Vacuum polarization and induced Maxwell and Kalb-Ramond effective action in very special relativity

This work investigates the implications of very special relativity (VSR) on the calculation of vacuum polarization for fermions in the presence of Maxwell and Kalb-Ramond gauge fields in four-dimensional spacetime. We derive the $SIM(2)$-covariant gauge theory associated with an Abelian antisymmetric 2-tensor and its corresponding field strength. We demonstrate that the free VSR-Kalb-Ramond electrodynamics is equivalent to a massive scalar field with a single polarization. Furthermore, we determine an explicit expression for the effective action involving Maxwell and Kalb-Ramond fields due to fermionic vacuum polarization at one-loop order. The quantum corrections generate divergences free of nonlocal terms only in the VSR-Maxwell sector. At the same time, we observe UV/IR mixing divergences due to the entanglement of VSR-nonlocal effects with quantum higher-derivative terms for the Kalb-Ramond field. However, in the lower energy limit, the effective action can be renormalized like in the Lorentz invariant case.


I. INTRODUCTION
In 2006, Cohen and Glashow proposed a modification to the theory of special relativity that preserves the familiar energy-momentum dispersion relation while breaking the invariance under the complete Lorentz group, SO (1,3) [1].They recognized that specific subgroups of the Lorentz group can still produce conservation laws and reproduce the wellknown effects of special relativity (SR).Among these subgroups, the HOM (2) and SIM (2) groups meet these requirements.The former subgroup, known as the Homothety group, consists of the boost generator K z and the generators T 1 = K x + J y , T 2 = K y − J x .These generators together form a group that is isomorphic to the group of translations in the plane.
The second subgroup, known as the similitude group SIM (2), is an enhanced version of the HOM (2) group obtained by including the generator J z .These subgroups do not admit invariant tensors that can act as constant background tensor fields, as seen in other theories of Lorentz violations [2].Hence, the breakdown of SO (3,1) into either HOM (2) or SIM (2) cannot be explained through local symmetry-breaking operators.This intriguing theory of relativity came to be known as "very special relativity" (VSR) [3].
A key attribute of VSR is that the SIM (2) generators preserve both the speed of light and the null 4-vector n µ = (1, 0, 0, 1), thereby establishing a preferred direction in space.
However, VSR algebras do not support discrete symmetry operators such as P , T , CP , and CT .Including any of these operators would result in the full Lorentz algebra [4].In this way, the lack of discrete symmetries may result in violations of unitarity and causality in quantum field theories.To circumvent this issue, Cohen and Glashow proposed the inclusion of non-local operators containing ratios of contractions of n µ in order to construct a unitary SIM (2)-invariant field theory.The non-local nature of VSR gives rise to a remarkable phenomenon of mass generation.Indeed, when a non-local operator like n µ /(n • ∂) is added in the momentum operator for a massless fermion, it results in a corresponding Klein-Gordon equation with a mass term proportional to the non-local coupling constant.This impressive property has led to proposals suggesting a VSR explanation for neutrino mass [5] and dark matter [6].
In particular, the VSR contributions to the induced effective action in the context of Maxwell-Chern-Simons electrodynamics have been studied in previous works [24,25].
In these studies, the authors employed the Mandelstam-Leibbrandt prescription [26][27][28], adapted to the VSR case by Alfaro [29,30], to handle the UV/IR mixing divergences that arise in the loop integrals.Another notable gauge field theory in four-dimensional Minkowski spacetime is built from an antisymmetric 2-tensor known as the Kalb-Ramond field [31].
The study of the classical action for antisymmetric tensor fields in the context of VSR was initially addressed in Ref. [49].However, a comprehensive analysis of the role of VSR in the quantum corrections of this field is still lacking in the literature.In this work, we propose a systematic procedure to construct an antisymmetric 2-tensor gauge field that incorporates VSR non-local operators and use it to derive the associated SIM (2)-invariant classical action.Subsequently, we obtain the free equation of motion in the VSR-Kalb-Ramond electrodynamics, and explicitly determine the resulting degrees of freedom.Furthermore, we explore, for the first time, the induced corrections to the effective action of the Maxwell and Kalb-Ramond gauge fields in the context of VSR.We obtain the effective Lagrangian density at the one-loop order by integrating the fermionic fields and calculating the vacuum polarization Feynman diagrams.We demonstrate that the divergent terms can be renormalized in the low-energy limit by appropriately rescaling the fields, masses, and coupling parameters in the model.Moreover, our results recover those obtained in the literature for the Lorentz invariant limit [50].
The present work is organized as follows.In Sec.II, we review the main aspects of VSR applied to the vector gauge field.In Sec.III, we propose a procedure to derive the SIM (2)invariant Lagrangian density for the Kalb-Ramond field and analyze its free physical modes.
In Sec.IV, we calculate the induced effective actions of the Maxwell and Kalb-Ramond fields, evaluate the Feynman diagrams for the two-point gauge functions, and examine the general tensorial form of the finite and divergent induced terms as well as the renormalization issues.Our final comments are presented in Section V.

