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provided by IUScholarWorks PHYSICAL REVIEW D 77, 065020 (2008) Spontaneous Lorentz and diffeomorphism violation, massive modes, and

Robert Bluhm,1 Shu-Hong Fung,1,2 and V. ´3 1Physics Department, Colby College, Waterville, Maine 04901, USA 2Physics Department, Chinese University of Hong Kong, Shatin, N.T., Hong Kong 3Physics Department, Indiana University, Bloomington, Indiana 47405, USA (Received 30 December 2007; published 21 March 2008) Theories with spontaneous local Lorentz and diffeomorphism violation contain massless Nambu- Goldstone modes, which arise as field excitations in the minimum of the symmetry-breaking potential. If the shape of the potential also allows excitations above the minimum, then an alternative gravitational can occur in which massive modes involving the metric appear. The origin and basic properties of the massive modes are addressed in the general context involving an arbitrary tensor vacuum value. Special attention is given to the case of bumblebee models, which are gravitationally coupled vector theories with spontaneous local Lorentz and diffeomorphism violation. Mode expansions are presented in both local and frames, revealing the Nambu-Goldstone and massive modes via decomposition of the metric and bumblebee fields, and the associated symmetry properties and gauge fixing are discussed. The class of bumblebee models with kinetic terms of the Maxwell form is used as a focus for more detailed study. The nature of the associated conservation laws and the interpretation as a candidate alternative to Einstein-Maxwell theory are investigated. Explicit examples involving smooth and Lagrange-multiplier potentials are studied to illustrate features of the massive modes, including their origin, nature, dispersion laws, and effects on gravitational interactions. In the weak static limit, the massive mode and Lagrange-multiplier fields are found to modify the Newton and Coulomb potentials. The nature and implications of these modifications are examined.

DOI: 10.1103/PhysRevD.77.065020 PACS numbers: 11.30.Cp, 04.50.h, 11.30.Er, 12.60.i

I. INTRODUCTION nonsinglet composites. The nonzero vacuum values are manifest both on the spacetime manifold and in local In relativistic quantum field theory, the nature of the field frames [4]. Their origin in spontaneous violation implies modes associated with the spontaneous breaking of an both local Lorentz violation and diffeomorphism violation, internal symmetry is now standard lore. When a global internal symmetry is spontaneously broken, one or more along with the existence of NG modes [5]. massless modes called Nambu-Goldstone (NG) modes At the level of an underlying Planck-scale theory, nu- appear [1]. If instead the symmetry is local, the Higgs merous proposals exist that involve spontaneous Lorentz mechanism can occur: the massless NG modes play the violation, including ones based on [6], non- role of additional components of the gauge fields, which commutative field theories [7], spacetime-varying fields then propagate as massive modes [2]. In either case, the [8], quantum gravity [9], random-dynamics models [10], spontaneous symmetry breaking is typically driven by a multiverses [11], braneworld scenarios [12], supersymme- potential term V in the Lagrange density. The vacuum field try [13], and massive gravity [14–16]. At experimentally accessible scales, the observable signals resulting from configuration lies in a minimum V0 of V. The massless NG modes can be understood as field excitations about the Lorentz breaking can be described using effective field theory [17]. The general realistic effective field theory vacuum that preserve the value V0, and they are associated with the broken generators of the symmetry. For many containing the Lagrange densities for both general relativ- potentials, there are also additional excitations involving ity and the standard model along with all scalar terms other values of V. These excitations, often called Higgs involving operators for Lorentz violation is called the modes, correspond to additional massive modes that are standard-model extension (SME) [4,18]. Searches for distinct from any massive gauge fields. low-energy signals of Lorentz violation represent a prom- This standard picture changes when the spontaneous ising avenue of investigation involving the phenomenology breaking involves a spacetime symmetry rather than an of quantum gravity [19]. Numerous experimental measure- internal one. In this work, we focus on spontaneous break- ments of SME coefficients for Lorentz violation have al- ing of Lorentz and diffeomorphism symmetries, for which ready been performed [20], including ones with the corresponding Higgs mechanisms exhibit some unique [21], electrons [22–24], protons and neutrons [25–27], features [3]. Spontaneous Lorentz violation occurs when mesons [28], muons [29], neutrinos [30], the Higgs [31], one or more Lorentz-nonsinglet field configurations ac- and gravity [32,33]. quire nonzero vacuum expectation values. The field con- For spontaneous Lorentz and diffeomorphism breaking, figurations of interest can include fundamental vectors or a general analysis of the nature of the NG modes and Higgs tensors, derivatives of scalars and other fields, and Lorentz- mechanisms is provided in Ref. [5]. One result is that the

1550-7998=2008=77(6)=065020(28) 065020-1 © 2008 The American Physical Society ROBERT BLUHM, SHU-HONG FUNG, AND V. ALAN KOSTELECKY´ PHYSICAL REVIEW D 77, 065020 (2008) spontaneous breaking of local Lorentz symmetry implies In the context of diffeomorphism symmetry, the role of spontaneous breaking of diffeomorphism symmetry and the gauge fields is played by the metric. For spontaneous vice versa. Since six local Lorentz transformations and diffeomorphism breaking in Riemann spacetime, a con- four diffeomorphisms can be broken, up to ten NG modes ventional diffeomorphism Higgs mechanism cannot gener- can appear. To characterize these, it is natural to adopt the ate a mass for the because the connection and vierbein formalism [34], in which the roles of local Lorentz hence the analogue of the usual mass term involve deriva- transformations and diffeomorphisms are cleanly distin- tives of the metric [3]. Also, since diffeomorphism NG guished. It turns out that the vierbein itself naturally in- modes are nonpropagating in a Lagrange density with corporates all ten modes. In an appropriate gauge, the six covariant kinetic terms for reasons mentioned above, the Lorentz NG modes appear in the antisymmetric compo- propagating NG degrees of freedom required to generate nents of the vierbein, while the four diffeomorphism NG massive fields via a conventional diffeomorphism Higgs modes appear along with the usual gravitational modes in mechanism are lacking. the symmetric components. In a conventional with a nonderivative The dynamical behavior of the various NG modes is potential V, the gauge fields are absent from V and so the determined by the structure of the action [5]. In a Lagrange potential cannot directly contribute to the gauge masses. density formed from tensor quantities and with However, in spontaneous Lorentz and diffeomorphism diffeomorphism-covariant kinetic terms, the diffeomor- violation, massive Higgs-type modes involving the vier- phism NG modes are nonpropagating. This feature, unique bein can arise via the alternative Higgs mechanism, which to diffeomorphism symmetry, can be viewed as arising involves the potential V [3]. The key point is that V because the diffeomorphism NG field excitations that pre- contains both tensor and metric fields, so field fluctuations serve V0 include metric excitations, and the combined about the vacuum value V0 can contain quadratic mass excitations cancel at propagation order in covariant deriva- terms involving the metric. tives and curvature. In contrast, the number of propagating In this work, we study the nature and properties of the Lorentz NG modes is strongly model dependent. For ex- additional massive Higgs-type modes arising from this ample, choosing the kinetic terms in the Lagrange density alternative Higgs mechanism. A general treatment is pro- of the original field theory to eliminate possible ghost vided for a variety of types of potentials V in gravitation- modes can also prevent the propagation of one or more ally coupled theories with Riemann geometry. We Lorentz NG modes, leaving them instead as auxiliary investigate the effects of the massive modes on the physical fields. properties of gravity. In certain theories with spontaneous Several types of possible Higgs mechanisms can be Lorentz violation, the NG modes can play the role of distinguished when spacetime symmetries are spontane- photons [5], and we also examine the effects of the massive ously broken. These have features distinct from the con- modes on electrodynamics in this context. ventional Higgs mechanism of gauge field theories [2] and The next section of this work provides a general dis- Higgs mechanisms involving gravity without Lorentz vio- cussion of the origin and basic properties of the massive lation [35]. The analysis of Higgs mechanisms can be Higgs-type modes. Section III analyzes the role of these performed either using the vierbein formalism, which per- modes in vector theories with spontaneous Lorentz viola- mits tracking of Lorentz and diffeomorphism properties, or tion, known as bumblebee models. In Sec. IV, the massive by working directly with fields on the spacetime manifold. modes in a special class of bumblebee models are studied The results of both approaches are equivalent. in more detail for several choices of potential in the For local Lorentz symmetry, the role of the gauge fields Lagrange density, and their effects on both the gravita- in the vierbein formalism is played by the . tional and electromagnetic interactions are explored. The Lorentz Higgs mechanism occurs when the Lorentz Section V summarizes our results. Some details about NG modes play the role of extra components of the spin transformation laws are provided in the appendix. connection [5]. Some components of the spin connection Throughout this work, the conventions and notations of then acquire mass via the covariant derivatives in the Refs. [4,5] are used. kinetic part of the Lagrange density. Explicit models dis- playing the Lorentz Higgs mechanism are known. For this II. MASSIVE MODES mechanism to occur, the components of the spin connec- tion must propagate as independent degrees of freedom, The characteristics of the massive Higgs-type modes which requires a theory based on Riemann-Cartan geome- that can arise from the alternative Higgs mechanism de- try. If instead the theory is based on Riemann geometry, pend on several factors. Among them are the type of field like , the spin connection is fixed non- configuration acquiring the vacuum expectation value and linearly in terms of the vierbein and its derivatives. The the form of the Lagrange density, including the choice of presence of these derivatives ensures that no mass terms potential V inducing spontaneous breaking of Lorentz and emerge from the kinetic part of the Lagrange density, diffeomorphism symmetries. In this section, we outline although the vierbein propagator is modified. some generic features associated with the alternative

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Higgs mechanism in a theory of a general tensor field given theory, then the value of t is specified up to T in a Riemann spacetime with metric g. We con- M N coset transformations, and so the vacuum is sider first consequences of the choice of potential V, then degenerate under M N additional gauge symmetries. discuss properties of vacuum excitations, and finally offer Note that these M N freedoms are distinct from comments on the massive modes arising from the alter- Lorentz and diffeomorphism transformations. native Higgs mechanism. It is useful to distinguish two classes of potentials V. The first consists of smooth functionals V of Xm that are mini- A. Potentials mized by the conditions Xm 0 for at least some m. These 0 potentials therefore satisfy V Vm 0 in the vacuum, The potential V in the Lagrange density is taken to where V0 denotes the derivative with respect to X , and trigger a nonzero vacuum expectation value m m they fix the vacuum value of T to t modulo

hTit (1) possible gauge freedoms. A simple example with the quan- tity X in Eq. (3)is for the tensor field, thereby spontaneously breaking local 1 2 Lorentz and diffeomorphism symmetries. In general, V VSX2X ; (5) varies with T, its covariant derivatives, the metric where is a constant. In this case, the vacuum value t g, and possibly other fields. However, for simplicity is a solution of V V0 0, where V0 denotes a derivative we suppose here that V has no derivative couplings and 00 with respect to X. If the matrix Vmn of second derivatives involves only T and g. We also suppose that V is everywhere positive except at its degenerate minima, has nonzero eigenvalues, the smooth functionals V can which have V 0 and are continuously connected via give rise to quadratic mass terms in the Lagrange density the broken Lorentz and diffeomorphism generators. The involving the tensor and metric fields. Potentials V of this vacuum is chosen to be the particular minimum in which type are therefore associated with the alternative Higgs T attains its nonzero value (1). mechanism, and they are the primary focus of our attention. Since the Lagrange density is an observer scalar, V must A second class of potentials introduces Lagrange- depend on fully contracted combinations of T and multiplier fields m for at least some m, to impose directly g. Provided T has finite rank, the number of inde- the conditions Xm 0 as constraint terms in the Lagrange pendent scalar combinations is limited. For example, for a density. We consider here both linear and quadratic func- symmetric two-tensor field C there are four independent tional forms for these constraints. An explicit linear ex- possibilities, which can be given explicitly in terms of ample is traces of powers of C and g [36]. It is convenient to

V ; XX; (6) denote generically these scalar combinations as Xm, where L m 1; 2; ...;Mranges over the number M of independent while a quadratic one is combinations. The functional form of the potential there- 1 2 fore takes the form VQ; X2X : (7)

V VX1;X2; ...;XM (2) In each example, is a Lagrange-multiplier field and has equation of motion with solution X 0, so the value t in terms of the scalars Xm. The definition of the scalars Xm can be chosen so that is a vacuum solution for the tensor. Potentials in the Xm 0 in the vacuum for all m. For example, a choice Lagrange-multiplier class are unlikely to be physical in involving a quadratic combination of the tensor with zero detail because they enforce singular slicings in the phase vacuum value is space for the fields. However, when used with care they can be useful as limiting approximations to potentials in the 2 X Tg g g T t ; (3) smooth class, including those inducing the alternative with t2 the norm Higgs mechanism [3]. Note that positivity of the potentials can constrain the range of the Lagrange multiplier field. 2 t thg ihg ihg it; (4) For example, the off-shell value of in VL must have the same sign as X, while that of in V must be non-negative. where hgi is the vacuum value of the inverse metric. The Q sign is introduced for convenience so that t2 can be chosen non-negative. In principle, t2 could vary with B. Excitations spacetime position, which would introduce explicit Field excitations about the vacuum solution (1) can be spacetime-symmetry breaking, but it suffices for present classified in five types: gauge modes, NG modes, massive purposes to take t2 as a real non-negative constant. modes, Lagrange-multiplier modes, or spectator modes. The M conditions Xm 0 fix the vacuum value t. Gauge modes arise if the potential V fixes only M N If only a subset of N of these M conditions is generated in a of the M conditions Xm 0, so that the vacuum is unspe-

