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

Renormalization in a Lorentz-violating model and higher-order operators J. R. Nascimento,1 A. Yu. Petrov,1 and Carlos M. Reyes2

arXiv:1706.01466v1 [hep-th] 5 Jun 2017

1 Departamento de F´ısica, Universidade Federal da Para´ıba Caixa Postal 5008, 58051-970, Jo˜ ao Pessoa, Para´ıba, Brazil∗ 2 Departamento de Ciencias B´ asicas, Universidad del B´ıo B´ıo, Casilla 447, Chill´ an, Chile†

The renormalization in a Lorentz-breaking scalar-spinor higher-derivative model involving φ4 selfinteraction 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 Lorentzbreaking 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 numbers: 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 [1]. 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 [2]. Second, one can start with the modified dispersion relations [3], 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 [4] and loop quantum gravity [5]. Finally, the Lorentz symmetry can be broken spontaneously, see f.e. [6]. 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 [7] 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 Θ(x) = kµ xµ [8], 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. [9]). Also, this term was shown to arise as a quantum correction in different Lorentz-breaking extensions of QED [10] and has been studied for causality and stability [11]. In the case of including higher time derivatives, it has been shown recently that the unitarity of the S-matrix can be preserved at the one-loop order in a Myers-Pospelov QED [12]. The proof has been accomplished using the Lee-Wick prescription for quantum field theories with negative metric [13]. For other studies on unitarity at tree level for minimal and nonminimal Lorentz violations, see [14, 15] respectively. It is important to notice that the Myers-Pospelov-like modifications of QED are actually experimentally studied as well within different contexts [16].

∗ Electronic † Electronic

address: jroberto, [email protected] address: [email protected]

2 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 [17], almost was not studied except of the QED [18] and superfield case [19]. 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 [20]. 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 S-matrix and the identification of the asymptotic state space [21]. 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 [22] and the LSZ reduction formalism [23] are modified in the presence of Lorentz symmetry violation [24]. An important finding is that spectral densities which in the standard case are functions of momentum-dependent observer scalars such as p2 , in the Lorentz violating scenario may depend on other scalars such as couplings of Lorentz-violating tensor coefficients with momenta [24]. 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 [21]; for other studies of the renormalization in Lorentz-breaking theories, see also [25]. 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 [26]. 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 nµ , the induced operators may involve contractions of nµ with matrices other than just γ µ , together with scalars such as (n · p). 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 ΣR (p / = mP ) = 0, with mP 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 λφ4 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 p0 -plane. In Sec. III, we compute the quantum corrections corresponding to the self interaction λφ4 . 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 A, B, 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 =

1 1 λ ∂µ φ∂ µ φ − M 2 φ2 + g1 φ(n · ∂)4 φ − φ4 , 2 2 4!

(2)

and the second one the fermionic Myers-Pospelov model [7], with dimension-five operators and the Yukawa coupling vertex ¯ /(n · ∂)2 ψ + g ψφψ ¯ . L2 = ψ¯ (i∂/ − m) ψ + g2 ψn

(3)

The constants g1 = Mκ2 and g2 = MηP l parametrize the higher-order Lorentz invariance violation with MP the Planck Pl mass representing itself as a natural mass scale, κ, η are dimensionless parameters, whose presence describes the intensity of the higher-derivative terms, and nµ is a dimensionless four-vector defining a preferred reference frame.

3

(s)

FIG. 1: The path of integration CF for the scalar propagator which encloses the poles −p1 and iP2 when closing the contour upward and encloses the poles p1 and −iP2 when closing the contour downward.

The propagators in momentum space read i , p2 − M 2 + 2g1 (n · p)4 i S(p) = . p/ − m − g2 n/(n · p)2

∆(p) =

(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 p0 -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 i prescription may lead to a wrong four-momentum representation for the propagator which may spoil any attempt to preserve unitarity or causality. Let us start with the scalar dispersion relation, namely p2 − M 2 + 2g1 (n · p)4 = 0 ,

(5)

which for a purely time-like four-vector nµ = (1, 0, 0, 0), the solutions are given by s p 1 −1 ± 1 + 8g1 E 2 p0 = ± , (6) 2 g1 p and where E(~ p) = p~2 + M 2 . The dispersion relation can also be written as (p20 − p21 )(p20 + P22 ) = 0, hence one has the solutions p0 = ±p1 and p0 = ±iP2 so that s s p p 1 −1 + 1 + 8g1 E 2 1 1 + 1 + 8g1 E 2 p1 = , P2 = . (7) 2 g1 2 g1 (s)

Their exact location in complex p0 -plane and also the contour of integration CF 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 ±p1 which are perturbative solutions to the usual ones ±E and two complex ones (and moreover, actually tachyonic) ±iP2 which diverge as g1 → 0. The extra solutions that appear ±P2 are associated to negative-metric states in Hilbert space and have been called Lee-Wick solutions [13]. Alternatively, we can write the scalar propagator as ∆(p) =

i , 2g1 (p20 + P22 )(p20 − p21 + i)

(8)

4

(f )

FIG. 2: The integration contour CF for the fermion propagator which is defined to round the negative pole from below and the positive poles from above. At higher energies than 1/(4g2 ), the two solutions ω1 and W1 become complex and move in opposite directions along the imaginary line starting at 1/(2g2 ), we deform the contour continuously avoiding any crossing or singularity with the poles.

which agrees with the usual propagator in the limit g1 → 0. In the fermion sector we have the dispersion relation (pµ − g2 nµ (n · p)2 )2 − m2 = 0 .

(9)

Again for the time-like nµ we have the equation (p0 − g2 p20 )2 − p~2 − m2 = 0 ,

(10)

whose standard, that is, non-singular at g2 → 0, solutions are √ √ 1 − 1 + 4g2 E 1 − 1 − 4g2 E , ω2 = , ω1 = 2g2 2g2

(11)

and the Lee-Wick ones W1 =

1+

√

1 − 4g2 E , 2g2

W2 =

1+

√

1 + 4g2 E , 2g2

(12)

p where E(~ p) = p~2 + m2 . In the region of energies satisfying the condition 4g2 E < 1 the four solutions are real and obey the inequality ω2 < ω1 < W1 < W2 , where ω2 is a negative number. However, beyond the critical energy 1/(4g2 ) both ω1 and W1 become complex and move in the opposite imaginary line at 1/(2g2 ) as shown in Fig. 2, while the other two solutions ω2 , W2 remain real. (f ) To define the contour CF 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 [27] which has been well tested to give a suitable correspondence with the normal theory when g2 → 0 and also to preserve the (f ) unitarity of the S matrix. In this effective region the integration contour CF 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 ω1 and W1 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
i (p0 − g2 p20 )γ 0 + pi γ i + m , (13) S(p) = 2 g2 (p0 − ω1 + i)(p0 − W1 + i)(p0 − ω2 − i)(p0 − W2 + i) which differs from the direct i prescription in the quadratic terms, but allows in particular to define a consistent Wick rotation which we use later.

5 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. p

k

k

FIG. 3: One-loop graph in the scalar model with self interaction λφ4 .

To proceed with it, we need to evaluate the basic integral Z 1 1 d4 p . Σ2 = λ 4 2 2 2 (2π) p − M + 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 ζ(s) for negative values of s. We go to d dimensions and choose the Lorentz-breaking four-vector to be timelike nµ = (1, 0, 0, 0) which yields Σ2 =

1 4−d µ λ 2

Z

dd p 1 , (2π)d 2g1 (p20 − p21 + i)(p20 + P22 )

(15) (s)

where p1 , P2 are given in (7). We perform the integration in the complex p0 -plane by closing the contour CF upward and enclosing the two poles −p1 and iP2 as depicted in Fig. 1, yielding Z d−1 d p (iF1 − F2 ) , (16) Σ2 = πiµ4−d λ (2π)d where √ g1 q F1 = p , p 1 + 8g1 E 2 1 + 1 + 8g1 E 2

F2 = p

√ g1 q . p 1 + 8g1 E 2 −1 + 1 + 8g1 E 2

(17)

1 Note that iF1 − F2 has the correct limit at g1 → 0, recovering the usual result − 2E . Now, it is convenient to change p 2 variables z = 1 + 8g1 E (p) yielding

p dp =

zdz 8g1

dd−1 p = |p|d−2 dp dΩd−1 ,

(18)

which allows to write the integral (16) as 4−d

Σ2 = −πµ

λ

√ 2π (d−1)/2 g1 (2π)d Γ( d−1 2 )(8g1 )

d−1 2

(I1 + I2 ) .

(19)

with Z

dz z0

where z0 =

d−3

∞

I1 =

(z 2 − z02 ) 2 √ , z+1

Z

d−3

∞

I2 = i

dz z0

(z 2 − z02 ) 2 √ , z−1

p 1 + 8g1 M 2 and we have used the definition of solid angle (A2).

