Angular distributions in $J/\psi\to p\bar{p}\pi^{0}(\eta)$ decays

The differential decay rates of the processes $J/\psi\to p\bar{p}\pi^{0}$ and $J/\psi\to p\bar{p}\eta$ close to the $p\bar{p}$ threshold are calculated with the help of the $N\bar{N}$ optical potential. The same calculations are made for the decays of $\psi(2S)$. We use the potential which has been suggested to fit the cross sections of $N\bar{N}$ scattering together with $N\bar{N}$ and six pion production in $e^{+}e^{-}$ annihilation close to the $p\bar{p}$ threshold. The $p\bar{p}$ invariant mass spectra is in agreement with the available experimental data. The anisotropy of the angular distributions, which appears due to the tensor forces in the $N\bar{N}$ interaction, is predicted close to the $p\bar{p}$ threshold. This anisotropy is large enough to be investigated experimentally. Such measurements would allow one to check the accuracy of the model of $N\bar{N}$ interaction.


I. INTRODUCTION
The cross section of the process e + e − → pp reveals an enhancement near the threshold [1][2][3][4]. The enhancement near the pp threshold has been also observed in the decays J/ψ → γpp, B + → K + pp, and B 0 → D 0 pp [5][6][7]. These observations led to numerous speculations about a new resonance [5], pp bound state [8][9][10] or even a glueball state [11][12][13] with the mass near two proton mass. This enhancement could appear due to the nucleonantinucleon final-state interaction. It has been shown that the behavior of the cross sections of NN production in e + e − annihilation can be explained with the help of Jülich model [14,15] or slightly modified Paris model [16,17]. These models also describe the energy dependence of the proton electromagnetic form factors ratio |G p E /G p M |. A strong dependence of the ratio on the energy close to the pp threshold is a consequence of the tensor part of the NN interaction.
Another phenomenon has been observed in the process of e + e − annihilation to mesons.
A sharp dip in the cross section of the process e + e − → 6π has been found in the vicinity of the NN threshold [18][19][20][21][22]. This feature is related to the virtual NN pair production with subsequent annihilation to mesons [23,24]. In Ref. [24] a potential model has been proposed to fit simultaneously the cross sections of NN scattering and NN production in e + e − annihilation. This model describes the cross section of the process e + e − → 6π near the NN threshold as well. A qualitative description of this process was also achieved using the Jülich model [23].
In this paper we investigate the decays J/ψ → ppπ 0 and J/ψ → ppη taking the pp finalstate interaction into account. Investigation of these processes has been performed in Ref. [25] using the chiral model. However, the tensor part of the pp interaction was neglected in that paper. To describe the pp interaction we use the potential model proposed in Ref. [24], where the tensor forces play an important role. The account for the tensor interaction allows us to analyze the angular distributions in the decays of J/ψ and ψ(2S) to ppπ 0 (η) near the pp threshold. The parameter of anisotropy is large enough to be studied in the experiments.

II. DECAY AMPLITUDE
Possible states for a pp pair in the decays J/ψ → ppπ 0 and J/ψ → ppη have quantum numbers J P C = 1 −− and J P C = 1 +− . The dominating mechanism of the pp pair creation is the following. The pp pair is created at small distances in the 3 S 1 state and acquires an admixture of 3 D 1 partial wave at large distances due to the tensor forces in the nucleonantinucleon interaction. The pp pairs have different isospins for the two final states under consideration (I = 1 for the ppπ 0 state, and I = 0 for the ppη state), that allows one to analyze two isospin states independently. Therefore, these decays are easier to investigate theoretically than the process e + e − → pp, where the pp pair is a mixture of different isospin states.
We derive the formulas for the decay rate of the process J/ψ → ppx, where x is one of the pseudoscalar mesons π 0 or η. The following kinematics is considered: k and ε k are the momentum and the energy of the x meson in the J/ψ rest frame, p is the proton momentum in the pp center-of-mass frame, M is the invariant mass of the pp system. The following relations hold: where m is the mass of the x meson, m J/ψ and m p are the masses of a J/ψ meson and a proton, respectively, and = c = 1. Since we consider the pp invariant mass region M − 2m p m p , the proton and antiproton are nonrelativistic in their center-of-mass frame, while ε k is about 1 GeV.
The spin-1 wave function of the pp pair in the center-of-mass frame has the form [17] ψ I λ = e λ u I wherep = p/p, e λ is the polarization vector of the spin-1 pp pair, u I 1 (r) and u I 2 (r) are the components of two independent solutions of the coupled-channels radial Schrödinger equations Here E = p 2 /2m p , V I S and V I D are the NN potentials in S-and D-wave channels, and V I T is the tensor potential. Two independent regular solutions of these equations are determined by their asymptotic forms at large distances [17] where S I ij are some functions of energy. The Lorentz transformation for the spin-1 wave function of the pp pair can be written as whereψ I λ is the wave function in the J/ψ rest frame,k = k/k, and γ is the γ-factor of the pp center-of-mass frame. The component collinear to k does not contribute to the amplitude of the decay under consideration because the amplitude is transverse to k. As a result, the dimensionless amplitude of the decay with the corresponding isospin of the pp pair can be written as Here G I is an energy-independent dimensionless constant, λ is the polarization vector where n is the unit vector collinear to the momentum of electrons in the beam.
The decay rate of the process J/ψ → ppx can be written in terms of the dimensionless amplitude T I λλ as (see, e.g., [26]) dΓ dM dΩ p dΩ k = pk 2 9 π 5 m 2 where Ω p is the proton solid angle in the pp center-of-mass frame and Ω k is the solid angle of the x meson in the J/ψ rest frame.
Substituting the amplitude (7) in Eq. (9) and averaging over the spin states, we obtain the pp invariant mass and angular distribution for the decay rate The invariant mass distribution can be obtained by integrating Eq. (10) over the solid angles Ω p and Ω k : The sum in the brackets is the so-called enhancement factor which equals to unity if the pp final-state interaction is turned off.
More information about the properties of NN interaction can be extracted from the angular distributions. Integrating Eq. (10) over Ω p we obtain where ϑ k is the angle between n and k. However, the angular part of this distribution does not depend on the features of the pp interaction. The proton angular distribution in the pp center-of-mass frame is more interesting. To obtain this distribution we integrate Eq. (10) over the solid angle Ω k : where ϑ p is the angle between n and p, P 2 (x) = 3x 2 −1 2 is the Legendre polynomial, and γ I is the parameter of anisotropy: Averaging (10) over the direction of n gives the distribution over the angle ϑ pk between p and k: dΓ dM dΩ pk = G 2 I pk 3 2 7 3π 4 m 4 Note that this distribution can be written in therms of the same anisotropy parameter (14).
The mass spectrum (11) and the anisotropy parameter (14) are sensitive to the tensor part of the NN potential and, therefore, gives the possibility to verify the potential model.

