DOE OSTI2012
The magnetic interactions between a fermion and an antifermion of opposite electric or color charges in the $^{1}S_{0}^{-+}$ and $^{3}P_{0}^{++}$ states with $J=0$ 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 $-\alpha ^{2}/r^{2}$, where $ \alpha $ is the coupling constant. Representing this quasipotential at short distances as $\lambda (\lambda +1)/r^{2}$ with $\lambda =(-1+\sqrt{1-4\alpha ^{2}})/2$, both $^{1}S_{0}^{-+}$ and $^{3}P_{0}^{++}$ states admit two types of eigenstates with drastically different behaviors for the radial wave function $u=r\psi $. One type of states, with $u$ growing as $r^{\lambda +1}$ at small $r$, will be called usual states. The other type of states with $u$ growing as $r^{-\lambda }$ will be called peculiar states. Both of the usual and peculiar eigenstates have admissible behaviors at short distances. Remarkably, the solutions for both sets of $^{1}S_{0}$ states can be written out analytically. The usual bound $^{1}S_{0}$ states possess attributes the same as those one usually encounters in QED and QCD, with bound state energies explicitly agreeing with the standard perturbative results through order $\alpha ^{4}$. In contrast, the peculiar bound $^{1}S_{0}$ states, yet to be observed, not only have different behaviors at the origin, but also distinctly different bound state properties (and scattering phase shifts). For the peculiar $^{1}S_{0}$ ground state of fermion-antifermion pair with fermion rest mass $m$, the root-mean-square radius is approximately $1/m$, binding energy is approximately $(2-\sqrt{2})m$, and rest mass approximately $\sqrt{2}m$. On the other hand, the $(n+1)$${}^{1}S_{0}$ peculiar state with principal quantum number $(n+1)$ is nearly degenerate in energy and approximately equal in size with the $n$$^{1}S_{0}$ usual states. For the $ {}^{3}P_{0}$ 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. As both sets of states are physically admissible, 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 that contains both the usual and peculiar states in different peculiarity sectors. Whether or not these newly-uncovered quantum-mechanically acceptable peculiar $^{1}S_{0}$ bound states and $^{3}P_{0}$ resonances for point fermion-antifermion systems correspond to physical states remains to be further investigated.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