Effective Theory Approach to the Spontaneous Breakdown of Lorentz Invariance
Cristian Armendariz-Picon, Alberto Diez-Tejedor, Riccardo Penco
aa r X i v : . [ h e p - ph ] A p r Effective Theory Approach to the Spontaneous Breakdown ofLorentz Invariance
Cristian Armendariz-Picon, Alberto Diez-Tejedor, and Riccardo Penco Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA Instituto de Ciencias Nucleares, Universidad NacionalAut´onoma de M´exico, M´exico D.F. 04510, M´exico (Dated: October 22, 2018)
Abstract
We generalize the coset construction of Callan, Coleman, Wess and Zumino to theories in whichthe Lorentz group is spontaneously broken down to one of its subgroups. This allows us to writedown the most general low-energy effective Lagrangian in which Lorentz invariance is non-linearlyrealized, and to explore the consequences of broken Lorentz symmetry without having to makeany assumptions about the mechanism that triggers the breaking. We carry out the constructionboth in flat space, in which the Lorentz group is a global spacetime symmetry, and in a generallycovariant theory, in which the Lorentz group can be treated as a local internal symmetry. As anillustration of this formalism, we construct the most general effective field theory in which therotation group remains unbroken, and show that the latter is just the Einstein-aether theory. . INTRODUCTION It is hard to overemphasize the central role that the Lorentz group plays in our presentunderstanding of nature. The standard model of particle physics, for instance, consists of allrenormalizable interactions invariant under Lorentz transformations and its internal symme-try gauge group, which act on the matter fields of the theory. While most standard modelextensions alter either its field content or gauge group, they rarely drop Lorentz invariance.Of course, such a reluctance has a well-established observational support. Elementary parti-cles appear in (irreducible) representations of the Lorentz group, and their interactions seemto be well described by Lorentz-covariant laws. Lorentz-breaking operators in the standardmodel of particle physics were first considered by Colladay and Kostelecky [1], and Colemanand Glashow [2]. Experimental and observational constraints on such operators are so strin-gent [3] that it is safe to assume that any violation of Lorentz invariance in the standardmodel must be extremely small.The status of the Lorentz group in theories of gravity is somewhat different. Becausethe group of diffeomorphisms does not admit spinor representations, in generally covarianttheories the Lorentz group is introduced as a local internal symmetry. Thus, in gravitationaltheories one formally deals with two distinct groups of transformations: diffeomorphisms andlocal Lorentz transformations. Even in the context of generally covariant theories, it is thusnatural to ask and inquire whether the gravitational interactions respect Lorentz invariance,and what constraints we can impose on any Lorentz-violating gravitational interactions.To date, experimental bounds still allow significant deviations from Lorentz invariance ingravitational interactions [3–5].In this article we mainly explore some consequences of broken Lorentz invariance in gener-ally covariant theories. We have in mind here theories that admit a generally covariant, butnot a Lorentz invariant formulation. Generically, the breaking of diffeomorphism invariancein non-trivial backgrounds also leads to Lorentz symmetry breaking [6–8], but the nature ofthe breaking in these cases is quite different from what we explore here, and indeed leadsto different phenomenology. The spontaneous breaking of Lorentz symmetry (in our sense)has been mostly explored by means of particular models in which vector fields [9–15] orhigher-rank tensors [16] develop a non-vanishing vacuum expectation value. In this articlewe follow a general approach and address consequences that merely follow from the symme-2ry breaking pattern, regardless of any specific model of Lorentz symmetry breaking. Such amodel-independent approach was first introduced by Weinberg to describe the breakdown ofchiral invariance in the strong interactions [17], and was subsequently generalized by Callan,Coleman, Wess and Zumino to the breaking of any internal symmetry group down to anyof its subgroups [18, 19]. Their approach was further broadened to the case of spontaneousbreaking of space-time symmetries [20–23] down to the Poincar´e group. Here, we extend allthese results to the case in which the Lorentz group itself is broken down to one of its sub-groups. A naive application of Goldstone’s theorem then implies the existence of masslessGoldstone bosons, which may in principle participate in long-ranged interactions and alterthe Newtonian and post-Newtonian limits of the theory. Equivalently, we may also think ofthese additional fields as additional polarizations of the graviton.These considerations are not a purely academic exercise. Motivated by cosmic accel-eration, several authors have devoted substantial attention to massive theories of gravity[24–27] and other modifications [28–30], even though the distinction between modificationsof gravity and theories with additional matter fields is often blurry. Within the last class,several groups have studied the cosmological dynamics induced by vector fields with non-zeroexpectation values (see for instance [31–38]), though the spontaneous breaking of Lorentzinvariance has not been the primary focus of their investigations. From this perspective,broken Lorentz invariance offers a new framework to study modifications of gravity, andmay cast some light onto theories that have been already proposed.The plan of the article is as follows. In Section II we generalize the coset constructionof Callan, Coleman, Wess and Zumino to theories in which the group of global Lorentztransformations is spontaneously broken. In Section III we briefly review the role of theLorentz group as an internal local symmetry group in generally covariant theories, andstudy the breaking of Lorentz invariance in this framework. Section IV is devoted to anillustration of our formalism in theories in which the rotation group remains unbroken. Wesummarize our results in Section V.
II. BROKEN LORENTZ INVARIANCE
In this section we explore how to construct theories in which the global symmetry ofthe action under a given Lorentz subgroup H is manifest (linearly realized), but the global3ymmetry under the “broken part” of the Lorentz group L ↑ + is hidden (non-linearly realized).After a brief review of the Lorentz group, we first consider how to parametrize the brokenpart of the Lorentz group, that is, the coset L ↑ + /H . The corresponding parameters are theGoldstone bosons of the theory. We define the action of the full Lorentz group on this set ofGoldstone bosons in such a way that they transform linearly under H , but non-linearly under L ↑ + /H . Initially, the transformation that we consider is internal, that is, does not affect thespacetime coordinates of the Goldstone bosons. This is the way the Lorentz group acts ingenerally covariant theories, which we discuss in Section III, but it is not the way it actsin theories in Minkowski spacetime, in which the Lorentz group is a spacetime symmetry.Hence, we subsequently extend our realization of the Lorentz group to a set of spacetimetransformations.In order to write down Lorentz-invariant theories in which the symmetry under H ismanifest, we need to come up with appropriate “covariant” derivatives that transform likethe Goldstone bosons themselves. As we shall see, once these covariant derivatives havebeen identified, the construction of actions invariant under the full Lorentz group becomesstraight-forward, and simply reduces to the construction of theories in which invarianceunder the linearly realized H is explicit. A. The Lorentz Group
The Lorentz group L is the set of transformations Λ ab that leave the Minkowski metric in-variant, η ab Λ ac Λ bd = η cd . Its component connected to the identity, the proper orthochronousLorentz group L ↑ + , is generated by rotations ~J and boosts ~K , with commutation relations[ J i , J j ] = iǫ ij k J k , (1a)[ J i , K j ] = iǫ ij k K k , (1b)[ K i , K j ] = − iǫ ij k J k . (1c)Any Lorentz transformation can be written as an orthochronous transformation timesa parity transformation P , time reversal T or a combination of the latter, P T . Thesetransformations define a discrete subgroup, V ≡ { , P, T, P T } , and the orthochronous group L ↑ + may be understood as the coset L ↑ + = L/V. (2)4he orthochronous group is an invariant subgroup of the Lorentz group. The elements of V define a map whose square is the identity, which also preserves the commutation relationsof the Lie-algebra of the proper orthochronous group L ↑ + , P : J i P J i P − = J i , K i P K i P − = − K i (3a) T : J i T J i T − = J i , K i T K i T − = − K i . (3b)For most of this article we are concerned with the spontaneous breaking of the properorthochronous Lorentz group L ↑ + . B. Coset Construction
