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Lorentz-Violating Extension of the Standard Model

D. Colladay and V. Alan KosteleckyPhysics Department, Indiana University, Bloomington, IN 47405, U.S.A.

(preprint IUHET 359 (1997); accepted for publication in Phys. Rev. D)

In the context of conventional quantum field theory, we present a general Lorentz-violatingextension of the minimal SU(3) × SU(2)× U(1) standard model including CPT-even and CPT-odd terms. It can be viewed as the low-energy limit of a physically relevant fundamental theorywith Lorentz-covariant dynamics in which spontaneous Lorentz violation occurs. The extensionhas gauge invariance, energy-momentum conservation, and covariance under observer rotations andboosts, while covariance under particle rotations and boosts is broken. The quantized theory ishermitian and power-counting renormalizable, and other desirable features such as microcausality,positivity of the energy, and the usual anomaly cancellation are expected. Spontaneous symmetrybreaking to the electromagnetic U(1) is maintained, although the Higgs expectation is shifted by asmall amount relative to its usual value and the Z

0 field acquires a small expectation. A generalLorentz-breaking extension of quantum electrodynamics is extracted from the theory, and someexperimental tests are considered. In particular, we study modifications to photon behavior. Onepossible effect is vacuum birefringence, which could be bounded from cosmological observations byexperiments using existing techniques. Radiative corrections to the photon propagator are examined.They are compatible with spontaneous Lorentz and CPT violation in the fermion sector at levelssuggested by Planck-scale physics and accessible to other terrestrial laboratory experiments.

I. INTRODUCTION

The minimal SU(3)×SU(2)×U(1) standard model, al-though phenomenologically successful, leaves unresolveda variety of issues. It is believed to be the low-energylimit of a fundamental theory that also provides a quan-tum description of gravitation. An interesting questionis whether any aspects of this underlying theory could berevealed through definite experimental signals accessiblewith present techniques.

The natural scale for a fundamental theory includ-ing gravity is governed by the Planck mass MP , whichis about 17 orders of magnitude greater than the elec-troweak scale mW associated with the standard model.This suggests that observable experimental signals from afundamental theory might be expected to be suppressedby some power of the ratio r ≈ mW /MP ≃ 10−17. De-tection of these minuscule effects at present energy scaleswould be likely to require experiments of exceptional sen-sitivity, preferably ones seeking to observe a signal for-bidden in conventional renormalizable gauge theories.

To identify signals of this type, one approach is to ex-amine proposed fundamental theories for effects that arequalitatively different from standard-model physics. Forexample, at present the most promising framework for afundamental theory is string (M) theory. The qualitativedifference between particles and strings means that qual-itatively new physics is expected at the Planck scale. Aninteresting challenge would be to determine whether thiscould lead to observable low-energy effects.

In the present work, we consider the possibility thatthe new physics involves a violation of Lorentz symmetry.

It has been shown that spontaneous Lorentz breakingmay occur in the context of string theories with Lorentz-covariant dynamics [1]. Unlike the conventional stan-dard model, string theories typically involve interactionsthat could destabilize the naive vacuum and trigger thegeneration of nonzero expectation values for Lorentz ten-sors. Note that some kind of spontaneous breaking of thehigher-dimensional Lorentz symmetry is expected in anyrealistic Lorentz-covariant fundamental theory involvingmore than four spacetime dimensions. If the breakingextends into the four macroscopic spacetime dimensions,apparent Lorentz violation could occur at the level of thestandard model. This would represent a possible observ-able effect from the fundamental theory, originating out-side the structure of conventional renormalizable gaugemodels.

A framework has been developed for treating the ef-fects of spontaneous Lorentz breaking in the context of alow-energy effective theory [2], where certain terms canbe induced that appear to violate Lorentz invariance ex-plicitly. It turns out that, from a theoretical perspective,the resulting effects are comparatively minimal.

An important point is that Lorentz symmetry remainsa property of the underlying fundamental theory becausethe breaking is spontaneous. This implies that various at-tractive features of conventional theories, including mi-crocausality and positivity of the energy, are expectedto hold in the low-energy effective theory. Also, en-ergy and momentum are conserved as usual, providedthe tensor expectation values in the fundamental theoryare spacetime-position independent. Moreover, standardquantization methods are unaffected, so a relativistic

1

Dirac equation and a nonrelativistic Schrodinger equa-tion emerge in the appropriate limits.

Another important aspect of the spontaneous breakingis that both the fundamental theory and the effective low-energy theory remain invariant under observer Lorentztransformations, i.e., rotations or boosts of an observer’sinertial frame [2]. The presence of nonzero tensor ex-pectation values in the vacuum affects only invarianceproperties under particle Lorentz transformations, i.e.,rotations or boosts of a localized particle or field thatleave unchanged the background expectation values.

This framework for treating spontaneous Lorentz vio-lation has been used to obtain a general extension of theminimal SU(3) × SU(2)× U(1) standard model that vio-lates both Lorentz invariance and CPT [2]. In addition tothe desirable features of energy-momentum conservation,observer Lorentz invariance, conventional quantization,hermiticity, and the expected microcausality and posi-tivity of the energy, this standard-model extension main-tains gauge invariance and power-counting renormaliz-ability. It would emerge from any fundamental theory(not necessarily string theory) that generates the stan-dard model and contains spontaneous Lorentz and CPTviolation.

The present work continues our previous theoretical in-vestigations of spontaneous Lorentz and CPT breaking.Working first at the level of the standard model, we pro-vide explicitly in section II the full Lorentz-violating ex-tension, including the CPT-even Lorentz-breaking termsdescribed implicitly in Ref. [2]. We also give some detailsof the modifications to the usual electroweak symmetrybreaking.

Since many sensitive measures of Lorentz and CPTsymmetry involve tests of quantum electrodynamics(QED), it is useful to extract from the standard-modelextension a generalized QED that allows for possibleLorentz and CPT violations. This extended QED, givenin section III, involves modifications of the usual QEDin both the fermion and the photon sectors. Some com-ments are also given in section III about the implicationsof this theory for experimental tests with electrons andpositrons.

In the remainder of this paper, we focus primarily onthe photon sector of the extended QED, presenting astudy of the theoretical and experimental implicationsof the modifications to photon properties arising fromthe possible Lorentz and CPT violations. Section IV dis-cusses changes in the basic theory, including the modifiedMaxwell equations and properties of their solutions. Onepossible effect is vacuum photon birefringence, and someassociated features are described. We show that feasi-ble measurements limiting birefringence on cosmologi-cal scales could tightly constrain the Lorentz-violatingterms. In section V, some important consistency checkson the theory at the level of radiative corrections are pre-sented, largely at the one-loop level. The types of Lorentz

violation that can be affected by radiative corrections areidentified, and explicit calculations are given. We showthat the effects are compatible with spontaneous Lorentzand CPT violation in the fermion sector at levels acces-sible to other QED experiments.

Since the standard-model extension provides a quanti-tative microscopic theory of Lorentz and CPT violation,it is feasible to identify potentially observable signals andto establish bounds from various experiments other thanones in the photon sector. Numerous tests of Lorentzinvariance and CPT exist. The present theory providesa single coherent framework at the level of the standardmodel and QED that can be used as a basis for the analy-sis and comparison of these tests. Although many exper-iments are insensitive to the suppressed effects motivat-ing our investigation, certain high-precision ones mighthave observable signals within this framework. In par-ticular, the results in the present paper have been usedto examine possible bounds on CPT and Lorentz vio-lations from measurements of neutral-meson oscillations[3–6], from tests of QED in Penning traps [7,8], and frombaryogenesis [9]. Several other investigations are under-way, including a study [10] of possible Lorentz and CPTeffects on hydrogen and antihydrogen spectroscopy [11]and another [12] of limits attainable in clock-comparisonexperiments [13].

The analyses of the standard-model and QED exten-sions performed in the present work leave unaddressed anumber of significant theoretical issues arising at scalesbetween the electroweak mass and the Planck mass.These include the ‘dimension problem’ of establishingwhether spontaneous Lorentz breaking in the fundamen-tal theory near the Planck scale indeed extends to thefour physical spacetime dimensions and, if so, the mech-anism for its suppression or, if not, why exactly fourspacetime dimensions are spared. Other issues includethe effects of mode fluctuations around the tensor expec-tation values and possible constraints and effects arisingfrom a nonminimal standard model or (super)unificationbelow the Planck scale.

Another potentially important topic is the implicationof spontaneous Lorentz violation for gravity at observableenergies. Like the usual standard model, the standard-model extension considered here disregards gravitationaleffects. The particle Lorentz symmetry that is brokenin this theory is therefore effectively a global symmetry,and so one might expect Nambu-Goldstone modes. Sincegravity is associated with local Lorentz invariance, it isnatural to ask about the role of these modes in a ver-sion of the standard-model extension that includes grav-ity. In a gauge theory, when a suitable scalar acquiresa nonzero expectation value, the Higgs mechanism oc-curs: the propagator for the gauge boson is modified,and a mass is generated. Similarly, in a theory withgravitational couplings, when a Lorentz tensor acquires anonzero expectation value, the graviton propagator can

2

be modified. However, no mass for the graviton is in-duced because the gravitational connection is related tothe derivative of the metric rather than to the metricitself [1]. In this sense, there is no gravitational Higgseffect.

The theory described here appears at present to be thesole candidate for a consistent extension of the standardmodel providing a microscopic theory of Lorentz viola-tion. A complete review of alternative approaches to pos-sible Lorentz and CPT violation lies beyond the scope ofthis paper. Works known to us of relevance in the presentcontext are referenced in the body of the text below.Among other ideas in the literature are several distinc-tive ones developed from perspectives very different fromours. Following early work by Dirac and Heisenberg, sev-eral authors have considered an unphysical spontaneousLorentz breaking in an effort to interpret the photonas a Nambu-Goldstone boson [14]. Nielsen and his col-leagues have suggested the converse of the philosophy inthe present work: that the observed Lorentz symmetry innature might be a low-energy manifestation of a funda-mental theory without Lorentz invariance. A discussionof this idea and a brief review of the literature on Lorentzbreaking prior to the establishment of the usual minimalstandard model may be found in Ref. [15]. Hawking hassuggested [16] the possibility that conventional quantummechanics is invalidated by gravitational effects and thatthis might lead to CPT violation, among other effects.The implications for experiments in the kaon system [17]are known to be entirely different from those arising inthe present standard-model extension, which is based onconventional quantum theory. There is also a body ofliterature pertaining to unconventional theories of grav-ity (without standard-model physics), among which aresome models containing various possible sources of localLorentz violation [18].

II. STANDARD-MODEL EXTENSION

In this section, we extend the minimal standard modelby adding all possible Lorentz-violating terms that couldarise from spontaneous symmetry breaking at a funda-mental level but that preserve SU(3) × SU(2)× U(1)gauge invariance and power-counting renormalizability.Terms that are odd under CPT are explicitly given inRef. [2] but are also included here for completeness.

The general form of a Lorentz-violating term involvesa part that acts as a coupling coefficient and a part con-structed from the basic fields in the standard model. Therequirements of the derivation impose various limitationson the possible structures of both parts. Taken together,these requirements place significant constraints on theform of terms in the standard-model extension.

The part acting as a coupling coefficient carries space-time indices reflecting the properties under observer

Lorentz transformations of the relevant nonzero expecta-tion values from the fundamental theory. The couplingcoefficient may be complex, but it is constrained by therequirement that the lagrangian be hermitian. For a cou-pling coefficient with an even number of spacetime in-dices, the pure trace component is irrelevant for presentpurposes because it maintains Lorentz invariance. A cou-pling coefficient of this type can therefore be taken trace-less.

The field part may involve covariant derivatives and,if fermions are involved, gamma matrices. Gauge invari-ance requires that the field part be a singlet under SU(3)× SU(2)× U(1), while power-counting renormalizabilityimplies that it must have mass dimension no greaterthan four. The requirement that the standard-modelextension originates from spontaneous Lorentz break-ing in a covariant fundamental theory implies the wholeLorentz-violating term must be a singlet under observerLorentz transformations, so the field part must have in-dices matching those of the coupling coefficient.

Following the discussion in the introduction, all cou-pling coefficients are assumed to be heavily suppressedby some power of the ratio r of the light scale to thePlanck scale. In the absence of a satisfactory explanationof the suppression mechanism, it would seem prematureto attempt specific detailed predictions about the relativesizes of different coupling coefficients. As a possible work-ing hypothesis, one might attribute comparable suppres-sion factors to all terms at the level of the standard-modelextension. Note that a term with field part having massdimension n must have a coupling coefficient with massdimension 4 − n, and the relevant scale for these effectsis roughly the Planck mass. The hypothesis would there-fore suggest that in the low-energy theory a term withfield part of mass dimension n + 1 would have couplingcoefficient suppressed by an additional power of r rela-tive to the coefficient of a field term of mass dimensionn. This scheme would be compatible with interpretingthe standard model as an effective field theory, in whicheach additional derivative coupling would involve an ad-ditional suppression factor in the coupling coefficient. Itwould imply a distinct hierarchy among the coupling co-efficients introduced below, and would suggest that cer-tain derivative couplings could be neglected relative tocomparable nonderivative ones. However, since this hy-pothesis presently has no basis in a detailed theory, inwhat follows we have chosen to retain on an equal foot-ing all renormalizable terms compatible with the gaugesymmetries of the standard model and with an origin inspontaneous Lorentz breaking.

In what follows, we denote the left- and right-handedlepton and quark multiplets by

LA =

(νAlA

)

L

, RA = (lA)R ,

3

QA =

(uAdA

)

L

, UA = (uA)R , DA = (dA)R , (1)

where

ψL ≡ 12 (1 − γ5)ψ , ψR ≡ 1

2 (1 + γ5)ψ , (2)

as usual, and where A = 1, 2, 3 labels the flavor: lA ≡(e, µ, τ), νA ≡ (νe, νµ, ντ ), uA ≡ (u, c, t), dA ≡ (d, s, b).We denote the Higgs doublet by φ, and in unitary gaugewe represent it as

φ =1√2

(0rφ

). (3)

The conjugate doublet is φc. The SU(3), SU(2), and U(1)gauge fields are denoted byGµ, Wµ, andBµ, respectively.The corresponding field strengths are Gµν , Wµν , andBµν , with the first two understood to be hermitian ad-joint matrices while Bµν is a hermitian singlet. The cor-responding couplings are g3, g, and g′. The electromag-netic U(1) charge q and the angle θW are defined throughq = g sin θW = g′ cos θW , as usual. The covariant deriva-

tive is denoted by Dµ, and A↔

∂µ B ≡ A∂µB − (∂µA)B.The Yukawa couplings are GL, GU , GD. Throughoutmost of this work we use natural units, which could be ob-tained from the SI system by redefining h = c = ǫ0 = 1,and we adopt the Minkowski metric ηµν with η00 = +1.

