No interactions for a collection of spin-two fields intermediated by a massive Rarita-Schwinger field
C. Bizdadea, E. M. Cioroianu, D. Cornea, S. O. Saliu, S. C. Sararu
aa r X i v : . [ h e p - t h ] A p r No interactions for a collection ofspin-two fields intermediated bya massive Rarita-Schwinger field
C. Bizdadea ∗ , E. M. Cioroianu † , D. Cornea ‡ ,S. O. Saliu § , S. C. S˘araru ¶ Faculty of Physics, University of Craiova13 A. I. Cuza Str., Craiova 200585, RomaniaNovember 1, 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 presenceof a massive Rarita-Schwinger field are investigated in the frameworkof the deformation theory based on local BRST cohomology. Un-der the hypotheses of locality, smoothness of the interactions in thecoupling constant, Poincar´e invariance, Lorentz covariance, and thepreservation of the number of derivatives on each field, we prove thatthere are no consistent cross-interactions among different gravitonswith a positively defined metric in internal space in the presence ofa massive Rarita-Schwinger field. The basic features of the couplingsbetween a single Pauli-Fierz field and a massive Rarita-Schwinger fieldare also emphasized.PACS number: 11.10.Ef ∗ e-mail address: [email protected] † e-mail address: [email protected] ‡ e-mail address: [email protected] § e-mail address: [email protected] ¶ e-mail address: [email protected] Introduction
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 (includ-ing itself) have been studied in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Inthis context the impossibility of cross-interactions among several Einsteingravitons under certain assumptions has been proved recently in [16] bymeans of a cohomological approach based on the Lagrangian BRST symme-try [17, 18, 19, 20, 21]. Moreover, in [16] the impossibility of cross-interactionsamong different Einstein gravitons in the presence of a scalar field has alsobeen shown.The main aim of this paper is to investigate the cross-couplings amongseveral massless spin-two fields (described in the free limit by a sum of Pauli-Fierz actions) in the presence of a massive Rarita-Schwinger field. Moreprecisely, under the hypotheses of locality, smoothness of the interactionsin the coupling constant, Poincar´e invariance, (background) Lorentz invari-ance, and the preservation of the number of derivatives on each field, weprove that there are no consistent cross-interactions among different gravi-tons with a positively defined metric in internal space in the presence of amassive Rarita-Schwinger field. This result is obtained by using the defor-mation technique [22] combined with the local BRST cohomology [23]. It iswell-known the fact that the spin-two field in metric formulation (Einstein-Hilbert theory) cannot be coupled to a spin-3/2 field. However, as it will beshown below, if we decompose the metric like g µν = σ µν + λh µν , where σ µν isthe flat metric and λ is the coupling constant, then we can indeed couple themassive spin-3/2 field to h µν in the space of formal series with the maximumderivative order equal to one in h µν . Thus, our approach envisages two differ-ent aspects. One is related to the couplings between the spin-two fields andone massive Rarita-Schwinger field, while the other focuses on proving theimpossibility of cross-interactions among different gravitons via a single mas-sive Rarita-Schwinger field. In order to make the analysis as clear as possible,we initially consider the case of the couplings between a single Pauli-Fierzfield [24] and a massive Rarita-Schwinger field [25]. In this setting we com-pute the interaction terms to order two in the coupling constant. Next, weprove the isomorphism between the local BRST cohomologies correspondingto the Pauli-Fierz theory and respectively to the linearized version of thevierbein formulation of the spin-two field. Since the deformation procedure2s controlled by the local BRST cohomology of the free theory (in ghostnumber zero and one), the previous isomorphism allows us to translate theresults emerging from the Pauli-Fierz formulation into the vierbein versionand conversely. In this manner we obtain that the first two orders of the in-teracting Lagrangian resulting from our setting originate in the developmentof the full interacting Lagrangian L (int) = e (cid:0) − i ¯ ψ µ e µa e νb e ρc γ abc D ν ψ ρ + m ¯ ψ µ e µa γ ab e νb ψ ν (cid:1) + λ (cid:2) eV ( X, Y, Z ) + d ( X, Y, Z ) e νa ¯ ψ ν γ a D µ ( eψ µ )+ ed ( X, Y, Z ) (cid:0) ¯ ψ µ γ b + e µa e b ρ ¯ ψ ρ γ a (cid:1) D µ ( e νb ψ ν ) (cid:3) . Here, e µa represent the vierbein fields, e is the inverse of their determinant, e = (det ( e µa )) − , D µ signifies the full covariant derivative, and γ a stand forthe flat Dirac matrices. The fields ψ ν denote the (curved) Rarita-Schwingerspinors ( ψ ν = e aν ψ a ). The quantities denoted by V , d , and d are arbitrarypolynomials of X ≡ ¯ ψ a ψ a , Y ≡ ¯ ψ a γ ab ψ b , and Z = i ¯ ψ a γ ψ a . Here and in thesequel λ is the coupling constant (deformation parameter). We observe thatthe first two terms in L (int) describe the standard minimal couplings betweenthe spin-two and massive Rarita-Schwinger fields. The last terms from L (int) ,namely those proportional with V , d , or d , produce non-minimal couplings.To our knowledge, these non-minimal interaction terms are not discussed inthe literature. However, they are consistent with the gauge symmetries ofthe 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 with a positively defined metric in internal space and amassive Rarita-Schwinger field, and prove that there are no consistent cross-interactions between different gravitons in the presence of such a fermionicmatter field.This paper is organized in seven sections. In Section 2 we constructthe BRST symmetry of a free model with a single Pauli-Fierz field and onemassive Rarita-Schwinger field. Section 3 briefly addresses the deformationprocedure based on BRST symmetry. In Section 4 we compute the firsttwo orders of the interactions between one graviton and one massive Rarita-Schwinger spinor. Section 5 presents the Lagrangian formulation of the in-teracting theory. Section 6 is devoted to the proof of the fact that there areno consistent cross-interactions among different gravitons in the presence ofa massive Rarita-Schwinger field. Section 7 exposes the main conclusions ofthe paper. The present paper also contains two appendix sections, in which3arious notations and conditions are listed and also some statements fromthe 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 Rarita-Schwinger field [25] S L0 [ h µν , ψ µ ] = 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 ) − i2 ¯ ψ µ γ µνρ ∂ ν ψ ρ + m ψ µ γ µν ψ ν (cid:19) ≡ Z d x (cid:16) L (PF) + L (RS)0 (cid:17) = S PF0 [ h µν ] + S RS0 [ ψ µ ] . (1)Everywhere in this paper we use the flat Minkowski metric of ‘mostly minus’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 µ are considered to be real(Majorana) spinors. We work with a representation of the Clifford algebra γ µ γ ν + γ ν γ µ = 2 σ µν (2)in which all the γ matrices are purely imaginary, so we have that γ ⊺ µ = − γ γ µ γ , µ = 0 , , (3)where here and in the sequel the notation N ⊺ signifies the transposed of thematrix N . In addition, γ is Hermitian and antisymmetric, while ( γ i ) i =1 , areanti-Hermitian and symmetric. The Dirac conjugation is defined as usuallythrough ¯ ψ µ = ( ψ µ ) † γ , (4)and the Majorana conjugation via ψ c = ( C ψ ) ⊺ , (5)4ith the corresponding charge conjugation given by C = − γ . (6)(The operation † signifies the Hermitian conjugation.) Action (1) possessesan irreducible and Abelian generating set of gauge transformations δ ǫ h µν = ∂ ( µ ǫ ν ) , δ ǫ ψ µ = 0 , (7)with ǫ µ bosonic gauge parameters. The parentheses 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 (cid:8) h ∗ µν , ψ ∗ µ (cid:9) and { η ∗ µ } . (The statistics of the antifields is opposite to that of the correlatedfields/ghosts.) The antifields of the Rarita-Schwinger fields are bosonic,purely imaginary spinors. Since the gauge generators of the free theory un-der study are field independent and irreducible, it follows that the BRSTdifferential simply decomposes into s = δ + γ, (8)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 isdefined 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 = ( h µν , ψ µ ) , Φ ∗ α = (cid:0) h ∗ µν , ψ ∗ µ (cid:1) , (9)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:0) Φ ∗ α (cid:1) = 1 , agh ( η ∗ µ ) = 2 , (10)pgh (Φ α ) = 0 , pgh ( η µ ) = 1 , pgh (cid:0) Φ ∗ α (cid:1) = pgh ( η ∗ µ ) = 0 . (11)5he actions of the differentials δ and γ on the generators from the BRSTcomplex are given by δh ∗ µν = 2 H µν , δψ ∗ µ = m ¯ ψ λ γ λµ − i ∂ ρ ¯ ψ λ γ ρλµ , (12) δη ∗ µ = − ∂ ν h ∗ µν , (13) δ Φ α = 0 = δη µ , (14) γ Φ ∗ α = 0 = γη ∗ µ , (15) γh µν = ∂ ( µ η ν ) , γψ µ = 0 , γη µ = 0 , (16)where H µν is the linearized Einstein tensor H µν = K µν − σ µν K, (17)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 ) , (18)via its trace and respectively double trace K µα = σ νβ K µναβ , K = σ µα σ νβ K µναβ . (19)The BRST differential is known to have a canonical action in a structurenamed antibracket and denoted by the symbol ( , ) ( s · = (cid:0) · , ¯ S (cid:1) ), which isobtained by decreeing the fields/ghosts respectively conjugated to the cor-responding antifields. The generator of the BRST symmetry is a bosonicfunctional of ghost number zero, which is solution to the classical masterequation (cid:0) ¯ S, ¯ S (cid:1) = 0. The full solution to the classical master equation forthe free model under study reads as¯ S = S L0 [ h µν , ψ µ ] + Z d x h ∗ µν ∂ ( µ η ν ) . (20) 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.6 S L0 δ Φ α Z α α = 0, and consider the problem of constructing consistent interac-tions among the fields Φ α such that the couplings preserve both the fieldspectrum and the original number of gauge symmetries. This matter is ad-dressed by means of reformulating the problem of constructing consistentinteractions as a deformation problem of the solution to the master equationcorresponding to the “free” theory [22]. Such a reformulation is possible dueto the fact that the solution to the master equation contains all the infor-mation on the gauge structure of the theory. If an interacting gauge theorycan be consistently constructed, then the solution ¯ S to the master equation (cid:0) ¯ S, ¯ S (cid:1) = 0 associated with the “free” theory can be deformed into a solution S ¯ S → S = ¯ S + λS + λ S + · · · = ¯ S + λ Z d D x a + λ Z d D x b + · · · , (21)of the master equation for the deformed theory( S, S ) = 0 , (22)such that both the ghost and antifield spectra of the initial theory are pre-served. The equation (22) splits, according to the various orders in thecoupling constant (deformation parameter) λ , into a tower of equations: (cid:0) ¯ S, ¯ S (cid:1) = 0 , (23)2 (cid:0) S , ¯ S (cid:1) = 0 , (24)2 (cid:0) S , ¯ S (cid:1) + ( S , S ) = 0 , (25) (cid:0) S , ¯ S (cid:1) + ( S , S ) = 0 , (26)...The equation (23) is fulfilled by hypothesis. The next equation requiresthat the first-order deformation of the solution to the master equation, S ,is a cocycle of the “free” BRST differential s · = (cid:0) · , ¯ S (cid:1) . However, only co-homologically non-trivial solutions to (24) should be taken into account, asthe BRST-exact solutions can be eliminated by some (in general non-linear)field redefinitions. This means that S pertains to the ghost number zerocohomological space of s , H ( s ), which is generically non-empty because itis isomorphic to the space of physical observables of the “free” theory. It7as been shown (by the triviality of the antibracket map in the cohomologyof the BRST differential) that there are no obstructions in finding solutionsto the remaining equations, namely (25), (26), etc. However, the resultinginteractions may be non-local, and there might even appear obstructions ifone insists on their locality. The analysis of these obstructions can be doneby 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 Rarita-Schwingerfield. This matter is addressed in the context of the antifield-BRST defor-mation procedure briefly addressed in the above and relies on computing thesolutions to the equations (24)–(26), etc., with the help of the free BRSTcohomology.For obvious reasons, we consider only smooth, local, (background) Lorentzinvariant quantities and, moreover, Poincar´e invariant quantities (i.e. we donot allow explicit dependence on the spacetime coordinates). The smooth-ness of the deformations refers to the fact that the deformed solution to themaster equation (21) is smooth in the coupling constant λ and reduces tothe original solution (20) in the free limit λ = 0. In addition, we require theconservation of the number of derivatives on each field (this condition is fre-quently met in the literature [16, 14]). If we make the notation S = R d x a ,with a a local function, then the equation (24), which we have seen thatcontrols the first-order deformation, takes the local form sa = ∂ µ m µ , gh ( a ) = 0 , ε ( a ) = 0 , (27)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 (27) is unique up to s -exact pieces8lus divergences a → a + sb + ∂ µ n µ , gh ( b ) = − , ε ( b ) = 1 , gh ( n µ ) = 0 , ε ( n µ ) = 0 . (28)At the same time, if the general solution of (27) 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 (27), 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 , (29)and take this decomposition to stop at some finite value I of the antighostnumber. The fact that I in (29) is finite can be argued like in [16]. Insertingthe above expansion into the equation (27) and projecting it on the variousvalues of the antighost number with the help of the split (8), we obtain thetower of equations γa I = ∂ µ ( I ) m µ , (30) δa I + γa I − = ∂ µ ( I − m µ , (31) δa i + γa i − = ∂ µ ( i − m µ , ≤ i ≤ I − , (32)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 [16] in the absence of the collection indices,the equation (30) can be replaced in strictly positive antighost numbers by γa I = 0 , I > . (33)Due to the second-order nilpotency of γ ( γ = 0), the solution to the equation(33) 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 . (34)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-order This is because the presence of the matter fields does not modify the general resultson H ( γ ) presented in [16]. 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 (27) (equivalent with (33) and (31)–(32)), weneed 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 differ-ential for a Pauli-Fierz field [16], as well as the definitions (15) and (16), wecan state that H ( γ ) is generated on the one hand by Φ ∗ α , η ∗ µ , ψ µ and K µναβ together with all of their spacetime derivatives and, on the other hand, by theghosts η µ and ∂ [ µ η ν ] . So, the most general (and non-trivial), local solution to(33) can be written, up to γ -exact contributions, as a I = α I (cid:0) [ ψ µ ] , [ K µναβ ] , (cid:2) Φ ∗ α (cid:3) , (cid:2) η ∗ µ (cid:3)(cid:1) ω I (cid:0) η µ , ∂ [ µ η ν ] (cid:1) , (35)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 in order to ensure that the ghost number of a I is equal to zero. Since theyhave a bounded number of derivatives and a finite antighost number, α I areactually polynomials in the linearized Riemann tensor, in the antifields, in allof their derivatives, as well as in the derivatives of the Rarita-Schwinger fields.The anticommuting behaviour of the vector-spinors induces that α I are alsopolynomials in the undifferentiated Rarita-Schwinger fields, so we concludethat these elements exhibit a 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 thelinearized Riemann tensor K µναβ , in the Rarita-Schwinger spinors, as well asin their derivatives.Inserting (35) in (31) we obtain that a necessary (but not sufficient)condition for the existence of (non-trivial) solutions a I − is that the invariantpolynomials 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 (cid:18) ( I − j µ (cid:19) = I − , pgh (cid:18) ( I − j µ (cid:19) = 0 . (36)10e recall that H ( δ | d ) is completely trivial in both strictly positive antighost and pure ghost numbers (for instance, see [23], Theorem 5.4 and [26]). Us-ing the fact that the Cauchy order of the free theory under study is equalto two together with the general results from [23], 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 > , (37)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 , (38)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 [16] 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 > . (39)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 [16]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 ) . (40)In contrast to the groups ( H J ( δ | d )) J ≥ and (cid:0) H inv J ( δ | d ) (cid:1) 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. 11he previous results on H ( δ | d ) and H inv ( δ | d ) in strictly positive anti-ghost numbers are important because they control the obstructions to re-moving the antifields from the first-order deformation. More precisely, basedon the formulas (36)–(39), one can successively eliminate all the pieces ofantighost number strictly greater that two from the non-integrated densityof the first-order deformation by adding only trivial terms, so one can take,without loss of 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(29) becomes a = a + a + a . (41)We can further decompose a in a natural manner as a sum between threekinds of deformations a = a (PF) + a (int) + a (RS) , (42)where a (PF) contains only fields/ghosts/antifields from the Pauli-Fierz sec-tor, a (int) describes the cross-interactions between the two theories (so iteffectively mixes both sectors), and a (RS) involves only the Rarita-Schwingersector. The component a (PF) is completely known (for a detailed analysissee [16]) and satisfies individually an equation of the type (27). It admits adecomposition similar to (41) a (PF) = a (PF)0 + a (PF)1 + a (PF)2 , (43)where a (PF)2 = 12 η ∗ µ η ν ∂ [ µ η ν ] , (44) a (PF)1 = h ∗ µρ (cid:0) ( ∂ ρ η ν ) h µν − η ν ∂ [ µ h ν ] ρ (cid:1) , (45)and a (PF)0 is the cubic vertex of the Einstein-Hilbert Lagrangian plus a cos-mological term . Due to the fact that a (int) and a (RS) involve different kinds The terms a (PF)2 and a (PF)1 given in (44) and (45) differ from the corresponding onesin [16] by a γ -exact and respectively a δ -exact contribution. However, the difference be-
12f fields, it follows that a (int) and a (RS) are subject to some separate equations sa (int) = ∂ µ m (int) µ , (46) sa (RS) = ∂ µ m (RS) µ , (47)for some local m µ ’s. In the sequel we analyze the general solutions to theseequations.Since the massive Rarita-Schwinger field does not carry gauge symmetriesof its own, it results that the massive gravitino sector can only occur inantighost number one and zero, so, without loss of generality, we can take a (int) = a (int)0 + a (int)1 (48)in (46), where the components involved in the right-hand side of (48) aresubject to the equations γa (int)1 = 0 , (49) δa (int)1 + γa (int)0 = ∂ µ (0) m (int) µ . (50)According to (35) in pure ghost number one and because ω is spanned by ω = (cid:0) η µ , ∂ [ µ η ν ] (cid:1) , we infer that the most general expression of a (int)1 as solution to the equation(49) is a (int)1 = ψ ∗ µ (cid:0) N ρµ η ρ + N ρλµ ∂ [ ρ η λ ] (cid:1) , (51)where N ρµ and N ρλµ are real, odd spinor-like functions, with N ρλµ antisym-metric in its upper indices. All the objects denoted by N are gauge-invariant,so they may depend on ψ µ , K µνρλ , and their spacetime derivatives. At thisstage we recall the hypothesis on the conservation of the number of derivatives tween our a (PF)2 + a (PF)1 and the corresponding sum from [16] 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 [16] belong to the same cohomological class from H ( s | d ). We remark that in principle we might have added to a (int)1 a component ˜ a (int)1 linearin the antifield of the Pauli-Fierz field, h ∗ µν . However, such terms cannot produce aconsistent component of the first-order deformation in antighost number zero, as it isshown in Appendix B.
