New Singlet Positronium Bound State

The Two-Body Dirac equations of constraint dynamics applied to QED yield an exact Sommerfeld-like solution for the spectrum of $^{1}J_{J}$ singlet positronium states which agrees with standard perturbative results through order $\alpha ^{4}$. At short distance the bound state equation is $ (-d^{2}/dr^{2}+(J(J+1)-\alpha ^{2})/r^{2})u=0,$ and the radial part of the wave function $u=r\psi $ has two solutions with probabilities near the origin of $\psi ^{2}d^{3}r=u^{2}drd\Omega =r^{(1\pm \sqrt{(2J+1)^{2}-4a^{2}})}drd\Omega $. For $J\neq 0$ only the `$+$' sign is allowable but both signs for $J=0$ are well behaved. The `$+$' sign corresponds to ordinary positronium (with a binding energy of about 6.8 eV). The `$-$' sign corresponds to a new positronium state with a binding energy of about 300 KeV and size about a electron Compton wave length. The ordinary $1S$ positronium state decays into this new $1S$ state with a life time on the order of $10^{-3} $ seconds by two photon emission with c.m. energy of about $700$ KeV. The peculiar $1S$ state then annihilates into two photons with c.m. energy of about $300$ KeV. Thus the existence of this new positronium state would be a distinctive 4 gamma decay signature of ordinary singlet positronium.