scholarly journals Two-loop integrals for planar five-point one-mass processes

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Samuel Abreu ◽  
Harald Ita ◽  
Francesco Moriello ◽  
Ben Page ◽  
Wladimir Tschernow ◽  
...  
Keyword(s):  
Scattering Amplitudes ◽  
Power Series ◽  
Differential Equations ◽  
Finite Fields ◽  
W Boson ◽  
Boson Production ◽  
Important Ingredient ◽  
Feynman Integrals ◽  
Generalized Power Series ◽  
Numerical Precision

Abstract We present the computation of a full set of planar five-point two-loop master integrals with one external mass. These integrals are an important ingredient for two-loop scattering amplitudes for two-jet-associated W-boson production at leading color in QCD. We provide a set of pure integrals together with differential equations in canonical form. We obtain analytic differential equations efficiently from numerical samples over finite fields, fitting an ansatz built from symbol letters. The symbol alphabet itself is constructed from cut differential equations and we find that it can be written in a remarkably compact form. We comment on the analytic properties of the integrals and confirm the extended Steinmann relations, which govern the double discontinuities of Feynman integrals, to all orders in ϵ. We solve the differential equations in terms of generalized power series on single-parameter contours in the space of Mandelstam invariants. This form of the solution trivializes the analytic continuation and the integrals can be evaluated in all kinematic regions with arbitrary numerical precision.

scholarly journals Integrating linear ordinary fourth-order differential equations in the MAPLE programming environment

2021 ◽  
Vol 3 (4 (111)) ◽  
pp. 51-57
Author(s):  
Irina Belyaeva ◽  
Igor Kirichenko ◽  
Oleh Ptashnyi ◽  
Natalia Chekanova ◽  
Tetiana Yarkho
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fourth Order ◽  
Eigenvalue Problems ◽  
Boundary Problems ◽  
Generalized Power Series ◽  
Electronic Computing ◽  
Santa Monica ◽  
Ordinary Power ◽  
Special Points

This paper reports a method to solve ordinary fourth-order differential equations in the form of ordinary power series and, for the case of regular special points, in the form of generalized power series. An algorithm has been constructed and a program has been developed in the MAPLE environment (Waterloo, Ontario, Canada) in order to solve the fourth-order differential equations. All types of solutions depending on the roots of the governing equation have been considered. The examples of solutions to the fourth-order differential equations are given; they have been compared with the results available in the literature that demonstrate excellent agreement with the calculations reported here, which confirms the effectiveness of the developed programs. A special feature of this work is that the accuracy of the results is controlled by the number of terms in the power series and the number of symbols (up to 20) in decimal mantissa in numerical calculations. Therefore, almost any accuracy allowed for a given electronic computing machine or computer is achievable. The proposed symbolic-numerical method and the work program could be successfully used for solving eigenvalue problems, in which controlled accuracy is very important as the eigenfunctions are extremely (exponentially) sensitive to the accuracy of eigenvalues found. The developed algorithm could be implemented in other known computer algebra packages such as REDUCE (Santa Monica, CA), MATHEMATICA (USA), MAXIMA (USA), and others. The program for solving ordinary fourth-order differential equations could be used to construct Green’s functions of boundary problems, to solve differential equations with private derivatives, a system of Hamilton’s differential equations, and other problems related to mathematical physics.

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.

scholarly journals The Wilson-loop d log representation for Feynman integrals

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Song He ◽  
Zhenjie Li ◽  
Yichao Tang ◽  
Qinglin Yang
Keyword(s):  
Scattering Amplitudes ◽  
Differential Equations ◽  
Integral Equations ◽  
Wilson Loop ◽  
Boundary Term ◽  
Feynman Integrals ◽  
Four Dimensions ◽  
Loop Integral ◽  
Six Dimensions

Abstract We introduce and study a so-called Wilson-loop d log representation of certain Feynman integrals for scattering amplitudes in $$ \mathcal{N} $$ N = 4 SYM and beyond, which makes their evaluation completely straightforward. Such a representation was motivated by the dual Wilson loop picture, and it can also be derived by partial Feynman parametrization of loop integrals. We first introduce it for the simplest one-loop examples, the chiral pentagon in four dimensions and the three-mass-easy hexagon in six dimensions, which are represented by two- and three-fold d log integrals that are nicely related to each other. For multi-loop examples, we write the L-loop generalized penta-ladders as 2(L − 1)-fold d log integrals of some one-loop integral, so that once the latter is known, the integration can be performed in a systematic way. In particular, we write the eight-point penta-ladder as a 2L-fold d log integral whose symbol can be computed without performing any integration; we also obtain the last entries and the symbol alphabet of these integrals. Similarly we study the symbol of the seven-point double-penta-ladder, which is represented by a 2(L − 1)-fold integral of a hexagon; the latter can be written as a two-fold d log integral plus a boundary term. We comment on the relation of our representation to differential equations and resumming the ladders by solving certain integral equations.

W± boson production at the e+e−, γe and γγ colliding beams

1983 ◽  
Vol 228 (2) ◽  
pp. 285-300 ◽  
Author(s):  
I.F. Ginzburg ◽  
G.L. Kotkin ◽  
S.L. Panfil ◽  
V.G. Serbo
Keyword(s):  
W Boson ◽  
Boson Production ◽  
Colliding Beams ◽  
W Boson Production
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