Alphabet of one-loop Feynman integrals *

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: Verification Through Differential Equations and Comparison W...mentioning

confidence: 78%

Section: Jhep10(2023)161mentioning

confidence: 99%

Symbol alphabets from the Landau singular locus

Dlapa,

Helmer,

Papathanasiou

et al. 2023

“…To translate the canonical differential equations into dlog form, we have constructed the hexagon alphabet from the previously known one-mass five-point alphabet [9,15] and the by now well-understood recursive structure of one-loop alphabets [51]. By imposing the absense of non-physical singularities, we have then determined the boundary values for the hexagon family for a judiciously chosen point in the Euclidean region.…”

Section: Discussion and Outlookmentioning

confidence: 99%

“…The additional ten letters, appearing for the first time in the hexagon integral, are obtained following the construction of one-loop alphabet letters from [51]. The letters involving square roots can be written in the form…”

Section: One-loop Hexagon Lettersmentioning

confidence: 99%

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

Henn

Matijašić

Miczajka

2023

The state-of-the-art in current two-loop QCD amplitude calculations is at five-particle scattering. Computing two-loop six-particle processes requires knowledge of the corresponding one-loop amplitudes to higher orders in the dimensional regulator. In this paper we compute analytically the one-loop hexagon integral via differential equations. In particular we identify its function alphabet for general D-dimensional external states. We also provide integral representations for all one-loop integrals up to weight four. With this, the one-loop integral basis is ready for two-loop amplitude applications. We also study in detail the difference between the conventional dimensional regularization and the four-dimensional helicity scheme at the level of the master integrals and their function space.

“…For applying the method of canonical differential equation, finding a corresponding UT basis is not a trivial task. There are many methods and algorithms designed for determining a UT basis, including the leading singularity analysis [21,22,27], the Magnus and Dyson Series [45], the dlog integrand construction [46][47][48][49], the intersection theory [50][51][52][53], the initial algorithm [54], the Lee's algorithm [55][56][57] and so on. Based on these methods or algorithms, some public available packages were designed including Canonica [58,59],…”

Section: Ut Integral Determinationmentioning

confidence: 99%

A study of Feynman integrals with uniform transcendental weights and their symbology

et al. 2022

Multi-loop Feynman integrals are key objects for the high-order correction computations in high energy phenomenology. These integrals with multiple scales may have complicated symbol structures, and we show that twistor geometries of closely related dual conformal integrals shed light on their alphabet and symbol structures. In this paper, first, as a cutting-edge example, we derive the two-loop four-external-mass Feynman integrals with uniform transcendental (UT) weights, based on the latest developments on UT integrals. Then we find that all the symbol letters of these integrals can be explained non-trivially by studying the so-called Schubert problem of certain dual conformal integrals with a point at infinity. Certain properties of the symbol such as first two entries and extended Steinmann relations are also studied from analogous properties of dual conformal integrals.