Angular distributions in the radiative decays of the $^3D_3$ state of charmonium originating from polarized $\bar{p}p$ collisions

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 $^3P_2\rightarrow \psi+\gamma_2$.


Introduction
The study of charmonium states above the open charm DD threshold of 3.73 GeV has captured much attention in the theoretical and experimental community recently [1][2][3][4][5][6][7]. Among the higher charmonium states, the unobserved 3 D 3 state is quite interesting as its decay width is expected to be narrow. Although the strong decay of the 3 D 3 state to DD is Zweig-allowed, it is suppressed by the F -wave centrifugal barrier factor. This dominant decay width is predicted to be less than 1 MeV [8][9][10] and thus the radiative transition of 3 D 3 → γ + 3 P 2 may be observable [3,4]. The measurement of the angular distributions in the radiative decay of this charmonium state can provide valuable information on the true dynamics of the charmonium system above the charm threshold. In fact, charminoum spectroscopy is a key element of the plannedPANDA experiments at GSI [11,12], which will carry out systematic high-precision study of charmonium states below and above the charm threshold inpp annihilation.
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 unpolarizedpp collisions, namely,pp → 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. By considering the sequential decay of 3 D 3 produced in polarizedpp collisions, one may also obtain unambiguously the sines of the relative phases. So in this paper we calculate the angular distributions of the final stable decay products, γ 1 , γ 2 and e − , in the above cascade process when bothp and p are arbitrarily polarized. Our final model-independent expressions for the angular distribution functions are valid in thepp center-of-mass frame and they are written as sums of terms involving products of the Wigner 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 angular-momentum helicity amplitudes which contain all the dynamics of the individual decay processes. They are also functions of the longitudinal and the transverse components of the polarization vector ofp and p in their respective rest frames.
Potential model calculations show that the helicity amplitudes are in general complex [14] and thus their relative phases are nontrivial. 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 angular-momentum helicity amplitudes in the radiative decay processes 3 D 3 → 3 P 2 + γ 1 and 3 P 2 → ψ + γ 2 . It is important that bothp and p are polarized to get this complete information. We will derive the angular distribution functions by means of density matrix formalism where the density matrix elements are given in terms of the polarization vectors defined for stationary antiproton and proton. Our results are valid even whenp and p have arbitrary momenta since the density matrix elements are Lorentz invariant [15].
The format of the rest of the paper is as follows: In section 2, we give the calculation for the combined angular distribution function of the electron and of the two photons in the cascade processpp → 3 D 3 → 3 P 2 + γ 1 → (ψ + γ 2 ) + γ 1 → (e − + e + ) + γ 2 + γ 1 , when p and p are arbitrarily polarized. We then show how the measurement of this combined angular distribution of γ 1 , γ 2 and e − enables us to obtain complete information on the helicity amplitudes in the two radiative transitions 3 D 3 → 3 P 2 + γ 1 and 3 P 2 → ψ + γ 2 . In sections 3, we present the results for the partially integrated angular distributions in six different cases where the combined angular distribution function of the three particles is integrated over the directions of one or two particles. We also show how the measurement of these simpler angular distributions will again give all the information there is to get on the helicity amplitudes. Finally, in section 4, we make some concluding remarks.

The combined angular distribution function of the photons and electron
We consider the cascade process,p( Greek symbols in the brackets represent the helicities of the particles except δ which represents the z component of the angular momentum of the stationary 3 D 3 resonance. We choose the z axis to be the direction of motion of 3 P 2 in the 3 D 3 rest frame. The x and y axes are arbitrary and the experimentalists can choose them according to their convenience. A symbolic sketch of the cascade process is shown in figure 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 where B λ 1 λ 2 ,A νµ , E σ,κ and C α 1 α 2 are the angular-momentum helicity amplitudes for the individual sequential processespp → 3 D 3 , 3 D 3 → 3 P 2 + γ 1 , 3 P 2 → ψ + γ 2 and ψ → e + + e − , respectively. In the D-functions, the angles (φ, θ) giving the direction ofp, the angles (φ , θ ) giving the direction of ψ and the angles (φ , θ ) giving the direction of e − are measured in the 3 D 3 , the 3 P 2 and the ψ rest frames, respectively. The angles of each decay particle observed in different rest frames can be calculated using the Lorentz transformation. The equations relating these angles are given in [16].
