Graviton Scattering and a Sum Rule for the c Anomaly in 4D CFT

TL;DR

This paper derives a sum rule for the c anomaly coefficient in 4D conformal field theories by analyzing graviton scattering amplitudes, establishing bounds on c based on operator product expansion data.

Contribution

It introduces a novel sum rule linking the c anomaly to OPE coefficients via graviton scattering, providing a new method to bound c in 4D CFTs.

Findings

Sum rule expresses c as a sum of positive terms.

Finiteness of correlation function limits verified for free theories.

Provides lower bounds on c from OPE coefficient bounds.

Abstract

4D CFTs have a scale anomaly characterized by the coefficient $c$, which appears as the coefficient of logarithmic terms in momentum space correlation functions of the energy-momentum tensor. By studying the CFT contribution to 4-point graviton scattering amplitudes in Minkowski space we derive a sum rule for $c$ in terms of $TT\mathcal{O}$ OPE coefficients. The sum rule can be thought of as a version of the optical theorem, and its validity depends on the existence of the massless and forward limits of the $\langle TTTT \rangle$ correlation functions that contribute. The finiteness of these limits is checked explicitly for free scalar, fermion, and vector CFTs. The sum rule gives $c$ as a sum of positive terms, and therefore implies a lower bound on $c$ given any lower bound on $TT\mathcal{O}$ OPE coefficients. We compute the coefficients to the sum rule for arbitrary operators of spin 0 and 2, including the energy-momentum tensor.