The Chiral Symmetry Breaking/Restoration in Dyonic Vacuum

TL;DR

This work builds a dyons-based framework to understand chiral symmetry breaking and restoration in QCD-like theories at finite temperature, where nonzero holonomy splits instantons into M and L dyons. By analyzing both classical and fermion-induced interactions, the authors develop three molecular models to describe the dyon ensemble and map their parameters to finite-$T$ QCD, showing how topological molecules form and how their fermionic zero modes drive chiral dynamics. A key finding is that high-temperature dyons form neutral, topological molecules, while lowering temperature or increasing the number of fermion flavors leads to stronger coupling and complex reorganization of the spectrum, including gaps whose existence depends on fermion boundary conditions and fermion representation. The results indicate a plausible, testable link between topological objects (dyons) and chiral restoration, and suggest lattice probes—e.g., holonomy dependence of Dirac spectra, zero-mode localization, and dyon clustering—to validate the scenario and illuminate the interplay between confinement and topology.

Abstract

We discuss the topological phenomena in the QCD-like theories with variable number of fundamental $N_f$ or adjoint $N_a$ fermions, focusing on the temperatures at or above the critical value $T_c$ of chiral symmetry restoration. Nonzero average of the Polyakov line, or holonomy, splits instantons into (anti)selfdual dyons, and we study both bosonic and fermionic interactions between them. The high temperature phase is a dilute gas of "molecules" made of $2N_c$ dyons, neutral in topological, electric and magnetic charges. At intermediate temperatures the diluteness reaches some critical level at which chiral symmetry gets restored: we explain why it is very different for the fundamental and adjoint fermions. At high density the ensemble is a strongly coupled liquid with crystal-like short range order: we speculate about its structure at small and large $N_f$. We finally explaine certain lattice observations and suggest a number of further lattice tests.

Figures (16)

• Figure 1: (Color online)The schematic picture of the dyonic molecule, for 2 colors and large $N_f$.
• Figure 2: (Color online) The critical lines for chiral restoration (solid line and diamonds) and deconfinement (dotted line and boxes) of the $N_c=3$ gauge theory. We plot the critical lattice coupling $\beta_c(T_c)=6/g^2_c(T_c)$ versus the number of fundamental quarks $N_f$ in (a) or a number of adjoint quarks $N_a$ in (b). Both paths of the infrared fixed point, calculated in the two-loop approximation, are shown by the thick (red) lines. The vertical dashed lines separate the "conformal window domain": its location is a guess. In Fig.(a) we also show, by the dash-dotted (red) line, our guess for the actual location of the fixed point. For the meaning of the data points see the text.
• Figure 3: The profile of (unnormalized) zero mode components $\alpha_{1,2}$ (the solid and the dashed curves) as a function of the distance from the dyon. We show four different values of $\phi=0,0.2 v/\beta,0.4 v/\beta, 0.5 v/\beta, 0.55 v/\beta$. Note that the zero mode delocalizes at $\phi=0.5 v/\beta$
• Figure 4: (Color online)A graphical interpretation of the weight in the background of the dyons. Note that the relative minus sign will always induce a repulsion between a dyon-antidyon pair.
• Figure 5: $\left\langle k\right\rangle/N$ as a function of parameter $A$ defined in (\ref{['eqn_A']}) .