SUSY QED with Lorentz-asymmetric fermionic matter and a glance at the
electron’s EDM

João Paulo S. Meloa ***E-mail address<EMAIL_ADDRESS>Wagno Cesar e
Silvab †††E-mail address<EMAIL_ADDRESS>and José A. Helayël-Netoa
‡‡‡E-mail address<EMAIL_ADDRESS>

${a)}$ Centro Brasileiro de Pesquisas Físicas,

Rua Dr. Xavier Sigaud 150, Urca, 22290-180, Rio de Janeiro, Brazil

${b)}$ Departamento de Física, ICE, Universidade Federal de Juiz de Fora,

Juiz de Fora, 36036-900, Minas Gerais, Brazil

###### Abstract

This contribution sets out to pursue the investigation of a supersymmetric
electrodynamics model with Lorentz-symmetry violation (LSV) manifested by a
space-time unbalance in the propagation of the fermionic charged matter.
Despite violation of Lorentz symmetry, the supersymmetry algebra is kept
untouched. We then adopt a superspace approach to build up an
$\mathcal{N}=1$-supersymmetric Abelian gauge theory in presence of a Lorentz-
violating background supermultiplet that accommodates the space-time asymmetry
parameter of the charged matter. We describe, in this scenario, how the
particular Lorentz-symmetry breaking, brought about by the fermionic matter,
affects its (matter) scalar partners and the photon/photino that minimally
couple to charged matter. From the (modified) Dirac, Klein-Gordon and Maxwell
field equations, we work out the corresponding dispersion relations to inspect
and discuss the physical effects of the LSV Majorana fermion condensates that
naturally emerge from the background supermultiplet. Finally, we target
efforts to investigate the Gordon decomposition of the charged lepton
electromagnetic current. This is carried out by iterating the (fermion and
scalar) matter field equations, which points to an effective contribution to
the electron’s electric dipole moment (EDM). This result allows us to attain
an estimate of the pseudo-vector condensate of the (LSV) Majorana background
fermion.

Keywords: Lorentz-symmetry violation, Supersymmetric QED, modified dispersion
relations, electron’s EDM.

##  1 Introduction

Lorentz invariance stands as an essential cornerstone of Quantum Field Theory,
manifesting as a fundamental symmetry within the framework of General
Relativity and the Standard Model of Particle Physics. However, in recent
decades, the possibility that some spacetime symmetries could be violated at
energies close to the Planck scale $(M_{\textrm{Planck}}\sim
10^{19}\,\textrm{GeV})$ has arisen in the context of different approaches,
such as, e.g., String Theory [1, 2], Loop Quantum Gravity [3, 4] and Modified
Electrodynamics [5]. At the same time, in this high-energy regime,
Supersymmetry (SUSY) is expected to keep unbroken [6]. In this way, it becomes
pretty natural to wonder about the relationship between the Lorentz-symmetry
violation (LSV) and SUSY breaking. Berger and Kostelecký were pioneers in
proposing a method for incorporating Lorentz-breaking terms into a SUSY
scenario, which is achieved by introducing background anisotropies into the
SUSY algebra [7, 6]. Subsequently, several approaches based on supersymmetric
extensions of Quantum Electrodynamics (QED) with LSV have emerged. These
approaches have led to intriguing phenomena such as vacuum birefringence [8],
as well as the manifestation of SUSY-breaking arising from the mass splitting
of superpartners [9].

On the other hand, the possibility of LSV in other scenarios gives rise to
interesting developments, e.g., in Condensed Matter Physics (CMP), where both
SUSY and Lorentz invariance may appear as emergent symmetries. The first
indication of SUSY in CMP has come out from the study of a two-dimensional
Ising model [14], where an emergent SUSY becomes apparent near the critical
point. In this context, the fundamental degrees of freedom may not exhibit
supersymmetry, but an effective description via a supersymmetric action can be
obtained from them. From this perspective, more recent works have contemplated
scenarios of emergent SUSY in a $(2+1)$-dimensional system composed of a
monolayer of carbon atoms arranged in a honeycomb lattice [15, 16, 17].
Moreover, such scenarios can also arise dynamically as pointed out, e.g., in
Refs. [18, 19, 20, 21]. It should also be mentioned that there are systems in
low-energy CMP where the emergent Lorentz-symmetry plays an important role; it
is the case of the so-called Dirac and Weyl materials [22, 23, 24, 25, 26, 27,
28, 29, 30], topological insulators [31, 32, 33, 34, 35] and graphene [36,
37]. In this regard, the interesting work [38] presents a comprehensive
exploration of quasi-particle excitations in the band structures of Dirac and
Weyl materials. According to the authors, the emergent Lorentz-symmetry
features in these materials shed light on Lorentz violation within the
Standard Model Extension (SME) framework [39, 40, 41].

Considering this perspective, the efforts of the present contribution are
directly motivated by the theoretical study guided by Isobe and Nagaosa of
quantum critical phenomena in the phase transition between the trivial and
topological insulators in $(3+1)$ dimensions, which is described by a Dirac
fermion coupled to the electromagnetic field [42]. In their model, LSV arises
due to a constant parameter present only in the purely fermionic piece of the
action. Our proposal consists in building up a $U(1)$ gauge theory from an
extended version of the Isobe-Nagaosa model, in which the LSV parameter may be
taken to be space-time dependent. The main purpose is to understand how the
LSV, initially manifested only in the fermionic sector, influences the other
sectors of the gauge-invariant theory.For this, we shall adopt an alternative
approach to implement the LSV, as was first introduced in [12, 13]. Unlike the
viewpoint presented by Berger and Kostelecký, this framework treats the
Lorentz-breaking parameters as components of a special supermultiplet while
SUSY algebra is kept unchanged. In this case, the central idea consists in
seeking a supersymmetric formulation for the background structures that
constitute the spacetime anisotropies responsible for the LSV. This allows us
to present a microscopic origin for the LSV background structures.

The paper is organized as follows: In Sect. 2, we briefly review the extended
Isobe-Nagaosa model, wherein the LSV parameter exhibits spacetime dependence,
and present a formulation of the superspace consistent with such a model. In
Sect. 3, we work to get a supersymmetric $U(1)$ gauge theory, based on the
extended Isobe-Nagaosa model, and investigate the modifications induced by
Lorentz-breaking background components. Next, in Section 4, we study the
modified field equations and dispersion relations for both the fermionic and
scalar matter sectors, and for the Maxwell sector as well. These results shall
be used, in what follows, to obtain the rest energy and group velocity for
some particular cases. In Sect. 5, we derive the effective electric dipole
interaction and, based on current experimental results for the electron EDM,
we present an estimate for one of the background Majorana fermion condensates.
Finally, in Sect. 6, we draw our Conclusive Considerations and Perspectives.
Two Appendices complement the main text. In the Appendix A, one can find an
alternative approach to generate the $v_{\mu}\bar{\Psi}\gamma^{\mu}\Psi$-term
from a vector superfield. Next, in Appendix B, we collect useful results
concerning the calculation of our dispersion relations worked out in in
Section 4.

The work with the Minkowski signature $(+,-,-,-)$. Also, we adopt the natural
system of units, with $c=1$ and $\hbar=1$.

##  2 The extended Isobe-Nagaosa model

We start off from the following extended version of the model proposed by
Isobe-Nagaosa,

$\displaystyle\mathcal{L}=i\bar{\Psi}\Big{[}\gamma^{0}\partial_{0}-\textrm{Re}(\xi)\gamma^{i}\partial_{i}\Big{]}\Psi-\dfrac{1}{2}\Big{\\{}\partial_{i}\textrm{Im}\big{[}\xi(x)\big{]}\Big{\\}}\bar{\Psi}\gamma^{i}\Psi-m\bar{\Psi}\Psi,$
(1)

where $\Psi$ is a Dirac fermion and $\xi$ is a factor that breaks Lorentz-
symmetry. In the extended model (1), we introduce a space-time dependence on
the background parameter $\xi$, since, in a general situation, such a
parameter may not be constant and/or homogeneous, and it would be pertinent to
exploit consequences of the non-constancy and non-homogeneity. In order to
guarantee the reality of the action associated with (1), it follows that the
real part of $\xi$, $\textrm{Re}(\xi)$, must be a real constant parameter,
while its imaginary part, $\textrm{Im}\big{[}\xi(x)\big{]}\equiv I(x)$,
carries all the eventual space-time dependence of the $\xi$ factor.

In what follows, it is convenient to introduce the covariant notation in the
form

$\displaystyle\tilde{\partial}_{\mu}$
$\displaystyle=(\partial_{0},\textrm{Re}(\xi)\partial_{i}),$ (2)
$\displaystyle v_{\mu}$ $\displaystyle=(0,-\partial_{i}I(x)).$ (3)

Then, the Lagrangian (1) can be brought into the form below:

$\displaystyle\mathcal{L}$
$\displaystyle=\dfrac{i}{2}\bar{\Psi}\gamma^{\mu}(\tilde{\partial}_{\mu}\Psi)-\dfrac{i}{2}(\tilde{\partial}_{\mu}\bar{\Psi})\gamma^{\mu}\Psi+\dfrac{1}{2}v_{\mu}\bar{\Psi}\gamma^{\mu}\Psi-m\bar{\Psi}\Psi.$
(4)

As a step further, considering a Dirac spinor in the Weyl representation, one
can arrange the $\Psi$-components according to their chirality,

$\displaystyle\Psi=\begin{pmatrix}\psi_{\alpha}\\\
\bar{\chi}^{\dot{\alpha}}\end{pmatrix},$ (5)

where the Van der Waerden notation has been adopted. Defining
$\sigma^{\mu}=(I_{2},\sigma^{i})$ and
$\bar{\sigma}^{\mu}=(I_{2},-\sigma^{i})$, the $\gamma$-matrices in Weyl
representation are written as

$\displaystyle\gamma^{\mu}=\begin{pmatrix}0&\sigma^{\mu}\\\
\bar{\sigma}^{\mu}&0\end{pmatrix}\;\;\;\;\;\textrm{and}\;\;\;\;\;\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=\begin{pmatrix}-I_{2}&0\\\
0&I_{2}\end{pmatrix}.$ (6)

Thus, the Lagrangian (1) acquires the form

$\displaystyle\mathcal{L}=$
$\displaystyle\;\dfrac{i}{2}\left[\bar{\psi}\bar{\sigma}^{\mu}(\tilde{\partial}_{\mu}\psi)-(\tilde{\partial}_{\mu}\bar{\psi})\bar{\sigma}^{\mu}\psi+\bar{\chi}\bar{\sigma}^{\mu}(\tilde{\partial}_{\mu}\chi)-(\tilde{\partial}_{\mu}\bar{\chi})\bar{\sigma}^{\mu}\chi\right]$
$\displaystyle+\,\dfrac{1}{2}v_{\mu}(\bar{\psi}\bar{\sigma}^{\mu}\psi-\bar{\chi}\bar{\sigma}^{\mu}\chi)-m(\bar{\psi}\bar{\chi}+\chi\psi),$
(7)

with two chiral sectors, left and right, described by the $\psi$\- and
$\chi$-spinors, respectively.

###  2.1 The SUSY-extended model in superspace

In order to incorporate SUSY in the framework of the extended fermionic
Lagrangian (2), and to carry out its coupling with the photon and photino, we
adopt an approach that keeps SUSY algebra intact. We call into question that,
instead of modifying SUSY algebra to include the LSV parameters, the latter
are rather accommodated as components of a background superfield. To implement
this point of view, one should remember the definition of four-divergence
operator, $\tilde{\partial}_{\mu}$, presented in (2) and define the position
four-vector as follows:

$\displaystyle\tilde{x}_{\mu}=\bigg{(}x_{0},-\dfrac{1}{\textrm{Re}(\xi)}x_{i}\bigg{)},$
(8)

in such a way that $\tilde{\partial}_{\mu}\tilde{x}^{\nu}=\delta^{\nu}_{\mu}$.
Then, we define the superspace coordinates, adopting the Salam-Strathdee
formulation [44] (see, e.g., [45] for the textbook level introduction), as
$\big{\\{}\tilde{x}^{\mu};\,\theta_{\alpha},\,\bar{\theta}_{\dot{\alpha}}\big{\\}}$
with $\mu$ being a space-time index and $\alpha=(1,2)$, as well as
$\dot{\alpha}=(1,2)$, being indices of Grassmannian nature. That said, one can
obtain the form of the SUSY generators,

$\displaystyle
Q_{\alpha}=-i\partial_{\alpha}-(\sigma^{\mu})_{\alpha\dot{\beta}}\bar{\theta}^{\dot{\beta}}\tilde{\partial}_{\mu},\;\;\;\;\;\bar{Q}_{\dot{\alpha}}=i\bar{\partial}_{\dot{\alpha}}+\theta^{\beta}(\sigma^{\mu})_{\beta\dot{\alpha}}\tilde{\partial}_{\mu},$
(9)

and establish the anti-commutation relations

$\displaystyle\left\\{D_{\alpha},Q_{\beta}\right\\}=\left\\{\bar{D}_{\dot{\alpha}},\bar{Q}_{\dot{\beta}}\right\\}=\left\\{D_{\alpha},\bar{Q}_{\dot{\beta}}\right\\}=\left\\{\bar{D}_{\dot{\alpha}},Q_{\beta}\right\\}=0,$
(10)

with the SUSY covariant derivative operators given by

$\displaystyle D_{\alpha}$
$\displaystyle=\partial_{\alpha}+i(\sigma^{\mu})_{\alpha\dot{\beta}}\bar{\theta}^{\dot{\beta}}\tilde{\partial}_{\mu},$
(11) $\displaystyle\bar{D}_{\dot{\alpha}}$
$\displaystyle=\bar{\partial}_{\dot{\alpha}}+i\theta^{\beta}(\sigma^{\mu})_{\beta\dot{\alpha}}\tilde{\partial}_{\mu}.$
(12)

From (9), it is clear that the SUSY algebra is left untouched, as already
mentioned.

##  3 $\mathcal{N}=1$-supersymmetric $U(1)$ gauge-field sector

In what follows, we shall apply the ideas presented in the previous Section to
build up a supersymmetric version of QED. First, let us investigate the
modifications the formulation of superspace, based on the fermionic matter
sector, in the the gauge field sector.

###  3.1 The Maxwell sector

Consider a real vector superfield, $A(\tilde{x},\theta,\bar{\theta})$, in the
Wess-Zumino gauge (see, e.g., the textbook [45] for more details),

$\displaystyle
A_{\textrm{WZ}}=i\theta^{2}\bar{\theta}\bar{\lambda}-i\bar{\theta}^{2}\theta\lambda+(\theta\sigma^{\mu}\bar{\theta})\tilde{a}_{\mu}+\dfrac{1}{2}\theta^{2}\bar{\theta}^{2}K,$
(13)

where $\lambda$ is the Weyl component of a Majorana spinor, $\tilde{a}_{\mu}$
is the four-vector and $K$ is an auxiliary field. To be consistent with the
definitions of superspace and superfields we are considering,
$\tilde{a}_{\mu}$ must have the form

$\displaystyle\tilde{a}_{\mu}=\left(A_{0},-\frac{1}{\textrm{Re}(\xi)}A_{i}\right).$
(14)

Here, it is clear that the Lorentz violation in the Maxwell sector is signaled
exclusively by the presence of $\textrm{Re}(\xi)$. Hence, we can expect that
this sector is not affected by the space-time dependence of the background.

