Breit Equation Research Articles

On Symmetry of the Generalized Breit Equation

Abstract In this paper we find the complete set of symmetry operators for the two-particle Breit equation in the class of first-order differential operators with matrix coefficients. A new integral of motion is obtained.

High-Order Expansion of the Energy Eigenvalues of a Relativistic Coulomb Equation

It is shown that the Herbst equation, i.e., a Coulomb potential wave equation including a free relativistic kinetic energy, presents a rather nontrivial series expansion of its eigenvalues in powers of α. In fact, in contrast to Klein-Gordon and Dirac equations, it presents not only odd powers of α but also nonanalytic 1n α terms. The first orders (α 4, α 5) are obtainable by a standard perturbation method on the Sommerfeld correction. A much more effective and systematic method is proposed to get higher orders (α 7). To appreciate the resulting expansion, we compare it to the known results coming from other approaches to relativistic bound states, namely Klein-Gordon and Dirac equations in an external field and for the two-body problem, the Breit and Sucher-type equations, and the positronium QED result.

How to obtain a covariant Breit type equation from relativistic constraint theory

It is shown that, by an appropriate modification of the structure of the interaction potential, the Breit equation can be incorporated into a set of two compatible manifestly covariant wave equations, derived from the general rules of constraint theory. The complementary equation to the covariant Breit type equation determines the evolution law in the relative time variable. The interaction potential can be systematically calculated in perturbation theory from Feynman diagrams. The normalization condition of the Breit wavefunction is determined. The wave equation is reduced, for general classes of potential, to a single Pauli-Schrodinger-type equation. As an application of the covariant Breit type equation, we exhibit massless pseudoscalar bound-state solutions, corresponding to a particular class of confining potentials.

Relativistic calculation of ground-state baryon masses

The masses of the ground-state baryons are calculated in a relativistic quark model by using the saddle-point variational method to solve the three-body Breit equation. The saddle-point variational method maximizes the energy with respect to small component parameters, while minimizing with respect to size parameters. This removes the problems usually caused by negative-energy states, without the need for positive-energy projection operators. The variational method, applied to an asymmetric trial wave function with naturally borken SU(3) and SU(6) symmetry, permits solution of the Breit equation for realistic QCD-inspired potentials without using perturbation theory. The calculated masses are in good agreement with experiment, and the asymmetric wave function gives reasonable values for the proton and neutron charge radii.

Solution of the Kemmer and Breit equations in a circularly polarized wave

Solution of the Kemmer and Breit equations in a circularly polarized wave

A MODEL OF THE DEUTERON BASED ON A BREIT TYPE EQUATION

A relativistic treatment of the deuteron and its observables based on a two-body Dirac (Breit) equation, with phenomenological interactions, associated to one-boson exchanges with cutoff masses, is presented. The 16-component wave function for the deuteron (Jπ=1+) solution contains four independent radial functions which obey a system of four coupled differential equations of first order. This radial system is numerically integrated, from infinity to the origin, by fixing the value of the deuteron binding energy and using appropriate boundary conditions at infinity. Specific examples of mixtures containing scalar, pseudoscalar and vector like terms are discussed in some detail and several observables of the deuteron are calculated. Our treatment differs from more conventional ones in that nonrelativistic reductions of the order c−2 are not used.

Relativistic corrections to Ps -

Explicit calculations were performed to verify that a previously derived relativistic three-fermion equation [Phys. Lett. A 158 (1991) 361] contains the complete O(α 4) physics. Perturbative solutions of the ground state eigenenergy, to O(α 4), are presented for equal mass systems, such as Ps -, as well as for helium-like systems. Results for the helium-like system agree with those obtained from the Breit equation in the Pauli approximation. The relativistic corrections to the Ps - energy are very small.

Relativistic quantum theory in momentum space for weak and strong coupling.

In this paper a relativistic quantum theory introduced by de Groot and Ruijgrok [Nucl. Phys. 101B, 95 (1975)] is reformulated. It is shown that it has the correct nonrelativistic limit and that it gives the same lowest-order relativistic corrections as the Breit equation. In this paper it is applied to bound systems of two charged particles with spin 1/2. For small coupling constants the lowest-order fine structure (up to order ${\mathrm{\ensuremath{\alpha}}}^{4}$) for all states and all mass ratios is calculated. No expansion in the mass ratio is made, so that the full recoil effects are obtained. It embraces as special cases the well-established fine structure of the hydrogen atom and positronium and the Fermi splitting of muonium. For the strong-coupling case the critical couplings are calculated. The convergence to the static limit for couplings in the neighborhood of the critical point is extremely slow. The bound-state masses for states with the same total angular momentum J but opposite parity are not equal at the critical point, so there is no parity doubling. The results are compared with a number of other theories.

