A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec

Abstract

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.

Two-loop mixed QCD-EW corrections to gg → Hg

“…Each master integral, with the exception of the weight four piece of the 7-propagator non-planar integrals, has been successfully checked using the Mathematica [85] package PolyLogTools [66,67,86] to numerically compare its expression obtained via differential equations to its expression calculated through integration over Feynman parameters in multiple points inside the Euclidean region. Furthermore, the results from Feynman parameters integration (including weight four for the 7-propagator integrals) have been checked numerically against PySecDec [87][88][89][90][91][92][93] both in the Euclidean and in the Minkowski region, finding excellent agreement in all points. Finally, also the results from the differential equations have been checked in random points in the Euclidean region against FIESTA [94], finding excellent numerical agreement.…”

Pentagon functions for scattering of five massless particles

“…One of the crucial ingredients in the analytic solution of the DEs for the master integrals are initial values, see section 4.4. For the double-pentagon family we use the initial values from [56], which have been already validated independently with pySecDec [88,89] for several permutations of this topology. 11 Multi-digit numerical values of the full set of initial values presented in this paper have been already employed in [26,59], where expected physical properties have been observed.…”

Pseudo-Goldstone dark matter: gravitational waves and direct-detection blind spots

“…We illustrate this in figure 4, where we plot the spin-independent cross section (again regardless of the relic-density contribution) at tree and one-loop levels as a function of m χ and δ for two representative values of µ 3 = 50, 500 GeV. For the numerical evaluation of the various loop functions, we use the pySecDec toolbox [50,51] with FORM optimization [52][53][54] and CUBA library for multi-dimensional integration [55,56]. Figure 4 shows that the suppression of the direct-detection cross section persists also at one-loop level for moderate values of the couplings µ 3 , µ 3 , but at larger values the interference with the symmetry-breaking mass term increases the one-loop cross section significantly at large m χ .…”