In this case, the generalization of the Abelian gauge condition for the
modified electromagnetic four-potential is given by

$\displaystyle\tilde{a}_{\mu}\rightarrow\tilde{a}^{\prime}_{\mu}=\tilde{a}_{\mu}-\tilde{\partial}_{\mu}\big{[}2\textrm{Im}(z)\big{]},$
(15)

with $\textrm{Im}(z)$ standing for the imaginary part of the complex scalar
component of some chiral superfield. The kinetic term for $\tilde{a}_{\mu}$
can be generated by the contractions

$\displaystyle W^{\alpha}W_{\alpha}\Big{|}_{\sim\theta^{2}}$
$\displaystyle=-\dfrac{1}{2}\mathcal{F}^{\mu\nu}\mathcal{F}_{\mu\nu}+\dfrac{i}{2}{{}^{*\\!\\!}\mathcal{F}^{\mu\nu}}\mathcal{F}_{\mu\nu}-2i\lambda\sigma^{\mu}\tilde{\partial}_{\mu}\bar{\lambda}+K^{2},$
(16)
$\displaystyle\bar{W}_{\dot{\alpha}}\bar{W}^{\dot{\alpha}}\Big{|}_{\sim\bar{\theta}^{2}}$
$\displaystyle=-\dfrac{1}{2}\mathcal{F}^{\mu\nu}\mathcal{F}_{\mu\nu}-\dfrac{i}{2}{{}^{*\\!\\!}\mathcal{F}^{\mu\nu}}\mathcal{F}_{\mu\nu}+2i(\tilde{\partial}_{\mu}\lambda)\sigma^{\mu}\bar{\lambda}+(K^{*})^{2},$
(17)

where $\mathcal{F}_{\mu\nu}$ is the electromagnetic field strength and
$\lambda$ is identified as the photino spinor field. In the expressions (16)
and (17), we define the spinorial superfield,
$W_{\alpha}=-\dfrac{1}{4}\bar{D}\bar{D}D_{\alpha}A_{\textrm{WS}}$, with
$\bar{W}_{\dot{\alpha}}$ representing its complex conjugate, and use the
compact notations

$\displaystyle\mathcal{F}_{\mu\nu}=(\tilde{\partial}_{\mu}\tilde{a}_{\nu}-\tilde{\partial}_{\nu}\tilde{a}_{\mu}),\;\;\;\;\;{{}^{*\\!\\!}\mathcal{F}^{\mu\nu}}=\dfrac{1}{2}\varepsilon^{\mu\nu\kappa\rho}\mathcal{F}_{\kappa\rho}.$
(18)

With all described above, the supersymmetric gauge field action reads as
follows:

$\displaystyle S_{\textrm{M}}$ $\displaystyle=\dfrac{1}{4}\int
d^{4}x\,\biggl{\\{}\int d^{2}\theta\,W^{2}+\int
d^{2}\bar{\theta}\,\bar{W}^{2}\biggr{\\}}$ (19)

or, in terms of the component fields,

$\displaystyle S_{\textrm{M}}$ $\displaystyle=\frac{1}{2}\int
d^{4}x\bigg{\\{}-\dfrac{1}{2}\mathcal{F}^{2}_{\mu\nu}+i\Big{[}(\tilde{\partial}_{\mu}\bar{\lambda})\bar{\sigma}^{\mu}\lambda-\bar{\lambda}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\lambda\Big{]}+|K|^{2}\bigg{\\}}.$
(20)

It is important to notice that, in our case, the modified Maxwell sector is
obtained from the matter sector as a consequence of supersymmetry. In Section
4, we shall explicitly see the modifications in Maxwell’s equations due to the
prescription (14).

###  3.2 The matter sector

Our next task is to supersymmetrize the matter sector based on the Lagrangian
(2). This procedure will lead to the kinetic and mass terms for the scalar
supersymmetric partners of $\psi$ and $\chi$.

To have a supersymmetric version of the left sector in (2), we define a chiral
superfield,

$\displaystyle\Phi_{\textrm{L}}$
$\displaystyle=e^{i(\theta\sigma^{\mu}\bar{\theta})\tilde{\partial}_{\mu}}\left(\varphi+\sqrt{2}\theta\psi+\theta^{2}f\right)$
$\displaystyle=\varphi+\sqrt{2}\theta\psi+\theta^{2}f+i(\theta\sigma^{\mu}\bar{\theta})\tilde{\partial}_{\mu}\varphi-\frac{i}{\sqrt{2}}\theta^{2}(\tilde{\partial}_{\mu}\psi)\sigma^{\mu}\bar{\theta}-\dfrac{1}{4}\theta^{2}\bar{\theta}^{2}\widetilde{\square}\varphi,$
(21)

and its anti-chiral associated superfield,

$\displaystyle\bar{\Phi}_{\textrm{L}}$
$\displaystyle=\varphi^{*}+\sqrt{2}\bar{\theta}\bar{\psi}+\bar{\theta}^{2}f^{*}-i(\theta\sigma^{\mu}\bar{\theta})\tilde{\partial}_{\mu}\varphi^{*}+\frac{i}{\sqrt{2}}\bar{\theta}^{2}\theta\sigma^{\mu}\tilde{\partial}_{\mu}\bar{\psi}-\dfrac{1}{4}\theta^{2}\bar{\theta}^{2}\widetilde{\square}\varphi^{*},$
(22)

where $\varphi$ is a charged scalar field, $\psi$ is the Weyl component of a
Dirac spinor and $f$ is an auxiliary field. Similarly, for the right sector,
we have

$\displaystyle\bar{\Phi}_{\textrm{R}}$
$\displaystyle=s+\sqrt{2}\bar{\theta}\bar{\chi}+\bar{\theta}^{2}g-i(\theta\sigma^{\mu}\bar{\theta})\tilde{\partial}_{\mu}s+\frac{i}{\sqrt{2}}\bar{\theta}^{2}\theta\sigma^{\mu}\tilde{\partial}_{\mu}\bar{\chi}-\dfrac{1}{4}\theta^{2}\bar{\theta}^{2}\widetilde{\square}s,$
(23) $\displaystyle\Phi_{\textrm{R}}$
$\displaystyle=s^{*}+\sqrt{2}\theta\chi+\theta^{2}g^{*}+i(\theta\sigma^{\mu}\bar{\theta})\tilde{\partial}_{\mu}s^{*}-\frac{i}{\sqrt{2}}\theta^{2}(\tilde{\partial}_{\mu}\chi)\sigma^{\mu}\bar{\theta}-\dfrac{1}{4}\theta^{2}\bar{\theta}^{2}\widetilde{\square}s^{*}.$
(24)

Here, $s$, $\chi$ and $g$ are the right component fields analogous to
$\varphi$, $\psi$ and $f$ in the left sector, respectively. It is important to
notice that the introduction of two chiral superfields with independent
fermionic components, $\psi$ and $\chi$, is necessary in order to compose the
(charged) Dirac fermion $\Psi$. The mass terms arise from the combinations

$\Phi_{\textrm{R}}\Phi_{\textrm{L}}\Big{|}_{\sim\theta^{2}}=s^{*}f+g^{*}\varphi-\chi\psi\;\;\;\textrm{and}\;\;\;\bar{\Phi}_{\textrm{L}}\bar{\Phi}_{\textrm{R}}\Big{|}_{\sim\bar{\theta}^{2}}=f^{*}s+\varphi^{*}g-\bar{\psi}\bar{\chi}.$
(25)

To implement the Abelian gauge symmetry in the matter sector, we consider the
transformation of chiral superfields under the global $U(1)$ gauge
transformation by the relations

$\displaystyle\Phi_{\textrm{L}}^{\prime}=e^{ie\Lambda}\Phi_{\textrm{L}},\;\;\;\;\bar{\Phi}_{\textrm{L}}^{\prime}=e^{-ie\bar{\Lambda}}\bar{\Phi}_{\textrm{L}},$
(26)
$\displaystyle\bar{\Phi}_{\textrm{R}}^{\prime}=e^{ie\bar{\Lambda}}\bar{\Phi}_{\textrm{R}},\;\;\;\;\Phi_{\textrm{R}}^{\prime}=e^{-ie\Lambda}\Phi_{\textrm{R}},$
(27)

where $e$ is the $U(1)$ charge of the superfield $\Phi$. Let us remark that
the mass term in the supersymmetric action is automatically gauge invariant.

After some algebra, one can verify that the kinetics terms in (2) are
generated by combinations like
$\big{(}\Phi_{\textrm{L}}e^{eA_{\textrm{WZ}}}\bar{\Phi}_{\textrm{L}}+\Phi_{\textrm{R}}e^{-eA_{\textrm{WZ}}}\bar{\Phi}_{\textrm{R}}\big{)}\big{|}_{\sim\theta^{2}\bar{\theta}^{2}}$.

###  3.3 The $v_{\mu}\bar{\Psi}\gamma^{\mu}\Psi$-term

The less immediate structure to supersymmetrize is the term

$\displaystyle\dfrac{1}{2}v_{\mu}\left(\bar{\psi}\bar{\sigma}^{\mu}\psi-\bar{\chi}\bar{\sigma}^{\mu}\chi\right).$
(28)

We have two alternatives to accomplish this program: $1)$ to introduce an
extra chiral-superfield and then identify $v_{\mu}$ as four-gradient of a
scalar field; $2)$ to define a vector superfield and consider $v_{\mu}$ as a
genuine four-vector.

From a conceptual point of view, we can consider that the anisotropy arises
from some kind of more fundamental field that may condensate to form the LSV
background. As we argue, it is important to look for a microscopic origin for
the anisotropy and, in this case, the first approach becomes more natural,
since the background field does not appear by itself; contrary, it is derived
from some more fundamental object; in our case, a scalar, as we shall soon
see. Let us remember that, in general, it is the scalars that trigger
transitions between different phases of the primordial Universe. Hence, they
may have played an important role in the formation of anisotropic structures
during the evolution of the Universe. These are motivations that support our
choice of the background as chiral superfield and not a four-vector
supermultiplet.

In what follows, we adopt the first scenario. For completeness, we add in the
Appendix A the results obtained from the alternative approach, i.e., the case
of a vector superfield.

####  3.3.1 Background chiral superfield

Let us introduce an additional chiral superfield $\Omega$, which is gauge
invariant from the perspective of the transformation in (26) and (27). The
$\theta$-expansion of $\Omega$ reads

$\displaystyle\Omega$
$\displaystyle=\omega+\sqrt{2}\theta\zeta+\theta^{2}h+i(\theta\sigma^{\mu}\bar{\theta})\tilde{\partial}_{\mu}\omega-\frac{i}{\sqrt{2}}\theta^{2}(\tilde{\partial}_{\mu}\zeta)\sigma^{\mu}\bar{\theta}-\dfrac{1}{4}\theta^{2}\bar{\theta}^{2}\widetilde{\square}\omega,$
(29)

where $\omega$ is a scalar field, $\zeta$ is the Weyl component of a Majorana
spinor and $h$ is an auxiliary field. One should also notice that, differently
from the physical propagating superfields, the background superfield $\Omega$
has trivial canonical dimension. The products of superfields which can
reproduce (28) are

$\displaystyle(\Omega+\bar{\Omega})\Phi_{\textrm{L}}e^{eA_{\textrm{WZ}}}\bar{\Phi}_{\textrm{L}}\Big{|}_{\sim\theta^{2}\bar{\theta}^{2}}=$
$\displaystyle-\frac{i}{2}\tilde{\partial}_{\mu}(\omega-\omega^{*})\bar{\psi}\bar{\sigma}^{\mu}\psi+(\omega+\omega^{*})\bigg{\\{}\frac{i}{2}\Big{[}(\tilde{\partial}_{\mu}\bar{\psi})\bar{\sigma}^{\mu}\psi-\bar{\psi}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\psi\Big{]}$
$\displaystyle-\dfrac{1}{2}\Big{[}(\widetilde{\square}\varphi^{*})\varphi+\varphi^{*}\widetilde{\square}\varphi\Big{]}+f^{*}f\bigg{\\}}+f^{*}(h\varphi-\zeta\psi)+f(h^{*}\varphi^{*}-\bar{\zeta}\bar{\psi})$
$\displaystyle+\dfrac{1}{2}\tilde{\partial}_{\mu}(\omega-\omega^{*})\Big{[}(\tilde{\partial}^{\mu}\varphi^{*})\varphi-\varphi^{*}\tilde{\partial}^{\mu}\varphi\Big{]}+\frac{i}{2}\Big{[}\varphi(\tilde{\partial}_{\mu}\bar{\psi})\bar{\sigma}^{\mu}\zeta-\varphi\bar{\psi}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\zeta$
$\displaystyle-\bar{\psi}\bar{\sigma}^{\mu}\zeta\tilde{\partial}_{\mu}\varphi\Big{]}-\frac{i}{2}\Big{[}\varphi^{*}\bar{\zeta}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\psi-\varphi^{*}(\tilde{\partial}_{\mu}\bar{\zeta})\bar{\sigma}^{\mu}\psi-\bar{\zeta}\bar{\sigma}^{\mu}\psi\tilde{\partial}_{\mu}\varphi^{*}\Big{]}$
$\displaystyle-\frac{e(\omega+\omega^{*})}{2}\bigg{\\{}i\sqrt{2}(\bar{\lambda}\bar{\psi}\varphi-\lambda\psi\varphi^{*})-i(\varphi^{*}\tilde{\partial}^{\mu}\varphi-\varphi\tilde{\partial}^{\mu}\varphi^{*})\tilde{a}_{\mu}$
$\displaystyle+(\bar{\psi}\bar{\sigma}^{\mu}\psi)\tilde{a}_{\mu}-\bigg{(}K+\frac{e}{2}\tilde{a}^{2}\bigg{)}\varphi^{*}\varphi\bigg{\\}}+\frac{e}{2}\bigg{\\{}i\tilde{\partial}^{\mu}(\omega-\omega^{*})\,\varphi^{*}\varphi\,\tilde{a}_{\mu}$
$\displaystyle-\big{[}(\bar{\psi}\bar{\sigma}^{\mu}\zeta)\varphi+(\bar{\zeta}\bar{\sigma}^{\mu}\psi)\varphi^{*}\big{]}\tilde{a}_{\mu}-i\sqrt{2}(\bar{\lambda}\bar{\zeta}-\lambda\zeta)\varphi^{*}\varphi\bigg{\\}},$
(30)

and an analog structure working to reproduce the right sector.