Effects of Negative-Energy Propagations in Deuteron-Nucleus Scattering

The effects of negative· energy propagations in the effective optical potentials for relativistic deuteron·nucleus elastic scattering are investigated utilizing the Breit approach. First, we consider the effects of the (+ -) and (- +) propagations. They represent that one of the nucleons in the deuteron propagates in its positive·energy state, while another propagates in negative·energy one. Second, the effects of the (- -) propagation, which represents that both the nucleons propagate in their negative· energy states, are studied. It has been found that the characteristic features of the optical potentials near 400 MeV - the wine·bottle·bottom·shape of the central part and the enhanced spin·orbit strength - are pure relativistic effects mainly due to the ( + -) and (- +) propagations, whereas the effects of the (- -) propagation are small. Recently, several efforts have been devoted to investigating the structures of the propagators and their effects in the relativistic nuclear scattering theories. 1 J-S) In the Dirac phenomenology for nucleon-nucleus (N-A) scattering, Cooper and Jennings have found that the negative-energy part of the Feynman propagator has short-ranged structure and plays an important role in producing the relativistic character of the effective optical potentials. 2 ) In the Breit 6 ) and Duffin-Kemmer-Petiau (DKP)1H) theories for relativistic deuteron-nucleus (d-A) scattering, the present author has found that the short-ranged structures of their propagators also contribute consider­ ably to the relativistic characters of d-A optical potentials. 5 ) However, in that work he has not been able to clarify another important aspect of the relativistic scattering theories, that is, the effect of negative-energy propagations. This is partially because that in the Breit or DKP approach short-ranged terms in the propagator are not identical with negative-energy terms, in contrast to the Feynman propagator. For the relativistic scattering of a composite particle, it becomes an important subject to study the effects of the negative-energy propagations of each constituent particle. 1OH2 ) Most of the recent studies of d-A elastic scattering at intermediate energies are inspired by the success of the Dirac phenomenology.9),13),14) Therefore, we want to clarify the effects of the negative-energy propagations of Dirac nucleons in the deuteron. For this purpose, in the present work we will treat d-A elastic scattering within a relativistic folding-model utilizing the Breit equation.

Charge asymmetry in non-relativistic nucleon-nucleon potential derived from charge symmetric relativistic interaction

Considers a relativistic model of the nucleon-nucleon (NN) system subject to an interaction which is a sum of a charge symmetric NN interaction and the Coulomb potential. When the model is rewritten in the form of a non-relativistic model, the resulting effective non-relativistic potential obtains a charge-symmetry-breaking (CSB) component. The authors illustrate this by means of the Breit equation. If the relativistic NN interaction contains a combination of a strongly attractive Lorentz scalar and a strongly repulsive vector, the non-relativistic potential obtains a large CSB component. When the magnetic interaction is included, however, the CSB effect may be reduced. This and other aspects, in which relativistic and non-relativistic models may exhibit significant differences, are discussed.

Useful radial equations for the dirac two-body problem

New second-order radial equations are derived from the two-body Dirac equation (called also the Breit equation). They are convenient for numerical calculations and may be also useful for a general discussion of the order of energy levels in quarkonia.

QED test of a Bethe-Salpeter solution method.

A method is described of solving Bethe-Salpeter equations for spin-\textonehalf{} particles using a standard quasipotential approximation with a correction series. A three-dimensional, 16-component bound-state equation is obtained whose form is different from the Breit equation. The equation has two equivalent first-order forms which show each particle obeying a Dirac equation in the presence of the other. The formalism also reconstitutes the four-dimensional bound-state vertex function from the three-dimensional wave function. The method is tested on positronium and the hydrogen atom, using a single-photon-exchange kernel in the Coulomb gauge. In the bound-state equation, in addition to the Coulomb potential and Breit interaction, the formalism gives a subtracted box potential $(\frac{1}{2E})\frac{{\ensuremath{\alpha}}^{2}}{{r}^{2}}$ where $E$ is the mass of the bound state. With this term included, the energy levels (fine structure and hyperfine structure) are correct to order ${\ensuremath{\alpha}}^{4}$. The reconstituted bound-state vertex function, when substituted in Feynman triangle diagrams, gives the lowest-order atomic dipole transition amplitudes correctly.

