Existence and order of the finite-temperature chiral phase transition in $N_f=8$ QCD

The $SU(3)$ gauge theory with $N_f=8$ mass-degenerate fundamental fermions is the benchmark for near-conformal, slowly running ("walking") strong dynamics and a candidate for the lower edge of the QCD conformal window; whether it undergoes a finite-temperature chiral phase transition, and of what order, bears on walking-technicolor model building and on electroweak baryogenesis, which requires a strongly first-order thermal transition. A LatKMI Collaboration study of its thermodynamics with Highly Improved Staggered Quark (HISQ) fermions (tree-level Symanzik gauge action, $24^3\times12$ and $32^3\times16$ lattices) found a chiral-susceptibility peak at bare gauge coupling $\beta_L=3.08$ and a step-function-like susceptibility ratio $R_\pi=\chi_{\mathrm{chiral}}/\chi_\pi$, suggesting a transition or crossover, but the authors stated that the existence of the transition was not conclusive: the strong-coupling runs used Hybrid Molecular Dynamics without Metropolis tests (unquantified systematics for $\beta_L\lesssim3.5$), no zero-temperature Wilson-flow scale setting converted the critical coupling into a physical temperature, small fermion masses were threatened by unphysical bulk lattice-artifact transitions, and no eigenvalue or finite-size evidence fixed the order. No subsequent study has completed these items: independent discretizations (nHYP-smeared and unimproved staggered, domain-wall, stout staggered) are obstructed by bulk transitions before the chiral limit is reached, and the most recent Roberge-Weiss analysis (2026) reports evidence for infrared conformality while not excluding the chirally broken alternative. The problem is to determine, from first principles and with controlled systematics, whether the chiral limit of $N_f=8$ QCD undergoes a finite-temperature chiral phase transition, and if so its critical temperature and order, or to establish infrared conformality and the absence of any such transition.

Background

QCD-like gauge theories with many fermion flavors probe strongly coupled dynamics beyond the Standard Model. In an $SU(3)$ gauge theory with $N_f$ massless fermions in the fundamental representation, asymptotic freedom makes the coupling weak at short distances, but for sufficiently large $N_f$ the running coupling instead approaches an infrared fixed point, above which the theory becomes scale-invariant (conformal) and chiral symmetry is never spontaneously broken; this range of $N_f$ is the conformal window, and its lower edge $N_f^*$ is a fully non-perturbative quantity. Just below $N_f^*$ the coupling runs very slowly over a broad range of scales ("walking" dynamics), which produces a light flavor-singlet scalar and makes such theories candidates for walking-technicolor and composite-Higgs models of electroweak symmetry breaking, in which the Higgs boson is a bound state of new strong fermions. $N_f=8$ has long been the prime candidate for sitting just below the conformal edge. Finite-temperature physics then provides a sharp diagnostic: if the theory is chirally broken at low temperature (nonzero quark condensate $\langle\bar\psi\psi\rangle$), heating must restore chiral symmetry, and the universality argument of Pisarski and Wilczek predicts a first-order chiral transition for three or more massless flavors. A strongly first-order thermal transition in a walking-technicolor electroweak sector would also supply the departure from thermal equilibrium required by the Sakharov conditions for electroweak baryogenesis, the scenario in which the cosmic baryon asymmetry is generated at the electroweak phase transition; this was the motivation stated by Aoki et al. (2018). If instead $N_f=8$ lies inside the conformal window, chiral symmetry is unbroken at