{r}_{\perp}^{2}}{2\ell^{2}}\right)L_{n^{\prime}}^{1}\left(\frac{\mathbf{r}_{\perp}^{2}}{2\ell^{2}}\right),$
(55) $\displaystyle\mathcal{I}_{2}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\frac{1}{\ell^{4}}\int_{0}^{\infty}r_{\perp}^{3}dr_{\perp}e^{-\mathbf{r}_{\perp}^{2}/(2\ell^{2})}J_{0}\left(r_{\perp}k_{\perp}\right)L_{n}^{1}\left(\frac{\mathbf{r}_{\perp}^{2}}{2\ell^{2}}\right)L_{n^{\prime}}^{1}\left(\frac{\mathbf{r}_{\perp}^{2}}{2\ell^{2}}\right),$
(56) $\displaystyle\mathcal{I}_{3}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\frac{1}{\ell^{4}}\int_{0}^{\infty}r_{\perp}^{3}dr_{\perp}e^{-\mathbf{r}_{\perp}^{2}/(2\ell^{2})}J_{2}\left(r_{\perp}k_{\perp}\right)L_{n}^{1}\left(\frac{\mathbf{r}_{\perp}^{2}}{2\ell^{2}}\right)L_{n^{\prime}}^{1}\left(\frac{\mathbf{r}_{\perp}^{2}}{2\ell^{2}}\right),$
(57)

where $\xi=k_{\perp}^{2}\ell^{2}/2$.

It should be noted that $\mathcal{I}_{0}^{n,n^{\prime}}(\xi)$,
$\mathcal{I}_{2}^{n,n^{\prime}}(\xi)$, and
$\mathcal{I}_{3}^{n,n^{\prime}}(\xi)$ are symmetric under the interchange of
indices $n$ and $n^{\prime}$. In contrast,
$\mathcal{I}_{1}^{n,n^{\prime}}(\xi)$ is not symmetric, but it can be
rewritten in terms of the other functions. Indeed, by making use of Eq. (53)
and the well-known identity for the Laguerre polynomials
$L_{n}^{\alpha}(x)=L_{n}^{\alpha+1}(x)-L_{n-1}^{\alpha+1}(x)$, one derives

$\mathcal{I}_{1}^{n,n^{\prime}}(\xi)=\sqrt{\frac{\xi}{2}}\left[\mathcal{I}_{2}^{n,n^{\prime}}(\xi)-\mathcal{I}_{2}^{n-1,n^{\prime}}(\xi)+\mathcal{I}_{3}^{n,n^{\prime}}(\xi)-\mathcal{I}_{3}^{n-1,n^{\prime}}(\xi)\right].$
(58)

One can also derive the following relations:

$\displaystyle\mathcal{I}_{1}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\frac{n+n^{\prime}+2}{2\sqrt{2\xi}}\left[\mathcal{I}_{0}^{n,n^{\prime}}(\xi)-\mathcal{I}_{0}^{n+1,n^{\prime}+1}(\xi)\right]+\frac{\sqrt{\xi}}{2\sqrt{2}}\left[\mathcal{I}_{0}^{n+1,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n,n^{\prime}+1}(\xi)\right]-\frac{n-n^{\prime}}{\sqrt{2\xi}}\mathcal{I}_{0}^{n,n^{\prime}}(\xi),$
(59) $\displaystyle\mathcal{I}_{2}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\frac{n+n^{\prime}+2}{2}\left[\mathcal{I}_{0}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n+1,n^{\prime}+1}(\xi)\right]-\frac{\xi}{2}\left[\mathcal{I}_{0}^{n+1,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n,n^{\prime}+1}(\xi)\right],$
(60) $\displaystyle\mathcal{I}_{3}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\frac{n+n^{\prime}+2}{2}\left[\mathcal{I}_{0}^{n+1,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n,n^{\prime}+1}(\xi)\right]-\frac{(n-n^{\prime})^{2}}{2\xi}\left[\mathcal{I}_{0}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n+1,n^{\prime}+1}(\xi)\right].$
(61)

Other random relations:

$\displaystyle\mathcal{I}_{1}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{\xi}}(n^{\prime}+1)\mathcal{I}_{0}^{n,n^{\prime}}(\xi)-\frac{1}{\sqrt{2\xi}}\mathcal{I}_{2}^{n,n^{\prime}}(\xi)$
(62) $\displaystyle=$
$\displaystyle\frac{n+n^{\prime}+2}{\sqrt{2\xi}}\mathcal{I}_{0}^{n,n^{\prime}}(\xi)-\frac{1}{\sqrt{2\xi}}\mathcal{I}_{2}^{n,n^{\prime}}(\xi)-\frac{n-n^{\prime}}{\sqrt{2\xi}}\mathcal{I}_{0}^{n,n^{\prime}}(\xi),$
$\displaystyle\mathcal{I}_{3}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{\xi}}\left[(n+1)\mathcal{I}_{1}^{n,n^{\prime}}(\xi)+(n^{\prime}+1)\mathcal{I}_{1}^{n^{\prime}+1,n}(\xi)\right],$
(63) $\displaystyle\mathcal{I}_{3}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\mathcal{I}_{2}^{n,n^{\prime}-1}(\xi)-\sqrt{\frac{2}{\xi}}(n-n^{\prime}-\xi)\mathcal{I}_{1}^{n^{\prime},n}(\xi),$
(64) $\displaystyle 2(n+n^{\prime}+1-\xi)\mathcal{I}_{0}^{n,n^{\prime}}(\xi)$
$\displaystyle=$
$\displaystyle\mathcal{I}_{2}^{n,n^{\prime}}(\xi)+\mathcal{I}_{2}^{n-1,n^{\prime}-1}(\xi),$
(65) $\displaystyle\sqrt{2\xi}\mathcal{I}_{0}^{n,n^{\prime}}(\xi)$
$\displaystyle=$
$\displaystyle\mathcal{I}_{1}^{n,n^{\prime}-1}(\xi)+\mathcal{I}_{1}^{n^{\prime},n}(\xi),$
(66) $\displaystyle 2(n-n^{\prime})\,\mathcal{I}_{0}^{n,n^{\prime}}(\xi)$
$\displaystyle=$
$\displaystyle\sqrt{2\xi}\left(\mathcal{I}_{1}^{n^{\prime},n}(\xi)-\mathcal{I}_{1}^{n,n^{\prime}}(\xi)\right),$
(67) $\displaystyle\mathcal{I}_{1}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\mathcal{I}_{1}^{n^{\prime},n}(\xi)+\sqrt{\frac{2}{\xi}}(n^{\prime}-n)\,\mathcal{I}_{0}^{n,n^{\prime}}(\xi),$
(68) $\displaystyle\mathcal{I}_{1}^{n^{\prime},n}(\xi)$ $\displaystyle=$
$\displaystyle\sqrt{2\xi}\mathcal{I}_{0}^{n,n^{\prime}}(\xi)-\mathcal{I}_{1}^{n,n^{\prime}-1}(\xi),$
(69) $\displaystyle\mathcal{I}_{1}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\sqrt{2\xi}\mathcal{I}_{0}^{n,n^{\prime}}(\xi)-\mathcal{I}_{1}^{n^{\prime},n-1}(\xi),$
(70)
$\displaystyle\frac{2(n^{\prime}-n)}{\xi}\mathcal{I}_{0}^{n,n^{\prime}}(\xi)$
$\displaystyle=$
$\displaystyle\mathcal{I}_{2}^{n,n^{\prime}-1}(\xi)-\mathcal{I}_{2}^{n-1,n^{\prime}}(\xi)+\mathcal{I}_{3}^{n,n^{\prime}-1}(\xi)-\mathcal{I}_{3}^{n-1,n^{\prime}}(\xi).$
(71)

