# Considerations on the modified Maxwell electrodynamics in the presence of an
electric and 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
Patricio Gaete<EMAIL_ADDRESS>Departamento de Física and Centro
Científico-Tecnológico de Valparaíso-CCTVal, Universidad Técnica Federico
Santa María, Valparaíso, Chile  L. P. R. Ospedal<EMAIL_ADDRESS>Instituto de Física, Universidade Federal do Rio Grande do Sul, Av. Bento
Gonçalves 9500, CEP 91501-970, Porto Alegre, Brazil  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

The properties of the modified Maxwell electrodynamics (ModMax) are
investigated in presence of external and uniform electric and magnetic fields.
We expand the non-linear theory around an electromagnetic background up to
second order in the propagating fields to obtain the permittivity and
permeability tensors, dispersion relations, group velocity and refractive
indices as functions of external fields. The case with perpendicular
background fields is contemplated. The phenomenon of birefringence is
discussed and the fundamental role of the ModMax parameter becomes clear. We
calculate the difference of the refractive indices in terms of this parameter
and the external fields. Finally, we set up a scenario where the axion is
present and compute the interaction energy for the coupled axion-ModMax
electrodynamics if a magnetic background field is considered. This calculation
is carried out in the framework of the gauge-invariant, but path-dependent
variables formalism. Our results show that the interaction energy contains a
linear component, leading to the confinement of static probe charges where the
interference between the ModMax parameter, the axion mass and the axion-photon
coupling constant is pointed out.

##  I Introduction

As widely-known, non-linear electrodynamics has a long history, beginning from
the pioneering paper by Born and Infeld (BI) [1], who introduced their theory
in order to overcome the intrinsic divergences in the Maxwell theory, at short
distances. In passing, we mention that, just like Maxwell electrodynamics, the
BI electrodynamics displays no vacuum birefringence. We also recall here that,
after the development of Quantum Electrodynamics (QED) [2, 3, 4, 5],
Heisenberg and Euler (HE) [6] proposed a new non-linear effective theory by
summing up the quantum effects of virtual electrons and positrons. Indeed,
this new theory contains a striking prediction of the QED, that is, the light-
by-light scattering arising from the interaction of photons with virtual
electron-positron pairs. It should be further noted that one of the most
interesting physical consequences of the HE result is vacuum birefringence. In
other words, when the quantum vacuum is stressed by external electromagnetic
fields, it behaves like a birefringent material medium. Let us also mention
that this physical effect has been emphasized from different viewpoints [7, 8,
9, 10, 11, 12]. In the context of non-linear electrodynamics, all the
solutions to the non-birefringence condition are shown in ref. [13].
Nevertheless, this optical phenomenon has not yet been confirmed [14, 15, 16,
17].

It is noteworthy to recall here that a few years ago the ATLAS and CMS
Collaborations at the Large Hadron Collider (LHC) have reported on the high-
energy photon-photon pair emission from virtual photon-photon scattering in
ultraperipheral Pb-Pb collisions [18, 19]. It is remarkable to notice that, in
these results, there is no modification of the optical properties of the
vacuum [20]. Besides, the new era of high-power LASER facilities also provides
us conditions to probe quantum vacuum non-linearities [21, 22]. Among various
experiments, a promising proposal is the DeLLight Project [23], based on the
induced change in the vacuum refractive index by virtue of non-linearities in
electrodynamics.

In the spirit of the foregoing remarks, we have considered the physical
effects presented by different models of $(3+1)$-D non-linear electrodynamics
in vacuum [24, 25, 26]. This has led us to fresh insights on quantum vacuum
non-linearities in different contexts. For instance, in the generalized Born-
Infeld and Logarithmic electrodynamics, our analysis reveals that the field
energy of a point-like charge is finite, apart from displaying the vacuum
birefringence phenomenon. Additionally, we have examined the lowest-order
modifications of the static potential within the framework of the gauge-
invariant but path-dependent variables formalism, which is an alternative to
the Wilson loop approach [27].

Moreover, a novel non-linear modification of Maxwell electrodynamics that
preserves all its symmetries including the electric-magnetic duality and the
conformal invariance has recently been proposed [28, 29, 30], which is
referred to as ModMax electrodynamics. Certainly, the interest in studying
non-linear electrodynamics is mainly due to its potentially significant
contributions to light-by-light scattering and in the description of novel
black hole solutions (see, for instance, refs. [31, 32, 33, 34, 35, 36]). It
is of relevance to point out that the ModMax model can be obtained from
$T\,\overline{T}$-like deformations of Maxwell electrodynamics [37, 38, 39,
40], and also within the context of string theory [41].

From the previous considerations, and given the interest and importance
related to photon-photon interaction physics, this work sets out to further
elaborate on a number of phenomenological consequences presented by the ModMax
electrodynamics. More specifically, we shall focus our attention on the
birefringence, dispersion relations, as well as the computation of the static
potential along the lines of the refs. [25, 26, 42, 43]. We shall present
results of the ModMax ED as a classical field theory and the continuity
equation for the energy-momentum tensor is derived . We also work out the
ModMax stress tensor and its angular momentum tensor split into its orbital
and spin components. The electric/magnetic properties of the vacuum as a
material medium ruled by the ModMax ED are studied in presence of uniform
electric and magnetic backgrounds.

We focus on the particular case with perpendicular electric and magnetic
backgrounds because the CP-symmetry of the theory is preserved under this
condition. Bearing this in mind, we study the wave propagation effects in the
linearized ModMax ED. We obtain the permittivity and permeability tensors, the
dispersion relations, the refractive index and group velocity as functions of
the ModMax parameter, and the external electric and magnetic fields. The
phenomenon of birefringence in the electromagnetic background is studied in
presence of an external magnetic field. Some particular cases are discussed
and the recent results in the literature of ModMax ED are correctly recovered.
As expected, all the results of the Maxwell ED are also correctly reproduced
whenever the ModMax parameter goes to zero.

In the final part of the paper, we present our motivations to introduce ModMax
ED coupled to an axion scalar field through an axion-(ModMax)photon
interaction term. In this case, we do not consider an electric background, and
focus on the axionic ModMax ED in presence of a magnetic background field. In
so considering, we study the confinement properties of this new model by
computing the interaction energy for a pair of static probe charges within the
framework of gauge-invariant but path-dependent variables formalism. In
particular, we shall be interested in the dependence of the confinement
properties interms of the axion mass, axion-photon coupling and the ModMax
parameter.

Our contribution is organized as follows : In Section II, we quickly review
the setup of ModMax theory, derive the field equations and conservation laws.
In Section III, we introduce the prescription of a uniform electromagnetic
background in order to expand the ModMax ED up to second order in the
propagating fields. In Section IV, the properties of the wave propagation
(dispersion relations, refractive index, and group velocity) are calculated in
the situation where the electric and magnetic background fields are
perpendicular. In Section V, we investigate the phenomenon of birefringence in
the presence of electromagnetic background fields. Subsequently, in Section
VI, we consider axionic ModMax electrodynamics and we show that the
interaction energy contains a linear potential, leading to the confinement of
static probe charges. Finally, our conclusions are cast in Section VII.

At this point, we place a caveat on the fact that Sections III, IV and V deal
essentially with the inspection of material properties of the vacuum in
presence of external electric and magnetic fields, by considering the
linearized approximation in the propagating fields. To our sense, it is
important to make clear to the reader that this issue has been fairly-well
investigated and discussed in a series of pioneer papers, as quoted in the
works cast in refs. [44, 45, 46, 47, 48, 49]. Nevertheless, in the above
mentioned Sections III, IV and V, we endeavor to propose an extended re-
assessment of the aforementioned classical papers while choosing the
particular ModMax ED as our object of investigation to illustrate the
propagation of the linear waves. We emphasize however that our attempt is not
a mere review. We work out new results, such as the explicit expressions for
refractive indices and group velocities, conditions to establish dichroism of
the vacuum, find conditions that involve both the ModMax $\gamma$-parameter
and the external fields to inspect the positivity of the permittivity and
permeability tensors in the special case where the external (electric and
magnetic) fields are orthogonal to each other. We would like to recall that in
the paper [28], the authors consider the specific situation of (anti) parallel
external fields. Moreover, we also present a discussion of the angular
momentum tensor and the corresponding spin vector for ModMax ED.

