Role of the low-lying nucleon resonances in the $p\bar{p} \to \psi \eta$ reaction

Within the effective Lagrangian approach, we study the $p\bar{p} \rightarrow \psi\eta$ ($\psi\equiv\psi(3686)$, $J/\psi$) reaction at the low energy where the contributions from nucleon pole and low-lying nucleon resonances, $N(1520)$, $N(1535)$ and $N(1650)$ are considered. All the model parameters are determined with the help of current experimental data on the decay of $\psi \to p \bar{p} \eta$. Within the model parameters, the total and differential cross sections of the $p\bar{p} \rightarrow \psi\eta$ reaction are predicted. We show that the relative phases between different amplitudes of different nucleon resonance will change significantly the angular distributions of the $\bar{p} p \to \psi \eta$ reaction. Therefore, we conclude that these reactions are suitable to study experimentally the properties of the low lying nucleon resonance and the reaction mechanisms. We hope that these theoretical calculations could be tested by the future experiments.


I. INTRODUCTION
The charmonium, H c , production in the pp → H c X (the X is a light meson) reaction is an interesting tool to gain a deeper understanding of the strong interaction and also of the nature of the hadrons [1]. There is a forthcoming experimental effort, the Anti-Proton Annihilations at Darmstadt (PANDA), dedicated to this reaction [2]. On the theoretical side, there exist several previous studies of this reaction. In Ref. [3], Gaillard and Maiani firstly estimated differential cross sections for the process of pp → ψπ 0 in the soft pion limit. While in Refs. [4][5][6], it was pointed out that, with these N * Nψ couplings extracted form the corresponding ψ →pN * decays, the contributions of intermediated N * resonances and nucleon pole to the process of pp → ψX can be investigated. Then, Lin, Xu and Liu [7] considered the contributions of the intermediate nucleon pole and the effect of form factors (FFs) to the charmonium production in the low energy pp interaction atPANDA. It was shown that the effect of the FFs is significant.
Since the experimental data on the ψ → ppX decays become rich, we can consider the contributions from nucleon resonances in the pp → ψX reaction where parameters can be fixed through the process of ψ → ppX. Indeed, in Refs. [8][9][10], the authors have calculated the cross sections of the processes pp → J/ψπ 0 , pp → ψ(3770)π 0 and pp → Y(4220)π 0 , respectively, where the contributions from the intermediate nucleon resonances were considered. And it was found that the contributions from these nucleon resonances are nonnegligible. Their contributions will significantly change the angular distributions of thepp → ψX reaction.
The experimental results of both CLEO and BESIII Collaboration [11,12] show that the nucleon resonance N(1535) * Electronic address: zhouqs13@lzu.edu.cn † Electronic address: wangjzh2012@lzu.edu.cn ‡ Electronic address: xiejujun@impcas.ac.cn § Electronic address: xiangliu@lzu.edu.cn have significant contribution in the decay of ψ(3686) → ppη. This may be because the large coupling of the N(1535) to the ηN channel. As a matter of course, we will consider that the N(1535) may have significant contribution in pp → ψη reaction. Along the above line, in this work, we will calculate the production cross sections of the process pp → ψη within the effective Lagrangian approach and also give the angular distributions, where the contributions from nucleon pole and three N * states are considered. We consider the contributions from nucleon resonances N(1520) (≡ D 13 ) with J P = 3 2 − , N(1535) (≡ S 11 ) and N(1650) (≡ S 11 ) with J P = 1 2 − , which have appreciable branching ratios for the decay into the ηN channel. On the other hand, there are unknown model parameters, which will be determined through fitting the experimental data of ψ → ppπ 0 and ψ → ppη decays.
This article is organized as follows. First, the formalism and ingredients of pp → ψη within the effective Lagrangian approach are presented in Sec. II. In Sec. II B, we fit the experimental data on the ψ → ppπ 0 and ψ → ppη decays to determine these unknown parameters. In Sec. III, we show the numerical results and make a detailed discussion. Finally, a short summery will be given in Sec. IV.

II. FORMALISM AND INGREDIENTS
In this section, we introduce the theoretical formalism and ingredients for investigating thepp → ψη reaction within the effective Lagrangian method, by including the contributions from the nucleon pole and the low lying nucleon excited states that have strong couplings to the ηN channel.

