The $\psi^{(\ast)}p$ scattering length based on near-threshold charmoniums photoproduction

Under the framework of Vector Meson Dominance model, the value of scattering length can be expressed as a function of the ratio between total cross section $\sigma (W)$ and $R (W)$, where $R(W)$ is the ratio between final momentum $|{\bf p}_3 |$ and initial momentum $|{\bf p}_1|$ and positively correlated with the center-of-mass energy. Based on the theoretical study of charmoniums photoproduction within two gluon exchange model and effective pomeron model, we research the scattering lengths of vector mesons and proton interaction in this work. Results show that the scattering length $\left|\alpha_{J/\psi p}\right|$ obtained from the two models are close and basically in agreement with the theoretical prediction of Strakovsky and co-workers. Additionally, we first calculate the scattering length of $\psi(2S) $-proton interaction in two gluon exchange model and effective pomeron model as $ 1.31\pm 0.92$ am (1 am = $10^{-3}$ fm) and $ 3.24 \pm 0.63$ am, respectively. This is a little bit different from the two models and requires precise measurements from subsequent experiments. In short, our results will provide a theoretical reference for future studies on characterizing the vector meson-proton scattering length.


I. INTRODUCTION
The evaluation of the scattering lengths may serve as a unique input for QCD-motivated models of vector mesonnucleon interactions [1][2][3][4]. The behavior of the near-threshold cross section is related to vector meson-proton (V p) scattering length [5]. Recently, scattering lengths for ωp, φp, J/ψp and Υp reactions have been reported using the recent photoproduction experiment data or quasi data [1][2][3][4]. However, there are also several V p scattering lengths that have not been studied for various reasons, such as one narrow vector meson: D * 0 (2007), since no charm conservation, the reaction γp → D * 0 (2007)p is impossible. Moreover, we find that there are no good threshold measurements for ρ, K * and D * meson photoproduction. As the excited states of J/ψ, it is of great interest to estimate the scattering length of vector meson ψ(2S )proton interaction.
On the experimental side, the J/ψ photoproduction off the proton was conducted with increasing precision over a large energy range [6][7][8][9][10][11][12][13], while the measurement of ψ(2S ) photoproduction data is very meager [14][15][16]. Considering ψ(2S ) and J/ψ have close mass, both have cc structure, the same quantum numbers, spin and parity as photon which is I G (J PC ) = 0 − (1 −− ), it is reasonable to study them with the same physical models and parameters. In our previous works [17], two gluon exchange model is applied to systematically analyze the J/ψ and ψ(2S ) photoproduction data from threshold to medium energy (near 400 GeV) [6][7][8][9][10][11][12][13]. In the literatures [18,19], the pomeron exchange process is considered to explain the photoproduction of charmoniums by JPAC collaboration. The numerical results from two gluon exchange model and effective pomeron model are both in agreement * xywang@lut.edu.cn † fczeng@yeah.net ‡ igor@gwu.edu with J/ψ experimental data, while the predicted photoproduction of ψ(2S ) in the two models at the threshold is basically consistent [17]. In this paper, we will use the predicted ψ(2S ) cross section data under the framework of the two models to extract the scattering lengths α ψ(2S )p . Many literatures give their scattering length results for proton interacts with vector meson by different methods. From a global fit to both differential and total cross section data, one work [20] extracted the scattering length α J/ψp is 0.046 fm from dσ/dt(s thr , t=0). Ref. [21] discussed the charmoniums bound states in nuclei from the method of multipole expansion and low-energy QCD theorems. Provided in the Vector Meson Dominance (VMD) model [22,23], Strakovsky and co-workers estimated the J/ψ-nucleon scattering length α J/ψp = 3.08 am (1 am = 10 −3 fm) by fitting the recent GlueX total cross section data [4]. To avoid additional uncertainty when extrapolating the differential cross section to the nonphysical point t=0, the approach of ref. [20] is not adopted. Compared to other methods, the VMD model does not contain free parameters in the process from γp to V p reaction. It is superior for us to obtain qualitative estimates when extracting scattering lengths α V p . The scattering length α J/ψp obtained by VMD model is smaller than other references. And this small value can be attributed to the size of the "Young" vector meson is smaller than that of the "old" one participating in the elastic V p → V p scattering, because cc pair lacks sufficient time to form the complete wave function of the vector meson [3].
The measurement of near-threshold ψ(2S ) photoproduction will give access to a variety of interesting physics aspects, e.g., trace anomaly, pentaquarks, cusp effects, vector-mesonnucleon scattering length and so on [22,24]. Nowadays, the Electron Ion Colliders (EIC) are proposed to be built for probing the deepest structure inside the hadron [25]. Meanwhile, the opportunities for Chinese EIC are now under discussion and will be an important and interesting future machine to collect ψ ( * ) data [26]. Relevant measurements will advance our understanding of QCD which governs the properties of hadrons and the interactions involving hadrons.
The paper is organized as follows. The formulas of effective pomeron model, two gluon exchange model and an expression for scattering length α V p are provided in Sec.II. Next section, we show the numerical result on the explanations of the current experimental data of J/ψ. Through the study of scattering length α J/ψp , the reliability of the two models and VMD method are determined. Then the results and discussion of α ψ(2S )p are obtained in Sec. III. A summary is given in Sec. IV.