(20)

6 Considering both contributions through the relation Σ2 = Σ(1) + Σ(2) , and after some algebra with d = 4 −  and expanding in , we find at lowest order
   ! 3π 2 F1 R(0,0,1,0) 14 , 34 , 2, 1 19 M 2λ 3 1 g1 M 2 √ − + 2γE + 6 ln(2) − + 1 − γE − ln(512) ln − , Σ(1) = 12π 3 3 8 8 8 8 2

λ M 2λ (−14 + 6γE + 3 ln(32g1 M 2 )) + 8 −17 + 6γE + 3 ln 32g1 M 2 3 3 144g1 π 192π     2
g1 M −2 3 + ln −14 + 6γE + 3 ln(32g1 M 2 ) . (21) 8
Here, 2 F1 R(0,0,1,0) 41 , 34 , 2, 1 is a hypergeometric function, whose exact value is −0, 71. Note that there is a fine tuning in this case, that is, the expression is singular at g1 → 0. However the correction to the two-point function Σ2 is UV finite. Σ(2) =

IV.

COUPLING OF SCALAR AND SPINOR FIELDS

λ 4 Let us consider the theory involving both the quartic interaction vertex V1 = − 4! φ and the Yukawa coupling ¯ vertex V2 = g ψψφ. 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 [28]. However, our theory essentially differs from that model. To classify the possible divergences, we should calculate the superficial degree of divergence ω of this theory. The naive result for it is

ω = 4 − 4V1 − 2V2 − Eψ ,

(22)

where Eψ 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. [29]). Since nµ is purely time-like, we can write (n · p)4 = p40 , so, we have from (4) i , p20 − p~2 − M 2 + 2g1 p40 i S(p) = . p/ − m − g2 γ 0 p20

∆(p) =

(23)

Following the methodology developed for the Horava-Lifshitz theories (see f.e. [29]), 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 p~ is 1. Taking into account only the leading degrees, we easily conclude that the canonical dimension of the momentum p0 is 1/2 (we note that this case does not occur in usual HoravaLifshitz-like theories where the canonical dimension of time momenta are always more than one, cf. [29]). Therefore, the spinor propagator has the canonical dimension (and the contribution to the superficial degree of divergence) equal to (−1), and the scalar one – equal to (−2) just as in the usual case. Nevertheless, the dimension of the integral measure, that is, d4 k = d3~kdk0 in this case is different from the usual one, being equal to 7/2 rather than 4. Hence the superficial degree in our theory is ω = (7/2)L − 2Pφ − Pψ ,

(24)

where L is a number of loops, and Pφ and Pψ are the numbers of scalar and spinor propagators respectively. Then, let V1 will be the number of φ4 vertices, and V2 – 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 Eφ , Eψ are the numbers of external scalar and spinor legs respectively. We use the topological identity L + V − P = 1, that is, L + V1 + V2 − Pψ − Pφ = 1. As a result, we eliminate numbers of loops and propagators from ω and rest with 7 1 1 3 5 ω = − V1 − V2 − Eφ − Eψ . (26) 2 2 4 4 4

7 A straightforward verification shows that the superficially divergent diagrams (that is, those ones with ω > 0) can be of the following types: (i): Eψ = 2, Eφ = 0, V2 = 2, ω = 1/2. This is the one-loop renormalization of the mass and kinetic terms for the spinor. (ii): Eψ = 2, Eφ = 0, V2 = 4, ω = 0. This is the two-loop renormalization of the mass and kinetic terms for the spinor. (iii): Eψ = 0, Eφ = 2, V2 = 2, ω = 3/2. This is the one-loop renormalization of the mass and kinetic terms for the scalar. (iv): Eψ = 0, Eφ = 2, V2 = 4, ω = 1. This is the two-loop renormalization of the mass and kinetic terms for the scalar. (v) Eψ = 0, Eφ = 2, V1 = 1, ω = 3/2. 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 Eφ 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 V1 = 1, that is, the purely scalar tadpole we studied above, and with V2 = 2 and Eψ = 0, Eφ = 2 or Eψ = 2, Eφ = 0 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 V1 → 0 and g1 → 0 in (1). The Lagrangian is L=

1 1 ¯ /(n · ∂)2 ψ + g ψφψ ¯ , ∂µ φ∂ µ φ − M 2 φ2 + ψ¯ (i∂/ − m) ψ + g2 ψn 2 2

(27)

and additionally, we impose the simplification of considering m = M and the preferred four-vector to be purely timelike n = (1, 0, 0, 0).

A.

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.

p+k p

p k

FIG. 4: Self-energy loop graph in the scalar sector.

It is represented by the integral Π(p) = −

g2 φ(−p)φ(p) 2

Z

d4 k Tr ((Qµ γ µ + m)(Rν γ ν + m)) , (2π)4 (Q2 − m2 )(R2 − m2 )

(28)

8 where we define Qµ = kµ − g2 nµ (n · k)2 , Rµ = kµ + pµ − g2 nµ (n · (k + p))2 .

(29)

Calculating the trace gives Π(p) = −2g 2 φ(−p)φ(p)

Q · R + m2 d4 k . 4 2 (2π) (Q − m2 )(R2 − m2 )

Z

(30)

e Let us write the corresponding contribution to the effective action as Π(p) = −2g 2 φ(−p)Π(p)φ(p) and study the typical low-energy behavior of this contribution by expanding it into Taylor series ! ! 2e e 1 ∂ Π ∂ Π e e + ... . (31) Π(p) = Π(0) + pµ pµ pν ∂pµ 2 ∂pµ ∂pν p=0

p=0

The zeroth-order contribution follows directly from (30) Z d4 k Q2 + m2 e Π(0) = . (2π)4 (Q2 − m2 )2

(32)

It is convenient to rewrite as e Π(0) = K + 2m2 P ,

(33)

where d4 k 1 , (2π)4 (Q2 − m2 )

(34)

d4 k 1 . 4 2 (2π) (Q − m2 )2

(35)

Z K= and Z P =

where the integrals (34) and (35) have been solved in the Appendix A. e ∂Π For the next term, it is clear that ( ∂p µ )|p=0 can be proportional to nµ only since there is no other vectors, the R corresponding contribution to the effective action will yield d4 xφ(n·∂)φ, that is, a surface term. So, we can disregard 2e Π it. Further, one would need then to find the second derivative, that is, ( ∂p∂µ ∂p )|p=0 which may contain naturally ν terms of higher-order in g2 . To find it, consider ! Z i h  1   ∂ ∂  1  e ∂ 2 Π(p) d4 k . (36) Tr = / − m p=0 ∂pµ ∂pν (2π)4 Q ∂pµ ∂pν R / −m p=0

Integrating by parts and neglecting the surface terms, we obtain the symmetric expression ! Z h ∂  1  ∂  1 i e ∂ 2 Π(p) d4 k = − Tr , ∂pµ ∂pν (2π)4 ∂kν Q / − m ∂kµ Q / −m

(37)

p=0

where we have used the identity



∂f (k+p) ∂pα



= p=0

∂f (k) ∂kα .

We consider ∂ ∂kµ



e ∂ 2 Π(p) ∂pµ ∂pν

!

1 Q / −m



 =

∂Qα ∂kµ



1 γα , (Q / − m)2

(38)

and after some algebra we arrive at Z =− p=0

d4 k (2π)4



∂Qα ∂kµ



∂Qσ ∂kν



T ασ ,

(39)

9 with T ασ =

32m2 4 η ασ + Qα Qσ . 2 2 2 −m ) (Q − m2 )4

(Q2

(40)

By using the relations 

∂Qα ∂kµ



∂Qα ∂kν



= η µν − 4g2 nµ nν (n · Q) ,      2 ∂Qα ∂Qσ ∂Q 1 ∂Q2 Qα Qσ = , ∂kµ ∂kν 4 ∂kµ ∂kν

(41)

one obtains e ∂ 2 Π(p) ∂pµ ∂pν

!

d4 k (2π)4

Z = −4

p=0



η µν − 4g2 nµ nν (n · Q) 2m2 + 2 2 2 2 (Q − m ) (Q − m2 )4



∂Q2 ∂kµ



∂Q2 ∂kν

 .

(42)

Considering the tensors available in our model, which are the flat metric ηµν and the preferred four-vector nµ we can write Z Qµ d4 k = nµ S , (43) (2π)4 (Q2 − m2 )2 and Z

d4 k 1 4 2 (2π) (Q − m2 )4



∂Q2 ∂kµ



∂Q2 ∂kν



= nµ nν L + η µν n2 M .

