Free energy of Lorentz-violating QED at high temperature
M. Gomes, T. Mariz, J. R. Nascimento, A. Yu. Petrov, A. F. Santos, A. J. da Silva
aa r X i v : . [ h e p - t h ] F e b Free energy of Lorentz-violating QED at high temperature
M. Gomes, T. Mariz, J. R. Nascimento, A. Yu. Petrov, A. F. Santos, and A. J. da Silva Instituto de F´ısica, Universidade de S˜ao PauloCaixa Postal 66318, 05315-970, S˜ao Paulo, SP, Brazil ∗ Instituto de F´ısica, Universidade Federal de Alagoas, 57072-270, Macei´o, Alagoas, Brazil † Departamento de F´ısica, Universidade Federal da Para´ıbaCaixa Postal 5008, 58051-970, Jo˜ao Pessoa, Para´ıba, Brazil ‡ Abstract
In this paper we study the one- and two-loop contribution to the free energy in QED with the Lorentzsymmetry breaking introduced via constant CPT-even Lorentz-breaking parameters at the high temperaturelimit. We find the impact of the Lorentz-violating term for the free energy and carry out a numericalestimation for the Lorentz-breaking parameter. ∗ Electronic address: mgomes,[email protected] † Electronic address: [email protected] ‡ Electronic address: alesandro,jroberto,petrov@fisica.ufpb.br . INTRODUCTION Nowadays the Lorentz symmetry breaking is treated as an important ingredient of field theorymodels for probing quantum gravity phenomena. In fact, Lorentz-violating theories have beenstudied in various contexts, such as string theory [1], noncommutative field theory [2], and, morerecently, Horava-Lifshitz gravity [3].Many of the effects observed for the Lorentz-breaking field theories at zero temperature areknown to persist also at finite temperatures. For example, the dependence of the loop correctionson the regularization scheme in the finite temperature case was shown to arise in the one-loop orderin the Lorentz-breaking quantum electrodynamics (QED) [4]. Analogous situation takes place inthe Lorentz-breaking Yang-Mills theory [5]. Different issues related to the Lorentz-violating QEDin the finite temperature case were considered in [6]. In [7] the finite temperature properties in theCPT-odd Lorentz-breaking extension of QED for a purely spacelike background were studied, andin [8] these properties were analysed for the CPT-even Lorentz-breaking extension.Most of the previous studies of the Lorentz-breaking theories, including those described in [4, 5],were based on the use of couplings which violate not only Lorentz symmetry but also the CPTsymmetry, although CPT-even Lorentz breaking interactions are also possible and certainly requiremore detailed study. In this paper, we will see that, at the high temperature regime, the linearcontribution in the Lorentz-breaking parameter may arise.The aim of this paper is the study of one and two-loop corrections to the free energy (secondorder in the coupling constant e ) of QED in the presence of Lorentz breaking terms, at finitetemperature. As it is known the free energy provides an important information to different physicalissues such as plasma behavior, solar interior and Big Bang nucleosynthesis (BBN) (see, [9–11]). Inthis context, some aspects of the corrections to the free energy in theories without Lorentz-breakingare discussed in [12, 13].For estimating bounds of the Lorentz-breaking parameter, we use information of the BBN (fora review, see [14]), which is one of the observational pillars of the standard cosmology. Note thatthe Lorentz-violating parameter can explain for the light elements abundance. In particular, thedifference between the theoretical and observational results can be understood as come from acontribution of the Lorentz-breaking parameter to the primordial helium abundance.The structure of the paper is as follows. In Sec. II we present the basic features of the Lorentz-violating QED in the regime of high temperature. In Sec. III we calculate the one and two-loopcontributions to the free energy for QED involving the Lorentz-breaking fermion coupling. In2ec. IV, by using information about the primordial helium abundance, we obtain a numericalestimation for the Lorentz violation parameter. A summary is presented in Sec. V. II. LORENTZ-VIOLATING QED AT HIGH TEMPERATURE
Let us start by considering the Lagrangian of the Lorentz- and CPT-violating QED extension[15] L = − F µν F µν −
