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In the context of gravity in a Riemann geometry, the permits a straightforward treatment of fermions in non- investigation of spontaneous Lorentz violation was initi- trivial spacetimes. Since this formalism distinguishes ated with a study of a class of vector theories [22], called cleanly between local Lorentz frames and coordinate bumblebee models, that are comparatively simple field frames on the spacetime manifold, it is also ideally suited theories in which spontaneous Lorentz violation occurs. for investigations of Lorentz and CPT breaking [5], in- These models and some versions with ghost modes have cluding the effects of spontaneous violation. since 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 A. General considerations is spontaneously broken by a timelike vector. If such a mode were to appear in a theory with second-order time a A basic object in the formalism is the vierbein eµ , derivatives, it has been shown that it would have an un- which can be viewed as providing at each point on the usual dispersion relation leading to interesting anomalous spacetime manifold a link between the covariant compo- spin-dependent forces [26]. nents Tλµν··· of a tensor field in a coordinate basis and the In this paper, we investigate the ultimate fate of the corresponding covariant components Tabc··· of the tensor NG modes associated with spontaneous violation of local field in a local Lorentz frame. The link is given by Lorentz and diffeomorphism symmetries. We perform a b c a generic analysis of theories formulated in Riemann- Tλµν··· = eλ eµ eν Tabc···. (1) Cartan spacetime and its limits, including the Riemann · · · spacetime of general relativity and the Minkowski space- In the coordinate basis, the components of the spacetime time of special relativity, and we illustrate the results metric are denoted gµν . In the local Lorentz frame, the within the bumblebee model. The standard vierbein for- metric components take the Minkowski form ηab, but the malism for gravity [27] offers a natural and convenient basis may be anholonomic. Expressions for contravariant framework within which to study the properties of the or mixed tensor components similar to Eq. (1) can be ob- NG modes, and we adopt it here. The basic gravita- tained by appropriate contractions with the components a µν tional fields can be taken as the vierbein eµ and the g of the inverse spacetime metric. ab The vierbein formalism permits the treatment of both spin connection ωµ . The associated field strengths are the curvature and torsion tensors. In a general theory of basic types of spacetime transformations relevant for gravity in a Riemann-Cartan spacetime [28], these fields gravitation theories: local Lorentz transformations, and are independent dynamical quantities. The usual Rie- diffeomorphisms. Consider a point P on the spacetime mann spacetime of general relativity is recovered in the manifold. Local Lorentz transformations at P act on zero-torsion limit, with the spin connection fixed in terms the tensor components Tabc··· via a transformation ma- a of the vierbein. Our focus here is on models in which trix Λ b applied to each index. For an infinitesimal trans- one or more tensor fields acquire vacuum values, a situ- formation, this matrix has the form ation that could potentially arise in the context of effec- Λa δa + ǫa , (2) tive field theories for a variety of quantum gravity frame- b ≈ b b works in which mechanisms exist for Lorentz violation. where ǫab = ǫba are the infinitesimal parameters car- These include, for example, string theory [2, 29], non- − commutative field theories [30], spacetime-varying fields rying the six Lorentz degrees of freedom and generating [31, 32, 33], loop quantum gravity [34], random-dynamics the local Lorentz group. In contrast, a diffeomorphism models [35], multiverses [36], and brane-world scenarios is a mapping of P to another point Q on the spacetime [37], so the results obtained in the present work are ex- manifold, with an associated mapping of tensors at P to pected to be widely applicable. tensors at Q. The pullback of a transformed tensor at Q to P differs from the original tensor at P . For infinitesi- The organization of this paper is as follows. A generic mal diffeomorphisms characterized in a coordinate basis discussion of spontaneous Lorentz violation in the vier- by the transformation bein formalism is presented in section II. Section III dis- cusses basic results for the bumblebee model. The three µ µ µ x x + ξ , (3) subsequent sections, IV, V, and VI, examine the fate of → the NG modes in Minkowski, Riemann, and Riemann- this difference is given by the Lie derivative of the ten- Cartan spacetimes, respectively. Section VII contains a µ sor Tλµν··· along the vector ξ . The four infinitesimal summary of the results. Throughout this work, we adopt parameters ξµ comprise the diffeomorphism degrees of the notation and conventions of Ref. [5]. freedom. The vierbein formalism is natural for studies of Lorentz violation. Spontaneous violation of local Lorentz invari- II. SPONTANEOUS LORENTZ VIOLATION ance occurs when the lagrangian of the theory is invariant under local Lorentz transformations but the vacuum so- For gravitational theories with a realistic matter sec- lution violates one or more of the symmetries. The key tor, the vierbein formalism [27] is widely used because it feature is the existence of a nonzero vacuum expectation 3 value for the components Tabc··· of a tensor field in a local transformations on the manifold are general coordinate Lorentz frame [5]: transformations, which leave invariant the action. The statement of observer invariance therefore contains no Tabc tabc =0. (4) h ···i≡ ··· 6 physical information other than the assumption of ob- server independence of the physics. The values tabc··· may be constants or specified functions, provided they solve the equations of motion of the theory. Particle transformations are defined to act on individ- Each such expectation value specifies one or more orien- ual particles or localized fields, while leaving unchanged tations within any local frame, which is the characteristic vacuum expectation values. A particle Lorentz trans- of spontaneous Lorentz violation. formation involves a rotation or boost only of localized The vacuum expectation value of the vierbein is also tensor fields. The components of the tensor are affected, a constant or a fixed function, either given by the solu- while the basis and any vacuum values are unchanged. tion to the gravitational equations or specified as a back- Similarly, particle diffeomorphisms with the pullback in- ground. For example, in a spacetime with Minkowski corporated can be viewed as changes only in localized a field distributions, with the tensor components trans- background the vacuum value of the vierbein is eµ = δ a in a suitable coordinate frame. It follows fromh Eq.i (1) forming via the Lie derivative but the basis and all vac- µ uum values unaffected. that the existence of a vacuum value tabc··· for a tensor in a local frame implies it also has a vacuum value tλµν··· Invariance of a system under particle transformations in the coordinate basis on the manifold. However, a non- has physical consequences, including notably the exis- trivial vacuum expectation value for tλµν··· also implies tence of conservation laws. Local Lorentz invariance spontaneous violation of diffeomorphism invariance. This implies a condition on the antisymmetric components µν shows that the spontaneous violation of local Lorentz in- of the energy-momentum tensor T , while diffeomor- variance implies spontaneous violation of diffeomorphism phism invariance implies a covariant conservation law for invariance. it. Thus, for example, in general relativity the laws are µν νµ µν In fact, the converse is also true: if diffeomorphism T = T and DµT = 0. Spontaneous breaking of invariance is spontaneously broken, so is local Lorentz these spacetime symmetries leaves unaffected the conser- invariance. It is immediate that any violation of dif- vation laws. In contrast, explicit breaking of these sym- feomorphism invariance via vacuum values of vectors or metries, which is described by noninvariant terms in the tensors breaks local Lorentz invariance, as above. An action, modifies the laws. For local Lorentz and diffeo- alternative source of diffeomorphism violations is possi- morphism transformations, the conservation laws in the ble via vacuum values of scalars provided the scalars are presence of spontaneous and explicit breaking are ob- nonconstant over the spacetime manifold, but this also tained in Ref. [5] in the context of a general gravitation leads to violations of local Lorentz invariance because the theory. derivatives of the scalar vacuum values provide an orien- In a theory with spontaneous breaking of a contin- tation within each local Lorentz frame. uous symmetry, one or more NG modes are expected. The NG modes can be identified with the virtual excita- tions around the vacuum solution that are generated by B. Identification of NG modes the particle transformations corresponding to the bro- ken symmetry. According to the above discussion, if the In discussing the consequences of spacetime-symmetry extremum of the action involves a nonzero vacuum ex- violations, it is useful to distinguish among several types pectation value tabc··· for a tensor in a local frame, both of transformations. Treatments of Lorentz-invariant the- local Lorentz invariance and diffeomorphism invariance ories in the literature commonly define two classes of are spontaneously broken. Since these invariances in- Lorentz transformations, called active and passive, which volve 10 generators for particle transformations [38, 39], act on tensor components essentially as inverses of each we conclude that up to ten NG modes can appear when an other. In a Lorentz-violating theory, however, the pres- irreducible Lorentz tensor acquires a vacuum expectation ence of vacuum expectation values with distinct proper- value. ties implies that there are more than two possible classes In the subsequent parts of this work, it is shown that of transformations [7]. For most purposes it suffices to the vierbein formalism is particularly well suited for de- limit attention to two possibilities, called observer trans- scribing these NG modes. A simple counting of modes a formations and particle transformations. illustrates the key idea. The vierbein eµ has 16 compo- Observer transformations involve changes of the ob- nents. In a Lorentz- and diffeomorphism-invariant the- server frame. It is standard to assume that any physically ory, 10 of these can be eliminated via gauge transforma- meaningful theory is covariant under observer transfor- tions, leaving six potentially physical degrees of freedom mations, and this remains true in the presence of Lorentz to describe the gravitational field. In general relativ- violation [7]. An observer local Lorentz transformation ity, four of these six are auxiliary and do not propagate, can be viewed as a rotation or boost of the basis vectors leaving only the two usual transverse massless graviton in the local tangent space. Tensor components are then modes; more general metric gravitational theories can expressed in terms of the new basis. Observer coordinate have up to six graviton modes [40]. However, in a the- 4 ory with local Lorentz and diffeomorphism violation, the There can be many more such excitations than NG 10 additional vierbein modes cannot all be eliminated by modes. The NG modes are distinguished by the require- gauge transformations and instead must be treated as ment that δTλµν··· maintains the extremum of the action dynamical fields in the theory. In short, the 10 potential and corresponds to broken symmetry generators. NG modes from spontaneous local Lorentz and diffeomor- For modes δTλµν··· excited via local Lorentz transfor- phism breaking are contained within the 10 components mations or diffeomorphisms, the magnitude of Tλµν··· at of the vierbein that are gauge degrees of freedom in the each point is preserved, Lorentz-invariant limit. λµν··· αβγ··· 2 T gλαgµβgνγ ...T = t , (10)

