Order of the QCD chiral phase transition and effective $U(1)_A$ restoration at the transition: the Pisarski–Wilczek open questions

Pisarski and Wilczek's 1984 renormalization-group analysis of the linear sigma model concluded that the finite-temperature chiral transition in QCD is first order for three or more massless flavors, and second order with $O(4)$ exponents for two flavors only if the axial $U(1)_A$ anomaly remains operative at the transition temperature $T_{\rm ch}$. The authors explicitly identified what their arguments could not settle: the actual magnitude of the anomaly coupling $c(T_{\rm ch})$, the possible existence of nonperturbative three-dimensional fixed points invisible to the $\epsilon$ expansion, the effect of physical quark masses on the first-order transition, the instanton density near $T_{\rm ch}$, and the need for dedicated numerical studies. Four decades of subsequent work have largely settled two of the five open issues: physical-mass QCD undergoes an analytic crossover rather than a phase transition, and topological/instanton density is strongly suppressed above the transition. The central core remains open and actively contested: the order of the chiral transition in the $N_f=2$ and $N_f=3$ chiral limits in the continuum (recent lattice, functional-RG and Dyson–Schwinger results challenge the first-order prediction, and a candidate three-dimensional fixed point has been found), the effective restoration of $U(1)_A$ at $T_{\rm ch}$ (different lattice collaborations reach opposite conclusions), and the resulting Columbia-plot critical surface.

Background

Quantum chromodynamics (QCD), the theory of the strong interaction, changes character at a temperature of order 155 MeV: the light quarks shed the chiral condensate $\langle\bar{q}q\rangle$ that binds them into hadrons, and the theory moves toward a chirally symmetric phase. With $N_f$ flavors of exactly massless quarks, the classical global flavor symmetry is $G_f=U(1)_A\times SU(N_f)_L\times SU(N_f)_R$, where the two $SU(N_f)$ factors rotate left- and right-handed quark fields independently; at low temperature it is spontaneously broken to the vectorial $SU(N_f)$, and its restoration at a temperature $T_{\rm ch}$ can be a genuine phase transition whose order is a property of QCD itself.

Two pieces of physics complicate this picture. First, the axial $U(1)_A$ factor is violated by the quantum chiral anomaly: instantons — classical, topologically nontrivial solutions of the Euclidean gauge-field equations carrying an integer topological charge — generate a $2N_f$-quark interaction of the form $c(\det\Phi+\det\Phi^\dagger)$, where $\Phi$ is the complex $N_f\times N_f$ order-parameter matrix $\Phi\sim\bar{q}(1+\gamma_5)q$ transforming as $\Phi\to e^{i\alpha}U_L\Phi U_R^\dagger$. This determinant interaction splits the $\eta'$ from the pion octet, and after removing $U(1)_A$ the exact quantum flavor symmetry is only $G_f'=Z_A(N_f)\times SU(N_f)_L\times SU(N_f)_R$, where $Z_A(N_f)$ is a discrete axial subgroup. Second, quarks have nonzero bare masses, which enter the low-energy theory as a symmetry-breaking term $\mathrm{tr}(M\Phi)$ acting like an external magnetic field on the order parameter.

Pisarski and Wilczek (1984) analyzed the order of this transition with renormalization-group (RG) methods. Near a continuous transition, static modes dominate, so the critical behavior is governed by a three-dimensional effective theory — the most general renormalizable (Ginzburg–Landau, or linear sigma model) Lagrangian for $\Phi$, containing two independent quartic couplings $g_1$ and $g_2$. They computed the one-loop beta functions of $g_1,g_2$ in $d=4-\epsilon$ dimensions (the $\epsilon$ expansion) and searched for infrared-stable fixed points: a stable fixed point means a second-order transition with universal critical exponents, while the absence of one means the transition is driven first order by fluctuations alone (a fluctuation-induced first-order transition, unanticipated by mean-field theory). Their conclusions: for $N_f\ge 3$ no infrared-stable fixed point exists at order $\epsilon$, so the chiral transition is first order even if the anomaly coupling $c(T)$ is strongly reduced near $T_{\rm ch}$; for $N_f=2$ the reduced symmetry $G_f'\cong O(4)$ gives a second-order transition in the familiar $O(4)$ universality class (that of the 3D Heisenberg-type magnet with four-component order parameter), but only if the anomaly is still operative at $T_{\rm ch}$, i.e. $c(T_{\rm ch})\approx c(0)$, keeping the $\eta'$ heavy; if instead instanton suppression makes $c(T_{\rm ch})$ small, the enlarged symmetry can change the order (possibly first order, or $O(2)\times O(4)$-type criticality).

