pySecDec: A toolbox for the numerical evaluation of multi-scale integrals

Abstract

We present pySecDec, a new version of the program SecDec, which performs the factorisation of dimensionally regulated poles in parametric integrals, and the subsequent numerical evaluation of the finite coefficients. The algebraic part of the program is now written in the form of python modules, which allow a very flexible usage. The optimization of the C++ code, generated using FORM, is improved, leading to a faster numerical convergence. The new version also creates a library of the integrand functions, such that it can be linked to user-specific codes for the evaluation of matrix elements in a way similar to analytic integral libraries. (2015) 470-491. Nature of the 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 (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 time. Running time: Between a few seconds and several days, depending on the complexity of the problem.

A numerical evaluation of planar two-loop helicity amplitudes for a W-boson plus four partons

“…We make use of the master integral solutions that are known analytically and readily available for evaluation in the Euclidean kinematics [6,69]. Tables 1 and 2, are numerically evaluated using pySecDec [68]. Note that there is a class of four-point master integrals with two off shell legs, that are not covered in [6], but available in [70][71][72], where the solutions are derived in the physical region.…”

“…Using a modular approach, recently presented as part of the FiniteFlow algorithms [5], we are able to numerically evaluate the diagrams and perform an integrand reduction, subsequently reducing the resulting integrals using integration-by-parts identities. Since the complete set of analytic master integrals is not known, some of the integrals were evaluated numerically using sector decomposition [66][67][68]. Analytic results for the following classes of master integral are available: one of the three families of the off-shell five-point pentagon-box [6] and fourpoint functions with one [69] and two off-shell [70][71][72][73] legs.…”

“…. The remaining master integrals, which contain either one-loop squared topologies or 5 propagator or fewer two-loop topologies, are evaluated directly using Fiesta[67] and pySecDec[68]. 22 The one-loop integrals, required to O( 2 ), are evaluated numerically.…”

Double-Higgs boson production in the high-energy limit: planar master integrals

“…In each case we display the approximations including m 2 t , m 4 t , m 8 t and m 16 t terms. In the panels showing G 6 (1, 1, 1, 1, 1, 1, 1, 0, 0) we compare the approximations to the exact result, which has been obtained numerically using pySecDec [49]. For this integral we observe a rapid convergence.…”

Automated Calculation of Dijet Soft Functions in the Presence of Jet Clustering Effects

“…We checked our results with an independent code that uses the public programs SecDec 3 [14] and pySecDec [15]. As the new python-based version of SecDec supports an arbitrary number of analytic regulators, it is particularly suited for SCET-2 problems.…”