No multi-graviton theories in the presence of a Dirac field
aa r X i v : . [ h e p - t h ] A p r No multi-graviton theories in the presence of aDirac field
C. Bizdadea ∗ , E. M. Cioroianu † , A. C. Lungu and S. O. Saliu ‡ Faculty of Physics, University of Craiova13 A. I. Cuza Str., Craiova 200585, RomaniaDecember 21, 2018
Abstract
The cross-couplings among several massless spin-two fields (de-scribed in the free limit by a sum of Pauli-Fierz actions) in the pres-ence of a Dirac field are investigated in the framework of the deforma-tion theory based on local BRST cohomology. Under the hypothesesof locality, smoothness of the interactions in the coupling constant,Poincar´e invariance, (background) Lorentz invariance and the preser-vation of the number of derivatives on each field, we prove that thereare no consistent cross-interactions among different gravitons in thepresence of a Dirac field. The basic features of the couplings betweena single Pauli-Fierz field and a Dirac field are also emphasized.PACS number: 11.10.Ef
Over the last twenty years there was a sustained effort for constructing the-ories involving a multiplet of spin-two fields [1, 2, 3, 4]. At the same time,various couplings of a single massless spin-two field to other fields (including ∗ e-mail address: bizdadea@central. ucv.ro † e-mail address: manache@central. ucv.ro ‡ e-mail addresses: [email protected] g µν = σ µν + gh µν , where σ µν is the flat metric and g is the couplingconstant, then we can indeed couple Dirac spinors to h µν in the space of for-mal series with the maximum derivative order equal to one in h µν , such thatthe final results agree with the usual couplings between the spin-1/2 and themassless spin-two field in the vierbein formulation [23]. Thus, our approachenvisages two different aspects. One is related to the couplings between thespin-two fields and the Dirac field, while the other focuses on proving the im-possibility of cross-interactions among different gravitons via Dirac spinors.In order to make the analysis as clear as possible, we initially consider thecase of the couplings between a single Pauli-Fierz field [24] and a Diracfield. In this setting we compute the interaction terms to order two in thecoupling constant. Next, we prove the isomorphism between the local BRSTcohomologies corresponding to the Pauli-Fierz theory and respectively to thelinearized version of the vierbein formulation of the spin-two field. Since thedeformation procedure is controlled by the local BRST cohomology of thefree theory (in ghost number zero and one), the previous isomorphism allowsus to translate the results emerging from the Pauli-Fierz formulation intothe vierbein version and conversely. In this manner we obtain that the firsttwo orders of the interacting lagrangian resulting from our setting originate2n the development of the full interacting lagrangian L (int) = e ¯ ψ (i e µa γ a D µ ψ − mψ ) + egM (cid:16) ¯ ψψ (cid:17) , where e µa are the vierbein fields, e is the inverse of their determinant, e =(det ( e µa )) − , D µ is the full covariant derivative and M (cid:16) ¯ ψψ (cid:17) is a polyno-mial in ¯ ψψ . Here and in the sequel g is the coupling constant (deformationparameter).The term eM (cid:16) ¯ ψψ (cid:17) is usually omitted in most of the textbookson General Relativity. However, it is consistent with the gauge symmetriesof the lagrangian L + L (int) , where L is the full spin-two lagrangian in thevierbein formulation. With this result at hand, we start from a finite sum ofPauli-Fierz actions and a Dirac field, and prove that there are no consistentcross-interactions between different gravitons in the presence of a Dirac field.This paper is organized in eight sections. In Section 2 we construct theBRST symmetry of a free model with a single Pauli-Fierz field and a Diracfield. Section 3 briefly addresses the deformation procedure based on BRSTsymmetry. In Section 4 we compute the first two orders of the interactionsbetween one graviton and a Dirac spinor. Section 5 is dedicated to the proofof the isomorphism between the local BRST cohomologies correspondingto the Pauli-Fierz theory and respectively to the linearized version of thevierbein formulation for the spin-two field. In Section 6 we connect the resultsobtained in Section 4 to those from the vierbein formulation. Section 7 isdevoted to the proof of the fact that there are no consistent cross-interactionsamong different gravitons in the presence of a Dirac field. Section 8 exposesthe main conclusions of the paper. The paper also contains two appendixsections, in which some statements from the body of the paper are proved. Our starting point is represented by a free model, whose lagrangian action iswritten like the sum between the action of the linearized version of Einstein-Hilbert gravity (the Pauli-Fierz action [24]) and that of a massive Dirac field S L0 h h µν , ψ, ¯ ψ i = Z d x (cid:18) −
12 ( ∂ µ h νρ ) ( ∂ µ h νρ ) + ( ∂ µ h µρ ) ( ∂ ν h νρ ) − ( ∂ µ h ) ( ∂ ν h νµ ) + 12 ( ∂ µ h ) ( ∂ µ h ) + ¯ ψ (i γ µ ( ∂ µ ψ ) − mψ ) (cid:19) Z d x (cid:16) L (PF) + L (D)0 (cid:17) . (1)Everywhere in the paper we use the flat Minkowski metric of ‘mostly plus’signature, σ µν = ( − + ++). In the above h denotes the trace of the Pauli-Fierz field, h = σ µν h µν , and the fermionic fields ψ and ¯ ψ are considered to becomplex (Dirac) spinors ( ¯ ψ = ψ † γ ). We work with the Dirac representationof the γ -matrices γ † µ = γ γ µ γ , µ = 0 , , (2)where † signifies the operation of Hermitian conjugation. Action (1) possessesan irreducible and Abelian generating set of gauge transformations δ ǫ h µν = ∂ ( µ ǫ ν ) , δ ǫ ψ = δ ǫ ¯ ψ = 0 , (3)with ǫ µ bosonic gauge parameters. The parantheses signify symmetrization;they are never divided by the number of terms: e.g., ∂ ( µ ǫ ν ) = ∂ µ ǫ ν + ∂ ν ǫ µ , andthe minimum number of terms is always used. The same is valid with respectto the notation [ µ · · · ν ], which means antisymmetrization with respect to theindices between brackets.In order to construct the BRST symmetry for (1) we introduce the fermionicghosts η µ corresponding to the gauge parameters ǫ µ and associate antifieldswith the original fields and ghosts, respectively denoted by n h ∗ µν , ψ ∗ , ¯ ψ ∗ o and { η ∗ µ } . (The statistics of the antifields is opposite to that of the correlatedfields/ghosts.) The antifields of the Dirac fields are bosonic spinors, assumedto satisfy the properties (cid:16) ¯ ψ ∗ (cid:17) † γ = − ψ ∗ , γ ( ψ ∗ ) † = − ¯ ψ ∗ . (4)Since the gauge generators of the free theory under study are field indepen-dent and irreducible, it follows that the BRST differential simply decomposesinto s = δ + γ, (5)where δ represents the Koszul-Tate differential, graded by the antighost num-ber agh (agh ( δ ) = − γ stands for the exterior derivative along thegauge orbits, whose degree is named pure ghost number pgh (pgh ( γ ) = 1).These two degrees do not interfere (pgh ( δ ) = 0, agh ( γ ) = 0). The overalldegree from the BRST complex is known as the ghost number gh and is4efined like the difference between the pure ghost number and the antighostnumber, such that gh ( δ ) = gh ( γ ) = gh ( s ) = 1. If we make the notationsΦ α = (cid:16) h µν , ψ, ¯ ψ (cid:17) , Φ ∗ α = (cid:16) h ∗ µν , ψ ∗ , ¯ ψ ∗ (cid:17) , (6)then, according to the standard rules of the BRST formalism, the degrees ofthe BRST generators are valued likeagh (Φ α ) = agh ( η µ ) = 0 , agh (cid:16) Φ ∗ α (cid:17) = 1 , agh ( η ∗ µ ) = 2 , (7)pgh (Φ α ) = 0 , pgh ( η µ ) = 1 , pgh (cid:16) Φ ∗ α (cid:17) = pgh ( η ∗ µ ) = 0 . (8)The actions of the differentials δ and γ on the generators from the BRSTcomplex are given by δh ∗ µν = 2 H µν , δψ ∗ = − (cid:16) m ¯ ψ + i ∂ µ ¯ ψγ µ (cid:17) , (9) δ ¯ ψ ∗ = − (i γ µ ∂ µ ψ − mψ ) , δη ∗ µ = − ∂ ν h ∗ µν , (10) δ Φ α = 0 = δη µ , (11) γ Φ ∗ α = 0 = γη ∗ µ , (12) γh µν = ∂ ( µ η ν ) , γψ = 0 = γ ¯ ψ, γη µ = 0 , (13)where H µν is the linearized Einstein tensor H µν = K µν − σ µν K, (14)with K µν and K the linearized Ricci tensor and respectively the linearizedscalar curvature, both obtained from the linearized Riemann tensor K µναβ = −
