1 Introduction
###### Abstract

The anomalous effective action describing the coupling of gravity to a non-abelian gauge theory can be determined by a variational solution of the anomaly equation, as shown by Riegert long ago. It is given by a nonlocal expression, with the nonlocal interaction determined by the Green’s function of a conformally covariant operator of fourth order. In recent works it has been shown that this interaction is mediated by a simple pole in an expansion around a Minkowski background, coupled in the infrared in the massless fermion limit. This result relies on the local formulation of the original action in terms of two auxiliary fields, one physical scalar and one ghost, which take the role of massless composite degrees of freedom. In the gravity case, the two scalars have provided ground in favour of some recent proposals of an infrared approach to the solution of the dark energy problem, entirely based on the behaviour of the vacuum energy at the QCD phase transition. As a test of this general result, we perform a complete one-loop computation of the effective action describing the coupling of a non-abelian gauge theory to gravity. We confirm the appearance of an anomaly pole which contributes to the trace part of the correlator and of extra poles in its trace-free part, in the quark and gluon sectors, describing the coupling of the energy momentum tensor () to two non abelian gauge currents ().

Trace Anomaly, Massless Scalars

and the Gravitational Coupling of QCD

Roberta Armillis, Claudio Corianò, Luigi Delle Rose 111roberta.armillis@le.infn.it, claudio.coriano@le.infn.it, luigi.dellerose@le.infn.it

Dipartimento di Fisica, Università del Salento

and INFN Sezione di Lecce, Via Arnesano 73100 Lecce, Italy

## 1 Introduction

The study of the effective action describing the coupling of a gauge theory to gravity via the trace anomaly [1] is an important aspect of quantum field theory, which is not deprived also of direct phenomenological implications. This coupling is mediated by the correlator involving the energy momentum tensor together with two vector currents (or vertex), which describes the interaction of a graviton with two photons or two gluons in QED and QCD, respectively. At the same time, the vertex has been at the center of an interesting case study of the renormalization properties of composite operators in Yang Mills theories [2], in the context of an explicit check of the violation of the Joglekar-Lee theorem [3] on the vanishing of S-matrix elements of BRST exact operators. In this second case it was computed on-shell, but only at zero momentum transfer. In this work we are going to extend this computation and investigate the presence of massless singularities in its expression. These contribute to the trace anomaly and play a leading role in fixing the structure of the effective action that couples QCD to gravity. The analysis of [2], which predates our study, unfortunately does not resolve the issue about the presence or the absence of the anomaly pole in the anomalous effective action of QCD because of the restricted kinematics involved in that analysis of the vertex, and for this reason we have to proceed with a complete re-computation.

Anomaly poles characterize quite universally (gravitational and chiral) anomalous effective actions, in the sense that account for their anomalies. They have been identified and discussed in the abelian case both by a dispersive analysis [4] and by a direct explicit computation of the related anomalous Feynman amplitudes quite recently [5, 6]. Extensive analysis in the case of chiral gauge theory for anomalous models have shown the close parallel between solutions of the Ward identities, the coupling of the poles in the ultraviolet and in the infrared region and the gravity case [7, 8].

It is therefore important to check whether similar contributions appear also in non-abelian gauge theories coupled to gravity. We recall that the same pole structure is found in the variational solution of the expression of the trace anomaly, where one tries to identify an action whose energy momentum tensor reproduces the trace anomaly. This action, found by Riegert long ago [9], is nonlocal and involves the Green function of a quartic (conformally covariant) operator. The action describes the structure of the singularities of anomalous correlators with any number of insertions of the energy momentum tensor and two photons (), which are expected to correspond both to single and to higher order poles, for a sufficiently high . For obvious reasons, explicit checks of this effective action using perturbation theory - as the number of external graviton lines grows - becomes increasingly difficult to handle. The correlator is the first (leading) contribution to this infinite sum of correlators in which the anomalous gravitational effective action is expanded.

