# Lorentz-Violating Extension of the Standard Model

D. Colladay and V. Alan Kostelecký Physics Department, Indiana University, Bloomington, IN 47405, U.S.A.
preprint IUHET 359 (1997); accepted for publication in Phys. Rev. D
###### Abstract

In the context of conventional quantum field theory, we present a general Lorentz-violating extension 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 theory with Lorentz-covariant dynamics in which spontaneous Lorentz violation occurs. The extension has gauge invariance, energy-momentum conservation, and covariance under observer rotations and boosts, while covariance under particle rotations and boosts is broken. The quantized theory is hermitian and power-counting renormalizable, and other desirable features such as microcausality, positivity of the energy, and the usual anomaly cancellation are expected. Spontaneous symmetry breaking to the electromagnetic U(1) is maintained, although the Higgs expectation is shifted by a small amount relative to its usual value and the field acquires a small expectation. A general Lorentz-breaking extension of quantum electrodynamics is extracted from the theory, and some experimental tests are considered. In particular, we study modifications to photon behavior. One possible effect is vacuum birefringence, which could be bounded from cosmological observations by experiments 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 levels suggested by Planck-scale physics and accessible to other terrestrial laboratory experiments.

###### pacs:
PACS numbers: 11.30.Er, 12.60.-i, 12.20.Fv, 41.20.Jb

## I Introduction

The minimal SU(3)SU(2)U(1) standard model, although phenomenologically successful, leaves unresolved a variety of issues. It is believed to be the low-energy limit of a fundamental theory that also provides a quantum description of gravitation. An interesting question is whether any aspects of this underlying theory could be revealed through definite experimental signals accessible with present techniques.

The natural scale for a fundamental theory including gravity is governed by the Planck mass , which is about 17 orders of magnitude greater than the electroweak scale associated with the standard model. This suggests that observable experimental signals from a fundamental theory might be expected to be suppressed by some power of the ratio . Detection of these minuscule effects at present energy scales would be likely to require experiments of exceptional sensitivity, preferably ones seeking to observe a signal forbidden in conventional renormalizable gauge theories.

To identify signals of this type, one approach is to examine proposed fundamental theories for effects that are qualitatively different from standard-model physics. For example, at present the most promising framework for a fundamental theory is string (M) theory. The qualitative difference between particles and strings means that qualitatively new physics is expected at the Planck scale. An interesting challenge would be to determine whether this could lead to observable low-energy effects.

In the present work, we consider the possibility that the new physics involves a violation of Lorentz symmetry. It has been shown that spontaneous Lorentz breaking may occur in the context of string theories with Lorentz-covariant dynamics . Unlike the conventional standard model, string theories typically involve interactions that could destabilize the naive vacuum and trigger the generation of nonzero expectation values for Lorentz tensors. Note that some kind of spontaneous breaking of the higher-dimensional Lorentz symmetry is expected in any realistic Lorentz-covariant fundamental theory involving more than four spacetime dimensions. If the breaking extends into the four macroscopic spacetime dimensions, apparent Lorentz violation could occur at the level of the standard model. This would represent a possible observable effect from the fundamental theory, originating outside the structure of conventional renormalizable gauge models.

A framework has been developed for treating the effects of spontaneous Lorentz breaking in the context of a low-energy effective theory , where certain terms can be induced that appear to violate Lorentz invariance explicitly. It turns out that, from a theoretical perspective, the resulting effects are comparatively minimal.

An important point is that Lorentz symmetry remains a property of the underlying fundamental theory because the breaking is spontaneous. This implies that various attractive features of conventional theories, including microcausality and positivity of the energy, are expected to hold in the low-energy effective theory. Also, energy and momentum are conserved as usual, provided the tensor expectation values in the fundamental theory are spacetime-position independent. Moreover, standard quantization methods are unaffected, so a relativistic Dirac equation and a nonrelativistic Schrödinger equation emerge in the appropriate limits.

Another important aspect of the spontaneous breaking is that both the fundamental theory and the effective low-energy theory remain invariant under observer Lorentz transformations, i.e., rotations or boosts of an observer’s inertial frame . The presence of nonzero tensor expectation values in the vacuum affects only invariance properties under particle Lorentz transformations, i.e., rotations or boosts of a localized particle or field that leave unchanged the background expectation values.

This framework for treating spontaneous Lorentz violation has been used to obtain a general extension of the minimal SU(3) SU(2) U(1) standard model that violates both Lorentz invariance and CPT . In addition to the desirable features of energy-momentum conservation, observer Lorentz invariance, conventional quantization, hermiticity, and the expected microcausality and positivity of the energy, this standard-model extension maintains gauge invariance and power-counting renormalizability. It would emerge from any fundamental theory (not necessarily string theory) that generates the standard model and contains spontaneous Lorentz and CPT violation.

The present work continues our previous theoretical investigations of spontaneous Lorentz and CPT breaking. Working first at the level of the standard model, we provide explicitly in section II the full Lorentz-violating extension, including the CPT-even Lorentz-breaking terms described implicitly in Ref. . We also give some details of the modifications to the usual electroweak symmetry breaking.

Since many sensitive measures of Lorentz and CPT symmetry involve tests of quantum electrodynamics (QED), it is useful to extract from the standard-model extension a generalized QED that allows for possible Lorentz and CPT violations. This extended QED, given in section III, involves modifications of the usual QED in both the fermion and the photon sectors. Some comments are also given in section III about the implications of this theory for experimental tests with electrons and positrons.

