1/Nccorrections to the hadronic matrix elements ofQ6andQ8inK→ππdecays

Abstract

We calculate long-distance contributions to the amplitudes ${A(K}^{0}\ensuremath{\rightarrow}2\ensuremath{\pi},I)$ induced by the gluon and the electroweak penguin operators ${Q}_{6}$ and ${Q}_{8},$ respectively. We use the ${1/N}_{c}$ expansion within the effective chiral Lagrangian for pseudoscalar mesons. In addition, we adopt a modified prescription for the identification of meson momenta in the chiral loop corrections in order to achieve a consistent matching to the short-distance part. Our approach leads to an explicit classification of the loop diagrams into non-factorizable and factorizable, the scale dependence of the latter being absorbed in the low-energy coefficients of the effective theory. Along these lines we calculate the one-loop corrections to the $\mathcal{O}{(p}^{0})$ term in the chiral expansion of both operators. In the numerical results, we obtain moderate corrections to ${B}_{6}^{(1/2)}$ and a substantial reduction of ${B}_{8}^{(3/2)}.$