The complete lagrangian for the Lorentz-breakingstandard-model extension can be separated into a sum ofterms. For completeness, we first provide the lagrangianterms in the usual SU(3) × SU(2) × U(1) minimal stan-dard model:

Llepton = 12 iLAγ

µ↔

Dµ LA + 12 iRAγ

µ↔

Dµ RA , (4)

Lquark = 12 iQAγ

µ↔

Dµ QA + 12 iUAγ

µ↔

Dµ UA

+ 12 iDAγ

µ↔

Dµ DA , (5)

LYukawa = −[(GL)ABLAφRB + (GU )ABQAφ

cUB

+(GD)ABQAφDB

]+ h.c. , (6)

LHiggs = (Dµφ)†Dµφ+ µ2φ†φ− λ

3!(φ†φ)2 , (7)

Lgauge = − 12Tr(GµνG

µν) − 12Tr(WµνW

µν) − 14BµνB

µν .

(8)

The usual θ terms have been omitted, and possible analo-gous total-derivative terms that break Lorentz symmetryare disregarded in this work.

In the fermion sector of the standard-model extension,the contribution to the lagrangian can be divided into

four parts according to whether the term is CPT even orodd and whether it involves leptons or quarks:

LCPT−evenlepton = 1

2 i(cL)µνABLAγµ

↔

Dν LB

+ 12 i(cR)µνABRAγ

µ↔

Dν RB , (9)

LCPT−oddlepton = −(aL)µABLAγ

µLB − (aR)µABRAγµRB ,

(10)

LCPT−evenquark = 1

2 i(cQ)µνABQAγµ

↔

Dν QB

+ 12 i(cU )µνABUAγ

µ↔

Dν UB

+ 12 i(cD)µνABDAγ

µ↔

Dν DB , (11)

LCPT−oddquark = −(aQ)µABQAγ

µQB − (aU )µABUAγµUB

−(aD)µABDAγµDB . (12)

In these equations, the various coupling coefficients cµνand aµ are understood to be hermitian in generationspace. The coefficients aµ have dimensions of mass. Thedimensionless coefficients cµν can have both symmetricand antisymmetric parts but can be assumed traceless.A nonzero trace would not contribute to Lorentz viola-tion and in any case can be absorbed by a conventionalfield normalization ensuring the usual kinetic operatorfor the matter fields.

The standard-model extension also contains Lorentz-violating couplings between the fermions and the Higgsfield, having the gauge structure of the usual Yukawa cou-plings but involving nontrivial gamma matrices. Theseterms are all CPT even:

LCPT−evenYukawa = − 1

2

[(HL)µνABLAφσ

µνRB

+(HU )µνABQAφcσµνUB

+(HD)µνABQAφσµνDB

]+ h.c. (13)

The coefficients Hµν are dimensionless and antisymmet-ric, but like the Yukawa couplings GL,U,D they are not

necessarily hermitian in generation space.The possible contributions in the Higgs sector can be

CPT even or CPT odd:

LCPT−evenHiggs = 1

2 (kφφ)µν(Dµφ)†Dνφ+ h.c.

− 12 (kφB)µνφ†φBµν

− 12 (kφW )µνφ†Wµνφ , (14)

LCPT−oddHiggs = i(kφ)

µφ†Dµφ+ h.c. . (15)

In Eq. (14), the dimensionless coefficient kφφ can havesymmetric real and antisymmetric imaginary parts. Theother coefficients in Eq. (14) have dimensions of mass and

4

must be real antisymmetric. The coefficient kφ for theCPT-odd term (15) also has dimensions of mass and canbe an arbitrary complex number.

The gauge sector has both CPT-even and CPT-oddcontributions. The CPT-even ones are:

LCPT−evengauge = − 1

2 (kG)κλµνTr(GκλGµν)

− 12 (kW )κλµνTr(WκλWµν)

− 14 (kB)κλµνB

κλBµν . (16)

In this equation, the dimensionless coefficients kG,W,Bare real. They must have the symmetries of the Rie-mann tensor and a vanishing double trace. The point isthat any totally antisymmetric part involves only a to-tal derivative in the lagrangian density, while a nonzerodouble trace can be absorbed into a redefinition of thenormalization of the corresponding kinetic term (8).

The CPT-odd gauge terms are given by the followingexpression [19]:

LCPT−oddgauge = (k3)κǫ

κλµνTr(GλGµν + 23 ig3GλGµGν)

+(k2)κǫκλµνTr(WλWµν + 2

3 igWλWµWν)

+(k1)κǫκλµνBλBµν + (k0)κB

κ . (17)

The coefficients k1,2,3 are real and have dimensions ofmass, while k0 is also real and has dimensions of masscubed. It turns out that, if any of these CPT-odd termsdo indeed appear, they would generate instabilities inthe minimal theory. They are all associated with neg-ative contributions to the energy, and in addition theterm with k0 would directly generate a linear instabil-ity in the potential. It might therefore seem desirablethat all the coefficients k0,1,2,3 vanish. While this couldbe imposed at the classical level, radiative quantum cor-rections from, say, the fermion sector might a priori beexpected to generate nonzero values. Remarkably, thestructure of the standard-model extension appears to besuch that no corrections arise, at least to one loop. Theseissues are discussed further in what follows, in particularin subsection IV A and section V.

It is known that some apparently CPT- and Lorentz-violating terms can be eliminated from the action viafield redefinitions [2]. Several types of redefinition can beconsidered. In the context of the present standard-modelextension, we have investigated a variety of possibilitiesfor each field. As a general rule, the more complex thetheoretical structure becomes, the less likely it is thata useful field redefinition exists. For instance, the pres-ence of Lorentz-violating CPT-even derivative couplingsin the standard-model extension complicates the analysisfor CPT-odd terms provided in Ref. [2], although it turnsout that the conclusions still hold. Here, we summarizea few methodological results and describe some specialcases of particular interest.

To eliminate a Lorentz-breaking term, a field redefi-nition must involve the associated coupling coefficient.

When derivative couplings play a role, the field redefini-tion may also involve spacetime-position variables. Theassumption that the coupling coefficients are small canbe helpful, in some cases directly assisting in derivationsand in others leading to a set of approximate field re-definitions. Under the latter, a theory with first-orderLorentz-breaking effects may be redefined into one witheffects appearing only at second or higher orders. Alter-natively, some first-order Lorentz-breaking terms may beabsorbed into others. A partial constraint on allowableredefinitions is provided by the transformation propertiesof the various Lorentz-violating terms under the discretesymmetries C, P, T. Only terms with identical discrete-symmetry properties can be absorbed into one anotherby first-order redefinitions.

Two types of redefinition that we have found of par-ticular value are linear phase redefinitions and linearnormalization redefinitions. For example, some termsinvolving the coefficients aL,R,Q,U,D can be eliminatedby position-dependent field-phase redefinitions, as de-scribed in Ref. [2]. Another example is provided by termsinvolving the coefficients HL,U,D, some of which canabsorb through field-normalization redefinitions certainother terms involving the coefficients cL,R,Q,U,D. Theseexamples have specific interesting implications for thequantum-electrodynamics limit of the standard-modelextension, and their explicit forms for that case are givenin section III below. Useful nonlinear field redefinitionsmight also exist in principle, but these are typically moredifficult to implement meaningfully because they mayrepresent (noncanonical) transformations between differ-ent physical systems rather than reinterpretations of thesame physics.

We next consider the issue of electroweak SU(2) × U(1)symmetry breaking. The static potential for the gaugeand Higgs fields can be extracted from the lagrangianterms given above for the standard-model extension. Itis possible to work in unitary gauge as usual, since theLorentz-breaking terms do not affect the gauge structureof the theory. The analysis is somewhat more compli-cated than the conventional case, as it involves additionalterms depending on the coupling coefficients kφφ, kφW ,kφ, kW , k2, and k0. In principle, there are also contri-butions from the SU(3) sector, but these decouple fromthe Higgs field and so the gluon expectation values can betaken to be zero as usual. As mentioned above, the termsk2 and k0 are expected to vanish for consistency of theminimal theory, and so we assume this in what follows.In fact, a nonzero k2 would have no effect on the vacuumvalues of the fields, but the linear instability that wouldbe introduced by a nonzero k0 would exclude a stablevacuum in the absence of other (nonlinear) effects.

Extremizing the static potential produces five simulta-neous equations. Three of these are satisfied if the ex-pectation values of W±

µ and the photon Aµ vanish. The

5

other two equations can be solved algebraically for theexpectation values of the Higgs and Z0

µ fields. In thegeneral case, both of these are nonzero and are given by

〈Z0µ〉 =

1

qsin 2θW (Re kφφ)

−1µν k

νφ , (18)

〈rφ〉 = a

(1 − 1

µ2(Re kφφ)

−1µν k

µφk

νφ

)1/2

, (19)

where kµνφφ ≡ ηµν + kµνφφ and a ≡√

6µ2/λ. Note that

the quantity (Re kφφ)−1µν always exists when the Lorentz

violation is small, |(kφφ)µν | ≪ 1. Note also that 〈rφ〉 isa scalar under both particle and observer Lorentz trans-formations, so quantities such as the fermion mass pa-rameters remain scalars despite the presence of Lorentzbreaking.

As might be anticipated, the above pattern of expec-tation values leaves unbroken the electromagnetic U(1)symmetry, and it can be shown that fluctuations aboutthe extremum are stable. When substituted into the la-grangian for the standard-model extension, the uncon-ventional nonzero expectation value for the field Z0

µ gen-erates some additional CPT- and Lorentz-violating con-tributions. However, these are all of the same form asother CPT- and Lorentz-violating terms already presentin the theory, so they can be absorbed into existing cou-pling coefficients.

Some analyses of experimental tests of the standard-model extension involving flavor-changing oscillations inneutral mesons have been performed in Refs. [3,5,6].Tests at the level of quantum electrodynamics are men-tioned below. Note that some bounds on both thefermion and the gauge sectors might be obtained fromavailable experimental information about the Z0

µ and per-haps the W±

µ . Such limits would be of interest in theirown right, although it seems likely that they would bemuch weaker than required to detect suppressed Lorentzviolation at the levels estimated in this work.

III. EXTENDED QUANTUM

ELECTRODYNAMICS

In much the same way that conventional quantumelectrodynamics (QED) can be obtained from the usualstandard model, a generalized quantum electrodynamicsincorporating Lorentz-breaking terms can be extractedfrom the standard-model extension given in section II.This is of particular interest because QED has beentested to high precision in a variety of experiments, someof which may tightly constrain the coupling coefficientsof the possible Lorentz-violating terms.

A straightforward way to obtain the extended QEDis as follows. After the SU(2) × U(1) symmetry break-ing, set to zero the fields Gµ for the gluons, W±

µ , Z0µ

for the weak bosons, and the physical Higgs field (butnot the expectation value of the Higgs doublet, whichgenerates fermion masses). The only remaining bosonis the photon, mediating the electromagnetic interac-tions. The neutrinos are charge neutral, so they de-couple and can be discarded. The resulting theory isan extended QED describing the electromagnetic inter-actions of quarks and (charged) leptons. It is expectedto inherit from the standard-model extension various at-tractive features mentioned in the introduction, includingU(1) gauge invariance, energy-momentum conservation,observer Lorentz invariance, hermiticity, microcausality,positivity of the energy, and power-counting renormaliz-ability.

Denote the standard four-component lepton fields bylA and their masses by mA, where A = 1, 2, 3 corre-sponds to electron, muon, tau, respectively. Then, thelagrangian for the conventional QED of leptons and pho-tons is

LQEDlepton−photon = 1

2 ilAγµ

↔

Dµ lA −mAlAlA − 14FµνF

µν .

(20)

In this equation and throughout what follows, Dµ ≡∂µ + iqAµ and the field strength Fαβ is defined by

Fαβ ≡ ∂αAβ − ∂βAα , (21)

as usual.The standard-model extension generates additional

terms that violate Lorentz symmetry. The CPT-eventerms involving the lepton fields are

LCPT−evenlepton = − 1

2 (Hl)µνAB lAσµν lB

+ 12 i(cl)µνABlAγ

µ↔

Dν lB

+ 12 i(dl)µνAB lAγ5γ

µ↔

Dν lB . (22)

In this equation, the coupling coefficients (Hl)µνAB areantisymmetric in spacetime indices and have dimensionsof mass. They arise from the coefficients in Eq. (13) fol-lowing gauge-symmetry breaking, and they are hermitianin generation space. The hermitian dimensionless cou-plings (cl)µνAB and (dl)µνAB could in principle have bothsymmetric and antisymmetric spacetime components butcan be taken traceless. They arise from the expressions(9).

The CPT-odd terms involving the lepton fields are

LCPT−oddlepton = −(al)µAB lAγ

µlB − (bl)µAB lAγ5γµlB .

(23)

The couplings (al)µAB and (bl)µAB are hermitian andhave dimensions of mass. They arise from Eq. (10). Notethat imposing individual lepton-number conservation in

6

both the above equations would make all the couplingcoefficients diagonal in flavor space.

In the pure-photon sector, there is one CPT-evenLorentz-violating term:

LCPT−evenphoton = − 1

4 (kF )κλµνFκλFµν . (24)

The coupling (kF )κλµν arises from Eq. (16) and is realand dimensionless. Without loss of generality it canbe taken as double traceless, since any trace componentwould serve merely to redefine the kinetic term and henceis just a field renormalization. We disregard a conceivableθ-type term proportional to Fκλǫ

κλµνFµν , which mightarise from a totally antisymmetric component of kF , onthe grounds that it is a total derivative. The couplingkF therefore can be taken to have the symmetries of theRiemann tensor.

There is also a CPT-odd pure-photon term:

LCPT−oddphoton = + 1

2 (kAF )κǫκλµνAλFµν , (25)

where the coupling coefficient (kAF )κ is real and has di-mensions of mass. This term arises from the CPT-oddgauge sector (17) of the standard-model extension. Asmentioned in the previous section, it has some theoreti-cal difficulties associated with negative contributions tothe energy and it therefore seems likely to be absent inpractice. It is included in what follows so that we candiscuss explicitly its difficulties and some related issuesinvolving radiative corrections. Note also that the ex-cluded destabilizing linear term in Bµ in the standard-model extension would, if present, generate a correspond-ing linear term −(kA)κA

κ in Eq. (25), where (kA)κ is areal coupling with dimensions of mass cubed. Certainissues involving this term are addressed in sections IV Aand V.