We adopt the natural units $\hbar=c=1$ with $4\pi\epsilon_{0}=1$, and the
Minkowski metric $\eta^{\mu\nu}=\mbox{diag}(+1,-1,-1,-1)$. The electric and
magnetic fields have squared-energy mass dimension in which 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 The modified Maxwell electrodynamics

We start off the classical description of the ModMax ED through the Lagrangian
density:

$\displaystyle{\cal L}_{MM}=\cosh\gamma\,{\cal F}+\sinh\gamma\sqrt{{\cal
F}^{2}+{\cal G}^{2}}+J_{\mu}A^{\mu}\;,$ (1)

where ${\cal F}$ and ${\cal G}$ denote the Lorentz and gauge invariants,

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

with $\gamma$ being a real parameter, that satisfies the condition $\gamma\geq
0$ to insure the causality and unitarity of the model [28], and
$J^{\mu}=(\rho,{\bf J})$ is a classical source of charge and current
densities. The
$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}=\left(\,-E^{i}\,,\,-\epsilon^{ijk}B^{k}\,\right)$
is the skew-symmetric field strength tensor, and
$\widetilde{F}^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}/2=\left(\,-B^{i}\,,\,\epsilon^{ijk}E^{k}\,\right)$
corresponds to the dual tensor. The usual Maxwell electrodynamics is recovered
when $\gamma\rightarrow 0$.

The action principle yields the field equations

$\partial_{\mu}\left[\,\cosh\gamma\,F^{\mu\nu}+\sinh\gamma\,\frac{F^{\mu\nu}\,{\cal
F}+\widetilde{F}^{\mu\nu}\,{\cal G}}{\sqrt{{\cal F}^{2}+{\cal
G}^{2}}}\,\right]=J^{\nu}\;,$ (3)

and the dual tensor satisfies the Bianchi identity
$\partial_{\mu}\widetilde{F}^{\mu\nu}=0$. The charge conservation obeys the
continuity equation $\partial_{\mu}J^{\mu}=0$, as in the Maxwell ED. The
equations in terms of the electric and magnetic fields can be written as :

$\displaystyle\nabla\cdot{\bf D}\\!$ $\displaystyle=$
$\displaystyle\\!\rho\hskip 8.5359pt,\hskip 8.5359pt\nabla\times{\bf
E}+\partial_{t}{\bf B}={\bf 0}\;,$ (4a) $\displaystyle\nabla\cdot{\bf B}\\!$
$\displaystyle=$ $\displaystyle\\!0\hskip 8.5359pt,\hskip
8.5359pt\nabla\times{\bf H}-\partial_{t}{\bf D}={\bf J}\;,\hskip 14.22636pt$
(4b)

where ${\bf D}$ and ${\bf H}$ are defined by

$\displaystyle{\bf D}=\cosh\gamma\,{\bf E}+\sinh\gamma\,\frac{{\cal F}\,{\bf
E}+{\cal G}\,{\bf B}}{\sqrt{{\cal F}^{2}+{\cal G}^{2}}}\;,$ (5a)
$\displaystyle{\bf H}=\cosh\gamma\,{\bf B}+\sinh\gamma\,\frac{{\cal F}\,{\bf
B}-{\cal G}\,{\bf E}}{\sqrt{{\cal F}^{2}+{\cal G}^{2}}}\;.$ (5b)

Multiplying the Bianchi identity by $F^{\mu\nu}$, and using the eq. (3), we
arrive at the conservation law

$\displaystyle\partial_{\mu}\Theta^{\mu\nu}=J_{\rho}\,F^{\rho\nu}\;,$ (6)

where the energy-momentum tensor of the ModMax ED is given by

$\Theta^{\mu\nu}=\left(\,F^{\mu\rho}F_{\rho}^{\;\;\nu}-\eta^{\mu\nu}\,{\cal
F}\right)\left[\cosh\gamma+\frac{\sinh\gamma\,{\cal F}}{\sqrt{{\cal
F}^{2}+{\cal G}^{2}}}\right]\;.$ (7)

This tensor has the properties of symmetry in the $(\mu\leftrightarrow\nu)$
indices and is gauge invariance. In the case of the ModMax in a free space
(with no charge and current densities), the expression (6) satisfies
immediately the continuity equation, $\partial_{\mu}\Theta^{\mu\nu}=0$, where
the $\Theta^{00}$\- and $\Theta^{0i}$-components denote the conserved energy
and momentum densities stored in the EM fields, namely,

$\displaystyle\Theta^{00}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2}\,({\bf E}^{\,2}\\!+\\!{\bf
B}^{2})\\!\left[\cosh\gamma\\!+\\!\frac{\sinh\gamma\,({\bf E}^{2}-{\bf
B}^{2})}{\sqrt{({\bf E}^{2}-{\bf B}^{2})^{2}+4({\bf E}\cdot{\bf
B})^{2}}}\right],$ $\displaystyle\Theta^{0i}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left({\bf E}\\!\times\\!{\bf
B}\right)^{i}\\!\left[\cosh\gamma\\!+\\!\frac{\sinh\gamma\,({\bf E}^{2}-{\bf
B}^{2})}{\sqrt{({\bf E}^{2}-{\bf B}^{2})^{2}+4({\bf E}\cdot{\bf
B})^{2}}}\right].$ (8b)

Notice that both components in eqs. (8) and (8b) are not defined if the
electric and magnetic fields, for example, satisfy the relation of null
electromagnetic fields, ${\bf E}^{2}-{\bf B}^{2}=0$ and ${\bf E}\cdot{\bf
B}=0$. To remove this singularity, it is convenient to define the Legendre
transformation :

$\displaystyle{\cal H}({\bf D},{\bf B})={\bf E}\cdot{\bf D}-{\cal L}_{MM}({\bf
E},{\bf B})\;,$ (9)

and using the ModMax Lagrangian (1), the Hamiltonian density turns out to be
given by

$\displaystyle{\cal H}_{MM}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{2}\,\cosh\gamma\left({\bf D}^{2}+{\bf
B}^{2}\right)$ (10) $\displaystyle-\frac{1}{2}\,\sinh\gamma\,\sqrt{\left({\bf
D}^{2}-{\bf B}^{2}\right)^{2}+4\left({\bf D}\cdot{\bf B}\right)^{2}}\;,$

which is positive-definite if the ModMax parameter obeys the condition

$\displaystyle\tanh\gamma\,<\,\sqrt{\,\frac{({\bf D}^{2}+{\bf
B}^{2})^{2}}{({\bf D}^{2}-{\bf B}^{2})^{2}+4({\bf D}\cdot{\bf B})^{2}}\,}\;.$
(11)

The quantity (10) is well-defined for any ${\bf D}$ and ${\bf B}$.
Furthermore, it is manifestly invariant under the duality transformations
${\bf D}^{\prime}+i\,{\bf B}^{\prime}=e^{i\alpha}\,({\bf D}+i\,{\bf B})$, with
$\alpha$ being a real parameter. The Poynting vector, given by the
$\Theta^{0i}$-component in terms of the ${\bf D}$\- and ${\bf B}$\- read as
below:

$\displaystyle{\bf S}_{P}={\bf D}\times{\bf B}\;.$ (12)

The spatial component $\nu=j$ in the conservation law (6) yields the
expression

$\displaystyle\nabla\cdot\overleftrightarrow{{\bf T}}-\partial_{t}{\bf
S}_{P}={\bf f}_{L}\;,$ (13)

where ${\bf f}_{L}=\rho\,{\bf E}\,+\,{\bf J}\times{\bf B}$ is the Lorentz
force density, while the components of the Maxwell stress tensor
$\overleftrightarrow{{\bf T}}$ are given by

$\displaystyle T^{ij}\\!$ $\displaystyle=$
$\displaystyle\\!\left[\,\cosh\gamma+\frac{\sinh\gamma\,{\cal F}}{\sqrt{{\cal
F}^{2}+{\cal G}^{2}}}\,\right]\times$ (14)
$\displaystyle\times\left[\,E^{i}\,E^{j}+B^{i}\,B^{j}-\delta^{ij}\,\frac{1}{2}\,({\bf
E}^{2}+{\bf B}^{2})\,\right]\;.$

The Lorentz force has the same definition as in Maxwell ED, namely, it is
obtained by integrating the corresponding density over a region of the space.

