Induced Maxwell-Chern-Simons Effective Action in Very Special Relativity

In this paper, we study the one-loop induced photon's effective action in the very special relativity electrodynamics in $(2+1)$ spacetime (VSR-QED$_{3}$). Due to the presence of new nonlocal couplings resulting from the VSR gauge symmetry, we have additional graphs contributing to the $\langle AA\rangle$ and $\langle AAA \rangle$ amplitudes. From these contributions, we shall discuss the VSR generalization of the Abelian Maxwell-Chern-Simons Lagrangian, consisting in the dynamical part and Chern-Simons-like self-couplings, respectively. Furthermore, this analysis allows us to verify explicitly the validity of the Furry's theorem in the VSR-QED$_{3}$. The VSR invariant Mandelstan-Leibbrant prescription plays an important role in our analysis by preserving all the model's symmetries.


Introduction
Over the past decades Lorentz violating field theories have gained considerable attention in the search for manifestations of physics beyond the Standard Model [1][2][3]. One might say that the main focus in this context corresponds to suitable extensions of spacetime symmetries, that are fundamental pieces of the Standard Model (SM) of particles and general relativity, thus any deviation from these symmetries is expected to be related with new physical phenomena [4]. In the description of these theories, we have a strong approach that is the covariant effective action. In recent years, we have seen interest in this formulation for different types of quantum fields, where new types of interactions that depend on the spin of the fields involved as well as the spacetime dimensionality can be obtained [5,6]. The effective action approach has as its canonical examples the Euler-Heisenberg effective action [7,8] and the quantum gravitational theory [9,10].
In the formulation of Lorentz violating field theories within the effective action framework, the current understanding is that the low energy Lorentz violating terms come as quantum corrections from heavy modes, where the energy scale Λ is related to the Planck energy scale E Pl (or length ℓ Pl ) [1,4]. Within the enormous class of models presenting Lorentz symmetry violation, the most interesting ones are those that preserve the basic elements of special relativity, because they are in agreement with well-known physics, but additionally these modified models present new and unexplored phenomena. A successful and phenomenologically rich scenario satisfying the above criteria is the Cohen and Glashow very special relativity (VSR) [11,12]. In the VSR framework, we have a modified gauge symmetry, admitting a variety of new gauge invariant interactions. Many interesting theoretical and phenomenological aspects of VSR effects have been extensively discussed [13][14][15][16][17][18][19].
The kinematics of the VSR framework, when it is defined in (3 + 1)-dims, have two subgroups satisfying the prior requirements, namely, the HOM(2) (with three parameters) and the SIM(2) (with four parameters). These symmetry groups SIM (2) and HOM (2) have the property of preserving the direction of a lightlike four-vector n µ by scaling, transforming as n → e ϕ n under boosts in the z direction. This feature implies that theories, which are invariant under one of these two subgroups, have a preferred direction in the Minkowski spacetime, where Lorentz violating terms can be constructed as ratios of contractions of the vector n µ with other kinematic vectors [11]. 1 Although many VSR features were discussed in the analysis of four-dimensional models, not much attention has been paid to the study of the influence of (nonlocal) VSR-effects into the behaviour of a lower-dimensional electromagnetic theory, for instance in a (2 + 1)-dimensional spacetime, where interesting phenomena are known within planar physics. In this case, we have in VSR the SIM(1) subgroup (of the SO(2, 1) Lorentz group) that preserves all the aforementioned conditions, in particular the existence of the invariant null-vector by a rescaling. 2 Since the proposal of the so-called topologically massive electrodynamics [21], also known as the Maxwell-Chern-Simons (MCS) electrodynamics describing a single massive gauge mode of helicity 1 The simplest example of VSR models is the dynamical part of scalar fields S = d ω x∂ µ φ∂ µ φ = d ω x φ − + m 2 φ, where the wiggle derivative operator is defined by∂ µ = ∂ µ + 1 2 m 2 n.∂ n µ . Observe that the Lorentz violation appears in a nonlocal form and the parameter m sets the scale for the VSR effects.
2 A detailed account of the SIM(1) subgroup can be found in Ref. [20].
±1, a great amount of attention has been paid into phenomenological application of this model. From the theoretical point of view, an interesting aspect of 3D field theories is the UV finiteness in some models. This feature, when applied to Lorentz violating models, provides an ambiguity free description of Lorentz violation, allowing a close contact of violating effects with planar phenomena. On the other hand, it is worth noting that massive modes of photons and their stability are a recurrent subject of analysis [22,23], and we can refer to some works about it on VSR [18,[24][25][26][27].
