Gravity, Lorentz Violation, and the Standard Model

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provided by IUScholarWorks PHYSICAL REVIEW D 69, 105009 ͑2004͒

Gravity, Lorentz violation, and the standard model

V. Alan Kostelecky´ Physics Department, Indiana University, Bloomington, Indiana 47405, USA ͑Received 8 January 2004; published 17 May 2004͒ The role of the gravitational sector in the Lorentz- and CPT-violating standard-model extension ͑SME͒ is studied. A framework is developed for addressing this topic in the context of Riemann-Cartan spacetimes, which include as limiting cases the usual Riemann and Minkowski geometries. The methodology is ﬁrst illustrated in the context of the QED extension in a Riemann-Cartan background. The full SME in this background is then considered, and the leading-order terms in the SME action involving operators of mass dimension three and four are constructed. The incorporation of arbitrary Lorentz and CPT violation into general relativity and other theories of gravity based on Riemann-Cartan geometries is discussed. The dominant terms in the effective low-energy action for the gravitational sector are provided, thereby completing the formulation of the leading-order terms in the SME with gravity. Explicit Lorentz symmetry breaking is found to be incompatible with generic Riemann-Cartan geometries, but spontaneous Lorentz breaking evades this difﬁculty.

DOI: 10.1103/PhysRevD.69.105009 PACS number͑s͒: 11.30.Cp, 04.50.ϩh, 11.30.Er, 12.60.Ϫi

I. INTRODUCTION these lines has been performed, and in fact the Lorentz- violating gravitational sector was among the first pieces of The combination of Einstein’s general relativity and the the SME to be studied ͓4͔. However, an explicit construction standard model ͑SM͒ of particle physics provides a remark- of all dominant gravitational couplings in the SME action ably successful description of nature. The former theory de- has been lacking to date. scribes gravitation at the classical level, while the latter en- The investigation of local Lorentz violation in non- compasses all other phenomena involving the basic particles Minkowski spacetimes requires a geometrical framework al- and forces down to the quantum level. These two field theo- lowing for nonzero vacuum quantities that violate local Lor- ries are expected to merge at the Planck scale, m P entz invariance but preserve general coordinate invariance. Ӎ1019 GeV, into a single unified and quantum-consistent The Riemann-Cartan geometry is well suited to this task, and description of nature. it also naturally handles minimal gravitational couplings of Uncovering experimental confirmation of this idea is spinors ͓5,6͔. The present work studies the SME in a general challenging because direct experiments at the Planck scale Riemann-Cartan spacetime, allowing for dynamical curva- are impractical. However, suppressed effects emerging from ture and torsion modes. The general-relativistic version of the underlying unified quantum gravity theory might be ob- this theory is readily recovered in the limit of zero torsion. servable in sensitive experiments performed at our presently The Lorentz-violating terms in the SME take the form of attainable low-energy scales. One candidate set of Planck- Lorentz-violating operators coupled to coefficients with Lor- scale signals is relativity violations, which are associated entz indices. Nonzero coefficients of this type could emerge with the breaking of Lorentz symmetry ͓1͔. in various ways. One attractive and generic mechanism is Any observable signals of Lorentz violation can be de- spontaneous Lorentz violation, studied in string theory and scribed using effective field theory ͓2͔. To ensure that known field theories with gravity ͓4,7͔. Noncommutative field theo- physics is reproduced, a realistic theory of this type must ries also contain Lorentz violation, with realistic models in- contain both general relativity and the SM, perhaps together volving a subset of SME operators of higher mass dimension with suppressed higher-order terms in the gravitational and ͓8͔. Other suggestions for sources of Lorentz violation in- SM sectors. Incorporating in addition terms describing arbi- clude, for example, various nonstring approaches to quantum trary coordinate-independent Lorentz violation yields an ef- gravity ͓9͔, random dynamics models ͓10͔, multiverses ͓11͔, fective field theory called the standard-model extension brane-world scenarios ͓12͔, and cosmologically varying ͑SME͒. At the classical level, the dominant terms in the SME fields ͓13,14͔. action include the pure-gravity and minimally coupled SM In the Minkowski-spacetime limit of the SME, the actions, together with all leading-order terms introducing Lorentz-violating terms can be classified according to their violations of Lorentz symmetry that can be constructed from properties under CPT. Indeed, since CPT violation implies gravitational and SM fields. Lorentz violation in this limit ͓15͔, the SME also incorpo- The SME has been extensively studied in the Minkowski- rates general CPT breaking. To determine the CPT properties spacetime limit, where all terms expected to dominate at low of a given operator in Minkowski spacetime, it suffices in energies are known ͓3͔. A primary goal of the present work is practice to count the number of indices on the corresponding to construct explicitly the modifications appearing in non- coefficient for Lorentz violation. A Lorentz-violating term Minkowski spacetimes, including both those in the pure- breaks CPT when this number is odd. However, in non- gravity sector and those involving gravitational couplings in Minkowski spacetimes, establishing a satisfactory definition the matter and gauge sectors. Some previous work along of CPT and its properties is challenging, and a complete

understanding is lacking at present. In this work, a practical vature and torsion. One well-known gravitation theory based definition is adopted: CPT-odd operators are taken to be on Riemann-Cartan geometry is the Einstein-Cartan theory, those with an odd total number of spacetime and local Lor- which has gravitational action of the Einstein-Hilbert form. entz indices. This suffices for present purposes and ensures a The torsion in this theory is static, and in the absence of smooth match to the Minkowski-spacetime limit. With this matter the solutions of the theory are equivalent to those of understanding, the SME serves as a realistic general basis for general relativity. However, more general gravitation theo- studies of Lorentz violation in Riemann-Cartan spacetimes, ries in Riemann-Cartan spacetime contain propagating vier- with or without CPT breaking. bein and spin-connection fields, describing dynamical torsion Since no compelling experimental evidence for Lorentz and curvature ͓6͔. violation has been uncovered as yet, it is plausible to assume The vierbein formalism has a close parallel to the descrip- that the SME coefficients for Lorentz violation are small in tion of local symmetry in gauge theory. A key feature is the any concordant frame ͓16͔. Indeed, sensitivity to the SME separation of local Lorentz transformations and general co- coefficients has attained Planck-suppressed levels in a num- ordinate transformations. At each spacetime point, the action ber of experiments, including ones with mesons ͓2,17–19͔, of the local Lorentz group allows three rotations and three baryons ͓20–22͔, electrons ͓23–25͔, photons ͓13,26–29͔, boosts, independent of general coordinate transformations. and muons ͓30͔, and discovery potential exists in experi- This situation is ideal for studies of local Lorentz violation in ments with neutrinos ͓3,31,32͔. Only a comparatively small which it is desired to maintain the usual freedom of choice of part of the coefficient space has been explored to date, and coordinates without affecting the physics. Within this frame- the present work is expected eventually to provide further work, local Lorentz violation is analogous to the violation of directions in which to pursue experimental searches for Lor- local gauge invariance. entz violation. The presence of Lorentz violation in a local Lorentz The organization of this paper is as follows. The frame- frame is signaled by a nonzero vacuum value for one or more work for local Lorentz violations is discussed in Sec. II A, quantities carrying local Lorentz indices, called coefficients while the structure of the action and the derivation of cova- for Lorentz violation. As a simple example, consider a toy riant conservation laws in the presence of Lorentz violation theory in which a nonzero timelike vacuum value ba are provided in Sec. II B. Section III considers the QED ex- ϭ(b,0,0,0) exists in a certain local Lorentz frame at some tension with gravitational couplings, and contains separate point P. One explicit theory of this type is the bumblebee subsections devoted to the fermion and photon actions. The model described in Appendix B. The presence of the coeffi- SME in a Riemann-Cartan background is presented in Sec. cient ba for Lorentz violation implies that a preferred direc- IV. The leading-order terms in the pure-gravity sector are tion is selected at P within the local Lorentz frame, leading constructed in Sec. V A, while the limiting Riemann- to equivalence-principle violations. Physical Lorentz break- spacetime case is considered in Sec. V B. Section V C ad- ing occurs at P whenever particles or fields have observable dresses the issue of the compatibility of explicit Lorentz vio- interactions with ba . lation with the underlying Riemann-Cartan geometry. The Rotations or boosts of particles or localized field distribu- body of the paper concludes with a summary in Sec. VI. tions in a given local Lorentz frame at P can be performed Appendix A lists conventions adopted in this work and some that leave ba unaffected. Lorentz transformations of this kind key results for Riemann-Cartan geometry. Appendix B pre- are called local particle Lorentz transformations, and under sents a class of models for Lorentz violation used to illustrate them ba behaves as a set of four scalars. However, the choice various concepts throughout this work. of the local Lorentz frame itself remains arbitrary up to spacetime rotations and boosts. Rotations or boosts changing II. FRAMEWORK the local Lorentz frame are called local observer Lorentz transformations, and under them ba behaves covariantly as a A. Local Lorentz violation four-vector. The theory thus maintains local observer Lorentz The classic description of gravity in a Riemann spacetime covariance, despite the presence of local particle Lorentz invokes a metric and a covariant derivative that acts on vec- violation. tor or tensor representations of Gl(4,R). However, Gl(4,R) The conversion from the local Lorentz frame to spacetime ϭ a has no spinor representations, whereas the fundamental con- coordinates is implemented via the vierbein, b␮ e␮ ba .A stituents of ordinary matter, leptons and quarks, are known to change of the observer’s spacetime coordinates x␮ induces a be spinors. One framework that incorporates spinors and dis- conventional general coordinate transformation on b␮ . The tinguishes naturally between local Lorentz and general coor- description of the physics is therefore invariant under general dinate transformations is the vierbein formalism ͓5͔, which is coordinate transformations, as is to be expected for adopted in the present work. coordinate-independent behavior. In the vierbein formalism, the basic gravitational fields Different local observer Lorentz frames can be reached a can be taken as the vierbein e␮ and the spin connection using different vierbeins, related by local observer Lorentz ab ␻␮ . The corresponding Riemann-Cartan spacetimes are transformations. In a local neighborhood containing P, b␮ is ␬ determined by the curvature tensor R ␭␮␯ and the torsion typically a function b␮(x) of position. Assuming for definite- ␭ ␮ tensor T ␮␯ . The usual Riemann spacetime of Einstein’s ness that b␮ has constant magnitude b b␮ , the local ob- a general relativity can be recovered in the zero-torsion limit, server Lorentz freedom in the vierbein e␮ (x) can be used to ϭ while Minkowski spacetime is a special case with zero cur- choose ba (b,0,0,0) everywhere in the neighborhood. This

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͒ ͑ Ϫ␻ b ϭ ץϵ defines a preferred set of frames over the neighborhood. D␮ba ␮ba ␮ abb 0. 1 Note that the existence of preferred frames is a special feature of this simple model. Extending the model to one However, the integrability conditions for this equation can be with a second nonzero coefficient for Lorentz violation ca satisfied globally only for special spacetimes, in particular, typically destroys the existence of preferred frames at P and for parallelizable manifolds. Such manifolds have zero cur- in the neighborhood. Observer Lorentz transformations have vature, are comparatively rare in four or more dimensions, only six degrees of freedom, which are used in selecting the and appear of lesser interest for theories of gravity. It is preferred frame for ba at P. In this preferred frame, ca ge- therefore reasonable to suppose that D␮b Þ0 at least in ϭ a nerically has the arbitrary form ca (c1 ,c2 ,c3 ,c4). More- some region of spacetime. This in turn implies nontrivial a over, once e␮ (x) has been selected to maintain the preferred consequences for the energy-momentum tensor. Section II B position-independent form of ba over a neighborhood of P, discusses these consequences and obtains the covariant con- ca can vary with position. Although another frame at P can servation law in the presence of Lorentz violation. In any ͑ be found in which ca does have a preferred timelike, space- case, an arbitrary a priori specification of b␮(x) in a given ͒ like, or lightlike form, then ba no longer has the preferred spacetime can be expected to be inconsistent with the simple ϭ form ba (b,0,0,0). The notion of preferred frame therefore condition ͑1͒. loses meaning in the generic case. A consistent prescription for determining b␮(x) and It is natural and convenient, although not necessary, to hence D␮ba exists in some cases. For example, this is true assume b␮(x) is a smooth vector field over the neighborhood when b␮(x) arises through a dynamical procedure, such as of P and over most of the spacetime, except perhaps for the development of a vacuum expectation value in the con- singularities. Since most applications involve second-order text of spontaneous Lorentz breaking. The dynamical equa- differential equations, C2 smoothness suffices. However, a tions for the spacetime curvature and torsion can then be smooth extension of b␮(x) over the entire spacetime may be solved simultaneously with the dynamical equations for b␮ , precluded by topological conditions analogous to the Hopf yielding a self-consistent solution. As usual, appropriate theorem, which states that smooth vector fields can exist on a boundary conditions are needed for all variables to fix the compact manifold if and only if its Euler characteristic ␹ solution. In the case of asymptotically Minkowski space- vanishes. Note that, if indeed singularities of b␮ occur, their times, which are relevant for many experimental purposes, it location can differ from those of singularities in the curva- may be physically reasonable to adopt as part of the bound- ture and torsion. Note also that some standard topological ary conditions the criterion ͑1͒ in the asymptotic limit where constraints on the spacetime itself are implied by the general the curvature and torsion vanish. Solutions of this form then framework adopted here. For example, the presence of spinor merge with those of the SME in Minkowski spacetime. More fields requires a spinor structure on the spacetime, so the complicated solutions involving asymptotic coefficients corresponding manifold must be a spin manifold and have varying with spacetime position could also be considered. trivial second Steifel-Whitney class. The corresponding potential experimental signals would in- Studies of Lorentz violation in the Minkowski-spacetime clude violations of energy-momentum conservation. In most limit commonly assume that the coefficients for Lorentz vio- of what follows, no particular special assumptions about the lation are constants over the spacetime, which ensures the global structure of the spacetime or about asymptotic prop- useful simplifying physical consequence that energy and mo- erties of the coefficients are made, and in particular Eq. ͑1͒ is mentum remain conserved. Various physical arguments can not assumed. be used to justify this assumption. For example, some For illustrative purposes, the above discussion uses a mechanisms for Lorentz violation may attribute higher over- simple toy model with a single coefficient ba(x) that behaves all energy to coefficients with nontrivial spacetime depen- like a vector under local observer Lorentz transformations. dence, so that constant coefficients are naturally preferred. More generally, there can be a ͑finite or infinite͒ number of More generally, if the Lorentz breaking originates at the coefficients for Lorentz violation, each transforming as a Planck scale and there is an inflationary period in cosmology, specific representation of the local observer Lorentz group. then a present-day configuration with constant coefficients In what follows, a generic coefficient with compound local x over the Hubble radius is a plausible consequence. Also, for Lorentz index x transforming in the representation (X͓ab͔) y sufficiently slow spacetime variation of the coefficients, the is denoted kx . The considerations presented above for ba assumption of constancy can be viewed as the leading ap- apply to the more general kx . In any case, the introduction of proximation in a series expansion. However, all arguments of coefficients for Lorentz violation suffices to encompass the this type are ultimately physical choices. From the formal description of Lorentz violation from any source that main- perspective, any vector or tensor field with smooth integral tains coordinate independence of physics. curves is also an acceptable candidate. The choice of constant coefficients for Lorentz violation can therefore be viewed as a kind of boundary condition for the theory. B. Action and covariant conservation laws For the simple toy model in the present example, the con- From the perspective of physics at our present compara- dition of constant coefficients in Minkowski spacetime can tively low energies, the underlying fundamental theory of .␮b␯ϭ0. In a more general Riemann-Cartan nature appears as a four-dimensional effective field theoryץ be written spacetime, it might seem natural to impose the covariant The action of this theory is expected to incorporate the stan- generalization of this, dard model ͑SM͒ of particle physics, including gravitational

