Automatic Integral Reduction for Higher Order Perturbative Calculations

TL;DR

The paper tackles the computational bottleneck of perturbative calculations at higher orders, where large systems of loop and phase-space integrals arise. It introduces AIR, a MAPLE-based implementation of Laporta's automated reduction using IBP and Lorentz-invariance identities, solved via Gauss elimination to express integrals in terms of master integrals. Two masking strategies are proposed to control expression growth: masking reduced expressions to master integrals and masking large integral coefficients with numerical checks, both supported by a simple database infrastructure. The authors demonstrate the approach on the massless one-loop box and the one-loop pentagon topologies, discuss performance and memory considerations, and outline directions for extending the method to more complex topologies and larger computations, highlighting AIR's potential to automate and accelerate multiloop perturbative calculations.

Abstract

We present a program for the reduction of large systems of integrals to master integrals. The algorithm was first proposed by Laporta; in this paper, we implement it in MAPLE. We also develop two new features which keep the size of intermediate expressions relatively small throughout the calculation. The program requires modest input information from the user and can be used for generic calculations in perturbation theory.