Infrared phase of massless twelve-flavor SU(3) QCD: infrared conformality or spontaneous chiral symmetry breaking?

The SU(3) gauge theory with twelve massless fundamental fermions ('twelve-flavor QCD') is the most heavily studied candidate lying just above the expected lower edge of the conformal window of QCD-like theories, yet its infrared phase is not determined. The originating lattice study (LatKMI Collaboration, 2012) found the meson spectrum and decay constants consistent with conformal hyperscaling with mass anomalous dimension $\gamma^*\approx0.4$-$0.5$, but could not exclude spontaneous chiral symmetry breaking because its simulated masses were too heavy for controlled chiral perturbation theory (chiral expansion parameter $X\simeq39$). Later work deepened rather than settled the question: gradient-flow step-scaling studies report an infrared fixed point - most recently with 'strong numerical evidence' from a Pauli-Villars-improved action - while an independent high-precision re-analysis of the same renormalization schemes finds no fixed point in the reported interval and refutes the earlier published claim; spectroscopy analyses split between hyperscaling-favored and chiral-fit-favored conclusions; strong-coupling lattice bulk phase transitions and discretization artifacts complicate continuum extrapolations; and analytic methods disagree among themselves. The problem is a decisive, systematics-controlled determination of the infrared phase of the massless theory - infrared fixed point with its mass anomalous dimension, or spontaneous chiral symmetry breaking with its breaking scale.

Background

QCD is the SU(3) gauge theory coupled to Dirac fermions, and although the physical theory has six quark flavors, the number of massless flavors $N_f$ is a parameter one may vary; the phase structure as a function of $N_f$ is a fundamental question about QCD-like gauge dynamics. For sufficiently many flavors the two-loop beta function develops an infrared zero, producing a Banks-Zaks infrared fixed point (IRFP) - a renormalization-group fixed point at which the continuum theory becomes a scale-invariant conformal field theory in the infrared (Banks and Zaks, 1982). The set of $N_f$ values for which an IRFP exists is the conformal window; its lower edge $N_f^*$ for fundamental-representation fermions is unknown and estimated to lie between roughly 8 and 12. Just below the edge the coupling is expected to run very slowly over a wide range of scales ('walking'), the basis of walking-technicolor and composite-Higgs scenarios of electroweak symmetry breaking, in which the mass anomalous dimension $\gamma^*$ (the exponent governing how a fermion mass grows under the renormalization group near the fixed point) controls fermion-mass generation and is desired to be sizable.

Inside the conformal window a fermion mass $m_f$ is a relevant perturbation of the fixed point: composite masses and decay constants obey hyperscaling, $M_H\propto m_f^{1/(1+\gamma^*)}$ and $F\propto m_f^{1/(1+\gamma^*)}$, so that ratios such as $F_\pi/M_\pi$ become approximately mass-independent, with an analogous finite-size hyperscaling on finite lattices. Outside the window the theory instead undergoes spontaneous chiral symmetry breaking (S$\chi$SB): a nonzero chiral-limit pion decay constant and condensate, Goldstone pions, and low-energy observables organized by chiral perturbation theory (ChPT), the effective field theory of the Goldstone modes. ChPT is self-consistent only when the expansion parameter $X=N_f(M_\pi/(4\pi F_\pi/\sqrt{2}))^2$ is below unity; near the conformal window $F_\pi$ is expected to be small and $N_f$ is large, so $X$ grows quickly and very light masses on very large lattices are needed for controlled chiral fits. The two hypotheses are hardest to distinguish exactly where they nearly meet.

$N_f=12$ is the standard template for many-flavor dynamics near the anticipated edge. Twelve degenerate fermions are simulated with staggered fermions (a lattice discretization in which one field describes multiple continuum 'tastes'; improved variants such as HISQ - highly improved staggered quarks - suppress taste-symmetry breaking and other discretization errors), with Wilson-type or domain-wall discretizations (alternative lattice fermion formulations that avoid taste doubling and control chiral-symmetry breaking differently, at higher computational cost), or with mixed setups. The originating study of this problem (Aoki et al., LatKMI Collaboration, 2012) simulated twelve-flavor QCD with a HISQ-type action at two lattice spacings, three volumes, and a range of masses. It found the data consistent with finite-size hyperscaling and concluded $\gamma^*\in[0.4,0.5]$ 'at IR fixed point if it exists'; but the chiral fits that would signal S$\chi$SB through a nonzero chiral-limit decay constant were not self-consistent ($X\simeq39$ at the lightest simulated mass), and the paper explicitly stated that 'A possibility of S$\chi$SB in Nf = 12 is not excluded yet' and that 'Further efforts would be required to arrive at a decisive conclusion'. A third observable class is the running coupling defined through the Yang-Mills gradient flow (a smoothing of the gauge field over a radius $\sqrt{8t}$) in finite volume, whose volume-to-volume step scaling gives a discrete beta function whose zero would signal an IRFP. A recurring obstacle for all approaches is the strong-coupling bulk phase transition - a first-order transition of the lattice action at coarse lattice spacing, absent from the continuum theory - which many twelve-flavor discretizations encounter before the infrared regime is reached, and whose location moves with the discretization.

Problem Statement

Determine, decisively and with controlled continuum-limit systematics, the infrared phase of the massless SU(3) gauge theory with $N_f=12$ fundamental Dirac fermions ('twelve-flavor QCD'): does its renormalization-group flow possess an infrared fixed point, so that the theory lies inside the conformal window and a fermion-mass deformation exhibits conformal hyperscaling $M_H,F\propto m_f^{1/(1+\gamma^*)}$; or does the theory undergo spontaneous chiral symmetry breaking, with nonzero chiral-limit pion decay constant and condensate and a Goldstone spectrum governed by chiral perturbation theory?

The determination is of (i) the phase in the continuum, massless limit - if conformal, the fixed-point coupling in a specified scheme together with the mass anomalous dimension $\gamma^*$; if broken, the breaking scale or chiral-limit order parameters - and (ii) quantified control of the systematics that currently make published determinations mutually contradictory - the continuum extrapolation at the strong couplings $g^2\sim5$-$7$ where the infrared dynamics appear, finite-volume and finite-mass effects in the regime $X=N_f(M_\pi/(4\pi F_\pi/\sqrt{2}))^2\gg1$ where neither chiral perturbation theory nor hyperscaling is self-consistent, and the discretization-dependent strong-coupling bulk first-order phase transitions - with the phase assignment and its quoted parameters resting on a continuum-limit extrapolation.

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