14 ( k F ) µνλρ F µν F λρ + 12 ( k AF ) µ ǫ µνλρ A ν F λρ + ¯ ψ ( i Γ µ D µ − M ) ψ + L gf + L gh , (1)where Γ µ = γ µ + Γ µ , M = m + M , withΓ µ = c µν γ ν + d µν γ γ ν + e µ + if µ γ + g λνµ σ λν (2) M = a µ γ µ + b µ γ γ µ + H µν σ µν , (3)and D µ = ∂ µ + ieA µ . L gf is the gauge fixing term and L gh is the ghost field term, which decouplesfrom the rest of the Lagrangian. The coefficients carrying an odd (even) number of Lorentz indicesare CPT-odd (-even).The two Lorentz-violating terms of the photon sector, the Chern-Simons-like (CPT-odd) and( k F ) µνλρ F µν F λρ (CPT-even) terms, can be induced by radiative corrections from the terms withthe coefficients b µ and c µν of the fermion sector, respectively, so that ( k AF ) µ ∝ b µ [16] and [17]( k F ) µνλρ ∝ g µλ ( c νρ + c ρν ) + 12 g νρ ( c µλ + c λµ ) − g µρ ( c νλ + c λν ) − g νλ ( c µρ + c ρµ ) . (4)The coefficients a µ , b µ , H µν , and ( k AF ) µ have dimensions of mass, while c µν , d µν , e µ , f µ , g λνµ , and ( k F ) µνλρ are dimensionless. In the high temperature regime ( T ≫ M ) the dimensionfulcoefficients may be neglected. For instance, the b µ -corrections to the free energy are observed tobe proportional to b T , similarly to the scenario occurring with the corrections stemming fromthe Chern-Simions-like term [7]. Thus both b µ and ( k AF ) µ are negligible at high temperature, aswell as a µ and H µν .Among the dimensionless coefficients, e µ , f µ , and g λνµ are expected to be much smaller than theother, because their terms cannot be obtained directly from the standard model extension [15] (formore details, see also [18]). Moreover, if we require that the theory at high temperature is invariantunder chiral transformations, these terms are ruled out, since { γ , e µ + if µ γ + g λνµ σ λν } 6 = 0.Therefore, the remaining coefficients are c µν , d µν , and ( k F ) µνλρ . Now, in order to get a Cliffordalgebra for Γ µ = γ µ + c µν γ ν + d µν γ γ ν , we must choose d µν = Q ( δ µν + c µν ), where Q is a constant319]. With this assumptions, the theory (1) becomes L high = − F µν F µν −
14 ( k F ) µνλρ F µν F λρ + ¯ ψ [ i∂ µ ( g µν + c µν )˜ γ ν − e A µ ( g µν + c µν )˜ γ ν ] ψ + L gf + L gh , (5)where ˜ γ µ = (1 + Qγ ) γ µ .We now assume rotational invariance, such that the coefficients in (5) may be reduced toproducts of a given unit timelike vector u µ , which describes the preferred frame (see [20], for moredetails). Proceeding in this way, we write c µν = κ u µ u ν , and( k F ) µνλρ = ˜ κ ( g µλ u ν u ρ + g νρ u µ u λ − g µρ u ν u λ − g νλ u µ u ρ ) , (6)see Eq. (4), where u µ = (1 , , ,
0) and now κ and ˜ κ are the coefficients that determine the scale ofLorentz violation. By choosing α = 1 (Feynman gauge) and ˜ κ = (1 + κ ) κ , which is convenient tokeep the Lagrangian (5) formally covariant, we get L = −
14 ˜ F µν ˜ F µν + i ¯ ψ ˜ ∂ µ ˜ γ µ ψ − e ¯ ψ ˜ A µ ˜ γ µ ψ + 12 ( ˜ ∂ µ ˜ A µ ) + ( ˜ ∂ µ ¯ C )( ˜ ∂ µ C ) , (7)where ˜ F µν = ˜ ∂ µ ˜ A ν − ˜ ∂ ν ˜ A µ , with ˜ ∂ µ = ((1 + κ ) ∂ , ∂ i ) and ˜ A µ = ((1 + κ ) A , A i ). The correspondingFeynman rules are shown in Fig. 1. FIG. 1: Feynman Rules. Continuous, wavy, and dashed lines represent the fermion, photon, and ghostpropagators, respectively, with momenta ˜ p µ = ((1 + κ ) p , p i ). The fermion-photon vertex is the usual one, − ie ˜ γ µ . III. THE FREE ENERGY
Let us now compute the free energy per unit of volume (pressure) as a function of the temper-ature T and of the coupling constant e , in the regime of high temperature. We shall calculate theexpression for the pressure to order e , which has the form P = T ln ZV = P + P , (8)where P is the zero order contribution in the coupling constant and P is the second order one.The free energy density is minus the expression (24).4 . One-loop contribution The lowest-order contributions are given by the three one-loop vacuum diagrams, displayed inFig. 