Residual Gauge Invariance in a Massive Lorentz-Violating Extension of QED

Marques, B. Alves; Scarpelli, A. P. Baêta; Felipe, J. C. C.; Brito, L. C. T.

doi:https://doi.org/10.1155/2022/7396078

Advances in High Energy Physics

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Research Article | Open Access

Volume 2022 | Article ID 7396078 | https://doi.org/10.1155/2022/7396078

Residual Gauge Invariance in a Massive Lorentz-Violating Extension of QED

B. Alves Marques,^1,2A. P. Baêta Scarpelli,¹J. C. C. Felipe,³and L. C. T. Brito⁴

Academic Editor: Enrico Lunghi

Received28 Jul 2021

Revised16 Dec 2021

Accepted01 Feb 2022

Published12 Feb 2022

Abstract

We reassess an alternative CPT-odd electrodynamics obtained from a Palatini-like procedure. Starting from a more general situation, we analyze the physical consistency of the model for different values of the parameter introduced in the mass tensor. We show that there is a residual gauge invariance in the model if the local transformation is taken to vary only in the direction of the Lorentz-breaking vector. This residual gauge invariance can be extended to all models whose only source of gauge symmetry breaking is such a mass term.

1. Introduction

It has been more than two decades that the investigation of Lorentz-violating models got the attention of the community of quantum field theory physicists. Indeed, it is believed that the usual symmetries of special relativity will be broken in the limit of very high energy physics as an effect of quantum gravity issues [1–5]. This possibility was first discussed in the papers of Kostelecký and Samuel [1, 2], but it gained great dimension when Carroll, Field, and Jackiw proposed a modification of the classical electrodynamics by the inclusion of a Chern-Simons-like term in the photon sector [6]. This term violates both CPT and Lorentz symmetries and gives rise to a wide range of interesting physical effects, which were intensively studied. Many papers treated features of the model like its physical consistency [7, 8], the possibility of the CFJ term being radiatively generated [9–19], and many other aspects [20–27].

Further, a general description of Lorentz violation in quantum field theories was provided by the Standard Model Extension (SME) [28–31]. The SME, which includes the Carroll-Field-Jackiw (CFJ) term, establishes a set of constant tensors as the parameters of Lorentz violation, whose small magnitudes are strongly constrained by experiments [32]. These constant tensors would arise from spontaneous Lorentz symmetry breaking at very high energies [28]. It is important to note that the SME preserves gauge symmetry and the renormalizability.

One aspect that has been gaining increasing attention is the inclusion of massive photons in models that violate Lorentz symmetry. In general, it is believed that the fact that photons are massless is a direct consequence of gauge invariance of quantum electrodynamics (QED). Nevertheless, many studies of the physics beyond the Maxwell electromagnetism have been performed, mainly in the context of Proca model [33–37]. Although studies using astrophysical sources and laboratory experiments put very strong constraints in the photon mass [38–51], it may be possible that it has a very tiny but nonvanishing rest mass . It is interesting to note that the usual CFJ model already accommodates an effective mass for the photon which is proportional to the absolute value of the Lorentz-breaking vector [52, 53]. Still, in the context of electrodynamics with the Carroll-Field-Jackiw term, an explicit mass term for the photon was used to repair unitarity problems for a time-like background vector [54].

On the other hand, models with Lorentz-violating (LVI) mass terms present remarkable peculiarities, like the ones pointed out in [55, 56], in which the gauge field has two massive degrees of freedom but the static force between charged particles is Coulomb-like. Many aspects like the quantum induction of Lorentz-breaking mass terms for the photon [57], the construction of a Stueckelberg Lagrangian in a generalized gauge [58], and spontaneous gauge symmetry breaking in a Lorentz-violating gauge-Higgs model [59] were investigated. In [60], some aspects of this kind of gauge symmetry breaking were focused in a study of dual models.

An alternative way of generating Lorentz-violating mass terms was investigated in [61], in which a Palatini-like formulation of the CFJ model was performed. Interestingly, this formulation, with the participation of the CFJ term, caused the emergence of an unusual mass contribution in the photon Lagrangian density. While the usual Proca mass term breaks only gauge invariance, the present mass contribution breaks Lorentz symmetry and, supposedly, gauge invariance. These results looked intriguing, since the model from which we begin, in which is treated as an independent field, has no any evidence of violation of gauge symmetry. In this paper, we study a more general version of the massive model obtained in [61] and show that there is a residual gauge invariance in the model obtained from the Palatini-like formulation.

