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We derive the conformal constraints satisfied by classical vertices of a (Einstein) Gauss-Bonnet theory around flat space, in general dimensions and at d=4 (4d EGB). In 4d EGB they are obtained by a singular limit of the integral of the Euler-Poincarè density. Our analysis exploits the relation between this theory and the conformal anomaly action, which allows to uncover some interesting features of the GB vertex at cubic and quartic level. If we introduce a conformal decomposition of the metric, the resulting theory can be formulated in two different versions, which are regularization dependent, a local one which is quartic in the dilaton field, and a nonlocal one, with a quadratic dilaton. The nonlocal version is derived by a finite redefinition of the GB density with the inclusion of a (d−4)R2 correction, before performing the singular d→4 limit. In the local version of the theory, we show how the independent dynamics of the metric and of the dilaton are interwinded by a classical trace identity. Three-gravitational wave interactions can be organised in a nontrivial way by using directly the nonlocal 4d EGB version of the theory. This is possible thanks to the consistency of such formulation - only up to 3-point functions - directly inherited from the conformal anomaly (Riegert) action. The constraints satisfied by the vertices are classical, hierarchical Ward identities. At quartic level, similar relations are derived, borrowing from the analysis of the counterterms of the 4T correlators of the conformal anomaly action, as defined by a perturbative expansion. For d≠4 these constraints hold also for Lovelock actions. They can be extended to higher order topological invariants in such class of theories.

Abstract We analyze the parity-odd correlators $$\langle JJO\rangle _{odd},$$ ⟨ J J O ⟩ odd , $$\langle JJT\rangle _{odd},$$ ⟨ J J T ⟩ odd , $$\langle TTO\rangle _{odd}$$ ⟨ T T O ⟩ odd and $$\langle TTT\rangle _{odd}$$ ⟨ T T T ⟩ odd in momentum space, constrained by conformal Ward identities, extending our former investigation of the parity-odd chiral anomaly vertex. We investigate how the presence of parity-odd trace anomalies affect such correlators. Motivations for this study come from holography, early universe cosmology and from a recent debate on the chiral trace anomaly of a Weyl fermion. In the current CFT analysis, O can be either a scalar or a pseudoscalar operator and it can be identified with the trace of the stress–energy tensor. We find that the $$\langle JJO\rangle _{odd}$$ ⟨ J J O ⟩ odd and $$\langle TTO\rangle _{odd}$$ ⟨ T T O ⟩ odd can be different from zero in a CFT. This occurs when the conformal dimension of the scalar operator is $$\Delta _3=4,$$ Δ 3 = 4 , as in the case of $$O=T^{\mu }_{\mu }.$$ O = T μ μ . Moreover, if we assume the existence of parity-odd trace anomalies, the conformal $$\langle JJT\rangle _{odd}$$ ⟨ J J T ⟩ odd and $$\langle TTT\rangle _{odd}$$ ⟨ T T T ⟩ odd are nonzero. In particular, in the case of $$\langle JJT\rangle _{odd}$$ ⟨ J J T ⟩ odd the transverse–traceless component is constrained to vanish, and the correlator is determined only by the trace part with the anomaly pole.

Abstract We illustrate how the Conformal Ward Identities (CWI) in momentum space for parity-odd correlators determine the structure of a chiral anomaly interaction, taking the example of the VVA (vector/vector/axial-vector) and AAA correlators in momentum space. Only the conservation and the anomalous WIs, together with the Bose symmetry, are imposed from the outset for the determination of the correlators. We use a longitudinal/transverse decomposition of tensor structures and form factors. The longitudinal (L) component is fixed by the anomaly content and the anomaly pole, while in the transverse (T) sector we define a new parameterization. We relate the latter both to the Rosenberg original representation of the VVA and to the longitudinal/transverse (L/T) one, first introduced in the analysis of $$g-2$$ g - 2 of the muon in the investigation of the diagram in the chiral limit of QCD. The correlators are completely identified by the conformal constraints whose solutions are fixed only by the anomaly coefficient, the residue of the anomaly pole. In both cases, our CFT result matches the one-loop perturbative expression, as expected. The CWIs for correlators of mixed chirality $$J_L J J_R$$ J L J J R generate solutions in agreement with the all-orders nonrenormalization theorems of perturbative QCD and in the chiral limit.