For the anomaly strength they relied on the semiclassical analysis of Gross, Pisarski and Yaffe (1981): the instanton density $d_I(T)$, to which $c(T)$ is proportional, vanishes as $T\to\infty$ because electric color fields are Debye screened, and decreases smoothly, falling below roughly $10\%$ of its zero-temperature value at $T\sim O(\Lambda)$. Pisarski and Wilczek stressed that this only 'suggests but does not prove' that $c(T_{\rm ch})\ll c(0)$, and that numerical simulations are required to settle it; they explicitly assumed the absence of any nonperturbative infrared-stable fixed point invisible to the $\epsilon$ expansion; and they left the effect of the observed quark masses as a 'prejudice' to be decided by detailed investigation.

The mass question evolved into the 'Columbia plot': the phase diagram of the order of the QCD transition in the plane of light-quark mass $m_{u,d}$ versus strange-quark mass $m_s$, so named after the Columbia group's early simulations (Brown et al. 1990). Modern work uses lattice QCD — Monte Carlo simulation on a discrete space-time grid — where the fermion discretization matters: staggered fermions carry a residual 'taste' symmetry breaking and Wilson fermions break chiral symmetry at finite lattice spacing, both of which can mimic or destroy first-order behavior at coarse resolution, while domain-wall and overlap fermions realize good or exact chiral symmetry at large computational cost. Key observables are the chiral condensate and susceptibility; the difference $\chi_\pi-\chi_\delta$ of susceptibilities in the pion and isovector-scalar meson channels, which vanishes when $U(1)_A$ is effectively restored; the density of near-zero eigenvalues of the Dirac operator (which, through Banks–Casher-type relations — identities relating the Dirac eigenvalue density at the origin to symmetry-breaking observables — controls both chiral-symmetry breaking and $U(1)_A$ breaking); and the topological susceptibility, measuring fluctuations of the topological charge carried by instanton-like configurations.

As of 2026, two things are settled: with physical quark masses the QCD transition is an analytic crossover, not a phase transition (Aoki et al. 2006) — so the first-order transition, if real, lives only at smaller masses — and the topological/instanton content is strongly suppressed above the transition. What remains open is the heart of the 1984 paper: the order of the chiral transition in the $N_f=2$ and $N_f=3$ chiral limits (the limits of exactly massless quarks) in the continuum, the fate of $U(1)_A$ at $T_{\rm ch}$, the possible existence of nonperturbative three-dimensional fixed points beyond the $\epsilon$ expansion, and the resulting shape of the Columbia plot's critical surface.

Problem Statement

Determine, nonperturbatively, the character of the finite-temperature chiral phase transition in QCD with $N_c=3$ colors and the role of the axial $U(1)_A$ anomaly in it, by resolving the following linked questions:

(1) Effective anomaly strength at the transition. Establish whether the $U(1)_A$-breaking coupling at the chiral transition temperature $T_{\rm ch}$ — the coefficient $c(T_{\rm ch})$ of the determinant interaction $c(\det\Phi+\det\Phi^\dagger)$, where $\Phi$ is the $N_f\times N_f$ chiral order-parameter field, also denoted $K(T_{\rm ch})$ in meson-field parameterizations of the same interaction — satisfies $c(T_{\rm ch})\ll c(0)$ or remains appreciable, the comparison being between the same effective coupling evaluated at the two temperatures in one fixed normalization and renormalization convention; that is, whether $U(1)_A$ is effectively restored at $T_{\rm ch}$, as probed by anomaly-sensitive observables such as the susceptibility difference $\chi_\pi-\chi_\delta$ and the density of near-zero Dirac eigenvalues in the window near $T_{\rm ch}$. In the strict chiral limit a caveat applies: by the Aoki–Fukaya–Taniguchi result (Aoki, Fukaya and Taniguchi 2012, ref. 7), the eigenvalue density at the origin and its first two derivatives vanish in the two-flavor chiral limit, making the anomaly invisible in scalar/pseudoscalar susceptibilities, so the restoration question cannot be decided by the vanishing of $\chi_\pi-\chi_\delta$ in that limit alone and must be posed at small nonzero quark mass near $T_{\rm ch}$ or through the effective coupling itself. Which physical configurations (a dilute gas of well-separated instantons, or near-zero Dirac modes of other origin) produce the near-zero-mode density near $T_{\rm ch}$ is a closely related open question.

(2) Order and universality class of the chiral limits. Determine the order of the chiral transition in continuum QCD with $N_f=2$ and $N_f=3$ massless quarks: first order, as the one-loop $\epsilon$ expansion predicts for $N_f\ge3$ (and as would also follow for $N_f=2$ if $U(1)_A$ is effectively restored at $T_{\rm ch}$), or second order — with $O(4)$ exponents for $N_f=2$ when the anomaly is operative, or possibly in a new universality class for $N_f=3$. The order is that of the continuum theory at exactly massless quarks; since the chiral and continuum limits do not commute, any determination must state its limiting procedure. This includes settling whether the three-dimensional $SU(N_f)_L\times SU(N_f)_R$ (or, with restored $U(1)_A$, enlarged) Ginzburg–Landau theories possess nonperturbative infrared-stable fixed points that the perturbative $\epsilon$ expansion cannot see.

