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

Abstract

We calculate the complete next-to-next-to-leading (NNLO) order QCD corrections to the charm contribution of the rare decay K+ → π+νν. 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) and in the determination of |Vtd|, sin 2β, and γ from the K → πνν system to be ±1.3%, ±1.0%, ±0.006, and ±1.2°, respectively. For the MS̄ 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 P c(X) on these and other parameters by an accurate approximate analytic formula. We find B(K+ → π+νν) = (8.0±1.1) × 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 and of the isospin breaking in the weak current matrix element will further sharpen the sensitivity of the two golden K→ πνν decays to new physics. © SISSA 2006.