12 ( ∂ µ ∂ α h νβ + ∂ ν ∂ β h µα − ∂ ν ∂ α h µβ − ∂ µ ∂ β h να ) , (15)via its simple and double traces K µα = σ νβ K µναβ , K = σ µα σ νβ K µναβ . (16)The BRST differential is known to have a canonical action in a structurenamed antibracket and denoted by the symbol ( , ) ( s · = (cid:16) · , ¯ S (cid:17) ), which isobtained by decreeing the fields/ghosts respectively conjugated to the cor-responding antifields. The generator of the BRST symmetry is a bosonic5unctional of ghost number zero, which is solution to the classical masterequation (cid:16) ¯ S, ¯ S (cid:17) = 0. The full solution to the master equation for the freemodel under study reads as¯ S = S L0 h h µν , ψ, ¯ ψ i + Z d x h ∗ µν ∂ ( µ η ν ) . (17) We begin with a “free” gauge theory, described by a lagrangian action S L0 [Φ α ],invariant under some gauge transformations δ ǫ Φ α = Z α α ǫ α , i.e. δS L0 δ Φ α Z α α =0, and consider the problem of constructing consistent interactions among thefields Φ α such that the couplings preserve both the field spectrum and theoriginal number of gauge symmetries. This matter is addressed by means ofreformulating the problem of constructing consistent interactions as a defor-mation problem of the solution to the master equation corresponding to the“free” theory [21]. Such a reformulation is possible due to the fact that thesolution to the master equation contains all the information on the gaugestructure of the theory. If an interacting gauge theory can be consistentlyconstructed, then the solution ¯ S to the master equation (cid:16) ¯ S, ¯ S (cid:17) = 0 associ-ated with the “free” theory can be deformed into a solution S ¯ S → S = ¯ S + gS + g S + · · · = ¯ S + g Z d D x a + g Z d D x b + · · · , (18)of the master equation for the deformed theory( S, S ) = 0 , (19)such that both the ghost and antifield spectra of the initial theory are pre-served. The equation (19) splits, according to the various orders in thecoupling constant (deformation parameter) g , into a tower of equations: (cid:16) ¯ S, ¯ S (cid:17) = 0 , (20)2 (cid:16) S , ¯ S (cid:17) = 0 , (21)2 (cid:16) S , ¯ S (cid:17) + ( S , S ) = 0 , (22)6 S , ¯ S (cid:17) + ( S , S ) = 0 , (23)...The equation (20) is fulfilled by hypothesis. The next one requires thatthe first-order deformation of the solution to the master equation, S , is acocycle of the “free” BRST differential s · = (cid:16) · , ¯ S (cid:17) . However, only cohomo-logically non-trivial solutions to (21) should be taken into account, as theBRST-exact ones can be eliminated by some (in general non-linear) fieldredefinitions. This means that S pertains to the ghost number zero co-homological space of s , H ( s ), which is generically non-empty due to itsisomorphism to the space of physical observables of the “free” theory. Ithas been shown (on behalf of the triviality of the antibracket map in thecohomology of the BRST differential) that there are no obstructions in find-ing solutions to the remaining equations, namely (22) and (23) and so on.However, the resulting interactions may be non-local, and there might evenappear obstructions if one insists on their locality. The analysis of theseobstructions can be done by means of standard cohomological techniques. H ( γ ) and H ( δ | d ) This section is devoted to the investigation of consistent cross-couplings thatcan be introduced between a spin-two field and a massive Dirac field. Thismatter is addressed in the context of the antifield-BRST deformation proce-dure briefly addressed in the above and relies on computing the solutions tothe equations (21)–(23), etc., with the help of the free BRST cohomology.For obvious reasons, we consider only smooth, local, (background) Lorentzinvariant and, moreover, Poincar´e invariant quantities (i.e. we do not allowexplicit dependence on the spacetime coordinates). The smoothness of thedeformations refers to the fact that the deformed solution to the master equa-tion (18) is smooth in the coupling constant g and reduces to the originalsolution (17) in the free limit g = 0. In addition, we require the conservationof the number of derivatives on each field (this condition is frequently met inthe literature; for instance, see the case of cross-interactions for a collection7f Pauli-Fierz fields [15] or the couplings between the Pauli-Fierz and themassless Rarita-Schwinger fields [14]). If we make the notation S = R d x a ,with a a local function, then the equation (21), which we have seen thatcontrols the first-order deformation, takes the local form sa = ∂ µ m µ , gh ( a ) = 0 , ε ( a ) = 0 , (24)for some local m µ and it shows that the non-integrated density of the first-order deformation pertains to the local cohomology of the BRST differentialin ghost number zero, a ∈ H ( s | d ), where d denotes the exterior spacetimedifferential. The solution to the equation (24) is unique up to s -exact piecesplus divergences a → a + sb + ∂ µ n µ , gh ( b ) = − , ε ( b ) = 1 , gh ( n µ ) = 0 , ε ( n µ ) = 0 . (25)At the same time, if the general solution of (24) is found to be completelytrivial, a = sb + ∂ µ n µ , then it can be made to vanish a = 0.In order to analyze the equation (24), we develop a according to theantighost number a = I X i =0 a i , agh ( a i ) = i, gh ( a i ) = 0 , ε ( a i ) = 0 , (26)and take this decomposition to stop at some finite value I of the antighostnumber. The fact that I in (26) is finite can be argued like in [15]. Insertingthe above expansion into the equation (24) and projecting it on the variousvalues of the antighost number with the help of the splitting (5), we obtainthe tower of equations γa I = ∂ µ ( I ) m µ , (27) δa I + γa I − = ∂ µ ( I − m µ , (28) δa i + γa i − = ∂ µ ( i − m µ , ≤ i ≤ I − , (29)where (cid:18) ( i ) m µ (cid:19) i =0 ,I are some local currents with agh (cid:18) ( i ) m µ (cid:19) = i . Moreover, ac-cording to the general result from [15] in the absence of the collection indices,the equation (27) can be replaced in strictly positive antighost numbers by γa I = 0 , I > . (30) This is because the presence of the matter fields does not modify the general resultson H ( γ ) presented in [15]. γ ( γ = 0), the solution to the equation(30) is clearly unique up to γ -exact contributions a I → a I + γb I , agh ( b I ) = I, pgh ( b I ) = I − , ε ( b I ) = 1 . (31)Meanwhile, if it turns out that a I reduces to γ -exact terms only, a I = γb I ,then it can be made to vanish, a I = 0. The non-triviality of the first-orderdeformation a is thus translated at its highest antighost number componentinto the requirement that a I ∈ H I ( γ ), where H I ( γ ) denotes the cohomologyof the exterior longitudinal derivative γ in pure ghost number equal to I . So,in order to solve the equation (24) (equivalent with (30) and (28) and (29)),we need to compute the cohomology of γ , H ( γ ), and, as it will be made clearbelow, also the local cohomology of δ in pure ghost number zero, H ( δ | d ).Using the results on the cohomology of the exterior longitudinal differen-tial for a collection of Pauli-Fierz fields [15], as well as the definitions (12) and(13), we can state that H ( γ ) is generated on the one hand by Φ ∗ α , η ∗ µ , ψ , ¯ ψ and K µναβ together with all of their spacetime derivatives and, on the otherhand, by the ghosts η µ and ∂ [ µ η ν ] . So, the most general (and non-trivial),local solution to (30) can be written, up to γ -exact contributions, as a I = α I (cid:16) [ ψ ] , h ¯ ψ i , [ K µναβ ] , h Φ ∗ α i , h η ∗ µ i(cid:17) ω I (cid:16) η µ , ∂ [ µ η ν ] (cid:17) , (32)where the notation f ([ q ]) means that f depends on q and its derivatives upto a finite order, while ω I denotes the elements of a basis in the space ofpolynomials with pure ghost number I in the corresponding ghosts and theirantisymmetrized first-order derivatives. The objects α I have the pure ghostnumber equal to zero and are required to fulfill the property agh ( α I ) = I inorder to ensure that the ghost number of a I is equal to zero. Since they have abounded number of derivatives and a finite antighost number, α I are actuallypolynomials in the linearized Riemann tensor, in the antifields, in all of theirderivatives, as well as in the derivatives of the Dirac fields. The anticommut-ing behaviour of the Dirac spinors induces that α I are polynomials also inthe undifferentiated Dirac fields, so we conclude that these elements exhibita polynomial character in all of their arguments. Due to their γ -closeness, γα I = 0, α I will be called “invariant polynomials”. In zero antighost number,the invariant polynomials are polynomials in the linearized Riemann tensor K µναβ , in the Dirac spinors, as well as in their derivatives.Inserting (32) in (28) we obtain that a necessary (but not sufficient)condition for the existence of (non-trivial) solutions a I − is that the invariant9olynomials α I are (non-trivial) objects from the local cohomology of theKoszul-Tate differential H ( δ | d ) in pure ghost number zero and in strictlypositive antighost numbers I > δα I = ∂ µ ( I − j µ , agh ( I − j µ ! = I − , pgh ( I − j µ ! = 0 . (33)We recall that H ( δ | d ) is completely trivial in both strictly positive antighost and pure ghost numbers (for instance, see [22], Theorem 5.4 and [25]). Us-ing the fact that the Cauchy order of the free theory under study is equalto two together with the general results from [22], according to which thelocal cohomology of the Koszul-Tate differential in pure ghost number zerois trivial in antighost numbers strictly greater than its Cauchy order, we canstate that H J ( δ | d ) = 0 for all J > , (34)where H J ( δ | d ) represents the local cohomology of the Koszul-Tate differen-tial in zero pure ghost number and in antighost number J . An interestingproperty of invariant polynomials for the free model under study is that if aninvariant polynomial α J , with agh ( α J ) = J ≥
2, is trivial in H J ( δ | d ), thenit can be taken to be trivial also in H inv J ( δ | d ), i.e. (cid:18) α J = δb J +1 + ∂ µ ( J ) c µ , agh ( α J ) = J ≥ (cid:19) ⇒ α J = δβ J +1 + ∂ µ ( J ) γ µ , (35)with both β J +1 and ( J ) γ µ invariant polynomials. Here, H inv J ( δ | d ) denotes theinvariant characteristic cohomology (the local cohomology of the Koszul-Tate differential in the space of invariant polynomials) in antighost number J . This property is proved in [15] in the case of a collection of Pauli-Fierzfields and remains valid in the case considered here since the matter fieldsdo not carry gauge symmetries, so we can write that H inv J ( δ | d ) = 0 for all J > . (36)For the same reason the antifields of the matter fields can bring only trivialcontributions to H J ( δ | d ) and H inv J ( δ | d ) for J ≥