Having established the energy-momentum tensor of the ModMax ED, we now proceed
to the angular momentum tensor. Let us adopt the canonical approach and split
it into an orbital (OAM) and spin (SAM) components as follows:

$\displaystyle
M^{\mu}_{\;\;\,\alpha\beta}=x_{\alpha}\,T^{\mu}_{\;\;\,\beta}-x_{\beta}\,T^{\mu}_{\;\;\,\alpha}+S^{\mu}_{\;\;\,\alpha\beta}\;,$
(15)

where $T^{\mu}_{\;\;\,\beta}$ stands for the canonical energy-momentum tensor,
whereas $S^{\mu}_{\;\;\,\alpha\beta}$ expresses the spin contribution:

$T^{\mu}_{\;\;\,\beta}=\left[\cosh\gamma\,F^{\mu\lambda}\\!+\\!\sinh\gamma\,\frac{{\cal
F}\,F^{\mu\lambda}+{\cal G}\,\tilde{F}^{\mu\lambda}}{\sqrt{{\cal F}^{2}+{\cal
G}^{2}}}\right]\\!\partial_{\beta}A_{\lambda}-\delta^{\mu}_{\;\,\,\beta}\,{\cal
L}_{MM}\,,\;\;$ (16)

and

$\displaystyle S^{\mu}_{\;\;\,\alpha\beta}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left[\cosh\gamma+\frac{\sinh\gamma\,{\cal F}}{\sqrt{{\cal
F}^{2}+{\cal
G}^{2}}}\right](F^{\mu}_{\;\;\,\beta}A_{\alpha}-F^{\mu}_{\;\;\,\alpha}A_{\beta})$
(17) $\displaystyle+\frac{\sinh\gamma\,{\cal G}}{\sqrt{{\cal F}^{2}+{\cal
G}^{2}}}\,(\widetilde{F}^{\mu}_{\;\;\,\beta}A_{\alpha}-\widetilde{F}^{\mu}_{\;\;\,\alpha}A_{\beta})\;.$

Consequently, the corresponding spin vector can be read from the expression
below:

$\displaystyle{\bf S}_{spin}\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\int
d^{3}{\bf x}\,\left\\{\left[\cosh\gamma+\frac{\sinh\gamma\,{\cal
F}}{\sqrt{{\cal F}^{2}+{\cal G}^{2}}}\right]({\bf E}\times{\bf A})\right.$
(18) $\displaystyle\left.+\frac{\sinh\gamma\,{\cal G}}{\sqrt{{\cal
F}^{2}+{\cal G}^{2}}}\,({\bf B}\times{\bf A})\right\\}\;,$

which can be recast as

$\displaystyle{\bf S}_{spin}=\int d^{3}{\bf x}\;{\bf D}\times{\bf A}\;.$ (19)

Both the OAM and SAM components are not gauge-invariant and a physically
unambiguous splitting into these two pieces is still controversial and object
of debate. Actually, the formal separation into orbital and spin parts of an
optical field first appeared in a paper by J. Humblet [50]. In 1932, C. G.
Darwin pioneered the investigation of the angular momentum tensor of
electromagnetic radiation, though he did not exploit the OAM-SAM splitting
[51]. For an updated discussion, we address the interested readers to the
papers [52, 53], where alternative decompositions into OAM and SAM components
are presented and discussed with a great deal of details.

##  III The linearized ModMax electrodynamics in an EM background

The ModMax ED can be linearized by expanding the $A^{\mu}$-potential around a
uniform and constant EM field, as $A_{\mu}=a_{\mu}+A_{0\mu}$, where $a_{\mu}$
is the excitation interpreted as the photon field, whereas $A_{0\mu}$ is the
potential associated with the EM background. We adopt a similar procedure and
notations of ref. [24]. The expansion is considered up to second order in the
photon fluctuation $a^{\mu}$. Thereby, the tensor $F_{\mu\nu}$ is also
decomposed as $F_{\mu\nu}=f_{\mu\nu}+F_{0\mu\nu}$, in which
$f^{\mu\nu}=\partial^{\mu}a^{\nu}-\partial^{\nu}a^{\mu}=\left(\,-e^{i}\,,\,-\epsilon^{ijk}\,b^{k}\,\right)$
is the EM field strength tensor of the propagating field, and
$F_{0}^{\;\,\mu\nu}=\partial^{\mu}A_{0}^{\;\,\nu}-\partial^{\nu}A_{0}^{\;\,\mu}=\left(\,-E_{0}^{\,\,i}\,,\,-\epsilon^{ijk}\,B_{0}^{\,\,k}\,\right)$
is the field strength associated with the background electric and magnetic
fields. Now, by expanding the Lagrangian density (1) around the external
field, we arrive at

$\displaystyle{\cal L}_{MM}^{(2)}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-\frac{1}{4}\,c_{1}\,f_{\mu\nu}^{\,2}-\frac{1}{4}\,c_{2}\,f_{\mu\nu}\,\widetilde{f}^{\mu\nu}-\frac{1}{2}\,G_{0\mu\nu}\,f^{\mu\nu}$
(20)
$\displaystyle+\,\frac{1}{8}\,Q_{0\mu\nu\kappa\lambda}\,f^{\mu\nu}\,f^{\kappa\lambda}\;,$

where $G_{0\mu\nu}=c_{1}\,F_{0\mu\nu}+c_{2}\,\widetilde{F}_{0\mu\nu}$ and
$Q_{0\mu\nu\kappa\lambda}=d_{1}\,F_{0\mu\nu}F_{0\kappa\lambda}+d_{2}\,\widetilde{F}_{0\mu\nu}\widetilde{F}_{0\kappa\lambda}+d_{3}\,F_{0\mu\nu}\widetilde{F}_{0\kappa\lambda}+d_{3}\,\widetilde{F}_{0\mu\nu}F_{0\kappa\lambda}$
are tensors that depend on the components of the EM background fields. The
$\widetilde{f}^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}f_{\alpha\beta}/2=\left(\,-b^{i}\,,\,\epsilon^{ijk}e^{k}\,\right)$
and
$\widetilde{F}_{0}^{\;\,\mu\nu}=\epsilon^{\mu\nu\alpha\beta}F_{0\alpha\beta}/2=\left(\,-B_{0}^{\;\,i}\,,\,\epsilon^{ijk}E_{0}^{\;\,k}\,\right)$
are the dual field strength tensors of the propagating and background fields,
respectively. The $\widetilde{f}^{\mu\nu}$ satisfies the Bianchi identity
$\partial_{\mu}\widetilde{f}^{\mu\nu}=0$ from which there emerge the
homogeneous field equations:

$\nabla\cdot{\bf b}=0\,\;,\,\;\nabla\times{\bf e}=-\partial_{t}{\bf b}\,.$
(21)

The coefficients $c_{1}$, $c_{2}$, $d_{1}$, $d_{2}$ and $d_{3}$ of this
expansion are defined as cast below:

$\displaystyle c_{1}=\left.\frac{\partial{\cal L}_{MM}}{\partial{\cal
F}}\right|_{{\bf E}_{0},{\bf B}_{0}}\,,\,\left.c_{2}=\frac{\partial{\cal
L}_{MM}}{\partial{\cal G}}\right|_{{\bf E}_{0},{\bf B}_{0}}\,,\,$
$\displaystyle\left.d_{1}=\frac{\partial^{2}{\cal L}_{MM}}{\partial{\cal
F}^{2}}\right|_{{\bf E}_{0},{\bf
B}_{0}}\,,\,\left.d_{2}=\frac{\partial^{2}{\cal L}_{MM}}{\partial{\cal
G}^{2}}\right|_{{\bf E}_{0},{\bf B}_{0}}\,,\,$
$\displaystyle\left.d_{3}=\frac{\partial^{2}{\cal L}_{MM}}{\partial{\cal
F}\partial{\cal G}}\right|_{{\bf E}_{0},{\bf B}_{0}},\hskip 11.38092pt$ (22)

that also depend only on the EM background. Substituting the ModMax Lagrangian
density (1), we obtain :

$\displaystyle c_{1}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\cosh\gamma+\frac{\sinh\gamma\,{\cal F}_{0}}{\sqrt{{\cal
F}_{0}^{2}+{\cal G}_{0}^{2}}}\;,\;c_{2}=\frac{\sinh\gamma\,{\cal
G}_{0}}{\sqrt{{\cal F}_{0}^{2}+{\cal G}_{0}^{2}}}\;,$ $\displaystyle
d_{1}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{\sinh\gamma}{\sqrt{{\cal F}_{0}^{2}+{\cal
G}_{0}^{2}}}\frac{{\cal G}_{0}^{2}}{{\cal F}_{0}^{2}+{\cal G}_{0}^{2}}\;,$
$\displaystyle d_{2}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{\sinh\gamma}{\sqrt{{\cal F}_{0}^{2}+{\cal
G}_{0}^{2}}}\frac{{\cal F}_{0}^{2}}{{\cal F}_{0}^{2}+{\cal G}_{0}^{2}}\;,$
$\displaystyle d_{3}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{\sinh\gamma}{\sqrt{{\cal F}_{0}^{2}+{\cal
G}_{0}^{2}}}\,\frac{{\cal F}_{0}\,{\cal G}_{0}}{{\cal F}_{0}^{2}+{\cal
G}_{0}^{2}}\;,$ (23)

in which ${\cal F}_{0}=({\bf E}_{0}^{2}-{\bf B}_{0}^{2})/2$ and ${\cal
G}_{0}={\bf E}_{0}\cdot{\bf B}_{0}$ are the gauge and Lorentz invariants
associated with the electric ${\bf E}_{0}$, and magnetic ${\bf B}_{0}$ fields.