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couplings and a purely gravitational sector. Assuming that Smatter,0 and a Lorentz-violating part Smatter,LV . In accordance gravity can be described using the vierbein and spin connec- with the above discussion, any term in the latter then has the tion, it is reasonable to suppose that the action of the effec- general form tive theory also contains the usual minimal gravitational couplings and the Einstein-Hilbert action among its terms. ϭ ͵ 4 x͑ y ␮ y͒ ͑ ͒ Whatever the underlying structure, the physics of the ef- Smatter,LV d xekxJ f ,e aD␮ f , 3 fective field theory is also expected to be coordinate independent. This corresponds to covariance under general coor- where the operator Jx can in this case be viewed as a current dinate and local observer Lorentz transformations. Then, formed from matter fields f y and their covariant derivatives, assuming the fundamental theory indeed incorporates a assuming minimal couplings for simplicity. The desired mechanism for Lorentz violation, it follows that the action energy-momentum conditions follow from the properties of contains terms involving operators with nontrivial local Lor- these terms under local Lorentz and general coordinate trans- entz transformations contracted with coefficients for Lorentz formations when the vierbein and spin connection are treated violation. The resulting effective field theory is the SME, as as background couplings fixing the Riemann-Cartan space- already mentioned in the Introduction. time. The present work considers the structure and some impli- Consider in particular a special variation of the action cations of the SME in Riemann-Cartan spacetime. As an ef- Smatter in which all fields and backgrounds are allowed to fective field theory, the SME action contains an infinite num- vary, including the coefficients for explicit Lorentz violation, but in which the equations of motion are obeyed for the ber of terms, but typically the physics is dominated by dynamical fields f x. The resulting change in the action takes operators of low mass dimension. In addition to the usual the form SM and Einstein-Hilbert terms, possible higher-order terms involving SM fields, and possible higher-order curvature and 1 torsion couplings, the terms of comparatively low mass di- ␦ ϭ ͵ 4 ͩ ␮␯ ␦ aϩ ␮ ␦␻ abϩ x␦ ͪ Smatter d xe Te e␯a e␮ S␻ ab ␮ eJ kx . mension include ones violating local Lorentz symmetry. 2 ͑4͒ Later sections of this work explicitly display the dominant Lorentz-violating terms involving the vierbein, spin connec- This expression can be taken to define the energy-momentum tion, and SM fields. It is straightforward to extend the analy- ␮␯ tensor Te associated with the vierbein and the spin-density sis to include Lorentz-violating couplings of other hypoth- ␮ tensor S␻ ab associated with the spin connection. The reader esized fields. ␮␯ is cautioned that in a Riemann-Cartan spacetime Te typi- The Lorentz-violating piece SLV of the SME effective ac- cally differs from the ͑Belinfante͒ energy-momentum tensor ␮␯ tion SSME consists of a series of terms, each of which can be Tg obtained by variation with respect to the metric, expressed as the observer-covariant integral of the product of whether or not Lorentz violation is present. Similarly, the x ␭ a coefficient kx for Lorentz violation with an operator J : definition of S␻ ab differs from those of the spin-density ␭ ␭ tensors ST ␮␯ and SK ␮␯ obtained by varying with respect to ʛ ͵ 4 x ͑ ͒ SLV d xekxJ . 2 the torsion and contortion, respectively. The tensors defined here are the most convenient for practical purposes because they are the sources in the equations of motion for the vierbein and the spin connection. The usual Einstein general The coefficient kx transforms in the covariant x representa- x relativity involving coupling to the symmetric energy- tion of the observer Lorentz group, while the operator J momentum tensor T ␮␯ is contained in this discussion as a transforms in the corresponding contravariant representation. g x special case with vanishing torsion. In the present context, J is understood to be formed from When the special variation ͑4͒ is induced by infinitesimal the vierbein, spin connection, and SM fields and is invariant local Lorentz transformations parametrized by ⑀ab, the rel- under general coordinate transformations. This structure of evant infinitesimal changes in the vierbein, spin connection, the effective action is independent of the origin of the Lor- and coefficients for Lorentz violation take the form entz violation, and in particular it is independent of whether the violation in the underlying theory is spontaneous or ex- a a b ␦e␮ ϭϪ⑀ e␮ , plicit. In practice, for many ͑but not all͒ calculations, the b coefficient kx can be treated as if it represents explicit viola- ab a cb cb a ab , ⑀␮ץ␻␮ ϭϪ⑀ ␻␮ ϩ⑀ ␻␮ ϩ␦ tion even when its origin lies in the development of a c c vacuum value. The covariant energy-momentum conservation law and 1 ␦k ϭϪ ⑀ab͑X ͒y k . ͑5͒ the symmetry property of the energy-momentum tensor are x 2 ͓ab͔ x y modified in the presence of explicit Lorentz violation. To obtain these conditions, separate the action SSME into a piece A suitable substitution of these results into Eq. ͑4͒ followed Sgravity involving only the vierbein and spin connection and a by some manipulation then yields the desired condition on ␮␯ piece Smatter containing the remainder. The matter action the symmetry of the energy-momentum tensor Te in the Smatter in turn can be split into a Lorentz-invariant part presence of coefficients for explicit Lorentz violation:

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␮␯Ϫ ␯␮ϭ Ϫ ␤ ͒ ␣␮␯ϩ ␮a ␯b ͒x y III. QED EXTENSION Te Te ͑D␣ T ␤␣ S␻ e e kx͑X͓ab͔ y J . ͑ ͒ 6 The basic nongravitational fields for the Lorentz- and CPT-violating QED extension in Riemann-Cartan spacetime In the Minkowski-spacetime limit, this equation becomes are a Dirac fermion ␺ and the photon A␮ . The action for the ␮␯ ␮␯ ␣␮␯ ͓␮␯͔ x y theory can be expressed as a sum of partial actions of the S ϩk ͑X ͒ J , ͑7͒␣ץϪ⌰ ϭ ⌰ c c c x y form ⌰ ␮␯ where c is the canonical energy-momentum tensor and ϭ ϩ ϩ ϩ ͑ ͒ ␭␮␯ S S␺ SA Sgravity . 11 Sc is the canonical spin-density tensor. With appropriate ¯ substitutions for the matter fields and coefficients for Lorentz The fermion part S␺ of the action S contains terms dominat- ͑ ͒ violation, Eq. 7 correctly reproduces the results in ing at low energies that involve fermions and their minimal ͓ ͔ Minkowski spacetime obtained in Ref. 3 . couplings to photons and gravity. The photon part S con- ͑ ͒ A If instead the special variation 4 is induced by a general tains terms dominating at low energies that involve only pho- ⑀␮ coordinate transformation with parameter , the relevant tons and their minimal couplings to gravity, while the pure- field variations are the Lie derivatives gravity part Sgravity involves only the vierbein and the spin connection. The ellipsis represents higher-order terms, in- a⑀␯ ץ␯ϩ⑀ ץaϭL aϭ a ␦ e␮ ⑀e␮ e␯ ␮ ␯e␮ , cluding ones involving fermions and photons that are nonrenormalizable in the Minkowski-spacetime limit, ones in- ab ab ab ␯ ab ␯ ␯␻␮ ⑀ , volving nonminimal and higher-order gravitationalץ␮⑀ ϩץ ␻␮ ϭL⑀␻␮ ϭ␻␯␦ couplings, and ones involving field operators of dimension ͒ ͑ ץϭL ϭ⑀␮ ␦ kx ⑀kx ␮kx . 8 greater than 4 that couple curvature and torsion to the matter and photon fields. Other possible nonminimal operators Substituting these expressions appropriately into Eq. ͑4͒, ma- formed from the fermion and photon fields, such as ones nipulating the result, and incorporating the condition ͑6͒ breaking U͑1͒ gauge invariance, may also be of interest for yields the covariant energy-momentum conservation law in certain considerations and can be included as appropriate. the presence of coefficients for explicit Lorentz violation: This section presents the explicit form of the two partial actions S␺ and SA and some of their basic physical implica- 1 ␭ ␮ ␭ ␮ ab ␮ x tions. Discussion of the gravity partial action is deferred to ͑D␮ϪT ␭␮͒T ␯ϩT ␮␯T ␭ϩ R ␮␯S␻ ϪJ D␯k ϭ0. e e 2 ab x Sec. V. ͑9͒ A. Fermion sector In the limiting case of Minkowski spacetime, where the curvature and torsion vanish, this equation becomes a modified The fermion partial action for the QED extension can be conservation law for the canonical energy-momentum tensor written as

␮␯ x ␯ 1 ␮ ͒ ͑ k . ͑10͒ ϭ ͵ 4 ͩ ¯␺⌫a ␺Ϫ ¯␺ ␺ͪ ץ ␮⌰ ϭJץ S␺ d x iee D␮ e M . 12 c x 2 a J Explicit substitution for the fields and currents shows that In this equation, the symbols ⌫a and M are defined by this result agrees with the Minkowski-spacetime results of Ref. ͓3͔, as expected. The interesting issue of the compatibil- a a ␯a ␮ b ␯a ␮ b ␮a ⌫ ϵ␥ Ϫc␮␯e e ␥ Ϫd␮␯e e ␥ ␥ Ϫe␮e ity of the relations ͑6͒, ͑9͒ with the underlying geometrical b b 5 assumptions of the Riemann-Cartan spacetime is discussed 1 ␮a ␯a ␭ ␮ bc Ϫif␮e ␥ Ϫ g␭␮␯e e e ␴ ͑13͒ in Sec. V. 5 2 b c A similar chain of reasoning can be adopted to obtain the symmetry property of the energy-momentum tensor and the and covariant energy-momentum conservation law relevant in the case of spontaneous Lorentz violation. Since spontaneous 1 ␮ a ␮ a ␮ ␯ ab Mϵmϩim ␥ ϩa␮e ␥ ϩb␮e ␥ ␥ ϩ H␮␯e e ␴ . violation of a symmetry leaves unaffected the associated 5 5 a a 5 2 a b conserved currents, it is to be expected that in this case the ͑14͒ ͑ ͒ ͑ ͒ terms involving kx in Eqs. 6 and 9 are absent. This is indeed confirmed by calculation. The basic point is that co- The first term of Eq. ͑13͒ leads to the usual Lorentz-invariant efficients originating from spontaneous breaking are vacuum kinetic term for the Dirac field. Similarly, the first two terms values of fields, and so they must obey the corresponding of Eq. ͑14͒ lead to a Lorentz-invariant mass. In the absence ␦ x equations of motion. Just as the variations f of other dy- of anomalies, the coefficient m5 can be chirally rotated to namical fields f x have vanishing coefficients in Eq. ͑4͒ and zero in Minkowski spacetime without loss of generality. The so provide no contributions to the covariant energy- same holds here provided suitable redefinitions of certain momentum and spin-density conservation laws, no contribu- coefficients are made. The coefficients for Lorentz violation ␦ tions arise from the variation kx when Lorentz symmetry is a␮ , b␮ , c␮␯ , d␮␯ , e␮ , f ␮ , g␭␮␯ , H␮␯ typically vary with spontaneously broken. position, in accordance with the discussion in Sec. II A. They

