Spontaneous Lorentz violation, Nambu-Goldstone modes, and gravity

PHYSICAL REVIEW D 71, 065008 (2005)

Spontaneous Lorentz violation, Nambu-Goldstone modes, and gravity

Robert Bluhm1 and V. Alan Kostelecky2

1Physics Department, Colby College, Waterville, Maine 04901, USA2Physics Department, Indiana University, Bloomington, Indiana 47405, USA

(Received 12 January 2005; published 22 March 2005)

1550-7998=20

The fate of the Nambu-Goldstone modes arising from spontaneous Lorentz violation is investigated.Using the vierbein formalism, it is shown that up to 10 Lorentz and diffeomorphism Nambu-Goldstonemodes can appear and that they are contained within the 10 modes of the vierbein associated with gaugedegrees of freedom in a Lorentz-invariant theory. A general treatment of spontaneous local Lorentz anddiffeomorphism violation is given for various spacetimes, and the fate of the Nambu-Goldstone modes isshown to depend on both the spacetime geometry and the dynamics of the tensor field triggering thespontaneous Lorentz violation. The results are illustrated within the general class of bumblebee modelsinvolving vacuum values for a vector field. In Minkowski and Riemann spacetimes, the bumblebee modelprovides a dynamical theory generating a photon as a Nambu-Goldstone boson for spontaneous Lorentzviolation. The Maxwell and Einstein-Maxwell actions are automatically recovered in axial gauge.Associated effects of potential experimental relevance include Lorentz-violating couplings in the matterand gravitational sectors of the Standard-Model Extension and unconventional Lorentz-invariant cou-plings. In Riemann-Cartan spacetime, the possibility also exists of a Higgs mechanism for the spinconnection, leading to the absorption of the propagating Nambu-Goldstone modes into the torsioncomponent of the gravitational field.

DOI: 10.1103/PhysRevD.71.065008 PACS numbers: 11.30.Cp, 04.50.+h, 11.30.Er, 12.60.–i

I. INTRODUCTION

Violations of Lorentz symmetry arising from new phys-ics at the Planck scale offer a potentially key signature forinvestigations of quantum-gravity phenomenology [1].One elegant possibility is that Lorentz symmetry is sponta-neously broken in an ultimate fundamental theory [2]. Thebasic idea is that interactions among tensor fields in theunderlying theory trigger the formation of nonzero vacuumexpectation values for Lorentz tensors. The presence ofthese background quantities throughout spacetime impliesthat Lorentz symmetry is spontaneously broken.

In general, spontaneous violation of a symmetry haswell-established consequences. The existence of a non-trivial vacuum expectation value directly modifies theproperties of fields that couple to it and can indirectlymodify them through interactions with other affectedfields. For example, the vacuum value h�i of the Higgsfield spontaneously breaks the SU�2� � U�1� symmetry ofthe standard model (SM), introducing effective fermionmasses via couplings of the fermion fields to h�i.Moreover, when a continuous global symmetry is sponta-neously broken, massless modes called Nambu-Goldstone(NG) modes appear [3]. If instead the local symmetry of agauge theory is spontaneously broken, then the gaugebosons for the broken symmetry become massive modesvia the Higgs mechanism [4].

For spontaneous Lorentz violation, the situation is simi-lar. The existence of a nontrivial vacuum value for a tensorfield affects the behavior of particles coupling to it, eitherdirectly or indirectly through other particles. These effectscan be comprehensively characterized in an effective the-ory for the gravitational and SM fields observed in nature[5–7], via the introduction of coefficients for Lorentz

05=71(6)=065008(17)$23.00 065008

violation carrying spacetime indices. This theory, calledthe Standard-Model Extension (SME), describes the phe-nomenological implications of spontaneous Lorentz viola-tion independently of the structure of the underlyingtheory. To date, Planck-scale sensitivity has been attainedto the dominant SME coefficients in many experiments,including ones with photons [8,9], electrons [10–12], pro-tons and neutrons [13], mesons [14], muons [15], neutrinos[16,17], and the Higgs [18], but a substantial portion of thecoefficient space remains to be explored.

As in the case of internal symmetries, the vacuum valuestriggering Lorentz violation are accompanied by NGmodes associated with the generators of the broken-symmetry transformations. The fate of these NG modesis relevant to gravitational and SM phenomenology. If themassless NG modes are present as such and propagate overlong distances, their phenomenology must be compatiblewith existing or hypothetical long-range forces. For ex-ample, it has been suggested that the NG modes in a vectortheory with spontaneous Lorentz violation may be equiva-lent to electrodynamics in a nonlinear gauge [19,20]. Ifinstead a mass were to develop for the graviton in analogywith the usual Higgs mechanism, other issues would arise.For example, even a small mass for the graviton canmodify the predictions of general relativity and disagreewith experiment [21]. In fact, in general relativity withspontaneous Lorentz breaking, it is known that a conven-tional Higgs mechanism cannot give rise to a mass for thegraviton since the analogue of the usual Higgs mass terminvolves derivatives of the metric [22]. Further complica-tions occur because the usual simple counting argumentsfor the number of NG modes can require modification inthe presence of Lorentz violation [23].

-1 2005 The American Physical Society

ROBERT BLUHM AND V. ALAN KOSTELECKY PHYSICAL REVIEW D 71, 065008 (2005)

In the context of gravity in a Riemann geometry, theinvestigation of spontaneous Lorentz violation was initi-ated with a study of a class of vector theories [22], calledbumblebee models, that are comparatively simple fieldtheories in which spontaneous Lorentz violation occurs.These models and some versions with ghost modes havesince been investigated in a variety of contexts [5,24,25].There has also been recent interest in the timelike diffeo-morphism NG mode that arises when Lorentz symmetry isspontaneously broken by a timelike vector. If such a modewere to appear in a theory with second-order time deriva-tives, it has been shown that it would have an unusualdispersion relation leading to interesting anomalous spin-dependent forces [26].

In this paper, we investigate the ultimate fate of the NGmodes associated with spontaneous violation of localLorentz and diffeomorphism symmetries. We perform ageneric analysis of theories formulated in Riemann-Cartanspacetime and its limits, including the Riemann spacetimeof general relativity and the Minkowski spacetime of spe-cial relativity, and we illustrate the results within thebumblebee model. The standard vierbein formalism forgravity [27] offers a natural and convenient frameworkwithin which to study the properties of the NG modes,and we adopt it here. The basic gravitational fields can betaken as the vierbein e�a and the spin connection !�

ab.The associated field strengths are the curvature and torsiontensors. In a general theory of gravity in a Riemann-Cartanspacetime [28], these fields are independent dynamicalquantities. The usual Riemann spacetime of general rela-tivity is recovered in the zero-torsion limit, with the spinconnection fixed in terms of the vierbein. Our focus here ison models in which one or more tensor fields acquirevacuum values, a situation that could potentially arise inthe context of effective field theories for a variety ofquantum-gravity frameworks in which mechanisms existfor Lorentz violation. These include, for example, stringtheory [2,29], noncommutative field theories [30],spacetime-varying fields [31–33], loop quantum gravity[34], random-dynamics models [35], multiverses [36],and brane-world scenarios [37], so the results obtained inthe present work are expected to be widely applicable.

The organization of this paper is as follows. A genericdiscussion of spontaneous Lorentz violation in the vierbeinformalism is presented in Sec. II. Section III discussesbasic results for the bumblebee model. The three subse-quent Secs. IV, V, and VI examine the fate of the NG modesin Minkowski, Riemann, and Riemann-Cartan spacetimes,respectively. Section VII contains a summary of the results.Throughout this work, we adopt the notation and conven-tions of Ref. [5].

II. SPONTANEOUS LORENTZ VIOLATION

For gravitational theories with a realistic matter sector,the vierbein formalism [27] is widely used because it

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permits a straightforward treatment of fermions in non-trivial spacetimes. Since this formalism distinguishescleanly between local Lorentz frames and coordinateframes on the spacetime manifold, it is also ideally suitedfor investigations of Lorentz and CPT breaking [5], in-cluding the effects of spontaneous violation.

A. General considerations

A basic object in the formalism is the vierbein e�a,

which can be viewed as providing at each point on thespacetime manifold a link between the covariant compo-nents T�� of a tensor field in a coordinate basis and thecorresponding covariant components Tabc�� of the tensorfield in a local Lorentz frame. The link is given by

T�� e�ae�be�c � � �Tabc��: (1)

In the coordinate basis, the components of the spacetimemetric are denoted g��. In the local Lorentz frame, themetric components take the Minkowski form �ab, but thebasis may be anholonomic. Expressions for contravariantor mixed tensor components similar to Eq. (1) can beobtained by appropriate contractions with the componentsg�� of the inverse spacetime metric.

