Strength of the $\Sigma$ mean field deduced from SCDW model analyses of $(\pi^-,K^+)$ inclusive spectra on $^{12}$C, $^{28}$Si, $^{58}$Ni and $^{209}$Bi

The semiclassical distorted wave model is developed to analyze $(\pi^-,K^+)$ $\Sigma$ formation inclusive spectra on various nuclear targets. We do not introduce a factorization approximation in terms of Fermi averaging of the elementary cross sections in nuclear medium. Confirming first that the model works well for the $(\pi^+,K^+)$ $\Lambda$ formation inclusive spectra, we apply it to the $\Sigma$ case. The shape and magnitude of the experimental spectra for various target nuclei taken at KEK [P.K. Saha {\it et al.}, Phys. Rev. C70, 044613 (2004)] are satisfactorily reproduced using a repulsive $\Sigma$-nucleus single-particle potential whose strength is of the order of 30$\sim$50 MeV. The isovector part of the $\Sigma$ single-particle potential is also determined from the data of heavier nuclei.