2 λµν··· αβγ··· C. Perturbative analysis where t = t ηλαηµβ ηνγ ...t . This holds, for example, in a theory with potential V having the simple functional form In general, each NG mode can be obtained by perform- λµν··· αβγ··· 2 ing on the vacuum a virtual particle transformation for a V = V (T gλαgµβgνγ ...T t ), (11) broken-symmetry generator and then elevating the cor- − responding spacetime-dependent parameter to the NG which can trigger a vacuum value tλµν··· when V is ex- field. To identify the NG modes and study their basic tremized. For instance, V could be a positive quar- properties, it therefore suffices to consider small excita- tic polynomial in T λµν··· with minima at zero, such as tions about the vacuum and to work in a linearized ap- V (x)= λx2/2, where λ is a coupling constant. The con- proximation. dition (10) is automatically satisfied by the choice If the vacuum solution of a given theory involves the α β γ vac Tλµν··· = eλ eµ eν ...tαβγ···, (12) metric gµν , then the metric gµν in the presence of small excitations can be written as which also reduces to the correct vacuum expectation vac value (8) when the vierbein excitations vanish. This im- gµν = gµν + hµν . (5) plies all the excitations in δTλµν··· associated with the NG In the general scenario, distinguishing the background modes are contained in the vierbein through Eq. (12). from gravitational fluctuations requires some care. For Using the expansion (7) of the vierbein in Eq. (12) instance, in the shortwave approximation [41] the dis- yields a first-order expression for the tensor excitations tinction is made in terms of the amplitude of hµν and δTλµν··· in terms of the 16 fields hµν and χµν : vac the scales on which gµν and hµν vary. For our purposes, 1 α however, the presence of a nontrivial background space- δTλµν··· ( hλα + χλα)t ≈ 2 µν··· time is unnecessary and serves to complicate the basic 1 α +( hµα + χµα)t + .... (13) study of the properties of the NG modes. We therefore 2 λ ν··· focus attention here on spacetimes in which the vacuum 1 Evidently, the combination ( 2 hµν + χµν ) contains the geometry is Minkowski. interesting dynamical degrees of freedom. Small metric fluctuations about the Minkowski back- We can observe the effects of local Lorentz and dif- ground can be written as feomorphism transformations by performing each sepa- rately. Under infinitesimal Lorentz transformations, the gµν = ηµν + hµν . (6) vierbein components transform as To linear order, the inverse metric is then gµν ηµν µα νβ ≈ − hµν hµν , η η hαβ. In this context, the distinction between co- → ordinate indices µ, ν, . . . on the manifold and the local χµν χµν ǫµν , (14) Lorentz indices a, b, . . . is diminished, and Greek letters → − can be used for both. The 16-component vierbein can be while their transformations under infinitesimal diffeomor- written as phisms are 1 eµν = ηµν + ( 2 hµν + χµν ), (7) hµν hµν ∂µξν ∂ν ξµ, → − 1 − χµν χµν (∂µξν ∂ν ξµ). (15) where the ten symmetric excitations hµν = hνµ are asso- → − 2 − ciated with the metric, while the six antisymmetric com- In these expressions, quantities of order (ǫh), (ǫχ), (ξh), ponents χµν = χνµ are the local Lorentz degrees of − (ξχ), etc. are assumed small and hence negligible in the freedom. linearized treatment. The vacuum expectation value (4) of an arbitrary ten- The excitation due to infinitesimal Lorentz transfor- sor becomes mations is

Tµν··· tµν···, (8) α α h i≡ δTλµν··· ǫλαt µν··· ǫµαtλ ν··· .... (16) ≈− − − and excitations about this vacuum value are denoted as Depending on the properties of the vacuum value tµν···, δTλµν = (Tλµν tλµν ). (9) up to six independent excitations associated with broken ··· ··· − ··· 5

