Numerical evaluation of one-loop integrals using quadrature rules

Abstract

We introduce a numerical quadrature algorithm with stretched exponential convergence to evaluate one-loop integrals in dimensional regularisation in the causal Loop Tree Dual (cLTD) representation. The key feature that makes this possible is the fact that for Euclidean kinematics all singularities of the cLTD integrand lie on a compact set in loop momentum space. We prove that our algorithm admits a stretched exponential convergence with the number of integrand evaluations, and we illustrate it by evaluating triangle, box and pentagon integrals at selected Euclidean kinematics points. We benchmark our implementation against analytic results as well as results obtained via Monte Carlo integration. We also provide a proof of concept that our algorithm can be extended to physical kinematics after subtraction of threshold singularities.