# Dispersion Relations in Non-Linear Electrodynamics and the Kinematics of the
Compton Effect in a Magnetic Background

M. J. Neves<EMAIL_ADDRESS>Departamento de Física, Universidade Federal
Rural do Rio de Janeiro, BR 465-07, 23890-971, Seropédica, RJ, Brazil  Jorge
B. de Oliveira<EMAIL_ADDRESS>Departamento de Física,
Universidade Federal Rural do Rio de Janeiro, BR 465-07, 23890-971,
Seropédica, RJ, Brazil  L. P. R. Ospedal<EMAIL_ADDRESS>Centro Brasileiro de
Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Urca, Rio de Janeiro, Brazil,
CEP 22290-180  J. A. Helayël-Neto<EMAIL_ADDRESS>Centro Brasileiro de
Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Urca, Rio de Janeiro, Brazil,
CEP 22290-180

###### Abstract

Non-linear electrodynamic models are re-assessed in this paper to pursue an
investigation of the kinematics of the Compton effect in a magnetic
background. Before considering specific models, we start off by presenting a
general non-linear Lagrangian built up in terms of the most general Lorentz-
and gauge-invariant combinations of the electric and magnetic fields. The
extended Maxwell-like equations and the energy-momentum tensor conservation
are presented and discussed in their generality. We next expand the fields
around a uniform and time-independent electric and magnetic backgrounds up to
second order in the propagating wave, and compute dispersion relations which
account for the effect of the external fields. We obtain thereby the
refraction index and the group velocity for the propagating radiation in
different situations. In particular, we focus on the kinematics of the Compton
effect in presence of external magnetic fields. This yields constraints that
relate the derivatives of the general Lagrangian with respect to the field
invariants and the magnetic background under consideration. We carry out our
inspection by focusing on some specific non-linear electrodynamic effective
models: Hoffmann-Infeld, Euler-Heisenberg, generalized Born-Infeld and
Logarithmic.

Non-linear electrodynamics, dispersion relations in a magnetic background,
Compton effect.

###### pacs:

11.15.-q; 11.10.Ef; 11.10.Nx

##  I Introduction

Maxwell Electrodynamics is a highly successful theory to describe properties
of the electromagnetic interaction at both the classical and the quantum
length scales. Photon-photon scattering in Quantum Electrodynamics (QED)
motivates the study of non-linear extensions of the Maxwell Electrodynamics
(MED), and phenomena like vacuum birefringence and vacuum dichroism may be a
guide to also inspect the consistency of non-linear extensions of MED Adler71
; Constantini71 ; Biswas07 ; Seco09 ; Seco07 ; Seco08 ; Kruglov07 . We cite
here the works by Plebanski, Boillat, Bialynicka-Birula and Bialynicki-Birula
as excellent articles for all those readers who wish to be introduced to the
issue of non-linear extensions of MED Plebanski ; Boillat ; Birula1 ; Birula2
. Different scenarios of the latter introduce new effects into the photon-
photon scattering process. To be more specific, we quote scenarios such as
Born-Infeld Electrodynamics, models with milli-charged particles and models of
axion-like particles that interact topologically with hidden-photons Born ;
Davila ; Ellis ; Gies ; Masso ; GaeteMPLA ; GaeteJPA ; Arias . The non-linear
Born-Infeld Electrodynamics was originally introduced to remove the
singularity of the electric field of point-like charges on their space
position. Nowadays, Born-Infeld effective actions emerge in diverse scenarios,
like superstring theory, quantum-gravitational models and theories with
magnetic monopoles Fradkin ; Pope ; Banerjee ; Mann ; Garcia ; NiauEPJC ;
NiauPRD . Furthermore, a number of new phenomena in Cosmology and black-hole
physics have been reported in connection with non-linear extensions of
electrodynamic systems coupled to gravity Hendi ; Pan ; Olea ; Halisoy ; Lopez
; Aiello ; Balart .

The introduction of external backgrounds in field-theoretic models is an old
procedure to reproduce effects of the vacuum polarization phenomenon. If the
external electromagnetic fields are strong enough (as compared to the so-
called Schwinger critical electric and magnetic fields), they can induce the
creation of real particle-antiparticle pairs in the non-trivial quantum
vacuum, such as, the electron-positron pairs of QED vacuum Dirac ; Schwinger .
Non-linear extensions of electrodynamics in connection with external
electromagnetic fields are able to describe effects of the vacuum structure on
waves that propagate in empty space, as already pointed out in the previous
paragraph, through the observation of birefringence and vacuum dichroism
phenomena Liao ; Valle1 ; Valle2 ; Potting ; Baier . A well-known case of non-
linear extension that stems from the vacuum polarization is the Euler-
Heisenberg Lagrangian EH , which is an effective photonic model with higher
powers in the electric and magnetic fields, attained upon integration over the
quantum effects of the (virtual) electron-positron pairs. We would like to
point out here the interesting paper of Reference GiesJHEP , where the EH
model is studied beyond the 1-loop level in a great deal of details. Let us
also recall that, in 1961, Franken et al. opened up the field of non-linear
optics with the celebrated experiment in which they successfully measured the
second-harmonic (optical) generation Franken .

Motivations that strongly justify the renewed interest in non-linear
extensions of MED are also coming from the recent high-intensity LASERs, which
are the best devices to test both Classical and Quantum Electrodynamics in the
strong-field regime, whose (field) scales are fixed by the critical intensity
$I_{crit}\sim 4.6\times 10^{29}\,\mbox{W}\cdot\mbox{cm}^{-2}$. The Station of
Extreme Light (SEL), the Europe’s Extreme Light Infrastructure (ELI Project)
and the ExaWatt Center for Extreme Light Studies (XCELS) shall, in a close
future, provide the facilities to pursue a very detailed inspection of
electromagnetic non-linearities and allow a real dive into the structure of
the quantum vacuum. Powers of these extremely-intense LASERs are expected to
reach $10^{2}$ PW, hopefully getting even to the Exa-W scale in a couple of
years. We bring to the reader’s attention that high-intensity LASER
experiments designed to inspect non-linearity effects in QED are presented and
discussed in Felix ; Lundin ; Battesti ; King .

Non-linearity means corrections to the Maxwell electrodynamics that, in
general, depend on the two Lorentz- and gauge-invariant quantities, ${\cal
F}=-F_{\mu\nu}^{2}/4$ and ${\cal G}=-F_{\mu\nu}\widetilde{F}^{\mu\nu}/4$. In
many examples, the Lagrangian density of the non-linear model depends
exclusively on powers of ${\cal F}$ kruglov17 ; GHL_2019 ; in other
situations, there may be dependence on even powers of ${\cal G}$ (even powers
avoid charge-parity (CP) symmetry violation) kruglov15 ; Gaete ; Gaete2 ; Haas
. In our present contribution, we start off with a general Lagrangian that is
a function of these two invariants to obtain the corresponding field equations
and the energy-momentum tensor. Next, we expand the field-strength tensor
around an electromagnetic background, initially considered non-uniform and
time-dependent. We keep the terms of the expansion in the non-linear
Lagrangian up to second order in the propagating excitation, and the
corresponding field equations in presence of an external electromagnetic field
are written down. The expansion displays coefficients that depend on the
external background fields. The components of the energy-momentum tensor are
calculated in the case of general space-time-dependent backgrounds. In the
absence of external sources, plane wave solutions are used to calculate the
allowed frequencies, the dispersion relations and the refraction index of the
photon in a uniform magnetic background. The dispersion relations, and
consequently, the refraction indices both depend on the relative direction of
the wave vector with respect to the external magnetic field.

The group velocity of the electromagnetic wave is determined from the
solutions of the frequency dictated by the dispersion relations . The
variation of the photon wavelength in the Compton effect is investigated in
terms of the new dispersion relations and depends, of course, on the magnetic
background field. We apply these results in some particular cases of non-
linear electrodynamics : Hoffmann-Infeld, generalized Born-Infeld, Logarithm
electrodynamics and the Euler-Heisenberg effective Lagrangian in the regime of
weak electromagnetic fields.

More recently, the use of astrophysical sources has shown to be a very
fruitful procedure to constrain modified dispersion relations (MDRs) for
photons propagating in the vacuum Bolmont ; SorokinPRD . We should also recall
that, early in 2019, the Major Atmospheric Gamma Imaging Cherenkov (MAGIC)
telescopes identified the GRB190114C above $0.2$ TeV. This corresponds to
photons with the highest frequencies detected so far in Gamma-Ray Bursts.
Photons at this energy scale may well probe the quantum structure of the
vacuum, so that non-linear electrodynamic effects should be taken into
account. These observations motivate the growing of interest in the activity
of photonic MDRs. Non-linearity in association with the presence of strong
background magnetic fields yields a rich class of MDRs which may, in turn,
unveil effects of new physics beyond the Standard Model (SM) of Particles and
Fundamental Interactions Acciari . Moreover, let us recall that, in the SM,
parity violation is verified in the weak-interaction sector. However, physics
beyond the SM may well be sensitive to parity transformations. In this
context, we could seek for evidence of parity-violating physics by inspecting
MDRs in a special class on non-linear extensions of electrodynamic models,
namely, the ones which explicitly depend on the special Lorentz- and gauge-
invariant quantity ${\cal G}$ which, appearing with an odd power in a
Lagrangian density, signals parity-symmetry breaking. This directly addresses
us to the set of Planck 2018 Polarization Data, which could become a rich
laboratory to constrain parity-violating non-linear extensions of
electromagnetism, which may, in turn, be opening up trends to see for new
physics beyond the SM Minami .

