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

Theories beyond the standard model involving a sub-GeV-scale vector $Z_d$ mediator have been largely studied as a possible explanation of the experimental values of the muon and electron anomalous magnetic moments. Motivated by the recent determination of the anomalous muon magnetic moment performed at Fermilab, we derive the constraints on such a model obtained from the magnetic moment determinations and the measurements of the proton and cesium 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 parity violation experiment, which depends critically on the value of the average neutron rms radius of $^{133}\text{Cs}$, by determining the latter from a practically model-independent extrapolation from the recent average neutron rms radius of $^{208}\text{Pb}$ performed by the PREX-2 Collaboration. From a combined fit of all the aforementioned experimental results, we obtain rather precise 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 $Z-Z_d$ mass mixing parameter $\delta$.