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

Authors: B. Alves Marques; A. P. Baêta Scarpelli; J. C. C. Felipe; L. C. T. Brito
Journal: Advances in High Energy Physics
 (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.


---

## Body

## 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 SU3×SU2×U1 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 mγ. 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 Rξ 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 Fμν 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 Section2, 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),(1)L=−14FμνFμν+αμκμAν∂βAσεμνβσ+12μ2AμMμνAν−JμAμ+LF,with
(2)Mμν=κ2ημν−ρκμκν,in whichα and ρ are dimensionless constants, μ is a mass parameter and κμ is a background constant pseudovector that causes the violation of Lorentz symmetry. Besides, Fμν=∂μAν−∂νAμ, Jμ=eψ¯γμψ is the conventional matter current and LF is the pure fermionic Lagrangian density. The gauge invariance of the model is apparently explicitly broken due to the presence of the mass term, 1/2μ2AμMμνAν, 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 α=−1, ρ=1, and μ2=1. This was accomplished through a Palatini-type procedure, in which the Fμν 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 U1Y [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 fieldAμ can be written as
(3)LG=12AμOμνAν,with
(4)Oμν=□+μ2κ2θμν+μ2κ2ωμν−2αμSμν−ρμ2Λμν,where θμν=ημν−∂μ∂ν/□ and ωμν=∂μ∂ν/□ are the transversal and longitudinal spin projectors, respectively, and the operators
(5)Sμν=εμναβκα∂βandΛμν=κμκνare dependent on the background vectorκμ. The wave operator Oμν is invertible. However, as we intend to define a closed algebra which includes the operators θμν, ωμν, Λμν, and Sμν, 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
(6)SμαSνα=κ2□−λ2θμν−λ2ωμν+λΣμν+Σνμ−□Λμν≡fμν,with
(7)Σμν=κμ∂ν,λ≡Σμμ=κμ∂μ,we include the operatorsΣμν and Σνμ in the algebra, which is displayed in Table 1.Table 1
Multiplicative table. The products are supposed to obey the order “row times column.”
θναωναSναΛναΣναΣναθμαθμν0SμνΛμν−λ/□ΣνμΣμν−λωμν0ωμα0ωμν0λ/□ΣνμλωμνΣνμSμαSμν0fμν000ΛμαΛμν−λ/□Σμνλ/□Σμν0κ2Λμνκ2ΣμνλΛμνΣμα0Σμν0λΛμνλΣμν□ΛμνΣαμΣνμ−λωμνλωμν0κ2Σνμκ2□ωμνλΣνμWe thus obtain, after a straightforward but lengthy calculation, the vector field propagator,<AμAν>=iO−1μν, in momentum-space:
(8)AμAν=iD−p2−μ2κ2θμν+1μ2κ2D+λ2D~p2H+4α2μ2D~ωμν+2iαμSμν−λHD~Σμν+Σνμ−μ2D~−κ2H+4α2D~Λμν,with
(9)D=p2−μ2κ22+4α2μ2κ2p2−λ2,D~=−κ2p2−ρλ2+μ2κ41−ρ,H=−ρp2−μ2κ2+4α2μ2κ21−ρ.The operators, in momentum-space, are given by(10)θμν=ημν−ωμν,ωμν=pμpνp2,Sμν=εμναβκαpβ,(11)Σμν=κμpνandΛμν=κμκν.We recognize above the propagator for the model of [61], if we fix α=−1, ρ=1, and μ2=1. Interesting that the dispersion relations extracted from the denominator D, responsible for the massive modes, are not affected by the constant ρ. This parameter only influences the modes coming from D~. 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 ρ on the model and, for comparison, let us fix α=−1 and μ2=1.


