Electroweak Penguin Contributions to Non-Leptonic Delta F = 1 Decays at NNLO

We calculate the O(alpha_s) corrections to the Z^0-penguin and electroweak box diagrams relevant for non-leptonic Delta F = 1 decays with F = S, B. This calculation provides the complete O(alpha_w alpha_s) and O(alpha_w alpha_s sin^2 theta_w m_t^2) corrections (alpha_w = alpha/sin^2 theta_w) to the Wilson coefficients of the electroweak penguin four quark operators relevant for non-leptonic K- and B-decays. We argue that this is the dominant part of the next-next-to-leading (NNLO) contributions to these coefficients. Our results allow to reduce considerably the uncertainty due to the definition of the top quark mass present in the existing NLO calculations of non-leptonic decays. The NNLO corrections to the coefficient of the color singlet (V-A)x(V-A) electroweak penguin operator Q_9 relevant for B-decays are generally moderate, amount to a few percent for the choice m_t(mu_t = m_t) and depend only weakly on the renormalization scheme. Larger NNLO corrections with substantial scheme dependence are found for the coefficients of the remaining electroweak penguin operators Q_7, Q_8 and Q_10. In particular, the strong scheme dependence of the NNLO corrections to C_8 allows to reduce considerably the scheme dependence of C_8 (Q_8)_2 relevant for the ratio epsilon'/epsilon.