An exploration of $B_s \to J/\psi s\bar{s}$

Measurements of the $B_s^0$--$\bar{B}_s^0$ mixing phase $\phi_s$ appear to be converging towards the SM prediction, suggesting that contributions from New Physics (NP), if present, are small. This poses the question of whether smallish NP in $\phi_s$ can be distinguished from the hadronic uncertainties present in the interfering $B_s$ decay mode. In this paper we discuss the potential of extracting $\phi_s$ from the decay modes $B_s\to J/\psi \eta^{(\prime)}$ and $B_s\to J/\psi f_0(980)$ and how their respective hadronic uncertainties can be controlled. In addition, we demonstrate how the branching ratios of the former decays can be used to estimate the $\eta$--$\eta'$ mixing angles. Finally, we point out that effective lifetime measurements of decay modes such as the $B_s\to J/\psi f_0(980)$ can be used to constrain both $\phi_s$ and the $B_s$ decay width difference $\Delta\Gamma_s$, complementary to the usual time-dependent tagged analysis.