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The ratio of the transverse and longitudinal component of polarizationtransfer to protons in quasi-elastic $(\vec{e}, e^{\prime} \vec{p}\,)$reaction, $P^{\prime}_x/P^{\prime}_z$, is sensitive to the proton'selectromagnetic form factor ratio, $G_E/G_M$. To explore density-dependentin-medium modifications, a comparison of polarization transfer ratios involvingprotons from distinct nuclear shells, each with different local nucleardensities, has been proposed. In this study, we present such comparisonsbetween 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 othermany-body effects that may differ between shells, we use state-of-the-artrelativistic distorted-wave impulse-approximation (RDWIA) calculation andpresent the double ratios, $(P^{\prime}_x/P^{\prime}_z)_{\rmData}/(P^{\prime}_x/P^{\prime}_z)_{\rm RDWIA}$ as well as the super ratios,$\left[(P^{\prime}_x/P^{\prime}_z)_{\rm A}/(P^{\prime}_x/P^{\prime}_z)_{\rmB}\right]_{\rm Data}/\left[(P^{\prime}_x/P^{\prime}_z)_{\rmA}/(P^{\prime}_x/P^{\prime}_z)_{\rm B}\right]_{\rm RDWIA}$, for chosen shells Aand B, as a function of effective local nuclear densities. We find that doubleratios for individual shells show a dependence on the probed effective nucleardensities. Studying the ratios, we observed a systematic variation betweenpairs of higher- and lower-density shells.

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