Chiral $g$-matrix model for reaction cross section of proton+$^{40}$Ca scattering: Matter radius $r_{\rm m}$ and skin $r_{\rm skin}$ for $^{40}$Ca

For $^{48}$Ca, we determined $r_{m}$fm and $r_{\rm skin}$fm from the central values of $\sigma_{\rm R}({\rm EXP})$ of p+$^{48}$Ca scattering, using the chiral (Kyushu) $g$-matrix folding model with the GHFB+AMP densities. For $^{40}$Ca, Zenihiro {\it et al.} determined $r_n({\rm RCNP})=3.375$~fm and $r_{\rm skin}({\rm RCNP})=-0.01 \pm 0.023$fm from the differential cross section and the analyzing powers for p+$^{40}$Ca scattering. For $^{40}$Ca, $\sigma_{\rm R}({\rm EXP})$ are available with high accuracy. Our aim is to determine matter radius $r_{m}^{40}$ and skin $r_{\rm skin}^{40}$ from $\sigma_{\rm R}({\rm EXP})$ by using the Kyushu $g$-matrix folding model with the GHFB+AMP densities. We first determine $r_m({\rm RCNP})=3.380$fm from the central value -0.01~fm of $r_{\rm skin}({\rm RCNP})$ and $r_p({\rm RCNP})=3.385$fm. The folding model with the GHFB+AMP densities reproduces $\sigma_{\rm R}({\rm EXP})$ in $30 \leq E_{\rm in} \leq 180$MeV, in 2-$\sigma$ level. We scale the GHFB+AMP densities so as to $r_p({\rm AMP})=r_p({\rm RCNP})$ and $r_n({\rm AMP})=r_n({\rm RCNP})$. The $\sigma_{\rm R}({\rm RCNP})$ thus obtained agrees with the original one $\sigma_{\rm R}({\rm AMP})$ for each $E_{\rm in}$. For $E_{\rm in}=180$MeV, we define $F$ as $F=\sigma_{\rm R}({\rm EXP})/\sigma_{\rm R}({\rm AMP})=0.929$. The $F\sigma_{\rm R}({\rm AMP})$ be much the same as the center values of $\sigma_{\rm R}({\rm EXP})$ in $30 \leq E_{\rm in} \leq 180$MeV. We then determine $r_{\rm m}^{40}({\rm EXP})$ from the center values of $\sigma_{\rm R}({\rm EXP})$, using $\sigma_{\rm R}({\rm EXP})=C r_{m}^{2}({\rm EXP})$ with $C=r_{m}^{2}({\rm AMP})/(F\sigma_{\rm R}({\rm AMP}))$. The $r_{m}({\rm EXP})$ are averaged over $E_{\rm in}$. The averaged value is $r_{m}({\rm EXP})=3.380$fm. Eventually, we obtain $r_{\rm skin}({\rm EXP})=-0.01$fm from the averaged $r_{\rm m}({\rm EXP})$~fm and $r_p({\rm PCNP})=3.385$fm.