Effect of the final state interaction of ${\eta }^{\prime }N$ on the ${\eta }^{\prime }$ photoproduction off the nucleon

We investigate the ${\eta }^{\prime }$ photoproduction off the nucleon with a particular interest in the effect of the final-state interaction (FSI) of the ${\eta }^{\prime }$ meson and nucleon $\left({\eta }^{\prime }N\right)$ based on the three-flavor linear $\sigma$ model. We find an enhancement in the cross section of the ${\eta }^{\prime }$ photoproduction near the ${\eta }^{\prime }N$-threshold energy owing to the ${\eta }^{\prime }N$ FSI. With the ${\eta }^{\prime }$ meson at forward angles, the energy dependence near the ${\eta }^{\prime }N$ threshold is well reproduced with the ${\eta }^{\prime }N$ FSI. The cross section at backward angles can also be a good probe to investigate the strength of the ${\eta }^{\prime }N$ interaction.

Figure 1

Diagrams for the scattering equation Eq. (2) for the ${\eta }^{\prime }$-photoproduction $T$ matrix. The white, small black, and shaded blobs are the $T$ matrix of the ${\eta }^{\prime }$ photoproduction, $T_{\gamma N\to {\eta }^{\prime }N}$, the ${\eta }^{\prime }$-photoproduction vertex, $V_{\gamma N\to {\eta }^{\prime }N}$, and the ${\eta }^{\prime }NT$ matrix, $T_{{\eta }^{\prime }N\to {\eta }^{\prime }N}$. The solid, dashed, and wavy lines stand for the propagations of the nucleon, ${\eta }^{\prime }$ meson, and photon, respectively.

Figure 2

A diagrammatic representation of the scattering matrix for the ${\eta }^{\prime }N$ scattering including the ${\eta }^{\prime }N$ and $\eta N$ channels, $T_{ij}(i,j={\eta }^{\prime }N,\eta N)$. The shaded and small white blobs represent the $T$ matrices, $T_{ij}$, and interaction kernels, $V_{ij}$. The solid line denotes the propagation of the nucleon, and the dashed one represents that of the $\eta$ or ${\eta }^{\prime }$ meson.

Figure 3

Diagrams for the interaction kernels $V_{ij}$ in Eq. (3) at the tree level. The thick solid line stands for the propagation of the scalar meson, and the other lines have the same notations as those in Fig. 2. The first term is the contribution from the scalar-meson exchange in the $t$ channel, and the second and third ones represent the Born terms in the $s$ and $u$ channels, respectively.

Figure 4

Diagrams for the ${\eta }^{\prime }$ photoproduction amplitude with the tree-level approximation, with the same notation as in Fig. 1. The first, second, and third terms are the contributions from the $s$, $u$, and $t$ channels, respectively.

Figure 5

Differential cross sections of the ${\eta }^{\prime }$ photoproduction without FSI at $cos{\theta }_{{\eta }^{\prime }}^{\mathrm{c}.\mathrm{m}.}=0.75$ (left) and $-0.75$ (right) as functions of the total energy $W$ in the c.m. frame. In the left figure, $s$- and $u$-channel contributions are multiplied by factors 150 and 5, respectively, and the $s$-channel one in the right figure is multiplied by a factor 100.

Figure 6

Cutoff dependence of the differential cross sections of the ${\eta }^{\prime }$ photoproduction off the nucleon without the ${\eta }^{\prime }N$ FSI. The cutoff parameter $\mathrm{\Lambda }$ is varied as $\mathrm{\Lambda }=500,700,$ and 900 MeV. Note that the results for $\mathrm{\Lambda }=500$ and 900 MeV are scaled by factors 11.5 and 0.2, respectively.

Figure 7

Differential cross sections at the forward (left, $cos{\theta }_{{\eta }^{\prime }}^{\mathrm{c}.\mathrm{m}.}=0.75$) and the backward (right, $cos{\theta }_{{\eta }^{\prime }}^{\mathrm{c}.\mathrm{m}.}=-0.75$) angles as functions of $W$ with and without the ${\eta }^{\prime }N$ FSI. The cases (a) to (e) in the legend follow those given in Table 2.

Figure 8

Differential cross sections of the ${\eta }^{\prime }$ photoproduction at $cos{\theta }_{{\eta }^{\prime }}^{\mathrm{c}.\mathrm{m}.}=0.75$ (left) and $-0.75$ (right) as functions of the total energy $W$. The cases (a) to (e) in the legend are the same as those in Fig. 7. The points with the error bar are the experimental data taken from Ref. [36].

Figure 9

Differential cross section of the ${\eta }^{\prime }$ photoproduction as functions of $cos{\theta }_{{\eta }^{\prime }}^{\mathrm{c}.\mathrm{m}.}$ with and without the ${\eta }^{\prime }N$ FSI at $W=1.925$ (upper left), 2.045 (upper right), 2.230 (lower left), and 2.420 (lower right) GeV. The legend is the same as that in Fig. 7.

Figure 10

Total cross sections of the ${\eta }^{\prime }$ photoproduction off the nucleon as functions of the total energy $W$. The legend is the same as that in Fig. 7.

Figure 11

Beam asymmetries $\mathrm{\Sigma }$ as functions of the scattering angle ${\theta }_{{\eta }^{\prime }}^{\mathrm{c}.\mathrm{m}.}$ with the total energy $W=1.903$ (left) and 1.912 (right) GeV. The cases (a) to (e) in the legend are the same as those in Fig. 7.