CPT Violation and the Standard Model

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 1997͒ Spontaneous CPT breaking arising in string theory has been suggested as a possible observable experimental 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. ͓S0556-2821͑97͒05211-9͔

PACS number͑s͒: 11.30.Er, 11.25.Ϫw, 12.60.Ϫi

I. INTRODUCTION a general methodology that is compatible with desirable features 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 ͓1–5͔. Numerous experiments have con- based on conventional quantum physics ͓15͔ and is dynami- firmed this result ͓6͔, including in particular high-precision cally CPT and Poincare´ invariant. The fundamental theory is tests using neutral-kaon interferometry ͓7,8͔. 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 ͓9,10͔. 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 ͓21͔. 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 ͓9͔ with accompanying partial Lorentz-symmetry issues with standard experimental tests of Lorentz symmetry. breaking ͓11͔. 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 ͓9,10͔, neutral Bd or Bs mesons ͓10,12͔,or ͑renormalization-group͒ stability of the suppression of CPT neutral D mesons ͓10,13͔. 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 ͓14͔. 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

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 Ј thus emerges, labeled by kϭ0,1,2,... . Omitting Lorentz indicesL for simplicity, the leading terms with kр2 have the schematic form II. BASICS ␭ k␺ϩH.c. ͑2͒͒ץA. Effective model for spontaneous CPT violation Јʛ ͗T͘ ¯␺⌫͑i L M k • We begin our considerations with a simple model within In this expression, the parameter ␭ is a dimensionless cou- which many of the basic features of spontaneous CPT vio- k represents k four-derivatives acting in (ץlation can be examined. The model involves a single massive pling constant, (i some combination on the fermion fields, and ⌫ represents Dirac field ␺(x) in four dimensions with Lagrangian density some gamma-matrix structure. Terms with kу3 and with ϭ Ϫ Ј, ͑1͒ more quadratic fermion factors also appear, but these are L L0 L further suppressed. Note that contributions of the form ͑2͒ arise in string theory ͓10͔. Note also that naive power count- where 0 is the usual free-field Dirac Lagrangian for a fer- ing indicates the dominant terms with kр1 are renormaliz- mion L␺ of mass m, and where Ј contains extra able. CPT-violating terms to be described below.L For the present For kϭ0, the above considerations indicate that the domi- discussion, we follow an approach in which the C, P, T and nant terms of the form ͑2͒ must have expectations ͗T͘ Lorentz properties of ␺ are assumed to be conventionally ϳm2/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 Ј ͓22͔. 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 kϭ1 and C, P, T and Lorentz properties that could encompass the full expectations ͗T͘ϳm or those with kϭ2 and expectations structure of . ͗T͘ϳM. Both these could in principle contribute leading We are interestedL in possible forms of Ј that could arise effects in the low-energy effective action. as effective contributions from spontaneousL CPT violation Each contribution to Ј from an expression of the form in a more complete theory. To our knowledge, string theory ͑2͒ is a fermion bilinearL involving a 4ϫ4 spinor matrix ⌫. forms the only class of ͑gauge͒ 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, ⌫ 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 ͓9͔ 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 Јis taken to be an tions to Ј. effective four-dimensional LagrangianL obtained from an un- For theL case kϭ0 of interest here, we find two possible derlying theory involving Poincare´-invariant interactions of types of CPT-violating term: ␺ with T. The discussion that follows is independent of any Јϵa ¯␺␥␮␺, Јϵb ¯␺␥ ␥␮␺. ͑3͒ specifics of string theory and should therefore be relevant to La ␮ Lb ␮ 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 kϭ1, where we find three types of relevant Even applying the stringent requirement of dynamical contribution: Poincare´ invariance, an unbroken realistic theory can in principle include terms with derivatives, powers of tensor fields, 1 ␣¯ 1 ␣¯ ,␺␣ץ␺, dЈϵ 2 d ␺␥5J␣ץcЈϵ 2 ic ␺J and powers of various terms quadratic in fermion fields. L L 1 ␣ ¯ ␮␯ ␺, ͑4͒␣ץHowever, any CPT-breaking term that is to be part of a eЈϵ 2 ie␮␯␺␴ J four-dimensional effective theory must have mass dimension L ␮A)B. In all these expressions, theץ)␮BϪץ␮BϵAץfour. In the effective Lagrangian, each combination of fields where AJ ␣ ␣ ␣ and derivatives of dimension greater than four therefore must quantities a␮ , b␮ , c , d , and e␮␯ must be real as conse- have a corresponding weighting factor of a negative power quences of their origins in spontaneous symmetry breaking Ϫk of at least one mass scale M that is large compared to the and of the presumed hermiticity of the underlying theory. scale m of the effective theory. In a realistic theory with the They are combinations of coupling constants, tensor expec- string scenario, M might be the Planck mass or perhaps a tations, mass parameters, and coefficients arising from the smaller mass scale associated with compactification and uni- decomposition of ⌫. fication. Moreover, since the expectations ͗T͘ of the tensors In keeping with their interpretation as effective coupling T are assumed to be Lorentz and possibly CPT violating, constants arising from a scenario with spontaneous symme- ␣ ␣ ␣ any terms that survive in Ј after the spontaneous symmetry try breaking, a␮ , b␮ , c , d , and e␮␯ are invariant under breaking must on physicalL grounds be suppressed, presum- CPT transformations. Together with the standard 6762 DON COLLADAY AND V. ALAN KOSTELECKY´ 55

CPT-transformation properties ascribed to ␺, this invariance rotations. Note also that the presence of the CPT-violating causes the terms in Eqs. ͑3͒ and ͑4͒ to break CPT ͓23͔.As terms in the Dirac equation destroys the usual symmetriz- discussed above, in the remainder of this work we restrict ability property of ⌰␮␯. The antisymmetric part ⌰͓␮␯͔ is ourselves largely to the expressions in Eq. ͑3͒. Allowing both kinds of term in Eq. ͑3͒ to appear in Ј ⌰͓␮␯͔ϵ⌰␮␯Ϫ⌰␯␮ produces a model Lagrangian of the form L 1 ¯ ␣ ␮␯ [␮ ␯] [␮ ␯] ␺͕␥ ,␴ ͖␺͔Ϫa j Ϫb j5 , ͑10͓͒␣ץ ϭϪ 1 ␺Ϫa ¯␺␥␮␺Ϫb ¯␺␥ ␥␮␺Ϫm¯␺␺. ͑5͒ 4 ץϭ i¯␺␥␮J L 2 ␮ ␮ ␮ 5 which is no longer a total divergence. The conventional con- The variational procedure generates a modified Dirac equa- struction of a symmetric energy-momentum tensor, involv- tion: ing a subtraction of this antisymmetric part from the canoni- ␮ ␮ ␮ cal energy-momentum tensor, would affect the conserved ␮Ϫa␮␥ Ϫb␮␥5␥ Ϫm͒␺ϭ0. ͑6͒ץ ␥i͑ energy and momentum and is therefore presumably inappli- Associated with this Dirac-type equation is a modified cable in the present case. The implications of this for a more Klein-Gordon equation. Proceeding with the usual squaring complete low-energy effective theory that includes gravity procedure, in which the Dirac-equation operator with oppo- remain to be explored. site mass sign is applied to the Dirac equation from the left, Next, consider the effect of Lorentz transformations, i.e., leads to the Klein-Gordon-type expression rotations and boosts. Conventional Lorentz transformations in special relativity relate observations made in two inertial 2 2 2 ␮␯ ␯Ϫa␯͔͒␺͑x͒ϭ0. frames with differing orientations and velocities. TheseץϪa͒ Ϫb Ϫm ϩ2i␥5␴ b␮͑iץi͓͑ ͑7͒ transformations can be implemented as coordinate changes, and we call them observer Lorentz transformations. It is also This equation is second order in derivatives, but unlike the possible to consider transformations that relate the properties usual Klein-Gordon case it contains off-diagonal terms in the of two particles with differing spin orientation or momentum spinor space. These may be eliminated by repeating the within a specific oriented inertial frame. We call these par- squaring procedure, this time applying the operator in Eq. ͑7͒ ticle Lorentz transformations. For free particles under usual with opposite sign for the off-diagonal piece. The result is a circumstances, the two kinds of transformation are ͑in- fourth-order equation satisfied by each spinor component of versely͒ related. However, this equivalence fails for particles any solution to the modified Dirac equation: under the action of a background field. Ϫa͒2 The reader is warned to avoid confusing observer LorentzץϪa͒2Ϫb2Ϫm2͔2ϩ4b2͑iץi͓͕͑ transformations ͑which involve coordinate changes͒ or par- ␮ 2 -␮Ϫa␮͔͒ ͖␺͑x͒ϭ0. ͑8͒ ticle Lorentz transformations ͑which involve boosts on parץϪ4͓b ͑i ticles or localized fields but not on background fields͒ with a B. Continuous symmetries third type of Lorentz transformation that within a specified inertial frame boosts all particles and fields simultaneously, Consider next the continuous symmetries of the model including background ones. The latter are sometimes called with Lagrangian ͑5͒. For definiteness, we begin with an ͑inverse͒ active Lorentz transformations. For the case of con- analysis in a given oriented inertial frame in which values of ventional free particles, they coincide with particle Lorentz the quantities a␮ and b␮ are assumed to have been specified. transformations. We have chosen to avoid applying the terms The effects of rotations and boosts are considered later. active and passive here because they are insufficient to dis- The CPT-violating terms in Eq. ͑5͒ leave unaffected the tinguish the three kinds of transformation and because in any usual global U͑1͒ gauge invariance, which has conserved case their interpretation varies in the literature. current j ␮ϭ¯␺␥␮␺. Charge is therefore conserved in the The distinction between observer and particle transforma- model. These terms also leave unaffected the usual breaking tions is relevant for the present model, where the ␮ ¯ ␮ of the chiral U͑1͒ current j5 ϭ␺␥5␥ ␺ due to the mass term. CPT-violating terms can be regarded as arising from con- In what follows, we denote the volume integrals of the cur- stant background fields a␮ and b␮ . The point is that these ␮ ␮ ␮ ␮ rent densities j and j5 by J and J5 , respectively. eight quantities transform as two four-vectors under observer The model is also invariant under translations provided Lorentz transformations and as eight scalars under particle the tensor expectations are assumed constant, i.e., provided Lorentz transformations, whereas they are coupled to cur- the possibility of CPT-breaking soliton-type solutions in the rents that transform as four-vectors under both types of trans- underlying theory is disregarded. This leads to a conserved formation. This means that observer Lorentz symmetry is canonical energy-momentum tensor ⌰␮␯ given by still an invariance of the model, but the particle Lorentz group is ͑partly͒ broken. 1 Physical situations with features like this can readily be ␮␯ϭ0, ͑9͒⌰ ץ ,␯␺ץ␮␯ϭ i¯␺␥␮J⌰ 2 ␮ identified. For example, an electron with momentum perpendicular to a uniform background magnetic field moves in a and a corresponding conserved four-momentum P␮. These circle. Suppose in the same observer frame we instanta- expressions have the same form as in the free theory. Note, neously increase the magnitude of the electron momentum however, that constancy of the energy and momentum does without changing its direction, causing the electron to move not necessarily imply conventional behavior under boosts or in a circle of larger radius. This ͑instantaneous͒ particle boost 55 CPT VIOLATION AND THE STANDARD MODEL 6763

