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

For certain dimensional ly-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 byproduct, 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. (C) 2001 Elsevier Science B.V. All rights reserved.