Renormalization in a Lorentz-violating model and higher-order operators

# Renormalization in a Lorentz-violating model and higher-order operators

J. R. Nascimento Departamento de Física, Universidade Federal da Paraíba
Caixa Postal 5008, 58051-970, João Pessoa, Paraíba, Brazil
A. Yu. Petrov Departamento de Física, Universidade Federal da Paraíba
Caixa Postal 5008, 58051-970, João Pessoa, Paraíba, Brazil
Carlos M. Reyes Departamento de Ciencias Básicas, Universidad del Bío Bío,
Casilla 447, Chillán, Chile
###### Abstract

The renormalization in a Lorentz-breaking scalar-spinor higher-derivative model involving self-interaction and the Yukawa-like coupling is studied. We explicitly demonstrate that the convergence is improved in comparison with the usual scalar-spinor model, so, the theory is super-renormalizable, and there is no divergences beyond two loops. We compute the one-loop corrections to the propagators for the scalar and fermionic fields and show that in the presence of higher-order Lorentz invariance violation, the poles that dominate the physical theory, are driven away from the standard on-shell pole mass due to radiatively induced lower dimensional operators. The new operators change the standard gamma-matrix structure of the two-point functions, introduce large Lorentz-breaking corrections and lead to modifications in the renormalization conditions of the theory. We found the physical pole mass in each sector of our model.

###### pacs:
11.55.Bq, 11.30.Cp, 04.60.Bc,11.10.Gh

## I Introduction

It is well known that the Lorentz-breaking field theory models can be introduced in several ways. We can list some of the most popular approaches. First, one can introduce small Lorentz-breaking modifications of the known theories through additive terms, thus implementing the Lorentz-breaking extensions of the standard model ColKost (). In principle, the most known extensions of the QED follow this way. A very extensive list of the possible Lorentz-breaking additive terms in different field theory models including QED is given by KosGra (). Second, one can start with the modified dispersion relations Amelino (), and, in principle, try to find a theory yielding such relations. Third, the Lorentz-breaking theories can be treated as a low-energy limit of some fundamental theories, for example string theory KostSam () and loop quantum gravity LQG (). Finally, the Lorentz symmetry can be broken spontaneously, see f.e. spont (). The main motivation behind all these approaches, however, is the same, and resides in the expectation that any experimental evidence of departure from Lorentz symmetry may provide the first germs towards the construction of a theory amalgamating both General Relativity and the Standard Model of particle physics.

At the same time, it is natural to consider one more aspect of studying the Lorentz-breaking extensions of the field theory models. It consists in introducing the essentially Lorentz-breaking terms, that is, those ones proportional to some constant vectors or tensors, involving higher derivatives. As a result, the corresponding theory will yield an essentially different quantum dynamics. The first known example of such a theory is the Myers-Pospelov extension of the electrodynamics MP () where the three-derivative term essentially involves the Lorentz symmetry breaking. Another important example of such a theory is a four-dimensional Chern-Simons modified gravity with the Chern-Simons coefficient chosen in special form  JaPi (), which, in the weak field limit, also involves third order in derivatives of the dynamical field (that is, the metric fluctuation). Moreover, the importance of the Myers-Pospelov-like term, and analogous terms for scalar and spinor fields which can be easily introduced, is also motivated by the fact that a special choice of the Lorentz-breaking vector will allow to eliminate the presence of higher time derivatives thus avoiding the arising of the ghosts which are typically present in theories with higher time derivatives (see f.e. ghosts ()). Also, this term was shown to arise as a quantum correction in different Lorentz-breaking extensions of QED MNP () and has been studied for causality and stability CMR0 (). In the case of including higher time derivatives, it has been shown recently that the unitarity of the -matrix can be preserved at the one-loop order in a Myers-Pospelov QED CMR (). The proof has been accomplished using the Lee-Wick prescription for quantum field theories with negative metric Lee-Wick (). For other studies on unitarity at tree level for minimal and nonminimal Lorentz violations, see Schreck1 (); Schreck2 () respectively. It is important to notice that the Myers-Pospelov-like modifications of QED are actually experimentally studied as well within different contexts MPexp ().

We emphasize that, up to now, the quantum impact of the Myers-Pospelov-like class of terms being introduced already at the classical level, where the higher-derivative additive term should carry a small parameter which can enforce large quantum corrections collins (), almost was not studied except of the QED Fine-Tuning () and superfield case CMP (). The presence of such effect raises the question how to define correctly the physical parameters in the renormalized theory. On the other hand, for studies in the context of semiclassical quantization it is natural to consider the presence of higher derivative terms in order to implement a consistent renormalization program shapiro (). With these considerations, the natural question is – what are the possible consequences of including the Lorentz-breaking higher-derivative terms into the classical action?

