Do we need Feynman diagrams for higher orders perturbation theory?

TL;DR

This paper presents an analytical computation of two and three loop corrections to the beta function in Yang-Mills theories without using Feynman diagrams, employing the background gauge field method.

Contribution

It introduces a novel diagram-free approach to higher-order perturbative calculations in gauge theories using the background field method.

Findings

Calculated two and three loop beta function corrections analytically.

Demonstrated the feasibility of diagram-free higher-order computations.

Provided insights into the structure of quantum corrections in Yang-Mills theories.

Abstract

We compute the two and three loop corrections to the beta function for Yang-Mills theories in the background gauge field method and using the background gauge field as the only source. The calculations are based on the separation of the one loop effective potential into zero and positive modes and are entirely analytical. No two or three loop Feynman diagrams are considered in the process.