An Algebraic Criterion for the Ultraviolet Finiteness of Quantum Field Theories

TL;DR

The paper develops an algebraic, BRST-cohomology based criterion for ultraviolet finiteness in renormalizable quantum field theories by exploiting descent equations from the Wess-Zumino consistency condition. A key result is a nonrenormalization theorem: if the one-loop beta function coefficient vanishes, then the beta function vanishes to all orders, with the bottom cocycle in the descent chain controlling the-perturbative corrections. This framework is illustrated through Chern-Simons coupled to matter, N=2 SYM, and N=4 SYM, where either full all-orders finiteness or one-loop exactness is established via vanishing anomalous dimensions of specific gauge-invariant insertions. The approach provides a systematic, algebraic method to identify finite theories by examining BRST cohomology and the associated descent structure, offering a parallel to Adler-Bardeen type nonrenormalization results and extendable to multiple couplings.

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 $β_g$, stating that if the one-loop order contribution vanishes, then $β_g$ will vanish to all orders of perturbation theory. As a by-product, the special case in which $β_g$ 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.