leaves the background field unaffected. However, if instead This current is conserved at the level of the underlying an observer boost perpendicular to the magnetic field is ap- theory with spontaneous symmetry breaking, but in the ef- plied, the electron no longer moves in a circle. This is fective low-energy theory where the spontaneous breaking viewed in the new inertial frame as an EϫB drift caused by appears as an explicit symmetry violation the conservation the presence of an electric field. In this example, the back- property is destroyed. In the latter case, the corresponding ground magnetic field transforms into a different electromag- Lorentz charges M ␮␯ obey netic field under observer boosts but ͑by definition͒ is unchanged by particle boosts, in analogy to the transformation dM␮␯ ϭϪa[␮J␯]Ϫb[␮J␯] . ͑12͒ of a␮ and b␮ in the CPT-violating model. dt 5 From the viewpoint of this example, the unconventional aspect of the CPT-violating model is merely that the con- Given explicit values of a␮ and b␮ in some inertial frame, stant fields a␮ and b␮ are a global feature of the model. They Eq. ͑12͒ can be used directly to determine which Lorentz cannot be regarded as arising from localized experimental symmetries are violated. Note that if either a␮ or b␮ van- conditions, which would cause them to transform under par- ishes, the Lorentz group is broken to the little group of the ticle Lorentz transformations as four-vectors rather than as nonzero four-vector. This means that the largest Lorentz- scalars. The behavior of a␮ and b␮ as background fields and symmetry subgroup that can remain as an invariance of the hence as scalars under particle Lorentz transformations is a model Lagrangian ͑5͒ is SO͑3͒,E͑2͒,orSO͑2,1͒. Since a consequence of their origin as nonzero expectation values of ␮ and b␮ represent two four-vectors in four-dimensional space- Lorentz tensors in the underlying theory. These Lorentz- time, they can define a two-dimensional plane. Transforma- tensor expectations break those parts of the particle Lorentz tions involving the two orthogonal dimensions have no effect group that cannot be implemented as unitary transformations on this plane. This means that the smallest Lorentz- on the vacuum. This is in parallel with other situations in- symmetry subgroup that can remain is a compact or noncom- volving spontaneous symmetry breaking, such as ones compact U͑1͒. monly encountered in the treatment of internal symmetries. In a realistic low-energy effective theory, CPT-violating The preservation of observer Lorentz symmetry is an im- terms would break the particle Lorentz group in a manner portant feature of the model. It is a consequence of observer related to the breaking given by Eq. ͑12͒. Since no zeroth- Lorentz invariance of the underlying fundamental theory. order CPT violation has been observed in experiments, This symmetry is unaffected by the appearance of tensor CPT-violating effects in the string scenario are expected to expectation values by virtue of its implementation via coor- be suppressed by at least one power of the Planck mass rela- dinate transformations. As an illustration of its use in the tive to the scale of the effective theory. However, the inter- effective model, we show that it permits a further classifica- esting and involved issue of exactly how small the magni- tion of types of CPT-violating term according to the ob- tudes of a␮ and b␮ ͑or their equivalents in a realistic model͒ server Lorentz properties of a␮ and b␮ . Thus, for example, must be to satisfy current experimental constraints lies be- if b␮ is future timelike in one inertial frame, it must be future yond the scope of the present work. We confine our remarks timelike in all frames. This implies that a class of inertial here to noting that the partial breaking of particle Lorentz frames can be found in which b␮ϭb(1,0,0,0), where calcu- invariance discussed above generates an effective boost de- lations are potentially simplified. A similar argument for the pendence in the CPT-breaking parameters. This could pro- lightlike or spacelike cases shows that the CPT-violating vide a definite experimental signature for our framework if physics of the four components of b␮ can in each case be CPT violation were detected at some future date. reduced to knowledge of its Lorentz type and a single number specifying its magnitude. Inertial frames within this ideal C. Field redefinitions class are determined by the little group of b␮ , which can in turn be used to simplify ͑partially͒ the form of a␮ . For the discussions in the previous subsections, we The reader is cautioned that the class of inertial frames adopted a practical approach to the definition of CPT and selected in this way may be distinct from experimentally Lorentz transformations. It involves treating C, P, T and relevant inertial frames such as, for example, those defined Lorentz properties of ␺ as being defined via the free-field using the microwave background radiation and interpreting theory 0 and subsequently using them to establish the sym- the dipole component in terms of the motion of the Earth. metry propertiesL of Ј. This approach requires caution, how- The point is that, given an inertial frame, the process of ever, because in principleL alternative definitions of the sym- spontaneous Lorentz violation in the underlying theory is metry transformations could exist that would leave the full assumed to produce some values of a␮ and b␮ . In this spe- theory invariant. cific inertial frame, there is no reason a priori why these ConsiderL first an apparently CPT- and Lorentz-violating values should take the ideal form described above. One is model formed with a␮ only, defined in a given inertial frame merely assured of the existence of some frame in which the by the Lagrangian ideal form can be attained. ␭␮␯ The current J for particle Lorentz transformations ͓␺͔ϭ 0͓␺͔Ϫ aЈ͓␺͔. ͑13͒ takes the usual form when expressed in terms of the energy- L L L momentum tensor: Introducing in this frame a field redefinition of ␺ by a spacetime-dependent phase,

␭␮␯ [␮ ␭␯] 1 ¯ ␭ ␮␯ J ϭx ⌰ ϩ 4 ␺͕␥ ,␴ ͖␺. ͑11͒ ␹ϭexp͑ia•x͒␺, ͑14͒ 6764 DON COLLADAY AND V. ALAN KOSTELECKY´ 55

the Lagrangian expressed in terms of the new field is equivalent to the original one under a field redefinition on ␺ ͓␺ϭexp(Ϫia•x)␹͔ϵ 0͓␹͔. This shows that the model is of the form discussed above for the ungauged model. equivalentL to a conventionalL free Dirac theory, in which To summarize, in the gauged theory the CPT-breaking

there is no CPT or Lorentz breaking, and thereby provides term aЈ can be interpreted as a background gauge choice and an example of redefining symmetry transformations to main- eliminatedL via a field redefinition as in the ungauged case. tain invariance ͓24͔. We note in passing that related issues arise for certain non- The connection between the Poincare´ generators in the linear gauge choices ͓25͔ and in the context of efforts to two forms of the theory can be found explicitly by substitut- interpret the photon as a Nambu-Goldstone boson arising ing ␺ϭ␺͓␹͔ in the Poincare´ generators for ͓␺͔ and ex- from ͑unphysical͒ spontaneous Lorentz breaking ͓26–32͔.In tracting the combinations needed to reproduceL the usual typical models of the latter type, a four-vector bilinear con- Poincare´ generators for . We find that the charge and ¯ 0͓␹͔ densate ͗␺␥␮␺͘ plays a role having some similarities to that ␮ ␮L chiral currents j and j5 take the same functional forms in of a␮ . both theories but that the form of the canonical energy- The model ͑5͒ involves only a single fermion field. All momentum tensor changes, CPT-violating effects can also be removed from certain theories describing more than one fermion field in which 1 ␯␹ϩa␯¯␹␥␮␹, ͑15͒ each fermion has a term of the form aЈ . For example, this isץ␮␯ϭ i¯␹␥␮J⌰ 2 possible if there is no fermion mixingL and each such CPT-violating term involves the same value of a␮ ,orifthe producing a corresponding change in the Lorentz current fermions have no interactions or mixings that acquire J␭␮␯. This means that in the original theory ͓␺͔ we could L spacetime-dependence upon performing the field redefini- introduce modified Poincare´ currents ⌰˜ ␮␯ and ˜J ␭␮␯ that tions. However, in generic multifermion theories with CPT have corresponding conserved charges generating an unbro- violation involving fermion-bilinear terms, it is impossible to ken Poincare´ algebra. These currents are given as functionals eliminate all CPT-breaking effects through field redefini- of ␺ by tions. Nonetheless, since Lagrangian terms that spontaneously break CPT necessarily involve paired fermion fields, ⌰˜ ␮␯ϭ⌰␮␯Ϫa␯ j␮, ˜J ␭␮␯ϭJ␭␮␯Ϫx[␮a␯] j␭. ͑16͒ at least one of the quantities a␮ can be removed. This means that only differences between values of a␮ are observable. The existence of this connection between the two theories Examples appear in the context of the CPT-violating exten- depends critically on the existence of the conserved current ␮ sion of the standard model discussed in Sec. V. j . In the model ͑5͒ with both a␮ and b␮ terms, the component Ј can be eliminated by a field redefinition as before La III. RELATIVISTIC QUANTUM MECHANICS but there is no similar transformation removing bЈ because ␮ L conservation of the chiral current j5 is violated by the mass. In this section, we discuss some aspects of relativistic In the massless limit of this model the chiral current is con- quantum mechanics based on Eq. ͑6͒, with ␺ regarded as a four-component wave function. The results obtained provide served, and we can eliminate both a␮ and b␮ via the field redefinition further insight into the nature of the CPT-violating terms and are precursors to the quantum field theory. The analo- ␹ϭexp͑ia•xϪib•xy5͒␺. ͑17͒ gous treatment in the context of the standard model involves several fermion fields, for which CPT-violating terms of the For the situation with mÞ0, however, this redefinition would form Ј cannot be altogether eliminated. We therefore ex- La introduce spacetime-dependent mass parameters. plicitly include the quantity a␮ in the following analysis, The term Ј in Eq. ͑3͒ is reminiscent of a local U͑1͒ even though it could be eliminated by a field redefinition for La coupling, although there is no local U͑1͒ invariance in the the simple one-fermion case. In fact, the reinterpretation of theory ͑5͒. It is natural and relevant to our later consider- negative-energy solutions causes the explicit effects of a␮ to ations of the standard model to ask how the above discussion be more involved than might otherwise be expected. of field redefinitions is affected if the U͑1͒ invariance of the The modified Dirac equation ͑6͒ can be solved by assum- original theory is gauged. Then, the term aЈ has the same ing the usual plane-wave dependence, form as a coupling to a constant backgroundL electromagnetic ␮ potential. At the classical level, this would be expected to ␺͑x͒ϭeϪi␭␮x w͑␭ជ ͒. ͑18͒ have no effect since it is pure gauge. However, a conventional quantum-field gauge transformation involving both ␺ In this equation, w(␭ជ ) is a four-component spinor satisfying and the electromagnetic potential A␮ cannot eliminate a␮ , ␮ ␮ ␮ since the theory is invariant under such transformations. In- ͑␭␮␥ Ϫa␮␥ Ϫb␮␥5␥ Ϫm͒w͑␭ជ ͒ϭ0. ͑19͒ stead, the electromagnetic field can be taken as the sum of a classical c-number background field and a quantum field For a nontrivial solution to exist, the determinant of the ma- A␮ A␮ , whereupon a␮ can be regarded as contributing to an trix acting on w(␭ជ ) in this equation must vanish. This means ␮ 0 0 0 effective ␮ . Conventional classical gauge transformations that ␭ ϵ(␭ ,␭ជ ), where ͓33͔␭ϭ␭(␭ជ), must satisfy the A can be performed on the c-number potential ␮ , while leav- requirement A ing the quantum fields ␺ and A␮ unaffected. This changes 2 2 2 2 2 2 ␮ 2 the Lagrangian but should not change the physics. In fact, ͓͑␭Ϫa͒ Ϫb Ϫm ͔ ϩ4b ͑␭Ϫa͒ Ϫ4͓b ͑␭␮Ϫa␮͔͒ ϭ0. the resulting gauge-transformed Lagrangian is unitarily ͑20͒ 55 CPT VIOLATION AND THE STANDARD MODEL 6765