The action principle applied to the Lagrangian density (20) yields the field
equations :

$\displaystyle\partial^{\mu}\left[\,c_{1}\,f_{\mu\nu}+c_{2}\,\widetilde{f}_{\mu\nu}-\frac{1}{2}\,Q_{0\mu\nu\kappa\lambda}\,f^{\kappa\lambda}\,\right]=0\;.$
(24)

The usual Maxwell ED is recovered when the EM background fields are turn-off
and the ModMax parameter goes to zero. The equations (24) can be written into
the same form of eqs. (4a) and (4b), with the auxiliary fields

$\displaystyle D_{i}\\!$ $\displaystyle=$
$\displaystyle\\!\varepsilon_{ij}\,e_{j}+\sigma_{ij}\,b_{j}\;,$ (25a)
$\displaystyle H_{i}\\!$ $\displaystyle=$
$\displaystyle\\!-\sigma_{ji}\,e_{j}+(\mu_{ij})^{-1}\,b_{j}\;,$ (25b)

where the permittivity symmetric tensor $\varepsilon_{ij}$, the inverse of the
permeability symmetric tensor $(\mu_{ij})^{-1}$, and $\sigma_{ij}$ are given
in terms of the electric and magnetic background components, as follows

$\displaystyle\varepsilon_{ij}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!c_{1}\,\delta_{ij}+d_{1}\,E_{0i}\,E_{0j}+d_{2}\,B_{0i}\,B_{0j}$
(26a) $\displaystyle+\,d_{3}\,E_{0i}\,B_{0j}+d_{3}\,B_{0i}\,E_{0j}\;,$
$\displaystyle\sigma_{ij}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!c_{2}\,\delta_{ij}-d_{1}\,E_{0i}\,B_{0j}+d_{2}\,B_{0i}\,E_{0j}$
(26b) $\displaystyle+\,d_{3}\,E_{0i}\,E_{0j}-d_{3}\,B_{0i}\,B_{0j}\;,$
$\displaystyle(\mu_{ij})^{-1}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!c_{1}\,\delta_{ij}-d_{1}\,B_{0i}\,B_{0j}-d_{2}\,E_{0i}\,E_{0j}$
(26c) $\displaystyle-d_{3}\,E_{0i}\,B_{0j}-d_{3}\,B_{0i}\,E_{0j}\;.$

Notice that the case in which $c_{2}\neq 0$ and $d_{3}\neq 0$, CP-symmetry is
violated in the linearized theory. To keep CP invariance, we just consider the
situation in which the electric and magnetic background fields are mutually
orthogonal.

##  IV Wave propagation for perpendicular external fields

If we consider the case with ${\bf E}_{0}$ perpendicular to ${\bf B}_{0}$, the
second Lorentz and gauge invariant quantity ${\cal G}_{0}=0$, so that the
coefficients from eq. (III) are given by

$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\cosh\gamma+\sinh\gamma\,\mbox{sgn}({\cal
F}_{0})\;,\;d_{2}=\frac{\sinh\gamma}{|{\cal F}_{0}|}\;,$ $\displaystyle c_{2}$
$\displaystyle=$ $\displaystyle d_{1}=d_{3}=0\;,$ (27)

in which $\mbox{sgn}({\cal F}_{0})$ is the signal function of ${\cal F}_{0}$,
where $\mbox{sgn}({\cal F}_{0})=1$ if $|{\bf E}_{0}|>|{\bf B}_{0}|$, and
$\mbox{sgn}({\cal F}_{0})=-1$ if $|{\bf B}_{0}|>|{\bf E}_{0}|$. Under these
conditions, the permittivity tensor, (26a), and the permeability tensor
related to eq. (26c) turn out to be

$\displaystyle\varepsilon_{ij}\\!$ $\displaystyle=$
$\displaystyle\\!c_{1}\,\delta_{ij}+d_{2}\,B_{0i}\,B_{0j}\;,$ (28a)
$\displaystyle\mu_{ij}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{c_{1}}\left[\,\delta_{ij}+\frac{d_{E}\,E_{0i}\,E_{0j}}{1-d_{E}\,{\bf
E}_{0}^{2}}\,\right]\;,$ (28b)

where the coefficient $d_{E}$ is

$\displaystyle d_{E}=\frac{d_{2}}{c_{1}}=\frac{\tanh(\gamma)}{|{\cal
F}_{0}|+\tanh(\gamma)\,{\cal F}_{0}}\;,$ (29)

in which

$\displaystyle d_{E}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1-e^{-2\gamma}}{E_{0}^{2}-B_{0}^{2}}\hskip
8.5359pt\mbox{if}\hskip 8.5359pt|{\bf E}_{0}|>|{\bf B}_{0}|\;,$ (30a)
$\displaystyle d_{E}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{e^{2\gamma}-1}{B_{0}^{2}-E_{0}^{2}}\hskip
8.5359pt\mbox{if}\hskip 8.5359pt|{\bf B}_{0}|>|{\bf E}_{0}|\;.$ (30b)

The eigenvalues of the matrices $\varepsilon_{ij}$ and $\mu_{ij}$ are given
by:

$\displaystyle\lambda_{1\varepsilon}\\!$ $\displaystyle=$
$\displaystyle\\!\lambda_{2\varepsilon}=c_{1}\;,\;\lambda_{3\varepsilon}=c_{1}+d_{2}\,{\bf
B}_{0}^{2}\;,$ (31a) $\displaystyle\lambda_{1\mu}\\!$ $\displaystyle=$
$\displaystyle\\!\lambda_{2\mu}=\frac{1}{c_{1}}\;,\;\lambda_{3\mu}=\frac{1}{c_{1}-d_{2}\,{\bf
E}_{0}^{2}}\;,$ (31b)

the electric permittivity and magnetic permeability are both positive-definite
if the eigenvalues satisfy the conditions $c_{1}>0$, $1+d_{E}\,{\bf
B}_{0}^{2}>0$, and $1-d_{E}\,{\bf E}_{0}^{2}>0$.

We turn now our attention to the dispersion relations. 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 eq. (24), the wave equation for the
components of the electric amplitude $e_{0i}$ is

$\displaystyle M_{ij}\,e_{0j}=0\;.$ (32)

The matrix elements $M_{ij}$ have the form

$M_{ij}=\left(\omega^{2}-{\bf
k}^{2}\right)\delta_{ij}+k_{i}\,k_{j}+u_{i}\,v_{j}\;,$ (33)

where the vectors are defined by $u_{i}=\omega\,B_{0i}-({\bf k}\times{\bf
E}_{0})_{i}$ and $v_{i}=d_{E}\,u_{i}$. The determinant of the $M$-matrix can
be calculated and yields the expression

$\mbox{det}M\\!=\\!(\omega^{2}-{\bf
k}^{2})\left[\,\omega^{2}\left(\omega^{2}-{\bf k}^{2}+{\bf u}\cdot{\bf
v}\right)-\left({\bf u}\cdot{\bf k}\right)\left({\bf v}\cdot{\bf
k}\right)\,\right]\;,\;\;\;$ (34)

The condition $\mbox{det}M=0$ leads to the usual photon dispersion relation
$\omega=|{\bf k}|$ as one of the solutions. The other solutions correspond to
the roots of the polynomial equation :

$\displaystyle\omega^{2}\left(\,P\,\omega^{2}+Q\,\omega+R\,\right)=0\;,$ (35)

where $P=1+d_{E}\,{\bf B}_{0}^{2}$, $Q=-2\,d_{E}\,{\bf B}_{0}\cdot({\bf
k}\times{\bf E}_{0})$, and $R=d_{E}\left({\bf k}\times{\bf
E}_{0}\right)^{2}-{\bf k}^{2}-d_{E}\,\left({\bf B}_{0}\cdot{\bf
k}\right)^{2}$. The zeroes of eq. (35) are $\omega=0$, and the non-trivial
solutions :

