PHYSICAL REVIEW D, VOLUME 65, 056006
One-loop renormalization of Lorentz-violating electrodynamics
V. Alan Kostelecky´ Physics Department, Indiana University, Bloomington, Indiana 47405
Charles D. Lane Physics Department, Berry College, Mount Berry, Georgia 30149
Austin G. M. Pickering Physics Department, Indiana University, Bloomington, Indiana 47405 ͑Received 10 October 2001; published 31 January 2002͒ We show that the general Lorentz- and CPT-violating extension of quantum electrodynamics is one-loop renormalizable. The one-loop Lorentz-violating beta functions are obtained, and the running of the coefficients for Lorentz and CPT violation is determined. Some implications for theory and experiment are discussed.
DOI: 10.1103/PhysRevD.65.056006 PACS number͑s͒: 11.30.Cp, 11.10.Hi, 12.20.Ϫm
I. INTRODUCTION regularization ͓6,7͔, although we have also checked their va- lidity in Pauli-Villars regularization ͓8͔. The standard model of particle physics is invariant under In the standard-model extension, all Lorentz- and Lorentz and CPT transformations. However, the possibility CPT-violating effects are controlled by a set of coefficients that nature exhibits small violations of Lorentz and CPT that can be regarded as originating in an underlying theory at symmetry appears compatible with quantum field theory and the Planck scale. For example, they might be associated with with existing experiments ͓1͔. A general description of the expectation values in string theory ͓9͔, and specific nonzero associated effects can be formulated at the level of quantum coefficients emerge in realistic noncommutative field theo- field theory as a Lorentz- and CPT-violating standard-model ries ͓10͔. Several of these coefficients in different sectors of extension ͓2͔. The Lagrangian of this theory includes all pos- the standard-model extension are now bounded by experi- sible operators that are observer Lorentz scalars and that are ments involving hadrons ͓11–14͔, protons and neutrons ͓15͔, formed from standard-model fields and coupling coefficients electrons ͓16,17͔, photons ͓18͔, and muons ͓19͔. Our results with Lorentz indices. Imposing the usual SU(3)ϫSU(2) in this work can be used to gain insight into the relationships ϫU(1) gauge invariance and restricting attention to low- among coefficients for Lorentz and CPT violation as the energy effects, the standard-model extension is well approxi- scale ranges between low and high energies. mated by the usual standard model together with all possible A basic tool for studying quantum physics over different Lorentz-violating terms of mass dimension four or less that scales is the renormalization group ͓20,21͔. Here, we discuss are constructed from standard-model fields. its relevance in the context of the Lorentz- and Among the interesting open issues associated with Lor- CPT-violating standard-model extension. We use our calcu- entz and CPT violation is the manner in which the low- lations of the one-loop divergences to extract the correspond- energy theory connects to the underlying Planck-level theory ing beta functions for all the coefficients for Lorentz and as the energy scale is increased. Some insight into this link CPT violation in the general QED extension. Solving the has been obtained through the study of causality and stability associated set of coupled partial-differential equations for the in Lorentz-violating quantum field theory ͓3͔. In the present renormalized coefficients yields their running as the scale is work, we study a different facet of this connection, involving changed. Knowledge of this running offers some insight into the role of radiative corrections and the renormalization the possible relative sizes of nonzero Lorentz- and group. CPT-violating effects. To provide a definite focus and a tractable scope, we limit This paper is organized as follows. Section II provides attention here to the special case of effects from one-loop some basic information about the general Lorentz- and divergences in the Lorentz- and CPT-violating quantum CPT-violating QED extension. Renormalizability of the electrodynamics ͑QED͒ of a single fermion. This QED ex- theory is considered in Sec. III. Some general issues are dis- tension can be regarded as a specific limit of the standard- cussed, following which we present the results of our one- model extension. Even in this simplified limit, relatively loop calculation. We establish the absence of divergent cubic little is known about loop effects. Some one-loop calcula- and quartic photon interactions, present explicit results for all tions have been performed in the photon sector ͓2,4͔, but a divergent radiative corrections to the Lagrangian, and show comprehensive treatment has been lacking. One goal of the that the Ward identities are preserved at this order. Section present work is to fill this gap. Tools such as a generalization IV begins with a discussion of the application of the of the Furry theorem ͓5͔ are developed, and all divergent renormalization-group method in the context of Lorentz and one-loop corrections are determined. We use these to prove CPT violation. The one-loop beta functions for all param- one-loop renormalizability and gauge invariance of the eters in the theory are then derived. The resulting coupled theory. The calculations are presented here in dimensional partial-differential equations are solved for the running pa-
0556-2821/2002/65͑5͒/056006͑12͒/$20.0065 056006-1 ©2002 The American Physical Society KOSTELECKY´ , LANE, AND PICKERING PHYSICAL REVIEW D 65 056006 rameters, and some implications for experiment are consid- TABLE I. Discrete-symmetry properties. ered. A summary is provided in Sec. V. The Feynman rules used in our analysis are presented in the Appendix. Through- CPTCPCTPTCPT out this work, our notation is that of Refs. ͓2,3͔. c00 ,(kF)0 j0k , ϩϩϩ ϩ ϩ ϩ ϩ c jk ,(kF) jklm II. BASICS ϩϩϪ ϩ Ϫ Ϫ Ϫ b j ,g j0l ,g jk0 ,(kAF) j ϩϪϩ Ϫ ϩ Ϫ Ϫ The Lagrangian L of the general Lorentz- and b0 ,g j00 ,g jkl ,(kAF)0 ϩϪϪ Ϫ Ϫ ϩ ϩ CPT-violating QED extension for a fermion field of mass c0 j ,c j0 ,(kF)0 jkl Ϫϩϩ Ϫ Ϫ ϩ Ϫ m in four spacetime dimensions can be written as ͓2͔ a0 ,e0 , f j ϪϩϪ Ϫ ϩ Ϫ ϩ H jk ,d0 j ,d j0 1 1 ϪϪϩ ϩ Ϫ Ϫ ϩ Lϭ ¯⌫ Ϫ¯ Ϫ H0 j ,d00 ,d jk i DJ M F F ϪϪϪ ϩ ϩ ϩ Ϫ 2 4 a j ,e j , f 0
1 1 Ϫ ͑k ͒F F ϩ ͑k ͒ ⑀A F , ͑1͒ 4 F 2 AF coefficients for Lorentz violation and hence change the phys- ics. The theory ͑1͒ therefore violates particle Lorentz invari- ⌫ϭ␥ϩ⌫ ϭ ϩ ance. where 1 and M m M 1, with Since the coefficients for Lorentz violation are trans- 1 formed by an observer Lorentz transformation, an appropri- ⌫ ϵc ␥ϩd ␥ ␥ϩe ϩif ␥ ϩ g , 1 5 5 2 ate boost can make at least some of them large. To avoid issues with perturbation theory, in this work we limit calcu- lations to concordant frames ͓3͔: ones in which the coeffi- 1 M ϵa␥ ϩb␥ ␥ ϩ H . ͑2͒ cients for Lorentz violation are small compared to the fer- 1 5 2 mion mass m or to the dimensionless charge q. Any frame in