The paper is organized as follows: in Section 2, general discussions on the model are presented and the gauge field propagator is obtained; in Section 3, we carry out an analysis of the field equations; we study tree-level unitarity of the model in Section 4; in Section 5, we investigate the presence of a residual gauge invariance in the massive model; conclusions and final remarks are presented in Section 6.

2. General Discussions on the Model

Let us consider the following Lagrangian density of a Lorentz-breaking extension of spinorial quantum electrodynamics (QED), with

in which and are dimensionless constants, is a mass parameter and is a background constant pseudovector that causes the violation of Lorentz symmetry. Besides, , is the conventional matter current and is the pure fermionic Lagrangian density. The gauge invariance of the model is apparently explicitly broken due to the presence of the mass term, , which also includes a Lorentz-breaking part. It is worth to note that such a term is usually called nonminimal, since it accommodates nonminimal couplings to the curvature in field theories in curved space-time [62–65].

This particular LVI extension of QED, in which the background vector of the Carroll-Field-Jackiw (CFJ) term is responsible for the mass tensor, was obtained in [61], with , , and . This was accomplished through a Palatini-type procedure, in which the tensor and the gauge field were initially considered to be independent. The result was an unusual situation, in which the breaking of Lorentz’s invariance induces a violation, at least apparent, of the gauge symmetry.

The breaking of gauge symmetry may cause problems to the consistency of the theory, as the violation of unitarity, with deleterious effects in the renormalizability of the model. A common procedure in cases like that is the searching for a hidden gauge symmetry, like the solution of Stueckelberg [66, 67], in which a mixed term that includes a scalar field is added to the Lagrangian density, such that gauge symmetry is restored. It is noteworthy that the idea of the Stueckelberg mechanism was extended to the Standard Model to give mass to the physical photon through the hypercharge [68, 69]. Another approach is the application of a dualization method in order to obtain a gauge theory which describes the same physics as the original model. In both cases, it is assumed that the model of interest is a gauge-fixed version of a gauge theory. We will perform further investigations on the possibility of finding a hidden gauge symmetry in the present model. Let us first analyze general features of the model (1).

The quadratic part of the Lagrangian density in the vectorial field can be written as with where and are the transversal and longitudinal spin projectors, respectively, and the operators

are dependent on the background vector . The wave operator is invertible. However, as we intend to define a closed algebra which includes the operators , , and , it is necessary to include new operators, as Equation (6) indicates. It should be noted that no new term, such as a gauge-fixing, is being added to the wave operator. Since we have with

we include the operators and in the algebra, which is displayed in Table 1.

We thus obtain, after a straightforward but lengthy calculation, the vector field propagator, , in momentum-space: with

The operators, in momentum-space, are given by

We recognize above the propagator for the model of [61], if we fix , and . Interesting that the dispersion relations extracted from the denominator , responsible for the massive modes, are not affected by the constant . This parameter only influences the modes coming from . In [61], the physical consistency of the modes of propagation of the vector field was studied in detail. For now, let us study the effects of the constant on the model and, for comparison, let us fix and .

3. Field Equations

Since in [61], it was shown that only a spacelike produces meaningful modes of propagation of the vector field, we restrict ourselves here to the analysis of this case. Let us consider a referential frame in which we have a purely spacelike background vector given by , with , and choose the direction of the -axis as the one towards which the vector is pointed. So, we have the poles

due to the denominator , and

due to the denominator , where is the component of orthogonal to the -axis.

We first examine the solutions of the field equation in momentum-space,

with . The contraction of the above equation with gives us the gauge condition , which, after the substitution in (14), furnishes us

in which , according to the definitions of (10) for the operators in momentum-space. For our spacelike , we have

In addition, let us define the -axis as the direction aligned with the component of which is orthogonal to , so that . For the -component of the field equation, we have

We see that the pole automatically satisfies the equation above. For , the poles are constrained to equal , which means, for a nonnull ,

It is not possible that these two poles satisfy these conditions at the same time for an arbitrary , if . The other equations of motion for , with and, consequently, , are written as

We try a solution with , for which, after some manipulation of equations (19), we obtain

If , we are left with the constraints and , which correspond to the two modes () and (). For the first mode, the electromagnetic field is polarized as and propagates in the direction , such that and . The gauge condition, however, imposes one more restriction:

So, this solution fixes . For the mode with , the conclusions are the same, only with the change in the sign of the component.

