Graviton Vertices and the Mapping of Anomalous Correlators to Momentum Space for a General Conformal Field Theory

TL;DR

This paper maps conformal correlators and their anomalies from position to momentum space in general dimensions, computing key vertex functions for free fields and clarifying the structure of anomalous correlators in conformal field theories.

Contribution

It provides a detailed mapping of anomalous conformal correlators from position to momentum space, including explicit computations for free fields and identification of tensor structures in various dimensions.

Findings

Explicit expressions for correlators in configuration space that are Fourier integrable.

Identification of tensor structures solving conformal Ward identities in momentum space.

Clarification of how anomalies manifest in different approaches to conformal correlators.

Abstract

We investigate the mapping of conformal correlators and of their anomalies from configuration to momentum space for general dimensions, focusing on the anomalous correlators $TOO$, $TVV$ - involving the energy-momentum tensor $(T)$ with a vector $(V)$ or a scalar operator ($O$) - and the 3-graviton vertex $TTT$. We compute the $TOO$, $TVV$ and $TTT$ one-loop vertex functions in dimensional regularization for free field theories involving conformal scalar, fermion and vector fields. Since there are only one or two independent tensor structures solving all the conformal Ward identities for the $TOO$ or $TVV$ vertex functions respectively, and three independent tensor structures for the $TTT$ vertex, and the coefficients of these tensors are known for free fields, it is possible to identify the corresponding tensors in momentum space from the computation of the correlators for free fields. This works in general $d$ dimensions for $TOO$ and $TVV$ correlators, but only in 4 dimensions for $TTT$, since vector fields are conformal only in $d=4$. In this way the general solution of the Ward identities including anomalous ones for these correlators in (Euclidean) position space, found by Osborn and Petkou is mapped to the ordinary diagrammatic one in momentum space. We give simplified expressions of all these correlators in configuration space which are explicitly Fourier integrable and provide a diagrammatic interpretation of all the contact terms arising when two or more of the points coincide. We discuss how the anomalies arise in each approach [...]