- which the Earth moves nonrelativistically is known experiץAs usual, we define the covariant derivative Dϵ mentally to be concordant, so this restriction offers no prac- AץϩiqA and the electromagnetic field strength Fϵ tical difficulty in applying our results. However, to maintain . AץϪ In the fermion sector, the coefficients for Lorentz viola- generality, we make no assumptions concerning the relative sizes of the different coefficients for Lorentz violation. tion are a , b , c , d , e , f , g , H . Of these, The hierarchy of scales between the coefficients for Lor- a , b , e , f , g govern CPT violation. The coeffi- entz violation and the parameters m,q has implications for cients a , b , H have dimensions of mass, while c , the structure of dominant one-loop Lorentz- and d , e , f , g are dimensionless. Both c and d can CPT-violating effects. In particular, since Lorentz and CPT be taken to be traceless, while H is antisymmetric and violation can be assumed small and since we are interested in g is antisymmetric on its first two indices. In the photon leading-order Lorentz- and CPT-violating effects, it suffices sector, the coefficients for Lorentz violation are (k ) , AF for the purposes of this work to define a one-loop diagram as (k ) . CPT violation is governed only by (k ) , F AF one that contains exactly one closed loop and is either zeroth which has dimensions of mass. The coefficient (k ) is F or first order in coefficients for Lorentz violation. All rel- dimensionless, has the symmetry properties of the Riemann O 2 tensor, and is double traceless: evant one-loop diagrams are therefore (q ) and at most linear in the coefficients for Lorentz violation. Note that it ͒ ϭ ͒ ϭϪ ͒ would be invalid to include nonlinear contributions from the ͑kF ͑kF ͑kF , coefficients for Lorentz violation without also considering ͒ ϩ ͒ ϩ ͒ ϭ ͑kF ͑kF ͑kF 0, multiloop contributions at high order in q, which could be the same order of magnitude. ͒ ϭ ͑ ͒ ͑kF 0. 3 Combined with symmetry arguments, the restriction to linear Lorentz- and CPT-violating effects enables some The requirement that the Lagrangian be Hermitian implies strong predictions about which terms in the Lagrangian ͑1͒ that all the coefficients for Lorentz violation are real. can contribute to the renormalization of any given coeffi- In the Lorentz-violating theory ͑1͒, two distinct types of cient. Since QED preserves C, P, and T invariance, any Lorentz transformation are relevant ͓2͔. The Lagrangian ͑1͒ Lorentz-violating terms mixing linearly under radiative cor- is invariant under observer Lorentz transformations: rota- rections must have identical C, P, and T transformation prop- tions and boosts of the observer inertial frame have no effect erties. Table I lists these properties for the field operators on the physics because both the field operators and the coef- appearing in the Lagrangian ͑1͒. For brevity, the correspond- ficients for Lorentz violation transform covariantly and be- ing coefficients for Lorentz violation are listed in the table cause each term in the Lagrangian ͑1͒ is an observer scalar. rather than the field operators themselves. These coordinate transformations are distinct from rotations The table reveals terms for which the C,P,T symmetries and boosts of a particle or localized field configuration allow mixing under renormalization group flow. Other re- within a fixed observer inertial frame. The latter are called strictions also exist. Since in what follows we adopt a mass- particle Lorentz transformations. They leave unchanged the independent renormalization scheme, operators associated
056006-2 ONE-LOOP RENORMALIZATION OF LORENTZ-... PHYSICAL REVIEW D 65 056006
FIG. 1. One-loop topologies for QED.
with dimensionless coefficients cannot receive corrections These exceptions arise because the mass dimensionality of from ones associated with massive coefficients. Thus, for the coefficients involved ensures a finite result. example, a can receive corrections from e on symmetry As an illustration, the QED diagram in Fig. 1͑e͒ leads to grounds, but e cannot receive corrections from a on di- the set of divergent contributions illustrated in Fig. 2. In mensional grounds. There are also restrictions arising from these diagrams, dimensionless coefficients for Lorentz viola- the rotational invariance of QED. For instance, rotational tion are represented as filled circles while the others are rep- symmetry prevents e0 and f j from mixing at this level of resented by crosses. The notation is detailed in the Appendix. approximation, even though this would be allowed by the As usual, each additional 1PI diagram involves a one-loop C, P, T properties of the corresponding field operators. integration. To evaluate the divergent contributions of these All these features are confirmed by the explicit calculations diagrams to the effective action, a regularization scheme for that follow. the loop integrations is needed. In this paper, we adopt di- mensional regularization. However, we have also repeated III. RENORMALIZABILITY AT ONE LOOP our calculations using Pauli-Villars regularization. It turns out that the usual correspondence between the two schemes In this section, we give an explicit demonstration of holds, supporting the expected scheme independence of the renormalizability at one loop for the QED extension ͑1͒. Fol- physical results. lowing some general considerations, we obtain a generaliza- In dimensional regularization, the presence of particle tion of the Furry theorem and establish the finiteness of the Lorentz violation has little effect on the standard evaluation photon vertices. The divergent propagator and vertex correc- of loop integrals. Although use is sometimes made of the tions are given along with the renormalization factors, and Lorentz properties of the integrand, the standard techniques the Ward identities are shown to hold. hold because the integrands involve momentum variables that behave covariantly under both observer and particle A. Setup transformations, as usual. Moreover, the linearity of Lorentz Renormalizability of a quantum field theory can be violation means that the role of the coefficients for Lorentz viewed as the requirements that the number of primitively violation is limited in this context to contraction with the divergent one-particle-irreducible ͑1PI͒ diagrams is finite and result of the integration. For similar reasons, no new issues that the number of parameters suffices to absorb the corre- arise with manipulations such as Wick rotations. We can sponding infinities. To establish renormalizability of the therefore perform the necessary regularization of divergent QED extension ͑1͒, we first determine the superficial degree of divergence of a general Feynman diagram. Using the Feynman rules for the theory provided in the Appendix, it follows that the superficial degree of divergence D of a gen- eral diagram in the QED extension is
3 Dϭ4Ϫ EϪE ϪV ϪV , ͑4͒ 2 A M 1 AF where E is the number of external fermion legs, EA is the number of external photon legs, V is the number of inser- M 1 tions of the M 1 operator in a fermion propagator, and VAF is the number of insertions of (kAF) in a photon propagator. The expression ͑4͒ shows that there are a finite number of potentially divergent 1PI diagrams at one loop. Their topolo- gies correspond to those of the divergent diagrams associated with conventional QED, displayed in Fig. 1. However, in addition to these usual diagrams, there is a set of diagrams obtained from them by single insertions of coefficients for Lorentz violation allowed by the Feynman rules. All such insertions lead to 1PI divergent diagrams, except for those involving the coefficients a , b , H ,(kAF) inserted into logarithmically divergent diagrams of conventional QED. FIG. 2. Fermion-photon vertices in the QED extension.
