Meson production in $J/\psi$ decays and $J/\psi\to N\bar{N}\gamma$ process

It is shown that an account for the final-state interaction of real or virtual nucleon and antinucleon produced in the processes $J/\psi\to p\bar{p}\gamma$, $\psi(2S)\to p\bar{p}\gamma$, $J/\psi\to p\bar{p}\omega$, and $J/\psi\to3\left(\pi^{+}\pi^{-}\right)\gamma$ near the threshold of $N\bar{N}$ pair production allows one to obtain self-consistent description of these processes. Predictions of our model are in good agreement with experimental data available. The proposed potential model also reproduces the corresponding partial cross sections of $p\bar{p}$ scattering.


I. INTRODUCTION
Invariant mass M of a nucleon-antinucleon pair N N in the decay J/ψ → N N + A, where A = γ, ρ, ω, π 0 , η, in the rest frame of J/ψ is determined by the energy E A of particle A, M 2 = m 2 J/ψ − 2m J/ψ E A .Therefore, measurement of E A allows one to fix the value of M .Anomalous behavior of the decay probabilities has been observed in the processes J/ψ → pp + A [1][2][3][4][5][6][7] and J/ψ → mesons + A [8,9], where the invariant mass of produced mesons is close to the double proton mass, M ≈ 2m p .Usually, this anomalous behavior is explained by the existence of a family of resonances X(1835), X(1840), X(1870), and others.However all available experimental data can be well explained within the approach based on an account for the interaction of real or virtual nucleon and antinucleon produced in J/ψ decays (see [10][11][12][13][14][15] and references therein).The same approach explains successfully experimental data for the cross sections of processes e + e − → mesons near the threshold of real N N pair production (see [16][17][18]).
In the approach based on the account for the final-state interaction, a quark-antiquark pair is produced at small distances r ∼ 1/2m p and then transforms into a nucleon-antinucleon pair at large distances r ∼ 1/Λ QCD as a result of hadronization.For small relative velocity of N and N , the N N interaction may significantly increase the modulus of wave function |ψ(0)| (here ψ(0) is the value of wave function of N N pair at distances r ∼ 1/Λ QCD ).Firstly, this happens when there is a loosely bound N N state with the binding energy |ε| ≪ U , ε < 0, where U is the characteristic value of the potential U (r) of N N interaction.Secondly, there is no loosely bound state, but a slight increase of the potential depth results in its appearance.We refer to the latter case as a virtual state with an energy ε ≪ U , ε > 0. In both cases, an energy ε is expressed in terms of the N N scattering length a, |ε| = 1/m p a 2 , where |a| is much larger than the characteristic size R of the potential.Moreover, a > 0 in the case of a loosely bound state and a < 0 for a virtual state.
We refer to the production of a real N N pair as an elastic process.A produced virtual N N pair can annihilate into a system of mesons, we refer to such process as inelastic.The sum of probabilities of elastic and inelastic processes is the total probability.The inelastic processes are possible above the threshold of real N N pair production as well as below this threshold (due to annihilation of a virtual pair).Therefore, the anomalous behavior of the probabilities of decays J/ψ → N N + A → mesons + A are determined by the energy dependence of the probability of virtual N N production.
The quantum numbers of a nucleon-antinucleon pair are determined by that of particle A and also by the quantum numbers of mesonic system.For instance, the main contribution to the probabilities of J/ψ → N N γ(ρ, ω) decays near the N N production threshold is given by the production of N N pair with the angular momentum l = 0, total spin S = 0, and charge parity C = +1.Then, the isospin of N N pair is I = 1 in the decay with ρ meson production, isospin I = 0 with ω meson production, and the isospin I is not fixed in the decay J/ψ → N N γ.In the process J/ψ → (6π)γ the G-parity of 6π state is G = +1, and the C-parity of N N pair in the process J/ψ → N N γ is C = +1.Therefore, the contribution to the probability of the decay J/ψ → (6π)γ is given by virtual N N pair with the isospin I = 0. Let's now consider the process J/ψ → (6π)π 0 , where 6π are produced through an intermediate N N state.In this case, the C-parity of N N pair is C = −1, total spin S = 1, and the isospin of the pair is I = 1, as well as the isospin of produced 6π system.Therefore, the effective nucleon-antinucleon potentials in the processes J/ψ → N N γ and J/ψ → N N π 0 are different, and the probabilities of the corresponding processes are also different.This is the reason why a large number of resonances X has been introduced for interpretation of anomalous behavior of probabilities in various processes with the meson production.