II. SIM(2)-INVARIANT MAXWELL GAUGE THEORY
To establish a consistent framework for our upcoming calculations, we will review the SIM (2)-invariant gauge vector theory described in Refs.[10,11,23].The VSR-modified Maxwell electrodynamics is a U (1) gauge theory that involves a 1-form gauge potential A µ (x) and a matter field ψ(x) that acts as the source of A µ (x).This theory obeys the following gauge transformations: where Λ(x) is an arbitrary 0-form field.The wiggle derivative operator is defined by where m A is a constant parameter with mass dimension, and n µ = (1, 0, 0, 1) is a fixed null vector present in VSR theories and select a preferred direction.
The covariant derivative for VSR-Maxwell electrodynamics is given by and it is constructed by demanding the fundamental property of transforming as ψ does under infinitesimal gauge transformations [11]: We can compute the field strength related to D µ as follows: This gives us the expression for the field strength tensor: where we introduce the notation N µ ≡ n µ /n • ∂ for the nonlocal vector operator.
It is noteworthy that F µν is invariant under the gauge transformation where we define the wiggle gauge vector Ãµ by such that We observe that by applying a field redefinition On the other hand, we can define a new SIM (2)-gauge invariant field strength Fµν , which is constructed using the wiggle derivative, namely, and it can be expressed explicitly as which shows that Fµν is also invariant under the gauge transformation (8).
The present analysis allows us to construct a SIM (2)-invariant Lagrangian density for the field A µ that is also invariant under standard gauge transformations.As pointed out in Ref. [23], this is an important result because this Lagrangian generates a mass term for the field A µ without breaking the original gauge symmetry of the theory.
According to the definition (11), Fµν is not Lorentz invariant but instead SIM (2) invariant.Moreover, as shown in result (12), it is also invariant under transformation (8).So, we can construct a VSR gauge-invariant Lagrangian density as follows: Therefore, from Eq. ( 12), this Lagrangian takes the form Finally, by applying a field redefinition from we obtain the desired result It is interesting to notice that if we start from −1/4F µν F µν to define our Lagrangian, the above field redefinition will withdraw the VSR effects.Furthermore, as shown in Ref. [23], we can apply the Lorentz gauge ∂ µ A µ = 0 plus the subsidiary gauge condition N • A = 0 into the equation of motion obtained from ( 15) and we find Hence, in the VSR scenario, we obtain a massive gauge field with two physical degrees of freedom, which is in contrast to the Proca case where the mass term m 2 A A µ A µ is not gauge invariant and has three degrees of freedom.

III. KALB-RAMOND ELECTRODYNAMICS IN VSR
In this section, we investigate the issue of constructing a SIM (2)-invariant action for the Kalb-Ramond field.As we will see, this is possible even when the Kalb-Ramond field does not carry matter charge, i.e., when it is not minimally coupled to matter fields and does not possess any associated covariant derivatives.

A. Setup
Let us start by defining the Lagrangian density that describes the dynamics for an antisymmetric 2-tensor B µν in 4D Minkowski spacetime, where is the field strength tensor associated with B µν , and J µν is an antisymmetric conserved current due to the coupling to the matter [38].The field strength H µνα corresponds to the components of an exact 3-form field H, which is constructed using the exterior derivative from the 2-form B associated with B µν .This field strength satisfies the identity which follows from the fact that an exact 3-form is closed [51].
The Lagrangian ( 17) is the simplest which can be constructed by demanding parity-even and invariance under the U (1) gauge transformation: where Σ µ is an arbitrary vector field.The field Σ µ also exhibits an extra gauge invariance given by with ϕ being an arbitrary scalar field.This latter transformation leaves Eq. ( 20) unchanged.
In general, the current J µν is constructed from other dynamical fields which involve extended objects of the type found in the string theory [31].For the sake of simplicity, we will not consider a string matter field for the source of B µν (x) in this work.In what follows, our attention will be focused only on the kinetic part of the Lagrangian density (17), such that the matter coupling, represented by J µν , will be turned off.