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integrals by extending spacetime to dϭ4Ϫ2⑀ dimensions in the conventional way, so that the one-loop divergent correc- tions to the effective Lagrangian take the form of poles in the infinitesimal parameter ⑀, as usual. ␥ Certain diagrams involve factors of 5, which introduces complications in dimensional regularization because the ␥ properties of 5 are dimension dependent. One possibility is to use the ’t Hooft–Veltman definition ͓6͔ of ␥ , in which the 5 FIG. 3. Two contributions to the cubic photon interaction. ␥-matrix algebra is infinite-dimensional in non-integer di- mensions and the first four ␥ are treated differently from ␥ practical or conceptual advantage over more conventional the others. In particular, 5 anticommutes with these four while commuting with all the others: regularizations. In any case, standard dimensional regulariza- tion suffices for our purposes here. ␥ ␥ ϭ ␥ ␥ ϭ у ͕ 5 , ͖ 0, ͕0,1,2,3͖, ͓ 5 , ͔ 0, 4. ͑5͒ B. Generalized Furry theorem and photon-interaction vertices This procedure introduces a technical breaking of Lorentz Renormalizability of the QED extension at one loop re- invariance in all but the first four dimensions, but without quires that no divergent contributions to the three- or four- introducing new physical features in our perturbation expan- point photon vertices arise, since these must be absent in an sion because the integrals to be regularized have conven- Abelian gauge theory. In conventional QED, the Furry theo- tional Lorentz properties. In any case, in the present context ͓ ͔ ␥ rem 5 plays a useful role in this regard. In this section, we it is simpler to adopt instead a naive 5 matrix that anticom- establish a generalized Furry theorem and use it and other ␥ mutes with all of the other matrices, calculations to prove the absence of divergent contributions ͕␥ ,␥͖ϭ0, у0, ͑6͒ to photon interactions at one loop. 5 In conventional QED, the Furry theorem relies on the ␥ which leads to errors of order ⑀ in ␥-matrix manipulations -matrix structure of the photon-fermion vertex, which leads and hence to errors in finite terms. Since we are interested to a cancellation between two nonzero loops differing only in here in the divergences at one loop, all of which are simple the direction of the charge flow. However, the QED exten- ⑀ ␥ sion ͑1͒ includes terms with more general ␥-matrix struc- poles in , the naive 5 leaves the poles unaffected while easing calculation. Determination of the finite radiative con- tures. In this case, corresponding loops with Lorentz- tributions at one loop would require more care but lies be- violating insertions either cancel or add, depending on the ␥ yond our present scope. charge-conjugation properties of the associated -matrix in- Another issue arises because the one-loop integrals span sertion. an infinite range of four-momentum. The theory ͑1͒ is known As an example, consider the cubic photon vertex at one ⌫ to violate stability or microcausality at sufficiently high en- loop with an insertion of 1 at one of the fermion-photon ergy and momentum, where unrenormalizable terms from vertices in Fig. 1͑c͒. This gives two contributions shown in Planck-scale physics become important and must be included Fig. 3. Take the loop momentum k to be positive in the in the analysis ͓3͔. The Feynman rules adopted here are clockwise direction, and assign the nth external photon line a ϭ ϩ therefore strictly valid only over a range of energy and mo- momentum pn and Lorentz index n . Define k1 k p1 and ϭ ϩ ϩ mentum lying below the Planck scale. We proceed in this k12 k p1 p2. Then, the two diagrams yield an expression section under the reasonable and customary assumption that proportional to any new physics entering at high scales has negligible effect d Tr͓͑kϩm͒␥ 1͑k ϩm͒␥ 2͑k ϩm͒⌫ 3͔ on the leading-order low-energy physics described by the ͵ d k ͫ ” ” 1 ” 12 1 Lagrangian ͑1͒. A definitive result concerning the validity of ͑ ͒d ͑ 2Ϫ 2͒͑ 2Ϫ 2͒͑ 2 Ϫ 2͒ 2 k m k1 m k12 m this assumption would be of interest. A technical point to note is that no external-leg propaga- Tr͓⌫ 3͑k Ϫm͒␥ 2͑k Ϫm͒␥ 1͑kϪm͔͒ 1 ” 12 ” 1 ” Ϫ ͬ. ͑7͒ tors appear because we are calculating corrections to the one- ͑k2Ϫm2͒͑k2Ϫm2͒͑k2 Ϫm2͒ loop effective action. External legs introduce additional com- 1 12 plications because the full propagator at all orders in coefficients for Lorentz violation is needed to establish the Taking the transpose of the argument of the trace in the sec- asymptotic Hilbert space ͓2͔. More attention would therefore ond term and inserting suitable factors of CCϪ1, where C is be required to extract the finite radiative corrections to physi- the charge-conjugation matrix, we can rewrite the numerator cal scattering cross sections or decay amplitudes at one loop. of the integrand as We finally remark in passing that, since Lorentz symmetry is no longer respected by the theory, certain Lorentz- noninvariant regularization schemes might in principle be Tr͓͑kϩm͒␥ 1͑k ϩm͒␥ 2͑k ϩm͒͑⌫ 3Ϫ˜⌫ 3͔͒, ͑8͒ ” ” 1 ” 12 1 1 envisaged instead. It is conceivable that a scheme chosen to respect both observer Lorentz invariance and any remaining subgroup of the particle Lorentz symmetry might offer some where
056006-4 ONE-LOOP RENORMALIZATION OF LORENTZ-... PHYSICAL REVIEW D 65 056006
⌫ FIG. 4. Cubic photon vertex with 1 propagator insertion.
