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A systematic algorithm for obtaining recurrence relations for dimensionally regularized Feynman integrals w.r.t. the space-time dimension d is proposed. The relation between d and d − 2 dimensional integrals is given in terms of a differential operator for which an explicit formula can be obtained for each Feynman diagram. We show how the method works for one-, two-and threeloop integrals. The new recurrence relations w.r.t. d are complementary to the recurrence relations which derive from the method of integration by parts. We find that the problem of the irreducible numerators in Feynman integrals can be naturally solved in the framework of the proposed generalized recurrence relations.

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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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.