On Conformal Field Theories With Extremal a/c Values

TL;DR

This paper investigates energy correlators in four-dimensional unitary conformal field theories, revealing that extremal a/c ratios correspond to trivial scattering and fixed energy correlator structures, thus constraining the space of such theories.

Contribution

It demonstrates that extremal a/c ratios in 4D CFTs lead to fixed energy correlator forms and trivial scattering, providing new constraints on the structure of these theories.

Findings

Extremal a/c ratios correspond to free theories.

Energy correlators are fixed for extremal ratios.

Three-point stress tensor functions are constrained.

Abstract

Unitary conformal field theories (CFTs) are believed to have positive (non-negative) energy correlators. Energy correlators are universal observables in higher-dimensional CFTs built out of integrated Wightman functions of the stress-energy tensor. We analyze energy correlators in parity invariant four-dimensional CFTs. The goal is to use the positivity of energy correlators to further constrain unitary CFTs. It is known that the positivity of the simplest one-point energy correlator implies that 1/3 <= a/c <= 31/18 where a and c are the Weyl anomaly coefficients. We use the positivity of higher point energy correlators to show that CFTs with extremal values of a/c have trivial scattering observables. More precisely, for a/c=1/3 and a/c=31/18 all energy correlators are fixed to be the ones of the free boson and the free vector theory correspondingly. Similarly, we show that the positivity and finiteness of energy correlators together imply that the three-point function of the stress tensor in a CFT cannot be proportional to the one in the theory of free boson, free fermion or free vector field.