Abstract
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.
