Plan B: New ${Z^\prime}$ models for $b\rightarrow sl^+l^-$ anomalies

Measurements of $b \rightarrow s \mu^+ \mu^-$ transitions indicate that there may be a new physics field coupling to di-muon pairs associated with the $b$ to $s$ flavour transition. Including the 2022 LHCb reanalysis of $R_K$ and $R_{K^\ast}$, one infers that there may also be associated new physics in $b\rightarrow e^+ e^-$ transitions. Here, we examine the extent of the statistical preference for $Z^\prime$ models coupling to di-electron pairs taking into account the relevant constraints, in particular from experiments at LEP-2. We identify an anomaly-free set of models which interpolates between the $Z^\prime$ not coupling to electrons at all, to one in which there is an equal $Z^\prime$ coupling to muons and electrons (but where in all models in the set, the $Z^\prime$ boson can mediate $b\rightarrow \mu^+ \mu^-$ transitions). A $3B_3-L_e-2L_\mu$ model provides a close-to-optimal fit to the pertinent measurements along the line of interpolation. We have (re-)calculated predictions for the relevant LEP-2 observables in terms of dimension-6 SMEFT operators and put them into the ${\tt flavio}$ computer program, so that they are available for global fits.