Given the presence of a quartic operator in Riegert’s nonlocal action, the proof that this action contains a single pole to lowest order (in the TJJ vertex), once expanded around flat space, has been given in [4] by Giannotti and Mottola, and provides the basis for the discussion of the anomalous effective action in terms of massless auxiliary fields contained in their work. The auxiliary fields are introduced in order to rewrite the action in a local form. We show by an explicit computation of the lowest order vertex that Riegert’s action is indeed consistent in the non-abelian case as well, since its pole structure is recovered in perturbation theory, similarly to the abelian case. Therefore, one can reasonably conjecture the presence of anomaly poles in each gauge invariant subsets of the diagrammatic expansion, as the computation for the non-abelian TJJ shows (here for the case of the single pole). In particular, this is in agreement with the result of [4], where it is shown that, after expanding around flat spacetime, the quartic operator in Riegert’s action becomes a simple nonlocal interaction (for the contribution), i.e. a pole term. We remark that the identification of a pole term in this and in others similar correlators, as we are going to emphasize in the following sections (at least in the case of QED and for the sector of QCD mediated by quark loops), requires an extrapolation to the massless fermion limit, and for this reason its interpretation as a long-range dynamical effect in the gravitational effective action requires some caution. In QCD, however, there is an extra sector that contributes to the same correlator, entirely due to virtual loops of gluons in the anomaly graphs, which remains unaffected by the massless fermion limit. The appearance of such a singularity in the effective action, however, does not necessarily imply that its contribution survives in the physical S-matrix. We will also establish the appearance of other singularities in the trace-free form factors which, obviously, are not part of Riegert’s action.

We will comment in our conclusions on the possible implications of these results and on some recent proposals to link this type of behaviour [10, 11] to cosmology and to the dark energy problem. We also remark that, in general, the coefficient in front of the trace anomaly, for a given theory, can be computed in terms of its massless fields content, and as such it is well known. However, the structure of the effective action and the characterization of its fundamental form factors at nonzero momentum transfer and its complete analytical structure is a novel result. In this respect, the classification of all the relevant tensor structures which appear in the computation of this correlator is rather involved and has been performed in the completely off-shell case. We remark that the complexity of the final expression, in the off-shell case, prevents us from presenting its form. For this reason we will give only the on-shell version of the complete vertex, which is expressed, as we have mentioned, only in terms of three fundamental form factors.

Concerning the phenomenological relevance of this vertex, we just mention that it plays an essential role in the study of NLO corrections to processes involving a graviton exchange. In fact, in theories with extra dimension, where a low-gravity scale and the presence of Kaluza-Klein excitations may enhance the rates for processes mediated by gluons and gravitons, the vertex appears in the hard scattering of the corresponding factorization formula [12] and has been computed in dimensional regularization. However, to our knowledge, in all cases, there has been no separate discussion of the general structure of the vertex (i.e. as an amplitude) nor of its Ward identities, which, in principle, would require a more careful investigation because of the trace anomaly. Anomalous amplitudes, in fact, are defined by the fundamental Ward identities imposed on them, that we are going to derive from general principles. We cover this gap and show, that both dimensional regularization and dimensional reduction reproduce the correct Ward identity satisfied by this vertex, showing at the same time that the use of these regularizations is indeed appropriate. Results for this vertex will be given only in the on-shell case, since in this case the result can be expressed in terms of just three form factors. We have computed also the off-shell effective action, but its expression is rather lengthy and will not be discussed here, since it is gauge dependent and of less significance compared to the on-shell result. Most of our work is concerned with a technical derivation of the leading contribution to the anomalous effective action of QCD coupled to gravity. We have summarized in our conclusions a brief discussion of the relevance of this study in the ongoing attempt to link the trace anomaly and QCD to a possible alternative solution of the problem of dark energy, using this effective action as an intermediate step [10, 11].

## 2 Anomalous effective actions and their variational solutions

In this section we briefly review the topic of the variational solutions of anomalous effective actions, and on the local formulations of these using auxiliary fields.

One well known result of quantum gravity is that the effective action of the trace anomaly is given by a nonlocal form when expressed in terms of the spacetime metric . This was obtained [9] from a variational solution of the equation for the trace anomaly [1]

 Tμμ=bF+b′(E−23□R)+b′′□R+cFaμνFaμν, (1)

(see also [13, 14] for an analysis of the gravitational sector) which takes in spacetime dimensions the form

 Sanom[g,A]= (2) 18∫d4x√−g∫d4x′√−g′(E−23□R)xΔ−14(x,x′)[2bF+b′(E−23□R)+2cFμνFμν]x′.

