Bs-Bs.bar Mixing, CP Violation and Extraction of CKM Phases from Untagged Bs Data Samples

A width difference of the order of 20\% has previously been predicted for the two mass eigenstates of the $B_s$ meson. The dominant contributor to the width difference is the $b\rightarrow c\bar c s$ transition, with final states common to both $B_s$ and $\overline{B}_s$. All current experimental analyses fit the time-dependences of flavor-specific $B_s$-modes to a single exponential, which essentially determines the average $B_s$ lifetime. We stress that the same data sample allows even the measurement of the width difference. To see that, this note reviews the time-dependent formulae for tagged $B_s$ decays, which involve rapid oscillatory terms depending on $\Delta mt$. In untagged data samples the rapid oscillatory terms cancel. Their time-evolutions depend only on the much more slowly varying exponential falloffs. We discuss in detail the extraction of the two widths, and identify the large (small) CP-even (-odd) rate with that of the light (heavy) $B_s$ mass eigenstate. It is demonstrated that decay length distributions of some \underline{untagged} $B_s$ modes, such as $\rho^0 K_S, \; D_s^{(*)\pm}K^{(*)\mp}$, can be used to extract the notoriously difficult CKM unitarity triangle angle $\gamma$. Sizable CP violating effects may be seen with such untagged $B_s$ data samples. Listing $\Delta\Gamma$ as an observable allows for additional important standard model constraints. Within the CKM model, the ratio $\Delta\Gamma/ \Delta m$ involves no CKM parameters, only a QCD uncertainty. Thus a measurement of $\Delta\Gamma \;(\Delta m)$ would predict $\Delta m \;(\Delta\Gamma )$, up to the QCD uncertainty. A large width difference would automatically solve the puzzle of the number of charmed hadrons per $B$ decay in favor of theory. We also derive an upper limit of $(| \Delta\Gamma | / \Gamma)_{B_s} <~ 0.3$. Further, we must abandon the notion of branching fractions of $B_s\rightarrow f$, and instead consider $B(B^0_{L(H)}\rightarrow f)$, in analogy to the neutral kaons.