Lorentz generators can appear in this expression. Their The ultimate fate of the NG modes, and in particular nature is determined by the six parameters ǫµν . It fol- whether some or all of them propagate as physical mass- lows that the corresponding NG modes µν for the broken less fields, depends on the specific dynamics of the theory. Lorentz symmetries stem from the antisymmetricE compo- At the level of the lagrangian, the linearized approxima- nents χµν of the vierbein. tion involves expanding all fields around their vacuum The excitation due to infinitesimal diffeomorphisms is values and keeping terms of quadratic order or less. The α α dominant terms in the effective lagrangian for the NG δTλµν··· (∂λξα)t µν··· (∂µξα)tλ ν··· . . . modes can then be obtained by replacing the tensor ex- ≈ − α − − ξ ∂αtλµν···. (17) citation with the appropriate NG modes and Ξ ac- − µν µ cording to Eqs. (16) and (17). The resultingE dynamics This can contain up to four independent excitations as- of the modes are determined by several factors, including sociated with broken diffeomorphisms, depending on the the basic form of the terms in the original action and the properties of t . Except for the case of a scalar T , the µν··· type of spacetime geometry in the theory. Disentangling four potential NG modes Ξµ corresponding to the four these issues is the subject of the following sections. parameters ξµ enter the vierbein accompanied by deriva- tives. This can potentially alter their dispersion relations and couplings to matter currents. III. BUMBLEBEE MODELS As a simple example, consider spontaneous breaking due to a nonzero vector vacuum value t , which could µ To study the behavior of the NG modes and distinguish be timelike, spacelike, or lightlike. Introduce a quantity dynamical effects from geometrical ones, it is valuable µ = µν t obeying t ν = 0. The three degrees of free- ν µ to consider a class of comparatively simple models for domE ofE µ correspondE to the three Lorentz NG modes Lorentz and diffeomorphism violation, called bumblebee associatedE with the three Lorentz generators broken by models, in which a vector field Bµ acquires a constant the direction t . Similarly, one can introduce a scalar ν expectation value bµ [5, 22]. These models contain many Ξµt , which plays the role of the NG mode correspond- µ of the interesting features of cases with more complicated ing to the diffeomorphism broken by t . In this example, µ tensor vacuum values. For example, all the basic types which is studied in more detail in the next section, there of rotation, boost, and diffeomorphism violations can be are four potential NG modes. If a second orthogonal vac- implemented, and the existence and properties of the cor- uum value t′ is also present, an additional two Lorentz µ responding NG modes can be studied for various space- NG modes appear because two additional Lorentz gen- time geometries. In this section, some general results for erators are broken. All six Lorentz NG modes enter µν these models are presented. the theory once a third orthogonal vacuum valueE exists. Similarly, as additional vacuum values are added, more components of the fields Ξµ enter as NG modes for diffeo- A. Projectors for NG modes morphisms, until all four are part of the broken theory. Examples with more complicated tensor representa- The characteristic feature of bumblebee models is that tions also provide insight. For instance, consider a the- µ a ory with an expectation value for a two-index symmetric a vector field B acquires a vacuum expectation value b tensor T µν = T νµ. In this case, the choice of vacuum in a local Lorentz frame. This breaks three Lorentz trans- formations and one diffeomorphism, so there are four po- value tµν can crucially affect the number and type of tential NG modes. According to Eq. (12) and the asso- NG modes. There is a choice among many possible sce- µ narios. A subset of the space of possible vacuum val- ciated discussion, the vector field B can be written in terms of the vierbein as ues consists of those tµν that can be made diagonal by a suitable choice of coordinate basis, but even if atten- µ µ a B = e ab , (18) tion is restricted to this subset there are many possibili- ties. For instance, a vacuum value with diagonal elements which holds in any background metric. The vierbein de- (3, 1, 1, 1) breaks three boosts and four diffeomorphisms grees of freedom include the NG modes of interest. for a total of seven NG modes, one with diagonal ele- As before, we proceed under the simplifying as- ments (4, 1, 1, 2) breaks five Lorentz transformations and sumption that the background spacetime geometry is four diffeomorphisms for a total of nine NG modes, while Minkowski. The vacuum solution then takes the form one with diagonal elements (6, 1, 2, 3) breaks all ten sym- metries for a total of ten NG modes. µ µ B = b , eµν = ηµν (19) In the general case, up to ten NG modes can appear h i h i when a tensor acquires a vacuum expectation value tµν···. in a suitable coordinate frame. The vierbein can be ex- The fluctuations of the tensor about the vacuum under panded in terms of hµν and χµν , as in Eq. (7). The virtual particle transformations are given as the sum of fluctuations about the vacuum can therefore be written the right-hand sides of Eqs. (16) and (17). The associated as NG modes consist of up to six Lorentz modes µν and E µ µ µ 1 µν µν up to four diffeomorphism modes Ξµ. δB = (B b ) ( h + χ )bν . (20) − ≈ − 2 6

The results of the previous subsection imply that three We see that the longitudinal excitation of Bµ can in- of the four potential NG modes are contained in fields µ deed be identified with the diffeomorphism NG mode, as µ E obeying bµ = 0, while one appears in a combination claimed above. Note that an associated fluctuation in µ E Ξ bµ. To identify these modes, it is convenient to sepa- the metric, given by rate the excitations (20) into longitudinal and transverse components using projection operators. Focusing for def- ηµν gµν ηµν ∂µξν ∂ν ξµ, (29) → ≈ − − initeness on the non-lightlike case (b2 = 0), we define the projectors 6 is also generated by the diffeomorphism.

µ µ b bν µ µ µ (Pk) ν = σ , (P⊥) ν = δ ν (Pk) ν . (21) B. Bumblebee dynamics b bσ −

The transverse and longitudinal projections of the fluc- µ µ The dynamical behavior of B and the associated NG tuations δB can then be identified as modes is determined by the structure of the action that µ µ ν 1 µν µν µ defines the specific model under study. In general, the = (P⊥) ν δB ( h + χ )bν b ρ (22) µ E ≈ − 2 − lagrangian B for a single bumblebee field B coupled to gravity andL matter can be written as a sum of terms and

µ µ ν µ B = g + gB + K + V + J . (30) ρ = (Pk) ν δB b ρ, (23) L L L L L L ≈ Here, g is the gravitational lagrangian, gB describes respectively, where we have introduced the quantity L L the gravity-bumblebee coupling, K contains the kinetic µ ν µ L b hµν b terms for B , V contains the potential, including terms ρ = . (24) triggering theL spontaneous Lorentz violation, and de- − 2bσb J σ termines the coupling of Bµ to the matter or other sectorsL In terms of these fluctuation projections, the field Bµ is in the model. Various forms for each of these partial lagrangians are Bµ (1 + ρ)bµ + µ. (25) possible, and for certain purposes some can be set to zero. ≈ E As an explicit example containing all types of terms, con- The reader is warned that at the same level of approxi- sider the lagrangian mation the covariant components Bµ are given by 1 µ ν 1 µν ν ν B = (eR + ξeB B Rµν ) 4 eBµν B Bµ gµν B (1 + ρ)bµ + µ + hµν b . (26) L 2κ − ≡ ≈ E µ 2 µ eV (BµB b ) eBµJ , (31) One effect of these projections is to disentangle in Bµ − ± − the NG modes associated with Lorentz and diffeomor- where κ = 8πG and e √ g is the determinant of phism breaking. To see this, start in the vacuum and the vierbein. The Lorentz-invariant≡ − limit of this theory perform a virtual local particle Lorentz transformation has been studied previously in the context of alternative µν µν with parameters ǫ satisfying ǫ bν = 0. This leaves theories of gravity in Riemann spacetime [42]. The sim- 6 unchanged the metric ηµν and the projection ρ. How- plified Lorentz-violating limit of the theory with ξ = 0 µν ever, nonzero transverse excitations ǫ bν are generated. was introduced in Ref. [2], while the theory (31) and some µν µν When ǫ is promoted to a field , these become the versions including ghost vectors have been explored more − µ µν E Lorentz NG modes bν . Note that they auto- recently in Refs. [5, 24, 25]. E ≡ E µ matically obey an axial-gauge condition, bµ = 0, the In the model (31) and its subsets, which are among E significance of which is elaborated in subsequent sections. those used in the sections below, the gravitational la- µ Similarly, the excitations ρ about the vacuum contain grangian g is that of general relativity. The specific L the diffeomorphism NG mode. To verify this, orient the nonminimal gravity-bumblebee interaction gB in this µ observer coordinate frame so that the parameter ξ for example is controlled by the coupling constantL ξ. The the broken diffeomorphism obeys last term in Eq. (31) represents a matter-bumblebee in- µ µ ν teraction J , involving the matter current J . The the- µ µ ν b ξ bν ory (31) isL written for a Riemann or Riemann-Cartan ξ = (Pk) ν ξ σ , (27) ≈ b bσ spacetime, but the limit of Minkowski spacetime is also of interest below, and the corresponding lagrangian can with ξµbν = ξν bµ. Then, a virtual diffeomorphism gener- be obtained by eliminating the first two terms and setting ates a nonzero value for ρ, but the field µ is unaffected. e = 1. Promoting ξµ to the NG field Ξµ, theE expression for ρ The partial lagrangian in the model (31) involves becomes V a potential V of the generalL form in Eq. (11), inducing µ ν b ∂µΞν b µ the spontaneous Lorentz and diffeomorphism violation. ρ = σ = ∂µΞ . (28) 2 b bσ The quantity b is a real positive constant, related to the 7

a 2 a b vacuum value b of the bumblebee field by b = b ηabb , to rotate and boost tensor components. These transfor- a | | ν while the sign in V determines whether b is timelike mations are global if ǫµ is independent of the spacetime or spacelike.∓ As before, V can be a polynomial such as point; otherwise, they are local. In any case, they main- 2 V (x)= λx /2, where λ is a coupling constant. An alter- tain the metric in the form ηµν at each point. It is in- native explicit form for V of particular value for studies structive to compare the expression (36) with the form of NG modes is the sigma-model potential of the vierbein when hµν = 0: µ 2 µ 2 V (BµB b )= λ(BµB b ), (32) e ν δ ν + χ ν . (37) ± ± µ ≈ µ µ where the quantity λ is now a Lagrange-multiplier field. The Lagrange multiplier acts to constrain the theory to It follows that a vierbein with hµν = 0 in Minkowski µ 2 spacetime can be identified with a local particle Lorentz the extrema of V obeying BµB b = 0, thereby elim- ± transformation. Also, starting from a vacuum solution inating fields other than the NG modes. This model is a ν ν with eµ = δµ , a local Lorentz transformation (36) gen- limiting case of the previous polynomial one in which the ν ν massive mode is frozen. Note that in any case the poten- erates χµ = ǫµ . tial V excludes the possibility of a U(1) gauge invariance In Minkowski spacetime, the diffeomorphisms main- µ taining hµν = 0 are global spacetime translations, corre- involving B , whatever the form of the bumblebee kinetic µ term in the action. sponding to constant ξ in cartesian coordinates. Under local diffeomorphisms with nonconstant ξµ, the metric The kinetic partial lagrangian K in the model of Eq. L transforms as [44] (31) involves a field strength Bµν for Bµ, defined as