It is well known that loop corrections in Lorentz-invariance violating quantum field theory may lead to new kinetic operators absent in the original Lagrangian. Recently, the consequences of these radiatively induced operators have been studied in relation with the finiteness of the -matrix and the identification of the asymptotic state space ralf (). These new terms introduce modifications in the propagation of free particles and change drastically the physical content of the space of in and out states. In particular, the Kallen-Lehmann representation KL () and the LSZ reduction formalism LSZ () are modified in the presence of Lorentz symmetry violation rob (). An important finding is that spectral densities which in the standard case are functions of momentum-dependent observer scalars such as , in the Lorentz violating scenario may depend on other scalars such as couplings of Lorentz-violating tensor coefficients with momenta rob (). This has led to modifications in the renormalization procedure, in the definition of the asymptotic Hilbert space and in general in the treatment for external-leg physics ralf (); for other studies of the renormalization in Lorentz-breaking theories, see also scarp (). A natural extension for these studies is to consider the nonminimal framework of Lorentz invariance violation, that is, when the Lorentz-breaking is performed with higher-order operators Kos-Mew (). It is well known that the inclusion of higher-order operators in quantum field theory will generate, via radiative corrections, all the lower dimensional operators allowed by the symmetries of the Lagrangian. For the case of breaking the Lorentz symmetry, let us say in QED and with a preferred four-vector , the induced operators may involve contractions of with matrices other than just , together with scalars such as . The new terms force to modify the renormalization conditions in order to extract the correct pole mass from the two-point functions. In particular, the renormalization condition for the renormalized fermion self-energy , with being the physical pole mass, has to be generalized, which ultimately will depend on the form of Lorentz breaking. In this work we continue these studies in order to carry out the renormalization in a theory with higher-order operators and in addition we study the possible effects of large Lorentz-violating corrections. Within our study, we consider the renormalization of the higher-derivative Lorentz-breaking generalizations of and Yukawa model.

The structure of the paper looks like follows. In Sec. II, we consider the classical actions of our models, write down the dispersion relations, find the poles and describe their analytical behavior in complex -plane. In Sec. III, we compute the quantum corrections corresponding to the self interaction . In Sec. IV we discuss the coupling of scalar and spinor fields and provide a study of the degree of divergencies in our model. In Sec. V, we compute the two-point functions in purely scalar and scalar-spinor sectors thus exhausting possible divergences and showing explicitly the radiatively induced operators with new gamma-matrix structure and large Lorentz-violating terms. In Sec. VI, we perform the mass renormalization in both sectors and find the physical masses in the theory. In the last section, we discuss our results, and in the Appendices AB, we provide some details of the calculations.

## Ii The effective models and pole structure

We are interested in the higher-order Lagrangian density describing two sectors of Lorentz-breaking theory:

 L=L1+L2. (1)

The first sector involves a scalar sector with a fourth derivative together with a self-interaction potential term

 L1=12∂μϕ∂μϕ−12M2ϕ2+g1ϕ(n⋅∂)4ϕ−λ4!ϕ4, (2)

and the second one the fermionic Myers-Pospelov model MP (), with dimension-five operators and the Yukawa coupling vertex

 L2=¯ψ(i∂/−m)ψ+g2¯ψn/(n⋅∂)2ψ+g¯ψϕψ. (3)

The constants and parametrize the higher-order Lorentz invariance violation with the Planck mass representing itself as a natural mass scale, , are dimensionless parameters, whose presence describes the intensity of the higher-derivative terms, and is a dimensionless four-vector defining a preferred reference frame.

The propagators in momentum space read

 Δ(p) = ip2−M2+2g1(n⋅p)4, S(p) = ip/−m−g2n/(n⋅p)2. (4)

We begin an analysis of the dispersion relations in both sectors. A further motivation for its study, and consequently, the finding of the poles and their analytical behavior in complex -plane, consists first in the fact that in our models namely using of the residues of the propagators is a most convenient approach for calculating the quantum corrections. Second, in the presence of higher-order time-derivative terms a direct implementation of the prescription may lead to a wrong four-momentum representation for the propagator which may spoil any attempt to preserve unitarity or causality.

 p2−M2+2g1(n⋅p)4=0, (5)

which for a purely time-like four-vector , the solutions are given by

 p0=±12√−1±√1+8g1E2g1, (6)

and where . The dispersion relation can also be written as , hence one has the solutions and so that

 p1=12√−1+√1+8g1E2g1,P2=12√1+√1+8g1E2g1. (7)

Their exact location in complex -plane and also the contour of integration are shown in Fig. 1.

