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 ($γ_1$ and $γ_2$) and the electron ($e^-$) in the triple cascade process $\bar{p}p\rightarrow{}^3D_3\rightarrow{}^3P_2+γ_1\rightarrow(ψ+γ_2) +γ_1 \rightarrow (e^- + e^+) +γ_2 +γ_1$, 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 $γ_1$ and $γ_2$ and that of $γ_2$ 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 ${}^3D_3\rightarrow{}^3P_2+γ_1$ and $^3P_2\rightarrow ψ+γ_2$.