II. FORMALISM
In quark-interchange mechanisms, some light quarks (such as π, ρ, ∆, etc.) are strongly suppressed in the heavy quarkonium photoproduction. So the channels are always dominated by two gluon exchange mechanisms or pomeron exchange contributions. Actually, the two gluon exchange and pomeron contributions both reproduce well the existing data [27]. The effective t−channel pomeron exchange contribution is shown in Fig.1. Proposed by JPAC collaboration, an effective pomeron exchange model is used to have a systematical analysis for J/ψ and ψ(2S ) photoproduction [18]. The differential cross section for γp → ψ ( * ) p reaction takes the form [19] The amplitude in low energy regions is given by [19,28] Here, α(t) = α 0 + α ′ t is the pomeron trajectory [29]. W 0 = 1 GeV is the energy scale parameter. W thr = M p + M ψ ( * ) is the energy threshold. The picture of the double gluon exchange between the nucleon state and the quark-antiquark pair is illustrated in Fig.2. The differential cross section of ψ ( * ) photoproduction is given as [30,31] where x = m 2 ψ ( * ) /W 2 ; α s = 0.5 is the strong coupling constant [32]; α is the fine-structure constant; m q is the mass of charm quark; Γ ψ ( * ) e + e − is the radiative decay [33]. xg x, m 2 defines the gluon distribution function at Q 2 = m 2 ψ ( * ) , which is parameterized using a simple function form xg x, m 2 [34]. The exponential slope parameter b can use the standard form based on the Regge phenomenology [16,35] for ψ(2S ) and similar form [30] for J/ψ.
The total cross section is obtained by integrating the differential cross section (Eq. (1) or (4)) over the allowed kinematical range from t min to t max , the total cross section can be written as, here, the limiting values t min and t max are The energies and momenta of the photon and meson in the center-of-mass (c.m.) frame are C. VMD model and scattering length α V p The ratio between the initial c.m. momentum and the final momentum R(W) is used as Note that, R(W) has a range of R(W) ∈ [0, 1) and is positively correlated with the c.m. energy.
The ratio of total cross section σ(R) and R is given by where the function a 1 (R) can be described by Eq. (1), (4) and (5). The V p scattering length is related to the near-threshold photoproduction of vector mesons. In this paper, the VMD model is used to connect the reaction γp → V p and V p → V p. Applying the effective VMD approach, the near-threshold cross section during the elastic scattering processes becomes [22] σ| thr (R) = here the VMD coupling constant g V is deduced from the leptonic decay width Γ V e + e − as [4] Combining Eq.(11), (12) and (13), scattering length α V p is given as The scattering length α V p can also be expressed by differential photoproduction cross section. When the c.m. energy W approaches the threshold, the total cross section is related to the differential cross section as [36] Combining Eq. (12) and (15), we can obtain the relation between the differential cross section of γp → V p reaction and scattering length as If setting dσ dt thr = b 1 , the scattering length α V p is given as Note that, b 1 and |p 1 | must satisfy the conditions that obtained from the threshold. The fitted values of the parameters A J/ψ , α 0 , α ′ and b 0 in effective pomeron model [18].

