Measurement of $\Gamma_s$ and $\Delta\Gamma_s$ from $B_s^0\rightarrow J/\psi \phi$ at CDF

We measure the mean lifetime, $\tau_s=2/(\Gamma_L+\Gamma_H)$, and the width difference, $\Delta\Gamma_s=\Gamma_L-\Gamma_H$, of the light and heavy mass eigenstates of the $B^0_s$~meson in $B_s^0 \rightarrow J/\psi \phi$ decays. The analysis is based on $\sim4$~fb$^{-1}$ of $p\bar{p}$~collisions collected with the CDF\,II detector at the Fermilab Tevatron.