Choosing the scalar component $\omega$ of the background supermultiplet
$\Omega$ as it is done in [12],

$(\omega+\omega^{*})=0\;\;\;\textrm{and}\;\;\;(\omega-\omega^{*})=2i\varpi,$
(31)

with $\varpi=v_{\nu}\tilde{x}^{\nu}$, we must impose that
$\tilde{\partial}_{\mu}v_{\nu}=0$, i.e., that the imaginary part of parameter
$\xi$ in Eq. (3) has an only linear dependence on the coordinates. From these
considerations, we can write our supersymmetric model as follows below:

$\displaystyle S=$ $\displaystyle\,\int d^{4}x\,\bigg{\\{}\int d^{2}\theta
d^{2}\bar{\theta}\,\bigg{[}\bar{\Phi}_{\textrm{L}}e^{eA_{\textrm{WZ}}}\Phi_{\textrm{L}}+\bar{\Phi}_{\textrm{R}}e^{-eA_{\textrm{WZ}}}\Phi_{\textrm{R}}+\frac{1}{2}(\Omega+\bar{\Omega})\big{(}\bar{\Phi}_{\textrm{L}}e^{eA_{\textrm{WZ}}}\Phi_{\textrm{L}}$
$\displaystyle-\,\bar{\Phi}_{\textrm{R}}e^{-eA_{\textrm{WZ}}}\Phi_{\textrm{R}}\big{)}\bigg{]}+\frac{1}{4}\bigg{(}\int
d^{2}\theta\,W^{2}+c.c.\bigg{)}+m\bigg{(}\int
d^{2}\theta\,\Phi_{\textrm{R}}\Phi_{\textrm{L}}+c.c.\bigg{)}\bigg{\\}}.$ (32)

Using the field equations for the auxiliary fields, (3.3.1) can be formulated
in terms of the physical fields:

$\displaystyle S_{1}=$ $\displaystyle\,\int
d^{4}x\,\bigg{\\{}-\frac{1}{4}\mathcal{F}^{2}_{\mu\nu}+\frac{i}{2}\Big{[}(\tilde{\partial}_{\mu}\bar{\lambda})\bar{\sigma}^{\mu}\lambda-\bar{\lambda}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\lambda\Big{]}-\dfrac{i}{2}\Big{[}\bar{\psi}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\psi-(\tilde{\partial}_{\mu}\bar{\psi})\bar{\sigma}^{\mu}\psi+\bar{\chi}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\chi$
$\displaystyle-\,(\tilde{\partial}_{\mu}\bar{\chi})\bar{\sigma}^{\mu}\chi\Big{]}+\dfrac{1}{2}v_{\mu}(\bar{\psi}\bar{\sigma}^{\mu}\psi-\bar{\chi}\bar{\sigma}^{\mu}\chi)-m(\bar{\psi}\bar{\chi}+\chi\psi)+\Big{[}(\tilde{\partial}^{\mu}\varphi^{*})\tilde{\partial}_{\mu}\varphi+(\tilde{\partial}^{\mu}s^{*})\tilde{\partial}_{\mu}s\Big{]}$
$\displaystyle+\,\dfrac{i}{2}v_{\mu}\Big{[}(\tilde{\partial}^{\mu}\varphi^{*})\varphi-\varphi^{*}\tilde{\partial}^{\mu}\varphi+(\tilde{\partial}^{\mu}s^{*})s-s^{*}\tilde{\partial}^{\mu}s\Big{]}-\bigg{(}m^{2}+\frac{1}{4}h^{*}h\bigg{)}(\varphi^{*}\varphi+s^{*}s)$
$\displaystyle+\,\frac{i}{2}\Big{[}\varphi(\tilde{\partial}_{\mu}\bar{\psi})\bar{\sigma}^{\mu}\zeta-\varphi^{*}\bar{\zeta}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\psi+s\bar{\zeta}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\chi-s^{*}(\tilde{\partial}_{\mu}\bar{\chi})\bar{\sigma}^{\mu}\zeta\Big{]}-\frac{1}{2}\bigg{[}\varphi\bigg{(}m\,\zeta\chi-\frac{1}{2}h\bar{\zeta}\bar{\psi}\bigg{)}$
$\displaystyle+\,\varphi^{*}\bigg{(}m\,\bar{\zeta}\bar{\chi}-\frac{1}{2}h^{*}\zeta\psi\bigg{)}-s\bigg{(}m\,\bar{\zeta}\bar{\psi}+\frac{1}{2}h^{*}\zeta\chi\bigg{)}-s^{*}\bigg{(}m\,\zeta\psi+\frac{1}{2}h\bar{\zeta}\bar{\chi}\bigg{)}\bigg{]}$
$\displaystyle-\,\frac{1}{4}\Big{(}\bar{\zeta}\bar{\psi}\zeta\psi+\bar{\zeta}\bar{\chi}\zeta\chi\Big{)}-\frac{e}{2}\Big{[}i\sqrt{2}(\varphi\bar{\lambda}\bar{\psi}-\varphi^{*}\lambda\psi+s\lambda\chi-s^{*}\bar{\lambda}\bar{\chi})+(\bar{\psi}\bar{\sigma}^{\mu}\psi-\bar{\chi}\bar{\sigma}^{\mu}\chi)\tilde{a}_{\mu}$
$\displaystyle+\,i(\varphi\,\tilde{\partial}^{\mu}\varphi^{*}-\varphi^{*}\tilde{\partial}^{\mu}\varphi+s\,\tilde{\partial}^{\mu}s^{*}-s^{*}\tilde{\partial}^{\mu}s)\tilde{a}_{\mu}\Big{]}-\frac{e}{4}\Big{[}(\bar{\psi}\bar{\sigma}^{\mu}\zeta)\varphi+(\bar{\zeta}\bar{\sigma}^{\mu}\psi)\varphi^{*}+(\bar{\chi}\bar{\sigma}^{\mu}\zeta)s^{*}$
$\displaystyle+\,(\bar{\zeta}\bar{\sigma}^{\mu}\chi)s\Big{]}\tilde{a}_{\mu}-\frac{e}{2}(\varphi^{*}\varphi+s^{*}s)\bigg{[}v^{\mu}\tilde{a}_{\mu}+\frac{i}{\sqrt{2}}(\bar{\lambda}\bar{\zeta}-\zeta\lambda)-\frac{e}{2}\tilde{a}^{2}\bigg{]}$
$\displaystyle-\,\frac{e^{2}}{8}(\varphi^{*}\varphi-s^{*}s)^{2}\bigg{\\}}.$
(33)

Despite the lengthy expression in $S_{1}$, it is immediate to identify the
fermionic model (1) in the first two lines. In order to speed up the analysis
of the contributions arising from the presence of the background, we group the
terms according to sectors. It proves much more useful to rewrite the result
(3.3.1) in terms of 4-component spinors.

In the gauge sector, we have the following standard structure

$\displaystyle S_{\textrm{gauge}}$ $\displaystyle=-\frac{1}{4}\int
d^{4}x\,\bigg{\\{}\mathcal{F}^{2}_{\mu\nu}-i\Big{[}(\tilde{\partial}_{\mu}\bar{\Lambda})\gamma^{\mu}\Lambda-\bar{\Lambda}\gamma^{\mu}\tilde{\partial}_{\mu}\Lambda\Big{]}\bigg{\\}},$
(34)

where the gaugino (photino) is a Majorana fermion written as

$\displaystyle\Lambda=\begin{pmatrix}\lambda_{\alpha}\\\
\bar{\lambda}^{\dot{\alpha}}\end{pmatrix}.$ (35)

It should be recalled that that the background component-fields are not
present in the photon/photino sector, in contrast to other scenarios with
Lorentz-symmetry violation, such as, e.g., the supersymmetric extensions of
the Carroll-Field-Jackiw model of Refs. [12, 43]. The reason for this is that,
in our case, the gauge sector is affected only by the real part of the
background parameter $\xi$. In fact, the modifications in the gauge sector,
induced by the background, only manifest as “dilations” in Maxwell’s
equations, as we shall see in the next Section.

Using (5) and defining the covariant derivative
$\tilde{D}_{\mu}\equiv\tilde{\partial}_{\mu}-\dfrac{ie}{2}\tilde{a}_{\mu}$, we
can organize the fermionic sector, including the gauge coupling field, in the
form

$\displaystyle S_{\textrm{fermion}}=$ $\displaystyle\,\int
d^{4}x\,\bigg{\\{}\dfrac{i}{2}\Big{[}(\tilde{D}^{*}_{\mu}\bar{\Psi})\gamma^{\mu}\Psi-\bar{\Psi}\gamma^{\mu}\tilde{D}_{\mu}\Psi\Big{]}+\dfrac{1}{2}v_{\mu}(\bar{\Psi}\gamma^{\mu}\Psi)-m\bar{\Psi}\Psi$
$\displaystyle-\frac{1}{16}(\bar{Z}\gamma_{\mu}\gamma_{5}Z)(\bar{\Psi}\gamma^{\mu}\gamma_{5}\Psi)\bigg{\\}},$
(36)

with the Majorana spinor $Z$ (the background fermion), given by

$\displaystyle Z=\begin{pmatrix}\zeta_{\alpha}\\\
\bar{\zeta}^{\dot{\alpha}}\end{pmatrix}.$ (37)

The term with dependence on the background fermionic condensate
$\bar{Z}\gamma_{\mu}\gamma_{5}Z$ that appears in the fermionic sector is an
additional contribution to the mass of the Dirac spinor $\Psi$. It is
interesting to notice that this correction can be identified as the
coefficient $b_{\mu}$ in one of the CPT-odd terms introduced in the Extended
Standard Model (SME) [39] in order to implement the violation of the Lorentz
symmetry.

The sector of scalars reads as given below:

$\displaystyle S_{\textrm{scalar}}=$ $\displaystyle\,\int
d^{4}x\,\bigg{\\{}(\tilde{D}^{\mu}\varphi^{*})\tilde{D}_{\mu}\varphi+(\tilde{D}^{\mu}s^{*})\tilde{D}_{\mu}s+\dfrac{i}{2}v_{\mu}\Big{[}(\tilde{\partial}^{\mu}\varphi^{*})\varphi-\varphi^{*}\tilde{\partial}^{\mu}\varphi+(\tilde{\partial}^{\mu}s^{*})s-s^{*}\tilde{\partial}^{\mu}s\Big{]}$
$\displaystyle-\bigg{(}m^{2}+\frac{1}{4}h^{*}h\bigg{)}(\varphi^{*}\varphi+s^{*}s)\bigg{\\}}.$
(38)

As in the fermionic sector, there is a correction to the scalar masses,
$\varphi$ and $s$; in this case, proportional to $h^{*}h$. This non-degeneracy
in the mass spectrum indicates a SUSY breaking due to the presence of the
background. Furthermore, there is also a contribution of the background
vector, $v_{\mu}$, to the dynamic part of the scalar sector. On the other
hand, in the fermionic case, $v_{\mu}$ only affects the mass term, as we can
see in the first line of (3.3.1).

Finally, we have the gaugino and matter self-interaction contributions to the
Lagrangian:

$\displaystyle S_{\textrm{mixing$+$int}}=$ $\displaystyle-\int
d^{4}x\,\bigg{\\{}\frac{ie}{\sqrt{2}}\Big{[}\bar{\Psi}(\varphi\mathcal{P}_{\textrm{R}}+s\mathcal{P}_{\textrm{L}})\Lambda-\bar{\Lambda}(\varphi^{*}\mathcal{P}_{\textrm{L}}+s^{*}\mathcal{P}_{\textrm{R}})\Psi\Big{]}$
$\displaystyle+\,\frac{i}{2}\Big{[}\bar{Z}\gamma^{\mu}(\varphi^{*}\mathcal{P}_{\textrm{L}}-s^{*}\mathcal{P}_{\textrm{R}})\tilde{D}_{\mu}\Psi-(\tilde{D}^{*}_{\mu}\bar{\Psi})\gamma^{\mu}(\varphi\mathcal{P}_{\textrm{L}}-s\mathcal{P}_{\textrm{R}})Z\Big{]}$
$\displaystyle+\,\frac{1}{2}\bar{\Psi}\bigg{[}\bigg{(}m\varphi-\frac{1}{2}h^{*}s\bigg{)}\mathcal{P}_{\textrm{L}}-\bigg{(}ms+\frac{1}{2}h\varphi\bigg{)}\mathcal{P}_{\textrm{R}}\bigg{]}Z$
$\displaystyle+\,\frac{1}{2}\bar{Z}\bigg{[}\bigg{(}m\varphi^{*}-\frac{1}{2}hs^{*}\bigg{)}\mathcal{P}_{\textrm{R}}-\bigg{(}ms^{*}+\frac{1}{2}h^{*}\varphi^{*}\bigg{)}\mathcal{P}_{\textrm{L}}\bigg{]}\Psi$
$\displaystyle+\,\frac{e}{2}\bigg{[}v^{\mu}\tilde{a}_{\mu}+\frac{i}{2\sqrt{2}}(\bar{\Lambda}\gamma_{5}Z+\bar{Z}\gamma_{5}\Lambda)\bigg{]}(\varphi^{*}\varphi+s^{*}s)+\frac{e^{2}}{8}(\varphi^{*}\varphi-s^{*}s)^{2}\bigg{\\}},$
(39)

with the parity operators,

$\displaystyle\mathcal{P}_{\textrm{R}}=\frac{I_{4}+\gamma_{5}}{2}=\begin{pmatrix}0&0\\\
0&I_{2}\end{pmatrix}\;\;\;\;\;\textrm{and}\;\;\;\;\;\mathcal{P}_{\textrm{L}}=\frac{I_{4}-\gamma_{5}}{2}=\begin{pmatrix}I_{2}&0\\\
0&0\end{pmatrix}.$ (40)

It is worth mentioning that the new couplings induced by the background
fermion, $Z$, except for those involving the degrees of freedom of the gauge
sector ($a_{\mu}$ and $\Lambda$), must be taken into account when calculating
the dispersion relations for the matter fields, because such couplings do not
characterize interaction terms (since the background is not dynamic).

##  4 Field equations and dispersion relations

With the supersymmetric action (3.3.1) in hand, we proceed, in this Section,
to a detailed analysis of the field equations in both the matter and gauge
sectors. Subsequently, in each case, we shall derive the modified dispersion
relations within this framework, aiming to understand the implications of LSV
in connection with SUSY.

###  4.1 The modified Maxwell’s equations

The modified Maxwell’s equations in the vacuum come from the
$\mathcal{F}^{2}_{\mu\nu}$ part of the action (34),

$\displaystyle\tilde{\partial}_{\mu}\mathcal{F}^{\mu\nu}=0.$ (41)

Let us recall that
$\mathcal{F}_{\mu\nu}=(\tilde{\partial}_{\mu}\tilde{a}_{\nu}-\tilde{\partial}_{\nu}\tilde{a}_{\mu})$.
From (41) we obtain the modified version of the Gauss’s law for the electric
field and the Ampère-Maxwell equation, in the simplified form,

$\displaystyle\textrm{Re}(\xi)\nabla\cdot\mathbfcal{E}=0,$ (42)
$\displaystyle\textrm{Re}(\xi)\nabla\times\mathbfcal{B}=\partial_{t}\mathbfcal{E},$
(43)

where, in an analogy with the usual electrodynamics, we define an effective
electric field given by
$\mathbfcal{E}=-\textrm{Re}(\xi)\nabla\phi-\textbf{A}/\textrm{Re}(\xi),$ while
the magnetic field remains unchanged in this scenario, i.e.
$\mathbfcal{B}=\nabla\times\textbf{A}$.