Abstract We present an analysis of the perturbative realization of the TTJJ correlator, with two stress energy tensors and two conserved currents, using free field theory, integrating out conformal sectors in the quantum corrections. This allows to define, around flat space, an exact perturbative expansion of the complete anomaly effective action – up to 4-point functions – whose predictions can be compared against those of the anomaly induced action. The latter is a variational solution of the conformal anomaly constraint at $$d=4$$ d = 4 in the form of a nonlocal Wess–Zumino action. The renormalization procedure and the degeneracies of the tensor structures of this correlator are discussed, valid for a generic conformal field theory, deriving its anomalous trace Ward identities (WIs). In this application, we also illustrate a general procedure that identifies the minimal number of tensorial structures and corresponding form factors for the TTJJ and any 4-point function. The approach is implemented for three, four and five dimensions, addressing the tensor degeneracies of the expansion in momentum space. We show that the renormalized TTJJ can be split into two contributions, a non anomalous and an anomalous part, each separately conserved. The first satisfies ordinary trace WIs, while the second satisfies anomalous trace WIs. The result of the direct computation is compared against the expression of the same 4-point function derived from the nonlocal anomaly induced action. We show that the prediction for the anomalous part of the TTJJ derived from such action, evaluated in two different conformal decompositions, the Riegert and Fradkin–Vilkovisky (FV) choices, differ from the anomaly part identified in the perturbative TTJJ, in the flat spacetime limit. The anomaly part of the correlator computed with the Riegert choice is affected by double poles, while the one computed with the FV choice does not satisfy the conservation WIs. We present the correct form of the expansion of the anomaly induced action at the second order in the metric perturbations around flat space that reproduces the perturbative result.

We elaborate on anomaly induced actions of the Wess-Zumino (WZ) form and their relation to the renormalized effective action, which is defined by an ordinary path integral over a conformal sector, in an external gravitational background. In anomaly - induced actions, the issue of scale breaking is usually not addressed, since these actions are obtained only by solving the trace anomaly constraint and are determined by scale invariant functionals. We investigate the changes induced in the structure of such actions once identified in dimensional renormalization (DR) when the ϵ=d−4→0 limit is accompanied by the dimensional reduction (DRed) of the field dependencies. We show that operatorial nonlocal modifications (∼□ϵ) of the counterterms are unnecessary to justify a scale anomaly. In this case, only the ordinary finite subtractions play a critical role in the determination of the scale breaking. This is illustrated for the WZ form of the effective action and its WZ consistency condition, as seen from a renormalization procedure. Logarithmic corrections from finite subtractions are also illustrated in a pure d=4 (cutoff) scheme. The interplay between two renormalization schemes, one based on dimensional regularization (DR) and the second on a cutoff in d=4, illustrates the ambiguities of DR in handling the quantum corrections in a curved background. Therefore, using DR in a curved background, the scale and trace anomalies can both be obtained by counterterms that are Weyl invariant only at d=4.

Abstract We investigate the gravitational backreaction, generated by coupling a general conformal sector to external, classical gravity, as described by a conformal anomaly effective action. We address the issues raised by the regularization of the topological Gauss–Bonnet and Weyl terms in these actions and the use of dimensional regularization (DR). We discuss both their local and nonlocal expressions, as possible IR and UV descriptions of conformal theories, below and above the conformal breaking scale. Our discussion overlaps with several recent studies of dilaton gravities – obtained via a certain singular limit of the Einstein–Gauss–Bonnet (EGB) theory – originally introduced as a way to bypass Lovelock’s theorem. We show that nonlocal, purely gravitational realizations of such EGB theories, quadratic in the dilaton field, beside their local quartic forms, are possible, by a finite renormalization of the Euler density. Such nonlocal versions, which are deprived of any scale, can be expanded, at least around flat space, in terms of the combination $$R \Box ^{-1}$$ R □ - 1 times multiple variations of the anomaly functional, as pointed out in recent studies at $$d=4$$ d = 4 . Similar conclusions can be drawn for the proposed nonlocal EGB theory. The expansion emerges from previous investigations of the anomalous conformal Ward identities that constrain such theories around the flat spacetime limit in momentum space.

