A study of the $p d \to p d \eta$ reaction

A study of the $p d \to p d \eta$ reaction in the energy range where the recent data from Uppsala are available, is done in the two-step model of $\eta$ production including the final state interaction. The $\eta -d$ final state interaction is incorporated through the solution of the Lippmann Schwinger equation using an elastic scattering matrix element, $T_{\eta d \to \eta d}$, which is required to be half off-shell. It is written in a factorized form, with an off-shell form factor multiplying an on-shell part given by an effective range expansion up to the fourth power in momentum. The parameters of this expansion have been taken from an existing recent relativistic Faddeev equation solution for the $\eta NN$ system corresponding to different $\eta-N$ scattering amplitudes. Calculations have also been done using few body equations within a finite rank approximation (FRA) to generate $T_{\eta d \to \eta d}$. The $p-d$ final state interaction is included in the spirit of the Watson-Migdal prescription by multiplying the matrix element by the inverse of the Jost function. The $\eta-d$ interaction is found to be dominant in the region of small invariant $\eta -d$ mass, $M_{\eta d}$. The $p-d$ interaction enhances the cross section in the whole region of $M_{\eta d}$, but is larger for large $M_{\eta d}$. We find nearly isotropic angular distributions of the proton and the deuteron in the final state. All the above observations are in agreement with data. The production mechanism for the entire range of the existing data on the $p d \to p d \eta$ reaction seems to be dominated by the two-step model of $\eta$ production.