If , taking into account the other equations, we necessarily have , such that . Since, the poles satisfy the -component of field equations for , we have that, if , the three poles are reduced to one , with the possible nonphysical consequences of a multiple pole.

In the case , the mode is not excluded. It is not possible, however, to fix . In order to obtain a solution which is valid for all values of , we obtain the constrainings and , which satisfy one of the two conditions: or .

In summary, in this section, we investigated how the solutions of the field equations are affected by the poles of the model. In the next section, we will look at the physical nature of these poles subject to specific conditions imposed by the field equations.

4. Tree-Level Unitarity Analysis

It is enlightening that we make an analysis of the physical nature of the poles. With this purpose, we investigate the tree-level unitarity of the model. It can be investigated through the propagator, when saturated by conserved currents,

The current conservation in momentum-space is written as , such that

Unitarity requires that the imaginary part of the residue of the saturated propagator in a physical pole is nonnegative (see [70, 71]). This requirement can be checked by calculating the residue matrix in the pole for the complete propagator and, then, verifying if its eigenvalues are nonnegative. Here, we opt to analyze directly the saturated propagator. Note that in [70], this technique of analysis was used in a model obtained from the dimensional reduction of the CPT-even sector of the Standard Model Extension.

Turning our attention to the saturated propagator, in consequence of current conservation, only terms on and remain, such that, for our spacelike , we stay with with

We have some potential problematic situations to analyze here: when , we have, in principle, double poles, with and , respectively; this situation repeats when , with an additional complication if , with , which, at first sight, brings us a triple pole (). We study below these cases.

4.1. The Cases

Since these two cases are similar, we analyze here the referential frame in which , such that we stay with the poles and . This situation is possible both for and , as we verified in the field equations. The key point in order to analyze the possibility of a double pole is the fact that in this referential frame. We stay with

and we see that, in order to eliminate the double pole, we need to have , in accordance with the result we obtained in the study of the field equations in conjunction with the gauge condition. If we fix , the saturated propagator reads

We are now in position to calculate the imaginary part of the residue in the two remaining poles. For the pole , the calculation is straightforward and gives

For the residue in the pole , the term in contributes. The residue gives us

We then remember that, for , we have and that, in this case, to obtain, after some calculation,

The result above, which can be diagonalized, indicates a massive mode with two degrees of freedom.

4.2. The Cases

We here analyze the case , since the two situations are similar. In this case, we have a possible double pole at

and, if , a triple pole. The saturated propagator, after algebraic manipulations, is given by

The residue in the pole gives

It is clear that a nonnegative imaginary part of the residue in the pole depends on the value of . In the case of , we are back to the case . The possible triple pole for and is not consistent, since the residue diverges. This physical inconsistency of such a case is also related to the stability of the model and will be made clear in the next section.

5. The Residual Gauge Invariance

All the above results present us clues of some special feature in the model when . First, let us rewrite the dispersion relations for the model in a more general way. From the factor in the denominator, taking and , we have which can be factorized as so that

For the purely spacelike background vector, , we stay with

Since the model is massive, we can go to the particle rest frame and get, for these two dispersion relations,

which represents a mode with a positive definite mass which propagates with two degrees of freedom. It is important to emphasize that these results do not depend on the value of the parameter. On the other hand, the factor is written as which furnishes us the dispersion relation

For the purely spacelike Lorentz-breaking vector, the result is written as

where is the angle between the vectors and . Let us consider the field propagates in the direction of the Lorentz-breaking vector, such that . In this situation, we stay with

which imposes some restrictions. First, if , we have a massless mode. For , the dispersion relation predicts a massive mode, but with the restriction , otherwise we have an unstable model, since in the particle rest frame we have . However, even with , the stability of the model imposes severe restrictions, as the magnitude of the field momentum would be limited by that of the background vector, which is very small. The propagation of the field would be practically restricted to the plane orthogonal to , although the component of the momentum in the direction of cannot be null (we must have ). Thus, in the special case in which , the model presents a massive mode, (coming from the factor in the denominator), with two degrees of freedom, and a massless mode, which corresponds to a propagation in the direction of the background vector (from ).