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cified up to N conditions. These modes can be disregarded T t T~ : (10) for most purposes here because they can be eliminated via gauge fixing without affecting the physics. The NG modes In a Minkowski background, the vacuum values in the two are generated by the virtual action of the broken Lorentz expansions (8) and (10) are related simply by and diffeomorphism generators on the symmetry-breaking t t: (11) vacuum. They preserve the vacuum condition V 0. Massive modes are excitations for which the symmetry However, the relationship between the two tensor excita- breaking generates quadratic mass terms. Lagrange- tions at leading order involves also the metric excitation: multiplier modes are excitations of the Lagrange multiplier T~ ht ht field. Finally, spectator modes are the remaining modes in the theory. h t ...: (12) For smooth potentials, field excitations preserving the In this expression, indices have been raised using the conditions Xm 0 for all m have potential V 0. They 0 Minkowski metric. also satisfy Vm 0. The NG modes are of this type. Any Lagrange multipliers m in the theory can also be Excitations with Xm Þ 0 having nonzero potential V Þ 0 0 expanded about their vacuum values m as and Vm Þ 0 are massive modes, with mass matrix deter- mined by the second derivatives V00 . Since the smooth ~ mn m m m: (13) potentials depend on the tensor T and the metric g, the corresponding massive modes involve combinations of For linear Lagrange-multiplier potentials, the equations of excitations of these fields. motion for T and g provide constraints on the In contrast, for Lagrange-multiplier potentials the con- vacuum values m. In a Minkowski background and a potential yielding X 0 for all m, the equations of ditions Xm 0 always hold on shell, which implies V 0 m 0 motion for T can be solved to give for all on-shell excitations. If also Vm 0, then the ex- citations remain in the potential minimum and include the 0: (14) NG modes. For linear functional forms of V, it follows that m 0 0 Vm m,soVm is nonzero only when the Lagrange- For quadratic Lagrange-multiplier potentials, the m are multiplier field m is excited. Any excitations of T absent from the equations of motion. Their vacuum values 0 and g must have V Vm 0. For quadratic functional are therefore physically irrelevant, and m 0 can also be 0 forms of V, one finds Vm 0 for all excitations, including adopted in this case. For the remainder of this work we take the field. We therefore can conclude that the combina- m 0, and for notational simplicity we write m for both ~ tions of T and g playing the role of massive modes the full field m and the excitation m. for smooth potentials are constrained to zero for Lagrange- multiplier potentials. Evidently, studies of the alternative C. Massive modes Higgs mechanism must be approached with care when the For a smooth potential V, the massive excitations typi- Lagrange-multiplier approximation to a smooth potential cally involve a mixture of tensor and gravitational fields. is adopted. As an example, consider the expression for V in Eq. (5). The tensor excitations about the vacuum can be ex- S This can be expanded in terms of the excitations and pressed by expanding T as h. Retaining only terms up to quadratic order in a

T t ; (8) Minkowski background gives 1 1 VS 2t ht ht where the excitation is defined as the difference 2 2 1 2 T T t between the tensor 2ht ... ; (15) and its vacuum value. We also expand the metric where index contractions are performed with the

g hgih (9) Minkowski metric . Evidently, the massive excitations in this example involve linear combinations of with in terms of the metric excitations h about the metric contractions of h and t. background value hgi. For simplicity and definiteness, in The explicit expressions for the massive modes can be much of what follows we take the background metric to be modified by local Lorentz and diffeomorphism gauge fix- that of Minkowski spacetime, hgi. We also sup- ing. The action is symmetric under 10 local Lorentz and pose t is constant in this background, so that diffeomorphism symmetries, so there are 10 possible @t 0 in cartesian coordinates. gauge conditions. For an unbroken symmetry generator, Other choices can be made for the expansion of the the effects of a gauge choice are conventional. For a broken tensor about its vacuum value. One alternative is to expand symmetry generator, a gauge choice fixes the location in the contravariant version of the tensor as field space of the corresponding NG mode. For example,

065020-4 SPONTANEOUS LORENTZ AND DIFFEOMORPHISM ... PHYSICAL REVIEW D 77, 065020 (2008) suitable gauge choices can place all the NG modes in the that might permit evading this obstacle is to relax the vierbein [5]. These gauge choices also affect the form of assumption of Lorentz invariance. Spontaneous Lorentz the massive modes. They can be used to isolate some or all violation in closed string theory has been proposed [14] of the massive modes as components of either the gravita- as a mechanism that might lead to graviton mass terms. tional or tensor fields by gauging other components to zero. Models involving infrared modifications of gravity have Alternatively, the gauge freedom can be used to simplify also been proposed that involve spontaneous Lorentz vio- the equations of motion, while the massive modes remain a lation with ghosts [15] and that have explicit Lorentz mixture of the excitations and h. violations [16]. Explicit Lorentz violation is incompatible The behavior of the massive modes depends on the form with Riemann and Riemann-Cartan geometries but may be of the kinetic terms in the Lagrange density as well as the compatible with Finsler or other geometries [4,39] or may form of the potential V. A gravitational theory with a be viewed as an approximation to spontaneous violation. In dynamical tensor field T has kinetic terms for both the present context, the possibility exists that the massive g and T and hence for both h and . Since in modes from the alternative Higgs mechanism could evade the alternative Higgs mechanism the potential V acts only the Veltman-van Dam-Zakharov constraint via their origin as a source of mass, the issue of whether the massive in spontaneous Lorentz violation and their nature as mix- modes propagate dynamically depends on the structure of tures of gravitational and tensor modes. For example, the these kinetic terms. In particular, propagating massive expansion (15) includes quadratic terms involving contrac- modes can be expected only if the theory without the tions of h with the vacuum value t, so a model with potential V contains the corresponding propagating mass- suitable vacuum values and incorporating also a ghost-free less modes. propagator for the corresponding modes would describe a It is desirable that any propagating modes be unitary and type of propagating massive gravity without conventional ghost-free. To avoid unitarity issues with higher-derivative Fierz-Pauli terms. propagation, the kinetic term for the metric excitation h In the alternative Higgs mechanism, the existence of can be taken to emerge as usual from the Einstein-Hilbert propagating massive modes involving the metric can be action. In the absence of the potential V, this is also a expected to affect gravitational physics. Effects can arise ghost-free choice. For the tensor field, higher-derivative directly from the modified graviton propagator and also propagation can be avoided by writing the general kinetic from the massive modes acting as sources for gravitational interactions. The latter can be understood by considering term LK as a weighted sum of all scalar densities formed the energy-momentum tensor T of the full theory, which from quadratic expressions in T that involve two can be found by variation of the action with respect to the covariant derivatives. For example, one such scalar density is metric g. The contribution TV to T arising from the potential V is L K eTDD T ; (16) p 0 Xm T g V 2V ; (17) where e g is the vierbein determinant. In the absence V m g of the potential V, the ghost-free requirement places strong constraints on allowed forms of LK and typically involves where a sum on m is understood. For the vacuum solution gauge symmetry for the tensor. For a vector field, for 0 Xm 0, which satisfies V Vm 0, TV remains zero example, the Maxwell action is the unique ghost-free and the gravitational sourcing is unaffected. The same is combination in the Minkowski spacetime limit. In any 0 true for excitations for which V and Vm both vanish, such case, since the massive modes are combinations of h as the NG modes. However, the massive modes arising 0 and , it follows that ghost-free propagation is pos- from a smooth potential have nonzero V and Vm and can sible only if the kinetic terms for these combinations are therefore act as additional sources for gravity. These con- ghost-free. Note that the potential V may explicitly break tributions can lead to a variety of gravitational effects the tensor gauge symmetry, so requiring ghost-free kinetic including, for example, modifications of the Newton gravi- terms is by itself insufficient to ensure ghost-free massive tational potential in the weak-field limit, which could have modes. relevance for dark matter, or cosmological features such as Under the assumption of Lorentz invariance, the Fierz- dark energy [4]. Pauli action involving quadratic terms for h is the unique Note that nonpropagating modes can also have similar ghost-free choice for a free massive spin-2 field [37]. significant physical effects on gravitational properties. However, when coupled to a covariantly conserved This possibility holds for any excitations appearing in V, energy-momentum tensor, the small-mass limit of a mas- whether they are physical, ghost, or Lagrange-multiplier sive spin-2 field includes modes that modify the gravita- modes. For example, any Lagrange-multiplier fields are tional bending of light in disagreement with observation auxiliary by construction and so the m excitations are [38]. This presents an obstacle to constructing a viable nonpropagating. Nonetheless, for linear Lagrange- theory of massive gravity. One avenue of investigation multiplier potentials these excitations can contribute to

065020-5 ROBERT BLUHM, SHU-HONG FUNG, AND V. ALAN KOSTELECKY´ PHYSICAL REVIEW D 77, 065020 (2008) 0 TV even though V 0 because Vm is nonvanishing. condition

However, a theory with a quadratic Lagrange-multiplier Z 4 1 potential has TV 0 and therefore leaves the gravita- d xeB B R 2eB B tional sourcing unaffected. eDBD B eDB DB 0 (21) III. BUMBLEBEE MODELS is identically satisfied, so only four of the corresponding five terms in LB are independent. No term of the form In this section, we focus attention on the special class of eB BR appears in the condition (21), so the parameter 2 theories in which the role of the tensor T is played by a vector B that acquires a nonzero vacuum expectation remains unaffected while the four parameters 1, 1, 2, 3 become linked. For example, the identity (21) implies that value b . These theories, called bumblebee models, are the action for the Lagrange density (19) with five nonzero among the simplest examples of field theories with sponta- parameters , , , , is equivalent to an action of neous Lorentz and diffeomorphism breaking. In what fol- 1 2 1 2 3 the same form but with only four nonzero parameters 0 lows, bumblebee models are defined, their properties under 1 , 0 , 0 2 , 0 2 , while local Lorentz and diffeomorphism transformations are pre- 1 3 2 2 1 1 3 2 2 3 0 0. Moreover, other factors may also constrain some sented, their mode content is analyzed, and issues involv- 3 of the five parameters. For example, certain models of the ing gauge fixing and alternative mode expansions are form yield equations of motion that imply additional considered. LB relationships among the parameters. Also, certain physical limits such as the restriction to Minkowski spacetime can A. Basics limit the applicability of the condition (21). Some cases The action SB for a single bumblebee field B coupled to with specific parameter values may be of particular interest gravity and matter can be written as for reasons of physics, geometry, or simplicity, such as the Z Z models with 1 1, 2 3 0 considered below, or the 4 4 SB d xLB d xLg LgB LK LV LM: model with 2 1=2 for which the bumblebee- curvature coupling involves the Einstein tensor, LB (18) 1eB B G. Since the most convenient choice of pa- In Riemann spacetime, the pure gravitational piece Lg of rameters depends on the specific model and on the physics the Lagrange density is usually taken to be the Einstein- being addressed, it is useful to maintain the five-parameter Hilbert term supplemented by the cosmological constant form (19) for generality. . The gravity-bumblebee couplings are described by LgB, Following the discussion in Sec. II A, we suppose the potential V in Eq. (19) has no derivative couplings and is while LK contains the bumblebee kinetic terms and any formed from scalar combinations of the bumblebee field self-interaction terms. The component LV consists of the B and the metric g . Only one independent scalar exists. potential VB, including terms triggering the spontane- It can be taken as the bumblebee version of X in Eq. (3): ous Lorentz violation. Finally, LM involves the bumblebee coupling to the matter or other sectors in the model. 2 X Bg B b ; (22) The forms of LgB and LK are complicated in the general case. However, if attention is limited to terms quadratic in where b2 is a real non-negative constant. The potential B involving no more than two derivatives, then only five VX itself can be smooth in X, like the form (5), or it possibilities exist. The Lagrange density LB can then be can involve Lagrange multipliers like the form (6)or(7). In written as any case, the vacuum is determined by the single condition

2 1 X B g B b 0: (23) L eR 2 eBBR eBB R B 16G 1 2 In the vacuum, the potential vanishes, VX0, and the 1 1 eBB eD B DB fields B , g acquire vacuum values 4 1 2 2

1 B !hBib;g !hgi: (24) 3eDB DB eV LM; (19) 2 2 The nonzero value b, which obeys bhg ib b , where G is the Newton gravitational constant and the field- spontaneously breaks both Lorentz and diffeomorphism strength tensor B is defined as symmetry. Note that the choice of the potential V can also have implications for the parameters in Eq. (19). For B @ B @ B : (20) example, if the potential takes the quadratic Lagrange- The five real parameters 1, 2, 1, 2, 3 in Eq. (19) are multiplier form (7), then 2 can be taken as zero because not all independent. Up to surface terms, which leave a nonzero value merely acts to rescale G and .However,a unaffected the equations of motion from the action, the nonzero value of 2 can have nontrivial consequences for

