scholarly journals Worldline master formulas for the dressed electron propagator. Part 2. On-shell amplitudes

2022 ◽  
Vol 2022 (1) ◽  
Author(s):  
N. Ahmadiniaz ◽  
V. M. Banda Guzmán ◽  
F. Bastianelli ◽  
O. Corradini ◽  
J. P. Edwards ◽  
...  
Keyword(s):  
Cross Sections ◽  
Drop Out ◽  
Matrix Elements ◽  
Fermion Propagator ◽  
Particle Path ◽  
Path Integral Representation ◽  
Fermion Line ◽  
Electron Propagator ◽  
The Matrix ◽  
Standard Linear

Abstract In the first part of this series, we employed the second-order formalism and the “symbol” map to construct a particle path-integral representation of the electron propagator in a background electromagnetic field, suitable for open fermion-line calculations. Its main advantages are the avoidance of long products of Dirac matrices, and its ability to unify whole sets of Feynman diagrams related by permutation of photon legs along the fermion lines. We obtained a Bern-Kosower type master formula for the fermion propagator, dressed with N photons, in terms of the “N-photon kernel,” where this kernel appears also in “subleading” terms involving only N − 1 of the N photons.In this sequel, we focus on the application of the formalism to the calculation of on-shell amplitudes and cross sections. Universal formulas are obtained for the fully polarised matrix elements of the fermion propagator dressed with an arbitrary number of photons, as well as for the corresponding spin-averaged cross sections. A major simplification of the on-shell case is that the subleading terms drop out, but we also pinpoint other, less obvious simplifications.We use integration by parts to achieve manifest transversality of these amplitudes at the integrand level and exploit this property using the spinor helicity technique. We give a simple proof of the vanishing of the matrix element for “all +” photon helicities in the massless case, and find a novel relation between the scalar and spinor spin-averaged cross sections in the massive case. Testing the formalism on the standard linear Compton scattering process, we find that it reproduces the known results with remarkable efficiency. Further applications and generalisations are pointed out.

Electromagnetic fields in the presence of cylindrical objects of arbitrary physical properties and cross sections

1970 ◽  
Vol 48 (15) ◽  
pp. 1789-1798 ◽  
Author(s):  
L. Shafai
Keyword(s):  
Physical Properties ◽  
Electromagnetic Fields ◽  
Cross Sections ◽  
Approximate Solutions ◽  
Scattering Matrix ◽  
Two Dimensional ◽  
Matrix Elements ◽  
Approximate Evaluation ◽  
Permittivity And Permeability ◽  
The Matrix

Approximate solutions for two-dimensional problems of electromagnetic fields in the presence of cylindrical objects have been found by approximate evaluation of a scattering matrix. The equations are derived for cylindrical objects of arbitrary physical properties and cross sections and a procedure for evaluation of the matrix elements is discussed. The elements of permittivity and permeability tensors are assumed to be analytic, but otherwise arbitrary functions of the transverse coordinates.

scholarly journals Matrix Elements of the Boltzmann Collision Operator in a Basis Determined by an Anisotropic Maxwellian Weight Function including Drift

1980 ◽  
Vol 33 (2) ◽  
pp. 449 ◽  
Author(s):  
Kailash Kumar
Keyword(s):  
Weight Function ◽  
Cross Sections ◽  
Collision Operator ◽  
Weight Functions ◽  
Anisotropic Pressure ◽  
Matrix Elements ◽  
Methods Of Calculation ◽  
Special Cases ◽  
The Matrix ◽  
Boltzmann Collision Operator

The matrix elements of the linear Boltzmann collision operator are calculated in a Burnett-function basis determined by a weight function which itself describes a velocity distribution with a net drift and an anisotropic pressure (or temperature) tensor. Three different methods of calculation are described, leading to three different types of formulae. Two of these involve infinite summations, while the third involves only finite sums, but at the cost of greater complications in the summands and the integrals over cross sections. Both elastic and inelastic collisions are treated. Special cases arising from particular choices of the parameters in the weight functions are pointed out. The structure of the formulae is illustrated by means of diagrams. The work is a contribution towards establishing efficient methods of calculation based upon a better understanding of the matrix elements in such bases.

Collisional depolarization of light alkali atoms excited to their lowest 2P levels

1970 ◽  
Vol 48 (24) ◽  
pp. 3047-3058 ◽  
Author(s):  
M. Elbel
Keyword(s):  
Cross Sections ◽  
Transition Matrix ◽  
Inner Product ◽  
Matrix Elements ◽  
Angular Momenta ◽  
Rank Tensor ◽  
Alkali Atoms ◽  
The Matrix ◽  
Transition Matrix Elements ◽  
Depolarization Of Light

Transition matrix elements connecting the Zeeman sublevels of the lowest p doublets in light alkali atoms have been derived using methods of steady-state collision theory. The matrix elements generally consist of two parts which, under rotations of the quantization axis with respect to the scattering plane, behave like components of a first rank and a second rank tensor, respectively. Only the second rank tensor components lead to the selection rule j, mJ↔j, −mj, whereas the first rank tensor components do not. The latter can be ascribed to a formal interaction term which is proportional to the inner product of the orbital angular momenta of the valence electron and the colliding atoms, respectively, thus accounting for molecular coupling phenomena during the collision. Finally, the transition matrix elements are used to calculate the depolarizing cross sections from the van der Waals potential.

