Universality class of the chiral phase transition in (2+1)-flavor QCD and the fate of effective $U_A(1)$ symmetry at $T_c^0$ in the chiral limit

In the chiral limit of (2+1)-flavor QCD — two massless light quarks and a strange quark at its physical mass — the chiral phase transition is expected to be second order in the 3D O(4) universality class provided the anomalous $U_A(1)$ symmetry remains effectively broken at the transition temperature $T_c^0 \approx 132$ MeV. Neither pillar is established: the critical exponent $\delta$ extracted from chiral-susceptibility peak scaling is too imprecise to confirm the universality class, and whether effective $U_A(1)$ restoration sets in at $T_c^0$ (disconnected chiral susceptibility vanishing) or only well above it is contested between lattice collaborations and discretizations. The problem asks for a first-principles, continuum-limit lattice QCD determination of the order and universality class of the chiral transition, including a controlled extraction of $\delta$ with quantified corrections-to-scaling and regular-term systematics, and a determination of the fate of effective $U_A(1)$ symmetry at $T_c^0$ together with the temperature range of its restoration above $T_c^0$.

Background

Quantum chromodynamics (QCD) with physical light-quark masses undergoes a smooth crossover, rather than a phase transition, from hadronic matter to a quark-gluon plasma at a pseudo-critical temperature of about 156 MeV. In the chiral limit — two exactly massless up and down quarks, strange quark at its physical mass — the spontaneous breaking of chiral symmetry, $SU(2)_L \times SU(2)_R \to SU(2)_V$, must end in a genuine phase transition, and its order and universality class have been debated since Pisarski and Wilczek. Their argument turns on the axial $U_A(1)$ symmetry: the classically massless QCD Lagrangian has a $U_A(1)$ symmetry that the quantum anomaly (topological, instanton-induced fluctuations) explicitly breaks. If $U_A(1)$ remains effectively broken at the transition temperature $T_c^0$ — meaning anomaly effects stay strong enough that $U_A(1)$-breaking observables stay nonzero in the chiral limit, even though the symmetry is never exact — the chiral transition of two massless flavors is predicted to be second order in the universality class of the three-dimensional $O(4)$ ferromagnet; if $U_A(1)$ is effectively restored at $T_c^0$, first-order behavior or a $U(2)\times U(2)$ universality class becomes possible.

The standard analysis uses the chiral condensate (order parameter) $M$ and the chiral susceptibility $\chi_M$. Near an $O(4)$ critical point they obey universal scaling laws $M = h^{1/\delta} f_1(z)$ and $\chi_M = h^{1/\delta-1} f_2(z)/h_0$, with reduced temperature $t=(T-T_c^0)/T_c^0$, external field $h \propto m_l$ (the light-quark mass), $z = t/h^{1/\beta\delta}$, universal critical exponents $\beta$, $\delta$, and universal scaling functions $f_1$, $f_2$. HotQCD (a major U.S.–German lattice-QCD collaboration) determined $T_c^0 = 132^{+3}_{-6}$ MeV from such scaling estimators with continuum extrapolation — but with the $O(4)$ universality class assumed as the scaling ansatz, not derived. The exponent extracted from peak scaling of $\chi_M$ came out as $\delta \simeq 4.8$ with an uncertainty too large to confirm the class, and the inverse-$\delta$ line positions of the $O(4)$, $O(2)$ and $Z_2$ classes differ by less than about 1%, so exponents alone barely discriminate. A further subtlety: staggered lattice fermions (a widely used lattice discretization of the Dirac operator, e.g. the highly improved staggered quark, HISQ, action) realize only an $O(2)$ subgroup of $O(4)$ at any finite lattice spacing; $O(4)$ is recovered only in the continuum limit.