065020-6 SPONTANEOUS LORENTZ AND DIFFEOMORPHISM ... PHYSICAL REVIEW D 77, 065020 (2008) models with other potentials, such as the smooth quadratic in covariant derivatives acting on local quantities. In form (5). Riemann-Cartan geometry, where the spacetime has both For generic choices of parameters, the Lagrange density curvature and torsion, the spin connection represents de- (19) is unitary because no more than two derivatives ap- grees of freedom independent of the vierbein. However, pear. However, as discussed in Sec. II C, the indefinite experimental constraints on torsion are tight [41]. In this metric and generic absence of gauge invariance typically work, we restrict attention to Riemann geometry, for which imply the presence of ghosts and corresponding negative- the torsion vanishes and the spin connection is fixed in energy problems, which can tightly constrain the viability terms of the vierbein. It therefore suffices for our purposes of various models [40]. If the gravitational couplings and to consider the vierbein degrees of freedom and their role the potential V are disregarded, there is a unique set of relative to the gravitational and NG modes. Bumblebee parameters ensuring the absence of ghosts: 1 1, 2 models in the more general context of Riemann-Cartan 3 0. With this choice, the kinetic term for the bumble- spacetime with nonzero torsion are investigated in bee becomes the Maxwell action, in which the usual U(1) Refs. [4,5]. gauge invariance excludes ghosts. When the gravitational There is a substantial literature concerning theories of terms and couplings are included, this gauge invariance is vacuum-valued vectors coupled to gravity. The five- maintained for the parameter choice 1 1, 2 3 parameter Lagrange density (19) excluding the potential 1 2 0. The further inclusion of the potential V V and the cosmological constant was investigated by Will breaks the U(1) gauge symmetry, but the form of the and Nordtvedt in the context of vector-tensor models of kinetic term ensures a remnant constraint on the equations gravity [42]. Kostelecky´ and Samuel (KS) [3] introduced of motion arising from the identity the potential V triggering spontaneous Lorentz violation and studied both the smooth quadratic potential (5) and the DDB 0: (25) linear Lagrange-multiplier case (6) for the class of models The action for these bumblebee models, introduced in given by Eq. (26). The presence of the potential introduces Ref. [3], is therefore of particular interest. The correspond- a variety of qualitatively new effects, including the neces- ing Lagrange density sary breaking of U(1) gauge invariance [4], the existence of massless NG modes and massive modes [5], and poten- 1 1 L eR 2 eB B eV L ; KS 16G 4 M tially observable novel effects for post-newtonian physics (26) [33] and for the matter sector [43]. The potential V also leads to a candidate alternative description of the is investigated in more detail in Sec. IV. The reader is [5] and the graviton [4,36]. In Minkowski spacetime, more warned that some confusion about the relationship between general potentials V of hypergeometric form are known to these models and ones with nonzero 2 and 3 exists in the satisfy the one-loop exact renormalization group [44]. literature. In particular, results for the models (26) can Models of the form (19) with 1 2 0, a unit timelike differ from those obtained in models with nonzero , 2 3 b, a linear Lagrange-multiplier potential VL, and an addi- via straightforward adoption of the limit , ! 0, due to 2 3 tional fourth-order interaction for B have been studied in the emergence of the remnant constraint (25). some detail as possible unconventional theories of gravity The discussion above defines bumblebee models on the [45,46]. Other works involving bumblebee models include spacetime manifold, without introducing a local Lorentz Refs. [40,47–51]. basis. In this approach, the Lorentz NG modes are hidden An aspect of bumblebee models of particular interest is within the bumblebee field B . Adopting instead a vierbein the appearance of massless propagating vector modes. This formulation reveals explicitly the local Lorentz properties feature suggests the prospect of an alternative to the usual of the models, and it also provides a natural way to incor- description of photons via U(1) gauge theory. The central porate spinor fields in the matter Lagrange density LM. idea is to identify the photon modes with the NG modes The vierbein e a converts tensors expressed in a local basis arising from spontaneous Lorentz violation. Early discus- to ones on the spacetime manifold. The spacetime metric sions along these lines centered on reinterpretating the g is related to the Minkowski metric ab in the local photon or electron in the context of frame as without physical Lorentz violation [52,53]. More recently, a b the Lorentz NG modes in certain bumblebee models with g e e ab; (27) physical Lorentz violation have been shown to obey the while the bumblebee spacetime vector B is related to the Einstein-Maxwell equations in Riemann spacetime in axial local bumblebee vector Ba as gauge [5]. These models are further considered below, with a the discussion initiated in Sec. IVA 3. One motivation of B e B : (28) a the present work is to investigate the role of massive modes A complete treatment in the vierbein formalism involves and Lagrange-multiplier fields in this context. In particular, ab also introducing the spin connection ! , which appears it is of interest to investigate possible modifications to

065020-7 ROBERT BLUHM, SHU-HONG FUNG, AND V. ALAN KOSTELECKY´ PHYSICAL REVIEW D 77, 065020 (2008) electrodynamics and gravity, with an eye to novel phe- of the system, but the physics cannot depend on this choice. nomenological applications of spontaneous Lorentz and In particular, this must remain true even when Lorentz diffeomorphism breaking. and diffeomorphism symmetry are broken, whether explic- itly or spontaneously. Viable candidate theories with B. Transformations spacetime-symmetry breaking must therefore be invariant under observer transformations, which are merely changes In considering theories with violations of Lorentz and of coordinate system. For example, the SME is formulated diffeomorphism symmetry, it is important to distinguish as a general effective field theory that is invariant under between observer and particle transformations [18]. Under observer general coordinate transformations and under either an observer general coordinate transformation or an observer local Lorentz transformations [4,17,18]. observer local , geometric quanti- In contrast, a theory with physical Lorentz and diffeo- ties such as scalars, vectors, tensors, and their derivatives morphism breaking cannot by definition remain fully in- remain unchanged, but the coordinate basis defining their variant under the corresponding particle transformations. components transforms. In contrast, particle transforma- tions can change geometric quantities, independently of For example, if the breaking is explicit, the action of the any coordinate system and basis. theory changes under particle transformations. If instead In theories without spacetime-symmetry breaking, the the breaking is spontaneous, the action remains invariant component forms of the transformation laws for particle and the equations of motion transform covariantly, as and observer transformations are inversely related. For usual. However, the vacuum solution to the equations of example, under infinitesimal particle diffeomorphisms de- motion necessarily contains quantities with spacetime in- scribed by four infinitesimal displacements , the com- dices that are unchanged by particle transformations. ponents of the bumblebee field transform according to the These vacuum values and the excitations around them Lie derivative as can lead to physical effects revealing the symmetry breaking. B ! B @ B @B; In bumblebee models, the relevant spacetime vacuum (29) values are those of the bumblebee and metric fields, de- B ! B @ B @B ; noted hBi and hgi. These are unaffected by particle while the metric transforms as diffeomorphisms, and their nonzero components thereby reveal the broken particle diffeomorphisms. In contrast,

g ! g @ g @ g @ g ; hBi and hgi both transform as usual under observer general coordinate transformations, which therefore are g ! g @ g @ g @g : unbroken. Analogous results hold for the local-frame vac- (30) uum value hBai of the local bumblebee field. The compo- Under infinitesimal observer general coordinate transfor- nents hBai remain unaffected by particle local Lorentz mations, which are the observer equivalent of diffeomor- transformations, with the invariance of the nonzero com- phisms, the formulas for the transformations of the ponents resulting from the breaking of local Lorentz gen- bumblebee and metric components take the same mathe- erators, while the components hBai transform under matical form up to a possible sign change in the arbitrary observer local Lorentz transformations in the usual way. a parameter , even though these transformations are only Similarly, the vacuum value he i of the vierbein is un- the result of a change of coordinates. Similarly, under changed by both particle diffeomorphisms and particle infinitesimal particle local Lorentz transformations with Lorentz transformations, but it transforms as a vector under six parameters ab ba related to the local Lorentz the corresponding observer transformations. b b b group element by a a a , the local components By virtue of the relation (28) between the spacetime and of the bumblebee field transform as local bumblebee fields, it follows that spontaneous local b Lorentz violation is necessarily accompanied by sponta-

B ! B B ; a a a b (31) neous diffeomorphism violation and vice versa [5]. The with the formula for Ba following from this by raising point is that the vacuum value of the vierbein is nonzero, so indices with ab. Under observer local Lorentz transfor- the existence of a nonzero hBai spontaneously breaking mations, the corresponding transformation formulas again local Lorentz symmetry also implies the existence of a take the same mathematical form. However, any form of nonzero hBi spontaneously breaking diffeomorphism physical Lorentz and diffeomorphism breaking destroys symmetry, and vice versa. Note, however, that this result the details of these equivalences. fails for explicit violation, where the analogue of the A fundamental premise in classical physics is that the relation (28) is absent. In general, explicit local Lorentz properties of a physical system are independent of the violation occurs when a nonzero quantity tabc... is exter- presence of a noninteracting observer. The observer is nally prescribed in the local frame, but the corresponding a b c free to select a coordinate system to describe the physics spacetime quantity t... e e e tabc... is defined

065020-8 SPONTANEOUS LORENTZ AND DIFFEOMORPHISM ... PHYSICAL REVIEW D 77, 065020 (2008) a a using the full vierbein and hence remains invariant under hBihe iba ba b: (38) diffeomorphisms. Similarly, explicit diffeomorphism vio- lation occurs when a nonzero quantity t is externally It is convenient to decompose the local vector excita- ... a prescribed on the spacetime manifold, but the correspond- tions a into transverse and longitudinal pieces ing local quantity defined via the inverse vierbein remains with respect to b. Excluding for simplicity the case of invariant under local Lorentz transformations. lightlike b, we write t ^ t b;b 0; (39) C. Mode expansions p ^ 2 To study the content and behavior of the modes, the where b b= b is a vector along the direction of b ^ ^ fields can be expanded as infinitesimal excitations about obeying b b 1. Using this decomposition, the bum- their vacuum values. At the level of the Lagrange density, it blebee mode expansions (36) become suffices for most purposes to keep only terms to second 1 t ^ order in the field excitations, which linearizes the equations B b h b b; 2 (40) of motion. B b 1h b t b^: Assuming for simplicity a Minkowski background 2 hgi, we expand the metric and its inverse as It is instructive to count degrees of freedom in the expressions (37) and (40). On the left-hand side, the vier- g h;g h : (32) bein has 16 components and the bumblebee field 4, for a For the local bumblebee vector, we write total of 20. On the right-hand side, the symmetric metric component h has 10, the antisymmetric component t Ba ba a; (33) has 6, the transverse bumblebee excitation has 3, and the longitudinal excitation has 1, producing the required where the vacuum value hB i is denoted b and assumed a a matching total of 20. Of these 20 degrees of freedom, 6 are constant, and the infinitesimal excitations are denoted . a metric modes, 4 are bumblebee modes, while 6 are asso- The vacuum condition (23) implies ciated with local Lorentz transformations and 4 with dif- ab 2 ba bb b : (34) feomorphisms. The explicit transformations of all the field components under particle diffeomorphisms and local In a Minkowski background, the vierbein vacuum value Lorentz transformations can be deduced from the full-field a a can be chosen to be he i in cartesian coordinates. expressions and from the invariance of the vacuum values. The expansions of the vierbein and its inverse are A list of formulas for both particle and observer trans-

a a 1 a a formations is provided in the appendix. e h ; 2 (35) Among all the excitations, only the longitudinal compo- 1 nent of the bumblebee field is invariant under both e a a 2h a a; particle diffeomorphisms and local Lorentz transforma- where ha and a are, respectively, the symmetric and tions. It is therefore a physical degree of freedom in any antisymmetric components of the vierbein. The covariant gauge. Moreover, using Eqs. (23) and (40) reveals that and contravariant components of the bumblebee field fol- exciting the mode alone produces a nonzero value of low from Eq. (28)as X, given at first order by a 1 a a 2 ^

B ba ah ba; X Bg B b 2b b: (41) 2 (36) a a 1 a B ab 2h a ab : As a result, the excitation is associated with a non- minimal value of the potential, and it therefore cannot be Since the local and spacetime background metrics are an NG mode. In fact, for the case of a smooth quadratic both Minkowski and since the excitations are infinitesimal, potential, is a massive mode. In contrast, in a theory with the distinction between the Latin local indices and the a Lagrange-multiplier potential where the constraint X 0 Greek spacetime indices can be dropped. We adopt is enforced as an equation of motion, the massive mode Greek indices for most purposes that follow, raising and identically vanishes. lowering indices on purely first-order quantities with and . The vierbein expansions (35) can then be rewrit- D. Gauge fixing and NG modes ten as Since the spacetime-symmetry breaking in bumblebee 1 e e h ; models is spontaneous, the four diffeomorphisms parame- 2 (37) 1 trized by and the six local Lorentz transformations e e 2h : parametrized by leave invariant the bumblebee action The vacuum value for the bumblebee vector becomes (19) and transform covariantly the equations of motion.