2, so the results from [15]concerning both H ( δ | d ) in pure ghost number zero and H inv2 ( δ | d ) remainvalid. These cohomological spaces are still spanned by the undifferentiatedantifields corresponding to the ghosts H ( δ | d ) and H inv2 ( δ | d ) : ( η ∗ µ ) . (37)10n contrast to the groups ( H J ( δ | d )) J ≥ and (cid:16) H inv J ( δ | d ) (cid:17) J ≥ , which are finite-dimensional, the cohomology H ( δ | d ) in pure ghost number zero, known tobe related to global symmetries and ordinary conservation laws, is infinite-dimensional since the theory is free. Moreover, H ( δ | d ) involves non-triviallythe antifields of the matter fields.The previous results on H ( δ | d ) and H inv ( δ | d ) in strictly positive antighostnumbers are important because they control the obstructions to removing theantifields from the first-order deformation. More precisely, based on the for-mulas (33)–(36), one can successively eliminate all the pieces of antighostnumber strictly greater that two from the non-integrated density of the first-order deformation by adding only trivial terms, so one can take, without lossof non-trivial objects, the condition I ≤ H inv2 ( δ | d ) for I = 2, and respectively from H ( δ | d )for I = 1. In the case I = 2 the non-integrated density of the first-order deformation(26) becomes a = a + a + a . (38)We can further decompose a in a natural manner as a sum between threekinds of deformations a = a (PF) + a (int) + a (Dirac) , (39)where a (PF) contains only fields/ghosts/antifields from the Pauli-Fierz sector, a (int) describes the cross-interactions between the two theories (so it effec-tively mixes both sectors), and a (Dirac) involves only the Dirac sector. Thecomponent a (PF) is completely known (for a detailed analysis see [15]) andsatisfies individually an equation of the type (24). It admits a decompositionsimilar to (38) a (PF) = a (PF)0 + a (PF)1 + a (PF)2 , (40)where a (PF)2 = 12 η ∗ µ η ν ∂ [ µ η ν ] , (41) a (PF)1 = h ∗ µρ (cid:16) ( ∂ ρ η ν ) h µν − η ν ∂ [ µ h ν ] ρ (cid:17) , (42)11nd a (PF)0 is the cubic vertex of the Einstein-Hilbert lagrangian plus a cos-mological term . Consequently, it follows that a (int) and a (Dirac) are subjectto some separate equations sa (int) = ∂ µ m (int) µ , (43) sa (Dirac) = ∂ µ m (Dirac) µ , (44)for some local m µ ’s. In the sequel we analyze the general solutions to theseequations.Since the Dirac field does not carry gauge symmetries of its own, it resultsthat the Dirac sector can only occur in antighost number one and zero, so,without loss of generality, we take a (int) = a (int)0 + a (int)1 (45)in (43), where the components involved in the right-hand side of (45) aresubject to the equations γa (int)1 = 0 , (46) δa (int)1 + γa (int)0 = ∂ µ (0) m (int) µ . (47)According to (32) in pure ghost number one and because ω is spanned by ω = (cid:16) η µ , ∂ [ µ η ν ] (cid:17) , we infer that the most general expression of a (int)1 as solution to the equation(46), which complies with all the general requirements imposed on the in-teracting theory (including the preservation of the number of derivatives oneach field with respect to the free theory), is a (int)1 = (cid:16) k ψ ∗ ( ∂ α ψ ) + k † (cid:16) ∂ α ¯ ψ (cid:17) ¯ ψ ∗ + k ψ ∗ γ α ψ + k † ¯ ψγ α ¯ ψ ∗ + k ψ ∗ γ α γ µ ( ∂ µ ψ ) + k † (cid:16) ∂ µ ¯ ψ (cid:17) γ µ γ α ¯ ψ ∗ (cid:17) η α + (cid:16) k ¯ ψ h γ α , γ β i ¯ ψ ∗ − k † ψ ∗ h γ α , γ β i ψ (cid:17) ∂ [ α η β ] . (48) The terms a (PF)2 and a (PF)1 given in (41) and (42) differ from the corresponding onesin [15] by a γ -exact and respectively a δ -exact contribution. However, the difference be-tween our a (PF)2 + a (PF)1 and the corresponding sum from [15] is a s -exact modulo d quantity.The associated component of antighost number zero, a (PF)0 , is nevertheless the same inboth formulations. As a consequence, the object a (PF) and the first-order deformationin [15] belong to the same cohomological class from H ( s | d ). k j ) j =1 , are arbitrary complex functions of ¯ ψ and ψ . If we representthem like k j = u j + i v j , j = 1 , , (49)with u j and v j real functions, then direct calculations, based on the definitions(9)–(13), lead to the elimination of some of these functions from (48). Forinstance, the pieces proportional with the real part of k are u (cid:16) ψ ∗ γ α γ µ ( ∂ µ ψ ) + (cid:16) ∂ µ ¯ ψ (cid:17) γ µ γ α ¯ ψ ∗ (cid:17) η α = s (cid:16) − i u ψ ∗ γ α ¯ ψ ∗ η α (cid:17) − m (cid:16) (i u ) ψ ∗ γ α ψ + (i u ) † ¯ ψγ α ¯ ψ ∗ (cid:17) η α . (50)However, we already have in a (int)1 terms proportional with ψ ∗ γ α ψη α and¯ ψγ α ¯ ψ ∗ η α . So, if we set k → k ′ = k − i mu in (48), then we can absorb thecomponents proportional with u into those containing k ′ and ( k ′ ) † since onecan always remove the s -exact terms from a (int) appearing in (50) through aredefinition of the type (25) corresponding to n µ = 0. The above analysisleads to the fact that we can safely take u = 0 (51)in (48), without loss of independent contributions to a (int)1 . Strictly speaking,one may add to a (int)1 given by (48) a term of the type ˜ a (int)1 = h ∗ µν η µ F ν (cid:16) ¯ ψ, ψ (cid:17) .On the one hand, we observe that by applying δ on (48), then a (int)1 , ifconsistent, would lead to some a (int)0 which contains a single field h µν (orone of its first-order derivatives). On the other hand, from the expressionof δ ˜ a (int)1 we notice that if consistent, it would give an ˜ a (int)0 with two h µν (or one h µν and one of its first-order derivatives). As a consequence, ˜ a (int)1 must satisfy, independently of a (int)1 , an equation of the type δ ˜ a (int)1 + γ ˜ a (int)0 = ∂ µ ρ µ . However, ˜ a (int)1 produces a consistent ˜ a (int)0 if and only if F ν (cid:16) ¯ ψ, ψ (cid:17) = ∂ ν F (cid:16) ¯ ψ, ψ (cid:17) . The proof of the last statement can be found in Appendix A.Under these conditions, it is easy to see that˜ a (int)1 = ∂ ν (cid:16) h ∗ µν η µ F (cid:16) ¯ ψ, ψ (cid:17)(cid:17) − γ (cid:18) h ∗ µν h µν F (cid:16) ¯ ψ, ψ (cid:17)(cid:19) + s (cid:18) η ∗ µ η µ F (cid:16) ¯ ψ, ψ (cid:17)(cid:19) , (52)so ˜ a (int)1 is trivial. 13n order to analyze the solution a (int)0 to the equation (47), it is useful todecompose δa (int)1 along the number of derivatives δa (int)1 = X k =0 (cid:16) δa (int)1 (cid:17) k , (53)where (cid:16) δa (int)1 (cid:17) k denotes the piece with k -derivatives from δa (int)1 . Accordingto this decomposition, it follows that each (cid:16) δa (int)1 (cid:17) k should be written in a γ -exact modulo d form, such that (47) is indeed satisfied. Using the definitions(9)–(11), we consequently obtain (cid:16) δa (int)1 (cid:17) = − m (cid:16) k + k † (cid:17) ¯ ψγ α ψη α . (54)Due to the fact that the right-hand side of (54) contains no derivatives, itresults that these terms neither reduce to a total divergence nor can notproduce γ -exact terms, so they must be made to vanish k + k † = 0 , (55)which is equivalent, by means of (49), with u = 0 . (56)The definitions (9)–(11) and the result (51) together with (56) further leadto (cid:16) δa (int)1 (cid:17) = − m (cid:18) k − v m + i v (cid:19) ¯ ψ ( ∂ α ψ ) + (cid:18) k − v m + i v (cid:19) † (cid:16) ∂ α ¯ ψ (cid:17) ψ ! η α + m (cid:18) v m + i v (cid:19) (cid:16) ∂ α ¯ ψ (cid:17) h γ α , γ β i ψ − (cid:18) v m + i v (cid:19) † ¯ ψ h γ α , γ β i ( ∂ α ψ ) ! η β − m (cid:16) k − k † (cid:17) ¯ ψ h γ α , γ β i ψ∂ [ α η β ] . (57)If we make the notations U (cid:16) ¯ ψ, ψ (cid:17) = mk − v + i mv , (58) V (cid:16) ¯ ψ, ψ (cid:17) = v + i mv , (59)14hen the formula (57) becomes (cid:16) δa (int)1 (cid:17) = ∂ α (cid:18) − U † ¯ ψψη α + 12 V ¯ ψ [ γ α , γ β ] ψη β (cid:19) + γ (cid:18) U † ¯ ψψh (cid:19) + (cid:18)(cid:16) U † − U (cid:17) ¯ ψ ( ∂ α ψ ) + 12 (cid:16) V + V † (cid:17) ¯ ψ h γ α , γ β i ( ∂ β ψ ) + 12 ¯ ψ h γ α , γ β i ψ ( ∂ β V )+ ¯ ψψ (cid:16) ∂ α U † (cid:17)(cid:17) η α − (cid:18) V + m (cid:16) k − k † (cid:17)(cid:19) ¯ ψ h γ α , γ β i ψ∂ [ α η β ] . (60)The right-hand side from (60) is γ -exact modulo d if the functions U and V satisfy the equations (cid:16) U † − U (cid:17) ¯ ψ ( ∂ α ψ ) + ¯ ψψ (cid:16) ∂ α U † (cid:17) + 12 ¯ ψ h γ α , γ β i ψ ( ∂ β V )+ 12 (cid:16) V + V † (cid:17) ¯ ψ h γ α , γ β i ( ∂ β ψ ) = ∂ β P βα , (61)where 12 (cid:16) P αβ − P βα (cid:17) = − (cid:18) V + 2 m (cid:16) k − k † (cid:17)(cid:19) ¯ ψ h γ α , γ β i ψ. (62)By direct computation we find that the left-hand side of (61) reduces to atotal derivative if − U ¯ ψ ( ∂ α ψ ) − U † (cid:16) ∂ α ¯ ψ (cid:17) ψ = 12 ∂ β (cid:16) P βα + P αβ − σ αβ U † ¯ ψψ (cid:17) , (63) − V (cid:16) ∂ β ¯ ψ (cid:17) h γ α , γ β i ψ + 12 V † ¯ ψ h γ α , γ β i ( ∂ β ψ )= 12 ∂ β (cid:16) P βα − P αβ − V ¯ ψ h γ α , γ β i ψ (cid:17) . (64)Now, the left-hand side from (63) is a total derivative if U = U † (65)and, in addition, U is a polynomial in ¯ ψψ with real coefficients. In thissituation we have that − U ¯ ψ ( ∂ α ψ ) − U † (cid:16) ∂ α ¯ ψ (cid:17) ψ = ∂ α W, (66)15here the function W is defined via the relation U = − dWd (cid:16) ¯ ψψ (cid:17) , (67)such that (63) can be written like ∂ β (cid:16) P βα + P αβ − σ αβ (cid:16) U ¯ ψψ + W (cid:17)(cid:17) = 0 . (68)Since the quantity P βα + P αβ − σ αβ (cid:16) U ¯ ψψ + W (cid:17) contains no derivatives,from (68) we obtain that P βα + P αβ = 2 σ αβ (cid:16) U ¯ ψψ + W (cid:17) . (69)Inserting (62) in (64), we arrive at − V (cid:16) ∂ β ¯ ψ (cid:17) h γ α , γ β i ψ + 12 V † ¯ ψ h γ α , γ β i ( ∂ β ψ )= 2 