Off-shell BCJ Relation in Nonlinear Sigma Model

TL;DR

This paper establishes and proves a generalized off-shell Bern-Carrasco-Johansson (BCJ) relation for tree-level currents in the nonlinear sigma model, revealing differences from Yang-Mills theory and providing a full off-shell correspondence.

Contribution

It introduces and proves a revised off-shell BCJ relation for nonlinear sigma model currents, extending the understanding of amplitude relations beyond on-shell cases.

Findings

Revised BCJ relation for even-point currents proved.

Off-shell BCJ relation differs from Yang-Mills theory.

On-shell limit recovers the known BCJ relation.

Abstract

We investigate relations among tree-level off-shell currents in nonlinear sigma model. Under Cayley parametrization, we propose and prove a general revised BCJ relation for even-point currents. Unlike the on-shell BCJ relation, the off-shell one behaves quite differently from Yang-Mills theory although the algebraic structure is the same. After performing the permutation summation in the revised BCJ relation, the sum is non-vanishing, instead, it equals to the sum of sub-current products with the BCJ coefficients under a specific ordering, which is presented by an explicit formula. Taking on-shell limit, this identity is reduced to the on-shell BCJ relation, and thus provides the full off-shell correspondence of tree-level BCJ relation in nonlinear sigma model.