065020-9 ROBERT BLUHM, SHU-HONG FUNG, AND V. ALAN KOSTELECKY´ PHYSICAL REVIEW D 77, 065020 (2008)

Fixing the gauge freedom therefore requires 10 gauge The properties of and E under various particle and conditions. observer transformations are given in the appendix. The For the diffeomorphism freedom, a choice common in projections b and b associated with the broken the literature is the harmonic gauge generators determine the NG modes, which are therefore b and Eb . @ h 0; (42) We can follow this procedure to elucidate the relation- 1 ship between the NG modes and the component fields in where h h 2 h and h h h . In the harmonic gauge, the Einstein tensor becomes G the decomposition (40)ofB. Consider first the diffeo- 1 ᮀ morphism NG mode, which is generated by a virtual trans- 2 h at linear order. An alternative choice for the diffeomorphism degrees of freedom is the axial gravita- formation acting on the spacetime vacuum value hBi and tional gauge involving the broken diffeomorphism generator. The rele- vant transformation is given in Eq. (A1), and it yields hb 0: (43) hBi!b @b : (48) Both these gauge choices represent four conditions. To fix the local Lorentz freedom, it is common to eliminate all six Comparison of this result to the form of Eq. (40) reveals antisymmetric vierbein components by imposing the six 1 that the four vierbein combinations 2 h b con- conditions tain the diffeomorphism NG mode. This agrees with the result obtained by combining virtual diffeomorphisms on

0: (44) the vacuum values of the component fields in Eq. (40):

Other choices are possible here too. Consider, for example, hhb i!@ @b ; the decomposition of in terms of projections along b, 1 h b i!@ @ b ; (49) tt t ^ t ^ 2 b b; t tt t (45) hi!0; hi!0: b b 0; Note also that the combinations 1 h b maintain which is the analogue of Eq. (39) for . The components 2 the potential minimum V 0, as is expected for an NG tt t ^ and b each contain 3 degrees of freedom. mode. Inspection of the transformation laws shows that an alter- In contrast, the three Lorentz NG modes Eb are native to fixing the local Lorentz gauge via Eq. (44) is the generated by virtual transformations acting on the local- set of six conditions frame vacuum value hBai and involving the broken local tt t 0: (46) Lorentz generators:

t b Note, however, that the combination b ap- hBai!ba Eabb ; (50) pearing in B in Eq. (40) is invariant and therefore cannot Comparison with Eqs. (33) and (39) shows that the trans- be gauged to zero. Evidently, the associated local Lorentz verse field t contains the three Lorentz NG modes degrees of freedom must remain somewhere in the theory. b. Note that there are exactly 3 degrees of freedom The bumblebee vacuum value hB ib breaks one of E in t , all of which maintain the potential minimum V 0. the four diffeomorphism symmetries. The broken genera- This result can be used to connect the component fields in tor is associated with the projected component b of the parameter . Analogously, the vacuum value hB ib the decomposition of B with the three Lorentz NG modes a a breaks three of the six local Lorentz symmetries. The Eb . The vacuum value hBi itself is invariant under broken generators are associated with the components local Lorentz transformations, so another component field b must also contain the Lorentz NG modes. Performing abb b of projected along b . In each case, the unbroken generators are associated with the orthogonal virtual local Lorentz transformations on the vacuum values complements to the projections. of the component fields gives Since the vacuum breaks one particle diffeomorphism hhb i!0; hb i!Eb ; and three local Lorentz transformations, four NG modes (51) t appear. A useful general procedure to identify these modes hi!Eb ; hi!0; is first to make virtual particle transformations using the broken generators acting on the appropriate vacuum values which shows that the three combinations hb i also for the fields in V, and then to promote the corresponding contain the three Lorentz NG modes. parameters and to field excitations [5]: Combining the results (49) and (51) reveals the follow- ing NG mode content for the component field combina-

! ;! E E : (47) tions in the decomposition (40)ofB:

065020-10 SPONTANEOUS LORENTZ AND DIFFEOMORPHISM ... PHYSICAL REVIEW D 77, 065020 (2008) 1 1 h b consist of the three Lorentz NG modes 2h b @b Eb ; 2 and the diffeomorphism NG mode, and all the NG modes t b; (52) E are contained in the vierbein [5]. The bumblebee field can 0: be written as 1 ^ B b @ b b b : We see that the four combinations 2 h b contain E (57) a mixture of the diffeomorphism NG mode and the three Note that only one projection of appears, even though Lorentz NG modes, while the three combinations t con- four diffeomorphism choices remain. The corresponding tain only the three Lorentz NG modes. expression for B includes additional metric contributions By fixing the 10 gauge freedoms in the theory, the above and is given by results can be used to determine the physical content of the ^ field B. Suppose first we adopt the 10 conditions (43) and B b @ b E b b : (58) (44). Then, the bumblebee field becomes This equation contains contributions from all four fields

t ^ B b b . However, if the diffeomorphism excitations are re- ^ stricted only to the one for the broken generator, for which b Eb b; (53) obeys b b, then B also reduces to an where the result (52) has been used. Note that the fixed expression depending only on the diffeomorphism NG gauge means that fields with different transformation prop- mode b . Related results are obtained in Ref. [5], which erties can appear on the left- and right-hand sides. In this investigates the fate of the NG modes using a decomposi- gauge, the four components of B are decomposed into tion of the vierbein into transverse and longitudinal com- t ponents along b . This decomposition leads to the same three Lorentz NG modes associated with and the one massive mode . The diffeomorphism NG mode is absent. relations as Eqs. (57) and (58) when the condition b It obeys b is applied. Even without gauge fixing, the fields cancel at linear @b @b (54) order in both G and B. As a result, propagating diffeo- and hence b @b 0, and it is locked to the Lorentz morphism NG modes cannot appear. This is a special case NG modes via the equation of a more general result. By virtue of their origin as virtual particle transformations, the diffeomorphism NG modes @b Eb : (55) appear as certain components of representation-irreducible For the diffeomorphism NG mode, this gauge is evidently fields with nonzero vacuum values. However, diffeomor- analogous to the unitary gauge in a non-Abelian gauge phism invariance ensures these modes enter in the metric theory. and bumblebee fields in combinations that cancel in a An alternative gauge choice could be to impose the 10 diffeomorphism-invariant action, including the general conditions (43) and (46) instead. This gives bumblebee action (19). In contrast, the Lorentz NG modes do play a role in the bumblebee action. They can be

t ^ B b b identified with a massless vector field A Eb in the ^ fixed axial gauge Ab 0. The fully gauge-fixed ex- b Eb b; (56) pression (53) for B then becomes where in this gauge the Lorentz NG modes b are E ^ t t B b A b; (59) identified with instead of . One way to understand this identification is to perform a local Lorentz transforma- where the transverse components have the form of photon tion with parameter E on the first result in fields in the axial gauge, and the longitudinal mode is the t Eq. (53). This changes the value of from Eb to massive mode . t zero, while simultaneously converting from zero to t Eb . In this gauge, the explicit decomposition E. Alternative expansions (56)ofB in terms of NG modes is the same as that in The analysis in the previous subsections is based on the Eq. (53), but the three Lorentz NG modes are now asso- local-frame expansion (33) of the bumblebee field Ba. t ciated with the components of the vierbein. The diffeo- However, other mode expansions are possible, including morphism mode remains absent. It still obeys the condition ones in which the vierbein makes no explicit appearance (54) and is locked to the Lorentz NG modes by Eq. (55). and the local Lorentz transformations are no longer mani- Partial gauge conditions can also be imposed. For ex- fest. This subsection offers a few comments on two alter- ample, suppose the choice (46) is made for the local native expansions used in some of the literature. Lorentz gauge, while the diffeomorphism gauge remains The first alternative mode expansion is specified using unfixed. Then, the 4 degrees of freedom in the combination the covariant spacetime components of the bumblebee field

065020-11 ROBERT BLUHM, SHU-HONG FUNG, AND V. ALAN KOSTELECKY´ PHYSICAL REVIEW D 77, 065020 (2008) [5,17], b 1 h b E 2 ^ 1 b E h b : (68) ^ B b E; (60) b b 2 where E represents the excitations of the spacetime bum- Notice that appears here as a diffeomorphism-invariant combination of field components that transform blebee field around the vacuum value b. It follows that the contravariant components are nontrivially. The choice of diffeomorphism gauge has interesting 0 B b E h b: (61) consequences for E and E. First, note that E cannot These fields are linked to the vierbein and the local-frame be gauged to zero, since only 1 degree of freedom b appears in its transformation law. However, it is possible to fluctuations via 0 gauge E to zero. The corresponding gauge condition for 1 t ^ E 2h b b: (62) E is E hb . In either of these gauges, the massive mode becomes The fields E are scalars under particle local Lorentz trans- 1 ^ ^ formations, but transform under particle diffeomorphisms 2b hb ; (69) as thereby becoming part of the metric. The bumblebee field E ! E @b ; strength reduces to (63) E E ! E @b : B @h @hb ; (70) Note that this usage of E differs from that in Ref. [5]. and so is also given by the metric. Evidently, in these The second alternative expansion [33,45,47–49] starts gauges the theory is defined entirely in terms of the metric instead with the contravariant bumblebee components excitations h. Requiring E to vanish or dropping the 0 term hb in Eq. (65) therefore improperly sets to zero B b E ; (64) the bumblebee field strength B, which alters the equa- where a prime is used to distinguish the excitations E0 tions of motion governing the Lorentz NG and massive from the previous case. The corresponding covariant com- modes [48]. It follows from Eq. (70) that the effective ponents are then Lagrange density can contain additional kinetic terms for 0 h, beyond those arising from the gravitational action, B b E hb : (65) that originate from the bumblebee kinetic terms LK. These The field redefinitions connecting these to the vierbein and additional terms propagate the Lorentz NG modes, dis- previous case are guised as massless modes contained in the metric. 0 1 ^ E h b b 2 t IV. EXAMPLES E hb : (66) In this section, some simple examples are developed to The fields E0 are scalars under particle local Lorentz illustrate and enrich the general results obtained in the transformations but transform under particle diffeomor- discussions above. We choose to work within the class of phisms as KS bumblebee models with Lagrange density given by

Eq. (26), which avoids a priori propagating ghost fields 0 0 E ! E @ b ; and the complications of nonminimal gravitational cou- 0 0 0 (67) E E ! E @b : plings. For definiteness, we set 0 and choose the matter-bumblebee coupling to be 0 Note that the fields E and E have different transforma- tion properties and contain different mixes of the bumble- L M eBJM; (71) 0 bee and metric excitations. However, E and E take the where JM is understood to be a current formed from matter same gauge-fixed form in the gauge (43). fields in the theory. In principle, this current could be The two alternative expansions are useful because the formed from dynamical fields in the theory or could be 0 excitations E and E are invariant under infinitesimal prescribed externally. For the latter case, diffeomorphism local particle Lorentz transformations. In these variables, invariance is explicitly broken unless the current satisfies a the local Lorentz symmetry is hidden and only the diffeo- suitable differential constraint. morphism symmetry is manifest. The Lorentz NG modes The section begins by presenting some general results 0 lie in the b-tranverse components of E or E. The for this class of models, including ones related to the massive mode joins the diffeomorphism mode in the equations of motion, conservation laws, and the connection longitudinal component of E. It can be identified from to Einstein-Maxwell electrodynamics. We then turn to a Eq. (62) as the combination more detailed discussion of three specific cases with differ-

065020-12 SPONTANEOUS LORENTZ AND DIFFEOMORPHISM ... PHYSICAL REVIEW D 77, 065020 (2008) ent explicit potentials VX, where the bumblebee field Note that this second conservation law is a special feature combination X is defined as in Eq. (22). The three cases of KS bumblebee models. It is a direct consequence of involve the smooth quadratic potential VSX of Eq. (5), the choosing the a priori ghost-free action (26). Note also that linear Lagrange-multiplier potential VL; X of Eq. (6), this conservation law holds even though the potential term and the quadratic Lagrange-multiplier potential VQ; X V excludes any local U(1) gauge symmetry in these of Eq. (7). The massive modes are studied for each poten- models. tial, and their effects on gravity and are explored. 2. Bumblebee currents The bumblebee current JB defined in Eq. (79) vanishes A. General considerations when V0 0. This situation holds for the vacuum solution 1. Equations of motion and conservation laws and for NG modes, and it then follows from Eq. (81) that the matter current JM is covariantly conserved. However, Varying the Lagrange density (26) with respect to the 0 metric yields the gravitational equations of motion V Þ 0 in the presence of a nonzero massive mode or a nonzero linear Lagrange multiplier, whereupon the bum- G 8GT : (72) blebee current JB acts as an additional source for the In this expression, the total energy-momentum tensor T bumblebee field equation (77). A nonzero massive mode or nonzero linear Lagrange multiplier contributes to the can be written as a sum of two terms, bumblebee component TB of the energy-momentum ten- T TM TB : (73) sor too, and it therefore also acts as an additional source for the gravitational field equations (72). The first is the energy-momentum tensor TM arising from The contributions to the energy-momentum tensor TB the matter sector. The second is the energy-momentum 0 stemming from V Þ 0 can be positive or negative. This is tensor T arising from the bumblebee kinetic and poten- B a generic feature, as can be seen from the expression for tial terms, TV in Eq. (17) and the general bumblebee Lagrange TB TK TV ; (74) density (19). The full energy-momentum tensor T is conserved, but the prospect of negative values for T where T and T are given by B K V implies stability issues for the models. Under suitable 1 T B B g B B (75) circumstances, for example, unbounded negative values K 4 of TB can act as unlimited sources of energy for the matter and sector. Stability is also a potential issue in the absence of 0 matter. For example, for the case where J is disregarded T Vg 2V B B : (76) M V and the potential involves a linear Lagrange multiplier Varying instead with respect to the bumblebee field producing Þ 0 on shell, at least one set of initial values generates the remaining equations of motion, is known that yields a negative-energy solution [54]. Whether instabilities occur in practice depends on the D B J : (77) form of the Lagrange density and on the initial conditions. Like the total energy-momentum tensor, the total current The situation is comparatively favorable for KS bumblebee J can be written as the sum of two terms models because the extra conservation law (81) can play a role. In principle, negative-energy contributions in the J J J : (78) M B initial state can be eliminated by choosing initial condi- 0 The partial current JM is associated with the matter sector tions such that V 0, while the conservation law can and acts as an external source for the bumblebee field. The prevent the eventual development of a destabilizing partial current JB arises from the bumblebee self- mode. One way to see this is to expand the conservation interaction, and it is given explicitly as law Eq. (81) at leading order in a Minkowski background. 0 For nonzero hB0i, we obtain JB 2V B : (79)

