# Scattering Equations and Factorization of Amplitudes II: Effective Field Theories

## TL;DR

This paper extends the scattering equation framework to effective field theories using a double-cover formalism, introducing new factorization relations and a recursion method to compute amplitudes algebraically without solving scattering equations.

## Contribution

It develops a double-cover formalism for effective field theories, especially the non-linear sigma model, with new factorization relations and a recursion relation for amplitude calculation.

## Key findings

- Derived new factorization relations for effective field theories.
- Established a recursion relation expressing amplitudes via off-shell three-point functions.
- Provided algebraic amplitude expressions without solving scattering equations.

## Abstract

We continue the program of extending the scattering equation framework by Cachazo, He and Yuan to a double-cover prescription. We discuss how to apply the double-cover formalism to effective field theories, with a special focus on the non-linear sigma model. A defining characteristic of the double-cover formulation is the emergence of new factorization relations. We present several factorization relations, along with a novel recursion relation. Using the recursion relation and a new prescription for the integrand, any non-linear sigma model amplitude can be expressed in terms of off-shell three-point amplitudes. The resulting expression is purely algebraic, and we do not have to solve any scattering equation. We also discuss soft limits, boundary terms in BCFW recursion, and application of the double-cover prescription to other effective field theories, like the special Galileon theory.