New results for the epsilon-expansion of certain one-, two- and three-loop Feynman diagrams

TL;DR

This paper investigates the epsilon-expansion and analytic continuation of specific multi-loop Feynman diagrams, providing new computational techniques and mathematical relations involving special functions.

Contribution

It introduces methods to compute arbitrary terms in epsilon-expansion for certain diagrams and establishes new relations among special functions.

Findings

Arbitrary epsilon-expansion terms can be calculated for some diagrams.

New relations involving Clausen functions and log-sine integrals are derived.

Results include useful formulas for hypergeometric functions at 1/4.

Abstract

For certain dimensionally-regulated one-, two- and three-loop diagrams, problems of constructing the epsilon-expansion and the analytic continuation of the results are studied. In some examples, an arbitrary term of the epsilon-expansion can be calculated. For more complicated cases, only a few higher terms in epsilon are obtained. Apart from the one-loop two- and three-point diagrams, the examples include two-loop (mainly on-shell) propagator-type diagrams and three-loop vacuum diagrams. As a by-product, some new relations involving Clausen function, generalized log-sine integrals and certain Euler--Zagier sums are established, and some useful results for the hypergeometric functions of argument 1/4 are presented.