Because of the C and P invariances [17], the angular-momentum helicity amplitudes in eq. (2.1) are not all independent. We have Making use of the symmetry relations of eq. (2.2), we now re-label the independent angularmomentum helicity amplitudes as follows: We will also make use of the following normalizations: The normalized angular distribution function for the cascade process when the initial p and p are arbitrary polarized and the final polarizations of γ 1 , γ 2 , e − and e + are not observed is given by where N is the normalization constant. It is determined by requiring that for the unpolarized case the integral of the angular distribution function W (θ, φ; θ , φ ; θ , φ ) over all the directions of γ 1 , γ 2 and e − or over all the angles, (θ, φ; θ , φ ; θ , φ ), is 1. In eq. (2.5) the symbols ρ 1 λ 1 λ 1 and ρ 2 λ 2 λ 2 represent the density matrices ofp and p, respectively. In the helicity basis states of the particles these matrix elements are [18] and where σ are the Pauli matrices. In eq. (2.6) and (2.7) P 1 and P 2 are the polarization vectors ofp and p and the two-component helicity eigenstates χ λ 1 ofp and β λ 2 of p satisfy wherep is the direction of the momentum ofp and λ 1 and λ 2 can take the values +1 or −1. In the coordinate system we defined in the beginning, we have χ + = cos(θ/2) sin(θ/2) exp(iφ) ; cos(θ/2) (2.10) and the phase of β is such that [17] β ∓ = χ ± . (2.11) Eq. (2.6) can be rewritten as where the unit vectors along the new x , y and z axes are related to the corresponding vectors of the xyz coordinate system bŷ i = (sin 2 φ + cos θ cos 2 φ)î − (sin φ cos φ − cos θ sin φ cos φ)ĵ − cos φ sin θk, j = (− cos φ sin φ + cos θ cos φ sin φ)î + (cos 2 φ + cos θ sin 2 φ)ĵ − sin φ sin θk, k = sin θ cos φî − sin θ sin φĵ + cos θk. (2.13) Similarly, eq. (2.7) can be rewritten as (2.14) In eq. (2.12) and (2.14), P 1z and −P 2z are the longitudinal components (components along the momenta of the respective particles) and the x and y components are the transverse compenets of the polarization vectors. Note that the angular distribution function W (θ, φ; θ , φ ; θ , φ ) is now given in terms of the density matrix elements defined for stationary proton and antiproton. But eq. (2.5) is of course valid in thepp c.m. frame, wherep and p are moving with relativistic velocities, since the density matrix elements are Lorentz invariant [15]. Substituting eq. (2.1) into eq. (2.5) and performing the various sums will give us a useful expression for the angular distribution function W (θ, φ; θ , φ ; θ , φ ) in terms of the Wigner D-functions. Before we do the sums we make use of the Clebsch-Gordan series relation for the D-functions, namely, and the relation After a long calculation, we obtain , which are independent of the angles in eq. (2.17), are defined as follows: In eq. (2.17) the components of the polarization vectors are contained in the coefficients defined as follows: where P ± = 1 ± P 1z P 2z , P C = P 1x P 2y + P 1y P 2x , P A = P 1x P 2x + P 1y P 2y , P D = P 1y P 2z + P 1z P 2y , (2.24) The explicit expressions for the non-zero coefficients, and β J 1 M , in eq. (2.17) are given in appendix A. Since the combined angular distribution in eq. (2.17) is expressed as a sum of products of the orthogonal Wigner D-functions, we can obtain the values for these coefficients from (2.25) In calculating eq. (2.25), we made use of the orthogonality relation: When we have sufficient experimental data for W (θ, φ; θ , φ ; θ , φ ), the integral on the right side of eq. (2.25) can be determined numerically for all possible allowed values of J 1 , J 2 , J 3 , d, d and M . A close examination of the explicit expressions for the coefficients γ J 3 , and β J 1 M shows that this will enable us to determine the relative magnitudes as well as the relative phases of all the angular-momentum helicity amplitudes A i and E j in the radiative decay processes 3 D 3 → 3 P 2 +γ 1 and 3 P 2 → ψ +γ 2 , respectively, when one or both of the incoming particles,p and p, are polarized. 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 processpp → 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 unpolarizedpp collisions given in [13].

Partially integrated angular distributions
The partially integrated angular distributions obtained from eq. (2.17) will look a lot simpler and we will gain greater insight from them. We calculate six different cases of partially integrated angular distributions. In deriving these results, we frequently make use of eq. (2.26) and the following property of the D-functions: We will express the final results for the three cases of single-particle angular distributions in terms of the orthogonal spherical harmonics by making use of the relation: Case 1: We will integrate over (θ ,φ ) and (θ ,φ ). Only the angular distribution of the first gamma photon γ 1 is measured. We obtain where the angles (θ,φ) represent the direction ofp measured from the z axis, which is taken to be the direction of the momentum of 3 P 2 . This angle is the same as that of γ 1 measured in the 3 D 3 rest frame with the z axis taken to be the direction of the proton. The x and y axes are arbitrary. With the normalization condition |B 0 | 2 + |B 1 | 2 = 1, eq. (3.4) allows us to determine the relative magnitude and the relative phase of the two helicity amplitudes in the processpp → 3 D 3 . There are also three equations relating the relative magnitudes of the A helicity amplitudes.