B. SIM (2)-covariant Kalb-Ramond gauge theory
For the SIM (2)-invariant generalization of the Kalb-Ramond Lagrangian (17), we expect that the gauge symmetry (20) modified by the nonlocal vector operator N µ ≡ n µ /n • ∂ will play a crucial role.As we saw in the Maxwell case, both F µν and Fµν are invariant under the standard gauge transformation, and the connection between the two kinds of field strengths is made through the vector potential Ãµ .
Hence, motivated by our earlier analysis, we will construct a Bµν field that satisfies the following requirements: (i) Bµν is a linear function that is first-order in B µν and secondorder in N µ ; (ii) Bµν has mass dimension one in 4D spacetime; (iii) Bµν transforms by After imposing these requirements, we arrive at the B-ansatz given by and it changes under the gauge transformation B µν → B µν + ∂µ Σ ν − ∂ν Σ µ as follows: It is worth noting that the gauge parameter Σµ = Σ µ − m 2 2 N µ (N • Σ) has the same form as Ãµ in Eq. ( 9), which was obtained in the Maxwell case.This is expected since Σ µ has the additional gauge symmetry (21), similar to the A µ field.Additionally, it is interesting to note that for a 0-form field, our prescription implies that φ Once we have found Bµν , we can define the tensor whose explicit form is given by: Also, the SIM (2)-covariant field strength tensor Hµνα can be defined as Taking the difference between the two kinds of field strengths Hµνα − H µνα , we obtain Furthermore, we can rewrite Hµνα solely in terms of H µνα , thereby guaranteeing the invariance of Hµνα under both SIM (2) and the wiggle gauge transformations, as required in condition (iii).To this end, it is easy to check the following identity: where using the properties of the operator N µ we have that [10] and the integration by parts rule holds: Besides, it would also be consistent to set N µ ϕ(x) ≡ 0 if ϕ is a constant.With all the above results, the SIM (2)-covariant tensor Hµνα can be cast as which is a natural generalization of the relation (12).
Finally, the SIM (2)-invariant action of the Kalb-Ramond field B µν is represented by Hµνα Hµνα , (32) and with help the of Eq. ( 31) we can write it as where we performed an additional change of field variables Therefore, similar to the Maxwell case, the SIM (2)-modified Kalb-Ramond action (32) is invariant under the standard gauge transformation (20).
The equation of motion follows from the action (33) by varying with respect to B να , Explicitly, we find By contracting Eq. (34) with N ν , we obtain the following constraint: Inserting this constraint back into the equation of motion, we find To find the physical modes, we must fix the gauge freedom.We can chose, analogous to the Maxwell case, the Lorentz gauge Then the equation of motion (36) and the constraint (35) become, respectively: and The form of Eq. ( 38) still contains redundant degrees of freedom.Indeed, the gauge condition ( 37) is insufficient to fix the gauge freedom completely since we can construct a solution , which preserves the Lorentz gauge (37) and satisfies the equation of motion (38).So, we can impose an additional condition on the field B ′να , namely, by choosing the gauge parameter Σ α as However, the last relation is invariant under the residual gauge symmetry ( 21), and we can use this fact to impose the condition by fixing the scalar gauge parameter as ϕ = N ν Σ ′ν .Thus, the gauge parameter is given by Now, we can show that the Lorentz gauge condition is valid to B ′να .From (37) we find On the other hand, the constraint condition (39) implies that where we used the relations N ν B ′να = 0 and N ν Σ ν = 0. Immediately, from (44) it follows that and by Eq. ( 43), we obtain the claimed result where in the last step we use again the Lorentz gauge ∂ µ B µν = 0.
Finally, by applying the subsidiary condition N ν B να = 0 to the equation of motion (38), it takes the simple form which represents a wave equation for a particle of mass m.At the end, the Kalb-Ramond field in VSR satisfies the standard Klein-Gordon equation under two gauge conditions The general solution to Eq. ( 48) takes the form: where ω p = √ p 2 + m 2 , and the associated 4-momenta p µ are on shell such that p µ = (ω p , p).
The Fourier coefficients B µν (p) can be expanded over a basis of polarization antisymmetric 2-tensors, labeled by λ = 1, • • • , 6: To find the physical polarization states, it is convenient to analyze the solution (50) in the rest frame where k µ = (m, 0).The solution for a general p µ can then be obtained by applying a VSR boost, i.e., p µ = L(p) µ ν k ν , with where T 1 , T 2 , and K 3 are the generators of the SIM (2) group in the vector representation [4].
In the rest frame, the gauge conditions (49) become The first condition implies that ϵ 0i (k, λ) = 0 with i = 1, 2, 3, which eliminates three polarizations.The second condition gives ϵ 3j (k, λ) = 0 with j = 1, 2, which eliminates two more polarizations.Therefore, there is only one non-zero polarization ϵ 12 (k, λ), which means that the free Kalb-Ramond field in VSR has only one degree of freedom, equivalent to a single massive scalar field.

