PHYSICAL REVIEW D VOLUME 55, NUMBER 11 1 JUNE 1997 CPT violation and the standard model Don Colladay and V. Alan Kostelecky´ Department of Physics, Indiana University, Bloomington, Indiana 47405 ~Received 22 January 1997! Spontaneous CPT breaking arising in string theory has been suggested as a possible observable experimen- tal signature in neutral-meson systems. We provide a theoretical framework for the treatment of low-energy effects of spontaneous CPT violation and the attendant partial Lorentz breaking. The analysis is within the context of conventional relativistic quantum mechanics and quantum field theory in four dimensions. We use the framework to develop a CPT-violating extension to the minimal standard model that could serve as a basis for establishing quantitative CPT bounds. @S0556-2821~97!05211-9# PACS number~s!: 11.30.Er, 11.25.2w, 12.60.2i I. INTRODUCTION a general methodology that is compatible with desirable fea- tures like microscopic causality while being sufficiently de- Among the symmetries of the minimal standard model is tailed to permit explicit calculations. invariance under CPT. Indeed, CPT invariance holds under We suppose that underlying the effective four- mild technical assumptions for any local relativistic point- dimensional action is a complete fundamental theory that is particle field theory @1–5#. Numerous experiments have con- based on conventional quantum physics @15# and is dynami- firmed this result @6#, including in particular high-precision cally CPT and Poincare´ invariant. The fundamental theory is tests using neutral-kaon interferometry @7,8#. The simulta- assumed to undergo spontaneous CPT and Lorentz breaking. neous existence of a general theoretical proof of CPT invari- In a Poincare´-observer frame in the low-energy effective ac- ance in particle physics and accurate experimental tests tion, this process is taken to fix the form of any CPT- and makes CPT violation an attractive candidate signature for Lorentz-violating terms. nonparticle physics such as string theory @9,10#. Since interferometric tests of CPT violation are so sensi- The assumptions needed to prove the CPT theorem are tive, we focus specifically on CPT violation and the associ- invalid for strings, which are extended objects. Moreover, ated Lorentz-breaking issues in a low-energy effective theory since the critical string dimensionality is larger than four, it without gravity @21#. For the most part, effects from deriva- is plausible that higher-dimensional Lorentz breaking would tive couplings and possible CPT-preserving but Lorentz- be incorporated in a realistic model. In fact, a mechanism is breaking terms in the action are disregarded, and any known in string theory that can cause spontaneous CPT vio- CPT-violating terms are taken to be small enough to avoid lation @9# with accompanying partial Lorentz-symmetry issues with standard experimental tests of Lorentz symmetry. breaking @11#. The effect can be traced to string interactions A partial justification for the latter assumption is that the that are absent in conventional four-dimensional renormaliz- absence of signals for CPT violation in the neutral-kaon able gauge theory. Under suitable circumstances, these inter- system provides one of the best bounds on Lorentz invari- actions can cause instabilities in Lorentz-tensor potentials, ance. thereby inducing spontaneous CPT and Lorentz breaking. If Our focus on the low-energy effective model bypasses in a realistic theory the spontaneous CPT and partial Lorentz various important theoretical issues regarding the structure of violation extend to the four-dimensional spacetime, detect- the underlying fundamental theory and its behavior at scales able effects might occur in interferometric experiments with above electroweak unification, including the origin and neutral kaons @9,10#, neutral Bd or Bs mesons @10,12#,or ~renormalization-group! stability of the suppression of CPT neutral D mesons @10,13#. For example, the quantities pa- breaking and the issue of mode fluctuations around Lorentz- rametrizing indirect CPT violation in these systems could be tensor expectation values. Since these topics involve the Lor- nonzero. There may also be implications for baryogenesis entz structure of the fundamental theory, they are likely to be @14#. related to the difficult hierarchy problems associated with In the present paper, our goal is to develop within an compactification and the cosmological constant. effective-theory approach a plausible CPT-violating exten- The ideas underlying our theoretical framework are de- sion of the minimal standard model that provides a theoreti- scribed in Sec. II. A simple model is used to illustrate con- cal basis for establishing quantitative bounds on CPT invari- cepts associated with CPT and Lorentz breaking, including ance. The idea is to incorporate notions of spontaneous the possibility of eliminating some CPT-violating effects CPT and Lorentz breaking while maintaining the usual through field redefinitions. The associated relativistic quan- gauge structure and properties like renormalizability. To tum mechanics is discussed in Sec. III. Section IV contains a achieve this, we first establish a conceptual framework and a treatment of some issues in quantum field theory. A procedure for treating spontaneous CPT and Lorentz viola- CPT-violating extension of the minimal standard model is tion in the context of conventional quantum theory. We seek provided in Sec. V, and the physically observable subset of 0556-2821/97/55~11!