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Symbol alphabets from the Landau singular locus

Dlapa, C., Helmer, M., Papathanasiou, G. ORCID: 0000-0002-2627-9906 & Tellander, F. (2023). Symbol alphabets from the Landau singular locus. Journal of High Energy Physics, 2023(10), article number 161. doi: 10.1007/jhep10(2023)161


We provide evidence through two loops, that rational letters of polylogarithmic Feynman integrals are captured by the Landau equations, when the latter are recast as a polynomial of the kinematic variables of the integral, known as the principal A-determinant. Focusing on one loop, we further show that all square-root letters may also be obtained, by re-factorizing the principal A-determinant with the help of Jacobi identities. We verify our findings by explicitly constructing canonical differential equations for the one-loop integrals in both odd and even dimensions of loop momenta, also finding agreement with earlier results in the literature for the latter case. We provide a computer implementation of our results for the principal A-determinants, symbol alphabets and canonical differential equations in an accompanying Mathematica file. Finally, we study the question of when a one-loop integral satisfies the Cohen-Macaulay property and show that for almost all choices of kinematics the Cohen-Macaulay property holds. Throughout, in our approach to Feynman integrals, we make extensive use of the Gel’fand, Graev, Kapranov and Zelevinskiĭ theory on what are now commonly called GKZ-hypergeometric systems whose singularities are described by the principal A-determinant.

Publication Type: Article
Additional Information: This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Publisher Keywords: Differential and Algebraic Geometry, Higher-Order Perturbative Calculations, Scattering Amplitudes
Subjects: Q Science > QA Mathematics
Q Science > QC Physics
Departments: School of Science & Technology
School of Science & Technology > Mathematics
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