Abstract We elaborate on the structure of the conformal anomaly effective action up to 4-th order, in an expansion in the gravitational fluctuations (h) of the background metric, in the flat spacetime limit. For this purpose we discuss the renormalization of 4-point functions containing insertions of stress-energy tensors (4T), in conformal field theories in four spacetime dimensions with the goal of identifying the structure of the anomaly action. We focus on a separation of the correlator into its transverse/traceless and longitudinal components, applied to the trace and conservation Ward identities (WI) in momentum space. These are sufficient to identify, from their hierarchical structure, the anomaly contribution, without the need to proceed with a complete determination of all of its independent form factors. Renormalization induces sequential bilinear graviton-scalar mixings on single, double and multiple trace terms, corresponding to $$R\square ^{-1}$$ R □ - 1 interactions of the scalar curvature, with intermediate virtual massless exchanges. These dilaton-like terms couple to the conformal anomaly, as for the chiral anomalous WIs. We show that at 4T level a new traceless component appears after renormalization. We comment on future extensions of this result to more general backgrounds, with possible applications to non local cosmologies.

Recently, the possibility of evading Lovelock's theorem at d=4, via a singular redefinition of the dimensionless coupling of the Gauss-Bonnet term, has been very extensively discussed in the cosmological context. The term is added as a quadratic contribution of the curvature tensor to the Einstein-Hilbert action, originating theories of “Einstein Gauss-Bonnet” (EGB) type. We point out that the action obtained by the dimensional regularization procedure, implemented with the extraction of a single conformal factor, correspond just to an ordinary Wess-Zumino anomaly action, even though it is deprived of the contribution from the Weyl tensor. We also show that a purely gravitational version of the EGB theory can be generated by allowing a finite renormalization of the Gauss-Bonnet topological contribution at d=4+ϵ, as pointed out by Mazur and Mottola. The result is an effective action which is quadratic, rather then quartic, in the dilaton field, and scale free, compared to the previous derivations. The dilaton, in this case can be removed from the spectrum, leaving a pure gravitational theory, which is nonlocal. We comment on the physical meaning of the two types of actions, which may be used to describe such topological terms both below and above the conformal breaking scale.

Abstract We derive and analyze the conformal Ward identities (CWI’s) of a tensor 4-point function of a generic CFT in momentum space. The correlator involves the stress–energy tensor T and three scalar operators O (TOOO). We extend the reconstruction method for tensor correlators from 3- to 4-point functions, starting from the transverse traceless sector of the TOOO. We derive the structure of the corresponding CWI’s in two different sets of variables, relevant for the analysis of the 1–3 (1 graviton $$\rightarrow $$ → 3 scalars) and 2–2 (graviton + scalar $$\rightarrow $$ → two scalars) scattering processes. The equations are all expressed in terms of a single form factor. In both cases we discuss the structure of the equations and their possible behaviors in various asymptotic limits of the external invariants. A comparative analysis of the systems of equations for the TOOO and those for the OOOO, both in the general (conformal) and dual-conformal/conformal (dcc) cases, is presented. We show that in all the cases the Lauricella functions are homogeneous solutions of such systems of equations, also described as parametric 4K integrals of modified Bessel functions.

