Document Type

Article

Publication Date

2019

Publication Title

2017 MATRIX Annals

Abstract

Integrals from Feynman diagrams with massive particles soon outgrow polylogarithms. We consider the simplest situation in which this occurs, namely for diagrams with two vertices in two space-time dimensions, with scalar particles of unit mass. These comprise vacuum diagrams, on-shell sunrise diagrams and diagrams obtained from the latter by cutting internal lines. In all these cases, the Feynman integral is a moment of n = a + b Bessel functions, of the form M(a,b,c) := ∫ 0 1a0(t)Kb0(t)tcdt. The corresponding L-series are built from Kloosterman sums over finite fields. Prior to the Creswick conference, the first author obtained empirical relations between special values of L-series and Feynman integrals with up to n = 8 Bessel functions. At the conference, the second author indicated how to extend these. Working together we obtained empirical relations involving Feynman integrals with up to 24 Bessel functions, from sunrise diagrams with up to 22 loops. We have related results for moments that lie beyond quantum field theory.

First Page

401

Last Page

403

DOI

https://doi.org/10.1007/978-3-030-04161-8_27

ISSN

978-3-030-04161-8

Comments

This is a pre-print version of an article published by Springer in 2017 MATRIX Annals. The definitive version can be found on the publisher's website.

Rights

© Springer Nature Switzerland AG 2019

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