Flavor dependence of chiral symmetry breaking and the conformal window

TL;DR

The paper investigates how dynamical chiral symmetry breaking in QCD depends on the number of quark flavors by solving coupled Dyson–Schwinger equations for quark and gluon propagators within a minimal, gauge-consistent truncation. Flavor effects enter primarily through the quark-loop backreaction on the gluon propagator, enabling a self-consistent analysis of DCSB and the approach to the conformal window. The key findings are a critical flavor number $N_f^c=6.81$, with the chiral condensate scaling as $-ig\uparrowar{ extpsi} extbar extpsiig angle \, extasymp\,|N_f-N_f^c|^{0.53(9)}$ and a related scaling exponent $\\Theta=1.88\pm0.032$ governing $\,ig\uparrowar{ extpsi} extbar extpsiig angle \, extasymp|N_f-N_f^c|^{1/\Theta}$. Beyond $N_f^c$, the gluon-dressing function flattens, signaling a walking regime toward the conformal window, with the dynamics dominated by gluon mass generation rather than chiral symmetry breaking. Limitations of the current scheme—namely the use of a quenched gluon baseline—restrict precise determination of the lower boundary of the conformal window, motivating future work with fully self-consistent gluon and ghost propagators and finite-temperature mappings.

Abstract

We investigate the phase structure of Quantum Chromodynamics (QCD) in the vacuum as a function of quark flavor number $N_f$ within the chiral limit. By self-consistently solving the coupled DSEs for the quark and gluon propagators in a minimal QCD scheme, we elucidate the nonperturbative dynamics governing dynamical chiral symmetry breaking. Our calculations determine a critical flavor number of $N_f^c=6.81$ which marks the chiral symmetry restoration of quarks. Further analysis reveals the critical exponents of the chiral condensate as $ -\langle\barψ ψ\rangle\sim |N_f-N_f^c|^{0.53(9)}$, characterized the second order feature of this phase transition of chiral symmetry. Additionally, we discuss the implications for the walking regime towards the conformal window at larger flavor.

Figures (7)

• Figure 1: The Dyson Schwinger equation for quark propagator, i.e., the quark gap equation.
• Figure 2: The Dyson-Schwinger equation for the gluon propagator, with the grey point standing for the quenched gluon propagator calculated in Yang-Mills theory and thus only the quark loop is left aside in this DSE of gluon propagator.
• Figure 3: Gluon propagator dressing function in the current work at $N_f=0+2$ and $0+3$, in comparison to the previous results of $N_f=2$ flavor Sternbeck:2005tkCyrol:2017ewj and $2+1$ flavor Boucaud:2018xupGao:2021wun.
• Figure 4: The obtained scalar function of quark propagator which determines the strength of the chiral symmetry breaking, in comparison with the previous results at $N_f=$ 2 and 2+1 flavors Gao:2020qsj.
• Figure 5: The evolution of quark mass function along with the change of flavor numbers. At $N_f^c=6.81$, the quark mass function vanishes.