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Unified Explanation of the Anomalies in Semi-Leptonic $B$ decays and the $W$ Mass

The discrepancies between the measurements of rare (semi-)leptonic $B$ decays and the corresponding Standard Model predictions point convincingly towards the existence of new physics for which a heavy neutral gauge boson ($Z^\prime$) is a prime candidate. However, the effect of the mixing of the $Z^\prime$ with the SM $Z$, even though it cannot be avoided by any symmetry, is usually assumed to be small and thus neglected in phenomenological analyses. In this letter we point out that a mixing of the naturally expected size leads to lepton flavour universal contributions, providing a very good fit to $B$ data. Furthermore, the global electroweak fit is affected by $Z-Z^\prime$ mixing where the tension in the $W$ mass, recently confirmed and strengthened by the CDF measurement, prefers a non-zero value of it. We find that a $Z^\prime$ boson with a mass between $\approx 1-5\,\rm {TeV}$ can provide a unified explanations of the $B$ anomalies and the $W$ mass. This strongly suggests that the breaking of the new gauge symmetry giving raise to the $Z^\prime$ boson is linked to electroweak symmetry breaking with intriguing consequences for model building.