Erratum: Reduction of the Breit Coulomb equation to an equivalent Schrödinger equation, and investigation of the behavior of the wave function near the origin

The Breit equation for two equal-mass spin-1/2 particles interacting through an attractive Coulomb potential is separated into its angular and radial parts, obtaining coupled sets of first-order differential equations for the radial wave functions. The radial equations for the /sup 1/J/sub J/, /sup 3/J/sub J/, and /sup 3/P/sub 0/ states are further reduced to a single, one-dimensional Schroedinger equation with a relatively simple effective potential. No approximations, other than the initial one of an instantaneous Coulomb interaction, are made in deriving this equation; it accounts for all relativistic effects, as well as for mixing between different components of the wave function. Approximate solutions are derived for this Schroedinger equation, which gives the correct O(..cap alpha../sup 4/) term for the 1 /sup 1/S/sub 0/ energy and for the n/sup 1/J/sub J/ energies, for J>0. The radial equations for the /sup 3/(J +- 1)/sub J/ states are reduced to two second-order coupled equations. At small r, the Breit Coulomb wave functions behave as r/sup ..nu..//sup -1/, where ..nu.. is either ..sqrt..J(J+1)+1-..cap alpha../sup 2//4 or ..sqrt..J(J+1)-..cap alpha../sup 2//4 . The /sup 1/S/sub 0/ and /sup 3/P/sub 0/ wave functions therefore diverge at the origin as r/sup //sup ..sqrt..//sup 1-//sup ..cap alpha..//sup <2//4 -1$.more » This divergence of the J = 0 states, however, does not occur when the spin-spin interaction, -(..cap alpha../r)..cap alpha..x..cap alpha.., is added to the Coulomb potential.« less

Normalization of bound-state solutions to the Breit equation.

We derive the spin-(1/2, two-particle Breit equation from the Bethe-Salpeter equation with instantaneous interaction kernel, and solve for the relative time dependence of the resulting Bethe-Salpeter amplitude \ensuremath{\psi}(x). As t\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}, \ensuremath{\psi}(x) increases exponentially in Dirac hole theory, but goes to zero in single-electron theory. Assuming single-electron theory, a normalization condition is derived for \ensuremath{\psi}(r,0), the solution to the Breit equation, from the normalization for the Bethe-Salpeter amplitude.

Reduction of the Breit Coulomb equation to an equivalent Schrödinger equation, and investigation of the behavior of the wave function near the origin

The Breit equation for two equal-mass spin-1/2 particles interacting through an attractive Coulomb potential is separated into its angular and radial parts, obtaining coupled sets of first-order differential equations for the radial wave functions. The radial equations for the /sup 1/J/sub J/, /sup 3/J/sub J/, and /sup 3/P/sub 0/ states are further reduced to a single, one-dimensional Schroedinger equation with a relatively simple effective potential. No approximations, other than the initial one of an instantaneous Coulomb interaction, are made in deriving this equation; it accounts for all relativistic effects, as well as for mixing between different components of the wave function. Approximate solutions are derived for this Schroedinger equation, which gives the correct O(..cap alpha../sup 4/) term for the 1 /sup 1/S/sub 0/ energy and for the n/sup 1/J/sub J/ energies, for J>0. The radial equations for the /sup 3/(J +- 1)/sub J/ states are reduced to two second-order coupled equations. At small r, the Breit Coulomb wave functions behave as r/sup ..nu..//sup -1/, where ..nu.. is either ..sqrt..J(J+1)+1-..cap alpha../sup 2//4 or ..sqrt..J(J+1)-..cap alpha../sup 2//4 . The /sup 1/S/sub 0/ and /sup 3/P/sub 0/ wave functions therefore diverge at the origin as r/sup //sup ..sqrt..//sup 1-//sup ..cap alpha..//sup <2//4 -1$.more » This divergence of the J = 0 states, however, does not occur when the spin-spin interaction, -(..cap alpha../r)..cap alpha..x..cap alpha.., is added to the Coulomb potential.« less

Relativistic wave equation for two dirac particles, suggested by the Duffin-Kemmer-Petiau algebra