All of the above integrals can be calculated exactly in terms of the
generalized Laguerre polynomials by using table integral Gradshtein . The
results read

$\displaystyle\mathcal{I}_{0}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle(-1)^{n+n^{\prime}}e^{-\xi}L_{n}^{n^{\prime}-n}\left(\xi\right)L_{n^{\prime}}^{n-n^{\prime}}\left(\xi\right),$
(72) $\displaystyle\mathcal{I}_{1}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\sqrt{2\xi}(-1)^{n+n^{\prime}}e^{-\xi}L_{n}^{n^{\prime}-n+1}\left(\xi\right)L_{n^{\prime}}^{n-n^{\prime}}\left(\xi\right),$
(73) $\displaystyle\mathcal{I}_{2}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle
2(-1)^{n+n^{\prime}}(n^{\prime}+1)e^{-\xi}L_{n}^{n^{\prime}-n}\left(\xi\right)L_{n^{\prime}+1}^{n-n^{\prime}}\left(\xi\right),$
(74) $\displaystyle\mathcal{I}_{3}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle 2(-1)^{n+n^{\prime}}\xi
e^{-\xi}L_{n}^{n^{\prime}-n+1}\left(\xi\right)L_{n^{\prime}}^{n-n^{\prime}+1}\left(\xi\right).$
(75)

Note that the above functions are expressed in terms of the exponential
function $e^{-\xi}$ and a product of the Laguerre polynomials of degrees $n$
and $n^{\prime}$ (or $n$ and $n^{\prime}+1$). In numerical calculations, it is
the product of two polynomials that produces the largest evaluation errors.
This is particularly detrimental when the values of $n$, $n^{\prime}$, and
$|n-n^{\prime}|$ are all large. As we argue below, the problem can be relaxed
partially by rewriting the corresponding functions equivalently as a product
of the power function $\xi^{|n-n^{\prime}|}$ and the Laguerre polynomials of
smaller degree(s) $\sim\mbox{min}(n,n^{\prime})$. This can be achieved by
utilizing the well-known identity for the Laguerre polynomials

$L_{n}^{\alpha}(\xi)=(-\xi)^{-\alpha}\frac{(n+\alpha)!}{n!}L_{n+\alpha}^{-\alpha}(\xi).$
(76)

By applying this to the first Laguerre polynomial in each product, the
functions in Eqs. (72) – (75) can be rewritten in the following equivalent
form:

$\displaystyle\mathcal{I}_{0}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\frac{(n^{\prime})!}{n!}e^{-\xi}\xi^{n-n^{\prime}}\left(L_{n^{\prime}}^{n-n^{\prime}}\left(\xi\right)\right)^{2},$
(77) $\displaystyle\mathcal{I}_{1}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle-\sqrt{2\xi}\frac{(n^{\prime}+1)!}{n!}e^{-\xi}\xi^{n-n^{\prime}-1}L_{n^{\prime}+1}^{n-n^{\prime}-1}\left(\xi\right)L_{n^{\prime}}^{n-n^{\prime}}\left(\xi\right),$
(78) $\displaystyle\mathcal{I}_{2}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle
2\frac{(n^{\prime}+1)!}{n!}e^{-\xi}\xi^{n-n^{\prime}}L_{n^{\prime}}^{n-n^{\prime}}\left(\xi\right)L_{n^{\prime}+1}^{n-n^{\prime}}\left(\xi\right),$
(79) $\displaystyle\mathcal{I}_{3}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle-2\frac{(n^{\prime}+1)!}{n!}e^{-\xi}\xi^{n-n^{\prime}}L_{n^{\prime}+1}^{n-n^{\prime}-1}\left(\xi\right)L_{n^{\prime}}^{n-n^{\prime}+1}\left(\xi\right).$
(80)

These representations are particularly useful in numerical calculations when
$n>n^{\prime}$. Similarly, by applying Eq. (76) to the second Laguerre
polynomial in each product, we can rewrite functions
$\mathcal{I}_{i}^{n,n^{\prime}}(\xi)$ as follows:

$\displaystyle\mathcal{I}_{0}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\frac{n!}{(n^{\prime})!}e^{-\xi}\xi^{n^{\prime}-n}\left(L_{n}^{n^{\prime}-n}\left(\xi\right)\right)^{2},$
(81) $\displaystyle\mathcal{I}_{1}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle\sqrt{2\xi}\frac{n!}{(n^{\prime})!}e^{-\xi}\xi^{n^{\prime}-n}L_{n}^{n^{\prime}-n+1}\left(\xi\right)L_{n}^{n^{\prime}-n}\left(\xi\right),$
(82) $\displaystyle\mathcal{I}_{2}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle
2\frac{(n+1)!}{(n^{\prime})!}e^{-\xi}\xi^{n^{\prime}-n}L_{n}^{n^{\prime}-n}\left(\xi\right)L_{n+1}^{n^{\prime}-n}\left(\xi\right),$
(83) $\displaystyle\mathcal{I}_{3}^{n,n^{\prime}}(\xi)$ $\displaystyle=$
$\displaystyle-2\frac{(n+1)!}{(n^{\prime})!}e^{-\xi}\xi^{n^{\prime}-n}L_{n}^{n^{\prime}-n+1}\left(\xi\right)L_{n+1}^{n^{\prime}-n-1}\left(\xi\right).$
(84)

These are suitable when $n<n^{\prime}$.

