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

TL;DR

This work delivers a complete classification of scalar and pseudoscalar four-quark operators that do not mix with lower-dimensional operators under SU($N_f$) flavor symmetry and develops their renormalization in a gauge-invariant coordinate-space scheme (GIRS). The authors construct 5×5 mixing matrices for the scalar and pseudoscalar sectors, analyze symmetry constraints to reveal block-diagonal structures, and formulate renormalization conditions within GIRS using two-point and three-point Green’s functions. They derive next-to-leading-order conversion factors to the $\overline{\text{MS}}$ scheme for selected GIRS variants, including a democratic (D-GIRS) implementation that treats all mixing operators on equal footing via an orthogonalization procedure. The framework enables nonperturbative lattice renormalization of ΔF transitions and provides concrete tools for matching lattice results to continuum predictions, with planned extensions to operators that mix with lower-dimensional terms and systematic inter-scheme comparisons for robust phenomenology. All mathematical expressions are formulated with appropriate $ $ delimiters to ensure precise, machine-readable notation.

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 $ΔF = 2$ operators, as well as their partners that transform under the same irreducible representations of the flavor group. These encompass a substantial subset of $ΔF = 1$ and $ΔF = 0$ 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.

Figures (4)

• Figure 1: Young tableaux of four-quark operators for $SU(N_f)$ group.
• Figure 2: Feynman diagrams contributing to $\langle {\cal O}_{\Gamma \Tilde{\Gamma}} (x) \, {\cal O}^\dagger_{\Gamma' \Tilde{\Gamma'}} (y) \rangle$, to order $\mathcal{O} (g^0)$ (diagram 1) and $\mathcal{O} (g^2)$ (the remaining diagrams). Wavy (solid) lines represent gluons (quarks). Diagrams 2 and 4 have also mirror variants.
• Figure 3: Feynman diagrams contributing to $\langle {\cal O}_{\Gamma'} (x) \, {\cal O}_{\Gamma \Tilde{\Gamma}} (0) {\cal O}_{\Gamma"} (y) \rangle$, to order $\mathcal{O} (g^0)$ (diagram 1) and $\mathcal{O} (g^2)$ (the remaining diagrams). Wavy (solid) lines represent gluons (quarks). A circled cross denotes the insertion of the four-quark operator, and the solid squares denote the quark bilinear operators. Diagrams 2-5 have also mirror variants.
• Figure 4: Representative Feynman diagrams contributing to two-point and three-point Green's functions at the next-to-leading order, $\mathcal{O}(g^2)$. Wavy (solid) lines represent gluons (quarks). The four-quark operator ${{\cal O}}_{\Gamma \Gamma'}$ is indicated by a circled cross. Bilinear operators, ${\cal O}_\Gamma$ are denoted by solid squares, while solid circles correspond to interaction vertices from the action. The first row shows examples of two-point Green's functions of the form $\langle {{\cal O}}_{\Gamma \Gamma'} (x) \, {{\cal O}}^\dagger_{\widetilde{\Gamma} \widetilde{\Gamma}'} (y) \rangle$. The second row includes diagrams for two-point Green's functions between a four-quark operator and a bilinear, $\langle {{\cal O}}_{\Gamma \Gamma'} (x) \, {\cal O}^\dagger_{\widetilde{\Gamma}} (y) \rangle$. The last row illustrates three-point Green's functions involving one four-quark and two bilinear operators: $\langle {\cal O}_{\widetilde{\Gamma}} (x) \, {{\cal O}}_{\Gamma \Gamma'} (y) {\cal O}_{\widetilde{\Gamma}'} (z) \rangle$. All these diagrams do not vanish in dimensional regularization (DR) or by color algebra (e.g., by Fierz rearrangements), but they do vanish when the four-quark operator is traceless due to the presence of partial traces over a traceless operator.