Renormalized symmetry probes for emergent symmetries and non-chiral lattice fermions

Abstract

The degeneracy of hadron correlators in symmetry-related channels signals symmetry restoration in high-temperature QCD. RG-invariant probes of this degeneracy need care when the regularization breaks chiral symmetry, or when the symmetry is not exact in the massless action, as for the emergent chiral-spin symmetry $SU(2{)}_{CS}$. We renormalize the thermal and screening meson correlators of every flavor sector by the thermal correlator of the same channel in the $\bar{u}d$ sector at a common reference point. The symmetry ratio ${\kappa }_{AB}$ then depends on the renormalization only through the ratio $R_{AB}$. For exact symmetries, $R_{AB}=1$ and ${\kappa }_{AB}$ is RG invariant and scheme independent in mass-independent schemes. For emergent symmetries, $R_{AB}\ne 1$ carries the relative logarithmic cutoff dependence of the two channels, which cancels in ${\kappa }_{AB}$, and the residual dependence on the reference scale is physical. The framework applies to Wilson, twisted-mass and staggered fermions, where $R_{AB}$ absorbs the channel-dependent renormalization, and the tensor pair $(T_k,X_k)$ keeps $R_{T_k{X}_k}=1$ on isotropic Wilson-type lattices. Lattice artifacts are a small correction to the chiral-spin probes ${\kappa }_{CS}$ and ${\kappa }_{fade}$, but as large as the signal of the exact-symmetry probes. For free quarks, we derive ${\kappa }_{fade}=2/3$ for every mass and separation, and ${\kappa }_{CS}\le 1/3$. We apply the renormalized probes to $N_f=2+1+1$ QCD with optimal domain-wall quarks at the physical point, with $a=0.064$ fm and $T=192$-$1539$ MeV. The hierarchical restoration of the chiral symmetries from the light to the charm sectors is confirmed, and is invariant under renormalization. The $\bar{u}c$ and $\bar{s}c$ sectors have the widest chiral-spin windows, also at a tighter threshold. Thus spin-1 mesons with one charm quark are the most attractive channels for the chiral-spin regime.

arXiv comment69 pages, 4 figures, 16 tables. Supplemental Material provided as the ancillary file supplemental_tables.pdf