An Algebraic Criterion for the Ultraviolet Finiteness of Quantum Field Theories

TL;DR

This paper introduces an algebraic criterion based on descent equations to determine when the beta function of certain quantum field theories vanishes, establishing a nonrenormalization theorem that applies to supersymmetric Yang-Mills theories.

Contribution

It presents a novel algebraic approach using descent equations to prove the vanishing of the beta function in renormalizable quantum field theories, extending nonrenormalization theorems.

Findings

Nonrenormalization theorem for the beta function $eta_g$

Conditions under which $eta_g$ vanishes to all orders

Application to N=2,4 supersymmetric Yang-Mills theories

Abstract

An algebraic criterion for the vanishing of the beta function for renormalizable quantum field theories is presented. Use is made of the descent equations following from the Wess-Zumino consistency condition. In some cases, these equations relate the fully quantized action to a local gauge invariant polynomial. The vanishing of the anomalous dimension of this polynomial enables us to establish a nonrenormalization theorem for the beta function $\beta_g$, stating that if the one-loop order contribution vanishes, then $\beta_g$ will vanish to all orders of perturbation theory. As a by-product, the special case in which $\beta_g$ is only of one-loop order, without further corrections, is also covered. The examples of the N=2,4 supersymmetric Yang-Mills theories are worked out in detail.