On the Superiority of the $|V_{cb}|-\gamma$ Plots over the Unitarity Triangle Plots in the 2020s

Abstract

The UT plots played already for three decades an important role in the tests of the SM and the determination of the CKM parameters. As of 2022, among the four CKM parameters, $V_{us}$ and $\beta$ are already measured with respectable precision, while this is not the case of $|V_{cb}|$ and $\gamma$. In the case of $|V_{cb}|$ the main obstacle are the significant tensions between its inclusive and exclusive determinations from tree-level decays. The present uncertainty in $\gamma$ of $4^{\circ }$ from tree-level decays will be reduced to $1^{\circ }$ by the LHCb and Belle II collaborations in the coming years. Unfortunately in the UT plots $|V_{cb}|$ is not seen and the experimental improvements in the determination of $\gamma$ from tree-level decays at the level of a few degrees are difficult to appreciate. In view of these deficiencies of the UT plots with respect to $|V_{cb}|$ and $\gamma$ and the central role these two CKM parameters will play in this decade, the recently proposed plots of $|V_{cb}|$ versus $\gamma$ extracted from various processes appear to be superior to the UT plots in the flavour phenomenology. We illustrate this idea with $\Delta M_s$, $\Delta M_d$, ${ϵ}_K$ and with rare decays $B_s\to {\mu }^{+}{\mu }^{-}$, $B_d\to {\mu }^{+}{\mu }^{-}$, $K^{+}\to {\pi }^{+}\nu \bar{\nu }$ and $K_L\to {\pi }^0\nu \bar{\nu }$. The power of ${ϵ}_K$, $K^{+}\to {\pi }^{+}\nu \bar{\nu })$ and $K_L\to {\pi }^0\nu \bar{\nu })$ in the determination of $|V_{cb}|$, due to their strong dependence on $|V_{cb}|$, is transparently exhibited in this manner. Combined with future reduced errors on $\gamma$ and $|V_{cb}|$ from tree-level decays such plots can better exhibit possible inconsistenices between various determinations of these two parameters, caused by new physics, than it is possible with the UT plots.

Comments: 14 pages, 3 Figures. V4: Fig. 4 and equation (35) modified because of new CDF data on Bs-> mumu. Equation (29) added. Conclusions unchanged