The a-function for gauge theories

TL;DR

This paper constructs the four-loop a-function for general gauge theories, providing a consistency check for three-loop beta-functions and deriving conditions for exact all-orders expressions in supersymmetric theories.

Contribution

It develops the four-loop a-function for gauge theories using lower-loop beta-functions and establishes conditions for exact all-orders a-function in N=1 supersymmetric theories.

Findings

Constructed the four-loop a-function for gauge theories.

Provided a consistency check for three-loop gauge beta-functions.

Derived conditions for exact all-orders a-function in supersymmetric theories.

Abstract

The a-function is a proposed quantity defined for quantum field theories which has a monotonic behaviour along renormalisation group flows, being related to the beta-functions via a gradient flow equation involving a positive definite metric. We construct the a-function at four loop order for a general gauge theory with fermions and scalars, using only one and two loop beta-functions; we are then able to provide a stringent consistency check on the general three-loop gauge beta-function. In the case of an N=1 supersymmetric gauge theory, we present a general condition on the chiral field anomalous dimension which guarantees an exact all-orders expression for the a-function; and we verify this up to fifth order (corresponding to the three-loop anomalous dimension).