On the $a$-theorem in the Conformal Window

TL;DR

This paper presents a new method to compute the $a$-function in four-dimensional gauge theories within the conformal window using 2-point functions, facilitating lattice simulations and confirming the strong $a$-theorem.

Contribution

It derives an expression for the $a$-function based on 2- and 3-point functions of the energy-momentum tensor trace, establishing the strong $a$-theorem for these theories.

Findings

The $a$-function can be expressed as an integral over 2- and 3-point functions.

A scheme is identified where the 3-point contribution vanishes.

The strong $a$-theorem is explicitly established for the class of theories studied.

Abstract

We show that for four dimensional gauge theories in the conformal window, the Euler anomaly, known as the $a$-function, can be computed from a $2$-point function of the trace of the energy momentum tensor making it more amenable to lattice simulations. Concretely, we derive an expression for the $a$-function as an integral over the renormalisation scale of quantities related to $2$- and $3$-point functions of the trace of the energy momentum tensor.The crucial ingredients are that the square of the field strength tensor is an exactly marginal operator at the Gaussian fixed point and that the relevant $3$-point correlation function is finite when resummed to all orders. This allows us to define a scheme for which the $3$-point contribution vanishes, thereby explicitly establishing the strong version of the $a$-theorem for this class of theories.