Continuum and chiral-limit status of eight-flavor SU(3) QCD: walking versus conformality, the light flavor-singlet scalar, and the mass anomalous dimension

Eight-flavor SU(3) QCD, the SU(3) gauge theory with $N_f=8$ degenerate fundamental fermions, is the canonical near-conformal theory of the lattice many-flavor program and the template of one-family walking technicolor. The LatKMI collaboration reported walking signals (finite-size hyperscaling with an anomalous dimension near 1, coexisting with a chirally broken spectrum) and a flavor-singlet scalar as light as the pion, proposed as a technidilaton that could account for a 125 GeV Higgs-like scalar, together with a broadly determined chiral-limit rho mass, all at a single lattice spacing and with large chiral-extrapolation uncertainties. The open question left by that work is whether the approximate conformality survives the continuum and thermodynamic limits with controlled systematics, whether a truly universal mass anomalous dimension emerges, and what the precise chiral-limit scalar and vector masses are. Later work, including independent discretizations by the Lattice Strong Dynamics collaboration, gradient-flow step scaling (a nonperturbative method that tracks the running coupling via smoothing the gauge field along its own gradient), dilaton effective theory analyses, strong-coupling phase studies, and mutually conflicting functional renormalization group (fRG) predictions for the critical flavor number, has sharpened but not resolved the question. The phase assignment (chirally broken walking just below the conformal window versus inside the window or at its opening), the chiral-limit scalar mass (whose squared-mass intercept is still consistent with zero or negative within errors), the anomalous dimension, and the TeV-scale phenomenology all remain open.

Background

Eight-flavor QCD is the SU(3) Yang-Mills gauge theory with $N_f=8$ degenerate Dirac fermions in the fundamental representation, investigated by Monte Carlo simulation of its lattice discretization. In lattice QCD the theory is regulated on a hypercubic grid of spacing $a$ and spatial extent $L$, and a physical answer requires three nested limits: the continuum limit $a\to 0$, the thermodynamic (infinite-volume) limit $L\to\infty$, and the chiral limit in which the fermion mass $m_f$ vanishes.

The theory belongs to the many-flavor program, which studies how the infrared behavior of QCD changes as the number of massless fermion flavors grows. Below the asymptotic-freedom bound ($N_f=16.5$ for fundamental fermions), theory predicts a conformal window: a range of $N_f$ in which the running gauge coupling, instead of growing without bound, approaches an infrared fixed point (IRFP), making the long-distance theory a scale-invariant conformal field theory. For smaller $N_f$ the theory behaves like QCD: chiral symmetry is spontaneously broken, producing a pion that is a Nambu-Goldstone boson (the massless boson of a spontaneously broken continuous symmetry) while other hadron masses and the pion decay constant $F_\pi$ stay nonzero in the chiral limit. Just below the edge of the conformal window the coupling is expected to run very slowly over a wide range of scales (walking). Walking enhances the fermion mass anomalous dimension $\gamma_m$, the exponent with which a fermion mass deforms the theory, since near an IRFP all masses scale as $m_f^{1/(1+\gamma_m)}$ (hyperscaling); $\gamma_m\simeq1$ is a key ingredient of walking technicolor and composite-Higgs model building, because it suppresses flavor-changing effects while generating Standard-Model fermion masses.

A second ingredient is the dilaton idea. If scale invariance is only approximate, its spontaneous breaking produces a light pseudo-Nambu-Goldstone scalar, the dilaton, in the same way chiral symmetry breaking gives the pion (Yamawaki, Bando and Matumoto 1986; Holdom 1985 proposed the walking mechanism). In a technicolor setting this state is the technidilaton. Since the observed Higgs boson has a mass of 125 GeV, a lattice theory that realizes walking with $\gamma_m\sim1$ and a flavor-singlet scalar as light as the pion is a candidate strongly interacting sector beyond the Standard Model. In the one-family technicolor normalization, the decay constant satisfies $\sqrt{N_D}\,F=\sqrt{2}\,v_{\rm EW}$ with $v_{\rm EW}=246$ GeV and $N_D=4$ electroweak doublets, so $F/\sqrt{2}\simeq123$ GeV; a chiral-limit scalar mass near $F/\sqrt{2}$ then corresponds to a 125 GeV Higgs-like composite scalar, and the vector (rho) meson mass sets the scale of TeV resonances.

