The One-Loop QCD $\beta$-Function as an Index

TL;DR

This paper demonstrates that the one-loop QCD beta function can be derived using an index theorem on twistor space, linking gauge theory anomalies to geometric structures.

Contribution

It introduces a novel geometric approach to compute the one-loop QCD beta function via an index theorem on twistor space.

Findings

Beta function derived from twistor space index theorem

Connection between gauge theory anomalies and geometric structures

Holomorphic reformulation of self-dual gauge theory

Abstract

In this letter we show that the one-loop QCD $\beta$-function can be obtained from an index theorem on twistor space. This is achieved by recalling that the $\theta$-angle of self-dual gauge theory flows according the one-loop $\beta$-function. Rewriting self-dual gauge theory as a holomorphic theory on twistor space this flow can be computed as the anomaly to scale invariance. The one-loop Weyl anomaly coefficient $a-c$ can be recovered similarly.