Asymptotic freedom in the BV formalism

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.