Exploring $B\to\pi\pi, \pi K$ Decays at the High-Precision Frontier

The $B\to\pi\pi,\pi K$ system offers a powerful laboratory to probe strong and weak interactions. Using the isospin symmetry, we determine hadronic $B\to\pi\pi$ parameters from data where new measurements of direct CP violation in $B^0_d\to\pi^0\pi^0$ resolve a discrete ambiguity. With the help of the $SU(3)$ flavour symmetry, the $B\to\pi\pi$ parameters can be converted into their $B\to\pi K$ counterparts, thereby allowing us to make predictions of observables. A particularly interesting decay is $B^0_d\to\pi^0 K_{\rm S}$ as it exhibits mixing-induced CP violation. Using an isospin relation, complemented with a robust $SU(3)$ input, we calculate correlations between the direct and mixing-induced CP asymmetries of $B^0_d\to\pi^0 K_{\rm S}$, which are the theoretically cleanest $B\to\pi K$ probes. Interestingly, they show tensions with respect to the Standard Model. Should this $B\to\pi K$ puzzle originate from New Physics, electroweak penguins offer an attractive scenario for new particles to enter. We present a strategy to determine the parameters characterising these topologies and obtain the state-of-the-art picture from current data. In the future, this method will allow us to reveal the $B\to\pi K$ dynamics and to obtain insights into the electroweak penguin sector with unprecedented precision.