A Appendix. Definitions and conventions for the scalar integrals

# Conformal Anomalies and the Gravitational Effective Action: The $TJJ$ Correlator for a Dirac Fermion

## Abstract

We compute in linearized gravity all the contributions to the gravitational effective action due to a virtual Dirac fermion, related to the conformal anomaly. This requires, in perturbation theory, the identification of the gauge-gauge-graviton vertex off mass shell, involving the correlator of the energy-momentum tensor and two vector currents (), which is responsible for the generation of the gauge contributions to the conformal anomaly in gravity. We also present the anomalous effective action in the inverse mass of the fermion as in the Euler-Heisenberg case.

Conformal Anomalies and the Gravitational Effective Action:

The Correlator for a Chiral Fermion

Roberta Armillis, Claudio Corianò and Luigi Delle Rose 1

Dipartimento di Fisica, Università del Salento

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

Department of Physics, University of Crete

Heraklion, Crete, Greece

## 1 Introduction

Investigations of conformal anomalies in gravity (see [1] for an historical overview and references) [2] and in gauge theories [3, 4, 5] as well as in string theory, have been of remarkable significance along the years. In cosmology, for instance, [6] (see also [7] for an overview) the study of the gravitational trace anomaly has been performed in an attempt to solve the problem of the “graceful exit” (see for instance [8, 9, 10, 11]). In other analysis it has been pointed out that the conformal anomaly may prevent the future singularity occurrence in various dark energy models [12, 13].

In the past the analysis of the formal structure of the effective action for gravity in four dimensions, obtained by integration of the trace anomaly [14, 15], has received a special attention, showing that the variational solution of the anomaly equation, which is non-local, can be made local by the introduction of extra scalar fields. The gauge contributions to these anomalies are identified at 1-loop level from a set of diagrams - involving fermion loops with two external gauge lines and one graviton line - and are characterized, as shown recently by Giannotti and Mottola in [16], by the presence of anomaly poles. Anomaly poles are familiar from the study of the chiral anomaly in gauge theories and describe the non-local structure of the effective action. In the case of global anomalies, as in QCD chiral dynamics, they signal the presence of a non-perturbative phase of the fundamental theory, with composite degrees of freedom (pions) which offer an equivalent description of the fundamental lagrangian, matching the anomaly, in agreement with ’t Hooft’s principle. Previous studies of the role of the conformal anomaly in cosmology concerning the production of massless gauge particles and the identification of the infrared anomaly pole are those of Dolgov [17, 18], while a discussion of the infrared pole from a dispersive derivation is contained in [19].

In a related work [20] we have shown that anomaly poles are typical of the perturbative description of the chiral anomaly not just in some special kinematical conditions, for instance in the collinear region, where the coupling of the anomalous gauge current to two (on-shell) vector currents (for the AVV diagram) involves a pseudoscalar intermediate state (with a collinear and massless fermion-antifermion pair) but under any kinematical conditions. They are the most direct - and probably also the most significant - manifestation of the anomaly in the perturbative diagrammatic expansion of the effective action. On a more speculative side, the interpretation of the pole in terms of composite degrees of freedom could probably have direct physical implications, including the condensation of the composite fields, very much like Bose Einstein (BE) condensation of the pion field, under the action of gravity. Interestingly, in a recent paper, Sikivie and Yang have pointed out that Peccei-Quinn axions (PQ) may form BE condensates [21]. With these motivations in mind, in this work, which parallels a previous investigation of the chiral gauge anomaly [20], we study the perturbative structure of the off-shell effective action showing the appearance of similar singularities under general kinematic conditions. Our investigation is a first step towards the computation of the exact effective action describing the coupling of the Standard Model to gravity via the conformal anomaly, that we hope to discuss in the future.