III. RESULTS AND DISCUSSION
In the present work we use the potential model suggested in Ref. [24]. The parameters of this model have been fitted using the pp scattering data, the cross section of NN pair production in e + e − annihilation near the threshold, and the ratio of the electromagnetic form factors of the proton in the timelike region. By means of this model and Eq. (11), we predict the pp invariant mass spectra in the processes J/ψ → ppπ 0 and J/ψ → ppη. The isospin of the pp pair is I = 1 and I = 0 for, respectively, a pion and η meson in the final state. The model [24] predicts the enhancement of the decay rates of both processes near the threshold of pp pair production (see the red band in Fig. 1). The invariant mass spectra predicted by our model are similar to those predicted in Ref. [25] with the use of the chiral model. Very close to the threshold the enhancement factor turned out to be slightly overestimated in comparison with the experimental data, as it is seen from Fig. 1. We have tried to refit the parameters of our model in order to achieve a better description of the invariant mass spectra of the decays considered. The predictions of the refitted model are shown in Fig. 1 with the green band. It is seen that the refitted model fits better the invariant mass spectra of J/ψ decays. However, the discrepancy in the cross sections of nn production in e + e − annihilation and the charge-exchange process pp → nn have slightly increased after refitting. Refs. [5,27,28]. The measurement of Ref. [5] is adopted for the scale of the left plot. The anisotropy (see Eqs. (13) and (15) Fig. 2. Note that the anisotropy in the distribution over the angle ϑ pk is expected to be two times larger than in the distribution over the angle ϑ p (compare Eqs. (13) and (15)).
There are some data on the angular distributions in the decays J/ψ → ppπ 0 [27] and J/ψ → ppη [28]. However, these distributions are obtained by integration over the whole The red/dark band corresponds to the model [24] and the green/light band corresponds to the refitted model. The phase space behavior is shown by the dashed curve. The experimental data are taken from Refs. [29][30][31]. The measurement of Ref. [29] is adopted for the scale of both plots.
pp invariant mass region. Unfortunately, our predictions are valid only in the narrow energy region above the pp threshold. Therefore, we cannot compare the predictions with the available experimental data. The measurements of the angular distributions at pp invariant mass close to the pp threshold would be very helpful. Such measurements would provide another possibility to verify the available models of NN interaction in the low-energy region.
The formulas written above are also valid for the decays ψ(2S) → ppπ 0 and ψ(2S) → ppη with the replacement of m J/ψ by the mass of ψ(2S). The invariant mass spectra for these decays are shown in Fig. 3. The angular distributions for these processes are the same as for the decays of J/ψ because they depend only on the invariant mass of the pp pair.

IV. CONCLUSIONS
Using the model proposed in Ref. [24], we have calculated the effects of pp final-state interaction in the decays J/ψ → ppπ 0 (η) and ψ(2S) → ppπ 0 (η). Our results for the pp invariant mass spectra close to the pp threshold are in agreement with the available experimental data.
The tensor forces in the pp interaction result in the anisotropy of the angular distributions.
The anisotropy in the decay J/ψ → ppπ 0 and especially in the J/ψ → ppη decay are large enough to be measured. The observation of such anisotropy close to the pp threshold would