In massive models with violation of gauge symmetry, it is usual to look for hidden symmetries through dualization processes, which could provide an equivalent gauge-invariant model. The gauge embedding method [72, 73] proved to be effective in this type of procedure in a wide range of models. The approach is based on the transformation of the model in a gauge theory by adding on mass-shell vanishing terms. This iterative embedding of Noether counterterms is based on the idea of local lifting a global symmetry and is reminiscent to procedures which were important in the construction of component-field supergravity actions [74–76]. It would be interesting to carry out the dualization of the present theory. However, the procedure needs the use of the inverse operator of the mass tensor, (defined in equation (2)), which is given by

Interestingly, there is no inverse for when . This is another compelling clue for the investigation of this particular case.

Last but not least, calculating the one-loop vacuum polarization tensor for the model up to second-order in gives us a transverse result [61], with

in which is the fermion mass and

All these results bring us the possibility of a residual gauge invariance. The gauge-violating term is given by

By considering a gauge transformation on the vectorial field, , we have

It is easy to see that if is proportional to , such that , with a dimensionless function of (assuming the -axis is parallel to the background vector), we get, explicitly using (2),

which is null for . In this situation, the variation of the Lagrangian is null. Just in order to illustrate, for our purely spacelike vector , this is accomplished if the gradient of is parallel to in the case . Thus, there is a class of gauge transformations which leaves the Lagrangian density invariant and, in this sense, we can affirm that the model accommodates a residual gauge invariance when .

Although we use a specific model in the discussion of the residual gauge invariance, it is easy to see that this is a general result for all the models whose only source of gauge symmetry violation is the above cited .

6. Conclusion

We studied an alternative CPT-odd electrodynamics which incorporates a general Lorentz-breaking mass term. Actually, it is an extended version of the one obtained from a Palatini-like procedure in [61]. We have showed that the Lorentz-breaking part of the mass tensor affects only one of the three modes of propagation of the gauge field. The parameter that controls this part of the mass tensor plays a fundamental role in preserving the essential physical properties of the model. It was shown, for a spacelike Lorentz-breaking vector , that if , the model presents serious problems with stability. Moreover, if this parameter is greater than one, a positive definite energy is required, and the field is restricted to propagate in a direction external to a cone defined by the rotation, around the -axis (defined as the direction of the background vector), of the lines , being the axis parallel to the component of the field momentum and the magnitude of the Lorentz-breaking vector . Moreover, the particular case in which accommodates a nonphysical triple pole when the component of the field momentum along the -axis is null.

The most interesting case, however, is the one with , which presents a massless mode of propagation along the -axis. This fact, together with the lack of an inverse for the mass tensor, led us to investigate the presence of a residual gauge invariance in the model. We then showed that a gauge transformation, , such that , leaves the action unchanged. For a purely spacelike background vector, this corresponds to have the gradiant of parallel to . Moreover, this is a general result for models whose only source of gauge symmetry violation is the mass term studied in the present paper.

For future work, it would be interesting to investigate how a dualization process would work in the limit . In other words, it would be worth to understand what kind of model comes up in this limit in a more general gauge-invariant model with undetermined, obtained from a gauge embedding procedure.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by Conselho Nacional de Desenvolvimento Cientfico e Tecnológico (CNPq). The authors acknowledge Prof. J. A. Helayël-Neto for elucidating discussions.