/6760~15!/$10.0055 6760 © 1997 The American Physical Society 55 CPT VIOLATION AND THE STANDARD MODEL 6761 CPT-breaking terms is established. We summarize in Sec. ably by at least one power of m/M relative to the scale of the VI. Some of the more technical results are presented in the effective theory. appendixes. A hierarchy of possible terms in 8 thus emerges, labeled by k50,1,2,... Omitting Lorentz indicesL for simplicity, the leading terms with k<2 have the schematic form II. BASICS l A. Effective model for spontaneous CPT violation 8. ^T& ¯cG~i]!kc1H.c. ~2! L M k • We begin our considerations with a simple model within In this expression, the parameter l is a dimensionless cou- which many of the basic features of spontaneous CPT vio- k lation can be examined. The model involves a single massive pling constant, (i]) represents k four-derivatives acting in some combination on the fermion fields, and G represents Dirac field c(x) in four dimensions with Lagrangian density some gamma-matrix structure. Terms with k>3 and with 5 2 8, ~1! more quadratic fermion factors also appear, but these are L L0 L further suppressed. Note that contributions of the form ~2! arise in string theory @10#. Note also that naive power count- where 0 is the usual free-field Dirac Lagrangian for a fer- ing indicates the dominant terms with k<1 are renormaliz- mion Lc of mass m, and where 8 contains extra able. CPT-violating terms to be described below.L For the present For k50, the above considerations indicate that the domi- discussion, we follow an approach in which the C, P, T and nant terms of the form ~2! must have expectations ^T& Lorentz properties of c are assumed to be conventionally ;m2/M. In the present work, we focus primarily on this determined by the free-field theory 0 and are used to estab- relatively simple case. Most of the general features arising lish the corresponding properties ofL 8 @22#. This method is from CPT and Lorentz violation together with some of our intrinsically perturbative, which isL particularly appropriate more specific results remain valid when terms with other here since any CPT-violating effects must be small. In Sec. values of k are considered, but it remains an open issue to II C, we consider the possibility of alternative definitions of investigate the detailed properties of terms with k51 and C, P, T and Lorentz properties that could encompass the full expectations ^T&;m or those with k52 and expectations structure of . ^T&;M. Both these could in principle contribute leading We are interestedL in possible forms of 8 that could arise effects in the low-energy effective action. as effective contributions from spontaneousL CPT violation Each contribution to 8 from an expression of the form in a more complete theory. To our knowledge, string theory ~2! is a fermion bilinearL involving a 434 spinor matrix G. forms the only class of ~gauge! theories in four or more Regardless of the complexity and number of the tensors T dimensions that are quantum consistent, dynamically Poin- inducing the breaking, G can be decomposed as a linear com- care´ invariant, and known to admit an explicit mechanism bination of the usual 16 basis elements of the gamma-matrix @9# for spontaneous CPT violation triggered by interactions algebra. Only the subset of these that produce in the Lagrangian. However, to keep the treatment as general CPT-violating bilinears are of interest for our present pur- as possible we assume only that the spontaneous CPT vio- poses, and they permit us to provide explicit and relatively lation arises from nonzero expectation values acquired by simple expressions for the possible CPT-violating contribu- one or more Lorentz tensors T,so 8is taken to be an tions to 8. effective four-dimensional LagrangianL obtained from an un- For theL case k50 of interest here, we find two possible derlying theory involving Poincare´-invariant interactions of types of CPT-violating term: c with T. The discussion that follows is independent of any 8[a ¯cgmc, 8[b ¯cg gmc. ~3! specifics of string theory and should therefore be relevant to La m Lb m 5 a nonstring model with spontaneous CPT violation, if such a For completeness, we provide here also the terms appearing model is eventually formulated. for the case k51, where we find three types of relevant Even applying the stringent requirement of dynamical contribution: Poincare´ invariance, an unbroken realistic theory can in prin- ciple include terms with derivatives, powers of tensor fields, 1 a¯ 1 a¯ c8[ 2 ic cJ]ac, d8[ 2 d cg5J]ac, and powers of various terms quadratic in fermion fields.