all temperatures and no thermal chiral transition exists. On the lattice, finite temperature is imposed by making the Euclidean time direction compact: for a lattice of size $N_s^3\times N_t$ with spacing $a(\beta_L)$, where $\beta_L=6/g^2$ is the bare gauge coupling, the temperature is $T=1/(N_t a(\beta_L))$, and varying $\beta_L$ at fixed $N_t$ traces a line of constant physics in temperature. Thermal transitions are located through the chiral condensate $\langle\bar\psi\psi\rangle$; the chiral susceptibility $\chi_{\mathrm{chiral}}$ (its derivative with respect to the quark mass), which peaks at a chiral transition; the pion susceptibility $\chi_\pi$ (the strength of the pseudoscalar correlation function); the dimensionless ratio $R_\pi=\chi_{\mathrm{chiral}}/\chi_\pi$, which tends to vanish in the chirally broken phase and to approach unity in the chirally restored phase, making it a sensitive chiral-transition diagnostic; and the Polyakov loop (the traced line operator wrapping the compact time direction), a deconfinement indicator sensitive to the liberation of quark degrees of freedom. Aoki et al. (2018) used the Highly Improved Staggered Quark (HISQ) fermion discretization, a staggered-fermion formulation with suppressed lattice artifacts, together with the tree-level Symanzik gauge action (a gluon action improved to reduce cutoff effects). In Hybrid Monte Carlo (HMC), a molecular-dynamics gauge-field proposal is followed by a Metropolis accept/reject step that makes the algorithm exact; in Hybrid Molecular Dynamics (HMD) that test is omitted, so the generated ensemble is only approximately correct unless the induced systematic error is quantified. The Wilson (gradient) flow is a smoothing evolution of the gauge fields in a fictitious flow time that defines renormalized reference scales such as $t_0$ and permits smearing before measuring Dirac eigenmodes. A bulk transition is a strongly coupled, unphysical lattice-artifact phase transition, distinct from the physical thermal transition, which competes with it at strong coupling and blocks small-mass simulations. The earlier finite-temperature literature on the $N_f=8$ theory was inconclusive and disagreed about the order. Brown et al. (1992), with unimproved staggered fermions, found a first-order transition at small mass but warned of strong-coupling lattice systematics; Jin and Mawhinney (2010), with an improved staggered action, also found evidence for a first-order transition. Deuzeman, Lombardo and Pallante (2008) established the zero-temperature walking regime and the bulk-transition hazard; Miura, Lombardo and Pallante (2012) and Miura and Lombardo (2013) collected thermal critical couplings for $N_f=0,4,6,8$ and estimated the conformal-window edge from their $N_f$-dependence as $N_f^*\simeq11.1\pm1.6$, placing $N_f=8$ below the edge, i.e. chirally broken (a view reiterated by Lombardo et al. in the CPOD2014 proceedings). The Lattice Strong Dynamics (LSD) Collaboration, in work contemporary with the LatKMI study (Schaich, Cheng, Hasenfratz and Petropoulos 2012; Schaich, Hasenfratz and Rinaldi, in the same SCGT15 proceedings volume), used nHYP-smeared staggered fermions (a hypercubic-blocking-smeared staggered formulation) and showed that finite-temperature transitions at nonzero mass run into the strongly coupled bulk phase as the mass decreases, requiring $N_t>24$ to directly confirm spontaneous chiral symmetry breaking. LatKMI's own zero-temperature program found walking signals and a light flavor-singlet scalar in the $N_f=8$ spectrum (Aoki et al. 2017). The LatKMI Collaboration's SCGT15 proceedings contribution "Thermodynamics in 8-Flavor QCD" (Aoki et al. 2018, presented by K. Miura) simulated $24^3\times12$ and $32^3\times16$ lattices with HISQ fermions, the tree-level Symanzik gauge action, and