Low-energy $\eta$-nucleon interaction studied with $\eta$ photoproduction off the deuteron

We develop a reaction model for $\eta$ photoproduction off the deuteron ($\gamma d\to \eta pn$) and study the reaction at a special kinematics, where the photon beam energy is $\sim 0.94$ GeV and the scattered proton is detected at $\sim 0^{\circ }$, for the purpose of determining the $\eta$-nucleon scattering length ($a_{\eta N}$) and effective range ($r_{\eta N}$). In this kinematics, the $\eta$-nucleon elastic rescattering is significantly enhanced while other background mechanisms are suppressed. We show that a ratio $R$, the $\gamma d\to \eta pn$ cross section divided by the $\gamma p\to \eta p$ cross section convoluted with the proton momentum distribution in the deuteron, has a very good resolving power of $a_{\eta N}$ and $r_{\eta N}$. We conclude that the $R$ data with 5% error, binned in 1 MeV width of the $\eta$-neutron invariant mass, can determine $\mathrm{Re}\left[a_{\eta N}\right]$ ($\mathrm{Re}\left[r_{\eta N}\right]$) at the precision of $\sim \pm 0.1$ fm ($\sim \pm 0.5$ fm), significantly narrowing down the previously estimated ranges of the parameters. To arrive at the conclusion, it is essential to use the $\gamma d\to \eta pn$ reaction model equipped with elementary amplitudes that are well constrained by $\pi N$ and $\gamma N$ reaction data through a sophisticated coupled-channel analysis. This result strongly motivates the Research Center for Electron Photon Science (ELPH) at Tohoku University to measure $R$.

Figure 1

Diagrammatic representation of reaction mechanisms considered in this work for $\gamma d\to \eta N_1{N}_2$: (a) impulse, (b) $\eta$-exchange, (c) $\pi$-exchange, and (d) $NN$-rescattering mechanisms. Labels for particles along with their momenta in the laboratory frame are indicated. The external lines are the same for all the diagrams and thus their labels are indicated in (a) only. Also, ${\mathit{k}}^{\prime }=\mathit{q}-{\mathit{p}}_{\mathit{2}}+\mathit{l}$ and ${\mathit{p}}^{\prime }=\mathit{q}-\mathit{k}+\mathit{l}$.

Figure 2

Differential cross sections for $\gamma p\to \eta p$ from the DCC model [15] in comparison with data [17] at selected invariant masses of the $\gamma p$ system. The corresponding values of the invariant mass are indicated in each panel.

Figure 3

Angular distributions of $\eta$ in $\gamma d\to \eta pn$ in the $\gamma d$ c.m. frame. The photon laboratory energy $E_{\gamma }$ is indicated in each panel. The solid curves are from the full calculation, while the dotted curves are obtained with the impulse mechanism only. The data are for the semi-inclusive $\gamma d\to \eta X$ process [21]; the coherent contribution is negligible here [22].

Figure 4

(Top) Threefold differential cross section, $d^3\sigma /d{M}_{\eta n}d{\mathrm{\Omega }}_p$, for $\gamma d\to \eta pn$ at $E_{\gamma }=0.94$ GeV and ${\theta }_p=0^{\circ }$, plotted as a function of $M_{\eta n}$. The results are from the full calculation (solid curve), the impulse mechanism only (dotted curve), the  impulse and $\eta$-exchange mechanisms (dashed curve), and the impulse, $\eta$- and $\pi$-exchange mechanisms (dash-dotted curve). The dash-dotted curve falls almost exactly on the solid curve. (Bottom) Ratios of the differential cross sections calculated with the various mechanisms to those from the full calculation.

Figure 5

(Top) $\mathrm{Re}\left[a_{\eta n}\right]$ dependence of $\gamma d\to \eta pn$ differential cross sections at $E_{\gamma }=0.94$ GeV and ${\theta }_p=0^{\circ }$ calculated with the full model. The curves are obtained with $\mathrm{Re}\left[a_{\eta n}\right]=0.2$, 0.5, 0.7, and 1.0 fm; $\mathrm{Im}\left[a_{\eta n}\right]=0.25$ fm, and $r_{\eta n}=0$. (Bottom) $R_{\mathrm{th}}$ defined in Eq. (4) for various values of $\mathrm{Re}\left[a_{\eta n}\right]$.

Figure 6

Similar presentation to Fig. 5, but using $\mathrm{Re}\left[r_{\eta n}\right]=0$, $-2.5$, $-3.5$, and $-6$ fm; $a_{\eta n}=0.75+0.26i$ fm and $\mathrm{Im}\left[r_{\eta n}\right]=0$ fm are fixed.