˜⌫ϵϪ͑ ⌫ Ϫ1͒T FIG. 5. Permutations for the cubic photon vertex. 1 C 1C
1 proportional to the other coefficients for Lorentz violation ϭc ␥Ϫd ␥ ␥Ϫe Ϫif ␥ ϩ g . ͑9͒ 5 5 2 occur either. For the quartic photon vertex, the generalized Furry theo- rem and the dimensionality of b together imply that the ⌫ In this case, it follows that only the terms in 1 associated only potential divergences arise from insertions of c or ␥ ␥ ␥ with C violation survive: those involving I, 5 , 5 . The g . In this case, there are four permutations of each type usual Furry theorem is a special case for the C-preserving of insertion. The contributions involving g are zero be- ⌫ ␥ QED interaction, with 1 replaced by . cause the trace of an odd number of ␥ matrices vanishes. ⌫ If instead a factor of 1 is inserted in a fermion propaga- Once again, explicit calculation reveals that no divergence tor, a related argument applies. See Fig. 4. Three conven- proportional to c occurs either, as required for renormaliz- tional fermion-photon vertices occur, but an extra fermion ability. propagator appears in the loop along with another momen- tum factor from the insertion. Since the propagator has no C. Propagator corrections net effect while the signs from the momentum insertion and the extra conventional vertex cancel, the surviving terms are In this subsection, we provide the results of our calcula- the same as before. tions for one-loop corrections to the photon and fermion For a four-point vertex, an extra propagator and ␥ matrix propagators. The calculational methods parallel the conven- appear relative to the three-point vertex. These combine to tional case, so for brevity we restrict the discussion largely to give an overall relative sign. It therefore follows that coeffi- the presentation of results. cients surviving in a three-point function are eliminated in For the photon propagator, the complete one-loop diver- the corresponding four-point function, and vice versa. These gence including the standard QED result is arguments can be generalized to include insertions of M in 1 4q2 ¯ ͒ϭ Ϫ 2 ͒Ϫ ϩ ͒ 2 fermion propagators and arbitrary numbers of photon legs ͑p I0͓͑pp p ͑c c p around the loop. 3 The generalization of the Furry theorem thus shows that ␣  ␣ Ϫ2c␣p p ϩ͑c␣ϩc␣͒p p there are no contributions proportional to b , c ,org ␣ for odd numbers of photon legs on a fermion loop, while ϩ͑c␣ϩc␣͒p p͔, ͑10͒ there are no contributions proportional to a , d , e , f , ϭ 2⑀ or H for even numbers of photon legs. The contributions where I0 i/16 . Only the symmetric part of c contrib- from other pairs of diagrams with opposing fermion loops utes. Corrections coming from diagrams involving vertex must be explicitly calculated and typically are nonzero. This and propagator insertions of g and the antisymmetric part applies to both finite and divergent corrections. For example, of c cancel. All other potential divergent corrections to this it is no longer necessarily the case that one-loop radiative propagator can be shown to vanish, using arguments similar corrections vanish for n-point photon S-matrix amplitudes to those in Sec. III B. Note that ¯ is symmetric and gauge with odd n. Even for the 3-point photon vertex, there could invariant, p ¯ ϭ0. The result ͑10͒ agrees with the original now be a nonzero amplitude. Although it lies outside our calculation in Ref. ͓2͔. present scope, it would be interesting to evaluate these radia- For the fermion propagator, the usual QED correction is tive effects and consider possible phenomenological implica- tions. 2 ⌺ ͑p͒ϭq I ͓͑1ϩ͒␥p Ϫ͑4ϩ͒m͔. ͑11͒ To investigate renormalizability, it is necessary to calcu- QED 0 late explicitly the divergent one-loop contributions to the ⌺ The divergent corrections x arising from all possible inser- three- and four-point photon vertices for those cases where tions of an x term in the standard one-loop QED diagram are the generalized Furry theorem allows a nonzero answer. For given by the cubic vertex there are three vertices and three propaga- tors, so any nonzero divergent contribution would occur ⌺ ͑ ͒ϭϪ 2 ͫ͑ ϩ͒͑ ϩ␥ ͒␥ϩ ͬ three times. The resulting permutations are illustrated in Fig. p q I 1 a b H , M 1 0 5 5 for the case of propagator insertions. Any diagram with an 2 H insertion is finite either because of the dimensionality of 2 H or because the trace of an odd number of ␥-matrices q ⌺ ͑p͒ϭ I ͓͑3Ϫ1͒cp ␥ Ϫ4cp ␥ ͔, vanishes. Explicit calculation reveals that no divergences c 3 0
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2 2 q ␣ q ␣␥ ⌺ ͑p͒ϭ I ͓͑3Ϫ1͒dp ␥ ␥ Ϫ4dp ␥ ␥ ͑Z ͒ b␣ϭbϪ ͓mg ⑀␣␥ d 3 0 5 5 b 322⑀ Ϫ ⑀␣ Ϫ ͒ 3m d␣ ͔, 6͑kAF ͔,
⌺ ͒ϭ 2 ϩ͒ Ϫ ␥ 2 e͑p q I0͓͑1 ep 3me ͔, ␣ q ͑Z ͒ c␣ϭcϩ ͓cϩc c 122⑀ 2 ⌺ ͑p͒ϭq I ͑1ϩ͒ifp ␥ , f 0 5 Ϫ ͒ ͑kF ͔, q2 ⌺ ͒ϭ Ϫ ͒ Ϫ 2 g͑p I0͓͑ 1 gp 2gp ␣ q 2 ͑Z ͒ d␣ϭdϩ ͑dϩd͒, d 122⑀ ␣ ␣␥ ϩ2g␣ p Ϫmg ⑀␣␥␥ ␥ ͔, 5 ͒ ϭ ͒ ϭ ͑Ze ͑Z f , 2 ⌺ ͑p͒ϭ3q I ͑k ͒␥ ␥ , kAF 0 AF 5 q2 ͑ ͒ ␣␥ ϭ ϩ ͑ Ϫ Zg g␣␥ g 2⑀ 2g g 4q2 16 ⌺ ͒ϭ ͒ ␣ ␥ ͑ ͒ k ͑p I0͑kF ␣ p . 12 ␦ ␦ F 3 ϩgϪg␦ ϩg␦ ͒,
q2 D. Quadratic-term renormalization factors ␣ ͑Z ͒ H␣ϭHϩ ͑H H 162⑀ To renormalize the quadratic terms, we must redefine the ␣ bare fields and the fermion mass in terms of renormalized Ϫ2md ⑀␣͒, ones, q2 ␣ ϭͱ ϭͱ ϭ ͑ ͒ ͑Z ͒ ͑k ͒␣ϭ͑k ͒ϩ ͑k ͒ , B Z , AB ZAA , mB Zmm, 13 kAF AF AF 122⑀ AF and the bare coefficients for Lorentz violation in terms of q2 ͑ ͒ ␣␥␦͑ ͒ ϭ͑ ͒ ϩ ͫ͑ ͒ renormalized ones, Z k ␣␥␦ k k kF F F 122⑀ F ϭ͑ ͒ ␣ ϭ͑ ͒ ␣ aB Za a␣ , bB Zb b␣ , 1 Ϫ ͑cϩc͒ ϭ ͒ ␣ ϭ ͒ ␣ 2 cB ͑Zc c␣ , dB ͑Zd d␣ ,
␣ ␣ 1 e ϭ͑Z ͒ e␣ , f ϭ͑Z ͒ f ␣ , ϩ ͑cϩc͒ B e B f 2 ϭ͑ ͒ ␣␥ ϭ͑ ͒ ␣ gB Zg g␣␥ , HB ZH H␣ , 1 ϩ ͑cϩc͒ ␣ 2 ͑k ͒ ϭ͑Z ͒ ͑k ͒␣ , AF B kAF AF 1 ␣␥␦ Ϫ ͑ ϩ ͒ͬ c c . ͑k ͒ ϭ͑Z ͒ ͑k ͒␣␥␦ . ͑14͒ F B kF F 2