The QED limit obtained from the standard-model ex-tension also has a quark sector. This has the same gen-eral form as the lepton sector given by Eqs. (20), (22),and (23), except that six quark fields replace the threeleptons and so twice as many Lorentz-violating couplingsoccur. Note that the lepton and quark sectors are cou-pled only through the photon: the gauge invariance ofthe standard-model extension excludes couplings mixingleptons and quarks.

The extended QED of leptons and photons given inEqs. (20) - (25) should suffice for certain applicationswhere the asymptotic states are leptons or photons andthe strong and weak interactions play a negligible role,including a variety of existing or proposed high-precisionexperiments involving leptons. Interesting options forsuch experiments are to establish the possible signals ofLorentz violation suggested by the extended QED and toplace bounds on the associated coupling coefficients. Forexample, promising possibilities involving the muon in-clude accurate measurements of g−2 such as those under-way at the Brookhaven muon ring [20] and sensitive tests

for the decay µ → eγ. There are also a variety of othercomparisons involving heavy leptons that are potentiallyof interest [21]. These issues lie beyond the scope of thepresent work and will be addressed elsewhere.

For certain experiments, it suffices to consider anotherlimiting case of the theory: the extended QED includ-ing only electrons, positrons and photons. This limit canbe extracted from the lagrangian terms for the extendedQED of leptons and photons by setting to zero the muonand the tau fields. Denoting the four-component electronfield by ψ and the electron mass by me, the usual QEDlagrangian for electrons and photons is

LQEDelectron = 1

2 iψγµ

↔

Dµ ψ −meψψ − 14FµνF

µν . (26)

In the Lorentz-violating sector, the pure-photon termsare still given by Eqs. (24) and (25). However, the CPT-even terms in the fermion sector become

LCPT−evenelectron = − 1

2Hµνψσµνψ

+ 12 icµνψγ

µ↔

Dν ψ

+ 12 idµνψγ5γ

µ↔

Dν ψ , (27)

while the CPT-odd ones become

LCPT−oddelectron = −aµψγµψ − bµψγ5γ

µψ . (28)

The real coupling coefficients a, b, c, d, and H arethe (1, 1)-flavor components of the corresponding coef-ficients in the extended QED of leptons and photonsand inherit the corresponding dimensions and Lorentz-transformation properties.

In addition to the expressions given in Eqs. (24) - (28)for the extended QED of electrons, positrons, and pho-tons, other Lorentz-violating terms can be envisaged thatare compatible with U(1) charge symmetry, renormaliz-ability, and an origin in spontaneous Lorentz breakingbut that cannot be obtained as a reduction from thestandard-model extension. All such terms would be CPTodd. They would have the form

Lextraelectron = 1

2 ieνψ↔

Dν ψ − 12fνψγ5

↔

Dν ψ

+ 14 igλµνψσ

λµ↔

Dν ψ , (29)

where the couplings eµ, fµ and gλµν are real and di-mensionless. The reason such terms are absent from theexpressions obtained above is that all putative renormal-izable terms in the standard-model extension that couldgenerate Eq. (29) are directly incompatible with the elec-troweak structure. However, it is possible that nonrenor-malizable higher-dimensional operators in the effectivelagrangian obeying SU(2) × U(1) symmetry and involv-ing the Higgs field might generate the expressions (29)when the Higgs field acquires its vacuum expectationvalue. According to standard lore and the discussion

7

in section II, such operators would be expected to behighly suppressed relative to those we have listed for thestandard-model extension. This suppression should re-main in force at the level of the extended QED, whichmeans any terms of the form (29) would be expected tohave coupling coefficients much smaller than the otherterms we consider. Similar considerations apply to pos-sible extra terms that might appear in the heavy-leptonand quark sectors of the extended QED.

Next, we address the issue of field redefinitions withinthe context of the extended QED of electrons, positrons,and photons. We have found several cases to be espe-cially useful. One is a linear phase redefinition of the formψ = exp(−ia · x)χ, which eliminates the term −aµψγµψfrom Eq. (28). This is equivalent to shifting the zerosof energy and momentum for electrons and positrons [2].We therefore expect no observable effects from a nonzeroaµ in any QED experiment.

Another useful class of redefinitions involves fieldrenormalizations depending on coupling coefficients. Fora fermion field ψ, consider the redefinition

ψ = (1 + v · Γ)χ , (30)

where Γ is one of γµ, γ5γµ, σµν and v is a combination

of coupling coefficients with appropriate spacetime in-dices. This set of redefinitions can be used to obtain sev-eral useful approximate results, valid to first order in the(small) coupling coefficients. One is that the combinationǫµναβHαβ + m(dµν − dνµ) can be eliminated and henceis unobservable at leading order in any QED experiment.Only the orthogonal linear combination is physical at thislevel. Another is that the antisymmetric component ofcµν can be eliminated to first order. Similarly, even ifthe extra terms with coefficients eµ and fµ in Eq. (29)should appear, they could be eliminated to first order bya combination of field redefinitions. The same is true ofthe trace components of the extra term with coefficientgλµν , while the totally antisymmetric component of thisterm can be absorbed into bµ to first order. Combiningall these results, it follows that at leading order in the ex-tended QED of electrons, positrons, and photons the onlyobservable coupling coefficients can be taken as bµ, Hµν ,the symmetric components of cµν and dµν , and possiblythe traceless mixed-symmetry components of the extracoefficient gλµν .

So far in this section we have considered various formsof extended QED that emerge as limits of the standard-model extension. For some purposes, it can be useful towork within an effective extended QED valid for a freefermion that is a composite of leptons and quarks, suchas a nucleon, atom, or ion. For a single fermion field ψof this type, the effective lagrangian would then have thesame form as that of the extended QED for electrons,positrons, and photons. A description of this type is use-ful for investigations of the implications of high-precision

experiments on composite fermions, such as compara-tive tests of proton properties or searches for a neutronelectric-dipole moment. In principle, extra terms of theform (29) could appear as a result of the interactionsamong the fermion constituents, but in the effective the-ory the coupling coefficients of such terms would involvecombinations of the constituent coupling coefficients withthe interaction coupling constants and might therefore beexpected to be absent at leading order in many cases.

Some possible experimental signals from extendedQED are investigated in Ref. [8]. Certain high-precisiontests that could be performed with present technologyare considered, and the attainable bounds on Lorentz-breaking coupling coefficients are estimated. The testsinvolve comparative measurements of anomalous mag-netic moments and charge-to-mass ratios for particlesand antiparticles confined in a Penning trap [7]. Theytypically have the potential to bound the coupling co-efficients of Lorentz- and CPT-violating terms at a levelclose to that expected from Planck-scale suppression. Forexample, the spacelike components of the coefficient bµcontrol the appropriate figure of merit for experimentscomparing the anomalous magnetic moments of the elec-tron and positron. This figure of merit can be boundedto about one part in 1020, which is comparable to theratio of me/M of the electron mass to the Planck scale.

IV. THE PURE-PHOTON SECTOR

In this section, we focus on the pure-photon sector ofthe extended QED. We examine some theoretical impli-cations of the existence of Lorentz- and CPT-violatingterms and address some experimental issues.

A. Lagrangian and Energy-Momentum Tensor

The lagrangian of interest, which is U(1) gauge invari-ant by construction, is a combination of the photon termin Eq. (26) with the expressions (24) and (25). It is

Ltotalphoton = − 1

4FµνFµν − 1

4 (kF )κλµνFκλFµν

+ 12 (kAF )κǫκλµνA

λFµν . (31)

Some properties of the coupling coefficients kF and kAFare described following Eqs. (24) and (25). For certaincalculations, it is useful to decompose the coefficient kFinto its two Lorentz-irreducible pieces, one with 10 in-dependent components analogous to the Weyl tensor ingeneral relativity and one with 9 components analogousto the trace-free Ricci tensor. Only one of the 19 to-tal independent components of kF (the 00 component ofthe trace-free Ricci-tensor analogue) and one of the fourindependent components of the coefficient kAF (the time-like component k0

AF ) are associated with terms invariantunder (particle) rotations.

8

Some insight into the structure of the lagrangian canbe obtained by expressing it in terms of the potentials φ,~A and the fields ~E, ~B. We find

Ltotalphoton = 1

2 ( ~E2 − ~B2) + 12α( ~E2 + ~B2)

+ 12β

jkE E

jEk + 12β

jkB B

jBk + 12β

jkEBE

jBk

+k0AF

~A · ~B − φ~kAF · ~B+~kAF · ( ~A× ~E) . (32)

Here and throughout this work, j, k, . . . = 1, 2, 3 are spa-tial indices. The real coefficients α and βjkE , βjkB , βjkEBare various combinations of the couplings (kF )κλµν ap-pearing in Eq. (31). Disregarding as before any total-

derivative effects, the βjkE , βjkB , βjkEB are traceless. Notethat all possible quadratic combinations of the electricand magnetic fields appear. Only two terms, involving αand k0

AF , preserve (particle) rotational invariance. Notealso that a rescaling without physical consequences canbe performed to obtain a standard normalization of theelectric field ~E. This produces a lagrangian of the samegeneral form as Eq. (32) except that the Lorentz-breaking

term proportional to ( ~E2 + ~B2) is replaced with one pro-

portional to ~B2 alone.The canonical energy-momentum tensor can be con-

structed following the standard procedure. This tensorcan be partially symmetrized, but complete symmetriza-tion is impossible because there is an antisymmetric com-ponent that cannot be written as a total derivative. Arelatively elegant expression can be obtained by addingjudiciously chosen total-derivative terms, which leave un-changed the physics. Denoting the resulting energy-momentum tensor by Θµν , we find

Θµν = −FµγF νγ + 14ηµνFαβF

αβ

−(kF )αβµγF νγFαβ + 14ηµν(kF )αβγδF

αβF γδ

+(kAF )νAαFαµ . (33)

Here, we define

Fµν = ǫκλµνFκλ/2 (34)

to be the dual field strength.The energy-momentum tensor obeys the usual conser-

vation relation,

∂µΘµν = 0 . (35)

In addition to the gauge-invariant and symmetric con-tributions to Θµν , which include the conventional piecesamong others, there are additional terms involving thecoefficient kF that are gauge invariant but asymmet-ric. The term with kAF is neither gauge invariant norsymmetric. Under a gauge transformation, an addi-tional total-derivative term appears. The presence ofan antisymmetric component in Θµν implies that care

is required in physical interpretations of the energy-momentum behavior. Although Θj0 can be regarded asthe components of a generalized Poynting vector, its vol-ume integral is no longer conserved and cannot be iden-tified with the conserved volume integral of the compo-nents Θ0j of the momentum density. These features area direct consequence of the presence of the backgroundexpectation values of tensor fields, represented in the low-energy theory by the coupling coefficients kF and kAF .

The energy density is given by the component Θ00.Inspection shows it can be written in the form:

Θ00 = 12 ( ~E2 + ~B2)

−(kF )0j0kEjEk + 14 (kF )jklmǫjkpǫlmqBpBq

−(kAF )0 ~A · ~B . (36)

If kAF vanishes and kF is small, Θ00 is nonnegative. Thiscan be seen as follows. The combination of the usual en-ergy density with the terms proportional to kF can beviewed as a bilinear form xTMx generated from a matrixM in the six-dimensional space xT ≡ ( ~E, ~B). The matrixM is symmetric and 3 × 3-block diagonal, since no crossterms in ~E and ~B appear in Θ00. Observer rotation in-variance can be used to diagonalize the upper 3×3 blockassociated with the electric field. Since kF is small, thethree diagonal entries are of the form 1

2 −O(kF ) > 0, sothe contribution to Θ00 from the electric field is nonneg-ative in any frame. A similar argument shows that thecontribution from the lower 3 × 3 block, associated withthe magnetic field, is also nonnegative. The conservedenergy E of a field configuration, obtained by integratingΘ00 over all space, is therefore also nonnegative.

If instead kF vanishes and kAF is small, the contribu-tion to E can be written in the form

E ≡∫d3x Θ00

= 12

∫d3x

(~E2 + [ ~B − (kAF )0 ~A]2

−[(kAF )0]2 ~A2)

. (37)

The last term is nonpositive and so can in principle intro-duce an instability in the theory [22]. Note that a similarsituation would hold for the linear term −(kA)κA

κ in thelagrangian that was discarded in section III, for whichthe energy density (kA)0φ − ~kA · ~A could also be neg-ative. The appearance of negative contributions to theenergy is unsatisfactory from a theoretical viewpoint.

It might seem tempting to resolve this issue by re-quiring that only the spacelike components of kAF arenonzero, so that the terms involving (kAF )0 are absentfrom Eqs. (36) and (37). However, this condition de-pends on the observer frame, so even an infinitesimalboost to another observer frame would reintroduce theinstability. A somewhat more interesting option might

9

be to combine the vanishing of (kAF )0 with the intro-duction of a (small) photon mass, perhaps arising froma hitherto unobserved spontaneous breaking of the elec-tromagnetic U(1) gauge symmetry. This would eliminatethe linear instability and in principle might also producea contribution canceling the negative term appearing inEq. (37), although perhaps only for a physically reason-able range of observer boosts determined by the size ofthe photon mass and the magnitude of the componentsof ~kAF . Although some form of this idea might be madephysically acceptable, we are restricting ourselves here tominimal modifications of the usual standard model andso we disregard this possibility in the present work.

In the absence of a complete demonstration of a con-sistent alternative interpretation, one option might be todiscard the term (25) depending on kAF . This is possibleat the classical level, but at the quantum level one mightexpect radiative corrections to induce it. We return tothis question in section V, meanwhile keeping the term(25) in the analysis for completeness.

B. Solution of Equations of Motion

The equations of motion arising from the lagrangian(31) are:

∂αFα

µ + (kF )µαβγ∂αF βγ + (kAF )αǫµαβγF

βγ = 0 .

(38)

These equations are the Lorentz-breaking extensions ofthe usual inhomogeneous Maxwell equations in the ab-sence of sources, ∂µF

µν = 0. By virtue of its conventionaldefinition in Eq. (21), the field strength Fµν satisfies theusual homogeneous Maxwell equations

∂µFµν = 0 , (39)

where Fµν is given in Eq. (34).An important feature of the equations (38) and (39)

is their linearity in Fµν and hence in Aµ. The Lorentz-violating terms thereby avoid the complications of non-linear modifications to the Maxwell equations, whichare known to occur in some physical situations such asnonlinear optics or when vacuum-polarization effects areincluded. Another feature is that the extra Lorentz-violating terms involve both the electric and the magneticfields, as well as their derivatives. As a result, the equa-tions (38) bear some resemblance to the usual Maxwellequations in moving media, for which the boost causesthe electric and magnetic fields to mix. Note that the co-efficients determining this mixing are directly dependenton the velocity of the medium and so change with the in-ertial frame. Similarly, for the Lorentz-violating case ofinterest here, a change of observer frame changes the cou-pling coefficients. Some other useful analogies between

the equations (38) and those of conventional electrody-namics in macroscopic media are described in section IVC.

