Two-photon width of singlet positronium and quarkonium with arbitrary total angular momentum

In this paper we derive ${\Gamma }_{\gamma \gamma }$ for $S=0\mathrm{}$ positronium and quarkonium states with all allowed quantum numbers ($J^{\mathrm{PC}}=0^{-+},2^{-+},4^{-+},\dots$), in both nonrelativistic and relativistic regimes. $\gamma \gamma$ partial widths have previously been published only for fermion-antifermion states with $J^{\mathrm{PC}}=0^{\pm ,+},1^{++}(\mathrm{o}\mathrm{f}\mathrm{f}-\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{s}), \mathrm{and} 2^{\pm ,+}$. The topic of higher-spin $\gamma \gamma$ partial widths is of current interest in part because a recent nonrelativistic $q\overline{q}$ calculation for $2^{-+}$ finds a much smaller $\gamma \gamma$ partial width than the ≅ 1 keV observed for the $I=1\mathrm{}$, $J^{\mathrm{PC}}=2^{-+}$ state ${\pi }_2\mathrm{}\left(1670\right)\mathrm{}$. We find very large relativistic corrections to the nonrelativistic (contact) approximation for this width as well as large systematic uncertainties in the procedure used to incorporate the physical resonance mass. Our relativistic $q\overline{q}$ estimate for ${\Gamma }_{\gamma \gamma }\left({\pi }_2\mathrm{}\right(1670\left)\right)$ is 0.1-0.3 keV, which is somewhat smaller than the experimental value. This discrepancy may only reflect inaccuracies in the theoretical technique. Finally, we quote relativistic numerical results for the $\gamma \gamma$ widths of various $0^{-+}$, $2^{-+}$, and $4^{-+}$ $\frac{(u\overline{u}-d\overline{d})}{\sqrt{2}}$, $c\overline{c}$ and $b\overline{b}$ states in a Coulomb-plus-linear potential model.