{'fraction_non_alphanumeric': 0.03784465669489998, 'fraction_numerical': 0.0130654171922869, 'mean_word_length': 4.372216844143272, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Revealing hidden symmetries and gauge invariance of the massive Carroll-Field-Jackiw model

In this paper we have analyzed the improved version of the Gauge Unfixing (GU) formalism of the massive Carroll-Field-Jackiw model, which breaks both the Lorentz and gauge invariances, to disclose hidden symmetries to obtain gauge invariance, the key stone of the Standard Model. In this process, as usual, we have converted this second-class system into a first-class one and we have obtained two gauge invariant models. We have verified that the Poisson brackets involving the gauge invariant variables, obtained through the GU formalism, coincide with the Dirac brackets between the original second-class variables of the phase space. Finally, we have obtained two gauge invariant Lagrangians where one of them represents the Stueckelberg form.


I. INTRODUCTION
The Caroll-Field-Jackiw (CFJ) model [1] is a variation of Maxwell's electrodynamics (ME) where, due to the presence of the CFJ term in the original Maxwell Lagrangian, the Lorentz invariance is broken. The CFJ term contains a Lorentz violating (LV) fixed background, V µ , which is possibly responsible for the phenomenon of birefringence [2]. This model is also known as the gauge sector of the extended standard CPT-odd model [3,4]. In the language of constrained systems, via Dirac formalism, both ME and CFJ models are gauge invariant systems with two first-class constraints. On the other hand, if we add a photon mass term in the original Maxwell Lagrangian, we have the Abelian Proca model which is not gauge invariant with two second-class constraints. Combining the CFJ theory with the Abelian Proca model we have a model where both Lorentz invariance and gauge invariance are broken. So, it is possible to disclose gauge symmetries in this new model by converting second-class system into a first-class one. In this context, one formalism which we can mention is the Batalin-Fradkin-Tuytin (BFT) method [5] which is formulated in an extended phase space with the introduction of the Wess-Zumino terms. The other one is the Gauge Unfixing (GU) formalism where gauge theories are obtained within the phase space of the original secondclass theory. The GU formalism was originally formulated by Mitra and Rajaraman [6] and continued by Vytheeswaran [7][8][9][10]. Since no extra variables in the phase space are not used, this procedure seems to be very attractive.
The purpose of this paper is to disclose the hidden gauge invariance and to discuss its consequences concerning the massive CFJ model. We have disclosed it through the improved GU formalism [11,12] in order to analyze the massive CFJ models which are not gauge invariant, obviously. We organized our ideas in the following manner. In section II we described a Hamiltonian formulation of the massive CFJ. In section III we presented a revision of an improved GU techniques. In section IV we applied our method to the massive CFJ model where two gauge theories were derived. Section V is devoted to the conclusions and final words.

II. THE CANONICAL STRUCTURE OF THE MASSIVE CFJ MODEL
The massive CFJ model is described by the following Lagrangian density where ε βαρϕ V β A α F ρϕ is the term responsible by the broken of the Lorentz symmetry, m is the mass of the photon and g µν = diag(+ − −−). The canonical momenta is given by From Eq.(2) we have the primary constraint In the phase space we have the fundamental Poisson bracket The momenta π k is given by After the Legendre transformation we can write the canonical Hamiltonian as being From the time stability condition of the constraint in Eq. (3), we can obtain the secondary constraintφ We can observe that no further constraints will be generated via this iterative procedure and φ 1 and φ 2 are the final group constraints of the model. Calculating the Poisson brackets of these constraints, we have and Eq. (8) shows that the constraints φ 1 and φ 2 have a second-class structure via Dirac constrained systems classification.

III. THE IMPROVED GU FORMALISM
Consider a Hamiltonian system with two second-class constraints, T 1 and T 2 . The basic idea of the GU formalism consists of selecting one of the two second-class constraints to be the gauge symmetry generator and the other one will be discarded as a gauge generator. For example, if we choose T 1 as the gauge symmetry generator then T 2 will be discarded as a gauge generator, obviously. The constraint T 1 will be redefined as The Poisson bracket between T and T 2 is { T , T 2 } = 1, so that T and T 2 are canonically conjugate. The second-class Hamiltonian needs to be modified in order to satisfy a first-class algebra and, consequently, to give rise to a first-class Hamiltonian. We can build a gauge invariant Hamiltonian by a series in powers of T 2 as where, by construction, we can show that {H,T } = 0. The improved GU formalism, introduced by one of us [11,13], modifies the original phase space variables in order to obtain a gauge invariant phase space. Consider the initial phase space variable written in the following form The gauge invariant variable F to be obtained must satisfy the variational condition where T is the constructed second-class constraint that was chosen to be the gauge symmetry generator and ǫ is an infinitesimal parameter. Any function of F will be gauge invariant since where Consequently, we can obtain a gauge invariant function from the following substitution