On the other hand, from the Bianchi identity,

$\displaystyle\tilde{\partial}_{\mu}{{}^{*\\!\\!}\mathcal{F}_{\mu\nu}}=0,\;\;\;\;\;{{}^{*\\!\\!}\mathcal{F}_{\mu\nu}}=\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}\mathcal{F}_{\alpha\beta}$
(44)

there follow the modified Faraday-Lenz (for $\nu=i$) and the Gauss’s law for
the magnetic field (for $\nu=0$):

$\displaystyle\textrm{Re}(\xi)\nabla\times\mathbfcal{E}=-\partial_{t}\mathbfcal{B},$
(45) $\displaystyle\textrm{Re}(\xi)\nabla\cdot\mathbfcal{B}=0.$ (46)

Following the standard procedure for obtaining the wave equation, let us apply
a time derivative on (45), and use the equations (42) and (43). For non-
trivial field configurations, the photonic scattering relation arises from the
requirement that

$\omega=\pm\textrm{Re}(\xi)|\textbf{k}|.$ (47)

For the photino, the field equation is
$\gamma^{\mu}\tilde{\partial}_{\mu}\Lambda=0$, the dispersion relation has the
same form as (47).

One of the main consequences of SUSY is the mass degeneracy between a particle
and its corresponding superpartner. Since the photon and photino share the
same modified dispersion relation, it is immediate to verify that they are
both massless. This is simply a confirmation that in the gauge sector there is
no SUSY breaking.

###  4.2 The modified Dirac and Klein-Gordon equations

As we have seen in Section 3.3, due to the fact that the SUSY background is
not dynamic, the mixing terms involving the spinors, $\Psi$ and $Z$, also
contribute to the extended Dirac action, in addition to the (pure) fermionic
sector,

$\displaystyle S_{\textrm{Dirac}}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\,\bigg{\\{}\dfrac{i}{2}\Big{[}(\tilde{\partial}_{\mu}\bar{\Psi})\gamma^{\mu}\Psi-\bar{\Psi}\gamma^{\mu}\tilde{\partial}_{\mu}\Psi\Big{]}+\dfrac{1}{2}v_{\mu}(\bar{\Psi}\gamma^{\mu}\Psi)-m\bar{\Psi}\Psi-\frac{1}{16}(\bar{\Psi}\gamma^{\mu}\gamma_{5}\Psi)(\bar{Z}\gamma_{\mu}\gamma_{5}Z)$
(48)
$\displaystyle-\frac{i}{2}\Big{[}\bar{Z}\gamma^{\mu}(\varphi^{*}\mathcal{P}_{\textrm{L}}-s^{*}\mathcal{P}_{\textrm{R}})\tilde{\partial}_{\mu}\Psi-(\tilde{\partial}_{\mu}\bar{\Psi})\gamma^{\mu}(\varphi\mathcal{P}_{\textrm{L}}-s\mathcal{P}_{\textrm{R}})Z\Big{]}-\frac{1}{2}\bar{\Psi}\bigg{[}\bigg{(}m\varphi-\frac{1}{2}h^{*}s\bigg{)}\mathcal{P}_{\textrm{L}}$
$\displaystyle-\bigg{(}ms+\frac{1}{2}h\varphi\bigg{)}\mathcal{P}_{\textrm{R}}\bigg{]}Z-\frac{1}{2}\bar{Z}\bigg{[}\bigg{(}m\varphi^{*}-\frac{1}{2}hs^{*}\bigg{)}\mathcal{P}_{\textrm{R}}-\bigg{(}ms^{*}+\frac{1}{2}h^{*}\varphi^{*}\bigg{)}\mathcal{P}_{\textrm{L}}\bigg{]}\Psi\bigg{\\}}.$

The modified Dirac equation in momentum space is given by

$\displaystyle\left[(\tilde{p}-\zeta)_{\mu}\gamma^{\mu}+m+R_{\mu}\gamma^{\mu}\gamma_{5}\right]\Psi_{0}=$
$\displaystyle-\dfrac{1}{2}\bigg{\\{}\left[(\tilde{p}_{\mu}\gamma^{\mu}+m)\varphi_{0}-\dfrac{1}{2}h^{*}s_{0}\right]\mathcal{P}_{\textrm{L}}$
$\displaystyle-\left[(\tilde{p}_{\mu}\gamma^{\mu}+m)s_{0}+\dfrac{1}{2}h\varphi_{0}\right]\mathcal{P}_{\textrm{R}}\bigg{\\}}Z,$
(49)

where $\zeta_{\mu}=\dfrac{1}{2}v_{\mu}$ and
$R_{\mu}=\dfrac{1}{16}(\bar{Z}\gamma_{\mu}\gamma_{5}Z)$. Let us remember that
the imaginary part of $\xi$ is a linear function of the coordinates which
yields a constant $v_{\mu}$.

Similarly, the action for the matter scalars reads as below:

$\displaystyle S_{\textrm{KG}}=$ $\displaystyle\,\int
d^{4}x\,\bigg{\\{}(\tilde{\partial}^{\mu}\varphi^{*})\tilde{\partial}_{\mu}\varphi+(\tilde{\partial}^{\mu}s^{*})\tilde{\partial}_{\mu}s+\dfrac{i}{2}v_{\mu}\Big{[}(\tilde{\partial}^{\mu}\varphi^{*})\varphi-\varphi^{*}\tilde{\partial}^{\mu}\varphi+(\tilde{\partial}^{\mu}s^{*})s-s^{*}\tilde{\partial}^{\mu}s\Big{]}$
$\displaystyle-\bigg{(}m^{2}+\frac{1}{4}|h|^{2}\bigg{)}(\varphi^{*}\varphi+s^{*}s)-\frac{i}{2}\Big{[}\bar{Z}\gamma^{\mu}(\varphi^{*}\mathcal{P}_{\textrm{L}}-s^{*}\mathcal{P}_{\textrm{R}})\tilde{\partial}_{\mu}\Psi-(\tilde{\partial}_{\mu}\bar{\Psi})\gamma^{\mu}(\varphi\mathcal{P}_{\textrm{L}}-s\mathcal{P}_{\textrm{R}})Z\Big{]}$
$\displaystyle-\frac{1}{2}\bar{\Psi}\bigg{[}\bigg{(}m\varphi-\frac{1}{2}h^{*}s\bigg{)}\mathcal{P}_{\textrm{L}}-\bigg{(}ms+\frac{1}{2}h\varphi\bigg{)}\mathcal{P}_{\textrm{R}}\bigg{]}Z-\frac{1}{2}\bar{Z}\bigg{[}\bigg{(}m\varphi^{*}-\frac{1}{2}hs^{*}\bigg{)}\mathcal{P}_{\textrm{R}}$
$\displaystyle-\bigg{(}ms^{*}+\frac{1}{2}h^{*}\varphi^{*}\bigg{)}\mathcal{P}_{\textrm{L}}\bigg{]}\Psi\bigg{\\}},$
(50)

which gives the following modified Klein-Gordon equations:

$\displaystyle\Theta\varphi_{0}=$
$\displaystyle-\frac{1}{2}\bar{Z}\bigg{[}\bigg{(}\tilde{p}_{\mu}\gamma^{\mu}-\frac{1}{2}h^{*}\bigg{)}\mathcal{P}_{\textrm{L}}+m\mathcal{P}_{\textrm{R}}\bigg{]}\Psi_{0},$
(51) $\displaystyle\Theta s_{0}=$
$\displaystyle\,\frac{1}{2}\bar{Z}\bigg{[}\bigg{(}\tilde{p}_{\mu}\gamma^{\mu}+\frac{1}{2}h\bigg{)}\mathcal{P}_{\textrm{R}}+m\mathcal{P}_{\textrm{L}}\bigg{]}\Psi_{0},$
(52)

where we introduce the useful notation
$\Theta\equiv-\bigg{[}\tilde{p}^{2}-2(\tilde{p}\cdot\zeta)-m^{2}-\dfrac{1}{4}|h|^{2}\bigg{]}$.

As expected from our analysis of the mixing part (3.3.1), the modified field
equations of the matter sector are coupled by means of the background fermion,
$Z$. In the next Section, we shall present a way to rewrite these equations
and their associated dispersion relations.

###  4.3 The matter sector and the associated dispersion relations

To obtain the fermionic dispersion relation, let us first substitute the
expressions for $\varphi_{0}$ and $s_{0}$, i.e., (51) and (52), into the
modified Dirac equation (4.2). The Fierz rearrangement,

$\displaystyle\left(\bar{Z}M\Psi_{0}\right)NZ$
$\displaystyle=-\frac{1}{4}\left[\left(\bar{Z}Z\right)NM+\left(\bar{Z}\gamma_{5}Z\right)N\gamma_{5}M+\left(\bar{Z}\gamma_{\alpha}\gamma_{5}Z\right)N\gamma^{\alpha}\gamma_{5}M\right]\Psi_{0},$
(53)

where $M$ and $N$ are arbitrary 4×4 matrices, allows us to rewrite the pieces
involving the mixing of $\Psi_{0}$ and $Z$ in terms of background fermionic
condensates, which we will denote just as $\theta=\bar{Z}Z$,
$\tau=\bar{Z}\gamma_{5}Z$ and $C_{\alpha}=\bar{Z}\gamma_{\alpha}\gamma_{5}Z$.
After some simplifications using the relations,
$\gamma^{\mu}\gamma^{\alpha}\gamma^{\nu}=\eta^{\mu\alpha}\gamma^{\nu}+\eta^{\alpha\nu}\gamma^{\mu}-\eta^{\nu\mu}\gamma^{\alpha}-i\epsilon^{\mu\alpha\nu\beta}\gamma_{\beta}\gamma_{5}$
and
$\Sigma^{\mu\alpha}=\dfrac{i}{4}\big{[}\gamma^{\mu},\,\gamma^{\alpha}\big{]}$,
the field equation for $\Psi_{0}$ can be written in compact form as

$\displaystyle\Big{[}(\tilde{p}-\zeta)_{\mu}\gamma^{\mu}+m-\mathcal{R}_{\mu}\gamma^{\mu}\gamma_{5}-W_{\mu\alpha}\Sigma^{\mu\alpha}\gamma_{5}\Big{]}\Psi_{0}$
$\displaystyle=0,$ (54)

where we define the nonlocal structures

$\displaystyle\mathcal{R}_{\mu}=\frac{1}{16\Theta}\bigg{[}2\tilde{p}_{\mu}\tilde{p}_{\nu}C^{\nu}-2\bigg{(}\tilde{p}\cdot\zeta+m^{2}+\frac{1}{4}|h|^{2}\bigg{)}C_{\mu}+\tilde{p}_{\mu}(h^{*}-h)\tau-\frac{1}{2}\tilde{p}_{\mu}(h^{*}+h)\theta\bigg{]},$
$\displaystyle W_{\mu\alpha}=\frac{im}{4\Theta}\tilde{p}_{\mu}C_{\alpha}.$
(55)

The derivation of dispersion relations for the matter sector is a more
involved task. In the fermionic case, in particular, the presence of the last
term, $\Sigma^{\mu\alpha}$, in Eq. (54) makes the structure of the Dirac-
equation operator more complicated to manipulate. Following the approach
presented in [39], we adopt the standard squaring procedure and apply the
Dirac operator with an opposite mass sign to the (54):

$\mathcal{O}_{2\textrm{-der}}\Psi_{0}=0,$ (56)

where the operator is given by

$\displaystyle\mathcal{O}_{2\textrm{-der}}=$
$\displaystyle\;(\tilde{p}-\zeta)^{2}-m^{2}-\mathcal{R}^{2}+\dfrac{1}{4}W_{\nu\beta}W_{\mu\alpha}\big{[}\left(\eta^{\nu\mu}\eta^{\beta\alpha}-\eta^{\nu\alpha}\eta^{\beta\mu}\right)+2i(\eta^{\nu\alpha}\Sigma^{\beta\mu}+\eta^{\beta\mu}\Sigma^{\nu\alpha})\big{]}$
$\displaystyle-i(\tilde{p}-\zeta-\mathcal{R})_{\nu}W_{\mu\alpha}(\eta^{\nu\mu}\gamma^{\alpha}-\eta^{\nu\alpha}\gamma^{\mu}).$
(57)

To eliminate the off-diagonal terms in spinor space, we repeat the squaring
procedure to Eq. (56). Now, let us consider the operator
$\mathcal{O}_{2\textrm{-der}}$ with opposite sign for the off-diagonal piece,
i.e.,
$+i(\tilde{p}-\zeta-\mathcal{R})_{\kappa}W_{\lambda\rho}(\eta^{\kappa\lambda}\gamma^{\rho}-\eta^{\kappa\rho}\gamma^{\lambda})$.
This procedure gives a fourth-order equation satisfied by each component of
the Dirac spinor $\Psi_{0}$, such that (for non-trivial field configurations)

$\displaystyle 0=$
$\displaystyle\;\tilde{p}^{4}-2\tilde{p}^{2}\bigg{[}4(\tilde{p}\cdot\zeta)+m^{2}-\zeta^{2}+\frac{1}{4}|h|^{2}\bigg{]}+8(\tilde{p}\cdot\zeta)\big{[}3(\tilde{p}\cdot\zeta)+m^{2}\big{]}$
$\displaystyle+m^{2}\bigg{(}m^{2}-2\zeta^{2}+\frac{1}{2}|h|^{2}\bigg{)},$ (58)

where we neglect higher-order terms in the Lorentz-breaking parameters, since
it is reasonable to consider them to be small (see, e.g., [46, 39] for
theoretical discussions). The result (4.3) is the most general expression up
to second order in Lorentz breaking for the fermionic dispersion relation of
model (3.3.1). It is clear its non-dependence on the background fermionic
condensates $\theta$, $\tau$ and $C_{\alpha}$. Therefore, we can see that, up
to second order in LSV, the explicit dependence on the background is
manifested only through the auxiliary complex scalar $h$ and vector
$\zeta_{\mu}=\dfrac{1}{2}v_{\mu}$ fields. An interesting consequence of this
is that the results obtained from (4.3) are automatically parity invariant.

