Three-Wave and Four-Wave Interactions in the $4d$ Einstein Gauss-Bonnet (EGB) and Lovelock Theories

TL;DR

This paper investigates the conformal properties and interaction vertices of 4D Einstein-Gauss-Bonnet and Lovelock theories, revealing regularization-dependent formulations and hierarchical Ward identities for gravitational interactions.

Contribution

It introduces a novel analysis of conformal constraints and interaction vertices in 4D EGB and Lovelock theories, including local and nonlocal formulations and their relation to the conformal anomaly.

Findings

Derived conformal constraints for classical vertices in 4D EGB.

Established local and nonlocal formulations of the theory with different dilaton dynamics.

Identified hierarchical Ward identities governing gravitational interactions.

Abstract

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-Poincar\`e 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 $(d-4)R^2$ correction, before performing the singular $d\to 4$ 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 $d\neq 4$ these constraints hold also for Lovelock actions. They can be extended to higher order topological invariants in such class of theories.