Photon sector analysis of Super and Lorentz symmetry breaking: effective photon mass, birefringence and dissipation
Abstract
Within the standard model extension (SME), we expand our previous findings on four classes of violations of SuperSymmetry (SuSy) and Lorentz Symmetry (LoSy), differing in the handedness of the Charge conjugationParityTime reversal (CPT) symmetry and in whether considering the impact of photinos on photon propagation. The violations, occurring at the early universe high energies, show visible traces at present in the Dispersion Relations (DRs). For the CPTodd classes (\(V_{\mu }\) breaking vector) associated with the Carroll–Field–Jackiw (CFJ) model, the DRs and the Lagrangian show for the photon an effective mass, gauge invariant, proportional to \({\mathbf {V}}\). The group velocity exhibits a classic dependency on the inverse of the frequency squared. For the CPTeven classes (\(k_{F}\) breaking tensor), when the photino is considered, the DRs display also a massive behaviour inversely proportional to a coefficient in the Lagrangian and to a term linearly dependent on \(k_{F}\). All DRs display an angular dependence and lack LoSy invariance. In describing our results, we also point out the following properties: (i) the appearance of complex or simply imaginary frequencies and superluminal speeds and (ii) the emergence of birefringence. Finally, we point out the circumstances for which SuSy and LoSy breakings, possibly in presence of an external field, lead to the nonconservation of the photon energymomentum tensor. We do so for both CPT sectors.
1 Introduction, motivation and structure of the work
For the most part, we base our understanding of particle physics on the standard model (SM). The SM proposes the Lagrangian of particle physics and summarises three interactions among fundamental particles, accounting for electromagnetic (EM), weak and strong nuclear forces. The model has been completed theoretically in the mid seventies, and has found several experimental confirmations ever since. In 1995, the top quark was found [1]; in 2000, the tau neutrino was directly measured [2]. Last, but not least, in 2012 the most elusive particle, the Higgs Boson, was found [3]. The associated Higgs field induces the spontaneous symmetry breaking mechanism, responsible for all the masses of the SM particles. Neutrinos and the photon remain massless, for they do not have a direct interaction with the Higgs field. Remarkably, massive neutrinos are not accounted for by the SM.
All ordinary hadronic and leptonic matter is made of Fermions, while Bosons are the interaction carriers in the SM. The force carrier for the electromagnetism is the photon. Strong nuclear interactions are mediated by eight gluons, massless but not free particles, described by quantum chromodynamics (QCD). Instead, the \(W^{+}\), \(W^{}\) and Z massive Bosons, are the mediators of the weak interaction. The charge of the Wmediators has suggested that the EM and weak nuclear forces can be unified into a single interaction called electroweak interaction.
We finally notice that the photon is the only massless nonconfined Boson; the reason for this must at least be questioned by fundamental physics.
SM considers all particles being massless, before the Higgs field intervenes. Of course, masslessness of particles would be in contrast with every day experience. In 1964, Higgs and others [4, 5, 6] came up with a mechanism that, thanks to the introduction of a new field  the Higgs field  is able to explain why the elementary particles in the spectrum of the SM, namely, the charged leptons and quarks, become massive. But the detected mass of the Higgs Boson is too light: in 2015 the ATLAS and CMS experiments showed that the Higgs Boson mass is \(125.09\pm 0.32\ \) GeV/c\(^2\) [3]. Between the GeV scale of the electroweak interactions and the Grand Unification Theory (GUT) scale (\(10^{16}\) GeV), it is widely believed that new physics should appear at the TeV scale, which is now the experimental limit up to which the SM was tested [7]. Consequently, we need a fundamental theory that reproduces the phenomenology at the electroweak scale and, at the same time, accounts for effects beyond the TeV scale.
An interesting attempt to go beyond the SM is for sure SuperSymmetry (SuSy); see [8] for a review. This theory predicts the existence of new particles that are not included in the SM. The interaction between the Higgs and these new SuSy particles would cancel out some SM contributions to the Higgs Boson mass, ensuring its lightness. This is the solution to the socalled gauge hierarchy problem. The SM is assumed to be Lorentz^{1} Symmetry (LoSy) invariant. Anyway, it is reasonable to expect that this prediction is valid only up to certain energy scales [9, 10, 11, 12, 13, 14, 15], beyond which a LoSy Violation (LSV) might occur. The LSV would take place following the condensation of tensor fields in the context of open Bosonic strings.