A new relativistic wave equation for two spin-1/2 particles is discussed which in the pointlike limit of tight bound states tends to the Duffin-Kemmer-Petiau equation for one spin-0 or spin-1 particle. The Bethe-Salpeter equation for two spin-1/2 particles can be rewritten in the new form. But, in the familiar case of an instantaneous interaction introduced through the minimal substitutionE→E−V of a «vector» potentialV, the new equation is neither equivalent to the Salpeter equation nor to the Breit equation. In particular, ifV is Coulombic atr→0, scalar states1S0 are (surprisingly) excluded from the new equation by the regularity condition atr=0, which may be a desirable feature in some composite models for W and Z bosons.

Comment on radial equations for two Dirac particles

The set of radial equations corresponding to the Breit equation is briefly commented on in connection with a recent article by Childers. The use of the Salpeter equation (generally in the form corrected according to Feldman, Fulton, and Townsend) is conservatively suggested as the proper remedy for the singular behavior of the Breit equation in the case of confining interactions.

Two-body Dirac equation for semirelativistic quarks

A semirelativistic quark model for mesons is proposed, based on a comprehensive treatment of the two-body Dirac equation introduced by Breit in 1929. Sixteen-component eigenstates of ${J}^{2}$, ${J}_{3}$, and parity are constructed. They are used to obtain radial equations for the most general local, nonderivative interaction Hamiltonian of order $\frac{{v}^{2}}{{c}^{2}}$ containing both scalar and four-vector terms. The Breit interaction is a special case of this interaction Hamiltonian, and the radial equations are considered first for that case. It has been known since 1930 that there is an ambiguity associated with the Breit equation. It arises because there are two different ways to reduce the Breit equation to four-component form. In one method the Breit interaction is regarded as a perturbation and in the other it is not. Only the first method gives correct results. This ambiguity is resolved by a consideration of the radial equations for the Breit equation. In addition, explicit solutions of the radial equations to order ${\ensuremath{\alpha}}^{4}$ are given for hydrogen and positronium. The conclusion is that the two-body Dirac equation is unambiguous and correct to order ${\ensuremath{\alpha}}^{4}$ in QED. The radial equations are used next in the context of QCD to determine which of the 24 possible combinations of scalar and vector potentials considered can give rise to quark confinement in a normalized theory. If the higher-order potentials are no larger than the zero-order potentials in magnitude, we show that there are only two possible forms for the confinement interaction: (1) the scalar interaction $S(r)$ $(1\ensuremath{-}\stackrel{^}{r}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}}_{1}\stackrel{^}{r}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}}_{2})$ ${\ensuremath{\beta}}_{1}{\ensuremath{\beta}}_{2}$ and (2) the combined scalar-vector interaction [$V(r)+S(r){\ensuremath{\beta}}_{1}{\ensuremath{\beta}}_{2}$] ($1\ensuremath{-}\stackrel{^}{r}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}}_{1}\stackrel{^}{r}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}}_{2}$) where $S\ensuremath{\ge}V$ for large $r$.

Relativistic recoil and spin corrections to atomic and Coulomb wave functions

A new wave equation for the relativistic two-body problem is derived and shown to be equivalent to the Breit equation and also to the “quasipotential” equation. Its unique advantage is that it provides analytic wave functions for arbitrary combinations of masses, spins, magnetic and higher moments of the two particles. Various applications are indicated, including an improvement of Dirac wave functions at small distances.

Klein paradox in the Breit equation

We demonstrate that in the Breit equation with a central potentialV(r) having the propertyV(r 0)=E there appears a Klein paradox atr=r 0. This phenomenon, besides the previously found Klein paradox arr→∞ appearing ifV(r)→∞ atr→∞, seems to indicate that in the Breit equation valid in the singleparticle theory the sea of particle-antiparticle pairs is not well separated from the considered two-body configuration. We conjecture that both phenomena should be absent from the Salpeter equation which is consistent with the hole theory. We prove this conjecture in the limit ofm (1)→∞ andm (2)→∞, where we neglect the terms ∼1/m (1) and 1/m (2). In Appendix I we show that in the Breit equation the oscillations accumulating atr=r 0 in the case ofm (1)≠m (2) are normalizable to the Dirac δ-function. In Appendix II the analogical statement is justified for the nonoscillating singular behaviour appearing atr=r 0 in the case ofm (1)=m (2).