This condition can also be obtained directly from Eq. ͑8͒ and positive-energy one, it leaves a hole appearing to be a par- the assumption ͑18͒. ticle with opposite energy, momentum, spin, and charge to The dispersion relation ͑20͒ is a quartic equation for that of the negative-energy state. In the present model, how- ␭0(␭ជ ). Although the Euler reducing cubic has a relatively ever, when a negative-energy state moving in a elegant form, in part because Eq. ͑20͒ contains no term cubic CPT-violating background with parameters a␮ and b␮ is in ␭0, the algebraic solutions to this equation are not particu- excited to a positive-energy one, it leaves a hole appearing to larly transparent. Even without examining the analytical re- be a particle with opposite values as before but moving in a sults, however, certain features of the solutions can be estab- CPT-violating background with parameters Ϫa␮ and b␮ in- lished. One is that all four roots must be real, due to stead. This is because the term Ј is odd under charge con- La Hermiticity of the quantum-mechanical Hamiltonian jugation. The same effect can be seen explicitly by construct- ing the charge-conjugate Dirac equation for the model. We ␺ findץ 0 ជ ជ 0 ␮ 0 ␮ 0 .H␺ϵi ϭ͑Ϫi␥ ␥•ٌϩa␮␥ ␥ Ϫb␮␥5␥ ␥ ϩm␥ ͒␺ t ␮ ␮ ␮ cץ ␮ϩa␮␥ Ϫb␮␥5␥ Ϫm͒␺ ϭ0, ͑23͒ץ ␥i͑ 21͒͑ c ¯T Another stems from the invariance of the quartic under the where as usual ␺ ϵC␺ and C is the charge-conjugation interchange (␭ Ϫa ) Ϫ(␭ Ϫa ), which implies that to matrix. ␮ ␮ → ␮ ␮ each solution ␭0 (␭ជ ) there corresponds a second solution The eigenfunctions corresponding to the two negative ei- ϩ 0 0 genvalues ␭ (␭ជ ) can be reinterpreted as positive-energy, ␭ (␭ជ ) given by Ϫ(␣) Ϫ reversed-momentum wave functions in the usual way. We (␣) (␣) 0 0 0 introduce momentum-space spinors u (pជ ), v (pជ) via the ␭ ͑␭ជ ͒ϭϪ␭ ͑Ϫ␭ជϩ2aជ͒ϩ2a . ͑22͒ Ϫ ϩ definitions This equation and the invariance of the quartic under the ␺͑␣͒͑x͒ϭexp͑Ϫip͑␣͒ x͒u͑␣͒͑pជ͒, interchange b Ϫb show that, unlike the conventional u • ␮→ ␮ Dirac case, the magnitudes of the eigenenergies of the four ͑␣͒ ͑␣͒ ͑␣͒ roots all differ generically as a direct consequence of the ␺ ͑x͒ϭexp͑ϩipv •x͒v ͑pជ͒, ͑24͒ CPT-violating terms ͓34͔. Another qualitatively different feature of the present where the four-momenta are given by 0 model is that under certain conditions the roots ␭ (␭ជ ) of the ͑␣͒ ͑␣͒ ͑␣͒ 0 p ϵ͑E ,pជ ͒, E ͑pជ ͒ϭ␭ ͑pជ ͒, dispersion relation can display cusps. For a conventional dis- u u u ϩ͑␣͒ persion relation, the energy is a smooth function of each ͑␣͒ ͑␣͒ ͑␣͒ 0 three-momentum component for both timelike and spacelike pv ϵ͑Ev ,pជ ͒, Ev ͑pជ ͒ϭϪ␭Ϫ͑␣͒͑Ϫpជ͒. ͑25͒ four-momenta, while there is a cusp at the origin for the (␣) (␣) lightlike case. By examining discontinuities in the deriva- The general forms of u (pជ ) and v (pជ ) are given in Ap- pendix A. The relation between the spinors u and v is deter- tives of the roots ␭0(␭ជ ) with respect to the components of mined by the charge-conjugation matrix and the charge- ␭ជ , we have demonstrated that the criterion for cusps to ap- conjugate Dirac equation 23 . For example, u(2)(p,a ,b ) 2 ͑ ͒ ជ ␮ ␮ pear in the present model with m Ͼ0 is that b␮ be timelike. (1)c ϰv (pជ,Ϫa␮ ,b␮), where the dependence on a␮ and b␮ The derivation is most straightforward using observer Lor- has been explicitly restored for clarity. The symmetry ͑22͒ of entz invariance to select one of the canonical frames listed in the dispersion relation then connects the two sets of energies Appendix B, for which exact solutions to the dispersion re- by lations can be found. The presence of cusps appears to have no directly observable consequences, in part because their ͑2,1͒ ͑1,2͒ 0 Ev ͑pជ ͒ϭEu ͑pជ ϩ2aជ ͒Ϫ2a . ͑26͒ size is governed by the magnitude of b␮ , which is highly suppressed in a realistic situation. The exact eigenenergies for the various canonical cases are The assumption that the CPT-violating quantities a␮ and provided in Appendix B, while Appendix C contains explicit b are small relative to the scale m of the low-energy theory ␮ solutions for the eigenspinors in the special case bជ ϭ0. implies the dispersion relation ͑20͒ must have two positive- The four spinors u(␣)(pជ ), v(␣)(pជ) are orthogonal. Their 0 ជ valued roots ␭ϩ(␣)(␭) and two negative-valued roots normalization can be freely chosen, although imposing the 0 ជ c c ␭Ϫ(␣)(␭), where ␣ϭ1,2. Since these roots are eigenvalues condition (␺ ) ϭ␺ provides a partial constraint. Our choice of the time-translation operator, the corresponding wave leads to orthonormality conditions given by functions can be termed positive- and negative-energy states, 0 ͑␣͒ respectively. Useful approximate solutions for ␭ (␭ជ ) that Eu Ϯ(␣) u͑␣͒†͑pជ ͒u͑␣Ј͒͑pជ ͒ϭ␦␣␣Ј , are valid to second order for arbitrary small a␮ and b␮ are m given in Eq. ͑62͒ of Appendix A. Some exact solutions valid for various important special cases are provided in Appendix ͑␣͒ Ev B. v͑␣͒†͑pជ͒v͑␣Ј͒͑pជ͒ϭ␦␣␣Ј , Within conventional relativistic quantum mechanics, m negative-energy states are deemed to be filled, forming the Dirac sea. When a negative-energy state is excited to a u͑␣͒†͑pជ͒v͑␣Ј͒͑Ϫpជ͒ϭ0, v͑␣͒†͑Ϫpជ ͒u͑␣Ј͒͑pជ ͒ϭ0. ͑27͒ 6766 DON COLLADAY AND V. ALAN KOSTELECKY´ 55

Note, however, that the Lorentz breaking precludes a simple It follows that the velocity is related to the energy and mo- generalization of the orthonormality relations involving the mentum by Dirac-conjugate spinors ¯u(␣)(pជ ) and ¯v(␣)(pជ ) instead of the (␣)† (␣)† Hermitian-conjugate spinors u (pជ) and v (pជ). Equa- ͑pជ Ϫaជ ͒ tion ͑27͒ produces the completeness relation. ␥mϭEϪa0, ␥mvជ ϭpជ Ϫaជ Ϫb0 , ͑32͒ ͉pជ Ϫaជ͉ 2 m ͑␣͒ ͑␣͒† 2 ͚ ͑␣͒ u ͑pជ ͒ u ͑pជ ͒ where ␥ϭ1/ͱ1Ϫv as usual. These are just the usual ␣ϭ1 E p͒ ͫ u ͑ ជ special-relativistic results shifted by ͑small͒ amounts con- m trolled by the CPT-violating terms. Note that four- ϩ v͑␣͒͑Ϫpជ ͒ v͑␣͒†͑Ϫpជ ͒ ϭI. ͑28͒ momentum conservation shows that the wave-packet veloc- E͑␣͒ Ϫp v ͑ ជ ͒ ͬ ity is constant, as usual. Even in conventional Dirac quantum mechanics, the We remark in passing that another useful result is the modi- above notion of velocity involves subtleties associated with ﬁed Gordon identity the presence of negative-energy solutions. For example, the velocity operator does not commute with the usual Dirac ͑␣Ј͒ ␮ ͑␣͒ ¯u ͑pជ Ј͒␥ u ͑pជ ͒ Hamiltonian. In the present CPT-violating model, additional subtleties arise. For example, it follows from the properties 1 ␣Ј ͑␣Ј͒␮ ͑␣͒␮ ␮ of the roots of the dispersion relation ͑20͒ or from the above ϭ ¯u ͑pជ Ј͓͒puЈ ϩpu Ϫ2a 2m discussion that for the special case of timelike b␮ the velocity near the origin is not in one-to-one correspondence with ␮␯ ͑␣Ј͒ ͑␣͒ ͑␣͒ ϩi␴ ͑puЈ␯ Ϫpu␯ Ϫ2␥5b␯͔͒u ͑pជ ͒. ͑29͒ the conserved momentum. Perhaps of more interest is that in the general case the velocity operator in the energy basis has The general solution to the modiﬁed Dirac equation ͑6͒ additional off-diagonal components, even in the positive- can be written as a superposition of the four spinors u(␣), energy sector. For the example above, these oscillate trans- (␣) Ϫ1 v : verse to (pជ Ϫaជ ) with relatively large period of order b0 . They provide time-independent corrections to the velocity 3 2 2 d p m ͑␣͒ eigenvalues, but only at order b0. The implications of these Ϫipu •x ͑␣͒ ␺͑x͒ϭ 3 ͚ ͑␣͒ b͑␣͒͑pជ ͒e u ͑pជ͒ features for possible bounds on b are therefore likely to be ͵ ͑2␲͒ ␣ϭ1 ͫ E ␮ u limited but remain to be explored.