The aforementioned facts show that there are valid reasons to undertake an investigation of physics beyond the SM and also consider LSV. There is a general framework where we can test the lowenergy manifestations of LSV, the socalled Standard Model Extension (SME) [16, 17, 18, 19]. Its effective Lagrangian is given by the usual SM Lagrangian, modified by a combination of SM operators of any dimensionality contracted with Lorentz breaking tensors of suitable rank to get a scalar expression for the Lagrangian.
For the Charge conjugationParityTime reversal (CPT) odd classes the breaking factor is the \(V_{\mu }\) vector associated with the Carroll–Field–Jackiw (CFJ) model [20], while for the CPTeven classes it is the \(k_{F}\) tensor.
In this context, LSV has been thoroughly investigated phenomenologically. Studies include electron, photon, muon, meson, baryon, neutrino and Higgs sectors [21]. Limits on the parameters associated to the breaking of relativistic covariance are set by quite a few experiments [21, 22, 23]. LSV has also been tested in the context of EM cavities and optical systems [24, 25, 26, 27, 28, 29, 30]. Also Fermionic models in presence of LSV have been proposed: spinless and/or neutral particles with a nonminimal coupling to a LSV background, magnetic properties in relation to Fermionic matter or gauge Bosons [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42].
More recently, [43, 44] present interesting results involving the electroweak sector of the SME.
Following [45, 46, 47, 48, 49, 50, 51, 52, 53, 54], LSV is stemmed from a more fundamental physics because it concerns higher energy levels of those obtained in particle accelerators. In Fig. 1, we show the energy scales at which the symmetries are supposed to break, referring to the model described in [55]. At Planck scale, \(10^{19}\) GeV, all symmetries are exact, unless LoSy breaking occurs. This latter may intervene at a lower scale of \(10^{17}\) GeV, but anyway above GUT. Between \(10^{11}\) and \(10^{19}\) GeV, we place the breaking of SuSy. In our analysis, we assume that the four cases of SuSy breaking occur only when LoSy has already being violated. Interestingly, at our energy levels, we can detect the reminiscences of these symmetry breakings.
Since gravitational wave astronomy is at its infancy, EM wave astronomy remains the main detecting tool for unveiling the universe. Thereby, testing the properties of the photons is essential to fundamental physics and astrophysics has just to interpret the universe accordingly.
A legitimate question addresses which mechanism could provide mass to the photon and thereby how the SM should be extended to accommodate such a conjecture. We have set up a possible scenario to reply to these two questions with a single answer.
NonMaxwellian massive photon theories have been proposed over the course of the last century. If the photon is massive, propagation is affected in terms of group velocity and polarisation.
This work is structured as follows. In Sect. 2, we summarise, complement and detail the results obtained in our letter [56], with some reminders to the appendix. Within the unique SME model, we consider four classes of models that exhibit LoSy and SuSy violations, varying in CPT handedness and in incorporating  or not  the effect of photino on the photon propagation. The violation occurs at very high energies, but we search for traces in the DRs visible at our energy scales. In the same Section, we confirm that a massive photon term emerges from the CPTodd Lagrangian. We discover that a massive photon emerges also for the CPTeven sector when the photino is considered. We also point out when i) complex or simply imaginary frequencies and superluminal speeds arise. In Sect. 3, we look for multifringence. In Sect. 4, we wonder if dissipation is conceivable for wave propagation in vacuum and find an affirmative answer. In Sect. 5, we propose our conclusions, discussion and perspectives. The appendix gives some auxiliary technical details.
1.1 Reminders and conventions
We shall encounter real frequencies sub and luminal velocities but also imaginary and complex frequencies, and superluminal velocities.^{2}
