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

This note presents an update of the B$^0$ $\rightarrow$ J/$\psi$($\mu\mu$)K$^0_S$($\pi^+\pi^-$) analysis in order to obtain the LHCb sensitivity to sin(2$/beta$). We present two selection criteria, one considering only lifetime unbiased cuts and another which includes lifetime biased cuts to reduce further the background contamination. For an integrated luminosity of 2 fb$^-1$ and after L0 trigger, we estimate a yield of 193k events for the B$^0$ $\rightarrow$ J/$\psi$($\mu\mu$)K$^0_S$ channel with the lifetime unbiased selection criteria, with a total background to signal ration 3.43. For the lifetime biased selection, the signal yield is 208k events and the background to signal ratio is to 0.63. The statistical sensitivity on sin(2$/beta$) is found to be 0.023 and 0.020 for the unbiased and biased selections, respectively.