$\displaystyle\omega^{(-)}({\bf k})\\!$ $\displaystyle=$
$\displaystyle\\!\frac{d_{E}{\bf B}_{0}\cdot({\bf k}\times{\bf
E}_{0})-\sqrt{{\bf k}^{2}-d_{E}\left({\bf k}\times{\bf
E}_{0}\right)^{2}-d_{E}^{2}\left({\bf E}_{0}^{2}-{\bf B}_{0}^{2}\right)({\bf
k}\cdot{\bf B}_{0})^{2}+d_{E}\left({\bf k}\cdot{\bf
B}_{0}\right)^{2}+d_{E}\,{\bf B}_{0}^{2}\,{\bf k}^{2}}}{1+d_{E}\,{\bf
B}_{0}^{2}}\;,$ (36a) $\displaystyle\omega^{(+)}({\bf k})\\!$ $\displaystyle=$
$\displaystyle\\!\frac{d_{E}{\bf B}_{0}\cdot({\bf k}\times{\bf
E}_{0})+\sqrt{{\bf k}^{2}-d_{E}\left({\bf k}\times{\bf
E}_{0}\right)^{2}-d_{E}^{2}\left({\bf E}_{0}^{2}-{\bf B}_{0}^{2}\right)({\bf
k}\cdot{\bf B}_{0})^{2}+d_{E}\left({\bf k}\cdot{\bf
B}_{0}\right)^{2}+d_{E}\,{\bf B}_{0}^{2}\,{\bf k}^{2}}}{1+d_{E}\,{\bf
B}_{0}^{2}}\;.$ (36b)

The limit of usual Maxwell ED $(\gamma\rightarrow 0)$ in eqs. (36b) and (36a)
yields the roots $\lim_{\gamma\rightarrow 0}\omega^{(\pm)}({\bf k})=\pm\,|{\bf
k}|$, so that the photon dispersion relation is recovered. Thereby, we choose
the $\omega^{(+)}$ frequency for the analysis of the refractive index of the
medium. Notice that the refractive index $n^{(+)}=|{\bf k}|/\omega^{(+)}({\bf
k})$ does not depend on the wavelengths, $(\lambda=2\pi/|{\bf k}|)$, so that
dispersion does not occur. It is important to remark that the frequencies in
eqs. (36a) and (36b) have an asymmetry: $|\omega^{(-)}({\bf
k})|\neq|\omega^{(+)}({\bf k})|$. In the regime of strong magnetic fields, i.
e., $|{\bf B}_{0}|\gg|{\bf E}_{0}|$, the previous frequencies can be
simplified according to the expressions that follow:

$\displaystyle\omega^{(\pm)}({\bf k})\simeq\pm\,|{\bf
k}|\,e^{-\gamma}\sqrt{1+2\,e^{\gamma}\,\sinh(\gamma)\,\cos^{2}\theta}\;,$ (37)

where $\cos\theta=\hat{{\bf k}}\cdot\hat{{\bf B}}_{0}$. It is interesting to
highlight that this frequency coincides with the results given in ref. [28],
where the authors investigated a different configuration of external fields,
namely, (anti)parallel external fields.

In this condition, the frequency depends on the direction of the magnetic
field relative to the ${\bf k}$-wave vector, and consequently, the refractive
index also depends on the $\theta$-angle. In addition, the frequency (37) does
not depend on the magnetic field magnitude when $|{\bf
B}_{0}|\rightarrow\infty$. We also point out that the frequencies (36a) and
(36b) are degenerate at $\omega^{(+)}=\omega^{(-)}=d_{E}\,{\bf
B}_{0}\cdot({\bf k}\times{\bf E}_{0})$ if the ModMax parameter and the EM
background field satisfy the condition

$\displaystyle 1+d_{E}(\hat{{\bf k}}\cdot{\bf B}_{0})^{2}+d_{E}\,{\bf
B}_{0}^{2}=d_{E}(\hat{{\bf k}}\times{\bf E}_{0})^{2}+$
$\displaystyle+\,d_{E}^{2}\left({\bf E}_{0}^{2}-{\bf
B}_{0}^{2}\right)(\hat{{\bf k}}\cdot{\bf B}_{0})^{2}\;.$ (38)

On the other hand, if the EM background fields satisfy the inequality

$\displaystyle 1+d_{E}(\hat{{\bf k}}\cdot{\bf B}_{0})^{2}+d_{E}\,{\bf
B}_{0}^{2}>d_{E}(\hat{{\bf k}}\times{\bf E}_{0})^{2}+$
$\displaystyle+\,d_{E}^{2}\left({\bf E}_{0}^{2}-{\bf
B}_{0}^{2}\right)(\hat{{\bf k}}\cdot{\bf B}_{0})^{2}\;,$ (39)

both the dispersion relations (36a) and (36b) are real. Otherwise, if the
expression (IV) has the opposite inequality $(<)$, the dispersion relations
have imaginary parts. The refractive index associated with the
$\omega^{(+)}$-frequency is defined by

$\displaystyle n=\frac{1+d_{E}\,{\bf B}_{0}^{2}}{d_{E}\,{\bf
B}_{0}\cdot(\hat{{\bf k}}\times{\bf E}_{0})+\sqrt{1-d_{E}(\hat{{\bf
k}}\times{\bf E}_{0})^{2}-d_{E}^{2}\left({\bf E}_{0}^{2}-{\bf
B}_{0}^{2}\right)(\hat{{\bf k}}\cdot{\bf B}_{0})^{2}+d_{E}\,(\hat{{\bf
k}}\cdot{\bf B}_{0})^{2}+d_{E}\,{\bf B}_{0}^{2}}}\;,$ (40)

where the condition (IV) can be imposed to obtain a real refractive index. It
is opportune to point out that eq. (IV) with the opposite inequality $(<)$
implies into a dichroism effect in the refractive index (40). In the regime of
a strong magnetic field $(|{\bf B}_{0}|\gg|{\bf E}_{0}|)$, the previous
refractive index leads to

$\displaystyle
n\simeq\frac{e^{\gamma}}{\sqrt{1+(e^{2\gamma}-1)\cos^{2}\theta}}\;,$ (41)

which depends on the $\theta$-angle and $\gamma$-parameter.

The group velocity steaming from the polynomial equation (35) is

$\displaystyle{\bf v}_{g}=\frac{\omega\,\left[\,\left(1-d_{E}\,{\bf
E}_{0}^{2}\right)\,{\bf k}+d_{E}\,({\bf E}_{0}\cdot{\bf k})\,{\bf
E}_{0}+d_{E}\,({\bf B}_{0}\cdot{\bf k})\,{\bf B}_{0}+d_{E}\,\omega\,({\bf
E}_{0}\times{\bf B}_{0})\,\right]}{2\omega^{2}-{\bf
k}^{2}+d_{E}\,(\omega\,{\bf B}_{0}-{\bf k}\times{\bf
E}_{0})^{2}+d_{E}\,\omega\,{\bf B}_{0}\cdot(\omega\,{\bf B}_{0}-{\bf
k}\times{\bf E}_{0})-d_{E}\,({\bf B}_{0}\cdot{\bf k})^{2}}\;,$ (42)

where $\omega$ must be evaluated at the solutions $\omega=0$, (36a) and (36b).
The external electric and magnetic fields correspond to space anisotropies and
the group velocity of the wave is no longer exclusively along k; it develops
components in other directions too. The limit $\gamma\rightarrow 0$ in eq.
(42) recovers the usual result from Maxwell ED, ${\bf v}_{g}=c\,\hat{{\bf k}}$
(with $c=1$ in natural units). As expected, the solution $\omega=0$ yields a
null group velocity. In the regime of a strong magnetic background $(|{\bf
B}_{0}|\gg|{\bf E}_{0}|)$, the group velocity (42) is reduced to

$\displaystyle{\bf v}_{g}\simeq\,e^{-\gamma}\frac{\hat{{\bf
k}}+2\,e^{\gamma}\,\sinh\gamma\,\cos\theta\,\hat{{\bf
B}}_{0}}{\sqrt{1+2\,e^{\gamma}\,\sinh(\gamma)\,\cos^{2}\theta}}\;.$ (43)

Observe that this result also depends on the $\theta$-angle and has a
component on the magnetic background direction.

For the sake of simplicity, we consider the vectors ${\bf k}$, ${\bf B}_{0}$
and ${\bf E}_{0}$ perpendicular among themselves, i.e., ${\bf k}\cdot{\bf
B}_{0}={\bf k}\cdot{\bf E}_{0}={\bf B}_{0}\cdot{\bf E}_{0}=0$. The refraction
index of this medium associated with the $\omega^{(+)}$-frequency is given by

$n_{1}=\frac{E_{0}^{2}-B_{0}^{2}+2\,e^{-\gamma}\sinh(\gamma)\,B_{0}^{2}}{\left(E_{0}^{2}-B_{0}^{2}\right)e^{-\gamma}+2\,e^{-\gamma}\sinh(\gamma)\,E_{0}\,B_{0}}\;,$
(44)

when $E_{0}>B_{0}$, and

$n_{2}=\frac{E_{0}^{2}-B_{0}^{2}-2\,e^{\gamma}\,\sinh(\gamma)\,B_{0}^{2}}{\left(E_{0}^{2}-B_{0}^{2}\right)e^{\gamma}-2\,e^{\gamma}\sinh(\gamma)\,E_{0}\,B_{0}}\;,$
(45)

for the condition $B_{0}>E_{0}$. Here, we denote $E_{0}=|{\bf E}_{0}|$ and
$B_{0}=|{\bf B}_{0}|$. The $\gamma\rightarrow 0$ limit leads to
$n_{1}=n_{2}=1$. Although the $d_{E}$ coefficient is not defined for
$E_{0}\rightarrow B_{0}$, this limit leads to $n_{1}=n_{2}=1$, if $E_{0}\neq
B_{0}$. For the condition of a strong magnetic field, $B_{0}\gg E_{0}$, the
refraction index $n_{2}\simeq e^{-\gamma}$ does not depend on the magnetic
field.