Bµν = DµBν Dν Bµ, (33) ηµν gµν = ηµν + hµν , (38) − → where Dµ are covariant derivatives appropriate for the where hµν = ∂µξν ∂ν ξµ. A corresponding term is chosen spacetime geometry. In a Riemann or Minkowski generated in the− vierbein,− given by Eq. (15). spacetime, where the torsion vanishes, this field strength To study the NG modes in Minkowski spacetime, con- reduces to Bµν = ∂µBν ∂ν Bµ. In either case, with sider first a theory with tensor field Tλµν··· satisfying the µ − B given by Eq. (25), it follows that the diffeomorphism condition (10). As before, the NG modes can be identi- mode contained in ρ cancels in the kinetic partial la- fied with small excitations δTλµν··· maintaining this con- grangian K in Eq. (31), and K propagates two modes. straint. The constraint is invariant under global Lorentz L L If an alternative kinetic partial lagrangian K is adopted transformations and translations, as is evident by writing L instead an additional mode can propagate. A simple ex- Eq. (10) in a coordinate frame in which Eq. (35) holds. ample is given by the choice [43] More importantly, the constraint is also invariant under 1 ν µ local Lorentz transformations and diffeomorphisms. It K, = eBµD Dν B , (34) L ghost 2 follows that even in Minkowski spacetime there are up which propagates three modes, although this kinetic par- to ten NG modes when spontaneous Lorentz violation tial lagrangian can be unphysical as such because it can occurs. contain ghost dynamics. As a more explicit example, consider a bumblebee The above discussion demonstrates that the fate of the model with the vacuum solution of the form (19). Lo- Lorentz and diffeomorphism NG modes depends on the cal Lorentz transformations and local diffeomorphisms µ geometry of the spacetime and on the dynamics of the change the components B and the metric gµν while µ ν 2 theory. To gain further insight into the nature and fate maintaining the equality B gµν B = b , but some of of the NG modes, we consider in turn the three cases of these transformations are spontaneously∓ broken by the Minkowski, Riemann, and Riemann-Cartan spacetimes, vacuum values. As before, there are potentially four NG each in a separate section below. modes that can appear, consisting of three Lorentz NG µ µν µ modes bν obeying bµ = 0, and one diffeomor- phism NGE ≡E mode contained inEρ and given by Eq. (28). IV. MINKOWSKI SPACETIME In a bumblebee model, the vacuum solution can break global Lorentz transformations while preserving transla- A. Role of the vierbein tions, so that energy-momentum is conserved. This type of assumption is sometimes adopted within the broader In Minkowski spacetime, the curvature and torsion context of the SME as a useful simplification for exper- µ vanish by definition, and global coordinate systems exist imental studies. Translation symmetry holds if b is a such that constant in a coordinate frame in which the metric takes the form (35). However, the vacuum solution bµ in this gµν = ηµν . (35) frame could in principle also be a smoothly varying func- µ ν 2 Particle Lorentz transformations can be performed using tion of spacetime obeying b ηµν b = b , such as a soli- ton solution. Then, bµ has constant∓ magnitude but dif- ferent orientation at different spacetime points, and both Λ ν δ ν + ǫ ν (36) global Lorentz and translation symmetries are broken. In µ ≈ µ µ 8 this case, a vierbein can be introduced at each point that by the broken particle Lorentz transformations and dif- a 2 transforms bµ into a local field ba obeying bab = b at feomorphisms, can be decomposed using the projector ∓ each point but having components that are constant over method. The result is Eq. (25), where µ and ρ contain the spacetime. The role of the vierbein in this context the NG modes. As before, the Lorentz NGE modes satisfy µ can be regarded as a link to a convenient anholonomic an axial-gauge condition bµ = 0 and so represent three E basis in which ba appears constant. Whatever the fate of degrees of freedom. the global transformations, however, local Lorentz trans- With the vacuum (40) and the choice of kinetic term in formations and diffeomorphisms are broken by the vac- Eq. (39), the diffeomorphism mode contained in ρ cancels uum solution (19), and the behavior of the four potential in Bµν . It therefore cannot propagate and is an auxiliary NG modes contained in µ and ρ is determined by the mode. In fact, the kinetic term in Eq. (39) reduces to dynamics of the theory. E the form of a U(1) gauge theory in an axial gauge, so one of the three Lorentz modes is auxiliary too. Adopting the suggestive notation µ Aµ and denoting the corre- E ≡ B. Fate of the NG modes sponding field strength by Fµν ∂µAν ∂ν Aµ, we find that the lagrangian (39) reduces≡ at leading− order to Since gravitational excitations are absent in Minkowski µν µ µ µ ν 1 F F A J b J + b ∂ Ξ J . spacetime, no kinetic terms for h can appear and there B NG 4 µν µ µ ν µ µν L → L ≈− − − (41) is no associated dynamics. Any propagation of NG This is the effective quadratic lagrangian determining the modes must therefore originate from lagrangian terms λµν··· propagators of the NG modes in the theory (39). Note involving T . Diffeomorphisms produce infinitesi- µ that the axial-gauge condition bµA = 0 includes the mal excitations of the vacuum solution given by (17), special cases of temporal gauge (A0 = 0) and pure axial which generate NG modes in the combination ∂ Ξ . It µ α gauge (A3 = 0), and it ensures the constraint term is might therefore seem that even nonderivative terms for λµν··· absent in NG. Note also that varying with respect to T in the lagrangian could generate derivative terms L µ Ξµ yields the current-conservation law, ∂µJ = 0. for some NG modes and hence possibly lead to their We see that the NG modes for the Minkowski- propagation. However, when a potential V drives the λµν··· spacetime bumblebee theory (39) have the basic prop- breaking, any nonderivative term in T is intrinsi- erties of the massless photon, described as a U(1) gauge cally part of V , so its presence may affect the specific theory in an axial gauge. This result is consistent with form of the vacuum solution (8) but cannot contribute an early analysis by Nambu [20], who investigated the to the propagator for the NG modes. Indeed, no contri- constraint B Bµ = b2 as a nonlinear gauge choice butions from V arise in the effective action for the NG µ that spontaneously breaks∓ Lorentz invariance. In the modes because this action is obtained via virtual particle linearized limit with Bµ bµ + µ, this gauge choice transformations leaving V invariant and at its extremum. reduces to an axial-gauge≈ conditionE b µ = 0 at leading This result can equivalently be obtained using the vier- µ order. The discussion here involves a Lagrange-multiplierE bein decomposition (13) of T λµν···, since this expansion constraint rather than a direct gauge choice and so the automatically extremizes V and also contains the NG theories differ, but the result remains unaffected. modes as shown before. It follows in this case that any The masslessness of the photon in the effective the- propagation of NG modes must be determined by kinetic ory (41) is a consequence of the spontaneously broken or derivative-coupling terms for T λµν···. Lorentz symmetry in the original theory (39). An inter- Next, we illustrate some of the possibilities for generat- esting question is whether this idea has experimentally ing propagators for the NG modes using kinetic terms in verifiable consequences. Indeed, a version of the theory a bumblebee model. Consider first a special Minkowski- (39) with an explicit matter sector has been presented in spacetime limit of the theory (31), for which the la- Ref. [24] as a model for quantum electrodynamics (QED) grangian is that generates a Lorentz-violating term in the SME limit. 1 µν µ 2 µ The latter appears in Eq. (41) as the Lorentz-violating B = 4 Bµν B λ(BµB b ) BµJ . (39) µ L − − ± − term bµJ , along with a conventional charged-current in- µ The ghost-free kinetic term chosen involves the zero- teraction AµJ . If the current J µ represents the usual electron current torsion limit of the field strength Bµν in Eq. (33), and the adoption of the sigma-model potential (32) ensures a in QED, then the term bµ can be identified with the co- e focus on the NG modes. efficient aµ for Lorentz violation in the QED limit of the For this theory, a coordinate frame can be chosen in SME [7]. If this coefficient is spacetime independent, it which the vacuum solution is is known to be unobservable in experiments restricted to the electron sector [45], but coefficients of this type µ µ B = b , eµν = ηµν , λ =0. (40) can generate signals in the quark [14, 46] and neutrino h i h i h i [7, 17] sectors. Moreover, various other possible sources For simplicity, in what follows we take bµ to be constant of experimental signals can be considered, such as space- in this frame. The relevant virtual fluctuations of the time dependence of the coefficients, the presence of mul- bumblebee field around the vacuum solution, generated tiple fields and other types of current, and interference 9 between several sources of Lorentz violation. Nonmin- V. RIEMANN SPACETIME imal couplings to other sectors, including the gravita- tional couplings discussed in the next section, can also In this section, we revisit for Riemann spacetimes the produce experimental signals. All these effects are con- results obtained in the Minkowski spacetime case. The tained within the SME. More radical options for interpre- general features obtained in the previous section apply tation of the NG modes from Lorentz and diffeomorphism to a nondynamical Riemann spacetime with fixed back- breaking in a general theory can also be envisaged, rang- ground metric. The primary interest here therefore lies ing from new long-range forces weakly coupled to matter instead with Riemann spacetimes having a dynamical with possible implications for dark matter and dark en- metric. ergy to the identification of many or all massless modes in nature with the NG modes. A careful investigation of these possibilities lies outside our present scope but A. Vierbein and spin connection would be of definite interest.