(44)

Now, consider the relation ∂Q2 = 2 (Qµ − 2g2 nµ (n · Q)(n · k)) , ∂kµ

(45)

and multiplying by nµ nν we to arrive at Z


d4 k 4(n · Q)2 1 − 4g2 n2 (n · k) + 4g22 (n2 )2 (n · k)2 = (n2 )2 (L + M ) , (2π)4 (Q2 − m2 )4

(46)

and by contracting with the metric ηµν Z


d4 k 4 Q2 − 4g2 (n · Q)2 (n · k) + 4g22 n2 (n · Q)2 (n · k)2 = n2 (L + 4M ) . (2π)4 (Q2 − m2 )4

Solving the algebraic equation we have   Z (n · Q)2 16 d4 k 1 Q2 2 2 2 2 2 − 3g (n · Q) (n · k) + 3g (n · Q) (n · k) n , L = − 2 2 3n2 (2π)4 (Q2 − m2 )4 n2 4   Z 4 d4 k 1 (n · Q)2 2 M = Q − . 3n2 (2π)4 (Q2 − m2 )4 n2

(47)

(48)

A similar analysis gives S=

1 n2

Z

d4 k (n · Q) . (2π)4 (Q2 − m2 )2

We find the second-order contribution ! e

∂ 2 Π(p) pµ pν = − 4P + 8m2 n2 M p2 + 16g2 n2 S − 8m2 L (n · p)2 . ∂pµ ∂pν p=0

(49)

(50)

10 Reorganizing this expression, we can write the correction to the scalar propagator up to second-order in p as e Π(p) = m2 q0 + p2 q1 + (n · p)2 qn ,

(51)

q0 = K + 2m2 P ,
q1 = −4 P + 2m2 n2 M ,
qn = 8 2g2 n2 S − m2 L .

(52)

where

Finally one has  g m  i  i 2 + 6γ − 0, 46 + 12iπ − 18 ln , E 48π 2 g22 m2 48π 2 2   g m 1 i 2 = − 2 iπ − ln − , 2π 2 3 i = 2. π

q0 = −

(53)

q1

(54)

qn

(55)

We provide details of the computation of q0 , q1 and qn in the Appendix A. The two-point function is finite and involves a fine-tuning term proportional to g2−2 . 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 [30]. B.

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 M = m. k−p

p

p

k

FIG. 5: The fermion self-energy graph.

The fermion self-energy graph is represented by the integral Z d4 k Q / +m Σ(p) = g 2 . 4 2 (2π) ((k − p) − m2 )(Q2 − m2 )

(56)

To find it, let us consider a Taylor expansion of the first denominator term up to second-order in p and rewrite the diagram as    Z d4 k 1 2(k · p) − p2 4(k · p)2 Q / +m Σ(p) ≈ g 2 + + + O(p3 , n) . (57) (2π)4 k 2 − m2 (k 2 − m2 )2 (k 2 − m2 )3 Q2 − m2 With the notation Σ(p) = g 2 I (0) + g 2 I (1) + g 2 I (2) , we introduce the the zeroth-order contribution    Z d4 k 1 Q / +m I (0) = , (2π)4 k 2 − m2 Q2 − m2

(58)

the linear-order contribution I (1) =

Z

d4 k (2π)4



2(k · p) (k 2 − m2 )2



Q / +m Q2 − m2

 ,

(59)

and the second-order contribution I

(2)

Z =

d4 k (2π)4



4(k · p)2 p2 − (k 2 − m2 )3 (k 2 − m2 )2



Q / +m Q2 − m2

 .

(60)

11 C.

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

Zeroth-order I (0)

Let us start with Eq. (58) and rewrite it as I (0) = γ µ Iµ(0) + mf0 ,

(61)

where we have defined Qµ d4 k , 4 2 2 (2π) (k − m )(Q2 − m2 )

Z

Iµ(0) =

(62)

and Z f0 =

1 d4 k . 4 2 2 (2π) (k − m )(Q2 − m2 )

From tensor analysis considerations one should have Z Qµ d4 k = nµ f1n . (2π)4 (k 2 − m2 )(Q2 − m2 )

(63)

(64)

Replacing the expression (64) in Eq. (61) produces the zeroth-order contribution I (0) = n / f1n + mf0 ,

(65)

with f1n =

1 n2

Z

d4 k (n · Q) . 4 2 (2π) (k − m2 )(Q2 − m2 )

(66)

We carry out the calculations of f0 and f1n following the lines given in Appendix B 2. The first coefficient f0 is naturally finite and the second one f1n is divergent and contains a large Lorentz-breaking correction term of the order of g2−1 . 2.

Linear-order I (1)

The linear-order integral (59) can be rewritten by introducing (1) I (1) = 2pµ γ ν Iµν + 2mpµ Iµ(1) ,

(67)

where Iµ(1) (1) Iµν

d4 k kµ , 4 2 2 (2π) (k − m )2 (Q2 − m2 ) Z d4 k kµ Qν = . 4 2 2 (2π) (k − m )2 (Q2 − m2 ) Z

=

(68)

By considering d4 k kµ = nµ D , (2π)4 (k 2 − m2 )2 (Q2 − m2 ) Z d4 k kµ Qν = nµ nν B + n2 ηµν C , (2π)4 (k 2 − m2 )2 (Q2 − m2 ) Z

(69)

and after some manipulations one finds d4 k (n · k) , (2π)4 (k 2 − m2 )2 (Q2 − m2 )

1 n2

Z

1 B= 2 3n

Z

4(n·k)(n·Q) − (k · Q) d4 k n2 , 4 2 2 2 (2π) (k − m ) (Q2 − m2 )

1 C= 2 3n

Z

(n·k)(n·Q) d4 k (k · Q) − n2 . (2π)4 (k 2 − m2 )2 (Q2 − m2 )

D=

(70)

12 Introducing the new notation we can write n I (1) = p / (n · p)f3n , /f1 + m(n · p)f2 + n

(71)

where f1 = 2n2 C , f2n = 2D , f3n = 2B .

(72) (73) (74)

The terms f1 , f2n and f3n are convergent and are explicitly calculated in Appendix B 3. 3.

Second-order I (2)

Following the same methodology the integral I (2) can be written as     (2) (2) , + mIµν I (2) = −p2 γ µ Iµ(2) + mf2 + 4pµ pν γ α Iµνα

(75)

with Qµ d4 k , (2π)4 (k 2 − m2 )2 (Q2 − m2 ) Z d4 k kµ kν = , (2π)4 (k 2 − m2 )3 (Q2 − m2 ) Z kµ kν Qα d4 k , = (2π)4 (k 2 − m2 )3 (Q2 − m2 ) Z d4 k 1 = . (2π)4 (k 2 − m2 )2 (Q2 − m2 )

Iµ(2) = (2) Iµν (2) Iµνα

f2

Z

(76)

(2)

We have that Iµν and f2 are of the order g22 so they can be neglected. For the remaining we define them with the following tensor structure Z Qµ d4 k = nµ C¯ , (77) 4 2 2 (2π) (k − m )2 (Q2 − m2 ) Z kµ kν Qα d4 k ¯ + n2 ηµν nα E ¯ + n2 ηνα nµ F¯ . = nµ nν nα D 4 2 (2π) (k − m2 )3 (Q2 − m2 ) After some algebra we find d4 k (n · Q) , (2π)4 (k 2 − m2 )2 (Q2 − m2 )   2 Z k 2 − (n·k) (n · Q) 4 n2 1 d k ¯ = E , 2 2 4 2 2 3 2 3(n ) (2π) (k − m ) (Q − m2 )   5(n·k)2 (n·Q) 2 Z (n · Q) − (n · k)(k · Q) − k 4 2 n 1 d k ¯ = D , 3(n2 )2 (2π)4 (k 2 − m2 )3 (Q2 − m2 )   (n·k)2 (n·Q) Z − (n · k)(k · Q) 4 n2 1 d k . F¯ = − 3(n2 )2 (2π)4 (k 2 − m2 )3 (Q2 − m2 ) 1 C¯ = 2 n

Z

(78)

Replacing the expressions (76) in (75) and using the integrals (78), up to linear-order in Lorentz violation and secondorder in p, we arrive at 2 ¯ ¯ + 4n2 (n · p)p I (2) = p2 n / (−C¯ + 4n2 E) /D . /F¯ + 4(n · p) n

(79)

13 We rewrite as n 2 I (2) = p2 n / f6n + (n · p)p / f5n , /f4 + (n · p) n

(80)

f4n = 4n2 F¯ , ¯, f5n = 4D n ¯, f6 = −C¯ + 4n2 E

(81) (82) (83)

with

where the explicit form of the relevant constants is given in the Appendix B 4. VI. A.

MASS RENORMALIZATION The on-shell subtraction scheme

Having computed the one-loop correction to the scalar and fermion propagators, we now proceed to determine the pole mass in our Yukawa-like model. The radiative corrections essentially involve extra terms such as the scalar (n · p), the matrix n / and their combinations. The theory is finite in the scalar sector and in the fermion sector the only divergency appears in the term f1n . Since the mass corrections are finite in both sectors, in some sense we are working in the scheme in which the finite parts of the renormalized mass correspond to the pole mass, and hence one can say that we are working in the on-shell subtraction scheme. Most of the ideas and method of calculation in the derivation of the pole mass, which are given below, are along the lines developed in the work of Ref. [21]. Let us remind the leading quantum corrections in the scalar sector Π(p) = −2g 2 m2 q0 − 2g 2 p2 q1 − 2g 2 (n · p)2 qn ,

(84)

where we have restored the constant −2g 2 . In the fermion sector we have found 2 n 2 n 2 2 Σ2 = g 2 n / f1n + g 2 mf0 + g 2 p / (n · p)f3n + g 2 p2 n / f6n + g 2 (n · p)p / f5n . /f1 + g m(n · p)f2 + g n /f4 + g (n · p) n

(85)

The coefficients q and f have been computed in the Appendix A and B. B.