Under the conditions of ${\bf k}\cdot{\bf B}_{0}={\bf k}\cdot{\bf E}_{0}={\bf
B}_{0}\cdot{\bf E}_{0}=0$, the group velocity associated with the previous
frequencies is obtained from eq. (42), namely,

${\bf v}_{g}=\frac{\hat{{\bf
k}}\,\omega\,\left[\,k-d_{E}\,B_{0}\,E_{0}\,\omega-
d_{E}\,E_{0}^{2}\,k\,\right]}{2\omega^{2}-k^{2}+d_{E}(\omega
B_{0}-k\,E_{0})^{2}+d_{E}\,\omega\,B_{0}(\omega B_{0}-kE_{0})}\;.$ (46)

For a strong magnetic field, we have ${\bf
v}_{g}\simeq\,e^{-\gamma}\,\hat{{\bf k}}$, the group velocity decays with the
ModMax parameter and propagates only on the $\hat{{\bf k}}$-direction. This
confirms the result (43) whenever $\theta=\pi/2$.

##  V Birefringence in presence of electric and magnetic backgrounds

In this Section, we investigate the birefringence phenomenon in the context of
ModMax ED. For this purpose, we need to obtain the variation of the refractive
index in relation to the magnetic background field. In what follows, we assume
the external fields as ${\bf B}_{0}=B_{0}\,\hat{{\bf z}}$ and ${\bf
E}_{0}=E_{0}\,\hat{{\bf y}}$.

Initially, let us consider the plane wave solution for the electric field into
the form ${\bf e}(x,t)=e_{03}\,\hat{{\bf z}}\,e^{i\left(kx-\omega t\right)}$.
Notice that the propagation direction is ${\bf k}=k\,\hat{{\bf x}}$ and the
wave amplitude is parallel to the magnetic background field. Under these
conditions, the wave equation (32) yields the parallel refractive index :

$\displaystyle
n_{\parallel}=\sqrt{\mu_{22}(E_{0},B_{0})\,\varepsilon_{33}(E_{0},B_{0})}=\sqrt{\frac{1+d_{E}\,B_{0}^{2}}{1-d_{E}\,E_{0}^{2}}}\;.$
(47)

In the second situation, the plane wave solution has the amplitude
perpendicular to the magnetic background field, ${\bf
e}(x,t)=e_{02}\,\hat{{\bf y}}\,e^{i\left(kx-\omega t\right)}$. In this case,
the perpendicular refractive index is

$\displaystyle
n_{\perp}=\sqrt{\mu_{33}(E_{0},B_{0})\,\varepsilon_{22}(E_{0},B_{0})}=1\;.$
(48)

The birefringence is defined by the difference of the refractive indices :

$\displaystyle\Delta
n=|n_{\parallel}-n_{\perp}|=\left|\,\sqrt{\frac{1+d_{E}\,B_{0}^{2}}{1-d_{E}\,E_{0}^{2}}}-1\,\right|\;.$
(49)

In the limit $\gamma\rightarrow 0$, we have $\Delta n=0$ and birefringence
disappears. Using the definitions of $d_{E}$ in eqs. (30a) and (30b), we
obtain the following difference

$\displaystyle\Delta n_{1}\\!$ $\displaystyle=$
$\displaystyle\\!\left|\,\sqrt{\frac{E_{0}^{2}\,e^{2\gamma}-B_{0}^{2}}{E_{0}^{2}-B_{0}^{2}\,e^{2\gamma}}}-1\,\right|\;,\;\;\mbox{if}\;\;E_{0}>B_{0}\;,\;$
(50a) $\displaystyle\Delta n_{2}\\!$ $\displaystyle=$
$\displaystyle\\!\left|\,\sqrt{\frac{B_{0}^{2}\,e^{2\gamma}-E_{0}^{2}}{B_{0}^{2}-E_{0}^{2}\,e^{2\gamma}}}-1\,\right|\;,\;\;\mbox{if}\;\;B_{0}>E_{0}\;.\;$
(50b)

The plot of $\Delta n_{1}$ versus $x\,(x=E_{0}/B_{0})$ (left panel), and
$\Delta n_{2}$ versus $x$ (right panel) are illustrated in fig. (1). The
vertical asymptote in these plots is located at $E_{0}=e^{\gamma}\,B_{0}$
(left panel), and at $E_{0}=e^{-\gamma}\,B_{0}$ in the right panel. Imposing
the causality and unitarity conditions on the ModMax ED [28], we choose the
$\gamma$-values with $\gamma>0$. In the plot of $\Delta n_{1}$, we choose the
values for the $\gamma$-ModMax parameter : $\gamma=0.5$ (black line),
$\gamma=0.8$ (blue line) and $\gamma=1.0$ (red line), in which $E_{0}>B_{0}$.
The $\gamma$-values in the plot of $\Delta n_{2}$ are $\gamma=0.5$ (black
line), $\gamma=1.0$ (blue line) and $\gamma=1.5$ (red line), that satisfies
the condition $B_{0}>E_{0}$.

Birefringence manifests in the region below the curves in fig. (1).

Figure 1: The variation of the refractive index $\Delta n_{1}$ (left panel)
and $\Delta n_{2}$ (right panel) as functions of the dimensionless variable
$x=E_{0}/B_{0}$. The left panel is plotted for $\gamma=0.5$ (black line),
$\gamma=0.8$ (blue line) and $\gamma=1.0$ (red line) in the range of
$E_{0}>B_{0}$. The right panel is for $\gamma=0.5$ (black line), $\gamma=1.0$
(blue line) and $\gamma=1.5$ (red line), with the condition of $B_{0}>E_{0}$,
respectively.

Under an intense magnetic background field, $B_{0}\gg E_{0}$ (or when
$E_{0}\rightarrow 0$) and $\gamma\ll 1$, the second solution (50b) reads as
given below:

$\displaystyle\Delta n_{2}\simeq\gamma\;,$ (51)

which agrees with the result given in ref. [29]. Similarly, if the electric
background field is intense, where $E_{0}\gg B_{0}$ (or when $B_{0}\rightarrow
0$) and $\gamma\ll 1$, the first solution in eq. (50a) leads to the same
result $\Delta n_{1}\simeq\gamma$. The birefringence curves go to infinity
when $x\rightarrow e^{\gamma}$ (left panel) and $x\rightarrow e^{-\gamma}$
(right panel).

##  VI Interaction energy for an Axionic ModMax electrodynamics under a
uniform magnetic field

The coupling of the axion to the photon has been widely investigated in
different scenarios. There is a rich literature on the issue, with special
attention to the Primakoff effect (see, for instance, the review [54] and
references therein). In the present Section, we pursue the investigation of
the low-energy interparticle potential that emerges from a scenario where the
axion is coupled to the photon as described by our object of study, ModMax ED.
It is true that we are taking an unusual viewpoint, by bringing together two
new physics: ModMax with its $\gamma$-parameter, and the axionic sector, with
the axion mass, $m_{a}$, and the axion-photon coupling constant,
$g_{a\gamma\gamma}$. We justify our attitude by calling into question our
interest in understanding how the mentioned three parameters interfere with
each other in the screening and the confining sectors of the particle-
antiparticle potential, after we integrate out the axion contributions to get
an effective photonic mediation of the interaction. Moreover, if an external
magnetic field is switched on, it would be interesting to understand how
${\gamma}$, $m_{a}$, $g_{a\gamma\gamma}$ and the external magnetic field, $\bf
B$ , combine to screen and to a confining tail to the interparticle potential.
This shall be the content of this Section.