For completeness, let us also discuss the properties of
$\mathcal{I}_{0}^{n,n^{\prime}}$ and $\mathcal{I}_{2}^{n,n^{\prime}}$ in the
limit of small $|\mathbf{k}_{\perp}|\ell$. This limit is of relevance for the
photon emission in the regime of small $\Omega$, as well as for the photon
emission in the direction of the magnetic field ($\theta\approx 0$) at an
arbitrary $\Omega$. By making use a closed form representation for the
generalized Laguerre polynomials Abramowitz

$\displaystyle L_{n}^{\alpha}(\xi)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{n}\frac{(n+\alpha)!(-\xi)^{i}}{i!(n-i)!(\alpha+i)!},$
(85)

which is valid for $\alpha>-1$, and keeping only the terms up to linear order
in $\xi=|\mathbf{k}_{\perp}|^{2}\ell^{2}/2$, one obtains the following
results:

$\displaystyle\mathcal{I}_{0}^{n,n^{\prime}}(\xi)$ $\displaystyle\simeq$
$\displaystyle\delta_{n,n^{\prime}}-\left[(2n+1)\delta_{n,n^{\prime}}-(n+1)\delta_{n,n^{\prime}-1}-(n^{\prime}+1)\delta_{n-1,n^{\prime}}\right]\xi+O\left[\xi^{2}\right],$
(86) $\displaystyle\mathcal{I}_{2}^{n,n^{\prime}}(\xi)$ $\displaystyle\simeq$
$\displaystyle
2(n+1)\delta_{n,n^{\prime}}-2(n+1)(n^{\prime}+1)\left(2\delta_{n,n^{\prime}}-\delta_{n,n^{\prime}-1}-\delta_{n-1,n^{\prime}}\right)\xi+O\left[\xi^{2}\right],$
(87) $\displaystyle\mathcal{I}_{3}^{n,n^{\prime}}(\xi)$ $\displaystyle\simeq$
$\displaystyle(n+1)(n^{\prime}+1)\left(2\delta_{n,n^{\prime}}-\delta_{n,n^{\prime}-1}-\delta_{n-1,n^{\prime}}\right)\xi+O\left[\xi^{2}\right].$
(88)

Note that a similar expansion for $\mathcal{I}_{1}^{n,n^{\prime}}(\xi)$ can be
obtained by using Eq. (58). As we can see, functions
$\mathcal{I}_{0}^{n,n^{\prime}}(\xi)$ and
$\mathcal{I}_{2}^{n,n^{\prime}}(\xi)$ are diagonal in the Landau-level indices
(i.e., $\sim\delta_{n,n^{\prime}}$) at the leading (zeroth) order in
$|\mathbf{k}_{\perp}|\ell$. On the other hand,
$\mathcal{I}_{3}^{n,n^{\prime}}(\xi)$ vanishes when
$|\mathbf{k}_{\perp}|\ell\to 0$.

##  Appendix C Explicit expressions for $T_{i}^{\mu\nu}$ and $I_{i}^{\mu\nu}$

In the expression for the photon polarization function, there are several
types of Dirac traces. For completeness, the corresponding results are
presented here. For brevity of notation, we will use the following shorthand
notations: $L_{n}^{m}\equiv L_{n}^{m}\left(\xi\right)$, $T_{i}^{\mu\nu}\equiv
T_{i}^{\mu\nu}(\mathbf{k}_{\perp})$, and $I_{i}^{\mu\nu}\equiv
I_{i}^{\mu\nu}(\mathbf{k}_{\perp})$, where $i=1,2,3,4$.

The four types of traces needed in the calculation are

$\displaystyle T_{1}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\,\mbox{tr}\left[\gamma^{\mu}\left(p_{\parallel}\gamma_{\parallel}+m\right)\left(\mathcal{P}_{+}L_{n}+\mathcal{P}_{-}L_{n-1}\right)\gamma^{\nu}\left((p_{\parallel}-k_{\parallel})\gamma_{\parallel}+m\right)\left(\mathcal{P}_{+}L_{n^{\prime}}+\mathcal{P}_{-}L_{n^{\prime}-1}\right)\right]$
(89) $\displaystyle=$ $\displaystyle
2\left(p_{\parallel}^{\mu}(p_{\parallel}-k_{\parallel})^{\nu}+(p_{\parallel}-k_{\parallel})^{\mu}p_{\parallel}^{\nu}-g_{\parallel}^{\mu\nu}\left[p_{\parallel}(p_{\parallel}-k_{\parallel})-m^{2}\right]\right)\left(L_{n}L_{n^{\prime}}+L_{n-1}L_{n^{\prime}-1}\right)$
$\displaystyle-$ $\displaystyle
2g_{\perp}^{\mu\nu}\left[p_{\parallel}(p_{\parallel}-k_{\parallel})-m^{2}\right]\left(L_{n}L_{n^{\prime}-1}+L_{n-1}L_{n^{\prime}}\right)+2i\ell^{2}qF^{\mu\nu}\left[p_{\parallel}(p_{\parallel}-k_{\parallel})-m^{2}\right]\left(L_{n}L_{n^{\prime}-1}-L_{n-1}L_{n^{\prime}}\right),$
$\displaystyle T_{2}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{i}{\ell^{2}}\,\mbox{tr}\left[\gamma^{\mu}\left(p_{\parallel}\gamma_{\parallel}+m\right)\left(\mathcal{P}_{+}L_{n}+\mathcal{P}_{-}L_{n-1}\right)\gamma^{\nu}(\mathbf{r}_{\perp}\cdot\bm{\gamma}_{\perp})L_{n^{\prime}-1}^{1}\right]$
(90) $\displaystyle=$
$\displaystyle-\frac{2i}{\ell^{2}}\left(p_{\parallel}^{\mu}r_{\perp}^{\nu}+r_{\perp}^{\mu}p_{\parallel}^{\nu}\right)\left(L_{n}+L_{n-1}\right)L_{n^{\prime}-1}^{1}+2q\left(p_{\parallel}^{\mu}F^{\nu\rho}r_{\perp,\rho}-F^{\mu\rho}r_{\perp,\rho}p_{\parallel}^{\nu}\right)\left(L_{n}-L_{n-1}\right)L_{n^{\prime}-1}^{1},$
$\displaystyle T_{3}^{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{i}{\ell^{2}}\,\mbox{tr}\left[\gamma^{\mu}(\mathbf{r}_{\perp}\cdot\bm{\gamma}_{\perp})L_{n-1}^{1}\gamma^{\nu}\left((p_{\parallel}-k_{\parallel})\gamma_{\parallel}+m\right)\left(\mathcal{P}_{+}L_{n^{\prime}}+\mathcal{P}_{-}L_{n^{\prime}-1}\right)\right]$
(91) $\displaystyle=$
$\displaystyle\frac{2i}{\ell^{2}}\left((p_{\parallel}-k_{\parallel})^{\mu}r_{\perp}^{\nu}+r_{\perp}^{\mu}(p_{\parallel}-k_{\parallel})^{\nu}\right)\left(L_{n^{\prime}}+L_{n^{\prime}-1}\right)L_{n-1}^{1}$
$\displaystyle+$ $\displaystyle
2q\left((p_{\parallel}-k_{\parallel})^{\mu}F^{\nu\rho}r_{\perp,\rho}-F^{\mu\rho}r_{\perp,\rho}(p_{\parallel}-k_{\parallel})^{\nu}\right)\left(L_{n^{\prime}}-L_{n^{\prime}-1}\right)L_{n-1}^{1},$
$\displaystyle T_{4}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{\ell^{4}}\,\mbox{tr}\left[\gamma^{\mu}(\mathbf{r}_{\perp}\cdot\bm{\gamma}_{\perp})L_{n-1}^{1}\gamma^{\nu}(\mathbf{r}_{\perp}\cdot\bm{\gamma}_{\perp})L_{n^{\prime}-1}^{1}\right]=\frac{4}{\ell^{4}}\left(g^{\mu\nu}\mathbf{r}_{\perp}^{2}+2r_{\perp}^{\mu}r_{\perp}^{\nu}\right)L_{n-1}^{1}L_{n^{\prime}-1}^{1},$
(92)

