Renormalization of meson susceptibilities and RG-invariant symmetry ratios in QCD

Abstract

We analyze the ultraviolet divergence structure of meson susceptibilities in finite-temperature QCD, for lattice formulations with exact chiral symmetry. The bare susceptibility separates into additive divergences and a multiplicative renormalization $Z_Γ^{2}$. The additive divergences are temperature-independent, and are removed by the temperature subtraction. They consist of the leading power divergence $α_Γ/(2a^2)$ from the identity operator, together with a mass-dependent logarithmic term $\propto m^2\ln(1/(am))$. Exact chiral symmetry forbids all mass-dependent \emph{power} divergences of the susceptibility. The multiplicative factor $Z_Γ^{2}$ has a logarithmic dependence on the lattice spacing, controlled by the operator anomalous dimension. We show that the symmetry ratio $κ_{AB} = (χ_A^{\rm reg} - χ_B^{\rm reg})/ (χ_A^{\rm reg} + χ_B^{\rm reg})$, built from temperature-subtracted susceptibilities of symmetry partners, is exactly renormalization-group invariant and scheme-independent. The additive divergence is removed by the subtraction, and the multiplicative factor cancels through the equality $Z_A = Z_B$. This equality holds for any number of flavors and any quark masses in a mass-independent scheme, unaffected by spontaneous symmetry breaking or the $U(1)_A$ anomaly. We derive the complete $Z$-factor chains for all meson channels and contrast the divergence structure with that of Wilson fermions, for which the explicit chiral-symmetry breaking induces a chiral-odd power-divergent mixing and spoils the equality $Z_A = Z_B$ on which the construction relies.