In this section, a subscript B is added to bare quantities We find that these renormalization factors suffice to render where needed to distinguish them from renormalized ones. finite at one loop all corrections to the quadratic fermion and An analysis of Eqs. ͑11͒ and ͑12͒ leads to the following photon terms. ͑ ͒ expressions for the above renormalization factors: The derivation of Eqs. 15 parallels the standard QED case. For example, the only correction to the Lorentz- and 2 ⌺ q CPT-invariant fermion kinetic term comes from QED(p), Zϭ1Ϫ ͑1ϩ͒, 162⑀ as usual. We therefore have
i¯ p ϩ¯ ⌺ ͑p͉͒ ϭ ϩ ... q2 B ” B B QED m 0 B ϭ Ϫ ͑ ͒ ZA 1 2⑀ , 15 ϭ ¯ ϩ 2 ¯͑ ϩ͒ ϩ ͑ ͒ 12 iZ p” q I0 1 p” ..., 16 3q2 where the right-hand side is written in terms of one-loop ϭ Ϫ renormalized quantities and the ellipsis refers to higher-order Zm 1 2⑀ , 16 terms that can be neglected. The right-hand side of this equa- tion must be finite, yielding the first of Eqs. ͑15͒. 3q2 ͑ ͒ ␣ ϭ Ϫ For the coefficients for Lorentz violation, similar methods Za a␣ a 2 me , 16 ⑀ apply. For example, for the e term we find that the only
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⌺ ͉ 2 relevant corrections come from e(p) mϭ0. Including the ef- q q ϭZ q, Z ϭ1ϩ . ͑19͒ fects arising from the wave-function renormalization Z re- B q q 242⑀ veals that no renormalization of e is needed at one loop. A more involved example is given by the corrections to c , This is in accordance with the usual QED Ward identity, which arise from both ⌺ (p) and ⌺ (p). Incorporating also c kF the wave-function renormalization Z leads to the above ex- Z ͱZ ϭ1. ͑20͒ ␣ q A pression for (Zc) c␣ . As a final example, consider ␣␥␦ (Z ) . Here, it is useful to note that the term Note that the Z factors for the couplings are all independent kF ␣  ␣ϭ Ϫ Ϫ2p p c␣ in Eq. ͑10͒ must cancel a correction to the of the gauge parameter (1 ), as expected. tree-level (k )␣ term, and hence the correction itself must At this stage no more parameters can be renormalized, so F ⌫ also have the symmetry of the Riemann tensor. Implement- all the remaining divergent corrections x must be made ing this requirement reveals that all the divergent terms can- finite by the Z factors already defined. Inspection shows that cel simultaneously provided ZA takes the same form as in this is indeed the case, provided the Ward identity holds. We conventional QED at one loop. can therefore conclude that the theory is multiplicatively renormalizable and that it remains gauge invariant at one E. Vertex corrections and Ward identities loop. The remaining 1PI diagrams arise in connection with the IV. RENORMALIZATION GROUP AND BETA FUNCTIONS one-loop three-point vertex. This section presents the results of our calculations of the associated diagrams. Given the results for the one-loop divergences presented For this vertex, the standard QED divergence at one loop in Sec. III, several interesting issues become amenable to is recovered: analysis. In particular, we can initiate a study of the behavior of the Lorentz- and CPT-violating QED extension over a ⌫ ϭϪ 3 ͑ ϩ͒␥ ͑ ͒ QED q I0 1 . 17 large range of energy scales. This offers, for example, partial insight about the absolute and relative sizes of the coeffi- ⌫ The divergent corrections x arising from all possible single cients for Lorentz violation and hence is of value from both insertions of an x term in the standard one-loop QED three- theoretical and experimental perspectives. point vertex are found to be In this section, we begin with some preliminary remarks delineating the framework for our analysis. We then obtain ⌫ ϭ0, M 1 the beta functions and solve the renormalization-group equa- tions for the running of the coefficients for Lorentz violation. 3 Some implications of these results are discussed. q ␣ ␣ ⌫ ϭϪ I ͓͑3Ϫ1͒c ␥␣Ϫ4c ␥␣͔, c 3 0 A. Framework 3 q ␣ ␣ Any regularization scheme naturally introduces a mass ⌫ ϭϪ I ͓͑3Ϫ1͒d ␥ ␥␣Ϫ4d ␥ ␥␣͔, d 3 0 5 5 scale . In the Pauli-Villars scheme it is the momentum cut- off, while in dimensional regularization it is the unit of mass required to maintain a dimensionless action in d dimensions. ⌫ ϭϪq3I ͑1ϩ͒e, e 0 In a conventional field theory, the bare Green functions are independent of the value chosen for . However, they are ⌫ϭϪ 3 ͑ ϩ͒ ␥ f q I0 1 if 5 , related by multiplicative renormalizability to the renormal- ized Green functions, which may depend on as a conse- 3 q ␣ ␣  quence of the regularization procedure. ⌫ ϭϪ I ͓͑Ϫ1͒g ␣ϩ2g  ␣ g 2 0 In perturbation theory, the coefficients of a renormalized Green function typically depend on logarithms ln(p2/2)of ␣ ϩ2g ␣͔, the momentum p. As a result, if ͱp2 is very different from , perturbation theory can become invalid even for small cou- ⌫ ϭ k 0, plings. To study the physics at momenta much larger than , AF the Green function must be expressed in terms of a new renormalization mass ЈϷͱp2 chosen to keep the loga- 4q3 ⌫ ϭϪ ͑ ͒ ␣␥ ͑ ͒ rithms small, so that it is justifiable to neglect higher-order k I0 kF ␣ . 18 F 3 terms. In the present case, we are interested in the behavior of The reader is cautioned not to confuse the standard notation the coefficients for Lorentz violation over a large range of ⌫ x for the divergent 3-vertex contributions with the quanti- scales. We are therefore interested in the dependence of the ⌫ ⌫ ͑ ͒ ͑ ͒ ties and 1 in Eqs. 1 and 2 . renormalized Green functions on the renormalization mass Taking into account the results ͑15͒, renormalization of scale . A standard procedure to obtain this is to solve the the QED vertex correction ͑17͒ yields renormalization-group equation at the appropriate order in