The equations of motion (38) and (39) depend only onFµν and so, as expected, they are gauge invariant underthe standard U(1) gauge transformations

Aµ → Aµ − 1

q∂µΛ . (40)

As in conventional electrodynamics, the presence ofgauge symmetry affects the interpretation and solutionof the equations of motion. We first consider a treat-ment in terms of the potentials Aµ and then one for the

field strengths ~E, ~B.Taking the potentials Aµ as basic, the four equations

(39) are directly satisfied. This appears to leave fourequations (38) for four unknowns Aµ. However, justas in conventional Maxwell electrodynamics, the conju-gate momentum to A0 vanishes because ∂0A

0 is missingfrom the lagrangian (31), so the theory has a Dirac pri-mary constraint. In the conventional case it then followsfrom the identity ∂ν∂µF

µν ≡ 0, which is associated withcurrent conservation when sources are present, that theequation of motion associated with A0 plays the role ofan initial condition. The same conclusion holds here be-cause when acted on by ∂µ the left-hand side of Eq. (38)also vanishes identically. This leaves three equations ofmotion and a constraint for four variables. One combi-nation of variables can be fixed by a gauge choice. Theconstraint then leaves two independent degrees of free-dom.

Despite the parallels with conventional electrodynam-ics, the gauge-fixing process involves some interesting dif-ferences. For example, there is normally an equivalencebetween the Coulomb gauge ~∇ · ~A = 0, the temporalgauge A0 = 0, and one of the members of the familyof Lorentz gauges ∂µA

µ = 0. When Lorentz-violatingeffects are included, these three gauge choices becomeinequivalent. For example, A0 typically is nonzero if theCoulomb gauge ~∇ · ~A = 0 is imposed.

More insight about the wave motion implied by Eq.(38) can be gained with the ansatz

Aµ(x) ≡ ǫµ(p) exp(−ipαxα) , (41)

where pµ ≡ (p0, ~p) can be regarded as the frequencyand wave vector of the mode or as the associated en-ergy and momentum (which can be distinct from the con-served energy and momentum obtained from the energy-momentum tensor). Note that taking the real part isunderstood, as usual. The equations of motion (38) gen-erate the momentum-space equation

Mαδ(p)Aδ = 0 , (42)

where the matrix Mαδ(p) is

10

Mαδ(p) ≡ ηαδp2 − pαpδ − 2(kF )αβγδpβpγ

−2i(kAF )βǫαβγδpγ . (43)

This 4 × 4 complex-valued matrix is hermitian becausethe first three terms are real and symmetric while the lastis imaginary and antisymmetric. Its determinant can beshown to vanish identically for all pµ, a feature related tothe gauge freedom. The conventional result is recoveredwhen the coefficients kF and kAF vanish.

Once a gauge choice is imposed, the relation (42) pro-vides a set of complex-valued equations for Aδ. The dif-ferences between various gauge choices that normally areequivalent can be seen explicitly at this stage. For exam-ple, Mαδ(p)Aδ is not proportional to Aα in the Lorentzgauges, the Coulomb gauge leaves a nontrival equationfor A0, and the temporal gauge generates an involvedconstraint on ~∇ · ~A. In practice, Eq. (42) then reducesto a (sub)set of equations involving an effective matrixMeff(p) with explicit form dependent on the gauge choice.The requirement for existence of nonzero solutions canbe obtained from a condition of the type detMeff(p) = 0.With fixed coefficients kF and kAF , this condition thendetermines p0 as a function of ~p. Since Mµα(p) is a 4× 4matrix with entries quadratic in pµ, a determinant of thistype can produce an eighth-order polynomial in p0.

In the conventional case in the Lorentz gauge, the poly-nomial reduces to one with two quadruply degenerateroots, p0 = ±|~p|. The apparent doubling of the rootsrelative to the number of variables can be understoodby the observation that in this case Mµα(p) is symmet-ric under pµ → −pµ, so for each solution p0(~p) there isanother solution −p0(−~p). These two solutions can beshown to be physically equivalent by examining the realpart of Aµ in Eq. (41).

In contrast, in the general extended electrodynamicsthe polynomial determining p0 may have eight distinctroots. Each of these could in principle produce a non-trivial solution for Aδ, double the expected number. Inthis case, Mµα(p) is symmetric under the simultane-ous operations pµ → −pµ and kAF → −kAF (leavingkF unchanged). Thus, for each solution p0(~p, kF , kAF )there is another solution −p0(−~p, kF ,−kAF ). The signchange for the coefficient kAF might appear to pre-clude the demonstration of the physical equivalence ofthese solutions. However, hermiticity of Mµα(p) im-plies its determinant is equivalent to the determinantof its complex conjugate. Since kAF appears onlyin the imaginary part of Mµα(p), it follows that foreach solution −p0(−~p, kF ,−kAF ) there is also a solution−p0(−~p, kF ,+kAF ). This is physically equivalent to thesolution p0(~p, kF , kAF ), as before. Thus, the number ofindependent roots is the same as the number of variablesas expected, despite the apparent complexity of the poly-nomial.

In the general extended electrodynamics, neither theCoulomb gauge nor the Lorentz gauges significantly sim-

plify the primary constraint. In contrast, the temporalgauge A0 = 0 immediately removes one degree of free-dom from Aµ. This gauge can be imposed by choosing

the function Λ in Eq. (40) as qΛ(t, ~x) =∫ tA0(t′, ~x)dt′.

Note that this choice breaks observer boost invariancebut leaves unaffected the observer rotation invariance. Itreduces the primary constraint to the form

M0jAj = 0 , (44)

where M0j are components in the temporal gauge of thematrix Mαδ given in Eq. (43).

At this stage an explicit solution could be found. Forexample, one could use two of the degrees of freedom ofobserver rotation invariance to select a convenient coordi-nate system, such as one in which pj ≡ (0, 0, p). Solvingfor A3 from the primary constraint (44) and substitut-ing into the remaining three equations of motion in (42)would then produce an identity and two simultaneouslinear equations for A1 and A2. A nontrivial solution ofthis pair of equations could be found by requiring the de-terminant of the system to vanish, which in turn wouldgenerate a relation between p0 and p. Solving this re-lation must give two independent dispersion relations,one for each of the two physical degrees of freedom. Thefull dispersion relations in an arbitrary coordinate systemcould then in principle be recovered by using argumentsbased on rotational covariance.

Rather than pursuing this approach, we return to theeight equations of motion (38) and (39) and reconsidertheir treatment taking as independent variables the sixelectric and magnetic fields. Here, we are interested inthe properties of electromagnetic radiation, so we workwith the standard ansatz

Fµν(x) ≡ Fµν(p) exp(−ipαxα) , (45)

where pµ ≡ (p0, ~p).The equations (39), which include the usual Faraday

law and the condition ensuring the absence of magneticmonopoles, are unaffected by the Lorentz breaking andfor radiation reduce as usual to

p0 ~B = ~p× ~E , ~p · ~B = 0 . (46)

The first of these can be regarded as defining the mag-netic field once the electric field is known. The secondof these equations follows from the first, and shows thatthe magnetic field remains transverse to ~p despite theLorentz violation.

The equations (38) generate modified Coulomb and

Ampere laws that are to be solved for ~E. A relativelystraightforward procedure is to substitute for ~B from Eq.(46). Using the Ampere law, we thereby obtain the vec-tor equation

M jkEk = 0 , (47)

11

where the 3×3 matrix M jk is identical to the (jk)-component submatrix of the matrix Mαδ in Eq. (43).The modified Coulomb law can be obtained from themodified Ampere law by taking the scalar product with ~p:pjM jkEk = 0. Note that this derivation provides someinsight about the temporal gauge A0 = 0 in a treatmentusing Aµ. Thus, there is a close parallel between the two

because ~E = ip0 ~A in this gauge.To obtain an explicit solution, it is helpful to take ad-

vantage of the observer rotation invariance to select a co-ordinate system in which key expressions are simplified.For example, a useful frame is the one with ~p = (0, 0, p).In it, the modified Coulomb equation can be solved forE3 in terms of E1 and E2. Substitution of this solutioninto the three component equations (47) produces oneidentity and two simultaneous linear equations for E1

and E2. The matrix of this system of equations is hermi-tian, and the condition for a nontrivial solution is that itsdeterminant vanishes. The determinant turns out to bea fourth-order polynomial for p0 as a function of p. Forreasons similar to those given for the Aµ case, there areonly two physically distinct solutions of this polynomial,one for each of the two independent degrees of freedom.

An explicit solution of the determinant condition in thegeneral case is involved because the fourth-order polyno-mial is homogeneous in the small coupling coefficients.Since kF is dimensionless while kAF has dimensions ofmass, to first order in the coupling coefficients and forp0 ≈ p ≫ |kAF | and 1 ≫ |kF | the solution for p0 as afunction of p must take the form of the sum of p with afunction of the quantities kF p and kAF . Indeed, we find

p0 = (1 + ρ)p±√

(σ2p2 + τ2) , (48)

where ρ and σ2 are functions of the components of kFand τ2 is a function of the components of kAF , given by

ρ = 12

[(kF )0101 + (kF )0202 + (kF )1313

+(kF )2323 + 2(kF )0113 + 2(kF )0223]

,

σ2 = 14

[(kF )0101 − (kF )0202 + (kF )1313

−(kF )2323 + 2(kF )0113 − 2(kF )0223]2

+[(kF )0102 − (kF )0123 + (kF )0213 + (kF )1323

]2,

τ2 =[(kAF )3 − (kAF )0

]2. (49)

The solution (48) entangles the components of kF andkAF in a way that cannot be separated without addi-tional information about their relative sizes. Note thatit reduces correctly to the result of Ref. [22] in the casekF = 0.

The corresponding general solutions for the vectors ofthe electric and magnetic fields are involved and providelittle insight for present purposes, so we omit them here.They exhibit two physical linear polarization vectors for~E, each obeying a different dispersion relation. This pro-

duces birefringence, among other effects. Note in par-ticular that, contrary to widespread assumption in theliterature, no circularly polarized solution to the equa-tions of motion typically exists. An electromagnetic waveprepared in a state of circular polarization would propa-gate as two linearly polarized components with distinctdispersion relations, so an initial circularly polarized con-figuration would gradually become elliptical. These andsome other interesting results about the wave propaga-tion are discussed further in subsections IV C and IV Dbelow.

In the remainder of this subsection, we present a sam-ple analytical solution to the equations of motion fora special case that provides further insight. We con-sider the lagrangian (31) with (kAF )µ = 0 and with theonly nonzero components of kF chosen to be (kF )0j0k =− 1

2βjβk, where the βj are three (small) real dimension-less quantities, and components related to these by thesymmetries of kF . In terms of the lagrangian (32) only

the term involving βjkE is nonzero, and it has a direct-

product structure: βjkE ≡ +βjβk. The lagrangian (32)therefore becomes

Lspecialphoton = 1

2 ( ~E2 − ~B2) + 12 (~β · ~E)2 . (50)

This example involves only CPT-even Lorentz violation.The lagrangian (50) generates modified inhomoge-

neous Maxwell equations in the absence of sources:

~∇ · ~E = −~β · ~∇(~β · ~E) ,

~∇× ~B − ∂0~E = ~β ∂0(~β · ~E) . (51)

In terms of the potentials Aµ of Eq. (41), appropriatefor describing radiation in momentum space, these areequivalent to the vector equation

p0[~p+ (~p · ~β)~β]A0 − (p0)2[ ~A+ ( ~A · ~β)~β]

+[~p 2 ~A− (~p · ~A)~p ] = 0 (52)

and its scalar product with ~p.For definiteness we proceed in Lorentz gauge, where

A0 = ~p · ~A/p0. According to the discussions above, inthe presence of Lorentz violation this gauge may requirenonzero A0 and ~p · ~A. For a nontrivial solution to Eq.(52), we find two possible dispersion relations:

(po)2 = 0 ,

(pe)2 = − (~β × ~pe)

2

1 + ~β2. (53)

The first corresponds to an ‘ordinary’ mode with four-momentum po obeying the conventional dispersion rela-tion, while the second is an ‘extraordinary’ mode withfour-momentum pe and a modified dispersion relation.

For a wave vector aligned along ~β, both modes reduceto the conventional case and exhibit normal behavior.However, for other alignments the properties of the two

12

modes differ. For simplicity, we restrict attention hereto the situation with wave vector orthogonal to ~β, so~p · ~β = 0. In this case, the ordinary mode Aµo can be

chosen to satisfy A0o = 0 with ~Ao parallel to ~p× ~β, while

the extraordinary mode must satisfy A0e = 0 and has

~Ae aligned along ~β. These two modes propagate withdifferent velocities. For example, their group velocities~vg ≡ ~∇pp

0 are

~vg,o = p , ~vg,e =1√

1 + ~β2

p . (54)

For each mode, the group and phase velocities are equal.One consequence of the difference between the two

modes is birefringence. For example, a plane-polarizedmonochromatic wave of frequency p0 that is initially ageneral combination of the two modes eventually be-comes elliptically polarized. For the electric field, we find

~E(t, ~x) = −p0

(coAo sin[p0(r − t)]

+ceAe sin[p0(

√1 + ~β2 r − t)]

), (55)

where r = |~x|, Ao is parallel to ~p × ~β and Ae is parallel

to ~β as before, and the weights co and ce are determinedby the initial polarization condition. This shows thatthe presence of ~β causes the wave to become ellipticallypolarized after it has traveled a distance

r ≃ π

2

(√1 + ~β2 − 1

)p0

. (56)

The magnetic field exhibits similar behavior.The explicit expressions for the electric and magnetic

fields can be used to derive the energy density Θ00e and

the Poynting vector Θj0e for the extraordinary mode. We

find

Θ00e = ~p 2c2e sin2 pµx

µ , Θj0e = p0pjc2e sin2 pµx

µ .

(57)

This shows that in the present case the velocity of en-ergy transport vje ≡ Θj0

e /Θ00e is identical to the group

and phase velocities, Eq. (54). Some comments aboutthe various velocities in the general case are made in thenext subsection.