II. RESULTS
In Ref. [12] we successfully described the experimental data for the energy dependence of J/ψ → ppγ(ω) decays probabilities.Therefore, we can predict the behavior of J/ψ → (6π)γ decay probability near the threshold of N N pair production.The probability of the process J/ψ → ppγ can be written as (see Ref. [12]) Here k is the photon momentum in the J/ψ rest frame, p is the nucleon momentum in the N N center of mass frame, G γ0 and G γ1 are some energy-independent constants related to the amplitudes of N N γ state production at small distances.The functions ψ (I) (r) are the regular solutions of the radial Schrödinger equations for N N pair with the corresponding isospin I.The probability of J/ψ → ppω decay reads (see Ref. [12]) where G ω is some constant.The total probability Γ (0) tot of J/ψ → N N + γ and J/ψ → N N + γ → mesons + γ processes, in which real or virtual N N pair has the isospin I = 0, is expressed via the Green's function D (0) (r, r ′ |E) of the radial Schrödinger equation for N N pair with quantum numbers l = 0, S = 0, and I = 0 (see Ref. [12]) The probability of elastic process J/ψ → N N γ, where N N has I = 0, reads The probability dΓ inel /dM of inelastic decays J/ψ → mesons + γ, in which the system of mesons has I = 0, is Note that the effects of isotopic invariance violation (the proton and neutron mass difference and the Coulomb interaction of proton and antiproton) only slightly affect dΓ (0) inel /dM .In a recent work [9], the distribution dΓ 6π /dM over the invariant mass of 3 (π + π − ) in the decay J/ψ → 3 (π + π − ) γ has been measured with high accuracy.To describe the probability of this process, it is necessary to take into account both the contribution of virtual N N pair with I = 0 in an intermediate state and the contributions of mechanisms not related to the annihilation of a virtual N N pair.The energy dependence of latter contributions is a smooth function of M near the threshold of real N N pair production, while the former contribution depends strongly on M in the near-threshold region.A smooth dependence on M of contributions, which are not related to the virtual N N pair production, can be approximated using few parameters [9].
Eq. ( 5) predicts the sum of probabilities of all inelastic processes, and the production of 3 (π + π − ) system is only one of possible channels.However, it is natural to assume that the annihilation amplitude of N N pair into mesons weakly depends on M near the threshold, and it can be considered a constant.Therefore, the contribution of virtual N N annihilation to the probability of J/ψ → 3 (π + π − ) γ decay is proportional to dΓ (0) inel /dM .This assumption was U (1)   V (MeV) −92 −24 W (MeV) 114 89 R (fm) 1.17 1.06 Table I.Parameters of potentials (7) of N N interaction in the states with isospin I = 0, 1.
fully justified when describing the anomalous behavior of meson production cross sections in e + e − annihilation near the threshold of real N N production [12,17,18].Using the experimental data [9] we found that the contribution of background processes in the near-threshold region can be approximated with good accuracy by a linear function of energy E. As a result, we describe the distribution dΓ 6π /dM by the formula where a, b and c are some parameters that have been determined by comparison of our predictions with experimental data.
In order to determine dΓ inel /dM it is necessary to find the parameters of N N interaction potential with quantum numbers l = 0, S = 0, and I = 0. We have used the experimental data on the production of pp in J/ψ → ppγ, ψ(2S) → ppγ and J/ψ → ppω decays [3][4][5][6][7], and also the results of partial-wave analysis of elastic and inelastic pp scattering data performed by the Nijmegen group [19].In processes J/ψ → ppγ and ψ(2S) → ppγ, pp pairs are produced both with I = 0 and I = 1 (see Eq. ( 1)).In the process J/ψ → ppω, the pp pair is produced with I = 0 (see Eq. ( 2)).However, the experimental data for this decay are limited as compared to the decay J/ψ → ppγ.Therefore, we have used the whole set of experimental data listed above to better fix the parameters of the potential.As a result, we found not only the effective interaction potential of N N with I = 0, but also with I = 1.
As shown in Refs.[20,21], the behavior of cross sections in the near-threshold region is determined by a small number of parameters (scattering lengths, effective ranges of interaction).Therefore, one can use any convenient parameterization of the effective potentials, which reproduces the required values of these parameters.We have used the parameterization of N N interaction potentials in states with l = 0, S = 0, and I = 0, 1 in the form of rectangular wells where V (I) , W (I) and R (I) -are some parameters, and θ(x) is the Heaviside function.The optical potential U (I) (r) contains an imaginary part that accounts for annihilation of N N pair into mesons.For such a parameterization one can obtain the analytical form of the wave functions ψ (I) (r) and the Green's functions D (I) (r, r ′ |E).We have (see Ref. [21]) = q e −ipR (I) q cos qR (I) − ip sin qR (I) , Im D (I) (0, 0|E) = Im q q sin qR (I) + ip cos qR (I) q cos qR (I) − ip sin qR (I) , The values of potential parameters, that provide the best fit of the experimental data, are given in the Table I.Fig. 1 shows the comparison of our predictions with experimental data for J/ψ → ppγ, ψ(2S) → ppγ, and J/ψ → ppω decays.In all cases, good agreement between the predictions and the experimental data is evident.We have checked that our model is consistent with the results of partial-wave analysis of pp scattering data performed by the Nijmegen group [19].

III. CONCLUSION
A simple model, based on the account for the final-state interaction, is proposed for self-consistent description of the processes J/ψ → ppγ , ψ(2S) → ppγ, J/ψ → ppω and J/ψ → 3 (π + π − ) γ near the threshold of N N pair production.It is shown that the nontrivial energy dependence of the probabilities of these processes is related to the interaction of real and virtual N and N .The proposed potential model is also consistent with the results of partial-wave analysis of pp scattering data.

Figure 2 .
Figure 2. The dependence of J/ψ → 3(π + π − )γ decay probability on the invariant mass M .The solid line is our predictions for dΓ6π/dM , the dashed line is the background contribution.The vertical dotted line indicates the N N threshold.Experimental points are recalculated from Ref. [9].