Four-quark operators with $\Delta F = 2$ in the GIRS scheme

TL;DR

This paper computes the mixing matrices and conversion factors for four-quark operators with elta F = 2 in the GIRS scheme, aiding precise determination of CKM matrix elements from lattice QCD simulations.

Contribution

It provides the first perturbative calculation of conversion factors between GIRS and elta F = 2 four-quark operators at next-to-leading order, including operator mixing effects.

Findings

Calculated mixing matrices for four-quark operators in GIRS.

Derived next-to-leading order conversion factors between GIRS and elta F = 2 schemes.

Analyzed both parity-conserving and parity-violating operators.

Abstract

We calculate the mixing matrices of four-quark operators that change flavor numbers by two units. Our approach employs two schemes: the coordinate-space Gauge Invariant Renormalization Scheme (GIRS) and the Modified Minimal Subtraction scheme. From our perturbative computations, we extract the conversion factors between these two renormalization schemes at the next-to-leading order. A significant challenge in the study of four-quark operators is that they mix among themselves upon renormalization. Additionally, computations in GIRS at a given order in perturbation theory require Feynman diagrams with at least one additional loop. The extraction of the conversion factors involves calculating two-point Green's functions, which include products of two four-quark operators, and three-point Green's functions, which involve one four-quark operator and two bilinear operators, with all operators located at distinct spacetime points. We investigate both parity-conserving and parity-violating four-quark operators. This calculation is relevant to the determination of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements from numerical simulations using the GIRS scheme.