105009-5 V. ALAN KOSTELECKY´ PHYSICAL REVIEW D 69, 105009 ͑2004͒ have no particular symmetry, except for the defining anti- and light-cone structure of the theory, which remains the symmetry of H␮␯ and of g␭␮␯ on two indices. By assump- subject of discussion even for Lorentz-invariant radiative tion, the action ͑12͒ is hermitian, which constrains the coef- corrections ͓35͔. ficients for Lorentz violation to be real. Relaxing the latter One difference between the QED extension in Minkowski constraint would permit the formalism to describe also non- and Riemann-Cartan spacetimes is that the presence of even hermitian Lorentz violation. Note the use of an uppercase weak gravitational couplings can change the effective prop- letter for H␮␯ , which avoids conflicts with the metric fluc- erties of certain coefficients for Lorentz violation. Adopting tuation h␮␯ . the weak-field form of the vierbein and spin connection The action ͑12͒ is also locally U͑1͒ invariant, by construc- given in Eq. ͑A20͒ of Appendix A and extracting from the tion. The covariant derivative D␮ appearing in it is under- Lagrangian only terms that are linear in small quantities, one stood to be a combination of the spacetime covariant deriva- finds tive, discussed in Appendix A, and the usual U͑1͒ covariant ͒ ͑ ␯␺Ϫ ͒ ¯␺␥ ␥␮␺ץL ʛϪ ͒ ¯␺␥␮ derivative: ␺ i͑ceff ␮␯ ͑beff ␮ 5 , 19

1 where ab ␮␺ϩ i␻␮ ␴ ␺ϪiqA␮␺. ͑15͒ץD␮␺ϵ 4 ab 1 ͒ ϵ Ϫ ϩ␹ ͑ceff ␮␯ c␮␯ h␮␯ ␮␯ , It is convenient to introduce the symbol (¯␺D¯ ␮) for the action 2 of the covariant derivative on a Dirac-conjugate field ¯␺: 1 ␣ ␤␥ 1 ␣␤␥ . ␹ ⑀␣␤␥␮ϩ T ⑀␣␤␥␮ ץ b ͒␮ϵb␮Ϫ͑ 1 eff 4 8 ab ͒ ͑ ␮¯␺Ϫ i␻␮ ¯␺␴ ϩiqA␮¯␺. ͑16͒ץ␺D¯ ␮͒ϵ¯͑ 4 ab 20 In this expression, leading-order terms arising from the scal- In terms of these quantities, the covariant derivative appears ing of the vierbein determinant e are neglected because they in the action ͑12͒ in a combination defined by are Lorentz invariant. ͑ ͒ a a a Equations 20 show that at leading order a weak back- ¯␹⌫ D␮␺ϵ¯␹⌫ D␮␺Ϫ͑¯␹D¯ ␮͒⌫ ␺. ͑17͒ J ground metric appears as a c␮␯ term, while the dual of the ⌫a antisymmetric part of the torsion behaves like a b␮ term, a This definition is understood to hold even when is result already noted elsewhere ͓36͔. The latter is a CPT- spacetime-position dependent. violating term, so the presence of background torsion can The generalized Dirac equation arising from the action S␺ mimic CPT violation. Experimental effects from these terms is have been estimated for some situations, including hydrogen spectral line shifts in the solar gravitational field ͓37͔ and 1 ␮ ⌫a ␺Ϫ ␺Ϫ ␭ ␮ ⌫a␺ reinterpretations of various recent results ͓38͔. Note, how- ie a D␮ M iT ␭␮e a 2 ever, that these gravitational couplings are flavor independent, whereas the values of b␮ and c␮␯ can depend on the 1 1 ϩ ␮ ␻ bcͩ ␩a ⌫ ϩ ͓␴ ⌫a͔ͪ ␺ϭ fermion species. This implies caution is required in interpret- ie ␮ i , 0. 2 a b c 4 bc ing the existing experimental sensitivities to b␮ in terms of ͑18͒ torsion, since some experiments are sensitive only to a nonzero difference in the value of b␮ for two fermion species. It As might be expected from nonderivative couplings, the further suggests that careful comparative experiments could Lorentz-violating terms involving M just add to the Dirac distinguish background curvature and torsion effects from equation in a minimal way. However, those involving ⌫a other sources of Lorentz and CPT violation. Note also that appear both minimally and through commutation with the the inclusion of subleading terms in the derivation would Lorentz generators in the covariant derivative. In particular, yield additional Lorentz-violating effects. For example, at the Lorentz-invariant parts of the last two terms in Eq. ͑18͒ this level all dimension-one effective coefficients for Lorentz cancel, but the terms involving coefficients for Lorentz vio- violation acquire a torsion dependence that can vary with lation yield nonzero results. flavor. Couplings of this type may play an important role in Many physical features of this theory are expected to be regions of possibly large torsion, such as spinning black similar to the QED extension in Minkowski spacetime intro- holes or the early Universe. duced in Ref. ͓3͔. Although beyond the scope of the present Another issue worth mention is the observability of vari- work, it would be of definite interest to investigate the cor- ous types of Lorentz violation. A given coefficient kx for rections to established results ͓3,13,16,27,29,33͔ arising from Lorentz violation leads to observable effects only when the the Riemann-Cartan couplings. A detailed study of quantum theory contains another conventional or Lorentz-violating corrections and renormalization issues may be particularly coupling that precludes the elimination of kx through field or challenging, since a satisfactory description of these is an coordinate redefinitions. In the Minkowski-spacetime limit open issue even for conventional Lorentz-invariant theories of the QED extension, the comparatively small number of in curved spacetime ͓34͔. Similar remarks apply to the causal couplings leaves the freedom to eliminate some Lorentz-

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violating terms ͓3,39,40͔. As might be expected, the presence In the QED extension there are comparatively few such non- of the additional curvature and torsion couplings in the minimal operators, and the only gauge-invariant ones are Riemann-Cartan spacetime reduces this freedom, but some products of the torsion with fermion bilinears. The Lorentz- options remain. invariant possibilities are As a ﬁrst example, consider a position-dependent redeﬁ- L ϭ ␭ ¯␺␥␮␺ϩ ␭ ¯␺␥ ␥␮␺ nition of the phase of the spinor: LI aeT ␭␮ beT ␭␮ 5 ␺͑x͒ϭexp͓if͑x͔͒␹͑x͒. ͑21͒ ϩ ␣␤␥⑀ ¯␺␥␮␺ϩ ␣␤␥⑀ ¯␺␥ ␥␮␺ a5eT ␣␤␥␮ b5eT ␣␤␥␮ 5 .

This is not a gauge transformation, since A␮ remains un- ͑23͒ changed. In the single-fermion Minkowski-spacetime limit ␮ with constant a␮ , the choice f (x)ϭa␮x can be used to The last of these already occurs in the minimal couplings. eliminate all four coefficients a␮ ,soa␮ is unphysical. How- The Lorentz-violating possibilities are ever, in Riemann-Cartan spacetime, the redefinition can typi- L ϭ ␣␤␥¯␺␺ϩ ␣␤␥¯␺␥ ␺ cally be used to eliminate only one of the four coefficients LV ek␣␤␥T ek5␣␤␥T 5 a␮ . An exception to this occurs for special models in which ␣␤␥ ␦ ␣␤␥ ␦ ϩek␣␤␥␦T ¯␺␥ ␺ϩek ␣␤␥␦T ¯␺␥ ␥ ␺ a␮ arises as the four-derivative of a scalar, in which case a␮ 5 5 is unphysical and can be removed. ␣␤␥ ␦⑀ ϩek␣␤␥␦⑀T ¯␺␴ ␺. ͑24͒ Another useful class of redefinitions consists of ones taking the general form If Lorentz violation is suppressed as expected and the torsion ␺͑x͒ϭ͓1ϩv͑x͒ ⌫͔␹͑x͒. ͑22͒ is also small, then all five of the latter are subdominant. Also, • if the torsion is constant or sufficiently slowly varying, only Here, v(x) is a set of complex functions with appropriate the last three are relevant. Nonetheless, all the above opera- local Lorentz indices and, for this equation only, ⌫ represents tors may be of interest in more exotic scenarios. Note that ␥a ␥ ␥a ␴ab one of , 5 , . These redefinitions can be regarded as the presence of fundamental scalars, such as the Higgs dou- position-dependent mixings of components in spinor space. blet in the SME, permits other types of nonminimal gravita- They can be used to show that, at leading order in coeffi- tional couplings of dimension four or less, including ones cients for Lorentz violation, there are no physical effects involving both curvature and torsion. Note also that any op- from the coefficients e␮ , f ␮ or from the antisymmetric parts erators of dimension greater than four must come with one or of c␮␯ , d␮␯ . However, attempting to remove the antisym- more inverse powers of mass, which may represent substan- metric and trace parts of g␭␮␯ generically introduces tial Planck-scale suppression. However, some care is re- spacetime-dependent mass terms proportional to the covari- quired in determining the relative dominance of operators. ant derivative of v, a feature absent in the Minkowski- For example, a dimension-five Lorentz-invariant operator spacetime limit. suppressed by the Planck mass m P would produce effects The freedom to redefine spacetime coordinates, perhaps comparable in magnitude to those of a dimension-four op- accompanied by field and coupling rescalings, can also be erator involving a coefficient for Lorentz violation sup- viewed as a means of eliminating or interrelating certain co- pressed by m P . efficients for Lorentz violation. The symmetric piece of the B. Photon sector coefficients c␮␯ and the 9 part of the photon-sector coeffi- s The photon part of the action for the QED extension in cient (kF)␬␭␮␯ , which is introduced in the next subsection, appear in the action in a form similar to parts of the metric Riemann-Cartan spacetime can be separated into two pieces, coupling. Appropriate coordinate choices can therefore ap- ϭ ͵ 4 ͑L ϩL ͒ ͑ ͒ pear to move the Lorentz violation from one sector to the SA d x F A , 25 other, or perhaps act to cancel effects between sectors. The coordinate frame used in reporting experimental results is where often implicitly fixed by the experimental setup, for example, by the choice of a standard clock or rod. Particular care is 1 1 L ϭϪ ␮␯Ϫ ͒ ␬␭ ␮␯ ͑ ͒ therefore required in claiming or interpreting sensitivities to F eF␮␯F e͑kF ␬␭␮␯F F , 26 these types of coefficients. An explicit example of this type 4 4 of redefinition is given for the case of Minkowski spacetime ͓ ͔ 1 in Sec. II C of Ref. 29 , where a constant coefficient of the L ϭ ͒␬⑀ ␭ ␮␯Ϫ ͒ ␬ ͑ ͒ A e͑kAF ␬␭␮␯A F e͑kA ␬A . 27 type c00 is converted into the combination (kF)0 j0 j . When 2 background curvature and torsion fields are present, the position dependence can complicate the analysis of these types The Lagrangian terms are hermitian provided the coefficients of redefinitions and can introduce other effects such as for Lorentz violation (kF)␬␭␮␯ ,(kAF)␮ , and (kA)␮ are real. spacetime-varying couplings. The electromagnetic field strength F␮␯ is defined by the lo- To conclude this subsection, here are a few remarks about cally U͑1͒-invariant form nonminimal gravitational couplings. For simplicity, attention ␭ ␯A␮ . ͑28͒ץ␮A␯Ϫץis restricted here to operators of mass dimension four or less. F␮␯ϵD␮A␯ϪD␯A␮ϩT ␮␯A␭ϭ

