Magnetic States at Short Distances

Abstract

The magnetic interactions between a fermion and an antifermion of opposite electric or color charges in the 1S0 and 3P0 states are very attractive and singular near the origin and may allow the formation of new bound and resonance states at short distances. In the two body Dirac equations formulated in constraint dynamics, the short-distance attraction for these states for point particles leads to a quasipotential that behaves near the origin as -α^2/r^2. Both 1S0 and 3P0 states admit two types of eigenstates with drastically different behaviors for the radial wave function u=rψ. One type of states, with u growing as r^{1+\sqrt(1-4*α^2)/2} at small r, will be called usual states. The other type of states with u growing as r^{(1-\sqrt(1-4*α^2))/2} will be called peculiar states. Both of the usual and peculiar eigenstates have admissible behaviors at short distances. The usual bound 1S0 states possess attributes the same as those one usually encounters in QED and QCD. In contrast, the peculiar bound 1S0 states, yet to be observed, have distinctly different bound state properties and scattering phase shifts. For the 3P0 states, the usual solutions lead to the standard bound state energies and no resonance, but resonances have been found for the peculiar states whose energies depend on the description of the internal structure of the charges, the mass of the constituent, and the coupling constant. The existence of both usual and peculiar eigenstates in the same system leads to the non-self-adjoint property of the mass operator and two non-orthogonal complete sets. The mass operator can be made self-adjoint with a single complete set of admissible states by introducing a new peculiarity quantum number and an enlarged Hilbert space.