On perturbative aspects of a nonminimal Lorentz-violating QED with CPT-even dimension-5 terms

In this paper, we explicitly calculate the lower CPT-even one-loop quantum corrections in nonminimal Lorentz-violating spinor QED with all possible CPT-even dimension-5 operators. Within our calculations, we restrict ourselves to the cases when these parameters are completely expressed in terms of one constant vector.


I. INTRODUCTION
Possibilities of breaking the Lorentz symmetry are actively studied now.The paradigmatic framework for describing the Lorentz-violating (LV) effects within the quantum field theory is the minimal Lorentz-violating Standard Model Extension (LV SME) originally formulated in [1,2].A detailed description of its electromagnetic sector is presented in [3].Further, the interest in nonminimal LV extensions of QED, being motivated by studies of phenomenology of elementary particles where dealing with higher-dimensional operators plays an important role for a long time, beginning from the Fermi Lagrangian, emerged naturally.An important role was played by the paper [4], where the simplest examples of such operators were introduced in the LV context for the first time.The first studies of their perturbative impact have been performed in [5][6][7][8][9].Further, all possible LV extensions of SME with dimensions up to 6 for fermion-dependent operators were listed in [10].
Therefore, the natural question consists in studying loop corrections generated by all these nonminimal operators, beginning from dimension 5 and so on.The lower quantum corrections generated by all possible CPT-odd dimension-5 additive terms have been obtained explicitly in our previous paper [11], so the natural continuation of that study consists in obtaining the quantum corrections generated by all CPT-even ones with this dimension.
Within this study, we restrict ourselves by the first, leading order of the nonminimal operators, taking in mind the aim to generate the CPT-even term similar to that one obtained in [5,7,9], and calculate such a term for all dimension-5 CPT-even operators.For the sake of the simplicity, we consider the case when CPT-even constant LV tensors are completely described in terms of one LV vector, similarly to the scenario considered in [11] for CPT-odd dimension-5 operators, and in [5,7,9] for the aether term.
The structure of the paper looks as follows.In section 2, we define our model and calculate the effective action.We perform the one-loop calculations in section 3 to study the generation of the aether term.Finally, in section 4, we discuss our results.

II. NONMINIMALLY EXTENDED LORENTZ-VIOLATING QED
In this paper, we are interested in analyzing the following nonminimal LV extended QED Lagrangian with dimension-5 terms: where The expression (1) includes all dimension-5 LV couplings defined in [10].In fact, our aim will consist of studying the CPT-even contributions to the one-loop effective action of the gauge field, e.g., the aether term.It is clear that, in the first order, exclusively the terms involving even-rank constant tensors must be considered since only contractions of such tensors with the Minkowski metric and the Levi-Civita symbol, in four-dimensional spacetime, could yield a CPT-even contribution.Therefore, within this study, we deal with all possible CPT-even dimension-5 couplings.This allows us to reduce our Lagrangian to We can also rewrite the expression (3) as follows: where and we introduced the definition Then, the corresponding effective action, after the integration over the spinor fields, is given by (7) with, in the momentum space, Here, Tr stands for the trace over the Dirac matrices, as well as the trace over the integration in momentum and coordinate spaces.
We can expand Eq. ( 7) in power series in external fields as where with Since, for this step, we are interested in the induced CPT-even terms, we need to work only with terms of first order in ( 5) and ( 6) and second order in A µ .
First, we write down the contributions from quartic vertices given by Fig. 1c, i.e., for n = 1 in (9).Their contribution to the effective action is with where we have considered We note that since the vertex is already of the first order in a corresponding LV parameter, the propagator is not needed to be modified now, i.e., we can use It is immediate to see that the contributions proportional to m (5)αβ 5 and H (5)µναβ from (12) for this Feynman diagram identically vanish since corresponding traces are equal to zero. Then, FIG. 1: General form of first order Lorentz-breaking contributions.
Thus, the only nontrivial contribution is proportional to m (5)αβ , which can be evaluated by taking into account It yields where It is important to note that when ζ = 1, m (5)αβ becomes traceless.This is a significant point that requires further exploration.Then, the effective action can be written as

The remaining LV contributions, which can contribute to graphs with triple vertices in
with Except of the H (5)µναβ and H (5)µναβ F parameters, all other LV parameters are given by second-rank tensors.We note again that our aim consists in obtaining the CPT-even contribution to the effective action given by the term If we use the second-rank constant tensors (let us denote them as c µν ), we must have so, L even ∝ c µν F µλ F ν λ .It is clear that this term vanishes identically if c µν is antisymmetric.Thus, the terms proportional to m after the calculation of the integrals, and, for m after the calculation of the traces. where in which our fourth-rank tensor is chosen to be described in terms of the unique vector similar to (15), i.e., by the expression with In the case where ζ = 1, the coefficient H µν F is traceless, leading to the expected outcome of H Actually, the divergent part of (23), denoted as has already been calculated in a previous study performed in [12].This term also has been discussed in [13], where a slightly different coupling (with σ µν γ 5 matrix instead of σ µν one was used) has been considered.In the limit of k 2 m 2 ≪ 0, the finite part of the Lagrangian (23) takes the form For other dimension-5 operators, those one proportional to m (5)αβ , m , and H (5)µναβ , we must take into account all the expressions (18), i.e., the graphs (a), (b), (c), and (d) of

