Angular distributions in the radiative decays of the $^3D_3$ state of charmonium originating from polarized $\bar{p}p$ collisions

Abstract

Using the helicity formalism, we calculate the combined angular distribution function of the two gamma photons ($\gamma_1$ and $\gamma_2$) and the electron ($e^-$) in the triple cascade process $\bar{p}p\rightarrow{}^3D_3\rightarrow{}^3P_2+\gamma_1\rightarrow(\psi+\gamma_2) +\gamma_1 \rightarrow (e^- + e^+) +\gamma_2 +\gamma_1$, when $\bar{p}$ and $p$ are arbitrarily polarized. We also derive six different partially integrated angular distribution functions which give the angular distributions of one or two particles in the final state. Our results show that by measuring the two-particle angular distribution of $\gamma_1$ and $\gamma_2$ and that of $\gamma_2$ and $e^-$, one can determine the relative magnitudes as well as the relative phases of all the helicity amplitudes in the two charmonium radiative transitions ${}^3D_3\rightarrow{}^3P_2+\gamma_1$ and $^3P_2\rightarrow \psi+\gamma_2$.