The vierbein formalism permits the treatment of bothbasic types of spacetime transformations relevant for gravi-tation theories: local Lorentz transformations and diffeo-morphisms. Consider a point P on the spacetime manifold.Local Lorentz transformations at P act on the tensor com-ponents Tabc�� via a transformation matrix �a

b applied toeach index. For an infinitesimal transformation, this matrixhas the form

�ab � �ab �ab; (2)

where �ab � �ba are the infinitesimal parameters carry-ing the six Lorentz degrees of freedom and generating thelocal Lorentz group. In contrast, a diffeomorphism is amapping of P to another point Q on the spacetime mani-fold, with an associated mapping of tensors at P to tensorsatQ. The pullback of a transformed tensor atQ to P differsfrom the original tensor at P. For infinitesimal diffeomor-phisms characterized in a coordinate basis by the trans-formation

x� ! x� ��; (3)

this difference is given by the Lie derivative of the tensorT�� along the vector ��. The four infinitesimal parame-ters �� comprise the diffeomorphism degrees of freedom.

The vierbein formalism is natural for studies of Lorentzviolation. Spontaneous violation of local Lorentz invari-ance occurs when the Lagrangian of the theory is invariantunder local Lorentz transformations but the vacuum solu-tion violates one or more of the symmetries. The keyfeature is the existence of a nonzero vacuum expectationvalue for the components Tabc�� of a tensor field in a local

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SPONTANEOUS LORENTZ VIOLATION, NAMBU-. . . PHYSICAL REVIEW D 71, 065008 (2005)

Lorentz frame [5]:

hTabc��i � tabc�� 0: (4)

The values tabc�� may be constants or specified functions,provided they solve the equations of motion of the theory.Each such expectation value specifies one or more orien-tations within any local frame, which is the characteristicof spontaneous Lorentz violation.

The vacuum expectation value of the vierbein is also aconstant or a fixed function, either given by the solution tothe gravitational equations or specified as a background.For example, in a spacetime with Minkowski backgroundthe vacuum value of the vierbein is he�ai � ��a in asuitable coordinate frame. It follows from Eq. (1) that theexistence of a vacuum value tabc�� for a tensor in a localframe implies it additionally has a vacuum value t�� inthe coordinate basis on the manifold. However, a non-trivial vacuum expectation value for t�� also impliesspontaneous violation of diffeomorphism invariance. Thisshows that the spontaneous violation of local Lorentzinvariance implies spontaneous violation of diffeomor-phism invariance.

In fact, the converse is likewise true: if diffeomorphisminvariance is spontaneously broken, so is local Lorentzinvariance. It is immediate that any violation of diffeo-morphism invariance via vacuum values of vectors ortensors breaks local Lorentz invariance, as above. Analternative source of diffeomorphism violations is possiblevia vacuum values of scalars provided the scalars are non-constant over the spacetime manifold, but this also leads toviolations of local Lorentz invariance because the deriva-tives of the scalar vacuum values provide an orientationwithin each local Lorentz frame.

B. Identification of NG modes

In discussing the consequences of spacetime-symmetryviolations, it is useful to distinguish among several types oftransformations. Treatments of Lorentz-invariant theoriesin the literature commonly define two classes of Lorentztransformations, called active and passive, which act ontensor components essentially as inverses of each other. Ina Lorentz-violating theory, however, the presence of vac-uum expectation values with distinct properties impliesthat there are more than two possible classes of trans-formations [7]. For most purposes it suffices to limit atten-tion to two possibilities, called observer transformationsand particle transformations.

Observer transformations involve changes of the ob-server frame. It is standard to assume that any physicallymeaningful theory is covariant under observer transforma-tions, and this remains true in the presence of Lorentzviolation [7]. An observer local Lorentz transformationcan be viewed as a rotation or boost of the basis vectorsin the local tangent space. Tensor components are thenexpressed in terms of the new basis. Observer coordinate

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transformations on the manifold are general coordinatetransformations, which leave invariant the action. Thestatement of observer invariance therefore contains nophysical information other than the assumption of observerindependence of the physics.

Particle transformations are defined to act on individualparticles or localized fields, while leaving unchanged vac-uum expectation values. A particle Lorentz transformationinvolves a rotation or boost only of localized tensor fields.The components of the tensor are affected, while the basisand any vacuum values are unchanged. Similarly, particlediffeomorphisms with the pullback incorporated can beviewed as changes only in localized field distributions,with the tensor components transforming via the Lie de-rivative but the basis and all vacuum values unaffected.

Invariance of a system under particle transformationshas physical consequences, including notably the existenceof conservation laws. Local Lorentz invariance implies acondition on the antisymmetric components of the energy-momentum tensor T��, while diffeomorphism invarianceimplies a covariant conservation law for it. Thus, forexample, in general relativity the laws are T�� T��

andD�T�� 0. Spontaneous breaking of these spacetime

symmetries leaves unaffected the conservation laws. Incontrast, explicit breaking of these symmetries, which isdescribed by noninvariant terms in the action, modifies thelaws. For local Lorentz and diffeomorphism transforma-tions, the conservation laws in the presence of spontaneousand explicit breaking are obtained in Ref. [5] in the contextof a general gravitation theory.

In a theory with spontaneous breaking of a continuoussymmetry, one or more NG modes are expected. The NGmodes can be identified with the virtual excitations aroundthe vacuum solution that are generated by the particletransformations corresponding to the broken symmetry.According to the above discussion, if the extremum ofthe action involves a nonzero vacuum expectation valuetabc�� for a tensor in a local frame, both local Lorentzinvariance and diffeomorphism invariance are spontane-ously broken. Since these invariances involve 10 genera-tors for particle transformations [38,39], we conclude thatup to ten NG modes can appear when an irreducibleLorentz tensor acquires a vacuum expectation value.

In the subsequent parts of this work, it is shown that thevierbein formalism is particularly well suited for describ-ing these NG modes. A simple counting of modes illus-trates the key idea. The vierbein e�a has 16 components. Ina Lorentz- and diffeomorphism-invariant theory, 10 ofthese can be eliminated via gauge transformations, leaving6 potentially physical degrees of freedom to describe thegravitational field. In general relativity, four of these six areauxiliary and do not propagate, leaving only the two usualtransverse massless graviton modes; more general metricgravitational theories can have up to six graviton modes[40]. However, in a theory with local Lorentz and diffeo-

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morphism violation, the ten additional vierbein modescannot all be eliminated by gauge transformations andinstead must be treated as dynamical fields in the theory.In short, the ten potential NG modes from spontaneouslocal Lorentz and diffeomorphism breaking are containedwithin the ten components of the vierbein that are gaugedegrees of freedom in the Lorentz-invariant limit.

C. Perturbative analysis

In general, each NG mode can be obtained by perform-ing on the vacuum a virtual particle transformation for abroken-symmetry generator and then elevating the corre-sponding spacetime-dependent parameter to the NG field.To identify the NG modes and study their basic properties,it therefore suffices to consider small excitations about thevacuum and to work in a linearized approximation.

If the vacuum solution of a given theory involves themetric gvac�� , then the metric g�� in the presence of smallexcitations can be written as

g�� gvac�� h��: (5)

In the general scenario, distinguishing the backgroundfrom gravitational fluctuations requires some care. Forinstance, in the shortwave approximation [41] the distinc-tion is made in terms of the amplitude of h�� and the scaleson which gvac�� and h�� vary. For our purposes, however, thepresence of a nontrivial background spacetime is unneces-sary and serves to complicate the basic study of the prop-erties of the NG modes. We therefore focus attention hereon spacetimes in which the vacuum geometry isMinkowski.

Small metric fluctuations about the Minkowski back-ground can be written as

g�� h��: (6)

To linear order, the inverse metric is then g�� h��. In this context, the distinction between coor-dinate indices �, �, . . . on the manifold and the localLorentz indices a, b, . . . is diminished, and Greek letterscan be used for both. The 16-component vierbein can bewritten as

e�� 1

2h��

�; (7)

where the ten symmetric excitations h�� h�� are asso-ciated with the metric, while the six antisymmetric com-ponents �� are the local Lorentz degrees offreedom.

