Note on Identities Inspired by New Soft Theorems

TL;DR

This work investigates identities arising from new soft theorems in gravity and gauge theories. It proves a nontrivial half-soft function identity using the CHY formulation and provides a simpler byproduct identity through an inductive BCFW-based argument, clarifying determinant representations via the $\psi$-matrix. For the KLT momentum kernel, the authors derive a first identity from CHY orthogonality and BCJ-basis transformations, and discuss a second identity cast as an angular-momentum conservation constraint whose general proof remains open. Together, these results deepen the connection between soft theorems, CHY representations, and BCJ/KLT structures, and point to promising directions for dimension-generalized proofs and further identities in scattering amplitudes.

Abstract

The new soft theorems, for both gravity and gauge amplitudes, have inspired a number of works, including the discovery of new identities related to amplitudes. In this note, we present the proof and discussion for two sets of identities. The first set includes an identity involving the half-soft function which had been used in the soft theorem for one-loop rational gravity amplitudes, and another simpler identity as its byproduct. The second set includes two identities involving the KLT momentum kernel, as the consistency conditions of the KLT relation plus soft theorems for both gravity and gauge amplitudes. We use the CHY formulation to prove the first identity, and transform the second one into a convenient form for future discussion.