Numerical Contour Integration for Loop Integrals

TL;DR

The paper introduces numerical contour integration (NCI) as a fully numerical method for loop integrals, recasting them as contour integrals in the complex plane to navigate multiple poles. It provides explicit NCI formulations for one-loop three- and four-point functions and two-loop two- and three-point functions, employing coordinate systems (including a wedge formulation) and techniques like the Good Lattice Point (GLP) method and SBT relations. The approach is validated against analytical results and established packages (FF, Kreimer, etc.), and extended to the numerical evaluation of hypergeometric functions in massless theories with high precision. Overall, NCI offers a stable, tensor-capable alternative to traditional analytic methods, broadening the toolkit for precision Standard Model calculations and multi-scale loop integrals.

Abstract

A fully numerical method to calculate loop integrals, a numerical contour-integration method, is proposed. Loop integrals can be interpreted as a contour integral in a complex plane for an integrand with multi-poles in the plane. Stable and efficient numerical integrations an along appropriate contour can be performed for tensor integrals as well as for scalar ones appearing in loop calculations of the standard model. Examples of 3- and 4-point diagrams in 1-loop integrals and 2- and 3-point diagrams in 2-loop integrals with arbitrary masses are shown. Moreover it is shown that numerical evaluations of the Hypergeometric function, which often appears in the loop integrals, can be performed using the numerical contour-integration method.

Figures (5)

• Figure 1: 1-loop/3-point diagram
• Figure 2: 1-loop/4-point diagram
• Figure 3: Parametrization of the wedge coordinate
• Figure 4: 2-loop/2-point diagram
• Figure 5: 2-loop vertex diagram (non-planer type)