Study of $\eta N$ scattering length by means of ($p,^{3}$He) nuclear reactions leading to the formation of $\eta$-mesic nucleus

Because $\eta N$ scattering length is not directly measurable, its value is strongly model dependent$^{1}$. A good knowledge of this scattering length is, however, valuable to the modeling of various hadronic interactions where $\eta$ is produced. Since low energy $\eta N$ interaction is attractive and $\eta$-mesic nuclei can exist$^{2}$, we suggest to use the $p+A\rightarrow ^{3}$He $+(A-2)_{\eta}$ reaction to probe the $\eta N$ scattering length. In our model, the $\eta$ is first produced via $p+d\rightarrow \eta + ^{3}$He reaction, and then captured by the residual nucleus of nucleon number $A-2$. As a stronger scattering length allows $\eta$ to be bound onto lighter nuclei, the smallest value of $(A-2)$ for which $\eta$-nucleus bound states can exist will set an upper limit for the scattering length. Our choice of having $^{3}$He in the final state is for the purpose of significantly reducing background events. This suggestion will soon be implemented in an experiment at Juelich, Germany. We will present our predictions for $^{16}$O and $^{12}$C as target nuclei, leading to $^{14}$N$_{\eta}$ and $^{10}$B$_{\eta}$, respectively. Sensitivity of the $\eta$-mesic nucleus formation to the $\eta N$ scattering length will also be discussed.\\ $^{1}$ Q. Haider and L.C. Liu, Phys. Rev. {\bf C 66}, 045208 (2002).\\ $^{2}$ Q. Haider and L.C. Liu, Phys. Lett. {\bf B 172}, 257 (1986).