How to tame penguins: advancing to high-precision measurements of $$\phi _d$$ and $$\phi _s$$

Abstract The complex phases $$\phi _d$$ ϕ d and $$\phi _s$$ ϕ s , associated with the mixing between neutral $$B_q^0$$ B q 0 and $$\bar{B}_q^0$$ B ¯ q 0 mesons ( $$q\in \{d,s\}$$ q ∈ { 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 $$B_d^0\rightarrow J/\psi K^0$$ B d 0 → J / ψ K 0 , $$B_s^0\rightarrow J/\psi \phi $$ B s 0 → J / ψ ϕ and $$B_s^0\rightarrow D_s^+D_s^-$$ B s 0 → D s + D s - . To reach the highest possible precision on $$\phi _d$$ ϕ d and $$\phi _s$$ ϕ 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 $$B_s^0\rightarrow D_s^+D_s^-$$ B s 0 → D s + D s - , $$B_d^0\rightarrow D^+D^-$$ <mm