Another interesting issue is whether a consistent the- In a Riemann spacetime with dynamical metric gµν , ory exists in Minkowski spacetime in which the diffeo- the nature and properties of the NG modes for sponta- morphism mode contained in ρ propagates. Consider, neous Lorentz and diffeomorphism violation can still be for example, substituting an alternative kinetic term of analyzed following the general approach of sections II and the form (34) in the lagrangian (39), yielding the model III. We assume that the solutions to a theory for a ten- sor Tλµν··· satisfy the condition (10), thereby inducing a 1 ν µ µ 2 µ B,ghost = Bµ∂ ∂ν B λ(BµB b ) BµJ (42) nonzero vacuum value t . This condition is automat- L 2 − ± − λµν··· ically satisfied by writing Tλµν··· in terms of the vierbein, in cartesian coordinates. This model may have a ghost, but the behavior of the NG modes can nonetheless be ex- a b c amined. Proceeding via the projector method as before, Tλµν··· = eλ eµ eν ...tabc···, (43) 1 µ ν we find the kinetic term becomes ∂ ∂ν µ, so the dif- 2 E E where tabc··· is the vacuum value of the tensor in a lo- feomorphism NG mode contained in ρ is auxiliary while cal Lorentz frame. For definiteness in what follows, we the three µ modes propagate. More generally, in the E suppose that tabc··· is constant over the spacetime man- covariant derivative DµBν relevant for a general coordi- ifold in the region of interest. This assumes appropriate nate system in Minkowski spacetime, the NG excitations smoothness of tλµν··· and compatibility with any bound- reduce to ∂µ ν when bν is constant in cartesian coor- E ary conditions. For example, if the Riemann spacetime dinates, so kinetic terms contain no propagation of the is asymptotically Minkowski and the vacuum value of diffeomorphism mode Ξµ in this case. the vierbein eµν is taken as ηµν , then the components of This example illustrates a general difficulty in forming tλµν··· must be asymptotically constant. a covariant kinetic term that permits propagation of the A primary difference in Riemann spacetime is that up diffeomorphism modes for the case of constant vacuum to six of the 16 independent components of the vierbein value tλµν···. To be observer independent, the Minkowski- a eµ can represent dynamical degrees of freedom of the spacetime lagrangian must be formed from contractions gravitational field. The lagrangian for the theory must λµν··· of a tensor T and its derivatives. Only terms with therefore contain dynamical terms for the vierbein. This one or more derivatives can contribute to the propaga- raises the issue of the effect of these additional terms on tion, as shown above. However, for covariant deriva- the other 10 components of the vierbein, all of which λµν··· tives of T with constant tλµν···, the diffeomorphism are potential NG modes for the spontaneous violation of modes always enter combined with a derivative, ∂µΞα, spacetime symmetries. while the connection acquires a corresponding change in- At first sight the situation might seem to be further duced by the metric fluctuation (29) that cancels them. ab complicated by the existence of the spin connection ωµ , Other possibilities would therefore need to be counte- which permits the construction of the covariant deriva- nanced, such as a nonconstant tλµν···. In any case, the tive and in principle can have up to 24 independent com- structure of terms containing derivatives of Ξµ in the ef- ponents. However, the requirement that the connection fective lagrangian for the NG modes represents a major be metric, difference between spontaneous breaking of internal and a spacetime symmetries. In the former, the relevant fields Dλeµ =0, (44) carry internal indices that are independent of spacetime derivatives, and so theories with compact internal sym- and the vanishing of the torsion tensor in a Riemann ab metry groups can be constructed that propagate all the spacetime imply that the spin connection ωµ can be NG modes without generating ghosts. In contrast, the specified completely in terms of the vierbein and its spontaneous violation of spacetime symmetries involves derivatives according to fields with spacetime indices, and for the diffeomorphism ab 1 νa b b 1 νb a a ω = e (∂µe ∂ν e ) e (∂µe ∂ν e ) NG modes this changes the derivative structure in the µ 2 ν − µ − 2 ν − µ 1 αa βb c effective lagrangian. e e e (∂αeβc ∂βeαc). (45) − 2 µ − 10

