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 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.

Comparison Of Finite-Difference And Analytic Microwave Calculation Methods

1996 ◽  
Vol 430 ◽  
Author(s):  
F. I. Friedlander ◽  
H. W. Jackson ◽  
M. Barmatz ◽  
P. Wagner
Keyword(s):  
Finite Difference ◽  
Numerical Evaluation ◽  
Normal Modes ◽  
Materials Processing ◽  
Calculation Methods ◽  
Power Absorption ◽  
Microwave Cavities ◽  
Analytic Expressions ◽  
Lossy Dielectric ◽  
Electromagnetic Solver

AbstractNormal modes and power absorption distributions in microwave cavities containing lossy dielectric samples were calculated for problems of interest in materials processing. The calculations were performed both using a commercially available finite-difference electromagnetic solver and by numerical evaluation of exact analytic expressions. Results obtained by the two methods applied to identical physical situations were compared. Our studies validate the accuracy of the finite-difference electromagnetic solver. Relative advantages of the analytic and finitedifference methods are discussed.

Calculation of electrostatic potentials and fields inside zeolite cavities

1988 ◽  
Vol 53 (10) ◽  
pp. 2308-2319 ◽  
Author(s):  
János G. Ángyán ◽  
György Ferenczy ◽  
Péter Nagy ◽  
Gábor Náray-Szabó
Keyword(s):  
Ab Initio ◽  
Lone Pair ◽  
Electrostatic Potentials ◽  
Basis Set ◽  
Mean Deviation ◽  
Minimal Basis ◽  
Approximate Methods ◽  
Monopole Approximation ◽  
Van Der Waals Surface ◽  
Modified Formula

We present a modification of our bond increment method for the calculation of molecular electrostatic potentials and fields inside zeolite cavities. Introducing a variant of the Mulliken approximation for the off-diagonal matrix elements of the potential and optimizing the parameters of the modified formula, we achieved much better agreement with ab initio STO-3G minimal basis set results than with the original version. For a representative set of 10 small molecules the standard mean deviation between potentials calculated on the van der Waals surface with the ab initio and approximate methods is 9·1 kJ/mol. The relative error decreases from 21 to 9 per cent for the lone-pair regions of molecules modelling zeolite cavities. Applying the modified bond increment method for a realistic faujausite model we have found that the potential and field are almost exclusively of long-range origin. This means that, if using appropriate atomic charges, the monopole approximation gives correct results for electrostatic potentials and fields inside zeolite cavities.

scholarly journals Random versus Systematic Errors in Reaction Enthalpies Computed Using Semiempirical and Minimal Basis Set Methods

ACS Omega ◽  
2018 ◽  
Vol 3 (4) ◽  
pp. 4372-4377 ◽  
Author(s):  
Jimmy C. Kromann ◽  
Alexander Welford ◽  
Anders S. Christensen ◽  
Jan H. Jensen
Keyword(s):  
Systematic Errors ◽  
Basis Set ◽  
Minimal Basis

The statistics of stiff chain molecules I. The particle scattering factor

1970 ◽  
Vol 316 (1525) ◽  
pp. 185-199 ◽  
Keyword(s):  
Distribution Functions ◽  
Chain Model ◽  
Particle Scattering ◽  
Hindered Rotation ◽  
Chain Molecules ◽  
Chain Stiffness ◽  
Angle Scattering ◽  
Analytic Expressions ◽  
Scattering Factor ◽  
The Difference

Analytic expressions for the particle scattering factor of stiff chains have been derived, both for the wormlike and a discrete chain model with an axial symmetric potential of hindered rotation. The angular distribution functions agree well with the results of Monte Carlo calculations by Heine, Kratky & Roeppert (1962), if the chains are longer than five persistence lengths. The particle scattering factor of short chains can be well represented by the simple Guinier approximation. A transition point from the behaviour of a coil to that of the rod-like short chain sections has been determined by graphical extrapolation and appears at X a = 2.87 ± 0.05. The difference between the wormlike and the discrete chain models turned out to be smaller than 14% even for an alkane type chain with free rotation of the chain elements and decreases with increasing chain stiffness. The influence of the cross-section has been taken into account by representing the chain by a pearl necklace. Comparison with X -ray small angle scattering measurements of a cellulosetricarbanilate reveals close similarities between calculated and experimental curves.

Ab initio calculations on tetramethoxymethane

1988 ◽  
Vol 66 (8) ◽  
pp. 2041-2044 ◽  
Author(s):  
R. J. McEachern ◽  
J. A. Weil ◽  
P. G. Mezey
Keyword(s):  
Dipole Moment ◽  
Ab Initio ◽  
Electron Diffraction Study ◽  
Energy Difference ◽  
The Other ◽  
Geometric Configuration ◽  
Basis Set ◽  
Mo Calculations ◽  
Minimal Basis ◽  
Atom System

Minimal basis set ab initio SCF-MO calculations were performed on the 21-atom system of tetramethoxymethane (tetramethyl orthocarbonate). The geometric configuration of this model was optimized in two conformations, one having quasi-S4 symmetry and the other D2d symmetry. The S4 conformation was found to be 8 kJ mol−1 lower in energy than the D2d conformation, at the STO-3G level. The calculated energy difference is consistent with the recently measured geometric configuration of crystalline tetrabenzyl orthocarbonate. The calculated values of the bond lengths and angles were compared to the results of an electron diffraction study of the methyl species, and agree well with experiment. The theoretical electric dipole moment was calculated to be 0.01 D.

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