m ␣͒ A related approach to the notion of velocity is to take the ip͑ x ϩ d* ͑pជ͒e v • v͑␣͒͑pជ͒ , ͑30͒ E͑␣͒ ͑␣͒ ͬ derivative of the energy with respect to the momentum. For v the special case bជ ϭ0, this definition produces the same result as above and moreover can be obtained without the ex- where b(␣)(pជ ), d(*␣)(pជ) are the usual complex weights for the momentum expansion. We remind the reader that in this plicit wave function. It therefore provides a relatively simple method of investigating velocity-related issues. For example, expression the a␮ and b␮ dependence of the energies and the spinors is understood. causality of the model is related to the restriction of the In the above expressions, the four-momenta are eigenval- group velocity to below the velocity of light. If causality is ues of the translation operators and hence are conserved satisfied, the criterion quantities. They therefore represent canonical energy and Eץ -momentum rather than kinetic energy and momentum. A dis ͉v j͉ϵ Ͻ1 ͑33͒ p jͯץͯ tinction of this type occurs in many physical systems, such as a charged particle moving in an electromagnetic field. In the present case this means, for example, that the canonical four- momenta are not related to velocity as are the usual kinetic should be obeyed for each jϭ1,2,3. Observer Lorentz in- four-momenta in special relativity. The actual relationship variance makes it sufficient to examine the various canonical can be explored by using the velocity operator, given in rela- cases. Calculating with the expressions in Appendix B, we find that the criterion ͑33͒ is satisfied for all values of a and tivistic quantum mechanics by vជ ϵdxជ/dtϭi͓H,xជ͔ϭ␥0␥ជ, ␮ b␮ . This supports the notion that causality is maintained. where H is the Hamiltonian ͑21͒. The expectation value of Although the Lorentz breaking does affect quantum wave this operator for a given wave packet is the vector-current propagation, it apparently is mild enough to avoid superlu- integral and gives the ͑group͒ velocity of the packet. minal signals. As an explicit example, consider the special case bជ ϭ0, Our treatment in this section of the relativistic quantum for which the eigenenergies and eigenspinors are provided in mechanics of a single fermion in the presence of Appendixes B and C, respectively. Suppose a wave packet of CPT-violating terms could be further developed to allow for energy E and momentum pជ is constructed as a superposition interactions with conventional applied fields, along the lines of positive-energy spin-up solutions. A short calculation proof the usual Dirac theory. In detail, this lies outside the scope duces of the present work. We remark, however, that standard Green-function methods should be applicable. In particular, 0 ͉͑pជ Ϫaជ͉Ϫb ͒ ͑pជ Ϫaជ ͒ we can introduce a generalized Feynman propagator ͗vជ ͘ϭ 0 . ͑31͒ ͳ ͑EϪa ͒ ͉pជ Ϫaជ͉ ʹ SF(xϪxЈ) satisfying 55 CPT VIOLATION AND THE STANDARD MODEL 6767

␮ ␮ ␮ 4 3 2 ␮Ϫa␮␥ Ϫb␮␥5␥ Ϫm͒SF͑xϪxЈ͒ϭ␦ ͑xϪxЈ͒ץ ␥i͑ d p m ͑␣͒ † ͑34͒ P␮ϭ 3 ͚ ͑␣͒ pu␮ b͑␣͒͑pជ ͒b͑␣͒͑pជ ͒ ͵ ͑2␲͒ ␣ϭ1 ͫ E u and obeying the usual Feynman boundary conditions. It has m integral representation ϩ p͑␣͒d† ͑pជ ͒d ͑pជ ͒ . ͑39͒ E͑␣͒ v␮ ͑␣͒ ͑␣͒ ͬ v d4p S ͑xϪxЈ͒ϭ eϪip•͑xϪxЈ͒ The reader can verify that the operator P generates space- F ͵ ͑2␲͒4 ␮ CF time translations by determining the commutation relation .(␮␺(xץwith the ﬁeld ␺: i͓P␮ ,␺(x)͔ϭ 1 ϫ , ͑35͒ The creation and annihilation operators can be written in p ␥␮Ϫa ␥␮Ϫb ␥ ␥␮Ϫm ␮ ␮ ␮ 5 terms of the ﬁelds as where CF is the direct analogue of the usual Feynman con- 3 ip͑␣͒ x 0 b ͑pជ ͒ϭ d xe u • ¯u͑␣͒͑pជ͒␥ ␺͑x͒, tour in p0 space, passing below the two negative-energy ͑␣͒ ͵ poles and above the positive-energy ones. Appendix D contains some remarks about this propagator, including a closed- † 3 Ϫip͑␣͒ x 0 d p͒ϭ d xe v • ¯v͑␣͒ p͒␥ ␺ x͒. ͑40͒ form integration for the case bជ ϭ0. ͑␣͒͑ជ ͵ ͑ជ ͑

IV. QUANTUM FIELD THEORY The vacuum state ͉0͘ of the Hilbert space obeys

In this section, we discuss a few aspects of the quantum b͑␣͒͑pជ ͉͒0͘ϭ0, d͑␣͒͑pជ ͉͒0͘ϭ0. ͑41͒ field theory associated with the model Lagrangian ͑5͒.Asin † † the usual Dirac case, direct canonical quantization is unsat- Acting on ͉0͘, the creation operators b(␣)(pជ ) and d(␣)(pជ ) isfactory, and the quantization condition is found instead by produce particles and antiparticles with four-momenta (␣)␮ (␣)␮ imposing positivity of the conserved energy. pu and pv , respectively. The reinterpretation ͑25͒, Promoting the Fourier coefficients in the expansion ͑30͒ which is based on the usual heuristic arguments in relativistic to operators on a Hilbert space, we can obtain from Eq. ͑9͒ quantum mechanics, therefore makes sense in the field- an expression for the normal-ordered conserved energy P0 theoretic framework. As expected, the usual fourfold degen- 3 0 ϭ͐d x:⌰ 0:. This expression is positive definite for eracy of the eigenstates of the Hamiltonian for a given three- a0Ͻm, provided the following nonvanishing anticommuta- momentum is broken by the CPT-violating terms. tion relations are imposed: The above expressions can be used to establish various results for the field theory with nonzero a␮ and b␮ . For ͑␣͒ example, the ͑time-dependent͒ commutation relation be- † Eu ͕b ͑pជ ͒,b ͑pជ Ј͖͒ϭ͑2␲͒3 ␦ ␦3͑pជϪpជЈ͒, tween the conserved charges P and the quantum operators ͑␣͒ ͑␣Ј͒ m ␣␣Ј ␮ M ␮␯ϭ͐d3x:J0␮␯:, obtained from the operator form of the currents ͑11͒, is found to be E͑␣͒ † 3 v 3 ͕d͑␣͒͑pជ͒,d ͑pជЈ͖͒ϭ͑2␲͒ ␦␣␣ ␦ ͑pជϪpជЈ͒. ␭ ␮␯ ␭[␮ ␯] ␭0 [␮ ␯] [␮ ␯] ͑␣Ј͒ m Ј i͓P ,M ͔ϭϪg P Ϫg ͑a J ϩb J5 ͒, ͑42͒ ͑36͒ ␮ 3 ␮ ␮ 3 ␮ where J ϭ͐d x: j : and J5 ϭ͐d x: j5 : are integrals of the charge and chiral currents. The ␭ϭ0 component of this For simplicity, the dependence on a␮ and b␮ is suppressed in these and subsequent equations. The corresponding equal- equation is the quantum-field analogue of Eq. ͑12͒. time field anticommutators are given by The generalizations of the equal-time anticommutation relations ͑37͒ to unequal times must be solutions to the modi- † 3 fied Dirac equation in each variable and must reduce to the ͕␺ j͑t,xជ ͒,␺k͑t,xជЈ͖͒ϭ␦ jk␦ ͑xជϪxជЈ͒, usual results in the limit where a␮ and b␮ vanish. The cor- † † rect expressions can in principle be derived by evolving for- ͕␺j͑t,xជ͒,␺k͑t,xជЈ͖͒ϭ͕␺j ͑t,xជ͒,␺k͑t,xជЈ͖͒ϭ0, ͑37͒ ward in time one of the two fields in each anticommutator of Eq. ͑37͒. We write where the spinor indices j,k are explicitly shown. Using these expressions, the normal-ordered conserved ͕␺͑x͒,¯␺͑xЈ͖͒ϭiS͑xϪxЈ͒, charge becomes ͕␺͑x͒,␺͑xЈ͖͒ϭ͕¯␺͑x͒,¯␺͑xЈ͖͒ϭ0, ͑43͒ 3 2 d p m † Qϭ 3 ͚ ͑␣͒ b͑␣͒͑pជ ͒b͑␣͒͑pជ ͒ where ͵ ͑2␲͒ ␣ϭ1 ͫ E u d4p Ϫip•͑xϪxЈ͒ m S͑xϪxЈ͒ϭ 4 e Ϫ d† ͑pជ ͒d ͑pជ ͒ . ͑38͒ ͵C ͑2␲͒ E͑␣͒ ͑␣͒ ͑␣͒ ͬ v 1 ϫ ␮ ␮ ␮ . ͑44͒ Similarly, the normal-ordered conserved four-momentum is p␮␥ Ϫa␮␥ Ϫb␮␥5␥ Ϫm 6768 DON COLLADAY AND V. ALAN KOSTELECKY´ 55

