Waiting for Precise Measurements of $\beta$ and $\gamma$

Abstract

During the last three decades the determination of the Unitarity Triangle (UT) was dominated by the measurements of its sides $R_b$ and $R_t$ through tree-level $B$ decays and the $\Delta M_d/\Delta M_s$ ratio, respectively, with some participation of the measurements of the angle $\beta$ through the mixing induced CP-asymmetries like $S_{\psi K_S}$ and $\varepsilon_K$. However, as pointed out already in 2002 by Fabrizio Parodi, Achille Stocchi and the present author, the most efficient strategy for a precise determination of the apex of the UT, that is $(\bar\varrho,\bar\eta)$, is to use the measurements of the angles $\beta$ and $\gamma$. The second best strategy would be the measurements of $R_b$ and $\gamma$. However, in view of the tensions between different determinations of $|V_{ub}|$ and $|V_{cb}|$, that enter $R_b$, the $(\beta,\gamma)$ strategy should be a clear winner once LHCb and Belle II will improve the measurements of these two angles. In this note we recall our finding of 2002 which should be finally realized in this decade through precise measurements of both angles by these collaborations. In this context we present two very simple formulae for $\bar\varrho$ and $\bar\eta$ in terms of $\beta$ and $\gamma$ which could be derived by high-school students, but to my knowledge never appeared in the literature on the UT, not even in our 2002 paper. We also emphasize the importance of precise measurements of both angles that would allow to perform powerful tests of the SM through numerous $|V_{cb}|$-independent correlations between $K$ and $B$ decay branching ratios $R_i(\beta,\gamma)$ recently derived by Elena Venturini and the present author. The simple findings presented here will appear in a subsection of a much longer contribution to the proceedings of KM50 later this year. I exhibited them here so that they are not lost in the latter.