2, and written in the imaginary time formalism as P = tr PZ { dp } ln ˜ p + 4( − ) PZ dp ln ˜ p + 2( ) PZ dp ln ˜ p , (9)respectively, where we have introduced the shorthand notation PZ { dp } = T X p =2 π ( n + ) T Z d p (2 π ) (10)for fermionic loop momenta and PZ dp = T X p =2 πnT Z d p (2 π ) (11)for bosonic loop momenta, and ˜ p = ˜ p µ ˜ γ µ .As usual, the bosonic contributions of Eq. (9) have four degrees of freedom for the gauge field(Fig. 2(b)) and two degrees of freedom for the ghost field (Fig. 2(c)). The fermionic contribution, ( a ) ( b ) ( c ) FIG. 2: One loop vacuum diagrams. after the calculation of the trace, P f = 2 PZ { dp } ln[(1 − Q )( p (1 + κ ) + p )] , (12)has the same form as the bosonic contributions, P b = − PZ dp ln( p (1 + κ ) + p ) . (13)In order to evaluate these expressions we proceed similarly to [21], such that P = Z d p (2 π ) (cid:20) | p | κ + 4 β ln (cid:18) e − | p | β κ (cid:19) − β ln (cid:18) − e − | p | β κ (cid:19)(cid:21) , (14)which is independent of the parameter Q . The zero temperature (divergent) part is absorbed in arenormalization of the vacuum energy [12]. Therefore, after integrating in the angular variables,we get P = 2 π β Z ∞ dr r ln (cid:16) e − rβ κ (cid:17) − π β Z ∞ dr r ln (cid:16) − e − rβ κ (cid:17) , (15)5nd finally, we obtain P = 11 π T (1 + κ ) . (16)Therefore we conclude that the only impact of the Lorentz symmetry breaking in the case ofthe CPT-even Lorentz-breaking parameter consists in the modification of a constant factor. B. Two-loop contribution
In this subsection we study the two-loop contributions to the free energy. The Lorentz violatingcontribution to free energy up to the two-loop order (which corresponds to the second order in e ,is given by Fig. 3), which can be written as FIG. 3: Two-loop diagram. P = 12 e PZ { dp } PZ { dq } tr (cid:20) ˜ γ µ ˜ p ˜ γ µ ˜ q p + ˜ q ) (cid:21) . (17)Note that, due to the fact that ˜ γ µ p α ˜ γ α = γ µ p α γ α , (18)the above second order contribution is also independent of the parameter Q , and so on for theother contributions. Thus, we can rewrite (17) as P = 12 e PZ { dp } PZ { dq } tr (cid:20) γ µ ˜ p α γ α ˜ p γ µ ˜ q β γ β ˜ q p + ˜ q ) (cid:21) , (19)so that, after calculating the trace, we get P = 2 e PZ { dp } PZ { dq } (cid:20) − p ˜ q + 1˜ q (˜ p + ˜ q ) + 1˜ p (˜ p + ˜ q ) (cid:21) . (20)Using the results for the sum-integrals, PZ { dp } PZ { dq } p ˜ q = 1576 T (1 + κ ) , (21) PZ { dp } PZ { dq } q (˜ p + ˜ q ) = − T (1 + κ ) , (22)6e arrive at the following Lorentz violating contribution to the free energy, P = − e T (1 + κ ) . (23)Thus, the modification in the free energy due to the Lorentz-breaking parameter for the two-looporder consists only in multiplying by a constant, just as in the one-loop order.Therefore, the expression for the pressure to order e in the presence of Lorentz symmetrybreaking looks like P = π T (cid:20) κ ) − e π (1 + κ ) (cid:21) . (24) IV. NUMERICAL ESTIMATIONS
In order to estimate a bound for the Lorentz violating parameter κ , we use the theoreticalpredictions of the primordial helium abundance Y developed in the references [9–11]. A way todetermine Y consists in the analysis of the change in thermodynamics quantities such as the energydensity ρ , the pressure P and the neutrino temperature T ν . Using the result (24), let us now studythese changes in the presence of the Lorentz-breaking.The contribution to the energy density can be found from the standard thermodynamic relation ρ = − P + T ( ∂P/∂T ), so that ρ = π T
15 ( N + δN ) , (25)where N = and δN ≈ − .
007 + 8 . κ . Thus, we obtain∆ ρρ ≈ − . × − + 2 . κ. (26)The fraction ∆ Y is affected by QED in several ways, as can be seen in [10]. The total effect isapproximately ∆ Y ≈ . × − + 0 .
15 ∆ T ν T ν + 0 .
07 ∆ ρρ , (27)where T ν depends on energy density ρ (for more details, see [22]).The usual theoretical result, without parameter κ , is ∆ Y ≈ − , whereas the experimental oneis ∆ Y ≈ − [14]. Therefore, an upper bound for κ necessary for the coincidence of the theoreticaland experimental results must be κ ∼ − . This result agrees with the value found in [23, 24].7 . SUMMARY We have calculated the contributions to the free energy in the rotationally invariant Lorentz-violating QED in one- and two-loop approximations at high temperature. The correspondingcorrection to the pressure was then determined.We also observe that Lorentz violation can be used to explain the difference between theoreticaland experimental predictions of the primordial helium abundance. By matching these predictions,we have estimated the Lorentz-breaking parameter κ ∼ − , which agrees with that obtained in[23]. Acknowledgements.
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