Hypergeometric Structures in Feynman Integrals

Blümlein, Johannes; Saragnese, Marco; Schneider, Carsten

doi:10.1007/s10472-023-09831-8

Journal Article

PUBDB-2023-07057

Hypergeometric Structures in Feynman Integrals

Blümlein, J.DESY* ; Saragnese, M.Extern* ; Schneider, C. (Corresponding author)Extern*

2023
Springer Science + Business Media B.V Dordrecht [u.a.]

Annals of mathematics and artificial intelligence 91(5), 591 - 649 (2023) [10.1007/s10472-023-09831-8]

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Please use a persistent id in citations: doi:10.1007/s10472-023-09831-8 doi:10.3204/PUBDB-2023-07057

Report No.: DESY-21-071; DO-TH 21/16; RISC Report Series 21-17; SAGEX-21-10-E; arXiv:2111.15501

Abstract: Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {\tt Sigma} in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code {\tt HypSeries} transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code {\tt solvePartialLDE} is designed. Generalized hypergeometric functions, Appell-,~Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton--type functions are considered. We illustrate the algorithms by examples.

Keyword(s): differential equations ; Feynman graph ; structure ; mathematical methods ; Taylor expansion ; topology ; master integral ; computer: algebra ; numerical methods

Classification:

ddc:004

Note: 55 pages, several anc. files

Contributing Institute(s):

Zeuthen Particle PhysicsTheory (Z_ZPPT)

Research Program(s):

611 - Fundamental Particles and Forces (POF4-611) (POF4-611)

Experiment(s):

HERA: ZEUS

Appears in the scientific report 2023

Database coverage:
Medline

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