As already expressed, we shall now discuss the interaction energy between
static point-like sources for an axionic ModMax electrodynamics under a
uniform magnetic field, along the lines of refs. [27, 25, 26, 42, 43]. This
can be done by computing the expectation value of the energy operator $H$ in
the physical state $\left|\Phi\right\rangle$, which we denote by $\langle
H\rangle_{\Phi}$. However, before going to the derivation of the interaction
energy, we will describe very briefly the model under consideration. The
initial point of our analysis is the Lagrangian density

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\cosh\left(\gamma\right){\cal
F}+\sinh\left(\gamma\right)\sqrt{{{\cal F}^{2}}+{{\cal
G}^{2}}}+\frac{1}{2}{\left({{\partial_{\mu}}a}\right)^{2}}$ (52)
$\displaystyle-\frac{1}{2}\,{m_{a}^{2}}\,a^{2}+{g_{a\gamma\gamma}}\,a\,{\cal
G}\;,$

where $a$ is the axion field and $g_{a\gamma\gamma}$ has dimension of
${(\mbox{mass})^{-1}}$.

Following our earlier procedure [42, 43], if we consider the model in the
limit of a very heavy $a$-field and we are bound to energies much below
$m_{a}$, we are allowed to integrate over $a$ and to speak about an effective
model for the $A^{\mu}$-ModMax field. Once this is done, we arrive at the
following effective Lagrangian density :

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\cosh\left(\gamma\right){\cal
F}+\sinh\left(\gamma\right)\sqrt{{{\cal F}^{2}}+{{\cal G}^{2}}}$ (53)
$\displaystyle+$ $\displaystyle\frac{{{g_{a\gamma\gamma}^{2}}}}{2}\,{\cal
G}\,\frac{1}{{\left({\Box+{m_{a}^{2}}}\right)}}\,{\cal G}\;,$

where we define the D’Alembertian operator
$\Box\equiv{\partial_{\mu}}{\partial^{\mu}}$.

Since we are interested in estimating the lowest-order interaction energy, we
will linearize the above effective theory following the procedure that led to
the equation (20). Thus, in the case of a pure magnetic background, we make
${\bf E}={\bf 0}$, in which the effective Lagrangian density simplifies to

$\displaystyle{\cal L}\\!$ $\displaystyle=$
$\displaystyle\\!-\frac{1}{4}\,{e^{-\gamma}}\,{f_{\mu\nu}}\,{f^{\mu\nu}}-\frac{1}{2}\,{e^{-\gamma}}\,F_{B\mu\nu}\,{f^{\mu\nu}}$
(54) $\displaystyle+\frac{1}{4}\,\frac{{\sinh\left(\gamma\right)}}{{{\bf
B}_{0}^{2}}}\,\tilde{F}_{B\mu\nu}\tilde{F}_{B\kappa\lambda}\,{f^{\mu\nu}}\,{f^{\kappa\lambda}}$
$\displaystyle+\frac{{{g_{a\gamma\gamma}^{2}}}}{8}\,\tilde{F}_{B\mu\nu}\,{f^{\mu\nu}}\,\frac{1}{{\left({\Box+{m_{a}^{2}}}\right)}}\,\tilde{F}_{B\kappa\lambda}\,{f^{\kappa\lambda}}\;.$

In passing, we note that the effective model described by the Lagrangian
density (54) is a theory with non-local time derivative. However, as we have
already explained in refs. [42, 43], we recall that this section is aimed at
studying the static potential, so that $\Box$ can be replaced by the spatial
operator $-{\nabla^{2}}$. As before, we will maintain $\Box$, but it should be
borne in mind that this paper essentially deals with the static case.

With this in hand, the canonical quantization of this effective theory from
the Hamiltonian analysis point of view is straightforward. The canonical
momenta reads

$\displaystyle{\Pi^{\mu}}\\!$ $\displaystyle=$
$\displaystyle\\!-{e^{-\gamma}}{f^{0\mu}}-{e^{-\gamma}}\,F_{B}^{\;\;\,0\mu}+\frac{{\sinh\left(\gamma\right)}}{{{\bf
B}_{0}^{2}}}\,\tilde{F}_{B\kappa\rho}\,{f^{\kappa\rho}}\,\tilde{F}_{B}^{\;\;\,0\mu}$
(55)
$\displaystyle+\frac{{{g_{a\gamma\gamma}^{2}}}}{2}\,\tilde{F}_{B\kappa\rho}{f^{\kappa\rho}}\,\frac{1}{{\left({\Box+{m_{a}^{2}}}\right)}}\,{\tilde{F}_{B}^{\;\;\,0\mu}}\,,$

which produces the usual primary constraint

$\Pi^{0}=0\;,$ (56)

and

$\Pi_{i}=-\left\\{{e^{-\gamma}}\,{{\delta_{ij}}+2{B_{i}}{B_{j}}\left[{\frac{{\sinh\left(\gamma\right)}}{{{\bf
B}_{0}^{2}}}+\frac{{{\raise
2.15277pt\hbox{$\scriptstyle{{g_{a\gamma\gamma}^{2}}}$}\kern-1.00006pt/\kern-1.49994pt\lower
1.07639pt\hbox{$\scriptstyle
2$}}}}{{\left({\Box+{m_{a}^{2}}}\right)}}}\right]}\right\\}{e_{j}}\;.$ (57)

Let us also mention here that the electric field due to the fluctuation takes
the form

${e_{i}}=\frac{1}{{u\det D}}\left({{\delta_{ij}}\det
D-\frac{1}{{{\Omega^{2}}}}{B_{i}}{B_{j}}}\right){\Pi_{j}}\;,$ (58)

where $u={e^{-\gamma}}$ and $\det D=1+\frac{{{{\bf B}^{2}}}}{{{\Omega^{2}}}}$,
whereas

$\displaystyle\frac{1}{{{\Omega^{2}}}}=2\,{e^{\gamma}}\left[\,{\frac{{\sinh\left(\gamma\right)}}{{{\bf
B}^{2}}}+\frac{{{\raise
2.15277pt\hbox{$\scriptstyle{{g_{a\gamma\gamma}^{2}}}$}\kern-1.00006pt/\kern-1.49994pt\lower
1.07639pt\hbox{$\scriptstyle
2$}}}}{{\left({\Box+{m_{a}^{2}}}\right)}}}\,\right]\;.$ (59)

Recalling again that ${\bf B}$ represents the external (background) magnetic
field around which the $a^{\mu}$-field fluctuates.

The canonical Hamiltonian can be worked as usual and is given by

$\displaystyle{H_{C}}\\!$ $\displaystyle=$
$\displaystyle\\!\int{{d^{3}}x\left[\,{{\Pi^{i}}\,{\partial_{i}}{a_{0}}+\frac{{{{\bf\Pi}^{2}}}}{{2u}}+\frac{1}{2}u\,{{\bf
b}^{2}}+u\,{\bf B}\cdot{\bf b}}\,\right]}$ (60)
$\displaystyle-\int{{d^{3}}x}\;\,\frac{1}{{2u\,{\Omega^{2}}\,\det
D}}\,\left({{\bf B}\cdot{\bf b}}\right)^{2}\,.$

Time conserving the primary constraint, ${\Pi}_{0}$, immediately yields the
secondary constraint, ${\Gamma_{1}}\equiv{\partial_{i}}{\Pi^{i}}=0$, which is
the Gauss constraint, and together displays the first-class structure of the
theory. The extended Hamiltonian that generates translations in time is now
found to be

$\displaystyle
H=H_{C}+\int{d^{3}}x\left[\,{c_{0}\left(x\right)\Pi_{0}\left(x\right)+c_{1}\left(x\right)\Gamma_{1}\left(x\right)}\,\right]\;,$
(61)

where $c_{0}\left(x\right)$ and $c_{1}\left(x\right)$ are the Lagrange
multipliers. As before, neither $a^{0}$ nor $\Pi^{0}$ are of interest in
describing the system and may be discarded of the theory. Thus we are left
with the following expression for the Hamiltonian

$H=\int{{d^{3}}x\left[\,{c\left(x\right){\partial_{i}}{\Pi^{i}}+\frac{{{{\bf\Pi}^{2}}}}{{2\,u}}-\frac{1}{{2\,u\,{\Omega^{2}}\det
D}}\,{{\left({{\bf B}\cdot{\bf b}}\right)}^{2}}}\,\right]}\,,$ (62)

we have defined $c(x)=c_{1}(x)-a_{0}(x)$.