056006-7 KOSTELECKY´ , LANE, AND PICKERING PHYSICAL REVIEW D 65 056006 perturbation theory ͓20,21͔. The solution can be expressed order in the string tension. In the present context, the exis- through the dependence on of the renormalized running tence of these operators suggests that the Lagrangian ͑1͒ can couplings x(), where x generically denotes parameters in be regarded as an effective low-energy theory. From this per- the theory. spective, the key assumption made in applying Eq. ͑22͒ at Under the assumption that the Lorentz- and one loop is that the theory remains valid as an effective field CPT-violating QED extension is a multiplicatively renor- theory at this order. Then, any problems arising from un- malizable theory to all orders, the usual derivation of the renormalizability at higher loops would be suppressed at low renormalization-group equation can be followed. For an energies, allowing one-loop calculations to be performed as n-point Green function ⌫(n), we can write though the theory were fully multiplicatively renormalizable. Unrenormalizable effects entering at some multiloop level ⌫ ͑ ͓͒ ͔ϭ Ϫn/2͓ ͑͒ ͔⌫͑ ͓͒ ͑͒ ͔ ͑ ͒ B n p,xB Z x , n p,x , , 21 might cause loss of predictability at that order in perturbation theory, but lower-order predictions would remain valid in the where x includes the coefficients for Lorentz violation as region where the corresponding running couplings are small. Ϫ well as m and q. The factor of Z n/2 arises from wave- For definiteness, we proceed in what follows under the function renormalization of the external legs of the Green reasonable and practical assumption that it is meaningful to function. Noting that the left-hand side of this equation is apply Eq. ͑22͒ at one loop. Although beyond our present independent of , differentiation with respect to yields scope, it would be interesting either to prove all-orders mul- tiplicative renormalizability of the QED extension or to de- d termine the formal regime of validity of the results to be 0ϭ ͕ZϪn/2͓x͑͒,͔⌫͑n͓͒p,x͑͒,͔͖. ͑22͒ d presented below.
This is the renormalization-group equation, which provides a B. The beta functions nontrivial constraint on ⌫(n) through the explicit depen- ϭ Given a theory with a set ͕x j͖, j 1,2,...,N, of running dence. parameters, the beta function ͓21͔ for a specific parameter x ͑ ͒ j For a one-loop calculation, it suffices to impose Eq. 22 is only to one-loop order. If the running couplings become large enough, the perturbative approach fails. However, in dxj  ϵ , ͑23͒ the region where the couplings remain small, the accuracy of x j d the perturbative approximation is improved compared to us- ing fixed couplings. This improvement can be attributed to a where is the renormalization mass parameter relevant to partial resummation of the perturbation series that includes the regularization method. In dimensional regularization with leading logarithmic corrections to all orders in perturbation minimal subtraction, the beta function for a given parameter theory. can be calculated directly from the simple ⑀ pole in the cor- Despite the link between higher loops and responding Z factor ͓22,23͔. In this subsection, we summa- renormalization-group improvement, the explicit one-loop rize the procedure and use it to obtain all the one-loop beta ͑ ͒ solution of Eq. 22 only uses multiplicative renormalizabil- functions for the Lorentz- and CPT-violating QED exten- ity at one loop. Since one-loop multiplicative renormalizabil- sion. ity of the Lorentz- and CPT-violating QED extension is For each parameter x , define the Z factor Z by proved in the previous section, it follows that we can per- j x j form one-loop renormalization-group calculations without ⑀ x ϭ x j Z x , ͑24͒ meeting practical obstacles. In effect, this procedure makes jB x j j the reasonable assumption that the couplings remain small where the factor x j ensures that the renormalized param- and hence that the perturbation approximation is valid. How- eter x has the same mass dimension as its corresponding ever, the derivation of the renormalization-group equation j bare parameter. In dϭ4Ϫ2⑀ dimensions, the fields and outlined above shows that adopting this calculational proce- Ϫ ⑀ Ϫ⑀ dure also tacitly assumes that the QED extension is multipli- A have mass dimension (3 2 )/2 and (1 ), respec- catively renormalizable at all orders. Resolving this multi- tively. We therefore find loop issue lies well beyond our present scope, so we restrict ϭ1, ϭ⌫ ϭ ϭ0. ͑25͒ ourselves here to a few pertinent remarks. q m 1 M 1 It is known that operators of mass dimension greater than four are required in the full QED extension for causality and In minimal subtraction, only the divergent terms in the ͓ ͔ regulated integrals are subtracted. Since these divergent stability at Planck-related energy scales 3 . Although such ⑀ operators are unnecessary at low energy, where the theory ͑1͒ terms are poles in , any given Z factor takes the generic holds sway, their presence implies that the full theory is un- form renormalizable in the usual sense. However, this may be ir- ϱ a j relevant in the underlying Planck-scale theory. For example, ϭ ϩ n ͑ ͒ Zx x j x j ͚ n . 26 difficulties with ultraviolet properties are absent in some j nϭ1 ⑀ string theories even though the corresponding particle field theories appear to have unrenormalizable terms at leading It can then be shown that ͓22͔
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-j ing the one-loop running of the coefficients for Lorentz vio ץ N a1  ϭ limͫ Ϫ a j ϩ ͚ x ͬ, ͑27͒ lation and other parameters in the theory. ץ x j x j 1 xk k ⑀→0 kϭ1 xk
j C. Running couplings which involves only the simple pole a1 . Among the , only is nonzero in the present case. To solve the coupled partial-differential equations ͑28͒ for x j q Equation ͑27͒ therefore implies that each beta function other the running parameters x j( ), boundary conditions are re-  quired. We provide these as the values than q is nontrivial only due to the q dependence of the corresponding Z factor. The running of all the coefficients for x ϵx ͑ ͒ ͑29͒ Lorentz violation at any loop order is therefore driven by the j0 j 0 standard QED running of the charge q. This result is ex- of the parameters x at the scale . In what follows, it is pected because the interactions in the theory ͑1͒ all arise j 0 convenient to define the quantity from the minimal coupling in the covariant derivative and are therefore controlled only by the charge q, even though q2 the perturbation theory is also an expansion in the coeffi- ͑͒ϵ Ϫ 0 ͑ ͒ Q 1 2 ln , 30 cients for Lorentz violation. 6 0 Using the expression ͑27͒, we obtain the following set of results for all the beta functions in the Lorentz- and which controls the running of the usual QED charge with the CPT-violating QED extension: scale . The first two of Eqs. ͑28͒ yield 3q2 q3  ϭϪ  ϭ Ϫ m 2 m, q 2 , ͑͒2ϭ 1 2 ͑͒ϭ 9/4 ͑ ͒ 8 12 q Q q0 , m Q m0 , 31