In this expression, C is the analogue of the usual closed The construction of the in and out fields and the definition contour in p0 space encircling all the poles in the anticlock- of the S matrix can be implemented in the normal manner. wise direction. Some comments about this anticommutator The Lehmann-Symanzik-Zimmermann ͑LSZ͒ reduction pro- function are given in Appendix E, along with a closed-form cedure for S-matrix elements generates expressions involv- integration for bជ ϭ0. For this case, we have checked explic- ing vacuum expectation values of time-ordered products of itly that the anticommutators ͑43͒ are detemined by the inte- the interacting fields, as usual, but with external-leg factors ␮ ␮ץ ␥gral ͑44͒. for fermions involving the modified Dirac operator (i ␮ ␮ ␮ឈ ␮ ␥␮ϩa␮ץ ␥The anticommutators ͑43͒ are relevant to the causal struc- Ϫa␮␥ Ϫb␮␥5␥ Ϫm) or its conjugate (i ␮ ture of the quantum theory. In particular, for the case bជ ϭ0 ϩb␮␥5␥ ϩm). the results in Appendix E can be used to show that the anti- The canonical Dyson formalism for the perturbation se- commutator of two fields separated by a spacelike interval ries of time-ordered products of interacting fields in terms of vanishes: in fields can be applied without encountering difficulties. Standard expressions emerge, including the vacuum bubbles. Wick’s theorem holds, reducing the time-ordered products ¯ 2 ͕␺␣͑x͒,␺␤͑xЈ͖͒ϭ0, ͑xϪxЈ͒ Ͻ0. ͑45͒ into normal-ordered products and pair contractions of in fields. For fermion in fields, the vacuum expectation value of Observables constructed out of bilinear products of field op- a pairwise contraction can be shown explicitly to be the generators and separated by a spacelike interval therefore com- eralized Feynman propagator introduced in the previous sec- ¯ mute. This means that the quantum field theory with timelike tion: ͗0͉T␺(x)␺(xЈ)͉0͘ϭiSF(xϪxЈ). In a momentum- b␮ preserves microscopic causality in all associated observer space Feynman diagram, the corresponding propagator is frames. The breaking of Lorentz invariance and the distortions relative to conventional propagation are apparently 1 SF͑p͒ϭ ␮ ␮ ␮ . ͑46͒ mild enough to exclude superluminal signals, in agreement p␮␥ Ϫa␮␥ Ϫb␮␥5␥ Ϫm with the result from relativistic quantum mechanics. The result might be anticipated since observer Lorentz invariance The momentum-space Feynman rules require this modified holds and the particle Lorentz breaking involves only local propagator for internal fermion lines and modified spinors on terms in the Lagrangian. A direct analytical proof of micro- external fermion lines, but are otherwise unchanged from scopic causality in the quantum field theory for the cases of those of a conventional theory. For example, translational lightlike or spacelike b␮ would be of interest but is hampered invariance insures that energy and momentum are conserved by the complexity of the integral ͑44͒. in any process and so the standard four-momentum ␦ func- We next turn to issues associated with interacting field tions emerge. theory. For the most part, since the CPT-violating terms in Since the CPT-violating terms are renormalizable by na- the model Lagrangian can be treated exactly, any added con- ive power counting, we anticipate no difficulties with the ventional interactions can be handled with standard methods. usual renormalization program. Details of loop calculations In what follows, we suppose that the Dirac fermion in the lie beyond the scope of the present work and remain an in- model has interactions with one or more other fields that are teresting open issue. We remark in passing that the form of a type acceptable within a conventional approach, and we ͑D3͒ of the propagator given in Appendix D shows that a␮ discuss effects from CPT-violating terms. cancels around all closed fermion loops in analogy with Fur- Essentially all the standard assumptions underlying treat- ry’s theorem. ments of conventional interacting field theories can reason- We also expect the unitarity of the S matrix to be unaf- ably be made in the present context. Thus, for example, the fected by CPT-violating terms. Since a complete Hilbert- property of observer Lorentz invariance ensures that consis- space solution exists in the pure fermion case, there are no tent quantization can be established in all observer frames hidden or inaccessible states that could generate nonunitarity once it is established in a given frame. Much of the usual of the type appearing in the first stage of Gupta-Bleuler analysis is performed in a given observer frame, which in the quantization of quantum electrodynamics, for example. present context means that the values of a␮ and b␮ are fixed. Moreover, the interaction Hamiltonian is Hermitian. In any Distinct effects are to be expected only in calculations for event, any nonunitarity appearing in a realistic model based which particle Lorentz covariance plays an essential role. For on a unitary fundamental theory would presumably be a sig- the most part, matters proceed in a straightforward manner at nal that the domain of validity of the effective low-energy the level of the general framework of interacting field theory. theory is being breached. One exception we have found is the explicit derivation of the In determining physically relevant quantities such as cross Ka¨lleń-Lehmann spectral representation for the vacuum ex- sections or transition rates, kinematic factors appear. For the pectation value of the field anticommutator, which normally most part, these are straightforward to obtain. A subtlety takes advantage of both CPT and particle Lorentz covari- arises in the calculation of a physical cross section because ance ͓35–38͔. The spectral representation could be used to the standard definition involves the notion of incident flux investigate microscopic causality of the interacting theory, defined in terms of incoming particle velocity. Since the although the conventional local interactions we consider velocity-momentum relation has corrections involving a␮ seem unlikely to introduce difficulties in this regard. In any and b␮ ͓cf. Eq. ͑32͔͒, there are corresponding small modifi- case, it remains an open issue to obtain the spectral represen- cations to the kinematic factors in standard cross-section for- tation in the present case, where additional four-vectors ap- mulas expressed in terms of conserved momenta. In a real- pear in the theory. istic case, these are unobservable because they are 55 CPT VIOLATION AND THE STANDARD MODEL 6769 suppressed. This should be contrasted with the CPT ¯ ␮ ¯ ␮ leptonϭϪ͑aL͒␮ABLA␥ LBϪ͑aR͒␮ABRA␥ RB . ͑49͒ CPT-violating corrections arising in an amplitude from the L modified fermion propagator, which are also suppressed but The constant flavor-space matrices aL,R are Hermitian. The might be detected in interferometric experiments using presence of the ␥␮ factor allows fields of only one handed- neutral-meson oscillations ͓10͔. ness to appear in a given term while maintaining gauge invariance. This contrasts with conventional Yukawa cou- V. STANDARD-MODEL EXTENSION plings, in which fields of both handedness appear and invariance is ensured by the presence of the Higgs doublet. In this section, we consider the possibility of generalizing After spontaneous symmetry breaking, the mass eigen- the minimal standard model by adding CPT-violating terms states ͑denoted with carets͒ are constructed with standard within a self-consistent framework of the type described in unitary transformations the previous sections. Since CPT violation has not been ob- ␯ l ˆ l ˆ served in nature, any CPT-violating constants appearing in ␯LAϭ͑UL͒AB␯ˆLB , lLAϭ͑UL͒ABlLB , lRAϭ͑UR͒ABlRB . an extension of the minimal standard model must be small. ͑50͒ In what follows, we assume that these constants are singlets under the unbroken gauge group, but as before behave under The CPT-violating term in Eq. ͑49͒ becomes particle Lorentz transformations as tensors with an odd num- CPT ¯ ␮ ˆ¯ ␮ˆ ber of Lorentz indices. Our primary goal is to obtain an leptonϭϪ͑aˆ␯L͒␮AB␯ˆLA␥ ␯ˆLBϪ͑aˆlL͒␮ABlLA␥ lLB L explicit and realistic model for CPT-violating interactions ˆ¯ ␮ˆ that could serve as a basis for establishing quantitative Ϫ͑aˆlR͒␮ABlRA␥ lRB , ͑51͒ CPT bounds. The discussion in the previous sections is limited to where each matrix of constants aˆ ␮ is obtained from the cor- CPT-breaking fermion bilinears. However, other types of responding a␮ via unitary rotation with the corresponding † terms violating CPT in the Lagrangian could in principle matrix U: aˆ ␮ϭU a␮U. originate from spontaneous symmetry breaking. We adopt Not all the couplings aˆ ␮ are observable. The freedom to here a general approach, investigating possible redefine fields allows some couplings to be eliminated, in CPT-violating extensions to the minimal standard model analogy to the discussion of Sec. II C for the model Lagrang- such that the SU͑3͒ϫSU͑2͒ϫU͑1͒ gauge structure is main- ian. Consider, for example, general field redefinitions of the tained. To preserve naive power-counting renormalizability form at the level of the unbroken gauge group, we restrict atten- ˜ ␯ ˆ ˜ l ˆ ˜ l ˆ tion to terms involving field operators of mass dimension ␯LAϭ͑VL͒AB␯LB , lLAϭ͑VL͒ABlLB , lRAϭ͑VR͒ABlRB , four or less. The simultaneous requirements of gauge invari- ͑52͒ ance, suitable mass dimensionality, and CPT violation allow where the matrices V(x␮) are unitary in generation space and relatively few new terms in the action ͓39͔. have the form Vϭexp(iH x␮) with H Hermitian. Then, in Any Lagrangian term must be formed from combinations ␮ ␮ ¯ˆ ␮ ˆ -␮␺, the redefiץ ␥of covariant derivatives and fields for leptons, quarks, gauge each kinetic term of the generic form i␺ bosons, and Higgs bosons. We consider first allowed nition ͑52͒ generates an apparent CPT-violating term of the CPT-violating Lagrangian extensions involving fermions. ˜¯ ␮ ␭ ␯ ˜ form Ϫ␺␥ exp(iH␭x )H␮ exp(ϪiH␯x )␺. A suitable choice Inspection shows that the only possibilities satisfying the of H might therefore remove a CPT-violating term involv- above criteria are pure fermion-bilinear terms without de- ␮ ing aˆ ␮ from Eq. ͑51͒. The matrices V must be chosen to rivatives ͓40͔. In the present context involving many fermi- leave unaffected the Yukawa couplings and the conventional ons, the analysis given in the previous sections of such terms nonderivative couplings defining the neutrino fields as weak requires some generalization. Since SU͑3͒ invariance pre- eigenstates. It can be shown that Vl must be a diagonal ma- cludes quark-lepton couplings, we can treat the lepton and L trix of phases, with Vl ϭVl and V␯ ϭ(Ul )†U␯ Vl . The free- quark sectors separately as usual. R L L L L L dom therefore exists to redefine the fields so as to eliminate, Consider first the lepton sector. Denote the left- and right- say, the three diagonal elements of the CPT-violating matrix handed lepton multiplets by in the neutrino sector. Note that the existence of this choice obviates possible theoretical issues arising from the combi- ␯A L ϭ , R ϭ l , ͑47͒ nation of massless fields at zero temperature and small nega- A ͩ l ͪ A ͑ A͒R A L tive energy shifts induced by CPT-breaking terms ͓41͔. Thus, omitting tildes and carets, the general where CPT-violating extension of the lepton sector of the minimal standard model has the form 1 1 ␺ ϵ ͑1ϩ␥ ͒␺, ␺ ϵ ͑1Ϫ␥ ͒␺, ͑48͒ 1 R 2 5 L 2 5 CPT ϭϪ͑a ͒ ¯␯ ͑1ϩ␥ ͒␥␮␯ Ϫ͑a ͒ ¯l ␥␮l Llepton ␯ ␮AB A 2 5 B l ␮AB A B as usual, and where Aϭ1,2,3 labels the lepton flavor: l ¯ ␮ A Ϫ͑bl͒␮ABlA␥5␥ lB , ͑53͒ ϵ(e,␮,␶), ␯Aϵ(␯e ,␯␮ ,␯␶). Then, the most general set of CPT-violating lepton bilinears consistent with gauge invari- where we have used Eq. ͑48͒ to replace left- and right- ance is handed couplings with vector and axial vector couplings, and 6770 DON COLLADAY AND V. ALAN KOSTELECKY´ 55

