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

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.

Comments: 49 pages, LaTeX, two figures (one in eps format), v3 is journal version; in v3 the typos in Egs.(2.17), (4.16), (B.12) corrected; in v4 the typos in Egs.(2.7),(4.10),(4.25),(B.11) corrected (see p.30 of present version)