Anomaly matching in QCD thermal phase transition

TL;DR

This work identifies a mixed 't Hooft anomaly between a Z2 center symmetry (emerging at the Roberge-Weiss point) and the chiral SU(N_f)_L × SU(N_f)_R symmetry in massless QCD at finite temperature. The anomaly persists under dimensional reduction to 3D and is reproduced in the chiral Lagrangian via a Wess-Zumino-Witten term tied to Skyrmion topology, ensuring anomaly matching across the confinement–deconfinement transition. The authors show that imaginary baryon chemical potential effects are subleading in the large-N limit, and argue that universality-based second-order transitions are disfavored by the anomaly, with a first-order transition at a single Tc emerging as a natural scenario for generic (N_c, N_f). They discuss μ_B = π and μ_B = 0 cases, draw parallels to pure Yang-Mills at θ angles, and outline implications for the QCD phase diagram, while highlighting the limits of universality arguments in this context. Overall, the paper provides a robust, anomaly-driven constraint on the nature of the QCD thermal phase transition and its dependence on large-N dynamics and imaginary chemical potential.

Abstract

We study an 't Hooft anomaly of massless QCD at finite temperature. With the imaginary baryon chemical potential at the Roberge-Weiss point, there is a $\mathbb{Z}_2$ symmetry which can be used to define confinement. We show the existence of a mixed anomaly between the $\mathbb{Z}_2$ symmetry and the chiral symmetry, which gives a strong relation between confinement and chiral symmetry breaking. The anomaly is a parity anomaly in the QCD Lagrangian reduced to three dimensions. It is reproduced in the chiral Lagrangian by a topological term related to Skyrmion charge, matching the anomaly before and after QCD phase transition. The effect of the imaginary chemical potential is suppresssed in the large $N$ expansion, and we discuss implications of the 't~Hooft anomaly matching for the nature of QCD phase transition with and without the imaginary chemical potential. Arguments based on universality alone are disfavored, and a first order phase transition may be the simplest possibility if the large $N$ expansion is qualitatively good.

Figures (5)

• Figure 1: How to define confinement. Here $Q_{\rm probe}$ is the probe quark representing the Polyakov loop, and $\overline{q}$ is a dynamical anti-quark. In the confinement phase (Right) the baryon number $B$ is always integer and hence $e^{i \pi B} = \pm 1 \in {\mathbb R}$, while in the deconfinement phase (Left) it is not necessarily integer because quarks have fractional baryon number $1/N_c$ and hence $e^{i \pi B} \in {\mathbb C}$.
• Figure 2: Mixed anomaly between ${\mathbb Z}_2^{\rm center}$ and $\mathrm{SU}(N_f)_L \times \mathrm{SU}(N_f)_R$ at finite temperature.
• Figure 3: The case $T_{\rm chiral} < T_{\rm center}$. We need massless degrees of freedom (DOF) in the region $T_{\rm chiral} \leq T \leq T_{\rm center}$ to match the 't Hooft anomaly. The required DOF is very complicated, as explained in the main text.
• Figure 4: The case $T_{\rm chiral} > T_{\rm center}$. In the range $T_{\rm chiral} > T > T_{\rm center}$, the chiral symmetry is broken even thougth the theory is in deconfinement phase.
• Figure 5: The case $T_{\rm chiral} = T_{\rm center}$. A first order phase transition is natural. If it is second order, some complicated massless DOF must appear at the critical temperature to match the 't Hooft anomaly.