Recursion Relations for Tree-level Amplitudes in the SU(N) Non-linear Sigma Model

TL;DR

The paper develops on-shell recursion relations for tree-level Goldstone-boson amplitudes in the $SU(N)$ non-linear sigma model, an effective field theory with infinitely many interaction terms. By introducing a semi-on-shell current and proving its favorable scaling under a tailored all-line shift, the authors adapt BCFW-like methods to this EFT. They derive a closed recursion that expresses $J_{2n+1}$ from lower-point data and extract on-shell amplitudes from the appropriate LSZ limit, with the four-point current already included in the recursion. This work demonstrates that on-shell techniques can extend to certain non-renormalizable theories and suggests avenues for loop extensions and purely on-shell reformulations.

Abstract

It is well-known that the standard BCFW construction cannot be used for on-shell amplitudes in effective field theories due to bad behavior for large shifts. We show how to solve this problem in the case of the SU(N) non-linear sigma model, i.e. non-renormalizable model with infinite number of interaction vertices, using scaling properties of the semi-on-shell currents, and we present new on-shell recursion relations for all on-shell tree-level amplitudes in this theory.