Angular distributions in the decays of the ψ' charmonium state directly produced in polarized proton-antiproton collisions

In this paper the authors calculate the combined angular distribution of the two gamma photons and of the electron in the triple cascade process p-p → ψ' → χj + γ1 → (ψγ2) + γ1→ (e+ e-) + γ1 + γ2 (J = 0, 1, 2), when the proton p and the antiproton p- are both arbitrarily polarized. Their final result is valid in the p-p c.m. frame and it is expressed in terms of the Wigner DJ functions whose arguments are the angles representing the various directions involved. The coefficients of the terms involving the Wigner DJ functions are functions of the angular-momentum helicity amplitudes or equivalently of the multipole amplitudes in the radiative processes. They are also functions of the components of the polarization vectors of the proton and of the antiproton in their respective rest frames. They also derive six different partially integrated angular-distribution functions where the combined angular-distribution function of the three particles is integrated over the directions of one or two particles. Their results show that by measuring the angular distribution of single particles, namely γ1 and γ2 and e-, alone at a time, they can obtain the relative magnitudes and the relative phases of all the helicity amplitudes in the individual processes, p-p → ψ', ψ' → χJ + γ1 and χJ → ψ + γ2 for the J = 0 and the J = 1 cases. For the J = 2 case, they can get complete information on the helicity amplitudes in the processes p-p → ψ' and ψ + χ2 + γ1. They can also obtain the relative magnitudes of the helicity amplitudes in the decay process χ2 → ψ + γ2. In order to obtain the relative phases of these amplitudes uniquely,, one has to measure the simultaneous angular distribution of the electron and of γ2