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Abstract In the early 1980s Sanda-san and collaborators wrote key papers on the direct and clean determination of the unitarity angle ϕ1 (β). This motivated many of us to analogously come up with ways for direct and clean determinations of the other two unitarity angles, ϕ2 (α) and ϕ3 (γ). This paper describes the current status of these direct determinations as well as our expectations for when Belle-II has 50 ab−1 of luminosity and LHCb is upgraded. In particular, it is emphasized that, for direct determination of ϕ3, Belle-II should be able to handle final states in D0 or D⁰ Dalitz decays, which contain one π0 (which are difficult for LHCb) ; then they may make further inroads in improving the accuracy of the ϕ3 determination. Early lattice inputs for constraining the unitarity triangle (UT) are briefly recalled. Its crucial role in supporting the Kobayashi–Maskawa theory of CP violation is emphasized. Over the years significant progress has been made in lattice methods and the latest constraints from these for the UT will be discussed, as well as the compatibility with current direct determinations and some comments on the future outlook.
Amarjit Soni (Sat,) studied this question.