scholarly journals Numerical Evaluation of Feynman Integrals by a Direct Computation Method

2009 ◽  
Author(s):  
Fukuko Yuasa ◽  
Tadashi Ishikawa ◽  
Junpei Fujimoto ◽  
Nobuyuki HAMAGUCHI ◽  
Elise de Doncker ◽  
...  
Keyword(s):  
Numerical Evaluation ◽  
Computation Method ◽  
Direct Computation ◽  
Feynman Integrals

scholarly journals Pentagon functions for scattering of five massless particles

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
D. Chicherin ◽  
V. Sotnikov
Keyword(s):  
Phase Space ◽  
Numerical Evaluation ◽  
Public Library ◽  
Basis Set ◽  
Particle Scattering ◽  
Minimal Basis ◽  
Feynman Integrals ◽  
Analytic Expressions ◽  
Transcendental Functions ◽  
Branch Cuts

Abstract We complete the analytic calculation of the full set of two-loop Feynman integrals required for computation of massless five-particle scattering amplitudes. We employ the method of canonical differential equations to construct a minimal basis set of transcendental functions, pentagon functions, which is sufficient to express all planar and nonplanar massless five-point two-loop Feynman integrals in the whole physical phase space. We find analytic expressions for pentagon functions which are manifestly free of unphysical branch cuts. We present a public library for numerical evaluation of pentagon functions suitable for immediate phenomenological applications.

scholarly journals A semi-analytic method to compute Feynman integrals applied to four-loop corrections to the $$ \overline{\mathrm{MS}} $$-pole quark mass relation

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Matteo Fael ◽  
Fabian Lange ◽  
Kay Schönwald ◽  
Matthias Steinhauser
Keyword(s):  
Quark Mass ◽  
Numerical Evaluation ◽  
Dimensionless Parameter ◽  
Series Expansions ◽  
Analytic Method ◽  
Coefficient Function ◽  
Mass Relation ◽  
Kinematic Range ◽  
Numerical Accuracy ◽  
Feynman Integrals

Abstract We describe a method to numerically compute multi-loop integrals, depending on one dimensionless parameter x and the dimension d, in the whole kinematic range of x. The method is based on differential equations, which, however, do not require any special form, and series expansions around singular and regular points. This method provides results well suited for fast numerical evaluation and sufficiently precise for phenomenological applications. We apply the approach to four-loop on-shell integrals and compute the coefficient function of eight colour structures in the relation between the mass of a heavy quark defined in the $$ \overline{\mathrm{MS}} $$ MS ¯ and the on-shell scheme allowing for a second non-zero quark mass. We also obtain analytic results for these eight coefficient functions in terms of harmonic polylogarithms and iterated integrals. This allows for a validation of the numerical accuracy.

New Computational Methods for Solving Problems of the Astronomical Vessel Position

2005 ◽  
Vol 58 (2) ◽  
pp. 315-335 ◽  
Author(s):  
Tien-Pen Hsu ◽  
Chih-Li Chen ◽  
Jiang-Ren Chang
Keyword(s):  
Coordinate System ◽  
Computational Methods ◽  
Matrix Method ◽  
Iteration Scheme ◽  
Computation Method ◽  
Direct Computation ◽  
Celestial Navigation ◽  
Vector Algebra ◽  
The Matrix ◽  
Triangle Method

In this paper, a simplified and direct computation method formulated by the fixed coordinate system and relative meridian concept in conjunction with vector algebra is developed to deal with the classical problems of celestial navigation. It is found that the proposed approach, the Simultaneous Equal-altitude Equation Method (SEEM), can directly calculate the Astronomical Vessel Position (AVP) without an additional graphical procedure. The SEEM is not only simpler than the matrix method but is also more straightforward than the Spherical Triangle Method (STM). Due to tedious computation procedures existing in the commonly used methods for determining the AVP, a set of optimal computation procedures for the STM is also suggested. In addition, aimed at drawbacks of the intercept method, an improved approach with a new computation procedure is also presented to plot the celestial line of position without the intercept. The improved approach with iteration scheme is used to solve the AVP and validate the SEEM successfully. Methods of solving AVP problems are also discussed in detail. Finally, a benchmark example is included to demonstrate these proposed methods.

scholarly journals Causal representation of multi-loop Feynman integrands within the loop-tree duality

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
J. Jesús Aguilera-Verdugo ◽  
Roger J. Hernández-Pinto ◽  
Germán Rodrigo ◽  
German F. R. Sborlini ◽  
William J. Torres Bobadilla
Keyword(s):  
Scattering Amplitudes ◽  
Finite Fields ◽  
Numerical Evaluation ◽  
Simple Structure ◽  
Space Time ◽  
Dual Representation ◽  
Feynman Integrals ◽  
Causal Representation ◽  
Analytic Expressions ◽  
Do So

Abstract The numerical evaluation of multi-loop scattering amplitudes in the Feynman representation usually requires to deal with both physical (causal) and unphysical (non-causal) singularities. The loop-tree duality (LTD) offers a powerful framework to easily characterise and distinguish these two types of singularities, and then simplify analytically the underling expressions. In this paper, we work explicitly on the dual representation of multi-loop Feynman integrals generated from three parent topologies, which we refer to as Maximal, Next-to-Maximal and Next-to-Next-to-Maximal loop topologies. In particular, we aim at expressing these dual contributions, independently of the number of loops and internal configurations, in terms of causal propagators only. Thus, providing very compact and causal integrand representations to all orders. In order to do so, we reconstruct their analytic expressions from numerical evaluation over finite fields. This procedure implicitly cancels out all unphysical singularities. We also interpret the result in terms of entangled causal thresholds. In view of the simple structure of the dual expressions, we integrate them numerically up to four loops in integer space-time dimensions, taking advantage of their smooth behaviour at integrand level.

Fermion Spin Amplitudes: a Direct Computation Method

2009 ◽  
Author(s):  
Matías Moreno ◽  
Rodolfo Leo ◽  
Genaro Toledo ◽  
Gabriel López-Castro
Keyword(s):  
Computation Method ◽  
Direct Computation

scholarly journals Matrix Approach To The Direct Computation Method For The Solution of Fredholm Integro-Differential Equations of The Second Kind With Degenerate Kernels

CAUCHY ◽  
2020 ◽  
Vol 6 (3) ◽  
pp. 100-108
Author(s):  
Nathaniel Mahwash Kamoh ◽  
Geoffrey Kumlengand ◽  
Joshua Sunday
Keyword(s):  
Differential Equations ◽  
Computation Method ◽  
Matrix Approach ◽  
Algebraic Equations ◽  
Direct Computation ◽  
Closed Form Solutions ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Sum Of Products ◽  
Finite Sum

In this paper, a matrix approach to the direct computation method for solving Fredholm integro-differential equations (FIDEs) of the second kind with degenerate kernels is presented. Our approach consists of reducing the problem to a set of linear algebraic equations by approximating the kernel with a finite sum of products and determining the unknown constants by the matrix approach. The proposed method is simple, efficient and accurate; it approximates the solutions exactly with the closed form solutions. Some problems are considered using maple programme to illustrate the simplicity, efficiency and accuracy of the proposed method.

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