C. Analogy to Macroscopic Media

In subsection IV B, an approximate analogy was notedbetween the equations of motion for the Lorentz-breakingextension of electrodynamics and those for electrodynam-ics in moving media. In this section, we introduce some

useful quantitative analogies between the extended elec-trodynamics and the electrodynamics of macroscopic me-dia. These can be used to gain further insight aboutthe nature of the extended electrodynamics with Lorentzbreaking.

Consider first the situation in position space, where therelevant equations are (38) and (39). We have alreadynoted that Eqs. (39) take the same form as in conven-tional electrodynamics. The idea is to define new quan-tities ~D and ~H such that the forms of Eqs. (38) becomesidentical to those of the Maxwell equations in materialmedia. It turns out that it suffices to introduce an ef-fective displacement current ~D and an effective magneticfield ~H having linear dependence on the electric field ~E,the magnetic induction ~B, the vector potential ~A, andthe scalar potential A0.

We find that the definitions

Dj = Ej − 2(kF )0j0kEk + (kF )0jklǫklmBm

+2ǫjkl(kAF )kAl ,

Hj = Bj + 12 (kF )pqrsǫpqjǫrskBk − (kF )0mklǫjklEm

−2(kAF )0Aj + 2(kAF )jA0 (58)

reproduce the usual Maxwell equations in material me-dia. The analogy can therefore be used to gain insightinto those properties of the extended electrodynamicsthat are directly associated with the equations of motion.However, caution is required in applying other conceptsof conventional electrodynamics. For example, it turnsout that if kAF 6= 0 then the conventional expressions forthe energy density and Poynting vector in terms of ~D and~H fail to reproduce completely the true energy densityΘ00 and Poynting vector Θj0 in the extended theory. IfkAF = 0, in contrast, the correct expressions are indeedreproduced by the analogy.

The above analogy is useful for general discussions ofthe properties of the extended electrodynamics. How-ever, it becomes somewhat cumbersome for certain con-siderations involving radiation. We have developed a sec-ond analogy that is of more direct use when the fields areconverted to momentum space through Eq. (45). It turnsout that the equations of motion can then be correctlyreproduced by defining an effective displacement current~D(p) through

Dj = ǫjkEk , (59)

where ǫjk is a hermitian effective permittivity given by

ǫjk ≡ δjk +2

(p0)2(kF )jβγkpβpγ +

2i

(p0)2(kAF )βǫ

jβγkpγ .

(60)

In particular, it is unnecessary to introduce an effectivemagnetic field ~H distinct from ~B. This second analogyis therefore different from the first. Note that again the

13

correct energy density and Poynting vectors cannot beobtained directly by substitution into the conventionalformulae. Nonetheless, the analogy is valuable becauseit permits insight into the effects of Lorentz violation onradiation. Note also that the effective permittivity (60)depends on the frequency p0 and wave vector ~p, whichimplies a nonlocal connection between ~D(x) and ~E(x).

The extended Maxwell equations for this analogy di-rectly yield

~p · ~B = ~p · ~D = ~E · ~B = ~D · ~B = 0 . (61)

The natural right-handed triad of orthonormal vectorsdescribing the vibration of the electromagnetic field istherefore (p, D, B). Unlike the case of conventional vac-

uum radiation, the electric-field vector ~E here is orthog-onal only to B and so lies off-axis in the p-D plane. Inthis analogy, the energy density is typically transportedneither in the direction p nor in the direction E × B.

It is useful to introduce a generalized refractive indexn(p) by n(p) ≡ |~p|/p0. Its inverse is the magnitude ofthe phase velocity of the mode (45). Using the extendedMaxwell equations, we can then deduce the result

~D = n2[~E − (p · ~E)p

]≡ n2 ~E⊥ , (62)

which determines the effective displacement current di-rectly in terms of the electric field and the momentum.Eliminating ~D via Eq. (59) produces a set of three linear

equations of the form (47) for ~E, with the matrix M nowgiven by

M jk ≡ n2δjk − pjpk

(p0)2− ǫjk . (63)

A nontrivial solution exists if det(M jk) = 0. We haveexplicitly verified that this condition is equivalent to thecondition of the vanishing of the determinant of the ma-trix M jk appearing in Eq. (43) in subsection IV B.

In conventional crystal optics, the permittivity is oftendiagonalized: ǫjk ≡ ǫjδjk (no sum), where the eigenval-ues ǫj are real. This means that the coordinate axes areidentified with the principle dielectric axes, which typ-ically represents a different coordinate system than thespecial one with ~p = (0, 0, p) used in section IV B. A di-agonalization of this type is also possible in the presentanalogy because the effective permittivity is hermitian.Substitution into Eq. (63) and expansion of the determi-nant then produces an expression with the form of theFresnel equation of crystal optics. The sixth-order termsin the determinant cancel, ultimately by virtue of the ex-istence of only two independent degrees of freedom in ~E.Solving the determinant condition provides the disper-sion relations for the independent degrees of freedom. Ifp is given, the condition specifies p0 as a function of |~p|.If p0 is given, the condition specifies |~p| as a function ofp.

The special choice of coordinate system in subsectionIV B permits a direct demonstration with this analogythat for a given momentum ~p the solutions for the ef-fective displacement current ~D are linearly polarized.First, substitute for Ej = (ǫ−1)jkDk in Eq. (62). Usingobserver rotation invariance to select a frame in which~p ≡ (0, 0, p) then yields the two simultaneous equations

(δab − n2(ǫ−1)ab

)Db = 0 , (64)

where a, b = 1, 2. The vanishing of the determinant ofthe expression in parentheses generates the analogue ofthe Fresnel equation in this special coordinate system. Itcan be seen directly from Eq. (64) that for fixed ~p the~D vectors for each of the two values of n must lie alongthe principal axes of symmetry of the two-dimensionalmatrix (ǫ−1)ab. These two ~D vectors are perpendicular,so an electromagnetic wave corresponding to either oneis necessarily linearly polarized.

Many other concepts of crystal optics can be appliedin the context of this analogy, including the wave-vectorand ray surfaces and the Fresnel and other ellipsoids.The presence of Lorentz violation means that the vac-uum as experienced by an electromagnetic wave behaveslike a special kind of crystal. Our results show that theeffective medium is optically anisotropic and gyrotropicand exhibits spatial dispersion of the axes. The earliestmention of effects of this type appeared in an 1878 paperof Lorentz [23], and they are now well established in avariety of physical systems [24]. Thus, the momentumdependence of the effective permittivity corresponds tospatial dispersion of the axes. A nonzero kAF produces acontribution to the effective permittivity analogous to theeffects of natural optical activity in a gyrotropic crystal,while a nonzero kF produces effects analogous to spatialdispersion in an optically inactive and anisotropic crys-tal. Partly on the basis of the hermiticity of the effectivepermittivity, we also anticipate that in the presence ofLorentz violation the vacuum behaves like a transparent(nonabsorptive) medium, although a complete and ele-gant demonstration of this remains an open issue.

The above analyses partially simplify if certain com-ponents of kF and kAF vanish. Suppose for definite-ness that kAF indeed vanishes and that the only nonzerocomponents of kF are (kF )0j0k and components relatedto these by the symmetries of kF . This makes the ef-fective permittivity real and independent of pµ: ǫjk =δjk − 2(kF )0j0k. It is then possible, for example, to solve

explicitly for the behavior if ~E is specified, which pro-vides yet another approach to the physics. In this case,~E⊥ is the component of ~E perpendicular to p in the E-pplane. It follows that ~E⊥ = ( ~E · D)D, from which onecan derive

p =( ~D)2 ~E − ( ~E · ~D) ~D√[

( ~E)2( ~D)2 − ( ~E · ~D)2]( ~D)2

(65)

14

provided ~D and ~E are not parallel. The phase velocityis given by

vp = 1/n =

√( ~E · ~D)/( ~D)2 . (66)

For instance, if ~E = (E, 0, 0) then to lowest order in kFwe find ~D ≈ | ~E|(1 − 2(kF )0101,−2(kF )0201,−2(kF )0301)and vp ≈ 1 + (kF )0101, which in the appropriate limitagrees with the result for the extraordinary mode of theexample at the end of subsection IV B. Even in thisrelatively simple case, Eq. (65) shows that the vector ~pcan have a complicated structure with components in allthree directions.

The above analysis uses the notion of the phase veloc-ity vp. However, even in conventional electrodynamicsthere are numerous possible definitions of the velocity vof an electromagnetic wave, including among others thegroup velocity, the velocity of energy-momentum trans-port, and the signal velocity [25]. The Lorentz violationadds further complications to this situation. In the re-mainder of this subsection, we comment on some aspectsof this issue.

An important feature is that the fundamental physi-cal constant c = 1 relating the space and time compo-nents of the metric is unaffected by the Lorentz viola-tion. The underlying spacetime structure of the theoryis the usual one because the apparent Lorentz breakingat the level of the standard model is merely a reflectionof the presence of nonzero tensor expectation values ina fundamental theory with Lorentz-covariant dynamics.Indeed, c is an invariant under both observer and particleboosts. However, the physical velocity of an electromag-netic wave can be affected. The situation is analogousto that of a fermion mass parameter m in the lagrangianfor the standard-model extension: although m remainsunchanged, the physical rest mass of a particle can beaffected [2].

As in conventional electrodynamics, the various defi-nitions of physical velocity are inequivalent in general.Any choice for the physical velocity v typically differsfrom c, although v ≈ c since kF and kAF are small.The analyses above indicate that, for any given defini-tion, the magnitude and direction of the velocity of anelectromagnetic wave can vary with the wave-vector ori-entation and the polarization. Incidentally, conventionalcrystal-optics experiments suggest that there is no gen-eral condition requiring the velocity of one type of po-larization to exceed the other. For example, the indicesof refraction no and ne for the ordinary and extraordi-nary rays, respectively, of the sodium D line are mea-sured to be no = 1.658 > ne = 1.486 in calcspar but areno = 1.544 < ne = 1.553 in quartz [26].

For the special case involving Eq. (66), vp may exceedc if the sign of kF is appropriate. In conventional electro-dynamics, a phase velocity exceeding c is known to occurin numerous physical situations, for example, for TE and

TM modes in wave guides. Indeed, both the phase andgroup velocities can simultaneously exceed c in certainrefractive materials. For the present theory, it is an openissue to demonstrate that a phase velocity exceeding c iscompatible with microcausality. It is possible in princi-ple that only certain sign choices for the components ofkF lead to physically acceptable microcausal theories. Ifthis occurs, it would be analogous to the usual require-ment of a particular sign for the mass-squared term in a(stable) scalar field theory. In any event, a satisfactoryproof of microcausality would involve a complete treat-ment at the level of quantum field theory and lies beyondthe scope of the present work.

Another issue involving the physical velocity v ofan electromagnetic wave is its behavior under Lorentztransformations. Since c is invariant under an observerLorentz transformation whereas kAF and kF change, vis expected to transform along with the frequency andwavelength. This is unlike the conventional case and is aconsequence of the presence of the background expecta-tion values. In contrast, a particle Lorentz transforma-tion, which for a fixed polarization mode involves remain-ing in the specified observer frame but changing ~p, hasno effect on c, kAF , or kF . Note that if ~p is changed whilethe polarization is fixed, the above analyses show that thefrequency p0 also changes in this case. One might insteadcountenance another kind of boost in which ~p is changedbut p0 is unaffected, in which case the polarization mustalso change.

D. Constraints from Birefringence

The existence of distinct dispersion relations for theindependent polarizations means that birefringence is amajor feature of the behavior of an electromagnetic wavein vacuum in the presence of Lorentz violation. In thissubsection, we investigate some of the theoretical andexperimental implications of a birefringent vacuum.

For definiteness, consider a monochromatic electro-magnetic wave of frequency p0. The electric field ~E(t, ~x)of this wave is formed in general from two independentpolarization components:

~E(t, ~x) =[~E1(~p1) exp(i~p1 · ~x)

+ ~E2(~p2) exp(i~p2 · ~x)]exp(−ip0t) . (67)

The wave vectors ~p1 and ~p2 must satisfy the appropri-ate dispersion relations for the specified frequency p0.Note that the direction of wave propagation must alsobe specified to fix completely the solution. One possibledetermining method could be to require that both com-ponent waves propagate their energy density in a givendirection.

15

Since the Lorentz violation is small, we expect ~p2 =~p1 + δ~p, where δ~p is small relative to ~p1 ≈ ~p2. Substitu-tion gives

~E(t, ~x) ≈[~E1(~p1) + ~E2(~p2) exp(iδ~p · ~x)

]

× exp(−ip0t+ i~p1 · ~x) . (68)

This equation shows that the birefringence length scaleis |δ~p|−1, which is large when |δ~p| is small. Since δ~p hasdimensions of mass and since it vanishes in the absenceof Lorentz violation, its dominant terms are expected tobe controlled by kAF , by a product of components of~p and kF , or by some combination of the two. This isin agreement with the discussion of the dispersion rela-tions in previous subsections. Note that the associatedphase shift ∆φ ≡ δ~p ·~x cannot correctly be regarded as aphase difference between two circular-polarization modesbecause typically no such modes exist as solutions of thedispersion relations.

In the remainder of this subsection, we consider pos-sible bounds on kF and kAF from some terrestrial, solarsystem, astrophysical, and cosmological experiments.

First, we summarize the case of nonzero kAF but zerokF . A term of the form (25) appears to have been in-troduced independently on several occasions, includingamong others in Ref. [27] and the review [15] mentionedearlier, although the observation that it is CPT violat-ing appears to have been overlooked prior to our earlierwork [2]. Given the theoretical difficulties arising fromnegative contributions to the energy as described in sub-section IV B, it seems possible that this term would needto be absent in nature even if Lorentz symmetry is vi-olated. However, this too is a suggestion that could bethe subject of tests.

In a pioneering work [22], Carroll, Field, and Jackiwinvestigated some properties of the term (25) and usedgeomagnetic constraints and limits on cosmological bire-fringence of radio waves to bound certain forms of thecoupling coefficient kAF . Their treatment of geomag-netic constraints is based on known bounds on the photonmass [28], and it constrains a term of the form (25) with(kAF )µ = (k,~0) to |k| ∼< 6 × 10−26 GeV. In contrast, theconstraints they obtain from cosmological birefringenceare considerably sharper, primarily because the distancescales are greater. Their investigation seeks a redshiftdependence in the established correlation [29] betweenthe intrinsic position angles and the polarization anglesof a set of radio galaxies and quasars at distances com-parable to the Hubble length. It constrains a particularcombination of the coefficients for a timelike (kAF )µ to

∼< 2 × 10−42 GeV. A more recent analysis [30] claims anonzero observed effect with a spacelike (kAF )µ at a scaleof approximately 10−41 GeV. This has been disputed byother authors [31].

