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Positivity and cluster structures in Landau analysis

Benjamin Hollering1, Elia Mazzucchelli2, Matteo Parisi3,2, and Bernd Sturmfels1

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Phys. Rev. D 114, 066014 – Published 22 September, 2026

DOI: https://doi.org/10.1103/cgg7-wyg7

Abstract

Landau analysis in momentum twistor space can be formulated as the study of varieties of lines in three-dimensional projective space, together with their projections and discriminants. Within this framework, we define enumerative invariants (leading singularity (LS) degrees) that count leading singularities. Leading Landau singularities (LS discriminants) arise as discriminants detecting the collision of leading singularities. We uncover a recursive mechanism underlying Landau singularities, governed by substitution maps between Grassmannians. Applying this framework, we prove positivity and factorization into cluster variables for the LS discriminant of a large class of Landau diagrams at arbitrary loop order. This provides a first-principles explanation for the emergence of positivity and cluster algebra structures in scattering amplitudes of planar N=4 super Yang-Mills theory.

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Phys. Rev. D 114, 066014– Published 22 September, 2026
  • Received 26 March 2026
  • Accepted 24 July 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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