Neutron skin thickness of ${}^{208}$Pb determined from reaction cross section for proton scattering

The reaction cross section $\sigma_R$ is useful to determine the neutron radius $R_n$ as well as the matter radius $R_m$. The chiral (Kyushu) $g$-matrix folding model for $^{12}$C scattering on $^{9}$Be, $^{12}$C, $^{27}$Al targets was tested in the incident energy range of $30 \lsim E_{\rm in} \lsim 400 $ MeV, and it is found that the model reliably reproduces the $\sigma_R$ in $30 \lsim E_{\rm in} \lsim 100 $ MeV and $250 \lsim E_{\rm in} \lsim 400$ MeV. \item[Aim] We determine $R_n$ and the neutron skin thickness $R_{\rm skin}$ of ${}^{208}{\rm Pb}$ by using high-quality $\sigma_R$ data for the $p+{}^{208}{\rm Pb}$ scattering in $30 \leq E_{\rm in} \leq 100$ MeV. The theoretical model is the Kyushu $g$-matrix folding model with the densities calculated with Gongny-D1S HFB (GHFB) with the angular momentum projection (AMP). \item[Results] The Kyushu $g$-matrix folding model with the GHFB+AMP densities underestimates $\sigma_{\rm R}$ in $30 \leq E_{\rm in} \leq 100$~MeV only by a factor of 0.97. Since the proton radius $R_p$ calculated with GHFB+AMP agrees with the precise experimental data of 5.444 fm, the small deviation of the theoretical result from the data on $\sigma_R$ allows us to scale the GHFB+AMP neutron density so as to reproduce the $\sigma_R$ data. In $E_{\rm in}$ = 30--100 MeV, the experimental $\sigma_R$ data can be reproduced by assuming the neutron radius of ${}^{208}{\rm Pb}$ as $R_n$ = $5.722 \pm 0.035$ fm. \item[Conclusion] The present result $R_{\rm skin}$ = $0.278 \pm 0.035$ fm is in good agreement with the recent PREX-II result of $r_{\rm skin}$ = $0.283\pm 0.071$ fm.