Some infinite series related to Feynman diagrams

TL;DR

This paper evaluates certain infinite series arising in Feynman diagram calculations, expressing their sums through special functions and constants like zeta values, Catalan's constant, and Clausen's function.

Contribution

It introduces methods to evaluate complex infinite series related to Feynman diagrams using integral representations of hypergeometric functions.

Findings

Series sums expressed in terms of zeta functions, Catalan's constant, and Clausen's function.

Applicable to one-, two-, and three-dimensional series in quantum field theory.

Provides integral representations for evaluating these series.

Abstract

Results are presented for some infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-, two- and three-dimensional series. The sums of these series can be evaluated with the help of various integral representations for hypergeometric functions, and expressed in terms of $\zeta(2)$, $\zeta(3)$, the Catalan constant $G$ and ${\rm Cl}_2(\pi/3)$ where ${\rm Cl}_2(\theta)$ is Clausen's function.