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Four-point functions of gravitons and conserved currents of CFT in momentum space: testing the nonlocal action with the TTJJ

Claudio Corian\`oMatteo Maria MaglioRiccardo Tommasi
Dec 2022
We present an analysis of the perturbative realization of the $TTJJ$correlator, with two stress energy tensors and two conserved currents, usingfree field theory realizations, integrating out conformal sectors in thequantum corrections. This allows defining, around flat space, an exactperturbative expansion of the complete anomaly effective action - up to 4-pointfunctions - whose predictions can be compared against those of the anomalyinduced action. The latter is a variational solution of the conformal anomalyconstraint at $d=4$ in the form of a nonlocal Wess-Zumino action. Therenormalization procedure and the degeneracies of the tensor structures of thiscorrelator are discussed, valid for a generic conformal field theory, derivingits anomalous trace Ward identities (WIs). In this application, we alsoillustrate a general procedure that identifies the minimal number of tensorialstructures and corresponding form factors for the $TTJJ$ and any $4$-pointfunction. The result of the direct computation is compared against theexpression of the same 4-point function derived from the nonlocal anomalyinduced action. We show that the prediction for the anomalous part of the$TTJJ$ derived from such action, evaluated in two different conformaldecompositions, the Riegert and Fradkin-Vilkovisky (FV) choices, differ fromthe anomaly part identified in the perturbative $TTJJ$, in the flat spacetimelimit. The anomaly part of the correlator computed with the Riegert choice isaffected by double poles, while the one computed with the FV choice does notsatisfy the conservation WIs. We present the correct form of the expansion ofthe anomaly induced action at the second order in the metric perturbationsaround flat space that reproduces the perturbative result.