Adaptive Integrand Decomposition

Abstract

We present a simplified variant of the integrand reduction algorithm for multiloop scattering amplitudes in d = 4 − 2ε dimensions, which exploits the decomposition of the integration momenta in parallel and orthogonal subspaces, d = d + d ⊥ , where d is the dimension of the space spanned by the legs of the diagrams. We discuss the advantages of a lighter polynomial division algorithm and how the orthogonality relations for Gegenbauer polynomilas can be suitably used for carrying out the integration of the irreducible monomials, which eliminates spurious integrals. Applications to one-and two-loop integrals, for arbitrary kinematics, are discussed. Loops and Legs in Quantum Field Theory

The pentabox master integrals with the simplified differential equations approach

“…In the recent years, a lot of progress has been made towards the extension of these reduction methods to the two-loop order at the integral [8,9]as well as the integrand [10,11,12,13,14] level. The master equation at the integrand level in four dimensions can be given schematically as follows [13] where an arbitrary contribution to the two-loop amplitude (left), can be reduced to a sum of terms (right) of all partitions S m;n , with up to eight denominators; l 1 , l 2 are the loop momenta, D i are the inverse scalar Feynman propagators, N (l 1 , l 2 ; {p i }) is a general numerator polynomial and…”

Interplay of colour kinematics duality and analytic calculation of multi-loop scattering amplitudes: one and two loops

“…This implementation automates, numerically and analytically, the method of [13]. We have also shown results for the analytic reduction of the one-and two-loop amplitudes of the µe-elastic scattering.…”

“…In this section, we explain the main features of the Adaptive Integrand Decomposition Algorithm (AIDA), the automation of the recent method proposed by Mastrolia, Primo and Peraro [13].…”

“…We focus our study on the automation of the recent algorithm proposed by Mastrolia, Peraro and Primo [13], that decomposes the space-time dimension into parallel and perpendicular components,…”

“…Apart from the examples provided in [13], this algorithm has also been applied to the leading color contribution to the two-loop all-plus five-gluon amplitude [15,16].…”