lattice fermion masses $a m_f=0.02$ and $0.04$, and varied $\beta_L\in[2.8,4.0]$ and measured $\langle\bar\psi\psi\rangle$, $\chi_{\mathrm{chiral}}$, $R_\pi$ and the Polyakov loop. They found a peak of $\chi_{\mathrm{chiral}}$ at $\beta_L=3.08$, suggesting a critical coupling $\beta_L^c\simeq3.08$; $R_\pi$ became more step-function-like as the fermion mass decreased, with an inflection point emerging at $a m_f=0.02$ on the $24^3\times12$ lattice; and the Polyakov loop began deviating from zero only at $\beta_L=3.16$, slightly above the chiral peak. They read the signals as consistent with a chirally broken $N_f=8$ theory undergoing a chiral phase transition or crossover, but stated explicitly that the results were preliminary, that the existence of the chiral phase transition in 8-flavor QCD would not be conclusive, and that four items were mandatory next steps: quantifying the HMD systematics in the strong-coupling region ($\beta_L\lesssim3.5$, where the HISQ HMC acceptance ratio collapses); zero-temperature Wilson-flow scale setting at $\beta_L^c$ to convert the critical coupling into a physical temperature; simulations at smaller fermion masses without contamination by unphysical bulk transitions; and the finite-temperature Dirac eigenvalue spectrum with Wilson-flow smearing. Subsequent work has not closed any of these items. The LSD Collaboration's systematic study (Appelquist et al. 2019) found the $N_f=8$ thermal transition line merging into the bulk transition at $a m\simeq0.0025$ on lattices up to $48^3\times24$, which prevents a direct confirmation of spontaneous chiral symmetry breaking. Cuteri, Philipsen and Sciarra (2021) mapped the chiral critical surface of unimproved-staggered $N_f=2$-$8$ theories using tricritical scaling (scaling governed by a tricritical point where first-order and second-order transition lines meet), finding evidence for a second-order chiral transition in the continuum for all $N_f\le6$, possibly extending to a conformal-window onset at $9\lesssim N_f^*\lesssim12$; Klinger, Kaiser, Philipsen and Schaible (2026) report that for $N_f\ge8$ the thermal critical structure merges into the bulk regime and that up to $N_\tau\le16$ no thermal transition in the chiral limit connects to the continuum. Hasenfratz, Rebbi and Witzel (2023) computed the gradient-flow step-scaling function (the lattice analog of the renormalization-group $\beta$-function) for $N_f=8$ with domain-wall fermions (a lattice fermion formulation with nearly exact chiral symmetry) but their strong-coupling reach is limited by bulk phenomena; Hasenfratz and Witzel (2025) map a strong-coupling symmetric-mass-generation phase (a chirally symmetric phase in which the fermions acquire mass without spontaneous chiral symmetry breaking) at $g_{\mathrm{GF}}^2\gtrsim25$ and develop stabilized setups to attack strong coupling. Rindlisbacher, Rummukainen and Salami (2023) constructed bulk-preventing gauge actions, a possible route past the obstruction, which however has not been applied to the $N_f=8$ thermal problem. Most pointedly, D'Elia, Nacci and Zambello (2026) proposed the Roberge-Weiss transition, a genuine thermal transition at imaginary baryon chemical potential $\mu_B/T=i\pi\;\mathrm{mod}\;2\pi/3$, as a conformality probe, and with stout-staggered (a smeared staggered-fermion discretization) $N_f=8$ simulations at $N_t=8$-$24$ found evidence that its transition temperature already vanishes in the chiral limit, implying $N_f^*\le8$ and no chiral phase transition at all, while noting that this does not completely exclude the chirally broken alternative. The existence and order of the thermal chiral transition in $N_f=8$ QCD therefore remain unsettled, and the four methodological prerequisites identified by Aoki et al. (2018) remain unexecuted for the HISQ formulation.

