$M_W$ helps select $Z^\prime$ models for $b \rightarrow s \ell \ell$ anomalies

As shown in Ref. \cite{Allanach:2021kzj}, the Third Family Hypercharge ($Y_3$) Model changes the Standard Model prediction for $M_W$ whilst simultaneously explaining anomalies in $b\to s\ell\ell$ transitions via a heavy $Z^\prime$ gauge boson which is spawned by a spontaneously broken gauged $U(1)_{Y_3}$ symmetry. The 2022 CDF II measurement of $M_W$, which is far from the Standard Model prediction, somewhat disfavours the $Y_3$ model. Here, we generalise the gauge charge assignments to the anomaly-free combination $s Y_3 + t (B_3-L_3)$ and show that incorporating the 2022 CDF II measurement of $M_W$ selects a viable domain of integers $s$ and $t$. For example, $s=1, t=-3$ yields a $p-$value of .12 in a two-parameter global fit to 277 electroweak and flavour changing $b$ data, much improving a SM $p-$value of $5\times 10^{-6}$.