where (a␯)␮AAϭ0. Note that Eq. ͑53͒ includes terms break- Omitting tildes and carets, the general CPT-violating ex- ing individual lepton numbers, although total lepton number tension of the quark sector of the minimal standard model is remains conserved. Flavor-changing transitions therefore ex- therefore ist in principle but are unobservable if the CPT-violating couplings are sufficiently suppressed. For example, a frac- CPT ϭϪ͑a ͒ ¯u ␥␮u Ϫ͑b ͒ ¯u ␥ ␥␮u Lquark u ␮AB A B u ␮AB A 5 B tional suppression of order 10Ϫ17 or smaller might occur in ¯ ␮ ¯ ␮ the string scenario ͓10͔. Ϫ͑ad͒␮ABdA␥ dBϪ͑bd͒␮ABdA␥5␥ dB , ͑59͒ Consider next the quark sector. The SU͑3͒ symmetry ensures that all three quark colors of any given flavor must where (au)␮11ϵ0. Again, these terms include small flavor- changing effects that are unobservable if the suppression is have the same CPT-violating current coupling. We can Ϫ17 therefore disregard the color space in what follows, and a sufficiently small, such as that of fractional order 10 or construction analogous to that for the lepton sector can be smaller possible in the string scenario. In contrast, the diag- applied. Denote the left- and right-handed components of the onal contributions might be detected in interferometric ex- quark fields by periments that measure the phenomenological parameters ␦ P for indirect CPT violation in oscillations of neutral-P mesons, where P is one of K, D, B ,orB . Each quantity uA d s Q ϭ , U ϭ͑u ͒ , D ϭ͑d ͒ , ͑54͒ A ͩ d ͪ A A R A A R ␦ P is proportional to the difference between the diagonal A L elements of the effective Hamiltonian governing the time evolution of the corresponding P-¯P system. Explicit expres- where Aϭ1,2,3 labels the quark flavors: uAϵ(u,c,t), dA ϵ(d,s,b). Then, at the unbroken-symmetry level, the most sions for ␦ P in terms of quantities closely related to those in general CPT-violating coupling is Eq. ͑59͒ have been given in Ref. ͓10͔. The two equations ͑53͒ and ͑59͒ represent allowed CPT ϭϪ͑a ͒ Q¯ ␥␮Q Ϫ͑a ͒ U¯ ␥␮U CPT-violating extensions of the fermion sector of the mini- Lquark Q ␮AB A B U ␮AB A B mal standard model. Next, we briefly consider other ¯ ␮ CPT-violating terms without fermions. Ϫ͑aD͒␮ABDA␥ DB . ͑55͒ The only CPT-violating term involving the Higgs field

As before, the constant flavor-space matrices aQ,U,D are Her- and satisfying our criteria is a derivative coupling of the mitian. form After spontaneous symmetry breaking the mass eigen- CPT ␮ † states are obtained with the standard unitary transformations Higgsϭik ␾ D␮␾ϩH.c., ͑60͒ L u u where k␮ is a CPT-violating constant, D is the covariant uLAϭ͑UL͒ABuˆLB , uRAϭ͑UR͒ABuˆRB , ␮ derivative, and ␾ is the usual SU͑2͒-doublet Higgs field. Let d d us proceed under the assumption that no self-consistency is- d ϭ͑U ͒ dˆ , d ϭ͑U ͒ dˆ . ͑56͒ LA L AB LB RA R AB RB sues arise for a scalar field that breaks CPT and Lorentz invariance, so that standard methods apply. Then, Eq. ͑60͒ The CPT-violating expression ͑55͒ becomes 0 represents a contribution to the Higgs-Z␮ sector of the CPT ¯ model. Disregarding possible CPT-preserving but Lorentz ϭϪ͑aˆ ͒ u¯ˆ ␥␮uˆ Ϫ͑aˆ ͒ dˆ ␥␮dˆ Lquark uL ␮AB LA LB dL ␮AB LA BL breaking contributions to the static potential, it can be shown that the term ͑60͒ produces a ͑stable͒ modification of the ¯ ␮ ¯ˆ ␮ˆ Ϫ͑aûR͒␮ABuˆRA␥ uˆRBϪ͑aˆdR͒␮ABdRA␥ dRB . standard symmetry-breaking pattern to include an expecta- 0 ͑57͒ tion value for the Z␮ field with magnitude proportional to k␮ . Several kinds of effect ensue but if, as expected, the ␮ As in the lepton sector, each constant matrix aˆ ␮ is obtained quantities k are sufficiently small then it can be shown that from the corresponding a␮ via the appropriate unitary rota- the results are either unobservable or produce additional con- tion. tributions to the fermion-bilinear terms already considered. Again, field redefinitions can be used to eliminate some It is also possible to find CPT-violating terms satisfying CPT violation. Consider field redefinitions of the form our criteria and involving only the gauge fields. They are of the form u u ˜uLAϭ͑VL͒ABuˆLB , ˜uRAϭ͑VR͒ABuˆRB , CPT ␬␭␮␯ 2 gaugeϭk3␬⑀ Tr(G␭G␮␯ϩ 3 G␭G␮G␯) ˜ d ˆ ˜ d ˆ L dLAϭ͑VL͒ABdLB , dRAϭ͑VR͒ABdRB , ͑58͒ ␬␭␮␯ 2 ϩk2␬⑀ Tr(W␭W␮␯ϩ 3 W␭W␮W␯) ␮ ␬␭␮␯ ␬ where as before the matrices Vϭexp(iH␮x ) are unitary in ϩk1␬⑀ B␭B␮␯ϩk0␬B , ͑61͒ generation space. In this case, invariance of the Yukawa and nonderivative couplings, including the Cabbibo-Kobayashi- where k3␬ , k2␬ , k1␬ , and k0␬ are CPT-violating constants. Maskawa mixings, requires the effect of the matrices V to Here, G␮ , W␮ , B␮ are the ͑matrix-valued͒ SU͑3͒,SU͑2͒, reduce to multiplication by a single phase. For example, we U͑1͒gauge bosons, respectively, and G␮␯ , W␮␯ , B␮␯ are can choose this phase so that the condition (auL)␮11 the corresponding field strengths. The first three of these ϭ(auR)␮11 holds. This removes the CPT-breaking vector terms can be shown to leave unaffected the symmetry- coupling from the u-quark sector. breaking pattern, and we expect only unobservable effects 55 CPT VIOLATION AND THE STANDARD MODEL 6771 for sufficiently small CPT-violating constants ͓42͔. The field Within the framework developed, we have constructed a entering the term with coupling k0␬ is of dimension one. It CPT-violating generalization of the minimal standard model appears to produce a linear instability in the theory because it that could be used in establishing quantitative CPT bounds. involves the photon, in which case it cannot emerge from a The criteria of gauge invariance and power-counting renor- fundamental theory with a stable ground state. malizability constrain the extension to a relatively simple form, involving the extra terms given in Eqs. ͑53͒, ͑59͒, ͑60͒, VI. SUMMARY and ͑61͒. It has been previously been suggested ͓9,10͔ that the properties of neutral-meson systems PP¯, where P is one In this paper, we have developed a framework for treating of K, D, Bd ,orBs, are well suited to interferometric tests spontaneous CPT and Lorentz breaking in the context of of spontaneous CPT violation, with the experimentally mea- conventional effective field theory. The underlying action is surable parameters for ͑indirect͒ CPT violation being explic- ´ assumed to be consistent and fully CPT and Poincare invari- itly related to certain diagonal elements of the quark-sector ant, with solutions exhibiting spontaneous CPT and Lorentz CPT-breaking matrices given in Eq. ͑59͒. Investigating the breaking. The effective low-energy field theory then remains current experimental constraints on the other translationally invariant and covariant under changes of ob- CPT-violating parameters introduced here is an interesting server inertial frame, but violates CPT and partially breaks open topic and could lead to additional signals for CPT vio- covariance under particle boosts. lation. Our focus has primarily been on Lagrangian terms that involve CPT-violating fermion bilinears, which are relevant for experiments bounding CPT in meson interferometry. In ACKNOWLEDGMENTS principle, these terms can be treated exactly because they are quadratic. We have investigated the relativistic quantum me- We thank Robert Bluhm for discussion. This work was chanics and the quantum field theory of a model for a Dirac supported in part by the United States Department of Energy fermion involving CPT violation. The analysis suggests that under Grant No. DE-FG02-91ER40661. effective field theories with spontaneous CPT breaking have desirable properties like microscopic causality and renormalizability. The existence of consistent theories of this type is APPENDIX A: EIGENSPINORS reasonable since they are analogous to conventional field OF THE DIRAC EQUATION theories in a nonvanishing background. Additional interactions appear minimally affected by the CPT violation, and Treating the CPT-violating parameters a␮ and b␮ as 0 the effects are largely restricted to modifications on fermion small relative to m, the four roots ␭Ϯ(␣)(␭ជ ), ␣ϭ1,2, of the lines. dispersion relation ͑20͒ are given to second order by

