Angular distribution functions in the decays of ψ′, ψ′′ and the singlet D charmonium state directly produced in p̄p collisions

We calculate the angular distribution of the two γ photons produced in the processes p̄p→ψ′, ψ′′→χJ+γ1→ψ+γ1+γ2(J=0,1,2), and p̄p→1D21→1P11+γ1→1S0+γ11+γ2 in terms of the helicity amplitudes of the individual processes. There is enough information in the angular distributions, even when p̄ and p are unpolarized, to extract the magnitudes of all the helicity amplitudes as well as the cosine of the relative phases of the helicity amplitudes or equivalently the multipole amplitudes in the processes ψ′ or ψ′′ →χJ+γ(J=0,1,2) and 1D21→1P11+γ. Finally we also derive the combined angular distribution of the electron and the photon in the cascade process p̄p→1D21→1S13+γ→e++e-+γ in terms of the helicity amplitudes of the individual processes. Here again, even when p̄ and p are unpolarized there is enough information in the angular distribution to determine the magnitudes of all the helicity amplitudes as well as the cosine of the relative phases of the helicity amplitudes or equivalently radiation multipole amplitudes in the parity-conserving one-photon radiative transitions 1D21(2-+)→1S13(1 - )+γ.