Suppose now that the proper orthochronous Lorentz group L ↑ + (“Lorentz group” for short)is spontaneously broken down to a subgroup H ⊂ L ↑ + . In the simplest models of this kind, thebreaking occurs because the potential energy of a vector field has a minimum at a non-zerovalue of the field, in analogy with spontaneous symmetry breaking in scalar field theories withMexican-hat potentials. Perhaps more interesting are cases in which Lorentz invariance isbroken “dynamically,” that is, when a strong interaction causes fermion bilinears to condenseinto spacetime vectors [40–42]. This is analogous to the way in which chiral invariance isbroken in QCD. The formalism we develop here however does not depend on the actualmechanism that triggers the symmetry breaking, and only relies on the unbroken group H .Let H be the Lie algebra of H , which we assume to be semisimple. Although the Lorentzgroup is not compact, it is simple, so the Killing form ( · , · ) is non-degenerate and may beregarded as a scalar product on H . We may then uniquely decompose the Lie algebra of L ↑ + into the algebra of H and its orthogonal complement, which we denote by C , L ↑ + = H ⊕ C . (4)Hence, by definition, for any t ∈ H and any x ∈ C , ( t, x ) = 0. In the following we assumethat the set of unbroken generators t i is a basis of H , and that the set of broken generators x m forms a basis of C . In any representation, l k collectively denotes the generators of theLorentz group, k = 1 , . . . , t ∈ H , the map f t : x ∈ C 7→ [ t, x ] is linear. Moreover, for any t ′ ∈ H we have( t ′ , [ t, x ]) = ([ t ′ , t ] , x ) = 0 , (5)5here we have used the properties of the Killing form and that [ t, t ′ ] ∈ H . Therefore, f t maps C into itself. In fact, the commutator defines a homomorphism of H into the linearmaps of C . Hence, the matrices C ( t ) with elements defined by[ t, x m ] = iC ( t ) nm x n (6)provide a representation of H . In particular, equation (6) implies that, for any element ofthe unbroken group h ∈ H and for any x ∈ C , h x h − ∈ C . (7)Following the standard coset construction of Callan, Coleman, Wess and Zumino [18, 19](see [43, 44] for brief reviews), we can write down realizations of the Lorentz group, in whichany given set of fields transform in a linear representation of the unbroken group H . Forthat purpose, let us first introduce a convenient parametrization of the coset space L ↑ + /H .Any element γ ∈ L ↑ + /H can be expressed as γ ( π ) = exp( iπ m x m ) , (8)where a sum over indices in opposite locations is always implied. The fields π m = π m ( x )correspond to the Goldstone bosons of the theory. If there are M broken generators of theLorentz group, there are M Nambu-Goldstone bosons π m . We may now introduce a realization of the group L ↑ + on this set of Goldstone bosons. Bydefinition, any g ∈ L ↑ + can be uniquely decomposed into the product of an element of theunbroken group h ∈ H and an element of the coset space γ ∈ L ↑ + /H , such that g = γ h .Therefore, the product g γ ( π ) ∈ L ↑ + also has a unique decomposition g γ ( π ( x )) = γ ( π ′ ( x )) h ( π ( x ) , g ) , with γ ( π ′ ) ∈ L ↑ + /H , h ( π, g ) ∈ H. (9)Equation (9) defines a non-linear realization of the Lorentz group by mapping π into π ′ for any given g ∈ L ↑ + . Notice however that this representation becomes linear when g It is at this point where the assumption of a semisimple group becomes necessary. As an illustration ofthis point, consider the case where the unbroken group is spanned by the single generator t ≡ K + J .Then, the commutation relations (1) imply [ t, K ] = it , which is not in C . See [45] for exceptions to this argument in the case of spontaneous breaking of translations. H . In fact, because of equation (7) we must have that ¯ h γ ( π ) ¯ h − = γ ( π ′ ) forevery ¯ h ∈ H , and a comparison with equation (9) implies h ( π, ¯ h ) = ¯ h. (10)In particular, use of equations (6), (8) and (10) shows that in this case the Goldstone bosonstransform in a linear representation of the unbroken group H , h ∈ H : π m π ′ m = R ( h ) mn π n , with R (exp it ) ≡ exp [ iC ( t )] . (11)Therefore, the Goldstone bosons have the same “quantum numbers” as the broken generators x m . For a compact , connected, semi-simple Lie group G broken down to H , the uniquenessof the transformation law (9), up to field-redefinitions, was proved in [18]. C. Covariant Derivatives
Thus far, the realization of the Lorentz group that we have defined in equation (9) treatsthe Lorentz group as an internal symmetry; the spacetime arguments on both sides of theequation coincide. This is going to be useful in our discussion of the Lorentz group ingenerally covariant theories, but it is not the way the Lorentz group acts in conventionalfield theories in Minkowski spacetime, in which the Lorentz group is a group of spacetimesymmetries. Following [21, 22], we define now a non-linear realization of the Lorentz groupas a spacetime symmetry by g : γ ( π ( x )) γ ( π ′ ( x ′ )) , where g e iP µ x µ γ ( π ( x )) = e iP µ x ′ µ γ ( π ′ ( x ′ )) h ( π ( x ) , g ) . (12)This implicitly defines a realization of the Lorentz group on the coordinates x µ and the fields π ( x ). In particular, under an arbitrary element g ∈ L ↑ + , equation (12) implies g : x µ x ′ µ = Λ µν ( g ) x ν , γ ( π ( x )) γ ( π ′ ( x ′ )) = γ ( π ′ ( x )) , (13)with gP µ g − = Λ νµ ( g ) P ν and γ ( π ′ ( x )) defined in equation (9).Because we are interested in theories in which the Lorentz group is a set of global sym-metries, any action constructed from the Goldstone bosons π can only depend on theirderivatives. In order to introduce appropriate covariant derivatives, in analogy with the7onventional prescription [19], we expand an appropriately modified [21, 22] Maurer-Cartanform in the basis of the Lie algebra, Ω µ ≡ i γ − e − iP · x ∂ µ ( e iP · x γ ) ≡ e µa P a + D µm x m + E µi t i ≡ e µ + D µ + E µ , (14)which immediately implies that e µa = Λ µa ( γ ) . (15)The field e µa is the analogue of the vierbein that we shall introduce in Section III. Bothtransform similarly under the Lorentz group, and this leads to formally identical expressionsin both cases. But the reader should nevertheless realize that the “vierbein” (15) and thevierbein of Section III are actually different objects.The transformation properties of e , D and E follow from the definition (12). Under anarbitrary g ∈ L ↑ + , they transform according to g : e µ ( x ) e ′ µ ( x ′ ) = Λ µν ( g ) h ( π, g ) e ν ( x ) h − ( π, g ) , (16a) D µ ( x ) D ′ µ ( x ′ ) = Λ µν ( g ) h ( π, g ) D ν ( x ) h − ( π, g ) , (16b) E µ ( x ) E ′ µ ( x ′ ) = Λ µν ( g ) (cid:2) h ( π, g ) E ν ( x ) h − ( π, g ) − ih ( π, g ) ∂ ν h − ( π, g ) (cid:3) , (16c)where h ( π, g ) is defined in equation (9). Therefore, none of these quantities really transformscovariantly, since the spacetime index µ and the components of the different fields transformunder different group elements. To define fully covariant quantities, let us introduce theinverse of the quantity defined in equation (15), e µa = Λ aµ ( γ − ) . (17)This is indeed the (transposed) inverse of e µa because it follows from equation (15) that e µa e µb = δ ab . Then, the quantities D a ≡ e µa D µ , E a ≡ e µa E µ , (18)do transform covariantly under the Lorentz group, D a ( x ) D ′ a ( x ′ ) = Λ( h ( π, g )) ab h ( π, g ) D b ( x ) h − ( π, g ) , (19a) E a ( x ) E ′ a ( x ′ ) = Λ( h ( π, g )) ab (cid:2) h ( π, g ) E b ( x ) h − ( π, g ) − ih ( π, g ) ∂ b h − ( π, g ) (cid:3) , (19b)where ∂ a ≡ e µa ∂ µ . We identify D a with the covariant derivative of the Goldstone bosons π m , and E a with a “gauge field” that will enter the couplings between the Goldstone bosons8nd other matter fields. The transformation rules (19) are again non-linear in general, but,because of equation (10), they reduce to a linear transformation if g ∈ H . Note that under g ∈ L ↑ + , the components of the covariant derivative D a transform as g : D am ( x )
7→ D ′ am ( x ′ ) = Λ ab ( h ( π, g )) R mn D bn ( x ) , (20)where the matrix R is the one we introduced in equation (11).For specific calculations, it is often required to have concrete expressions for the covariantderivatives. It follows from the definitions (8) and (14) that D am = ∂ a π m − iπ n ( x n (4) ) ab ∂ b π m + 12 π n ∂ a π p C npm + O ( π ) , (21)where x n (4) is the fundamental (form) representation of the Lorentz generator x n , and the C npm are the structure constants of the Lie algebra H in our basis of generators. Parity and Time Reversal
In certain cases, we can also define the transformation properties of the Goldstonebosons under parity and time reversal, or, in general, under an appropriate subgroup of V ≡ { , P, T, P T } . Let V H denote the “stabilizer” of H , that is, the set of all elements v ∈ V that leave H invariant, v h v − ∈ H for all h ∈ H . This is a subgroup of V, whichmay contain just the identity, either P or T , or V itself. Because H is invariant under V H ,the latter defines an homomorphism on C by conjugation, v x m v − = V nm x n . (22)The two sets L ↑ + V H and HV H are two subgroups of L , and, by definition, HV H is asubgroup of L ↑ + V H . Thus, just as in Section II B , we may define a realization of V H (whichis now contained in L ↑ + V H ) on the coset L ↑ + V H HV H = L ↑ + H . (23)In particular, for g ∈ L ↑ + V H and γ ( π ) ∈ L ↑ + /H we set gγ ( π ) = γ ( π ′ ) h ( γ, g ) v ( γ, g ) , with h ( γ, g ) ∈ H and v ( γ, g ) ∈ V H . (24)If g ∈ L ↑ + , this definition reduces to that of equation (9). For v ∈ V H it leads to v : γ ( π ) γ ( π ′ ) = v γ ( π ) v − , (25)9hich can be extended to include the arguments of the Goldstone boson fields as before, v : γ ( π ( x )) γ ( π ′ ( x ′ )) , where v e iP · x γ ( π ( x )) v − = e iP · x ′ γ ( π ′ ( x ′ )) . (26)Under these group elements the Goldstone bosons change according to v : π m π ′ m ( x ′ ) = V mn π n ( x ) , (27)and, from equation (20), their covariant derivatives according to v : D am ( x ) D ′ am ( x ′ ) = V ab V mn D bn ( x ) , (28)where vP a v − = V ab P b . D. Invariant Action
If we are interested in the low-energy limit of theories in which Lorentz-invariance isbroken, we can restrict our attention to their massless excitations. This is a restatementof the Appelquist-Carazzone theorem [39], though the latter has been actually proven onlyfor renormalizable Lorentz-invariant theories in flat spacetime. Typically, massless fieldsare those protected by a symmetry, and always include the Goldstone bosons, since invari-ance under the broken symmetry prevents them from entering the action undifferentiated.Therefore, the low-energy effective action of any theory in which Lorentz invariance is bro-ken must contain the covariant derivatives of the Goldstone bosons. To leading order in thelow-energy expansion, we can restrict our attention to the minimum number of spacetimederivatives, namely, two.The tensor product representation in equation (20) under which the covariant derivativestransform is in general reducible. Let Λ ⊗ R = ⊕ i R ( i ) be its Clebsch-Gordan series, and let D ( i ) be the linear combination of covariant derivatives that furnishes the i -th irreduciblerepresentation. Some of these representations may be singlets, and we shall label themby s . Because the unbroken group is not necessarily compact, the non-trivial irreduciblerepresentations are generally not unitary. In any case, if G ( i ) is invariant under the i -threpresentation of the unbroken group H , i.e. R ( i ) T G ( i ) R ( i ) = G ( i ) , then the Lagrangian L = X s F s D ( s ) + X i F i D ( i ) T G ( i ) D ( i ) (29)10ransforms as a scalar under the Lorentz-group L ↑ + . Here, F s and F i are free parametersin the effective, which remain undetermined by the symmetries of the theory. In orderto construct a Lorentz-invariant action, we just need a volume element that transformsappropriately under our realization of the Lorentz group. This is in general given by [22] d V ≡ d x det e µa , (30)which, because of equation (15), results in d V = d x . (Inside the determinant, the vierbeinshould be regarded as a 4 × µ and columns labeled by a .)The functional S = Z d V L (31)is then invariant under the action of the Lorentz group defined by equation (12). E. Couplings to Matter
