Models of $J/\varPsi$ photo-production reactions on the nucleon

Abstract

The $J/\varPsi$ photo-production reactions on the nucleon can provide information on the roles of gluons in determining the $J/\varPsi$-nucleon ($J/\varPsi$-N) interactions and the structure of the nucleon. The information on the $J/\varPsi$-N interactions is needed to test lattice QCD (LQCD) calculations and to understand the nucleon resonances such as $N^*(P_c)$ recently reported by the LHCb Collaboration. In addition, it is also needed to investigate the production of nuclei with hidden charms and to extract the gluon distributions in nuclei. The main purpose of this article is to review six models of the $\gamma + p \rightarrow J/\varPsi +p$ reaction which have been and can be applied to analyze the data from Thomas Jefferson National Accelerator Facility (JLab). The formulae for each model are given and used to obtain the results to show the extent to which the available data can be described. The models presented include the Pomeron-exchange model of Donnachie and Landshoff (Pom-DL) and its extensions to include $J/\varPsi$-N potentials extracted from LQCD (Pom-pot) and to also use the constituent quark model (CQM) to account for the quark substructure of $J/\varPsi$ (Pom-CQM). The other three models are developed from applying the perturbative QCD approach to calculate the two-gluon exchange using the generalized parton distribution (GPD) of the nucleon (GPD-based), two- and three-gluon exchanges using the parton distribution of the nucleon ($2g+3g$), and the exchanges of scalar ($0^{++}$) and tensor ($2^{++}$) glueballs within the holographic formulation (holog). The results of investigating the excitation of the nucleon resonances $N^*(P_c)$ in the $\gamma + p \rightarrow J/\varPsi +p$ reactions are also given. We demonstrate that the differences between these six models can be unambiguously distinguished and the $N^*$ can be better studied by using the forthcoming JLab data at large |t| and at energies very near the $J/\varPsi$ production threshold. Possible improvements of the considered models are discussed.

Appendix A: Regge phenomenology

There exists extensive literature [14,15,16] on Regge phenomenology. For our purposes, we will only give sufficiently self-contained explanations which are needed to present the formulas of Pomeron-exchange models.

Fig. 33

Momentum variables of the s-channel (left) and t-channel scattering (right)

Considering the two-body scattering, the amplitudes of s-channel $1(p_1)+2(p_2) \rightarrow 3(p_3)+4(p_4)$ and t-channel $1(p_1)+ 3(-p_3) \rightarrow 2(-p_2)+4(p_4)$ scattering, as shown in Fig. 33, are written in terms of the usual Mandelstam variables defined by

$$\begin{aligned} s= & {} (p_1+p_2)^2 , \end{aligned}$$

(A.1)

$$\begin{aligned} t= & {} (p_1-p_3)^2 \end{aligned}$$

(A.2)

for the s-channel scattering of Fig. 33(left) and

$$\begin{aligned} s_t= & {} (p_1+(-p_3))^2=(p_1-p_3)^2=t, \end{aligned}$$

(A.3)

$$\begin{aligned} t_t= & {} (p_1-(-p_2))^2=(p_1+p_2)^2=s \end{aligned}$$

(A.4)

for the t-channel of Fig. 33(right). One of the steps in developing Regge phenomenology is to assume the crossing symmetry that the scattering amplitudes T(t, s) for the s-channel and $T_t(t_t,s_t)$ for the t-channel are related by

$$\begin{aligned} T(t,s) =T_t(s,t)=T_t(t_t,s_t)\,. \end{aligned}$$

(A.5)

It is important to note here that s ($s_t$) is the total collision energy in the s-channel (t-channel) CM frame, and t ($t_t$) defines the corresponding momentum-transfer of the scattering. Thus the crossing symmetry implies that a bound or resonance state, called R, in the t-channel scattering $1+3\rightarrow R \rightarrow 2+4$ can be an exchanged particle R in the s-channel $1+2\rightarrow 3+4$ scattering.

We now describe the essential steps in getting the s-channel scattering amplitude T(t, s) from the t-channel scattering amplitude $T_t(t_t,s_t)$ by using the crossing symmetry relation of Eq. (A.5). Considering $1(p_1) + 3(-p_3) \rightarrow 2(-p_2) + 4(p_4)$ in the CM system of the t-channel, we then have the following definitions of the momentum variables:

$$\begin{aligned} p= & {} |\textbf{p}_1|=|\textbf{p}_3| ,\nonumber \\ q= & {} |\textbf{p}_2|=|\textbf{p}_4| , \end{aligned}$$

(A.6)

and

$$\begin{aligned} s_t= & {} t= [E_{1}(p)+E_{3}(p)]^2=[E_2(q)+E_4(q)]^2, \end{aligned}$$

(A.7)

$$\begin{aligned} t_t= & {} s=m^2_1+m^2_2+2E_1(p)E_2(q)-2\,qp \cos \theta _t , \end{aligned}$$

(A.8)

where $\cos \theta _t=\hat{p}_1\cdot (-\hat{p}_2)$ defines the scattering angle $\theta _t$ in t-channel. Eq. (A.8) then leads to

$$\begin{aligned} \cos \theta _t=\frac{1}{2pq} \left[ s-m^2_1-m^2_2+2E_1(p)E_2(q) \right] . \end{aligned}$$

(A.9)

Note that p and q are function of t as can be seen from Eq. (A.7) and hence for a given t, $\cos \theta _t$ depends linearly on s.

