Horace W. Crater, Cheuk-Yin Wong
The magnetic interaction between a fermion and an antifermion of 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 $2/r^{2}$ at short distances (of about $10^{-2}$-$10^{-3}$ fermis), leading to a barrier between the short-distance region and the long-distance region. In the two body Dirac equations formulated in constraint dynamics, such a short-distance attraction for this ${}^{3}P_{0}$ state leads to a quasipotential that behaves near the origin as $-\alpha ^{2}/r^{2}$, where $ \alpha $ is the coupling constant. Representing this quasipotential as $\lambda (\lambda+1)/r^{2}$ with $\lambda =(-1+\sqrt{1-4\alpha ^{2}})/2$, the solution of the $^3P_0$ states admits two solutions for the radial part of the relative wave function $u=r\psi $. One solution, which we call the usual solution, grows as $r^{\lambda +1}$ while the other solution, which we call the peculiar solution, grows as $r^{-\lambda}$. Both of these solutions have admissible behaviors at short distances. While the usual solution leads to no resonant behavior, we find a resonance for the peculiar solution whose energy depends on the description of the internal structure of the charges, the mass of the constituent, and the coupling constant. Whether or not these quantum-mechanically acceptable resonances correspond to physical states remains to be further investigated.
View original:
http://arxiv.org/abs/1203.0687
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