The scalar pole mass

Let us start with the renormalized scalar Lagrangian L =

1 1 ∂µ φR ∂ µ φR − m2R φ2R , 2 2

(86)

and without loss of generality we absorb a term proportional to g2−2 in the definition of renormalized mass m2R as appears in the correction of q0 in (53). Consider the renormalized two-point function (2)

(ΓR )−1 = p2 − m2R + ΠR (p) ,

(87)

ΠR (p) = Π(p) + O(g 4 ) .

(88)

ΠR (p) = p2 Aφ + m2R Bφ + (n · p)2 Cφ ,

(89)

for which

Let us write

where Aφ , Bφ and Cφ are constants that can be deduced from the expressions (53) , (54), (55) being Aφ = −2g 2 q1 , Bφ = −2g 2 q0 , Cφ = −2g 2 qn .

(90)

14 In order to find the pole defined by the condition at P¯φ2 = 0 on the renormalized two-point function (2) (ΓR )−1 (P¯φ2 = 0) = 0 ,

(91)

2 P¯φ2 = p2 − Mph + y¯(n · p)2 ,

(92)

we define

where Mph and y¯ are unknown constants we should find. From (87) replacing the value of p2 given in (92) and using the condition (91), we arrive at the equation
2 2 0 = Mph − y¯(n · p)2 − m2R + Aφ Mph − y¯(n · p)2 + Bφ m2R + Cφ (n · p)2 .

(93)

Due to the independence of each term, and after some algebra, we find 2 Mph = m2R

(1 − Bφ ) , (1 + Aφ )

(94)

and Cφ . 1 + Aφ

y¯ =

C.

(95)

The fermion pole mass

In the fermion sector we consider the renormalized Lagrangian and a corresponding counterterm in order to make the quantum corrections finite and we associate the counterterm with the interaction term L = iψ¯R ∂/ψ R − mR ψ¯R ψ R + g2 ψ¯R n / (n · ∂)2 ψ R + δg2 ψ¯R n / ψR .

(96)

The renormalized two-point function is (2)

(ΓR )−1 = p / − mR + ΣR ,

(97)

ΣR = Σ2 + δg2 n / + O(g 4 ) .

(98)

with

We can write the finite contribution to the renormalized two-point function up to second-order in p as ΣR = p / Cψ , /Aψ + mR Bψ + n

(99)

where the explicit Lorentz violation coefficients at linear order in g2 , and coded in Σ2 , are given by Aψ = A(0) + g2 A(1) (n · p) , Bψ = B (0) + g2 B (1) (n · p) , Cψ =

C (0) + C (1) (n · p) + g2 C (2) (n · p)2 + g2 C (3) p2 . g2

(100)

We work in the minimal subtraction scheme and hence we fix the divergent term to be δg2 =

ig 2 . 4g2 π 2 

(101)

Within this scheme and according to the previous calculations we make the identification A(0) = g 2 f1 , g 2 f2n , g2 g 2 f5n = , g2

A(1) =

g 2 f4n , g2

B (0) = g 2 f0 ,

B (1) =

C (0) = g2 (g 2 f1n − δg2 ) ,

C (2)

C (3) =

g 2 f6n . g2

C (1) = g 2 f3n , (102)

15 To find the pole we consider the ansatz P¯ = p ¯ +x ¯n /, /−m

(103)

m ¯ = mph + g2 mn (n · p) ,

(104)

where

and x ¯=

x ¯(0) +x ¯(1) (n · p) + g2 x ¯(2) (n · p)2 + g2 x ¯(3) p2 . g2

(105)

We include the large Lorentz violating terms in the pole extraction process which has been included explicitly in x ¯. Considering the condition for P¯ to be a pole, that is to say, (2)

(ΓR )−1 (P¯ = 0) = p / − mR + ΣR = 0 ,

(106)

provides us with the six terms mph , mn , x ¯(0) , x ¯(1) , x ¯(2) , x ¯(3) . We proceed as follows: we replace P¯ in (97), and arrive at (2) (ΓR )−1 = P¯ + m ¯ −x ¯n ¯ −x ¯n / − mR + A(P¯ + m / ) + BmR + C n /.

(107)

With P¯ = 0 we write    (2) (ΓR )−1 (P¯ = 0) = 0 = mph + g2 mn (n · p) − x ¯n ¯n / − mR + A(0) + gA(1) (n · p) mph + gmn (n · p) − x /     1 (0) (0) (1) (1) (2) 2 (3) 2 + B + g2 B (n · p) mR + C + C (n · p) + g2 C (n · p) + g2 C p n (108) /. g2 At lowest-order we have the first condition mph − mR + A(0) mph + B (0) mR = 0 ,

(109)

which has the solution mph = mR

(1 − B (0) ) . (1 + A(0) )

(110)

Considering that all coefficients are independent, we have at linear-order for the factor accompanying (n · p): h i g2 (n · p) mn + A(0) mn + A(1) mph + B (1) mR = 0 , (111) and consequently we have  mn = −mR

(A(1) (1 − B (0) ) B (1) + (0) 2 (1 + A ) 1 + A(0)

 .

The next conditions determine x ¯ which follows from the equations with factor accompanying 2 g2 p n / respectively. Therefore, one arrives at

(112) 1 /, g2 n

(n · p)/ n , g2 (n · p)2 n /,

C (0) , 1 + A(0) A(1) C (0) C (1) = − + , (0) 2 (1 + A ) 1 + A(0)

x ¯(0) = x ¯(1)

x ¯(2) =

(A(1) )2 C (0) A(1) C (1) C (2) − + , (1 + A(0) )3 (1 + A(0) )2 1 + A(0)

x ¯(3) =

C (3) . 1 + A(0)

This finishes the pole extraction in both sectors.

(113)

16 VII.

SUMMARY

We considered the Myers-Pospelov-like higher-derivative extensions of the Yukawa model which incorporates possible new physics from the Planck scale through dimension five operators coupled to a preferred four vector nµ which breaks the Lorentz symmetry. We have selected a particular configuration of Lorentz symmetry violation in which the preferred four-vector nµ is purely timelike. This choice produces higher-order time-derivative terms and leads to extra solutions and new poles in the model. We have found and identified the poles associated to standard solutions and those ones corresponding to negative-metric states or Lee-Wick solutions. Some of these poles move in the real axis, as in the usual Yukawa model, however, above a specific energy called the critical energy some solutions become complex introducing a extra difficulty to define a consistent prescription for the contour of integration in the complex plane [31]. To solve this problem we have analyzed the motion of the poles in the complex plane. In the scalar sector we have defined the usual Feynman integration contour which rounds the negative pole from below and the positive from above; the new poles that appear are located in the imaginary axis and hence they do not present problems in this sense. However, in the fermion sector we have defined the prescription with two considerations in mind: first, the ability to recover the standard location of the poles relative to the real axis when the Lorentz invariance (f ) violation is turned off. With this consideration we have imposed the minimal requirement on the contour CF to round the negative pole ω2 from below and ω1 from above. Second, to account for the other poles we have used the prescription putted forward in [27], which has been shown to lead to a unitary S-matrix in the regime of real solutions. Using this Lee-Wick prescription, the three positive solutions W1 , W2 and ω1 are defined to lie below the contour and the negative one ω2 above. At higher energies beyond the critical energy, we have called for the following heuristic construction. We start at lower energies in which all the poles are real and where the basic requirements for consistency of the theory are satisfied. Next, we begin to increase the energy at which complex solutions appear and define the new contour as the one obtained by continuously deforming the first one in such a way to avoid any crossing and singularity with the complex poles. A central part of this work has been the study of mass renormalization in the presence of both Lorentz-invariance violation and higher-order operators. We have computed the one-loop corrections in the model and shown how they lead to modifications in the renormalization procedure by pushing the pole mass to a sector involving other gamma matrices besides of p /. The significance of our results, in particular, is based on the fact that this is one of the first works on the renormalization of higher-derivative Lorentz-breaking theories within the context of modifications of asymptotic Hilbert space, whereas even without introducing the higher derivatives the renormalization issues in Lorentz-breaking theory were considered only in a few papers, in particular [25, 32, 33]. For other approaches in renormalization of higher-order Lorentz breaking theories, see [34]. We found that the theory is super-renormalizable (in principle, increasing the value of N in the added kinetic operator (n · ∂)N , we can get a completely finite theory, the same result can be achieved through a supersymmetric extension of the theory, see [19]). The kinetic terms of the both fields are explicitly finite. As a by-product, we conclude that the aether-like terms originally introduced in [30, 35] are generated within our studies both in scalar and spinor sectors, and they are essentially finite. It is interesting to note that in the spinor sector, the aether terms dominate in the low-energy limit in comparison with the second-derivative term we introduced into the classical action. At the same time, the large quantum corrections or, in other words, fine-tuning arise in our theory. We have found in the scalar sector a large correction which in principle should be added to the usual fine tuning proportional to the square of the fermion mass. In consequence the effect of UV sensitivity of the scalar mass is maintained in our model. In the fermion sector we have that the large corrections affect a term which do not involve any physical observable protecting the theory in this sector to a genuine fine tuning. In a certain sense, this effect of large quantum corrections taking place in our theory resembles the effect of UV/IR mixing taking place in noncommutative field theories responsible for arising infrared singularities in a small noncommutativity limit [36]. Let us mention that in principle these large corrections are natural to expect since the higher-derivative extensions of the classical action can be treated as a kind of the higher-derivative regularization, hence, removing the regularization we return to singular results. We note that this effect is rather generic since the large quantum corrections are present even in the supersymmetric extensions of the Myers-Pospelov-like theories [19]. The complete elimination of large quantum corrections could consist in employing the extended supersymmetry which is well known to achieve complete finiteness of the field theory models as occurs for example for N = 4 super-Yang-Mills theory. Therefore, the natural problem could consist in study of some Myers-Pospelov-like extensions of N = 4 super-Yang-Mills theories. Another relevant problem consists in studying of the two-loop approximation thus exhausting the possible divergences. We plan to carry out these studies in one of our next papers.