In our study we follow closely the work of [16]. There the authors have presented a complete off-shell classification of the invariant amplitudes of the relevant correlator responsible for the conformal anomaly, which involves the energy momentum tensor (T) and two vector currents (J), , and have thoroughly investigated it in the QED case, drawing on the analogy with the case of the chiral anomaly. The analysis of [16] is based on the use of dispersion relations, which are sufficient to identify the anomaly poles of the amplitude from the spectral density of this correlator, but not to characterize completely the off-shell effective action of the theory and the remaining non-conformal contributions, which will be discussed in this paper. The poles that we extract from the complete effective action include both the usual poles derived from the spectral analysis of the diagrams, which are coupled in the infrared (IR) and other extra poles which account for the anomaly but are decoupled in the same limit. These extra poles appear under general kinematic configurations and are typical of the off-shell as well as of the on-shell effective action, both for massive and massless fermions.

We also show, in agreement with those analysis, that the pole terms which contribute to the conformal anomaly are indeed only obtained in the on-shell limit of the external gauge lines, and identify all the mass corrections to the correlator in the general case. This analysis is obtained by working out all the relevant kinematical limits of the perturbative corrections. We present the complete anomalous off-shell effective action describing the interaction of gravity with the photons, written in a form in which we separate the non-local contribution due to the anomaly pole from the rest of the action (those which are conformally invariant in the massless fermion limit). Away from the conformal limit of the theory we present a expansion of the effective action as in the Euler-Heisenberg approach. This expansion, naturally, does not convey the presence of non-localities in the effective action due to the appearance of massless poles.

The computation of similar diagrams, for the on-shell photon case, appears in older contributions by Berends and Gastmans [22] using dimensional regularization, in their study of the gravitational scattering of photons and by Milton using Schwinger’s methods [23]. The presence of an anomaly pole in the amplitude has not been investigated nor noticed in these previous analysis, since they do not appear explicitly in their results, nor the expansion of the three form factors of the on-shell vertex, contained in [22], allows their identification in the S-matrix elements of the theory. Two related analysis by Drummond and Hathrell in their investigation of the gravitational contribution to the self-energy of the photon [24] and the renormalization of the trace anomaly [25] included the same on-shell vertex. Later, this same vertex has provided the ground for several elaborations concerning a possible superluminal behaviour of the photon in the presence of an external gravitational field [26].

## 2 The conformal anomaly and gravity

In this section we briefly summarize some basic and well known aspects of the trace anomaly in quantum gravity and, in particular, the identification of the non-local action whose variation generates a given trace anomaly.

We recall that the gravitational trace anomaly in 4 spacetime dimensions generated by quantum effects in a classical gravitational and electromagnetic background is given by the expression

 Tμμ=−18[2bC2+2b′(E−23□R)+2cF2] (1)

where , and are parameters that for a single fermion in the theory result , , and ; furthermore denotes the Weyl tensor squared and is the Euler density given by

 C2 = CλμνρCλμνρ=RλμνρRλμνρ−2RμνRμν+R23 (2) E = ∗Rλμνρ∗Rλμνρ=RλμνρRλμνρ−4RμνRμν+R2. (3)

The effective action is identified by solving the following variational equation by inspection

 −2√ggμνδΓδgμν=Tμμ. (4)

Its solution is well known and is given by the non-local expression

 Sanom[g,A]= (5) 18∫d4x√−g∫d4x′√−g′(E−23□R)xG4(x,x′)[2bC2+b′(E−23□R)+2cFμνFμν]x′.

Notice that we are omitting terms which are not necessary at one loop level. The notation denotes the Green’s function of the differential operator defined by

 Δ4≡∇μ(∇μ∇ν+2Rμν−23Rgμν)∇ν=□2+2Rμν∇μ∇ν+13(∇μR)∇μ−23R□ (6)

and requires some boundary conditions to be specified. This operator is conformally covariant, in fact under a rescaling of the metric one can show that

 gμν=eσ¯gμν→Δ4=e−2σ¯Δ4. (7)