The solutions can be classified according to their perturbative behavior when taking the Lorentz violation to zero. We identify two standard solutions which are perturbative solutions to the usual ones and two complex ones (and moreover, actually tachyonic) which diverge as . The extra solutions that appear are associated to negative-metric states in Hilbert space and have been called Lee-Wick solutions Lee-Wick ().

Alternatively, we can write the scalar propagator as

 Δ(p) = i2g1(p20+P22)(p20−p21+iϵ), (8)

which agrees with the usual propagator in the limit .

In the fermion sector we have the dispersion relation

 (pμ−g2nμ(n⋅p)2)2−m2=0. (9)

Again for the time-like we have the equation

 (p0−g2p20)2−→p2−m2=0, (10)

whose standard, that is, non-singular at , solutions are

 ω1=1−√1−4g2E2g2,ω2=1−√1+4g2E2g2, (11)

and the Lee-Wick ones

 W1=1+√1−4g2E2g2,W2=1+√1+4g2E2g2, (12)

where .

In the region of energies satisfying the condition the four solutions are real and obey the inequality , where is a negative number. However, beyond the critical energy both and become complex and move in the opposite imaginary line at as shown in Fig. 2, while the other two solutions remain real.

To define the contour we use an heuristic argument, specially to go beyond the critical energy at which complex solutions appear. We implement a correct low energy limit by considering the prescription given in QEDunitarity () which has been well tested to give a suitable correspondence with the normal theory when and also to preserve the unitarity of the matrix. In this effective region the integration contour is defined to round the negative pole from below and the three positive ones from above. Now we increase the energy to values at which the two solutions and become complex, and define the new contour as the one obtained by continuously deforming the curve by avoiding any crossing and singularity with the poles, as shown in Fig. 2.

With this consideration in mind, the fermion propagator reads

 S(p) = i((p0−g2p20)γ0+piγi+m)g22(p0−ω1+iϵ)(p0−W1+iϵ)(p0−ω2−iϵ)(p0−W2+iϵ), (13)

which differs from the direct prescription in the quadratic terms, but allows in particular to define a consistent Wick rotation which we use later.

## Iii The interaction λϕ4

In this section we explore the potentially divergent one-loop radiative correction in the scalar propagator which is generated by the well-known tadpole graph given by Fig. 3.

To proceed with it, we need to evaluate the basic integral

 Σ2=12λ∫d4p(2π)41p2−M2+2g1(n⋅p)4. (14)

Note that a naive power counting gives a logarithmic divergence for the integral which, however, as shown below using dimensional regularization is found to have a finite result in four dimensions; in a similar fashion of what happens with the Riemann zeta function for negative values of .

We go to dimensions and choose the Lorentz-breaking four-vector to be timelike which yields

 Σ2 = 12μ4−dλ∫ddp(2π)d12g1(p20−p21+iϵ)(p20+P22), (15)

where are given in (7). We perform the integration in the complex -plane by closing the contour upward and enclosing the two poles and as depicted in Fig. 1, yielding

 Σ2=πiμ4−dλ∫dd−1p(2π)d(iF1−F2), (16)

where

 F1=√g1√1+8g1E2√1+√1+8g1E2,F2=√g1√1+8g1E2√−1+√1+8g1E2. (17)

Note that has the correct limit at , recovering the usual result . Now, it is convenient to change variables yielding

 pdp=zdz8g1dd−1p=|p|d−2dpdΩd−1, (18)

which allows to write the integral (16) as

 Σ2=−πμ4−dλ2π(d−1)/2√g1(2π)dΓ(d−12)(8g1)d−12(I1+I2). (19)

with

 I1=∫∞z0dz(z2−z20)d−32√z+1,I2=i∫∞z0dz(z2−z20)d−32√z−1, (20)

where and we have used the definition of solid angle (115).