References

V. A. Kostelecký and S. Samuel, “Spontaneous breaking of Lorentz symmetry in string theory,” Physical Review D, vol. 39, no. 2, pp. 683–685, 1989.
View at: Publisher Site | Google Scholar
V. A. Kostelecký and S. Samuel, “Gravitational phenomenology in higher-dimensional theories and strings,” Physical Review D, vol. 40, no. 6, pp. 1886–1903, 1989.
View at: Publisher Site | Google Scholar
S. Doplicher, K. Fredenhagen, and J. E. Roberts, “The quantum structure of spacetime at the Planck scale and quantum fields,” Communications in Mathematical Physics, vol. 172, no. 1, pp. 187–220, 1995.
View at: Publisher Site | Google Scholar
J. Collins, A. Perez, D. Sudarsky, L. Urrutia, and H. Vucetich, “Lorentz invariance and quantum gravity: an additional fine-tuning problem?” Physical Review Letters, vol. 93, no. 19, article 191301, 2004.
View at: Publisher Site | Google Scholar
P. Horava, “Quantum gravity at a Lifshitz point,” Physical Review D, vol. 79, no. 8, article 084008, 2009.
View at: Publisher Site | Google Scholar
S. M. Carroll, G. B. Field, and R. Jackiw, “Limits on a Lorentz- and parity-violating modification of electrodynamics,” Physical Review D, vol. 41, p. 1231, 1990.
View at: Publisher Site | Google Scholar
A. P. Baêta Scarpelli, H. Belich, J. L. Boldo, and J. A. Helayël-Neto, “Aspects of causality and unitarity and comments on vortexlike configurations in an Abelian model with a Lorentz-breaking term,” Physical Review D, vol. 67, no. 8, article 085021, 2003.
View at: Publisher Site | Google Scholar
A. P. Baêta Scarpelli and J. A. Helayël-Neto, “Lorentz-violating SO(3) model: discussing unitarity, causality, and ’t Hooft-Polyakov monopoles,” Physical Review D, vol. 73, no. 10, article 105020, 2006.
View at: Publisher Site | Google Scholar
R. Jackiw and V. A. Kosteleckỳ, “Radiatively induced Lorentz and CPT violation in electrodynamics,” Physical Review Letters, vol. 82, no. 18, pp. 3572–3575, 1999.
View at: Publisher Site | Google Scholar
J.-M. Chung and B. K. Chung, “Induced Lorentz- and CPT-violating Chern-Simons term in QED: Fock-Schwinger proper time method,” Physical Review D, vol. 63, no. 10, article 105015, 2001.
View at: Publisher Site | Google Scholar
J.-M. Chung, “Lorentz- and CPT-violating Chern-Simons term in the functional integral formalism,” Physical Review D, vol. 60, no. 12, article 127901, 1999.
View at: Publisher Site | Google Scholar
G. Bonneau, “Regularisation: many recipes, but a unique principle: ward identities and normalisation conditions. The case of CPT violation in QED,” Nuclear Physics B, vol. 593, no. 1-2, pp. 398–412, 2001.
View at: Publisher Site | Google Scholar
M. Perez-Victoria, “Exact calculation of the radiatively induced Lorentz and CPT Violation in QED,” Physical Review Letters, vol. 83, no. 13, pp. 2518–2521, 1999.
View at: Publisher Site | Google Scholar
J. High, “Motion on moduli spaces with potentials,” Journal of High Energy Physics, vol. 2001, no. 12, p. 032, 2001.
View at: Publisher Site | Google Scholar
O. A. Battistel and G. Dallabona, “Role of ambiguities and gauge invariance in the calculation of the radiatively induced Chern–Simons shift in extended QED,” Nuclear Physics B, vol. 610, no. 1-2, pp. 316–330, 2001.
View at: Publisher Site | Google Scholar
O. A. Battistel and G. Dallabona, “Ambiguities versus gauge invariance in the calculation of the radiatively induced Chern-Simons shift in extended QED,” Journal of Physics G, vol. 27, no. 7, pp. L53–L59, 2001.
View at: Publisher Site | Google Scholar
O. A. Battistel and G. Dallabona, “Consistency in perturbative calculations and radiatively induced Lorentz and CPT violations,” Journal of Physics G, vol. 28, no. 8, pp. L23–L32, 2002.
View at: Publisher Site | Google Scholar
A. P. Baêta Scarpelli, M. Sampaio, M. C. Nemes, and B. Hiller, “Chiral anomaly andCPTinvariance in an implicit momentum space regularization framework,” Physical Review D, vol. 64, no. 4, article 046013, 2001.
View at: Publisher Site | Google Scholar
