Toolkit for staggered $\mathrm{\Delta }S=2$ matrix elements

A recent numerical lattice calculation of the kaon mixing matrix elements of general $\mathrm{\Delta }S=2$ four-fermion operators using staggered fermions relied on two auxiliary theoretical calculations. Here we describe the methodology and present the results of these two calculations. The first concerns one-loop matching coefficients between staggered lattice operators and the corresponding continuum operators. Previous calculations with staggered fermions have used a nonstandard regularization scheme for the continuum operators, and here we provide the additional matching factors needed to connect to the standard regularization scheme. This is the scheme in which two-loop anomalous dimensions are known. We also observe that all previous calculations of this operator matching using staggered fermions have overlooked one matching step in the continuum. This extra step turns out to have no impact on three of the five operators (including that relevant for $B_K$), but it does affect the other two operators. The second auxiliary calculation concerns the two-loop renormalization group (RG) evolution equations for the $B$ parameters of the $\mathrm{\Delta }S=2$ operators. For one pair of operators, the standard analytic solution to the two-loop RG equations fails due to a spurious singularity introduced by the approximations made in the calculation. We give a nonsingular expression derived using analytic continuation and check the result using a numerical solution to the RG equations. We also describe the RG evolution for “golden” combinations of $B$ parameters and give numerical results for RG evolution matrices needed in the companion lattice calculation.

Figure 1

Classes of one-loop diagrams, with labeling as in Ref. [23]. Each filled circle represents one of the two bilinears composing the four-fermion operator. For each diagram shown, there is a second one (not shown) in which the gluon connects the other two fermion propagators.