Fig. 34

Contour in l-plane

The next step is to examine the partial-wave expansion of t-channel amplitude. By using the relations Eqs. (A.7)–(A.9), we then have

$$\begin{aligned} T_t(t_t,s_t)= & {} T_t(t_t(\cos \,\theta _t),t) \nonumber \\= & {} \sum _{l=0}^{\infty }(2l+1)P_l(\cos \,\theta _t) A(l,t) , \end{aligned}$$

(A.10)

where $P_l(x)$ is a Legendre polynomial in x. In the complex-l plane, we apply the Watson-Sommerfeld transformation [115,116,117] to write the above expression as

$$\begin{aligned} T(t,s)= & {} T_t(t_t,s_t) \nonumber \\= & {} \frac{1}{2\pi i}\int _C\frac{\pi }{\sin l\pi }P_l(-\cos \theta _t)A(l,t)\, dl , \end{aligned}$$

(A.11)

where C is the contour indicated in Fig. 34. The denominator $\sin l\pi$ generates the poles (solid circles) indicated in Fig. 34. Within the non-relativistic quantum mechanics, Regge [83,84,85] showed that if the s-channel amplitude T(t, s) is defined by a local potential like Yukawa potential $(\sim e^{-\mu r}/r$), A(l, t) is analytic in the complex l-plane, aside from poles in the $\text{ Re }(l) \ge -1/2$. It can therefore be written in the following form:

$$\begin{aligned} A(l,t)=\sum _{n}\frac{\beta _n(t)}{l-\alpha _n(t)} . \end{aligned}$$

(A.12)

Closing the contour C at infinity and through the $\text{ Re }(J)=-1/2$ line, as indicated in Fig. 34, and using the Cauchy’s theorem, Eq. (A.11) then becomes

$$\begin{aligned} T(t,s)= & {} T_t(t_t,s_t) \nonumber \\= & {} \int _{-1/2-i\infty }^{-1/2+i\infty }\frac{\pi }{\sin \,l\pi } P_l(-\cos \,\theta _t)A(l,t)\, dl \nonumber \\{} & {} +\sum _{n}\beta _n(t)P_{\alpha _n(t)}(-\cos \theta _t) \frac{1}{\sin \pi \alpha _n(t)} . \end{aligned}$$

(A.13)

Here $\alpha _n(t)$ is called the Regge trajectory which leads to poles of the amplitude at

$$\begin{aligned} \alpha _n(t= M_{L_n}^2)= L_n\,;\quad L_n = 0,1,2, \dots . \end{aligned}$$

(A.14)

At these pole positions, the usual Legendre polynomial has the property $P_{\alpha _n(t)}(-\cos \theta ) \rightarrow (-1)^{L_n}P_{L_n}(\cos \theta )$. Thus it is suggestive that $L_n$ can be interpreted as the angular momentum of the particle formed in the t-channel process with mass $M_{L_n}$ because $t=s_t=[E_1(p)+E_2(p)]^2$. These particles are interpreted as the exchanged particle in s-channel scattering. If this interpretation is correct, we can use the particle spectrum found in t-channel scattering to define the Regge trajectory. Thus the main feature of the Regge phenomenology is: the particle spectrum can define the scattering amplitudes.

The first term in Eq. (A.13) is neglected in practice. It is also extended to define natural-parity exchange from the unnatural-parity exchange. The amplitude of s-channel scattering amplitude is then of the following form:

$$\begin{aligned} T(s,t)=\sum _{n}\beta _n(t)\frac{P_{\alpha _n(t)}(-\cos \theta _t) +s_n\,P_{\alpha _n(t)}(+\cos \theta _t)}{2\sin [\pi \alpha _n(t)]} , \end{aligned}$$

(A.15)

where the signature of the trajectory, $s_n=+1$ $(-1)$ corresponds to even (odd) parity exchanges. In the high energy limit with very large s and $s \gg |t|$, $\cos \theta _t \sim -s/(2q(t)p(t))$, as can be seen from Eq. (A.9). It follows that

$$\begin{aligned} P_{\alpha _n(t)}(-\cos \theta _t) \sim \left( \frac{s}{2p(t)q(t)}\right) ^{\alpha _n(t)} . \end{aligned}$$

(A.16)

Here we recall Eq. (A.7) to note that the momenta p and q of t-channel as functions of t of the s-channel scattering. We then have

$$\begin{aligned} T(s,t)= & {} \sum _{n}F_{t}(t) \frac{1+s_ne^{-i\pi \alpha _n(t)}}{2\sin [\pi \alpha _n(t)]} \left[ \alpha _{1,n}(t)s \right] ^{\alpha _n(t)} , \end{aligned}$$

(A.17)

where

$$\begin{aligned} F_{t}(t)= \beta _n(t) \left( \frac{\alpha _{1,n}}{2p(t)q(t)} \right) ^{-\alpha _n(t)} . \end{aligned}$$

(A.18)

If we write $F_{f}(t)=\beta ^{13}_n(t)\beta ^{24}_n(t)$ and assume that $\beta ^{13}_n(t)$ and $\beta ^{24}_n(t)$ characterize the hadron structure, we then have the following form

$$\begin{aligned} T(s,t)=\sum _{n}\beta ^{13}_n(t) \beta ^{24}_n(t) \frac{1+s_ne^{-i\pi \alpha _n(t)}}{2\sin [\pi \alpha _n(t)]} \left[ \alpha _{1,n}(t)s \right] ^{\alpha _n(t)} . \end{aligned}$$

(A.19)

The amplitude can then be interpreted as the exchange of particles with masses defined by $\alpha _n(t=M^2_{L_n})= L_n$. This is an intuitively very attractive interpretation of the scattering. However, there exists no successful derivation of Eq. (A.19) from relativistic quantum field theory and the form factors $\beta ^{13}_n(t)$ and $\beta ^{24}_n(t)$ are determined experimentally or calculated from a theoretical model.