Effects of Quenching and Partial Quenching on Penguin Matrix Elements

Abstract

In the calculation of non-leptonic weak decay rates, a "mismatch" arises when the QCD evolution of the relevant weak hamiltonian down to hadronic scales is performed in unquenched QCD, but the hadronic matrix elements are then computed in (partially) quenched lattice QCD. This mismatch arises because the transformation properties of penguin operators under chiral symmetry change in the transition from unquenched to (partially) quenched QCD. Here we discuss QCD-penguin contributions to $\Delta S=1$ matrix elements, and show that new low-energy constants contribute at leading order in chiral perturbation theory in this case. In the partially quenched case (in which sea quarks are present), these low-energy constants are related to electro-magnetic penguins, while in the quenched case (with no sea quarks) no such relation exists. As a simple example, we give explicit results for $K^+\to\pi^+$ and $K^0\to vacuum$ matrix elements, and discuss the implications for lattice determinations of $K\to\pi\pi$ amplitudes from these matrix elements.