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Recent $b\to s\ell\ell$ anomalies may imply the existence of a new $Z'$ bosonwith left-handed $Z'bs$ and $Z'\mu\mu$ couplings. Such a $Z'$ may be directlyobserved at LHC via $b \bar s \to Z' \to \mu^+\mu^-$, and its relevance to$b\to s\ell\ell$ may be studied by searching for the process $gs \to Z'b \to\mu^+\mu^- b$. In this paper, we analyze the capability of the 14 TeV LHC toobserve the $Z'$ in the $\mu^+ \mu^-$ and $\mu^+\mu^- b$ modes based on aneffective model with major phenomenological constraints imposed. We find thatboth modes can be discovered with 3000 fb$^{-1}$ data if the $Z'bs$ couplingsaturates the latest $B_s-\bar B_s$ mixing limit from UTfit at around$2\sigma$. Besides, a tiny right-handed $Z'bs$ coupling, if it exists, opens upthe possibility of a relatively large left-handed counterpart, due tocancellation in the $B_s-\bar B_s$ mixing amplitude. In this case, we show thateven a data sample of $\mathcal{O}(100)$ fb$^{-1}$ would enable discovery ofboth modes. We further study the impact of a $Z'bb$ coupling as large as the$Z'bs$ coupling. This scenario enables discovery of the $Z'$ in both modes withmilder effects on the $B_s-\bar B_s$ mixing, but obscures the relevance of the$Z'$ to $b \to s\ell\ell$. Discrimination between the $Z'bs$ and $Z'bb$couplings may come from the production cross section for the $Z'b\bar{b}$ finalstate. However, we do not find the prospect for this to be promising.

Identifying a $Z'$ behind $b \to s \ell \ell$ anomalies at the LHC