Here, the parameters and are the coefficients of the Weyl tensor squared,

 F=CλμνρCλμνρ=RλμνρRλμνρ−2RμνRμν+R23 (3)

and the Euler density

 E=∗Rλμνρ∗Rλμνρ=RλμνρRλμνρ−4RμνRμν+R2 (4)

respectively of the trace anomaly in a general background curved spacetime. Notice that the last term in (2) is the contribution generated in the presence of a background gauge field, with coefficient . For a Dirac fermion in a classical gravitational () and abelian () background, the values of the coefficients are , and , and , with being the electric charge of the fermion. One crucial feature of this solution is its origin, which is purely variational. Obtained by Riegert long ago, the action was derived by solving the variational equation satisfied by the trace of the energy momentum tensor. denotes the Green’s function inverse of the conformally covariant differential operator of fourth order, defined by

 Δ4≡∇μ(∇μ∇ν+2Rμν−23Rgμν)∇ν=□2+2Rμν∇μ∇ν+13(∇μR)∇μ−23R□. (5)

Given a solution of a variational equation, it is mandatory to check whether the solution is indeed justified by a perturbative computation. One specific feature of these solutions is the presence of anomaly poles. In previous works we have elaborated on the significance of these interactions, extracted from a direct perturbative computation, by a painstaking analysis of anomaly graphs under general kinematical conditions, and not just by a dispersive approach. The dispersive approach allows to connect this behaviour of the spectral density to a very specific infrared configuration.

### 2.1 The kinematics of an anomaly pole

In our conventions we will denote with and the outgoing momenta of the two photons/gluons and with the incoming momentum of the graviton. denotes the invariant mass of the external graviton line. A computation of the spectral density of the amplitude in QED shows that this takes the form . The configuration responsible for the appearance of a pole is illustrated in Fig. 1 (a). It describes the decay of a graviton line into two on-shell photons. The decay is mediated by a collinear and on-shell fermion-antifermion pair and can be interpreted as a spacetime process. The corresponding interaction vertex, described as the exchange of a pole, is instead shown in Fig. 1 (b). The actual process depicted in Fig. 1 (a) is obtained at diagrammatic level by setting on-shell the fermion/antifermion pair attached to the graviton line. This configuration, present in the spectral density of the diagram only for on-shell photons, generates a pole contribution which can be shown to be coupled in the infrared. This means that if we compute the residue of the amplitude for we find that it is non-vanishing. In the general expression of the vertex, a similar configuration is extracted in the high energy limit, not by a dispersive analysis, but by an explicit (off-shell) computation of the diagrams. Clearly, the pole, in this second case, has a vanishing residue as , but is nevertheless a signature of the anomaly at high energy. Either for virtual or for real photons, a direct computation of the vertex allows to extract the pole term, without having to rely on a dispersive analysis. This point has been illustrated in our previous computations of the chiral anomaly vertex [8] and in the computation of the TJJ vertex for QED [5]. The identification of this singularity in the case of QCD is in perfect agreement with those previous results.

### 2.2 The single pole from Δ4

In the case of the gravitational effective action, the appearance of the inverse of operator seems to be hard to reconcile with the simpler interaction which is predicted by the perturbative analysis of the correlator, which manifests a single anomaly pole. In [4], Giannotti and Mottola show step by step how a single pole emerges from this quartic operator, by using the auxiliary field formulation of the same effective action. Clearly, more computations are needed in order to show that the nonlocal effective action consistently does justice of all the poles (of second order and higher) which should be present in the perturbative expansion. Obviously, the perturbative computations - being either based on dispersion theory or on complete evaluations of the vertices, as in our case - become rather hard as we increase the number of external lines of the corresponding perturbative correlator. For instance, this check becomes almost impossible for correlators of the form or higher, due to the appearance of a very large number of tensor structure in the reduction to scalar form of the tensor Feynman integrals. In the case of the computation is still manageable, since it does not require Feynman integrals beyond rank-4.