Considering both contributions through the relation , and after some algebra with and expanding in , we find at lowest order

 Σ(1) = M2λ12π3⎛⎜ ⎜⎝−193+2γE+6ln(2)−3π2F1R(0,0,1,0)(14,34,2,1)8√2+(1−38γE−18ln(512))ln(−g1M28)⎞⎟ ⎟⎠, Σ(2) = λ144g1π3(−14+6γE+3ln(32g1M2))+M2λ192π3(8(−17+6γE+3ln(32g1M2)) (21) −2(3+ln(g1M28))(−14+6γE+3ln(32g1M2))).

Here, is a hypergeometric function, whose exact value is . Note that there is a fine tuning in this case, that is, the expression is singular at . However the correction to the two-point function is UV finite.

## Iv Coupling of scalar and spinor fields

Let us consider the theory involving both the quartic interaction vertex and the Yukawa coupling vertex . We note, that, in principle, the second time derivatives in a free action of a spinor field are present also in specific Lorentz invariant theories, for example, the known ELKO model ELKO (). However, our theory essentially differs from that model. To classify the possible divergences, we should calculate the superficial degree of divergence of this theory. The naive result for it is

 ω=4−4V1−2V2−Eψ, (22)

where is a number of spinor legs. However, this manner yields incorrect results because of the strong anisotropy between time and space components of the momenta (for example, in this case one can naively suggest that the two-point function of the spinor field can yield only the renormalization of the mass of the spinor field). So, let us proceed in the manner similar to that one used for Horava-Lifshitz-like theories (cf. Anselmi ()). Since is purely time-like, we can write , so, we have from  (II)

 Δ(p) = ip20−→p2−M2+2g1p40, S(p) = ip/−m−g2γ0p20. (23)

Following the methodology developed for the Horava-Lifshitz theories (see f.e. Anselmi ()), we suggest that the denominators of the propagators are the homogeneous functions with respect to higher orders in corresponding momenta, and the canonical dimension of the spatial momentum is 1. Taking into account only the leading degrees, we easily conclude that the canonical dimension of the momentum is (we note that this case does not occur in usual Horava-Lifshitz-like theories where the canonical dimension of time momenta are always more than one, cf. Anselmi ()). Therefore, the spinor propagator has the canonical dimension (and the contribution to the superficial degree of divergence) equal to , and the scalar one – equal to just as in the usual case. Nevertheless, the dimension of the integral measure, that is, in this case is different from the usual one, being equal to rather than . Hence the superficial degree in our theory is

 ω=(7/2)L−2Pϕ−Pψ, (24)

where is a number of loops, and and are the numbers of scalar and spinor propagators respectively. Then, let will be the number of vertices, and – of Yukawa-like vertices. One has the identities for numbers of scalar and spinor fields in an arbitrary Feynman diagram:

 Nϕ = 4V1+V2=2Pϕ+Eϕ, Nψ = 2V2=2Pψ+Eψ, (25)

where , are the numbers of external scalar and spinor legs respectively. We use the topological identity , that is, . As a result, we eliminate numbers of loops and propagators from and rest with

 ω=72−12V1−14V2−34Eϕ−54Eψ. (26)

A straightforward verification shows that the superficially divergent diagrams (that is, those ones with can be of the following types:

(i): , , , . This is the one-loop renormalization of the mass and kinetic terms for the spinor.

(ii): , , , . This is the two-loop renormalization of the mass and kinetic terms for the spinor.

(iii): , , , . This is the one-loop renormalization of the mass and kinetic terms for the scalar.

(iv): , , , . This is the two-loop renormalization of the mass and kinetic terms for the scalar.

(v) , , , . This is the one-loop renormalization of the mass term for the scalar. Actually, we already showed in the previous section that, due to the specific structure of poles of the propagator, this contribution is finite.

Actually, in the cases (iii) and (iv) the divergence will be not linear but logarithmic, by the reasons of symmetry of integrals over momenta. The diagrams with odd numbers of will vanish due to an analogue of the Furry theorem. So, our theory is super-renormalizable. Moreover, we note that since the kinetic term for the scalar involves two derivatives acting to the external fields, its superficial degree of freedom should be decreased at least by 1, if these derivatives are the time ones, and by two for space derivatives; actually, in one-loop case in a purely scalar sector the kinetic term simply does not arise. Also, in the cases (i) and (ii) one will have the only divergent contribution to the mass of the spinor. So, taking into account the previous section as well, we conclude that at the one-loop order one could have only the renormalization of the masses of the spinor and the scalar arisen from the Yukawa-like coupling.

We note that namely this degree of divergence correctly explains why the self-energy of the fermion diverges, as we will see further (indeed, the naive calculation yields a finite result for it). To study the renormalization, we can restrict ourselves by the lower order, that is, one loop.