The LatKMI collaboration (the lattice group of the Kobayashi-Maskawa Institute, Nagoya University) simulated this theory with the Highly Improved Staggered Quark (HISQ) action, a staggered-fermion discretization whose dominant lattice artifact is the breaking of Taste symmetry (an exact lattice remnant of flavor symmetry), at a single bare gauge coupling $\beta=3.8$ on volumes up to $48^3\times64$ with fermion masses down to $m_f=0.009$ in lattice units. They found that near the chiral limit the data are described by chiral perturbation theory (ChPT, the effective field theory of pions) with nonzero $F_\pi$, a nonzero chiral-limit rho mass and chiral condensate, indicating spontaneous chiral symmetry breaking; while at larger $m_f$ the same spectrum obeys finite-size hyperscaling with $\gamma_m\sim1$, interpreted as a remnant of infrared conformality, i.e. walking just below the conformal window. Most strikingly, the flavor-singlet scalar $\sigma$ (extracted from quark-line-disconnected correlator diagrams, which are noisy and demand high statistics) is as light as the pion, unlike the heavy broad $\sigma$ of real QCD. Independently, the Lattice Strong Dynamics (LSD) collaboration studied the same continuum theory with domain wall fermions (a discretization with exact chiral symmetry up to an exponentially small residual mass) and nHYP-smeared staggered fermions (staggered fermions with n-step hypercubic smearing of the gauge links, which reduces lattice artifacts), finding the light scalar degenerate with the pseudo-Goldstone states but reporting that finite-temperature transitions run into a strongly coupled lattice phase as the mass decreases, preventing direct confirmation of the chiral-limit behavior. Matsuzaki and Yamawaki proposed dilaton chiral perturbation theory (dChPT) to extract the technidilaton mass and decay constant from the LatKMI data.

The unresolved target is that none of these results yet exist with controlled continuum, thermodynamic, and chiral-limit systematics: the phase of the massless theory, the universality and value of $\gamma_m$, and the precise chiral-limit scalar and vector masses, which decide the technidilaton and TeV-resonance phenomenology, are all still open.

Problem Statement

Determine, from the available lattice evidence, the infrared behavior of SU(3) gauge theory with $N_f=8$ fundamental fermions with respect to the continuum ($a\to0$), thermodynamic ($L\to\infty$), and chiral ($m_f\to0$) limits, as the question left open by the LatKMI walking and light-scalar studies. The target is the phase of the massless theory, whether chirally broken with a slowly running (walking) coupling just below the conformal window, mass-deformed infrared-conformal inside the conformal window, or at the window's opening, decided on the evidence for and against each option, together with the value and universality of the mass anomalous dimension and the chiral-limit scalar and vector masses, each with its statistical error and dominant uncontrolled systematic and with the extent to which the continuum, thermodynamic, and chiral limits are controlled. Concretely:

1. Phase and approximate conformality. Establish whether the massless theory is chirally broken with a slowly running (walking) coupling just below the conformal window, the LatKMI interpretation in which small-$m_f$ spectra follow chiral perturbation theory while larger-$m_f$ data obey hyperscaling $M_H\propto m_f^{1/(1+\gamma_m)}$, or whether it is a mass-deformed infrared-conformal theory inside the conformal window, or sits at the window's opening. All existing spectrum data are at single lattice spacings (LatKMI at $\beta=3.8$, LSD at $\beta=4.8$), so the observed approximate conformality is not yet established in the continuum limit; a valid answer must confront or remove the strong-coupling lattice artifacts (bulk transitions, i.e. unphysical first-order transitions in the bare gauge coupling that appear at strong lattice coupling, and strongly coupled lattice phases) that currently block the chiral limit.

