Prospects for CP violation in $B^0_s \to J/\psi\phi$ from first LHCb data

The determination of the $CP$-violating phase in $B^0_S \to J/\psi\phi$ decays is one of the key goal of the LHCb experiment. Its value is predicted to be very small in the Standard Model but can be significantly enhanced in many models of New Physics. The steps towards a precise determination of this phase with a flavour-tagged, time-dependent angular analysis of the decay $B^0_S \to J/\psi\phi$ are reviewed and first studies performed with data collected in 2010 at LHC in pp collisions at 7 TeV center-of-mass energy are presented for the first time. In particular, we report the first LHCb measurements of lifetime in the channels $B^+ \to J/\psi K^+, B^0 \to J/\psi K^{*0}, B^0 \to J/\psi K^0_S, B^0_s \to J/\psi\phi, \Lambda_b \to J/\psi \Lambda$; the polarization amplitudes in $B^0 \to J/\psi K^{*0}$ and $B^0_s \to J/\psi\phi$; the width and mass differences of the $B^0_S$ mass eigenstates, $\Delta\Gamma_s$ and $\Delta m_S$. The data sampled used corresponds to an integrated luminosity of 36 pb$^{-1}$.