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

G. Spanoudes1,*, M. Costa2,†,∥, K. Mitsidi1,‡,¶, and H. Panagopoulos1,§

  • *Contact author: spanoudes.gregoris@ucy.ac.cy
  • †Contact author: marios.costa@cut.ac.cy
  • ‡Contact author: kmitsidi@tudelft.nl
  • §Contact author: haris@ucy.ac.cy
  • ∥Present address: Department of Mechanical Engineering and Material Science and Engineering, Cyprus University of Technology, Limassol, CY-3036, Cyprus.
  • Present address: Faculty of Applied Sciences, Delft University of Technology, Lorentzweg 1, 2628 CK, Delft, The Netherlands.
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Phys. Rev. D 113, 034504 – Published 13 February, 2026

DOI: https://doi.org/10.1103/h9p1-6cbw

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 MS¯ scheme at next-to-leading order.

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Phys. Rev. D 113, 034504– Published 13 February, 2026
  • Received 11 November 2025
  • Accepted 15 January 2026
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