All-order splits and multi-soft limits for particle and string amplitudes

TL;DR

This paper reveals a geometric origin for new split factorizations in particle and string amplitudes, enabling all-order multi-soft limit computations and extending the understanding of amplitude structures beyond poles.

Contribution

It introduces a geometric framework for all-order splits in scattering amplitudes, generalizing factorization properties and enabling loop-level multi-soft limit calculations.

Findings

Derived a geometric origin for amplitude splits from binary curve integral geometry.

Extended split factorizations to all orders in the topological expansion.

Enabled computation of loop-integrated multi-soft limits for various amplitudes.

Abstract

The most important aspects of scattering amplitudes have long been thought to be associated with their poles. But recently a very different sort of "split" factorizations for a wide range of particle and string tree amplitudes have been discovered away from poles. In this paper, we give a simple, conceptual origin for these splits arising from natural properties of the binary geometry of the curve integral formulation for scattering amplitudes for Tr$(\Phi^3)$ theory. The most natural way of "joining" smaller surfaces to build larger ones directly produces a choice of kinematics for which higher amplitudes factor into lower ones. This gives a generalization of splits to all orders in the topological expansion. These splits allow us to access and compute loop-integrated multi-soft limits for particle and string amplitudes, at all loop orders. This includes split factorizations and multi-soft limits for pion and gluon amplitudes, that are related to Tr$(\Phi^3)$ theory by a simple kinematical shift.