Walking or conformal: the infrared phase of SU(3) gauge theory with eight fundamental fermions ($N_f=8$ QCD)

The LatKMI collaboration's 2013 proceedings chapter "KMI Lattice Project on 8-Flavor QCD — Exploration of the Walking Technicolor" reported lattice simulations of SU(3) gauge theory with eight fundamental fermions in which small-mass data looked chirally broken while larger-mass data obeyed hyperscaling with an anomalous dimension near one, and posed the question: is $N_f=8$ QCD in the chirally broken, walking phase, or inside the conformal window? Thirteen years of larger-scale follow-up — multi-action spectroscopy, gradient-flow step scaling, dilaton effective-field-theory fits, finite-temperature studies, and most recently symmetric-mass-generation and Roberge-Weiss (imaginary-chemical-potential, finite-temperature) analyses — have produced conflicting answers: several collaborations find a light flavor-singlet scalar and dilaton-EFT behavior consistent with walking, while recent studies conclude that the theory sits at or inside the opening of the conformal window. The problem posed here is: a definitive determination of the infrared phase of $N_f=8$ QCD that resolves the systematic ambiguities (single gauge coupling, fit-window dependence and large $\chi^2$) which motivated the question. The question remains open: no consensus resolution exists in the literature through 2026.

Background

SU(3) gauge theory with $N_f$ massless Dirac fermions in the fundamental representation — many-flavor QCD — changes character as $N_f$ grows. For small $N_f$ it is QCD-like: confinement with spontaneous chiral symmetry breaking (SχSB, the mechanism that makes the pions light and gives fermions their masses). Asymptotic freedom holds up to $N_f=16.5$, and near that value two-loop perturbation theory predicts a weak infrared fixed point (the Banks-Zaks fixed point), making the theory infrared (IR) conformal. The interval of $N_f$ admitting an infrared fixed point is the conformal window; its lower edge $N_f^\ast$ is a nonperturbative quantity that perturbation theory cannot fix, and estimates place it between roughly 8 and 12. Just below the edge the theory is expected to be walking: the coupling runs very slowly over a wide range of scales, the mass anomalous dimension $\gamma_m$ (the exponent governing the running of the fermion mass) can become of order one, the chiral condensate is enhanced, and the approximate scale invariance may produce a light flavor-singlet scalar, a pseudo-Nambu-Goldstone boson of broken scale invariance called a technidilaton. Walking dynamics underlies walking technicolor and composite-Higgs model building, and $N_f=8$ QCD is the standard lattice caricature of the one-family technicolor model.

On the lattice the two phases are told apart by spectroscopy. If the theory is chirally broken, the pseudo-Nambu-Goldstone "pion" sector obeys chiral perturbation theory (ChPT), the low-energy effective theory of the pions: $M_\pi^2 \propto m_f$, the pion decay constant $F_\pi$ tends to a nonzero constant $F$, and the condensate satisfies the Gell-Mann-Oakes-Renner relation $M_\pi^2 F_\pi^2 = 2 m_f \langle\bar\psi\psi\rangle$. If the theory is mass-deformed IR conformal, dimensionless ratios such as $M_\pi/F_\pi$ or $M_\pi L$ (with $L$ the box size) obey finite-size hyperscaling: they depend on the fermion mass only through the combination $m_f L^{y_m}$ with $y_m = 1/(1+\gamma_m)$, so fits return the anomalous dimension at the fixed point. The difficulty near the window edge is that both patterns can appear in different fermion-mass windows of the same data set, and slow walking over a finite scale range can mimic mass-deformed conformality.

Before the source work, lattice studies disagreed: the staggered-fermion study by Appelquist, Fleming and Neil (arXiv:0901.3766) found $N_f=8$ chirally broken and $N_f=12$ conformal, placing the window edge between 8 and 12; a Monte Carlo renormalization group study by Hasenfratz (arXiv:1004.1004) found the $N_f=8$ system QCD-like; and a finite-temperature study by Jin and Mawhinney (arXiv:1011.1511) found a first-order thermal transition in eight-flavor QCD. The source of the present question is the LatKMI collaboration's 2012-2013 study of $N_f=8$ using the Highly Improved Staggered Quark (HISQ) action — a staggered-fermion discretization with reduced lattice artifacts whose three-link Naik term improves the covariant derivative — at a single gauge coupling $\beta = 6/g^2 = 3.8$, fixed aspect ratio $T/L = 4/3$ (with $T$ the temporal lattice extent), fermion masses $m_f = 0.015$-$0.16$, and volumes up to $36^3 \times 48$, without the mass correction to the Naik term. Their data near the chiral limit were described by ChPT and consistent with SχSB ($F \ne 0$, $M_\pi \to 0$, nonzero mass for the $\rho$ meson (the light spin-one vector meson), condensate consistent with GMOR), while data at larger masses exhibited finite-size hyperscaling with effective anomalous dimensions up to $\gamma \simeq 0.97$ ($\gamma \simeq 0.78$-$0.93$ in fits including mass corrections), suggesting a walking theory with $\gamma_m \sim 1$. Yet several fits carried large $\chi^2/\mathrm{dof}$ and the results were obtained at a single $\beta$, so the collaboration stated that more data at various fermion masses and couplings on larger lattices, an infinite-volume extrapolation, and a study of the fermion-mass-range dependence were needed before the phase of $N_f=8$ could be decided.

Problem Statement

Determine the infrared phase of SU(3) gauge theory with $N_f=8$ massless Dirac fermions in the fundamental representation: does the theory exhibit spontaneous chiral symmetry breaking with walking dynamics, or is it inside the conformal window, flowing to an infrared fixed point?

The single scientific objective is this phase determination. A definitive determination resolves the systematic issues that motivated the question — a single gauge coupling, dependence on the fit window, and large $\chi^2/\mathrm{dof}$ in several fits — and characterizes whichever phase is found: for a walking, chirally broken phase, the chiral-limit mass anomalous dimension $\gamma_m$; for an infrared-conformal phase, the anomalous dimension $\gamma_m$ at the infrared fixed point.

References

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