Abstract
We calculate the combined angular distribution functions of the polarized photons ($\gamma_1$ and $\gamma_2$) and electron ($e^-$) produced in the cascade process $\bar{p}p\rightarrow$ $^3D_3\rightarrow$ $^3P_2+\gamma_1\rightarrow$ $(\psi+\gamma_2)+\gamma_1\rightarrow(e^++e^-)+\gamma_1+\gamma_2$, when the colliding $\bar{p}$ and $p$ are unpolarized. Our results are independent of any dynamical models and are expressed in terms of the spherical harmonics whose coefficients are functions of the angular-momentum helicity amplitudes of the individual processes. Once the joint angular distribution of ($\gamma_1$, $\gamma_2$) and that of ($\gamma_2$, $e^-$) with the polarization of either one of the two particles are measured, our results will enable one to determine the relative magnitudes as well as the relative phases of all the angular-momentum helicity amplitudes in the radiative decay processes $^3D_3\rightarrow$ $^3P_2+\gamma_1$ and $^3P_2\rightarrow\psi+\gamma_2$.
Highlights
There has been great interest in charmonium spectroscopy above the open charm D Dthreshold of 3.73 GeV [1,2,3,4]
The observation of the radiative decays of the charmonium states below and above the charm threshold is an important component of the planned PANDA experiments at FAIR
We have derived the model-independent expressions for the combined angular distribution of the final photons (γ1 and γ2)
Summary
In our previous paper [9], it is shown that by measuring the combined angular distribution of the two photons (γ1, γ2) and that of the second photon and electron (γ2, e−), regardless of their polarizations, in the sequential decay process originating from unpolarized pp collisions, namely, pp → 3 D3 → 3 P2 + γ1 → (ψ + γ2) + γ1 → (e+ + e−) + γ1 + γ2, 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 D3 → 3 P2 + γ1 and 3 P2 → ψ + γ2. In this paper we calculate the combined angular distributions of the final particles (γ1, γ2 and e−) with the determination of the polarization of one particle in the above cascade process when pand p are unpolarized. Once the combined angular distribution of γ1, γ2 and e− and the polarization of any one of the particles in unpolarized pp collisions are experimentally measured, our expressions will enable one to calculate the relative magnitudes as well as the relative phases of all the angular-momentum helicity amplitudes in the two radiative decay processes 3 D3 → 3 P2 + γ1 and 3 P2 → ψ + γ2. We show how the measurement of this joint angular distribution of polarized γ1, γ2 and e− enables us to obtain complete information on the helicity amplitudes in the two radiative decay processes 3 D3 → 3 P2 + γ1 and 3 P2 → ψ + γ2.
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