The vacuum expectation value (4) of an arbitrary tensorbecomes

hT��i � t��; (8)

and excitations about this vacuum value are denoted as

065008

�T�� T�� t��: (9)

There can be many more such excitations than NG modes.The NG modes are distinguished by the requirement that�T�� maintains the extremum of the action and corre-sponds to broken-symmetry generators.

For modes �T�� excited via local Lorentz transfor-mations or diffeomorphisms, the magnitude of T�� ateach point is preserved,

T��g��g��g�� . . .T�� t2; (10)

where t2 � t�� . . . t��. This holds, forexample, in a theory with potential V having the simplefunctional form

V � V�T��g��g��g�� . . .T�� t2�; (11)

which can trigger a vacuum value t�� when V is extre-mized. For instance, V could be a positive quartic poly-nomial in T�� with minima at zero, such asV�x� � �x2=2, where � is a coupling constant. The condi-tion (10) is automatically satisfied by the choice

T�� e��e��e�� . . . t��; (12)

which also reduces to the correct vacuum expectation value(8) when the vierbein excitations vanish. This implies allthe excitations in �T�� associated with the NG modesare contained in the vierbein through Eq. (12).

Using the expansion (7) of the vierbein in Eq. (12) yieldsa first-order expression for the tensor excitations �T��in terms of the 16 fields h�� and ��:

�T�� 1

2h��

�t��

�1

2h��

�t�� . . . : (13)

Evidently, the combination �12 h�� contains the in-teresting dynamical degrees of freedom.

We can observe the effects of local Lorentz and diffeo-morphism transformations by performing each separately.Under infinitesimal Lorentz transformations, the vierbeincomponents transform as

h�� ! h��;

�� ! �� ;(14)

while their transformations under infinitesimal diffeomor-phisms are

h�� ! h�� @�� @��;

�� ! �� 1

2�@�� @��:

(15)

In these expressions, quantities of order ��h�, ��, ��h�,��, etc. are assumed small and hence negligible in thelinearized treatment.

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The excitation due to infinitesimal Lorentz transforma-tions is

�T�� t�� t�� . . . : (16)

Depending on the properties of the vacuum value t��, upto six independent excitations associated with brokenLorentz generators can appear in this expression. Theirnature is determined by the six parameters ��. It followsthat the corresponding NG modes E�� for the brokenLorentz symmetries stem from the antisymmetric compo-nents �� of the vierbein.

The excitation due to infinitesimal diffeomorphisms is

�T�� @��t�� @��t�� . . .

��@�t��: (17)

This can contain up to four independent excitations asso-ciated with broken diffeomorphisms, depending on theproperties of t��. Except for the case of a scalar T, thefour potential NG modes �� corresponding to the fourparameters �� enter the vierbein accompanied by deriva-tives. This can potentially alter their dispersion relationsand couplings to matter currents.

As a simple example, consider spontaneous breakingdue to a nonzero vector vacuum value t�, which could betimelike, spacelike, or lightlike. Introduce a quantity E� �E��t� obeying t�E� � 0. The 3 degrees of freedom of E�

correspond to the three Lorentz NG modes associated withthe three Lorentz generators broken by the direction t�.Similarly, one can introduce a scalar ��t�, which playsthe role of the NG mode corresponding to the diffeomor-phism broken by t�. In this example, which is studied inmore detail in the next section, there are four potential NGmodes. If a second orthogonal vacuum value t0� is alsopresent, an additional two Lorentz NG modes appear be-cause two additional Lorentz generators are broken. All sixLorentz NG modes E�� enter the theory once a thirdorthogonal vacuum value exists. Similarly, as additionalvacuum values are added, more components of the fields�� enter as NG modes for diffeomorphisms, until all fourare part of the broken theory.

Examples with more complicated tensor representationsalso provide insight. For instance, consider a theory withan expectation value for a two-index symmetric tensorT�� T��. In this case, the choice of vacuum value t��can crucially affect the number and type of NG modes.There is a choice among many possible scenarios. A subsetof the space of possible vacuum values consists of thoset�� that can be made diagonal by a suitable choice ofcoordinate basis, but even if attention is restricted to thissubset there are many possibilities. For instance, a vacuumvalue with diagonal elements (3,1,1,1) breaks three boostsand four diffeomorphisms for a total of seven NG modes,one with diagonal elements (4,1,1,2) breaks five Lorentz

065008

transformations and four diffeomorphisms for a total ofnine NG modes, while one with diagonal elements(6,1,2,3) breaks all ten symmetries for a total of ten NGmodes.

In the general case, up to ten NG modes can appear whena tensor acquires a vacuum expectation value t��. Thefluctuations of the tensor about the vacuum under virtualparticle transformations are given as the sum of the right-hand sides of Eqs. (16) and (17). The associated NG modesconsist of up to six Lorentz modes E�� and up to fourdiffeomorphism modes ��.

The ultimate fate of the NG modes, and, in particular,whether some or all of them propagate as physical masslessfields, depends on the specific dynamics of the theory. Atthe level of the Lagrangian, the linearized approximationinvolves expanding all fields around their vacuum valuesand keeping terms of quadratic order or less. The dominantterms in the effective Lagrangian for the NG modes canthen be obtained by replacing the tensor excitation with theappropriate NG modes E�� and �� according to Eqs. (16)and (17). The resulting dynamics of the modes are deter-mined by several factors, including the basic form of theterms in the original action and the type of spacetimegeometry in the theory. Disentangling these issues is thesubject of the following sections.

III. BUMBLEBEE MODELS

To study the behavior of the NG modes and distinguishdynamical effects from geometrical ones, it is valuable toconsider a class of comparatively simple models forLorentz and diffeomorphism violation, called bumblebeemodels, in which a vector field B� acquires a constantexpectation value b� [5,22]. These models contain many ofthe interesting features of cases with more complicatedtensor vacuum values. For example, all the basic types ofrotation, boost, and diffeomorphism violations can be im-plemented, and the existence and properties of the corre-sponding NG modes can be studied for various spacetimegeometries. In this section, some general results for thesemodels are presented.

A. Projectors for NG modes

The characteristic feature of bumblebee models is that avector field B� acquires a vacuum expectation value ba in alocal Lorentz frame. This breaks three Lorentz transforma-tions and one diffeomorphism, so there are four potentialNG modes. According to Eq. (12) and the associateddiscussion, the vector field B� can be written in terms ofthe vierbein as

B� � e�aba; (18)

which holds in any background metric. The vierbein de-grees of freedom include the NG modes of interest.

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As before, we proceed under the simplifying assumptionthat the background spacetime geometry is Minkowski.The vacuum solution then takes the form

hB�i � b�; he��i � �� (19)

in a suitable coordinate frame. The vierbein can be ex-panded in terms of h�� and ��, as in Eq. (7). Thefluctuations about the vacuum can therefore be written as

�B� � �B� b��

1

2h��

�b�: (20)

The results of the previous subsection imply that three ofthe four potential NG modes are contained in fields E�

obeying b�E� � 0, while one appears in a combination��b�. To identify these modes, it is convenient to separatethe excitations (20) into longitudinal and transverse com-ponents using projection operators. Focusing for definite-ness on the nonlightlike case (b2 � 0), we define theprojectors

�Pk��

b�b�b!b!

; �P?�� Pk�

��: (21)

The transverse and longitudinal projections of the fluctua-tions �B� can then be identified as

E � � �P?��B� �

�

1

2h��

�b� b�"; (22)

and

"� � �Pk��B� � b�"; (23)

respectively, where we have introduced the quantity

" � b�h��b�

2b!b!: (24)

In terms of these fluctuation projections, the field B� is

B� � �1 "�b� E�: (25)

The reader is warned that at the same level of approxima-tion the covariant components B� are given by

B� � g��B� � �1 "�b� E� h��b�: (26)

One effect of these projections is to disentangle in B�

the NG modes associated with Lorentz and diffeomor-phism breaking. To see this, start in the vacuum andperform a virtual local particle Lorentz transformationwith parameters �� satisfying ��b� � 0. This leavesunchanged the metric �� and the projection ".However, nonzero transverse excitations ��b� are gener-ated. When �� is promoted to a field E��, these becomethe Lorentz NG modes E� � E��b�. Note that they auto-matically obey an axial-gauge condition, b�E� � 0, thesignificance of which is elaborated in subsequent sections.

