We suggest that the $\\gamma p \\rightarrow \\phi \\eta p$ reaction cross section, in the kinematics where the $\\eta p$ invariant mass in the final state
lies between the threshold value (m$_p$+m$_\\eta$) and the N*(1535) resonance mass, is largely determined by the $\\eta N$ scattering amplitude close to threshold. The initial photon energy is chosen in the range $4\\, <\\,E_\\gamma^{Lab}\\,<\\,5$ GeV, in order to reach low (absolute) values of the squared 4-momentum transfer from the initial photon
to the final $\\phi$-meson. In these conditions, we expect the t-channel $\\pi^0$- and $\\eta$-meson exchanges to drive the dynamics underlying the $\\gamma p \\rightarrow \\phi \\eta p$ process. We show that the $\\eta$-exchange is the dominating contribution to the cross section while the $\\pi^0$-exchange is negligible. The $\\eta$-$\\pi^0$ interference is of the order of $20\\,-\\,30 \\%$. The sign of this term is not known and alters significantly our results. Data on the
$\\gamma p \\rightarrow \\phi \\eta p$ process would be therefore very useful to help unravelling
the behavior of the $\\eta p$ scattering amplitude close to threshold and assessing the possibility of producing $\\eta$-nucleus bound states. |
