## 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 P_{c}(X), implying the leftover scale uncertainties in B(K^{+} → π^{+}νν) and in the determination of |V_{td}|, 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 m_{c}(m_{c}) = (1.30±0.05)GeV and |V_{us}| = 0.2248 the NLO value P_{c}(X) = 0.37±0.06 is modified to P_{c}(X) = 0.38±0.04 at NNLO and the error is fully dominated by the uncertainty in m_{c}(m_{c}). We tabulate P_{c}(X) in terms of m_{c}(m_{c}) and α_{s}(M_{Z}) 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 m_{c}(m_{c}) 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.

Original language | English |
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Article number | 002 |

Journal | Journal of High Energy Physics |

Volume | 2006 |

Issue number | 11 |

DOIs | |

State | Published - 1 Nov 2006 |

## Keywords

- Kaon Physics
- NLO Computations
- Rare Decays
- Renormalization Group

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