The contracted Bianchi identities for G lead to con- 0 1 0 j 0 @ V @ J 2V @ B 2B @ V : (82) servation of the total energy-momentum tensor, 0 2B0 M j D T D T T 0: (80) M B Suppose the matter current is independently conserved, 0 The antisymmetry of the bumblebee field strength B DJM @JM 0. The initial condition V 0 then 0 implies the remnant constraint (25) and hence a second implies @0V 0 initially. Taking further derivatives conservation law, shows that all time derivatives of V0 vanish initially and so V0 remains zero for all time, indicating stability is DJ DJM JB 0: (81) maintained.

065020-13 ROBERT BLUHM, SHU-HONG FUNG, AND V. ALAN KOSTELECKY´ PHYSICAL REVIEW D 77, 065020 (2008) It is interesting to note that the matter-current conserva- This modified Gauss law is unavailable to other bumblebee M tion law D J 0 emerges naturally in the limit of models. vanishing NG and massive modes, where the bumblebee In a suitable limit, the equations of motion (72) and (77) field reduces to its vacuum value B !hBib. This for the KS bumblebee models reduce to the Einstein- agrees with a general conjecture made in Ref. [4]. The line Maxwell equations in a fixed gauge for the metric and of reasoning is as follows. In the limit, the bumblebee photon fields. To demonstrate this, we work in asymptoti- theory with spontaneous Lorentz violation takes the form cally Minkowski spacetime and choose the gauge in which of a theory with explicit Lorentz violation with couplings the bumblebee field is given by Eq. (59). The limit of ! only to b. However, theories with explicit Lorentz viola- interest is 0, corresponding to zero massive mode. tion contain an incompatibility between the Bianchi iden- For the case with the Lagrange-multiplier potential (6), where is constrained to zero, an equivalent result also tities and the energy-momentum conservation law. The incompatibility is avoided for spontaneous Lorentz viola- follows in the limit ! 0. Since TV acquires nonzero tion by the vanishing of a particular variation in the action, contributions only from the massive mode or linear Lagrange multiplier, it follows that as ! 0 or ! 0 which reduces in the limit to the requirement of current M the bumblebee energy-momentum tensor T reduces to conservation D J 0. Nonetheless, if B is excited B away from this limit, current conservation in the matter that of Einstein-Maxwell electrodynamics, sector may fail to hold. 1 T ! T T F F g F F : (84) In practice, the potential instability may be irrelevant for B K EM 4 physics. For example, if a bumblebee model is viewed as In this equation, the bumblebee field strength is reinter- an effective field theory emerging from an underlying preted as the electromagnetic field strength stable quantum theory of gravity, any apparent instabilities may reflect incomplete knowledge of constraints on the B F @A @A; (85) massive modes or of countering effects that come into play at energy scales above those of the effective theory. Under via the mode expansion (59), and the gravitational field these circumstances, in KS bumblebee models it may equations (72) reduce to their Einstein-Maxwell counter- suffice for practical purposes simply to postulate that the parts. Similarly, the bumblebee current JB in Eq. (79) matter and bumblebee currents do not mix and hence obey vanishes as ! 0 or ! 0 because V0 ! 0. This leaves separate conservation laws, only the conventional matter current JM, which by virtue of Eq. (81) obeys covariant conservation. The bumblebee

D J 0;DJ 0: (83) M B field equations (77) therefore also reduce to their

For instance, one can impose that only matter terms LM Einstein-Maxwell counterparts. with a global U(1) symmetry are allowed, as is done for the Even when vanishes, the interpretation of the model as case of Minkowski spacetime in Ref. [50]. Another option bumblebee electrodynamics can in principle be distin- M might be to disregard J altogether and allow couplings guished from conventional Einstein-Maxwell electrody- between the matter and bumblebee fields only through namics through nontrivial SME-type couplings involving gravity. In what follows, we adopt Eqs. (83) as needed to the vacuum value b in the matter equations of motion. investigate modifications of gravity arising from massive Depending on the types of couplings appearing in the modes and to study the conditions under which Einstein- matter-sector Lagrange density LM, a variety of effects Maxwell solutions emerge in bumblebee models. could arise. At the level of the minimal SME, for example, axial vector couplings involving b can be sought in 3. Bumblebee electrodynamics numerous experiments, including ones with electrons [22,23], protons and neutrons [25,26], and muons [29]. An interesting aspect of the KS bumblebee models is Other types of SME coefficients can also be generated their prospective interpretation as alternative theories of from the vacuum value b . For example, the nonzero electrodynamics with physical signatures of Lorentz vio- symmetric traceless SME coefficients c might emerge lation [5]. In this approach, basic electrodynamics proper- ties such as the masslessness of photons are viewed as via the symmetric traceless product bb b b=4, consequences of spontaneous local Lorentz and diffeomor- and experimental searches for these include ones with phism breaking rather than of exact local U(1) symmetry. electrons [22,24], protons, and neutrons [25,26]. A key point is that KS bumblebee models have no depen- If the model conditions permit the field to be nonzero, further deviations from Einstein-Maxwell electrodynamics dence on second time derivatives of B0. They therefore feature an additional primary constraint relative to the can arise. For example, in the weak-field regime where ^ bumblebee models (19) with more general kinetic terms. b is small compared to b, the bumblebee current JB is The primary constraint generates a secondary constraint in linear in and acts as an additional source in the equation the form of a modified Gauss law, which permits a physical of motion (77) for the field strength B. Effects of this interpretation of the theory in parallel to electrodynamics. type are investigated in the next few subsections. For large

065020-14 SPONTANEOUS LORENTZ AND DIFFEOMORPHISM ... PHYSICAL REVIEW D 77, 065020 (2008) fields, nonlinear effects in the bumblebee current and component of the energy-momentum tensor becomes energy-momentum tensor may also play a role. However, the field value of is typically suppressed when the mass ^ TB TEM 4b bb b ; (87) of is large, so if the mass is set by the Planck scale then the linear approximation is likely to suffice for most real- where TEM is the zero- limit of TK given in Eq. (84) and istic applications. the other term arises from T. The bumblebee current V In the limit of zero , the physical fields are the field reduces to strengths B and the gravitational field. Excitations of the bumblebee field B in the classical theory are then unob- ^ JB 4b bb : (88) servable, in parallel with the potential A in classical electromagnetism. This may have consequences for leading-order effects in the weak-field limit. For example, The constraint obtained by assuming current conservation the weak static limit in general bumblebee models (19) in the matter sector becomes requires B itself to be time independent. However, at b @ 0: (89) leading order in the KS bumblebee model, it can suffice to require that only the field strength B be time indepen- These expressions reveal that at leading order the massive dent while B itself has time dependence, in analogy with Maxwell electrodynamics in Minkowski spacetime. For mode acts as a source for gravitation and electrodynam- ics, subject to the constraint (89). example, the Coulomb electric field E~ @ A~ for a static 0 Note that the linearization procedure can alter the time point charge q emerges in temporal gauge A 0 from a 0 behavior and dynamics of the fields in the presence of a time-dependent A 0;t@ , where is the j q q nonzero massive mode . Suppose, for example, that b is Coulomb potential. Likewise, in certain leading-order timelike and we adopt the global observer frame in which weak static solutions for which acts as a source of b b; 0; 0; 0. Nonlinearities in the current J associ- charge, the bumblebee field B may naturally exhibit B ated with nonzero generate time dependence for most potential lines converging at the source of charge, similar solutions because the spatial current J~ does work on the to the physical configurations with singular derivatives of B field strength B . However, the linearization (88) implies A that occur in classical electrodynamics. J is nonzero only along the direction of b, reducing to a Another interesting issue is the role of quantum effects. B pure charge density J ; 0; 0; 0 with Since bumblebee electrodynamics involves gravity, it faces B B B b2 the same quantization challenges as other theories of grav- 4 , and the current-conservation law (89) then re- ity and electrodynamics, including Einstein-Maxwell the- quires B and hence to be static. Nonetheless, the ory. Some work on the renormalization structure at one linearization procedure captures the dominant effects of loop has been performed [44,55], and in the limit of zero nonzero . massive mode and in Minkowski spacetime the usual properties of quantum electrodynamics are expected to 1. Propagating modes hold. Addressing this issue in detail is an interesting To study free propagation of the gravitational and bum- open problem that lies beyond the scope of this work. blebee fields in the absence of charge and matter, set JM TM 0 in the linearized equations of motion. The gravi- B. Smooth quadratic potential tational equations (72) then become

This subsection considers the specific KS bumblebee ᮀ model (26) having smooth quadratic potential (5) with h @ @ h @ @h @ @h ^ the definition (22): 64Gb bbb: (90) 1 2 2 V VSX2Bg B b : (86) Contributions from TEM are second-order in E and can be For definiteness, we adopt the mode expansion B neglected in this context. The bumblebee equations (77) b E of Eq. (60) in a Minkowski background and reduce to assume weak fields h and E so that the gravitational ᮀ ^ and bumblebee equations of motion (72) and (77) can be E @@ E 4b bb: (91) linearized. One goal is to investigate deviations from Einstein- Note that the massive mode is defined by Eq. (68)in Maxwell theory arising from the presence of a weak non- terms of E and h, so it is not an independent field in zero massive mode . We therefore focus on dominant these equations. Note also that the linearized current- corrections to the linearized Einstein-Maxwell equations conservation law (89) follows by taking a derivative of arising from . In this approximation, the bumblebee Eq. (91).

065020-15 ROBERT BLUHM, SHU-HONG FUNG, AND V. ALAN KOSTELECKY´ PHYSICAL REVIEW D 77, 065020 (2008) To investigate the behavior of the massive mode ,itis duce to those of electrodynamics in axial gauge [5]. A convenient to combine the above two equations to obtain nonzero value of modifies the equations, but the usual propagation-transverse components of A and hence the ᮀ 4b b1 4Gb b physical photon modes remain unaffected. Since and A are independent fields, the dispersion 1 b @@ E hb : (92) properties of the massive mode can be identified from bb^ Eq. (96). Any solutions of the theory describing freely propagating modes are required to obey the boundary At first glance, this equation might suggest that can be a conditions that the spacetime be asymptotically flat and propagating massive field. However, depends on the that the bumblebee fields vanish at infinity. At linear order, fields appearing on the right-hand side, so the modes are these solutions are formed from harmonic plane waves still coupled and care is required in determining the dis- with energy-momentum vectors p E; p~ obeying suit- persion properties, including the mass value. able dispersion relations. The modes satisfying the asymp- A suitable choice of diffeomorphism gauge clarifies totic boundary conditions are then constructed as Fourier matters. The modes remain entangled in the harmonic superpositions of the plane-wave solutions. For the mas- gauge (42) and also in the barred axial gravitational gauge sive mode , the constraint equation (89) imposes the h b 0. However, a sufficient decoupling can be additional requirement achieved by adopting the axial gravitational gauge (43) and by decomposing into pieces parallel and perpen- E b p 0: (97) dicular to b. In this gauge, the longitudinal component E of E reduces to E , so that Any freely propagating massive mode is therefore con- strained to have an energy-momentum vector orthogonal t ^ ^ E E Eb A b: (93) to the vacuum value b. For harmonic-mode solutions to the equations of motion, this requirement implies that the t In this equation, we write E A for the transverse term on the right-hand side of Eq. (96) vanishes in Fourier components of E to make easier the task of tracking space. The resulting dispersion law for the massive mode -dependent deviations from conventional electrodynam- is then ics. These components satisfy the axial condition Ab p p 4b b ; (98) 0. We emphasize that the axial condition is not a gauge choice. It is a consequence of projecting along b and is which involves the squared-mass parameter independent of the gravitational gauge fixing. It does, however, have the same mathematical form as an axial- 2 M 4b b : (99) gauge condition in electrodynamics, even though the bum- blebee models have no U(1) symmetry. The sign of M2 depends on whether b is timelike or With these choices, the usual Einstein equations for h spacelike. The scale of M2 depends on both and b. in axial gravitational gauge are recovered from the result (90) when vanishes [5]. For nonzero , only one linearly Inspection of Eqs. (94)or(96) shows that the limits j j2 !1 !1 independent combination of the Einstein equations is M or with b fixed are equivalent to changed. It can be written taking the limit of vanishing massive mode, ! 0. The discussion in Sec. IVA 3 then implies that the limit of large 2 ᮀ ^ jMj approximates Einstein-Maxwell theory. h 2@ @ h 64Gb bb b: (94) For the case of a timelike vacuum value b, the parame- ter M2 4b2 has the wrong sign for a physical mass. Since the usual propagation-transverse components of h remain unaffected, the physical graviton modes propagate Adopting the global observer frame in which b normally. The bumblebee equations (91) become b; 0; 0; 0, we see that the constraint (97) forces the energy E of this mode to vanish, leaving the condition that the 1 ᮀ magnitude of the momentum must remain fixed at jp~j A @@ A bb @@ A 0 (95) b b jMj. A time-independent mode with fixed spatial wave- length cannot be Fourier superposed to form a physical and wave packet satisfying the asymptotic boundary condi- tions. It follows that no physical propagating massive 1 ᮀ mode appears when b is timelike. 4b b b @ @ A : (96) ^ 2 b b If instead the vacuum value b is spacelike, then M 4b2 is positive and is a candidate for a physical mass. When the massive mode vanishes, these equations re- Choose the global observer coordinate system such that the

