Angular distribution in the decay of the singlet D state of charmonium directly produced in unpolarized p̄p collisions

We calculate the combined angular distribution of the final electron, of the γ photon, and of the π0 meson produced in the cascade process p̄p→1D2→1P1+γ→(ψπ0) +γ→(e+e-)+π0+γ, 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 cosines of the relative phases of all the angular momentum helicity amplitudes or equivalently of all the multipole amplitudes in the decay processes 1D2→1P1+γ and 1P1→ψ+π0. We also derive six different partially integrated angular distribution functions which give the angular distributions of one or two particles in the final state. They can all be expressed entirely in terms of the spherical harmonics. By measuring these simpler angular distributions in the six different cases we get as much information on the helicity amplitudes as we obtained by measuring the combined angular distribution of the three particles, namely, the electron, the γ photon, and the π0 meson.