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Extended optimal Fermi averaging for near-recoilless $Λ$ production in the ($K^-$, $π^-$) reaction on nuclei

We propose to extend the optimal Fermi averaging procedure theoretically in order to calculate an in-medium $K^-n\to π^- Λ$ amplitude for the exothermic ($K^-$, $π^-$) reaction on nuclei in the framework of a distorted-wave impulse approximation, taking into account the local momentum transfer generated by semiclassical distorted waves for $K^-$ and $π^-$ mesons. Angular distributions for the $^{12}$C($K^-$, $π^-$)$^{12}_Λ$C reaction in which a momentum transfer is $q \lesssim$ 80 MeV/$c$ at $p_K=$ 800 MeV/$c$ in the $π^-$ forward direction, are estimated by applying the extended procedure. The result shows that the calculated angular distributions are in good agreement with those of the data, and this extension is a successful prescription making it possible to describe the reaction cross sections in near-recoilless reactions such as ($K^-$, $π^-$) in our framework.