The exclusive vision of rare K and B decays and of the quark mixing in the standard model

Abstract The most common predictions for rare K and B decay branching ratios in the Standard Model in the literature are based on the CKM elements $$|V_{cb}|$$ | V cb | and $$|V_{ub}|$$ | V ub | resulting from global fits, that are in the ballpark of their inclusive and exclusive determinations, respectively. In the present paper we follow another route, which to our knowledge has not been explored for $$\Delta M_{s,d}$$ Δ M s , d and rare K and B decays by anybody to date. We assume, in contrast to the prevailing inclusive expectations for $$|V_{cb}|$$ | V cb | , that the future true values of $$|V_{cb}|$$ | V cb | and $$|V_{ub}|$$ | V ub | will be both from exclusive determinations; in practice we use the most recent averages from FLAG. With the precisely known $$|V_{us}|$$ | V us | the resulting rare decay branching ratios, $$\varepsilon _K$$ ε K , $$\Delta M_d$$ Δ M d , $$\Delta M_s$$ Δ M s and $$S_{\psi K_S}$$ S ψ K S depend then only on the angles $$\beta $$ β and $$\gamma $$ γ in the unitarity triangle that moreover are correlated through the CKM unitarity. An unusual pattern of SM predictions results from this study with some existing tensions being dwarfed and new tensions being born. In particular using HPQCD $$B^0_{s,d}-{{\bar{B}}}^0_{s,d}$$ B s , d 0 - B ¯ s , d 0 hadronic matrix elements a $$3.1\sigma $$ 3.1 σ tension in $$\Delta M_s$$ Δ M s independently of $$\gamma $$ γ is found. For $$60^\circ \le \gamma \le 75^\circ $$ 60 ∘ ≤ γ ≤ 75 ∘ the tension in $$\Delta M_d$$ Δ M d between $$4.0\sigma $$ 4.0 σ and $$1.1\sigma $$ 1.1 σ is found and in the case of $$\varepsilon _K$$ ε K between $$5.2\sigma $$ 5.2 σ and $$2.1\sigma $$ 2.1 σ . Moreover, the room for new physics in $$K^+\rightarrow \pi ^+\nu {\bar{\nu }}$$ K + → π + ν ν ¯ , $$K_L\rightarrow \pi ^0\nu {\bar{\nu }}$$ K L → π 0 ν ν ¯ and $$B\rightarrow K(K^*)\nu {\bar{\nu }}$$ B → K ( K ∗ ) ν ν ¯ decays is significantly increased. We compare the results in this EXCLUSIVE scenario with the HYBRID one in which $$|V_{cb}|$$ | V cb | in the former scenario is replaced by the most recent inclusive $$|V_{cb}|$$ | V cb | and present the dependence of all observables considered by us in both scenarios as functions of $$\gamma $$ γ . As a byproduct we compare the determination of $$|V_{cb}|$$ | V cb | from $$\Delta M_s$$ Δ M s , $$\Delta M_d$$ Δ M d , $$\varepsilon _K$$ ε K and $$S_{\psi K_S}$$ S ψ K S using $$B^0_{s,d}-{{\bar{B}}}^0_{s,d}$$ B s , d 0 - B ¯ s , d 0 hadronic matrix elements from LQCD with $$2+1+1$$ 2 + 1 + 1 flavours, $$2+1$$ 2 + 1 flavours and their average. Only for the $$2+1+1$$ 2 + 1 + 1 case values for $$\beta $$ β and $$\gamma $$ γ exist for which the same value of $$|V_{cb}|$$ | V cb | is found: $$|V_{cb}|=42.6(4)\times 10^{-3}$$ | V cb | = 42.6 ( 4 ) × 10 - 3 , $$\gamma =64.6(16)^\circ $$ γ = 64.6 ( 16 ) ∘ and $$\beta =22.2(7)^\circ $$ β = 22.2 ( 7 ) ∘ . This in turn implies a $$2.7\sigma $$ 2.7 σ anomaly in $$B_s\rightarrow \mu ^+\mu ^-$$ B s → μ + μ -