Chiral phase structure of QCD with many flavors

TL;DR

The paper investigates the chiral phase structure of QCD with many flavors using the functional renormalization group (FRG) while including all pointlike four-fermion interactions allowed by the symmetries. By combining a complete fermionic truncation with gauge dynamics constrained by Ward-Takahashi identities and regulator-dependent threshold functions, the authors locate an infrared conformal window defined by $N_f^{cr} < N_f < N_f^{af}$ and provide a quantitative prediction for SU(3): $N_f^{cr} = 10.0 \pm 0.29$ (fermion) $^{+1.55}_{-0.63}$ (gluon). The results show that in the symmetric phase the four-fermion couplings approach finite IR fixed points, while crossing $\alpha_{cr}$ leads to chiral symmetry breaking, implying a non-standard, continuous transition without light scalars. Overall, the work verifies the existence of a conformal phase in many-flavor QCD and demonstrates a robust nonperturbative FRG framework with regulator-based error estimates, informing both fundamental strong-coupling physics and beyond-Standard-Model model building.

Abstract

We investigate QCD with a large number of massless flavors with the aid of renormalization group flow equations. We determine the critical number of flavors separating the phases with and without chiral symmetry breaking in SU(Nc) gauge theory with many fermion flavors. Our analysis includes all possible fermionic interaction channels in the pointlike four-fermion limit. Constraints from gauge invariance are resolved explicitly and regulator-scheme dependencies are studied. Our findings confirm the existence of an Nf window where the system is asymptotically free in the ultraviolet, but remains massless and chirally invariant on all scales, approaching a conformal fixed point in the infrared. Our prediction for the critical number of flavors of the zero-temperature chiral phase transition in SU(3) is Nf^{cr}=10.0\pm 0.29(fermion)[+1.55;-0.63](gluon), with the errors arising from approximations in the fermionic and gluonic sectors, respectively.

Figures (3)

• Figure 1: Sketch of a typical $\beta$ function for the fermionic self-interactions $\lambda_i$: at zero gauge coupling, $\alpha=0$ (solid curve), the Gauß ian fixed point $\lambda_i=0$ is IR attractive (the second fixed point at $\lambda_i>0$ corresponds to the IR repulsive critical coupling of NJL type). For small $\alpha\gtrsim 0$ (dashed curve), the fixed-point positions are shifted on the order of $\alpha^2$. For gauge couplings larger than the critical coupling $\alpha>\alpha_{cr}$ (dot-dashed curve), no fixed points remain and the self-interactions quickly grow large, signaling $\chi$SB. We emphasize that both fixed-points values remain finite until the fixed points eventually vanish at $\alpha_{cr}$.
• Figure 2: Critical coupling for the four-fermion system. From top to bottom, the number of colors increases from $\textrm{N}_{\textrm{c}}=2$ to $\textrm{N}_{\textrm{c}}=8$ (thick line corresponds to $\textrm{N}_{\textrm{c}}=3$). The red/dark-grey (green/light-grey) line shows the fixed-point gauge coupling for $\textrm{N}_{\textrm{c}}=3$ at four (two) loop. At the crossing, the critical number of flavors can be read off.
• Figure 3: Critical number of flavors for $\textrm{SU}(\textrm{N}_{\textrm{c}})$ gauge theory. The result based on the four-loop $\overline{\text{MS}}$ beta function is denoted by black circles which lie almost on top of stars, representing the three-loop result; black boxes correspond to the two-loop beta function. The (inner) green/dark-grey shaded region around the four-loop result displays our error estimate for the fermionic sector. The (outer) turquois/light-grey shaded region shows the approximate gluonic error, estimated by a variation of the higher-loop coefficients. The red/grey line shows the number of flavors above which asymptotic freedom is lost.