2. Mass anomalous dimension. Determine whether a universal mass anomalous dimension $\gamma_m$ exists in the infrared and what its value is, reconciling the hyperscaling estimate $\gamma^*\simeq1$ (most recently $1.0830(11)$), the running $\gamma_m$ found in dilaton-EFT analyses of the same data, and the slow running with no infrared fixed point up to $g^2\approx10$ to $14$ seen in gradient-flow step scaling.

3. Flavor-singlet scalar. Determine the chiral-limit scalar mass $M_\sigma$ and decay constant $F_\sigma$ precisely. At present $M_\sigma$ remains lighter than the pion at the lightest simulated masses, and the squared-mass intercept $d_0=M_\sigma^2|_{m_f\to0}$ has a negative central value with uncertainties that still admit zero or positive values. Under the one-family technicolor normalization $F/\sqrt{2}\simeq123$ GeV, the test is whether the chiral-limit scalar can robustly account for a 125 GeV Higgs-like state as a technidilaton, with the current decay-constant estimate $F_\sigma/(F_\pi/\sqrt{2})\simeq4$ to be made precise.

4. Vector channel. Determine the chiral-limit rho mass precisely, replacing the current broad range $\sqrt2 M_\rho/F=8.91(66)(^{+4.44}_{-1.00})$, so that the predicted TeV-scale resonance phenomenology of the model becomes a quantitative statement rather than a suggestive one.