In the remainder of this paper, we focus primarily on the photon sector of the extended QED, presenting a study of the theoretical and experimental implications of the modifications to photon properties arising from the possible Lorentz and CPT violations. Section IV discusses changes in the basic theory, including the modified Maxwell equations and properties of their solutions. One possible effect is vacuum photon birefringence, and some associated features are described. We show that feasible measurements limiting birefringence on cosmological scales could tightly constrain the Lorentz-violating terms. In section V, some important consistency checks on the theory at the level of radiative corrections are presented, largely at the one-loop level. The types of Lorentz violation that can be affected by radiative corrections are identified, and explicit calculations are given. We show that the effects are compatible with spontaneous Lorentz and CPT violation in the fermion sector at levels accessible to other QED experiments.

Since the standard-model extension provides a quantitative microscopic theory of Lorentz and CPT violation, it is feasible to identify potentially observable signals and to establish bounds from various experiments other than ones in the photon sector. Numerous tests of Lorentz invariance and CPT exist. The present theory provides a single coherent framework at the level of the standard model and QED that can be used as a basis for the analysis and comparison of these tests. Although many experiments are insensitive to the suppressed effects motivating our investigation, certain high-precision ones might have observable signals within this framework. In particular, the results in the present paper have been used to examine possible bounds on CPT and Lorentz violations from measurements of neutral-meson oscillations [3, 4, 5, 6], from tests of QED in Penning traps [7, 8], and from baryogenesis . Several other investigations are underway, including a study  of possible Lorentz and CPT effects on hydrogen and antihydrogen spectroscopy  and another  of limits attainable in clock-comparison experiments .

The analyses of the standard-model and QED extensions performed in the present work leave unaddressed a number of significant theoretical issues arising at scales between the electroweak mass and the Planck mass. These include the ‘dimension problem’ of establishing whether spontaneous Lorentz breaking in the fundamental theory near the Planck scale indeed extends to the four physical spacetime dimensions and, if so, the mechanism for its suppression or, if not, why exactly four spacetime dimensions are spared. Other issues include the effects of mode fluctuations around the tensor expectation values and possible constraints and effects arising from a nonminimal standard model or (super)unification below the Planck scale.

Another potentially important topic is the implication of spontaneous Lorentz violation for gravity at observable energies. Like the usual standard model, the standard-model extension considered here disregards gravitational effects. The particle Lorentz symmetry that is broken in this theory is therefore effectively a global symmetry, and so one might expect Nambu-Goldstone modes. Since gravity is associated with local Lorentz invariance, it is natural to ask about the role of these modes in a version of the standard-model extension that includes gravity. In a gauge theory, when a suitable scalar acquires a nonzero expectation value, the Higgs mechanism occurs: the propagator for the gauge boson is modified, and a mass is generated. Similarly, in a theory with gravitational couplings, when a Lorentz tensor acquires a nonzero expectation value, the graviton propagator can be modified. However, no mass for the graviton is induced because the gravitational connection is related to the derivative of the metric rather than to the metric itself . In this sense, there is no gravitational Higgs effect.

The theory described here appears at present to be the sole candidate for a consistent extension of the standard model providing a microscopic theory of Lorentz violation. A complete review of alternative approaches to possible Lorentz and CPT violation lies beyond the scope of this paper. Works known to us of relevance in the present context are referenced in the body of the text below. Among other ideas in the literature are several distinctive ones developed from perspectives very different from ours. Following early work by Dirac and Heisenberg, several authors have considered an unphysical spontaneous Lorentz breaking in an effort to interpret the photon as a Nambu-Goldstone boson . Nielsen and his colleagues have suggested the converse of the philosophy in the present work: that the observed Lorentz symmetry in nature might be a low-energy manifestation of a fundamental theory without Lorentz invariance. A discussion of this idea and a brief review of the literature on Lorentz breaking prior to the establishment of the usual minimal standard model may be found in Ref. . Hawking has suggested  the possibility that conventional quantum mechanics is invalidated by gravitational effects and that this might lead to CPT violation, among other effects. The implications for experiments in the kaon system  are known to be entirely different from those arising in the present standard-model extension, which is based on conventional quantum theory. There is also a body of literature pertaining to unconventional theories of gravity (without standard-model physics), among which are some models containing various possible sources of local Lorentz violation .

## Ii Standard-Model Extension

In this section, we extend the minimal standard model by adding all possible Lorentz-violating terms that could arise from spontaneous symmetry breaking at a fundamental 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 in Ref.  but are also included here for completeness.

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

The part acting as a coupling coefficient carries spacetime indices reflecting the properties under observer Lorentz transformations of the relevant nonzero expectation values from the fundamental theory. The coupling coefficient may be complex, but it is constrained by the requirement that the lagrangian be hermitian. For a coupling coefficient with an even number of spacetime indices, the pure trace component is irrelevant for present purposes because it maintains Lorentz invariance. A coupling coefficient of this type can therefore be taken traceless.

The field part may involve covariant derivatives and, if fermions are involved, gamma matrices. Gauge invariance requires that the field part be a singlet under SU(3) SU(2) U(1), while power-counting renormalizability implies that it must have mass dimension no greater than four. The requirement that the standard-model extension originates from spontaneous Lorentz breaking in a covariant fundamental theory implies the whole Lorentz-violating term must be a singlet under observer Lorentz transformations, so the field part must have indices matching those of the coupling coefficient.