## 3. Field Equations

Since in [61], it was shown that only a spacelike κμ 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 κμ=0,κ⟶, with κ⟶2=t2, and choose the direction of the z-axis as the one towards which the κ⟶ vector is pointed. So, we have the poles
(12)p02=p⊥2+pz+t2≡m+2andp02=p⊥2+pz−t2≡m−2,due to the denominatorD, and
(13)p02=p⊥2+1−ρpz2−t2≡m~2,due to the denominatorD~, where p⟶⊥ is the component of p orthogonal to the z-axis.We first examine the solutions of the field equation in momentum-space,(14)−p2+κ2Aμ+p·Apμ−ρκ·Aκμ+2iεμναβAνκαpβ=0,withAμ=ϕ,A. The contraction of the above equation with pμ gives us the gauge condition p·A=ρλ/κ2κ·A, which, after the substitution in (14), furnishes us
(15)−p2+κ2Aμ+ρλκ2pμ−κμκ·A+2iεμναβAνκαpβ=0,in whichλ=Σμμ, according to the definitions of (10) for the operators in momentum-space. For our spacelike κμ, we have
(16)−p2+t2Aμ−ρpzpμ+t2δμ3Az+2itεμν3βAνpβ=0.In addition, let us define thex-axis as the direction aligned with the component of A which is orthogonal to κ⟶, so that Ay=0. For the z-component of the field equation, we have
(17)p02−p⊥2−1−ρpz2−t2Az=0.We see that the polep02=m~2 automatically satisfies the equation above. For Az≠0, the poles p02=m±2 are constrained to equal m~2, which means, for a nonnull ρ,
(18)pz+t2=1−ρpz2−t2⇒pz=−t,if0<ρ1ρ−2ρt,ifρ>1orρ<0,form~2=m+2,pz−t2=1−ρpz2−t2⇒pz=t,if0<ρ12−ρρt,ifρ>1orρ<0,form~2=m−2.It is not possible that these two poles satisfy these conditions at the same time for an arbitraryρ, if t≠0. The other equations of motion for Az≠0, with p02=m~2 and, consequently, p2+t2=−ρpz2−t2, are written as
(19)−ρpz2−t2Ax+ρpxpzAz+2itpyϕ=0,ρpypzAz+2itp0Ax−pxϕ=0,ρpz2−t2ϕ−ρp0pzAz−2itpyAx=0.We try a solution withϕ=0, for which, after some manipulation of equations (19), we obtain
(20)p02−py2Ax=0.IfAx≠0, we are left with the constraints px=0 and pz=±t, which correspond to the two modes m~2=m+2 (pz=−t) and m~2=m−2 (pz=t). For the first mode, the electromagnetic field is polarized as A=Ax,0,Az and propagates in the direction p=0,py,−t, such that p·A=−tAz and p·A=tAz. The gauge condition, however, imposes one more restriction:
(21)p·A=ρλκ2κ·A⇒p·A=ρtAz.So, this solution fixesρ=1. For the mode with m~2=m−2, the conclusions are the same, only with the change in the sign of the pz component.IfAx=0, taking into account the other equations, we necessarily have pz=0, such that m~2=p⊥2−1−ρt2. Since, the poles m±2 satisfy the 3-component of field equations for pz=±ρ−2t/ρ, we have that, if ρ=2, the three poles are reduced to one p02=m~2=m±2=p⊥2+t2, with the possible nonphysical consequences of a multiple pole.In the caseAz=0, the mode p02=m~2 is not excluded. It is not possible, however, to fix ϕ=0. In order to obtain a solution which is valid for all values of ρ, we obtain the constrainings py=0 and pz=±t, which satisfy one of the two conditions: p02=m~2=m+2 or p02=m~2=m−2.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,(22)SP=JμAμAνJ∗ν.The current conservation in momentum-space is written aspμJμ=0, such that
(23)p·J=p0J0−p·J=0⇒J0=p·Jp0,JμJμ∗=J02−J⊥2−Jz2=1p02p·J2−p02J⊥2+Jz2.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 onημν and Λμν remain, such that, for our spacelike κμ, we stay with
(24)SP=ip02D−p2+t2p·J2−p02J⊥2+p02Jz2D~−t4H+p2−3t2D~,with
(25)D=p02−m+2p02−m−2,D~=t2p02−m~2,H=−ρp2+t2−4t21−ρ.We have some potential problematic situations to analyze here: whenpz=±t, we have, in principle, double poles, with m~2=m+2 and m~2=m−2, respectively; this situation repeats when pz=±ρ−2/ρt, with an additional complication if ρ=2, with pz=0, which, at first sight, brings us a triple pole (m~2=m+2=m−2). We study below these cases.


