Continuum-limit status of the light flavor-singlet scalar in many-flavor SU(3) gauge theory: dilatonic physics or lattice artifact?

In 2013 the LatKMI collaboration observed, in lattice simulations of SU(3) gauge theory with $N_f=12$ fundamental fermions using the highly improved staggered quark (HISQ) action at a single lattice spacing, a flavor-singlet scalar meson ($\sigma$) lighter than the pion, and interpreted its lightness as a signal of infrared (near-)conformal dynamics — a candidate technidilaton, the pseudo-Nambu-Goldstone boson of approximate broken scale invariance predicted in walking technicolor. The authors themselves flagged the central unresolved issue: the simulations used one bare coupling, so whether the lightness survives the continuum limit, rather than being a finite-lattice-spacing artifact, was unknown. More than a decade of follow-up work — including the extension to the more phenomenologically viable $N_f=8$ theory, independent confirmation of a light scalar there by two collaborations, continuum-extrapolated determinations of the infrared fixed point in the $N_f=12$ running coupling, and accumulating evidence for bulk-transition lattice artifacts in 12-flavor staggered simulations — has not produced a controlled multi-lattice-spacing extrapolation of the scalar. The problem is to determine, with controlled discretization, volume, and phase-structure systematics, whether the light scalar of $N_f=12$ (and $N_f=8$) many-flavor QCD is a property of the continuum theory, what its fermionic-versus-gluonic composition is, and whether its scaling is consistent with the dilatonic (technidilaton) interpretation.

Background

Walking technicolor is a hypothetical new strongly interacting gauge theory proposed as a dynamical origin of electroweak symmetry breaking, in which the gauge coupling runs very slowly ('walks') over a wide range of energy scales. This walking behavior is expected when the theory sits just below the edge of the conformal window — the range of the number of fermion flavors $N_f$, at fixed number of colors $N_c=3$, inside which the coupling flows to an infrared fixed point (IRFP) and chiral symmetry is unbroken. Because the beta function nearly vanishes in the walking regime, scale invariance is only weakly broken, and the theory can contain a technidilaton: a light scalar bound state emerging as the pseudo-Nambu-Goldstone boson of the approximately broken scale symmetry, with mass of order the electroweak scale set by the pseudoscalar decay constant $F_\pi$. After the 2012 discovery of a 125 GeV Higgs boson, a technidilaton light enough to mimic it became one of the few viable composite-Higgs scenarios for such theories, and the mass anomalous dimension $\gamma_m$ (which controls how hadron masses scale with the fermion mass, $M_H \propto m_f^{1/(1+\gamma_m)}$, a relation called hyperscaling when it holds universally) became a key diagnostic, since several phenomenological models require $\gamma_m \approx 1$.

Lattice gauge theory regulates spacetime on a grid of spacing $a$; continuum physics is recovered by extrapolating $a \to 0$ at fixed physical volume and masses, which requires simulating at several bare couplings $\beta = 6/g^2$ (larger $\beta$ means finer $a$). This extrapolation is the unavoidable correctness check for any lattice claim, and it is exactly what the staggered-fermion discretization makes expensive near the conformal window: staggered fermions represent four continuum tastes per lattice field, and at finite $a$ the residual 'taste breaking' splits what should be degenerate states, an artifact that must be tracked as $a$ decreases. The highly improved staggered quark (HISQ) action suppresses taste breaking, which is why it was chosen for these studies. The flavor-singlet scalar ($\sigma$, quantum numbers $0^{++}$) is the hardest hadron to measure: its correlator receives disconnected fermion-line contributions whose stochastic noise grows rapidly at large time separations, and it can mix with non-singlet scalar mesons ($a_0$), the scalar glueball (a bound state of pure gauge fields), and two-pion $s$-wave states.

In the 2013 Phys. Rev. Lett. 111, 162001 paper, Aoki et al. (LatKMI collaboration) simulated $N_f=12$ with the HISQ action at a single bare coupling $\beta=4.0$, three volumes, and four fermion masses, and extracted $m_\sigma$ from both fermion bilinear and gluonic interpolating operators. They found $m_\sigma < m_\pi$ (ratio $\approx 0.7$–$0.9$), with the mass described by hyperscaling using the anomalous dimension $\gamma = 0.414$ fixed from their earlier pion analysis. Interpreting $N_f=12$ as consistent with being inside the conformal window, they attributed the lightness of the scalar to infrared conformality and proposed it as a promising signal of a technidilaton-like state. They closed by stating that investigation of the lattice-spacing (continuum) dependence of the scalar state is important, and that the most pressing future direction is the more phenomenologically viable $N_f=8$ theory, where the scalar could be identified with a technidilaton of mass of order $F_\pi$ in the chiral limit. Dilaton chiral perturbation theory (dChPT) — an effective field theory incorporating a light scalar through a dilaton-like Ward identity, predicting $M_\sigma^2 \approx d_0 + d_1 M_\pi^2$ with $d_0 \to 0$ in the chiral limit of a (near-)conformal theory — provides the standard framework later used to extrapolate the scalar mass.

