Lifetime Difference and $CP$ Asymmetry in the $B_s^0 \to J/\psi \phi$ Mode

We report on the progress of the analysis of $B^0_s \to J/\psi \phi$ decays to extract the parameters $\Delta \Gamma_s / \Gamma_s$ and $\sin 2 \beta_s$ using 1.4~fb$^{-1}$ of data taken with the CDF\,II detector. The use of a time-dependent angular analysis can separate the $CP$~eigenstates of the $B^0_s$~meson to determine their separate lifetimes. Further tagging of the $B^0_s$ meson at time $t=0$ as $B^0_s$ or $\bar{B}^0_s$, allows for the analysis of the $CP$ asymmetry, which then yields the $\sin 2 \beta_s$ parameter. The $\Delta \Gamma_s$ measurement is an improved analysis using additional data, neural network selection for better signal/background, and reduced systematic errors. The $\sin 2 \beta_s$ measurement is the first such analysis at CDF.