We illustrate how the conformal Ward identities (CWI) and the gravitational chiral anomaly completely determine the structure of the <math><mo>⟨</mo><mi>T</mi><mi>T</mi><msub><mi>J</mi><mn>5</mn></msub><mo>⟩</mo></math> (graviton-graviton-chiral gauge current) correlator in momentum space. This analysis extends our previous results on the anomaly vertices <math><mo>⟨</mo><mi>A</mi><mi>V</mi><mi>V</mi><mo>⟩</mo></math> and <math><mo>⟨</mo><mi>A</mi><mi>A</mi><mi>A</mi><mo>⟩</mo></math>, as well as the <math><mo>⟨</mo><mi>T</mi><mi>J</mi><mi>J</mi><mo>⟩</mo></math> parity-odd conformal anomaly vertex in general CFTs. The <math><mo>⟨</mo><mi>T</mi><mi>T</mi><msub><mi>J</mi><mn>5</mn></msub><mo>⟩</mo></math> plays a fundamental role in the analysis of the conformal backreaction in early Universe cosmology, affecting the particle content and the evolution of the primordial plasma. Our approach is nonperturbative and not Lagrangian based, requiring the inclusion of a single anomaly pole in the solution of the anomaly constraint. The pole and its residue, along with the CWIs, determine the entire correlator in all of its sectors (longitudinal/transverse), all of which are proportional to the same anomaly coefficient. The method does not rely on a specific expression of the <math><mi>C</mi><mi>P</mi></math>-odd anomalous current, which in free field theory can be represented either by a bilinear fermion current or by a gauge-dependent Chern-Simons current; it relies solely on the symmetry constraints. We compute the correlator perturbatively at one loop in free field theory and verify its exact agreement with the nonperturbative result. A comparison with the perturbative analysis confirms the presence of a sum rule satisfied by the correlator, similar to the parity-even <math><mo>⟨</mo><mi>T</mi><mi>J</mi><mi>J</mi><mo>⟩</mo></math> and the chiral <math><mo>⟨</mo><mi>A</mi><mi>V</mi><mi>V</mi><mo>⟩</mo></math>.

Italy

Claudio Corianò

Stefano Lionetti

Germany

Matteo Maria Maglio

Parity-violating CFT and the gravitational chiral anomaly

We derive the conformal constraints satisfied by classical vertices of a (Einstein) Gauss-Bonnet theory around flat space, in general dimensions and at <math altimg='si1.svg'><mi>d</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>4</mn></math> (4d EGB). In 4 <italic>d</italic> EGB they are obtained by a singular limit of the integral of the Euler-Poincar&#232; 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 <math altimg='si2.svg'><mo stretchy='false'>(</mo><mi>d</mi><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>4</mn><mo stretchy='false'>)</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math> correction, before performing the singular <math altimg='si3.svg'><mi>d</mi><mo stretchy='false'>&#8594;</mo><mn>4</mn></math> 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 <math altimg='si4.svg'><mi>d</mi><mo>&#8800;</mo><mn>4</mn></math> these constraints hold also for Lovelock actions. They can be extended to higher order topological invariants in such class of theories.

Mario Cretì

Three-wave and four-wave interactions in the 4 <italic>d</italic> Einstein Gauss-Bonnet (EGB) and Lovelock theories

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 <math><mi>ϵ</mi><mo>=</mo><mi>d</mi><mo>−</mo><mn>4</mn><mo>→</mo><mn>0</mn></math> limit is accompanied by the dimensional reduction (DRed) of the field dependencies. We show that operatorial nonlocal modifications <math><mo>(</mo><mo>∼</mo><msup><mrow><mo>□</mo></mrow><mrow><mi>ϵ</mi></mrow></msup><mo>)</mo></math> 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 <math><mi>d</mi><mo>=</mo><mn>4</mn></math> (cutoff) scheme. The interplay between two renormalization schemes, one based on dimensional regularization (DR) and the second on a cutoff in <math><mi>d</mi><mo>=</mo><mn>4</mn></math>, 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 <math><mi>d</mi><mo>=</mo><mn>4</mn></math>.

Broken scale invariance and the regularization of a conformal sector in gravity with Wess-Zumino actions

Recently, the possibility of evading Lovelock's theorem at <math><mi>d</mi><mo>=</mo><mn>4</mn></math>, 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 <math><mi>d</mi><mo>=</mo><mn>4</mn><mo>+</mo><mi>ϵ</mi></math>, 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.

Einstein Gauss-Bonnet theories as ordinary, Wess-Zumino conformal anomaly actions

The conformal anomaly in curved spacetime generates a nontrivial anomalous vertex, given by the three-point correlation function TTT of the energy-momentum tensor <math><msup><mrow><mi>T</mi></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msup></math>. 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.

Russia

France

M.N. Chernodub

Anomalous gravitational TTT vertex, temperature inhomogeneity, and pressure anisotropy

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 <math><mo>〈</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><mo>〉</mo></math> in <math><mi>d</mi><mo>=</mo><mn>4</mn></math> curved space. This action implies the existence of massless propagator poles in three and higher point correlators of <math><msup><mrow><mi>T</mi></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msup></math>. 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 <math><mo>〈</mo><mi>T</mi><mi>T</mi><mi>T</mi><mo>〉</mo></math> 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 <math><mi>d</mi><mo>=</mo><mn>4</mn></math> CFT.

United States

Emil Mottola

TTT in CFT: Trace identities and the conformal anomaly effective action

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 <math><mi>d</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn></math> 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 <math><msub><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msub></math> and <math><msub><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msub></math> of 2- and 3-point functions, with significant simplifications.

Exact correlators from conformal Ward identities in momentum space and the perturbative Vertex

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 <math><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math> 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 <math><mi>d</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn></math> prove the consistency of this correspondence. By matching the 3 constants of the CFT solution with the 3 free field theory sectors available in <math><mi>d</mi><mo>=</mo><mn>4</mn></math>, the general solutions of the conformal constraints is expressed just in terms of ordinary scalar 2- and 3-point functions <math><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math>. We show how the renormalized <math><mi>d</mi><mo>=</mo><mn>4</mn></math> 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 general 3-graviton vertex ( ) of Conformal Field Theories in Momentum Space in = 4

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 <math><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>13</mn></mrow></msub><mo>)</mo></math>, 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.