The formalism can be also extended to capture the effects of Lorentz breaking on thematter sector. As mentioned above, at low-energies we can restrict our attention to massless(or light) fields, which are typically those that are prevented from developing a mass by asymmetry like chiral or gauge invariance. We consider couplings to the graviton in SectionIII.Let ψ be any matter field that transforms under any (possibly reducible) representation R ( h ) of the unbroken group H , with generators t i . Let us now define the transformationlaw under the full Lorentz group to be [18] g : ψ ( x ) ψ ′ ( x ′ ) = R ( h ( π, g )) ψ ( x ) , (32)where x ′ and h ( π, g ) is given in equation (12). We can also construct covariant derivativesunder the Lorentz group by setting, D a ψ ≡ e µa [ ∂ µ ψ + i E µ ψ ] = ∂ a ψ + i E a ψ, (33)where E µ is defined in equation (14). The covariant derivative transforms just as the fielditself, under a representation of the same group element, g : D a ψ ( x ) D ′ a ψ ′ ( x ′ ) = Λ( h ( π, g )) ab R ( h ( π, g )) D b ψ ( x ) . (34)11herefore, any Lagrangian built out of d V , ψ , D a ψ and D am that is invariant under theunbroken group H is then invariant under the full Lorentz group.With these ingredients we could develop a formulation of the standard model in whichthe Lorentz group is spontaneously broken to any subgroup. If the unbroken group is trivial, H = 1, this construction would be analogous to the standard model extension consideredby Colladay and Kostelecky [1]. Our article mainly focuses on the general formalism ofbroken Lorentz invariance, so we shall not carry out this program here. For the purpose ofillustration however, and in order to establish the connection to previous work on the subject,let us consider a formulation of QED (quantum electro-dynamics) in which the Lorentz groupis completely broken. For simplicity we consider a theory with a single “spinor” ψ α of charge q coupled to a “photon” A a . We use quotation marks because, according to (32), we assumethat under the (completely) broken Lorentz group both fields are invariant. On the otherhand, we require that the theory be invariant under gauge transformations, that is, wedemand invariance under ψ α → e iqχ ψ α , A a → A a + ∂ a χ, (35)where χ is an arbitrary spacetime scalar. If the Lorentz group is broken down to H = 1,there are six Goldstone bosons in the theory, and γ becomes γ ≡ exp( iπ k l k ), which, underthe Lorentz group transforms as g : γ γ ′ ( x ′ ) = g γ ( x ) . Following (33) we introduce nowthe covariant derivatives D a A b ≡ Λ( γ − ) aµ ∂ µ A b , D a ψ α ≡ Λ( γ − ) aµ ∂ µ ψ α , (36)which by construction are Lorentz-invariant (if the Lorentz group is completely broken, E µ ≡ F ab ≡ D a A b − D a A b , ∇ a ψ α ≡ ( D a − iqA a ) ψ α . (37)Any gauge invariant combination of these elements, such as L QED = M abcd F ab F cd + N αβ ψ † α ∇ a ψ β + P αβ ψ † α ψ β , (38)is also Lorentz invariant (for simplicity, we have not written down all the terms compatiblewith the two symmetries). In equation (38), the dimensionless matrices M , N and P are12 rbitrary , up to the restrictions imposed by permutation symmetry and hermiticity. TheLagrangian (38) is thus the analogue of the extension of QED described in [1]. From aphenomenological perspective, its coefficients can be regarded as quantities to be determinedor constrained by experiment, as in the standard model extension of [1]. But of course, asopposed to the latter, the Lagrangian (38) contains couplings to the Goldstone bosons, andshould be supplemented with the Goldstone boson Lagrangian, which for a trivial H is L π = G am D am + F mnab D am D bn , (39)where D am is given in equation (14), and m, n = 1 , . . . ,
6. As we shall see in the next section,in a gravitational theory these covariant derivatives should be included in the Lagrangian too,but in that case they reduce to appropriate components of the spin connection. Note thatin our conventions the Goldstone bosons are dimensionless. Thus the coefficients in G havemass dimension three, and those in F mass dimension two. In theories in which an internalsymmetry is spontaneously broken, Lorentz invariance and invariance under the unbrokengroup severely restrict the possible different mass scales appearing in the Lagrangian. In ourcase however, the values of G and F are (up to symmetry under permutations) completelyarbitrary. In particular, the unbroken symmetries do not imply that there is a single energyscale at which the Lorentz group is broken.The obvious problem with this approach is that the Lorentz group seems to be an un-broken symmetry in the matter sector. A generic Lagrangian like (38), constructed out ofthe standard model fields ψ , their covariant derivatives D a ψ and the covariant derivativesof the Goldstones D am would clearly violate Lorentz invariance, in flagrant conflict withexperimental constraints [3]. Thus, we are forced to assume that these “Lorentz-violating”terms are sufficiently suppressed, which in our context requires specific relations betweenthe coefficients in the effective Lagrangian.To illustrate this point, let us briefly discuss how to construct scalars under linearlyrealized Lorentz transformations out of the ingredients at our disposal, namely, ψ, D a ψ and D am . Imagine that the matter fields ˜ ψ actually fit in a representation of the Lorentz group R ( g ). It is then more convenient to postulate that under the full Lorentz group, these fieldstransform as g : ˜ ψ ˜ ψ ′ ( x ′ ) = R ( g ) ˜ ψ ( x ) . (40)13hen, any Lagrangian that is invariant (a scalar) under global Lorentz transformations, L inv [ ˜ ψ, ∂ µ ˜ ψ ] = L inv [ R ( g ) ˜ ψ, R ( g )Λ( g ) µν ∂ ν ˜ ψ ] , g ∈ L ↑ + , (41)is clearly invariant under the unbroken subgroup H of global transformations, and can thusbe part of the effective Lagrangian in the spontaneously broken phase. Note that theseLorentz invariant terms would not contain any couplings to the Goldstone bosons. Butgiven the transformation law (40) we can also construct a new quantity that transformsunder the non-linear realization of the Lorentz group (32), ψ ≡ R ( γ − ) ˜ ψ, (42)and whose covariant derivative can again be defined by equation (33). In this case, however,the field ψ is to be understood simply as a shorthand for the right hand of equation (42),which contains the Goldstone bosons γ ( π ). Given any Lagrangian L break that is invariantunder the linearly realized unbroken group H , but not invariant under linear representationsof the full Lorentz group L ↑ + , L break [ ψ, ∂ µ ψ ] = L break [ R ( h ) ψ, R ( h )Λ( h ) µν ∂ ν ψ ] , h ∈ H, (43)we can then construct further invariants under Lorentz transformations, L break [ R ( γ − ) ˜ ψ, D µ ( R ( γ − ) ˜ ψ )] . (44)Here, the appearance of the Goldstone bosons in those terms that violate the full Lorentzsymmetry is manifest.It seems now that the Lagrangians (41) and (44) do not fit into the general prescriptionto construct invariant Lagrangians that we described at the beginning of this subsection,but this is just an appearance. Suppose we perform a field redefinition R ( γ − ) ˜ ψ → ψ , andassume that the new field ψ transforms as in equation (32). This field redefinition turns theLagrangian in equation (44) into L break [ ψ, D µ ψ ], and takes the Lagrangian (41) into L inv [ ψ, D µ ψ + iD µm x m ψ ] . (45)Both Lagrangians are invariant under the linearly realized symmetry group H (and the non-linearly realized Lorentz group L ↑ + ), and both are solely constructed in terms of ψ, D µ ψ and D am . 14f course, a general Lagrangian invariant under H will have the form of equation (45) onlyfor very particular choices of the coefficients that remain undetermined under the unbrokensymmetry. From the point of view of the effective theory, this particular choice cannotbe explained, though it is certainly compatible with the symmetries we are enforcing. Toaddress it we would have to rely on specific models. Say, if Lorentz symmetry is brokenin a hidden sector which is completely decoupled from the standard model, the breakingin the hidden sector should not have any impact on the visible sector. But of course, thetwo sectors must couple at least gravitationally. Then, if the scale of Lorentz-symmetrybreaking is sufficiently small compared to the Planck mass, we expect a double suppressionof Lorentz-violating terms in the matter sector: from the weakness of gravity, and from thesmallness of the symmetry breaking scale. We defer the discussion of gravitation to thenext section. Radiative corrections to Lorentz-violating couplings in the matter sector ofEinstein-aether models [12] have been calculated in [46]. F. Broken Rotations
As an example of the formalism discussed so far, we shall briefly study a pattern ofsymmetry breaking in which the unbroken group H is non-compact. This is an interestingcase since, for internal non-compact symmetry groups, the theory contains ghosts in thespectrum of Goldstone bosons [41, 43]. We show that, instead, it is certainly possibleto have a well-behaved spectrum in a theory in which the Lorentz group is broken downto a non-compact subgroup. We consider the widely-studied case of unbroken rotations, H = SO (3), in Section IV.Suppose that the Lorentz group L ↑ + is broken down to the group of transformationsthat leave the vector field A µ = (0 , , , F ) invariant. This breaking pattern was studiedin references [40, 41], in which the “photon” of electromagnetism is identified with theGoldstone bosons associated with the breaking. The Lie algebra of the unbroken group H is then H = Span { K , K , J } , (46)which is simple, and isomorphic to the Lie algebra of the group of Lorentz transformationsin three-dimensional spacetime so (1 , C = Span { J , J , K } . (47)15ecause dim( C ) = 3, there are three Goldstone bosons in the theory. It follows from thecommutation relations (1) and equations (6) and (11) that π m ≡ ( π , π , π ) transforms likea Lorentz three-vector. It is thus convenient to let m run from 0 to 2 and identify π ≡ π .The covariant derivative D am transforms in a reducible representation of H = SO (1 , D m ≡ D m (48)is an SO (1 ,
2) three-vector. The remaining irreducible spaces are spanned by the scalar ϕ ,the vector a mn and the tensor s mn defined by ϕ ≡ D mm , a mn ≡
12 ( D mn − D nm ) , s mn ≡
12 ( D mn + D nm ) − ϕ η mn , (49)where indices are raised with the (inverse) of the Minkowski metric in three dimensions, η mn = diag( − , ,
1) and m = 0 , ,
