Classification of four-quark operators with $\Delta F\le 2$ under flavor symmetry and their renormalization in a gauge-invariant scheme

TL;DR

This paper classifies four-quark operators with specific flavor-changing properties, analyzes their renormalization within a gauge-invariant scheme, and provides conversion matrices to the MS scheme at NLO.

Contribution

It offers a comprehensive classification of four-quark operators under flavor symmetry and studies their renormalization in GIRS, including conversion to MS scheme at NLO.

Findings

Complete classification of scalar and pseudoscalar four-quark operators.

Analysis of operator mixing patterns and symmetry properties.

Provision of conversion matrices from GIRS to MS scheme at NLO.

Abstract

In this paper we study a complete set of scalar and pseudoscalar four-quark operators, with a particular emphasis on their renormalization within a Gauge-Invariant Renormalization Scheme (GIRS). We focus on operators that do not mix with lower-dimensional operators by virtue of their transformation properties under the flavor-symmetry group. This class includes all $\Delta F = 2$ operators, as well as their partners that transform under the same irreducible representations of the flavor group. These encompass a substantial subset of $\Delta F = 1$ and $\Delta F = 0$ operators. The present analysis provides a detailed classification of all four-quark operators, exploring their Fierz identities, symmetry properties, and mixing patterns. Different variants of GIRS are explored, including a democratic version that treats all mixing operators uniformly. For selected variants, which exhibit smaller mixing effects, we present the conversion matrices from GIRS to the $\overline{\text{MS}}$ scheme at next-to-leading order.