# Asymptotic Freedom in the BV Formalism

## TL;DR

This paper rigorously defines the beta-function within the BV formalism using Costello's framework, demonstrating its properties and computing it for Yang--Mills theory, thereby mathematically confirming asymptotic freedom.

## Contribution

It introduces a cohomological definition of the beta-function in the BV formalism and computes it explicitly for Yang--Mills theory, establishing a rigorous mathematical foundation for asymptotic freedom.

## Key findings

- The one-loop beta-function is a well-defined local deformation class.
- The beta-function is locally constant on the space of classical interactions.
- The computed beta-function confirms asymptotic freedom in Yang--Mills theory.

## Abstract

We define the beta-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 element in the obstruction-deformation complex. We show that the one-loop beta-function is a well-defined element of the local 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 beta-function in first-order Yang--Mills theory, recovering the famous asymptotic freedom for Yang--Mills in a mathematical context.