It is therefore sufficient to consider the behavior of the lagrangian B to quadratic order, keeping couplings to vierbein in studying the properties of the NG modes. matter currentsL and curvature, and using the decomposi- For example, a Higgs mechanism is excluded for the spin tion (25) of Bµ. Disregarding total derivative terms, we connection in a Riemann spacetime, since no independent find propagating massless modes for ω ab exist to absorb the µ 1 µ ν µ ν NG degrees of freedom. In the next section, we revisit NG [eR + ξeb b Rµν + ξeA A Rµν L ≈ 2κ this issue in the context of the more general Riemann- µ ν µ ν +ξeρ(ρ + 2)b b Rµν +2ξe(ρ + 1)b A Rµν ] Cartan geometry, for which the spin connection is an 1 µν µ µ µ ν eFµν F eAµJ ebµJ + eb ∂ν ΞµJ , independent dynamical field. − 4 − − Covariant derivatives acting on Tλµν··· in the la- (49) grangian can also generate propagators for the vierbein µ and hence for the NG modes. The covariant derivative Here, we again relabel Aµ µ, which obeys bµA = 0, and write F ∂ A ∂ A≡E, which is the field strength D T is given by µν µ ν ν µ α λµν··· of a gauge-fixed≡ U(1)− field. In Eq. (49), the gravita- µ D T = e ae be c ...D t . (46) tional excitations hµν obey hµν b = 0. In the absence α λµν··· λ µ ν α abc··· of curvature sources, Eq. (49) reduces to the Minkowski- spacetime result (41). The term Dαtabc··· in this equation contains products of The form of reveals that only two of the four po- the spin connection with the vacuum value tabc···, which LNG according to Eq. (45) generates expressions involving a tential NG modes propagate. The propagating modes single derivative of the vierbein. In the presence of spon- are transverse Lorentz NG modes, while the longitudinal taneous violation of spacetime symmetries, it follows that Lorentz and the diffeomorphism NG modes are auxiliary. any piece of the lagrangian involving a quadratic power In particular, with the kinetic term given by the square of the field strength B in Eq. (47), the diffeomor- of DαTλµν··· can produce quadratic-derivative terms in- µν volving the vierbein. phism NG mode in ρ again cancels, as in the Minkowski- The above discussion shows that in a Riemann space- spacetime case. Moreover, the curvature terms in B also yield no contributions for ρ in . This is because,L in an time the fate of the NG modes can depend on both the LNG gravitational terms in the lagrangian and the kinetic or asymptotically Minkowski spacetime, metric excitations of the required NG form hµν = ∂µΞν ∂ν Ξµ produce other derivative-coupling terms for the tensor field. In − − what follows, we consider implications of these results only a vanishing contribution to the curvature tensor at for bumblebee models. linear order and contribute only as total derivatives at second order when contracted with ηµν or bµ. As in the Minkowski-spacetime case, it is interesting in B. Bumblebee and photon the present context with gravity to consider the possibil- ity that the photon observed in nature can be identified For a bumblebee model in an asymptotically flat Rie- with the NG mode resulting from spontaneous Lorentz mann spacetime, the vacuum structure is similar to the violation. We see that, in a Riemann spacetime, the the- Minkowski case. We take the vacuum values for Bµ ory of the form (49) produces a free propagator for the and the vierbein to be those of Eq. (19). The projec- Lorentz NG mode consistent with this idea at the lin- tor method can be applied, leading to the decomposition earized level. Also, the interaction with the charged cur- (25) of Bµ. There are four potential NG modes con- rent Jµ has an appropriate form. Indeed, the effective µ action contains as a subset the standard Einstein- tained in the fields and ρ, and the axial-gauge condi- LNG µ E Maxwell electrodynamics in axial gauge. tion bµ = 0 holds. Note that the field strength Bµν in Eq. (33)E can be rewritten to give The issue of possible experimental signals can be re- visited for the present Riemann-spacetime case. The a a discussion in the previous section about potential SME Bµν = (∂µeν ∂ν eµ )ba, (47) − and related signals in Minkowski spacetime still applies, where ba is taken constant as in the previous subsection. but further possibilities exist. In particular, there are in- The properties of the NG modes depend on the ki- teresting unconventional couplings of the curvature with netic terms for Bµ and the gravitational terms in the la- Aµ, ρ, and bµ. The photon acquires Lorentz-invariant µ ν grangian. To gain further insight, consider the lagrangian curvature couplings of the form eA A Rµν , which are (31) with a Lagrange-multiplier potential, forbidden by gauge invariance in conventional Einstein- Maxwell electrodynamics but are consistent here with µ ν 1 µ ν 1 µν the axial-gauge condition. The term ξeb b Rµν /2κ cor- B = (eR + ξeB B Rµν ) eBµν B L 2κ − 4 responds to a nonzero coefficient of the sµν type in 1 µ 2 µ the pure-gravity sector of the SME [5]. The other 2 eλ(BµB b ) eBµJ . (48) − ± − terms with curvature also represent Lorentz-violating The vacuum solution for this theory is that of Eq. (40). couplings. This lagrangian therefore gives rise to addi- The effective lagrangian NG determining the proper- tional effects that could serve to provide experimental ties of the NG modes can beL obtained by expanding the evidence for the idea that the photon is an NG mode for 11 spontaneous Lorentz violation. The analysis of the asso- by expressing Tλµν··· in terms of the vierbein according ciated signals is evidently of interest but lies outside our to Eq. (43). This result can be used to calculate the present scope. covariant derivative of the field Tλµν···, which enters the It is also of interest to ask whether there exists a the- kinetic lagrangian for Tλµν··· and therefore affects the NG ory in Riemann spacetime with a nontrivial propagator modes for the spontaneous Lorentz violation. A key fea- for the diffeomorphism mode contained in ρ. Indeed, for ture of Riemann-Cartan spacetime is that this covari- 0 a purely timelike coefficient bµ, for which ρ = ∂0Ξ , it ant derivative now involves the spin connection as an has been shown that if a kinetic term for the diffeomor- independent degree of freedom. For instance, assuming phism NG mode were to appear with second-order time constant tλµν···, the linearized expression in a Minkowski derivatives, then an unusual dispersion relation would fol- background is low with potentially interesting phenomenological conse- ρ ρ quences [26]. In general, the presence of curvature makes DαTλµν··· ω tρµν··· + ω tλρν··· + .... (50) ≈ α λ α µ this question more challenging than in Minkowski space- time. It follows that a kinetic term involving a quadratic ex- In the context of the bumblebee model (48) with the pression in the covariant derivative of Tλµν··· generates field strength (47) having constant ba, we have seen that a nonderivative quadratic expression in the spin connec- the diffeomorphism NG mode fails to propagate. At- tion. This could represent a mass for the spin connec- tempting to change this by modifying the gravitational tion, so the spontaneous violation of Lorentz symmetry terms in the lagrangian (48) to any combination of co- in Riemann-Cartan spacetime could incorporate a gravi- κ tational version of the Higgs mechanism. Note that this variant contractions of the curvature tensor R λµν , in- cluding theories with general quadratic curvature terms Higgs mechanism cannot occur in a Riemann spacetime, [47, 48], also fails to yield a nontrivial propagator for where the spin connection is identically the derivative the diffeomorphism NG mode for the same reason as expression (45) for the vierbein, because the same calcu- above. However, possibilities exist that might overcome lation produces instead a kinetic term for the NG modes, this difficulty, such as allowing for nonconstant ba. An- as shown in the previous section. other interesting option is to relax the requirement of an The lagrangian for a generic theory with spontaneous asymptotically Minkowski spacetime, perhaps by adding Lorentz violation in Riemann-Cartan spacetime can be a cosmological-constant term to the theory. This leads written as to modifications in the projector analysis and changes in = + , (51) the effective action for the NG modes. For example, in a L L0 LSSB curved background a term of the form ξeρ(ρ+2)bµbν R µν where describes the unbroken theory and in- in Eq. (49) would generate quadratic terms for ρ in the 0 SSB duces spontaneousL Lorentz violation. We supposeL for effective lagrangian, as needed for the propagation of Ξµ. simplicity that contains only gravitational terms A cosmological term eΛ also contains quadratic terms 0 formed from theL curvature and torsion, while the la- h hµν 1 h2 in the weak-field limit, which might serve µν 2 grangian for a tensor field T contains a kinetic as a suitable− source of quadratic terms because a virtual SSB λµν··· piece andL a potential driving the spontaneous Lorentz diffeomorphism generates time derivatives for the space- violation. like components of Ξµ. Exploration of these issues is of A priori, it might seem that a large range of mod- definite interest but lies beyond the scope of this work. els could implement this Higgs mechanism, since numer- ous types of tensors could acquire a vacuum expectation VI. RIEMANN-CARTAN SPACETIME value. However, a complete Higgs mechanism requires a theory that has a massless propagating spin con- L0 a nection prior to the spontaneous Lorentz violation. A In a Riemann-Cartan spacetime, the vierbein eµ and fully satisfactory example also requires the theory to be ab the spin connection ωµ represent independent degrees free of ghosts. It turns out that these conditions severely of freedom determined by the dynamics [28]. It follows restrict the possibilities for model building. that the effects of spontaneous Lorentz breaking can be General studies exist of theories 0 with a propagating strikingly different from the cases examined above. In spin connection [47, 48], including onesL with both massive particular, we focus in this section on the possibility that and massless propagating modes. However, the subset of the NG modes are absorbed into the spin connection via ghost-free models is relatively small, especially for the a Higgs mechanism. case of a massless spin connection. The total number of propagating modes in these models depends on the pres- ence of certain accidental symmetries. Our investigations A. Higgs mechanism for the spin connection reveal that the symmetry-breaking lagrangian SSB typ- ically breaks one or more of the accidental symmetriesL As in the Riemann spacetime case, we suppose that of when the tensor field acquires a vacuum expecta- L0 a tensor Tλµν··· obeys the condition (10) and acquires a tion value. This significantly complicates the analysis of nonzero vacuum value. This condition can be satisfied models, but also offers interesting new avenues by which 12 spontaneous Lorentz violation could affect the physical fields in this case include the projections 2+,2−,1+,1−, + − modes in a realistic theory. 0 , 0 , while the six auxiliary fields ω0µν contain two We are primarily interested in ghost-free lagrangians triplet projections we denote by e1+, e1−. Again, we stress 0 formed from powers of the curvature and torsion with that these projections involve the connection rather than L at most two derivatives in the equations of motion for a tensor, so the notation fails to reflect the true transfor- the vierbein and spin connection. In this case, up to 18 mation properties. For example, the 1+ projection yields ab of the 24 components of the spin connection ωµ can in a triplet of scalars under spatial rotations. principle behave as propagating degrees of freedom. The Explicit expressions for each of these projections can six components ω0ab are auxiliary fields, irrespective of be found. For example, we find any gauge choices imposed for the Lorentz symmetry or − [0+] [0 ] 1 any simplifications from accidental symmetries. ω = ωj0j , ω = 2 ǫjklωjkl, − The behavior of the 18 modes can be studied by assum- [1+] [1 ] ing a background Minkowski spacetime and linearizing ωl = ǫjklωj0k, ωk = ωjkj , e+ e− the equations of motion along the lines discussed in sec- [1 ] 1 [1 ] ωk = 2 ǫklmω0lm, ωk = ω00k, tion II. Note that, at linear order in infinitesimal quanti- + [2 ] 1 1 ties, the spin connection transforms under Lorentz trans- ω = (ωj k + ωk j ) δjkωl l, jk 2 0 0 − 3 0 formations according to − [2 ] 1 1 ωjk = 4 (ǫklmωjlm + ǫjlmωklm) 6 δjkǫlmnωnlm, ab ab ab − ω ω ∂µǫ , (52) µ → µ − (55) ab while infinitesimal diffeomorphisms leave ωµ invariant where spatial components are denoted by j,k,.... at lowest order. The vacuum solution now takes the form The two sets of projections can be related. We find