In this paper, we examine in details the modifications of the photon's dynamics within the SIM(1) VSR effective action framework. Moreover, this analysis also allows us to verify the validity of Furry's theorem explicitly at one-loop order. We start Sec. 2 by reviewing some aspects of the fermionic electrodynamics in the VSR context. There we explain the subtle points related to the nonlocal gauge couplings introduced by VSR effects, and also present the respective vertex Feynman rules. We also discuss charge-conjugation symmetry in the Lagrangian level, showing that the Cinvariance is preserved in the VSR setting. In Sec. 3 we compute the 2-point function AA at one-loop, corresponding to the dynamical part of the photon's effective action, and present the VSR modifications to the photon's polarization tensor. We analyze whether the VSR nonlocal couplings are sufficient to generate Chern-Simons-like self-couplings (in the Abelian VSR theory), and also check the Furry's theorem, by computing the 3-point function AAA in Sec.4. Finally, we summarize the results, and present our final remarks and prospects in Sec.5.

Gauge fields in VSR
In order to discuss the one-loop photon's effective action in VSR in the (2 + 1) spacetime, we start by considering Dirac fermions formulated in the VSR framework interacting with an external gauge field as below (2.1) The gauge and fermion fields are minimally coupled through the VSR covariant derivative [25] ∇ µ ψ = D µ ψ + 1 2 which is written in terms of ordinary covariant derivative D µ = ∂ µ − ieA µ , and the preferred null direction is chosen as n µ = (1, 0, 1). Observe that the expression (2.2) recovers the wiggle derivativẽ n.∂ n µ in the noninteracting limit when A µ → 0. This new operator obeys the known transformation law for a charged field δ (∇ µ ψ) = iΛ (∇ µ ψ) and also satisfies the required properties VSR gauge transformation δA µ =∂ µ Λ.
The fermionic propagator can readily be computed wherep µ = p µ − 1 2 m 2 n.p n µ , and we also have the fermionic dispersion relationp 2 = m 2 e , or equivalently as p 2 = µ 2 , where µ 2 = m 2 e + m 2 is the fermionic mass. On the vertex functions, we should analyze the expression (2.2) for the VSR covariant derivative. About the perturbative analysis, the presence of the term 1/ (n.D) in (2.2) shows that there are now an infinite number of nonlocal interactions (in the coupling e). The Feynman rules for these interactions can be obtained by means of use of Wilson lines, which express the respective terms in a convenient form with n = 1, 2, 3, ... legs of photon fields [13], making perturbative analysis workable. Since we are interested in computing vevs AA and AAA at one-loop order, we should consider the Feynman rules with one, two and three photon legs. Already the ψψA vertex signals minimal deviation from the usual QED, while the ψψAA and ψψAAA are new vertices resulting from VSR effects. The 1PI vertex Feynman rules of interest can readily be obtained [13,27], these are depicted in Fig. 1, and they are listed as below n n µ n ν (n.p)(n.q) n n µ n ν n ρ (n.p)(n.q) In these vertex functions, we have assumed that all the photon momenta are inward, implying the energy-momentum conservation law p − q + i k i = 0.
In regard to discrete symmetries in this Lorentz violating setting, one should recall that the VSR context does not permit the inclusion of parity (P) and time-reversal (T) symmetries, as well as composed symmetries CP and CT, since these are sufficient to restore the full Lorentz invariance [11]. On the other hand, we can speak about charge-conjugation symmetry (C) in regard of the nonlocal VSR couplings, because it does not depend on spacetime. Since one of the main points we wish to verify in this paper is the validity of Furry's theorem in the VSR setting, we shall first study the charge-conjugation symmetry at the Lagrangian level (classical level). With this purpose, we first consider the behavior of the free part of the Lagrangian (2.1) under C transformation which is given by Using the anticommuting property of the spinor components and the identity C −1 γ µ C = −(γ µ ) T , it is easy to see that the first and second terms are C-invariant, by a total derivative term in the action. About the last term, including the VSR effects, it transforms asψ Finally, with this result we can conclude that the free part of the action The remaining part of the analysis consists in check the behavior of the interaction part of the Lagrangian (2.1) under C. It is well known that the usual QED interaction term, i.e.ψγ µ A µ ψ, is explicitly C-invariant. However, the nonlocal interaction terms, arising from VSR effects, are generated by the perturbative expansion of the term Similarly to the study of the free part under C, we have that by considering A c µ = −A µ it is straightforward to show that the whole VSR couplings will be C-invariant under the action integral, added by total derivatives. Hence, we can conclude that C-invariance is preserved in the VSR setting at the classical level. We shall return to the C-invariance in Sec. 4 through the explicit verification of the Furry's theorem at one-loop order in the VSR context.