Problem Statement

Determine, from first principles and with controlled statistical and systematic errors, the finite-temperature chiral phase structure of $SU(3)$ gauge theory with $N_f=8$ mass-degenerate fundamental fermions. Specifically: (1) establish whether the chiral limit of the theory is chirally broken at low temperature and undergoes a finite-temperature chiral phase transition, or whether the theory is infrared conformal so that chiral symmetry is never spontaneously broken and no thermal chiral transition exists; (2) in the chirally broken case, determine the critical temperature $T_c$ in physical units with controlled scale setting, and the order of the transition, first order, second order with its universality class, or crossover, supported by chiral and continuum ($N_t\to\infty$) extrapolations together with finite-size scaling, hysteresis/metastability, or Binder-cumulant evidence (dimensionless moment ratios of the order-parameter distribution whose volume dependence distinguishes transition orders); (3) in the conformal case, demonstrate infrared conformality by a reproducible observable that distinguishes conformal from chirally broken dynamics, so that the absence of the chiral transition is established rather than inferred from non-observation. In every case the strongly coupled bulk (lattice-artifact) transition must be avoided or explicitly separated from the physical signal. The systematics flagged as future work by Aoki et al. (2018) for its own HISQ formulation — the systematic error of Hybrid Molecular Dynamics without Metropolis tests in the strong-coupling region $\beta_L\lesssim3.5$ relative to exact Hybrid Monte Carlo, contamination by bulk transitions at small fermion mass, and scale setting at the critical couplings — remain documented context for that program; these HMD-specific systematics are specific to the HISQ formulation and do not apply to other fermion discretizations or to exact algorithms.

References

1. Yasumichi Aoki, Tatsumi Aoyama, Ed Bennett, Masafumi Kurachi, Toshihide Maskawa, Kohtaroh Miura, Kei-ichi Nagai, Hiroshi Ohki, Enrico Rinaldi, Akihiro Shibata, Koichi Yamawaki, Takeshi Yamazaki (LatKMI Collaboration), "Thermodynamics in 8-Flavor QCD", in Origin of Mass and Strong Coupling Gauge Theories (SCGT15), World Scientific, 2018, pp. 331-334. DOI: 10.1142/9789813231467_0046. https://doi.org/10.1142/9789813231467_0046
2. Yasumichi Aoki, Tatsumi Aoyama, Ed Bennett, Masafumi Kurachi, Toshihide Maskawa, Kohtaroh Miura, Kei-ichi Nagai, Hiroshi Ohki, Enrico Rinaldi, Akihiro Shibata, Koichi Yamawaki, Takeshi Yamazaki (LatKMI Collaboration), "Light flavor-singlet scalars and walking signals in Nf = 8 QCD on the lattice", Physical Review D 96, 014508 (2017). arXiv:1610.07011. https://arxiv.org/abs/1610.07011
3. Robert D. Pisarski, Frank Wilczek, "Remarks on the Chiral Phase Transition in Chromodynamics", Physical Review D 29, 338 (1984). DOI: 10.1103/PhysRevD.29.338. https://doi.org/10.1103/PhysRevD.29.338
4. Frank R. Brown, Hong Chen, Norman H. Christ, Zhihua Dong, Robert D. Mawhinney, Wendy Schaffer, Alessandro Vaccarino, "Lattice QCD with 8 Light Quark Flavors", Physical Review D 46, 5655 (1992). DOI: 10.1103/PhysRevD.46.5655. arXiv:hep-lat/9206001. https://arxiv.org/abs/hep-lat/9206001
5. Xiao-Yong Jin, Robert D. Mawhinney, "Evidence for a First Order, Finite Temperature Phase Transition in 8 Flavor QCD", Proceedings of Science, LATTICE2010:055 (2010). arXiv:1011.1511. https://arxiv.org/abs/1011.1511
