The Rare Decay K + → π + νν̄ at the Next - to - Next - to - Leading Order in QCD

We calculate the complete next-to-next-to-leading order QCD correction of the charm quark contribution to the branching ratio for the rare decay K + → π + ν ¯ ν in the standard model. The inclusion of these O(αs) contributions leads to a significant reduction of the theoretical uncertainty from ±10.1% down to ±2.4% in the relevant parameter Pc, implying the left over scale uncertainties in B(K + → π + ν ¯ ν) and in the determination of |V td |, sin 2β and γ from the K → πν ¯ ν system to be ±1.3%, ±1.0%, ±0.006 and ±1.2 • , respectively. For the charm quark MS mass mc(mc) = (1.30 ± 0.05) GeV and |Vus| = 0.2248 the next-to-leading order value Pc = 0.37 ± 0.06 is modified to Pc = 0.37 ± 0.04 at the next-to-next-to-leading order level with the latter error fully dominated by the uncertainty in mc(mc). Adding the recently calculated long-distance contributions we find B(K + → π + ν ¯ ν) = (8.0 ± 1.1) × 10 −11 with the quoted error almost entirely due to the present uncertainties in mc(mc) and the Cabibbo-Kobayashi-Maskawa elements. The rare process K + → π + ν ¯ ν belongs to the theoretically cleanest decays in the field of K-and B-mesons. As it offers in conjunction with K L → π 0 ν ¯ ν a very clean determination of the standard unitarity triangle (UT) [1], a comparison of the information obtained from the K → πν ¯ ν system with the one from B-decays provides a critical and truly unique test of the Cabibbo-Kobayashi-Maskawa (CKM) mechanism in the standard model (SM) [2, 3]. Even if these K-and B-physics predictions agree, K + → π + ν ¯ ν will allow to discriminate between different extensions of the SM [2, 3], by probing effective scales of new physics operators of up to a several TeV or even higher [4]. In the SM the decay K + → π + ν ¯ ν proceeds through Z-penguin and electroweak box diagrams, that are sensitive to short-distance dynamics. As the required hadronic matrix elements can be extracted, including isospin breaking corrections [5], from the accurately measured leading semileptonic decay K + → π 0 e + ν, and the remaining long-distance contributions turn out to be small [6], and in principle calculable by means of lattice QCD [7], theoretical …