The generating functional for hadronic weak interactions and its quenched approximation

We derive the generating functional of $\mathrm{\mid }\mathrm{\Delta }S\mathrm{\mid }=1,2$ hadronic weak interactions at one loop for a generic number of flavours and its counterpart in the quenched approximation. A systematic analysis of the ultraviolet divergences in the full theory (with and without a singlet dynamical field) and in the quenched case is performed. We show that the quenched chiral logarithms in the presence of weak interactions amount to a redefinition of the weak mass term in the $\mathrm{\Delta }S=\pm 1$ weak effective Lagrangian at leading order. Finally, we apply the results to $B_K$ and $K\to \pi \pi$ matrix elements with $\mathrm{\Delta }I=1\mathrm{/}2,3\mathrm{/}2$ to analyze the modifications induced by quenching on the coefficients of chiral logarithms in the one-loop corrections.