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

Summary

In this paper we calculate the combined angular distribution of the two gamma photons and of the electron in the triple cascade process$\bar pp \to \psi ' \to \chi _J + \gamma _1 \to (\psi \gamma _2 ) + \gamma _1 \to (e^ + e^ - ) + \gamma _1 + \gamma _2$ (J = 0, 1, 2), when the protonp and the antiprotonp are both arbitrarily polarized. Our final result is valid in the$\bar pp$ c.m. frame and it is expressed in terms of the WignerD ^J functions whose arguments are the angles representing the various directions involved. The coefficients of the terms involving the WignerD ^J 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. We also derive six different partially integrated angular-distribution functions where the combined angulardistribution function of the three particles is integrated over the directions of one or two particles. Our results show that by measuring the angular distribution of single particles, namely γ₁, γ₂ ande ⁻, alone at a time, we can obtain the relative magnitudes and the relative phases of all the helicity amplitudes in the individual processes,$\bar pp \to \psi ', \psi ' \to \chi _J + \gamma _1$ and$\chi _J \to \psi + \gamma _2$ for the J=0 and the J=1 cases. For theJ = 2 case, we can get complete information on the helicity amplitudes in the processes$\bar pp \to \psi '$ and$\psi ' \to \chi _2 + \gamma _1$. We can also obtain the relative magnitudes of the helicity amplitudes in the decay process$\chi _2 \to \psi + \gamma _2$. 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.