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.
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BibTeX
@misc{dolz2026numerical,
title = {{Numerical evaluation of one-loop integrals using quadrature rules}},
author = {Jürgen Dölz and Claude Duhr and Bartul Kovačić and Cathrin Semper},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41872},
eprint = {2610.41872},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41872},
}
RIS
TY - UNPB
TI - Numerical evaluation of one-loop integrals using quadrature rules
AU - Jürgen Dölz
AU - Claude Duhr
AU - Bartul Kovačić
AU - Cathrin Semper
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41872
UR - https://arxiv.org/abs/2610.41872
AB - 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.
ER -