Symbol Alphabets in QCD and Flag Cluster Algebras

TL;DR

This paper connects recent advances in the symbol alphabet for two-loop six-point Feynman integrals with cluster algebra structures, showing how algebraic letters emerge from infinite mutation sequences in flag cluster varieties.

Contribution

It demonstrates that most rational symbol letters are expressible via flag cluster variables and that all algebraic letters originate from infinite mutation sequences.

Findings

Most rational symbol letters are expressible in terms of flag cluster variables.

All algebraic symbol letters arise from infinite mutation sequences.

The work bridges Feynman integral symbol alphabets with cluster algebra structures.

Abstract

The full 245-letter symbol alphabet for all planar massless two-loop six-point Feynman integrals was recently determined in arXiv:2412.19884 and arXiv:2501.01847. In a parallel mathematical development, it was shown in arXiv:2408.14956 that there is an embedding of the cluster algebra associated to the partial flag variety $Fl_{2,n-2;n}$, which describes the kinematics of $n$ massless particles, into that of the Grassmannian Gr$(n{-}2,2n{-}4)$. In this paper we connect these developments by showing that most of the rational symbol letters can be expressed in terms of flag cluster variables, and that all of the algebraic symbol letters arise from infinite mutation sequences.