065020-16 SPONTANEOUS LORENTZ AND DIFFEOMORPHISM ... PHYSICAL REVIEW D 77, 065020 (2008) spacelike b takes the form b 0; 0; 0;b. The con- Its combination with TB given in Eq. (74) forms the total straint (97) then requires the vanishing of the momentum energy-momentum tensor, which is conserved. The matter- ~ along the spatial direction b3, and the dispersion law (98) sector current JM q; J is becomes X Z 2 2 2 2 1 dxn E p p M : (100) 4

x y JM qn d x xn: (103) e n d This suggests that could propagate as a harmonic plane wave along spatial directions transverse to b . However, It is conserved, D J 0. these harmonic plane waves are constant in z and hence fail M The standard route to adopting the weak static limit is to to satisfy the asymptotic boundary conditions as z !1 linearize the equations of motion in the fields and then unless their amplitude is zero. Since all harmonics propa- discard all time derivatives. This meets no difficulty under gate under the same constraint (97), no Fourier superposi- suitable circumstances, such as the vacuum value b being tion can be constructed to obey the boundary conditions. It follows that no physical propagating massive mode exists purely spacelike, and we follow this route in the present for the case of a spacelike vector either. subsection. However, in certain cases additional care is required because the b-transverse components of the The above results can also be confirmed by obtaining the t eigenvalues of the Fourier-transformed equations of mo- bumblebee field obey the axial condition b E 0, which tion, which form a 14 14 matrix determining the nature can imply time dependence even for static fields. For of the modes. Consider, for example, the case of purely example, as discussed in Sec. IVA 3, the case of purely timelike b and zero is equivalent to electrodynamics in timelike b. In the axial gravitational gauge, four of the modes are zero, corresponding to the four fixed degrees of temporal gauge, for which the static Coulomb potential freedom. Another four eigenvalues describe modes prop- involves a linear time dependence in A~. The corresponding agating via a massless dispersion law, corresponding to the weak static limit with nonzero can therefore be expected two photon and two graviton modes. One eigenvalue has to exhibit this feature. We return to this issue in the next zero energy and sets jp~jjMj, corresponding to the subsection. massive mode . The remaining 5 modes are auxiliary, To proceed, it is useful to choose a diffeomorphism and all have zero energy. Related results can be obtained gauge that simplifies the Einstein tensor and allows direct for the case of spacelike b. comparison between the geodesic equation and the Newton force law. One possible choice is the harmonic gauge (42), in which the static gravitational potential g is related to 2. Weak static limit 1 1 1 the metric by g 2 h00 2 h00 4 h. The linear- Although the massive mode is nonpropagating, it ized gravitational equations become plays the role of an auxiliary field and can thereby produce various effects. For example, its presence can affect the 1ᮀ ^ weak static limits of the gravitational and bumblebee in- 2 h 8GTM TEM 4b bb b ; teractions. In particular, a nonzero can modify the forms (104) of both the Newton and Coulomb potentials. To demonstrate this, we consider the weak static limit of the field equations with an external matter sector contain- where is the massive mode given in terms of E and h ing massive point charges. The matter Lagrange density in Eq. (68), and TEM is the zero- form of TK given in can be taken as Eq. (84). We can now extract the relevant equations in the weak X Z 1=2 dxn dxn 4 static limit. The equation for the gravitational potential g LM mn d g x xn n d d is the 00 component of the trace-reversed form of X Z Eq. (104), dxn 4 qn dBx x xn; (101) n d 2 00 00 ^ 2 ~2

r g 8GTM TEM 2b bb0 b ; where mn and qn are the masses and charges of the point (105) particles, respectively. The energy-momentum tensor for the matter sector is where the trace-reversed energy-momentum tensors are X Z defined by T T T and are evaluated in 1 dxn dxn 4 TM mn d x xn: (102) the static limit. The weak static bumblebee equations e n d d become

065020-17 ROBERT BLUHM, SHU-HONG FUNG, AND V. ALAN KOSTELECKY´ PHYSICAL REVIEW D 77, 065020 (2008)

~ 2 ^ from the massive mode of nine of the 10 gravitational r E0 q 4b bb0; (106) equations. The transverse gauge is defined by ~ 2 ~ ~ ~ ~ ~ ^ ~ r E rrEJ 4b bb; j j 1 j @ h0j 0;@hjk 3@kh j; (107) ~ where q and J are static charge and current sources. where j, k range over the spatial directions. In this gauge, The above equations show that the massive mode can 2 2 G00 rh00 2r g, where g is the gravitational be viewed in the weak static limit as a simultaneous source potential. of energy density, charge density, and current density. The Taking the weak static limit of the gravitational and relative weighting of the contributions is controlled by the bumblebee equations of motion in this gauge yields relative sizes of the components of b. In fact, the same is also true of contributions to gravitomagnetic and higher- 2 00 00 3 r g 4GTM TEM 4b ; order gravitational effects arising from other components ~ 2 ~ ~ 2 of the gravitational equations. In the interpretation as r E0 r@0E q 4b ; (108) bumblebee electrodynamics, the equations show that we ᮀE~ r~ r~ E~J:~ can expect deviations from the usual Newton and Coulomb force laws when is nonzero. Since the effects are con- The other nine components of h obey equations that are b trolled by components of , they are perceived differently identical to those of general relativity in the weak static from different observer frames, much like charge and 00 limit. In Eq. (108), TM is the static energy density in the current in ordinary electrodynamics. Also, the unconven- 00 matter sector, and TEM is the static 00 component of the tional source terms are linear in , so the solutions can be zero- limit of T given in Eq. (84). The charge density interpreted as superpositions of conventional gravitational K and current density J~ are also understood to be time- and electrodynamic fields with effects due to . However, q since itself is formed as the combination (68) of the independent distributions. metric and bumblebee fields, the extra source terms in fact The explicit form of the massive mode follows from reflect contributions from the gravitational and electrody- Eq. (68) and is found to be namic fields that are absent in the Einstein-Maxwell limit. E0 bg: (109)

3. Timelike case For purely timelike b, the constraint (89) reduces to In this subsection, we consider in more detail the weak

@ 0; (110) static limit with timelike b. We solve explicitly the weak 0 static equations in the special observer frame for which so is time independent at this order. As described in the b b; 0; 0; 0 is purely timelike, and we provide some previous subsection, can be interpreted as a source for remarks about the more general case. The special frame and E . Here, acts as a static source of energy density might be identified with the natural frame of the cosmic g 0 and charge density with microwave background. However, in laboratory searches

for modifications to weak static gravitation and electro- T b 00; dynamics, this frame can at best be an approximation B B (111) because the Earth rotates and revolves about the Sun, JB B; 0; 0; 0; which in turn is moving with respect to the cosmic micro- 2 wave background. Nonetheless, the explicit solutions ob- where B 4b . tained here offer useful insight into the modifications to the To find the explicit modifications to the gravitational and Newton and Coulomb potentials that are introduced by the electromagnetic potentials caused by , we focus on the massive mode. case of a point particle of mass m and charge q located at For the purely timelike case, both h and B are the origin. The source terms in this case have the form

expected to be time independent in the weak static limit. 00 3 ~ T m m x~; However, time dependence must appear in E even in the M (112) static limit because as ! 0 the match to electrodynamics 0 3 JM q q x~; arises in the temporal gauge, for which the standard Coulomb solution has a linear time dependence. For this and all other components of TM and JM are zero. In the reason, in taking the weak static limit in what follows, we absence of Lorentz violation, the weak static solution to the keep time derivatives acting on the vector components E~. Einstein-Maxwell equations with these source terms con- The analysis for purely timelike b could be performed sists of the linearized Reissner-Nordstro¨m metric and cor- in the harmonic gauge, but it turns out that choosing the responding electromagnetic fields. Denote by m the transverse gauge instead results in the complete decoupling associated gravitational potential. It obeys

065020-18 SPONTANEOUS LORENTZ AND DIFFEOMORPHISM ... PHYSICAL REVIEW D 77, 065020 (2008)

~ 2 00 ~ r m 4Gm TEM: (113) rB~ 0; r~ ~ (120) In the limit q ! 0, it reduces to the Newton gravitational E 0; 2 potential and obeys the usual Poisson equation r m r~ B~ 0: 4Gm. Similarly, denote the conventional Coulomb po- tential by q. It obeys the Poisson equation These equations demonstrate for the purely timelike case the ways in which a nonzero massive mode modifies the 2 r q q: (114) conventional static gravitational and electrodynamic fields. Compared to electrodynamics, the purely timelike KS It is also convenient to express the massive mode in bumblebee model introduces an additional degree of free- terms of a potential B. We define dom into the problem of determining the static electric field E~. This extra degree of freedom is the massive mode 1 ~ 2

, and the extent of its effects depends on the initial 2 r Bx~; (115) jMj conditions for . In the absence of a satisfactory under- lying theory predicting or of direct experimental obser- 2 where jMj is the absolute value of the squared mass in vation of , these initial conditions are undetermined. Eq. (99). Note that B is time independent due to the One natural choice is to adopt B 0 as an initial constraint (110). Since acts as a source of static charge condition, which implies Bx~0 for all time and hence density given by Eq. (111), the definition of B can be corresponds to zero massive mode . The solution (117) interpreted as a third Poisson equation, then reduces to the weak static limit of Einstein-Maxwell theory, as expected. The gravitational potential becomes ~ 2 r B B: (116) g m. The bumblebee field reduces to E bm;t@jq bm, which despite the appearance of ~ ~ The time independence of both and B means they are the gravitational potential implies E rq and the determined by their initial values. usual Gauss and Poisson equations r~ E~ r~ 2 For the point particle of mass m and charge q, the q . We note in passing that conventional electrodynamics general solution for the gravitational potential and the q g is also recovered in the limit jM j2 !1because taking bumblebee field E in the weak static limit can be ex- this limit in Eq. (108) implies ! 0 and hence ! 0. pressed in terms of the conventional potentials and B m q Thus, even with a nonzero massive mode , a theory with a and the bumblebee potential B. We obtain large mass parameter jMj approximates Einstein-

4Gb ; Maxwell theory in the low-energy regime, in agreement g m B with the discussion following Eq. (99). 2 1 ~ 2 Other choices of initial conditions on might appear E0 bm 4Gb B r B; B jM j2 in the context of a more fundamental theory for which a 1 bumblebee model provides an effective partial low-energy t@ b Gb2 r~ 2 : Ej j q m 1 4 B 2 B theory. One set of examples with nonzero B consists of jMj solutions with zero matter current, JM 0. Imposing (117) JM 0 by hand is a common approach in the literature. It implies the bumblebee field has no couplings to matter The corresponding static gravitational field g~, static elec- and hence is unrelated to electrodynamics, being a field tric field E~, and static magnetic field B~ are found to be that interacts only gravitationally. The massless Lorentz

NG modes then propagate freely as sterile fields that carry ~ ~ g~ rm 4GbrB; energy and momentum but convey no direct forces on charged particles. We emphasize here that a theory of E~ r~ r~ ; (118) q B this type can nonetheless lead to modifications of the B~ 0: gravitational interaction, contrary to some claims in the literature. This is readily illustrated in the context of the The static Maxwell equations include a modified Gauss purely timelike example considered in this subsection. law With JM 0, the linearized solutions for a static point

particle of mass m are still given by Eqs. (117), but ~ ~ ~ 2 ~ 2 q rE r q r B 0 and the fields E~ and B~ are largely irrelevant. However, the

q B; (119) gravitational potential g is modified by the potential B for the massive mode , with the value of B fixed by together with the usual static laws initial conditions.

