Selection of the $B_{s}^{0} \to J/\psi(\mu^{+}\mu^{-})\;\eta'(\rho^{0}\gamma)$ decay at LHCb and its sensitivity to the $B_{s}^{0}$ mixing phase $\phi_{s}$

The $B_{s}^{0}\to J/\psi\;\eta'$ is a decay to a pure $CP$ eigenstate through a $\overline{b} \to \overline{c}c\overline{s}$ quark-level transition which is dominated by a tree diagram. By measuring the time-dependent $CP$ asymmetry, the $B_{s}^{0}$ mixing phase $\phi_{s}$ can be extracted without theoretical uncertainties. The results of the $B_{s}^{0} \to J/\psi(\mu^{+}\mu^{-})\;\eta'(\rho^{0}\gamma)$ selection studies at LHCb based on a full Monte Carlo simulation are summarised in this note. In addition, the sensitivity to measure $\phi_{s}$ with $B_{s}^{0} \to J/\psi\;\eta'$ is presented.