13n each field, which allows us to simplify the solution (51) to the equation(49) by imposing that the following requirements are simultaneously satisfied:i) the interaction vertices present in a (int)0 as solution to (50), assuming a (int)0 exists, contain at most two derivatives of the fields;ii) the deformed field equations associated with a (int)0 involve at mostthe first-order derivatives of the spinor fields and at most the second-orderderivatives of the Pauli-Fierz field.By applying the differential δ on (51) and using the definitions (12)–(16),we infer that δa (int)1 = ∂ µ m µ + γb + c , (52)where m µ = − i ¯ ψ β γ µβν (cid:0) N ρν η ρ + N ρλµ ∂ [ ρ η λ ] (cid:1) , (53) b = i2 ¯ ψ β γ αβµ (cid:0) N ρµ h αρ + 2 N ρλµ ∂ [ ρ h λ ] α (cid:1) , (54) c = (cid:0) m ¯ ψ α γ αµ N ρµ + i ¯ ψ β γ αβµ ∂ α N ρµ (cid:1) η ρ + (cid:0) m ¯ ψ α γ αµ N ρλµ + i ¯ ψ β γ αβµ ∂ α N ρλµ + i2 ¯ ψ β γ ρβµ N λµ (cid:19) ∂ [ ρ η λ ] . (55)Taking into account the previous two requirements on the derivative be-haviour of a (int)0 , from (54) we get that the spinor-tensor N ρµ may contain atmost one derivative of the spinor ψ µ , while the spinor-tensor N ρλµ can onlydepend on the undifferentiated Rarita-Schwinger field. As a consequence, wehave that N ρµ = ¯ N ρλµ ψ λ + ¯ N ρλσµ ∂ λ ψ σ , N ρλµ = N ρλσµ ψ σ , (56)and hence a (int)1 = ψ ∗ µ (cid:0) ¯ N ρλµ ψ λ + ¯ N ρλσµ ∂ λ ψ σ (cid:1) η ρ + ψ ∗ µ N ρλσµ ψ σ ∂ [ ρ η λ ] , (57)where ¯ N ρλµ , ¯ N ρλσµ , and N ρλσµ are real, bosonic 4 × ψ µ . Inserting (56) in theformulas (54)–(55), we get b = i2 ¯ ψ β γ αβµ (cid:0)(cid:0) ¯ N ρλµ ψ λ + ¯ N ρλσµ ∂ λ ψ σ (cid:1) h αρ +2 N ρλσµ ψ σ ∂ [ ρ h λ ] α (cid:1) , (58)14 = (cid:0) m ¯ ψ α γ αµ (cid:0) ¯ N ρλµ ψ λ + ¯ N ρλσµ ∂ λ ψ σ (cid:1) +i ¯ ψ β γ αβµ ∂ α (cid:0) ¯ N ρλµ ψ λ + ¯ N ρλσµ ∂ λ ψ σ (cid:1)(cid:1) η ρ + (cid:0) m ¯ ψ α γ αµ N ρλσµ ψ σ + i ¯ ψ β γ αβµ ∂ α (cid:0) N ρλσµ ψ σ (cid:1) + i2 ¯ ψ β γ ρβµ (cid:0) ¯ N λσµ ψ σ + ¯ N λασµ ∂ α ψ σ (cid:1)(cid:19) ∂ [ ρ η λ ] . (59)The condition that δa (int)1 should be written like in (50) restricts c expressedin (59) to be a γ -exact modulo d quantity, i.e. c = γm + ∂ µ n µ . (60)At this stage it is useful to split c like c = X k =0 ( c ) k , (61)where ( c ) k denotes the piece from c with k -derivatives. According to thisdecomposition, it follows that each ( c ) k should be written in a γ -exact mod-ulo d form, such that (50) is indeed satisfied. Using (59), we obtain that( c ) = m ¯ ψ α γ αµ ¯ N ρλµ ψ λ η ρ . (62)As the right-hand side of (62) is derivative-free, it follows that these termsneither reduce to a total derivative nor can be expressed in a γ -exact form,so they must vanish ¯ ψ α γ αµ ¯ N ρλµ ψ λ = 0 . (63)Simple computation exhibit that (63) is checked if γ γ αµ ¯ N ρλµ = (cid:0) γ γ λµ ¯ N ραµ (cid:1) ⊺ , (64)whose general solution is expressed by¯ N ρλµ = c δ ρµ γ λ + c δ λµ γ ρ + c σ ρλ γ µ + 12 ( c + 2 c + 3 c ) γ ρλµ , (65)with c , c , and c some arbitrary functions depending on ψ µ . As it has beenshown in Appendix B, the functions c , c , and c from (65) can be made tovanish by adding some trivial, s -exact terms and by conveniently redefiningthe functions ¯ N ρλσµ . In consequence, we can take¯ N ρλµ = 0 . (66)15he equation (60) for k = 1 becomes m ¯ ψ α γ αµ ¯ N ρλσµ ( ∂ λ ψ σ ) η ρ + m ¯ ψ α γ αµ N ρλσµ ψ σ ∂ [ ρ η λ ] = γm + ∂ µ n µ , (67)where γm = ( ∂m /∂h ρλ ) ∂ ( ρ η λ ) . By taking the Euler-Lagrange deriva-tives of the relation (67) with respect to η ν we obtain that the quantity m ¯ ψ α γ αµ ¯ N ρλσµ ( ∂ λ ψ σ ) should reduce to a total derivative m ¯ ψ α γ αµ ¯ N ρλσµ ( ∂ λ ψ σ ) = ∂ λ M ρλ . (68)The left-hand side of (68) is a full divergence if the following conditions ∂ λ ¯ N ρλσµ = 0 , (69) γ γ αµ ¯ N ρλσµ = − (cid:0) γ γ σµ ¯ N ρλαµ (cid:1) ⊺ (70)are simultaneously satisfied. The general solution to (69)–(70) takes the form¯ N ρλσµ = k (cid:18) σ λσ (cid:18) δ ρµ + 12 γ ρµ (cid:19) + σ ρσ (cid:18) δ λµ + 12 γ λµ (cid:19)(cid:19) + k σ ρλ (cid:18) δ σµ + 12 γ σµ (cid:19) + k σ ρλ δ σµ + k (cid:0) σ λσ δ ρµ − σ ρσ δ λµ − δ σµ γ ρλ + γ ρλσµ − (cid:0) δ λµ γ ρσ − δ ρµ γ λσ (cid:1) + 12 (cid:0) σ σλ γ ρµ − σ ρσ γ λµ (cid:1)(cid:19) = ¯ N ρλσ µ + ¯ N ρλσ µ , (71)with ¯ N ρλσ µ = k (cid:18) σ λσ (cid:18) δ ρµ + 12 γ ρµ (cid:19) + σ ρσ (cid:18) δ λµ + 12 γ λµ (cid:19)(cid:19) + k σ ρλ (cid:18) δ σµ + 12 γ σµ (cid:19) + k σ ρλ δ σµ , (72)and ( k i ) i =1 , some arbitrary constants. Under these circumstances (if theequations (69)–(70) are verified), we find that m ¯ ψ α γ αµ ¯ N ρλσµ ( ∂ λ ψ σ ) η ρ + m ¯ ψ α γ αµ N ρλσµ ψ σ ∂ [ ρ η λ ] = γ (cid:18) − m ¯ ψ α γ αµ ¯ N ρλσ µ ψ σ h ρλ (cid:19) + ∂ λ (cid:18) m ¯ ψ α γ αµ ¯ N ρλσµ ψ σ η ρ (cid:19) m ¯ ψ α γ αµ (cid:18) N ρλσµ + 14 ¯ N ρλσ µ (cid:19) ψ σ ∂ [ ρ η λ ] . (73)By comparing the last equation to (67) we observe that the last term fromthe right-hand side of (73) must be γ -exact modulo d . This takes place if¯ ψ α γ αµ (cid:18) N ρλσµ + 14 ¯ N ρλσ µ (cid:19) ψ σ = 0 , (74)from which we further deduce N ρλσµ = −
14 ¯ N ρλσ µ + ˆ N ρλσµ , (75)where ˆ N ρλσµ is solution to the equation¯ ψ α γ αµ ˆ N ρλσµ ψ σ = 0 . (76)It is simple to see that (76) holds if γ γ αµ ˆ N ρλσµ = (cid:16) γ γ σµ ˆ N ρλαµ (cid:17) ⊺ , (77)whose general solution is given byˆ N ρλσµ = ¯ k (cid:0) σ λσ δ ρµ − σ ρσ δ λµ (cid:1) + ¯ k δ σµ γ ρλ +¯ k (cid:0) δ λµ γ ρσ − δ ρµ γ λσ (cid:1) + ¯ k γ ρλσµ + 12 (cid:0) ¯ k − k + ¯ k (cid:1) (cid:0) σ σλ γ ρµ − σ ρσ γ λµ (cid:1) , (78)with (cid:0) ¯ k i (cid:1) i =1 , some arbitrary functions depending on ψ µ .Next, we analyze the solution to the equation (60) for k = 2. It takes theconcrete form i2 ¯ ψ β (cid:0) γ αβµ ¯ N ρλσµ + γ λβµ ¯ N ρασµ (cid:1) ( ∂ α ∂ λ ψ σ ) η ρ +i ¯ ψ β γ αβµ ∂ α (cid:0) N ρλσµ ψ σ (cid:1) ∂ [ ρ η λ ] + i2 ¯ ψ β γ ρβµ ¯ N λασµ ( ∂ α ψ σ ) ∂ [ ρ η λ ] = γm + ∂ µ n µ , (79)with ¯ N ρλσµ and N ρλσµ determined previously. By taking the Euler-Lagrangederivatives of (79) with respect to η ν and by using the result that γm =17 δm /δh ρλ ) ∂ ( ρ η λ ) + ∂ λ v λ , with δm /δh ρλ the variational derivative of m with respect to h ρλ , it follows thati2 ¯ ψ β (cid:0) γ αβµ ¯ N ρλσµ + γ λβµ ¯ N ρασµ (cid:1) ( ∂ α ∂ λ ψ σ ) = ∂ λ P ρλ , (80)for some P ρλ . The left-hand side of the last equation is written as a fulldivergence if (cid:0) ∂ λ ¯ ψ β (cid:1) (cid:0) γ αβµ ¯ N ρλσµ + γ λβµ ¯ N ρασµ (cid:1) ( ∂ α ψ σ ) = 0 , (81)which further produces k = k = k = 0 , (82)such that we havei2 ¯ ψ β (cid:0) γ αβµ ¯ N ρλσµ + γ λβµ ¯ N ρασµ (cid:1) ( ∂ α ∂ λ ψ σ ) η ρ = − i k γ (cid:0) ¯ ψ β (cid:0) γ αβσ ( ∂ α ψ σ ) h + γ λβσ ( ∂ ρ ψ σ ) h λρ (cid:1)(cid:1) + i k ψ β (cid:0) σ αρ γ λβσ − σ αλ γ ρβσ (cid:1) ( ∂ α ψ σ ) ∂ [ ρ η λ ] + ∂ λ u λ . (83)On the other hand, it is easy to see thati ¯ ψ β γ αβµ ∂ α (cid:0) N ρλσµ ψ σ (cid:1) ∂ [ ρ η λ ] = − γ (cid:16) i ¯ ψ β γ αβµ ˆ N ρλσµ ψ σ ∂ [ ρ h λ ] α (cid:17) + ∂ λ ¯ u λ − i (cid:0) ∂ α ¯ ψ β (cid:1) γ αβµ ˆ N ρλσµ ψ σ ∂ [ ρ η λ ] . (84)Inserting (83)–(84) in (79) and taking into account the result (82), the equa-tion (79) reduces to − i (cid:0) ∂ α ¯ ψ β (cid:1) γ αβµ ˆ N ρλσµ ψ σ ∂ [ ρ η λ ] − i k ψ β (cid:0) σ αρ γ λβσ − σ αλ γ ρβσ (cid:1) ( ∂ α ψ σ ) ∂ [ ρ η λ ] = γ ¯ m + ∂ µ ¯ n µ . (85)Now, we decompose γ αβµ ˆ N ρλσµ like γ αβµ ˆ N ρλσµ = (cid:16) γ αβµ ˆ N ρλσµ (cid:17) + (cid:16) γ αβµ ˆ N ρλσµ (cid:17) , (86)18ith (cid:16) γ αβµ ˆ N ρλσµ (cid:17) = 12 (cid:18)
12 ¯ k + ¯ k − k −
12 ¯ k (cid:19) × (cid:0) σ λσ γ αβρ + σ λβ γ ασρ − σ ρσ γ αβλ − σ ρβ γ ασλ (cid:1) +¯ k (cid:0) σ σβ γ ραλ − σ σα γ ρβλ − σ βα γ ρσλ (cid:1) + (cid:0) ¯ k − ¯ k + ¯ k + ¯ k (cid:1) (cid:0) σ σρ σ λβ − σ σλ σ ρβ (cid:1) γ α + 12 (cid:0) ¯ k + ¯ k (cid:1) (cid:0)(cid:0) σ σλ σ αρ − σ σρ σ λα (cid:1) γ β + (cid:0) σ βρ σ αλ − σ βλ σ αρ (cid:1) γ σ (cid:1) + 12 (cid:0) ¯ k + ¯ k (cid:1) (cid:0)(cid:0) σ βρ σ ασ − σ σρ σ αβ (cid:1) γ λ + (cid:0) σ σλ σ αβ − σ βλ σ ασ (cid:1) γ ρ (cid:1) , (87) (cid:16) γ αβµ ˆ N ρλσµ (cid:17) = 12 (cid:18)
12 ¯ k − ¯ k −
12 ¯ k (cid:19) × (cid:0) σ λσ γ αβρ − σ λβ γ ασρ − σ ρσ γ αβλ + σ ρβ γ ασλ (cid:1) + (cid:0) ¯ k − ¯ k (cid:1) (cid:0) σ ρα γ βσλ − σ λα γ βσρ (cid:1) + ¯ k (cid:0) σ βα γ ρσλ − σ σα γ ρβλ (cid:1) + 12 (cid:0) ¯ k − k + ¯ k + 2¯ k (cid:1) (cid:0)(cid:0) σ σλ σ αρ − σ σρ σ λα (cid:1) γ β + (cid:0) σ βλ σ αρ − σ βρ σ αλ (cid:1) γ σ (cid:1) + 12 (cid:0) ¯ k + ¯ k (cid:1) (cid:0) σ βσ (cid:0) σ αλ γ ρ − σ αρ γ λ (cid:1) + (cid:0) σ βρ σ ασ + σ σρ σ αβ (cid:1) γ λ − (cid:0) σ ασ σ βλ + σ αβ σ σλ (cid:1) γ ρ (cid:1) . (88)By direct computation it can be shown that the two components of γ αβµ ˆ N ρλσµ satisfy the properties γ (cid:16) γ αβµ ˆ N ρλσµ (cid:17) = − (cid:16) γ (cid:16) γ ασµ ˆ N ρλβµ (cid:17) (cid:17) ⊺ , (89) γ (cid:16) γ αβµ ˆ N ρλσµ (cid:17) = (cid:16) γ (cid:16) γ ασµ ˆ N ρλβµ (cid:17) (cid:17) ⊺ . (90)By means of the formulas (89)–(90) we can write − i (cid:0) ∂ α ¯ ψ β (cid:1) γ αβµ ˆ N ρλσµ ψ σ ∂ [ ρ η λ ] = γ (cid:18) i2 ¯ ψ β (cid:16) γ αβµ ˆ N ρλσµ (cid:17) ψ σ ∂ [ ρ h λ ] α (cid:19)