m∂ β (cid:16)(cid:16) k − k † (cid:17) ¯ ψ h γ α , γ β i ψ (cid:17) . (70)At this stage we observe that the left-hand side of the previous formula leadsto a total derivative if V is a purely imaginary constant V = const , V + V † = 0 , (71)in which case the relation (70) takes the form ∂ β (cid:18) − V ¯ ψ h γ α , γ β i ψ (cid:19) = 2 m∂ β (cid:16)(cid:16) k − k † (cid:17) ¯ ψ h γ α , γ β i ψ (cid:17) , (72)and thus V = − m (cid:16) k − k † (cid:17) . (73)Relying on the last result, by means of (62) we obtain P αβ − P βα = 0 , (74)such that (69) gives P αβ = σ αβ (cid:16) U ¯ ψψ + W (cid:17) . (75)16et us analyze the results deduced so far. The relations (59) and (71) allowus to state that v must be a true constant v = const , (76)while v must vanish v = 0 . (77)In the meantime, the formula (73) implies that v = − v . (78)Using (58), (65) and (77) we arrive at v = − v , (79)such that U = mu . (80)Introducing all the above results into (60), we infer that (cid:16) δa (int)1 (cid:17) = ∂ α (cid:18) W (cid:16) ¯ ψψ (cid:17) η α + i mv ψ h γ α , γ β i ψη β (cid:19) − γ (cid:18) W (cid:16) ¯ ψψ (cid:17) h (cid:19) . (81)Taking into account the results (77)–(79), the pieces containing two deriva-tives from δa (int)1 can be written like (cid:16) δa (int)1 (cid:17) = ∂ µ (cid:16) i u ¯ ψ ( γ α ( ∂ α ψ ) η µ − γ µ ( ∂ α ψ ) η α )+ (cid:18) i u − v (cid:19) ¯ ψγ µ h γ α , γ β i ψ∂ [ α η β ] + v ψ (cid:16) γ α h γ µ , γ β i − γ µ h γ α , γ β i(cid:17) ( ∂ α ψ ) η β (cid:19) + γ (cid:18) − i u ψ (cid:16) γ µ ( ∂ µ ψ ) h − γ α (cid:16) ∂ β ψ (cid:17) h αβ (cid:17) − (cid:18) i u − v (cid:19) ¯ ψγ µ h γ α , γ β i ψ∂ [ α h β ] µ + v ψγ µ h γ α , γ β i ( ∂ α ψ ) h βµ (cid:19) + (cid:16) i ( ∂ µ u ) ¯ ψγ µ ( ∂ α ψ ) − i ( ∂ α u ) ¯ ψγ µ ( ∂ µ ψ )172 v (cid:16)(cid:16) ∂ µ ¯ ψ (cid:17) γ µ ( ∂ α ψ ) − (cid:16) ∂ µ ¯ ψ (cid:17) γ α ( ∂ µ ψ ) (cid:17)(cid:17) η α + (cid:18) i2 ( u + 16 u + i v ) ¯ ψγ α ∂ β ψ − ¯ ψγ µ h γ α , γ β i (cid:18) ψ i ( ∂ µ u ) + v ∂ µ ψ ) (cid:19)(cid:19) ∂ [ α η β ] . (82)In consequence, (cid:16) δa (int)1 (cid:17) is γ -exact modulo d if v = 0 , (83) u (cid:16) ¯ ψψ (cid:17) + 16 u (cid:16) ¯ ψ, ψ (cid:17) + i v = 0 , (84) ∂ µ u (cid:16) ¯ ψ, ψ (cid:17) = ∂ µ u (cid:16) ¯ ψψ (cid:17) = 0 . (85)By means of (84)–(85) we get that the functions u and u are some constants u = const , u = const , (86)related via the formula u = − u . (87)As u is constant, from (67) and (80) we find that W = − u m ¯ ψψ, (88)such that (81) becomes (cid:16) δa (int)1 (cid:17) = ∂ α (cid:16) − u m ¯ ψψη α (cid:17) + γ (cid:18) u m ¯ ψψh (cid:19) . (89)Introducing the results (83) and (87) in (82), it follows that (cid:16) δa (int)1 (cid:17) = ∂ µ (cid:16) i u ¯ ψ ( γ α ( ∂ α ψ ) η µ − γ µ ( ∂ α ψ ) η α − γ µ h γ α , γ β i ψ∂ [ α η β ] (cid:19)(cid:19) + γ (cid:18) − i u ψ (cid:16) γ µ ( ∂ µ ψ ) h − γ α (cid:16) ∂ β ψ (cid:17) h αβ − γ µ h γ α , γ β i ψ∂ [ α h β ] µ (cid:19)(cid:19) . (90)Putting together the formulas (49), (51), (56), (77)–(79), (83), and (86)–(87), we conclude that the most general expression of a (int)1 that produces a18onsistent component in antighost number zero as solution to the equation(47) and complies with all the general requirements imposed at the beginningof this section can be expressed in terms of a single arbitrary, real constant, u . From now on we will denote this constant by k , such that the resulting a (int)1 becomes a (int)1 = − k (cid:18) (cid:16) ¯ ψ h γ α , γ β i ¯ ψ ∗ − ψ ∗ h γ α , γ β i ψ (cid:17) ∂ [ α η β ] − (cid:16) ψ ∗ ( ∂ α ψ ) + (cid:16) ∂ α ¯ ψ (cid:17) ¯ ψ ∗ (cid:17) η α (cid:17) . (91)Then, using (47), (53), (54)–(55), (89) and (90) we find that the correspond-ing a (int)0 is a (int)0 = k (cid:16) ¯ ψ (i γ µ ( ∂ µ ψ ) − mψ ) h − i ¯ ψγ α (cid:16) ∂ β ψ (cid:17) h αβ − i8 ¯ ψγ µ h γ α , γ β i ψ∂ [ α h β ] µ (cid:19) + ¯ a (int)0 . (92)In the above, ¯ a (int)0 represents the general local solution to the homoge-neous equation γ ¯ a (int)0 = ∂ µ ¯ m (int) µ , (93)for some local ¯ m (int) µ . Such solutions correspond to ¯ a (int)1 = 0 and thus theycannot deform either the gauge algebra or the gauge transformations, butsimply the lagrangian at order one in the coupling constant. There are twomain types of solutions to (93). The first one corresponds to ¯ m (int) µ = 0and is given by gauge-invariant, non-integrated densities constructed fromthe original fields and their spacetime derivatives. According to (32) forboth pure ghost and antighost numbers equal to zero, they are given by¯ a ′ (int)0 = ¯ a ′ (int)0 (cid:16) [ ψ ] , h ¯ ψ i , [ K µναβ ] (cid:17) , up to the conditions that they effectivelydescribe cross-couplings between the two types of fields and cannot be writ-ten in a divergence-like form. Unfortunately, this type of solutions mustdepend on the linearized Riemann tensor (and possibly of its derivatives) inorder to provide cross-couplings, and thus would lead to terms with at leasttwo derivatives of the Dirac spinors. So, by virtue of the derivative orderassumption, they must be discarded by setting ¯ a ′ (int)0 = 0. The second kindof solutions is associated with ¯ m (int) µ = 0 in (93), being understood thatthey lead to cross-interactions, cannot be written in a divergence-like form19nd contain at most one derivative of the fields. Consequently, we obtainthat γ ¯ a (int)0 = ∂ ρ ∂ ¯ a (int)0 ∂ ( ∂ ρ h µν ) ∂ ( µ η ν ) + δ ¯ a (int)0 δh µν ∂ ( µ η ν ) = ∂ ρ ∂ ¯ a (int)0 ∂ ( ∂ ρ h µν ) ∂ ( µ η ν ) + 2 δ ¯ a (int)0 δh ρµ η µ − ∂ ρ δ ¯ a (int)0 δh ρµ η µ . (94)Thus, this ¯ a (int)0 fulfills (93) if and only if its Euler-Lagrange derivatives withrespect to the Pauli-Fierz fields satisfy the equations ∂ ρ δ ¯ a (int)0 δh ρµ = 0 . (95)Since ¯ a (int)0 may contain at most one derivative, it follows that the solutionto (95) reads as δ ¯ a (int)0 δh µν = ∂ ρ D ρµν , (96)where D ρµν contains no derivatives and is antisymmetric in its first two in-dices D ρµν = − D µρν . (97)We insist on the fact that a solution of the type δ ¯ a (int)0 /δh µν = ∂ α ∂ β D µανβ ,with D µανβ possessing the symmetry properties of the Riemann tensor, isnot allowed in our case due to the hypothesis on the derivative order, andhence (96) is the most general solution to the equation (95). Moreover, from(96) we have that D ρµν must be symmetric in its last two indices D ρµν = D ρνµ . (98)The properties (97) and (98) imply that D ρµν = − D µρν = − D µνρ = D νµρ = D νρµ = − D ρνµ = − D ρµν , (99) Strictly speaking, we might have added to the right-hand side of (96) the contribution cσ µν , with c an arbitrary real constant. It would not have led to cross-interactions, butto the cosmological term ch , which has already been considered in a (PF) . D ρµν = 0. Consequently, (96) yields δ ¯ a (int)0 δh µν = 0 , (100)and thus we can write that ¯ a (int)0 = L (cid:16) [ ψ ] , h ¯ ψ i(cid:17) + ∂ µ g µ (cid:16) ψ, ¯ ψ, h αβ (cid:17) . Since weare interested only in cross-interactions, we must set L = 0. At this stage weremain with the trivial solutions¯ a (int)0 = ∂ µ g µ (cid:16) ψ, ¯ ψ, h αβ (cid:17) , (101)which can be completely removed from the first-order deformation via atransformation of the form (25) with b = 0. In conclusion, we can take,without loss of generality ¯ a (int)0 = 0 (102)in the solution (92). As a consequence of the above discussion, we can statethat the antighost number zero component of a (int) reads as a (int)0 = k (cid:16) ¯ ψ (i γ µ ( ∂ µ ψ ) − mψ ) h − i ¯ ψγ α (cid:16) ∂ β ψ (cid:17) h αβ − i8 ¯ ψγ µ h γ α , γ β i ψ∂ [ α h β ] µ (cid:19) . (103)After some computation, we find that a (int)1 + a (int)0 = − k (cid:18) (cid:16) ¯ ψ h γ α , γ β i ¯ ψ ∗ − ψ ∗ h γ α , γ β i ψ (cid:17) ∂ [ α η β ] − (cid:16) ψ ∗ ( ∂ α ψ ) + (cid:16) ∂ α ¯ ψ (cid:17) ¯ ψ ∗ − ( ∂ α ψ ∗ ) ψ − ¯ ψ (cid:16) ∂ α ¯ ψ ∗ (cid:17)(cid:17) η α (cid:19) − k µν h µν + s Λ + ∂ µ v µ , (104)where Θ µν = i2 (cid:16) ¯ ψγ ( µ ∂ ν ) ψ − (cid:16) ∂ ( µ ¯ ψ (cid:17) γ ν ) ψ (cid:17) , (105)represents the stress-energy tensor of the Dirac field, while Λ is given byΛ = − k (cid:16) ψ ∗ ψ + ¯ ψ ¯ ψ ∗ (cid:17) h. (106)Obviously, the term s Λ+ ∂ µ v µ from (104) is cohomologically trivial, and hencecan be discarded. Thus, the coupling between a Dirac field and one graviton21t the first order in the deformation parameter takes the form Θ µν h µν . Wecannot stress enough that is not an assumption, but follows entirely from thedeformation approach developed here. However, for subsequent purposes itis useful to work with the expressions (91) and (103) of a (int)0 and a (int)1 .Finally, we analyze the component a (Dirac) from (39). As the Dirac actionfrom (1) has no non-trivial gauge invariance, it follows that a (Dirac) can onlyreduce to its component of antighost number zero a (Dirac) = a (Dirac)0 (cid:16) [ ψ ] , h ¯ ψ i(cid:17) , (107)which is automatically solution to the equation sa (Dirac) ≡ γa (Dirac)0 = 0.It comes from a (Dirac)1 = 0 and does not deform the gauge transformations(3), but merely modifies the Dirac action. The condition that a (Dirac)0 is ofmaximum derivative order equal to one is translated into a (Dirac)0 = f (cid:16) ¯ ψ, ψ (cid:17) + (cid:16) ∂ µ ¯ ψ (cid:17) g µ (cid:16) ¯ ψ, ψ (cid:17) + g µ (cid:16) ¯ ψ, ψ (cid:17) ( ∂ µ ψ ) , (108)where f , g µ and g µ are polynomials in the undifferentiated spinor fields (sincethey anticommute). The first polynomial is a scalar (bosonic and real), whilethe one-tensors g µ and g µ are fermionic and spinor-like. They are