Similarly, the excitations "� about the vacuum containthe diffeomorphism NG mode. To verify this, note that the

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components �� of the broken diffeomorphism obey

�� Pk��

� �b��b�b!b!

; (27)

with ��b� � ��b�. Then, a virtual diffeomorphism gen-erates a nonzero value for ", but the field E� is unaffected.Promoting �� to the NG field ��, the expression for "becomes

" �b�@��b

�

b!b!� @��: (28)

We see that the longitudinal excitation of B� can indeed beidentified with the diffeomorphism NG mode, as claimedabove. Note that an associated fluctuation in the metric,given by

�� ! g�� @�� @��; (29)

is also generated by the diffeomorphism.

B. Bumblebee dynamics

The dynamical behavior of B� and the associated NGmodes is determined by the structure of the action thatdefines the specific model under study. In general, theLagrangian LB for a single bumblebee field B� coupledto gravity and matter can be written as a sum of terms

L B � Lg LgB LK LV LJ: (30)

Here, Lg is the gravitational Lagrangian, LgB describesthe gravity-bumblebee coupling, LK contains the kineticterms for B�, LV contains the potential, including termstriggering the spontaneous Lorentz violation, and LJ de-termines the coupling of B� to the matter or other sectorsin the model.

Various forms for each of these partial Lagrangians arepossible, and for certain purposes some can be set to zero.As an explicit example containing all types of terms,consider the Lagrangian

L B �1

2%�eR �eB�B�R��

1

4eB��B��

eV�B�B� � b2� eB�J

�; (31)

where % � 8'G and e ��g

pis the determinant of the

vierbein. The Lorentz-invariant limit of this theory hasbeen studied previously in the context of alternative theo-ries of gravity in Riemann spacetime [42]. The simplifiedLorentz-violating limit of the theory with � � 0 was in-troduced in Ref. [2], while the theory (31) and someversions including ghost vectors have been exploredmore recently in Refs. [5,24,25].

In the model (31) and its subsets, which are among thoseused in the sections below, the gravitational Lagrangian Lg

is that of general relativity. The specific nonminimalgravity-bumblebee interaction LgB in this example is con-trolled by the coupling constant �. The last term in Eq. (31)

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represents a matter-bumblebee interaction LJ, involvingthe matter current J�. The theory (31) is written for aRiemann or Riemann-Cartan spacetime, but the limit ofMinkowski spacetime is also of interest below, and thecorresponding Lagrangian can be obtained by eliminatingthe first two terms and setting e � 1.

The partial Lagrangian LV in the model (31) involves apotential V of the general form in Eq. (11), inducing thespontaneous Lorentz and diffeomorphism violation. Thequantity b2 is a real positive constant, related to the vacuumvalue ba of the bumblebee field by b2 � jba�abbbj, whilethe � sign in V determines whether ba is timelike orspacelike. As before, V can be a polynomial such asV�x� � �x2=2, where � is a coupling constant. An alter-native explicit form for V of particular value for studies ofNG modes is the sigma-model potential

V�B�B� � b2� � ��B�B

� � b2�; (32)

where the quantity � is now a Lagrange-multiplier field.The Lagrange multiplier acts to constrain the theory to theextrema of V obeying B�B� � b2 � 0, thereby eliminat-ing fields other than the NG modes. This model is alimiting case of the previous polynomial one in whichthe massive mode is frozen. Note that in any case thepotential V excludes the possibility of a U(1) gauge invari-ance involving B�, whatever the form of the bumblebeekinetic term in the action.

The kinetic partial Lagrangian LK in the model ofEq. (31) involves a field strength B�� for B�, defined as

B�� D�B� D�B�; (33)

where D� are covariant derivatives appropriate for thechosen spacetime geometry. In a Riemann or Minkowskispacetime, where the torsion vanishes, this field strengthreduces to B�� @�B� @�B�. In either case, with B�

given by Eq. (25), it follows that the diffeomorphism modecontained in " cancels in the kinetic partial Lagrangian LKin Eq. (31), and LK propagates two modes. If an alternativekinetic partial Lagrangian LK is adopted instead an addi-tional mode can propagate. A simple example is given bythe choice [43]

L K;ghost �1

2eB�D�D�B�; (34)

which propagates three modes, although this kinetic partialLagrangian can be unphysical as such because it can con-tain ghost dynamics.

The above discussion demonstrates that the fate of theLorentz and diffeomorphism NG modes depends on thegeometry of the spacetime and on the dynamics of thetheory. To gain further insight into the nature and fate ofthe NG modes, we consider in turn the three cases ofMinkowski, Riemann, and Riemann-Cartan spacetimes,each in a separate section below.

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IV. MINKOWSKI SPACETIME

A. Role of the vierbein

In Minkowski spacetime, the curvature and torsion van-ish by definition, and global coordinate systems exist suchthat

g�� : (35)

Particle Lorentz transformations can be performed using

��

� �� (36)

to rotate and boost tensor components. These transforma-tions are global if �� is independent of the spacetimepoint; otherwise, they are local. In any case, they maintainthe metric in the form �� at each point. It is instructive tocompare the expression (36) with the form of the vierbeinwhen h�� 0:

e�� : (37)

It follows that a vierbein with h�� 0 in Minkowskispacetime can be identified with a local particle Lorentztransformation. Also, starting from a vacuum solution withe�

� � ��, a local Lorentz transformation (36) generates

�� .In Minkowski spacetime, the diffeomorphisms main-

taining h�� 0 are global spacetime translations, corre-sponding to constant �� in Cartesian coordinates. Underlocal diffeomorphisms with nonconstant ��, the metrictransforms as [44]

�� ! g�� h��; (38)

where h�� @�� @��. A corresponding term isgenerated in the vierbein, given by Eq. (15).

To study the NG modes in Minkowski spacetime, con-sider first a theory with tensor field T�� satisfying thecondition (10). As before, the NG modes can be identifiedwith small excitations �T�� maintaining this constraint.The constraint is invariant under global Lorentz transfor-mations and translations, as is evident by writing Eq. (10)in a coordinate frame in which Eq. (35) holds. Moreimportantly, the constraint also is invariant under localLorentz transformations and diffeomorphisms. It followsthat even in Minkowski spacetime there are up to ten NGmodes when spontaneous Lorentz violation occurs.

As a more explicit example, consider a bumblebeemodel with the vacuum solution of the form (19). LocalLorentz transformations and local diffeomorphismschange the components B� and the metric g�� whilemaintaining the equality B�g��B

� � �b2, but some ofthese transformations are spontaneously broken by thevacuum values. As before, there are potentially four NGmodes that can appear, consisting of three Lorentz NGmodes E� � E��b� obeying b�E

� � 0, and one diffeo-morphism NG mode contained in " and given by Eq. (28).

In a bumblebee model, the vacuum solution can breakglobal Lorentz transformations while preserving transla-

-7


tions, so that energy-momentum is conserved. This type ofassumption is sometimes adopted within the broader con-text of the SME as a useful simplification for experimentalstudies. Translation symmetry holds if b� is a constant in acoordinate frame in which the metric takes the form (35).However, the vacuum solution b� in this frame in principlealso could be a smoothly varying function of spacetimeobeying b��b

� � �b2, such as a soliton solution.Then, b� has constant magnitude but different orientationat different spacetime points, and both global Lorentz andtranslation symmetries are broken. In this case, a vierbeincan be introduced at each point that transforms b� into alocal field ba obeying baba � �b2 at each point but hav-ing components that are constant over the spacetime. Therole of the vierbein in this context can be regarded as a linkto a convenient anholonomic basis in which ba appearsconstant. Whatever the fate of the global transformations,however, local Lorentz transformations and diffeomor-phisms are broken by the vacuum solution (19), and thebehavior of the four potential NG modes contained in E�

and " is determined by the dynamics of the theory.

B. Fate of the NG modes

Since gravitational excitations are absent in Minkowskispacetime, no kinetic terms for h�� can appear and there isno associated dynamics. Any propagation of NG modesmust therefore originate from Lagrangian terms involvingT��. Diffeomorphisms produce infinitesimal excitationsof the vacuum solution given by (17), which generate NGmodes in the combination @��. It might therefore seemthat even nonderivative terms for T�� in the Lagrangiancould generate derivative terms for some NG modes andhence possibly lead to their propagation. However, when apotential V drives the breaking, any nonderivative term inT�� is intrinsically part of V, so its presence may affectthe specific form of the vacuum solution (8) but cannotcontribute to the propagator for the NG modes. Indeed, nocontributions from V arise in the effective action for theNG modes because this action is obtained via virtualparticle transformations leaving V invariant and at itsextremum. This result can equivalently be obtained usingthe vierbein decomposition (13) of T��, since this ex-pansion automatically extremizes V and also contains theNG modes as shown before. It follows in this case that anypropagation of NG modes must be determined by kinetic orderivative-coupling terms for T��.

