Neutron skin in $^{48}$Ca determined from p+$^{48}$Ca and $^{48}$Ca+$^{12}$C scattering

In our previous paper, we determined $r_{\rm skin}^{208}({\rm exp})=0.278 \pm 0.035$~fm from $σ_{\rm R}$ for p+$^{208}$Pb scattering, using the Kyushu (chiral) $g$-matrix folding model with the densities calculated with D1S-GHFB with the angular momentum projection (AMP). The value agrees with that of PREX2. Reaction cross sections $σ_{\rm R}$ are available for p+$^{48}$Ca scattering, whereas interaction cross sections $σ_{\rm I}$ are available for $^{48}$Ca + $^{12}$C scattering. As for $^{48}$Ca, the high-resolution $E1$ polarizability experiment ($E1$pE) yields $r_{\rm skin}^{48}(E1{\rm pE}) =0.14 \sim 0.20~{\rm fm}$. We determine $r_{\rm skin}^{48}({\rm exp})$ from the data on $σ_{\rm R}$ for p+$^{48}$Ca scattering and from the data on $σ_{\rm I}$ for $^{48}$Ca+$^{12}$C scattering. We use the Kyushu $g$-matrix folding model with the densities calculated with the D1M-GHFB+AMP densities. The D1M-GHFB+AMP proton and neutron densities are scaled so as to reproduce the data under the condition that the radius $r_{\rm p}$ of the scaled proton density equals the data $r_{\rm p}({\rm exp})$ determined from the electron scattering. We deduce skin values $r_{\rm skin}=r_{\rm n}({\rm exp})-r_{\rm p}({\rm exp})$ from the resulting $r_{\rm n}({\rm exp})$ and the $r_{\rm p}({\rm exp})$ determined from electron scattering. The same procedure is taken for D1S-GHFB+AMP. We regard $r_{\rm skin}^{48}(E1{\rm pE})$ as a reference skin value. Using the reference skin value and taking D1M-GHFB+AMP, we determine $r_{\rm skin}^{48}({\rm exp})=0.158 \pm 0.025$~fm for p+$^{48}$Ca scattering and $0.160 \pm 0.058$~fm for $^{48}$Ca + $^{12}$C scattering. We take the weighted mean and its error for the two skin values. The result is $r_{\rm skin}^{48}({\rm exp})=0.158 \pm (0.023)_{\rm exp} \pm (0.012)_{\rm th}~{\rm fm}$.