So, we rest with only three potentially divergent graphs – with , that is, the purely scalar tadpole we studied above, and with and or we study below.

## V The Yukawa-like theory

In the next subsections we compute the radiative corrections to the scalar and fermion two-point function in the Yukawa-like theory which arises by considering the self-interaction term and in (1). The Lagrangian is

 L=12∂μϕ∂μϕ−12M2ϕ2+¯ψ(i∂/−m)ψ+g2¯ψn/(n⋅∂)2ψ+g¯ψϕψ, (27)

and additionally, we impose the simplification of considering and the preferred four-vector to be purely timelike .

### v.1 Scalar self-energy Π(p)

As a first example of quantum corrections in our Yukawa-like model, we study the contribution with two external scalar legs depicted at Fig. 3.

It is represented by the integral

 Π(p)=−g22ϕ(−p)ϕ(p)∫d4k(2π)4Tr((Qμγμ+m)(Rνγν+m))(Q2−m2)(R2−m2), (28)

where we define

 Qμ = kμ−g2nμ(n⋅k)2, Rμ = kμ+pμ−g2nμ(n⋅(k+p))2. (29)

Calculating the trace gives

 Π(p)=−2g2ϕ(−p)ϕ(p)∫d4k(2π)4Q⋅R+m2(Q2−m2)(R2−m2). (30)

Let us write the corresponding contribution to the effective action as and study the typical low-energy behavior of this contribution by expanding it into Taylor series

 ˜Π(p) = ˜Π(0)+pμ(∂˜Π∂pμ)p=012pμpν(∂2˜Π∂pμ∂pν)p=0+…. (31)

The zeroth-order contribution follows directly from (30)

 ˜Π(0)=∫d4k(2π)4Q2+m2(Q2−m2)2. (32)

It is convenient to rewrite as

 ˜Π(0)=K+2m2P, (33)

where

 K=∫d4k(2π)41(Q2−m2), (34)

and

 P=∫d4k(2π)41(Q2−m2)2. (35)

where the integrals (34) and (35) have been solved in the Appendix A.

For the next term, it is clear that can be proportional to only since there is no other vectors, the corresponding contribution to the effective action will yield , that is, a surface term. So, we can disregard it. Further, one would need then to find the second derivative, that is, which may contain naturally terms of higher-order in . To find it, consider

 (∂2˜Π(p)∂pμ∂pν)p=0 = ∫d4k(2π)4Tr[(1⧸Q−m)[∂∂pμ∂∂pν(1⧸R−m)]p=0]. (36)

Integrating by parts and neglecting the surface terms, we obtain the symmetric expression

 (∂2˜Π(p)∂pμ∂pν)p=0 = −∫d4k(2π)4Tr[∂∂kν(1⧸Q−m)∂∂kμ(1⧸Q−m)], (37)

where we have used the identity .

We consider

 ∂∂kμ(1⧸Q−m)=(∂Qα∂kμ)1(⧸Q−m)2γα, (38)

and after some algebra we arrive at

 (∂2˜Π(p)∂pμ∂pν)p=0=−∫d4k(2π)4(∂Qα∂kμ)(∂Qσ∂kν)Tασ, (39)

with

 Tασ=4(Q2−m2)2ηασ+32m2(Q2−m2)4QαQσ. (40)

By using the relations

 (∂Qα∂kμ)(∂Qα∂kν)=ημν−4g2nμnν(n⋅Q), (∂Qα∂kμ)(∂Qσ∂kν)QαQσ=14(∂Q2∂kμ)(∂Q2∂kν), (41)

one obtains

 (∂2˜Π(p)∂pμ∂pν)p=0=−4∫d4k(2π)4(ημν−4g2nμnν(n⋅Q)(Q2−m2)2+2m2(Q2−m2)4(∂Q2∂kμ)(∂Q2∂kν)). (42)

Considering the tensors available in our model, which are the flat metric and the preferred four-vector we can write

 ∫d4k(2π)4Qμ(Q2−m2)2=nμS, (43)

and

 ∫d4k(2π)41(Q2−m2)4(∂Q2∂kμ)(∂Q2∂kν)=nμnνL+ημνn2M. (44)