19i ¯ ψ β (cid:16) γ αβµ ˆ N ρλσµ (cid:17) ( ∂ α ψ σ ) ∂ [ ρ η λ ] + ∂ α (cid:18) − i2 ¯ ψ β (cid:16) γ αβµ ˆ N ρλσµ (cid:17) ψ σ ∂ [ ρ η λ ] (cid:19) , (91)such that − i (cid:0) ∂ α ¯ ψ β (cid:1) γ αβµ ˆ N ρλσµ ψ σ ∂ [ ρ η λ ] − i k ψ β (cid:0) γ λβσ σ αρ − γ ρβσ σ αλ (cid:1) ( ∂ α ψ σ ) ∂ [ ρ η λ ] = γ (cid:18) i2 ¯ ψ β (cid:16) γ αβµ ˆ N ρλσµ (cid:17) ψ σ ∂ [ ρ h λ ] α (cid:19) + ∂ α (cid:18) − i2 ¯ ψ β (cid:16) γ αβµ ˆ N ρλσµ (cid:17) ψ σ ∂ [ ρ η λ ] (cid:19) − i ¯ ψ β (cid:18) k (cid:0) σ αρ γ λβσ − σ αλ γ ρβσ (cid:1) − (cid:16) γ αβµ ˆ N ρλσµ (cid:17) (cid:17) ( ∂ α ψ σ ) ∂ [ ρ η λ ] . (92)Comparing (92) with (85) it results that the last term in (92) has to be γ -exact modulo d . This holds ifi ¯ ψ β (cid:18)(cid:18) k (cid:0) σ αρ γ λβσ − σ αλ γ ρβσ (cid:1) − (cid:16) γ αβµ ˆ N ρλσµ (cid:17) (cid:19) ( ∂ α ψ σ ) (cid:19) = ∂ α θ α (93)for some θ α or, in other words, if M αβρλσ = γ (cid:18) k (cid:0) σ αρ γ λβσ − σ αλ γ ρβσ (cid:1) − (cid:16) γ αβµ ˆ N ρλσµ (cid:17) (cid:19) (94)fulfills the condition M αβρλσ = − (cid:0) M ασρλβ (cid:1) ⊺ . (95)With the help of (90) we obtain the relations M αβρλσ = (cid:0) M ασρλβ (cid:1) ⊺ , (96)which indicate that (95) cannot be satisfied, and hence neither (93). As aconsequence, the term − i ¯ ψ β M αβρλσ ( ∂ α ψ σ ) ∂ [ ρ η λ ] from (92) must be canceled,which implies M αβρλσ = 0 . (97)20he solution to the above equation reads as¯ k = 14 k , ¯ k = 18 k , ¯ k = 0 , ¯ k = 0 . (98)Redenoting k by k , we finally find the relations¯ N ρλσµ = kσ ρλ δ σµ , N ρλσµ = ˆ N ρλσµ = 14 k (cid:18) σ λσ δ ρµ − σ ρσ δ λµ + 12 δ σµ γ ρλ (cid:19) . (99)Replacing (66) and (99) in (57), we get that a (int)1 = kψ ∗ µ ( ∂ ν ψ µ ) η ν + k ψ ∗ µ ψ ν ∂ [ µ η ν ] + k ψ ∗ ρ γ µν ψ ρ ∂ [ µ η ν ] . (100)Meanwhile, if we insert (99) in (58), (73), (83)–(84), and (92) and the result-ing expressions in (52), we deduce that the component of antighost numberzero from the first-order deformation is given by a (int)0 = k (cid:18) σ ρλ L (RS)0 − i2 ¯ ψ µ γ µνρ ∂ λ ψ ν (cid:19) h ρλ + i k (cid:18)
12 ¯ ψ µ γ ρ ψ ν + σ µρ ¯ ψ ν γ σ ψ σ + ¯ ψ σ γ σρµ ψ ν (cid:19) ∂ [ µ h ν ] ρ + ¯ a (int)0 , (101)where ¯ a (int)0 represents the general, local solution to the homogeneous equa-tion γ ¯ a (int)0 = ∂ µ ¯ m (int) µ , (102)with some local ¯ m (int) µ .Such solutions correspond to ¯ a (int)1 = 0 and thus they cannot deform eitherthe gauge algebra or the gauge transformations, but simply the Lagrangianat order one in the coupling constant. There are two main types of solutionsto (102). The first one corresponds to ¯ m (int) µ = 0 and is given by gauge-invariant, non-integrated densities constructed from the original fields andtheir spacetime derivatives. According to (35) for both pure ghost and anti-ghost numbers equal to zero, they are given by ¯ a ′ (int)0 = ¯ a ′ (int)0 ([ ψ µ ] , [ K µναβ ]),up to the conditions that they effectively describe cross-couplings between thetwo types of fields and cannot be written in a divergence-like form. Unfortu-nately, this type of solutions must depend on the linearized Riemann tensor21and possibly of its derivatives) in order to provide cross-couplings, and thuswould lead to terms with at least two derivatives of the Rarita-Schwingerspinors in the deformed field equations. 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 (102) and will be approachedbelow.We split the solution to the equation (102) for ¯ m (int) µ = 0 along thenumber of derivatives present in the interaction vertices¯ a (int)0 = X i =0 ( i ) ω , (103)where ( i ) ω contains i derivatives of the fields. The decomposition (103) yields asimilar splitting with respect to the equation (102), which becomes equivalentto three independent equations γ ( i ) ω = ∂ µ ( i ) m µ , i = 0 , . (104)Let us solve (104) for i = 0. With the help of the definitions of γ actingon the generators from the BRST complex we get γ (0) ω = − ∂ ν ∂ (0) ω∂h µν η µ + ∂ µ π µ . (105)Thus, (0) ω is solution to (104) for i = 0 if and only if ∂ ν ∂ (0) ω∂h µν = 0 . (106)Since (0) ω has no derivatives, the equation (106) implies that ∂ (0) ω /∂h ρµ mustbe constant. As the only constant and symmetric tensor in four spacetimedimensions is the flat metric, we can write ∂ (0) ω∂h µν = pσ µν , (107)22ith p a real constant. Integrating (107), it results that the solution to theequation (104) for i = 0 reads as (0) ω = ph + F ( ψ µ ) , but since it provides no cross-interactions, we can take (0) ω = 0 . (108)Next, we pass to the equation (104) for i = 1. We obtain that γ (1) ω = − ∂ ν δ (1) ωδh µν η µ + ∂ µ β µ , (109)so (1) ω checks (104) for i = 1 if and only if ∂ ν δ (1) ωδh µν = 0 . (110)Because (1) ω includes just one spacetime derivative, the solution to (110) is δ (1) ωδh µν = ∂ ρ D ρµν , (111)where D ρµν depends only on the undifferentiated fields and is antisymmetricin its first two indices D ρµν = − D µρν . (112)Since D ρµν is derivative-free and h µν is symmetric, (111) implies that D ρµν must be symmetric in its last two indices D ρµν = D ρνµ . (113)The properties (112) and (113) further lead to D ρµν = − D µρν = − D µνρ = D νµρ = D νρµ = − D ρνµ = − D ρµν , (114)23o D ρµν = 0. Consequently, (111) reduces to δ (1) ωδh µν = 0 , (115)whose solution is expressed by (1) ω = L ([ ψ µ ]) + ∂ µ G µ ( ψ µ , h αβ ) (116)and is not suitable as the first term provides no cross-interactions, while thesecond is trivial, so we have that (1) ω = 0 . (117)In the end, we solve (104) for i = 2. From the relation γ (2) ω = − ∂ ν δ (2) ωδh µν η µ + ∂ µ ξ µ , (118)we observe that (2) ω verifies (104) for i = 2 if and only if ∂ ν δ (2) ωδh µν = 0 . (119)The solution to the last equation reads as δ (2) ωδh µν = ∂ α ∂ β U µανβ , (120)where U µανβ displays the symmetry properties of the Riemann tensor and in-volves only the undifferentiated fields ψ µ and h µν . At this stage it is useful tointroduce a derivation in the algebra of the fields h µν and of their derivativesthat counts the powers of the fields and their derivatives, defined by N = X k ≥ ( ∂ µ ··· µ k h µν ) ∂∂ ( ∂ µ ··· µ k h µν ) . (121)24hen, it is easy to see that for every nonintegrated density χ , we have that N χ = h µν δχδh µν + ∂ µ s µ . (122)If χ ( l ) is a homogeneous polynomial of order l > N χ ( l ) = lχ ( l ) . Using (120), and (122), we find that N (2) ω = − K µανβ U µανβ + ∂ µ v µ . (123)We expand (2) ω like (2) ω = X l> ω ( l ) , (124)where N (2) ω ( l ) = l (2) ω ( l ) , such that N (2) ω = X l> l (2) ω ( l ) . (125)Comparing (123) with (125), we reach the conclusion that the decomposition(124) induces a similar decomposition with respect to U µανβ , i.e. U µανβ = X l> U µανβ ( l − . (126)Substituting (126) into (123) and comparing the resulting expression with(125), we obtain that (2) ω ( l ) = − l K µανβ U µανβ ( l − + ∂ µ ¯ v µ ( l ) . (127)Introducing (127) in (124), we arrive at (2) ω = − K µανβ ¯ U µανβ + ∂ µ ¯ v µ , (128)where ¯ U µανβ = X l> l U µανβ ( l − . (129)25ven if consistent, an (2) ω of the type (128) would produce field equationswith two spacetime derivatives acting on the Rarita-Schwinger spinors, whichbreaks the hypothesis on the derivative order of the interacting theory, so wemust take (2) ω = 0 . (130)The results (108), (117), and (130) enable us to take, without loss of gener-ality ¯ a (int)0 = 0 (131)in (101).Finally, we analyze the component a (RS) from (42). As the massive Rarita-Schwinger action from (1) has no non-trivial gauge invariance, it follows that a (RS) can only reduce to its component of antighost number zero a (RS) = a (RS)0 ([ ψ µ ]) , (132)which is automatically solution to the equation sa (RS) ≡ γa (RS)0 = 0. Itcomes from a (RS)1 = 0 and does not deform the gauge transformations (9),but merely modifies the massive spin-3 / a (RS)0 isof maximum derivative order equal to one is translated into a (RS)0 = V ( ψ µ ) + V αβ ( ψ µ ) ∂ α ψ β , (133)where V and V αβ are polynomials in the undifferentiated spinor fields (sincethey anticommute). The first polynomial is a scalar (bosonic and real), whilethe tensor V αβ is fermionic and anti-Majorana spinor-like.The general conclusion of this subsection is that the first-order deforma-tion associated with the Pauli-Field theory plus the massive Rarita-Schwingerfield can be written like S = S (PF)1 + S (int)1 , (134)with S (PF)1 = Z d x (cid:16) a (PF)0 + a (PF)1 + a (PF)2 (cid:17) , (135)and S (int)1 = Z d x (cid:16) a (int)0 + a (int)1 + a (RS)0 (cid:17) . (136)The first two components of (136) are expressed by (100) and (101) with¯ a (int)0 = 0, while a (RS)0 is given by (133). This is the most general form thatcomplies with all the hypotheses that must be satisfied by the deformations,including that related to the derivative order of the deformed Lagrangian.26 .3 Second-order deformation In this subsection we are interested in determining the complete expression ofthe second-order deformation for the solution to the master equation, whichis known to be subject to the equation (25). Proceeding in the same mannerlike during the first-order deformation procedure, we can write the second-order deformation of the solution to the master equation like the sum betweenthe Pauli-Fierz and the interacting parts S = S (PF)2 + S (int)2 . (137)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 , (138)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) (139)and S (int)1 is presented in (136). If 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(138) becomes ∆ (int) = − sb (int) + ∂ µ n µ , (140)with gh (cid:0) ∆ (int) (cid:1) = 1 , gh (cid:0) b (int) (cid:1) = 0 , gh ( n µ ) = 1 , (141)for some local current n µ . Direct computation shows that ∆ (int) decomposeslike ∆ (int) = ∆ (int)0 + ∆ (int)1 , agh (cid:16) ∆ (int) I (cid:17) = I, I = 0 , , (142)with ∆ (int)1 = γ (cid:18) k (cid:18) − (cid:18) ψ ∗ [ µ ψ σ ] + 12 ψ ∗ ρ γ µσ ψ ρ (cid:19) ∂ [ σ η λ ] σ νλ + kψ ∗ σ ( ∂ µ ψ σ ) η ν ) h µν + k (2 − k )2 (cid:18) ψ ∗ µ ψ ν + 14 ψ ∗ σ γ µν ψ σ (cid:19) η ρ ∂ [ µ h ν ] ρ (cid:19) k (1 − k ) (cid:18) ψ ∗ µ ( ∂ ν ψ µ ) η ρ ∂ [ ν η ρ ] + 14 (cid:0) ψ ∗ [ µ ψ ν ] + 12 ψ ∗ σ γ µν ψ σ (cid:19) ∂ [ µ η ρ ] ∂ [ ν η λ ] σ ρλ (cid:19) , (143)and ∆ (int)0 = γ (cid:18) k L (RS)0 h µν h µν (cid:19) + k (cid:18) −L (RS)0 η µ + i k η σ ∂ σ (cid:0) ¯ ψ µ γ ρ ψ ρ (cid:1) + i k ψ µ γ ρ ψ σ ∂ [ ρ η σ ] + i k ψ σ γ ρ ψ ρ ∂ [ µ η σ ] + i k
