Penguin Diagrams for the Hyp Staggered Fermions *

The low energy effective Hamiltonian of the standard model includes ∆S = 1 four-fermion operators with corresponding Wilson coefficients, which contains all the short-distance physics. The low energy effects of the electroweak and strong interactions can be expressed in terms of matrix elements of the four-fermion operators between hadronic states. Lattice QCD is wellsuited to calculate these matrix elements nonperturbatively at low energy. One essential step in using lattice QCD is to find the relationship between the continuum and lattice operators, which is often called “matching formula”. There are two classes of Feynman diagrams at the one-loop level: (1) current-current diagrams and (2) penguin diagrams. At the one loop level, it is possible to treat the penguin contribution and the currentcurrent contribution separately. In the case of the current-current diagrams, the matching formula at the one-loop level is given in [1]. Here, we focus on penguin diagrams in which one of the quarks in the four-fermion operator is contracted with one of the anti-quarks to form a closed loop. Hence, the main goal is to calculate the penguin diagrams for improved staggered operators constructed using various fat links and to provide the corresponding matching formula. Here, we adopt the same notation and Feynman rules outlined in [1].