0 ជ 2 ជ 2 ␣ 2 ជ ជ 2 ជ2 2 ជ ជ ជ 2 ជ ជ ជ 2 ជ 2 1/2 ␭Ϯ͑␣͒͑␭͒ϭϮ͕m ϩ͑␭Ϫaជ͒ ϩ2͑Ϫ1͒ ͓b0͑␭Ϫa͒ ϩb m ϩ„b•͑␭Ϫa͒… ϯ2b0b•͑␭Ϫa͒ͱm ϩ͑␭Ϫaជ͒ ͔

1/2 2b bជ ជ Ϫaជ 2 2 0 •͑␭ ͒ ϩb0ϩbជ ϯ ϩa0 . ͑A1͒ ͱm2ϩ͑␭ជ Ϫaជ ͒2ͮ

This equation produces exact solutions to the dispersion re- ͑E͑␣͒Ϫa ϩmϩbជ ␴ជ ͓͒͑pជϪaជ͒ ␴ជϪb ͔ ជ ជ ជ ͑␣͒ u 0 • • 0 lation in any of the special cases for which b0b•(␭Ϫa) Xu ϭ . ͑A3͒ (␣) (␣) ͑␣͒ 2 2 ϭ0. The eigenenergies of the four spinors u (pជ ), v (pជ) ͑Eu Ϫa0ϩm͒ Ϫbជ deﬁned in Eq. ͑24͒ can be obtained by combining Eq. ͑A1͒ with Eq. 25 . (␣) ͑ ͒ The analogous quantity X for the second equation in ͑A2͒ In the general case, the four spinor eigensolutions can be v can be found by replacing all subscripts u by v in Eq. ͑A3͒ written in the Pauli-Dirac representation as and implementing the substitutions a␮ Ϫa␮ , b␮ Ϫb␮ wherever these quantities explicitly appear.→ The quantities→ ␾͑␣͒ (␣) (␣) ͑␣͒ ͑␣͒ ␾ and ␹ are two-component spinors satisfying the ei- u ͑pជ ͒ϭN ␣͒ , u ͩ X͑ ␾͑␣͒ ͪ genvalue equations u

X͑␣͒␹͑␣͒ ␬ជ ͑␣͒ ␴ជ ␾͑␣͒ϭ␩͑␣͒␾͑␣͒, ␬ជ ͑␣͒ ␴ជ ␹͑␣͒ϭ␩͑␣͒␹͑␣͒, v͑␣͒͑pជ ͒ϭN͑␣͒ v . ͑A2͒ u • u v • v v ͩ ␹͑␣͒ ͪ ͑A4͒

(␣) (␣) 2 (␣) 2 (␣) In the ﬁrst of these equations, Nu is an arbitrary spinor with ͓␬ជ u ͔ ϭ͓␩u ͔ . Here, the vector ␬ជ u and the scalar (␣) (␣) normalization factor and Xu is a spinor matrix deﬁned by ␩u are given by 6772 DON COLLADAY AND V. ALAN KOSTELECKY´ 55

͑␣͒ ␮ ͑␣͒ 2 2 ␬ជ u ϭ2͓͑pu␮Ϫa␮͒b ϩmb0͔͑pជϪaជ͒ Eu ϭ͓m ϩ͑pជ Ϫaជ ͒

2 2 2 ͑␣͒ ϩ͑Ϫ1͒␣2ͱm2bជ 2ϩ bជ ͑pជ Ϫaជ ͒ 2ϩbជ 2͔1/2ϩa , Ϫ͓͑puϪa͒ ϩb ϩm ϩ2m͑Eu Ϫa0͔͒bជ, „ • … 0

␩͑␣͒ϭ2 E͑␣͒Ϫa bជ2Ϫ2b bជ pϪa ͑␣͒ 2 2 u ͑ u 0͒ 0 •͑ជ ជ͒ Ev ϭ͓m ϩ͑pជ ϩaជ ͒ ͑␣͒ 2 2 2 Ϫ͑Eu Ϫa0ϩm͓͒͑puϪa͒ Ϫb Ϫm ͔. ␣ 2 ជ 2 ជ 2 ជ 2 1/2 Ϫ͑Ϫ1͒ 2ͱm b ϩ„b•͑pជ ϩaជ ͒… ϩb ͔ Ϫa0 . ͑A5͒ ͑B2͒ The analogous quantities with subscripts v are given by the ␮ same substitutions as before. Finally, for the lightlike case b␮b ϭ0 the exact eigenvalues of the dispersion relation after reinterpretation are APPENDIX B: EXACT EIGENENERGIES FOR CANONICAL CASES ͑␣͒ 2 ␣ ជ 2 1/2 ␣ Eu ϭ͓m ϩ„pជ Ϫaជ Ϫ͑Ϫ1͒ b… ͔ ϩa0ϩ͑Ϫ1͒ b0 ,

For the case where b␮ is timelike, observer Lorentz invariance can be used to select a canonical frame in which ͑␣͒ 2 ␣ ជ 2 1/2 ␣ Ev ϭ͓m ϩ„pជ ϩaជ ϩ͑Ϫ1͒ b… ͔ Ϫa0Ϫ͑Ϫ1͒ b0 . bជ ϭ0. In this frame, we ﬁnd the exact eigenenergies after ͑B3͒ reinterpretation are

͑␣͒ 2 ␣ 2 1/2 These last expressions hold in all observer frames. Eu ϭ͓m ϩ„͉pជ Ϫaជ͉ϩ͑Ϫ1͒ b0… ͔ ϩa0 ,

͑␣͒ 2 ␣ 2 1/2 ᠬ 0؍Ev ϭ͓m ϩ„͉pជ ϩaជ͉Ϫ͑Ϫ1͒ b0… ͔ Ϫa0 , ͑B1͒ APPENDIX C: EXPLICIT SOLUTION FOR b where ␣ϭ1,2 as usual. For the special case bជ ϭ0, the eigenenergies are given by For the case of spacelike b␮ , an observer frame can be Eq. ͑B1͒ and the eigenspinors can be written in a relatively chosen in which b0ϭ0. After reinterpretation the exact simple form. Introducing momentum-space spinors via Eq. eigenenergies become ͑24͒, we ﬁnd in the Pauli-Dirac basis the expressions

͑␣͒ ͑␣͒ ͑␣͒ 1/2 ␾ ͑pជ Ϫaជ ͒ Eu ͑Eu Ϫa0ϩm͒ ␣ u͑␣͒͑pជ ͒ϭ Ϫ͑Ϫ1͒ ͉pជ Ϫaជ͉Ϫb0 , 2m͑E͑␣͒Ϫa ͒ ␾͑␣͒͑pជ Ϫaជ ͒ ͩ u 0 ͪ ͩ E͑␣͒Ϫa ϩm ͪ u 0

Ϫ͑Ϫ1͒␣͉pជ ϩaជ͉ϩb E͑␣͒͑E͑␣͒ϩa ϩm͒ 1/2 0 ͑␣͒ ͑␣͒ v v 0 ͑␣͒ ␾ ͑pជ ϩaជ ͒ v ͑pជ ͒ϭ ͑␣͒ Ev ϩa0ϩm , ͑C1͒ ͩ 2m͑E ϩa0͒ ͪ v ͩ ␾͑␣͒͑pជ ϩaជ ͒ ͪ

where ␣ϭ1,2 as usual and where we have chosen the normalization of the spinors so that Eq. ͑27͒ is satisﬁed. In Eq. ͑C1͒, the (␣) ˆ ␣ two two-component spinors ␾ (␭ជ ) are the eigenvectors of ␴ជ •␭ with eigenvalues Ϫ(Ϫ1) . If the spherical-polar angles that ␭ជ subtends are speciﬁed as ͑␪,␾͒, then the spinors ␾(␣)(␭ជ ) are given explicitly by

␪ ␪ cos Ϫsin eϪi␾ 2 2 ␾͑1͒͑␭ជ ͒ϭ , ␾͑2͒͑␭ជ ͒ϭ . ͑C2͒ ␪ ␪ sin ei␾ cos ͩ 2 ͪ ͩ 2 ͪ

Note that the structure of the CPT-violating terms forces the spinors ͑C1͒ and their generalizations in Appendix A to involve helicity-type states. In the limit of vanishing CPT violation, the solutions ͑C1͒ reduce to standard Dirac spinors in the helicity basis.

APPENDIX D: PROPAGATOR FUNCTIONS It can be shown that the generalized Feynman propagator determined by Eqs. ͑34͒ and ͑35͒ has the form

␭ ␭ ␭ ␮ ␮ ␮ ␯ ␯ ␯ ,␯Ϫa␯␥ ϩb␯␥5␥ Ϫm͒⌬F͑xϪxЈ͒ץ ␥␮Ϫa␮␥ ϩb␮␥5␥ ϩm͒͑iץ ␥␭Ϫa␭␥ Ϫb␭␥5␥ ϩm͒͑iץ ␥SF͑xϪxЈ͒ϭ͑i ͑D1͒

where 55 CPT VIOLATION AND THE STANDARD MODEL 6773

d4p 1 ⌬ ͑y͒ϭ eϪip•y , ͑D2͒ F ͵ 4 Ϫ 2Ϫ 2Ϫ 2 2ϩ 2 Ϫ 2Ϫ ␮ Ϫ 2 CF ͑2␲͒ ͓͑p a͒ b m ͔ 4b ͑p a͒ 4͓b ͑p␮ a␮͔͒

with CF the same contour in the p0 plane as that in Eq. ͑35͒. Direct integration for the special case bជ ϭ0 gives

Ϫia•͑xϪxЈ͒ ␮ 0 2 2 2 0 j j͒⌬F͑xϪxЈ͒, ͑D3͒ץ ␥ ␥Ϫm Ϫb0ϩ2ib0␥5 ץ␮ϩb0␥5␥ ϩm͒͑Ϫץ ␥SF͑xϪxЈ͒ϭe ͑i where

2 2 2 2 1 sin b0r 2iK0͑mͱr Ϫt ͒, r Ͼt , ⌬F͑xϪxЈ͒ϭ 2 ͑2͒ 2 2 2 2 ͑D4͒ 16␲ b0r ͭ ␲H0 ͑mͱt Ϫr ͒, r Ͻt ,

(2) where r is the radial spherical-polar coordinate. In this expression, K0 is a modiﬁed Bessel function and H0 is a Hankel function of the second kind. The result ͑D3͒ reduces to the standard one in the limit a␮ϭb0ϭ0. Note that the propagator is singular on the light cone, as usual.

