Gravitational Axial Anomaly for Four Dimensional Conformal Field Theories

TL;DR

This paper constructs and analyzes a three-point function involving axial currents and energy-momentum tensors in four-dimensional conformal field theories, revealing how gravitational axial anomalies influence these correlators.

Contribution

It provides a unique determination of the three-point function form under conformal symmetry and explicitly relates its coefficient to the gravitational axial anomaly scale.

Findings

Derived the three-point function form using conformal symmetry and conservation laws.

Calculated the anomaly contribution directly using differential regularisation.

Validated results with fermion and photon axial current examples.

Abstract

We construct the three point function involving an axial vector current and two energy-momentum tensors for four dimensional conformal field theories. Conformal symmetry determines the form of this three point function uniquely up to a constant factor if the necessary conservation conditions are imposed. The gravitational axial anomaly present on a curved space background leads to a non-zero contribution for the divergence of the axial current in this three point function even on flat space. Using techniques related to differential regularisation which guarantee that the energy-momentum tensor is conserved and traceless, we calculate the anomaly in the three point function directly. In this way we relate the overall coefficient of the three point function to the scale of the gravitational axial anomaly. We check our results by applying them to the examples of the fermion and photon axial currents.