We organize our paper as follows. In Section II, we give highlights of the
non-linear electrodynamics framework, the corresponding field equations and
energy-momentum tensor in the presence of general electric and magnetic
background fields. Section III focus on the frequencies for the plane wave
solutions, the photonic dispersion relations in presence of a uniform magnetic
field and the consequences of this background on the kinematical description
of the Compton effect. In Section IV, we apply the results of the previous
Section for a special non-linear ED depending only on the ${\cal
F}$-invariant. In Section V, we discuss the results by exploiting other cases
of non-linear ED known in the literature that depend on both the ${\cal F}$\-
and ${\cal G}$-invariants. Our Conclusions and Final Remarks are cast in
Section VI.

The convention for the metric we adopt is
$\eta^{\mu\nu}=\mbox{diag}\left(+1,-1,-1,-1\right)$. We choose to work with
natural units: $\hbar=c=1$ and $4\pi\epsilon_{0}=1$. In this unit system, the
electric and magnetic fields have squared-energy dimension. The conversion of
Volt/m and Tesla (T) to the natural system is as follows:
$1\,\mbox{Volt/m}=2.27\times 10^{-24}\,\mbox{GeV}^{2}$ and
$1\,\mbox{T}=6.8\times 10^{-16}\,\mbox{GeV}^{2}$, respectively.

##  II A quick glance at a general non-linear electrodynamic model

We start off the description of the non-linear electrodynamics through the
most general Lagrangian, $\mathcal{L}$, written as a function of the Lorentz-
and gauge-invariant bilinears, ${\cal F}$ and ${\cal G}$, defined,
respectively, as follows below Plebanski ; Boillat ; Birula1 ; Birula2  :

$\displaystyle{\cal F}\\!$ $\displaystyle=$
$\displaystyle\\!-\frac{1}{4}\,F_{\mu\nu}^{2}=\frac{1}{2}\,\left(\,{\bf
E}_{0}^{2}-{\bf B}_{0}^{2}\,\right)\;,$ (1a) $\displaystyle{\cal G}\\!$
$\displaystyle=$
$\displaystyle\\!-\frac{1}{4}\,F_{\mu\nu}\widetilde{F}^{\mu\nu}={\bf
E}_{0}\cdot{\bf B}_{0}\;,$ (1b)

where
$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}=\left(\,-E_{0}^{\,\,i}\,,\,-\epsilon^{ijk}B_{0}^{\,\,k}\,\right)$
is the skew-symmetric field-strength tensor,
$\widetilde{F}^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}/2=\left(\,-B_{0}^{\,\,i}\,,\,\epsilon^{ijk}E_{0}^{\,\,k}\,\right)$
its corresponding dual tensor, which satisfies the Bianchi identity
$\partial_{\mu}\widetilde{F}^{\mu\nu}=0$. We decompose the $A^{\mu}$ potential
as $A^{\mu}=a^{\mu}+A_{B}^{\;\;\,\,\mu}$, where $a^{\mu}$ is identified as the
photon field, and $A_{B}^{\;\;\,\,\mu}$ is a background potential. As
consequence of this decomposition, the tensor $F^{\mu\nu}$ is written as
$F^{\mu\nu}=f^{\mu\nu}+F_{B}^{\;\;\,\mu\nu}$, in which
$f^{\mu\nu}=\partial^{\mu}a^{\nu}-\partial^{\nu}a^{\mu}=\left(\,-e^{i}\,,\,-\epsilon^{ijk}\,b^{k}\,\right)$
is the electromagnetic field-strength tensor of the propagating excitation,
whereas
$F_{B}^{\;\,\mu\nu}=\partial^{\mu}A_{B}^{\;\,\nu}-\partial^{\nu}A_{B}^{\;\,\mu}=\left(\,-E^{i}\,,\,-\epsilon^{ijk}\,B^{k}\,\right)$
corresponds to the field-strength associated with the electric and magnetic
background fields. These fields, in general, depend on the space-time
coordinates. Therefore, by expanding the Lagrangian ${\cal L}\left({\cal
F},{\cal G}\right)$ around the background fields and keeping terms up to the
second-order in the propagating field, we get

$\displaystyle{\cal L}^{(2)}\\!$ $\displaystyle=$
$\displaystyle\\!-\frac{1}{4}\,c_{1}\,f_{\mu\nu}^{\,2}-\frac{1}{4}\,c_{2}\,f_{\mu\nu}\widetilde{f}^{\mu\nu}-\frac{1}{2}\,f_{\mu\nu}G_{B}^{\;\;\;\mu\nu}$
(2)
$\displaystyle+\frac{1}{8}\,Q_{B}^{\;\;\;\mu\nu\kappa\lambda}f_{\mu\nu}f_{\kappa\lambda}-J_{\mu}\,a^{\mu}-J_{\mu}\,A_{B}^{\;\;\,\,\mu}\;,$

where the background tensors are defined by

$\displaystyle G_{B}^{\;\;\;\mu\nu}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!c_{1}\,F_{B}^{\,\;\;\mu\nu}+c_{2}\,\widetilde{F}_{B}^{\,\;\;\mu\nu}\;,$
$\displaystyle Q_{B}^{\;\;\,\,\mu\nu\kappa\lambda}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!d_{1}\,F_{B}^{\;\;\,\mu\nu}F_{B}^{\;\;\,\kappa\lambda}+d_{2}\,\widetilde{F}_{B}^{\,\;\;\mu\nu}\widetilde{F}_{B}^{\,\;\;\kappa\lambda}+$
(3)
$\displaystyle+d_{3}\,F_{B}^{\,\;\;\mu\nu}\widetilde{F}_{B}^{\,\;\;\kappa\lambda}+d_{3}\,\widetilde{F}_{B}^{\,\;\;\mu\nu}F_{B}^{\,\;\;\kappa\lambda}\;,\hskip
14.22636pt$

and $J^{\mu}$ is a classical source. By construction,
$G_{B}^{\;\;\;\mu\nu}=-G_{B}^{\;\;\;\nu\mu}$ and the tensor
$Q_{B}^{\;\;\mu\nu\kappa\lambda}$ is antisymmetric under the exchange
$\mu\leftrightarrow\nu$ or $\kappa\leftrightarrow\lambda$, and symmetric in
$\mu\nu\leftrightarrow\kappa\lambda$. The coefficients $c_{1}$, $c_{2}$,
$d_{1}$, $d_{2}$ and $d_{3}$ are evaluated at the background fields ${\bf E}$
, ${\bf B}$ :

$\displaystyle c_{1}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left.\frac{\partial{\cal L}}{\partial{\cal
F}}\right|_{{\bf E},{\bf B}}\;,\;\left.c_{2}=\frac{\partial{\cal
L}}{\partial{\cal G}}\right|_{{\bf E},{\bf B}}\;,\;$ $\displaystyle
d_{1}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left.\frac{\partial^{2}{\cal L}}{\partial{\cal
F}^{2}}\right|_{{\bf E},{\bf B}}\,,\,\left.d_{2}=\frac{\partial^{2}{\cal
L}}{\partial{\cal G}^{2}}\right|_{{\bf E},{\bf
B}}\,,\,\left.d_{3}=\frac{\partial^{2}{\cal L}}{\partial{\cal F}\partial{\cal
G}}\right|_{{\bf E},{\bf B}}\;,\hskip 8.5359pt$ (4)

that, in a general situation, are space-time-dependent.

Using the second-order expanded Lagrangian (2), we give below the energy-
momentum, up to the second order in the photon field strength, $f_{\mu\nu}$,
for a general space-time-dependent background, $F_{B\mu\nu}$. Let us consider
again the Lagrangian (2) whose corresponding field equations are as follows:

$\partial_{\mu}\left[c_{1}\,f^{\mu\nu}+c_{2}\,\widetilde{f}^{\mu\nu}-\frac{1}{2}\,Q_{B}^{\;\;\;\mu\nu\kappa\lambda}f_{\kappa\lambda}\right]=-\partial_{\mu}G_{B}^{\;\;\,\mu\nu}+J^{\nu}\,.$
(5)

The dual tensor $\widetilde{f}^{\mu\nu}$ satisfies the Bianchi identity :
$\partial_{\mu}\widetilde{f}^{\mu\nu}=0$. We contract the field equations with
$f_{\nu\alpha}$, and using the Bianchi identity for $f_{\nu\alpha}$, we obtain
the continuity equation

$\displaystyle\partial_{\mu}\Theta_{ph}^{\;\;\;\;\mu\alpha}=h^{\alpha}\;,$ (6)

where the energy-momentum tensor of the photon field is given by

$\displaystyle\Theta_{ph}^{\;\;\;\;\mu\alpha}\\!$ $\displaystyle=$
$\displaystyle\\!c_{1}\,f^{\mu\nu}f_{\nu}^{\;\;\alpha}-\frac{1}{2}\,Q_{B}^{\;\;\,\,\mu\nu\kappa\lambda}f_{\kappa\lambda}f_{\nu}^{\;\;\alpha}$
(7)
$\displaystyle+\,\eta^{\mu\alpha}\,\left(\,\frac{1}{4}\,c_{1}\,f_{\rho\sigma}^{2}-\frac{1}{8}\,Q_{B}^{\;\;\,\,\rho\sigma\omega\tau}f_{\rho\sigma}f_{\omega\tau}\,\right)\;,\;\;$

and the vector $h^{\alpha}$ is

$\displaystyle h^{\alpha}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!J_{\nu}f^{\nu\alpha}-\left(\partial^{\mu}G_{B\mu\nu}\right)f^{\nu\alpha}+\frac{1}{4}\left(\partial^{\alpha}c_{1}\right)f_{\mu\nu}^{\,2}$
(8)
$\displaystyle+\frac{1}{4}\left(\partial^{\alpha}c_{2}\right)\widetilde{f}_{\mu\nu}f^{\mu\nu}-\frac{1}{8}\,\left(\partial^{\alpha}Q_{B}^{\;\;\,\,\mu\nu\kappa\lambda}\right)f_{\mu\nu}f_{\kappa\lambda}\;.$