16 ¯ ψ µ (cid:2) γ ρ , γ αβ (cid:3) ψ ρ ∂ [ α η β ] (cid:19) ( ∂ ν h µν − ∂ µ h )+ i k (cid:0) η σ ∂ σ (cid:0) ¯ ψ µ γ α ψ ν − ψ β γ αβµ ψ ν (cid:1) + ¯ ψ µ γ α ψ σ ∂ [ ν η σ ] − ¯ ψ β γ αβµ ψ σ ∂ [ ν η σ ] − ¯ ψ σ γ αβµ ψ ν ∂ [ β η σ ] + (cid:18)
18 ¯ ψ µ (cid:2) γ α , γ ρλ (cid:3) ψ ν −
14 ¯ ψ β (cid:2) γ αβµ , γ ρλ (cid:3) ψ ν (cid:19) ∂ [ ρ η λ ] (cid:19) ∂ [ µ h ν ] α + k (cid:18) η σ ∂ σ L (RS)0 − i2 ¯ ψ µ γ µνρ ( ∂ σ ψ ρ ) ∂ ν η σ + m ψ µ γ µν ψ σ ∂ [ ν η σ ] − i2 ¯ ψ µ γ µνρ ∂ ν (cid:0) ψ σ ∂ [ ρ η σ ] (cid:1) − i2 ¯ ψ σ γ µνρ ( ∂ ν ψ ρ ) ∂ [ µ η σ ] + m (cid:0) ¯ ψ µ (cid:2) γ µν , γ αβ (cid:3) ψ ν − i ¯ ψ µ (cid:2) γ µνρ , γ αβ (cid:3) ∂ ν ψ ρ (cid:1) ∂ [ α η β ] − i16 ¯ ψ µ γ µνρ γ αβ ψ ρ ∂ ν (cid:0) ∂ [ α η β ] (cid:1)(cid:19) h − i k (cid:0) η σ ∂ σ (cid:0) ¯ ψ µ γ µνρ ∂ λ ψ ν (cid:1) + ¯ ψ µ γ µνρ ( ∂ σ ψ ν ) ∂ λ η σ + 12 ¯ ψ µ γ µνρ (cid:0) ∂ λ ψ σ ∂ [ ν η σ ] (cid:1) + 12 ¯ ψ σ γ µνρ (cid:0) ∂ λ ψ ν (cid:1) ∂ [ µ η σ ] + 18 ¯ ψ µ (cid:2) γ µνρ , γ αβ (cid:3) (cid:0) ∂ λ ψ ν (cid:1) ∂ [ α η β ] + 18 ¯ ψ µ γ µνρ γ αβ ψ ν ∂ λ (cid:0) ∂ [ α η β ] (cid:1)(cid:19) h ρλ − i k ψ µ γ µν ( ρ ∂ λ ) ψ ν ×× ( h λσ ∂ ρ η σ − η σ ( ∂ ρ h λσ − ∂ σ h ρλ )) + i k ψ µ γ ( ρ ψ λ ) ×× ∂ µ ( h λσ ∂ ρ η σ − η σ ( ∂ ρ h λσ − ∂ σ h ρλ )) + i k ψ µ γ ρ ψ ρ × ∂ ν (cid:0) h σ ( µ ∂ ν ) η σ − η σ (cid:0) ∂ ( µ h ν ) σ − ∂ σ h µν (cid:1)(cid:1) − i k ψ µ γ ρ ψ ρ ×× ∂ µ (cid:0) h αβ ∂ α η β − η α (cid:0) ∂ β h αβ − ∂ α h (cid:1)(cid:1) − i k ψ µ γ µν ( ρ ψ λ ) ×× ∂ ν ( h λσ ∂ ρ η σ − η σ ( ∂ ρ h λσ − ∂ σ h ρλ ))+2 k (cid:0) ∂ µ V + V αβ ∂ α ψ β (cid:1) η µ + 2 kV µν ( ∂ σ ψ ν ) ∂ µ η σ + k ∂ R V∂ψ µ ψ ν ∂ [ µ η ν ] + kV µν ∂ µ (cid:0) ψ σ ∂ [ ν η σ ] (cid:1) + k ¯ ψ σ ∂ L V µν ∂ ¯ ψ ρ ( ∂ µ ψ ν ) ∂ [ ρ η σ ] + k (cid:18) ∂ R V∂ψ ρ γ αβ ψ ρ − ¯ ψ ρ γ αβ ∂ L V µν ∂ ¯ ψ ρ ∂ µ ψ ν (cid:19) ∂ [ α η β ] ++ k V µν γ αβ ∂ µ (cid:0) ψ ν ∂ [ α η β ] (cid:1) . (144)Since the first-order deformation in the interacting sector starts in anti-ghost number one, we can take, without loss of generality, the correspondingsecond-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 , , , (145) n µ = n µ + n µ + n µ , agh ( n µI ) = I, I = 0 , , . (146)By projecting the equation (140) on various antighost numbers, we obtain γb (int)2 = ∂ µ (cid:18) n µ (cid:19) , (147)∆ (int)1 = − (cid:16) δb (int)2 + γb (int)1 (cid:17) + ∂ µ n µ , (148)∆ (int)0 = − (cid:16) δb (int)1 + γb (int)0 (cid:17) + ∂ µ n µ . (149)The equation (147) can always be replaced, by adding trivial terms, with γb (int)2 = 0 . (150)Looking at ∆ (int)1 given in (143), it results that it can be written like in (148)if χ = k (1 − k ) (cid:18) ψ ∗ µ ( ∂ ν ψ µ ) η ρ ∂ [ ν η ρ ] + 14 (cid:0) ψ ∗ [ µ ψ ν ]
29 12 ψ ∗ σ γ µν ψ σ (cid:19) ∂ [ µ η ρ ] ∂ [ ν η λ ] σ ρλ (cid:19) (151)can be expressed like χ = δϕ + γω + ∂ α l α . (152)Supposing that (152) holds and applying δ on it, we infer that δχ = γ ( − δω ) + ∂ α ( δl α ) . (153)On the other hand, using the concrete expression of χ , we have that δχ = γ (cid:18) k (1 − k )2 δ ( ψ ∗ ρ ψ ρ η ν ( ∂ µ h µν − ∂ ν h )) (cid:19) + ∂ µ (cid:18) k (1 − k ) δ (cid:0) ψ ∗ ρ ψ ρ η ν ∂ [ µ η ν ] (cid:1)(cid:19) + γ (cid:18) i4 k (1 − k ) (cid:0)(cid:0) ¯ ψ β γ αβσ ( ∂ µ ψ σ ) h ρα − (cid:0) ¯ ψ β γ αβ [ µ ψ ν ] − ¯ ψ µ γ α ψ ν − σ α [ µ ¯ ψ ν ] γ σ ψ σ (cid:1) σ ρλ ∂ [ ν h λ ] α (cid:1) ∂ [ µ η ρ ] − ψ β γ αβµ ( ∂ ν ψ µ ) η ρ ∂ [ ν h ρ ] α (cid:1)(cid:1) + ∂ α (cid:18) i2 k (1 − k ) (cid:18) ¯ ψ β γ αβσ ( ∂ µ ψ σ ) η ρ − (cid:0) ¯ ψ β γ αβ [ µ ψ ν ] − ¯ ψ µ γ α ψ ν − σ α [ µ ¯ ψ ν ] γ σ ψ σ (cid:1) σ ρλ ∂ [ ν η λ ] (cid:1) ∂ [ µ η ρ ] (cid:1) . (154)The right-hand side of (154) can be written like in the right-hand side of(153) if the following conditions are simultaneously satisfied δω ′ = (cid:0) ¯ ψ β γ αβσ ( ∂ µ ψ σ ) h ρα − (cid:0) ¯ ψ β γ αβ [ µ ψ ν ] − ¯ ψ µ γ α ψ ν − σ α [ µ ¯ ψ ν ] γ σ ψ σ (cid:1) σ ρλ ∂ [ ν h λ ] α (cid:1) ∂ [ µ η ρ ] − ψ β γ αβµ ( ∂ ν ψ µ ) η ρ ∂ [ ν h ρ ] α , (155) δl ′ α = (cid:18) ¯ ψ β γ αβσ ( ∂ µ ψ σ ) η ρ − (cid:0) ¯ ψ β γ αβ [ µ ψ ν ] − ¯ ψ µ γ α ψ ν − σ α [ µ ¯ ψ ν ] γ σ ψ σ (cid:1) σ ρλ ∂ [ ν η λ ] (cid:1) ∂ [ µ η ρ ] . (156)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 ¯ ψ β γ αβσ ( ∂ µ ψ σ ) = δ Ω αµ , (157)30 ψ β γ αβ [ µ ψ ν ] − ¯ ψ µ γ α ψ ν − σ α [ µ ¯ ψ ν ] γ σ ψ σ = δ Γ µνα (158)take place simultaneously. Assuming that both the equations (157) and (158)are valid, they further give ∂ α (cid:0) ¯ ψ β γ αβσ ( ∂ µ ψ σ ) (cid:1) = δ (cid:0) ∂ α Ω αµ (cid:1) , (159) ∂ α (cid:0) ¯ ψ β γ αβ [ µ ψ ν ] − ¯ ψ µ γ α ψ ν − σ α [ µ ¯ ψ ν ] γ σ ψ σ (cid:1) = δ ( ∂ α Γ µνα ) . (160)On the other hand, by direct computation we obtain that ∂ α (cid:0) ¯ ψ β γ αβσ ( ∂ µ ψ σ ) (cid:1) = δ (cid:0) − i (cid:0) ψ ∗ σ ( ∂ µ ψ σ ) − ¯ ψ σ (cid:0) ∂ µ ¯ ψ ∗ σ (cid:1)(cid:1)(cid:1) , (161) ∂ α (cid:0) ¯ ψ β γ αβ [ µ ψ ν ] − ¯ ψ µ γ α ψ ν − σ α [ µ ¯ ψ ν ] γ σ ψ σ (cid:1) = δ ( − i ψ ∗ σ γ µν ψ σ − ψ ∗ [ µ ψ ν ] (cid:1) − ¯ ψ α γ αβ [ µ ∂ ν ] ψ β , (162)so the right-hand sides of (161)–(162) cannot be written like in the right-handsides of (159)–(160). This means that the relations (157)–(158) are not valid,and therefore neither are (155)–(156). As a consequence, χ must vanish, andhence we must set k (1 − k ) = 0 . (163)Using (163), we conclude that k = 1 . (164)Inserting (164) in (143), we obtain that∆ (int)1 = γ (cid:18)(cid:18) − (cid:18) ψ ∗ [ µ ψ σ ] + 12 ψ ∗ ρ γ µσ ψ ρ (cid:19) ∂ [ σ η λ ] σ νλ + ψ ∗ σ ( ∂ µ ψ σ ) η ν ) h µν + 12 (cid:18) ψ ∗ µ ψ ν + 14 ψ ∗ σ γ µν ψ σ (cid:19) η ρ ∂ [ µ h ν ] ρ (cid:19) . (165)Comparing (165) with (148), we find that b (int)2 = 0 , (166) b (int)1 = 18 (cid:18) ψ ∗ [ µ ψ σ ] + 12 ψ ∗ ρ γ µσ ψ ρ (cid:19) h λµ ∂ [ σ η λ ] ψ ∗ σ ( ∂ µ ψ σ ) η ν h µν − (cid:18) ψ ∗ µ ψ ν + 14 ψ ∗ σ γ µν ψ σ (cid:19) η ρ ∂ [ µ h ν ] ρ . (167)Substituting (164) in (144) and using (167), we deduce∆ (int)0 + 2 δb (int)1 = ∂ µ n µ + γ (cid:18) − L (RS)0 (cid:0) h − h µν h µν (cid:1) + i8 ¯ ψ µ γ λ ψ ν (cid:0)(cid:0) h λρ − hδ λρ (cid:1) ∂ [ µ h ν ] λ + h σν (cid:0) ∂ [ µ h σ ] λ + ∂ λ h µσ (cid:1)(cid:1) + i4 ¯ ψ µ γ σ ψ σ (cid:0) h ( ∂ µ h − ∂ ν h µν ) + h ρµ (cid:0) ∂ λ h ρλ − ∂ ρ h (cid:1) − h αβ ∂ µ h αβ + 32 h ρλ ∂ ρ h µλ + 12 h µν ∂ ρ h ρν (cid:19) + i4 ¯ ψ µ γ µνβ ( ∂ α ψ ν ) (cid:18) hh αβ − h ασ h σβ (cid:19) − ( V + V µν ∂ µ ψ ν ) h + i8 ¯ ψ β γ βµα ψ ν (cid:18)(cid:18) hδ ρν − h ρν (cid:19) ∂ [ µ h α ] ρ + h ρµ (3 ∂ α h νρ − ∂ ρ h αν ) (cid:19) + V µν (cid:18) h µσ ∂ σ ψ ν + ψ σ ∂ [ ν h σ ] µ + 14 γ αβ ψ ν ∂ [ α h β ] µ (cid:19)(cid:19) + Π µν ∂ [ µ η ν ] , (168)where Π µν = V µρ ∂ ν ψ ρ + ∂ R V∂ψ µ ψ ν + V ρµ ∂ ρ ψ ν + ¯ ψ ν ∂ L V ρλ ∂ ¯ ψ µ ∂ ρ ψ λ + 14 (cid:18) ∂ R V∂ψ ρ γ µν ψ ρ + V ρλ γ µν ∂ ρ ψ λ − ¯ ψ θ γ µν ∂ L V ρλ ∂ ¯ ψ θ ∂ ρ ψ λ (cid:19) . (169)We observe that (168) can be written like in (149) if and only ifΠ µν − Π νµ = ∂ ρ U ρµν . (170)The right-hand side of (169) splits according to the number of derivativesinto Π µν = Π µν + Π µν , (171)where we made the notationsΠ µν = ∂ R V∂ψ µ ψ ν + 14 ∂ R V∂ψ ρ γ µν ψ ρ , (172)32 µν = V µρ ∂ ν ψ ρ + V ρµ ∂ ρ ψ ν + ¯ ψ ν ∂ L V ρλ ∂ ¯ ψ µ ∂ ρ ψ λ + 14 (cid:18) V ρλ γ µν ∂ ρ ψ λ − ¯ ψ θ γ µν ∂ L V ρλ ∂ ¯ ψ θ ∂ ρ ψ λ (cid:19) . (173)As Π µν has no derivatives, it cannot bring to (170) a divergence-like contri-bution, and Π µν contains just one derivative, so in principle it may lead toa total derivative, as required by (170). As a consequence, from (170) pro-jected on the number of derivatives equal to zero we find that Π µν is subjectto the equation Π µν − Π νµ = 0 , (174)which is, via (172), equivalent to ∂ R V∂ψ µ ψ ν − ∂ R V∂ψ ν ψ µ = − ∂ R V∂ψ ρ γ µν ψ ρ . (175)If we generically represent ∂ R V /∂ψ µ under the form ∂ R V∂ψ µ = ¯ ψ α M αµ ( ψ ν ) , (176)then the equation (175) requires that γ V µνασ = (cid:0) γ V µνσα (cid:1) ⊺ , (177)where V µνασ = M αµ σ νσ − M αν σ µσ + 12 M ασ γ µν = − V νµασ . (178)If we decompose V µνασ like V µνασ = V µνασ + V µνασ τ γ τ + V µνασ τγ γ τγ + V µνασ τγρ γ τγρ + V µνασ τγρλ γ τγρλ , (179)then the condition (177) implies the relations V µνασ = − V µνσα , V µνασ τ = V µνσα τ , V µνασ τγ = V µνσα τγ , (180) V µνασ τγρ = − V µνσα τγρ , V µνασ τγρλ = − V µνσα τγρλ . (181)In a similar manner, if we expand M αµ along the basis in the space of con-stant, 4 × M αµ = M αµ + M αµ τ γ τ + M αµ τγ γ τγ + M αµ τγρ γ τγρ + M αµ τγρλ γ τγρλ , (182)33ubstitute (182) in (178), and take into account the relations (180)–(181),then we finally find that M αµ = m ( ψ ν ) σ αµ , M αµ τ = 0 , M αµ τγ = m ( ψ ν ) δ α [ τ δ µγ ] , (183) M αµ τγρ = 0 , M αµ τγρλ = m ( ψ ν ) ε τγρλ σ αµ , (184)where m ( ψ ν ), m ( ψ ν ), and m ( ψ ν ) are arbitrary functions. Replacing now(183)–(184) in (182) and then the resulting expression in (176), we find that ∂ R V∂ψ µ = m ( ψ ν ) ¯ ψ µ + 2 m ( ψ ν ) ¯ ψ α γ αµ + 24i m ( ψ ν ) ¯ ψ µ γ = 12 m ( ψ ν ) ∂ R X∂ψ µ + m ( ψ ν ) ∂ R Y∂ψ µ + 12 m ( ψ ν ) ∂ R Z∂ψ µ , (185)with X ≡ ¯ ψ µ ψ µ , Y ≡ ¯ ψ α γ αµ ψ µ , Z ≡ i ¯ ψ µ γ ψ µ . (186)The equation (185) shows that the solution to (175) is nothing but an arbi-trary polynomial of X , Y , and Z , i.e. V = V ( X, Y, Z ) . (187)In order to complete the analysis of the equation (170), we need to solve itscomponent of order one in the spacetime derivativesΠ µν − Π νµ = ∂ ρ U ρµν , (188)with Π µν given in (173) and U ρµν containing no derivatives. Taking intoconsideration the formula (173), it follows that the equation (188) restricts V µλ to satisfy the equation V µλ σ νρ + V ρµ σ νλ − V νλ σ µρ − V ρν σ µλ + ¯ ψ ν ∂ L V ρλ ∂ ¯ ψ µ − ¯ ψ µ ∂ L V ρλ ∂ ¯ ψ ν + 12 (cid:18) V ρλ γ µν − ¯ ψ θ γ µν ∂ L V ρλ ∂ ¯ ψ θ (cid:19) = ∂ R U ρµν ∂ψ λ . (189)The last equation is fulfilled if there exist some objects Q µ such that thefollowing conditions take place simultaneously: V µλ = − ∂ R Q µ ∂ψ λ , (190)34 R Q ρ ∂ψ µ σ νλ − ∂ R Q ρ ∂ψ ν σ µλ = 0 . (191)On the other hand, by adding to and subtracting from the left-hand side of(189) the quantity (1 / (cid:0) ∂ R Q ρ /∂ψ λ (cid:1) γ µν = ∂ R (1 / Q ρ γ µν )) /∂ψ λ , we canstate that (189) is checked if (190) and ∂ R Q ρ ∂ψ µ σ νλ − ∂ R Q ρ ∂ψ ν σ µλ + 12 ∂ R Q ρ ∂ψ λ γ µν = 0 (192)are simultaneously verified. By multiplying (192) from the right with ψ λ weget the equation ∂ R Q ρ ∂ψ µ ψ ν − ∂ R Q ρ ∂ψ ν ψ µ + 12 ∂ R Q ρ ∂ψ λ γ µν ψ λ = 0 , (193)which shows that (see (175) and (187)) Q ρ = Q ρ ( X, Y, Z ) . (194)Since Q µ like in (194) must provide V µλ via taking its right derivative withrespect to ψ λ (see (190)), it results that Q µ = Q ( X, Y, Z ) γ µ , (195)with Q ( X, Y, Z ) an arbitrary polynomial. Formulas (190) and (195) togetherwith some appropriate Fierz identities further yield V µν = ¯ ψ ρ P ρµν ( X, Y, Z ) , (196)where P ρµν ( X, Y, Z ) = ( P ρµν ) α ( X, Y, Z ) γ α + ( P ρµν ) αβγ ( X, Y, Z ) γ αβγ . (197)The dependence on X , Y , and Z of the functions ( P ρµν ) α and ( P ρµν ) αβγ enables us to conclude that the most general form of these coefficients readsas ( P ρµν ) α ( X, Y, Z ) = d δ ρα σ µν + d δ µα σ ρν + d δ να σ ρµ , (198)( P ρµν ) αβγ ( X, Y, Z ) = d δ ρ [ α δ µβ σ νγ ] , (199)35here ( d i ) i =1 , , , are arbitrary polynomials in X , Y , and Z . We remark that(199) gives in (133), and thus in S (int)1 , a contribution (up to a trivial, s -exactterm) that is already contained in (187) since¯ ψ ρ ( P ρµν ) αβγ γ αβγ ∂ µ ψ ν = 6 d ¯ ψ µ γ µνρ ∂ ν ψ ρ = s (cid:0) − d ¯ ψ µ ¯ ψ ∗ µ (cid:1) − d ¯ ψ µ γ µν ψ ν , (200)so we can take, without loss of generality d = 0 (201)in (199). Taking into account the last result and inserting (198) in (197) andthen in (196), we infer that V µν ∂ µ ψ ν = d ¯ ψ ρ γ ρ ∂ µ ψ µ + d ¯ ψ ν γ µ ∂ µ ψ ν + d ¯ ψ ν γ µ ∂ ν ψ µ = d ¯ ψ ρ γ ρ ∂ µ ψ µ + 12 ( d + d ) ¯ ψ ν γ µ ∂ ( µ ψ ν ) + 12 ( d − d ) ¯ ψ µ γ ν ∂ [ µ ψ ν ] = d ¯ ψ ρ γ ρ ∂ µ ψ µ + 12 ( d + d ) ¯ ψ ν γ µ ∂ ( µ ψ ν ) + s (cid:18) − i4 ( d − d ) (cid:0) ¯ ψ µ γ µν ¯ ψ ∗ ν − ¯ ψ µ ¯ ψ ∗ µ (cid:1)(cid:19) − i m d − d ) (cid:0) ¯ ψ µ ψ µ + ¯ ψ µ γ µν ψ ν (cid:1) . (202)Thus, up to an irrelevant, s -exact term, V µν ∂ µ ψ ν contains, beside the firsttwo pieces, the last component, which is a contribution already consideredin (187). We can thus forget about it by setting d − d = 0 . (203)At this stage, from (201) and (203) replaced in (198)–(199) and the resultingrelations further substituted in (197), with the help of the representation(196) we determine the relevant part of V µν under the form V µν = d ( X, Y, Z ) ¯ ψ α γ α σ µν + d ( X, Y, Z ) (cid:0) ¯ ψ ν γ µ + ¯ ψ µ γ ν (cid:1) . (204)Consequently, we find that V µν ∂ µ ψ ν no longer contains the unwanted (trivialor redundant) contributions, being precisely given by V µν ∂ µ ψ ν = d ( X, Y, Z ) ¯ ψ ρ γ ρ ∂ µ ψ µ + d ( X, Y, Z ) ¯ ψ ν γ µ ∂ ( µ ψ ν ) . (205)36ased on the relations (204) and (205), we deduce that the antisymmetricpart of Π µν must vanish Π µν − Π νµ = 0 . (206)As a consequence of this step of the deformation procedure, on the onehand the results (164), (187), and (205) completely determine the component(133), and hence the cross-coupling part of the first-order deformation (136)like S (int)1 = Z d x (cid:18) ψ ∗ µ ( ∂ ν ψ µ ) η ν + 12 ψ ∗ µ ψ ν ∂ [ µ η ν ] + 18 ψ ∗ ρ γ µν ψ ρ ∂ [ µ η ν ] + 12 (cid:18) σ ρλ L (RS)0 − i2 ¯ ψ µ γ µνρ ∂ λ ψ ν (cid:19) h ρλ + i4 (cid:18)
12 ¯ ψ µ γ ρ ψ ν + σ µρ ¯ ψ ν γ σ ψ σ + ¯ ψ σ γ σρµ ψ ν (cid:19) ∂ [ µ h ν ] ρ + V + d ¯ ψ ρ γ ρ ∂ µ ψ µ + d ¯ ψ ν γ µ ∂ ( µ ψ ν ) (cid:1) . (207)On the other hand, (168), (174), (187), (204), and (206) offer us the concreteform of b (int)0 as solution to the equation (149) like b (int)0 = 18 L (RS)0 (cid:0) h − h µν h µν (cid:1) − i16 ¯ ψ µ γ λ ψ ν (cid:0)(cid:0) h λρ − hδ λρ (cid:1) ∂ [ µ h ν ] λ + h σν (cid:0) ∂ [ µ h σ ] λ + ∂ λ h µσ (cid:1)(cid:1) − i8 ¯ ψ µ γ σ ψ σ (cid:0) h ( ∂ µ h − ∂ ν h µν ) + h ρµ (cid:0) ∂ λ h ρλ − ∂ ρ h (cid:1) − h αβ ∂ µ h αβ + 32 h ρλ ∂ ρ h µλ + 12 h µν ∂ ρ h ρν (cid:19) − i8 ¯ ψ µ γ µνβ ( ∂ α ψ ν ) (cid:18) hh αβ − h ασ h σβ (cid:19) − i16 ¯ ψ β γ βµα ψ ν (cid:18)(cid:18) hδ ρν − h ρν (cid:19) ∂ [ µ h α ] ρ + h ρµ (3 ∂ α h νρ − ∂ ρ h αν ) (cid:19) + h V + d ψ ρ γ ρ (cid:0) h∂ µ ψ µ − ( ∂ µ ψ ν ) h µν − σ µν ψ σ ∂ [ ν h σ ] µ −− σ µν γ αβ ψ ν ∂ [ α h β ] µ (cid:19) + d ψ ρ ( hγ µ ∂ µ ψ ρ − h µν γ µ ∂ ν ψ ρ − γ µ ψ λ ∂ [ ρ h λ ] µ − γ µ γ αβ ψ ρ ∂ [ α h β ] µ (cid:19) + d (cid:0) h ¯ ψ ρ γ µ ∂ ρ ψ µ h µν ¯ ψ µ γ ρ ∂ ν ψ ρ − ¯ ψ µ γ ν ψ ρ ∂ [ ν h ρ ] µ −
14 ¯ ψ µ γ ν γ αβ ψ ν ∂ [ α h β ] µ (cid:19) . (208)At this moment, the components (cid:16) b (int) I (cid:17) I =0 , , expressed by (166), (167), and(208) yield the cross-coupling part of the second-order deformation S (int)2 = R d x (cid:16) b (int)0 + b (int)1 + b (int)2 (cid:17) as S (int)2 = Z d x (cid:18) (cid:18) ψ ∗ [ µ ψ σ ] + 12 ψ ∗ ρ γ µσ ψ ρ (cid:19) h λµ ∂ [ σ η λ ] − ψ ∗ σ ( ∂ µ ψ σ ) η ν h µν − (cid:18) ψ ∗ µ ψ ν + 14 ψ ∗ σ γ µν ψ σ (cid:19) η ρ ∂ [ µ h ν ] ρ + 18 L (RS)0 (cid:0) h − h µν h µν (cid:1) − i16 ¯ ψ µ γ λ ψ ν (cid:0)(cid:0) h λρ − hδ λρ (cid:1) ∂ [ µ h ν ] λ + h σν (cid:0) ∂ [ µ h σ ] λ + ∂ λ h µσ (cid:1)(cid:1) − i8 ¯ ψ µ γ σ ψ σ (cid:0) h ( ∂ µ h − ∂ ν h µν ) + h ρµ (cid:0) ∂ λ h ρλ − ∂ ρ h (cid:1) − h αβ ∂ µ h αβ + 32 h ρλ ∂ ρ h µλ + 12 h µν ∂ ρ h ρν (cid:19) − i8 ¯ ψ µ γ µνβ ( ∂ α ψ ν ) (cid:18) hh αβ − h ασ h σβ (cid:19) − i16 ¯ ψ β γ βµα ψ ν (cid:18)(cid:18) hδ ρν − h ρν (cid:19) ∂ [ µ h α ] ρ + h ρµ (3 ∂ α h νρ − ∂ ρ h αν ) (cid:19) + h V + d ψ ρ γ ρ (cid:0) h∂ µ ψ µ − ( ∂ µ ψ ν ) h µν − σ µν ψ σ ∂ [ ν h σ ] µ −− σ µν γ αβ ψ ν ∂ [ α h β ] µ (cid:19) + d ψ ρ ( hγ µ ∂ µ ψ ρ − h µν γ µ ∂ ν ψ ρ − γ µ ψ λ ∂ [ ρ h λ ] µ − γ µ γ αβ ψ ρ ∂ [ α h β ] µ (cid:19) + d (cid:0) h ¯ ψ ρ γ µ ∂ ρ ψ µ − h µν ¯ ψ µ γ ρ ∂ ν ψ ρ − ¯ ψ µ γ ν ψ ρ ∂ [ ν h ρ ] µ −
14 ¯ ψ µ γ ν γ αβ ψ ν ∂ [ α h β ] µ (cid:19)(cid:19) . (209)This ends the second step of the deformation procedure for the Pauli-Fierzfield and the massive Rarita-Schwinger field.38 Lagrangian formulation of the interactingtheory
The main aim of this section is to give an appropriate interpretation of theLagrangian formulation of the interacting theory obtained in the previoussection from the deformation of the solution to the master equation. In viewof this, we initially prove that the linearized versions of first- and second-order formulations of spin-two field theory possess isomorphic local BRSTcohomologies. We start from the first-order formulation of spin-two fieldtheory S [ e µa , ω µab ] = − λ Z d x (cid:0) ω abν ∂ µ ( ee µa e νb ) − ω abµ ∂ ν ( ee µa e νb )+ 12 ee µa e νb (cid:0) ω acµ ω bν c − ω acν ω bµ c (cid:1)(cid:19) , (210)where 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 )) − . (211)In order to linearize action (210), we develop the vierbein like e µa = δ µa − λ f µa , e = 1 + λ f, (212)where f is the trace of f µa . Consequently, we find that the linearized formof (210) 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:0) ω αβα ω λλβ − ω µαβ ω αµβ (cid:1)(cid:19) . (213)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, (214)39here the latter gauge parameters are antisymmetric, ǫ αβ = − ǫ βα . Eliminat-ing the spin connection components on their equations of motion (auxiliaryfields) from (213) ω µαβ ( f ) = 12 (cid:0) ∂ [ µ f α ] β − ∂ [ µ f β ] α − ∂ [ α f β ] µ (cid:1) , (215)we obtain the second-order action S ′ [ f µν , ω µαβ ( f )] = S ′′ [ f µν ] = − Z d x (cid:18) (cid:0) ∂ [ µ f ν ] α (cid:1) (cid:0) ∂ [ µ f ν ] α (cid:1) + 14 (cid:0) ∂ [ µ f ν ] α (cid:1) (cid:0) ∂ [ µ f α ] ν (cid:1) −
