Exact Consequences of the Trace Anomaly in Four Dimensions

TL;DR

This paper derives the exact form of the stress-tensor three-point function in four dimensions, exploring its properties, relation to trace anomalies, and implications for renormalization group flows, highlighting parallels and differences with two-dimensional theories.

Contribution

It provides the explicit form of the three-point function in four dimensions and relates anomaly coefficients to finite amplitudes, offering new insights into trace anomalies and RG flow sum rules.

Findings

Relation of anomaly coefficients to finite amplitudes

Sum rules for RG flow of a and c

Discussion of scheme independence and properties of correlators

Abstract

The general form of the stress-tensor three-point function in four dimensions is obtained by solving the Ward identities for the diffeomorphism and Weyl symmetries. Several properties of this correlator are discussed, such as the renormalization and scheme independence and the analogies with the anomalous chiral triangle. At the critical point, the coefficients a and c of the four-dimensional trace anomaly are related to two finite, scheme-independent amplitudes of the three-point function. Off-criticality, the imaginary parts of these amplitudes satisfy sum rules which express the total renormalization-group flow of a and c between pairs of critical points. Although these sum rules are similar to that satisfied by the two-dimensional central charge, the monotonicity of the flow, i.e. the four-dimensional analogue of the c-theorem, remains to be proven.