Phase structure of lattice QCD for general number of flavors

TL;DR

The paper nonperturbatively investigates how the number of quark flavors $N_F$ shapes confinement and chiral symmetry on the lattice, using Wilson fermions and the one-plaquette gauge action. It demonstrates a bulk first-order transition for $N_F\ge7$ separating confined and deconfined phases, with a massless line occurring only in the deconfined region. Based on strong-coupling results and a proposed weak-coupling picture, it proposes a flavor-number–dependent beta-function: confinement for $N_F\le6$, a nontrivial IR fixed point without confinement for $7\le N_F\le16$, and a trivial IR fixed point for $N_F\ge17$. The work thus constrains the continuum limits of QCD-like theories and highlights a possible mechanism linking flavor content to confinement and fixed-point structure, with implications for interpreting nondegenerate quark masses and beyond-Standard-Model scenarios.

Abstract

We investigate the phase structure of lattice QCD for the general number of flavors in the parameter space of gauge coupling constant and quark mass, employing the one-plaquette gauge action and the standard Wilson quark action. Performing a series of simulations for the number of flavors $N_F=6$--360 with degenerate-mass quarks, we find that when $N_F \ge 7$ there is a line of a bulk first order phase transition between the confined phase and a deconfined phase at a finite current quark mass in the strong coupling region and the intermediate coupling region. The massless quark line exists only in the deconfined phase. Based on these numerical results in the strong coupling limit and in the intermediate coupling region, we propose the following phase structure, depending on the number of flavors whose masses are less than $Λ_d$ which is the physical scale characterizing the phase transition in the weak coupling region: When $N_F \ge 17$, there is only a trivial IR fixed point and therefore the theory in the continuum limit is free. On the other hand, when $16 \ge N_F \ge 7$, there is a non-trivial IR fixed point and therefore the theory is non-trivial with anomalous dimensions, however, without quark confinement. Theories which satisfy both quark confinement and spontaneous chiral symmetry breaking in the continuum limit exist only for $N_F \le 6$.

Figures (23)

• Figure 1: Renormalization group beta function. (a) Conjecture by Banks and Zaks Banks1 assuming confinement in the strong coupling limit for all $N_F$. (b) Our conjecture deduced from the results of lattice simulations.
• Figure 2: Phase diagram for a chirally symmetric case. (a) $N_F \le 6$, (b) $N_F \ge 17$, (c) $7\le N_F \le 16$.
• Figure 3: (a) $m_q$ at $\beta=\infty$. (b) $m_\pi$ at $\beta=\infty$. Results with an anti-periodic boundary condition (apbc) in the $t$-direction and those with the periodic boundary condition (pbc) are compared on $N_t=8$ and 4 lattices.
• Figure 4: The phase structure for $N_F \leq 6$; (a) at zero temperature, and (b) at finite temperatures. The chiral limit (massless quark limit) is shown by thick curves labeled by "$m_q=0$", and the finite temperature QCD transition at a fixed finite $N_t$ is shown by a shaded curve.
• Figure 5: Phase diagram for $N_F=240$. Dark shaded (green) lines represent our conjecture for the bulk transition line in the limit $N_t=\infty$. Points connected by dashed lines are the measured points where quarks are massless. Light shaded lines are for the finite temperature transition at $N_t=4$ and 8.