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␣ By definition, all curvature and torsion contributions cancel shift of . The couplings of the remaining 9s and 10s in the field strength. Gravitational effects in the photon- Lorentz-violating terms are similar to those in Minkowski sector Lagrangian therefore are associated with the appear- spacetime ͓3,29͔ but now typically vary with position. These ance of the metric in the index contractions and with the 19 coefficients control the leading-order CPT-even Lorentz scaling by the vierbein determinant e. violation in the photon sector. L ͑ ͒ The generalized Maxwell equations obtained from the ac- Next, consider the Lagrangian A in Eq. 27 , which con- tion ͑25͒ are conveniently written using the standard sists of CPT-odd terms. The corresponding partial action is ͑ ͒ Riemann-spacetime covariant derivative D˜ ␮ , described in U 1 gauge invariant only under special circumstances. As- Appendix A. They consist of the homogeneous equations suming no monopoles, as before, the coefficients for Lorentz violation must obey D˜ ␭F␮␯ϩD˜ ␮F␯␭ϩD˜ ␯F␭␮ϭ0, ͑29͒ ˜ ͒ Ϫ ˜ ͒ ϭ D␮͑kAF ␯ D␯͑kAF ␮ 0, which follow from the definition ͑28͒ of the field strength, ˜ ͒␮ϭ ͑ ͒ and the inhomogeneous equation obtained by varying the D␮͑kA 0, 32 sum of the fermion action ͑12͒ and the photon action ͑25͒: where the tilde again indicates the zero-torsion limit. These ˜ ␣ϩ ˜ ͒ ␤␥ ϩ ͒␣⑀ ␤␥ϩ ͒ D␣F␮ D␣͓͑kF ␮␣␤␥F ͔ ͑kAF ␮␣␤␥F ͑kA ␮ conditions must be satisfied in addition to any dynamical or other equations determining the form of (kAF)␮ and (kA)␮ . ϭ j␮ . ͑30͒ For (kAF)␮ , an example of this is known: the mechanism for Lorentz violation in the supergravity cosmology of Ref. ͓13͔ ␾ ␾ ץIn this equation, the current j␮ is ϵ enforces (kAF)␮ ␮ for an axion scalar , which satisfies the requirement ͑32͒. However, for the coefficient (k )␮ , ␮ϭ ␮ ¯␺⌫a␺ ͑ ͒ A j qe a . 31 ͑ ͒ ϭ Eq. 32 implies (kA)␮ (k0)␮ /e, where (k0)␮ is a constant 4-vector. Generic manifolds do not admit such vectors, so These results correctly reduce to the usual QED extension in (k )␮ must typically vanish. This is consistent with other the Minkowski-spacetime limit. A requirements emerging in the Minkowski-spacetime limit Consider first the Lagrangian L , which is invariant un- F ͓3͔. der local U͑1͒ transformations by construction. The first term As in the fermion sector, the presence of weak gravita- in L is the Lorentz-invariant action for photons in a F tional couplings can affect the interpretation of certain coef- Riemann-Cartan background, while the second term violates ficients for Lorentz violation. The leading-order weak-field Lorentz invariance. Both terms are CPT even. The coeffi- couplings can be extracted from the Lorentz-invariant part of cient (k )␬␭␮␯ for Lorentz violation is antisymmetric on the F the Lagrangian L using the expression ͑A20͒ of Appendix first two and on the last two indices, and it is symmetric F A. The result is a contribution that has the operator structure under interchange of the first and last pair of indices. These of the (k )␬␭␮␯ term, with an effective coefficient (k )␬␭␮␯ symmetries reduce the number of independent components F F,eff given by of (kF)␬␭␮␯ to 21. Decomposing into irreducible Lorentz ϭ ϩ ϩ ϩ multiplets gives 21 1a (1 9 10)s . 1 ͒ ϭ ͒ ϩ ␩ ϩ␩ The antisymmetric singlet 1a provides a Lorentz-invariant ͑kF,eff ␬␭␮␯ ͑kF ␬␭␮␯ ͑ ␬␮h␭␯ ␭␯h␬␮ ϵ⑀␬␭␮␯ 2 parity-odd coupling k1 (kF)␬␭␮␯ . Its coupling in the ˜ ␮␯ ˜ Ϫ␩ Ϫ␩ ͑ ͒ Lagrangian is therefore proportional to ek1F␮␯F , where F ␬␯h␭␮ ␭␮h␬␯͒. 33 is the dual field strength. Integrating by parts and discarding the surface term under the usual assumption of no monopoles A weak-field background metric can therefore partially simu- converts this into an expression proportional to late the effect of the coefficient (kF)␬␭␮␯ for Lorentz viola- ␮␯ tion. Some of the physical implications of this coupling can e(D␮k )A␯˜F . In the Minkowski-spacetime limit with con- 1 be appreciated by converting to the notation of Ref. ͓29͔. stant (kF)␬␭␮␯ , no net effect results. In the present more Setting (k )␮ and (k )␬␭␮␯ to zero for simplicity, only the general case with position-dependent (k )␬␭␮␯ , the expres- AF F F ␬ jk ␬ jk ␬ sion can instead be absorbed into the term involving the coefficients (˜ eϪ) ,(˜ oϩ) , ˜ tr acquire nonzero contribu- L tions, given by coefficient (kAF)␮ in A . This conversion of a scalar into a Lorentz-violating coefficient has features in common with 1 the generation of a nonzero (k )␮ through the gradient of ␬ ͒ jkϭϪ jkϩ ll␩ jk AF ͑˜ eϪ h h , the axion in supergravity cosmology ͓13͔. 3 Of the remaining 20 independent coefficients, the sym- ͑␬ ͒ jkϭϪ⑀ jkl 0l metric singlet 1s is the irreducible double trace, which is ˜ oϩ h , Lorentz invariant. It can be regarded as renormalizing the Lorentz-invariant kinetic term. If (k )␬␭␮␯ varies with posi- 2 F ˜␬ ϭ hll. ͑34͒ tion, this renormalization corresponds to a spacetime varia- tr 3 ␣ tion of the fine structure constant . If instead (kF)␬␭␮␯ is constant, as is usually assumed in the Minkowski-spacetime One consequence is that both polarizations of light are af- limit, then the 1s generates only an unobservable constant fected in the same way, so no birefringence occurs. Experi-

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ϭ ϭ ments with sensitivity to these coefficients could therefore be UA (uA)R , DA (dA)R . The left-handed leptons and ͑ ͒ ϭ ␯ T adapted to study the background-metric fluctuation h␮␯ , pro- quarks form SU 2 doublets, LA ͓( A)L ,(lA)L͔ , QA ϭ T vided the signals in question involve no complete cancella- ͓(uA)L ,(dA)L͔ . tion of the effects. In the boson sector, the Higgs doublet ␾ is taken to have T As a final remark, note that the combined action ͑12͒ and the form ␾ϭ(0,r␾) /& in unitary gauge, and the conjugate ͑25͒ for the leading-order QED extension can be used to doublet is denoted ␾c. The color gauge fields are denoted by obtain a general classical action for the Lorentz-violating be- the hermitian SU͑3͒ adjoint matrix G␮ . The SU͑2͒ gauge havior of point test particles and electrodynamic fields in a fields also form a hermitian adjoint matrix denoted W␮ , Riemann-Cartan background. Although its explicit form lies while the hermitian singlet hypercharge gauge field is B␮ . beyond the scope of the present work, the resulting theory The associated field strengths are G␮␯ , W␮␯ , and B␮␯ . would represent a useful test model for Lorentz-violating They are defined by expressions of the standard form in physics. For example, it could be used to provide insight into Minkowski spacetime, except that the Riemann-Cartan cova- the interpretation of classical concepts such as mass, veloc- riant derivative is used and a torsion term is added in analogy ity, and geodesic trajectories, each of which typically is split to Eq. ͑28͒. This ensures that all spacetime curvature and by Lorentz violation into distinct notions that merge in the torsion contributions cancel in the field strengths, which Lorentz-invariant limit ͓3͔. It would also be of interest to therefore have conventional SU(3)ϫSU(2)ϫU(1) proper- obtain the connection between this theory and the TH⑀␮ ties. formalism ͓41,42͔, which is a widely used model involving a The covariant derivative D␮ and its conjugate D¯ ␮ are now four-parameter action with modified classical test particles understood to be both spacetime covariant and SU(3) and electrodynamic fields in a conventional static and spheri- ϫSU(2)ϫU(1) covariant, in parallel with the cally symmetric Riemann background. electromagnetic-U͑1͒ and spacetime covariant derivative ͑15͒ and its conjugate ͑16͒. The definition ͑17͒ is maintained. IV. STANDARD-MODEL EXTENSION As usual, the coupling strengths for the three groups SU͑3͒, ͑ ͒ ͑ ͒ SU 2 , and U 1 are g3 , g, and gЈ, respectively. Also, the The action SSME for the full SME in a Riemann-Cartan charge q for the electromagnetic U͑1͒ group and the angle spacetime can conveniently be expressed as a sum of partial ␪ are defined through qϭg sin ␪ ϭgЈ cos ␪ . actions W W W Consider first the action SSM for the SM in a Riemann- L ϭ ϩ ϩ ϩ ͑ ͒ Cartan background. The corresponding Lagrangian SM is SSME SSM SLV Sgravity ¯ . 35 SU(3)ϫSU(2)ϫU(1) gauge invariant, and it is convenient to separate it into five parts: The term SSM is the SM action, modified by the addition of gravitational couplings appropriate for a background L ϭL ϩL ϩL ϩL ϩL ͑ ͒ SM lepton quark Yukawa Higgs gauge . 36 Riemann-Cartan spacetime. The term SLV contains all L Lorentz- and CPT-violating terms that involve SM fields and The lepton sector has Lagrangian lepton given by dominate at low energies, including minimal gravitational 1 1 couplings. The term Sgravity represents the pure-gravity sec- ␮ a ␮ a L ϭ iee ¯L ␥ D␮L ϩ iee ¯R ␥ D␮R , tor, constructed from the vierbein and the spin connection lepton 2 a A J A 2 a A J A and incorporating possible Lorentz and CPT violation. The ͑37͒ ellipsis represents contributions to SSME that are of higher L order at low energies, some of which violate Lorentz sym- while the quark sector Lagrangian quark is metry. It includes terms nonrenormalizable in the 1 1 Minkowski-spacetime limit, nonminimal and higher-order ␮ a ␮ a L ϭ iee Q¯ ␥ D␮Q ϩ iee U¯ ␥ D␮U gravitational couplings, and operators of mass dimension quark 2 a A J A 2 a A J A greater than four coupling curvature and torsion to SM fields. 1 Other possible nonminimal operators formed from SM fields, ␮ a ϩ iee D¯ ␥ D␮D . ͑38͒ such as ones that break the SU(3)ϫSU(2)ϫU(1) gauge 2 a A J A invariance, can be included as needed. For example, these could play a significant role in the neutrino sector ͓32͔. The Yukawa couplings are In this section, the explicit forms of SSM and SLV are L ϭϪ ͒ ¯ ␾ ϩ ͒ ¯ ␾c presented, while discussion of the gravity action Sgravity is Yukawa ͓͑GL ABeLA RB ͑GU ABeQA UB deferred to Sec. V. The notation adopted for the basic SM ϩ ͒ ¯ ␾ ϩ ͑ ͒ fields is as follows. First, consider the fermion sector. Intro- ͑GD ABeQA DB͔ H.c., 39 duce the generation index Aϭ1,2,3, so that the three charged ϵ ␮ ␶ ␯ leptons are denoted lA (e, , ), the three neutrinos are A where (GL)AB ,(GU)AB ,(GD)AB are the Yukawa-coupling ϵ ␯ ␯ ␯ ϵ ( e , ␮ , ␶), and the six quark flavors are uA (u,c,t), matrices. The Higgs sector has Lagrangian d ϵ(d,s,b). The color index on the quarks is suppressed for A ␭ simplicity. Define as usual the left- and right-handed spinor L ϭϪ ͑ ␾͒† ␮␾ϩ␮2 ␾†␾Ϫ ͑␾†␾͒2 ␺ ϵ 1 Ϫ␥ ␺ ␺ ϵ 1 ϩ␥ ␺ Higgs e D␮ D e e , components L 2 (1 5) , R 2 (1 5) . The right- 3! ͑ ͒ ϭ handed leptons and quarks are SU 2 singlets, RA (lA)R , ͑40͒

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while the gauge sector is Remarks analogous to those for the lepton-sector coefficients for Lorentz violation also hold for the quark-sector coeffi- 1 ␮␯ 1 ␮␯ 1 ␮␯ cients in these equations. L ϭϪ eTr͑G␮␯G ͒Ϫ eTr͑W␮␯W ͒Ϫ eB␮␯B . gauge 2 2 4 The CPT-even Lorentz-violating Yukawa-type operators ͑41͒ have the usual Yukawa gauge structure but involve different fermion bilinears. The Lagrangian for these terms is Possible ␪ terms are omitted in the latter for simplicity. 1 Next, consider the partial action SLV containing Lorentz- CPTϩ ␮ ␯ ab L ϭϪ ͓͑H ͒␮␯ ee e ¯L ␾␴ R and CPT-violating operators constructed from SM fields of Yukawa 2 L AB a b A B mass dimension four or less. In parallel with Eq. ͑36͒, the L ϩ͑ ͒ ␮ ␯ ¯ ␾c␴ab corresponding Lagrangian LV can be decomposed as a sum HU ␮␯ABee ae bQA UB of terms separating the contributions from the lepton, quark, ϩ͑ ͒ ␮ ␯ ¯ ␾␴ab ͔ϩ ͑ ͒ Yukawa, Higgs, and gauge sectors. The Lagrangians for HD ␮␯ABee ae bQA DB H.c. 47 these five sectors can be further split into pieces that are CPT even and odd, except for the Yukawa-type couplings for The dimensionless coefficients (HL,U,D)␮␯AB are antisym- which no CPT-odd terms arise: metric in the spacetime indices. Like the conventional Yukawa couplings (GL,U,D)AB , they can violate hermiticity L ϭLCPTϩϩLCPTϪϩLCPTϩϩLCPTϪϩLCPTϩ ϩLCPTϩ in generation space. LV lepton lepton quark quark Yukawa Higgs The CPT-even Lagrangian in the Higgs sector is ϩLCPTϪϩLCPTϩϩLCPTϪ ͑ ͒ Higgs gauge gauge . 42 1 CPTϩ ␮␯ † L ϭ ͑k␾␾͒ e͑D␮␾͒ D␯␾ϩH.c. The Lagrangian for the CPT-even lepton sector is Higgs 2