2. Scalar invariants are constructed then by appropriatecontraction of indices, L π = G ϕ ϕ + F ϕ ϕ + F D D m D m + F a a mn a mn + F ǫ ǫ mnp a mn D p + F s s mn s mn . (50)For simplicity, let us now consider the case where G ϕ = 0. Because to lowest orderin the Goldstone bosons D mn = ∂ m π n + · · · , inspection of (50) reveals the lower-dimensionanalogue of a generalized vector field theory in which the vector field consists of the Goldstonebosons π m . This analogy can be further strengthened by dimensionally reducing the fourdimensional theory from four to three spacetime dimensions. Expanding the Goldstonebosons in Kaluza-Klein modes π m ( t, x, y, z ) = X k z π ( k ) m ( t, x, y ) e ikz , (51)and inserting into the action we obtain, to quadratic order, S = X k S k , where S k [ π ( k ) m ] = Z dt d x (cid:20) F a + F s a mn a mn + (cid:18) F ϕ + 2 F s (cid:19) ( ∂ m π m ) + F D k π m π m (cid:21) . (52)Note that we have suppressed the index k of the Kaluza-Klein modes on the right handside of equation (52). The Kaluza-Klein modes π ( k =0) are massless, and transform likean SO (1 ,
2) vector. They can be thought of as the Goldstone bosons associated with thebreaking L ↑ + ∼ SO (1 , → SO (1 ,
2) induced by the compactification.16he spectrum of excitations in the theory described by (52), and the conditions thatstability imposes on the free parameters F a , F ϕ , F s and F D can be derived by relying on thesimilarity of the action S k with the four-dimensional models analyzed in [38]. Since theirstability analysis does not crucially depend on the dimensionality of spacetime, their resultsalso apply in the case at hand. Following the analysis in Section V of [38] we find:i) If both F a + F s and F ϕ + 2 F s / SO (1 ,
2) vector and an SO (1 ,
2) scalar. There is always a ghost at high spatialmomenta ( k x + k y ≫ k ).ii) For F a + F s = 0 the theory is stable if F ϕ + 2 F s / > F D <
0. The spectrumconsists of a scalar under SO (1 , F D k = 0, there are no dynamical fields in thespectrum.iii) For F ϕ + 2 F s / SO (1 ,
2) vector, with two polarizations.The theory is stable for F a + F s > F D <
0. If F D k = 0 the vector is massless,with only one polarization. This last cast corresponds to electrodynamics in threespacetime dimensions.Hence, as we anticipated there are theories in which the low-energy theory is free of ghosts.These are however non-generic, in the sense that they require the coefficients of certain termsotherwise allowed by Lorentz invariance to be zero. III. COUPLING TO GRAVITY
In the previous section we have explored spontaneous symmetry breaking of Lorentzinvariance in Minkowski spacetime, in which the Lorentz group is a global symmetry. Thoughthis approach should appropriately capture the local physical implications of the breakingin non-gravitational phenomena, it certainly does not suffice to study arbitrary spacetimebackgrounds, or the gravitational interactions themselves. There is just one difference between the four-dimensional and the three-dimensional case: In four dimen-sions, the vector sector (under spatial rotations) contains two modes, while in three dimension the vectorsector (under spatial rotations) only contains one mode.
17n order to couple gravity to the Goldstone bosons, it is convenient to exploit the formalanalogy between gravity and gauge theories. For that purpose, one introduces the Lorentzgroup L ↑ + as an “internal” group of symmetries, in addition to the symmetry under generalcoordinate transformations [47]. In theories with fermions (such as the standard model) thisis actually mandatory, as the group of general coordinate transformations does not admitspinor representations. In the first part of this section we review the standard formulationof general relativity as a gauge theory of the Lorentz group [48]. In the second part we thenextend this standard formulation to theories in which Lorentz invariance is broken. Readersalready familiar with the vierbein formalism can skip directly to Subsection III B. A. General Formalism
In any generally covariant theory defined on a spacetime manifold in which the metrichas Lorentzian signature, and regardless of whether the Lorentz group is spontaneouslybroken or not, it is always possible to introduce a vierbein, an orthonormal set of formsˆ e ( a ) = e µa dx µ in the cotangent space of the spacetime manifold, g µν e µa e ν b = η ab . (53)Greek indices µ, ν, . . . now denote cotangent space indices in a coordinate basis, while latinindices a, b, . . . label the different vectors in the orthonormal set. Thus, the order of thevierbein indices is important. The first one is always a spacetime index, and the secondone is always a Lorentz index. Spacetime indices are raised and lowered with the metric ofspacetime, and Lorentz indices are raised and lowered with the Minkowski metric. Undercoordinate transformations, the vierbein e µa transforms like a vector,diff : e µa ( x ) e ′ µa ( x ′ ) = ∂x ν ∂x ′ µ e νa ( x ) . (54)The freedom to choose a vierbein whose sixteen components satisfy the orthonormalitycondition (53) does not add anything to the original ten metric components if the theoryremains invariant under the six parameter group of local Lorentz transformations, g ( x ) : e µa ( x ) e ′ µa ( x ) = Λ ab ( g ) e µb ( x ) , g ( x ) ∈ L ↑ + . (55)Note that this transformation does not affect the coordinates of the vectors, that is, theLorentz group acts as an “internal” symmetry.18he derivatives of the vierbein do not transform covariantly under these local Lorentztransformations. We thus introduce the spin connection ω µ , which plays the role of thegauge field of the Lorentz group. Let l k , k = 1 , . . .
6, denote the generators of the Lorentzgroup L ↑ + (in any representation), and let us define the components of the spin connectionby ω µ ≡ ω µk l k , (56)which transforms like a one-form under general coordinate transformations,diff : ω µ ( x ) ω ′ µ ( x ′ ) = ∂x ν ∂x ′ µ ω ν ( x ) . (57)In complete analogy with gauge field theories, let us assume that under local Lorentz trans-formations the spin connection transforms as g ( x ) : ω µ ( x ) g ω µ ( x ) g − + g ∂ µ g − . (58)In that case, it is then easy to verify that the covariant derivative ∇ µ e νa = ∂ µ e νa − Γ ρνµ e ρa + ω µk [ l k (4) ] ab e ρb (59)transforms covariantly both under coordinate and local Lorentz transformations. Here,Γ µνρ are the Christoffel symbols associated with the spacetime metric g µν , and the l k (4) arethe Lorentz group generators in the fundamental representation, under which the vierbeintransforms. In our convention, these matrices are purely imaginary. Similarly, given anymatter field ψ that transforms as a scalar under diffeomorphisms, and in a representationof the Lorentz group with generators l k , we can construct its covariant derivative by ∇ µ ψ ≡ ∂ µ ψ + ω µ ψ, (60)which also transforms covariantly both under diffeomorphisms and local Lorentz transfor-mations.In any generally covariant theory defined on a Riemannian spacetime manifold, the co-variant derivative is compatible with the metric, that is, ∇ µ g νρ = 0. Moreover, because the To recover expressions fully analogous to those found in gauge theories, the reader should replace ω µ by − i ω µ . ∇ ν e µa = 0 . (61)Equation (59), in combination with equation (61) allows us to express the spin connectionin terms of the vierbein, ω µk [ l k (4) ] ab = 12 [ e ν a ( ∂ µ e ν b − ∂ ν e µb ) − e ν b ( ∂ µ e νa − ∂ ν e µa ) − e ρa e σb ( ∂ ρ e σc − ∂ σ e ρc ) e µc ] , (62)and it is readily verified that this connection indeed transforms as in equation (57).Equation (62) is what sets gauge theories and gravity apart. In gauge theories, the gaugefields are “fundamental” fields on which the action functional depends. In gravity the spinconnection can be expressed in terms of the vierbein, which constitute the fundamental fieldsin the gravitational sector. In particular, the metric can be also expressed in terms of thevierbein, g µν = e µa e νa , (63)where, as in Subsection II C, e µa is the (transposed) inverse of e µa , that is, e µa e ν a = δ µν .Because the covariant derivative of the vierbein vanishes by construction, one can use thevierbein to freely alter the transformation properties of any field under diffeomorphisms andLorentz transformations. For instance, ∇ µ A a ≡ e ν a ∇ µ A ν , so one can use the vierbein tofreely convert diffeomorphism vectors into Lorentz vectors and vice versa.Since the spin connection transforms like a gauge field, the curvature tensor R µν ≡ ∂ µ ω ν − ∂ ν ω µ + [ ω µ , ω ν ] (64)transforms like a two-form under general coordinate transformations, and in the adjointrepresentation under local Lorentz transformations g ( x ) ∈ L ↑ + , g ( x ) : R µν g R µν g − . (65)This transformation law is particularly simple in the fundamental (form) representation ofthe Lorentz group. In that case, for fixed µ and ν the curvature R µν is a matrix [ R µν ] ab whose elements transform according to g ( x ) : [ R µν ] ab → [ R ′ µν ] ab = Λ ac ( g )Λ bd ( g )[ R µν ] cd . (66)20ote that the curvature tensor is antisymmetric in the coordinate and Lorentz indices, R µνab = − R νµab = − R µνba . (67)Recall that spacetime indices are raised and lowered with the spacetime metric g µν , andLorentz indices are raised and lowered with the Minkowski metric η ab .With these ingredients it is then possible to construct invariants under both generalcoordinate and Lorentz transformations. In particular, the combination d V in equation(30) is invariant under both coordinate and Lorentz transformations, and thus provides anappropriate volume element for the integration of appropriate field invariants. If we weredealing with an actual gauge theory, the appropriate kinetic term for the spin connectionwould be the curvature squared, but in the case of gravity the situation is slightly different.In fact, in this case the spin connection is not an independent field, but is determinedinstead by the vierbein. The only scalar invariant under coordinate transformations andlocal transformations which contains up to two derivatives of the vierbein is the Ricci scalar, R ≡ e µ a e ν b R µνab . (68)Recall that ∇ µ e νa vanishes by construction. B. Broken Lorentz Symmetry
The extension of this formalism to theories with broken Lorentz invariance is relativelystraight-forward, and parallels the standard construction in flat spacetime. We begin byconstructing the most general theory invariant under (linearly realized) local transformationsin a Lorentz subgroup H and general coordinate transformations, and then we show that,by introducing Goldstone bosons, the theory can be made explicitly invariant under the full(non-linearly realized) Lorentz group.
