Standard Model Predictions for Rare $K$ and $B$ Decays without $|V_{cb}|$ and $|V_{ub}|$ Uncertainties

Abstract: The persistent tensions between inclusive and exclusive determinations of $|V_{cb}|$ and $|V_{ub}|$ weaken the power of theoretically clean rare $K$ and $B$ decays in the search for new physics (NP). We demonstrate how this uncertainty can be practically removed by considering within the SM suitable ratios of various branching ratios. This includes the branching ratios for $K^+\to\pi^+\nu\bar\nu$, $K_{L}\to\pi^0\nu\bar\nu$, $K_S\to\mu^+\mu^-$, $B_{s,d}\to\mu^+\mu^-$ and $B\to K(K^*)\nu\bar\nu$. Also $\epsilon_K$, $\Delta M_d$, $\Delta M_s$ and the mixing induced CP-asymmetry $S_{\psi K_S}$, all measured already very precisely, play an important role in this analysis. The highlights of our analysis are 16 $|V_{cb}|$ and $|V_{ub}|$ independent ratios that often are independent of the CKM arameters or depend only on the angles $\beta$ and $\gamma$ in the Unitarity Triangle with $\beta$ already precisely known and $\gamma$ to be measured precisely in the coming years by the LHCb and Belle II collaborations. Once $\gamma$ Once $\gamma$ is measured precisely these 16 ratios taken together are expected to be a powerful tool in the search for new physics. Assuming no NP in $|\epsilon_K|$ and $S_{\psi K_S}$ we determine independently of $|V_{cb}|$: $\mathcal{B}(K^+\to\pi^+\nu\bar\nu)_\text{SM}= (8.60\pm0.42)\times 10^{-11}$ and $\mathcal{B}(K_L\to\pi^0\nu\bar\nu)_\text{SM}=(2.94\pm 0.15)\times 10^{-11}$. This are the most precise determinations to date. Assuming no NP in $\Delta M_{s,d}$ allows to obtain analogous results for all $B$ decay branching ratios considered in our paper without any CKM uncertainties.