12 ( ∂ µ f − ∂ ν f µν ) ( ∂ µ f − ∂ α f µα ) (cid:19) , (216)subject to the gauge invariances δ ǫ f µν = ∂ ( µ ǫ ν ) − ǫ µν . (217)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 , (218)the action (216) 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) , (219)while the accompanying gauge transformations are given by δ ǫ h µν = ∂ ( µ ǫ ν ) , δ ǫ B µν = − ǫ µν . (220)It is easy to see that the right-hand side of (219) is nothing but the Pauli-Fierzaction S ′′ [ h µν , B µν ] = S PF0 [ h µν ] . (221)As we have previously mentioned, we pass from (213)–(214) to (219)–(220)via the elimination of the auxiliary fields ω µαβ , such that the general theoremsfrom Section 15 of the first reference in [23] ensure the isomorphism H ( s ′ | d ) ≃ H ( s ′′ | d ) , (222)40ith s ′ and s ′′ the BRST differentials corresponding to (213)–(214) and re-spectively to (219)–(220). On the other hand, we observe that the field B µν does not appear in (219) 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 as well as 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 [23]). As aconsequence, we have that H ( s ′′ | d ) ≃ H ( s | d ) , (223)where s is the Pauli-Fierz BRST differential. Combining (222) and (223), wearrive at H ( s ′ | d ) ≃ H ( s ′′ | d ) ≃ H ( s | d ) . (224)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.It is easy to see that one can go from (219)–(220) to the Pauli-Fierzversion through the partial gauge-fixing B µν = 0. This gauge-fixing is aconsequence of the more general gauge-fixing condition [27] σ µ [ a e µb ] = 0 . (225)In the context of the larger partial gauge-fixing (225) simple computationleads to the vierbein fields e µa , their inverse e aµ , the inverse of their deter-minant e , and the components of the spin connection ω µab up to the secondorder in the coupling constant in terms of the Pauli-Fierz field as e µa = (0) e µa + λ (1) e µa + λ e µa + · · · = δ µa − λ h µa + 3 λ h ρa h µρ + · · · , (226) e aµ = (0) e aµ + λ (1) e aµ + λ e aµ + · · · = δ aµ + λ h aµ − λ h aρ h ρµ + · · · , (227) e = (0) e + λ (1) e + λ e + · · · = 1 + λ h + λ (cid:0) h − h µν h µν (cid:1) + · · · , (228) ω µab = λ (1) ω µab + λ ω µab + · · · , (229)41here (1) ω µab = − ∂ [ a h b ] µ , (230) (2) ω µab = − (cid:0) h c [ a (cid:0) ∂ b ] h c µ (cid:1) − h ν [ a ∂ ν h b ] µ − (cid:0) ∂ µ h ν [ a (cid:1) h b ] ν (cid:1) . (231)Based on the isomorphisms (224), we can further pass to the analysis of thedeformed theory obtained in the previous sections.The component of antighost number equal to zero in S (int)1 is preciselythe interacting Lagrangian at order one in the coupling constant L (int)1 = a (int)0 + a (RS)0 L (int)1 = (cid:20)
14 ¯ ψ µ ( − i γ µνρ ∂ ν ψ ρ + mγ µν ψ ν ) h (cid:21) + (cid:20) i4 ¯ ψ µ γ µνρ (cid:0) ∂ λ ψ ρ (cid:1) h νλ (cid:21) + (cid:20) i4 ¯ ψ µ γ µνρ (cid:0) ∂ ν ψ λ (cid:1) h ρλ (cid:21) + (cid:20) i8 (cid:0) ¯ ψ µ γ λ ψ ν − σ νλ ¯ ψ µ γ ρ ψ ρ (cid:1) ∂ [ µ h ν ] λ (cid:21) + (cid:20) − i8 (cid:0) ψ µ γ µνρ (cid:0) ∂ ν ψ λ (cid:1) h ρλ + ¯ ψ ρ γ ρµν ψ λ ∂ [ µ h ν ] λ (cid:1)(cid:21) + [ V ] + (cid:2) d ¯ ψ ρ γ ρ ∂ µ ψ µ (cid:3) + (cid:2) d ¯ ψ ( µ γ ν ) ∂ µ ψ ν (cid:3) ≡ (1) e L (RS)0 + (0) e (1) e µb (0) e νc (cid:18) − i2 ¯ ψ a γ abc (0) D µ (0) ψ ν (cid:19) + (0) e µb (1) e νc (cid:18) − i2 ¯ ψ a γ abc (0) D µ (0) ψ ν (cid:19) + (0) e (0) e µb (0) e νc (cid:18) − i2 ¯ ψ a γ abc (1) D µ (0) ψ ν (cid:19) + (0) e (0) e µb (0) e νc (cid:18) − i2 ¯ ψ a γ abc (0) D µ (1) ψ ν (cid:19) + (0) e V + d ¯ ψ a γ a (0) D µ (0) e (0) ψ µ ! + d e ¯ ψ ( a γ b )(0) e µb (0) e νc (0) D µ (cid:18) (0) e (0) ψ ν (cid:19) , (232)where (0) D µ = ∂ µ , (233)and (1) D µ = 18 (1) ω µab γ ab , (234)with (1) ω µab given in (230). Along the same line, the piece of antighost numberequal to zero from the second-order deformation offers us the interactingLagrangian at order two in the coupling constant L (int)2 = b (int)0 L (int)2 = b (int)0 = (cid:20)
116 ¯ ψ µ ( − i γ µνρ ∂ ν ψ ρ + mγ µν ψ ν ) (cid:0) h − h αβ h αβ (cid:1)(cid:21) (cid:20) i8 ¯ ψ µ (cid:0) γ µαν (cid:0) ∂ β ψ ν (cid:1) h αβ + γ µνρ (cid:0) ∂ ν ψ λ (cid:1) h ρλ (cid:1) h (cid:21) + (cid:20) i h (cid:0) − ¯ ψ µ γ µνρ (cid:0) (cid:0) ∂ ν ψ λ (cid:1) h ρλ + ψ λ ∂ [ ν h ρ ] λ (cid:1) + (cid:0) ¯ ψ α γ ρ ψ β − σ βρ ¯ ψ α γ µ ψ µ (cid:1) ∂ [ α h β ] ρ (cid:1)(cid:3) + (cid:20) − i8 ¯ ψ µ γ µνρ ( ∂ α ψ β ) h αν h βρ (cid:21) + (cid:20) i8 (cid:0) ¯ ψ α γ αβγ (cid:0) h µβ ∂ µ (cid:0) h σγ ψ σ (cid:1) + h µγ ∂ β (cid:0) h σµ ψ σ (cid:1)(cid:1) − (cid:0) ¯ ψ µ γ ρ ψ ν h ρσ − ψ µ γ ρ ψ ρ h νσ (cid:1) ∂ [ µ h ν ] σ (cid:19)(cid:21) + (cid:20) i8 (cid:18) ¯ ψ α γ αβγ ∂ β (cid:0) h µγ h σµ ψ σ (cid:1) −
12 ¯ ψ µ (cid:0) γ ρ ψ ρ (cid:0) h µλ ∂ σ h λσ + h λσ ∂ λ h µσ − h µσ ∂ σ h − h αβ ∂ µ h αβ (cid:1) − γ λ ψ ν (cid:0) h ρµ ∂ ν h ρλ − h ρµ ∂ ρ h νλ − h νρ ∂ λ h ρµ (cid:1)(cid:1)(cid:1)(cid:3) + (cid:20) ψ µ γ µνβ ( ∂ α ψ ν ) h ασ h σβ (cid:21) + (cid:20) − ψ µ γ µνρ ( ∂ ν ψ λ ) h ρσ h σλ (cid:21) + (cid:20) h V (cid:21) + (cid:20) d ¯ ψ ρ γ ρ ∂ µ (cid:18) h ψ µ (cid:19)(cid:21) + (cid:20) − d ψ ρ γ ρ ∂ µ ( ψ ν h µν ) (cid:21) + (cid:20) − d ψ ρ γ ρ γ αβ ψ µ ∂ [ α h β ] µ (cid:21) + (cid:20) d h ¯ ψ ( µ γ ν ) ∂ µ ψ ν (cid:21) + (cid:20) − d h µα ¯ ψ ( α γ ν ) ∂ µ ψ ν (cid:21) + (cid:20) − d ¯ ψ ( µ γ ν ) (cid:18) ψ ρ ∂ [ ν h ρ ] µ + 18 γ αβ ψ ν ∂ [ α h β ] µ (cid:19)(cid:21) ≡ (cid:20) (2) e L (RS)0 (cid:21) + (cid:20) (1) e (cid:18) (0) e µb (1) e νc + (1) e µb (0) e νc (cid:19) (cid:18) − i2 ¯ ψ a γ abc (0) D µ (0) ψ ν (cid:19)(cid:21) + (cid:20) (1) e (0) e µb (0) e νc (cid:18) − i2 ¯ ψ a γ abc (cid:18) (0) D µ (1) ψ ν + (1) D µ (0) ψ ν (cid:19)(cid:19)(cid:21) + (cid:20) (0) e (1) e µb (1) e νc (cid:18) − i2 ¯ ψ a γ abc (0) D µ (0) ψ ν (cid:19)(cid:21) + (cid:20) (0) e (1) e µb (0) e νc (cid:18) − i2 ¯ ψ a γ abc (cid:18) (0) D µ (1) ψ ν + (1) D µ (0) ψ ν (cid:19)(cid:19) (0) e (0) e µb (1) e νc (cid:18) − i2 ¯ ψ a γ abc (cid:18) (0) D µ (1) ψ ν + (1) D µ (0) ψ ν (cid:19)(cid:19)(cid:21) + (cid:20) (0) e (0) e µb (0) e νc (cid:18) − i2 ¯ ψ a γ abc (cid:18) (0) D µ (2) ψ ν + (1) D µ (1) ψ ν + (2) D µ (0) ψ ν (cid:19)(cid:19)(cid:21) + (cid:20) (0) e (2) e µb (0) e νc (cid:18) − i2 ¯ ψ a γ abc (0) D µ (0) ψ ν (cid:19)(cid:21) + (cid:20) (0) e (0) e µb (2) e νc (cid:18) − i2 ¯ ψ a γ abc (0) D µ (0) ψ ν (cid:19)(cid:21) + (cid:20) (1) e V (cid:21) + " d ¯ ψ a γ a (0) D µ (1) e (0) ψ µ ! + " d ¯ ψ a γ a (0) D µ (0) e (1) ψ µ ! + " d ¯ ψ a γ a (1) D µ (0) e (0) ψ µ ! + (cid:20) d e (0) e µa ¯ ψ ( a γ b )(0) D µ ψ b (cid:21) + (cid:20) d e (1) e µa ¯ ψ ( a γ b )(0) D µ ψ b (cid:21) + (cid:20) d e (0) e µa ¯ ψ ( a γ b )(1) D µ ψ b (cid:21) , (235)where (2) D µ = 18 (2) ω µab γ ab (236)and (2) ω µab like in (231). With the help of (226) and (228) we deduce that L (RS)0 + λ L (int)1 + λ L (int)2 + · · · comes from expanding the fully deformedLagrangian written in terms of either the original flat Rarita-Schwinger spinor ψ a L (int) = e (cid:0) − i ¯ ψ a e νb e ρc γ abc D ν (cid:0) e dρ ψ d (cid:1) + m ¯ ψ a γ ab ψ b (cid:1) + λ (cid:2) eV ( X, Y, Z ) + d ( X, Y, Z ) ¯ ψ a γ a D µ (cid:0) ee µb ψ b (cid:1) + ed ( X, Y, Z ) e µa ¯ ψ ( a γ b ) D µ ψ b (cid:3) (237)or the curved Rarita-Schwinger spinor ψ µ L (int) = e (cid:0) − i ¯ ψ µ e µa e νb e ρc γ abc D ν ψ ρ + m ¯ ψ µ e µa γ ab e νb ψ ν (cid:1) + λ (cid:2) eV ( X, Y, Z ) + d ( X, Y, Z ) e νa ¯ ψ ν γ a D µ ( eψ µ )+ ed ( X, Y, Z ) (cid:0) ¯ ψ µ γ b + e µa e b ρ ¯ ψ ρ γ a (cid:1) D µ ( e νb ψ ν ) (cid:3) . (238)44he notations D µ ψ a and D µ ψ ρ denote the full covariant derivatives of ψ a andrespectively of ψ ρ D µ ψ a = ∂ µ ψ a + 12 ω µab ψ b + 18 γ bc ψ a ω µbc , (239) D µ ψ ρ = ∂ µ ψ ρ + 18 ω µab γ ab ψ ρ . (240)The pieces linear in the antifields ψ ∗ µ from the deformed solution to themaster equation give us the deformed gauge transformations for the Rarita-Schwinger fields as δ ǫ ψ µ = λ (cid:18) ( ∂ α ψ µ ) ǫ α + 12 ψ ν ∂ [ µ ǫ ν ] + 18 γ αβ ψ µ ∂ [ α ǫ β ] (cid:19) + λ (cid:18) −
12 ( ∂ α ψ µ ) ǫ β h αβ + 116 γ ρλ ψ µ h σρ ∂ [ λ ǫ σ ] + 18 ψ ρ (cid:0) h λµ ∂ [ ρ ǫ λ ] − h λρ ∂ [ µ ǫ λ ] (cid:1) − ψ ν ǫ ρ ∂ [ µ h ν ] ρ − γ αβ ψ µ ǫ ρ ∂ [ α h β ] ρ (cid:19) = λ (1) δ ǫ ψ µ + λ δ ǫ ψ µ + · · · . (241)The first two orders of the gauge transformations can be put under the form (1) δ ǫ ψ m = ( ∂ µ ψ m ) (0) ¯ ǫ µ + 12 (0) ǫ mn ψ n + 14 γ ab ψ m (0) ǫ ab , (242) (2) δ ǫ ψ m = ( ∂ µ ψ m ) (1) ¯ ǫ µ + 12 (1) ǫ mn ψ n + 14 γ ab ψ m (1) ǫ ab , (243)where we used the notations (0) ¯ ǫ µ = ǫ µ = ǫ a δ µa , (1) ¯ ǫ µ = − ǫ a h µa , (244) (0) ǫ ab = 12 ∂ [ a ǫ b ] , (245) (1) ǫ ab = − ǫ c ∂ [ a h b ] c + 18 h c [ a ∂ b ] ǫ c + 18 (cid:0) ∂ c ǫ [ a (cid:1) h cb ] . (246)Based on these notations, the gauge transformations of the spinors take theform δ ǫ ψ m = λ (cid:18) ( ∂ µ ψ m ) (cid:18) (0) ¯ ǫ µ + λ (1) ¯ ǫ µ + · · · (cid:19) (cid:18) (0) ǫ mn + λ (1) ǫ mn + · · · (cid:19) ψ n + 14 γ ab ψ m (cid:18) (0) ǫ ab + λ (1) ǫ ab + · · · (cid:19)(cid:19) . (247)The gauge parameters (0) ǫ ab and (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 (225). Indeed, (225) leads to¯ δ ǫ σ µ [ a e µb ] = 0 , (248)where ¯ δ ǫ e µa = ¯ ǫ ρ ∂ ρ e µa − e ρa ∂ ρ ¯ ǫ µ + ǫ ba e µb . (249)Substituting (226) together with the expansions¯ ǫ µ = (0) ¯ ǫ µ + λ (1) ¯ ǫ µ + · · · = (cid:18) δ µa − λ h µa + · · · (cid:19) ǫ a (250)and ǫ ab = (0) ǫ ab + λ (1) ǫ ab + · · · (251)in (248), we arrive precisely to (245)–(246). At this point it is easy to seethat the gauge transformations (247) come from the perturbative expansionof the full gauge transformations δ ǫ ψ m = λ (cid:18) ( ∂ µ ψ m ) ¯ ǫ µ + ǫ mn ψ n + 14 γ ab ψ m ǫ ab (cid:19) . (252)Moreover, based on (252) and (249), it is easy to see that δ ǫ ψ µ = λ (cid:18) ( ∂ σ ψ µ ) ¯ ǫ σ − ψ σ ∂ σ ¯ ǫ µ + 14 γ ab ψ µ ǫ ab (cid:19) . (253)In conclusion, under the above mentioned hypotheses we have shown thatthe interactions between a massive Rarita-Schwinger field and a spin-twofield are described by the coupled Lagrangian (237) or (238), while the gaugetransformations of the Rarita-Schwinger spinors are given by (252) or (253).If we require in addition that the interacting model remains PT-invariant,then the results (237)–(238) remain valid up to the point that the functions V , d , and d must depend only on X and Y (and not on Z ).46 Impossibility of cross-interactions betweengravitons in the presence of the massiveRarita-Schwinger field