17 Acknowledgments

This work was partially supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq). The work by A. Yu. P. has been supported by the CNPq project No. 303783/2015-0. CMR wants to thank the hospitality of the Universidade Federal da Paraiba (UFPB) and acknowledges support by FONDECYT grant 1140781. A. Yu. P. thanks the group of F´ısica de Altas Energ´ıas of UBB for the hospitality. We also want to thank Markos Maniatis for valuable comments and helpful discussions. Appendix A: The calculation of q0 , q1 , qn

We start to compute q0 in (52), so we focus on K and P as given in Eqs. (34) and (35) and promote the integrals to d dimensions and consider dd k = Ωd−1 |~k|d−2 d|~k| dk0 = Ωd−1 E(E 2 − m2 )

d−3 2

dE dk0 ,

(A1)

where we have performed an integration in the angles producing the solid angle in d − 1 dimensions Ωd−1 =

2π (d−1)/2
. Γ d−1 2

(A2)

We obtain 4−d

K = µ

Z

∞

Ωd−1 m

P = µ4−d Ωd−1

Z

∞

m

E(E 2 − m2 ) (2π)d

d−3 2

E(E 2 − m2 ) (2π)d

d−3 2

dE

Z (f ) CF

dE

dk0 , − m2 )

(Q2

Z (f ) CF

(Q2

(A3)

dk0 . − m2 )2

We rewrite the contour integrals making explicit their poles, and to compute them we use the method of residues (f ) with the contour CF closed from above enclosing the pole ω2 , see Fig. 2. We find Z dk0 πi = − √ , (A4) (f ) g 2 (k − ω )(k − ω )(k − W )(k − W ) E 1 + 4g2 E 1 0 2 0 1 0 2 CF 2 0 Z dk0 πi(1 + 6g2 E) . = 3 (1 + 4g E)3/2 (f ) g 4 (k − ω )2 (k − ω )2 (k − W )2 (k − W )2 2E 1 0 2 0 1 0 2 2 CF 2 0 The next integral in the variable E is direct, and gives at lowest order in  = 4 − d for both pieces  g m  i im2  2 + 6γ − 0, 46 − 6 ln , E 48π 2 g22 48π 2 2  gm  i  P = iπ − ln . 8π 2 2

K = −

It is simple to show that by combining the two contributions through q0 = K + 2m2 P , produces  g m  i i  2 q0 = − + 6γ − 0, 46 + 12iπ − 18 ln . E 48π 2 g22 m2 48π 2 2

(A5)

(A6)

The second-order contributions q1 and qn are given in terms of P , M , S and L as shown in Eqs. (52). We begin with M , and since it is obtainable in four dimension, we just set d = 4, giving Z Z 16π ∞ E(E 2 − m2 )3/2 dE dk0 M =− . (A7) (f ) (Q2 − m2 )4 3 m (2π)4 CF Solving the first integral gives Z (f ) CF

g28 (k0

−

ω1 )4 (k0

dk0 πi (5 + 2g2 E (35 + 4g2 E (43 + 77g2 E))) = , 4 4 4 − ω2 ) (k0 − W1 ) (k0 − W2 ) 16E 7 (1 + 4g2 E)7/2

(A8)

18 and calculating then the E-integral, we arrive at M = −

i . 48π 2 m2

(A9)

With the result of P given in (A5), one has q1

i = − 2 2π

 g m 1 2 iπ − ln − . 2 3

(A10)

We continue to compute qn . From (49) we have 4−d

S = µ

∞

Z Ωd−1

m

E(E 2 − m2 )(d−3)/2 dE (2π)d

Z (f ) CF

k0 (1 − g2 k0 ) dk0 . g24 (k0 − ω1 )2 (k0 − ω2 )2 (k0 − W1 )2 (k0 − W2 )2

(A11)

In the same way, we arrive at  g m  ig2 m2  i 2 + −4 + γ − 0, 15 − 2 ln . E 16g2 π 2 16π 2 2

S =

(A12)

To compute L given in the first equation (48),   Z 16 1 (n · Q)2 Q2 d4 k 2 2 2 2 2 L = − − 3g (n · Q) (n · k) + 3g (n · Q) (n · k) n , 2 2 3n2 (2π)4 (Q2 − m2 )4 n2 4

(A13)

we note that L = L2 − M ,

(A14)

with Z

∞

L2 = 16π m

E(E 2 − m2 )1/2 dE (2π)4

Z (f )

CF

k02 (1 − g2 k0 )2 (1 − 2g2 k0 )2 dk0 . g28 (k0 − ω1 )4 (k0 − ω2 )4 (k0 − W1 )4 (k0 − W2 )4

(A15)

We find L2 = −

i , 48π 2 m2

(A16)

and therefore L = 0. Considering the leading contributions for qn we obtain qn =

i . π2

(A17)

Appendix B: The calculation of Σ(p) 1.

The general strategy

In this subsection, we describe the main steps to derive the basic integral that will appear in the computation of the fermion self-energy correction (57). Let us start to consider the integral below where each order is labelled with α = 1, 2, 3 Z M(α) =

d4 k F (k0 , ~k) , 4 2 (2π) (k − m2 )α (Q2 − m2 )

(B1)

and where F (k0 , ~k) is an arbitrary function of k0 and ~k. We promote the integrals to d dimensions and recall the phase space measure given in (A1) in order to write M(α) = µ4−d Ωd−1

Z

∞

m

d−3 dE E(E 2 − m2 ) 2 d (2π)

~

Z (f ) CF

(k02

−

F (k0 , k) dk0 2 E )α ((k0 − g2 k02 )2

− E2)

.

(B2)

19 Working the denominator, we can rewrite the last contour integral as   Z 1 1 1 1  −  . dk0 F (k0 , ~k) 2 − k0 (−1+g2 k0 )+E k0 (−1+g2 k0 )−E 2Eg2 CF(f ) (k0 − E 2 )α g2

(B3)

g2

Now, using the Feynman parametrization

and using β = 1 we write the integral as  Z Z 1 α  dx  h − dk0 F (k0 , ~k) 2Eg2 CF(f ) 0 k02 −

1

Z

Γ(α + β) 1 = Aα B β Γ(α)Γ(β)

dx 0

xα−1 (1 − x)

β−1

(Ax + B (1 − x))

α+β

,

(B4)

 x

α−1

iα+1 − h − E 2 x + E(1−x) k02 − g2

k0 (1−x) g2

x k0 (1−x) g2

α−1

− E2x −

E(1−x) g2

 iα+1  . (B5)

0 0 Next, we carry out the change of variables k00 = k0 − (1−x) 2g2 and perform a subsequent Wick rotation k0 → ik0E , as a result, and dropping the tilde, we arrive at   Z Z ∞ i(−1)α α 1 (1 − x) ~ dx dk0E F ik0E + ,k 2Eg2 2g2 0 −∞  

xα−1  × h 2 + E 2 x − E(1−x) + k0E g2

−h i 2 α+1

(1−x) 4g22

xα−1 2 + E2x + k0E

E(1−x) g2

+

(1−x)2 4g22

 iα+1  .

(B6)

Replacing the above expression in (B2) and again changing variables by the rule t = g2 E produces the final expression M(α)

i(−1)α αΩd−1 µ4−d = 2

Z

∞

g2 m

(d−3) dt 1 (t2 − g22 m2 ) 2 l(α) , d−1 d (2π) g2

(B7)

with Z

∞

l(α) =

1

Z dk0E

−∞

0

   (1 − x) ~  dxF ik0E + , k h 2g2

xα−1 2 + k0E

t2 x g22

−

t(1−x) g22

+

(1−x)2 4g22

iα+1

 − h 2 + k0E

xα−1 t2 x g22

+

t(1−x) g22

+

(1−x)2 4g22

 iα+1  ,

(B8)

which is the basic integral we need to solve in the next subsections.

2.

The zeroth-order terms f0 and f1n

We start to compute f0 in d dimensions which follows from (63) f0 = µ4−d

dd k 1 . d 2 2 (2π) (k − m )(Q2 − m2 )

Z

(B9)

We use the Eqs. (B7) and (B8) with the identification α = 1 and F (k0 , ~k) = 1, which yield iΩd−1 µ4−d f0 = − 2

Z

∞

g2 m

(d−3) (1) dt 1 (t2 − g22 m2 ) 2 l1 , d−1 d (2π) g2

(B10)

20 with 

 (1)

l1

Z

∞

=

1

Z dk0E

−∞

 dx  h

0

1 t2 x g22

2 + k0E

−

To proceed, we consider some useful integrals Z ∞ dk0E

t(1−x) g22

+

(1−x)2 4g22

i2 − h 2 + k0E

1 t2 x g22

+

t(1−x) g22

+

(1−x)2 4g22

 i2  .

(B11)

√ 1 π Γ(r − 12 ) = , + M 2 )r Γ(r) (M 2 )r− 21

(B12)

√ 2 k0E π Γ(r − 32 ) dk0E 2 , = (k0E + M 2 )r 2Γ(r) (M 2 )r− 32 −∞

(B13)

√ 4 k0E 3 π Γ(r − 52 ) dk0E 2 = . (k0E + M 2 )r 4Γ(r) (M 2 )r− 52 −∞

(B14)

−∞

Z

Z

2 (k0E

∞

∞

By using the integral (B12) for r = 2, we are able to solve the time integral in (B11) arriving at   Z 1 π 1 1 (1) . l1 = dx  2 − 2 2 2 2 0 ( t 2 x − t(1−x) ( t 2 x + t(1−x) + (1−x) )3/2 + (1−x) )3/2 2 2 2 2 g2

g2

4g2

g2

g2

(B15)

4g2

Introducing the new variable ε = t(1 − x), we write # " Z 1 g23 π 1 1 (1) − . l1 = dx 2 2 2 0 )3/2 )3/2 (t2 x − ε + (1−x) (t2 x + ε + (1−x) 4 4

(B16)

We use the approximation 1 (t2 x

−ε+

(1−x)2 n/2 ) 4

−

1 (t2 x

+ε+

(1−x)2 n/2 ) 4

=

nε (t2 x

+

(1−x)2 n/2+1 ) 4

+ O(ε3 ) ,

(B17)

for n = 3 followed by replacing d = 4 in all explicitly finite terms, as a result, we arrive at the expression f0 = −

3i 16π 2

Z

1

Z

∞

dx 0

dt g2 m

t(1 − x)(t2 − g22 m2 ) (t2 x +

d−3 2

.

(1−x)2 5/2 ) 4

Integrating in t and then in x we arrive at the finite expression      i 1 1 1 3 f0 = AppellF1 − , 1, 1, , a, b + AppellF1 , 1, 1, , a, b , 2π 2 2 2 2 2

(B18)

(B19)

with the notation a=

1 1 − 2g22 m2 − 2g2 m

p , −1 + g22 m2

b=

1 1 − 2g22 m2 + 2g2 m

p . −1 + g22 m2

(B20)

The function AppellF1 (a, b1 , b2 , c, x1 , x2 ) that appear above, is the hypergeometric function of two variables defined by AppellF1 (a, b1 , b2 , c, x1 , x2 ) =

∞ X ∞ X (a)m+n (b1 )m (b2 )n m n x1 x2 , m! n! (c)m+n m=0 n=0

(B21)

where (q)n = q(q + 1) . . . (q + n − 1), or equivalently (q)n =

Γ(q + n) . Γ(q)

(B22)

21 We continue with f1n , given in (66), and again promote to d dimensions Z k0 − g2 k02 dd k , f1n = µ4−d (2π)d (k 2 − m2 )(Q2 − m2 )

(B23)

and using our Eqs. (B7) and (B8), with α = 1 and F (k0 , ~k) = k0 − g2 k02 , we write Z d−3 (1) 1 iΩd−1 µ4−d ∞ dt (t2 − g22 m2 ) 2 l2 , f1n = − d d−1 2 (2π) g gm 2

(B24)

with  (1)

l2

Z

1

Z

=

dk0E

 dx

0

2

1−x (1 − x) 2 + g2 k0E − 2g2 4g2

  h 2 + k0E

1 t2 g22

x−

t(1−x) g22

+

(1−x)2 4g22

i2

 −h

1 2 + k0E

t2 x g22

+

t(1−x) g22

+

(1−x)2 4g22

 i2  .

(B25)

Using the integrals (B12) and (B13) we arrive at  Z 1 (1−x)2 (1−x)2 ( 1−x ( 1−x π 2g2 − 4g2 ) 2g2 − 4g2 ) (1)  − 2 + 2 l2 = dx 2 2 2 2 0 ( gt 2 x − t(1−x) + (1−x) )3/2 ( gt 2 x + t(1−x) + (1−x) )3/2 ( gt 2 x − 2 2 2 2 g 4g g 4g 2 2 2 2 2 2 2  g2 . − 2 t(1−x) t2 1/2 ) ( g2 x + g2 + (1−x) 2 4g 2

2

g2 2 t(1−x) )1/2 + (1−x) g2 4g22 (B26)

2

Just as in the previous calculation, we define ε = t(1 − x) and consider the approximation (B17) with n = 3 for the first and second term and n = 1 for the third and fourth term in (B26). Together with this, we replace d = 4 in all 2 (1−x)2 ) the finite terms and consider the identity 1−x = (1−x to arrive at the simpler expression 2 − 4 4 f1n

i = − 16π 2

Z

1

dx 0

µ g2d−3

Z

∞

dt g2 m

(d−3) 3(1−x2 )(1−x) t(t2 − g22 m2 ) 2 4 2 (t2 x + (1−x) )5/2 4

+

t(1 − x)(t2 − g22 m2 ) (t2 x +

(d−3) 2

(1−x)2 3/2 ) 4

! .

(B27)

Solving the x integrals and then expanding in , we are able to isolate the divergence, so to arrive at the final result        i 1 3 1 5 1 3 i 1 n + , 1, 1, , a, b + AppellF1 , 1, 1, , a, b f1 = AppellF1 − , 1, 1, , a, b + AppellF1 4g2 π 2  8g2 π 2 2 2 2 2 3 2 2      1 5 7 i 3 − AppellF1 , 1, 1, , a, b − γE − 2 log(2gµ) + PolyGamma 0, , (B28) 5 2 2 8g2 π 2 2
where a, b are given by (B20) and PolyGamma 0, 32 = 0, 036. 3.

The linear-order terms f1 , f2n and f3n .

Now we compute the linear-order corrections given by the expressions (72), (73) and (74). We start with 2 f1 = − µ4−d 3

Z

~k 2 dd k , (2π)d (k02 − E 2 )2 ((k0 − g2 k02 )2 − E 2 )

(B29)

and rewrite it with α = 2 and F (k0 , ~k) = ~k 2 , as f1 = −

2iΩd−1 µ4−d 3

Z

∞

g2 m

d−3 (2) dt 1 (t2 − g22 m2 ) 2 l1 , d−1 d (2π) g2

(B30)

22 where 

 (2)

l1

Z =

1

Z

 dx ~k 2  h 2 + k0E

dk0E 0

x t2 g22

x−

t(1−x) g22

+

(1−x)2 4g22

i3 − h 2 + k0E

x t2 g22

x+

t(1−x) g2

+

(1−x)2 4g22

 i3  .

(B31)

Using (B12) with r = 3, we arrive at (2)

l1

3π 8

=

With ε = t(1 − x), ~k 2 =



1

Z

dx ~k 2 

0

1 (t2 g22

 x 2 ( gt 2 x 2

−

t(1−x) g22

+

(1−x)2 5/2 ) 4g22

−

x 2 ( gt 2 x 2

+

t(1−x) g22

+

(1−x)2 5/2 ) 4g22

.

(B32)

− g22 m2 ) and using the approximation (B17) gives

5i f1 = − 16π 2

Z

1

dx 0

µ

Z

∞

dt

g2d−4

g2 m

t(1 − x)x(t2 − g22 m2 ) (t2 x +

d−1 2

! .

