Production of $J/\Psi$ on the nucleon and on deuteron targets

A coupled-channel model with $\pi N$, $\rho N$, and $J/\Psi N$ channels is developed to predict the $\pi +N\to J/\Psi +N$ cross sections. The $J/\Psi$-$N$ interaction is parameterized in a form related to what has been predicted by the effective field theory approach and lattice QCD. The other interactions within the model are constrained by the decay width of $J/\Psi \to \rho +\pi$ and the total cross-sectional data of $\pi N$ reactions. The calculated meson-baryon amplitudes are then used to predict the cross sections of the $J/\Psi$ production on the deuteron target by including the contributions from the impulse term and the one-loop calculations of the final $NN$ and $J/\Psi N$ rescattering effects. Predictions of the dependence of the cross sections of ${\pi }^{-}+p\to J/\Psi +n$, $\gamma +d\to J/\Psi +n+p$, and ${\pi }^{+}d\to J/\Psi +p+p$ on the $J/\Psi$-$N$ potentials are presented to examine the feasibility of experimental determinations of the $J/\Psi$-$N$ interaction. Within the vector meson dominance model, we have also applied the constructed coupled-channel model to predict the $\gamma +p\to J/\Psi +p$ cross sections near the $J/\Psi$ production threshold.

Figure 1

Reaction mechanisms of $\pi +N\to J/\psi +N$ (left) and $\rho +N\to J/\psi +N$ (right).

Figure 2

Graphical representation of $J/\psi$ photoproduction on the deuteron. Panels (a), (b), and (c) correspond, respectively, to the impulse contribution, Eq. (43); $NN$ rescattering, Eq. (50); and $J/\psi N$ rescattering, Eq. (49).

Figure 3

Reaction mechanisms of $\gamma +N\to J/\psi +N$: (a) Pomeron exchange and (b) pion exchange.

Figure 4

The total cross section of $\gamma +p\to J/\psi +p$. $W$ is the invariant mass of the $\gamma p$ system. The red solid curves are calculated from the model of Ref. [7]. The data is from [16, 17, 18, 19, 20, 21, 22, 23, 24].

Figure 5

The fits to the data of the total cross sections ${\sigma }_{{\pi }^{\pm }p}^{\mathrm{tot}}$ of ${\pi }^{\pm }p$ reactions, and the total elastic ${\pi }^{\pm }p\to {\pi }^{\pm }p$ cross section ${\sigma }_{{\pi }^{\pm }p,{\pi }^{\pm }p}^{el}$. The data are from PDG [25].

Figure 6

The fits to the data of the total cross sections ${\sigma }_{{\pi }^{\pm }p\to {\rho }^{\pm }p}$ of ${\pi }^{\pm }p\to {\rho }^{\pm }p$ (left) and ${\sigma }_{\gamma p\to {\rho }^{0}p}$ for $\gamma p\to {\rho }^{0}p$ (right). The data of ${\sigma }_{{\pi }^{\pm }p\to {\rho }^{\pm }p}$ are from Ref. [26], and the data of ${\sigma }_{\gamma p\to {\rho }^{0}p}$ are from Refs. [27, 28, 29].

Figure 7

The cross sections of ${\pi }^{-}+p\to J/\Psi +n$ (left) and $J/\Psi +p\to J/\Psi +p$ (right). Black solid, full calculation; dashed (red online), coupled channel, but setting $V_{i,J/\Psi }=0$ for $i,j=\pi N,\rho N$; and dotted (blue online), Born calculation $T_{i,j}=V_{i,j}$. $W$ is the total energy in the c.m. frame.

Figure 8

The total cross sections of ${\pi }^{-}+p\to J/\Psi +n$ (left) and $J/\psi +p\to J/\Psi +p$ (right). $W$ is the total energy in the c.m. frame. The solid, dotted, and dashed curves are calculated by using the $J/\Psi$-$N$ potential, Eq. (11), with $\mu =0.6$ GeV and $\alpha =0.20,0.09,0.06$, respectively.

Figure 9

The differential cross section $d\sigma /\left(d{\Omega }_{p}d{\kappa }_{J/\Psi }\right)$ of $\gamma +d\to J/\Psi +p+n$ vs the momentum ${\kappa }_{J/\Psi }$ of $J/\Psi$ in the c.m. frame of the $J/\Psi +n$ subsystem. The results are from coupled-channel calculations using the $J/\Psi$-$N$ potential, Eq. (11), with $\mu =0.6$ GeV and $\alpha =0.20$. ${\theta }_p$ in panels (a) and (b) are the angles between the incoming photon and the outgoing proton. The dashed (red online), dotted (blue online), and dashed-dotted (pink online) lines are the contributions from the amplitudes of the impulse term, $J/\Psi N$ rescattering, and $NN$ rescattering, respectively. The black solid lines are the coherent sum of these three amplitudes.

Figure 10

The differential cross section $d\sigma /\left(d{\Omega }_{p}d{\kappa }_{J/\Psi }\right)$ of $\gamma +d\to J/\Psi +p+n$ vs the momentum ${\kappa }_{J/\Psi }$ of $J/\Psi$ in the c.m. frame of the $J/\Psi +n$ subsystem. ${\theta }_p$ is the angle between the incoming photon and the outgoing proton. The solid, dotted (blue online), and dashed-dotted (red online) lines are from the coupled-channel calculations using the $J/\Psi N\to J/\Psi N$ potential, Eq. (11), with $\mu =0.6$ GeV and $\alpha =0.20,0.09,0.06$, respectively.

Figure 11

The differential cross section $d\sigma /\left(d{\Omega }_{p}d{\kappa }_{J/\Psi }\right)$ of $\gamma +d\to J/\Psi +p+n$ vs the angle ${\theta }_p$ between outgoing proton and incoming photon. The results are from coupled-channel calculations using the $J/\Psi$-$N$ potential, Eq. (11), with $\mu =0.6$ GeV and $\alpha =0.20$. $T_{J/\psi -n}$ is kinematic energy of $J/\psi -n$ system. The sets of lines are the same as in Fig. 9.

Figure 12

The differential cross section $d\sigma /\left(d{\Omega }_{p}d{\kappa }_{J/\Psi }\right)$ of ${\pi }^{+}+d\to J/\Psi +p+p$ vs the momentum ${\kappa }_{J/\Psi }$ of $J/\Psi$ in the c.m. frame of the $J/\Psi +p$ subsystem. The black solid, dotted (blue online), and dashed-dotted (red online) lines are from the coupled-channel calculations using the $J/\Psi$-$N$ potential, Eq. (11), with $\mu =0.6$ GeV and $\alpha =0.20,0.09,0.06$. ${\theta }_p$ is the angle between the incoming pion and the outgoing proton.

Figure 13

The cross section of $\gamma +N\to J/\Psi +N$ reaction. $W$ is the total energy in the c.m. frame. In the left side, the black solid, dotted (blue online), dashed (red online) lines are calculated by using coupled-channel model with $\mu =0.6$ GeV and $\alpha =0.20,0.09,0.06$ and the dashed-dotted (green online) line is from Pomeron exchange. On the right side, the dashed (red online) and dashed-dotted (blue online) lines are from the full calculation and the Born approximation calculation $T_{i,j}=V_{i,j}$. The parameters of $v_{J/\Psi N,J/\Psi N}$ of the coupled-channel model are $\mu =0.6$ GeV and $\alpha =0.06$. The data are from Refs. [16, 17, 18].