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We propose a first implementation of the integrand-reduction method fortwo-loop scattering amplitudes. We show that the residues of the amplitudes onmulti-particle cuts are polynomials in the irreducible scalar productsinvolving the loop momenta, and that the reduction of the amplitudes in termsof master integrals can be realized through polynomial fitting of theintegrand, without any apriori knowledge of the integral basis. We discuss howthe polynomial shapes of the residues determine the basis of master integralsappearing in the final result. We present a four-dimensional constructivealgorithm that we apply to planar and non-planar contributions to the 4- and5-point MHV amplitudes in N=4 SYM. The technique hereby discussed extends thewell-established analogous method holding for one-loop amplitudes, and can beconsidered a preliminary study towards the systematic reduction at theintegrand-level of two-loop amplitudes in any gauge theory, suitable for theirautomated semianalytic evaluation.

On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes