Precise prediction for the mass of the $W$ boson in gauged U(1) extensions of the standard model

We present the one-loop radiative corrections to the muon decay in U(1)${}_z$ extensions of the standard model. We compute the mass of the $W$ boson using those corrections and compare it to an approximation of the complete one-loop prediction implemented in automated computational tools. We point out that the truncation of the complete formulas become unreliable if the mass of the $Z^{\mathrm{\prime }}$ boson, corresponding to the new U(1)${}_z$ gauge group is larger than about 1 TeV.