1 1 ␮␯ 1 ␮␯ CPTϩ ␮ a ␯ Ϫ ͑ ͒ ␾† ␾Ϫ ͑ ͒ ␾†␾ L ϭϪ i͑c ͒␮␯ ee ¯L ␥ D L k␾W e W␮␯ k␾B e B␮␯ . lepton 2 L AB a A J B 2 2 ͑ ͒ 1 48 Ϫ ͑ ͒ ␮ ¯ ␥a ␯ ͑ ͒ i cR ␮␯ABee aRA DJ RB , 43 2 All the coefficients for Lorentz violation in this equation are ␮␯ dimensionless. The coefficient (k␾␾) can be taken to have where the dimensionless coefficients (cL)␮␯AB and (cR)␮␯AB symmetric real and antisymmetric imaginary parts, while can be taken to be hermitian in generation space. The space- ␮␯ ␮␯ (k␾W) and (k␾B) are real antisymmetric. The last two time traces of these coefficients preserve Lorentz symmetry. terms directly couple the Higgs scalar to the SU(2)ϫU(1) In the Minkowski-spacetime limit with conserved energy and field strengths. They have no analogue in the usual SM. The momentum, these traces act to renormalize the fermion fields CPT-odd Higgs Lagrangian is and are unobservable, but in the present context the spacetime dependence can correspond to spacetime-varying cou- LCPTϪϭ ͑ ͒␮ ␾† ␾ϩ ͑ ͒ Higgs i k␾ e D␮ H.c. 49 plings. The Lagrangian for the CPT-odd lepton sector is ␮ The coefficient (k␾) is complex valued and has dimensions LCPTϪϭϪ͑ ͒ ␮ ¯ ␥a Ϫ͑ ͒ ␮ ¯ ␥a lepton aL ␮ABee aLA LB aR ␮ABee aRA RB , of mass. ͑44͒ The Lagrangian for the CPT-even gauge sector is

where the coefficients (a )␮ and (a )␮ are also hermit- L AB R AB CPTϩ 1 ␬␭ ␮␯ L ϭϪ ͑k ͒␬␭␮␯eTr͑G G ͒ ian in generation space but have dimensions of mass. gauge 2 G The quark-sector Lagrangians take a similar form: 1 Ϫ ͑ ͒ ͑ ␬␭ ␮␯͒ 1 kW ␬␭␮␯eTr W W CPTϩ ␮ a ␯ 2 L ϭϪ i͑c ͒␮␯ ee Q¯ ␥ D Q quark 2 Q AB a A J B 1 Ϫ ͑ ͒ ␬␭ ␮␯ ͑ ͒ 1 kB ␬␭␮␯eB B . 50 ␮ a ␯ 4 Ϫ i͑c ͒␮␯ ee U¯ ␥ D U 2 U AB a A J B All the coefficients for Lorentz violation in this equation are 1 ␮ a ␯ real. Each is antisymmetric on the first two and on the last Ϫ i͑c ͒␮␯ ee D¯ ␥ D D , ͑45͒ 2 D AB a A J B two indices, and each is symmetric under interchange of the first and last pair of indices. Their spacetime properties are Ϫ ␮ LCPT ϭϪ͑ ͒ ¯ ␥a similar to those of the coefficient (kF)␬␭␮␯ in the photon quark aQ ␮ABee aQA QB sector of the QED extension, discussed in Sec. III B, which is Ϫ ͒ ␮ ¯ ␥a ͑aU ␮ABee aUA UB itself a combination of (kW)␬␭␮␯ and (kB)␬␭␮␯ . Note that possible total-derivative terms analogous to the usual ␪ terms Ϫ ͒ ␮ ¯ ␥a ͑ ͒ ͑ ͒ ͑aD ␮ABee aDA DB . 46 in the SM are neglected in Eq. 50 for simplicity.

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It is also possible to construct some CPT-odd Lagrangian As in the case of the QED extension, care is required in terms that under special circumstances are invariant under determining the observability of a given coefficient for Lor- ϫ ϫ infinitesimal SU(3) SU(2) U(1) transformations. They entz violation in SLV because there is freedom to eliminate have the Chern-Simons form certain coefficients by appropriate field and coordinate redefinitions. For example, for each fermion field there is a LCPTϪϭ͑ ͒ ⑀␬␭␮␯ ͑ ϩ 2 ͒ gauge k3 ␬ eTr G␭G␮␯ 3 ig3G␭G␮G␯ phase degree of freedom of the form ͑21͒ and possible rein-

␬␭␮␯ 2 terpretations of the spinor-space components of the form ϩ͑k ͒␬⑀ eTr͑W␭W␮␯ϩ igW␭W␮W␯͒ 2 3 ͑22͒. There is also freedom in the Higgs sector, including the ϩ ͒ ⑀␬␭␮␯ ϩ ͒ ␬ ͑ ͒ phase redefinition ͑k1 ␬ eB␭B␮␯ ͑k0 ␬eB . 51 All the coefficients for Lorentz violation in these equations ␾͑x͒ϭexp͓Ϫig͑x͔͒␳͑x͒. ͑52͒ can be taken real. The coefficients (k1,2,3)␬ have dimensions ␮ For instance, the choice g(x)ϭ(k␾)␮x can be used to ab- of mass, while (k0)␬ has dimensions of mass cubed. These terms are the SM analogues of those in Eq. ͑27͒ of the QED sorb part of the effects from the coefficient (k␾)␮ . Also, extension, which they contain as a limiting case. Their in- suitable coordinate redefinitions can interrelate some of the variance requires that subsidiary conditions generalizing fermion coefficients c␮␯ , Higgs coefficients (k␾␾)␮␯ , and those in Eq. ͑32͒ be satisfied, so Eq. ͑51͒ is relevant only in 9s Lorentz-irreducible pieces of the gauge coefficients special circumstances. (kG)␬␭␮␯ ,(kW)␬␭␮␯ ,(kB)␬␭␮␯ . However, the presence of cross couplings between generations means that some types The above equations describe the actions SSM and SLV prior to the breaking of the electroweak SU(2)ϫU(1) sym- of coefficient unobservable in the QED extension are now metry to the electromagnetic U͑1͒ subgroup. In the minimal physical under suitable experimental circumstances. For ex- SM in Minkowski spacetime, arguments based on energetics ample, the presence of flavor-changing weak interactions in make this breaking plausible, at least for some range of the the SME quark sector means that differences between con- couplings ␮ and ␭ in Eq. ͑40͒. However, it is an open issue stant coefficients of the a␮ type become observable in inter- whether the Higgs potential in Eq. ͑40͒ suffices to drive elec- ferometric experiments with neutral-meson oscillations, a ͓ ͔ troweak symmetry breaking to the charge subgroup in the feature absent in the QED extension 19 . SM in a curved spacetime background ͓43͔. Suppose this is indeed the case for at least some types of background, per- V. GRAVITATIONAL SECTOR haps such as weak gravitational fields. Then, the presence in A. Action SLV of small Lorentz-violating terms involving the Higgs and charge-neutral fields changes the pattern of expectation val- It is convenient to write the pure-gravity action as ues that break the SU(2)ϫU(1) symmetry. A small Lorentz- 0 1 violating expectation value emerges for the neutral Z␮ field, ϭ ͵ 4 L ͑ ͒ Sgravity d x gravity , 53 and the expectation value of the Higgs is shifted slightly. It 2␬ has been shown that this breaking pattern preserves the elec- ␬ tromagnetic U͑1͒ in the Minkowski-spacetime limit ͓3͔.A where the usual gravitational coupling constant 1/2 ϵ ␲ Ӎ ϫ 36 2 careful study of this issue in Riemann-Cartan spacetime 1/16 GN 3 10 GeV has been factored outside the L would be of interest. Note also that the standard procedure of integral for convenience. The Lagrangian gravity can then be separated as expanding the terms in SSM and SLV about the vacuum expectation values generates additional effective contributions L ϭLLI ϩLLV ϩ ͑ ͒ to some of the coefficients for Lorentz violation. gravity e,␻ e,␻ ¯ , 54 The presence of weak curvature and torsion couplings in LLI the actions S and S can modify the interpretation of where the Lorentz-invariant piece e,␻ and the Lorentz- SM LV LLV a certain coefficients for Lorentz violation. The contributions violating piece e,␻ are constructed using the vierbein e␮ ab and the spin connection ␻␮ . Following Sec. II A, the latter of this type from SLV are proportional to the product of weak fields and coefficients for Lorentz violation, so they are sup- are viewed as basic dynamical objects for the gravitational ͑ ͒ field. The ellipsis represents possible dependence on other pressed relative to those from SSM . The expansions A20 of dynamical gravitational fields, which could be fundamental Appendix A can be used to extract from SSM the dominant effects. The analysis follows a pattern similar to that in the or composite and could have both Lorentz-invariant and ͑ ͒ QED extension leading to Eqs. ͑20͒ and ͑33͒, with the sym- Lorentz-violating parts. The Lagrangian 54 is assumed to metric part of the metric generating effective contributions to combine with the matter and gauge sectors of the SME, per- certain CPT-even Lorentz-violating terms and the torsion haps along with other modes as yet unobserved, to yield a generating contributions to CPT-odd ones. The effects of the smooth connection to the underlying theory at the Planck vierbein and the torsion are independent of flavor at leading scale. LLI order, but the sign of the torsion contribution depends on the The Lorentz-invariant Lagrangian e,␻ can be written as a handedness of the fermion. This is reflected in Eq. ͑20͒ for series in powers of the curvature, torsion, and covariant de- the fermion sector of the QED extension, where the coeffi- rivatives: cient b␮ϳ(a )␮ Ϫ(a )␮ is affected but a␮ϳ(a )␮ L AB R AB L AB LI ϩ L ␻ϭeRϪ2e⌳ϩ . ͑55͒ (aR)␮AB is unchanged. e, ¯

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The first term in this expression is the Einstein-Hilbert La- efficients for Lorentz violation with an even number of indi- L grangian EH in Riemann-Cartan spacetime, while the sec- ces can contribute to a position-dependent term of the same ond contains the cosmological constant ⌳. When coupled to general form as the cosmological-constant term. The net ef- matter and gauge fields with energy-momentum and spin- fective cosmological constant may therefore be partially or density tensors defined as in Eq. ͑4͒, these two terms gener- entirely due to Lorentz violation and may vary with space- ate field equations of the form time position. It is conceivable that a simple model could be found featuring a realistically small cosmological constant G␮␯ϩ⌳g␮␯ϭ␬T ␮␯, e tied to small Lorentz violation. The Lagrangian series ͑55͒ and ͑57͒ can be organized ˆ ␭␮␯ϭ␬ ␭␯␮ ͑ ͒ T S␻ 56 according to the mass dimension of the operators or directly in powers of the fields. In any case, several potential simpli- for the Riemann-Cartan spacetime, where the trace-corrected fications can be considered. First, appropriate use of the Bi- ˆ ␭␮␯ ͑ ͒ torsion T is defined in Eq. A10 of Appendix A. In the anchi identities for the curvature and torsion may eliminate ͑ ͒ Lorentz-invariant Lagrangian 55 , the ellipsis represents some combinations of operators. Second, partial integrations possible higher-order terms in curvature, torsion, and cova- on operators with covariant derivatives can be used to inter- riant derivatives. These terms generate corrections to the relate terms if total derivatives are disregarded. In this way, field equations ͑56͒, and they can produce independently ␬␭␮␯ ͑ ͒ for instance, the coefficient (kDT) in Eq. 57 can be propagating vierbein and spin-connection modes correspond- ␣␤␥␭␮␯ converted into a special case of the coefficient (kTT) . ing to dynamical torsion and curvature. Note that terms with Also, general topological results such as the Gauss-Bonnet mass dimension greater than two typically lead to higher- theorem imply that under suitable circumstances some com- derivative conditions. The complexity of the Lagrangian se- binations of terms form topological invariants and so could ries is already considerable at second order in the curvature be removed in the classical action. ͓ ͔ and torsion 44 . However, the explicit form of the higher- The Lorentz-violating terms in the Lagrangian ͑57͒ intro- order Lorentz-invariant terms is unnecessary for present pur- duce spacetime anisotropies in the gravitational field equa- poses. tions, which in turn could trigger various physical conse- Following the discussion in Sec. II A, each term in the quences of theoretical and experimental relevance. Standard LLV Lorentz-violating Lagrangian e,␻ is constructed by combin- gravitational solutions such as those for black holes, cosmol- ing coefficients for Lorentz violation with gravitational field ogy, gravitational waves, and post-Newtonian physics are all operators to produce a quantity that is both local observer expected to be corrected by terms depending on the coeffi- Lorentz invariant and general observer coordinate invariant. cients for Lorentz violation in Eq. ͑57͒. These effects would The relevant field operators are formed from the vierbein, the be independent of ones induced by Lorentz violation in the spin connection, and their derivatives. It is convenient to matter and gauge sectors of the SME. Both for gravitational express these operators in terms of the curvature, torsion, and quanta and for other fundamental particles in the SME, the covariant derivatives wherever possible. The Lagrangian ensuing Lorentz-violating behavior can depend on momen- LLV e,␻ can then also be written as a series: tum magnitude and orientation, spin magnitude and orientation, and the particle species and CPT properties. LLV ϭ ͑ ͒␭␮␯ ϩ ͑ ͒␬␭␮␯ e,␻ e kT T␭␮␯ e kR R␬␭␮␯ The effects of Lorentz violation are likely to be large only ␣␤␥␭␮␯ in regions of large curvature and torsion, such as near black ϩe͑k ͒ T␣␤␥T␭␮␯ TT holes or in the early Universe, or in certain cosmological ϩ ͑ ͒␬␭␮␯ ϩ ͑ ͒ e kDT D␬T␭␮␯ ¯ . 57 contexts such as those involving the cosmological constant, dark matter, or dark energy. Nonetheless, Lorentz-violating In this equation, all the coefficients for Lorentz violation are effects could be detectable in various situations. For ex- real, and they inherit the symmetries of the associated ample, the homogeneous Friedman-Robertson-Walker cos- ␭␮␯ Lorentz-violating operators. The coefficient (kT) has di- mological solutions may acquire anisotropic corrections, po- mensions of mass, while the others listed are dimensionless. tentially leading to a realistic anisotropic cosmology with The ellipsis represents higher-order terms in the curvature, observable signals. Candidate Lorentz-violating cosmologi- torsion, and covariant derivatives, along with other possible cal effects include the alignment anomalies on large angular higher-order terms such as the gravitational analogue of the scales reported in the Wilkinson microwave anisotropy probe Chern-Simons terms ͑51͒ in the SME gauge sector ͓45͔.At ͑WMAP͒ data ͓46͔, which are theoretically problematic in low energies, the leading-order terms displayed explicitly in conventional scenarios ͓47͔. Another example is provided by Eq. ͑57͒ describe dominant effects of Lorentz violation. As the gravitational-wave equations, which acquire corrections the relevant energies increase towards the Planck scale, from the coefficients for Lorentz violation in Eq. ͑57͒. The higher-order terms represented by the ellipsis in Eq. ͑57͒ are resulting effects are compounded in certain scenarios for expected to play an increasingly significant role. Lorentz violation. For instance, the Goldstone modes arising LLV Note that any coefficients for Lorentz violation in e,␻ from spontaneous Lorentz violation are known to affect the with an even number of indices can also yield Lorentz- propagating degrees of freedom ͓4,48͔. Spacetime- invariant contributions to the Lagrangian ͑54͒, since they can anisotropic features of gravitational modes may eventually contain pieces proportional to products of g␮␯ and ⑀␬␭␮␯. be detectable in Earth- or space-based gravitational-wave ex- Similarly, by direct contraction with g␮␯ and ⑀␬␭␮␯, any co- periments ͓49͔. For suitable astrophysical sources, compari-