Notice that the topological term, the one in $c_{2}$, is cancelled in the
expression for (7); as expected, it does not contribute to the stress-tensor
by virtue of its topological nature. In the general case, the background
fields are non-homogeneous over space and time-dependent. Whenever
$J^{\nu}=0$, the term of $h^{\alpha}$ is not zero and, as consequence, the
components of the energy-momentum tensor are not conserved if the background
fields are neither uniform nor constant in time. If we consider the background
fields to be constant and uniform, $h^{\alpha}$ is vanishing and, in this
case, the energy-momentum tensor (7) satisfies a continuity equation with the
conserved energy density given in what follows below:

$\displaystyle\Theta_{ph}^{\;\;\;\;00}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2}\,c_{1}\left({\bf e}^{2}+{\bf
b}^{2}\right)+\frac{1}{2}\,d_{1}\,\left({\bf e}\cdot{\bf
E}\right)^{2}+\frac{1}{2}\,d_{2}\,\left({\bf e}\cdot{\bf B}\right)^{2}$ (9)
$\displaystyle-\frac{1}{2}\,d_{1}\left({\bf b}\cdot{\bf
B}\right)^{2}-\frac{1}{2}\,d_{2}\left({\bf b}\cdot{\bf
E}\right)^{2}+d_{3}\left({\bf e}\cdot{\bf E}\right)\left({\bf e}\cdot{\bf
B}\right)$ $\displaystyle+d_{3}\left({\bf b}\cdot{\bf E}\right)\left({\bf
b}\cdot{\bf B}\right)\;,$

where all the coefficients depend on the external fields, ${\bf E}$ and ${\bf
B}$. We recover the Maxwell limit by turning off the background fields and by
taking $c_{1}=1$.

The energy density can be written as

$\displaystyle\Theta_{ph}^{\;\;\;\;00}=\frac{1}{2}\,K_{ij}\,e_{i}\,e_{j}\,+\,\frac{1}{2}\,\Lambda_{ij}\,b_{i}\,b_{j}\;,$
(10)

where $K_{ij}$ and $\Lambda_{ij}$ are, respectively, defined by

$\displaystyle K_{ij}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!c_{1}\,\delta_{ij}+\,d_{1}\,E_{i}\,E_{j}+d_{2}\,B_{i}\,B_{j}+$
$\displaystyle+\,d_{3}\left(E_{i}\,B_{j}+E_{j}\,B_{i}\right)\;,$
$\displaystyle\Lambda_{ij}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!c_{1}\,\delta_{ij}-d_{1}\,B_{i}\,B_{j}-d_{2}\,E_{i}\,E_{j}+$
(11) $\displaystyle+\,d_{3}\left(E_{i}\,B_{j}+E_{j}\,B_{i}\right)\;.$

The energy density (10) is positive-definite whenever the eigenvalues of the
symmetric matrices $K_{ij}$ and $\Lambda_{ij}$ are non-negative. Let us
contemplate the case $d_{3}=0$ and assume a purely magnetic background, i. e.,
$E_{i}=0$. These conditions shall actually be the ones we are going to work
with in the forthcoming Sections. Therefore, with these assumptions, the
eigenvalues of $K_{ij}$ are $c_{1}$, $c_{1}+\left(d_{2}-|d_{2}|\right){\bf
B}^{2}/2$ and $c_{1}+\left(d_{2}+|d_{2}|\right){\bf B}^{2}/2$, and the
eigenvalues of $\Lambda_{ij}$ are $c_{1}$, $c_{1}$ and $c_{1}-d_{1}\,{\bf
B}^{2}$, respectively. If $d_{2}<0$ or $d_{2}>0$, to ensure positive
eigenvalues, the following conditions should be fulfilled:

$\displaystyle c_{1}>0\;\;,\;\;c_{1}-d_{1}\,{\bf
B}^{2}>0\;\;\mbox{and}\;\;c_{1}+d_{2}\,{\bf B}^{2}>0\;.$ (12)

To elaborate more on the positivity of the energy density, let us recall that
a general non-linear Lagrangian can be expanded as an asymptotic series in
powers of ${\cal F}$ and G :

$\displaystyle{\cal L}=a_{ij}\,{\cal F}^{i}\,{\cal
G}^{j}\;,\;i,j=0,1,2,...\;.$ (13)

MED corresponds to $i=1,j=0$. However, except for $a_{10}=1$, all the
coefficients of the expansion above are small, for they describe tiny non-
linear effects, even for strong external fields, such as, for example,
magnetic fields in the neighborhood of magnetized astrophysical objects. So,
in the energy density (9) and in (II), $c_{1}=1+\delta_{1}$, with
$\delta_{1}\ll 1$, since the latter stems from the coefficients $a_{ij}$ that
extend the Maxwellian version. We are then arguing that, in (9) and (II),
$\delta_{1}$ and the coefficients $d_{i}$ correspond all to tiny corrections,
so that it is expected that the eigenvalues above are, in a wide range of
situations, but not generally, are all positive. In these cases, the energy
density (9) is consequently non-negative, once the contribution given by the
Maxwellian term, ${\bf e}^{2}+{\bf b}^{2}$, dominates over the other terms.
Nevertheless, we shall consider, further on, specific cases of non-linear
models and the argument above may not work if the external fields become
stronger than some critical value. In all situations, we are going to point
out critical values of the external magnetic fields above which the average
energy density of plane waves become negative, which is, to our sense, void of
physical meaning. Therefore, for all models we shall discuss, we will be bound
to consider external fields below the critical values we are going to derive,
so as to undertake that the average energy density of the radiation be
positive.

After the previous considerations, in the incoming Sections IV and V, we are
going to apply these results and conditions to the particular cases of
Hoffman-Infeld, the generalized Born-Infeld, Logarithm and Euler-Heisenberg
electrodynamics.

##  III Photon dispersion relations and the kinematics of the Compton effect

In this Section, we consider the expansion, up to second order in
$f^{\mu\nu}$, to compute the dispersion relations and the propagation of
photons in presence of external electromagnetic fields. Writing the components
of the field-strength tensors in terms of the photon electric and magnetic
fields, $\left(\,{\bf e}\,,\,{\bf b}\,\right)$, the background given by
$\left(\,{\bf E}\,,\,{\bf B}\,\right)$, and the source components
$J^{\mu}=\left(\,\rho\,,\,{\bf J}\,\right)$, the field equations (5) read as
follows below:

$\displaystyle\nabla\cdot{\bf e}+\left(\frac{d_{1}}{c_{1}}\,{\bf
E}+\frac{d_{3}}{c_{1}}\,{\bf B}\right)\cdot\nabla({\bf E}\cdot{\bf e}-{\bf
B}\cdot{\bf b})+$ $\displaystyle+\,\left(\frac{d_{2}}{c_{1}}\,{\bf
B}+\frac{d_{3}}{c_{1}}\,{\bf E}\right)\cdot\nabla({\bf B}\cdot{\bf e}+{\bf
E}\cdot{\bf b})=$ $\displaystyle=-\nabla\cdot{\bf
E}+\frac{\rho}{c_{1}}\;,\;\;\;\;\;$ (14a) $\displaystyle\nabla\times{\bf
e}+\partial_{t}{\bf b}={\bf 0}\;\;,\;\;\nabla\cdot{\bf b}=0\;,\;\hskip
28.45274pt$ (14b) $\displaystyle\nabla\times{\bf
b}+\left(-\frac{d_{1}}{c_{1}}\,{\bf B}+\frac{d_{3}}{c_{1}}\,{\bf
E}\right)\times\nabla({\bf E}\cdot{\bf e}-{\bf B}\cdot{\bf b})$
$\displaystyle+\left(\frac{d_{2}}{c_{1}}\,{\bf E}-\frac{d_{3}}{c_{1}}\,{\bf
B}\right)\times\nabla({\bf B}\cdot{\bf e}+{\bf E}\cdot{\bf
b})=\partial_{t}{\bf e}+\hskip 14.22636pt$
$\displaystyle+\left(\frac{d_{1}}{c_{1}}\,{\bf E}+\frac{d_{3}}{c_{1}}\,{\bf
B}\right)\,\partial_{t}({\bf E}\cdot{\bf e}-{\bf B}\cdot{\bf b})+\hskip
14.22636pt$ $\displaystyle+\left(\frac{d_{2}}{c_{1}}\,{\bf
B}+\frac{d_{3}}{c_{1}}\,{\bf E}\right)\,\partial_{t}({\bf B}\cdot{\bf e}+{\bf
E}\cdot{\bf b})-\nabla\times{\bf B}+\,\frac{{\bf J}}{c_{1}}\;.\hskip
11.38092pt$ (14c)

Since the Bianchi identity remains valid in non-linear electrodynamics, the
divergent of ${\bf b}$ and the rotational of ${\bf e}$ keep like in Maxwell
electrodynamics.

