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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <msub> <mi mathvariant="script">Fl</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo>;</mo> <mi>n</mi> </mrow> </msub> </math></EquationSource> <EquationSource Format="TEX">\( {\mathcal{Fl}}_{2,n-2;n} \)</EquationSource> </InlineEquation>, which describes the kinematics of <i>n</i> massless particles, into that of the Grassmannian Gr(<i>n</i>–2, 2<i>n</i>–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.</p>

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Symbol alphabets in QCD and flag cluster algebras

  • 期刊论文

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  • 发表日期：

    2026年3月12日

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  • Andrzej Pokraka,
  • Marcus Spradlin,
  • Anastasia Volovich,
  • He-Chen Weng

摘要

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 \( {\mathcal{Fl}}_{2,n-2;n} \) , 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.