+ e+ [1 ] [1 ] 1 0lm 2 ab ωk = ωk Vk = 2 ǫk0lmM 3 Vk, Tµν··· = tµν···, eµν = ηµν , ωµ = 0 (53) − −e− − h i h i h i [1 ] [1 ] 2 ωk = ωk Tk = M00k 3 Tk, ab − − −− − in a suitable coordinate frame. The vanishing of ωµ [0+] [0 ] ab h i ω = T0, ω = V0. (56) and the invariance of ωµ under diffeomorphisms sug- − gests that the fate of the diffeomorphism NG mode is unlikely to be appreciably altered by the new role of the C. Bumblebee spin connection in the present context, and this is con- firmed in what follows. To gain further insight, we investigate a definite form for the lagrangian , namely, the simple bumblebee LSSB B. Decompositions model 1 µν µ 2 SSB = 4 eBµν B eλ(BµB b ) (57) The investigation of various models is facilitated by L − − ± introducing two different decompositions of the 24 fields with a Lagrange-multiplier potential freezing any non- ωλµν . The first is a decomposition according to Lorentz NG modes. In a Riemann-Cartan spacetime, the field indices, strength Bµν in the kinetic term of this theory is defined in Eq. (33). Its expression in terms of the vierbein and 1 ωλµν = Aλµν + Mλµν + (ηλµTν ηλν Tµ), (54) spin connection is 3 − b a b a where Aλµν is totally antisymmetric, Mλµν has mixed Bµν = (e ω e ω )ba. (58) µ ν b − ν µ b symmetry, and Tµ is the trace piece. The antisymmet- λµν ric components define a dual Vκ = ǫκλµν A /2 that Note that this form reduces to Eq. (47) in the limits of has four independent components. The mixed compo- Riemann and Minkowski spacetimes, for which the spin nents Mλµν satisfy eight identities, which can be written connection is given in terms of derivatives of the vierbein ν M = 0 and Mλµν Mνµλ = Mµλν , and they therefore by Eq. (45). νλ − contain sixteen degrees of freedom. The trace Tµ con- When Bµν is squared to yield the kinetic term, a tains the remaining four degrees of freedom. The reader quadratic terms in ωµ b appear in the lagrangian SSB. is cautioned that Eq. (54) is not a Lorentz-irreducible For example, for a Minkowski background we find L decomposition in the usual sense because the field being 1 µν decomposed is a connection rather than a tensor. K eBµν B L ≡ − 4 1 µσν νσµ ρ The second useful decomposition involves the spin- (ωµρν ωνρµ)(ω ω )b bσ. (59) P 4 parity projections J of the fields ωλµν . These are par- ≈ − − − ticularly appropriate for the case of timelike Lorentz vi- The appearance of these quadratic terms again suggests olation induced by tµν···, such as a bumblebee vacuum that a Higgs mechanism can occur involving the absorp- value of the form bµ = (b, 0, 0, 0). The 18 dynamical tion of the NG modes by the spin connection. 13

In terms of the Lorentz decomposition in the previous The idea behind the second model is to start with a µ subsection, the kinetic term K for B becomes special theory 0 in which accidental symmetries exclude L all propagatingL physical modes, but chosen such that 2 µ ν µ ν 1 ρ σµν K (bµb Vν V bµV bν V ) M M bρbσ L ≈ 9 − − 4 µν physical propagating modes emerge when the lagrangian 1 µ ν µ ν 1 ρ σλµ ν (bµb Tν T bµT bν T )+ ǫλµνρV M b bσ SSB triggering spontaneous Lorentz violation is added. − 18 − 3 L 1 λ µ ν ν µ The appearance of the physical modes via this ‘phoenix’ M (b T b T )bλ. (60) − 6 µν − mechanism can be traced to the breaking of some acci- µ dental symmetries of by . This result holds for any vacuum value b , but its phys- L0 LSSB ical interpretation can be involved in the general case. A number of models in which all modes are auxiliary For the special case of timelike Lorentz violation in- or gauge are known [48]. Here, we consider one explicit example, with lagrangian given by duced by a vacuum expectation value bµ = (b, 0, 0, 0), P the J decomposition provides a more convenient ex- 1 µν 1 νµ , = Rµν R Rµν R , (64) pression. With this assumption, we find L0 2 2 − 2

