Perturbative Corrections for Staggered Four-Fermion Operators

TL;DR

This paper computes one-loop matching coefficients for four-fermion operators in lattice QCD using staggered fermions, enabling more accurate calculations of kaon mixing and decay amplitudes with perturbative corrections.

Contribution

It provides the first comprehensive calculation of one-loop matching coefficients for staggered four-fermion operators, including penguin diagrams and operator mixing effects.

Findings

Calculated one-loop matching coefficients for various operators.

Included effects of penguin diagrams and operator mixing.

Facilitated improved lattice QCD calculations of kaon processes.

Abstract

We present results for one-loop matching coefficients between continuum four-fermion operators, defined in the Naive Dimensional Regularization scheme, and staggered fermion operators of various types. We calculate diagrams involving gluon exchange between quark lines, and ``penguin'' diagrams containing quark loops. For the former we use Landau gauge operators, with and without $O(a)$ improvement, and including the tadpole improvement suggested by Lepage and Mackenzie.For the latter we use gauge-invariant operators. Combined with existing results for two-loop anomalous dimension matrices and one-loop matching coefficients, our results allow a lattice calculation of the amplitudes for $K\bar K$ mixing and $K\to\pi\pi$ decays with all corrections of $O(g^2)$ included. We also discuss the mixing of $\Delta S=1$ operators with lower dimension operators, and show that, with staggered fermions, only a single lower dimension operator need be removed by non-perturbative subtraction.