where $p_{\parallel}\gamma_{\parallel}\equiv
p_{0}\gamma^{0}-p_{z}\gamma^{3}=i\omega_{k}\gamma^{0}-p_{z}\gamma^{3}$ and
$(p_{\parallel}-k_{\parallel})\gamma_{\parallel}\equiv(p_{0}-k_{0})\gamma^{0}-(p_{z}-k_{z})\gamma^{3}=(i\omega_{k}-i\Omega_{m})\gamma^{0}-(p_{z}-k_{z})\gamma^{3}$.
The only nonzero components of the field strength tensor are
$F^{12}=-F^{21}=-B$ or, equivalently, $F^{\mu\nu}=-\varepsilon^{0\mu\nu 3}B$.
Thus, $F^{\mu\rho}r_{\perp,\rho}=(\mathbf{r}_{\perp}\times\mathbf{B})^{\mu}$.

By making use of the results in Appendix B, one can also calculate the
corresponding integrals over the transverse spatial coordinates,
$\mathbf{r}_{\perp}$. The results are given by

$\displaystyle I_{1}^{\mu\nu}$ $\displaystyle=$ $\displaystyle\int
d^{2}\mathbf{r}_{\perp}e^{-i\mathbf{r}_{\perp}\cdot\mathbf{k}_{\perp}}e^{-\mathbf{r}_{\perp}^{2}/(2\ell^{2})}T_{1}^{\mu\nu}=-4\pi\ell^{2}g_{\perp}^{\mu\nu}\left[p_{\parallel}(p_{\parallel}-k_{\parallel})-m^{2}\right]\left[\mathcal{I}_{0}^{n,n^{\prime}-1}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}}(\xi)\right]$
(93) $\displaystyle+$ $\displaystyle
4\pi\ell^{2}\left[p_{\parallel}^{\mu}(p_{\parallel}-k_{\parallel})^{\nu}+(p_{\parallel}-k_{\parallel})^{\mu}p_{\parallel}^{\nu}-g_{\parallel}^{\mu\nu}\left[p_{\parallel}(p_{\parallel}-k_{\parallel})-m^{2}\right]\right]\left[\mathcal{I}_{0}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}-1}(\xi)\right]$
$\displaystyle+$ $\displaystyle 4\pi
i\ell^{4}qF^{\mu\nu}\left[p_{\parallel}(p_{\parallel}-k_{\parallel})-m^{2}\right]\left[\mathcal{I}_{0}^{n,n^{\prime}-1}(\xi)-\mathcal{I}_{0}^{n-1,n^{\prime}}(\xi)\right],$
$\displaystyle I_{2}^{\mu\nu}$ $\displaystyle=$ $\displaystyle\int
d^{2}\mathbf{r}_{\perp}e^{-i\mathbf{r}_{\perp}\cdot\mathbf{k}_{\perp}}e^{-\mathbf{r}_{\perp}^{2}/(2\ell^{2})}T_{2}^{\mu\nu}=-4\pi\ell\left(p_{\parallel}^{\mu}\hat{k}_{\perp}^{\nu}+\hat{k}_{\perp}^{\mu}p_{\parallel}^{\nu}\right)\left[\mathcal{I}_{1}^{n,n^{\prime}-1}(\xi)+\mathcal{I}_{1}^{n-1,n^{\prime}-1}(\xi)\right]$
(94) $\displaystyle-$ $\displaystyle 4\pi
i\ell^{3}q\left(p_{\parallel}^{\mu}F^{\nu\rho}\hat{k}_{\perp,\rho}-F^{\mu\rho}\hat{k}_{\perp,\rho}p_{\parallel}^{\nu}\right)\left[\mathcal{I}_{1}^{n,n^{\prime}-1}(\xi)-\mathcal{I}_{1}^{n-1,n^{\prime}-1}(\xi)\right],$
$\displaystyle I_{3}^{\mu\nu}$ $\displaystyle=$ $\displaystyle\int
d^{2}\mathbf{r}_{\perp}e^{-i\mathbf{r}_{\perp}\cdot\mathbf{k}_{\perp}}e^{-\mathbf{r}_{\perp}^{2}/(2\ell^{2})}T_{3}^{\mu\nu}=4\pi\ell\left((p_{\parallel}-k_{\parallel})^{\mu}\hat{k}_{\perp}^{\nu}+\hat{k}_{\perp}^{\mu}(p_{\parallel}-k_{\parallel})^{\nu}\right)\left[\mathcal{I}_{1}^{n^{\prime},n-1}(\xi)+\mathcal{I}_{1}^{n^{\prime}-1,n-1}(\xi)\right]$
(95) $\displaystyle-$ $\displaystyle 4\pi
i\ell^{3}q\left((p_{\parallel}-k_{\parallel})^{\mu}F^{\nu\rho}\hat{k}_{\perp,\rho}-F^{\mu\rho}\hat{k}_{\perp,\rho}(p_{\parallel}-k_{\parallel})^{\nu}\right)\left[\mathcal{I}_{1}^{n^{\prime},n-1}(\xi)-\mathcal{I}_{1}^{n^{\prime}-1,n-1}(\xi)\right],$
$\displaystyle I_{4}^{\mu\nu}$ $\displaystyle=$ $\displaystyle\int
d^{2}\mathbf{r}_{\perp}e^{-i\mathbf{r}_{\perp}\cdot\mathbf{k}_{\perp}}e^{-\mathbf{r}_{\perp}^{2}/(2\ell^{2})}T_{4}^{\mu\nu}=8\pi\left[g_{\parallel}^{\mu\nu}\mathcal{I}_{2}^{n-1,n^{\prime}-1}(\xi)-\left(g_{\perp}^{\mu\nu}+2\hat{k}_{\perp}^{\mu}\hat{k}_{\perp}^{\nu}\right)\mathcal{I}_{3}^{n-1,n^{\prime}-1}(\xi)\right].$
(96)