Subsequent work established two facts that frame the open question. First, both LatKMI and the Lattice Strong Dynamics (LSD) collaboration observed, in independent discretizations (HISQ and nHYP-smeared staggered fermions respectively), a light flavor-singlet scalar in the $N_f=8$ theory at finite lattice spacing, degenerate with or lighter than the pion; LatKMI's chiral extrapolation via dChPT was consistent, within large errors, with the scalar being the 125 GeV Higgs of a one-family walking technicolor model. Second, continuum-extrapolated step-scaling studies of the gradient-flow running coupling found strong evidence for an IRFP in the massless $N_f=12$ theory, supporting the conformality prerequisite of the dilatonic reading. However, a persistent complication is that 12-flavor staggered simulations at strong coupling encounter first-order bulk (zero-temperature) phase transitions — discretization artifacts whose critical endpoints can themselves drive the scalar mass toward zero, and it remains contested whether the observed light scalar reflects continuum conformal dynamics or proximity to such a lattice phase.

Problem Statement

Determine whether the light flavor-singlet scalar observed in many-flavor SU(3) gauge theory with fundamental fermions is a property of the continuum theory or a finite-lattice-spacing artifact, and whether its properties support the dilatonic (technidilaton) interpretation. Concretely, for the $N_f=12$ theory — and for the more phenomenologically viable $N_f=8$ theory, where a scalar as light as the pion has already been observed at single lattice spacings by two independent groups — determine, from the cited single-lattice-spacing results together with the phase-structure, continuum-extrapolation, and dilatonic-interpretation literature, what the current evidence implies — accounting for the controlling systematics (finite volume, staggered-fermion taste breaking, proximity to strong-coupling/bulk-transition lattice phases, and disconnected-correlator noise) — and determine: (1) the continuum-limit ($a \to 0$) behavior of the scalar mass in physical units, e.g. the ratios $m_\sigma/F_\pi$ and $m_\sigma/m_\pi$ extrapolated at fixed physical fermion mass and then in the chiral limit — in particular whether the observed lightness ($m_\sigma$ comparable to or below $m_\pi$) survives, or whether the state becomes heavy or disappears, or is instead driven by proximity to a bulk-transition critical endpoint; (2) the composition of the light scalar, i.e. the fermionic-bound-state versus gluonic (glueball) content and how the fermion-bilinear/glueball operator mixing is expected to evolve with the lattice spacing; and (3) whether the scalar's mass and its fermion-mass scaling are consistent with the dilatonic picture — hyperscaling with an anomalous dimension $\gamma_m$ consistent across channels and the dilaton-chiral-perturbation-theory relation $M_\sigma^2 \approx d_0 + d_1 M_\pi^2$ with $d_0 \to 0$, giving a technidilaton of mass of order $F_\pi$ in the chiral and continuum limits. The three parts jointly determine whether the light scalar of near-conformal many-flavor QCD is real continuum physics and a viable dilaton.

