Two-body Wave Equations Research Articles

Symmetries of the Relativistic Two-Boson System in External Field

We investigate the survival of symmetries in a relativistic system of two mutually interacting bosons coupled with an external field, when this field is “strongly” translation invariant in some directions and additionally remains unchanged by other isometries of spacetime. Since the relativistic interactions cannot be composed additively, it is not a priori garanteed that the two-body system inherits all the symmetries of the external potential. However, using an ansatz which permits to preserve the compatibility of the mass-shell constraints in the presence of the field, we show how the “surviving isometries” can actually be implemented in the two-body wave equations.

Variational wave equations of two fermions interacting via scalar, pseudoscalar, vector, pseudovector and tensor fields

Abstract We consider a method for deriving relativistic two-body wave equations for fermions in the coordinate representation. The Lagrangian of the theory is reformulated by eliminating the mediating fields by means of covariant Green's functions. Then, the nonlocal interaction terms in the Lagrangian are reduced to local expressions which take into account retardation effects approximately. We construct the Hamiltonian and two-fermion states of the quantized theory, employing an unconventional “empty” vacuum state, and derive relativistic two-fermion wave equations. These equations are a generalization of the Breit equation for systems with scalar, pseudoscalar, vector, pseudovector and tensor coupling.

Mass spectra and leptonic decay widths of heavy quarkonia by using psi function

In this study, a non-relativistic two-body wave equation is used to describe the properties of heavy quark–antiquark systems with a potential proportional to the ψ-function. The wave equation is transformed into a true eigenvalue equation and solved numerically. Both the resonance masses and the leptonic decay widths of and mesons are calculated. The obtained results showed that the quark–antiquark interaction in these systems could be described adequately by using this simple potential form which contains one adjustable parameter besides the quark masses.

Incorporation of anomalous magnetic moments in the two-body relativistic wave equations of constraint theory

Using a Dirac-matrix substitution rule, applied to the electric charge, the anomalous magnetic moments of fermions are incorporated in local form in the two-body relativistic wave equations of constraint theory. The structure of the resulting potential is entirely determined, up to magnetic type form factors, from that of the initial potential describing the mutual interaction in the absence of anomalous magnetic moments. The wave equations are reduced to a single eigenvalue equation in the sectors of pseudoscalar and scalar states (j=0). The requirement of a smooth introduction of the anomalous magnetic moments imposes restrictions on the behavior of the form factors near the origin, in x-space. These conditions are satisfied at the one-loop level of perturbation theory. The nonrelativistic limit of the eigenvalue equation is also studied.

Relativistic wave equation for a boson-fermion system

We present a two-body relativistic wave equation for a system composed of a boson and a fermion. One-body equations such as the Dirac and the Klein-Gordon equations are often used as an approximate equation for relativistic two-body systems. However, when the masses of two particles are not very different, the use of one-body equations comes into question. We use the Feshbach-Villars formalism for the boson so that the wave equation can be given in the form of an eigenvalue equation for the Hamiltonian. Differences between our equation and the one-body equations are examined and illustrated in a numerical example of a two-body system with scalar and vector potentials.

Two-body quantum transport theory of interacting electrons in laterally patterned semiconductor 2DEG structures

A previous coupled-mode formalism for quantum transport in semiconductor two-dimensional electron gases is extended to interacting electrons. The monomode case has similar effective potentials to the single-carrier theory arising from the curvature of the actual confinement potential. In the case of the tunnelling of two electrons through a constriction in monomode a new phenomenon of Coulomb-assisted resonant tunnelling on single barriers is proposed and confirmed by numerical solutions to the two-body wave equation.

Two-body relativistic wave equations in external potentials admitting directions of strong translation invariance

A two-body relativistic system of scalar particles, obeying a pair of coupled Klein-Gordon equations, is embedded into prescribed external potentials. The composition of these interactions can be explicitly carried out for a wide class of potentials, characterized by invariance properties along one or more space-time directions. Various possibilities are unified in a geometric treatment which contains the Bijtebier ansatz as a special case. Compatible equations for the whole system are written down in closed form. Our method is especially appropriate for the degenerate situations where the external field is not static in a unique laboratory frame. More attention is devoted to the case of a uniform constant magnetic field and, for neutral systems, we obtain a covariant law of conservation of the pseudomomentum. This result enables us to eliminate the centre-of-mass variables and the relative time.

On the occurrence of particle-antiparticle resonances in scalar QED

We derive a relativistic two-body wave equation from scalar QED (the Wick-Cutkosky model) in the one-photon exchange approximation, using a technique proposed earlier for QED. This involves the study of the equation of motion for a matrix element of the “electron” field operator between a state with two particles and another with one particle, and the application of well defined approximations. We study the L = 0 partial wave in some detail. Using a complex scaling technique we show that within the current approximation no resonances are found that can explain the long-lived states observed in heavy-ion scattering at the GSI. We comment on other calculations that have reported positive results for the same problem.

Electron-proton resonances at low energy from a relativistic two-body wave equation

>The K-matrix for electron-proton scattering is evaluated in the 0− channel from 0.05 to 5.0 eV using a relativistic wave equation derived from QED. We find very narrow resonances at Ecm=0.748, 1.347, 2.095, 3.032, and 4.707 eV. Since these are short-distance phenomena, these resonances may provide a super screening mechanism for fusion reactions.