Notice that the general solution of (4) involves, in principle, also a conformally invariant part that is not identified by this method. As in ref. [16], we concentrate on the contribution proportional to and perform an expansion of this term for a weak gravitational field and drop from this action all the terms which are at least quadratic in the deviation of the metric from flat space

 gμν=ημν+κhμνκ2=16πG, (8)

with the gravitational constant. The non-local action reduces to

 Sanom[g,A]=−c6∫d4x√−g∫d4x′√−g′R(1)x□−1x,x′[FαβFαβ]x′, (9)

valid for a weak gravitational field. In this case

 R(1)x≡∂xμ∂xνhμν−□h,h=ημνhμν. (10)

The presence of the Green’s function of the operator in Eq. (9) is the clear indication that the solution of the anomaly equation is characterized by an anomaly pole. In the next sections we are going to perform a direct diagrammatic computation of this action and reobtain from it the pole contribution identified in the dispersive analysis of [16] and the conformal invariant extra terms which are not present in (9). We start with an analysis of the correlator following an approach which is close to that followed in ref. [16]. The crucial point of the derivation presented in that work is the imposition of the Ward identity for the correlator (see Eq. (42) below) which allows to eliminate all the Schwinger (gradients) terms which otherwise plague any derivation based on the canonical formalism and are generated by the equal-time commutator of the energy momentum tensor with the vector currents. In reality, this approach can be bypassed by just imposing at a diagrammatic level the validity of an operatorial relation for the trace anomaly, evaluated at a nonzero momentum transfer, together with the conservation of the vector currents on the other two vector vertices of the correlator.

## 3 The construction of the full amplitude Γμναβ(p,q)

We consider the standard QED lagrangian

 L=−14FμνFμν+i¯ψγμ(∂μ−ieAμ)ψ−m¯ψψ, (11)

with the energy momentum tensor 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), (12)
 Tμνfp=−eJ(μAν)+egμνJλAλ, (13)

and

 Tμνph=FμλFν  λ−14gμνFλρFλρ, (14)

where the current is defined as

 Jμ(x)=¯ψ(x)γμψ(x). (15)

In the coupling to gravity of the total energy momentum tensor

 Tμν≡Tμνf+Tμνfp+Tμνph (16)

we keep terms linear in the gravitational field, of the form , and we have introduced some standard notation for the symmetrization of the tensor indices and left-right derivatives and . It is also convenient to introduce a partial energy momentum tensor , corresponding to the sum of the Dirac and interaction terms

 Tμνp≡Tμνf+Tμνfp (17)

which satisfies the inhomogeneous equation

 ∂νTμνp=−∂νTμνph. (18)

Using the equations of motion for the e.m. field , the inhomogeneous equation becomes

 ∂νTμνp=FμλJλ. (19)

There are two ways to identify the contributions of and in the perturbative expansion of the effective action. In the formalism of the background fields, the correlator can be extracted from the defining functional integral

 ⟨Tμνp(z)⟩A ≡ ∫DψD¯ψTμνp(z)ei∫d4xL+∫J⋅A(x)d4x (20) = ⟨Tμνpei∫d4xJ⋅A(x)⟩

expanded through second order in the external field . The relevant terms in this expansion are explicitly given by

 ⟨Tμνp(z)⟩A=12!⟨Tμνf(z)(J⋅A)(J⋅A)⟩+⟨Tμνfp(J⋅A)⟩+..., (21)

with . The corresponding diagrams are extracted via two functional derivatives respect to the background field and are given by

 Γμναβ(z;x,y)≡δ2⟨Tμνp(z)⟩AδAα(x)δAβ(y)∣∣∣A=0=Vμναβ+Wμναβ (22)
 Vμναβ=(ie)2⟨Tμνf(z)Jα(x)Jβ(y)⟩A=0 (23)
 Wμναβ = δ2⟨Tμνfp(z)(J⋅A)⟩δAα(x)δAβ(y)∣∣∣A=0 = δ4(x−z)gα(μΠν)β(z,y)+δ4(y−z)gβ(μΠν)α(z,x)−gμν[δ4(x−z)−δ4(y−z)]Παβ(x,y)

These two contributions are of . Alternatively, one can directly compute the matrix element

 Mμν=⟨0|Tμνp(z)∫d4wd4w′J⋅A(w)J⋅A(w′)|γγ⟩, (25)

which generates the diagrams (b) and (c) shown in Fig.1, respectively called the “triangle” and the “t-p-bubble” (“t-” stays for tensor), together with the two ones obtained for the exchange of with and with .

