Lepton-flavour violating $B$ decays in generic $Z^\prime$ models

LHCb has reported deviations from the Standard Model in $b\to s\mu^+\mu^-$ transitions for which a new neutral gauge boson is a prime candidate for an explanation. As this gauge boson has to couple in a flavour non-universal way to muons and electrons in order to explain $R_K$, it is interesting to examine the possibility that also lepton flavour is violated, especially in the light of the CMS excess in $h\to\tau^\pm\mu^\mp$. In this article, we investigate the perspectives to discover the lepton-flavour violating modes $B\to K^{(*)}\tau^\pm\mu^\mp$, $B_s\to \tau^\pm\mu^\mp$ and $B\to K^{(*)} \mu^\pm e^\mp$, $B_s\to \mu^\pm e^\mp$. For this purpose we consider a simplified model in which new-physics effects originate from an additional neutral gauge boson ($Z^\prime$) with generic couplings to quarks and leptons. The constraints from $\tau\to3\mu$, $\tau\to\mu\nu\bar{\nu}$, $\mu\to e\gamma$, $g_\mu-2$, semi-leptonic $b\to s\mu^+\mu^-$ decays, $B\to K^{(*)}\nu\nu$ and $B_s$--$\overline{B}_s$ mixing are examined. From these decays, we determine upper bounds on the decay rates of lepton flavour violating $B$ decays. $Br(B\to K\nu\nu)$ limits the branching ratios of LFV $B$ decays to be smaller than $8\times 10^{-5} (2\times 10^{-5})$ for vectorial (left-handed) lepton couplings. However, much stronger bounds can be obtained by a combined analysis of $B_s$--$\overline{B}_s$, $\tau\to3\mu$, $\tau\to\mu\nu\bar{\nu}$ and other rare decays. The bounds depend on the amount of fine-tuning among the contributions to $B_s$--$\overline{B}_s$ mixing. Allowing for a fine-tuning at the percent level we find upper bounds of the order of $10^{-6}$ for branching ratios into $\tau\mu$ final states, while $B_s\to \mu^\pm e^\mp$ is strongly suppressed and only $B\to K^{(*)} \mu^\pm e^\mp$ can be experimentally accessible (with a branching ratio of order $10^{-7}$).