Whether $U_A(1)$ is effectively restored at $T_c^0$ is probed by the disconnected part of the light chiral susceptibility, $\chi_{l,\mathrm{disc}}$ (the piece of $\chi_M$ arising from quark-line disconnected diagrams): in a chirally restored phase $\chi_\pi - \chi_\delta = \chi_{l,\mathrm{disc}}$, where $\chi_\pi$ and $\chi_\delta$ are the susceptibilities of the pion and of the scalar isovector $\delta$ meson, which $U_A(1)$ relates; $\chi_{l,\mathrm{disc}}$ vanishes in the chiral limit exactly when $U_A(1)$ is effectively restored. Equivalently one studies the eigenvalue density $\rho(\lambda)$ of the Dirac operator near zero eigenvalue: the Banks–Casher relation ties the chiral condensate to $\rho(0)$, and near-zero eigenvalues control $\chi_\pi-\chi_\delta$. The current lattice evidence is contradictory: staggered-fermion and overlap-valence studies find $U_A(1)$ still broken at and above $T_c$ toward the chiral limit, while chiral-fermion formulations (Möbius domain-wall and overlap fermions, which realize chiral symmetry essentially exactly at finite lattice spacing) find $U_A(1)$-breaking consistent with zero above $T_c$. Resolving this conflict and confirming or refuting the $O(4)$ scenario in the continuum limit is the open problem.

Problem Statement

Determine, from first-principles lattice QCD with controlled thermodynamic, continuum and chiral extrapolations, for (2+1)-flavor QCD with the strange quark at its physical mass and the light quarks taken to zero mass:

(i) the order and universality class of the chiral phase transition — in particular a controlled determination of the critical exponent $\delta$, with quantified statistical and systematic uncertainties including corrections-to-scaling (sub-leading deviations from the leading critical scaling laws) and regular (non-singular, analytic) background contributions, sufficient to confirm or refute the assumed $O(4)$ universality class and to solidify the continuum-extrapolated $T_c^0$; and

(ii) the fate of effective $U_A(1)$ symmetry at the chiral transition — whether the disconnected chiral susceptibility $\chi_{l,\mathrm{disc}}$ (equivalently $\chi_\pi - \chi_\delta$, or the near-zero Dirac eigenvalue density) vanishes or remains nonzero at $T_c^0$ as $m_l \to 0$ — together with the temperature range above $T_c^0$ over which effective $U_A(1)$ restoration sets in.

These two determinations are two faces of one objective: by the Pisarski–Wilczek mechanism the $U_A(1)$ fate at $T_c^0$ is the deciding ingredient of the universality class, so the order and universality class of the chiral-limit transition and the role of the axial anomaly in it are linked, and both hold in the continuum and thermodynamic limits.