The particular case of a Lagrangian that depends only on the invariant ${\cal
F}$ in the presence of the background fields ${\bf E}$ and ${\bf B}$, we have
$c_{1}\neq 0$, $d_{1}\neq 0$ and $c_{2}=d_{2}=d_{3}=0$. The simplest case is
Maxwell electrodynamics, where the Lagrangian is given by ${\cal F}$, the
first coefficient is $c_{1}=1$, and the other coefficients of the expansion
are all vanishing. We shall consider in this paper the case of a purely
magnetic background, i. e., ${\bf E}=0$. Whenever the non-linear model also
exhibits dependence on ${\cal G}$, this dependence must be quadratic (or an
even power) in ${\cal G}^{2}$ to insure the charge-parity symmetry. This fact
happens in non-linear electrodynamics such as the generalized Born-Infeld,
Logarithm and ArcSinh theories. If there is no electric background, we obtain
$c_{2}=0$ and the CP-symmetry is recovered. We work with a uniform external
magnetic field, ${\bf B}$, and, as consequence, the coefficients are also
uniform and constant in time. Other important fact is that whenever the
electric background field is not present, $d_{3}=0$ for all the examples of
non-linear electrodynamics in the literature. Thereby, using that $d_{3}=0$,
the equations (14a) - (14c) with no classical sources read as below :

$\displaystyle\nabla\cdot{\bf e}+\,\frac{d_{2}}{c_{1}}\,{\bf
B}\cdot\nabla\left({\bf B}\cdot{\bf e}\right)=0\;,$ (15a)
$\displaystyle\nabla\times{\bf e}+\partial_{t}{\bf b}={\bf 0}\hskip
8.5359pt,\hskip 8.5359pt\nabla\cdot{\bf b}=0\;,\hskip 8.5359pt$ (15b)
$\displaystyle\nabla\times{\bf b}+\frac{d_{1}}{c_{1}}\,{\bf
B}\times\nabla\left({\bf B}\cdot{\bf b}\right)=\partial_{t}{\bf e}+$
$\displaystyle+\frac{d_{2}}{c_{1}}\,{\bf B}\,\partial_{t}\left({\bf
B}\cdot{\bf e}\right)\;.\hskip 14.22636pt$ (15c)

The usual Maxwell equations are obtained for $d_{1}=d_{2}=0$ and $c_{1}=1$,
which is equivalent to taking $|{\bf B}|\rightarrow 0$ in (15a) and (15c).

Considering plane wave solutions, ${\bf e}({\bf x},t)={\bf
e}_{0}\,e^{i\,\left({\bf k}\cdot{\bf x}-\omega t\right)}$ and ${\bf b}({\bf
x},t)={\bf b}_{0}\,e^{i\,\left({\bf k}\cdot{\bf x}-\omega t\right)}$ in (15a),
(15b) and (15c), the relation between the frequency, $\omega$, and the wave
vector, ${\bf k}$, can be written in a matrix form:

$\displaystyle M_{ij}\,e_{0j}=0\;,$ (16)

where $e_{0j}\,(j=1,2,3)$ are the components of the amplitude of the electric
field, ${\bf e}_{0}$. The matrix elements $M_{ij}$ take the form

$M_{ij}=\alpha\,\delta_{ij}+u_{i}\,v_{j}+w_{i}\,B_{j}\;,$ (17)

where the coefficients $\alpha$, $u_{i}$, $v_{i}$ and $w_{i}$ are,
respectively, defined by

$\displaystyle\alpha\\!$ $\displaystyle=$ $\displaystyle\\!\omega^{2}-{\bf
k}^{2}\hskip 5.69046pt,\hskip 5.69046pt$ $\displaystyle{\bf u}$
$\displaystyle=$ $\displaystyle\frac{d_{1}}{c_{1}}\,{\bf B}\times{\bf
k}\;,\;{\bm{v}}={\bf B}\times{\bf k}\;,$ $\displaystyle{\bm{w}}\\!$
$\displaystyle=$ $\displaystyle\\!\frac{d_{2}}{c_{1}}\,\omega^{2}\,{\bf
B}-\frac{d_{2}}{c_{1}}\,\left({\bf B}\cdot{\bf k}\right){\bf k}\;.$ (18)

The matrix equation (16) has non-trivial solutions only if the $M$-matrix is
singular. It can be cast in the form

$\displaystyle\mbox{det}M=\alpha\left[\left(\alpha+{\bf u}\cdot{\bf
v}\right)\left(\alpha+{\bf w}\cdot{\bf B}\right)-\left({\bf u}\cdot{\bf
B}\right)\left({\bf v}\cdot{\bf w}\right)\right]\;,\;\;\;$ (19)

and the condition $\mbox{det}M=0$ leads to the usual photon dispersion
relation $\omega^{2}=|{\bf k}|^{2}$ as one of the solutions. This is so by
virtue of gauge invariance, which, in a particle scenario, corresponds to the
presence of the genuine (zero mass) photon. Along with this possibility, there
appear other solution as the zeroes of the polynomial equation that follows:

$\displaystyle P\,\omega^{4}+Q\,\omega^{2}+R=0\;,$ (20)

where

$\displaystyle P\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!1+\frac{d_{2}}{c_{1}}\,{\bf B}^{2}\;,$ $\displaystyle
Q\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!-2{\bf
k}^{2}+\frac{d_{1}}{c_{1}}\,\left({\bf B}\times{\bf
k}\right)^{2}-\frac{d_{2}}{c_{1}}\,\left[{\bf B}^{2}\,{\bf k}^{2}+\left({\bf
B}\cdot{\bf k}\right)^{2}\right]$
$\displaystyle+\frac{d_{1}d_{2}}{c_{1}^{2}}\,{\bf B}^{2}\left({\bf
B}\times{\bf k}\right)^{2}\;,$ $\displaystyle R\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!{\bf k}^{4}-\frac{d_{1}}{c_{1}}\,{\bf k}^{2}\left({\bf
B}\times{\bf k}\right)^{2}+\frac{d_{2}}{c_{1}}\,{\bf k}^{2}\left({\bf
B}\cdot{\bf k}\right)^{2}$ (21)
$\displaystyle-\frac{d_{1}d_{2}}{c_{1}^{2}}\,\left({\bf B}\cdot{\bf
k}\right)^{2}\left({\bf B}\times{\bf k}\right)^{2}\;.$

The roots of (20) are $\omega_{1}^{(\pm)}=\pm\,\omega_{1}({\bf k})$ and
$\omega_{2}^{(\pm)}=\pm\,\omega_{2}({\bf k})$, whose frequencies are shown
below:

$\displaystyle\omega_{1}({\bf k})\\!$ $\displaystyle=$ $\displaystyle\\!|{\bf
k}|\,\sqrt{1-\frac{d_{1}}{c_{1}}\,({\bf B}\times\hat{{\bf k}})^{2}}\;,$ (22a)
$\displaystyle\omega_{2}({\bf k})\\!$ $\displaystyle=$ $\displaystyle\\!|{\bf
k}|\,\sqrt{1-\frac{d_{2}\,({\bf B}\times\hat{{\bf k}})^{2}}{c_{1}+d_{2}\,{\bf
B}^{2}}}\;.$ (22b)

The usual photon frequencies are recovered in the limit $d_{1}\rightarrow 0$
and $d_{2}\rightarrow 0$, or, equivalently, whenever $|{\bf B}|\rightarrow 0$.
Notice that, if the non-linear theory only depends on the ${\cal
F}$-invariant, $d_{2}=0$ and the second solution recovers the usual dispersion
relation. The frequencies (22a) and (22b) are real if $c_{1}>d_{1}({\bf
B}\times\hat{{\bf k}})^{2}$ and $c_{1}+d_{2}({\bf B}\cdot\hat{{\bf
k}})^{2}>0$, respectively. The (magnetized) vacuum refraction index is given
as the inverse of the phase velocity, $\omega_{i}/|{\bf k}|\,(i=1,2)$ :

$\displaystyle n_{1}^{-1}\\!$ $\displaystyle=$
$\displaystyle\\!\sqrt{1-\frac{d_{1}}{c_{1}}\,({\bf B}\times\hat{{\bf
k}})^{2}}\;,$ (23a) $\displaystyle n_{2}^{-1}\\!$ $\displaystyle=$
$\displaystyle\\!\sqrt{1-\frac{d_{2}\,({\bf B}\times\hat{{\bf
k}})^{2}}{c_{1}+d_{2}\,{\bf B}^{2}}}\;.$ (23b)

From the De Broglie duality correspondence, the energy-momentum relations for
the propagating excitations read as below:

$\displaystyle E_{1}^{2}\\!$ $\displaystyle=$ $\displaystyle\\!{\bf
p}^{2}\left[\,1-\frac{d_{1}}{c_{1}}\,({\bf B}\times\hat{{\bf
p}})^{2}\,\right]\;,$ (24a) $\displaystyle E_{2}^{2}\\!$ $\displaystyle=$
$\displaystyle\\!{\bf p}^{2}\left[\,1-\frac{d_{2}\,({\bf B}\times\hat{{\bf
p}})^{2}}{c_{1}+d_{2}\,{\bf B}^{2}}\,\right]\;.$ (24b)

The group velocity associated with the previous frequencies can be read off
from the equations cast in what follows:

