Magnetic Resonances at Short Distances

The magnetic interaction at short distances between two opposite electric or color charges in the $^{3}P_{0}$ state $(J=0,L=1,S=1,P=1$ and $C=1)$ is very attractive and can overwhelm the centrifugal barrier at short distances (of about $10^{-2}$-$10^{-3}$ fermis), with a barrier between the short-distance region and the long-distance region. In the two body Dirac equations formulated in constraint dynamics, the short-distance attraction for this ${}^{3}P_{0}$ state admits a peculiar solution for the radial part of the relative wave function $u=r\psi $ that grows with distance as $r^{(1-\sqrt{1-4\alpha^{2}})/2}$, in addition to the usual solution that behaves as $r^{(1+\sqrt{1-4\alpha^{2}})/2}$. Both solutions have admissible behaviors at short distances and the usual solution leads to no resonant behavior. However, for the peculiar solution we find a resonance at about 28 MeV for $e^{+}e^{-}$ and a width of about 15 KeV.