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Theories beyond the standard model involving a sub-GeV-scale vector $Z_d$mediator have been largely studied as a possible explanation of theexperimental values of the muon and electron anomalous magnetic moments.Motivated by the recent determination of the anomalous muon magnetic momentperformed at Fermilab, we derive the constraints on such a model obtained fromthe magnetic moment determinations and the measurements of the proton andcesium weak charge, $Q_W$, performed at low-energy transfer. In order to do so,we revisit the determination of the cesium $Q_W$ from the atomic parityviolation experiment, which depends critically on the value of the averageneutron rms radius of $^{133}\text{Cs}$, by determining the latter from apractically model-independent extrapolation from the recent average neutron rmsradius of $^{208}\text{Pb}$ performed by the PREX-2 Collaboration. From acombined fit of all the aforementioned experimental results, we obtain ratherprecise limits on the mass and the kinetic mixing parameter of the $Z_d$ boson,namely $m_{Z_d} = 47{}^{+61}_{-16} \, \mathrm{MeV}$ and $\varepsilon =2.3{}^{+1.1}_{-0.4} \times 10^{-3}$, when marginalizing over the $Z-Z_d$ massmixing parameter $\delta$.

Muon and electron g-2 and proton and cesium weak charges implications on dark $\mathbf{Z_d}$ models