Update to the LHCb sensitivity to $\sin 2\beta$ from the CP-asymmetry in $B^0 \to J\psi(\mu\mu)K^0_S$ decays

This note presents an update of the \BdJmmKS\ analysis in order to obtain the LHCb sensitivity to $\sin(2\beta)$. An update of the analysis of \BdJmmKst, the control channel for tagging, is also shown here. We estimate an annual yield of 236k for \BdJmmKS\ channel, with a background to signal ratio (B/S) of 8.4. A yield of 228k with B/S of about 0.9 is obtained by applying a cut to reduce prompt \Jpsi\ decays. We estimate an annual yield of 932k with B/S$= 0.155$ for \BdJmmKst. We measure the mistag rate to be $38.7\pm 0.1$\% from \BdJmmKst\ and find that this value can be used without corrections for the \BdJmmKS\ channel. The statistical sensitivity on $\sin(2\beta)$ is found to be 0.020 for 2 fb$^{-1}$ of data.