Cross section Measurements for $^{12}$C$(K^-, K^+\Xi^-)$ and $^{12}$C$(K^-, K^+\Lambda\Lambda)$ Reactions at 1.8 GeV$/c$

We present a measurement of the production of $\Xi^-$ and $\Lambda\Lambda$ in the $^{12}$C$(K^-, K^+)$ reaction at an incident beam momentum of 1.8 GeV/$\mathit{c}$, based on high-statistics data from J-PARC E42. The cross section for the $^{12}$C$(K^-, K^+\Xi^-)$ reaction, compared to the inclusive $^{12}$C$(K^-, K^+)$ reaction cross section, indicates that the $\Xi^-$ escaping probability peaks at 70\% in the energy region of $E_\Xi=$100 to 150 MeV above the $\Xi^-$ emission threshold. A classical approach using eikonal approximation shows that the total cross sections for $\Xi^-$ inelastic scattering ranges between 42 mb and 23 mb in the $\Xi^-$ momentum range from 0.4 to 0.6 GeV/c. Furthermore, based on the relative cross section for the $^{12}$C$(K^-, K^+\Lambda\Lambda)$ reaction, the total cross section for $\Xi^-p\to\Lambda\Lambda$ is estimated in the same approach to vary between 2.2 mb and 1.0 mb in the momentum range of 0.40 to 0.65 GeV/c. Specifically, a cross section of 1.0 mb in the momentum range of 0.5 to 0.6 GeV/c imposes a constraint on the upper bound of the decay width of the $\Xi^-$ particle in infinite nuclear matter, revealing $\Gamma_\Xi< \sim 0.6$ MeV.