The conformal anomaly appears in the perturbative expansion of and involves these four diagrams. The electromagnetic contribution is responsible for other two diagrams whose invariant amplitudes are well-defined and will be used to fix the ill-defined amplitudes present in the tensor expansion of by using a Ward identity.

The lowest order contribution is obtained, in the background field formalism, from Maxwell’s e.m. tensor, and is given by

 Sμναβ = δ2⟨Tμνph(z)⟩δAα(x)δAβ(y)∣∣∣A=0. (26)

Equivalently, it can be obtained from the matrix element

 ⟨0|Tμνph|γγ⟩ (27)

which allows to identify the vertex in momentum space. Using (26) we easily obtain

 Sμναβ(z,x,y) = 2gαβ∂(μδxz∂ν)δyz−2gβ(μ∂ν)δxz∂αδyz−2gα(ν∂μ)δyz∂βδxz +gαμgβν∂λδyz∂λδxz+gανgβμ∂λδyz∂λδxz+gμν∂βδxz∂αδyz−∂ρδyz∂ρδxzgαβgμν

where and so on. In momentum space this lowest order vertex is given by

 Sμναβ = (pμqν+pνqμ)gαβ+p⋅q(gανgβμ+gαμgβν)−gμν(p⋅qgαβ−qαpβ) (29) −(gβνpμ+gβμpν)qα−(gανqμ+gαμqν)pβ.

The corresponding vertices which appear respectively in the triangle diagram and on the t-bubble at are given by

 V′μν(k1,k2) = 14[γμ(k1+k2)ν+γν(k1+k2)μ]−12gμν[γλ(k1+k2)λ−2m], (30) W′μνα = −12(γμgνα+γνgμα)+gμνγα, (31)

where is outcoming (incoming).

Using the two vertices and we obtain for the diagrams (b) and (c) of Fig.1

 Vμναβ(p,q)=−(−ie)2i3∫d4l(2π)4tr{V′μν(l+p,l−q)(l/−q/+m)γβ(l/+m)γα(l/+p/+m)}[l2−m2][(l−q)2−m2][(l+p)2−m2],

and

 Wμναβ(p,q) = −(ie2)i2∫d4l(2π)4tr{W′μνα(l/+m)γβ(l/−q/+m)}[l2−m2][(l−q)2−m2], (33)

so that the one-loop amplitude in Fig. 1 results

 Γμναβ(p,q)=Vμναβ(p,q)+Vμνβα(q,p)+Wμναβ(p,q)+Wμνβα(q,p). (34)

The bare Ward identity which allows to define the divergent amplitudes that contribute to the anomaly in in terms of the remaining finite ones is obtained by re-expressing the classical equation

 ∂νTμνph=−FμνJν (35)

as an equation of generating functionals in the background electromagnetic field

 ∂ν⟨Tμνph⟩A=−Fμν⟨Jν⟩A, (36)

which can be expanded perturbatively as

 ∂ν⟨Tμνph⟩A=−Fμν⟨Jν∫d4w(ie)J⋅A(w)⟩+.... (37)

Notice that we have omitted the first term in the Dyson’s series of , shown on the r.h.s of (37) since . The bare Ward identity then takes the form

 ∂νΓμναβ=δ2(Fμλ(z)⟨Jλ(z)⟩A)δAα(x)δAβ(y)∣∣∣A=0 (38)

which takes contribution only from the first term on the r.h.s of Eq. (37). This relation can be written in momentum space. For this we use the definition of the vacuum polarization

 Παβ(x,y)≡−ie2⟨Jα(x)Jβ(y)⟩, (39)

or

 Παβ(p) = Missing or unrecognized delimiter for \left (40) = (p2gαβ−pαpβ)Π(p2,m2)

with

 Missing or unrecognized delimiter for \right (41)

which obviously satisfies the Ward identity . The expressions of the and contributions are given in Appendix A.

