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

Abstract

We develop a reaction model for $η$ photoproduction off the deuteron ($γd\toηpn$), 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 $η$-nucleon scattering length ($a_{ηN}$) and effective range ($r_{ηN}$). In this kinematics, the $η$-nucleon elastic rescattering is significantly enhanced while other background mechanisms being suppressed. We show that a ratio $R$, the $γd\toηpn$ cross section divided by the $γp\toηp$ cross section convoluted with the proton momentum distribution in the deuteron, has a very good resolving power of $a_{ηN}$ and $r_{ηN}$. We conclude that the $R$ data with 5% error, binned in 1 MeV width of the $η$-neutron invariant mass, can determine ${\rm Re}[a_{ηN}]$ (${\rm Re}[r_{ηN}]$) 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 $γd\toηpn$ reaction model equipped with elementary amplitudes that are well constrained by $πN$ and $γN$ reaction data through a sophisticated coupled-channels analysis. This result strongly motivates the Research Center for Electron Photon Science (ELPH) at Tohoku University to measure $R$.