Four-quark operators with $\hat{I}”F=2$ in the GIRS scheme

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.

8 pages, 2 tables, 2 figures, Contribution to The 41st International Symposium on Lattice Field Theory (LATTICE2024), 28 July - 3 August 2024, Liverpool, UK. arXiv admin note: text overlap with arXiv:2406.08065