In this work, see the title, we intend photon mass as an effective mass. The photon is dressed of an effective mass, that we shall see, depends on the perturbation vector or tensor. Nevertheless, we are cautious in differentiating an effective from a real mass. The Higgs mechanism gives masses to the charged leptons and quarks, the W and Z bosons, while the composite hadrons (baryons and mesons), built up from the massive quarks, have most of their masses from the mechanism of Chiral Symmetry (Dynamical) Breaking (CSB). It would be epistemologically legitimate to consider such mechanisms as producing an effective mass to particles which, without such dressing mechanisms, would be otherwise massless. What is then real or effective? The feature of being frame dependent renders surely the concept of mass unusual, but still acceptable to our eyes, being the dimension indeed that of a mass.
We adopt natural units for which \(c=\hbar =1/4\pi \varepsilon _0 = \mu = 1\), unless otherwise stated. We adopt the metric signature as \((+, , , )\). Although more recent literature adopts \(k_{AF}^\mu \) and \(k_{F}^{\mu \nu \rho \sigma }\) for LSV vector and tensor, respectively, we drop the former in favour of \(V^\mu \) for simplicity of notation especially when addressing time or space components and normalised units.
Finally, we omit to use the adjective angular, when addressing the angular frequency \(\omega \).
1.2 Upper limits on \(V_{\mu }\) vector and photon mass \(m_\gamma \)
Ground based experiments indicate that \({\mathbf {V}}\), the space components, must be smaller than \(10^{10}\) eV \( = 1.6 \times 10^{29}\) J from the bounds given by the energy shifts in the spectrum of the hydrogen atom [67]; else smaller than \(8\times 10^{14}\) eV \( = 1,3 \times 10^{32}\) J from measurements of the rotation in the polarisation of light in resonant cavities [67]. The time component of \(V_\mu \) is smaller than \(10^{16}\) eV \( = 1.6 \times 10^{35}\) J [67] Instead, astrophysical observations lead to \({\mathbf {V}}< 10^{34}\) eV \( = 1.6 \times 10^{53}\) J. We cannot refrain to remark that such estimate is equivalent to the Heisenberg limit (\(\varDelta m\varDelta t >1\)) on the smallest measurable energy or mass for a given time t, set equal to the Universe age. The actual Particle Data Group (PDG) limit on photon mass [68] refers to values obtained in [69, 70] of \(10^{54}\) kg or \(5.6 \times 10^{19}\) eV/c\(^2\), to be taken with some care, as motivated in [71, 72, 73].
2 LSV and two classes of SuSy breaking for each CPT sector
We summarise and complement in this section the results obtained in [56].
2.1 CPTodd sector and the \(V_{\mu }\) vector: classes 1 and 2
The CFJ proposition [20] introduced LSV by means of a ChernSimons (CS) [74] term in the Lagrangian that represents the EM interaction. It was conceived and developed outside any SuSy scenario. The works [75] and later [55] framed the CFJ model in a SuSy scenario. The LSV is obtained through the breaking vector \(V_{\mu }\), the observational limits of which are considered in the CFJ framework. For the origin, the microscopic justification was traced in the fundamental Fermionic condesates present in SuSy [55]. In other words, the Fermionic fields present in the in SuSy background may condensate (that is, take a vacuum expectation value), thereby inducing LSV.
In the following, the implications of the CS term on the propagation and DR of the photon are presented.
2.1.1 Class 1: CFJ model
2.1.2 Class 2: Supersymmetrised CFJ model and SuSy breaking
2.1.3 Group velocities and time delays for Classes 1 and 2
Zero time component of the breaking vector.
For \({ p} = 1\) and \(\cos \theta \ne 0\), we get \(k^\mu k_\mu < 0\), that is \(k_\mu \) spacelike and tachyonic velocities. Still for \({ p}= 1\), but \(\cos \theta = 0\), that is the wave propagating orthogonally to \(\mathbf {V}\), we obtain \(\omega ^2 = {\mathbf {k}}^2\) and thus a Maxwellian propagation, luxonic velocities, in this specific direction.
Instead, \({ p} = 1\) leads to \(k^\mu k_\mu = m_\gamma ^2\), that is \(k_\mu \) timelike and bradyonic velocities associated to a massive photon.
Exploring the general DRs
Nonzero time component of the breaking vector
Here we obtain similar solutions to Eqs. (23, 24), differing by a factor depending on the time component of the CFJ breaking vector. However, this coefficient is not trivial, and it offers some quite interesting features.
Presence of all breaking vector components and \(V^{\mu }\) lightlike. When all parameters differ from zero in Eq. (25), it is obviously the most complex case. Nevertheless, we can comment specific solutions.
We suppose the vector \(V^{\mu }\) being lightlike.

