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The decays $B_d^0\to J/\psi K^0_{\text{S}}$and $B_s^0\to J/\psi\phi$ play akey role for the determination of the $B^0_q$-$\bar B^0_q$ ($q=d,s$) mixingphases $\phi_d$ and $\phi_s$, respectively. The theoretical precision of theextraction of these quantities is limited by doubly Cabibbo-suppressed penguintopologies, which can be included through control channels by means of the$SU(3)$ flavour symmetry of strong interactions. Using the currently availabledata and a new simultaneous analysis, we discuss the state-of-the-art pictureof these effects and include them in the extracted $\phi_q$ values. We have acritical look at the Standard Model predictions of these phases and explore theroom left for new physics. Considering future scenarios for the high-precisionera of flavour physics, we illustrate that we may obtain signals for physicsbeyond the Standard Model with a significance well above five standarddeviations. We also determine effective colour-suppression factors of $B_d^0\toJ/\psi K^0$, $B_d^0\to J/\psi K^0_{\text{S}}$ and $B_d^0\to J/\psi\pi^0$decays, which serve as benchmarks for QCD calculations of the underlying decaydynamics, and present a new method using information from semileptonic$B_d^0\to\pi^-\ell^+\nu_{\ell}$ and $B_s^0\to K^-\ell^+\nu_{\ell}$ decays.

In Pursuit of New Physics with $B_d^0\to J/\psi K^0$ and $B_s^0\to J/\psi\phi$ Decays at the High-Precision Frontier