### 4.1. The Casespz=±t

Since these two cases are similar, we analyze here the referential frame in whichpz=−t, such that we stay with the poles p02=m~2=m+2=p⊥2 and p02=m−2=p⊥2+4t2. This situation is possible both for Az=0 and Az≠0, as we verified in the field equations. The key point in order to analyze the possibility of a double pole is the fact that p2+t2=p02−m~2 in this referential frame. We stay with
(26)SP=ip02p02−m~2−4t2−p·J2−p02J⊥2+p02Jz2p02−m~22p02−m~2p02−m~2+ρ−4t2+4t41−ρ,and we see that, in order to eliminate the double pole, we need to haveρ=1, in accordance with the result we obtained in the study of the field equations in conjunction with the gauge condition. If we fix ρ=1, the saturated propagator reads
(27)SP=ip02p02−m~2−4t2−p·J2−p02J⊥2+p02Jz2p02−m~2p02−m~2−3t2.We are now in position to calculate the imaginary part of the residue in the two remaining poles. For the polep02=m~2, the calculation is straightforward and gives
(28)ImRp02=m~2SP=34Jz2.For the residue in the polep02=m−2, the term in p·J2 contributes. The residue gives us
(29)ImRp02=p⊥2+4t2SP=14m~2+4t2−4p·J2+4m~2+4t2J⊥2+m~2+4t2Jz2.We then remember that, forpz=−t, we have px=0 and that, in this case, m~2=py2 to obtain, after some calculation,
(30)ImRp02=py2+4t2SP=pyJz+4tJy24py2+4t2+Jx2>0.The result above, which can be diagonalized, indicates a massive mode with two degrees of freedom.


### 4.2. The Casespz=±ρ−2/ρt

We here analyze the casepz=ρ−2/ρt, since the two situations are similar. In this case, we have a possible double pole at
(31)p02=m~2=m+2=p⊥2+41−ρ2t2ρ2,and, ifρ=2, a triple pole. The saturated propagator, after algebraic manipulations, is given by
(32)SP=ip02p02−m~2p02−m~2−42−ρt2/ρ−p02−m~2+4ρ−1t2ρp·J2−p02J⊥2+p02∣Jz2p02−m~2+ρ2−4t2ρ.The residue in the polep02=m~2 gives
(33)ImRp02=m~2SP=ρ+24Jz2+1−ρ2−ρ1m~2p·J2−m~2J⊥2.It is clear that a nonnegative imaginary part of the residue in the polep02=m~2 depends on the value of ρ. In the case of ρ=1, we are back to the case pz=−t. The possible triple pole for ρ=2 and pz=0 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.


## 4.1. The Casespz=±t

Since these two cases are similar, we analyze here the referential frame in whichpz=−t, such that we stay with the poles p02=m~2=m+2=p⊥2 and p02=m−2=p⊥2+4t2. This situation is possible both for Az=0 and Az≠0, as we verified in the field equations. The key point in order to analyze the possibility of a double pole is the fact that p2+t2=p02−m~2 in this referential frame. We stay with
(26)SP=ip02p02−m~2−4t2−p·J2−p02J⊥2+p02Jz2p02−m~22p02−m~2p02−m~2+ρ−4t2+4t41−ρ,and we see that, in order to eliminate the double pole, we need to haveρ=1, in accordance with the result we obtained in the study of the field equations in conjunction with the gauge condition. If we fix ρ=1, the saturated propagator reads
(27)SP=ip02p02−m~2−4t2−p·J2−p02J⊥2+p02Jz2p02−m~2p02−m~2−3t2.We are now in position to calculate the imaginary part of the residue in the two remaining poles. For the polep02=m~2, the calculation is straightforward and gives
(28)ImRp02=m~2SP=34Jz2.For the residue in the polep02=m−2, the term in p·J2 contributes. The residue gives us
(29)ImRp02=p⊥2+4t2SP=14m~2+4t2−4p·J2+4m~2+4t2J⊥2+m~2+4t2Jz2.We then remember that, forpz=−t, we have px=0 and that, in this case, m~2=py2 to obtain, after some calculation,
(30)ImRp02=py2+4t2SP=pyJz+4tJy24py2+4t2+Jx2>0.The result above, which can be diagonalized, indicates a massive mode with two degrees of freedom.