One-loop 2-point function A µ A ν
In order to compute the dynamical part of the photon's effective action we shall discuss the 2-point function A µ A ν at one-loop order. The respective contributions are depicted in Fig. 2. The first graph (a) corresponds to the usual photon polarization of QED, where the fermion propagator (2.3) and vertex (2.4) are modified by VSR nonlocal terms, resulting into There is a second VSR contribution at this order, graph (b), corresponding to the quartic VSR vertex (2.5), which gives Then, the full contribution to the A µ A ν part is given by Π µν = Π µν (a) + Π µν (b) . In the first part of the computation, we make use of the known algebra for the Dirac's gamma matrices in the two-component representation in the (2 + 1) spacetime, After simplifying the trace part of the polarization tensor, we obtain where we have defined by notation the nonlocal tensor In order to solve the momentum integrals, it is convenient to use the property wheren is a new null vector with the property (n.n) = 1, which explicit form must be provided in the VSR framework [17]. We discuss the conditions to determine the specific form ofn below. One main result for momentum integrals with (n · q) −1 involved is [17,30] where ∆ = m 2 + p 2 − 2(n.p)(n.p)t. Hence, making use of the identity (3.5) and inserting back the results Eqs.
We can explicitly see that the Mandelstam-Leibbrandt prescription preserves the VSR gauge invariance, since (3.8) satisfies the Ward identity p µ Π µν = p ν Π µν = 0. Now, in order to fully determine the integrals in (3.8), we must present an explicit form for the vectorn. Taking into account properties as reality, right scaling (n,n) → (λn, λ −1n ) and dimensionlessness [17], we find a SIM (1) , with a, b, c pure numbers. However, notice that this fails to be real for q 2 < 0. Thus, to preserve reality it is necessary to consider c = 0. Finally, in our prescriptionn µ = − p 2 2(n.p) 2 n µ + pµ n.p , thenn.p = p 2 2(n.p) . Hence, replacing this result back into (3.8), solving the integration over the variable t, we arrive at Π µν (p) = ie 2 2π p µ p ν − η µν p 2 I 1 + e 2 4π m e ε αµν p α I 2 + ie 2 m 2 4π n µ p ν + n ν p µ n.p − η µν − n µ n ν (n.p) 2 p 2 I 3 + e 2 m e m 2 4π ε ασν p α n σ n µ (n.p) 2 + ε αµσ p α n σ n ν (n.p) 2 + ε αµν n α n.p where we have considered ω → 3 + , since the expression (3.8) is UV finite in this limit, and also defined the integrals I i by simplicity in Eqs. In order to determine the VSR contributions to the Maxwell-Chern-Simons kinetic terms for the photon's effective action, we consider the low-momentum limit p 2 ≪ m 2 e of the expression (3.9). In that case, the integrals behave as (3.10) Finally, the polarization tensor in the low-momentum limit p 2 ≪ m 2 e can be written in the form −iΠ µν (p) We observe that in the absence of VSR framework i.e. m → 0, we obtain the ordinary Maxwell-Chern-Simons term and its higher-derivative correction, as expected. The low-energy VSR photon's effective action can be obtained by using the result (3.11) into (3.12) We can compare the tensor structure of the quantum effective action, arising from (3.11), to the classical action, which is a VSR and gauge invariant generalization of the ordinary Maxwell-Chern-Simons Lagrangian density Furthermore, we can cast the Lagrangian (3.13) as L 2+1 = 1 2 A λ O λα A α , where O λα in momentum space is written as O λα = p λ p α − η λα p 2 + im e ǫ µαλ p µ + m 2 n λ n α p 2 (n.p) 2 − n λ p α + n α p λ (n.p) + η αλ + im e m 2 2 ǫ ανλ n ν (n.p) − ǫ µνλ p µ n ν n α (n.p) 2 + ǫ µνα p µ n ν n λ (n.p) 2 . (3.14) We can easily observe that the quantum effective action (3.11) and the classical action (3.14) have the same tensor structure in the VSR framework, showing that both VSR invariance and gauge symmetry are preserved. However, there is a major difference between the results, the VSR effects in the quantum counterpart (3.11) all come as higher-derivative terms.