To fix gauge symmetry we adopt the gauge discussed previously [27], that is,

$\Gamma_{2}\left(x\right)\equiv\int\limits_{C_{\zeta
x}}{dz^{\nu}}\,a_{\nu}\left(z\right)\equiv\int\limits_{0}^{1}{d\lambda\,x^{i}}a_{i}\left({\lambda
x}\right)=0\,.$ (63)

Here $\lambda$ $(0\leq\lambda\leq 1)$ is the parameter describing the space-
like straight path $x^{i}=\zeta^{i}+\lambda\left({x-\zeta}\right)^{i}$, and
$\zeta^{i}$ is a fixed point (reference point). There is no essential loss of
generality if we restrict our considerations to $\zeta^{i}=0$. With such a
choice, the fundamental Dirac bracket is given by

$\displaystyle\left\\{{a_{i}\left({\bf x}\right),\Pi^{j}\left({\bf
y}\right)}\right\\}^{\ast}\\!$ $\displaystyle=$
$\displaystyle\\!\delta_{i}^{\;\,j}\,\delta^{\left(3\right)}\left({{\bf
x}-{\bf y}}\right)$ (64)
$\displaystyle-\partial_{i}^{x}\int\limits_{0}^{1}{d\lambda\,x^{j}}\delta^{\left(3\right)}\left({\lambda{\bf
x}-{\bf y}}\right)\,.$

Next, we recall that the physical states $\left|\Phi\right\rangle$ are gauge-
invariant [42, 43]. In that case we consider the stringy gauge-invariant state

$\displaystyle\left|\Phi\right\rangle\\!$ $\displaystyle\equiv$
$\displaystyle\\!\left|\,{\overline{\Psi}\left({\bf y}\right)\Psi\left({{{\bf
y}^{\prime}}}\right)}\,\right\rangle$ (65) $\displaystyle=$
$\displaystyle\overline{\Psi}\left({\bf y}\right)\exp\left({iq\int_{{{\bf
y}^{\prime}}}^{\bf y}{d{z^{i}}{a_{i}}\left(z\right)}}\right)\Psi\left({{{\bf
y}^{\prime}}}\right)\left|0\right\rangle\,,$

where the line integral is along a space-like path on a fixed time slice, $q$
is the fermion charge and $\left|0\right\rangle$ is the physical vacuum state.

This leads us to the expectation value ${\left\langle H\right\rangle_{\Phi}}$

${\left\langle H\right\rangle_{\Phi}}={\left\langle
H\right\rangle_{0}}+\left\langle H\right\rangle_{\Phi}^{\left(1\right)}\;,$
(66)

where ${\left\langle H\right\rangle_{0}}=\left\langle
0\right|H\left|0\right\rangle$, whereas the $\left\langle
H\right\rangle_{0}^{\left(1\right)}$ term is given by

$\left\langle
H\right\rangle_{\Phi}^{\left(1\right)}=-\frac{{{e^{-\gamma}}}}{2}\left\langle\Phi\right|\int{{d^{3}}x}\,{\Pi^{i}}\,\frac{{\left({{\nabla^{2}}-{m_{a}^{2}}}\right)}}{{\left({{\nabla^{2}}-{M^{2}}}\right)}}\,{\Pi_{i}}\,\left|\Phi\right\rangle\;,$
(67)

where ${M^{2}}={m_{a}^{2}}+{g_{a\gamma\gamma}^{2}}\,{{\bf
B}^{2}}\,{e^{-\gamma}}$.

Following our earlier procedure [42, 43], when $g_{a\gamma\gamma}\to 0$, the
static potential profile for two opposite charges located at ${\bf y}$ and
${\bf y}^{\prime}$ then reads

$V(L)=-\frac{{{q^{2}}}}{{4\pi}}\,{e^{-\gamma}}\,\frac{1}{L}\,,$ (68)

where $L\equiv|{\bf y}-{{\bf y}^{\prime}}|$ is the distance that separates the
two charges. It is also, up to the ${e^{-\gamma}}$ factor, just the Coulomb
potential. This result agrees with that of ref. [30], and finds here an
independent derivation. While in the case $g_{a\gamma\gamma}\neq 0$, the
interaction energy takes the form

$V(L)=-\frac{{{q^{2}}}}{{4\pi}}\,{e^{-\gamma}}\,\frac{{{e^{-ML}}}}{L}+\frac{{{q^{2}}{m_{a}^{2}}}}{{8\pi}}\,{e^{-\gamma}}\left[{\ln\left({1+\frac{{{\Lambda^{2}}}}{{{M^{2}}}}}\right)}\right]L\,,$
(69)

where $\Lambda$ is a cutoff. The next step is to give a physical meaning to
the cutoff $\Lambda$. Proceeding in the same way as we did before [42, 43], we
recall that our effective model for the electromagnetic field is an effective
theory that arises from the integration over the $a$-field, whose excitation
is massive. In this case, the Compton wavelength of this excitation
($\ell=m_{a}^{-1}$) defines a correlation distance. In view of this situation,
we see that physics at distances of the order or lower than $m_{a}^{-1}$ must
necessarily take into account a microscopic description of the $a$-fields. By
this we mean that, if we work with energies of the order or higher than
$m_{a}$, our effective description with the integrated effects of $a$ is no
longer sensible. As a consequence of this, we can identify $\Lambda$ with
$m_{a}$. This then implies that the static potential profile assumes the form

$V(L)=-\frac{{{q^{2}}}}{{4\pi}}\,{e^{-\gamma}}\,\frac{{{e^{-ML}}}}{L}+\frac{{{q^{2}}{m_{a}^{2}}}}{{8\pi}}\,{e^{-\gamma}}\left[{\ln\left({1+\frac{{{m_{a}^{2}}}}{{{M^{2}}}}}\right)}\right]L\,.$
(70)

Again, up to the ${e^{-\gamma}}$ factor, we mention that similar forms of
interaction potentials have been reported before from different viewpoints.
For example, in the context of the Standard Model with an anomalous triple
gauge boson couplings [42], in connection with anomalous photon and $Z$-boson
self couplings from the Born-Infeld weak hypercharge action [43], also in a
theory of antisymmetric tensor fields that results from the condensation of
topological defects [55], and in a Higgs-like model [56].

##  VII Conclusions

In this contribution, we study the modified Maxwell electrodynamics (ModMax
ED) and its propagation properties in uniform electric and magnetic background
fields. By expanding the ModMax Lagrangian density up to second order in the
propagating photon field, we obtain the permittivity and permeability tensors,
dispersion relations, refractive index and group velocity as functions of the
wave propagation direction, ModMax parameter and external fields. In the
regime of a strong magnetic field, the dispersion relations for plane wave
solutions depend only on the ModMax parameter and $\theta$-angle between the
wave propagation direction and magnetic background field. In this regime, we
also find that the refractive index and group velocity decay with the ModMax
parameter in the situation where the wave vector, the electric and magnetic
background fields are perpendicular to each other.

We discuss the birefringence phenomenon by taking into account the difference
of the refractive indices for the wave amplitude parallel and perpendicular to
the external magnetic field. We obtain the region of birefringence in the
plots of fig. (1). In both regimes of strong magnetic or electric fields,
birefringence is approximately given by the ModMax parameter, which confirms
the result obtained previously in the ref. [29]. The well-known results of
Maxwell ED are recovered in the limit where the ModMax parameter goes to zero.

Finally, using the gauge-invariant but path-dependent variables formalism, we
have computed the static potential profile for an axionic ModMax
electrodynamics under a uniform magnetic field. Once again, we have exploited
a correct identification of field degrees of freedom with observable
quantities. Interestingly, the static potential profile contains a linear
potential leading to the confinement of static charges. As already expressed
similar forms of interaction potentials have been reported before from
different viewpoints [42, 43, 55, 56].

Having in mind the possible relevance of the ModMax model to describe non-
linear electromagnetic effects, we call into question its application to study
a number of physical properties of Dirac materials. In the work of ref. [57],
the authors show that the latter may display electromagnetic non-linearities
at magnetic fields as low as $1T$. We point out that reassessing the
inspection of magnetic enhancement of the dielectric constant of insulators
and, on the other hand, possible electric modulation of magnetization could be
a good path to further investigate the potentialities of ModMax.

A direct contact we might establish between ModMax and electroweak physics
could be through the issue of the photon and $Z$-boson self-couplings by
associating the weak hypercharge symmetry to a ModMax description. In the work
of Ref. [43], we adopt a Born-Infeld description for the weak hypercharge and
consider the $Z$-decay channel into three photons to constrain the Born-
Infeld parameter. By going along the same lines with ModMax, we could get a
bound on the $\gamma$-parameter by considering the $Z$-three photon anomalous
vertex and the decay of the $Z$-boson into three photons.

## Acknowledgments

L. P. R. Ospedal is grateful to the CNPq - Conselho Nacional de
Desenvolvimento Científico e Tecnológico (Brazil) - supported by Grant
166386/2020-0 for his postdoctoral fellowship. P. Gaete was partially
supported by Grant No. ANID PIA/APOYO AFB220004 (Chile).

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