Algebraic reduction of one-loop Feynman graph amplitudes Original Research Article

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 [A.I. Davydychev, Phys. Lett. B 263 (1991) 107] and reduce these by recurrence relations to integrals in generic dimension [O.V. Tarasov, Phys. Rev. D 54 (1996) 6479]. Also the integration-by-parts method [F.V. Tkachov, Phys. Lett. B 100 (1981) 65; K.G. Chetyrkin, F.V. Tkachov, Nucl. Phys. B 192 (1981) 159] 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.