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Structure of QC2D ground state fields at nonzero matter densities

Ragib F. Hasan1, Matthew Cummins1, Waseem Kamleh1, Dale Lawlor2, Derek B. Leinweber1, Ian van Schalkwyk1, and Jon-Ivar Skullerud2

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Phys. Rev. D 114, 094504 – Published 5 November, 2026

DOI: https://doi.org/10.1103/31cw-ym32

Abstract

A quantitative investigation into the modification of ground-state field structures in two-color QCD (QC2D) is presented at finite chemical potential. Using lattice simulations with Wilson gauge and fermion actions, we explore the chromo-electromagnetic field strengths under varying matter densities. To ensure accurate measurements, we develop and calibrate two highly improved topological charge operators and evaluate four gradient flow actions. Our results reveal a finite-volume crossover in the regime of the anticipated phase boundary at μ=mπ/2, with both chromoelectric and chromomagnetic field strengths suppressed before recovering and exceeding vacuum values at higher chemical potentials. We find the difference between the squared chromoelectric and chromomagnetic field strengths, E2−B2, to increase in magnitude monotonically with increasing chemical potential. At aμ=0.7, we find an 11% suppression of E2, a relatively small effect. A systematic analysis using sigmoid fits of lattice simulations in the crossover regime is performed to confirm the critical chemical potential obtained from the field structure is in agreement with the phase boundary at mπ/2. These findings provide new insight into non-Abelian ground-state vacuum field structures and offer a foundation for future studies in real QCD.

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References (57)

