Standard Model Predictions for Rare K and B Decays without New Physics Infection

The Standard Model (SM) does not contain by definition any new physics (NP) contributions to any observable but contains four CKM parameters which are not predicted by this model. We point out that if these four parameters are determined in a global fit that includes processes which are infected by NP, the resulting SM contributions to rare decay branching ratios cannot be considered as true SM contributions to the latter. On the other hand true SM predictions, that are free from the CKM dependence, can be obtained for suitable ratios of the $K$ and $B$ rare decay branching ratios to $\Delta M_s$, $\Delta M_d$ and $|\varepsilon_K|$, all calculated within the SM. These three observables contain by now only small hadronic uncertainties and are already well measured so that rather precise true SM predictions for the ratios in question can be obtained. In this context the rapid test of NP infection in the $\Delta F=2$ sector is provided by a $|V_{cb}|-\gamma$ plot that involves $\Delta M_s$, $\Delta M_d$, $|\varepsilon_K|$, and the mixing induced CP-asymmetry $S_{\psi K_S}$. As with the present hadronic matrix elements this test turns out to be negative, assuming negligible NP infection in the $\Delta F=2$ sector and setting the values of these four observables to the experimental ones, allows to obtain SM predictions for all $K$ and $B$ rare decay branching ratios that are most accurate to date and as a byproduct to obtain the full CKM matrix on the basis of $\Delta F=2$ transitions alone. Using this strategy we obtain SM predictions for 26 branching ratios for rare semileptonic and leptonic $K$ and $B$ decays with the $\mu^+\mu^-$ pair or the $\nu\bar\nu$ pair in the final state. Most interesting turn out to be the anomalies in the low $q^2$ bin in $B^+\to K^+\mu^+\mu^-$ ($4.4\sigma$) and $B_s\to \phi\mu^+\mu^-$ ($4.8\sigma$).