Search

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 larger by roughly a factor of two 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 $$|V_{ub}|$$ .

Motivated by the recently improved results from the Fermilab Lattice and MILC Collaborations on the hadronic matrix elements entering $\Delta M_{s,d}$ in $B_{s,d}^0-\bar B_{s,d}^0$ mixing, we determine the Universal Unitarity Triangle (UUT) in models with Constrained Minimal Flavour Violation (CMFV). Of particular importance are the very precise determinations of the ratio $|V_{ub}|/|V_{cb}|=0.0864\pm0.0025$ and of the angle $\gamma=(63.0\pm 2.1)^\circ$. They follow in this framework from the experimental values of $\Delta M_{d}/\Delta M_s$ and of the CP-asymmetry $S_{\psi K_S}$. As in CMFV models the new contributions to meson mixings can be described by a single flavour-universal variable $S(v)$, we next determine the CKM matrix elements $|V_{ts}|$, $|V_{td}|$, $|V_{cb}|$ and $|V_{ub}|$ as functions of $S(v)$ using the experimental value of $\Delta M_s$ as input. The lower bound on $S(v)$ in these models, derived by us in 2006, implies then upper bounds on these four CKM elements and on the CP-violating parameter $\varepsilon_K$, which turns out to be significantly below its experimental value. This strategy avoids the use of tree-level determinations of $|V_{ub}|$ and $|V_{cb}|$ that are presently subject to considerable uncertainties. On the other hand if $\varepsilon_K$ is used instead of $\Delta M_s$ as input, $\Delta M_{s,d}$ are found significantly above the data. In this manner we point out that the new lattice data have significantly sharpened the tension between $\Delta M_{s,d}$ and $\varepsilon_K$ within the CMFV framework. This implies the presence of new physics contributions beyond this framework that are responsible for the breakdown of the flavour universality of the function $S(v)$. We also present the implications of these results for $K^+\to\pi^+\nu\bar\nu$, $K_L\to\pi^0\nu\bar\nu$ and $B_{s,d}\to\mu^+\mu^-$ within the Standard Model.