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Renormalizing the quark-meson-diquark model

Hosein Gholami1,*, Lennart Kurth1,†, Ugo Mire2,‡, Michael Buballa1,3,§, and Bernd-Jochen Schaefer2,4,∥

  • 1Technische Universität Darmstadt, Fachbereich Physik, Institut für Kernphysik, Theoriezentrum, Schlossgartenstrasse 2, D-64289 Darmstadt, Germany
  • 2Institut für Theoretische Physik, Justus-Liebig-Universität Gießen, 35392 Gießen, Germany
  • 3Helmholtz Forschungsakademie Hessen für FAIR (HFHF), GSI Helmholtzzentrum für Schwerionenforschung, Campus Darmstadt, D-64289 Darmstadt, Germany
  • 4Helmholtz Forschungsakademie Hessen für FAIR (HFHF), GSI Helmholtzzentrum für Schwerionenforschung, Campus Gießen, 35392 Gießen, Germany

  • *Contact author: mohammadhossein.gholami@tu-darmstadt.de
  • †Contact author: lennart.kurth@stud.tu-darmstadt.de
  • ‡Contact author: ugo.louis.tryphon.mire@physik.uni-giessen.de
  • §Contact author: michael.buballa@tu-darmstadt.de
  • ∥Contact author: bernd-jochen.schaefer@uni-giessen.de
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Phys. Rev. D 113, 114020 – Published 11 June, 2026

DOI: https://doi.org/10.1103/bx12-6tt6

Abstract

We present a comprehensive study of the two-flavor quark-meson-diquark (QMD) model by comparing a renormalization approach with a renormalization-group (RG) consistent mean-field formulation based on the functional renormalization group (FRG). The renormalized QMD model allows analytical investigations of key quantities such as the zero-temperature diquark gap and the critical temperature for color superconductivity, ultimately reproducing the exact BCS relation in the high-density limit. We carry out the same analysis for different schemes of RG-consistent QMD models. We show that the RG-consistent approach yields a phase diagram and thermodynamic properties qualitatively similar to those of the renormalized model, provided both are embedded within a unified scheme that ensures consistent vacuum properties. In particular, both treatments recover the Stefan-Boltzmann limit at high densities. On the other hand, whether the BCS relation for the critical temperature is satisfied depends on the details of the RG-consistent setup. Our results highlight the relevance of renormalization and RG-consistent methods for accurately capturing the thermodynamics of QMD and related effective models with diquark degrees of freedom.

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Phys. Rev. D 113, 114020– Published 11 June, 2026
  • Received 28 July 2025
  • Accepted 6 May 2026
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