A. P. Baêta Scarpelli, M. Sampaio, M. C. Nemes, and B. Hiller, “Gauge invariance and the CPT and Lorentz violating induced Chern–Simons-like term in extended QED,” The European Physical Journal C, vol. 56, no. 4, pp. 571–578, 2008.
View at: Publisher Site | Google Scholar
C. Adam and F. R. Klinkhamer, “Causality and CPT violation from an Abelian Chern–Simons-like term,” Nuclear Physics B, vol. 607, no. 1-2, pp. 247–267, 2001.
View at: Publisher Site | Google Scholar
C. Adam and F. R. Klinkhamer, “Photon decay in a CPT-violating extension of quantum electrodynamics,” Nuclear Physics B, vol. 657, pp. 214–228, 2003.
View at: Publisher Site | Google Scholar
A. A. Andrianov and R. Soldati, “Lorentz symmetry breaking in Abelian vector-field models with Wess-Zumino interaction,” Physical Review D, vol. 51, no. 10, pp. 5961–5964, 1995.
View at: Publisher Site | Google Scholar
A. A. Andrianov and R. Soldati, “Patterns of Lorentz symmetry breaking in QED by CPT-odd interaction,” Physics Letters B, vol. 435, no. 3-4, pp. 449–452, 1998.
View at: Publisher Site | Google Scholar
A. A. Andrianov, R. Soldati, and L. Sorbo, “Dynamical Lorentz symmetry breaking from a(3+1)-dimensional axion-Wess-Zumino model,” Physical Review D, vol. 59, no. 2, article 025002, 1998.
View at: Publisher Site | Google Scholar
A. A. Andrianov, D. Espriu, P. Giacconi, and R. Soldati, “Anomalous positron excess from Lorentz-violating QED,” Journal of High Energy Physics, vol. 909, p. 057, 2009.
View at: Google Scholar
J. Alfaro, A. A. Andrianov, M. Cambiaso, P. Giacconi, and R. Soldati, “Bare and induced lorentz and cpt invariance violations in qed,” International Journal of Modern Physics A, vol. 25, no. 16, pp. 3271–3306, 2010.
View at: Publisher Site | Google Scholar
V. C. Zhukovsky, A. E. Lobanov, and E. M. Murchikova, “Radiative effects in the standard model extension,” Physical Review D, vol. 73, no. 6, article 065016, 2006.
View at: Publisher Site | Google Scholar
D. Colladay and V. A. Kostelecky, “CPT violation and the standard model,” Physical Review D, vol. 55, p. 6760, 1997.
View at: Publisher Site | Google Scholar
D. Colladay and V. A. Kostelecky, “Lorentz-violating extension of the standard model,” Physical Review D, vol. 58, article 116002, 1998.
View at: Publisher Site | Google Scholar
S. Coleman and S. L. Glashow, “Cosmic ray and neutrino tests of special relativity,” Physical Review D, vol. 405, no. 3-4, pp. 249–252, 1997.
View at: Publisher Site | Google Scholar
S. Coleman and S. L. Glashow, “High-energy tests of Lorentz invariance,” Physical Review D, vol. 59, no. 11, article 116008, 1999.
View at: Publisher Site | Google Scholar
V. A. Kosteleckỳ and N. Russell, “Data tables for Lorentz and CPT violation,” Reviews of Modern Physics, vol. 83, no. 1, pp. 11–31, 2011.
View at: Publisher Site | Google Scholar
P. Robles and F. Claro, “Can there be massive photons? A pedagogical glance at the origin of mass,” European Journal of Physics, vol. 33, p. 1217, 2012.
View at: Google Scholar
L. C. Tu, J. Luo, and G. T. Gillies, “The mass of the photon,” Reports on Progress in Physics, vol. 68, no. 1, pp. 77–130, 2005.
View at: Publisher Site | Google Scholar
A. S. Goldhaber and M. M. Nieto, “Photon and graviton mass limits,” Reviews of Modern Physics, vol. 82, p. 939, 2010.
View at: Publisher Site | Google Scholar
G. Spavieri, J. Quintero, G. T. Gillies, and M. Rodriguez, “Atomic, molecular, optical and plasma physics,” European Physical Journal D, vol. 61, p. 531, 2011.
View at: Google Scholar
M. V. S. Fonseca and A. V. Paredes, “Is it possible to accommodate massive photons in the framework of a gauge-invariant electrodynamics?” Brazilian Journal of Physics, vol. 40, p. 319, 2010.
View at: Publisher Site | Google Scholar
J.-J. Wei and X.-F. Wu, “Robust limits on photon mass from statistical samples of extragalactic radio pulsars,” Journal of Cosmology and Astroparticle Physics, vol. 2018, no. 7, p. 045, 2018.
View at: Publisher Site | Google Scholar