+ + e− e− where R is the Ricci tensor in Riemann-Cartan space- = 1 b2ω[1 ]ω[1 ]j + 1 b2ω[1 ]ω[1 ]j . (61) µν LK − 2 j 2 j time. Since our focus is on the spin connection, we re- We see that this expression contains an apparent physi- strict attention for simplicity to solutions in background cal mass term for the 1+ and a wrong-sign mass term for Minkowski spacetime. With this choice, the vierbein dis- the e1−. Since the e1− is an auxiliary field, it cannot prop- appears from the linearized theory, so the spin connection agate independently. However, the 1+ is an independent is the only relevant dynamical field. dynamical field, so interpreting its apparent mass term The unbroken lagrangian can be written in terms of requires a study of the dynamical content of . the Lorentz decomposed fields as L0 1 µν 1 µν 1 ρ σµν , = Fµν F Gµν G + ∂ρM ∂σM L0 2 9 − 9 4 µν 1 1 ρσ λµν D. Illustrative models (Fµν + ǫµνρσ G )∂λM . (65) − 3 2 Next, we present three different sample models , all The corresponding equations of motion are L0 containing dynamical terms for the spin connection, to σ 2 1 ρσ ∂λ[∂ Mσµν (Fµν + ǫµνρσG )]=0, (66) illustrate some of the possible effects and issues emerg- − 3 2 µ 3 µ σ ing from the presence of the Lorentz-breaking term SSB. ∂ F = ∂ ∂ M , (67) L µν 2 σµν In the first model, denoted 0,1, ghosts are present but µ 3 σ λ ρµ L ∂ Gµν = ǫµνσρ∂ ∂ M . (68) an analysis shows that a Higgs mechanism occurs when − 4 λ is added. The second, , initially has only aux- SSB 0,2 In these equations, F = ∂ T ∂ T and G = ∂ V Liliary or gauge degrees of freedom,L but the addition of µν µ ν ν µ µν µ ν ∂ V are the field strengths for−T and V , respectively.− the Lorentz-violating term breaks some accidental ν µ µ ν SSB A cursory inspection might suggest that this theory symmetries and hence causesL some modes to propagate. has at least two sets of massless fields, T and V , which The third model, , is ghost free and has a massless µ ν 0,3 correspond to the 1+ and the 1− modes in the J P de- propagating spin connection.L composition. However, there are a number of accidental The lagrangian for the first example is symmetries in this theory associated with the projection 1 λκµν operators for the 2+, 2−, 0+, and 0− fields. These and 0,1 = Rλκµν R . (62) L 4 Lorentz transformations can be used to remove all phys- To lowest order in the spin connection, the curvature ten- ical propagating degrees of freedom [48]. In particular, − sor becomes Rλκµν ∂κωλµν ∂λωκµν . In this model, all the 1 mode can be gauged away using only rotations, ≈ − + the fields ωλµν with λ = 0 propagate as massless modes. while the 1 mode can be gauged away using only boosts: However, when resolved6 into J P projections, the second- − − [1 ] [1 ] k derivative terms in the equations of motion for the even- ωj ωj ∂ εjk and odd-parity states have opposite signs, so the theory → − [1+] [1+] lm contains ghosts. When is combined with , the ω ω + ǫ ∂lε0m. (69) 0,1 SSB j → j j0 linearized equations of motionL become L The net result is that the Lorentz-invariant theory (64) ρ ρ 1 σ ∂ρ∂ ωλµν ∂λ∂ ωρµν = (ωλσν ωνσλ)bµb has no physical content. − − 2 − 1 σ Suppose now the term SSB in Eq. (61) for the case + 2 (ωλσµ ωµσλ)bν b . (63) L − of a timelike vacuum expectation value bµ is added to These 24 equations can be diagonalized to determine the the lagrangian (64). This spontaneously breaks boosts nature of the modes in the combined theory, and we find while maintaining rotation symmetry. The 1− mode can that among the propagating modes is a massive one. This still be gauged away via rotations, but the 1+ mode can confirms the existence of a Higgs mechanism for the spin no longer be removed using boosts and so might be ex- connection in this model. pected to propagate as a massive mode. However, the 14 mass term in SSB also affects the structure of the gauge and SM fields are included in the gravitational couplings and auxiliaryL fields in the theory by breaking some of of the Lorentz- and CPT-violating SME in Riemann- the accidental symmetries, so other field combinations Cartan spacetime [5], which therefore provides the ap- now become physical. We find two such massless modes, propriate framework for investigating phenomenological involving superpositions of the J P projections. implications of these models. For our third example, we take for 0 a ghost-free model with a massless propagating spinL connection. A general analysis under the assumption of Lorentz invari- VII. SUMMARY ance finds only four ghost-free possibilities [48]. All share the property of the previous example that the propagat- In this paper, we have examined the fate of the Nambu- ing modes consist of mixtures of J P projections. In two Goldstone modes when Lorentz symmetry is sponta- models, the massless propagating mode incorporates con- neously broken. The analysis is performed in the con- tributions from the 1+ projections, while in the other two text of the vierbein formalism, which is well suited for it includes contributions from the 1− projections. It is this purpose because it admits a clear separation be- therefore of interest to adopt for either of the first 0 tween local Lorentz and coordinate frames on the space- two models and investigate the effectL on the propagating time manifold. Within this formalism, we have demon- modes of adding the Lorentz-violating term . SSB strated in section II that spontaneous particle Lorentz As an explicit example, consider the lagrangianL [48] violation is accompanied by spontaneous particle diffeo- µν 1 2 morphism violation and vice versa, and that up to 10 0,3 = Rµν R R . (70) L − 3 NG modes can appear. These modes can naturally be It turns out that the propagating modes of this Lorentz- matched to those 10 of the 16 modes of the vierbein that invariant model are a mixture of 1+ and 2+ projections. in a Lorentz-invariant theory are gauge degrees of free- + In contrast, SSB contains quadratic terms for the 1 and dom. This match provides further evidence for the value the auxiliaryL1˜− states. In the theory resulting from the of the vierbein formalism in studies of spontaneous viola- combination of the two, the nature of the modes can be tions of spacetime symmetries. We have also provided a determined by diagonalizing the 24 linearized equations generic treatment for background Minkowski spacetimes. for the spin connection. We find that the propagation The fate of the NG modes is found to depend both on of the massless modes is altered, but there is no massive the spacetime geometry and also on the dynamics of the propagating 1+. The incompatibility between the mix- tensor field triggering the spontaneous violation of local P ture of J states appearing in , and that appearing in Lorentz and diffeomorphism symmetries. L0 3 SSB prevents the occurrence of a clean Higgs mechanism As illustrative models for the analysis, we have adopted Lfor the J P modes in this example. a general class of bumblebee models, involving vacuum In the context of these ideas, a number of issues of in- values for a vector field that break some of the local terest remain open for future investigation. Studies of the Lorentz and diffeomorphism symmetries. Some proper- large variety of possible Lorentz-invariant lagrangians 0 ties of these models have been presented in section III, could lead to additional features beyond those identifiedL where projectors are constructed that permit separation in the three examples above. It would also be of inter- of the Lorentz and diffeomorphism NG modes. est to explore more explicitly the effects of lightlike and In the later sections of this work, we have studied the P spacelike bµ in the Riemann-Cartan spacetimes. The J behavior of the NG modes in Minkowski, Riemann, and decomposition is less appropriate for these cases, so alter- Riemann-Cartan spacetimes. Each of these offers dis- native decompositions with respect to the corresponding tinctive general features, which can be illustrated within little group are likely to be useful. Different choices for bumblebee models. In Minkowski and Riemann space- SSB, including ones in which the spontaneous Lorentz times, Lorentz NG modes exist that can propagate as Lviolation involves one or more tensor fields, can also be massless modes, with effective lagrangians containing the expected to affect the dynamics of the NG modes. From Maxwell and Einstein-Maxwell theories in axial gauge. a broader theoretical perspective, the incorporation of Suitable bumblebee models thereby provide dynamical Lorentz violation opens an arena for the search for ghost- methods of generating the photon as a Nambu-Goldstone free theories with dynamical curvature and torsion. boson for spontaneous Lorentz violation. Various possi- Various implications for phenomenology in the context bilities exist for experimental signals in these models, in- of Riemann-Cartan spacetime also merit exploration. cluding both unconventional Lorentz-invariant couplings The scale of the emergent mass in the models consid- and Lorentz-breaking couplings in the matter and grav- ered here is set by b2. Even if this is of order of itational sectors of the SME. In Riemann-Cartan space- the Planck mass, the existence of fields with Lorentz- times, the interesting possibility exists that the spin violating physics could have effects on cosmology and connection could absorb the propagating NG modes in in regions with strong gravitational fields such as black a gravitational version of the Higgs mechanism. This holes. The couplings to other known fields also merit unique feature of gravity theories with torsion may offer attention and could lead to interesting signals for exper- another phenomenologically viable route for constructing iments. All relevant terms associated with gravitational realistic models with spontaneous Lorentz violation. 15

Acknowledgments the National Aeronautics and Space Administration un- der grant numbers NAG8-1770 and NAG3-2194, and by This work was supported in part by the Department the National Science Foundation under grant number of Energy under grant number DE-FG02-91ER40661, by PHY-0097982.

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