Asymptotic freedom in the BV formalism

Abstract

We define the β-function of a perturbative quantum field theory in the mathematical framework introduced by Costello – combining perturbative renormalization and the BV formalism – as the cohomology class of a certain functional measuring scale dependence of the effective interaction. We show that the one-loop β-function is a well-defined element of the obstruction–deformation complex for translation-invariant and classically scale-invariant theories, and furthermore that it is locally constant as a function on the space of classical interactions and computable as a rescaling anomaly, or as the logarithmic one-loop counterterm. We compute the one-loop β-function in first-order Yang–Mills theory, recovering the famous asymptotic freedom for Yang–Mills in a mathematical context.