28Si + 28Si elastic scattering at energies above the Coulomb barrier

1988 ◽  
Vol 66 (9) ◽  
pp. 813-817 ◽  
Author(s):  
Ashok Kumar ◽  
B. B. Srivastava
Keyword(s):  
Elastic Scattering ◽  
Cross Sections ◽  
Simple Procedure ◽  
Matrix Elements ◽  
Nucleon Potential ◽  
Generator Coordinate ◽  
Group Methods ◽  
The Matrix ◽  
Sharp Cutoff ◽  
Good Agreement

A relatively simple procedure has been used to calculate the differential cross sections for 28Si + 28Si elastic scattering at Elab = 67, 74, 77, and 120 MeV. The nuclear interactions have been calculated microscopically using the equivalence relation between the matrix elements of the generator coordinate and the resonating group methods from the two-nucleon potential, which explains the two-nucleon data fairly well. The absorption effects due to the opening of the nonelastic channels are taken into account approximately by the classical sharp cutoff of partial waves. The calculated results are in quite good agreement with the available experimental data at most of these energies.

ELECTRON – HYDROGEN ATOM SCATTERING IN BORN APPROXIMATION

1960 ◽  
Vol 38 (12) ◽  
pp. 1654-1660 ◽  
Author(s):  
Ta-You Wu
Keyword(s):  
Hydrogen Atom ◽  
Cross Sections ◽  
Born Approximation ◽  
Matrix Elements ◽  
Total Cross Sections ◽  
Scattering Cross ◽  
Atom Scattering ◽  
The Matrix ◽  
Scattering Cross Sections

The elastic (1s–1s) and the inelastic (1s–2s, 1s–2p) scattering cross sections in the Born approximation at energies of 1, 4, 9, 16 rydbergs have been calculated exactly from the closed formulas of the matrix elements for these transitions. Both the differential and the total cross sections are given here.

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

scholarly journals Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

2021 ◽  
Author(s):  
Mariusz Pawlak ◽  
Marcin Stachowiak
Keyword(s):  
Spherical Harmonics ◽  
Interaction Potential ◽  
Chem Phys ◽  
Basis Set ◽  
Matrix Elements ◽  
Trigonometric Functions ◽  
Variational Theory ◽  
Molecular Collision ◽  
The Matrix ◽  
Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

Comparative Isolation of Cilia and Flagella from the Lamellibranch Mollusc, Aequipecten irradians

1973 ◽  
Vol 12 (2) ◽  
pp. 345-367
Author(s):  
R. W. LINCK
Keyword(s):  
Atpase Activity ◽  
Cross Sections ◽  
Basal Body ◽  
Electron Dense Material ◽  
The Matrix ◽  
Fine Structures ◽  
Dynein Arms ◽  
Cilia And Flagella ◽  
Triton X ◽  
Ciliary Ultrastructure

Gill cilia and sperm flagella from the lamellibranch mollusc Aequipecten irradians were compared with respect to their ultrastructures and adenosinetriphosphatase activities. Cilia were isolated from excised gills using 3 different solutions: twice-concentrated seawater, 10 % ethanol-10 mM CaCl2 and 60% glycerol. In each case deciliation occurs by the severance of the cilium at the junction of the transition zone and the basal body, and in each case the ciliary ultrastructure is maintained. Sperm flagella were purified by mechanical decapitation. Cilia and sperm flagella have similar fine structures, except that the matrix of the cilia contains substantially more electron-dense material than that of flagella. The ATPase activity of purified cilia is approximately 0.09,µmol P1/min/mg protein; that of flagella is 0.13. Ciliary and flagellar axonemes were prepared by repeated extraction of the membranes with 1% Triton X-100. Ciliary axonemes maintain their 9 + 2 cylindrical orientation, whereas flagellar axonemes often appear as opened or fragmented arrays of the 9 + 2 structure, due to the partial breakdown of the flagellar nexin fibres. A-subfibre arms which were obvious in whole organelles are rarely seen in axoneme preparations. Again the ciliary matrix is considerably more amorphous than in flagellar axonemes. The ATPase activities of ciliary and flagellar axonemes are 0.13 and 0.12 µmol P1/min/mg protein respectively; however, activities of ciliary axonemes may vary by a factor of 2, depending on the method of isolation. The difficulty in observing A-subfibre arms in cross-sections of ciliary and flagellar axonemes is discussed in terms of random, non-reinforcing arrangements of the dynein arms.

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