Theoretical study of photoproduction of an ${\eta }^{\prime }N$ bound state on a deuteron target with forward proton emission

Possibilities of observing a signal of an ${\eta }^{\prime }n$ bound state are investigated by considering photoproductions of the $\eta$ and ${\eta }^{\prime }$ mesons on a deuteron target with forward proton emission. For this purpose, we take the ${\eta }^{\prime }n$ interaction from the linear $\sigma$ model with a coupling to $\eta n$, in which an $s\text{-wave} {\eta }^{\prime }n$ bound state can be dynamically generated, and we fix the $\gamma p\to \eta p$ and ${\eta }^{\prime }p$ scattering amplitudes so as to reproduce the experimental cross sections with forward proton emission. By using these $\gamma p\to {\eta }^{(\prime )}p$ and ${\eta }^{(\prime )}n\to {\eta }^{(\prime )}n$ amplitudes, we calculate cross sections of the $\gamma d\to \eta np$ and ${\eta }^{\prime }np$ reactions with forward proton emission in single and ${\eta }^{(\prime )}\text{-exchange}$ double-scattering processes. As a result, we find that the signal of the ${\eta }^{\prime }n$ bound state can be seen below the ${\eta }^{\prime }n$ threshold in the $\eta n$ invariant mass spectrum of the $\gamma d\to \eta np$ reaction and is comparable with the contribution from the quasifree ${\eta }^{\prime }$ production above the ${\eta }^{\prime }n$ threshold. We also discuss the behavior of the signal of the ${\eta }^{\prime }n$ bound state in several experimental conditions and model parameters.

Figure 1

Feynman diagrams for the $\gamma d\to mnp$ reaction with $m=\eta$ or ${\eta }^{\prime }$. Here $T_1$ and $T_2$ are the $\gamma p\to mp$ and $mn\to mn$ scattering amplitudes, respectively.

Figure 2

Feynman diagrams for the $\gamma p\to mp$ reaction with $m=\eta$ or ${\eta }^{\prime }$. Here the solid, dashed, and wavy lines represent the proton, $m$, and photon, respectively. The open and shaded circles correspond to the $\gamma p\to mp$ and $mp\to mp$ amplitudes, respectively.

Figure 3

Differential cross sections of $\gamma p\to \eta p$ and ${\eta }^{\prime }p$ reactions calculated with the amplitudes in Eq. (20) and comparison with the experimental data in Refs. [32, 34]. The experimental data are taken with the scattering angle $-0.8<cos{\theta }_m^{\mathrm{c}.\mathrm{m}.}<-0.7$ for both $\eta$ and ${\eta }^{\prime }$ photoproductions, with ${\theta }_m^{\mathrm{c}.\mathrm{m}.}$ being the meson angle in the center-of-mass frame.

Figure 4

Invariant mass spectra for the $\gamma d\to Xp$ reactions with $X=\eta n$ and ${\eta }^{\prime }n$. The exchanged meson energy $q^0$ is fixed as in Eq. (8) [Eq. (9)] for the option A (B). The initial photon energy is fixed as $E_{\gamma }^{\mathrm{lab}}=2.1\text{GeV}$ and the proton scattering angle in the global center of mass is ${\theta }_p^{\mathrm{c}.\mathrm{m}.}=0^{\circ }$.

Figure 5

Invariant mass spectrum for the $\gamma d\to \eta np$ reaction. The initial photon energy is fixed as $E_{\gamma }^{\mathrm{lab}}=2.1\text{GeV}$ and the proton scattering angle in the global center of mass is ${\theta }_p^{\mathrm{c}.\mathrm{m}.}=0^{\circ }$. The inset represents an enlarged figure.

Figure 6

Invariant mass spectrum for the $\gamma d\to {\eta }^{\prime }np$ reaction. The initial photon energy is fixed as $E_{\gamma }^{\mathrm{lab}}=2.1\text{GeV}$ and the proton scattering angle in the global center of mass is ${\theta }_p^{\mathrm{c}.\mathrm{m}.}=0^{\circ }$. The inset represents an enlarged figure.

Figure 7

Invariant mass spectrum for the $\gamma d\to Xp$ reaction with $X=\eta n$ and ${\eta }^{\prime }n$ as the sum of the two contributions. The interaction strength is controlled by the parameter $g$ in Eq. (12). The initial photon energy is fixed as $E_{\gamma }^{\mathrm{lab}}=2.1\text{GeV}$ and the proton scattering angle in the global center of mass is ${\theta }_p^{\mathrm{c}.\mathrm{m}.}=0^{\circ }$.

Figure 8

Invariant mass spectrum for the $\gamma d\to {\eta }^{\prime }np$ reaction. The interaction strength is controlled by the parameter $g$ in Eq. (12). The initial photon energy is fixed as $E_{\gamma }^{\mathrm{lab}}=2.1\text{GeV}$ and the proton scattering angle in the global center of mass is ${\theta }_p^{\mathrm{c}.\mathrm{m}.}=0^{\circ }$.

Figure 9

Invariant mass spectrum for the $\gamma d\to Xp$ reaction with $X=\eta n$ and ${\eta }^{\prime }n$ as the sum of the two contributions. The interaction strength is controlled by the parameter $m_{\sigma 8}$ in Eq. (12). The initial photon energy is fixed as $E_{\gamma }^{\mathrm{lab}}=2.1\text{GeV}$ and the proton scattering angle in the global center of mass is ${\theta }_p^{\mathrm{c}.\mathrm{m}.}=0^{\circ }$.

Figure 10

Invariant mass spectrum for the $\gamma d\to Xp$ reaction with $X=\eta n$ and ${\eta }^{\prime }n$ as the sum of the two contributions. Solid and dashed lines represent the invariant mass spectra with and without inclusion of the $\pi N$ channel, respectively. The initial photon energy is fixed as $E_{\gamma }^{\mathrm{lab}}=2.1\text{GeV}$ and the proton scattering angle in the global center of mass is ${\theta }_p^{\mathrm{c}.\mathrm{m}.}=0^{\circ }$.

Figure 11

Invariant mass spectrum for the $\gamma d\to Xp$ reaction with $X=\eta n$ and ${\eta }^{\prime }n$ as the sum of the two contributions. The initial photon energy is taken from $E_{\gamma }^{\mathrm{lab}}=2.0$ to $2.4\text{GeV}$ in intervals of $0.1\text{GeV}$. The proton scattering angle in the global center of mass is ${\theta }_p^{\mathrm{c}.\mathrm{m}.}=0^{\circ }$.

Figure 12

Invariant mass spectrum for the $\gamma d\to Xp$ reaction with $X=\eta n$ and ${\eta }^{\prime }n$ as the sum of the two contributions. The scattering angle of the final-state proton is taken from ${\theta }_p^{\mathrm{c}.\mathrm{m}.}=0^{\circ }$ to ${30}^{\circ }$ in the global center-of-mass frame. The initial photon energy is fixed as $E_{\gamma }^{\mathrm{lab}}=2.1\text{GeV}$.

Figure 13

Invariant mass spectrum for the $\gamma d\to Xp$ reaction with $X=\eta n$ and ${\eta }^{\prime }n$ as the sum of the two contributions. The scattering angle of the final-state proton in the laboratory frame, ${\theta }_p^{\mathrm{lab}}$, is integrated in the ranges [$0^{\circ }, 5^{\circ }$], [$0^{\circ }, {10}^{\circ }$], and [$0^{\circ }, {20}^{\circ }$]. The initial photon energy is fixed as $E_{\gamma }^{\mathrm{lab}}=2.1\text{GeV}$.