RG-Invariant Symmetry Ratio for QCD: A Study of $U(1)_A$ and Chiral Symmetry Restoration

TL;DR

This work introduces the RG-invariant symmetry strength ratio $κ_{AB}$ to quantify symmetry breaking in QCD and applies it to finite-temperature $N_f=2+1+1$ lattice QCD with optimal domain-wall fermions. By analyzing three nonsinglet channels—$κ_{PS}$ for $U(1)_A$, $κ_{VA}$ for $SU(2)_L\times SU(2)_R$, and $κ_{TX}$ for a secondary $U(1)_A$ channel—the study demonstrates a finite-$a$ hierarchy ($κ_{PS} \gg κ_{TX} \sim κ_{VA}$) that collapses in the continuum, yielding statistically indistinguishable symmetry strengths near the chiral crossover. This indicates a two-stage restoration: initial effective restoration in the nonsinglet sector around the crossover, followed by full restoration including singlet channels only at higher temperatures once topological fluctuations are suppressed. The results provide a model-independent continuum benchmark for symmetry restoration and motivate future tensor-singlet and flavor-singlet investigations to map the complete restoration landscape of chiral and axial symmetries in QCD. The framework also offers a quantitative bridge between lattice observables and continuum effective descriptions, with implications for the structure of the quark-gluon plasma and the role of topology at high temperature.

Abstract

We introduce a renormalization-group invariant observable, the symmetry strength parameter $κ_{AB}$, for the quantitative characterization of symmetry breaking in QCD. As a first application, we employ $κ_{AB}$ to investigate the relative strength of $SU(2)_L \times SU(2)_R$ chiral symmetry and $U(1)_A$ axial symmetry breaking in $N_f=2+1+1$ lattice QCD using optimal domain-wall fermions at the physical point. Our study covers three lattice spacings and twelve temperatures in the range 164-385~MeV. We examine three independent symmetry-breaking channels in the nonsinglet sector with connected correlators: the $U(1)_A$-sensitive scalar-pseudoscalar channel ($κ_{PS}$), probing the $π$-$δ$ system; the $SU(2)_L \times SU(2)_R$-sensitive vector--axial-vector channel ($κ_{VA}$), probing the $ρ$-$a_1$ system; and an additional $U(1)_A$-sensitive tensor--axial-tensor channel ($κ_{TX}$), probing the $ρ$-$b_1$ system. At finite lattice spacing, we observe a clear hierarchy $κ_{PS} > κ_{TX} \sim κ_{VA}$. A controlled continuum extrapolation reveals that this hierarchy collapses, with all three symmetry-breaking strengths becoming statistically indistinguishable within our precision. This result provides a new, model-independent benchmark from a chirally symmetric lattice action. Our findings indicate that the effective restoration scales for $SU(2)_L \times SU(2)_R$ and $U(1)_A$ in the nonsinglet sector converge closely near the chiral crossover, placing stringent quantitative constraints on the temperature window for chiral and axial symmetry manifestation in connected channels. These results support a two-stage restoration scenario, in which full symmetry restoration -- including the singlet sector -- occurs only at significantly higher temperatures once topological fluctuations are suppressed.

Figures (4)

• Figure 1: $t$-correlators of $\bar{u} \Gamma d$ at the three lowest temperatures. The dashed lines connecting the data points in each channel are shown only to guide the eye.
• Figure 2: Regularized susceptibilities of $\bar{u} \Gamma d$ for three lattice spacings $a$=(0.075, 0.069, 0.064) fm and twelve temperatures from 164-385 MeV. The dashed lines connecting the data points in each channel are shown only to guide the eye.
• Figure 3: RG-invariant symmetry ratios $\kappa_{VA}$ and $\{ \kappa_{PS}, \kappa_{TX} \}$ for the $SU(2)_L \times SU(2)_R$ and $U(1)_A$ chiral symmetries of $(u,d)$ quarks, computed at three lattice spacings $a$ = (0.075, 0.069, 0.064) fm and twelve temperatures in the range 164-385 MeV. The solid line and its band denote the continuum-extrapolated value and its uncertainty from 2D global fit.
• Figure 4: Schematic illustration of the two-stage hierarchical restoration scenario for light $(u, d)$ quarks. Stage 1 (nonsinglet restoration) occurs around $T_c^{ns} \sim 156$ MeV, while Stage 2 (full effective restoration including singlets) requires $\chi_t \to 0$ and $\kappa_{TX}^s \to 0$ at a much higher temperature $T_c^s \gg T_c^{ns}$. The $\kappa_{AB}$ ratios for nonsinglet channels ($\kappa_{PS}$, $\kappa_{VA}$, $\kappa_{TX}$) probe Stage 1, while tensor singlet ratios ($\kappa_{TX}^s$) and direct $\chi_t$ measurements probe Stage 2.