As it has been proved in [16], there are no direct cross-couplings that canbe introduced among a finite number of gravitons and also no intermedi-ate cross-couplings between different gravitons in the presence of a scalarfield. In this section, under the hypotheses of locality, smoothness of theinteractions in the coupling constant, Poincar´e invariance, Lorentz covari-ance, and the preservation of the number of derivatives on each field, wewill prove that there are no intermediate cross-couplings between differentgravitons intermediated by a massive spin-3 / δ AB by a linear redefinition of the Pauli-Fierz fields. This is theconvention we will work with in the sequel.In view of this we start from a finite sum of Pauli-Fierz actions and amassive Rarita-Schwinger action S L0 (cid:2) h Aµν , ψ µ (cid:3) = Z d x (cid:18) − (cid:0) ∂ µ h Aνρ (cid:1) ( ∂ µ h νρA ) + ( ∂ µ h µρA ) (cid:0) ∂ ν h Aνρ (cid:1) − (cid:0) ∂ µ h A (cid:1) ( ∂ ν h νµA ) + 12 (cid:0) ∂ µ h A (cid:1) ( ∂ µ h A ) (cid:19) + Z d x ¯ ψ (cid:18) − i2 ¯ ψ µ γ µνρ ∂ ν ψ ρ + m ψ µ γ µν ψ ν (cid:19) , (254)where h A denotes the trace of the field h µνA ( h A = σ µν h µνA ), with A thecollection index, running from 1 to n . The gauge transformations of theaction (254) read as δ ǫ h Aµν = ∂ ( µ ǫ Aν ) , δ ǫ ψ µ = 0 . (255)The BRST complex comprises the fields/ghosts φ α = (cid:0) h Aµν , ψ µ (cid:1) , η Aµ , (256)and respectively their antifields φ ∗ α = ( h ∗ µνA , ψ ∗ µ ) , η ∗ µA . (257)47he BRST differential splits in this situation like in (8), while the actions of δ and γ on the BRST generators are defined by δh ∗ µνA = 2 H µνA , δψ ∗ µ = m ¯ ψ λ γ λµ − i ∂ ρ ¯ ψ λ γ ρλµ , (258) δη ∗ µA = − ∂ ν h ∗ µνA , (259) δφ α = 0 , δη Aµ = 0 , (260) γφ ∗ α = 0 , γη ∗ µA = 0 , (261) γh Aµν = ∂ ( µ η Aν ) , γψ µ = 0 , γη Aµ = 0 , (262)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 (cid:2) h Aµν , ψ µ (cid:3) + Z d x (cid:0) h ∗ µνA ∂ ( µ η Aν ) (cid:1) . (263)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 (RS) . (264)The first-order deformation in the Pauli-Fierz sector, α (PF) , is of the form [16] α (PF) = α (PF)2 + α (PF)1 + α (PF)0 , (265)with α (PF)2 = 12 f ABC η ∗ µA η Bν ∂ [ µ η Cν ] . (266)In (266) 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 [16] f ABC = f ACB . (267)With (267) at hand, we find that α (PF)1 = f ABC h ∗ µρA (cid:0)(cid:0) ∂ ρ η Bν (cid:1) h Cµν − η Bν ∂ [ µ h Cν ] ρ (cid:1) . (268) The term (266) differs from that corresponding to [16] through a γ -exact term, whichdoes not affect (267). α (PF)1 leads to a consistent α (PF)0 implies that f ABC must be symmetric [16] f ABC = 13 f ( ABC ) , (269)where, by definition, f ABC = δ AD f DBC . Based on (269), we obtain that theresulting α (PF)0 reads as in [16] (where this component is denoted by a and f ABC by a abc ).If one goes along exactly the same line like in the subsection 4.2, we getthat α (int) = α (int)1 + α (int)0 , where α (int)1 = k A ψ ∗ µ ( ∂ ν ψ µ ) η Aν + k A ψ ∗ µ ψ ν ∂ [ µ η Aν ] + k A ψ ∗ ρ γ µν ψ ρ ∂ [ µ η Aν ] , (270) α (int)0 = k A (cid:18) σ ρλ L (RS)0 − i2 ¯ ψ µ γ µνρ ∂ λ ψ ν (cid:19) h Aρλ + i k A (cid:18)
12 ¯ ψ µ γ ρ ψ ν + σ µρ ¯ ψ ν γ σ ψ σ + ¯ ψ σ γ σρµ ψ ν (cid:19) ∂ [ µ h Aν ] ρ , (271)and k A are some real constants. Meanwhile, we find in a direct manner that α (RS) = a (RS)0 , (272)with a (RS)0 given in (133).Let us investigate next the consistency of the first-order deformation. Ifwe perform the notationsˆ S (PF)1 = Z d xα (PF) , (273)ˆ S (int)1 = Z d x (cid:0) α (int) + α (RS) (cid:1) , (274)ˆ S = ˆ S (PF)1 + ˆ S (int)1 , (275)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 two independent equations (cid:16) ˆ S (PF)1 , ˆ S (PF)1 (cid:17) + 2 s ˆ S (PF)2 = 0 , (276) The piece (268) differs from that corresponding to [16] through a δ -exact term, whichdoes not change (269). (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 , (277)where ˆ S = ˆ S (PF)2 + ˆ S (int)2 . The equation (276) requires that the constants f CAB satisfy the supplementary conditions [16] f DA [ B f EC ] D = 0 , (278)so they are the structure constants of a finite-dimensional, commutative,symmetric, and associative real algebra A . The analysis realized in [16]shows us that such an algebra has a trivial structure (being expressed like adirect sum of some one-dimensional ideals). So, we obtain that f CAB = 0 if A = B. (279)Let us analyze now the equation (277). If we denote by ˆ∆ (int) and β (int) thenon-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 (277) takes the local formˆ∆ (int) = − sβ (int) + ∂ µ k µ , (280)with gh (cid:16) ˆ∆ (int) (cid:17) = 1 , gh (cid:0) β (int) (cid:1) = 0 , gh ( k µ ) = 1 . (281)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 , , (282)with ˆ∆ (int)1 = γ (cid:18)(cid:18) − k A f ABC (cid:18) ψ ∗ [ µ ψ σ ] + 12 ψ ∗ ρ γ µσ ψ ρ (cid:19) ∂ [ σ η Bλ ] σ νλ + ψ ∗ σ ( ∂ µ ψ σ ) η Bν (cid:1) h Cµν + (cid:18) k B k C − k A f ABC (cid:19) (cid:18) ψ ∗ µ ψ ν + 14 ψ ∗ σ γ µν ψ σ (cid:19) η Bρ ∂ [ µ h Cν ] ρ (cid:19) + (cid:0) k A f ABC − k B k C (cid:1) (cid:18) ψ ∗ µ ( ∂ ν ψ µ ) η Bρ ∂ [ ν η Cρ ] + 14 (cid:0) ψ ∗ [ µ ψ ν ] + 12 ψ ∗ σ γ µν ψ σ (cid:19) ∂ [ µ η Bρ ] ∂ [ ν η Cλ ] σ ρλ (cid:19) . (283)50he concrete form of ˆ∆ (int)0 is not important in what follows and thereforewe will skip it. Due to the expansion (282), we have that β (int) and k µ from(280) split like β (int) = β (int)0 + β (int)1 + β (int)2 , agh (cid:16) β (int) I (cid:17) = I, I = 0 , , , (284) k µ = k µ + k µ + k µ , agh ( k µI ) = I, I = 0 , , . (285)By projecting the equation (280) on the various decreasing values of theantighost number, we obtain the equivalent tower of equations γβ (int)2 = ∂ µ (cid:18) k µ (cid:19) , (286)ˆ∆ (int)1 = − (cid:16) δβ (int)2 + γβ (int)1 (cid:17) + ∂ µ k µ , (287)ˆ∆ (int)0 = − (cid:16) δβ (int)1 + γβ (int)0 (cid:17) + ∂ µ k µ . (288)By a trivial redefinition, the equation (286) can always be replaced with γβ (int)2 = 0 . (289)Analyzing the expression of ˆ∆ (int)1 in (283) we observe that it can be writtenlike in (287) if the quantityˆ χ = (cid:0) k A f ABC − k B k C (cid:1) (cid:18) ψ ∗ µ ( ∂ ν ψ µ ) η Bρ ∂ [ ν η Cρ ] + 14 (cid:0) ψ ∗ [ µ ψ ν ] + 12 ψ ∗ σ γ µν ψ σ (cid:19) ∂ [ µ η Bρ ] ∂ [ ν η Cλ ] σ ρλ (cid:19) (290)can be put in the form ˆ χ = δ ˆ ϕ + γ ˆ ω + ∂ µ j µ . (291)Assume that (291) holds. Then, by applying δ on this equation we infer δ ˆ χ = γ ( − δ ˆ ω ) + ∂ µ ( δj µ ) . (292)On the other hand, if we use the concrete expression (290) of ˆ χ , by directcomputation we are led to δ ˆ χ = γ (cid:18) (cid:0) k A f ABC − k B k C (cid:1) δ (cid:0) ψ ∗ ρ ψ ρ η Bν (cid:0) ∂ µ h Cµν − ∂ ν h C (cid:1)(cid:1)(cid:19) ∂ µ (cid:18) (cid:0) k A f ABC − k B k C (cid:1) δ (cid:0) ψ ∗ ρ ψ ρ η Bν ∂ [ µ η Cν ] (cid:1)(cid:19) + γ (cid:18) i4 (cid:0) k A f ABC − k B k C (cid:1) (cid:0)(cid:0) ¯ ψ β γ αβσ ( ∂ µ ψ σ ) h Bρα − (cid:0) ¯ ψ β γ αβ [ µ ψ ν ] − ¯ ψ µ γ α ψ ν − σ α [ µ ¯ ψ ν ] γ σ ψ σ (cid:1) σ ρλ ∂ [ ν h Bλ ] α (cid:1) ∂ [ µ η Cρ ] − ψ β γ αβµ ( ∂ ν ψ µ ) η Bρ ∂ [ ν h Cρ ] α (cid:1)(cid:1) + ∂ α (cid:18) i2 (cid:0) k A f ABC − k B k C (cid:1) (cid:18) ¯ ψ β γ αβσ ( ∂ µ ψ σ ) η Bρ − (cid:0) ¯ ψ β γ αβ [ µ ψ ν ] − ¯ ψ µ γ α ψ ν − σ α [ µ ¯ ψ ν ] γ σ ψ σ (cid:1) σ ρλ ∂ [ ν η Bλ ] (cid:1) ∂ [ µ η Cρ ] (cid:1) . (293)The right-hand side of (293) can be written like in the right-hand side of(292) if the following conditions are simultaneously fulfilledi4 (cid:0) k A f ABC − k B k C (cid:1) (cid:8)(cid:2) ¯ ψ β γ αβσ ( ∂ µ ψ σ ) h ρα − (cid:0) ¯ ψ β γ αβ [ µ ψ ν ] − ¯ ψ µ γ α ψ ν − σ α [ µ ¯ ψ ν ] γ σ ψ σ (cid:1) σ ρλ ∂ [ ν h Bλ ] α (cid:3) ∂ [ µ η Cρ ] − ψ β γ αβµ ( ∂ ν ψ µ ) η Bρ ∂ [ ν h Cρ ] α (cid:9) = − δ ˆ ω ′ , (294)i2 (cid:0) k A f ABC − k B k C (cid:1) (cid:18) ¯ ψ β γ αβσ ( ∂ µ ψ σ ) η Bρ − (cid:0) ¯ ψ β γ αβ [ µ ψ ν ] − ¯ ψ µ γ α ψ ν − σ α [ µ ¯ ψ ν ] γ σ ψ σ (cid:1) σ ρλ ∂ [ ν η Bλ ] (cid:1) ∂ [ µ η Cρ ] = δj ′ µ . (295)However, from the action of δ on the BRST generators we observe that noneof h Aµβ , ∂ [ α h Aβ ] µ , η Aβ , or ∂ [ λ η Aβ ] are δ -exact. In consequence, the relations(294)–(295) hold if the equations¯ ψ β γ αβσ ( ∂ µ ψ σ ) = δ Ω αµ , (296)and ¯ ψ β γ αβ [ µ ψ ν ] − ¯ ψ µ γ α ψ ν − σ α [ µ ¯ ψ ν ] γ σ ψ σ = δ Γ µνα (297)take place simultaneously. The last equations are precisely the equations(157) and respectively (158). Due to the fact that they do not involve (Pauli-Fierz) collection indices, some arguments identical to those employed in sub-section 4.3 ensure that (296) and (297) cannot be satisfied. As a consequence,ˆ χ must vanish, which further implies that k D f DAB − k A k B = 0 . (298)52sing (298) and (279) we obtain that for A = Bk A k B = 0 , (299)which shows that the Rarita-Schwinger field can couple to only one graviton,so the assertion from the beginning of this section is finally proved. To conclude with, in this paper we have investigated the couplings between acollection of massless spin-two fields (described in the free limit by a sum ofPauli-Fierz actions) and a massive Rarita-Schwinger field using the powerfulsetting based on local BRST cohomology. Initially, we have shown that if wedecompose the metric like g µν = σ µν + gh µν , then we can couple the massiveRarita-Schwinger field to h µν in the space of formal series with the maxi-mum derivative order equal to one in h µν . The interacting Lagrangian L (int) obtained here contains, besides the standard minimal couplings, also threetypes of non-minimal couplings, which are not discussed in the literature,but are nevertheless consistent with the gauge symmetries of the Lagrangian L + L (int) , where L is the full spin-two Lagrangian in the vierbein formula-tion. Next, 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 massive Rarita-Schwinger field if the metric in internalspace is positively defined. 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.). 53
