Note on graph-based BCJ relation for Berends-Giele currents

TL;DR

This paper systematically proves a graph-based BCJ relation for Berends-Giele currents in bi-adjoint scalar and Yang-Mills theories, using recursive and graph-theoretic methods, extending previous proposals.

Contribution

It provides a rigorous proof of the graph-based BCJ relation for arbitrary tree graphs and extends it to Yang-Mills theory via off-shell numerators.

Findings

Proved BCJ relations for simple chains and star graphs.

Established general BCJ relations for arbitrary tree graphs.

Connected relations to Yang-Mills amplitudes through on-shell limits.

Abstract

Graph-based Bern-Carasso-Johansson (BCJ) relation for Berends-Giele currents in bi-adjoint scalar (BS) theory, which is characterized by connected tree graphs, was proposed in an earlier work. In this note, we provide a systematic study of this relation. We first prove the relations based on two special types of graphs: simple chains and star graphs. The general graph-based BCJ relation established by an arbitrary tree graph is further proved, through Berends-Giele recursion. When combined with proper off-shell extended numerators, this relation induces the graph-based BCJ relation for Berends-Giele currents in Yang-Mills theory. The corresponding relations for amplitudes are obtained via on-shell limits.