The explicit expressions for functions $\mathcal{I}_{i}^{n,n^{\prime}}$
($i=1,2,3,4$) are given in Eqs. (72) – (75) in Appendix B.

After performing the Matsubara sums, substituting
$i\Omega_{m}\to\Omega+i\epsilon$, and using the energy conservation condition
(8), one finds that the $I_{i}^{\mu\nu}$ functions in the final result will be
replaced by similar expressions where only the following replacements should
be made:

$\displaystyle p_{\parallel}^{\mu}$ $\displaystyle\to$
$\displaystyle\bar{p}_{\parallel}^{\mu}\Big{|}_{p_{z}=p_{z}^{(\pm)}}=-\eta
E_{n,p_{z}}\delta^{\mu}_{0}+p_{z}\delta^{\mu}_{3}\Big{|}_{p_{z}=p_{z}^{(\pm)}}$
(97) $\displaystyle=$
$\displaystyle\frac{1}{2}k_{\parallel}^{\mu}\left(\frac{2(n-n^{\prime})|qB|}{\Omega^{2}-k_{z}^{2}}+1\right)\pm\frac{1}{2}\tilde{k}_{\parallel}^{\mu}\sqrt{\left(1-\frac{k_{-}^{2}}{\Omega^{2}-k_{z}^{2}}\right)\left(1-\frac{k_{+}^{2}}{\Omega^{2}-k_{z}^{2}}\right)},$
$\displaystyle(p_{\parallel}-k_{\parallel})^{\mu}$ $\displaystyle\to$
$\displaystyle(\bar{p}_{\parallel}-k_{\parallel})^{\mu}\Big{|}_{p_{z}=p_{z}^{(\pm)}}=-\eta\lambda
E_{n^{\prime},p_{z}-k_{z}}\delta^{\mu}_{0}+(p_{z}-k_{z})\delta^{\mu}_{3}\Big{|}_{p_{z}=p_{z}^{(\pm)}}$
(98) $\displaystyle=$
$\displaystyle\frac{1}{2}k_{\parallel}^{\mu}\left(\frac{2(n-n^{\prime})|qB|}{\Omega^{2}-k_{z}^{2}}-1\right)\pm\frac{1}{2}\tilde{k}_{\parallel}^{\mu}\sqrt{\left(1-\frac{k_{-}^{2}}{\Omega^{2}-k_{z}^{2}}\right)\left(1-\frac{k_{+}^{2}}{\Omega^{2}-k_{z}^{2}}\right)},$
$\displaystyle p_{\parallel}(p_{\parallel}-k_{\parallel})$ $\displaystyle\to$
$\displaystyle\bar{p}_{\parallel}(\bar{p}_{\parallel}-k_{\parallel})\Big{|}_{p_{z}=p_{z}^{(\pm)}}=\lambda
E_{n,p_{z}}E_{n^{\prime},p_{z}-k_{z}}-p_{z}(p_{z}-k_{z})\Big{|}_{p_{z}=p_{z}^{(\pm)}}$
(99) $\displaystyle=$ $\displaystyle
m^{2}+(n+n^{\prime})|qB|-\frac{1}{2}k_{\parallel}^{2}.$

where $k_{\parallel}^{\mu}=\Omega\delta^{\mu}_{0}+k_{z}\delta^{\mu}_{3}$ and
$\tilde{k}_{\parallel}^{\mu}=k_{z}\delta^{\mu}_{0}+\Omega\delta^{\mu}_{3}$.
Note that $k_{\parallel,\mu}\tilde{k}_{\parallel}^{\mu}=0$ and
$\tilde{k}_{\parallel,\mu}\tilde{k}_{\parallel}^{\mu}=-k_{\parallel}^{2}$. The
definition of the transverse threshold momenta $k_{-}$ and $k_{+}$ are given
in Eqs. (11) and (13), respectively.

By using these results, we can derive the alternative expressions for tensors
$I_{i}^{\mu\nu}$ tensors. In particular, one obtains

$\displaystyle I_{1}^{\mu\nu}\Big{|}_{p_{z}=p_{z}^{(\pm)}}$ $\displaystyle=$
$\displaystyle-4\pi\ell^{2}g_{\perp}^{\mu\nu}\left[(n+n^{\prime})|qB|-\frac{1}{2}k_{\parallel}^{2}\right]\left[\mathcal{I}_{0}^{n,n^{\prime}-1}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}}(\xi)\right]$
(100) $\displaystyle-$ $\displaystyle
4\pi\ell^{2}g_{\parallel}^{\mu\nu}\left[(n+n^{\prime})|qB|-\frac{1}{2}k_{\parallel}^{2}\right]\left[\mathcal{I}_{0}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}-1}(\xi)\right]$
$\displaystyle+$ $\displaystyle 4\pi
i\ell^{4}qF^{\mu\nu}\left[(n+n^{\prime})|qB|-\frac{1}{2}k_{\parallel}^{2}\right]\left[\mathcal{I}_{0}^{n,n^{\prime}-1}(\xi)-\mathcal{I}_{0}^{n-1,n^{\prime}}(\xi)\right]$
$\displaystyle+$ $\displaystyle
4\pi\ell^{2}A_{\pm}^{\mu\nu}\left[\mathcal{I}_{0}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}-1}(\xi)\right],$

where we used the following shorthand notation:

