Interaction of real and virtual NN̄ pairs in J / ψ decays

The differential decay rates of the processes J/ψ→ pp̄π0, J/ψ→ pp̄η, J/ψ→ pp̄ω, J/ψ→ pp̄ρ, and J/ψ→ pp̄γ close to the pp̄ threshold are calculated with the help of the NN̄ optical potential. We use the potential which has been suggested to fit the cross sections of NN̄ scattering together with all other NN̄ experimental data available. The pp̄ invariant mass spectra of J/ψ decays are in agreement with the available experimental data. The anisotropy of the angular distributions of the decays J/ψ→ pp̄π0(η), which appears due to the tensor forces in the NN̄ interaction, is predicted close to the pp̄ threshold. This anisotropy is large enough to be investigated experimentally. Such measurements would allow one to check the accuracy of the model of NN̄ interaction. Using our potential and the Green’s function approach we also describe the peak in the η′π+π− invariant mass spectrum in the decay J/ψ→ γη′π+π− in the energy region near the NN̄ threshold.

We describe the nucleon-antinucleon interaction by means of an optical potential model. Several optical nucleon-antinucleon potentials [24][25][26] are usually used to describe the interaction in the low-energy region. All these nucleon-antinucleon potentials have been proposed to fit the nucleon-antinucleon scattering data. These data include elastic, charge-exchange, and annihilation cross sections of pp scattering, as well as some single-spin observables. There were attempts to describe the processes of NN production in e + e − annihilation using these potential models. For instance, using the Paris [27] and Jülich [28] * e-mail: S.G.Salnikov@inp.nsk.su models, it has been shown that the near-threshold enhancement of the cross sections of these processes can be explained by the final-state nucleon-antinucleon interaction. The strong dependence of the ratio of electromagnetic form factors of the proton on the energy in the timelike region near the threshold has been explained by the influence of the tensor part of the nucleon-antinucleon interaction.
In our recent papers [29,30], to fit the parameters of the potentials, we have suggested to include all available experimental data in addition to the nucleon-antinucleon scattering data. A simple potential model of NN interaction in the partial waves 3 S 1 − 3 D 1 , coupled by the tensor forces, has been suggested [29]. The parameters of this model has been obtained by fitting simultaneously the nucleon-antinucleon scattering data, the cross sections of pp and nn production in e + e − annihilation, and the ratio of electromagnetic form factors of the proton in the timelike region. This model has allowed us to calculate also the contribution of virtual NN intermediate state to the processes of meson production in e + e − annihilation and to describe the sharp dip in the cross section of 6π production in the vicinity of the NN threshold [29]. Similar results have also been obtained in Ref. [31] within the chiral model [26] but without the tensor NN interaction taken into account.
The potential [29] has also been used to explain the enhancement observed in the pp invariant mass spectra of the decays J/ψ(ψ ) → ppπ 0 (η) near the pp threshold [32]. Note that in these decays in the near-threshold region the most important contribution is also given by the partial waves 3 S 1 − 3 D 1 . The spectra of these decays, as well as the decays J/ψ(ψ ) → ppω(ρ, γ), have also been studied in Refs. [33,34] using the chiral model [26].
In the paper [30] we follow our idea and construct a simple optical potential model of the NN interaction in the 1 S 0 partial wave. This partial wave gives the most important contribution to the final-state pp interaction in the decays J/ψ(ψ ) → ppω(ρ, γ) in the energy region close to the pp threshold. The parameters of this potential has been obtained by fitting simultaneously the nucleonantinucleon scattering data and the pp invariant mass spectra of the decays J/ψ(ψ ) → ppγ and J/ψ → ppω. We have showed that it is possible to describe the pronounced peak in the pp invariant mass spectrum of the decay J/ψ → ppγ using a simple model of the NN interaction. Moreover, in contrast to the results of Ref. [33], our model doesn't predict such peak in the spectrum of the decay J/ψ → ppρ which has not been observed yet.
We have used our model to calculate the contribution of virtual NN pair to the J/ψ → γη π + π − decay rate in the energy region near the NN threshold. Our model describes a peak in the η π + π − invariant mass spectrum. It has been pointed out in Ref. [17] that a contribution of virtual pp state may be one of possible origins of the peak in the spectrum. However, in Ref. [17] any models of the NN interaction have not been applied.
The present paper is a compilation of the results of Refs. [30,32], and some parts are identical to the corresponding sections of these papers.

