Nobuyuki Hamaguchi

Numerical Evaluation of Feynman Integrals by a Direct Computation Method

A purely numerical method, Direct Computation Method is applied to evaluate Feynman integrals. This method is based on the combination of an efcient numerical integration and an efcient extrapolation. In addition, high-precision arithmetic and parallelization technique can be used in this method if required. We present the recent progress in development of this method and show results such as one-loop 5-point and two-loop 3-point integrals.

Quadpack computation of Feynman loop integrals

Abstract The paper addresses a numerical computation of Feynman loop integrals, which are computed by an extrapolation to the limit as a parameter in the integrand tends to zero. An important objective is to achieve an automatic computation which is effective for a wide range of instances. Singular or near singular integrand behavior is handled via an adaptive partitioning of the domain, implemented in an iterated/repeated multivariate integration method. Integrand singularities possibly introduced via infrared (IR) divergence at the boundaries of the integration domain are addressed using a version of the Dqags algorithm from the integration package Quadpack, which uses an adaptive strategy combined with extrapolation. The latter is justified for a large class of problems by the underlying asymptotic expansions of the integration error. For IR divergent problems, an extrapolation scheme is presented based on dimensional regularization.

Transformation, reduction and extrapolation techniques for feynman loop integrals

We address the computation of Feynman loop integrals, which are required for perturbation calculations in high energy physics, as they contribute corrections to the scattering amplitude for the collision of elementary particles. Results in this field can be used in the verification of theoretical models, compared with data measured at colliders. We made a numerical computation feasible for various types of one and two-loop Feynman integrals, by parametrizing the integral to be computed and extrapolating to the limit as the parameter introduced in the denominator of the integrand tends to zero. In order to handle additional singularities at the boundaries of the integration domain, the extrapolation can be preceded by a transformation and/or by a sector decomposition. With the goal of demonstrating the applicability of the combined integration and extrapolation methods to a wide range of problems, we give a survey of earlier work and present additional applications with new results. We aim for an automatic or semi-automatic approach, in order to greatly reduce the amount of analytic manipulation required before the numeric approximation.

Numerical approach to multi-loop integrals

For the calculation of multi-loop Feynman integrals, a novel numerical method, the Direct Computation Method (DCM) is developed. It is a combination of a numerical integration and a series extrapolation. In principle, DCM can handle diagrams of arbitrary internal masses and external momenta, and can calculate integrals with general numerator function. As an example of the performance of DCM, a numerical computation of two-loop box diagrams is presented. Further discussion is given on the choice of control parameters in DCM. This method will be an indispensable tool for the higher order radiative correction when it is tested for a wider class of physical parameters and the separation of divergence is done automatically.

Interdisciplinary applications of mathematical modeling

We demonstrate applications of numerical integration and visualization algorithms in diverse fields including psychological modeling (biometrics); in high energy physics for the study of collisions of elementary particles; and in medical physics for regulating the dosage of proton beam radiation therapy. We discuss the problems and solution methods, as supported by numerical results.