(3) Critical region in the quark-mass plane (related consequence). Given that the physical point is known to be a crossover, whether, and for which quark masses, a first-order chiral transition region or chiral critical endpoint survives at small quark masses, as approached through the Columbia plot in the $(m_{u,d},m_s)$ plane (Brown et al. 1990). This descends from the source's remark on the observed quark masses and is a closely related open consequence of questions (1)–(2); any first-order region must terminate consistently on the chiral-limit order.

Questions (1) and (2) are one problem because they are different projections of a single question — what is the order and universality class of the QCD chiral transition, and what (anomaly strength, critical fluctuations, quark masses) determines it — and they constrain one another: the order for $N_f=2$ depends on the anomaly strength at the same transition, and any small-mass first-order region terminates on the chiral-limit order. The order and universality class in both chiral limits must be established nonperturbatively for the continuum theory at exactly massless quarks, with controlled systematics and a stated limiting procedure, and the fate of $U(1)_A$ at $T_{\rm ch}$ must be settled by anomaly-sensitive observables — all consistently with one another. Locating the small-mass first-order region of question (3) in the continuum limit is a related open problem whose resolution must be consistent with the answers to (1)–(2).

References

1. R. D. Pisarski and 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 (primary source)
2. D. J. Gross, R. D. Pisarski and L. G. Yaffe, 'QCD and Instantons at Finite Temperature', Rev. Mod. Phys. 53, 43 (1981). DOI: 10.1103/RevModPhys.53.43, https://doi.org/10.1103/RevModPhys.53.43
3. J. B. Kogut, M. Snow and M. Stone, 'Mean Field and Monte Carlo Studies of SU(N) Chiral Models in Three-dimensions', Nucl. Phys. B200 [FS4], 211 (1982). DOI: 10.1016/0550-3213(82)90065-7, https://doi.org/10.1016/0550-3213(82)90065-7
4. F. R. Brown, F. P. Butler, H. Chen, N. H. Christ, Z. Dong, W. Schaffer, L. I. Unger and A. Vaccarino, 'On the existence of a phase transition for QCD with three light quarks', Phys. Rev. Lett. 65, 2491 (1990). DOI: 10.1103/PhysRevLett.65.2491, https://doi.org/10.1103/PhysRevLett.65.2491
5. P. Calabrese and P. Parruccini, 'Five-loop $\epsilon$ expansion for $U(n)\times U(m)$ models: finite-temperature phase transition in light QCD', JHEP 05 (2004) 018. DOI: 10.1088/1126-6708/2004/05/018, arXiv:hep-ph/0403140, https://arxiv.org/abs/hep-ph/0403140
6. Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo, 'The Order of the quantum chromodynamics transition predicted by the standard model of particle physics', Nature 443, 675 (2006). DOI: 10.1038/nature05120, https://doi.org/10.1038/nature05120
7. S. Aoki, H. Fukaya and 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). DOI: 10.1103/PhysRevD.86.114512, arXiv:1209.2061, https://arxiv.org/abs/1209.2061
8. A. Bazavov, T. Bhattacharya, M. I. Buchoff, M. Cheng, N. H. Christ, H.-T. Ding, R. Gupta, P. Hegde, C. Jung, F. Karsch, Zh. Lin, R. D. Mawhinney, S. Mukherjee, P. Petreczky, R. A. Soltz, P. M. Vranas and H. Yin (HotQCD Collaboration), 'Chiral transition and $U(1)_A$ symmetry restoration from lattice QCD using domain wall fermions', Phys. Rev. D 86, 094503 (2012). DOI: 10.1103/PhysRevD.86.094503, arXiv:1205.3535, https://arxiv.org/abs/1205.3535
9. G. Cossu, S. Aoki, H. Fukaya, S. Hashimoto, T. Kaneko, H. Matsufuru and J.-I. Noaki, 'Finite temperature study of the axial U(1) symmetry on the lattice with overlap fermion formulation', Phys. Rev. D 87, 114514 (2013). DOI: 10.1103/PhysRevD.87.114514, arXiv:1304.6145, https://arxiv.org/abs/1304.6145
