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We develop a reaction model for $\eta$ photoproduction off the deuteron($\gamma d\to\eta pn$), and study the reaction at a special kinematics, wherethe photon beam energy is $\sim 0.94$ GeV and the scattered proton is detectedat $\sim 0^\circ$, for the purpose of determining the $\eta$-nucleon scatteringlength ($a_{\eta N}$) and effective range ($r_{\eta N}$). In this kinematics,the $\eta$-nucleon elastic rescattering is significantly enhanced while otherbackground mechanisms being suppressed. We show that a ratio $R$, the $\gammad\to\eta pn$ cross section divided by the $\gamma p\to\eta p$ cross sectionconvoluted with the proton momentum distribution in the deuteron, has a verygood 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 ${\rm Re}[a_{\eta N}]$ (${\rm Re}[r_{\eta N}]$) at the precisionof $\sim\pm$0.1 fm ($\sim\pm$0.5 fm), significantly narrowing down thepreviously estimated ranges of the parameters. To arrive at the conclusion, itis essential to use the $\gamma d\to\eta pn$ reaction model equipped withelementary amplitudes that are well constrained by $\pi N$ and $\gamma N$reaction data through a sophisticated coupled-channels analysis. This resultstrongly motivates the Research Center for Electron Photon Science (ELPH) atTohoku University to measure $R$.

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