Determination of $r_{\rm skin}^{208}$ from reaction cross section for proton scattering

Reaction cross sections are available for $p$ scattering on $^{208}$Pb. We reanalyzed interaction cross sections for $^{42-51}$Ca+$^{12}$C scattering, using the chiral (Kyushu) $g$-matrix folding model with the densities calculated with GHFB and GHFB+AMP. We determined $r_{\rm skin}$ from the $\sigma_{\rm I}$, using the folding model. One of the results is $r_{\rm skin}^{48}=0.164$fm for $^{48}$Ca. Our aim is to determine $r_{\rm skin}^{208}$ from the central values of measured $\sigma_{\rm R}$ for $p$+$^{208}$Pb scattering, using the Kyushu $g$-matrix folding model with the GHFB+AMP densities. The Kyushu $g$-matrix folding model with the GHFB+AMP densities reproduces $\sigma_{\rm R}$ in $40 \leq E_{\rm in} \leq 180$MeV. The $r_{\rm m}({\rm AMP})$ calculated with GHFB+AMP slightly overestimates the lower bound 5.54 fm of $r_{\rm m}({\rm PREX})$ by 0.3\%. We then identify the $\sigma_{\rm R}({\rm AMP})$ with the lower bound of $\sigma_{\rm R}({\rm PREX})$. The $r_{\rm p}({\rm PREX})$ and $r_{\rm n}({\rm PREX})$ in PREX are most reliable. The $r_{\rm p}({\rm AMP})$ and $r_{\rm n}({\rm AMP})$ slightly underestimate the central value $r_{\rm p}({\rm PREX})=$5.45fm by 0.1\% and the central value $r_{n}({\rm PREX})=$5.78 fm by 3.6\%, respectively. The small deviation allows us to scale the GHFB+AMP densities to the central values of $r_{\rm p}({\rm PREX})$ and $r_{\rm n}({\rm PREX})$. The Kyushu $g$-matrix folding model with the scaled densities almost reproduces the central values of $\sigma_{\rm R}$ when $E_{\rm in}=40-81$ MeV, so that both the $\sigma_{\rm R}({\rm AMP})$ and $\sigma_{\rm R}({\rm PREX})$ are in one $\sigma$ of $\sigma_{\rm R}$ there. In $E_{\rm in}=40-81$MeV, we determine the $r_{m}$ from the central values of $\sigma_{\rm R}$ and take the average for the $r_{m}$. The averaged value is $r_{m}=5.61$fm, leading to $r_{\rm skin}^{208}=0.25$fm.