Case 1: \(V^{0}\omega \mathbf {V}\cdot \mathbf {k}\ge 0\Rightarrow \omega ^{2}\mathbf {k}^{2}=V^{0}\omega \mathbf {V}\cdot \mathbf {k}\),

Case 2: \(V^{0}\omega \mathbf {V}\cdot \mathbf {k}\le 0\Rightarrow \omega ^{2}\mathbf {k}^{2}=V^{0}\omega +\mathbf {V}\cdot \mathbf {k}\).
We now discuss the current bounds on the value of the breaking vector in Sect. 1.2 in SI units. In the yet unexplored low radio frequency spectrum [77], a frequency of \(10^5\) Hz and a wavelength \(\lambda \) of \(3\times 10^{3}\) m results in \(\mathbf {k} \hbar c={\displaystyle \frac{2\pi }{\lambda }} \hbar c \sim 6.3 \times 10^{30}\) J, while in the gammaray regime, a wavelength \(\lambda \) of \(3\times 10^{11}\) m results in \(\mathbf {k} \hbar c={\displaystyle \frac{2\pi }{\lambda }} \hbar c \sim 6.3 10^{16}\) J. Spanning the domains of the parameters \(\mathbf {V}\) and \({\mathbf {k}}\), we cannot assure the positiviness of the factor \(\mathbf {k}\mathbf {V}\cos \theta \) in Eq. (48). Moving toward smaller but somewhat less reliable astrophysical upper limits, we insure such positiveness. The nonnegligible price to pay is that the photon effective mass and the perturbation vector decrease and their measurements could be confronted with the Heisenberg limit, see Sect. 1.2. This holds especially for low frequencies around and below \(10^5\) Hz.
The most general case represented by Eq. (25) should be possibly dealt with a numerical treatment.
Time delays.
Finally, given the prominence of the delays of massive photon dispersion, either of dBP or CFJ type, at low frequencies, a swarm of nanosatellites operating in the subMHz region [77] appears a promising avenue for improving upper limits through the analysis of plasma dispersion.
2.1.4 A quaside BroglieProcalike massive term
A quasidBPlike term from the CPTodd Lagrangian has been extracted [56], but without giving details. Indeed, the interaction of the photon with the background gives rise to an effective mass for the photon, depending on the breaking vector \(V^{\mu }\). As we will show, this can be linked to the results we obtained from the DR applied to polarised fields.
2.2 The CPTeven sector and the \(k_{F}\) tensor: classes 3 and 4
For the CPTeven sector, in [55] the authors investigate the \(k_{F}\)term from SME, focusing on how the Fermionic condensates affect the physics of photons and photinos.
2.2.1 Class 3: \(k_F\) model
This shows a nonMaxwellian behaviour, \({\mathbf {v}}_{g} \ne 1 \), whenever the second lefthand side term differs from zero. We observe that there is a frequency dependency, but absence of mass since, Eq. (96), \({\mathbf {k}} = 0\) implies \(\omega = 0\). The frequency never becomes complex, while superluminal velocities may appear if \(C t_i k_i^2\) in Eq. (99) is positive. The parameters \(t_{}\) are suppressed by powers of the Planck energy, so they are very small. This justifies the truncation in Eq. (99). The value depends on the constraints of such parameters.
2.2.2 Class 4: \(k_F\) model and SuSy breaking
3 Birefringence in CPTodd classes
For CPTodd classes, the determination of the DRs in terms of the fields provides a fruitful outcome, since it relates the solutions to the physical polarisations of the fields themselves. This approach must obviously reproduce compatible results with those obtained with the potentials. However, the physical interpretation of said results should be clearer in this new approach.
One might be persuaded, as we initially were, that this result entails the property of trirefringence, because with the same wave vector as in the case of circular polarization, we get a different \(v_g\), namely, \(v_g = 1\). And trirefringence actually means three distinct refraction indices for the same wave vector. However, the linear polarisation and the result \(v_g = 1\) correspond to a lightlike \(V_\mu \), whereas for the circular polarization and birefringence, we have considered \(V_\mu \) spacelike. We then conclude that, since we are dealing with different spacetime classes of \(V_\mu \), triple refraction is not actually taking place.
4 Wave energy loss
4.1 CPTodd classes
Since in Eq. (146) the background vector \(V_\mu \), and the external field, which is treated nondynamically, are both spacetimeindependent, they are not expected to contribute to the nonconservation of the energymomentum tensor, for they do not introduce any explicit \(x_{\mu }\) dependence in the CFJ Lagrangian, Eq. (3). However, there is here a subtlety. The LSV term, which is of the CS type, depends on the fourpotential, \(A_\mu \). By introducing the constant external fields, \(E_B\) and \(B_B\), and performing the splittings of Eqs. (134, 135), an explicit dependence on the background potentials, \(\phi _B\) and \({\mathbf {A}}_B\), appear now in the Lagrangian. But, if the background fields are constant, the background potentials must necessarily display linear dependence on \(x_\mu \) (\(A^\mu _B = {\displaystyle \frac{1}{2}}F_B^{\mu \nu }x_\nu \)); the translation invariance of the Lagrangian is thereby lost. Then the LSV term triggers the appearance of the term \(V_{0}\mathbf {B}_{B}\mathbf {V}\times \mathbf {E}_{B}\) in the righthand side of Eq. (146).
4.2 CPTodd and CPTeven classes