065020-19 ROBERT BLUHM, SHU-HONG FUNG, AND V. ALAN KOSTELECKY´ PHYSICAL REVIEW D 77, 065020 (2008) In the absence of additional theoretical or experimental ~ @ v~B r 0; (123) information, the possibilities for model building are vast. 0 One could, for example, consider initial conditions with which implies B proportional to m,

x~ v~Bt: (124)

; (121) B 4Gb m We see that the massive mode moves in the frame O with a velocity v~B equal to the relative boost velocity linking b to its purely timelike form. The gravitational and electro- with a constant. For this class of solutions, g has the usual form Gm=r for a point mass but with a dynamic equations are therefore modified by time- g dependent source terms moving with velocity v~ relative scaled value of the Newton gravitational constant B to the point particle. An equivalent result is obtained by 0 working directly in the frame O , in which b is purely G 1 GN; (122) timelike and is therefore static, but in which all compo- nents of the matter energy-momentum tensor and current where GN is the value of G for B 0. This rescaling has are nonzero and represent a particle moving with velocity no observable effects in the special observer frame because v~B relative to . As before, the form of the massive mode laboratory measurements would determine G instead of is set by initial conditions. Since the equations are linear GN, although sidereal, annual, and solar motions might in either frame, the solutions consist of a superposition of introduce detectable effects in realistic experiments. One conventional potentials and the massive-mode contribu- could equally well consider instead examples in which B tions with relative motion. has nonlinear functional dependence on m or has other coordinate dependence. Various forms could be proposed C. Linear Lagrange-multiplier potential as candidate models to explain phenomena such as dark In this subsection, we discuss the KS bumblebee model matter or possibly dark energy. With the choice of B (26) with the linear Lagrange-multiplier potential (6) and being conjecture at the level of the linearized equations, the definition (22): this approach is purely phenomenological. However, if a bumblebee model appears as part of the effective theory in 2 V VL; XBg B b : (125) a complete theory of quantum gravity, the form of B could well be fixed and predictions for dark matter and Bumblebee models with this potential have been widely perhaps dark energy could become possible. studied at the linearized level for about two decades [3]. We conclude this subsection with some comments about Here, we compare and contrast this theory to the case with the case where b takes a more general timelike form in the smooth quadratic potential VS. We show the Lagrange the given observer frame O. To investigate this, one can multiplier produces effects in the VL model that are very either study directly the weak static limit of the gravita- similar to those of the massive mode in the VS model. tional and bumblebee equations with a charged massive Paralleling the treatment of the VS model in Sec. IV B, the point particle at rest at the origin in O as before, or one can mode expansion B b E of Eq. (60)ina boost the observer frame O to another frame O0 in which Minkowski background is adopted, and the fields h b is purely timelike but the point particle is moving. The and E are assumed weak so that linearization can be two pictures are related by observer Lorentz transforma- performed. tions and are therefore physically equivalent. Variation of the action produces the gravitational and In the frame O in which the particle is at rest, the matter bumblebee equations of motion (72) and (77), together energy-momentum tensor and current are given by with the Lagrange-multiplier constraint Eq. (112) as before, but the bumblebee energy-momentum tensor T and current J have additional nonzero com- 2 B B X Bg B b 0: (126) ponents due to the nonzero spacelike components of b. The complete solution therefore involves components of This condition enforces the vanishing of the massive mode, h other than the gravitational potential g.However,we as discussed in Sec. II B, which leaves only the Lorentz NG can gain insight into the behavior of the massive mode by modes as possible excitations of the bumblebee field B. considering the constraint (89) and performing an observer However, there is also an additional degree of freedom, the ~ 0 Lorentz boost with velocity v~B b=b0 to a frame O in Lagrange multiplier itself, that appears in the equations which b becomes purely timelike. Since this transforma- of motion. tion must reduce the constraint (89) to the purely timelike One might naively expect the VL model to yield solu- form (110), it follows that in the original frame O the tions identical to those obtained in the infinite-mass limit constraint takes the form jMj!1of the VS model. For example, one might reason

065020-20 SPONTANEOUS LORENTZ AND DIFFEOMORPHISM ... PHYSICAL REVIEW D 77, 065020 (2008) that an infinite mass would make energetically impossible implies that must be time independent at leading order, any field excitations away from the minimum of VS, lead- ing to the constraint (126). However, the potential VL is a @0 0: (131) function of two combinations of fields, VL VL; X, whereas VSX involves only one. The infinite-mass limit A nonzero therefore acts as an additional static source of indeed suppresses X excitations away from X 0 in VS, energy density and charge density, which can modify the but it contains no match for . There is thus an extra field static potentials. We again adopt the transverse gauge degree of freedom in VL relative to the infinite-mass limit (107), for which the gravitational potential is g 0 of VS. For example, VS ! 0 in the infinite-mass limit h00=2. Linearizing the constraint (126) for the purely 0 because no excitations of X are allowed, whereas VL timelike case yields , which need not vanish. We therefore conclude that the

correspondence between the infinite-mass limit of the VS E 0 bg 0: (132) model and the VL model can occur only when 0. To gain intuition about the effects associated with the As expected, this condition corresponds in the context of Lagrange-multiplier field , consider its role in the equa- the VS model to enforcing the vanishing of the massive tions of motion. Since the constraint (126) ensures no mode . In the context of Einstein-Maxwell theory, it massive excitations can appear, the bumblebee energy- corresponds as before to a gauge-fixing condition that momentum tensor (74) reduces to reduces to the usual temporal gauge in the absence of gravity. TB TEM TV : (127) For a single point particle of charge q and mass m at rest at the origin, the energy density m and charge density q We find are given by Eq. (112). The equations of motion in the weak static limit become T 2B B 2b b ; (128) V 2 00 2 r g 4Gm TEM 2b ; where the last form is the leading-order contribution in the r~ 2 r~ ~ (133) linearized limit. Similarly, the bumblebee current (79) E0 @0E q 2b; becomes ᮀE~ r~ r~ E~0:

J 2B 2b : (129) B Comparison of these expressions with Eqs. (108) for the VS model again reveals the correspondence between the roles Conservation of the matter current JM implies the con- of the Lagrange-multiplier field and the massive mode . straint With acting as an effective source of charge and

energy density, we can introduce a potential B, b @ 0: (130)

1 2 r ; (134) When ! 0, all these equations reduce to those of 2b B Einstein-Maxwell theory, in agreement with the discussion of bumblebee electrodynamics in Sec. IVA 3. Moreover, which equivalently can be viewed as the solution to the by comparison with Eqs. (87)–(89), we see that the Poisson equation Lagrange multiplier in the VL model plays a role very 2 comparable to that of the massive mode in the VS model. r B B;B 2b: (135) In effect, acts as an additional source of energy- momentum density and current density in the equations As before, let m denote the usual gravitational potential of motion. obeying Eq. (113), and let q denote the usual Coulomb The propagating modes for the KS bumblebee model potential obeying Eq. (114). In terms of these three poten- with the linear Lagrange-multiplier potential VL have been tials, the weak-field static solutions to the equations of investigated elsewhere [3,5]. The usual two graviton motion (133) can be written as

modes propagate as free transverse massless modes inde- pendently of , as do the usual two photon modes emerg- g m 4GbB; ing from the Lorentz NG modes. Since has no kinetic E b 4Gb2 ; (136) terms, it is auxiliary and cannot propagate. 0 m B 2 The weak static limit of the VL model can also be Ej t@jq bm 1 4Gb B: studied, following the lines of the discussion for the VS model in Sec. IV B 3. Consider, for example, the case of The associated static gravitational field g~, static electric ~ ~ purely timelike b b; 0; 0; 0. The constraint (130) then field E, and static magnetic field B are all given by the

065020-21 ROBERT BLUHM, SHU-HONG FUNG, AND V. ALAN KOSTELECKY´ PHYSICAL REVIEW D 77, 065020 (2008) same mathematical expressions as Eqs. (118), but the Ref. [45] as potentially flawed due to the formation of potential B is now defined in terms of according to shock discontinuities in E. Whether or not is zero, Eq. (134). Similarly, a nonzero Lagrange-multipler field this solution is unusual. Both the field strength F and yields the same modified mathematical form (119) of the the energy-momentum tensor TEM vanish because the ef- Gauss law, but the potential B and charge B are now fective charge density B associated with the Lagrange- defined in terms of . multiplier field cancels the matter charge density q. The bumblebee potential B in Eqs. (136) is undeter- Although the field strengths are zero, the bumblebee field mined by the equations of motion (133) and must be E must be nonzero because the constraint (132) implies specified as an initial value. This specification also fixes E0 bg, which cannot vanish in the presence of gravity. the initial value of the Lagrange multiplier , and the For a point charge the solution E does indeed contain a condition (131) then ensures that remains unchanged singularity, but this is physically unremarkable as it merely for all time. The situation here parallels that of the V S reflects the usual 1=r dependence in . The same behav- model, where initial conditions must be imposed that sub- g ior arises in the standard solutions of Einstein-Maxwell sequently fix the bumblebee potential and the massive theory in a gauge fixed by Eq. (132), to which the V model mode for all time. These results are special cases of a L is equivalent. more general fact often overlooked in the literature: to be well defined, all bumblebee models require explicit initial conditions on the massive modes and Lagrange-multiplier D. Quadratic Lagrange-multiplier potential fields. The subsequent development of the massive modes As a final example, we consider in this subsection the and Lagrange multipliers can then be deduced from the specific KS bumblebee model (26) with the quadratic equations of motion or from derived constraints. In the Lagrange-multiplier potential (7) absence of specified initial conditions, physical interpreta- 1 2 2

tions and predictions cannot be reliable. Moreover, in the V VQ ; X 2 Bg B b ; (137) absence of direct experimental observation or prediction where X is defined in Eq. (22). As before, we adopt the from an underlying theory, the choice of initial conditions mode expansion B b E of Eq. (60)ina is largely unrestricted and can lead to widely differing Minkowski background, and where useful we assume effects. weak fields h and E . One natural initial condition is 0, 0. The B Bumblebee models with a quadratic Lagrange- above results then reduce to Einstein-Maxwell theory in multiplier potential have not previously been considered the weak static limit, as expected. This V model has only L in detail in the literature. We introduce them here partly as field excitations maintaining both V 0 and V0 0.It L L a foil for the V case, in which the Lagrange multiplier corresponds to the infinite-mass limit of V , which is itself L S plays a key role in the physics of the model. In contrast, the equivalent to Einstein-Maxwell theory. Lagrange multiplier for the potential V decouples entirely Another possibility is to consider models without direct Q from the classical dynamics. The point is that variation coupling to matter, J 0, which implies 0, M q q with respect to yields the constraint X2 0, which is 0 in the above equations. Like their V counterparts dis- S equivalent to X 0 and forces the massive mode to vanish. cussed in Sec. IV B 3, these models are purely gravitational However, the quadratic nature of V means that always and are unrelated to electrodynamics. As always, initial Q appears multiplied by X in the gravitational and bumblebee conditions on must be specified. One simple possibil- B field equations, so the on-shell condition X 0 forces the ity, for example, is to choose b . This V model B g L field to decouple. Also, the potential obeys both V 0 is contained in the analysis of Ref. [47]. The spatial com- Q and V0 0 on shell, so the bumblebee energy-momentum ponents of the bumblebee vanish, E 0, so the bumble- Q j tensor and bumblebee current reduce to the standard ex- bee field B is parallel to a timelike Killing vector. The pressions for electrodynamics in curved spacetime. The modifications to in this example involve a rescaling of g equations of motion are therefore equivalent to those of the Newton gravitational constant. Einstein-Maxwell electrodynamics in the gauge X 0. Examples with both B and q nonzero can also be This correspondence holds both in the linearized limit considered. These incorporate direct charge couplings to and in the full nonlinear theory. matter, but the static weak-field limit yields modified Since the equations of motion generated by a theory with gravitational and Coulomb potentials given by Eq. (136). a quadratic Lagrange-multiplier potential are the equations One simple example is the choice B q. Since in the absence of the constraint plus the constraint itself, g Þ m, this VL model has a modified gravitational introducing this type of potential is equivalent at the clas- potential. The solution for the bumblebee field has the sical level to imposing the constraint by hand on the form of a total derivative, E @tbg. In the limit equations of motion. The only distinction between the q ! 0 and ! 0, it provides an example of a solution that two approaches is the presence of the decoupled with 0 and hence B 0 has been identified in Lagrange-multiplier field. Bumblebee models with the