III. RESULTS AND DISCUSSIONS
In our previous work, the free parameters A 0 , A 1 and A 2 in two gluon exchange model were obtained by a global analysis of both the total cross section data below medium energy [6][7][8][9][10][11][12][13] and the near-threshold (W = 4.58 GeV) differential cross section data of J/ψ [6]. JPAC collaboration determined the free parameters A J/ψ , α 0 , α ′ and b 0 by fitting the total cross section of J/ψ experimental data [6,9]. The fitted parameters from the two models are listed in Tab. I and II. The numerical results of γp → J/ψp from two gluon exchange model and effective pomeron model are shown in Fig.3 and 4, compared with GlueX , SLAC and HERMES experiments [6,9,37]. We perceive that the two models are reliable to explain J/ψp photoproduction.
According to the obtained total cross section from models, Eq. (11) and (14), the J/ψp scattering length as a function of R is shown in Fig.5 (blue-solid curve). Listed in Table III,    We also compared our results with the phenomenological result [36] and odd-polynomial fitted result [4]. Our results are in agreement with the above phenomenological results. We also extract the scattering lengths from the total cross section GlueX data, as shown in the black circles in Fig.5. Ref. [36] obtained α J/ψp = 3.83 ± 0.98 am (the green circle in Fig.5) derived from the differential cross section GlueX data. Although this energy (W = 4.59 GeV) is a little far from the threshold, the value of scattering length is close to our estimation from two models and other works [4,36]. Therefore, the scattering lengths extracted from the differential cross section experimental data may be more advantageous, compared with the instability extracted from total cross section data.
In the above study, we found that the scattering lengths α J/ψp given by the two models are approximate and consistent with other literatures [4,36]. The reliability of the two models and VMD method was determined. Next, we will extend the study to ψ(2S )p scattering length based on the above The black circles show the results derived from the GlueX total cross section data. The green circle shows the results derived from the GlueX differential cross section data [36]. The curves have the same meaning as in Fig.3. study. The photoproduction of ψ(2S ) is obtained by using the same parameters listed in Tab. I and II. Note that, the parameter A ψ(2S ) in Eq. (3) can be written as A ψ(2S ) = R ψ(2S ) ·A J/ψ , and the relative strength R ψ(2S ) = 0.55 is obtained from the extraction by CLEO [18,38]. Then the ψ(2S )p scattering length as a function of R is shown in Fig.6 (Red-dashed curve). We obtained calculation results from two gluon exchange model and effective pomeron model to be 1.31 ± 0.92 am and 3.24 ± 0.63 am, respectively (table IV). Note that, the value of α J/ψp is bigger than α ψ(2S )p in both models. Because of the difference in the size of the total cross section predicted by the two models, the α ψ(2S )p scattering length is a little bit different. Precise results require more measurement from sub-  [3] sequent experiments. Based on the recent threshold measurements of the photoproduction of ω and φ mesons off the proton by the A2 (MAMI) [1] and CLAS (JLab) [39], one can determine vector meson proton scattering lengths α V p using the VMD model [1,2]. What is more, the absolute value of the Υp scattering length is studied using quasi data generated from the QCD model [3,40]. The corresponding results for the scattering lengths are shown in table V and Fig.7 as a function of the inverse vector meson mass. Concretely, the relationship in- [1]), φ (Strakovsky 2020: [2]) and Υ ( Strakovsky 2021: [3]). The magenta-circles show the analysis from two gluon exchange model, black-squares from effective pomeron model. The red-solid line is hypothetical [3].
Actually, the binding energy E b in nuclear matter can be determined by scattering length α V p . In a linear density approximation, the binding energy E b can be written as [21] in which the nuclear matter density ρ nm ⋍ 0.17 fm −3 . Usually, we define the initial stage of meson formation as the "young" stage of meson, and there is also a binding energy between meson and nucleon. The ψ(2S ) binding energy in nuclear matter is calculated as 0.073 MeV. Therefore, the smallness of the binding energy may be related to the "Young" age of the vector mesons participating in the interaction with the proton. As a primitive meson, its properties show some differences. The weak combination between ψ(2S ) and proton can promote the reaction of ψ(2S )p → ψ(2S )p. Moreover, if it is assumed that the interaction between vector meson and nucleon needs more time to reach equilibrium specifically for slow heavy quarkoniums (J/ψ, ψ(2S ), and Υ), this means that the "Young" vector meson effect is more pronounced for heavy quarkoniums. For light vector mesons (such as ω, φ, etc.), this "Young" vector meson effect may be relatively weak.

IV. SUMMARY
In this paper, the value of the scattering length is expressed as a function of the ratio between σ(W) and R(W) within the VMD model. This description can avoid delivering unnecessary inaccuracy in numerical calculation. It is not only suitable for extracting the scattering length from the experimental data directly, but also convenient for observing the near-threshold overall situation for theoretical model. In this paper, we research the J/ψ and ψ(2S ) scattering lengths according to the theoretical study of charmoniums photoproduction within two gluon exchange model and effective pomeron model. The scattering lengths α J/ψp from the two models are basically consistent with several other theoretical predictions. Moreover, the scattering length of ψ(2S )-proton interaction extracted from two gluon exchange model is 1.31 ± 0.92 am. Additionally, α ψ(2S )p = 3.24 ± 0.63 am extracted from effective pomeron model. The value of α J/ψp is bigger than α ψ(2S )p in both models. And these results satisfy the nonlinear exponential increase α V p ∝ exp(1/m V ) basically. According to the present results, it can be roughly concluded that one of the main factors affecting the scattering length α V p is the size of the corresponding cross section of vector meson photoproduction. For example, the cross section of the φ meson photoproduction is more than two orders of magnitude higher than the cross section of J/ψ photoproduction, and correspondingly, the scattering length α φp is nearly twenty times higher than the α J/ψp . In addition, the cross section at the threshold is generally more complicated, and the results given by the two models also show differences. However, due to the lack of experimental data, especially the experimental data of ψ(2S ) photoproduction, the scattering length of ψ(2S )-proton cannot be determined very accurately. Therefore, to better determine the scattering lengths of vector mesons and proton interaction, more high-precision experimental measurements for the photo/electro-production of charmoniums are highly needed, which can not only be realized in the JLab experiment [6], but also within the capabilities of EicC and US-EIC facility [25,26].