Angular distribution of photons in the parity-changing one-photon radiative decays of quarkonia.

We derive a general expression, correct to order $\frac{{v}^{2}}{{c}^{2}}$ and valid in any potential model, for the angular distribution of photons in the parity-changing one-photon radiative decays of quarkonia, in terms of its different multipole contributions $E1$, $M2$, and $E3$. The expression is given in terms of Clebsch-Gordan coefficients and six reduced matrix elements. We make detailed numerical calculations for the radiative decays of charmonium using Buchm\"uller-Tye and Gupta-Radford-Repko potential models and find that there is not much difference in the angular distributions between the two models. It is interesting to note that the $E3$ contributions to the decays ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{\rightarrow}{\ensuremath{\chi}}_{2}+\ensuremath{\gamma}$ and ${\ensuremath{\chi}}_{2}\ensuremath{\rightarrow}\ensuremath{\psi}+\ensuremath{\gamma}$ are nonvanishing only ${\ensuremath{\psi}}^{\ensuremath{'}}$ and $\ensuremath{\psi}$ have admixtures of $D$ states in them.