IV. RAISING GAUGE THEORIES IN THE MASSIVE CFJ MODEL
We will now compute the underlying symmetries of the massive CFJ model by using the improved GU formalism [11,13]. From the two second-class constraints, Eqs. (3) and (7), we have two possible choices for the gauge symmetry generator. We consider these ones separately.
Case (i) (φ 1 is the gauge symmetry generator) We will begin by redefining the constraint φ 1 , Eq. (3), which was initially chosen to be the symmetry gauge generator, asφ The gauge invariant variable A 0 is constructed by the power series of the integral in φ 2 The coefficients C n in Eq. (17) are then determined by the variational condition Using and in Eq. (19) we find that The linear equation in φ 2 is Using Eqs. (21) and (22) in Eq. (23) we can obtain the coefficient C 2 (x, y, z) Since C 2 (x, y, z) = 0 then all the correction terms C n with n ≥ 2 are zero. Using Eq.(22) into Eq. (17) we obtain the gauge invariant fieldÃ 0 given bỹ The others fields A i , π 0 , π i , are not modified due to relation Therefore we have thatπ The Poisson brackets between the gauge invariant fields are Ã 0 (x),Ã 0 (y) = {π 0 (x),π 0 (y)} = Ã 0 (x),π 0 (y) Here it is important to comment that the Poisson brackets between the gauge invariant variables, Eqs. (30), (31), (32) and (33), are the same obtained by the Dirac brackets between the original phase space variables. Therefore the GU variables could be an alternative to the usual algorithm that calculates the Dirac brackets in a particular constrained second-class system. For more details see ref. [13]. Using Eqs.
It is clear that, by construction, we have { H,φ} = 0. The invariant Lagrangian can be found by using the functional form of the original Lagrangian, Eq. (1). For more detail about this procedure see ref. [11]. So, the gauge invariant Lagrangian can be initially written in the following formL Using Eq. (25) into (35) we can derive the gauge invariant Lagrangiañ Here it is important to mention that the gauge invariant Lagrangian, Eq. (36), can not be reduced to a covariant form.
Case (ii) (φ 2 is the gauge symmetry generator) We will redefine the constraint φ 2 asφ so that Then, we can use the procedure applied in case (i) again. For example, the gauge invariant variableÃ i is constructed by the power series of the integral in φ 1 As a result we have where we have used Eq. (38) and We can show that all the correction terms C n with n ≥ 2 are zero. So, using Eqs. (40) and (41) into (39) we findÃ Repeating this same iterative process in order to obtain the other gauge invariant variables we findπ Using Eqs. (42), (44) and (45) into the Hamiltonian in Eq. (6) and noticing thatF ij = F ij so that we can derive the gauge invariant Hamiltoniañ The Poisson brackets between the gauge invariant variables are Ã 0 (x),Ã 0 (y) = {π 0 (x),π 0 (y)} = Ã 0 (x),π 0 (y) The GU variable,F αν , in Eq. (51) is defined as We can generalize the gauge invariant variable, Eq. (42), using the result of the equation of motion, ∂ 0 π 0 = −∂ k π k − 1 4 ǫ 0kij V k F ij + m 2 A 0 = 0, in such way that we can defineÃ µ as Substituting Eq.(53) into Eq. (51) and using the fact thatF αν = F αν so the gauge invariant Lagrangian can be reduced to a covariant form where θ in Eq. (54) is defined as θ ≡ π 0 m 2 (Stueckelberg trick [8]). Here it is important to mention that we can identified the θ field with the so-called Stueckelberg scalar whose gauge transformation cancels that of the A µ field, therefore making L invariant. This Lagrangian is the same found by Vytheeswaran [8] with an extra Lorentz symmetry violation term.

V. CONCLUSIONS AND FINAL REMARKS
In theoretical physics, one of its key improvements was the construction of gauge invariant systems and its corresponding symmetries, which made the framework of Standard Model possible. In other words, gauge field theories have an underlying rôle in the discussion of the physical fundamental interactions. It is also important to mention that, due to the presence of symmetries, gauge invariant systems can describe the theoretical models in a more complete approach.
Having said that, in this paper, we have used the improved GU formalism to discuss gauge invariance and to disclose hidden symmetries that dwell inside the massive CFJ model, which breaks both Lorentz and gauge invariances. The method provide these results by converting the massive CFJ, which is a second-class system, into a first-class one, namely, into a gauge invariant system. This method has the advantage of not introducing extra variables. Only variables of the original phase space are utilized. The GU conversion method preserves the degrees of freedom of the initial system. This fact can be verified by using the following physical degrees of freedom counting formula [14], N d = N t − N sc − 2N f c , where N d is the number of degrees of freedom, N t is the total number of canonical variables and N sc and N f c are the number of second and first-class constraints, respectively. It is worth to mention that the obtainment of gauge invariant variables simplifies the derivation of both the gauge invariant Hamiltonian and Lagrangian that correspond to the massive CSF model.
To sum up the results obtained here, we have obtained two gauge invariant actions dual to the massive CFJ model, where the Stückelberg trick was not necessary, of course. Besides, we have shown precisely that the improved GU procedure can lead us to the same brackets as the ones obtained via Dirac constrained systems method. The other positive point is that no extra variables were used in the calculations, as we said before.