Recursion Relations for One-Loop Goldstone Boson Amplitudes

TL;DR

This paper develops recursion relations for one-loop planar integrands in the SU(N) non-linear sigma model, extending amplitude methods to effective field theories with unique soft limits.

Contribution

It introduces a novel approach to construct one-loop integrands using soft limits and unitarity cuts, advancing amplitude techniques for effective field theories.

Findings

Recursion relations for one-loop planar integrands are derived.

Additional constraints on soft behavior are necessary for fixing integrands.

This work extends modern amplitude methods to effective field theories.

Abstract

In this letter, we construct the recursion relations for one-loop planar integrands in the SU(N) non-linear sigma model. This generalizes the soft recursions for tree-level amplitudes in a variety of quantum field theories with special soft limits. The main ingredient is the definition of the one-loop planar integrand, which is fixed by cuts in the sense of generalized unitarity and by requiring the Adler zero on all external legs. We show that this does not uniquely fix the integrand, so additional constraints on the soft behavior of the loop momentum have to be imposed. Our work is the first step in extending modern amplitudes methods for loop amplitudes to effective field theories with special soft limits.