Exact scheme independence at two loops

TL;DR

This paper develops a general computational framework for exact renormalization groups that confirms the scheme independence of the two-loop beta function in massless scalar field theory.

Contribution

It introduces an algorithmic approach that handles arbitrary regularization schemes and demonstrates scheme independence at two loops in scalar field theory.

Findings

Beta function computed up to two loops matches universal coefficients.

The method confirms scheme independence of results.

Effective propagators and vertices simplify calculations.

Abstract

We further develop an algorithmic and diagrammatic computational framework for very general exact renormalization groups, where the embedded regularisation scheme, parametrised by a general cutoff function and infinitely many higher point vertices, is left unspecified. Calculations proceed iteratively,by integrating by parts with respect to the effective cutoff, thus introducing effective propagators, and differentials of vertices that can be expanded using the flow equations; many cancellations occur on using the fact that the effective propagator is the inverse of the classical Wilsonian two-point vertex. We demonstrate the power of these methods by computing the beta function up to two loops in massless four dimensional scalar field theory, obtaining the expected universal coefficients, independent of the details of the regularisation scheme.