Expanding around flat space, the local formulation of Riegert’s action, as shown in [4, 15], can be rewritten in the form

 Sanom[g,A]→−c6∫d4x√−g∫d4x′√−g′Rx□−1x,x′[FαβFαβ]x′, (6)

which is valid to first order in the fluctuation of the metric around a flat background, denoted as

 gμν=ημν+κhμν,κ=√16πGN (7)

with being the 4-dimensional Newton’s constant. The formulation in terms of auxiliary fields of this axion gives

 Sanom[g,A;φ,ψ′]=∫d4x√−g[−ψ′□φ−R3ψ′+c2FαβFαβφ], (8)

where and are the auxiliary scalar fields. They satisfy the equations

 ψ′≡b□ψ, (9) □ψ′=c2FαβFαβ, (10) □φ=−R3. (11)

In order to make contact with the amplitude, one needs the expression of the energy momentum extracted from (8) to leading order in , or, equivalently, from (6) that can be shown to take the form

 Tμνanom(z)=c3(gμν□−∂μ∂ν)z∫d4x′□−1z,x′[FαβFαβ]x′. (12)

Notice that is the expression of the energy momentum tensor of the theory in the background of the gravitational and gauge fields. We recall, in fact, that in the QED case, for instance, the energy momentum tensor of the theory is split into the free fermionic part , the interacting fermion-photon part and the photon contribution which are given by

 Tμνf=−i¯ψγ(μ\lx@stackrel↔∂ν)ψ+gμν(i¯ψγλ\lx@stackrel↔∂λψ−m¯ψψ), (13)
 Tμνfp=−eJ(μAν)+egμνJλAλ, (14)

and

 Tμνph=FμλFν  λ−14gμνFλρFλρ, (15)

where the current is defined as

 Jμ(x)=¯ψ(x)γμψ(x). (16)

The connected components of can be obtained directly from the quantum average of , defined as the sum of the fermion contribution and its interaction part with the photon field,

 Tμνp≡Tμνf+Tμνfp. (17)

In the formalism of the background fields, the correlator then can be extracted from the defining functional integral

 ⟨Tμνp(z)⟩A ≡ ∫DψD¯ψTμνp(z)ei∫d4xL+∫J⋅A(x)d4x (18) = ⟨Tμνpei∫d4xJ⋅A(x)⟩

via two functional derivatives respect to the background field and generates the effective action

 Γμναβ(z;x,y)≡δ2⟨Tμνp(z)⟩AδAα(x)δAβ(y)∣∣∣A=0=Γμναβanom+~Γμναβ. (19)

We have separated in (19) the pole contribution from the rest of the amplitude (), which does not contribute to the trace part. Notice that , derived from either the classical generating functional (12) given by Riegert’s action or from the direct perturbative expansion of (19), should nevertheless coincide, for the pole term not to be a spurious artifact of the variational solution. In particular, a computation performed in QED shows that the pole term extracted from via functional differentiation

 Γμναβanom(p,q) = ∫d4x∫d4yeip⋅x+iq⋅yδ2Tμνanom(0)δAα(x)Aβ(y)=e218π21k2(gμνk2−kμkν)uαβ(p,q)

with

 uαβ(p,q)≡(p⋅q)gαβ−qαpβ, (21)

indeed coincides with the result of the perturbative expansion, as defined from the first term on the rhs of (19). Thus, the entire contribution to the anomaly is extracted form as

 gμνTμνanom=cFαβFαβ=−e224π2FαβFαβ. (22)

As we have already mentioned, the full action (2), varied several times with respect to the background metric and/or the background gauge fields gives those parts of the correlators of higher order, such as and , which contribute to the trace anomaly. In particular, the anomalous contributions of the ’s vertices are obtained by varying the local action both respect to the metric and to the gauge fields.

## 3 The energy momentum tensor and the Ward identities

Moving to the QCD case, we introduce the definition of the QCD energy-momentum tensor, which is given by

 Tμν = −gμνLQCD−FaμρFaρν−1ξgμν∂ρ(Aaρ∂σAaσ)+1ξ(Aaν∂μ(∂σAaσ)+Aaμ∂ν(∂σAaσ)) (23) + i4[¯¯¯¯ψγμ(→∂ν−igTaAaν)ψ−¯¯¯¯ψ(←∂ν+igTaAaν)γμψ+¯¯¯¯ψγν(→∂μ−igTaAaμ)ψ − ¯¯¯¯ψ(←∂μ+igTaAaμ)γνψ]+∂μ¯¯¯ωa(∂νωa−gfabcAcνωb)+∂ν¯¯¯ωa(∂μωa−gfabcAcμωb),

where is the non-abelian field strength of the gauge field

 Faμν=∂μAaν−∂νAaμ+gfabcAbμAcν (24)

and we have denoted with the Faddeev-Popov ghosts and with the antighosts, while is the gauge-fixing parameter. The ’s are the gauge group generators in the fermion representation and are the antisymmetric structure constants. For later use, it is convenient to isolate the gauge-fixing and ghost dependent contributions from the entire tensor

The coupling of QCD to gravity in the weak gravitational field limit is given by the interaction Lagrangian

 Lint=−12κhμνTμν. (27)

Notice that as defined in Eq. (23) is symmetric, while traceless for a massless theory. The symmetric expression can be easily found as suggested in [16], by coupling the theory to gravity and then defining it via a functional derivative with respect to the metric, recovering (23) in the flat spacetime case.