References

1. O. Kaczmarek, F. Karsch, A. Lahiri, L. Mazur, C. Schmidt, 'QCD phase transition in the chiral limit' (2020), arXiv:2003.07920, doi:10.48550/arXiv.2003.07920, https://arxiv.org/abs/2003.07920
2. R. D. Pisarski, F. Wilczek, 'Remarks on the chiral phase transition in chromodynamics', Phys. Rev. D 29, 338 (1984), doi:10.1103/PhysRevD.29.338, https://doi.org/10.1103/PhysRevD.29.338
3. H.-T. Ding, P. Hegde, O. Kaczmarek, F. Karsch, A. Lahiri, S.-T. Li, S. Mukherjee, H. Ohno, P. Petreczky, C. Schmidt, P. Steinbrecher [HotQCD], 'Chiral phase transition temperature in (2+1)-Flavor QCD', Phys. Rev. Lett. 123, 062002 (2019), arXiv:1903.04801, doi:10.1103/PhysRevLett.123.062002, https://arxiv.org/abs/1903.04801
4. H.-T. Ding, S.-T. Li, S. Mukherjee, A. Tomiya, X.-D. Wang, Y. Zhang, 'Correlated Dirac Eigenvalues and Axial Anomaly in Chiral Symmetric QCD', Phys. Rev. Lett. 126, 082001 (2021), arXiv:2010.14836, doi:10.1103/PhysRevLett.126.082001, https://arxiv.org/abs/2010.14836
5. O. Kaczmarek, L. Mazur, S. Sharma, 'Eigenvalue spectra of QCD and the fate of U_A(1) breaking towards the chiral limit', Phys. Rev. D 104, 094518 (2021), arXiv:2102.06136, doi:10.1103/PhysRevD.104.094518, https://arxiv.org/abs/2102.06136
6. H.-T. Ding, O. Kaczmarek, F. Karsch, S.-T. Li, S. Mukherjee, A. Tomiya, Y. Zhang, 'Dirac Eigenvalue spectrum of Nf=2+1 QCD towards the chiral limit using HISQ fermions', PoS LATTICE2019, 251 (2020), arXiv:2001.05217, doi:10.22323/1.363.0251, https://arxiv.org/abs/2001.05217
7. S. Aoki, Y. Aoki, G. Cossu, H. Fukaya, S. Hashimoto, T. Kaneko, C. Rohrhofer, K. Suzuki [JLQCD], 'Study of the axial U(1) anomaly at high temperature with lattice chiral fermions', Phys. Rev. D 103, 074506 (2021), arXiv:2011.01499, doi:10.1103/PhysRevD.103.074506, https://arxiv.org/abs/2011.01499
8. G. Aarts, C. Allton, R. Bignell, B. Jäger, S. Kim, J.-I. Skullerud, A. Smecca, 'On the effective restoration of U(1)_A symmetry at finite temperature' (2026), arXiv:2604.11916, doi:10.48550/arXiv.2604.11916, https://arxiv.org/abs/2604.11916
9. S. Aoki, H. Fukaya, Y. Taniguchi, 'Chiral symmetry restoration, the eigenvalue density of the Dirac operator, and the axial U(1) anomaly at finite temperature', Phys. Rev. D 86, 114512 (2012), arXiv:1209.2061, doi:10.1103/PhysRevD.86.114512, https://arxiv.org/abs/1209.2061
10. M. Giordano, 'Comment on "Chiral symmetry restoration, the eigenvalue density of the Dirac operator, and the axial U(1) anomaly at finite temperature"', Phys. Rev. D 114, 018501 (2026), arXiv:2510.24403, doi:10.1103/xs2n-c7t9, https://arxiv.org/abs/2510.24403
11. S. Aoki, H. Fukaya, 'Reply to "Comment on "Chiral symmetry restoration, the eigenvalue density of the Dirac operator, and the axial U(1) anomaly at finite temperature""', Phys. Rev. D 114, 018502 (2026), arXiv:2606.10462, doi:10.1103/dk71-9mzv, https://arxiv.org/abs/2606.10462
12. S. Mitra, F. Karsch, S. Sharma, 'Towards a parameter-free determination of critical exponents and chiral phase transition temperature in QCD', PoS LATTICE2024, 187 (2025), arXiv:2411.15988, doi:10.22323/1.466.0187, https://arxiv.org/abs/2411.15988
13. S. Mitra, F. Karsch, 'Towards a Parameter-Free Analysis of the QCD Chiral Phase Transition and Its Universal Critical behavior', Springer Proc. Phys. 432, 781 (2026), arXiv:2501.13653, doi:10.1007/978-981-95-1513-4_178, https://arxiv.org/abs/2501.13653
14. S. Mitra, J. Goswami, F. Karsch, 'Universal and non-universal finite-volume effects in the vicinity of chiral phase transition in (2+1)-flavor QCD' (2026), arXiv:2602.23118, doi:10.48550/arXiv.2602.23118, https://arxiv.org/abs/2602.23118
15. A. Yu. Kotov, M. P. Lombardo, A. Trunin, 'QCD transition at the physical point, and its scaling window from twisted mass Wilson fermions', Phys. Lett. B 823, 136749 (2021), arXiv:2105.09842, doi:10.1016/j.physletb.2021.136749, https://arxiv.org/abs/2105.09842
16. H.-T. Ding, 'Lattice QCD at finite temperature and density' (2026), arXiv:2603.16230, doi:10.48550/arXiv.2603.16230, https://arxiv.org/abs/2603.16230
17. S. Aoki, Y. Aoki, H. Fukaya, S. Hashimoto, I. Kanamori, T. Kaneko, Y. Nakamura, K. Suzuki, D. Ward [JLQCD], 'Chiral susceptibility and axial U(1) anomaly near the (pseudo-)critical temperature', PoS LATTICE2023, 184 (2024), arXiv:2401.06459, doi:10.22323/1.453.0184, https://arxiv.org/abs/2401.06459
18. Y. Aoki, H. Fukaya, S. Hashimoto, I. Kanamori, Y. Nakamura, C. Rohrhofer, K. Suzuki, D. Ward [JLQCD], 'Symmetry of screening masses of mesons in two-flavor lattice QCD at high temperatures', Phys. Rev. D 111, 114506 (2025), arXiv:2501.12675, doi:10.1103/yh8j-q93c, https://arxiv.org/abs/2501.12675