$\displaystyle\left.{\bf v}_{g}\right|_{\omega=\omega_{1}}\\!$
$\displaystyle=$ $\displaystyle\\!\frac{c_{1}\,\hat{{\bf k}}+d_{1}\,{\bf
B}\times({\bf B}\times\hat{{\bf k}})}{c_{1}\,\sqrt{1-\frac{d_{1}}{c_{1}}({\bf
B}\times\hat{\bf k})^{2}}}\;,$ (25a) $\displaystyle\left.{\bf
v}_{g}\right|_{\omega=\omega_{2}}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{\hat{{\bf k}}\,c_{1}+d_{2}\,{\bf B}\,({\bf
B}\cdot\hat{{\bf k}})}{(c_{1}+d_{2}\,{\bf B}^{2})\sqrt{1-\frac{d_{2}({\bf
B}\times\hat{\bf k})^{2}}{c_{1}+d_{2}{\bf B}^{2}}}}\;.$ (25b)

The group velocity vectors have components in the directions of $\hat{{\bf
k}}$ and $\hat{{\bf B}}$. Both results go to ${\bf v}_{g}=\hat{{\bf
k}}\,\omega/|{\bf k}|$, in the limit $|{\bf B}|\rightarrow 0$, or if we
consider the Maxwellian limit. If the magnetic background, ${\bf B}$, is
perpendicular to the direction of propagation, $\hat{{\bf k}}$, the solutions
also reduce to ${\bf v}_{g}=(1-d_{1}\,{\bf B}^{2}/c_{1})^{1/2}\hat{{\bf k}}$
and ${\bf v}_{g}=(1+d_{2}\,{\bf B}^{2}/c_{1})^{-1/2}\hat{{\bf k}}$,
respectively.

The Compton effect is a scattering process in which the dispersion relations
(24a) and (24b) can be applied to study the increasing of the photon
wavelength after being scattered by the electron, taken as the target, in
presence of a magnetic background. The photon initial state has wavelength
$\lambda=1/|{\bf p}|$, with energy $E$, where the relation between $E$ and the
momentum, ${\bf p}$, must now follow from (24a) or (24b). The physical
scenario consists of a photon that propagates in the magnetic background and
collides with an electron in an atom at rest. So, before the photon hits the
atom, we consider that the external magnetic field affects only the photon
dispersion relation. The collision causes the electron to recoil with a given
energy and momentum. Then, after the scattering process takes place, the
outgoing electron couples to the magnetic field and, clearly, the electron
dispersion relation is no longer of a free electron. But, we are focusing on
the wavelength shift and scattering angle of the photon; this is why we are
not discussing the effect of the magnetic field on the dispersion relation of
the emergent electron. The setup corresponding to the Compton effect whose
kinematics we are studying here is depicted in Figure 1.

[coilaspect=0,coilarm=0,coilwidth=0.25,coilheight=1.3,linecolor=black](-2.3,1)(-0.5,1)
$(E,{\bf
p})$[coilaspect=0,coilarm=0,coilwidth=0.25,coilheight=1.3,linecolor=black](0,1)(2,3)
$(E^{\prime},{\bf p}^{\prime})$$\theta_{c}$atom$e^{-}$${\bf B}$ Figure 1:
Setup for the study of the kinematics of the Compton effect. We consider the
external magnetic field orthogonal to the plane of the Compton array and
coming out from the plane.

After the collision, the photon trajectory is deviated by an angle
$\theta_{c}$, with wavelength $\lambda^{\prime}=1/|{\bf p}^{\prime}|$ and
energy $E^{\prime}$. From the energy and linear momentum conservation, the
variation of the photon wavelength, after the collision process, turns out to
be

$\lambda_{i}^{\prime}-\lambda=2\,\lambda_{e}\sin^{2}\left(\frac{\theta_{c}}{2}\right)+a_{i}\left[\lambda_{i}^{\prime}-\lambda+\frac{\lambda_{e}}{2}\frac{\left(\lambda_{i}^{\prime}-\lambda\right)^{2}}{\lambda_{i}^{\prime}\,\lambda}\right]\;,$
(26)

where $\lambda_{e}=m_{e}^{-1}=2\,\left(\mbox{MeV}\right)^{-1}$ is the Compton
wavelength of the electron, $a_{i}\,(i=1,2)$ means $a_{1}:=\left|\,{\bf
B}\times\hat{{\bf p}}\,\right|^{2}d_{1}/c_{1}$ and $a_{2}:=d_{2}\left|\,{\bf
B}\times\hat{{\bf p}}\,\right|^{2}/\left(c_{1}+d_{2}{\bf B}^{2}\right)$ for
both the cases of (24a) and (24b), respectively, and $\lambda_{i}^{\prime}$
are the wavelengths for both cases $i=1,2$ after the collision. We take the
initial photon wavelength in the range of the X-ray spectrum
$\lambda=1.52\times\left(\,10^{10}-10^{13}\,\right)\,\mbox{MeV}^{-1}$. The
corresponding solutions to (26) yielding the final wavelength of the photon
are given by

$\displaystyle\lambda_{i}^{\prime(\pm)}$ $\displaystyle=$
$\displaystyle\frac{\lambda+2\lambda_{e}\sin^{2}(\theta_{c}/2)-a_{i}\left(\lambda+\lambda_{e}\right)}{2-2a_{i}-a_{i}\lambda_{e}/\lambda}\,\pm\,\frac{\sqrt{\left(\lambda+2\lambda_{e}\sin^{2}(\theta_{c}/2)-a_{i}\lambda-
a_{i}\lambda_{e}\right)^{2}+a_{i}\lambda_{e}\left(2\lambda-2a_{i}\lambda-
a_{i}\lambda_{e}\right)}}{2-2a_{i}-a_{i}\lambda_{e}/\lambda}\;.\;\;$ (27)

If we assume $\lambda\gg\lambda_{e}$ in (27) , the real and positive solutions
lead us to the two wavelength variations below:

$\displaystyle\Delta\lambda_{1}\\!$ $\displaystyle\simeq$
$\displaystyle\\!2\lambda_{e}\sin^{2}\left(\frac{\theta_{c}}{2}\right)\left[\,1-\frac{d_{1}}{c_{1}}\,|{\bf
B}\times\hat{{\bf p}}|^{2}\,\right]^{-1},$ (28a)
$\displaystyle\Delta\lambda_{2}\\!$ $\displaystyle\simeq$
$\displaystyle\\!2\lambda_{e}\sin^{2}\left(\frac{\theta_{c}}{2}\right)\left[\,1-\frac{d_{2}\,|{\bf
B}\times\hat{{\bf p}}|^{2}}{c_{1}+d_{2}\,{\bf B}^{2}}\,\right]^{-1},$ (28b)

that are positive if $c_{1}>d_{1}\,|{\bf B}\times\hat{{\bf p}}|^{2}$ and
$c_{1}+d_{2}\left({\bf B}\cdot\hat{{\bf p}}\right)^{2}>0$. Whenever
$d_{1},d_{2}\rightarrow 0$, the standard variation of the photon wavelength in
the Compton effect is recovered. The non-linear contribution depends on the
external magnetic field, ${\bf B}$, and the parameters $c_{1}$, $d_{1}$ and
$d_{2}$. These parameters, in turn, also depend on the external magnetic field
and the specific dependence is governed by the non-linear electrodynamics
under consideration.

##  IV  Example of an ${\cal F}$-dependent electrodynamic model

The Hoffmann-Infeld (HI) model is an example of non-linear electrodynamics
with interesting application in the study of special black-hole solutions
Aiello . The Lagrangian is given by

$\displaystyle{\cal L}_{HI}({\cal
F})=\frac{\beta^{2}}{4}\,\left[\,1-\eta({\cal F})-\ln\eta({\cal
F})\,\right]\;,$ (29)

where $\eta({\cal F})$ is defined by

$\displaystyle\eta({\cal F})=\frac{4{\cal
F}}{\beta^{2}-\beta\,\sqrt{\beta^{2}-8{\cal F}}}\;.$ (30)

The parameter $\beta$ is introduced to guarantee a finite electrostatic field
configuration in the case the particle-like charges. Maxwell Electrodynamics
is recovered when $\beta\rightarrow\infty$. In this Section, since the model
depends exclusively on the invariant ${\cal F}$, the coefficient vanishes,
$d_{2}=0$. Thereby, the non-trivial dispersion relation in a magnetic
background corresponds to (24a), which depends on $d_{1}$ and $c_{1}$. The
non-trivial coefficients, $c_{1}$ and $d_{1}$, are given by

$\displaystyle\left.c_{1}^{HI}\right|_{{\bf E}=0,{\bf B}}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{\beta}{4{\bf B}^{2}}\frac{-\beta^{2}+2{\bf
B}^{2}+\beta\,\sqrt{\beta^{2}+4{\bf B}^{2}}}{\sqrt{\beta^{2}+4{\bf
B}^{2}}}\;,\hskip 19.91684pt$ (31a)
$\displaystyle\left.d_{1}^{HI}\right|_{{\bf E}=0,{\bf B}}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{\beta}{{\bf B}^{2}}\frac{2{\bf
B}^{2}-3\beta^{2}+2\beta\sqrt{\beta^{2}+4{\bf B}^{2}}}{\left(\beta^{2}+4{\bf
B}^{2}\right)^{3/2}}$ (31b) $\displaystyle-\frac{\beta^{4}}{2{\bf
B}^{4}}\frac{\beta-\sqrt{\beta^{2}+4{\bf B}^{2}}}{\left(\beta^{2}+4{\bf
B}^{2}\right)^{3/2}}\;.$

The corresponding refraction index as function of the angle $\theta$ between
the magnetic background and the direction of propagation, $\hat{{\bf k}}$, is
shown in figure (2). We choose the following values for the magnetic
background: $|{\bf B}|=1.0\,\mbox{MeV}^{2}$ (black line), $|{\bf
B}|=4.0\,\mbox{MeV}^{2}$ (blue line) and $|{\bf B}|=8.0\,\mbox{MeV}^{2}$ (red
line).

