High precision in CKM unitarity tests in $b$ and $c$ decays

This article surveys the important questions attached to unitarity tests of the Cabibbo-Kobayashi-Maskawa matrix, with a focus on the direct determinations of the magnitude of CKM matrix elements. The current CKM anomalies are discussed and the clear-cut prospects at LHCb and Belle II envisaged. The anticipated precision on $C\!P$-conserving and $C\!P$-violating observables at the projected upgrades of the LHCb and Belle~II experiments is examined from the point of view of the search for New Physics in $B$-meson mixing, addressing the bottlenecks in precision for the interpretation of the measurements for future experiments and highlighting the related theoretical and experimental challenges. The vibrant prospects at future $e^+e^-$ colliders running at the $Z^0$ pole and $W^+W^-$ are eventually discussed.

Comment: 18 pages, 5 figures, submitted to the Proceedings of the US Community Study on the Future of Particle Physics (Snowmass 2021)