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

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.