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Angular distributions of the polarized photons and electron in the decays of the $^3D_3$ state of charmonium

We calculate the combined angular distribution functions of the polarized photons ($γ_1$ and $γ_2$) and electron ($e^-$) produced in the cascade process $\bar{p}p\rightarrow$ $^3D_3\rightarrow$ $^3P_2+γ_1\rightarrow$ $(ψ+γ_2)+γ_1\rightarrow(e^++e^-)+γ_1+γ_2$, when the colliding $\bar{p}$ and $p$ are unpolarized. Our results are independent of any dynamical models and are expressed in terms of the spherical harmonics whose coefficients are functions of the angular-momentum helicity amplitudes of the individual processes. Once the joint angular distribution of ($γ_1$, $γ_2$) and that of ($γ_2$, $e^-$) with the polarization of either one of the two particles are measured, our results will enable one to determine the relative magnitudes as well as the relative phases of all the angular-momentum helicity amplitudes in the radiative decay processes $^3D_3\rightarrow$ $^3P_2+γ_1$ and $^3P_2\rightarrowψ+γ_2$.