Nucleon form factors at threshold

The pp̄ invariant mass spectra in the processes e+e− → pp̄, e+e− → nn̄, J/ψ→ pp̄ω, J/ψ → pp̄ρ, and J/ψ → pp̄γ close to the pp̄ is discussed. The optical potentials for NN̄ pair in the superposition of S and D waves (due to tensor forces) at spin unity of the pair, as well as for NN̄ pair in the S wave at spin zero of the pair, are proposed. The parameters of the potentials are obtained by fitting the cross sections of NN̄ scattering together with the pp̄ invariant mass spectra in e+e− → pp̄ annihilation and J/ψ decays. Good agreement with the available experimental data is achieved. Using our potential and the Green’s function approach we also describe the cross section of e+e− → 6π and the η′π+π− invariant mass spectrum in the decay J/ψ → γη′π+π− in the energy region near the NN̄ threshold.


Introduction
Investigation of the nucleon-antinucleon interaction in the low-energy region is an actual topic today. Unusual behavior of the cross sections of several processes has been discovered in recent years. For instance, the cross sections of the processes e + e − → pp and e + e − → nn reveal an enhancement near the threshold [1][2][3][4]. The enhancement near the pp threshold is also observed in the decays J/ψ(ψ ) → ppπ 0 (η) [5][6][7], J/ψ(ψ ) → ppω(γ) [5,[8][9][10][11]. The sharp peak in the vicinity of NN threshold has been observed in the cross sections of several processes, i.e., e + e − → 6π [12][13][14][15][16] and J/ψ → γη π + π − [17]. Strong enhancement of decay probability at low invariant mass of pp is observed in the processes B + → K + pp and B 0 → D 0 pp, B + → π + pp and B + → K 0 pp, Υ → γpp... These effect is similar to that in e + e − annihilation. One of the most natural explanation of this enhancement is final state interaction of nucleon and antinucleon. This interaction can be taken into account by means of an optical potential model. Several optical nucleon-antinucleon potentials [18][19][20] 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. In our recent paper [21], to fit the parameters of the potential, 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. 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. We have also constructed a simple optical potential model of the NN interaction in the 1 S 0 partial wave [22]. This partial wave should give 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. In my report I discus comparison of our results, obtained on the basis of our optical potentials, with the available experimental data.
2 Annihilation e + e − → NN near the threshold The Hamiltonian H I of NN interaction for the isospin I = 0, 1, spin S = 1 of the produced pair, and the total angular momentum J = 1 is where S is the spin operator, L = 0, 2 denotes the orbital angular momentum, and n = r/r. The radial wave functions u I nR (r) and w I nR (r), n = 1, 2, are the regular solutions of the equations Here M is the proton mass, E = k 2 /(2M). Near the threshold, the amplitude T I λµ of e + e − → NN annihilation in the non-relativistic approximation (in units 4πα/Q 2 , Q = 2(M + E)) for the certain isospin channel I = 0, 1 reads [23]: where G I s is an energy-independent constant, e µ is a virtual photon polarization vector, corresponding to the projection of spin J z = µ = ±1, λ is the spin-1 function of NN pair, λ = ±1, 0 is the projection of spin on the nucleon momentum k, andk = k/k. In terms of the form factor G I s , the electromagnetic Sachs form factors have the form .
The ratio G I E //G I M is independent of G I s and not zero only because of d-wave ( u I 2R (0) 0). Our predictions for the prosess e + e − → NN near the threshold are shown in Fig. 1 3 Virtual NN pair production near the threshold.
The total cross section (elasic+inelastic) σ I tot can be written as [24] σ where D(r, r |E) is the Green's function. The solution of this equation has the form  where χ T denotes transposition of χ and χ nN (r) is the non-regular solutions of the wave equation [21]. The inelastic cross section is the difference between the total cross section and elastic cross section: Data on e + e − → 4π do not demonstrate strong energy dependence in the vicinity of the NN threshold [25,26]. Data on e + e − → 5π are not accurate enough for all channels [13]. The cross section of e + e − → 6π near the threshold in the energy region between 1.7 GeV and 2.1 GeV, see Fig. 2, is approximated by the formula σ 6π = Aσ 1 ann + B · E + C, where the best coincidence is for A = 0.56, B = 0.012 nb/MeV, C = 4.96 nb. The coefficient A agrees with the data of pp → pions annihilation at rest, where 6π give ∼ 55% of I = 1 contribution [27]. 4 Decays J/ψ → ppπ 0 (η), J/ψ → ppρ(ω), J/ψ, ψ(2S ) → ppγ, J/ψ → γη π + π − Notations: k is the momentum of meson (photon) 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 pair. Dominant contribution to the amplitudes of J/ψ → ppπ 0 (η) decays is given by the state of pp pair with J PC = 1 −− ( 3 S 1 ). Then [22] dΓ I dMdΩ k = G 2 I pk 3 F 2 9 π 4 m 4 where ϑ k is the angle between n and k, n is directed along the electron momentum in the beam. The angular part of this distribution does not depend on the features of the pp interaction. The distributions over the angle ϑ p between p and n and over the angle ϑ pk between p and k read: where P 2 (x) = 3x 2 −1 2 is the Legendre polynomial. The invariant mass spectra of J/ψ → ppπ 0 and J/ψ → ppη decays are shown in Fig. 3. Dominant contribution to the amplitude of J/ψ → ppρ(ω) and J/ψ, ψ(2S ) → ppγ decays is given by the state of pp pair with J PC = 1 −+ ( 1 S 0 ) [28]. Only one function u I R (0) contributes: The corresponding invariant mass spectra are shown in Figs. 4 and 5.
The η π + π − invariant mass spectrum in the decay J/ψ → γη π + π − decay near the NN threshold is given by the formulas  where A, B and C are some fitting parameters. The coefficient A ≈ 4 · 10 −3 is 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. Comparison with the experiment is shown in Fig. 6.

Conclusion
• Unusual phenomena are related to interaction at large distances ("nuclear physics" of elementary particles).
• The results of SND and CMD-3 obtained at e + e − collider VEPP-2000 give an important contribution to our understanding of these phenomena.