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sons of the speed of gravitational waves with the speed of constant terms, together with the curvature-linear Lorentz- light and neutrinos may also eventually be feasible, which violating piece of Eq. ͑57͒. In fact, the resulting action could would represent direct sensitivity to a combination of coef- also be obtained directly by starting from general relativity ficients for Lorentz violation in the gravitational, photon, and and imposing plausible constraints on the form of allowed matter sectors of the SME. Similarly, Lorentz violation may Lorentz-violating terms. It is convenient to expand the coef- ␬␭␮␯ ͑ ͒ be detectable in laboratory and space-based experiments ficient (kR) for Lorentz violation in Eq. 57 and to write studying post-Newtonian gravitational physics, such as tests the action in the form of the inverse square law ͓50͔ or of gravitomagnetic effects, 1 including geodetic precession and the dragging of inertial 4 ␮␯ S ␻ ⌳ϭ ͵ d x͓e͑1Ϫu͒RϪ2e⌳ϩes R␮␯ frames ͓51͔. The detailed exploration of all these effects e, , 2␬ would be of definite interest but lies beyond the scope of the ϩ ␬␭␮␯ ͔ ͑ ͒ present work. et R␬␭␮␯ . 58 Experiments sensitive to Lorentz violation in the matter ␮␯ ␬␭␮␯ The introduction of the coefficients s , t , u explicitly and gauge sectors of the SME ͓13,17–30͔ suggest that the distinguishes unconventional effects involving the Riemann, coefficients for Lorentz violation are minuscule, which is Ricci, and scalar curvatures and so can simplify the consid- consistent with the notion that they arise as Planck- eration of certain special models. As an example, consider suppressed effects. If this feature extends to the gravitational the action ͑B3͒ of the curvature-coupled bumblebee model sector as expected, it is likely that the many existing standard ␮ ␮ ␮ described in Appendix B. With the field B ϭb ϩ␦B ex- experimental tests of gravity ͓42͔ would lack sufficient sen- panded about its Lorentz-violating vacuum value, this theory sitivity to detect Lorentz violation, although a few may ex- incorporates only a coefficient for Lorentz violation of the hibit the necessary exceptional sensitivity. For the analysis of ␮␯ s type: these experiments in the context of metric theories of gravity, a widely applicable test framework exists, called the param- ␮␯ 1 etrized post-Newtonian ͑PPN͒ formalism ͓52,53͔. A standard s ϭ␰b␮b␯Ϫϩ ␰b2g␮␯. ͑59͒ B 4 version of this formalism ͓42͔ that is relevant for solar sys- tem experiments assumes a Riemann spacetime asymptotic In this equation, the trace has been absorbed into a u rescal- to Minkowski spacetime, a perfect fluid obeying conven- ing of R, although this could be avoided by adding an extra tional equations for the covariant conservation of energy mo- Ϫ 1 ␰ 2 ͑ ͒ term 4 eB R to the Lagrangian B3 . In general, if indeed mentum and for electrodynamic fields, and conventional there is Lorentz violation in nature, coefficients for Lorentz geodesic equations for test particles. This PPN formalism violation of only the s␮␯ or only the t␬␭␮␯ type might well contains ten parameters, and bounds on them have been ob- emerge as the result of a comparatively simple mechanism at tained in a variety of experiments. Under suitable assump- the Planck scale. tions on the SME matter sector and in the zero-torsion limit, The coefficients for Lorentz violation s␮␯ and t␬␭␮␯ ap- an explicit connection between the SME coefficients for Lor- pearing in the action ͑58͒ are real and dimensionless. By entz violation and the PPN parameters should exist. Al- definition, s␮␯ inherits the symmetries of the Ricci tensor though beyond the scope of the present work, determining and t␬␭␮␯ inherits those of the Riemann tensor. In consider- this connection would also be of definite interest. ing the full theory ͑58͒, the saturated traces of these coeffi- ␮ ␮␯ cients could be assumed to vanish, s ␮ϭt ␮␯ϭ0, since any nonzero values could be absorbed into the Lorentz-invariant B. Riemannian limit ␬␭␮␯ ␭␮ ␯ coefficient u. Moreover, single traces of t such as t ␭ The Lorentz-violating extension of Einstein’s theory of could also be assumed zero, since nonzero contributions general relativity is contained in the results of the previous could be absorbed into s␮␯. It follows that the theory ͑58͒ subsection as the limit in which the torsion vanishes. This involves 19 independent Lorentz-violating degrees of free- Riemann-spacetime limit is of interest both its own right and dom, nine controlled by the trace-free coefficient s␮␯ and ten also as a case in which the field equations remain compara- controlled by the trace-free t␬␭␮␯. Only one combination of 0 ϵϪ j tively simple. Even in a Riemann-Cartan spacetime with these 19 coefficients, given in a local frame by s 0 s j ,is nonzero torsion, the relevant dominant Lorentz-violating ef- locally rotation invariant. Note that the vanishing-trace as- fects can under suitable circumstances be extracted from the sumptions are equivalent to replacing s␮␯ and t␬␭␮␯ with zero-torsion limit because in realistic situations torsion ef- their irreducible Ricci and Weyl pieces, whereupon the fects are typically heavily suppressed compared to curvature Lorentz-violating part of the Lagrangian for the action ͑58͒ effects. could be written in the form The remainder of this subsection assumes that quantities L ʛ ␮␯ T ϩ ␬␭␮␯ ͑ ͒ such as the curvature tensor, its contractions, covariant de- e,␻,⌳ es R␮␯ et C␬␭␮␯ , 60 rivatives, and the Einstein tensor are all evaluated in the T zero-torsion limit. For simplicity, the tilde notation for these where R␮␯ is the trace-free Ricci tensor and C␬␭␮␯ is the quantities adopted elsewhere in the present work is sup- Weyl tensor. ␮␯ ␬␭␮␯ pressed throughout this subsection. The above properties of s and t are reminiscent of The leading-order Lagrangian terms for this zero-torsion those for the coefficient (kF)␬␭␮␯ in the QED extension or theory consist of the Einstein-Hilbert and cosmological- the CPT-even coefficients in the gauge sector of the SME.

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This is because s␮␯ and t␬␭␮␯ are extracted from the coeffi- symmetries of s␮␯ and t␬␭␮␯. The result ͑64͒ in turn can be ␬␭␮␯ ͑ ͒ ͑ ͒ cient (kR) in Eq. 57 , which like (kF)␬␭␮␯ has the sym- used to obtain the trace-reversed version of Eq. 63 : metries of the Riemann tensor. Among the consequences is ␮␯ that the coefficient s can under suitable circumstances be ␮␯ ␮␯ 1 ␮␯ ␮␯ R ϭ␬ͩ T Ϫ g T ͪ ϩ͑TRst͒ moved to other sectors of the SME by redefining the coordi- g 2 g nates and fields, following the discussion at the end of Sec. III A. 1 ␮␯ ␣␤ ␣␤␥␦ ϩ g ͑D␣D␤s ϩR␣␤␥␦t ͒. ͑65͒ Since the theory ͑58͒ is torsion free, the gravitational field 2 equations can be obtained directly by varying with respect to ␮␯ ␬␭␮␯ the metric while treating the spin connection as a dependent The presence of nonzero s and t also allows some variable. Restricting attention for simplicity on the case with qualitatively different types of trace condition. For example, ␮␯ ␮␯ ͑ ͒ uϭ⌳ϭ0 but making no assumptions about the traces of s contracting s with Eq. 63 yields ␬␭␮␯ and t , the variation of the action can be written as ␮␯Ϸ␬ ␮␯ ͑ ͒ s␮␯G s␮␯Tg 66 1 ␦ ϭ ͵ 4 ͓Ϫ ␮␯ϩ͑ Rst͒␮␯͔␦ ϩ ␮␯␦ to first order in the small coefficients for Lorentz violation. Se,␻ ␬ d xe G T g␮␯ eR s␮␯ 2 Acting with D␮ on the extended Einstein equations ͑63͒ ␮␯ ␬␭␮␯ and imposing the trace Bianchi identity D␮G ϭ0 yields ϩeR ␦t␬␭␮␯ . ͑61͒ the condition

The variations ␦s␮␯ and ␦t␬␭␮␯ are included in this expres- ␮ Rst ␮ ␬D␮T ␯ϭϪD␮͑T ͒ ␯ sion for completeness. They contribute to the variational g ␮␯ ␬␭␮␯ equations fixing the coefficients s , t for Lorentz vio- 1 ␣␤ ␣␤ 1 ␣ ͑ ͒ Rst ␮␯ ϭϪ R D␯s␣␤ϩR D␤s␣␯ϩ s␣␯D R lation. In Eq. 61 , the quantity (T ) is defined by 2 2 1 1 Rst ␮␯ ␣␤ ␮␯ ␮␣ ␯ ␯␣ ␮ ␮ ␣␯ 1 ␣␤␥␦ ␣␤␥␦ ͑T ͒ ϵ s R␣␤g Ϫs R␣ Ϫs R␣ ϩ D␣D s Ϫ R D␯t␣␤␥␦ϩ2R D␦t␣␤␥␯ 2 2 2 1 1 1 Ϫ ␣ ␤␥ ͑ ͒ ␯ ␣␮ 2 ␮␯ ␮␯ ␣␤ 4t␣␤␥␯D R . 67 ϩ D␣D s Ϫ D s Ϫ g D␣D␤s 2 2 2 This condition can be interpreted as the statement of covari- 3 3 ant conservation of total energy-momentum, including both ␣␤␥␮ ␯ ␣␤␥␯ ␮ ␮␯ Ϫ t R␣␤␥ Ϫ t R␣␤␥ 2 2 the matter energy-momentum tensor Tg and the energy- momentum contribution from the curvature couplings asso- ␮␯ ␬␭␮␯ 1 ␣␤␥␦ ␮␯ ␮␣␯␤ ␯␣␮␤ ciated with s , t . The same result would also follow by ϩ t R␣␤␥␦g ϪD␣D␤t ϪD␣D␤t . ␮␯ 2 direct calculation of D␮Tg using the matter-sector action, followed by substitution of the complete variational equa- ͑62͒ ␮␯ ␬␭␮␯ ␮␯ tions for s and t . Since by definition Tg is indepen- ␮␯ dent of the Lorentz-violating curvature couplings involving Then, denoting by T the symmetric energy-momentum g s␮␯ and t␬␭␮␯, all the terms on the right-hand side of Eq. tensor arising from varying the matter sector with respect to ͑67͒ would then arise from the latter step. Note that Eq. ͑67͒ the metric g␮␯ , the field equations following from the varia- ͑ ͒ implies the matter energy-momentum tensor can be covari- tion 61 are found to be ␮␯ϭ antly conserved by itself, D␮Tg 0, under suitable circum- ␮␯Ϫ Rst͒␮␯ϭ␬ ␮␯ ͑ ͒ stances. For example, this is the case for any solution to the G ͑T Tg . 63 equations of motion obeying the conditions R␮␯ϭ0 and These 10 extended Einstein equations incorporate the D␣s␤␥ϭD␣t␤␥␦⑀ϭ0. leading-order effects of Lorentz violation in general relativ- An illustrative example of the above considerations is ity, and they reduce as expected to the usual Einstein equa- provided by the zero-torsion limit of the curvature-coupled tions when s␮␯ and t␬␭␮␯ vanish. Although beyond the scope bumblebee model described in Appendix B. This model in- ␮␯ ͑ ͒ of the present work, it would be of interest and appears fea- volves a traceless coefficient sB given in Eq. 59 , but the sible to study the Cauchy initial-value problem for these ex- relevant calculations in this case can be performed for the B tended equations. The presence of coefficients for Lorentz full theory. The matter energy-momentum tensor T␮␯ ob- violation can be expected to modify the conventional analy- tained from the action ͑B3͒ is sis ͓54͔. The extended Einstein equations ͑63͒ imply several other B ␣ 1 ␣␤ T ϭϪB␮␣B ␯Ϫ B␣␤B g␮␯ϪVg␮␯ϩ2VЈB␮B␯ , results. Tracing with the metric gives ␮␯ 4 ͑68͒ Ϫ ␣␤Ϫ ␣␤␥␦ϭϪ␬ ͑ ͒ R DaD␤s R␣␤␥␦t Tg , 64 where the prime denotes differentiation with respect to the ϵ ␮␯ where Tg g␮␯Tg . This expression is comparatively argument, as usual. The equations of motion are the extended simple because several terms vanish as a consequence of the Einstein equations,

