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Topological observables and domain wall tension from finite temperature chiral perturbation theory

Zhen-Yan Lu, Quan Tang, Shu-Peng Wang, Yang Huang, Zhen Zhang, Bonan Zhang

Abstract

Within the framework of SU(2) chiral perturbation theory, we derive the general solution of the QCD $θ$-vacuum for an arbitrary vacuum phase, explicitly incorporating isospin-breaking effects from the light quark mass difference, and compute the temperature dependence of the topological susceptibility, higher-order cumulants, and the domain wall tension up to next-to-leading order. We find that the topological susceptibility agrees with lattice data at low temperatures but deviates at higher temperatures as expected from the breakdown of the chiral expansion; moreover, we demonstrate that the normalized fourth-order cumulant and the domain wall tension decrease monotonically with increasing temperature, while the normalized sixth-order cumulant exhibits the opposite behavior. These results extend earlier analyses by showing how isospin breaking reshapes the full hierarchy of topological charge cumulants and the dynamics of $θ$-vacuum domain walls, thereby offering new theoretical input on the $θ$-vacuum properties, which are relevant for axion-related effective theories in hot QCD matter.

Topological observables and domain wall tension from finite temperature chiral perturbation theory

Abstract

Within the framework of SU(2) chiral perturbation theory, we derive the general solution of the QCD -vacuum for an arbitrary vacuum phase, explicitly incorporating isospin-breaking effects from the light quark mass difference, and compute the temperature dependence of the topological susceptibility, higher-order cumulants, and the domain wall tension up to next-to-leading order. We find that the topological susceptibility agrees with lattice data at low temperatures but deviates at higher temperatures as expected from the breakdown of the chiral expansion; moreover, we demonstrate that the normalized fourth-order cumulant and the domain wall tension decrease monotonically with increasing temperature, while the normalized sixth-order cumulant exhibits the opposite behavior. These results extend earlier analyses by showing how isospin breaking reshapes the full hierarchy of topological charge cumulants and the dynamics of -vacuum domain walls, thereby offering new theoretical input on the -vacuum properties, which are relevant for axion-related effective theories in hot QCD matter.
Paper Structure (7 sections, 22 equations, 5 figures, 1 table)

This paper contains 7 sections, 22 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Vacuum energy at finite temperature
  3. Leading order
  4. Next-to-leading order
  5. Topological observables at finite temperature
  6. $\theta$-dependent effective potential and domain wall tension
  7. Conclusions

Figures (5)

  • Figure 1: The temperature dependence of the topological susceptibility, $\chi_t^{1/4}$, calculated within CHPT. For comparison, lattice QCD results from Ref. Borsanyi:2016ksw are shown as the green band at finite temperature, with the zero-temperature value indicated by the data point with error bar.
  • Figure 2: The normalized fourth-order cumulant of the QCD topological charge distribution as a function of the temperature, with (solid curve) and without (dashed curve) isospin breaking. The green curve denotes the asymptotic value $b_2^{\text{inst}}$ predicted by the dilute instanton gas model Bonati:2015uga.
  • Figure 3: The normalized sixth-order cumulant of the QCD topological charge distribution as a function of the temperature, with (solid curve) and without (dashed curve) isospin breaking.
  • Figure 4: Free energy density of the $\theta$-vacuum, $\Delta V(\theta, T)$, as a function of the angle $\theta$ for different temperatures.
  • Figure 5: Normalized domain wall tension, $\sigma/\sigma_0$, of the QCD $\theta$-vacuum as a function of temperature, calculated with (solid curve) and without (dashed curve) isospin breaking.