A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec

• Published in: Computer physics communications Vol. 240; pp. 120 - 137
• Main Authors: Borowka, S., Heinrich, G., Jahn, S., Jones, S.P., Kerner, M., Schlenk, J.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.07.2019
• Subjects: Feynman diagrams; Multi-loop; Numerical integration; Perturbation theory; Perturbation theory; Multi-loop; Numerical integration; Feynman diagrams
• ISSN: 0010-4655; 1879-2944; 1879-2944
• DOI: 10.1016/j.cpc.2019.02.015

The purely numerical evaluation of multi-loop integrals and amplitudes can be a viable alternative to analytic approaches, in particular in the presence of several mass scales, provided sufficient accuracy can be achieved in an acceptable amount of time. For many multi-loop integrals, the fraction of time required to perform the numerical integration is significant and it is therefore beneficial to have efficient and well-implemented numerical integration methods. With this goal in mind, we present a new stand-alone integrator based on the use of (quasi-Monte Carlo) rank-1 shifted lattice rules. For integrals with high variance we also implement a variance reduction algorithm based on fitting a smooth function to the inverse cumulative distribution function of the integrand dimension-by-dimension. Additionally, the new integrator is interfaced to pySecDec to allow the straightforward evaluation of multi-loop integrals and dimensionally regulated parameter integrals. In order to make use of recent advances in parallel computing hardware, our integrator can be used both on CPUs and CUDA compatible GPUs where available. Program Title: pySecDec, qmc Program Files doi:http://dx.doi.org/10.17632/dnrkf5jxzh.2 Licensing provisions: GNU General Public License v3 Programming language: python, FORM, C++, CUDA External routines/libraries: catch [1], gsl [2], numpy [3], sympy [4], Nauty [5], Cuba [6], FORM [7], Normaliz [8]. The program can also be used in a mode which does not require Normaliz. Journal reference of previous version: Comput. Phys. Commun. 222 (2018) 313–326. Does the new version supersede the previous version?: Yes Nature of problem: Extraction of ultraviolet and infrared singularities from parametric integrals appearing in higher order perturbative calculations in quantum field theory. Numerical integration in the presence of integrable singularities (e.g. kinematic thresholds). Solution method: Algebraic extraction of singularities within dimensional regularization using iterated sector decomposition. This leads to a Laurent series in the dimensional regularization parameter ϵ (and optionally other regulators), where the coefficients are finite integrals over the unit-hypercube. Those integrals are evaluated numerically by Monte Carlo integration. The integrable singularities are handled by choosing a suitable integration contour in the complex plane, in an automated way. The parameter integrals forming the coefficients of the Laurent series in the regulator(s) are provided in the form of libraries which can be linked to the calculation of (multi-) loop amplitudes. Restrictions: Depending on the complexity of the problem, limited by memory and CPU/GPU time. References: [1] https://github.com/philsquared/Catch/. [2] http://www.gnu.org/software/gsl/. [3] http://www.numpy.org/. [4] http://www.sympy.org/. [5] http://pallini.di.uniroma1.it/. [6] T. Hahn, “CUBA: A Library for multidimensional numerical integration,” Comput. Phys. Commun. 168 (2005) 78 [hep-ph/0404043], http://www.feynarts.de/cuba/. [7] J. Kuipers, T. Ueda and J. A. M. Vermaseren, “Code Optimization in FORM,” Comput. Phys. Commun. 189 (2015) 1 [arXiv:1310.7007], http://www.nikhef.nl/ form/. [8] W. Bruns, B. Ichim, B. and T. Römer, C. Söger, “Normaliz. Algorithms for rational cones and affine monoids.” http://www.math.uos.de/normaliz/.