Following the discussion in the introduction, all coupling coefficients are assumed to be heavily suppressed by some power of the ratio of the light scale to the Planck scale. In the absence of a satisfactory explanation of the suppression mechanism, it would seem premature to attempt specific detailed predictions about the relative sizes of different coupling coefficients. As a possible working hypothesis, one might attribute comparable suppression factors to all terms at the level of the standard-model extension. Note that a term with field part having mass dimension must have a coupling coefficient with mass dimension , and the relevant scale for these effects is roughly the Planck mass. The hypothesis would therefore suggest that in the low-energy theory a term with field part of mass dimension would have coupling coefficient suppressed by an additional power of relative to the coefficient of a field term of mass dimension . This scheme would be compatible with interpreting the standard model as an effective field theory, in which each additional derivative coupling would involve an additional suppression factor in the coupling coefficient. It would imply a distinct hierarchy among the coupling coefficients introduced below, and would suggest that certain derivative couplings could be neglected relative to comparable nonderivative ones. However, since this hypothesis presently has no basis in a detailed theory, in what follows we have chosen to retain on an equal footing all renormalizable terms compatible with the gauge symmetries of the standard model and with an origin in spontaneous Lorentz breaking.

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

 LA=(νAlA)L,RA=(lA)R,
 QA=(uAdA)L  ,  UA=(uA)R  ,  DA=(dA)R, (1)

where

 ψL≡12(1−γ5)ψ,ψR≡12(1+γ5)ψ, (2)

as usual, and where labels the flavor: , , , . We denote the Higgs doublet by , and in unitary gauge we represent it as

 ϕ=1√2(0rϕ). (3)

The conjugate doublet is . The SU(3), SU(2), and U(1) gauge fields are denoted by , , and , respectively. The corresponding field strengths are , , and , with the first two understood to be hermitian adjoint matrices while is a hermitian singlet. The corresponding couplings are , , and . The electromagnetic U(1) charge and the angle are defined through , as usual. The covariant derivative is denoted by , and . The Yukawa couplings are , , . Throughout most of this work we use natural units, which could be obtained from the SI system by redefining , and we adopt the Minkowski metric with .

The complete lagrangian for the Lorentz-breaking standard-model extension can be separated into a sum of terms. For completeness, we first provide the lagrangian terms in the usual SU(3) SU(2) U(1) minimal standard model:

 Llepton=12i¯¯¯¯LAγμ↔DμLA+12i¯¯¯¯RAγμ↔DμRA, (4)
 Lquark = 12i¯¯¯¯QAγμ↔DμQA+12i¯UAγμ↔DμUA (5) +12i¯¯¯¯¯DAγμ↔DμDA,
 LYukawa = −[(GL)AB¯¯¯¯LAϕRB+(GU)AB¯¯¯¯QAϕcUB (6) +(GD)AB¯¯¯¯QAϕDB]+h.c.,
 LHiggs=(Dμϕ)†Dμϕ+μ2ϕ†ϕ−λ3!(ϕ†ϕ)2, (7)
 Lgauge=−12Tr(GμνGμν)−12Tr(WμνWμν)−14BμνBμν . (8)

The usual terms have been omitted, and possible analogous total-derivative terms that break Lorentz symmetry are 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 or odd and whether it involves leptons or quarks:

 LCPT−evenlepton = 12i(cL)μνAB¯¯¯¯LAγμ↔DνLB (9) +12i(cR)μνAB¯¯¯¯RAγμ↔DνRB,
 LCPT−oddlepton=−(aL)μAB¯¯¯¯LAγμLB−(aR)μAB¯¯¯¯RAγμRB, (10)
 LCPT−evenquark = 12i(cQ)μνAB¯¯¯¯QAγμ↔DνQB (11) +12i(cU)μνAB¯¯¯¯UAγμ↔DνUB +12i(cD)μνAB¯¯¯¯¯DAγμ↔DνDB,

In these equations, the various coupling coefficients and are understood to be hermitian in generation space. The coefficients have dimensions of mass. The dimensionless coefficients can have both symmetric and antisymmetric parts but can be assumed traceless. A nonzero trace would not contribute to Lorentz violation and in any case can be absorbed by a conventional field normalization ensuring the usual kinetic operator for the matter fields.

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

 LCPT−evenYukawa=−12 [(HL)μνAB¯¯¯¯LAϕσμνRB (13) +(HU)μνAB¯¯¯¯QAϕcσμνUB +(HD)μνAB¯¯¯¯QAϕσμνDB]+h.c.

The coefficients are dimensionless and antisymmetric, but like the Yukawa couplings they are not necessarily hermitian in generation space.

The possible contributions in the Higgs sector can be CPT even or CPT odd:

 LCPT−evenHiggs = 12(kϕϕ)μν(Dμϕ)†Dνϕ+h.c. (14) −12(kϕB)μνϕ†ϕBμν −12(kϕW)μνϕ†Wμνϕ,
 LCPT−oddHiggs=i(kϕ)μϕ†Dμϕ+h.c.. (15)

In Eq. (14), the dimensionless coefficient can have symmetric real and antisymmetric imaginary parts. The other coefficients in Eq. (14) have dimensions of mass and must be real antisymmetric. The coefficient for the CPT-odd term (15) also has dimensions of mass and can be an arbitrary complex number.

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

 LCPT−evengauge = −12(kG)κλμνTr(GκλGμν) (16) −12(kW)κλμνTr(WκλWμν) −14(kB)κλμνBκλBμν.

In this equation, the dimensionless coefficients are real. They must have the symmetries of the Riemann tensor and a vanishing double trace. The point is that any totally antisymmetric part involves only a total derivative in the lagrangian density, while a nonzero double trace can be absorbed into a redefinition of the normalization of the corresponding kinetic term (8).

The CPT-odd gauge terms are given by the following expression :

 LCPT−oddgauge = (k3)κϵκλμνTr(GλGμν+23ig3GλGμGν) (17) +(k2)κϵκλμνTr(WλWμν+23igWλWμWν) +(k1)κϵκλμνBλBμν+(k0)κBκ.

