Emerging $\Delta M_{d}$-Anomaly from Tree-Level Determinations of $|V_{cb}|$ and the Angle $\gamma$

Abstract

We point out that the recently increased value of the angle $\gamma$ in the Unitarity Triangle (UT), determined in tree-level decays to be $\gamma=(74.0^{+5.0}_{-5.8})^\circ$ by the LHCb collaboration, combined with the most recent value of $|V_{cb}|$ implies an enhancement of $\Delta M_{d}$ over the data in the ballpark of $30\%$. This is roughly by a factor of two larger than the enhancement of $\Delta M_{s}$ that is independent of $\gamma$. This disparity of enhancements is problematic for models with Constrained Minimal Flavour Violation (CMFV) and also for $U(2)^3$ models. In view of the prospects of measuring $\gamma$ with the precision of $\pm 1^\circ$ by Belle II and LHCb in the coming years, we propose to use the angles $\gamma$ and $\beta$ together with $|V_{cb}|$ and $|V_{us}|$ as the fundamental parameters of the CKM matrix until $|V_{ub}|$ from tree-level decays will be known precisely. Displaying $\Delta M_{s,d}$ as functions of $\gamma$ clearly demonstrates the tension between the value of $\gamma$ from tree-level decays, free from new physics (NP) contributions, and $\Delta M_{s,d}$ calculated in CMFV and $U(2)^3$ models and thus exhibits the presence of NP contributions to $\Delta M_{s,d}$ beyond these frameworks. We calculate the values of $|V_{ub}|$ and $|V_{td}|$ as functions of $\gamma$ and $|V_{cb}|$ and discuss the implications of our results for $\varepsilon_K$ and rare $K$ and $B$ decays. We also briefly discuss a future strategy in which $\beta$, possibly affected by NP, is replaced by $|V_{ub}|$.