Efficient Numerical Evaluation of Feynman Integral
Zhao Li, Jian Wang, Qi-Shu Yan, Xiaoran Zhao
TL;DR
The paper tackles the challenge of efficiently evaluating Feynman loop integrals for higher-order QCD effects by coupling sector decomposition with a shifted rank-1 lattice quasi-Monte Carlo method on CUDA/GPU hardware. It demonstrates that the approach yields high accuracy with substantial speedups over traditional CPU-based Monte Carlo methods, validated against FIESTA3 across Euclidean and physical kinematic regions. Key results include $\mathcal{O}(10^{-4})$ accuracy for one-loop integrals in tens of milliseconds and $\mathcal{O}(10^{-3})$ accuracy for two-loop integrals in seconds (even for non-planar topologies), enabling practical NLO virtual corrections for Higgs pair production with finite top mass. This work indicates that direct numerical evaluation can play a viable role in precision multi-loop computations relevant for LHC phenomenology.
Abstract
Feynman loop integrals are a key ingredient for the calculation of higher order radiation effects, and are responsible for reliable and accurate theoretical prediction. We improve the efficiency of numerical integration in sector decomposition by implementing a quasi-Monte Carlo method associated with the CUDA/GPU technique. For demonstration we present the results of several Feynman integrals up to two loops in both Euclidean and physical kinematic regions in comparison with those obtained from FIESTA3. It is shown that both planar and non-planar two-loop master integrals in the physical kinematic region can be evaluated in less than half a minute with $\mathcal{O}(10^{-3})$ accuracy, which makes the direct numerical approach viable for precise investigation of higher order effects in multi-loop processes, e.g. the next-to-leading order QCD effect in Higgs pair production via gluon fusion with a finite top quark mass.