A. Thepp → ψη reaction
The production of charmonium (ψ ≡ ψ(3686) and J/ψ) plus a light meson η in the low energy pp interaction can be achieved by exchanging intermediate nucleon and nucleon excited states. There are two type of Feynman diagrams to depict the pp → ψη reaction on the tree-level as shown in Fig. 1 In this calculation, we use the effective interaction Lagrangian densities for each vertex in Fig. 1. For the ψNN and ηNN vertexes, we use the effective Lagrangians as where V µ donates the vector field of ψ. For the N * Nη and ψN * N vertexes, we adopt the Lagrangian densities as used in Refs. [13][14][15][16][17][18][19]: where R denotes the N * field. Then, we can write the scattering amplitudes of process pp → ψη as, The F (u) and F (t) stand for the form factor of uand t-channel, respectively. Besides, we adopt the expression as used in Refs. [20][21][22][23] where the cutoff parameter Λ N * can be parameterized as with Λ QCD = 220 MeV, and the β will be determined by fitting the experimental data on the ψ(3686) → ppη decay. The Breit-Wigner form of the propagator G J (p) for the J = Note that we take the energy-dependent form for the decay width Γ N * of N(1535) resonance, we take the energydependent form, which is given by [25] Γ N * (q 2 ) = Γ N * →πN (q 2 ) + Γ N * →ηN (q 2 ) + Γ 0 , (2.13) We take the g πN * N = 0.62 and g ηN * N = 1.85, which are determined from the partial widths of N(1535) decay to Nπ and Nη. With these values we can get Γ N * →Nπ = 54.9 MeV and Γ N * →Nη = 55.1 MeV if we take q 2 = 1524 MeV. To agree with experimental result, we choose Γ 0 = 19 MeV for Γ N * (q 2 ) = 130 MeV. Here, the mass and width of N(1535) are adopted in Ref. [12]. The other coupling constants in the above Lagrangian densities can be also determined from their partial decay widths. The obtained numerical results for these relevant coupling constants are listed in Tables II and I. The coupling constants g ψNN * are obtained from the decay process of ψ →NN * + NN * → ppπ 0 , while the coupling constant g J/ψpp = 1.63 is extracted from the J/ψ →pp. In addition, for the coupling constants g ηNN and g ψ(3686)NN we will discuss them below. are estimated from the branching fraction (B.F.) of each intermediate nucleon resonance of J/ψ → NN * + N * N →ppπ 0 (second column), the width of J/ψ is 92.9 KeV. The last column is the parameter of g J/ψη N * which are estimated by formula g J/ψη Finally, in the center of mass (cm) frame, the differential cross section of pp → ψη process can be written as where θ is the scattering angle of outgoing η relative to direction of antiproton beam in the center of mass frame, while the − → p cm 1 and − → p cm 3 are the three-momentum of proton and ψ in the center of mass frame, respectively. The M tot is the total invariant scattering amplitude of thepp → ψη reaction, which can be written as 19) where M N and M N * are the contributions from the nucleon pole and the nucleon resonances, respectively. Besides, we introduce relative phase φ N * between M N * and M N .
B. Determine the model parameters from the analysis of the ψ(3686) →ppη decay On the tree level, the process ψ →ppη is described by the Feynman diagrams as shown in Fig. 2. With the effective interaction Lagrangian densities given above, we can easily obtain the decay width of ψ →ppη, which can be written as where p * 2 (Ω * 2 ) stands for the three momentum (solid angle) of the proton in the rest frame of the p and η system, p 3 (Ω 3 ) is the three momentum (solid angle) of the antiproton in the rest frame of ψ, and m pη is the invariant mass of p and η system. On the other hand, the amplitude M ψ→ppη tot is easily obtained just by applying the substitution to M pp→ψη tot : The ψ(3686) →ppη decay are experimentally studied by the CLEO and BESIII Collaborations [11,12], and they found that the most contributions are from nucleon excited state N(1535), which has large coupling to the Nη channel. However, since the N(1520) and N(1650) have significant couplings to the Nη channel, in this work, we will also take the contributions from them into account. Then, we perform fiveparameter (g ψ(3686)η N ≡ g ψ(3686)NN × g ηNN , φ N(1520) , φ N(1535) , φ N(1650) and β) 1 χ 2 fits to the experimental data [12] on the pη invariant mass distributions for the ψ(3686) → NN * + N * N → ppη decay.

