Algebraic branch points at all loop orders from positive kinematics and wall crossing

TL;DR

This paper explores how algebraic branch points in planar $ ext{N}=4$ SYM amplitudes emerge from boundary structures using wall crossing and scattering diagrams, proposing a new coordinate system and conjecturing all algebraic letters for 8-point amplitudes.

Contribution

It introduces a systematic approach using wall crossing and scattering diagrams to understand algebraic branch points and proposes a new coordinate system for kinematic space.

Findings

Asymptotic chambers explain algebraic branch points.

A new coordinate system rationalizes algebraic letter relations.

Conjecture of all algebraic letters for 8-point MHV amplitude.

Abstract

There is a remarkable connection between the boundary structure of the positive kinematic region and branch points of integrated amplitudes in planar $\mathcal{N}=4$ SYM. A long standing question has been precisely how algebraic branch points emerge from this picture. We use wall crossing and scattering diagrams to systematically study the boundary structure of the positive kinematic regions associated with MHV amplitudes. The notion of asymptotic chambers in the scattering diagram naturally explains the appearance of algebraic branch points. Furthermore, the scattering diagram construction also motivates a new coordinate system for kinematic space that rationalizes the relations between algebraic letters in the symbol alphabet. As a direct application, we conjecture a complete list of all algebraic letters that could appear in the symbol alphabet of the 8-point MHV amplitude.