scholarly journals Bound states of dtμ, pdμ and t pμ mesomolecules

2019 ◽  
Vol 204 ◽  
pp. 05006 ◽  
Author(s):  
A. V. Eskin ◽  
V. I. Korobov ◽  
A. P. Martynenko ◽  
V. V. Sorokin
Keyword(s):  
Numerical Calculation ◽  
Bound States ◽  
Computer Code ◽  
Bound State ◽  
Wave Functions ◽  
Matrix Elements ◽  
Muonic Molecule ◽  
Gaussian Form ◽  
Stochastic Variational Method ◽  
The Matrix

The energy spectrum of excited bound states of muonic molecules ptμ, pdμ, and dtμ is calculated on the basis of the stochastic variational method. The basis wave functions of the muonic molecule are taken in the Gaussian form. The matrix elements of the Hamiltonian are calculated analytically. For numerical calculation, a computer code was written in the MATLAB system. As a result, the numerical values of bound state energies for excited P-states of muonic molecules ptμ, pdμ and dtμ were obtained.

scholarly journals Energy levels in muonic helium

2019 ◽  
Vol 222 ◽  
pp. 03009
Author(s):  
A.V. Eskin ◽  
V.I. Korobov ◽  
A.P. Martynenko ◽  
V.V. Sorokin
Keyword(s):  
Hyperfine Structure ◽  
Bound States ◽  
Energy Levels ◽  
Computer Code ◽  
Bound State ◽  
Relativistic Correction ◽  
Matrix Elements ◽  
Gaussian Form ◽  
Stochastic Variational Method ◽  
The Matrix

The energy spectrum of bound states and hyperfine structure of muonic helium is calculated on the basis of stochastic variational method. The basis wave functions of muonic helium are taken in the Gaussian form. The matrix elements of the Hamiltonian are calculated analytically. For numerical calculation a computer code is written in the MATLAB system. As a result, numerical values of bound state energies and hyperfine structure are obtained. We calculate also correction to the structure of the nucleus, vacuum polarization and relativistic correction.

scholarly journals Hyperfine structure of muonic mesomolecules tdµ, tpµ, dpµ in variational method

2019 ◽  
Vol 222 ◽  
pp. 03011
Author(s):  
A.V. Eskin ◽  
V.I. Korobov ◽  
A.P. Martynenko ◽  
V.V. Sorokin
Keyword(s):  
Variational Method ◽  
Hyperfine Structure ◽  
Vacuum Polarization ◽  
Energy Levels ◽  
Computer Code ◽  
Wave Functions ◽  
Matrix Elements ◽  
Gaussian Form ◽  
Stochastic Variational Method ◽  
The Matrix

The hyperfine structure of energy levels of muonic molecules tdµ, tpµ and dpµ is calculated on the basis of stochastic variational method. The basis wave functions are taken in the Gaussian form. The matrix elements of the Hamiltonian are calculated analytically. Vacuum polarization, relativistic and nuclear structure corrections are taken into account to increase the accuracy. For numerical calculation, a computer code is written in the MATLAB system. Numerical values of energy levels of hyperfine structure in muonic molecules tdµ, tpµ and dpµ are obtained.

Mikroskopische Theorie der Kernreaktionen für einen Mehrteilchen-Aufbruch / Microscopic Theory of Nuclear Reactions for a Multi-particle-break-up

1974 ◽  
Vol 29 (6) ◽  
pp. 859-866 ◽  
Author(s):  
A. Grauel
Keyword(s):  
Nuclear Reactions ◽  
Microscopic Theory ◽  
Bound State ◽  
Wave Functions ◽  
Matrix Elements ◽  
S Matrix ◽  
The Matrix ◽  
Nucleon Emission ◽  
The Continuum ◽  
Continuum Wave

Introducing correlated continuum wave functions for the two- and re-particle-continuum a microscopic theory of nuclear reactions based on a method of Fano is developed. The S-matrix-elements are given by the matrix-elements between correlated continuum wave functions and bound state wave functions. The antisymmetrization of the continuum wave functions with more than one particle in the continuum is included. The theory can be straightforwardly applied on the n-nucleon-emission process following photo- and particle excitations.