Case 2: We will integrate over (θ,φ) and (θ ,φ ). Only the angular distribution of the second gamma photon γ 2 is measured. We get Here, (θ ,φ ) are the angles between 3 D 3 and γ 2 in the 3 P 2 rest frame. As we can obtain one equation relating the relative magnitudes of the E helicity amplitudes from case 3 and also three equations relating the relative magnitudes of the A helicity amplitudes from case 1, the measurement of the single-particle angular distribution of γ 2 allows us to determine the relative magnitudes of the E and A helicity amplitudes and also the cosines of the relative phases of the A helicity amplitudes. It should be noted that β J 1 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. Case 3: We will integrate over (θ,φ) and (θ ,φ ). Only the angular distribution of the electron is measured. We have (3.6) where (θ ,φ ) are the angles between the directions of the momenta of e − and 3 P 2 in the ψ rest frame. From the measurement of the angular distribution of the electron alone we find that we cannot get any more useful information on the helicity amplitudes. So from cases 1-3, we see that we can obtain the relative magnitudes of all the helicity amplitudes in the processes 3 D 3 → 3 P 2 + γ 1 and 3 P 2 → ψ + γ 2 by measuring only the single-particle angular distributions of γ 1 , γ 2 and e − . We can also get the cosines of the relative phases of the helicity amplitudes in the process 3 D 3 → 3 P 2 + γ 1 . In order to obtain complete information on the relative phases of the helicity amplitudes in the radiative processes, we need to measure the simultaneous angular distributions of two particles.
Case 4: We will integrate over the angles (θ ,φ ), the direction of the final electron. The combined angular distribution of the two photons γ 1 and γ 2 is measured. We get Since the explicit expressions for the partially inegrated angular distributions of two particles are rather long, we only give the results in terms of the sums of the coefficients defined in appendix A. In eq. (3.7), however, we can obtain the coefficients of the angular functions from shows that eq. (3.8) enables us to obtain the sines and the cosines of the relative phases of all the A and B helicity amplitudes. It also enables us to determine the relative magnitudes of all the A, E and B helicity amplitudes. Only the relative phases among the E helicity amplitudes remain undetermined. We can get these phases only by measuring the simultaneous angular distribution of γ 2 and of e − as we will see in case 6.
Case 5: Here we integrate over (θ ,φ ) or the direction of γ 2 to get the combined angular distribution of γ 1 and e − . We get The great advantage of studying polarizedpp collisions is that one can obtain not only the relative magnitudes of the helicity amplitudes but also both the cosines and the sines of the relative phases of the helicity amplitudes from the measurement of the simultaneous angular distributions of two particles. This is important because the helicity amplitudes are in general complex. Therefore, we can get complete information on all the helicity amplitudes in the process 3 D 3 → 3 P 2 + γ 1 from the simultaneous angular distribution of γ 1 and γ 2 and also in the process 3 P 2 → ψ+γ 2 from the simultaneous angular distribution of γ 2 and e − , when bothp and p are polarized with both transverse and longitudinal polarization vector components in their respective frames. Moreover, we can also obtain the relative magnitude and the relative phase of the helicity amplitudes in the processpp → 3 D 3 . Polarizations of bothp and p are necessary to get all this information. Alternatively, one can also consider the polarizations of the final decay products γ 1 , γ 2 and e − [19,20].
We should also emphasize that the angular distributions alone will not give the absolute strengths of the helicity amplitudes. We get the magnitudes of all the helicity amplitudes only with the arbitrary normalization convention of eq. (2.4). In order to get the true absolute values which are physically significant one has to measure the branching ratios of each of the above processes and the parent particle's lifetime or decay width. The measurement of the angular distributions alone will only give the relative magnitudes and the relative phases of the helicity amplitudes in each radiative decay process.
Both the theorists and the experimentalists would like to express their results in terms of the multipole amplitudes in the radiative transitions 3 D 3 → 3 P 2 + γ 1 and 3 P 2 → ψ + γ 2 . The relationship between the helicity and the multipole amplitudes are given by the orthogonal transformations [21,22] A i = where a k and e k are the radiative multipole amplitudes in 3 D 3 → 3 P 2 +γ 1 and 3 P 2 → ψ+γ 2 , respectively. Since the transformations of eq. (4.1) and eq. (4.2) are orthogonal,