Field-theoretic ramifications of Barut’s method

Justification of Barut’s method for constructing two-body relativistic wave equations in the framework of quantum field theory is given. A possible way of generalization of the method is indicated.

Electron-positron scattering resonances from relativistic two-body wave equations

The K-matrix for electron-positron scattering is evaluated in the 0 + channel for each of three relativistic wave equations derived from QED. When all the transverse single-photon-exchange contributions are treated we obtain zero width resonances or continuum bound states at mass values of 1.351, 1.498, 1.659, 1.830, 2.009, and 2.195 MeV. We assume that neglected processes are not crucial to the complete problem and therefore we compare these results with electron-positron coincidence peaks found in heavy ion experiments performed at GSI. We suggest additional types of experiments to search for these resonances.

Three-dimensional relativistic two-body wave equation

It is shown by a simple method that the relativistic two-body wave equation can be written rigorously in a three-dimensional form without making use of the four-dimensional Bethe-Salpeter equation and its reduction. A closed formula for the effective interaction is derived and a perturbation expansion for its evaluation developed in such a way that each order of the approximation not only is relativistic but also takes account of the retardation effect properly, no ambiguity being involved. As an illustration, an explicit expression is given for the ladder approximation.

Color vector potential effects on the glueball mass spectrum

We present a nonperturbative variational method for calculating the glueball mass with an explicit color vector potential present. Using a two-body relativistic wave equation for bound states in quantum chromodynamics we examine the gluonium state including a Darwin-type color vector potential with a QCD-motivated scalar potential. Our equations are solvable variationally for the energy, in the Coulomb gauge, with spin interactions. We find that for theS-states, where the vector potential contributions are the largest, the effect on the bound-state energy is not negligible. In particular the ground state 0++ glueball has a mass of 1215 MeV.

ENERGY LEVELS OF QUARKONIA IN POTENTIAL MODELS

We review the subject of the energy levels of light and heavy quark-antiquark bound states in potential models, with emphasis on potentials which are motivated from quantum chromodynamics. In addition to pointing out the successes of potential models, we discuss problems with the potential approach, including uncertainties in the form of the potential, its spin dependence, its transformation properties, and its nondiagonal terms. We consider two-body wave equations with both nonrelativistic and relativistic kinematics, pointing out that in the latter case, a wide variety of different wave equations have been used in the literature.

A NEW TWO-BODY WAVE EQUATION WITH RELATIVISTIC KINEMATICS

We propose a new phenomenological two-body wave equation with relativistic kinematics which has certain advantages for treating quarkonium, including quarkonium decay processes.

Observations on the two-body relativistic wave equation

Observations on the two-body relativistic wave equation

Action at a distance and relativistic wave equations for spinless quarks

Within a manifestly covariant formulation of classical dynamics, we characterize the relativistic two-body wave equations admitting a classical analog in which the interaction is realized by means of action at a distance. As an application (and for example) the mass spectrum of Gunion and Li can be obtained from such a classical system, by quantization.

Solutions to a gauge-invariant, equal-time two-body wave equation. Light-mass quark-antiquark system

We study systematically the equal-time relativistic equation with a confinement potential, its boundary conditions, and the resulting bag-type solutions and eigenvalues. We show that the flux is continuous across the bag boundary. The chiral symmetry of the equation for vanishing quark masses is spontaneously broken for $J=0$ states because of the boundary condition. We take a model potential consisting of a linear potential and a modified Coulomb potential that vanishes at the origin and show that the Nambu-Goldstone condition on the $\ensuremath{\pi}$ meson can be implemented, and that the known $I=1$ meson spectrum can be accounted for with remarkable success, provided the Coulomb coupling constant ${g}^{2}$ is chosen above the critical value 2.

Relativistic Two-Body Wave Equation and Meson Spectrum

The meson spectrum is studied with use of an equal-time relativistic two-body equation with electrostatic confinement potentials. A normalizability condition excludes all but a bag solution which gives a spectral pattern in between the SU(6) and the chiral SU(3) \ensuremath{\bigotimes} SU(3) limits. Orbital mixing creates a pair of $^{3}S_{1}+^{3}D_{1}$ states sandwiching the $^{1}P_{1}$ level. $\ensuremath{\rho}\ensuremath{-}{\ensuremath{\rho}}^{\ensuremath{'}}$ and $\frac{\ensuremath{\psi}}{J}\ensuremath{-}{\ensuremath{\psi}}^{\ensuremath{'}}$ pairs fit this picture very well.

Perturbation Theory, Bethe-Goldstone Equation, and the Binding Energy of Solid Helium

Perturbation diagrams of the Brueckner-Goldstone expansion are partially summed to derive a Bethe-Goldstone equation for the short-range correlations in solid helium. The three corresponding two-body wave equations used in binding-energy calculations by Guyer, by Horner, and by Ebner and Sung are critically compared with the present one by means of the theory of $t$-matrix propagator perturbations. This provides qualitative explanations for the discrepancies between these published calculations and experiment.