Main conventions and properties of the γ -matrices Here, we collect the main conventions and properties of the representation ofthe γ -matrices employed in this paper. We work with the charge conjugationmatrix C = − γ (300)and with that representation of the Clifford algebra γ µ γ ν + γ ν γ µ = 2 σ µν (301)for which all the γ -matrices are purely imaginary. In addition, γ is Hermitianand antisymmetric, while ( γ i ) i =1 , are anti-Hermitian and symmetric. Wetake a basis in the space of spinor matrices of the form , γ µ , γ µ µ , γ µ µ µ , γ µ µ µ µ , (302)where γ µ ··· µ k = 1 k ! X σ ∈ S k ( − ) σ γ µ σ (1) γ µ σ (2) · · · γ µ σ ( k ) . (303)In the above definition S k is the set of permutations of { , , . . . , k } and( − ) σ denotes the signature of a given permutation σ . This means that any4 × M with purely spinor indices can be expressed in terms of thematrices (302) via M = 14 X k =0 ( − ) k ( k − / k ! Tr ( γ µ ··· µ k M ) γ µ ··· µ k . (304)We list below some Fierz identities that are useful at the construction ofconsistent interactions between the Pauli-Fierz field and the massive Rarita-Schwinger spinor. They provide the products of the various elements from(302) in terms of their linear combinations γ µν γ ρ = − δ ρ [ µ γ ν ] + γ ρµν , (305) γ µν γ ρλ = − δ ρ [ µ δ λν ] − δ [ ρ [ µ γ λ ] ν ] + γ ρλµν , (306) γ µν γ ρλσ = − δ [ ρµ δ λν γ σ ] − δ [ ρ [ µ γ λσ ] ν ] , (307) γ µν γ ρλσξ = − δ [ ρµ δ λν γ σξ ] , (308)54 µνρ γ α = δ α [ µ γ νρ ] + γ αµνρ , (309) γ µνρ γ αβγ = − δ [ αµ δ βν δ γ ] ρ − δ [ α [ µ δ βν γ γ ] ρ ] . (310)Moreover, in the chosen representation of the γ -matrices the elements of thebasis (302) display the following symmetry/antisymmetry properties: γ γ µ , γ γ µν (311)are symmetric and γ γ µνρ , γ γ µνρλ , γ γ (312)are antisymmetric. If we take γ = i γ γ γ γ and work with ε = − ε =1, then γ µνρλ = ε µνρλ γ γ γ γ = i ε µνρλ γ , (313) γ µνρλ = − ε µνρλ γ γ γ γ = i ε µνρλ γ . (314) B Proof of some assertions made in the sub-section 4.2
Initially, we show that our statement from footnote 3 is indeed valid. Theterms linear in the Pauli-Fierz antifield h ∗ µν that can be in principle addedto a (int)1 have the generic form˜ a (int)1 = h ∗ µν (cid:0) M ρµν η ρ + M ρλµν ∂ [ ρ η λ ] (cid:1) ≡ ˜ a ′ (int)1 + ˜ a ′′ (int)1 , (315)where M ρµν and M ρλµν are bosonic, real, gauge-invariant functions. Imposingthat (315) satisfies the requirements i)–ii) from the subsection 4.2, then thefunctions M ρµν and M ρλµν are restricted to depend at most on the undifferenti-ated Rarita-Schwinger field. The consistency equation for ˜ a (int)1 in antighostnumber zero δ ˜ a (int)1 + γ ˜ a (int)0 = ∂ µ ˜ j (int)0 , (316)is independent of that for a (int)1 of the form (57) since the former piece pro-duces in ˜ a (int)0 components quadratic in the Pauli-Fierz field, while the latterintroduces in a (int)0 terms linear in h µν . Moreover, the consistency equationof ˜ a ′ (int)1 is independent of that implying ˜ a ′′ (int)1 due to the different number55f derivatives contained in these two types of terms, so (316) is equivalent tothe equations δ ˜ a ′ (int)1 + γ ˜ a ′ (int)0 = ∂ µ ˜ j ′ (int)0 , (317) δ ˜ a ′′ (int)1 + γ ˜ a ′′ (int)0 = ∂ µ ˜ j ′′ (int)0 . (318)Now, we prove that (315) is not consistent in antighost number zero, i.e.,there are no solutions ˜ a ′ (int)0 or ˜ a ′′ (int)0 to the equations (317)–(318). To thisend we use the fact that the linearized Einstein tensor (17) can be writtenlike H µν = ∂ α ∂ β φ µανβ , (319)with φ µανβ = 12 (cid:0) − h µν σ αβ + h αν σ µβ + h µβ σ αν − h αβ σ µν + h (cid:0) σ µν σ αβ − σ µβ σ αν (cid:1)(cid:1) . (320)By direct computation, we find that δ ˜ a ′ (int)1 = − ∂ α ∂ β φ µανβ M ρµν η ρ = ∂ α (cid:0) − (cid:0) ∂ β φ µανβ (cid:1) M ρµν η ρ (cid:1) + ∂ β (cid:0) φ µανβ ∂ α (cid:0) M ρµν η ρ (cid:1)(cid:1) + φ µανβ ∂ [ µ M ρα ] ν ∂ [ β η ρ ] + 12 φ µανβ ∂ [ µ M ρα ][ ν,β ] η ρ + γ (cid:18) φ µανβ (cid:18) ∂ [ µ M ρα ] ν h βρ − M ρµν (1) Γ ραβ (cid:19)(cid:19) − (cid:0) γφ µανβ (cid:1) (cid:18) ∂ [ µ M ρα ] ν h βρ − M ρµν (1) Γ ραβ (cid:19) , (321)where (1) Γ ραβ = 12 ( ∂ α h βρ + ∂ β h αρ − ∂ ρ h αβ ) . (322)Comparing (321) with (317) and observing that the term in (321) involv-ing (cid:0) γφ µανβ (cid:1) comprises the symmetric derivatives ∂ ( β η ρ ) , it follows that thispiece, which is constrained to contribute to a full divergence, can only realizethis task together with the part proportional with ∂ [ µ M ρα ][ ν,β ] . Accordingly,the γ -exactness modulo d of the right-hand side of (321), which is demandedby the equation (317), requires that the functions M ρµν are subject to theequations ∂ [ µ M ρα ] ν = 0 , (323)56ossessing the trivial solution M ραν = 0 (324)since M ραν are derivative-free (they depend only on the undifferentiated spinor-vector ψ µ ). In an identical manner, starting with δ ˜ a ′′ (int)1 = − ∂ α ∂ β φ µανβ M ρλµν ∂ [ ρ η λ ] = ∂ α (cid:0) − (cid:0) ∂ β φ µανβ (cid:1) M ρλµν ∂ [ ρ η λ ] (cid:1) + ∂ β (cid:0) φ µανβ ∂ α (cid:0) M ρλµν ∂ [ ρ η λ ] (cid:1)(cid:1) + 12 φ µανβ ∂ [ µ M ρλα ][ ν,β ] ∂ [ ρ η λ ] + γ (cid:16) φ µανβ (cid:16) ∂ [ µ M ρλα ] ν ∂ [ ρ h λ ] β − M ρµν ∂ α ∂ [ ρ h λ ] β (cid:17)(cid:17) − (cid:0) γφ µανβ (cid:1) (cid:16) ∂ [ µ M ρλα ] ν ∂ [ ρ h λ ] β − M ρµν ∂ α ∂ [ ρ h λ ] β (cid:17) , (325)we argue that the functions M ρλµν must obey the equations ∂ [ µ M ρλα ][ ν,β ] = 0 , (326)which, due to the fact that M ρλµν are derivative-free, possess only the trivialsolution M ρλµν = 0 . (327)If we substitute the results (324) and (327) into (315), we conclude thatthere is no term linear in the Pauli-Fierz antifield h ∗ µν that can be added to a (int)1 such as to give a consistent component of antighost number zero in thefirst-order deformation of the solution to the master equation.Finally, we show that we can always make the functions c , c , and c from(57) vanish via adding some trivial terms and making some redefinitions ofthe functions ¯ N ρλσµ . In view of this, we insert (65) in (57), such that thepart from a (int)1 proportional with c , c , or c reads as T ( c , c , c ) = (cid:20) c (cid:18) ψ ∗ λ γ µ ψ µ − ψ ∗ µ γ µνλ ψ ν (cid:19) + c (cid:0) ψ ∗ µ γ λ ψ µ − ψ ∗ µ γ µνλ ψ ν (cid:1) + c (cid:18) ψ ∗ µ γ µ ψ λ − ψ ∗ µ γ µνλ ψ ν (cid:19)(cid:21) η λ . (328)Based on the second definition in (12) related to the Koszul-Tate differentialand on the Fierz identities from the previous appendix section, we obtainthat δ (cid:0) ψ ∗ λ γ µ ¯ ψ ∗ µ (cid:1) = − mψ ∗ λ γ µ ψ µ + mψ ∗ µ γ λ ψ µ + mψ ∗ µ γ µνλ ψ ν (cid:0) ψ ∗ λ γ µν + ψ ∗ [ µ γ ν ] λ (cid:1) ∂ µ ψ ν + i ψ ∗ µ γ µνρλ ∂ ν ψ ρ , (329) δ (cid:0) ψ ∗ µ γ λ ¯ ψ ∗ µ (cid:1) = − mψ ∗ λ γ µ ψ µ + 2 mψ ∗ µ γ µ ψ λ − mψ ∗ µ γ µνλ ψ ν +2i (cid:0) ψ ∗ λ γ µν ∂ µ ψ ν + ψ ∗ µ γ ρµ ∂ [ λ ψ ρ ] (cid:1) + 2i ψ ∗ µ γ µνρλ ∂ ν ψ ρ , (330) δ (cid:0) ψ ∗ µ γ µνλ ¯ ψ ∗ ν (cid:1) = 4 mψ ∗ µ γ µ ψ λ − mψ ∗ µ γ λ ψ µ − mψ ∗ µ γ µνλ ψ ν +4i ψ ∗ µ ∂ [ µ ψ λ ] + 2i ψ ∗ µ γ λν ∂ [ µ ψ ν ] − ψ ∗ µ γ µν ∂ [ λ ψ ν ] . (331)Relying on the above results, we can rewrite the three terms present in (328)in the form c (cid:18) ψ ∗ λ γ µ ψ µ − ψ ∗ µ γ µνλ ψ ν (cid:19) η λ = s h c m (cid:0) ψ ∗ ρ γ µ ¯ ψ ∗ µ − ψ ∗ µ γ ρ ¯ ψ ∗ µ + ψ ∗ µ γ µνρ ¯ ψ ∗ ν (cid:1) η ρ (cid:3) + i c m (cid:20)(cid:18) ψ ∗ λ γ µν + 12 ψ ∗ [ µ γ ν ] λ (cid:19) ∂ µ ψ ν + 12 ψ ∗ µ γ µρ ∂ [ λ ψ ρ ] + ψ ∗ µ ∂ [ µ ψ λ ] + 12 ψ ∗ µ γ µνρλ ∂ ν ψ ρ (cid:21) η λ , (332) c (cid:0) ψ ∗ µ γ λ ψ µ − ψ ∗ µ γ µνλ ψ ν (cid:1) η λ = s h c m (cid:0) ψ ∗ ρ γ µ ¯ ψ ∗ µ − ψ ∗ µ γ ρ ¯ ψ ∗ µ + ψ ∗ µ γ µνρ ¯ ψ ∗ ν (cid:1) η ρ (cid:3) + i c m (cid:2) − (cid:0) ψ ∗ λ γ µν + ψ ∗ [ µ γ ν ] λ (cid:1) ∂ µ ψ ν +2 ψ ∗ µ γ µρ ∂ [ λ ψ ρ ] + 4 ψ ∗ µ ∂ [ µ ψ λ ] − ψ ∗ µ γ µνρλ ∂ ν ψ ρ (cid:3) η λ , (333) c (cid:18) ψ ∗ µ γ µ ψ λ − ψ ∗ µ γ µνλ ψ ν (cid:19) η λ = s h c m (cid:0) ψ ∗ ρ γ µ ¯ ψ ∗ µ − ψ ∗ µ γ ρ ¯ ψ ∗ µ + ψ ∗ µ γ µνρ ¯ ψ ∗ ν (cid:1) η ρ (cid:3) + i c m (cid:2)(cid:0) − ψ ∗ λ γ µν + 2 ψ ∗ [ µ γ ν ] λ (cid:1) ∂ µ ψ ν +14 ψ ∗ µ γ µρ ∂ [ λ ψ ρ ] + 4 ψ ∗ µ ∂ [ µ ψ λ ] − ψ ∗ µ γ µνρλ ∂ ν ψ ρ (cid:3) η λ . 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