L. Shao and B. Zhang, “Bayesian framework to constrain the photon mass with a catalog of fast radio bursts,” Physical Review D, vol. 95, no. 12, article 123010, 2017.
View at: Publisher Site | Google Scholar
L. Bonetti, J. Ellis, N. E. Mavromatos, A. S. Sakharov, E. K. Sarkisyan-Grinbaum, and A. D. A. M. Spallicci, “FRB 121102 casts new light on the photon mass,” Physics Letters B, vol. 768, pp. 326–329, 2017.
View at: Publisher Site | Google Scholar
Y.-P. Yang and B. Zhang, “Tight constraint on photon mass from pulsar spindown,” The Astrophysical Journal, vol. 842, no. 1, p. 23, 2017.
View at: Publisher Site | Google Scholar
J.-J. Wei, E.-K. Zhang, S.-B. Zhang, and X.-F. Wu, “New limits on the photon mass with radio pulsars in the Magellanic Clouds,” Research in Astronomy and Astrophysics, vol. 17, no. 2, p. 13, 2017.
View at: Publisher Site | Google Scholar
B. Zhang, Y.-T. Chai, Y.-C. Zou, and X.-F. Wu, “Constraining the mass of the photon with gamma-ray bursts,” Journal of High Energy Astrophysics, vol. 11-12, pp. 20–28, 2016.
View at: Publisher Site | Google Scholar
L. Bonetti, J. Ellis, N. E. Mavromatos, A. S. Sakharov, E. K. G. Sarkisyan-Grinbaum, and A. D. A. M. Spallicci, “Photon mass limits from fast radio bursts,” Physics Letters B, vol. 757, pp. 548–552, 2016.
View at: Publisher Site | Google Scholar
X. F. Wu, S. B. Zhang, H. Gao et al., “Constraints on the photon mass with fast radio bursts,” The Astrophysical Journal Letters, vol. 822, p. L15, 2016.
View at: Google Scholar
A. Retinò, A. D. A. M. Spallicci, and A. Vaivads, “Solar wind test of the de Broglie-Proca massive photon with Cluster multi-spacecraft data,” Astroparticle Physics, vol. 82, p. 49, 2016.
View at: Publisher Site | Google Scholar
Particle Data Group, “Review of particle physics,” Physical Review D, vol. 98, no. 3, article 030001, 2018.
View at: Publisher Site | Google Scholar
D. D. Ryutov, “The role of finite photon mass in magnetohydrodynamics of space plasmas,” Plasma Physics and Controlled Fusion, vol. 39, no. 5A, pp. A73–A82, 1997.
View at: Publisher Site | Google Scholar
D. D. Ryutov, “Using plasma physics to weigh the photon,” Plasma Physics and Controlled Fusion, vol. 49, no. 12B, pp. B429–B438, 2007.
View at: Publisher Site | Google Scholar
L. Davis, A. S. Goldhaber, and M. M. Nieto, “Limit on the photon mass deduced from Pioneer-10 observations of Jupiter's magnetic field,” Physical Review Letters, vol. 35, p. 1402, 1975.
View at: Publisher Site | Google Scholar
G. Chibisov, “Astrophysical upper limits on the photon rest mass,” Uspekhi Fizicheskikh Nauk, vol. 119, no. 7, p. 551, 1976.
View at: Publisher Site | Google Scholar
L. Bonetti, L. R. dos Santos Filho, J. A. Helayël-Neto, and A. D. A. M. Spallicci, “Effective photon mass by Super and Lorentz symmetry breaking,” Physics Letters B, vol. 764, pp. 203–206, 2017.
View at: Publisher Site | Google Scholar
L. Bonetti, L. R. dos Santos, J. A. Helayël-Neto, and A. D. A. M. Spallicci, “Photon sector analysis of Super and Lorentz symmetry breaking: effective photon mass, bi-refringence and dissipation,” European Physical Journal C: Particles and Fields, vol. 78, no. 10, p. 811, 2018.
View at: Publisher Site | Google Scholar
M. M. Ferreira, J. A. Helayël-Neto, C. M. Reyes, M. Schreck, and P. D. S. Silva, “Unitarity in Stückelberg electrodynamics modified by a Carroll-Field-Jackiw term,” Physics Letters B, vol. 804, article 135379, 2020.
View at: Publisher Site | Google Scholar
G. Gabadadze and L. Grisa, “Lorentz-violating massive gauge and gravitational fields,” Physics Letters B, vol. 617, p. 124, 2005.
View at: Publisher Site | Google Scholar
G. Dvali, M. Papucci, and M. D. Schwartz, “Infrared Lorentz violation and slowly instantaneous electricity,” Physical Review Letters, vol. 94, article 191602, 2005.
View at: Publisher Site | Google Scholar
B. Altschul, “Radiatively induced Lorentz-violating photon masses,” Physical Review D, vol. 73, article 036005, 2006.
View at: Publisher Site | Google Scholar
M. Cambiaso, R. Lehnert, and R. Potting, “Massive photons and Lorentz violation,” Physical Review D, vol. 85, article 085023, 2012.