(1−x)2 7/2 ) 4

(B33)

Again, performing the t integral followed by the x integral, we arrive at      i 1 3 1 1 f1 = AppellF , 1, 1, , a, b + AppellF , 1, 1, , a, b , − 1 1 2π 2 2 2 2 2

(B34)

with a, b are given by (B20). Now we consider f2n

4−d

Z

= 2µ

dd k k0 , 2 d 2 2 (2π) (k0 − E ) ((k0 − g2 k02 )2 − E 2 )

(B35)

which we rewrite as f2n = 2iΩd−1 µ4−d

Z

∞

g2 m

d−3 (2) 1 dt (t2 − g22 m2 ) 2 l2 . d−1 d (2π) g2

(B36)

(2)

After the same change of variables in l2 , a linear term in k00 vanishes and we are left with  Z Z 1 Z x x (1 − x)  (2) l2 = dk0E −h dx h i 3 2g2 2 + t2 x − t(1−x) + (1−x)2 2 + t2 x + t(1−x) + 0 k0E k0E g2 g2 4g 2 g2 g2 2

2

2

2

2

 (1−x)2 4g22

 i3  . (B37)

Using (B12), we have (2)

l2

3π = 8

Z 0

1

 (1 − x)  dx 2 2g2 ( t2 x − g2

 x t(1−x) g22

+

(1−x)2 5/2 ) 4g22

−

x 2

( gt 2 x + 2

t(1−x) g22

+

(1−x)2 5/2 ) 4g22

.

(B38)

Again, we define ε = t(1 − x) and consider again the approximation (B17) and replace d = 4 in all the finite terms. We arrive at the simpler expression ! (d−3) Z 1 Z ∞ 15i µ (1 − x)2 tx(t2 − g22 m2 ) 2 n f2 = dx d−5 dt . (B39) 2 32π 2 0 g2 (t2 x + (1−x) )7/2 g2 m 4 Integrating in t and then in x at lowest order, we arrive at    2ig2 1 3 f2n = AppellF , 1, 1, , a, b , 1 π2 2 2 with the same a, b given by (B20).

(B40)

23 To find f3n we take advantage of having already calculated the piece f1 and by considering f3n = 2B Z (n · k)(n · Q) 2 d4 k f3n = 2 2 − f1 ≡ T1 + T2 − f1 , (n ) (2π)4 (k 2 − m2 )2 (Q2 − m2 )

(B41)

we focus on the pieces 4−d

k02 dd k , (2π)d (k 2 − m2 )2 (Q2 − m2 )

Z

T1 = µ

(B42)

and T2 = −g2 µ4−d

Z

dd k

(k 2

−

k03 2 m )2 (Q2

− m2 )

.

(B43)

Following the same technique we find      i 1 3 1 3 5 T1 = − 2 AppellF1 , 1, 1, , a, b − AppellF1 , 1, 1, , a, b 4π 2 2 3 2 2        1 i 3 3 5 3 5 7 AppellF1 + , 2, 2, , a, b − AppellF1 , 2, 2, , a, b + AppellF1 , 2, 2, , a, b 2π 2 2 2 2 2 5 2 2   9 1 7 , 2, 2, , a, b − AppellF1 , 7 2 2

(B44)

and         3 5 i 3 4 3 5 3i 4g 2 m2 1 AppellF , 1, 1, , a, b − , 2, 2, , a, b − AppellF , 2, 2, , a, b 1 − AppellF 1 1 1 8π 2 3 2 2 4π 2 2 2 3 2 2       6 7 4 9 1 11 5 7 9 + AppellF1 , 2, 2, , a, b − AppellF1 , 2, 2, , a, b + AppellF1 , 2, 2, , a, b , (B45) 5 2 2 7 2 2 9 2 2

T2 =

and therefore f3n

       i 3 3 3 1 5 3i 1 1 3 + 2 −AppellF1 − , 1, 1, , a, b − AppellF1 , 1, 1, , a, b + AppellF1 , 1, 1, , a, b = 8π 2 2π 2 2 2 2 2 6 2 2       1 1 3 2 3 5 1 7 9 , 2, 2, , a, b − AppellF1 , 2, 2, , a, b + AppellF1 , 2, 2, , a, b + AppellF1 2 2 2 3 2 2 7 2 2   1 11 9 − AppellF1 , 2, 2, , a, b . (B46) 18 2 2

with a, b given by (B20). The second-order terms f4n , f5n , f6n .

4.

For the second-order contributions we start with f4n in Eq.(81) and consider Z ~k 2 k0 1 dd k F¯ = − µ4−d . d 2 2 3 (2π) (k − m )3 (Q2 − m2 )

(B47)

We write i F¯ = Ωd−1 µ4−d 2

Z

∞

g2 m

d−3 (3) 1 dt (t2 − g22 m2 ) 2 l1 , d−1 d (2π) g2

(B48)

where (3) l1

Z =

  2 (1 − x) ~ dk0E k 2g2

 Z  × 

0

 1

dx h

x2 2 k0E

+

t2 x g22

−

t(1−x) g22

Z +

i − 2 4

(1−x) 4g22

0

1

dx h

x2 2 k0E

+

t2 x g22

+

t(1−x) g22

+

(1−x)2 4g22

 i4  .

(B49)

24 Using (B12), we arrive at (3)

l1

5π = 16

1

Z 0

 (1 − x)  dx(~k 2 ) 2 2g2 ( t2 x −

x2 t(1−x) g22

g2

+

(1−x)2 7/2 ) 4g22

−



x2 2

( gt 2 x + 2

t(1−x) g22

+

(1−x)2 7/2 ) 4g22

.

(B50)

We again use ε = t(1 − x) and write (3) l1

5πg26 = 32

Z

1

 dx

0

t2 − g22 m2 g22

"

(1 − x)x2 (t2 x − ε +

(1−x)2 7/2 ) 4

−

#

(1 − x)x2 (t2 x + ε +

(1−x)2 7/2 ) 4

.

(B51)

We consider the approximation (B17) and replace d = 4 in all the finite terms. We arrive at ! (d−1) Z 1 Z ∞ µ (1 − x)2 tx2 (t2 − g22 m2 ) 2 35i ¯ . dx d−5 dt F = 2 256π 2 0 g2 )9/2 (t2 x + (1−x) gm 4

(B52)

Integrating in t and then in x and considering f4n = 4F¯ we have    ig2 1 3 f4n = AppellF , 1, 1, , a, b , 1 4π 2 2 2

(B53)

with a, b given by (B20). ¯ From the integrals (78) one can show that Now we compute f5n and focus on D. Z f4n d4 k (3k03 − 3g2 k04 + k0~k 2 − g2 k02~k 2 ) 1 ¯ − , D= 3 (2π)4 (k 2 − m2 )3 ((k0 − g2 k02 )2 − E 2 ) 4

(B54)

and write f5n = P − T − Y − 2f4n ,

(B55)

by considering the definitions of the d dimensions integrals Z dd k k03 4−d P = 4µ , (2π)d (k 2 − m2 )3 ((k0 − g2 k02 )2 − E 2 ) Z k04 dd k , T = 4g2 µ4−d (2π)d (k 2 − m2 )3 ((k0 − g2 k02 )2 − E 2 ) Z ~k 2 k 2 µ4−d dd k 0 Y = 4g2 . d 2 2 3 3 (2π) (k − m ) ((k0 − g2 k02 )2 − E 2 )

(B56)

Let us compute P , with α = 3, we have P =−

3i Ωd−1 µ 2

Z

∞

gm

(d−3) (3) dt 1 (t2 − g22 m2 ) 2 l2 , d (d−1) (2π) g

(B57)

2

with (3)

l2

Z =

  (1 − x)3 2 (1 − x) dk0E −3k0E + 2g2 8g23

 Z  × 

0

(B58) 

1

dx h

x 2 + k0E

t2 x g22

−

2

t(1−x) g22

Z +

(1−x)2 4g22

i4 −

0

1

dx h

x 2 + k0E

t2 x g22

+

2

t(1−x) g22

+

(1−x)2 4g22

 i4  .

After some algebra we find at lowest order        3 5 4 3 5 16 5 7 ig2 P = 2 AppellF1 , 1, 1, , a, b − AppellF1 , 3, 3, , a, b + AppellF1 , 3, 3, , a, b (B59) π 2 2 3 2 2 5 2 2       24 7 9 16 9 11 4 11 13 − AppellF1 , 3, 3, , a, b + AppellF1 , 3, 3, , a, b − AppellF1 , 3, 3, , a, b . 7 2 2 9 2 2 11 2 2

25 In the same way we find        3 5 2 3 5 10 7 ig2 1 AppellF1 , 1, 1, , a, b + AppellF1 , 3, 3, , a, b − 2AppellF1 , 3, 3, , a, b T = − 2 π 4 2 2 3 2 2 2 2       20 7 9 20 9 11 10 11 13 + AppellF1 , 3, 3, , a, b − AppellF1 , 3, 3, , a, b + AppellF1 , 3, 3, , a, b 7 2 2 9 2 2 11 2 2        3 2 13 15 ig2 5 18 5 7 − AppellF1 , 3, 3, , a, b + 2 2AppellF1 , 2, 2, , a, b − AppellF1 , 2, 2, , a, b 13 2 2 2π 2 2 5 2 2     18 7 9 1 1 11 + AppellF1 , 2, 2, , a, b − AppellF1 , 2, 2, , a, b , (B60) 7 2 2 3 3 2 and      1 ig2 3 1 3 5 Y = 2 AppellF1 , 1, 1, , a, b − AppellF1 , 1, 1, , a, b π 2 2 3 2 2      1 ig2 3 3 5 6 5 7 − 2 2AppellF1 ( , 2, 2, , a, b) − 2AppellF1 , 2, 2, , a, b + AppellF1 , 2, 2, , a, b π 2 2 2 2 5 2 2   2 7 9 − AppellF1 , 2, 2, , a, b , 7 2 2

(B61)

with a, b are given by (B20). To find f5n we just have to add the above contributions. ¯ we need the two pieces C¯ and E. ¯ We have To compute f6n = −C¯ + 4E fn C¯ = 2 − g2 T1 , 2

(B62)

and from (78) ¯=1 E 3

Z

d4 k −~k 2 k0 + g2~k 2 k02 , (2π)4 (k 2 − m2 )3 (Q2 − m2 )

(B63)

which can be given in terms of coefficients we have computed before, from (B56), (B47) and f4n = 4F¯ , as n ¯ = Y + f4 . E 4 4

(B64)

f2n + g2 T1 + Y + f4n , 2

(B65)

Finally we can write f6n = −

which can be derived with the expressions we have calculated before.