1. Unitary Gauge
Let us first postulate the existence of a vierbein e µa that transforms linearly under localtransformations in a subgroup of the Lorentz group, h ( x ) : e µa ( x ) e ′ µa ( x ) = Λ ab ( h ) e µb , h ( x ) ∈ H ⊂ L ↑ + . (69)21t is the existence of an orthonormal frame in any spacetime with Minkowski signature whatforces us to introduce the vierbein in such a representation. Given this vierbein, we define the spacetime metric to be g µν ≡ e µa e ν b η ab . (70)It follows then from the definition of the metric that the vierbein forms a set of orthonormalvectors, as in equation (53), and that the volume element (30) is invariant both under generalcoordinate and Lorentz transformations.In order to construct derivatives that transform covariantly under local transformationsin H , we need to postulate the existence of an appropriate connection ω µ . If we want toavoid introducing extraneous ingredients into the gravitational sector, we should constructsuch a gauge field solely in terms of the vierbein, as in the standard construction. Inspectionof equations (58) and (62) reveals that if we define ω µ by equation (62), under an elementof H the connection transforms like h ( x ) : ω µ h ω µ h − + h ∂ µ h − . (71)But as opposed to the original construction in which we demanded invariance under the fullLorentz group, the reduced symmetry in the broken case allows us to introduce additionalcovariant quantities. In particular, expanding the connection in the basis of broken andunbroken generators, ω µ ≡ i (cid:0) D µm x m + E µi t i (cid:1) ≡ i ( D µ + E µ ) , (72)it is then easy to verify that D µ transforms covariantly (under H ), while E µ transforms likea gauge field, h ( x ) : D µ ( x ) h D µ ( x ) h − , (73a) E µ ( x ) h E µ ( x ) h − − i h ∂ µ h − . (73b)These transformation laws are analogous to those in equations (16). The only difference,setting g = h and using equation (10), is that in the latter the Lorentz group acts a trans-formation in spacetime, which changes the spacetime coordinates of the fields, while herethe Lorentz group acts internally, and thus leaves the spacetime dependence of the fieldsunchanged. 22he transformation properties of E µ allow us to define another covariant derivative of thevierbein, ¯ ∇ ρ e µa = ∂ ρ e µa − Γ νµρ e νa − iE ρi ( t i ) ab e µb . But because ∇ ρ e µa = 0, this derivativeequals − iD νm ( x m ) ab e µb , and therefore does not yield any additional covariant quantity.Finally, from the connection ω µ we define the curvature (64), which under (69) transformslike h ( x ) : R µν h R µν h − . (74)In order to construct invariants under both diffeomorphisms and local Lorentz transforma-tions, it is convenient to consider quantities that transform as scalars under diffeomorphisms,and tensors under the unbroken Lorentz subgroup H . We thus define, in full analogy withequations (18), D a ≡ e µa D µ , E a ≡ e µa E µ , R ab ≡ e µa e ν b R µν . (75)The quantities D a and E a are the appropriate generalization of the covariant derivativesdefined in equation (18), since they also transform like in equation (19), the only differencebeing again that here the Lorentz group acts as an internal transformation. As before, thecovariant derivatives of any diffeomorphism scalar ψ that transforms in a representation ofthe unbroken group with generators t i are defined by equation (33), where E µi is now givenby equation (72).By construction, any term solely built from the covariant quantities d V , D am , R abcd , ψ and D a ψ , which is invariant under global H transformations is also invariant under localtransformations in H and diffeomorphisms. In particular, because the covariant derivatives D am defined in (18) and the the covariant derivatives in equation (75) transform in the sameway under H , the unbroken symmetries now allow us to write down linear and quadraticterms for the components of the connection ω µ along the directions of the broken generators,as in equation (29). In an ordinary gauge theory, the quadratic terms give mass to somegauge bosons, but in our context, because the spin connection depends on derivatives ofthe vierbein, these quadratic terms cannot be properly considered as mass terms for thegraviton. Since the spacetime metric is g µν = e µa e νa , a graviton mass term should be aquartic polynomial in the vierbein. But the only invariants one can construct from thevierbein e µa are field-independent constants.23 . Manifestly Invariant Formulation Let us assume now that we have constructed an H invariant action, S [ e, ψ ] = S [Λ( h ) e, R ( h ) ψ ] , h ( x ) ∈ H, (76)where the functional dependence emphasizes that only e and ψ are the “fundamental” fieldsof the theory, from which the remaining covariant quantities are constructed, as discussedabove. We show next that by introducing the corresponding Goldstone bosons in the theory γ ≡ γ ( π m ), this symmetry can be extended to the full Lorentz group. To that end, let usassume that the vierbein e µa transforms in a linear representation of the Lorentz group, asin equation (55), and let us define ˜ e µa ≡ Λ ab ( γ − ) e µb , (77)where ˜ e µa is to be regarded as a shorthand for the expression on the right hand side, and γ is a function of the Goldstone bosons defined in equation (8). Let us postulate that underlocal Lorentz transformations, γ ( π ) transforms as in equation (9), while under g ( x ) ∈ L ↑ + , g ( x ) : ψ R ( h ( π, g )) ψ. (78)In that case, it follows from the definition (77) that ˜ e transforms analogously, g ( x ) : ˜ e µa Λ ab ( h ( π, g )) ˜ e µb . (79)The transformation properties (78) and (79) and the invariance of the action (76) implythat a theory with ˜ S [ γ, e, ψ ] ≡ S [Λ( γ − ) e, ψ ] (80)is invariant under the full Lorentz group. In the Lorentz-invariant formulation of the theoryin equation (80) the action appears to depend on the Goldstone bosons γ ( π ). However,inspection of the right hand side of the equation reveals that such a dependence can beremoved by the field redefinition (77). By a “field redefinition” we mean here a change ofvariables in the theory, which replaces the combination of two fields Λ( γ − ) e by a singlefield, which we may call again e . Since the field variables we use do not have any impact onthe physical predictions of a theory, we may thus replace S [Λ( γ − ) e, ψ ] by S [ e, ψ ]. In this24unitary gauge” we have effectively set γ = 1, and returned back to the original action inequation (76).It is instructive to show how the introduction of the Goldstone bosons would make thetheory manifestly invariant under local transformations. For simplicity, let us just focus onthe gravitational sector. As mentioned above, the modified vierbein (77) transforms non-linearly under the action (55) of the Lorentz group, g ( x ) ∈ L ↑ + . When we substitute thismodified vierbein into the expression for the spin connection (62) we obtain˜ ω µ = γ − ( ∂ µ + ˜ ω µ ) γ, (81)which is just the covariant generalization of the Maurer-Cartan form γ − ∂ µ γ , and transformsnon-linearly under (55), g ( x ) : ˜ ω µ h ( π, g ) ω µ h − ( π, g ) + h ( π, g ) ∂ µ h − ( π, g ) , (82)with h ( π, g ) defined in equation (9). Therefore, if we expand this connection in the basis ofthe Lie algebra, ˜ ω µ ≡ i h ˜ D µm x m + ˜ E µi t i i , (83)we obtain covariant derivatives ˜ D a ≡ ˜ e µa ˜ D µ and gauge fields ˜ E a ≡ ˜ e µa ˜ E µ that transformlike in equations (19), but with x ′ = x . The curvature tensor ˜ R µν associated with ˜ ω µ is infact given by ˜ R µν = γ − R µν γ, (84)where R µν is the curvature tensor associated with the spin connection ω µ , derived itself from e µa . Under the action of elements g ( x ) ∈ L ↑ + on the vierbein (55), this curvature transformsnon-linearly too, g ( x ) : ˜ R µν h ( π, g ) ˜ R µν h − ( π, g ) . (85)It is thus clear from the transformation properties of these new quantities that if theoriginal action S is invariant under H , the new action ˜ S defined in equation (80) will beinvariant under L ↑ + . In fact, we could have reversed the whole construction. We could havestarted by defining a modified vierbein ˜ e µa , a modified covariant derivative ˜ D µ and a mod-ified curvature tensor ˜ R µν according to equations (77), (83) and (84). Then, any invariantaction under H , solely constructed out of these ingredients would have been automaticallyand manifestly invariant under L ↑ + . 25 V. UNBROKEN ROTATIONS
We turn now our attention to cases in which the unbroken group is the rotation group, H = SO (3), which is the maximal compact subgroup of L ↑ + . This pattern of symme-try breaking is analogous to the spontaneous breaking of chiral invariance in the twoquark model. In the latter, the chiral symmetry of QCD with two massless quarks, SU (2) L × SU (2) R , is broken down to the isospin subgroup SU (2), while in the former,the Lorentz group SO (1 , ∼ SU (2) × SU (2) is broken down to the diagonal subgroup ofrotations SO (3) ∼ SU (2). Hence, the construction of rotationally invariant Lagrangianswith broken Lorentz invariance is formally analogous to the construction of isospin invariantLagrangians with broken chiral symmetry.As in the two-quark model, the case for unbroken rotations can be motivated phenomeno-logically. If rotations were broken, we would expect the expansion of the universe to beanisotropic, in conflict with observations, which are consistent with a nearly isotropic cos-mic expansion all the way from the initial stages of inflation. Our main goal here howeveris not to consider the phenomenology of theories with unbroken rotations, as this has beenalready extensively studied, but simply to illustrate how our formalism applies to theorieswith gravity. We shall see in particular how in this case our construction directly leadsto the well-known Einstein-aether theories, which we show to be the most general class oftheories in which rotations remain unbroken. A. Coset Construction
