Matching the Electroweak Penguins $Q_7$, $Q_8$ and Spectral Correlators

Abstract

Exact analytical expressions for the $\Delta S=1$ coupling $\im G_E$ in terms of observable spectral functions are given. This coupling determines the size of the $\Delta I=3/2$ contribution to $\epsilon'$. We show analytically how the scheme-dependence and scale dependences vanish to all orders in $1/N_c$ and NLO in $\alpha_S$ explicitly both for $Q_7$ and $Q_8$. Numerical results are derived for both $Q_7$ and $Q_8$ from the $\tau$-data and known results on the scalar spectral functions. In particular we study the effect of all higher dimension operators. The coefficients of the leading operators in the OPE of the needed correlators are derived to NLO in $\alpha_S$.