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compatible with the covariant conservation laws for the ϭ␬ B ϩ␰ͫ 1 ␣ ␤ Ϫ ␣ Ϫ ␣ G␮␯ T␮␯ 2 B B R␣␤g␮␯ B␮B R␣␯ B␯B R␣␮ energy-momentum and spin-density tensors. To demonstrate this, it is convenient to start with the Bi- 1 1 1 anchi identities in the form given in Eq. ͑A14͒ of Appendix ␣ ␣ 2 ϩ D␣D␮͑B B␯͒ϩ D␣D␯͑B B␮͒Ϫ D ͑B␮B␯͒ 2 2 2 A. Some manipulation, which includes taking traces, converts the ﬁrst of these into the form 1 Ϫ ͑ ␣ ␤͒ͬ ͑ ͒ g␮␯D␣D␤ B B , 69 ␮␯ 1 ␣␤␮␯ ␭␮␯ 2 D␮G ϭ T␮␣␤R ϪT R␮␭ . ͑73͒ 2 and the equations for the bumblebee ﬁeld, From this expression, it is straightforward to prove the iden- ␰ tity ␮␯ϭ Ј ␯Ϫ ␮␯ ͑ ͒ D␮B 2V B ␬ B␮R . 70 ␭ ␮␯ ␯ ␮␭ 1 ␣␤␮␯ ͑D␮ϪT ␭␮͒G ϩT␭␮ G ϩ R Tˆ ␮␤␣ϭ0, ͑74͒ The latter imply the covariant current-conservation law 2

␯ ␮␯ ␭␮␯ D␯͑2␬VЈB ͒ϭD␯͑␰B␮R ͒. ͑71͒ where the trace-corrected torsion Tˆ is defined in Eq. ͑A10͒ of Appendix A. Similarly, tracing the second Bianchi The covariant conservation law for the energy-momentum identity and extracting the antisymmetric part of the Einstein tensor is tensor yields ␬ ␮ B ϭ␰ ␤͑ ␣ ͒Ϫ 1 ␰ ␣␤ ͑ ͒ ͑ ͒ ␣ ␣ ␣ ␤ ␣ D T␮␯ D R␣␤B B␯ 2 R D␯ B␣B␤ , 72 G␮␯ϪG␯␮ϭD␮T ␣␯ϪD␯T ␣␮ϪD T␣␮␯ϩT ␤␣T ␮␯ , ͑75͒ and it can be obtained at least two ways. One follows the derivation of Eq. ͑67͒, taking the covariant derivative of the from which follows the identity ͑ ͒ extended Einstein equations 69 and applying the trace Bi- ␮␯ ␯␮ ␤ ␣␮␯ anchi identity. The other applies the procedure outlined be- G ϪG ϭϪ͑D␣ϪT ␤␣͒Tˆ . ͑76͒ low Eq. ͑67͒, involving the direct calculation of D␮TB from ␮␯ Note that the results ͑74͒ and ͑76͒ are a strict consequence of the defining equation ͑68͒, followed by substitution of the the original two Bianchi identities ͑A14͒, following from ba- equations of motion ͑70͒. sic tensorial manipulation alone. The identities ͑74͒ and ͑76͒ have been written so that C. Geometry direct substitution of the field equations yields conditions on This subsection contains some remarks about the compat- the sources in the form of covariant conservation laws. Tak- ibility of explicit Lorentz violation with the geometry of a ing ⌳ to be zero for simplicity, the field equations ͑56͒ be- ␮␯ϭ␬ ␮␯ ˆ ␭␮␯ϭ␬ ␭␯␮ Riemann-Cartan spacetime. For simplicity, the arguments are come G Te and T S␻ . Substitution imme- presented allowing for torsion but restricting Lorentz viola- diately gives tion to the matter sector. They can be extended to other situ- 1 ations, including the presence of Lorentz-violating curvature ␭ ␮ ␭ ␮ ab ␮ ͑D␮ϪT ␭␮͒T ␯ϩT ␮␯T ␭ϩ R ␮␯S␻ ϭ0, and torsion couplings, and they contain as a special limit the e e 2 ab case of general relativity coupled to a Lorentz-violating matter sector. ␮␯Ϫ ␯␮Ϫ Ϫ ␤ ͒ ␣␮␯ϭ Te Te ͑D␣ T ␤␣ S␻ 0. The basic chain of reasoning is as follows. The geometry ͑77͒ of a Riemann-Cartan theory with local Lorentz and general coordinate invariance can be regarded as a bundle of frames These two equations have the same form as the covariant over a base spacetime manifold endowed with a metric and conservation laws ͑6͒, ͑9͒, except that the terms in the latter with structure group being the Lorentz group. This frame- two that depend on the coefficients kx for explicit Lorentz work offers the freedom to define certain geometrical quan- violation are missing in Eq. ͑77͒. The two sets of equations tities, notably the curvature and torsion, prior to specification are therefore incompatible unless these terms vanish identi- of the equations of motion that fix the spacetime. The curva- cally. ture and torsion are required by the geometrical structure to The incompatibility arises from the special geometrical satisfy two sets of Bianchi identities. The curvature and tor- structure of the gravitational bundle of frames, which ties the sion and hence the Riemann-Cartan spacetime are fixed by Bianchi identities to the equations of motion in a nontrivial demanding that they also solve certain other differential way. This can already be seen in the context of conventional equations, the field equations. The Bianchi identities impose general relativity without torsion, where the Bianchi identi- ␮␯ ␮␯ ␮␯ certain conditions on the sources of the field equations, and ties are D␮G ϭ0, the Einstein equations are G ϭ␬T , the compatibility of these conditions with properties of the and substitution of the Einstein equations into the Bianchi ␮␯ sources is a necessary requirement for the theory to be self- identities yields the constraint D␮T ϭ0 on the energy- consistent. However, for sources exhibiting explicit Lorentz momentum source. In contrast, the geometrical description violation, it turns out that these conditions are typically in- of a local gauge theory lacks this feature. For example, the

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geometry of a theory such as QED with U͑1͒ gauge invari- VI. SUMMARY ance is based on a principal fiber bundle with U͑1͒ structure In this work, the gravitational couplings in the Lorentz- group over a base spacetime manifold. The curvature of the and CPT-violating standard-model extension ͑SME͒ have bundle is the antisymmetric field strength F␮␯ , obeying the been studied. A general framework is discussed for treating ϭ ץϩ ץϩ ץ Bianchi identities ␭F␮␯ ␮F␯␭ ␯F␭␮ 0. The field Lorentz violation in the context of a Riemann-Cartan space- strength and hence the bundle geometry are fixed by impos- ␮␯ ␯ time with curvature and torsion. This allows the description ␮F ϭ j . In this instance, of gravitational couplings involving matter fields for bosonsץ ing equations of motion, say direct attempts to substitute the equations of motion into the and fermions, with the general-relativistic and Minkowski- Bianchi identities fail to yield the current-conservation law spacetime cases recovered as special limits. ␯ -␯ j ϭ0, which instead follows immediately from the equa- The Lorentz- and CPT-violating QED extension incorpoץ tions of motion by virtue of the antisymmetry of the curva- rating gravitational couplings is constructed, and the domi- ture F␮␯. The current source j␯ can therefore incorporate nant terms in the low-energy effective action are explicitly explicit Lorentz violation without incompatibility. given. The partial action in the fermion sector can be found The above clash between geometry and symmetry viola- in Eq. ͑12͒. Many of the properties and physical implications tion occurs for explicit Lorentz breaking but not for sponta- are similar to those of the Minkowski-spacetime limit, but neous Lorentz breaking. As discussed in Sec. II B, Eq. ͑77͒ is some new features emerge in the presence of nonzero curva- indeed valid when Lorentz symmetry is spontaneously bro- ture and torsion. The leading terms in the photon partial ac- ͑ ͒ ken. For example, no difficulties are encountered in the treat- tion for the QED extension are given in Eq. 25 , and some ment of the bumblebee model in the previous subsection. consequences of the gravitational coupling are deduced. Since in a suitable limit the effects of spontaneous symmetry The action for the matter and gauge sector of the SME breaking can be approximated by terms in the action with with gravitational couplings is considered in Sec. IV. First, the conventional standard model of particle physics is em- explicit symmetry breaking, it is interesting to consider how bedded in a Riemann-Cartan spacetime. Then, the Lagrang- in this limit the results ͑6͒ and ͑9͒ are recovered from the law ian terms expected to dominate Lorentz- and CPT-violating ͑77͒. Suppose the spontaneous Lorentz violation occurs physics at low energies are explicitly given for the case of when a set of fields f x acquire nonzero vacuum values kx . SU(3)ϫSU(2)ϫU(1) invariance. Up to possible coordinate The limit in question requires discarding all modes of f x and field redefinitions, each term in the SME offers a distinct representing fluctuations about kx , including massive modes way for Lorentz symmetry to be violated. The presence of and Goldstone modes or their Higgs equivalents. Discarding gravitational couplings enhances the options for experimen- the massive modes has no untoward consequences in the tal tests. low-energy limit. However, in the case of spontaneous Lor- The pure-gravity sector of the SME is considered in Sec. entz violation, it is known that the Goldstone modes are ab- V. The leading-order terms in the Lagrangian are given in sorbed into the gravitational fields without generating a mass Eqs. ͑55͒ and ͑57͒. These terms suggest several interesting for the graviton h␮␯ ͓4,48͔. Discarding the Goldstone modes directions for theoretical and experimental study. The special therefore changes certain degrees of freedom in the curvature limit of zero torsion, which is the Lorentz-violating exten- and torsion, and so it is unsurprising that the condition ͑77͒ sion of general relativity, is comparatively simple. The becomes modified in this limit. It would be of some interest Lorentz-violating physics is dominated by the action ͑58͒, to demonstrate this limiting procedure in a simple model, which contains 19 independent coefficients for Lorentz vio- including the explicit recovery of Eqs. ͑6͒ and ͑9͒, but this lation. The presence of Lorentz-violating curvature couplings lies outside the scope of the present work. has several physical implications, such as curvature- Another interesting question is whether there exists an dependent modifications to the covariant conservation law. In alternative to the geometry of the Riemann-Cartan bundle of Sec. V C, some geometrical issues associated with explicit frames that would yield consistent Bianchi identities in the Lorentz breaking in the effective field theory are addressed. presence of explicit Lorentz violation. Intuitively, the clash Explicit Lorentz breaking is shown to clash with the geometry of Riemann-Cartan spacetime, but spontaneous Lorentz described above arises because the Riemann-Cartan geom- violation encounters no difficulty. etry is predicated upon the existence throughout the bundle In conclusion, relativity violations provide candidate low- of certain geometrical quantities like the curvature and tor- energy signals for a unified quantum theory of gravity and sion. Incorporating a coefficient for Lorentz violation corre- other forces. The SME is the appropriate general framework sponds geometrically to introducing another quantity that for describing the associated Lorentz- and CPT-violating ef- couples to the existing ones but that originates outside the fects. The gravitational couplings presented in this work of- Riemann-Cartan framework and hence disrupts it. However, fer promising directions for exploration, with the potential it is reasonable to conjecture that a more general geometrical ultimately to offer insight into physics at the Planck scale. framework can be constructed in which the basic geometrical entities implement directional dependences at each space- ACKNOWLEDGMENTS time point corresponding to nonzero coefficients for explicit Lorentz violation. One option might be to generalize the no- I thank Y. Nambu for valuable correspondence. This tion of metric to include a dependence on direction, as occurs work was supported in part by the National Aeronautics and in Finsler geometries ͓55͔. Space Administration under Grant Nos. NAG8-1770 and

