ϵ-finite basis of master integrals for the integration-by-parts method

It is shown that for every problem within dimensional regularization, using the integration-by-parts method, one is able to construct a set of master integrals such that each corresponding coefficient function is finite in the limit of dimension equal to four. We argue that the use of such a basis simplifies and stabilizes the numerical evaluation of the master integrals. As an example we explicitly construct the ϵ-finite basis for the set of all QED-like four-loop massive tadpoles. Using a semi-numerical approach based on Padé approximations we evaluate analytically the divergent and numerically the finite part of this set of master integrals. The calculations confirm the recent results of Schröder and Vuorinen. All the contributions found there by fitting the high precision numerical results have been confirmed by direct analytical calculation without using any numerical input.