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The $B^0_{s,d} \to J/\psi \eta^{(\prime)}$ decays provide new terrain forexploring CP violation. After briefly discussing $\eta$-$\eta'$ mixing, weanalyse the effective lifetimes and CP-violating observables of the $B_s$channels, which allow us to probe New-Physics effects in $B^0_s$-$\bar B^0_s$mixing. We have a critical look at these observables and show how hadroniccorrections can be controlled by means of the $B_d$ decays. Using measurementsof the $B^0_{s,d}\to J/\psi \eta^{(\prime)}$ branching ratios by the Bellecollaboration, we discuss tests of the $SU(3)_F$ flavour symmetry of stronginteractions, obtain the first constraints on the hadronic parameters enteringthe $B^0_{s,d} \to J/\psi \eta$ system, and predict the $B^0_d\to J/\psi \eta'$branching ratio at the $5\times10^{-6}$ level. Furthermore, we presentstrategies for the determination of the $\eta$-$\eta'$ mixing parameters fromthe $B^0_{s,d} \to J/\psi \eta^{(\prime)}$ observables. We also observe thatthe $B^0_{s,d} \to J/\psi \eta$ and $B^0_{s,d} \to J/\psi \eta'$ decays are -from a formal point of view - analogous to the quark-antiquark and tetraquarkdescriptions of the $f_0(980)$ in the $B^0_{s,d} \to J/\psi f_0(980)$ channels,respectively.

Exploring CP Violation and $\eta$-$\eta'$ Mixing with the $B^0_{s,d} \to J/\psi \eta^{(\prime)}$ Systems