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

Abstract

We introduce a renormalization-group invariant observable, the symmetry strength parameter $\kappa_{AB}$, for the quantitative characterization of symmetry breaking in QCD. As a first application, we employ $\kappa_{AB}$ 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 quark-connected correlators: the $U(1)_A$-sensitive scalar-pseudoscalar channel ($\kappa_{PS}$), probing the $\pi$-$\delta$ system; the $SU(2)_L \times SU(2)_R$-sensitive vector--axial-vector channel ($\kappa_{VA}$), probing the $\rho$-$a_1$ system; and an additional $U(1)_A$-sensitive tensor--axial-tensor channel ($\kappa_{TX}$), probing the $\rho_T$-$b_1$ system. At finite lattice spacing, we observe a clear hierarchy $\kappa_{PS} > \kappa_{VA} > \kappa_{TX}$. 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 breaking in quark-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 sufficiently suppressed.