Integrand-level reduction of loop amplitudes by computational algebraic geometry methods

TY - JOUR

T1 - Integrand-level reduction of loop amplitudes by computational algebraic geometry methods

AU - Zhang, Yang

PY - 2012/9/1

Y1 - 2012/9/1

N2 - 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) the Gr̈obner basis method to determine the basis for integrand-level reduction, (2) the 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∈ dimensions, and we present some two and three-loop examples of applications of this algorithm.

AB - 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) the Gr̈obner basis method to determine the basis for integrand-level reduction, (2) the 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∈ dimensions, and we present some two and three-loop examples of applications of this algorithm.

U2 - 10.1007/jhep09(2012)042

DO - 10.1007/jhep09(2012)042

M3 - Journal article

SN - 1126-6708

VL - 2012

SP - 042

JO - Journal of High Energy Physics (Online)

JF - Journal of High Energy Physics (Online)

IS - 9

ER -