The coefficients are real and have dimensions of mass, while is also real and has dimensions of mass cubed. It turns out that, if any of these CPT-odd terms do indeed appear, they would generate instabilities in the minimal theory. They are all associated with negative contributions to the energy, and in addition the term with would directly generate a linear instability in the potential. It might therefore seem desirable that all the coefficients vanish. While this could be imposed at the classical level, radiative quantum corrections from, say, the fermion sector might a priori be expected to generate nonzero values. Remarkably, the structure of the standard-model extension appears to be such that no corrections arise, at least to one loop. These issues are discussed further in what follows, in particular in subsection IV A and section V.

It is known that some apparently CPT- and Lorentz-violating terms can be eliminated from the action via field redefinitions . Several types of redefinition can be considered. In the context of the present standard-model extension, we have investigated a variety of possibilities for each field. As a general rule, the more complex the theoretical structure becomes, the less likely it is that a useful field redefinition exists. For instance, the presence of Lorentz-violating CPT-even derivative couplings in the standard-model extension complicates the analysis for CPT-odd terms provided in Ref. , although it turns out that the conclusions still hold. Here, we summarize a few methodological results and describe some special cases of particular interest.

To eliminate a Lorentz-breaking term, a field redefinition must involve the associated coupling coefficient. When derivative couplings play a role, the field redefinition may also involve spacetime-position variables. The assumption that the coupling coefficients are small can be helpful, in some cases directly assisting in derivations and in others leading to a set of approximate field redefinitions. Under the latter, a theory with first-order Lorentz-breaking effects may be redefined into one with effects appearing only at second or higher orders. Alternatively, some first-order Lorentz-breaking terms may be absorbed into others. A partial constraint on allowable redefinitions is provided by the transformation properties of the various Lorentz-violating terms under the discrete symmetries C, P, T. Only terms with identical discrete-symmetry properties can be absorbed into one another by first-order redefinitions.

Two types of redefinition that we have found of particular value are linear phase redefinitions and linear normalization redefinitions. For example, some terms involving the coefficients can be eliminated by position-dependent field-phase redefinitions, as described in Ref. . Another example is provided by terms involving the coefficients , some of which can absorb through field-normalization redefinitions certain other terms involving the coefficients . These examples have specific interesting implications for the quantum-electrodynamics limit of the standard-model extension, and their explicit forms for that case are given in section III below. Useful nonlinear field redefinitions might also exist in principle, but these are typically more difficult to implement meaningfully because they may represent (noncanonical) transformations between different physical systems rather than reinterpretations of the same physics.

We next consider the issue of electroweak SU(2) U(1) symmetry breaking. The static potential for the gauge and Higgs fields can be extracted from the lagrangian terms given above for the standard-model extension. It is possible to work in unitary gauge as usual, since the Lorentz-breaking terms do not affect the gauge structure of the theory. The analysis is somewhat more complicated than the conventional case, as it involves additional terms depending on the coupling coefficients , , , , , and . In principle, there are also contributions from the SU(3) sector, but these decouple from the Higgs field and so the gluon expectation values can be taken to be zero as usual. As mentioned above, the terms and are expected to vanish for consistency of the minimal theory, and so we assume this in what follows. In fact, a nonzero would have no effect on the vacuum values of the fields, but the linear instability that would be introduced by a nonzero would exclude a stable vacuum in the absence of other (nonlinear) effects.

Extremizing the static potential produces five simultaneous equations. Three of these are satisfied if the expectation values of and the photon vanish. The other two equations can be solved algebraically for the expectation values of the Higgs and fields. In the general case, both of these are nonzero and are given by

 ⟨Z0μ⟩=1qsin2θW(Re^kϕϕ)−1μνkνϕ, (18)
 ⟨rϕ⟩=a(1−1μ2(Re^kϕϕ)−1μνkμϕkνϕ)1/2, (19)

where and . Note that the quantity always exists when the Lorentz violation is small, . Note also that is a scalar under both particle and observer Lorentz transformations, so quantities such as the fermion mass parameters remain scalars despite the presence of Lorentz breaking.

As might be anticipated, the above pattern of expectation values leaves unbroken the electromagnetic U(1) symmetry, and it can be shown that fluctuations about the extremum are stable. When substituted into the lagrangian for the standard-model extension, the unconventional nonzero expectation value for the field generates some additional CPT- and Lorentz-violating contributions. However, these are all of the same form as other CPT- and Lorentz-violating terms already present in the theory, so they can be absorbed into existing coupling coefficients.

Some analyses of experimental tests of the standard-model extension involving flavor-changing oscillations in neutral mesons have been performed in Refs. [3, 5, 6]. Tests at the level of quantum electrodynamics are mentioned below. Note that some bounds on both the fermion and the gauge sectors might be obtained from available experimental information about the and perhaps the . Such limits would be of interest in their own right, although it seems likely that they would be much weaker than required to detect suppressed Lorentz violation at the levels estimated in this work.

## Iii Extended Quantum Electrodynamics

In much the same way that conventional quantum electrodynamics (QED) can be obtained from the usual standard model, a generalized quantum electrodynamics incorporating Lorentz-breaking terms can be extracted from the standard-model extension given in section II. This is of particular interest because QED has been tested to high precision in a variety of experiments, some of which may tightly constrain the coupling coefficients of the possible Lorentz-violating terms.

A straightforward way to obtain the extended QED is as follows. After the SU(2) U(1) symmetry breaking, set to zero the fields for the gluons, , for the weak bosons, and the physical Higgs field (but not the expectation value of the Higgs doublet, which generates fermion masses). The only remaining boson is the photon, mediating the electromagnetic interactions. The neutrinos are charge neutral, so they decouple and can be discarded. The resulting theory is an extended QED describing the electromagnetic interactions of quarks and (charged) leptons. It is expected to inherit from the standard-model extension various attractive features mentioned in the introduction, including U(1) gauge invariance, energy-momentum conservation, observer Lorentz invariance, hermiticity, microcausality, positivity of the energy, and power-counting renormalizability.

