Insights into nuclear saturation density from parity violating electron scattering

The saturation density of nuclear matter $rho_0$ is a fundamental nuclear physics property that is difficult to predict from fundamental principles. The saturation density is closely related to the interior density of a heavy nucleus, such as $^{208}$Pb. We use parity violating electron scattering to determine the average interior weak charge and baryon densities in $^{208}$Pb. This requires not only measuring the weak radius $R_{rm wk}$ but also determining the surface thickness of the weak charge density $a$. We obtain $rho_0=0.150pm0.010$ fm$^{-3}$, where the 7% error has contributions form the PREX error on the weak radius, an assumed 10% uncertainty in the surface thickness $a$, and from the extrapolation to infinite nuclear matter. These errors can be improved with the upcoming PREX II results and with a new parity violating electron scattering experiment, at a somewhat higher momentum transfer, to determine $a$.