IV. MAXWELL-KALB-RAMOND VACUUM POLARIZATION IN VSR
In this section, we calculate the effective action for the case of a fermion field interacting with Maxwell and Kalb-Ramond fields within the context of VSR, which has been developed in the preceding sections.As we proceed, we will obtain exact solutions to the one-loop vacuum polarization amplitudes involving the external gauge fields.
Let us start by recalling that Kalb-Ramond quantum electrodynamics is a U (1) gauge theory that involves a 2-form gauge potential B µν (x) and a string matter field ψ(x(σ)) serving as the source for B µν (x) [31].Considering the complexities inherent in string theorybased systems, let us focus on exploring a simplified scenario within the framework of VSR.
Specifically, we investigate an interaction model involving a point-like fermion field and the Maxwell and Kalb-Ramond fields in four-dimensional Minkowski spacetime.As we will see below, this type of interaction is only viable if ψ does not carry any Kalb-Ramond charge and couples nonminimally with it [50].
We consider SIM (2)-covariant gauge theories under the U (1) gauge transformations where we have defined the wiggle operators as where m A and m B represent the VSR-mass associated with the Maxwell and Kalb-Ramond fields, respectively.
The simplest Lagrangian density that can be constructed, invariant under the aforementioned gauge transformations, is given by [50]: where m represents the usual fermion mass, m ψ the VSR-mass associated to the ψ field, and g is a coupling constant with mass dimension [g] = M −2 (in natural units).Also, the operator D µ denotes the standard covariant derivative, given by and σ µνλ represents the fully antisymmetrized product of two gamma matrices normalized to unit strength, defined as where ϵ µνλα is the Levi-Civita symbol.It satisfies the commutation relation It is important to note that the Lagrangian density (58) involves a nonminimal coupling of the Kalb-Ramond field to point-like fermions.We could interpret this kind of coupling as similar to that involving neutral particles, such as the neutron, interacting with the electromagnetic field through their anomalous magnetic moments.
In order to determine the effective Lagrangian resulting from fermionic vacuum polarization at the one-loop level, we consider the generating functional defined as where N is a normalization constant which will be used to absorb field-independent factors.
By performing the fermionic integration, we obtain (up to a field-independent factor that can be absorbed in the normalization): where Tr stands for the trace over Dirac matrices as well as the trace over the integration in coordinate space.At this point, it is worth mentioning that the Lorentz covariant calculation of vacuum polarization for the model under study was carried out in Ref. [50] for the simplified case of constant fields using Schwinger's approach.In this work, we extend this calculation to the SIM (2)-covariant case using Feynman techniques without the restriction to constant external gauge fields.Furthermore, we can verify whether our results can reproduce those obtained in the literature by taking the limit m A,B,ψ → 0 and assuming constant Therefore, we can obtain the corresponding new types of vertices involving more than one external gauge vector field from the perturbative Lagrangian density: Thus, we can write the nontrivial part of the effective action as where the operator Ô can be determined from Eq. ( 65).
The formal contributions of this formula will give rise to the n-point vertex functions of the fields A µ and B µν .At this point, a graphical representation may be helpful.Following the conventions depicted in Fig. 1, the contributions to the tadpole and the self-energy are illustrated in Figs. 2 and 3 up to one-loop order.For n = 1 the expression (66) gives rise to three contributions (67) up to second order in the coupling constant e, as graphically indicated in Fig. 2. We found that the tadpole diagrams 1(a) and 1(b) vanish, as expected by Furry's theorem [24].The remaining contribution, i.e., the self-energy 1(c), gives and we can write it in momentum space.The result is where with u = p + q, and µ 2 = m 2 ψ + m 2 representing the modified fermion mass.Besides, we define the wiggle momentum by After calculating the traces of the Dirac matrices in four dimensions, the expression (70) yields This integral is ultraviolet divergent and also exhibits an infrared divergence when (n•p) = 0.
To deal with these types of divergences in VSR, we use the Mandelstan-Leibbrant pre- where nµ is a extra null vector which obeys (n • n) = 1.Moreover, we employ a useful decomposition formula to isolate factors as 1/(n • p) in each momentum integration.The resulting loop integrals can be evaluated using the formulas introduced by Alfaro in Ref. [29].Here, we will quote the most basic of them: where dp is the integration measure in d-dimensional space with ω = d/2.
