1209.4319 (Pierpaolo Mastrolia et al.)

Integrand-Reduction for Two-Loop Scattering Amplitudes through
Multivariate Polynomial Division    [PDF]

Pierpaolo Mastrolia, Edoardo Mirabella, Giovanni Ossola, Tiziano Peraro

We describe the application of a novel approach for the reduction of scattering amplitudes, based on multivariate polynomial division, which we have recently presented. This technique yields the complete integrand decomposition for arbitrary amplitudes, regardless of the number of loops. It allows for the determination of the residue at any multi-particle cut, whose knowledge is a mandatory prerequisite for applying the integrand-reduction procedure. By using the division modulo Groebner basis, we can derive a simple integrand recurrence relation that generates the multi-particle pole decomposition for arbitrary multi-loop amplitudes. We apply the new reduction algorithm to the two-loop five-point planar and non-planar scattering amplitudes in N = 4 SYM and N = 8 SUGRA in four dimensions, whose numerator functions contain up to rank-two terms in the integration-momenta. We determine all polynomial residues parametrizing the cuts of the corresponding topologies and sub-topologies. Moreover, we obtain the integral basis for the amplitude decomposition, defined by the polynomial form of the residues. Our approach is well suited for a semi-numerical implementation, and its general mathematical properties provide an effective algorithm for the generalization of integrand reduction method to all orders in perturbation theory.

View original: http://arxiv.org/abs/1209.4319