related viathe relation ( g µ ) † γ = g µ (109)in order to ensure that a (Dirac)0 is indeed a scalar. In the previous part of the paper we have seen that the first-order deformationof the theory can be written like the sum between the first-order deformationof the master equation for the Pauli-Fierz theory S (PF)1 , and the ‘interacting’part S (int)1 S (int)1 = Z d x (cid:16) a (int)1 + a (int)0 + a (Dirac)0 (cid:17) , (110)with a (int)1 , a (int)0 and a (Dirac)0 respectively given by (91), (92) and (108).As shown in Appendix B, the first-order deformation is consistent at ordertwo in the coupling constant if the constant k that parametrizes both a (int)1 and a (int)0 is equal to unit k = 1 , (111)22nd the functions appearing in a (Dirac)0 are of the form f (cid:16) ¯ ψ, ψ (cid:17) = M (cid:16) ¯ ψψ (cid:17) , g µ = 0 , g µ = 0 , (112)with M (cid:16) ¯ ψψ (cid:17) a polynomial in ¯ ψψ . Under these circumstances, we have that S = S (PF)2 + S (int)2 , where S (PF)2 can be deduced from [15], and S (int)2 = Z d x (cid:18) − (cid:16) ψ ∗ ( ∂ α ψ ) + (cid:16) ∂ α ¯ ψ (cid:17) ¯ ψ ∗ (cid:17) η β h αβ + 132 (cid:16) ¯ ψ h γ α , γ β i ¯ ψ ∗ − ψ ∗ h γ α , γ β i ψ (cid:17) (cid:18) η σ ∂ [ α h β ] σ − h σ [ α (cid:16) ∂ β ] η σ − ∂ σ η β ] (cid:17)(cid:19) − i4 ¯ ψγ µ ( ∂ ν ψ ) (cid:18) hh µν − h µσ h σν (cid:19) − (cid:16) ¯ ψ i γ µ ( ∂ µ ψ ) − m ¯ ψψ (cid:17) ×× (cid:18) h αβ h αβ − h (cid:19) − i32 ¯ ψγ µ h γ α , γ β i ψ (cid:16) h∂ [ α h β ] µ − h σµ ∂ [ α h β ] σ + h σα (cid:16) ∂ [ β h σ ] µ + ∂ µ h βσ (cid:17)(cid:17) + 12 M (cid:16) ¯ ψψ (cid:17) h (cid:19) . (113)The concrete expression of S (int)2 is inferred also in Appendix B. Making useof (111)–(112), it results that S (int)1 takes the final form S (int)1 = Z d x (cid:18) − (cid:16) ¯ ψ h γ α , γ β i ¯ ψ ∗ − ψ ∗ h γ α , γ β i ψ (cid:17) ∂ [ α η β ] + (cid:16) ψ ∗ ( ∂ α ψ ) + (cid:16) ∂ α ¯ ψ (cid:17) ¯ ψ ∗ (cid:17) η α + 12 (cid:16) ¯ ψ (i γ µ ( ∂ µ ψ ) − mψ ) h − i ¯ ψγ α (cid:16) ∂ β ψ (cid:17) h αβ − i8 ¯ ψγ µ h γ α , γ β i ψ∂ [ α h β ] µ (cid:19) + M (cid:16) ¯ ψψ (cid:17)(cid:19) . (114) In this section we correlate the linearized versions of first- and second-orderformulations of spin-two field theory via the local BRST cohomology. In viewof this, we start from the first-order formulation of spin-two field theory S [ e µa , ω µab ] = − λ Z d x (cid:16) ω abν ∂ µ ( ee µa e νb ) − ω abµ ∂ ν ( ee µa e νb )+ 12 ee µa e νb (cid:16) ω acµ ω bν c − ω acν ω bµ c (cid:17)(cid:19) , (115)23here e µa is the vierbein field and ω µab are the components of the spin con-nection, while e is the inverse of the vierbein determinant e = (det ( e µa )) − . (116)In order to linearize action (115), we develop the vierbein like e µa = δ µa − λ f µa , e = 1 + λ f, (117)where f is the trace of f µa . Consequently, we find that the linearized formof (115) reads as (we come back to the notations µ , ν , etc. for flat indices) S ′ [ f µν , ω µαβ ] = Z d x (cid:18) ω αµα ( ∂ µ f − ∂ ν f µν ) + 12 ω µαβ ∂ [ α f β ] µ − (cid:16) ω αβα ω λλβ − ω µαβ ω αµβ (cid:17)(cid:19) . (118)We mention that the field f µν contains a symmetric, as well as an anti-symmetric part. The above linearized action is invariant under the gaugetransformations δ ǫ f µν = ∂ µ ǫ ν + ǫ µν , δ ǫ ω µαβ = ∂ µ ǫ αβ, (119)where the latter gauge parameters are antisymmetric, ǫ αβ = − ǫ βα . Eliminat-ing the spin connection components on their equations of motion (auxiliaryfields) from (118) ω µαβ ( f ) = 12 (cid:16) ∂ [ µ f α ] β − ∂ [ µ f β ] α − ∂ [ α f β ] µ (cid:17) , (120)we obtain the second-order action S ′ [ f µν , ω µαβ ( f )] = S ′′ [ f µν ] = − Z d x (cid:18) (cid:16) ∂ [ µ f ν ] α (cid:17) (cid:16) ∂ [ µ f ν ] α (cid:17) + 14 (cid:16) ∂ [ µ f ν ] α (cid:17) (cid:16) ∂ [ µ f α ] ν (cid:17) −
12 ( ∂ µ f − ∂ ν f µν ) ( ∂ µ f − ∂ α f µα ) (cid:19) , (121)subject to the gauge invariances δ ǫ f µν = ∂ ( µ ǫ ν ) + ǫ µν . (122)If we decompose f µν in its symmetric and antisymmetric parts f µν = h µν + B µν , h µν = h νµ , B µν = − B νµ , (123)24he action (121) becomes S ′′ [ f µν ] = S ′′ [ h µν , B µν ] = Z d x (cid:18) −
12 ( ∂ µ h νρ ) ( ∂ µ h νρ ) + ( ∂ µ h µρ ) ( ∂ ν h νρ )( − ∂ µ h ) ( ∂ ν h νµ ) + 12 ( ∂ µ h ) ( ∂ µ h ) (cid:19) , (124)while the accompanying gauge transformations are given by δ ǫ h µν = ∂ ( µ ǫ ν ) , δ ǫ B µν = ǫ µν . (125)It is easy to see that the right-hand side of (124) is nothing but the Pauli-Fierzaction S ′′ [ h µν , B µν ] = S (PF)0 [ h µν ] . (126)Now, we show that the local BRST cohomologies associated with the formu-lations (118)–(119), (124)–(125) and the Pauli-Fierz model are isomorphic.As we have previously mentioned, we pass from (118)–(119) to (124)–(125)via the elimination of the auxiliary fields ω µαβ , such that the general theoremsfrom Section 15 of the first reference in [22] ensure the isomorphism H ( s ′ | d ) ≃ H ( s ′′ | d ) , (127)with s ′ and s ′′ the BRST differentials corresponding to (118)–(119) and re-spectively to (124)–(125). On the other hand, we observe that the field B µν does not appear in (124) and is subject to a shift gauge symmetry. Thus,in any cohomological class from H ( s ′′ | d ) one can take a representative thatis independent of B µν , the shift ghosts and all of their antifields. This isbecause these variables form contractible pairs that drop out from H ( s ′′ | d )(see the general results from Section 14 of the first reference in [22]). As aconsequence, we have that H ( s ′′ | d ) ≃ H ( s | d ) , (128)where s is the Pauli-Fierz BRST differential. Combining (127) and (128), wearrive at H ( s ′ | d ) ≃ H ( s ′′ | d ) ≃ H ( s | d ) . (129)Because the local BRST cohomology (in ghost number equal to zero andone) controls the deformation procedure, it results that the last isomorphismsallow one to pass in a consistent manner from the Pauli-Fierz version to thefirst- and second-order ones (in vierbein formulation) during the deformationprocedure. 25 Analysis of the deformed theory
It is easy to see that one can go from (124)–(125) to the Pauli-Fierz versionthrough the partial gauge-fixing B µν = 0. This gauge-fixing is a consequenceof the more general gauge-fixing condition [23] σ µ [ a e µb ] = 0 . (130)In the context of this partial gauge-fixing simple computation leads to thevierbein fields and the inverse of their determinant up to the second order inthe coupling constant as e µa = (0) e µa + g (1) e µa + g e µa + · · · = δ µa − g h µa + 3 g h ρa h µρ + · · · , (131) e = (0) e + g (1) e + g e + · · · = 1 + g h + g (cid:16) h − h µν h µν (cid:17) + · · · . (132)Based on the isomorphisms (129), we can further pass to the analysis ofthe deformed theory obtained in the previous sections. The component ofantighost number equal to zero in S (int)1 is precisely the interacting lagrangianat order one in the coupling constant L (int)1 = a (int)0 + a (Dirac)0 = (cid:20)
12 ¯ ψ (i γ µ ( ∂ µ ψ ) − mψ ) h (cid:21) + (cid:20) − i2 ¯ ψγ α (cid:16) ∂ β ψ (cid:17) h αβ (cid:21) + (cid:20) − i16 ¯ ψγ µ h γ α , γ β i ψ∂ [ α h β ] µ (cid:21) + h M (cid:16) ¯ ψψ (cid:17)i ≡ (1) e L (D)0 + (0) e (1) e µa ¯ ψ i γ a (0) D µ ψ + (0) e (0) e µa ¯ ψ i γ a (1) D µ ψ + (0) e M (cid:16) ¯ ψψ (cid:17) , (133)where (0) D µ = ∂ µ , (134)and (1) D µ = 116 (1) ω µab h γ a , γ b i , (135)with (1) ω µab = − ∂ [ a h b ] µ (136)the linearized form of the full spin-connection ω µab = e νb ∂ ν e aµ − e νa ∂ ν e bµ + e aν ∂ µ e νb − e bν ∂ µ e νa + e ρ [ a e νb ] e cµ ∂ ν e c ρ = g (1) ω µab + g ω µab + · · · . (137)26n (137) e aµ represents the inverse of the vierbein field. Along the sameline, the piece of antighost number equal to zero from the second-order de-formation offers us the interacting lagrangian at order two in the couplingconstant L (int)2 = b (int)0 = (cid:20) (cid:16) ¯ ψ i γ µ ( ∂ µ ψ ) − m ¯ ψψ (cid:17) (cid:16) h − h µν h µν (cid:17)(cid:21) + (cid:20) hM (cid:16) ¯ ψψ (cid:17)(cid:21) + (cid:20) − i4 ¯ ψγ µ ( ∂ ν ψ ) hh µν (cid:21) + (cid:20) − i32 ¯ ψγ µ h γ α , γ β i ψh∂ [ α h β ] µ (cid:21) + (cid:20) ψγ µ ( ∂ ν ψ ) h µσ h σν (cid:21) + (cid:20) i32 ¯ ψγ µ h γ α , γ β i ψh σµ ∂ [ α h β ] σ (cid:21) + (cid:20) − i64 ¯ ψγ µ h γ α , γ β i ψ (cid:16) h ρ [ α (cid:16) ∂ β ] h ρµ (cid:17) + 2 (cid:16) ∂ ρ h µ [ α (cid:17) h ρβ ] − (cid:16) ∂ µ h ρ [ α (cid:17) h ρβ ] (cid:17)(cid:21) ≡ (2) e L (D)0 + (1) e M (cid:16) ¯ ψψ (cid:17) + (1) e (1) e µa ¯ ψ i γ a (0) D µ ψ + (1) e (0) e µa ¯ ψ i γ a (1) D µ ψ + (0) e (2) e µa ¯ ψ i γ a (0) D µ ψ + (0) e (1) e µa ¯ ψ i γ a (1) D µ ψ + (0) e (0) e µa ¯ ψ i γ a (2) D µ ψ, (138)where (2) D µ = 116 (2) ω µab h γ a , γ b i , (139)while (2) ω µab = − (cid:16) h c [ a (cid:16) ∂ b ] h c µ (cid:17) − h ν [ a ∂ ν h b ] µ − (cid:16) ∂ µ h ν [ a (cid:17) h b ] ν (cid:17) (140)is the second-order approximation of the spin-connection. With the help of(133) and (138) we deduce that L (D)0 + g L (int)1 + g L (int)2 + · · · comes fromexpanding the fully deformed lagrangian L (int) = e ¯ ψ (i e µa γ a D µ ψ − mψ ) + geM (cid:16) ¯ ψψ (cid:17) , (141)where D µ ψ = ∂ µ ψ + 116 ω µab h γ a , γ b i ψ (142)is the full covariant derivative of ψ .The pieces linear in the antifields ψ ∗ and ¯ ψ ∗ from the deformed solutionto the master equation give us the deformed gauge transformations for theDirac fields as δ ǫ ψ = g ( ∂ α ψ ) ǫ α + g h γ α , γ β i ψ∂ [ α ǫ β ] − g ∂ α ψ ) ǫ β h αβ g h γ α , γ β i ψ (cid:18) ǫ σ ∂ [ α h β ] σ − h σ [ α (cid:16) ∂ β ] ǫ σ − ∂ σ ǫ β ] (cid:17)(cid:19) + · · · = g (1) δ ǫ ψ + g δ ǫ ψ + · · · , (143) δ ǫ ¯ ψ = g (cid:16) ∂ α ¯ ψ (cid:17) ǫ α − g