In order to build the most general theory in which the rotation group remains unbroken,let us assume first that spacetime is flat, as in Section II B. In the case at hand, then,the generators of the unbroken group are the generators of rotations J i , and the remaining“broken” generators are the boosts K m . Therefore, the theory contains three Goldstonebosons π m . Of particular relevance are the transformation properties of these Goldstonebosons under rotations. For an infinitesimal rotation t = ω i J i , equations (6) and (11) leadto t : π m π ′ m = π m + ( ω × π ) m . (86)26n addition, since P K m P − = − K m and T K m T − = − K m we have, from (25) that π m → − π m under parity and time reversal. Therefore, the set of Goldstone bosons transformlike a 3-vector under spatial rotations. These are analogous to the pions of spontaneouslybroken chiral invariance.The restriction of the four-vector representation Λ( g ) to the subgroup of rotations H isreducible, = ⊕ , so the tensor product representation of the rotation group in equation(20) is also reducible, ( ⊕ ) ⊗ = ⊕ ⊕ ⊕ . (87)(The different representations of the rotation group are labeled by their dimension. Thedimension N of the representation is N = 2 S + 1, where S is the spin of the representation.)More precisely, the covariant derivative D m ≡ D m (88)transforms like a spatial vector under rotations (spin one, ), while D mn transforms in thetensor product representation of rotations ⊗ . Defining D mn = 13 ϕ δ mn + a mn + s mn , (89)with a antisymmetric and s symmetric and traceless, leads to a scalar ϕ (spin zero, ),a vector a mn ≡ ǫ mnp a p (spin one, ), and a traceless symmetric tensor s mn (spin two, ). Therefore, the most general Lagrangian density at most quadratic in the covariantderivatives, and invariant under the full Lorentz group is L π = 12 (cid:0) F ϕ ϕ + F D D m D m + F a a mn a mn + F s s mn s mn (cid:1) , (90)where indices are raised with the (inverse) metric of Euclidean space, δ mn . Note that wehave omitted a linear term proportional to ϕ , and the parity-violating expression ǫ mnp a mn D p in the Lagrangian. As we show below, these terms are just total derivatives.Let us now address the new ingredients that gravity introduces into the theory. Aswe discussed in Section III B, in a generally covariant theory we may choose to work inunitary gauge, in which the Goldstone bosons identically vanish. In this gauge, the covariantderivatives D am defined above simply reduce to the spin connection along the appropriategenerators, as in equations (72). Therefore, using the explicit form of the rotation generators27n the fundamental representation, and tr( x m (4) · x n (4) ) = − δ mn , we find D m = ω m , D mn = ω mn . (91)Recall that there are three broken generators which transform like vectors under rotations,which we label by m, n , and that the derivatives defined in equations (75) transform in thesame way as the covariant derivatives defined in equation (18), with x ′ = x . Therefore, theLagrangian (90) already contains all the rotationally invariant terms constructed from theundifferentiated spin connection.To complete the most general gravitational action invariant under general coordinate andlocal Lorentz transformations, with at most two derivatives acting on the vierbein, we justneed to add all invariant terms that can be constructed from the curvature alone. Withoutloss of generality, we may restrict ourselves to the components of the Riemann tensor inan orthonormal frame, R abcd . Then, indices along spatial direction transform like vectors,while indices along the time direction transform like scalars under rotations. Most of theinvariants one can construct out of the Riemann tensor vanish because of antisymmetry.For instance, the term R mnp ǫ mnp is identically zero because of the antisymmetry of thecurvature tensor in the last three indices. In addition, the identity [ ∇ µ , ∇ ν ] A ρ = R µν ρσ A σ ,in an orthonormal frame and up to boundary terms, implies the relation Z d V (cid:2) R m m − D mn D mn + ( D mm ) (cid:3) = 0 , (92)which can be used to eliminate a scalar term proportional to R m m from the action. As wementioned earlier a term linear in the covariant derivative, ϕ ≡ D mm , is a total derivative,since from equations (59) and (61) ω m m = ∂ µ e µ + Γ µνµ e ν = 1det e ∂ µ (det e e µ ) . (93)Similarly, one can show that ǫ mnp a mn D p is a total derivative too, since the latter equals ǫ mnpq ∇ m A n ∇ p A q , for A m = δ m . We therefore conclude that the most general diffeomor-phism invariant action invariant under local rotations is S = M P Z d V [ R + L π ] + S M , (94)where R ≡ R abab is the Ricci scalar, the “Goldstone” Lagrangian L π is given by equation(90), and S M denotes the matter action. Tests of the equivalence principle [5] and constraints28n Lorentz-violating couplings in the standard model [3] suggest that any Lorentz-violatingterm in the matter action S M is very small. Hence, for phenomenological reasons, we assumethat the breaking of Lorentz invariance is restricted to the gravitational sector. Therefore, S M is taken to be invariant under Lorentz transformations, and the action (94) defines ametric theory of gravity. B. The Einstein-aether
For unbroken rotations, the matrix γ that we introduced in Section II B is a boost, γ = exp( iπ m K m ). Hence, instead of characterizing the Goldstone bosons by the set of threescalars π m , we may simply describe them by the transformation matrix Λ a of the boostitself. The latter has four components, u a ≡ Λ a , (95)but not all of them are independent, because Lorentz transformations preserve the Minkowskimetric. In particular, the vector field u a has unit norm u a u a ≡ η ab Λ a Λ b = η = − . (96)In the conventional approach to the formulation of the most general theory in which rotationsremain unbroken, one would solve the constraint (96) by introducing an appropriate set ofthree parameters, and then identify their transformation properties under the Lorentz group[15]. One would then proceed to define covariant derivatives of these parameters, and usethem to construct the most general theory compatible with the unbroken symmetry, just aswe did.In this case however, a simpler approach leads to the same general theory, but avoidsintroducing coset parametrizations and covariant derivatives altogether. Since the Lorentztransformation of a boost can be described by a the vector field (95), one may simply expectthat the problem of constructing the most low-energy effective theory in which the rotationgroup remains unbroken just reduces to the problem of writing down the most generaldiffeomorphism invariant theory with the least numbers of derivatives acting on a unit normvector field. This was precisely the problem that Jacobson and Mattingly studied in [12],which resulted in what they called the “Einstein-aether”. The most general action in this29lass of theories is S = M G Z d V h R − c ∇ a u b ∇ a u b − c ( ∇ a u a ) − c ∇ a u b ∇ b u a ++ c u a u b ∇ a u c ∇ b u c + λ ( u a u a + 1) i , (97)where the parameters c i are constant, and we have written down all the components of the“aether” vector field u µ in an orthonormal frame, u a ≡ e µa u µ , with covariant derivativesgiven by ∇ a u b ≡ e µa (cid:0) ∂ µ u b + ω µbc u c (cid:1) . (98)The constraint u a u a = − λ .Hence, the action (97) is analogous to the linear σ -model in which chiral symmetry breakingwas originally studied. In this formulation, the Lorentz group acts linearly on the vector field u a , and, as we shall see, the fixed-norm constraint can be understood as limit in which thepotential responsible for Lorentz symmetry breaking is infinitely steep around its minimum.To establish the connection between the Einstein-aether (97) and the rotationally in-variant action (94), we simply need to impose unitary gauge. We can solve the unit normconstraint in (97) by expressing the vector field u a as a Lorentz transformation acting on anappropriately chosen vector ˜ u a , u a = Λ ab ( π )˜ u b , with ˜ u a = δ a , (99)which is just a restatement of equation (95). Then, invariance under local Lorentz trans-formations implies that the aether action (97) can be equally thought of as a functional of˜ u b and the transformed vierbein ˜ e µa = (Λ − ( π )) ab e µb . If we now redefine the vierbein field,˜ e µb → e µa , the Goldstone bosons π disappear from the action, and we are left with thetheory in unitary gauge. In this gauge the vierbein is arbitrary, but (dropping the tildes) wecan assume that u a = δ a . In that case equation (98) gives in addition ∇ a u b = ω ab , which,when substituted into the Einstein-aether action (97) precisely yields the action (94). Thecorresponding parameters M P and F i are expressed in terms of five linearly independentcombinations of aether parameters, M P = M G , F ϕ = −
13 ( c +3 c + c ) , F D = c + c , F a = c − c , F s = − ( c + c ) , (100)and, therefore, the Einstein-aether is the most general low-energy theory in which the ro-tation group remains unbroken. The correspondence (100) also explains then why these30articular combinations of the Einstein-aether parameters enter the predictions of the the-ory. In our language, they map into the different irreducible representations in which one canclassify the covariant derivatives of the Goldstone bosons. The phenomenology of Einstein-aether theories is nicely reviewed in [49]. C. General Vector Field Models
