Phases of $SU(3)$ gauge theories with fundamental quarks via Dirac spectral density

We propose that, in $SU(3)$ gauge theories with fundamental quarks, confinement can be inferred from spectral density of the Dirac operator. This stems from the proposition that its possible behaviors are exhausted by three distinct types (Fig. 1). The monotonic cases are standard and entail confinement with valence chiral symmetry breaking (A) or the lack of both (C,C’). The bimodal (anomalous) option (B) was frequently regarded as an artifact (lattice or other) in previous studies, but we show for the first time that it persists in the continuum limit, and conclude that it informs of a nonconfining phase with broken valence chiral symmetry. This generalization rests on the following. $(\alpha )$ We show that bimodality in ${\mathrm{N}}_f=0$ theory past deconfinement temperature $T_c$ is stable with respect to removal of both infrared and ultraviolet cutoffs, indicating that anomalous phase is not an artifact. $(\beta )$ We demonstrate that transition to bimodality in ${\mathrm{N}}_f=0$ is simultaneous with the loss of confinement: anomalous phase occurs for $T_c<T<T_{ch}$, where $T_{ch}$ is the valence chiral restoration temperature. $(\gamma )$ Evidence is presented for thermal anomalous phase in ${\mathrm{N}}_f=2+1$ QCD at physical quark masses, whose onset too coincides with the conventional “crossover $T_c$.” We conclude that the anomalous regime $T_c<T<T_{ch}$ is very likely a feature of nature’s strong interactions. $(\delta )$ Our past studies of zero-temperature ${\mathrm{N}}_f=12$ theories revealed that bimodality also arises via purely light-quark effects. As a result, we expect to encounter the anomalous phase on generic paths to valence chiral restoration. We predict its existence also for $N_f$ massless flavors ($T=0$) in the range ${\mathrm{N}}_f^c<{\mathrm{N}}_f<{\mathrm{N}}_f^{ch}$, where ${\mathrm{N}}_f^c$ could be quite low. Conventional arguments would associate ${\mathrm{N}}_f^{ch}$ with the onset of conformal window.

Figure 1

Confinement/vSChSB structure of $\mathcal{T}$ via infrared behavior of Dirac spectral density.

Figure 2

Anomalous phase in ${\mathrm{N}}_f=0$ theory at $T=1.12T_c$ and lattice spacing $a=0.085\mathrm{fm}$.

Figure 3

Ultraviolet cutoff and anomalous behavior: ${\mathrm{N}}_f=0$ theory at $T=1.12T_c$, $L=2.0\mathrm{fm}$.

Figure 4

Thermal transitions to deconfined and anomalous phase coincide in ${\mathrm{N}}_f=0$ theory.

Figure 5

Anomalous phase across the thermal crossover in ${\mathrm{N}}_f=2+1$ QCD at the physical point.

Figure 6

The structure of set $\mathcal{T}$ proposed here (left) and the conventional one (right).