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NAG3-2194 and by the United States Department of Energy Curved-spacetime indices are corrected with the Cartan ␭ under Grant No. DE-FG02-91ER40661. connection ⌫ ␮␯ , while mixed objects acquire both types of correction. For example, APPENDIX A: CONVENTIONS a a a a a b ␮e␯ Ϫ⌫ ␮␯e␣ ϩ␻␮ e␯ . ͑A8͒ץD␮e␯ ϭ ␩ b The Minkowski metric ab in a local Lorentz frame is The Cartan connection is a combination of the Levi-Civita Ϫ1000 connection and the torsion tensor 0 ϩ10 0 ␩ ϭ ͑ ͒ ab . A1 ␭ ͩ ϩ ͪ ␭ ␭ ␭ ␭ ␭ 00 10 ⌫ ϭ⌫ ϩ 1 ϭͭ ͮ Ϫ ϩ 1 ␮␯ ͑␮␯͒ 2 T ␮␯ ␮␯ T͑␮␯͒ 2 T ␮␯ , 000ϩ1 ͑A9͒ Note that this metric convention involves a sign relative to where the first term after the second equality is the Christof- that adopted for the original discussion of the SME in Ref. ␭ ␭ ͓ ͔ ⑀ fel symbol and T ␮␯ϭϪT ␯␮ is the torsion tensor. Parenthe- 3 . The antisymmetric tensor in this frame is fixed by 0123 ϭϩ ses enclosing pairs of indices denote symmetrization with a 1. The Dirac matrices in this frame are taken to satisfy 1 factor of 2. ͕␥a,␥b͖ϭϪ2␩ab, ͑A2͒ In practical applications, the trace-corrected torsion tensor defined by with the additional definition ␭␮␯ ␭␮␯ ␣ ␮ ␭␯ ␣ ␯ ␭␮ Tˆ ϵT ϩT ␣ g ϪT ␣ g ͑A10͒ 1 ␴abϭ i͓␥a,␥b͔. ͑A3͒ 2 is often useful. Also, equations involving torsion are sometimes more profitably expressed in terms of the contortion Latin indices are used to label local Lorentz coordinates, ␭ while Greek indices are used for spacetime coordinates. tensor K ␮␯ , defined as ͑ ͒ However, x,y denote generic composite indices spanning an ␭ ϭ 1 ͑ ␭ Ϫ ␭Ϫ ␭͒ ͑ ͒ x K ␮␯ 2 T ␮␯ T␮␯ T␯␮ . A11 irreducible representation (X͓ab͔) y of the local Lorentz group. The commutation relations for the Lorentz algebra are ␭ ␭ ␭ The inverse relation is T ␮␯ϭK ␮␯ϪK ␯␮ . The contortion ϭ␩ Ϫ␩ Ϫ␩ ϩ␩ ϭϪ ␭ ϭ ␭ ͓X͓ab͔ ,X͓cd͔͔ acX͓bd͔ adX͓bc͔ bcX͓ad͔ bdX͓ac͔ . tensor obeys K␭␮␯ K␯␮␭ . Note that K ␭␯ T ␭␯ . ͑A4͒ The curvature tensor is defined as

ϭϪ ␴ ␬ ␬ ␬ ␣ ␮⌫ ␯␭ϩ⌫ ␮␣⌫ ␯␭͒Ϫ͑␮↔␯͒ץFor example, for the spinor representation X͓ab͔ i ab/2, R ␭␮␯ϵ͑ c ϭϪ␩ c␩ while for the vector representation (X͓ab͔) d a bd ϩ␩ ␩ c ␬ ␬ ␣ ␬ ␣ ␬ ad b . ϭ˜R ␭␮␯ϩ͓͑D␮K ␯␭ϩK ␮␯K ␣␭ϩK ␮␭K ␯␣͒ The Minkowski metric is related to the curved-spacetime a Ϫ͑␮↔␯͔͒ ͑ ͒ metric g␮␯ by the vierbein e␮ : , A12

ϭ a b␩ ͑ ͒ ␬ g␮␯ e␮ e␯ ab . A5 where ˜R ␭␮␯ is the usual Riemann curvature tensor in the absence of torsion, given by replacing the Cartan connec- The determinant of the vierbein is denoted e. To avoid con- tions in the first expression above with the corresponding fusion, the charge on the electron is denoted by Ϫq. The Christoffel symbols. The Ricci tensor R␮␯ , the curvature symbol D is used for all covariant derivatives, including scalar R, and the Einstein tensor G␮␯ are defined as spacetime, internal, and mixed covariant derivatives, with the meaning understood from the context or otherwise speci- ␬ R␮␯ϵR ␮␬␯ , fied. For the spacetime covariant derivative, the connection is assumed to be metric: ␮␯ Rϵg R␮␯ , a D␭g␮␯ϭ0, D␭e␮ ϭ0. ͑A6͒ 1 The spacetime covariant derivative corrects local Lorentz G␮␯ϵR␮␯Ϫ g␮␯R. ͑A13͒ ab 2 indices with the spin connection ␻␮ . Thus, acting on a field f y, it takes the matrix form The reader is cautioned that the presence of nonzero torsion 1 in a generic Riemann-Cartan spacetime means that these ͒ ͑ Ϫ ␻ ab͑ ͒x ͬ y ץ x yϭͫ␦x͒ ͑ D␮ f ␮ ␮ X f . A7 y y 2 ͓ab͔ y three quantities also differ from their Riemann-spacetime counterparts ˜R, ˜R␮␯ , and G˜ ␮␯ . The covariant derivative of the conjugate representation f x is The curvature and torsion tensors satisfy symmetry prop- y given by the same equation with f replaced by f x and the erties that follow directly from their definition. They also minus sign replaced by a plus sign. obey the two sets of Bianchi identities

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␰ ␣ ␰ where the metric fluctuation h␮␯ is symmetric. At leading ͚ ͓D␯R ␬␭␮ϩT ␭␮R ␬␣␯͔ϭ0, order, spacetime and local Lorentz indices can be treated as ͑␭␮␯͒ equivalent, and the vierbein and spin connection can be expressed in terms of small quantities: ␬ ␣ ␬ ␬ ͚ ͓D␯T ␭␮ϩT ␭␮T ␣␯ϪR ␯␭␮͔ϭ0. ϭ␩ ϩ⑀ Ϸ␩ ϩ 1 ϩ␹ ͑␭␮␯͒ e␮a ␮a ␮a ␮a 2 h␮a ␮a , ͑A14͒ Ϸ ϩ 1 e 1 2 h, In these equations, the summation symbol is understood to ␹ ϩ ץϩ ץ ϩ 1 ץ ␻ ϷϪ 1 represent the sum over cyclic permutations of the indices in ␮ab 2 ah␮b 2 bh␮a ␮ ab Ka␮b . parentheses. ͑A20͒ The definition ͑A5͒ and the condition ͑A6͒ fix the rela- Here, the antisymmetric part of the vierbein fluctuation is tionship between the spin connection and the torsion or con- ␹ tortion. The basic variables can be taken as the vierbein and denoted ␮a . This variable can be viewed as containing the the spin connection, and all other variables such as curvature six extra degrees of freedom in the vierbein relative to the metric that transform under local Lorentz rotations, so fixing and torsion can then be expressed in terms of these. For ␹ example, the Cartan connection is ␮a can be regarded as a gauge choice. Throughout most of this work, natural units with បϭc ␭ ␭ ϭ⑀ ϭ .Ϫ␻ b ͒ ͑ ͒ 0 1 are adopted ץ ϭ a ⌫ ␮␯ e ͑ ␮e␯a ␮ ae␯b , A15

while the torsion is APPENDIX B: BUMBLEBEE MODEL

a b Models in which the Lorentz violation arises from the ␮e␯ ϩ␻␮ e␯ ͒Ϫ͑␮↔␯͔͒, ͑A16͒ץ͓͑ T␭␮␯ϭe␭ a ab dynamics of a single vector or axial-vector field B␮ , called the bumblebee field, are of particular interest because they and the curvature is have a comparatively simple form but encompass interesting features, including rotation, boost, and CPT violations. In a ␻ a ϩ␻ a ␻ c ͒Ϫ͑␮↔␯͔͒ ץ␬ ϭ ␬ b͓͑ R ␭␮␯ e ae␭ ␮ ␯ b ␮ c ␯ b . Riemann-Cartan spacetime, the field strength corresponding ͑A17͒ to B␮ can be defined either as ␭ ␯B␮ ͑B1͒ץ␮B␯ϪץAnother useful expression is the relationship between the B␮␯ϵD␮B␯ϪD␯B␮ϩT ␮␯B␭ϭ spin connection and the vierbein: or as 1 1 ab ␯a b b ␯b a a ␯e␮ ͒ B␮␯ϵD␮B␯ϪD␯B␮ . ͑B2͒ץ␮e␯ Ϫץ͑ ␯e␮ ͒Ϫ eץ␮e␯ Ϫץ͑ ␻␮ ϭ e 2 2 The former is U͑1͒ gauge invariant even in the presence of 1 ϩ ␯a ␭b͒ ץϪ ץ Ϫ ␣a ␤b c e e e␮ ͑ ␣e␤c ␤e␣c K␯␮␭e e . torsion while the latter is not, so the two definitions involve 2 qualitatively different physics. However, they coincide in ͑A18͒ Riemann or Minkowski spacetimes. As an example, consider the simple model with action In the limiting case of Riemann geometry relevant for Ein- 1 stein gravity, the torsion and contortion are zero. This equa- ϭ ͵ 4 ͫ ͑ ϩ␰ ␮ ␯ ͒Ϫ 1 ␮␯ SB d x eR eB B R␮␯ 4 eB B␮␯ tion then fixes the spin connection in terms of the metric. 2␬ Using these expressions, the standard Riemann-spacetime ˜ Ϫ ͑ ␮ Ϯ 2͒ͬ ͑ ͒ covariant derivative D␮ involving a symmetric connection eV B B␮ b , B3 and the Christoffel symbols emerges as the zero-torsion limit of the covariant derivative in Eq. ͑A7͒. where ␰ is a real coupling constant controlling a nonminimal Various special cases of the general Riemann-Cartan curvature-coupling term, and b2 is a real positive constant. ͑ ␬ ␭ ͒ spacetimes which have R ␭␮␯ , T ␮␯ both nonzero are of The potential V driving Lorentz and CPT violation can be interest. They include the Riemann spacetimes of general ␮ Ϯ 2ϭ ␭ chosen to have a minimum at B B␮ b 0. A simple ϭ 1 relativity mentioned above, with T ␮␯ 0. The Weitzenbo¨ck choice for V(x)isV(x)ϭ ␭x2, where ␭ is a real coupling ͓ ͔ ␬ ϭ 2 spacetimes 56 are defined by R ␭␮␯ 0. The term ‘‘flat’’ is constant. Another simple choice with similarities to a sigma ␬ reserved for spacetimes with ˜R ␭␮␯ϭ0, which may have model is V(x)ϭ␭x, where now ␭ is a Lagrange-multiplier nonzero torsion. Finally, the Minkowski spacetimes have field. Note that the form of the potential ensures breaking of ␬ ␭ R ␭␮␯ϭT ␮␯ϭ0. the U͑1͒ symmetry, irrespective of the definition ͑B1͒ or ͑B2͒ It is sometimes useful to work in a Minkowski-spacetime adopted for B␮␯ . background containing weak gravitational fields. Then, the In a region where the curvature and torsion vanish, the metric can be written as potential drives a nonzero vacuum value B␮ϭb␮, where ␮ 2 b b␮ϭϯb . The quantity b␮ is a coefficient for Lorentz and g␮␯ϭ␩␮␯ϩh␮␯ , ͑A19͒ CPT violation. In a local Lorentz frame the condition be-

105009-18 GRAVITY,LORENTZVIOLATION,ANDTHE... PHYSICAL REVIEW D 69, 105009 ͑2004͒

aϭ 2 ␰ comes BaB b , and the local Lorentz coefficient ba can be V but with nonzero has been used as an alternative theory taken to have a preferred form as discussed in Sec. II A. This of gravity in a Riemann spacetime by Will and Nordtvedt holds in an asymptotically flat spacetime and also in the ͓42,53,58͔. The theories with ␰ϭ0 were introduced in Ref. Minkowski-spacetime limit, although the effects of the po- ͓4͔ to illustrate some ideas about spontaneous Lorentz viola- tential may be masked for certain matter couplings and in tion, and these and related models have been explored fur- regions of strong curvature and torsion. ther in recent works ͓16,59,60͔. In particular, if one or more The physical insights offered by this theory are remark- fermion fields also appear in the action, the covariant axial ably rich. The special limit of Minkowski spacetime and the coupling to the bumblebee field induces terms with coeffi- Lagrange-multiplier potential is equivalent to a theory stud- cients for Lorentz and CPT violation of the type b␮ in the ied many years ago by Nambu ͓57͔, who obtained an elegant fermion sector of the SME ͓16͔. The action ͑B3͒ with a proof that it is equivalent to electrodynamics in a nonlinear potential V and nonzero curvature coupling ␰ is used as an gauge. The case without Lorentz violation and zero potential illustrative example in parts of the present work.

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