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One-Loop Renormalization of Lorentz-Violating Electrodynamics

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One-Loop Renormalization of Lorentz-Violating Electrodynamics 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 D␮ϵ mentally to be concordant, so this restriction offers no prac- ␮A␯ץϩiqA␮ and the electromagnetic field strength F␮␯ϵ tical difficulty in applying our results. However, to maintain . ␯A␮ץϪ 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.
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