Figure 2: The vacuum refraction index of the HI ED appears as a function of
the $\theta$-angle between ${\bf B}$ and the direction of $\hat{{\bf k}}$. We
choose $\beta=10.0\,\mbox{MeV}^{2}$, for the values $|{\bf
B}|=1.0\,\mbox{MeV}^{2}$, $|{\bf B}|=4.0\,\mbox{MeV}^{2}$ and $|{\bf
B}|=6.0\,\mbox{MeV}^{2}$.

For a strong magnetic background, i. e., $|{\bf B}|\gg\beta$, the refraction
index of the HI model is $n_{HI}=|\sec\theta|$; this refraction index vanishes
if ${\bf B}$ is perpendicular to the direction $\hat{{\bf k}}$, and $n_{HI}=1$
if ${\bf B}$ is parallel to $\hat{{\bf k}}$.

The components of the group velocity (25b) for the HI ED as functions of
$\theta$-angle are plotted in the figure (3). We choose
$\beta=50.0\mbox{MeV}^{2}$ and $|{\bf B}|=10.0\mbox{MeV}^{2}$ in this case.

Figure 3: The components of the group velocity as functions of $\theta$ in the
HI ED. The black line is the velocity component in the direction of $\hat{{\bf
k}}$, while the red line stands for the $\hat{{\bf B}}$ component.

The black line stands for the component in the direction of $\hat{{\bf k}}$,
whereas the red line represents the component in the direction of magnetic
field $\hat{{\bf B}}$. This component is negative for the range
$\pi/2\leq\theta\leq\pi$. When $|{\bf B}|\gg\beta$, the group velocity has the
behaviour

$\displaystyle{\bf v}_{gHI}\simeq-\frac{\beta}{|{\bf
B}|}\,|\sec\theta|\,\hat{{\bf k}}+\mbox{sgn}(\cos\theta)\,\hat{{\bf B}}\;.$
(32)

The next analysis refers to the energy density of the HI ED in the presence of
the background, ${\bf B}$. We use the result (9) with the coefficients (31a)
and (31b). Under the conditions (12), the energy density in the HI ED is
positive if $|{\bf B}|<0.972\,\beta$, or $|{\bf B}|>\sqrt{2}\,\beta$. To
illustrate the energy density (9), we consider the case in which the plane
wave for the electric and magnetic fields ${\bf e}({\bf x},t)={\bf
e}_{0}\,e^{i\,\left({\bf k}\cdot{\bf x}-\omega t\right)}$ and ${\bf b}({\bf
x},t)={\bf b}_{0}\,e^{i\,\left({\bf k}\cdot{\bf x}-\omega t\right)}$,
respectively, propagate in a medium with an uniform and constant magnetic
background. We obtain thereby the time average of the energy density of the HI
ED per unit of the squared electric field :

$\frac{\langle\Theta_{ph}^{00}\rangle^{HI}}{{\bf
e}_{0}^{2}}=\frac{1}{4}\,c_{1}^{HI}\left[1+\frac{{\bf
k}^{2}}{\omega^{2}}-\frac{d_{1}^{HI}}{c_{1}^{HI}}\frac{{\bf
k}^{2}}{\omega^{2}}\left(\hat{{\bf e}}_{0}\cdot(\hat{{\bf k}}\times{\bf
B})\right)^{2}\right]\;.\;\;\;$ (33)

Notice that this result depends on the frequency solutions of $\omega$, that
in the HI case just (22a) depend on the external magnetic field. We consider
the vectors $\hat{{\bf k}}$, $\hat{{\bf e}}_{0}$ and ${\bf B}$ perpendicular
to each other in this analysis, and we also choose $\beta=50\mbox{MeV}^{2}$.
Under these conditions, the result (33) is plotted as function of the magnetic
background in the figure (4).

Figure 4: The average (in time) of the energy density per unit of the squared
amplitude of the electric field versus the external magnetic field in the HI
ED. We choose $\beta=50\,\mbox{MeV}^{2}$, and the vectors $\hat{{\bf k}}$,
$\hat{{\bf e}}_{0}$ and ${\bf B}$ are perpendicular to one another.

In this case, the energy density is positive for all values of the ${\bf B}$
magnitude, and goes to zero if $|{\bf B}|\rightarrow\infty$. The limit $|{\bf
B}|\rightarrow 0$ recovers the known result in Maxwell ED.

##  V Some ${\cal F}$\- and ${\cal G}$-dependent electrodynamic models

Many examples of non-linear electrodynamics with dependence on both ${\cal F}$
and ${\cal G}$ are discussed in the literature. In these cases, the
coefficient $d_{2}\neq 0$ and the two frequency solutions, (22a) and (22b),
must be considered. The models, in general, depend on ${\cal G}^{2}$ (simplest
case) or an even power of ${\cal G}$, so as to ensure CP-invariance.

Let us start off by contemplating the well-known case of the Euler-Heisenberg
(EH) electrodynamics, described by the effective Lagrangian

$\displaystyle{\cal L}_{EH}({\cal F},{\cal G})={\cal
F}-\frac{1}{8\pi^{2}}\int_{0}^{\infty}\frac{ds}{s^{3}}\,\,e^{-m^{2}\,s}\,\times\hskip
14.22636pt$ $\displaystyle\times\,\left[(es)^{2}\,{\cal
G}\,\frac{\Re\cosh\left(es\sqrt{-{\cal F}+i\,{\cal
G}}\right)}{\Im\cosh\left(es\sqrt{-{\cal F}+i\,{\cal
G}}\right)}+\frac{2}{3}(es)^{2}{\cal F}-1\right]\;,\;\;\;$ (34)

where $\Re$ and $\Im$ stand for the real and imaginary parts, respectively,
and $m=0.5\,\mbox{MeV}$ is the electron mass. In the weak field approximation,
the EH Lagrangian (V) is reduced to the form

$\displaystyle{\cal L}_{EH}({\cal F},{\cal G})\simeq{\cal
F}+\frac{2\alpha^{2}}{45m^{4}}\left(\,4\,{\cal F}^{2}\,+\,7\,{\cal
G}^{2}\,\right)\;,$ (35)

in which $\alpha=e^{2}=(137)^{-1}=0.00729$ is the fine structure constant.
Below, we quote the coefficients $c_{1}$, $d_{1}$ and $d_{2}$ of the expansion
corresponding to the truncation given by (35) :

$\displaystyle\left.c_{1}^{EH}\right|_{{\bf E}=0,{\bf B}}\\!$ $\displaystyle=$
$\displaystyle\\!1-\frac{8\alpha^{2}\,{\bf B}^{2}}{45m^{4}}\;,\;$
$\displaystyle\left.d_{1}^{EH}\right|_{{\bf E}=0,{\bf B}}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{16\alpha^{2}}{45m^{4}}\;,$
$\displaystyle\left.d_{2}^{EH}\right|_{{\bf E}=0,{\bf B}}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{28\alpha^{2}}{45m^{4}}\;.$ (36)

Using the results (22a) and (22b) with the coefficients (V), the two solutions
for the frequencies are

$\displaystyle\omega_{1}^{EH}({\bf k})\\!\\!$ $\displaystyle\simeq$
$\displaystyle\\!\\!|{\bf k}|\left[1-\frac{8\alpha^{2}}{45m^{4}}({\bf
B}\times\hat{{\bf k}})^{2}\right]\;,$ (37a) $\displaystyle\omega_{2}^{EH}({\bf
k})\\!\\!$ $\displaystyle\simeq$ $\displaystyle\\!\\!|{\bf
k}|\left[1-\frac{14\alpha^{2}}{45m^{4}}({\bf B}\times\hat{{\bf
k}})^{2}\right]\;,$ (37b)

and, then, the corresponding vacuum refraction index for the EH effective
model, in the approximation we are working, turn out given by

$\displaystyle n_{1}^{EH}\simeq 1+\frac{8\alpha^{2}}{45m^{4}}({\bf
B}\times\hat{{\bf k}})^{2}\;,\hskip 14.22636pt$ (38a) $\displaystyle
n_{2}^{EH}\simeq 1+\frac{14\alpha^{2}}{45m^{4}}({\bf B}\times\hat{{\bf
k}})^{2}\;.\hskip 14.22636pt$ (38b)

It is worthy to highlight that these results are in agreement with Reference
Adler71  after suitable changes in the unit system.

The second case of a ED non-linear is the generalized Born-Infeld (BI)
Lagrangian Gaete2 ,

$\displaystyle{\cal L}_{BI}({\cal F},{\cal
G})=\beta^{2}\left[\,1-\left(1-2\frac{{\cal F}}{\beta^{2}}-\frac{{\cal
G}^{2}}{\beta^{4}}\right)^{p}\,\right]\;,$ (39)

where $\beta$ is a scale parameter with dimension of squared energy (in
natural units), and $p$ is a real parameter that satisfies $0<p<1$. The usual
Born-Infeld theory is obtained for $p=1/2$. For $\beta\gg(|{\bf E}_{0}|,|{\bf
B}_{0}|)$, the Lagrangian (39) leads to

${\cal L}_{BI}\simeq 2\,p\,{\cal
F}+\frac{1}{\beta^{2}}\left[\phantom{\frac{1}{2}}\\!\\!\\!2p\left(1-p\right)\,{\cal
F}^{2}+p\,{\cal G}^{2}\,\right]\;.\;\;$ (40)

Here, we recall that Maxwell electrodynamics is recovered in the limit
$\beta\rightarrow\infty$, and $p=1/2$.