APPENDIX E: ANTICOMMUTATOR FUNCTIONS The anticommutator function S(xϪxЈ) deﬁned in Eq. ͑44͒ can be shown to be given by

␭ ␭ ␭ ␮ ␮ ␮ ␯ ␯ ␯ ␯Ϫa␯␥ ϩb␯␥5␥ Ϫm͒⌬͑xϪxЈ͒, ͑E1͒ץ ␥␮Ϫa␮␥ ϩb␮␥5␥ ϩm͒͑iץ ␥␭Ϫa␭␥ Ϫb␭␥5␥ ϩm͒͑iץ ␥S͑xϪxЈ͒ϭ͑i where

d4p 1 i⌬͑y͒ϭ eϪip•y , ͑E2͒ ͵ 4 Ϫ 2Ϫ 2Ϫ 2 2ϩ 2 Ϫ 2Ϫ ␮ Ϫ 2 C ͑2␲͒ ͓͑p a͒ b m ͔ 4b ͑p a͒ 4͓b ͑p␮ a␮͔͒

with C being the contour of Eq. ͑44͒ in the p0 plane. For the special case bជ ϭ0, direct integration gives

Ϫia•͑xϪxЈ͒ ␮ 0 2 2 2 0 j j͒⌬͑xϪxЈ͒, ͑E3͒ץ ␥ ␥Ϫm Ϫb0ϩ2ib0␥5 ץ␮ϩb0␥5␥ ϩm͒͑Ϫץ ␥S͑xϪxЈ͒ϭe ͑i where

2 2 J0͑mͱt Ϫr ͒, tϾr, 1 sin b0r i⌬͑xϪxЈ͒ϭϪ 0, ϪrϽtϽr, ͑E4͒ 8␲ b0r 2 2 ͭ ϪJ0͑mͱt Ϫr ͒, tϽϪr,

where r is the radial spherical-polar coordinate and J0 is a Bessel function. The expression ͑E3͒ reduces to the standard result in the limit a␮ϭb0ϭ0.

͓1͔ J. Schwinger, Phys. Rev. 82, 914 ͑1951͒. in Proceedings of the Joint International Lepton-Photon Sym- ͓2͔ G. Lu¨ders, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 28 ͑5͒ posium and Europhysics Conference on High Energy Physics, ͑1954͒; Ann. Phys. ͑N.Y.͒ 2,1͑1957͒. Geneva, Switzerland, 1991, edited by S. Hegarty, K. Potter, ͓3͔ J. S. Bell, Birmingham University thesis ͑1954͒; Proc. R. Soc. and E. Quercigh ͑World Scientiﬁc, Singapore, 1992͒;V.A. London A231, 479 ͑1955͒. Kostelecky´ and R. Potting, in Gamma Ray-Neutrino Cosmol- ͓4͔ W. Pauli, in Niels Bohr and the Development of Physics, ed- ogy and Planck Scale Physics, edited by D. B. Cline ͑World ited by W. Pauli ͑McGraw-Hill, New York, 1955͒,p.30. Scientiﬁc, Singapore, 1993͒. ͓5͔G. Lu¨ders and B. Zumino, Phys. Rev. 106, 385 ͑1957͒. ͓11͔ V. A. Kostelecky´ and S. Samuel, Phys. Rev. D 39, 683 ͑1989͒; ͓6͔ Particle Data Group, R. M. Barnett et al., Phys. Rev. D 54,1 Phys. Rev. Lett. 63, 224 ͑1989͒; Phys. Rev. D 40, 1886 ͑1989͒; ͑1996͒. Phys. Rev. Lett. 66, 1811 ͑1991͒. ͓7͔ L. K. Gibbons et al., Phys. Rev. D 55, 6625 ͑1997͒;B. ͓12͔ D. Colladay and V. A. Kostelecky´, Phys. Lett. B 344, 259 Schwingenheuer et al., Phys. Rev. Lett. 74, 4376 ͑1995͒. ͑1995͒; V. A. Kostelecky´ and R. Van Kooten, Phys. Rev. D ͓8͔ R. Carosi et al., Phys. Lett. B 237, 303 ͑1990͒. 54, 5585 ͑1996͒. ͓9͔ V. A. Kostelecky´ and R. Potting, Nucl. Phys. B359, 545 ͓13͔ D. Colladay and V. A. Kostelecky´, Phys. Rev. D 52, 6224 ͑1991͒; Phys. Lett. B 381, 389 ͑1996͒. ͑1995͒. ͓10͔ V. A. Kostelecky´ and R. Potting, Phys. Rev. D 51, 3923 ͓14͔ O. Bertolami, D. Colladay, V. A. Kostelecky´, and R. Potting, ͑1995͒. See also V. A. Kostelecky´, R. Potting, and S. Samuel, Phys. Lett. B 395, 178 ͑1997͒. 6774 DON COLLADAY AND V. ALAN KOSTELECKY´ 55

͓15͔ A different source of CPT violation than that discussed here, ͓30͔ G. S. Guralnik, Phys. Rev. 136, B1404 ͑1964͒. involving unconventional quantum mechanics, has been sug- ͓31͔ Y. Nambu, Prog. Theor. Phys. Suppl. Extra 190 ͑1968͒. gested in the context of quantum gravity ͓16–19͔. The experi- ͓32͔ T. Eguchi, Phys. Rev. D 14, 2755 ͑1976͒. 0 mental signature it might produce in the kaon system is known ͓33͔ The roots ␭ (␭ជ ) also have arguments a␮ and b␮ , which for to be different from that of the sources discussed in the present brevity are omitted throughout. work ͓20͔. ͓34͔ These differences can intuitively be regarded as particle- ͓16͔ S. W. Hawking, Phys. Rev. D 14, 2460 ͑1976͒; Commun. antiparticle and spin splittings. Although details lie outside our Math. Phys. 87, 395 ͑1982͒. present scope, we note that features of this type suggest a ͓17͔ D. Page, Phys. Rev. Lett. 44, 301 ͑1980͒. variety of feasible experiments that could be used to bound ͓18͔ R. M. Wald, Phys. Rev. D 21, 2742 ͑1980͒. CPT-violating parameters in an extension of the standard ͓19͔ J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, Int. J. Mod. model ͑cf. Sec. V͒. Phys. A 11, 146 ͑1996͒. ͓35͔ G. Ka¨lleń, Helv. Phys. Acta 25, 417 ͑1952͒. ͓20͔ J. Ellis, J. L. Lopez, N. E. Mavromatos, and D. V. Nanopoulos, ͓36͔ H. Lehmann, Nuovo Cimento 11, 342 ͑1954͒. Phys. Rev. D 53, 3846 ͑1996͒. ͓37͔ K. Sekine, Nuovo Cimento 11,87͑1959͒; C. H. Albright, R. ͓21͔ We disregard the Nambu-Goldstone modes associated with Haag, and S. B. Trieman, ibid. 13, 1282 ͑1959͒. spontaneous breaking of global Lorentz invariance, since in a ͓38͔ K. Hiida, Phys. Rev. 132, 1239 ͑1963͒. complete theory including gravity these modes would have the ͓39͔ Allowed terms preserving CPT but spontaneously breaking same quantum numbers as the graviton and so would generate particle Lorentz invariance are also restricted. They are given distortions ͑but not a mass term͒ in its propagator. These and in D. Colladay and V. A. Kostelelecky´, Indiana Univ. Report other gravitational effects of spontaneous Lorentz breaking are No. IUHET359, 1997 ͑unpublished͒. discussed in Ref. ͓11͔. ͓40͔ If dimension-five operators are admitted at the unbroken- ͓22͔ R. G. Sachs, Prog. Theor. Phys. Suppl. 86, 336 ͑1986͒; The symmetry level, then among the additional terms appearing Physics of Time Reversal ͑University of Chicago Press, Chi- after the SU͑2͒ϫU͑1͒ breaking could be corresponding cago, 1987͒. dimension-five CPT-violating terms involving the Higgs field ͓23͔ The effective coupling constants are also invariant under the with derivative couplings to fermion-bilinear terms. If such individual discrete transformations C, P, T. The C, P, T terms are deemed acceptable, the Lagrangian could then also properties of the various terms in Eqs. ͑3͒ and ͑4͒ are therefore contain dimension-four CPT-violating fermion-bilinear terms determined by the standard C, P, T transformation properties with ͑covariant͒ derivatives and couplings proportional to the of ␺. dimensionless ratio of the Higgs expectation value to a large ͓24͔ At the quantum level, there is a unitarily implementable mass scale. Such derivative terms would be standard-model equivalence between the Hilbert spaces of the two theories. It generalizations of those appearing in Eq. ͑4͒ and we expect 0 is generated by Uϭexp(ia•xJ ). This Hilbert-space map acts to their treatment to be relatively straightforward, although de- modify the phase of each state in the original CPT- and tails of the associated framework along lines discussed for the Lorentz-violating theory ͓␺͔ in a position- and charge- nonderivative case in previous sections would need to be es- L dependent way. Neutral states, including the vacuum, remain tablished. unaffected. ͓41͔ Neutrino-mass terms in a nonminimal standard model would ͓25͔ P. A. M. Dirac, Proc. R. Soc. London A209, 291 ͑1951͒. exclude some of the redefinitions but would also avoid any ͓26͔ W. Heisenberg, Rev. Mod. Phys. 29, 269 ͑1957͒. zero-mass issues. ͓27͔ P. G. O. Freund, Acta Phys. Austriaca 14, 445 ͑1961͒. ͓42͔ The term involving k1␬ modifies the photon propagator, so k1␬ ͓28͔ J. D. Bjorken, Ann. Phys. ͑N.Y.͒ 24, 174 ͑1963͒. can be bounded by terrestrial, astrophysical, and cosmological ͓29͔ I. Bialynicki-Birula, Phys. Rev. 130, 465 ͑1963͒. experiments. See Colladay and Kostelelcky´ ͓39͔.