Off-shell recursion for all-loop planar integrands in Yang-Mills theory

TL;DR

This paper develops an off-shell recursion method for all-loop planar Yang-Mills integrands, clarifying their structure and including ghost contributions, with a focus on 2-loop recursion strategies.

Contribution

It introduces a matrix formalism for pure gluon integrands and extends the recursion to include ghost contributions, enhancing understanding of higher-loop amplitude relations.

Findings

Matrix formalism clarifies off-shell structure of integrands.

Recursion method includes ghost contributions.

Special case analysis for 2-loop integrand recursion.

Abstract

In this letter, we focus on the application of the off-shell recursion method proposed in \cite{Tao:2025fch} in the Yang-Mills planar loop integrands, which starts with solving the classical equation of motion via the perturbiner method. Following the recursion steps, we point out that the pure gluon sector of the planar loop integrands can be written in matrix formalism. This matrix formalism not only makes the off-shell structure of the Yang-Mills planar integrands clearer, but also has potential use in finding amplitude relations at higher-loop levels. Furthermore, we add the ghost contribution and write down the whole recursion step of the Yang-Mills planar loop integrands with ghost contributions. Finally, we consider the 2-loop planar integrand recursion as a special case and conclude a recursion strategy in this case.