In the case of scalar fields, $\varphi_{0}$ and $s_{0}$, the right-hand side
of their respective equations, (51) and (52), involves a mixing of Dirac and
background fermions. To express these equations in terms of the background
fermionic condensates, $\theta$, $\tau$ and $C_{\alpha}$, we first employ the
squaring procedure to Eq. (4.2). This allows us to compose an expression for
$\Psi_{0}$, which can then be substituted into the modified Klein-Gordon
equations. After some algebra, we get

$\displaystyle\Theta\varphi_{0}=$
$\displaystyle\,\frac{1}{4\Xi}\big{[}X^{(1)}(\varphi_{0},s_{0})\theta+\mathcal{M}^{(1)}(\varphi_{0},s_{0})\tau+Y^{(1)}_{\alpha}(\varphi_{0},s_{0})C^{\alpha}\big{]},$
(59) $\displaystyle\Theta s_{0}=$
$\displaystyle-\frac{1}{4\Xi}\big{[}X^{(2)}(\varphi_{0},s_{0})\theta+\mathcal{M}^{(2)}(\varphi_{0},s_{0})\tau+Y^{(2)}_{\alpha}(\varphi_{0},s_{0})C^{\alpha}\big{]},$
(60)

where
$\Xi=\big{[}(\tilde{p}-\zeta)^{2}-m^{2}-R^{2}\big{]}^{2}+4(\tilde{p}-\zeta)^{2}R^{2}-4\big{[}(\tilde{p}-\zeta)\cdot
R\big{]}^{2}$, and the explicit form of $X$’s, $\mathcal{M}$’s and
$Y_{\alpha}$’s can be found in Appendix B. Finally, from (59) and (60), is
possible to obtain the full dispersion relation for the scalars, up to second
order in LSV,

$\displaystyle 0=$
$\displaystyle\;\tilde{p}^{4}-4\tilde{p}^{2}\bigg{[}3(\tilde{p}\cdot\zeta)+\frac{1}{2}m^{2}-\zeta^{2}+\frac{1}{8}|h|^{2}\bigg{]}+60(\tilde{p}\cdot\zeta)^{2}$
$\displaystyle+4m^{2}\bigg{[}3(\tilde{p}\cdot\zeta)+\dfrac{1}{32}(\tilde{p}\cdot
C)+\frac{1}{4}m^{2}-\zeta^{2}+\frac{1}{8}|h|^{2}\bigg{]}.$ (61)

In contrast to the fermionic case, Eq. (4.3), the modified dispersion relation
for the scalar sector has a non-trivial dependence on the background fermionic
condensate $C_{\alpha}$, in addition to terms proportional to $h$ and
$v_{\mu}$ components.

Now, we are in a position to exploit the effects of this LSV background on the
mass spectrum of the fermionic and scalar matter sectors. Let us take the rest
frame, where the linear momentum is zero, $\bm{p}=0$, and the rest energy is
nothing but the rest mass. From (4.3), we verify that the fermionic rest
energy results in

$\displaystyle
E_{0,\,\textrm{fermion}}=\pm\sqrt{m^{2}+\frac{1}{2}\big{(}\bm{v}^{2}+|h|^{2}\big{)}}.$
(62)

In this scenario, the fermion in question does not allow the existence of
tachyonic mass modes. Therefore, considering that LSV terms are very small
compared to the mass scale of the particles, we can take the limit
$m^{2}\rightarrow\infty$, which corresponds to the regime $m^{2}\gg\bm{v}^{2}$
and $m^{2}\gg|h|^{2}$, in the positive solution of the spectra (62) in order
to obtain a LSV shift in the fermion mass. The result is

$\displaystyle E_{0,\,\textrm{fermion}}\Big{|}_{m^{2}\rightarrow\infty}\approx
m\left(1+\dfrac{\bm{v}^{2}+|h|^{2}}{4m^{2}}\right).$ (63)

On the other hand, for the scalar sector, Eq. (4.3), let us consider two
particular cases, for simplicity: $i)$ in the previous Section, we have got
that the induced SUSY breaking occurs even when the background auxiliary
component is zero. So, assuming $h=0$, we arrive at the following non-parity
invariant results

$\displaystyle
E_{0,\,\textrm{scalar}}^{(1)}\Big{|}_{h=0}=\,\dfrac{\sqrt{B}}{2}\pm\frac{1}{2}\sqrt{-B+2(2m^{2}+\bm{v}^{2})-\dfrac{m^{2}C_{0}}{4\sqrt{B}}},$
(64) $\displaystyle
E_{0,\,\textrm{scalar}}^{(2)}\Big{|}_{h=0}=-\,\dfrac{\sqrt{B}}{2}\pm\frac{1}{2}\sqrt{-B+2(2m^{2}+\bm{v}^{2})+\dfrac{m^{2}C_{0}}{4\sqrt{B}}}.$
(65)

In the solutions (64) and (65) we defined the notations

$\displaystyle A$
$\displaystyle=G+\sqrt{G^{2}-4\left(16m^{4}+16m^{2}\bm{v}^{2}+\bm{v}^{4}\right)^{3}},$
(66) $\displaystyle G$
$\displaystyle=m^{4}\bigg{[}\dfrac{27}{64}\,C_{0}^{2}+64(2\,m^{2}+3\bm{v}^{2})\bigg{]}+2\bm{v}^{4}(30\,m^{2}-\bm{v}^{2}),$
(67) $\displaystyle B$
$\displaystyle=\dfrac{1}{3}\left\\{\left(\dfrac{A}{2}\right)^{\\!\\!-\frac{1}{3}}\left[\left(\dfrac{A}{2}\right)^{\\!\\!\frac{2}{3}}+16m^{4}+16m^{2}\bm{v}^{2}+\bm{v}^{4}\right]+2(2m^{2}+\bm{v}^{2})\right\\},$
(68)

and $C_{0}=\bar{Z}\gamma_{0}\gamma_{5}Z$ is the zeroth component of the
background fermionic condensate, responsible for the parity breaking in both
(64) and (65) rest mass modes.

$ii)$ In other situation, we assume that the background is invariant under
parity transformations. In this case, the effects of the background fermionic
condensate disappear, i.e., $C_{\alpha}=0$,

$\displaystyle
E_{0,\,\textrm{scalar}}\Big{|}_{C_{\alpha}=0}=\pm\,\sqrt{m^{2}+\bm{v}^{2}+\dfrac{1}{2}|h|^{2}}.$
(69)

In this situation, the scalar mass spectrum is quite similar to the fermionic
case and does not allow for the presence of tachyonic mass modes. As before,
assuming that $m^{2}\gg\bm{v}^{2}$ and $m^{2}\gg|h|^{2}$, (69) reduces to

$\displaystyle E_{0,\,\textrm{scalar}}\Big{|}_{C_{\alpha}=0}\approx
m\left(1+\dfrac{2\bm{v}^{2}+|h|^{2}}{4m^{2}}\right).$ (70)

Unlike from the photon-photino sector, the fermion-scalar sector exhibits a
SUSY breaking due the difference between the scalar and fermion rest masses in
both cases, parity invariant and parity breaking LSV backgrounds.

Furthermore, another interesting result which can be expressed from the
modified dispersion relations is the group velocity,

$\displaystyle\bm{v}_{g}=\bm{\nabla}_{\textrm{Re}(\xi)\bm{p}}E,$ (71)

where $\bm{\nabla}_{\textrm{Re}(\xi)\bm{p}}$ is a gradient of linear momentum
$\textrm{Re}(\xi)\bm{p}$ and $E$ is the energy that comes from the zeroth
component of $\tilde{p}_{\mu}$. According to the conditions $i)$ and $ii)$, we
arrive at the following group velocity for the scalars

$\displaystyle v^{i}_{g,\,\textrm{scalar}}\Big{|}_{h=0}=$
$\displaystyle\,\frac{\tilde{p}^{\,i}\big{[}\tilde{\bm{p}}^{\,2}+m^{2}-E^{2}-3(\tilde{\bm{p}}\cdot\bm{v})+\bm{v}^{2}/2\big{]}}{E\big{[}\tilde{\bm{p}}^{\,2}+m^{2}-E^{2}-3(\tilde{\bm{p}}\cdot\bm{v})+\bm{v}^{2}/2\big{]}-m^{2}C_{0}/32}$
$\displaystyle-\frac{3v^{i}\big{[}\tilde{\bm{p}}^{\,2}+m^{2}-E^{2}-5(\tilde{\bm{p}}\cdot\bm{v})\big{]}+m^{2}C^{i}/16}{2E\big{[}\tilde{\bm{p}}^{\,2}+m^{2}-E^{2}-3(\tilde{\bm{p}}\cdot\bm{v})+\bm{v}^{2}/2\big{]}-m^{2}C_{0}/16},$
(72)

in the case of $h=0$, and

$\displaystyle v^{i}_{g,\,\textrm{scalar}}\Big{|}_{C_{\alpha}=0}=$
$\displaystyle\,\frac{\tilde{p}^{\,i}}{E}-\frac{3v^{i}\big{[}\tilde{\bm{p}}^{\,2}+m^{2}-E^{2}-5(\tilde{\bm{p}}\cdot\bm{v})\big{]}}{2E\big{[}\tilde{\bm{p}}^{\,2}+m^{2}-E^{2}-3(\tilde{\bm{p}}\cdot\bm{v})+(2\bm{v}^{2}+|h|^{2})/4\big{]}},$
(73)

for a parity invariant background.

And finally, the group velocity for the fermionic case is given by

$\displaystyle v^{i}_{g,\,\textrm{fermion}}=$
$\displaystyle\,\frac{\tilde{p}^{\,i}}{E}-\frac{v^{i}\big{[}\tilde{\bm{p}}^{\,2}+m^{2}-E^{2}-3(\tilde{\bm{p}}\cdot\bm{v})\big{]}}{E\big{[}\tilde{\bm{p}}^{\,2}+m^{2}-E^{2}-2(\tilde{\bm{p}}\cdot\bm{v})+(\bm{v}^{2}+|h|^{2})/4\big{]}},$
(74)

where $\tilde{p}_{i}=\textrm{Re}(\xi)p_{i}$, with $i=1,2,3$. Both cases reduce
to the usual situation, $v^{i}_{g}=p^{i}/E$, if we turn off the LSV
background. Let us notice that, even in the presence of LSV terms, the
expressions for the group velocities (4.3), (73) and (74) admit the condition
$|v^{i}_{g}|<1$, indicating in this case that the causal structure of the
theory is maintained (see, e.g., [39, 47, 48, 13] for discussions on causality
in other scenarios with Lorentz violation).

##  5 The electron’s EDM

We are, at this point, ready to start off the task of inspecting the structure
of the fermionic current, as the modifications that arise due to the
background may induce a contribution to the charged particle electric dipole
moment interaction (see, e.g., [49, 50] for the textbook level introduction).
First, due to non-local structure, it proves useful to rewrite the modified
equation (54) as

$\displaystyle\Big{\\{}(1-s)\big{[}(\tilde{p}-\zeta)_{\mu}\gamma^{\mu}+m\big{]}-r_{\mu}\gamma^{\mu}\gamma_{5}-w_{\mu\alpha}\Sigma^{\mu\alpha}\Big{\\}}\Psi_{0}$
$\displaystyle=0,$ (75)

where we introduce the following definitions

$\displaystyle
r_{\mu}=\frac{1}{16m^{2}}\bigg{[}2\tilde{p}_{\mu}\tilde{p}_{\nu}C^{\nu}-2\bigg{(}\tilde{p}\cdot\zeta+m^{2}+\frac{1}{4}|h|^{2}\bigg{)}C_{\mu}+\tilde{p}_{\mu}(h^{*}-h)\tau-\frac{1}{2}\tilde{p}_{\mu}(h^{*}+h)\theta\bigg{]},$
$\displaystyle w_{\mu\alpha}=\frac{i}{4m}\tilde{p}_{\mu}C_{\alpha},$
$\displaystyle
s=\dfrac{1}{m^{2}}\big{[}\tilde{p}^{2}-2(\tilde{p}\cdot\zeta)\big{]}.$ (76)

By iterating modified equation (75) and taking the Dirac conjugate, we get the
Gordon decomposition below

$\displaystyle\bar{\Psi}_{0}(\tilde{p}^{\prime})\gamma^{\nu}\Psi_{0}(\tilde{p})=$
$\displaystyle\,-\dfrac{1}{2m}\bar{\Psi}_{0}(\tilde{p}^{\prime})\bigg{\\{}\Big{[}(\tilde{p}+\tilde{p}^{\prime}-2\zeta)^{\nu}-s(\tilde{p}-\zeta)^{\nu}-s^{\prime}(\tilde{p}^{\prime}-\zeta)^{\nu}\Big{]}$
$\displaystyle\,+\dfrac{i}{2}\left[{\left(w^{\prime}+w\right)_{\mu}}^{\nu}-{\left(w^{\prime}+w\right)^{\nu}}_{\mu}+2im(s+s^{\prime})\delta_{\mu}^{\nu}\right]\gamma^{\mu}$
$\displaystyle\,-2i\Big{[}(\tilde{p}^{\prime}-\tilde{p})_{\mu}+s(p-\zeta)_{\mu}-s^{\prime}(p^{\prime}-\zeta)_{\mu}\Big{]}\Sigma^{\mu\nu}+(r^{\prime}-r)^{\nu}\gamma_{5}$
$\displaystyle\,+\dfrac{1}{2}\left(w^{\prime}-w\right)_{\mu\alpha}\epsilon^{\mu\alpha\nu\tau}\gamma_{\tau}\gamma_{5}\Psi_{0}(\tilde{p})-2i(r^{\prime}+r)_{\mu}(\tilde{p}^{\prime})\Sigma^{\mu\nu}\gamma_{5}\bigg{\\}}\Psi_{0}(\tilde{p})\,.$
(77)

Here, the primes denote the structures carrying the momentum
$\tilde{p}^{\prime}$. This result, besides providing information on the vector
and axial charge densities in terms proportional to the identity and
$\gamma_{5}$, also brings to light the possible interactions that the
electromagnetic current may induce in this scenario, i.e., interactions such
as electric and magnetic dipole moments associated with the terms
$\Sigma^{\mu\nu}$ and $\Sigma^{\mu\nu}\gamma_{5}$, respectively, as well as
other terms that require further attention.

From the last term of (77), one can identify the piece from which the electric
dipole moment (EDM) emerges,

$\displaystyle\dfrac{i}{8m^{3}}C^{\tau}(p^{\prime}_{\mu}p^{\prime}_{\tau}+p_{\mu}p_{\tau})\bar{\Psi}_{0}(\tilde{p}^{\prime})\Sigma^{\mu\nu}\gamma_{5}\Psi_{0}(\tilde{p}).$
(78)

When minimally coupled to the gauge field $\tilde{a}_{\nu}$, the term in Eq.
(78) corresponds to the following structure for the effective electric dipole
interaction

$\displaystyle\mathcal{L}_{\textrm{EDM}}=-i\,\dfrac{\mu_{\textrm{B}}}{8m^{2}}\,C^{\tau}\tilde{\partial}_{\tau}\big{(}\bar{\Psi}\Sigma^{\mu\nu}\gamma_{5}\Psi\big{)}\mathcal{F}_{\mu\nu},$
(79)

where $\mu_{\textrm{B}}=e/(2m)$ is the Bohr magneton and
$\mathcal{F}_{\mu\nu}$ is defined in Eq. (18). Following Ref.[51], one can
verify that, in our case, the EDM is given by

$\displaystyle
d=-\,\dfrac{F_{3}(\tilde{q}^{2}=0)}{2m}=\dfrac{\mu_{\textrm{B}}}{4m^{2}}C^{\tau}\tilde{q}_{\tau},$
(80)

with $\tilde{q}=\tilde{p}^{\prime}-\tilde{p}$ being the transfer momentum of
the interaction. $F_{3}(\tilde{q}^{2})$ is the third electromagnetic form
factor associated with the EDM.

