The One-Loop QCD $β$-Function as an Index

TL;DR

This work recasts the one-loop QCD β-function as an index-theoretic, twistor-space computation, linking RG flow to the anomaly of scale invariance in a holomorphic (twistor) setting. By evaluating determinant lines over families of twistor spaces and GH backgrounds via Grothendieck–Riemann–Roch, the authors recover the standard one-loop β-function coefficient and express the anomaly data in terms of geometric invariants such as c1(L) and the Pontryagin class p. The approach also yields a twistorial account of the Weyl anomaly through the difference a−c, with explicit index-theoretic checks on GH spaces and a discussion of higher-loop implications and independent recovery of a and c. The results illuminate a deep connection between algebro-geometric structures in twistor theory and perturbative QCD data, suggesting a unified twistorial framework for quantum anomalies. Practical impact lies in offering a conceptually transparent, geometry-driven route to fundamental RG data and gravitational anomalies in gauge theories.

Abstract

In this letter we show that the one-loop QCD $β$-function can be obtained from an index theorem on twistor space. This is achieved by recalling that the $θ$-angle of self-dual gauge theory flows according the one-loop $β$-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.