Next, we illustrate some of the possibilities for generat-ing propagators for the NG modes using kinetic terms in abumblebee model. Consider first a special Minkowski-spacetime limit of the theory (31), for which theLagrangian is

L B � 1

4B��B�� B�B� � b2� B�J�: (39)

The ghost-free kinetic term chosen involves the zero-

065008

torsion limit of the field strength B�� in Eq. (33), and theadoption of the sigma-model potential (32) ensures a focuson the NG modes.

For this theory, a coordinate frame can be chosen inwhich the vacuum solution is

hB�i � b�; he��i � ��; h�i � 0: (40)

For simplicity, in what follows we take b� to be constant inthis frame. The relevant virtual fluctuations of the bumble-bee field around the vacuum solution, generated by thebroken particle Lorentz transformations and diffeomor-phisms, can be decomposed using the projector method.The result is Eq. (25), where E� and " contain the NGmodes. As before, the Lorentz NG modes satisfy an axial-gauge condition b�E� � 0 and so represent 3 degrees offreedom.

With the vacuum (40) and the choice of kinetic term inEq. (39), the diffeomorphism mode contained in " cancelsin B��. It therefore cannot propagate and is an auxiliarymode. In fact, the kinetic term in Eq. (39) reduces to theform of a U(1) gauge theory in an axial gauge, so one of thethree Lorentz modes is auxiliary too. Adopting the sugges-tive notation E� � A� and denoting the correspondingfield strength by F�� @�A� @�A�, we find that theLagrangian (39) reduces at leading order to

LB !LNG �1

4F��F��A�J� b�J�b�@��J�:

(41)

This is the effective quadratic Lagrangian determining thepropagators of the NG modes in the theory (39). Note thatthe axial-gauge condition b�A� � 0 includes the specialcases of temporal gauge (A0 � 0) and pure axial gauge(A3 � 0), and it ensures the constraint term is absent inLNG. Note also that varying with respect to �� yields thecurrent-conservation law, @�J� � 0.

We see that the NG modes for the Minkowski-spacetimebumblebee theory (39) have the basic properties of themassless photon, described as a U(1) gauge theory in anaxial gauge. This result is consistent with an early analysisby Nambu [20], who investigated the constraint B�B� �

�b2 as a nonlinear gauge choice that spontaneously breaksLorentz invariance. In the linearized limit with B� � b� E�, this gauge choice reduces to an axial-gauge conditionb�E

� � 0 at leading order. The discussion here involves aLagrange-multiplier constraint rather than a direct gaugechoice and so the theories differ, but the result remainsunaffected.

The masslessness of the photon in the effective theory(41) is a consequence of the spontaneously broken Lorentzsymmetry in the original theory (39). An interesting ques-tion is whether this idea has experimentally verifiableconsequences. Indeed, a version of the theory (39) withan explicit matter sector has been presented in Ref. [24] as

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a model for quantum electrodynamics (QED) that gener-ates a Lorentz-violating term in the SME limit. The latterappears in Eq. (41) as the Lorentz-violating term b�J�,along with a conventional charged-current interactionA�J

�.If the current J� represents the usual electron current in

QED, then the term b� can be identified with the coeffi-cient ae� for Lorentz violation in the QED limit of the SME[7]. If this coefficient is spacetime independent, it is knownto be unobservable in experiments restricted to the electronsector [45], but coefficients of this type can generatesignals in the quark [14,46] and neutrino [7,17] sectors.Moreover, various other possible sources of experimentalsignals can be considered, such as spacetime dependenceof the coefficients, the presence of multiple fields and othertypes of current, and interference between several sourcesof Lorentz violation. Nonminimal couplings to other sec-tors, including the gravitational couplings discussed in thenext section, also can produce experimental signals. Allthese effects are contained within the SME. More radicaloptions for interpretation of the NG modes from Lorentzand diffeomorphism breaking in a general theory likewisecan be envisaged, ranging from new long-range forcesweakly coupled to matter with possible implications fordark matter and dark energy to the identification of manyor all massless modes in nature with the NG modes. Acareful investigation of these possibilities lies outside ourpresent scope but would be of definite interest.

Another interesting issue is whether a consistent theoryexists in Minkowski spacetime in which the diffeomor-phism mode contained in " propagates. Consider, forexample, substituting an alternative kinetic term of theform (34) in the Lagrangian (39), yielding the model

LB;ghost �1

2B�@�@�B� ��B�B� � b2� B�J� (42)

in Cartesian coordinates. This model may have a ghost, butthe behavior of the NG modes can nonetheless be exam-ined. Proceeding via the projector method as before, wefind the kinetic term becomes 1

2 E�@�@�E�, so the diffeo-

morphism NG mode contained in " is auxiliary while thethree E� modes propagate. More generally, in the covariantderivative D�B� relevant for a general coordinate systemin Minkowski spacetime, the NG excitations reduce to@�E� when b� is constant in Cartesian coordinates, sokinetic terms contain no propagation of the diffeomor-phism mode �� in this case.

This example illustrates a general difficulty in forming acovariant kinetic term that permits propagation of thediffeomorphism modes for the case of constant vacuumvalue t��. To be observer independent, the Minkowski-spacetime Lagrangian must be formed from contractions ofa tensor T�� and its derivatives. Only terms with one ormore derivatives can contribute to the propagation, asshown above. However, for covariant derivatives of

065008

T�� with constant t��, the diffeomorphism modesalways enter combined with a derivative, @��, while theconnection acquires a corresponding change induced bythe metric fluctuation (29) that cancels them. Other possi-bilities would therefore need to be countenanced, such as anonconstant t��. In any case, the structure of termscontaining derivatives of �� in the effective Lagrangianfor the NG modes represents a major difference betweenspontaneous breaking of internal and spacetime symme-tries. In the former, the relevant fields carry internal indicesthat are independent of spacetime derivatives, and so theo-ries with compact internal symmetry groups can be con-structed that propagate all the NG modes withoutgenerating ghosts. In contrast, the spontaneous violationof spacetime symmetries involves fields with spacetimeindices, and for the diffeomorphism NG modes thischanges the derivative structure in the effectiveLagrangian.

V. RIEMANN SPACETIME

In this section, we revisit for Riemann spacetimes theresults obtained in the Minkowski-spacetime case. Thegeneral features obtained in the previous section apply toa nondynamical Riemann spacetime with fixed backgroundmetric. The primary interest here therefore lies insteadwith Riemann spacetimes having a dynamical metric.

A. Vierbein and spin connection

In a Riemann spacetime with dynamical metric g��, thenature and properties of the NG modes for spontaneousLorentz and diffeomorphism violation still can be analyzedfollowing the general approach of Secs. II and III. Weassume that the solutions to a theory for a tensor T��satisfy the condition (10), thereby inducing a nonzerovacuum value t��. This condition is automatically sat-isfied by writing T�� in terms of the vierbein,

T�� e�ae�be�c . . . tabc��; (43)

where tabc�� is the vacuum value of the tensor in a localLorentz frame. For definiteness in what follows, we sup-pose that tabc�� is constant over the spacetime manifold inthe region of interest. This assumes appropriate smooth-ness of t�� and compatibility with any boundary con-ditions. For example, if the Riemann spacetime isasymptotically Minkowski and the vacuum value of thevierbein e�� is taken as��, then the components of t��must be asymptotically constant.

A primary difference in Riemann spacetime is that up tosix of the 16 independent components of the vierbein e�a

can represent dynamical degrees of freedom of the gravi-tational field. The Lagrangian for the theory must thereforecontain dynamical terms for the vierbein. This raises theissue of the effect of these additional terms on the other 10components of the vierbein, all of which are potential NG

-9


modes for the spontaneous violation of spacetimesymmetries.