  1. G. Aarts, Introductory lectures on lattice QCD at nonzero baryon number, J. Phys. Conf. Ser. 706, 022004 (2016).
  2. J. N. Guenther, An overview of the QCD phase diagram at finite T and μ, Proc. Sci. LATTICE2021 (2022) 013. [arXiv:2201.02072].
  3. E. Itou, Lattice results for the equation of state in dense QCD-like theories, Universe 11, 380 (2025).
  4. J. B. Kogut, M. A. Stephanov, D. Toublan, J. J. M. Verbaarschot, and A. Zhitnitsky, QCD—like theories at finite baryon density, Nucl. Phys. B582, 477 (2000).
  5. M. Creutz, Monte Carlo study of quantized SU(2) gauge theory, Phys. Rev. D 21, 2308 (1980).
  6. S. Hands, J. B. Kogut, M.-P. Lombardo, and S. E. Morrison, Symmetries and spectrum of SU(2) lattice gauge theory at finite chemical potential, Nucl. Phys. B558, 327 (1999).
  7. V. V. Braguta, Phase diagram of dense two-color QCD at low temperatures, Symmetry 15, 1466 (2023).
  8. T. Boz, S. Cotter, L. Fister, D. Mehta, and J.-I. Skullerud, Phase transitions and gluodynamics in 2-colour matter at high density, Eur. Phys. J. A 49, 87 (2013).
  9. T. Boz, P. Giudice, S. Hands, and J.-I. Skullerud, Dense two-color QCD towards continuum and chiral limits, Phys. Rev. D 101, 074506 (2020).
  10. N. Astrakhantsev, V. V. Braguta, E. M. Ilgenfritz, A. Y. Kotov, and A. A. Nikolaev, Lattice study of thermodynamic properties of dense QC2D, Phys. Rev. D 102, 074507 (2020).
  11. K. Iida, E. Itou, K. Murakami, and D. Suenaga, Lattice study on finite density QC2D towards zero temperature, J. High Energy Phys. 10 (2024) 022.
  12. A. Begun, V. G. Bornyakov, V. A. Goy, A. Nakamura, and R. N. Rogalyov, Study of two color QCD on large lattices, Phys. Rev. D 105, 114505 (2022).
  13. S. Hands, S. Kim, and J.-I. Skullerud, Deconfinement in dense 2-color QCD, Eur. Phys. J. C 48, 193 (2006).
  14. K. Iida and E. Itou, Velocity of sound beyond the high-density relativistic limit from lattice simulation of dense two-color QCD, Prog. Theor. Exp. Phys. 2022, 111B01 (2022).
  15. R. Abbott, W. Detmold, M. Illa, A. Parreño, R. J. Perry, F. Romero-López, P. E. Shanahan, and M. L. Wagman (NPLQCD Collaboration), QCD constraints on isospin-dense matter and the nuclear equation of state, Phys. Rev. Lett. 134, 011903 (2025).
  16. E. Annala, T. Gorda, A. Kurkela, J. Nättilä, and A. Vuorinen, Evidence for quark-matter cores in massive neutron stars, Nat. Phys. 16, 907 (2020).
  17. L. Brandes, W. Weise, and N. Kaiser, Inference of the sound speed and related properties of neutron stars, Phys. Rev. D 107, 014011 (2023).
  18. K. Fukushima and S. Minato, Speed of sound and trace anomaly in a unified treatment of the two-color diquark superfluid, the pion-condensed high-isospin matter, and the 2SC quark matter, Phys. Rev. D 111, 094006 (2025).
  19. F. Gross et al., 50 years of quantum chromodynamics, Eur. Phys. J. C 83, 1125 (2023).
  20. A. Cucchieri, T. Mendes, O. Oliveira, and P. J. Silva, SU(2) meets SU(3) in lattice-Landau-gauge gluon and ghost propagators, Proc. Sci. LATTICE2007 (2007) 322. [arXiv:0710.0344].
  21. A. Sternbeck, L. von Smekal, D. B. Leinweber, and A. G. Williams, Comparing SU(2) to SU(3) gluodynamics on large lattices, Proc. Sci. LATTICE2007 (2007) 340. [arXiv:0710.1982].
  22. R. Alkofer and L. von Smekal, The Infrared behavior of QCD Green’s functions: Confinement dynamical symmetry breaking, and hadrons as relativistic bound states, Phys. Rep. 353, 281 (2001).
  23. T. Boz, O. Hajizadeh, A. Maas, and J.-I. Skullerud, Finite-density gauge correlation functions in QC2D, Phys. Rev. D 99, 074514 (2019).
  24. B. Alles, M. D’Elia, and M. P. Lombardo, Behaviour of the topological susceptibility in two colour QCD across the finite density transition, Nucl. Phys. B752, 124 (2006).
  25. S. Hands and P. Kenny, Topological fluctuations in dense matter with two colors, Phys. Lett. B 701, 373 (2011).
  26. N. Y. Astrakhantsev, V. V. Braguta, A. Y. Kotov, D. D. Kuznedelev, and A. A. Nikolaev, Lattice study of QCD at finite chiral density: Topology and confinement, Eur. Phys. J. A 57, 15 (2021).
  27. K. Iida, E. Itou, and T.-G. Lee, Two-colour QCD phases and the topology at low temperature and high density, J. High Energy Phys. 01 (2020) 181.
  28. M. P. Lombardo, Topological aspects of dense matter: Lattice studies, Universe 7, 336 (2021).
  29. K. Ishiguro, E. Itou, and K. Iida, Flux tube profiles in two-color QCD at low temperature and high density, Proc. Sci. LATTICE2021 (2022) 063. [arXiv:2111.13067].
