Gauge invariance induced relations and the equivalence between distinct approaches to NLSM amplitudes

TL;DR

This paper establishes gauge invariance relations for NLSM and related theories, demonstrating the equivalence of three different methods for computing NLSM amplitudes through generalized BCJ relations.

Contribution

It derives generalized BCJ relations for multiple theories and proves the equivalence of three approaches to NLSM amplitudes using gauge invariance.

Findings

Gauge invariance relations hold across several theories.

Different approaches to NLSM amplitudes yield identical results.

Unified understanding of NLSM amplitude calculations.

Abstract

In this paper, we derive generalized Bern-Carrasco-Johansson relations for color-ordered Yang-Mills amplitudes by imposing gauge invariance conditions and dimensional reduction appropriately on the new discovered graphic expansion of Einstein-Yang-Mills amplitudes. These relations are also satisfied by color-ordered amplitudes in other theories such as color-scalar theory, bi-scalar theory and nonlinear sigma model (NLSM). As an application of the gauge invariance induced relations, we further prove that the three types of BCJ numerators in NLSM , which are derived from Feynman rules, Abelian Z-theory and Cachazo-He- Yuan formula respectively, produce the same total amplitudes. In other words, the three distinct approaches to NLSM amplitudes are equivalent to each other.