Abstract
The $B0_s$ mixing phase $\beta_s$ is a key probe of physics
beyond the standard model (SM). The first Tevatron measurements hinted
at a $\approx 2\sigma$ departure from the SM expectations. We present a
new analysis using $B_s0 \rightarrow J/\psi\,\phi$ decays reconstructed
in 5.2~fb$^{-1}$ of data collected by CDF detector, where tightened
bounds on the mixing phase are obtained along with an improved
measurement of the decay-width difference $\Delta \Gamma_s$ and of
$B0_s$ lifetime.