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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 theintegral of the Euler-Poincar\`e density. Our analysis exploits the relationbetween this theory and the conformal anomaly action, which allows to uncoversome interesting features of the GB vertex at cubic and quartic level. If weintroduce a conformal decomposition of the metric, the resulting theory can beformulated in two different versions, which are regularization dependent, alocal one which is quartic in the dilaton field, and a nonlocal one, with aquadratic dilaton. The nonlocal version is derived by a finite redefinition ofthe GB density with the inclusion of a $(d-4)R^2$ correction, before performingthe singular $d\to 4$ limit. In the local version of the theory, we show howthe independent dynamics of the metric and of the dilaton are interwinded by aclassical trace identity. Three-gravitational wave interactions can beorganised in a nontrivial way by using directly the nonlocal 4d EGB version ofthe 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 theconformal anomaly action, as defined by a perturbative expansion. For $d\neq 4$these constraints hold also for Lovelock actions. They can be extended tohigher order topological invariants in such class of theories.

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