[1] D. Colladay and V. A. Kostelecky, Phys. Rev. D55, 6760 (1997), [hep-ph/9703464]; Phys. Rev. D58, 116002 (1998), hep-ph/9809521. [2] V. A. Kostelecky, Phys. Rev. D69, 105009 (2004), hep-th/0312310. [3] G. Amelino-Camelia, J. R. Ellis, N. E. Mavromatos, D. V. Nanopoulos and S. Sarkar, Nature 393, 763 (1998), astroph/9712103. [4] V. A. Kostelecky and S. Samuel, Phys. Rev. D39, 683 (1989). [5] R. Gambini and J. Pullin, Phys. Rev. D59, 124021 (1999); J. Alfaro, H. A. Morales-Tecotl and L. F. Urrutia, Phys. Rev. Lett. 84, 2318 (2000). [6] A. Jenkins, Phys. Rev. D9, 105007 (2004), hep-th/0311127; O. Bertolami, J. Paramos, Phys. Rev. D72, 044001 (2005), hep-th/0504215. [7] R. C. Myers and M. Pospelov, Phys. Rev. Lett. 90, 211601 (2003), hep-ph/0301124. [8] R. Jackiw and S. Y. Pi, Phys. Rev. D68, 104012 (2003), gr-qc/0308071. [9] S. W. Hawking, T. Hertog, Phys. Rev. D65, 105315 (2002), hep-th/0107088; I. Antoniadis, E. Dudas, D. M. Ghilencea, Nucl. Phya. B767, 29 (2007), hep-th/0608094.

26 [10] T. Mariz, Phys. Rev. D83, 045018 (2011), arXiv: 1010.5013; T. Mariz, J. R. Nascimento, A. Yu. Petrov, Phys. Rev. D85, 125003 (2012), arXiv: 1111.0198; A. Celeste, T. Mariz, J. R. Nascimento, A. Yu. Petrov, Phys. Rev. D93, 065012 (2016), arXiv: 1602.02570. [11] C. Marat Reyes, Phys. Rev. D82, 125036 (2010), arXiv: 1011.2971. [12] M. Maniatis, C. Marat Reyes, Phys. Rev. D89, 056009 (2014), arXiv: 1401.3572; J. Lopez-Sarrion, C. Marat Reyes, Eur. Phys. J. C72, 2140 (2012), arXiv: 1109.5927; Eur. Phys. J. C73, 2391 (2013), arXiv: 1304.4966. [13] T. D. Lee and G. C. Wick, Nucl. Phys. B9, 209 (1969); T. D. Lee, G. C. Wick, Phys. Rev. D2, 1033 (1970). [14] F. R. Klinkhamer and M. Schreck, Nucl. Phys. B 848, 90 (2011); M. Schreck, Phys. Rev. D 86, 065038 (2012). [15] M. Schreck, Phys. Rev. D 89, no. 10, 105019 (2014); Phys. Rev. D 90, no. 8, 085025 (2014). [16] T. Jacobson, S. Liberati, D. Mattingly, F. W. Stecker, Phys. Rev. Lett. 93, 021101 (2003), astro-ph/0309681; O. Bertolami, J. G. Rosa, Phys. Rev. D71, 097901 (2005), hep-ph/0412289. [17] J. Collins, A. Perez, D. Sudarsky, L. Urrutia and H. Vucetich, Phys. Rev. Lett. 93, 191301 (2004), gr-qc/0403053. [18] C. M. Reyes, S. Ossandon and C. Reyes, Phys. Lett. B746, 190 (2015), arXiv: 1409.0508. [19] M. Cvetic, T. Mariz, A. Yu. Petrov, Phys. Rev. D92, 085041 (2015), arXiv: 1509.00251. [20] F. de O. Salles and I. L. Shapiro, Phys. Rev. D89, 084054 (2014), arXiv: 1401.4583; G. Cusin, F. de O. Salles and I. L. Shapiro, Phys.Rev. D93, 044039 (2016), arXiv:1503.08059. [21] M. Cambiaso, R. Lehnert and R. Potting, Phys. Rev. D90, 065003 (2014), arXiv: 1401.0317. [22] G. Kallen, Helv. Phys. Acta 25, 417 (1952); H. Lehmann, Nuovo Cim. 11, 342 (1954). [23] H. Lehmann, K. Symanzik, W. Zimmermann, Nuovo Cim. 1, 205 (955); Nuovo Cim. 6, 319 (1957). [24] R. Potting, Phys. Rev. D85, 045033 (2012), arXiv: 1201.3045. [25] L. C. T. Brito, H. G. Fargnoli and A. P. Baeta Scarpelli, Phys. Rev. D87, no. 12, 125023 (2013), arXiv:1304.6016. [26] V. A. Kostelecky and M. Mewes, Phys. Rev. D 80 (2009) 015020; A. Kostelecky and M. Mewes, Phys. Rev. D 85, 096005 (2012); A. Kosteleck and M. Mewes, Phys. Rev. D 88, no. 9, 096006 (2013). [27] C. M. Reyes and L. F. Urrutia, Phys. Rev. D95, 015024 (2017), arXiv: 1610.06051. [28] D. V. Ahluwalia and D. Grumiller, JCAP 0507, 012 (2005), hep-th/0412080. [29] D. Anselmi, JHEP 0802, 051 (2008), arXiv: 0801.1216. [30] S. Carroll, H. Tam, Phys. Rev. D78, 044047 (2008), arXiv: 0802.0521; M. Gomes, J. R. Nascimento, A. Yu. Petrov, A. J. da Silva, Phys. Rev. D81, 045018 (2010), arXiv: 0911.3548; ”On the Aether-like Lorentz-breaking action for the electromagnetic field”, arXiv: 1008.0607; G. Gazzola, H. G. Fargnoli, A. P. Baeta Scarpelli, M. Sampaio, M. C. Nemes, J. Phys. G39, 035002 (2012), arXiv: 1012.3291; T. Mariz, J. R. Nascimento, A. Yu. Petrov, W. Serafim, Phys.Rev. D90, 045015 (2014), arXiv: 1406.2873; R. Casana, M. M. Ferreira, R. Maluf, F. E. P. dos Santos, Phys. Lett. B726, 815 (2013), arXiv: 1302.2375. [31] D. Anselmi and M. Piva, arXiv:1703.04584 [hep-th]; arXiv:1703.05563 [hep-th]. [32] V. A. Kostelecky, C. D. Lane and A. G. M. Pickering, Phys. Rev. D 65, 056006 (2002). [33] A. Ferrero and B. Altschul, Phys. Rev. D 84, 065030 (2011). [34] D. Anselmi and M. Taiuti, Phys. Rev. D81, 085042 (2010), arXiv: 0912.0113; D. Anselmi and E. Ciuffoli, Phys. Rev. D81, 085043 (2010), arXiv: 1002.2704. [35] R. Casana, M. M. Ferreira, J. S. Rodrigues, M. R. O. Silva, Phys. Rev. D80, 085026 (2009), arXiv: 0907.1924; R. Casana, A. R. Gomes, M. M. Ferreira, P. R. D. Pinheiro, Phys. Rev. D80, 125040 (2009), arXiv: 0909.0544; F. R. Klinkhammer, M. Schreck, Nucl. Phys. B856, 666 (2012), arXiv: 1110.4101; R. Casana, E. S. Carvalho, M. M. Ferreira, Phys. Rev. D84, 045008 (2011), arXiv: 1107.2664; R. Casana, M. M. Ferreira, R. P. M. Moreira, Phys. Rev. D84, 125014 (2011), arXiv: 1108.6193. [36] S. Minwalla, M. van Raamsdonk, N. Seiberg, JHEP 02, 020 (2000), hep-th/9912072.