065020-22 SPONTANEOUS LORENTZ AND DIFFEOMORPHISM ... PHYSICAL REVIEW D 77, 065020 (2008) 00 potential VQ therefore incorporate a variety of models in where the energy-momentum tensor TEM has the usual which constraints are imposed by hand. For example, Will Einstein-Maxwell form (84). The solution for g is there- and Nordtvedt have considered solutions involving a non- fore the standard gravitational potential for a static point zero background value for a vector in bumblebee-type charge in a curved spacetime in the weak-field limit. At models with Lagrange density of the form (19) but without quadratic order, the solution E E0; Ej satisfying the a potential term [42]. This approach produces equations of constraint (138) and the field equations is found to be motion identical to those of the corresponding VQ model. b t2 The V models are also related to theories in which the 3 2 2 Q E0 bg g @jq bg ; constraint is substituted into the action before varying, 2 2b although the correspondence is inexact. For example, Ej t@jq bg3btg@jg (140) Nambu has investigated the Maxwell action in t3 Minkowski spacetime using the constraint X 0 for the @kq bg@j@kq bg; purely timelike case as a nonlinear gauge condition sub- 3b stituted into the action prior to the variation [53]. This where q is the conventional Coulomb potential obeying represents gauge fixing at the level of the action, and it the Poisson equation (114). These expressions are both yields a total of four equations for the fields: the original time dependent and nonlinear, but they nonetheless gen- constraint and three equations of motion from the varia- erate the usual static electric field E~ and magnetic field B~ tion. However, the corresponding VQ model yields five for a point charge. Explicit calculation to quadratic order equations of motion instead, one of which is the constraint. yields The extra equation is the Gauss law, which in the Nambu ~ ~ ~ approach is imposed as a separate initial condition that E rq; B 0; (141) subsequently holds at all times by virtue of the three ~ ~ equations of motion and the constraint. which implies the usual form rE q of the Gauss law. For the specific KS bumblebee model (26) with potential From these equations, we see again that Einstein-Maxwell theory is recovered despite the absence of U(1) symmetry. VQ in Eq. (137), the freely propagating modes can be found by linearizing the gravitational and bumblebee equations In effect, the nonlinear condition (138) plays the role of a of motion (72) and (77) and the constraint X 0. The nonlinear gauge-fixing condition in a U(1)-invariant the- linearization generates the same equations as emerge for ory, removing a degree of freedom and leaving only NG modes that propagate as photons and generate the usual the VL model with potential (125) in the limit ! 0. The propagating modes consist of the usual graviton and pho- Coulomb potential in the weak-field limit. ton in an axial gauge [5], as expected. Similarly, the weak static limit of the VQ model with potential (137) produces V. SUMMARY equations identical at linear level to those of the VL model with potential (125) in the limit ! 0. The usual gravita- In this paper, we have investigated the properties of the tional and electromagnetic Poisson equations therefore massive modes that can emerge from spontaneous local emerge, and the correct Newton and Coulomb potentials Lorentz and diffeomorphism breaking. In Riemann space- hold at the linearized level. time, no massive modes of the conventional Higgs-type The exact correspondence between the static limit of the can appear because covariant kinetic terms involve con- V bumblebee model and the Newton and Coulomb po- nections with derivatives. However, an alternative form of Q the Higgs mechanism can occur instead, in which massive tentials involves the nonlinear constraint (126). The ex- modes originate from quadratic couplings in the potential plicit forms of the solutions are therefore also nonlinear. V inducing the symmetry breaking. For example, consider again the case of a point particle of Section II provides an analysis of this alternative Higgs mass m and charge q at rest at the origin in the presence of mechanism in the general context of an arbitrary tensor a purely timelike vacuum value b b; 0; 0; 0. The field. Both smooth potentials and Lagrange-multiplier po- equation of motion for is the constraint (126), which tentials are considered. Massive modes appear for a smooth represents a nonlinear condition relating and h . E potential when excitations with V0 Þ 0 exist, and they are Expanding this constraint through quadratic order gives formed as combinations of field and metric fluctuations. 2 2 2 For Lagrange-multiplier potentials the massive modes are E 0 2bE0 2b g 4bmE0 Ej 0; (138) constrained to vanish, but the Lagrange-multiplier fields where m obeys the conventional Poisson equation (113) can play a related role. The propagation of the massive for the point source with mass and charge density given by modes depends on the nature of the kinetic terms in the Eq. (112). In the transverse gauge, the gravitational poten- theory, and the requirements of unitarity and ghost-free tial g obeys propagation constrain possible models. Even if the massive modes are nonpropagating they can influence gravitational ~ 2 00 r g 4G m TEM ; (139) phenomena through, for example, effects on the static

065020-23 ROBERT BLUHM, SHU-HONG FUNG, AND V. ALAN KOSTELECKY´ PHYSICAL REVIEW D 77, 065020 (2008) gravitational potential. These modifications are of poten- persion law is unconventional and no localized physical tial interest in alternative theories of gravity and descrip- solutions satisfying suitable asymptotic boundary condi- tions of phenomena such as dark matter or dark energy. tions exist, so the VS model has no freely propagating Following the general treatment, we investigate in massive modes. Nonetheless, the massive mode has a Sec. III a broad class of theories called bumblebee models physical impact as an auxiliary field, acting as an addi- that involve gravitationally coupled vector fields with tional source of energy-momentum density and current spontaneous local Lorentz and diffeomorphism breaking. density. In the weak static limit, for example, the solutions For arbitrary quadratic kinetic terms, the Lagrange density to the equations of motion in the presence of mass and is given in Eq. (19). Along with the symmetry-breaking charge describe modified Newton and Coulomb potentials potential V and a matter sector, this Lagrange density according to Eq. (117). These may be of phenomenological involves five parameters, four of which can be linearly interest in the context of dark matter and perhaps also dark independent in specific models. A particularly attractive energy. The effects are controlled by the massive mode, but class of theories are the KS bumblebee models, which have its form is dynamically undetermined and must be imposed kinetic term of the Maxwell form and hence an additional via initial conditions. constraint that minimizes problems with unitarity and Many of the results obtained for the VS model apply also ghosts. These models also offer candidate alternatives to to the VL and VQ models that have Lagrange-multiplier Einstein-Maxwell electrodynamics. fields. For example, all the models contain four massless In a series of subsections and the associated appendix, propagating modes behaving like and photons. we provide some results valid for general bumblebee mod- Although the VL and VQ models generate an additional els. The observer and particle forms of local Lorentz trans- constraint that eliminates the massive mode on shell, the formations and diffeomorphisms are presented. Using the Lagrange-multiplier field appears instead as a extra degree vierbein formalism, some decompositions of the bumble- of freedom playing a similar role to that of the massive bee field and metric are given that are suitable for the mode in the VS model. The form of the Lagrange multiplier identification of Lorentz and diffeomorphism NG modes. must be set by initial conditions, and if it is chosen to The effects of various choices of Lorentz and diffeomor- vanish then Einstein-Maxwell theory is recovered. The key phism gauges are described. Alternative decompositions difference between the VL and VQ models lies in the role of used in some of the literature are also discussed, in which the Lagrange multiplier, which can affect the physics as an the Lorentz NG modes are hidden and only spacetime auxiliary mode in the VL model but which decouples from variables are used. the theory in the VQ model. To provide explicit examples and to gain insight via a In the context of bumblebee electrodynamics, the mas- more detailed analysis, we focus attention in Sec. IVon the sive mode or Lagrange-multiplier field acts as an additional class of KS bumblebee models. The basic equations of degree of freedom relative to Einstein-Maxwell theory. motion and conservation laws are obtained, and some The extra freedom arises because the bumblebee model properties of the bumblebee currents are considered. The has no U(1) gauge symmetry, and the structure of the interpretation of these bumblebee models as theories of kinetic terms implies that the freedom must be fixed as electromagnetism and gravity, known as bumblebee elec- an initial condition. In the absence of experimental evi- trodynamics, is discussed. They contain four transverse dence for a massive mode or of guidance from an under- massless modes, two of which are massless gravitons and lying theory, the choice of initial condition is largely two of which are massless photons, along with a massive arbitrary. A natural choice sets the massive mode or mode or Lagrange-multiplier mode. When the massive Lagrange multiplier field to zero, reducing the theory to mode or Lagrange multipler vanishes, or in the limit of Einstein-Maxwell electrodynamics up to possible SME infinite mass, conventional Einstein-Maxwell theory in an matter-sector couplings. Bumblebee electrodynamics axial gauge is recovered. therefore provides a candidate alternative explanation for Section IValso contains subsections considering in more the existence of massless photons, based on the massless- detail various types of potentials, including the smooth ness of NG modes instead of the usual gauge symmetry. In potential V in Eq. (86), the linear Lagrange-multiplier S any case, the possibility of spontaneous breaking of local potential V in Eq. (125), and the quadratic Lagrange- L Lorentz and diffeomorphism symmetry remains a promis- multiplier potential V in Eq. (137). For the V model, Q S ing avenue for exploring physics emerging from the Planck the gravitational and bumblebee equations of motion are scale. investigated to determine whether a physical massive mode can propagate as a free field. The sign of the squared-mass ACKNOWLEDGMENTS term depends on whether the bumblebee vacuum value b is timelike or spacelike. In the timelike case, the massive This work was supported in part by DOE Grant No. DE- mode is a ghost, while in the spacelike case the squared FG02-91ER40661, NASA Grant No. NAG3-2914, and mass has the usual sign. However, in both cases the dis- NSF Grant No. PHY-0554663.

065020-24 SPONTANEOUS LORENTZ AND DIFFEOMORPHISM ... PHYSICAL REVIEW D 77, 065020 (2008) APPENDIX: TRANSFORMATIONS 2. Observer transformations In this appendix, we provide explicit transformation The primary difference between observer and particle formulas for the vierbein, bumblebee, and NG field com- transformations is that the components of vacuum-valued ponents under particle diffeomorphisms, particle local fields transform under observer transformations. However, Lorentz transformations, observer general coordinate additional differences arise when the distinction between transformations, and observer local Lorentz transforma- spacetime and local indices is dropped. In particular, care tions. The vierbein decomposition is given in Eq. (37), is required in expressing observer transformation laws for the bumblebee decomposition is given in Eq. (40), and the metric and the vierbein. the NG fields are defined in Eq. (47). The transformation Three versions of the Minkowski metric appear in the formulas can be deduced from these expressions and from formalism: the spacetime metric , the local metric ab, the behavior of the vacuum expectation values. All the and the mixed metric a used with the vierbein. All three formulas given here are linearized assuming infinitesimal are numerically equal in cartesian coordinates, but they fields and transformation parameters. behave differently under observer transformations. In the customary notation using only Greek indices, these three 1. Particle transformations metrics must be labeled differently. Since they represent Under infinitesimal particle diffeomorphisms, vacuum values of the metric, we write them as expectation

values and define

b ! b; hi abjab!; ! ; h i j ; (A3) e a a! ! ; h i : t t g ! ; The inverse metrics are denoted with upper indices and are ! ; defined so that h ! h @ @; (A1) high ig hieh ie hih i : ! 1@ @ 2 (A4) e e ! e @ In terms of these vacuum values, the metric and its inverse e e ! e @ ; involve infinitesimal excitations: B ! B @b ; g hig h; (A5) B ! B @ b ; g h ig h :

! ; The transformation laws for the vierbein also depend on the component basis. We define E ! E:

Under infinitesimal local particle Lorentz transformations, heihie hi; heihi hi; b ! b; e (A6) h ih i h i ! ; e e ; h ih i h i ! ; e e : t t ! b ; For example, these vacuum values obey ! ; hig heiheih i hiehieh i; h ! h ; (A7) (A2) ! as expected from the relationship between the metric and the vierbein. The vierbein field itself can be written e e ! e 1 e e ! e ; e hie hi2h ; (A8) e hi hi1h : B ! B; e 2

! ; In much of the literature, the vacuum value for hi is assumed to vanish in cartesian coordinates. However, this E ! E : choice is observer dependent. Under either observer gen-

065020-25 ROBERT BLUHM, SHU-HONG FUNG, AND V. ALAN KOSTELECKY´ PHYSICAL REVIEW D 77, 065020 (2008) eral coordinate transformations or observer local Lorentz The formulas for infinitesimal observer local Lorentz transformations, a zero value of hi transforms into a transformations are

nonzero value. We therefore include the vacuum value h i !h i ; hi in the formulas below. g g The transformation rules for infinitesimal observer gen- hie !hie; eral coordinate transformations are h i !h i ; hig !hig @ @; hi!hi; h ig !h ig @ @ ; hei!hei; 1 h i !h i @ @ ; b ! b b; e e 2 t ! t 1 ; (A10) h ie !h ie @ @ ; 2 ! ; hi !hi; h ! h; h i !h i; ! ; 1 h i!h i @ @ ; e ! e ; 2 1 B ! B ; hi!hi @ @; 2 ! ; hei!hei@; E ! E: hei!hei@; In these expressions, indices can be raised and lowered b ! b @b ; (A9) using hi. b h igb ! b @ b ; For some purposes, it may be desirable for an observer t ! t ; transformation to take the appearance of the inverse of a particle transformation. This can be achieved by changing ! ; the signs of the parameters, ! and !, either in the formulas for the observer transformations or in h ! h; those for the particle transformations. For example, with ! ; signs chosen appropriately the bumblebee field B can be e ! e @; arranged to be invariant, B ! B, under the composite transformation consisting of a particle diffeomorphism e ! e @; followed by the corresponding inverse general coordinate B ! B @b ; transformation. However, the reader is cautioned that in- variance of this type may fail for the vacuum value and B ! B @ b ; excitations of B. For instance, the vacuum value b is ! ; invariant under particle diffeomorphisms but transforms as E ! E : ordinary components of a vector under observer general coordinate transformations.

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