Before proceeding with the calculations of the dispersion relations, it is
interesting to consider the electrostatic case, with ${\bf B}_{0}=0$, the
corresponding field equation in presence of charges is

$\displaystyle\nabla\cdot{\bf D}_{0}=\rho\;,$ (41)

where $\rho$ denotes the charge density, and ${\bf D}_{0}$ is defined by

$\displaystyle{\bf D}_{0}=\frac{{\bf E}_{0}}{\left(1-{\bf
E}_{0}^{\,2}/\beta^{2}\right)^{1-p}}\;.$ (42)

In the point-like particle case, $\rho({\bf r})=Q\,\delta^{3}({\bf r})$,
equation (41) yields ${\bf D}_{0}=\hat{{\bf r}}\,Q/r^{2}$. So, the solutions
of the electric field in (41) are difficult to obtain due to the polynomial
equation with degree $(1-p)^{-1}$. The case $p=3/4$ is chosen and the solution
for the electric field is Gaete2

$\displaystyle{\bf E}_{0}=\frac{\sqrt{3}\,\beta\,Q\,\hat{{\bf
r}}}{\sqrt{Q^{2}+\sqrt{Q^{4}+9\beta^{4}r^{8}}}}\;.$ (43)

The magnitude of the electrostatic field goes to zero whenever
$r\rightarrow\infty$. In the limit $r\rightarrow 0$, the electric field is
finite at the origin, i. e., $E_{0}(r=0)=\sqrt{3/2}\,\beta\,\mbox{sgn}(Q)$, in
which $\mbox{sgn}(Q)$ denotes the signal function. If the charge is positive,
the electric field does not blow up at the charge position and the maximum
value it reaches is $\left.E_{0}\right|_{max}=\sqrt{3/2}\,\beta$. Otherwise,
if the charge is negative, the electric field has a minimum at
$\left.E_{0}\right|_{min}=-\sqrt{3/2}\,\beta$. The electric potential
associated with (43) is given by

$\displaystyle
V_{BI}(r)=\frac{3Q}{4(27)^{1/4}}\sqrt{\frac{\beta}{2Q}}\left\\{\frac{64\sqrt{2\pi}}{15}\frac{\Gamma(9/4)\cos(\pi/8)}{\Gamma(3/4)}\right.$
$\displaystyle\left.+\frac{4}{3}\left[\,\frac{\sin\phi_{0}\,\cos\phi_{0}}{\cos^{2}\left(\phi_{0}/2\right)}\,\right]^{1/4}\\!\\!-\frac{16}{3}\left[\,\frac{\sin\phi_{0}}{\cos^{2}\left(\phi_{0}/2\right)}\,\right]^{1/4}\times\right.$
$\displaystyle\left.\times\,_{2}F_{1}\left(\frac{1}{8},\frac{3}{4},\frac{9}{8},\tan^{2}\frac{\phi_{0}}{2}\right)\right\\}\;,\hskip
14.22636pt$ (44)

in which $\phi_{0}=\tan^{-1}(3\beta^{2}r^{4}/Q^{2})$. In addition, this
potential reduces to the Maxwellian case for $\beta\rightarrow\infty$. Using
the result (V), the electrostatic potential of an electron, with $Q=e=0.085$,
evaluated at the origin $r\rightarrow 0$ is finite given by

$\displaystyle\lim_{r\rightarrow 0}V_{BI}(r)=0.62\,\sqrt{\beta}\;.$ (45)

Now, returning to the dispersion relation analysis, the derivations from (II)
in the Lagrangian (39) yield the coefficients $c_{1}$, $d_{1}$ and $d_{2}$ for
the generalized BI theory:

$\displaystyle\left.c_{1}^{BI}\right|_{{\bf E}=0,{\bf B}}$ $\displaystyle=$
$\displaystyle\frac{2p}{(1+{\bf B}^{2}/\beta^{2})^{1-p}}\;,\;$
$\displaystyle\left.d_{1}^{BI}\right|_{{\bf E}=0,{\bf B}}$ $\displaystyle=$
$\displaystyle\frac{4p(1-p)}{\beta^{2}(1+{\bf B}^{2}/\beta^{2})^{2-p}}\;,$
$\displaystyle\left.d_{2}^{BI}\right|_{{\bf E}=0,{\bf B}}$ $\displaystyle=$
$\displaystyle\frac{2p}{\beta^{2}(1+{\bf B}^{2}/\beta^{2})^{1-p}}\;.$ (46)

The dispersion relations in this case are cast below:

$\displaystyle\omega_{1}^{BI}({\bf k})\\!$ $\displaystyle=$
$\displaystyle\\!|{\bf k}|\,\,\sqrt{1-2(1-p)\,\frac{({\bf B}\times\hat{{\bf
k}})^{2}}{{\bf B}^{2}+\beta^{2}}}\;,$ (47a) $\displaystyle\omega_{2}^{BI}({\bf
k})\\!$ $\displaystyle=$ $\displaystyle\\!|{\bf k}|\,\,\sqrt{1-\frac{({\bf
B}\times\hat{{\bf k}})^{2}}{{\bf B}^{2}+\beta^{2}}}\;.$ (47b)

Notice that the second frequency is always real (since $\beta^{2}+({\bf
B}\cdot\hat{{\bf k}})^{2}>0$) and independent of the $p$-parameter of the
generalized BI theory. On the other hand, we need to be more carefully with
the first frequency, which is real when

$\beta^{2}+(2p-1){\bf B}^{2}+2(1-p)({\bf B}\cdot\hat{{\bf k}})^{2}>0\,\;.$
(48)

For the values $1/2\leq p<1$, this condition is fulfilled. However, for
$0<p<1/2$, will be necessary to impose some constraints between $\beta$ and
${\bf B}$. In addition, for the particular case of the BI theory $(p=1/2)$, we
obtain $\omega_{1}^{BI}=\omega_{2}^{BI}$ and this frequency leads to the well-
known result in the literature of non-birefringence Birula2 . Whenever the
magnetic field is strong, that is $|{\bf B}|\gg\beta$, the first frequency
depends on the $p$ and the $\theta$ angle $\omega_{1}^{BI}\simeq|{\bf
k}|\,\sqrt{p+(1-p)\cos(2\theta)}$ and the correspondent refraction index is
$n_{1}^{BI}\simeq[\,p+(1-p)\cos(2\theta)\,]^{-1/2}$, where $\theta$ is the
angle between ${\bf B}$ and the direction of $\hat{{\bf k}}$. The second
frequency for $|{\bf B}|\gg\beta$ is $\omega_{2}^{BI}\simeq|{\bf
k}||\cos\theta|$, and the refraction index is $n_{2}^{BI}\simeq|\sec\theta|$.
Thereby, when the medium is under a strong magnetic background, the refraction
index changes with the angle $\theta$ and it does not depend on the magnitude
of the magnetic field. Notice that the result of $\omega_{2}^{BI}$, for $|{\bf
B}|\gg\beta$, coincides with the dispersion relation of SorokinPRD  (eq. 48)
when the propagation is not superluminal. The refraction index related to the
generalized and usual BI theory as function of angle $\theta$ are plotted in
the figure (5). The top panel is the case of the generalized BI theory with
$p=3/4$ and $\beta=5.0\,\mbox{MeV}^{2}$, for the background values $|{\bf
B}|=8.0\,\mbox{MeV}^{2}$ (red line), $|{\bf B}|=4.0\,\mbox{MeV}^{2}$ (blue
line) and $|{\bf B}|=1.0\,\mbox{MeV}^{2}$ (black line), respectively. The
bottom panel is the usual BI theory $(p=1/2)$ with
$\beta=5.0\,\mbox{MeV}^{2}$, for the background values $|{\bf
B}|=10\,\mbox{MeV}^{2}$ (red line), $|{\bf B}|=5.0\,\mbox{MeV}^{2}$ (blue
line) and $|{\bf B}|=1.0\,\mbox{MeV}^{2}$ (black line), respectively, with
$\beta=5.0\,\mbox{MeV}^{2}$. Notice that, in the limit $|{\bf B}|\rightarrow
0$, both the refraction index approach the value one.

Figure 5: The top panel : The refraction index of the generalized Born-Infeld
theory (47a) as function of the angle $\theta$ (between ${\bf B}$ and
$\hat{{\bf k}}$), when $\beta=5.0\,\mbox{MeV}^{2}$ and $p=3/4$, for the values
of $|{\bf B}|=1.0\,\mbox{MeV}^{2}$, $|{\bf B}|=4.0\,\mbox{MeV}^{2}$ and $|{\bf
B}|=8.0\,\mbox{MeV}^{2}$. The bottom panel : The refraction index of the Born-
Infeld theory (47b) for $|{\bf B}|=1.0\,\mbox{MeV}^{2}$, $|{\bf
B}|=5.0\,\mbox{MeV}^{2}$ and $|{\bf B}|=10.0\,\mbox{MeV}^{2}$, when
$\beta=5.0\,\mbox{MeV}^{2}$.