We next consider the case with kF 6= 0 but kAF = 0.In its general form, this possibility appears to have been

largely disregarded in the literature. However, the ro-tationally invariant term of the form 1

2α( ~B2 + ~E2) inthe extended-QED lagrangian (32) has been consideredby several authors, usually in the rescaled form involvingonly ~B2. In particular, this term has a counterpart in theTHǫµ formalism [32]. This formalism is a phenomenolog-ical parametrization for the motion and electromagneticinteractions of charged pointlike test particles in an exter-nal spherically symmetric and static gravitational field.It has been extensively used for quantitative tests of thefoundations of gravity, including local Lorentz invariance.In this context, clock-comparison experiments have con-strained the analogue of the parameter α to better thanabout one part in 1021 [18]. An improvement over thisbound of about an order of magnitude may be possiblebased on the existence and properties of high-energy cos-mic rays [33].

In the general case with nonzero kF and violation ofrotational invariance, the sharpest bounds are likely toemerge once again from observational constraints on cos-mological birefringence. However, the discussion follow-ing Eq. (68) shows there is a significant difference in thekF case: the phase shift ∆φ here depends on a product ofcomponents of kF and ~p, whereas in the kAF case ∆φ de-pends only on kAF . This behavior can be seen explicitly,for example, in the special analytical solution presentedat the end of subsection IV B.

The linear dependence of ∆φ on momentum or wavenumber implies an inverse dependence on wavelength.The rotation measures and intrinsic position angles of ra-dio sources [29] are obtained by a fitting procedure thatassumes a quadratic dependence on wavelength (propor-tional to the rotation measure and attributed to Faradayrotation) with a wavelength-independent zero offset (theintrinsic position angle). This procedure is suitable forobtaining constraints on kAF , which would generate anextra wavelength-independent effect, but may be inade-quate to place a reliable bound on kF or to detect theassociated wavelength-dependent effects. It therefore ap-pears somewhat involved to obtain an accurate estimateof the constraints on kF from cosmological birefringence.

Although a complete treatment lies outside our presentscope, a crude estimate of an attainable bound on kFcan readily be found. It is plausible to suppose that theresults of a careful analysis would provide a limit on aproduct of certain components of kF and ~p comparableto that of order 10−42 GeV obtained for kAF in Ref. [22].The radio sources typically involve wavelengths of order10 cm, which corresponds to an inverse wavelength ofabout 10−15 GeV. This suggests that an upper bound ofapproximately 10−27 could be placed on at least someof the (dimensionless) coefficients kF . The tightness ofthis constraint and the apparent feasibility of the analysissuggests this investigation would be worthwhile to pur-sue. Ideally, a complete study would obtain combinedbounds on both of the coupling coefficients kF and kAF .

16

An interesting implication of the (inverse) wavelengthdependence of the birefringence is that shorter wave-lengths are more sensitive to the effects. Although it maybe infeasible in practice, a measurement of cosmologicalbirefringence comparable to the above but obtained with,say, optical sources would be much more sensitive to pos-sible effects from kF . Optical wavelengths are a factor ofabout 10−6 of radio wavelengths, which would correspondto a millionfold improvement in sensitivity to kF .

Other bounds on Lorentz violation could be deduced.In the next section, we show that one-loop radiative cor-rections induce a dependence of kF on the coefficients cµνin Eq. (22) for the extended QED. This suggests that ifa tight bound were obtained on kF as above, an indirectconstraint might also be inferred on cµν . The latter con-straint would be weaker by a factor of the fine-structureconstant, but the limits deduced would nonetheless prob-ably be comparable to the best ones attainable in othertests of Lorentz symmetry.

If a nonzero effect is detected in the future, it might beof some theoretical interest to investigate the possibilityof a correlation between the particular coupling coeffi-cients involved and the motion of the Earth relative tothe cosmic microwave background radiation. The pointis that the apparent Lorentz violation induces boost (andorientation) dependence in experiments [6]. Although thestandard-model extension strictly has no preferred frame,the coupling coefficients must take a canonical form insome observer frame [2]. If the latter is at rest with re-spect to the cosmic microwave background radiation, asmall deviation from the canonical form might arise fromthe Earth’s motion. Although the Earth’s speed in thisframe is about 10−3c, the sensitivity of the birefringencemeasurements might nonetheless be sufficient to detectits effects.

V. RADIATIVE CORRECTIONS

We next examine some radiative corrections to thepure-photon sector. In subsection V A, CPT-odd termsare investigated. Of particular interest is whether thetree-level vanishing of the coefficient kAF in Eq. (25) forthe QED extension, which would eliminate negative con-tributions to the energy, is reasonable in the light of quan-tum effects. The point is that the latter might in principleinduce a nonzero coefficient through radiative correctionsfrom another sector of the theory. Other quantum cor-rections that might generate an instability through thelinear term −(kA)µA

µ are considered at the end of thissubsection. In subsection V B, we study quantum cor-rections in the CPT-even sector, involving the coefficientkF .

The analysis in this section is based on the quantiza-tion discussed in Ref. [2]. It is largely at the one-looplevel and for leading-order Lorentz-violating effects, and

it is primarily limited to issues involving radiative cor-rections to the pure-photon sector. A few results are alsopresented for higher loops and effects in other parts ofthe standard-model extension.

An interesting issue indirectly related to some calcu-lations in subsection V A is whether the anomaly can-cellations occurring in the conventional standard modelstill hold for its extension presented in section II. Threeknown types of chiral gauge anomaly are relevant [34]. Itlies beyond our present scope to provide a complete anal-ysis of all these in the presence of the Lorentz-violatingterms, and it is certainly conceivable that the latterwould modify the standard derivations. We do expect,however, that the usual cancellations of the (abelian,singlet, and nonabelian) triangular gauge anomalies andthe nonperturbative global SU(2) anomaly indeed remainvalid. The point is that the standard-model extensionhas the same multiplets and the same gauge structure asthe conventional case, so the group-theoretic underpin-nings of the usual analyses are unaffected. The situationfor the third type of anomaly, which is the mixed gauge-gravitational chiral anomaly associated in part with localLorentz transformations, is less clear. The presence ofLorentz violations might appear to suggest a potentiallynonzero contribution to this anomaly. However, a carefulanalysis is needed because observer Lorentz invariance isin fact maintained in the standard-model extension.

A. CPT-Odd Terms

As discussed above, the possible difficulties with neg-ative contributions to the energy and the tight experi-mental constraints suggest that the coefficient kAF van-ishes. If it is set to zero at tree level, the issue arises asto whether it acquires radiative corrections from quan-tum loop corrections. If so, there could be both theoret-ical and experimental arguments suggesting associatedconstraints on certain other coefficients in the standard-model extension. The issue of the vanishing of radiativecorrections to kAF therefore has the potential to pro-vide a nontrivial consistency check on the theory. In thepresent subsection we investigate this, assuming that kAFis zero at tree level and beginning with one-loop effectsat leading order in the Lorentz-breaking terms. Remark-ably, as we show next, the structure of the standard-model extension is such as to preserve a vanishing coeffi-cient kAF at this level.

A radiative contribution to kAF would represent acorrection to the photon propagator. In the standard-model extension, the Feynman rules for leading-order ef-fects from Lorentz-violating terms take the form of in-sertions on propagators or at vertices already existing inthe conventional theory [2]. Also, the photon interactswith charged particles as usual, so the only possible dia-grams modifying the photon propagator at the one-loop

17

level are those of the standard one-loop vacuum polariza-tion but with an insertion either on an internal charged-particle line or at one of the vertices.

The apparently daunting task of examining every pos-sible insertion implied by the extra terms in the standard-model extension can be simplified by taking advantage ofthe discrete operations C, P, and T. A radiative term pur-porting to contribute to the coefficient kAF must haveappropriate transformation properties under these dis-crete symmetries. In particular, it must be C even andPT (and CPT) odd, although either of the two possi-ble combinations of P and T could occur. At the levelof the QED extension with electrons and positrons, theonly term of this type is the one with coupling coefficientbµ in Eq. (28). This is true even if the discarded linearterm −(kA)κA

κ in the lagrangian were present, which isC and CPT odd. At any loop order, contributions musttherefore involve an odd number of line insertions aris-ing from the term with coefficient bµ. At the one-looplevel in the full standard-model extension, similar termsinvolving the other lepton and quark fields would alsocontribute to appropriate internal lines. However, onlyone additional distinct type of one-loop contribution ap-pears, involving a vertex correction proportional to thecoefficient (k2)µ in Eq. (17) in a diagram with a W+-W− loop. The demonstration that no net contributionsto kAF arise at one loop therefore involves considerationof only terms involving the bµ-type and the (k2)µ coeffi-cients.

Excluding the external photon legs, any contributionsto the vacuum polarization must have dimensions of masssquared. The leading-order contribution to kAF must in-volve both a momentum factor from the necessary deriva-tive on an external leg and one power of either bµ or(k2)µ. Since these factors already give the correct di-mensionality, any others must appear in dimensionlesscombinations of the photon momentum pµ and the massm of the particle in the loop. This is confirmed by theexplicit calculation below.

We first consider corrections to the one-loop vacuumpolarization involving bµ. Each such two-point diagramhas the usual form except for an insertion of the factor−ibµγ5γ

µ on one internal fermion line. From the per-spective of the fundamental theory, a one-loop two-pointdiagram with a fermion-line insertion is closely related toa one-loop three-point diagram containing the same twophoton legs together with a third leg involving a cou-pling to an axial vector. A fermion-line insertion in thetwo-point diagram can then be viewed as a limit of thisthree-point diagram in which there is zero momentumtransfer to the axial-vector leg and the axial vector isreplaced with a vacuum expectation value.

This line of reasoning is interesting because a one-loopthree-point diagram with an axial-vector and two pho-ton couplings is directly related to a triangular gaugeanomaly. If the axial vector is a gauge field in the under-

lying theory, such anomalies must cancel for the theoryto be renormalizable. One might therefore conjecturethat the cancellation of these anomalies could also im-ply cancellation of the limiting two-point diagrams in thestandard-model extension. If true, this provides anotherlink relating consistency of the standard-model extensionto the spontaneous nature of the Lorentz violation in theunderlying theory. Next, we develop a line of reasoningthat provides insight into this question.

Independently of the issue of corrections to the photonpropagator, the requirement that the triangular anoma-lies cancel in the underlying theory implies a constrainton coefficients of the type bµ that is of interest in its ownright. It turns out that this constraint is relevant to thephoton propagator, so we begin by deriving it.

Consider first the origin in the underlying theory ofthe axial-vector coupling in the triangle diagram. Priorto spontaneous symmetry breaking, the fundamentallagrangian may contain several terms of the generalform gaψ(T a)µν...ρψ(Γa)µν...ρψ for each fermion species ψ,where T a is a tensor field, Γa is a gamma-matrix struc-ture, gaψ is the associated coupling constant, and a is alabel ranging over the set of tensor fields that couple tothe species ψ. Note that the only acceptable line inser-tions in the two-point one-loop diagram are flavor diag-onal, so in the present context contributions are possibleonly from the diagonal components of Eqs. (10) and (12).We therefore disregard possible cross-couplings betweenfermion species in this derivation.

Each of these lagrangian terms can be decomposed interms of the usual 16 basis gamma matrices in four di-mensions. Collecting terms produces for each fermionspecies a lagrangian separated into five parts, one foreach of the five types of fermion bilinear: scalar, pseu-doscalar, vector, axial-vector, and tensor. The particularcomponents of the fields T a that multiply the axial-vectorbilinear can be regarded as a set of effective axial vectorsAa5µ with associated coupling constants gaψ. These axialvectors are the fields relevant for the one-loop three-pointdiagrams of interest. When the axial vectors Aa5µ acquirevacuum expectation values 〈Aa5µ〉, their net contribution

generates the coupling coefficient bψµ ≡ ∑a g

aψ〈Aa5µ〉 at

the level of the standard-model extension for this speciesof fermion.

The triangle diagram for one axial vector Aa5µ and twophotons Aµ has an anomaly proportional to the prod-uct of gaψ and q2ψ, where the latter is the charge of thefermion ψ. When summed over all fermion species, theanomaly-cancellation condition is therefore

∑

ψ

q2ψgaψ = 0 (69)

for each a. Multiplying this equation by 〈Aa5µ〉 and sum-ming over a yields the constraint

18

∑

ψ

q2ψbψµ = 0 (70)

on coupling coefficients of the bµ type. Note that, at thelevel of the standard-model extension, the sum over allfermion species would include the leptons and the quarks.Also, in contrast to the usual anomaly-cancellation mech-anism which produces a single condition, Eq. (70) is a setof four constraints. This is a direct consequence of spon-taneous Lorentz breaking, in which for each a the vacuumexpectation value 〈Aa5µ〉 involves four numbers.

Next, we present the results of an explicit calculationof the bµ-linear one-loop corrections to the photon propa-gator involving a fermion of mass m and charge q. Thereare two diagrams to consider, since a factor −ibµγ5γ

µ

can be inserted on either of the two internal lines. Us-ing an argument similar to the standard one proving theFurry theorem, the two diagrams can be shown to giveidentical contributions to the amplitude. Omitting theexternal photon legs, the correction to the two-point am-plitude for a photon of four-momentum pµ then becomes

ωµν(p,m, b) = −2iq2bλ

×∫

d4l

(2π)4Tr[γµSF (l − p)γνSF (l)γ5γ

λSF (l)], (71)

where lµ is the momentum of the fermion in the loop andSF (l) = i(6 l−m+ iǫ)−1 is the usual fermion propagator.

As anticipated above, the expression (71) is related toone appearing in the calculation of the triangular gaugeanomaly. It can directly be verified that

ωµν(p,m, b) ≡ q2bλTµνλ(−p, p) , (72)

where T µνλ(p1, p2) is the standard amplitude for the tri-angle diagram with one axial-vector coupling in conven-tional QED. The full anomaly amplitude T µνλ(p1, p2) canbe regularized in the Pauli-Villars scheme and reduced toa set of integral expressions [35,36]. These can be evalu-ated in closed form for the present case of interest. Forp2 < 4m2, we find

ωµν(p,m, b) =q2bλ2π2

pκǫκλµν

[1 − 4√

(p2/m2)(4 − p2/m2)tan−1

(√p2/m2

4 − p2/m2

)]. (73)

Note that this expression is gauge invariant,

pµωµν = pνω

µν = 0 , (74)

as expected.At this stage, the issue of radiative corrections to kAF

can be addressed. The result (73) is finite. Since no di-vergence cancellation is necessary, a zero value of kAFat tree level is consistent with a renormalizable theory.Moreover, ωµν vanishes for the on-shell condition p2 = 0,as is to be expected in a renormalizable theory without aradiatively induced phase transition. Thus, none of thefinite radiative corrections have the form needed to mod-ify the coefficient kAF , and they are therefore irrelevantto the analysis of cosmic birefringence in subsection IVD.

