Regularization of energy-momentum tensor correlators and parity-odd terms

TL;DR

This work develops a momentum-space framework for regularizing energy-momentum tensor correlators in conformal field theories, using differential and dimensional regularization within Feynman-diagram techniques. It presents a thorough comparison between 2d and 4d cases for 2-point functions, and a detailed treatment of the parity-odd part of the 3-point function in 4d, uncovering a covariant Pontryagin-density trace anomaly that coexists with a diffeomorphism anomaly. The authors show that regularization can introduce contact terms and that covariance requires carefully chosen counterterms, with the surprising outcome that parity-odd effects in the 3-point function are subtle and demand separate treatment of tensor components. Overall, the paper clarifies how regularization interfaces with diffeomorphism and Weyl symmetries, and how parity-odd anomalies arise in gravity-CFT contexts, notably via the Pontryagin density.

Abstract

We discuss the problem of regularizing correlators in conformal field theories. The only way to do it in coordinate space is to interpret them as distributions. Unfortunately except for the simplest cases we do not have tabulated mathematical results. The way out we pursue here is to go to momentum space and use Feynman diagram techniques and their regularization methods. We focus on the energy-momentum tensor correlators and, to gain insight, we compute and regularize 2-point functions in 2d with various techniques both in coordinate space and in momentum space, obtaining the same results. Then we do the same for 2-point functions in 4d. Finally we turn to 3-point function in 4d, and concentrate on the parity-odd part. We derive in particular the regularized trace and divergence of the energy-momentum tensor in a chiral fermion model. We discuss the problems related to the parity-odd trace anomaly.

Figures (1)

• Figure 1: The relevant Feynman diagram for the computation.