The rare decays K → πνν, B → xνν and B → l⁺l^-: An update

We update the Standard Model predictions for the rare decays K⁺ → π⁺νν and K_L → π⁰νν. In view of improved limits on B_s-B_s mixing we derive a stringent and theoretically clean Standard Model upper limit on B(K⁺ → π⁺νν), which is based on the ratio of B_d-B_d to B_s-B_s mixing, ΔM_d/ΔM_s, alone. This method avoids the large hadronic uncertainties present in the usual analysis of the CKM matrix. We find B(K⁺ →⁺νν) < 1.67 × 10^-10, which can be further improved in the future. In addition we consider the extraction of V_td from a future measurement of B(K⁺ → π⁺νν), discussing the various sources of uncertainties involved. We also investigate theoretically clean constraints on B(K_L → π⁰νν). We take the opportunity to review the next-to-leading order (NLO) QCD corrections to K → πνν, K_L → μ⁺μ^-, B → Xνν and B → l⁺l^-, including a small additional term that had been missed in the original publications. The phenomenological impact of this change is negligible, the corresponding numerical shift being essentially within the small perturbative uncertainties at the NLO level.