Taming Penguins: Towards High Precision Measurements in $\phi_d$ and $\phi_s$

Experimentally, the phases $\phi_d$ and $\phi_s$ are determined from CP asymmetry measurements in the "golden modes" $B_d^0\to J/\psi K_{\mathrm{S}}^0$ and $B_s^0\to J/\psi\phi$. At leading order, the theoretical interpretation of these measurements is straightforward. However, to reach high precision determinations of $\phi_d$ and $\phi_s$, which is essential in view of the searches for signs of beyond the SM physics, corrections from next-to-leading order effects need to be accounted for. These corrections primarily originate from so-called penguin topologies. Using the $SU(3)$ flavour symmetry, these corrections can be determined using suitably chosen control modes. Recent new CP asymmetry measurements from LHCb in $B\to DD$ and Belle-II in $B_d^0\to J/\psi\pi^0$ decays greatly improve our knowledge on the parameters describing the contribution from penguin topologies. These proceedings will discuss the current constraints on the penguin parameters in $B\to J/\psi X$ and $B\to DD$ decays, provide corrected determinations for $\phi_d$ and $\phi_s$, and highlight what can be expected at the end of the HL-LHC and Belle-II programmes.