The conservation equation of the energy-momentum tensor takes the following form off-shell [17, 18]

 ∂μTμν = −δSδψ∂νψ−∂ν¯ψδSδ¯ψ+12∂μ(δSδψσμνψ−¯ψσμνδSδ¯ψ)−∂νAaμδSδAaμ (28) + ∂μ(AaνδSδAaμ)−δSδωa∂νωa−∂ν¯ωaδSδ¯ωa

where . It is indeed conserved by using the equations of motion of the ghost, antighost and fermion/antifermion fields. The off-shell relation is particularly useful, since it can be inserted into the functional integral in order to derive some of the Ward identities satisfied by the correlator. In fact, the implications of the conservation of the energy-momentum tensor on the Green’s functions can be exploited through the generating functional, obviously defined as

where is the standard QCD action and we have added the coupling of the energy-momentum tensor of the theory to the background gravitational field , which is the typical expression needed in the study of QCD coupled to gravity with a linear deviation from the flat metric. We have denoted with the sources of the gauge field (), the source of the fermion and antifermion fields () and of the ghost and antighost fields (, ) respectively. The generating functional of the connected Green’s functions is, as usual, denoted by

 expiW[J,η,¯η,χ,¯χ,h]=Z[J,η,¯η,χ,¯χ,h]Z[0,0,0,0,0,0] (30)

(normalized to the vacuum functional) and the effective action, defined as the generating functional of the 1-particle irreducible and truncated amplitudes. This is obviously obtained from by a Legendre transformation respect to all the sources, except, in our case, , which is taken as a background external field

 Γ[Ac,¯ψc,ψc,¯ωc,ωc,h]=W[J,η,¯η,χ,¯χ,h]−∫d4x(JμAμc+¯ηψc+¯ψcη+¯χωc+¯ωcχ). (31)

The source fields are eliminated from the right hand side of Eq. (31) inverting the relations

 Aμc=δWδJμ,ψc=δWδ¯η,¯ψc=δWδη,ωc=δWδ¯χ,¯ωc=δWδχ (32)

so that the functional derivatives of the effective action with respect to its independent variables are

 δΓδAμc=−Jμ,δΓδψc=−¯η,δΓδ¯ψc=−η,δΓδωc=−¯χ,δΓδ¯ωc=−χ, (33)

and for the source we have instead

 δΓδhμν=δWδhμν. (34)

The conservation of the energy-momentum tensor summarized in Eq. (28) in terms of classical fields, can be re-expressed in a functional form by a differentiation of with respect to and the use of Eq. (28) under the functional integral. We obtain

 ∂μδWδhμν = ¯η∂νδWδ¯η+∂νδWδηη−12∂μ(¯ησμνδWδ¯η−δWδησμνη)+∂νδWδJμJμ−∂μ(δWδJμJν) (35) + ¯χ∂νδWδ¯χ+∂νδWδχχ,

and finally, for the one particle irreducible generating functional, this gives

 ∂μδΓδhμν = −δΓδψc∂νψc−∂ν¯ψcδΓδ¯ψc+12∂μ(δΓδψcσμνψc−¯ψcσμνδΓδ¯ψc)−∂νAμcδΓδAμc+∂μ(AνcδΓδAμc) (36) − δΓδωc∂νωc−∂ν¯ωcδΓδ¯ωc,

obtained from Eq. (35) with the help of Eqs. (32, (33), (34). We summarize below the relevant Ward identities that can be used in order to fix the expression of the correlator.

