Cluster algebras for Feynman integrals

TL;DR

This paper explores the connection between cluster algebras and Feynman integrals, revealing how algebraic structures can restrict the function space in scattering amplitude calculations.

Contribution

It demonstrates that certain Feynman integrals are described by specific cluster algebras and relates these to known physical constraints like the Steinmann relations.

Findings

Four-point Feynman integrals with one off-shell leg are described by a C2 cluster algebra.

Cluster adjacency relations restrict the allowed function space for scattering amplitudes.

Identification of one-loop alphabets with cluster algebras for multi-particle scattering.

Abstract

We initiate the study of cluster algebras in Feynman integrals in dimensional regularization. We provide evidence that four-point Feynman integrals with one off-shell leg are described by a $C_{2}$ cluster algebra, and we find cluster adjacency relations that restrict the allowed function space. By embedding $C_{2}$ inside the $A_3$ cluster algebra, we identify these adjacencies with the extended Steinmann relations for six-particle massless scattering. The cluster algebra connection we find restricts the functions space for vector boson or Higgs plus jet amplitudes, and for form factors recently considered in $\mathcal{N}=4$ super Yang-Mills. We explain general procedures for studying relationships between alphabets of generalized polylogarithmic functions and cluster algebras, and use them to provide various identifications of one-loop alphabets with cluster algebras. In particular, we show how one can obtain one-loop alphabets for five-particle scattering from a recently discussed dual conformal eight-particle alphabet related to the $G(4,8)$ cluster algebra.