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 Rsquare ^{-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.
Highlights
The analysis of multi-point correlation functions in conformal field theory (CFT) in higher dimensions (D > 2) plays an important role in the attempt to understand such symmetry at quantum level
If we require that the metric background allows conformal Killing vectors (CKVs) and the effective action is invariant under such transformations, we have the possibility to take into account the anomaly contribution to the equations
In this work we have shown that a previous analysis of the 3-point correlator (3T) vertex, in which some special features of the anomaly action, emerging at O(h3) in the gravitational fluctuations, had been identified, can be extended to the 4-graviton vertex 4T
Summary
The analysis of multi-point correlation functions in conformal field theory (CFT) in higher dimensions (D > 2) plays an important role in the attempt to understand such symmetry at quantum level. Examples are the classification of the minimal number of form factors present in their expansions in the external momenta, the identification of the arbitrary functions which appear in the solution of the corresponding CWIs for n > 3, [11,12,13,14], or the search for exact solutions in the presence of dual conformal symmetry [15] Among these correlators, an important role is played by those involving the stress energy tensor, due to the appearance of a conformal (trace) anomaly in even spacetime dimensions (see [16] for a general discussion).
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