Denote the standard four-component lepton fields by and their masses by , where corresponds to electron, muon, tau, respectively. Then, the lagrangian for the conventional QED of leptons and photons is

 LQEDlepton−photon=12i¯lAγμ↔DμlA−mA¯lAlA−14FμνFμν. (20)

In this equation and throughout what follows, and the field strength is defined by

 Fαβ≡∂αAβ−∂βAα, (21)

as usual.

The standard-model extension generates additional terms that violate Lorentz symmetry. The CPT-even terms involving the lepton fields are

 LCPT−evenlepton = −12(Hl)μνAB¯lAσμνlB (22) +12i(cl)μνAB¯lAγμ↔DνlB +12i(dl)μνAB¯lAγ5γμ↔DνlB.

In this equation, the coupling coefficients are antisymmetric in spacetime indices and have dimensions of mass. They arise from the coefficients in Eq. (13) following gauge-symmetry breaking, and they are hermitian in generation space. The hermitian dimensionless couplings and could in principle have both symmetric and antisymmetric spacetime components but can be taken traceless. They arise from the expressions (9).

The CPT-odd terms involving the lepton fields are

 LCPT−oddlepton=−(al)μAB¯lAγμlB−(bl)μAB¯lAγ5γμlB. (23)

The couplings and are hermitian and have dimensions of mass. They arise from Eq. (10). Note that imposing individual lepton-number conservation in both the above equations would make all the coupling coefficients diagonal in flavor space.

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

 LCPT−evenphoton=−14(kF)κλμνFκλFμν. (24)

The coupling arises from Eq. (16) and is real and dimensionless. Without loss of generality it can be taken as double traceless, since any trace component would serve merely to redefine the kinetic term and hence is just a field renormalization. We disregard a conceivable -type term proportional to , which might arise from a totally antisymmetric component of , on the grounds that it is a total derivative. The coupling therefore can be taken to have the symmetries of the Riemann tensor.

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

 LCPT−oddphoton=+12(kAF)κϵκλμνAλFμν, (25)

where the coupling coefficient is real and has dimensions of mass. This term arises from the CPT-odd gauge sector (17) of the standard-model extension. As mentioned in the previous section, it has some theoretical difficulties associated with negative contributions to the energy and it therefore seems likely to be absent in practice. It is included in what follows so that we can discuss explicitly its difficulties and some related issues involving radiative corrections. Note also that the excluded destabilizing linear term in in the standard-model extension would, if present, generate a corresponding linear term in Eq. (25), where is a real coupling with dimensions of mass cubed. Certain issues involving this term are addressed in sections IV A and V.

The QED limit obtained from the standard-model extension also has a quark sector. This has the same general form as the lepton sector given by Eqs. (20), (22), and (23), except that six quark fields replace the three leptons and so twice as many Lorentz-violating couplings occur. Note that the lepton and quark sectors are coupled only through the photon: the gauge invariance of the standard-model extension excludes couplings mixing leptons and quarks.

The extended QED of leptons and photons given in Eqs. (20) - (25) should suffice for certain applications where the asymptotic states are leptons or photons and the strong and weak interactions play a negligible role, including a variety of existing or proposed high-precision experiments involving leptons. Interesting options for such experiments are to establish the possible signals of Lorentz violation suggested by the extended QED and to place bounds on the associated coupling coefficients. For example, promising possibilities involving the muon include accurate measurements of such as those underway at the Brookhaven muon ring  and sensitive tests for the decay . There are also a variety of other comparisons involving heavy leptons that are potentially of interest . These issues lie beyond the scope of the present work and will be addressed elsewhere.

For certain experiments, it suffices to consider another limiting case of the theory: the extended QED including only electrons, positrons and photons. This limit can be extracted from the lagrangian terms for the extended QED of leptons and photons by setting to zero the muon and the tau fields. Denoting the four-component electron field by and the electron mass by , the usual QED lagrangian for electrons and photons is

 LQEDelectron=12i¯¯¯¯ψγμ↔Dμψ−me¯¯¯¯ψψ−14FμνFμν. (26)

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

 LCPT−evenelectron = −12Hμν¯¯¯¯ψσμνψ (27) +12icμν¯¯¯¯ψγμ↔Dνψ +12idμν¯¯¯¯ψγ5γμ↔Dνψ,

while the CPT-odd ones become

 LCPT−oddelectron=−aμ¯¯¯¯ψγμψ−bμ¯¯¯¯ψγ5γμψ. (28)

The real coupling coefficients , , , , and are the -flavor components of the corresponding coefficients in the extended QED of leptons and photons and 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 photons, other Lorentz-violating terms can be envisaged that are compatible with U(1) charge symmetry, renormalizability, and an origin in spontaneous Lorentz breaking but that cannot be obtained as a reduction from the standard-model extension. All such terms would be CPT odd. They would have the form

 Lextraelectron = 12ieν¯¯¯¯ψ↔Dνψ−12fν¯¯¯¯ψγ5↔Dνψ (29) +14igλμν¯¯¯¯ψσλμ↔Dνψ,

where the couplings , and are real and dimensionless. The reason such terms are absent from the expressions obtained above is that all putative renormalizable terms in the standard-model extension that could generate Eq. (29) are directly incompatible with the electroweak structure. However, it is possible that nonrenormalizable higher-dimensional operators in the effective lagrangian obeying SU(2) U(1) symmetry and involving the Higgs field might generate the expressions (29) when the Higgs field acquires its vacuum expectation value. According to standard lore and the discussion in section II, such operators would be expected to be highly suppressed relative to those we have listed for the standard-model extension. This suppression should remain in force at the level of the extended QED, which means any terms of the form (29) would be expected to have coupling coefficients much smaller than the other terms we consider. Similar considerations apply to possible extra terms that might appear in the heavy-lepton and quark sectors of the extended QED.

Next, we address the issue of field redefinitions within the context of the extended QED of electrons, positrons, and photons. We have found several cases to be especially useful. One is a linear phase redefinition of the form , which eliminates the term from Eq. (28). This is equivalent to shifting the zeros of energy and momentum for electrons and positrons . We therefore expect no observable effects from a nonzero in any QED experiment.

Another useful class of redefinitions involves field renormalizations depending on coupling coefficients. For a fermion field , consider the redefinition

 ψ=(1+v⋅Γ)χ, (30)

where is one of , , and is a combination of coupling coefficients with appropriate spacetime indices. This set of redefinitions can be used to obtain several useful approximate results, valid to first order in the (small) coupling coefficients. One is that the combination can be eliminated and hence is unobservable at leading order in any QED experiment. Only the orthogonal linear combination is physical at this level. Another is that the antisymmetric component of can be eliminated to first order. Similarly, even if the extra terms with coefficients and in Eq. (29) should appear, they could be eliminated to first order by a combination of field redefinitions. The same is true of the trace components of the extra term with coefficient , while the totally antisymmetric component of this term can be absorbed into to first order. Combining all these results, it follows that at leading order in the extended QED of electrons, positrons, and photons the only observable coupling coefficients can be taken as , , the symmetric components of and , and possibly the traceless mixed-symmetry components of the extra coefficient .

So far in this section we have considered various forms of extended QED that emerge as limits of the standard-model extension. For some purposes, it can be useful to work within an effective extended QED valid for a free fermion that is a composite of leptons and quarks, such as a nucleon, atom, or ion. For a single fermion field of this type, the effective lagrangian would then have the same form as that of the extended QED for electrons, positrons, and photons. A description of this type is useful for investigations of the implications of high-precision experiments on composite fermions, such as comparative tests of proton properties or searches for a neutron electric-dipole moment. In principle, extra terms of the form (29) could appear as a result of the interactions among the fermion constituents, but in the effective theory the coupling coefficients of such terms would involve combinations of the constituent coupling coefficients with the interaction coupling constants and might therefore be expected to be absent at leading order in many cases.

Some possible experimental signals from extended QED are investigated in Ref. . Certain high-precision tests that could be performed with present technology are considered, and the attainable bounds on Lorentz-breaking coupling coefficients are estimated. The tests involve comparative measurements of anomalous magnetic moments and charge-to-mass ratios for particles and antiparticles confined in a Penning trap . They typically have the potential to bound the coupling coefficients of Lorentz- and CPT-violating terms at a level close to that expected from Planck-scale suppression. For example, the spacelike components of the coefficient control the appropriate figure of merit for experiments comparing the anomalous magnetic moments of the electron and positron. This figure of merit can be bounded to about one part in , which is comparable to the ratio of of the electron mass to the Planck scale.

## Iv The Pure-Photon Sector

In this section, we focus on the pure-photon sector of the extended QED. We examine some theoretical implications of the existence of Lorentz- and CPT-violating terms and address some experimental issues.

### iv.1 Lagrangian and Energy-Momentum Tensor

The lagrangian of interest, which is U(1) gauge invariant by construction, is a combination of the photon term in Eq. (26) with the expressions (24) and (25). It is

 Ltotalphoton = −14FμνFμν−14(kF)κλμνFκλFμν (31) +12(kAF)κϵκλμνAλFμν.

Some properties of the coupling coefficients and are described following Eqs. (24) and (25). For certain calculations, it is useful to decompose the coefficient into its two Lorentz-irreducible pieces, one with 10 independent components analogous to the Weyl tensor in general relativity and one with 9 components analogous to the trace-free Ricci tensor. Only one of the 19 total independent components of (the 00 component of the trace-free Ricci-tensor analogue) and one of the four independent components of the coefficient (the timelike component ) are associated with terms invariant under (particle) rotations.

Some insight into the structure of the lagrangian can be obtained by expressing it in terms of the potentials , and the fields , . We find

 Ltotalphoton = 12(→E2−→B2)+12α(→E2+→B2) (32) +12βjkEEjEk+12βjkBBjBk+12βjkEBEjBk +k0AF→A⋅→B−ϕ→kAF⋅→B +→kAF⋅(→A×→E).

Here and throughout this work, are spatial indices. The real coefficients and , , are various combinations of the couplings appearing in Eq. (31). Disregarding as before any total-derivative effects, the , , are traceless. Note that all possible quadratic combinations of the electric and magnetic fields appear. Only two terms, involving and , preserve (particle) rotational invariance. Note also that a rescaling without physical consequences can be performed to obtain a standard normalization of the electric field . This produces a lagrangian of the same general form as Eq. (32) except that the Lorentz-breaking term proportional to is replaced with one proportional to alone.

The canonical energy-momentum tensor can be constructed following the standard procedure. This tensor can be partially symmetrized, but complete symmetrization is impossible because there is an antisymmetric component that cannot be written as a total derivative. A relatively elegant expression can be obtained by adding judiciously chosen total-derivative terms, which leave unchanged the physics. Denoting the resulting energy-momentum tensor by , we find

 Θμν = −FμγFννγ+14ημνFαβFαβ (33) −(kF)αβμγFννγFαβ+14ημν(kF)αβγδFαβFγδ +(kAF)νAα˜Fαμ.