## 4.2. The Casespz=±ρ−2/ρt

We here analyze the casepz=ρ−2/ρt, since the two situations are similar. In this case, we have a possible double pole at
(31)p02=m~2=m+2=p⊥2+41−ρ2t2ρ2,and, ifρ=2, a triple pole. The saturated propagator, after algebraic manipulations, is given by
(32)SP=ip02p02−m~2p02−m~2−42−ρt2/ρ−p02−m~2+4ρ−1t2ρp·J2−p02J⊥2+p02∣Jz2p02−m~2+ρ2−4t2ρ.The residue in the polep02=m~2 gives
(33)ImRp02=m~2SP=ρ+24Jz2+1−ρ2−ρ1m~2p·J2−m~2J⊥2.It is clear that a nonnegative imaginary part of the residue in the polep02=m~2 depends on the value of ρ. In the case of ρ=1, we are back to the case pz=−t. The possible triple pole for ρ=2 and pz=0 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 whenρ=1. First, let us rewrite the dispersion relations for the model in a more general way. From the D factor in the denominator, taking α=1 and μ=1, we have
(34)p2−κ22+4κ2p2−λ2=0,which can be factorized as
(35)p2+κ2+2λp2+κ2−2λ=0,so that
(36)ω2≡p02=p2−κ2±2λ.For the purely spacelike background vector,κμ=0,κ⟶, we stay with
(37)ω2=p±κ⟶2.Since the model is massive, we can go to the particle rest frame and get, for these two dispersion relations,(38)ω2=κ⟶2,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 theρ parameter. On the other hand, the D~ factor is written as
(39)D~=ρλ2−κ2p2+1−ρκ4,which furnishes us the dispersion relation
(40)ω2=p2+1κ2ρλ2+1−ρκ4.For the purely spacelike Lorentz-breaking vector, the result is written as(41)ω2=p2−1κ⟶2ρp·κ⟶2+1−ρκ⟶4=p21−ρcos2θ−1−ρκ⟶2,whereθ is the angle between the vectors p and κ⟶. Let us consider the field propagates in the direction of the Lorentz-breaking vector, such that θ=0. In this situation, we stay with
(42)ω2=1−ρp2−κ⟶2,which imposes some restrictions. First, ifρ=1, we have a massless mode. For ρ≠1, the dispersion relation predicts a massive mode, but with the restriction ρ>1, otherwise we have an unstable model, since in the particle rest frame we have ω2=ρ−1κ⟶2. However, even with ρ>1, 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 κ⟶, although the component of the momentum in the direction of κ⟶ cannot be null (we must have ρ≠2). Thus, in the special case in which ρ=1, the model presents a massive mode, ω2=κ⟶2 (coming from the factor D 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 D~).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, Mμν (defined in equation (2)), which is given by
(43)Lμν=1κ2ημν+ρκμκν1−ρκ2.Interestingly, there is no inverse forMμν when ρ=1. 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 inκμ gives us a transverse result [61], with
(44)Πμν2=−ie26π2m21+25p2m2+Op4m4Tμν,in whichm is the fermion mass and
(45)Tμν=κ2pμpν−p2ημν+p2κμκν+p·κ2ημν−p·κpμκν+pνκμ.All these results bring us the possibility of a residual gauge invariance. The gauge-violating term is given by(46)LM=12AμMμνAν.By considering a gauge transformation on the vectorial field,Aμ,
(47)Aμ⟶Aμ+∂μχ,we have
(48)δL=AμMμν∂νχ+12∂μχMμν∂νχ.It is easy to see that if∂μχ is proportional to κμ, such that ∂μχ=βzκμ, with βz a dimensionless function of z (assuming the z-axis is parallel to the background vector), we get, explicitly using (2),
(49)∂μχMμν=βκ2κν1−ρ,which is null forρ=1. In this situation, the variation of the Lagrangian is null. Just in order to illustrate, for our purely spacelike vector κμ=0,κ⟶, this is accomplished if the gradient of χ is parallel to κ⟶ in the case ρ=1. 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 ρ=1.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 citedLM.


## 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 ρ 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 κμ, that if ρ<1, 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 z-axis (defined as the direction of the background vector), of the lines z=±t/p⊥u, being u the axis parallel to the component p⟶⊥ of the field momentum and t the magnitude of the Lorentz-breaking vector κ⟶. Moreover, the particular case in which ρ=2 accommodates a nonphysical triple pole when the component of the field momentum along the z-axis is null.The most interesting case, however, is the one withρ=1, which presents a massless mode of propagation along the z-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, Aμ⟶Aμ+∂μχ, such that χ=χz, leaves the action unchanged. For a purely spacelike background vector, this corresponds to have the gradiant of χ parallel to κ⟶. 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 limitρ⟶1. 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 ρ undetermined, obtained from a gauge embedding procedure.



---

*Source: 7396078-2022-02-12.xml*