In the case of the electron EDM, denoted by $d_{e}$, the most recent
experimental results, obtained by using electrons confined inside molecular
ions subject to a huge intramolecular electric field, provide an upper bound
of $d_{e}<4.1\times 10^{-30}$ e$\cdot$cm [52, 53]. We can use these results to
estimate a bound for the modulus of the fermionic background condensate
$C^{\tau}$.

Let us remember that $\tilde{q}^{2}=\tilde{q}^{\mu}\tilde{q}_{\mu}=0$ for the
EDM form factor configuration. Consequently, assuming a typical energy scale
of the order of $1$ MeV and considering the special case where $C^{\tau}$ is
timelike, i.e., $\bm{C}=0$, we arrive at

$\displaystyle|C^{0}|<2.2\times 10^{-13}\,\textrm{eV}.$ (81)

For a purely spacelike $C^{\tau}$, a reference frame can be chosen in which
$C^{0}=0$. In this case, it is possible to establish a bound on the magnitude
of $\bm{C}$ component,

$\displaystyle|\bm{C}|<\frac{1}{\textrm{Re}(\xi)\cdot\cos(\alpha)}\,2.2\times
10^{-13}\,\textrm{eV},$ (82)

where $\alpha$ is the angle between the spatial part of the four-momentum
transfer, $\tilde{\bm{q}}$, and $\bm{C}$. We justify that the choice of
splitting the two cases of $C^{\tau}$ time- or space-like is a common
procedure in the study of LSV models. The reason is that either choice (time-
or space-like) may induce different effects as far as physical observables are
concerned.

Once the bound in (82) depends on both $\textrm{Re}(\xi)$ and $\alpha$, we can
collect all upper bounds that within a range $\alpha=\big{[}0,\pi/2\big{)}$ by
plotting a graph of $|\bm{C}|$ as a function of $\alpha$ for a specific value
of $\textrm{Re}(\xi)$. As an example, Figure 1 represents the plot of the
upper bounds on $|\bm{C}|$ for $\textrm{Re}(\xi)=0.5,\;1.0$ and $1.5$.

> Figure 1: Plot of the upper bounds on the spatial components of $C^{\tau}$
> as a function of $\alpha$ for $\textrm{Re}(\xi)=0.5,\;1.0$ and $1.5$, in
> purple, orange, and blue, respectively.

##  6 Conclusive Considerations and Perspectives

In this contribution, we have studied the implications of a Lorentz-violating
background supermultiplet in an $\mathcal{N}=1$-supersymmetric $U(1)$ gauge
model that extends the Isobe-Nagaosa model [42]. In our endeavour, we consider
that the LSV parameters may have their microscopic origin traced back to a
background SUSY multiplet, namely, a chiral superfield. Instead of taking the
LSV parameters for granted and extending the SUSY algebra, we keep the
ordinary SUSY structure and assume the existence of the background mentioned
above. Condensates of the fermionic component of the background appear that
are in the backstage of the scalar, pseudoscalar and pseudovector objects that
parametrize LSV.

The modifications in the gauge sector that arise as a consequence of the
Lorentz breaking in the fermionic matter consist of nothing but a dilation
factor in Maxwell’s equations. Indeed, the photon-photino sector is influenced
only by the real component of the Lorentz-violating parameter and, therefore,
we do not observe any physically relevant modifications in this sector, at
least initially. On the other hand, the scalar matter is modified in both its
dynamic and mass terms. Furthermore, the fermionic sector also receives an
additional contribution that is proportional to the background fermionic
condensate, $\bar{Z}\gamma_{\mu}\gamma_{5}Z$. In this case, we believe that
such a contribution arises due to the SUSY breaking induced by the Lorentz-
symmetry violation. In this sense, it is interesting to recall that our model
successfully reproduces the CPT-odd terms of the Standard Model Extension,
with one of these terms appearing naturally as a direct consequence of SUSY
breaking.

Considering the dispersion relations, we verify that, in the case of fermions,
the terms proportional to the background fermionic condensates appear only as
higher-order contributions of the Lorentz violation. Therefore, by assuming
that the LSV parameters are small compared to the mass $m$, such condensates
do not provide relevant contributions to the rest energy nor to the fermionic
group velocity. On the other hand, the scalar dispersion relation receives
contribution from the background condensate $\bar{Z}\gamma_{\mu}\gamma_{5}Z$
already in the second order.

Finally, our results from Section 5 indicate a possible connection between the
electric dipole moment and the SUSY breaking induced by the Lorentz-symmetry
violation. In this scenario, it is shown that SUSY matter scalars play a
supportive role in the process of generating the electric dipole moment of the
charged fermion.

The introduction of additional Lorentz-violating terms in the gauge sector,
which induce significant modifications in the photon-photino sector, is a path
to be followed as a step forward. The motivation to do that is to keep track
of the interference between the effects of the different LSV terms present in
the matter and the gauge sectors.

Furthermore, based on the theoretical framework developed here, we aim to
explore potential applications to those condensed matter systems that exhibit
(low-energy) SUSY and Lorentz symmetry as emergent phenomena in the dynamics
of quasiparticle excitations. On the other hand, the modified photonic and
electronic dispersion relations including both SUSY and LSV effects may be
studied in connection with photon-photon, Breit-Wheeler and Compton
scatterings which may be used to impose upper limits on the parameters with
the help of astrophysical data, such as measurements of (non-)attenuation of
gamma-rays from active galactic nuclei (AGNs), gamma-ray bursts (GRBs) and
extra galactic background light (EBL). Also, we keep in mind that QED high-
precision tests can also provide estimates and constraints on the parameters
of the model.

## Acknowledgments

W.C.S. and J.P.S.M. express their gratitude to CAPES-Brazil and CNPq-Brazil,
respectively, for granting their PhD Fellowships.

## Appendices

###  A The alternative case of a vector superfield

In this Appendix, we show the result of supersymmetrization following the
second approach cited in the Section 3.3, in which we define a vector
superfield in the form

$\displaystyle V$
$\displaystyle=C+\theta\zeta+\bar{\theta}\bar{\zeta}+(\theta\sigma^{\mu}\bar{\theta})v_{\mu}+\theta^{2}M+\bar{\theta}^{2}M^{*}+\theta^{2}\bar{\theta}\bar{\rho}+\bar{\theta}^{2}\theta\rho+\theta^{2}\bar{\theta}^{2}D.$
(83)

Here, the terms in (28) comes out from the combination
$V\big{(}\Phi_{\textrm{L}}e^{eA_{\textrm{WZ}}}\bar{\Phi}_{\textrm{L}}-\Phi_{\textrm{R}}e^{-eA_{\textrm{WZ}}}\bar{\Phi}_{\textrm{R}}\big{)}\big{|}_{\sim\theta^{2}\bar{\theta}^{2}}$.
Then, after some algebra, one can verify that

$\displaystyle S=$ $\displaystyle-\int d^{4}x\,\bigg{\\{}\int d^{2}\theta
d^{2}\bar{\theta}\,\bigg{[}\bar{\Phi}_{\textrm{L}}e^{eA_{\textrm{WZ}}}\Phi_{\textrm{L}}+\bar{\Phi}_{\textrm{R}}e^{-eA_{\textrm{WZ}}}\Phi_{\textrm{R}}+V\big{(}\bar{\Phi}_{\textrm{L}}e^{eA_{\textrm{WZ}}}\Phi_{\textrm{L}}-\bar{\Phi}_{\textrm{R}}e^{-eA_{\textrm{WZ}}}\Phi_{\textrm{R}}\big{)}\bigg{]}$
$\displaystyle-\frac{1}{4}\bigg{(}\int
d^{2}\theta\,W^{2}+c.c.\bigg{)}-m\bigg{(}\int
d^{2}\theta\,\Phi_{\textrm{R}}\Phi_{\textrm{L}}+c.c.\bigg{)}\bigg{\\}}.$ (84)

where

$\displaystyle
V\Phi_{\textrm{L}}e^{eA_{\textrm{WZ}}}\bar{\Phi}_{\textrm{L}}\Big{|}_{\sim\theta^{2}\bar{\theta}^{2}}=$
$\displaystyle-\frac{1}{2}v_{\mu}(\bar{\psi}\bar{\sigma}^{\mu}\psi)-\frac{i}{2}v_{\mu}\Big{[}(\tilde{\partial}^{\mu}\varphi^{*})\varphi-\varphi^{*}\tilde{\partial}^{\mu}\varphi\Big{]}+\varphi^{*}\varphi
D+f^{*}\varphi M+f\varphi^{*}M^{*}$
$\displaystyle-\frac{1}{\sqrt{2}}\Big{[}\varphi\bar{\psi}\bar{\rho}+\varphi^{*}\psi\rho+f^{*}\psi\zeta+f\bar{\psi}\bar{\zeta}\Big{]}+\frac{i}{2\sqrt{2}}\Big{[}\varphi(\tilde{\partial}_{\mu}\bar{\psi})\bar{\sigma}^{\mu}\zeta+(\bar{\zeta}\bar{\sigma}^{\mu}\psi)\tilde{\partial}_{\mu}\varphi^{*}$
$\displaystyle-(\bar{\psi}\bar{\sigma}^{\mu}\zeta)\tilde{\partial}_{\mu}\varphi-\varphi^{*}\,\bar{\zeta}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\psi\Big{]}+C\Big{[}f^{*}f+(\tilde{\partial}^{\mu}\varphi^{*})\tilde{\partial}_{\mu}\varphi\Big{]}-\frac{i}{2}C\Big{[}\bar{\psi}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\psi$
$\displaystyle-(\tilde{\partial}_{\mu}\bar{\psi})\bar{\sigma}^{\mu}\psi\Big{]}+e\bigg{\\{}\frac{iC}{\sqrt{2}}\Big{[}\varphi^{*}(\lambda\psi)-\varphi(\bar{\lambda}\bar{\psi})\Big{]}-\frac{1}{2}C\Big{[}i(\varphi\,\tilde{\partial}^{\mu}\varphi^{*}-\varphi^{*}\tilde{\partial}^{\mu}\varphi)$
$\displaystyle+\bar{\psi}\bar{\sigma}^{\mu}\psi\Big{]}\tilde{a}_{\mu}-\frac{1}{2\sqrt{2}}\Big{[}\varphi(\bar{\psi}\bar{\sigma}^{\mu}\zeta)+\varphi^{*}(\bar{\zeta}\bar{\sigma}^{\mu}\psi)\Big{]}\tilde{a}_{\mu}+\frac{1}{2}\varphi^{*}\varphi\,\bigg{[}i(\lambda\zeta-\bar{\lambda}\bar{\zeta})$
$\displaystyle+v^{\mu}\tilde{a}_{\mu}+C\bigg{(}K+\frac{e}{2}\tilde{a}^{2}\bigg{)}\bigg{]}\bigg{\\}},$
(85)

and right sector contribution is

$\displaystyle
V\Phi_{\textrm{R}}e^{-eA_{\textrm{WZ}}}\bar{\Phi}_{\textrm{R}}\Big{|}_{\sim\theta^{2}\bar{\theta}^{2}}=$
$\displaystyle-\frac{1}{2}v_{\mu}(\bar{\chi}\bar{\sigma}^{\mu}\chi)+\frac{i}{2}v_{\mu}\Big{[}(\tilde{\partial}^{\mu}s^{*})s-s^{*}\tilde{\partial}^{\mu}s\Big{]}+s^{*}sD+gs^{*}M+g^{*}sM^{*}$
$\displaystyle-\frac{1}{\sqrt{2}}\Big{[}s^{*}\bar{\chi}\bar{\rho}+s\chi\rho+g\chi\zeta+g^{*}\bar{\chi}\bar{\zeta}\Big{]}+\frac{i}{2\sqrt{2}}\Big{[}s^{*}(\tilde{\partial}_{\mu}\bar{\chi})\bar{\sigma}^{\mu}\zeta+(\bar{\zeta}\bar{\sigma}^{\mu}\chi)\tilde{\partial}_{\mu}s$
$\displaystyle-(\bar{\chi}\bar{\sigma}^{\mu}\zeta)\tilde{\partial}_{\mu}s^{*}-s\,\bar{\zeta}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\chi\Big{]}+C\Big{[}g^{*}g+(\tilde{\partial}^{\mu}s^{*})\tilde{\partial}_{\mu}s\Big{]}-\frac{i}{2}C\Big{[}\bar{\chi}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\chi$
$\displaystyle-(\tilde{\partial}_{\mu}\bar{\chi})\bar{\sigma}^{\mu}\chi\Big{]}-e\bigg{\\{}\frac{iC}{\sqrt{2}}\Big{[}s(\lambda\chi)-s^{*}(\bar{\lambda}\bar{\chi})\Big{]}-\frac{1}{2}C\Big{[}i(s^{*}\tilde{\partial}^{\mu}s-s\,\tilde{\partial}^{\mu}s^{*})$
$\displaystyle+\bar{\chi}\bar{\sigma}^{\mu}\chi\Big{]}\tilde{a}_{\mu}-\frac{1}{2\sqrt{2}}\Big{[}s^{*}(\bar{\chi}\bar{\sigma}^{\mu}\zeta)+s(\bar{\zeta}\bar{\sigma}^{\mu}\chi)\Big{]}\tilde{a}_{\mu}+\frac{1}{2}s^{*}s\,\bigg{[}i(\lambda\zeta-\bar{\lambda}\bar{\zeta})$
$\displaystyle+v^{\mu}\tilde{a}_{\mu}+C\bigg{(}K-\frac{e}{2}\tilde{a}^{2}\bigg{)}\bigg{]}\bigg{\\}}.$
(86)

In this case, using the constraints obtained from the field equations for the
auxiliary fields, namely $f,\,g$ and $D$, we get

