$p,Lambda$ emission in coincidence following $K^-$ absorption at rest in nuclei is studied using quantum mechanical scattering theory and nuclear wave functions. $K^-$ absorption is assumed to occur on two protons in the nucleus. In the formalism, em
phasis is put on the study of the final state interaction (FSI) effects of $p$ and $Lambda$ with the recoiling nucleus. We include elastic scattering and single nucleon knock-out (KO) channels in the FSI. Calculations are presented for the $^{12}$C nucleus, using shell model wave functions, and without any extra mass modification of the $K^-,pp$ system in the nucleus. Calculated results are presented for the angular correlation distribution between $p$ and $Lambda$, their invariant mass distribution and the momentum spectra of $p$ and $Lambda$. These results are compared with the corresponding experimental measurements cite{agnello}. With only elastic scattering FSI included, the angular correlation distribution and the momentum spectra are found to be in good accord with the corresponding measurements. With full FSI the calculated $p,Lambda$ invariant mass distribution is found to have two peaks, one corresponding to the elastic scattering FSI and another to single nucleon KO FSI. The KO peak agrees fully, in position and shape, with the peak observed in Ref. cite{agnello}. The peak corresponding to elastic scattering FSI does not seem to exist in the measured distribution. Considering that such a two peak structure is always seen in the inclusive ($p$, $p^prime $) and ($e$, $e^prime $) reactions in nuclei at intermediate energies, absence of the elastic scattering peak in the $p,Lambda$ reaction is intriguing.
A model for the p d --> p d eta reaction published earlier, including the final state interaction (FSI) of all particles, is revisited to investigate the low energy data on this reaction. The three body problem of p-d-eta scattering in the final stat
e is approximated in terms of pairwise interactions between the three particles in the final state. Apart from a comparison with some preliminary data, two new findings relevant to the near threshold data analysis are reported. The first one points toward the limitations of an FSI factor used conventionally to extract the eta-deuteron scattering length and infer subsequently on the existence of eta-mesic states. The second result emphasizes the role of the $p-d$ FSI and the strong Coulomb repulsion near threshold. Finally, a comparison of the above model calculation with low energy data, excludes very large eta-nucleon scattering lengths.