At first sight the situation might seem to be furthercomplicated by the existence of the spin connection!�

ab, which permits the construction of the covariantderivative and in principle can have up to 24 independentcomponents. However, the requirement that the connectionbe metric,

D�e�a � 0; (44)

and the vanishing of the torsion tensor in a Riemann space-time imply that the spin connection !�

ab can be specifiedcompletely in terms of the vierbein and its derivativesaccording to

!�ab �

1

2e�a�@�e�

b @�e�b�

1

2e�v�@�e�

a @�e�a�

1

2e�ae�be�c�@�e�c @�e�c�: (45)

It is therefore sufficient to consider the behavior of thevierbein in studying the properties of the NG modes. Forexample, a Higgs mechanism is excluded for the spinconnection in a Riemann spacetime, since no independentpropagating massless modes for !�

ab exist to absorb theNG degrees of freedom. In the next section, we revisit thisissue in the context of the more general Riemann-Cartangeometry, for which the spin connection is an independentdynamical field.

Covariant derivatives acting on T�� in the Lagrangiancan also generate propagators for the vierbein and hencefor the NG modes. The covariant derivative D�T�� isgiven by

D�T�� e�ae�be�c . . .D�tabc��: (46)

The term D�tabc�� in this equation contains products of thespin connection with the vacuum value tabc��, which ac-cording to Eq. (45) generates expressions involving asingle derivative of the vierbein. In the presence of sponta-neous violation of spacetime symmetries, it follows thatany piece of the Lagrangian involving a quadratic power ofD�T�� can produce quadratic-derivative terms involv-ing the vierbein.

The above discussion shows that in a Riemann space-time the fate of the NG modes can depend on both thegravitational terms in the Lagrangian and the kinetic orother derivative-coupling terms for the tensor field. In whatfollows, we consider implications of these results for bum-blebee models.

B. Bumblebee and photon

For a bumblebee model in an asymptotically flatRiemann spacetime, the vacuum structure is similar tothe Minkowski case. We take the vacuum values for B�

and the vierbein to be those of Eq. (19). The projector

065008

method can be applied, leading to the decomposition (25)of B�. There are four potential NG modes contained in thefields E� and ", and the axial-gauge condition b�E� � 0holds. Note that the field strength B�� in Eq. (33) can berewritten to give

B�� @�e�a @�e�a�ba; (47)

where ba is taken constant as in the previous subsection.The properties of the NG modes depend on the kinetic

terms for B� and the gravitational terms in the Lagrangian.To gain further insight, consider the Lagrangian (31) with aLagrange-multiplier potential,

L B �1

2%�eR �eB�B�R��

1

4eB��B

��

1

2e��B�B

� � b2� eB�J�: (48)

The vacuum solution for this theory is that of Eq. (40).The effective Lagrangian LNG determining the proper-

ties of the NG modes can be obtained by expanding theLagrangian LB to quadratic order, keeping couplings tomatter currents and curvature, and using the decomposition(25) of B�. Disregarding total derivative terms, we find

LNG �1

2%�eR �eb�b�R�� eA�A�R��

�e"�" 2�b�b�R�� 2�e�" 1�b�A�R��

1

4eF��F�� eA�J� eb�J� eb�@��J�;

(49)

Here, we again relabel A� � E�, which obeys b�A� � 0,and write F�� @�A� @�A�, which is the field strengthof a gauge-fixed U(1) field. In Eq. (49), the gravitationalexcitations h�� obey h��b� � 0. In the absence of curva-ture sources, Eq. (49) reduces to the Minkowski-spacetimeresult (41).

The form of LNG reveals that only two of the fourpotential NG modes propagate. The propagating modesare transverse Lorentz NG modes, while the longitudinalLorentz and the diffeomorphism NG modes are auxiliary.In particular, with the kinetic term given by the square ofthe field strength B�� in Eq. (47), the diffeomorphism NGmode in " again cancels, as in the Minkowski-spacetimecase. Moreover, the curvature terms in LB also yield nocontributions for " in LNG. This is because, in an asymp-totically Minkowski spacetime, metric excitations of therequired NG form h�� @�� @�� produce only avanishing contribution to the curvature tensor at linearorder and contribute only as total derivatives at secondorder when contracted with �� or b�.

As in the Minkowski-spacetime case, it is interesting inthe present context with gravity to consider the possibilitythat the photon observed in nature can be identified with

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the NG mode resulting from spontaneous Lorentz viola-tion. We see that, in a Riemann spacetime, the theory of theform (49) produces a free propagator for the Lorentz NGmode consistent with this idea at the linearized level. Also,the interaction with the charged current J� has an appro-priate form. Indeed, the effective action LNG contains as asubset the standard Einstein-Maxwell electrodynamics inaxial gauge.

The issue of possible experimental signals can be revis-ited for the present Riemann-spacetime case. The discus-sion in the previous section about potential SME andrelated signals in Minkowski spacetime still applies, butfurther possibilities exist. In particular, there are interestingunconventional couplings of the curvature with A�, ", andb�. The photon acquires Lorentz-invariant curvature cou-plings of the form eA�A�R��, which are forbidden bygauge invariance in conventional Einstein-Maxwell elec-trodynamics but are consistent here with the axial-gaugecondition. The term �eb�b�R��=2% corresponds to a non-zero coefficient of the s�� type in the pure-gravity sector ofthe SME [5]. The other terms with curvature also representLorentz-violating couplings. This Lagrangian thereforegives rise to additional effects that could serve to provideexperimental evidence for the idea that the photon is an NGmode for spontaneous Lorentz violation. The analysis ofthe associated signals is evidently of interest but lies out-side our present scope.

It also is of interest to ask whether there exists a theory inRiemann spacetime with a nontrivial propagator for thediffeomorphism mode contained in ". Indeed, for a purelytimelike coefficient b�, for which " � @0�0, it has beenshown that if a kinetic term for the diffeomorphism NGmode were to appear with second-order time derivatives,then an unusual dispersion relation would follow withpotentially interesting phenomenological consequences[26]. In general, the presence of curvature makes thisquestion more challenging than in Minkowski spacetime.

In the context of the bumblebee model (48) with the fieldstrength (47) having constant ba, we have seen that thediffeomorphism NG mode fails to propagate. Attemptingto change this by modifying the gravitational terms in theLagrangian (48) to any combination of covariant contrac-tions of the curvature tensor R%��, including theories withgeneral quadratic curvature terms [47,48], also fails toyield a nontrivial propagator for the diffeomorphism NGmode for the same reason as above. However, possibilitiesexist that might overcome this difficulty, such as allowingfor nonconstant ba. Another interesting option is to relaxthe requirement of an asymptotically Minkowski space-time, perhaps by adding a cosmological-constant term tothe theory. This leads to modifications in the projectoranalysis and changes in the effective action for the NGmodes. For example, in a curved background a term of theform �e"�" 2�b�b�R�� in Eq. (49) would generatequadratic terms for " in the effective Lagrangian, as

065008

needed for the propagation of ��. A cosmological terme� also contains quadratic terms h��h��

12h

2 in theweak-field limit, which might serve as a suitable sourceof quadratic terms because a virtual diffeomorphism gen-erates time derivatives for the spacelike components of��. Exploration of these issues is of definite interest butlies beyond the scope of this work.

VI. RIEMANN-CARTAN SPACETIME

In a Riemann-Cartan spacetime, the vierbein e�a and thespin connection !�

ab represent independent degrees offreedom determined by the dynamics [28]. It follows thatthe effects of spontaneous Lorentz breaking can be strik-ingly different from the cases examined above. In particu-lar, we focus in this section on the possibility that the NGmodes are absorbed into the spin connection via a Higgsmechanism.

A. Higgs mechanism for the spin connection

As in the Riemann-spacetime case, we suppose that atensor T�� obeys the condition (10) and acquires anonzero vacuum value. This condition can be satisfied byexpressing T�� in terms of the vierbein according toEq. (43). This result can be used to calculate the covariantderivative of the field T��, which enters the kineticLagrangian for T�� and therefore affects the NG modesfor the spontaneous Lorentz violation. A key feature ofRiemann-Cartan spacetime is that this covariant derivativenow involves the spin connection as an independent degreeof freedom. For instance, assuming constant t��, thelinearized expression in a Minkowski background is

D�T�� !�"�t"�� !�

"�t�"�� . . . : (50)

It follows that a kinetic term involving a quadratic expres-sion in the covariant derivative of T�� generates a non-derivative quadratic expression in the spin connection. Thiscould represent a mass for the spin connection, so thespontaneous violation of Lorentz symmetry in Riemann-Cartan spacetime could incorporate a gravitational versionof the Higgs mechanism. Note that this Higgs mechanismcannot occur in a Riemann spacetime, where the spinconnection is identically the derivative expression (45)for the vierbein, because the same calculation producesinstead a kinetic term for the NG modes, as shown in theprevious section.

