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

Cheuk-Ping Wong,Wai-Yu Sit,Alex W K Mok

doi:10.1140/epjc/s10052-015-3322-z

Abstract

Using the helicity formalism, we calculate the combined angular distribution function of the two gamma photons ($\gamma _1$ and $\gamma _2$) and the electron ($e^-$) in the triple cascade process $\bar{p}p\rightarrow {}^3D_3\rightarrow {}^3P_2+\gamma _1\rightarrow (\psi +\gamma _2) +\gamma _1 \rightarrow (e^- + e^+) +\gamma _2 +\gamma _1$, when $\bar{p}$ and $p$ are arbitrarily polarized. We also derive six different partially integrated angular distribution functions which give the angular distributions of one or two particles in the final state. Our results show that by measuring the two-particle angular distribution of $\gamma _1$ and $\gamma _2$ and that of $\gamma _2$ and $e^-$, one can determine the relative magnitudes as well as the relative phases of all the helicity amplitudes in the two charmonium radiative transitions ${}^3D_3\rightarrow {}^3P_2+\gamma _1$ and $^3P_2\rightarrow \psi +\gamma _2$.

Highlights

In our previous paper [13], it is shown that by measuring the joint angular distribution of the two photons (γ1, γ2) and that of the second photon and electron (γ2, e−), in the sequential decay process originating from unpolarized pp collisions, namely, pp → 3 D3 → 3 P2 + γ1 → (ψ + γ2) + γ1 → (e− + e+) + γ2 + γ1, 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 D3 → 3 P2 + γ1 and 3 P2 → ψ + γ2
In this paper we calculate the angular distributions of the final stable decay products, γ1, γ2, and e−, in the above cascade process when both pand p are arbitrarily polarized
Our final modelindependent expressions for the angular distribution functions are valid in the pp center-of-mass frame and they are written as sums of terms involving products of the Wigner

Summary

Introduction

In our previous paper [13], it is shown that by measuring the joint angular distribution of the two photons (γ1, γ2) and that of the second photon and electron (γ2, e−), in the sequential decay process originating from unpolarized pp collisions, namely, pp → 3 D3 → 3 P2 + γ1 → (ψ + γ2) + γ1 → (e− + e+) + γ2 + γ1, 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 D3 → 3 P2 + γ1 and 3 P2 → ψ + γ2. In this paper we calculate the angular distributions of the final stable decay products, γ1, γ2, and e−, in the above cascade process when both pand p are arbitrarily polarized. D-functions whose arguments are the angles representing the directions of the final electron and of the two photons The coefficients in these expansions are functions of the angularmomentum helicity amplitudes which contain all the dynamics of the individual decay processes. Once the angular distributions in polarized pp collisions are experimentally measured, our expressions will enable one to determine the relative magnitudes as well as the relative phases of all the complex angularmomentum helicity amplitudes in the radiative decay pro-.

The combined angular distribution function of the photons and electron

Partially integrated angular distributions

Concluding remarks