$eta$ photoproduction off the deuteron and low-energy $eta$-nucleon interaction

We study $eta$ photoproduction off the deuteron ($gamma dtoeta pn$) at a special kinematics: $sim 0.94$ GeV of the photon beam energy and $sim 0^circ$ of the scattering angle of the proton. This kinematics is ideal to extract the low-energy $eta$-nucleon scattering parameters such as $a_{eta N}$ (scattering length) and $r_{eta N}$ (effective range) because the $eta$-nucleon elastic scattering is significantly enhanced. We show that if a ratio $R$, the $gamma dtoeta pn$ cross section divided by the $gamma ptoeta p$ cross section convoluted with the proton momentum distribution in the deuteron, is measured with 5% error, ${rm Re}[a_{eta N}]$ (${rm Re}[r_{eta N}]$) can be determined at the precision of $simpm$0.1 fm ($simpm$0.5 fm), significantly narrowing down the currently estimated range of the parameters. The measurement is ongoing at the Research Center for Electron Photon Science (ELPH), Tohoku University.