Two-body Dirac equation for semirelativistic quarks

Abstract

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 $r$.