Abstract We discuss specific hypergeometric solutions of the conformal Ward identities (CWI’s) of scalar 4-point functions of primary fields in momentum space, in d spacetime dimensions. We determine such solutions using various dual conformal ansätze (DCA’s). We start from a generic dual conformal correlator, and require it to be conformally covariant in coordinate space. The two requirements constrain such solutions to take a unique hypergeometric form. They describe correlators which are at the same time conformal and dual conformal in any dimension. These specific ansätze also show the existence of a link between 3- and 4-point functions of a CFT for such class of exact solutions, similarly to what found for planar ladder diagrams. We show that in d = 4 only the box diagram and its melonic variants, in free field theory, satisfies such conditions, the remaining solutions being nonperturbative. We then turn to the analysis of some approximate high energy fixed angle solutions of the CWI’s which also in this case take the form of generalized hypergeometric functions. We show that they describe the behaviour of the 4-point functions at large energy and momentum transfers, with a fixed −t/s. The equations, in this case, are solved by linear combinations of Lauricella functions of 3 variables and can be rewritten as generalized 4K integrals. In both cases the CWI’s alone are sufficient to identify such solutions and their special connection with generalized hypergeometric systems of equations.

Stress-energy correlation functions in a general Conformal Field Theory (CFT) in four dimensions are described in a fully covariant approach, as metric variations of the quantum effective action in an arbitrary curved space background field. All Conservation, Trace and Conformal Ward Identities (CWIs), including contact terms, are completely fixed in this covariant approach. The Trace and CWIs are anomalous. Their anomalous contributions may be computed unambiguously by metric variation of the exact 1PI quantum effective action determined by the conformal anomaly of 〈Tμν〉 in d=4 curved space. This action implies the existence of massless propagator poles in three and higher point correlators of Tμν. The metric variations of the anomaly effective action in its local form in terms of a scalar conformalon field are carried out explicitly for the case of the correlator of three CFT stress-energy tensors, and the result is shown to coincide with the algebraic reconstruction of 〈TTT〉 from its transverse, tracefree parts, determined independently by the solution of the CWIs in d dimensional flat space in the momentum representation. This demonstrates that the specific analytic structure and massless poles predicted by the general curved space anomaly effective action are in fact a necessary feature of the exact solution of the anomalous CWIs in any d=4 CFT.

We present a general study of 3-point functions of conformal field theory in momentum space, following a reconstruction method for tensor correlators, based on the solution of the conformal Ward identities (CWI's), introduced in recent works by Bzowski, McFadden and Skenderis (BMS). We investigate and detail the structure of the CWI's, their non-perturbative solutions and the transition to momentum space, comparing them to perturbation theory by taking QED as an example. We then proceed with an analysis of the TJJ correlator, presenting independent and detailed re-derivations of the conformal equations in the reconstruction method of BMS, originally formulated using a minimal tensor basis in the transverse traceless sector. A careful comparison with a second basis introduced in previous studies shows that this correlator is affected by one anomaly pole in the graviton (T) line, induced by renormalization. The result shows that the origin of the anomaly, in this correlator, should be necessarily attributed to the exchange of a massless effective degree of freedom. Our results are then exemplified in massless QED at one-loop in d-dimensions, expressed in terms of perturbative master integrals. An independent analysis of the Fuchsian character of the solutions, which bypasses the 3K integrals, is also presented. We show that the combination of field theories at one-loop – with a specific field content of degenerate massless scalar and fermions – is sufficient to generate the complete non-perturbative solution, in agreement with a previous study in coordinate space. The result shows that free conformal field theories, in specific dimensions, arrested at one-loop, reproduce the general result for the TJJ. Analytical checks of this correspondence are presented in d=3,4 and 5 spacetime dimensions. This implies that the generalized 3K integrals of the BMS solution can be expressed in terms of the two single master integrals B0 and C0 of 2- and 3-point functions, with significant simplifications.