View at: Publisher Site | Google Scholar
B. Altschul, “Lorentz violation and the Higgs mechanism,” Physical Review D, vol. 86, article 045008, 2012.
View at: Publisher Site | Google Scholar
H. G. Fargnoli, L. C. T. Brito, A. P. Baêta Scarpelli, and M. Sampaio, “Dual embedding of extended models with a Lorentz-breaking mass term,” Physical Review D, vol. 90, no. 8, article 085016, 2014.
View at: Publisher Site | Google Scholar
J. C. C. Felipe, H. G. Fargnoli, A. P. Baêta Scarpelli, and L. C. T. Brito, “Consistency of an alternative CPT-odd and Lorentz-violating extension of QED,” International Journal of Modern Physics A, vol. 34, no. 25, article 1950139, 2019.
View at: Publisher Site | Google Scholar
D. J. Toms.“Quantization of the minimal and non-minimal vector field in curved space,” http://arxiv.org/abs/:1509.05989.
I. L. Buchbinder, T. P. Netto, and I. L. Shapiro, “Massive vector field on curved background: nonminimal coupling, quantization, and divergences,” Physical Review D, vol. 95, no. 8, article 085009, 2017.
View at: Publisher Site | Google Scholar
M. S. Ruf and C. F. Steinwachs, “Renormalization of generalized vector field models in curved spacetime,” Physical Review D, vol. 98, article 025009, 2018.
View at: Publisher Site | Google Scholar
C. Garcia-Recio and L. L. Salcedo, “Renormalization of vector fields with mass-like coupling in curved spacetime,” European Physical Journal C: Particles and Fields, vol. 79, no. 5, p. 438, 2019.
View at: Publisher Site | Google Scholar
E. C. G. Stueckelberg, “Interaction energy in electrodynamics and in the field theory of nuclear forces,” Helvetica Physica Acta, vol. 11, pp. 225–244, 1938.
View at: Publisher Site | Google Scholar
E. C. G. Stueckelberg, “Forces of interaction in electrodynamics and in the field theory of nuclear forces.(parts ii and iii),” Helvetica Physica Acta, vol. 11, p. 299, 1938.
View at: Google Scholar
B. Körs and P. Nath, “A Stueckelberg extension of the standard model,” Physics Letters B, vol. 586, p. 366, 2004.
View at: Publisher Site | Google Scholar
H. Ruegg and M. Ruiz-Altaba, “The stueckelberg field,” International Journal of Modern Physics A, vol. 19, pp. 3265–3347, 2004.
View at: Publisher Site | Google Scholar
R. Casana, M. M. Ferreira Jr., and R. P. M. Moreira, “Consistency analysis of a nonbirefringent Lorentz-violating planar model,” The European Physical Journal C, vol. 72, article 2070, 2012.
View at: Publisher Site | Google Scholar
M. Veltman, “Quantum theory of gravitation,” in Methods in Field Theory, R. Bailian and J. Zinn-Justin, Eds., North-Holland Publising Company and World Scientific Publising Co Ltd, Singapore, 1981.
View at: Google Scholar
A. Ilha and C. Wotzasek, “Duality equivalence between self-dual and topologically massive non-abelian models,” Nuclear Physics B, vol. 604, p. 426, 2001.
View at: Publisher Site | Google Scholar
M. A. Anacleto, A. Ilha, J. R. S. Nascimento, R. F. Ribeiro, and C. Wotzasek, “Dual equivalence between self-dual and Maxwell–Chern–Simons models coupled to dynamical U(1) charged matter,” Physics Letters B, vol. 504, p. 268, 2001.
View at: Publisher Site | Google Scholar
D. Z. Freedman, P. van Nieuwenhuizen, and S. Ferrara, “Progress toward a theory of supergravity,” Physical Review D, vol. 13, no. 12, pp. 3214–3218, 1976.
View at: Publisher Site | Google Scholar
S. Ferrara, D. Z. Freedman, P. van Nieuwenhuizen, P. Breitenlohner, F. Gliozzi, and J. Scherk, “Scalar multiplet coupled to supergravity,” Physical Review D, vol. 15, no. 4, pp. 1013–1018, 1977.
View at: Publisher Site | Google Scholar
S. Ferrara, J. Scherk, and P. van Nieuwenhuizen, “Locally Supersymmetric Maxwell-Einstein Theory,” Physical Review Letters, vol. 37, no. 16, pp. 1035–1037, 1976.
View at: Publisher Site | Google Scholar

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Copyright © 2022 B. Alves Marques et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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