Towards New Frontiers in the Exploration of Charmless Non-Leptonic $B$ Decays

Non-leptonic $B$ decays into charmless final states offer an important laboratory to study CP violation and the dynamics of strong interactions. Particularly interesting are $B_s^0\to K^{-}{K}^{+}$ and $B_d^0\to {\pi }^{-}{\pi }^{+}$ decays, which are related by the $U$-spin symmetry of strong interactions, and allow for the extraction of CP-violating phases and tests of the Standard Model. The theoretical precision is limited by $U$-spin-breaking corrections and innovative methods are needed in view of the impressive future experimental precision expected in the era of Belle II and the LHCb upgrade. We have recently proposed a novel method to determine the $B_s^0$-${\bar{B}}_s^0$ mixing phase ${\varphi }_s$ from the $B_s^0\to K^{-}{K}^{+}$, $B_d^0\to {\pi }^{-}{\pi }^{+}$ system, where semileptonic $B_s^0\to K^{-}{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$, $B_d^0\to {\pi }^{-}{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ decays are a new ingredient and the theoretical situation is very favourable. We discuss this strategy in detail, with a focus on penguin contributions as well as exchange and penguin-annihilation topologies which can be probed by a variety of non-leptonic $B$ decays into charmless final states. We show that a theoretical precision as high as $\mathcal{O}({0.5}^{\circ })$ for ${\varphi }_s$ can be attained in the future, thereby offering unprecedented prospects for the search for new sources of CP violation.