Using these definitions, the unrenormalized Ward identity which allows to completely characterize the form of the correlator in momentum space becomes

 kνΓμναβ(p,q) = (qμpαpβ−qμgαβp2+gμβqαp2−gμβpαp⋅q)Π(p2) (42) +(pμqαqβ−pμgαβq2+gμαpβq2−gμαqβp⋅q)Π(q2).

### 3.1 Tensor expansion and invariant amplitudes of Γ

The full one-loop amplitude can be expanded on the basis provided by the 43 monomial tensors listed in Tab.1

 Γμναβ(p,q)=43∑i=1Ai(k2,p2,q2)lμναβi(p,q). (43)

Since the amplitude has total mass dimension equal to it is obvious that not all the coefficients are convergent. They can be divided into groups:

• - multiplied by a product of four momenta, they have mass dimension and therefore are UV finite;

• - these have mass dimension since the four Lorentz indices of the amplitude are carried by two metric tensors

• - they appear next to a metric tensor and two momenta, have mass dimension and are divergent.

The way in which the invariant amplitudes will be managed in order to reduce them to the named is the subject of this section. The reduction is accomplished in 4 different steps and has as a guiding principle the elimination of the divergent amplitudes in terms of the convergent ones after imposing some conditions on the whole amplitude. We require

• the symmetry in the two indices and of the symmetric energy-momentum tensor ;

• the conservation of the two vector currents on and ;

• the Ward identity on the vertex with the incoming momentum defined above in Eq. (LABEL:WI).

Condition a) becomes

 Γμναβ(p,q)=Γνμαβ(p,q), (44)

giving a linear system of equations; of them being identically satisfied because the tensorial structures are already symmetric in the exchange of and , while the remaining conditions are

 A5=A6,A8=A9,A10=A11,A13=A14,A18=A19, Missing dimension or its units for \hskip Missing dimension or its units for \hskip (45)

where all are thought as functions of the invariants . After substituting (45) into the invariant tensors of the decomposition are multiplied by only invariant amplitudes. Condition b), which is vector current conservation on the two vertices with indices and allows to re-express some divergent in terms of other finite ones

 pαΓμναβ(p,q)=qβΓμναβ(p,q)=0. (46)

This constraint generates two sets of independent tensor structures each, so that in order to fulfill (46) each coefficient is separately set to vanish. The first Ward identity leads to a linear system composed of equations

 pαΓμναβ(p,q)=0⇒⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩A19+A36p⋅p+A37p⋅q=0,A38p⋅p+A39p⋅q=0,A17+A40p⋅p+A42p⋅q=0,A41p⋅p+A43p⋅q=0,A20+2A28+A1p⋅p+A4p⋅q=0,2A30+A3p⋅p+A7p⋅q=0,A22+A29+A6p⋅p+A11p⋅q=0,A31+A9p⋅p+A14p⋅q=0,A23+A12p⋅p+A16p⋅q=0,A15p⋅p+A2p⋅q=0; (47)

we choose to solve it for the set in which only the first one is convergent and the others are UV divergent. The set would not include all the divergent since in the last equations appear two convergent coefficients, and .
Following our choice the result is

 A15=−A2p⋅qp⋅p,A17=−A40p⋅p−A42p⋅q, (48) A19=−A36p⋅p−A37p⋅q,A23=−A12p⋅p−A16p⋅q, (49) A28=12[−A20−A1p⋅p−A4p⋅q],A29=−A22−A6p⋅p−A11p⋅q, (50) A30=−12[A3p⋅p+A7p⋅q],A31=−A9p⋅p−A14p⋅q, (51) A39=−A38p⋅pp⋅q,A43=−A41p⋅pp⋅q. (52)

In an analogous way we go on with the second Ward identity (WI) after replacing the solution of the previous system in the original amplitude. The new one is indicated by , where the subscript is there to indicate that we have applied condition on . The constraint gives