In Einstein-aether theories, Lorentz invariance is broken because the vector field u a de-velops a time-like vacuum expectation value. In this context, it is then natural to considergeneric vector field theories in which a vector field develops a non-zero expectation value,and to study how the latter reduce to the Einstein-aether in the limit of low energies. Thiswill also help us to illustrate our formalism in cases in which the spectrum of excitationscontains a massive field, and how the latter disappears from the low-energy predictions ofthe theory.The most general low energy effective action for a vector field non-minimally coupled togravity which contains at most two derivatives and is invariant under local Lorentz trans-formations and general coordinate transformations reads S = 12 Z d V (cid:20) M G R + α F ab F ab + β ( ∇ a A a ) + β R A a A a + β R ab A a A b + (101)+ A a A b Λ ( α ∇ a A c ∇ b A c + α ∇ c A a ∇ c A b + α ∇ a A b ∇ c A c ) ++ γ A a A b A c A d Λ ∇ a A b ∇ c A d + δ A b A b ∇ a A a − Λ V (cid:21) . Here, F ab ≡ ∂ a A b − ∂ b A a , A a are the components of the vector field in an arbitrary or-thonormal frame, and the various coefficients α , α i , β , β i , γ , δ and V should be regardedas arbitrary (dimensionless) functions of A a A a / Λ . Finally, M G and Λ are the two charac-teristic energy scales of the effective theory, which is valid at energies E ≪ min(Λ , M G ). Inorder to generate spontaneous breaking of Lorentz symmetry down to rotations we assume,without loss of generality, that the potential V is minimized by field configurations with A a A a = − Λ . Other low energy terms that do not appear in the expression (101) can be re-duced to linear combinations of the terms above after integrations by parts. An action verysimilar to (101) has been already considered in [13], though the latter did not include theterms proportional to β and δ , and all the other couplings were assumed to be constants31ather than arbitrary functions of A a . Models involving fewer terms have been studied forinstance in [50–53] under the name of “bumblebee models,” and in [38] under the name of“unleashed aether models.”In order to make contact with the formalism developed in the previous sections, we shallparametrize again the vector field as a Lorentz transformation acting on A a ( x ) = δ a (Λ + σ ( x )) , (102)where the field σ is just a singlet under rotations. This is the same we did for the aether,the only difference being that there the fixed-norm constraint forced the field σ to vanish.As before, invariance under local Lorentz transformations then implies that the vector fieldcan be taken to be given by (102). In this unitary gauge, the covariant derivative of A a is ∇ a A b = δ b ( e µa ∂ µ σ ) + η bm (Λ + σ ) D am , (103)where we have used equations (91). Thus, the action (101) can be expressed in terms ofrotationally invariant operators that solely involve R abcd , D am , the scalar σ and its covariantderivative D a σ = e µa ∂ µ σ .It shall prove to be useful to expand the action (101) in powers of σ. To quadratic order,and to leading order in derivatives, this results is S = 12 Z d V (cid:20) ( M G − ¯ β Λ ) R + Λ (cid:18) ¯ β + 2 ¯ β (cid:19) ϕ + Λ ( ¯ α − ¯ α ) D m D m + (104)+Λ (2 ¯ α + ¯ β ) a mn a mn − Λ ¯ β s mn s mn + σ ( − δ Λ ϕ + · · · ) + σ ( − V ′′ Λ + · · · ) + O ( σ ) (cid:21) , where the dots stands for the subleading terms in the derivative expansion and ¯ V ′′ denotesthe second derivative of the potential function with respect to its argument, evaluated atits minimum, where A a A a = − Λ . Similarly, ¯ α, ¯ β, ¯ β , ¯ β , ¯ α and ¯ δ stand for the valuesof the couplings at the minimum of the potential. Apart from the additional rotationallyinvariant terms involving the field σ , the action (104) has manifestly the form (94) with M P ≡ (1 − ¯ β ) M G .We study the spectrum of this class of theories in Appendix A. Their scalar sector consistsof a massless excitation, one of the Goldstone bosons, and a massive field, whose mass islinear in ¯ V ′′ . We show in the appendix that in the low-momentum limit, the field σ has avanishing matrix element between the massless scalar particle and the vacuum,lim p → h m = 0 | σ ( p ) | i = 0 . (105)32ence, if we are interested in low momenta and massless excitations, the field σ can be simplyintegrated out. At tree level, this can be easily done by solving the classical equations ofmotion to express σ in terms of the covariant derivatives D am . From (104), we see that tolowest order in derivatives the result is completely determined by the two terms proportionalto σ and σϕ . Thus, solving the corresponding linear equation, σ = − ¯ δ V ′′ ϕ + O ( ∂ / Λ) , (106)and plugging back into the action (104) we get, to leading order in derivatives, S = 12 Z d V (cid:20) ( M G − ¯ β Λ ) R + Λ (cid:18) ¯ β + ¯ δ V ′′ + 2 ¯ β (cid:19) ϕ + Λ ( ¯ α − ¯ α ) D m D m ++Λ (2 ¯ α + ¯ β ) a mn a mn − Λ ¯ β s mn s mn (cid:21) . (107)As expected the low energy action (107) has the form of (94). Integrating out the field sigmahas simply renormalized the coefficients of the low energy theory, which are now given by M P = M G − ¯ β Λ , F ϕ = (cid:18) ¯ β + ¯ δ V ′′ + 2 ¯ β (cid:19) Λ M P , F D = ( ¯ α − ¯ α ) Λ M P ,F a = (2 ¯ α + ¯ β ) Λ M P , F s = − ¯ β Λ M P . (108)By combining these relations with equations (100), one can easily derive the dispersionrelations and residues of the massless excitations in the model (101) from the known aethertheory results [49]. Equations (108) show from the very beginning that the couplings γ, α and α will not enter the low-energy phenomenology. A “brute force” calculation based onthe action (101) tends to obscure this fact, as shown explicitly in Appendix A, although thefinal results are of course identical.Alternatively, if we are interested only in the low energy phenomenology of the theory,we can choose to drop the field σ from the onset, as massive excitations will not give anyobservable contribution at low energies [39]. In the limit ¯ V ′′ → ∞ where the massive modebecomes infinitely heavy, the potential may be replaced by a fixed-norm constraint, as inEinstein-aether theories. In fact, when ¯ V ′′ → ∞ , equation (106) implies that σ can be simplyset to zero, and the general class of vector field models described by (101) directly reducesto the Einstein-aether. After introducing a rescaled vector A a ≡ Λ u a and integrating someterms by parts, the coefficients c i in (97) can be easily mapped onto the couplings in (101)33s follows: α = − c M G Λ , β = − ( c + c + c ) M G Λ , β = ( c + c ) M G Λ , α = c M G Λ ,α = α = β = γ = δ = 0 . (109)Once again, equations (109) can be easily combined with the known Einstein-aether re-sults [49] to immediately obtain the dispersion relations and the residues for the masslesspropagating modes in the specific model (101). V. SUMMARY AND CONCLUSIONS
In this article we have generalized the effective Lagrangian construction of Callan, Cole-man, Wess and Zumino to the Lorentz group. In flat spacetime, the Lorentz group is aglobal symmetry, and its breaking implies the existence of Goldstone bosons, one for eachbroken Lorentz generator. The broken global symmetry is not lost, and is realized non-linearly in the transformation properties of these Goldstone bosons and the matter fields ofthe theory. Because the Lorentz group is a spacetime symmetry, the Goldstone bosons trans-form non-trivially under the Lorentz group, and can be classified in linear representationsof the unbroken subgroup. The same non-linearly realized global symmetry prevents theGoldstone bosons from entering the Lagrangian undifferentiated, which allows us to identifythem as massless excitations. Because spacetime derivatives transform non-trivially underthe Lorentz group, the covariant derivatives of Goldstone bosons typically furnish reduciblerepresentations of the unbroken Lorentz subgroup. The Lorentz group does not seem to bebroken in the standard model sector, so any eventual breaking of this symmetry must beconfined to a hidden sector of the theory. In that respect, phenomenologically realistic the-ories must resemble models of gravity-mediated supersymmetry breaking [54–56]. In bothcases, a spacetime symmetry is broken in a hidden sector, the breaking is communicatedto the standard model by the gravitational interactions, and, for phenomenological reasons,the symmetry breaking scale has to be sufficiently low.Given an internal symmetry group, one always has a choice to make it global or local.But in the case of the Lorentz group this choice does not seem to exist. Any generallycovariant theory that contains spinor fields, such as the standard model coupled to generalrelativity, requires that Lorentz transformations be an internal local symmetry, very much34ike a group of internal gauge symmetries. We have therefore extended the constructionof actions in which global Lorentz invariance is broken to generally covariant formulationsin which the group of local Lorentz transformations is non-linearly realized on the fields ofthe theory, which at the very least must contain the covariant derivatives of the Goldstonebosons and the vierbein, which describes the gravitational field. But in this case, since theLorentz group is a local symmetry, it is possible and simpler to work in a formulation inwhich the Goldstone bosons are absent, and Lorentz symmetry is explicitly broken. In this“unitary gauge,” the theory remains generally covariant, but Lorentz symmetry is lost. Eventhough the lost invariance under the Lorentz group can always be restored by introducingthe appropriate Goldstone bosons, this restored symmetry is merely an artifact.Generally covariant theories with broken Lorentz invariance differ significantly from theirfully symmetric counterparts. In unitary gauge for instance, the covariant derivatives of theGoldstone bosons that the unbroken symmetry allows us to write down simply become thespin connection along the broken generators. This is just the Higgs mechanism. But in agenerally covariant theory without extraneous additional fields, this connection is expressedin terms of the vierbein, so these terms actually represent kinetic terms for some of itscomponents. Thus, instead of a massive theory of gravity, when Lorentz invariance is brokenwe obtain a theory with additional massless excitations (in Minkowski spacetime), which wecan interpret as extra graviton polarizations in unitary gauge, or simply as the Goldstonebosons of the theory in general.We have illustrated these issues for cases in which the rotation group remains unbroken.In particular, we have rigorously shown that the most general low-energy effective theorywith unbroken spatial rotations is the Einstein-aether, and how generic vector field theoriesreduce to the latter at low energies.The construction of low-energy effective theories that we have described here provides uswith a tool to explore Lorentz symmetry breaking systematically and in a model-independentway. It identifies first how the Lorentz group acts on the field of the theory, it removes theclutter of particular models by focusing on the relevant fields at low energies, and it uniquelyenumerates all the invariants under the unbroken symmetries.35 cknowledgments
The work of CAP and RP is supported in part by the National Science Foundationunder Grant No. PHY-0855523. The work of ADT is supported by a UNAM postdoctoralfellowship.
Appendix A: Dispersion Relations for Vector-Tensor Effective Theories
In this appendix we study the spectrum of excitations in the vector-tensor theories in-troduced in Section IV C, in which Lorentz symmetry is spontaneously broken down torotations. Although such a study is usually carried out in the standard metric formulation(see for example [13]), in what follows we adopt instead the vierbein formulation, which isthe one we employ in the main body of this paper.