The Lagrangian for a generic theory with spontaneousLorentz violation in Riemann-Cartan spacetime can bewritten as

L � L0 LSSB; (51)

where L0 describes the unbroken theory and LSSB inducesspontaneous Lorentz violation. We suppose for simplicitythat L0 contains only gravitational terms formed from thecurvature and torsion, while the Lagrangian LSSB for a

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tensor field T�� contains a kinetic piece and a potentialdriving the spontaneous Lorentz violation.

A priori, it might seem that a large range of modelscould implement this Higgs mechanism, since numeroustypes of tensors could acquire a vacuum expectation value.However, a complete Higgs mechanism requires a theoryL0 that has a massless propagating spin connection prior tothe spontaneous Lorentz violation. A fully satisfactoryexample also requires the theory to be free of ghosts. Itturns out that these conditions severely restrict the possi-bilities for model building.

General studies exist of theories L0 with a propagatingspin connection [47,48], including ones with both massiveand massless propagating modes. However, the subset ofghost-free models is relatively small, especially for thecase of a massless spin connection. The total number ofpropagating modes in these models depends on the pres-ence of certain accidental symmetries. Our investigationsreveal that the symmetry-breaking Lagrangian LSSB typi-cally breaks one or more of the accidental symmetries ofL0 when the tensor field acquires a vacuum expectationvalue. This significantly complicates the analysis of mod-els, but also offers interesting new avenues by whichspontaneous Lorentz violation could affect the physicalmodes in a realistic theory.

We are primarily interested in ghost-free LagrangiansL0 formed from powers of the curvature and torsion withat most two derivatives in the equations of motion for thevierbein and spin connection. In this case, up to 18 of the24 components of the spin connection !�

ab in principlecan behave as propagating degrees of freedom. The sixcomponents !0ab are auxiliary fields, irrespective of anygauge choices imposed for the Lorentz symmetry or anysimplifications from accidental symmetries.

The behavior of the 18 modes can be studied by assum-ing a background Minkowski spacetime and linearizing theequations of motion along the lines discussed in Sec. II.Note that, at linear order in infinitesimal quantities, thespin connection transforms under Lorentz transformationsaccording to

!�ab ! !�

ab @��ab; (52)

while infinitesimal diffeomorphisms leave !�ab invariant

at lowest order. The vacuum solution now takes the form

hT��i � t��; he��i � ��; h!�abi � 0

(53)

in a suitable coordinate frame. The vanishing of h!�abi

and the invariance of !�ab under diffeomorphisms sug-

gests that the fate of the diffeomorphism NG mode isunlikely to be appreciably altered by the new role of thespin connection in the present context, and this is con-firmed in what follows.

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B. Decompositions

The investigation of various models is facilitated byintroducing two different decompositions of the 24 fields!��. The first is a decomposition according to Lorentzindices,

!�� A�� M�� 1

3��T� ��T��; (54)

where A�� is totally antisymmetric, M�� has mixedsymmetry, and T� is the trace piece. The antisymmetriccomponents define a dual V% � �%��A

��=2 that has fourindependent components. The mixed components M��

satisfy eight identities, which can be written M�� 0

and M�� M�� M��, and they therefore contain16 degrees of freedom. The trace T� contains the remain-ing 4 degrees of freedom. The reader is cautioned thatEq. (54) is not a Lorentz-irreducible decomposition inthe usual sense because the field being decomposed is aconnection rather than a tensor.

The second useful decomposition involves the spin-parity projections JP of the fields !��. These are particu-larly appropriate for the case of timelike Lorentz violationinduced by t��, such as a bumblebee vacuum value of theform b� � �b; 0; 0; 0�. The 18 dynamical fields in this caseinclude the projections 2, 2, 1, 1, 0, 0, while thesix auxiliary fields !0�� contain two triplet projections wedenote by ~1, ~1. Again, we stress that these projectionsinvolve the connection rather than a tensor, so the notationfails to reflect the true transformation properties. For ex-ample, the 1 projection yields a triplet of scalars underspatial rotations.

Explicit expressions for each of these projections can befound. For example, we find

!�0� � !j0j; !�0� �1

2�jkl!jkl;

!�1�l � �jkl!j0k; !�1�

k � !jkj;

!�~1�k �

1

2�klm!0lm; !�~1�

k � !00k;

!�2�jk �

1

2�!j0k !k0j�

1

3�jk!l0l;

!�2�jk �

1

4��klm!jlm �jlm!klm�

1

6�jk�lmn!nlm;

(55)

where spatial components are denoted by j; k; . . . .The two sets of projections can be related. We find

!�1�k �!�~1�

k Vk �1

2�k0lmM

0lm 2

3Vk;

!�1�k � !�~1�

k Tk � M00k 2

3Tk;

!�0� � T0; !�0� � V0:

(56)

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C. Bumblebee

To gain further insight, we investigate a definite form forthe Lagrangian LSSB, namely, the simple bumblebeemodel

L SSB � 1

4eB��B

�� e��B�B� � b2� (57)

with a Lagrange-multiplier potential freezing any non-NGmodes. In a Riemann-Cartan spacetime, the field strengthB�� in the kinetic term of this theory is defined in Eq. (33).Its expression in terms of the vierbein and spin connectionis

B�� e�b!�

ab e�

b!�ab�ba: (58)

Note that this form reduces to Eq. (47) in the limits ofRiemann and Minkowski spacetimes, for which the spinconnection is given in terms of derivatives of the vierbeinby Eq. (45).

When B�� is squared to yield the kinetic term, quadraticterms in !�

ab appear in the Lagrangian LSSB. For ex-

ample, for a Minkowski background we find

L K � 1

4eB��B��

� 1

4�!�"� !�"��!

�!� !�!��b"b!: (59)

The appearance of these quadratic terms again suggeststhat a Higgs mechanism can occur involving the absorptionof the NG modes by the spin connection.

In terms of the Lorentz decomposition in the previoussubsection, the kinetic term LK for B� becomes

LK �2

9�b�b�V�V� b�V�b�V��

1

4M"

��M!��b"b!

1

18�b�b�T�T� b�T�b�T��

1

3��"V

"M!��b�b!

1

6M�

��b�T� b�T��b�: (60)

This result holds for any vacuum value b�, but its physicalinterpretation can be involved in the general case.

For the special case of timelike Lorentz violation in-duced by a vacuum expectation value b� � �b; 0; 0; 0�, theJP decomposition provides a more convenient expression.With this assumption, we find

L K � 1

2b2!�1�

j !�1�j 1

2b2!�~1�

j !�~1�j: (61)

We see that this expression contains an apparent physicalmass term for the 1 and a wrong-sign mass term for the~1. Since the ~1 is an auxiliary field, it cannot propagateindependently. However, the 1 is an independent dynami-

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cal field, so interpreting its apparent mass term requires astudy of the dynamical content of L0.

D. Illustrative models

Next, we present three different sample models L0, allcontaining dynamical terms for the spin connection, toillustrate some of the possible effects and issues emergingfrom the presence of the Lorentz-breaking term LSSB. Inthe first model, denoted L0;1, ghosts are present but ananalysis shows that a Higgs mechanism occurs when LSSB

is added. The second L0;2 initially has only auxiliary orgauge degrees of freedom, but the addition of the Lorentz-violating term LSSB breaks some accidental symmetriesand hence causes some modes to propagate. The thirdmodel L0;3 is ghost free and has a massless propagatingspin connection.

The Lagrangian for the first example is

L 0;1 �1

4R�%��R

�%��: (62)

To lowest order in the spin connection, the curvature tensorbecomes R�%�� @%!�� @�!%��. In this model, allthe fields !�� with � � 0 propagate as massless modes.However, when resolved into JP projections, the second-derivative terms in the equations of motion for the even-and odd-parity states have opposite signs, so the theorycontains ghosts. When L0;1 is combined with LSSB, thelinearized equations of motion become

@"@"!��@�@

"!"��1

2�!�!�!�!��b�b

!

1

2�!�!�!�!��b�b!: (63)

These 24 equations can be diagonalized to determine thenature of the modes in the combined theory, and we findthat among the propagating modes is a massive one. Thisconfirms the existence of a Higgs mechanism for the spinconnection in this model.