It is well known that infrared fluctuations can generate nonlocal terms in the quantum effective action. This generation of nonlocal terms can be traced back to the presence of massless particles in the fundamental theory, this is the case for example of QED and gravity [31]. On the other hand, higher derivative terms are also discussed as having a quantum nature [32]. Hence, from this point of view, we understand that in the VSR framework the nonlocal and higher-derivative are intermingled in the quantum effective action (3.11). It is important to emphasize that this statement is not valid on the (2 + 1) VSR electrodynamics only, in fact if we analyze the generation of VSR gauge terms in the (3 + 1) spacetime [17] and (1 + 1) spacetime [33] we observe that all the nonlocal effects are entangled to higher-derivative terms. Hence, this mixture of nonlocal effects and higher-derivative terms in the gauge sector of quantum effective action seems to be a common feature, and thus we can signal that we have the presence of UV/IR mixing in the VSR quantum effective action.

One-loop 3-point function A µ A ν A ρ
A strong result in standard QED is the C invariance known as the Furry's theorem, and that states that the total amplitude of the graphs containing a closed fermion loop with an odd number of external photon legs vanishes. Although the VSR electrodynamics (2.1) is also invariant under charge conjugation, we have a number of additional couplings in relation to QED, and we wish to analyze further these couplings. Hence, in this section, we shall explicitly verify whether Furry's theorem is valid at one-loop order in the presence of the nonlocal VSR couplings. The four contributing graphs are depicted in Fig. 3. Observe that the triangle graphs (a) and (b) have the same structure as ordinary QED, whereas graphs (c) and (d) are VSR effects, coming from the new vertices (2.5) and (2.6), respectively. An underlying aim from the analysis is to check if the Furry's theorem is violated in VSR, a topological Chern-Simons self coupling AAA can be dynamically generated.
In order to verify whether the total amplitude consisting of the diagrams in Fig. 3 vanishes, we start by discussing the triangle graphs (a) and (b). Following the set of Feynman rules, we have where, the external legs are denoted by the inward momenta (p µ 1 , p ν 2 , p ρ 3 ), and k = q − p 2 , s = q + p 3 . The second graph (b) is given by We can add these two contributions, and separate the matrix parts conveniently as where we have defined We can show that every term from these contributions cancel mutually by straightforward manipulations using Tr γ µ 1 γ µ 2 . . . γ µ n−1 γ µn = (−1) n Tr γ µn γ µ n−1 . . . γ µ 2 γ µ 1 , (4.5) that follows from charge conjugation invariance, C −1 γ µ C = −(γ µ ) T , and it is valid for any number of gamma matrices. Hence, after some manipulations, we can show that This result for VSR electrodynamics is the same as in the ordinary QED, and the nonlocal VSR couplings do not change the outcome for the triangle graphs. However, we have two additional graphs due to the VSR couplings for the one-loop vev AAA , which we must explicitly evaluate to verify the validity of Furry's theorem at this order. We shall show next that each one of these two contributions vanish independently.
We start from the graph (c), which consist of a diagram with the cubic and quartic vertex, which can be written as where we have defined the quantities a µνρ i in Appendix B, and introduced the integrals J i conveniently as n.q , (4.11) in terms of The last piece we shall discuss is the contribution from graph (d), which is written as After computing the trace of gamma matrices, we are left with . (4.18) In order to solve the momentum integrals, it is convenient to use the identity (3.5) to simplify the expression. In this process, we make use of the result that follows from (3.7). Thus, after some manipulations with (3.5) and simplifications, we finally arrive at the expression where we have introduced by simplicity the notation , i = 1, 2, 3. This final result shows that similarly to the ordinary QED, the VSR electrodynamics also satisfies Furry's theorem (at least in the one-loop order), and that no Chern-Simons-like self-coupling is dynamically generated. Although VSR changed the photon's dynamics in the Maxwell-Chern-Simons action, its Abelian structure and additional couplings are not sufficient to engender new self-couplings. Of course, if we increase the number of external photon legs to four, i.e. the vev AAAA , we could check the VSR contribution to the Euler-Heisenberg effective action in (2 + 1)-dim., similarly to the SIM(2) invariant analysis for a (3 + 1) spacetime [27].

