[Paper Review] The fate of $U_A(1)$ and topological features of QCD at finite temperature

Sayantan Sharma|arXiv (Cornell University)|Jan 25, 2018

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This lattice QCD study uses overlap fermions and domain wall fermions to investigate the fate of the $U_A(1)$ symmetry at finite temperature, finding that anomalous $U_A(1)$ violation persists through the chiral crossover, with non-analytic near-zero Dirac eigenvalue modes surviving in the continuum limit. The results support a dilute instanton gas description of topological susceptibility above $2T_c$, with implications for axion dark matter models.

ABSTRACT

The nature of chiral phase transition for QCD with two light quark flavors is not yet completely resolved. This is primarily because one has to understand whether or not the anomalous U(1) symmetry in the flavor sector is effectively restored along with the chiral symmetry. Since the physics near the chiral phase transition is essentially non-perturbative, we employ first principles lattice techniques to address this issue. We use overlap fermions, which have exact chiral symmetry on the lattice, to probe the anomalous U(1) symmetry violation of 2+1 flavor dynamical QCD configurations with domain wall fermions. The latter also optimally preserves chiral and flavor symmetries on the lattice. We observe that the anomalous U(1) is not effectively restored in the chiral crossover region. We perform a systematic study of the finite size and cut-off effects since the signals of U(1) violation are sensitive to it. For the same reasons we also compare our results from the continuum extrapolated results of the QCD Dirac spectrum obtained from a different lattice discretization called Highly Improved Staggered Quarks. Our studies also provide a glimpse of the microscopic topological structures of the QCD medium that are responsible for the strongly interacting nature of the quark gluon plasma phase and related to the physics of confinement and chiral symmetry breaking.

Motivation & Objective

• To resolve the unresolved question of whether the anomalous $U_A(1)$ symmetry is effectively restored during the chiral crossover in QCD with two light quarks.
• To determine if the near-zero eigenvalues of the QCD Dirac operator—responsible for $U_A(1)$ violation—are robust against lattice artifacts and finite-volume effects.
• To assess the validity of the dilute instanton gas model in describing topological susceptibility at high temperatures.
• To provide a first-principles determination of the topological susceptibility and its temperature dependence for axion mass constraints.
• To compare results across different fermion discretizations (HISQ and Möbius domain wall fermions) to ensure robustness and perform continuum extrapolation.

Proposed method

• Employed overlap fermions with exact chiral symmetry to compute the Dirac spectrum and detect $U_A(1)$ violation via near-zero eigenvalues.
• Used Möbius domain wall fermions with Iwasaki gauge action to generate QCD configurations with physical and slightly heavier pion masses.
• Measured low-lying eigenvalues of $D_{ ext{ov}}^ au D_{ ext{ov}}$ using the Kalkreuter-Schmidt Ritz algorithm on 100–150 configurations per ensemble.
• Performed continuum extrapolation of topological susceptibility using both gluonic and fermionic definitions, verifying chiral Ward identities.
• Compared results from two distinct discretizations (HISQ and domain wall fermions) to control for finite cutoff and volume effects.
• Fitted the temperature dependence of topological susceptibility to test agreement with dilute instanton gas model predictions.

Experimental results

Research questions

• RQ1 Is the anomalous $U_A(1)$ symmetry effectively restored in the chiral crossover region of 2+1 flavor QCD?
• RQ2 Are the non-analytic near-zero modes in the Dirac spectrum robust against lattice discretization and finite-volume effects?
• RQ3 Does the dilute instanton gas model accurately describe the topological susceptibility of QCD at temperatures above $2T_c$?
• RQ4 What is the impact of the scale factor $K$ in the topological susceptibility amplitude on axion mass predictions?
• RQ5 Do the observed topological structures in the QCD medium persist in the continuum limit and reflect genuine non-perturbative physics?

Key findings

• The anomalous $U_A(1)$ symmetry is not effectively restored in the chiral crossover region; near-zero Dirac eigenvalue modes persist up to $1.5T_c$.
• The exponent $\gamma$ in the non-analytic part of the eigenvalue spectrum changes from 1 at $T_c$ to 2 at $1.2T_c$, indicating strong temperature dependence.
• The non-analytic peak in the eigenvalue spectrum survives in the continuum limit for HISQ ensembles, suggesting it is not a lattice artifact.
• The topological susceptibility $\chi_t$ shows a temperature dependence consistent with the dilute instanton gas model, with exponent $b = 1.85(15)$ matching the predicted $b \sim 2$ for $N_f=3$.
• The amplitude of $\chi_t^{1/4}$ requires a scale factor $K = 1.9$ to match leading-order dilute instanton gas predictions, indicating higher-order corrections are significant.
• The axion decay constant is predicted to be $f_a \sim 10^{12}$ GeV, within the range probed by the ADMX experiment, with uncertainty from $K$ affecting the prediction by only $\sim 15\%$.