$\displaystyle S_{2}=$ $\displaystyle\int
d^{4}x\,\bigg{\\{}-\frac{1}{4}\mathcal{F}^{2}_{\mu\nu}+\frac{i}{2}\Big{[}(\tilde{\partial}_{\mu}\bar{\lambda})\bar{\sigma}^{\mu}\lambda-\bar{\lambda}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\lambda\Big{]}+\dfrac{i}{2}(1+C)\Big{[}\bar{\psi}\bar{\sigma}^{\mu}(\tilde{\partial}_{\mu}\psi)-(\tilde{\partial}_{\mu}\bar{\psi})\bar{\sigma}^{\mu}\psi\Big{]}$
$\displaystyle+\dfrac{i}{2}(1-C)\Big{[}\bar{\chi}\bar{\sigma}^{\mu}(\tilde{\partial}_{\mu}\chi)-(\tilde{\partial}_{\mu}\bar{\chi})\bar{\sigma}^{\mu}\chi\Big{]}+\dfrac{1}{2}v_{\mu}(\bar{\psi}\bar{\sigma}^{\mu}\psi-\bar{\chi}\bar{\sigma}^{\mu}\chi)-m(\bar{\psi}\bar{\chi}+\chi\psi)-(1+C)$
$\displaystyle\times(\tilde{\partial}^{\mu}\varphi^{*})\tilde{\partial}_{\mu}\varphi-(1-C)(\tilde{\partial}^{\mu}s^{*})\tilde{\partial}_{\mu}s+\frac{i}{2}v_{\mu}\Big{[}(\tilde{\partial}^{\mu}\varphi^{*})\varphi-\varphi^{*}\tilde{\partial}^{\mu}\varphi+(\tilde{\partial}^{\mu}s^{*})s-s^{*}\tilde{\partial}^{\mu}s\Big{]}$
$\displaystyle+\bigg{[}\frac{m^{2}}{(1-C)}+\frac{M^{*}M}{(1+C)}\bigg{]}\varphi^{*}\varphi+\bigg{[}\frac{m^{2}}{(1+C)}+\frac{M^{*}M}{(1-C)}\bigg{]}s^{*}s-\dfrac{m}{\sqrt{2}}\bigg{[}\frac{1}{(1-C)}(\varphi\chi\zeta+\varphi^{*}\bar{\chi}\bar{\zeta})$
$\displaystyle-\frac{1}{(1+C)}(s\bar{\psi}\bar{\zeta}+s^{*}\psi\zeta)\bigg{]}-\frac{i}{2\sqrt{2}}\Big{[}\varphi(\tilde{\partial}_{\mu}\bar{\psi})\bar{\sigma}^{\mu}\zeta-(\bar{\psi}\bar{\sigma}^{\mu}\zeta)\tilde{\partial}_{\mu}\varphi-\varphi^{*}\bar{\zeta}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\psi+(\bar{\zeta}\bar{\sigma}^{\mu}\psi)\tilde{\partial}_{\mu}\varphi^{*}$
$\displaystyle+s\bar{\zeta}\bar{\sigma}^{\mu}\tilde{\partial}_{\mu}\chi-(\bar{\zeta}\bar{\sigma}^{\mu}\chi)\tilde{\partial}_{\mu}s-s^{*}(\tilde{\partial}_{\mu}\bar{\chi})\bar{\sigma}^{\mu}\zeta+(\bar{\chi}\bar{\sigma}^{\mu}\zeta)\tilde{\partial}_{\mu}s^{*}\Big{]}+\frac{1}{\sqrt{2}}\Big{[}(\varphi\bar{\psi}-s^{*}\bar{\chi})\bar{\rho}$
$\displaystyle+(\varphi^{*}\psi-s\chi)\rho\Big{]}+\dfrac{1}{2(1+C)}\Big{[}\bar{\zeta}\bar{\psi}\zeta\psi-\sqrt{2}(M\varphi\bar{\zeta}\bar{\psi}+M^{*}\varphi^{*}\zeta\psi)\Big{]}+\dfrac{1}{2(1-C)}\Big{[}\bar{\zeta}\bar{\chi}\zeta\chi$
$\displaystyle-\sqrt{2}(Ms^{*}\bar{\zeta}\bar{\chi}+M^{*}s\zeta\chi)\Big{]}-(\varphi^{*}\varphi-s^{*}s)D-\frac{e}{2}(\varphi^{*}\varphi+s^{*}s)\Big{[}i(\lambda\zeta-\bar{\lambda}\bar{\zeta})+v^{\mu}\tilde{a}_{\mu}\Big{]}$
$\displaystyle+\frac{e}{2\sqrt{2}}(\varphi\bar{\psi}\bar{\sigma}^{\mu}\zeta+\varphi^{*}\bar{\zeta}\bar{\sigma}^{\mu}\psi+s^{*}\bar{\chi}\bar{\sigma}^{\mu}\zeta+s\bar{\zeta}\bar{\sigma}^{\mu}\chi)\tilde{a}_{\mu}-\frac{e}{2}(1+C)\Big{[}i\sqrt{2}(\varphi^{*}\lambda\psi-\varphi\bar{\lambda}\bar{\psi})$
$\displaystyle+(\bar{\psi}\bar{\sigma}^{\mu}\psi)\tilde{a}_{\mu}-i(\varphi\tilde{\partial}^{\mu}\varphi^{*}-\varphi^{*}\tilde{\partial}^{\mu}\varphi)\tilde{a}_{\mu}\Big{]}+\frac{e}{2}(1-C)\Big{[}i\sqrt{2}(s\lambda\chi-s^{*}\bar{\lambda}\bar{\chi})$
$\displaystyle+i(s\tilde{\partial}^{\mu}s^{*}-s^{*}\tilde{\partial}^{\mu}s)\tilde{a}_{\mu}-(\bar{\chi}\bar{\sigma}^{\mu}\chi)\tilde{a}_{\mu}\Big{]}-\frac{e^{2}}{4}\Big{[}(1+C)\varphi^{*}\varphi+(1-C)s^{*}s\Big{]}\tilde{a}^{2}$
$\displaystyle-\frac{e^{2}}{8}\Big{[}(1+C)\varphi^{*}\varphi-(1-C)s^{*}s\Big{]}^{2}\bigg{\\}}.$
(87)

As in the first approach, the gauge sector is given by (34). For the other
sectors, we have

  * •

fermionic sector:

$\displaystyle S_{\textrm{fermion}}=$ $\displaystyle\int
d^{4}x\,\bigg{\\{}\dfrac{i}{2}\Big{[}\bar{\Psi}\gamma^{\mu}\tilde{\partial}_{\mu}\Psi-(\tilde{\partial}_{\mu}\bar{\Psi})\gamma^{\mu}\Psi\Big{]}+\dfrac{1}{2}v_{\mu}(\bar{\Psi}\gamma^{\mu}\Psi)-m\bar{\Psi}\Psi$
$\displaystyle+\dfrac{e}{2}\big{(}\bar{\Psi}\gamma^{\mu}\gamma_{5}\Psi-C\bar{\Psi}\gamma^{\mu}\Psi\big{)}\tilde{a}_{\mu}-\dfrac{i}{2}C\Big{[}\bar{\Psi}\gamma^{\mu}\gamma_{5}\tilde{\partial}_{\mu}\Psi-(\tilde{\partial}_{\mu}\bar{\Psi})\gamma^{\mu}\gamma_{5}\Psi\Big{]}\bigg{\\}}.$
(88)

  * •

scalar sector:

$\displaystyle S_{\textrm{scalar}}=$ $\displaystyle-\int
d^{4}x\,\bigg{\\{}(1+C)(\tilde{D}^{\mu}\varphi^{*})\tilde{D}_{\mu}\varphi+(1-C)(\tilde{D}^{\mu}s^{*})\tilde{D}_{\mu}s+(\varphi^{*}\varphi-s^{*}s)D$
$\displaystyle-\frac{i}{2}v_{\mu}\Big{[}(\tilde{\partial}^{\mu}\varphi^{*})\varphi-\varphi^{*}\tilde{\partial}^{\mu}\varphi+(\tilde{\partial}^{\mu}s^{*})s-s^{*}\tilde{\partial}^{\mu}s\Big{]}-\bigg{[}\frac{m^{2}}{(1-C)}+\frac{M^{*}M}{(1+C)}\bigg{]}\varphi^{*}\varphi$
$\displaystyle-\bigg{[}\frac{m^{2}}{(1+C)}+\frac{M^{*}M}{(1-C)}\bigg{]}s^{*}s\bigg{\\}}.$
(89)

  * •

gaugino and matter self-interaction sector:

$\displaystyle S_{\textrm{int}}=$ $\displaystyle\int
d^{4}x\,\bigg{\\{}\frac{ie}{\sqrt{2}}\bigg{[}(1+C)(\varphi\bar{\Psi}\mathcal{P}_{\textrm{R}}\Lambda-\varphi^{*}\bar{\Lambda}\mathcal{P}_{\textrm{L}}\Psi)+(1-C)(s\bar{\Psi}\mathcal{P}_{\textrm{L}}\Lambda-s^{*}\bar{\Lambda}\mathcal{P}_{\textrm{R}}\Psi)\bigg{]}$
$\displaystyle+\frac{1}{\sqrt{2}}\Big{[}\bar{\Psi}(\varphi\mathcal{P}_{\textrm{R}}-s\mathcal{P}_{\textrm{L}})R+\bar{R}(\varphi^{*}\mathcal{P}_{\textrm{L}}-s^{*}\mathcal{P}_{\textrm{R}})\Psi\Big{]}-\frac{i}{2\sqrt{2}}\Big{[}(\tilde{D}^{*}_{\mu}\bar{\Psi})\gamma^{\mu}(\varphi\mathcal{P}_{\textrm{L}}-s\mathcal{P}_{\textrm{R}})Z$
$\displaystyle-\bar{Z}\gamma^{\mu}(\varphi^{*}\mathcal{P}_{\textrm{L}}-s^{*}\mathcal{P}_{\textrm{R}})\tilde{D}_{\mu}\Psi-\bar{\Psi}\gamma^{\mu}(\tilde{\partial}_{\mu}\varphi\mathcal{P}_{\textrm{L}}-\tilde{\partial}_{\mu}s\mathcal{P}_{\textrm{R}})Z$
$\displaystyle+\bar{Z}\gamma^{\mu}(\tilde{\partial}_{\mu}\varphi^{*}\mathcal{P}_{\textrm{L}}-\tilde{\partial}_{\mu}s^{*}\mathcal{P}_{\textrm{R}})\Psi\Big{]}-\dfrac{m}{\sqrt{2}}\bigg{[}\frac{1}{(1-C)}(\varphi\bar{\Psi}\mathcal{P}_{\textrm{L}}Z+\varphi^{*}\bar{Z}\mathcal{P}_{\textrm{R}}\Psi)$
$\displaystyle-\frac{1}{(1+C)}(s\bar{\Psi}\mathcal{P}_{\textrm{R}}Z+s^{*}\bar{Z}\mathcal{P}_{\textrm{L}}\Psi)\bigg{]}+\dfrac{1}{2(1+C)}\Big{[}(\bar{\Psi}\mathcal{P}_{\textrm{R}}Z)(\bar{Z}\mathcal{P}_{\textrm{L}}\Psi)$
$\displaystyle-\sqrt{2}(M\varphi\bar{\Psi}\mathcal{P}_{\textrm{R}}Z+M^{*}\varphi^{*}\bar{Z}\mathcal{P}_{\textrm{L}}\Psi)\Big{]}+\dfrac{1}{2(1-C)}\Big{[}(\bar{Z}\mathcal{P}_{\textrm{R}}\Psi)(\bar{\Psi}\mathcal{P}_{\textrm{L}}Z)$
$\displaystyle-\sqrt{2}(Ms^{*}\bar{Z}\mathcal{P}_{\textrm{R}}\Psi+M^{*}s\bar{\Psi}\mathcal{P}_{\textrm{L}}Z)\Big{]}-\frac{e}{2}(\varphi^{*}\varphi+s^{*}s)\bigg{[}v^{\mu}\tilde{a}_{\mu}-\frac{i}{2}(\bar{\Lambda}\gamma_{5}Z+\bar{Z}\gamma_{5}\Lambda)\bigg{]}$
$\displaystyle-\frac{e^{2}}{8}\Big{[}(1+C)\varphi^{*}\varphi-(1-C)s^{*}s\Big{]}^{2}\bigg{\\}},$
(90)

where

$\displaystyle R=\begin{pmatrix}\rho_{\alpha}\\\
\bar{\rho}^{\dot{\alpha}}\end{pmatrix}.$ (91)

###  B Useful results

We collect here some useful results related to the calculation of dispersion
relations in Sect. 4.3.

$\displaystyle X^{(1)}(\varphi_{0},s_{0})=$
$\displaystyle\,\dfrac{1}{16}\,\bigg{\\{}\Big{[}2m^{2}(\tilde{p}\cdot\zeta)-2(\tilde{p}^{2}-\tilde{p}\cdot\mathcal{R}-\tilde{p}\cdot\zeta)(\tilde{p}^{2}+2\tilde{p}\cdot\mathcal{R}-2\tilde{p}\cdot\zeta$
$\displaystyle+\mathcal{R}^{2}-2\zeta\cdot\mathcal{R}+\zeta^{2})\Big{]}h-2\Big{[}2m^{2}\tilde{p}^{2}-3m^{2}\tilde{p}\cdot\zeta-\tilde{p}^{4}-2(\tilde{p}\cdot\mathcal{R})^{2}$
$\displaystyle-(\tilde{p}\cdot\zeta)(2\tilde{p}\cdot\zeta+3\mathcal{R}^{2}-2\zeta\cdot\mathcal{R}-\zeta^{2})-(\tilde{p}\cdot\mathcal{R})(\mathcal{R}^{2}-2\zeta\cdot\mathcal{R}+\zeta^{2})$
$\displaystyle+\tilde{p}^{2}(\tilde{p}\cdot\mathcal{R}+3\tilde{p}\cdot\zeta+3\mathcal{R}^{2}-2\zeta\cdot\mathcal{R}-\zeta^{2})\Big{]}h^{*}+2m^{2}\big{(}m^{2}+\tilde{p}\cdot\mathcal{R}$
$\displaystyle+\mathcal{R}^{2}-\tilde{\zeta}^{2}\big{)}(h+h^{*})\bigg{\\}}\varphi_{0}+\dfrac{1}{16}\,\bigg{\\{}4m^{5}+4m^{3}\big{(}2\tilde{p}\cdot\tilde{\zeta}+\mathcal{R}^{2}-\tilde{\zeta}^{2}\big{)}$
$\displaystyle-4m\Big{[}\tilde{p}^{2}\big{(}\tilde{p}^{2}-2\tilde{p}\cdot\tilde{\zeta}+3\mathcal{R}^{2}+\tilde{\zeta}^{2}\big{)}-4(\tilde{p}\cdot\mathcal{R})\big{(}\tilde{p}\cdot\mathcal{R}-\zeta\cdot\mathcal{R}\big{)}$
$\displaystyle-4\tilde{p}\cdot\zeta\mathcal{R}^{2}\Big{]}-m\big{(}m^{2}-\tilde{p}^{2}+2\tilde{p}\cdot\tilde{\zeta}+\mathcal{R}^{2}-\tilde{\zeta}^{2}\big{)}h^{*2}\bigg{\\}}s_{0},$
(92)

$\displaystyle X^{(2)}(\varphi_{0},s_{0})=$
$\displaystyle\,-\dfrac{1}{16}\,\bigg{\\{}4m^{3}(m^{2}-2\tilde{p}^{2}+2\tilde{p}\cdot\mathcal{R}+4\tilde{p}\cdot\zeta+\mathcal{R}^{2}-\zeta^{2})+4m\Big{[}\tilde{p}^{2}(\tilde{p}^{2}-2\tilde{p}\cdot\mathcal{R}$
$\displaystyle-4\tilde{p}\cdot\zeta-5\mathcal{R}^{2}+\zeta^{2})+2(\tilde{p}\cdot\mathcal{R})(2\tilde{p}\cdot\mathcal{R}+2\tilde{p}\cdot\zeta+\mathcal{R}^{2}-2\zeta\cdot\mathcal{R}-\zeta^{2})$
$\displaystyle+2(\tilde{p}\cdot\zeta)(2\tilde{p}\cdot\zeta+3\mathcal{R}^{2}-\zeta^{2})+m(m^{2}-\tilde{p}^{2}+2\tilde{p}\cdot\zeta+\mathcal{R}^{2}-\zeta^{2})h^{*}h\Big{]}\bigg{\\}}\varphi_{0}$
$\displaystyle+\,\dfrac{1}{16}\,\bigg{\\{}4m^{2}(m^{2}-2\tilde{p}^{2}+3\tilde{p}\cdot\zeta+\mathcal{R}^{2}-\zeta^{2})+4\Big{[}\tilde{p}^{2}(\tilde{p}^{2}-3\tilde{p}\cdot\zeta-3\mathcal{R}^{2}$
$\displaystyle-2\zeta\cdot\mathcal{R}+\zeta^{2})+2(\tilde{p}\cdot\mathcal{R})(\tilde{p}\cdot\mathcal{R}+\tilde{p}\cdot\zeta-\zeta\cdot\mathcal{R}-\zeta^{2})$
$\displaystyle+(\tilde{p}\cdot\zeta)(2\tilde{p}\cdot\zeta+3\mathcal{R}^{2}+2\zeta\cdot\mathcal{R}-\zeta^{2})\Big{]}\bigg{\\}}h^{*}s_{0},$
(93)

