Kinematic singularities of Feynman integrals and principal A-determinants

Abstract

We consider the analytic properties of Feynman integrals from the perspective of general A-discriminants and A-hypergeometric functions introduced by Gelfand, Kapranov and Zelevinsky (GKZ). This enables us, to give a clear and mathematically rigour description of the singular locus, also known as Landau variety, via principal A-determinants. We also comprise a description of the various second type singularities. Moreover, by the Horn-Kapranov-parametrization we give a very efficient way to calculate a parametrization of Landau varieties. We furthermore present a new approach to study the sheet structure of multivalued Feynman integrals by use of coamoebas.

FeynGKZ: A Mathematica package for solving Feynman integrals using GKZ hypergeometric systems

“…For unimodular regular triangulations, each simplex of the triangulation can lead to only one such series, and hence, we can label the -series by their corresponding simplices. It can be shown that in the unimodular case, the -series associated to some simplex σ ∈ T takes the following form: (19) Using the Knudsen-Mumford theorem [41,42], it could be argued [15] that by adding monomials to the toric G-polynomial such that G z → G z (or equivalently, by scaling the underlying Newton polytope G z with some integer scale-factor such that G z → G z…”

“…For example, one can derive series solutions using two alternative algorithms as discussed in [14] and [15]. Alternatively, the GKZ framework is also useful to study the singular locus of Feynman integrals [19], vector spaces generated by Feynman integrals [20], Cohen-Macaulay property of Feynman integrals [21], matroids attached to Feynman integrals [22], etc. GKZ theory, along with ideas from studies of Calabi-Yau manifolds, has been useful in understanding the analytic structure of a certain class of amplitudes to all orders of perturbation [23,24].…”

Hypergeometric Feynman Integrals

“…Moreover, by means of the results in A-hypergeometric theory, the most considerations can be reduced to polytopal combinatorics instead of algebraic topology as done in [47,128,173,197]. We also want to draw attention to the very interesting work of [170], published shortly after the article [151] that constitutes the basis of this chapter.…”

“…Shortly afterwards, a series of articles considered Feynman integrals by use of GKZ methods developed in string theory [36,37,152], which were applied mainly to the banana graph family. Moreover, the Landau variety of Feynman integrals was considered from the A-hypergeometric perspective for banana graphs in [37] and for arbitrary Feynman graphs in [151].…”

“…This thesis builds on the two mentioned articles [150] and [151]. However, we will include several major generalizations of these works, and we also want to provide a more substantial overview of the mathematical theory behind it.…”