16 ¯ ψ h γ α , γ β i ∂ [ α ǫ β ] − g (cid:16) ∂ α ¯ ψ (cid:17) ǫ β h αβ + g
32 ¯ ψ h γ α , γ β i (cid:18) ǫ σ ∂ [ α h β ] σ − h σ [ α (cid:16) ∂ β ] ǫ σ − ∂ σ ǫ β ] (cid:17)(cid:19) + · · · = g (1) δ ǫ ¯ ψ + g δ ǫ ¯ ψ + · · · . (144)The first two orders of the gauge transformations can be put under the form (1) δ ǫ ψ = ( ∂ µ ψ ) (0) ¯ ǫ µ + 18 h γ a , γ b i ψ (0) ǫ ab , (145) (2) δ ǫ ψ = ( ∂ µ ψ ) (1) ¯ ǫ µ + 18 h γ a , γ b i ψ (1) ǫ ab , (146) (1) δ ǫ ¯ ψ = (cid:16) ∂ µ ¯ ψ (cid:17) (0) ¯ ǫ µ −
18 ¯ ψ h γ a , γ b i (0) ǫ ab , (147) (2) δ ǫ ¯ ψ = (cid:16) ∂ µ ¯ ψ (cid:17) (1) ¯ ǫ µ −
18 ¯ ψ h γ a , γ b i (1) ǫ ab , (148)where we used the notations (0) ¯ ǫ µ = ǫ µ = ǫ a δ µa , (1) ¯ ǫ µ = − ǫ a h µa , (149) (0) ǫ ab = 12 ∂ [ a ǫ b ] , (150) (1) ǫ ab = − ǫ c ∂ [ a h b ] c + 18 h c [ a ∂ b ] ǫ c + 18 (cid:16) ∂ c ǫ [ a (cid:17) h cb ] . (151)Based on these notations, the gauge transformations of the spinors take theform δ ǫ ψ = g ( ∂ µ ψ ) (0) ¯ ǫ µ + g (1) ¯ ǫ µ + · · · ! + 18 h γ a , γ b i ψ (cid:18) (0) ǫ ab + g (1) ǫ ab + · · · (cid:19)(cid:19) , (152)28 ǫ ¯ ψ = g (cid:16) ∂ µ ¯ ψ (cid:17) (0) ¯ ǫ µ + g (1) ¯ ǫ µ + · · · ! −
18 ¯ ψ h γ a , γ b i (cid:18) (0) ǫ ab + g (1) ǫ ab + · · · (cid:19)(cid:19) . (153)The gauge parameters (0) ǫ ab si (1) ǫ ab are precisely the first two terms fromthe Lorentz parameters expressed in terms of the flat parameters ǫ a via thepartial gauge-fixing (130). Indeed, (130) leads to δ ǫ (cid:16) σ µ [ a e µb ] (cid:17) = 0 , (154)where δ ǫ e µa = ¯ ǫ ρ ∂ ρ e µa − e ρa ∂ ρ ¯ ǫ µ + ǫ ba e µb . (155)Substituting (131) together with the expansions¯ ǫ µ = (0) ¯ ǫ µ + g (1) ¯ ǫ µ + · · · = (cid:18) δ µa − g h µa + · · · (cid:19) ǫ a (156)and ǫ ab = (0) ǫ ab + g (1) ǫ ab + · · · (157)in (154), we arrive precisely to (150)–(151). At this point it is easy to see thatthe gauge transformations (152)–(153) come from the perturbative expansionof the full gauge transformations δ ǫ ψ = g (cid:18) ( ∂ µ ψ ) ¯ ǫ µ + 18 h γ a , γ b i ψǫ ab (cid:19) , (158) δ ǫ ¯ ψ = g (cid:18)(cid:16) ∂ µ ¯ ψ (cid:17) ¯ ǫ µ −
18 ¯ ψ h γ a , γ b i ǫ ab (cid:19) . (159)The full gauge transformations can be suggestively written like δ ǫ ψ = g (cid:18) ( ∂ µ ψ ) ¯ ǫ µ + 12 Σ ab ψǫ ab (cid:19) , (160) δ ǫ ¯ ψ = g (cid:18)(cid:16) ∂ µ ¯ ψ (cid:17) ¯ ǫ µ −
12 ¯ ψ Σ ab ǫ ab (cid:19) , (161)where Σ ab = 14 h γ a , γ b i (162)29re the spin operators, whose commutators read as h Σ ab , Σ cd i = σ a [ c Σ d ] b − σ b [ c Σ d ] a . (163)In conclusion, the interaction between a Dirac field and one spin-two fieldleads to the interacting lagrangian (141), while the gauge transformations ofthe Dirac spinors are given by (160) and (161). As it has been proved in [15], there are no direct cross-couplings that can beintroduced in a finite collection of gravitons and also no intermediate cross-couplings between different gravitons in the presence of a scalar field. In thissection, under the hypotheses of locality, smoothness of the interactions inthe coupling constant, Poincar´e invariance, (background) Lorentz invarianceand the preservation of the number of derivatives on each field, we will provethat there are no intermediate cross-couplings between different gravitons inthe presence of a Dirac field.Now, we start from a sum of Pauli-Fierz actions and a Dirac action S L0 h h Aµν , ψ, ¯ ψ i = Z d x (cid:18) − (cid:16) ∂ µ h Aνρ (cid:17) ( ∂ µ h νρA ) + ( ∂ µ h µρA ) (cid:16) ∂ ν h Aνρ (cid:17) − (cid:16) ∂ µ h A (cid:17) ( ∂ ν h νµA ) + 12 (cid:16) ∂ µ h A (cid:17) ( ∂ µ h A ) (cid:19) + Z d x ¯ ψ (i γ µ ( ∂ µ ψ ) − mψ ) , (164)where h A is the trace of the field h µνA ( h A = σ µν h µνA ), while A = 1 , , · · · , n .The gauge transformations of the action (164) are δ ǫ h Aµν = ∂ ( µ ǫ Aν ) , δ ǫ ψ = δ ǫ ¯ ψ = 0 . (165)The BRST complex comprises the fields/ghosts φ α = (cid:16) h Aµν , ψ, ¯ ψ (cid:17) , η Aµ , (166)and respectively the antifields φ ∗ α = (cid:16) h ∗ µνA , ψ ∗ , ¯ ψ ∗ (cid:17) , η ∗ µA . (167)30he BRST differential splits in this situation like in (5), while the actions of δ and γ on the BRST generators are defined by δh ∗ µνA = 2 H µνA , δψ ∗ = − (cid:16) m ¯ ψ + i ∂ µ ¯ ψγ µ (cid:17) , (168) δ ¯ ψ ∗ = − (i γ µ ∂ µ ψ − mψ ) , δη ∗ µA = − ∂ ν h ∗ µνA , (169) δφ α = 0 , δη Aµ = 0 , (170) γφ ∗ α = 0 , γη ∗ µA = 0 , (171) γh Aµν = ∂ ( µ η Aν ) , γψ = γ ¯ ψ = 0 , γη Aµ = 0 , (172)where H µνA = K µνA − σ µν K A is the linearized Einstein tensor for the field h µνA . In this case the solution to the master equation reads as¯ S = S L0 h h Aµν , ψ, ¯ ψ i + Z d x (cid:16) h ∗ µνA ∂ ( µ η Aν ) (cid:17) . (173)The first-order deformation of the solution to the master equation maybe decomposed in a manner similar to the case of a single graviton α = α (PF) + α (int) + α (Dirac) . (174)The first-order deformation in the Pauli-Fierz sector, α (PF) , is of the form [15] α (PF) = α (PF)2 + α (PF)1 + α (PF)0 , (175)with α (PF)2 = 12 f ABC η ∗ µA η Bν ∂ [ µ η Cν ] . (176)In (176), all the coefficients f ABC are constant. The condition that α (PF)2 indeedproduces a consistent α (PF)1 implies that these constants must be symmetricin their lower indices [15] f ABC = f ACB . (177)With (177) at hand, we find that α (PF)1 = f ABC h ∗ µρA (cid:16)(cid:16) ∂ ρ η Bν (cid:17) h Cµν − η Bν ∂ [ µ h Cν ] ρ (cid:17) . (178) The term (176) differs from that corresponding to [15] through a γ -exact term, whichdoes not affect (177). α (PF)1 leads to a consistent α (PF)0 implies that [15] f ABC = 13 f ( ABC ) , (179)where, by definition, f ABC = δ AD f DBC . Based on (179), we obtain that theresulting α (PF)0 reads as in [15] (where this component is denoted by a and f ABC by a abc ).If we go along exactly the same line like in the subsection 4.2, we get that α (int) = α (int)1 + α (int)0 , where α (int)1 = k A (cid:16) ψ ∗ ( ∂ α ψ ) + (cid:16) ∂ α ¯ ψ (cid:17) ¯ ψ ∗ (cid:17) η Aα − k A (cid:16) ¯ ψ h γ α , γ β i ¯ ψ ∗ − ψ ∗ h γ α , γ β i ψ (cid:17) ∂ [ α η Aβ ] , (180) α (int)0 = k A (cid:16) ¯ ψ (i γ µ ( ∂ µ ψ ) − mψ ) h A − i ¯ ψγ α (cid:16) ∂ β ψ (cid:17) h Aαβ (cid:17) − i k A
16 ¯ ψγ µ h γ α , γ β i ψ∂ [ α h Aβ ] µ , (181)and k A are some real constants. Meanwhile, we find in a direct manner that α (Dirac) = a (Dirac)0 , (182)with a (Dirac)0 given in (108).Let us investigate next the consistency of the first-order deformation. Ifwe perform the notationsˆ S (PF)1 = Z d xα (PF) , (183)ˆ S (int)1 = Z d x (cid:16) α (int) + α (Dirac) (cid:17) , (184)ˆ S = ˆ S (PF)1 + ˆ S (int)1 , (185)then the equation (cid:16) ˆ S , ˆ S (cid:17) + 2 s ˆ S = 0 (expressing the consistency of thefirst-order deformation) equivalently splits into the equations (cid:16) ˆ S (PF)1 , ˆ S (PF)1 (cid:17) + 2 s ˆ S (PF)2 = 0 , (186)2 (cid:16) ˆ S (PF)1 , ˆ S (int)1 (cid:17) + (cid:16) ˆ S (int)1 , ˆ S (int)1 (cid:17) + 2 s ˆ S (int)2 = 0 , (187) The piece (178) differs from that corresponding to [15] through a δ -exact term, whichdoes not change (179). S = ˆ S (PF)2 + ˆ S (int)2 . The equation (186) requires that the constants f CAB satisfy the supplementary conditions [15] f DA [ B f EC ] D = 0 , (188)so they are the structure constants of a finite-dimensional, commutative,symmetric and associative real algebra A . The analysis realized in [15] showsus that such an algebras has a trivial structure (being expressed like a directsum of some one-dimensional ideals). So we obtain that f CAB = 0 if A = B. (189)Let us analyze now the equation (187). If we denote by ˆ∆ (int) and β (int) the non-integrated densities of the functionals 2 (cid:16) ˆ S (PF)1 , ˆ S (int)1 (cid:17) + (cid:16) ˆ S (int)1 , ˆ S (int)1 (cid:17) and respectively of ˆ S (int)2 , then the equation (187) in local form becomesˆ∆ (int) = − sβ (int) + ∂ µ k µ , (190)with gh (cid:16) ˆ∆ (int) (cid:17) = 1 , gh (cid:16) β (int) (cid:17) = 0 , gh ( k µ ) = 1 . (191)The computation of ˆ∆ (int) reveals in our case the following decompositionalong the antighost numberˆ∆ (int) = ˆ∆ (int)0 + ˆ∆ (int)1 , agh (cid:16) ˆ∆ (int) I (cid:17) = I, I = 0 , , (192)withˆ∆ (int)1 = γ (cid:16) k A k B (cid:16) ψ ∗ ( ∂ α ψ ) + (cid:16) ∂ α ¯ ψ (cid:17) ¯ ψ ∗ (cid:17) η Aβ h Bαβ − M αβ (cid:16)(cid:16) k A k B − k D f DAB (cid:17) η Aσ ∂ [ α h Bβ ] σ − k D f DAB h Aρα ∂ [ β η Bρ ] (cid:17)(cid:19) + (cid:16) k D f DAB − k A k B (cid:17) (cid:16)(cid:16) ψ ∗ ( ∂ α ψ ) + (cid:16) ∂ α ¯ ψ (cid:17) ¯ ψ ∗ (cid:17) η Aβ ∂ [ α η Bβ ] − M αβ σ µν ∂ [ α η Aµ ] ∂ [ β η Bν ] (cid:19) , (193)where we used the notation M αβ = ¯ ψ h γ α , γ β i ¯ ψ ∗ − ψ ∗ h γ α , γ β i ψ. (194)33he concrete form of ˆ∆ (int)0 is not important in what follows and thereforewe will skip it. Due to the decomposition (192), we have that β (int) and k µ from (190) can be decomposed like β (int) = β (int)0 + β (int)1 + β (int)2 , agh (cid:16) β (int) I (cid:17) = I, I = 0 , , , (195) k µ = k µ + k µ + k µ , agh ( k µI ) = I, I = 0 , , . (196)By projecting the equation (190) on various values of the antighost number,we obtain the tower of equations γβ (int)2 = ∂ µ (cid:18) k µ (cid:19) , (197)ˆ∆ (int)1 = − (cid:16) δβ (int)2 + γβ (int)1 (cid:17) + ∂ µ k µ , (198)ˆ∆ (int)0 = − (cid:16) δβ (int)1 + γβ (int)0 (cid:17) + ∂ µ k µ . (199)By a trivial redefinition, the equation (197) can always be replaced with γβ (int)2 = 0 . (200)Analyzing the expression of ˆ∆ (int)1 in (193) we observe that it can be expressedas in (198) ifˆ χ = (cid:16) k