Low-energy η-nucleon interaction studied with η photoproduction off the deuteron

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