• Single derivative general Ward identity
The Ward identities describing the conservation of the energy-momentum tensor for the one-particle irreducible Green’s functions with an insertion of can be obtained from the functional equation (36) by taking functional derivatives with respect to the classical fields. For example, the Ward identity for the graviton - gluon gluon vertex is obtained by differentiating Eq. (36) with respect to and and then setting all the external fields to zero

 ∂μ⟨Tμν(x)Aaα(x1)Abβ(x2)⟩trunc = −∂νδ4(x1−x)D−1αβ(x2,x)−∂νδ4(x2−x)D−1αβ(x1,x) + ∂μ(gανδ4(x1−x)D−1βμ(x2,x)+gβνδ4(x2−x)D−1αμ(x1,x))

where is the inverse gluon propagator defined as

 D−1αβ(x1,x2)=⟨Aα(x1)Aβ(x2)⟩trunc=δ2ΓδAαc(x1)δAβc(x2) (38)

and where we have omitted, for simplicity, both the colour indices and the symbol of the -product. The first Ward identity (3) becomes

 kμ⟨Tμν(k)Aα(p)Aβ(q)⟩trunc=qμD−1αμ(p)gβν+pμD−1βμ(q)gαν−qνD−1αβ(p)−pνD−1αβ(q). (39)
• Trace Ward identity at zero momentum transfer

It is possible to extract a Ward identity for the trace of the energy-momentum tensor for the same correlation function using just Eq. (39). In fact, differentiating it with respect to (or ) and then evaluating the resulting expression at zero momentum transfer () we obtain the Ward identity in spacetime dimensions

 ⟨Tμμ(0)Aα(p)Aβ(−p)⟩trunc=(2−d+p⋅∂∂p)D−1αβ(p) (40)

where the number counts the number of external gluon lines. For and using the transversality of the one-particle irreducible self-energy, namely

 D−1αβ(p)=(p2gαβ−pαqβ)Π(p2), (41)

the Ward identity in Eq. (40) simplifies to

 ⟨Tμμ(0)Aα(p)Aβ(−p)⟩trunc=2p2(p2gαβ−pαqβ)dΠdp2(p2). (42)

The trace Ward identity in Eq. (40) at zero momentum transfer can also be explicitly related to the function and the anomalous dimensions of the renormalized theory. These enter through the renormalization group equation for the two-point function of the gluon. Defining the beta function and the anomalous dimensions as

 β(g)=μ∂g∂μ,γ(g)=μ∂∂μlog√ZA,mγm(g)=μ∂m∂μ (43)

and denoting with the wave function renormalization constant of the gluon field, with the renormalized coupling, and with the renormalized mass, the trace Ward identity can be related to these functions by the relation

 ⟨Tμμ(0)Aα(p)Aβ(−p)⟩trunc=[β(g)∂∂g−2γ(g)+m(γm(g)−1)∂∂m]D−1αβ(p). (44)
• Two-derivatives Ward identity via BRST symmetry

We can exploit the BRST symmetry of the gauge-fixed lagrangian in order to derive some generalized Ward (Slavnov-Taylor) identities. We start by computing the BRST variation of the energy-momentum tensor, which is given by

 δAaμ = λDabμωb, (45) δωa = −12gλfabcωbωc, (46) δ¯ωa = −1ξ(∂μAaμ)λ, (47) δψ = igλωataψ, (48) δ¯ψ = −ig¯ψtaλωa, (49)

where is an infinitesimal Grassmann parameter.
A careful analysis of the energy-momentum tensor presented in Eq. (23) shows that the fermionic and the gauge part are gauge invariant and therefore invariant also under BRST. Instead the gauge-fixing and the ghost contributions must be studied in more detail. Using the transformation equations (45) and (47) in (26) one can prove the two identities

 λTg.f.μν = −Aaν∂μδ¯ωa−Aaμ∂νδ¯ωa+gμν[12∂⋅Aaδ¯ωa+Aaρ∂ρδ¯ωa], (50) λTghμν = −∂μ¯ωaδAaν−∂ν¯ωaδAaμ+gμν∂ρ¯ωaδAaρ, (51)

which show that the ghost and the gauge-fixing parts of the energy-momentum tensor (times the anticommuting factor ) can be written as an appropriate BRST variation of ghost/antighost and gauge contributions. Their sum, instead, can be expressed as the BRST variation of a certain operator plus an extra term which vanishes when using the ghost equations of motion

 (52)

which shows explicitly the structure of the gauge-variant terms in the energy-momentum tensor. Using the nilpotency of the BRST operator (), the BRST variation of is given by