III. THE TOTAL CROSS SECTIONS AND ANGULAR DISTRIBUTIONS OF pp → ψη
In this section, we show theoretical results on the total cross sections and angle distributions of the pp → ψη reaction near reaction threshold.
A. The total cross sections and angular distributions of pp → ψ(3686)η In Fig. 4, we show the numerical cross sections of pp → ψ(3686)η as a function of the energy of center of mass E cm = √ s. It is shown that the nucleon pole contribution is predominant in the whole energy region, but the contributions of the N * resonances gradually become significant when the E cm is increasing, especially for the contribution from N(1520). The contributions from N(1535) and N(1650) are mainly reflected in forepart and begin to decreasing around the E cm = 4.4 GeV.
Although the contribution of N(1535) is very predominant in the decay process ψ(3686) →ppη, but its contribution for the pp → ψ(3686)η is not so important. The contributions of the N * resonances are suppressed due to the highly off-shell effect of their propagators in the t-and u-channel. We also calculate the angular distribution of the pp → ψ(3686)η reaction at E cm = 4.3, 4.4, 4.5, 4.7, 4.9, 5.1, 5.3 and 5.5 GeV. The numerical results of dσ/d cos θ as a function of θ are shown in Fig. 5, where the red solid line stands for the total contribution, and the blue dashed line is result of only considering the contribution of nucleon pole. The every gray concentric circle means same value of dσ/d cos θ, and these concentric circles are evenly spaced from inside to outside, whose difference value is labeled in the bottom left corner. From Fig. 5 one sees that the blue dashed lines are symmetry with respect to θ = 90 • or θ = 270 • , which is because the contributions from u-channel and t-channel have the same weights for all of E cm if the only contribution from nucleon pole is considered. However, the shape of red solid lines are not symmetry with respect to θ = 90 • . The symmetry behavior of angular distribution is helpful to identify the role of excited nucleon resonances in the pp → ψ(3686)η reaction in futurePANDA experiment.  ative phase φ parameters. Firstly, we calculate the invariant mass distribution of J/ψ →ppη without considering the interference between different N * resonance. The numerical results obtained with β = 1.42 are shown in Fig. 6, where one can see that the N(1535) and N(1650) have significant contributions. Secondly, we consider only the contributions from N(1535) and N(1650), and we choose four typical values of 0, π/2, π and 3π/2 for the relative phase between them. In Fig.  7, we can see that phase interference will greatly change the line shape of the pη invariant mass distribution. These different line shape behaviors can provide valuable information for future experimental analyses on the process J/ψ →ppη. Next, we pay attention on the pp → J/ψη reaction. Although the present existing experimental data are not enough to determine the relative phases between different scattering amplitudes, we can still accurately estimate the absolute magnitude of cross sections from different nucleon resonance.  Fig. 4, but for the case ofpp → J/ψη reaction. .
In Fig. 8, we how the numerical results of the total cross sections of pp → J/ψη reaction as a function of E cm , where the relative phases between different nucleon states are not taken into account. The results indicate that the total cross section have maximum of about 80.5 pb at E cm = 5.5 GeV. From Fig. 8, one can also clearly see that contributions from excited nucleon resonances are significant, and the contribution from N(1520) even exceeds the contribution of nucleon pole when E cm > 5.0 GeV.
In addition, we also calculate the angular distributions of the process pp → J/ψη, which are presented in Fig. 9. Compared with the results for the process of pp → ψ(3686)η, on can see that the angular distributions of the pp → J/ψη reaction are always symmetry with respect to θ = 90 • or θ = 270 • for all the energies that we take. This is because that we did not consider the possible interference contributions among different scattering amplitudes of nucleons. Combining with the angular distributions of pp → ψ(3686)η and pp → J/ψη reactions, we can conclude that the weight difference between u-channel and t-channel is also due to the interference amplitudes from relative phases between different nucleon states.

C. Comparison with other work
The pp → ψη reactions are also studied in Refs. [1,7]. In Table III, we make a comparison between our results and other theoretical results of Refs. [1,7]. For the reaction of pp → ψ(3686)η and pp → J/ψη, our results are smaller than those of Ref. [1], but larger than the ones of Ref. [7].  In Ref. [1], the authors estimated the total cross sections of pp → ψX assuming a constant amplitude for the ψ → ppX decay. Under this approximation, it implies that the contributions of these intermediate resonances in the decay and production process are same. Hence, this approximation may lead to overestimation of the cross sections of pp → ψη. In this work, one can clear see that the contributions from different intermediate resonances are very different in decay and production processes, especially for N(1535). The N(1535) have extremely significant contribution in the decay process of ψ → ppη, but it is not important for the production process of pp → ψη. However, the constant amplitude approximation provides a good idea that the information of pp → ψη can be extracted from the decay process ψ → ppη.
In Ref. [7], the authors firstly introduced the form factor in predicting the cross sections of pp → ψη, where the only nucleon pole contribution is included. Their results are shown in fourth column of Table III, which are smaller than our results because the contributions from nucleon resonances are considered in our numerical results.
Finally, it needs to emphasize that we take the same form factors for both the pp → ψη reaction and the decay process ψ →ppη, although they should be different in these two different processes. However, since the hadron structure is still an open question, the hadronic form factors are generally adopted phenomenologically. Of course, the reliability of the treatment here can be left to future experiment to test.

IV. SUMMARY
The forthcomingPANDA will an ideal platform to carry out the study of hadron physics. Among these running facilities of particle physics, BESIII can provide abundant experimental data to the physics of charm tau physics. In fact, these studies onPANDA and BESIII can be borrowed with each other, which was indicated in Refs. [9,10].
In this work, based on the studies of the process ψ →ppη, we have calculated the total cross sections and angular distributions of the pp → ψη reaction within an effective Lagrangian approach. These contributions from nucleon resonances N(1520), N(1535) and N(1650) in pp → ψη reaction are considered for the first time. Our results show that these contributions from excited nucleon resonances are very important for estimating the cross section of pp → ψη, and the relative phases between different amplitudes will influence the total cross section and change the shape of the angular distributions. Hence, the ψ →ppη reactions are suitable for investigating the properties of the low lying nucleon resonance.
We hope that these theoretical calculations presented in this work may stimulate experimentalist's interest in exploring the pp → ψη reaction through thePANDA experiment. Meanwhile, we also suggest our colleague to pay more attentions to the theoretical issue around the charmonium production via the pp scattering processes since the present work is only a start point.