World-line approach to the Bern-Kosower formalism in two-loop Yang-Mills theory

TL;DR

This paper develops a world-line formalism approach to derive two-loop Yang-Mills effective actions, simplifying the computation of gluon scattering amplitudes and encapsulating complex interactions in a unified framework.

Contribution

It introduces a novel world-line sewing method to derive two-loop Yang-Mills effective actions, simplifying the inclusion of four-gluon interactions into three-gluon vertices.

Findings

Derived a master formula for two-loop gluon scattering amplitudes.

Unified four-gluon and three-gluon interactions in a simple action.

Extended the formalism to ghost and fermion loops in an $su(2)$ background.

Abstract

Based on the world-line formalism with a sewing method, we derive the Yang-Mills effective action in a form useful to generate the Bern-Kosower-type master formulae for gluon scattering amplitudes at the two-loop level. It is shown that four-gluon ($\Phi^4$ type sewing) contributions can be encapsulated in the action with three-gluon ($\Phi^3$ type) vertices only, the total action thus becoming a simple expression. We then derive a general formula for a two-loop Euler-Heisenberg type action in a pseudo-abelian $su(2)$ background. The ghost loop and fermion loop cases are also studied.