Following the calculation procedure mentioned above to evaluate the integral in Eq. (72), we obtain: As we can note, the Mandelstam-Leibbrandt prescription (73) adopted in the formula for the loop integral (75) introduces a new null vector nµ , which a priori could break the SIM (2) symmetry.To preserve the SIM (2) covariance in this calculation, Alfaro proposes in Ref. [30] to fix the vector nµ as a linear combination of the original null vector n µ and the external momentum of the diagram.By imposing certain conditions, such as reality, appropriate scaling (n, n) → (λn, λ −1 n), and being dimensionless [24,25], the vector nµ can be expressed in the following form: which leads to n • q = q 2 2(n•q) .Hence, we can substitute this result into Eq.( 76), and after integrating over the variable t, we find where we have performed an expansion around ω = 2 and retained only the 1/(2 − ω) pole and the finite terms when ω → 2 + .We note that this result is exclusively an effect of the VSR and goes to zero when we take the limit m 2 ψ → 0. For n = 2 the expression (66) yields four 2-point amplitudes up to one-loop order as depicted in Fig. 3.We have explicitly verified that graphs 2(b) and 2(c) vanish after momentum integration.The only nontrivial contributions come from graphs 2(a) and 2(d).
The Feynman diagram 2(a) corresponds to the usual vacuum polarization of the photon observed in QED, with the additional VSR nonlocal corrections incorporated into the fermion propagator and vertex.Its analytical expression is given by which written in momentum space result in where Following the procedure outlined earlier, we find where we have defined the integrals: Expression (84), by itself, does not have a transverse structure as required by U (1) gauge invariance of the photon field.To bring the photon self-energy into its desired form, we must include the full diagram contributions Π (Total) µν = Π (1,c)  µν + Π (2,a)  µν .Let us separate this sum into two parts.The one with the simplest structure is the divergent contribution, namely, Π (Total)   µν div = Π (1,c)   µν div + Π (2,a)   µν div which has the same form present in the usual QED.The second part is UV finite and exhibits a more intricate structure.To simplify the integrals I i , we expand the corresponding contributions as a power series in the external momenta before integrating over the parameters x and t.The resulting expression can be written in the following form: where • • • indicates terms of higher order in the power of q 2 /µ 2 .In the end, the full one-loop vacuum polarization of the photon in the VSR-QED is manifestly transverse, as required by the Ward identity q µ Π µν (q) = 0.But only the UV finite part receives corrections due to the VSR-nonlocal terms.Note that a similar result was obtained recently in Ref. [25] for the low-energy limit, q 2 ≪ µ 2 , in the context of Maxwell-Chern-Simons electrodynamics within the VSR framework.
The results obtained so far allow us to write the VSR effective Lagrangian for the gauge field A µ as follows: where 1 with In order to yield a divergence-free effective Lagrangian we perform a suitable change of scale, by defining the renormalized quantities: Using the above scale transformations we get This effective Lagrangian retains the exact form of its classical counterpart, incorporating finite terms encompassing higher-derivative and VSR-nonlocal corrections.In particular, no additional counterterms were required to cancel the divergences, and the tensorial structure of these finite terms preserves both VSR and gauge symmetry.
To determine the tensor structure of the quantum effective action and the necessary counterterms to renormalize it, we can utilize the identity where H ⋆ µ is the dual tensor corresponding to H µνλ and is given by So we can write the effective Lagrangian density associated with the Kalb-Ramond field as where 1 with C div defined in Eq. (94).
We note that from the second and third terms in Eq. ( 108), it is necessary to introduce new counterterms into the classical action in order to renormalize it within the minimal subtraction scheme.This is an expected result since it is well-known that nonminimal couplings can lead to higher-derivative divergences in the quantum effective action.Hence, at this point, we may interpret our model as an effective theory that is valid in the lowmomentum limit, where q 2 ≪ m 2 .In this regime, we can make the approximations µ 2 ≈ m 2 with m 2 ψ ≪ 1.Thus we have 1 and by defining a renormalized field B µν R = Z

Figure 1 :
Figure 1: Feynman rules.Continuous, wave, and double wave lines represent the fermion propagator, the gauge vector A µ , and the field strength≁H

− 1 2 B 2 B
B µν and a renormalized coupling constant g R = Z 1 g, we getL eff[B] one can eliminate the modification by the VSR-nonlocal terms.After this redefinition, we obtain the same field strength tensor F µν → F µν = ∂ µ A ν − ∂ ν A µ and the same covariant derivative gauge transformation as in standard electrodynamics, given by