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

The decays $B_d^0\to J/\psi K^0_{\text{S}}$and $B_s^0\to J/\psi\phi$ play a key role for the determination of the $B^0_q$-$\bar B^0_q$ ($q=d,s$) mixing phases $\phi_d$ and $\phi_s$, respectively. The theoretical precision of the extraction of these quantities is limited by doubly Cabibbo-suppressed penguin topologies, which can be included through control channels by means of the $SU(3)$ flavour symmetry of strong interactions. Using the currently available data and a new simultaneous analysis, we discuss the state-of-the-art picture of these effects and include them in the extracted $\phi_q$ values. We have a critical look at the Standard Model predictions of these phases and explore the room left for new physics. Considering future scenarios for the high-precision era of flavour physics, we illustrate that we may obtain signals for physics beyond the Standard Model with a significance well above five standard deviations. We also determine effective colour-suppression factors of $B_d^0\to J/\psi K^0$, $B_d^0\to J/\psi K^0_{\text{S}}$ and $B_d^0\to J/\psi\pi^0$ decays, which serve as benchmarks for QCD calculations of the underlying decay dynamics, 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.