 Missing or unrecognized delimiter for \left (53)

We solve these equations determining the amplitudes in the set in terms of the remaining ones, obtaining

 A38=−A12p⋅pp⋅q−A2p⋅qq⋅q2p⋅p,A40=−A41q⋅qp⋅q, (54) Missing dimension or its units for \hskip (55) Missing dimension or its units for \hskip (56)

The manipulations above have allowed a reduction of the number of invariant amplitudes from the initial to using the symmetry ( equations), the first WI on ( equations) and the second WI on ( equations). The surviving invariant amplitudes in which the amplitude can be expanded using the form factors are . This set still contains divergent amplitudes, . The amplitude is indeed ill-defined until we impose on it condition c), that is Eq. (42). This condition gives

 Eq.(???)⇒⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩\vspace0.2cm−A3[1+p⋅p2p⋅q]+A6+12A7−A9−A41p⋅q=0,\vspace0.2cmA37+A42+A4[p⋅p+p⋅q]+A11p⋅q+12A7q⋅q+\vspace0.2cm+A11q⋅q+12A3p⋅pq⋅qp⋅q=0,\vspace0.2cm12A2p⋅qq⋅qp⋅p−A41p⋅p+q⋅qp⋅q−12A3p⋅p+A7(p⋅p+12p⋅q)+A6p⋅q\vspace0.2cm+A12(12p⋅q+q⋅q)+A14�(p⋅q+2q⋅q)+2�A37−Π(p2)−Π(q2)=0

From this condition we obtain

 A37=−A24p⋅qq⋅qp⋅p+14A3p⋅p−14A7(�2p⋅p+p⋅q)+12A41(p⋅p+q⋅qp⋅q) −12A6Êp⋅q−14A12(Êp⋅q+2q⋅q)−12A14(Êp⋅q+2Êq⋅q)+12[Π(p2)+ΠÊ(q2)] (57) A41=−A32p⋅p−(A3−A6−A7+A9)p⋅q (58) A42=A32p⋅p(p⋅pp⋅q+1−q⋅qp⋅q)+12A7(p⋅p+p⋅q−q⋅q)−A4(p⋅p+p⋅q) −(A6−A9)p⋅p+(A14−A11)(q⋅q+p⋅q). (59)

After these steps we end up with an expression for written in terms of only invariant amplitudes, that are , significantly reduced respect to the original . Further reductions are possible (down to independent invariant amplitudes), however, since these reductions just add to the complexity of the related tensor structures, it is convenient to select an appropriate set of reducible (but finite) components characterized by a simpler tensor structure and present the result in that form. The 13 amplitudes introduced in the final decomposition are, in this respect, a good choice since the corresponding tensor structures are rather simple. These tensors are combinations of the monomials listed in Tab.1.

The set is very useful for the actual computation of the tensor integrals and for the study of their reduction to scalar form. To compare with the previous study of Giannotti and Mottola [16] we have mapped the computation of the components of the set into their structures . Also in this case, the truly independent amplitudes are 8. One can extract, out of the 13 reducible amplitudes, a consistent subset of 8 invariant amplitudes. The remaining amplitudes in the 13 tensor structures are, in principle, obtainable from this subset.

### 3.2 Reorganization of the amplitude

Before obtaining the mapping between the amplitudes in and the structures , we briefly describe the tensor decomposition introduced in [16] which defines these 13 structures. We define the rank-2 tensors

 uαβ(p,q)≡(p⋅q)gαβ−qαpβ, (60) wαβ(p,q)≡p2q2gαβ+(p⋅q)pαqβ−q2pαpβ−p2qαqβ, (61)

which are Bose symmetric,

 uαβ(p,q)=uβα(q,p), (62) wαβ(p,q)=wβα(q,p), (63)

and conserve vector current,

 pαuαβ(p,q)=qβuαβ(p,q)=0, (64) pαwαβ(p,q)=qβwαβ(p,q)=0. (65)