$p,Lambda$ 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, emphasis 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.
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