$\displaystyle A_{\pm}^{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}k_{\parallel}^{\mu}k_{\parallel}^{\nu}\left(1-\frac{4(n-n^{\prime})^{2}(qB)^{2}}{(\Omega^{2}-k_{z}^{2})^{2}}\right)+\frac{1}{2}\tilde{k}_{\parallel}^{\mu}\tilde{k}_{\parallel}^{\nu}\left(1-\frac{4\left[m^{2}+(n+n^{\prime})|qB|\right]}{\Omega^{2}-k_{z}^{2}}+\frac{4(n-n^{\prime})^{2}(qB)^{2}}{(\Omega^{2}-k_{z}^{2})^{2}}\right)$
(101) $\displaystyle\pm$
$\displaystyle\left(k_{\parallel}^{\mu}\tilde{k}_{\parallel}^{\nu}+k_{\parallel}^{\nu}\tilde{k}_{\parallel}^{\mu}\right)\frac{(n-n^{\prime})|qB|}{\Omega^{2}-k_{z}^{2}}\sqrt{1-\frac{4\left[m^{2}+(n+n^{\prime})|qB|\right]}{\Omega^{2}-k_{z}^{2}}+\frac{4(n-n^{\prime})^{2}(qB)^{2}}{(\Omega^{2}-k_{z}^{2})^{2}}}.$

Note that the upper and lower signs correspond to $p_{z}^{(+)}$ and
$p_{z}^{(-)}$, respectively.

Similarly, by combining the next two tensor functions together, one finds

$\displaystyle I_{2}^{\mu\nu}+I_{3}^{\mu\nu}\Big{|}_{p_{z}=p_{z}^{(\pm)}}$
$\displaystyle=$ $\displaystyle-4\pi\ell
B_{\pm}^{\mu\nu}\left[\mathcal{I}_{1}^{n,n^{\prime}-1}(\xi)+\mathcal{I}_{1}^{n-1,n^{\prime}-1}(\xi)\right]$
(102) $\displaystyle-$ $\displaystyle
4\pi\ell\left(k_{\parallel}^{\mu}\hat{k}_{\perp}^{\nu}+k_{\parallel}^{\nu}\hat{k}_{\perp}^{\mu}-B_{\pm}^{\mu\nu}\right)\left[\mathcal{I}_{1}^{n^{\prime},n-1}(\xi)+\mathcal{I}_{1}^{n^{\prime}-1,n-1}(\xi)\right]$
$\displaystyle-$ $\displaystyle 4\pi
i\ell^{3}\frac{qB}{k_{\perp}}C_{\pm}^{\mu\nu}\left[\mathcal{I}_{1}^{n,n^{\prime}-1}(\xi)-\mathcal{I}_{1}^{n-1,n^{\prime}-1}(\xi)\right]$
$\displaystyle-$ $\displaystyle 4\pi
i\ell^{3}\frac{qB}{k_{\perp}}\left[C_{\pm}^{\mu\nu}-\left(k_{\parallel}^{\mu}\tilde{k}_{\perp}^{\nu}-k_{\parallel}^{\nu}\tilde{k}_{\perp}^{\mu}\right)\right]\left[\mathcal{I}_{1}^{n^{\prime},n-1}(\xi)-\mathcal{I}_{1}^{n^{\prime}-1,n-1}(\xi)\right],$

where, by definition,

$B_{\pm}^{\mu\nu}=\frac{1}{2}\left(k_{\parallel}^{\mu}\hat{k}_{\perp}^{\nu}+k_{\parallel}^{\nu}\hat{k}_{\perp}^{\mu}\right)\left(\frac{2(n-n^{\prime})|qB|}{\Omega^{2}-k_{z}^{2}}+1\right)\pm\frac{1}{2}\left(\tilde{k}_{\parallel}^{\mu}\hat{k}_{\perp}^{\nu}+\tilde{k}_{\parallel}^{\nu}\hat{k}_{\perp}^{\mu}\right)\sqrt{1-\frac{4\left[m^{2}+(n+n^{\prime})|qB|\right]}{\Omega^{2}-k_{z}^{2}}+\frac{4(n-n^{\prime})^{2}(qB)^{2}}{(\Omega^{2}-k_{z}^{2})^{2}}},$
(103)

and

$C_{\pm}^{\mu\nu}=\frac{1}{2}\left(k_{\parallel}^{\mu}\tilde{k}_{\perp}^{\nu}-k_{\parallel}^{\nu}\tilde{k}_{\perp}^{\mu}\right)\left(\frac{2(n-n^{\prime})|qB|}{\Omega^{2}-k_{z}^{2}}+1\right)\pm\frac{1}{2}\left(\tilde{k}_{\parallel}^{\mu}\tilde{k}_{\perp}^{\nu}-\tilde{k}_{\parallel}^{\nu}\tilde{k}_{\perp}^{\mu}\right)\sqrt{1-\frac{4\left[m^{2}+(n+n^{\prime})|qB|\right]}{\Omega^{2}-k_{z}^{2}}+\frac{4(n-n^{\prime})^{2}(qB)^{2}}{(\Omega^{2}-k_{z}^{2})^{2}}}.$
(104)

Finally, the last one is given by

$\displaystyle I_{4}^{\mu\nu}\Big{|}_{p_{z}=p_{z}^{(\pm)}}$ $\displaystyle=$
$\displaystyle
8\pi\left[g_{\parallel}^{\mu\nu}\mathcal{I}_{2}^{n-1,n^{\prime}-1}(\xi)-\left(g_{\perp}^{\mu\nu}+2\hat{k}_{\perp}^{\mu}\hat{k}_{\perp}^{\nu}\right)\mathcal{I}_{3}^{n-1,n^{\prime}-1}(\xi)\right].$
(105)

When calculating the Lorentz contracted expression for the polarization tensor
$\mbox{Im}\left[\Pi^{\mu}_{\mu}\right]\sim\sum_{i=1}^{4}g_{\mu\nu}I^{\mu\nu}_{i}$,
it is convenient to introduce the following scalar functions:
$\mathcal{F}_{i}=g_{\mu\nu}I^{\mu\nu}_{i}$. By making use of the original
definition for $I^{\mu\nu}_{i}$, one finds that

$\displaystyle\mathcal{F}_{1}$ $\displaystyle=$ $\displaystyle
8\pi\left[\frac{k_{\parallel}^{2}\ell^{2}}{2}-(n+n^{\prime})\right]\left(\mathcal{I}_{0}^{n-1,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n,n^{\prime}-1}(\xi)\right)+8\pi\ell^{2}m^{2}\left(\mathcal{I}_{0}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}-1}(\xi)\right),$
(106) $\displaystyle\mathcal{F}_{2}$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}I_{2}^{\mu\nu}=0,$ (107) $\displaystyle\mathcal{F}_{3}$
$\displaystyle=$ $\displaystyle g_{\mu\nu}I_{3}^{\mu\nu}=0,$ (108)
$\displaystyle\mathcal{F}_{4}$ $\displaystyle=$
$\displaystyle=16\pi\,\mathcal{I}_{2}^{n-1,n^{\prime}-1}(\xi).$ (109)