scholarly journals Mirror channel eigenvectors of the d-dimensional fishnets

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Sergey Derkachov ◽  
Gwenaël Ferrando ◽  
Enrico Olivucci
Keyword(s):  
Integral Operators ◽  
Bound State ◽  
Internal Symmetry ◽  
Wave Functions ◽  
Matrix Elements ◽  
Feynman Integrals ◽  
R Matrix ◽  
Quantum Numbers ◽  
The Matrix ◽  
State Index

Abstract We present a basis of eigenvectors for the graph building operators acting along the mirror channel of planar fishnet Feynman integrals in d-dimensions. The eigenvectors of a fishnet lattice of length N depend on a set of N quantum numbers (uk, lk ), each associated with the rapidity and bound-state index of a lattice excitation. Each excitation is a particle in (1 + 1)-dimensions with O(d) internal symmetry, and the wave-functions are formally constructed with a set of creation/annihilation operators that satisfy the corresponding Zamolodchikovs-Faddeev algebra. These properties are proved via the representation, new to our knowledge, of the matrix elements of the fused R-matrix with O(d) symmetry as integral operators on the functions of two spacetime points. The spectral decomposition of a fishnet integral we achieved can be applied to the computation of Basso-Dixon integrals in higher dimensions.

RETARDATION EFFECTS IN COVARIANT THREE-DIMENSIONAL BOUND STATE WAVE FUNCTIONS

2005 ◽  
Vol 14 (06) ◽  
pp. 931-947 ◽  
Author(s):  
F. PILOTTO ◽  
M. DILLIG
Keyword(s):  
Bound States ◽  
Three Dimensional ◽  
Bound State ◽  
Relative Energy ◽  
Wave Functions ◽  
Three Dimensions ◽  
State Wave ◽  
Scalar Diquark ◽  
Bound State Wave Function ◽  
Retardation Effects

We investigate the influence of retardation effects on covariant 3-dimensional wave functions for bound hadrons. Within a quark-(scalar) diquark representation of a baryon, the four-dimensional Bethe–Salpeter equation is solved for a 1-rank separable kernel which simulates Coulombic attraction and confinement. We project the manifestly covariant bound state wave function into three dimensions upon integrating out the non-static energy dependence and compare it with solutions of three-dimensional quasi-potential equations obtained from different kinematical projections on the relative energy variable. We find that for long-range interactions, as characteristic in QCD, retardation effects in bound states are of crucial importance.

THE BETHE–SALPETER EQUATION APPROACH TO BOUND STATE: SCALAR–FERMION CASE AND SCALAR–SCALAR CASE

2007 ◽  
Vol 22 (39) ◽  
pp. 2979-2992 ◽  
Author(s):  
JIAO-KAI CHEN ◽  
ZHENG-XIN TANG ◽  
QING-DONG CHEN
Keyword(s):  
Bound States ◽  
Bound State ◽  
Wave Functions ◽  
Scalar Case ◽  
Equation Approach

The general form of the Bethe–Salpeter wave functions for bound states comprising one scalar constituent and one fermion, or two scalars is presented. Using the reduced Salpeter equation obtained, we can work out the effective nonrelativistic potentials. And one new version of reduced Bethe–Salpeter equation is proposed by extending Gross approximation.

scholarly journals ACCURACY OF APPROXIMATE EIGENSTATES

2000 ◽  
Vol 15 (20) ◽  
pp. 3221-3235 ◽  
Author(s):  
WOLFGANG LUCHA ◽  
FRANZ F. SCHÖBERL
Keyword(s):  
Eigenvalue Problem ◽  
Mass Spectra ◽  
Bound States ◽  
Main Tool ◽  
Bound State ◽  
Variational Techniques ◽  
The Matrix ◽  
Reasonable Choice ◽  
Unperturbed Solution ◽  
New Criteria