1. Perturbations
Our starting point is the action (101), which is a functional of the vierbein e µa and thevector field A a , and describes the behavior of both light and heavy modes. Perturbationsof the vierbein around the Minkowski solution e µa = δ µa can be decomposed into scalars,vectors and tensors under spatial rotations as follows: δe = φ, (A1a) δe i = ∂ i B + S i , (A1b) δe i = − ∂ i C − T i , (A1c) δe ij = − δ ij ψ + ∂ i ∂ j E + ǫ ijk ∂ k D − ∂ ( i F j ) + ǫ ijk W k + 12 h ij . (A1d)In this decomposition φ, B, C, ψ, E, D are scalars, S i , T i , F i , W i are transverse vectors, ∂ i S i = · · · = ∂ i W i = 0, and h ij is a transverse and traceless tensor, h ii = ∂ i h ij = 0. Here, i = 1 , , δ ij .Scalars, vectors and tensors transform in different irreducible representations of the ro-tation group and therefore do not couple from each other in the free theory. As we showin Section IV C, no matter what the spacetime background is, we can always use invariance36nder local boosts to impose the “unitary gauge” condition (102), namely A a ( x ) = δ a (Λ + σ ( x )) . (A2)The field σ is a scalar under rotations. Gauge fixing
At this point, not all the scalars and vectors in equations (A1) and (A2) describe in-dependent degrees of freedom, because of the residual gauge invariance associated withgeneral coordinate transformations and the unbroken group of local rotations. In fact, un-der infinitesimal coordinate transformations ( x µ → x µ + ξ µ ) and local Lorentz rotations( e iµ → e iµ + ω k ǫ ij k e j µ ) the fluctuations of the vierbein around a Minkowski background(A1) transform in the following way: δe → δe − ∂ t ξ , (A3a) δe i → δe i − ∂ t ∂ i ξ − ∂ t ξ iT , (A3b) δe i → δe i − ∂ i ξ , (A3c) δe ij → δe ij − ∂ i ∂ j ξ − ∂ i ξ jT + ǫ ijk ∂ k ω + ǫ ij k ω kT , (A3d)where we have decomposed ξ µ and ω i into the scalars ξ , ξ , ω and the transverse vectors ξ iT and ω iT ( ∂ i ξ iT = ∂ i ω iT = 0). Comparison of equations (A1) and (A3) then shows that,by performing an appropriately chosen rotation together with a general coordinate trans-formation, one can set for instance F i = W i = 0 and C = D = E = 0 = 0. Thus, we areeventually left with only four scalars ( φ, B, ψ and σ ), two vectors ( S i and T i ) and one tensor( h ij ). This is the same number of degrees of freedom one obtains in the metric formulationof the theory, after completely fixing the gauge.
2. Tensor Sector
As we mention above, in the free theory, scalars, vectors and tensors decouple from eachother. Let us therefore start by considering the tensor sector, which is described by thequadratic Lagrangian L t = 14 n(cid:2) M G − (cid:0) ¯ β + ¯ β (cid:1) Λ (cid:3) ˙ h ij ˙ h ij − (cid:2) M G − ¯ β Λ (cid:3) ∂ k h ij ∂ k h ij o , (A4)37rom which we can immediately read off the residue and the speed of sound of the tensormodes, Z − t = M G − (cid:0) ¯ β + ¯ β (cid:1) Λ , c t = M G − ¯ β Λ M G − ( ¯ β + ¯ β )Λ . (A5)Once again, ¯ β and ¯ β stand for the values of the couplings at the minimum of the potential,and a similar notation applies in what follows to the other couplings too. The tensor sectoris ghost free provided ( ¯ β + ¯ β ) ≪ ( M G / Λ) . We should also impose ¯ β ≪ ( M G / Λ) in orderto ensure classical stability. The results (A5) agree with the ones of aether models withparameters given by equation (108), and they also reduce to the ones found by Gripaios [13]in the limit where Λ ≪ M G .
3. Vector Sector
The Lagrangian for the vector modes is only slightly more complicated, and reads L v = 12 n(cid:2) M G − ¯ β Λ (cid:3) ∂ i ( T j + S j ) ∂ i ( T j + S j ) + 2( ¯ α − ¯ α )Λ ˙ T i ˙ T i + (A6)+( ¯ β + 2 ¯ α )Λ ∂ i T j ∂ i T j − ¯ β Λ ∂ i S j ∂ i S j o . The field S i only appears in the Lagrangian density through the combination ∂ i S j and doesnot propagate. Its equation of motion can be easily solved to get (cid:2) M G − ( ¯ β + ¯ β )Λ (cid:3) S i = − (cid:2) M G − ¯ β Λ (cid:3) T i , (A7)which, when substituted back in (A6) gives L v = ( ¯ α − ¯ α )Λ ˙ T i ˙ T i + ¯ α − ¯ β Λ (cid:2) M G − ( ¯ β + ¯ β )Λ (cid:3) ! ∂ i T j ∂ i T j . (A8)Therefore, only two massless vector modes propagate, with residue and a speed of soundgiven by Z − v = 2( ¯ α − ¯ α )Λ , c v = 1¯ α − ¯ α ¯ α − ¯ β Λ (cid:2) M G − ( ¯ β + ¯ β )Λ (cid:3) ! . (A9)In empty space, the vector sector of general relativity is non-dynamical. However, thebreakdown of Lorentz invariance gives dynamics to this sector, even in the absence of matterfields. Of course, these two vector modes correspond to two of the Goldstone bosons ofthe spontaneously broken phase. They are well behaved in the limit Λ ≪ M G provided( ¯ α − ¯ α ) > α <
0. Notice that this result does not agree with [13], though it doesagree with the result found in aether theories [49], upon the identification in equations (108).38 . Scalar Sector
Let us finally consider the scalar sector, which now contains both massive and masslessfields. To quadratic order in the perturbations, its Lagrangian density is given by L s = 12 n M G − ¯ β Λ )( ∂ i ψ∂ i ψ − ∂ i φ∂ i ψ ) + ¯ β Λ (∆ B ) +(3 ¯ β Λ + 2 ¯ β Λ + 2 ¯ β Λ − M G )(3 ˙ ψ + 2 ˙ ψ ∆ B )+( ¯ α − ¯ α )Λ ∂ i φ ∂ i φ + ( ¯ β − ¯ α − ¯ α − ¯ α + ¯ γ ) ˙ σ − ( ¯ α − ¯ α ) ∂ i σ∂ i σ + − V ′′ Λ σ + [( − β + 4 ¯ β ′ − β + 2 ¯ β ′ + ¯ α − β ) ˙ σ + 2¯ δ Λ σ ](∆ B + 3 ˙ ψ ) + − ∂ i σ∂ i [(4 ¯ β − β ′ + 2 ¯ β − β ′ + 2 ¯ α ) φ −
8( ¯ β − ¯ β ′ ) ψ ] o . (A10)The scalars φ and B only appear in the Lagrangian trough the combinations ∂ i φ and ∆ B ,so they can be easily eliminated by solving their classical equations of motion. At this point,it is more convenient to switch to Fourier space, and write the action for the two remainingscalars in the form S s = − Z d k X † DX, with X ≡ σ ( k ) ψ ( k ) (A11)and D ≡ a ω + a k + a Λ a ω + a k + ia Λ ωa ω + a k − ia Λ ω a ω + a k . (A12)Here, the (dimension two) coefficients a i are some complicated functions of the variouscoupling constants of the model. In particular, a and a are the only couplings that breakthe Z symmetry A a → − A a .The inverse of the matrix D is just the field propagator. In order to find the propagatingmodes we just have to find the values of ω at which its eigenvalues have poles, or, equiv-alently, the values of ω at which the eigenvalues of D have zeros. Requiring that det( D )vanish we thus arrive at the frequencies of the two propagating modes, ω = m Λ + c k + O ( k / Λ ) , ω = c k + O ( k / Λ ) , (A13)39ith m = a − a a a a − a , (A14a) c = a ( a a − a a ) + ( a − a a )(2 a a − a a )( a − a a )( a a − a ) , (A14b) c = a a a − a a . (A14c)In the absence of fine-tuning, the first mode has a mass of order Λ and can be excluded fromthe low-energy theory. On the other hand, the speed of sound of the massless mode, c = (2 ¯ V ′′ ¯ β + ¯ δ ) (cid:2) M G − (2 ¯ β − ¯ α + ¯ α )Λ (cid:3) (cid:2) M G − ¯ β Λ (cid:3) ( ¯ α − ¯ α ) (cid:2) M G − ( ¯ β − ¯ β )Λ (cid:3) (cid:2) V ′′ (cid:0) M G − (2 ¯ β + 2 ¯ β + 3 ¯ β )Λ (cid:1) − δ Λ (cid:3) , (A15)coincides with the speed of sound of the scalar mode in aether theories [49], after substitutionof equations (108). Note that the terms O ( k / Λ ) in equation (A13) cannot be trustedsince our starting point was an effective action in which all the terms with more than twoderivatives were excluded.As in the vector sector, in the absence of matter fields the scalar sector of general relativityis non-dynamical. But again, the breakdown of Lorentz invariance gives dynamics to thissector. This captures of course the existence of a Goldstone boson in the scalar sector of thetheory, which, together with the two massless modes we found in the vector sector, play therole of the three Goldstone bosons associated with the broken boost generators.The residues of the scalar modes can be determined using the general result [38]1 Z , = − D ) ∂∂ω det( D ) (cid:12)(cid:12)(cid:12)(cid:12) ω = ω , , (A16)which, in our case, yields Z − = a ( a + a ) − a ( a + a )( a a − a )( a − a a ) + O ( k / Λ ) , Z − = a a a − a + O ( k / Λ ) . (A17)Like for the speed of sound, the residue of the massless mode Z − = 2 (cid:2) M G − ( ¯ β + ¯ β )Λ (cid:3) (cid:2) δ Λ − V ′′ (2 M G − (2 ¯ β + 2 ¯ β + 3 ¯ β )Λ ) (cid:3) (¯ δ + 2 ¯ V ′′ ¯ β )Λ + O ( k / Λ )(A18)agrees with that obtained in aether theories [49], upon the identification (108). Once again,the terms O ( k / Λ ) in the residues are out of the reach of validity of the effective theory wewrote down. 40o conclude, it is interesting to point out that none of the results concerning the masslessmodes depend on α , α , γ , nor on the derivatives of β and β . A brute-force approach likethe one we just followed makes this look like the result of accidental cancellations. Noticefor instance that in fact the free scalar Lagrangian (A10) does depend on α , α , γ , as wellas on the derivatives of β and β . The low-energy effective action (107) on the other handmakes this manifest from the very beginning.
5. The field σ We obtained the low energy effective Lagrangian (107) by integrating out the field σ . Inthat context, we claimed that this procedure was justified because that the matrix element of σ between the vacuum and a state with one massless particle vanishes in the low-momentumlimit (see equation (105)). We are now in a position to prove this result.As we have seen above, the scalar spectrum consists of a massive field s and a masslessfield s . We can thus express the field σ as a linear combination of the two canonicallynormalized fields, σ = κ s + κ s , (A19)in which κ and κ are momentum-dependent coefficients. 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