10. M. I. Buchoff, M. Cheng, N. H. Christ, H.-T. Ding, C. Jung, F. Karsch, Zh. Lin, R. D. Mawhinney, S. Mukherjee, P. Petreczky, D. Renfrew, C. Schroeder, P. M. Vranas and H. Yin (HotQCD Collaboration), 'QCD chiral transition, U(1)A symmetry and the dirac spectrum using domain wall fermions', Phys. Rev. D 89, 054514 (2014). DOI: 10.1103/PhysRevD.89.054514, arXiv:1309.4149, https://arxiv.org/abs/1309.4149
11. A. Tomiya, G. Cossu, S. Aoki, H. Fukaya, S. Hashimoto, T. Kaneko and J. Noaki, 'Evidence of effective axial U(1) symmetry restoration at high temperature QCD', Phys. Rev. D 96, 034509 (2017). DOI: 10.1103/PhysRevD.96.034509, arXiv:1612.01908, https://arxiv.org/abs/1612.01908
12. A. Bazavov, H.-T. Ding, P. Hegde, F. Karsch, E. Laermann, S. Mukherjee, P. Petreczky and C. Schmidt, 'Chiral phase structure of three flavor QCD at vanishing baryon number density', Phys. Rev. D 95, 074505 (2017). DOI: 10.1103/PhysRevD.95.074505, arXiv:1701.03548, https://arxiv.org/abs/1701.03548
13. Y. Kuramashi, Y. Nakamura, H. Ohno and S. Takeda, 'Nature of the phase transition for finite temperature $N_f=3$ QCD with nonperturbatively O(a) improved Wilson fermions at $N_t=12$', Phys. Rev. D 101, 054509 (2020). DOI: 10.1103/PhysRevD.101.054509, arXiv:2001.04398, https://arxiv.org/abs/2001.04398
14. R. D. Pisarski and F. Rennecke, 'Multi-instanton contributions to anomalous quark interactions', Phys. Rev. D 101, 114019 (2020). DOI: 10.1103/PhysRevD.101.114019, arXiv:1910.14052 (preprint posted under the title 'The Myriad Uses of Instantons'), https://arxiv.org/abs/1910.14052
15. F. Cuteri, O. Philipsen and A. Sciarra, 'On the order of the QCD chiral phase transition for different numbers of quark flavours', JHEP 11 (2021) 141. DOI: 10.1007/JHEP11(2021)141, arXiv:2107.12739, https://arxiv.org/abs/2107.12739
16. G. Fejös, 'Second-order chiral phase transition in three-flavor quantum chromodynamics?', Phys. Rev. D 105, L071506 (2022). DOI: 10.1103/PhysRevD.105.L071506, arXiv:2201.07909, https://arxiv.org/abs/2201.07909
17. L. Dini, P. Hegde, F. Karsch, A. Lahiri, C. Schmidt and S. Sharma, 'Chiral phase transition in three-flavor QCD from lattice QCD', Phys. Rev. D 105, 034510 (2022). DOI: 10.1103/PhysRevD.105.034510, arXiv:2111.12599, https://arxiv.org/abs/2111.12599
18. Y.-C. Chen, T.-W. Chiu and T.-H. Hsieh, 'Topological susceptibility in finite temperature QCD with physical (u/d, s, c) domain-wall quarks', Phys. Rev. D 106, 074501 (2022). DOI: 10.1103/PhysRevD.106.074501, arXiv:2204.01556, https://arxiv.org/abs/2204.01556
19. J. Bernhardt and C. S. Fischer, 'QCD phase transitions in the light quark chiral limit', Phys. Rev. D 108, 114018 (2023). DOI: 10.1103/PhysRevD.108.114018, arXiv:2309.06737, https://arxiv.org/abs/2309.06737
20. R. D. Pisarski and F. Rennecke, 'Conjectures about the Chiral Phase Transition in QCD from Anomalous Multi-Instanton Interactions', Phys. Rev. Lett. 132, 251903 (2024). DOI: 10.1103/PhysRevLett.132.251903, arXiv:2401.06130 (preprint posted under the title 'The chiral phase transition and the axial anomaly'), https://arxiv.org/abs/2401.06130
21. S. Aoki, Y. Aoki, H. Fukaya, S. Hashimoto, I. Kanamori, T. Kaneko, Y. Nakamura, C. Rohrhofer, K. Suzuki and D. Ward (JLQCD Collaboration), 'Axial U(1) symmetry near the pseudocritical temperature in $N_f=2+1$ lattice QCD with chiral fermions', PoS LATTICE2023 (2024) 185. DOI: 10.22323/1.453.0185, arXiv:2401.14022, https://arxiv.org/abs/2401.14022
22. F. Giacosa, Gy. Kovács, P. Kovács, R. D. Pisarski and F. Rennecke, 'Anomalous $U(1)_A$ couplings and the Columbia plot', Phys. Rev. D 111, 016014 (2025). DOI: 10.1103/PhysRevD.111.016014, arXiv:2410.08185, https://arxiv.org/abs/2410.08185
23. A. Sciarra, 'The chiral phase transition in the 3D Columbia plot', arXiv:2512.18393 (2025). https://arxiv.org/abs/2512.18393