Kinematics, cluster algebras and Feynman integrals

TL;DR

This paper explores the connection between cluster algebras and the singularities of conformal Feynman integrals in four dimensions, providing new insights into their structure and applications to specific integrals.

Contribution

It identifies specific cluster algebras for planar kinematics, demonstrates their role in encoding integral singularities, and applies this to a complex three-loop wheel integral example.

Findings

Cluster algebras encode singularities of Feynman integrals.

Application of D3 cluster algebra to an 8-point three-loop integral.

Bootstrap of the integral's symbol constrained by cluster adjacency.

Abstract

We identify cluster algebras for planar kinematics of conformal Feynman integrals in four dimensions, as sub-algebras of that for top-dimensional $G(4,n)$ corresponding to $n$-point massless kinematics. We provide evidence that they encode information about singularities of such Feynman integrals, including all-loop ladders with symbol letters given by cluster variables and algebraic generalizations. As a highly-nontrivial example, we apply $D_3$ cluster algebra to a $n=8$ three-loop wheel integral, which contains a new square root. Based on the $D_3$ alphabet and three new algebraic letters essentially dictated by the cluster algebra, we bootstrap its symbol, which is strongly constrained by the cluster adjacency. By sending a point to infinity, our results have implications for non-conformal Feynman integrals, e.g., up to two loops the alphabet of two-mass-easy kinematics is given by limit of this generalized $D_3$ alphabet. We also find that the reduction to three dimensions is achieved by folding and the resulting cluster algebras may encode singularities of amplitudes and Feynman integrals in ABJM theory, at least through $n=7$ and two loops.