Symbol Alphabets in QCD and Flag Cluster Algebras

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.

Figures (5)

• Figure 1: The initial cluster of $\mathcal{F\ell}_{2,3;5}$ with nodes labeled by the Plücker coordinates of the partial flag variety (left) or by the corresponding spinor helicity variables (right), with frozen nodes boxed.
• Figure 2: The initial cluster of $\mathcal{F\ell}_{2,4;6}$ with nodes labeled by the Plücker coordinates of the partial flag variety (left) or by the corresponding spinor helicity variables (right), with frozen nodes boxed.
• Figure 3: The starting point of the infinite mutation sequence construction described in Drummond:2019cxm. The boxed nodes are frozen and nodes not connected to $z_0$ or $w_0$ are omitted.
• Figure 4: The infinite mutation sequence that starts from mutating on $w_0$. For clarity we omit arrows and nodes that are not connected to the two nodes in the first cluster where $z_0$ and $w_0$ sit.
• Figure 5: Left: The initial cluster of $\mathcal{F\ell}_{2,4;6}$, with nodes labeled in red. Right: The cluster obtained after performing the mutation sequence $10\rightarrow 8\rightarrow 9\rightarrow 13 \rightarrow 10$, highlighting the embedded cluster of the type shown in Figure \ref{['fig: Gr48 infinite start']} with the nodes and edges not connected to $w_0$ or $z_0$ dimmed.