Decay amplitudes
We give only the resulting formulas here and the reader is referred to the Refs. [30,32] for more details.
At first let us discuss the J/ψ decay into a pp pair and a pseudoscalar meson. Possible states for a pp pair in the decays J/ψ → ppπ 0 and J/ψ → ppη have quantum numbers J PC = 1 −− and J PC = 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 nucleon-antinucleon 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 centerof-mass frame has the form [35] 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 for 3 S 1 − 3 D 1 partial waves (see Ref. [32] for more details). 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 and λ is the polarization vector of J/ψ. 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., [36]) where Ω p is the proton solid angle in the pp center-ofmass frame and Ω k is the solid angle of the x meson in the J/ψ rest frame. Substituting the amplitude (3) in Eq. (4) 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. (5) 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. (5) 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. (5) 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 (5) over the direction of n gives the distribution over the angle ϑ pk between p and k: Note that this distribution can be written in terms of the same anisotropy parameter (9). The mass spectrum (6) and the anisotropy parameter (9) are sensitive to the tensor part of the NN potential and, therefore, give the possibility to verify the potential model. Now let us discuss other J/ψ decays where a pp pair and a vector particle (ρ, ω, γ) are produced. Due to the C-parity conservation law, possible states for a pp pair in such decays are 1 S 0 and 3 P j . The S -wave state dominates in the near-threshold region where the relative velocity of the nucleons is small. The pp pairs have different isospins for the final states containing a vector meson (I = 1 for the ppρ state, and I = 0 for the ppω state). In the case of ppγ final state, the pp pair is a mixture of two isospin states.
The same kinematics relations (1) hold, as well as the relation between the decay rate and the amplitude (4). The dimensionless amplitude of these decays can be written via the radial wave function of the pp pair corresponding to the 1 S 0 wave, ψ I R (r), as Here G I is an energy-independent dimensionless constant, e λ and λ are the polarization vectors of the final vector particle and J/ψ, respectively. Substituting the amplitude (11) in Eq. (4), averaging over the spin states and integrating over the solid angles, we obtain the pp invariant mass distribution The wave function module squared is the so-called enhancement factor which equals to unity if the pp final-state interaction is turned off.
The optical NN potential can also be used to calculate the decay rates of the processes with a virtual NN pair in the intermediate state. In Ref. [29] it is shown that the total cross section of NN production, which is a sum of the cross section of real NN pair production (the elastic cross section) and the cross section of the meson production via annihilation of a virtual NN pair (the inelastic cross section), can be written in terms of the Green's function of the NN pair. According to Ref. [29], in order to switch from the elastic cross section to the total one, we should replace where M is the invariant mass of the mesons, E = M/2 − m p . The Green's function can be written in terms of regular, ψ I R (r), and non-regular, ψ I N (r), solutions of the Schrödinger equation: where θ(x) is the Heaviside function.