References

1. Y. Aoki, T. Aoyama, E. Bennett, M. Kurachi, T. Maskawa, K. Miura, K. Nagai, H. Ohki, E. Rinaldi, A. Shibata, K. Yamawaki, T. Yamazaki (LatKMI Collaboration), 'Lattice Studies on 8-Flavor QCD in The Light of Physics Beyond The Standard Model', Nucl. Part. Phys. Proc. 270-272, 242-246 (2016), DOI 10.1016/j.nuclphysbps.2016.02.048, https://doi.org/10.1016/j.nuclphysbps.2016.02.048
2. Y. Aoki et al. (LatKMI Collaboration), 'Walking signals in $N_f=8$ QCD on the lattice', Phys. Rev. D 87, 094511 (2013), arXiv:1302.6859, DOI 10.1103/PhysRevD.87.094511, https://arxiv.org/abs/1302.6859
3. Y. Aoki et al. (LatKMI Collaboration), 'Light composite scalar in eight-flavor QCD on the lattice', Phys. Rev. D 89, 111502 (2014), arXiv:1403.5000, DOI 10.1103/PhysRevD.89.111502, https://arxiv.org/abs/1403.5000
4. T. Appelquist et al. (Lattice Strong Dynamics Collaboration), 'Lattice simulations with eight flavors of domain wall fermions in SU(3) gauge theory', Phys. Rev. D 90, 114502 (2014), arXiv:1405.4752, DOI 10.1103/PhysRevD.90.114502, https://arxiv.org/abs/1405.4752
5. A. Hasenfratz, D. Schaich, A. Veernala, 'Nonperturbative beta function of eight-flavor SU(3) gauge theory', JHEP 06, 143 (2015), arXiv:1410.5886, DOI 10.1007/JHEP06(2015)143, https://arxiv.org/abs/1410.5886
6. D. Schaich, A. Hasenfratz, E. Rinaldi (Lattice Strong Dynamics Collaboration), 'Finite-temperature study of eight-flavor SU(3) gauge theory', in 'Origin of Mass and Strong Coupling Gauge Theories (SCGT15)' (World Scientific, 2017), pp. 351-354, arXiv:1506.08791, DOI 10.1142/9789813231467_0051, https://arxiv.org/abs/1506.08791
7. T. Appelquist et al. (Lattice Strong Dynamics Collaboration), 'Nonperturbative investigations of SU(3) gauge theory with eight dynamical flavors', Phys. Rev. D 99, 014509 (2019), arXiv:1807.08411, DOI 10.1103/PhysRevD.99.014509, https://arxiv.org/abs/1807.08411
8. M. Golterman, Y. Shamir, 'Explorations beyond dilaton chiral perturbation theory in the eight-flavor SU(3) gauge theory', Phys. Rev. D 102, 114507 (2020), arXiv:2009.13846, DOI 10.1103/PhysRevD.102.114507, https://arxiv.org/abs/2009.13846
9. T. Appelquist et al. (Lattice Strong Dynamics Collaboration), 'Hidden conformal symmetry from the lattice', Phys. Rev. D 108, L091505 (2023), arXiv:2305.03665, DOI 10.1103/PhysRevD.108.L091505, https://arxiv.org/abs/2305.03665
10. R. C. Brower et al. (Lattice Strong Dynamics Collaboration), 'Light scalar meson and decay constant in SU(3) gauge theory with eight dynamical flavors', Phys. Rev. D 110, 054501 (2024), arXiv:2306.06095, DOI 10.1103/PhysRevD.110.054501, https://arxiv.org/abs/2306.06095
11. A. Hasenfratz, C. Rebbi, O. Witzel, 'Gradient flow step-scaling function for SU(3) with $N_f=8$ fundamental flavors', Phys. Rev. D 107, 114508 (2023), arXiv:2210.16760, DOI 10.1103/PhysRevD.107.114508, https://arxiv.org/abs/2210.16760
12. A. Hasenfratz, 'Emergent strongly coupled ultraviolet fixed point in four dimensions with eight Kähler-Dirac fermions', Phys. Rev. D 106, 014513 (2022), arXiv:2204.04801, DOI 10.1103/PhysRevD.106.014513, https://doi.org/10.1103/PhysRevD.106.014513
13. A. Hasenfratz, O. Witzel, 'Investigating SU(3) with $N_f=8$ fundamental fermions at strong renormalized coupling', PoS LATTICE2024, 146 (2025), arXiv:2412.10322, DOI 10.22323/1.466.0146, https://arxiv.org/abs/2412.10322
14. Y. Aoki et al. (LatKMI Collaboration), 'Novel view of the flavor-singlet spectrum from multi-flavor QCD on the lattice', Phys. Rev. D 112, 114503 (2025), arXiv:2505.08658, DOI 10.1103/vnml-g6nx, https://arxiv.org/abs/2505.08658
15. Z. Davoudi, E. T. Neil, et al. (Snowmass 2021 Topical Group TF05, community report), 'Report of the Snowmass 2021 Topical Group on Lattice Gauge Theory', arXiv:2209.10758 (2022), https://arxiv.org/abs/2209.10758
16. A. Pastor-Gutierrez, 'Non-perturbative news from the conformal window', arXiv:2608.23674 (2026), https://arxiv.org/abs/2608.23674
17. F. Goertz, A. Pastor-Gutierrez, J. M. Pawlowski, 'Gauge-fermion cartography: From confinement and chiral symmetry breaking to conformality', Phys. Rev. D 112, 034029 (2025), arXiv:2412.12254, DOI 10.1103/7dzj-k6k8, https://arxiv.org/abs/2412.12254
18. S. Matsuzaki, K. Yamawaki, 'Dilaton Chiral Perturbation Theory: Determining the Mass and Decay Constant of the Technidilaton on the Lattice', Phys. Rev. Lett. 113, 082002 (2014), arXiv:1311.3784, DOI 10.1103/PhysRevLett.113.082002, https://doi.org/10.1103/PhysRevLett.113.082002
19. K. Yamawaki, M. Bando, K. Matumoto, 'Scale-Invariant Hypercolor Model and a Dilaton', Phys. Rev. Lett. 56, 1335 (1986), DOI 10.1103/PhysRevLett.56.1335, https://doi.org/10.1103/PhysRevLett.56.1335
20. B. Holdom, 'Techniodor', Phys. Lett. B 150, 301 (1985), DOI 10.1016/0370-2693(85)91015-9, https://doi.org/10.1016/0370-2693(85)91015-9