Fig. (2).
Then, it is easy to see that if we consider the m The graphs (18c) and (18d) yield contributions proportional to tr m)γ µ γ 5 γ ν , which also vanish.Thus, we have So, we rest with the insertions m (5)αβ and H (5)µναβ .First, we find the contribution with H (5)µναβ , which is given by with, as in (24), where we again have done the choice allowing us to describe our fourth-rank tensor by a single vector, namely, and, similarly to (15), we required which, as expected, we have a double traceless coefficient, i.e., H (5)κµ κµ = 0, when ζ = 1.It is easy to check that H (5)κµλν k κ k λ k µ = H (5)κµλν k κ k λ k ν = 0 which guarantees the transversality of the operator H (5)κµλν k κ k λ and hence the gauge symmetry of our result (30).The divergent part of Eq. ( 30) can be rewritten as Additionally, if we take the limit of k 2 m 2 ≪ 0, we can express the finite part as It is interesting to note that for the special relation of our LV parameters, namely, H µ = 1 2 H µ F , the one-loop corrections (23) and (30) mutually cancel.The similar effect occurs also in LV QED with CPT-odd dimension-5 parameters [11] and, in principle, can be treated as one more example of similarity of perturbative corrections generated by different LV parameters, analogous to the scenario discussed in [14] for the minimal LV QED.Now, it remains to consider m (5)αβ , which we assume to have the form given by (15).In this case, we have contributions generated both by Fig. 1c and Fig. 2.Then, we obtain where or, returning to the coordinate space, we can rewrite the divergent contribution as i.e. we have Maxwell-like and aether-like divergent terms.We note that if we introduce the traceless LV tensor m(5)µν = m µ m ν − 1 4 m 2 g µν and choose ζ = 1 in (38), the divergent result (37) will take the form i.e. in this case the trace part of the tensor m µ m ν is completely ruled out, and the result is completely described by its traceless part.
The finite part of the CPT-even contribution of the Lagrangian (36) is with It is worth noting that the above Lagrangian L m is invariant under gauge transformations, which requires that D = B + C.This identity is checked straightforwardly.In the IR limit of k 2 m 2 ≪ 0, we find After implementing a Fourier transform and removing the coefficient B, it is possible to simplify the finite contribution (40) into the form where A(−□), C(−□), and D(−□) are Fourier transforms for A(k 2 ), C(k 2 ), and D(k 2 ).
Rewriting this expression in terms of the traceless tensor m(5)µν = m µ m ν − 1 4 m 2 g µν , we find that, at ζ = 1, this expression takes the form i.e. in this case the finite part is completely described by the traceless LV tensor m(5)µν , and Lorentz symmetric terms are ruled out.
Taking into account (45), we see that in the IR limit we arrive at the finite part of our result given by the linear combination of four-derivative CPT-even terms m(5)αβ F µν ∂ α ∂ β F µν and m(5)µλ F µν □F λ ν .Explicitly, the one-loop contribution to the effective Lagrangian takes the form We see that, for this LV additive term, terms with only two derivatives are canceled out in the finite part (actually, since µ dependence is related to the divergent part, we can conclude that absence of the second-derivative terms in the finite part is a consequence of our renormalization prescription).Thus, the first nontrivial contributions are the four-derivative ones.

III. SUMMARY
We calculated the CPT-even term for the nonminimal LV extended QED with all possible CPT-even dimension-5 operators, with the corresponding LV constant tensors are chosen to be expressed in terms of one LV vector.It turns out to be that in this case, unlike the schemes given in [5,9], for all cases where a CPT-even contribution differs from zero, it is divergent.
Nevertheless, it is a rather reasonable result explained by the non-renormalizability of our theory.Also, we found the finite part of the CPT-even term which involves fourth derivatives.
Actually this is the second example of calculating four-derivative LV terms while the first one was performed in [12,13] where only one of the operators treated in this paper, that one proportional to H (5)µναβ F , has been considered, however, in this paper we calculate fourderivative terms from a contribution of first order in the LV parameter only, where in [12,13] the second order was studied.In a certain sense our result is more advantageous since corrections of the second order in LV parameters are naturally expected to be suppressed in comparsion with the first-order ones, due to the well-known smallness of LV vectors and tensors, cf.[15].
We conclude that we succeeded to complete the study of lower-order contributions generated by dimension-5 LV operators that was started in [11].Possible continuations of this study could consist, first, in development of some its applications in phenomenologically interesting models, second, in studying of perturbative impacts of dimension-6 operators.
We plan to perform these studies in our next papers.
) and (18b) will not yield a nontrivial contribution to the desired term.In fact, for m (5)αβ F ,

Now, we need
to calculate the remaining insertions of Eq. (18).Initially, let us analyze the coefficient H (5)µναβ F , whose contributions are present in both Eqs.(18a) and (18b).The results are