Angular distribution in the decays of the triplet D2 state of charmonium directly produced in unpolarized p̄p collisions

We calculate the combined angular distribution of the final electron and of the two γ photons in the cascade process p̄p→3D2→χJ+γ1 (J=0,1,2)→(ψγ2)+γ1→(e+e-)+γ2+γ1, where p̄ and p are unpolarized. Our final result is valid in the p̄p c.m. frame and it is expressed in terms of the Wigner DJ functions and the spherical harmonics whose arguments are the angles representing the various directions involved. The coefficients of the terms involving the spherical harmonics and the Wigner DJ functions are functions of the angular momentum helicity amplitudes or equivalently of the multipole amplitudes of the individual processes. Once the combined angular distribution is measured, our expressions will enable one to calculate the relative magnitudes as well as the relative phases of all the helicity amplitudes in the processes 1 3D2→1 3PJ+γ1 and 1 3PJ→ψ+γ2 for the J=2 case. For the J=1 case, we can determine the relative magnitudes of all the helicity amplitudes as well as the cosines of all their relative phases. The sines are not completely determined. If the sine of the relative phase between any two amplitudes is known, then the sines of the relative phases among other amplitudes can be determined. For the J=0 case, there is only one helicity amplitude in each decay and that is fixed by our normalization. We also present the partially integrated angular distributions in six different cases, which can all be expressed in terms of the spherical harmonics. We also calculate the angular distribution of the γ photon in the process p̄p→3D2→1S0+γ where again p̄ and p are unpolarized. In this case, the angular distribution has a very simple form: namely W(θ)=(1/ 4π)[Y00+(5/14) Y20(θ)+8/21Y40(θ)], where θ is the angle γ makes with the p̄ direction. So the observation of a γ photon with an energy of about 840 MeV and with the above angular distribution can be used as a signal for the formation of the 3D2 state in p̄p collisions.