6. Albert Deuzeman, Maria Paola Lombardo, Elisabetta Pallante, "The physics of eight flavours", Physics Letters B 670, 41-48 (2008). DOI: 10.1016/j.physletb.2008.10.039. arXiv:0804.2905. https://arxiv.org/abs/0804.2905
7. Kohtaroh Miura, Maria Paola Lombardo, Elisabetta Pallante, "Chiral phase transition at finite temperature and conformal dynamics in large Nf QCD", Physics Letters B 710, 676-682 (2012). DOI: 10.1016/j.physletb.2012.03.017. arXiv:1110.3152. https://arxiv.org/abs/1110.3152
8. Kohtaroh Miura, Maria Paola Lombardo, "Lattice Monte Carlo study of pre-conformal dynamics in strongly flavoured QCD in the light of the chiral phase transition at finite temperature", Nuclear Physics B 871, 52-81 (2013). DOI: 10.1016/j.nuclphysb.2013.02.008. arXiv:1212.0955. https://arxiv.org/abs/1212.0955
9. Maria Paola Lombardo, Kohtaroh Miura, Tiago J. Nunes da Silva, Elisabetta Pallante, "Chiral symmetry restoration in QCD with many flavours", Proceedings of Science, CPOD2014:059 (2015). DOI: 10.22323/1.217.0059. arXiv:1506.05946. https://arxiv.org/abs/1506.05946
10. David Schaich, Anqi Cheng, Anna Hasenfratz, Gregory Petropoulos, "Bulk and finite-temperature transitions in SU(3) gauge theories with many light fermions", Proceedings of Science, LATTICE2012:028 (2012). DOI: 10.22323/1.164.0028. arXiv:1207.7164. https://arxiv.org/abs/1207.7164
11. David Schaich, Anna Hasenfratz, Enrico Rinaldi (for the LSD Collaboration), "Finite-temperature study of eight-flavor SU(3) gauge theory", in Origin of Mass and Strong Coupling Gauge Theories (SCGT15), World Scientific, 2018, pp. 351-354. DOI: 10.1142/9789813231467_0051. arXiv:1506.08791. https://arxiv.org/abs/1506.08791
12. T. Appelquist, R. C. Brower, G. T. Fleming, A. Gasbarro, A. Hasenfratz, X. Y. Jin, E. T. Neil, J. C. Osborn, C. Rebbi, E. Rinaldi, D. Schaich, P. Vranas, E. Weinberg, O. Witzel (Lattice Strong Dynamics (LSD) Collaboration), "Nonperturbative investigations of SU(3) gauge theory with eight dynamical flavors", Physical Review D 99, 014509 (2019). DOI: 10.1103/PhysRevD.99.014509. arXiv:1807.08411. https://arxiv.org/abs/1807.08411
13. Francesca Cuteri, Owe Philipsen, Alessandro Sciarra, "On the order of the QCD chiral phase transition for different numbers of quark flavours", Journal of High Energy Physics 2021, 141 (2021). DOI: 10.1007/JHEP11(2021)141. arXiv:2107.12739. https://arxiv.org/abs/2107.12739
14. Anna Hasenfratz, Claudio Rebbi, Oliver Witzel, "Gradient flow step-scaling function for SU(3) with Nf = 8 fundamental flavors", Physical Review D 107, 114508 (2023). arXiv:2210.16760. https://arxiv.org/abs/2210.16760
15. Tobias Rindlisbacher, Kari Rummukainen, Ahmed Salami, "Bulk-transition-preventing actions for SU(N) gauge theories", Physical Review D 108, 114511 (2023). DOI: 10.1103/PhysRevD.108.114511. arXiv:2306.14319. https://arxiv.org/abs/2306.14319
16. Oliver Witzel, Anna Hasenfratz (for the Lattice Strong Dynamics Collaboration), "Investigating SU(3) with Nf = 8 fundamental fermions at strong renormalized coupling", Proceedings of Science, LATTICE2024:146 (2025). DOI: 10.22323/1.466.0146. arXiv:2412.10322. https://arxiv.org/abs/2412.10322
17. Jan Philipp Klinger, Reinhold Kaiser, Owe Philipsen, Jonas Schaible, "On the Phase Structure of Massless Many-Flavour QCD with Staggered Quarks", Proceedings of Science, LATTICE2025:119 (2026). DOI: 10.22323/1.518.0119. arXiv:2603.20099. https://arxiv.org/abs/2603.20099
18. Massimo D'Elia, Marco Nacci, Kevin Zambello, "Roberge-Weiss transition as a probe for conformality in many-flavor QCD", Physical Review D 114, 034502 (2026). DOI: 10.1103/d8jw-ydq7. arXiv:2604.07284. https://arxiv.org/abs/2604.07284