Importance of $Z-Z^\prime$ Mixing in $b\to s\ell^+\ell^-$ and the $W$ mass

The discrepancies between $b\to s\ell^+\ell^-$ data and the corresponding Standard Model predictions point convincingly towards the existence of new physics. In particular, a heavy neutral gauge boson ($Z^\prime$) is a prime candidate for explaining these tensions. 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 negligibly small. In this letter we point out that a mixing of the order of $m_Z^2/m_{Z^\prime}^2$, as expected in generic models, leads to lepton flavour universal contributions, providing a good fit to $b\to s\ell^+\ell^-$ data. Furthermore, the global electroweak fit is affected by $Z-Z^\prime$ mixing where the tension in the $W$ mass prefers a non-zero value of it. This implies that $Z-Z^\prime$ mixing is welcome by data such that the breaking of the new gauge symmetry giving raise to the $Z^\prime$ could be linked to electroweak symmetry breaking with intriguing consequences for model building.Comment: 10 pages, 3 figures, 3 tables; factor 1/2 in Eq. (8) correcte