The background time derivative terms \(\left( \partial _{t}F_{B}^{\mu \nu }\right) f_{\mu \nu }\) and \(k_{F}^{\mu \nu \kappa \lambda }\left( \partial _{t}F_{B\kappa \lambda }\right) f_{\mu \nu }\) may account for a deviation from the conservation of the energymomentum tensor of the propagating wave, whenever one of the fields \(\mathbf {E}_{B}\), \(\mathbf {B}_{B}\) is not constant.
4.2.1 Varying breaking vector \(V_{\mu }\) and tensor \(k_F\) without an external EM field
Recalling that \(\theta ^{\mu \nu }\) is no longer symmetric in presence of a LSV background, if we consider the continuity equation for the momentum density of the field, described by \(\theta ^{0i}\), it can be readily checked that the space component of \(V_\mu \), \({\mathbf {V}}\), through its space and time dependencies, and the space dependency of the \(k_F\) components will be also responsible for the nonconservation of the momentum density carried by the electromagnetic signals.
4.2.2 The most general situation: LSV background and external field \(x_{\mu }\)dependent
In Eq. (165), the first two righthand side terms are purely Maxwellian. Further, since \(\theta ^\mu _{~~\nu }\) is not symmetric in presence of LSV terms, when taking its fourdivergence with respect to its second index, namely \(\partial ^\nu \theta ^\mu _{~~\nu }\), contributions of the forms \(\partial ^\nu k_{F\kappa \lambda \nu \rho }F^{\kappa \lambda }f^{\rho \mu }\) and \(\partial ^\nu k_{F}^{\kappa \lambda \mu \rho }F_{\kappa \lambda }f_{\rho \nu }\) appear. Thus, even when \(k_{F}^{\kappa \lambda \mu \rho }\) is only space dependent, though not contributing to \(\partial _\nu \theta ^{\nu 0}\), it does contribute to \(\partial _\nu \theta ^{0\nu }\). We observe that the roles of the perturbation vector and tensor differ, the latter demanding a spacetime dependence of the tensor or of the external field, conversely to the former.
As final remark, the energy losses would presumably translate into frequency damping if the excitation were a photon. Whether such losses could be perceived as ’tired light’ needs an analysis of the waveparticle relation.
5 Conclusions, discussion and perspectives
We have approached the question of nonMaxwellian photons from a more fundamental perspective, linking their appearance to the breaking of the Lorentz symmetry. Despite massive photons have been proposed in several works, few hypothesis on the mass origin have been published, see for instance [82], and surely there is no comprehensive discussion taking form of a review on such origin, see for instance [83]. It is our belief that answering this question is a crucial task in order to truly understand the nature of the electromagnetic interaction carrier and the potential implications in interpreting signals from the Universe. Given the complexity of the subject, we intend to carry on our research in future works.
The chosen approach concerns well established SuSy theories that go beyond the Standard Model. Some models originated from SuSy:^{5} see for instance [55, 75, 86] determined dispersion relations, but the analysis of the latter was unachieved. We also derived the dispersion relations for those cases not present in the literature and also for those we charged ourselves with the task of studying the consequences in some detail. We did not intend to cover all physical cases, and we do not have any pretense of having done so. Nevertheless, we have explored quite a range of both odd and even CPT sectors.
We stand on the conviction that a fundamental theory describing nature should include both CPT sectors. The understanding of the interaction between the two sectors is far from being unfolded and one major question remains open. If we are confronted with a nonMaxwellian behaviour for one sector, or worse for two sectors, how would a twosector theory narrate the propagation? Would the two contributions be simply additive or would there be more interwoven relations? The answers to these questions would prompt other stimulating future avenues of research.
Starting from the actions representing odd and even CPT sector, for both we have analysed whether the photon propagation is impacted by its SuSy partner, the photino. Though the SuSy partners have not been experimentally detected yet, it is possible to assess their impact. Indeed, the actions of Eqs. (10, 100), describe effective photonic models for which the effects of the photino have been summed up at the classical level, that is without loop corrections. Thus, the corresponding DRs include SuSy through the background of the Fermionic sector accompanying the \(V^\mu \) and \(k_F\) breaking vector and tensor, respectively. It would be worth to draw from the constraints on the SME coefficients the estimates of the background SuSy condensates. The latter when related to the SuSy breaking scale and thereby to the masses of the SuSy partners, and specifically the photino. This is a relevant issue for investigating the connection between the SuSy breaking scale, associated to the condensates of the Fermionic partners in the LSV background, and the constraints on the SME.
For the CPTodd case, we study the supersymmetrised [55, 75] Carroll–Field–Jackiw model [20], where the LorentzPoincaré symmetry violation is determined by the \(V_\mu \) fourvector. The resulting dispersion relation is of the fourth order.
For the next conclusions, we do not distinguish between classes with respect to photino integration.

