E. de Doncker

Loop integration results using numerical extrapolation for a non-scalar integral

Abstract Loop integration results have been obtained using numerical integration and extrapolation. An extrapolation to the limit is performed with respect to a parameter in the integrand which tends to zero. Results are given for a non-scalar four-point diagram. Extensions to accommodate loop integration by existing integration packages are also discussed. These include: using previously generated partitions of the domain and roundoff error guards.

Numerical computation of two-loop box diagrams with masses

Abstract A new approach is presented to evaluate multi-loop integrals, which appear in the calculation of cross-sections in high-energy physics. It relies on a fully numerical method and is applicable to a wide class of integrals with various mass configurations. As an example, the computation of two-loop planar and non-planar box diagrams is shown. The results are confirmed by comparisons with other techniques, including the reduction method, and by a consistency check using the dispersion relation.

Adaptive integration using evolutionary strategies

Multivariate integration problems arising in the real world often lead to computationally intensive numerical solutions. If the singularities and/or peaks in the integrand are not known a priori, the use of adaptive methods is recommended. The efficiency of adaptive methods depends heavily on focusing on the sub-regions that contain singularities or peaks in the integrands. In this paper, we present techniques based on evolutionary strategies that can be used to identify such subregions. Adaptive integration algorithms and evolutionary strategies can be parallelized easily and hence combining the parallel implementations of these result in efficient parallel adaptive integration algorithms.

Regularization of IR divergent loop integrals

We report results of a new numerical regularization technique for infrared (IR) divergent loop integrals using dimensional regularization, where a positive regularization parameter ?, satisfying that the dimension d = 4 + 2?, is introduced in the integrand to keep the integral from diverging as long as ? > 0. A sequence of integrals is computed for decreasing values of ?, in order to carry out a linear extrapolation as ? ? 0. Each integral in the sequence is calculated according to the Direct Computation Method (DCM) to handle (threshold) integrand singularities in the interior of the domain. The technique of this paper is applied to one-loop N-point functions. In order to simplify the computation of the integrals for small ?, particularly in the case of a threshold singularity, a reduction of the N-point function is performed numerically to a set of 3-point and 4-point integrals, and DCM is applied to the resulting vertex and box integrals.

Grid-based numerical integration and visualization

A grid service, called integration service (PI), is used to solve numerical integration problems which are computationally intensive. Remote visualization helps monitor the progress of the computation, and gives a vivid view of the shape of a multidimensional function. The data are filtered by the server and transferred to the client, which is responsible for visualization mapping and rendering. Both numerical integration and its visualization are Web service-based to ensure interoperability.

Distributed and multi-core computation of 2-loop integrals

For an automatic computation of Feynman loop integrals in the physical region we rely on an extrapolation technique where the integrals of the sequence are obtained with iterated/repeated adaptive methods from the QUADPACK 1D quadrature package. The integration rule evaluations in the outer level, corresponding to independent inner integral approximations, are assigned to threads dynamically via the OpenMP runtime in the parallel implementation. Furthermore, multi-level (nested) parallelism enables an efficient utilization of hyperthreading or larger numbers of cores. For a class of loop integrals in the unphysical region, which do not suffer from singularities in the interior of the integration domain, we find that the distributed adaptive integration methods in the multivariate PARINT package are highly efficient and accurate. We apply these techniques without resorting to integral transformations and report on the capabilities of the algorithms and the parallel performance for a test set including various types of two-loop integrals.

Regularization with numerical extrapolation for finite and UV-divergent multi-loop integrals

Abstract We give numerical integration results for Feynman loop diagrams such as those covered by Laporta (2000) and by Baikov and Chetyrkin (2010), and which may give rise to loop integrals with UV singularities. We explore automatic adaptive integration using multivariate techniques from the ParInt package for multivariate integration, as well as iterated integration with programs from the Quadpack package, and a trapezoidal method based on a double exponential transformation. ParInt is layered over MPI (Message Passing Interface), and incorporates advanced parallel/distributed techniques including load balancing among processes that may be distributed over a cluster or a network/grid of nodes. Results are included for 2-loop vertex and box diagrams and for sets of 2-, 3- and 4-loop self-energy diagrams with or without UV terms. Numerical regularization of integrals with singular terms is achieved by linear and non-linear extrapolation methods.

Automatic numerical integration methods for Feynman integrals through 3-loop

We give numerical integration results for Feynman loop diagrams through 3-loop such as those covered by Laporta [1]. The methods are based on automatic adaptive integration, using iterated integration and extrapolation with programs from the QUADPACK package, or multivariate techniques from the ParInt package. The Dqags algorithm from QuadPack accommodates boundary singularities of fairly general types. PARINT is a package for multivariate integration layered over MPI (Message Passing Interface), which runs on clusters and incorporates advanced parallel/distributed techniques such as load balancing among processes that may be distributed over a network of nodes. Results are included for 3-loop self-energy diagrams without IR (infra-red) or UV (ultra-violet) singularities. A procedure based on iterated integration and extrapolation yields a novel method of numerical regularization for integrals with UV terms, and is applied to a set of 2-loop self-energy diagrams with UV singularities.

Scalable Software for Multivariate Integration on Hybrid Platforms

The paper describes the software infrastructure of the PARINT package for multivariate numerical integration, layered over a hybrid parallel environment with distributed memory computations (on MPI). The parallel problem distribution is typically performed on the region level in the adaptive partitioning procedure. Our objective has been to provide the end-user with state of the art problem solving power packaged as portable software. We will give test results of the multivariate ParInt engine, with significant speedups for a set of 3-loop Feynman integrals. An extrapolation with respect to the dimensional regularization parameter (e) is applied to sequences of multivariate ParInt results Q(e) to obtain the leading asymptotic expansion coefficients as e → 0. This paper further introduces a novel method for a parallel computation of the Q(e) sequence as the components of the integral of a vector function.

Multi-threaded adaptive extrapolation procedure for Feynman loop integrals in the physical region

Feynman loop integrals appear in higher order corrections of interaction cross section calculations in perturbative quantum field theory. The integrals are computationally intensive especially in view of singularities which may occur within the integration domain. For the treatment of threshold and infrared singularities we developed techniques using iterated (repeated) adaptive integration and extrapolation. In this paper we describe a shared memory parallelization and its application to one- and two-loop problems, by multi-threading in the outer integrations of the iterated integral. The implementation is layered over OpenMP and retains the adaptive procedure of the sequential method exactly. We give performance results for loop integrals associated with various types of diagrams including one-loop box, pentagon, two-loop self-energy and two-loop vertex diagrams.