One-loop hexagon integral to higher orders in the dimensional regulator

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.

Section: Pentagon Graphs and Beyondsupporting

confidence: 87%

Symbol alphabets from the Landau singular locus

Dlapa,

Helmer,

Papathanasiou

et al. 2023

“…Comparison with CDE. The one-loop differential equations and the letters has been investigated from various aspects, such as from direct computation [69], diagrammatic coaction [51,70], hyperbolic geometries [53], dual forms [68], A-determinants [71], and also from Baikov d log representations [54,55] and intersection theory [72], etc. There are also papers that focus on the finite truncation of one-loop DE [7,11,52,73].…”

Section: Jhep12(2023)140mentioning

confidence: 99%

On symbology and differential equations of Feynman integrals from Schubert analysis

He,

Jiang,

Liu

et al. 2023

We take the first step in generalizing the so-called “Schubert analysis”, originally proposed in twistor space for four-dimensional kinematics, to the study of symbol letters and more detailed information on canonical differential equations for Feynman integral families in general dimensions with general masses. The basic idea is to work in embedding space and compute possible cross-ratios built from (Lorentz products of) maximal cut solutions for all integrals in the family. We demonstrate the power of the method using the most general one-loop integrals, as well as various two-loop planar integral families (such as sunrise, double-triangle and double-box) in general dimensions. Not only can we obtain all symbol letters as cross-ratios from maximal-cut solutions, but we also reproduce entries in the canonical differential equations satisfied by a basis of d log integrals.

Module intersection and uniform formula for iterative reduction of one-loop integrals

Chen

Feng

2023

In this paper, we develop an iterative sector-level reduction strategy for Feynman integrals, which bases on module intersection in the Baikov representation and auxiliary vector for tensor structure. Using this strategy we have studied the reduction of general one-loop integrals, i.e., integrals having arbitrary tensor structures and arbitrary power for propagators. Inspired by these studies, a uniform and compact formula that iteratively reduces all one-loop integrals has been written down, where messy polynomials in integration-by-parts (IBP) relations have organized themselves to Gram determinants.