Algebraic singularities of scattering amplitudes from tropical geometry

TL;DR

This paper explores how tropical geometry and cluster algebras can identify algebraic singularities and finite symbol alphabets in scattering amplitudes, including those with square roots, extending beyond six and seven points.

Contribution

It introduces a tropical geometry framework to predict finite symbol alphabets and square root letters in scattering amplitudes, connecting algebraic singularities to geometric structures.

Findings

Tropical fans generate finite symbol alphabets for scattering amplitudes.

The minimal fan captures all square root letters in a two-loop eight-point NMHV calculation.

Connections between cluster algebras and tropical geometry provide natural language for amplitude singularities.

Abstract

We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $\mathcal{N}=4$ super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.