Charm quark contribution to K+→π+νν-bar at next-to-next-to-leading order

We calculate the complete next-to-next-to-leading (NNLO) order QCD corrections to the charm contribution of the rare decay K+→π+νν-bar . We present the results for the two-loop matching conditions of the Wilson coefficients, the three-loop anomalous dimensions, and the two-loop matrix elements of the relevant operators entering the NNLO renormalization group analysis of the Z-penguin and the electroweak box contribution. The NNLO QCD corrections reduce the theoretical uncertainty from ±9.8% at NLO to ±2.4% in the relevant parameter Pc(X), implying the leftover scale uncertainties in B(K+→π+νν-bar ) and in the determination of Vtd, sin 2β, and γ from the K→πνν-bar system to be ±1.3%, ±1.0%, ±0.006, and ±1.20, respectively. For the M-bar S-bar charm quark mass mc(mc) (1.30±0.05)GeV and |Vus| = 0.2248 the NLO value Pc(X) = 0.37±0.06 is modified to Pc(X) = 0.38±0.04 at NNLO and the error is fully dominated by the uncertainty in mc(mc). We tabulate Pc(X) in terms of mc(mc) and αs(MZ) and express the dependences of Pc(X) on these and other parameters by an accurate approximate analytic formula. We find B(K+→π+νν-bar ) = (8.0±1.1) x 10-11 and the quoted uncertainty mainly stems from mc(mc) and the Cabibbo-Kobayashi-Maskawa elements. We also emphasize that improved calculations of the long-distance contributions to K+→π+νν-bar and of the isospin breaking in the weak current matrix element will further sharpen the sensitivity of the two golden K→πνν-bar decays to new physics