The above results might make it seem tempting toconclude that there are no bµ-linear one-loop radiativecorrections affecting kF . However, such a conclusionwould be premature. The integral T µνλ(−p, p) in Eq.(72) is superficially linearly divergent. As usual, this in-troduces an ambiguity because a shift in the loop mo-mentum lµ produces a shift in the value of the inte-gral: T µνλ(−p, p) → T µνλ(−p, p) + ζpκǫ

κλµν , where ζis a constant. Certain choices of regularization schemecould therefore generate an additional term of the form

δωµν(p, b) = ζq2bλpκǫκλµν (75)

to the result (73), which would represent a regularization-dependent radiative correction to kAF . Note that this

does not occur in the Pauli-Villars scheme because theterm (75) is mass independent, so in this case the regu-larization automatically subtracts it.

Ambiguities of the general form (75) involving com-binations of the external momenta arise in the stan-dard triangular-anomaly diagram with a finite momen-tum transfer from the two photons to the axial vector.In this case, the ambiguity is conventionally fixed by im-posing U(1) current conservation. However, in the gen-eral context the presence of an ambiguity is independentof the issue of anomalies. Under certain circumstances,anomalies can appear in superficially convergent (non-abelian pentagon) diagrams that are ambiguity-free [37].Also, ambiguities originating in loop-momentum shiftsfor divergent amplitudes other than those associated withanomalies are a standard feature of quantum field theo-ries. For example, the usual vacuum-polarization dia-gram has an ambiguity. Similarly, results such as theFurry theorem rely on a consistent assignment of loopmomenta. Typically, these ambiguities either appear asfinite constant modifications to divergent constants orcan be eliminated by imposing gauge invariance.

A striking feature of the ζ ambiguity is that it ariseswithout an associated divergence and is gauge invariant.It therefore cannot be fixed by the usual methods. Thus,gauge invariance ensures the Ward identities are satis-fied, so vector-current conservation holds for any ζ. Also,the ambiguity fails to produce an anomaly in the axial-current conservation law because there is zero momentum

19

transfer away from the loop at the axial vertex for any ζ.However, the mass independence of the term (75) impliesthat the fermion mass circulating in the loop could inprinciple be arbitrarily large without affecting the valueof ζ, which intuitively seems unphysical and would ap-pear to suggest that ζ must vanish.

In the standard triangular-anomaly diagram, fixingthe ambiguity by requiring vector-current conservationplaces the anomaly in the axial Ward identity. If the axialvector is ungauged, chiral-current conservation is then vi-olated and the anomaly may have physical consequences.An example of this occurs in the decay π → 2γ. If in-stead the axial vector is a gauge field, then the anomalydestroys renormalizability unless the total anomaly con-tribution from all fermion species vanishes. A cancella-tion of this type, which is widely used in model build-ing, implicitly assumes the ambiguity has been fixed ina standard way in all contributing diagrams. This couldbe regarded as a (reasonable) choice made to obtain asatisfactory theory.

If a similar choice is made for the present case, sothat the same regularization scheme is adopted for allthe contributing diagrams and therefore the same con-stant ζ appears in each, then it can be argued that theanomaly cancellation in the underlying theory causes theambiguity to disappear. Thus, suppose as above we as-sume gauged axial vectors in a renormalizable underly-ing theory, so that the anomaly-cancellation condition(70) must hold. Then, the net contribution to the pho-ton propagator from the ambiguous terms is given by∑

ψ ζq2ψbψλpκǫ

κλµν , which vanishes by Eq. (70). This con-firms the conjecture made in the first part of this sub-section: the anomaly cancellation implies the absence ofbµ-linear one-loop radiative corrections to kAF .

Note that this argument presupposes that the axialvectors Aa5µ are gauged and that a consistent choice ofregularization is used. To demonstrate the absence ofnegative-energy contributions to the theory at this level,it suffices that a natural procedure of this kind exists. Ifan ambiguity ζ did remain in the theory, it would seemto suggest that at the quantum level there would be aspectrum of physically allowed theories. The issue of de-termining the correct one would then become experimen-tal, much as the values of the renormalized couplings andmasses are experimentally determined. However, in thepresent case there are both theoretical and experimentalreasons to believe that ζ vanishes.

We next address some issues arising in higher pertur-bation orders. Consider first the case of the photon prop-agator in the extended QED. At any loop order but withonly one CPT-violating insertion of bµ, all diagrams aresuperficially divergent and hence can be expected to haveambiguities. In parallel with the previous case, these di-agrams can be related to higher-loop three-point trian-gle diagrams with one axial-vector and two photons onthe external legs. The Adler-Bardeen theorem [38] shows

that the anomalies arising from the one-loop triangle di-agram are unaffected at higher loops. This implies thatthe constraint (70) holds at arbitrary loop order. How-ever, it follows as before that the total ambiguity is pro-portional to this constraint and so vanishes. If this argu-ment holds, then there can be no bµ-linear contributionsto kAF at any order in the fine-structure constant.

Diagrams that involve higher-order Lorentz viola-tion may also be of potential theoretical importance.Their transformation properties under discrete symme-tries place strong constraints on their possible contribu-tions to kAF , as in the lowest-order case. For example,in the extended QED at the quadratic level of Lorentzviolation, only a product of the coefficients bµ and cµνcan appear. At the one-loop level, all higher-order dia-grams are related to polygonal diagrams in the underly-ing theory that couple two photons to a variety of vector,axial-vector, and tensor fields. At least one factor of bµis required, so a chiral coupling must be involved and acancellation mechanism may still apply. The implicationof the consistency of the underlying theory for correctionsto kAF at all orders in Lorentz violation and includingpossible higher-loop corrections remains an open issue.We remark, however, that the effects at the cubic lev-els and above are at most of theoretical interest as theywould be well below experimental detection for the levelsof Lorentz violation considered in the present work.

At the level of the standard-model extension, a possi-ble lowest-order one-loop correction to the photon prop-agator could in principle also arise from the coefficient k2

when aW+-W− pair circulates in the loop. Indeed, therewould be a contribution from insertions on the gauge-boson lines and another, related to the first by gaugeinvariance, involving a modified vertex. However, if theterm involving k2 were to exist, it would exhibit difficul-ties with negative contributions to the energy, as doeskAF . One option is therefore that k2 vanishes at treelevel, which eliminates the possible associated radiativecorrections to the term involving kAF . An issue thenarises concerning possible radiatively induced contribu-tions to k2 from the fermion sector. We conjecture that,if all the terms k0, k1, k2, k3 vanish at tree level, thenno radiative corrections to any of these coefficients arisefrom the fermion sector. An anomaly-cancellation mech-anism would again play a role, although nonabelian fieldswould now be involved and so the singlet and nonabeliananomalies would also be relevant. The situation in thestandard-model extension at higher loops remains open.

Finally, we present a few remarks about the possibil-ity of radiative corrections to a hypothetical linear termof the form −(kA)κA

κ in the absence of a photon mass.This term is C and CPT odd. In the extended QED, onlyterms of the type aµ have this symmetry. As discussed insection III, a field redefinition can be used to eliminatethese (flavor-diagonal) terms, and hence they are unob-servable in any experiment. Since the electromagnetic in-

20

teractions are C even, at lowest order in Lorentz-violatingcoefficients there cannot be any radiative corrections tokA at any order in QED loops. Any contributions thatmight arise at higher orders in Lorentz violation wouldagain be related to polygonal diagrams in the underly-ing theory. It would be of some theoretical interest toinvestigate the possible contributions to these terms.

B. CPT-Even Terms

In contrast to the situation for the CPT-odd terms,a nonzero tree-level value for the CPT-even term (24)with coefficient kF presents no immediate theoretical dif-ficulty. We have shown in section IV D that it is ex-perimentally feasible to place relatively tight bounds onkF from measurements of cosmological birefringence, al-though this has not yet been done and the wavelengthdependence may result in constraints somewhat weakerthan those on kAF . Nonetheless, the attainable limitson kF are of interest because they might in principle besufficiently sharp to be sensitive to effects at a scale com-parable to finite radiative corrections from the fermionsector. It is therefore of interest to determine whetherthe coupling kF must be present for renormalizabilityand, if so, which fermion-sector coupling coefficients areinvolved. In this subsection, we investigate this issue inthe context of the extended QED.

At the one-loop level and to leading order in Lorentzviolation, the possible radiative contributions to the co-efficient kF in the term (24) are significantly constrainedby the requirements of discrete symmetries. This term isboth C and CPT even, and an inspection shows that theonly other type of term in the fermion sector with theseproperties is the term with coefficient cµν in Eq. (27). Itcontributes both on the loop through fermion-line inser-tions with a derivative and at the vertices through theextra gauge coupling.

The form of the cµν -linear correction ωµν(p,m, c) tothe two-point amplitude for a photon of four-momentumpµ is strongly constrained by its discrete-transformationproperties, observer Lorentz covariance, and the require-ments (74) of gauge invariance. Thus, invariance underCPT implies ωµν(p,m, c) is an even function of pµ. Also,by virtue of the definition of the photon propagator asa vacuum expectation value of a time-ordered product,ωµν(p,m, c) is symmetric under the combination of a signchange pλ → −pλ of the momentum and an interchangeµ ↔ ν of the spacetime indices. These conditions implythat the correction to the photon propagator at any or-der in the fine-structure constant but at linear order incµν must take the form

ωµν(p,m, c) = icαβ

(Agαβ(p2gµν − pµpν)

+B[(p2gαµgβν − gαµpβpν − gανpβpµ)

+(α↔ β)]

+Cgµνpαpβ +D

m2pαpβpµpν

). (76)

Here, A, B, C, and D are (possibly divergent) scalarfunctions of p2/m2 obeying the relationship

C − 2B +p2

m2D = 0 (77)

to ensure gauge invariance.Some information about photon propagation under

specified circumstances can be deduced from Eq. (76)under the assumption that the scalar functions A, B, C,D have been regularized as needed and divergent contri-butions have been removed by the renormalization pro-cedure. For example, in the case of cosmological bire-fringence of interest in subsection IV D, the photon mo-mentum can be taken as on shell and the Lorentz gaugecondition can be applied. In Eq. (76), this corresponds tosetting to zero both p2 and the momentum factors pµ andpν with specific indices µ and ν. This leaves only the termcαβC(0)gµνpαpβ . This is precisely of the form needed forradiative corrections to the coefficient kF , which can thusbe seen to be governed in this gauge by the on-shell valueof C.

To obtain the explicit result and as a check on therenormalization procedure when Lorentz violations areinvolved, we have directly performed the one-loop calcu-lation. This also verifies the structure of Eq. (76). Theterms in Eq. (27) associated with the coupling coeffi-cient cµν lead to four new cµν -linear one-loop vacuum-polarization diagrams. The possibility of fermion-lineinsertions arising from the derivative coupling leads totwo diagrams, each with one insertion on one of the twointernal fermion lines. The appearance of modified ver-tices from the extra gauge coupling leads to another two,each with one normal and one modified vertex. Thesetwo types of contribution are related by gauge invari-ance. Indeed, we anticipate this gauge invariance leadsto Ward-type identities valid at arbitrary loop order, al-though an explicit demonstration of this remains an openissue.

The sum of the four additional diagrams generates aone-loop correction to the photon propagator of

ωµν(p,m, c) = iq2cαβ

×∫

d4l

(2π)4

{lβTr[γµSF (l − p)γνSF (l)γαSF (l)]

+(l− p)βTr[γµSF (l − p)γαSF (l − p)γνSF (l)]

−igβµTr[γαSF (l − p)γνSF (l)]

−igβνTr[γµSF (l − p)γαSF (l)]

}, (78)

where the first two terms arise from line insertions andthe last two from the modified vertices.

21

The integral in Eq. (78) is superficially quadraticallydivergent. It has the standard ambiguity, arising fromthe possibility of shifting the integration variable, thatis (largely) fixed by imposing gauge invariance. The de-nominators arising from the fermion propagators SF canbe combined with the usual Feynman parametrization.

All the necessary shifts performed in the resulting inte-gration variables must be the same, so that the contri-butions from the surface terms remain gauge invariant.To accomplish this, it is convenient to separate ωµν intotwo pieces that can be parametrized so as to maintainthe equivalence of shifts.

We define ωµν = ωµν(1) + ωµν(2), where

ωµν(1) = 4q2cαβ

∫d4l

(2π)41

∆

{gµν

[(l − p)αlβ + (α↔ β)

]+[gµβ [l · (l − p) −m2]gνα + (µ↔ ν)

]

−([gµα[lβ(l − p)ν + (β ↔ ν)] + (µ ↔ ν)

]+ (α↔ β)

)}(79)

and

ωµν(2) = 8q2cαβ

∫d4l

(2π)41

∆2

{(lαlβ[(l − p)2 −m2] + (l − p)α(l − p)β(l2 −m2)

)

×(m2gµν − l · (l − p)gµν + (l − p)µlν + lµ(l − p)ν

)}. (80)

In these expressions, ∆ = (l2 −m2)[(l − p)2 −m2]. Thesame shift is introduced in all the integrals, thereby pre-serving gauge invariance, with the substitution in Eq.(79) of

1

∆=

∫ 1

0

dz1

[k2 + z(1 − z)p2 −m2]2(81)

and the substitution in Eq. (80) of

1

∆2=

∫ 1

0

dz6z(1 − z)

[k2 + z(1 − z)p2 −m2]4, (82)

where k = l− pz is the new integration variable.The divergences in the resulting integrals can be

treated using dimensional regularization in D = 4 − ǫdimensions. Performing various partial integrations, weobtain for p2 < 4m2 a radiative correction of the form(76) with

A = −B = ω(p2/m2) ,

C = −2ω(p2/m2) − p2

m2D ,

D = 2∂

∂(p2/m2)ω(p2/m2) . (83)

Here, ω(p2/m2) is the standard vacuum-polarization re-sult, given by

ω(p2/m2) =q2

4π2

(1

3

(2

ǫ− γ

)

−2

∫ 1

0

dz z(1 − z) ln[1 − z(1 − z)(p2/m2)]

), (84)

where γ is the Euler constant. Note that the results (83)satisfy the gauge-invariance condition (77).

