In all mass cases needed for quark and gluon self-energies, the two-loop master diagram is expanded at large and small q 2 , in d dimensions, using identities derived from integration by parts. Expansions are given, in terms of hypergeometric series, for all gluon diagrams and for all but one of the quark diagrams; expansions of the latter are obtained from differential equations. Padé approximants to truncations of the expansions are shown to be of great utility. As an application, we obtain the two-loop photon selfenergy, for all d, and achieve highly accelerated convergence of its expansions in powers of q 2 /m 2 or m 2 /q 2 , for d = 4.

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For a large class of two-loop selfenergy-and vertex-type diagrams with only one nonzero mass (M ) and the vertices also with only one non-zero external momentum squared (q 2 ) the first few expansion coefficients are calculated by the large mass expansion. This allows to 'guess' the general structure of these coefficients and to verify them in terms of certain classes of 'basis elements', which are essentially harmonic sums. Since for this case with only one non-zero mass the large mass expansion and the Taylor series in terms of q 2 are identical, this approach yields analytic expressions of the Taylor coefficients, from which the diagram can be easily evaluated numerically in a large domain of the complex q 2 −plane by well known methods. It is also possible to sum the Taylor series and present the results in terms of polylogarithms.

An algorithm for the reduction of one-loop n-point tensor integrals to basic integrals is proposed. We transform tensor integrals to scalar integrals with shifted dimension [1,2] and reduce these by recurrence relations to integrals in generic dimension [3]. Also the integration-by-parts method [4] is used to reduce indices (powers of scalar propagators) of the scalar diagrams. The obtained recurrence relations for one-loop integrals are explicitly evaluated for 5-and 6-point functions. In the latter case the corresponding Gram determinant vanishes identically for d = 4, which greatly simplifies the application of the recurrence relations.