Numerical integration of loop integrals through local cancellation of threshold singularities

D Kermanschah

doi:10.1007/jhep01(2022)151

Abstract

We propose a new approach that allows for the separate numerical calculation of the real and imaginary parts of finite loop integrals. We find that at one-loop the real part is given by the Loop-Tree Duality integral supplemented with suitable counterterms and the imaginary part is a sum of two-body phase space integrals, constituting a locally finite representation of the generalised optical theorem. These expressions are integrals in momentum space, whose integrands were specially designed to feature local cancellations of threshold singularities. Such a representation is well suited for Monte Carlo integration and avoids the drawbacks of a numerical contour deformation around remaining singularities. Our method is directly applicable to a range integrals with certain geometric properties but not yet fully generalised for arbitrary one-loop integrals. We demonstrate the computational performance with examples of one-loop integrals with various kinematic configurations, which gives promising prospects for an extension to multi-loop integrals.

Highlights

The endeavor of automating perturbative calculations in Quantum Field Theories beyond next-to-leading order has lead to a continuous development of new methods and computational techniques
We focus on methods based on the numerical integration of loop integrals
We derived a new representation of one-loop integrals that is locally free of poles in the integration domain

Summary

Introduction

The endeavor of automating perturbative calculations in Quantum Field Theories beyond next-to-leading order has lead to a continuous development of new methods and computational techniques. Note that as an alternative to our method, the Cauchy principal value of onedimensional integrals can be numerically evaluated by symmetrising the integrand in the neighbourhood of each of its poles Such a symmetrisation cannot be generalised for higher-order poles and multi-dimensional integrals with complicated intersecting singular surfaces in the integration domain. We can further simplify the expression eq (2.21) in the case, where no locally pinched poles (with respect to the integration contour of r) are present in the LTD integrand of eq (3.12) In this case, all intersections of E-surfaces and their corresponding poles in residues and counterterms will locally cancel amongst themselves, such that the sum of fractional residues is free of poles in the integration domain.

Im i k k

D3 D4 D1 D2 D4

Conclusion