References

1. Y. Aoki, T. Aoyama, M. Kurachi, T. Maskawa, K. Nagai, H. Ohki, E. Rinaldi, A. Shibata, K. Yamawaki, T. Yamazaki (LatKMI), 'Light composite scalar in twelve-flavor QCD on the lattice', Phys. Rev. Lett. 111, 162001 (2013), DOI 10.1103/PhysRevLett.111.162001, https://arxiv.org/abs/1305.6006 (arXiv:1305.6006).
2. Y. Aoki, T. Aoyama, M. Kurachi, T. Maskawa, K. Miura, K. Nagai, H. Ohki, E. Rinaldi, A. Shibata, K. Yamawaki, T. Yamazaki (LatKMI), 'Composite flavor-singlet scalar in twelve-flavor QCD', PoS(LATTICE 2013)077, DOI 10.22323/1.187.0077, https://arxiv.org/abs/1311.6885 (arXiv:1311.6885).
3. Y. Aoki, T. Aoyama, M. Kurachi, T. Maskawa, K. Miura, K. Nagai, H. Ohki, E. Rinaldi, A. Shibata, K. Yamawaki, T. Yamazaki (LatKMI), 'Gluonic observables and the scalar spectrum of twelve-flavour QCD', PoS(LATTICE 2013)073, DOI 10.22323/1.187.0073, https://arxiv.org/abs/1310.0963 (arXiv:1310.0963).
4. 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), 'Conformality in twelve-flavor QCD', PoS(LATTICE 2014)256, DOI 10.22323/1.214.0256, https://arxiv.org/abs/1501.06660 (arXiv:1501.06660).
5. X.-Y. Jin, R. D. Mawhinney, 'Lattice QCD with 12 Quark Flavors: A Careful Scrutiny', in Strong Coupling Gauge Theories in the LHC Perspective (SCGT12), pp. 96-102, World Scientific (2014), DOI 10.1142/9789814566254_0011, https://arxiv.org/abs/1304.0312 (arXiv:1304.0312).
6. Z. Gelzer, Y. Liu, Y. Meurice, 'Exploring the phase structure of 12-flavor SU(3)', PoS(LATTICE 2014)255, DOI 10.22323/1.214.0255, https://arxiv.org/abs/1411.3360 (arXiv:1411.3360).
7. A. Saleh, M. Hite, D. Floor, Y. Meurice, 'Zeros of the partition function for 12 flavor QCD', arXiv:2606.14642 (2026, preprint), DOI 10.48550/arXiv.2606.14642, https://arxiv.org/abs/2606.14642.
8. A. Hasenfratz, D. Schaich, 'Nonperturbative beta function of twelve-flavor SU(3) gauge theory', JHEP 2018 (02) 132, DOI 10.1007/JHEP02(2018)132, https://arxiv.org/abs/1610.10004 (arXiv:1610.10004).
9. C. T. Peterson, A. Hasenfratz, 'Infrared fixed point in the massless twelve-flavor SU(3) gauge-fermion system', Phys. Rev. D 109, 114507 (2024), DOI 10.1103/PhysRevD.109.114507, https://arxiv.org/abs/2402.18038 (arXiv:2402.18038).
10. Y. Aoki, T. Aoyama, M. Kurachi, T. Maskawa, K. Miura, K. Nagai, H. Ohki, E. Rinaldi, A. Shibata, K. Yamawaki, T. Yamazaki (LatKMI), 'Light composite scalar in eight-flavor QCD on the lattice', Phys. Rev. D 89, 111502(R) (2014), DOI 10.1103/PhysRevD.89.111502, https://arxiv.org/abs/1403.5000 (arXiv:1403.5000).
11. E. Rinaldi (for the LSD Collaboration), 'Investigation of the scalar spectrum in SU(3) with eight degenerate flavors', Int. J. Mod. Phys. A 32, 1747002 (2017) (LATTICE 2015 proceedings), DOI 10.1142/S0217751X17470029, https://arxiv.org/abs/1510.06771 (arXiv:1510.06771).
12. 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), 'Light flavor-singlet scalars and walking signals in $N_f=8$ QCD on the lattice', Phys. Rev. D 96, 014508 (2017), DOI 10.1103/PhysRevD.96.014508, https://arxiv.org/abs/1610.07011 (arXiv:1610.07011).
13. 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 (LSD), 'Nonperturbative investigations of SU(3) gauge theory with eight dynamical flavors', Phys. Rev. D 99, 014509 (2019), DOI 10.1103/PhysRevD.99.014509, https://arxiv.org/abs/1807.08411 (arXiv:1807.08411).
14. T. Appelquist, R. C. Brower, K. K. Cushman, G. T. Fleming, A. D. Gasbarro, A. Hasenfratz, X.-Y. Jin, E. T. Neil, J. C. Osborn, C. Rebbi, E. Rinaldi, D. Schaich, P. Vranas, O. Witzel (LSD), 'Near-conformal dynamics in a chirally broken system', Phys. Rev. D 103, 014504 (2021), DOI 10.1103/PhysRevD.103.014504, https://arxiv.org/abs/2007.01810 (arXiv:2007.01810).
15. T. Appelquist, R. C. Brower, K. K. Cushman, G. T. Fleming, A. Gasbarro, A. Hasenfratz, J. Ingoldby, X.-Y. Jin, J. Kiskis, E. T. Neil, J. C. Osborn, C. Rebbi, E. Rinaldi, D. Schaich, P. Vranas, E. Weinberg, O. Witzel (LSD), 'Goldstone boson scattering with a light composite scalar', Phys. Rev. D 105, 034505 (2022), DOI 10.1103/PhysRevD.105.034505, https://arxiv.org/abs/2106.13534 (arXiv:2106.13534).
16. Y. Aoki, T. Aoyama, E. Bennett, T. Maskawa, K. Miura, H. Ohki, E. Rinaldi, A. Shibata, K. Yamawaki, T. Yamazaki (LatKMI), 'A novel view of the flavor-singlet spectrum from multi-flavor QCD on the lattice', Phys. Rev. D 112, 114503 (2025), DOI 10.1103/vnml-g6nx, https://arxiv.org/abs/2505.08658 (arXiv:2505.08658).
17. A. Pomarol, L. Salas, 'Exploring the conformal transition from above and below', JHEP 2024 (08) 149, DOI 10.1007/JHEP08(2024)149, https://arxiv.org/abs/2312.08332 (arXiv:2312.08332).