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

TL;DR

This paper uses helicity formalism to derive angular distribution functions for photons and electrons in polarized antiproton-proton collisions producing charmonium states, enabling detailed analysis of helicity amplitudes.

Contribution

It provides new analytical formulas for angular distributions in charmonium radiative decays from polarized collisions, facilitating amplitude determination.

Findings

Derived six partially integrated angular distribution functions.

Showed how to extract helicity amplitude magnitudes and phases.

Demonstrated the method's potential for experimental analysis.

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 $^3P_2\rightarrow \psi+\gamma_2$.