QCD corrections to inclusive $ΔS=1,2$ transitions at the next-to-leading order

Abstract

The 2--point functions for $ΔS=1$ current-current and QCD-penguin operators, as well as for the $\ds$ operator, are calculated at the next-to-leading order. The calculation is performed in two different renormalization schemes for $γ_5$, and the compatibility of the results obtained in the two schemes is verified. The scale- and scheme-invariant combinations of spectral 2--point functions and corresponding Wilson-coefficients are constructed and analyzed. For $ΔS=1$, the QCD corrections to the CP-conserving part, dominated by current-current operators, are 40\%--120\% at $q^2=\left(1-3\,\mbox{\rm GeV}\right)^2$, whereas the correction to the imaginary part, mainly coming from the penguin operator $Q_6$, are 100\%--240\%. The large size of the gluonic corrections to current-current operators provides a qualitative understanding of the observed enhancement of $ΔI=1/2$ transitions. In the $ΔS=2$ sector the QCD corrections are quite moderate ($\approx -20\%$).