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<p>We revisit the conjectural method called Schubert analysis for generating the alphabet of symbol letters for Feynman integrals, which was based on geometries of intersecting lines associated with corresponding cut diagrams. We explain the effectiveness of this somewhat mysterious method by relating such geometries to the corresponding Landau singularities, which also amounts to “uplifting” Landau singularities of a Feynman integral to its symbol letters. We illustrate this <i>Landau-based Schubert analysis</i> using various multi-loop Feynman integrals in four dimensions and present an automated Mathematica notebook for it. We then apply the method to a simplified problem of studying alphabets of physical quantities such as scattering amplitudes and form factors in planar <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = 4 super-Yang-Mills. By focusing on a small set of Landau diagrams (as opposed to all relevant Feynman integrals), we show how this method nicely produces the two-loop alphabet of <i>n</i>-point MHV amplitudes and that of the <i>n</i> = 4 MHV form factors. A byproduct of our analysis is an explicit representation of any symbol alphabet obtained this way as the union of various type-<i>A</i> cluster algebras.</p>

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Landau-based Schubert analysis

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月12日

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  • Song He,
  • Xuhang Jiang,
  • Jiahao Liu,
  • Qinglin Yang

摘要

We revisit the conjectural method called Schubert analysis for generating the alphabet of symbol letters for Feynman integrals, which was based on geometries of intersecting lines associated with corresponding cut diagrams. We explain the effectiveness of this somewhat mysterious method by relating such geometries to the corresponding Landau singularities, which also amounts to “uplifting” Landau singularities of a Feynman integral to its symbol letters. We illustrate this Landau-based Schubert analysis using various multi-loop Feynman integrals in four dimensions and present an automated Mathematica notebook for it. We then apply the method to a simplified problem of studying alphabets of physical quantities such as scattering amplitudes and form factors in planar N \( \mathcal{N} \) = 4 super-Yang-Mills. By focusing on a small set of Landau diagrams (as opposed to all relevant Feynman integrals), we show how this method nicely produces the two-loop alphabet of n-point MHV amplitudes and that of the n = 4 MHV form factors. A byproduct of our analysis is an explicit representation of any symbol alphabet obtained this way as the union of various type-A cluster algebras.