1205.5707 (Yang Zhang)

Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic
Geometry Methods    [PDF]

Yang Zhang

We present an algorithm for the integrand-level reduction of multi-loop amplitudes of renormalizable field theories, based on computational algebraic geometry. This algorithm uses (1) Gr\"obner basis method to determine the basis for integrand-level reduction, (2) primary decomposition of an ideal to classify all inequivalent solutions of unitarity cuts. The resulting basis and cut solutions can be used to reconstruct the integrand from unitarity cuts, via polynomial fitting techniques. The basis determination part of the algorithm has been implemented in the Mathematica package, BasisDet. The primary decomposition part can be readily carried out by algebraic geometry softwares, with the output of the package BasisDet. The algorithm works in both D=4 and $D=4-2\epsilon$ dimensions, and we present some two and three-loop examples of this algorithm.

View original: http://arxiv.org/abs/1205.5707