Here, we define

 ˜Fμν=ϵκλμνFκλ/2 (34)

to be the dual field strength.

The energy-momentum tensor obeys the usual conservation relation,

 ∂μΘμν=0. (35)

In addition to the gauge-invariant and symmetric contributions to , which include the conventional pieces among others, there are additional terms involving the coefficient that are gauge invariant but asymmetric. The term with is neither gauge invariant nor symmetric. Under a gauge transformation, an additional total-derivative term appears. The presence of an antisymmetric component in implies that care is required in physical interpretations of the energy-momentum behavior. Although can be regarded as the components of a generalized Poynting vector, its volume integral is no longer conserved and cannot be identified with the conserved volume integral of the components of the momentum density. These features are a direct consequence of the presence of the background expectation values of tensor fields, represented in the low-energy theory by the coupling coefficients and .

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

 Θ00 = 12(→E2+→B2) (36) −(kF)0j0kEjEk+14(kF)jklmϵjkpϵlmqBpBq −(kAF)0→A⋅→B.

If vanishes and is small, is nonnegative. This can be seen as follows. The combination of the usual energy density with the terms proportional to can be viewed as a bilinear form generated from a matrix in the six-dimensional space . The matrix is symmetric and -block diagonal, since no cross terms in and appear in . Observer rotation invariance can be used to diagonalize the upper block associated with the electric field. Since is small, the three diagonal entries are of the form , so the contribution to from the electric field is nonnegative in any frame. A similar argument shows that the contribution from the lower block, associated with the magnetic field, is also nonnegative. The conserved energy of a field configuration, obtained by integrating over all space, is therefore also nonnegative.

If instead vanishes and is small, the contribution to can be written in the form

 E ≡ ∫d3x Θ00 (37) = 12∫d3x(→E2+[→B−(kAF)0→A]2 −[(kAF)0]2→A2).

The last term is nonpositive and so can in principle introduce an instability in the theory . Note that a similar situation would hold for the linear term in the lagrangian that was discarded in section III, for which the energy density could also be negative. The appearance of negative contributions to the energy is unsatisfactory from a theoretical viewpoint.

It might seem tempting to resolve this issue by requiring that only the spacelike components of are nonzero, so that the terms involving are absent from Eqs. (36) and (37). However, this condition depends on the observer frame, so even an infinitesimal boost to another observer frame would reintroduce the instability. A somewhat more interesting option might be to combine the vanishing of with the introduction of a (small) photon mass, perhaps arising from a hitherto unobserved spontaneous breaking of the electromagnetic U(1) gauge symmetry. This would eliminate the linear instability and in principle might also produce a contribution canceling the negative term appearing in Eq. (37), although perhaps only for a physically reasonable range of observer boosts determined by the size of the photon mass and the magnitude of the components of . Although some form of this idea might be made physically acceptable, we are restricting ourselves here to minimal modifications of the usual standard model and so we disregard this possibility in the present work.

In the absence of a complete demonstration of a consistent alternative interpretation, one option might be to discard the term (25) depending on . This is possible at the classical level, but at the quantum level one might expect radiative corrections to induce it. We return to this question in section V, meanwhile keeping the term (25) in the analysis for completeness.

### iv.2 Solution of Equations of Motion

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

 ∂αFμαμ+(kF)μαβγ∂αFβγ+(kAF)αϵμαβγFβγ=0. (38)

These equations are the Lorentz-breaking extensions of the usual inhomogeneous Maxwell equations in the absence of sources, . By virtue of its conventional definition in Eq. (21), the field strength satisfies the usual homogeneous Maxwell equations

 ∂μ˜Fμν=0, (39)

where is given in Eq. (34).

An important feature of the equations (38) and (39) is their linearity in and hence in . The Lorentz-violating terms thereby avoid the complications of nonlinear modifications to the Maxwell equations, which are known to occur in some physical situations such as nonlinear optics or when vacuum-polarization effects are included. Another feature is that the extra Lorentz-violating terms involve both the electric and the magnetic fields, as well as their derivatives. As a result, the equations (38) bear some resemblance to the usual Maxwell equations in moving media, for which the boost causes the electric and magnetic fields to mix. Note that the coefficients determining this mixing are directly dependent on the velocity of the medium and so change with the inertial frame. Similarly, for the Lorentz-violating case of interest here, a change of observer frame changes the coupling coefficients. Some other useful analogies between the equations (38) and those of conventional electrodynamics in macroscopic media are described in section IV C.

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

 Aμ→Aμ−1q∂μΛ. (40)

As in conventional electrodynamics, the presence of gauge symmetry affects the interpretation and solution of the equations of motion. We first consider a treatment in terms of the potentials and then one for the field strengths , .

Taking the potentials as basic, the four equations (39) are directly satisfied. This appears to leave four equations (38) for four unknowns . However, just as in conventional Maxwell electrodynamics, the conjugate momentum to vanishes because is missing from the lagrangian (31), so the theory has a Dirac primary constraint. In the conventional case it then follows from the identity , which is associated with current conservation when sources are present, that the equation of motion associated with plays the role of an initial condition. The same conclusion holds here because when acted on by the left-hand side of Eq. (38) also vanishes identically. This leaves three equations of motion and a constraint for four variables. One combination of variables can be fixed by a gauge choice. The constraint then leaves two independent degrees of freedom.

Despite the parallels with conventional electrodynamics, the gauge-fixing process involves some interesting differences. For example, there is normally an equivalence between the Coulomb gauge , the temporal gauge , and one of the members of the family of Lorentz gauges . When Lorentz-violating effects are included, these three gauge choices become inequivalent. For example, typically is nonzero if the Coulomb gauge is imposed.