By adding together the nonvanishing functions $\mathcal{F}_{i}$, we derive

$\mathcal{F}_{1}+\mathcal{F}_{4}=8\pi\left(n+n^{\prime}+m^{2}\ell^{2}\right)\left[\mathcal{I}_{0}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}-1}(\xi)\right]+8\pi\left(\frac{k_{\parallel}^{2}-k_{\perp}^{2}}{2}\ell^{2}-(n+n^{\prime})\right)\left[\mathcal{I}_{0}^{n,n^{\prime}-1}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}}(\xi)\right].$
(110)

Note that the second term simplifies when photons satisfy the on-shell
condition $k_{\parallel}^{2}=k_{\perp}^{2}$.

##  Appendix D Tensor structure of $\mbox{Im}\left[\Pi^{\mu\nu}\right]$

By making use of the expression for the polarization tensor in Eq. (19) and
the definition of tensors $I_{i}^{\mu\nu}$ in Appendix C, we find that the
imaginary part of $\Pi_{R}^{\mu\nu}(\Omega;\mathbf{k})$ has the following
structure:

$\displaystyle\mbox{Im}\left[\Pi_{R}^{\mu\nu}(\Omega;\mathbf{k})\right]$
$\displaystyle=$
$\displaystyle\left(\frac{k_{\parallel}^{\mu}k_{\parallel}^{\nu}}{k_{\parallel}^{2}}-g_{\parallel}^{\mu\nu}\right)\mbox{Im}\left[\Pi_{1}\right]+\left(g_{\perp}^{\mu\nu}+\frac{k_{\perp}^{\mu}k_{\perp}^{\nu}}{k_{\perp}^{2}}\right)\mbox{Im}\left[\Pi_{2}\right]$
(111) $\displaystyle+$
$\displaystyle\left(\frac{k_{\parallel}^{\mu}\tilde{k}_{\parallel}^{\nu}+\tilde{k}_{\parallel}^{\mu}k_{\parallel}^{\nu}}{k_{\parallel}^{2}}+\frac{\tilde{k}_{\parallel}^{\mu}k_{\perp}^{\nu}+k_{\perp}^{\mu}\tilde{k}_{\parallel}^{\nu}}{k_{\perp}^{2}}\right)\mbox{Im}\left[\Pi_{3}\right]+\left(\frac{k_{\parallel}^{\mu}k_{\perp}^{\nu}+k_{\perp}^{\mu}k_{\parallel}^{\nu}}{k_{\parallel}^{2}}+\frac{k_{\perp}^{2}}{k_{\parallel}^{2}}g_{\parallel}^{\mu\nu}-g_{\perp}^{\mu\nu}\right)\mbox{Im}\left[\Pi_{4}\right]$
$\displaystyle+$
$\displaystyle\left(\frac{F^{\mu\nu}}{B}+\frac{k_{\parallel}^{\mu}\tilde{k}_{\perp}^{\nu}-\tilde{k}_{\perp}^{\mu}k_{\parallel}^{\nu}}{k_{\parallel}^{2}}\right)\mbox{Im}\left[\tilde{\Pi}_{5}\right]+\frac{\tilde{k}_{\parallel}^{\mu}\tilde{k}_{\perp}^{\nu}-\tilde{k}_{\perp}^{\mu}\tilde{k}_{\parallel}^{\nu}}{k_{\parallel}^{2}}\mbox{Im}\left[\tilde{\Pi}_{6}\right],$

where we utilized the shorthand notations introduced in Eq. (22). Note that
$k_{\mu}\tilde{k}_{\perp}^{\mu}=k_{\mu}\tilde{k}_{\parallel}^{\mu}=0$ and
$F^{\mu\nu}k_{\perp,\nu}=B\tilde{k}_{\perp}^{\mu}$. One can also check that
$\tilde{k}_{\perp,\mu}\tilde{k}_{\perp}^{\mu}=k_{\perp,\mu}k_{\perp}^{\mu}=-k_{\perp}^{2}$
and
$\tilde{k}_{\parallel,\mu}\tilde{k}_{\parallel}^{\mu}=-k_{\parallel,\mu}k_{\parallel}^{\mu}=-k_{\parallel}^{2}$.

To simplify the representation of the six component functions in the imaginary
part of polarization tensor (111), it is convenient to introduce the following
operator:

$\hat{\cal X}(\ldots)=\frac{\alpha
N_{f}}{4\pi\ell^{4}}\sum_{n,n^{\prime}=0}^{\infty}\sum_{\lambda,\eta=\pm
1}\sum_{s=\pm
1}\Theta_{\lambda,\eta}^{n,n^{\prime}}(\Omega,k_{z})\frac{n_{F}(E_{n,p_{z}})-n_{F}(\lambda
E_{n^{\prime},p_{z}-k_{z}})}{\eta\lambda\sqrt{\left(\Omega^{2}-k_{z}^{2}-k_{-}^{2}\right)\left(\Omega^{2}-k_{z}^{2}-k_{+}^{2}\right)}}(\ldots).$
(112)

Note that, with the help of the operator $\hat{\cal X}$, the polarization
tensor in Eq. (19) can be rewritten in a compact form as
$\mbox{Im}\left[\Pi_{R}^{\mu\nu}\right]=\hat{\cal
X}\sum_{i=1}^{4}I_{i}^{\mu\nu}$.