The idea behind the second model is to start with aspecial theory L0 in which accidental symmetries excludeall propagating physical modes, but chosen such thatphysical propagating modes emerge when the LagrangianLSSB triggering spontaneous Lorentz violation is added.The appearance of the physical modes via this ‘‘phoenix’’mechanism can be traced to the breaking of some acciden-tal symmetries of L0 by LSSB.

A number of models in which all modes are auxiliary orgauge are known [48]. Here, we consider one explicitexample, with Lagrangian given by

L 0;2 �1

2R��R��

1

2R��R��; (64)

where R�� is the Ricci tensor in Riemann-Cartan space-time. Since our focus is on the spin connection, we restrictattention for simplicity to solutions in background

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Minkowski spacetime. With this choice, the vierbein dis-appears from the linearized theory, so the spin connectionis the only relevant dynamical field.

The unbroken Lagrangian can be written in terms of theLorentz decomposed fields as

L 0;2 �1

9F��F��

1

9G��G��

1

4@"M"

��@!M!��

1

3

�F��

1

2��"!G

"!�@�M

��:

(65)

The corresponding equations of motion are

@�

�@!M!��

2

3

�F��

1

2��"!G"!

�� 0; (66)

@�F�� 3

2@�@!M!��; (67)

@�G�� 3

4��!"@

!@�M�"�: (68)

In these equations, F�� @�T� @�T� and G��

@�V� @�V� are the field strengths for T� and V�,respectively.

A cursory inspection might suggest that this theory hasat least two sets of massless fields, T� and V�, whichcorrespond to the 1 and the 1 modes in the JP decom-position. However, there are a number of accidental sym-metries in this theory associated with the projectionoperators for the 2, 2, 0, and 0 fields. These andLorentz transformations can be used to remove all physicalpropagating degrees of freedom [48]. In particular, the 1

mode can be gauged away using only rotations, while the1 mode can be gauged away using only boosts:

!�1�j ! !�1�

j @k"jk;

!�1�j ! !�1�

j �j0lm@l"0m:

(69)

The net result is that the Lorentz-invariant theory (64) hasno physical content.

Suppose now the term LSSB in Eq. (61) for the case of atimelike vacuum expectation value b� is added to theLagrangian (64). This spontaneously breaks boosts whilemaintaining rotation symmetry. The 1 mode can still begauged away via rotations, but the 1 mode can no longerbe removed using boosts and so might be expected topropagate as a massive mode. However, the mass term inLSSB also affects the structure of the gauge and auxiliaryfields in the theory by breaking some of the accidentalsymmetries, so other field combinations now becomephysical. We find two such massless modes, involvingsuperpositions of the JP projections.

For our third example, we take for L0 a ghost-freemodel with a massless propagating spin connection. Ageneral analysis under the assumption of Lorentz invari-

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ance finds only four ghost-free possibilities [48]. All sharethe property of the previous example that the propagatingmodes consist of mixtures of JP projections. In two mod-els, the massless propagating mode incorporates contribu-tions from the 1 projections, while in the other two itincludes contributions from the 1 projections. It is there-fore of interest to adopt for L0 either of the first two modelsand investigate the effect on the propagating modes ofadding the Lorentz-violating term LSSB.

As an explicit example, consider the Lagrangian [48]

L 0;3 � R��R�� 1

3R2: (70)

It turns out that the propagating modes of this Lorentz-invariant model are a mixture of 1 and 2 projections. Incontrast, LSSB contains quadratic terms for the 1 and theauxiliary ~1 states. In the theory resulting from the combi-nation of the two, the nature of the modes can be deter-mined by diagonalizing the 24 linearized equations for thespin connection. We find that the propagation of the mass-less modes is altered, but there is no massive propagating1. The incompatibility between the mixture of JP statesappearing in L0;3 and that appearing in LSSB prevents theoccurrence of a clean Higgs mechanism for the JP modesin this example.

In the context of these ideas, a number of issues ofinterest remain open for future investigation. Studies ofthe large variety of possible Lorentz-invariant LagrangiansL0 could lead to additional features beyond those identi-fied in the three examples above. It also would be ofinterest to explore more explicitly the effects of lightlikeand spacelike b� in the Riemann-Cartan spacetimes. TheJP decomposition is less appropriate for these cases, soalternative decompositions with respect to the correspond-ing little group are likely to be useful. Different choices forLSSB, including ones in which the spontaneous Lorentzviolation involves one or more tensor fields, similarly canbe expected to affect the dynamics of the NG modes. Froma broader theoretical perspective, the incorporation ofLorentz violation opens an arena for the search for ghost-free theories with dynamical curvature and torsion.

Various implications for phenomenology in the contextof Riemann-Cartan spacetime also merit exploration. Thescale of the emergent mass in the models considered here isset by b2. Even if this is of order of the Planck mass, theexistence of fields with Lorentz-violating physics couldhave effects on cosmology and in regions with stronggravitational fields such as black holes. The couplings toother known fields also merit attention and could lead tointeresting signals for experiments. All relevant terms as-sociated with gravitational and SM fields are included inthe gravitational couplings of the Lorentz- andCPT-violating SME in Riemann-Cartan spacetime [5],which therefore provides the appropriate framework forinvestigating phenomenological implications of thesemodels.

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VII. SUMMARY

In this paper, we have examined the fate of the Nambu-Goldstone modes when Lorentz symmetry is spontane-ously broken. The analysis is performed in the context ofthe vierbein formalism, which is well suited for this pur-pose because it admits a clear separation between localLorentz and coordinate frames on the spacetime manifold.Within this formalism, we have demonstrated in Sec. II thatspontaneous particle Lorentz violation is accompanied byspontaneous particle diffeomorphism violation and viceversa, and that up to 10 NG modes can appear. Thesemodes can naturally be matched to those 10 of the 16modes of the vierbein that in a Lorentz-invariant theoryare gauge degrees of freedom. This match provides furtherevidence for the value of the vierbein formalism in studiesof spontaneous violations of spacetime symmetries. Wehave also provided a generic treatment for backgroundMinkowski spacetimes. The fate of the NG modes is foundto depend both on the spacetime geometry and also on thedynamics of the tensor field triggering the spontaneousviolation of local Lorentz and diffeomorphism symmetries.

As illustrative models for the analysis, we have adopteda general class of bumblebee models, involving vacuumvalues for a vector field that break some of the localLorentz and diffeomorphism symmetries. Some propertiesof these models have been presented in Sec. III, whereprojectors are constructed that permit separation of theLorentz and diffeomorphism NG modes.

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In the later sections of this work, we have studied thebehavior of the NG modes in Minkowski, Riemann, andRiemann-Cartan spacetimes. Each of these offers distinc-tive general features, which can be illustrated within bum-blebee models. In Minkowski and Riemann spacetimes,Lorentz NG modes exist that can propagate as masslessmodes, with effective Lagrangians containing the Maxwelland Einstein-Maxwell theories in axial gauge. Suitablebumblebee models thereby provide dynamical methodsof generating the photon as a Nambu-Goldstone bosonfor spontaneous Lorentz violation. Various possibilitiesexist for experimental signals in these models, includingboth unconventional Lorentz-invariant couplings andLorentz-breaking couplings in the matter and gravitationalsectors of the SME. In Riemann-Cartan spacetimes, theinteresting possibility exists that the spin connection couldabsorb the propagating NG modes in a gravitational ver-sion of the Higgs mechanism. This unique feature ofgravity theories with torsion may offer another phenom-enologically viable route for constructing realistic modelswith spontaneous Lorentz violation.

ACKNOWLEDGMENTS

This work was supported in part by the Department ofEnergy under Grant No. DE-FG02-91ER40661, by theNational Aeronautics and Space Administration underGrant Nos. NAG8-1770 and NAG3-2914, and by theNational Science Foundation under Grant No. PHY-0097982.

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[39] Under special circumstances more than 10 generators areinvolved. For example, in a theory based on specialrelativity with multiple noninteracting particle species,there are distinct particle Lorentz transformations foreach field. The particle symmetry group of the theorythen contains a direct product of several Lorentz groups,and so there could be multiple sets of NG modes.However, in typical gravitational theories with the metriccoupling to all irreducible tensors, the full particle invari-ance involves only one copy of the particle Lorentz anddiffeomorphism groups, so only ten corresponding NGmodes can appear.

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Spontaneous Lorentz violation, Nambu-Goldstone modes, and gravity