Whenever an explicit solution is determined, at least one solution shows a massive photon behaviour. It is characterised by a frequency dependency of the type \(\omega ^{2}\) like the classic de BroglieProca photon.

The mass is effective and proportional to the absolute value of the Lorentz symmetry breaking vector. The ground based upper limits [67] are compatible with state of the art experimental findings on photon mass [68].

The group velocity is almost always subluminal. Superluminal speeds may appear if the time component of the breaking vector differs from zero. They appear beyond a frequency threshold.

The photon mass is gauge invariant as drawn by the Carroll–Field–Jackiw model, conversely to the de BroglieProca photon.

Birefringence accompanies the CPTodd sector.

When the time component of the LSV breaking vector differs from zero, imaginary and complex frequencies may arise.

We have determined group velocities in the following cases: when the time component or the along the line of sight component of the breaking vector vanishes. The most general case, all components being present, was analysed for \(V^\mu \) lightlike.

The solutions feature anisotropy and lack of Lorentz invariance, due to the dependency on the angle between the breaking vector and the propagation direction, or else on the chosen reference frame.

Since two group velocities for the CPTodd handedness were found except for \(V^\mu \) lightlike, we pursued an analysis of the dispersion relation in terms of the fields, in well defined polarisations. We have determined the existence of birefringence.

Generally, being the propagation of the photon affected by the action of the breaking tensor, we have a tensorial anisotropy and thereby a patent lack of Lorentz invariance. The main consequence is that the speed of light depends on the direction. The correction goes like the breaking components squared. As the components are tiny, since they represent the deviation from the Lorentz invariance, also the correction to c will be limited to small values.