$\displaystyle\mathcal{M}^{(1)}(\varphi_{0},s_{0})=$
$\displaystyle\,\dfrac{1}{16}\,\bigg{\\{}\Big{[}2m^{2}(\tilde{p}\cdot\zeta)-2(\tilde{p}^{2}-\tilde{p}\cdot\mathcal{R}-\tilde{p}\cdot\zeta)(\tilde{p}^{2}+2\tilde{p}\cdot\mathcal{R}-2\tilde{p}\cdot\zeta+\mathcal{R}^{2}$
$\displaystyle-2\zeta\cdot\mathcal{R}+\zeta^{2})\Big{]}h+\Big{[}4m^{2}\tilde{p}^{2}-6m^{2}\tilde{p}\cdot\zeta-2\big{[}\tilde{p}^{4}+2(\tilde{p}\cdot\mathcal{R})^{2}$
$\displaystyle+(\tilde{p}\cdot\zeta)(2\tilde{p}\cdot\zeta+3\mathcal{R}^{2}-2\zeta\cdot\mathcal{R}-\zeta^{2})\big{]}+(\tilde{p}\cdot\mathcal{R})(\mathcal{R}^{2}-2\zeta\cdot\mathcal{R}+\zeta^{2})$
$\displaystyle+\tilde{p}^{2}(-\tilde{p}\cdot\mathcal{R}-3\tilde{p}\cdot\zeta-3\mathcal{R}^{2}+2\zeta\cdot\mathcal{R}+\zeta^{2})\Big{]}h^{*}+2m^{2}\big{(}m^{2}+\tilde{p}\cdot\mathcal{R}$
$\displaystyle+\mathcal{R}^{2}-\tilde{\zeta}^{2}\big{)}(h-h^{*})\bigg{\\}}\varphi_{0}+\dfrac{1}{16}\,\bigg{\\{}4m^{5}+4m^{3}\big{(}2\tilde{p}\cdot\tilde{\zeta}+\mathcal{R}^{2}-\tilde{\zeta}^{2}\big{)}$
$\displaystyle-4m\Big{[}\tilde{p}^{2}\big{(}\tilde{p}^{2}-2\tilde{p}\cdot\tilde{\zeta}+3\mathcal{R}^{2}+\tilde{\zeta}^{2}\big{)}-4(\tilde{p}\cdot\mathcal{R})\big{(}\tilde{p}\cdot\mathcal{R}-\zeta\cdot\mathcal{R}\big{)}$
$\displaystyle-4(\tilde{p}\cdot\zeta)\mathcal{R}^{2}\Big{]}+m\big{(}m^{2}-\tilde{p}^{2}+2\tilde{p}\cdot\tilde{\zeta}+\mathcal{R}^{2}-\tilde{\zeta}^{2}\big{)}h^{*2}\bigg{\\}}s_{0},$
(94)

$\displaystyle\mathcal{M}^{(2)}(\varphi_{0},s_{0})=$
$\displaystyle\,\dfrac{1}{16}\,\bigg{\\{}4m^{3}(m^{2}-2\tilde{p}^{2}+2\tilde{p}\cdot\mathcal{R}+4\tilde{p}\cdot\zeta+\mathcal{R}^{2}-\zeta^{2})+4m\Big{[}\tilde{p}^{2}(\tilde{p}^{2}-2\tilde{p}\cdot\mathcal{R}$
$\displaystyle-4\tilde{p}\cdot\zeta-5\mathcal{R}^{2}+\zeta^{2})+2(\tilde{p}\cdot\mathcal{R})(2\tilde{p}\cdot\mathcal{R}+2\tilde{p}\cdot\zeta+\mathcal{R}-2\zeta\cdot\mathcal{R}-\zeta^{2})$
$\displaystyle+2(\tilde{p}\cdot\zeta)(2\tilde{p}\cdot\zeta+3\mathcal{R}^{2}-\zeta^{2})\Big{]}+m(m^{2}-\tilde{p}^{2}+2\tilde{p}\cdot\zeta+\mathcal{R}^{2}-\zeta^{2})h^{*}h\bigg{\\}}\varphi_{0}$
$\displaystyle-\,\dfrac{1}{16}\,\Big{[}4m^{2}(\tilde{p}\cdot\mathcal{R})-4(\tilde{p}\cdot\mathcal{R})(\tilde{p}^{2}-2\tilde{p}\cdot\zeta)+\mathcal{R}^{2}-\zeta^{2})\Big{]}h^{*}s_{0},$
(95)

$\displaystyle Y^{(1)}_{\alpha}(\varphi_{0},s_{0})=$
$\displaystyle\,-\dfrac{1}{16}\,\Big{[}4m\big{(}m^{2}-\tilde{p}^{2}+2\tilde{p}\cdot\zeta-\mathcal{R}^{2}-\zeta^{2}\big{)}\tilde{p}_{\alpha}+8m\mathcal{R}^{2}\zeta_{\alpha}-4m\big{(}m^{2}-\tilde{p}^{2}$
$\displaystyle+2\tilde{p}\cdot\mathcal{R}+2\tilde{p}\cdot\zeta-\mathcal{R}^{2}-\zeta^{2}\big{)}\mathcal{R}_{\alpha}\Big{]}h^{*}s_{0}+\dfrac{1}{16}\,\bigg{\\{}\Big{[}4m^{2}\big{(}m^{2}-2\tilde{p}^{2}+4\tilde{p}\cdot\zeta$
$\displaystyle-\mathcal{R}^{2}-2\zeta\cdot\mathcal{R}-\zeta^{2}\big{)}+4\tilde{p}^{2}\big{(}\tilde{p}^{2}-4\tilde{p}\cdot\mathcal{R}-4\tilde{p}\cdot\zeta+\mathcal{R}^{2}-2\zeta\cdot\mathcal{R}+\zeta^{2}\big{)}$
$\displaystyle+8(\tilde{p}\cdot\mathcal{R})\big{(}-2\tilde{p}\cdot\mathcal{R}+4\tilde{p}\cdot\zeta+\mathcal{R}^{2}-\zeta^{2}\big{)}+8(\tilde{p}\cdot\zeta)\big{(}2\tilde{p}\cdot\zeta+\mathcal{R}^{2}-\zeta^{2}\big{)}\Big{]}\tilde{p}_{\alpha}$
$\displaystyle+\Big{[}4m^{2}\big{(}m^{2}-2\tilde{p}^{2}+4\tilde{p}\cdot\zeta+3\mathcal{R}^{2}+2\zeta\cdot\mathcal{R}-\zeta^{2}\big{)}+4\tilde{p}^{2}\big{(}\tilde{p}^{2}+2\tilde{p}\cdot\mathcal{R}$
$\displaystyle-2\tilde{p}\cdot\zeta-3\mathcal{R}^{2}+2\zeta\cdot\mathcal{R}+\zeta^{2}\big{)}+8(\tilde{p}\cdot\mathcal{R})\big{(}2\tilde{p}\cdot\mathcal{R}-2\tilde{p}\cdot\zeta\big{)}\Big{]}\zeta_{\alpha}$
$\displaystyle+4\Big{[}m^{2}\big{(}m^{2}+2\tilde{p}\cdot\mathcal{R}+2\tilde{p}\cdot\zeta+\mathcal{R}^{2}-2\zeta\cdot\mathcal{R}-3\zeta^{2}\big{)}+\tilde{p}^{2}\big{(}3\tilde{p}^{2}-2\tilde{p}\cdot\mathcal{R}$
$\displaystyle+6\tilde{p}\cdot\zeta+\mathcal{R}^{2}-\zeta\cdot\mathcal{R}-\zeta^{2}\big{)}+4(\tilde{p}\cdot\zeta)\big{(}\tilde{p}\cdot\zeta-\tilde{p}\cdot\mathcal{R}\big{)}\Big{]}\mathcal{R}_{\alpha}$
$\displaystyle-16i\big{(}\tilde{p}^{2}-\tilde{p}\cdot\mathcal{R}-\tilde{p}\cdot\zeta\big{)}{\varepsilon_{\alpha}}^{\mu\nu\kappa}\tilde{p}_{\mu}\mathcal{R}_{\nu}\zeta_{\kappa}\bigg{\\}}\varphi_{0},$
(96)

$\displaystyle Y^{(2)}_{\alpha}(\varphi_{0},s_{0})=$
$\displaystyle\,\dfrac{1}{16}\,\bigg{\\{}m\Big{[}4\big{(}m^{2}-\tilde{p}^{2}+2\tilde{p}\cdot\mathcal{R}+2\tilde{p}\cdot\zeta+2\mathcal{R}^{2}-\zeta\cdot\mathcal{R}-\zeta^{2}\big{)}\tilde{p}_{\alpha}-2\big{(}m^{2}-\tilde{p}^{2}$
$\displaystyle+4\tilde{p}\cdot\mathcal{R}+2\tilde{p}\cdot\zeta+3\mathcal{R}^{2}-2\zeta\cdot\mathcal{R}-\zeta^{2}\big{)}\zeta_{\alpha}+2\big{(}m^{2}-5\tilde{p}^{2}-2\tilde{p}\cdot\mathcal{R}$
$\displaystyle+8\tilde{p}\cdot\zeta+\mathcal{R}^{2}+\zeta\cdot\mathcal{R}-3\zeta^{2}\big{)}\mathcal{R}_{\alpha}-4i{\varepsilon_{\alpha}}^{\mu\nu\kappa}\tilde{p}_{\mu}\mathcal{R}_{\nu}\zeta_{\kappa}\Big{]}h-\Big{[}4\big{(}\mathcal{R}^{2}+\zeta\cdot\mathcal{R}\big{)}\tilde{p}_{\alpha}$
$\displaystyle-2\big{(}m^{2}-\tilde{p}^{2}+2\tilde{p}\cdot\zeta+3\mathcal{R}^{2}+2\zeta\cdot\mathcal{R}-\zeta^{2}\big{)}\zeta_{\alpha}-2\big{(}m^{2}-\tilde{p}^{2}+2\tilde{p}\cdot\mathcal{R}$
$\displaystyle+4\tilde{p}\cdot\zeta+\mathcal{R}^{2}-2\zeta\cdot\mathcal{R}-3\zeta^{2}\big{)}\mathcal{R}_{\alpha}-4i{\varepsilon_{\alpha}}^{\mu\nu\kappa}\tilde{p}_{\mu}\mathcal{R}_{\nu}\zeta_{\kappa}\Big{]}h^{*}\bigg{\\}}\varphi_{0}$
$\displaystyle+\,\dfrac{1}{16}\,\bigg{\\{}\Big{[}\big{(}m^{2}-\tilde{p}^{2}-2\tilde{p}\cdot\mathcal{R}+2\tilde{p}\cdot\zeta+3\mathcal{R}^{2}+2\zeta\cdot\mathcal{R}-\zeta^{2}\big{)}\tilde{p}_{\alpha}-\big{(}m^{2}-\tilde{p}^{2}$
$\displaystyle-2\tilde{p}\cdot\mathcal{R}+2\tilde{p}\cdot\zeta-3\mathcal{R}^{2}+2\zeta\cdot\mathcal{R}-\zeta^{2}\big{)}\zeta_{\alpha}-\big{(}m^{2}-3\tilde{p}^{2}+2\tilde{p}\cdot\mathcal{R}$
$\displaystyle+6\tilde{p}\cdot\zeta+\mathcal{R}^{2}-2\zeta\cdot\mathcal{R}-3\zeta^{2}\big{)}\mathcal{R}_{\alpha}\Big{]}{h^{*}}^{2}+4\Big{[}3m^{4}+\tilde{p}^{4}+4(\tilde{p}\cdot\mathcal{R})^{2}$
$\displaystyle+4(\tilde{p}\cdot\zeta)^{2}-2(\tilde{p}\cdot\mathcal{R}-\tilde{p}\cdot\zeta)\mathcal{R}^{2}+2(\tilde{p}\cdot\mathcal{R}-\tilde{p}\cdot\zeta)\zeta^{2}-m^{2}(\tilde{p}^{2}-8\tilde{p}\cdot\zeta$
$\displaystyle-5\mathcal{R}^{2}+2\zeta\cdot\mathcal{R}+3\zeta^{2})-\tilde{p}^{2}(4\tilde{p}\cdot\zeta+\mathcal{R}^{2}+2\zeta\cdot\mathcal{R}-\zeta^{2})\Big{]}\tilde{p}_{\alpha}$
$\displaystyle-4\Big{[}m^{4}-\tilde{p}^{4}+4(\tilde{p}\cdot\mathcal{R})^{2}+4(\tilde{p}\cdot\mathcal{R})(\tilde{p}\cdot\zeta)+m^{2}(2\tilde{p}\cdot\mathcal{R}+2\tilde{p}\cdot\zeta+3\mathcal{R}^{2}$
$\displaystyle-2\zeta\cdot\mathcal{R}-\zeta^{2})-\tilde{p}^{2}(2\tilde{p}\cdot\mathcal{R}-2\tilde{p}\cdot\zeta+\mathcal{R}^{2}+2\zeta\cdot\mathcal{R}+\zeta^{2})\Big{]}\zeta_{\alpha}$
$\displaystyle+4\Big{[}m^{4}+\tilde{p}^{4}+4(\tilde{p}\cdot\zeta)(\tilde{p}\cdot\mathcal{R}+\tilde{p}\cdot\zeta)-m^{2}(2\tilde{p}^{2}+2\tilde{p}\cdot\mathcal{R}-6\tilde{p}\cdot\zeta-\mathcal{R}^{2}$
$\displaystyle-2\zeta\cdot\mathcal{R}+3\zeta^{2})-\tilde{p}^{2}(2\tilde{p}\cdot\mathcal{R}+2\tilde{p}\cdot\zeta-\mathcal{R}^{2}+2\zeta\cdot\mathcal{R}-\zeta^{2})\Big{]}\mathcal{R}_{\alpha}$
$\displaystyle-16i(\tilde{p}^{2}-\tilde{p}\cdot\mathcal{R}-\tilde{p}\cdot\zeta){\varepsilon_{\alpha}}^{\mu\nu\kappa}\tilde{p}_{\mu}\mathcal{R}_{\nu}\zeta_{\kappa}\bigg{\\}}s_{0}.$
(97)

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