The decays J/ψ → ppπ 0 (η)
In the present work we use the potential model suggested in Ref. [29] to describe the pp interaction in the 3 S 1 − 3 D 1 coupled channels. 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. We have slightly refitted the parameters of the model in order to achieve a better description of the invariant mass spectra of the decays considered. By means of this model and Eq. (6), 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 [29] predicts the enhancement of the decay rates of both processes near the threshold of pp pair production (see Fig. 1). The invariant mass spectra predicted by our model are similar to those predicted in Ref. [33] with the use of the chiral model. An important prediction of our model is the angular anisotropy of these J/ψ decays. This anisotropy is the result of D-wave admixture due to the tensor forces in NN interaction. The anisotropy (see Eqs. (8) and (10)) is characterized by the parameters γ 1 and γ 0 (9) for the ppπ 0 and ppη final states, respectively. The dependence of the parameters γ I on the invariant mass of the pp pair is shown in Fig. 2. For pp invariant mass about 100 − 200 MeV above the threshold, significant anisotropy of the angular distributions is predicted. 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. (8) and (10)). There are some data on the angular distributions in the decays J/ψ → ppπ 0 [6] and J/ψ → ppη [7]. However, these distributions are obtained by integration over the whole 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 decays J/ψ → ppγ(ρ, ω)
In order to describe the pp interaction in the decays J/ψ → ppρ(ω, γ) we use an NN optical potential for the 1 S 0 partial wave proposed in Ref. [30]. The data used for fitting the parameters of the potential include the partial contributions of 1 S 0 wave to the elastic, chargeexchange, and total cross sections of pp scattering, and the pp invariant mass spectra of the decays J/ψ → ppω, J/ψ → ppγ, and ψ(2S ) → ppγ. The partial cross sections of pp scattering are calculated from the Nijmegen partial wave S -matrix (Table V of Ref. [25]).
By means of the potential model and Eq. (12), we calculate the pp invariant mass spectra in the processes J/ψ → ppω and J/ψ → ppρ (see Fig. 3). The isospin of the pp pair is I = 0 and I = 1 for ω meson and ρ meson in the final state, respectively. Therefore, the decay rates for these processes are given by Eq. (12) with the corresponding constants G I and wave functions ψ I R (0). Our model fits the experimental data for the decay J/ψ → ppω quite well. There are no experimental data for the decay J/ψ → ppρ, therefore, the predictions for the invariant mass spectrum are especially important. The pp spectrum in the decay J/ψ → ppρ, calculated in Ref. [33] with the use of the chiral model [26], has a pronounced peak close to the pp threshold, while our model predicts a monotonically increasing spectrum without any peak.
The decay amplitude of the process J/ψ → ppγ is a sum of two isospin contributions. Therefore, the decay   Figure 3. The invariant mass spectra of J/ψ decays to ppω, ppρ, ppγ, and ψ(2S ) decay to ppγ. The invariant mass spectra without NN interaction taken into account are shown by the dashed curves. The experimental data are taken from Refs. [5,[8][9][10][11]. The earliest measurements are adopted for the scale of the plots. This Figure has been copied from Ref. [30].
rate reads Our model describes with good accuracy the pronounced peak, seen fairly well in the experimental data for the decay J/ψ → ppγ (see Fig. 3). For the best fit, the ratio of the constants is G γ1 /G γ0 = −1.29 − 0.6 i. We have investigated in details the origin of this peak and found out that it arises because of a significant compensation of two isospin amplitudes at energy above 10 MeV per nucleon, though each isospin amplitude has no peak. This leads to another interesting prediction. The decay rate of the process J/ψ → nnγ, given by the formula should be much larger than that for the process J/ψ → ppγ. The experimental investigation of the decay J/ψ → nnγ would provide important information about the NN interaction. For completeness, we also consider the decay ψ(2S ) → ppγ (the corresponding ratio of the constants is G γ1 /G γ0 = −1.06 − 0.09 i), see Fig. 3.
Making use of our potential model and Eq. (13), we obtain also the predictions for the decay rates of the processes with the interaction of virtual nucleon-antinucleon pairs in the intermediate state (see Fig. 4). A peak in the total and inelastic invariant mass spectra exists near the pp threshold, especially in the isoscalar channel. This behavior seems to be the consequence of the existence of a quasi-bound state near the pp threshold. Our analysis shows that such state does exist in the isoscalar channel, and its energy is E B = (22 − 33 i) MeV. This is an unstable bound state in the classification of Ref. [37] because its energy moves to E B = −3 MeV when the imaginary part of the NN potential is turned off.
Let us discuss the exotic behavior of the decay rate of the process J/ψ → γη π + π − near the NN threshold observed in Ref. [17]. An approximate description of the spectrum of this decay was suggested in Ref. [38]. However, due to poor accuracy of the experimental data available at that time, only general resonance structure of the spectrum has been discussed in Ref. [38], but not the sharp dip in the vicinity of NN threshold. The new experimental data [17] allows us to investigate this spectrum in more details. The G-parity of the intermediate NN state, G NN = C NN (−1) I , should be equal to that of the final η π + π − state, G η π + π − = 1. Taking into account C-parity conservation we obtain C NN = 1, thus the isospin of the NN pair is I = 0. Possible NN states with positive C-parity are 1 S 0 and 3 P j , and the former one is expected to dominate in the near-threshold region. Therefore, we believe that the peak in the η π + π − invariant mass spectrum could occur because of the interaction of virtual nucleons in the isoscalar 1  of non-NN channels should be a smooth function in the vicinity of the NN threshold. Therefore, we approximate the invariant mass spectrum of the decay J/ψ → γη π + π − by the function where A, B and C are some fitting parameters. The comparison of the experimental data and our fitting formula in Fig. 5 demonstrates good agreement in the near-threshold region.
One can expect that the γpp intermediate state plays an important role in the decay J/ψ → γη π + π − near the threshold of pp pair production. Therefore, the branching ratio of this channel should be related to the branching ratio for the pp annihilation at rest into the η π + π − final state. To check the validity of this statement, we should take into account that the scale of Fig. 4 for the decay J/ψ → ppγ differs from that of Fig. 5 for the decay J/ψ → γη π + π − because of different total numbers of J/ψ events in the experiments [5] and [17]. After that we find the absolute value of the coefficient A in Eq. (17). The coefficient A ≈ 4 · 10 −3 can be considered as the estimation of the branching ratio of the decay pp → η π + π − at rest. This value is very close to the branching ratio 3.46 · 10 −3 measured in the experiment [39].

Conclusions
Using the model proposed in Ref. [29], we have calculated the effects of pp final-state interaction in the decays J/ψ → 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 is large enough to be measured. The observation of such anisotropy close to the pp threshold would allow one to refine the model of NN interaction.
Using the model of NN interaction in 1 S 0 partial wave we have calculated the effects of pp final-state interaction  Figure 5. The η π + π − invariant mass spectrum for the decay J/ψ → γη π + π − . The thin line shows the contribution of non-NN channels. Vertical dashed line is the NN threshold. The experimental data are taken from Ref. [17]. This Figure has been copied from Ref. [30].
in other J/ψ decays. Our model describes the pp invariant mass spectra of the decays J/ψ → ppω, J/ψ → ppγ, and ψ(2S ) → ppγ with good precision. We have also obtained the predictions for the pp invariant mass spectrum in the decay J/ψ → ppρ which has not been measured yet. Our prediction for this spectrum differs from the theoretical results obtained earlier. Therefore, the experimental study of the decay rate of this process would help to discriminate different models of the nucleon-antinucleon interaction.
We have used the Green's function approach to calculate the contribution of the interaction of virtual NN pairs in the 1 S 0 state to the cross sections of the processes. In particular we have calculated the contribution of the NN intermediate state to the η π + π − invariant mass spectrum for the decay J/ψ → γNN → γη π + π − in the energy region near the NN threshold. Our results are in good agreement with the available experimental data and describe the peak in the invariant mass spectrum just below the NN threshold.