The correction to the Compton effect (26) for the usual BI theory $(p=1/2)$ is
shown in figure (6). We plot the curves for the magnetic field values $|{\bf
B}|=1.0\,\mbox{MeV}^{2}$ (black line), $|{\bf B}|=5.0\,\mbox{MeV}^{2}$ (blue
line) and $|{\bf B}|=10.0\,\mbox{MeV}^{2}$ (red line). The variation of the
photon wavelength in the Compton effect increases with the magnetic field
magnitude.

Figure 6: The Compton effect in the usual Born-Infeld theory $(p=1/2)$. The
variation of the photon wavelength as function of the Compton angle,
$\theta_{c}$, for the values of $|{\bf B}|=1.0\,\mbox{MeV}^{2}$, $|{\bf
B}|=5.0\,\mbox{MeV}^{2}$ and $|{\bf B}|=10.0\,\mbox{MeV}^{2}$. We choose here
$\beta=5.0\,\mbox{MeV}^{2}$.

Using the plane wave for ${\bf e}$ and ${\bf b}$, the time average of the
energy density in generalized BI theory is given by

$\displaystyle\frac{\langle\Theta_{ph}^{00}\rangle^{BI}}{{\bf
e}_{0}^{2}}=\frac{1}{4}\,c_{1}^{BI}\left[1+\frac{{\bf
k}^{2}}{\omega^{2}}-\frac{d_{1}^{BI}}{c_{1}^{BI}}\frac{{\bf
k}^{2}}{\omega^{2}}\left(\hat{{\bf e}}_{0}\cdot(\hat{{\bf k}}\times{\bf
B})\right)^{2}\right.$
$\displaystyle\left.+\frac{d_{2}^{BI}}{c_{1}^{BI}}\left(\hat{{\bf
e}}_{0}\cdot{\bf
B}\right)^{2}-\left(\frac{d_{2}^{BI}}{c_{1}^{BI}}\right)^{2}\frac{{\bf
k}^{2}}{\omega^{2}}\left({\bf B}\cdot\hat{{\bf k}}\right)^{2}\left({\bf
B}\cdot\hat{{\bf e}}_{0}\right)^{2}\right]\;.\;\;\;$ (49)

In this case, both frequencies (47a) and (47b) depend on the external magnetic
field, and just $\omega_{1}^{BI}$ depends on the $p$-parameter. Therefore, the
time average of energy density associated with the frequency $\omega_{1}^{BI}$
changes with the values of the $p$-parameter. We plot the time average of
energy density by unit of ${\bf e}_{0}^{2}$ associated with $\omega_{1}^{BI}$
for $p=0.5$ (usual BI ED in the top panel) and $p=0.75$ (generalized BI ED in
the bottom panel), when $\beta=50\,\mbox{MeV}^{2}$, in the figure (7).

Figure 7: The time average of energy density by unit of ${\bf e}_{0}^{2}$ for
the usual BI ED $p=0.5$ (top panel), and the generalized BI ED with $p=0.75$
(bottom panel) as function of the background magnitude. We choose
$\beta=50\,\mbox{MeV}^{2}$ in this plot.

In both plots, the time average of the the energy density becomes negative for
$|{\bf B}|>|{\bf B}_{n}|$, where $|{\bf B}_{n}|=59.46\,\mbox{MeV}^{2}$ is a
magnetic critical field for the case of the usual BI ED (top panel), and
$|{\bf B}_{n}|=89.94\,\mbox{MeV}^{2}$ is a magnetic critical field for the
case of the generalized BI ED (bottom panel).

Another interesting model is Logarithm ED Gaete . The corresponding Lagrangian
is given by

$\displaystyle{\cal L}_{ln}({\cal F},{\cal
G})=-\beta^{2}\,\ln\left[1-\frac{{\cal F}}{\beta^{2}}-\frac{{\cal
G}^{2}}{2\beta^{4}}\right]\;.$ (50)

As in the previous case, Maxwell electrodynamics is recovered for
$\beta\rightarrow\infty$. In this model, we obtain the following coefficients
$c_{1}$, $d_{1}$ and $d_{2}$ :

$\displaystyle\left.c_{1}^{ln}\right|_{{\bf E}=0,{\bf B}}$ $\displaystyle=$
$\displaystyle\frac{2\beta^{2}}{2\beta^{2}+{\bf B}^{2}}\;,\;$
$\displaystyle\left.d_{1}^{ln}\right|_{{\bf E}=0,{\bf B}}$ $\displaystyle=$
$\displaystyle\frac{4\beta^{2}}{(2\beta^{2}+{\bf B}^{2})^{2}}\;,$
$\displaystyle\left.d_{2}^{ln}\right|_{{\bf E}=0,{\bf B}}$ $\displaystyle=$
$\displaystyle\frac{2}{2\beta^{2}+{\bf B}^{2}}\;.$ (51)

Using these results, the combination of the coefficients $c_{1}^{ln}$,
$d_{1}^{ln}$ and $d_{2}^{ln}$ in (22b) yields the same result (47b) for the
second frequency in logarithm theory. The first solution (22a) in this case is

$\displaystyle\omega_{1}^{ln}({\bf k})=|{\bf k}|\,\sqrt{1-\frac{2|{\bf
B}\times\hat{{\bf k}}|^{2}}{2\beta^{2}+{\bf B}^{2}}}\;.$ (52)

For $|{\bf B}|\rightarrow\infty$, the correspondent refraction index is
$n_{1}^{ln}=\sqrt{\sec(2\theta)}$. The energy density is positive-definite if
the magnetic background satisfies the condition $|{\bf B}|<\sqrt{2}\,\beta$.

##  VI Conclusions and Final Remarks

Our contribution sets out to pursue a study of general non-linear models of
electrodynamics in presence of external electric and magnetic fields. The
energy and linear momentum of the electromagnetic field fulfill a continuity
equation if the background fields are uniform and constant. We have consider
only the case of (uniform and constant) magnetic backgrounds for the analysis
of the non-linear models we have picked out. The energy density is positive if
the (external) field-dependent coefficients satisfy the conditions (12).
Otherwise, the energy density may assume negative values for sufficiently
strong external fields, as our calculations point out. Plane wave solutions
are considered to describe the non-linear photon. Two frequency solutions come
out as it happens in the case of the usual photon; the other two frequencies
exhibit a dependence on the uniform background magnetic field and the angle
between the latter and the direction of the wave propagation. As a
consequence, the refraction index also changes with the direction of the field
${\bf B}$ relative to the direction of propagation of the wave, $\hat{{\bf
k}}$. We have also obtained the contribution of the uniform magnetic
background to the kinematics of the Compton effect by employing the modified
dispersion relations. We have applied these results to four examples of non-
linear electrodynamics : Hoffmann-Infeld, generalized Born-Infeld, Logarithm
and the Euler-Heisenberg effective Lagrangian. In all these cases, the
refraction index depends on the angle, $\theta$, between ${\bf B}$ and
$\hat{{\bf k}}$, and on the magnitude of ${\bf B}$ as well. The case of $|{\bf
B}|\gg\beta$ is interesting because the results of the dispersion relation and
refraction index are independent on the ${\bf B}$ magnitude. For the
generalized Born-Infeld $n_{1}^{BI}\simeq[\,p+(1-p)\cos(2\theta)\,]^{-1/2}$,
and for the usual Born-Infeld $n_{2}^{BI}\simeq|\sec\theta|=|\hat{{\bf
B}}\cdot\hat{{\bf k}}|^{-1}$. Notice that $n_{1}^{BI}=n_{2}^{BI}$, when
$p=1/2$. Furthemore, we have shown that the electrostatic potential is finite
at the origin for $p=3/4$. The result for the Logarithm electrodynamics is
$n_{1}^{ln}\simeq\sqrt{\sec(2\theta)}$, when $|{\bf B}|\gg\beta$.

Further research has been initiated in a scenario involving models of scalar
axions in connection with particular non-linear models. The purpose of this
particular investigation is to try to understand how the present
phenomenological astrophysical data known for the axions may dictate
restrictions on the form of non-linear Lagrangian densities. On the other
hand, we are also particularly interested in contemplating CP-breaking non-
linear models in connection with the Planck 2018 Polarization Data to
eventually use the latter to derive constraints on CP- violating non-
linearities. We intend to report on the progress of these two lines of
investigation in further papers.

Also, we would like to bring the reader’s attention that the study we report
in this contribution, mainly the investigation we carry out on dispersion
relations, may be applied to extract bounds on the parameters of the non-
linear models here inspected (and others we have not contemplated in this
paper), if we consider current experiments, such as PVLAS and BMV, and the
future high-power LASERs mentioned in the introduction, namely, SEL, ELI
Project and XCELS. The work of Ref. Davila  raises the interesting possibility
that large-scale LASERs may also be used in the LIGO, VIRGO and GEO
interferometers to constrain the parameters of non-linear models by measuring
the birefringence of the QED vacuum.

## Acknowledgments

The authors express their gratitude to P. Gaete and A. Spallicci for
stimulating discussions on diverse aspects of non-linear electrodynamic
models. F. Karbstein and D. Sorokin are also deeply acknowledged for pointing
out important references and drawing our attention to relevant aspects of the
Euler-Heisenberg and Born-Infeld Electrodynamics, respectively. M. J. Neves
thanks CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico),
Brazilian scientific support federal agency, for partial financial support,
Grant number 313467/2018-8. L.P.R. Ospedal is grateful to the Ministry for
Science, Technology and Innovations (MCTI) and CNPq for his Post-Doctoral
Fellowship under the Institutional Qualification Program (PCI).

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