We develop a reaction model for η photoproduction off the deuteron (γd→ηpn) and study the reaction at a special kinematics, where the photon beam energy is ∼0.94 GeV and the scattered proton is detected at ∼0°, for the purpose of determining the η-nucleon scattering length (aηN) and effective range (rηN). In this kinematics, the η-nucleon elastic rescattering is significantly enhanced while other background mechanisms are suppressed. We show that a ratio R, the γd→ηpn cross section divided by the γp→ηp cross section convoluted with the proton momentum distribution in the deuteron, has a very good resolving power of aηN and rηN. We conclude that the R data with 5% error, binned in 1 MeV width of the η-neutron invariant mass, can determine Re[aηN] (Re[rηN]) at the precision of ∼±0.1 fm (∼±0.5 fm), significantly narrowing down the previously estimated ranges of the parameters. To arrive at the conclusion, it is essential to use the γd→ηpn reaction model equipped with elementary amplitudes that are well constrained by πN and γN 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.
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