On the phase structure of massless many-flavour QCD with staggered fermions

Abstract

When the number of massless fermions exceeds a critical value $N_f^*$, QCD enters the conformal window and becomes chirally symmetric already in the vacuum. Determining $N_f^*$ from lattice simulations is challenging, since calculations are performed at finite lattice spacing, quark mass, and temporal lattice size, where both a thermal transition and an unphysical bulk transition obscure the conformal behaviour. In this work, we present results on the chiral phase boundaries in the bare lattice parameter space $(N_τ,\;β,\;am,\;N_f)$ of unimproved staggered fermions. Our analysis indicates that the chiral transition in continuum QCD is of second order for all $N_f$ up to the onset of the conformal window. By systematically studying the thermal chiral transition and its interplay with the bulk transition, we obtain a coherent picture of the lattice phase structure and suggest how the onset of the conformal window can be identified from simulations performed away from the chiral and continuum limits.

Figures (7)

• Figure 1: Chiral phase diagram for massless continuum QCD. Our analysis (see Section \ref{['sec: thermal']}) suggests a thermal phase transition of second order for all $N_f<N_f^*$Cuteri_2021Klinger:2025mU.
• Figure 2: Chiral phase diagram for massless QCD at non-zero coupling Miransky. A zero-temperature bulk transition separates the $(N_f,g)$-plane into a continuum-connected weak-coupling regime and a strong-coupling regime. The weak-coupling regime divides into a symmetric phase ($S$) and a thermally broken phase ($B_\text{th}$), whereas the bulk regime is chirally broken ($B_B$) by lattice artifacts.
• Figure 3: Bulk transition between weak- and strong-coupling phases for different $N_f$. The right panel is the extension of Fig. \ref{['fig: Mirsanky Nf g']} towards finite bare mass $am$. The bulk transition appears to be non-analytic for $N_f>6$.
• Figure 4: Thermal phase transition for different temporal extent $N_\tau$. The 1st order region is bounded by a 2nd order wing line, which terminates at a tricritical point $(\beta^{\mathrm{tric}}, N_\tau^{\mathrm{tric}})$ as $N_\tau$ increases. For $N_\tau \geq N_\tau^{\mathrm{tric}}$, the thermal transition in the chiral limit is of 2nd order, while for $am>0$ it turns into a crossover. As a consequence, the chiral continuum limit ($am=0$, $\beta \to \infty$, $N_\tau \to \infty$) is governed by a 2nd order transition.
• Figure 5: Thermal and bulk transition for $N_f=8$. Within the weak-coupling regime, the thermal transition depends on $N_\tau$ and separates the symmetric phase $S$ from the thermally broken phase $B_\text{th}$. As $N_\tau$ increases from $N_\tau=8$ to $N_\tau=10$, the first-order thermal transition (orange) vanishes into the bulk regime (green).