The above calculation shows that the scalar functionC contains a momentum-independent divergence. As de-scribed above, the on-shell value of C determines thecoefficient kF in Lorentz gauge, so the appearance ofthis divergence shows that a bare coefficient kF mustbe present in the original theory for renormalizability.The renormalization procedure then removes the infiniteand (ambiguous) constant pieces, leaving a physical coef-ficient kF (to be determined by experiment) and a set offinite radiative corrections governed by the ratio p2/m2.

We have seen in subsection IV D that a nonzerovalue of the coefficient kF induces cosmological birefrin-gence. The above calculation shows that imposing a zerovalue of this coefficient at tree level is incompatible withrenormalizability. It is therefore reasonable to expect anonzero physical value of kF . Although nonrigorous, aheuristic argument might also be used to provide a rela-tionship between the physical values of kF and cµν : forconsistency of perturbation theory, it is plausible that thephysical value of kF should be larger than the expectedfinite quantum corrections of order αcµν , where α is thefine-structure constant. If kF is eventually bounded toabout 10−27 as estimated in subsection IV D, then thiswould suggest the components of cµν might be expectedto be smaller than about 10−25.

As in the CPT-odd case, the momentum-dependentradiatively induced corrections in Eq. (83) are irrelevantin the context of cosmological birefringence. However,these radiative corrections do modify the off-shell propa-gator and might therefore be expected to generate smalleffects under suitable circumstances. For example, theremay be contributions to electromagnetic scattering crosssections, governed by the ratio αme/MP ≈ 10−25. Sim-ilarly, a small correction to the Coulomb law might ap-

22

pear. These issues lie beyond the scope of the presentwork.

In addition to the cµν -linear one-loop contribution ob-tained above, there are also cµν -linear higher-loop cor-rections in the extended QED. The general structure ofthe contributions at any loop order is given by Eq. (76).Although the detailed form of the scalar functions A, B,C, and D will differ, the physically relevant correctionsshould also depend on p2/m2, since any terms indepen-dent of p are expected to be absorbed by the renormal-ization procedure. The above conclusions about cosmo-logical birefringence are therefore likely to remain valid.Effects from higher-order Lorentz violation should alsoarise in the extended QED but are probably of a size thatis physically irrelevant. Much of the above discussionshould also hold for radiative effects in the full standard-model extension. In this context, note that off-diagonalterms in generation space cannot contribute at leadingorder.

VI. SUMMARY

In this paper, we presented a general Lorentz-violatingextension of the minimal SU(3) × SU(2)× U(1) standardmodel including both CPT-even and CPT-odd terms,and we discussed some of its theoretical and experimen-tal properties. The analysis was performed within thecontext of a framework previously described [2], which isbased on spontaneous Lorentz and CPT violation occur-ring in an underlying theory of nature.

Despite the existence of terms causing a certain typeof Lorentz breaking, the resulting theory preserves var-ious desirable features of standard quantum field theo-ries such as gauge invariance, energy-momentum conser-vation, observer Lorentz invariance, hermiticity, the va-lidity of conventional quantization methods, and power-counting renormalizability. Other important featuressuch as positivity of the energy, microcausality, and theusual anomaly cancellation are also expected. We havedemonstrated that the usual breaking of SU(2) × U(1)symmetry to the electromagnetic U(1) is maintained, al-though the expectation value of the Higgs is slightlychanged and the Z0 field acquires a small expectationvalue. The theory presented here appears at present to bethe sole candidate for a consistent extension of the stan-dard model providing a microscopic theory of Lorentzviolation.

We have extracted extensions of several of the con-ventional varieties of QED by considering limiting casesof the standard-model extension. Part of the motiva-tion for investigating extended QED is the existence ofhigh-precision tests of Lorentz and CPT invariance thatinvolve electrodynamics. A summary was provided ofsome recent studies of possible experimental constraints.

Another major focus of this work is the effect of theLorentz violations on the photon sector. The generalpure-photon lagrangian can be written in a form con-taining only two additional terms, one CPT odd and oneCPT even. This lagrangian and the associated energy-momentum tensor were discussed, and it was found thatthe CPT-even component has positive conserved energybut that in the absence of a photon mass the CPT-oddcomponent can generate negative contributions to the en-ergy. Despite this theoretical difficulty, the two termswere retained for the whole analysis so that the implica-tions for the full quantum theory could be examined.

The equations of motion generalizing the Maxwellequations in the presence of Lorentz violation were ob-tained, and their solution was outlined both using po-tentials and using fields. Some technical complicationsarise relative to the case of conventional electrodynam-ics in vacuo, but they can largely be overcome. A keyfeature is that, although there are still two independentpropagating degrees of freedom, in the typical situationthe two modes obey different dispersion relations. Thisimplies a variety of interesting effects, including birefrin-gence in the vacuum. We presented a few quantitativeanalogies with crystal optics and showed that the pres-ence of Lorentz violation means that the vacuum as ex-perienced by an electromagnetic wave behaves like anoptically anisotropic and gyrotropic transparent crystalexhibiting spatial dispersion of the axes.

A variety of terrestrial, astrophysical and cosmologicalbounds on photon properties are known. Sharp experi-mental limits on the photon sector of the extended QEDcan be obtained from the absence of birefringence on cos-mological scales. It has been shown in earlier work [22]that the problematic CPT-odd term is experimentallylimited to scales comparable to the Hubble length.

A significant result of this paper is that most of thecomponents of the CPT-even term could also be boundedexperimentally from cosmological birefringence with ex-isting techniques. This case is particularly interestingas it has no evident theoretical difficulties and appearsto have been overlooked in the previous literature. Also,unlike the CPT-odd term, the CPT-even contribution ex-hibits a dependence on wavelength that might provide auseful signature of the effect. We have crudely estimatedthe attainable bounds, which would be sensitive to sup-pressed Lorentz violation in the general range consideredhere.

The paper also contains a series of consistency checkson the theory, primarily at the level of one-loop radiativecorrections. We discussed the cancellation of the variousconventional anomalies in the standard-model extensionand considered other anomaly cancellations that mightoccur in the underlying theory. The latter were used toobtain a constraint on a set of coupling coefficients forLorentz violation in the standard-model extension.

We have investigated the feasibility of imposing tree-

23

level vanishing of the problematic CPT-odd terms, inlight of possible radiative corrections that could be in-duced from the non-photon sectors in the standard-modelextension. We have shown that the radiative correctionsat one loop are finite, so it is unnecessary at this level ofrenormalization to include a CPT-odd term in the orig-inal theory. The finite corrections are gauge invariantbut ambiguous, a situation somewhat reminiscent of theusual anomaly calculations. However, if the theory un-derlying the standard-model extension is anomaly free,the CPT-odd effects in the photon sector can be ne-glected at this level. Generalizations of this argumentmay apply at higher loops.

For the CPT-even sector, we have demonstrated byexplicit one-loop calculation that divergent radiative cor-rections appear. A term of this type therefore must bepresent in the original theory. An experimental search forthe associated renormalized coupling based, for example,on cosmological birefringence could be performed. It isremarkable that physics associated with the Planck scalemight produce observable effects in measurements madeat the largest scales in the Universe.

ACKNOWLEDGMENTS

We thank R. Bluhm and P. Kronberg for discussions.This work was supported in part by the United StatesDepartment of Energy under grant number DE-FG02-91ER40661.

[1] V.A. Kostelecky and S. Samuel, Phys. Rev. Lett. 63

(1989) 224; ibid., 66 (1991) 1811; Phys. Rev. D 39 (1989)683; ibid., 40 (1989) 1886; V.A. Kostelecky and R. Pot-ting, Nucl. Phys. B 359 (1991) 545; Phys. Lett. B 381

(1996) 89.[2] D. Colladay and V.A. Kostelecky, Phys. Rev. D 55

(1997) 6760.[3] V.A. Kostelecky and R. Potting, in D.B. Cline, ed.,

Gamma Ray–Neutrino Cosmology and Planck Scale

Physics (World Scientific, Singapore, 1993) (hep-th/9211116); Phys. Rev. D 51 (1995) 3923; D. Colladayand V. A. Kostelecky, Phys. Lett. B 344 (1995) 259;Phys. Rev. D 52 (1995) 6224; V.A. Kostelecky and R.Van Kooten, Phys. Rev. D 54 (1996) 5585.

[4] B. Schwingenheuer et al., Phys. Rev. Lett. 74 (1995)4376; L.K. Gibbons et al., Phys. Rev. D 55 (1997) 6625;R. Carosi et al., Phys. Lett. B 237 (1990) 303.

[5] OPAL Collaboration, R. Ackerstaff et al., Z. Phys. C76 (1997) 401; DELPHI Collaboration, M. Feindt et al.,preprint DELPHI 97-98 CONF 80 (July 1997).

[6] V.A. Kostelecky, Phys. Rev. Lett. 80 (1998) 1818.

[7] P.B. Schwinberg, R.S. Van Dyck, Jr., and H.G. Dehmelt,Phys. Lett. A 81 (1981) 119; Phys. Rev. D 34 (1986) 722;L.S. Brown and G. Gabrielse, Rev. Mod. Phys. 58 (1986)233; R.S. Van Dyck, Jr., P.B. Schwinberg, and H.G.Dehmelt, Phys. Rev. Lett. 59 (1987) 26; G. Gabrielseet al., ibid., 74 (1995) 3544.

[8] R. Bluhm, V.A. Kostelecky and N. Russell, Phys. Rev.Lett. 79 (1997) 1432; Phys. Rev. D 57 (1998) 3932.

[9] O. Bertolami et al., Phys. Lett. B 395 (1997) 178.[10] R. Bluhm, V.A. Kostelecky and N. Russell, preprint

IUHET 388 (1998).[11] M. Charlton et al., Phys. Rep. 241 (1994) 65; J. Eades,

ed., Antihydrogen, J.C. Baltzer, Geneva, 1993.[12] V.A. Kostelecky and C.D. Lane, in preparation.[13] V.W. Hughes, H.G. Robinson, and V. Beltran-Lopez,

Phys. Rev. Lett. 4 (1960) 342; R.W.P. Drever, Philos.Mag. 6 (1961) 683; J.D. Prestage et al., Phys. Rev. Lett.54 (1985) 2387; S.K. Lamoreaux et al., ibid., 57 (1986)3125; T.E. Chupp et al., ibid., 63 (1989) 1541.

[14] P.A.M. Dirac, Proc. Roy. Soc. Lon. 209 (1951) 291;W. Heisenberg, Rev. Mod. Phys. 29 (1957) 269; P.G.O.Freund, Acta Physica Austriaca 14 (1961) 445; J.D.Bjorken, Ann. Phys. 24 (1963) 174; I. Bialynicki-Birula,Phys. Rev. 130 (1963) 465; G.S. Guralnik, Phys. Rev.136 (1964) B1404; Y. Nambu, Prog. Theor. Phys. Suppl.Extra (1968) 190; T. Eguchi, Phys. Rev. D 14 (1976)2755.

[15] H.B. Nielsen and I. Picek, Nucl. Phys. B 211 (1983) 269.[16] S.W. Hawking, Phys. Rev. D 14 (1976) 2460; Commun.

Math. Phys. 87 (1982) 395.[17] J. Ellis, J.L. Lopez, N.E. Mavromatos, and D.V.

Nanopoulos, Phys. Rev. D 53 (1996) 3846.[18] For a review, see, for example, C.M. Will, Theory and Ex-

periment in Gravitational Physics, Cambridge UniversityPress, Cambridge, England, 1993.

[19] Note that the definitions of these terms differ slightlyfrom those in Ref. [2].

[20] See, for example, D.W. Hertzog et al., to appear in P.Nath, ed., Proceedings of the 1998 International Confer-

ence on Particles, Fields and Cosmology.

[21] See, for example, R.M. Barnett et al., Review of ParticleProperties, Phys. Rev. D 54 (1996) 1.

[22] S.M. Carroll, G.B. Field, and R. Jackiw, Phys. Rev. D41 (1990) 1231.

[23] H.A. Lorentz, Verh. Kon. Akad. Wetensch. 18 (1878).[24] See, for example, M. Born and E. Wolf, Principles of

Optics, 5th Ed., Pergamon Press, New York, 1975; L.D.Landau, E.M. Lifshitz and L.P. Pitaevskii, Electrodynam-

ics of Continuous Media, 2nd Ed., Pergamon Press, NewYork, 1984.

[25] A. Sommerfeld, Ann. Physik 44 (1914) 177; L. Brioullin,Ann. Physik 44 (1914) 203.

[26] See, for example, R.C. Weast, ed., Handbook of Chem-

istry and Physics, 56th Ed., CRC Press, Cleveland, Ohio,1975.

[27] W.-T. Ni, Phys. Rev. Lett. 38 (1977) 301.[28] The existing limit on the photon mass based on direct

measurements is mγ < 6× 10−25 GeV. See L. Davis, Jr.,A.S. Goldhaber and M.M. Nieto, Phys. Rev. Lett. 35

(1975) 1402. This bound, obtained from observation ofthe Jovian magnetosphere during the Pioneer-10 mission,

24

might eventually be improved through measurements onthe solar magnetosphere. See V.A. Kostelecky and M.M.Nieto, Phys. Lett. B 317 (1993) 223.

[29] P. Haves and R.G. Conway, Mon. Not. R. Astr. Soc. 173

(1975) 53P; J.N. Clarke, P.P. Kronberg and M. Simard-Normandin, ibid., 190 (1980) 205.

[30] B. Nodland and J.P. Ralston, Phys. Rev. Lett. 78 (1997)3043; Phys. Rev. Lett. 79 (1997) 1958; astro-ph/9706126.

[31] S.M. Carroll and G.B. Field, Phys. Rev. Lett. 79 (1997)2394; D.J. Eisenstein and E.F. Bunn, Phys. Rev. Lett.79 (1997) 1957; J.P. Leahy, astro-ph/9704285; J.F.C.Wardle, R.A. Perley and M.H. Cohen, Phys. Rev. Lett.

79 (1997) 1801; T.J. Loredo, E.E. Flanagan and I.M.Wasserman, Phys. Rev. D 56 (1997) 7507.

[32] A.P. Lightman and D.L. Lee, Phys. Rev. D 8 (1973) 64.[33] S. Coleman and S. Glashow, Phys. Lett. B 405 (1997)

249.[34] See, for example, C.Q. Geng and R.E. Marshak, Com-

ments Nucl. Part. Phys. 18 (1989) 331.[35] L. Rosenberg, Phys. Rev. 129 (1963) 2786.[36] S.L. Adler, Phys. Rev. 177 (1969) 2426.[37] W.A. Bardeen, Phys. Rev. 184 (1969) 1848.[38] S.L. Adler and W.A. Bardeen, Phys. Rev. 182 (1969)

1517.

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