3q2 which is the standard QED result. Substituting these expres- ͑ ͒ϭϪ me , sions in the remaining equations permits a complete integra- a 82 tion. As an example, consider the coefficient d . Construct- 2 ing the beta function for dϩd gives q ␣␥ ͑ ͒ϭϪ ͓mg ⑀␣␥Ϫ6͑k ͔͒, b 162 AF 2 d q0 ͑dϩdT͒ϭ QϪ1͑dϩdT͒, ͑32͒ d ln 32 q2 ͑ ͒ ϭ ͓ ϩ Ϫ͑ ͒ ␣͔ c 2 c c kF ␣ , 6 which can be rewritten as
q2 ͑ ͒ ϭ ͑ ϩ ͒ ͑ ͒ ϭ͑ ͒ ϭ d d 2 d d , e f 0, ͓Q2͑dϩdT͔͒ϭ0. ͑33͒ 6 d ln
2 Ϫ2 q ␣ Therefore, the symmetric part of d runs like Q . Simi- ͑ ͒ϭ ͑2gϪgϩgϪg␣ g 82 larly, we find the antisymmetric part has no running. Com- bining the two gives the running of d . ϩ ␣͒ g␣ , In this way, we find that the coefficients for Lorentz vio- lation run as follows: q2 ͑ ͒ ϭ ͑ Ϫ ␣⑀ ͒ H 2 H 2md ␣ , ϭ Ϫ Ϫ 9/4͒ 8 a a0 m0͑1 Q e0 ,
q2 1 9/4 ␣␥ ͑ ͒ϭ ͑k ͒ , bϭb Ϫ m ͑1ϪQ ͒g ⑀␣␥ kAF 62 AF 0 6 0 0
2 9 q 1 Ϫ ͑ ͒ ͑ ͒ ϭ ͫ Ϫ ͑ ϩ ͒ ln Q kAF 0 , k c c kF 62 F 2 4
1 1 1 ϩ ͑ ϩ ͒ϩ ͑ ϩ ͒ Ϫ3 ␣ c c c c cϭc Ϫ ͑1ϪQ ͓͒c ϩc Ϫ͑k ͒ ␣ ͔, 2 2 0 3 0 0 F 0 1 Ϫ ͑ ϩ ͒ͬ ͑ ͒ 1 c c . 28 Ϫ2 2 dϭd Ϫ ͑1ϪQ ͒͑d ϩd ͒, 0 2 0 0 Through the definition ͑23͒, these beta functions specify a ϭ ϭ complete set of coupled partial-differential equations govern- e e0 , f f 0 ,
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1 Ϫ9/4 gϭg Ϫ ͑1ϪQ ͒͑2g Ϫg ϩg 0 3 0 0 0 Ϫ ␣ϩ ␣͒ g0␣ g0␣ ,
1 9/4 ␣ HϭH ϩ m ͑1ϪQ ͒d ⑀␣ , 0 2 0 0
͒ ϭ Ϫ1 ͒ ͑kAF Q ͑kAF 0 ,
1 Ϫ3 ͑k ͒ϭ͑k ͒ ϩ ͑1ϪQ ͒ F F 0 6 n ϫ ϩ Ϫ ͒ ␣ FIG. 6. Variation of the function Q() . ͕ ͓c c ͑kF 0␣ ͔ Ϫ ϩ ϩ ͒ ␣ ͓c c ͑kF 0␣ ͔ also be necessary to account for the possible embedding of the charge U͑1͒ group in a larger unification gauge group. Ϫ ͓ ϩ ϩ͑ ͒ ␣͔ c c kF 0␣ Since the function Q() is sensitive to the coefficient of ϩ ϩ Ϫ ͒ ␣ ͑ ͒ the logarithm, the combination of the above factors implies ͓c c ͑kF 0␣ ͔͖. 34 that different physically realistic theories can produce a va- Some coefficients increase or decrease with Q while some riety of behaviors for Q(). For the standard-model fermi- are unaffected, including irreducible combinations such as ons alone, the sum of the squared charges is about 0.7, while the totally antisymmetric part of g . Note that the mixings the Landau pole for running between the electroweak scale Ӎ Ӎ ϫ 19 of coefficients for Lorentz violation displayed in these results mew 250 GeV and the Planck scale M P 2 10 GeV are consistent with the symmetry-based predictions given at occurs at a corresponding factor of about 1.5. For simplicity the end of Sec. II. and to gain basic insight, we consider explicitly the case where this factor is chosen to be 1. Figure 6 shows the various powers of the function Q() D. Some implications 2ϭ for this case, q0 1, plotted as a logarithmic function of The expressions ͑34͒ display a range of behavior for the from mew to M P . In a simple scenario for Lorentz and CPT running of the coefficients for Lorentz violation. All the run- violation, such as might arise in spontaneous symmetry ning is controlled by the single function Q() given in Eq. breaking ͓9͔, it is conceivable that near the Planck scale all ͑30͒, but the powers of Q() involved range from Ϫ3to nonzero coefficients for Lorentz violation are the same order 9/4. In this section, we comment on some implications of of magnitude in Planck units. Then, Fig. 6 can be directly this behavior. interpreted in terms of the divergence of renormalization- Note first that the calculations above are performed for group trajectories at low energies. If instead the coefficients the Lorentz- and CPT-violating QED extension with a single for Lorentz violation start at different sizes at the Planck Dirac fermion. However, the full standard-model extension scale, then matching the curves in Fig. 6 to the trajectories of involves additional interactions, three generations of chiral the running couplings requires shifts. In any case, the figure fermion multiplets, a scalar field, and several intermeshed shows that some coefficients for Lorentz violation increase sets of coefficients for Lorentz violation. All these would while others decrease as the energy scale changes, with pos- affect the structure of the solutions ͑34͒. A definitive under- sible relative changes of several orders of magnitude. standing of the physical implications of the running of the The rate of running of the coefficients for Lorentz viola- coefficients for Lorentz violation must therefore await a tion is relatively small, as is to be expected from the loga- complete analysis within the standard-model extension. rithmic scale dependence and from the slow running of the Nonetheless, despite these caveats, some meaningful physi- QED coupling q. This running would therefore be insuffi- cal insight can be obtained. cient by itself to account for the heavy suppression of coef- We focus here on the behavior of Q() from low energies ficients for Lorentz violation necessary for compatibility to the Planck scale. A key factor in determining this behavior with existing experimental bounds. If not already present at is the value of the fermion charge q0 at the reference scale the Planck scale, any suppression must be driven by some 0, which controls the size of the coefficient of the logarithm other factor relevant between the electroweak and the Planck