D f DAB − k A k B (cid:17) (cid:16)(cid:16) ψ ∗ ( ∂ α ψ ) + (cid:16) ∂ α ¯ ψ (cid:17) ¯ ψ ∗ (cid:17) η Aβ ∂ [ α η Bβ ] − M αβ σ µν ∂ [ α η Aµ ] ∂ [ β η Bν ] (cid:19) , (201)can be put in the form ˆ χ = δ ˆ ϕ + γ ˆ ω + ∂ µ j µ . (202)Assume that (202) holds. Then, by applying δ on this equation we infer δ ˆ χ = γ ( − δ ˆ ω ) + ∂ µ ( δj µ ) . (203)On the other hand, if we use the concrete expression (201) of ˆ χ , by directcomputation we are led to δ ˆ χ = γ (cid:16) δ (cid:16) − (cid:16) k D f DAB − k A k B (cid:17) ¯ ψ ¯ ψ ∗ η Aµ (cid:16) ∂ µ h B − ∂ ν h Bµν (cid:17)(cid:17) + (cid:16) k D f DAB − k A k B (cid:17) (cid:18) i ¯ ψγ µ ( ∂ α ψ ) (cid:18) h Aµβ ∂ [ α η Bβ ] − η Aβ ∂ [ α h Bµβ ] (cid:19) i8 ¯ ψγ µ h γ α , γ β i ψσ ρλ ∂ [ ρ η Aα ] ∂ [ λ η Bβ ] (cid:19)(cid:19) + ∂ µ (cid:16) δ (cid:16)(cid:16) k D f DAB − k A k B (cid:17) ¯ ψ ¯ ψ ∗ η Aν (cid:16) ∂ µ η Bν − ∂ ν η Bµ (cid:17)(cid:17) +i (cid:16) k D f DAB − k A k B (cid:17) (cid:16) ¯ ψγ µ ( ∂ α ψ ) η Aβ ∂ [ α η Bβ ] + 116 ¯ ψγ µ h γ α , γ β i ψσ ρλ ∂ [ ρ η Aα ] ∂ [ λ η Bβ ] (cid:19)(cid:19) . (204)The right-hand side of (204) can be written like in the right-hand side of(203) if the following conditions are simultaneously fulfilled (cid:16) k D f DAB − k A k B (cid:17) (cid:18) i ¯ ψγ µ ( ∂ α ψ ) (cid:18) h Aµβ ∂ [ α η Bβ ] − η Aβ ∂ [ α h Bµβ ] (cid:19) − i8 ¯ ψγ µ h γ α , γ β i ψσ ρλ ∂ [ ρ η Aα ] ∂ [ λ η Bβ ] (cid:19) = − δ ˆ ω ′ , (205)+i (cid:16) k D f DAB − k A k B (cid:17) (cid:16) ¯ ψγ µ ( ∂ α ψ ) η Aβ ∂ [ α η Bβ ] + 116 ¯ ψγ µ h γ α , γ β i ψσ ρλ ∂ [ ρ η Aα ] ∂ [ λ η Bβ ] (cid:19) = δj ′ µ . (206)However, from the action of δ on the BRST generators we observe that noneof h Aµβ , ∂ [ α h Aβ ] µ , η Aβ and ∂ [ λ η Aβ ] are δ -exact. In consequence, the relations(205)–(206) hold if the equations¯ ψγ µ ( ∂ α ψ ) = δ Ω µα , (207)and ¯ ψγ µ [ γ α , γ β ] ψ = δ Γ µαβ (208)take place simultaneously. Let us suppose that the relations (207)–(208) areindeed satisfied. Acting with ∂ µ on (207)–(208) we arrive at ∂ µ (cid:16) ¯ ψγ µ ( ∂ α ψ ) (cid:17) = δ ( ∂ µ Ω µα ) , (209)and ∂ µ (cid:16) ¯ ψγ µ [ γ α , γ β ] ψ (cid:17) = δ (cid:16) ∂ µ Γ µαβ (cid:17) . (210)On the other hand, by direct computation we arrive at ∂ µ (cid:16) ¯ ψγ µ ( ∂ α ψ ) (cid:17) = δ (cid:16) − i (cid:16) ψ ∗ ( ∂ α ψ ) − ¯ ψ (cid:16) ∂ α ¯ ψ ∗ (cid:17)(cid:17)(cid:17) , (211) ∂ µ (cid:16) ¯ ψγ µ [ γ α , γ β ] ψ (cid:17) = δ (i M αβ ) − ψγ [ α ∂ β ] ψ. (212)35he right-hand sides of (211)–(212) are not of the same type like the corre-sponding ones in (209)–(210). This means that the relations (207)–(208) arenot valid, and therefore neither are (205)–(206). As a consequence, ˆ χ mustvanish, which further implies that k D f DAB − k A k B = 0 . (213)Using (213) and (189) we obtain that for A = Bk A k B = 0 , (214)which shows that the Dirac fields can couple to only one graviton, whichproves the assertion from the beginning of this section. To conclude with, in this paper we have investigated the couplings betweena collection of massless spin-two fields (described in the free limit by a sumof Pauli-Fierz actions) and a Dirac field using the powerful setting based onlocal BRST cohomology. Initially, we have shown that, if we decompose themetric like g µν = σ µν + gh µν , then we can couple Dirac spinors to h µν in thespace of formal series with the maximum derivative order equal to one in h µν ,such that the final results agree with the usual couplings between the spin-1/2 and the massless spin-two field in the vierbein formulation. Based onthis result, we have proved, under the hypotheses of locality, smoothness ofthe interactions in the coupling constant, Poincar´e invariance, (background)Lorentz invariance and the preservation of the number of derivatives on eachfield, that there are no consistent cross-interactions among different gravitonsin the presence of a Dirac field. Acknowledgment
Three of the authors (C.B., E.M.C. and S.O.S.) are partially supported bythe European Commission FP6 program MRTN-CT-2004-005104 and by thetype A grant 305/2004 with the Romanian National Council for AcademicScientific Research (C.N.C.S.I.S.) and the Romanian Ministry of Educationand Research (M.E.C.). One of the authors (A.C.L.) is supported by theWorld Federation of Scientists (WFS) National Scholarship Programme. Use-ful discussions with Glenn Barnich and Ion I. Cotaescu are also acknowledged.36
Proof of a statement made in subsection4.2
Here, we prove that a term of the type˜ a (int)1 = h ∗ µν η µ F ν (cid:16) ¯ ψ, ψ (cid:17) (215)is consistent in antighost number zero, δ ˜ a (int)1 + γ ˜ a (int)0 = ∂ µ ρ µ , (216)if and only if F ν (cid:16) ¯ ψ, ψ (cid:17) = ∂ ν F (cid:16) ¯ ψ, ψ (cid:17) . (217)Indeed, by applying δ on ˜ a (int)1 we obtain that δ ˜ a (int)1 = − H µν η µ F ν (cid:16) ¯ ψ, ψ (cid:17) . (218)It is easy to see that, if F ν (cid:16) ¯ ψ, ψ (cid:17) if the form (217), then (218) implies δ ˜ a (int)1 = γ (cid:16) H µν h µν F (cid:16) ¯ ψ, ψ (cid:17)(cid:17) + ∂ µ (cid:16) − H µν η ν F (cid:16) ¯ ψ, ψ (cid:17)(cid:17) , (219)and therefore ˜ a (int)1 indeed checks an equation of the type (216). Let ussuppose now that ˜ a (int)1 satisfies the equation (216). Inserting the relations γ ˜ a (int)0 = 2 δ ˜ a (int)0 δh µν ∂ µ η ν + ∂ µ t µ , (220)and (218) in (216), we get that − H µν η µ F ν (cid:16) ¯ ψ, ψ (cid:17) + 2 δ ˜ a (int)0 δh µν ∂ ν η µ = ∂ µ p µ . (221)The left-hand side of the last relation reduces to a total derivative if H µν F ν (cid:16) ¯ ψ, ψ (cid:17) = − ∂ ν δ ˜ a (int)0 δh µν . (222)In order to investigate under what conditions the left-hand side of (222) alsoprovides a total derivative, we start from the fact that H µν = ∂ α ∂ β φ µανβ , (223)37here φ µανβ = 12 (cid:16) − h µν σ αβ + h αν σ µβ + h µβ σ αν − h αβ σ µν + h (cid:16) σ µν σ αβ − σ µβ σ αν (cid:17)(cid:17) . (224)By means of (223) we further deduce that H µν F ν = ∂ ν (cid:16) ∂ β φ µναβ F α − φ µβαν ∂ β F α (cid:17) + 12 φ µανβ ∂ α ∂ [ β F ν ] . (225)Thus, the right-hand side of (225) gives a total derivative if and only if φ µανβ ∂ α ∂ [ β F ν ] = 0 , which further yields F ν = ∂ ν F . This completes the proof. B Complete computation of the second-orderdeformation
In this appendix we are interested in determining the complete expressionof the second-order deformation for the master equation, which is knownto be subject to the equation (22). Proceeding in the same manner likeduring the first-order deformation procedure, we can write the second-orderdeformation of the master equation like the sum between the Pauli-Fierz andthe interacting parts S = S (PF)2 + S (int)2 . (226)The piece S (PF)2 describes the second-order deformation in the Pauli-Fierzsector and we will not insist on it since we are merely interested in the cross-couplings. The term S (int)2 results as solution to the equation12 ( S , S ) (int) + sS (int)2 = 0 , (227)where ( S , S ) (int) = (cid:16) S (int)1 , S (int)1 (cid:17) + 2 (cid:16) S (PF)1 , S (int)1 (cid:17) . (228)38f we denote by ∆ (int) and b (int) the non-integrated densities of ( S , S ) (int) and respectively of S (int)2 , the local form of (227) becomes∆ (int) = − sb (int) + ∂ µ n µ , (229)with gh (cid:16) ∆ (int) (cid:17) = 1 , gh (cid:16) b (int) (cid:17) = 0 , gh ( n µ ) = 1 , (230)for some local currents n µ . Direct computation shows that ∆ (int) decomposeslike ∆ (int) = ∆ (int)0 + ∆ (int)1 , agh (cid:16) ∆ (int) I (cid:17) = I, I = 0 , , (231)with∆ (int)1 = γ (cid:16) k (cid:16) ψ ∗ ( ∂ α ψ ) + (cid:16) ∂ α ¯ ψ (cid:17) ¯ ψ ∗ (cid:17) η β h αβ + k M αβ (cid:18) (1 − k ) η σ ∂ [ α h β ] σ + 12 h σ [ α (cid:16) ∂ β ] η σ − ∂ σ η β ] (cid:17)(cid:19)! + k (1 − k ) (cid:16) ψ ∗ ( ∂ α ψ ) + (cid:16) ∂ α ¯ ψ (cid:17) ¯ ψ ∗ (cid:17) η β ∂ [ α η β ] − k
16 (1 − k ) M αβ σ µν ∂ [ α η µ ] ∂ [ β η ν ] , (232)and∆ (int)0 = k L (D)0 (cid:16) h αβ ∂ α η β + η β ( ∂ β h − ∂ α h αβ ) (cid:17) + i k ψγ ( α (cid:16) ∂ β ) ψ (cid:17) (cid:16) − h σα ∂ β η σ + η σ (cid:16) ∂ ( α h β ) σ − ∂ σ h αβ (cid:17)(cid:17) + i k ψγ µ [ γ α , γ β ] ψ∂ α (cid:16) η σ (cid:16) ∂ ( β h µ ) σ − ∂ σ h µβ (cid:17) − (cid:16) ∂ ( β η σ (cid:17) h µ ) σ (cid:17) + k h (cid:16)(cid:16) ∂ σ L (D)0 (cid:17) η σ + ¯ ψ i γ α (cid:16) ∂ β ψ (cid:17) ∂ α η β (cid:17) − i k h αβ (cid:16)(cid:16) ∂ σ ¯ ψ (cid:17) γ α (cid:16) ∂ β ψ (cid:17) η σ + ¯ ψγ α ∂ β ( ∂ σ ψη σ ) (cid:17) − i k (cid:16) ∂ σ (cid:16) ¯ ψγ µ [ γ α , γ β ] ψ (cid:17)(cid:17) η σ ∂ [ α h β ] µ + i k
16 ¯ ψγ µ [ γ α , γ β ] ψh∂ µ (cid:16) ∂ [ α η β ] (cid:17) + i k (cid:16) ¯ ψ [ γ µ , γ ν ] γ α (cid:16) ∂ β ψ (cid:17) h αβ ∂ [ µ η ν ] − ¯ ψγ α [ γ µ , γ ν ] (cid:16) ∂ β (cid:16) ψ∂ [ µ η ν ] (cid:17)(cid:17) h αβ (cid:17) + i k
