In pursuit of new physics with $B_s$ decays

The presence of a sizable CP-violating phase in $B_s-B_s$ mixing would be an unambiguous signal of physics beyond the standard model. We analyze various possibilities to detect such a new phase considering both tagged and untagged decays. The effects of a sizable width difference $\Delta \Gamma$ between the $B_s$ mass eigenstates, on which the untagged analyses rely, are included in all formulas. A novel method to find this phase from simple measurements of lifetimes and branching ratios in untagged decays is proposed. This method does not involve two-exponential fits, which require much larger statistics. For the tagged decays, an outstanding role is played by the observables of the time-dependent angular distribution of the $B_s\to J/\psi \overrightarrow{[}{l}^{+}{l}^{-}\left]\phi \overrightarrow{[}{K}^{+}{K}^{-}\right]$ decay products. We list the formulas needed for the angular analysis in the presence of both a new CP-violating phase and a sizable $\Delta \Gamma ,$ and propose methods to remove a remaining discrete ambiguity in the new phase. This phase can therefore be determined in an unambiguous way.