Nuclear density dependence of polarization transfer in quasi-elastic $\mathrm{A}(\vec{e},e^{\prime }\vec{p})$ reactions

Abstract

The ratio of the transverse and longitudinal component of polarization transfer to protons in the quasi-elastic $(\vec{e},e^{\prime }\vec{p})$ reaction, $P_x^{\prime }/P_z^{\prime }$, is sensitive to the proton's electromagnetic form factor ratio, $G_E/G_M$. To explore density-dependent in-medium modifications, a comparison of polarization transfer ratios involving protons from distinct nuclear shells, each with different local nuclear densities, has been proposed. In this study, we present such comparisons between four shells, $1s_{1/2}$, $1p_{3/2}$ in ${}^{12}\mathrm{C}$ and $1d_{3/2}$, $2s_{1/2}$ in ${}^{40}\mathrm{Ca}$. In an effort to account for other many-body effects that may differ between shells, we use a state-of-the-art relativistic distorted-wave impulse-approximation (RDWIA) calculation and present the double ratios, $(P_x^{\prime }/P_z^{\prime }{)}_{\mathrm{Data}}/(P_x^{\prime }/P_z^{\prime }{)}_{\mathrm{RDWIA}}$ as well as the super ratios, ${\left[(P_x^{\prime }/P_z^{\prime }{)}_{\mathrm{A}}/(P_x^{\prime }/P_z^{\prime }{)}_{\mathrm{B}}\right]}_{\mathrm{Data}}/{\left[(P_x^{\prime }/P_z^{\prime }{)}_{\mathrm{A}}/(P_x^{\prime }/P_z^{\prime }{)}_{\mathrm{B}}\right]}_{\mathrm{RDWIA}}$, for chosen shells A and B, as a function of effective local nuclear densities. We find that double ratios for individual shells show a dependence on the probed effective nuclear densities. Studying the ratios, we observed a systematic variation between pairs of higher- and lower-density shells.