Angular distribution functions in the decays of the 3 D 3 state of charmonium originating from unpolarized $\overline{p}p$ collisions

Angular distributions in the radiative decays of the $$^3\varvec{D}_3$$ 3 D 3 state of charmonium originating from polarized $$\varvec{\bar{p}p}$$ p ¯ p collisions

“…Moreover, we can also obtain the relative magnitude and the relative phase of the two independent helicity amplitudes B 0 and B 1 in the initial process p p → 3 D 3 . It should be noted that the coefficients β J 1 M are functions of the longitudinal (P z ) and the transverse (P x ,P y ) components of the polarization vectors ofp and p. If the polarization vectors P 1 and P 2 go to zero, then β L 1 M = 0 when M is nonzero or when J 1 is odd, and we will recover the results of the unpolarizedp p collisions given in [13].…”

“…M (M = 0) will vanish if there is no polarization in the p andp beams, and we will not get any information on the helicity amplitudes [13]. So the polarization of the proton or the antiproton is crucial for extracting this information from the single-particle angular distributions.…”

“…1. Following the conventions of our previous paper [13], the probability amplitude for the cascade process can be expressed in terms of the Wigner D-function as…”

“…In our previous paper [13], it is shown that by measuring the joint angular distribution of the two photons (γ 1 , γ 2 ) and that of the second photon and electron (γ 2 , e − ), in the sequential decay process originating from unpolarizedp p collisions, namely,p p → 3 D 3 → 3 P 2 + γ 1 → (ψ + γ 2 ) + γ 1 → (e − + e + ) + γ 2 + γ 1 , one can extract the relative magnitudes as well as the cosines of the relatives phases of all the angular-momentum helicity amplitudes in the radiative decay processes 3 D 3 → 3 P 2 + γ 1 and 3 P 2 → ψ + γ 2 . The sines of the relative phases of these helicity amplitudes, however, cannot be determined uniquely.…”

Angular distributions of the polarized photons and electron in the decays of the $$^3D_3$$ 3 D 3 state of charmonium

“…In our previous paper [9], it is shown that by measuring the combined angular distribution of the two photons (γ 1 , γ 2 ) and that of the second photon and electron (γ 2 , e − ), regardless of their polarizations, in the sequential decay process originating from unpolarizedp p collisions, namely,p p → 3 D 3 → 3 P 2 + γ 1 → (ψ + γ 2 ) + γ 1 → (e + + e − ) + γ 1 + γ 2 , one can extract the relative magnitudes as well as the cosines of the relative phases of all the angular-momentum helicity amplitudes in the radiative decay processes 3 D 3 → 3 P 2 + γ 1 and 3 P 2 → ψ + γ 2 . The sines of the relative phases of these helicity amplitudes, however, cannot be determined uniquely.…”