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

LHCb has reported deviations from the Standard Model in $b\to sμ^+μ^-$ 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 $h\toτ^\pmμ^\mp$. In this article, we investigate the perspectives to discover the lepton-flavour violating modes $B\to K^{(*)}τ^\pmμ^\mp$, $B_s\to τ^\pmμ^\mp$ and $B\to K^{(*)} μ^\pm e^\mp$, $B_s\to μ^\pm e^\mp$. 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 $τ\to3μ$, $τ\toμν\barν$, $μ\to eγ$, $g_μ-2$, semi-leptonic $b\to sμ^+μ^-$ decays, $B\to K^{(*)}νν$ 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. $Br(B\to Kνν)$ 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 $B_s$--$\overline{B}_s$, $τ\to3μ$, $τ\toμν\barν$ 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 $τμ$ final states, while $B_s\to μ^\pm e^\mp$ is strongly suppressed and only $B\to K^{(*)} μ^\pm e^\mp$ can be experimentally accessible (with a branching ratio of order $10^{-7}$).