 δTμν=δ(Tg.f.μν+Tghμν)=λξ[Aaμ∂ν∂ρDabρωb+Aaν∂μ∂ρDabρωb−gμν∂σ(Aaσ∂ρDabρωb)], (53)

where it is straightforward to recognize the equation of motion of the ghost field on its right-hand side. Using this last relation, we are able to derive some constraints on the Green’s functions involving insertions of the energy-momentum tensor. In particular, we are interested in some identities satisfied by the correlator in order to define it unambiguously. For this purpose, it is convenient to choose an appropriate Green’s function, in our case this is given by , and then exploit its BRST invariance to obtain

 δ⟨Tμν∂αAaα¯ωb⟩=⟨δTμν∂αAaα¯ωb⟩+λ⟨Tμν∂αDacαωc¯ωb⟩−λξ⟨Tμν∂αAaα∂βAbβ⟩=0, (54)

where the first two correlators, built with operators proportional to the equations of motion, contribute only with disconnected amplitudes, that are not part of the one-particle irreducible vertex function. From Eq. (54) we obtain the identity

 ∂αx1∂βx2⟨Tμν(x)Aaα(x1)Abβ(x2)⟩trunc=0, (55)

which in momentum space becomes

 pαqβ⟨Tμν(k)Aaα(p)Abβ(q)⟩trunc=0. (56)

A subtlety in these types of derivations concerns the role played by the commutators, which are generated because of the T-product and can be ignored only if they vanish. In general, in fact, the derivatives are naively taken out of the correlator, in order to arrive at Eq.  (56) and this can generate an error. In this case, due to the presence of an energy momentum tensor, the evaluation of these terms is rather involved. For this reason one needs to perform an explicit check of (56) to ensure the consistency of the formal result in a suitable regularization scheme. As we are going to show in the next sections, these three Ward identities turn out to be satisfied in dimensional regularization.

## 4 The perturbative expansion

The perturbative expansion is obtained by taking into account all the diagrams depicted in Figs. 2, 3, 4, where an incoming graviton appears in the initial state and two gluons with momenta and characterize the final state. The different contributions to the total amplitude are identified by the nature of the internal lines and are computed with the aid of the Feynman rules defined in Appendix A. Each amplitude is denoted by , with a superscript in square brackets indicating the figure of the corresponding diagram.

The contributions with a massive fermion running in the loop are depicted in Fig. 2; for the triangle in Fig. 2a we obtain

 −iκ2Γ[2a]abμναβ(p,q)=−κ2g2Tr(TbTa)∫d4l(2π)4tr{V′μν(l−q,l+p)1l/−q/−mγβ1l/−mγα1l/+p/−m} (57)

where the color factor is given by ; the diagram in Fig. 2c contributes as

 −iκ2Γ[2c]abμναβ(p,q)=−κ2g2Tr(TaTb)∫d4l(2π)4tr{W′μνα1l/−q/−mγβ1l/−m}, (58)

with the vertices and defined in Appendix A, Eqs. (LABEL:VGff) and (LABEL:WGffg) respectively. The remaining diagrams in Fig. 2 are obtained by exchanging and

 −iκ2Γ[2b]abμναβ(p,q) = −iκ2Γ[2d]abμναβ(p,q) = (60)

Moving to the gauge sector we find the four contributions in Fig. 3: the first one with a triangular topology is given by

where the color factor is . Those in Figs. 3b and 3c, containing gluon loops attached to the graviton vertex, are called “t-bubbles” and can be obtained one from the other by the exchange of and . The first “t-bubble” is given by

which is multiplied by an additional symmetry factor . There is another similar contribution obtained from the previous one after exchanging and

 (63)

The last diagram with gluons running in the loop is the one in Fig. 3d which is given by

 −iκ2Γ[3d]abμναβ(p,q)=12κ2g2∫d4l(2π)4VGggμνρσ(−l,l−p−q)δdfV4abcdρασβl2(l−p−q)2, (64)

where is the four gluon vertex defined as

and therefore

 δdfV4abcdρασβ = −CAδab~V4ρασβ=−CAδab(gασgβρ+gαρgβσ−2gαβgσρ), (66)

so that the amplitude in Eq. (64) becomes

 −iκ2Γ[3d]abμναβ(p,q)=−12κ2g2CAδab∫d4l(2π)4VGggμνρσ(−l,l−p−q)~V4ρασβl2(l−p−q)2. (67)

In the expression above we have explicitly isolated the color factor and the symmetry factor .

Finally, the ghost contributions shown in Fig. 4 are given by the sum of

 −iκ2Γ[4a]abμ