We present a study of the correlation function of three stress-energy tensors in d dimensions using free field theory realizations, and compare them to the exact solutions of their conformal Ward identities (CWI's) obtained by a general approach in momentum space. The identification of the corresponding form factors is performed within a reconstruction method, based on the identification of the transverse traceless components (Ai) of the same correlator. The solutions of the primary CWI' s are found by exploiting the universality of the Fuchsian indices of the conformal operators and a re-arrangement of the corresponding inhomogenous hypergeometric systems. We confirm the number of constants in the solution of the primary CWI's of previous analysis. In our comparison with perturbation theory, we discuss scalar, fermion and spin 1 exchanges at 1-loop in dimensional regularization. Explicit checks in d=3,4,5 prove the consistency of this correspondence. By matching the 3 constants of the CFT solution with the 3 free field theory sectors available in d=4, the general solutions of the conformal constraints is expressed just in terms of ordinary scalar 2- and 3-point functions (B0,C0). We show how the renormalized d=4 TTT vertex separates naturally into the sum of a traceless and an anomaly part, the latter determined by the anomaly functional and generated by the renormalization of the correlator in dimensional regularization. The result confirms the emergence of anomaly poles and effective massless exchanges as a specific signature of conformal anomalies in momentum space, directly connected to the renormalization of the corresponding gravitational vertices, generalizing the behaviour found for the TJJ vertex in previous works.

The conformal anomaly in curved spacetime generates a nontrivial anomalous vertex, given by the three-point correlation function TTT of the energy-momentum tensor Tμν. We show that a temperature inhomogeneity in a gas of charged massless particles generates, via the TTT vertex, a pressure anisotropy with respect to the axis of the temperature variation. This very particular signature may provide an experimental access to the elusive gravitational coefficient b which determines the anomaly contribution of the Weyl tensor to the trace of the energy-momentum tensor in curved spacetime. We present an estimate of the pressure anisotropy both for fermionic quasiparticles in the solid-state environment of Dirac semimetals as well as for a quark-gluon plasma in relativistic heavy-ion collisions. In both cases, the pressure anisotropy is small compared to the mean thermal pressure. Keywords: Conformal anomaly, Thermodynamics, Curved spacetime

We investigate the emergence of a conformal anomaly pole in conformal field theories in the case of the TJJ correlator. We show how it comes to be generated in dimensional renormalization, using a basis of 13 form factors (the F-basis), where only one of them requires renormalization (F13), extending previous studies. We then combine recent results on the structure of the non-perturbative solutions of the conformal Ward identities (CWI's) for the TJJ in momentum space, expressed in terms of a minimal set of 4 form factors (A-basis), with the properties of the F-basis, and show how the singular behaviour of the corresponding form factors in both basis can be related. The result proves the centrality of such massless effective interactions induced by the anomaly, which have recently found realization in solid state, in the theory of topological insulators and of Weyl semimetals. This pattern is confirmed in massless abelian and nonabelian theories (QED and QCD) investigated at one-loop.

We review the emergence of hypergeometric structures (of F4 Appell functions) from the conformal Ward identities (CWIs) in conformal field theories (CFTs) in dimensions d > 2. We illustrate the case of scalar 3- and 4-point functions. 3-point functions are associated to hypergeometric systems with four independent solutions. For symmetric correlators, they can be expressed in terms of a single 3K integral—functions of quadratic ratios of momenta—which is a parametric integral of three modified Bessel K functions. In the case of scalar 4-point functions, by requiring the correlator to be conformal invariant in coordinate space as well as in some dual variables (i.e., dual conformal invariant), its explicit expression is also given by a 3K integral, or as a linear combination of Appell functions which are now quartic ratios of momenta. Similar expressions have been obtained in the past in the computation of an infinite class of planar ladder (Feynman) diagrams in perturbation theory, which, however, do not share the same (dual conformal/conformal) symmetry of our solutions. We then discuss some hypergeometric functions of 3 variables, which define 8 particular solutions of the CWIs and correspond to Lauricella functions. They can also be combined in terms of 4K integral and appear in an asymptotic description of the scalar 4-point function, in special kinematical limits.