Besides perturbation theory, which requires the knowledge of the exact unperturbed solution, variational techniques represent the main tool for any investigation of the eigenvalue problem of some semibounded operator H in quantum theory. For a reasonable choice of the employed trial subspace of the domain of H, the lowest eigenvalues of H can be located with acceptable precision whereas the trial-subspace vectors corresponding to these eigenvalues approximate, in general, the exact eigenstates of H with much less accuracy. Accordingly, various measures for the accuracy of approximate eigenstates derived by variational techniques are scrutinized. In particular, the matrix elements of the commutator of the operator H and (suitably chosen) different operators with respect to degenerate approximate eigenstates of H obtained by the variational methods are proposed as new criteria for the accuracy of variational eigenstates. These considerations are applied to that Hamiltonian the eigenvalue problem of which defines the spinless Salpeter equation. This bound-state wave equation may be regarded as the most straightforward relativistic generalization of the usual nonrelativistic Schrödinger formalism, and is frequently used to describe, e.g. spin-averaged mass spectra of bound states of quarks.

Dynamical variables in the Bethe-Salpeter formalism

1955 ◽  
Vol 233 (1193) ◽  
pp. 248-266 ◽  
Keyword(s):  
Matrix Element ◽  
Bound States ◽  
Dynamical Variable ◽  
Wave Functions ◽  
Coupling Constant ◽  
Scattering States ◽  
The Matrix ◽  
The Masses

It is shown that a knowledge of the behaviour of the propagators around their singularities enables one to determine not only the masses of bound states, but also the matrix element of any dynamical variable between two bound states. One is thus enabled to find such a matrix element, to any order in the coupling constant, by the integration of certain expressions over the corresponding Bethe-Salpeter wave-functions. As a consequence, it is possible to find normalization and orthogonality properties of these wave-functions, which in turn lead to the condition which must be imposed on their singularities a t the origin. More light is thus shed on Goldstein’s difficulty concerning the existence of a continuous infinity of bound states. The formalism is extended to scattering states in which some of the particles may be composite—in particular, an expression for the S -matrix is obtained

scholarly journals TOWARDS A THREE-DIMENSIONAL SOLUTION FOR 3N BOUND STATES WITH 3NFs

2009 ◽  
Vol 24 (11n13) ◽  
pp. 816-822 ◽  
Author(s):  
M. R. HADIZADEH ◽  
S. BAYEGAN
Keyword(s):  
Partial Wave ◽  
Integral Equations ◽  
Bound States ◽  
Three Dimensional ◽  
Bound State ◽  
Matrix Elements ◽  
Dimensional Solution ◽  
Dimensional Integral

After a brief discussion about the necessity of using the 3D approach, we present the non partial wave (PW) formalism for 3N bound state with the inclusion of 3N force (3NF). As an example the evaluation of 3NF matrix elements, which appear in the obtained coupled three-dimensional integral equations, for 2π-exchange Tucson–Melbourne 3NF show how would be this formalism efficient and less cumbersome in comparison with the PW formalism.

scholarly journals Quarkonium Production: Velocity-Scaling Rules and Long-Distance Matrix Elements

1997 ◽  
Vol 12 (22) ◽  
pp. 3951-3963 ◽  
Author(s):  
Gerhard A. Schuler
Keyword(s):  
Heavy Quark ◽  
Bound States ◽  
Potential Model ◽  
Distance Matrix ◽  
Bound State ◽  
Matrix Elements ◽  
Long Distance ◽  
Quarkonium Production ◽  
Quark Potential ◽  
Colour Octet

The hierarchy of long-distance matrix elements (MEs) for quarkonium production depends on their scaling with the velocity v of the heavy quark in the bound state. Ranges for the velocities in various bound states and uncertainties of colour-singlet MEs are estimated in a quark-potential model. Different possibilities for the scaling with v of the MEs are discussed; they depend on the actual values of v and the QCD scale. As an application, J/ψ polarization in e+e- annihilation is discussed. The first non-perturbative estimates of colour-octet MEs are presented and compared with phenomenological determinations. Finally, various predictions of prompt quarkonium production at LEP are compared.

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