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Gluon condensate via Dirac spectral density: IR phase, scale anomaly, and IR decoupling

Ivan Horváth*

  • *Contact author: ihorv2@g.uky.edu
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Phys. Rev. D 112, 094505 – Published 12 November, 2025

DOI: https://doi.org/10.1103/w33k-gr8k

Abstract

Quark and gluon scalar densities, ⟨ψ¯ψ⟩ and ⟨F2⟩, reflect the degree of scale-invariance violations in SU(N) gauge theories with fundamental quarks. It is known that ⟨ψ¯ψ⟩ can be usefully scale-decomposed via the spectral density ρ(λ) of Dirac modes. Here, we give such formula for ⟨F2⟩, which reveals that the gluon condensate is a strictly UV quantity. For the recently found IR phase [1,2], where the infrared (IR) degrees of freedom separate out and become independent of the system’s bulk, it implies that ⟨F2⟩ due to this IR part vanishes. Its glue, thus, does not contribute to the scale anomaly of the entire system and is, in this sense, scale invariant consistently with the original claim. Associated formulas are used to define the IR decoupling of glue, which may serve as an alternative indicator of IR phase transition. Using the simplest form of coherent lattice QCD, we express the effective action of full QCD entirely via Dirac spectral density.

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References (29)

  1. A. Alexandru and I. Horváth, Phases of SU(3) gauge theories with fundamental quarks via Dirac spectral density, Phys. Rev. D 92, 045038 (2015).
  2. A. Alexandru and I. Horváth, Possible new phase of thermal QCD, Phys. Rev. D 100, 094507 (2019).
  3. M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, QCD and resonance physics: Theoretical foundations, Nucl. Phys. B147, 385 (1979).
  4. A. Alexandru and I. Horváth, Unusual features of QCD low-energy modes in the infrared phase, Phys. Rev. Lett. 127, 052303 (2021).
  5. A. Alexandru and I. Horváth, Anderson metal-to-critical transition in QCD, Phys. Lett. B 833, 137370 (2022).
  6. X.-L. Meng, P. Sun, A. Alexandru, I. Horváth, K.-F. Liu, G. Wang, and Y.-B. Yang (χQCD, CLQCD Collaborations), Separation of infrared and bulk in thermal QCD, J. High Energy Phys. 12 (2024) 101.
  7. A. Alexandru, I. Horváth, and N. Bhattacharyya, Localized modes in the IR phase of QCD, Phys. Rev. D 109, 014501 (2024).
  8. A. Alexandru, C. Bonanno, M. D’Elia, and I. Horváth, Dirac spectral density in Nf=2+1 QCD at T=230  MeV, Phys. Rev. D 110, 074515 (2024).
  9. N. Nielsen, Gauge invariance and broken conformal symmetry, Nucl. Phys. B97, 527 (1975).
  10. J. C. Collins, A. Duncan, and S. D. Joglekar, Trace and dilatation anomalies in gauge theories, Phys. Rev. D 16, 438 (1977).
  11. N. K. Nielsen, The energy-momentum tensor in a non-Abelian quark gluon theory, Nucl. Phys. B120, 212 (1977).
  12. T. Banks and A. Casher, Chiral symmetry breaking in confining theories, Nucl. Phys. B169, 103 (1980).
  13. I. Horváth, Coherent lattice QCD, Proc. Sci., LAT2006 (2006) 053 [arXiv:hep-lat/0610121].
  14. I. Horvath, A framework for systematic study of QCD vacuum structure: II. Coherent lattice QCD, arXiv:hep-lat/0607031.
  15. A. Alexandru, I. Horváth, and K.-F. Liu, Classical limits of scalar and tensor gauge operators based on the overlap Dirac matrix, Phys. Rev. D 78, 085002 (2008).
  16. P. H. Ginsparg and K. G. Wilson, A remnant of chiral symmetry on the lattice, Phys. Rev. D 25, 2649 (1982).
  17. I. Horvath, Ginsparg-Wilson relation and ultralocality, Phys. Rev. Lett. 81, 4063 (1998).
  18. I. Horvath, Ginsparg-Wilson-Luscher symmetry and ultralocality, Phys. Rev. D 60, 034510 (1999).
  19. I. Horvath, C. T. Balwe, and R. Mendris, Strong nonultralocality of Ginsparg-Wilson fermionic actions, Nucl. Phys. B599, 283 (2001).
  20. H. Neuberger, Exactly massless quarks on the lattice, Phys. Lett. B 417, 141 (1998).
  21. L. Del Debbio, L. Giusti, M. Luscher, R. Petronzio, and N. Tantalo, Stability of lattice QCD simulations and the thermodynamic limit, J. High Energy Phys. 02 (2006) 011.
  22. L. Giusti and M. Luscher, Chiral symmetry breaking and the Banks-Casher relation in lattice QCD with Wilson quarks, J. High Energy Phys. 03 (2009) 013.
  23. J. C. Osborn, D. Toublan, and J. J. M. Verbaarschot, From chiral random matrix theory to chiral perturbation theory, Nucl. Phys. B540, 317 (1999).
  24. I. Horváth, The Infrared Phase of QCD and Anderson Localization, in 16th Conference on Quark Confinement and the Hadron Spectrum (2025), 6, arXiv:2506.0411.
  25. A. Alexandru and I. Horváth, Chiral symmetry breaking and chiral polarization: Tests for finite temperature and many flavors, Nucl. Phys. B891, 1 (2015).
  26. A. Alexandru and I. Horváth, Broken valence chiral symmetry and chiral polarization of Dirac spectrum in Nf=12 QCD at small quark mass, AIP Conf. Proc. 1701, 030008 (2016).
  27. A. M. Garcia-Garcia and J. C. Osborn, Chiral phase transition in lattice QCD as a metal-insulator transition, Phys. Rev. D 75, 034503 (2007).
  28. T. G. Kovacs and F. Pittler, Anderson localization in quark-gluon plasma, Phys. Rev. Lett. 105, 192001 (2010).
  29. M. Giordano, T. G. Kovacs, and F. Pittler, Universality and the QCD Anderson transition, Phys. Rev. Lett. 112, 102002 (2014).

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Phys. Rev. D 112, 094505– Published 12 November, 2025
  • Received 22 September 2025
  • Accepted 21 October 2025
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