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 - .

Standard Model predictions for rare K and B decays without new physics infection

“…Therefore the question arises which of these two values should be used in a global fit if any. 4 As shown recently in [ 31 ], the SM predictions for the branching ratios in question and observables are drastically different for these two values of . This problem existed already in 2015 in the context of the widely cited paper in [ 34 ] as stressed recently in a short note in [ 28 ].…”

“…The present paper suggests a possible solution to these problems and studies its implications. It is based on the ideas developed in collaboration with Elena Venturini [ 30 , 31 ] and extends them in a significant manner. The short note in [ 28 ] by the present author, in which some critical comments about the literature have been made, can be considered as an overture to the present paper.…”

“…This peculiar situation has been already addressed in the context of B physics anomalies by other authors and we will add a few additional remarks at the end of our paper. However, in none of the related papers in the literature the suggestion has been made to use this fact for the determintation of the CKM parameters without NP infection from observables alone, so that the strategies developed in [ 30 , 31 ] and used extensively in the present paper open a new route to phenomenology of flavour violating processes, not only in the SM but also beyond it.…”

“…In Sect. 3 I will argue that the strategy developed recently in collaboration with Elena Venturini [ 30 , 31 ] is presently the most efficient method for obtaining CKM-independent SM predictions for various suitable ratios of rare decay branching ratios to the observables in ( 2 ). In Sect.…”

“…Setting the values of observables to their experimental values and using the CKM-independent ratios found above, allows to obtain SM predictions for all very rare K and B branching ratios that are most accurate to date [ 30 , 31 ]. Another bonus of this strategy is the determination of the CKM parameters from processes alone , that allows in turn to make accurate predictions for a number of -independent ratios that depend on and [ 30 ].…”