Classification of four-quark operators with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>F</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>2</mml:mn> </mml:math> under flavor symmetry

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="normal">Δ</a:mi> <a:mi>F</a:mi> <a:mo>=</a:mo> <a:mn>2</a:mn> </a:math> operators, as well as their partners that transform under the same irreducible representations of the flavor group. These encompass a substantial subset of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mi mathvariant="normal">Δ</d:mi> <d:mi>F</d:mi> <d:mo>=</d:mo> <d:mn>1</d:mn> </d:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi mathvariant="normal">Δ</g:mi> <g:mi>F</g:mi> <g:mo>=</g:mo> <g:mn>0</g:mn> </g:math> 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 <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mover accent="true"> <j:mi>MS</j:mi> <j:mo stretchy="true">¯</j:mo> </j:mover> </j:math> scheme at next-to-leading order.