The complex phases (phi _d) and (phi _s), associated with the mixing between neutral (B_q^0) and (bar{B}_q^0) mesons ((qin {d,s})), are key observables to test the Standard Model and search for contributions from new physics. They are conventionally determined from the measurements of mixing-induced CP violation in the decays (B_d^0rightarrow J/psi K^0), (B_s^0rightarrow J/psi phi ) and (B_s^0rightarrow D_s^+D_s^-). To reach the highest possible precision on (phi _d) and (phi _s), it is crucial that corrections from next-to-leading order effects – primarily associated with penguin decay topologies – are accounted for. The strategy adopted in this paper uses the SU(3) flavour symmetry of QCD to relate the unknown contributions from penguin topologies to their counterparts in suitably-chosen control modes, where their effects are enhanced. Utilising new CP asymmetry measurements from LHCb on the decays (B_s^0rightarrow D_s^+D_s^-), (B_d^0rightarrow D^+D^-), (B^+rightarrow J/psi K^+) and (B^+rightarrow J/psi pi ^+), as well as from Belle-II on the decay (B_d^0rightarrow J/psi pi ^0), we present the current state-of-the-art picture on controlling the penguin contributions and extract (phi _d) and (phi _s) from the corresponding observables. We explore the prospects for the end of the Belle-II and HL-LHC flavour physics programmes, and demonstrate the importance of measuring the control modes with future data.
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