Nevertheless, if the breaking tensor is proportional to the Kroeneker’s delta, the dispersion relation looks as a light ray propagating through a medium. The vacuum assumes an effective refraction index due to the interaction of the photon with the background.

From the Class 3 Lagrangian, it follows that no mass can be generated for the photon. Indeed, the dispersion relation yields \(\omega = 0\) whenever \({\mathbf {k}} = 0\). Instead, for Class 4, there may take place a photon mass generation, due to the bterm which represents higher derivatives in the Lagrangian. Thus, the DR includes the possibility of a nontrivial \(\omega \)solution even if we take a trivial wave vector.

In the odd sector, the coupling of a constant external field, with a constant breaking vector, determines an energy loss even in absence of an external current. This is revealed by the breaking of the continuity equation (or conservation) of the photon energymomentum tensor. If the photon is coupled to the LSV background and/or an EM external field which explicitly depend on the spacetime coordinates, then translational symmetry is broken and the energymomentum tensor is no longer conserved. This means that the system under consideration is exchanging energy (loosing or even receiving) with the environment.

Still in the odd sector, in absence of an external field, but in presence of a space and/or time dependency of the time component of the breaking vector, energy loss occurs.

Finally, we have considered odd and even CPT sectors together. We found if \(V_\mu \) and \(k_F\) are coordinate dependent, there is dissipation in absence of an external EM field.
Dissipation occurs in both odd and even CPT sectors when the associated breaking factors are not constant over spacetime (for the following considerations, we neglect any external field). However, in the odd sector, even if \(V_\mu \) is constant, complex frequencies may arise since the dispersion relation is quartic in frequency. This is due to the Carroll–Field–Jackiw model which does not ensure a positivedefinite energy, and thereby we may have unstable configurations. This leads to complex frequencies. Imaginary frequencies imply damping which is associated to dissipation, and we don’t feel having cleared the issue sufficiently.
The CPTeven sector does not get in trouble with the positiveness of the energy, and thereby complex frequencies associated to unstable excitations are absent. So, the CPTeven sector may yield dissipation, when \(k_F\) is nonconstant, even if it does not exhibit complex frequencies.
In short, future analysis of dissipation will have to tackle and possibly set boundaries towards imaginary frequencies and superluminal velocities, knowing that dissipation might very well occur for subluminal propagation.
We shall be analysing these and related issues, in connection with the conjectures of tired light in forthcoming works, also in the frame of a classic nonlinear formulation of electromagnetism. We take note of different but otherwise possibly converging efforts [87].
Footnotes
 1.
Usually, the Lorentz transformations describe rotations in space (J symmetry) and boosts (K symmetry) connecting uniformly moving bodies. When they are complemented by translations in space and time (symmetry P), the transformations include the name of Poincaré.
 2.A velocity v larger than c is associated to the concept of tachyon [57, 58] and implies an imaginary relativistic factor \(\gamma \). If wishing (relativistic) energy E and (relativistic) mass m to remain real, rest mass \(m_0\) must be imaginary Similarly, wishing measured frequency f to remain real, frequency \(f_0\) must be imaginary in the rest frame
Alternatively, letting rest mass and rest frequency real, mass and energy become imaginary. In the particle view, recalling that \(E= h \nu \), we recover both interpretations. An imaginary frequency implies an evanescent wave amplitude, and thereby tachyonic modes are associated to transitoriness. Complex frequencies present the features above for the imaginary part, and usual properties for the real part. Finally, few scholars consider causality not necessarily incompatible with tachyons [59, 60, 61, 62, 63, 64, 65, 66].
 3.If we take \(V_{0}=0\) in Eq. (27), the solution reads
 4.
Setting \(V_{0}=0\), this result equals that of Eq. (14) for \({ p} = 1\) and \(\theta =\pi /2\) that is propagation along the z axis.
 5.
Notes
Acknowledgements
LB and ADAMS acknowledge CBPF for hospitality, while LRdSF and JAHN are grateful to CNPqBrasil for financial support. All authors thank the referee for the very detailed comments that improved the work.
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