in Eq. ͑30͒. In a realistic theory, the running of the QED scales. In fact, it is known that unrenormalizable terms be- 2 charge q involves all charged fermions, so the factor q0 in come important for consistency as the Planck scale is ap- Eq. ͑30͒ must be replaced with a quantity involving a sum proached ͓3͔. We conjecture that, in mixing with other coef- over squared charges of all fermions participating in the ficients in the renormalization-group equations, these terms loops. At sufficiently high energies, this includes all known could suffice to provide the necessary suppression at the fermions in the standard model and possibly others predicted electroweak scale. A rapid running of coefficients for Lorentz by the theory but as yet unobserved. Any charged scalars in violation with negative mass dimension is consistent with the theory could also play a role. In many theories, it may expectations from the results ͑34͒, which indicate that di-
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mensionless coefficients increase towards the Planck scale ACKNOWLEDGMENTS while the massive ones decrease or remain unchanged. Al- This work was supported in part by the United States though it lies outside our present scope, it would be interest- Department of Energy under grant No. DE-FG02- ing to investigate further this line of reasoning in the context 91ER40661. A.P. wishes to thank John Gracey and D.R.T. of explicit models. Jones for useful discussions. In the standard-model extension, the coefficients for Lor- entz violation may vary with the particle species. Figure 6 shows that the running of Q() with the scale can suffice APPENDIX: FEYNMAN RULES to induce a significant range of values for the various coef- This appendix provides the Feynman rules appropriate for ficients specific to a given species, even if all these coeffi- our one-loop calculations using the theory ͑1͒ with the stan- ␣ 2 ץ cients are comparable at some large scale. The figure also Ϫ dard gauge-fixing term ( •A) /2 . They are linear in the suggests that conventional running alone cannot separate co- coefficients for Lorentz and CPT violation. efficients for a given species by more than several orders of The fermion propagator is magnitude. In contrast, for different species of the same non- ͑␥p ϩm͒ zero charge, no separation is induced between coefficients ϭi , ͑A1͒ p2Ϫm2 for Lorentz violation of a specific type. Note also that the running of the coefficients in the full standard-model extension must also be controlled by factors where the momentum p is understood to travel in the direc- Q () and Q () associated with the SU͑2͒ and SU͑3͒ tion of the charge arrow shown in the diagram. The coeffi- 2 3 ⌫ gauge groups, respectively. In some theories, the SU͑3͒ cou- cients for Lorentz and CPT violation contained in 1 and pling runs faster than the others, which suggests a larger M 1 lead to insertions in the fermion propagator: spread in the quark-sector coefficients for Lorentz violation ϭϪ ͑ ͒ and emphasizes the value of a range of tests sensitive to iM1, A2 different coefficients for hadrons ͓11–15͔. In other theories, the U͑1͒ coupling runs the fastest, so the greatest spread may ϭ ⌫ ͑ ͒ be in the charged-lepton sector, emphasizing the need for i 1 p. A3 different measurements with electrons ͓16,17͔ and muons ͓19͔. Taken together, all these considerations underline the For a photon of momentum p, the propagator is importance of performing experiments measuring a wide va- i pp ϭϪ ͩ ϩ ͪ ͑ ͒ riety of different coefficients, both within a given species and 2 2 , A4 across different sectors of the standard-model extension. p p ϭ␣Ϫ V. SUMMARY where 1. The coefficients for Lorentz and CPT vio- lation yield two types of insertion on this propagator: In this paper we have shown the one-loop multiplicative renormalizability of the general Lorentz- and CPT-violating ϭϪ ␣  ͒ ͑ ͒ QED extension ͑1͒. A generalized Furry theorem has been 2ip p ͑kF ␣, A5 obtained and used to prove the absence of one-loop diver- gences in the cubic and quartic photon vertices. The diver- gent one-loop corrections to the photon propagator are given ϭ ͒␣⑀  ͑ ͒ 2͑kAF ␣ p . A6 in Eq. ͑10͒, while those to the fermion propagator are in Eqs. ͑11͒ and ͑12͒. The associated renormalization factors are in ␣ The momentum p for the (k ) insertion is taken to travel Eqs. ͑15͒ and ͑19͒. The divergent one-loop corrections to the AF from the index to the index. Note that the Feynman fermion-photon vertex are given in Eqs. ͑17͒ and ͑18͒. The diagram is symmetric: the antisymmetry of ⑀␣ under , usual Ward identities are found to hold at this order. interchange is compensated by the reversal of the momentum We have also initiated an investigation into the running of direction. the coefficients for Lorentz violation and the The usual fermion-photon vertex is renormalization-group equations. Following some discussion of the framework, the one-loop beta functions are obtained as Eq. ͑28͒. The associated partial-differential equations are ϭϪiq␥, ͑A7͒ solved, and the running couplings are provided in Eqs. ͑30͒, ͑31͒, and ͑34͒. We show that these equations imply that dif- ferent coefficients for Lorentz violation typically run differ- where q is the fermion charge and is the space-time index ently between the electroweak and Planck scales and can on the photon line. The dimensionless coefficients for Lor- lead to a spread of several orders of magnitude over this entz and CPT violation lead to the additional vertex range. Our work emphasizes the value of developing tests to measure many different coefficients for Lorentz and CPT ϭϪ ⌫ ͑ ͒ violation, both for specific field species and across all sectors iq 1 . A8 of the standard-model extension.
056006-11 KOSTELECKY´ , LANE, AND PICKERING PHYSICAL REVIEW D 65 056006
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