Gauge-invariant renormalization of four-quark operators

TL;DR

This paper introduces a gauge-invariant renormalization scheme for four-quark operators, providing next-to-leading order conversion factors to the MS scheme, which aids nonperturbative lattice QCD studies and understanding of operator mixing.

Contribution

It develops a coordinate-space gauge-invariant renormalization scheme for four-quark operators and computes conversion factors to the MS scheme at NLO, addressing operator mixing complexities.

Findings

Derived conversion factors between GIRS and MS schemes at NLO.

Analyzed mixing patterns of parity-conserving and violating four-quark operators.

Provided tools for improved nonperturbative QCD calculations and CKM matrix element determination.

Abstract

We study the renormalization of four-quark operators in one-loop perturbation theory. We employ a coordinate-space Gauge-Invariant Renormalization Scheme (GIRS), which can be advantageous compared to other schemes, especially in nonperturbative lattice investigations. From our perturbative calculations, we extract the conversion factors between GIRS and the modified Minimal Subtraction scheme ($\overline{\rm MS}$) at the next-to-leading order. A formidable issue in the study of the four-quark operators is that operators with different Dirac matrices mix among themselves upon renormalization. We focus on both parity-conserving and parity-violating four-quark operators, which change flavor numbers by two units ($\Delta F = 2$). The extraction of the conversion factors entails the calculation of two-point Green's functions involving products of two four-quark operators, as well as three-point Green's functions with one four-quark and two bilinear operators. The significance of our results lies in their potential to refine our understanding of QCD phenomena, offering insights into the precision of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and shedding light on the nonperturbative treatment of complex mixing patterns associated with four-quark operators.