Now, consider the relation

 ∂Q2∂kμ=2(Qμ−2g2nμ(n⋅Q)(n⋅k)), (45)

and multiplying by we to arrive at

 ∫d4k(2π)44(n⋅Q)2(Q2−m2)4(1−4g2n2(n⋅k)+4g22(n2)2(n⋅k)2)=(n2)2(L+M), (46)

and by contracting with the metric

 ∫d4k(2π)44(Q2−m2)4(Q2−4g2(n⋅Q)2(n⋅k)+4g22n2(n⋅Q)2(n⋅k)2)=n2(L+4M). (47)

Solving the algebraic equation we have

 L = 163n2∫d4k(2π)41(Q2−m2)4((n⋅Q)2n2−Q24−3g2(n⋅Q)2(n⋅k)+3g22(n⋅Q)2(n⋅k)2n2), M = 43n2∫d4k(2π)41(Q2−m2)4(Q2−(n⋅Q)2n2). (48)

A similar analysis gives

 S=1n2∫d4k(2π)4(n⋅Q)(Q2−m2)2. (49)

We find the second-order contribution

 (∂2˜Π(p)∂pμ∂pν)p=0pμpν=−(4P+8m2n2M)p2+(16g2n2S−8m2L)(n⋅p)2. (50)

Reorganizing this expression, we can write the correction to the scalar propagator up to second-order in as

 ˜Π(p) = m2q0+p2q1+(n⋅p)2qn, (51)

where

 q0 = K+2m2P, (52) q1 = −4(P+2m2n2M), qn = 8(2g2n2S−m2L).

Finally one has

 q0 = −i48π2g22m2+i48π2(6γE−0,46+12iπ−18ln(g2m2)), (53) q1 = −i2π2(iπ−ln(g2m2)−13), (54) qn = iπ2. (55)

We provide details of the computation of , and in the Appendix A. The two-point function is finite and involves a fine-tuning term proportional to . The Lee-Wick modes improve the convergence of the theory such to make the two-point function of the scalar field essentially UV finite and involves the aether term ouraether ().

### v.2 Fermion self-energy Σ(p)

Now we focus on the contribution of the fermion self-energy graph depicted in Fig. 4, and recall we are considering .

The fermion self-energy graph is represented by the integral

 Σ(p)=g2∫d4k(2π)4⧸Q+m((k−p)2−m2)(Q2−m2). (56)

To find it, let us consider a Taylor expansion of the first denominator term up to second-order in and rewrite the diagram as

 Σ(p) ≈ g2∫d4k(2π)4(1k2−m2+2(k⋅p)−p2(k2−m2)2+4(k⋅p)2(k2−m2)3)(⧸Q+mQ2−m2)+O(p3,n). (57)

With the notation , we introduce the the zeroth-order contribution

 I(0)=∫d4k(2π)4(1k2−m2)(⧸Q+mQ2−m2), (58)

the linear-order contribution

 I(1)=∫d4k(2π)4(2(k⋅p)(k2−m2)2)(⧸Q+mQ2−m2), (59)

and the second-order contribution

 I(2) = ∫d4k(2π)4(4(k⋅p)2(k2−m2)3−p2(k2−m2)2)(⧸Q+mQ2−m2). (60)

### v.3 The gamma-matrix structure of I(0), I(1)I(2)

#### v.3.1 Zeroth-order I(0)

 I(0) = γμI(0)μ+mf0, (61)

where we have defined

 I(0)μ = ∫d4k(2π)4Qμ(k2−m2)(Q2−m2), (62)

and

 f0 = ∫d4k(2π)41(k2−m2)(Q2−m2). (63)

From tensor analysis considerations one should have

 ∫d4k(2π)4Qμ(k2−m2)(Q2−m2)=nμfn1. (64)

Replacing the expression (64) in Eq. (61) produces the zeroth-order contribution

 I(0)=⧸nfn1+mf0, (65)

with

 fn1 = 1n2∫d4k(2π)4(n⋅Q)(k2−m2)(Q2−m2). (66)

We carry out the calculations of and following the lines given in Appendix B.2. The first coefficient is naturally finite and the second one is divergent and contains a large Lorentz-breaking correction term of the order of .

#### v.3.2 Linear-order I(1)

The linear-order integral (59) can be rewritten by introducing

 I(1)=2pμγνI(1)μν+2mpμI(1)μ, (67)

where

 I(1)μ = ∫d4k(2π)4kμ(k2−m2)2(Q2−m2), (68) I(1)μν = ∫d4k(2π)4kμQν(k2−m2)2(Q2−m2).

By considering

 ∫d4k(2π)4kμ(k2−m2)2(Q2−m2)