  30. S. Cotter, P. Giudice, S. Hands, and J.-I. Skullerud, Towards the phase diagram of dense two-color matter, Phys. Rev. D 87, 034507 (2013).
  31. K. Murakami, E. Itou, and K. Iida, Chemical potential (in)dependence of hadron scatterings in the hadronic phase of QCD-like theories and its applications, J. High Energy Phys. 02 (2024) 152.
  32. S. Hands, Lattice matter, Nucl. Phys. B, Proc. Suppl. 106, 142 (2002).
  33. P. Hasenfratz and F. Karsch, Chemical potential on the lattice, Phys. Lett. 125B, 308 (1983).
  34. J. B. Kogut, H. Matsuoka, M. Stone, H. W. Wyld, S. H. Shenker, J. Shigemitsu, and D. K. Sinclair, Chiral symmetry restoration in baryon rich environments, Nucl. Phys. B225, 93 (1983).
  35. S. Hands, S. Kim, D. Lawlor, A. Lee-Mitchell, and J.-I. Skullerud, Dense QC2D: What’s up with that?!? Proc. Sci. LATTICE2024 (2025) 165. [arXiv:2412.15872].
  36. S. O. Bilson-Thompson, D. B. Leinweber, and A. G. Williams, Highly improved lattice field strength tensor, Ann. Phys. (Amsterdam) 304, 1 (2003).
  37. J. A. Mickley, W. Kamleh, and D. B. Leinweber, Numerical evidence for fractional topological objects in SU(3) gauge theory, Phys. Rev. D 109, 094507 (2024).
  38. P. de Forcrand, M. García Pérez, and I.-O. Stamatescu, Topology of the SU(2) vacuum: A Lattice study using improved cooling, Nucl. Phys. B499, 409 (1997).
  39. P. de Forcrand, M. García Pérez, and I.-O. Stamatescu, Improved cooling algorithm for gauge theories, Nucl. Phys. B, Proc. Suppl. 47, 777 (1996).
  40. M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, J. High Energy Phys. 08 (2010) 071; 03 (2014) 92(E).
  41. C. Morningstar and M. J. Peardon, Analytic smearing of SU(3) link variables in lattice QCD, Phys. Rev. D 69, 054501 (2004).
  42. F. D. R. Bonnet, D. B. Leinweber, and A. G. Williams, General algorithm for improved lattice actions on parallel computing architectures, J. Comput. Phys. 170, 1 (2001).
  43. Y. Tanizaki, A. Tomiya, and H. Watanabe, Lattice gradient flows (de-)stabilizing topological sectors, J. High Energy Phys. 04 (2025) 123.
  44. K. Symanzik, Continuum limit and improved action in lattice theories. 2. O(N) Nonlinear sigma model in perturbation theory, Nucl. Phys. B226, 205 (1983).
  45. Y. Iwasaki, Renormalization group analysis of lattice Theories and improved lattice action. II. Four-dimensional non-Abelian SU(N) gauge model, arXiv:1111.7054.
  46. T. Takaishi, Heavy quark potential and effective actions on blocked configurations, Phys. Rev. D 54, 1050 (1996).
  47. M. García Pérez, A. González-Arroyo, J. R. Snippe, and P. van Baal, Instantons from over-improved cooling, Nucl. Phys. B413, 535 (1994).
  48. P. J. Moran and D. B. Leinweber, Over-improved stout-link smearing, Phys. Rev. D 77, 094501 (2008).
  49. M. Luscher and P. Weisz, Perturbative analysis of the gradient flow in non-abelian gauge theories, J. High Energy Phys. 02 (2011) 051.
  50. S. Datta, S. Gupta, and A. Lytle, Using Wilson flow to study the SU(3) deconfinement transition, Phys. Rev. D 94, 094502 (2016).
  51. S. Borsanyi, R. Kara, Z. Fodor, D. A. Godzieba, P. Parotto, and D. Sexty, Precision study of the continuum SU(3) Yang-Mills theory: How to use parallel tempering to improve on supercritical slowing down for first order phase transitions, Phys. Rev. D 105, 074513 (2022).
  52. R. V. Harlander, Y. Kluth, and F. Lange, The two-loop energy–momentum tensor within the gradient-flow formalism, Eur. Phys. J. C 78, 944 (2018); 79, 858(E) (2019).
  53. T. Iritani, M. Kitazawa, H. Suzuki, and H. Takaura, Thermodynamics in quenched QCD: Energy–momentum tensor with two-loop order coefficients in the gradient-flow formalism, Prog. Theor. Exp. Phys. 2019, 023B02 (2019).
  54. Y. Taniguchi, S. Ejiri, K. Kanaya, M. Kitazawa, H. Suzuki, and T. Umeda (WHOT-QCD Collaboration), Nf=2+1 QCD thermodynamics with gradient flow using two-loop matching coefficients, Phys. Rev. D 102, 014510 (2020); 102, 059903(E) (2020).
  55. www.dirac.ac.uk.
  56. D. Lawlor and J.-I. Skullerud, Dense QC2D. What’s up with that?!? The dataset (2024), 10.5281/zenodo.14201453.
  57. R. F. Hasan, M. Cummins, W. Kamleh, D. Lawlor, D. Leinweber, I. van Schalkwyk, and J.-I. Skullerud, Dataset for “Structure of QC2D ground state fields at nonzero matter densities” (2026), 10.5281/zenodo.19324947.

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Phys. Rev. D 114, 094504– Published 5 November, 2026
  • Received 30 March 2026
  • Accepted 9 October 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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