128 ¯ ψ (cid:16) [ γ ρ , γ λ ] γ µ [ γ α , γ β ] − γ µ [ γ α , γ β ][ γ ρ , γ λ ] (cid:17) ψ (cid:16) ∂ [ α h β ] µ (cid:17) ∂ [ ρ η λ ] + γ (cid:16) − k (cid:16) f (cid:16) ¯ ψ, ψ (cid:17) + (cid:16) ∂ µ ¯ ψ (cid:17) g µ (cid:16) ¯ ψ, ψ (cid:17) + g µ (cid:16) ¯ ψ, ψ (cid:17) ( ∂ µ ψ ) (cid:17) h kh µν (cid:16)(cid:16) ∂ ν ¯ ψ (cid:17) g µ (cid:16) ¯ ψ, ψ (cid:17) + g µ (cid:16) ¯ ψ, ψ (cid:17) ( ∂ ν ψ ) (cid:17)(cid:17) + k ∂ R f∂ψ [ γ µ , γ ν ] ψ − ¯ ψ [ γ µ , γ ν ] ∂ L f∂ ¯ ψ ! ∂ [ µ η ν ] + k ∂ [ µ η ν ] (cid:16) ¯ ψ [ γ µ , γ ν ] ( ∂ ρ g ρ ) − ( ∂ ρ g ρ ) [ γ µ , γ ν ] ψ + ∂ R (cid:16)(cid:16) ∂ ρ ¯ ψ (cid:17) g ρ + g ρ ( ∂ ρ ψ ) (cid:17) ∂ψ [ γ µ , γ ν ] ψ − ¯ ψ [ γ µ , γ ν ] ∂ L (cid:16)(cid:16) ∂ ρ ¯ ψ (cid:17) g ρ + g ρ ( ∂ ρ ψ ) (cid:17) ∂ ¯ ψ − (cid:16)(cid:16) ∂ [ µ ¯ ψ (cid:17) g ν ]1 − g [ µ (cid:16) ∂ ν ] ψ (cid:17)(cid:17)(cid:17) , (233)where we used the notation M αβ = ¯ ψ h γ α , γ β i ¯ ψ ∗ − ψ ∗ h γ α , γ β i ψ. (234)Since the first-order deformation in the interacting sector starts in antighostnumber one, we can take, without loss of generality, the corresponding second-order deformation to start in antighost number two b (int) = b (int)0 + b (int)1 + b (int)2 , agh (cid:16) b (int) I (cid:17) = I, I = 0 , , , (235) n µ = n µ + n µ + n µ , agh ( n µI ) = I, I = 0 , , . (236)By projecting the equation (229) on various antighost numbers, we obtain γb (int)2 = ∂ µ (cid:18) n µ (cid:19) , (237)∆ (int)1 = − (cid:16) δb (int)2 + γb (int)1 (cid:17) + ∂ µ n µ , (238)∆ (int)0 = − (cid:16) δb (int)1 + γb (int)0 (cid:17) + ∂ µ n µ . (239)The equation (237) can always be replaced, by adding trivial terms, with γb (int)2 = 0 . (240)Looking at ∆ (int)1 given in (232), it results that it can be written like in (238)if χ = k (1 − k ) (cid:16)(cid:16) ψ ∗ ( ∂ α ψ ) + (cid:16) ∂ α ¯ ψ (cid:17) ¯ ψ ∗ (cid:17) η β ∂ [ α η β ] − M αβ σ µν ∂ [ α η µ ] ∂ [ β η ν ] (cid:19) , (241)40an be expressed like χ = δϕ + γω + ∂ µ l µ . (242)Supposing that (242) holds and applying δ on it, we infer that δχ = γ ( − δω ) + ∂ µ ( δl µ ) . (243)On the other hand, using the concrete expression of χ , we have that δχ = γ (cid:16) δ (cid:16) k (1 − k ) ¯ ψ ¯ ψ ∗ (cid:16) ∂ µ h µβ − ∂ β h (cid:17) η β (cid:17)(cid:17) + ∂ µ (cid:16) δ (cid:16) k (1 − k ) ¯ ψ ¯ ψ ∗ η β ∂ [ µ η β ] (cid:17)(cid:17) + γ (cid:18) i k (1 − k ) ¯ ψγ µ ( ∂ α ψ ) (cid:18) h µβ ∂ [ α η β ] − (cid:16) ∂ [ α h β ] λ (cid:17) σ λµ η β (cid:19) + i8 k (1 − k ) ¯ ψγ µ h γ α , γ β i ψ∂ [ α h λ ] µ ∂ [ β η ν ] σ λν (cid:19) + ∂ µ (cid:16) i k (1 − k ) ¯ ψγ µ ( ∂ α ψ ) η β ∂ [ α η β ] + i16 k (1 − k ) ¯ ψγ µ h γ α , γ β i ψ∂ [ α η λ ] ∂ [ β η ν ] σ λν (cid:19) . (244)The right-hand side of (244) can be written like in the right-hand side of(243) if the following conditions are simultaneously satisfied − δω ′ = ¯ ψγ µ ( ∂ α ψ ) (cid:18) h µβ ∂ [ α η β ] − (cid:16) ∂ [ α h β ] λ (cid:17) σ λµ η β (cid:19) + 18 ¯ ψγ µ h γ α , γ β i ψ∂ [ α h λ ] µ ∂ [ β η ν ] σ λν , (245) δl ′ µ = ¯ ψγ µ ( ∂ α ψ ) η β ∂ [ α η β ] + 116 ¯ ψγ µ h γ α , γ β i ψ∂ [ α η λ ] ∂ [ β η ν ] σ λν . (246)Since none of the quantities h µβ , ∂ [ α h β ] λ , η β or ∂ [ α η β ] are δ -exact, the lastrelations hold if the equations¯ ψγ µ ( ∂ α ψ ) = δ Ω µα , ¯ ψγ µ [ γ α , γ β ] ψ = δ Γ µαβ (247)take place simultaneously. Assuming that both the equations (247) are valid,they further give ∂ µ (cid:16) ¯ ψγ µ ( ∂ α ψ ) (cid:17) = δ ( ∂ µ Ω µα ) , (248) ∂ µ (cid:16) ¯ ψγ µ [ γ α , γ β ] ψ (cid:17) = δ (cid:16) ∂ µ Γ µαβ (cid:17) . (249)41n the other hand, by direct computation we obtain that ∂ µ (cid:16) ¯ ψγ µ ( ∂ α ψ ) (cid:17) = δ (cid:16) − i (cid:16) ψ ∗ ( ∂ α ψ ) − ¯ ψ (cid:16) ∂ α ¯ ψ ∗ (cid:17)(cid:17)(cid:17) , (250) ∂ µ (cid:16) ¯ ψγ µ [ γ α , γ β ] ψ (cid:17) = δ (i M αβ ) − ψγ [ α (cid:16) ∂ β ] ψ (cid:17) , (251)so the right-hand sides of (250)–(251) cannot be written like in the right-hand sides of (248)–(249). This means that the relations (247) are not valid,and therefore neither are (245)–(246). As a consequence, χ must vanish, andhence we must set k (1 − k ) = 0 . (252)Using (252), we conclude that k = 1 . (253)Inserting (253) in (232), we obtain that∆ (int)1 = γ (cid:16)(cid:16) ψ ∗ ( ∂ α ψ ) + (cid:16) ∂ α ¯ ψ (cid:17) ¯ ψ ∗ (cid:17) η β h αβ − M αβ (cid:18) η σ ∂ [ α h β ] σ − h σ [ α (cid:16) ∂ β ] η σ − ∂ σ η β ] (cid:17)(cid:19)(cid:19) . (254)Comparing (254) with (238), we find that b (int)2 = 0 , (255) b (int)1 = − (cid:16) ψ ∗ ( ∂ α ψ ) + (cid:16) ∂ α ¯ ψ (cid:17) ¯ ψ ∗ (cid:17) η β h αβ + 132 (cid:16) ¯ ψ h γ α , γ β i ¯ ψ ∗ − ψ ∗ h γ α , γ β i ψ (cid:17) (cid:18) η σ ∂ [ α h β ] σ − h σ [ α (cid:16) ∂ β ] η σ − ∂ σ η β ] (cid:17)(cid:19) . (256)Substituting (253) in (233) and using (256), we deduce∆ (int)0 + 2 δb (int)1 = ∂ µ n µ + γ (cid:18) i2 ¯ ψγ µ ∂ ν ψ (cid:18) hh µν − h µσ h σν (cid:19) + 12 (cid:16) ¯ ψ i γ µ ( ∂ µ ψ ) − m ¯ ψψ (cid:17) (cid:18) h αβ h αβ − h (cid:19) + i16 ¯ ψγ µ h γ α , γ β i ψ (cid:16) h∂ [ α h β ] µ − h σµ ∂ [ α h β ] σ + h σα (cid:16) ∂ [ β h σ ] µ + ∂ µ h βσ (cid:17)(cid:17) − (cid:16) f (cid:16) ¯ ψ, ψ (cid:17) + (cid:16) ∂ µ ¯ ψ (cid:17) g µ (cid:16) ¯ ψ, ψ (cid:17) + g µ (cid:16) ¯ ψ, ψ (cid:17) ( ∂ µ ψ ) (cid:17) h h µν (cid:16)(cid:16) ∂ ν ¯ ψ (cid:17) g µ (cid:16) ¯ ψ, ψ (cid:17) + g µ (cid:16) ¯ ψ, ψ (cid:17) ( ∂ ν ψ ) (cid:17)(cid:17) + 18 ∂ R f∂ψ [ γ µ , γ ν ] ψ − ¯ ψ [ γ µ , γ ν ] ∂ L f∂ ¯ ψ ! ∂ [ µ η ν ] + 18 ∂ [ µ η ν ] (cid:16) ¯ ψ [ γ µ , γ ν ] ( ∂ ρ g ρ ) − ( ∂ ρ g ρ ) [ γ µ , γ ν ] ψ + ∂ R (cid:16)(cid:16) ∂ ρ ¯ ψ (cid:17) g ρ + g ρ ( ∂ ρ ψ ) (cid:17) ∂ψ [ γ µ , γ ν ] ψ − ¯ ψ [ γ µ , γ ν ] ∂ L (cid:16)(cid:16) ∂ ρ ¯ ψ (cid:17) g ρ + g ρ ( ∂ ρ ψ ) (cid:17) ∂ ¯ ψ − (cid:16)(cid:16) ∂ [ µ ¯ ψ (cid:17) g ν ]1 − g [ µ (cid:16) ∂ ν ] ψ (cid:17)(cid:17)(cid:17) . (257)The right-hand side of (257) can be written like in (239) if18 ∂ R f∂ψ [ γ µ , γ ν ] ψ − ¯ ψ [ γ µ , γ ν ] ∂ L f∂ ¯ ψ ! ∂ [ µ η ν ] + 18 ∂ [ µ η ν ] (cid:16) ¯ ψ [ γ µ , γ ν ] ( ∂ ρ g ρ ) − ( ∂ ρ g ρ ) [ γ µ , γ ν ] ψ + ∂ R (cid:16)(cid:16) ∂ ρ ¯ ψ (cid:17) g ρ + g ρ ( ∂ ρ ψ ) (cid:17) ∂ψ [ γ µ , γ ν ] ψ − ¯ ψ [ γ µ , γ ν ] ∂ L (cid:16)(cid:16) ∂ ρ ¯ ψ (cid:17) g ρ + g ρ ( ∂ ρ ψ ) (cid:17) ∂ ¯ ψ − (cid:16)(cid:16) ∂ [ µ ¯ ψ (cid:17) g ν ]1 − g [ µ (cid:16) ∂ ν ] ψ (cid:17)(cid:17)(cid:17) = γθ + ∂ µ ρ µ . (258)The term (cid:16) ∂ R f∂ψ [ γ µ , γ ν ] ψ − ¯ ψ [ γ µ , γ ν ] ∂ L f∂ ¯ ψ (cid:17) ∂ [ µ η ν ] is neither γ -exact nor a to-tal derivative (as f (cid:16) ¯ ψ, ψ (cid:17) has no derivatives), and hence we must requirethat ∂ R f∂ψ [ γ µ , γ ν ] ψ − ¯ ψ [ γ µ , γ ν ] ∂ L f∂ ¯ ψ = 0 . (259)The solution to (259) reads as f (cid:16) ¯ ψ, ψ (cid:17) = M (cid:16) ¯ ψψ (cid:17) , (260)where M is a polynomial in ¯ ψψ . With (259) at hand, the equation (258)becomes 18 (cid:16) ∂ [ µ η ν ] (cid:17) Π µν = γθ + ∂ µ ρ µ , (261)43here Π µν ≡ ¯ ψ [ γ µ , γ ν ] ( ∂ ρ g ρ ) − ( ∂ ρ g ρ ) [ γ µ , γ ν ] ψ − (cid:16)(cid:16) ∂ [ µ ¯ ψ (cid:17) g ν ]1 − g [ µ (cid:16) ∂ ν ] ψ (cid:17)(cid:17) + ∂ R (cid:16)(cid:16) ∂ ρ ¯ ψ (cid:17) g ρ + g ρ ( ∂ ρ ψ ) (cid:17) ∂ψ [ γ µ , γ ν ] ψ − ¯ ψ [ γ µ , γ ν ] ∂ L (cid:16)(cid:16) ∂ ρ ¯ ψ (cid:17) g ρ + g ρ ( ∂ ρ ψ ) (cid:17) ∂ ¯ ψ . (262)The left-hand side of (261) is γ -exact modulo d if there exists a real, bosonicfunction, involving only the undifferentiated Dirac fields, M ρµν , which isantisymmetric in its last two indices M ρµν = − M ρνµ , (263)such that Π µν = ∂ ρ M ρµν . (264)From (262), we observe that the existence of such functions M ρµν is controlledby the functions g µ and g µ . The most general form of the functions g µ is g µ = ψg µ + X n =1 γ ν · · · γ ν n ψg µν ...ν n , (265)where g µ and g µν ...ν n are some real, bosonic functions in the undifferentiatedDirac fields. Now, from (109) it results that g µ = g µ ¯ ψ + X n =1 g µν ...ν n ¯ ψγ ν n · · · γ ν . (266)Inserting (265)–(266) in (262), we arrive atΠ µν = (cid:16) ∂ ρ (cid:16) ¯ ψψ (cid:17)(cid:17) ∂ R g ρ ∂ψ [ γ µ , γ ν ] ψ − ¯ ψ [ γ µ , γ ν ] ∂ L g ρ ∂ ¯ ψ − σ ρ [ µ g ν ] ! + ∂ ρ − X n =1 g ρν ...ν n (cid:16) ¯ ψγ ν n · · · γ ν [ γ µ , γ ν ] ψ − ¯ ψγ ν · · · γ ν n [ γ µ , γ ν ] ψ (cid:17)! + X n =1 ∂ R g ρν ...ν n ∂ψ [ γ µ , γ ν ] ψ − ¯ ψ [ γ µ , γ ν ] ∂ L g ρν ...ν n ∂ ¯ ψ − σ ρ [ µ g ν ] ν ...ν n ! × (cid:16)(cid:16) ∂ ρ ¯ ψ (cid:17) γ ν · · · γ ν n ψ + ¯ ψγ ν n · · · γ ν ( ∂ ρ ψ ) (cid:17) +4 X n =1 g ρν ...ν n n X k =1 (cid:16)(cid:16) ∂ ρ ¯ ψ (cid:17) γ ν · · · γ ν k − σ ν k [ µ γ ν ] γ ν k +1 · · · γ ν n ψ + ¯ ψγ ν n · · · γ ν k +1 σ ν k [ µ γ ν ] γ ν k − · · · γ ν ( ∂ ρ ψ ) (cid:17) . (267)The right-hand side of (267) is of the form ∂ ρ M ρµν if ∂ R g ρ ∂ψ [ γ µ , γ ν ] ψ − ¯ ψ [ γ µ , γ ν ] ∂ L g ρ ∂ ¯ ψ − σ ρ [ µ g ν ] = C ρµν , (268) n = 1 , g ρν = δ ρν Q (cid:16) ¯ ψψ (cid:17) , (269)where C ρµν = − C ρνµ are some constants (and not some 4 × Q is an arbitrary polynomial in ¯ ψψ . Since in four spacetime dimensions thereare no such constants, they must vanish, which imply that ∂ R g ρ ∂ψ [ γ µ , γ ν ] ψ − ¯ ψ [ γ µ , γ ν ] ∂ L g ρ ∂ ¯ ψ − σ ρ [ µ g ν ] = 0 . (270)The solution to the last equation reads as g ρ = ¯ ψγ ρ ψN (cid:16) ¯ ψψ (cid:17) , (271)where N is an arbitrary polynomial in ¯ ψψ . In consequence, Π µν expressedby (262) can be written like in (264) if the functions (265)–(266) are of theform g µ = ψ (cid:16) ¯ ψγ µ ψ (cid:17) N (cid:16) ¯ ψψ (cid:17) + γ µ ψQ (cid:16) ¯ ψψ (cid:17) , (272) g µ = (cid:16) ¯ ψγ µ ψ (cid:17) N (cid:16) ¯ ψψ (cid:17) ¯ ψ + ¯ ψγ µ Q (cid:16) ¯ ψψ (cid:17) . (273)By means of (272)–(273) we deduce (cid:16) ∂ µ ¯ ψ (cid:17) g µ (cid:16) ¯ ψ, ψ (cid:17) + g µ (cid:16) ¯ ψ, ψ (cid:17) ( ∂ µ ψ ) = ∂ µ (cid:16) ¯ ψγ µ ψP (cid:16) ¯ ψψ (cid:17)(cid:17) + s (cid:16) i (cid:16) ψ ∗ ψ − ¯ ψ ¯ ψ ∗ (cid:17) (cid:16) P (cid:16) ¯ ψψ (cid:17) − Q (cid:16) ¯ ψψ (cid:17)(cid:17)(cid:17) , (274)where P is a polynomial in ¯ ψψ defined by N (cid:16) ¯ ψψ (cid:17) = dP (cid:16) ¯ ψψ (cid:17) /d (cid:16) ¯ ψψ (cid:17) . Therelation (274) shows that the last two terms from (108) produce a trivialdeformation, which can always be removed by setting g µ (cid:16) ¯ ψ, ψ (cid:17) = 0 , g µ (cid:16) ¯ ψ, ψ (cid:17) = 0 . (275)45hen, with the help of (260) it follows that a (Dirac)0 = M (cid:16) ¯ ψψ (cid:17) . 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