Final remarks
In this paper we have discussed the photon's effective action in the context of VSR electrodynamics in the (2 + 1) spacetime, with special interest in analyzing the validity of Furry's theorem in the context of VSR. Initially, we revised the main aspects regarding the VSR gauge symmetry, and how this invariance introduced a new covariant derivative, which imply in an infinite series of nonlocal couplings among the fermionic and gauge fields. After deriving the respective Feynman rules for the new VSR couplings, we proceeded to the evaluation of the respective graphs contributing to the one-loop amplitudes AA and AAA , corresponding to the dynamical and Chern-Simons-like parts of the photon's effective action.
In the analysis of the two-point function AA , we have in addition to the usual polarization graph a second coming solely from the new VSR quartic coupling. When solving the momentum integration, we used the Mandelstan-Leibbrant prescription extended to the SIM(2) invariant case, where a new vectorn is introduced [17]. In order to present an explicit form for this vector, it is necessary to consider some properties: reality, symmetry, etc, which ultimately implied in a form preserving all of these features. Furthermore, this prescription preserves gauge symmetry, which is explicitly verified in terms of the Ward identity. Finally, we consider the low-momentum limit in order to determine the dynamical part of the effective action. There, we have obtained the usual Maxwell-Chern-Simons terms, added by VSR contributions both to the parity even and parity odd sectors. However, the VSR effects are different in the classical and quantum realm, since the nonlocal and higher-derivative terms are intermingled in the quantum effective action. This entanglement of nonlocal effects and higher-derivative terms, in the gauge sector, signal that we have the presence of UV/IR mixing in the VSR quantum effective action.
The last piece we have analyzed was the three-point function AAA , where we have the same two triangle graphs from ordinary QED, but due to the VSR couplings two new graphs are present in the evaluation of the complete one-loop amplitude. On one side, the two triangle graphs canceled mutually by using simply properties of the gamma matrices trace, which are based in the charge conjugation symmetry. Actually, this result for the triangle graphs is independent of the VSR coupling, i.e. the parameter m 2 , since Ξ µνρ (a+b) has the same matrix structure of the ordinary QED. In relation to the two additional graphs, present due to the VSR couplings, we explicitly showed by evaluation that these contributions vanish individually. Hence, based in the fact that the amplitude AAA vanished, we concluded Furry's theorem (at least in the one-loop order), and that no Chern-Simonslike self-coupling is dynamically generated. Although VSR changed the photon's dynamics in the Maxwell-Chern-Simons action, its Abelian structure and additional couplings are not sufficient to engender new self-couplings.
Since the induced one-loop effective action for the non-Abelian gauge fields interacting with Dirac fermions in d = 3 leads to the known Yang-Mills (even-parity) and non-Abelian Chern-Simons (oddparity) terms, it would be interesting to generalize our analysis to a non-Abelian VSR gauge theory [25]. This analysis would allow us to observe, in particular, how the parity odd terms change under VSR effects, possibly resulting in different type of self couplings [34]. Another point of interest, is the study of the general tensorial structure of the photon self-energy in different dimensions (d = 3, 4) at any order of perturbation and accordingly find the full tensorial form of the photon propagator. This study permits us to investigate the physical pole structure of the photon propagator and also to check whether the topological mass in the Chern-Simons theory (which is proportional to Hall conductivity) receives any corrections from VSR effects [35].

A Useful integrals
In order to cope with the momentum integral in VSR, we use the Mandelstan-Leibbrant prescription, and further useful results can be obtained from (3.7) by taking a derivative in relation to p µ d ω q q µ (q 2 + 2q.p − m 2 ) a+1 where ∆ = m 2 + p 2 − 2(n · p)(n · p)t, another useful result is found by by taking a derivative of (A.1) in relation to p ν and contracting with the metric, implying Furthermore, in the evaluation of vev AA in section 3 we have defined some integrals by simplicity of notation Afterwards we have considered the low-momentum limit of these expressions in (3.10) to determine the dynamical part of the photon's effective action.

B Tensor quantities
We present here some tensor quantities we have introduced in the evaluation of one-loop contribution for the vev AAA in section 4. In particular, we have the tensor quantities present in the graph (c) expression a µνρ 1 m 2 , p i = m 2 e 3 µ 2 n µ n ν n ρ (n.p 2 ) n. (p 1 + p 2 ) + im 2 m e e 3 ε µαβ p 1α n β n ν n ρ (n.p 2 ) n.