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

 Aμ(x)≡ϵμ(p)exp(−ipαxα), (41)

where can be regarded as the frequency and wave vector of the mode or as the associated energy and momentum (which can be distinct from the conserved energy and momentum obtained from the energy-momentum tensor). Note that taking the real part is understood, as usual. The equations of motion (38) generate the momentum-space equation

 Mαδ(p)Aδ=0, (42)

where the matrix is

 Mαδ(p) ≡ ηαδp2−pαpδ−2(kF)αβγδpβpγ (43) −2i(kAF)βϵαβγδpγ.

This complex-valued matrix is hermitian because the first three terms are real and symmetric while the last is imaginary and antisymmetric. Its determinant can be shown to vanish identically for all , a feature related to the gauge freedom. The conventional result is recovered when the coefficients and vanish.

Once a gauge choice is imposed, the relation (42) provides a set of complex-valued equations for . The differences between various gauge choices that normally are equivalent can be seen explicitly at this stage. For example, is not proportional to in the Lorentz gauges, the Coulomb gauge leaves a nontrival equation for , and the temporal gauge generates an involved constraint on . In practice, Eq. (42) then reduces to a (sub)set of equations involving an effective matrix with explicit form dependent on the gauge choice. The requirement for existence of nonzero solutions can be obtained from a condition of the type . With fixed coefficients and , this condition then determines as a function of . Since is a matrix with entries quadratic in , a determinant of this type can produce an eighth-order polynomial in .

In the conventional case in the Lorentz gauge, the polynomial reduces to one with two quadruply degenerate roots, . The apparent doubling of the roots relative to the number of variables can be understood by the observation that in this case is symmetric under , so for each solution there is another solution . These two solutions can be shown to be physically equivalent by examining the real part of in Eq. (41).

In contrast, in the general extended electrodynamics the polynomial determining may have eight distinct roots. Each of these could in principle produce a nontrivial solution for , double the expected number. In this case, is symmetric under the simultaneous operations and (leaving unchanged). Thus, for each solution there is another solution . The sign change for the coefficient might appear to preclude the demonstration of the physical equivalence of these solutions. However, hermiticity of implies its determinant is equivalent to the determinant of its complex conjugate. Since appears only in the imaginary part of , it follows that for each solution there is also a solution . This is physically equivalent to the solution , as before. Thus, the number of independent roots is the same as the number of variables as expected, despite the apparent complexity of the polynomial.

In the general extended electrodynamics, neither the Coulomb gauge nor the Lorentz gauges significantly simplify the primary constraint. In contrast, the temporal gauge immediately removes one degree of freedom from . This gauge can be imposed by choosing the function in Eq. (40) as . Note that this choice breaks observer boost invariance but leaves unaffected the observer rotation invariance. It reduces the primary constraint to the form

 M0jAj=0, (44)

where are components in the temporal gauge of the matrix given in Eq. (43).

At this stage an explicit solution could be found. For example, one could use two of the degrees of freedom of observer rotation invariance to select a convenient coordinate system, such as one in which . Solving for from the primary constraint (44) and substituting into the remaining three equations of motion in (42) would then produce an identity and two simultaneous linear equations for and . A nontrivial solution of this pair of equations could be found by requiring the determinant of the system to vanish, which in turn would generate a relation between and . Solving this relation must give two independent dispersion relations, one for each of the two physical degrees of freedom. The full dispersion relations in an arbitrary coordinate system could then in principle be recovered by using arguments based on rotational covariance.

Rather than pursuing this approach, we return to the eight equations of motion (38) and (39) and reconsider their treatment taking as independent variables the six electric and magnetic fields. Here, we are interested in the properties of electromagnetic radiation, so we work with the standard ansatz

 Fμν(x)≡Fμν(p)exp(−ipαxα), (45)

where .

The equations (39), which include the usual Faraday law and the condition ensuring the absence of magnetic monopoles, are unaffected by the Lorentz breaking and for radiation reduce as usual to

 p0→B=→p×→E,→p⋅→B=0. (46)

The first of these can be regarded as defining the magnetic field once the electric field is known. The second of these equations follows from the first, and shows that the magnetic field remains transverse to despite the Lorentz violation.

The equations (38) generate modified Coulomb and Ampère laws that are to be solved for . A relatively straightforward procedure is to substitute for from Eq. (46). Using the Ampère law, we thereby obtain the vector equation

 MjkEk=0, (47)

where the 33 matrix is identical to the -component submatrix of the matrix in Eq. (43). The modified Coulomb law can be obtained from the modified Ampère law by taking the scalar product with : . Note that this derivation provides some insight about the temporal gauge in a treatment using . Thus, there is a close parallel between the two because in this gauge.

To obtain an explicit solution, it is helpful to take advantage of the observer rotation invariance to select a coordinate system in which key expressions are simplified. For example, a useful frame is the one with . In it, the modified Coulomb equation can be solved for in terms of and . Substitution of this solution into the three component equations (47) produces one identity and two simultaneous linear equations for and . The matrix of this system of equations is hermitian, and the condition for a nontrivial solution is that its determinant vanishes. The determinant turns out to be a fourth-order polynomial for as a function of . For reasons similar to those given for the case, there are only 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 the general case is involved because the fourth-order polynomial is homogeneous in the small coupling coefficients. Since is dimensionless while has dimensions of mass, to first order in the coupling coefficients and for and the solution for as a function of must take the form of the sum of with a function of the quantities and . Indeed, we find

 p0=(1+ρ)p±√(σ2p2+τ2), (48)

where and are functions of the components of and is a function of the components of , 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−(
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