By using the operator in Eq. (112), we can also write down the explicit
expressions for the individual tensor component functions. In particular, the
first four components, defining the symmetric tensor structures, are

$\displaystyle\mbox{Im}\left[\Pi_{1}\right]$ $\displaystyle=$ $\displaystyle
8\pi\hat{\cal
X}\left(\frac{2(n-n^{\prime})^{2}}{k_{\parallel}^{2}\ell^{2}}-m^{2}\ell^{2}-(n+n^{\prime})\right)\left[\mathcal{I}_{0}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}-1}(\xi)\right],$
(113) $\displaystyle\mbox{Im}\left[\Pi_{2}\right]$ $\displaystyle=$
$\displaystyle-16\pi\hat{\cal X}\,\mathcal{I}_{3}^{n-1,n^{\prime}-1}(\xi)$
(114) $\displaystyle=$ $\displaystyle-8\pi\hat{\cal
X}\Bigg{\\{}(n+n^{\prime})\left[\mathcal{I}_{0}^{n,n^{\prime}-1}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}}(\xi)\right]-\frac{(n-n^{\prime})^{2}}{\xi}\left[\mathcal{I}_{0}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}-1}(\xi)\right]\Bigg{\\}},$
$\displaystyle\mbox{Im}\left[\Pi_{3}\right]$ $\displaystyle=$
$\displaystyle\pm 4\pi\hat{\cal
X}(n-n^{\prime})\sqrt{\left(1-\frac{k_{-}^{2}}{k_{\parallel}^{2}}\right)\left(1-\frac{k_{+}^{2}}{k_{\parallel}^{2}}\right)}\left[\mathcal{I}_{0}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}-1}(\xi)\right],$
(115) $\displaystyle\mbox{Im}\left[\Pi_{4}\right]$ $\displaystyle=$
$\displaystyle-2\pi\hat{\cal
X}\Bigg{\\{}\frac{k_{\parallel}^{2}\ell^{2}+2(n-n^{\prime})}{k_{\perp}\ell}\left[\mathcal{I}_{1}^{n,n^{\prime}-1}(\xi)+\mathcal{I}_{1}^{n-1,n^{\prime}-1}(\xi)\right]+\frac{k_{\parallel}^{2}\ell^{2}-2(n-n^{\prime})}{k_{\perp}\ell}\left[\mathcal{I}_{1}^{n^{\prime},n-1}(\xi)+\mathcal{I}_{1}^{n^{\prime}-1,n-1}(\xi)\right]\Bigg{\\}}$
(116) $\displaystyle=$ $\displaystyle-2\pi\hat{\cal
X}\left\\{k_{\parallel}^{2}\ell^{2}\left[\mathcal{I}_{0}^{n,n^{\prime}-1}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}}(\xi)\right]-\frac{2(n-n^{\prime})^{2}}{\xi}\left[\mathcal{I}_{0}^{n,n^{\prime}}(\xi)+\mathcal{I}_{0}^{n-1,n^{\prime}-1}(\xi)\right]\right\\}.$

Similarly, the component functions of the antisymmetric contributions read

$\displaystyle\mbox{Im}\left[\tilde{\Pi}_{5}\right]$ $\displaystyle=$
$\displaystyle-2\pi is_{\perp}\hat{\cal
X}\left(k_{\parallel}^{2}\ell^{2}-2(n+n^{\prime})\right)\left[\mathcal{I}_{0}^{n,n^{\prime}-1}(\xi)-\mathcal{I}_{0}^{n-1,n^{\prime}}(\xi)\right],$
(117)

and

$\displaystyle\mbox{Im}\left[\tilde{\Pi}_{6}\right]$ $\displaystyle=$
$\displaystyle\mp 2\pi i\hat{\cal
X}\frac{s_{\perp}k_{\parallel}^{2}\ell^{2}}{k_{\perp}\ell}\sqrt{\left(1-\frac{k_{-}^{2}}{k_{\parallel}^{2}}\right)\left(1-\frac{k_{+}^{2}}{k_{\parallel}^{2}}\right)}\left[\mathcal{I}_{1}^{n,n^{\prime}-1}(\xi)-\mathcal{I}_{1}^{n-1,n^{\prime}-1}(\xi)+\mathcal{I}_{1}^{n^{\prime},n-1}(\xi)-\mathcal{I}_{1}^{n^{\prime}-1,n-1}(\xi)\right]$
(118) $\displaystyle=$ $\displaystyle\mp 4\pi
i\frac{s_{\perp}}{k_{\perp}^{2}}\hat{\cal
X}(n+n^{\prime})\sqrt{\left(k_{\parallel}^{2}-k_{-}^{2}\right)\left(k_{\parallel}^{2}-k_{+}^{2}\right)}\left[\mathcal{I}_{0}^{n,n^{\prime}}(\xi)-\mathcal{I}_{0}^{n-1,n^{\prime}-1}(\xi)\right].$

##  Appendix E Lowest Landau level approximation

In the lowest Landau level approximation (i.e., $n=n^{\prime}=0$), the
explicit result for the Lorentz-contracted imaginary part of the polarization
tensor reads

$\displaystyle\mbox{Im}\left[\Pi^{\mu}_{R,\mu}(\Omega;\mathbf{k})\right]$
$\displaystyle=$ $\displaystyle-\frac{4\alpha
N_{f}m^{2}\theta\left(\Omega^{2}-k_{z}^{2}-4m^{2}\right)}{(\Omega^{2}-k_{z}^{2})\ell^{2}R_{m}}\frac{\sinh\left(\frac{\Omega}{2T}\right)e^{-k_{\perp}^{2}\ell^{2}/2}}{\cosh\left(\frac{\Omega}{2T}\right)+\cosh\left(\frac{k_{z}R_{m}}{2T}\right)},$
(119)

where $R_{m}=\sqrt{1-4m^{2}/(\Omega^{2}-k_{z}^{2})}$. Note that the solutions
to the energy conservation equation are given by

$\displaystyle p_{z,0}^{(\pm)}$ $\displaystyle=$
$\displaystyle\frac{k_{z}}{2}\pm\frac{\Omega}{2}\sqrt{1-\frac{4m^{2}}{\Omega^{2}-k_{z}^{2}}}.$
(120)

On these solutions, the charged particle energies are

$\displaystyle
E_{0,p_{z}^{s+}}=E_{0,p_{z}^{s-}-k_{z}}=\frac{\Omega+k_{z}R_{m}}{2},$ (121)
$\displaystyle E_{0,p_{z}^{s+}-k_{z}}=E_{0,p_{z}^{s-}}=\frac{\Omega-
k_{z}R_{m}}{2}.$ (122)

In deriving the result in Eq. (119), we used $L_{0}^{0}(x)=L_{0}^{-1}(x)=1$
and took into account that, in the lowest Landau level approximation,

$\mathcal{F}_{1}=8\pi m^{2}\ell^{2}e^{-k_{\perp}^{2}\ell^{2}/2},$ (123)

and $\mathcal{F}_{4}=0$. As is easy to check, the result in Eq. (119) is
consistent with the spectral function obtained in the lowest Landau level
approximation in Ref. Bandyopadhyay:2016fyd . With that being said, it is
important to remember that, as emphasized in the main text, this approximation
is not very reliable for calculating the photon production rate even in very
strong magnetic fields.

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